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http://arxiv.org/abs/2307.03003v2	20230706140623	Improving the Efficiency of Human-in-the-Loop Systems: Adding Artificial to Human Experts	[ "Johannes Jakubik", "Daniel Weber", "Patrick Hemmer", "Michael Vössing", "Gerhard Satzger" ]	cs.LG	[ "cs.LG" ]	Karlsruhe Institute of Technology, Karlsruhe, Germany {johannes.jakubik,daniel.weber,patrick.hemmer,michael.voessing,gerhard.satzger}@kit.edu Improving the Efficiency of Human-in-the-Loop Systems: Adding Artificial to Human Experts Johannes Jakubik Daniel Weber Patrick Hemmer Michael VössingGerhard Satzger ^a Center fo Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya ulitsa 46, Moscow 119361, Russia ^b Institute of Gravitation and Cosmology, RUDN University, ulitsa Miklukho-Maklaya 6, Moscow 117198, Russia ^c National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, Moscow 115409, Russia ^d N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kremlyovskaya ulitsa 18, Kazan 420008, Russia ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Information systems increasingly leverage artificial intelligence (AI) and machine learning (ML) to generate value from vast amounts of data. However, ML models are imperfect and can generate incorrect classifications. Hence, human-in-the-loop (HITL) extensions to ML models add a human review for instances that are difficult to classify. This study argues that continuously relying on human experts to handle difficult model classifications leads to a strong increase in human effort, which strains limited resources. To address this issue, we propose a hybrid system that creates artificial experts that learn to classify data instances from unknown classes previously reviewed by human experts. Our hybrid system assesses which artificial expert is suitable for classifying an instance from an unknown class and automatically assigns it. Over time, this reduces human effort and increases the efficiency of the system. Our experiments demonstrate that our approach outperforms traditional HITL systems for several benchmarks on image classification. Keywords: Human-in-the-Loop Systems, Artificial Experts, Human-AI Collaboration, Unknown Data. WI_footer § INTRODUCTION AI-based information systems have become increasingly prevalent and powerful over the last decade <cit.>. Yet, predictions (i.e., classification) of the incorporated machine learning (ML) models are subject to uncertainty and may be incorrect. As a result, human experts are often tasked with reviewing the predictions of these models to identify and override errors <cit.>. In these so-called human-in-the-loop (HITL) systems, a model might classify the majority of instances automatically. However, instances that are difficult for the model are assigned to a human expert for manual review. This concept makes HITL systems a viable means for human-AI collaboration and has promoted their applications in domains such as medicine <cit.> and manufacturing <cit.>. However, these systems typically still require a significant amount of human effort <cit.>. In HITL systems, detecting data from unknown classes to forward them to a human expert is essential to guarantee high levels of accuracy—especially in high-stake decision-making like the medical domain, finance, or autonomous driving <cit.>. For example, when a model encounters an image of an “unfamiliar” traffic sign, it should treat it as unknown rather than attempting to match it to an existing class. However, this approach requires human experts to “build” knowledge regarding the novel class the instance belongs to. To address this issue and to reduce the required human effort, an approach is required that can incrementally learn from unknown data <cit.>. In this work, we aim to improve the efficiency of HITL systems. We train additional ML models—called artificial experts—that mimic the human expert's knowledge regarding classes that are unknown to the general ML model (i.e., the model trained to solve the task). Our so-called AI-in-the-Loop (AIITL) system creates (i.e., trains) multiple artificial experts based on the knowledge incrementally acquired from human expert in traditional HITL systems. The AIITL system then allocates instances that are recognized as unknown to suitable artificial experts. To achieve this, we reinterpret out-of-distribution (OOD) detectors as an allocation mechanism. Each artificial expert is responsible for a separate set of classes unknown to the general model and uses an OOD detector to independently “claim” instances for classification that it considers to stem from one of the classes in their respective set. Data instances that the OOD detector deems to originate from an unknown class are rejected by the individual artificial expert. The human expert is only consulted when none or multiple artificial experts claim an instance for “their” set of classes. The human expert then provides his or her knowledge by assigning the instance to the correct artificial expert or instantiates a new one responsible for the novel class. The overall objective of the system is to reduce human effort while maintaining classification accuracy. Following <cit.>, we optimize a utility metric represented by the combination of classification accuracy and human effort. Overall, our contributions are as follows. First, we propose a novel technique for capturing human expert knowledge about unknown data instances and make it accessible by “creating” artificial experts. Second, we show that our approach outperforms traditional HITL systems by a large margin in terms of their utility—the combination of classification accuracy and human effort. Third, we reinterpret OOD detectors as allocation mechanisms for the collaboration of artificial and human experts—allowing for a coordination between artificial experts as ML models that can “claim” unknown instances. We provide our code at <https://github.com/jhnnsjkbk/AIITL>. § BACKGROUND AND RELATED WORK In the following, we review related work on HITL systems, the detection of data from unknown classes, and incremental learning. §.§ Human-in-the-Loop systems AI-based systems including a "human-in-the-loop" enable a machine learning model to consult a human expert for instances that are difficult to classify <cit.>. For example, in the medical domain, it is essential that machine learning models forward x-ray images that are difficult to classify (i.e., the model is uncertain about the prediction) to physicians for manual inspection. In the literature, several setups exist in which human experts augment and complement ML models: HITL systems are employed in supervised learning <cit.>, semi-supervised learning <cit.>, and reinforcement learning <cit.>. However, these approaches generally require repetitive human effort that is growing with the number of unknown instances and the inaccuracy in detecting such instances. The resulting strain placed on human resources often renders HITL systems inefficient (see ). Therefore, any approach that can reduce human effort required in these systems is highly desirable. A literature search identified two recent works pursuing this objective, however with slightly different approaches: First, <cit.> consider a HITL system where the classifier is incrementally re-trained on novel, unseen data. For this re-training, the ML model acquires human expert knowledge for instances with low model confidence. Thus, the overall goal is to incrementally mimic several human experts with a single ML model in order to reduce subsequent effort. In contrast, we propose various seperate artificial experts that individually learn to classify instances from specific domains and collaboratively classify novel instances in order to reduce human effort. Unlike the approach of <cit.>, our technique does not require to train an additional deferral model for the collaboration of multiple agents in the system. Second, <cit.> have recently proposed the collaboration between two ML models in an HITL system. Their approach defers instances to human experts based on the alignment of these ML models as part of a multi-model collaboration. However, their work is specifically tailored to document layout analysis, while we propose a system to reduce the effort in HITL systems in general. §.§ Detection of data instances from unknown classes In our work, we make use of out-of-distribution detectors from the field of computer science and reinterpret them as allocation mechanisms. These OOD detectors allow to determine whether a data instance originates from an unknown class. The differentiation of known and unknown data is achieved based on whether unknown data originates from the same underlying distribution as known data or not. For example, everyday images of cars follow the same (or very similar distribution) but these images follow a different distribution than x-ray images from the medical domain. This approach can also be applied to distinguish more similar classes. In our case, we will consider classes from one dataset to be known, while classes from a range of other datasets are unknown and incrementally learned over time. In general, OOD detection aims at identifying data from unknown distributions <cit.>). Among the most widespread approaches for OOD detection are ODIN <cit.> and Mahalanobis-based OOD detection <cit.>. These two approaches are popular benchmarks utilized in recent studies <cit.>. In the context of HITL systems, OOD detection is typically considered during the training in active learning and few-shot learning <cit.> or for explainability <cit.>. Importantly, current literature finds a lack of handling detected unknown data in general <cit.>. We later address this by using OOD detectors as allocation mechanisms and by processing detected data from unknown classes within our AIITL system. §.§ Incremental learning Deep neural networks face severe difficulties when learning from evolving streams of training data. This phenomenon is often referred to as catastrophic forgetting <cit.> as—while adapting to the new classes—the performance of the ML model on the original classes deteriorates. That means the accuracy of ML models drops strongly, when the model is incrementally trained on new data. In the context of reducing human effort in HITL systems, this implies that we cannot simply fine-tune our model incrementally based on acquired human expert knowledge without facing catastrophic forgetting. Retraining the model on the entire set of known and unknown data each time unknown instances emerge would result in high computational cost and, thus, does not represent a suitable alternative either. To tackle catastrophic forgetting, class-incremental learning aims at making models more robust against new data that becomes available over time. Researchers have proposed approaches such as exemplar selection <cit.>, forgetting-constraint <cit.>, and bias removal methods <cit.>. One of the most popular approaches in incremental learning is Deep Model Consolidation <cit.>. The approach consists of a general model for old (known) data and a separate model that is responsible for the new classes. We later follow a similar idea by generating separate models for novel data from unknown classes to reduce the human effort in HITL systems. § METHODOLOGY The task of the AIITL system is to classify data from both known and unknown classes. We train a general model on data from the known classes. Unknown data originates from a set of unknown classes. The classification of an instance is then either conducted by the general model, one of n artificial experts, or a human expert. Our objective is to improve the utility of the system that is influenced by the level of human effort and the overall classification accuracy. The hybrid system is detailed in Figure <ref> and will be explained in the following. §.§ Notation The objective of the AIITL system is to reduce repetitive manual reviewing effort of human experts. For this, artificial experts need to mimic human experts on data instances that are unknown for the general model. In line with our research questions, we developed a system of human and artificial experts that aid a general ML model in classifying images. For this, we design artificial experts as an optional support that actively try to classify instances—and only forward instances to a human expert that both the general model and none of the artificial experts could classify. We formalize our approach in the following. Let 𝒳∈ℝ^N and 𝒴∈ℕ denote the input and target space of the AIITL system. The input space consists of the images that need to be classified by our system, while the target space refers to the classication outcomes for each of the input images. In our approach, we refer to known data by 𝐗_known and define data from unknown classes as 𝐗_unknown. The overall task of the AIITL system is to provide a set of classifications 𝐲̂ for the input data instances carried out by either the general ML model, a maximum of n supplementary artificial experts, or a human expert. Each of the ML models (the general ML model and the artificial experts) are iteratively trained to classify data from a specific domain. This results in individual classification functions f_p: 𝒳^j →𝒴^j, 𝐗^j ↦𝐲^j for each of the ML models in the AIITL system, where j refers to the index of the model. Note, that these functions are unknown a priori and are approximated by convolutional neural networks as f̂_p(𝐗^j) = 𝐲̂^j when training the AIITL system. Following <cit.>, we use a utility score that combines the system's accuracy ϕ(𝐗, f̂) and the human effort ρ(𝐗, f̂) to approximate the efficiency of the AIITL system for our evaluation. It is defined as: U(𝐗, f̂) = α·ϕ(𝐗, f̂) - β·ρ(𝐗, f̂) with 𝐗 = ⋃_j 𝐗^j and f̂ = ⋃_p f̂_p. Accuracy quantifies the classification quality of the AIITL system. Human effort refers to the number of instances that are classified by the human expert in relation to the total number of instances. For the calculation of the utility score, we follow <cit.> and set α = 1 and β = 0.5. That means, that we consider costs of inaccuracies to be two times higher than cost of manual review by human experts. In our experiments, we later demonstrate that our results are robust against changes in these parameters. §.§ Allocation mechanisms We implement two allocation stages that guide instances through the AIITL system. The Expert Consultancy Decision is part of the traditional HITL system, deciding whether an instance is known or unknown utilizing OOD detection. Instances from known classes are then classified by the general model, while unknown instances are passed to an expert for review. The Expert Selection allows us to defer detected unknown data to artificial experts that are incrementally trained using data and labels from the manual review of the human expert. For the Expert Selection, we reinterpret OOD detectors as allocation mechanisms. To this end, we fit an OOD detector for each of the artificial experts. The OOD detectors then claim an instance for the respective expert if the instance is considered to originate from a known class (the instance is drawn from the experts' training distribution). If an instance is claimed by none or multiple artificial experts, the instance is allocated to a human expert for manual review. In the following, we briefly introduce the employed allocation mechanisms: * ODIN is an OOD detector based on temperature scaling and perturbations of the input data <cit.>. Model- and domain-specific scores are compared to thresholds. Based on this score, an instance is either defined as originating from a known class or from an unknown class. * Mahalanobis-based OOD (MAHA) computes scores based on input perturbation and the Mahalanobis-distance between the input and the closest class-conditional Gaussian distribution <cit.>. These scores are leveraged to classify an instance to part of the set of known classes or stem from an unknown class. * Gating model is a separate model from the field of so-called “Mixture of Experts” that is trained to allocate instances to the artificial experts <cit.>. In contrast to the OOD detectors, this model is explicitly trained on allocating data instances to artificial experts. § EXPERIMENTS In the following, we describe our experimental setup. For the evaluation, we use popular datasets for image classification (e.g., CIFAR-10, SVHN). We evaluate the AIITL system under incremental data availability over 30 discrete steps. After training the general model, known and unknown data incrementally appears over the 30 steps. We train the “candidate” artificial experts on the labels that are iteratively generated by human experts during manual reviewing. In our implementations, we simulate a single human expert and assume a perfect classification accuracy for all input instances following related literature <cit.>. We evaluate the performance of the AIITL system in a separate test batch (after the 30 steps of incremental learning). We use a small proportion of the data from manual reviewing for validation and test of the artificial experts. We split the remaining data in 80% data for training and 20% for test. We then utilize the accuracy of artificial experts on the test set as indicator when a specific artificial experts should be included in the AIITL system. This is in line with <cit.>, who utilize the model confidence, but more directly related to the utility metric. When the test accuracy of an artificial expert exceeds a threshold of 95%, the system includes the artificial expert and considers it during the allocation of detected unknown data. In Section <ref>, we conduct sensitivity analyses on this threshold. §.§ Benchmarks Our evaluation procedure uses three benchmarks. First, we employ the general model from the HITL system without the option of a manual review from the human experts. This benchmark can be considered as a full automation baseline and prevents human effort but is likely to have a low classification accuracy in the presence of unknown data. Second, we utilize a traditional HITL system, where the general model classifies known instances and the human expert is consulted for manual reviewing of the unknown instances. This HITL system also constitutes the initial version of each AIITL system (step 1) and allows us to assess the merit of introducing artificial experts over the steps. Finally, we compare the AIITL system with the HITL system under perfect allocation of unknown data, which is an upper bound in terms of classification accuracy. §.§ Datasets For our experiments, we build a dataset out of four well-known image classification datasets. While CIFAR-10 <cit.> is used to train the general model and, thus, represents the known classes, three others represent unknown data classes: SVHN <cit.>, MNIST <cit.> and Fashion-MNIST <cit.>. CIFAR-10 consists of 60,000 images of ten different classes (e.g., airplane, car, truck). Fashion-MNIST contains 60,000 images of Zalando's articles from ten different classes. MNIST consists of 60,000 images of handwritten digits and SVHN includes 600,000 images of printed digits from pictures of house number plates from Google Street View. We utilize the predefined train-test splits for the datasets. §.§ Training We employ a Wide-ResNet-28-10 architecture <cit.> for the general model and a total of three artificial experts, where each artificial expert will be responsible for a separate set of unknown data. For the gating model, we make use of a DenseNet-121 <cit.> pretrained on ImageNet <cit.>. We train the general model for 200 epochs with a batch size of 256 and SGD as the optimizer with a learning rate of 0.1. The artificial experts are iteratively trained for 300 epochs with a 128 batch size using the SGD optimizer with a learning rate of 0.1. All models were trained on a NVIDIA A100 GPU. §.§ Results We present the utility scores of the AIITL system and the proposed baselines in Figure <ref>. After 30 steps of incremental learning, we observe that the proposed hybrid system outperforms the perfect HITL system with all allocation mechanisms. While the performance of the HITL system under perfect allocation results in an utility of 0.51, our system achieves scores of 0.92 based on the allocation using the gating model, 0.73 with allocation based on the Mahalanobis method, and 0.61 for allocation based on the ODIN method. This suggests that even when we assume a perfect allocation of known and unknown images (i.e., the general model only classifies known images and the human expert only reviews unknown images), the HITL system is less efficient than our hybrid system. Table <ref> reports the accuracy and human effort of the initial HITL system from step 1 (i.e., a traditional HITL system with imperfect allocation using allocation mechanisms) and the resulting AIITL system after 30 steps. The AIITL system outperforms the HITL system (in the absence of artificial experts) both in terms of accuracy and human effort for allocation based on the gating model and MAHA. The accuracy increases by 17pp from 75% to 92%. The human effort is reduced by 73pp and 34pp by the gating model and MAHA, respectively. For the allocation based on ODIN the accuracy slightly declines from 75% to 74%, while the human effort is reduced significantly from 0.73 to 0.27. Overall, this demonstrates that the suggested hybrid system improves the efficiency strongly compared to HITL systems. §.§ Analyzing the influence of varying weights for human effort on the system efficiency We conduct sensitivity analyses to better comprehend the influence of the weight of human effort in Eq. <ref>. For that, we vary the weight of human effort and depict examples in Figure <ref>. Across experiments, we find consistent improvements of our hybrid system over traditional HITL systems. In addition, we evaluate when traditional HITL systems become preferable in our experiments and found that HITL systems improve over our hybrid system only when human effort is more than ten times less important than accuracy (β < 0.1). In our experiments, AIITL systems consistently outperform HITL systems when accuracy is less than ten times more important than human effort (β > 0.1). Overall, our results demonstrate strong increases in the efficiency of the proposed hybrid AIITL system compared to traditional HITL systems. §.§ Applying the hybrid system in the context of semantically similar classes In the previous section, we focused on unknown classes that originate from other datasets than the training dataset for the general model (e.g., the general model was trained on CIFAR-10 images, while unknown data originates from SVHN). In this section, we show that our hybrid system improves the efficiency of HITL systems even when we sample a range of classes of the training set to be known, while other classes of this dataset are treated as unknown. For this, we select the first six classes of CIFAR-10 to be known, while we consider the remaining four as unknown classes. For example, the classes “car” and “deer” represent known classes, while the semantically similar classes “truck” and “horse” represents unknown classes (see Figure <ref>). We then train the general model on the known classes, while a single artificial expert and the human expert are later responsible for classifying the four unknown classes. Following common practice for detecting unknown classes that are part of the same set as known classes, we utilize softmax thresholding as a simple baseline to detect unknown instances <cit.>. We present our results in Figure <ref>. Overall, we observe that our hybrid system outperforms the traditional HITL system, the HITL system under perfect allocation, and the general model by a large margin. Our results are in line with the previous findings and demonstrate that the AIITL system can be leveraged in contexts where known and unknown classes are very similar. § DISCUSSION Our work results in a range of implications for research and practice in the field of HITL systems and hybrid systems that leverage both ML and human expert knowledge. First, we observe inefficiencies in traditional HITL systems that are created by allocation of tasks to a human experts that are difficult to classify for the ML model. This results in substantial human effort that ultimately can inhibit the application of HITL systems in general. We demonstrate that a hybrid team of artificial and human experts can significantly reduce the human effort and thereby increase the efficiency of the overall system. For practitioners, this implies that knowledge of domain experts can continuously be ingested into artificial experts (i.e., ML models) that can support human experts on repetitive tasks. This frees human capacities for creative, and more complex tasks. For research, our approach demonstrates that a hybrid team of human and artificial experts can successfully support a general ML model in the presence of data that originates from unknown classes. Especially in real-world applications, well-defined processes to handle data from unknown classes is essential to ensure a required level of system accuracy. Second, we observe two interesting trade-offs in the design of our hybrid system that need to be taken account for when utilizing our approach in real-world applications: (a) the choice of the allocation mechanism and (b) the required accuracy level of artificial experts to include them in the hybrid team. For the choice of the allocation mechanism, we observe a trade-off between the efficiency level (i.e., utility score) and the capability of adapting to data from unknown classes. For example, the gating model achieves a very strong utility, while it is limited in the ability to detect data from novel classes. The human expert first needs to feed the gating model with a sufficient number of instances from the encountered unknown class so that the gating model is able to allocate instances from that class correctly. On the other hand, ODIN and MAHA techniques achieve slightly lower performances (still outperforming traditional HITL systems by a large margin), but are adapting more quickly to a novel set of unknown classes. When designing hybrid systems in practice, it is critical to carefully select a suitable allocation mechanism. The second trade-off refers to the accuracy of artificial experts that is required to include the expert in the hybrid system. Generally, the higher the accuracy of an artificial expert, the better the overall performance of the system. However, achieving an increased performance of the artificial experts typically requires a higher amount of labeled data and, therefore, of manual reviews by the human expert. Defining the required level of accuracy of the artificial expert in applications in research and practice will require an in depth understanding of the costs of misclassifications (i.e., inaccuracies) and the costs of manual reviews by the human expert. Third, we see exciting parallels between our hybrid system and the field of human-AI collaboration as well as complementary team performance. In the latter, human and AI achieve a joint accuracy that none of them would have achieved individually. These approaches focus on maximizing the accuracy, while we add an additional dimension by focusing on the human effort. Coupling these approaches may result in even more promising performances in terms of the overall efficiency of the system. As any research, ours is not free of limitations. While we evaluated our approach on a wide range of image classification datasets, evidence from the field of, for example, natural language and from structured data is missing. Moreover, the approach needs to be validated in real-world situations with domain experts representing the “human-in-the-loop” to assess actual cost savings and efficiency improvements. Thus, as a next step, we want to explore variations of AIITL systems in different use cases and on real-world datasets. Finally, while our approach is general, we evaluated our approach in a linear setting due to computational limitations, meaning that the number of artificial experts linearly increased with the number of unknown domains. It will be interesting for future research to investigate how e.g., a constant number of artificial experts performs in settings with an increasing number of unknown domains. Thus, additional research on the optimal number of artificial experts is necessary. To increase the practical applicability, we are interested in exploring additional mechanisms to assess when an artificial expert should be included in the system. § CONCLUSION In this work, we argue that consulting human experts for manual review of difficult model classifications leads to a strong increase in human effort. Retraining an ML model on incrementally appearing novel classes is not an alternative as it leads to a significantly declining performance on the base classes—a phenomenon called “catastrophic forgetting”. To improve the efficiency of HITL systems, we introduce several artificial experts that learn to classify data from unknown classes based on the manual review from human experts. The resulting hybrid system of human and artificial experts improves the efficiency of HITL systems by reducing human effort and is robust against catastropic forgetting. We additionally introduce allocation mechanisms within our hybrid system that allow us to automatically assign detected unknown data to a suitable artificial expert. Overall, we find that our hybrid system outperforms traditional HITL systems by a large margin in terms of their utility—the combination of classification accuracy and human effort—across a range of benchmarks. agsm
http://arxiv.org/abs/2307.01865v1	20230704182452	Phase separation on varying surfaces and convergence of diffuse interface approximations	[ "Heiner Olbermann", "Matthias Röger" ]	math.AP	[ "math.AP", "49Q20, 49J45, 92C10" ]	fancy NRAuxiliary Calculation [RO,LE,LO,RE] [CO] [CE] theoremTheorem lemmaLemma propositionProposition conjectureConjecture corollaryCorollary definitionDefinition assumptionAssumption definition hypoHypothesis exampleExample noteNoteremarkRemark av hom iso =1 [Heiner Olbermann]UCLouvain, Belgium [email protected] [Matthias Röger]Department of Mathematics, Technische Universität Dortmund [email protected] Phase separation on varying surfaces and convergence of diffuse interface approximations Matthias Röger Received June 2023; accepted y ======================================================================================== Phase separation on varying surfaces and convergence of diffuse interface approximations Matthias Röger August 1, 2023 ======================================================================================== In this paper we consider phase separations on (generalized) hypersurfaces in the Euclidian space. We consider a diffuse surface area (line tension) energy of Modica–Mortola type and prove a compactness and lower bound estimate in the sharp interface limit. We use the concept of generalized BV functions over currents as introduced by Anzellotti et. al. [Annali di Matematica Pura ed Applicata, 170, 1996] to give a suitable formulation in the limit and achieve the necessary compactness property. We also consider an application to phase separated biomembranes where a Willmore energy for the membranes is combined with a generalized line tension energy. For a diffuse description of such energies we give a lower bound estimate in the sharp interface limit. AMS Classification. 49Q20, 49J45, 92C10 Keywords. Sharp interface limit, phase separation on generalized surfaces, Multiphase biomembranes. § INTRODUCTION Phase separation processes are ubiquitious in many applications from material sciences, physics and biology. The mathematical analysis is in many cases well understood and has seen many contributions from different communities. As two fundamental classes of descriptions we can distinguish sharp and diffuse interface models. In the sharp interface approach each material point belongs to exactly one of different phases or to lower dimensional interfaces between the distinct phases. Mathematically this can be described by a family of phase indicator functions (characteristic functions of subsets) that are generically discontinuos. In the simplest case of an open domain Ω⊂^n and two phases it is sufficient to consider one phase indicator function u:Ω→{0,1}. The relevant energy is in many cases proportional to the surface area of the phase interface. This in particular allows for a formulation of a surface energy functional that is finite only if u is of bounded variation and given by (u) = ∫_Ω \|∇ u\|. In the diffuse interface (or phase field) approach one allows for mixtures between phases and considers concentration fields of the different phases, which are generically continuous on the considered domain. A typical example of a diffuse interface energy is the Van der Waals–Cahn-Hilliard energy, given by _(u) := ∫_Ω(/2 \|∇ u\|^2 + 1/W(u)) dŁ^n, where W is a suitable double-well potential and u is a smooth function on Ω⊆^n. To achieve low energy values, the function u has to be close to the wells of the potential except for thin transition layers with thickness of order . One key question is whether a given diffuse interface model reduces to a sharp interface model in the sharp interface limit → 0. In the case of the energy (<ref>) this has been made rigorous in the celebrated result by Modica and Mortola <cit.>: the functionals _ converge in the sense of Γ-convergence to the perimeter functional, _→ c_0 , c_0 = ∫_0^1 √(2W). In the present contribution we consider phase separation processes on varifolds and currents, which present classes of generalized hypersurfaces. We are interested in applications, where both the location of the (generalized) hypersurfaces and the separation into phases represent degrees of freedom. Such a situation for example occurs in the modeling of biomembranes but again could be motivated by a variety of applications from different fields. Biomembranes consist of a large number of lipids and other ingredients and their shape and internal organization can adapt to the environment and dynamical changes of the structure are key to many functions. Variational models for multiphase membranes therefore depend on both the shape of the membrane and the internal composition. Typical ingredients are a bending energy with phase-dependent parameters and a line tension energy between phases. In the simplest situation of two phases, extensions of the classical one-phase shape energies of Canham and Helfrich <cit.> have been proposed in the form of a Jülicher-Lipowsky energy <cit.> ℰ(S_1,S_2) = ∑_j=1,2∫_S_j(k_1^j (H-H_0^j)^2+ k_2^j K) d^2 + σ∫_Γ 1 d^1. Here the biomembrane is represented by a closed surface S⊂^3 that is decomposed into a disjoint union of open subsets S_1,S_2 of S representing the phases, and their common boundary Γ. The first integral represents a bending energy that involves in general phase-dependent bending constants k_1^j,k_2^j and the spontaneous curvature H_0^j, j=1,2. The second integral in (<ref>) describes a phase separation (line tension) energy. A simple prototype of the bending contribution is the Willmore energy, that is obtained in the case H_0^j=0=k_2^j, j=1,2. Corresponding diffuse interface descriptions consider a smooth field u_ that describes the phase decomposition on the hypersurface S, and replace the line tension energy ∫_Γ 1 d^1 by a Modica–Mortola type approximation, which leads to energies of the form (S_,u_) = ∫_S_(k_1(u_) (H-H_0(u_))^2+ k_2(u_) K) d^2 + σ∫_S_(/2 \|∇ u\|^2 + 1/W(u)) d^2. In a regular setting, S_ would be assumed to be a hypersurface of class C^2 and u_ to be a smooth function on S_, with ∇ u_ denoting the tangential gradient on S_. A rigorous mathematical understanding of such energies is rather difficult and very little seems to be known in a general situation. Reductions to rotational symmetry have been studied in <cit.>. The only variational analysis in the general case seems to be the recent work of Brazda et.al. <cit.>. One of the challenges is that bounds on curvature energies alone do not induce good compactness properties in classes of smooth surfaces. Therefore it is necessary to consider generalized concepts of surfaces S_, such as (oriented) integer rectifiable varifolds with a weak second fundamental form as introduced by Hutchinson <cit.>. Such concepts have been used rather successfully to study Willmore or Canham–Helfrich type functionals. In the case of phase separated membranes an additional challenge is to describe the decomposition into distinct phases and phase interfaces. In <cit.> phases are characterized by oriented curvature varifolds with boundary, and suitable conditions are posed that guarantee an appropriate global structure. Below we will present an approach that in contrast describes phases by smooth phase fields or generalized indicator functions. This may offer a more concise description and embeds diffuse and sharp interface formulations in a common framework. On the other hand already the question of suitable concepts of Sobolev spaces on varifolds is non-trivial and it seems that it is not possible to guarantee all the `good' properties that are present in the case of Sobolev spaces over open domains in ^n, see the discussion in <cit.>. In the present work we use the concept of Sobolev spaces with respect to a given Radon measure as introduced by <cit.>, which are characterized as an appropriate closure of smooth test functions in the ambient space. Even less understood are concepts of BV-functions on generalized surfaces. Best suited for our purposes are BV-functions on currents as introduced by Anzellotti, Delladio and Scianna <cit.> (see also <cit.>) that are defined in terms of a generalized graph of a function over a current, see below for a precise definition. The authors there in particular present compactness and closure properties in such spaces of generalized BV-functions. Still, such results require additional (and rather restrictive assumptions) on the convergences of the underlying currents. These are in particular expressed by a suitable strict convergence property of currents (similar to the strict convergence of BV functions). For an alternative formulation of BV function with respect to measures see <cit.>. The main goal of the present paper is the derivation of rigorous sharp interface limits for diffuse phase separation energies of Modica–Mortola type for phase fields on varying (generalized) surfaces. We consider a generalization of the energy (<ref>) for pairs of currents and phase fields. The currents are assumed to represent boundaries of finite perimeter sets in ^n, and the phase fields are taken from the space of generalized H^1,p-functions with respect to the generalized surface area measure. Using such generalizations leads to a suitable formulation of a Modica–Mortola energy in the form I_(u_,μ_)=∫_^n(\|∇_μ_ u_\|^2+^-1W(u_)) μ̣_ , where μ_ is the generalized surface area measure, see Definition <ref> for a precise definition. The limit sharp interface energy is a function of pairs consisting of a current and a phase indicator function. We again consider currents S that arise from integration over the boundary of finite perimeter sets. The phase indicator functions u belong to the space BV(S) in the sense of <cit.>. We show that we can associate a generalized jump set J_u to u that consists of a ^n-2-rectifiable set. This in particular allows to assign a generalized phase interface area to this set, of the form I(u,S)=^n-2(J_u) . Our first main result is a compactness and lower bound estimate of the functionals I_ that correspond to a compactness and lower bound statement in the spirit of Gamma-convergence, see Theorem <ref> below. As we rely on a compactness result from <cit.> we in particular need to impose a crucial strict convergence assumption for the approximating currents. As a second main contribution we present an application of this result to a Jülicher–Lipowsky type two-phase biomembrane energy of the form (<ref>), where for simplicity we restrict ourselves to a bending energy of Willmore type. Here the strict convergence property needed for the application of Theorem <ref> is enforced by the assumption that the Willmore energy of the approximating varifolds is small, allowing to deduce a unit density property by the Li-Yau inequality <cit.>. Under these assumptions we are able to prove a compactness and lower bound result, see Theorem <ref> below. The paper is organized as follows. In the next section we present some notations and recall some relevant concepts of generalized surfaces and generalized Sobolev and BV-functions. The main result on a sharp interface limit of our generalized Modica–Mortola energy for varying surfaces is contained in Section <ref>, the application to two-phase biomembrane energies is content of the final Section <ref>. § NOTATION AND AUXILIARY RESULTS §.§ Currents and BV functions on currents We briefly recall the definition of (rectifiable) currents and in particular the notion of BV functions on currents from <cit.>. We denote by Λ_k(^N), 0≤ k≤ N and by Λ^k(^N) the spaces of all k-vectors and k-covectors, respectively, in ^N. We call v a simple k-vector if v can be written as v=v_1∧…∧ v_k. With Λ(N,k)={α=(α_1,…,α_k) : 1≤α_1<…<α_k≤ N}, (e_1,…,e_N) the standard orthonormal basis of ^N and (^1,…,^N) the corresponding dual basis of Λ^1(^N) we can represent any v∈Λ_k(^N) and any ω∈Λ^k(^N) uniquely as v=∑_α∈Λ(N,k) a_α e_α, ω=∑_α∈Λ(N,k) a_α e_α^α, with a_α∈, e_α=e_α_1∧…∧ e_α_k, ^α=^α_1∧…^α_k for all α∈Λ(N,k). This representation induces a scalar product and an induced norm \|·\| on Λ^k(^N) and Λ_k(^N). The canonical Hodge star isomorphism is denoted by :Λ_k(^N)→Λ_N-k(^N). For U⊂^N open and k ∈{0,…,N} we denote by ^k(U) the space of all infinitely differentiable k-differential forms U→Λ^k(^N) with compact support in U, equipped with usual topology of distributions. The space _k(U) of k-currents on U is the dual of ^k(U). We denote by ∂ T∈_k-1(U) the boundary of T ∈_k(U), defined by ∂ T,ω = T,dω for all ω∈^k-1(U). We say a k-current T on U⊂^N is representable by integration if [U]T:=sup{T,φ : φ∈^k(U),φ_L^∞≤ 1}<∞ . In that case by the Riesz representation theorem there exists a Radon measure T on U and a T-measurable function T⃗:U→Λ_k^N satisfying \|T⃗\|=1 a.e. on U such that T,ω :=∫_U ω,T⃗ dT for all ω∈^k(U). We call T the mass measure of T and T=[U]T the total mass (in U). Given a k-rectifiable set M⊂^N for ^k-almost any p ∈ M there is a well-defined measure-theoretic tangent space T_p M. We say that a map τ: M→Λ_k(^N) is an orientation on M if such a map is ^k-measurable and τ(p) is a unit simple k-vector on ^N that spans T_pM for ^k-almost any point p∈ M. Let ρ M →^+ be a ^k-locally summable function. Then, if M⊂ U with U open in ^N we can define a current T=M,ρ,τ∈_k(U) by ⟨ T,ω⟩:=∫_M ⟨ω,τ⟩ ρ d^k for all ω∈^k(U). The set _k(U) of currents T∈_k(U) which can be written in the form T=M,τ,ρ as above are called rectifiable currents, the function ρ is then called the multiplicity of T. If in addition ρ is integer-valued we call T integer-rectifiable and write T∈ℐ_k(U). A current T∈ℐ_k(U) with ∂ T∈ℐ_k-1(U) is called integral. In the context of graphs over sets in ^n it is often useful to consider the variable (x,y)∈^n+1=^n×=^n_x⊗_y with ^n_x=^n×{0}, _y={0_^n}×. We denote by e_y:=e_n+1 the (n+1)th vector and by the (n+1)th covector of the standard bases in ^n+1 and Λ^1^n+1, respectively. The stratification of a k-vector ξ∈Λ_k(^n_x⊗_y) is given by the unique decomposition ξ=ξ_0+ξ_1, ξ_0 ∈ Λ_k(^n_x), ξ_1∈Λ_k-1(^n_x)∧Λ_1(_y), that is ξ_0 = ∑_α∈Λ(n,k)^α,ξ e^α, ξ_1 = ∑_β∈Λ(n,k-1)^β∧,ξ e^β∧. The corresponding stratification of a current T∈Λ_k(^n_x⊗_y) is given by []T_0,∑_α∈Λ(n,k) a_α^α+ ∑_β∈Λ(n,k-1)a_β^β∧ = []T,∑_α∈Λ(n,k) a_α^α, []T_1,∑_α∈Λ(n,k) a_α^α+ ∑_β∈Λ(n,k-1)^β∧ = []T,∑_β∈Λ(n,k-1)a_β^β∧, for a_α,a_β∈ C^∞_c(^n_x⊗_y). In the case that T=M,ξ,ρ is rectifiable we obtain for j=0,1 T_j,ω = ∫_M ω,ξ_jρ d^k for all ω∈^k(^n_x⊗_y). Let M⊂^n_x⊂^n+1 be a k-rectifiable set, and let p:^n+1→^n_x denote the projection on ^n_x. For the rectifiable k-current S=M,τ,ρ and a function u:M→ we consider the set between the graph of u and ^n_x, E_u,S = {(x,y)∈ M⊗_y : 0<y<u(x) if u(x)>0, u(x)<y<0 if u(x)<0} and define for (x,y)∈ E_u,S an induced orientation and induced multiplicity by α(x,y) = e_y∧τ(x) if y>0 -e_y∧τ(x) if y<0 θ(x,y) =ρ(x). We then obtain the generalized graph of u over S as T_u,S = -∂ [[E_u,S,α,θ]]+S_0 . Let a rectifiable k-current S=M,τ,ρ, M⊂^n_x and a function u:M→ be given. Then we say that u is a function of bounded variation over S and write u∈ BV(S) if the total mass of T_u,S is bounded, T_u,S<∞. From the definition, we straightforwardly get T_u,S(φ_α^α)= -∫_M⟨^α,τ⟩φ_α(x,u(x))̣̋^k(x) T_u,S(φ_β^β∧ỵ) =-∫_M∫_0^u(x)∑_i=1^n _i∧^β,τ∂φ_β(x,y)/∂ x_iỵ̣̋^n-1(x) for α∈Λ(n,k) and β∈Λ(n,k-1). If S=M,τ,ρ is in ℐ_k(_x^n) and u∈ BV(S) then the boundary rectifiability theorem <cit.> implies that ∂E_u,S,α,θ is integer rectifiable and hence T_u,S∈ℐ_k(_x^n⊗_y). Therefore there exists a k-rectifiable set ⊂^n+1, a multiplicity function θ:→ and an orientation ξ:→Λ_k(_x^n⊗_y) such that T_u,S = ,ξ,θ. Let T_j be a sequence of n-dimensional integer multiplicity rectifiable currents in ^n+1 such that (i)T_j T (ii) sup_j(yT_j,0+T_j)<∞ (iii) lim_j→∞p_# T_j=p_#T Then we say that T_j converges strictly to T, T_jc^T. In the case that T_j=T_u_j,S_j, T=T_u,S are generalized graphs with S_j=M_j,τ_j,ρ_j and S=M,τ,ρ being k-rectifiable currents we obtain yT_j,0 = ∫_M_j \|u_j\|ρ_j d^k, see <cit.>. Moreover, in this case the convergence T_j T implies S = p_#T_u,S = lim_j→∞ p_#T_u_j,S_j = lim_j→∞S_j, and the convergence in the third item is equivalent to S_j→S, i.e. ∫_M_jρ_j d^k → ∫_M ρ d^k. §.§ Oriented varifolds We introduce oriented varifolds as in <cit.>. Let G^o(n,m) denote the set of m-dimensional oriented subspaces of ^n. This set may be identified with the unit simple elements of Λ_m^n. An oriented m-varifold is an element of ℳ(^n× G^o(n,m)). If M is m-rectifiable with orientation τ, and if θ_±:M→^+_0 are locally ^m-summable multiplicity functions with θ_++θ_->0, then we write v(M,τ,θ_±) for the associated rectifiable oriented varifold v(M,τ,θ_±)(φ)=∫_^n× G^o(n,m)(θ_+(x)φ(x,τ(x))+θ_-(x)φ(x,-τ(x)))̣̋^m(x) . We then can pass to a representation M,τ̃,θ̃_± such that θ̃_+≥θ̃_-. In the following we always assume this additional property. If the multiplicity functions θ_± are _0 valued, then we say that v(M,τ,θ_±) is an integral oriented varifold. The class of m-dimensional oriented (resp. rectifiable oriented, resp. integral oriented) varifolds is denoted by V^o_m(^n) (resp. RV^o_m(^n), resp. IV^o_m(^n)). To V∈ V^o_m(^n), we associate the m-dimensional current c(V)(φ)=∫_^n× G^o(n,m)⟨φ(x),ξ⟩Ṿ(x,ξ) and observe that in the case of V∈ RV^0_m(^n), V=v(M,τ,θ_±) c(V)(φ)=∫_M⟨τ(x),φ(x)⟩ (θ_+-θ_-)(x)^m(x). We remark that convergence as oriented varifolds implies convergence of the associated currents, V_j V c(V_j) c(V) . §.§ Sobolev functions with respect to measures For a Radon measure μ on an open set Ω⊂^n, we introduce the Sobolev space H^1,p_μ(Ω) as in <cit.>: We consider the linear operator A:(^n)→ L^p_μ(Ω;^n) defined by (Aφ)(x)=P_μ(x)∇φ(x) , where P_μ(x) is the projection onto the tangent space of μ at x. We do not give the definition of tangent spaces of measures from <cit.> here, but only note that in the case that μ=V is the mass measure of an integer rectifiable k-varifold with locally bounded first variation, then P_μ(x) is the projection onto the k-dimensional tangent space of μ in x, which exists for μ almost every x. With this definition the operator A is closable in the norm given by φ_L^p_μ(Ω)+Aφ_L^p_μ(Ω;^n), and H^1,p_μ(Ω) is defined as the domain of the unique closed extension A̅, which is denoted by ∇_μ. For the case that μ=V is the mass measure of an integer rectifiable k-varifold with locally bounded first variation an equivalent definition for Sobolev spaces has been given in <cit.>, based on the previous work <cit.>. §.§ Measure-function pairs We use the definitions of measure-function pairs from <cit.>, which in turn are based on <cit.>. Let Ω⊂^n. If μ∈ℳ(^n) and f∈ L^1_loc,μ(Ω;^m), then we say that (μ,f) is a measure-function pair over Ω with values in ^m. Let {(μ_k,f_k):k∈} and (μ,f) be measure-function pairs over Ω with values in ^m, and 1≤ p<∞. (i) We say that (μ_k,f_k) converges weakly in L^p to (μ,f) and write (μ_k,f_k) (μ,f) in L^p if μ_kμ in ℳ(Ω), μ_k f_kμ f in ℳ(Ω;^m), and f_k_L^p_μ_k(Ω;^m) is uniformly bounded. (ii) We say that (μ_k,f_k) converges strongly in L^p to (μ,f) and write (μ_k,f_k)→ (μ,f) in L^p if for all φ∈ C_c^0(Ω×^m), lim_k→∞∫_Ωφ(x, f_k(x))μ̣_k(x)=∫_Ωφ(x, f(x))μ̣(x) , and lim_j→∞∫_S_kj\|f_k\|^pμ̣_k=0 uniformly in k , where S_kj={x∈Ω:\|x\|≥ j or \|f_k(x)\|≥ j}. §.§ Convergence of BV functions over varying currents Consider a sequence (S_j)_j of (n-1)-rectifiable currents in ^n. Moreover, let a sequence (u_j)_j be given with u_j∈ BV(S_j), and denote the associated generalized graphs by T_j=T_u_j,S_j Assume that sup_j u_j_L^p(S_j)<∞ for some 1<p<∞ , and that for some (n-1)-rectifiable current S and some u∈ BV(S) T_j T=T_u,S , S_j→S . Then the strong measure-function pair convergence (S_j,u_j) →(S,u) holds in any L^q, 1≤ q<p. We first deduce from (<ref>) for some Λ>0 and any j∈, 1≤ q<p, R>0 Λ^p ≥∫ \|u_j\|^pS_j≥ R^pS_j({\|u_j\|>R}) and ∫_{\|u_j\|>R}\|u_j\|^qS_j ≤Λ^q (S_j({\|u_j\|>R}))^1-q/p ≤Λ^p R^-p+q → 0 (R→∞). Similarly, ∫_{\|x\|>R}\|u_j\|^qS_j ≤Λ^q sup_j∈(S_j({\|x\|>R}))^1-q/p → 0 (R→∞) by Prokhorov's Theorem <cit.> and (<ref>). This verifies the second condition in the definition of strong measure-function pair convergence in L^q. Next, for any α∈Λ(n,n-1), any φ_α∈ C^0_c(^n), and any ψ∈ C^0_c(^n×) we let η_α(x,y)=φ_α(x)ψ(x,y) and deduce from (<ref>) and (<ref>) ∫_^nS⃗(x),^αφ_α(x)ψ(x,u(x)) S(x) =S,ψ(·,u)φ_α^α = -T_u,S,η_α^α = -lim_j→∞T_u_j,S_j,η_α^α = lim_j→∞S_j,ψ(·,u_j)φ_α^α = lim_j→∞∫_^nS⃗_⃗j⃗,^αφ_α(x)ψ(x,u_j(x)) S_j(x). Since α∈Λ(n,n-1), φ_α are arbitrary we arrive at (ψ(·,u_j)S⃗_⃗j⃗, S_j) (ψ(·,u)S⃗, S). Next, by the Reshetnyak continuity theorem <cit.> and (<ref>) we have for any φ∈ C^0_c(^n×^n-1) lim_j→∞∫_^nφ(·,S⃗_⃗j⃗)S_j = ∫_^nφ(·,S⃗)S, which implies the strong measure-function pair convergence (S⃗_⃗j⃗,S_j) →(S⃗,S) in any L^r, 1≤ r<∞. Together with (<ref>) we deduce from <cit.> that lim_j→∞(ψ(·,u_j),S_j) =lim_j→∞(ψ(·,u_j)S⃗_⃗j⃗·S⃗_⃗j⃗, S_j) = (ψ(·,u)S⃗·S⃗, S) = (ψ(·,u), S) in the sense of weak measure-function-pair convergence in any L^r, 1≤ r<∞. Since ψ was arbitrary (<ref>) follows. § COMPACTNESS AND LOWER SEMICONTINUITY FOR A MODICA-MORTOLA FUNCTIONAL ON VARYING SURFACES We first define a generalized Modica–Mortola functional. Fix a nonnegative continuous double-well potential W such that {W=0}={0,1} and such that for some T,c>0, p≥ 2 c\|t\|^p ≤ W(t)≤1/c\|t\|^p for all \|t\|≥ T . Below it will be convenient to fix a first integral of √(W), ψ(r)=∫_0^r √(W(t))ṭ for r∈ and to define the surface tension constant k=∫_0^1√(W(r))ṛ . For >0, a Radon measure μ_ on ^n and a function u_∈ H^1,p_μ_(^n) we define the Modica–Mortola-type functional I_(u_,μ_)=∫_^n(\|∇_μ_ u_\|^2+^-1W(u_)) μ̣_∈ [0,∞] . In the following we consider a more restrictive setting where μ_ is the area measure of the reduced boundary of a finite perimeter set. In the sharp interface limit of phase fields we will obtain phase indicator functions with a generalized BV-regularity. In Proposition <ref> below we will justify the following notion of jump sets for such phase indicator functions. Consider a set E⊂^n of finite perimeter with inner unit normal ν=ν_E:∂_* E→^n-1 of E, and set S=∂_E,ν,1. When u∈ BV(S) with u(x)∈{a,b} ^n-1-almost everywhere for some a≠ b we write J_u:=(∂u^-1(b),ν,1) and call this set the jump set of u. We will now state our main result in this section, which is a generalization of the lim inf statement in the Modica–Mortola Gamma-convergence statement. Consider p≥ 2 with (<ref>). Let a family (E_)_ of finite perimeter sets in ^n with associated perimeter measures μ_=^̋n-1∂_ E_ and a sequence (u_)_ in H^1,p_μ_(^n) be given. Assume that for some set E of finite perimeter χ_E_→χ_E strictly in BV(^n), that is χ_E_→χ_E in L^1(^n), ∇χ_E_∇χ_E, and lim_→ 0^̋n-1(∂_* E_)=^̋n-1(∂_* E), and let ν=ν_E:∂_* E→^n-1 denote the inner unit normal of E, and set S=[[∂_E,ν_E,1]], μ=^̋n-1∂_* E. Let us further assume that for some Λ>0 I_(u_,μ_) < Λ. Then there exists u∈ BV(S) and a subsequence → 0 such that the following holds: u(x)∈{0,1} for ^̋n-1-almost all x∈∂_E, (μ_,u_) →(μ,u) as measure-function pairs in L^q for any 1≤ q<p. Moreover, {u=1}, ν,1 is an integral (n-1)-current and we have the lower estimate lim inf_→ 0 I_(u_,μ_) ≥ 2k^̋n-2(J_u), with the generalized jump set J_u as in Definition <ref>. We first prove that we can assume u_∈(^n) for all >0. In fact, by the definition of H^1,p_μ_(^n) we can approximate (u_)_ by a family (ũ_)_ in (^n) such that (<ref>) holds for (ũ_)_, such that lim inf_→ 0 I_(ũ_,μ_) =lim inf_→ 0 I_(u_,μ_) , and u_-ũ__L^p(μ_)→ 0 (→ 0) . Let us assume we have proved (μ_,ũ_)→ (μ, u) as measure-function pairs in L^q for some u as in the statement of the present theorem. For any Lipschitz-continuous function ψ∈ C^0_c(^n×) we then deduce \|∫(ψ(·,u_)-ψ(·,ũ_)) dμ_\| ≤ψ_C^0,1(^n+1)u_-ũ__L^p(μ_)μ_(^n)^1-1/p → 0 with → 0. An approximation argument yields for all ψ∈ C^0_c(^n×) lim_→ 0∫ψ(·,u_) dμ_ =lim_→ 0∫ψ(·,ũ_) dμ_ =∫ψ(·,u) dμ and therefore the strong measure-function pair convergence of the original sequence (μ_,u_) to (μ,u). Therefore it is sufficient to prove the Theorem for sequences (u_)_ in (^n), which we assume in the remainder of the proof. We next consider the modified phase fields v_:=ψ∘ u_ and the generalized graphs T_:=T_v_,E_. Since u_∈(^n) we can apply the chain rule ∇_μ_v_=√(W(u_))∇_μ_u_ and obtain the representation T_v_,S_ = (Φ_)_# (S_), Φ_(x):=(x,v_(x)) for x∈∂_* E_ . Therefore we may apply the usual Modica-Mortola trick to obtain ∫_∂_* E_ \|∇_μ_v_\|̣̋^n-1 ≤∫_∂_* E_√(W(u_))\|∇_μ_ u_\|̣̋^n-1 ≤1/2 I_(u_,μ_)≤1/2Λ . Furthermore we have T_ =(Φ_)_# (S_) =∫_∂_E_√(1+\|∇_μ_v_\|^2)̣̋^n-1 ≤^̋n-1(∂_E_)+ ∫_∂_* E_ \|∇_μ_v_\|̣̋^n-1 ≤^̋n-1(∂_E_)+ 1/2 I_(u_,μ_)≤ C(Λ) and y T_,0 = v__L^1(μ_)≤ C(Λ,W). This implies that the currents T_ are uniformly bounded in mass and that (v_)_ is bounded in an L^1 sense. Since ∂ T_=0, we may use the compactness theorem for integral currents, to obtain a weak limit T of T_. By (<ref>), (<ref>) and (<ref>) we may use <cit.> to conclude that there exists v∈ BV(S) such that T=T_v,S . Furthermore, the assumptions on W induce that (v_)_ is bounded in L^p(μ_). Together with the strict BV-convergence of χ_E_ and the convergence T_ T_u,S we obtain by Proposition <ref> that we have the strong measure-function pair convergence (μ_,v_) →(μ,v) in any L^q, 1≤ q<p. Since ψ is continuous we deduce that (<ref>) holds for u:=ψ^-1(v). Using the measure-function pair convergence (<ref>) and the energy bound (<ref>) it follows that for any η∈ C^0_c(^n×) ∫_^nη(·,u)W(u) μ̣=lim_→ 0∫_^nη(·,u_)W(u_) μ̣_ =0 . Since η was arbitrary (<ref>) holds. Clearly this implies that v(x)∈{0,k} for ^̋n-1 almost all x∈∂ E_. Using Proposition <ref> below we next obtain that the generalized jump set J_u is (n-2)-rectifiable with T_v,S-S =k^̋n-2(J_u)=k(T_u,S-S) . To prove the lower estimate (<ref>), we note that mass is weakly lower semicontinuous under weak convergence, and that by strict BV-convergence (<ref>) lim_→ 0S_=lim_→ 0^̋n-1(∂_* E_)=^̋n-1(∂_* E)=S. We then deduce from (<ref>), (<ref>) and (<ref>) that k^̋n-2(J_u) =T_v,S-S ≤lim inf_→ 0(T_-S_) ≤lim inf_→ 0((^̋n-1(∂_* E_)+1/2I_(u_,μ_)) - ^̋n-1(∂_* E_)) = lim inf_→ 01/2I_(u_,μ_) , which proves (<ref>). Let a set E⊂^n of finite perimeter and a constant k>0 be given, let ν=ν_E:∂_E→^n-1 denote the inner unit normal, and consider the associated current S=∂_E,τ,1. If w∈ BV(S) and w∈{0,k} ^̋n-1∂_* E almost everywhere, then {w=k},ν,1 is an integral (n-1)-current, and the generalized jump set J_w=(∂{w=k},ν,1) , satisfies k^̋n-2(J_w)=T_w,S-^̋n-1(∂_* E) . Since T:=T_w,S is representable by integration, we obtain T = T(^n+1) = T(^n×{0}) + T(^n× (0,k)) + T(^n×{k}), where we have also used that the definition of the graph implies T=T (^n× [0,k]). Using (<ref>), we observe that T{(x,k):x∈∂_E} =sup{∫_{x∈∂_ E:w(x)=k}∑_α∈Λ(n,n-1)⟨^α,ν⟩φ̃_α(x)̣̋^n-1(x): φ̃_α∈ C^1_c(^n), ∑_α∈Λ(n,n-1)φ̃_α^α≤ 1} =^̋n-1({x∈∂_ E:w(x)=k}) , with an analogous equality for k replaced by 0. Therefore T=^̋n-1(∂_E) + T(^n× (0,k)) . Let Y:^n+1→ be given by Y(x,y)=y, and let ⟨·,·,·⟩ denote the slicing operation from <cit.>. Consider an arbitrary φ∈^n-2(^n+1) and the stratification φ=φ_0+φ_1 with φ_0=∑_β∈Λ(n,n-2)φ_β^β, φ_1=∑_γ∈Λ(n,n-3)φ_γ^γ∧ỵ . Then, using (<ref>) and the definition of the slicing operation we calculate that for all 0<s<k ⟨ T,Y,s⟩(φ) =lim_ρ↓ 0(-1)/2ρ(Tχ_{y:s-ρ<y<s+ρ}ỵ)(φ) =lim_ρ↓ 0(-1)/2ρ∫_∂_E∫_0^w(x)⟨_̣xφ_0(x,y),ν(x)⟩χ_(s-ρ,s+ρ)(y)ỵ̣̋^n-1(x) = - ∫_{w=k}⟨_̣x(φ_0 (x,s)),ν(x)⟩̣̋^n-1(x) = - {w=k},ν,1(_̣x φ_0 (·,s)) . This implies that for all 0<s<k ⟨ T,Y,s⟩(φ) = -∂{w=k},ν,1(φ_0(·,s)) =∂{w=0},ν,1(φ_0(·,s)) , where in the last equality we have used that ∂_E,ν,1=∂E,e_x,1 for e_x=e_1∧…∧ e_n, hence 0=∂∂_E,ν,1 =∂{w=k},ν,1-∂{w=0},ν,1 . The representation (<ref>) in particular implies that ∂{w=k},ν,1 is an integral (n-2)-current and that {w=k},ν,1 is an integal (n-1)-current. Furthermore, we have ⟨ T,Y,s ⟩ =∂({w=k},ν,1)×{s} =N×{s}⊂^n+1 , where N⊂^n is some (n-2)-rectifiable set. It follows that ⟨ T,Y,s⟩=^̋n-2(N) for 0<s<k . By <cit.> this yields (k-2)^̋n-2(N)=∫_^k-⟨ T_w,S,Y,s⟩ṣ = T{<y<k-} and hence we obtain in the limit → 0 that T(^n× (0,k)) =T{(x,y):0<y<k}=k ^̋n-2(N) . Together with (<ref>) this proves the claim. For a corresponding Gamma-convergence result one would need to complement Theorem <ref> by an upper bound estimate. It is rather straightforward to prove such a statement in a more regular setting: Assume that M⊂^n is a smooth oriented n-1-dimensional submanifold, V=v(M,ν,1), S=[[M,ν,1]], u∈ BV(S) with u∈{0,1} ^̋n-1 M almost everywhere, J_u a smooth curve on M. Then there exists a sequence (u_)_ in H^1,p_^̋n-1 M(^n) such that T_u_,S T_u,S and lim sup_→ 0 I_(S,u_)≤ 2k^̋n-2(J_u) . Indeed, one can adapt the proof from <cit.> rather straightforwardly to obtain the above statement. The proof of the upper bound becomes non-trivial as soon as one allows for non-smooth surfaces M= S. We do not discuss this question here and leave it open for future research. § APPLICATION TO A TWO-PHASE MEMBRANE In this section we consider a class of two-phase membrane energies that consist of a bending contribution given by a phase-dependent Willmore functional and a line tension energy. Such kind of energies in particular appear as reductions of the Jülicher–Lipowsky energy discussed in the introduction. The main result of this section connects diffuse and sharp interface description of such energies. Let V∈ IV_2^o(^3) be an integer-rectifiable oriented 2-varifold in ^3 with weak mean curvature H_V∈ L^2_V(^3;^3) and consider a V-measurable function u on M=(V) with u(x)∈{0,1} for ^2-almost every x∈ M. We then define for given constants a_1,a_2,k>0 (u,V) = 1/4∫_^3(a_1u + a_2(1-u)) \|H_V\|^2V , (u,V) := (u,V) + 2k^1(J_u) , where J_u denotes the jump set of u as introduced in Definition <ref>. In the corresponding diffuse interface description we will use the perimeter approximation from Section <ref>. In particular we assume a given nonnegative double-well potential W with {W=0}={0,1} that satisfies (<ref>). We define ψ as in (<ref>), the constant k=ψ(1) as in (<ref>), and the Modica–Mortola type functional I_ as in (<ref>). Finally, we fix a smooth interpolation between the phase dependent constants a_1,a_2 from Definition <ref> of the form a^ω̅(r) = ω̅(r)a_1+(1-ω̅(r))a_2 , where ω̅∈ C^∞_c() satisfies 0≤ω̅≤ 1 and ω̅(0)=0, ω̅(1)=1. Let V_∈ IV_2^o(^3) be an integer-rectifiable oriented 2-varifold in ^3 with weak mean curvature H_:=H_V_∈ L^2_V_(^3;^3) and let u_∈ H^1,p_V_(^3) be given. Then we define _(u_,V_) := 1/4∫ a^ω̅∘ u_ \|H_\|^2 dV_, _(u_,V_) := _(u_,V_) + I_(u_,V_) , where a^ω̅ is as in (<ref>) and I_ as in Definition <ref>. Again we will consider in our main result a more restrictive setting, in particular to enforce the crucial strict convergence property that is needed in Theorem <ref>. Here we use the Li-Yau inequality <cit.>, that guarantees for any V∈ IV_2^o(^3) θ_2(·,V)≤max(a_1^-1,a_2^-1)(u,V)/4π , where θ_2(·,V) denotes the two dimensional density of V, θ_2(x,V)=lim_r→ 0V(B(x,r))/π r^2 . Since V is rectifiable, the limit in this defintion exists V almost everywhere. From (<ref>) we deduce in particular that V has unit density if (u,V)<8πmin{a_1,a_2}. Let p≥ 2 be as in (<ref>). Suppose (E_)_ is a sequence of finite perimeter sets in ^3 and let μ_=^2∂_E_, ν_:∂_E_→^2 the inner normal, and V_=∂_E_,ν_,1,0. Consider in addition a sequence (u_)_ of phase fields u_∈ H^1,p_μ_(^3). Assume that for some Λ>0 ^n-1(∂_E_) + _(u_,V_) ≤Λ for all >0 , _(u_,V_) < 8πmin{a_1,a_2} . Then there exists a subsequence → 0, a finite perimeter set E⊂^3 and a function u:∂_E→{0,1} such that with V=∂_* E,ν_E,1,0, S=c(V) the following holds: u belongs to BV(S) and V_ → V in IV^o_2(^3) , (μ_,u_) → (μ,u) as measure-function pairs in L^q , for any 1≤ q<p. Moreover, it holds the lower bound estimate lim inf_→ 0_(u_,V_) ≥(u,V) . The proof will be given below after some preparations. In the proof we will obtain some additional properties. In particular, we show the lower semicontinuity of both the bending and the line tension energy contribution to _ separately, and we obtain the strict convergence of the graphs of ψ∘ u_ to the generalized graph of ku. The next lemma is used to show the strict convergence of generalized graphs associated to u_. Let V_→ V in IV^o_m(^n) such that θ^_m(x,V) =1 for V-almost every x∈^n . Then we have that c(V_)→c(V) . By (<ref>) convergence as oriented varifolds implies weak convergence as currents, c(V_) c(V). The lower semicontinuity of the mass under weak convergence of currents yields c(V)≤lim inf_→ 0c(V_) . Let us write V_ =M_,τ_,θ_,±, V =M,τ,θ_± . By assumption θ_++θ_- =θ_+-θ_-=1 V-almost everywhere. Since mass is continuous under varifold convergence we deduce lim sup_→ 0c(V_) ≤lim sup_→ 0V_(^n) =V(^n) =∫_M (θ_++θ_-) ̣̋^m =∫_M (θ_+-θ_-) ̣̋^m =c(V) . Together with (<ref>) this completes the proof. By (<ref>) we may pass to a further subsequence such that there exists a set of finite perimeter E⊂^3 and an integral varifold V∈ IV_2^o(^3) with χ_E_ →χ_E in L^1(^3), χ_E_χ_E in BV , V_ V as oriented varifolds, V≥ \|∇χ_E\| . By (<ref>) and the Li-Yau inequality (<ref>), we have that V almost everywhere θ_2(·,V)=1. Hence S:=c(V)=[[∂_ E,ν_E,1]] and by Lemma <ref> it follows that lim_→ 0^̋n-1(∂_ E_)=lim_→ 0c(V_)=c(V)=^̋n-1(∂_* E) , which shows the strict BV-convergence (<ref>) of the sets E_, >0. Therefore, we can apply Theorem <ref> (see also its proof) and obtain the existence of some u∈ BV(S) with u∈{0,1} S-almost everywhere such that T_v_,S_c^* T_ψ(u),S and such that the measure-function pair convergence (<ref>) and the lower estimate lim inf_→ 0 I_(u_,V_)≥ 2k^̋n-2(J_u) holds. It remains to show the lower semicontinuity statement for . By V_ V as varifolds and by the uniform bound on H__L^2_μ_(^3;^3), we have the weak convergence of measure-function pairs (V_,H_) (V,H) in L^2 . By (<ref>) we further deduce the strong measure-function pair convergence (V_,√(a^ω̅(u_))) →(V,√(a^ω̅(u))) in L^2 . By <cit.>, (<ref>) and (<ref>) may be combined to yield the weak convergence (V_, H_√(a^ω̅(u_)))(V,√(a^ω̅(u))) in L^1 . But clearly H_√(a^ω̅(u_))_L^2_μ_(^3;^3)≤√(a^ω̅)_L^∞H__L^2_μ_(^3;^3) yields a uniform bound on the L^2 norms, and the weak convergence in L^1 can be upgraded to the weak convergence in L^2 (V_, H_√(a^ω̅(u_)))(V,√(a^ω̅(u))) in L^2 . By <cit.> we obtain (u,V) =∫(a_1u + a_2(1-u))\|H_V\|^2 dV = ∫ \|√(a^ω̅(u))H_V\|^2 dV ≤lim inf_→ 0∫ \|√(a^ω̅(u_))H_\|^2 dV_ =lim inf_→ 0_(u_,V_). This completes the proof of the theorem. alpha
http://arxiv.org/abs/2307.05382v1	20230702142812	Protecting the Future: Neonatal Seizure Detection with Spatial-Temporal Modeling	[ "Ziyue Li", "Yuchen Fang", "You Li", "Kan Ren", "Yansen Wang", "Xufang Luo", "Juanyong Duan", "Congrui Huang", "Dongsheng Li", "Lili Qiu" ]	eess.SP	[ "eess.SP", "cs.AI", "cs.LG" ]	A novel multi-step method for the partial pole assignment in symmetric quadratic pencil with time delay [ August 1, 2023 ======================================================================================================== empty empty A timely detection of seizures for newborn infants with electroencephalogram (EEG) has been a common yet life-saving practice in the Neonatal Intensive Care Unit (NICU). However, it requires great human efforts for real-time monitoring, which calls for automated solutions to neonatal seizure detection. Moreover, the current automated methods focusing on adult epilepsy monitoring often fail due to (i) dynamic seizure onset location in human brains; (ii) different montages on neonates and (iii) huge distribution shift among different subjects. In this paper, we propose a deep learning framework, namely , to address the exclusive challenges with exquisite designs at the temporal, spatial and model levels. The experiments over the real-world large-scale neonatal EEG dataset illustrate that our framework achieves significantly better seizure detection performance. § INTRODUCTION As neurological disorders caused by epilepsy, seizures are associated with high morbidity and mortality for new born infants <cit.>. And the timely detection and proper treatments become a common yet vital practice in the Neonatal Intensive Care Unit (NICU). Although it can be observed and detected through electroencephalogram (EEG) of an individual which has been considered as the “golden standard", it requires significant human efforts from experts for monitoring, caring and intensive diagnosis. Thus, building an accurate automated framework to detect seizure events in real time can help free experts from tedious works to better focus on the treatments. Automated seizure detection attracts lots of attention in both signal processing and machine learning (ML) communities. Signal processing methods <cit.> mainly focus on statistical or hand-crafted features of EEG, requiring lots of expert knowledge. Besides, ML approaches <cit.> rely on data-driven paradigm to process EEG for seizure detection. But these solutions are mainly focusing on adult epilepsy monitoring and can not be directly applied to neonates considering several clinical differences. While some deep learning methods <cit.> were proposed for EEG seizure detection in neonates, they did not explicitly handle the critical challenge in this problem, i,e, dynamics in the number of electrodes and seizure patterns, which often introduces more artifacts as burdens of detecting informative seizure events. To address these exclusive challenges, we propose a deep learning framework, SpaTiAl-Temporal EEG Network (). In , we incorporate a channel-level temporal modeling component for fine-grained brain signal processing, which is more flexible when tackling varying yet limited EEG channels on neonates. After the temporal modeling process, we leverage a spatial fusion module to comprehensively synthesize channel-level temporal patterns for detection. This process has been optimized through an end-to-end manner without explicitly signal preprocessing or human-crafted artifact removal. Moreover, we propose a model-level ensemble by dynamically aggregating the outcomes of diverse spatial-temporal deep models to better generalize among different neonates. We conduct experiments on a real-world large-scale neonatal dataset, to compare our model with several competitive baselines focusing on adult seizure detection and illustrate that our method has achieved significantly better seizure detection performance. Furthermore, we limit the number of channels to simulate a different clinical scenario and transfer the learned model directly, and we observe little performance drop in the detection quality of , which illustrates the robustness of our method. § PRELIMINARIES §.§ Materials In this paper, we utilize the neonatal EEG recording dataset <cit.>, which collected multi-channel EEG signals from a cohort of 79 term neonates admitted to the NICU at the Helsinki University Hospital. The recordings have been annotated by three experts individually and an average of 460 seizures were annotated per expert in the dataset; Among them, 39 neonates had seizures and 22 were seizure free, by consensus of all experts. The dataset provides a standard 18-channel bipolar montage <cit.>, with the electrode graph of solid blue arrows illustrated in Fig. <ref>. Note that, the head size of neonates is relatively smaller, which may make the full montage, that is successfully tested on adults, inappropriate for neonates. Thus, additionally, we select 3-channel bipolar montage: C3-P3, C4-P4 and P3-P4, following the findings of human experts <cit.>, as illustrated in red dotted line annotation of Fig. <ref>. As an example shown in Fig. <ref>, the pieced EEG waves of the selected three bipolar channels contain normal state and epileptic waving state. In seizure state (with orange area), the brain signal waves of all the channels present apparent disorder; while in the non-seizure state, the signal waves recover to normal fluctuation. §.§ Solution Formulation The whole dataset consists of N EEG samples { (^(i), y^(i)) }_i=1^N with seizure labels. Without cause of confusion, we omit the index notation i for clearer clarification. The goal of seizure detection is to estimate the probability p̂=Pr(y=1 \| ) that there exists a seizure state (y=1) in the current time piece given the EEG wave sample , where y∈{0,1}. The input =[_1, …, _c, …, _C]^⊤ is a multivariate time-series instance containing totally C channels of EEG time-series signals, as illustrated in Fig. <ref>. Each univariate time series is _c ∈ℝ^L where L is the overall timestep number. Concretely, L=6000 for the 30-second EEG signal recorded in 200Hz sampling frequency. Formally, given the sample input , each model f(·; θ) with parameter θ estimates the seizure probability as p̂=f_θ(). The learning objective is to minimize the cross-entropy loss w.r.t. the model parameter θ as Ł = - 1/N∑_i=1^N [ y^(i)logp̂^(i) + (1-y^(i)) . . log(1-p̂^(i)) ] + λ1/2θ^2 , with the regularization term weighted by hyperparameter λ. §.§ Challenges of Neonatal Seizure Detection Here we describe the existed challenges in processing and modeling the EEG signal data for neonatal seizure detection, which also motivates our model design detailed later. Challenge 1: Epilepsy seizure events occur dynamically in different channels. In Fig. <ref>, when in the seizure state, the EEG waves illustrate typical disorder with high-frequency spike and wave discharges. However, the signals _c at different channels have shown different patterns or even do not illustrate seizure, which corresponds to different causes of epileptic seizure and their paroxysm location in the brain. The existing methods such as <cit.> often overlook the fine-grained seizure pattern situation and takes the signals of all channels as a whole which may confound various patterns thus degenerate the detection performance. Thus, fine-grained EEG signal processing is required. Challenge 2: The channel number is variant even limited on neonatal brain health monitoring. Since the head size of neonates is much smaller than adults, which results in the limited sensing electrodes limited in the real scenario. And the sensing devices in different centers can be quite different, which leads to variant channel information in the data. Several related works <cit.> have also studied neonatal seizure detection with limited channels, e.g., only two channels. All the observations encourage researchers to conduct dynamic modeling techniques for dynamically modeling various even limited EEG signal channels for seizure detection. Challenge 3: Seizure patterns vary among neonates resulting in model generalization issue. The EEG dataset often contains the recordings from a cohort of people and a large variance of data distributions from different subjects have been observed, as shown in Fig. <ref>. From the figure, we can find that different clusters representing the EEG signal distributions of different neonates diverge in a large margin, which places obstacles for machine learning models to generalize from the training dataset to the test dataset which violates the independently identical distributional assumption of machine learning. § METHOD §.§ : Spatial-Temporal EEG Network For each input EEG montage , we first conduct a channel-level temporal modeling that independently models the signal of each channel. To get the final prediction p̂ and dynamically detect seizure from different channels, we then utilize a multi-channel spatial fusion module to fuse the information from multiple channels, as described below. This proposed framework manages to conduct fine-grained modeling on single channel while remains feasible to handle the input with various channel numbers, tackling above Challenges 1 and 2. Channel-level Temporal Modeling. We utilize temporal convolutional network (TCN) <cit.> for channel-level temporal modeling. By stacking L dilated convolution layers with 1-d convolution filters, we independently model each channel with relatively low time complexity. The process can be formulated as _c^0 = _c, _c^l = ReLU(h_c^l-1∗𝐰_l), _c = _c^L , where 𝐰_l is the convolution filter of the l-th layer, which is shared among all channels, ∗ is the dilated convolution operation, _c^l is the extracted hidden representation of the c-th channel at the l-th layer, and ReLU(x) = max(x, 0) is the activation function. Multi-channel Spatial Fusion. We utilize a multi-channel spatial fusion module to fuse the information from each channel as p̂ = f_s(g([_1, _2, ..., _C])) , where g:ℝ^C × d↦ℝ^d is an aggregation function and f_s is a three-layer multi-layer perception (MLP) with ReLU as activation function. The EEG signal of each channel reflects the brain activity in a certain brain region. It is important to model the spatial relations between these channels to better detect unusual EEG patterns and locate the source of seizure. However, existing methods either overlook the spatial relations between channels or relies on a predefined graph to describe the connections between channels <cit.>, which is unavailable under neonatal EEG scenario when the number and positions of channels are not fixed. To solve this problem, we utilize Graph Neural Networks (GNN) as our aggregation function g, which are commonly used for spatial-temporal modeling <cit.> including EEG modeling <cit.>. Specifically, we implement the aggregation function g as a graph attention network (GAT) <cit.> to dynamically decide the strength of spatial connectivity between channels. The process is denoted as _c^0 = _c, α_i, j^l = e^𝐖_i^l ·𝐖_j^l/∑_k=1^C e^𝐖^l_i^l 𝐖·_k^l, 1 ≤ i,j ≤ C, _c^l+1 = ∑_k=1^C α_c,k^l𝐖_k^l, g[_1, _2, ..., _C] = 1/C∑_c _c^L, where 0 ≤ l ≤ L is the layer of GAT, _c^l is the node representation vector of the c-th channel at the l-th GAT layer, 𝐖∈ℝ^d × d are learnable parameters, and α is the edge weights dynamically decide by the node representations of the last layer. This module flexibly models the spatial relations between changing number of channels and efficiently aggregates the information from different channels. The dynamic spatial relation mining of g also helps locating EEG seizure source, as we will exhibit in Sec. <ref>. §.§ Mixture of Experts for Cross-Person Generalization To tackle the generalization issue of Challenge 3 described in Sec. <ref>, we incorporate diverse models which own different specialties and utilize a mixture-of-expert (MoE) framework <cit.> to specify their contributions to predicting each test sample. Ensemble learning is known as an effective method to improve the generalization ability of neural networks <cit.>. Meanwhile, MoE has demonstrated excellent performance in computer vision <cit.>, natural language processing <cit.>, etc. In addition, although multiple models are included, the computational cost of ensemble is manageable and can even be compared to or lower than that of a single model <cit.>. Specifically, each sample ^(i) is assigned to K models, and their predictions [p̂_1^(i), …, p̂_K^(i)] are weighted by normalized sample-level ensemble weights ^(i)∈ℝ^K to output the ensemble prediction, which is obtained by p̂^(i) = ∑^K_k=1^(i)_kp̂_k^(i) where ∑^K_k=1^(i)_k = 1. To obtain its ensemble weights, each sample ^(i) is first embedded using a standard GRU network <cit.>, and the output is then fed into a single-layer MLP. The MLP output is normalized using the softmax operation Softmax(𝐳)_k=e^z_k/∑_j=1^K e^z_j to obtain the sample-level ensemble weights ^(i) = Softmax[MLP(GRU(^(i)))]. Note that the integrated base models are not fine-tuned, and only the parameters of the GRU network and the single-layer MLP are updated. In Sec. <ref>, we demonstrate that despite incurring additional computational costs, our proposed ensemble yields significant performance gains. Our proposed ensemble method offers several advantages over existing methods. Firstly, our approach leverages a diverse set of models, allowing for greater flexibility in capturing a wide range of seizure patterns. Secondly, our ensemble method dynamically dispatches models based on a given neonatal sample, enhancing the adaptability of ensembles to individual differences. Moreover, our approach does not require complex calibration or training phases, making it easy to implement in existing clinical settings. § EXPERIMENT §.§ Experimental Setup Dataset Preparation. The dataset <cit.> presented in Sec.<ref> consists of the EEG records of 79 neonates, and we randomly split the dataset into training and test sets by patients following <cit.>, which is realistic since the automated seizure detection service is trained on existing data and predicts for future coming patients. Based on the data split, we conduct four-fold cross-validation for evaluation. We further obtain 30-s EEG clips using non-overlapping sliding windows and overall sample number is more than 40 thousand. And each clip is annotated as a positive sample if all the three experts annotated the presence of seizure. Compared Methods. We compare our method with existing seizure detection and time series classification methods, including GBDT <cit.>, a gradient boosting decision tree model utilizing spike, temporal feature extracted from EEG data; ROCKET <cit.>, which utilizes random convolution kernels to extract feature vectors from EEG data and uses a ridge regression to get final predictions; GRU <cit.>, a recurrent neural network (RNN) with a gating mechanism to efficiently capture information in long signals; TCN <cit.>, a dilated convolutional neural network (CNN) designed for time-series modeling; MLSTM-FCN <cit.> combining CNN and RNN with squeeze-and-excitation blocks; InceptionTime <cit.> enhancing CNN and RNN with time-series Inception modules; DCRNN <cit.> utilizing correlation between variables to build a graph to capture relations between brain activities and leveraging Fourier transform to capture meaningful information in EEGs for seizure detection and classification. All methods are evaluated using area under precision-recall curve (AUPRC) and area under receiver operating characteristic curve (AUROC). For both metrics, higher value indicates better performance. Each reported number is averaged from three runs of different random seeds. We will publish the codes upon the acceptance of this paper. Setting of Ensemble. To enhance the generalization capability of our neural network models, we propose an ensemble method that leverages diverse network architectures. Specifically, our ensemble consists of four independently trained models, each based on a different network architecture: GRU, DCRNN, TCN, and . These models are dispatched to different test samples by the MoE framework. §.§ Experimental Results The experiment results of all cross-validation folds and the average performance on both 18-channel and 3-channel datasets are presented in Table <ref>. We have the following observations from the results. (1) Superiority of individual model: Our proposed achieves the best performance compared with other baseline models without ensemble on most folds of two datasets, which illustrates that the fine-grained channel-level temporal modeling and spatial fusion offer great capacity for EEG modeling. (2) Advances of ensemble: Over a diverse set of trained models, the ensemble model further boosts the performances, which results from the better generalization ability brought by the ensemble learning process. (3) Transferability: achieves a comparable performance on 3-channel datasets to that on 18-channel datasets, indicating that manage to adapt to limited channel scenario, thus more suitable for neonatal seizure detection. This observation is also consistent with clinical observations <cit.>. §.§ Extended Investigation §.§.§ Transfer Across Montages Note that in Eq. (<ref>), the filters of our dilated convolution layers are shared across channels, and the spatial fusion operation in Eq. (<ref>) can adapt to montages with variant number of channels. As a result, our method can be easily transferred to EEG data with different channels without retraining. Table <ref> presents the results of the transferred models. The results show that although the transferred models suffer performance drop compared with the results in Table <ref>, the performances are still comparable and outperform all our baselines. It indicates that our spatial-temporal modeling framework offers great flexibility and is perfectly suitable for the neonatal EEG scenarios where the channel number is variant or even limited. §.§.§ Occlusion Map Based Localization We leverage occlusion map techniques to analyze the localization ability of our models. Figure <ref> shows the occlusion map of a same sample from 3-channel and 18-channel datasets, using the prediction of trained on 18-channel dataset. We observe that both occlusion maps indicate that the seizure occurs at the early phase of this sample, which accord with the annotations of experts. The occlusion maps show that our method has good interpretability and can help for seizure localization, and the ability is also transferable across data with varying number of channels. §.§.§ Model Suitability Motivated by the evidence that data distributions differ significantly from people, we design a mixture-of-expert based framework that dynamically assigns models to each sample. Here we analyze its dispatching behaviors to reveal whether models are diversified to different distributions in terms of people. Specifically, we randomly select 4 neonates, on which we compute ensemble weights assigned to each model averagely by our best-performed ensemble, with results shown in Fig. <ref>. As can be seen, we dynamically assign weights for predicting samples of different neonates, and our promising ensemble performance indicating that in this way we can achieve better cross-person generalization. Furthermore, our analysis reveals that is assigned the highest weight for all neonates, further validating its usefulness in modeling neonatal physiological patterns. § CONCLUSION This paper aims at neonatal seizure detection task based on EEG signals, which has been recognized as a common problem in NICU of the date while lack of enough attention. We propose a spatial-temporal deep learning architecture which applies multi-channel spatial fusion with channel-level temporal modeling on EEG signal waves, tackling the challenges in practical scenario of neonatal brain health caring. We also utilize an ensemble of diverse models to alleviate the generalization issue. The experimental results on a large-scale real-world dataset of neonatal seizure detection have illustrated superior performance of our proposed solution with promising transferring ability on different EEG monitoring montages. IEEEtran
http://arxiv.org/abs/2307.00247v1	20230701062214	Safe Screening for Unbalanced Optimal Transport	[ "Xun Su", "Zhongxi Fang", "Hiroyuki Kasai" ]	math.OC	[ "math.OC", "cs.LG" ]	[ Christopher F. Parmeter August 1, 2023 =========================== This paper introduces a framework that utilizes the Safe Screening technique to accelerate the optimization process of the Unbalanced Optimal Transport (UOT) problem by proactively identifying and eliminating zero elements in the sparse solutions. We demonstrate the feasibility of applying Safe Screening to the UOT problem with ℓ_2-penalty and KL-penalty by conducting an analysis of the solution's bounds and considering the local strong convexity of the dual problem. Considering the specific structural characteristics of the UOT in comparison to general Lasso problems on the index matrix, we specifically propose a novel approximate projection, an elliptical safe region construction, and a two-hyperplane relaxation method. These enhancements significantly improve the screening efficiency for the UOT's without altering the algorithm's complexity. § INTRODUCTION Optimal transport (OT), as a metric, has gained significant attention in the field of machine learning in recent years due to its remarkable ability to capture geometric relationships between data distributions. It has demonstrated impressive achievements in many fields <cit.>. To overcome the limitation of OT in handling data with unequal quantities, researchers introduced unbalanced optimal transport (UOT) <cit.> by relaxing the constraints using penalty functions. UOT has been found extensive applications in computational biology <cit.>, machine learning <cit.>, and deep learning domains <cit.>. However, compared to traditional metrics, the computational burden associated with OT, including UOT, has impeded their widespread adoption on large-scale problems. The state-of-the-art linear programming algorithms suffer from cubic computational complexity and are challenging to parallelize on GPUs <cit.>. To address these challenges, Sinkhorn's algorithm <cit.> has been introduced for entropy-regularized OT and UOT <cit.>, reducing the computational complexity <cit.> . Furthermore, due to the instability of optimization and loss of sparsity in the solutions of Sinkhorn on entropy regularization <cit.>, researchers have shifted their attention towards exploring alternative regularization terms. This has led to the proposal of a series of new optimization algorithms <cit.>. Apart from algorithmic advancements, low-rank approximations in Sinkhorn's iterations have accelerated OT and UOT <cit.>, by further sacrificing solution accuracy to obtain dense approximate solutions. However, considering the inherent sparsity of OT's solutions, transportation mess tend to concentrate on a small number of transportation pairs. Researchers have attempted to reduce the computational burden by early pruning of infeasible pairs or by neglecting small solution elements in advance <cit.> . However, these methods have remained empirical in nature, lacking rigorous theoretical foundations. Recently, researchers has revealed the connection between Lasso-like problems <cit.> and UOT <cit.>. This finding has motivated us to consider applying the well-known Safe Screening technique <cit.> in the lasso domain to the UOT. Safe screening can theoretically identify and freeze the zero elements in the solution before computing the sparse optimum, leveraging the sparsity to shrink the dimension of the problem. Moreover, the sparsity of OT and UOT increases along the dimension, which would benefit a lot for large-scale problems. However, the lack of uniform weights caused by the cost matrix in UOT leads to poor performance when using traditional Safe Screening algorithms based on ℓ_1-norm regularization, even resulting in theoretical degeneration and failure. Moreover, for the commonly used KL-penalized UOT, the existing method requires the index matrix to be invertible, which is not satisfied by the KL-penalized UOT <cit.>. This paper presents a theoretical framework for applying Safe Screening to UOT. Additionally, we propose a novel projection method and Safe Screening techniques based on the unique sparse structure of the index matrix in UOT. For an overview of our contributions, please refer to Table <ref>. Our specific contributions can be summarized as follows: * This paper presents the first feasible theoretical framework for Safe Screening on UOT, including the dual problem, the projection method, and dual feasible region construction. * Leveraging the specific structural characteristics of the index matrix in UOT, we introduce the Shifting Projection in Section <ref> as a new approximate projection method that significantly reduces projection errors without increasing computational complexity, thereby improving screening efficiency. * For the KL-penalized UOT, we establish the local blockwise strong convexity property through a bound analysis for a new dual feasible region. Based on the analysis, we propose to construct smaller Ellipse-based safe regions to exploit the anisotropy of the dual variables in the dual space in Section <ref> . To the best of our knowledge, this is the first utilization of ellipses to construct safe regions in the Safe Screening community. * Considering the sparse correlation of the index matrix, we propose the Cruciform Two-hyperPlane (CTP) method to further shrink the safe region in Section <ref> . Using the Lagrangian method, we obtain closed-form solutions without significantly increasing the computational burden. § PRELIMINARIES ℝ^n denotes n-dimensional Euclidean space, and ℝ^n_+ denotes the set of vectors in which all elements are non-negative. ℝ^m × n represents the set of n × m matrices. In addition, ℝ^n × m_+ stands for the set of n × m matrices in which all elements are non-negative. We present vectors as bold lower-case letters a⃗,b⃗,c⃗,… and matrices as bold-face upper-case letters A,B,C,…. The i-th element of a⃗ and the element at the (i,j) position of A are stated respectively as a_i and A_i,j. The i-th column of A is represented as a⃗_i. In addition, _n ∈ℝ^n is the n-dimensional vector in which all elements are one. For two matrices of the same size A and B, ⟨A,B⟩= tr(A^TB) is the Frobenius dot-product. We use a⃗_2, a⃗_1, and a⃗_∞ to represent the ℓ_2-norm, ℓ_1-norm, and ℓ_∞ norm of a⃗, respectively. D_ϕ is the Bregman divergence with the strictly convex and differentiable function ϕ, i.e., D_ϕ(a⃗,b⃗)=∑_i d_ϕ(a_i, b_i)=∑_i [ϕ(a_i) - ϕ(b_i) - ϕ'(b_i)(a_i -b_i)]. Additionally, we suggest vectorization for A∈ℝ^n × m as lowercase letters a⃗∈ℝ^nm and a⃗=vec(A)=[A_1,1, A_1,2, ⋯, A_m,n-1, A_m,n]^T, i.e., the concatenated vector of the transposed row vectors of A. §.§ Optimal Transport and Unbalanced Optimal Transport Optimal Transport (OT): Given two discrete probability measures a⃗∈^m and b⃗∈^n that a⃗_1 = b⃗_1, the standard OT problem seeks a corresponding transport matrix T∈_+^m × n minimizing the total transport cost <cit.>. This can be formulated as OT(a⃗,b⃗) := min_T∈_+^n × m⟨C, T⟩, subject to T_m= a⃗, T^T_n = b⃗, where C∈ℝ_+^n × m is the cost matrix. The famous Wasserstein q-distance is obtained for C_i,j = a_i - b_j_q <cit.>. As t⃗=vec(T) ∈ℝ^nm and c⃗=vec(C) ∈ℝ^nm, we reformulate Eq. (<ref>) in a vector format as <cit.> OT(a⃗,b⃗) := min_t⃗∈_+^nmc⃗^Tt⃗, subject to Nt⃗ = a⃗, Mt⃗ = b⃗, where N∈^n × nm and M∈^m × nm are two indices matrices composed of “0" and “1" to compute row-sum and column-sum of T. Each column of N and M has only a single non-zero element equal to 1. Examples are listed in the supplementary. We denote t̂⃗̂ as the optimal primal solution such that OT(a⃗,b⃗) = c⃗^Tt̂⃗̂. Unbalanced Optimal Transport (UOT): The UOT relaxes the marginal constraints in Eq. (<ref>) by replacing the equality constraints with penalty functions on the marginals with divergence <cit.>. Formally, defining y⃗ = [a⃗^T, b⃗^T]^T ∈ℝ^n+m and the concatenation index matrix X = [M^T,N^T]^T ∈ℝ^(n+m) × nm, the UOT can be formulated by introducing a penalty function for the discrete distributions as <cit.> UOT(a⃗,b⃗) := min_t⃗∈_+^nmλc⃗^Tt⃗ + D_ϕ(Xt⃗,y⃗). §.§ Duality and Strong Concavity We consider an optimization problem as min_t⃗∈R^n{ f(t⃗) := g(t⃗) + h(Xt⃗) }, where g: R^n→ (-∞, ∞] and h: R^m→ (-∞, ∞] represent proper convex functions, t⃗∈R^n is the primal optimization variable, and X∈R^m× n. To derive the dual problem of Eq. (<ref>), we rely on the Fenchel–Rockafellar Duality: Consider the problem Eq. (<ref>). Assuming that all the assumptions are satisfied, we have min_t⃗ g(t⃗) + h(Xt⃗) = max_θ⃗⃗⃗ -h^(-θ⃗)-g^(X^Tθ⃗), where θ⃗∈R^m, and where g^: R^n → [-∞, ∞] and h^: R^m → [-∞, ∞] respectively stand for the convex conjugate of the extended real-valued functions g: R^n→ (-∞, ∞] and h: R^m → (-∞, ∞]. We call the former primal problem P(t⃗) and the latter dual problem D(θ⃗). Next, we introduce the concepts of strong concavity and L-blockwise strong concavity, the latter being a more generalized form. Function f(x⃗) is an L-strongly concave function if there exists a positive constant L that, for ∀y⃗∈R^n: f(y⃗) ≤ f(x⃗) +∇ f(x⃗)^T(y⃗-x⃗) - L/2y⃗ -x⃗_2^2. A function f(x⃗) is L-blockwise strongly concave if there exists a diagonal positive-definite matrix L such that f(y⃗) ≤ f(x⃗) +∇ f(x⃗)^T(y⃗-x⃗) - 1/2(y⃗ -x⃗)^T L (y⃗ -x⃗). The L of f(x⃗) can be obtained by computing a lower bound on the Hessian matrix of the f(x⃗) <cit.>. Assuming that L exhibits no anisotropy. In this case, L = LI. When Eq. (<ref>) and Eq. (<ref>) only hold for y⃗∈𝒢⊂R^n, they referred to as 𝒢-locally L-strongly concave and 𝒢-locally L-blockwise strongly concave, respectively. § UOT SAFE SCREENING In high-dimensional regression tasks, regularization terms like ℓ_1-norm bring desirable sparsity to solutions. Safe Screening seeks to extend this benefit by identifying and discarding zero values in the solution before the computationally expensive optimization process concludes. It was applied to accelerate Lasso problems <cit.> and SVMs <cit.> by eliminating unnecessary features. In recent years, researchers have continuously proposed new methods to narrow down the safe region while ensuring acceptable computational costs <cit.>. Furthermore, they actively extend the application of Safe Screening to a broader spectrum of functions and regularizations. <cit.> This section presents the Safe Screening framework proposed for the UOT. We begin by formulating the dual form D(θ⃗) and discussing the dual feasible region, denoted as ℛ^D, we then propose Shifting Projection based on the special index matrix X. We discuss the strong concavity under different penalty functions and explain how they influence the design of the safe region ℛ^S. Next, we focus on the KL-penalty function, proving its L-blockwise local strong concavity and introducing the Gap-Ellipse (Ell) method. Finally, considering the unique structural characteristic of X in the UOT, we design and propose the Cruciform two-hyperplane (CTP) method to further enhance the effectiveness of Safe Screening. Concrete proofs are provided in the supplementary material. §.§ Dual Formulation and Feasible Region Applying Theorem <ref>, we can derive the dual problem for UOT as follows: max_θ⃗∈^n+m D(θ⃗) = -D_ϕ^(-θ⃗) subject to x⃗_p^Tθ⃗ -λ c_p ≤ 0, ∀ p ∈ [nm], Different choices of penalty functions result in different dual functions (see Table <ref>), but the dual feasible region, denoted as ℛ^D, composed of the constraints in Eq. (<ref>), remains consistent, and the optimal dual solution θ̂⃗̂∈ℛ^D. According to the KKT conditions of the Fenchel–Rockafellar duality in Theorem <ref>, a connection exists between the t̂⃗̂ and θ̂⃗̂ as presented below. (Primal-Dual Relationship of Optimal Solution in the UOT) For the optimal primal solution t̂⃗̂ and the optimal dual solution θ̂⃗̂, we have the following relationship for ∀ p, p ∈ [nm]: x⃗_p^Tθ̂⃗̂ -λ c_p { < 0, ⟹ t̂_p = 0, = 0, ⟹ t̂_p ≥ 0. . It is important to note that x⃗_p^Tθ̂⃗̂ on the left-hand side (LHS) of Eq. (<ref>) differs from \|x⃗_p^Tθ̂⃗̂\| in the standard Lasso problem due to the constraint t_p > 0 in the UOT problem. Hence, the fundamental concept behind the Safe Screening is to identify a region ℛ^S such that θ̂⃗̂∈ℛ^S. If the following inequality holds: max_θ⃗∈ℛ^Sx⃗_p^Tθ⃗ -λ c_p < 0, then we have x⃗_p^Tθ̂⃗̂ -λ c_p < 0, and the corresponding t̂_p = 0. Consequently, t_p can be screened out. Safe Screening seeks a safe region ℛ^S containing θ̂⃗̂ as small as possible to make accurate and anticipatory judgments. The tool we rely on to construct this is the strong convexity in Definition <ref>. §.§ Shifting Projection To construct ℛ^S, a feasible dual variable, denoted θ̃⃗̃, belonging to the set ℛ^D is required. The variable θ⃗, derived from t⃗ through the Fenchel–Rockafellar relationship in Eq. (<ref>), may not satisfy the dual constraints, resulting in θ⃗∉ℛ^D and necessitating projections. As illustrated in Algorithm <ref>, as t⃗^k approaches t̂⃗̂ during optimization, θ⃗^k similarly converges to θ̂⃗̂. A poor projection may significantly deviate from θ̂⃗̂, leading to an expansion of ℛ^S and adversely affecting the screening outcome. This necessitates a high degree of accuracy in the projection. However, since ℛ^D in Eq. (<ref>) is a polytope composed of nm hyperplanes, regular projections can be computationally expensive. To address this issue, researchers have employed Residuals Rescaling <cit.> to achieve cost-effective approximate projections <cit.> <cit.>. One extension of it to UOT can be achieved by considering θ̃⃗̃ = θ⃗/max(1, X^Tθ⃗/λc⃗_∞), where the division (· /·) is performed element-wise. The computational complexity is O(max(n,m)(n+m)). However, Residuals Rescaling in UOT can exhibit significant fluctuations due to the varying range of values c_p in the denominator term, and it may degenerate when there exists a p such that c_p = 0. To overcome this issue, we propose the Shifting Projection: Let θ⃗ = [α⃗^T,β⃗^T]^T, where α⃗∈^m and β⃗∈^n. For u ∈ [m] and v ∈ [n], the Shifting Projection is defined as: {[ α̃_u = α_u - max_0≤ j < nα_u +β_j - λc_un+j/2; β̃_v = β_v - max_0 ≤ i < mα_i +β_v - λc_in+v/2. ]. Since x⃗_p is an index vector containing only two elements with a value of "1", we can express the constraint as x⃗_p^Tθ⃗= α_u + β_v < λ c_p, where (u,v)=(p \| n,p n). In Figure <ref>, it can be observed that the variables need to be shifted by half of the maximum value among the row and column constraints. The Shifting Projection is particularly suitable for UOT as it avoids degeneracy issues and takes into account differences in coordinate dimensions. §.§ Safe Region Construction After obtaining θ̃⃗̃∈ℛ^D, we construct ℛ^S to include θ̂⃗̂ if the dual function satisfies Definition <ref>. We choose simple geometries, such as balls and domes for testing Eq. (<ref>) due to their computational efficiency requirements. For a ℛ^D-locally L-strongly concave function D(θ⃗), finding a point θ̃⃗̃∈ℛ^𝒟, then we can construct the following region that guarantees θ̂⃗̂∈ℛ^G: ℛ^G(θ̃⃗̃) :={θ⃗ \|(θ⃗-θ̃⃗̃)_2^2 ≤2( D(θ⃗) - D(θ̃⃗̃))/ L}. The famous Gap-safe method <cit.> uses a constant bound to relax the right-hand side (RHS): D(θ⃗) - D(θ̃⃗̃) < P(t⃗) - D(θ̃⃗̃) = (t⃗, θ̃⃗̃), which is known as weak duality. Sasvi method in <cit.> computes ℛ^S(θ̃⃗̃) :={θ⃗ \| (θ⃗-θ̃⃗̃)^T(θ⃗-y⃗)≤ 0}, which is proved in Theorem 6 of <cit.>, are both balls. Considering the anisotropy at different coordinates of the dual KL-penalized UOT problem, we propose a new screening method called Gap-Ellipse Safe Screening method: For a L-blockwise strongly concave function D(θ), finding a point θ̃⃗̃∈ℛ^𝒟, then we can construct the following ellipse region that guarantees θ̂⃗̂∈ℛ^G: ℛ^G(θ̃⃗̃) :={θ⃗ \| (θ⃗-θ̃⃗̃)^T L(θ⃗-θ̃⃗̃) ≤ 2(t⃗, θ̃⃗̃)}. Considering the dual function of the UOT, shown in Table <ref>, the TV-penalty results in linearity, while the L2-penalty leads to strong convexity. However, the KL-penalty does not exhibit strong convexity. Only the ℓ_2-penalty can apply the Theorem <ref>. Therefore, we consider proving the local strong concavity of the dual function in a new region under the conditions of the KL-penalty. If the dual function exhibits local strong concavity on ℛ^D, then Theorem <ref> remains applicable. The dual function is to be ℛ^D-locally strongly concave when the index matrix X has a right inverse <cit.>. However, in the case of UOT, the index matrix X does not possess a right inverse, as proven in the supplementary material. This prompts us to seek an alternative approach to establish its locally strong concave characteristic by leveraging the unique structure of UOT. We propose to construct a new optimal feasible region, ℛ^B, such that θ̂⃗̂∈ℛ^B. We then demonstrate that the KL-penalized UOT has a locally strongly concave dual function on ℛ^B∩ℛ^D. For θ̂⃗̂ = [α̂⃗̂^T,β̂⃗̂^T]^T and (u,v)=(p \| n,p n) as stated in Theorem <ref>, considering the symmetry of the rows and columns, we focus on α_u and use Low(θ⃗,v+n) to represent a lower bound of β_v derived from ∀θ⃗, without loss of generality. Thus, we have α̂_̂û < α_u < min_v (λ c_p -β_v ). Here, we use α_u to represent its upper bound, and β_v := Low(θ⃗, j), with j = n+v, to represent the lower bound. We define the lower bound obtained from θ⃗ as: Low(θ⃗, j) = ln(ϵ - y_j/K^j - y_j +ϵ), where K^j= D(θ⃗) - P^j(t⃗), and P^j is the primal problem of ∑_i≠ j^n+m D^i(θ_i). The dual problem has a decomposable structure where D(θ⃗)=∑_i=1^n+m D^i(θ_i), and D^i(θ_i) represents the value of the i-th coordinate of the dual problem. We construct a computable primal function P^j(t⃗) and utilize it to provide new bounds θ⃗ and θ⃗ for θ̂⃗̂ as θ⃗ updates during optimization. Detailed proof is provided in the supplementary material. By using these bounds, we construct a Box region ℛ^B(θ⃗,θ⃗^k) = θ⃗\| θ_i≤θ_i ≤θ_i, i ∈ [n+m], and ensure that θ̂⃗̂∈ℛ^B. We can find θ̃⃗̃∈ℛ^B∩ℛ^D as projecting on Box region is a coordinate-wise shifting, given that the Hessian function is an increasing function, we can demonstrate that L = (y⃗ e^-θ⃗), and the dual function of the KL-penalized UOT problem is (ℛ^B∩ℛ^D)-locally L-blockwise strongly concave. The LHS in Eq. (<ref>) represents an ellipse rather than a ball with the center of θ̃⃗̃. Theorem <ref> enables us to take into account the anisotropic property of the function as diagnose elements in Hessian hugely vary, which leads to considerable performance improvement in the screening process. Two-hyperplane Safe Screening Region. We can adapt the ball or ellipse ℛ^G as ℛ^S, and combine it with the ℛ^D to construct a smaller ℛ^S. As ℛ^D is a (n+m)-polytope with nm facets, finding the maximum value within it is expensive <cit.>. A simple approach is to relax all the hyperplanes into a single hyperplane, transforming the ℛ^S into the intersection of a sphere or ellipse with the half-space <cit.>. This allows us to obtain the maximum value at an acceptable cost. We denote the relaxed region by ℛ^Re. and then the new region can be constructed by ℛ^S = ℛ^G∩ℛ^Re. For every primal variable t_p ≥ 0, p ∈ [mn], we construct a specific region ℛ^Re_p as presented below. ℛ^Re(θ⃗, t⃗):= {θ⃗ \|∑_p∈ [nm](x⃗_p^Tθ⃗ - λc_p)t_p≤ 0 . }. Considering the specific structure of X in the UOT, we propose to relax the polytope into the intersection of half-spaces by combining the nm dual constraints with a positive weight t_p ≥ 0. For every primal variable t_p ≥ 0, p ∈ [mn], let I_p = { i \| 0≤ i<nm, u = i\| n and v = i n}, and I^C_p = { i \| 0≤ i<nm, i ∉ I_p}. Then we construct a specific region ℛ^Re_p as presented below. ℛ^Re_p(θ⃗, t⃗):= {θ⃗ \|∑_p∈ I_p(x⃗_p^Tθ⃗ - λc_p)t_p≤ 0, ∑_p∈I^C_p(x⃗_p ^Tθ⃗- λc_p)t_p≤ 0 . }. To elaborate, we partition the polytope constraints into a primary group I_p and a secondary group I^C_p. For each p, ℛ^Re_p is the intersection of two relaxed half-spaces. The effectiveness of ℛ^Re_p can be intuitively understood: while more hyperplanes result in a tighter region, in the standard Lasso problem, given that all optimization directions in x⃗_p^Tθ are of equal importance, an arbitrary split is likely to result in nearly parallel high-dimensional hyperplanes. This phenomenon is illustrated in Figure <ref> in Section <ref>, where using random hyperplanes does not lead to improved screening. In contrast, we can incorporate n+m -1 constraints, depicted in light yellow in Figure <ref>, to form the primary hyperplane. These constraints are directly associated with the optimization direction α_u and β_v. The remaining constraints are grouped as the secondary hyperplane, whose coefficients of α_u and β_v would be zero. This arrangement avoids parallelism with the primary hyperplane and significantly shrinks the safe region in comparison with the Dome method in Theorem <ref>. It is worth noting that, unlike <cit.>, we design two specific hyperplanes for each element in the UOT and maximize Eq. (<ref>) on them. Consequently, as in Figure <ref>, the two green lines represent two hyperplanes. Optimizing on ℛ^Re_p∩ℛ^G has a closed-form solution by using the Lagrangian method. Its computational process is in the supplementary material. §.§ Optimization Algorithm and Computational Cost Analysis Safe Screening Algorithm. We define the screening mask vector, denoted as s⃗∈ℝ^mn at first, for the transport vector t⃗. We remove t_p if s_p = 0 before sending it to the optimizer. The entire optimization process is summarized as in Algorithm <ref>. The update(t⃗,c⃗, X) operator in the algorithm indicates that the optimizer updates t⃗^k into t⃗^k+1. The dimension of t⃗^k would gradually decrease as well as s⃗. w ∈ℕ specifies the screening period to update the mask s⃗ according to the intensity of the change on the screening ratio, decided by the user. Faster convergence of the optimizer is associated with a smaller period w. It must be emphasized that the Safe Screening is independent of the adopted optimization algorithm. Computational Cost Analysis. The Shifting Projection has the same computational burden as Residuals Rescaling for the UOT. The Gap-Ellipse method has the same expense as Gap as O(nm) because only two dual coordinates are activated each time. The CTP method must apply the Lagrangian method for every primal element t_p. However, under the special structure of the X, for every t_p, the data required during the Lagrangian method can be divided as row summation and column summation of the transport matrix T, which can be computed together and be reused for other elements that have the same t m or t \| m. This process of summarization helps to preserve the overall optimization complexity to O(mn). The detailed steps are in the supplementary material. § EXPERIMENTS We conduct three experiments on randomly generated high dimensional Gaussian distributions and MNIST dataset. We first validate the effectiveness of the Shifting Projection in Eq. (<ref>). Next, we compare the screening ratio of different methods for the ℓ_2 and KL-penalized problems. Finally, we apply our Safe Screening on some popular optimization solvers, i.e., FISTA <cit.>, Majorization-Minimization(MM) <cit.>, coordinate descent, and BFGS for the ℓ_2-norm penalized UOT, and MM, and BFGS method for the KL-penalized problem to elucidate its computation speed-up ratios. All the code was written in Matlab. Due to space constraints, we have included the experiments involving coordinate descent and BFGS in the supplementary materials. Projection Method Comparison. We evaluate the dual gap of two approximate projection methods as depicted in Figure <ref>. We utilize the FISTA algorithm to solve 10 pairs of ℓ_2-penalized 100-dimensional Gaussian distribution transport problems. θ⃗_S and θ⃗_L are obtained by Shifting Projection and Residuals Rescaling with λ = 10^-2, respectively. We then calculate the corresponding dual gap. Our method is more accurate and stable. However, Residuals Rescaling exhibits slow screening progress due to the subpar dual gap. Screening Method Comparison. We evaluate the screening ratios of different methods on the ℓ_2-penalized and KL-penalized UOT. For the KL-penalized problem, we select λ∈{10^-1, 10^0, 10^1}, and for the ℓ_2-penalized problem, we select λ∈{10^0, 10^-1, 10^-2}. All methods compared employ our Shifting Projection as it has been demonstrated to be more suitable for the UOT than other techniques in previous experiments. We conducted experiments on the MNIST dataset 5 times, solving them with the FISTA and MM algorithms. For the KL-penalized problem, we compare the Gap (Gap), Gap with CTP (Gap-CTP), Gap-Ellipse (Ell), and Gap-Ellipse with CTP (Ell-CTP). For the ℓ_2-penalized problem, we select the Gap (Gap), Dynamic Sasvi-Dome (Sa), Sasvi with CTP (Sa-CTP), and Sasvi with randomly-selected two-hyperplanes (Sa-Ran) methods. The results are depicted in Figure <ref>. The blue shadow almost overlaps with the yellow Sa method. This suggests that the simple addition of more planes is not the key factor contributing to the success of the CTP method. Rather, the advantage arises from the specific composition of the planes. Computational Speed-up Ratio. In this section, we conduct experiments on both the MNIST and Gaussian datasets using different screening methods 5 times to demonstrate the speed-up ratio of these methods on ℓ_2 and KL-penalized UOT. We adopt two stopping conditions, namely (t⃗, θ̃⃗̃) ≤ε, where ε∈{10^-5, 10^-7}. For the KL-penalized problem, we choose λ∈{10^0, 10^-1}, and for the ℓ_2-penalized problem, we select λ∈{10^-1, 10^-2}. § CONCLUSION In this paper, we propose the first Safe Screening method specifically tailored for the UOT, leveraging its unique structure. We introduce a series of novel methods for projection and safe region construction, designed to improve performance without adding significant computational burden. For the KL-penalized UOT, we present a new theoretical analysis framework, from which we derive optimal bounds and prove the property of local blockwise strong convexity. We illustrate the substantial improvements of these methods through numerical experiments. With its capacity to exploit the gradually increasing sparsity, our method holds considerable potential for efficiently solving large-scale problems. plain § NOTATIONS §.§ The example of the index matrix As we illustrated in Section <ref>, the index matrices N and M follow a distinct format. N signifies the indices tied to the computation of each row sum for the matrix T when presented in vector form, while M embodies the column sums. Let us illustrate this with an example where m=n=3: N =[ 1 1 1 0 0 0 0 0 0; 0 0 0 1 1 1 0 0 0; 0 0 0 0 0 0 1 1 1; ], M =[ 1 0 0 1 0 0 1 0 0; 0 1 0 0 1 0 0 1 0; 0 0 1 0 0 1 0 0 1; ]. § PROOFS §.§ Proofs of Theorem <ref> and Theorem <ref> We replace f in Definitions <ref> with the dual function D(θ) since D(y⃗) ≤ D(x⃗) +∇ D(x⃗)^T(y⃗-x⃗) - L/2y⃗ -x⃗^2 By choosing y = θ⃗ and x = θ̂⃗̂, assuming θ̂⃗̂ is the optimum for the convex function D with ∇ D(θ̂⃗̂) = 0⃗, we have D(θ⃗) ≤ D(θ̂⃗̂) - L/2θ⃗-θ̂⃗̂^2 ⟺ θ⃗-θ̂⃗̂^2 ≤2/L( D(θ⃗) - D(θ̂⃗̂)) ⟺ θ⃗-θ̂⃗̂^2 ≤2/L( P(t⃗) - D(θ̂⃗̂)) ⟺ θ⃗-θ̂⃗̂^2 ≤2(t⃗,θ⃗)/L. This proves Theorem <ref>. As for the proof of Theorem <ref>, we can deduce it in the same way by replacing L/2θ⃗ - θ̂⃗̂^2_2 with 1/2(θ̂⃗̂ - θ⃗)^T 𝐋(θ̂⃗̂ - θ⃗) in Definitions <ref>. §.§ Proofs of the dual property in Table <ref> Considering the conjugate function of the UOT function with different penalties: For the ℓ_2 penalty h(t⃗) = t⃗-y⃗_2^2: h^(θ⃗) = max _∀ p, t_p ≥ 0 (θ⃗^Tt⃗- t⃗ - y⃗^2_2 ) = 1/2θ⃗_2^2 +y⃗^Tθ⃗ For the KL penalty, we have: h^(θ⃗) = max _∀ p, t_p ≥ 0 (θ⃗^Tt⃗-(t⃗ +ϵ)lnt⃗ + ϵ/y⃗+(t⃗ +ϵ)^T-y⃗^T) = y⃗^T e^θ⃗ -y⃗^T -ϵθ⃗^T For the TV penalty, we have: h^(θ⃗) = max _∀ p, t_p ≥ 0 (θ⃗^Tt⃗- t⃗ - y⃗_1) = {θ⃗^Ty⃗, ∀ p, if \|θ_p\| < 1, +∞, otherwise. . As for g(x) = c⃗^Tt⃗, which does not change with the choice of the penalty function, we have: g^(θ⃗) = max _∀ p, t_p ≥ 0 (θ⃗^Tt⃗-λc⃗^Tt⃗) = { 0 , ∀ p ∈ [n+m], θ_p ≤λ c_p ∞, otherwise. . The h^ function works as the main dual function and the g^* function works as the constraints. By adding the dual function with the appropriate variable, we can use the Fenchel-Rockafellar Duality to obtain the dual form. During the duality computation, we can derive the primal-dual relationship for θ̂⃗̂ and t̂⃗̂, which is used for computing θ⃗^k from t⃗^k. §.§ Proofs of Theorem. <ref> For any θ⃗, we have P(t̂⃗̂) = D(θ̂⃗̂). Substituting θ̂⃗̂ into the dual function yields P(t⃗) ≥ P(t̂⃗̂) = D(θ̂⃗̂) ≥ D(θ⃗). Now let's consider a single dual variable θ_q: D(θ⃗) =∑_i≠ q^n+m D^i(θ_i) + D^q(θ_q), where D^q(θ_q) = -y_q e^-θ_q + y_q - ϵθ_q. We can view ∑_i ≠ q^n+mD^i(θ_i) as the dual problem of a new UOT problem P^q(t⃗). In essence, by excluding the element q-th element from the dual problem, the corresponding primal problem remains a UOT problem without the element y_q. This is akin to modifying the original P problem by setting y_q = 0, rendering the corresponding X_q invalid, along with the indices t_i, c_i, that ∀ i ∈ I_q. We denote this new y⃗ as ỳ⃗̀^q and establish new c̀⃗̀^q, t̀⃗̀^q, and new X̀^q by deleting y_q corresponding elements. Consequently, we derive the corresponding Primal problem for ∑i ≠ q^n+mD^i(θ_i), where θ⃗∈ℛ^D: P^q(t̀⃗̀^q) = λ̀⃗c⃗^qTt̀⃗̀^q + D_ϕ(X̀^qt̀⃗̀^q,ỳ⃗̀^q) Then we have D(θ⃗) ≤ D(θ̂⃗̂) = ∑_i≠ q^n+m D^i(θ̂_i) + D^q(θ̂_q) ≤ P^q(t̂̀̂⃗̂̀̂^q) + D^q(θ̂_q) ≤ P^q(t̀⃗̀^q) + D^q(θ̂_q) where t̂̀̂⃗̂̀̂^q is the optimum for P^q. Setting K^q= D(θ⃗) - P^q(t⃗), we have D^q(θ̂_q) > K^q. Then, this implies - y_q e^-θ̂_q - θ̂_q ϵ + y_q > K^q. Noting that D^q(θ_q) is an increasing function, we have (ϵ - y_p)e^-θ̂_q - ϵ > D^q(θ̂_q)- y_q > K^q - y_q. Finally, we obtain θ̂_p > ln(ϵ - y_p/K_q - y_q +ϵ) = Low(θ⃗, q). We now have the lower bound for θ̂_q. As θ̂_p has a lower bound, we can get its upper bound based on the constraints x⃗_p^Tθ⃗< λ c_p, if the p-th elements of the flattened vector are in row q or column q-n. Hence, we derive an upper bound for θ̂_q <min_p ∈ I_q (λ c_p - θ_i)< min_p ∈ I_q (λ c_p -Low(θ⃗, i))from all the relevant constraints. If 0< q ≤ n, then i ∈{n+1,n+2, ..., n+m }, otherwise, if n< q, then i ∈{1,2, ..., n }. This completes the proof. we can further enhence the bound to θ̅= lny_i /y⃗^T e^ - θ̂⃗̂_k - C, based on the dual constraints ℛ^D, we know that, for (u,v)=(p \| n,p n), we rewrite S_p(θ⃗) , and we have α_u + β_v < c_p. Then we can divide the θ_i into groups, the first group has pair relationships between every pair, and the second one has no pair relationships. If ∃ i, θ̂_i <θ̅, D(θ̂⃗̂) + -y⃗^T = - ∑_j^min(n,m)-1(y_j_1 e^ - θ̂_j_1 + y_j_1 e^ - θ̂_j_2) - (∑_p≠ j_1, j_2, iy_p e^ - θ̂_p) - y_i e^ - θ̂_i < - ∑_i^min(n,m)-12e^c_j/2√(y_j_1y_j_2) - 0- y_i e^ - θ̂_i = - C - y_i e^ - θ̂_i < - C - y_i e^ - θ̅_i = V Since we get the lower bound for every θ̂_i, we can turn to get an upper bound because, for every θ_i, we have some x⃗_p^T θ≤ c_p we have θ̂_i <min_p c_i_p - θ̅ then we can get a hyper box for the θ̂⃗̂, we call it ℛ^V, to get the maximum upper bound for local strong concavity, we have to compute the max_imin_p(c_i_p) - θ̅, we call this θ̅̅̅ §.§ Proof of Theorem. <ref> The proposed CTP method constructs a region ℛ^Re_p for every element t_p and makes sure it that includes ℛ^C∩ℛ^D, and is also inside the region ℛ^S proposed in (<cit.>), which is ℛ^S(θ⃗, t⃗):= {θ⃗ \| ∑_p∈ I_p(x⃗_p^Tθ⃗ - λc_p)t_p≤ 0, ∑_p∈I^C_p(x⃗_p ^Tθ⃗- λc_p)t_p≤ 0, θ⃗∈ℛ^G. . }. The dual feasible region consists of two parts: one is the circular or elliptical region ℛ^G in Theorem <ref> and Theorem <ref>, which remains unchanged in the construction of the dual feasible area in Sa and CTP. Therefore, it does not require any further consideration, and we focus on the ℛ^D as follows: θ⃗∈ℛ^D ⟹ x⃗_p^Tθ⃗- λ c_p ≤ 0, ∀ p ∈ [nm] ⟹ ∑_p ∈ I_p ⊂ [nm] ( x⃗_p^Tθ⃗- λ c_p)t_p ≤ 0, (t_p ≥ 0) ⟹ θ⃗∈ℛ^Re_p By proving that CTP is a relaxation of ℛ^D, since θ⃗∈ℛ^G, we know that the optimum θ̂⃗̂∈ℛ^S = ℛ^G∩ℛ^Re_p. This completes the proof. §.§ Proof of Maximizing On ℛ^Re_p∈ℛ^G For any p ∈ [nm], we have: x⃗_p^Tθ̃⃗̃ = α̃_u + β̃_v = α_u + β_v - max_0≤ j< nα_u +β_j - λc_un+j/2 - max_0 ≤ i < mα_i +β_v - λc_in+v/2 ≤α_u + β_v - α_u +β_v - λc_p/2 - α_u +β_v - λc_p/2 ≤λc_p. This proves that ∀ p, we have x⃗_p^Tθ̃⃗̃≤λc_p, then θ̃⃗̃∈ℛ^D. This completes the proof. §.§ Proof of Maximizing on ℛ^Re_p∩ℛ^G By rewriting θ⃗:= θ⃗_o + q⃗, and η,μ,ν (≥ 0) are Lagrange multipliers. Optimizing on the intersection of region. (<ref>) ℛ^Re_p with ℛ^G, whose Lagrangian function is given as min_q⃗max_η,μ,ν≥ 0{ L(q⃗,η,μ,ν) := - x⃗_p^Tθ⃗_o - q_u - q_u+v+1 + η( q⃗^T Lq⃗ - 1) +μ( (∑_l∈ I_px⃗_l)^T(q⃗+θ⃗_o) - λ∑_l∈ I_pc_lt_l) + ν( (∑_l∈ I^C_px⃗_l)^T(q⃗+θ⃗_o) -λ∑_l∈ I^C_pc_lt_l) }, is the Lagrangian optimization problem of max_θ⃗∈ℛ^S_pθ_u +θ_u+v+1, ∀ p ∈ [nm], where u = i\| n, v = i n. Noticing that, when ℛ^G is a circle, it is just one specific condition by replacing L to a constant L. Considering the center of the circle ℛ^C as θ⃗_o, we define θ⃗ := θ⃗_o + q⃗. Also, defining i_1:=u, and i_2:=u+v+1, since x⃗_p^Tθ⃗_o is a constant, Eq.(<ref>) can be simplified to min_(θ⃗_⃗o⃗ + q⃗ )∈ℛ^S_p- ( q_i_1 +q_i_2 ). Here, we define the followings: v⃗:=∑_l∈ I_px⃗_l, e_v :=λ∑_l∈ I_pc_lt_l, w⃗:=∑_l∈ I^C_px⃗_l, e_w :=λ∑_l∈ I^C_pc_lt_l. Then, we re-write the Lagrangian function as min_q⃗max_η,μ,ν≥ 0{ L(q⃗,η,μ,ν) := - q_i_1 - q_i_2 + η( q⃗^T Lq⃗ - 1)+μ( v⃗^Tq⃗ - e_v ) + ν( w⃗^Tq⃗ - e_w )}. Similarly, considering the derivative with respect to q_i yields ∂ L/∂q_i = { -1 + 2ηq_iL_i +μ v_i + ν w_i, i = i_1, i_2, 2ηq_iL_i +μ v_i + ν w_i, i ≠ i_1, i_2. . Consequently, we obtain q_i^* = {1- μ v_i - ν w_i/2η L_i, i = i_1, i_2, -μ v_i + ν w_i/2η L_i, i ≠ i_1, i_2. . Consequently, plugging this into Eq .(<ref>), we obtain the following Lagrangian dual problem: max_η,μ,ν≥0{ L(η,μ,ν) := μ v_i_1 + ν w_i_1-1/2η L_i +μ v_i_2 + ν w_i_2-1/2η L_i+ η((q⃗^)^T Lq⃗^-1 )+μ( v⃗^Tq⃗^* - e_v ) + ν( w⃗^Tq⃗^* - e_w ) }. From the KKT conditions, we know that if η ((q⃗^)^T Lq⃗^ -1) = 0, μ( v⃗^Tq⃗^* - e_v) = 0, ν(w⃗^Tq⃗^* - e_w) = 0. We set η^, μ^, ν^* as the solution of the equations, which are also the solutions to the dual problem. Firstly, we assume that η^, μ^, ν^≠ 0, then the solution is equal to computing the following equations: (1-μ v_i_1-ν w_i_1/√(L_i_1))^2 + (1-μ v_i_2-ν w_i_2/√(L_i_2))^2 + ∑^n+m_i≠ i_1,i_2(v_iμ+w_iν/√(L_i))^2 - 4η^2 = 0, v_i_1-μ v_i_1^2-ν w_i_1v_i_1/L_i_1 + v_i_2-μ v_i_2^2-ν w_i_2v_i_2/L_i_2 - ∑^n+m_i≠ i_1,i_2(v_i^2μ +w_i v_iν/L_i) - 2ηe_v = 0, w_i_1-ν w_i_1^2-μ w_i_1v_i_1/L_i_1 + w_i_2-ν w_i_2^2-μ w_i_2v_i_2/L_i_2 - ∑^n+m_i≠ i_1,i_2(w_i^2ν +w_i v_iμ/L_i) - 2ηe_w = 0. Set l⃗ = vec( L), and ṽ = v/l then they are rearranged into (1/L_i_1+1/L_i_2)-2μ (ṽ_̃ĩ_̃1̃+ṽ_̃ĩ_̃2̃)-2ν(w̃_̃ĩ_̃1̃+w̃_̃ĩ_̃2̃)+ ṽ⃗̃^Tv⃗μ^2+ w̃⃗̃^Tw⃗ν^2+2μνṽ⃗̃^Tw⃗- 4η^2 = 0, (ṽ_̃ĩ_̃1̃+ ṽ_̃ĩ_̃1̃) - ṽ⃗̃^Tv⃗μ - ṽ⃗̃^Tw⃗ν - 2ηe_v = 0, (w̃_̃ĩ_̃1̃+ w̃_̃ĩ_̃2̃) - w̃⃗̃^Tw⃗ν - ṽ⃗̃^Tw⃗μ - 2ηe_w = 0. From these results, we obtain μ = 2( e_wṽ⃗̃^Tw⃗ - e_vw̃⃗̃^Tw⃗ )η + (ṽ_i_1+ṽ_i_2)w̃⃗̃^Tw⃗ - (w̃_i_1 + w̃_i_2) (ṽ⃗̃^Tw̃⃗̃)/ṽ⃗̃^Tv⃗w̃⃗̃^Tw⃗ -(ṽ⃗̃^Tw⃗)^2, ν = 2( e_vṽ⃗̃^Tw⃗ - e_wṽ⃗̃^Tv⃗ )η + (w̃_i_1+w̃_i_2) ṽ⃗̃^Tv⃗ - (ṽ_i_1 + ṽ_i_2) (ṽ⃗̃^Tw̃⃗̃)/ṽ⃗̃^Tv⃗w̃⃗̃^Tw⃗ -(ṽ⃗̃^Tw⃗)^2. Denoting μ := s_1 η + s_2, and ν := u_1 η + u_2, we can solve η by the following quadratic equation: 0 = aη^2+bη+c, a = 4 - s_1^2ṽ⃗̃^Tv⃗ - u_1^2w̃⃗̃^Tw⃗ -2s_1 u_1ṽ⃗̃^Tw⃗, b = 2(ṽ_i_1 + ṽ_i_2)s_1 +2(w̃_i_1 + w̃_i_2)u_1 - 2s_1s_2 ṽ⃗̃^Tv⃗ - 2u_1u_2w̃⃗̃^Tw⃗ - 2(s_1u_2+s_2u_1)ṽ⃗̃^Tw⃗, c = 2(ṽ_i_1 + ṽ_i_2)s_2 +2(w̃_i_1 + w̃_i_2)u_2 -s_2^2ṽ⃗̃^Tv⃗ -u_2^2w̃⃗̃^Tw⃗ - 2s_2u_2ṽ⃗̃^Tw⃗ -(1/L_i_1+1/L_i_2). Putting these equations back into Eq.(<ref>), we obtain μ and ν. If the solution satisfies the constraints η^, μ^, ν^ > 0, then it is exactly the solution for the optimization problem. However, if one of the dual variables is less than 0, the problem would degenerate into a simpler question as one of the constraints is not activated. 1) If only the circle is activated:, then min_θ⃗∈ℛ^S_I- ( q_i_1 +q_i_2 ) = -2/√(L_1 + L_2). 2) If only one plane and one circle are activated, we can optimize on a spherical cap or dome, the solution is the same with (<cit.>). Or just setting the ν = 0 in our formula. As the hyperplane w⃗^Tq⃗ - e_w is parallel to the optimization direction, it is impossible to be the activated plane with the circle. Thanks to the distinctive structure of the index matrix of the UOT problem, the summation involved in solving this Lagrangian problem, with a complexity of 𝒪(m) or 𝒪(n), can be precomputed. Subsequently, this precomputed summation can be reused for elements within the same row or column, thereby maintaining the overall complexity of the screening method at O(nm). This complexity is consistent with other screening methods for the UOT problem. §.§ Computational complexity The complexity of Safe Screening for ℓ_2-penalized UOT primarily consists of three components: 1) duality and projection, 2) computation of the maximum value within the safe region. Regarding the duality and projection part, we have discussed it in the main text, where the complexity of the Shifting Projection is nearly identical to that of Residuals Rescaling. As for the computation of the maximum value, we have addressed it in Section <ref>. Although the CTP method involves additional computations compared to the Gap method, it shares the same complexity as the Dome method and the projection process. However, for KL-penalized OT, an additional step is introduced in the Safe Screening procedure: constructing a ℛ^B with local strong concavity, which necessitates the computation of bounds for optimizing the objective function. Based on the insights from the proof presented in Section <ref>, we observe that the computation of bounds is similar to finding the optimum in the CTP method. Due to the specific structure of the UOT problem, variables related to row and column summations can be reused, resulting in a complexity of O(nm). Thus, there is no increase in computational complexity in terms of the order of magnitude. §.§ Proof for Optimal Stepsize of FISTA in Section <ref> The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)<cit.> is a popular solver for regularized problems and can be applied to ℓ_1 regularized problems. It is a variant of the Mirror Descent algorithm, specifically when the smoothing part in the proximal operator is chosen as x⃗_2^2. The theoretical optimal step size for the Mirror Descent method is 1/L, where L is the constant of L-relatively smoothness for the function H(t⃗) = h(Xt⃗) <cit.>. In the context of applying FISTA to the UOT problem, where the proximal function is ϕ(t⃗) = t⃗_2^2, we aim to compute the minimum constant L that ensures the relative smoothness of the UOT problem. It is important to note that if a function H is L-relatively smooth with respect to function ϕ, then the function Lϕ - H is convex, or equivalently, D_H(x⃗, y⃗) < LD_ϕ(x⃗, y⃗) <cit.> We prove that Lϕ - H is convex by proving its positive semi-definiteness, i.e., d⃗^T ∇^2(Lϕ - H)d⃗≽ 0 for any d⃗∈ℝ^nm. Here, we denote that the i-th row vector of X, i.e., x⃗^i = [X_i,1,X_i,2,X_i,1,...,X_i,nm]. Then, because ∇^2 H(t⃗) = X^T X where X∈^(n+m) × nm, we have d⃗^T∇^2 H(t⃗)d⃗ = ∑_i=1^n+m(d⃗^Tx⃗^i)^2 = 2∑_j=1^nm d_j^2 + 2∑_p∈ I_p^mnd_u d_v ≤ 2∑_j=1^nm d_j^2 + (n+m - 2)∑_j^nmd_j^2 = (n+m) ∑_j=1^nmd_j^2 ≤ L ∑_j^nmd_j^2 = L d⃗^T∇^2 ϕ(t⃗)d⃗. Thus, we have proven the convexity of the Lϕ - H as L≥ n+m, and the theoretical stepsize is 1/n+m. As for the MM algorithm in <cit.> following the same method, we can get a fixed stepsize as 1/2, which is the same as the parameter in the paper. § ADDITIONAL EXPERIMENTS This section provides some additional experimental results. §.§ Comparison on coordinate descent (CD) method We have organized our experiments on CD with ℓ_2-penalized UOT, which has the same parameters setting with Para <ref>. Our experiments show great improvement, especially on the MNIST dataset. The outcome is illustrated in Table <ref>. §.§ Comparison on BFGS method We have organized our experiments on L-BFGS-B solver <cit.> with both ℓ_2-penalized and KL-penalized UOT, which has the same parameters setting with Para <ref>. The outcome is illustrated in Table <ref>. It is worth noting that the original code for the L-BFGS-B algorithm is encapsulated in Fortran, making direct modifications challenging for us. Consequently, we choose to call the L-BFGS-B Fortran code via mex and restart the algorithm after the Safe Screening process. This approach, albeit interrupting the update of the Hessian matrix, could potentially impact the convergence speed of the algorithm. In our comparison, both methods, with and without screening, utilized this interrupt-and-restart optimization process. This was undertaken to ensure the consistency of our experimentation. We only add BFGS experiments on KL-penalized UOT with λ = 1 because the convergence is too slow. § DISCUSSION AND PERSPECTIVES §.§ Large-scale OT Problem and Sparsity The sparsity of the OT and UOT problems is 1 - 1/(n+m) <cit.>. This indicates that a sparse-aware algorithm could potentially achieve a speed-up factor of n+m. Employing Safe Screening for OT and UOT problems is therefore promising for large-scale problems. §.§ Aggressive Screening Our Shifting Projection and CTP methods exploit the index matrix structure characteristics, and the Ellipse method utilizes the anisotropy of the dual variables in the dual space. However, there is potential for the development of more aggressive Safe Screening methods. For instance, the locally strongly concave function we proved on ℛ^B indicates that the Box bound is quite loose compared to the real dual optimum. This suggests that an improved method could enhance the screening performance on KL-penalized UOT by several orders of magnitude. Additionally, in real applications, we do not require a completely safe screening, which means that we could utilize more relaxed bounds to screen more elements, thereby achieving a faster algorithm with a minor sacrifice in accuracy. §.§ Limitations and Shortcomings Due to the specific structure of UOT, we can calculate its theoretical Lipschitz constant to compute an theoretically optimal stepsize. However, Safe Screening changes the dimension and structure of the problem, causing the optimal stepsize to change dynamically. Moreover, an effective Safe Screening method would identify theoretically zero elements before they reach zero. This behavior could detrimentally impact the error (like dual gap), and temporarily slow down the screening process. Such oscillation is difficult to manage. We prevent the algorithm from screening elements larger than zero to maintain the stability of our algorithm, facilitating a straightforward comparison of the speed-up ratio. This limitation also inhibits the potential of Safe Screening. Further research is necessary to facilitate its application.
http://arxiv.org/abs/2307.01680v1	20230704122240	Robust Hate Speech Detection in Social Media: A Cross-Dataset Empirical Evaluation	[ "Dimosthenis Antypas", "Jose Camacho-Collados" ]	cs.CL	[ "cs.CL", "I.2.7" ]	Serving Graph Neural Networks With Distributed Fog Servers For Smart IoT Services Liekang Zeng, Xu Chen, Peng Huang, Ke Luo, Xiaoxi Zhang, and Zhi Zhou The authors are with the School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, Guangdong, 510006 China (e-mail: [email protected], [email protected], {huangp57, luok7}@mail2.sysu.edu.cn, {zhangxx89, zhouzhi9}@mail.sysu.edu.cn). August 1, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================= The automatic detection of hate speech online is an active research area in NLP. Most of the studies to date are based on social media datasets that contribute to the creation of hate speech detection models trained on them. However, data creation processes contain their own biases, and models inherently learn from these dataset-specific biases. In this paper, we perform a large-scale cross-dataset comparison where we fine-tune language models on different hate speech detection datasets. This analysis shows how some datasets are more generalisable than others when used as training data. Crucially, our experiments show how combining hate speech detection datasets can contribute to the development of robust hate speech detection models. This robustness holds even when controlling by data size and compared with the best individual datasets. § INTRODUCTION Social media has led to a new form of communication that has changed how people interact across the world. With the emergence of this medium, hateful conduct has also found a place to propagate online. From more obscure online communities such as 4chan <cit.> and Telegram rooms <cit.> to mainstream social media platforms such as Facebook <cit.> and Twitter <cit.>, the spread of hate speech is an on going issue. Hate speech detection is a complex problem that has received a lot of attention from the Natural Language Processing (NLP) community. It shares a lot of challenges with other social media problems (emotion detection, offensive language detection, etc), such as an increasingly amount of user generated content, unstructured <cit.> and constantly evolving text <cit.>, and the need of efficient large scale solutions. When dealing with hate speech in particular, one has to consider the sensitivity of the topics, their wide range (e.g. sexism, sexual orientation, racism), and their evolution through time and location <cit.>. Understanding the extent of the problem and tracking hate speech online through automatic techniques can therefore be part of the solution of this ongoing challenge. One way to contribute to this goal is to both improve the current hate speech detection models and, crucially, the data used to train them. The contributions of this paper are twofold. First, we provide a summary and unify existing hate speech detection datasets from social media, in particular Twitter. Second, we analyse the performance of language models trained on all datasets, and highlight deficiencies in generalisation across datasets, including the evaluation in a new independently-constructed dataset. Finally, as a practical added value stemming from this paper, we share all the best models trained on the unification of all datasets, providing a relatively small-size hate speech detection model that is generalisable across datasets.[The best binary hate speech detection model is available at <https://huggingface.co/cardiffnlp/twitter-roberta-base-hate-latest>; the multiclass hate speech detection model identifying target groups is available at <https://huggingface.co/cardiffnlp/twitter-roberta-base-hate-multiclass-latest>. These models have been integrated into the TweetNLP library <cit.>.] Content Warning The article contains examples of hateful and abusive language. The first vowel in hateful slurs, vulgar words, and in general profanity language is replaced with an asterisk (). § RELATED WORK Identifying hate speech in social media is an increasingly important research topic in NLP. It is often framed as a classification task (binary or multiclass) and through the years various machine learning and information sources approaches have been utilised <cit.>. A common issue of supervised approaches lies not necessarily with their architecture, but with the existing hate speech datasets that are available to train supervised models. It is often the case that the datasets are focused on specific target groups <cit.>, constructed using some specific keyword search terms <cit.>, or have particular class distributions <cit.> that leads to a training process that may or may not generalise. For instance, <cit.> analysed the temporal aspect of hate speech, and demonstrate how brittle hate speech models are when evaluated on different periods. Recent work has also shown that there is a need to both focus on the resources available and also try to expand them in order to develop robust hate speech classifiers that can be applied in various context and in different time periods <cit.>. In this paper, we perform a large-scale evaluation to analyse how generalisable supervised models are depending on the underlying training set. Then, we propose to mitigate the relative lack of generalisation by using datasets from various sources and time periods aiming to offer a more robust solution. § DATA In this section, we describe the data used in our experiments. First, we describe existing hate speech datasets in Section <ref>. Then, we unify those datasets and provide statistics of the final data in Section <ref> §.§ Hate Speech datasets In total, we collected 13 datasets related to hate speech in social media. The datasets selected are diverse both in content, different kind of hate speech, and in a temporal aspect. Measuring hate speech (MHS) MHS <cit.> consists of 39,565 social media (YouTube, Reddit, Twitter) manually annotated comments. The coders were asked to annotate each entry on 10 different attributes such as the presence of sentiment, respect, insults and others; and also indicate the target of the comment (e.g. age, disability). They use Rasch measurment theory <cit.> to aggregate the annotators' rating in a continuous value that indicates the hate score of the comment. Call me sexist, but (CMS) This dataset of 6,325 entries <cit.> focuses on the aspect of sexism and includes social psychology scales and tweets extracted by utilising the "Call me sexist, but" phrase. The authors also include two other sexism datasets <cit.> which they re-annotate. Each entry is annotated by five coders and is labelled based on its content (e.g. sexist, maybe-sexist) and phrasing (e.g. civil, uncivil). Hate Towards the Political Opponent (HTPO) HTPO <cit.> is a collection of 3,000 tweets related to the 2020 USA presidential election. The tweets were extracted using a set of keywords linked to the presidential and vice presidential candidates and each tweet is annotated for stance detection (in favor of/against the candidate) and whether it contains hateful language or not. HateX HateX <cit.> is a collection of 20,148 posts from Twitter and Gab extracted by utilising relevant hate lexicons. For each entry, three annotators are asked to indicate: (1) the existence of hate speech, offensive speech, or neither of them, (2) the target group of the post (e.g. Arab, Homosexual), and (3) the reasons for the label assigned. Offense The Offense dataset <cit.> contains 14,100 tweets extracted by utilising a set of keywords and categorises them in three levels: (1) offensive and non-offensive; (2) targeted/untargeted insult; (3) targeted to individual, group, or other. Automated Hate Speech Detection (AHSD) In this dataset, <cit.> the authors utilise a set of keywords to extract 24,783 tweets which are manually labelled as either hate speech, offensive but not hate speech, or neither offensive nor hate speech. Hateful Symbols or Hateful People? (HSHP) This is a collection <cit.> of 16,000 tweets extracted based on keywords related to sexism and racism. The tweets are annotated as on whether they contain racism, sexism or neither of them by three different annotators.[A subset of the dataset is included in the Call me sexist, but and is not considered.] Are You a Racist or Am I Seeing Things? (AYR) This dataset <cit.> is an extension of Hateful Symbols or Hateful People? and adds the "both" (sexism and racism) as a potential label. Overlapping tweets were not considered. Multilingual and Multi-Aspect Hate Speech Analysis (MMHS) MMHS <cit.> contains hateful tweets in three different languages (English, French, Arabic). Each tweet has been labelled by three annotators on five different levels: (1) directness, (2) hostility (e.g. abusive, hateful), (3) target (e.g. origin, gender), (4) group (e.g. women, individual) and (5) annotator emotion (disgust, shock, etc). A total of 5,647 tweets are included in the dataset. HatE HatE <cit.> consists of English and Spanish tweets (19,600 in total) that are labelled on whether they contain hate speech or not. The tweets in this dataset focus on hate speech towards two groups: (1) immigrants and (2) women. HASOC This dataset <cit.> contains 17,657 tweets in Hindi, German and English which are annotated on three levels: (1) whether they contain hate-offensive content or not; (2) in the case of hate-offensive tweets, whether a post contains hate, offensive, or profane content/words; (3) on the nature of the insult (targeted or un-targeted). Detecting East Asian Prejudice on Social Media (DEAP) This is a collection of 20,000 tweets <cit.> focused on East Asian prejudice, e.g. Sinophobia, in relation to the COVID-19 pandemic. The annotators were asked to labelled each entry based on five different categories (hostility, criticism, counter speech, discussion, non-related) and also indicate the target of the entry (e.g. Hong Kongers, China). Large Scale Crowdsourcing and Characterization of Twitter Abusive Behavior (LSC) The dataset <cit.> consists of 80,000 tweets extracted using a boosted random sample technique. Each tweet is labelled as either offensive, abusive, hateful, aggressive, cyberbullying or normal. §.§ Unification Even though all of the datasets that were collected revolve around hate speech, there are major differences among them in terms of both format and content. We attempt to unify the datasets by standarizing their format and combining the available content into two settings: (1) binary hate speech classification and (2) a multiclass classification task including the target group. We note that in cases where the original annotation results were provided, we decided to assign a label if at least two of the coders agree on it and not necessarily the majority of them. This approach can lead to a more realistic dataset and contribute in creating more robust systems <cit.>. §.§.§ Initial preprocessing For each dataset collected, a simple preprocessing pipeline is applied. Firstly, any non-Twitter content is removed; despite the similarities between the content shared in various social media (e.g. internet slang, emojis), Twitter displays unique characteristics, such as the concept of retweets and shorter texts, which differentiate it from other platforms such as Reddit or Youtube <cit.>. Moreover, as our main consideration is hate speech in the English language, we exclude any non-English subset of tweets, and also verify the language by using a fastText based language identifier <cit.>. Finally, considering that some datasets in this study utilise similar keywords to extract tweets, we remove near duplicated entries to avoid any overlap between them. This is accomplished by applying a normalisation step where entries that are considered duplicated based on their lemmatised form are ignored. Also, all URLs and mentions are removed. As a final note, three of the datasets (HSHP, AYR, LSC) were dehydrated using the Twitter API since only their tweet IDs and their labels were publicly available. Unfortunately, a significant number of tweets (≈10,000) were no longer available from the API. §.§.§ Binary Setting The majority of the datasets collected are either set as a binary hate classification task and no further preprocessing is applied (HTPO), or offer a more fine-grained classification of hate speech (e.g. HateX, CMS) where we consider all "hate" subclasses as one. In general, when a dataset focuses on a specific type of hate speech (e.g. sexism) we map it as hate speech. Notable exceptions are: (1) The MSH dataset, where a continues hate score is provided which is transformed into a binary class according to the mapping proposed by the original authors. (2) Datasets that consist of offensive speech but also provide information about the target of the tweet. In these cases, (Offense), we consider only entries that are classified as offensive and are targeting a group of people and not individuals. Our assumption is that offensive language towards a group of people is highly likely to target protected characteristics and thus be classified as hate speech. (3) Finally, only entries classified as hate speech were considered in datasets where there is a clear distinction between hate, offensive, or profound speech (LSC, AHSD, HASOC). All data labelled as normal or not-hateful are also included as not-hate speech. §.§.§ Multiclass Setting Having established our binary setting, we aggregated the available datasets aiming to construct a more detailed hate speech classification task. As an initial step, all available hate speech sub-classes present were considered. However, this led to a very detailed but sparse hate taxonomy, with 44 different hate speech categories, but with only a few entries for some of the classes (e.g. "economic" category with only four tweets present). Aiming to create an easy-to-use and extendable data resource, several categories were grouped together. All classes related to ethnicity (e.g. Arab, Hispanic) or immigration were grouped under racism, while religious categories (e.g. Muslim, Christian) were considered separately. Categories related to sexuality and sexual orientation (e.g. heterosexual, homosexual) were also grouped in one class, and tweets with topics regarding gender (men, women) constitute the sexism class. Finally, all entries labelled as "not-hate" speech were also included. To keep our dataset relatively balanced we also ignored classes that constitute less than 1% of the total hate speech data. Overall, the multiclass setting proposed consists of 7 classes: Racism, Sexism, Disability, Sexual orientation, Religion, Other, and Not-Hate. It is worth noting that tweets falling under the Other class do not belong to any of the other five hate speech classes. §.§.§ Statistics and Data Splits In total, we collected 83,230 tweets, from 13 different datasets (Table <ref>), of which only 33% are classified as hate speech. This unified dataset may seem imbalanced but it is commonly assumed that only around 1% of the content shared on social media contains hate speech <cit.>. When considering the multiclass setting, the hate speech percentage decreases even more with only 26% of tweets labelled as a form of hate speech, with the religion class being the least popular with only 709 entries. The data in both settings (binary & multiclass) are divided into train and test sets using a stratified split to ensure class balance between the splits (Table <ref>). In general, for each dataset present, we allocate 70% as training data, 10% as validation, and 20% as test data. Exceptions to the aforementioned approach are datasets where the authors provide a preexisting data split which we use. § EVALUATION We present our main experimental results comparing various language models trained on single datasets and in the unified dataset presented in the previous section. §.§ Experimental Setting Models. For our experiments we rely on four language models of a similar size, two of them being general-purposes and the other two specialized on social media: BERT-base <cit.> and RoBERTa-base <cit.> as general-purpose models; and BERTweet <cit.> and TimeLMs-21 <cit.> as language models specialized on social media, and particularly Twitter. There is an important difference between BERTweet and TimeLMs-21: since BERTweet was trained from scratch, TimeLMs-21 used the RoBERTa-base checkpoint as initialization and then continued training on a Twitter corpus. An SVM classifier is also utilized as a baseline model. Settings. Aiming to investigate the effect of a larger and more diverse hate speech training corpus on various types of hate speech, we perform an evaluation on both the binary and multiclass settings described in Section <ref>. Specifically, for the binary setting we fine-tune the models selected first on each individual dataset, and secondly while using the unified dataset created. For the multiclass setting, we considered the unified and the HateX dataset, which includes data for all classes. In total, we fine-tuned 54 different binary[MMHS dataset was used only for the training/evaluation of the unified dataset as it is lacking the not-hate class] and 8 multiclass models. Training. The implementations provided by Hugging Face <cit.> are used to train and evaluate all language models, while we utilise Ray Tune <cit.> along with HyperOpt <cit.> and Adaptive Successive Halving <cit.> for optimizing the learning rate, warmup steps, number of epochs, and batch size, hyper-parameteres of each model.[Optimal hyperparameters can be found in Table <ref> in the Appendix] Evaluation metrics. The macro-averaged F1 score is reported and used to compare the performance of the different models. Macro-F1 is commonly used in similar tasks <cit.> as it provides a more concrete view on the performance of each model. §.§ Datasets For training and evaluation, we use the splits described in Section <ref>. As described above, for each language model we trained on each dataset training set independently, and in the combination of all dataset-specific training sets. The results on the combination of all datasets are averaged across each dataset-specific test set (AVG), i.e., each dataset is given the same weight irrespective of its size. In addition to the datasets presented in Section <ref>, we constructed an independent test set (Indep) to test the robustness of models outside existing datasets. Independent test set (Indep). This dataset was built by utilising a set of keywords related to the International Women's Day and International Day Against Homophobia, Transphobia and Biphobia and extracting tweets from the respected days of 2022. Then, these tweets were manually annotated by an expert. In total 200 tweets were annotated as hateful, not-hateful, or as "NA" in cases where the annotator was not sure whether a tweet contained hate speech or not. The Indep test set consists of 151 non-hate and 20 hate tweets and due to its nature (specific content & expert annotation) can be leveraged to perform a targeted evaluation on models trained on similar and unrelated data. While we acknowledge the limitations of the Indep test set (i.e., relative small number of tweets and only one annotator present), our aim is to use these tweets, collected using relatively simple guidelines[Annotator guidelines are available in Appendix <ref>.], to test the overall generalisation ability of our models and how it aligns to what people think of hate speech. §.§ Results §.§.§ Binary Setting Table <ref> displays the macro-F1 scores achieved by the models across all test sets when fine-tuned: (1) on all available datasets (All), (2) on the best overall performing model trained on a single dataset, and (3) on a balanced sample of the unified dataset of the same data size as (2). When looking at the average performance of (1) and (2), it is clear that when utilising the combined data, all models perform considerably better overall. This increased performance may not be achieved across all the datasets tested, but it does provide evidence that the relatively limited scope of the individual datasets hinder the potential capabilities of our models. An even bigger constrast is observed when considering the performance difference on the DEAP subset, which deals with a less common type of hate speech (prejudice towards Asian people), where even the best performing single dataset model achieves barely 19.79% F1 compared to the worst combined classifier with 49.27% F1 (BERT All / BERT HTPO). To further explore the importance of the size and diversity of the training data we train and evaluate our models in an additional settings. Considering the sample size of the best performing dataset for each model, an equally sized training set is extracted from all available data while enforcing a balanced distribution between hate and not-hate tweets (All). Finally, we make sure to sample proportionally across the available datasets. The results (Table <ref>) reveal the significance that a diverse dataset has in the models' performance. All models tested perform on average better when trained on the newly created subsets (All) when compared to the respective models trained only on the best performing individual dataset. Interestingly, this setting also achieves the best overall scores on the Indep. set, which reinforces the importance of balancing the data. Nonetheless, all the transformers models still achieve their best score when trained on all the combined datasets (All) which suggests that even for these models, the amount of available training data remains an important factor of their performance. §.§.§ Multiclass Setting Similarly to our binary setting, utilising the combined datasets in the multiclass setting enhances the models' performance. As can be observed from Table <ref>, all the models struggle to function at a satisfactory degree when trained on the HateX subset only. In particular, when looking at the "disability" class, none of the models manage to classify any of the entries correctly. This occurs even though "disability" entries exist in the HateX training subset, albeit in a limited number (21). This behaviour suggests that even when information about a class is available in the training data, language models may fail to distinguish and utilise it. Imbalanced datasets are a common challenge in machine learning applications. This issue is also present in hate speech, in this case exacerbated given the nature of the problem (including a potential big overlap of features between classes) and the lack of resources available. § ANALYSIS In this section, we dissect the results presented in the previous section by performing a cross-dataset comparison and a qualitative error analysis. §.§ Cross-dataset Analysis Figure <ref> presents a cross-dataset comparison of the language models used for the evaluation. The heatmap presents the results of the models fine-tuned and tested for all dataset pair combinations. All models evaluated tend to perform better when they are trained and tested on specific subsets (left diagonal line on the heat-maps). Even when we evaluate models on similar subsets, they tend to display a deterioration in performance. For example both CMS and AYR datasets deal with sexism but the models trained only on CMS perform poorly when evaluated on AYR (e.g. BERTweet-CSM achieves 87% F1 on CSM, but only 52% on AYR). Finally, it is observable again that the models trained on the combined datasets (column "all") display the best overall performance and attain consistently high results in each individual test set. When analysing the difficulty of each individual dataset when used as a test set, DEAP is clearly the most challenging one overall. This may be due to the scope of the dataset, dealing with East Asian Prejudice during the COVID-19 pandemic, which is probably not well captured in the rest of the datasets. When used as training sets, none of the individual datasets is widely generalisable, with the results of the model fine-tuned on them being over 10 points lower than when fine-tuned on the unified dataset in all cases. §.§ Qualitative Error Analysis Aiming to better understand the models' results we perform a qualitative analysis focusing on entries miss-classified by our best performing model, TimeLMs-All. Multiclass. When considering the multiclass setting, common errors are tweets that have been labelled as hateful, e.g. "U right, probably some old ngga named Clyde" is labelled as racism and "@user @user she not a historian a jihadi is the correct term" as religion, but the model classifies them as not-hate. However, depending on the context and without having access to additional information (author/target of the tweet) these entries may not actually be hateful. It is also interesting to note the limitations that arise when training only on a single dataset, particularly if the data collection is done by utilising specific keywords. For example the tweets "Lana i love you btch. Put that flag back up he #lustfoflife" and "happy birthday btch, hope you have a great one he! @user" are correctly classified as not-hate by TimeLMs-All but are miss-classified as sexism by TimeLMs-HateX, despite sexism being present in the HateX dataset. Binary In the binary setting, the model seems to struggle with entries such as "Meanwhile in Spain..#stopimmigration" and "This is outrageous. Congress should be fired on the spot. #BuildThatWall #stopwastingmytaxdollars" where both entries are classified as hate but are labelled as not-hate. Similarly to the previous case, the classification of such tweets without additional context is a difficult task. While these tweets have hateful undertones, they may not be necessarily hate speech without considering them in their broader context. Finally, when looking at the classification errors of TimeLMs-AYR (trained only on sexist and racist tweets) the need of diverse training data becomes apparent. For example, TimeLM-AYR fails to classify as hate speech the tweets "@user that rtarded guy should not be a reporter" and "I'm going to sell my iPhone and both my Macs, I don't support fggots." as hate speech in contrast to TimeLMs-All which classifies the tweets correctly as hateful. § CONCLUSION In this paper, we presented a large-scale analysis of hate speech detection systems based on language models. In particular, our goal was to show the divergences across datasets and the importance of having access to a diverse and complete training set. Our results show how the combination of datasets make for a robust model performing competitively across all datasets. This is not a surprising finding given the size of the corresponding training sets, but the considerable gap (e.g. 70.7% to 61.0% in Macro-F1 for the best TimeLMs-21 performing model) shows that models trained on single datasets have considerable room for improvement. Moreover, even when controlling for data size, a model trained on a diverse set instead of a single dataset leads to better overall results. As future work, we are planning to extend this analysis beyond English, in the line of previous multilingual approaches <cit.>, and masked language models by including, among others, generative and instruction-tuning language models. In addition to the extensive binary-level evaluation, recognising the target group is a challenging area of research. While in Section <ref>, we provided some encouraging results, the results could be expanded with a unified taxonomy. § ETHICS STATEMENT Our work aims to contribute and extend research regarding hate speech detection in social media and particular in Twitter. We believe that our efforts to contribute on the ongoing concerns around the status of hate speech on social medial. We acknowledge the importance of the ACM Code of Ethics, and are committed on following it's guidelines. Our current work, uses either publicly available tweets under open licence and does not infringe any of the rules of Twitter's API. Moreover, given that our task includes user generated content we are committed to respect the privacy of the users, by replacing each user mention in the texts with a placeholder. § LIMITATIONS In this paper, we have focused on existing datasets and a unification stemming from their features. The decisions taken to this unification, particularly in the selection of dataset and target groups, may influence the results of the paper. We have focused on social media (particularly Twitter) and on the English language. While there has been extensive work on this medium and language, the conclusions that we can take from this study can be limiting, as the detection of hate speech involves other areas, domains and languages. In general, we studied a particular aspect of hate speech detection which may or not be generalizable. Finally, due to computational limitations, all our experiments are based on base-sized language models. It is likely that larger models, while exhibiting similar behaviours, would lead to higher results overall. § ACKNOWLEDGEMENTS The authors are supported by a UKRI Future Leaders Fellowship. They also acknowledge the collaboration with the Spanish National Office Against Hate Crimes and the support of the EU Citizens, Equality, Rights and Values (CERV) programme. However, the authors have the exclusive responsibility for the contents of this publication. Finally, the authors thank Nina White for her help annotating the independent test set. acl_natbib § ANNOTATION GUIDELINES In the following we present the guidelines provided to the annotator for the independent test set (Section <ref>). A tweet is: * labelled as "1" ("hate speech") if it contains any “discriminatory” (biased, bigoted or intolerant) or “pejorative” (prejudiced, contemptuous or demeaning) speech towards individuals or group of people. * labelled as "0" ("not-hate-speech") if it does not contain hate speech as defined above. * labelled "NA" if the coder is not sure whether the tweet contains hate speech or not. The annotation should be based only on the text content of the tweet. This means that the coder should not follow any URL/media links if present. § HYPERPARAMETER TUNING Table <ref> lists the best hyperparameters for each of the models used in the evaluation.
http://arxiv.org/abs/2307.02329v2	20230705143947	Data-driven Predictive Latency for 5G: A Theoretical and Experimental Analysis Using Network Measurements	[ "Marco Skocaj", "Francesca Conserva", "Nicol Sarcone Grande", "Andrea Orsi", "Davide Micheli", "Giorgio Ghinamo", "Simone Bizzarri", "Roberto Verdone" ]	cs.NI	[ "cs.NI", "cs.LG" ]	IEEEexample:BSTcontrol Data-driven Predictive Latency for 5G: A Theoretical and Experimental Analysis Using Network Measurements Marco Skocaj1, Francesca Conserva1, Nicol Sarcone Grande1, Andrea Orsi2, Davide Micheli2, Giorgio Ghinamo2, Simone Bizzarri2 and Roberto Verdone1 1 DEI, University of Bologna, & WiLab, CNIT, Italy 2 TIM, Italy This work has been accepted for publication at IEEE PIMRC 2023.© 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= The advent of novel 5G services and applications with binding latency requirements and guaranteed qos hastened the need to incorporate autonomous and proactive decision-making in network management procedures. The objective of our study is to provide a thorough analysis of predictive latency within 5G networks by utilizing real-world network data that is accessible to mobile network operators (MNOs). In particular, (i) we present an analytical formulation of the user-plane latency as a Hypoexponential distribution, which is validated by means of a comparative analysis with empirical measurements, and (ii) we conduct experimental results of probabilistic regression, anomaly detection, and predictive forecasting leveraging on emerging domains in ml, such as bl and gml. We test our predictive framework using data gathered from scenarios of vehicular mobility, dense-urban traffic, and social gathering events. Our results provide valuable insights into the efficacy of predictive algorithms in practical applications. Predictive Quality of Service, Latency, Machine Learning, Bayesian Learning, Machine Learning on Graphs, 5G. § INTRODUCTION The 5g wireless technology allows the virtual connection of everyone and everything together, including machines and devices. urllc is one of the leading pillars of the 5g standard, which aims to provide extremely low latency values and reliability up to 99.99% <cit.>. Industries, transportation, precision agriculture, and v2x communications are some of the driving applications for the development of urllc. The rise of such new services and applications with binding latency requirements and guaranteed qos, together with recent advancements in ai, are paving the way to the deployment of autonomous connected systems. Within this context, the idea of pqos has been introduced as a means of equipping autonomous systems with proactive notification regarding imminent changes in qos. The experienced qos is affected by various elements, such as interference, mobility, network conditions, and terminal characteristics (e.g., number of antennas). Although different services have different qos constraints in terms of latency and reliability, being able to prevent a session interruption due to qos degradation becomes a key requirement <cit.>. In this respect, being able to predict qos changes, becomes a crucial aspect to preventively adjust the application behavior <cit.>. Nowadays, minimizing latency means providing aid for real-time applications (e.g., online games, autonomous driving, etc.), ensuring greater interactivity and smoother experiences, increasing the energy efficiency of 5g networks, and improving reliability in mission-critical applications. Furthermore, latency is critical to foster a range of new applications, such as virtual and augmented reality, smart cities, and connected cars. mno have access to a vast amount of ran measurements, including network kpi and counters, measuring uplink/downlink data volumes, transmission parameters, monitoring of radio resources, as well as accessibility/handover requests/failures, among many others. Such data availability offers significant opportunities: different levels of granularity at both a spatial and temporal level boost the network analysis capability, enabling the true potential of pqos. The present study aims to provide a thorough investigation of predictive latency within 5g networks. This is achieved through the utilization of kpi ran measurements obtained at each gnb, combined with the development of a predictive framework based on cutting-edge ml methodologies. Our objective is to offer a comprehensive analysis of the key factors affecting predictive latency in 5G networks and to assess the potential benefits of employing advanced ml techniques in this context. In this regard, we collected measurements on three clusters characterized by diverse traffic patterns, among which vehicular and dense-urban traffic. §.§ State of the art Various studies have focused on the subject of pqos, with particular emphasis on its relevance to v2x communications. Authors in <cit.> model the qos prediction as a binary classification problem to determine whether a packet can be delivered within a defined latency window with the use of standard ml techniques such as rrf and mlp. In the framework of aqosa, a qos adjustment assistance mechanism has been developed to predict and notify qos changes at application level <cit.>. Another aspect of pqos concerns the identification of the relationship between features. In this regard, a work of noticeable importance is <cit.>, which proposes a network model based on gnn capable of understanding the connection between topology and input traffic to estimate the per-packet delay distribution using deep learning techniques. With respect to urllc, the authors of <cit.> attempt to monitor and forecast the rapid fluctuations in channel conditions caused by fast fading, in order to facilitate advanced scheduling. Finally, the research community has shown significant interest in reducing latency in 5G networks. Various analytical models have been developed to evaluate e2e latency by implementing different scheduling configurations and observing several 5G features <cit.>. §.§ Contributions Our work aims to offer a comprehensive analysis of predictive latency in 5G networks using real-world network data available to mno and developing a solid framework leveraging recent advancements in ml. Our contributions can be summarized as follows: * Starting from 3GPP definitions, we present an analytical formulation of the U-plane (User-plane) latency, proving the latter can be modeled as a Hypoexponential distribution. We ascertain the validity of our analytical outcomes by means of a comparative analysis with empirical network measurements. * We discuss the use of emerging domains within the field of ml, such as bl and gml to tackle three distinct PQoS use cases: probabilistic regression, anomaly detection, and predictive forecasting. * We conduct numerical experiments using kpi collected from three distinct traffic scenarios, namely vehicular mobility, dense-urban environment, and social gathering events. Our objective is to evaluate the performance of predictive models under diverse and representative traffic conditions. § PROBLEM FORMULATION Our reference scenario comprises network kpi gathered from three clusters of cells scattered throughout the entire area of the city of Bologna, Italy (Fig. <ref>). The first cluster gathers gnb from the city center area, the second one encompasses the highway and the ring road, whereas the last one covers an industrial area hosting concerts and social gathering events. The network kpi exploited for the latency prediction are obtained as a statistical average of the measurements gathered with a periodic interval of 15 minutes, for a total of one entire month of data. Details about the feature selection and the individual indicators are discussed in section <ref>. 3gpp employs the qci scalar value to assess the quality of packet communication. The QCI refers to a particular packet forwarding behavior (e.g.: resource type, priority, and packet loss rate) to be delivered to a sdf <cit.>. For the scope of this work, we focus our attention on kpi data collected from two qci classes: * qci1, i.e. conversational voice service that requires gbr resource type and packet error loss rate of 10^-2. * qci7, i.e. voice, video streaming, and interactive gaming services that represent the majority of traffic nowadays; they require non-gbr resource type and packet error loss rate of 10^-3. The remainder of this section delves into the details of latency formulation and derives a probabilistic interpretation of the U-plane latency within 5g communications systems. This model is derived from the definitional framework established by the 3gpp, described in section <ref>. §.§ 3GPP Overview According to 3gpp, latency can be formalized as the sum of C-plane (Control-plane) and U-plane (User-plane) latency<cit.>. The former measures the time elapsed from a ue's rach preamble transmission and the successful reception at the gnb of a rrc Connection Complete message; in other words, it measures the transition time of a ue from rrc-idle state to rrc-connected state. On the other hand, the U-plane latency is a measure of the transit time between a packet being available at the ue (or ran gnb) IP layer, and the availability of this packet at the IP layer of the ran gnb (or ue) <cit.>. Besides the processing delays, the tti duration, and harq loop needed to receive the packet correctly, the U-plane latency also accounts for the number of packet retransmissions occurring with probability equal to the bler. Within the scope of our work, we direct our attention towards the dl U-plane latency for a two-fold reason: (i) the user plane latency offers greater degrees of freedom in terms of optimization compared to the C-plane latency, which depends primarily on the random access procedure; (ii) network measurements are collected uniquely for users in RRC-connected mode, for whom U-plane latency is the sole quantifiable delay because it involves only the ran, whereas the C-plane latency includes delay contributions that impact the cn as well. Furthermore, the U-plane dl represents the majority of generated traffic. As a final remark, we consider the case of dynamic (grant-based) scheduling, for which the gnb needs to forward scheduling information to the ue before transmitting data on the PDSCH. §.§ Latency formulation Leveraging on the 3gpp definitions introduced above, we define the U-plane latency, denoted as L, as per (<ref>): L = τ_ radio + τ_ HARQ + N * (τ_ radio' + τ_ HARQ), with N={0, 1, …, N_max}. In (<ref>), τ_ radio is a random variable accounting for the radio latency over the Uu interface related to the first gnb-ue transmission. Similarly, τ_ radio' accounts for the same delay when a packet is re-scheduled for transmission upon reception of a negative acknowledgment. For the sake of generality, we account for the two terms as separate and independent random variables, assuming that prioritization mechanisms take place for the dynamic scheduling of previously discarded packets, i.e., 𝔼[τ_ radio'] ≤𝔼[τ_ radio]. Finally, τ_ HARQ accounts for the delay introduced by the HARQ mechanisms, and N denotes the total number of re-transmissions. Notice that Eq. (<ref>) is a measure of latency at the IP layer, consistent with the 3gpp definition discussed in Sec. <ref>. The present analysis excludes the transport layer due to the tendency to introduce varying additional delays based on the particular protocol employed (e.g., tcp introduces delays due to connection establishment between two end-point, or error-checking and re-transmissions of corrupted packets). Similarly to previous works <cit.>, we can hereby decompose τ_ radio (τ_ radio') into the sum of two independent quantities related to the scheduling and the transmission time of the packet, namely τ_ sch (τ_ sch') and τ_ pack. Downlink scheduling information on the PDSCH is delivered to the ue by the dci on the PDCCH. Differently from <cit.>, we denote with τ_sch the whole interval of time between the packet generation at the gnb and the instant when the packet is transmitted on the PDSCH, as reported in Fig. <ref>. Accordingly, τ_sch coincides with the waiting time of a M/M/1 system, as per traditional queuing theory. On the other hand, τ_pack, which is fully determined by 5g numerology, average ue's mcs, number of available rb, etc., is equivalent to the service time. In Appendix A, leveraging on queuing theory, we show that the sum of τ_sch and τ_pack in an M/M/1 system can be modeled as a negative exponential distribution. On the other hand, for the sake of simplicity and without loss of generality, let us assume τ_HARQ as a fixed delay. Consequently, it is possible to re-formulate (<ref>) as a sum of independent random variables, as per (<ref>): L = τ_sch + τ_pack + τ_HARQ_= C_τ_tx + N · (τ_sch' + τ_pack + τ_HARQ_= C_τ_rtx), where N is geometrically distributed with success parameter p equivalent to the complementary bler, i.e., N ∼ geom(1 - BLER), and τ_sch + τ_pack∼ exp(λ_1), τ_sch' + τ_pack∼ exp(λ_2). As a result, τ_tx and τ_rtx are still negative exponential distribution with rate parameters λ_1, λ_2 and mean value 1/λ_i + C. Writing N in explicit form, we can reformulate (<ref>) as: L = ∑_j=0^N_max P_j(τ_tx + j ·τ_rtx)= τ_tx∑_j=0^N_max P_j_=1 + τ_rtx∑_j=1^N_maxj · P_j = = τ_tx + τ_rtx P_1 + 2τ_rtx P_2 + 3τ_rtxP_3 + … + o(P_n), where P_j = P(N=j) = BLER^j· (1-BLER). Thus, L can be approximated at the n-th order as the sum between n independent negative exponential random variables with monotonically increasing rate values ∝ 1/P_j, i.e., τ_tx∼ exp(λ_1) and τ_rtx∼ exp(λ_2/P_j). As analytically shown in Appendix B, this results in a Hypoexponential distribution L ∼ hexp(λ_1, …, λ_N). Our theoretical formulation is confirmed by empirical data, as depicted in Fig. <ref>. § MEASURING LATENCY FROM NETWORK KPIS In this section, we will discuss the kpi employed as the ground truth (L) in our analysis, as well as the designed feature space. In order to identify a suitable set of kpi for measuring and predicting L, it is essential to have a comprehensive understanding of the factors that influence its behavior. As per (<ref>), L is influenced by both traffic and radio channel conditions. Indeed, situations of high network congestion may adversely affect τ_sch, which is dependent on the number of total ue in the queue. Similarly, unfavorable radio conditions and high levels of interference can lead to an increased bler, requiring a potentially higher number of packet retransmissions to achieve a successful transmission. §.§ Ground truth evaluation The identified kpi for estimating L provides a measure of the delay in transmitting a pdcp sdu in the downlink given a specific qci value, as defined in the 3gpp TS 36.314, whose definition is reported in (<ref>): P_delay(T, QCI) = ⌊∑_it_ack(i)-t_arriv(i)/I(T)⌋, where t_arriv(i) is the point in time when the pdcp sdu reaches the pdcp layer at the transmitter side (i.e., at the gnb in case of DL transmission); t_ack(i) represents the instant corresponding to the last piece of the i-th pdcp sdu received by the gnb according to received harq feedback. Finally, I(T) indicates the total number of pdcp sdu, and T represents the period during which the measurement is performed. The elected kpi is provided as the average sum of two network counters: the first accounts for the retention delay within the gnb; the second considers the average delay introduced by harq loop. Although this measure is taken at the pdcp level, thus excluding the IP layer latency, without loss of generality, it can still be considered a good estimation of L. Indeed, the missing inter-layer processing delay can be neglected if compared to the other delay contributions. §.§ Feature selection Here, we delve into the feature selection process, where we identify the kpi to be utilized for predicting L. Feature selection has been performed following both numerical investigations, such as correlation analysis, and logical criteria. Specifically, we considered the dependency of L on traffic conditions and network quality. The first group includes the average number of active users in the dl, the traffic volume (expressed in terms of pdcp sdu) in dl, and the average prb usage during the tti in the dl. On the other hand, the second group leverages the average cqi, the average values of rssi and sinr on the on pusch, and the average values of mcs on both the pusch and pdsch. As an additional feature, the temporal information of data acquisition is also incorporated. Table <ref> displays the results of a Pearson correlation analysis between the selected features and the latency measure intended for prediction. It is noteworthy that the features related to traffic exhibit a stronger correlation in comparison to those pertaining to the quality of the radio channel. As a matter of fact, the kpi related to the utilization of resources in the dl shows a correlation value of approximately 0.8 with the average pdcp sdu latency in dl. § ALGORITHMS AND EXPERIMENTAL RESULTS This section presents exemplary experimental results for three use cases of interest in the context of pqos. The subsequent subsections introduce each use case, elucidate the underlying theoretical aspects of the proposed algorithms, and subsequently present the numerical outcomes. To ensure the preservation of sensitive information of the mno, the numerical findings are displayed in a standardized format. §.§ Use case 1: Bayesian probabilistic regression Accurate evaluation of network performance in mobile networks requires the application of regression techniques to qos indicators. For instance, these can be employed by mno to assess network performance using simulated data prior to on-field deployment. Unlike non-probabilistic regression methods that provide only a single-point estimate of the predicted value, probabilistic regression allows for the estimation of the probability distribution of the predicted values, providing a complete picture of the underlying uncertainty associated with the predictions. bnn <cit.>, in particular, are a powerful tool for modeling aleatoric and epistemic uncertainty in a principled way. While the former refers to the intrinsic randomness of the observed data, the latter is captured by the posterior distribution P(θ\| D) of the bnn's parametrized model weights, which is updated by means of Bayesian inference as new data D becomes available. In practice, bnn are usually trained via svi by minimizing a Monte-Carlo estimate of the variational free energy cost function (<ref>)<cit.>: min_λ{KL[q_λ(θ)‖ P(θ)]- 𝔼_θ∼ q_λ[log(P(𝐲\|𝐱, θ))]}. In (<ref>), the left-hand side term refers to the KL divergence between q_λ, a variational distribution parametrized by a set of parameters λ (typically modeled as a multi-variate normal with learnable diagonal covariance matrix), and P(θ), the true prior distribution of the model weights. On the right-hand side, 𝔼_θ∼ q_λ[log(P(𝐲\|𝐱, θ))] refers to the statistical average of the model likelihood, obtained via Monte Carlo sampling of the bnn. Minimizing (<ref>) embodies the tradeoff between maximizing the likelihood over the training data and minimizing the KL divergence with respect to a known prior, which acts as a regularization term. In our experiments, we aim to reflect the latency probability distribution derived in section <ref>. To this end, we explicitly model the last layer of a bnn as a Hypoexponential distribution that is parametrized based on the output of the preceding layer. Specifically, the output dimension of the previous layer reflects an n-th order approximation of the Hypoexponential distribution. In Fig. <ref>, we provide exemplary results on a regression task performed on the dense-urban scenario. As noticeable, the true latency values (blue samples) trustfully lie within the 95% confidence intervals of the probabilistic model, which achieves an overall R2 score of 0.77 on a held-out test set. It is important to notice that Fig. <ref> portrays latency measurements obtained from distinct cells captured at various points in time. Consequently, the depicted data is not arranged in a temporal sequence. §.§ Use case 2: Anomaly detection Anomaly detection refers to the identification of events significantly differing from the expected behavior of a system. These can manifest by unusual patterns that can be captured or not by network kpi. Typical examples may include network congestion, hardware failures, or jamming attacks. Based on the assumption that anomalies are often unlikely, the latter can be formulated as a density estimation problem. Given any point {𝐱, 𝐲}∈{ℝ^n, ℝ^l}, if we can estimate a probability density function f̂_θ(𝐱,𝐲), parametrized by θ, indicating the latency distribution for any given point of the feature space ∈ℝ^n, then we can detect an anomaly as per (<ref>): {𝐱_i, 𝐲_i}∈𝒜⟺f̂(𝐱_i, 𝐲_i\|θ) ≤Γ, where 𝒜 denotes the set of anomalies and Γ indicates a likelihood threshold, which is fine-tuned a-posteriori based on a cost model devised as a function of the confusion matrix. When targeting anomaly detection of latency patterns, two distinct methodologies can be pursued, as elaborated upon subsequently: (i) The establishment of a threshold on the conditional probability distribution of y given x, i.e., f̂(𝐱_i, 𝐲_i\|θ) = P(𝐲\|𝐱, θ), which can be suitably modeled using either a pnn or a bnn, or (ii) The establishment of a threshold based on the reconstruction error of an ae on the whole set {𝐱,𝐲}. In the latter case, the joint probability distribution of x,y, i.e., f̂(𝐱_i, 𝐲_i\|θ) = P(𝐱, 𝐲\|θ) is modeled by the latent space of the ae. Both methods are rational as a network's kpi in the feature space 𝐱 can detect abnormal situations like network congestion. Conversely, such kpi may not be able to recognize anomalies such as malfunctioning antenna hardware. Empirical findings resulting from the application of the second approach are depicted in Fig. <ref>. To construct our test set, we adopt the following method: utilizing the social gathering events dataset, we designate as anomalous the samples obtained from the cellular network coverage encompassing the stadium during the concert event from 7 pm to 11:30 pm on the 16th and 17th of March 2023, yielding a total of 38 anomalies. We subsequently train our ae on a set of non-anomalous samples, obtaining a confusion matrix yielding 717 true negatives, 2 false negatives, 0 false positives, and 36 true positives on the test set. §.§ Use case 3: Predictive forecasting As a final use case, we focus here on predictive forecasting leveraging temporal and spatial information. Predictive forecasting refers to the prediction of future latency given a set of instantaneous and past observations gathered at different locations in the network. This is at the core of pqos, as it allows for a proactive optimization approach. In this section, we aim to show how predictive latency forecasting can effectively be achieved by leveraging spatial and temporal information with the use of rnn and gnn. While lstm networks <cit.> have the ability to capture long-term dependencies in time-series by utilizing a memory cell and three gating mechanisms, gnn afford a strong relational inductive bias beyond that which convolutional and recurrent layers can provide <cit.>. In the remainder of the section, we present the numerical outcomes achieved by utilizing a probabilistic lstm and GraphSAGE <cit.> on Key Performance Indicators (KPIs) collected in the two distinct scenarios of vehicular mobility and social gathering events. §.§.§ Time-series forecasting We leverage a lstm model equipped with a probabilistic layer at its final stage and trained via minimization of negative log-likelihood. For the sake of simplicity, and without loss of generality, we focus on the prediction task of instant t+1, i.e. 15 min ahead. The vehicular traffic dataset was partitioned into two sets, with 20 consecutive days designated for training purposes and the subsequent 10 days employed for testing (Fig. <ref>), obtaining an overall R2 score of 0.75. §.§.§ Spatial forecasting Lastly, we compare the performance of a GraphSAGE model, composed of 3 graph convolutional layers, against a baseline dnn trained on the entire dataset with samples from individual cells. The obtained results, depicted in Fig. <ref>, demonstrate that the former yields superior performance with an R2 score of 0.77 compared to the baseline dnn with an R2 score of 0.62. As expected, the obtained results suggest that incorporating spatial information from neighboring data points can significantly enhance the model's predictive capability. § CONCLUSION In this work, we provided a comprehensive theoretical and experimental analysis of predictive latency using real-world measurements available to mno. The principal outcomes of our study demonstrate that the latency observed in the U-plane conforms to a Hypoexponential probability distribution. This valuable insight was utilized in our experimental assessment of state-of-the-art ml techniques in the context of probabilistic regression, anomaly detection and predictive forecasting. § SOJOURN TIME DISTRIBUTION The sojourn time S of a packet in a system can be calculated as the sum of its waiting time W in the queue and the service time B required by the server to process the request <cit.>. Here, τ_sch and τ_pack represent W and B, respectively. Under the hypothesis of a M/M/1 queue, the packets' arrivals are Poisson distributed with parameter β, B∼ exp(μ), and the utilization factor, ρ :=β / μ≤ 1. According to the Pollaczek –Khinchine formula for a M/G/1 queue <cit.>, the Laplace-Stieltjes transform, S̃(s) of S is expressed as: S̃(s) = (1-ρ) ·B̃(s) · s/β·B̃(s)+s-β. In a M/M/1 model, B̃(s)=μ /(μ+s) <cit.>. Therefore, Eq. (<ref>), becomes: S̃(s) = μ· (1-ρ)/μ· (1-ρ)+s) Applying the definition of Laplace-Stieltjes transform of a non-negative r.v X, i.e.,∫_x=0^∞ e^-sx· f(x) dx with s ≥ 0, we obtain: ∫_x=0^∞ e^-st·λ e^-λ· t dt=μ· (1-ρ)/μ· (1-ρ)+s= S̃(s). Hence, S is exponentially distributed with rate parameter λ = μ· (1-ρ), that is f_S(t)=λ e^-λ· t. § DERIVATION OF L'S PDF For sufficiently small values of BLER, and without loss of generality, let us consider the case for which (<ref>) can be approximated at the 2-nd order, i.e. L ≈τ_tx + τ_rtx· P_1_τ_rtx'. The probability distribution of the sum of two independent continuous random variables can be computed as the convolution between the two individual distributions. Therefore, considering τ∼τ_tx and τ_L∼ L = τ_tx + τ_rtx', we have: p_L_2(τ_L) = ∫_-∞^∞p_τ_tx(τ)p_τ_rtx(τ_L - τ) dx = ∫_0^τ_Lλ_1 e^-λ_1 τλ_2/P_1 e^-λ_2/P_1 (τ_L - τ) dx = λ_1 λ_2/P_1 e^-λ_2/P_1τ_L∫_0^τ_L e^(λ_2/P_1 - λ_1) τ dx = λ_1λ_2/λ_2 - λ_1/P_1 e^-λ_2/P_1τ_L[e^(λ_2/P_1 - λ_1)τ]_0^τ_L = λ_1λ_2/λ_2 - λ_1/P_1(e^-λ_1 τ_L - e^-λ_2/P_1τ_L), which is equivalent to the probability distribution of L ∼ hexp(λ_1, λ_2/P_1). § ACKNOWLEDGMENTS This work was partially supported by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, partnership on “Telecommunications of the Future” (PE00000001 - program “RESTART”), by the Italian MUR PON 2014-2020 under Project “reCITY - Resilient City - Everyday Revolution” (cod. ARS01 00592, CUP B69C21000390005), and by the European Union’s Horizon Europe program through the project CENTRIC. IEEEtran
http://arxiv.org/abs/2307.00809v2	20230703074756	Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation	[ "Lucas Huysmans", "Edriss S. Titi" ]	math.AP	[ "math.AP", "76F25 (Primary) 35A02, 35D30, 35Q35, 60J60, 76R10, 76R50 (Secondary)" ]	We consider the transport equation of a passive scalar f(x,t)∈ℝ along a divergence-free vector field u(x,t)∈ℝ^2, given by ∂ f/∂ t + ∇· (u f) = 0; and the associated advection-diffusion equation of f along u, with positive viscosity/diffusivity parameter ν>0, given by ∂ f/∂ t + ∇· (u f) -νΔ f = 0. We demonstrate failure of the vanishing viscosity limit of advection-diffusion to select unique solutions, or to select entropy-admissible solutions, to transport along u. First, we construct a bounded divergence-free vector field u which has, for each (non-constant) initial datum, two weak solutions to the transport equation. Moreover, we show that both these solutions are renormalised weak solutions, and are obtained as strong limits of a subsequence of the vanishing viscosity limit of the corresponding advection-diffusion equation. Second, we construct a second bounded divergence-free vector field u admitting, for any initial datum, a weak solution to the transport equation which is perfectly mixed to its spatial average, and after a delay, unmixes to its initial state. Moreover, we show that this entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of the corresponding advection-diffusion equation. A Comprehensive Survey of Artificial Intelligence Techniques for Talent Analytics Chuan Qin, Member, IEEE, Le Zhang, Rui Zha, Dazhong Shen, Qi Zhang, Ying Sun, Member, IEEE, Chen Zhu, Member, IEEE, Hengshu Zhu, Senior Member, IEEE, Hui Xiong, Fellow, IEEE This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. C. Qin, C. Zhu, and H. Zhu are with the Career Science Lab, BOSS Zhipin, Beijing, China. E-mail: [email protected], [email protected], [email protected]. L. Zhang is with the Business Intelligence Lab, Baidu Inc, Beijing, China. E-mail: [email protected]. R. Zha is with the University of Science and Technology of China, Anhui, China. E-mail: [email protected]. D. Shen and Q. Zhang are with the Shanghai Artificial Intelligence Laboratory. E-mail: [email protected], [email protected]. Y. Sun and H. Xiong are with the Hong Kong University of Science and Technology (Guangzhou), china. E-mail: [email protected], [email protected] H. Zhu and H. Xiong are the corresponding authors. August 1, 2023 =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION §.§ Background We are concerned with the fundamental problem of selection among non-unique weak solutions in fluid mechanics. Consider a divergence-free vector field u:𝕋^d×[0,T]→ℝ^d, on a d-dimensional spatial torus 𝕋^d and a time interval [0,T]. For a passive scalar f:𝕋^d×[0,T]→ℝ, we may write the passive transport of f along u in conservation-form as ∂ f/∂ t + ∇· (u f) = 0. TE This equation is understood in the distributional sense, and may exhibit non-uniqueness of weak solutions, see e.g. diperna1989ordinary,depauw2003non or modena2018non,cheskidov2021nonuniqueness for higher regularity of the vector field. Meanwhile, the advection-diffusion of f along u with viscosity/diffusivity parameter ν>0 is given by ∂ f/∂ t + ∇· (uf) - νΔ f = 0. ν-ADE In contrast to the transport equation, equations of this type are well-posed, see e.g. evans2010partial,flandoli2010well. The inviscid limit ν→0 is of great interest to many problems in fluid mechanics, including also anomalous dissipation in turbulence (both hydrodynamic kolmogorov1941local,brue2023anomalous,brue2022onsager, and passive scalar corrsin1951spectrum,obukhov1949,drivas2019anomalous), and enhanced dissipation constantin2008diffusion,bedrossian2021almost,elgindi2023optimal which arises in the general study of mixing, see e.g. <cit.>. We restrict our attention to its use as a selection principle, see e.g. <cit.>, and also as a justification of the inviscid model. That is, we concern ourselves with the nature of the limiting solution. When the inviscid equation exhibits non-uniqueness, it becomes an important question which solutions arise as the vanishing viscosity limit. In general one expects these to be more regular, physically pertinent, and perhaps unique, and indeed this is verified in specific cases, see e.g. dafermos2005hyperbolic,Bianchini2005,bardos2012vanishing,bardos2013stability,Nussenzveig_Lopes_2021. To date there have been very few counterexamples. The first non-uniqueness result for the case of passive scalars was announced recently, in <cit.>, to which we add an entirely novel construction, see Theorem <ref> below. We also ask another fundamental question - whether or not the resulting vanishing viscosity solutions are indeed physical. The presence of molecular diffusion leads one to conclude that vanishing viscosity should be taken as the definition of physical admissibility. However, the second law of thermodynamics implies that any physical solution should satisfy the entropy-inequalities introduced in <cit.>. Since energy (or entropies) are weakly lower semi-continuous, the vanishing viscosity limit may only dissipate energy/entropy. It is then generally believed that this ensures agreement between the above selection rules, and indeed the previously mentioned works support this conjecture. We give the first known counterexample for passive scalars, see Theorem <ref> below. Specifically, we show that entropy-admissibility fails despite even uniqueness of the vanishing viscosity limit, and that this occurs for any (non-constant) initial datum. This result is extremely surprising since it implies disagreement between the above physically motivated laws. Moreover, in the context of active scalars such as the incompressible Euler equations, entropy-inadmissible solutions are termed wild solutions. There is a long history of construction and research into such solutions, starting with the early works scheffer1993inviscid,shnirelman1997nonuniqueness,constantin1994onsager, and then the introduction of the convex-integration method, see e.g. <cit.>. It is widely believed that these wild solutions may not be the limit, let alone unique limit, of strong or weak Leray-Hopf <cit.> solutions of the incompressible Navier-Stokes equations. Our construction highlights a possible mechanism by which this may fail. While weak lower semi-continuity of energy ensures that energy may only dissipate in the vanishing viscosity limit, it fails to ensure that the amount of dissipation increases with time. It is therefore possible that energy cascades to infinite frequencies only for short time, before cascading back to lower frequencies. There is therefore no known mechanism preventing wild solutions to the Euler equations, in particular those constructed by the convex-integration machinery (see e.g. de2013dissipative,buckmaster2017onsager,buckmaster2019nonuniqueness), arising as the late-time behaviour of vanishing viscosity limits of strong solutions to the incompressible Navier-Stokes equations, given sufficient initial energy. Moreover, our construction also highlights that the general existence of entropy-admissible solutions of passive scalar transport, and perhaps other inviscid hyperbolic conservation laws, is an open problem. Whether or not other regularisation schemes may be found which recover entropy-admissibility is an important question for future work. §.§ Main results We point to Section <ref> for notation of function spaces, and for rigorous definitions of weak solutions of the transport equation (Definition <ref>, (<ref>)), weak renormalised solutions of the transport equation (Definition <ref>), and weak solutions of the advection-diffusion equation with viscosity ν>0 (Definition <ref>, (<ref>)). We point to Section <ref>, Theorem <ref>, for well-posedness of the advection-diffusion equation, in particular the uniqueness stated in the below Theorems. There exists a divergence-free vector field u ∈ L^∞([0,1];L^∞(𝕋^2;ℝ^2)), and a sequence {ν_n}_n∈ℕ with ν_n > 0 and ν_n0, such that for any initial data f_0∈ L^∞(𝕋^2), and for f^ν the unique solution to (<ref>) along u with initial data f_0, one has f^ν_2nf^even, f^ν_2n+1f^odd, with the above convergence in weak-* L^∞([0,1];L^∞(𝕋^2)), and strong in L^p([0,1];L^p(𝕋^2)) for all p ∈ [1,∞). Furthermore, the limit functions f^even, f^odd are renormalised weak solutions to (<ref>) along u with initial data f_0. Moreover, if f_0 is not constant, then f^even f^odd. There exists a divergence-free vector field u ∈ L^∞([0,100];L^∞(𝕋^2;ℝ^2)), such that for any initial data f_0∈ L^∞(𝕋^2), and for f^ν the unique solution to (<ref>) along u with initial data f_0, one has f^νf, with the above convergence in weak-* L^∞([0,100];L^∞(𝕋^2)), strong in L^p([0,42];L^p(𝕋^2)), C^0([0,42-ϵ];L^p(𝕋^2)), L^p([58,100];L^p(𝕋^2)), and C^0([58+ϵ,100];L^p(𝕋^2)) for all p ∈ [1,∞), and all ϵ >0. The limit function f ∈ C_weak-^0([0,100];L^∞(𝕋^2)) is a weak solution to (<ref>) along u with initial data f_0. Moreover, for all t ∈ [42,58] f(·, t) ≡∫_𝕋^2f_0(y) dy, is perfectly mixed to its spatial average. Furthermore, for all t ∈ [0,100], f(·, t)=f(·,100-t) and in particular, f(·, 100) = f_0, is perfectly unmixed. In particular, if f_0 is not constant, any L^p(𝕋^2) norms of f(·, t) (for p∈(1,∞]) increase after t=58, contrary to the entropy-admissibility criterion of Dafermos in <cit.>. §.§ Outline of paper Section <ref> contains the notation and definitions used throughout this paper. Section <ref> contains the necessary uniqueness, existence, and regularity theory for solutions of the transport equation and advection-diffusion equation. In section <ref> we develop the main techniques and intuition used to construct the examples in Theorems <ref>, <ref>. Section <ref> is then devoted to the proof of Theorem <ref>. Section <ref> is instead devoted to the proof of Theorem <ref>, and does not build on Section <ref>. § NOTATION AND DEFINITIONS §.§ Notation Denote by 𝕋^d = ℝ^d / ℤ^d the d-dimensional unit torus. Throughout this paper we work on the spatial domain 𝕋^d, or on the spatio-temporal domain 𝕋^d× [0,T] for some T>0. The gradient operator ∇ and Laplacian Δ will act on spatial coordinates only. If the dimension d and time interval [0,T] are implicit we use the shorthand notation L^pL^q for L^p([0,T];L^q(𝕋^d)) and similar spaces. Meanwhile, L^q(𝕋^d) is shorthand for scalar functions L^q(𝕋^d;ℝ). For integrability exponents p,q,... ∈ [1,∞] (which are allowed to be infinite unless otherwise stated), denote by p^ the Hölder conjugate. For a positive integer n∈ℕ we denote the homogeneous Sobolev space Ḣ^n(𝕋^d)={f∈ L^2(𝕋^d):∫_𝕋^df(x) dx=0, and (-Δ)^n/2f∈ L^2(𝕋^d)}, where the fractional Laplacian (-Δ)^n/2 is defined in terms of the Fourier transform/series ℱ by ℱ((-Δ)^n/2f) = \|ξ\|^n ℱ(f). We denote the norm in Ḣ^n(𝕋^d) by f_Ḣ^n(𝕋^d) = (-Δ)^n/2f_L^2(𝕋^d), for all f ∈Ḣ^n(𝕋^d). Moreover, we define the Sobolev space H^n(𝕋^d) = {f∈ L^2(𝕋^d):(-Δ)^n/2f∈ L^2(𝕋^d)}, with the norm f_H^n(𝕋^d) = f_L^2(𝕋^d) + (-Δ)^n/2f_L^2(𝕋^d). Furthermore, we denote by H^-n(𝕋^d) the dual space of H^n(𝕋^d). For each p ∈ [1,∞] we define also the space C^0L^p = {f:[0,T]→ L^p(𝕋^d) such that f is continuous}⊂ L^∞ L^p, with the norm f_C^0L^p = sup_0≤ t ≤ Tf(·,t)_L^p(𝕋^d). Moreover, we define the linear space C_weak^0L^p = {f:[0,T]→ L^p(𝕋^d) such that f is continuous in L^p_weak}⊂ L^∞ L^p, and for p ∈ (1,∞] the linear space C_weak-^0L^p = {f:[0,T]→ L^p(𝕋^d) such that f is continuous in L^p_weak-}⊂ L^∞ L^p. For p ∈ (1,∞), L^p(𝕋^d) is reflexive and so these two definitions coincide. However, we commonly take p=∞, hence we choose the notation C_weak-^0L^p and take care of the case when p=1. All results will be stated and proved for the compact domains 𝕋^d and [0,T]. The exact statements do not necessarily hold when we replace 𝕋^d with ℝ^d, or [0,T] with [0,∞), though analogues can surely be found. Consider a vector field u∈ L^1([0,T];L^1(𝕋^d;ℝ^d)) with ∇· u=0 in the distributional sense. We say f∈ L^1([0,T];L^1(𝕋^d;ℝ)) with uf ∈ L^1L^1 is a weak solution to the transport equation along u ∂ f/∂ t + ∇· (u f) = 0, TE with initial data f_0 ∈ L^1(𝕋^d) if, [Equivalently we may take ϕ to be Lipschitz in time and space with compact support in 𝕋^d × [0,T). This is done by finding a sequence of smooth functions ϕ_n ∈ C_c^∞(𝕋^d×[0,T)) such that ϕ_n are bounded in C^1 by the Lipschitz norm of ϕ, and ∂ϕ_n/∂ t∂ϕ/∂ t, ∇ϕ_n ∇ϕ pointwise almost everywhere.]for any ϕ∈ C_c^∞(𝕋^d×[0,T)), ∫_𝕋^d×[0,T) f (∂ϕ/∂ t + u·∇ϕ) dxdt = - ∫_𝕋^d f_0 ϕ_0 dx, where ϕ_0(x) = ϕ(x, 0). Meanwhile, we say the transport equation is satisfied on an open interval I⊂(0,T) if, for any ϕ∈ C_c^∞(𝕋^d× I), ∫_𝕋^d× I f (∂ϕ/∂ t + u·∇ϕ) dxdt = 0. Following the definition introduced in <cit.>, suppose f is a weak solution to (<ref>) along u with initial data f_0. If, for any β∈ C_b^0(ℝ), β(f) is a weak solution to (<ref>) along u with initial data β(f_0), then we say f is a renormalised weak solution of (<ref>). This definition is well motivated by the expression ( ∂/∂ t + u ·∇) β(f) = β'(f)( ∂/∂ t + u ·∇) f when β is a differentiable function. Indeed some authors require β∈ C^1(ℝ) as in <cit.>, or even with decay at infinity as in <cit.>. In our case (∇· u = 0) it is straightforward to show that these give equivalent definitions. For a vector field u ∈ L^1([0,T];L^1(𝕋^d;ℝ^d)), we say a family (t ∈ [0,T]) of Lebesgue-measure preserving bijections y_t : 𝕋^d →𝕋^d is a Lagrangian flow along u if for a.e. x∈𝕋^d the map t ↦ y_t(x) is absolutely continuous and the derivative satisfies d y_t(x)/dt = u(y_t(x),t) as a function class in L^1([0,T];L^1(𝕋^d;ℝ^d)). [Notice, without loss of generality we may take y_0=Id.]We say that a function f ∈ L^1([0,T]; L^1(𝕋^d;ℝ)) is a Lagrangian solution to (<ref>) along u (with initial data f_0 ∘ y_0^-1) if f(·, t) = f_0 ∘ y_t^-1 for f_0∈ L^1(𝕋^d), and {y_t}_t∈[0,T] a Lagrangian flow along u. [We reduce the problem to bounded f and f_0 by the point-wise approximation with f 1_\|f\|≤ k, so that u f 1_\|f\|≤ k uf in L^1L^1 by dominated convergence. Observe then that f 1_\|f\|≤ k are already Lagrangian solutions for the initial data f_0 1_\|f_0\|≤ k, since (f_0 ∘ y_t^-1) 1_\|f_0 ∘ y_t^-1\| ≤ k = (f_0 1_\|f_0\|≤ k) ∘ y_t^-1.]It is straightforward to show that, if additionally uf ∈ L^1 L^1, then a Lagrangian solution to (<ref>) along u with initial data f_0 is also a weak solution in the sense of Definition <ref>. The result for bounded f and f_0 follows from changing variables in the integral ∫_𝕋^d×[0,T) f_0(y_t^-1(x)) (∂ϕ/∂ t + u·∇ϕ) dxdt, and using that (∂ϕ/∂ t + u·∇ϕ)(y_t(x)) = ∂/∂ t(ϕ(y_t(x), t)) by the chain rule for absolutely continuous functions. Moreover, since for any β∈ C_b^0(ℝ) we can rewrite β(f_0 ∘ y_t^-1) = β(f_0) ∘ y_t^-1, these solutions are then also renormalised weak solutions in the sense of Definition <ref>. Consider a vector field u∈ L^1([0,T];L^1(𝕋^d;ℝ^d)) with ∇· u=0 in the distributional sense, and some positive constant viscosity ν > 0 (also called diffusivity). We say f∈ L^1([0,T];L^1(𝕋^d;ℝ)) with uf ∈ L^1L^1 is a weak solution to the advection-diffusion equation along u ∂ f/∂ t + ∇· (u f) - νΔ f = 0 ν-ADE, with initial data f_0 ∈ L^1(𝕋^d), if for any ϕ∈ C_c^∞(𝕋^d×[0,T)), ∫_𝕋^d×[0,T] f (∂ϕ/∂ t + u·∇ϕ + νΔϕ) dxdt = - ∫_𝕋^d f_0 ϕ_0 dx, where ϕ_0(x) = ϕ(x,0). Meanwhile, we say the advection-diffusion equation (<ref>) is satisfied on an open interval I⊂(0,T) if, for any ϕ∈ C_c^∞(𝕋^d× I), ∫_𝕋^d× I f (∂ϕ/∂ t + u·∇ϕ + νΔϕ) dxdt = 0. Consider a vector field u∈ L^1([0,T];L^1(𝕋^d;ℝ^d)) with ∇· u=0 in the distributional sense. We say a weak solution f to (<ref>) along u with initial data f_0 ∈ L^1(𝕋^d) is a vanishing viscosity solution, if there exists a positive sequence ν_n→0 and corresponding weak solutions f^(n) to (<ref>) with viscosity ν_n along u with initial data f_0, such that f^(n)n →∞ f, converges as distributions in 𝒟'(𝕋^d×[0,T)). In the literature f^(n) is often taken to have, in addition, non-constant initial data f_0^(n), where f_0^(n) converges to f_0 in a suitable topology. We do not consider such a more general definition here. § NECESSARY BACKGROUND The following is by no means a full exposition of standard existence, uniqueness, and regularity theory for (<ref>) and (<ref>), but contains some of the more salient points, and in particular those relevant to this paper. The exact statements are adapted for this paper, with similar results, and similar methods of proof, found in the literature. We fix a finite time interval [0,T] throughout. Suppose f is a weak solution to (<ref>) or (<ref>) (i.e. ν=0) along u ∈ L^1 L^1 with initial data f_0. Then for any ϕ∈ C^∞(𝕋^d×[0,T]), for a.e. t ∈ [0,T], (Trace Formula) ∫_𝕋^d f(·, t) ϕ(·, t) dx = ∫_𝕋^d f_0 ϕ_0 dx + ∫_𝕋^d×[0,t] f(∂ϕ/∂ t+u·∇ϕ +νΔϕ) dxdt, [An analogous result holds when p=1 if we additionally assume f(·,t): [0,T] → L^1(𝕋^d) is uniformly integrable in the indexing variable t (for a.e. t∈[0,T]).]Suppose further that f ∈ L^∞ L^p for p ∈ (1, ∞], then there is a (unique) representation of f ∈ C_weak-^0L^p, such that (<ref>) holds for all t ∈ [0,T]. In particular f(·, 0) = f_0 in L^p(𝕋^d). For any weak solution of (<ref>) or (<ref>) we have f, uf ∈ L^1L^1. Fix some ϕ∈ C^∞(𝕋^d × [0,T]) and consider the following L^1([0,T]) function of t ∈ [0,T], ∫_𝕋^d f(·, t) ϕ(·, t) dx. Then for all ψ∈ C_c^∞([0,T)), by Definitions <ref>, <ref>, ∫_𝕋^d × [0,T) f ϕdψ/ds dxds [t] = ∫_𝕋^d × [0,T) f ϕdψ/ds dxds - (∫_𝕋^d f_0 ϕ_0 ψ_0 dx + ∫_𝕋^d×[0,T) f(∂/∂ s+u·∇ + νΔ)(ϕψ(s)) dxds ) = - ∫_𝕋^d f_0 ϕ_0 ψ_0 dx - ∫_𝕋^d×[0,T) f(∂ϕ/∂ s+u·∇ϕ + νΔϕ) ψ(s) dxds, where ψ_0=ψ(0) and ϕ_0(x)=ϕ(x,0). And so the function defined in (<ref>) is an absolutely continuous function of t ∈ [0,T], with derivative ∫_𝕋^d f(x, t)(∂ϕ/∂ s+u·∇ϕ + νΔϕ)(x, t) dx, and the initial value at t=0 is ∫_𝕋^d f_0 ϕ_0 dx, see Chapter 3, Lemma 1.1 in <cit.>. That is (<ref>) holds for a.e. t ∈ [0,T]. Suppose now, for some p ∈ (1,∞], that f ∈ L^∞ L^p. Define for each t ∈ [0,T] the distribution F_t∈𝒟'(𝕋^d) acting on test functions χ∈ C^∞(𝕋^d), given by ⟨ F_t,χ⟩ = ∫_𝕋^d f_0 χ dx + ∫_𝕋^d×[0,t] f(u·∇χ +νΔχ) dxdt. Then thanks to (<ref>), for a.e. t ∈ [0,T], F_t = f(·, t) as distributions in 𝒟'(𝕋^d), which is assumed uniformly bounded in L^p(𝕋^d). We therefore have for a.e. t ∈ [0,T], for all χ∈ C^∞(𝕋^d), the bound \|F_t(χ)\| ≤f_L^∞ L^pχ_L^p^. By the absolute continuity of t↦⟨ F_t,χ⟩ the bound in fact holds for all t ∈ [0,T]. Since p∈(1,∞] this implies further that F_t can be extended to a function f̅(·, t) ∈ L^p(𝕋^d) for all t ∈ [0,T], and the absolute continuity of ⟨ F_t, χ⟩ implies f̅∈ C_weak-^0L^p with (<ref>) holding for the representative f̅ now for all t ∈ [0,T], as required. An important application of this is that for t ∈ [0, T], and any ϕ∈ C_c^∞(𝕋^d × [t,T)), ∫_𝕋^d×[t,T] f(∂ϕ/∂ t+u·∇ϕ +νΔϕ) dxdt = - ∫_𝕋^d f(·, t) ϕ(·, t) dx. So if f∈ C_weak-^0L^p is a weak solution to (<ref>)/(<ref>) along u with initial data f_0, then it is also a weak solution to (<ref>)/(<ref>) along u on the time interval [t, T] with initial data f(·, t). This in particular allows us to say that f∈ C_weak-^0L^p is a weak solution to (<ref>)/(<ref>) with initial data f_0 if and only if it is a weak solution to (<ref>)/(<ref>) on [0,t] with initial data f_0, and on [t, T] with initial data f(·, t). That is we have transitivity of the transport equation. Suppose f_0 ∈ L^∞(𝕋^d), and u∈ L^1([0,T];L^1(𝕋^d;ℝ^d)) is divergence-free in the distributional sense, then there exists a weak solution f ∈ C_weak-^0 L^∞ to (<ref>)/(<ref>) along u with initial data f_0. Moreover, for all p ∈ [1,∞], t ∈ [0,T], (Initial L^p-Inequality) f(·, t)_L^p(𝕋^d)≤f_0_L^p, (Conservation of Mass) ∫_𝕋^d f(·, t) dx = ∫_𝕋^d f_0 dx . More generally, for f_0 ∈ L^q(𝕋^d), u ∈ L^1([0,T];L^q^(𝕋^d;ℝ^d)), existence of a weak solution f with f_L^∞ L^p≤f_0_L^p (for all p∈[1,∞], permitting infinite values of the norms) follows from a standard approximation scheme (regularisation of u and f_0), see <cit.> for details; though the proof considers only ν = 0 (i.e. (<ref>)), and the spatial domain ℝ^d instead of 𝕋^d, an identical argument goes through here. By assumption u ∈ L^1L^1, and so we may apply the above result with q=∞. By Theorem <ref> the solution is in C_weak-^0 L^∞, and so the bound f_L^∞ L^p≤f_0_L^p implies the Initial L^p-Inequality (<ref>). Conservation of Mass (<ref>) follows from the Trace Formula (<ref>) in Theorem <ref> with ϕ≡ 1 on 𝕋^d × [0,T]. In fact Conservation of Mass (<ref>) will hold for a.e. t ∈ [0,T] for every weak solution on the torus, and not only those constructed in Theorem <ref>. This does not hold in ℝ^d. Though a large topic of interest, we do not mention further regularity results for the transport equation, as they require further assumptions on u. Instead we point to <cit.> for an exposition of the Cauchy-Lipschitz theory, and other more standard results, including the DiPerna-Lions theory of renormalised weak solutions (introduced in <cit.>), and Ambrosio's extension of this well-posedness class to u ∈ L^1([0,T];BV) (originally in <cit.>). Next, we give the following well-posedness and regularity result for (<ref>). To illustrate the stark contrast between (<ref>) and its regularisation (<ref>) we show well-posedness for any f_0 ∈ L^1(𝕋^d). When in addition f_0 is more integrable we may obtain further regularity. Therefore, when stating our main results, we shall later assume f_0∈ L^∞. Suppose u∈ L^∞ L^∞, then for any initial data f_0 ∈ L^1(𝕋^d) any weak solution (in the class L^1 L^1) to (<ref>) along u with initial data f_0 is unique. Moreover this solution exists, f∈ C^0L^1 with f_C^0L^1≤f_0_L^1(𝕋^d) and becomes immediately bounded, f ∈ C^0([ϵ,T];C^0(𝕋^d)) for all ϵ∈(0,T]. Furthermore, when f_0 ∈ L^∞ we have the following additional regularity (for all p∈[1,∞)) f ∈ (C^0 L^p) ∩ (L^2 H^1), f ∈ C_weak-^0L^∞, (L^p-Inequality) 0 ≤ s ≤ t f(·, t)_L^p(𝕋^d)≤f(·, s)_L^p(𝕋^d), (Energy Identity) ∫_𝕋^d\|f(x, t)\|^2 dx + 2ν∫_𝕋^d × [0,t]\|∇ f(x,s)\|^2 dxds = ∫_𝕋^d\|f_0(x)\|^2 dx, (Equicontinuity) ∂ f/∂ t_L^2H^-1≤(u_L^2 L^∞ + √(ν/2))f_0_L^2. The standard well-posedness result is for f_0∈ L^2, and is done via energy estimates, see for example Chapter 7 in <cit.>. However, if we wish to obtain uniqueness in the class L^1 L^1 we must be more careful. [Given by its spatial Fourier series ℱ(K_ν(·,t))(ξ) = e^-νξ^2 t.]Denote by K_ν∈ C^∞(𝕋^d×(0,∞))∩ L^∞((0,∞); L^1) the heat kernel for the heat equation with diffusivity ν, that is for any ϕ∈ C_c^∞(𝕋^d × (-∞, ∞)) we have (∂/∂ t - νΔ)(ϕ _x,t (K_ν 1_t>0)) = ϕ, where _x,t denotes convolution is space and time. Denoting by K̅_ν∈ C^∞(𝕋^d×(-∞,0)) the backwards heat kernel K̅_ν(x,t) = K_ν(-x,-t), then we have for the backwards heat equation, for any ϕ∈ C_c^∞(𝕋^d × (-∞, ∞)) (∂/∂ t + νΔ)(ϕ _x,t (K̅_ν 1_t<0)) = -ϕ. For ϕ∈ C_c^∞(𝕋^d×[0,T)) we may take ϕ _x,t (K̅_ν 1_t<0) as a test function in (<ref>), which (by expanding out all convolutions) can be rewritten as ∫_𝕋^d×[0,T] fϕ dx dt = ∫_𝕋^d f_0(x) (ϕ _x,t (K̅_ν 1_t<0))(x,0) dx + ∫_𝕋^d×[0,T] fu·(ϕ _x,t (∇K̅_ν 1_t<0)) dx dt = ∫_𝕋^dϕ(x,t) (f_0 _x (K_ν(·, t)) - (fu 1_t ∈ [0,T]) _x,t (∇ K_ν 1_t>0))(x,t) dx dt, where _x denotes convolution over the spatial variable x ∈𝕋^d only. That is we have shown, indeed for any weak solution f to (<ref>) along u with initial data f_0 ∈ L^1(𝕋^d), f = f_0 _x K_ν(·, t) - (fu1_t∈[0,T]) _x,t (∇ K_ν 1_t>0). When f_0 = 0 we have by Young's convolution inequality, f1_t∈[0,ϵ]_L^1 L^1≤f1_t∈[0,ϵ]_L^1 L^1u_L^∞ L^∞∇ K_ν 1_t∈(0,ϵ]_L^1 L^1. It is straightforward to check that ∇ K_ν 1_t ∈ (0,T]∈ L^1 L^1, and so for ϵ small enough (depending only on u_L^∞ L^∞ and ν) the above implies that f1_t∈[0,ϵ] = 0. Repeating the argument then shows that for all n ∈ℕ, f1_[0,nϵ]=0, and so indeed f=0, proving uniqueness. Existence of a weak solution f ∈ L^∞ L^1 with f_L^∞ L^1≤f_0_L^1 follows by solving the equation for mollified u and f_0 as in Theorem <ref>. It can be checked that this produces a sequence of smooth functions f_n converging to the solution f weakly, and moreover, by the argument in <cit.>, with an a-priori bound on f_n_L^∞([ϵ,T];L^∞(𝕋^d)) depending only on f_0_L^1(𝕋^d), u_L^1L^∞, ϵ > 0, and ν. We therefore have f ∈ L^∞ L^1 and f∈ L^∞([ϵ,T];L^∞(𝕋^d)) for any ϵ>0. [We do not elaborate further on this point as it is not essential for the remainder of the paper.]The further regularity f ∈ C^0 L^1, and f ∈ C^0([ϵ,T];C^0(𝕋^d)) for all ϵ∈(0,T], then follow from the formula (<ref>) and the regularity of K_ν. When in addition f_0 ∈ L^2(𝕋^d) the regularised sequence f_n can further be shown to converge in (C^0L^2) ∩ (L^2H^1). Statements (<ref>) and (<ref>) then follow from their counterparts for smooth f_0 and u. If f_0 ∈ L^∞(𝕋^d) we have, say by (<ref>), f ∈ L^∞ L^∞, and hence by Theorem <ref> we prove (<ref>). The continuity f ∈ C^0L^p then follows from f∈ C^0L^2, for p∈ [1,2) by compactness of 𝕋^d, and for p∈(2,∞) by interpolation with f∈ L^∞ L^∞. It remains to show (<ref>). From (<ref>) we have the bounds f_L^∞ L^2, √(2ν)∇ f_L^2L^2≤f_0_L^2, and hence for any ϕ∈ C_c^∞(𝕋^d×(0,T)), by the equation (<ref>), \|∫_𝕋^d×(0,T)f∂ϕ/∂ t dxdt\| = \|∫_𝕋^d×(0,T)f (u·∇ϕ + νΔϕ) dxdt\| ≤f_L^∞ L^2u_L^2L^∞∇ϕ_L^2L^2 + ν∇ f_L^2L^2∇ϕ_L^2L^2 ≤f_0_L^2u_L^2L^∞∇ϕ_L^2L^2 + √(ν/2)f_0_L^2∇ϕ_L^2L^2, which proves (<ref>). It is likely possible to extend the first part of Theorem <ref> to the Prodi-Serrin class <cit.>, u ∈ L^pL^q, 2/p + d/q≤ 1 (d<q), though we do not attempt to prove such a result here as it is immaterial to this paper. In fact, such a result is available in the stochastic setting due to Flandoli, see <cit.>. When f_0 ∈ L^1 explicit decay rates of the L^∞(𝕋^d) norm (and other L^p norms) are provable. See <cit.> for details when working on the whole space ℝ^d. For general existence and regularity results concerning similar parabolic PDEs, we point to <cit.>, and also Chapter 7 in <cit.>. § CONTROLLING THE VANISHING VISCOSITY LIMIT The main purpose of this section is to prove Theorem <ref> below, which allows us to construct bounded divergence-free vector fields in a way that permits control of the corresponding vanishing viscosity limit of (<ref>). This relies on two Propositions. The first, Proposition <ref> below, gives a general criterion for which the vanishing viscosity limit of (<ref>) converges strongly to (<ref>). That is, for a suitable divergence-free vector field u:𝕋^d×[0,T]→ℝ^d, and small viscosity ν>0, that solutions of (<ref>) along u are well-approximated by a weak solution of (<ref>) along u. This result is a generalisation of the Selection Theorem in <cit.>. The second, Proposition <ref> below, uses a similar argument to show, for fixed viscosity ν>0, how solutions of (<ref>) depend little on the small spatial scales of the vector field u. We quantify these scales through the weak- topology of vector fields in L^∞([0,T];L^∞(𝕋^d;ℝ^d)). The intuition is that the viscosity `blurs' these small spatial scales. The key idea of Theorem <ref> is then that solving (<ref>) with reduced viscosity ν>0, is akin to adding small spatial scales while solving (<ref>). The advantage of solving (<ref>) is then that Lagrangian solutions (Definition <ref>) can be designed rather explicitly. Consider a vector field u ∈ L^1( [0,T]; L^1(𝕋^d;ℝ^d)) with ∇· u=0 in the distributional sense. Fix some initial data f_0 ∈ L^∞(𝕋^d). [Since TE is linear this implies uniqueness for all initial data in L^∞.]Suppose that there is a unique weak solution f (in the class L^∞ L^∞) to (<ref>) along u with initial data f_0, [This is not uncommon, see Remark 4.3 in <cit.>, and is true for example when u ∈ L^1([0,T];BV) in <cit.>.]and that additionally f is a renormalised weak solution (Definition <ref>). For each ν > 0 denote by f^ν any weak solution to (<ref>) along u with initial data f_0. Suppose in addition that f^ν∈ C_weak-^0L^∞ and satisfies the Initial L^p-Inequality (<ref>) for all p ∈ [1,∞]. (This would be the case if say u ∈ L^∞ L^∞, see Theorem <ref>.) Then, for each p ∈ [1,∞), f^νf converges strongly in L^p L^p and also in weak- L^∞ L^∞. [By compactness of 𝕋^d, and interpolation with the bound in L^∞ L^∞, f ∈ C^0L^1 is equivalent to f ∈ C^0L^p for any p ∈ [1,∞).]If additionally f ∈ C^0 L^1, then for each p ∈ [1,∞), f^νf also converges strongly in L^∞ L^p. Suppose that f^ν does not converge in weak-* L^∞ L^∞ to f as ν→0. Then there exists some g ∈ L^1 L^1, and a sequence ν_i 0 and c>0 such that for all i∈ℕ \|∫_𝕋^d×[0,T](f^ν_i-f)g dxdt\|≥ c. By the Initial L^p-Inequality (<ref>) f^ν_i is uniformly bounded for all i∈ℕ in L^∞ L^∞, and so by taking a subsequence if necessary, we may assume f^ν_ii→∞f̅ converges in weak-* L^∞ L^∞ to some f̅∈ L^∞ L^∞. Then, for any ϕ∈ C_c^∞(𝕋^d × [0,T)), ∫_𝕋^d×[0,T)f̅(∂ϕ/∂ t + u·∇ϕ) dxdt = lim_i→∞∫_𝕋^d×[0,T) f^ν_i(∂ϕ/∂ t + u·∇ϕ + ν_i Δϕ) dxdt = - ∫_𝕋^d f_0 ϕ_0 dx, and so the limit f̅ is a weak solution to (<ref>) along u with initial data f_0. Moreover, it is in L^∞ L^∞ so by assumption must be the unique weak solution f, contradicting (<ref>). Therefore f^νf converges in weak-* L^∞ L^∞. Recall by Theorem <ref> that f, f^ν∈ C_weak-^0L^∞. Since by assumption f^ν satisfies the Initial L^p-Inequality (<ref>) for all p ∈ [1,∞], we may bound f^ν_L^p L^p≤ T^1/pf_0_L^p. If p∈(1,∞], weak- convergence in L^∞ L^∞ implies weak-* convergence in L^p L^p. [This is a standard result following from uniform convexity of L^p(𝕋^d) for p ∈ (1,∞).]Whenever p ∈ (1,∞), weak-* convergence in L^pL^p is also strong in L^pL^p if and only if ν→0lim supf^ν_L^pL^p≤f_L^pL^p. To show that this is satisfied we use that f ∈ C_weak-^0L^∞ is a renormalised weak solution to (<ref>) (Definition <ref>). [Here a∧ b = min{a,b}.]Let M ∈ℕ and β(x) = M∧\|x\|^p, then β(f) is a weak solution to (<ref>) along u with initial data β(f_0) ∈ L^∞. Taking ϕ≡ 1 in the Trace Formula (<ref>) shows that there exists a subset E_M⊂[0,T] with zero Lebesgue-measure, such that for all t ∈ [0,T]∖ E_M we have, ∫_𝕋^d M ∧ \|f(·, t)\|^p dx = ∫_𝕋^d M ∧ \|f_0\|^p dx. In particular, the above holds for all t ∈ [0,T]∖⋃_M∈ℕE_M. By the Lebesgue monotone convergence Theorem, taking M→∞ shows that f(·, t)_L^p(𝕋^d) = f_0_L^p for all t ∈ [0,T]∖⋃_M∈ℕE_M, which implies f_L^p L^p = T^1/pf_0_L^p. Combined with (<ref>) this implies that ν→0lim supf^ν_L^pL^p≤f_L^pL^p, and hence f^ν f in L^pL^p as required. Convergence of f^ν f in L^1 L^1 follows from convergence in L^p L^p for any p ∈ (1,∞) and the compactness of the domain 𝕋^d × [0,T]. We now assume that f ∈ C^0 L^1 and wish to upgrade to convergence in L^∞ L^p for each p ∈ [1,∞). The idea is to use the Trace Formula (<ref>) to show for each t ∈ [0,T] that f^ν(·, t) ν→0 f(·, t) converges in weak- L^∞, and then upgrade to (uniform in time) strong convergence in L^p(𝕋^d) by convergence of the norm f^ν(·,t)_L^p(𝕋^d) for p ∈ (1,∞). [Convergence in L^∞ L^p for p ∈ (1,∞) can also be shown directly, though is more technical. This is useful if say f_0 ∉ L^2(𝕋^d).]Since the L^2(𝕋^d)-inner product makes life easier, we will only prove convergence in L^∞ L^2, and notice that convergence in L^∞ L^p follows for p ∈ [1,2) by compactness of 𝕋^d, and for p∈(2,∞) by interpolation with the existing uniform bound in L^∞ L^∞. We have already shown that f^ν f in L^1L^1, with uniform bound in L^∞ L^∞. Suppose that uf^ν does not converge strongly in L^1 L^1 to uf as ν→0. Then there exists a sequence ν_i 0 and c>0 such that for all i∈ℕ uf^ν_i - uf_L^1 L^1≥ c . Now f^ν_i f strongly in L^1 L^1, and so by taking a further subsequence if necessary, we may assume that f^ν_i f point-wise a.e. in 𝕋^d×[0,T]. But then uf^ν_i uf converge point-wise a.e. in 𝕋^d×[0,T], and are also uniformly bounded for all i∈ℕ in L^1 L^1 by u_L^1 L^1f_0_L^∞(𝕋^d), and so the dominated convergence Theorem yields a contradiciton to (<ref>). Therefore the product uf^ν uf converges strongly in L^1 L^1. We now use that f ∈ C^0 L^1, and therefore f∈ C^0 L^2 (by interpolation with the existing bound f ∈ L^∞ L^∞), to take a smooth approximation ϕ_ϵ∈ C^∞(𝕋^d×[0,T]), such that ϕ_ϵ-f_L^∞ L^2≤ϵ. Then by the Trace Formula (<ref>), and that f^ν(·, t)_L^2(𝕋^d)≤f_0_L^2 = f(·, t)_L^2(𝕋^d) for a.e. t∈[0,T], we may write for a.e. t ∈ [0,T] f^ν(·, t) - f(·, t)_L^2(𝕋^d)^2 = f^ν(·, t)_L^2(𝕋^d)^2 + f(·, t)_L^2(𝕋^d)^2 - 2 ∫_𝕋^d f^ν(·, t) f(·, t) dx ≤ 2f(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2(𝕋^d) - 2 ∫_𝕋^d f^ν(·, t) ϕ_ϵ(·, t) dx ≤[t] 2f(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2(𝕋^d) - 2∫_𝕋^df_0ϕ_ϵ(·, 0) dx - 2∫_𝕋^d × [0,t]f^ν(∂ϕ_ϵ/∂ t + u ·∇ϕ_ϵ + νΔϕ_ϵ) dxdt ≤[t] 2f(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2(𝕋^d) - 2∫_𝕋^df_0ϕ_ϵ(·, 0) dx - 2∫_𝕋^d × [0,t]f (∂ϕ_ϵ/∂ t + u ·∇ϕ_ϵ) dxdt + 2f^ν - f_L^1L^1∂ϕ_ϵ/∂ t_L^∞ L^∞ + 2νf^ν_L^1L^1Δϕ_ϵ_L^∞ L^∞ + 2uf^ν - uf_L^1L^1∇ϕ_ϵ_L^∞ L^∞ = [t] 2f(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2(𝕋^d) - 2∫_𝕋^df(·, t)ϕ_ϵ(·, t) dx + 2f^ν - f_L^1L^1∂ϕ_ϵ/∂ t_L^∞ L^∞ + 2νf^ν_L^1L^1Δϕ_ϵ_L^∞ L^∞ + 2uf^ν - uf_L^1L^1∇ϕ_ϵ_L^∞ L^∞ ≤ 4ϵf_0_L^2(𝕋^d) + C_ϵ(f^ν - f_L^1L^1 + νf^ν_L^1L^1 + uf^ν - uf_L^1L^1), with ϵ>0 arbitrary, and C_ϵ a constant that depends only on ϵ and f, and not on t ∈ [0,T]. Then since f^ν f, and uf^ν uf in L^1 L^1 we see that f^ν f in L^∞ L^2 as required. Consider a sequence of uniformly bounded, divergence-free vector fields u_n ∈ L^∞( [0,T];L^∞(𝕋^d;ℝ^d)), such that u_n n→∞ u converges in weak-* L^∞ L^∞ to some u ∈ L^∞ L^∞. Fix some initial data f_0 ∈ L^∞(𝕋^d). For each n ∈ℕ, and ν > 0 denote by f^n,ν the unique (by Theorem <ref>) weak solution to (<ref>) along u_n with initial data f_0. Similarly denote by f^ν the unique weak solution to (<ref>) along u with initial data f_0. Then for each 0<a≤ b, p ∈ [1,∞) sup_ν∈[a,b]f^n,ν-f^ν_(L^2H^1) ∩ (L^∞ L^p) 0. The case p ∈ (2,∞) follows from the same result for p=2 by interpolation with the existing uniform bound on f^n,ν, f^ν in L^∞ L^∞ from the L^p-Inequality (<ref>), while for the case p ∈ (1,2) it follows from the case for p=2 by compactness of 𝕋^d. Therefore, we may restrict to p=2. Assume to the contrary that for some 0<a≤ b that there exists c > 0 and sequences {n_i}_i ∈ℕ, {ν_i}_i ∈ℕ with n_i ∈ℕ increasing, a≤ν_i ≤ b, such that for all i ∈ℕ f^n_i,ν_i - f^ν_i_L^2H^1 + f^n_i,ν_i - f^ν_i_L^∞ L^2≥ c. By taking a subsequence if necessary we may assume ν_i ν for some ν∈[a,b]. Since ν_i are bounded above and below, by Energy Identity (<ref>) f^n_i, ν_i, f^ν_i are uniformly bounded for all i∈ℕ in L^2 H^1, and by Equicontinuity (<ref>) ∂/∂ tf^n_i, ν_i, ∂/∂ tf^ν_i are uniformly bounded for all i∈ℕ in L^2 H^-1. Since H^1 ⋐ L^2 ⊂ H^-1 we may apply the Aubin-Lions compactness Lemma (see Chapter 3, Theorem 2.1 in <cit.>) to deduce that the set {f ∈ L^2H^1 : ∂ f/∂ t_L^2 H^-1≤ C} is compactly embedded into L^2 L^2. Hence, again taking a subsequence if necessary, we may assume that both f^n_i, ν_i, f^ν_i converge, to some limits, strongly in L^2L^2 (and weakly in L^2H^1) as i→∞. We next show that both f^n_i, ν_i, and f^ν_i converge to f^ν strongly in L^2H^1 and L^∞ L^2 as i→∞, contradicting the assumption (<ref>). We will only show the required convergence for f^n_i, ν_i f^ν, since the same result for f^ν_i follows the same proof by additionally assuming that u_n_i are a constant sequence u_n_i = u for all i ∈ℕ. Since u_n_ii →∞ u are uniformly bounded and converge in weak-* L^∞ L^∞, and as we have already shown that f^n_i,ν_i converges to some f̅ strongly in L^2L^2 as i→∞, then the product f^n_i, ν_iu_n_i converges to f̅u weakly in L^2 L^2 as i→∞. Then by the weak formulation of (<ref>), for any ϕ∈ C_c^∞(𝕋^d × [0,T)), one has ∫_𝕋^d×[0,T]f̅( ∂ϕ/∂ t + u·∇ϕ + νΔϕ) dxdt = lim_i→∞∫_𝕋^d×[0,T] f^n_i,ν_i(∂ϕ/∂ t + u_n_i·∇ϕ + ν_i Δϕ) dxdt = - ∫_𝕋^d f_0 ϕ_0 dx. Consequently, we see that f̅ is a weak solution to (<ref>) along u with initial data f_0. By Theorem <ref> this solution is unique and so must be f^ν. We are left to upgrade the convergence f^n_i,ν_if̅ from strong in L^2 L^2 and weak in L^2 H^1, to strong in L^2 H^1 and strong in L^∞ L^2. In light of the Trace Formula (<ref>), and the uniform bound on f^n_i,ν_i for all i∈ℕ in C^0L^2 (by Theorem <ref>), strong convergence f^n_i,ν_i f^ν in L^2 L^2 implies f^n_i,ν_i(·, T) i→∞ f^ν(·, T) converges weakly in L^2(𝕋^d) (note that f^n_i,ν_i, f^ν are uniformly bounded in C^0L^2 by Theorem <ref>). In particular lim inf_i→∞∫_𝕋^d\|f^n_i,ν_i(·, T)\|^2 dx ≥∫_𝕋^d\|f^ν(·, T)\|^2 dx. In light of the Energy Identity (<ref>) this implies lim sup_i→∞∫_𝕋^d × [0,T]\|∇ f^n_i,ν_i\|^2 dx dt ≤∫_𝕋^d × [0,T]\|∇ f^ν\|^2 dx dt. Since also f^n_i,ν_ii→∞ f^ν weakly in L^2H^1 we have lim inf_i→∞∫_𝕋^d × [0,T]( \|f^n_i,ν_i\|^2 + \|∇ f^n_i,ν_i\|^2 ) dx dt ≥∫_𝕋^d × [0,T]( \|f^ν\|^2 + \|∇ f^ν\|^2 ) dx dt. However, since f^n_i,ν_i f^ν strongly in L^2L^2, from the above we must have convergence of the norms f^n_i,ν_i_L^2H^1^2 f^ν_L^2H^1^2. Thus the weak convergence f^n_i,ν_ii→∞ f^ν in L^2H^1 is in fact strong, as required. To extend to strong convergence in L^∞ L^2, we use the fact that f^ν∈ C^0L^2 to take a smooth approximation ϕ_ϵ∈ C^∞(𝕋^d× [0,T]), such that ϕ_ϵ - f^ν_L^∞ L^2≤ϵ. Then by the Trace Formula (<ref>), and the Energy Identity (<ref>), for all t ∈ [0,T] we see that f^n_i,ν_i(·, t) - f^ν(·, t)_L^2(𝕋^d)^2 = [t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 + f^ν(·, t)_L^2(𝕋^d)^2 - 2 ∫_𝕋^d f^n_i,ν_i(·, t) f^ν(·, t) dx ≤[t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 + f^ν(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2 - 2 ∫_𝕋^d f^n_i,ν_i(·, t) ϕ_ϵ(·, t) dx = [t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 + f^ν(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2 - 2 ∫_𝕋^d f_0 ϕ_ϵ(·, 0) dx - 2∫_𝕋^d × [0,t]f^n_i,ν_i(∂ϕ_ϵ/∂ t + u_n_i·∇ϕ_ϵ + ν_i Δϕ_ϵ) dxdt ≤[t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 + f^ν(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2 - 2 ∫_𝕋^d f_0 ϕ_ϵ(·, 0) dx - 2∫_𝕋^d × [0,t]f^ν(∂ϕ_ϵ/∂ t + u ·∇ϕ_ϵ + νΔϕ_ϵ) dxdt + 2f^n_i,ν_i - f^ν_L^2L^2∂ϕ_ϵ/∂ t_L^2 L^2 + 2\|ν_i-ν\|f^n_i,ν_i_L^2L^2Δϕ_ϵ_L^2 L^2 + 2\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| = [t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 + f^ν(·, t)_L^2(𝕋^d)^2 + 2ϵf_0_L^2 - 2 ∫_𝕋^d f^ν(·, t) ϕ_ϵ(·, t) dx + 2f^n_i,ν_i - f^ν_L^2L^2∂ϕ_ϵ/∂ t_L^2 L^2 + 2\|ν_i-ν\|f^n_i,ν_i_L^2L^2Δϕ_ϵ_L^2 L^2 + 2\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| ≤[t] f^n_i,ν_i(·, t)_L^2(𝕋^d)^2 - f^ν(·, t)_L^2(𝕋^d)^2 + 4ϵf_0_L^2 + 2f^n_i,ν_i - f^ν_L^2L^2∂ϕ_ϵ/∂ t_L^2 L^2 + 2\|ν_i-ν\|f^n_i,ν_i_L^2L^2Δϕ_ϵ_L^2 L^2 + 2\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| = [t] 4ϵf_0_L^2 - ∫_𝕋^d × [0,t](2ν_i \|∇ f^n_i,ν_i\|^2 - 2ν\|∇ f^ν\|^2) dxdt + 2f^n_i,ν_i - f^ν_L^2L^2∂ϕ_ϵ/∂ t_L^2 L^2 + 2\|ν_i-ν\|f^n_i,ν_i_L^2L^2Δϕ_ϵ_L^2 L^2 + 2\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| ≤[t] 4ϵf_0_L^2 + 2ν_i^1/2f^n_i,ν_i - ν^1/2f^ν_L^2 H^1(ν_i^1/2f^n_i,ν_i_L^2 H^1 + ν^1/2f^ν_L^2 H^1) + C_ϵ(f^n_i,ν_i - f^ν_L^2L^2 + \|ν_i-ν\|f^n_i,ν_i_L^2L^2) + 2\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \|, with ϵ>0 arbitrary, and C_ϵ a constant that depends only on ϕ_ϵ, and not on t ∈ [0,T]. Therefore by the convergence ν_i ν, and f^n_i,ν_i f^ν strongly in L^2H^1, to show strong convergence of f^n_i,ν_if^ν in L^∞ L^2 we need only show, for fixed ϕ_ϵ sup_t ∈[0,T]\|∫_𝕋^d × [0,t] (f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| 0. Suppose not, then there exists a sequence t_i ∈ [0,T] for i∈ℕ such that \|∫_𝕋^d×[0,t_i](f^n_i,ν_iu_n_i - f^ν u)·∇ϕ_ϵ dxdt \| ≥ c, for some c>0. By compactness of [0,T], and taking a subsequence if necessary, we may further assume that t_i t̅ for some t̅∈[0,T]. However, there is strong convergence 1_t∈[0,t_i]f^n_i,ν_i, 1_t∈[0,t_i]f^ν 1_t∈[0,t̅]f^ν in L^2 L^2, and boundedness of the sequence u_n_i in L^∞ L^∞, and weak-* convergence u_n_ii→∞ u in L^∞ L^∞. Therefore, the products 1_t∈[0,t_i]f^n_i,ν_i u_n_i - 1_t∈[0,t_i]f^ν u i→∞ 0 converge weakly in L^2 L^2, contradicting (<ref>), and hence proving the claim. We now give the following corollary of the previous two propositions, which allows us to construct bounded divergence-free vector fields u_∞ in a way that permits control of the corresponding vanishing viscosity limit of (<ref>). Fix some M>0, and a metric d_* inducing the weak-* topology on X = {u∈ L^∞( [0,T];L^∞(𝕋^d;ℝ^d)) : u_L^∞ L^∞≤ M }. [Note that Y is non-empty, e.g. 0∈ Y and any divergence-free, in the distribution sense, vector field in X and L^1W^1,1 is in Y, see <cit.>. Moreover, by the work of Ambrosio in <cit.>, we may relax W^1,1 to the space of bounded variation BV.]Let Y⊂ X be the set of all divergence-free vector fields admitting a unique renormalised weak solution (unique in the class of all L^∞ L^∞ weak solutions) to (<ref>) for any initial data f_0 ∈ L^∞(𝕋^d). Let {u_i}_i∈ℕ⊂ Y, f_0 ∈ L^∞(𝕋^d). For each n∈ℕ, and ν > 0, denote by f^n,ν, respectively f^n, the unique weak solution to (<ref>), respectively (<ref>), along u_n with initial data f_0. Then, * For all n∈ℕ, there exists ν_n>0, ϵ_n>0 depending only on {u_i}_i=1^n (and in particular not on f_0), with ν_n0 monotonically, such that the following hold true: * For all p∈[1,∞), sup_0<ν≤ν_nf^n,ν-f^n_L^∞ L^p0. * If d_(u_n+1,u_n) ≤ϵ_n for all n∈ℕ, then u_n n→∞ u_∞ converges in weak- L^∞ L^∞ to some divergence-free vector field u_∞∈ L^∞ L^∞, * and if we denote by f^∞,ν the unique weak solution to (<ref>) along u_∞ with initial data f_0, then for all p ∈ [1,∞), sup_ν_n ≤ν≤ν_1f^n,ν-f^∞,ν_L^∞ L^p0. * In particular, for all p∈[1,∞), f^∞,ν_n-f^n_L^∞ L^p0. By separability of L^1(𝕋^d), take a countable dense sequence {f_0^m}_m ∈ℕ in L^1(𝕋^d), satisfying f_0^m ∈ L^∞(𝕋^d). For all m∈ℕ, n ∈ℕ, and ν > 0, denote by f^m;n,ν, respectively f^m;n, the unique weak solution to (<ref>), respectively (<ref>), along u_n with initial data f^m_0. If in addition u_n n→∞ u_∞ in weak-* L^∞ L^∞, denote by f^m;∞,ν the unique weak solution to (<ref>) along u_∞ with initial data f_0^m. For the given f_0 ∈ L^∞(𝕋^d), for each n ∈ℕ, and ν > 0, denote by f^n,ν, respectively f^n, the unique weak solution to (<ref>), respectively (<ref>), along u_n with initial data f_0. If in addition u_n n→∞ u_∞ in weak-* L^∞ L^∞, denote by f^∞,ν the unique weak solution to (<ref>) along u_∞ with initial data f_0. Notice, by linearity of (<ref>), (<ref>), and the Initial L^p-Inequality (<ref>), that for each m ∈ℕ, n∈ℕ, ν >0, we have f^n,ν-f^m;n,ν_L^∞ L^1≤f_0 - f^m_0_L^1, f^n - f^m;n_L^∞ L^1≤f_0 - f^m_0_L^1, f^∞,ν-f^m;∞,ν_L^∞ L^1≤f_0 - f^m_0_L^1. We will use this and L^1(𝕋^d)-density to only consider the countable set of initial data {f_0^m}_m∈ℕ. Given n∈ℕ and {u_i}_i=1^n⊂ Y we need to find ν_n>0, ϵ_n>0 such that (<ref>)-(<ref>) hold. For the purpose of the proof we additionally find δ_n>0 such that, for all m∈{1,...,n}, ν∈[ν_n,ν_1], and w∈ Y with d_(w,u_n)≤δ_n, f^m;n,ν - g^m;ν_L^∞ L^1≤1/n, where g^m,ν is the unique weak solution to (<ref>) along w with initial data f_0^m. We do so by induction. Given {u_i}_i=1^n as in the statement of the theorem assume {(ν_i, ϵ_i, δ_i)}_i=1^n-1 are already chosen (an empty set if n=1). By applying Proposition <ref> for each initial data {f_0^m}_m=1^n there exists some ν_n > 0 such that for all m∈{1,...,n}, ν∈ (0,ν_n], f^m;n,ν - f^m;n_L^∞ L^1≤1/n. One may choose ν_n to be smaller, so that ν_n ≤1/n, and if n1, ν_n < ν_n-1. We next find δ_n > 0 satisfying (<ref>). Assume to the contrary that no such δ_n exists, then there is a sequence of divergence-free vector fields {w_k}_k ∈ℕ⊂ Y with w_k k→∞ u_n converging in weak- L^∞ L^∞, which violate Proposition <ref> for at least one of the initial data {f_0^m}_m=1^n, where we have substituted a=ν_n, b = ν_1 into Proposition <ref>. One may choose δ_n smaller so that δ_n ≤1/n. Finally take ϵ_n = min_k∈{0,...,n-1}{δ_n-k2^-k-1}. (<ref>) follows immediately from our choice of ν_n. Meanwhile (<ref>) with p=1 follows from (<ref>), (<ref>), (<ref>). Interpolation with the existing uniform bound on f^n,ν, f^n in L^∞ L^∞ then gives (<ref>) for p∈(1,∞). We are now left to show (<ref>)-(<ref>) are satisfied when in addition d_(u_n+1,u_n) ≤ϵ_n for all n∈ℕ. By (<ref>), ∑_i=n^∞ϵ_i ≤δ_n, and also δ_n → 0, so {u_i}_i∈ℕ is a d_-Cauchy sequence, uniformly bounded in L^∞ L^∞ by M, and so converges in weak-* L^∞ L^∞ to some limit u_∞, proving (<ref>). Moreover, for all n∈ℕ, d_(u_∞, u_n) ≤δ_n, so taking in (<ref>) w=u_∞, g^m;ν=f^m;∞,ν then (<ref>) follows from (<ref>), (<ref>), and interpolation with the existing uniform bound on f^n,ν, f^∞,ν in L^∞ L^∞. Finally (<ref>) is a simple corollary of (<ref>) and (<ref>). § FRACTAL SHEAR FLOW - NON-UNIQUENESS §.§ Notation For x ∈ℝ^2 we set x=(x_1, x_2), and define the corresponding unit vectors e_1 = (1,0), e_2 = (0,1). For i ∈{1,2} an index, we define by î the other index, that is î∈{1,2}∖{i}. When working on the 2-torus, [x] ∈𝕋^2 = ℝ^2/ℤ^2 shall instead always be written in terms of a representative x ∈ℝ^2, usually x ∈ [0,1)^2. For notational convenience functions on the torus 𝕋^2 will often be considered periodic functions on ℝ^2, and vice versa. Integrals over 𝕋^2 are then strictly speaking over [0,1)^2, such as when taking norms. First, we define the shear flows that form the building block of our construction. We shall later apply these shears in a carefully designed order, reminiscent of a cantor-subset of [0,T]. The arising self-cancellation of these shear flows will produce non-uniqueness of renormalised weak solutions (Definition <ref>) for any initial data to (<ref>). Moreover, the small spatial-scale of the shears, see (<ref>) below, will allow us to approximate the vanishing viscosity limit of (<ref>). For L∈ℕ, we divide ℝ into disjoint intervals ⋃_m ∈ℤ[m/2L,m+1/2L). We define for each i∈{1,2}, L∈ℕ, the periodic vector field u^(i;L): 𝕋^2 →ℝ^2, u^(i;L)(x) = e_i if x_î∈[m/2L,m+1/2L) for even m, -e_i if x_î∈[m/2L,m+1/2L) for odd m, where, since 2L is even, the definition is a periodic function of x∈ℝ^2, and so is well-defined on 𝕋^2. u^(i;L) is bounded and divergence-free in the distributional sense since it is of the form u^(1;L)(x)=(g(x_2),0), or u^(2;L)(x)=(0,g(x_1)) for some g ∈ L^∞(𝕋^2). We refer to this vector field as the (i;L)-Lagrangian shear. [Strictly speaking a Lagrangian flow along a divergence-free vector field u∈ L^1([0,T];L^1(𝕋^d;ℝ^d)) need not be unique, though it is in this case, e.g. by Proposition <ref>, below.]We shall denote by {y_t^(i;L)}_t ∈ (-∞, ∞) the corresponding Lagrangian flow (Definition <ref>) for the transport problem along u^(i;L), y_t^(i;L) : [ 𝕋^2 → 𝕋^2; x ↦ x+t e_i ℤ^2 if x_î∈[m/2L,m+1/2L) for even m, x-t e_i ℤ^2 if x_î∈[m/2L,m+1/2L) for odd m. ] As per Definition <ref> of Lagrangian flows, this preserves the Lebesgue-measure on 𝕋^2, e.g. for i=1 we can decompose a Lebesgue-measureable subset A ⊂𝕋^2 into A_m = A ∩(𝕋×[m/2L,m+1/2L)), for which each (y_t^(1;L))^-1(A_m) is a translation. Moreover, it is invertible with inverse given by (y_t^(i;L))^-1(x) = y_-t^(i;L)(x), and is absolutely continuous since with respect to t since it is differentiable with derivative ∂/∂ t y_t^(i;L) (x) = u^(i;L)(x) = u^(i;L)(y_t^(i;L)(x)). Therefore, {y_t^(i;L)}_t ∈ (-∞, ∞) is a Lagrangian flow along u^(i;L). Finally, for each i ∈{1,2}, we show u^(i;L)L→∞ 0, with convergence in weak- L^∞(𝕋^2). The proof of (<ref>) is similar to that of the Riemann-Lebesgue lemma. Notice that u^(i;L)(x+1/2Le_î) = - u^(i;L)(x) and so for any test function f ∈ L^1(𝕋^2), by changing variables we have that ∫_𝕋^2 f(x)u^(i;L)(x) dx = ∫_𝕋^2f(x) - f(x+1/2Le_î)/2 u^(i;L)(x) dx. Now f(x+1/2Le_î) f converges strongly in L^1(𝕋^2). Together with the uniform bound u^(i;L)_L^∞≤ 1 this implies the above integral converges to zero as L→∞, as required. Next, we prove well-posedness of (<ref>) along these vector fields. We follow the notation introduced in Definition <ref>. Suppose f∈ L^1L^1 is a weak solution to (<ref>) along u^(i;L) on an open interval I⊂(0,T). Then there exists some g ∈ L^1(𝕋^2) such that f is (a.e. in 𝕋^2× I) equal to the following Lagrangian solution (Definition <ref>) associated with g, f(·,t)=g ∘(y^(i;L)_t)^-1. In particular f is a renormalised weak solution to (<ref>) (Definition <ref>). Moreover, g ∘(y^(i;L)_t)^-1∈ C^0((-∞,∞);L^1(𝕋^2)). Without loss of generality suppose i=1, and use the shorthand y_t=y_t^(1;L) for each t ∈ (-∞, ∞), that is y_t(x) = x+t e_1 if x_2∈[m/2L,m+1/2L) for even m, x-t e_1 if x_2∈[m/2L,m+1/2L) for odd m. Since the e_2-component of y_t(x) is unchanged, that is [y_t(x)]_2 = x_2, we see u^(1;L)(y_t(x),t) = u^(1;L)(x,t). Fix m ∈{0,1,...2L-1}. Take a test function ϕ∈ C_c^∞((𝕋×(m/2L,m+1/2L)) × I), and let ψ(x,t)=ϕ(y_t^-1(x), t). This is well-defined since y_t^-1 preserves the set 𝕋×(m/2L,m+1/2L). [Notice also that ψ is Lipschitz (moreover it is smooth). This will not be true for a general test function ϕ∈ C_c^∞(𝕋^2× I). For an arbitrary Lagrangian flow this issue is why existence of a Lagrangian flow does not prove uniqueness of weak solutions to (<ref>) .]Then ψ∈ C_c^∞((𝕋×(m/2L,m+1/2L)) × I) is bounded, and supported and smooth on 𝕋×(m/2L,m+1/2L) × I since y_t^-1 is smooth there. By chain rule, the following point-wise equality holds (∂ψ/∂ t + u^(1;L)·∇ψ)(x,t) = ∂ϕ/∂ t(y_t^-1(x),t). Since f ∈ L^1L^1 is assumed to be a weak solution to (<ref>) along u^(1;L) on the open interval I⊂ (0,T), ∫_𝕋×(m/2L,m+1/2L) × I f(x,t) (∂ψ/∂ t + u^(1;L)·∇ψ)(x,t) dxdt = 0, which implies ∫_𝕋×(m/2L,m+1/2L) × I f(x,t) ∂ϕ/∂ t(y_t^-1(x),t) dxdt = 0, and after changing variables, ∫_𝕋×(m/2L,m+1/2L) × I f(y_t(x), t) ∂ϕ/∂ t(x,t) dxdt = 0. [See for example Theorem 3.1.4' in <cit.>]This holds for all ϕ∈ C_c^∞((𝕋×(m/2L,m+1/2L)) × I), and so as a distribution f(y_t(x), t) ∈ L^1L^1 is independent of t∈ I, so there must exists some g_m ∈ L^1(𝕋×(m/2L,m+1/2L)) such that (for a.e. t ∈ I) f(y_t(x), t) = g_m(x) a.e. on 𝕋×(m/2L,m+1/2L). Repeating for each m ∈{0,1,...2L-1} gives some g ∈ L^1(𝕋^2) such that, f(y_t(x), t) = g(x) a.e. on 𝕋^2. And so f(·, t) = g ∘ y_t^-1 is a Lagrangian solution to (<ref>) (Definition <ref>), and hence also a renormalised weak solution (Definition <ref>, Remark <ref>). To prove that g∘ y_t^-1∈ C^0((-∞,∞);L^1(𝕋^2)) we shall use the fact that y_t is bijective, measure preserving, and 1-Lipschitz in time, i.e. for all x ∈𝕋^2, t,s ∈ℝ \|y_t(x) - y_s(x)\| ≤ \|t-s\|, and so, by replacing x with y_s^-1(x), \|y_t ∘ y_s^-1(x) - x\| ≤ \|t-s\|. The following follows a standard argument. We fix t ∈ I and write g∘ y_s^-1 = ( g∘ y_t^-1) ∘ y_t ∘ y_s^-1. Now mollify g∘ y_t^-1 in L^1(𝕋^2), that is for each ϵ > 0 take some ϕ_ϵ∈ C^∞(𝕋^2) such that ϕ_ϵ - g∘ y_t^-1_L^1(𝕋^2)≤ϵ. But also, since y_t, y_s^-1 are measure preserving, ϕ_ϵ∘ y_t ∘ y_s^-1 - g∘ y_s^-1_L^1(𝕋^2)≤ϵ. Then since ϕ_ϵ is uniformly continuous on 𝕋^2, by (<ref>) there exists δ∈ (0,1) such that if \|t-s\| < δ then ϕ_ϵ∘ y_t ∘ y_s^-1 - ϕ_ϵ_L^1≤ϵ, and so g∘ y_t^-1 - g∘ y_s^-1_L^1(𝕋^2)≤ 3 ϵ, which completes the proof. Next, we demonstrate how these shear flows self-cancel. It is this property that gives rise to non-uniqueness in our later construction. Let L_1, L_2 ∈ℕ, and i_1, i_2 ∈{1,2} such that i_1 i_2. Suppose that 2τ_2 = 1/2L_1, and that 2L_2τ_1 is an odd integer. [Here we denote by f^2 := f ∘ f.]Then the composition (y_τ_2^(i_2;L_2))^2 ∘ y_τ_1^(i_1;L_1)∘(y_τ_2^(i_2;L_2))^2 ∘ y_τ_1^(i_1;L_1) = Id. Without loss of generality suppose i_1=1,i_2=2. Since (y_τ_2^(2;L_2))^2 = y_2τ_2^(2;L_2) we instead prove the more general result y_τ_2^(2;L_2)∘ y_τ_1^(1;L_1)∘ y_τ_2^(2;L_2)∘ y_τ_1^(1;L_1) = Id, whenever both 2L_1τ_2 and 2L_2τ_1 are odd integers. To this end we divide 𝕋^2 into tiles as follows. Given x ∈ [0,1)^2 we find the unique integers m_1, m_2 ∈ℤ such that x ∈[m_1/2L_2,m_1+1/2L_2) ×[m_2/2L_1,m_2+1/2L_1). We are concerned with the parity of m_1 and m_2 (even or odd), that is we have four `colours' on our tiling, two for each coordinate. This defines two equivalence relations [·]_m_1, [·]_m_2 on 𝕋^2, each with two equivalence classes corresponding to the parity of either m_1 or m_2 in (<ref>). Since 2L_1τ_2, 2L_2τ_1 are odd integers, the action of y_τ_1^(1;L_1) changes the parity of m_1 and not m_2, while the action of y_τ_2^(2;L_2) changes the parity of m_2 and not m_1. Introduce the shorthand x=:x^(0), y_τ_1^(1;L_1)(x^(0))=:x^(1), y_τ_2^(2;L_2)(x^(1))=:x^(2), y_τ_1^(1;L_1)(x^(2))=:x^(3), y_τ_2^(2;L_2)(x^(3))=:x^(4). Then by the above discussion, [x^(2)]_m_2[x^(0)]_m_2, [x^(3)]_m_1[x^(1)]_m_1. From the definitions of y_τ_1^(1;L_1), y_τ_2^(2;L_2) this implies x^(3) - x^(2) = -(x^(1) - x^(0)), x^(4) - x^(3) = -(x^(2) - x^(1)), and hence x^(4) = x^(0) as required. Next, we define the times at which to apply these shear flows y_τ^(i;L). We use a double index (k,m)∈𝒟⊂ℤ^2, where k shall index a Lagrangian shear y_τ_k^(i_k;L_k), while m denotes the mth occurrence of that shear. To exploit Cancellation <ref>, we shall ensure that there are exactly two occurances of y_τ_k+1^(i_k+1;L_k+1) between each occurance of y_τ_k^(i_k;L_k). If we denote by t_k,m∈[0,1] the times at which the shears are first applied we may illustrate this construction in the following diagram: backgroundbackground,main[scale=12] [] (0,0) – (1,0); [anchor=east] (t0) at (0,0) t=0; [anchor=west] (t1) at (1,0) t=1; [yshift=20mm] (t10) at (0,0) t_1,0; [<-] ([yshift=6/12mm]0,0) – (t10); [anchor=base,yshift=20mm] (t11) at (1/2,0) t_1,1; [<-] ([yshift=6/12mm]1/2,0) – (t11); [anchor=base,yshift=15mm] (t20) at (2/8,0) t_2,0; [<-] ([yshift=6/12mm]2/8,0) – (t20); [anchor=base,yshift=15mm] (t21) at (3/8,0) t_2,1; [<-] ([yshift=6/12mm]3/8,0) – (t21); [anchor=base,yshift=15mm] (t22) at (6/8,0) t_2,2; [<-] ([yshift=6/12mm]6/8,0) – (t22); [anchor=base,yshift=15mm] (t23) at (7/8,0) t_2,3; [<-] ([yshift=6/12mm]7/8,0) – (t23); [anchor=base,yshift=10mm] (t305) at (10.5/32,0) t_3,0 t_3,1; [anchor=base,yshift=5mm] (n1) at (10.5/32,0) ...; [<-] ([yshift=6/12mm]10/32,0) – ([shift=(-0.5/32,0)]t305.south); [anchor=base,yshift=5mm] (n1) at (11.5/32,0) ...; [<-] ([yshift=6/12mm]11/32,0) – ([shift=(0.5/32,0)]t305.south); [anchor=base,yshift=10mm] (t325) at (14.5/32,0) t_3,2 t_3,3; [anchor=base,yshift=5mm] (n1) at (14.5/32,0) ...; [<-] ([yshift=6/12mm]14/32,0) – ([shift=(-0.5/32,0)]t325.south); [anchor=base,yshift=5mm] (n1) at (15.5/32,0) ...; [<-] ([yshift=6/12mm]15/32,0) – ([shift=(0.5/32,0)]t325.south); [anchor=base,yshift=10mm] (t345) at (26.5/32,0) t_3,4 t_3,5; [anchor=base,yshift=5mm] (n1) at (26.5/32,0) ...; [<-] ([yshift=6/12mm]26/32,0) – ([shift=(-0.5/32,0)]t345.south); [anchor=base,yshift=5mm] (n1) at (27.5/32,0) ...; [<-] ([yshift=6/12mm]27/32,0) – ([shift=(0.5/32,0)]t345.south); [anchor=base,yshift=10mm] (t365) at (30.5/32,0) t_3,6 t_3,7; [anchor=base,yshift=5mm] (n1) at (30.5/32,0) ...; [<-] ([yshift=6/12mm]30/32,0) – ([shift=(-0.5/32,0)]t365.south); [anchor=base,yshift=5mm] (n1) at (31.5/32,0) ...; [<-] ([yshift=6/12mm]31/32,0) – ([shift=(0.5/32,0)]t365.south); in 0,1[] (/2,0) circle (1/12pt); i̱n 0,1[] (/2+2/8+/̱8,0) circle (1/12pt); i̧n 0,1[] (/2+2/8+/̱8+2/32+/̧32,0) circle (1/12pt); ịn 0,1[] (/2+2/8+/̱8+2/32+/̧32+2/128+/̣128,0) circle (1/12pt); in 0,1 [anchor=base,yshift=-20pt] (n1) at (0,0) f_0; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (1/4,0) – (0,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_1^(i_1;L_1); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (3/4,0) – (2/4,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_1^(i_1;L_1); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (5/16,0) – (4/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_2^(i_2;L_2); [anchor=base,yshift=-20pt] (n1) at (5.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (7/16,0) – (6/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_2^(i_2;L_2); [anchor=base,yshift=-20pt] (n1) at (7.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (13/16,0) – (12/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_2^(i_2;L_2); [anchor=base,yshift=-20pt] (n1) at (13.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (15/16,0) – (14/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]∘ y_-τ_2^(i_2;L_2); [anchor=base,yshift=-20pt] (n1) at (15.5/16,0) ...; [anchor=base west,yshift=-20pt] (n1) at (1,0) =f(·, 1); If we include only the shears y_τ_k^(i_k;L_k) for k≤ K, it can now be seen how Cancellation <ref> may create two different behaviours of the trace f(·, 1) as 2K →∞, and 2K+1 →∞. The frequencies of the shears, L_k∈ℕ, will later be chosen to grow sufficiently quickly, so that advection-diffusion (<ref>) along a viscosity subsequence ν_k>0 will only include the effect of the first k shears, as per Theorem <ref>. To this end, we first define a total order <_time on the indexing set 𝒟, and then define t_k,m so that they respect this ordering. We define the following set of `dyadic' pairs 𝒟={(k,m):k∈ℕ, m ∈ℤ, 0≤ m<2^k}, and define a total order <_time on 𝒟 via (k_1,m_1) <_time (k_2,m_2) if and only if m_1 2^-k_1 < m_2 2^-k_2, or m_1 2^-k_1 = m_2 2^-k_2 and k_1<k_2. Define also for each K ∈ℕ the finite subset 𝒟_K = {(k,m) ∈𝒟 : k ≤ K}, which inherits the total order (𝒟,<_time), which is now a finite order. [The bound on t_k,m can be found by direct calculation of ∑_k∈ℕ2^k2^-2k=1.]We define for each k ∈ℕ, and m ∈ℤ with 0≤ m < 2^k, t_k,m = ∑_(k',m')<_time(k,m) 2^-2k' < 1, where an empty sum is zero. We illustrate this in the following diagram: backgroundbackground,main[scale=12] [] (0,0) – (1,0); [anchor=east] (t0) at (0,0) t=0; [anchor=west] (t1) at (1,0) t=1; [yshift=20mm] (t10) at (0,0) t_1,0; [<-] ([yshift=6/12mm]0,0) – (t10); [anchor=base,yshift=20mm] (t11) at (1/2,0) t_1,1; [<-] ([yshift=6/12mm]1/2,0) – (t11); [anchor=base,yshift=15mm] (t20) at (2/8,0) t_2,0; [<-] ([yshift=6/12mm]2/8,0) – (t20); [anchor=base,yshift=15mm] (t21) at (3/8,0) t_2,1; [<-] ([yshift=6/12mm]3/8,0) – (t21); [anchor=base,yshift=15mm] (t22) at (6/8,0) t_2,2; [<-] ([yshift=6/12mm]6/8,0) – (t22); [anchor=base,yshift=15mm] (t23) at (7/8,0) t_2,3; [<-] ([yshift=6/12mm]7/8,0) – (t23); [anchor=base,yshift=10mm] (t305) at (10.5/32,0) t_3,0 t_3,1; [anchor=base,yshift=5mm] (n1) at (10.5/32,0) ...; [<-] ([yshift=6/12mm]10/32,0) – ([shift=(-0.5/32,0)]t305.south); [anchor=base,yshift=5mm] (n1) at (11.5/32,0) ...; [<-] ([yshift=6/12mm]11/32,0) – ([shift=(0.5/32,0)]t305.south); [anchor=base,yshift=10mm] (t325) at (14.5/32,0) t_3,2 t_3,3; [anchor=base,yshift=5mm] (n1) at (14.5/32,0) ...; [<-] ([yshift=6/12mm]14/32,0) – ([shift=(-0.5/32,0)]t325.south); [anchor=base,yshift=5mm] (n1) at (15.5/32,0) ...; [<-] ([yshift=6/12mm]15/32,0) – ([shift=(0.5/32,0)]t325.south); [anchor=base,yshift=10mm] (t345) at (26.5/32,0) t_3,4 t_3,5; [anchor=base,yshift=5mm] (n1) at (26.5/32,0) ...; [<-] ([yshift=6/12mm]26/32,0) – ([shift=(-0.5/32,0)]t345.south); [anchor=base,yshift=5mm] (n1) at (27.5/32,0) ...; [<-] ([yshift=6/12mm]27/32,0) – ([shift=(0.5/32,0)]t345.south); [anchor=base,yshift=10mm] (t365) at (30.5/32,0) t_3,6 t_3,7; [anchor=base,yshift=5mm] (n1) at (30.5/32,0) ...; [<-] ([yshift=6/12mm]30/32,0) – ([shift=(-0.5/32,0)]t365.south); [anchor=base,yshift=5mm] (n1) at (31.5/32,0) ...; [<-] ([yshift=6/12mm]31/32,0) – ([shift=(0.5/32,0)]t365.south); in 0,1[] (/2,0) circle (1/12pt); i̱n 0,1[] (/2+2/8+/̱8,0) circle (1/12pt); i̧n 0,1[] (/2+2/8+/̱8+2/32+/̧32,0) circle (1/12pt); ịn 0,1[] (/2+2/8+/̱8+2/32+/̧32+2/128+/̣128,0) circle (1/12pt); in 0,1 [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (1/4,0) – (0,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-2; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (3/4,0) – (2/4,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-2; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (5/16,0) – (4/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-4; [anchor=base,yshift=-20pt] (n1) at (5.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (7/16,0) – (6/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-4; [anchor=base,yshift=-20pt] (n1) at (7.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (13/16,0) – (12/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-4; [anchor=base,yshift=-20pt] (n1) at (13.5/16,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (15/16,0) – (14/16,0) node[anchor=base,pos=0.5,yshift=-20pt,black,font=]2^-4; [anchor=base,yshift=-20pt] (n1) at (15.5/16,0) ...; Next, we construct the `fractal' vector fields exploiting the above. Consider a sequence of tuples {(i_k, L_k, τ_k)}_k ∈ℕ with i_k ∈{1,2}, L_k ∈ℕ, and τ_k > 0. * We say {(i_k, L_k, τ_k)}_k ∈ℕ satisfies the Finiteness Condition if for all k∈ℕ we have (Finiteness Condition) τ_k < 2^-2k. * We say {(i_k, L_k, τ_k)}_k ∈ℕ satisfies the Cancellation Conditions if for all k∈ℕ we have (Cancellation Conditions) i_k+1 i_k, 2τ_k+1 = 1/2L_k, 2L_k+1τ_k an odd integer. Following the notation of Definition <ref>, in particular (<ref>), when {(i_k, L_k, τ_k)}_k ∈ℕ satisfy the Finiteness Condition (<ref>), we may define for each K ∈ℕ the following fractal shear flows on 𝕋^2 × [0, 1] →ℝ^2, u_K^{(i_k, L_k, τ_k)}_k=1^K(x,t) = u^(i_k;L_k)(x) if t ∈ [t_k,m, t_k,m + τ_k] for some (k,m) ∈𝒟_K, 0 otherwise, u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ(x,t) = u^(i_k;L_k)(x) if t ∈ [t_k,m, t_k,m + τ_k] for some (k,m) ∈𝒟, 0 otherwise. By (<ref>), this is well-defined since the Finiteness Condition (<ref>) implies that the intervals [t_k,m, t_k,m + τ_k] are disjoint subsets of [0,1]. Next, we show how this construction exploits Cancellation <ref>, to create non-uniqueness of renormalised solutions to (<ref>). We follow the notation introduced in Definitions <ref>, <ref>. Let f_0 ∈ L^∞(𝕋^2), and suppose we have an infinite sequence of tuples {(i_k,L_k,τ_k)}_k∈ℕ with i_k ∈{1,2}, L_k ∈ℕ, and τ_k > 0, satisfying the Finiteness Condition (<ref>). Then for each K ∈ℕ there exists a unique weak solution f^K to (<ref>) along u_K^{(i_k,L_k,τ_k)}_k=1^K with initial data f_0. Moreover f^K∈ (C^0L^1) ∩ (L^∞ L^∞), is a Lagrangian solution, and hence also a renormalised weak solution (Definitions <ref>, <ref>). If in addition {(i_k,L_k,τ_k)}_k∈ℕ satisfy the Cancellation Conditions (<ref>) then f^2Kf^even, f^2K+1f^odd, [Moreover, the sequences f^2K∈ C^0L^1 and f^2K+1∈ C^0L^1 are eventually constant when restricted to the spatio-temporal domain 𝕋^2×[t_k,m,t_k,m+2^-2k] for any (k,m)∈𝒟, or to the spatio-temporal domain 𝕋^2×{1}, as K→∞.]with the above convergence in weak-* L^∞ L^∞, and strong in L^p L^∞ for any p∈[1,∞). Furthermore, the limit functions f^even, f^odd are renormalised weak solutions to (<ref>) along u_∞^{(i_k,L_k,τ_k)}_k∈ℕ with initial data f_0. Moreover, if f_0 is not constant, then f^even f^odd. In particular, f^even(·, 1) = f_0, f^odd(·, 1) = f_0 ∘ y_-2τ_1^(i_1;L_1). Recall the language and notation introduced in Definitions <ref>, <ref>, as well as Definitions <ref>, <ref> of renormalised and Lagrangian solutions to (<ref>). By Theorems <ref>, <ref>, there exists some weak solution f^K ∈ C_weak-^0L^∞ to (<ref>) along u_K^{(i_k,L_k,τ_k)}_k=1^K with initial data f_0. We aim to show f ∈ C^0 L^1 with f^K(·, t) = f_0 ∘(ỹ_t^K)^-1, for {ỹ_t^K}_t∈[0,1] a Lagrangian flow along u_K^{(i_k,L_k,τ_k)}_k=1^K which does not depend on f_0, thus proving uniqueness. From the definition of u_K^{(i_k,L_k,τ_k)}_k=1^K, finiteness of the set 𝒟_K, and disjointness of the intervals [t_k,m, t_k,m+τ_k] for all (k,m) ∈𝒟_K, we may piecewise apply Proposition <ref> to f^K. [We also use that any weak solution to (<ref>) along u≡ 0 on an open time interval I is a constant function of t∈ I, see Theorem 3.1.4' in <cit.>.]We use that f^K ∈ C_weak-^0L^∞ to glue together the pieces. This constructs for each K ∈ℕ a Lagrangian flow ỹ_t^K satisfying (<ref>), and moreover gives an expression for ỹ_t^K in terms of the Lagrangian shear flows y_t^(i_k;L_k), see (<ref>) below. For any (k,m) ∈𝒟_K, define t_suc_K(k,m) = 1 if (k,m) is maximal in (𝒟_K,<_time), else t_𝒮 for 𝒮∈𝒟_K the successor of (k,m) in (𝒟_K,<_time). Note from (<ref>), and the Finiteness Condition (<ref>), that τ_k < 2^-2k≤ t_suc_K(k,m) - t_k,m. Therefore, for each t ∈ [t_k,m,t_suc_K(k,m)], ỹ_t^K ∘(ỹ_t_k,m^K)^-1 = y_t-t_k,m^(i_k;L_k) if t ∈ [t_k,m,t_k,m+τ_k], y_τ_k^(i_k;L_k) if t ∈ [t_k,m+τ_k,t_suc_K(k,m)]. This expression defines ỹ_t^K for all t ∈ [0,1], since t_1,0 = 0 and ỹ_0^K = Id. This completes the proof of uniqueness. Next, by considering (<ref>), (<ref>), we may show that (see the illustration (<ref>)) t_suc_K(k,m) = t_k,m+2^-2k if k < K, t_k,m + 2^1-2k if k=K. Moreover, for all (k,m) ∈𝒟, t_k,m+2^-2k = t_k+1,2m, t_k,m+2^1-2k = t_k,m+1 when m even. Assume that additionally {(i_k,L_k,τ_k)}_k∈ℕ satisfy the Cancellation Conditions (<ref>). Fix some K ∈ℕ. We claim that, for each (k,m) ∈𝒟_K, and for all .t ∈ [t_k,m,t_k,m+2^-2k]., one has as maps 𝕋^2→𝕋^2, ỹ_t^K+2 = ỹ_t^K. By (<ref>), (<ref>), it is sufficient to show, for all (k,m)∈𝒟_K, that as maps 𝕋^2→𝕋^2, ỹ_t_suc_K(k,m)^K+2∘(ỹ_t_suc_K+2(k,m)^K+2)^-1 = ỹ_t_suc_K(k,m)^K ∘(ỹ_t_suc_K+2(k,m)^K)^-1. By (<ref>), for all k<K this is immediate. Meanwhile, by (<ref>), (<ref>), for k=K we may rewrite this, and now need to prove that for all m ∈ℤ with 0≤ m<2^K, ỹ_t_K+1,2m+2^-2K^K+2∘(ỹ_t_K+1,2m^K+2)^-1 = ỹ_t_K+1,2m+2^-2K^K∘(ỹ_t_K+1,2m^K)^-1. In particular, it is certainly sufficient to show, for any (k,m) ∈𝒟_K+1 with m even, that as maps 𝕋^2→𝕋^2, ỹ_t_k,m+2^2-2k^K∘(ỹ_t_k,m^K)^-1[t] = y_2τ_k^(i_k;L_k) if k = K (mod 2), Id otherwise. = : Y_k,m^K, where we have defined the shorthand Y_k,m^K:𝕋^2→𝕋^2. Fixing an even m∈ℤ with 0≤ m < 2^K+1, we prove this by induction. For k=K+1, by (<ref>), (<ref>) we see that as maps 𝕋^2→𝕋^2, ỹ_t_suc_K(k,m)^K ∘(ỹ_t_suc_K+2(k,m)^K)^-1 = Id. After rewriting this using (<ref>), (<ref>) this proves (<ref>) for k=K+1. Now let k≤ K. By (<ref>), and since by assumption m is even, we may rewrite (it may be helpful to consider the illustration (<ref>)) Y_k,m^K = [t] (ỹ_t_k+1,2m+2+2^-2k^K∘(ỹ_t_k+1,2m+2^K)^-1) ∘(ỹ_t_k,m+1+2^-2k^K∘(ỹ_t_k,m+1^K)^-1) ∘(ỹ_t_k+1,2m+2^-2k^K∘(ỹ_t_k+1,2m^K)^-1) ∘(ỹ_t_k,m+2^-2k^K∘(ỹ_t_k,m^K)^-1). Then by the definition of Y_k+1,m'^K, (<ref>), and (<ref>), the right hand side may be rewritten again as Y_k,m^K = Y_k+1,2m+2^K ∘ y_τ_k^(i_k;L_k)∘ Y_k+1,2m^K ∘ y_τ_k^(i_k;L_k). By Cancellation <ref>, the result (<ref>) for k now follows from the same result for k+1. This completes the induction, and thus the proof of (<ref>), and so also the proof of (<ref>). Next, note by (<ref>), that E = (⋃_(k,m)∈𝒟 [t_k,m,t_k,m+2^-2k]) ∪{t_k,m+2^2-2k:(k,m)∈𝒟 with m even} ⊂ [0,1], has Lebesgue-measure 1. Moreover, by (<ref>), (<ref>), the sequences f^2K, f^2K+1 are eventually constant on 𝕋^2 × [t_k,m,t_k,m+2^-2k], and each 𝕋^2 ×{t_k,m+2^2-2k} with m even. Therefore, for all t∈ E, and so for a.e. t ∈ [0,1], we see that f^2K(·, t) f_0 ∘(ỹ^even_t)^-1, f^2K+1(·, t) f_0 ∘(ỹ^odd_t)^-1, for some Lebesgue-measure preserving {ỹ^even_t}_t∈ E, {ỹ^odd_t}_t∈ E independent of f_0, with the convergence strong in L^∞(𝕋^2). Moreover, by the dominated convergence Theorem, we see that (<ref>) converges strongly in L^p(E;L^∞(𝕋^2)) for all p∈[1,∞), and hence also in weak-* L^∞(E;L^∞(𝕋^2)). By Definition <ref>, for all t∈[0,1] (in particular t∈ E), u_K^{(i_k, L_k, τ_k)}_k=1^K(·, t) is bounded by 1 in L^∞(𝕋^2;ℝ^2), and as K→∞ the sequence is eventually equal to u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ(·,t). Therefore it too converges strongly in L^p(E;L^∞(𝕋^2;ℝ^2)) for all p∈[1,∞). Fix t ∈ E, and apply the Trace Formula (<ref>) to f^2K, f^2K+1. Taking now the limit K→∞ then gives that for all t∈ E, for any ϕ∈ C^∞(𝕋^d×[0,T]), ∫_𝕋^2 f_0 ∘(ỹ^even_t)^-1ϕ(·, t) dx = ∫_𝕋^2 f_0 ϕ_0 dx + ∫_𝕋^2×(E∩[0,t]) f_0 ∘(ỹ^even_t)^-1(∂ϕ/∂ t+u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ·∇ϕ +νΔϕ) dxdt, ∫_𝕋^2 f_0 ∘(ỹ^odd_t)^-1ϕ(·, t) dx = ∫_𝕋^2 f_0 ϕ_0 dx + ∫_𝕋^2×(E∩[0,t]) f_0 ∘(ỹ^odd_t)^-1(∂ϕ/∂ t+u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ·∇ϕ +νΔϕ) dxdt, where ϕ_0=ϕ(·,0). This implies that f_0 ∘(ỹ^even_t)^-1, f_0 ∘(ỹ^odd_t)^-1∈ C^0_weak-(E;L^∞(𝕋^2)) and so may be extended to f^even, f^odd∈ C^0_weak-([0,1];L^∞), as argued in Theorem <ref>. For any ϕ∈ C_c^∞(𝕋^2×[0,1)) let t=1 in the above (noting that 1=t_1,0+2^2-2∈ E). This proves that f^even, f^odd are both weak solutions to (<ref>) along u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ with initial data f_0. Moreover, for any β∈ C_b^0(ℝ), we may rewrite β(f_0∘(ỹ^even_t)^-1) = β(f_0)∘(ỹ^even_t)^-1. By repeating the above arguments we have also that β(f_0)∘(ỹ^even_t)^-1 (can be extended to) a weak solution to (<ref>) along u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ with initial data β(f_0). Therefore, we see that f^even is a renormalised weak solution to (<ref>) along u_∞^{(i_k, L_k, τ_k)}_k ∈ℕ with initial data f_0. Similarly for f^odd. Next, observe that by (<ref>), (<ref>), for all (k,m)∈𝒟 with m even, that f^even(·,t_k,m+2^2-2k) = f^even(·,t_k,m) ∘(y_2τ_k^(i_k;L_k))^-1 for even k, Id for odd k. f^odd(·,t_k,m+2^2-2k) = f^odd(·,t_k,m) ∘Id for even k, (y_2τ_k^(i_k;L_k))^-1 for odd k. In particular, since t_1,0=0, we have proved (<ref>). It remains to show, when f_0 is not constant, that from (<ref>) we deduce f^even f^odd. Assume to the contrary that they are equal and call this solution f. Then by (<ref>) (noting that by (<ref>), (<ref>), 1-2^2-2k = t_k,2^k-2, see for example the illustration (<ref>)), we have for all k∈ℕ, that f(·, 1) = f(·,1-2^2-2k), f(·, 1) = f(·,1-2^2-2k) ∘(y_2τ_k^(i_k;L_k))^-1. In particular, taking k=1, we see f(·,1)=f_0. Substituting this back in gives f_0 = f(·,1-2^2-2k) for all k∈ℕ. Again substituting this back in gives that for all k∈ℕ, f_0 = f_0 ∘ y_2τ_k^(i_k;L_k). In terms of the unit vectors e_1, e_2, we have that for a.e. x∈𝕋^2, f_0(x) = f_0 (x+2τ_k e_i_k). Convolving with a smooth function ρ∈ C^∞(𝕋^2) then gives that for all x ∈𝕋^2, (ρ * f_0)(x) = (ρ * f_0)(x+2τ_k e_i_k). By the Finiteness and Cancellation Conditions (<ref>), (<ref>), we see that τ_k 0, and i_k+1 i_k (so equal to both 1 and 2 infinitely often). Now (ρ * f_0) ∈ C^∞(𝕋^2), and so by (<ref>) we must have that ρf_0 is a constant. This holds for all ρ∈ C^∞(𝕋^2), and therefore implies also that f_0 is a constant, reaching the required contradiction. Finally, for a suitably fast growing sequence {L_k}_k∈ℕ⊂ℕ, we apply Theorem <ref> to the sequence of vector fields u_n^{(i_k,L_k,τ_k)}_k=1^n(x,t). This allows us to control the vanishing viscosity limit along the vector field u_∞^{(i_k,L_k,τ_k)}_k∈ℕ. There exists a divergence-free vector field u ∈ L^∞([0,1];L^∞(𝕋^2;ℝ^2)), and a sequence {ν_n}_n∈ℕ with ν_n > 0 and ν_n0, such that for any initial data f_0∈ L^∞(𝕋^2), and for f^ν the unique solution to (<ref>) along u with initial data f_0, one has f^ν_2nf^even, f^ν_2n+1f^odd, where the above convergence in weak- L^∞([0,1];L^∞(𝕋^2)), and strong in L^p([0,1];L^p(𝕋^2)) for all p ∈ [1,∞). Furthermore, the limit functions f^even, f^odd are renormalised weak solutions to (<ref>) along u with initial data f_0. [In this case one may even show that uncountably many vanishing viscosity solutions exist. We very briefly sketch this. Work in the weak-* topology on L^∞ L^∞. First, any weak-* limit point of f^ν as ν→0 is a vanishing viscosity solution. Next, the set of limit points is bounded, and so a metric space. Moreover, the map ν→ f^ν is continuous, which implies the set of limit points is connected. If it is a connected metric space, and not unique, then it is uncountable.]Moreover, if f_0 is not constant, then f^even f^odd. Recall the language and notation introduced in Definitions <ref>, <ref>, as well as Definitions <ref>, <ref> of renormalised and Lagrangian solutions to (<ref>). Consider any infinite sequence of tuples {(i_k, L_k, τ_k)}_k ∈ℕ with i_k ∈{1,2}, L_k ∈ℕ, τ_k >0, and satisfying the Finiteness Condition (<ref>). By Proposition <ref> there exists a Lagrangian solution f^n∈ C^0L^1, unique in the class of all weak solutions, to (<ref>) along u_n^{(i_k,L_k,τ_k)}_k=1^n for any initial data f_0 ∈ L^∞(𝕋^2). Moreover, u_n^{(i_k,L_k,τ_k)}_k=1^n is bounded by 1 in L^∞([0,1];L^∞(𝕋^2;ℝ^2)). Let d_* be a metric inducing the weak-* topology on X = {u∈ L^∞( [0,1];L^∞(𝕋^2;ℝ^2)) : u_L^∞ L^∞≤ 1 }. Let f_0 ∈ L^∞(𝕋^2), and denote for each n∈ℕ, ν > 0, by f^n,ν, respectively f^n, the unique weak solution to (<ref>), respectively (<ref>), along u_n with initial data f_0. Moreover denote by f^∞,ν the unique weak solution to (<ref>) along u_∞^{(i_k,L_k,τ_k)}_k∈ℕ with initial data f_0. Then by Theorem <ref>, * For all n∈ℕ, there exists ν_n>0, ϵ_n>0 depending only on {(i_k,L_k,τ_k)}_k=1^n (and in particular not on f_0), with ν_n0 monotonically, such that the following holds true: * If d_(u_n+1,u_n) ≤ϵ_n for all n∈ℕ, then for all p ∈ [1,∞) f^∞,ν_n-f^n_L^∞ L^p0. We now construct such a sequence {(i_k,L_k,τ_k)}_k∈ℕ. Let (i_1, L_1, τ_1)=(1,2^2,2^-2). We proceed by induction on n∈ℕ. Assume {(i_k,L_k,τ_k)}_k=1^n are given, and satisfy for k∈{1,...,n}. i_k ∈{1,2}, L_k ∈ℕ, L_k ≥ 2^2k, 0 < τ_k ≤ 2^-2k, It is straightforward to check that this is satisfied for the base case (i_1, L_1, τ_1)=(1,2^2,2^-2). If n ≥ 2, we assume in the inductive hypothesis that also, for k∈{1,...,n-1}, i_k+1 i_k, 2τ_k+1 = 1/2L_k, 2L_k+1τ_k is an odd integer, d_(u_k+1^{(i_l,L_l,τ_l)}_l=1^k+1,u_k^{(i_l,L_l,τ_l)}_l=1^k) ≤ϵ_k. We then choose (i_n+1, L_n+1, τ_n+1) as follows. Let i_n+1∈{1,2}∖{i_n}. Let τ_n+1=1/4L_n which by the inductive hypothesis satisfies τ_n+1≤ 2^-2k-2. Let L_n+1=2L_n-1(2M+1), for some large M∈ℕ to be chosen, where when n=1 we take L_0 = 1. This will ensure 2L_n+1τ_n is an odd integer. By (<ref>) and Definition <ref>, we see for n∈ℕ fixed, that d_(u_n+1^{(i_k,L_k,τ_k)}_k=1^n+1,u_n^{(i_k,L_k,τ_k)}_k=1^n)0. Therefore, by taking M∈ℕ large enough in (<ref>), we have that both L_n+1≥ 2^2n+2, and d_(u_n+1^{(i_k,L_k,τ_k)}_k=1^n+1,u_n^{(i_k,L_k,τ_k)}_k=1^n) ≤ϵ_n, completing the induction. We have constructed {(i_k, L_k, τ_k)}_k ∈ℕ satisfying the Finiteness and Cancellation Conditions (<ref>), (<ref>), and also (<ref>), (<ref>). The statement of the theorem is now a straightforward corollary of Proposition <ref>. § GRADUAL PERFECT MIXING - INADMISSIBILITY §.§ Notation We continue with the same notation for the 2-torus introduced in Section <ref>. However we no longer make use of the notation in Definitions <ref>, <ref>, <ref>. In addition, we define the binary expansion of some x=(x_1,x_2) ∈𝕋^2 as follows. For the representative (x_1,x_2)∈[0,1)^2 denote for i∈{1,2}, k∈ℕ, by x_i,k the kth binary digit of the ith coordinate of x. That is, for each i ∈{1,2}, x_i = ∑_k=1^∞ x_i,k 2^-k, where x_i,k∈{0,1} and x_i,k 1 (as is standard to ensure uniqueness of the binary expansion). First, we define vector fields and corresponding Lagrangian flows which swap points in 𝕋^2 according to their binary expansion. These `binary swaps' form the building block of our construction. Subsequently we shall give a divergence-free vector field in L^∞([0,100];L^∞(𝕋^2;ℝ^2)) with L^∞ L^∞-norm equal to 1, which perfectly mixes the transported scalar to its spatial average, and subsequently unmixes, any initial data to (<ref>). Our aim is then to show that this behaviour is the unique limit point of vanishing viscosity of the associated solution to (<ref>). In order to ensure uniqueness of the limit points it is necessary to `gradually' perform these binary swaps, that is they must be restricted to gradually smaller regions of space, see (<ref>) below. Suppose i ∈{1,2}, k ∈ℕ, n∈{1,...,2^⌊ k/2⌋}, L ∈ℕ, with L ≥ k+1. In the proof below will define a time-dependent, divergence-free vector field, called the (i,k,n;L)-binary swap u^(i,k,n;L) : 𝕋^2 × [0,3·2^-k] →ℝ^2, and a corresponding Lagrangian flow map (Definition <ref>) {y^(i,k,n;L)_t}_t ∈ [0,3·2^-k] with the properties (<ref>)-(<ref>) below. Define J_k,n=[(n-1)2^-⌊ k/2⌋,n2^-⌊ k/2⌋]⊂𝕋. Then at time t=3·2^-k, for a.e. x=(x_1,x_2) ∈𝕋^2, y^(i,k,n;L)_3·2^-k will swap the kth and (k+1)th binary digits of x_i if x_i∈ J_k,n. That is y_0^(i,k,n;L) = Id. For a.e. x=(x_1,x_2)∈𝕋^2 denote by (x'_1,x'_2)=y^(i,k,n;L)_3·2^-k(x). Following the notation for binary expansions in (<ref>), for j∈{1,2} the coordinate, and for l∈ℕ the binary digit, x'_j = x_j if j≠ i, x'_i,l = x_i,l for l∉{k,k+1}, x'_i,k+1 = x_i,k if x_i ∈ J_k,n, x_i,k+1 otherwise, x'_i,k = x_i,k+1 if x_i ∈ J_k,n, x_i,k otherwise. Additionally, the vector field u^(i,k,n;L) will satisfy u^(i,k,n;L)_L^∞ L^∞≤ 1, and for all t∈[0,3·2^-k], and all x=(x_1,x_2)∈𝕋^2 with x_i ∉ J_k,n, u^(i,k,n;L)(x,t)=0. Moreover, for i∈{1,2}, k∈ℕ fixed, as L →∞, u^(i,k,n;L)L →∞ 0, in weak-L^∞ L^∞. Finally, for any r, r' ∈{1,...,2^k-1}, for the spatial intervals J=[(r-1)2^1-k,r2^1-k], and J'=[(r'-1)2^1-k,r'2^1-k], the Lagrangian-flow y^(i,k,n;L)_t preserves the squares y^(i,k,n;L)_t:J× J'↔ J× J'. We now construct the above vector field and Lagrangian flow. We shall give the construction for i=1, and then define for i=2 its coordinate reflection. That is, for each (x_1,x_2) ∈𝕋^2, t ∈ [0,3·2^-k] u^(2,k,n;L)((x_1,x_2),t) = u^(1,k,n;L)((x_2,x_1),t), y_t^(2,k,n;L)((x_1,x_2)) = y_t^(1,k,n;L)((x_2,x_1)). so that (<ref>), (<ref>), (<ref>) with i=2 follow from the same result for i=1. We shall achieve the required binary-swaps (<ref>) by piecing together particular rotating vector fields which perform half rotations of rectangular regions of 𝕋^2. Suppose W,H>0, and define the following 1-Lipschitz stream-function ψ:(-W/2,W/2)×(-H/2,H/2)⊂ℝ^2 →ℝ, ψ((x_1,x_2)) = min{W,H}·max{(x_1/W)^2,(x_2/H)^2}. This defines a 1-bounded, time-independent, divergence-free, vector field ∇^⊥ψ = (-∂ψ/∂ x_2, ∂ψ/∂ x_1) on the open rectangle (-W/2,W/2)×(-H/2,H/2). ∇^⊥ψ has rectangular flow lines, as illustrated in the following diagram: [scale=3/2] backgroundbackground,main[scale=1/2,decoration= markings, mark=at position 1/36 with [xshift=-2pt]latex[length=4pt], mark=at position 10/36 with [xshift=-2pt]latex[length=4pt], mark=at position 19/36 with [xshift=-2pt]latex[length=4pt], mark=at position 28/36 with [xshift=-2pt]latex[length=4pt], ] [dashed,postaction=decorate] (3,3/8) rectangle (5,5/8); [dashed,postaction=decorate] (2,1/4) rectangle (6,3/4); [dashed,postaction=decorate] (1,1/8) rectangle (7,7/8); [thick] (0,0) – (8,1); [thick,] (0,1) – (8,0); [thick,red] (0,0) rectangle (8,1); (n3) at (0,0); (n4) at (0,1); (n5) at (8,0); (n6) at (8,1); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (0,1) – (8,1) node[pos=0.5,above=10pt,black]W; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=2.5pt, aspect=0.5 ] (0,0) – (0,1) node[pos=0.5,left=7pt,black]H; Moreover, on each triangular segment, ∇^⊥ψ is a linear function of space, and each flow line within each segment has time period equal to max{W,H}. [That is for all t∈(-∞,∞), y_t:(-W/2,W/2)×(-H/2,H/2)↔(-W/2,W/2)×(-H/2,H/2) is a Lebesgue-measure preserving bijection, y_0= Id, and for all x∈(-W/2,W/2)×(-H/2,H/2), and all t∈(-∞,∞), d/dty_t(x)=∇^⊥ψ(y_t(x)).]Therefore, ∇^⊥ψ admits a Lagrangian flow {y_t}_t∈(-∞,∞) on (-W/2,W/2)×(-H/2,H/2). [Vector fields achieving the same half rotation property with more regularity exist in the literature. See for example Section 3 in <cit.>.]Moreover, we have y_0= Id, while y_2max{W,H} is exactly a half rotation of (-W/2,W/2)×(-H/2,H/2) around its centre. [This may be easier to see by first using a different metric.]Finally, both y_t and y_t^-1 are Lipschitz in both x and t. That is, for all x,x' ∈(-W/2,W/2)×(-H/2,H/2), and all t,s ∈ (-∞,∞), \|y_t(x) - y_s(x')\| ≤ C(\|x-x'\| + \|t-s\|), \|y_t^-1(x) - y_s^-1(x')\| ≤ C(\|x-x'\| + \|t-s\|), for some constant C>0. Consider now any open rectangle Q⊂ℝ^2 with width W and height H. By translating (<ref>) to Q we have the same vector field on Q, which to simplify the following presentation we symbolically notate by [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4); : Q →ℝ^2. We iterate that this is a 1-bounded, time-independent, divergence-free vector field, admitting a Lipschitz Lagrangian flow {y_t}_t∈(-∞,∞), as in (<ref>). Moreover, y_0=Id, while y_2max{W,H} is exactly a half rotation of Q around its centre. We also denote the same vector field multiplied by -1 by the same diagram with the arrows reversed, [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4); = - [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4); . This admits the Lagrangian flow {y_t^-1}_t∈(-∞,∞), and so enjoys the same properties as above. Next, we piece these half-rotations together to perform orientation-preserving swaps. This is exactly the building block required to perform the binary swaps (<ref>). Let Q⊂ℝ^2 be again an open rectangle with width W>0, and height H>0. Let L ∈ℕ and suppose in addition that H is an integer multiple of 2^1-L, and W ≥ 2^1-L. Subsequently, we define a time-dependent vector field, symbolically notated by [scale=1/2] backgroundbackground,main[dashed] (1,0) – (1,1/2); [thick,-latex,yshift=3pt] (0.5,1/4) – (1.5,1/4); [thick,latex-,yshift=-3pt] (0.5,1/4) – (1.5,1/4); [thick,red] (0,0) rectangle (2,1/2); : Q ×[0,3W] →ℝ^2, by [scale=1] backgroundbackground,main[dashed] (1,0) – (1,1); [thick,-latex,yshift=3pt] (0.5,1/2) – (1.5,1/2); [thick,latex-,yshift=-3pt] (0.5,1/2) – (1.5,1/2); [thick,red] (0,0) rectangle (2,1); (·,t) = [scale=2/3] backgroundbackground,main[scale=2,shift=(0.75,0.5)] in 0,...,3 int(mod(,2))=0 [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (1,/4+1/8) ellipse (1/2 and 1/16); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (1,/4+1/8) ellipse (1/2 and 1/16); in 1,...,3[dashed] (0,/4) – (2,/4); in 1,...,1[thick,dashed] (0,/2) – (2,/2); [thick,red] (0,0) rectangle (2,1); (n3) at (0,0); (n4) at (0,1); [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,1) – (2,1) node[pos=0.5,above=5pt,black]W; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,0) – (0,1) node[pos=0.5,left=5pt,black]H; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=2pt, aspect=0.5 ] (2,0.25) – (2,0) node[pos=0.5,right=2pt,black]2^-L; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=2pt, aspect=0.5 ] (2,1) – (2,0.5) node[pos=0.5,right=2pt,black]2^1-L; [right=10pt] (time0) at (2,1) ; if t ∈ [0,2W), [scale=2/3] backgroundbackground,main[scale=2,shift=(3.25,0.5)] in 0,...,3int(mod(,2)) in 0,1 =0 [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (+1/2,/4+1/8) ellipse (1/4 and 1/16); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (+1/2,/4+1/8) ellipse (1/4 and 1/16); in 1,...,3[dashed] (0,/4) – (2,/4); in 1,...,1[thick,dashed] (0,/2) – (2,/2); [thick,dashed] (1,0) – (1,1); [thick,red] (0,0) rectangle (2,1); [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,1) – (1,1) node[pos=0.5,above=5pt,black]1/2W; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,0) – (0,1) node[pos=0.5,left=5pt,black]H; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=2pt, aspect=0.5 ] (2,1) – (2,3/4) node[pos=0.5,right=2pt,black]2^-L; [right=10pt] (time1) at (2,1) ; if t ∈ [2W,3W]. This has the property that at time t=3W the left and right halves are swapped in an orientation preserving manner. This is illustrated below: [scale=2/3] backgroundbackground,main[scale=2,shift=(-2,0.5)] [left color=black!80,right color=white,shading angle=0] (0,0) rectangle (2,1); [opacity=0.6,left color=black,right color=white,shading angle=90] (0,0) rectangle (2,1); [thick,red] (0,0) rectangle (2,1); [scale=2,shift=(0.75,0.5)] [left color=black!80,right color=white,shading angle=0] (0,0) rectangle (2,1); [opacity=0.6,left color=black,right color=white,shading angle=90] (0,0) rectangle (2,1); in 0,...,3 int(mod(,2))=0 [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (1,/4+1/8) ellipse (1/2 and 1/16); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (1,/4+1/8) ellipse (1/2 and 1/16); in 1,...,3[dashed] (0,/4) – (2,/4); [thick,red] (0,0) rectangle (2,1); (n3) at (0,0); (n4) at (0,1); [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,1) – (2,1) node[pos=0.5,above=5pt,black]W; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=2pt, aspect=0.5 ] (0,3/4) – (0,1) node[pos=0.5,left=2pt,black]2^-L; [right=10pt] (time0) at (2,1) ; [scale=2,shift=(3.25,0.5)] in 0,...,380/480/4+20[left color=black!,right color=black!,̱shading angle=0] (0,/4) rectangle (2,/4+1/4); [opacity=0.6,left color=white,right color=black,shading angle=90] (0,0) rectangle (2,1); in 0,...,3int(mod(,2)) in 0,1 =0 [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (+1/2,/4+1/8) ellipse (1/4 and 1/16); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (+1/2,/4+1/8) ellipse (1/4 and 1/16); in 1,...,3[dashed] (0,/4) – (2,/4); [dashed] (1,0) – (1,1); [thick,red] (0,0) rectangle (2,1); [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (0,1) – (1,1) node[pos=0.5,above=5pt,black]1/2W; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=2pt, aspect=0.5 ] (2,1) – (2,3/4) node[pos=0.5,right=2pt,black]2^-L; [right=10pt] (time1) at (2,1) ; [scale=2,shift=(6,0.5)] [left color=black!80,right color=white,shading angle=0] (0,0) rectangle (2,1); [opacity=0.6,left color=black!50,right color=white,shading angle=90] (0,0) rectangle (1,1); [opacity=0.6,left color=black,right color=black!50,shading angle=90] (1,0) rectangle (2,1); [thick, red] (0,0) rectangle (2,1); [right=10pt] (time2) at (2,1) ; [thick, -latex] (1,0) – (11,0); (1,0) circle (2pt); (6,0) circle (2pt); (11,0) circle (2pt); [below=5pt] (t0) at (1,0) t=0; [below=5pt] (t1) at (6,0) t=2W; [below=5pt] (t2) at (11,0) t=3W; [very thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (1,0) – (6,0) node[pos=0.5,above=5pt,black]; [very thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (6,0) – (11,0) node[pos=0.5,above=5pt,black]; Finally, suppose k ∈ℕ, n ∈{1,...,2^⌊ k/2⌋}, and L∈ℕ, with L≥ k+1. Let J_k,n=[(n-1)2^-⌊ k/2⌋,n2^-⌊ k/2⌋]⊂𝕋. For W=2^-k, H=1, and L≥ k+1 we may use [scale=1/2] backgroundbackground,main[dashed] (1,0) – (1,1/2); [thick,-latex,yshift=3pt] (0.5,1/4) – (1.5,1/4); [thick,latex-,yshift=-3pt] (0.5,1/4) – (1.5,1/4); [thick,red] (0,0) rectangle (2,1/2); to define the time-dependent vector field u^(1,k,n;L) : 𝕋^2 × [0,3·2^-k] →ℝ^2, backgroundbackground,main[scale=3/2] [thick,postaction=decorate,decoration= markings, mark=at position 1/8 with [xshift=2pt]>[length=4pt], mark=at position 3/8 with [xshift=4pt]>[length=4pt]>[length=4pt], mark=at position 5/8 with [xshift=2pt]<[length=4pt], mark=at position 7/8 with [xshift=4pt]<[length=4pt]<[length=4pt], ] (0,0) rectangle (4,4); in 1,...,15[dashed] (/4,0) – (/4,4); in 1,...,3[thick,dashed] (,0) – (,4); in 2,...,3 [thick, red] (+1/4,0) rectangle (+3/4,4); [thick,-latex,yshift=2pt] (+3/8,2) – (+5/8,2); [thick,latex-,yshift=-2pt] (+3/8,2) – (+5/8,2); (n1) at (3,0); (n2) at (3,4); [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (3,0) – (2,0) node[pos=0.5,below=5pt,black]2^1-k; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=3pt, aspect=0.5 ] (0,4) – (0.25,4) node[pos=0.5,above=3pt,black]2^-k-1; [thick,decorate,decoration = calligraphic brace, raise=2pt, amplitude=5pt, aspect=0.5 ] (2.25,4) – (2.75,4) node[pos=0.5,above=5pt,black]2^-k; [thick,decorate,decoration = calligraphic brace, raise=20pt, amplitude=5pt, aspect=0.5 ] (4,0) – (2,0) node[pos=0.5,below=25pt,black]J_k,n; [thick,decorate,decoration = calligraphic brace, raise=20pt, amplitude=5pt, aspect=0.5 ] (0,4) – (4,4) node[pos=0.5,above=25pt,black]𝕋; [thick,decorate,decoration = calligraphic brace, raise=20pt, amplitude=5pt, aspect=0.5 ] (0,0) – (0,4) node[pos=0.5,left=25pt,black]𝕋; . Note that u^(1,k,n;L) is piecewise given by (<ref>), and so is 1-bounded and divergence-free. By construction (<ref>), (<ref>) are satisfied, while (<ref>) follows from the illustration (<ref>) after noting the bound 2^-L≤ 2^1-k. Finally, to prove (<ref>), notice in (<ref>) we alternate between [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4); and [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4);, and so for all x ∈𝕋^2, t ∈ [0,3·2^-k] u^(1,k,n;L)(x,t) = - u^(1,k,n;L)(x+2^-Le_2,t). The proof of (<ref>) now follows the same argument as the proof of (<ref>). Next, we prove well-posedness of (<ref>) along these vector fields. Suppose i ∈{1,2}, k∈ℕ, n∈{1,...,2^⌊ k/2⌋} L ∈ℕ, with L ≥ k+1. Suppose f∈ L^1L^1 is a weak solution to (<ref>) along u^(i,k,n;L) on the open time interval (0,3·2^-k). Then there exists some f_0 ∈ L^1(𝕋^2) such that f is (a.e. in 𝕋^2×(0,3·2^-k)) equal to the Lagrangian solution (Definition <ref>) f(·, t)=f_0 ∘(y^(i,k,n;L)_t)^-1. In particular, f is a renormalised weak solution to (<ref>) (Definition <ref>) along u^(i,k,n;L) with initial data f_0. Moreover f_0 ∘(y^(i,k,n;L)_t)^-1∈ C^0([0,3·2^-k];L^1(𝕋^2)). We shall give only a brief proof, since this essentially follows the proof of Proposition <ref>. Alternatively one may be able prove the result via the L^1([0,T]; BV)-theory of transport developed by Ambrosio in <cit.>. The case i=2 follows from the same result for i=1 and the definitions (<ref>), (<ref>). Hence we may assume i=1. Notice, from the construction (<ref>), that u^(1,k,n;L) is stationary/time-independent on the time intervals I_1=[0,2·2^-k], and I_2=[2·2^-k,3·2^-k]. We claim it is sufficient to prove (<ref>) for some f_0∈ L^1(𝕋^2) for a.e. t ∈ I_1, and for some f_0' ∈ L^1(𝕋^2) for a.e. t ∈ I_2. This is because, after we prove the continuity f_0 ∘(y^(1,k,n;L)_t)^-1∈ C^0([0,3·2^-k];L^1(𝕋^2)), we apply Theorem <ref> to deduce f_0=f_0'. Restrict now to one of the time intervals t∈ I_1 or t∈ I_2. Then we may divide 𝕋^2 into open rectangles Q⊂𝕋^2 (and a set of Lebesgue-measure zero between rectangles) on which u^(1,k,n;L) is given either by zero, [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4);, or [scale=1] backgroundbackground,main[scale=1/4] [red, thick] (-8,0) rectangle (0,1); [thick,postaction=decorate,decoration= markings, mark=at position 1/4 with [scale=1]latex, mark=at position 3/4 with [scale=1]latex, ] (-4,1/2) ellipse (2 and 1/4);. As in the proof of Proposition <ref> we take a test function ϕ∈ C_c^∞(Q × I_1), supported on Q × I_1 (or Q × I_2 respectively), and let ψ(x, t) = ϕ((y^(1,k,n;L)_t)^-1(x),t). From the Lipschitz bound (<ref>), ψ∈ L^∞(Q× I_1) is then Lipschitz (so may be taken as a test function in (<ref>)), supported on Q, and by chain rule the following point-wise equality holds (∂ψ/∂ t + u^(1,k,n;L)·∇ψ)(x,t) = ∂ϕ/∂ t((y^(1,k,n;L)_t)^-1(x),t). Following the argument as in Proposition <ref> then proves (<ref>) holds a.e. on each Q × I_1, for some f_0. By gluing together the f_0 for each Q, we show that (<ref>) holds a.e. on 𝕋^2 × I_1 (or 𝕋^2 × I_2 respectively). We are left to show f_0 ∘(y^(1,k,n;L)_t)^-1∈ C^0([0,3·2^-k];L^1(𝕋^2)). As in the proof of Proposition <ref>, this will follow from the global Lipschitz in time bound. That is we must show, for all x ∈𝕋^2, and t,s ∈ [0,3·2^-k], \|y^(1,k,n;L)_t∘(y^(1,k,n;L)_s)^-1(x) - x \| ≤ C \|t-s\|. This follows by taking x=x'=(y^(1,k,n;L)_s)^-1(z) and applying the local Lipschitz bound (<ref>). As discussed, by Theorem <ref> we may glue together 𝕋^2 × I_1 and 𝕋^2 × I_2, proving (<ref>). f is now a Lagrangian solution to (<ref>) along u^(1,k,n;L) with initial data f_0, and so a renormalised weak solution by Remark <ref>. Next, we define the times at which to apply the binary swaps y^(i,k,n;L)_3·2^-k. We use a quadruple index (k,m,i,n)∈𝒩⊂ℕ^4 where k,i,n shall determine which binary swap y^(i,k,n;L)_3·2^-k to perform, while m denotes the mth occurrence of that binary swap. To define the times t_(k,m,i,n)∈[0,1] at which the binary swaps will be applied we first define an ordering <_time on the indexing set 𝒩, and then define t_(k,m,i,n) so that they respect this ordering. The resulting time ordering is illustrated in the diagram (<ref>) below, with the ordering designed to iteratively make 4^m copies of the initial data on a square lattice with widths 2^-m, thus mixing the initial data to its spatial average as m→∞. We additionally define a well-order <_lex on the indexing set 𝒩. This will inform which binary swaps y^(i,k,n;L)_3·2^-k will be included when approximating the vanishing viscosity limit of (<ref>), as in Proposition <ref>. For any p ∈ℕ, we define the lexicographic well-order <_lex on ℕ^p via a <_lex b (∃ i ∈{1,...,p}), (∀ j < i), a_j = b_j, a_i < b_i. Let 𝒩={(k,m,i,n) ∈ℕ^4: m≤ k,i∈{1,2}, n≤ 2^⌊ k/2⌋}, then note that the well-order (𝒩,<_lex) is order isomorphic to ℕ with the usual order. We also define another total-order <_time on 𝒩, via (k_1,m_1,i_1,n_1) <_time (k_2,m_2,i_2,n_2) if (m_1,k_2,i_2,n_2) <_lex (m_2,k_1,i_1,n_1), which is not a well-order. Define also for each 𝒦∈𝒩 the finite subset 𝒩_𝒦 = {(k,m,i,n) ∈𝒩: (k,m,i,n) ≤_lex𝒦}, which inherits the finite (and therefore well-) orders (𝒩_𝒦,<_lex), (𝒩_𝒦,<_time). [The bound on T_(k,m,i,n) can be found by direct calculation of ∑_k∈ℕ 2k2^⌊ k/2⌋·3·2^-k = 42.]For each (k,m,i,n) ∈𝒩 we define T_(k,m,i,n) = ∑_(k',m',i',n')<_time(k,m,i,n)3·2^-k' < 42, where an empty sum is zero. Moreover, for each m ∈ℕ, let T_m = ∑_m' ≤ m (k',m',i',n') ∈𝒩3·2^-k'<42. Writing also T_0 = 0, we illustrate the time ordering <_time in the following diagram, [scale=0.86] backgroundbackground,main[scale=2/3] [] (0,0) – (18.5,0); [anchor=south,yshift=7mm] (t0) at (0,0) T_0; [] (0,-1) – (0,1); [anchor=south,yshift=7mm] (t1) at (8,0) T_1; [] (8,-1) – (8,1); [anchor=south,yshift=7mm] (t2) at (14,0) T_2; [] (14,-1) – (14,1); [anchor=south,yshift=7mm] (t3) at (18,0) T_3; [] (18,-1) – (18,1); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (6,0) – (8,0) node[pos=0.5,above=7pt,black,font=]T_(1,1,i,n); [] (6,-0.2) – (6,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (6,0) – (4,0) node[pos=0.5,below=7pt,black,font=]T_(2,1,i,n); [] (4,-0.2) – (4,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (3,0) – (4,0) node[pos=0.5,above=7pt,black,font=]T_(3,1,i,n); [] (3,-0.2) – (3,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (3,0) – (2,0) node[pos=0.5,below=7pt,black,font=]T_(4,1,i,n); [] (2,-0.2) – (2,0.2); [] (1.5,-0.2) – (1.5,0.2); [] (1,-0.2) – (1,0.2); [anchor=south,font=] (l1) at (0.5,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (14,0) – (12,0) node[pos=0.5,below=7pt,black,font=]T_(2,2,i,n); [] (12,-0.2) – (12,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (11,0) – (12,0) node[pos=0.5,above=7pt,black,font=]T_(3,2,i,n); [] (11,-0.2) – (11,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (11,0) – (10,0) node[pos=0.5,below=7pt,black,font=]T_(4,2,i,n); [] (10,-0.2) – (10,0.2); [] (9.5,-0.2) – (9.5,0.2); [] (9,-0.2) – (9,0.2); [anchor=south,font=] (l2) at (8.5,0) ...; [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (17,0) – (18,0) node[pos=0.5,above=7pt,xshift=-2.5mm,black,font=]T_(3,3,i,n); [] (17,-0.2) – (17,0.2); [thick,decorate,decoration = calligraphic brace, raise=5pt, amplitude=5pt, aspect=0.5 ] (17,0) – (16,0) node[pos=0.5,below=7pt,black,font=]T_(4,3,i,n); [] (16,-0.2) – (16,0.2); [] (15.5,-0.2) – (15.5,0.2); [] (15,-0.2) – (15,0.2); [anchor=south,font=] (l3) at (14.5,0) ...; [anchor=west] (l4) at (18.5,0) ...; where for fixed k,m ∈ℕ with m ≤ k, the bracket [thick,decorate,decoration = calligraphic brace, raise=0pt, amplitude=5pt, aspect=0.5 ] (0,0) – (1,0) node[pos=0.5,above=2pt,black,font=]T_(k,m,i,n); contains for all i∈{1,2}, n∈ℕ with n ≤ 2^⌊ k/2⌋, precisely the time intervals [T_(k,m,i,n),T_(k,m,i,n)+3·2^-k). Next, we apply the divergence-free vector fields u^(i,k,n;L) (for some L∈ℕ to be chosen) in each time interval [T_(k,m,i,n),T_(k,m,i,n)+3·2^-k), with m∈{1,...,k} an index denoting the mth occurrence of this swap. The order is chosen so that the solution to (<ref>) at time t=T_m creates 4^m identical copies of the initial data, on a square lattice with widths 2^-m. To see how this can be achieved, i∈{1,2} denotes which coordinate the binary swap (<ref>) acts on, while n∈{1,...,2^⌊ k/2⌋} is an integer denoting which region of 𝕋^2 the binary swap is completed on. For fixed k∈ℕ, m∈{1,...,k}, these binary swap commute, and together the bracket [thick,decorate,decoration = calligraphic brace, raise=0pt, amplitude=5pt, aspect=0.5 ] (0,0) – (1,0) node[pos=0.5,above=2pt,black,font=]T_(k,m,i,n); swaps the kth and (k+1)th binary digit of both coordinates of every x=(x_1,x_2)∈𝕋^2. The order ≤_lex is chosen such that between T_m-1 and T_m, an `undefined' binary digit passes all the way from infinitely far up the binary expansion, down to the mth position. Since it's values is undefined, this creates 4 copies (2 in each coordinate) of the t=T_m-1 data at t=T_m. See Proposition <ref> below for the details. But first, we define this vector field and its finite approximations, which by Theorem <ref> will control the vanishing viscosity limit of (<ref>). We follow the notation introduced in Definition <ref>. Notice that a particular 𝒫=(k,m,i,n)∈𝒩 fixes k∈ℕ, i∈{1,2}, n∈{1,...,2^⌊ k/2⌋}. For 𝒦∈𝒩, and a finite sequence {L_𝒫}_𝒫∈𝒩_𝒦⊂ℕ with L_(k,m,i,n)≥ k+1 for all (k,m,i,n)∈𝒩_𝒦, we define the following time-dependent vector field. u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦:𝕋^2 × [0,50] →ℝ^2, u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦(x,t) = u^(i,k,n;L_𝒫)(x,t-T_𝒫) for 𝒫 = (k,m,i,n) ∈𝒩_𝒦, and t ∈[T_𝒫,T_𝒫+3·2^-k), 0 otherwise, in particular for t ≥ 42. Meanwhile, for an infinite sequence {L_𝒫}_𝒫∈𝒩⊂ℕ with L_(k,m,i,n)≥ k+1 for all (k,m,i,n)∈𝒩, we define the following time-dependent vector field. u_∞^{L_𝒫}_𝒫∈𝒩:𝕋^2 × [0,50] →ℝ^2, u_∞^{L_𝒫}_𝒫∈𝒩(x,t) = u^(i,k,n;L_𝒫)(x,t-T_𝒫) for 𝒫=(k,m,i,n) ∈𝒩, and t ∈[T_𝒫,T_𝒫+3·2^-k), 0 otherwise, in particular for t ≥ 42. These vector fields are well-defined since by (<ref>) the time intervals [T_(k,m,i,n),T_(k,m,i,n)+3·2^-k), are disjoint subsets of [0,42]. We follow the notation introduced in Definitions <ref>, <ref>. Let f_0 ∈ L^∞(𝕋^2), and suppose we have an infinite sequence {L_𝒫}_𝒫∈𝒩⊂ℕ with L_(k,m,i,n)≥ k+1 for all (k,m,i,n)∈𝒩. Then for each 𝒦∈𝒩 there exists a unique weak solution f^𝒦 to (<ref>) along u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦 with initial data f_0. Moreover, f^𝒦∈ (C^0L^1) ∩ (L^∞ L^∞), is a Lagrangian solution, and hence also a renormalised weak solution (Definitions <ref>, <ref>). By the well-order (𝒩,<_lex) (which is order isomorphic to (ℕ,<)), as 𝒦∞, f^𝒦f^∞, with the above convergence in weak- L^∞ L^∞, strong in L^p([0,42]; L^p) and .C^0([0,42-ϵ];L^p). for all p ∈ [1,∞), and all ϵ > 0. The limit function f^∞∈ C_weak-^0L^∞ is a weak solution to (<ref>) along u_∞^{L_𝒫}_𝒫∈𝒩 with initial data f_0. Moreover, for all t ∈ [42,50] f^∞(·,t) ≡∫_𝕋^2 f_0(y) dy, i.e., f^∞(·, t) is perfectly mixed after t=42. Recall the language and notation introduced in Definitions <ref>, <ref>, <ref>, as well as Definitions <ref>, <ref> of renormalised and Lagrangian solutions to (<ref>). By Theorems <ref>, <ref>, there exists some weak solution f^𝒦∈ C_weak-^0L^∞ to (<ref>) along u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦 with initial data f_0. We aim to show f ∈ C^0 L^1 with f^𝒦(·, t) = f_0 ∘(ỹ_t^𝒦)^-1, for {ỹ_t^𝒦}_t ∈ [0,50] a Lagrangian flow along u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦 which does not depend on f_0, thus proving uniqueness. From the definition of u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦, finiteness of the set 𝒩_𝒦, and disjointness of the intervals [T_(k,m,i,n),T_(k,m,i,n)+3·2^-k) for all (k,m,i,n) ∈𝒩_𝒦, we may piecewise apply Proposition <ref> to f^𝒦. [We also use that any weak solution to (<ref>) along u≡ 0 on an open time interval I is a constant function of t∈ I, see Theorem 3.1.4' in <cit.>.]We use that f^𝒦∈ C_weak-^0L^∞ to glue together the pieces. This constructs for each 𝒦∈𝒩 a Lagrangian flow ỹ_t^𝒦 satisfying (<ref>), and moreover gives an expression for ỹ_t^𝒦 in terms of the binary swaps y_t^(i,k,n;L), see (<ref>) below. For each 𝒫=(k,m,i,n) ∈𝒩_𝒦, define T_suc(𝒫) = 50 if 𝒫 is maximal in (𝒩_𝒦,<_time), else T_𝒫' for 𝒫' the successor of 𝒫 in (𝒩_𝒦,<_time). With each 𝒫=(k,m,i,n)∈𝒩_𝒦 fixed, and therefore fixed k∈ℕ, by (<ref>) we have that T_𝒫 + 3·2^-k≤ T_suc(𝒫). With each 𝒫=(k,m,i,n)∈𝒩_𝒦 fixed, and therefore fixed k∈ℕ, i∈{1,2}, n∈{1,...,2^⌊ k/2⌋}, we have that for all t ∈ [T_𝒫, T_suc(𝒫)] ỹ_t^𝒦∘(ỹ_T_𝒫^𝒦)^-1 = y_t-T_𝒫^(i,k,n;L_𝒫) if t ∈ [T_𝒫, T_𝒫 + 3·2^-k], y_3·2^-k^(i,k,n;L_𝒫) if t ∈ [T_𝒫 + 3·2^-k, T_suc(𝒫)]. Meanwhile, if 𝒫_min is minimal in (𝒩_𝒦,<_time), we have that for all t ∈ [0,T_𝒫_min], ỹ_t^𝒦 = Id. This completes the proof of uniqueness. Next, we wish to show, for any m ∈ℕ, that the sequence f^𝒦∈ C^0L^1 is Cauchy in C^0([T_m-1,T_m];L^1) as 𝒦∞, as follows. We proceed by induction on m ∈ℕ. [Recall the intuition that as 𝒦∞, f^𝒦(·, T_m-1) is suppose to create 4^m-1 copies of the initial data f_0, on a square lattice of widths 2^1-m.]By the inductive hypothesis we have that f^𝒦(·, T_m-1)∈ L^1(𝕋^2) is Cauchy in L^1(𝕋^2) as 𝒦∞ (for the base case m=1 this follows by f^𝒦(·, T_0)≡ f_0). From this we wish to deduce that f^𝒦∈ C^0L^1 is Cauchy in C^0([T_m-1,T_m];L^1) as 𝒦∞. Before we proceed with the induction we will need some properties of the Lagrangian flow ỹ_t^𝒦 on the interval t∈[T_m-1,T_m]. Fix any (k,m,i,n) ∈𝒩, which fixes k∈ℕ, m∈{1,...,k}, i∈{1,2}, n∈{1,...,2^⌊ k/2⌋}. Recall the definition (<ref>) of T_m and T_m-1, where we set T_0=0 if necessary. By (<ref>), we have (it may be helpful to consider the illustration (<ref>)) T_m-1 < T_(k,m,i,n), T_(k,m,i,n) + 3·2^-k≤ T_m, and also for each 𝒫 = (k',m',i',n') >_lex (k,m,i,n), which fixes k'∈ℕ, m' ∈{1,...,k'}, T_𝒫 + 3·2^-k'≤ T_(k,m,i,n) if m' ≤ m, T_m < T_𝒫 if m<m'. Therefore, by the piecewise structure (<ref>), for each t ∈ [T_(k,m,i,n),T_m], and for each 𝒦' >_lex𝒦≥_lex (k,m,i,n), we have that ỹ_t^𝒦'∘(ỹ_T_(k,m,i,n)^𝒦')^-1 = ỹ_t^𝒦∘(ỹ_T_(k,m,i,n)^𝒦)^-1, is unchanged. We shall need the inverse of these Lagrangian maps, ỹ_T_(k,m,i,n)^𝒦'∘(ỹ_t^𝒦')^-1 = ỹ_T_(k,m,i,n)^𝒦∘(ỹ_t^𝒦)^-1. [For k'<k, y_t^(i',k',n';L_𝒫) may not preserve the squares lattice of widths 2^1-k in (<ref>).]Next, for each 𝒫 = (k',m',i',n') ∈𝒩 with k' < k, which fixes k'∈ℕ, m' ∈{1,...,k'}, we have that T_𝒫 +3· 2^-k'≤ T_m-1 if m' < m, T_(k,m,i,n) < T_𝒫 if m ≤ m'. Therefore, we may apply (<ref>) and the piecewise structure (<ref>) to show the following. For any r, r' ∈{1,...,2^k-1}, denote the intervals J=[(r-1)2^1-k,r 2^1-k], J'=[(r'-1)2^1-k,r' 2^1-k]. For all 𝒦∈𝒩, for all t ∈ [T_m-1,T_(k,m,i,n)], we see that the Lagrangian-flow ỹ_t^𝒦∘(ỹ_T_m-1^𝒦)^-1 preserves the square lattice ỹ_t^𝒦∘(ỹ_T_m-1^𝒦)^-1 : J × J' ↔ J × J'. We are now ready to proceed with the induction. Recall we have assumed that for some m∈ℕ, f^𝒦(·, T_m-1)∈ L^1(𝕋^2) is Cauchy in L^1(𝕋^2) as 𝒦∞. We wish to deduce f^𝒦∈ C^0L^1 is Cauchy in C^0([T_m-1,T_m];L^1(𝕋^2)) as 𝒦∞. Denote by f_m-1∈ L^1(𝕋^2) the limit f^𝒦(·, T_m-1) f_m-1 in L^1(𝕋^2). Note, by (<ref>), for all 𝒦∈𝒩, and for all t ∈ [T_m-1,T_m], we may write f^𝒦(·, t) = f^𝒦(·, T_m-1) ∘ỹ_T_m-1^𝒦∘(ỹ_t^𝒦)^-1. Fix some ϵ > 0. Let ϕ_ϵ∈ C^∞(𝕋^2) be a smooth approximation of f_m-1 in L^1(𝕋^2) so that ϕ_ϵ - f_m-1_L^1(𝕋^2)≤ϵ. Fixing also some k ∈ℕ, let 𝒦∈𝒩 be large enough that we have 𝒦≥_lex (k,m,1,1), and that for all 𝒦'≥_lex𝒦, f^𝒦'(·,T_m-1) - f_m-1_L^1(𝕋^2)≤ϵ. Then, it also follows that f^𝒦'(·, T_m-1) - ϕ_ϵ_L^1(𝕋^2)≤ 2ϵ. Therefore, by (<ref>), and since Lagrangian flows are Lebesgue-measure preserving, for any 𝒦' ≥_lex𝒦, and for all t ∈ [T_m-1,T_m], we have that f^𝒦'(·, t) - f^𝒦(·, t) _L^1(𝕋^2)≤ 4ϵ + ϕ_ϵ∘ỹ_T_m-1^𝒦'∘(ỹ_t^𝒦')^-1 - ϕ_ϵ∘ỹ_T_m-1^𝒦∘(ỹ_t^𝒦)^-1_L^1(𝕋^2). We now split into two cases, t ∈ [T_m-1,T_(k,m,1,1)], and t ∈ [T_(k,m,1,1),T_m]. We first consider the former. For the square lattice of widths 2^1-k, J× J'⊂𝕋^2 defined in (<ref>), we have for all x,y∈ J × J', \|ϕ_ϵ(x) - ϕ_ϵ(y) \| ≤∇ϕ_ϵ_L^∞(𝕋^2) \|x-y\| ≤√(2)∇ϕ_ϵ_L^∞(𝕋^2) 2^1-k. So by (<ref>), for all t ∈ [T_m-1,T_(k,m,1,1)], and for all 𝒦'≥_lex𝒦, we have that ϕ_ϵ∘ỹ_T_m-1^𝒦'∘(ỹ_t^𝒦')^-1 - ϕ_ϵ∘ỹ_T_m-1^𝒦∘(ỹ_t^𝒦)^-1_L^1(𝕋^2)≤√(2)∇ϕ_ϵ_L^∞(𝕋^2)2^1-k. Next, by (<ref>), for all t ∈ [T_(k,m,1,1),T_m], we have that ϕ_ϵ∘ỹ_T_m-1^𝒦'∘(ỹ_t^𝒦')^-1 - ϕ_ϵ∘ỹ_T_m-1^𝒦∘(ỹ_t^𝒦)^-1_L^1(𝕋^2) [t] = ϕ_ϵ∘ỹ_T_m-1^𝒦'∘(ỹ_T_(k,m,1,1)^𝒦')^-1∘ỹ_T_(k,m,1,1)^𝒦∘(ỹ_t^𝒦)^-1 - ϕ_ϵ∘ỹ_T_m-1^𝒦∘(ỹ_t^𝒦)^-1_L^1(𝕋^2) = ϕ_ϵ∘ỹ_T_m-1^𝒦'∘(ỹ_T_(k,m,1,1)^𝒦')^-1 - ϕ_ϵ∘ỹ_T_m-1^𝒦∘(ỹ_T_(k,m,1,1)^𝒦)^-1_L^1(𝕋^2), where the last line is already bounded in (<ref>), by √(2)∇ϕ_ϵ_L^∞(𝕋^2)2^1-k. Putting everything together we see that f^𝒦' - f^𝒦_L^∞([T_m-1,T_m];L^1(𝕋^2))≤ 4ϵ + √(2)∇ϕ_ϵ_L^∞(𝕋^2)2^1-k. ∇ϕ_ϵ_L^∞(𝕋^2) depends only on ϵ and f_m-1, and in particular not on k. Thus for k sufficiently large, i.e. for 𝒦',𝒦∈ (𝒩,<_lex) sufficiently large, we can make the right hand side arbitrarily small. Therefore f^𝒦 is Cauchy in L^∞([T_m-1,T_m];L^1(𝕋^2)) as 𝒦∞, as required. This completes the induction. Observe that T_m 42, and so we have proven that f^𝒦 converges in C^0([0,42-ϵ];L^1) as 𝒦∞, for any ϵ > 0. f^𝒦 are Lagrangian (and weak) solutions to (<ref>) along u_𝒦^{L_𝒫}_𝒫∈𝒩 with initial data f_0. Therefore, since f_0 ∈ L^∞(𝕋^2), f^𝒦 are uniformly bounded in L^∞ L^∞. Moreover, by Definition <ref>, u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦(·, t) is eventually constant, and equal to u_∞^{L_𝒫}_𝒫∈𝒩(·, t) as 𝒦∞ for each t ∈ [0,50], and are uniformly bounded in L^∞ L^∞. Denote by f̅ a weak- limit point of {f^𝒦}_𝒦∈𝒩 in L^∞([0,50];L^∞), that is there exists a subsequence 𝒦_n ∞ with f^𝒦_nf̅ in weak-* L^∞ L^∞. Then for all ϕ∈ C_c^∞(𝕋^2 × [0,50)), we see that ∫_𝕋^2×[0,50]f̅(∂ϕ/∂ t + u_∞^{L_𝒫}_𝒫∈𝒩·∇ϕ) dxdt = lim_n→∞∫_𝕋^2×[0,50] f^𝒦_n(∂ϕ/∂ t + u_∞^{L_𝒫}_𝒫∈𝒩·∇ϕ) dxdt = lim_n→∞∫_𝕋^2×[0,50] f^𝒦_n(∂ϕ/∂ t + u_𝒦_n^{L_𝒫}_𝒫∈𝒩_𝒦_n·∇ϕ) dxdt = - ∫_𝕋^2 f_0 ϕ_0 dx. Therefore, f̅ is a weak solution to (<ref>) along u_∞^{L_𝒫}_𝒫∈𝒩 with initial data f_0. Moreover, it is bounded in L^∞ L^∞ and so by Theorem <ref> the limit is in C_weak-^0([0,50];L^∞). However, u_∞^{L_𝒫}_𝒫∈𝒩(·, t) ≡ 0 for each t ∈ [42,50] and so (say by (<ref>)), for all t∈[42,50], f̅(·, t) is determined by f̅(·, s) for s∈[0,42). However, we have already shown 𝒦→∞limf^𝒦 converges in C^0([0,42-ϵ];L^1) for any ϵ > 0. Therefore, the limit f̅ is unique. Assume then that 𝒦→∞lim f^𝒦 does not converge in weak- L^∞([0,50];L^∞). Then by the uniform bound in L^∞([0,50];L^∞) there exist at least two limit points, contradicting uniqueness. Denote now the limit by f^∞, that is f^𝒦f^∞ in weak-* L^∞([0,50];L^∞) and strongly in C^0([0,42-ϵ];L^p) for all ϵ > 0. We are left to show convergence in L^p([0,42];L^p) and C^0([0,42-ϵ];L^p) for all p ∈ [1,∞), and all ϵ >0. The later of these follows by interpolation between convergence in C^0([0,42-ϵ];L^1) and the existing uniform bound in L^∞ L^∞. When combined with the uniform bound in L^2([0,42];L^2) this further implies convergence of the norm f^𝒦_L^2([0,42];L^2). Convergence in L^2([0,42];L^2) then follows from the already proved weak convergence in L^2([0,42];L^2). The analogous result for p∈[1,2) follows from compactness of the domain 𝕋^2 × [0,42]. Moreover, the convergence for p ∈ (2,∞) follows by interpolation with the existing uniform bound in L^∞([0,42]; L^∞). Finally, we show the mixing formula (<ref>). For K∈ℕ denote by f^K = f^(K,K,2,2^⌊ K/2⌋), 𝒩_K = 𝒩_(K,K,2,2^⌊ K/2⌋), and by ỹ_t^K = ỹ_t^(K,K,2,2^⌊ K/2⌋). By (<ref>), observe that (k,m,i,n) ∈𝒩_K if and only if k ≤ K, and moreover (K,K,2,2^⌊ K/2⌋) ∞ as K→∞. Fix k,m∈ℕ with k ≤ K, m∈{1,...,k}. For all i ∈{1,2}, for all n∈{1,...,2^⌊ k/2⌋}, we see that (k,m,i,n) ∈𝒩_K, and so consider the maps ỹ_T_(k,m,i,n)+3·2^-k^K∘(ỹ_T_(k,m,i,n)^K)^-1. By (<ref>), (<ref>), (<ref>), these maps commute for all i ∈{1,2}, for all n∈{1,...,2^⌊ k/2⌋}. Moreover, by (<ref>), (<ref>) (illustrated in (<ref>)), their composition is equal to ỹ_T_(k,m,1,1)+3·2^-k^K ∘(ỹ_T_(k,m,2,2^⌊ k/2 ⌋)^K)^-1. Furthermore, we have the following expression for this map in terms of binary expansions. For x=(x_1,x_2)∈𝕋^2 denote by (x'_1,x'_2)=ỹ_T_(k,m,1,1)+3·2^-k^K ∘(ỹ_T_(k,m,2,2^⌊ k/2 ⌋)^K)^-1(x). For a.e. x∈𝕋^2, for all j∈{1,2}, x'_j has swapping the kth and (k+1)th binary digits of x_j, illustrated by the map Y_k below. For the convenience of the reader we use colour to denote swaps in the binary expansion. Following the notation for binary expansions introduced in (<ref>), we have that for j∈{1,2} the coordinate, and for l∈ℕ the binary digit, (ỹ_T_(k,m,1,1)+3·2^-k^K ∘(ỹ_T_(k,m,2,2^⌊ k/2 ⌋)^K)^-1(x))_j = ( [baseline=(n1.base)] backgroundbackground,main[scale=1/4] [rectangle] (n1) at (0,0) 0 . x_j,1; [rectangle] (n5) [right=0mm of n1.south east, anchor=south west, yshift=2.5pt] ...; [rectangle] (n7) [right=0mm of n5.south east, anchor=south west, yshift=-2.5pt] x_j,k; [rectangle] (n8) [right=-1mm of n7.south east, anchor=south west] x_j,k+1; [rectangle] (n9) [right=0mm of n8.south east, anchor=south west, yshift=2.5pt] ...; [<->] ([xshift=10mm] n8.north west) arc (0:180:15mm); [rectangle,text=red] (l0) at ([xshift=-2.5mm, yshift=25mm] n8.north west) Y_k; ), (ỹ_T_(k,m,1,1)+3·2^-k^K ∘(ỹ_T_(k,m,2,2^⌊ k/2 ⌋)^K)^-1(x))_j,l = x_j,k+1 if l=k, x_j,k if l=k+1, x_j,l otherwise. Next, we fix m ∈{1,...,K}, and use (<ref>), (<ref>), (<ref>), (illustrated in (<ref>)), to piece together ỹ_T_(k,m,1,1)+3·2^-k^K ∘(ỹ_T_(k,m,2,2^⌊ k/2 ⌋)^K)^-1 for k ∈{m,m+1,...}. From this, we deduce that for a.e. x∈𝕋^2, for all m ∈{1,...,K}, and for j∈{1,2} the coordinate, and l∈ℕ the binary digit, we have that (ỹ_T_m^K∘(ỹ_T_m-1^K)^-1 (x))_j = ( [baseline=(n1.base),scale=1] backgroundbackground,main[scale=1/4] [rectangle] (n1) 0 . x_j,1; [rectangle] (n2) [right=0mm of n1.south east, anchor=south west, yshift=2.5pt] ...; [rectangle] (n3) [right=0mm of n2.south east, anchor=south west, yshift=-2.5pt] x_j,m; [rectangle] (n5) [right=3mm of n3.south east, anchor=south west, yshift=2.5pt] ...; [rectangle] (n7) [right=3mm of n5.south east, anchor=south west, yshift=-2.5pt] x_j,K; [rectangle] (n8) [right=-1mm of n7.south east, anchor=south west] x_j,K+1; [rectangle] (n9) [right=0mm of n8.south east, anchor=south west, yshift=2.5pt] ...; [->] ([xshift=10mm] n8.north west) arc (0:180:10mm); [->] ([xshift=-20mm] n8.north west) arc (0:180:10mm); [->] ([xshift=-50mm] n8.north west) arc (0:180:10mm); [->] ([xshift=-80mm] n8.north west) arc (0:180:10mm); [rectangle,text=red] (l0) at ([xshift=0mm, yshift=20mm] n8.north west) Y_K; [rectangle,text=red] (l1) at ([xshift=-30mm, yshift=20mm] n8.north west) Y_K-1; [rectangle,text=red] (l1) at ([xshift=-60mm, yshift=20mm] n8.north west) ...; [rectangle,text=red] (l2) at ([xshift=-90mm, yshift=20mm] n8.north west) Y_m; ), (ỹ_T_m^K∘(ỹ_T_m-1^K)^-1 (x))_j,l = x_j,K+1 if l=m, x_j,l-1 if m < l≤ K+1, x_j,l otherwise. Therefore, since ỹ_T_0^K=Id, we see that for a.e. x∈𝕋^2, for all m ∈{1,...,K}, and for j∈{1,2} the coordinate, and l∈ℕ the binary digit, ((ỹ_T_m^K)^-1(x))_j = ( [baseline=(n1.base),scale=1] backgroundbackground,main[scale=1/4] [rectangle] (n1) 0 . x_j,m+1; [rectangle] (n2) [right=-1mm of n1.south east, anchor=south west, yshift=0pt] x_j,m+2; [rectangle] (n3) [right=0mm of n2.south east, anchor=south west, yshift=2.5pt] ...; [rectangle] (n5) [right=0mm of n3.south east, anchor=south west, yshift=-2.5pt] x_j,K+1; [rectangle] (n6) [right=-1mm of n5.south east, anchor=south west, yshift=0pt] x_j,m; [rectangle] (n7) [right=-1mm of n6.south east, anchor=south west, yshift=0pt] x_j,m-1; [rectangle] (n8) [right=0mm of n7.south east, anchor=south west, yshift=2.5pt] ...; [rectangle] (n10) [right=0mm of n8.south east, anchor=south west, yshift=-2.5pt] x_j,1; [rectangle] (n11) [right=-1mm of n10.south east, anchor=south west, yshift=0pt] x_j,K+2; [rectangle] (n12) [right=-1mm of n11.south east, anchor=south west, yshift=0pt] x_j,K+3; [rectangle] (n12) [right=0mm of n12.south east, anchor=south west, yshift=2.5pt] ...; ), ((ỹ_T_m^K)^-1(x))_j,l = x_j,m+l if l ≤ K+1-m, x_j,K+2-l if K+2-m ≤ l ≤ K+1, x_j,l if l ≥ K+2. For each m∈ℕ define now the map z_m:𝕋^2 →𝕋^2 by, for all x∈𝕋^2, for j∈{1,2} to coordinate, and l∈ℕ the binary digit, (z_m(x))_j = ( [baseline=(n1.base),scale=1] backgroundbackground,main[scale=1/4] [rectangle] (n1) 0 . x_j,m+1; [rectangle] (n2) [right=-1mm of n1.south east, anchor=south west, yshift=0pt] x_j,m+2; [rectangle] (n3) [right=-1mm of n2.south east, anchor=south west, yshift=0pt] x_j,m+3; [rectangle] (n4) [right=0mm of n3.south east, anchor=south west, yshift=2.5pt] ...; ), (z_m(x))_j,l = x_j,m+l for all l∈ℕ. This map is an approximation of (<ref>). Notice that for the initial data f_0 ∈ L^∞(𝕋^2), f_0∘ z_m contains 4^m scaled copies of f_0, one on each tile in the square lattice with widths 2^-m. For any r, r' ∈{1,...,2^k-1}, define the intervals J=[(r-1)2^1-k,r 2^1-k], J'=[(r'-1)2^1-k,r' 2^1-k]. Then z_m is a bijection from this tile J× J' to 𝕋^2, that is z_m:J× J'↔𝕋^2. Moreover, for dμ the Lebesgue-measure on 𝕋^2, .z_m\|_J× J'∘ dμ = 4^-m dμ. In particular z_m:𝕋^2 →𝕋^2 is measure preserving, and ∫_J× J' f_0∘ z_m dx = 1/4^m∫_𝕋^2f_0 dx. Therefore, we deduce that f_0∘ z_m m→∞∫_𝕋^2f_0(y) dy, with the above convergence in weak-* L^∞(𝕋^2). By (<ref>), for all K∈ℕ, m∈{1,...,L}, we may write f^K(·, T_m) = f_0∘(ỹ_T_m^K)^-1. We now wish to approximate f^K(·, T_m) by f_0∘ z_m to prove the mixing formula (<ref>). Fix some ϵ > 0. Take a smooth approximation ϕ_ϵ∈ C^∞(𝕋^2) of f_0 in L^1(𝕋^2), so that ϕ_ϵ - f_0_L^1(𝕋^2)≤ϵ. Since (ỹ_T_m^K)^-1, z_m are measure preserving, we see that also ϕ_ϵ∘(ỹ_T_m^K)^-1 - f_0 ∘(ỹ_T_m^K)^-1_L^1(𝕋^2)≤ϵ, ϕ_ϵ∘ z_m - f_0 ∘ z_m_L^1(𝕋^2)≤ϵ. Next, consider the square lattice with widths 2^m-K-1, that is for r, r' ∈{1,...,2^1+K-m} define J=[(r-1)2^m-K-1,r2^m-K-1], and J'=[(r-1)2^m-K-1,r2^m-K-1]. Then, by (<ref>), we see that for a.e. x∈𝕋^2, (ỹ_T_m^K)^-1(x) ∈ J× J' if and only if z_m (x) ∈ J × J'. But then by the Lipschitz bound on ϕ_ϵ, \|ϕ_ϵ∘(ỹ_T_m^K)^-1(x) - ϕ_ϵ∘ z_m (x)\| ≤√(2)∇ϕ_ϵ_L^∞ 2^m-K-1. Therefore, ϕ_ϵ∘(ỹ_T_m^K)^-1 - ϕ_ϵ∘ z_m _L^∞(𝕋^2)≤√(2)∇ϕ_ϵ_L^∞ 2^m-K-1. In light of (<ref>), we deduce that f^K(·,T_m) - f_0∘ z_m _L^1(𝕋^2)≤ 2ϵ + √(2)∇ϕ_ϵ_L^∞ 2^m-K-1. Recall that f^Kf^∞ strongly in C^0([0,42-ϵ];L^1(𝕋^2)) for all ϵ > 0. Therefore, for m ∈ℕ fixed, f^K(·, T_m)f^∞(·, T_m) strongly in L^1(𝕋^2). [That is f^∞(·, T_m) contains 4^m scaled copies of f_0, one on each tile in the square lattice with widths 2^-m.]So, by (<ref>), for all m∈ℕ, we see that f^∞(·, T_m) = f_0∘ z_m. Hence, by (<ref>), f^∞(·, T_m) m→∞∫_𝕋^2f_0(y) dy converges in weak-* L^∞(𝕋^2). Since f^∞∈ C_weak-^0L^∞, with f^∞(·, t) independent of t∈ [42,50], and T_m 42, we have proved the mixing formula (<ref>). Finally, it remains to find a suitably fast growing sequence {L_𝒫}_𝒫∈𝒩⊂ℕ in Definition <ref>. Then to apply Theorem <ref> to the sequence of vector fields u_𝒦^{L_𝒫}_𝒫∈𝒩_𝒦, indexed by 𝒦∈𝒩. This will allow us to approximate the vanishing viscosity limit to (<ref>) along u_∞^{L_𝒫}_𝒫∈𝒩. Additionally, by subsequently time-reversing the vector field u_∞^{L_𝒫}_𝒫∈𝒩 on the time interval [50,100], we will obtain the inadmissible behaviour below. There exists a divergence-free vector field u ∈ L^∞([0,100];L^∞(𝕋^2;ℝ^2)), such that for any initial data f_0∈ L^∞(𝕋^2), and for f^ν the unique solution to (<ref>) along u with initial data f_0, one has f^νf, with the above convergence in weak- L^∞([0,100];L^∞(𝕋^2)), strong in L^p([0,42];L^p(𝕋^2)), C^0([0,42-ϵ];L^p(𝕋^2)), L^p([58,100];L^p(𝕋^2)), and C^0([58+ϵ,100];L^p(𝕋^2)) for all p ∈ [1,∞), and all ϵ >0. The limit function f ∈ C_weak-^0([0,100];L^∞(𝕋^2)) is a weak solution to (<ref>) along u with initial data f_0. Moreover, for all t ∈ [42,58] f(·, t) ≡∫_𝕋^2f_0(y) dy, is perfectly mixed to its spatial average. Furthermore, for all t ∈ [0,100], f(·, t)=f(·,100-t) and in particular, f(·, 100) = f_0, is perfectly unmixed. In particular, if f_0 is not constant, any L^p(𝕋^2) norms of f(·, t) (for p∈(1,∞]) increase after t=58, contrary to the entropy-admissibility criterion of Dafermos in <cit.>. Recall the language and notation introduced in Definitions <ref>, <ref>, as well as Definitions <ref>, <ref> of renormalised and Lagrangian solutions to (<ref>). For each n ∈ℕ we denote by 𝒦_n ∈𝒩 the isomorphism between the well orders (ℕ, <) and (𝒩,<_lex). That is n↦𝒦_n is a bijection from ℕ to 𝒩, and for all n_1, n_2 ∈ℕ, we have that n_1 < n_2 if and only if 𝒦_n_1 <_lex𝒦_n_2. For an infinite sequence {L_𝒫}_𝒫∈𝒩⊂ℕ with L_(k,m,i,q)≥ k+1 for all (k,m,i,q) ∈𝒩, we define u_n:𝕋^2 × [0,100] →ℝ^2 by u_n(x,t) = u_𝒦_n^{L_𝒫}_𝒫∈𝒩_𝒦_n(x,t) if t ∈ [0,50], -u_𝒦_n^{L_𝒫}_𝒫∈𝒩_𝒦_n(x,100-t) if t ∈ [50,100]. u_n ∈ L^∞ L^∞ is then bounded by 1. Let f_0 ∈ L^∞(𝕋^2), then for f^𝒦_n given by Proposition <ref>, we have the following Lagrangian solution f^n∈ C^0L^1 to (<ref>) along u_n with initial data f_0, f^n(x,t) = f^𝒦_n(x,t) if t ∈ [0,50], f^𝒦_n(x,100-t) if t ∈ [50,100]. We now apply Theorem <ref>. Let d_ be a metric inducing the weak-* topology on {u∈ L^∞( [0,100];L^∞(𝕋^2;ℝ^2)) : u_L^∞ L^∞≤ 1 }. Let f_0 ∈ L^∞(𝕋^2), and denote for each n∈ℕ, ν > 0, by f^n,ν, respectively f^n, the unique weak solution to (<ref>), respectively (<ref>), along u_n with initial data f_0. Moreover denote by f^∞,ν the unique weak solution to (<ref>) along u_∞^{L_𝒫}_𝒫∈𝒩 with initial data f_0. Then by Theorem <ref>, * For all n ∈ℕ there exists ν_n > 0, ϵ_n > 0 depending only on {L_𝒫}_𝒫∈𝒩_𝒦_n (and in particular not on f_0), with ν_n0 monotonically, such that the following hold true: * For all p∈[1,∞) sup_ν≤ν_nf^n,ν-f^n_L^∞ L^p0, * If d_(u_n+1,u_n) ≤ϵ_n for all n∈ℕ, then for all p ∈ [1,∞) sup_ν_n ≤ν≤ν_1f^n,ν-f^∞,ν_L^∞ L^p0. We now choose {L_𝒫}_𝒫∈𝒩. Proceeding by induction on N∈ℕ, assume there exists a sequence {L_𝒫}_𝒫∈𝒩_𝒦_N⊂ℕ, so that with {u_n}_n=1^N ⊂ L^∞([0,100];L^∞(𝕋^2;ℝ^2)) given by (<ref>), and {ϵ_n}_n=1^N⊂ (0,∞) given by (<ref>), we have that for all n ∈{1,...,N-1}, d_(u_n+1,u_n) ≤ϵ_n. We next pick L_𝒦_N+1∈ℕ. Note that for any L_𝒦_N+1∈ℕ we obtain by (<ref>) a vector field u_N+1. By (<ref>) and Definition <ref>, we see that as L_𝒦_N+1→∞, d_(u_N+1,u_N)0, and so we may pick L_𝒦_N+1 large enough that d_(u_N+1,u_n) ≤ϵ_N. This completes the inductive step. That is, there exists a sequence {L_𝒫}_𝒫∈𝒩⊂ℕ such that for all n∈ℕ, with u_n given by (<ref>), we have that d_*(u_n+1,u_n) ≤ϵ_n. Therefore, (<ref>), and (<ref>) are satisfied. Next, for all n ∈ℕ, and for all p∈[1,∞), we see that sup_ν_n+1≤ν≤ν_nf^∞,ν-f^n_L^∞ L^p≤ sup_ν_n+1≤ν≤ν_nf^∞,ν-f^n+1,ν_L^∞ L^p + sup_ν_n+1≤ν≤ν_nf^n+1,ν-f^n,ν_L^∞ L^p + sup_ν_n+1≤ν≤ν_nf^n,ν-f^n_L^∞ L^p, Therefore, by (<ref>), (<ref>), if for all p∈[1,∞), sup_ν_n+1≤ν≤ν_nf^n+1,ν-f^n,ν_L^∞ L^p0, then also for all p∈[1,∞), sup_ν_n+1≤ν≤ν_nf^∞,ν-f^n_L^∞ L^p0. Therefore, the statement of Theorem <ref> follows from Proposition <ref> applied to (<ref>). Notice that for all p∈(1,∞), (<ref>) follows from the case p=1 by interpolation with the existing uniform bound in L^∞ L^∞. We therefore only prove (<ref>) for p=1. Let n∈ℕ. Express 𝒦_n+1∈𝒩 as 𝒦_n+1 = (k,m,i,q) with k∈ℕ, m∈{1,...,k}, i∈{1,2}, and q∈{1,...,2^⌊ k/2⌋}. Then, by the expression (<ref>), and Definitions <ref>, <ref>, we have that for all t ∉ [T_𝒦_n+1,T_𝒦_n+1+3·2^-k] ∪ [100-T_𝒦_n+1-3·2^-k,100-T_𝒦_n+1], that u_n+1(·, t) = u_n(·, t). Therefore, for all ν>0, f^n+1,ν-f^n,ν∈ C^0L^1 is a solution to (<ref>) on the time interval [0,T_𝒦_n+1] along the same u_n with initial data 0∈ L^∞(𝕋^2). Similarly, on the time interval [T_𝒦_n+1+3·2^-k,100-T_𝒦_n+1-3·2^-k] with initial data (f^n+1,ν-f^n,ν)(·,T_𝒦_n+1+3·2^-k)∈ L^∞(𝕋^2). Similarly, on the time interval [100-T_𝒦_n+1,100] with initial data (f^n+1,ν-f^n,ν)(·,100-T_𝒦_n+1)∈ L^∞(𝕋^2). Applying the L^p-Inequality (<ref>) to these three cases shows that f^n+1,ν-f^n,ν_L^∞([0,T_𝒦_n+1]; L^1(𝕋^2)) = 0, f^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1+3·2^-k,100-T_𝒦_n+1-3·2^-k]; L^1(𝕋^2))≤(f^n+1,ν-f^n,ν)(·, T_𝒦_n+1+3·2^-k)_L^1(𝕋^2), f^n+1,ν-f^n,ν_L^∞([100-T_𝒦_n+1,100]; L^1(𝕋^2))≤(f^n+1,ν-f^n,ν)(·, 100-T_𝒦_n+1)_L^1(𝕋^2). Therefore, using the continuity f^n+1,ν-f^n,ν∈ C^0L^1, we have the bound f^n+1,ν-f^n,ν_L^∞ L^1 ≤f^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1,T_𝒦_n+1+3·2^-k]∩[100-T_𝒦_n+1-3·2^-k,100-T_𝒦_n+1]; L^1(𝕋^2)). Next, for k∈ℕ given in terms of n∈ℕ by the expression 𝒦_n+1=(k,m,i,q), we aim to prove the bound, for all n∈ℕ, and for all ν>0, f^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1,T_𝒦_n+1+3·2^-k];L^1(𝕋^2)) ≤ 2f_0 _L^∞ 2^-⌊ k/2 ⌋ + Cf_0_L^∞√(ν 2^-k), with C>0 independent of n∈ℕ and ν>0. In the expression 𝒦_n+1 = (k,m,i,q), we assume, without loss of generality, that i=1. The case i=2 then follows the same argument with the coordinates reversed. Let J=[(q-1)2^-⌊ k/2 ⌋,q2^-⌊ k/2 ⌋]⊂𝕋. Then, by (<ref>), for all t∈[0,3·2^-k], and for all x ∉ J×𝕋, we see that the binary swap vector field u^(i,k,q,L_𝒦_n+1)(x,t)=0. Therefore, for all t ∈ [T_𝒦_n+1,T_𝒦_n+1+3·2^-k]∩[100-T_𝒦_n+1-3·2^-k,100-T_𝒦_n+1], and for all x ∉ J×𝕋, we have that u_n+1(x, t), u_n(x, t) = 0. We first tackle the case t∈[T_𝒦_n+1,T_𝒦_n+1+3·2^-k]. Let Ω=[0,1-2^-⌊ k/2 ⌋]×𝕋⊂𝕋^2, a periodic strip. Then, on the spatio-temporal domain ((x_1,x_2),t)∈Ω×[0,3·2^-k], we see that (f^n+1,ν-f^n,ν)((x_1+q2^-⌊ k/2 ⌋,x_2),t+T_𝒦_n+1) is a solution to the heat equation, and by (<ref>) has the initial data 0. However, its boundary data is unknown, so we will construct a second solution to the heat equation on the same domain, with initial data 0, which is an upper bound on the boundary, and then apply the maximum principle. We have the a-priori bound f^n+1,ν-f^n,ν_L^∞ L^∞≤ 2f_0_L^∞. Therefore, we may apply hypo-ellipticity for the heat equation (see for example Section 4.4 in <cit.>) to deduce that (f^n+1,ν-f^n,ν)((x_1+q2^-⌊ k/2 ⌋,x_2),t+T_𝒦_n+1) is smooth on the interior of the domain Ω×[0,3·2^-k]. We introduce erf(x) = ∫_-∞^x e^-y^2 dy, C_0=erf(0), a = 1-2^-⌊ k/2 ⌋. We define the following solution to the heat equation. θ:Ω×[0,3·2^-k]→ℝ, θ((x_1,x_2),t) = 2f_0_L^∞C_0^-1(erf(-x_1/√(4ν t)) + erf(x_1-a/√(4ν t))). Observe that θ has initial data 0, and its value on the boundary ∂Ω = {0,a}×𝕋 is greater than or equal to 2f_0_L^∞. Also for all (x_1,x_2)∈Ω, we have that θ((x_1,x_2),t) is an increasing function of t∈[0,3·2^-k]. Therefore, by maximum principle we have the following point-wise bound on the interior of the domain, for all (x_1,x_2) ∈ (0,1-2^-⌊ k/2 ⌋)×𝕋, and for all t∈(0,3·2^-k), \|(f^n+1,ν-f^n,ν)((x_1+q2^-⌊ k/2 ⌋,x_2),t+T_𝒦_n+1)\| ≤θ((x_1,x_2),3·2^-k). Hence, f^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1,T_𝒦_n+1+3·2^-k];L^1((𝕋∖ J)×𝕋)) ≤ 4f_0_L^∞C_0^-1∫_0^a erf(-x/√(4ν· 3· 2^-k)) dx. After changing variables, for C=8√(3)C_0^-1∫_0^∞erf(-x) dx, we have that f^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1,T_𝒦_n+1+3·2^-k];L^1((𝕋∖ J)×𝕋)) ≤ Cf_0_L^∞√(ν 2^-k). Since additionally J×𝕋 has Lebesgue-measure 2^-⌊ k/2 ⌋, and we have the a-priori bound f^n+1,ν-f^n,ν_L^∞ L^∞≤ 2f_0_L^∞, we deduce (<ref>). Next, since k in determined by n through the map n↦𝒦_n+1=(k,i,m,q), denote this now by k_n ∈ℕ. Since n↦𝒦_n+1 respects the well-orders (ℕ,<), (𝒩,<_lex), we see that k_n∞. Moreover, by (<ref>), we have that ν_n0. Therefore, by (<ref>), we see that sup_ν_n+1≤ν≤ν_nf^n+1,ν-f^n,ν_L^∞([T_𝒦_n+1,T_𝒦_n+1+3·2^-k_n];L^1(𝕋^2))0. By (<ref>), (<ref>), the proof of (<ref>) will then be complete if also sup_ν_n+1≤ν≤ν_nf^n+1,ν-f^n,ν_L^∞([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1];L^1(𝕋^2))0. To this end denote by g^n,ν∈ C^0([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1],L^1(𝕋^2)) the solution to (<ref>) along u_n on the time interval [100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1] with initial data f^n+1,ν(·,100-T_𝒦_n+1-3·2^-k_n)∈ L^∞(𝕋^2). Then, arguing as in (<ref>), we have the same bound. For all n∈ℕ, for all ν>0, f^n+1,ν-g^n,ν_L^∞([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1];L^1(𝕋^2)) ≤ 2f_0 _L^∞ 2^-⌊ k_n/2 ⌋ + Cf_0_L^∞√(ν 2^-k_n). Therefore, as before we deduce that sup_ν_n+1≤ν≤ν_nf^n+1,ν-g^n,ν_L^∞([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1];L^1(𝕋^2))0. By the L^p-Inequality (<ref>), and (<ref>), we have that, for all n∈ℕ, and for all ν>0, g^n,ν-f^n,ν_L^∞([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1];L^1(𝕋^2)) ≤(f^n+1,ν-f^n,ν)(·, T_𝒦_n+1+3·2^-k_n)_L^1(𝕋^2). So, by (<ref>), sup_ν_n+1≤ν≤ν_ng^n,ν-f^n,ν_L^∞([100-T_𝒦_n+1-3·2^-k_n,100-T_𝒦_n+1];L^1(𝕋^2))0. Therefore, by also (<ref>), we deduce (<ref>). § ACKNOWLEDGEMENTS Lucas Huysmans acknowledges support from the UK Engineering and Physical Sciences Research Council (EPSRC) under grant numbers EP/T517847/1 and EP/V52024X/1. The work of E.S.T. has benefited from the inspiring environment of the CRC 1114 “Scaling Cascades in Complex Systems”, Project Number 235221301, Project C06, funded by Deutsche Forschungsgemeinschaft (DFG). The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Mathematical aspects of turbulence: where do we stand?” where part of this work was undertaken and supported by EPSRC grant no. EP/K032208/1. apa
http://arxiv.org/abs/2307.03101v1	20230706161814	Contextual Affinity Distillation for Image Anomaly Detection	[ "Jie Zhang", "Masanori Suganuma", "Takayuki Okatani" ]	cs.CV	[ "cs.CV" ]	Contextual Affinity Distillation for Image Anomaly Detection Jie Zhang^1 Masanori Suganuma^1 Takayuki Okatani^1,2 ^1Graduate School of Information Sciences, Tohoku University ^2RIKEN Center for AIP {jzhang,suganuma,okatani}@vision.is.tohoku.ac.jp August 1, 2023 =================================================================================================================================================================================================================== empty Previous works on unsupervised industrial anomaly detection mainly focus on local structural anomalies such as cracks and color contamination. While achieving significantly high detection performance on this kind of anomaly, they are faced with logical anomalies that violate the long-range dependencies such as a normal object placed in the wrong position. In this paper, based on previous knowledge distillation works, we propose to use two students (local and global) to better mimic the teacher's behavior. The local student, which is used in previous studies mainly focuses on structural anomaly detection while the global student pays attention to logical anomalies. To further encourage the global student's learning to capture long-range dependencies, we design the global context condensing block (GCCB) and propose a contextual affinity loss for the student training and anomaly scoring. Experimental results show the proposed method doesn't need cumbersome training techniques and achieves a new state-of-the-art performance on the MVTec LOCO AD dataset. § INTRODUCTION The task of anomaly detection and localization aims to identify whether an image is normal or anomalous and localize the anomalies <cit.>. It has a wide range of real-world applications including industrial inspection of products <cit.>. As anomalous samples rarely appear in manufacturing product lines and the unpredictable nature of anomalies, most of the efforts are paid to unsupervised anomaly detection methods, in which we have only anomaly-free samples for training. Recent studies showed that using intermediate features of a deep pre-trained model is representative enough to achieve state-of-the-art performance <cit.>. Knowledge distillation <cit.> is a straightforward and effective way to achieve this goal. Recent knowledge distillation-based anomaly detection approaches <cit.> try to transfer the knowledge of normal samples from a teacher which is pre-trained on a large-scale natural image dataset, e.g. ImageNet <cit.> into a student model. The use of per-pixel <cit.> or local patch-based regression loss <cit.> further improves the fine-grained knowledge transfer and anomaly detection performance. The teacher model acts as a knowledgeable feature extractor that could extract representative feature embeddings for both normal and anomalous samples, while the student is trained exclusively on anomaly-free samples that it is expected to only mimic the teacher's behavior for normal features. During inference, the anomaly scores are derived from the discrepancy between student and teacher features. As anomalies could be of any size and at any abstract level, using multi-layer features from the teacher could better cover more types of anomalies. Previous unsupervised anomaly detection and localization datasets focus on concise scenes where each image consists of only one product object, e.g. a capsule or one kind of texture. In this case, most of the anomalies are local structural anomalies such as cracks and scratches. The above-mentioned methods are effective for this structural anomaly detection. However, in complex scenarios with global contextual constraints, logical anomalies violating long-range dependencies such as a normal object appearing in the wrong position or a missing object are likely to be identified as normal. Figure <ref> shows structural and logical anomaly examples from MVTec LOCO <cit.> dataset. The first image from the screw bag category contains contamination as a structural anomaly, while cereals missing in the breakfast box and two additional pushpins in one compartment are defined as logical anomalies. However, directly using deep high-semantic level features from a pre-trained model cannot address this issue well, as deeper features are more source domain biased <cit.> and could generalize to anomaly features <cit.>. To better detect both structural and logical anomalies, we propose to further use a global student in reverse distillation method <cit.> to comprise the dual-student knowledge distillation framework (DSKD). Unlike conventional ensemble methods <cit.> in which each model has the same role as each other, we explicitly divide the student models into two models: local and global students. The objective of the local student is to detect structural anomalies, and it is trained to reconstruct the low-level features of those of the teacher model. The global student is trained to capture global contextual information for detecting logical anomalies. Since the logical anomalies are harder to detect than the structural anomalies, we introduce a global context condensing block into the global student, aiming to capture global information from images effectively <cit.>. The training paradigm of our method is based on the reverse distillation <cit.> where the teacher acts as an encoder and the students play the role of decoders for the feature reconstruction. Moreover, we propose a contextual affinity loss to promote further capturing the global contextual information by the global student. Specifically, we compute the cosine similarity between each feature vector of the global student and all feature vectors of the teacher, and then the cosine similarity maps are converted into the probability distribution by a softmax function. We minimize the discrepancy between the probability distributions of the global student and the teacher. It should be noted that this differs from simply choosing several neighboring feature representations such as pair-wise distillation <cit.> in that such a pair-wise loss treats all features equally, but ours can capture important contextual information at the whole image level. We conduct extensive experiments on public unsupervised anomaly detection and localization datasets and achieve state-of-the-art performance. Our contributions are threefold: * We introduce a new dual-student knowledge distillation framework for anomaly detection. The local student aims for accurate local feature reconstruction and the global student focuses on global contextual information. The students play different roles, enabling the detection ability for both structural and logical anomalies. * We propose the global contextual condensing block and contextual affinity loss, further enforcing the global contextual learning ability. * We demonstrate the effectiveness of our proposed method and report new state-of-the-art performance. § RELATED WORKS We briefly review recent research on unsupervised anomaly detection as well as related knowledge distillation works on supervised dense prediction tasks. The recent works on anomaly detection could be classified into three prototypes: generative models, anomaly synthesis-based methods, and methods leveraging features extracted by pre-trained networks. Generative models aim to reconstruct normal samples from the encoded feature space. Autoencoders (AEs) and Gnenerative Adversarial Nets <cit.> are popularly used for sample reconstruction. These models are trained exclusively on normal images. Since the input image is encoded into a compact feature space to only keep the most useful information, the unseen anomalies are expected to be abandoned during inference and thus reconstruct the anomaly-free images for anomaly samples <cit.>. However, deep models could generalize well to anomaly patterns and fail to detect anomalies. To overcome this issue, normal representation searching <cit.> in the encoded continuous feature space, iterative reconstruction approaches <cit.> and memory-guied autoencoders <cit.> are proposed to limit the model's generalization ability. Anomaly synthesis-based methods <cit.> focus on addressing the issue of the lack of anomaly samples so as to train the models in a supervised manner. However, the detection ability is heavily affected by the synthesizing strategies, and their performance shows a strong bias to the synthesized kind of anomalies <cit.>. To generate more realistic anomalies, DSR <cit.> tries to generate near-in-distribution low-level anomalies from a vector-quantized feature space. However, it is still challenging for high semantic-level anomaly generation. There is also a lot of attention paid to employing pre-trained models to extract representative features from images. The kNN-based approaches <cit.> construct a feature gallery for normal representations and derive anomaly scores by computing the distances between input and its nearest neighbors in the feature space. They suffer from computational complexity <cit.> and can not utilize high-semantic level features well as they as more source-domain biased <cit.>. The knowledge distillation-based approaches try to transfer knowledge of normal samples to student networks. US <cit.> distills knowledge from a pre-trained teacher network to an ensemble of students for each patch scale. MKD <cit.> directly distills multi-level features into one compact student model. STFPM <cit.> uses a vector-wise cosine similarity loss for both student training and anomaly scoring. The reverse distillation <cit.> proposed the encoder-decoder architecture to distill the knowledge from a bottleneck feature space. These methods are capable of learning local features or patches but are likely to ignore global contextual constraints. GCAD <cit.> designed a two-branch framework based on US <cit.> for both structural and logical anomaly detection but it is still a two-step distillation framework where the teacher is trained with a deep pre-trained model and a large number of cropped image patches from ImageNet <cit.>. To insure training stability, multi-step training and skip connections with linearly decreased weights are added to the student. There are also some knowledge distillation methods applied to dense prediction tasks such as semantic segmentation <cit.> and object detection <cit.> trying to transfer the knowledge to a compact student network via fully exploring the rich information within the intermediate features from the teacher. MIMIC <cit.> samples features from feature regions for object detection. Pair-wise knowledge distillation <cit.> was proposed to distill the structured knowledge from the feature. Channel-wise knowledge distillation <cit.> converts the features of each channel into probability distributions leveraging the prior that the activations from each channel tend to encode specific scene categories. It is pointed out that strictly applying the per-pixel loss which means each pixel or correlation is treated equally may enforce overly strict constraints on the student model and lead to sub-optimal solutions <cit.>. However, the most important guidance for training the student model comes from the ground truth labels that are not available for unsupervised anomaly detection. Also, in conventional knowledge distillation applications where only the student model is deployed after training, the structured knowledge is computed or evaluated separately for each student and teacher feature map, while both the teacher and student are used for knowledge distillation-based anomaly detection methods. We distinguish our proposed contextual affinity loss from prior arts that the contextual affinity for student features is computed using both student and teacher features for better guidance and to make training stable. § PROPOSED METHOD §.§ Problem Formulation Given a set of anomaly-free training images 𝒮^t={I_1^t, ..., I_n_t^t} and a validation set 𝒮^v={I_1^v, ..., I_n_v^v} that consists of also anomaly-free images, we aim to detect if a test image from the test set 𝒮^q={I_1^q, ..., I_n_q^q} is anomalous or not, and also localize the defected area if it is anomalous. §.§ Overview of the Dual-student Framework As shown in Fig <ref>, our dual-student distillation framework consists of five parts: a deep neural network pre-trained on ImageNet as the fixed teacher T to extract multi-level representative features, a one-class bottleneck embedding module OCBE_loc for the local student, a local student decoder S_loc, a OCBE_glo for the global student that contains a global context condensing block GCCB, and a global student decoder S_glo. The OCBE_loc is designed for fusing multi-level features into a compact feature space followed by a local student decoder S_loc to reconstruct the feature representations, especially low-level features accurately. The first three modules compose the reverse distillation <cit.> which is effective for structural anomaly detection and we change nothing for it in order to preserve the accurate low-level feature reconstruction ability. To better capture global contextual correlations which we expect to have the benefit of logical anomaly detection, we additionally design the S_glo with a GCCB that can keep the most condensed contextual information, and the output is then decoded by the global student decoder S_glo. Since the teacher T is pre-trained on a large natural image dataset, it is expected to extract representative features for both normal and anomaly inputs. However, the two students are trained solely on anomaly-free samples, and they fail to mimic the teacher's behavior for either low-level structural anomalies or global logical anomalies during inference. The pixel-level anomaly scores are computed by comparing the decoded features from both students and the features extracted by the teacher. The local student is primarily responsible for structural anomaly detection and the global student pays more attention to the global contextual constraints. §.§ Local Knowledge Distillation Different from the conventional discriminative paradigm where the student is also a feature extractor, reverse distillation <cit.> is in a generative manner to reconstruct the features extracted by the teacher. The student decoder receives dense encoded features and decodes the features from high-semantic levels to low-semantic levels. The dense feature space is likely to abandon unseen anomalous feature representations at inference to encourage feature discrepancies for anomalies. We use the reverse distillation <cit.> method for accurate feature reconstruction and structural anomaly detection. Given an image I, the output of the first three residual stages of a pre-trained WideResNet50 <cit.> T extracts multi-layer intermediate features F_T^l ∈ℝ^h_l × w_l × c_l, where l ∈{1, 2, 3}. The OCBE_loc encodes the features into the embedding ϕ_loc. The local student S_loc then generates the corresponding feature maps F_S_loc^l from ϕ_loc. The student decoder has a symmetrical architecture with the teacher T while the input is high abstract feature representations and the down-sampling operations used in original ResNets <cit.> are replaced by up-samplings. The vector-wise cosine distance is used as the loss function for training the local student. A 2-D anomaly score map could be obtained at each layer scale M_loc^l = 1 - F_T^l · F_S_loc^l/F_T^l F_S_loc^l The final loss for training the local student is ℒ_loc = ∑_l=1^3{1/h_l · w_l∑_i=1^h_l · w_l M_loc^l } §.§ Global Contextual Affinity Distillation Although the encoder-decoder architecture naturally can keep the most important information and the multi-scale feature distillation paradigm could take both low-level information and high-level information into account, the student still cannot learn globally. Furthermore, the OCBE_loc fuses low-level features into the final embedding space, which is beneficial for accurate low-level feature reconstruction but decreases the global contextual learning ability. We design the global context condensing block to keep the most important global information as shown in Fig. <ref>. It is realized by compressing the high-semantic level feature F_T^3 into a one-dimensional feature space with g channels and then restoring it to the original feature size. The output ϕ_glo of GCCB is then decoded by a student S_glo that has the identical architecture as S_loc. To further encourage the global student to better learn the global contextual information, different from the per-pixel cosine similarity loss used for the local student, we propose the contextual affinity loss for the global student. To learn the local feature embedding f_S_loc, i^l, the local student S_loc can only learn from f_f_T, i^l, which is a benefit for accurately reconstruct local features. However, it fails to learn the contextual information from the whole image. For example, if a normal feature appears in the wrong position, the local student can't detect it as anomalous since the feature itself is a representation of a normal structure. We are inspired to propose the contextual affinity loss aiming to enable the student to learn a local feature f_S_glo, i^l from the whole feature map. For a feature vector f_T, i^l from a feature map extracted by the teacher F_T^l, we first compute the cosine similarity between f_T, i^l and all feature vectors to get a similarity list A_T, i^l =[a_i, 1^t, l, ..., a_i, h_l · w_l^t, l], where a_i, j^t, l=f_T, i^l · f_T, j^l/f_T, i^lf_T, j^l We define the contextual affinity for a feature vector in the whole feature map as the probability distribution P_T, i^l = [p_T, i^1, l, ..., p_T, i^h_l · w_l, l] where p_T, i^i, l = exp(a_i, j^t, l/𝒯)/∑_j=1^h_l · w_lexp(a_i, j^t, l/𝒯) where 𝒯 is the temperature. By converting the similarity list into a probability distribution, the scales of the contextual affinity for each feature vector are normalized and the large spatial similarity relations that we believe are the most important elements are paid more attention to, while the less similar relations are ignored. By using a small 𝒯, the probability distribution becomes harder, which means we only focus on a small portion of spacial relations in the feature map. Similarly, for the student feature embedding, it is intuitive to compute the corresponding contextual affinity probability distribution P_S_glo, i^l for f_S_glo, i^l within the student feature map F_S_glo^l and minimize the difference between P_T, i^l and P_S_glo, i^l. However, in this case, the optimization of one feature vector f_S_glo, i^l is coupled with all the feature vectors in F_S_glo^l, making the optimization difficult <cit.>. For unsupervised knowledge distillation where we only have the knowledge distillation training target, we experimentally found the model can't converge. Considering that we also use the teacher model for inference, we compute the contextual similarity list A_S_glo, i^l and affinity probability distribution P_S_glo, i^l using the corresponding student feature embedding f_S_glo, i^l and the whole teacher feature map F_T^l, where a_i, j^s_glo, l=f_S_glo, i^l · f_T, j^l/f_S_glo, i^lf_T, j^l We then use KL divergence to evaluate the discrepancy between the contextual affinity distributions from the teacher and global student 𝒦ℒ(P_T, i^l, P_S_glo, i^l) =𝒯^2 ∑_j=1^h_l · w_l p_T, i^j, l·log[p_T, i^j, l/p_S_glo, i^j, l] The p_T, i^j, l in Equation (<ref>) could be interpreted as a weighting factor. Large p_T, i^j, l values that indicate the spatial relations with high similarities are paid more attention to, while the KL divergence tends to neglect less similar relations. Note that since each feature vector always has the largest similarity with itself, it is not a contradiction with accurately reconstructing the feature vector. The high-similarity relations are the guiding signposts for training the student feature vector from the global context. During inference, the global student fails to capture the global contextual information for logical anomalies. Similarly, the final loss for training the global student is ℒ_glo = ∑_l=1^3{1/h_l · w_l∑_i=1^h_l · w_l𝒦ℒ(P_T, i^l, P_S_glo, i^l) } §.§ Pixel and Image Anomaly Scoring Following Equation (<ref>) and Equation (<ref>), we could get anomaly score maps M_loc^l and M_glo^l for the l-th layer from the local and global student. Each element in the score map indicates the feature or contextual affinity discrepancy. To get precise multi-scale anomaly detection and localization, we first up-sample each score map to the image resolution and conduct element-wise addition for each student. The final score map for an input image I is the combination of the two students' normalized detection results M(I) = M_loc - μ_loc/σ_loc + M_glo - μ_glo/σ_glo, M_loc = ∑_l = 1^3Ψ(M_loc^l), M_glo = ∑_l = 1^3Ψ(M_glo^l) Where Ψ is the bilinear up-sampling operation, μ and σ are the mean and standard deviation values, respectively. They are computed on the validation set 𝒮^v or the training set 𝒮^t if 𝒮^v is not available. The image-level anomaly score is derived by choosing the maximum score from the final score map. We apply a Gaussian filter before image-level anomaly scoring to remove local noises. § EXPERIMENTAL RESULTS §.§ Experimental Settings Datasets. We use two public unsupervised anomaly detection and localization benchmarks: MVTec LOCO AD <cit.> and the modified MVTec AD <cit.>. The recently introduced MVTec LOCO AD covers both local structural anomalies and logical anomalies that violate long-range dependencies. It contains 5 object categories and 1772 anomaly-free images in total for training. 304 anomaly-free images are also provided for validation. Each of the 1568 test images is either anomaly-free or contains at least one structural or logical anomaly. Pixel-level annotations are provided for testing. The MVTec AD dataset has 15 single object or texture categories, consisting mainly of anomaly-free and structural anomalous samples but 37 test images that are better defined as logical anomaly samples <cit.> are split out as the logical anomaly test subset. Model training. All images are resized to 256 × 256 resolution. We follow the one-model-per-category setting of previous studies. The two students could be trained simultaneously or separately. For each student, we use the same training configuration as <cit.>. We use Adam optimizer using β=(0.5,0.999) with a fixed learning rate 0.005. Each student is trained for 200 epochs with the same batch size of 16. The channel dimension g of GCCB is set to 1024 by default and the temperature 𝒯 is set to 1. Evaluation metrics. We take the area under the receiver operating characteristic (AUROC) score as the threshold-free image-level anomaly detection evaluation metric. For anomaly localization, it is also suitable to use AUROC for structural anomaly detection evaluation. However, logical anomalies, e.g. a missing object, are difficult to annotate and segment for each pixel. We use the saturated per-region overlap (sPRO)<cit.>, a generalized version of the PRO metric <cit.> to evaluate the anomaly localization performance. The metric score saturates once the overlap with the ground truth meets a predefined saturation threshold. All thresholds are also provided by LOCO dataset. §.§ Experimental Results on LOCO We compare our proposed method against autoencoders including a vanilla autoencoder (AE), a variational autoencoder (VAE), a memory-guided autoencoder (MNAD) <cit.>, f-AnoGAN <cit.>, Variation Model (VM)<cit.>, Uninformed Students (US) <cit.>, SPADE <cit.>, Reverse Distillation (RD) <cit.> and Global Context Anomaly Detection (GCAD) <cit.>. The same data augmentations are used as GCAD <cit.> throughout our experiments. We first report the anomaly localization results in Table <ref>. The average score of our proposed method over 5 categories reaches 0.73 at a very low integration limit of 0.05. Especially on the breakfast box, screw bag, and pushpins that most of our counterparts can only reach relatively low scores because of the complex contextual logical constraints, we outperformed their performance by a large margin. The image-level anomaly detection results are shown in Tab. <ref>. Most of the existing methods could already achieve relatively high performance on structural anomaly detection, especially the knowledge distillation-based US <cit.> and RD <cit.> that use patch-based and per-pixel training targets. However, the logical anomaly detection scores drop fast compared to their good performance on structural anomaly detection. Although GCAD <cit.> is based on US <cit.> aiming to improve the logical anomaly detection ability and achieves the best score on it, the performance on structural anomaly detection is severely cumbered. The proposed method improves the logical anomaly detection performance by a large margin, without decreasing RD <cit.> 's structural anomaly detection ability and also achieves a new state-of-the-art (0.84 score). We give some qualitative visualization results for each category in Fig. <ref>. For each category, we give one structural anomaly image on the left and one logical anomaly image on the right. The local student is mainly responsible for low-semantic level structural anomaly detection but fails to capture long-range dependencies, while the global student could better learn the global contextual constraints but can not perform well on fine-grained local structural anomaly detection. Finally, the dual-student knowledge distillation framework enables our method to detect both structural anomalies and logical anomalies. §.§ Experimental Results on modified MVTec AD We report the anomaly detection and localization results on the modified MVTec AD dataset in Tab. <ref>. The results show that some of the existing works perform well on structural anomaly detection, while still showing the ability for logical anomaly detection. The SPADE <cit.>achieves a 0.906 image-level AUROC score which is even higher than structural anomaly detection. An underlying assumption is that SPADE uses the high-semantic level feature, e.g., the output of the 4-th residual block of a pre-trained ResNet for image-level anomaly scoring. RD <cit.> also achieves a high score of 0.914 because it benefits from the compact one-class bottleneck embedding space that also contains high-semantic level information. PatchCore achieves SOTA image-level anomaly detection results with the help of using locally aware patch features. Similar to the results on LOCO, GCAD <cit.> is capable of logical anomaly detection, at the cost of an obvious performance drop for structural anomaly detection. The proposed DSKD showed better logical and average anomaly localization performance. We visualize two logical anomaly samples from the transistor and cable category in Fig <ref>, where one example is better detected by the local student while the remaining one is better detected by the global student. Although our proposed method achieves the second-best overall performance by improving the logical anomaly detection ability without obviously affecting the structural anomaly detection performance, we observed similar limitations with results on the LOCO dataset. Our global student could capture global logical constraints but is not sensitive to small-sized defects and ambiguous anomalies violating both low-level and long-range dependencies. In Fig. <ref>, a blue cable replaced by a green one may also be defined as a kind of color contamination. But the global student could identify a missing object or an object in the wrong place while identifying the right position. §.§ Ablation Studies We investigate the effectiveness of the dual-student architecture, and the contextual affinity loss and assess the sensitivity of hyperparameters. The performance of a single student trained with different losses is reported in Tab. <ref>. Benefiting from the RD <cit.> architecture which has a low-level feature bias, and our contextual affinity learning scheme, the local student is capable of logical anomaly detection and yields the best overall performance. The global student trained with per-pixel cosine similarity is enhanced for better low-level feature reconstruction which in turn improved both structural and logical anomaly detection performance. Tab. <ref> gives qualitative comparisons of our dual-student architecture trained with different loss pairs. Although the global student trained with per-pixel loss outperforms the one trained with contextual affinity loss, however, the dual-student architecture design releases the constraint of accurate low-level feature reconstruction for the global student and encourages the global student to focus on global contextual information. We also investigate the impact of GCCB along with its channel dimensions g. The results are shown in Tab. <ref>. The use of GCCB improved the performance by a large margin and performs well with various g values. The results with different 𝒯 are shown in Fig. <ref>. A large 𝒯 makes the distribution softer and covers wider relations. Although it may confront the low-level feature reconstruction ability, our method is stable for a wide range of 𝒯. § CONCLUSION We proposed the dual-student knowledge distillation framework and contextual affinity loss for anomaly detection. The two students play different roles and the contextual affinity computation for the student model uses both teacher and student features. The design and use of global context condensing block GCCB and the contextual affinity loss enable our method capable of both structural and logical anomaly detection. Experiments showed that the proposed method outperformed previous studies and achieved SOTA performance on public benchmarks. ieee_fullname
http://arxiv.org/abs/2307.02188v1	20230705102746	Improving Algorithms for Fantasy Basketball	[ "Zach Rosenof" ]	stat.ME	[ "stat.ME" ]	Heralded Three-Photon Entanglement from a Single-Photon Source on a Photonic Chip Jian-Wei Pan August 1, 2023 ================================================================================= < g r a p h i c s > Fantasy basketball has a rich underlying mathematical structure which makes optimal drafting strategy unclear. The most commonly used heuristic, “Z-score" ranking, is appropriate only with the unrealistic condition that weekly player performance is known exactly beforehand. An alternative heuristic is derived by adjusting the assumptions that justify Z-scores to incorporate uncertainty in weekly performance. It is dubbed the “G-score", and it outperforms the traditional Z-score approach in simulations. A more sophisticated algorithm is also derived, which dynamically adapts to previously chosen players to create a cohesive team. It is dubbed “H-scoring" and outperforms both Z-score and G-score. § INTRODUCTION Fantasy basketball is a popular past-time. According to the Fantasy Sports & Gaming Association, 62.5 million Americans and Canadians played at least one fantasy sport in 2022, and roughly 22% of them played fantasy basketball <cit.>. In fantasy basketball, participants draft players to their teams before the NBA season, then compete via their proxies' in-game performances during the season. Participants can also substitute some of their players for unselected players during the season, though this adds a level of complexity which is best ignored for drafting purposes. There are many different ways to have participants compete throughout the fantasy season. As described by https://support.espn.com/hc/en-us/articles/360003913632-Scoring-FormatsESPN, the three most popular are * Rotisserie: Rotisserie, or “Roto", is the most common way to play fantasy basketball. In this scoring type, teams are ranked from first to last in each statistical category. Points are then awarded according to the order in each category and totaled to determine an overall score and league rank. * Head-to-Head: Each Category: H2H Each Category is the most common type of head-to-head play in fantasy basketball. For each scoring period, team totals are accumulated and a win, loss or tie is credited in each category based on the matchup results (i.e. 6-3-1 in a 10 category league). * Head-to-Head: Most Categories: For each scoring period, team totals are accumulated in each of the categories. At the end of the scoring period, the winner is determined by which team wins the most number of categories. The end result is a win (1-0-0), loss (0-1-0) or tie (0-0-1). These results correspond directly to each team's overall record. <cit.> Common settings specify nine categories: points, rebounds, assists, steals, blocks, three-pointers, field goal %, free throw %, and turnovers. Each team has the same number of players, often thirteen. Player performances are aggregated across teams to get overall team metrics for each scoring period. Points, rebounds, assists, steals, blocks, three-pointers, and turnovers, henceforth dubbed “counting statistics" are aggregated by taking the sum. Field goal % and free throw %, henceforth dubbed as “percentage statistics" are aggregated by taking the total number of successes over the total number of attempts for the category. Categories are won by the team with the higher metric, except for the turnovers category which is won by the team with the lower metric. Scoring periods are usually weeks, and after twenty or so either a winner is declared or an elimination bracket is run to determine the winner. One might note that from a mathematical perspective the first two formats are similar. In Rotisserie, if a team performs better in a category than five other teams and worse than six, it earns five points. That is the same reward that it would have earned for the category if it had faced each team individually. It is therefore only necessary to analyze the two H2H formats, with the understanding that the strategy for Each Category will generalize well to Rotisserie. Another relevant feature of NBA fantasy drafts is that participants are limited in the number of players that can contribute each day in each position. For example, a participant might not be allowed to have four players at “center" on the same day, forcing them to place a center “on the bench" where their performance for the day does not count. Fortunately, many leagues are loose on positions, and the trend is for position requirements to become looser over time <cit.>. For example, as of this writing James Harden is officially listed as a shooting guard by the NBA but is also allowed to count as a point guard in Yahoo fantasy leagues. While fantasy eligibility can change during the season and is hard to keep track of, the general looseness means that it does not matter much. A reasonable simplification is to presume that all players will be able to have their games count, so long as the overall team composition is balanced. For all analysis herein, the fantasy basketball problem will be simplified in the following way: * Each team must have two centers, one point guard, one shooting guard, two guards (either point guard or shooting guard), one small forward, one power forward, two forwards (either shooting forward or power forward), and three utilities (any position) * Every game played by every drafted player counts towards totals § COMPUTATIONAL COMPLEXITY With complete information on probability distributions for each players' performances, probabilities of victory can be calculated exactly based on which players are taken by which drafters. This conception of the fantasy basketball drafting problem is a perfect-information sequential game, indicating that a subgame-perfect equilibrium can be derived through backwards induction <cit.>. However, the game has a high state-space complexity which makes applying backwards induction practically difficult. If there are J available players to start, the first drafter's initial pick breaks into J subgames. Each of those players leads to J-1 subgames for the next drafter, or J * (J-1) in total. In general, the number of subgames at step x of the draft is J!/(J - x)! If N drafters each choose P players, then the total number of subgames is ∑_x = 1^NPJ!/(J - x)! With small numbers, the sum is reasonable. For example with P=1, N=2, and J=3, the result is 3!/2! + 3!/1! = 9. However, the sum gets much larger with larger values for N, P, and J. Since all terms are positive, the final term serves as a lower bound on the sum. The other terms will also contribute to the sum, though at an exponentially diminishing rate of roughly 1/NP, which will make them generally inconsequential. The final term with J=500 (≈ the number of NBA players), N=12, and P=13 is 500!/(500 - 156)! Which is above 10^409, far beyond the limits of current computers. Therefore, crafting practical strategies will require heuristics. § DEFINING Z-SCORES One simple drafting strategy is ranking players by statistical performance, then choosing the player with the highest performance available at each step of the draft. Many drafters use this strategy or something close to it in practice, applying a measure called “Z-score" to rank players. Some terminology will be helpful for defining Z-scores. To start, define Q to be a set of players of size \|Q\|. Then, for counting statistics define the following: m_μ_p Mean performance for player p in a week m_μ Mean of m_μ_p across players: ∑_p ∈ Q m_μ_p/\|Q\| m_σ Standard deviation of player means: √(∑_p ∈ Q (m_μ_p - m_μ)^2/\|Q\|) And for percentage statistics: a_μ_p Mean number of attempts for player p per week r_μ_p Rate at which player p's attempts succeed a_μ Mean of a_μ_p across players: ∑_p ∈ Q a_μ_p/\|Q\| r_μ Aggregate success rate across players: ∑_p ∈ Q a_μ_p * r_μ_p/∑_p ∈ Q a_μ_p r_σ Standard deviation of a_μ_p/a_μ * (r_μ_p - r_μ) across players The most basic version of Z-score is made by normalizing NBA-wide statistics. That is to say, Q is set to be all ≈ 500 NBA players, all categories are considered counting statistics, and scores per category are calculated as Z = m_μ_p - m_μ/m_σ. Then the overall score for a player is calculated as ∑_c ∈ C Z_c , where C is the set of all categories and Z_c is the Z-score for a category. The name “Z-score" comes from the fact that the scores per category can be conceived of as statistical Z-scores. The most basic version of Z-score has significant flaws. Two adjustments are required to make a more proper version of the ranking. * The percentage categories are modified from the counting statistics to take into account attempt volume per player, through three procedural changes * Unlike m_μ_p, r_μ is defined as the success rate across players weighted by attempts rather than the simple mean of success rate. * Unlike m_σ, r_σ is defined as the standard deviation of a_μ_p/a_μ * (r_μ_p - r_μ) rather than of r_μ_p or r_μ_p - r_μ * During the ultimate calculation of Z-score, the differential between r_μ_p and r_μ is multiplied by a_μ_p/a_μ before being divided by standard deviation * Q is set to the top N players instead of the entire NBA, where N is the total number of players that will be drafted by participants (e.g. with 12 drafters and 13 players per drafter, N = 12 * 13 = 156). “Top" is often defined by Z-score where Q is the entire league, since many drafters use that for their player rankings. With Q set to be the top N players, proper Z-scores can then be defined in the following way: Points m_μ_p - m_μ/m_σ Rebounds " Assists " Steals " Blocks " Three-pointers " Turnovers m_μ - m_μ_p/m_σ Field goal % a_μ_p/a_μ * (r_μ_p - r_μ)/r_σ Free throw % " + Total = Z-score Note that Z-scores are often defined with metrics per game instead of per week. The two bases are equivalent, since standard deviations and means both scale to the same degree when switching between them. The popular website Basketball Monster serves as an example of how Z-scores are implemented. It calculates empirical Z-scores based on a historical time period specified by the user. <cit.> § JUSTIFYING Z-SCORES There is a sense in which Z-scores are close to optimal. Consider the problem statement: Team A has N players, chosen randomly from a player pool. Team B has N + 1 players, chosen in the same way. Individual player performances are known to be their long-term means, but it is not known which players were chosen by which team. In order to maximize the expected value of number of categories won against team B, what single statistic ought team A maximize in their next player? Team T's total score for a counting statistic can be written as ∑_w ∈ T m_w Where m_w is player w's category mean. The differential between team B and team A is then ∑_w ∈ T_B m_w + ∑_w ∈ T_A - m_w With 13 players per team, the score differential is the sum of 132 = 26 random variables. That is roughly large enough to invoke the law of large numbers, which simplifies the distribution to a normal distribution. The differential can then be described as 𝒩( (N+1) m_μ - N * m_μ - m_μ_p, N * m_σ^2 + (N+1) * m_σ^2) Where p is the unchosen player, and all of the other generic players are simplified to their expected values and variances. The equation can be further simplified to 𝒩(m_μ - m_μ_p, (2N + 1) * m_σ^2) If this value is below zero, than team A scores more points than team B and therefore wins the category. The probability that this value is below zero is the cumulative distribution function of the normal distribution at zero. Therefore, the probability of victory is the cumulative distribution function of the above normal distribution at zero. For an arbitrary normal distribution it is 1/2[1 + ( 0 - μ/√(2 σ^2))] In this specific case, it becomes 1/2[1 + ( m_μ_p - m_μ/√(2 * (2N + 1) * m_σ^2 ))] In most circumstances, the number going into the error function is quite small. The exception would be if the differential in the denominator is exceptionally large compared to m_σ. Otherwise the first-order Taylor approximation, which is 2X/√(π), can be used for the error function. This simplifies the expression to 1/2[1 + 2/√(π) m_μ_p - m_μ/√(2 * (2N + 1) * m_σ^2)] Or 1/2[1 +1/√(π * (N+1/2)) * m_μ_p - m_μ/ m_σ] The chance of winning is linearly dependent on m_μ_p - m_μ/ m_σ, the Z-score. Since the equations are equivalent for each category, it is then fair to say that maximizing total win probability across counting statistics is achieved by maximizing their Z-scores. The percentage statistics are more complicated because they involve division between two numbers, each with variance. For the purposes of justifying the Z-score, two reasonable assumptions will be invoked to simplify the problem. * If the unchosen player makes the same average number of attempts as the average between players, then the category should be handled exactly like a counting statistic. This is reasonable because if attempts were held constant, the percentage categories would be equivalent to counting statistics * The effect of the unchosen player's differential from the aggregate success rate is linearly dependent on the average number of attempts made by the player. This is reasonable because a player that makes two free throws per game should be roughly twice as important as a player who makes one free throw per game to the result of the category These assumptions lead to the adjustments that were defined earlier for the percentage categories. They imply a score of the form a_μ_p/a_μ(r_μ_p - r_μ)/r_σ Where r_μ is weighted by number of attempts in accordance with the second assumption, and r_σ is the standard deviation of the numerator analogous to m_σ. This is the Z-score for percentage statistics. When comparing between two players or teams, a_μ and r_σ can be removed and the percentage statistic score simplifies to a_μ_p (r_μ_p - r_μ) With deterministic performance, the team with a higher total for this value across players should be winning the category. The first product is the expected successes for player p and the second is the expected successes if they performed at the aggregate rate. Across two teams, this heuristic then implicitly assumes that the team with more successful attempts over the baseline expected wins the category. Intuitively it is clear that this heuristic is reasonable when both teams shoot approximately the same number of attempts, and that will be the case when most players are chosen randomly as was specified in the problem statement. § A FLAW OF Z-SCORES The problem statement which justified Z-scores allowed for variations in performance between players, but not between different weeks for the same players. One might conjecture that this omission is lossless, because in the long term weekly variations balance out. Unfortunately for Z-scores, the omission is not lossless. This is because categories with high week-to-week variation are fundamentally more difficult to reliably win, which is relevant. For intuition, consider a hypothetical category with arbitrarily high week-to-week standard deviation and arbitrarily low intra-player standard deviation. Clearly a player's long-term mean for the category would be largely irrelevant, because the winner of the category would be close to a coin flip in all circumstances. A Z-score, which smooths out intra-player variance via normalization, would put significant weight on drafting players with relatively high long-term means for the category. But this would only negligibly improve winning probability for the category, while potentially sacrificing significant success in other categories. Even in less extreme scenarios, week-to-week variance could be important. Event count data is generally modeled with the Poisson distribution, and the counting statistics are event count data. It stands to reason that they follow a distribution close to Poisson, which means that standard deviations are roughly the square roots of the means. The ratio between the standard deviation and the mean, or the coefficient of variation, then decreases as the mean increases. Therefore rare categories, like steals that only happen a few times per game, likely have higher coefficient of variations than common categories like points. This could matter for optimal strategy. § FORMULATING G-SCORES A novel variant of the Z-score, which accounts for uncertainty, can be derived. To start, define V to be a set of weeks in a fantasy season of size \|V\|. Then, for counting statistics define the following additional terms: m_n_p Performance for player p in week n m_τ_p Standard deviation of m_n_p across weeks: √(∑_n ∈ V (m_n_p - m_μ_p)^2/\|V\|) m_τ Root-mean-square of m_τ_p: √(∑_p ∈ Q m_τ_p^2/\|Q\|) And for the percentage statistics: a_n_p Number of attempts made by player p in week n [5pt] r_n_p Success rate for player p in week n r_τ_p Standard deviation of a_n_p/a_μ_p * (r_n_p - r_μ_p) across weeks r_τ Root-mean-square of r_τ_p: √(∑_p ∈ Q r_τ_p^2/\|Q\|) The new score, dubbed G-score, is calculated in the following way: Points m_μ_p - m_μ/√((m_σ^2 + m_τ^2)) Rebounds " Assists " Steals " Blocks " Three-pointers " Turnovers m_μ - m_μ_p/√((m_σ^2 + m_τ^2)) Field goal % a_μ_p/a_μ * (r_μ_p - r_μ)/√(r_σ^2 +r_τ^2) Free throw % " + Total = G-score Justification for the G-score arises after reframing the Z-score problem to: Team A has N players, chosen randomly from a player pool. Team B has N + 1 players, chosen in the same way. Probability distributions for individual player performance are known, but it is not known which players were chosen. In order to maximize total winning probability against team B, what single statistic ought team A maximize in their next player? Weekly totals are the sum of many individual actions, so they can be simplified as Gaussians. Performance of player w is then the result of a normal distribution with parameters that are randomly distributed. It can be written as: m_w = 𝒩(m_μ_w, m_τ_w^2) Writing out the outer normal into mean and variance terms: m_w = m_μ_w + 𝒩(0, m_τ_w^2) As many normal distributions with different variances are added together, it can be estimated that the variance equals the mean variance multiplied by the number of players. Equivalently, the standard deviation converges to the root-mean-square of standard deviation. Since m_τ is defined as the root-mean-square of standard deviations, it follows that the distribution for team B can be written as ∑_w ∈ T_B m_μ_w + 𝒩(0, (N+1) * m_τ^2) Where T_B is the set of N+1 players already chosen by team B. Similarly, team A's distribution is ∑_w ∈ T_A m_μ_w + m_μ_p + 𝒩(0, N * m_τ^2 + m_τ_p^2) Where T_A is the set of N players already chosen by team A and p is the unchosen player. The differential between team B and team A is ((∑_w ∈ T_B m_μ_w) + 𝒩(0, (N+1) * m_τ^2)) - ((∑_w ∈ T_A m_μ_w) + m_μ_p + 𝒩(0, N * m_τ^2 + m_τ_p^2)) Or, after adding the two normal distributions, (∑_w ∈ T_B m_μ_w) - (∑_w ∈ T_A m_μ_w) - m_μ_p + 𝒩(0, (2N+1) * m_τ^2 + m_τ_p^2) Remembering that m_μ_w is itself a random distribution with mean m_μ and standard deviation m_σ except in the case of player p, and invoking the law of large numbers, this becomes 𝒩((N+1) * m_μ, (N+1) * m_σ^2) - 𝒩(N * m_μ, N * m_σ^2) - m_μ_p + 𝒩(0, (2N+1) * m_τ^2 + m_τ_p^2) Or simplified into one normal distribution: 𝒩( m_μ - m_μ_p, (2N+1) * (m_σ^2 + m_τ^2) + m_τ_p^2) Note that this expression is similar to what was derived for the Z-score differential. One difference is that there is now a m_τ_p^2 term. It is relatively inconsequential to total variance, so it can be dropped in the pursuit of a simple heuristic. The only remaining difference then is that there is an extra term of m_τ^2. Following the same steps as before, the resultant statistic is 1/2[1 +1/√(π * (N+1/2)) * m_μ_p - m_μ/√((m_σ^2 + m_τ^2))] This equation has the same constants as the equivalent that was derived for Z-score. The non-constant part has changed to incorporate the extra m_τ^2 term, becoming m_μ_p - m_μ/√((m_σ^2 + m_τ^2)). This is the G-score for counting statistics. The percentage statistics can again be handled heuristically. Where m_τ is the root-mean-square of m_τ_p across players, the analogous r_τ is the root-mean-square of r_τ_p. r_τ_p is defined in such a way that it weighs high-volume weeks more than low-volume weeks, which is appropriate for the same reason that r_σ weighs high-volume players more than low-volume players. Incorporating the r_τ term to the Z-score for percentage categories in the same way that m_τ was included for the percentage categories yields a_μ_p/a_μ * (r_μ_p - r_μ)/√(r_σ^2 +r_τ^2) Which is the G-score for percentage categories. Is is thus demonstrated that the G-score optimizes for a more nuanced version of the fantasy basketball problem: winning categories with week-to-week variance considered. It is important to keep in mind the assumptions made by the G-score calculation, which determine when the scores will be applicable. They are * Probability distributions per player are known exactly beforehand. Incorporating uncertainty in player distributions would result in a different scoring system * Drafters are picking players that look random in aggregate, in terms of category distribution. This assumption will not hold if other drafters are using advanced strategies like de-prioritizing particular categories * The simplification of modeling the percentage statistics as weighted counting statistics does not cause significant disruptions. This will be the case so long as the number of free throws between drafted teams is relatively similar; distortions happen when they are very different * It is safe to assume that total variance of a counting statistic over multiple players is equal to the number of players multiplied by the average variance in that statistic of all players. Equivalently, the standard deviation is the number of players times the the root-mean-square of the standard deviation * The m_τ_p term can be safely ignored * The numbers being plugged into the error function are small enough that the first-order approximation can be used without significant issue. This is not the case if the differential is very large or the variation is very small. * The probability of winning against an arbitrary team is monotonically equivalent to winning a league. In real leagues teams have multiple matchups against multiple teams, and need to do consistently well to win. This could impact optimal strategy § A FLAW OF G-SCORES The G-score was justified under the assumption that no information is available about previously drafted players. This is convenient because it allows for the creation of a static ranking list, which can be used to advise selections in any context. The algorithm needs to be run only once, to create the list, and then any drafter can use it at any step of their draft. If it took previously drafted players into account, then it would need to be re-run for every pick. While convenient, the assumption of ignorance about other players is sub-optimal. Consider a situation where a drafter is average in eight categories and arbitrarily strong in the ninth, based on players they had already chosen. Investing more in the ninth category would not improve the drafter's overall probability of victory, since victory for the category would be guaranteed anyway. Instead, the drafter would be best off optimizing for strength strength in the other eight categories- a strategy which cannot be captured by any model ignorant about existing team composition. The common strategy known as 'punting' also relies on information about previously drafter players. When a fantasy drafter punts a category, it means that they purposely don't invest into the category with the implicit belief the resulting edge in other categories outweighs the sacrifice of one category <cit.>. The mathematical rationale for this strategy is that when a team is on the lower edge of the distribution for a category, the probability density function and therefore marginal benefit of investing into the category is low. It is possible to construct an algorithm which understands the differences in marginal utility between categories, and chooses players accordingly. § FORMULATING H-SCORES To start, define a new scoring system called X-score, which is simply the G-score with the m_σ and r_σ terms set to zero. For counting statistics, that is X_p = m_μ_p - m_μ/m_τ And for percentage statistics X_p = a_μ_p/a_μ(r_μ_p - r_μ)/r_τ It is helpful to simplify the drafting problem to a random process depending on draft round number. That is to say, all unknown first-round picks are chosen according to one distribution of players, all second round-picks are chosen from another, etc. The mean and standard deviation of m_p in draft round i are m_μ_i and m_σ_i. The mean and standard deviation of X_p by draft round are X_μ_i and X_σ_i. Consider the problem statement: Team A has chosen K of their J players. Mean category values for the chosen players and candidates for the (k+1)th player on team A are known. Probability distributions for mean category values of players chosen by team B, as well as additional players that will be chosen by team A, are known as functions of draft round number. In order to maximize total winning probability across categories against team B, what single statistic ought team A maximize in their next player? A total of 2J players are relevant to the problem statement. They can be broken down into five groups: * w ∈ A_c, already chosen players on team A. Size = K * w = p, the single player to be chosen by team A * w ∈ A_u unknown remaining team A players. Size = J - K - 1 * w ∈ B_m, unknown players on team B, matching up to known players on team A. Size = K + 1 * w ∈ B_u, unknown players on team B, matching up to unknown players on team A. Size = J - K - 1 Leveraging existing nomenclature, the point differential for a counting statistic between an arbitrary team B and team A in week n can be written as ∑_w ∈ B_m m_n_w + ∑_w ∈ B_u m_n_w - (m_n_p + ∑_w ∈ A_c m_n_w + ∑_w ∈ A_u m_n_w) The expected value of this differential is ∑_w ∈ B_m m_μ_w + ∑_w ∈ B_u m_μ_w - ( m_μ_p + ∑_w ∈ A_c m_μ_w + ∑_w ∈ A_u m_μ_w ) B_u and A_u have the same number of players and are drawn from the same pool. With the simplifying assumption that the drafters are choosing unknown players without sophisticated strategies, the two sets cancel out in the mean. The expression becomes ∑_w ∈ B_m m_μ_w - ( m_μ_p + ∑_w ∈ A_c m_μ_w ) The players in B_m are not known, and so their means must be imputed by the function for unknown players. The expression for the means becomes ∑_i=0^K+1 m_μ_i - ( m_μ_p + ∑_w ∈ A_c m_μ_w ) The variance must also be calculated. It is known that the distribution for a single player can be described as m_w = m_μ_w + 𝒩(0, m_τ_w^2) If the player is known, the only variance contribution is m_τ_w. If the player is unknown, m_μ_w is itself a distribution and contributes variance. Dubbing the standard deviation for an unknown player m_σ_w, the overall variance can then be written as ∑_w ∈ B_m (m_τ_w^2 + m_σ_w^2) + ∑_w ∈ B_u (m_τ_w^2 + m_σ_w^2) + ( m_τ_p + ∑_w ∈ A_c m_τ_w^2 + ∑_w ∈ A_u(m_τ_w^2 + m_σ_w^2)) Standard deviations for the unknown players are known as a function of draft number. They simplify the expression to Jm_τ_w^2 + ∑_i=0^J m_σ_i^2 + Jm_τ^2 + ∑_i=K+1^J m_σ_i^2 Or 2Jm_τ^2 + ∑_i=0^K m_σ_i^2 + 2 ∑_i=K+1^J m_σ_i^2 Altogether, the point differential can be described as the following distribution 𝒩( ∑_i=0^K+1 m_μ_i - m_μ_p - ∑_w ∈ A_c m_μ_w , 2Jm_τ^2 + ∑_i=0^K m_σ_i^2 + 2 ∑_i=K+1^J m_σ_i^2 ) The distribution can be used to calculate probabilities of victories for each player and category. However, raw values are slightly difficult to interpret, since scales vary across categories. For ease of understanding, it is helpful to substitute X-scores. Remember that the CDF of a normal distribution at zero is 1/2[1 + ( - μ/√(2 σ^2))]. Clearly, it would not change if the standard deviation and mean were both scaled by the same factor. So it is possible to divide the mean by m_τ and the variance by m_τ^2 without changing the value of the CDF, the ultimate quantity of interest. This yields 𝒩( ∑_i=0^K+1 m_μ_i - m_μ_p - ∑_w ∈ A_c m_μ_w/m_τ , 2J + ∑_i=0^K m_σ_i^2 + ∑_i=K+1^J 2m_σ_i^2/m_τ^2 ) Adding and subtracting (K+1) * m_μ to the mean changes it to ∑_i=0^K+1 (m_μ_i - m_μ) - (m_μ_p - m_μ) - ∑_w ∈ A_c (m_μ_w - m_μ) /m_τ Or ∑_i=0^K+1m_μ_i - m_μ/m_τ - m_μ_p - m_μ/m_τ - ∑_w ∈ A_cm_μ_w - m_μ/m_τ These are X-scores, as defined previously. The mean can then be simplified to ∑_i=0^K+1 X_μ_i - X_p - ∑_w ∈ A_c X_w The variance as described before is 2J + ∑_i=0^Km_σ_i^2/m_τ^2 + ∑_i=K+1^J2m_σ_i^2/m_τ^2 Imputing from its definition, the variance of the X-score is the variance of m_μ_p across players, divided by m_τ^2. The overall variance can therefore be rewritten to 2J + ∑_i=0^K X_σ_i^2 + ∑_i=K+1^J 2X_σ_i^2 All in all, the resulting distribution is 𝒩(∑_i=0^K+1 X_μ_i - X_p - ∑_w ∈ A_c X_w,2J + ∑_i=0^K X_σ_i^2 + 2∑_i=K+1^J X_σ_i^2) This CDF of this distribution at zero can be used to estimate winning probabilities with all candidate players, so long as reasonable estimates of X_μ_i and X_σ_i exist. The distribution was defined with counting statistics in mind, but it can also be heuristically applied to the percentage categories for simplicity of the process. Optimizing player selection for Each Category is simple via the derived distributions. Since the goal is expected value of category wins, one only needs to sum up the resultant victory probabilities per player and choose the player with the highest sum. Call the sum a H_e score. However, the Most Categories format is more complicated, since winning at least five categories in one match-up is needed for an overall win. Assuming for simplicity that categories are independent, the distribution of the number of total categories won is a Poisson binomial, with a component binomial for each category parameterized by n = 1 and p = probability of winning the category. Unfortunately the Poisson binomial has a CDF involving quite a bit of calculation- there is no way to simplify the function besides enumerating all possibilities. Fortunately, the number of winning scenarios (for example, winning points/rebounds/assists/steals/blocks and losing the others) is only 9 5 + 9 6 + 9 7 + 9 8 + 9 9 = 256 This is because there are 9 5 scenarios where five categories are won and four are lost, 9 6 scenarios where six categories are won and three are lost, etc. Manually checking each of these 256 scenarios is tractable. Each winning scenario involves calculating 8 multiplication steps so the total number of operations is no more than 2048 per player. The resultant probability of winning scenarios can be dubbed the H_m score, analogous to the H_e score. The efficiency of the operation can improved by computing probabilities along a tree with the following structure: [. [.Pts-W Reb-W Reb-L ] [.Pts-L Reb-W Reb-L ] ] Each node stores the probability of the scenario occurring, and the children can be computed with one multiplication step each. Any node that represents five or more losses can be pruned. At layer 6 for example, there are 6 5 nodes representing 5 losses and one win, and 6 6 nodes representing 6 losses, all of which can be ignored. The total nodes requiring multiplication in each layer, starting from the second layer, are Layer Operations 2 2^2 = 4 3 2^3 = 8 4 2^4 = 16 5 2^5- 5 5= 31 6 2^6 - 6 5 - 6 6 = 57 7 2^7 - 7 5 - 7 6 - 7 7 = 99 8 2^8 - 8 5 - 8 6 - 8 7 - 8 8 = 163 9 256 as calculated earlier These numbers add up to 634, for a 69% reduction in multiplication operations. This produces a relatively tractable operation, albeit a complicated one, to calculate the predicted win probability between two teams. A full process for the H-scoring procedure is outlined as follows Estimate X_μ_i and X_σ_i for all i. This can be done by applying a ranking metric such as Z-score to all players, then dividing them into the top J buckets of P players, where J is the number of players per drafter and P is the number of drafters. Then calculate the mean and variance of X-scores for all i from 0 to J-1 * For each available player, calculate the projected win probabilities per category * For every possibility from step 2, calculate either the H_e or H_m score, depending on whether the format is Each Category or Most Categories * Choose the player with the highest metric from step 3 It is worth keeping in mind that H-scores have a significant theoretical limitation. They are justified with the assumption that remaining players for team A are chosen randomly from the same round-based distributions that all other drafters are using for selecting players. This assumption is definitely incorrect, because drafters using the procedure will continue to weigh categories differently based on the existing team composition. This is particularly notable in cases where the team is already weak in a category, expects to lose it, and therefore does not invest in it. It will continue to invest less and less into the category, creating a snowball effect which significantly biases the remaining selections away from purely random. Perhaps in the future, a more advanced version of H-scores could account for this effect. § ADJUSTING FOR POSITION When drafting, the sensible strategy of taking the available player with the highest score in G, H_e, or H_m is complicated by the position requirement. This can be solved for with a simple modification to the ranking procedure- after ranking available players, take the first player who can fit on the team. This simple modification is not trivial to implement, because there are flex spots that can accommodate players of any position. It is necessary to construct an integer linear program and determine if it is feasible. The linear program is formulated as: min 0 s.t. x_p,k≤ E_p,k ∀ p,k ∑_p x_p,k <= T_k ∀ k ∑_k x_p,k = 1 ∀ p Where * x_p,k is a binary decision variable which is 1 if player p is assigned to position k and 0 otherwise * E_p,k is 1 if player p is eligible for position k and 0 otherwise * T_k is the limit on the number of players at position k, all of which were defined in the introduction Minimizing zero is an odd construction. It is used because there is no need to actually solve the optimization problem; only a feasibility check is required. This strategy is a “greedy" heuristic, because it takes the highest short-term reward in all situations without considering the future. It is not necessarily optimal. An alternative to the greedy heuristic is “value above replacement" or VAR score. To demonstrate by example, if each of twelve teams need exactly one running back in fantasy football, true running back value can be calculated by subtracting the score of the twelfth-best running back from the scores of other running backs. The rationale is that even if running backs theoretically had low utility on average, teams would still need to select one of them, so getting one with medium utility would be valuable. This strategy works well for fantasy football which has tight position requirements, but is awkward for fantasy basketball because there are many flex spots which do not translate well into VAR scores. In that context, the simple greedy heuristic is more natural. The lack of a sophisticated handling of position is relatively unimportant for G-scores, because they don't change dynamically and are generally spread out between positions. For example, the top 156 G-scores in 2023 were distributed as follows: Center PG SG PF SF Rank 1-52 13 12 10 9 8 53-104 10 11 13 11 6 105-156 7 10 11 15 8 Overall 30 34 34 36 22 There are many players of each position available at each step of the draft, allowing drafters to select the ones they need to meet their position requirements without much hassle. The greedy heuristic should not often force drafters to take severely suboptimal players from a G-score perspective. Still, a truly sophisticated algorithm for optimizing G-score would use a more complicated strategy than either the greedy heuristic or VAR. It would prioritize positions deferentially in order to preserve flexibility and avoid situations where position requirements force a severely sub-optimal pick. Such an algorithm could be of interest to future work. H-scores could benefit even more from a sophisticated position strategy, since they are unbalanced in category weighing and therefore can value some positions over others. A VAR-like adjustment accounting for flex spots could be useful, though that is complicated and out of the scope of this work. § TESTING THE ALGORITHMS Since the algorithms utilize many simplifying assumptions, it cannot be taken for granted that they perform well with real NBA data. Fortunately, simulations suggest that they do. Several seasons were used to simulate fantasy seasons, with drafters using empirically correct scores and the greedy heuristic for positions. Z-scores based on the full league were used to choose Q, as well as estimate X_μ_i and X_σ_i. Drafters were tested at all “seats", that is, their position in the initial drafting order. For each simulated week, player weeks were chosen randomly from their actual weeks in the NBA season. 10,000 25-week simulated seasons were run for every draft seat. The results for win rates were as follows: G vs 11 Z drafters Z vs 11 G drafters Most Categories 2021 23.1% 1.7% 2022 21.6% 1.5% 2023 37.2% 0.5% Overall 27.3% 1.2% Each Category 2021 13.3% 1.8% 2022 15.1% 1.7% 2023 26.0% 0.8% Overall 18.2% 1.4% H_m vs 11 Z drafters Z vs 11 H_m drafters Most Categories 2021 41.9% 1.2% 2022 31.0% 1.5% 2023 42.2% 0.2% Overall 38.8% 1.0% H_e vs 11 Z drafters Z vs 11 H_e drafters Each Category 2021 28.9% 1.2% 2022 19.8% 1.5% 2023 28.1% 0.5% Overall 25.6% 1.1% H_m vs 11 G drafters G vs 11 H_m drafters Most Categories 2021 31.5% 2.1% 2022 17.3% 3.4% 2023 23.6% 3.4% Overall 24.1% 3.0% H_e vs 11 G drafters G vs 11 H_e drafters Each Category 2021 17.9% 4.6% 2022 12.7% 4.9% 2023 14.2% 5.8% Overall 15.0% 5.1% The G-score drafter consistently did well against the field of Z-score drafters, scoring above the baseline expected rate of 1/12 or 8.33%. The Z-score drafter did not fare as well against a field of G-score drafters, scoring far below the baseline. This is direct evidence that G-scores do well against Z-scores in Most Categories and Each Category. It also implies that G-score would do well in Rotisserie, since Rotisserie is similar to Each Category as discussed earlier. The reason for this broad success is clear upon inspecting the G-score win rates per category against a field of Z-score drafters: G-score win rate Points 74.4% Rebounds 82.7% Assists 78.0% Steals 28.6% Blocks 51.8% Three-pointers 70.7% Turnovers 18.3% Field goal % 41.2% Free throw % 26.0% Overall 52.3% The baseline win rate for a head-to-head competition is 50%. The G-score performs well above that benchmark in categories that are abundant and relatively stable, like rebounds and assists, and worse in volatile categories like steals and turnovers. The advantage in the more stable categories empirically outweighs the disadvantage in volatile categories as expected. The H-drafter did well against both G-score and Z-score drafters. This is evidence that the on-the-fly adjustments are appropriate and elevate team cohesion above what can be expected from ranking-based strategies. The validity of the winner-take-all algorithm can be verified by inspecting average victory rates per category against Z-drafters across both the winner-take-all strategy and the total categories strategy, and observing which converts to the highest probability of winning a majority of categories. Majority victory Average victory Most Cat's Each Cat' Most Cat's Each Cat' 2021 60.7% 60.3% 53.1% 53.2% 2022 56.7% 56.3% 52.1% 52.1% 2023 59.8% 59.5% 53.2% 53.1% Overall 59.0% 58.7% 52.8% 52.8% While there is no discernible difference between the two algorithms on average category victory probability, the algorithm for Most Categories performed slightly better at winning a majority of categories It should be noted that in practice, a participant using the advanced algorithms cannot expect to win at the elevated rates observed in simulations, for several reasons * The tests' opposing drafter(s) were not using advanced strategies, like ignoring certain categories to prioritize others * Real fantasy basketball involves substituting players based on dynamic predictions of performance, which has little to do with the starting team * True scores are not known before a season begins. In practice estimates will be used, decreasing the reliability of all strategies * Simulated drafters used the greedy position heuristic, which may not be realistic * It is possible for the G-score to backfire in certain scenarios. For example, if a G-score drafter values rebounds more than others drafters do, they might end up taking too many high-rebound players and sacrificing other categories as a result § CONCLUSION Simple assumptions lead to the formulation of three novel scoring systems: G, H_m, and H_e, all of which which seem to outperform Z-scores in the backwards-looking context. With future work, these algorithms could be superceded by more complex versions which eliminate assumptions from the algorithms' justifications. unsrt 5 FSGA The Fantasy Sports and Gaming Association (2023). “Industry Demographics.” FSGA. Accessed May 1, 2023. Available at: https://thefsga.org/industry- demographics/ ESPN2 Scoring formats – ESPN fan support. (n.d.). ESPN. Accessed May 1, 2023 Available at: https://support.espn.com/hc/en-us/articles/360003913632-Scoring- Formats ESPN3 Carpenter, Tom (2020). “The best strategies to dominate your fantasy basketball drafts." ESPN. Accessed May 9, 2023. Available at: https://www.espn.com/fantasy /basketball/story/_/id/20607451/the-best-strategies-dominate-your-fantasy- basketball-drafts MIT Drew Fudenberg and Jean Tirole, “Game Theory", Section 3.5, page 92. MIT Press, 1991. BBM Lloyd, Josh (n.d.). Questions Answered - Basketball Monster. Basketball Monster. Accessed May 8, 2023 Available at: basketballmonster.com/questionsanswered.aspx HTBJoseph Mamone (2022). An Introduction to Punting in Fantasy Basketball. Hashtag Basketball. Accessed June 20,2023. Available at hashtagbasketball.com/introduction- punting-fantasy-basketball Disclaimer: The views and opinions expressed in this article are those of the independent author and do not represent those of any organization, company or entity
http://arxiv.org/abs/2307.04821v1	20230706115345	Amplifying Limitations, Harms and Risks of Large Language Models	[ "Michael O'Neill", "Mark Connor" ]	cs.CL	[ "cs.CL", "cs.AI", "cs.CY" ]	A Meta-Evaluation of C/W/L/A Metrics Tetsuya Sakai August 1, 2023 ==================================== We present this article as a small gesture in an attempt to counter what appears to be exponentially growing hype around Artificial Intelligence (AI) and its capabilities, and the distraction provided by the associated talk of science-fiction scenarios that might arise if AI should become sentient and super-intelligent. It may also help those outside of the field to become more informed about some of the limitations of AI technology. In the current context of popular discourse AI defaults to mean foundation and large language models (LLMs) such as those used to create ChatGPT. This in itself is a misrepresentation of the diversity, depth and volume of research, researchers, and technology that truly represents the field of AI. AI being a field of research that has existed in software artefacts since at least the 1950's. We set out to highlight a number of limitations of LLMs, and in so doing highlight that harms have already arisen and will continue to arise due to these limitations. Along the way we also highlight some of the associated risks for individuals and organisations in using this technology. § INTRODUCTION With at least one author of this article being a researcher in Artificial Intelligence (AI) <cit.> for over 25 years, and all having undertaken PhDs in an evolutionary form of generative AI, based on formal generative grammars <cit.>, the recent advances made in the development of large language models (LLMs) and their associated tools are perhaps the most impressive and surprising results we have witnessed during our careers. These advances and the provision of access to the general public to use the arising technology has resulted in a lot of media interest and hype, which is tending to over-emphasise the possibility of futuristic, science fiction-like scenarios occurring. This is often at the expense of masking the fundamental limitations of the existing technology, the resulting harms which it can and has caused, and the future long-term risks associated with continuing to blindly adopt the technology. On a more positive note, it is nice to see the field of AI emerge from its relative obscurity into the mainstream where it seems everyone now has an interest and opinion. Briefly, for the benefit of readers without a background in AI, and at the risk of introducing many many more confusing (three letter) acronyms, let's say a few words about the relationship between LLMs and AI. Large language (LLM) and foundation model technology in their current form are based on deep learning, which belongs to a sub-field of AI known as machine learning (ML) <cit.>. Deep learning is effectively a re-branding and development of what were traditionally called multi-layer perceptrons (MLP) <cit.>, a form of artificial neural network (ANN). ANNs and MLPs are inspired by the biological brain, and have developed out from a model of a single brain cell (a neuron) called a perceptron <cit.>. There are in fact many AI technologies that have been inspired by nature, with researchers also attempting to distil the computational essence underpinning, for example, the biological process of evolution, the social behaviour of insects, birds and fish, the immune system, the grammar underpining language, physics (classical and quantum) and even chemistry <cit.>. One can visualise a deep learning model as being a very large network of computational units (analagous to "brain cells") comprised of many connected layers, with each layer itself being comprised of many computational units. The machine learning process adjusts the importance (weights) of the connections between the different computational units in response to network outputs generated by the data upon which the network is trained. Researchers in machine learning have and continue to explore, for example, how to implement the computational units, how to learn (or adjust the weights on the network connections), and how to design the overall architecture of the network. Coupled to massive amounts of training data, and to increasingly fast computer number crunching power upon which these deep networks can run, a pivotal transition moment in the development of the deep learning technology that underpins the LLM (GPT) behind ChatGPT was the invention of the transformer network architecture <cit.>. For the interested reader in finding out more about the field of AI, as an accessible and recent introduction we would recommend Mitchell's book <cit.>. In terms of gaining some perspectives on the history of the field, a nice collection of articles can be found in <cit.>. In the remainder of this article we attempt to amplify the limitations, harms and risks of LLMs, and foundation models more generally, in an attempt to re-balance the current discourse, and ideally help to inform and provide those outside of the machine learning and AI community (and perhaps some within) with a more holistic perspective and awareness that this technology should not be blindly accepted and adopted. Before discussing the limitations of LLM technology let's first say a few words about ChatGPT and the LLM that underpins it. § A BRIEF FOCUS ON THE LARGE LANGUAGE MODEL OF CHATGPT We think it is fair to say that the coherence of the text generated by the likes of ChatGPT has surprised many researchers. Particularly the apparent jump in performance from November 2022 (GPT-3.5) to March 2023 (GPT-4). Unfortunately, we do not have all the necessary information required to properly analyse how ChatGPT is working under the hood. Since Microsoft invested a billion dollars in OpenAI in 2019, and then ten billion dollars more in January 2023, they have been less transparent, but GPT-4 does appear to have raised its game over and above previous LLM-based tools. Why? Working through the GPT-4 Technical Report <cit.> and various media reports, some factors to explain performance gains include: * the model may be bigger than GPT-3's 175 billion parameters <cit.>, with numbers speculated to be somewhere around the one trillion region. An alternative hypothesis is that GPT-4 is a mixture of models (8 models each containing 220 billion parameters) not one monolithic model. As an aside, it is reported to have cost more than US$100 Million to train it. * GPT-4 can be given a larger context of text to set the scene. The pay-walled GPT-4 has 32K tokens vs 8K for standard GPT-4, while for the GPT-3 generation it started with a context of 2k, which then grew to 4k tokens. * GPT-4 may have been trained using significantly more tokens, a trend that has also been seen in Google's PaLM2 model which was reportedly trained on 3.6 trillion tokens, 2.82 trillion more than its predecessor PaLM. * sitting on top of the GPT-4 large language model is a stack of added layers of rules and engineering, which implement guard rails to try and filter the generated output to avoid some of the factually incorrect, harmful, and biased text that can be generated. * there is also an additional layer of learning sitting on top of the LLM, called reinforcement learning through human feedback (RLHF). The purpose of this layer is to align the model's output with human preferences. On that last point, it is worth highlighting that our (the public) interactions with ChatGPT can be used to refine any current or future OpenAI models based on the feedback we provide. Currently, this feedback is discretely collected in the form of a thumbs up or down vote next to all ChatGPT responses. Users in that case are effectively working (for free) for OpenAI, or if they are using the pay-walled version, users are paying to work, enabling OpenAI and Microsoft to create a better product by way of refinement based on the user's feedback. This also raises a potentially serious security and privacy risk. There exists the possibility that users are sharing sensitive and/or personal identity information or perhaps even sensitive government information <cit.> with OpenAI. For example, what if ChatGPT is behind the customer service chatbot on your utilities provider, insurance broker, or bank web interface or app? You might think that you're engaging with a trusted service provider, and as such, you are comfortable sharing dates of births, payment card details, your address, etc. If ChatGPT, or whatever LLM, is being used by your service provider as a "software-as-a-service" (SaaS) by the developers of the LLM, there is the possibility that all your details have automatically been passed to that third party! From an EU GDPR perspective, the LLM SaaS provider is now a data processor. Have you given them permission to process your data? What are they doing with your data? Are they storing it? Do the developers of the LLM see examples of the prompts coming in to the LLM in order to improve its guardrails etc? Can those developers be trusted with your data? Is your data being incorporated into the RLHF layer of the model? With an appropriately designed prompt could your data then be retrieved and regurgitated back to another party, perhaps a bad actor, who might then use your data for fraud? Clearly, without having delved too deep into the technology behind the likes of ChatGPT we start to encounter some uncomfortable questions and risks. Given the significant hype that currently exists around ChatGPT and LLMs generally, in this article we set out to provide a more sober perspective on the technology, by highlighting a number of issues with LLMs. In so doing, we are not claiming novelty in our perspective. Rather in many cases we are simply amplifying issues that have already been raised by other researchers drawn from our field's wider core set of disciplines (e.g., Bender et al <cit.>, Melanie Mitchell <cit.>, Abeba Birhane <cit.>, Judea Pearl <cit.>, Rodney Brooks <cit.>, etc.,). This is also by no means intended to be an attack on ChatGPT, LLM and foundation model technology. It would be foolish to dismiss the potential utility of this technology, particularly as a productivity tool when used by a human, where the human is filtering, correcting, and taking responsibility for the output. But given the level of hype that currently exists, coupled with the wide accessibility of the technology, there is a real and growing danger that unsuspecting users will be caught off guard (e.g., recent reports on a US Lawyer using ChatGPT for legal research <cit.>), that harm will continue to arise, and that risks will not be recognised without a number of researchers joining in to counter the hype by amplifying what might be considered a more grounded scientific perspective that identifies the limitations of this technology (e.g., for discussions on their risk of use, see Birhane et al <cit.> on use in Science, Morley & Floridi <cit.> on their use in Medicine, and Gros et al <cit.> on their use in Software Engineering). At this point it is also important to mention that ChatGPT is but one example of a LLM-based technology. There exists a rapidly expanding ecosystem of large language and foundation models that can generate text, code, images, video and audio (e.g., <cit.>). Also, recognising that in terms of foundation models such as LLM's that the "genie is out of the bottle". That is, various models are in the public domain, and are becoming increasingly accessible with variants emerging that are smaller and capable of running on personal devices. As such these models are not going to disappear if the big corporations switch off access to them. As many of the commercial organisations deploying LLM and foundation model technology are more focused on effective marketing hype and sales, it unfortunately falls upon users to educate themselves about this technology. To this end, in the remainder of this article we outline some of the main limitations that we and others have observed about LLMs and foundation models. § WHAT ARE THE LIMITATIONS, HARMS AND RISKS OF LARGE LANGUAGE MODELS? A number of fundamental limitations can be associated with LLM-based technology and more broadly foundation models, which in turn can give rise to harms and risks. These include but are not limited to: * Meaning arises from expression with intent * Generative stochasticity * Anthropomorphism * The training data contains biases, nonsense and harmful content * Security and Privacy * Correlation is not causation * All models are wrong, some are useful * Leaky data makes it hard to evaluate true performance * Emergent abilities are misleading * Rights-laundering * Sustainability * Regulatory non-compliance * Transparency * Equality, Diversity & Inclusion Let's briefly touch upon each of the limitations of LLM technology listed above. §.§ Meaning arises from expression with intent Put simply, LLMs do not understand the prompts given to them, and do not understand the text they generate. From a Linguistics perspective, researchers like Bender & Koller <cit.>, highlight that the data upon which LLMs are trained does not contain intent that is present in human communication. This lack of intent means that only very weak semantics, or meaning, can be captured by a LLM trained on purely expression (structural) data. To think of this another way, when humans communicate they do so in such a way that there is (hopefully) a shared understanding of the intent and meaning of the communication. This arises in part due to joint attention, where those engaged in the communication are focused on the same subject. They also have (shared) life experience and an associated world model (including practical knowledge of concepts, such as gravity, that arise due to the physics our world exists in, as well as other cultural and social signals), which would include a model of self (knowledge of our own sensorimotor space), a theory of mind, and those also are likely sensing other signals outside of the raw text of the message such as body language, intonation, a shared physical context etc. These are simply not present in a ChatGPT or similar technology. Human understanding and meaning appears to arise from a grounding of language in concepts through joint attention, embodiment and associated world model, and perhaps multimodality may also be required (see literature on embodied cognition e.g., <cit.>). In other words, on their own LLMs at present don't understand, and they are not sentient. For LLMs to gain understanding, either more sophisticated LLMs will need to be developed that successfully capture intent, and/or additional "layers" of capability will be required that will somehow implement understanding. How that might be achieved is very much an open research problem. For the current generation of LLM-based technology, the human user is the language model layer that is implementing all the understanding by conferring meaning onto the prompts and generated text. §.§ Generative stochasticity The nature of the process used to generate the text outputted by a large language model is that it includes a degree of stochasticity (i.e., randomness). Bender et al <cit.> have referred to LLMs as Stochastic Parrots. We can consider a LLM such as ChatGPT to be a complex “auto-complete" tool. That is, given a list of words (the prompt) provided by the user, the LLM passes that prompt through its model to generate the next word that might follow the prompt, and then the next word following that, and the next word following that word, and so on. Note that we simplify this description to ease discussion, so that, for example, we draw an equivalence between tokens (the lowest level primitives that LLMs process and generate) and words. When selecting the next word to output, there may be a number of words available to the LLM that might be equal or similar in likelihood to occur. This is where the randomness comes into play. One of these words is chosen at random. If you have ever used ChatGPT, this explains why if you give the same prompt over and over again, each time the response will be different. This is generative stochasticity at work. Coupled with the inability to understand meaning, generative stochasticity also plays its part in the generation of nonsensical output. LLM output is strongly linked to the training data used to create the LLM. That is, an LLM will never make a next-word prediction that it has not seen in the training data, and conversely, it favours those words which are more strongly represented in the training data. A truly intelligent and creative agent/system should be capable of selecting a word based on first-order logical and deductive reasoning thus reducing the generation of nonsensical output to nearly zero, as opposed to the current mechanism LLMs employ which is based on probabilistic stochastic selection, adding weight to our categorisation of LLMs as complex “auto-complete" tools. §.§ Anthropomorphism We humans are flawed, we have many cognitive biases <cit.>. Perhaps AI might be useful in the future to assist us in counteracting our cognitive biases. When engaging with a chatbot (underpinned by an LLM or even some other language model) through an interface where we enter a prompt, and the bot replies by seemingly typing out letter by letter, word by word a response, perhaps even pausing before starting to respond. The delay in response may be deliberately engineered by the chatbot designers, or it could arise as a result of the large amount of processing that might be required to generate a response due to the size of the LLM. The chatbot interface may even provide a human-like name to the chatbot agent. In such an environment, even if we are told the chatbot is a computer program and not a human, we can still succumb to an "Eliza effect" <cit.>. Our learned ability to process and find meaning in conversational text means that we slip, and often forget that we are not dealing with a human, and very quickly we anthropomorphise the bot. We may attribute personality traits to it. We may attribute an intended meaning to the generated text. Anthropomorphism of this nature is not trivial and can have serious consensuses, indeed previous research has highlighted people's tendency to “overtrust" robotic systems with potentially disastrous outcomes <cit.>. Arguably the designers of tools such as ChatGPT should provide interfaces that make it much clearer (and perhaps in a repetitive, ongoing basis) to the human user that this is a bot. They should also be careful with the language they use to describe their models, systems and services, avoiding terms that might be associated with human abilities. However, the tools are designed to be appealing to the human user by being human-like. It also does not help that anthropomorphic terms are used to describe nonsensical output, such as describing this as the model sometimes "hallucinates", rather than just clearly stating that it can make stuff up, generating text that is "factually incorrect" or simply "nonsense". §.§ The training data contains biases, nonsense and harmful content Due to the data upon which many of the LLMs that currently exist are trained, the output generated by LLMs may contain harmful, biased, intolerant, hate and factually incorrect content. The reality is, that in the case of ChatGPT, we simply do not know what the underlying training data set is that was used to develop the GPT-4 LLM, as OpenAI have not shared this information. We know that GPT-3 was trained on a combination of sources drawn from the wilds of the internet, including Common Crawl data from 2016 to 2019, Webtext, Books and Wikipedia <cit.>. Analysis of Common Crawl data has shown that it contains harmful content, such as hate speech and sexually explicit content (e.g., <cit.>). We're basically talking about the training data including the content on web pages drawn from across the world wide web. As such it is not surprising if we find mis- and dis-information, stereotypes and bias, falsehoods, hate speech, sexist, and racist content, etc., in the training data. Given the content of the training data, coupled with the power of these models to faithfully capture the training data, and the stochastic nature of these models, it is always possible that harmful content, bias, mis- and dis-information, and outright fabrications will be output. As Kidd & Birhane <cit.> highlight, these models then have the potential to distort human beliefs. And, this is reinforced by the hype that is associated with the technology, in its overstating its capability. And, further driven by our human tendency to anthropomorphise the technology and its ability, as discussed earlier. To quote Kidd & Birhane "People form stronger, longer-lasting beliefs when they receive information from agents that they judge to be confident and knowledgeable..." <cit.> Attempts to refine the outputs resulting from the LLM training data have been made using techniques such as RLHF. Although effective in part, the RLHF process manifests its own issues regarding biases and content. There exists a subtle trade off between the inconsiderate curating of free speech and the moderation of harmful content. The lines separating these two actions are not always so black and white, they likely depend heavily on ones own beliefs, culture, religion and heritage, etc. The creators of LLMs and technologies derived from them are currently the ones who draw these lines, thus making it likely that these models will inherently reflect their own morals and beliefs, given that the four most prominent models in production today have been developed by companies founded and head quartered in the United States of America. One might postulate that attempts to remove biases, nonsense and harmful content from the training data may also come with certain degrees of diversity and cultural suppression by way of amplifying these companies own beliefs and traditions. §.§ Security & Privacy As discussed in the introduction, there is the possibility of sensitive data being passed to a third party providing the LLM as SaaS, and/or sensitive data being ingested into the model during model training and later massaged out of it through prompt engineering. Recent work by <cit.> has highlighted the propensity of Large Pre-Trained Language Models to leak private or sensitive information contained within their training data corpus. These leaks are possible because of the model's ability to "memorize" sequences of tokens at training time <cit.>. Ironically, Huang et al, also found that the models' inability to associate related pieces of information (e.g. Name, Email Address, etc) made it more difficult for attackers to extract the information. Nonetheless, this fragility poses the real and significant threat of exposing the personal information of individuals contained in an LLM training data set, information that individuals may not have agreed to share for the original or otherwise intended purpose of training LLMs. §.§ Correlation is not causation The language of statistics is not amenable to capturing causation (e.g., see <cit.>), and correlations do not imply causation. The current generation of LLMs are effectively statistical models of token/word associations, and as such are models of correlation. These and other foundation models do not capture the Why or causation of association. As such they are not capable of causal reasoning. §.§ All models are wrong, some are useful All models are simplifications of the reality they attempt to represent. LLMs are not an exception to this. Simplifications can arise in the representative capacity of the deep learning approach (the computation being undertaken at each node in the network, the parameter ranges and algorithm strategies e.g., the type and rate of learning), and as highlighted earlier there are clearly issues with the representative nature of the training data. The training data does not provide perfect coverage of all possible sentences that might be expressed in every human language, in fact, this is impossible given that is an infinite space! Another interesting feature of human language, is that it changes over time, it might be said to evolve through the generations of humans that live the language in an evolving society <cit.>. To be relevant in terms of the latest developments in, for example, science and society, and to embrace the adaptive nature of the language itself, LLMs will need to be periodically re-trained. Given the estimated cost of training ChatGPT being in the order of 100 Million US dollars, and the duration that it takes to run the training (months), this is not a task that will be undertaken lightly or frequently, and as such the relevance of the underlying language model will start to decay from the moment training stops. As to the rate of decay and when it would stop being useful, will likely depend on the context in which it is being used, and also in proportion to the rate of the change language, the rate of generation of scientific knowledge, the rate of generation of artistic and cultural output, and the rate of generation of news, etc. §.§ Leaky data makes it hard to evaluate true performance We often do not know what comprises the data used to train LLMs. For example, OpenAI have not revealed the training data for GPT-4. This presents a significant challenge to rigorously evaluate the performance of these models in a scientific manner. The traditional way to measure the performance of a (machine learning) model is to test the model on "unseen data". That is data that the model has not been trained on. We refer to this as the test (data) set. This test set performance measurement sets out to determine the level of ability the model might have to generalise beyond what it has been trained on. For example, has the model learned general concepts and principles that can be applied to a wider set of circumstances? But if the test set contains data that is included in the training set, we have what is known as "leaky data", and we cannot be sure if the model is simply regurgitating what it has already been exposed to, akin to a photographic memory without necessarily understanding the meaning of what it has stored in its memory. Interestingly, the proposed amendments to the EU AI Act, which are currently working their way through the European Parliament, will require those who deploy LLMs to reveal the training data used by the LLM. §.§ Emergent Abilities are misleading Recently the term "emergence" has started to be strongly associated with the abilities of large language models. This term has many interpretations but in the context of LLMs emergence generally refers to abilities or behaviours that are not present in smaller models but are present in larger ones, such that these abilities or behaviours could not have been predicted by simply extrapolating the performance of smaller models <cit.>. The idea that LLMs have surpassed some necessary scale needed to give rise to unpredictable and inherently uncontrollable emergent abilities has fuelled much of the rhetoric and hysteria surrounding their future potential to cause catastrophic harm by acting autonomously and adaptability through an agent of some kind. At the moment this is the stuff of science fiction and distracts from the very real potential for harm that LLMs present. Whilst the claims of LLM emergence continue to prevail little scientific analysis has been conducted to evaluate the type and true extent of emergence in LLMs. Initial work has postulated that emergence in LLMs can be largely explained by the choice of metric a researcher uses to evaluate fixed model outputs <cit.>. The author's findings concluded that when using the InstructGPT/GPT3 family of models, the presence of emergent abilities or behaviours could be modulated through the use of different metrics and improved statistical analysis, thus pouring cold water on the theory that the presence of emergence in LLMs is a property of their scale. Other theories relating to the "emergence" of the human-like reasoning and understanding abilities sometimes exhibited by LLMs also support a much more grounded statistical explanation suggesting that LLMs exploit the distributional properties of language in latent space <cit.>. While research is ongoing to explain the emergent-like behaviour of LLMs, all be it hampered by the lack of access to propriety models and data, it is important not to be misled by claims that LLMs do indeed have inherent emergent abilities and by extension could be capable of developing enhanced capabilities in a random uncontrollable fashion or produce outputs which are the result of an “artificial intelligence" which has emerged during the models development process. Another perspective on emergence, arises when a technology such as a LLM becomes part of a larger technology solution or system. What might be the unintended consequences that might emerge, especially in light of the various limitations of the underlying LLM itself? How might these limitations be expressed and ripple through the behaviour of the larger system of which it is part of? §.§ Rights-laundering The training data of many foundation and large language models include copyrighted material and material published under licences with varying permissiveness. It is entirely possible, and many examples have been reported, where copyrighted or licensed material is retrieved or reproduced as the output of a LLM. Lawsuits are even being filed as a result <cit.>. Who owns the output of a LLM or other foundation model? Is it the person entering the prompt? Is it the people or organisation responsible for the foundation model? Does the foundation model own it? Or is it jointly owned? If the original copyright or licence holder is not protected this effectively amounts to licence laundering or copyright laundering. In a disturbing twist, it is possible to use some of these foundation models to generate voice, image, and video versions of individuals (deep fakes). Effectively an individual person can be impersonated through media whether or not they give permission for this. For what purpose might this be adopted, and by whom? How will society at large be capable of distinguishing the virtual from the reality? This potential for rights-laundering does suggest that regulations are required as soon as possible and that greater responsibility be placed on the developers of these technologies to limit misuse. If for a moment we are to overlook the potential breaches of law that might arise in the use of this technology. In light of the potential for rights laundering, we should be asking ourselves if it is good ethical practice to automatically assume that we can adopt the output of a LLM as if we own it and have the right to use it. Just because we can use the tool and its output, we should be asking ourselves, should we use the tool and its output? §.§ Sustainability Bender et al <cit.> highlight the unsustainable nature of the current generation of LLM technology, given the significantly increasing amount of energy and resources required to train and deploy LLM-technology. They also note the environmental racism that arises when the world's most marginalised communities are the ones most deeply impacted by climate change arising from technology development and use by more privileged communities. We have to ask ourselves are we prepared to accept the environmental and societal cost? Or should we be focused on developing more sustainable AI technology? §.§ Regulatory non-compliance We mentioned earlier the potential of foundation model technology for rights-laundering in the output it might generate. Thankfully, efforts are already underway to move to regulate AI technology. The EU are one of the early movers in this regard having initially developed “Ethics guidelines for trustworthy AI" <cit.> in 2019, and in 2021 proposed a draft for an AI Act, which sets out to regulate AI technology such as foundation models. If the current draft of EU AI Act, which passed through the European Parliament in June 2023, comes into effect some LLM's (e.g., ChatGPT) will not be in compliance with the regulation due in part to non-disclosure of the training data that is copyrighted <cit.>. What then happens to the various services that organisations are building on top of the likes of ChatGPT? §.§ Transparency In terms of the trustworthiness of foundation model AI technology, there are a number of deficits that we have already touched upon related to security and privacy, rights violations, harmful and factually incorrect content generation etc. Here we focus on transparency, or lack there of. This lack of transparency exists in a number of ways. Firstly, in terms of the lack of transparency around the data upon which many of these models are trained upon, secondly in terms of the associated practices (e.g., potentially exploitative labour practices) that might be used building guard rails and in training the RLHF. Thirdly, in terms of the lack of documentation and disclosures describing many of the commercial systems that are currently available. And fourthly, in terms of the inherent nature of the underlying foundation model technology of deep learning. In the fourth case this exposes a fundamental limitation of deep learning technology, which has given rise to a significant open area of research (e.g., the sub-field of explainable AI). At one level we can describe the current generation of foundation model technology to be black-box in nature. That is, they receive some input (e.g., a user prompt) and generate a response (e.g., generated text), but we have no human-interpretable explanation of how the underlying model generated the response from the output. It is technically possible, and extremely laborious, to observe each of the millions or billions of low-level calculations that occur in generating the response, but these are so abstracted away from concepts in the real world that they are effectively meaningless and reveal no true explanatory insight. This raises serious questions and limitations in how the technology should be deployed, and touches again on the issue of regulatory non-compliance. If a foundation model is used as part of some decision-making system (whether automated or not) this means it will be extremely challenging, if not impossible, to extract an explanation for each decision or recommendation made. By way of example, if we are to consider existing regulations such as the EU GDPR <cit.>, European citizens have a right to explanation of automated decisions that significantly affects them or produces legal effects, and they have the right to challenge those decisions. If a foundation model is part of such a decision-making process how can a citizen be provided with a meaningful explanation? §.§ Equality, Diversity & Inclusion Finally, we also need to be aware that the field of AI is (or at least, should be) inherently multidisciplinary. It draws upon a number of foundational disciplines including computer science, robotics, neuroscience, statistics and mathematics, psychology, linguistics, philosophy, biology, evolution and genetics, complex systems, humanities and social sciences, ethics and law, to name a few. At its best AI is an inherently interdisciplinary science, bringing researchers from all of these communities together in a collaborative effort. To truly understand intelligence, to build an artificial intelligence equivalent to (or even superior to) that of living organisms, and to understand the limitations and implications of the current state of the technology arising from the field of AI we need to draw upon the expertise of researchers from these core disciplines. In terms of large language models in particular, we would suggest that at a minimum we should listen to experts in linguistics (e.g., Prof Emily Bender and collaborators), statistics and causal inference (e.g., Prof Judea Pearl), and of course machine learning generally. This points to another potential limitation of many existing LLMs. They and their associated tools have not always been developed, tested and deployed by diverse teams. Diversity in this case meaning inclusion of the diverse set of core disciplines of expertise that should make up AI, in addition to the other desirable diversity attributes of a team designed to better represent a global society, including ethnicity, race, gender, sexual orientation, age, religious beliefs, socio-economic status, civil status, physical impairment, learning difficulties, and neurodiversity. We might argue, that if the teams in organisations responsible for developing and deploying AI technology such as LLMs were truly diverse, that many of the limitations and resulting issues of the technology might be more successfully addressed. § CONCLUSIONS Given the level of hype surrounding Large Language Models and ChatGPT in particular, it is important to maintain a sense of balance and perspective. We have to zoom out and think of the bigger picture here. If a single company invests over 11 Billion dollars in a technology, what are the expectations of that company? For context, the Irish Governments Budget package for 2023 was 11 Billion Euro. In other words, you could run a small country with that kind of money. Arguably, with that level of investment, an organisation will want to maximise their return. Naturally, any signs of utility are going to be exploited and marketed. For example, perhaps they might allude to the magical properties of the technology, suggesting that everyone needs to get their hands on it to succeed in business and life more generally. The fact that the technology being hyped is itself capable of generating stories, blogs, and tweets at the speed of light is probably helping to feed the hype. They might allude to the potential of the technology to lead to what are science fiction Hollywood scenarios of AI being a threat to the existence of humanity and taking over the universe. If these organisations truly believe this to be a potential reality, then (at this moment in time) they have the power to stop it from happening. As they alone have the computing and data resources available at sufficient scale to realise an AI technology at the level of a ChatGPT, never mind at the level of what might be an artificial super intelligence. They could choose not to develop the technology, rather than call for others to stop them through regulation. However, whether we like it or not, the genie has left the bottle, and the technology is out in the wild. As such, we need to adapt to live with it and equip ourselves with some understanding of both the opportunities and the limitations of this technology as it evolves. Ironically, LLM technology also has the potential to pollute the future training data sets of the next generation of LLMs, and even the re-training of existing model architectures with even more garbage. This might result in a negatively reinforcing feedback loop that further propagates the generation of increasing amounts of misinformation, bias, hate speech etc. As the machine learning adage goes, "garbage in, garbage out". In other words, if you train your model on data that is inappropriate for the task at hand, the resulting behaviour or output of the model will be useless. We have to constantly remind ourselves, that for the current generation of foundation model technology, no company that sells or uses this technology can guarantee that all the output generated will be factually correct, that it won't contain bias and/or harmful content WITHOUT having a HUMAN read it. Everyone NEEDS to read the small print. The creators of these models basically admit this to be the case, for example, see OpenAI's Usage Policy <cit.>, which, does not permit ChatGPT use for legal and financial advice, or in health for diagnosis, treatment advice or disease management, or for government decision-making. We asked ChatGPT itself what it should not be used for, and the generated response is provided in Figure <ref>. In other words, we need a working language model (e.g., a human) to check the output of the AI language model. It will take an extraordinary effort and perhaps an infinite amount of time for, for example, reinforcement learning with human feedback to provide the necessary "guard rails" with an equivalence to a humans language model, or for humans to manually codify all the rules required! 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http://arxiv.org/abs/2307.00228v1	20230701052328	InferTurbo: A Scalable System for Boosting Full-graph Inference of Graph Neural Network over Huge Graphs	[ "Dalong Zhang", "Xianzheng Song", "Zhiyang Hu", "Yang Li", "Miao Tao", "Binbin Hu", "Lin Wang", "Zhiqiang Zhang", "Jun Zhou" ]	cs.LG	[ "cs.LG" ]	InferTurbo: A Scalable System for Boosting Full-graph Inference of Graph Neural Network over Huge Graphs Dalong Zhang, Xianzheng Song, Zhiyang Hu, Yang Li, Miao Tao, Binbin Hu, Lin Wang, Zhiqiang Zhang, Jun Zhou^∗ Corresponding author {dalong.zdl, xianzheng.sxz, zhiyang.hzhy, ly120983, taotao.tm, bin.hbb, fred.wl, lingyao.zzq, jun.zhoujun}@antfin.com Ant Group, Hangzhou, China August 1, 2023 ================================================================================================================================================================================================================================================================================================= With the rapid development of Graph Neural Networks (GNNs), more and more studies focus on system design to improve training efficiency while ignoring the efficiency of GNN inference. Actually, GNN inference is a non-trivial task, especially in industrial scenarios with giant graphs, given three main challenges, i.e., scalability tailored for full-graph inference on huge graphs, inconsistency caused by stochastic acceleration strategies (e.g., sampling), and the serious redundant computation issue. To address the above challenges, we propose a scalable system named InferTurbo to boost the GNN inference tasks in industrial scenarios. Inspired by the philosophy of “think-like-a-vertex", a GAS-like (Gather-Apply-Scatter) schema is proposed to describe the computation paradigm and data flow of GNN inference. The computation of GNNs is expressed in an iteration manner, in which a vertex would gather messages via in-edges and update its state information by forwarding an associated layer of GNNs with those messages and then send the updated information to other vertexes via out-edges. Following the schema, the proposed InferTurbo can be built with alternative backends (e.g., batch processing system or graph computing system). Moreover, InferTurbo introduces several strategies like shadow-nodes and partial-gather to handle nodes with large degrees for better load balancing. With InferTurbo, GNN inference can be hierarchically conducted over the full graph without sampling and redundant computation. Experimental results demonstrate that our system is robust and efficient for inference tasks over graphs containing some hub nodes with many adjacent edges. Meanwhile, the system gains a remarkable performance compared with the traditional inference pipeline, and it can finish a GNN inference task over a graph with tens of billions of nodes and hundreds of billions of edges within 2 hours. Graph Neural Networks, Distributed System, Full-graph Inference, Big Data § INTRODUCTION Graph Neural Networks (GNNs) generalize deep learning over regular grids (e.g., images) to highly unstructured data and have emerged as a powerful learning tool for graph data. Recently, GNNs have demonstrated impressive success on various tasks, ranging from traditional graph mining tasks (e.g., node classification<cit.> and link prediction<cit.>) to the fields of natural language processing<cit.> and computer vision<cit.>. Meanwhile, graph data forms the basis of innumerable industrial systems owing to its widespread utilization in real-world applications (e,g., recommendation<cit.>, marketing<cit.>, fraud detection<cit.>), further facilitating the flourishing of GNNs in industrial communities. The key idea of current GNNs is taking advantage of information aggregation schema, which can effectively summarize multi-hop neighbors into representations via stacking multiple GNN layers. Essentially, the high computation cost exponentially growing with the number of GNN layers poses the non-trivial challenges for GNN Training and Inference in industrial scenarios with huge graphs, containing billions of nodes and trillions of edges. With the aim of the possibility of performing the highly scalable and efficient GNN training in real industrial scenarios, several recent efforts have been made in industrial-purpose graph learning systems, including single-machine systems with the cooperation between GPUs and CPUs (i.e., DGL<cit.> and PyG<cit.>), distributed systems based on localized convolutions (i.e., Aligraph<cit.>) and pre-processed information-complete sub-graphs (i.e., AGL<cit.>). Unfortunately, the main focus of these systems is training GNNs efficiently on graphs at scale, while few attention has been paid to the inference phase of GNNs [Different from online inference whose goal is guaranteeing the low latency of real-time online serving <cit.>, the focus of GNN inference in our paper aims at efficiently performing full-graph inference in the offline environment, which is ubiquitous in financial applications.], which is also the core of an industrial graph learning system. In current graph learning systems<cit.>, the inference phase is conducted by blindly imitating the pipeline of the training phase. However, the inference phase of GNN has its unique characteristics, making it distinct from the training phase, such that the pure design of inference stage in current graph learning systems is unsuitable for performing large-scale inference tasks in industrial scenarios. The large gap of data scale between training and inference phases. In a general graph learning task, only a small number of nodes (e.g., 1% or even less in actual scenarios) are labeled for GNN training, while the inference tasks are usually expected to be taken over the entire graph. Such a gap is steadily worsening in the industrial scenarios with huge graphs, containing hundreds of millions or even billions of nodes<cit.> (e.g., Facebook[https://en.wikipedia.org/wiki/Facebook,_Inc.], Ant Group[https://en.wikipedia.org/wiki/Alipay]). Both observations encourage us to re-consider the scalability issue encountered in the inference tasks with the careful thought of memory storage and communication bandwidth. * No guarantee for the consistency of predictions in inference phase. As a basic requirement in industrial scenarios, the prediction score for a certain node should keep consistency at multiple runs, which unfortunately gets no guarantee in the GNN inference of current systems. To extend GNNs to large-scale graphs, the k-hop neighbor sampling strategy is widely adopted in current graph learning systems<cit.>. Although the training phase could greatly benefit from the efficiency-oriented strategy for generalized GNNs, the inherent stochasticity for prediction is unacceptable in inference phase, especially for financial applications (e.g., fraud detection and loan default prediction<cit.>). * Serious redundant computation issue in the inference phase. Current graph learning systems<cit.> follow the procedure that performs forward and backward of GNNs over k-hop neighborhoods in a mini-batch manner, and would obtain a well-trained GNN by repeating this procedure enough times. In spite of its potential capability of enjoying mature optimization algorithms and data-parallelism in training phase, k-hop neighbor sampling with the mini-batch training manner would introduce the undesirable redundant computation issue in the inference phase since only one forward pass of GNNs is required. Addressing the limitations of current graph learning systems, we aim to facilitate GNN inference in industrial scenarios through a collaborative setting of mini-batch-based training and full-batch-based inference. The significance of the cooperative setting is not trivial considering two major challenges that are not explored in current graph learning systems: * C1: How to unify the mini-batch based training and full-batch based inference in a graph learning system? As mentioned above, the mini-batch strategy is the promising way for efficient GNNs training, while the full-batch strategy is suitable for inference to alleviate computation issues. To combine those two advantages, it is impressive to abstract a new schema of graph learning that a GNN trained in the mini-batch manner could naturally perform inference in the full-batch manner. * C2: How to efficiently perform GNN inference in distributed environments, especially for huge graphs with extremely skew degree distribution? To ensure the consistency of prediction in inference tasks, we discard the strategies related to neighbor sampling, which put forward the urgent request for efficient GNN inference on huge graphs. In addition, due to the ubiquity of Power-Law, it is critically important to handle long-tailed nodes with extremely large degrees in terms of time cost and resource utilization, as well as avoiding the unexpected Out Of Memory (OOM) issues. To this end, we design a system named InferTurbo to boost the inference tasks on industrial-scale graphs. We propose a GAS-like (Gather-Apply-Scatter) schema<cit.> together with an annotation technique to describe different stages of the data flow and the computation flow of GNNs, which could be used to unify the mini-batch training and full-graph inference phases. The training and inference phases share the same computation flow but use different implementations of data flow. The data flow parts are made as built-in functions currently, and users would only focus on how to write models in our computation flow. Moreover, to handle the corner cases in natural graphs in industrial scenarios, such as nodes with extremely large amount of in-edges or out-edges, a set of strategies without any sampling like partial-gather, broadcast, and shadow-nodes are proposed to balance the load of computation and communication and mitigate the long tail effect caused by those nodes. As a result, consistency prediction results could be guaranteed at different runs, and the system enjoys a better load-balancing. At last, the system is implemented on both the batch processing system<cit.> and the graph process system<cit.>, which makes it gain good system properties (e.g., scalability, fault tolerance) of those mature infrastructures. The main contributions of this work are summarized as follows: * We propose a GAS-like schema together with an annotation technique to describe the data and computing flow of GNNs, and make it possible to hierarchically conduct the inference phase over a full graph. In this way, the redundant computation caused by k-hops neighborhood is eliminated in inference phase. * We describe implementation details of the system on both batch processing system<cit.> and graph computing system<cit.>, which are mature infrastructures and widely available in industrial communities. * We design a set of strategies such as partial-gather, broadcast, and shadow-nodes for inference tasks to handle the power-law problem in industrial graphs. * Compared with some state-of-the-art GNN systems (DGL, PyG), the proposed system demonstrates remarkable results in efficiency and scalability while achieving comparable prediction performance, and it could finish a node classification task over a graph with tens of billions of nodes and hundreds of billions of edges within 2 hours. § PRELIMINARIES §.§ K-hop Neighborhood and Sampling A directed, weighted, and attributed graph is denoted as 𝒢={𝒱,ℰ,𝐗,𝐄}, where 𝒱 and ℰ⊆𝒱×𝒱 are the node set and edge set of 𝒢, respectively. 𝐗 and 𝐄 are node features and edge features. The k-hop neighborhood 𝒢_v^k of node v, is defined as the induced attributed subgraph of 𝒢 whose node set is 𝒱_v^k = {v}∪{u \| d(v,u) ≤ k}, where d(v,u) denotes the length of the shortest path from u to v , and edge set is ℰ_v^k = {(u,u') \| (u,u')∈ℰ u∈𝒱_v^k u'∈𝒱_v^k }. Additionally, 𝒢_v^k also contains the feature vectors of the nodes and edges. Without loss of generality, node v itself is treated as its 0-hop neighborhood. It is proved that k-hop neighborhood would provide sufficient and necessary information for k-layer GNNs<cit.>. However, the size of k-hop neighborhood grows exponentially with the number of hops, making the computations performed on it memory-intensive and time-consuming. K-hop sampling is proposed to address those problems. It would sample neighbors in a top-down manner, i.e., sample neighbors in the k-th hop given the nodes in the (k-1)-th hop recursively. In particular, different sampling approaches<cit.> employ different ways to select the neighbors, and one typical method is to randomly choose a fixed number of neighbors in each iteration. §.§ Graph Neural Networks Graph neural network is a popular way to learn over graph data, and it aggregates neighbors' information for a certain node in an iterative way. Given an attributed graph 𝒢, a GNN layer can be formulated in a message passing paradigm<cit.>, which first computes messages via edges and then updates the state of the target node with those messages: [ message: m_v,u^k+1 = ℳ(𝐡_v^k, 𝐡_u^k, e_v,u),; ; update: 𝐡_v^k+1= 𝒰(𝐡_v^k, ℛ({m_v,u^k+1}_u ∈𝒩^+_v)),; ] where 𝐡_v^k denotes intermediate embedding of node v at the k-th layer and 𝐡_v^0 = 𝐱_v, e_v,u and m_v,u^k+1 indicate edge features and messages associated with v's in-edge from u respectively, ℳ represents the message function that generates messages according to adjacent node features and edge features on each edge, 𝒰 means the update function that updates node embedding by aggregating incoming messages based on the reduce function ℛ. The message function and update function are usually neural networks, while the reduce function could be a certain pooling function (such as sum, mean, max, min) or neural networks. Most GNN algorithms (e.g., GCN<cit.>, GraphSAGE<cit.>, GAT<cit.>) could be expressed in this paradigm with a few changes in those functions. For example, message functions for GCN, GraphSAGE, and GAT, only take the source node's hidden state (or with edge features) as input. In addition, reduce functions for GCN and GraphSAGE are set to pooling functions, while for GAT, the reduce function would perform a weighted sum over messages based on the attention mechanism. § RELATED WORKS §.§ Graph Processing System The idea of message passing could be traced back to the graph processing field, which focuses on understanding the structure properties or topology of graphs, such as finding out the most important vertices (e.g., PageRank<cit.>), seeking the shortest path between nodes (e.g., SSSP<cit.>), and so on. In the following, some gifted works will be introduced as they motivate our work a lot. In the graph processing field, with the rapid growth of graph data, many efforts<cit.> have been paid to conduct graph processing tasks distributedly to handle extremely large graphs. Pregel<cit.> adopts a “think like a vertex" programming model, in which each node receives messages from its neighbors in the previous iteration, modifies its own state or mutates the graph topology, and sends messages to other nodes via its out-edges. PowerGraph<cit.> further develops the vertex-centric programming model with three fine-grained conceptual phases to describe the data flow, i.e., Gather, Apply, and Scatter (GAS). The gather and scatter are a pair of symmetrical operations that demonstrate the information flowing in and out a node and perform edge-wise processing on the corresponding in-edges and out-edges. The apply stage is used to update the state of the node itself. It enables a node-level paralleling by assigning nodes to partitions, along with their states and adjacent edges. Many graph processing algorithms (e.g., PageRank, SSSP) could be expressed in this paradigm. The following works<cit.> expand the abstraction and propose many talented strategies to optimize graph processing systems such as partitioning, communication, and so on. The key difference between graph processing algorithm and GNN is that the former mainly focuses on graph structure, while the latter usually models graph structure and rich attribute information together and is optimized by gradient-based algorithms (e.g., SGD<cit.>). That is, GNNs are more compute-intensive and communication-intensive, and data dependency exists in both forward and backward pass. §.§ Graph Learning System Inspired by the development in deep learning communities, researchers also design and build systems to perform graph learning algorithms based on some deep learning frameworks, such as TensorFlow<cit.> and PyTorch<cit.>. Many works<cit.> build graph learning systems following the message-passing paradigm to unify different GNN variants, which help researchers design new GNN algorithms in an easy way. The development of graph learning systems evolves from optimizing systems on a single machine<cit.> to providing an efficient way to learn over large graphs distributedly<cit.>. Inspired by the k-hop sampling strategy<cit.>, recent works<cit.> perform localized convolutions on sampled k-hop neighborhoods to mitigate the inherent data dependency problem and enjoy the mature data-parallelism architecture in graph learning. Specially, Aligraph<cit.> implements distributed in-memory graph storage engine, and workers will query subgraphs for a batch of training samples and do the training workloads. DGL<cit.> proposes to use a partitioning algorithm when building the graph storage engine to reduce the number of edge cuts between different partitions, which is helpful to reduce the communication overhead when querying subgraphs of training samples. PyG<cit.> provides lightweight, user-friendly sampling APIs, whereas users usually need to prepare an extra graph storage engine for distributed training. AGL<cit.> proves that the k-hop neighborhood could provide sufficient and necessary information for k-layer GNN models and pre-generates the k-hop neighborhoods with a MapReduce pipeline to solve the inherent data dependency problem. It also provides a way to estimate the errors caused by random sampling strategy, and errors would decrease as the sampling number increases. Users might trade off computing speed and prediction accuracy by changing the number of neighbors for each layer. However, little attention has been paid to the inference phase of GNNs. Many works thought that inference tasks could be perfectly conducted in the same way as the training phase, which is hard to meet the requirements of inference tasks in industrial scenarios. The inference should be efficiently conducted over huge graphs with skewed degree distribution, and the prediction result should keep consistent at different runs. A few existing works<cit.> mainly focus on the efficiency problem in the inference phase on a single machine and propose some techniques, such as operation fusion and graph-specific caching, to reduce intermediate results and achieve better IO speed. AGL<cit.> proposes a full-graph inference module to mitigate the redundant computation problem caused by k-hop neighborhoods but misses to solve the consistency problem and straggler problems caused by `hub' nodes in industrial scenarios. It is still challenging to build an efficient and scalable inference system for industrial applications. § SYSTEM This section will first provide an overview of the InferTurbo inference system. The system is then detailed from three aspects: the programming model abstraction, implementations using various backends, and large-graph optimization techniques. §.§ System Overview The key motivation of the work is to build an efficient GNN inference system on top of scalability for huge graphs in industrial scenarios. Therefore, instead of designing and optimizing inference system over a single monster machine, we'd rather build the system on mature, scalable infrastructures with high throughput capacity, such as batch processing systems and graph processing systems. In addition, to boost the inference efficiency, the system is expected to address the following issues: serious redundant computation caused by inferring on k-hop neighborhoods and the straggler problem brought on by skewed degree distribution. Furthermore, as the consistency of prediction results at different runs is a fundamental requirement for industrial applications, some optimization strategies with randomness should be avoided. To this end, we propose InferTurbo to boost the inference tasks over industrial graphs. The overall architecture is illustrated in Fig. 1. First, a GAS-like abstraction is proposed to describe the data flow and computation flow of GNNs by combining the classical message-passing schema in (<ref>) and the GAS schema in graph processing system. This abstraction is utilized to integrate the mini-batch training phase and full-batch inference phase. In this way, the inference phase would not rely on k-hop neighborhoods and thus avoids redundant computation, while we could still enjoy the mature optimization algorithms in mini-batch manner and the data parallelism in training phase. In addition, by designing and implementing adaptors, a well-trained GNN model in our abstraction could be deployed on batch processing systems or graph processing systems, which are mature industrial infrastructures with properties of high throughput and scalability. Applications could choose one of them as the backend by trading off the efficiency and resource cost. Furthermore, a set of strategies without dropping any information are proposed to handle problems caused by nodes with a large degree, since the degree distribution could be skewed for industrial graphs. With those strategies, our system could achieve consistent prediction results at different runs and gain better load-balancing by mitigating the stragglers caused by those “hub” nodes. §.§ InferTurbo Abstraction In industrial scenarios, there are some key distinctions between the training and inference phases of GNN algorithms. In the training phase, the labeled nodes may be one percent of all nodes or even less, and the optimization procedure would be repeated several times on them to obtain a converged model. It is a wise decision to conduct localized GNNs on those labeled nodes based on their k-hop neighborhoods since we can benefit from data parallelism and sophisticated optimization methods in the mini-batch setting. The cost is also reasonable since labeled nodes would be scattered throughout the whole graph, and there would not be many overlaps between k-hop neighborhoods of corresponding labeled nodes. On the contrary, inference tasks usually should be conducted on the entire graph. Forwarding localized GNNs over the k-hop neighbors of all those nodes might result in the redundant computation problem. A good way to solve it is to design a full-batch distributed inference pipeline and bridge the gap between those two different modes in training and inference phases. Inspired by the philosophy<cit.> of “think-like-a-vertex" in the graph processing field, we re-express the classical message-passing schema of GNNs to a GAS-like abstraction to unify the mini-batch training and full-graph inference phases. As shown in Fig. <ref>, for a certain GNN layer, the GAS-like abstraction can be defined as follows: * Gather. * gather_nbrs (Data Flow): A “vertex" receives messages from its in-edges and then vectorizes the collected information into tensors. * aggregate (Computation Flow): The “vertex" would preliminarily process the messages from its in-edges, which is quite similar to the reduce function in ℛ in (<ref>). The difference is that this process should obey the commutative law and associative law (like max/min/sum/mean pooling or union) for further optimization in inference phase. Otherwise, the computation should be placed in the next stage. * Apply. * apply_node (Computation Flow): The “vertex" then updates its state information by combing the former state of itself and the “gathered" message from the former stage. * Scatter. * apply_edge (Computation Flow): The “vertex" would generate messages according to the updated state information together with edge features. * scatter_nbrs (Data Flow): The “vertex" sends messages via out-edges. Those five stages are used to describe the data flow and computation flow in GNNs, and their roles are emphasized by underlining and annotating the corresponding part. Compared with the classical GAS schema, both the Gather and Scatter are expanded to two sub-stages to distinguish the data flow and computation flow in those stages. In general, gather_nbrs and scatter_nbrs, a pair of symmetry operations in the data flow, are used to receive and send messages via in-edges or out-edges, respectively. We make them as built-in methods since they are the same among a variety of commonly used GNNs. Computation stages would vary for different GNNs, and users should override those stages according to their specific requirements. Specially, a rule is defined to set a boundary between aggregate and apply_node stages: the computation of the aggregate should obey the commutative law and associative law. Oherwise, related operations should be placed in the apply_node stage. For example, many pooling functions (such as sum, mean, max, min), used as the reduce function in GCN and GraphSAGE, should be placed in the aggregate stage following the rule. For GAT, the computation of attention would break the rule, and thus, we simply union messages in the aggregate stage and perform the reduce function in the apply_node stage. This rule also facilitates further optimizations, which will be detailed in the next several sections. The mini-batch training and full-graph inference are unified with such abstraction: §.§.§ Training In the training phase, following the traditional training-inference pipeline, the system still takes k-hop neighborhoods as input and train GNNs in the mini-batch manner. As shown in Fig. <ref>, we take two widely used GNN algorithms (i.e., GraphSAGE, GAT) as examples to demonstrate how to organize codes in such abstraction. The data flow in training phase is quite simple. Since k-hops neighborhoods provide sufficient information for k-layer GNNs<cit.> and are locally available for a certain training worker, the data flow is just accessing and updating related local tensors. Meanwhile, a model doesn't need much change and only should be expressed in our schema. As shown in Fig. <ref>, a certain GNN algorithm should override three methods (gather, apply_node, and apply_edge) from the base class. In addition, we also provide an interface named scatter_and_gather in case the scatter and gather stages could be fused together to avoid storing intermediate edge-level messages in training phase<cit.>. For example, the scatter and gather processes in GraphSAGE are fused by a generalized sparse-dense matrix multiplication operation. It's worth noting that since the gather_nbrs is just accessing local tensors in training phase, it is ignored here for simplicity. The gather interface in Fig. <ref> represents the computation of aggregate stage. Furthermore, some function decorators are developed to mark the beginning point and end point of functions, as shown in Fig. <ref>. Meanwhile, we would generate layer-wise signature files to record those information at the time to save a well-trained model (parameters and so on). In this way, different parts of the computation flow could be reorganized and deployed in corresponding stages in the inference phase. Note that parameters in those decorators indicate whether to enable optimization strategies and also would be recorded in signature files. Those information would be loaded in the inference phase to avoid excessive manual configurations. §.§.§ Inference Different from the training phase, the inference task is conducted in the full-graph manner. Since the InferTurbo abstraction is expressed from the perspective of a node, the forward pass of GNNs could be treated as a “vertex" program over a certain node. By partitioning nodes in a graph into different machines, the total inference task could be conducted distributedly. The data flow in the inference phase is quite different from that in the training phase. Neighbors of a certain node could be placed on different machines, and data should be exchanged among those machines to prepare sufficient information to perform the computation flow on the node. Therefore, rather than simply accessing local tensors, the data flow in inference phase mainly plays a role in communicating with other nodes on remote machines: the gather_nbrs would receive information from remote nodes via in-edges and vectorize those information into adjacency matrix, node/edge feature matrix, and so on. The scatter_nbrs would send messages to other machines according to the destination node's id for the next iteration. In contrast, the computation flow could be shared in training and inference phases. In general, a certain computation stage of a well-trained model would be attached to the corresponding part in inference phase. Specially, the computation flow would be reorganized for optimization. For example, the aggregate function may be performed in the Scatter stage in advance to reduce messages sent to the same destination node. The implementation details about how to conduct the inference tasks with specific backends and the optimization strategies will be presented in the following sections. §.§ Alternative Backends and Implementation Details Scalability is the key property that should be taken into consideration when implementing the InferTurbo in the inference phase. Except for scalability, different industrial applications also have some specific requirements. Inference tasks for some time-sensitive applications should be finished as fast as possible, even with relatively high costs on expensive and exclusive resources (like memory). Others may be cost-sensitive and seek a way to conduct the inference over large graphs with limited commodity computation resources since the resource cost is also critical in industrial scenarios. InferTurbo provides two alternative backends on graph processing systems (i.e., Pregel-like systems) and batch processing systems (i.e., MapReduce, Spark) to trade off computation efficiency and resource cost. In general, InferTurbo on the graph processing system could be more efficient than that on batch processing system but with more strict requirements on stable and exclusive resources, such as, memory, CPU cores. In contrast, InferTurbo on batch processing systems is more flexible in memory and CPU requirements as it processes data from external storage. For example, the batch processing system could handle large graphs even with a single reducer if sufficient external storage is available. The implement details of those two backends are introduced respectively as follows: §.§.§ InferTurbo on Graph Processing System We implement the InferTurbo abstraction on a Pregel-like graph processing system, which is one of the most popular distributed graph processing systems. Note that it could be easily migrated to other graph processing systems as the abstraction originates from the graph processing field. Graph Partition. Following Pregel<cit.>, a graph is divided into partitions according to node ids by a partitioning function (like, mod N), and each partition contains a set of nodes and all out-edges of those nodes. Each partition is assigned to a worker to perform the “vertex” program (i.e., a layer of GNN). As shown in Fig. <ref>, a certain node would also maintain the node state and edge state (out-edges), such as raw features, intermediate embeddings, or even historical embeddings. In this way, structure information and feature information could be stored in one place to avoid redundant storage. It's worth mentioning that, with this partitioning strategy, the forward pass for a layer of GNNs over a certain node could be finished in one superstep by receiving messages from other nodes (maybe on other machines), which avoids synchronizing between different workers within a superstep. Execution. As shown in Fig. <ref>, we mainly perform the computation flow of GNN models in the compute() function of the Pregel-like system. At first, we implement the gather stage based on the message iterator in Pregel, which is used to collect all messages sent to a certain node. Meanwhile, we provide a built-in vectorization function to transfer the messages and local state information into tensors, including structure information (i.e., adjacent matrix), node-level information (i.e., node state), and edge-level information (i.e., edge-wise messages or state). Then, we perform the computation flow based on those matrices and provide functions to update node and edge states maintained in the machine. At last, the messages generated by the model would be sent to other nodes via the send_message function in Pregel. Specially, we design an optimization strategy for performing the aggregation part of the GNN model in the combiner stage of Pregel to reduce the communication cost, as shown in Fig. <ref>. After k iterations (supersteps), we would finally get the result of a k-layer GNN model. Note that the first and the last supersteps play special roles in the full pipeline. The first superstep could be treat as an initialization step since there are no messages received at this stage. It mainly transforms the raw node states and edge features into initial embeddings and then calls the Scatter stage to send those information to other nodes to start the following supersteps. For the last superstep, we attach a prediction part after the apply_node stage to get prediction scores for nodes (if needed) and then output the results (node embeddings or scores). §.§.§ InferTurbo on MapReduce We also provide an alternative backend on MapReduce (or Spark), as shown in Fig. <ref>. Compared with the backend on the graph processing system, the MapReduce backend could not maintain some information (e.g., node states, out-edges) in memory. Owing to this, the communication between different “supersteps" could be quite different: all necessary information should be treated as messages and sent to the next round. The MapReduce pipeline takes a node table and an edge table as input. The node table consists of node id, node features, and ids of all out-edge neighbors, while the edge table contains source node id, destination node id, and edge features. The overall pipeline on MapReduce is as follows: * Map. The phase acts as the initialization step. At first, it transforms the raw features of nodes and edges to initial embeddings. Then, for a certain mapper, the initial embeddings of a certain node would be sent to itself and all its out-edge neighbors, while the edge information would be sent to both the source and the destination nodes. In short, the Map phase generates three kinds of information for each node: self-state information, in-edge information (i.e., edge features of in-edges, node features of the in-edge neighbors), and out-edge information (i.e., edge features of out-edges). * Reduce. The reduce phase performs the forward pass for a certain GNN layer. Compared with the backend on graph processing system, the input and output in this phase are a little different. In the Gather stage, a node needs to receive edge messages and its own state. However, in the Scatter stage, the node also sends the updated state as an additional message to itself. That is, messages (e.g., node state, updated edge embeddings via out-edges) would be sent not only to destination nodes but also to itself. The shuffle keys for those messages are set as ids of destination nodes or itself, respectively. Messages for destination nodes represent in-edge information in the next round, while for itself, indicate new self-state information and out-edge information. Similar to the implementation on Pregel, the prediction slice of a well-trained model will be merged to the last reduce phase, and after k times Reduce stages, we could finish the forward propagation of a k-layer GNN. Though the number of messages could be larger than that in the backends of graph processing system, this backend is promising for industrial applications since messages and state information usually are stored and exchanged with external storage (HHD or SSD). As a result, a reducer could load a part of nodes together with their information into memory instead of loading all the data in the partition. The peak usage of memory could be far less than that in the backends of graph processing system, as the latter backend usually should maintain all necessary information belonging to it in memory. Therefore, it could be more suitable for scaling to extremely large graphs. Moreover, many applications would leverage the ability of the powerful GNNs on various specific graphs in industrial scenarios. The solution based on graph processing system is strict on stable and exclusive resources and may lead to resource competition for different applications. The implementation on the batch processing system could release the heavy resource competition by sacrificing a little efficiency. §.§ Optimization Strategies It is still challenging to conduct GNN inference tasks for industrial applications on those implementations since a hallmark property of graphs is skewed power-law degree distribution<cit.>. The power law means a few nodes act as hubs and have many neighbors, while most nodes have few neighbors. Those “hub" nodes could increase the cost of computation and communication in inference phase, and substantially slow down the machines that contain them. What's worse, the inference pipeline could crash as those nodes may lead to the OOM problem. The neighbor-sampling strategy<cit.> may mitigate those problems but lead to unstable embeddings or prediction scores, which is unacceptable for industrial applications. By diving into those problems, we classify them into two categories: problems caused by the skewed in-degree and the skewed out-degree, and propose several strategies to address them respectively: Partial-Gather For nodes with many in-edges, the time cost of receiving and computing over messages could increase significantly and lead workers containing them in the long tail. We propose a partial-gather strategy to address this problem, and the schematic diagram is illustrated in Fig. <ref>. The basic idea of this strategy is to conduct the Gather stage in advance to balance the computation of messages on the sender side while reducing the total amount of messages to be sent out. In addition, the five-stage GAS-like abstraction and the annotation technique described above make it possible to call related computation modules at any time in the pipeline. Therefore, in a certain round, the aggregate part of the model (i.e., gather function, partial=True) could be conducted using partially available messages when they are sent to the same destination. Then in the next round, the receiver receives those aggregated messages from different workers and conducts the Gather stage as usual. Since the aggregate function satisfies the commutative law and associative law, this strategy would not lead to any difference in final results. We implement the strategy on both the Pregel-like system and the MapReduce (Spark) system with their built-in “combining" function. With this strategy, the communication complexity for a particular node is reduced to a constant level and only depends on the number of workers. Additionally, the computation of Gather is carried out relatively uniformly across source workers, which also avoids the computation hubs and may reduce the long-tail effect caused by large in-degree nodes. This strategy involves almost no overhead and can be applied to all nodes regardless of their degrees. Broadcast For many GNN algorithms, all or part of messages sent via out-edges are usually the same. For example, the intermediate embeddings of a certain source node could be sent to all its neighbors via out-edges, which leads to overhead in communication. For nodes with many out-edges, the communication would be the bottleneck, which should be avoided by “compressing" those repeated messages. To address this problem, we propose a broadcast strategy as shown in Fig. <ref>. In detail, instead of sending edge-level messages, nodes will send “one" unique message to each machine together with a unique identifier (e.g., UUID or the source node id) as the index. Meanwhile, identifiers would be sent as messages via out-edges. Then, in the next round, the receiver should look up real messages by those identifiers and then perform the inference tasks as usual. We implement this strategy with the built-in “aggregator" class in batch processing systems and graph processing systems. Furthermore, this strategy could be easily migrated on the latest graph processing systems which provide a native “broadcast" mechanism. It's obvious that the broadcast could mitigate the overhead caused by the repeated messages and thus release the heavy communication pressure for nodes with large out-degree. Shadow Nodes A strategy named shadow nodes is designed as an alternative solution for nodes with large out-degree, since the broadcast strategy may not work well when messages vary with out-edges and cannot be compressed. The basic idea of this strategy is to average the communication load caused by out-edges, as shown in Fig. <ref>. First, the out-edges of a node are divided into n groups evenly. Then the node is duplicated n times, and each mirror is associated with a group of out-edges and all in-edges. Note that the id of each mirror would be appended with the group information. This process would be conducted in the preprocessing phase, and out-edges will be averaged to each mirror which could be placed on different machines. Since each mirror holds all the in-edges of the original node and the out-edges of all the duplicates are equal to that of the original node, conducting GNN algorithms with this strategy is just the same as the original pipeline and would not change the result. This strategy is simple but more general than the Broadcast strategy. It could re-balance the computation load whether messages could be compressed or not. Note that, in those two strategies for large out-degree problems, a heuristic formula is designed to estimate the threshold: threshold = λ× total_edges/total_workers. That is, those strategies should be activated when the number of out-edges for a node exceeds a certain percentage of edges on the worker. In our experiment, the hyper-parameter λ is set to an empirical value of 0.1. § EXPERIMENT In this section, we conduct extensive experiments to evaluate our inference system and compare it with two state-of-the-art GNN systems, PyG<cit.> and DGL<cit.>. §.§ Experiment Settings Datasets. Four datasets are used in the experiments, including PPI<cit.>, OGB-Products<cit.>, OGB-MAG240M<cit.>, and Power-Law, whose data scales range from small, medium, large, to extremely large. The properties of those datasets are shown in Tab. <ref>. The first three datasets are collected from the real world, and following the experimental settings in <cit.>, they are divided into three parts as the training, validation, and test sets. Note that settings of MAG240M follow the baseline examples provided by OGB team[https://ogb.stanford.edu/], and only a part of the graph is used in the examples, and thus we count what we used in Tab. <ref>. The Power-Law dataset is synthesized following the power-law<cit.> for the following two considerations. Firstly, to verify the scalability of the system, graphs at different scales with the same degree distribution should be used in experiments, since the time cost could be affected a lot by different degree distributions even with the same scale. Secondly, experiments for analyzing large in-degree and out-degree problems should be conducted on datasets with in-degree and out-degree following the power law respectively, for variable-controlling purposes. In our experiment, we synthesize datasets following the same rule with different scales and Tab. <ref> only records the largest one we used. Specially, all nodes in Power-Law datasets are used in inference task, while millesimal are used in training phase. Evaluation. We design a series of experiments to evaluate our system in terms of effectiveness, efficiency, consistency, and scalability. We first evaluate two widely used GNNs (GraphSAGE<cit.>, GAT<cit.>) on three real-world datasets to verify the effectiveness of our system. Then, some experiments are conducted to show the unstable prediction scores caused by neighbor-sampling and prove the consistency of our inference system. Moreover, we also record the time cost and compare it with traditional inference pipelines to see the efficiency of our system. We verify the scalability of the system and analyze the effectiveness of optimization strategies on the Power-Law dataset. We use cpumin to measure the usage of resources and record it at different data scales to prove the scalability of the system. The data scales range from 100 million nodes (about 1 billion edges), 1 billion nodes (10 billion edges), to 10 billion nodes (100 billion edges). Specially, when analyzing the effectiveness of different optimization strategies individually, power-law datasets are generated with in-degree and out-degree following the power-law, respectively. The different backends of our system are deployed on different clusters. The backend on graph processing system consists of about 1000 instances, and each is powered by 2 CPU (Intel Xeon E5-2682 [email protected]) and 10GB memory, while the one on MapReduce contains about 5000 instances (2-CPU, 2GB memory). The network bandwidth is about 20 Gb/s. For fairness, only 1000 instances are used in experiments to compare those two backends. §.§ Results and Analysis §.§.§ Comparisons We conduct a set of experiments to compare the system with traditional training-inference pipelines and present experimental results to demonstrate the effectiveness, efficiency, consistency, and scalability of the system. Effectiveness. Effectiveness is always in the first place. Table <ref> illustrates comparisons of performance between our system and traditional training-inference pipelines powered by PyG and DGL. We report the results of GraphSAGE and GAT on PPI, OGB-Product, and OGB-MAG240M in different inference pipelines. Configurations of those GNN algorithms follow examples in OGB leaderboard[https://ogb.stanford.edu/docs/lsc/leaderboards/]. In general, results achieved by our system are comparable with those on PyG and DGL across different datasets and algorithms. It's reasonable since our system just changes the way to perform the inference but never changes the formula of GNNs or introduces any approximation strategy in inference phase. As those two algorithms represent the two most widely used algorithms while the scale of datasets ranges from small, medium, and large, those comparisons and results prove the effectiveness of our system in different scenarios. Efficiency. We evaluate the efficiency of our system by comparing it with the traditional inference pipelines powered by PyG and DGL, and record the resource cost measured by cpumin together with time cost on the OGB-MAG240M dataset. Note that, in traditional pipelines, we use a distributed graph store (20 workers) to maintain the graph data and 200 workers (10-CPU, 10G memory) for inference tasks. Increasing the number of workers would exacerbate the communication problem between workers and the graph store, reducing the efficiency of traditional pipelines. For fairness, the total CPU cores of inference workers are equal to our system with different backends, and the CPU utilization remains 90%+ during the inference tasks. Table <ref> demonstrates the experimental results compared with traditional inference pipelines. Generally, both the two backends of our system demonstrate superior efficiency. Our system achieves 30× ∼ 50× speedup compared with traditional inference pipelines, while in terms of the total resource cost, traditional inference pipelines take about 40× ∼ 50× more than our system. Those results show that our system is quite promising for industrial scenarios: it not only could finish inference tasks faster but also is more economy friendly for industrial applications as it takes fewer resources. The results we achieved mainly benefit from the new inference pattern in our system. That is, with the InferTurbo abstraction, it could conduct layer-wise inference over the entire graph while enjoying the good property of parallelism. In this way, redundant computation caused by k-hop neighborhoods could be avoided. Notably, even ignoring the bottleneck of communication in traditional pipelines and assuming they could achieve 5× speedup by scaling up 5× resource (i.e., 1000 workers), our system still gains up to 10× speedup over them. In addition, we also design a set of experiments to demonstrate the relationship between time cost and the number of GNN layers for further analysis, and results are presented in Tab. <ref>. Without loss of generality, the comparisons are mainly conducted between PyG (traditional pipeline) and the MapReduce backend. nbr50 and nbr10000 mean the number of neighbors for a certain node is limited to 50 and 10000 respectively with neighbor-sampling strategy for experiments on PyG, while there is no sampling for our system. Both the time cost and resource usage of our system increase nearly linearly by varying hops of GNNs, while they increase exponentially in the traditional pipeline. Specially, the traditional pipeline even crashes with the OOM problem when the number of neighbors is set to 10000. Because, in the forward pass of the traditional pipeline, the k-hop neighborhoods of nodes increase exponentially with the hop count, resulting in the exponential growth of communication and computation. However, k-hops neighborhoods for different target nodes could overlap with each other, and the communication and computation on the overlaps are unnecessary. Our system avoids those redundant problem by conducting inference tasks over the entire graph in the full-batch manner. For each node on each layer, it will be used only once in our system. Therefore, the time cost and resources are only related to the hop count. Consistency. We take the GraphSAGE model in PyG on the OGB-MAG240M dataset as baselines to verify the consistency of our system and traditional pipelines. The number of neighbors for a node in traditional pipelines is limited to 10, 50, 100, 1000 in different experiments. We count the number of node classes predicted in the traditional pipeline at 10 runs and summarize such information for about 130,000 nodes to see whether predicting scores would keep unchanged. The statistical results are presented in Fig. <ref>. The x-axis denotes how many classes a certain node belongs to, while the y-axis indicates how many nodes would be predicted to a certain number of classes at 10 runs. For example, when the number of neighbors is set to 10, 30840 nodes are predicted into 2 different classes at 10 runs. In all, about 30% nodes would be predicted to at least 2 different classes when the sampling number is set to 10, which indicates that the prediction score is not trusted. Even increasing the sampling number to 1000, there are still about 0.1% nodes in trouble, which is unacceptable in industrial scenarios, especially in financial applications. Note that when the sampling number increases to 10000, such risk is mitigated, but the time cost increases a lot and even causes the OOM problem as shown in Tab. <ref>. In contrast, our system achieves the same results at different runs since it conducts the inference phase over the full graph without sampling. And we propose a series of strategies to avoid the OOM problem caused by nodes with a large degree. Therefore, the system is suitable for inference tasks on large graphs, even with skewed degree distribution. Scalability. Scalability is a key issue for industrial scenarios. We design a series of experiments and illustrate the relationship between the resource usage (cpumin) and the data scale to evaluate the scalability of our system. The experiments are conducted over Power-Law datasets, and the data scale includes three orders of magnitude as mentioned in Sec. <ref>. A 2-layer GAT model with an embedding size of 64 is conducted on those datasets, and we record the time and resource cost to see how they vary with different data scales. Note that the backend of our system is set to MapReduce as the resource on the cluster for graph processing backend is not enough for the largest dataset. Figure <ref> demonstrates the experimental results. The y-axis on the left and right represent resource usage (cpumin) and time cost (s), respectively, while the x-axis is the data scale. Both the resource and the time-cost curves demonstrate a nearly linear relationship between them and the data scale, which proves the linear scalability of our system. It is worth noting that our system could finish the inference task for all 10 billion nodes (with about 100 billion edges) within 2 hours (6765 seconds), which shows that our system could scale to extremely large graphs and finish inference tasks efficiently. §.§.§ Analysis for Optimization Strategies For further analysis of our systems, we also design a set of experiments to verify the effectiveness of the optimization strategies for hub nodes proposed in Sec. <ref>. We enable those strategies on the GraphSAGE model over Power-law datasets (about 100 million nodes, 1.4 billion edges) respectively for variable-controlling purposes. Figure <ref> illustrates experimental results on the large in-degree problem and the effect of the partial-gather strategy. The x-axis indicates the number of in-edges for nodes in one instance, while the y-axis denotes the instance latency, and each point represents an instance. Results show that latency is positively associated with the number of in-edges. It's reasonable since the time cost for receiving messages and the computation to aggregate those messages are related to the number of in-edges. Therefore, nodes with large in-degree would lead the time cost of the related instance in the long tail and become the bottleneck of the whole inference task. In contrast, the partial-gather strategy helps release the stress caused by a large number of in-edges: the variance of time cost on different instances shrinks a lot, and points are close to the mean line. The reason is from two aspects: firstly, with the partial-gather strategy, messages would be partially aggregated on the sender side, which brings down the time cost in communication. Secondly, the computation of in-edge messages is uniformly conducted in advance on different instances, which balances the computation load for hub nodes. We also make an analysis of the large out-degree problem, and the results are illustrated in Fig. <ref>. Base denotes the experiment without optimization strategy, while SN, BC, and SN+BC indicate experiments with shadow node strategy, broadcast strategy, and the combination of them respectively. The y-axis represents the variance of time cost in all instances. Compared with the baseline, both the shadow node and broadcast strategies are helpful in mitigating the problem caused by nodes with large out-edges, and the latter one is slightly better than the former one since shadow node strategy would lead to some overhead by duplicating in-edges to average the communication over different mirror nodes. Specially, for GraphSAGE, we could achieve better results by combining the two strategies since messages for different out-edges are the same. Furthermore, we develop some experiments to evaluate our system's IO costs. Figures <ref>, <ref>, and <ref> show the results of strategies for large in- and out-degree problems, respectively. Their y-axises show the input or output bytes of each instance. For Fig. <ref> and <ref>, x-axises show the initial number of input or output records for different instances. Specially, the x-axis for Fig. <ref> is worker index sorted according to the output bytes since the shadow node strategy would re-arrange the placement of records. Figure <ref> demonstrates how our system, using the partial-gather approach, reduces the input IO cost to a constant level. This approach is effective for all nodes regardless of degree and performs particularly well for nodes with large in-degree since there is no overhead involved. The partial-gather strategy reduces the total communication cost of all works by roughly 25% while saving up to 73% IO cost for the 10% workers in the tail. The reason is that each instance would only send one message for a particular node in the upcoming round since the messages have already been pre-aggregated. Each node would only receive as many as instance-number messages in the following round. Moreover, we also vary the threshold for those strategies to verify the heuristic formula in Sec. <ref>. The hyper-parameter λ is set to 0.1 in our system, and the threshold to distinguish whether a node is a `hub' node then becomes 100,000 (1 billion edges, 1000 workers). Compared with the baselines, both the broadcast and shadow nodes strategies in this setting could decrease the IO cost for “hub” nodes significantly. For instance, the communication costs for the last 10% workers are reduced by 42% and 53% when activating those two techniques, respectively. Furthermore, the efficacy of those strategies would increase as the threshold decreases. However, there is no significant difference for the threshold in the range of [10,000, 100,000], and the difference in IO cost is less than 5%. This is because a low threshold would enable more nodes to benefit from those techniques, but their overhead could not be disregarded. For example, the memory cost is nearly doubled. Therefore, although the threshold determined by the heuristic formula may not be the optimal one, it is reasonable to use it to quickly estimate the threshold, and in this setting, those strategies successfully reduce the IO cost for straggler workers. In summary, the experimental results demonstrate that those strategies proposed in Sec. <ref> help eliminate the stragglers caused by the power law, and as a result, our system achieves better load-balancing. § CONCLUSION In this paper, we propose InferTurbo, to boost the inference tasks at industrial-scale graphs. The design follows the GAS-like paradigm underlying the computation of GNNs, with which we could specify the data flow and computation flow of GNNs and hierarchically conduct the inference phase in full-graph manner. In this way, our system could avoid the redundant computation problem. We provide alternative backends of MapReduce and Pregel for different industrial applications, and both could scale to large graphs. Furthermore, a set of strategies without dropping any information is used to solve problems caused by nodes with large in-degree or out-degree. The experimental results demonstrate that our system achieves 30× ∼ 50× speedup compared with some state-of-the-art graph learning systems. Our system could finish the inference task over a graph with 10 billion nodes and 100 billion edges within 2 hours, which shows the superior efficiency and scalability of our system. 00 b1 Welling, Max, and Thomas N. Kipf. "Semi-supervised classification with graph convolutional networks." J. 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http://arxiv.org/abs/2307.02543v2	20230705180003	Discovery of Dipolar Chromospheres in Two White Dwarfs	[ "J. Farihi", "J. J. Hermes", "S. P. Littlefair", "I. D. Howarth", "N. Walters", "S. G. Parsons" ]	astro-ph.SR	[ "astro-ph.SR" ]	From the tabletop to the Big Bang: Analogue vacuum decay from vacuum initial conditions Silke Weinfurtner August 1, 2023 ======================================================================================= This paper reports the ULTRACAM discovery of dipolar surface spots in two cool magnetic white dwarfs with Balmer emission lines, while a third system exhibits a single spot, similar to the prototype GD 356. The light curves are modeled with simple, circular, isothermal dark spots, yielding relatively large regions with minimum angular radii of 20. For those stars with two light curve minima, the dual spots are likely observed at high inclination (or colatitude), however, identical and antipodal spots cannot simultaneously reproduce both the distinct minima depths and the phases of the light curve maxima. The amplitudes of the multi-band photometric variability reported here are all several times larger than that observed in the prototype GD 356; nevertheless, all DAHe stars with available data appear to have light curve amplitudes that increase toward the blue in correlated ratios. This behavior is consistent with cool spots that produce higher contrasts at shorter wavelengths, with remarkably similar spectral properties given the diversity of magnetic field strengths and rotation rates. These findings support the interpretation that some magnetic white dwarfs generate intrinsic chromospheres as they cool, and that no external source is responsible for the observed temperature inversion. Spectroscopic time-series data for DAHe stars is paramount for further characterization, where it is important to obtain well-sampled data, and consider wavelength shifts, equivalent widths, and spectropolarimetry. stars: evolution— stars: magnetic field— white dwarfs § INTRODUCTION The origin of magnetism in white dwarf stars is an outstanding astrophysical puzzle more than a half century old, but recent and ongoing developments are now shedding light on this fundamental, and still poorly understood aspect of stellar evolution. The first signatures of white dwarf magnetism resulted from the detection of circular polarization in spectra that were quasi-featureless or with unidentified absorption bands <cit.> that were later understood to be shifted hydrogen and (neutral) helium lines, mostly consistent with centered or offset dipole field geometries <cit.>. A summary of magnetic white dwarf research over the first several decades can be found in two published reviews <cit.>. One of the key developments was the recognition that magnetic white dwarfs are nearly exclusively found as isolated stars, or in cataclysmic variables <cit.>. This empirical finding led to the hypothesis that fields are generated during common envelope evolution <cit.>, a process that may function effectively for stars, brown dwarfs, and giant planets that are engulfed during the post-main sequence <cit.>. And while fast-spinning and massive magnetic white dwarfs are known, and thus consistent with a stellar merger origin <cit.>, it is also clear that magnetism, high remnant mass, and rapid rotation are far from tightly correlated <cit.>. It has been suspected for decades that cooler white dwarfs are more often found to be magnetic <cit.>. However, luminosity and sensitivity biases exist, where the coolest white dwarfs essentially require metal pollution to detect Zeeman splitting <cit.>. Despite these uncertainties, the possibility that magnetic fields first emerge in cool and isolated white dwarfs is intriguing, as substantial cooling is necessary for core crystallization, which has been hypothesized to be a source of an internal dynamo powered by the liquid-solid phase separation at the core boundary <cit.>. In this scenario, magnetic field generation is decoupled from external sources of mass and angular momentum, but nevertheless, all else being equal, more rapidly rotating remnants should have stronger fields. In a pioneering effort to overcome the aforementioned biases, and determine the actual frequency of magnetism as a function of white dwarf characteristics, <cit.> carried out a nearly complete census of (N≈150) white dwarfs within 20 pc. This volume-limited survey used sensitive circular spectropolarimetry and resulted in the first unbiased study of white dwarf magnetism, where the principal findings can be summarized as follows. * All spectral classes have similar incidences of magnetism, regardless of atmospheric composition. * The field strength distribution is uniform over four orders of magnitude from 40 kG to 300 MG. * Magnetism is detected more frequently in white dwarfs with higher than average mass. * White dwarfs with cooling ages younger than 0.5 Gyr – prior to core crystallization – are rarely magnetic. * There is no evidence of field strength decay over time. It is within this background of recent developments that emerged the relatively new and small class of DAHe white dwarfs (D: degenerate, A: Balmer lines strongest, H: magnetic line splitting, e: emission). The prototype is GD 356, an isolated T_ eff≈7500 K star with Balmer emission lines split in a B≈13 MG field. There are deep, multi-wavelength, non-detections that yield stringent upper limits on an X-ray corona, ongoing accretion, and low-mass companions <cit.>. This apparently single white dwarf has a 1.927 h rotation period, based on a nearly sinusoidal light curve that is well modeled by single dark spot, whose size is consistent with that of the magnetic and heated region <cit.>. These enigmatic properties led to the hypothesis that, analogous to the Jupiter-Io system, the relatively cool white dwarf surface could be heated by Ohmic dissipation of a current loop set up by the orbital motion of a conducting planet <cit.>; referred to as the unipolar inductor model. GD 356[Previously thought to have a helium-rich atmosphere <cit.>.] was the only known DAHe white dwarf for 35 years, until 2020 when second and third cases were reported <cit.>. In addition to their shared spectral morphology and strong magnetism implied from Zeeman splitting, these three cool white dwarfs with emission lines all share commonalities with some magnetic white dwarfs: relatively rapid rotation, masses only slightly above average, and no evidence for low-mass stellar or substellar (detached) companions. A detailed time-series study of the prototype has shown that (i) the spin period is stable over two decades, with no other independent frequency signals as would be expected from a unipolar inductor, (ii) the emission line strength oscillates in anti-phase with the broad-band stellar brightness, and (iii) so far, DAHe stars share a tightly correlated set of effective temperatures and luminosities <cit.>. This clustering is potentially related to core crystallization and magnetic field diffusion toward the stellar surface <cit.>. This paper reports detailed light curves for three DAHe white dwarfs: the second known example, SDSS J125230.93-023417.7 (; hereafter SDSS J1252), and two recently identified members of this class, LP 705-64 and WD J143019.29-562358.3 (; hereafter WD J1430). Two of the three stars reveal light curves with asymmetric dimming events that are 180 out-of-phase, and thus consistent with dipolar star spots. These data are inconsistent with a unipolar inductor model, and instead support the generation of intrinsic chromospheres in some isolated, magnetic white dwarfs. The observations and data are discussed in Section 2, the time-series analysis is presented in Section 3, followed by a summary and discussion. § OBSERVATIONS This study focuses on light curves and the resulting periodicities of three DAHe white dwarfs, using both ground- and space-based photometric monitoring as described below. §.§ Target properties and selection SDSS J1252 is the second discovered example of a DAHe white dwarf, reported to have emission lines split in a B≈5 MG field, and with a sinusoidal light curve dominated by a period of 317.3 s <cit.>. The fast rotation of this star makes it an attractive target for high-cadence photometric monitoring from the ground, with a goal to obtain a detailed light curve. LP 705-64 and WD J1430 are two newer members of the DAHe spectral class with indications from TESS data that their full spin cycles could each be readily covered in a single night of ground-based photometry <cit.>. The initial observational goals were similar to those achieved by <cit.>, to establish robust ephemerides against which future period changes might be investigated (e.g. within a unipolar inductor and orbiting planet model), and to constrain the nature of the emitting and magnetic regions. §.§ ULTRACAM observations All three stars were observed with ULTRACAM, a frame-transfer CCD imaging camera (24 ms dead time between exposures; ) that is permanently mounted on the 3.6 m NTT telescope at the La Silla Observatory in Chile. The instrument has three independent channels that enable the use of independent filters simultaneously, and data were taken with filters similar to standard u, g, and one of r or i bandpasses, but with higher throughput. In each case, the blue channel was co-added every three frames to improve the effective signal-to-noise on the target. The observation details, including exposure times (same as the cadences for ULTRACAM), and durations of the resulting light curve segments, are summarized in Table <ref>. Images were corrected for bias and flat fielded with normalized sky images obtained during evening twilight (taken in a continuous spiral to remove stars). Differential brightnesses were measured relative to field stars with dedicated software[https://github.com/HiPERCAM/hipercam.] using photometric apertures that were typically scaled to 2× the mean full width at half maximum of the stellar profiles for each exposure. The sky annuli were fixed to span the region 8.75–15.75 arcsec from the stars, where a clipped mean was used to determine the background. For all stars in all observations, the same sets of comparison stars were used to generate light curves, consisting of two or three stars in the gri frames, and one to two stars in the u-band images. Light curves were constructed by dividing the science target flux by the sum of the comparison star fluxes, and normalizing the result. Measurement errors were propagated from the aperture photometry, by summing in quadrature the fractional flux errors of all stars measured for a given light curve. All ULTRACAM times were converted to Barycentric Julian Day (BJD) using Barycentric Dynamical Time (TDB), following <cit.>. §.§ TESS data Data for each of the three DAHe targets are available from TESS <cit.>, and were downloaded from the MAST archive, where the pdcsap processed light curves were retained for analysis. Time stamps were were corrected to BJD = TESS_BJD + 2457000. LP 705-64 (= TIC 136884288) was observed in Sector 30, while data were collected for WD J1430 (= TIC 1039012860) within Sector 38, and for SDSS J1252 (= TIC 953086708) during Sector 46. All three stars have 120 s cadence observations. These data were further cleaned of NaN flux entries, but with no other processing based on data quality flags, yielding light curves that retained between 80 and 90 per cent of their pdcsap array values. Lastly, outliers beyond ±5 of the local time average (or phase average) flux were removed, which were fewer than five points in total for each source. It is worth noting that these data are not all equally useful in subsequent analysis. The following TESS benchmarks summarize their relative quality: SDSS J1252 has G=17.5 mag, a mean flux of 19.2±5.3 e^- s^-1 (28 per cent scatter); LP 705-64 has G=16.9 mag, a mean flux of 38.0±5.6 e^- s^-1 (15 per cent scatter), while WD J1430 has G=17.4 mag, a mean flux of 9.6±5.4 e^- s^-1 (73 per cent scatter), and lies within the Galactic plane. § TIME-SERIES ANALYSIS AND RESULTS All light curves were analyzed using period04 <cit.>, where a Lomb-Scargle periodogram was constructed using ULTRACAM data, TESS pdcsap light curves, or a combination of the two datasets, with a goal to identify that which produces the most precise ephemerides for each target. Monte Carlo simulations, run within period04, were used to determine errors in frequency and phase for the strongest periodogram peak for each star and set of light curves, then propagated to determine the error in T_0 corresponding to photometric minimum. The frequency and phase were allowed to vary independently during the simulations to determine errors, which were typically repeated 1000 times. §.§ SDSS J1252 For SDSS J1252, there are sufficient ULTRACAM data to uniquely determine the photometric period and provide an improved ephemeris. Light curves cover more than 30 epochs of its previously reported 317.3 s periodicity (frequency 272.3 d^-1, ), with an observational baseline of 136 d, spanning several 10^4 cycles at this frequency. In Figure <ref> are shown the first and second g-band light curves obtained for this white dwarf, from which can be discerned that there are two distinct set of minima (and maxima), each manifesting every 317.3 s, and thus revealing an actual photometric period of 634.6 s. The ULTRACAM g+r co-added light curves were analyzed using data from all three observing runs. The resulting best periodogram is plotted in Figure <ref>, where the strongest peak is identical to the frequency reported in the discovery paper; however, there is a second outstanding signal near 408 d^-1. This secondary peak is not as well-determined as the 272.3 d^-1 signal, but these two frequencies appear to have a near an exact ratio of 3:2. Additionally, the periodogram also reveals a weak-amplitude peak at roughly 816 d^-1, and the ratio of these three frequencies is 6:3:2. These periodicities are far shorter than any possible range of orbital signals originating from non-degenerate companions, as the lowest-frequency periodogram signal, at 272.3 d^-1, corresponds to a Keplerian orbit near 7 R_⋆ (seven white dwarf radii), deep within the nominal Roche limit. Only a compact object could survive at this orbital distance, such as those in close but detached, double white dwarf binaries. And while there are a few such systems known to have orbital periods comparable to the frequencies exhibited by SDSS J1252, their light curves reveal ellipsoidal modulation owing to tidal distortions in the primary white dwarfs <cit.>. Furthermore, these rare, deformed degenerates are all helium-core white dwarfs less massive than 0.3 M_⊙, and thus significantly more prone to tidal distortion than SDSS J1252 and the DAHe stars, which are considerably more compact <cit.>. Therefore, the light curve and resulting periodogram of SDSS J1252 are interpreted as arising from a single star. It is reasonable to assume the T_ eff≈8000 K white dwarf has a fixed spin period (no differential rotation), and no stellar pulsations, as it is far from the hydrogen atmosphere instability strip <cit.>. The observed signals are then interpreted as the 2^ nd, 3^ rd, and 6^ th harmonics of the stellar rotation frequency. The periodogram signals and their amplitudes reflect the fitting of sinusoids to the light curve, where the two highest have a 3:2 ratio in order to generate both the principal flux variation at 272.3 d^-1, and the alternating minima via the interference with the 408 d^-1 frequency. The revised stellar rotation period and associated uncertainty were determined by dividing the 2^ nd harmonic frequency by two, yielding 136.15032(2) d^-1 (equivalent to a period of 634.59273(9) s). Although essentially no amplitude (or power) is seen in the periodogram at the frequency inferred to be the fundamental, this is an expected consequence of the light curve morphology and sinusoidal fitting <cit.>. A similar analysis was attempted using the TESS light curve, both on its own and in combination with ULTRACAM data. While the TESS 120 s cadence is 2.6× faster than the peak periodogram frequency for SDSS J1252, and thus above the Nyquist rate, the data quality are relatively poor (Section 2.3). No time-series analysis utilizing TESS led to any improvement in frequency or phase precision, and therefore all calculations for SDSS J1252 are based solely on the ULTRACAM observations. §.§ LP 705-64 and WD J1430 TESS data were the initial means of identifying the stellar rotation rates in these two DAHe white dwarfs <cit.>. However, similar to as observed in SDSS J1252, the ULTRACAM light curve of LP 705-64 exhibits two unequal minima in a single cycle, and thus the period determined by TESS represents one half its spin period (see Figure <ref>). For this source, the ULTRACAM data alone do not span a sufficient number of cycles to determine the photometric period with precision comparable to TESS. A significant improvement in the TESS ephemeris is achieved using the combination of ULTRACAM g+r co-added light curves and TESS, resulting in a periodogram with a single peak at 39.65325(3) d^-1 [cf. 39.653(2) d^-1; ], and a corresponding higher precision in phase. However, the true spin period must be calculated from this frequency by recognizing it is the 2^ nd harmonic of the fundamental, which is 19.82662(1) d^-1. For WD J1430, the ULTRACAM light curves reveal a single maximum and minimum with one of the largest amplitudes observed to date for a DAHe white dwarf (5.8 per cent in the g band). Similar to LP 705-64, there are insufficient ULTRACAM data from which to derive a precise ephemeris for this source, and thus the combination of ULTRACAM and TESS Sector 38 pdcsap data were utilized for this goal. Initially, the analysis of these combined datasets improved the precision of the periodogram frequency, but resulted in phase errors that were larger than those based on TESS alone. Subsequently, these TESS data were re-scaled (see Section 3.4) to more closely match those of the co-added g+i-band ULTRACAM data, and the resulting analysis marginally improved the uncertainty in phase. Ultimately for this star, the best constraints were achieved by adding a third set of light curves into the periodogram analysis, using full-frame data from TESS Sector 11, where fluxes were extracted based on PSF-subtracted images following <cit.>. §.§ Light curve morphologies, ephemerides, spectral phases Based on the preceding analysis, and to reveal light curve structures more precisely as a function of phase, the ULTRACAM multi-band data were phase-folded and binned using a weighted average onto regular grids. The resulting light curves are shown in Figure <ref>, where there are 80 phase bins for LP 705-64 and WD J1430 in all three channels. In the case of SDSS J1252, the short spin period indicates that a single ULTRACAM frame has a phase width of 0.016 in the green and red channels, but 0.048 in the blue channel (owing to three co-adds). For this reason, the light curves of SDSS J1252 were re-sampled into 60 phase bins in g and r, but only 20 bins in u band. The folded light curves for SDSS J1252 and LP 705-64 both exhibit alternating minima that are indicative of two distinct star spots 180 out-of-phase during rotation. While this behavior is not novel among magnetic white dwarfs, there appear to be only a few documented examples of white dwarf light curves where dipolar spots are suggested or required <cit.>. In contrast, the majority of magnetic white dwarf light curves seem to be broadly consistent with sinusoidal (single spot) morphologies <cit.>, including the prototype DAHe star GD 356 <cit.>. However, it should be noted that incomplete phase coverage and modest photometric precision can inhibit the detection of subtle light curve features (e.g. the discovery light curve of SDSS J1252, and the TESS light curve of LP 705-64; ). To calculate accurate ephemerides based on the best precision achieved here, T_0 was chosen from an ULTRACAM light curve located nearest to the middle of the temporal coverage for each star, and where a feature could be unambiguously identified as a true photometric minimum. The periodogram analysis of the preceding sections then results in the following best ephemerides for all three DAHe white dwarfs, where zero phase corresponds to actual photometric minimum, and the periods are accurate and precise determinations of their spins: BJD_ TDB (SDSS J1252) = 2459313.809921(6) + 0.007344823(1) E BJD_ TDB (LP 705-64) = 2459444.92339(6) + 0.05043723(3) E BJD_ TDB (WD J1430) = 2459696.8239(3) + 0.05999529(3) E As mentioned earlier, the ephemeris of SDSS J1252 is based solely on ULTRACAM, whereas those of LP 705-64 and WD J1430 are based on the combination of TESS and ULTRACAM. From these ephemerides, forward and backward extrapolations can be made and compared with published, time-varying spectra of LP 705-64 and WD J1430, but in the case of SDSS J1252 there is insufficient time resolution to compare its spectroscopic variations in phase with photometry <cit.>. Starting with the more straightforward case of WD J1430 which exhibits a single spot, the photometric minimum (phase 0) occurred at BJD_ TDB=2459049.056±0.001 nearest the two epochs of the published spectroscopy. This notably falls close to halfway in time between the two spectra plotted and described by <cit.> as 'emission' and 'absorption'. Specifically, and taking the reported epochs of observation at face value, these spectra correspond to photometric phases 0.720±0.007 and 0.221±0.007, respectively, and thus both occur close to the average stellar flux. While these two spectral phases are reported as potentially representing a maximum and minimum magnetic field strength, this interpretation seems uncertain, especially if other spectral phases exhibit weaker emission or absorption, where Zeeman splitting is not a robust diagnostic. Superficially interpreting these spectral phases of WD J1430 as the highest and lowest field strength would be somewhat inverse to that observed for the prototype DAHe GD 356, where there are multiple, full-phase coverage observations using both spectroscopy and spectropolarimetry. For this well-studied case, the magnetic field variations, both from the observed parallel component and using Zeeman splitting, display a peak and trough near phases 0.3 and 0.8, respectively, from photometric minimum <cit.>. For WD J1430, existing data may not probe the magnetic field with sufficient sensitivity or phase coverage, and hence these comparative results should be considered preliminary at best. In the case of LP 705-64, the situation is more complex. Depending on the spot sizes, one might expect two minima and maxima in both equivalent width and magnetic field variations, one pair associated with each spot. However, there are only two epochs of spectroscopy plotted and described by <cit.>, and here again a comparison must be considered not only preliminary, but possibly inapt for the aforementioned reasons. Again taking the published epochs at face value, and where the the deeper of the two light curve minima is zero phase, the spectrum shown with the broader Zeeman splitting corresponds to photometric phase 0.048±0.001. The two reported spectral epochs were chosen as to be separated by exactly one half spin cycle <cit.>, so that further interpretation would reflect the selection. While the updated photometric ephemeris is sufficient to predict precise spin phases for spectroscopic observations of LP 705-64, their potential correlation is not yet straightforward. It has not yet been demonstrated that high and low Zeeman splitting might be in-phase with photometric extrema (cf. GD 356 ). The two published spectra may not represent precise peak behavior, and there may be some uncertainty in the epoch dates reported. Measurements of both equivalent width and magnetic field strength at all rotational phases would eliminate these ambiguities. The sparse set of published spectroscopic measurements of DAHe white dwarfs, currently prevents a more robust correlation of photometric and spectroscopic phases. §.§ Multi-band light curve amplitudes To better understand the nature of the spots and their associated magnetic regions, multi-band light curves for DAHe white dwarfs were used to calculate the photometric amplitude for each star in each observed bandpass. For the three stars observed by ULTRACAM as well as GD 356, this was done by taking only the strongest signal in the best periodogram for each star (Figures 2 and 3), and determining the sinusoidal amplitude for each light curve in each bandpass at that frequency. In this way, all light curve amplitudes are evaluated by their strongest sinusoidal components, including those stars with little or no periodogram power at their true rotation frequency. Light curve amplitudes and uncertainties were determined using period04, using fixed frequencies and Monte Carlo simulations. Table <ref> lists the multi-band, photometric amplitudes for the four DAHe white dwarfs with available data, the three stars reported here and the prototype GD 356 <cit.>. It is interesting to note that the amplitude of photometric variation is relatively small in GD 356 (on the order of 1 per cent; ), compared to the newer DAHe stars with several percent variations in their light curves. Although there is not yet any published multi-band photometry of SDSS J1219, its light curve amplitude is around 3 per cent in the B band (; roughly halfway between the u and g filters), and thus at least double that of GD 356 at similar wavelengths. It is also possible that WD J161634.36+541011.51 (; hereafter WD J1616) has a photometric amplitude comparable to the strongest found here. These are likely the results of observational bias, which enhances their detection as variables in surveys such as Gaia and ZTF (e.g. ). It should be noted that WD J1430 is positioned in a crowded field at Galactic latitude <4 degr, and the TESS fluxes are dominated by other sources in the photometric aperture (pipeline keyword crowdsap = 0.032). This pipeline metric implies that only 3.2 per cent of the flux in the extracted aperture is likely from the white dwarf, and subsequently the extracted flux has been dramatically reduced to recover a more accurate stellar brightness in the pdcsap light curve <cit.>. While the ULTRACAM observations independently confirm the stellar spin frequency identified by periodogram analysis of the TESS light curve, the pipeline fluxes are simply too noisy (see Section 2.3) and likely offset significantly from the true mean flux. Thus, no reliable variability amplitude can be deduced for the TESS bandpass (see footnote to Table <ref>). The relative strengths of the photometric variations in DAHe stars appear to follow a trend as a function of wavelength, with increasing amplitudes towards the blue. Figure <ref> plots the strengths of the multi-band variability for the four white dwarfs, where the photometric amplitude for each bandpass is plotted relative to the g band for each star. Three of the four stars have data in ugr (or similar) bandpasses, and all three exhibit a relatively tight correlation in their amplitude ratios as a function of these three wavelength ranges. Three stars have reliable TESS amplitudes where again the same behavior is evident, and suggesting a phenomenon associated with this emerging spectral family. Based on the narrow range of T_ eff among DAHe white dwarfs, Figure <ref> implies their spots have similar spectral properties. This indication is remarkable given the range of DAHe rotation periods and especially magnetic field strengths. To date, only relatively weak Balmer features have been detected blueward of H in any DAHe star, and yet the photometric variability remains strongest at shorter wavelengths. This is consistent with the previous finding that the photometric variability arises from changes in the stellar continuum, and not from fluctuations in the Balmer emission lines <cit.>. In 2002-2003, the V-band (5500 Å) light curve amplitude of the GD 356 was recorded as 0.2 per cent <cit.>. But in 2020 the photometric variations observed using an SDSS g-band filter (4700 Å), and a V+R filter (6200 Å) were found to be 4× - 6× higher (Table <ref>). It thus seems possible that the starspot has evolved during this time frame; however, all six emission features within H and H seem consistent over at least 35 years <cit.>. §.§ Simple spot modeling A basic set of spot models and corresponding light curves were constructed to better constrain the observed stellar surfaces as a function of rotation, with a particular motivation towards those stars where two spots appear to be necessary. Each white dwarf was treated as a T_ eff=8000 K blackbody, with one or two circular, isothermal spots whose single temperature is controlled by a scaling factor f(T). Where required by the light curve morphology, two identical spots were placed on the surface at antipodal points. The other model parameters are the inclination (i) of the stellar rotation axis to the observer, the spot colatitude (), and the spot angular radius (). It is commonly acknowledged that such models are potentially degenerate when the values of i and are interchanged (e.g. ). For each star, a grid of models was generated with the following parameter ranges and step sizes as given in Table <ref>. For small angular radii (<20) the spot temperature range was expanded because in order to reproduce a fixed photometric amplitude, the smallest spots must be darkest. The root mean square (RMS) difference between the model and observed fluxes in a given bandpass as a function of spin phase was computed and used to identify a best-fit model for each , although in practice there is nearly always a range of models that yield similarly satisfactory results. While the modeling is relatively simple, a few basic results emerge. Small spots with ≲15 cannot generate a sufficiently large photometric amplitude for any of the three white dwarfs with ULTRACAM data, but otherwise the spot size is mostly unconstrained. Beyond this small angular size threshold, the combination of adjustable geometry and spot temperature permits sufficient model flexibility to achieve comparably good fits within a range of the other three parameters. Nevertheless, the spots must be at least modestly large; in terms of solid angle, covering several per cent or more of one hemisphere. In the case of WD J1430, which has the largest photometric variability amplitude, only spots with ≥20 can reproduce the observed flux changes. However, the models with the smallest RMS differences for the light curves with two minima exhibit clear shortcomings. The resulting inclinations (or colatitudes) tend toward 90, and consequently the model fits have equally deep, light curve minima. These fitted parameters are driven in the direction of maximum inclination (or colatitude) because, while the lower i (or ) solutions can reproduce a more shallow secondary minimum as observed, this particular shape exhibits light curve maxima whose phase positions are shifted towards the secondary minimum. Examples of these modeling outcomes are illustrated in Figure <ref>. Despite these limitations, the modeling demonstrates that the basic geometry of antipodal spots is essentially correct. However, in the context of the simple model assumptions, it is unlikely that there is a centered but tilted, symmetric dipolar arrangement for the dual-spotted stars. Instead, the spots may not be circular, and where two opposing spots are necessary each may be distinct in size, shape, or temperature; alternatively, the spots may be in a dipolar configuration that is offset from the rotational center of the star. § DISCUSSION AND SUMMARY The discovery of DAHe white dwarfs whose light curves require two spots in a basic dipolar configuration, now totaling at least three systems <cit.>, is a modest breakthrough in their characterization. It raises the immediate question of whether all DAHe stars have dual spots, which manifest as light curves with either one or two minima, depending on the viewing angle and spot orientation. As of this publication, there are now just over two dozen known DAHe stars, but where only six have robustly measured light curves <cit.>. There is a seventh DAHe candidate with a well-measured TESS light curve, SDSS J041246.85+754942.26, but which currently lacks any type of magnetic field indication or upper limit <cit.>. Of these seven objects, three of their light curves exhibit two photometric minima, and four are consistent with a single minimum. With such small numbers and weak constraints on spot properties, a statistical assessment of the inferred viewing geometry is not possible, but the data to date are likely consistent with all class members having dipolar magnetic and spotted regions. If the simple modeling performed here is any indication, it may be that magnetic (spot) axes must be highly inclined towards the viewer (or equivalently have similar colatitudes) for both spots to transit, i.e. have ingress and egress as opposed to being partly visible at all times. The detection of photometric variability in GD 356 yielded limited results on its spot properties, where the size of the temperature-contrast surface was assumed to be identical to that of the magnetic region inferred from modeling of spectropolarimetry, around 40 <cit.>. Otherwise the modeling followed the same assumptions as those described in Section 3.5, and the mostly sinusoidal light curve was ultimately fitted to two sets of models, one with a dark spot near the rotational pole (low ) viewed at high inclination, and a second viewed near the axis of rotation (low i), but with high colatitude; an example of the degeneracy between i and . In the case that GD 356 has antipodal spots, in the former scenario the secondary spot can remain hidden from the observer at all spin phases, and in the latter scenario, it is possible for both spots to be partly visible at all times. If the prototype does indeed have two spots, the previous photometric modeling would disfavor the latter orientation, as it would result in some light curve impact on from both spots. As with the DAHe prototype, it is tempting to co-identify the sizable spots with their magnetic and chromospherically active regions <cit.>. In one such picture, the spots are dark and magnetic regions underlying the chromospheric activity, so that the Balmer emission lines are at maximum brightness when the stellar continuum yields photometric minimum <cit.>. This behavior may also be seen in SDSS J1212 and SDSS 1219 <cit.>, but insufficient phase coverage and sampling prevent any certainty at present, and equivalent widths have not been determined for those stars. The results here for LP 705-64 and WD J1430 are currently ambiguous for similar reasons, and owing to additional complications. Interestingly, time-series spectroscopy for SDSS J1252, WD J1430, and WD J1616 suggest that their emission lines may effectively disappear at some phases <cit.>, presumably when a spot or spots (and associated magnetic region) are out of view or have minimum visibility. However, as discussed in Section 3.3, this is likely an oversimplified picture; with the exception of GD 356, there is a distinct lack of magnetic field determinations across the entire spin phases of DAHe white dwarfs, and Zeeman splitting may be an ineffective tool for weak or transient emission features. At present, empirical metrics associated with published, DAHe time-series spectroscopy are sparse, and it would be ideal for observers to provide both equivalent widths and central wavelengths for emission features over at least one full cycle with sufficient sampling. In contrast to magnetic field strength estimates, which may not be possible to measure at all spin phases via Zeeman splitting if magnetic regions rotate in and out of view, only the phase behavior of equivalent width has been robustly characterized, and only in the prototype <cit.>. It is thus essential that full spectroscopic phase coverage of DAHe white dwarfs is carried out with these measurements in mind, and where spectropolarimetry will be more sensitive to magnetic field strength, particularly when emission or absorption features are weak. The dual-spotted nature of at least three DAHe white dwarfs has direct bearing on the hypothesis that a heated region can be caused by star-planet interactions such that a current loop is dissipated in one region of the star (e.g. the unipolar inductor ). If such planetary interactions are in fact taking place within the strong magnetospheres of DAHe white dwarfs, they are unlike the interactions that lead to unipolar, Jupiter-Io footprint mechanisms <cit.>. Going forward, models that require the presence of closely orbiting and interacting planets, which at present lack no empirical support in observations of DAHe stars, should require strong evidence to be re-considered. Given the lack of additional periodic signals and the compelling evidence of DAHe white dwarf clustering in the HR diagram <cit.>, an intrinsic mechanism is the most likely source for the spotted regions and chromospheric activity. § ACKNOWLEDGEMENTS J. Farihi is grateful to the Laboratory for Atmospheric and Space Physics at the University of Colorado Boulder, and the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, for hosting during extended visits, and to J. S. Pineda for an illuminating discussion on the nature of chromospheres in sun-like and low-mass stars. The authors acknowledge the European Southern Observatory for the award of telescope time via program 105.209J. J. Farihi acknowledges support from STFC grant ST/R000476/1 and National Science Foundation Grant No. NSF PHY-1748958. S. P. Littlefair acknowledges the support of the STFC grant ST/V000853/1. N. Walters has been supported by a UK STFC studentship hosted by the UCL Centre for Doctoral Training in Data Intensive Science. S. G. Parsons acknowledges the support of a STFC Ernest Rutherford Fellowship. For the purpose of open access, the authors have applied a creative commons attribution (CC BY) license to any author accepted manuscript version arising. This paper includes data collected by the TESS mission, which is funded by the NASA Explorer Program. § DATA AVAILABILITY ULTRACAM data are available on reasonable request to the instrument team, while TESS data are available through the Mikulski Archive for Space Telescopes. mnras
http://arxiv.org/abs/2307.01657v2	20230704113104	An experimental access to observation of decay from extremely long-lived metastable electronic states via Penning trap spectrometry	[ "Bingsheng Tu", "Ran Si", "Yang Shen", "Jiarong Wang", "Ke Yao", "Baoren Wei", "Chongyang Chen", "Yaming Zou" ]	physics.atom-ph	[ "physics.atom-ph" ]	Pre-main sequence stars Evgeni H. Semkov1 ^1Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411, Tartu, Estonia ===================================================================================== Long-lived ionic quantum states known as metastable electronic states in highly-charged ions (HCIs) are of great interest in fundamental physics. Especially, it generates transitions with very narrow natural linewidth which is a promising candidate for use in the next generation HCI atomic clocks to reach an accuracy below 10^-19. A recent experiment reported in [Nature,581(7806) 2020] <cit.>, used Penning trap mass spectrometry to measure the energy of an extremely long-lived metastable electronic state, thus opening doors to search for HCI clock transitions. This work is an extension of the previous experiment, presenting an experiment proposal to determine its lifetime which is another key factor in the development of HCI clocks. By means of an in situ state regeneration method and an updated pules-and-phase measurement scheme, the decay from a metastable electronic state can be directly detected in a Penning trap, enabling a determination of the lifetime of long-lived metastable electronic state on the order of over seconds. This is a highly challenging task for any other conventional technique such as decay curve of fluorescence detections. To demonstrate the effectiveness of this method, a comprehensive simulation is carried out under real experimental conditions. The predicted result shows advancements over existing techniques in the specific case for determining both energy and lifetime of a metastable electronic state. Finally, two suitable candidates are proposed for testing this method, and the state-of-the-art MCDHF theory are employed for accurate energy levels and transition rate calculations. The future prospects in the experimental determinations of a wide range of energies and lifetimes of long-lived metastable electronic states, probing hyperfine and magnetic quenching effects on high-order forbidden transitions and search for highly quality HCI clock transitions are discussed. § INTRODUCTION Atomic electron transition between two electronic energy levels is a fundamental quantum mechanical process in atoms and is responsible for most of the light observed in daily life. The rate of transition was first described by Einstein coefficients in 1916 and later expressed by Fermi's golden rule within the framework of quantum electrodynamics (QED). According to the selection rule, transitions via electronic dipole (E1) radiation cause excited states to decay rapidly, with natural lifetimes below nanoseconds. However, if the transitions are forbidden, then excited states have significant longer lifetimes and become metastable. These metastable electronic states are of great interest in modern clocks and frequency standards due to their transitions of high-quality factors <cit.>. On top of that, it provides an isolated quantum system for quantum computing and quantum simulation <cit.>, allowing for stringent tests of building blocks of standard model and fundamental theory (for example Lorentz symmetry <cit.> and general relativity <cit.>) and search for new physics <cit.>. Optical clocks, such as Yb^+ <cit.> and Ca^+ <cit.> clocks, typically exhibit lifetimes of seconds and have sub-hertz nature linewidths. One promising candidate that can be used for building next generation of frequency standard is highly charged ions. In HCIs, the outer electrons are strongly bound and thus the sensitivity to external electromagnetic perturbations is highly suppressed in comparison with clocks of singly charged ions or neutral atoms. While inter-shell transitions of HCIs are typically in the x-ray region, some intra-shell spectral lines - such as fine structure transitions - occur in the visible or ultraviolet (UV) range that frequency combs can in principle access <cit.>. As the first realization of this new clock, a group at Physikalisch-Technische Bundesanstalt (PTB) detected the 2s^22p ^2P_3/2→^2P_1/2 magnetic-dipole (M1) fine-structure transition of boron-like Ar^13+ <cit.>. What’s more, as the nuclear number increases, the energy of electron shells will rearrange giving rise to a level crossing feature and thus the visible inter-shell optical transitions take place <cit.>. In some specific isoelectronic configurations such as 5p4f and 4f^-15s, several metastable electronic states and their transitions have been predicted for use in HCI clocks <cit.>. A few optical experiments have been performed in an electron beam ion trap (EBIT) to detect those metastable electronic states with lifetimes on the order of ms <cit.>, providing knowledges on understanding the level systems. However, through the conventional florescence detections of the decay photons from the ions in an EBIT, the most suitable clock transitions with lifetimes over seconds such as 5p4f ^3F_2 to 5p^2 ^3P_0 of Pr^9+ <cit.> can not be directly measured due to the very rare spontaneous emissions. Consequently, the energy levels and lifetimes of these transitions remain unknown. Recently, an experimental group at the Max-Planck-Institut für Kernphysik (MPIK) performed a direct detection of metastable electronic state 4d^94f ^3H_5 of ^187Re^29+ by Penning trap mass spectrometry which opens a new door to search for high quality HCI clock transitions <cit.>. Apart from the transition frequency ν, the lifetime τ or its natural linewidth 1/τ is another key parameter in the determination of clock instability <cit.>. Owing to the challenges in dealing with electron correlations in such complex system, theoretical calculations of the transition rates can have large discrepancy to each other, even if the calculated energy levels show a very good agreement. For example, a metastable electronic state 3s^23p^53d ^3F_4 of Fe^8+ has been predicted in many theoretical calculations since last two decades (see <cit.> and references therein). From different groups, the reported lifetimes are 578 s by Tayal et al. <cit.>, 493 s by Kynienė, et al. <cit.>, 970 s by Hahn et al. <cit.> and 1085 s from CHIANTI database <cit.> while the predicted energy levels agree within 1%. For the proposed HCI atomic clock transitions, the calculated lifetime of 5p4f ^3F_2 state of Pr^9+ was 59 s by Safronova et al. <cit.>, while the calculation result from Bekker et al. <cit.> is 155 s. To detect lifetimes of metastable electronic states in HCIs, fluoresces detection technique has been widely used in ion traps <cit.> and others measure the population losses via electron-ion recombination in storage rings (see <cit.> and reference therein). These experiments have observed a number of metastable electronic states with lifetimes ranging from microseconds to hundreds of milliseconds. However, because of the detection efficiency on time resolved photons or particles and the loss mechanism of metastable electronic states arising from ion escape and ion collisions, to measure longer lifetimes over seconds is experimentally challenging. In this paper, we propose a Penning trap spectrometry-based experimental approach to observing the decay of extremely long-lived metastable electronic states. Instead of detecting decay photons, our approach entails continuously weighing a single ion in a Penning trap. Once the ion emits a photon that carries energy away, an equivalent mass change can be measured, where weighing ion's mass by cyclotron frequency ratio (CFR) measurement technique has been successfully implemented to resolve the metastable electronic state by MPIK group <cit.>. This technique enables the determination of the transition frequency by measuring the mass difference and determining the lifetime of the ion by recording the time when ion mass change occurs. This way, we learn the transition frequency by measuring the mass difference and know the lifetime via recording the time when the ion mass changes. This method offers a unique opportunity to observe a quantum process in an isolated single-atom system, which is likely to arouse the interest of researchers in fundamental research. §.§ Summary of proposal At PENTATRAP (see ref. <cit.>), an elaborated measurement scheme was employed in the cyclotron frequency ratio (CFR) measurement of two electronic states of ^187Re^29+ ions by transporting three ions (one of which stays at metastable electronic state) between two cylindrical Penning traps such that the influence from the magnetic field drift is largely suppressed, allowing for a CFR measurement at a precision of a few 10^-11 level. This way, the energy of this metastable electronic state was determined with an accuracy below 2 or 500 via a direct measure of mass difference of ions in the ground state and metastable electronic state. For lifetime determinations, if keep this measurement running until the measured ion mass ratio changes, the decay of metastable electronic state could be observed. Since the transition occurs randomly with a mean rate, the measurement must be repeated for hundreds of cycles to obtain the lifetime. In principle, this can be performed, but it is not such easy in practice. Besides the predicted lifetime of about 130 days for ^3H_5 of ^187Re^29+ is too long to measure, there are still a few technical reasons to hinder lifetime measurements of other metastable electronic states at the current PENTATRAP setup. Firstly, the metastable-state ions were produced in an external electron beam ion source (EBIS) and injected into the Penning trap setup. After that, it normally takes tens of minutes to hours to prepare a single ion before the measurement starts and therefore it is impossible to measure any lifetime which is shorter than the preparation time. Secondly, one run of CFR measurement including ion transport can take tens of minutes, which also limits the measurable lifetimes. Lastly, because the trap setup has to connect to a room-temperature beamline allowing for ion injection, the vacuum in the cryogenic Penning trap is strongly impacted resulting in a high charge exchange rate between ions and background gases and therefore the ion can only stay hours to days for the measurement. Once the ion loses its right charge state, the traps need to be emptied and a new set of ions is loaded. This is very time-consuming since hundreds of cycles need to be performed to determine the mean lifetime which involves lots of efforts in ion loading. To carry out measurements of metastalbe states with lifetimes of a wide range from a few seconds to days, we propose some improvements based on the state-of-the-art Penning trap spectrometry. First and foremost, the metastable electronic states need to be internally reproducible. After a single ion is prepared, a field emission array (FEA) mounted on one end of the trap tower can be employed to emit electrons that hits the trapped ion with kinetic energy such that the ion can be excited to its metastable electronic state by electron impact excitation (EIE). If the trap is accessible to optical system, photon excitation (PE) is an alternative way as long as the excitation energy is suitable for lasers. But, in comparison, EIE can be used for most of the case, because the collisional energy can be tuned up to keV and furthermore the excitation energy is not necessary to know in such high accuracy as is needed for PE process. The advantages of this method are twofold. On one hand, it enables metastable electronic states to be reproduced in sub-seconds without reloading new ions which significantly improve the measurement efficiency. On the other hand, once ions are injected into the trap, a dedicated mechanical cryo-valve can isolate the trap vacuum from room-temperature beamline leading to an ultra-high vacuum (UHV) environment in which HCIs can stay for months <cit.>. Secondly, we propose a sequential pulse-and-phase (Seq-PnP) measurement scheme to measure the modified cyclotron frequency variance of a single ion continuously. Once the frequency changes by a certain value due to the decay, the time can be recorded. Compared to the typical PnP method used in <cit.>, there is no need to take different evolution time in the phase measurement, or dip, double dip spectra followed by ions transport between different traps for a standard CFR measurement. Each run of modified cyclotron frequency measurement in Seq-PnP only takes a few seconds to tens of seconds which significantly extends the lower limit for the measurable lifetimes. Further, we will show by measuring extra dip and double dip spectra just after the metastable electronic states decay, the mass ratio can be determined as well, which gets even less impacted from the magnetic field instability. As a result, the two key factors – the frequency and lifetime of a clock transition can be precisely determined in one measurement. The transitions from metastable electronic states can be strongly impacted by an electromagnetic field generated from either the superconducting magnet or the nucleus. The latter effect is also known as hyperfine interaction. The magnetic and hyperfine induced transitions have been observed in some previous experiments <cit.> via fluorescence detections. For extremely long-lived metastable electronic states of which the decay photon is barely detectable, the proposed technique also allows to probe hyperfine and magnetic quenching effect by measuring a reduction on their lifetimes in Penning traps. The rest of this paper is organized as follows. In section <ref>, the measurement principle of penning trap spectrometry is described. In section <ref>, a simulation of measurement scheme based on a Seq-PnP method is presented on how the proposed technique works in the determination of energies and lifetimes of metastable electronic states. In section <ref>, an accurate theoretical calculation using MCDHF method is implemented to give proper candidates of which the energies and lifetimes can be measured by the proposed method. In section <ref>, we discuss the future prospects in measuring long-lived metastable electronic states and search for highly quality clock transitions. In section <ref>, we conclude this proposal work. § MEASUREMENT PRINCIPLE §.§ Penning Trap Principle The principle of Penning trap experiments has been well described in depth by other pioneers of Penning trap groups <cit.>. The experimental setup consists of a homogeneous magnetic field superimposed with an electrostatic quadrupole potential. The magnetic field provides a radial confinement of charged particles while the electrostatic potential prevents ions’ escape in axial direction. The resulting ion motion is a superposition of three harmonic oscillations: the radial modes split up into a fast modified cyclotron motion with frequency ν_+ and slow magnetron motion with frequency ν_-, and an axial harmonic oscillation with frequency ν_z: ν_+ = 1/2(ν_c + √(ν_c^2 - 2ν_z^2)) ν_- = 1/2(ν_c - √(ν_c^2 - 2ν_z^2)) ν_z = 1/2π√(q/MU_0C_2/d_char^2), where q and M are the charge and mass of the ion, respectively. C_2 and d_char represent geometrical parameters defined by the trap size. ν_c=Bq/2π M is free cyclotron frequency which can be obtained by the so-called invariance theorem: ν_c= √((ν_+^2+ν_z^2+ν_-^2)) <cit.>. With the typical B field of a few T and electrostatic potential U_0 of about -10, the frequencies of eigen motions follows the hierarchy: ν_c≈ν_+≫ν_z≫ν_-. The proposed experiment on metastable electronic states is planned to be performed at Shanghai Penning Trap (SH-Trap) which is under construction. The trap tower is structured by using a cylindrical stack of electrodes like the predecessor experiments <cit.> (Fig. <ref>), located in the worm bore of the magnet and cooled down to 4 with the help of an ultra-low-vibration cryostat. The electrostatic potential produced by this cylindrical trap design could deviate from the ideal quadrupole potential which is called electric field imperfections. The general potential in the center of the measurement trap (MT) can be written as a series expansion: U(r,θ,z) = U_0/2∑_n=0^∞ C_n(r/d_char)^n P_n(cosθ), where, C_n represents a dimensionless expansion coefficient and P_n(cosθ) is the Legendre polynomials. To achieve the desired quadrupole potential with only a non-zero C_2, pairs of correction electrodes can be stacked on both sides of the (center) ring electrode. However, due to the limited space and manufacture imperfections, typically only the main high order coefficients C_4 and C_6 are optimized. With a 7-mm-diameter five-electrode trap design the electric potential coefficients \|C_2\|=0.55, \|C_4\|<10^-5 and \|C_6\|<10^-2 can be achieved as a similar design has already described in Mainz <cit.>. A superconducting solenoid magnet will be manufactured to produce a homogeneous magnetic field upto 7. The main magnetic field imperfections can be expressed by a linear magnetic field gradient B_1 as well as the quadratic dependency B_2: B(r,z) =B_0e_z+ B_1 (ze_z - r/2e_r) + B_2[(z^2-r^2/2)e_z-zre_r]. Both the electric and magnetic field imperfections can cause undesirable frequency shifts on eigen motions, which contributes systematic uncertainties in many precision measurements at Penning trap facilities <cit.>. Therefore, more care should be taken to evaluate the effects on the energy and lifetime measurement of metastable electronic states proposed in this work, for details see chapters below. §.§ Detection method Highly charged ions can be produced by an in-trap EBIS on the bottom side of the trap tower, see Fig. <ref>. The electron beam which emits from a field emission point or array (FEP/FEA) can be reflected by the reflector electrodes and then hit on a target to sputter the atoms out. Those atoms will continuously collide with electrons such that they become highly charged and trapped in the Creation Trap. This procedure is successfully tested in Mainz <cit.>. However, the charge state is limited by the maximum bias on the electrodes due to the compact structure of the trap and feedthroughs. For ions of higher charge states, they must be produced in an external ion source. A room temperature permanent magnet electron beam ion trap called CUBIT has been developed for HCIs extraction, for details see <ref>. After the HCIs inject into the capture trap with a kinetic energy of about a few hundreds eV, the voltage on the electrodes can be quickly pulsed in order to capture the incoming ions. With the help of cryogenic pumping and a dedicated cryo-valve, most residual gas in the trap chamber is frozen resulting in a very good vacuum of 1E-16 and thus HCIs can stay long, e.g. a few months for Ar^13+ for measurements (see the ALPHATRAP experiment <cit.>). Once the ions are captured in CT or created in CreaT, they will be transported to MT. Then, a single HCI will be produced after a dedicated purification procedure (see <cit.>). The ion's oscillation frequencies can be detected by a so-called image current detection method. The image current induced by ion's oscillation on one of the trap electrodes is on the order of a few fA that can be then picked up by a superconducting tank circuit (resonator) which is tuned close to ion's oscillation frequencies. In this way, the weak current signal is converted into a detectable voltage signal across the equivalent parallel resistance R_p, which is written as R_p = Q ω_R L, where Q, ω_R and L are the quality factor, resonance frequency and inductance of the tank circuit, respectively. This voltage signal is then amplified by a self-made low-noise cryogenic amplifier, which features an equivalent input noise u_n < 1 /√(Hz) at 600 and an amplification factor of about 15 dB. Before the Fast Fourier Transformation (FFT), the signal usually needs to be further amplified via a commercial low-noise room-temperature amplifier and down-mixed with a local oscillator to around 10. Fig. 2(a) shows the typical dip signal (simulated) of the axial motion of a single ion represented by Fe^8+ when it is in the thermal equilibrium with the detector circuit. While the ion is axially excited, the peak signal instead of the dip signal appears upon the resonator noise, see Fig. 2(b). The (-3 dB) width of the dip is written as (R_p q^2)/(2π MD_eff^2 ), where D_eff is the effective distance of the electrode for the signal pickup which is about 8 according to the trap design. Two radial mode frequencies can be measured by coupling either modified cyclotron motion or magnetron motion to axial motion via a quadrupole excitation with radio frequency at ν_rf = ν_+ - ν_z or ν_rf = ν_- + ν_z. During the coupling, the axial and radial motion start to exchange energy undergoing a rabi-type oscillation. In this case, two radial motions are cooled and eventually thermalized with the tank circuit as well and the axial dip splits into two dips which is so-called double dip spectrum (see Fig. 2(c)). By measuring the left and right dips, ν_l and ν_r, the modified cyclotron frequency and magnetron frequency can be obtained by ν_+ = ν_rf - ν_z + ν_l + ν_r and ν_- = ν_rf + ν_z - ν_l - ν_r. § MEASUREMENT SCHEME One typical double dip spectrum enables an accurate determination of the cyclotron frequency at a few 10^-9 level. It is still hard to distinguish some long-lived metastable electronic states, for instance 3s^23p^53d ^3F_4 state of ^56Fe^8+ of which the energy is about 52.79. In addition, recording FFT spectrum takes around 100 seconds, which is not suitable for measuring short lifetimes. For a better frequency resolution, a pulse and phase sensitive method <cit.> can be used, and a relative cyclotron phase jitter of below 2E-10 from pulse to pulse can be achieved. To observe decay from metastable electronic states, instead of taking tens of minutes to implement normal PnP method introduced in <cit.>, we propose a sequential PnP method continuously measuring the variance of modified cyclotron frequency. Because of ν_+ ≈ν_c , if a phase jump on the modified cyclotron frequency is observed, it indicates that the ion mass has changed, or in other words, the decay occurs. The measurement scheme (see Fig. <ref>) starts when the electronic state is repopulated for instance by EIE, and ^56Fe^8+ ion is quickly thermalized with the tank circuit. Then, it includes several steps in one run: (1) Excite the modified cyclotron motion with a starting phase φ_0. (2) Wait for cyclotron phase evolution with t_evol. The phase ends with φ(t_evol)=φ_0+360^∘ν_+ t_evol. For a well-defined φ_0 and fixed t_evol, the end phase only depends on ν_+ corresponding to which electronic state the ion stays. (3) Implement a red sideband ν_rf=ν_+ - ν_z quadrupole pi-pulse coupling between modified cyclotron motion and axial motion to transfer the cyclotron amplitude and phase to axial motion for readout. (4) Measure the axial peak (axially hot ion) signal to obtain the transferred cyclotron phase. (5) Cool axial motion and start next cycle. According to the envisaged electromagnetic field settings in SH-Trap (B = 7 and U_0 = -5.7) and the mass value of ^56Fe^8+ from AME 2020 <cit.>, the frequencies are ν_+ ∼ 15, ν_z ∼ 350 and ν_- ∼ 4. In the first step, the ion is excited to 50 in cyclotron radius. Then it freely evolutes for t_evol ∼ 10 giving rise to a phase difference Δφ∼ 54 degrees if ion stays in difference electronic states. With a careful control of the quadrupole pi pulse, cyclotron energy can be fully transferred to axial mode. Finally, the axial motion with an amplitude of about 330 is detected and then cooled by the tank circuit. By sequentially implementing this measurement scheme, the modified cyclotron phase for a fixed t_evol is continuously recorded such that the variance of the cyclotron frequency can be observed immediately once the ion mass changes due to the decay. To perform CFR measurement in the same measurement scheme, we refer to an alternative solve of mass ratio R <cit.>: R = ν_z1^2/2ν_c1^2+(1-ν_z1^2/2ν_c1^2)(1+Δν_+(Δν_+ - 2 ν_+1)/(ν_c1-ν_z1^2 /2 ν_c1)^2)^1/2 where, we define subscript 1 the ground state. Δν_+ = ν_+,g - ν_+,m = -Δφ/360^∘t_evol is the modified cyclotron frequency difference, which is measured from the variance of the cyclotron phase. Just after the decay, PnP measurement with short evolution time can be implemented to determine ν_+1 or ν_+,g. Dip and double dip spectra can be recorded to measure ν_z1 and ν_-1 and the resulting ν_c1. Because Δν_+ and ν_+1 are measured back-to-back, the effect from magnetic field instability is significantly suppressed, allowing for an even higher accuracy in the comparison to the standard CFR measurement involving ions transport. §.§ Numerical implementation Here, we present a numerical implementation for the proposed Seq-PnP measurement scheme by solving the equations of ion motion in the external electromagnetic field from first principles. The numerical model consists of a single ion oscillating in a static electromagnetic trapping field, a tank circuit for image current detection and external electric pulses for ion motion operation. The differential equations of this model can be expressed by: ẍ = q/M(-∂ U/∂ x + B_z ẏ - B_y ż + E_x(x,y,z ) ) ÿ = q/M(-∂ U/∂ y - B_z ẋ + B_x ż + E_y(x,y,z ) ) z̈ = q/M(-∂ U/∂ z + B_y ẋ - B_x ẏ + E_z(x,y,z ) + IL̇_̇İ/D_eff) Ï_̈L̈ = 1/LC( -q/D_effż - I_L -I_noise -L/R_pİ_̇L̇) Here U and B are the electric potential and magnetic field strength, respectively, and the main field imperfection coefficients C_4, C_6 and B_2 are included as described in eq. (<ref>) and (<ref>), while the effect from the odd terms like C_3, C_5 can be neglected due to the trap symmetry. On top of that, a voltage noise and magnetic field fluctuation are included by adding a random walk noise upon the main field. E_x,y,z(x,y,z) denotes the external electric excitation. To take account the interaction between a single ion and a superconducting tank circuit, a treatment of image current detection has been described in ref <cit.>. C is the equivalent parallel capacitance of the detection circuit. I_L is the current and I_noise = N(1,0) √(2k_B T_LC L/R t) is the thermal Johnson noise current through the inductance of the tank circuit, where T_LC is the environmental temperature and N(1,0) is a number sampling from a Gaussian distribution. An appropriate simulation time step should be chosen for example t= 0.2 for ^56Fe^8+ with ν_+ ∼ 15 so that the effect on the accuracy of numerical solution attributed to the finite time step has negligible influence on the cyclotron frequency (or phase) determinations while a reasonable time spend on the numerical computation is accessible. The set of the differential equations is solved by employing a fourth-order Runge-Kutta method (RK4) which is a widely-used numerical integration algorithms for finding solutions in initial value problems. For an efficient calculation, the RK4 solver is written by C++ language and compiled into a package which is then called by the main script written by Python. In the main script, all the variables in the simulation are initialized and then sent to RK4 solver for calculation. After the differential equations are solved, the time dependent data is down mixed by a sinusoidal signal with a frequency at ν_z- 11.8 and then sampled with a rate of 200 for FFT spectrum in order to determine the motional frequencies and phases. §.§ Simulation results and benefits The simulation results of a Seq-PnP measurement scheme of metastable electronic state ^3F_4 of ^56Fe^8+ is shown in Fig. <ref>. The equivalent mass value of this excited state is converted from its excitation energy. The modified cyclotron phase evolution time is 10 seconds which causes a phase difference of about 54 degrees between the metastable and ground state. Once the decay takes place at measurement cycle N, a phase jump can be resolved and the simulated phase jitter if around 11∼ 12 degrees equivalent to about 2E-10 relative fluctuations. In order to verify the feasibility of this measurement technique, the main systematic uncertainties are taken into account for a simulation close to real experimental conditions. (1) The thermal motion can affect the modified cyclotron phase from several aspects. Firstly, the imprinted phase from the radial dipole excitation can be disturbed by the thermal motion with a random phase. Secondly, after this excitation the cyclotron amplitude can be written as r_+ = r_exc + δr, where δr denotes a jitter attributed to the thermal motion. Then, this amplitude jitter causes the frequency jitter due to the field imperfections and special relativity. In the present model, the ion is thermalized with the detection circuit to about 4.2, and thus the initialized ion motional amplitudes and phases are drawn from a Boltzmann distribution at this temperature. The field imperfections are conservative set with C_4 = 1E-5, C_6 = 1E-2 and B_2 = 0.2 T/m^2 in our model. The special relativity is not included in the simulation but its effect is calculated to a relative uncertainty at 2E-11 level. (2) During the free evolution time, any fluctuation on the electromagnetic field can disturb the modified cyclotron phase. The simulation model has already included a random walk noise based on an achievable noise feature of voltage stability of 4E-8 (UM1-14 power supply by Stahl Electronics) and B field stability of 2E-10 in a time span of 10 seconds. The effect from a magnetic field drift typically <1E-9 per hour can cause a frequency drift only at 1E-12 level in one measurement run and therefore it is not included in the simulation model for simplicity. (3) The technical phase jitter during the axial peak signal detection is also simulated in our model. The signal-to-noise ratio (SNR) of the axial peak spectrum Fig. 2(b) which is synthesized by employing a FFT analysis leads to a major phase jitter to a few degrees. The axial amplitude can be optimized to reduce the SNR jitter until the phase jitter caused by the field imperfections or special relativity becomes sizeable. In the present model, the axial amplitude is about 330 um after a quadrupole pi-pulse coupling corresponding to a SNR of about 25 dB and the phase jitter from the field imperfection effect is still small within a readout time of 0.1 second which is about three times of the cooling time constant of the ion. In addition, the impact on the phase determination from the axial thermal motion is negligible due to the relatively small amplitude of about 10 um. The axial frequency fluctuates because of the voltage supply stability, which in principle leads to another readout phase jitter, but this effect is also small in the comparison with other sources of jitter. (4) Some other negligible effects we do not include in our model: i) The recoil effect from the decay photon to the heavy ion like ^56Fe^8+. ii) The quantum state jump in cyclotron motion due to the electric field noise. iii) The image charge effect is not included since it cancels for the ion of identical charge and mass. As a result, we have run the simulation model including all the systematics discussed above for a short evolution time of 5 and a long evolution time of 10. The simulated short-term phase jitter about 5 degrees comes from the axial peak signal readout, while the long-term phase jitter about 11∼ 12 degrees (also see the prediction band in Fig. <ref>) is mainly attributed to the magnetic field fluctuation. From the simulation verification, the Seq-PnP method enables to search for decay of long-lived metastable electronic states like ^3F_4 of ^56Fe^8+ with equivalent mass difference at 1E-9 level in 10 seconds. What's more, in comparison to other lifetime measurements by storing lots of ions, the accuracy of this single-particle measurement technique is only limited by the statistics, giving opportunities to test the atomic structure theory and check other existed lifetime measurements in high precision. § ATOMIC STRUCTURE CALCULATIONS §.§ MCDHF method For the theoretical studies on energies and lifetimes of long-lived metastable electronic states, a state-of-the-art theoretical method called the multiconfiguration Dirac–Hartree–Fock (MCDHF) method <cit.> has been employed. In the MCDHF method, as implemented in the Grasp package <cit.>, the atomic state function (ASF), is expanded in antisymmetrized and coupled configuration state functions (CSFs), \|Γ J ⟩=∑_γ c_γ\|γ J⟩, where γ represent all the coupling tree quantum numbers needed to uniquely define the CSF. The CSFs are four component spin-angular coupled, antisymmetric products of one-electron Dirac orbitals, ϕ _nκ m = 1/r( [ P_nκ(r)χ _κ m(θ ,ϕ ); iQ_nκ(r)χ _ - κ m(θ ,ϕ ) ]), where χ_± km(θ,ϕ) are two-component spin-orbit functions. The radial functions P_nκ( r) and Q_nκ( r) are represented numerically on a grid. The radial parts of the Dirac orbitals together with the mixing coefficients are obtained in a relativistic self-consistent field (RSCF) procedure in the extended optimal level (EOL) scheme <cit.>. The angular integrations needed for the construction of the energy functional are based on the second quantization method in the coupled tensorial form <cit.>. The transverse photon (Breit) interaction and the leading quantum electrodynamic (QED) corrections (vacuum polarization and self-energy) can be accounted for in subsequent relativistic configuration interaction (RCI) calculations. In the RCI calculations, the Dirac orbitals from the RSCF step are fixed and only the mixing coefficients of the CSFs are determined by diagonalizing the Hamiltonian matrix. The transition rate A (s^-1) between two states \|Γ'J'⟩ and \|Γ J⟩ are expressed in terms of reduced matrix elements of the relevant transition operators <cit.>, where the reduced transition matrix element is the square root of the line strength S multiplied with a factor. For the electric dipole (E1) transitions, A(Γ' J' →Γ J)=2.0261×10^18/(2J'+1)λ^3\|⟨Γ J\|\| P^(1) \|\| Γ' J' ⟩\|^2, where, λ is the transition wavelength in Å. In isotopes with non-zero nuclear spin, the interaction between the nuclear electric and magnetic multipole moments and the electrons couples the nuclear spin I and the electronic angular momentum J to a new total angular momentum F and splits each fine structure level into several hyperfine levels. The corresponding Hamiltonian may be represented as a multipole expansion: H_hfs=∑_k≥1T^(k)·M^(k), where T^(k) and M^(k) are spherical tensor operators of rank k, operating on the electronic and nuclear parts of the wave function, respectively. The hyperfine interaction will introduce a mixing between states with different J quantum numbers, and can open new decay channels for some transitions. The wavefunction of the atomic system can be written as an expansion: \| Γ̃ IF ⟩= ∑_Γ J d_Γ J \| Γ IJF ⟩. The E1 transition rate between two hyperfine states \|Γ̃'̃ IF'⟩ and \|Γ̃ IF⟩ is given by A(Γ̃'̃ IF' →Γ̃ IF)= 2.0261×10^18×(2F+1)/λ^3\| ∑_Γ J∑_Γ' J'd_Γ J d_Γ' J' × (-1)^I+J'+F+1×{[ J F I; F' J' 1 ]}⟨Γ J\|\| P^(1) \|\| Γ' J' ⟩\|^2. In presence of the magnetic field, choosing the direction of the magnetic field as the z-direction and neglecting all diamagnetic contributions, the interaction between the magnetic moment of an atom in an external field B can be written as H_M=(N_0^(1)+Δ N_0^(1))B, where the first term is an operator of the same tensorial form as the magnetic dipole hyperfine operator, and the last term is the Schwinger QED correction due to QED corrected electronic g-factor value g_s = 2.00232. When the field is included, only M_J or M_F remains a good quantum number, depends on the nuclear spin. For isotopes with zero nuclear spin, only M_J is a good quantum number, we can represent the wavefunction of the atomic system by \| Γ̃ M_J ⟩= ∑_Γ J d_Γ J \| Γ JM_J ⟩. The E1 transition rate between two states \|Γ̃' M_J'⟩ and \|Γ̃ M_J⟩ is given by A(Γ̃' M_J' →Γ̃ M_J) = 2.0261×10^18/λ^3∑_q \| ∑_Γ J∑_Γ' J' (-1)^J-M_J d_Γ J d_Γ' J' ×( [ J 1 J'; -M_J q M_J' ]) ⟨Γ J\|\| P^(1) \|\| Γ' J' ⟩\|^2. For isotopes with non-zero nuclear spin, the wavefunction of the atomic system can be represented as, \| Γ̃ IM_F ⟩= ∑_Γ JF d_Γ JF \| Γ IJFM_F ⟩. The transition rate for an E1 transition between \|Γ̃' I M_F' ⟩ and \|Γ̃ I M_F ⟩ is given by A(Γ̃' IM_F' →Γ̃ IM_F) = 2.0261×10^18/λ^3∑_q \| ∑_Γ JF∑_Γ' J'F' (-1)^F-M_F d_Γ JF d_Γ' J'F' ×√((2F+1)(2F'+1))( [ F 1 F'; -M_F q M_F' ]) (-1)^I+J'+F+1 ×{[ J F I; F' J' 1 ]}⟨Γ J\|\| P^(1) \|\| Γ' J' ⟩\|^2. By diagonalizing the interaction matrix using the HFSZEEMAN program <cit.>, hyperfine structure substates and Zeeman energy splittings are obtained together with the expansion coefficients of the basis functions, the transition rates can thus be computed. The other multiplolar transition rates between fine structure/hyperfine structure/Zeeman levels are not present here, and can be obtained in a similar way. §.§ Calculations and Results Our calculations are based on the restricted active space (RAS) method, where the CSF expansions are obtained by allowing single and double (SD) substitutions from selected reference configurations to given orbitals to an active set (AS) <cit.>. The active set increases by increasing the number of layers, specifically, a set of virtual orbitals specified by its principal quantum number. We sort the electron correlation effects into three types: i) substitutions only from the outermost valence subshells, valence-valence (VV) correlation is included; ii) at the most one substitution from the core subshell, core-valence (CV) correlation is accounted for; iii) double substitutions from the core subshells are allowed, core-core (CC) correlation is included. The low-lying levels of interest in this work for ^56Fe^8+ and ^76,77Se^6+ are shown in Fig. <ref> and <ref>. As a starting point RSCF calculations were done for 3s^22p^6, 3s^23p^53d configurations of ^56Fe^8+, and for the 3d^10, 3d^94s configurations of ^76,77Se^6+, these configurations are treated as reference configurations in the subsequent calculations. The 3s, 3p, 3d subshells in ^56Fe^8+ and 3d, 4s subshells in ^76,77Se^6+ are defined as valance subshells, and the other inner subshells are core subshells. By allowing restricted single (S) and double (D) substitutions from the reference configurations to active sets with principal quantum numbers up to n = 7 and with orbital angular momenta up to l = 6, we include the VV correlations for both ^76,77Se^6+ and ^56Fe^8+, CV correlations of 2s, 2p subshells for ^56Fe^8+, CV correlations of 3s, 3p subshells for ^76,77Se^6+. The RSCF calculations were followed by RCI calculations, including the Breit interaction and leading QED effects. By using the resulted ASFs, hyperfine and Zeeman interactions were computed by using the HFSZEEMAN program <cit.>. In Table <ref>, our calculations on the energies of metastable electronic state 3s^23p^53d ^3F_4 of ^56Fe^8+ and 3d^94s ^3D_3 of ^77,76Se^6+ are in good consistency with other theoretical values. For lifetime calculations of ^3F_4 state, it can be noticed that there are several decay channels to the ground state (see Fig. <ref>). Among them, the transition rate of ^3F_4 to ^3P_2 determines the lifetime, because other decay channels are even higher-order forbidden transitions, of which the transition rates are much smaller. Further, the transition of the intermediate state ^3P_2 to the ground state is much faster compared with that of ^3F_4 to ^3P_2. With a typical magnetic field in Penning trap, although the wavefunction of ^3F_4 state can mixed with other states like ^3F_3, however, due to its low transition rate the lifetime of ^3F_4 state is not affected. For calculations of ^77,76Se^6+, this M3 forbidden transition of 3d^94s ^3D_3 to ground state can be impacted through hyperfine interaction and external magnetic field because of the strong mixing with shorted-lived ^3D_2 and ^1D_2 states whose decay rates are 4.3E+4 s^-1 and 1.6E+5 s^-1, respectively. As a result, for ^76Se^6+ ion with zero nuclear spin the lifetime of ^3D_3 state reduces from 4518 to 563 and 115 in the external magnetic field from zero to 3 and 7, respectively, while for ^77Se^6+ ion with a nuclear spin of 1/2, the lifetime of ^3D_3 state is reduced to 865 without external magnetic field. In a magnetic field upto 7, the interaction between the electronic states and the external field becomes much stronger than the interaction with nucleus, which is so-called Paschen-Back regime, and the resulting lifetimes of two isotopes are similar. Note that, in a strong magnetic field only M_J and M_F are good quantum numbers instead of J and F. Each run of the measurements, a decay from one of the M-states is recorded via Penning trap spectrometry. After a number of runs, the lifetime of the J states can be determined by averaging values. The detailed transition rate calculations on M_J and M_F states of ^3D_3 of ^77,76Se^6+ are listed in the Appendix <ref>: Tables. As long as decay from different M-states is time-resolved, it gives a unique opportunity to identify individual event of transitions from each M-states. In table <ref>, the listed lifetimes of ^3D_3 state are averaged values assuming a uniform-populated M_J and M_F states. Via a polarized electron beam for instance in a homogeneous strong magnetic field, the population on magnetic sub-states can be non-uniformed. According to a population calculation based on a collisional radiative model (CRM) on magnetic sub-states, the relative population of M_J=±3,±2,±1,0 for ^3D_3 is in a range of 12.2-15.9%. § DISCUSSION AND IMPLICATIONS In this section, we will discuss the implementation of energy and lifetime measurements according to the proposal and its future prospect. As a single ^56Fe^8+ ion is prepared normally in the ground state, it needs to be excited to its metastable electronic state ^3F_4 for measurement. To this end, an electron beam emitted from an FEA (by HeatWave Labs) with a size of 1 and current up to mA can be used for EIE. Through an EIE process, the ion is usually excited to some states with higher energy and then followed by radiative decays it has a chance to arrive at the long-lived metastable electronic state. The probability to find an ion staying at this metastable electronic state can be calculated by solving a set of rate equations of excitation, radiation and other process between all the atomic states, which is described by a collisional radiative model. Based on CRM calculations integrated in a flexible atomic code <cit.> (for details, see <ref>), the metastable electronic states can be produced in sub-seconds with the typical electron density produced by an FEA and the resulting populations are 19% and 9% for ^3F_4 of Fe^8+ and ^3D_3 of Se^6+, with the impact electron energy of 170 and 130, respectively. It is true that the ion decays to the ground state after EIE for most times, however, to check whether the ion stay in the metastable electronic state is pretty fast because the modified cyclotron phase is continuously measured and compared with the last run. The time between two measurement cycle contains the spend on EIE and ion cooling, which can be controlled to such short that the frequency drift in this time interval is negligible. Besides a thermal heating by the electrons, the ion can reduce the charge by radiative recombination (RR). With a careful control of electron density, this can be avoided since the cross section of RR is 3-4 orders of magnitude lower than EIE. Even if this RR process happened, the electron beam energy can be tuned over the ionization threshold to reproduce the right charge state in short time. Once the electron beam collides with background gas, contamination ions could be created. In this case, a quick axial pulse can be employed to 'kick' other ions out of the trap. To reduce the chance of producing contamination ions, the a short EIE time and a small beam size are favourable. The visibility in the observation of decay from metastable electronic states via the proposed Penning trap spectrometry is mainly limited by the experimental phase jitter of the modified cyclotron motion. This uncertainty in the phase determination consists of cyclotron frequency fluctuations and a technical jitter due to thermal motion and FFT readout. According to our simulations, the technical jitter is about 5 degrees if the signal with a reasonable SNR is tuned, while, the other jitter depends on the magnetic field fluctuation and is proportional to the evolution time. By setting a 2σ confidence level for the observations of the decay, the corresponding visible phase jump needs to be: Δφ > 2√(δφ_0^2 + δφ_B^2), here, Δφ =360^∘Δ M/Mν_+t_evol, Δ M is the equivalent mass difference between the metastable electronic state and the ground state. δφ_B = 360^∘δ B/Bν _+t_evol and φ_0 represents the technical phase jitter. In Fig.<ref>, the visibility bounds with 2σ confidence level (C.L.) are shown with different magnetic field fluctuations and assuming ν_c= 15. For short measurement time, the measurable mass difference is limited by the technical jitter and free cyclotron frequency. Lower technical jitter and higher cyclotron frequency will extend this visibility bound. Taking account of the ion cooling time as well as the measurement time, the shortest lifetime that can be measured needs to be longer than 1. In principle, to observe decay with even shorter lifetimes is possible, but the probability of occurrence of the decay is reduced as P(t)=exp(- t/τ). For a long measurement time, the measurable mass difference mostly depends on the magnetic field fluctuation. Thus, to extend this limit the stability of the field needs to be improved. The longest lifetime can be measured by this method is basically limited by the time that the ion can be stored, which depends on the charge exchange cross section and the vacuum in the trap. In Table <ref>, the lifetime of metastable electronic state ^3F_4 of ^56Fe^8+ according to our MCDHF calculations is 1110 s which agrees with CHIANTI database <cit.> but shows large discrepancy to S. S. Tayal et al. <cit.> and Kynienė, et al. <cit.>. Since this state is barely affected by the typical magnetic field strength in a Penning trap, its lifetime can be directly measured allowing to distinguish between existing predictions. For metastable electronic state ^3D_3 of ^77,76Se^6+, its life time does get impacted by the hyperfine effect and the external magnetic field, but in turn, it provides an opportunity to probe the hyperfine and magnetic quenching. Since both hyperfine and magnetic quenching result in reduction of the lifetime, which means the two effects are in competition. Therefore, if we want to observe the hyperfine quenching, the magnetic quenching effect must be controlled to be small, and vice versa. To this end, the even-even nuclei is a perfect candidate for probing magnetic quenching effecte, e.g. ^76Se^6+ . To measure its lifetime in different fields, changing the main field is one solution but it will take lots of effort and furthermore lowering the magnetic field will always has limit due to the visibility of the phase jump. Another way is to build storage traps on the outer side where the field is low. Once the metastable electronic state is produced and measured in strong-field measurement trap, the ion is then transported to low-field storage trap. This way, the wavefunction of the metastable electronic state turns from a stronger mixing M-state to a weaker mixing state whose lifetime is much longer. Then, the electronic state can be determined in MT again after a waiting time. During the waiting time an identical ion can be used in MT to monitor the main magnetic field drift by measuring its cyclotron frequency continuously. In order to probe the hyperfine quenching, isotope nuclei with nuclear spin should be used and strong magnetic field is unfavourable. Since, it is not possible to totally get rid of magnetic field, an alternative solution is still to use storage traps described above and thus the magnetic quenching effect scaling with B^2 becomes small while the hyperfine interaction is sizeable. Turn to the future prospect, a direct detection on states such as 5s ^2S_1/2 of Sm^15+, 4f ^2F_5/2 of Nd^13+ and 5p4f ^3F_2 of Pr^9+ (marked in Fig.<ref>) for HCI clock transitions proposed by Safronova et al. <cit.> can be expected, after installing dedicated self-shielding coils and pressure regulation system in the liquid helium dewar of superconducting magnet, which is already mounted in many Penning trap facilities, improves the magnetic field stability to a few 1E-11 level. Further, by combining with a two-ion balance technique <cit.> to control two ions with very close charge to mass ratio rotating in a common magnetron orbit, we could measure relative phases of cyclotron motion of the two ions to balance the inherent fluctuations of the magnetic field on the individual ions. Based on this two-ion balance system, an optical laser can be employed to excite one of the ion. Once the ion is pumped from one state to the other, the mass variance can be observed by applying a simultaneous modified cyclotron phase measurement, which allows for a precision towards 10^-12 level and paves the way to distinguish most long-lived metastable electronic states to search for suitable HCI clock transitions. § CONCLUSION In summary, we present an experimental access to observe the decay of long-lived metastable electronic states of highly charged ions using a Penning trap. To measure the lifetimes of long-lived metastable electronic states, which is very challenging by any other conventional techniques, becomes in principle feasible through the proposed techniques based on a single-ion mass spectrometry. The method's implementation and benefits are described in detail. A dedicated simulation study has been conducted to validate the method's effectiveness and discuss expected results in a realistic setting. For the theoretical studies, we calculated the energy levels and transition rates of 3p^53d ^3F_4 state of ^56Fe^8+ and 3d^94s ^3D_3 state of ^77,76Se^6+ which are suitable candidates for testing the proposed technique. Furthermore, the expected outcomes of measuring their lifetimes to distinguish the existed theories and to probe hyperfine and magnetic field quenching effects are previewed. This method will be potentially extended to use in any precision Penning trap mass spectrometer to detect metastable electronic states with a broad range of energies and lifetimes for fundamental research purposes. § ACKNOWLEDGMENTS The author thanks Prof. Klaus Blaum, Dr. Sven Sturm, Dr. Fabian Heißer, Dr. Wenxian Li and Dr. Xiangjin Kong for kind discussion. This work was supported by the National Key R&D Program of China under Grant No. 2022YFA1602504 and No. 2022YFA1602303, the National Natural Science Foundation of China under Grant No. 12204110, No. 12074081 and No. 12104095, Sponsored by Shanghai Pujiang Program under Grant No. 22PJ1401100 and Max-Planck Partner Group Project. § TABLES Here, we listed the calculated transition rates in M_J and M_F states for ^3D_3 of ^77,76Se^6+ ions with magnetic field of 0, 3 and 7 in tables <ref>-<ref>. § CUBIT AND HCIS PRODUCTION So far, the most commonly used HCI source is electron beam ion trap or sources (EBIT/S). An EBIT produces HCIs by means of successive collisions with a compressed mono-energetic electron beam. In an EBIT, the electron beam is emitted by a hot cathode of a Piece-type electron gun. The beam is accelerated to desired energy by static electric potential applied between the electron gun cathode and an assembly of drift tubes, in the meanwhile it is compressed to a diameter of about 100 either by a superconducting magnet field or permanent magnet filed. In the central drift tube region, externally injected neutral atoms or low charge state ions are subsequently ionized to required charge state by electron bombardment. After passing through the trap region, the electrons are decelerated and diverge in the decreasing magnetic field, finally collected by a collector, while the ions are remaining trapped. The ions are confided by a potential well generated by a static voltage biased on the outer drift tubes, and radially trapped by the combination effects of electron space charge and the magnetic field. Not only the electron impact ionization, the high energy electron can also excite the ions, populate them in metastable electronic states. The ions can be extracted in a leaky mode in which hot ions overcome the axial trapping potential and escape from the trap, or in a pulse mode by quickly ramping down the voltage at the collector side drift tube, which mode is usually used in ion re-trapping. The extracted ions are then charge-to-mass separated, transported, decelerated and re-stored into the ion trap for further analysis. In the present planned setup, CUBIT, a home developed room temperature permanent magnet EBIT, will be used as an ion source. One prototype of the CUBIT has already been commissioned for spectrometry studies of highly charged ions <cit.>. In these studies, ions were also extracted and the charge state distribution were analyzed. The CUBIT uses a 0.3 diameter LaB_6 cathode to emit the electrons. The electron beam is compressed by a 0.56 cylinder shape NbFeB permanent magnet installed in the vacuum vessel. The magnet has a 60 bore, where the drift tubes and other electron transmission electrodes are mounted. The electron gun and the collector are mounted at the two sides of the magnet bore. Four ports are available at the lateral side of the magnet cylinder for light detection and gas injection. Till now, more than 20 elements including W, Pr, Rh, Ru, Mo, Y, Ni, Fe, as well as some noble gases were injected. Ions with charge state up to W^38+ was successfully produced at an electron beam energy of 2, and extracted from the CUBIT in a pulsed mode at 0.5. The extracted HCIs with different charge states typically have a kinetic energy of a few keV which cannot be directly captured by Penning trap setup, and therefore, those ions are state-separated by a Wien velocity filter, decelerated by a pulsed drift tube, focused by electronic optical lens and then guided towards the Penning trap. § CRM CALCULATION To calculate the probability of producing metastable electronic states by EIE in a specific atomic level system like ^56Fe^8+, a sophisticated method known as collisional-radiative model can be used <cit.>. In this model, atomic processes such as spontaneous radiation, electron impact excitation and de-excitation, recombination, ionization and charged exchange are taken to form a set of differential rate equations expressing the population and de-population of atomic levels. In the case of ^56Fe^8+, the energy of metastable electronic state ^3F_4 is 52.79 which is much lower than the ionization threshold and thus by tuning the electron impact energy below this threshold the ionization process can be excluded. The recombination process can reduce the charge state. However, the cross section of the radiative recombination is 3-4 order of magnitude lower than the EIE process. With a pulsed electron beam, it is possible to avoid an incident of radiative recombination. In contrast to radiative recombination, the resonant dielectric recombination has a much large cross section comparable to EIE. Nevertheless, by tuning the electron beam energy off the resonance, dielectric recombination can also be avoided. In a measurement time of a range from hundreds of seconds to weeks, the charged exchange can hardly happen due to the good vacuum under the cryogenic environment. In the end, the differential rate equations can be expressed as: dN_i/dt =∑_j>iA_j→ i^rN_j+∑_j<iC_j→ i^eN_jn_e+∑_j>iC_j→ i^dN_jn_e -∑_j<iA_i→ j^rN_i-∑_j>iC_i→ j^eN_in_e-∑_j<iC_i→ j^dN_in_e, where, the subscript i,j represent the initial stat and final state and N_i denotes the population of state i. A_j→ i^r, C_j→ i^e and C_j→ i^d represent the radiative decay rate, the cross sections of electron impact excitation and electron impact de-excitation, respectively. n_e denotes the electron density. According to the equilibrium condition dN_i/dt=0 and normalized condition ∑ N_i =1 the population of each state can be calculated. unsrtnat
http://arxiv.org/abs/2307.01505v1	20230704063154	Kinetic inductance and voltage response dependence on temperature: Asymmetric dc SQUID case study	[ "M. A. Gali Labarias", "O. A. Nieves", "S. T. Keenan", "E. E. Mitchell" ]	cond-mat.supr-con	[ "cond-mat.supr-con", "physics.app-ph" ]	Kinetic inductance and voltage response dependence on temperature: Asymmetric dc SQUID case study 1st M. A. Galí Labarias CSIRO Manufacturing Lindfield, NSW, Australia. [email protected] 2nd O. A. Nieves CSIRO Manufacturing Lindfield, NSW, Australia. [email protected] 3rd S. T. Keenan CSIRO Manufacturing Lindfield, NSW, Australia. [email protected] 4th E. E. Mitchell CSIRO Manufacturing Lindfield, NSW, Australia. [email protected] August 1, 2023 ====================================================================================================================================================================================================================================================================================================================================================================================================== Inductance plays a crucial role in the design and optimization of superconducting quantum interference devices (SQUIDs) for quantum sensing applications, since it dictates the sensitivity and coupling ratio with other circuit elements. In high-temperature superconductors the kinetic inductance, which depends on both geometry and temperature, becomes a dominant part of the device's total self-inductance, since their London penetration depth is considerably larger compared to low-temperature superconductors. In this work, we use an asymmetric SQUID to investigate the kinetic self-inductance ratio and voltage modulation depth at different operating temperatures, device geometries and bias currents. We first validate our approach by comparing our modelled data with experimental measurements. Then, through numerical simulations, we show: (i) kinetic inductance dominates for thin superconducting films, while for thicker films the inductance is less sensitive to temperature changes; (ii) the voltage modulation depth decreases exponentially with the total inductance independent of the asymmetry ratio; (iii) narrower superconducting tracks lead to a broader temperature operation range, Δ T ∼ 30 K, while wider tracks operate in a smaller temperature range, Δ T ∼ 10 K, but are more sensitive to temperature changes; and (iv) the device performance versus temperature strongly depends on the bias current used. § INTRODUCTION Superconducting Quantum Interference Devices (SQUID) have been studied for many years for their high magnetic field sensitivity <cit.> and applications <cit.>. The voltage-to-magnetic field response is greatly dependent on the inductance of the superconducting elements that constitute the SQUID, but also on the operating temperature. Therefore, a better understanding of these parameters is crucial for designing an optimal SQUID. For instance, at low-temperatures SQUID technologies have been used in thermometry <cit.> for mK measurements, and also Johnson noise based thermometers <cit.>. Thermometry applications have also been used with high-temperature superconductors (HTS) as devices with a broader applicability range <cit.>. Other applications of HTS include mineral exploration <cit.>, medical imaging <cit.> or gradiometry <cit.>. Proper understanding and characterisation of the inductance and temperature dependence is fundamental for all these applications, since the SQUID inductance determines the coupling with amplifier or pick-up loop, but also affects the device flux noise, thus its performance. This is even more important for HTS, since due to larger London penetration depths, the kinetic inductance can be the dominant term of the total SQUID self-inductance. Experimental measurement of the inductance and transfer function of YBCO SQUIDs have been previously done for certain SQUID geometries and Josephson junction (JJ) technologies, e.g. step-edge JJs <cit.> or nano-slit JJs <cit.>. Investigation into the kinetic inductance contribution and SQUID optimization of YBCO have been reported <cit.>. Also, a method to estimate the temperature-dependent London penetration depth <cit.>. In this work, we investigate the kinetic inductance contribution to the total inductance and its dependence on film thickness and operating temperature. Then, for different geometries and bias currents, we analyse the effect of changing the operating temperature on the device performance. To do so, we introduce a mathematical model for an asymmetric SQUID and validate it by comparing our simulations with experimental data. § MATHEMATICAL FRAMEWORK In this Section we present a mathematical model for computing the voltage response of a geometrical asymmetric SQUID (see Fig. <ref>(a)). This model assumes homogeneous current densities across superconducting tracks, which is valid when the widths of the superconductive tracks w are smaller than twice the Pearl penetration depth <cit.> Λ = λ_L^2 / d, where d is the film thickness and λ_L the London penetration depth <cit.>. We also assume short Josephson junctions where the RSJ model can be used. §.§ Asymmetric SQUID modelling In this Section we introduce the main equations that model an asymmetric dc-SQUID as shown in Fig. <ref>. Similar to the model introduced in <cit.>, we define the total current flowing through each junction as I_k = I_csin ( φ_k) + I_n_k + Φ_0/2π R_nd φ_k/dt , where I_k is the total current going through the k^th junction, I_c is the critical current and φ_k the gauge-invariant phase difference across the k^th junction. I_n_k is the current created due to thermal noise, R_n is the junction normal resistance and Φ_0 is the flux quantum. In previous works <cit.>, the effects of non-identical Josephson junctions, i.e. R_n,1≠ R_n,2 and I_c,1≠ I_c,2, have been studied. In this work, we assume identical junctions, i.e. R_n,1 = R_n,2=R_n and I_c,1 = I_c,2=I_c, and focus on the effect of inductance and temperature on the device performance. Using Kirchhoff's law, we can express the currents I_1 and I_2 in terms of the input currents I and I_b, and the current at the upper part of the loop, I_up. I_1 = I - I_up , I_2 = I_b - I + I_up . Using the second Ginzburg-Landau equation, we can express the gauge-invariant phase differences as Φ_0 /2π(φ_2 - φ_1 ) = Φ_a - L_s I_up + L_1 (2I - I_b ), where L_s=L_up +2 L_b is the total SQUID self-inductance, and L_1 = L_0 + L_b, where L_b is the inductance of the bottom part of the asymmetric SQUID. Both L_s and L_b contain kinetic and geometric inductance contributions, and L_0 is the inductance of the horizontal bias leads (see Fig. <ref>(b)). For thin superconducting tracks where the supercurrent density can be assumed homogeneous across the track, the kinetic inductance is defined as L_k(x)=μ_0 λ_L^2/d wx where μ_0 is the material permeability, λ_L the London penetration depth, d and w the film thickness and track width, and x is the length element <cit.>. For the SQUID under study the kinetic part of L_s and L_b are L_s^k =μ_0 λ_L^2/d( 2l_1 + a/w + 2l_2 + a/w_J) , L_b^k = μ_0 λ_L^2/d w_J( l_2 + a/2) . Here, we assume that the two horizontal bias leads are identical, so their inductances L_0 are the same. Using Eqs. (<ref>)-(<ref>), we obtain a set of two coupled differential equations that describe the dynamics of the SQUID φ̇_1 = -sin(φ_1) - i_n_1 + i + f(φ_1, φ_2) , φ̇_2 = -sin(φ_2) - i_n_2 + 2i_b - i - f(φ_1, φ_2) , where i=I/I_c and i_b=I_b/(2I_c) are the normalised input currents, f(φ_1, φ_2)= ( φ_2 -φ_1 - 2πϕ_nf)/(πβ_L) with ϕ_nf = ϕ_a + L_1 (2I - I_b )/Φ_0 is a time-independent term, β_L=2 I_c L_s/Φ_0 is the screening parameter and φ̇_̇k̇=d φ_k / dτ indicates the time-derivative where τ is the normalised time defined as τ=2π R_n I_c t / Φ_0. The normalized thermal noise current i_n_k=I_n_k/I_c is generated at each time-step using a random number generator which follows a Gaussian distribution with zero mean ⟨ i_n_k⟩=0, and mean-square-deviation ⟨ i_n,k^2⟩=σ(i_n_k)^2=2Γ/Δτ, where Γ=2π k_B T/(I_cΦ_0) is the thermal noise strength with k_B the Boltzmann constant and T the operating temperature, and Δτ is the time-step used in the numerical simulations (see <cit.> for details). §.§ Penetration Depth and Critical Current Temperature Dependence In this paper we use the well-known empirical formula <cit.> to determine the values of the London penetration depth at different temperatures, λ_L(T) = λ_L(0)/√(1 - ( T/T_c )^2) , where T_c is the critical temperature of the superconducting material. We use measured T_c and experimental data of YBCO films at T=77K to estimate the penetration depth at zero temperature, namely λ_L (0). We then use this value and Eq. (<ref>) to calculate λ_L(T) at different operating temperatures. Since the critical current I_c and junction resistance R_n also depend on the operating temperature, we use the following empirical formulas which are fitted to experimental data in <cit.> as well as our measured data for I_c and R_n at T=77K: J_c(T) = 2.079× 10^5(0.9 - T/T_c) T<75 4.145× 10^5(1 - T/T_c)^2 T≥ 75 , where J_c has units of A/cm^2, and for I_c(T) we have I_c(T) = J_c(T)· w_J · d , where w_J is the junction width and d the film thickness. The normal resistance R_n is computed from the phenomenological relation I_c R_n ∝ J_c^p with p=0.5 <cit.>. § COMPARISON WITH EXPERIMENTAL MEASUREMENTS In this section we compare our model with experimental measurements. First, we verify that our method to calculate inductances for thin structures agrees with experimental measurements. Next, we test our asymmetric dc-SQUID model by comparing simulated and experimental data of the voltage response depending on the current I and also the I-V characteristics of the device. §.§ Inductance To measure the inductance of these asymmetric SQUIDs (Fig. <ref>(b)), the current I is increased by a ramp current Δ I until a multiple of a flux quantum n Φ_0 is measured, with n an integer. This same method was used previously <cit.>. Then, the measured inductance is obtained from the following expression: L_exp = n ·Φ_0/Δ I . Following the same method and adding a ramp signal Δ I to Eq. (<ref>), we obtain ( L_s - 2L_1 ) Δ I = n Φ_0 . Therefore, the experimentally measured inductance (L_exp) corresponds to L_exp = L_up - 2 L_0 , where L_up is the inductance of the upper part of the asymmetric SQUID (Fig. <ref>(b)). Figure <ref> shows the comparison between the calculated and measured inductances for different asymmetric SQUID designs and upper loop lengths l_1 (the experimental measurements of these devices have been previously reported in Fig. 2 of <cit.>). The devices S3-A, B and C are from the same batch with d=113nm, but with different geometry designs, while S1-A has the same design as S3-A but is from a different batch with d=220nm. The geometric parts of these inductances have been calculated using the expressions for rectangular bar conductors introduced elsewhere <cit.>. These results demonstrates that our model and method to calculate L_exp is appropriate for this kind of device. §.§ Voltage response and I-V characteristics The asymmetric dc-SQUID that was used for this experiment is shown in the inset of Fig. <ref>(b), and it has the following dimensions: a=4m, l_1=21.2m, l_2=10m, upper-loop track width of w=4m, the junction width is w_J=2m and the film thickness is d=113nm. From the inset in Fig. <ref>(b) we can see that the geometry of the bottom part of the SQUID is different from our model diagram (Fig. <ref>). Here the bottom bias lead is relatively wide which produces flux focusing due to the Meissner currents, as seen previously in wide YBCO structures <cit.>. In order to account for this, we need to introduce an effective area which creates the equivalent total flux, i.e. we have to estimate an effective l_2. For our simulations we need to determine the I_c, R_n and l_2,eff which best fits with the experimental voltage response and I-V characteristics. For these simulations l_2,eff=20m, I_c=29.5A, R_n=10.2 Ω and T=77K. Figure <ref>(a) shows the time-averaged voltage response V versus the ramp-current I of experimental (blue line) and simulated (red dots) data. Figure <ref>(b) depicts the I-V characteristics of experimental (blue) and simulated data (circles) at two different applied magnetic fields: ϕ_a=0.4 (red) and ϕ_a=0.85 (black). Our results show a remarkable agreement between our simulation and the experimental measurements. The experimental data shown in Fig. <ref> has been obtained using Magicon SEL-1 electronics. § MODELLING: RESULTS AND DISCUSSION In this section we discuss the kinetic self-inductance depending on the film thickness and operating temperature. Then, we investigate the voltage response and voltage modulation depth for different L_s, SQUID asymmetric ratios l_1/l_2, operating temperatures and bias currents. The simulation parameters used in this paper are Δτ =0.1, N_t=4× 10^5 (total time-steps used) and the device geometry is defined by w_J=w, a=4m and l_2=10m and the film critical temperature used is T_c=86.2K, which is a common T_c for YBCO thin films. At different temperatures I_c, R_n and λ_L are determined using Eq. (<ref>). For example, at T=77K, λ_L= 392.4nm, I_c=20A and R_n=10 Ω. As shown in Fig. <ref>(a), modulating I creates different induced fluxes, thus without loss of generality we can fix I to a certain value, here we use I=I_b, and then only vary the applied magnetic field Φ_a. Unless specified otherwise, w=2m, d= 200nm, T=77K and i_b=I_b/(2I_c)=0.75. §.§ Kinetic self-inductance dependence on geometry and temperature In Figs. <ref> and <ref>, we plot the normalized kinetic self-inductance κ = L^k_s/L_s where L^k_s is the kinetic self-inductance, Eq. (<ref>), and L_s is the total SQUID self-inductance <cit.>. Figures <ref> and <ref> show that κ is larger for designs with thinner tracks w and it decreases increasing the film thickness d. Thus, κ decreases for devices with wide tracks and thick films. This means that designs with a smaller kinetic inductance contribution are more robust to temperature fluctuations, since it is the only component of the inductance which is temperature dependent. On the other hand, to design devices that are temperature sensitive, large κ values are needed and thus thinner films are more suitable. §.§ Voltage dependence on the SQUID self-inductance Similarly to the method used in <cit.>, Eq. (<ref>) can be solved numerically to obtain the time-averaged voltage response of the SQUID V. Figure <ref>(a) shows V versus ϕ_a of asymmetric SQUIDs operating at T=77K and biased at i_b=0.75 for three different upper-loop lengths and therefore different asymmetry ratios: l_1=20 m (blue line, l_1/l_2=2), l_1=40 m (green dashed line, l_1/l_2=4) and l_1=80m (red dotted line, l_1/l_2=8). These results show that large inductances decrease the device performance, but also that larger l_1 produce more fluxoid which shifts the voltage response (clearly seen in the red dotted line in Fig. <ref>(a)). Figure <ref>(b) shows the dependence of voltage modulation depth on L_s and the colored points shows the devices studied in Fig. <ref>(a). This figure shows more clearly the exponential decay of the device sensitivity with increasing L_s, independent of the asymmetry ratio l_1/l_2. §.§ Voltage modulation depth dependence on the operating temperature Using the same approach for generating Fig. <ref>, we can study the effect of changing the temperature and bias current on the device's performance. We see from Fig. <ref> that for a fixed i_b=0.75, loop length l_1 and film thickness d, the voltage modulation depth dependence on T, ΔV(T), has a clear maximum which shifts with the upper track width w. Also, the width of the ΔV(T) curve increases with decreasing w. At i_b=0.75, the max (ΔV) values appear when β_L / Γ≈ 1. This suggests that one can optimise the geometry of the device depending on which temperature we want to operate it. Another approach is to use a SQUID for thermometry. The results in Fig. <ref> show that fixing the inputs i_b and ϕ_a we can indirectly estimate the temperature based on ΔV. For instance, the device's temperature sensitivity can be finely-tuned for two different applications: (i) using narrower tracks ΔV presents a broader response in T with a measurable voltage modulation depth in the range T ∈ [40, T_c). (ii) On the other hand, wider tracks present narrower but steeper ΔV(T), which is suitable when higher device temperature sensitivity is required. Figure <ref> shows ΔV(T) for different i_b values. As reported previously <cit.>, i_b=0.75 shows an optimum response when operating close to T=77K. Below and above i_b=0.75, the response becomes narrower in T and the magnitude of ΔV also decreases substantially, indicating that the choice of i_b plays a crucial role in the device's performance. Figure <ref> displays the dependence of ΔV on both i_b and T and shows that the optimal bias current at lower temperatures is i_b^≲ 1, while at operating temperatures close to T_c ΔV optimizes at smaller i_b. Thus, the optimal bias current depends also on the operating temperature i_b^(T). For instance, at T=77K then i_b^≈ 0.75 as shown by the white dashed lines in Fig. <ref>, which agrees with previously reported values <cit.>. § CONCLUSION In this work we have used a mathematical model of an asymmetric SQUID to study the effect of the inductance on the device performance. By comparing our calculated inductances with experimental measurements, we have shown that the method used is adequate for these kind of structures, showing great accuracy. From our modelling, as shown in Fig. <ref>, we found that thinner films produce a stronger dependence on temperature of the kinetic self-inductance ratio κ, while thicker films have smaller κ and therefore are more robust to temperature changes. We also found that the device response V and ΔV with respect to the applied magnetic field is controlled by the total self-inductance, but is unaffected by the device's asymmetry aside from an off-set in the voltage response. Finally, we analysed the device temperature dependence for different track widths and bias currents. Our results show that biasing the device at the optimal point is essential for maximising the voltage response, achieving higher sensitivities and broader temperature operation ranges. We see that the voltage modulation depth shows a bell-like curve, in which the maximum coincides with β_L/Γ≈ 1, when biased at i_b=0.75 which is close to the optimal bias current at T=77K. These curves indicate that the sensitivity of devices with narrow tracks show a broader temperature response, while wider tracks give a narrower temperature range but higher temperature-sensitivity. For instance, Fig. <ref> showed that devices with w=2m present voltage modulation depth variations of tens of micro-volts when temperature changes by only a few Kelvin. On the other hand, devices with w=0.5m show a slower variation of ΔV for a wider temperature range. Our investigation of ΔV depending on T and i_b has shown that at lower temperatures the device voltage modulation depth maximises for i_b^ ≈ 1, also showing a very large and quick decrease of ΔV for bias current away from the optimal. This sharp decrease of ΔV with i_b emphasises the importance of correct biasing, which becomes more important as temperature decreases since junction activation due to thermal noise becomes less probable. § ACKNOWLEDGMENTS The authors would like to thank K.-H. Müller and K. E. Leslie for insightful discussions.
http://arxiv.org/abs/2307.02296v1	20230705135229	Bayesian evidence for spectral lag transition due to Lorentz Invariance Violation for 32 Fermi/GBM Gamma-ray Bursts	[ "Vibhavasu Pasumarti", "Shantanu Desai" ]	astro-ph.HE	[ "astro-ph.HE", "astro-ph.CO" ]	P[1]>p#1 =1 E-mail:[email protected] E-mail: [email protected] We use the spectral lag data of 32 long GRBs detected by Fermi/GBM, which has been recently collated in <cit.> to carry out a search for Lorentz Invariance violation (LIV) using Bayesian model selection. We use two different parametric functions to model the null hypothesis of only intrinsic emission: a smooth broken power law model (SBPL) (proposed in <cit.>) as well as a simple power law model, which has been widely used before in literature. We find that using the SBPL model as the null hypothesis, only three GRBs show decisive evidence for linear LIV, of which only one shows decisive evidence for quadratic LIV. When we use the simple power-law model as the null hypothesis, we find 15 and 16 GRBs showing decisive evidence for linear and quadratic LIV, respectively. Finally, when we apply the SBPL model to model the intrinsic emission in GRB 1606025B, the evidence for LIV (which was previously reported using the simple power law model) disappears. This underscores the importance of adequately modelling the intrinsic emission while searching for evidence of LIV using spectral lags. Dept of Physics, IIT Hyderabad, Kandi, Telangana-502284, India Bayesian evidence for spectral lag transition due to Lorentz Invariance Violation for 32 Fermi/GBM Gamma-ray Bursts Shantanu Desai August 1, 2023 =================================================================================================================== § INTRODUCTION In various theoretical scenarios beyond the Standard Model of Particle Physics, Lorentz Invariance is not an exact symmetry at energies close to the Planck scale (E_pl∼ 10^19 GeV), and the speed of light v(E) varies as a function of energy according to <cit.>: v(E) = c[1 - s_±n+1/2(E/E_QG)^n], where s_± = ± 1 corresponds to either sub-luminal (s_±=+1) or super-luminal (s_±=-1) Lorentz Invariance Violation (LIV); E_QG denotes the energy scale where LIV effects dominate, and n represents the order of the modification of the photon group velocity. In all LIV searches in literature, the series expansion is usually restricted to linear (n=1) or quadratic corrections (n=2). Both linear and quadratic LIV models are predicted by different theoretical approaches <cit.>. For more than two decades Gamma-Ray Bursts (GRBs) have been a very powerful probe of LIV searches. GRBs are single-shot explosions located at cosmological distances, which were first detected in 1960s and have been observed over ten decades in energies from keV to over 10 TeV range <cit.>. They are located at cosmological distances, although a distinct time-dilation signature in the light curves is yet to be demonstrated <cit.>. GRBs are traditionally divided into two categories based on their durations, with long (short) GRBs lasting more (less) than two seconds <cit.>. Long GRBs are usually associated with core-collapse SN <cit.> and short GRBs with neutron star mergers <cit.>. There are however many exceptions to this conventonal dichotomy, and many claims for additional GRB sub-classes have also been made <cit.> (and references therein). The observable used in almost all the LIV searches with GRBs consists of spectral lags, defined as the arrival time difference between high energy and low energy photons, and is positive if the high energy photons precede the low energy ones. Searches for LIV with spectral lags have been done using single lags from different GRBs (for example <cit.>), multiple spectral lags from the same GRB (GRB 1606025B, GRB 1901114 C, GRB 1905030A) <cit.>, as well as stacking multiple spectral lags from multiple GRBs <cit.>. A comprehensive uptodate review of all searches for LIV using GRB spectral lags can be found in <cit.>. Most recently, Liu et al <cit.> (L22, hereafter) did a comprehensive study of LIV (assuming sub-luminal propagation with s_±=+1) using spectral lags of 32 long GRBs detected by Fermi/GBM. Most of the GRBs studied in L22 contained had a turn-over in the spectral lag data. The intrinsic model which they used consisted of a smooth broken power law as a function of energy. L22 then obtained limits on LIV for both a linear and quadratic model of LIV for each of the 32 GRBs. The characteristic limits they obtained were E_QG≳ 1.5 × 10^14 GeV and E_QG≳ 8 × 10^5 GeV for linear and quadratic LIV, respectively. In this work, we supplement the analysis in L22, by calculating the significance for both the models of LIV compared to only the intrinsic astrophysical emission using Bayesian model comparison, similar to our past works <cit.>. We also test the efficacy of the more prosaic power law model which has been extensively used previously in mainly LIV searches starting from Ref. <cit.> and compare the two sets of results. The outline of this manuscript is as follows. We discuss the GRB dataset used for this work and the analysis procedure in Sect. <ref>. A brief primer on Bayesian model comparison is given in Sect. <ref>. We recap our analysis procedure in Sect. <ref>. Our results are outlined in Sect. <ref> and we conclude in Sect. <ref>. § DATASET We briefly describe the data analysis procedure in L22, where more details can be found. The sample chosen in L22 consists of 32 long GRBs in the redshift range z ∈ [0.54 , 4.35] chosen from the Fermi/GBM catalogue <cit.> We use the spectral lag data of 32 GRB collated in L22, which has been kindly provided to us. These data consists of the observed energy (in keV) and the corresponding spectral lag observed along with the uncertainty in the lag (s). The extraction of light curves followed by the spectral lag calculation were done using the methods described in <cit.>. The spectral lags were calculated using distinct energy bands as well as bin sizes. The data for all the 32 GRBs including bin size, number of energy bands, energy range used for spectral lag calculation, redshift, and time interval can be found in Table 1 of L22. Each GRB contains about 15-20 spectral lag data in the 10-1000 keV energy range. We note that spectral lags for two of the GRBs in this catalog, namely GRB 1606025B and GRB 190114C have been reported before and used to search for LIV <cit.>. However, in the previous works the spectral lag data for GRB 1606025B and GRB 190114C were binned in energy, whereas the L22 spectral lag data have been provided at specific energies. § ANALYSIS In this analysis, we follow the same analysis as L22 to predict the spectral lag of GRBs. The observed spectral lag is given by: Δ t_obs = Δ t_int + Δ t_LIV, where Δ t_int is the intrinsic lag from the GRB radiation which is purely astrophysical and Δ t_LIV is the contribution from LIV. Since the exact physical process for the intrinsic lag is not yet found and could vary according based on the GRB, we use two different models for the intrinsic emission. The first model which we use is the Smoothly broken power-law (SBPL) to model, proposed in L22 and can be written as: Δ t_int = ζ(E - E_0E_b)^α_1(12[1 + (E - E_0Eb)^1/μ])^(α_2 - α_1)μ , where ζ is the normalization parameter, E_b is the transition energy, α_1 and α_2 are the slopes before and after E_b, and μ is the transition smoothness. This SBPL transforms into a single power law for α_1 = α_2. The SBPL enables us to account for the negative lags observed in the data <cit.>. We also use the power-law model first proposed in <cit.>, which has been used in a number of works on LIV <cit.> and was motivated based on the analysis of single-pulse properties of about 50 GRBs <cit.>: Δ t_int = (1+z) τ[ (E/keV)^α-(E_0/keV)^α], The model for the spectral lag originating from LIV is sames as the one used in <cit.> and is given by: Δ t_LIV=-1+n/2 H_0E^n-E_0^n/E_QG, n^n∫_0^z(1+z^')^n d z^'/√(Ω_m(1+z^')^3+Ω_Λ), Note the above equation assumes that the expansion history of the universe is described by the ΛCDM model. Other expansion histories as well as non-parametric methods to model the expansion histories have also been considered <cit.> (and references therein). However, it has been found that the final results do not change much compared to using ΛCDM expansion history <cit.>. Therefore, for this work we use Eq. <ref> to calculate the lag due to LIV. The cosmological parameters which we use are H_0 = 67.36 km/sec/Mpc, Ω_m=0.315, Ω_Λ=0.695, which are the same as that used in L22 and based on Planck 2020 cosmological parameters <cit.>. § BAYESIAN MODEL COMPARISON We evaluate the significance of any LIV using Bayesian Model Comparison. We provide a very brief prelude to Bayesian model comparison, and more details can be found in recent reviews <cit.>. To evaluate the significance of a model (M_2) as compared to another model (M_1), one usually calculates the Bayes factor (B_21) given by: B_21= ∫ P(D\|M_2, θ_2)P(θ_2\|M_2) dθ_2/∫ P(D\|M_1, θ_1)P(θ_1\|M_1) dθ_1 , where P(D\|M_2,θ_2) is the likelihood for the model M_2 given the data D and P(θ_2\|M_2) denotes the prior on the parameter vector θ_2 of the model M_2. The denominator in Eq. <ref> denotes the same for model M_1. If B_21 is greater than one, then M_2 is preferred over M_1 and vice-versa. The significance can be qualitatively assessed using the Jeffreys' scale <cit.>. In the present paper, the model M_1 corresponds to the hypothesis, where the spectral lags are produced only by intrinsic astrophysical emission (Eq. <ref> or Eq. <ref>), whereas M_2 corresponds to the lags being described by Eq <ref>, consisting of both intrinsic and LIV delays. To calculate the Bayes factor, we need a model for the likelihood (ℒ), which we define as: ℒ=∏_i=1^N 1/σ_t√(2π)exp{-[Δ t _i-f(Δ E_i,θ)]^2/2σ_t^2}, where N is the total number of spectral lags per GRB; Δ t_i denotes the observed spectral lag data, and σ_t denotes the uncertainty in the observed spectral lags. In this expression, f corresponds to the particular model being tested, which could either be the two LIV models or the null hypothesis of only astrophysical emission; σ_t is the uncertainty on the spectral lag. Finally, to evaluate Eq. <ref>, we need the priors for the three models. We have used uniform priors for all the intrinsic parameters, and log-uniform priors on E_QG. The prior ranges for all these parameters for both the LIV and the two intrinsic models considered can be found in Table <ref>. § RESULTS We now calculate the Bayes factors assuming that the spectral lags can described using a superposition of intrinsic emission along with a model of LIV compared to only intrinsic emission. To evaluate the Bayesian evidence we use the Dynesty Nested Sampler <cit.>. Along with the Bayes factor, we also check the efficacy of our fits based on the reduced χ^2, where the reduced χ^2 is the χ^2 divided by the total degrees of freedom. These values of the Bayes factor and reduced χ^2 for all the 32 GRBs considered can be found in Table <ref> for the SPBL model as the null hypothesis. For linear LIV, we find 3 GRBs with Bayes factor > 100. These are GRB 190114C, GRB 130925A, and GRB 131231A. For all other GRBs, the Bayes factors are close to 1 for linear LIV. For quadratic LIV, only one of these GRBs (viz. GRB 131213A) has a Bayes factor >100. For GRB 190114C and GRB 130925A, the Bayes factors reduce to 40 and 66, respectively which roughly correspond to very strong evidence for LIV. We also find that for most GRBs, the Bayes factor for quadratic model of LIV is smaller than compared to the linear model. The corresponding results for the more simple power law model in Eq. <ref> can be found in Table <ref>. In contrast to the SBPL intrinsic model, now we find 15 GRBs with Bayes factors > 100 for linear LIV model. On the other hand for the quadratic model of LIV, we find 16 GRBs with Bayes factor > 100. Also for most GRBs, the Bayes factor is greater for quadratic LIV as compared to linear LIV. We also find that the reduced χ^2 for the null hypothesis is larger while using Eq. <ref>, as compared to the SBPL parametrization. In order to validate this with Bayesian model comparison, we compare the Bayesian evidence for both these intrinsic models. We again use the same priors for both these models as before. The Bayes factors for the SBPL model compared to Eq. <ref> can be found in Table <ref> for all the 32 GRBs. We find that 22 GRBs show decisive evidence in favor of the SBPL model and two others show very strong evidence. The more simple power law mode (Eq. <ref>) is favored compared to the SBPL for only about six GRBs. This agrees with the conclusions in L22, who pointed out that the SBPL model is more accurate than the simple power model used before in literature. Finally, it is instructive to compare the significance of LIV for GRB 190114C and GRB 1606025B with previous works on the subject, which have used Eq. <ref> for the null hypothesis. For this GRB, the natural log of Bayes factor was approximately 175 for linear and quadratic LIV <cit.>. We find the same to be about 148 and 101 for linear and quadratic LIV respectively using the same intrinsic model. However, when we use the SBPL as the null hypothesis, the corresponding natural log of the Bayes factors reduce to 9.9 and 3.7, corresponding to Bayes factors of around 20,000 and 40, respectively. Therefore, we find that the quadratic LIV model no longer shows decisive evidence compared to the null hypothesis for the SBPL null hypothesis. For GRB 1606025B, frequentist, information theory, and Bayesian model comparison techniques have been used to evaluate the statistical significance of LIV <cit.>. <cit.> reported the natural log of Bayes factor of 16 and 20, for linear and quadratic LIV using the intrinsic model in Eq. <ref>. The corresponding values for the natural lag of the Bayes factors, which we get for the same null hypothesis are comparable with values of 12 and 10 for linear and quadratic LIV, respectively. These Bayes factors correspond to decisive evidence for LIV. However, when we consider the SBPL model as the null hypothesis, evidence of LIV completely disappears and the null hypothesis is now favored compared to both the LIV models. This underscores the importance of correctly modelling the intrinsic emission, while drawing conclusions about LIV from spectral lag data. This also agrees with the conclusions in <cit.>, who showed that some of the turn-over in spectral lag data which was previously attributed to LIV could be explained using spectral evolution. Note that we have also uploaded the plots for the best-fits for both sets of models along with the spectral lag data for all the 32 GRBs on github, whose link is provided in Sect. <ref> § CONCLUSIONS In a recent work, L22 carried out a comprehensive study of the spectral lags of 32 long GRBs detected by Fermi-GBM, which had a transition from positive to negative lags. They fit the intrinsic lags using an empirical SBPL (Eq. <ref>). L22 used this data to constrain LIV and obtained constraints on E_QG≥ 1.5 × 10^14 GeV and E_QG≥ 8 × 10^5 GeV. In this work, we extended the original analysis in L22 by evaluating the significance based on Bayesian model selection, that the spectral lags can be adequately modelled by a mixture of intrinsic emission and LIV-induced compared to only intrinsic emission. For the intrinsic emission, we consider two models. One of them is the SBPL model considered in L22. The second intrinsic model we consider is the simple power law model (cf. Eq. <ref>) first used in <cit.> and which was used in some of our past works <cit.>. Our results for the Bayes factor and reduced χ^2 can be found for the Table <ref> and Table <ref> for the SBPL model and Eq. <ref>, respectively. To evaluate the relative efficacy of the two models of intrinsic emission, we also calculate the Bayes factor for the SBPL model compared to that in Eq. <ref>. These Bayes factors can be found for all the GRBs in Tab. <ref>. Our conclusions are as follows: * We find 3 GRBs (GRB 190114C, GRB 130925A, and GRB 131231A) with Bayes factor > 100 (corresponding to decisive evidence) for a model consisting of a superposition of SBPL + linear LIV compared to only the SBPL model. For quadratic LIV, only one of these GRBs, namely GRB 131213A has a Bayes factor >100. * When we replace the SBPL model with Eq. <ref> as the null hypothesis, we find 15 and 16 GRBs with decisive evidence for linear and quadratic LIV, respectively. * When the SBPL model is used as the null hypothesis, the Bayes factor for the quadratic LIV model is mostly smaller than the linear LIV model, whereas the opposite is true while using Eq. <ref> for the null hypothesis. * For most GRBs, Bayesian model comparison decisively favors the SBPL model compared to Eq. <ref>. This is in accord with the conclusions in L22. * Previous works <cit.> have also studied and reported decisive evidence for both linear and quadratic LIV for GRB 190114C and GRB 1606025B, while using Eq. <ref> as the null hypothesis. However, when we use SBPL as the null hypothesis to model the intrinsic emission, we find the evidence for LIV for GRB 1606025B completely vanishes and the null hypothesis is preferred. For GRB 190114C, we still get decisive evidence for linear LIV. However, for quadratic LIV, the Bayes factor reduces to about 40, corresponding to “very strong” evidence according to Jeffrey's scale. This underscores the importance of the intrinsic emission model while making any claims for evidence of LIV. Our analysis codes along with supplementary plots showing comparison of the different models on top of data have been uploaded on github and can be found in <https://github.com/DarkWake9/Project-QG> § ACKNOWLEDGEMENTS We are grateful to Zik Liu and Binbin Zhang for generously sharing the spectral lag data used in L22 with us. We acknowledge National Supercomputing Mission (NSM) for providing computing resources of ‘PARAM SEVA’ at IIT, Hyderabad, which is implemented by C-DAC and supported by the Ministry of Electronics and Information Technology (MeitY) and Department of Science and Technology (DST), Government of India.
http://arxiv.org/abs/2307.00415v1	20230701194154	Inertial-like waves in rigidly-rotating odd viscous liquids	[ "E. Kirkinis", "M. Olvera de la Cruz" ]	physics.flu-dyn	[ "physics.flu-dyn" ]	#1 #1(<ref>) #̨1κ#1 '
http://arxiv.org/abs/2307.00767v1	20230703061344	The Number of Ribbon Tilings for Strips	[ "Yinsong Chen", "Vladislav Kargin" ]	math.CO	[ "math.CO" ]	DifFSS: Diffusion Model for Few-Shot Semantic Segmentation Bo Yan August 1, 2023 ========================================================== First, we consider order-n ribbon tilings of an M-by-N rectangle R_M,N where M and N are much larger than n. We prove the existence of the growth rate γ_n of the number of tilings and show that γ_n ≤ (n-1) ln 2. Then, we study a rectangle R_M,N with fixed width M=n, called a strip. We derive lower and upper bounds on the growth rate μ_n for strips as ln n - 1 + o(1) ≤μ_n ≤ln n. Besides, we construct a recursive system which enables us to enumerate the order-n ribbon tilings of a strip for all n ≤ 8 and calculate the corresponding generating functions. Keywords Key words: tilings, superadditivity, the leftmost tiling process, growth rate. § INTRODUCTION Let an integer n ≥ 2 be fixed. We say a square [x, x+1] × [y, y+1] in the two-dimensional integer lattice has level l = x+ y and color c ≡ x+y n where 0 ≤ c < n. An order-n ribbon tile is a set of squares connected along an edge and containing exactly one square of each color. In other words, a ribbon tile is a sequence of n adjacent squares, each of which is located above or to the right of its predecessor. An order-n ribbon tiling is a covering of a region with non-overlapping order-n ribbon tiles. (See Figure <ref> for an example of ribbon tiles and a ribbon tiling when n=4.) Two natural questions about ribbon tilings are whether there exists a ribbon tiling of a given region, and if it does, then how many different ribbon tilings there are. The existence question for every simply connected region and arbitrary order n was settled in <cit.>, who provided an algorithm, linear in the area of a region, that checks if the region has a ribbon tiling. The enumeration question for domino tilings, which are ribbon tilings for n=2, goes back to 1960's. The papers <cit.> and <cit.> provided a formula for rectangular regions. <cit.> and <cit.> studied domino tilings of rectangular regions with fixed width from a different perspective. In particular, they studied the generating function for the number of domino tilings. Later, <cit.> considered domino tilings on a torus. A technique to analyze domino tilings of more general regions was invented by <cit.> and <cit.>. A related enumeration problem of tilings with T-tetrominoes was studied by <cit.>. Ribbon tilings for n > 2 were first studied in <cit.>. Pak introduced the term tile counting group and made a conjecture about ribbon tiling, which was proved in <cit.>. These techniques have been extended in <cit.> who discovered a one-to-one correspondence between ribbon tilings and acyclic orientations of a certain partially oriented graph. This discovery led him to the development of the algorithm verifying the existence of ribbon tilings. <cit.> calculated the number of ribbon tilings for n>2 of a special rectangle with dimension n × 2n. In this paper, we extend these results by studying the number of ribbon tilings of rectangular regions for n > 2. Specially, we study the n × N strips with arbitrary large N. For N = 2n, our numerical results are in agreement with the results in <cit.>. §.§ Main results Let R_M,N be an M-by-N rectangle. It is known that R_M,N has an order-n ribbon tiling if and only if n \| M or n \| N. In this paper, we always suppose n \| M. Let f_M,N be the number of ribbon tilings of R_M,N. Define the growth rate of f_M,N to be γ_n = lim_M,N →∞ln (f_M,N )/ A where A = MN / n is the number of tiles of R_M,N, provided the limit in Equation <ref> exists. Our first result is demonstrating the existence of the growth rate γ_n. For any order n ≥ 2, the growth rate γ_n exists and γ_n ≤ (n-1) ln 2. The proof of Theorem <ref> will be given in Section 2. Now, we consider a special rectangle R_N with fixed width n and length N. We call R_N a strip with length N. It is evident that R_N can be tiled by N rectangles with dimension n × 1. Hence, the region R_N always has at least one order-n ribbon tiling. We are interested in the growth rate of the number f_N of ribbon tilings of R_N, that is μ_n = lim_N →∞ln f_N/N. The existence of μ_n can be proved by an argument similar to that in the proof of Theorem <ref>. We give upper and lower bounds on μ_n in Theorem <ref>. The growth rate μ_n satisfies the following inequalities: 1/nln (n!) ≤μ_n ≤ln (n). For growing n, ln (n) - 1 + o(1) ≤μ_n ≤ln (n). The proof of Theorem <ref> and Corollary <ref> will be given in Section 4. For the number of tilings f_N, we define the leftmost tiling process in order to produce a recursive system. In the leftmost tiling process, there are (n-1)! specific types of regions with different left boundaries that re-appear repeatedly in the process. We call these (n-1)! types the fundamental regions. One of these fundamental regions, with a vertical line as the left boundary, has the same shape as the original n-by-N strip. Let 𝐟(N) be a vector of which each component is the number of ribbon tilings of the corresponding fundamental region with size N-i where 0 ≤ i < n. Theorem <ref> shows that the vector 𝐟(N) satisfies a recursive system. Consider order-n ribbon tilings of an n-by-N strip. The number of ribbon tilings of fundamental regions satisfies a recursive system 𝐟(N) = A_n 𝐟(N-1) . The transfer matrix A_n has the following properties: (1) The elements of A_n are either 0 or ± 1. (2) A_n has the form A_n = [ A_n' I_2 I_1 0 ] . where A_n ' has dimension (n-1)! × (n-1)(n-1)!, I_1 is an identity matrix with dimension (n-1)(n-1)! × (n-1)(n-1)! and I_2 is an identity matrix with dimension (n-1)! × (n-1)!. The construction of the recursive system and the proof of Theorem <ref> are in Section 5. There is some interesting structure in the eigenvalues of A_n shown in Figure <ref>. Numeric results for the number of ribbon tilings f_N complement our result in Theorem <ref>. Table <ref> lists the first ten terms of f_N for the cases n = 2, 3,⋯, 8. Let λ_n be the eigenvalue with the largest absolute value of the transfer matrix A_n in Theorem <ref>. Then, μ_n = ln(λ_n). Numeric values of λ_n's are listed in Table <ref>; they are rounded to six digits after decimal. Figure <ref> shows the asymptotic behavior of f_N. Figure <ref> compares the growth rates with their bounds in Theorem <ref>. The rest of the paper is organized as follows. In Section 2, we use the superadditivity argument to show the existence of the growth rate γ_n. In Section 3, we introduce the leftmost tiling process in order to enumerate all ribbon tilings of a strip. In Section 4, we prove Theorem <ref>. In Section 5, we construct the recursive system to calculate the number of tilings of a strip. § SUPERADDITIVITY Each pair of tilings of non-overlapping regions R_1 and R_2 corresponds to a tiling of the region R_1 ∪ R_2. From this fact, it follows that the logarithm of the number of tilings is superadditive. We formalize this argument in Lemma <ref>. Lemma <ref> is a multidimensional version of Fekete's lemma. In all statments, it is assumed that n \| M. Let f_M,N be the number of tilings in an M-by-N rectangle. Then, f_M,N≤ 2^MN(n-1)/n . There are M N / n tiles in an M-by-N rectangle. By using the result of <cit.>, they can be canonically ordered. Let the tiles be labeled as 1, 2, ⋯, M N / n, which is invariant for all tilings. In different tilings, each labeled tile may have different types. By the definition of a ribbon tile, there are 2^n-1 different types for a tile. Hence, a tiling corresponds to a sequence of types, which consist of MN/n elements and this map is injective. It is not necessary for each sequence of types to be a valid tiling. Then, the number of tilings can be upper bounded as f_M,N≤ (2^n-1)^MN/n = 2^MN(n-1)/n . Suppose M_2 = p M_1 and N_2 = q N_1. Then, ln f_M_2, N_2≥ p q ln f_M_1, N_1. A M_2-by-N_2 rectangle can be separated into pq number of M_1-by-N_1 rectangles. The union of the tilings of small rectangles is a valid tiling of a big rectangle. This map is injective. The following equality holds lim_M, N →∞ln ( f_M,N ) /MN = sup_M,N ≥ 1ln ( f_M,N ) /MN < ∞. Let S = sup_M,N ≥ 1ln (f_M,N)/MN. Then S is finite by Lemma <ref>. For any ϵ > 0, the definition of S gives M_0 and N_0 such that ln (f_M_0,N_0)/M_0 N_0 > (S-ϵ). Consider an M-by-N rectangle R and pick integers p, q such that p M_0 ≤ M ≤ (p+1) M_0 and q N_0 ≤ N ≤ (q+1) N_0. Since every ribbon tiling of the left lower p M_0-by-q N_0 sub-rectangle of R can be extended to a ribbon tiling of R, we get f_M,N≥ f_p M_0, q N_0. Then Lemma <ref> gives ln f_M,N/MN≥ln f_p M_0, q N_0/MN≥p q ln f_M_0,N_0/(p+1)M_0 (q+1)N_0≥p/p+1q/q+1 (S - ϵ) . Since ϵ is arbitrary, we obtain lim_M,N →∞ln ( f_M,N ) /MN = S. Now, we are ready to prove Theorem <ref>. Lemma <ref> establishes that the growth rate γ_n exists. Lemma <ref> implies the upper bound on γ_n. § THE LEFTMOST TILING PROCESS First, let us explain how ribbon tilings are related to acyclic orientations on partially oriented graphs. The paper <cit.> introduced a “left of” relation for both tiles and squares, denoted as ≺. Let s_x,y be a square [x, x+1] × [y, y+1]. We say s_x,y≺ s_x',y' if one of the following two conditions holds: (1) x + y = x' + y' and x < x'; (2) \|(x+y)-(x'+y')\| = 1, x ≤ x' and y ≥ y'. Let t be a tile and s be a square. We write s ≺ t if s ≺ s' for some square s' ∈ t, and t ≺ s if s' ≺ s for some square s' ∈ t. If t_1 and t_2 are two tiles in a tiling, we write t_1 ≺ t_2 if there exist a square s_1 ∈ t_1 and a square s_2 ∈ t_2 with s_1 ≺ s_2. It is not possible that both t_1 ≺ t_2 and t_2 ≺ t_1 unless t_1 = t_2. The relation ≺ is not transitive. However, if t_1 ≺ t_2 ≺…≺ t_k, then either t_1 ≺ t_k or t_1 and t_k are incomparable. It cannot happen that t_k ≺ t_1. We say that a tile has level l if l is the lowest level of the squares in this tile. <cit.> showed that every tiling of a region R has the same number of tiles in a given level. Hence tiles can be enumerated independently of a tiling as “a tile number i in level l”. Note that in different tilings a tile with the same label can have different locations and different shapes. Given this enumeration, we identify tiles with vertices in a graph G_R. The set of vertices of G_R consists of all tiles and boundary squares (squares outside R, but having an edge in R), and two vertices are connected by an edge if they are comparable with respect to the “left of” relation. Further, Sheffield showed that there is a one-to-one correspondence between ribbon tilings and acyclic orientations of the graph G_R with a fixed partial orientation. For an n-by-N strip region R, the graph G_R can be described as follows. Note that the “left of” relations between boundary squares and tiles are the same in all tilings and a boundary square can never be between two tiles, so we can restrict G_R to the vertices that represent tiles. Sheffield’s results imply that there is exactly one tile in each level for a strip R. Therefore, we label the vertices of G_R with the level of their corresponding tiles as 0, 1, ⋯, N-1. There is an edge between two vertices i and j if and only if \|i-j\| ≤ n. The edge is oriented from i to j, denoted as i → j, if i ≺ j. Note that the orientation i → j is forced if j-i=n since in all tilings the “left of” relation i ≺ j is fixed for j - i = n. This fixed orientation on the edge (i, j) is a part of the fixed partial orientation on G_R, and these are the only forced orientations on G_R. The other orientations depend on the particular tiling of R. See Figure <ref> for an example of the “left of” relation and the acyclic orientation correspondence. Let 0, 1, ⋯, N-1 be the labels of tiles that will be used to cover an n-by-N strip R. It was shown by <cit.> that building a tiling is equivalent to determining the “left of" relations among these labeled tiles. We will build these relations step by step using the concept of the “leftmost” tile. Let α be a tiling of R and G_R(α) be the acyclic orientation of G_R corresponding to α. Every tile that corresponds to a source of G_R(α) is called a source tile. The leftmost tile is a source tile with the smallest label. Now, we introduce the leftmost tiling process to determine relations among tiles step by step. At each step, we choose an appropriate label and declare that the tile with this label will be the leftmost tile in a tiling of the untiled region. We put this tile in place and continue the procedure. We keep choosing a label from the set of remaining labels and declaring the corresponding tile the leftmost among the remaining tiles until all the relations among all tiles are determined. We say a sequence T of tile labels is a tiling sequence if it is a valid sequence for the leftmost tiling process. It is evident that there is a bijection between the set of ribbon tilings and the set of tiling sequences. We can enumerate ribbon tilings of R with the help of the leftmost tiling process. In Figure <ref>, the tiling corresponds to the tiling sequence T = [2, 0, 1, 3]. Let us define an operation on tiling sequences, which will be useful later. For any sequence T = [ t_1, ⋯, t_k, 0, s_1, ⋯, s_l] containing a unique 0, define the return operator ζ_0 by the formula ζ_0 (T) = [ 0, t_1, ⋯, t_k, s_1, ⋯, s_l]. That is, the operator ζ_0 moves the 0 to the front of the sequence. See Figure <ref> for an example of the return operator. This operator will be used in the proof of both Theorem <ref> and Theorem <ref>. Let T be a tiling sequence of a strip R. Then ζ_0 (T) is a valid tiling sequence of R. Let G_R be the partially ordered graph associated to the rectangle R. Let T = [ t_1, ⋯, t_k, 0, t_k+2, ⋯, t_N ] correspond to a tiling of R with associated orientation α on G_R. If G_R(α) has edges between some of t_1, ⋯, t_k and 0, they are directed towards 0. Otherwise, 0 would be chosen before one of these t_1, ⋯, t_k in the tiling sequence. By reversing directions of these edges, we define another orientation β of G_R. This reversal does not affect the forced orientations on G_R, because no forced edge is directed towards 0. Moreover, G_R(β) is acyclic since 0 is a source in G_R(β), so there are no cycles that go through 0, and cycles that do not include 0 do not exist, too, because G_R(α) is acyclic. So β represents a ribbon tiling of R. To see that ζ_0(T) is the tiling sequence of that tiling, note first that 0 is a source in G_R(β). Moreover, since t_s, 1 ≤ s ≤ k, was a source in G_R(α) ∖{ t_1, ⋯, t_s-1}, it is a source in G_R(β) ∖{ 0, t_1, ⋯, t_s-1}. So the new sequence can start with [0; t_1, ⋯, t_k]. The final part [ t_k+2, ⋯, t_N ] of the tiling sequence of G_R(α) agrees with that of G_R(β), because G_R(α) ∖{ t_1, ⋯, t_k, 0 } = G_R(β) ∖{ t_1, ⋯, t_k, 0 }. § UPPER AND LOWER BOUNDS ON THE NUMBER OF TILINGS In this section, we prove Theorem <ref> and Corollary <ref>. For a fixed integer n ≥ 2, let f_N be the number of order-n ribbon tilings of a strip with length N. First, the function ln(f_N) is superadditive, that is, ln f_N ≥ln f_N_1 + ln f_N_2 where N = N_1 + N_2. This is because every pair of tilings of two non-overlapping regions R_1 and R_2 forms a tiling of R_1 ∪ R_2. By Fekete's Lemma, we have the existence of the following limit lim_N →∞ln f_N/N = μ_n where μ_n = sup_N {ln f_N/N} could be a constant or infinity. We call μ_n the growth rate of the number of ribbon tilings of a strip. In order to prove the upper bound on μ_n stated in Theorem <ref>, we first prove the following lemma. Let f_N be the number of ribbon tilings of a strip R with length N. Then, f_N ≤ n f_N-1 . Let 𝕋 be the set of all tiling sequences of R and 𝕋_0 be the set of all tiling sequences of R starting with 0. Note that f_N = \| 𝕋 \| and f_N-1 = \| 𝕋_0 \| by the one-to-one correspondence between tilings and tiling sequences. We need to show that \| 𝕋 \| ≤ n \| 𝕋_0 \|. By Lemma <ref>, for every T ∈𝕋, ζ_0(T) is a valid tiling sequence of R. It follows that the image set ζ_0(𝕋) is a subset of 𝕋_0. Define C_i = { T ∈𝕋: ζ_0(T) = T_i } where { T_i } is an enumeration of ζ_0(𝕋). It is clear that { C_i } are mutually exclusive and ∪ C_i = 𝕋. We claim that the size of each C_i is less than or equal to n. Indeed, in order to recover a tiling sequence T ∈ζ_0^-1 (T_i) = C_i from T_i ∈ζ_0(𝕋_0), it is enough to consider the possible positions at which the tile 0 can be embedded into T_i. By the definition of the leftmost tiling process, either tile 0 is the first element, or it follows a comparable tile t. Otherwise, if t > 0 and 0 were not comparable, then 0 would have been declared the leftmost before t. It follows that the possible tiles that can be followed by 0 are 1,2,⋯,n-1. Then, there are n possible positions for 0 to be embedded into T_i to recover a tiling sequence T ∈ C_i. Hence, \| C_i \| ≤ n for all i. Thus, we have \| 𝕋 \| ≤ n \| 𝕋_0 \|. Now, we are ready to prove Theorem <ref> and Corollary <ref>. For the upper bound, by using the inequality in Lemma <ref>, we obtain f_N ≤ n f_N-1≤ n^2 f_N-2≤⋯≤ n^N . It follows that μ_n = lim_N →∞ln f_N/N≤ln n. For the lower bound, consider a strip R_m that has length m for m ≤ n. The corresponding graph of R_m is the complete graph with m vertices and no forced edges. Then, tilings of R_m bijectively correspond to permutations of { 0, 1, ⋯, m-1 }. It follows that the number of tilings f_m of R_m equals m!. Next, we can divide an n-by-N strip R into ⌊N/n⌋ squares with dimensions n × n and a small remainder rectangle with dimension n × m where m = N n. By superadditivity, we have the inequality ln f_N ≥⌊N/n⌋ln (n!) + ln m! By the definition of μ_n, we obtain the lower bound in Theorem <ref>. By Stirling's approximation ln(n!) = n ln (n) - n + O( ln(n) ), we obtain Corollary <ref> from Theorem <ref>. § ENUMERATION This section explains the enumeration process in detail and proves Theorem <ref>. At each step of the leftmost tiling process, it is important to know which tiles are valid candidates for the leftmost tile. Lemma <ref> allows one to find all valid choices for the leftmost tile at each step of the process. The leftmost tiling process builds a sequence of labels S=[ w_1, w_2, ⋯, w_N ] as an output. At each step of the process, let X be the sequence of labels that have been declared at the previous steps and let Y = S \ X be the set of remaining labels. Let w_X ∈ X be the final element of X and max(X) be the largest element of X. Note that max(X) can be different from w_X. For example, in Figure <ref> (d), X= [ 1, 5, 2 ], so max(X)=5 and w_X = 2. Obviously, w_X ≤max(X). Let C̅ = { w ∈ Y : w_X - n < w ≤max(X) + n and w - n ∉ Y} in the case when X is not empty, and let C̅ = { w ∈ Y: 0 ≤ w ≤ n-1 } when X is empty (that is, at the first step). Sort C̅ as an increasing sequence [ w_1,w_2, ⋯, w_k ]. Let i be the smallest index such that w_i+1 - w_i ≥ n for 1 ≤ i ≤ k-1 where k is the largest index of the sequence C̅. Let C={ w_1, ⋯, w_i } if i exists, and C=C̅ otherwise. At each step of the leftmost tiling process, w_c ∈ Y can be the leftmost tile of a tiling of the remaining region if and only if w_c ∈ C. Initially, the tile covering the north-western corner of the strip is always the leftmost tile. It follows that the candidate set is { 0, 1, ⋯, n-1} at the first step. The complete proof of Lemma <ref> is based on the study of the corresponding acyclic orientation of G_R and is relegated to Appendix. In order to give an intuitive idea, we illustrate the lemma for n=4 in an example shown in Figure <ref>. Diagram (a) shows that the candidate labels are 0,1,2,3 at the first step of the leftmost tiling process. Lemma <ref> says that 4 is not a valid candidate because 4 - n = 0 ∈ Y. (Intuitively, at the first step there are no tilings such that 4 is a source tile with the smallest label.) Diagram (b) shows that the candidate labels are 0,2,3,5 at the second step if 1 was declared at the first step. Note that tile 5 is a valid candidate since 5-n=1 ∈ X and so this choice is not ruled out by Lemma <ref>. (Intuitively, if one declares 5 a source tile with the smallest label, one can build a tiling such that this declaration is true.) In Diagram (c), we see that 7 is not a valid candidate since 7 - 2 = 5 > n, so that 7 is in C̅ but not in C. Again, intuitively this is because 7 can never be a leftmost tile in the tiling of the remaining region. Diagram (d) shows a case that C = C̅ = { 0, 3, 6, 9 }. In Diagram (e), 0 is not a valid candidate since 5 - 0 = 5 > n, so the first inequality in the definition of C̅ is not satisfied. Also note that 9 is in C̅ but not in C. (Intuitively, 0 is not valid candidate in diagram (e) because declaring it the leftmost tile would contradict the choice of 5 as the source tile with the smallest label in the previous step.) Lemma <ref> provides us an efficient way to find out all the leftmost tiles at each step. We can enumerate all tiling sequences using Lemma <ref>. §.§ The Deducting Moment and Fundamental Regions In the leftmost tiling process of a strip R, we can stop at any specific step and obtain a sequence J. We call J initial segment of a tiling sequence. The region R is separated into two parts: a partial region covered by the tiling corresponding to J and a remainder region that is not yet tiled. Define a residual region R_J to be the untiled region which is obtained from R by removing the tiling corresponding to J. The region R_J has N-\|J\| tiles. See Figure <ref> for an example. More generally, we use the notation R_J(N-l) for 0 ≤ l ≤ N to denote a residual region of the strip with length N - l + \|J\|. That is, R_J(N-l) contains N-l tiles and its left boundary is a horizontal translate of that of R_J. Clearly, R_J(N-\|J\|) = R_J since R_J contains N-\|J\| tiles. For any two residual regions R_J(N-l) and R_J'(N-l'), we say that they are similar if their left boundaries are translates of each other. Obviously, R_J(N-l) is similar to R_J for any valid N-l ∈ℕ. The next lemma will show that a residual region R_J is invariant under any valid permutation of J. Let J be an initial segment. If a permutation π(J) of J is also a valid initial segment, then R_J = R_π(J). Let Y be the label set of tiles in the region R_J. For the graph G_R, consider the cut set of (J, Y). The two sequences J and π(J) give the same orientation for this cut set, in which each edge is directed from J (or π(J)) to Y. So we can use the same orientations on the subgraph induced by Y and be sure that they will lead to acyclic orientations whether we add them to orientations defined by J or to orientations defined by π(J). In particular, we can use orientations on Y defined by a sequence [ w_\|J\| + 1, ⋯, w_N ]. Then we can build the tiling starting the construction from the end of this sequence and proceeding by going backwards to the beginning and adding new tiles. This method will determine the identical tilings on R_J and R_π(J). In particular, this shows that these regions coincide. In the leftmost tiling process, define the deducting moment to be the step at which the tile with label 0 (which is unique) is declared to be the leftmost. At the deducting moment, we call a column of the strip a full column if it is fully covered by the current tiling. We will show later in Lemma <ref> that the deducting moment will be achieved in O(n^2/2) steps. For any sequence J = [0, j_1, ⋯, j_k] starting with 0, define the deduction operator δ by the formula δ(J) = [ j_1 - 1, ⋯, j_k -1 ] . That is, the operator δ removes the zero from the sequence and decreases each element by 1. For an initial segment J starting with 0, δ can be regarded as an operator that removes the fist column of a strip and relabel the tiles. That is, δ(J) is a tiling sequence of a strip with length N-1 and its corresponding ribbon tiling is the same as that of J with the first column removed. See Figure <ref> for an example. A residual region obtained at the deducting moment can be deducted to a similar residual region by removing initial full columns. Let J be an initial segment achieving the deducting moment with d full columns. Then 0 is the last element of J, and 0, 1, ⋯, d-1 ∈ J. We use the composite operator δ∘ζ_0 repeatedly d times and denote it as (δ∘ζ_0)^d. Let J' = (δ∘ζ_0)^d (J). The deducted region R_J' is similar to R_J and their size difference is d. Now, we are ready to define fundamental regions. For an initial tiling sequence J achieving the deducting moment with d full columns, we call the sequence F = ϕ (J) = π( (δ∘ζ_0 )^d (J) ) a fundamental sequence, where π by definition permutes every sequence as an increasing sequence. Let 𝔽 be the set of all fundamental sequences, and R_F(N-l) be a fundamental region corresponding to F ∈𝔽 with size N-l. Note that the empty set is a valid fundamental sequence. For notational convenience, we use [0] to represent the empty fundamental sequence since their corresponding fundamental regions are similar. By convention, we will keep 𝔽 ordered first by the increasing length of the fundamental sequences and then by the lexicographic order if the sequences have the same length. Algorithm <ref> is provided to obtain all fundamental sequences by running the leftmost tiling process. We list the results for n=3 and n=4 as examples. For n=3, the fundamental sequences are [0] and [1]. For n=4, the fundamental sequences are [0], [1], [2], [1,2], [1,3] and [1,2,5]. Every fundamental sequence F is obtained from an initial segment J ending with 0 and having d ≥ 1 full columns. Namely, F = π( (δ∘ζ_0 )^d (J) ) = δ^d ( π(J) ). Hence, it will be useful to discuss the properties of π(J). We will show that π(J) is a valid initial segment in Lemma <ref>. This implies that a fundamental sequence F = δ^d ( π(J) ) is a valid initial segment. In order to prepare for the proof of Lemma <ref>, we introduce an auxiliary sequence A_J. Let A_J = {w ∈ J: w+n ∉ J }. By the definition of the leftmost tiling process, it is not possible that w+n ∈ J if w ∉ J. It follows that A_J is a subset of J obtained by keeping the largest element of each modulo-n equivalence class. We permute A_J = [a_1, ⋯, a_k] as an increasing sequence and call it the essential sequence of J. Since J ends with 0, it follows that m n ∉ J for any positive integer m. Thus, we have 0 ∈ A_J and a_1 = 0. (For example, in Figure <ref> (a), the initial sequence [1,2,5,3,0] has the essential sequence [0,2,3,5].) We say that an increasing sequence W = [w_1, ⋯, w_k] has connectivity if w_i+1 - w_i < n for i=1,2, ⋯, k-1. Note that this is stronger than the theoretical connectivity of subgraph [w_1, ⋯, w_k] in G_R: we rule out w_i+1 - w_i =n although w_i and w_i+1 are connected by an edge. The reason for our definition is that: only if W has connectivity, then it is possible to have a directed path w_k → w_k-1→⋯→ w_1 without generating any directed cycle in the graph G_R. Since J is an initial sequence ending with 0, for every w ∈ J ∖{0}, we must have a directed path from w to 0 corresponding to the tiling decided by J. From the existence of these paths, it follows that π(J) has connectivity. Otherwise, let π(J) = [j_1=0, ⋯, j_s, j_s+1, ⋯, j_k] such that j_s+1 - j_s ≥ n, then it is not possible for the directed edge j_s+1→ j_s. Together with the fact that there is no edge between j_s+1 and j_1, ⋯, j_s-1, it follows that it is not possible to have a directed path from j_s+1 to j_1 in the graph G_J. This contradicts the assumption that J is an initial segment ending with 0. Lemma <ref> shows that the essential sequence A_J also has connectivity. Let J be an initial segment ending with 0 and let A_J be the essential sequence of J. Then, A_J has connectivity. In order to seek contradiction, suppose a_m-1, a_m ∈ A_J such that a_m - a_m-1 > n. (It is not possible a_m - a_m-1 = n by the definition of A_J.) Let D_0 = { w ∈ J: a_m - n ≤ w < a_m } and D_1 = { w+n: w ∈ D_0 }. By our assumption, D_0 ∩ A_J = ∅. It follows that if w ∈ D_0, then w+n ∈ J, otherwise w ∈ J would be the largest in its equivalence class and would belong to A_J. Thus, we have D_1 ⊆ J. From the definition of D_0 and D_1, all elements of D_1 are comparable, thus G_D_1 is a complete graph. Let s be the unique sink of G_D_1 corresponding to the orientation induced by the initial segment J. We have already noted that for every w ∈ J there is a directed path from w to 0. In particular, it follows that there is a directed path p_s in G_J from s to 0. Since D_0 contains all elements of J between a_m-n and a_m, it follows that the directed path p_s must contain a directed edge s → w_0 for some w_0 ∈ D_0. However, s → w_0 → w_0+n → s is a directed cycle where w_0+n ∈ D_1, contradiction. Now, we are ready to prove Lemma <ref>. Let J be an initial segment ending with 0 and let π(J) be the increasing permutation of J. Then, π(J) is a valid initial segment. For the increasing sequence π(J), 0 is declared to be the leftmost tile at the first step. From Lemma <ref>, it is clear that 0 is a valid candidate at the first step. Let u be the l-th element and v be the (l+1)-th element of the sequence π(J), and π(J)_u be the sub-sequence of π(J) containing all elements before and including u. We will prove the lemma by induction. Suppose π(J)_u is a valid initial segment. We need to show that v is a valid candidate for the leftmost tile at the (l+1)-th step. At the (l+1)-th step, we have X = π(J)_u and w_X = max(X) = u. Recall that C̅ = { w ∈ Y : w_X - n < w ≤max(X) + n and w - n ∉ Y}. Then, in our situation, we have C̅ = { w ∈ Y : u - n < w ≤ u + n and w - n ∉ Y }. Let A_J be the essential sequence of J, and let A_u be the sub-sequence of A_J such that A_u = { w ∈ A_J: u-2n < w ≤ u }, then for every w ∈ A_u we have u-n < w+n ≤ u+n. By the definition of essential sequence, it follows that for every w ∈ A_u we have w+n ∉ J. Let A'_u = {w+n: w∈ A_u }. By the definition of C̅, we have A'_u ⊆C̅. Let w_0 be the smallest element of C̅. It is clear that w_0 > u-n by the definition of C̅. By Lemma <ref>, A_J has connectivity and 0 ∈ A_J, it follows that min(A_u) - (u-2n) < n by the definition of A_u, and thus min(A'_u) - (u-n) = min(A_u) + n - (u-n) < n. Therefore, we have min(A'_u) - w_0 < n. Note that the connectivity of A'_u is inherited from A_u. It follows that A'_u ⊆ C by the definition of C. By the definition of C̅, it is clear that u+n ∈C̅. From the connectivity of A_J, it follows that 0 ≤ u - max(A_u) < n by the definition of A_u, and thus (u+n) - max(A'_u) < n. Since max(A'_u) ∈ C and u+n ∈C̅, we have u+n ∈ C by the definition of C. Note that u+n is the largest element of C̅. Therefore, we have C = C̅. From the connectivity of π(J), it follows that u < v < u+n. By the definition of C̅, it follows that v ∈C̅, and thus v ∈ C since C = C̅. Therefore, v ∈π(J) is a valid candidate at the (l+1)-th step. By induction, π(J) is a valid initial segment. Lemma <ref> shows that every fundamental sequence F is a valid initial segment. Let C^0_F be the candidate set at the first step of the leftmost tiling process in a fundamental region R_F. We obtain C^0_F by setting w_X = 0 and X=F in Lemma <ref>. Note that the order of F does not make any difference for C^0_F except through setting w_X = 0, since the fundamental region R_F is invariant under any valid permutation of F by Lemma <ref>. Note, however, that in the leftmost tiling process of the strip R, if J is an initial segment, the order of J (and in particular, the last element of J) plays an important role in the determination of the candidate set C_J of the leftmost tiles by Lemma <ref>. In particular, if F is a fundamental sequence and R_F is similar to R_J then it might happen that C_J ≠ C^0_F, which will lead to an issue for our enumeration method. This issue will be discussed and rectified in the next section. §.§ The Recursive System We think about the tiling process as the sequence of transitions between fundamental regions (up to similarity). By running the leftmost tiling process in a fundamental region R_F, we will obtain for the first time another fundamental region in one of the following two situations. (One of the two situations must happen.) Let J be a sequence obtained by running the leftmost tiling process in R_F. Then, * Case (1): J achieves the deducting moment with d full columns. * Case (2): J does not achieve the deducting moment, but π ([F, J]) is a fundamental sequence. We consider each of these cases as a transfer between two fundamental sequences. Define F F' to be a tiling transition from F to F' such that F' = { ϕ ( [F, J]) if Case (1); π ([F, J]) if Case (2). . where ϕ is as defined in Equation <ref>. For a tiling transition F F', we call J a transition sequence. Note that R_F'(N-l-\|J\|) is the first fundamental region hit by the leftmost tiling process in the region R_F(N-l). (We assume here that N-l is not too small so that the region R_F(N-l) can be tiled by J.) Figure <ref> shows an example of a Case (1) tiling transition. Suppose we start from the region R_F with F = [ 1,2,5 ] shown in (a) with black bold boundary. The transition sequence is J = [ 3,0 ]. Then, we obtain F' = ϕ ( [F, J] ) = ϕ ( [ 1,2,5,3,0 ] ) = [ 1 ] and the corresponding region is shown in (b). In Figure <ref>, we show an example of a Case (2) tiling transition from F=[1,2] to F'=[1,2,5] with transition sequence J=[5]. Let f_F(N-j) be the number of tilings of the fundamental region R_F(N-j). In order to calculate f_F(N-\|F\|), we run the leftmost tiling process in R_F and consider all possible tiling transitions F F'. By the definition of fundamental sequence, R_F' and R_[F,J] are similar regions, hence R_F'(N-\|F\|-\|J\|) and R_[F,J](N-\|F\| - \|J\|) are identical. Therefore, it is reasonable to collect f_F'(N-\|F\|-\|J\|) as a positive term in the equation calculating f_F(N-\|F\|). By collecting f_F'(N-\|F\|-\|J\|) in the equation, we are trying to use f_F'(N-\|F\|-\|J\|) to represent the number of tiling sequences of R_F starting with J. This might lead to an error since f_F'(N-\|F\|-\|J\|) is obtained by ignoring the order of the sequence [F,J], as mentioned in the last paragraph in Section 5.1. Taking the summation over all possible tiling transitions τ: F F' in the fundamental region R_F, we have f_F(N-\|F\|) = ∑_τ f_F'(N-\|F\|-\|J\|) + e where e is an error term. Given the initial segment J in the fundamental region R_F, let C_[F,J] be the candidate set of the leftmost tiles at the next step. Let C^0_F' be the candidate set of the leftmost tiles at the first step of the leftmost tiling process for the fundamental region R_F'. (Recall that C^0_F' is obtained by setting w_X = 0 where X=F' with the notation as in Lemma <ref>.) In order to calculate the error term e, we need to clarify the difference between C^0_F' and C_[F,J]. As there are two possible cases for a tiling transition, they will be discussed in the following two lemmas, respectively. We will use the notation C̅_[F,J] and C̅^0_F' for the set C̅ in Lemma <ref> corresponding to C_[F,J] and C^0_F', respectively. Suppose [F,J] achieves the deducting moment with d full columns and F' = ϕ([F,J]). Then, w ∈ C_[F,J] if and only if w - d ∈ C^0_F'. From the definition of the tiling transition F' = ϕ([F,J]), it follows that R_[F,J] and R_F' are similar regions. Therefore, we have w ∈C̅_[F,J] if and only if w - d ∈C̅^0_F' by the definition of C̅_[F,J] and C̅^0_F'. From the definition of the candidate set C in Lemma <ref>, it follows that w ∈ C_[F,J] if and only if w - d ∈ C^0_F'. Suppose [F,J] does not achieve the deducting moment, but F' = π([F,J]) is a fundamental sequence. Then, C_[F,J]⊆ C^0_F'. Since F' is a fundamental sequence, hence by the definition of a fundamental sequence, there is an initial segment J' ending with 0 such that ϕ(J') = F'. Let A_J' be the essential sequence of J'. By Lemma <ref>, A_J' has connectivity. From the proof of Lemma <ref>, we have already noted that the connectivity of C̅_J' is inherited from A_J'. (C̅_J' corresponds to C̅ in the notation of Lemma <ref> applied to J'.) Since R_F' is similar to R_J', it follows that the connectivity of C̅^0_F' is guaranteed by C̅_J', thus C^0_F' = C̅^0_F'. From the definition of C̅_[F,J] and C̅^0_F', it follows that C̅_[F,J]⊆C̅^0_F'. Note that C_[F,J]⊆C̅_[F,J] holds by the definition of C in Lemma <ref>. Therefore, C_[F,J]⊆C̅_[F,J]⊆C̅^0_F' = C^0_F'. The following example shows that it is possible to obtain C_[F,J]⊊ C^0_F'. For n=4, let J = [5] and J transfers F = [1,2] to F'=[1, 2, 5]. We see that C_[F,J] = { 3, 6, 9 } and C^0_F' = { 0, 3, 6, 9}. (See Figure <ref>.) Then, 0 ∈ C^0_F'∖ C_[F, J]. The situation C_[F,J]⊊ C^0_F' results in the error e in Equation <ref>. When [F,J] achieves the deducting moment, Lemma <ref> shows that there is a bijection between C_[F,J] and C^0_F'. Hence, this case will not contribute to the error term e in Equation <ref>. When [F,J] does not achieve the deducting moment but F' = π([F,J]) is a fundamental sequence, Lemma <ref> indicates that the error term e could come from an over-counting issue, and the previous example shows that the over-counting issue can really happen. This over-counting issue can be corrected by considering the set C^0_F'∖ C_[F, J]. For every w ∈ C^0_F'∖ C_[F, J], let f_w be the number of tiling sequences of R_F'(N-\|F\|-\|J\|) that start with w. By seeking all possible tiling transitions F' F” where J' starts from w, we can represent f_w to be the summation of f_F”(N-\|F'\|-\|J'\|) over all possible F”. We switch the sign of these terms and add them to Equation <ref>. If the over-counting issue happens again in the calculation of f_w, we repeat the procedure by considering the set C^0_F”∖ C_[F', J']. In the whole procedure, the sign of the terms may be switched multiple times based on the inclusion-exclusion principle. In our numeric calculation, the over-counting issue happens rarely. In the previous example, we have C_[F,J] = { 3, 6, 9 } and C^0_F' = { 0, 3, 6, 9}, so C^0_F'∖ C_[F, J] = { 0 }. Note that [0] transfers F'=[1,2,5] to F”=[2]. Therefore, f_[2](N-4) is an additional negative term in the equation calculating f_[1,2](N-2). We now construct the recursive system. As we will show later in Lemma <ref>, there are (n-1)! fundamental sequences. In addition, it will be shown in Lemma <ref> that it is sufficient to consider n regions of different sizes corresponding to each fundamental sequence. Thus, define an n! dimensional vector 𝐟(N) = ( f_F_i(N-j) )^⊺ for i=1,2,⋯,(n-1)! and j=0,1,⋯,(n-1) such that each component f_F_i(N-j) is the number of tilings of the fundamental region R_F_i(N-j). We order the elements of 𝐟(N) first by decreasing size for fundamental regions, then for each size by the order of the sequence in 𝔽. Note that the first component of 𝐟(N) is exactly the number of tilings of the n-by-N strip. For example in the case n=3, the corresponding vector is 𝐟(N) = ( f_[0](N), f_[1](N), f_[0](N-1), f_[1](N-1), f_[0](N-2), f_[1](N-2) )^⊺ . Algorithm <ref> in Appendix is used to obtain the transfer matrix. We report the results for n=3 and n=4. The corresponding transfer matrices A_3 and A_4 have the form declared in Theorem <ref>. In particular, we have A_3^' = [ 1 1 0 1; 1 0 1 0 ] and A_4^' = [ 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0; 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0; 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1; 1 0 0 0 0 1 1 0 -1 0 0 0 0 0 1 0 0 0; 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0; 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0; ] . §.§ The Generating Function Define the generating function for a square matrix A to be the matrix G(x) with entries G_i,j(x) = ∑_N=0^∞ (A^N)_i,j x^N . By Theorem 4.7.2 in <cit.>, we have G_i,j(x) = (-1)^i+j(I - xA : i, j)/(I-xA) where (B: i, j) denotes the matrix obtained by removing the i-th row and j-th column of B. For the number of tilings, let g_n(x) = ∑_N=0^∞ f_N x^N be the generating function for f_N. Using above formula, we have g_n(x) = G_1,1(x) = (I - xA_n : 1, 1)/(I-xA_n). From Equation <ref>, we are able to calculate the generating function of transfer matrix A_n for small n. For the case n=3, the generating function is g_3(x) = 1-x^3/1-x-x^2-4x^3+x^6 . For the case n=4, the generating function is g_4(x) = p(x)/q(x) where p(x) = 1 - x^2 - 13 x^4 - 2 x^5 + 5 x^6 - x^7 + 39 x^8 + 6 x^9 + 6 x^11 -37 x^12 - 5 x^14 - x^15 + 11 x^16 + x^18 - x^20 and q(x) = 1 - x - 2 x^2 - 2 x^3 - 25 x^4 + 3 x^5 + 12 x^6 + 4 x^7 + 109 x^8 + 5 x^9 - 9 x^10 + 7 x^11 - 159 x^12 - 4 x^13 - 16 x^14 - 7 x^15 + 82 x^16 + 10 x^18 + x^19 - 16 x^20 - x^22 + x^24 . §.§ Proof of Theorem <ref> We start this section by describing fundamental regions using another construction. This will allow us determine the number of fundamental regions explicitly. Define a boundary sequence B= [ b_0, b_1, ⋯, b_n-1 ] as a sequence of n tiles with dimension 1 × n such that b_0 = 0 and b_k satisfies the following conditions for k=1,⋯,n-1: (1) b_k-1+1 ≤ b_k ≤ b_k-1 + n; (2) b_k ≠ b_i + ln for i=0,⋯,k-1 and every l ∈ℕ. See Figure <ref> (a) for an example of boundary sequence [0,3,6,9] and (b) for a counter example [0,2,7,9]. Let 𝔹 be the set of all boundary sequences. Define a map σ: 𝔽→𝔹 from fundamental sequences to boundary sequences as follows. Given F ∈𝔽, let p be the left boundary of the fundamental region R_F, then σ(F) = [ b_0, b_1, ⋯, b_n-1 ] such that b_i is a tile with dimension 1 × n whose left boundary belongs to p. First, we need to show σ(F) ∈𝔹. By the definition of a fundamental sequence, we have 0 ∉ F. It follows that b_0 = 0. Let B' = [ b_n-1, ⋯, b_1, b_0 ]. Using Lemma <ref>, we can check that [F, B'] is a valid tiling sequence for the region R̅^F which is the union of R^F, formed by the tiles given by F, and n horizontal tiles in rows 0,1,2, ⋯, n-1. It follows that σ(F) satisfies condition (1) and (2) in the definition of boundary sequence. Hence, σ is well defined. Let B be a boundary sequence and R^B be the region of the strip on the left of the tiles of B. Then R^B is either empty or it has a tiling containing the tile T_0 with label b_n-1 - n, composed of the rightmost squares at the levels b_n-1 - n, b_n-1 - n +1, ⋯, b_n-1 - 1 in R^B. We proceed by induction on b_n-1. The situation is trivial if b_n-1 is smallest possible, i.e. if b_n-1 = n-1, since then R^B = ∅. Now let b_n-1≥ n. Then there exists i ∈{ 1, ⋯, n-1 } such that b_i-1 < b_n-1 - n < b_i. The strictness of the inequalities comes from the second part of the definition of a boundary sequence. We define b̃_k = { b_k, 0 ≤ k ≤ i-1, b_n-1 - n, k = i, b_k-1, i+1 ≤ k ≤ n-1. . Then B̃ = [b̃_0, ⋯, b̃_n-1] is a boundary sequence: the first condition is evident and the second one is satisfied, because the sets of integers { b_0, ⋯, b_n-1} and {b̃_0, ⋯, b̃_n-1} agree modulo n. Note that R^B splits into R^B̃ and T_0 without overlap. Using the tiling of R^B̃ given by the induction hypothesis, we obtain the required tiling of R^B. The map σ is a bijection between the set of fundamental sequences 𝔽 and the set of boundary sequences 𝔹. First, we show that σ is an injection. For any two different fundamental sequences F_1 and F_2, their corresponding fundamental regions have different left boundaries. It follows that the corresponding boundary sequences σ (F_1) and σ (F_2) are different. Now, we show that σ is a surjection. Suppose B = [ b_0, b_1,⋯, b_n-1 ] is a boundary sequence. By Lemma <ref>, let J = [j_1, ⋯, j_k = b_n-1 - n] be an initial sequence corresponding to a tiling of R^B. In order to prove that σ is a surjection, it is sufficient to show that J represents a fundamental sequence. Note that J̃ = J ∪ [ b_n-1, ⋯, b_1, b_0 ] is a tiling sequence achieving the deducting moment with n full columns. It follows that R_J̃ is a fundamental region. Note that R_J and R_J̃ are similar regions since b_i's are tiles with dimension 1 × n. By the definition of fundamental sequences, J is a fundamental sequence. Running the leftmost tiling process in a strip, the deducting moment will appear in at most L = n(n-1)/2 + 1 steps. Suppose X is a tiling sequence at the deducting moment. Maximizing the number of elements in X is equivalent to maximizing the area of the tiled region by X. By Lemma <ref>, it is equivalent to building a boundary sequence B such that the area of R̅^B is maximized where R̅^B is the union of R^B and the region covered by the tiles in B. It follows that B is an arithmetic progression with initial term 0 and common difference n-1. (See Figure <ref> (a) for an example.) Then the maximal area of R̅^B is n^2(n-1)/2 + n, which contains L = n(n-1)/2 + 1 tiles. Lemma <ref> gives an upper bound on the number of the steps for the appearance of the deducting moment. It guarantees that Algorithm <ref> and Algorithm <ref> are feasible. By the bijection in Lemma <ref>, we can also obtain all fundamental sequences using the construction of boundary sequences. There are exactly (n-1)! fundamental sequences. By Lemma <ref>, it is sufficient to show that there are (n-1)! different boundary sequences. Let [ b_0, b_1, ⋯, b_n-1 ] be a boundary sequence. By definition, there is only one choice for b_0, that is, b_0 = 0. Consider b_k for 1 ≤ k ≤ n-1. There are n candidates for b_k satisfying the condition (1) in the definition of boundary sequence. For these n candidates of b_k, consider the following statement corresponding to condition (2) in the definition. For each i, 0 ≤ i ≤ k-1, we can find an l_i ∈ℕ such that b_k-1+1 ≤ b_i + l_i n ≤ b_k-1 + n. It follows that condition (2) will remove one of the n candidates of b_k for each i, 0 ≤ i ≤ k-1. Therefore, there are n-k possible values for b_k. Then, the number of possible choices for [b_0, b_1, ⋯, b_n-1] is 1, n-1, n-2, ⋯, 1. Hence, there are (n-1)! possible boundary sequences. Let J be an initial segment such that 0 ∉ J, that is, the deducting moment has not been achieved. For the residual region R_J, let W_J= [ w_0, w_1, ⋯, w_n-1 ] be an increasing sequence of 1 × n tiles such that the left boundary of these tiles coincides with the left boundary of R_J. We call W_J the quasi-boundary sequence of R_J. (For example, in Figure <ref> (b), [0,2,7,9] is a quasi-boundary sequence of R_[1,5, 3].) Since W_J is adjacent to a tilable region that is covered by J, it follows that W_J satisfies Condition (2) in the definition of boundary sequence. We observe that if W_J has connectivity (satisfies Condition (1) in the definition of boundary sequence), then W_J is a valid boundary sequence. From the bijection between 𝔽 and 𝔹 shown in Lemma <ref>, it follows that the residual region R_J is a fundamental region if and only if W_J has connectivity. Now we are going to prove the crucial property of tiling transitions. Let F F' be a tiling transition between fundamental sequences F and F' where F and F' are fixed. Then, J is a unique decreasing sequence. Moreover, \|J\| ≤ n for every tiling transition and \|J\| = n if and only if J transits from a fundamental sequence F to F itself. By definition, J is a tiling sequence from R_F(N-l) to R_F'(N-l'). It is clear that \|J\| = l' - l. Let D be the difference region between R_F'(N-l') and R_F(N-l) and D be covered by the tiling corresponding to J. From <cit.>, it follows that the elements of J, which are the labels of tiles, are determined by the region D. For these elements of J, we will show that J = [ j_1, j_2, ⋯, j_l'-l ] is a decreasing sequence. If l' - l=1, then there is only one tile j_1 between F and F'. Thus, J = [j_1] is uniquely determined by F and F'. Suppose l'-l > 1. For induction, suppose J^(m) = [j_1, j_2, ⋯, j_m] is a decreasing sequence where m < l'-l. Let W^(m) = W_[F, J^(m)] = [w_0, ⋯, w_n-1] be the quasi-boundary sequence of the residual region R_[F, J^(m)], w_0 < w_1 < ⋯ < w_n-1. Since m < l'-l, it follows that R_[F, J^(m)] is not a fundamental region by the definition of tiling transition. It follows that W^(m) is not a valid boundary sequence, thus the connectivity of W^(m) does not hold. We are going to prove that this implies that j_m+1 < j_m. By our induction assumption, J^(m) is a decreasing sequence. From the definition of the leftmost tiling process, we must have a directed path j_1 → j_2 →⋯→ j_m. It follows that J^(m) has connectivity. For every j ∈ J^(m), let j+n be the tile with dimension 1 × n and adjacent to the last square of the tile j. (For example, in Figure <ref>, let J^(m)=[6,3] and j_1+n = 9. The tile labeled by 9, which is adjacent to the last square of the tile labeled by 5, is shown in the picture.) The tile j+n will not overlap any existing tile since J^(m) is a decreasing sequence. It follows that j+n ∈ W^(m) by definition. At the m-th step, W^(m) can be obtained from W^(m-1) by removing j_m and including j_m + n, that is, W^(m) = (W^(m-1)∖{ j_m } ) ∪{ j_m + n } where W^(0) is the boundary sequence of F. It is clear that j_m ∉ W^(m) and j_m + n ∈ W^(m). Let W^(m)_* = { w ∈ W^(m) : w > j_m } be the tail of W^(m) where W^(0)_* = ∅, and W^(m-1)_ = { w ∈ W^(m-1): j_m < w < j_m-1} where W^(0)_ = { w ∈ W^(0) : w > j_1 }. (See Figure <ref> for an example.) From W^(m) = (W^(m-1)∖{ j_m } ) ∪{ j_m + n }, we obtain that W^(m)_* = { w ∈ W^(m) : w > j_m } = { w ∈ W^(m-1) : w > j_m }∪{ j_m + n } = { w ∈ W^(m-1) : w > j_m-1}∪{ w ∈ W^(m-1) : j_m < w < j_m-1}∪{ j_m + n } = W^(m-1)_∪ W^(m-1)_∪{ j_m + n } . We show that W^(m)_ has connectivity by induction. (By default, we permute W^(m)_* to be an increasing sequence.) At the first step (m=1), we have W^(1)_* = W^(0)_∪{ j_1 + n }. The sequence W^(0)_ = { w ∈ W^(0) : w > j_1 } has connectivity since it is the tail of the boundary sequence W^(0). The connectivity is not broken if j_1 + n is added. Hence, W^(1)_* has connectivity. Suppose, for induction, that W^(m-1)_* has connectivity. From the connectivity of J^(m), it follows that for every w' ∈ W^(m-1)_ we have 0 < (j_m + n) - w' < n. Thus, the connectivity of W^(m-1)_∪{ j_m + n } is proved. Note that j_m-1 + n ∈ W^(m-1)_. It follows that the connectivity between W^(m-1)_ and W^(m-1)_*∪{ j_m + n } is guaranteed by the fact 0 < j_m-1 - j_m < n. Therefore, W^(m)_ = W^(m-1)_∪ W^(m-1)_∪{ j_m + n } has connectivity. Now we consider the sequence W^(m)∖ W^(m)_. We see that W^(m)∖ W^(m)_* is the same as the head of the boundary sequence of R_F(N-l). (For example, in Figure <ref>, W^(m)∖ W^(m)_* = [0] for m=2, which is unchanged as the head of the boundary sequence W^(0).) From the definition of a boundary sequence, it follows that the connectivity of W^(m)∖ W^(m)_* holds. Let w_i be the largest element of W^(m) such that w_i < j_m, then W^(m)∖ W^(m)_* = [w_0, ⋯, w_i] and W^(m)_* = [w_i+1, ⋯, w_n-1]. Since the connectivity of both W^(m)_* and W^(m)∖ W^(m)_* hold, and W^(m) does not have connectivity, we must have w_i+1 - w_i > n. (See Figure <ref> for an example. ) Let w_i be the largest element of W^(m) such that w_i < j_m. It is clear that j_m+n > j_m > w_i. Since j_m+n ∈ W^(m), it follows that j_m+n ≥ w_i+1. Thus, [j_m+n, ⋯, j_1+n] is a sub-sequence of [w_i+1, ⋯, w_n-1] ⊆ W^(m). By the definition of w_i, it follows that w_i+1≥ j_m. Note that j_m ∉ W^(m), thus w_i+1 > j_m. Therefore, we have 0 ≤ (j_m + n) - w_i+1 < n. For every w' ∈ [w_i+2, ⋯, w_n-1] such that w' ≤maxJ^(m)+n, it is clear that there is j' ∈ J^(m) such that 0 ≤ (j'+n) - w' < n. Let W^* = [ w ∈ W^(0): w ≥max(J^(m)) ] where W^(0) is boundary sequence of F. From our induction assumption, we have max(J^(m)) = j_1, then W^* is unchanged after the first step, that is, W^* = [ w ∈ W^(0): w ≥ j_1 ]. It is clear that W^* has connectivity since W^* is the tail of W^(0). It follows that the connectivity of [w_i+1, ⋯, w_n-1] is inherited from J^(m) and W^. Since w_i < j_m by definition, it follows that [w_0, ⋯, w_i] is the same as the head of the boundary sequence of R_F(N-l). (For example, in Figure <ref> (b), let F=[1] and J^(m) = [5, 3]. The boundary sequence of R_[1] is [0,2,3,5], and W^(m) = [0,2,7,9]. We see that j_m = 3 and w_i = 2, thus [0,2] = {w ∈ W^(m): w < j_m } is unchanged.) From the definition of a boundary sequence, it follows that the connectivity of [w_0, ⋯, w_i] holds. Since the connectivity of both [w_0, ⋯, w_i] and [w_i+1, ⋯, w_n-1] hold, and W^(m) does not have connectivity, we must have w_i+1 - w_i > n. (For example in Figure <ref> (b), let F=[1] and J^(m) = [5,3], then W^(m) = [0,2,7,9]. We have j_m = 3, w_i = 2 and w_i+1=7 satisfying w_i+1 - w_i = 5 > n. ) We apply Lemma <ref> to J^(m). Note that j_m is the last element of J^(m). The definition of w_i ensures that w_i and w_i+1 satisfy the inequalities of C̅. It is also clear that w_i - n, w_i+1-n ∉ Y where X = [F, J^(m)]. Therefore, we have w_i, w_i+1∈C̅ by definition. Since the connectivity is broken from w_i to w_i+1, it follows that w_i is the largest element of the candidate set C of the leftmost tile at the (m+1)-th step of the leftmost tiling process. Therefore, j_m+1≤ w_i since j_m+1∈ C. From w_i < j_m, it follows that j_m+1 < j_m. Thus, J is a decreasing sequence by induction. The maximal value of \|J\| happens when the sequence J equals W_F listed in reverse order where W_F is the quasi-boundary sequence of R_F(N-l). (In fact, W_F is a boundary sequence since R_F(N-l) is a fundamental region.) In this case, we see that R_F(N-l) and R_F'(N-l') are similar, thus F'=F. (See Figure <ref> (a) for an example.) Now we prove Theorem <ref>. We have constructed the recursive system for the number of tilings of fundamental regions in Section 5.1 and Section 5.2. By the uniqueness of tiling transitions proved in Lemma <ref>, it follows that the transfer matrix A_n only contains elements 0, ±1. By Lemma <ref>, there are (n-1)! fundamental sequences. By Lemma <ref>, the size difference of the recursive system is n. Note that the difference between the candidate sets of C̃ based on the setup with w̃_X = 0 and C based on the original w_X may lead to an extension of tiling transitions. For every element w ∈C̃∖ C, we have w < min (C). Therefore, Lemma <ref> holds for tiling transitions including extended transitions. The upper-right block of A_n must be identity matrix I_2 with dimension (n-1)! × (n-1)!. Therefore, the block A_n^' has dimensions (n-1)! × (n-1)(n-1)!. The identity matrix I_1 results from the construction of the recursive system automatically. § APPENDIX §.§ Proof of Lemma <ref> Proof of Necessity. We will prove w_c ∈C̅ assuming that w_c is the leftmost. We first show that w_c - n ∉ Y. Otherwise, if w_c - n ∈ Y then the relation w_c - n ≺ w_c is forced. It is not possible that w_c is the leftmost, contradiction. In order to prove the inequality w_c ≤max(X) + n in the definition of C̅, by seeking contradiction, we suppose that w_c -n > max(X). We have already shown that w_c - n ∉ Y. It follows that w_c - n ∈ X. However, w_c - n > max(X) contradicts the assumption that max(X) is the largest element in X. We prove the other inequality w_c > w_X - n in the definition of C̅ by considering the following two cases and showing that they both lead to a contradiction. Case (1) w_c = w_X - n. In this case, the relation w_c ≺ w_X is forced. However, in the previous step w_X was declared to be the leftmost. It contradicts the relation w_c ≺ w_X. * Case (2) w_c < w_X - n. In this case, w_c and w_X are not comparable. Consider the previous step when w_X is declared to be the leftmost. One of the following two statements must be true at this step. (i) The relation w_X ≺⋯≺ w_c holds. (ii) There exists a source v ∈ Y such that v ≺⋯≺ w_c and w_X < v. For statement (i), there exists a non-empty sequence u_1, ⋯, u_j ∈ Y such that w_X ≺ u_1 ≺⋯≺ u_j ≺ w_c. (This sequence cannot include any of vertices in X since then one of these vertices would not be a source at the moment it is chosen.) Hence, w_c is not a source in Y and cannot be leftmost. For statement (ii), we see w_c cannot be the leftmost in Y since v ≺⋯≺ w_c and v ∈ Y. Both statements (i) and (ii) contradict that w_c is the leftmost. To finish the proof of the necessity, we show that w_c ∈ C assuming the existence of i in the definition of C. (If i does not exist, then C=C̅ and there is nothing to prove.) Let C_0 = { w ∈ Y : w ≤ w_i } and D_0 = { w ∈ Y: w_i < w ≤ w_i+n }. We claim that w-n ∈ C_0 for every w ∈ D_0. Now we prove the claim. For every w ∈ D_0, it is obvious that w-n ≤ w_i, thus it is sufficient to show that w-n ∈ Y. In order to show w-n ∈ Y, we suppose w-n ∈ X (or w-n < 0) seeking contradiction. From w-n ∈ X (or w-n < 0), it follows that w-n ≤max(X), so w ≤max(X) + n. Since w_i ∈ C ⊆C̅, it follows that w_X - n < w_i by the definition of C̅. (Recall that C̅ = { w ∈ Y : w_X - n < w ≤max(X) + n and w - n ∉ Y}.) For w ∈ D_0, we have w_i < w. Hence, w_X - n < w_i < w. From the inequality w_X-n < w ≤max(X) + n shown above and the assumption w-n ∈ X (or w-n < 0), it follows that w ∈C̅. Since w ∈ D_0, we have w_i < w ≤ w_i + n. However, this inequality w_i < w ≤ w_i + n together with w ∈C̅ contradicts the definition of the index i, since w > w_i +n must hold for every w ∈C̅ and w > w_i by the definition of i. Therefore, it is not possible that w-n ∈ X (or w-n < 0), thus w-n ∈ Y. Hence we proved the claim. For any tiling, let G_C_0 be the corresponding graph restricted to C_0, and w_s be a source of G_C_0. Let E_0 be the set of all edges uv where u ∈ C_0 and v ∈ D_0. By the definition of C_0 and D_0, E_0 is the cut-set of the cut (C_0, Y ∖ C_0) for the graph G_Y. For any w ∈ Y ∖ C_0, we show that there is no directed path from w to w_s. In order to seek contradiction, suppose that there is a directed path p from w to w_s. Since w_s is a source of G_C_0 and E_0 is the cut-set, it follows that p must contain a directed edge w̃→ w_s where w̃∈ D_0. From the claim we have already proved, it follows that w̃ - n ∈ C_0 for w̃∈ D_0. Since w̃→ w_s is a directed edge, and w̃ > w_s for w̃∈ D_0 and w_s ∈ C_0, it follows that w̃ - n and w_s are comparable. We have w_s →w̃ - n, since w_s is a source and w̃ - n ∈ C_0. Then, w_s →w̃ - n →w̃→ w_s generates a directed cycle, contradiction. Therefore, there is no directed path from w to w_s where w ∈ Y ∖ C_0. For any w ∈ Y ∖ C_0, it is clear that w > w_i, and then w > w_s. We have already shown that there is no directed path from w to w_s for any tiling. It follows that w cannot be a candidate of the leftmost tile by definition. Thus, for every candidate w_c the inequality w_c ≤ w_i holds, then w_c ∈ C. Proof of Sufficiency. Let C̃ = { w ∈ Y: w ≤ w_i and w - n ∉ Y } where w_i is the largest element in C. So, C is a subset of C̃ obtained by adding the requirement w > w_X - n. It is clear that C ⊆C̃, and C̃ = C at the first step of the leftmost tiling process. We order C̃ as an increasing sequence [ w_m ]_m=-j^i where w_m ∈ C for 1 ≤ m ≤ i. We claim that C̃ has connectivity, that is, w_m+1 - w_m < n for m = -j, ⋯, i-1. (We will prove the claim later.) Let w_c ∈ C ⊆C̃ be the tile that will be declared to be the leftmost tile. We build an acyclic orientation α_C̃ of G_C̃ by setting w_s → w_t for all w_c ≤ w_s < w_t, w_s → w_t for all w_s < w_c < w_t, and w_t → w_s for all w_s < w_t ≤ w_c where w_s and w_t are comparable in C̃. Note that there are no forced edges in G_C̃, hence α_C̃ is possible and acyclic. Let C̃ = [v_j, ⋯, v_1, w_c, u_1, ⋯, u_k]. From our claim that C̃ has connectivity, we have two directed paths w_c → u_1 →⋯→ u_k and w_c → v_1 →⋯→ v_j in α_C̃. It follows that w_c is the unique source of G_C̃ for the acyclic orientation α_C̃. Now we extend α_C̃ to G_Y by setting u → v for all free edges uv where u ∈C̃ and v ∈ Y ∖C̃, and s → t if s < t for all free edges st where s, t ∈ Y ∖C̃. We call this extended orientation α_Y. It is already known that α_C̃ is acyclic, and it is clear that there is no directed cycle in G_Y ∖C̃ for α_Y by the construction. Moreover, there is no edge directed into C̃ for α_Y since all forced edges directed into C̃ have been ruled out by the fact that w-n ∉ Y for every w ∈C̃. Therefore, the orientation α_Y is acyclic. Consider the acyclic orientation α_Y. By construction, w_c is the unique source of G_C̃. Since there is no edge directed into C̃, it follows that w_c is a source of G_Y. For any other source w_s of G_Y, if w_s ∉C̃, then w_c < w_s by the definition of C̃. Thus, w_c is the leftmost tile for α_Y. In summary, for an arbitrary tile w_c ∈ C, we can build an acyclic orientation α_Y such that w_c is the leftmost tile. This proves the sufficiency. To complete the proof, we now prove the claim that C̃ has connectivity. First, the connectivity holds for C by the definition of the index i, thus the connectivity holds for C̃ at the first step when C̃ = C. We will apply induction and use the notation ·^(l) to represent an object · at the (l)-th step of the leftmost tiling process. For induction, suppose the connectivity holds for C̃^(l) at the l-th step. Recall that C̃^(l) = { w ∈ Y^(l): w ≤ w_i^(l) and w - n ∉ Y^(l)}. Let w_c^(l) be the leftmost tile declared at the l-th step. At the (l+1)-th step, we have w_X^(l+1) = w_c^(l), which is the final element in X^(l+1) declared at the l-th step. Let D^(l) := { w ∈ Y^(l): w - n ∉ Y^(l)}, A^(l) := { w ∈ D^(l): w < w_c^(l)} and A'^(l+1) := { w ∈ D^(l+1): w < w_c^(l)}. Note that A^(l)⊆C̃^(l)⊆ D^(l). By definition, D^(l+1) differs from D^(l) by excluding w_c^(l) and including w_c^(l) + n, that is, D^(l)∖ D^(l+1) = { w_c^(l)} and D^(l+1)∖ D^(l) = { w_c^(l) + n }. It follows that A^(l) = A'^(l+1). In the following proof, let A = A^(l) = A'^(l+1) for notation convenience and order A by increasing labels. Note that A is an initial segment of C̃^(l) by the definition of A^(l). Since C̃^(l) has connectivity by the inductive assumption, it follows that A has connectivity. If A = A'^(l+1) = ∅, then by the definition of A'^(l+1), we must have w > w_c^(l) for every w ∈ D^(l+1). It is clear that C̃^(l+1)⊆ D^(l+1), so w > w_c^(l) for every w ∈C̃^(l+1). It follows that every w ∈C̃^(l+1) satisfies the condition w > w_X^(l+1) - n since w_X^(l+1) = w_c^(l). From C^(l+1)⊆C̃^(l+1) and C̃^(l+1)∖ C^(l+1) = {w ∈C̃^(l+1): w ≤ w_X^(l+1) - n }, we have C̃^(l+1)∖ C^(l+1) = ∅, and therefore C̃^(l+1) = C^(l+1). Therefore, the connectivity of C̃^(l+1) holds by the connectivity of C^(l+1). Let A ≠∅. At the (l+1)-th step, let w_i^(l+1) be the largest element of C^(l+1). We consider two alternatives depending on whether w_i^(l+1)∈ A or not. If w_i^(l+1)∉ A, then w_i^(l+1)∈ D^(l+1)∖ A'^(l+1). By the definition of A'^(l+1), it follows that w_i^(l+1) > w_c^(l), thus w < w_i^(l+1) for every w ∈ A. If w_i^(l+1)∈ A, we show that w ≤ w_i^(l+1) for every w ∈ A. In order to seek contradiction, suppose there is w_i'^(l+1)∈ A such that w_i'^(l+1) > w_i^(l+1). Since A has connectivity, we can further assume that w_i'^(l+1) < w_i^(l+1) + n. We show that w_i'^(l+1)∈C̅^(l+1) by checking the inequalities in the definition of C̅^(l+1) in the following paragraph. (Recall that C̅^(l+1) = { w ∈ Y^(l+1) : w_X^(l+1) - n < w ≤max(X^(l+1)) + n and w - n ∉ Y^(l+1)}.) First, since w_i^(l+1)∈ C^(l+1), it follows that w_i^(l+1) > w_X^(l+1) - n by the definition of C^(l+1), then w_i'^(l+1) > w_i^(l+1) > w_X^(l+1) - n. Secondly, since w_c^(l)∈ C^(l), it follows that w_c^(l)≤max(X^(l)) + n. Since w_i'^(l+1)∈ A, then by the definition of A we have w_i'^(l+1) < w_c^(l). Thus, we have w_i'^(l+1) < w_c^(l)≤max(X^(l)) + n ≤max(X^(l+1)) + n. Therefore, w_i'^(l+1) satisfies the inequalities in the definition of C̅^(l+1). From w_i'^(l+1)∈C̅^(l+1) and the inequality w_i^(l+1) < w_i'^(l+1) < w_i^(l+1) + n, we have a contradiction with the assumption that w_i^(l+1) is the largest element of C^(l+1). Hence, it is not possible that there is w_i'^(l+1)∈ A such that w_i'^(l+1) > w_i^(l+1). Thus, w ≤ w_i^(l+1) for every w ∈ A. For both alternatives of w_i^(l+1)∈ A and w_i^(l+1)∉ A, we have shown that w ≤ w_i^(l+1) for every w ∈ A. It follows that A ⊆C̃^(l+1) by the definition of C̃^(l+1). Let A and C̃^(l+1) be ordered by increasing labels, then A is an initial segment of C̃^(l+1). We consider the following two cases by comparing the elements of A with w_c^(l) - n. * Case (1) w > w_c^(l) - n for every w ∈ A. Since A is an initial segment of C̃^(l+1), it follows that w > w_c^(l) - n for every w ∈C̃^(l+1). By the definition of C̃^(l+1) and C^(l+1), we have C̃^(l+1)∖ C^(l+1) = { w ∈C̃^(l+1): w ≤ w_X^(l+1) - n }. Since w_X^(l+1) = w_c^(l), it follows that C̃^(l+1)∖ C^(l+1) = ∅, thus C̃^(l+1) = C^(l+1). Hence, the connectivity of C̃^(l+1) holds by the connectivity of C^(l+1). * Case (2) There is at least one element w_m_0^(l)∈ A such that w_m_0^(l) < w_c^(l) - n. We have w_m_0^(l), w_c^(l)∈C̃^(l) by the definition of C̃^(l). From the connectivity of C̃^(l) ensured by the inductive assumption, it follows that there is an element w_m_1^(l)∈C̃^(l) such that w_m_0^(l) < w_c^(l) - n < w_m_1^(l) < w_c^(l). From w_m_1^(l)∈C̃^(l) and the inequality w_m_1^(l) < w_c^(l), we have w_m_1^(l)∈ A by the definition of A^(l). It follows that w_m_1^(l)∈C̃^(l+1) since A = A'^(l+1)⊆C̃^(l+1). Let C̃_1^(l+1) = { w ∈C̃^(l+1): w ≤ w_m_1^(l)} and C̃_2^(l+1) = { w ∈C̃^(l+1): w ≥ w_m_1^(l)}. We have C̃_1^(l+1) is an initial segment of A since w_m_1^(l) < w_c^(l), thus the connectivity of C̃_1^(l+1) is preserved from A. We also have C̃_2^(l+1) is a tail of C^(l+1) since w_m_1^(l) > w_c^(l) - n = w_X^(l+1) - n, thus the connectivity of C̃_2^(l+1) is preserved from C^(l+1). It is clear that C̃^(l+1) = C̃_1^(l+1)∪C̃_2^(l+1) and C̃_1^(l+1)∩C̃_2^(l+1) = { w_m_1}. Therefore, the connectivity of C̃^(l+1) holds. §.§ Algorithms Remark. For every J ∈𝕁, it is sufficient to store J as an increasing sequence π(J) and the last element of J. This information will be enough to generate candidate set of the leftmost tiles by Lemma <ref>. Then, the duplication in 𝕁 can be removed. For example, we can store both [1,2,5] and [2,1,5] as a two-tuple ( [1,2,5], 5). asa.bst
http://arxiv.org/abs/2307.02900v2	20230706102114	Meta Federated Reinforcement Learning for Distributed Resource Allocation	[ "Zelin Ji", "Zhijin Qin", "Xiaoming Tao" ]	eess.SP	[ "eess.SP", "cs.SY", "eess.SY" ]	lemmaLemma IEEEtran Meta Federated Reinforcement Learning for Distributed Resource Allocation Zelin Ji, Graduate Student Member, IEEE, Zhijin Qin, Senior Member, IEEE, and Xiaoming Tao Part of this work was presented at the IEEE International Conference on Communications 2022 <cit.>. Zelin Ji is with School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (email: [email protected]). Zhijin Qin and Xiaoming Tao are with Department of Electronic Engineering, Tsinghua University, Beijing, China. (email: [email protected]; [email protected]). August 1, 2023 ====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In cellular networks, resource allocation is usually performed in a centralized way, which brings huge computation complexity to the base station (BS) and high transmission overhead. This paper explores a distributed resource allocation method that aims to maximize energy efficiency (EE) while ensuring the quality of service (QoS) for users. Specifically, in order to address wireless channel conditions, we propose a robust meta federated reinforcement learning (MFRL) framework that allows local users to optimize transmit power and assign channels using locally trained neural network models, so as to offload computational burden from the cloud server to the local users, reducing transmission overhead associated with local channel state information. The BS performs the meta learning procedure to initialize a general global model, enabling rapid adaptation to different environments with improved EE performance. The federated learning technique, based on decentralized reinforcement learning, promotes collaboration and mutual benefits among users. Analysis and numerical results demonstrate that the proposed MFRL framework accelerates the reinforcement learning process, decreases transmission overhead, and offloads computation, while outperforming the conventional decentralized reinforcement learning algorithm in terms of convergence speed and EE performance across various scenarios. Federated learning, meta learning, reinforcement learning, resource allocation. § INTRODUCTION The inexorable progression of wireless networks is the trend. The 3rd Generation Partnership Project (3GPP) has standardized the access technique and physical channel model for the fifth-generation new radio (5G NR) network, which enables dynamic switching of user equipment (UE) between resource blocks (RBs) possessing varying bandwidths and supports multiple subcarrier spacing <cit.>. Building upon the foundation established by 5G, the sixth generation (6G) and beyond networks aspire to provide the enhanced and augmented services of 5G NR, while transitioning toward decentralized, fully autonomous, and remarkably flexible user-centric systems <cit.>. These emerging techniques impose more stringent requirements on decentralized resource allocation methods, emphasizing the significance of optimizing RB assignments to enhance the overall quality of service (QoS) within the systems. Nevertheless, the fast variations and rapid fluctuations in channel conditions render conventional resource allocation approaches reliant on perfect channel state information (CSI) impractical <cit.>. The inherent non-convexity of the resource allocation problem resulting from discrete resource block association necessitates computationally demanding solutions. Furthermore, the coupled variables further exacerbate the complexity of the problem. Traditionally, resource allocation problems have been addressed through matching algorithms executed at the central base station (BS), resulting in substantial computational burdens on the cloud server. All of the aforementioned challenges require a brand-new optimization tool capable of effectively operating in unstable wireless environments. Machine learning (ML) methods, especially deep learning (DL) approaches, have become promising tools to address mathematically intractable and high-computational problems. However, artificial neural networks (NNs) require massive amounts of training data, even for a simple binary classification task. Moreover, the overfitting issue makes artificial NNs hard to adapt and generalize when facing new environments, hence requiring additional data to retrain the models and affecting the training data efficiency. Particularly, the fast channel variations and the flexible network structure in 5G beyond network services restrict the application of conventional ML algorithms. To enable fast and flexible learning, meta learning has been proposed to enable the model to adapt to new tasks with faster convergence speed by taking the input of experience from different training tasks <cit.>. For instance, model-agnostic meta-learning (MAML) <cit.> is a meta-learning technique that can integrate prior experience and knowledge from the new environment, empowering the models with the ability to generalization and fast adaptation to new tasks. Another way to improve data efficiency is to enable experience sharing among models, which is known as federated learning. By the periodic local model averaging at the cloud BS, federated learning enables the local users, to collectively train a global model using their raw data while keeping these data locally stored on the mobile devices <cit.>. In this paper, we focus on meta learning enabled federated reinforcement learning, to improve the performance of the reinforcement learning algorithm for resource allocation tasks in wireless communications. Through the implementation of periodic local model averaging at the cloud-based base station (BS), federated learning facilitates collaborative training of a global model by enabling local users to utilize their respective raw data, which remains stored locally on their mobile devices <cit.>. This paper investigates the application of meta learning within the context of federated reinforcement learning, with the aim of enhancing the performance of the reinforcement learning algorithm in resource allocation tasks within wireless communication systems. §.§ Related work §.§.§ Energy-Efficient Resource Allocation for Cellular Networks Presently, most cellular user equipment (UE) operates on battery power, and the use of rate maximization-oriented algorithms <cit.> may result in unnecessary energy consumption, which is unfavorable for the advancement of massive capacity and connectivity in 5G and beyond communications. Existing literature on energy-efficient resource allocation primarily focuses on optimizing transmit power and channel assignment <cit.>. Robat Mili et al. <cit.> concentrate on maximizing energy efficiency (EE) for device-to-device communications. While numerous studies have investigated resource allocation in wireless communication systems, most of them rely on centralized approaches, which are considered as complex and not easily scalable<cit.>. In such centralized approaches, the central entity needs to obtain global channel state information (CSI) to assign channels to UEs, leading to significant communication overhead and latency. Consequently, distributed low-complexity algorithms are preferable over centralized ones. Game theory has been adopted for decentralized resource allocation <cit.>. However, these approaches typically assume a static radio environment and require multiple iterations for UEs to converge to the Nash Equilibrium (NE) point. In the practical environment, the performance of game theory based algorithms is impacted by the rapid fluctuations in the wireless channel. Yang et al. <cit.> and Dominic et al. <cit.> integrate the game theory and stochastic learning algorithm (SLA) to enable local users to learn from past experience and adapt to channel variations. Nevertheless, game theory based algorithms do not fully explore the advantages of collaboration and communication among users, potentially affecting system-level performance. §.§.§ Decentralized Reinforcement Algorithms in Wireless Communications A promising solution to address concerns regarding complexity and signaling cost concerns involves establishing a decentralized framework for resource allocation and extending the intelligent algorithms to encompass cooperative large-scale networks. The adoption of multi-agent reinforcement learning (MARL) algorithm presents an opportunity to tackle the challenges associated with complexity and enhance the intelligence of local UEs. MARL algorithms rely solely on real-time local information and observations, thereby significantly reducing communication overhead and latency. Mathematically, MARL can be formulated as a Markov decision process (MDP), where training agents observe the current state of the environment at each step and determine an action based on the current policy. Agents receive corresponding rewards that evaluate the immediate impact of the chosen state-action pair. The policy updates are based on the received rewards and the specific state-action pair, and the environment transitions to a new state subsequently. The application of MARL approaches in wireless communications has been extensive <cit.>. Wang et al. <cit.> have demonstrated that such a decentralized optimization approach can achieve near-optimal performance. However, local user equipment (UE) cannot directly access global environmental states, and UEs are unaware of the policies adopted by other UEs. Consequently, there is a possibility that UEs may select channels already occupied by other UEs, leading to transmission failures in the orthogonal frequency-division multiple access (OFDMA) based schemes. §.§.§ Reinforcement Algorithm for Jointly Resource Optimization It is noted that the resource block association problem is a discrete optimization problem, which is usually solved by value-based methods, e.g., Q-learning, SARSA, and Deep Q-learning. Meanwhile, the transmit power is the continuous variable, and only policy-based algorithm can deal with the continuous optimization. Hence, how to jointly optimize the transmit power and channel assignment becomes a challenge. In some work, the transmit power is approximated to discrete power levels, and the user can only transmit by these presetting power levels <cit.>. However, discrete transmit power with large intervals means performance reduction.On the other hand, the complexity could be very high if the number of power levels is significant. To address these concerns, Yuan et al. <cit.> proposed a framework with a combination of value-based network and policy-based network. Similarly, Hehe et al. <cit.> also proposed a combination framework with different components to address the discrete user association problem and continuous power allocation problem. However, in such works the different networks are trained simultaneously, which leads to an unstable framework and makes the NNs hard to be trained and converge. §.§ Motivations and Contributions §.§.§ Federated Reinforcement Learning The primary obstacle faced by MARL algorithms is the instability and unpredictability of actions taken by other user equipment (UEs), resulting in an unstable environment that affects the convergence performance of MARL <cit.>. Consequently, a partially collaborative MARL structure with communication among UEs becomes necessary. In this structure, each agent can share its reward, RL model parameters, action, and state with other agents. Various collaborative RL algorithms may employ different information-sharing strategies. For instance, some collaborative MARL algorithms require agents to share their state and action information, while others necessitate the sharing of rewards. The training complexity and performance of a collaborative MARL algorithm are influenced by the data size that each agent needs to share. This issue becomes severer when combining neural networks (NN) with reinforcement learning. In a traditional centralized reinforcement algorithm, e.g., deep Q-network (DQN), the environment's interactive experiences and transitions are stored in the replay memory and utilized to train the DQN model. However, in multi-agent DQN, local observations fail to represent the global environment state, significantly diminishing the effectiveness of the replay memory. Although some solutions have been proposed to enable replay memory for MARL, these approaches lack scalability and fail to strike a suitable balance between signaling costs and performance. To address the issue of non-stationarity, it is necessary to ensure the sharing of essential information among UEs, which can be facilitated by federated learning <cit.>. Federated learning has demonstrated successful applications in tasks such as next-word prediction <cit.> and system-level design <cit.>. Specifically, federated reinforcement learning (FRL) enables UEs to individually explore the environment while collectively training a global model to benefit from each other's experiences. In comparison to MARL approaches, the FRL method enables UEs to exchange their experiences, thereby enhancing convergence performance <cit.>. This concept has inspired the work of Zhang et al.<cit.> in improving WiFi multiple access performance and Zhong et al.<cit.> in optimizing the placement of reconfigurable intelligent surfaces through the application of FRL. §.§.§ Meta Reinforcement Technique for Fast Adaptation and Robustness Another main challenge of the reinforcement learning algorithm is the demand for massive amounts of training data. Since the training data can only be acquired by interacting with the environment, the agent usually needs a long-term learning process until it can learn from a good policy. Moreover, using such a large amount of data to train an agent also may lead to overfitting and restrict the scalability of the trained model. In the scope of the wireless environment, the fast fading channels and unstable user distributions also put forward higher requirements on robustness and generalization ability. Particularly, the previous resource allocation algorithms usually set a fixed number of users, which makes the algorithm lack scalability to various wireless environments in practical implementation. Meta learning is designed to optimize the model parameters using less training data, such that a few gradient steps will produce a rapid adaptation performance on new tasks. During the meta learning training process, the model takes a little training data from different training tasks to initialize a general model, which reduces the model training steps significantly. The meta learning can be implemented in different ways. Wang et al. <cit.> and Duan et al. <cit.> have applied recurrent NN and the long short-term memory to integrate the previous experience into a hidden layer, and NNs have been adopted to learn the previous policy. Finn et al. <cit.> have leveraged the previous trajectories to update the NNs, and further extended the meta learning to reinforcement learning. In this paper, we consider the meta learning for initializing the NNs for MARL. In the scope of wireless communications, Yuan et al. <cit.> have adopted the meta reinforcement learning for different user distributions and confirm that the meta reinforcement leaning is a better initialization approach and can achieve better performance in new wireless environments. Another challenge caused by federated learning is the heterogeneity in systems and the non-identical data distributions in RL may slow down or even diverge the convergence of the local model. Inspired by the meta learning, Fallah et al. <cit.> have developed a combined model, in which the global training stage of the federated learning can be considered as the initialization of the model for meta learning, and the personalized federated learning stage can be seen as the adaptation stage for meta learning. Due to the similar mathematical expression, we can combine federated learning and meta learning naturally, so that training and adapting the models from statistically heterogeneous local RL replay memories. The aforementioned studies serve as valuable inspiration for us to explore the application of meta learning and FRL in addressing the challenges of channel assignment and power optimization. By leveraging these techniques, we aim to distribute the computational load to local user equipment (UEs), reduce transmission overhead, and foster collaboration among UEs. This paper introduces a novel framework that combines meta learning and FRL for distributed solutions to the channel assignment and power optimization problem. To the best of our knowledge, this is the first endeavor to integrate meta learning and FRL in the context of resource allocation in wireless communications. The contributions of this paper are summarized as follows: * A meta federated reinforcement learning framework, named MFRL, is proposed to jointly optimize the channel assignment and transmit power. The optimization is performed distributed at local UEs to lower the computational cost at the BS and the transmission overhead. * To improve the robustness of the proposed algorithm, we leverage the meta learning to initialize a general model, which can achieve fast adaptation to new resource allocation tasks and guarantee the robustness of the proposed MFRL framework. * To address the joint optimization of the discrete and continuous variables, we redesign the action space for the RL algorithm and design the corresponding proximal policy optimization (PPO) network to optimize the real-time resource allocation for each UE. * To explore the collaboration among cellular users, we propose a global reward regarding the sum EE and the successful allocation times for all UEs and apply the MFRL framework for enabling experience sharing among UEs. The remainder of the paper is organized as follows. In Section 2, the system model is presented and an EE maximization problem is formulated. The meta federated reinforcement learning algorithm is presented in Section 3. The proposed MFRL framework is illustrated in Section 4. The numerical results are illustrated in Section 5. The conclusion is drawn in Section 6. § SYSTEM MODEL In this paper, we assume that the set of UEs is denoted as UE = {UE_1, …, UE_I }, where I is the total number of UEs. For UE_i, the binary channel assignment vector is given by ρ_i = [ρ_i,1, …,ρ_i,n, …, ρ_i,N], i ∈ I, n∈ N, where N is the number of subchannels. The channel assignment parameter ρ_i,n=1 indicates that the n-th subchannel is allocated to UE_i, otherwise ρ_i,n = 0. Each UE can only accesses one channel, i.e., ∑ ^N_n=1ρ_i,n = 1, ∀ i ∈ I. Meanwhile, we consider a system with OFDMA, which means a channel can be accessed by at most one UE within a cluster, i.e., ∑^I_i=1ρ_i,n∈{0,1}, ∀ n ∈ N. In the case of each user equipment (UE), successful transmission with the base station (BS) is achieved when the UE accesses a specific subchannel without any other UEs within the same cluster accessing the same subchannel. Consequently, if each UE is allocated a channel that does not conflict with other UEs within the cluster, this allocation is considered a successful channel assignment. The pathloss of a common urban scenario with no line of sight link between UE_i and the BS can be denoted by <cit.> PL_i,n = 32.4 + 20 log_10 (f_n ) + 30 log_10 (d_i,n) (dB), where d_i,n represents the 3D distance between UE_i and the BS, f_n represents the carrier frequency for n-th subchannel. Considering the small-scale fading, the overall channel gain can be thereby denoted by h_i,n = 1/10^(PL_i,n/10)ψ m_n, where ψ is the log-normally distributed shadowing parameter. According to the aforementioned pathloss model, there is no line of sight between UEs and the BS, and m_n represents the Rayleigh fading power component of the n-th subchannel. Hence, the corresponding signal-to-noise ratio (SNR) between the BS and UE_i transmitting over the n-th subchannel is represented as γ_i,n = ρ_i,nh_i,np_i/N_n, where N_n = W_nσ_n^2 represents the Gaussian noise power on the n-th subchannel. The uplink EE for a successful channel assignment of UE_i is given by u_i,n = BW_n/p_ilog_2 (1+γ_i,n), if ∑ ^N_n=1ρ_i,n = 1; 0, else. where BW_n = k× b_n is the bandwidth of the n-th subchannel, k represents the number of subcarriers in each subchannel, and b_n denotes the subcarriers spacing for n-th subchannel. Meanwhile, for the unsuccessful assignment, i.e., the UE cannot access any subchannel, the uplink rate is set to 0 as it is unacceptable for the OFDMA system. The problem is formulated as \beginmaxi!\|l\| {ρ, p}∑^I_i=0 ∑^N_n=0 u_i,n (P0) p_i ≤p_max, ∀i ∈I γ_i,n > γ_min, ∀i ∈I ∑^N_n=1 ρ_i,n = 1 , ∀i ∈I ∑^I_i=1 ρ_i,n ∈{0, 1}, ∀n ∈N. \endmaxi! where p = {p_1, …, p_I } denotes the transmit power vector of UEs, γ_min represents the minimum SNR requirement to guarantee the QoS for UEs. Constraint (<ref>) and (<ref>) make the EE maximization problem a non-convex optimization problem and cannot be solved by mathematical convex optimization tools. In the literature, channel allocation problems are usually formed as linear sum assignment programming (LSAP) problems. To solve this problem, local CSI or the UE related information, e.g., location and velocity should be uploaded to the BS, then the centralized Hungarian algorithm <cit.> can be invoked to solve the problem with computational complexity O (I^3). The computational complexity grows exponentially with the number of UEs, and the mobility of UEs causes the variable CSI, which means the high-complexity algorithm needs to be executed frequently, leading to high transmission overhead and high computational pressure to the BS. Moreover, due to the transmission latency, the current optimized resource allocation scheme by the BS may not be optimal for UEs anymore, and a distributed and low complexity resource allocation approach on the UE side is more than desired. According to the constraint (<ref>) and (<ref>), each UE can only access one subchannel, and it is clear that the subchannel assignment is a discrete optimization problem. As aforementioned concerns in Section 1, it is hard to train different types of neural networks simultaneously. In another way, the discrete assignment problem can be described by different probabilities to choose different subchannels, and then one-dimensional discrete choice can be mapped to high-dimensional probability distributions. Overall, the joint optimization problem can be solved by a simple policy-based framework with a specific output design. § PROPOSED META FEDERATED REINFORCEMENT LEARNING FOR RESOURCE ALLOCATION In this section, we will first introduce the proposed MFRL framework from an overall perspective. Then we will design the NN structure to solve this EE maximization problem, and propose a meta reinforcement learning scheme for the NN initialization. We also demonstrate the meta-training and meta-adapting algorithms in detail. Finally, we will present the federated learning algorithm and procedures. The proposed algorithm starts from the meta-training for initializing the global generalized model at the BS. The initial model is meta-trained using the BS data set. After the initial global model is trained, it will be broadcast to the local UEs for adapting to the new environments. During the meta-adapting, i.e., the fine-tuning process, the local models are trained using a local database, i.e., local CSI, and the local models can be reunited as a global model so that the UEs could learn the knowledge from the experiences of other UEs and improve the global EE. One popular way is to average the distributed models and form a global model, which is called federated learning <cit.>. After the local models are averaged by the BS, it would be broadcast to the local UEs which will fine-tune the global model and adapt to the local scenarios. This process will be repeated until the meta-adaptation stage finishes. The overall procedure is shown in Fig. <ref> §.§ Neural Network Structure Design As the aforementioned description, the resource allocation problem can be modeled as a multi-agent markov decision process (MDP), which is mathematically expressed by a tuple, ⟨ I, O, A, R, P ⟩, where I is the number of agents, N = 1 degenerates to a single-agent MDP, O is the combination set of all observation state, A = A_0 ×…× A_I is the set of actions for each agent, R is the reward function, which is related to current observation O_t = {o_0, …, o_I}∈ O, A_t = {a_0, …, a_I}∈ A, and O_t+1∈ O. Transition probability function is defined as P : O× A→𝒫( O), with P(O_t+1\|O_t,A_t) being the probability of transitioning into state O_t+1 if the environment start in state O_t and take joint action A_t. One of the challenges of using deep reinforcement learning algorithms to solve the problem (P0) is that the resource allocation of the transmit power and subchannel association is the hybrid optimization of the continuous and discrete variables. As the analysis above, the discrete subchannel association parameter can be described by different probabilities to choose different subchannels, thus the discrete variable can be expressed by probability distributions on subchannels, which is generated by a categorical layer. Meanwhile, continuous power optimization is performed by the Gaussian layer, where the mean and variance of the transmit power can be trained. In fact, any deep reinforcement learning algorithms with continuous action space can be applied for training the proposed network structure. Specifically, we apply the PPO algorithm because of its ease of use and robustness, which make it the default algorithm by OpenAI <cit.>. It is noted that the NN architecture shares parameters between the policy and value function, so that the actor network and critic network share the underlying features in the NN, and simplify the meta learning initialization and model broadcast costs. The corresponding network structure of the local models is illustrated in Fig. <ref>. In this paper, we define the observation state at training step t for the UEs, which are considered as the agents in the MFRL framework, as o_t, i = {{h_i,n}_∀ n ∈ N, t} with dimension \|o_i\|, where t represents the number of epoch. The variables t can be treated as a low-dimensional fingerprint information to contain the policy of other agents <cit.>, thus enhancing the stationary and the convergence performance of the MFRL algorithm. The action a_t, i for the UE_i including the subchannel and the transmit power choice with dimension \|a\| = 2. The Actor network contains a categorical layer with N neurons to determine which subchannel the local UE should access. The continuous transmit power is optimized by a σ layer and a μ layer, and the power is sampled according to the probability distribution N(μ,σ^2). Since we aim to maximize the sum EE of the cellular network, here we design a global reward r_t, according to the joint action a_t such that encouraging collaboration of UEs. The global reward at training step t can be defined as r_t= ∑^I_i=0 r_i(t) if∑^I_i=0ρ_i,n∈{0, 1}, ∀ i ∈ I, ∀ n ∈ N; I^suc-I/I, Otherwise, where I^suc denotes the number of UEs that satisfies the subchannel assignment constraints, i.e., ∑^I_i=0ρ_i,n∈{0, 1}, ∀ n ∈ N. For the assignment that fails to meet the subchannel access requirements, a punishment is set to proportional to the number of failure UEs. Meanwhile, the reward for a successful subchannel assignment is expressed by r_i(t)= ξ u_i,n(t), if γ_i,n > γ_min; ξ u^p_max_i,n(t), Otherwise, where ξ is a constant coefficient, u^p_max_i,n(t) denotes the EE by the maximum transmit power, which means if the UE fails to meet the SNR constraint, it need to use the maximum transmit power to avoid transmission failure. The success rate of UE_i can be defined as η_i = β_i/T, where β_i represents the successful resource assignment counts for UE_i, and T represents the number of resource allocation counts since the initialization the system.[Please note that reward is designed as a sum of EE and the punishment, which makes it a dimensionless parameter and we only need to focus on the value of it.] The objective of the proposed MFRL framework is to enable UEs to learn a strategy that maximizes the discount reward, which can be expressed by R(τ) = ∑_t=0^∞ξ^t r_t, where τ = (o_0, a_0, ..., o_T+1) is a trajectory, T is the current timestamp, ξ∈ (0, 1) represents the discount rate, which denotes the impact of the future reward to the current action. §.§ Policy Gradient in Meta-training In the previous work <cit.>, the number of UEs in each cluster is fixed, and the training and testing performance are implemented in the same environment. Particularly, the local model is trained by each UE individually for the MFRL algorithm, which limits its application, making it hard to adapt to more complicated practical scenarios. The resource allocation model should have the ability to adapt and generalize to different wireless communication environments with different cluster sizes. Hence, the meta reinforcement learning algorithm can be considered to meet the requirement of the generalization. The meta learning can be implemented in different ways, and in this paper we apply the MAML method for reinforcement learning <cit.>. The meta-training process takes the experience from different tasks, i.e., the resource allocation for different cluster sizes, to initialize a model which can be adopted by UEs in different scenarios and achieve fast adaptation. To take the number of UEs into account, the local observation should include the total number of UEs, i.e., o_t, i = {{h_i,n}_∀ n ∈ N, I, t}. The task set of resource allocation for UEs is defined as T = { T^I_k}, ∀ k ∈ K, where K is the number of tasks, I_k is the number of UEs for task k. The meta-training process is implemented at the BS, which can use the previous resource allocation experience for different number of UEs to meta-train an initial model. At the end of each training epoch, the BS stores the transitions e^k_t, i = { (o^k_t, i, a^k_t, i, r^k_t, o^k_t+1, i)\| i = 0, 1, …, I_k-1} acquired from T^I_k in the central dataset. The transitions e_t, i = (o_t, i, a_t, i, r_t, o_t+1, i) are sampled from B for calculating the advantage function and the estimated state value function, which are introduced in the following paragraphs. The objective function for training the reinforcement model is to maximize the expected reward for each trajectory as J (π_θ) = 𝔼_τ∼π_θ(τ) [R(τ)]= ∫_τ P(τ\|π_θ) R(τ), where π_θ is the parameterized policy, P(τ\|π_θ) =P (o_0) ∏_t=0^T-1 P(o_t+1, i \| o_t,i, a_t,i) π_θ(a_t,i \| o_t,i) represents the probability of the trajectory τ, P(o_t+1, i \| o_t,i, a_t,i) is the state transformation probability, π_θ(a_t,i \| o_t,i) is the action choice probability, and P (o_0) is the probability of the initial state o_0. To optimize the policy, the policy gradient needs to be calculated, i.e., θ_j+1 = θ_j + α. ∇_θ J(π_θ) \|_θ_j, where α is the learning rate or the learning step. The gradient of the policy can be expressed by a general form as ∇_θJ(π_θ) = 𝔼_τ∼π_θ(τ)[∑_t=0^T∇_θlogπ_θ(a_t,i \|o_t,i) Φ_t,i], where Φ_t,i could be denoted as the action-value function Q^π_θ(o,a) = 𝔼_τ∼π_θ(τ)[R(τ)\| o_0 = o, a_0 = a], which is the expectation reward for taking action a at state o. Although we can use the action-value function to evaluate the action is good or bad, the action-value function Q^π_θ(o,a) relies on the state and the action, which means an optimal policy under a bad state may have less action-value than an arbitrary action under a better state. To address this issue, we need to eliminate the influence caused by the state. First, we prove that the state influence elimination will not affect the value of the policy gradient <cit.>. Given P^π_θ is a parameterized probability distribution over a random variable o, then 𝔼_o∼ P^π_θ[∇_θlog P^π_θ(o)] = 0. For all probability distributions, we have ∫_o P^π_θ(o) = 1. Take the gradient of both side ∇_θ∫_o P^π_θ(o) = ∇_θ1 = 0. Thus 𝔼_o∼ P^π_θ[∇_θlog P^π_θ(o)] = ∫_o P^π_θ(o) ∇_θlog P^π_θ(o) = ∫_o ∇_θ P^π_θ(o) = ∇_θ∫_o P^π_θ(o) = 0. According to Lemma <ref>, we can derive that for any function b(o_t) that only depends on the state, 𝔼_a ∼π_θ[∇_θlogπ_θ(a\|o) b(o)]= 0. Hence, it would cause the same expected value of the policy gradient ∇_θJ(π_θ) if we substitute the b(o) into the action-value function Q^π_θ(o,a). In fact, we can use the state-value function V^π_θ(o) which represents whether the state is good for a higher reward or not. Instead of comparing the action-value function Q^π_θ(o,a) of the action a directly, it is more reasonable to substitute the influence of the state into the action-value function. We define the substitution A^π_θ(o,a) = Q^π_θ(o,a) - V^π_θ(o) as the advantage function, which represents whether an action good or bad compared with other actions relative to the current policy. Hence, the value function Φ_t,i can be also denoted as Φ_t,i = Q^π_θ(o_t,i,a_t,i)-V^π_θ(o_t,i) = A^π_θ(o_t,i,a_t,i). §.§ Advantage Estimation and Loss Function Design Although we express the policy gradient by introducing the advantage function, the challenge is, the action-value function and the state-value function cannot be acquired directly from the experience e_t,i. Instead, the action-value function can be expressed by the temporal difference form <cit.> as Q^π_θ(o_t,i,a_t,i) = r_t + ξ V^π_θ(o_t+1, i). In deep reinforcement learning approaches, NNs can be used to estimate the state-value function as V̂^π_θ, then the estimated advantage function Â^π_θ(o_t,i,a_t,i) = δ^V_t,i = r_t + ξV̂^π_θ(o_t+1,i) -V̂^π_θ(o_t,i) can be derived. However, the bias for this estimation is high, which restricts the training and convergence performance. To overcome this issue, generalized advantage estimation (GAE) <cit.> can be applied to estimate the advantage function for multi-steps and strike a tradeoff between the bias and variance. The GAE advantage function is denoted by A^GAE(o_t,i,a_t,i) = ∑ _l=0^T-t(λξ)^lδ^V_t+l,i, where λ∈ (0,1] is the discount factor for reducing the variance of the future advantage estimation. The actor network is optimized by maximising L_AC = 𝔼_τ∼π_θ(τ)[ratio_t,i× A^GAE(o_t,i,a_t,i)], where ratio_t,i =π_θ(a_t,i \|o_t,i)/π_θ_old(a_t,i \|o_t,i) is the action step. However, too large action step could lead to an excessively large policy update, hence we can clip this step and restrict it. The clipped actor objective function is expressed by L^Clip_t = min( ratio_t,i× A^GAE(o_t,i,a_t,i), g(ϵ, A^GAE(o_t,i,a_t,i))), where g(ϵ, A) = (1 + ϵ) A, A ≥ 0; (1 - ϵ) A A < 0, in which the ϵ is a constant value representing the clip range. The clip operation have been proved to improve the robustness <cit.>. The loss L_CR for the critic network is to minimize the gap between the estimated state-value function and discount sum reward, which can be expressed by L^CR_t =‖ r_t + V̂^π_θ(o_t+1,i) - V̂^π_θ(o_t,i) ‖^ 2. Combining the objective of the actor network and critic network, we can express the overall objective as L=min_θ𝔼_t[L^Clip_t-c_1L^CR_t+c_2E_t], where E_t represents an entropy bonus to ensure sufficient exploration, θ is the weights for the PPO network, c_1 and c_2 are weight parameters for the estimation of value function and entropy, respectively. Then the initial model will be updated by the stochastic gradient (SG) ascent approach. The details of the meta-training algorithm is shown in Algorithm <ref>. §.§ Meta-Adapting Process Unlike the meta-training process where the BS stores the transitions and uses these experiences to train a global model, the local UE can train its own model based on its own observations and experience during the meta-adaptation process. Compared with supervised learning which requires sufficient data set and pre-knowledge of the system, the proposed MFRL framework can train the local model with the local CSI data which is required by interacting with the environment, thus not only offloading the computational pressure to the UEs, but also lower the transmission overhead significantly. As the local models are inherited from the global model, the network structure, the observation state space, the action, and the reward are defined the same as Section 3. Considering that the i-th UE interacts with the environment at adapting epoch j, i.e., observes the state o_j,i, and takes action according to current policy π(θ_j,i). Then the i-th UE receives the reward r_j and observes the next state o_j+1,i. The transition e_j,i = (o_j,i, a_j,i, r_j, o_j+1,i) is stored in its local memory M_i which can be sampled in the batch to train the local models. The advantage is estimated using the GAE method and the loss function is the same as the meta-training process. The details of the meta-adapting process are described in Algorithm <ref>. §.§ Global Averaging of Local Models Unlike the meta-training process that the BS uses the centralized replay memory that collects from all UEs to train the global model, the local UEs can only access their local memories during the meta-adaptation process, which affects the robustness of the local models when encountering unfamiliar scenarios. To enable the individual models at each UE can be benefited from other UEs, the federated learning technique can be applied. The local model is averaged to a global model, then the global model is broadcast to UEs and the UEs will continue to train the new global model locally. By averaging the models, each UE is able to benefit from the experience of other UEs, since the weights direct correspond to the experience and memory. Mathematically, the model averaging process at the central BS can be denoted as W = ∑^I_i=1 \| B_i\| W_i/∑^I_i=1 \| B_i\| , where \| B_i\| represents the number of number of elements in B_i. The average algorithm shows that the averaged model will learn more from the model with more training cases. However, in the proposed MFRL framework, we assume that UEs share the team stage reward, which means the replay memory of each UE has an equivalent size. To ensure that the averaged model can benefit from the model that caters to the needs of QoS, we further revised the averaging algorithm that considers the success rate, which is denoted by Ŵ = ∑^I_i=1η_i W_i/∑^I_i=1η_i, where η_i is the resource allocation success rate for UE_i as defined in Section 2. § NUMERICAL RESULTS We consider a communication scenario underlying a single cellular network. For the meta-training process, we adopt the urban micro (street canyon) scenario in <cit.>. For the meta-adaptation process, the pre-trained models are trained and fine-tuned in the indoor scenario, the urban macro scenario, and the rural macro scenario. For all of the scenarios, the BS is fixed at the center of the considered square. We also adopt the simulation assumptions in <cit.> to model the channels. To enable the mobility of UEs, we assume that the UEs can move with the speed from 0 meters per second (m/s) to 1 m/s within the square. Each subcarrier has Δ f = 2^ψ· 15 kHz spacing, where ψ denotes an integer. A resource block usually consists of 12 consecutive subcarriers <cit.>, hence we set the bandwidth set of the subchannels as [0.18, 0.18, 0.36, 0.36, 0.36, 0.72, 0.72, 0.72, 1.44, 1.44] M. The rest of the parameters of the proposed simulation environment are listed in Table <ref>. The network structure of local models is shown in Fig. <ref>. The state information is fed in two fully connected feed-forward hidden layers, which contain 512 and 256 neurons respectively. Then the PPO network diverges to actor networks and critic networks. The actor branch contains two layers for channel choice and power optimization independently, while the critic branch includes an additional hidden layer with 128 neurons, following which is the value layer for estimating the advantage function for the output of the actor network. The meta-training rate for different number of users is 5e^-7, while the learning rate for meta adaptation is 1e^-6. The meta learning rate is set relatively small to avoid the overfitting of the meta model for some specific tasks. The weight for the loss of the value function c_1 and entropy c_2 are set as 0.5 and 0.01, respectively. The sample batch size is 256, and the discount rate for the future reward ξ is set to 0.9. The discount factor for the advantage function λ = 0.98 in Eq. (11) is set according to <cit.>. To verify the performance of the proposed MFRL framework with the following benchmarks: * MRL: Meta reinforcement learning benchmark. The local models are pre-trained and inherited from the global model, but the local models are not averaged by federated learning. * FRL: Federated reinforcement leanring benchmark. The local models are trained from the random initialization and averaged by the federated learning every 100 episodes. * MFRL_early: The early model of the proposed MFRL framework. The models are stored at half of the meta-adaptation period, i.e., store the local models at 500 episodes to evaluate the fast-adaptation performance of the proposed framework at the early stage. * MARL: The multi-agent reinforcement learning benchmark <cit.>. The local models are trained from random initialization and are not averaged by the federated learning technique. Each UE learns the policy according to the local observations and receives the global reward, but cannot communicate the model with the centralized cloud or other UEs. Fig. <ref> demonstrates the reward for different tasks (with different amounts of users) during the meta-training process. Particularly, the meta reward is the sum of the reward of the resource allocation tasks for 2, 4, and 8 UEs in the urban micro scenario. The increase in the meta reward demonstrates the effectiveness of the meta-training. It is also noted that with the meta-training step increasing over 100 episodes, the sum reward keeps stable. This is because the meta-training process is to train a global and generalized model which can be adapted to different tasks, but the performance of the generalized model itself cannot be as well as the models for the specific tasks. Fig. <ref> shows the training reward comparison over different episodes of meta-training, from which we can see that the meta-training could lead to faster convergence and higher rewards. Due to the punishment, the reward for all schemes is low at the beginning of the training period. With the execution of the training progress, the proposed algorithms with meta learning can achieve faster convergence and higher training reward, while the conventional benchmark needs more iterations to find the appropriate actions to converge. The improved training performance verifies the fast adaptation by the meta learning is robust to different scenarios. To further verify the robustness of the trained local models, we set different simulation settings under each scenario. At each random testing user distribution, the system EE is averaged by 100 testing steps with fast-fading channel updates. Fig. <ref> illustrates the testing performance for 10 random user distributions. The proposed algorithm outperforms other reinforcement learning benchmarks in terms of average system EE. We also store the local models at 500 episodes to test the performance of the algorithms at the early training stage. As expected, the proposed MFRL framework outperforms the MRL and FRL algorithms. Moreover, even if MFRL_early models are only trained half of the whole training period, they still provide good performances compared with the models that are not pre-trained, which verifies the fast adaptation ascendancy of the meta learning. To evaluate the convergence speed and the stability of the policy, and verify the fast adaptation performance of the proposed MFRL framework, we use the policy entropy as the measure. The policy entropy is an dimensionless index in policy gradient based reinforcement learning algorithms, to measure the randomness of a policy. As shown in Fig. <ref>, the lower entropy of the MFRL algorithm verifies that meta learning can speed up the training process and achieve convergence earlier. The MFRL framework also achieves a similar lower entropy and faster convergence compared with the benchmarks in other scenarios, and the results are omitted due to space limitations. Fig. <ref> concludes the sum EE in different scenarios. The results are averaged according to 100 random user distributions. It is clear that the proposed MFRL framework achieves the highest sum EE in all of the scenarios, which verifies the robustness of the proposed scheme. Additionally, although the models for the MFRL_early benchmarks are trained half of the whole adapting period, it still achieves better performance compared with the FRL and MARL models. The MFRL framework and the FRL scheme enable the UEs to corporate with each others and benefit the local models, hence also improving the overall system EE. Fig. <ref> shows the testing sum EE of the system over a different number of users. Note that for different users, the training parameters may differ slightly for the best performance. It is obvious that as the number of UEs increases, more subchannels can be accessed and the sum system EE can be improved. However, the improvement slows down as the number of UEs increases, since the bandwidth of subchannels in the proposed scenario is not equal, and when the number of UEs is less than the subchannels, it would access the subchannel with larger bandwidth for higher EE. § CONCLUSION In this paper, a distributed energy-efficient resource allocation scheme was developed. The system energy efficiency was maximized by jointly optimizing the channel assignment and the transmit power of user equipments. The formulated non-convex problem was solved by the proposed robust meta federated reinforcement learning framework to overcome the challenge of the computational complexity at the base station and the transmission cost by the local data. Quantity analysis and numerical results showed that the meta training model has good generalization ability under different scenarios, even if the scenarios and tasks are different. Meanwhile, the combination of federated learning and meta learning with reinforcement learning enables the decentralized algorithm a better performance on convergence and robustness.
http://arxiv.org/abs/2307.01277v1	20230703180622	X-ray metal line emission from the hot circumgalactic medium: probing the effects of supermassive black hole feedback	[ "Nhut Truong", "Annalisa Pillepich", "Dylan Nelson", "Ákos Bogdán", "Gerrit Schellenberger", "Priyanka Chakraborty", "William R. Forman", "Ralph Kraft", "Maxim Markevitch", "Anna Ogorzalek", "Benjamin D. Oppenheimer", "Arnab Sarkar", "Sylvain Veilleux", "Mark Vogelsberger", "Q. Daniel Wan", "Norbert Werner", "Irina Zhuravleva", "John Zuhone" ]	astro-ph.GA	[ "astro-ph.GA", "astro-ph.CO" ]	firstpage–lastpage Metallicity Dependence of Molecular Cloud Hierarchical Structure at Early Evolutionary Stages [ August 1, 2023 ============================================================================================= We derive predictions from state-of-the-art cosmological galaxy simulations for the spatial distribution of the hot circumgalactic medium (CGM, [0.1-1]R_200c) through its emission lines in the X-ray soft band ([0.3-1.3] keV). In particular, we compare IllustrisTNG, EAGLE, and SIMBA and focus on galaxies with stellar mass 10^10-11.6 at z=0. The three simulation models return significantly different surface brightness radial profiles of prominent emission lines from ionized metals such as OVII(f), OVIII, and FeXVII as a function of galaxy mass. Likewise, the three simulations predict varying azimuthal distributions of line emission with respect to the galactic stellar planes, with IllustrisTNG predicting the strongest angular modulation of CGM physical properties at radial range ≳0.3-0.5 R_200c. This anisotropic signal is more prominent for higher-energy lines, where it can manifest as X-ray eROSITA-like bubbles. Despite different models of stellar and supermassive black hole (SMBH) feedback, the three simulations consistently predict a dichotomy between star-forming and quiescent galaxies at the Milky-Way and Andromeda mass range, where the former are X-ray brighter than the latter. This is a signature of SMBH-driven outflows, which are responsible for quenching star formation. Finally, we explore the prospect of testing these predictions with a microcalorimeter-based X-ray mission concept with a large field-of-view. Such a mission would probe the extended hot CGM via soft X-ray line emission, determine the physical properties of the CGM, including temperature, from the measurement of line ratios, and provide critical constraints on the efficiency and impact of SMBH feedback on the CGM. galaxies: evolution – galaxies: formation – galaxies: haloes – galaxies: circumgalactic medium – galaxies: supermassive black holes — X-ray: galaxies § INTRODUCTION In the standard ΛCDM model of structure formation, dark matter (DM) haloes form first and the baryonic component follows into these already established gravitational potential wells. Unlike DM, gas component is collisional, and cosmic gas infalling into high-mass halos is shock-heated to approximately the virial temperature, leading to the formation of hot gaseous atmospheres (, see for a recent review). These hot atmospheres – the circumgalactic medium (CGM) – play an essential role in regulating the growth and evolution of galaxies. However, despite being predicted long ago, observational detection of the extended hot CGM at galactic mass scales (M_200c≳10^12)[M_200c is defined as the total mass within the radius of R_200c, within which the average matter density is 200 times the cosmological critical density at a given redshift, namely it is defined by the relation: M_200c=4/3π×200ρ_c(z)R_200c^3.] remains difficult, if not elusive. Observationally, a broad, multi-wavelength view of the multi-phase CGM has been established (see e.g. for a general review). For individual objects, several works have reported the detection of diffuse, emitting gas in the X-ray, observed by Chandra and XMM-Newton. These observations have targeted both early-type galaxies (e.g. ) and late-type galaxies, at the Milky-Way (MW) mass range or below (e.g. ), and a few more massive merger galaxies (e.g. ). Nonetheless, due to the rapid drop of density and thus emissivity with distance, existing observations with CCD-based instruments such as Chandra/XMM-Newton primarily probe the X-ray emission within the central regions of halos (up to 60-70 kpc). Moreover, these CCD-based observations unable to yield substantial constraints on the thermodynamcis and metal content of the CGM. Emission from hot gas at larger radii is measurable by stacking X-ray photons at the positions of galaxies from large-scale surveys. <cit.> demonstrate the potential of this technique by combining hundreds of thousands of galaxies from local (z < 0.5) galaxy surveys, including the Sloan Digital Sky Survey (SDSS), and X-ray data from the ROSAT All-Sky Survey, finding extended X-ray emission down to galaxy stellar masses as low as ∼10^10.8. More recently, <cit.> and <cit.> use X-ray data from the eROSITA Final Equatorial Depth Survey (eFEDS). By stacking X-ray events around thousand of galaxies identified in the Galaxy And Mass Assembly (GAMA) and SDSS surveys, respectively, they characterize the radial profile of extended X-ray emission from the CGM. They both detect X-ray emission in the CGM out to galactocentric distances of ∼100 kpc for galaxies with ≲10^11. However, the quantitative result depends on details of the stacking methodology, including the selected galaxy sample, removal of point sources including background AGN, contaminating X-ray photons from PSF effects, and so on. Regardless, stacking can only measure the mean (averaged) profile of a galaxy population, which can be biased high by a few brightest galaxies. Additionally, these stacking experiments cannot directly constrain the physical properties of the CGM such as temperature or metal abundance. These limitations in current observations with X-ray CCD detectors of the hot CGM preclude a comprehensive understanding of its crucial role in galaxy formation and evolution. Theoretically, recent cosmological hydrodynamical simulations of galaxies predict that the CGM of massive galaxies (∼ 10^10-11.5) is strongly affected by feedback activity, especially due to energy released from supermassive black hole (SMBH) accretion. Three key phenomenological CGM features have been recently predicted and further quantified with modern galaxy simulations: * Diversity between star-forming versus quiescent galaxies in hot gas content. Recent studies based on the IllustrisTNG and EAGLE simulations suggest that central massive galaxies are quenched, at least in part, by ejective outflows produced by SMBH feedback (e.g. ). As a consequence, these simulations predict that star-forming galaxies are significantly more X-ray luminous compared to their quiescent counterparts at the transitional mass scale of ∼ 10^10.5-11.2 in stars (), and overall have different CGM properties <cit.>. Chandra observations of nearby galaxies tentatively support this finding (although see for a different result). However, <cit.> show that, as current X-ray observations are limited to the central region of galaxies (within at most a few times R_e), it is challenging to distinguish the X-ray emission from diffuse halo gas versus contamination from the hot interstellar medium and point-like sources such as X-ray binary stars, especially in star-forming galaxies. Simultaneously, recent stacking studies using eFEDS <cit.> find discrepant results for the X-ray surface brightness profiles of star-forming versus quiescent galaxies, highlighting the difficulty of this measurement. * Angular/azimuthal distribution of the CGM. SMBH feedback can cause an angular dependence in properties of the CGM around massive galaxies, such as temperature, density, and metallicity, with respect to the orientation of the central galaxy (). According to the IllustrisTNG simulations, this produces galaxy population-wide observable signatures such as broad-band X-ray luminosity or X-ray hardness, which could be detectable via stacking of eROSITA all-sky data. At the individual galaxy level, this angular dependence of the CGM can manifest itself even in the form of X-ray emitting bubbles, shells, and cavities in the CGM above and below the stellar disks of MW and Andromeda-like simulated galaxies, similar in morphology to the eROSITA/Fermi bubbles in our own Galaxy <cit.>. * Thermal content of the CGM. Observations (e.g. ) and simulations (e.g. ) both suggest that the CGM temperature in M_200c∼10^12 halos is higher than the self-similar prediction. This indicates that in addition to gravitational heating there is a contribution from other energy sources, i.e. SMBH feedback. However, current measurements of CGM temperature are mainly available for massive early-type galaxies, and limited to the hot ISM within their central regions (e.g. ). Extended emission from the hot CGM can also be observed in narrow band X-ray emission. The warm-hot phase of the CGM has a temperature T ∼ 10^5-7 K such that it emits abundantly in the soft X-ray band ([0.3-2.0] keV), where the emission is expected to be dominated by individual metal lines (). Metal line emission has the potential to probe the extended hot CGM without significant confusion due to foreground emission from the CGM of our Milky Way, unlike current low spectral resolution observations. With a high spectral resolution instrument (∼ eV), line emission from a distant galaxy is observable as it redshifts out of the energy range occupied by the same lines in the MW foreground (see Section <ref>). This technique will be possible with the next generation of X-ray instruments equipped with microcalorimeters, such as XRISM Resolve () and the mission concepts ATHENA X-IFU (), HUBS (), and Line Emission Mapper (LEM, ). In order to effectively study the CGM in nearby galaxies through line emission, we need both high (∼ eV) spectral resolution and a large (∼ 10s of arcmin) field of view. High spectral resolution is necessary to resolve individual lines, while a large field of view is required to map the CGM out to the virial radius with a single pointing. In this regards, ATHENA X-IFU and XRISM Resolve are limited by their small field of view, which are of 5 and 3 arcminutes, respectively (see Table 1 in for a detailed comparison between future X-ray spectrometers). Although this limitation can be overcome by observing galaxies with multiple pointings or at higher redshifts, it will be challenging to observe such galaxies given the needed observing time or the large cosmological dimming factor. This paper explores observational opportunities to uncover the connection between galactic feedback and CGM properties with the aforementioned Line Emission Mapper (LEM), an X-ray mission concept that has been recently developed with the main scientific goal to study the formation of cosmic structures (). LEM has 1-2 eV spectral resolution and a large field of view (30'x30'), making it an ideal instrument to observe the CGM in the local Universe. We especially provide quantitative theoretical predictions to probe the effects of SMBH-driven feedback on the spatial distribution and azimuthal anisotropy of the CGM, via narrow-band X-ray imaging and high-resolution X-ray spectroscopy. This work is based on the outcome of current state-of-art cosmological simulations including IllustrisTNG (), EAGLE (), and SIMBA (), which implement different models for SMBH feedback (see Section <ref>). Our main objective is therefore to examine the impact of SMBH feedback on the spatial distribution of line emission in the CGM, as predicted by these three simulations. In doing so we explore the potential that future microcalorimeter-based missions such as LEM have in constraining the physics of galactic formation, feedback, and quenching. The paper is arranged as follows. In Section <ref> we introduce the three simulations as well as our methodology for computing CGM observables. Section <ref> presents our main results and the comparison across the three models. In Section <ref> we present the key observational considerations for the CGM science case of LEM. Finally, we summarize and conclude in Section <ref>. § METHODOLOGY §.§ Cosmological galaxy formation simulations In this paper, we extract quantitative predictions from three current state-of-the-art cosmological simulation models for the formation and evolution of galaxies: TNG100 of the IllustrisTNG suite (TNG hereafter), EAGLE, and SIMBA.[The data of these simulation are all publicly available. Herein we use versions of EAGLE and SIMBA that have been re-processed to enable an apples-to-apples comparison with TNG. For all three simulations we therefore identify structures (i.e. haloes, subhaloes, and hence galaxies) and their properties in the same manner, with the data of EAGLE and SIMBA having been rewritten exactly in the TNG format <cit.>, enabling identical analysis routines.] These three simulations assume the ΛCDM cosmological model with similar parameters, and have similar simulated volumes. Below are brief descriptions of these simulation projects, which have been presented, analyzed and discussed in a large number of previous scientific works. Of relevance for this paper, all the considered simulation models evolve cosmic gas, cold dark matter, stellar populations and SMBHs from high redshift (z > 100) to the current epoch. TNG follows also the dynamics and amplification of magnetic fields. They all account for the heating and cooling of gas, assume a uniform UV/X-ray background arising since the epoch of Reionization, include numerical subgrid implementations for star formation, stellar evolution and chemical enrichment through SNIa, SNII and AGB channels, stellar feedback and AGN feedback, and are fully cosmological. The result of solving the coupled equations of gravity, (magneto)hydrodynamics, and galaxy astrophysics in a large cosmological volume is thousands of galaxies. These span a wide range of halo and stellar masses, stellar morphologies and cosmic environments, including their surrounding dark matter and gaseous haloes. As a cautionary note, these models do not simulate gas colder than 10^4 K (see e.g. for a review on this topic) and, among other processes, do not explicitly account for photon propagation and for the explicit radiation-gas interaction. * IllustrisTNG (hereafter TNG)[<https://www.tng-project.org/>] is a project of cosmological magneto-hydrodynamical simulations of galaxies and their evolution over cosmic time (). The TNG simulations were performed with the moving-mesh code () and with a galaxy formation model that includes a variety of astrophysical processes: see for a detailed description. TNG includes three main flagship simulations: TNG50, TNG100 and TNG300. Here we mostly use the TNG100 run (aka TNG100-1). TNG100 has a simulated volume of ∼(110.7 cMpc)^3 and baryon mass resolution m_ baryon=1.4×10^6. In Section <ref>, we also showcase galaxies simulated with TNG50, with 16 (2.5) times better mass (spatial) resolution than TNG100. * EAGLE[<https://eagle.strw.leidenuniv.nl/>] is also a suite of cosmological hydrodynamical simulations of galaxy formation and evolution (). The simulations are run with a modified version of the smoothed particle hydrodynamics (SPH) code (). Here we employ the flagship run referred as 'Ref-L0100N1504', which encompasses a comoving volume of about 100 Mpc per side and has a baryonic mass resolution of m_ baryon=1.81×10^6. * SIMBA[<http://simba.roe.ac.uk/>] is the next generation of the MUFASA cosmological galaxy formation simulations () and it is performed with GIZMO's meshless finite mass hydrodynamics (). We employ the main run of the SIMBA simulations that has a simulated box of ∼(147 Mpc)^3 and baryon mass resolution m_ baryon=1.82×10^7. Although they are run with different numerical codes for the solution of gravity and hydrodynamics, the key aspect making these three simulation projects highly complementary is their vastly different underlying implementations of the galaxy astrophysics processes. In particular and of relevance here, TNG, EAGLE and SIMBA rely on different physical models and numerical implementations of SMBH feedback. For example, EAGLE employs a single thermal channel, whereas in TNG and SIMBA different energy injection modes occur depending on SMBH accretion rate, including in both cases a kinetic energy injection. In EAGLE and TNG such energy is always distributed isotropically at the injection scales (≲ kpc) whereas SIMBA adopts bipolar jet-like kicks. Despite these differences, feedback from SMBHs is responsible for quenching star formation in massive galaxies in all the three simulation suites. It also drives gas out of the central regions of haloes <cit.>, and even far beyond haloes <cit.>, while heating gaseous halos and offsetting the cooling times of these CGM reservoirs <cit.>. In fact, SMBH-related model choices in the underlying galaxy formation models have been calibrated, either quantitatively or approximately, to reproduce reasonably realistic samples of galaxies, particularly with respect to observations of their stellar component in the local Universe. The degree to which such a model tuning has been performed varies enormously across the three simulation projects, which in fact produce, in detail, different galaxy populations <cit.>. On the other hand, none of these models have been calibrated with respect to detailed properties of the CGM or IGM, making their outcomes in this regime highly predictive <cit.>. §.§ Galaxy selection and definition of CGM For the analysis in this paper, we focus on simulated galaxies with stellar mass >10^10, and we select only galaxies which are the centrals of their host halos, i.e. excluding satellite galaxies. Throughout, we measure within twice the half-stellar mass radius (R_e). There are a few thousand galaxies at z=0 above this mass limit in each of the simulated volumes. As motivated in Section <ref>, we particularly aim to quantify the differences in the gas properties around galaxies at the Milky Way and Andromeda mass scale (= 10^10.5-11.2). This range encompasses the stellar mass estimates, including 2-σ and systematic uncertainties, of our Milky Way and Andromeda <cit.>, and thus represents a mass scale of reference and importance. Furthermore, ∼ 10^10-11 is the range where the galaxy population transitions from being dominated by star-forming galaxies to quenched systems, in both the real Universe and in the models <cit.>. As a result, in all three models, the physical properties of the circumgalactic gas around galaxies at this mass scale encode information about the effects and functioning of feedback energy injection from SMBHs. We elect to use a galaxy selection based on galaxy stellar mass because this is accessible via observations. As a result, differences among simulations will be, at least partially, driven also by underlying differences in the stellar-to-halo mass relation. We hence focus on the physical and observable properties of CGM gas. Throughout this paper, by CGM we mean all non-starforming gas within a 3-dimensional distance from a galaxy center of [0.1-1], where is the radius within which the average density is 200 times the present cosmological critical density. While the outer limit is approximately the virial radius of haloes, the inner limit (>0.1) is larger than twice the stellar effective radius for our sample, and is adopted to avoid the hot interstellar medium, whose physical interpretation requires more care. In all cases we include gas within each friends-of-friends (FoF) halo as determined by the halo finding algorithm, which is identical in all analyzed simulations. §.§ Computation of the gas thermodynamical properties and intrinsic X-ray emission For each galaxy, we extract CGM properties predicted by the simulations (density, temperature and metallicity) by neglecting star-forming gas cells/particles, e.g. with non-zero instantaneous star formation rate, but without imposing any additional cut on temperature or phase. Unless otherwise stated, gas temperature and metallicity are mass weighted and are meant to convey the true physical state of the gas. To connect to X-ray observations, we compute the intrinsic emission in the soft band from the diffuse gas, without accounting for observational effects nor telescope responses and without adding the possible contamination from point-like sources such as X-ray binaries and AGNs. In particular, we focus on either the full X-ray spectrum in the [0.3-1.3] or [0.3-2] keV bands (continuum + emission lines) or on a few selected soft X-ray emission lines from key metal species, as listed in Table <ref>. X-ray emission is computed using the simulated physical properties of the gas on a cell-by-cell basis. We pre-calculate emissivity tables using a single-temperature APEC model (version 3.0.9) implemented in the XSPEC package (). The APEC model assumes an optically-thin plasma in collisional ionization equilibrium (CIE). We use the density, temperature, and elemental abundances of each gas cell/particle to derive its emission. For the solar abundance values we employ <cit.>. Our calculations exclude the effects of photo-ionization, due to the presence of UV/X-ray photons, local or otherwise. We also focus on non-resonant lines so that we can safely neglect X-ray resonant scattering <cit.>. In Fig. <ref>, we show the emissivity as a function of temperature for the 9 lines listed in Table <ref> assuming an APEC model. The emissivity of selected lines peaks in the temperature range of T∼10^5.5-7.0 K, which corresponds to virial temperatures of halos across a broad mass range of M_200c∼10^11.5-13.7M_⊙. When quoting the emission from specific metal lines, the emission is extracted from an energy range of ∼0.4 eV, while mock spectra are shown at a resolution of ∼2 eV. The total X-ray emission from a galaxy, from its CGM or a portion thereof, is obtained by summing up the contribution of all gas cells/particles in the given region. § PREDICTIONS FOR SOFT X-RAY LINE EMISSION OF THE CGM AT Z=0 §.§ From large spatial scales to the haloes Fig. <ref> shows the large-scale distribution of OVIII line emission from TNG100, EAGLE, and SIMBA at redshift z=0. The three simulations consistently predict that, as expected, most of the OVIII emission concentrates within gravitationally-bound DM haloes, marked by the bright regions in the maps, where most hot gas is also found. However, they differ significantly from each other on how distant from the halo centers the emission extends. As clearly visible, the SIMBA simulation predicts significantly more extended OVIII emission outside of haloes in comparison to TNG100 and EAGLE. This is qualitatively consistent with the finding that different feedback models lead to different baryon re-distribution within and beyond haloes: the `closure radius' in SIMBA is much larger than in the other considered models (), because of the differently far-reaching effects of feedback <cit.>. To better quantify the OVIII emission from haloes, in Fig. <ref> we show the OVIII luminosity as a function of central galaxy stellar mass for the three simulations. The luminosity is measured within the radial range [0.1-1] for each galaxy, representing the integrated luminosity of the CGM. Solid curves denote the median across galaxies and the shaded areas visualize the galaxy-to-galaxy variation at fixed galaxy stellar mass. Across the considered mass range, the OVIII CGM luminosity predicted by SIMBA is consistently lower than the other two simulations, especially at the low-mass end. The differences between TNG100 and EAGLE depend on galaxy stellar mass, and is larger for high-mass haloes: for galaxies more massive than the Milky Way, EAGLE predicts OVIII brighter haloes than TNG100, whereas the opposite is true at the lower-mass end. The predictions of Fig. <ref> are the result of the different physical properties of the CGM gas and hence depend on the typical density, temperature and Oxygen abundance in the gaseous haloes and on how these change, on average, with halo mass. Overall, CGM properties depend on the combined effects of hierarchical assembly of structures and feedback, which unfold differently in the different simulations and differently so in galaxies with different halo masses, SMBHs, etc. §.§ Most important soft X-ray emission lines in the CGM The OVIII line at 18.9709Å of the previous Section is only one of the prominent metal lines at soft X-ray wavelengths that characterize the CGM of normal galaxies according to current cosmological simulations. Fig. <ref>, top panel, shows the soft X-ray spectra from the CGM of all TNG100 galaxies in the 10^10-12 stellar mass range. Each horizontal row in the figure represents one galaxy and its spectrum, whereby emission at each energy is shown via the colorbar. Across the considered mass range, there are several prominent metal lines, whose intensity generally depend on galaxy mass. For example, according to TNG100, high-energy lines such as FeXVIII or NeX mostly appear in massive galaxies (≳11.5), while lower-energy lines such as FeXVII, OVIII, or OVII (triplets) are visible virtually at all considered mass scale. This finding is expected as high-energy ions are primarily associated with high-temperature CGM, as indicated by Fig. <ref>, which is more abundant in massive galaxies. In low-mass galaxies the CGM temperature is not high enough to effectively ionise highly-charged ions, limiting their line emission. A qualitatively-similar picture also holds in EAGLE and SIMBA, albeit with different quantitative predictions, as seen in Fig. <ref> for the case of OVIII. In the bottom panel of Fig. <ref> we show that the soft X-ray emission from the CGM of galaxies with stellar mass below a few 10^11 is strongly dominated by metal lines. This is quantified with TNG100, but holds also for EAGLE and SIMBA. The contribution of nine prominent metal lines to the CGM X-ray emission relative to the broad-band emission (in [0.3-2] keV) is given as a function of galaxy stellar mass: this varies from a few percent to over 10 per cent in the case of the resonant OVII line, the brightest line below = 10^11M_⊙ among the considered lines. Low-energy lines such as CVI, NVII, OVII are more prominent in low-mass galaxies (∼10^10M_⊙), whereas the contribution of higher-energy lines such as OVIII, FeXVII, or NeX peaks at a higher mass scale (≳10^11M_⊙). Notably, in galaxies with stellar mass in the MW-mass range the combined emission from CVI, OVII(f), OVIII, and FeXVII lines approaches the level of the continuum contribution, accounting for nearly 20% of the total broad-band emission. These lines are promising targets for LEM detection of the CGM in MW-mass range (See Section <ref>). §.§ Radial dependence §.§.§ Physical properties of the gas. We turn to radial trends, and inspect the thermodynamics as well as metal content of the CGM predicted by TNG100, EAGLE and SIMBA. In Fig. <ref> we show the radial profiles of the hot gas surface density, temperature, and metallicity in 4 different stellar mass bins, from left to right. Overall, the three simulations produce significant diversity in the radial profiles of gas properties across the considered galactic mass range (10^10</<10^11.6), most notably in gas column density. The SIMBA simulation consistently has lower gas density profiles compared to the other two simulations across the considered radial and mass range. For instance, at the MW-mass range (=10^10.8-11.2) and at the radius r=0.5R_200c, SIMBA Σ_gas profile is below TNG100 (EAGLE) by a factor of ∼3 (∼6). Comparing TNG100 and EAGLE, the former has a larger gas density profile at the low-mass end (<10^10.4) but smaller densities at the high-mass end (>10^11.2). At the Milky Way-mass range, the TNG100 density profile is comparable within the galaxy central region (r<0.1R_200c), but smaller at larger radii, in comparison to the EAGLE profile. Concerning the shape i.e. slope, TNG100 produces significantly steeper profiles compared to EAGLE and SIMBA, in particular at the low-mass end (<10^10.8). For the mass-weighted temperature profiles, the differences between the three simulations are less significant than in gas density, especially at large radii and at the high-mass end where the three simulations produce consistent profiles. In low-mass galaxies (below the MW-mass range) and within small radii (r≲0.5 R_200c), EAGLE has flatter temperature profiles compared to the other simulations. For the mass-weighted metallicity profiles, TNG100 and EAGLE profiles are broadly consistent within the sample uncertainty, whereas SIMBA has more metal-enriched CGM than the other two simulations (within a factor of 3) especially for galaxies with < 10^11.0M_⊙. It is worth noticing that SIMBA produces a significantly larger scatter in gas properties than the other two simulations, especially at the low-mass end. For example, the average scatter in the SIMBA gas column density at the MW-mass range is about 0.5 dex in comparison to 0.1-0.2 dex in TNG100 or EAGLE. We note that while the three simulations are designed to reproduce realistic samples of galaxies, the degree to which they do so differs. In addition, they predict considerable variation in in the thermodynamics and metal content of the CGM, due in large part to their distinctive models for stellar and SMBH feedback. To zeroth order, stellar feedback serves as the dominant channel in galaxies with mass substantially below the MW mass range, whereas SMBH feedback dominates in galaxies within or above the MW mass range (see e.g. ). §.§.§ Observable properties of the gas. The diversity in the gas properties across the three simulations is reflected in the line emission profiles. In Fig. <ref> we show the surface brightness radial profiles for a selected sample of four metal lines from low to high energy: CV, OVIIf, OVIII, and FeXVII. The results for other lines in the soft band ([0.3-1.3] keV) are qualitatively similar. These lines, especially OVIIf, OVIII, and FeXVII, are crucial for probing the CGM in nearby galaxies (e.g. z∼0.01) through LEM observations (see Section <ref> for more details). Overall, SIMBA predicts significantly less line emission compared to TNG100 and EAGLE, which are more similar depending on galaxy mass. Moreover, at the low-mass end the line emission profiles in TNG100 are significantly steeper than the EAGLE profiles. These results emphasize the observational manifestation of the diversity in the gas density profiles predicted by the three simulations. In Fig. <ref>, we also illustrate the LEM detectability threshold for three metal lines: OVIIf, OVIII, and FeXVII. To do so, we show the 20 per cent level of the background+foreground (BG+FG), and use it as an approximate detection threshold.[We refer to a companion paper by Schellenberger et al. (in prep) for a systematic study on the LEM detectability.] The LEM BG+FG level is derived assuming that each galaxy is observed at z=0.01, and the extent of the horizontal lines indicate the galactocentric distance covered by the LEM field of view, which is 30 arcmin. For galaxies more massive than MW/M31 (≳10^11.2M_⊙), a single LEM pointing will cover a spherical region out to r≲0.4 R_200c, whereas for less massive galaxies (≲10^11.0M_⊙), LEM can comfortably cover the CGM out to R_500c or even in a single pointing. Detectability for LEM depends on various factors, such as the line energy, the considered mass range, and the simulation model. For instance, for galaxies in the MW/M31 mass range (middle columns of Figs. <ref> and <ref>), SIMBA predicts that LEM could detect the CGM emission in OVIIf or OVIII only out to ∼0.3 R_200c. On the other hand, TNG100 and EAGLE predict that LEM will detect OVIIf or OVIII line emission out to ∼0.6-0.7 R_200c, which is similar to R_500c for galaxies at this mass range. §.§ Dichotomy between star-forming versus quiescent galaxies in X-ray line emission We now investigate the diversity between star-forming and quiescent galaxies. To do so we restrict our study to a sample of galaxies within a narrow stellar mass range: =10^10.7±0.1, which corresponds approximately the MW-mass range. At this mass scale, galaxies can be both star-forming as well as quenched in all three simulations, as well as in the Universe <cit.>. To identify these two classes we use the specific star formation rate sSFR=SFR/, where SFR is measured within the aperture of 2R_e and is the stellar mass within the same aperture. The galaxies are subdivided into 2 subsamples: star-forming versus quiescent galaxies, with the division lines specified as sSFR=10^-11yr^-1. At these redshifts the exact definition of `quenched' is unimportant. Different tracers for SFR e.g. instantaneous vs. averaged over the last 100-200 million years result in negligible differences in quenched fractions <cit.>. In Fig. <ref> we show the X-ray line emission from the CGM for star-forming versus quiescent galaxies across the three simulations. After dividing the sample into the two populations, we find that the star-forming population is significantly more luminous in line emission compared to the quenched counterparts <cit.>. Remarkably, at the MW/M31 transitional mass scale, star-forming galaxies are X-ray brighter than quenched ones for each line we consider and for all three simulations, despite the vastly different implementations of SMBH feedback and of the quenching mechanisms. However, there are also more subtle quantitative differences. The difference between the two galaxy populations is most significant in SIMBA, and in OVIIf. For OVIIf in particular, the three simulations agree that the dichotomy is strongest at small radii, for example at r∼0.1R_200c the difference between star-forming and quiescent OVIIf emission is 1-2 orders of magnitudes. However, they differ in the extent out to which the dichotomy persists. TNG100 and EAGLE predict that the dichotomy can be seen out to r∼0.7-0.8 R_200c, while in SIMBA the dichotomy exists all the way out to . §.§.§ Physical origin of the CGM X-ray dichotomy between star-forming and quiescent galaxies. To identify the physical origin of this dichotomy, we quantify thermodynamical and metallicity differences between the two galaxy populations (shown in Fig. <ref>). There are a number of qualitative similarities among the three simulations at fixed galaxy stellar mass: i) star-forming galaxies contain more CGM gas compared to quiescent galaxies; ii) their CGM temperature is lower than of quiescent counterparts; and iii) their CGM is slightly more metal enriched compared to the quenched galaxies on average, although this feature is less clear in the SIMBA simulation. These trends suggest that the line-emission dichotomy between star-forming versus quiescent galaxies is mainly due to the offset in the gas content between the two populations. The difference in metallicity can in principle also increase line emission from star-forming galaxies, thereby partly explaining the emission dichotomy, however the density effect dominates. Several studies have reported that SMBH feedback is the primary mechanism for quenching star formation in both TNG100 and EAGLE simulations (e.g. ). Quenching is chiefly driven by ejection of gas from the central region of galaxies caused by feedback from SMBH, which leads to a depletion of gas in quiescent galaxies compared to their star-forming counterparts in the transitional mass range <cit.>. To further explore this phenomenon, we investigate the impact of SMBH feedback on CGM gas content in the three simulations by examining the relationship between the gas content and central SMBH mass, which serves as a proxy for the amount of feedback energy injected in surrounding environment. The key idea for interpreting the diagnostics below is that, in all three simulations, galaxies would continue to form stars in the absence of SMBH feedback <cit.>. Thus, the differences in the physical and observable properties of the CGM between quiescent and star-forming galaxies are due to the impact of SMBH feedback. We refer the reader to previous investigations about the interrelation between the star formation with SMBH feedback (), the effects of SMBH feedback on the CGM properties (), and how the X-ray emission of the gaseous halo is impacted by different models for SMBH and stellar feedback <cit.>. In the top row of Fig.<ref>, we show the radial profile of the gas column density for the sample of MW-like galaxies, dividing it into three subsets based on percentiles of SMBH mass. Overplotted are also the loci of star-forming and quiescent galaxies as per Fig. <ref>. We find that galaxies with higher M_BH percentiles have lower density profiles compared to those with smaller SMBH masses, indicating an anti-correlation between the CGM gas content and the SMBH feedback activity integrated across time, of which SMBH mass is a proxy. The significance and the radial extent of this anti-correlation varies among the three simulations. In the TNG100 and EAGLE simulations the anti-correlation is mostly found within 0.3-0.5R_200c, while in SIMBA it extends all the way to . All this is consistent with the finding that SMBH mass (and not SMBH accretion rate, see below) is unanimously the most predictive parameter of central galaxy quenching at all epochs, in both current galaxy simulations and in observations <cit.>. The bottom row of Fig. <ref> explores how halo gas density varies with the amount of feedback energy that is injected by the central SMBH into the surrounding gas, for TNG100 (see a similar quantification in Figure 8 of ). On the left panel, we show the dependence of the gas density profile on the total amount of kinetic feedback energy that is injected by the central SMBH over its lifetime. TNG100 galaxies that have a larger amount of accumulated SMBH feedback energy have lower densities, i.e. contain less CGM gas. On the right panel, we show the dependence of the gas density profiles on the instantaneous accretion rate onto SMBH. In the case of the instantaneous SMBH accretion rate (or luminosity), the trend is reversed: this is expected in the context of the considered simulations, as in the TNG model (as well as in SIMBA) SMBHs in the high accretion rate mode (yellow) exert less efficient feedback (thermal mode) in comparison to the low accretion rate mode i.e. kinetic wind mode (purple). That is, the instantaneous accretion rate is not expected to explain the rarefaction of halo gas density for quenched galaxies. Rather, it is the energy injected in the kinetic wind mode, which is able to efficiently couple to the surrounding gas, which impacts and thus correlates with global CGM properties <cit.>. Overall, our results suggest the picture that SMBH feedback has an ejective effect on the CGM, as has been suggested and demonstrated in earlier studies of TNG100 and EAGLE simulations. This effect is also observed in the SIMBA simulation, highlighting a commonality. Despite the differences in their implementations of SMBH growth and feedback, the three simulations all produce a large-scale ejective effect via SMBH feedback. This leads to the dichotomy between star-forming and quiescent galaxies in X-ray line emission. This prediction is in line with previous studies that found a dichotomy in broad band X-ray emission for the inner halos in TNG100 <cit.> and EAGLE <cit.>, which also manifests as a dichotomy in OVI column density i.e. tracing the warm-hot phase of the CGM <cit.>. §.§ Azimuthal dependence We now explore the azimuthal distribution of the CGM properties, which has been found to exhibit striking anisotropic features in a number of recent simulation studies <cit.>. In general, there is a difference in the CGM physical properties along the minor versus major axes, i.e. as a function of azimuthal angle with respect to the orientation of the stellar body of the central galaxy. We evaluate the angular anisotropy of X-ray line emission with a sample of simulated galaxies with stellar masses within the MW mass range, =10^10.7±0.1. Surface brightness profiles of CGM line emission are computed for separate regions along the minor and major axis. The minor axis is defined based on the total angular momentum of stellar particles within twice the half-stellar mass radius (R_e), and the CGM is subdivided into evenly-sized quadrants, with two quadrants along the minor axis and two quadrants along the major axis. The profiles are computed along each axis using the two quadrants along that axis. In Fig. <ref> we present a comparison between the surface brightness profiles computed along the minor axis versus those computed along the major axis in the three simulations. All three simulations predict that the emission in low-energy lines, such as CV, NVII, OVIIf, is largely isotropic, while the emission in higher-energy lines, including OVIII, FeXVII and NeX, is enhanced along the minor axis. However, the level of this CGM angular dependence varies quantitatively among the three simulations. For instance, TNG100 and SIMBA predict that the emission in FeXVII at R_500c is significantly higher along the minor axis – the minor-to-major ratio is a factor of ∼ 3 – while the corresponding ratio in EAGLE is approximately half that value. The increased emission of high-energy lines along the minor axis may be due to the higher CGM temperature along that axis, caused by SMBH feedback <cit.>. The outflows driven by SMBH feedback can evacuate, heat, and metal enrich the gas along the way, and these outflows occur preferentially along the minor axis, where the outflows propagate along the path of least resistance <cit.>. However, the predicted level of the anisotropy among the simulations varies, and we speculate this is due to two main factors. First, the efficiency of SMBH feedback in driving outflows: the TNG100 and SIMBA models for SMBH feedback appear to be more efficient in this regard than EAGLE model, which is more efficient in heating up the gas at small radii (e.g r<0.3R_200c, as shown in Fig. <ref>). Second, the degree of collimation of SMBH-driven outflows: SIMBA injects SMBH feedback energy explicitly along a bi-polar direction, whereas TNG100 and EAGLE inject all (time-averaged) SMBH energy isotropically. Together with the morphological distribution of gas in the surroundings of SMBHs, this undoubtedly affects the degree of outflow collimation, and hence the degree of anisotropy arising in the CGM. Finally, we also detect an inverse anisotropy of greater emission along the major axis, in the interior (r≲0.3 R_200c) of the TNG100 halos, and to a lesser extent in the EAGLE and SIMBA halos. The major axis anisotropy in TNG100 is most pronounced for lower energy lines, implying an augmented presence of cooler gas in this direction. <cit.> found that EAGLE star-forming galaxies above M_=10^10.7 exhibit a major axis enhancement inside 30 kpc, which corresponds to 0.1 R_200c, and is only weakly apparent just above this radius. This elevated major axis emission may be associated with the accretion of ≥ 10^6 K gas onto disc galaxies <cit.>, or the overall density structure of the inner disk <cit.>, and is explored in more detail for LEM in (Zuhone et al. in prep). § OBSERVABILITY CONSIDERATIONS To potentially distinguish between the model differences highlighted previously, we proceed to explore the prospects of observationally detecting X-ray line emission with the LEM mission. Compared to other planned X-ray missions with microcalorimeters onboard, such as XRISM and ATHENA, LEM has a significantly larger field of view and a greater grasp, making it ideal for probing physical properties of the CGM in nearby galaxies <cit.>. Our objective is not to conduct a comprehensive assessment of the LEM capabilities, which would necessitate analysis of LEM mock observations (Schellenberger et al. in prep). Rather, we aim to point out potential observations that LEM can make to illuminate the physics of feedback as traced through CGM observables. §.§ Observing galaxies in narrow X-ray bands To illustrate the advantage of observing the CGM via line emission, we compare the intrinsic spectrum of an extragalactic CGM with the emission spectrum of the Milky Way CGM. The latter is the main foreground component that contaminates the intrinsic CGM emission of an extragalactic galaxy. In Fig. <ref> we show X-ray spectra of different components: the cosmic X-ray background (CXB) + Milky-Way foreground, and the intrinsic CGM emission from galaxies of MW-like mass from TNG100, EAGLE, and SIMBA. The CXB background is estimated assuming emission from a power-law spectral source with a spectral index of 1.47 (see e.g. ). The MW foreground emission is modelled as a sum of 2 thermal components: i) the local hot bubble as a thermal plasma with k_BT∼0.99 keV and metallicity z∼1.0 Z_⊙; ii) the hot halo as a 2-temperature plasma with k_BT∼0.225 keV and k_BT∼0.7 keV. Both are assumed to have a metallicity of Z∼1.0 Z_⊙ (see e.g. and ). The CGM emission of external galaxies is taken from a thin annulus at r=(0.5±0.1)R_ 200c and computed for a sample of simulated galaxies with stellar mass =10^10.6-10.8 placed at a redshift z=0.01. Of these, we show the sample-median spectra of their CGM emission predicted by each of the three simulations plus the foreground (MW-CGM) and background (CXB) emission. The total emission is largely dominated by the CXB background and the MW-CGM foreground aross the soft X-ray band. However, thanks to the line redshifting, the intrinsic CGM emission is visible at certain lines including CVI, OVIIf, OVIII, and FeXVII (denoted by shaded areas in the plot). The strength of those lines varies between the simulations. For instance, as expected from the previous figures, the OVIIf strength in the EAGLE simulation is the highest and it is larger than the CXB background+MW-CGM foreground level by a factor of ∼4. The TNG100 line strength is about 50 per cent of the background+foreground level. SIMBA produces almost no significant OVIIf line compared to the background+foreground level (a few percent). This result implies that the detectability of CGM in emission via metal lines strongly depends on the underlying simulation model. In this illustrating exercise our primary focus is to study galaxies at a redshift of z=0.01, where the redshifting of lines starts to separate few of them from the MW foreground. Observations at higher redshifts could reveal additional lines. For instance, observations of galaxies at z=0.035 could unveil the entire OVII triplets. However, this would be at the expense of a corresponding decrease in surface brightness. On the other hand, observing galaxies at lower redshifts would limit the number of detectable lines. Nonetheless, such scenario may permit studies of the inner part of the CGM, since its X-ray emission could be bright enough to overcome the MW foreground. Such observations would be useful to study the dynamical and chemical properties of the inner CGM as well as its interplay with the galactic disks. §.§ The dichotomy between star-forming and quiescent galaxies As shown in Section <ref>, there is a predicted dichotomy between star-forming versus quiescent galaxies in intrinsic X-ray line emission. With a future large field of view X-ray telescope like LEM, this prediction can be observationally tested via surface brightness mapping on a sample of galaxies each imaged individually. For illustration, in Fig. <ref> and Fig. <ref> we show surface brightness maps of OVIIf for a set of 30 star-forming and 30 quiescent TNG100 galaxies, respectively. Overall, it is visually clear from the maps that in star-forming galaxies, the OVII emission is remarkably higher than quiescent counterparts. More importantly, the maps clearly show that while star-forming galaxies have centrally peaked distributions of OVIII abundance, the quiescent counterparts exhibit much less concentrated distributions. The contrast between the two populations is a visual reflection of the ejective effect of SMBH feedback at 100-kpc scales, as discussed in Section <ref>. §.§ Detecting X-ray bubbles in the CGM The result presented in Section <ref> suggests that the CGM exhibits a notable degree of anisotropy in its line emission, with an increasing level of angular dependence at higher energies. More specifically, the three simulations predict that the emission of heavy-element lines, such as FeXVII and NeX, is more pronounced along the minor axis compared to the major axis of the central galaxy's stellar body. We note that this result is a statistical finding, that is manifest at the galaxy population level. In this Section, our objective is to explore how the anisotropic signals might be observed at the individual galaxy level. For this purpose, we utilise simulated galaxies taken from TNG model, as it predicts the most robust signal among the three simulation especially at extended radii. Additionally, we employ the TNG50 simulation <cit.> due to its superior resolution among TNG simulations (2.5 times better spatial resolution than TNG100), making it ideally suited for examining the spatial distribution of the CGM. To explore whether this anisotropic signal can be seen also for individual galaxies, we again use a sample of MW-like galaxies with stellar masses ∼10^10.7 M_⊙, in this case from the TNG50 simulation. In Fig. <ref> we show X-ray surface brightness maps of a selection of 6 TNG50 MW-like galaxies that exhibit bubbles in their CGM reminiscent of the eROSITA/Fermi bubbles in the Milky Way <cit.>. A comprehensive study of X-ray bubbles in TNG50 can be found in <cit.>. The maps are presented for an edge-on orientation and displayed in various lines, from low to high energy. Each map covers an area of 200 kpc per side, which falls well within the LEM field of view when observing a galaxy at redshift z=0.01. It is evident from the maps that the bubbles appear to be more prominent at progressively high energy lines. The existence of the bubbles is a strong manifestation of the anisotropic signal in CGM line emission, but at the individual galaxy level. However, the presence of an anisotropy signal does not necessarily imply the existence of bubbles, as the latter requires a sufficiently strong outflow to create shock fronts, which are the defining boundaries where a discontinuity in gas temperature and other gas properties occurs (see e.g. Figure 6 in ). The observation of X-ray bubbles beyond our own Milky Way <cit.> will be an important target for LEM <cit.>. The presence of bubbles provides direct, spatially resolved evidence for galactic feedback processes impacting the CGM. The morphology and extent of bubbles encodes information on the physical nature of feedback processes operating within the CGM (see also for a comprehensive discussion). Therefore, the detection (or non-detection) of these X-ray bubbles represents a critical test for current models of galaxy formation. §.§ Inferring gas temperature from line ratios As soft X-ray lines of different transition energies depend on the temperature of the emitting gas, the ratios of different emission lines constraints the thermal properties of the CGM. That is, emission line ratios are an observable proxy for temperature (e.g. ). To study this probe, we take the single TNG50 galaxy for which X-ray emission maps were shown in Fig. <ref> (ID 535410, third row from the top). In the top left panel of Fig. <ref> we present the map of the OVIII/OVIIf ratio for this galaxy, which is oriented edge-on and covers an area similar to that of the maps in Fig. <ref>. It is apparent that the line ratio map qualitatively resembles the emission maps in heavy-element lines such as FeXVII and NeX. Specifically, there are dome-like features located above and below the galactic disk, within which the line ratio is significantly higher than surrounding values. Additionally, we find that the ratio map correlates strongly with the map of intrinsic mass-weighted gas temperature (not shown). To examine how accurately the line ratio reflects the CGM temperature, we show in the top right panel of Fig. <ref> the line ratio-temperature diagram for all individual pixels of the map. We weight by gas mass T_mw (blue), OVIIf emission (orange), and OVIII emission (green). In addition, the predicted relation between the OVIII/OVIIf ratio and gas temperature based on the APEC 1-temperature model is shown as the solid gray line in the plot. The plot reveals that the APEC-predicted temperature, which is derived from the OVIII/OVIIf ratio, overestimates the gas mass-weighted temperature, and underestimates the OVIII emission-weighted temperature, by roughly 0.2 dex in each case. On the other hand, the APEC-predicted temperature tracks the OVIIf emission-weighted estimate reasonably well. This result arises from the multi-temperature nature of the CGM, also within small probed areas, by biasing T_mw low, whereas the line ratio OVIII/OVIIf is mainly sensitive to the gas component responsible for most of the OVIIf emission. In the bottom panels of Fig. <ref> we show the line emission (orange and green), as well as gas mass (blue), versus temperature for gas cells obtained from the two extraction regions indicated by star symbols on the ratio map, one along the minor axis and the other along the major axis, each region covers a (10kpc x 10kpc x 400kpc) volume. We find that the difference between the gas mass and line emission distributions determines the ability of the APEC-predicted temperatures to recover the mass-weighted values. In the selected minor-axis region (bottom left panel), the line emission distribution, both in OVIIf and OVIII, peaks around a temperature similar to that of the gas mass distribution. As a consequence, the APEC-predicted temperature is quite close to the mass-weighted value (within 0.1 dex). On the other hand, for the major axis region (bottom right panel), the gas mass distribution is skewed toward lower temperatures (T∼ 10^4K) resulting in a much lower mass-weighted temperature than the APEC-predicted or emission-weighted temperatures. To conclude, this exercise demonstrates the potential of deriving fairly precise measurements of the underlying physical temperature of the X-ray emitting CGM through the extraction of individual line ratios in spatially-resolved regions. The temperature of the hot phase CGM, as characterised by emission-weighted estimates, can be well recovered within 0.2 dex. It is worth noting that the results in this Section are obtained based on the assumption that the CGM is in the state of collisional ionisation equilibrium and without considering potential impacts of photoionisation. Studies on non-equilibrium ionisation effects, e.g. <cit.>, have indicated that the cooling efficiency of both OVII and OVIII ion species could be significantly impacted depending on the specific background radiation present. This implies that in cases of non-equilibrium ionisation, a CIE-based derived temperature based on the OVIII/OVIIf ratio could potentially be misestimated. We postpone a quantitative investigation of the implications of non-equilibrium ionisation as well as photoionisation on the line diagnostics of CGM temperature for a future work. § SUMMARY AND CONCLUSIONS Future X-ray imaging microcalorimeter missions with high spatial and spectral resolution such as the Line Emission Mapper (LEM) will, for the first time, detect and characterize the hot circumgalcatic medium (CGM) of galaxies via X-ray metal line emission. In this paper, we utilise three modern cosmological hydrodynamical simulations of galaxy formation (TNG100, EAGLE, and SIMBA) to make predictions for the spatial distribution of soft band X-ray metal line emission in the CGM of galaxies over a wide range of stellar masses 10^10-11.6 M_⊙ at z=0. We focus on the effects of supermassive black hole (SMBH) feedback on the CGM, the observability of the signal, and its ability to constrain the physics of galactic feedback and gaseous atmospheres. Our main results are: The three simulations show significant diversity in the thermodynamical properties of the CGM, as well as in the radial profiles of X-ray line emission (Fig. <ref>, <ref>). SIMBA predicts the lowest overall emission, due to lower CGM gas content than in TNG100 or EAGLE. This reflects the different efficiencies and ejective strengths of the SMBH feedback models across these three simulations. * For galaxies at the Milky-Way mass scale, all three simulations consistently indicate a clear correlation between CGM line emission and star formation activity: star-forming galaxies are significantly more luminous compared to their quiescent counterparts (Fig. <ref>) over a broad radial range up to R_500c. This dichotomy between star-forming versus quiescent galaxies is caused by SMBH feedback that expels gas from the central regions of galaxies and halos as the quenching process proceeds (Fig. <ref>). * At the population level, the three simulations predict that the spatial distribution of CGM line emission is largely anisotropic, differing along the major versus minor axis directions with respect to the orientation of the central galaxy. This anisotropy is maximal for high-energy lines, such as FeXVII or NeX: emission is enhanced along the minor axis (Fig. <ref>). However, the magnitude of the angular signal varies quantitatively between the simulations: TNG100 predicts the strongest signal, while EAGLE predicts the weakest one. Finally, we show that future X-ray missions will provide sufficient spectral resolution (∼ eV) to disentangle an extragalactic CGM signal from the Milky-Way foreground, thanks to the line redshifitng (Fig. <ref>). In particular, the currently planned LEM mission, which is designed to have a large field of view and advanced spectral resolution <cit.>, will have excellent mapping capability to explore many physical aspects of the CGM: * Star-forming versus quiescent dichotomy. By mapping the surface brightness in X-ray line emission around star-forming and quiescent galaxies at similar masses, LEM will test the dichotomy between the two populations predicted to exist out to large radii (∼100-200 kpc) by these cosmological simulations (Fig. <ref>, <ref>). * Detection of X-ray bubbles. The anisotropic signal observed at the population level may manifest as X-ray bubbles around individual galaxies at the Milky-Way mass scale, as predicted by TNG. LEM will detect, map, and measure the physical properties of these bubbles for the first time. The whole extent of the bubbles and inner CGM (∼40-50 kpc) are comfortably within its single pointing field of view (Fig. <ref>). * Mapping the CGM thermal structure. The ratio between emission lines, such as OVIII/OIIf, offers a powerful diagnostic of CGM temperature (Fig. <ref>). The ratio serves as a good proxy for the emission-weighted temperature. Although the cosmological simulations studied in this paper employ different physical models for SMBH feedback, they predict a number of qualitative similarities for the impact of SMBH feedback on the CGM. First, gas is physically ejected. Feedback can expel gas out of galaxies and even their halos, particularly as a result of the quenching of star formation in massive galaxies. Second, halo gas experiences anisotropic outflows. Feedback-driven outflows tend to move and heat gas preferentially along the minor axis of galaxies. This angular anistropy is reflected in observable X-ray line emission. The next generation of imaging microcalorimeter-based X-ray missions, including LEM, will provide new opportunities to study the CGM through its narrow-band emission. These will reveal the fundamental connection between galactic feedback and the circumgalactic medium, a key missing link in our understanding of the cosmic baryon cycle and thus galaxy evolution as a whole. § DATA AVAILABILITY The IllustrisTNG simulations are publicly available and accessible at <www.tng-project.org/data> <cit.>. Similarly, data from the Illustris and EAGLE projects are available at <www.illustris-project.org> <cit.> and e.g. <eagle.strw.leidenuniv.nl> <cit.>, respectively. The SIMBA data is publicly available at <simba.roe.ac.uk>. We note that for this study we have not used the original versions of EAGLE and SIMBA, but instead have used versions thereof re-processed into a TNG-like format, enabling an apples-to-apples comparison. Data directly related to this publication and its figures are available on request from the corresponding author. § ACKNOWLEDGEMENTS AP and NT thank Elad Zinger for useful conversations. NT thanks Nastasha Wijers for useful discussions. DN acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) through an Emmy Noether Research Group (grant number NE 2441/1-1) and NT and AP acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”, subproject A01). NW is supported by the GACR grant 13491X. AB, WF acknowledge support from the Smithsonian Institution and the Chandra High Resolution Camera Project through NASA contract NAS8-03060. The material is based upon work supported by NASA under award number 80GSFC21M0002. The primary TNG simulations were carried out with compute time granted by the Gauss Centre for Supercomputing (GCS) under Large-Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of the supercomputer Hazel Hen at the High Performance Computing Center Stuttgart (HLRS). Additional simulations and analyses had been carried out on the Isaac machine of the Max Planck Institute for Astronomy (MPIA) and on the other systems at the Max Planck Computing and Data Facility (MPCDF). mnras § CGM PROPERTIES IN STAR-FORMING VS QUIESCENT GALAXIES In Fig. <ref> we show the physical properties of CGM gas that are responsible for the dichotomy in X-ray emission between star-forming and quiescent galaxies (Section <ref>). From top to bottom: radial profiles of gas mass surface density, mass-weighted temperature, and mass-weighted metallicity. In all cases we restrict our galaxy sample to the same MW-mass selection as previously. The three columns show the three simulations separately: TNG100 (left), EAGLE (center), and SIMBA (right). In all panels, the star-forming subset is shown with the blue lines and shaded bands, while the quiescent subset is shown in red. Star-forming galaxies, across all three simulations, have higher gas densities, lower temperatures, and higher metallicities throughout their gaseous halos. As discussed in the main text, these differences collectively result in observable differences in X-ray line emission.
http://arxiv.org/abs/2307.01431v2	20230704015044	Reconciling the magnetic field in central disc galaxies with the dynamical mass using the cosmological simulations	[ "Mohammad Hosseinirad", "Fatemeh Tabatabaei", "Mojtaba Raouf", "Mahmood Roshan" ]	astro-ph.GA	[ "astro-ph.GA" ]	=1 eq:Eq.Eqs. fig:FigFigs
http://arxiv.org/abs/2307.02434v1	20230705165459	The tension in the absolute magnitude of Type Ia supernovae	[ "David Camarena", "Valerio Marra" ]	astro-ph.CO	[ "astro-ph.CO" ]	David Camarena () Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87106, USA, [email protected] Valerio Marra Núcleo de Astrofísica e Cosmologia & Departamento de Física, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil, [email protected] The tension in the absolute magnitude of Type Ia supernovae David Camarena and Valerio Marra Received ; accepted =========================================================== This study aims to elucidate the tension in the Hubble constant (H_0), a key metric in cosmology representing the universe's expansion rate. Conflicting results from independent measurements such as the Planck satellite mission and the SH0ES collaboration have sparked interest in exploring alternative cosmological models. We extend the analysis by SH0ES to an arbitrary cosmographic model, obtaining a competitive local H_0 determination which only assumes the standard flat ΛCDM model (73.14 ± 1.10 km/s/Mpc), and another which only assumes the FLRW metric (74.56 ± 1.61 km/s/Mpc). The study also stresses the importance of the supernova magnitude calibration (M_B) in cosmological inference and highlights the tension in M_B when supernovae are calibrated either by CMB and BAO observations or the first two rungs of the cosmic distance ladder. This discrepancy, independent of the physics involved, suggests that models solely changing the Hubble flow and maintaining a sound horizon distance consistent with CMB, fail to explain the discrepancy between early- and late-time measurements of H_0. § INTRODUCTION In the study of cosmology, one of the key quantities of interest is the Hubble constant, denoted as H_0. It represents the current rate of expansion of the universe and plays a crucial role in determining its age and evolution. Over the years, various methods have been employed to measure H_0, including observations of the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and the use of astronomical distance indicators such as Type Ia supernovae and Cepheids variables. In recent years, the tension between different measurements of H_0 has garnered significant attention. While the Planck satellite mission <cit.>, which analyzed the CMB, reported a value of H_0 around 67 km/s/Mpc, independent measurements using the local distance ladder technique, such as the SH0ES collaboration <cit.>, have consistently obtained higher values, closer to 73 km/s/Mpc. This discrepancy, significant at the 5σ level and often referred to as the “Hubble tension,” has led to considerable interest in exploring cosmological models that can reconcile these conflicting results. Here, we aim to understand the implications of the local calibration of the luminosity of Type Ia supernovae. After reviewing the basic background in Section <ref>, in Section <ref> we will start with the 46-dimensional analysis by SH0ES and extend it to an arbitrary cosmographic model. This will allow us to obtain a determination of local H_0 that only assumes the validity of the standard flat ΛCDM model, and another one that only assumes the FLRW metric, that is, large-scale homogeneity and isotropy. We proceed in Section <ref> to highlight the importance of considering the calibration of the supernova magnitude M_B during cosmological inference, emphasizing the necessity to test the consistency of a given cosmological model with respect to M_B. Finally, in Section <ref> we discuss the “M_B tension,” the fact that the supernova luminosity either gets calibrated by CMB and BAO observations or by the first two rungs of the cosmic distance ladder, providing inconsistent constraints. Our conclusions are presented in Section <ref>. § CURVED WCDM COSMOGRAPHY Given the luminosity distance d_L, the apparent magnitude is given by: m_B(z) = 5log_10[d_L(z)/1 Mpc] +25 + M_B , and the distance modulus by: μ(z)= m_B(z) - M_B , where M_B is the absolute magnitude of Type Ia supernovae. The luminosity distance is a prediction of a cosmological model and, assuming the FLRW metric, is: d_L(z) = (1+z) c/ H_0Ω_k0^-1/2sinh [ Ω_k0^1/2∫_0^z ̣̅z/E(z̅) ] , where H=ȧ(t)/a(t) describes the expansion history of the universe and a(t) is the scale factor. For a curved wCDM model it is: H^2(z)/H^2_0 = E^2(z) = Ω_ m0 (1+z)^3 + Ω_ de 0(1+z)^3(1+w) + Ω_k 0 (1+z)^2 , where we neglected radiation as it is inconsequential at redshifts lower than 100. In the previous equation the subscript 0 denotes the present-day value of the corresponding quantity, Ω_ m0 is the density parameter for cold dark matter (zero pressure), Ω_ de0 the one for a dark energy fluid with equation of state parameter w, and Ω_k0=-k c^2/H_0^2 represents the spatial-curvature contribution to the Friedmann equation (a(t_0)=1). It is Ω_ m0+Ω_ de0+Ω_k0=1. Alternatively, using a cosmographic approach one has <cit.>: d_L(z) = c z/H_0 [ 1 + 1-q_0/2 z -1- q_0-3 q_0^2+j_0 -Ω_k0/6 z^2 + O(z^3) ] , where the Hubble constant H_0, the deceleration parameter q_0 and the jerk parameter j_0 are defined, respectively, according to: H_0=. ȧ(t)/a(t)\|_t_0 , q_0=. - ä(t)/H^2(t) a(t)\|_t_0 , j_0=.⃛ a(t)/H^3(t) a(t)\|_t_0 . Cosmography is a model-independent approach in the sense that it does not assume a specific model as it is based on the Taylor expansion of the scale factor. However, this does not mean that its parameters do not contain cosmological information. For example, in the case of curved wCDM cosmologies, q_0 and j_0 are given by: q_0 = Ω_ m 0/2 + 1+3 w/2Ω_ de 0flat=1/2+ 3/2 w(1-Ω_ m 0) Λ=3/2Ω_ m 0 -1 fid= -0.55 , j_0 = Ω_ de 0+ Ω_ m 0 + 9/2w(1+w) Ω_ de 0flat=1 + 9/2w(1+w) (1-Ω_ m 0) Λ= 1 , where in the last equality of (<ref>) the concordance value of Ω_ m 0=0.3 was adopted. Finally, when determining the luminosity distance, one should adopt both the peculiar velocity-corrected CMB-frame redshift z (also called Hubble diagram redshift) and the heliocentric redshift z_ hel so that the luminosity distance is given by <cit.>: d_L(z, z_ hel)= 1+z_ hel/1+zd_L(z) . § LOCAL DETERMINATION OF H_0 The SH0ES collaboration measures the Hubble constant in the redshift range 0.023<z<0.15, where the minimum redshift is large enough in order to reduce the impact of cosmic variance by the local large-scale structure <cit.> and the maximum redshift is small enough to provide a late-time local measurement of the Hubble constant and reduce the impact of cosmology in its determination. Ideally, one would want a cosmology-independent local constraint. SH0ES traditionally fits the value of H_0 in (<ref>), but fixes the values of the higher-order parameters to their fiducial values of q^ fid_0=-0.55 and j^ fid_0=1, using information from observations beyond the local universe, like higher-z supernovae. In <cit.> we obtained a local determination of H_0 without fixing the deceleration parameter q_0. Here, we will improve upon our previous results by extending the analysis of <cit.> using their latest data.[<https://github.com/PantheonPlusSH0ES/DataRelease>] Specifically, we adopt the χ^2 statistic: χ^2(θ, q_0, j_0) = (y(q_0, j_0)-L θ )^ T C_y^-1 (y(q_0, j_0)-L θ ) , where y is the 3492-dimensional data vector, θ is the 46-d parameter vector considered by SH0ES, L is the 46×3492 equation matrix, so that the model is Lθ, and C_y is the 3492×3492 covariance matrix. The last elements of y are about the 277 Hubble flow supernovae (nhf) that are used to determine H_0: y^nhf_i = m_B, i - 5 log_10 H_0 d_L - 25 = m_B, i - 5 log_10 c z_i [ 1 + 1-q_0/2 z_i -1- q_0-3 q_0^2+j_0 -Ω_k0/6 z_i^2 ] -25 , and in Eq. (<ref>) we let the components y^nhf_i be functions of q_0 and j_0.[We set Ω_k0=0, but the latter is degenerated with j_0, see discussion below.] The baseline analysis by SH0ES adopts χ^2(θ, q^ fid_0, j^ fid_0), so that y is constant and the model is linear. In this case the best-fit model and its covariance matrix are: θ_ bf(q_0, j_0) = ( L^ T C_y^-1 L )^-1L^ T C_y^-1 y(q_0, j_0) , C_θ = ( L^ T C_y^-1 L )^-1 , from which we can note that the best fits depend on q_0 and j_0, but their covariance matrix does not. Using the properties of multi-variate Gaussian distributions (the model is linear), it is then straightforward to plot the marginalized constraints on any combination of parameters by considering the eigenvectors and eigenvalues of the covariance matrix. This is equivalent to performing a Bayesian analysis with wide flat priors, as done in <cit.>. In Fig. <ref> we show the marginalized constraints on M_B and H_0. If we now do not fix q_0 and j_0, the model used in Eq. (<ref>) becomes nonlinear and a full Bayesian analysis is necessary. One possible approach is to approximate the posterior as a Gaussian distribution via its Fisher approximation, F_α, ij = 1/2∂^2 χ^2/∂α_i ∂α_j, where α= {θ, q_0, j_0 } is the 48-d parameter vector, and C_α=F_α^-1 is the covariance matrix on the parameters (the inverse of the Fisher matrix).[Note that 1/2∂^2 χ^2/∂θ_i ∂θ_j exactly gives C^-1_θ as, in this case, the model is linear.] However, the posterior will likely show a non-Gaussian character in the j_0 direction as the latter is expected to be poorly constrained by supernovae at z<0.15. Consequently one should perform a computationally demanding 48-d Bayesian analysis. Fortunately, we can exploit the fact that the model is linear in all parameters but q_0 and j_0 to substantially speed up the estimation of the posterior. We can indeed marginalize analytically on all the parameters we are not interested in: χ^2_ marg (M_B, H_0, q_0, j_0) = χ^2 (θ_ bf(q_0, j_0), q_0, j_0 ) + V C_M_B, H_0^-1 V , V = { M_B, H_0 } -{M_B^ bf(q_0, j_0) , H_0^ bf(q_0, j_0) } , where M_B^ bf and H_0^ bf comes from the corresponding entries of θ_ bf(q_0, j_0) <cit.>, and C_M_B, H_0 is obtained by removing all the rows and columns of C_θ expect the ones relative to M_B and H_0, as this operation is equivalent to marginalizing the posterior over all the other linear parameters. In Figure <ref> and Table <ref> we compare the baseline results by SH0ES with our results. It is interesting to note that the analysis relative to q_0 free and j_0=1 provides a very competitive constraint that is independent of the particular ΛCDM model as it assumes just flat FLRW metric and cosmological constant. It features a 1.5% uncertainty, as compared with the 1.4% uncertainty of the baseline analysis. In particular, marginalizing over q_0 is equivalent to marginalizing over Ω_ m 0. Indeed, their relation is linear, so that the same wide flat prior is adopted, and the relative difference of ΛCDM with respect to the corresponding cosmographic model, see Eq. (<ref>), is ≈ 10^-6 at z=0.15. Next, by adopting an improper flat prior for j_0, one obtains a measurement that only assumes the FLRW metric, that is, large-scale homogeneity and isotropy. It is interesting to note that in Eq. (<ref>) the degenerate combination j_0-Ω_k0 appears so that by marginalizing over j_0, we are effectively marginalizing also over spatial curvature. The data poorly constrain this combination, giving j_0 - Ω_k0= 28^+39_-28. Finally, we found that the deceleration parameter is in good agreement with the value relative to the standard model, contrary to what we found in <cit.> with the Pantheon and Supercal datasets. The difference is likely due to the improved modeling of uncertainty and redshift corrections in the Pantheon+ dataset <cit.>. § COSMOLOGICAL INFERENCE WITH THE LOCAL PRIOR ON M_B The latest combined data release by the SH0ES and Pantheon+ collaborations significantly improved the robustness and consistency of cosmological analyses that include a local determination of the Hubble constant. Now, the distance calibration from 37 Cepheids is provided for the 42 supernovae in the Cepheid host galaxies,[Some galaxies hosted more than one supernova. Note also that some supernovae have been observed by more than one telescope so that in Pantheon+ there are actually 47 supernova entries that are calibrated by Cepheid distances.] together with the combined supernova and Cepheid covariance matrix <cit.>. This provides an absolute calibration for the supernova Ia absolute magnitude M_B as the likelihood is now given by: V_i = m_B,i-M_B - μ(z_i) if Hubble diagram supernova, m_B,i-M_B - μ_ ceph,i if supernova in Cepheid host, χ_sne^2 = V^ T ( Σ_sne+Σ_ceph )^-1 V , where Σ_ceph has nonzero entries only for the supernovae in Cepheid hosts. The “Hubble diagram” (HD) supernovae are the 1466 supernovae – corresponding to 1580 measurements – that are used to constrain cosmological models but not to calibrate M_B. These have a redshift z>0.01, again to remove the cosmic variance from the local structure. The 42 supernovae (47 light curves) in Cepheid host galaxies are instead used to calibrate M_B, whose calibration is then propagated to the HD supernovae. This breaks the degeneracy between M_B and the 5 log_10 H_0 term inside the distance modulus μ(z), see Eq. (<ref>). The above methodology is ideal as it includes the correlations between Cepheid distances and calibrating and HD supernovae, ensuring an accurate representativeness of the cosmic distance ladder in cosmological inference <cit.>. Indeed, if one just uses HD supernovae, together with a local prior on H_0, one will i) double count the 277 low-redshift supernova measurements that were used to obtain the prior on H_0, ii) assume the validity of cosmography, in particular fixing the deceleration parameter to the standard model value of q_0=-0.55, iii) not include in the analysis the fact that M_B is constrained by the local calibration, an information which would otherwise be neglected in the analysis, biasing both model selection and parameter constraints <cit.>. The approach of Eq. (<ref>) allows to carry out a variety of analyses. For example, one can only consider the 238 HD supernovae (but 277 supernova measurements) that are used to determine the Hubble constant and determine H_0 according to various fiducial models, that is, varying q_0 and j_0 in the case of cosmography or varying Ω_ m 0 and w in the case of wCDM models. This approach is equivalent to the one of the previous Section, which only uses the quantities L, y and C from SH0ES. Here, we preferred to adopt the methodology of the previous Section because we noticed that the Pantheon+ dataset produces a value of H_0 which is 0.5 km/s/Mpc higher than the SH0ES baseline value when considering only the 277 low-redshift supernova measurements that are used to determine the Hubble constant.[We contacted the authors of <cit.> and <cit.>; according to Adam Riess there may be an inconsistency between the production and application of the covariance matrix of Cepheid/SN Ia host distances in the Pantheon+ fitting code. This issue is under investigation.] The most interesting use of Eq. (<ref>) is the possibility of constraining the Hubble constant conditionally to a particular cosmological model, bypassing completely the use of cosmography. In this case the tension in the high- and low-redshift constraint on H_0 is traduced into a tension between the local calibration of M_B and the one induced by the CMB via BAO observations. This highlights the important of using the new Pantheon+ & SH0ES likelihood. In order to offer an easy way to implement the absolute calibration of M_B, here we apply the demarginalization method introduced in <cit.> to obtain an effective prior on M_B. As target for the demarginalization procedure, we consider both the constraint H_0 = 73.6 ± 1.1 km/s/Mpc for the flat ΛCDM model from <cit.> and a value 0.5 km/s/Mpc lower, because of the discrepancy mentioned above when using the Pantheon+ & SH0ES likelihood.[This approach was suggested by Adam Riess.] We use the 1580 HD supernova entries and obtain: M_B = -19.243 ± 0.032 mag for H_0 = 73.6 ± 1.1 km/s/Mpc , M_B = -19.258 ± 0.032 mag for H_0 = 73.1 ± 1.1 km/s/Mpc . As a cross-check, using Eq. (<ref>) together with the Pantheon+ likelihood, we obtain a good agreement with the results from <cit.>:[Ref. <cit.> adopted a prior Ω_ m 0>0.1, affecting their results in that parameter space region.] H_0 = 73.6 ± 1.1 Ω_ m 0 =0.333 ± 0.018 (ΛCDM) , H_0 =73.5 ± 1.1 Ω_ m 0 =0.306^+0.063_-0.076 w=-0.88±0.15 (wCDM) . Fig. <ref> shows a full comparison between the analysis that uses the Pantheon+ & SH0ES likelihood and the one that uses the Pantheon+ likelihood with the Gaussian prior on M_B. Constraints and correlations are reproduced accurately. Furthermore, the use of the Gaussian prior has the advantage that one can marginalize analytically over M_B, see <cit.> for details. § M_B TENSION BETWEEN SH0ES AND PLANCK Several theoretical models have been proposed to resolve the discrepancy between the CMB determination of the Hubble constant by Planck <cit.> and the local measurement by SH0ES <cit.>. These models modify the ΛCDM cosmology either in the late or early universe. Early-time modifications of the standard cosmological model typically accommodate a larger H_0 value by reducing the sound horizon scale r_s during recombination, whereas late-time modifications typically increase the Hubble rate at low redshifts <cit.>. While no compelling evidence for physics beyond the ΛCDM model has been found, late-time modifications of the standard paradigm face an additional problem: the M_B tension. The local determination of the Hubble constant provided by the SH0ES collaboration relies on a set of astrophysical distances that are used to calibrate the absolute magnitude of supernovae. In <cit.> we built a model-independent version of the inverse distance ladder, which uses BAO distances and a prior on r_s, coming from the Planck 2018 data release <cit.>, to effectively calibrate the supernovae. Our analysis demonstrated that the absolute magnitude obtained from the cosmic distance ladder <cit.> is in tension with the underlying calibration on M_B provided by the combination of CMB + BAO, see Tab. <ref>. Since the latter constitutes a standard ruler that relies on the value adopted for r_s, the discrepancy in M_B suggests that, independent of the physics, models that solely change the Hubble flow, while keeping a sound horizon distance consistent with the one inferred from CMB under the assumption of a ΛCDM cosmology, will fail at explaining the discrepancy between the early and late times. Indeed, the inability of late-time models to adequately explain the Hubble tension has been clearly demonstrated by several analyses <cit.>, including the analysis presented in <cit.>. This particular analysis employs a toy model, referred to as the `hockey-stick' model, wherein the dark energy component rapidly undergoes a phantom transitions at low redshift. This illustrates that the M_B tension must be addressed in order to resolve the Hubble discrepancy effectively. As shown in <cit.>, when analyzed with a prior on H_0 together with CMB and BAO data, the hockey-stick model, hsCDM, provides a value of the Hubble constant consistent with the local observation. Nevertheless, the underlying calibration of M_B that is obtained in the analysis still is in tension with the calibration inferred from the cosmic distance ladder, showing that hsCDM does not really restore the agreement between early and late times distances. This issue, and spurious conclusions, can be avoided if instead a prior on M_B is assumed.[Alternatively, one should build a likelihood containing all the rungs of the cosmic distance ladder <cit.>. For a more detailed discussion, see Sec. <ref>.] Indeed, the analysis of the hsCDM model under the assumption of this prior explicitly shows the failure of the model to alleviate the H_0 (M_B) tension. Similar conclusions hold for void models, which try to explain the local higher expansion rate by placing the observer inside a large void. As far as the observed luminosity distance-redshift relation is concerned, these models are very similar to the hsCDM model and are ruled out for the very same reasons <cit.>. A category of models that potentially resolve the M_B tension encompasses those that feature a transition in supernova brightness at the extremely low redshifts of the first two rungs – specifically, before entering the Hubble flow with supernovae in the range 0.023<z<0.15. This transition could be realized by a sudden shift in the effective gravitational constant by approximately 10% around 50–150 million years ago, consequently altering the Chandrasekhar mass and thereby the supernova luminosity <cit.>. While this proposal conveniently resolves the Hubble tension, it also yields a series of testable predictions that could either support or disprove it <cit.>. Physical mechanisms that might trigger such an ultra-late gravitational transition include a first-order scalar-tensor theory phase transition from an early false vacuum, corresponding to the measured value of the cosmological constant, to a new vacuum with lower or zero vacuum energy <cit.>. We update the current significance of the M_B tension using the latest data and adopting the fiducial ΛCDM model. Using BAO data from BOSS DR12 <cit.>, eBOSS ELG <cit.>, eBOSS LRG <cit.>, SDSS MGS <cit.>, and WiggleZ <cit.>, high-ℓ TT+TE+EE, low-ℓ TT, and low-ℓ EE CMB data from Planck 2018 <cit.>,[For simplicity, we used the compressed version of high-ℓ CMB data.] and the latest Pantheon+ & SH0ES release, we constrain the ΛCDM model considering the cosmological equivalents to the cosmic distance ladder (Pantheon+ & SH0ES data) and the inverse distance ladder (Pantheon+ in combination with BAO and CMB). The results of this analysis are presented in Fig. <ref> and Tab. <ref>. We used CosmoSIS and CAMB <cit.> to carry out the analyses. We find that the tension on M_B exceeds the 6σ level. § CONCLUSIONS In this study, we have examined three distinct yet interconnected topics. Firstly, we extended the analysis conducted by the SH0ES collaboration to determine the local value of the Hubble constant. We broadened their 46-dimensional analysis to cosmography with arbitrary values of the deceleration and jerk parameters. We found that the latest Pantheon+ & SH0ES data provides a competitive constraint that solely assumes the validity of the standard flat ΛCDM model: 73.14 ± 1.10 (1.5%) km/s/Mpc, and a model-independent constraint that only assumes the FLRW metric: 74.56 ± 1.61 (2.2%) km/s/Mpc. Next, we underscored the importance of considering the calibration of the supernova magnitude M_B by the first two rungs of the distance ladder when conducting cosmological inference. Indeed, M_B is a crucial astrophysical parameter and should not be integrated out, but examined to understand the consistency of a given cosmological model. This discussion led us to the issue of the “M_B tension.” The supernova luminosity either gets calibrated by CMB and BAO observations or by the first two rungs of the cosmic distance ladder. We found that, assuming the standard flat ΛCDM model, the two constraints on M_B are in tension at the 6.5σ level. The discrepancy in M_B suggests that, independent of the physics involved, models that solely change the Hubble flow, while maintaining a sound horizon distance consistent with the one inferred from CMB, fail to explain the discrepancy between the early- and late-time measurements of H_0. We are grateful to Adam Riess for his valuable feedback regarding the SH0ES data release. DC thanks the Robert E. Young Origins of the Universe Chair fund for its generous support. VM thanks CNPq (Brazil, 307969/2022-3) and FAPES (Brazil, TO 365/2022, 712/2022, 976/2022, 1020/2022, 1081/2022) for partial financial support. The authors acknowledge the use of the computational resources of the Sci-Com Lab of the Department of Physics–UFES. utcaps
http://arxiv.org/abs/2307.02608v1	20230703133610	Resonant screening in dense and magnetized QCD matter	[ "Guojun Huang", "Jiaxing Zhao", "Pengfei Zhuang" ]	hep-ph	[ "hep-ph", "nucl-th" ]	Department of Physics, Tsinghua University, Beijing 100084, China SUBATECH, Université de Nantes, IMT Atlantique, IN2P3/CNRS, 4 rue Alfred Kastler, 44307 Nantes cedex 3, France Department of Physics, Tsinghua University, Beijing 100084, China We calculate the Debye screening mass in thermal, dense and magnetized QCD matter in the frame of resummed perturbation theory. In the limit of zero temperature, when the Landau energy level and Fermi surface of quarks match each other μ_q^2=2n\|qB\|, where q, μ_q and B are respectively the quark electric charge, chemical potential and external magnetic field, the screening mass diverges and the system is in the state of weakly interacting parton gas, which is very different from the known result of strongly interacting quark-gluon plasma at high temperature. The divergence disappears in thermal medium, but the screening mass oscillates with clear peaks at the matched magnetic field. Resonant screening in dense and magnetized QCD matter Pengfei Zhuang August 1, 2023 ===================================================== Introduction.– An immediate consequence of quarks being fermions is that quarks must satisfy the Pauli exclusion principle, which states that no two quarks can occupy the same state. The application of this pure quantum effect in quark matter at finite baryon density means a boundary separating occupied and unoccupied states in momentum space, it is called the Fermi surface controlled by the baryon chemical potential. The sharp Fermi surface leads to the expectation that there should a first-order phase transition in Quantum ChromoDynamics (QCD) systems at high baryon density. While lattice QCD simulations meet the sign problem, many effective models obtain the first-order phase transition for the chiral symmetry restoration at high density <cit.>. A hot topic in the experimental and theoretic study of QCD thermodynamics in relativistic heavy ion collisions is the search for the critical point connecting the crossover at high temperature and the first-order transition at high baryon density <cit.>. A strong external electromagnetic field can also induce QCD phase transitions and change the QCD medium properties, like magnetic catalysis <cit.>, inverse magnetic catalysis <cit.> and chiral magnetic effect <cit.>. It is widely believed that the strongest electromagnetic field in nature can be created in non-central nuclear collisions <cit.>. A familiar quantum phenomenon for a fermion moving in an external magnetic field is the Landau energy level: The fermion's moving in the transverse plane perpendicular to the magnetic field is like a harmonic oscillator which leads to a discrete transverse energy <cit.>. A question we ask ourselves is what will happen when the sharp Fermi surface meets the discrete Landau levels. The related physics systems with high baryon density and strong magnetic field might be realized in compact stars and intermediate energy nuclear collisions <cit.>. We study in this work the color screening in thermal, dense and magnetized QCD matter. Analog to the Debye screening in electrodynamics, the color interaction between a pair of quarks is screened by the surrounding quarks and gluons <cit.>. The Debye mass m_D or the Debye length r_D∼ 1/m_D can be used to effectively describe the QCD phase transition: When the Debye length is shorter than the hadron averaged radius, the constituents inside the hadron cannot see each other through the color interaction and the hadron as a whole particle disappears. In finite temperature field theory, the Debye mass is defined as the static and long wavelength limit of the gluon self-energy <cit.>. In a hot QCD medium, the Hard Thermal Loop (HTL) method gives a temperature-dependent Debye screening mass m_D(T) <cit.>. For a dense QCD medium, the Hard Dense Loop (HDL) provides a similar baryon chemical potential dependence m_D(μ_B) <cit.>. Recently, the screening effect is investigated in a thermal and magnetized QCD matter <cit.>, and the calculated Debye mass m_D(T,B) recovers the previously obtained limits of weak and strong magnetic fields <cit.>. The paper is organized as follows. We first derive the Debye screening mass m_D(T,μ_B,B) at finite temperature, baryon chemical potential and magnetic field in the frame of loop-resumed perturbation theory, and compare it with the previously obtained limits of dilute quark gas, weak and strong magnetic field. We then focus on the case at zero temperature and analyze the new phenomenon induced by the quantum Fermi surface and Landau levels. We will show the numerically calculated Debye mass at zero and low temperature to see clearly the new phenomenon. We summarize the work in the end. Screening mass.– The magnetic field breaks down the translation invariance, which leads to the separation of the quark momentum p into a longitudinal and a transverse part p_\|\| and p_⊥, parallel and perpendicular to the magnetic field. Using the Schwinger propagator for massless quarks with electric charge q <cit.>, G(p) = -∫_0^∞dv \|qB\|[(γ· p)_\|\|(1-i sgn(q)γ_1γ_2tanh v) -(γ· p)_⊥cosh^2 v]e^v \|qB\|(p_\|\|^2-tanh v v p_⊥^2), where the magnetic field B is explicitly shown, and the temperature T and baryon chemical potential μ_B enter the calculation through the Matsubara frequency p_0=iω_m=i(2m+1)π T and energy shift p_0→ p_0-μ_q with quark chemical potential μ_q=μ_B/3, one can calculate the gluon polarization induced by a quark loop, Π_μν(k)=g^2∫d^4p (2π)^4Tr[γ_μ G(p)γ_ν G(p-k)], where k is the gluon momentum, and the quark momentum integration includes a three-momentum integration and a Matsubara frequency summation. Note that, we have neglected the Schwinger phase factor in the propagator (<ref>), since in the calculation of the polarization the two-phase factors for the quark and anti-quark cancel to each other. Here we have also fixed the quark flavor, we will consider the flavor summation in the end. Taking the usually used summation over quark loops on a chain, one can derive a non-perturbative gluon propagator <cit.>, and the Debye mass is defined as the pole of the propagator. In the cases of high temperature and/or high density, the screening mass is determined only by the polarization in the limit of zero gluon momentum Π_μν(k_0=0, k→ 0), called as HTL and HDL approximations <cit.>. Since in these cases, the polarization depends only on the external parameters, one can explicitly express the dependence as Π_μν(T,μ_B,B). It is easy to see that all the off-diagonal elements (μ≠ν) of the polarization vanish automatically, one needs to consider the diagonal elements only. One further divides the diagonal polarization into a parallel and a perpendicular part Π_μμ^ with μ∈{0,3} and Π_μμ^⊥ with μ∈{1,2}, only the parallel part is related to the color screening mass <cit.>. Including the gluon-loop and ghost-loop contribution Π_μν to the gluon polarization which is only temperature dependent, since gluons and ghosts do not carry charge and chemical potential, the Debye screening mass is expressed as m_D^2(T,μ_B,B)=-Π_00^(T,μ_B,B)-Π_00^(T). Since the pure temperature and density dependence in the absence of magnetic field is known <cit.>, m_D^2(T,μ_B,0) = g^2[(N_c 3+N_f 6)T^2+N_fμ_q^2 2π^2] with numbers N_c and N_f of color and flavor degrees of freedom, we focus in the following on the shift of the squared Debye screening mass induced by B. From the calculation at zero baryon density <cit.>, it is δ m_D^2(T,μ_B,B) = m_D^2(T,μ_B,B)-m_D^2(T,μ_B,0) = 2g^2T∑_p_z,mϵ̅_-^2𝒦(ϵ̅_+^2), where the chemical potential is included in the dimensionless variables ϵ̅_±^2=[p_z^2± (ω_m+iμ_q)^2]/(2\|qB\|), and the function 𝒦 is defined as 𝒦(x)=1/(2x^2)+1/x-ψ'(x) with ψ(x)=Γ'(x)/Γ(x) controlled by the Gamma function. The quark frequency summation ∑_m=-∞^∞ and the longitudinal momentum integration ∑_p_z=∫ dp_z/(2π)^2 are explicitly shown in (<ref>), and the summation over Landau levels for the transverse momentum p_n=√(2n\|qB\|) is reflected in the recurrence relation of the function 𝒦, 𝒦(x) = 1/(2x^2)+1/x-1/(2(x+N)^2)-1/(x+N) +𝒦(x+N)-∑_n=0^N-11/(x+n)^2 with N =[Floor(μ_q^2-π^2T^2 2\|qB\|)+1]θ(μ_q-π T). To understand the physics of the summation here, we consider the Fermi surface for quarks moving in an external magnetic field. It is determined by the quark Fermi-energy ϵ_n=√(p_z^2+p_n^2) = μ_q which restricts the quark momentum p_z^2+2n\|qB\|≤μ_q^2. Since p_z^2 is positive, the maximum Landau level is Floor[μ_q^2/(2\|qB\|)]. Extending the analysis to finite temperature, the restriction condition becomes p_z^2+2n\|qB\|+ω_m^2≤μ_q^2. Considering again the minimum longitudinal momentum p_z=0 and the minimum thermal energy π T, the maximum Landau level becomes Floor(μ_q^2-π^2T^2)/(2\|qB\|) under the condition of μ_q > π T, which is expressed as N-1 in (<ref>) and (<ref>). When the Fermi surface is not high enough to overcome the thermal energy μ_q < π T, there is N=0 and the summation in (<ref>) vanishes automatically. The recurrence relation (<ref>) leads to 𝒦(ϵ̅_+^2) =1ϵ̅_+^2 -1ϵ̅_N^2-1 2∑_n=0^NΔ^n_N(ϵ̅_n^2)^2+𝒦(ϵ̅_N^2) with Δ^n_N=2-δ_0n-δ_nN and ϵ̅_n^2=ϵ̅_+^2+n. Substituting 𝒦 into the mass shift (<ref>), after a straightforward but a little tedious calculation, see the details shown in Suppl. <ref>, the total mass shift is separated into two components, δ m_D^2 = δ m_1^2+δ m_2^2, δ m_1^2 = -g^2 4T∑_p_z,s[p_z^2𝒮_N^s-\|qB\| 2∑_n=0^NΔ^n_Nsech^2(ϵ̅_n^s)], δ m_2^2 = 2g^2T^2π^1/2∫_0^∞dξξ^2e^cξ^2ϑ_2(aξ^2,e^-ξ^2)ℳ(bξ^2) with the constants a^2=μ_q^2/(4π^2T^2), b=\|qB\|/(4π^2T^2), c=a^2-2Nb and ϵ̅_n^s=(p_z^2+p_n^2+sμ_q)/(2T) and the functions 𝒮_N^s = sech^2(ϵ̅_0^s) -sech^2(ϵ̅_N^s), ϑ_2(u,x) = 2x^1/4∑_i=0^∞ x^i(i+1)cos((2i+1)u) and ℳ(x) = 1-x^2/sinh^2 x+2Nx(1-x x). The summation ∑_s=± is over quarks and anti-quarks. Note that the first component δ m_1^2 disappears automatically for N=0 due to 𝒮_0^s=0 and Δ^n_0=0, the often used approximation <cit.> of taking only the lowest Landau level is included in the second component δ m_2^2. Weak and strong magnetic field.– We now start to consider some physics limits of the general mass shift (<ref>). We first consider a dilute quark gas with μ_q→ 0. In this case with N=0, the component δ m_1^2 vanishes automatically, δ m_D^2(T,0,B)=δ m_2^2(T,0,B) goes back to the known result in hot medium <cit.>. We then consider the limit of weak magnetic field with \|qB\|→ 0. In this limit, the second component δ m_2^2 disappears due to ℳ(0)=0. From the derivation provided in Suppl.II the first component δ m_1^2 vanishes too. Therefore, we obtain the Taylor expansion of the mass shift in terms of \|qB\| for a weak magnetic field at any temperature and baryon chemical potential, δ m_D^2 = 14ζ(3) 9(2π)^4g^2 T^2\|eB\|^2+𝒪(\|eB\|^4), where we have taken into account the quark flavor summation. While the quark chemical potential is flavor independent, the electric charge q depends on the flavor q_u=2e/3, q_d=-e/3 and q_s=-e/3 . The other limit we consider is the strong magnetic field limit with √(\|qB\|)≫ T,μ_B. Taking a variable transformation ξ=Tη/\|qB\|^1/2 and employing the limit lim_\|qB\|→∞ e^a^2ξ^2ϑ_2(aξ^2,e^-ξ^2)=√(π\|qB\|)/(Tη), the component δ m_2^2 becomes δ m_2^2 =g^2 4π^2\|qB\|θ(π T-μ_q). This covers the familiar result <cit.> including only the contribution from the lowest Landau level. More details about the derivation of (<ref>) are provided in Suppl.III. For the component δ m_1^2, in the limit of strong magnetic field with N=1 at μ_q > π T, by using the delta function δ(x) = lim_T/\|qB\|^1/2→ 0[1 4T/\|qB\|^1/2sech^2(x 2T/\|qB\|^1/2)], we have δ m_1^2 = g^2 4π^2(\|qB\|-2μ_q^2-23π^2T^2)θ(μ_q-π T). Considering the summation over all quark flavors, the total mass shift in a strong magnetic field is δ m_D^2 =g^2 3π^2\|eB\|-N_fg^2 2π^2(μ_q^2+π^2 3T^2)θ(μ_q-π T). The magnetic field dependence of the scaled screening mass m_D/g (<ref>) and the comparison with the approximations of weak (<ref>) and strong (<ref>) magnetic field are shown in Fig.<ref> at high baryon chemical potential μ_B=1 GeV and low and high temperatures T=20 and 100 MeV. In the frame of loop resummation, m_D/g is no longer coupling constant dependent. The integrated function in δ m_2^2 (<ref>) looks like divergent at ξ=0, but it is convergent due to the limits lim_ξ→ 0e^a^2ξ^2ϑ_2(aξ^2,e^-ξ^2)∼ 1/ξ and lim_ξ→ 0 M_N(bξ^2)∼ξ^4+⋯. The color and flavor numbers are chosen as N_c=N_f=3. Naturally, the full calculation is always in between the two approximations. Because the external magnetic field breaks down the isotropy symmetry but random thermal motion restores the symmetry, the approximation of weak magnetic field becomes better and the approximation of strong magnetic field becomes worse with increasing temperature, as a consequence of the competition of the two effects. Resonant screening.– An interesting phenomenon shown in Fig.<ref> is the oscillation behavior of the screening mass at low temperature. To understand the physics behind it we turn to discuss the zero-temperature limit of Eq.(<ref>). Similar to the treatment for the integration over ξ in a strong magnetic field, after taking the variable transformation from ξ to η, the integrated function can be expressed as a complete differential, and it is zero at both the lower and upper limit of the integration, which leads to δ m_2^2=0 at T→ 0. By taking the limits lim_T→ 0tanh (x/T) = 2θ(x)-1, ∂ x(tanh(x/(2T)) = sech^2(x/(2T))/(2T)→ 2δ(x), the total mass shift becomes δ m_D^2 = δ m_1^2 = -g^2μ_q^2 2π^2+g^2μ_q\|qB\| (2π)^2∑_n=0^N-12-δ_0n√(μ_q^2-2n\|qB\|) with the maximum Landau level N-1=Floor[μ_q^2/(2\|qB\|)]. The details to derive (<ref>) are shown in Suppl.VI. Considering the result (<ref>) in the absence of magnetic field, and taking into account all the flavors, the total mass square in zero temperature limit is m_D^2 = g^2μ_q (2π)^2∑_f∑_n=0^N-1\|q_fB\|(2-δ_0n)√(μ_q^2-2n\|q_fB\|). When the Fermi surface matches the Landau energy level, μ_q^2=2n\|q_fB\|, controlled by the chemical potential and magnetic field, the Debye mass diverges m_D→∞ and the screening length approaches to zero r_D→ 0. That means a full screening: the color interaction between a pair of quarks is completely screened, and the medium becomes a weakly interacting quark gas. This phenomenon at high baryon density is different from the known result that the QCD medium is a strongly interacting quark-gluon plasma at high temperature <cit.>. The numerical calculation in the limit of zero temperature and the modification by the thermal motion are shown in Fig.<ref>. For fixed chemical potential μ_B=1 GeV, the magnetic fields where the mass diverges are \|eB\|=1/6, 1/12,⋯ GeV^2 for d and s quarks and \|eB\|=1/12, 1/24,⋯ GeV^2 for u quarks, corresponding to the first, second, third and other divergences counted from the right. When the magnetic field is too strong to satisfy the matching condition, \|q_fB\|>μ_q^2/2=μ_B^2/18, the mass becomes chemical potential independent and is linear in the magnetic field, m_D^2 = g^2 3π^2\|eB\|, see the black dashed line in the region of strong magnetic field shown in Fig.<ref>. The above divergence induced by the matched Fermi surface and Landau energy level is similar to the well-known resonant transmission in quantum mechanics <cit.>. For the particle tunneling through a potential well, when the particle energy and the well structure (the depth and width of the well) meet certain condition, the particle completely penetrates the well with transmission coefficient T=1 without any reflection R=0. From the comparison with this resonant transmission, we call the above divergence of the screening mass as resonant screening. When the thermal motion is turned on, the sharp Fermi surface becomes a smooth distribution, and the divergence is washed away. When the temperature is not so high, the thermal motion is not yet strong enough, the mass oscillates with the magnetic field, and the divergence is changed to a peak, see the red lines in Figs. (<ref>) and (<ref>) at T=20 MeV. Summary.– We calculated in this paper the Debye screening mass in QCD matter at finite temperature, baryon density and magnetic field in the resummed perturbation theory. The Landau energy levels of moving quarks in the external magnetic field are explicitly shown in the mass. Three limits of the general result, namely the dilute quark gas, and weak and strong magnetic field, are discussed in detail. We focused on the dense and magnetized medium in the limit of vanishing temperature. In this case, when the Landau energy level matches the Fermi surface of quarks, the screening mass goes to infinity, indicating that the color interaction between a pair of quarks is completely screened. We call this full screening as resonant screening in comparison with the resonant transmission in quantum mechanics, the location of the resonance is determined by the matching condition μ_q^2=2n\|qB\|. While the random thermal motion smears the divergence, there is still oscillation with clear peaks at low temperatures. The resonant screening or screening peaks may have applications in dense and magnetized matter created in compact stars and intermediate energy nuclear collisions. 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Phys. A 951, 1-30 (2016). Supplemental Materials We provide supplemental materials in the sequence according to the contents in the main material. We provide in Sections <ref>, <ref> and <ref> the details to derive the full mass shift δ m_D^2(T,μ_B,B) (<ref>) in general case with nonzero temperature, baryon density and magnetic field and its approximations (<ref>) and (<ref>) in weak and strong magnetic field. We focus on the screening mass (<ref>) in dense and magnetized matter at zero temperature in Section <ref>. § SCREENING MASS IN GENERAL CASE Substituting the recurrence relation (<ref>) into the mass shift (<ref>), we have δ m_D^2 = 2g^2T∑_p_z,mϵ̅_-^2[1ϵ̅_+^2 -1ϵ̅_N^2-1 2∑_n=0^NΔ^n_N(ϵ̅_n^2)^2+𝒦(ϵ̅_N^2)]. After the Matsubara frequency summation and longitudinal momentum integration by parts, the first two terms becomes 2T∑_p_z,mϵ̅_-^2(1ϵ̅_+^2-1ϵ̅_N^2) = ∑_p_z,s,s'∂∂ p_z[p_z (ϵ_N+s'p_z)/4][s'tanh(ϵ_0^s)-tanh(ϵ_N^s)] = -2N\|qB\| (2π)^2-1 4T∑_p_z,sp_z^2𝒮_N^s. For the third term, doing again the frequency summation and momentum integration by parts, it becomes -T∑_p_z,m∑_n=0^NΔ_N^nϵ̅_-^2(ϵ̅_n^2)^2 = \|qB\|∑_n=0^NΔ_N^n(1+1 8T∑_p_z,ssech^2(ϵ̅_n^s)) = 2N\|qB\| (2π)^2+\|qB\| 8T∑_p_z,s∑_n=0^NΔ_N^nsech^2(ϵ̅_n^s), where we have used the summation ∑_n=0^NΔ_N^n=2N. By using the integral representation for 𝒦, 𝒦(x) = 4b∫_0^∞ dξ e^-2bxξ^2ξ[1-bξ^2(bξ^2)], doing the ξ-integration by parts, and then taking the frequency summation ∑_m e^-2bϵ̅_N^2ξ^2=e^(c-p_z^2/(4π^2T^2))ξ^2ϑ_2(aξ^2,e^-ξ^2) and momentum integration, the last term in (<ref>) is exactly the second component δ m_2^2. Considering the contributions from all the four terms, we obtain the total mass shift shown in (<ref>). § WEAK MAGNETIC FIELD In this Section we prove that the first component δ m_1^2 disappears in the limit of weak magnetic field, namely our calculation in general case goes back to the usually known result without magnetic field. By taking into account the limit lim_\|qB\|→ 0 2N\|qB\| = (μ_q^2-π^2T^2)θ(μ_q-π T) and the change from Landau level summation ∑_n into Riemann integration by using the transformation p_n^2=2n\|qB\|→ζ in the definition of 𝒮_μ̅^2^s and ϵ̅_ζ^s, the first component becomes δ m_1^2 = θ(μ_q-π T)g^2 8T∑_p_z,s[-2p_z^2𝒮_μ̅^2^s+∫_0^μ̅^2dζ sech^2(ϵ̅_ζ^s)] with μ̅^2=μ_q^2-π^2 T^2. By writing the function sech^2(ϵ̅_ζ^s) in terms of two partial differentials, sech^2(ϵ̅_ζ^s) = ∂_p_z[p_z sech^2(ϵ̅_ζ^s)]-2p_z^2∂_ζsech^2(ϵ̅_ζ^s), the integration of the first term over p_z vanishes, and the integration of the second term over ζ cancels exactly the first term with 𝒮 in Eq.(<ref>). Therefore, the component δ m_1^2 and in turn the total mass shift δ m_D^2 disappear in the limit of weak magnetic field. § STRONG MAGNETIC FIELD In strong magnetic field, the maximum Landau level is a small integral and the lowest Landau level becomes the dominant contribution to the QCD thermodynamics. Taking the transformation ξ=Tη/\|qB\|^1/2 and employing the limit (<ref>), the component δ m_2^2 becomes δ m_2^2 = 2g^2\|qB\|∫_0^∞ dη 1η^3e^-N 2π^2η^2ℳ_N(η^2 4π^2) = g^2\|qB\| 4π^2∫_0^∞ dη∂∂η[e^-N 2π^2η^2((η^24π^2)-(η^24π^2)^-1)] = g^2\|qB\| 4π^2δ_0N which is zero unless N=0 at μ_q <π T. § RESONANT SCREENING In the limit of zero temperature, the QCD thermodynamics is controlled by the sharp Fermi surface, and the calculation is largely simplified. By taking the limits (<ref>), the total mass shift becomes δ m_D^2 =-g^2∑_p_z,s[p_z^2(δ(ϵ_0+sμ_q)-δ(ϵ_N+sμ_q)) -\|qB\| 2∑_n=0^NΔ_N^nδ(ϵ_n+sμ_q)]. Taking the integration over the momentum and summation over quarks and anti-quarks, it becomes the result (<ref>) showing the resonant screening.
http://arxiv.org/abs/2307.02047v1	20230705060536	CAME: Confidence-guided Adaptive Memory Efficient Optimization	[ "Yang Luo", "Xiaozhe Ren", "Zangwei Zheng", "Zhuo Jiang", "Xin Jiang", "Yang You" ]	cs.CL	[ "cs.CL" ]	Emoji Prediction using Transformer Models 1st Muhammad Osama Nusrat Dept of Computing Fast Nuces Islamabad, Pakistan [email protected] 2nd Zeeshan Habib Dept of Computing Fast Nuces Islamabad, Pakistan [email protected] 2nd Mehreen Alam Dept of Computing Fast Nuces Islamabad, Pakistan [email protected] 2nd Saad Ahmed Jamal Department of GeoInformatics ZGIS University of Salzburg Salzburg, Austria [email protected] ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Work was done when Yang Luo was an intern at Huawei Noah’s Ark Lab. Adaptive gradient methods, such as Adam and LAMB, have demonstrated excellent performance in the training of large language models. Nevertheless, the need for adaptivity requires maintaining second-moment estimates of the per-parameter gradients, which entails a high cost of extra memory overheads. To solve this problem, several memory-efficient optimizers (e.g., Adafactor) have been proposed to obtain a drastic reduction in auxiliary memory usage, but with a performance penalty. In this paper, we first study a confidence-guided strategy to reduce the instability of existing memory efficient optimizers. Based on this strategy, we propose CAME to simultaneously achieve two goals: fast convergence as in traditional adaptive methods, and low memory usage as in memory-efficient methods. Extensive experiments demonstrate the training stability and superior performance of CAME across various NLP tasks such as BERT and GPT-2 training. Notably, for BERT pre-training on the large batch size of 32,768, our proposed optimizer attains faster convergence and higher accuracy compared with the Adam optimizer. The implementation of CAME is publicly available[<https://github.com/huawei-noah/Pretrained-Language-Model/tree/master/CAME>]. § INTRODUCTION Robust training of large language models (LLMs) often relies on adaptive gradient-based optimization methods <cit.>. Through the use of cumulative second-order statistics, these methods adapt the per-parameter learning rate and demonstrate superior convergence speed during the training process of LLMs. However, the remarkable performance of adaptive methods incurs an extra cost of memory usage indeed. For example, Adam requires to preserve the first moment estimate and second raw moment estimate of each gradient in order to tune the learning rate for each parameter, which inevitably triples the memory usage concerning the optimizer states. Besides, with the growing size of the model, LLMs are becoming increasingly expensive in terms of memory, and the limitation of memory is gradually emerging as a main bottleneck for training LLMs. Many existing memory-efficient optimizers attempt to store second-order statistics with sublinear memory requirement while retaining the exceptional convergence property of adaptivity <cit.>. Adafactor optimizer achieves remarkable memory cost reduction by applying the non-negative matrix factorization algorithm <cit.> to factorize the accumulator matrix for squared gradients into two rank-1 factors as shown in Figure <ref>, where the memory requirement for the original matrix V decreases from O(nm) to O(n+m). Whereas, it is observed that Adafactor suffers a performance degradation in the training of large language models universally compared with conventional adaptive gradient-based optimization methods. The reason for this phenomenon is Adafactor inevitably introduces some errors that cause instability in training deep networks due to the operation of non-negative matrix factorization. In addition, in the case of large-batch training that aims to accelerate the training of deep neural networks, the memory consumption of each machine (GPU/TPU) is much higher than general batch size training, which further imposes a grave constraint on the performance of the trained model. In comparison to standard training tasks, large-batch training presents more challenges for optimizers. Empirically, when the mini-batch size increases after a certain point (e.g. 1024), the test accuracy of the converged solution decreases significantly compared with the baseline <cit.>. To our knowledge, there is currently no work related to memory-efficient optimizers for large-batch training. Motivated by these challenges, we firstly study a confidence-guided strategy catered to alleviate the instability of Adafactor by calculating the confidence of the generated update at each training step. On the basis of the adaptation strategy, we propose a novel CAME optimizer that saves nearly the same memory footprint as existing memory-efficient optimizers while attaining faster convergence and superior generalization performance. To further assess the scalability of our proposed algorithm, we consider an additional challenging experiment - performing large-batch training on BERT using CAME optimizer. Contributions of our paper can be summarized in the following: Inspired by training instability of Adafactor, we explore a confidence-guided strategy centered on the existing error in the raw updates of Adafactor for parameters of large language models. * In light of the dedicated strategy, we propose a novel optimization algorithm, CAME, for achieving faster convergence and less performance degradation catered at memory-efficient optimization. We further investigate the effect of the proposed memory-efficient optimization algorithm in large-batch training settings. * We demonstrate the powerful performance of CAME with extensive NLP experiments: CAME shows faster convergence and better generalization capability than Adam in BERT pre-training task with two different batch sizes (32k and 8k); in the training of GPT-2 model and T5 model, CAME achieves fast convergence speed as Adam without degrading of performance. Notably, in the large-batch training of the BERT model, CAME obtains comparable validation accuracy with LAMB using around 15% less memory usage. § RELATED WORK Memory Efficient Adaptive Optimization Memory efficient optimizers maintain the benefits of standard per-parameter adaptivity while significantly reducing memory footprint. Adafactor <cit.> proposes to reconstruct a low-rank approximation of the exponentially smoothed accumulator at each training step that is optimal with respect to the generalized Kullback-Leibler divergence. SM3 <cit.> divides the elements in the second-order gradient matrix into sets by the observed similarity of the elements, and each item in the generated approximation matrix is the minimum of the maximum value of each set in which it is located. The methods mentioned above behave poorly in the training of large language models and converge slowly, which raises a significant challenge for memory-efficient optimization methods. Large Batch Training A large-batch training scheme is preferred in distributed machine learning because of its ability to increase parallelism by enhancing large-scale cluster utilization. It has seen growing interest in recent years in large-batch training <cit.>. In particular, a layer-wise adaptive learning rate algorithm LARS <cit.> is proposed to scale the batch size to 32k for ResNet-50. Based on LARS, LAMB optimizer <cit.> can finish the BERT training in 76 minutes through TPU v3 Pod. Despite the success of these approaches for BERT models, the much larger batch size highly boosts the GPU usage which is prohibitively expensive and inaccessible to most researchers. Moreover, training with a large batch size incurs additional challenges <cit.>. Large-batch training is prone to converge to sharp local minima, since the number of interactions will decrease when the batch size is increased if the number of epochs is fixed, which causes a wide gap in generalization of the model<cit.>. Traditional methods seek to narrow the generalization gap by carefully tuning hyperparameters, such as learning rate, momentum, and label smoothing, to narrow the generalization gap <cit.>. Yet there have been few attempts to reduce memory usage in large-batch training, and the underlying challenge remains unclear. § METHOD In this section, we firstly provide a brief description of the Adafactor optimizer and discuss the errors contained in the update of Adafactor (erroneous update). We further study a confidence-guided strategy and introduce the proposed CAME in detail in light of the strategy. §.§ An overview of Adafactor The ℒ(θ) ∈ℝ represents the loss function that we plan to minimize, where θ∈ℝ^n × m is the parameter of the model. g_t is the gradient at step t, η is the learning rate, r_t and c_t are the exponential moving average of two low-rank factors for the second moments of the gradient. ϵ_1 is a small regularization constants and u_t is the current approximate update. TrainingAdversarial HARDer-Net Learningend DQNHardness-Guided Learningend [t] Initial parameters θ_0, learning rate η, momentum of update m_0, r_0, c_0, step t, regularization constant ϵ_1, exponential moving average parameters β_1, β_2, clipping threshold d θ_t not converge Compute g_t = ∇ f(θ_t-1) r_t = β_2 r_t-1 + (1-β_2) (g_t^2 + ϵ_1 1_n 1^T_m)1_m c_t = β_2 c_t-1 + (1-β_2) 1^T_n (g_t^2 + ϵ_1 1_n 1^T_m) v_t = r_t c_t / 1^T_n r_t u_t = g_t / √(v_t) û_t = u_t/max(1, RMS(u_t)/d) m_t = β_1 m_t-1 + (1 - β_1) û_t θ_t = θ_t-1 - η m_t Adafactor Optimizer In the training of large language models, Adafactor is required to apply momentum to ensure the convergence <cit.>, and the corresponding pseudocode is illustrated in Algorithm <ref>. The problem setting is as follows. Assume that we aim to minimize the expected value of an objective function f(θ). At each training step, we receive the loss derived from a mini-batch of data, and calculate the gradient g_t of the function based on the previous parameters. Subsequently, we update the exponential running averages of two factors for second moments of the gradient r_t and c_t, compute approximations for the second moments of the gradient v_t, and adjust the generated update (u_t) when RMS(u_t) surpasses a specific threshold value d as in: û_t = u_t/max(1, RMS(u_t)/d) where RMS(u_t) refers to the root-mean-square calculation of the components of u_t. Finally, the first moment of the adjusted update m_t is utilized to update the parameter, resulting in a new iteration θ_t. The optimization continues until the parameters converge and returns the final iteration θ_T as our approximate solution. Adafactor derives an effective solution for non-negative matrix factorization in the special case of rank-1 factors, which obtains the minimal Kullback–Leibler divergence <cit.> between the matrix V and the approximated matrix WH. The formulation of the solution is as follows, in which 1_m =(1,...,1) ∈ℝ^m represents a column vector of m ones: W = V1_m, H = 1_n^TV/1_n^TV1_m. It should be noted that Adafactor stores only the moving averages of these factors rather than the entire matrix V, yielding considerable memory savings and requiring memory usage proportional to O(n + m) instead of O(nm). §.§ Erroneous Update The non-negative matrix factorization operation in Adafactor will inevitably incur erroneous update in the training of deep neural networks. As shown in Figure <ref>, Adafactor always converge slower than Adam due to the existing error in calculated updates, which further limits the application scenarios of memory-efficient optimizers. As shown in Figure <ref>, two scenarios demonstrate how two types of erroneous updates are supposed to be handled in the ideal case. In Figure <ref>(a), the difference between the momentum of updates m_t and the current update u_t is large, illustrating that the historical experience for the update of original Adafactor contains high level of errors that will inevitably influence the stability of the training process. If we utilize the raw m_t to take an optimization step, the direction of optimization will deviate increasingly from the desired direction, which is reflected by the slow convergence and performance degradation of existing memory-efficient optimizers. By contrast, when the difference between m_t and u_t is small as shown in Figure <ref>(b), the momentum m_t is stable with limited errors and high confidence therefore a large optimization step is required with the updating direction close to m_t. Inspired by the erroneous update that is universal in existing memory-efficient optimizers, we firstly consider an efficient approach to decrease the side effect caused by insecure updating. Given m_t and u_t, we take the residual between them as the instability in the preserved momentum and set generated instability as the denominator of original m_t to more adaptively take an update step. Following is the formulation of the adjusted update u', where ϵ is the regularization constant: u'_t = m_t/√((m_t-u_t)^2 + ϵ) Extending further on the plain method, we propose a confidence-guided strategy that enables self-adjusted updates by taking the confidence of the raw update of Adafactor into consideration. The intuition behind the proposed strategy is to calculate the residual between the exponential moving average (EMA) of the update and the current update, which represents the deviation of the approximated update. The larger the deviation of the EMA value from the current generated update, the wider the error EMA of update contains, resulting in a lower level of confidence in the EMA of update. Obviously, we expect the optimizer to take a small update when it incorporates huge error (a large residual from the present update), while updating parameters more when the optimization process is stable (involved error of EMA is limited). Specifically, the EMA of update m_t is directly used to take an update step in Adafactor, while in our proposed strategy, m_t is divided by √(U_t), where U_t is the calculated instability matrix. Therefore, 1/√(U_t) is the confidence in the observation: viewing m_t as the prediction of the update, if m_t deviates greatly from u_t (U_t is large), which indicates a weak confidence in m_t, the optimizer performs a small optimization step; if u_t closely matches m_t, we have solid confidence in m_t, and correspondingly take a large optimization step. §.§ CAME Algorithm [t] Initial parameters θ_0, learning rate η, momentum of update m_0 = 0, r_0 = 0, c_0 =0, step t = 0, regularization constants ϵ_1, ϵ_2, exponential moving average parameters β_1, β_2, β_3, clipping threshold d θ_t not converge Compute g_t = ∇ f(θ_t-1) r_t = β_2 r_t-1 + (1-β_2) (g_t^2 + ϵ_1 1_n 1^T_m)1_m c_t = β_2 c_t-1 + (1-β_2) 1^T_n (g_t^2 + ϵ_1 1_n 1^T_m) v_t = r_t c_t / 1^T_n r_t u_t = g_t / √(v_t) û_t = u_t/max(1, RMS(u_t)/d) m_t = β_1 m_t-1 + (1 - β_1) û_t U_t = (û_t - m_t)^2 R_t = β_3 R_t-1 + (1-β_3) (U_t + ϵ_2 1_n 1^T_m)1_m C_t = β_3 C_t-1 + (1-β_3) 1^T_n (U_t + ϵ_2 1_n 1^T_m) S_t = R_t C_t / 1^T_n R_t θ_t = θ_t-1 - η/√(S_t) m_t CAME Optimizer Based on the proposed confidence-guided strategy, we develop a brand-new variant of memory-efficient optimization methods with faster convergence. Our proposed CAME optimization method successfully obtains the same rate of convergence as prevailing first-order optimization algorithms (e.g., Adam) and with almost equal memory cost to available memory-efficient optimizers (e.g., Adafactor). The pseudocode of CAME algorithm is specified in Algorithm <ref>. By calculating U_t at each training step, we employ non-negative matrix factorization on the instability matrix U_t following <cit.> where the generalized Kullback-Leibler divergence between V and WH is minimal. With U_t factorized into R_t and C_t, it is sufficient to store only the moving averages of these factors rather than the full matrix U_t, thus saving considerable memory footprint. We simply validate intuitions and the corresponding example is shown in Figure <ref>, in which the proposed CAME reaches the optimal point much faster than Adafactor. Learning rate is 10^-3 for all optimizers. In the example, we set the parameters of CAME to be the same as the default in Adafactor, β_1=0.9, β_2=0.999 and set extra β_3=0.9999 for CAME. § EXPERIMENTS In this section, we present extensive comparisons with existing optimizers on training tasks of three important large language models: BERT <cit.>, GPT-2 <cit.> and T5 <cit.>. §.§ Setup Dataset We perform experiments on the BookCorpus <cit.> and English Wikipedia with 800M and 2.5B words respectively. Furthermore, we focus on the GLUE benchmark <cit.>, SQuAD v1.1 dataset <cit.> and SQuAD v2.0 dataset <cit.> to demonstrate the performance of pre-trained BERT models with CAME optimizer. Model We evaluate the efficiency of our proposed CAME on three trending large language models: BERT, GPT-2 and T5. We further test the performance of CAME for large-batch training with BERT-Large. Compared methods The main baselines comprise two widely-used optimizers: classic optimizer Adam and memory-efficient optimizer Adafactor. With regard to large-batch training, LAMB optimizer is additionally considered when setting baselines. Implementation Detail We implement our optimization algorithm in Pytorch <cit.>. The parameters β_1 and β_2 in Algorithm <ref> are set as 0.9 and 0.999 respectively, and we search for optimal β_3 among {0.9, 0.99, 0.999, 0.9999, 0.99999}. We use 8 Tesla V-100 GPUs and set ϵ_1, ϵ_2 as 10^-30, 10^-16 in all experiments with gradient accumulation and model parallelism. Besids, we set η as 2 × 10^-4, 6 × 10^-4, 3 × 10^-4 for BERT-Large (32K), GPT-2, T5 training and apply learning rate warmup scheduling <cit.> to avoid divergence due to the large learning rate, by starting with a smaller learning rate η and gradually increasing to the large learning rate η. To make sure we are comparing with solid baselines, we use grid search to tune the hyperparameters for Adafactor, Adam and LAMB. We further improve the performance of large-batch training by applying Mixup <cit.> to scale the batch size up to 32,768. §.§ BERT Training We firstly present empirical results in the training task of BERT model to evaluate the performance of our proposed CAME optimizer, focusing on its larger variant, BERT-Large, which has 340M parameters in all. Following the default setting, we pre-train the BERT-Large model (L=24, H=1024) with a sequence length of 128 on 8 Tesla V-100 GPUs. The experiments were implemented with the code from NVIDIA [<https://github.com/NVIDIA/DeepLearningExamples>] and mainly include two types of batch sizes: 8k and 32k, one of which represents the widely used setting for pre-training BERT and the other denotes the training scenario under large-batch training. The empirical results are presented in Figure <ref> and Figure <ref>. As illustrated in Figure <ref>, CAME achieves a significant improvement compared with Adam and Adafactor. To be specific, CAME (66.5%) increases validation accuracy at with an increment 3.4% in comparison to Adafactor (63.1%) using same number of training steps (20k). Apart from Adafactor, our proposed CAME achieves better performance than Adam in the pre-training of BERT-Large model with a huge reduction of memory cost. To evaluate the performance of our proposed CAME for large-batch training, we scale the batch size for BERT-Large training to 32,768 on Wikipedia dataset. As illustrated in Figure <ref>, CAME consistently reaches a more remarkable improvement compared with Adafactor. We notice that the accuracy of CAME on BERT-Large pre-training is 68.0%, which is highly over the original Adafactor (61.9%) with same number of training steps. In addition, CAME reaches comparable accuracy with only half the training steps required for Adafactor. With batch size getting larger from 8k to 32k, CAME brings more enhancements to the training of BERT-Large in comparison with Adam and Adafactor. Compared with LAMB in large-batch training, CAME saves a high-level of memory footprint with slight training performance degradation. Memory Usage Comparison We set batch size to 1 to measure the memory usage of each optimizer more efficiently. As shown in Table <ref>, the two optimizers (Adam and LAMB) frequently employed for training large language models consume the highest amount of memory usage. Meanwhile, our proposed CAME optimizer exhibits a reduced memory footprint over the existing SM3 memory-efficient optimizer. As a consequence of our confidence-guided strategy in CAME, there is no doubt that CAME will introduce an increased memory footprint in comparison with Adafactor. However, the extra memory footprint incurred of CAME is almost negligible (1%) with a substantial performance improvement. For further demonstration of the memory saving effect of CAME, we expand BERT model to BERT-4B with 4 billion weights using the scaling method of GPT-3 <cit.>. We set the mini-batch size to 64 and the accumulation steps to 16 in this experiment. In Figure <ref>, we train BERT-4B with three different optimizers using PyTorch framework. As a result, CAME can save 47% memory footprint about optimizer states compared with Baseline (Adam) when the weights number of a model get to 4 billion. §.§ Downstream Tasks We select a representative set of downstream tasks to further demonstrate the performance of BERT models pre-trained by our proposed CAME. In this part we adopt BERT-Base model for the fine-tuning task and follow the originally published BERT-Base results in <cit.> and <cit.> as the main baseline. The learning rate is tuned on the dev set for each setting and each task is fine-tuned for three epochs. We compare the end-task performance of BERT-Base with the baseline on typical downstream tasks and the empirical results are presented in Table <ref>. The experimental results demonstrate the efficiency of our proposed CAME optimizer by showing that BERT-Base model trained with CAME on two batch sizes both achieve comparable performance to the baseline with less memory cost. In particular, we observe that BERT-Base model trained with large batch (32k) presents no performance degradation and even attains higher evaluation metrics scores on some downstream tasks. Specifically, the BERT-Base model trained on CAME improves on average by 0.5 across five metrics compared to the baseline, proving the feasibility of CAME for the large-batch training task. §.§ GPT-2 Training In addition to BERT pre-training task, we perform CAME-based training task on another typical large language model, GPT-2. Using the original structure of GPT-2 <cit.>, we specifically adopt GPT-medium (L=24, H=1024) with 345M parameters in our experiment. This implementation is based on the code provided by Megatron[<https://github.com/NVIDIA/Megatron-LM>]. Identically, we take English Wikipedia as the training dataset for this section. Unlike the pre-training of BERT in Section <ref>, we only concentrate on standard training batch size (128) for GPT-2 pre-training. The empirical results of validation loss are shown in Figure <ref>. We are able to find that CAME achieves similar convergence and final accuracy compared to Adam, which reveals an impressive improvement over the performance of Adafactor with comparable training steps. Moreover, as indicated in Figure <ref>, the validation perplexity of CAME presents the same convergence performance as Adam but faster convergence speed than Adafactor, which clearly supports the validity of CAME that has fast convergence as in traditional adaptive methods and low memory usage as in existing memory-efficient methods. For instance, the converged validation perplexity of CAME and Adafactor is 50.1 and 56.9 respectively, which yields a considerable improvement of 12.0%. §.§ T5 Training Finally, we report empirical results from a different large language model training task: Text-to-Text Transfer Transformer, T5. Concretely, we follow the architecture of T5 <cit.> and choose T5-Base (L=24, H=1024) with 220M parameters for the experiment. All of our implementations are also based on the code provided by Megatron. Similarly, we consider Wikipedia with 2.5B words as the training dataset in this part. As with the training of GPT-2 in Section <ref>, we only concentrate on standard training batch size (128) for T5. The comparison of CAME with Adafactor and Adam is conducted in the same manner as Section <ref>, and corresponding results of validation loss and validation perplexity are illustrated in Figure <ref> and Figure <ref> seperately. Note that CAME consistently obtains comparable convergence performance for validation loss and validation perplexity on par with Adam, while reducing similar memory usage as Adafactor. § CONCLUSION In this paper we propose a novel memory-efficient optimizer called CAME, which supports adaptive confidence-based updating guided by the residual between predicted update and generated update. CAME achieves a considerable improvement compared to existing memory-efficient optimizers in the training of large language models, with an ignorable extra memory footprint. Moreover, CAME shows comparable convergence to Adam and LAMB with huge memory reduction. In particular, CAME has proven effective for large-batch training, which serves as an advantageous extension to memory-efficient optimizers. We hope our work will provide insight into memory reduction of optimizers in future exploration. § LIMITATIONS Despite the success of our CAME optimizer in training large language models with memory efficiency, there are still some limitations that need to be addressed in the future. Our proposed memory-efficient optimizer introduces additional computation costs for the non-negative matrix factorization of the instability matrix in comparison with Adafactor. We observe, however, that the training time of CAME increases only slightly in our experiments. Beyond that, CAME exhibits minor performance degradation in large-batch training of the BERT-Large model versus LAMB, which allows for further improvement in the future. Meanwhile, it is possible to conduct further experiments on other models in other fields, such as Computer Vision and Reinforcement Learning, thereby exploring the effectiveness of CAME training under more application scenarios. As a final point, it would be much more helpful to provide an in-depth theoretical analysis of CAME to improve comprehensiveness of the paper. § ACKNOWLEDGEMENTS Yang You's research group is being sponsored by NUS startup grant (Presidential Young Professorship), Singapore MOE Tier-1 grant, ByteDance grant, ARCTIC grant, SMI grant and Alibaba grant. We also thank Huawei Noah's Ark Lab for providing the necessary computing resources and support for datasets. acl_natbib
http://arxiv.org/abs/2307.11759v2	20230701125352	Progress Towards Untethered Autonomous Flight of Northeastern University Aerobat	[ "Adarsh Salagame" ]	cs.RO	[ "cs.RO", "cs.SY", "eess.SY" ]	[1]Covercover To my family. [1]Table of Contentscontents CHAPTER: LIST OF ACRONYMS tocchapterList of Acronyms FWMAVFlapping Wing Micro Aerial Vehicle BLDC MotorBrushless DC Motor ESCElectronic Speed Controller Used to control the three-phase voltage to a BLDC motor to control its speed. IMUInertial Measurement Unit VIOVisual Inertial Odometry PWMPulse Width Modulation FPVFirst Person View GPUGraphics Processing Unit ROSRobotic Operating System This work has been supported by a lot of people in a number of different ways. I would first like to acknowledge the mentorship and guidance provided to me by Dr. Alireza Ramezani over the course of this thesis, opening up new opportunities and perspectives for research. I would also like to acknowledge Dr. Hanumanth Singh and Dr. Milad Ramezani for their guidance. As a collaborative project involving multiple specialities, a number of students have contributed to make this work possible. I would like to acknowledge the foundational work done by Eric Sihite at Caltech University, and Xintao Hu and Bozhen Li at Northeastern University. I would also like to acknowledge the work done by Roman Snegatch, Hamza Iqbal, Arunbhaarthi Anbu, Rohit Rajput, Yizhe Xu and Xuejian Niu, and the brainstorming and troubleshooting support from the rest of the Silicon Synapse Lab. Finally, I would like to acknowledge my parents and everyone who made this work possible through their constant motivation and heartfelt support. State estimation and control is a well-studied problem in conventional aerial vehicles such as multi-rotors. But multi-rotors, while versatile, are not suitable for all applications. Due to turbulent airflow from ground effects, multi-rotors cannot fly in confined spaces. Flapping wing micro aerial vehicles have gained research interest in recent years due to their lightweight structure and ability to fly in tight spaces. Further, their soft deformable wings also make them relatively safer to fly around humans. This thesis will describe the progress made towards developing state estimation and controls on Northeastern University's Aerobat, a bio-inspired flapping wing micro aerial vehicle, with the goal of achieving untethered autonomous flight. Aerobat has a total weight of about 40g and an additional payload capacity of 40g, precluding the use of large processors or heavy sensors. With limited computation resources, this report discusses the challenges in achieving perception on such a platform and the steps taken towards untethered autonomous flight. headings CHAPTER: INTRODUCTION Flapping Wing aerial locomotion is an interesting field of study that is gaining a lot of research interest <cit.>. Flapping robots offer a number of advantages over conventional aerial robots such as quad-copters, which rely on propeller based lift generation. The biggest of these is their ability to fly in confined spaces. Quad-copters and other multi-rotor vehicles are heavily affected by turbulent air flow when flying in confined spaces or close to the ground <cit.>. On the other hand, flapping wing robots have the opposite effect, not only being able to fly in tight spaces aided by their high agility, but also showing higher efficiency when flying close to the ground, a phenomenon well studied in birds <cit.>. This makes flapping wing robots a huge potential asset for applications in disaster management, for example flying through the narrow spaces inside a collapsed building, for applications in inspection such as flying through sewers or air vents that are inaccessible to humans and other types of robots, or even for data collection for scientific research in previously inaccessible areas. A further advantage of flapping wing robots is their relative safety to operate. With soft deformable wings and significantly smaller weight density, they are not only safer than propeller based aerial robots to operate around people, they are less affected by crashes into walls or ceilings and can continue flying. And finally, flapping wing robots are extremely agile, able to perform zero momentum turns, and are more efficient in their agility when compared with multi-rotor systems that rely on thrust vectoring for their agility, which is very power hungry. <cit.> For all these advantages, however, flapping wing robots still pose a number of challenges that must be solved before they may fully reach the impact that multi-rotors have had. Flapping wing systems generate much less thrust when compared to multi-rotors of similar size. This severely impacts the available payload for sensors and other electronics that would enable the robot to be fully autonomous. Further, these are highly dynamic platforms, with flapping motions causing vibrations that an onboard perception system must deal with <cit.>. Also, unlike multi-rotors, flapping systems have a constantly shifting center of mass, affected not only by the wing position, but also by the variable deformations in the wings and any inherent compliance in their structure due to their lightweight designs. These factors make localization and autonomous control of the robot a challenge. In order to develop autonomous flight, two things are required: * Low Level Control: The ability to track any desired trajectory and accurately execute any desired motion * High level control: The ability to decide what trajectory or motion to execute based on knowledge about the robot state and it's surroundings. High level control may be further divided into two sub-goals: * Perception and State Estimation: Understand the surrounding environment and localize the robot within this space * Trajectory Planning: Decide a trajectory to follow based on the perception and state estimation All of these are eventual goals for Aerobat. However, this work focuses on making progress towards Perception, State Estimation and Low-level control. The thesis is organized according to these goals as follows. Chapter <ref> goes through contemporary works on aerial and flapping wing systems, focusing specifically on works that have had success with autonomous flight. Chapter <ref> describes initial results with open loop untethered flight and the development made towards safe and controlled testing of untethered flight. Chapter <ref> describes the progress made towards low level control of Aerobat, describing the aerodynamic model of Aerobat and validation of the aerodynamic model. Chapter <ref> describes the progress made towards developing onboard perception and state estimation, with a special focus on the limited payload capacity available and the challenges in implementation on limited computation hardware. Finally, Chapter <ref> presents an overview of the milestones reached, challenges faced and future development to take place towards untethered autonomous flight. § ABOUT AEROBAT Northeastern University's Aerobat is a tail-less flapping wing robot that, unlike existing examples, is capable of significantly morphing it's wing structure during each gait cycle. The robot, with a weight of 40g (when carrying a battery and a basic microcontroller) and a wingspan of 30 cm, was initially developed to study the flapping-wing flight of bats. Aerobat utilizes a computational structure, called the Kinetic Sculpture (KS) <cit.>, that introduces computational resources for wing morphing. The KS is designed to actuate the robot's wings as it is split into two wing segments: the proximal and distal wings, which are actuated by what is the equivalent of shoulder and elbow joints, respectively. The gait captures the wing folding during the upstroke motion, which is one of the key modes in bat flight. The wing folding reduces the wing surface area and minimizes the negative lift during the upstroke and results in a more efficient flight. Aerobat is capable of flapping at a frequency of up to 8 Hz. Without a tail, Aerobat is unstable in its longitudinal (pitch dynamics) and frontal (roll dynamics) planes of flight. CHAPTER: RELATED WORK Flapping Wing Micro-Aerial Vehicle (FWMAV) platforms in the literature may be broadly divided into two categories based on the size of the robot. Insect-scale platforms such as Harvard's Robobee <cit.> and University of Washington's RoboFly <cit.> range in weight from a few milligrams to a few grams (<10g). These typically implement offboard processing with little to no payload budget for onboard sensors. Other examples of platforms in this category are Robo Moth <cit.>, Delfly Micro <cit.>, Jellyfish Flier <cit.> and Insectothopter <cit.>. Larger-scale platforms such as TU Delft's DelFly <cit.> and Purdue's Hummingbird <cit.> weigh in the order of tens of grams and are capable of carrying sensors and sufficient processing onboard for basic perception. Other platforms in this category are UC Berkeley's DASH and BOLT <cit.> and KUBeetle-S <cit.>. On the extreme end of this scale are large ornithopters such as the University of Seville's GRIFFIN <cit.>, RoboRaven <cit.>, FESTO Smart Bird, Pidgeonbot <cit.>, MIT Phoenix <cit.> and EPFL's morphing wing robot <cit.> which all weigh in the order of a few hundred grams, comparable in weight and payload capacity to multi-rotors. All these platforms are compared in Figure <ref>by plotting them on a logarithmic scale of mass and wingspan. Northeastern University's Aerobat sits uniquely in the middle of the range of FWMAV sizes. With a wing span of 30 cm and weighing 40g (when carrying a battery and a basic microcontroller), it is small enough that it can be agile, but also is capable of carrying an additional 40g of payload which can be used for sensors and processing to develop autonomy, which is significantly larger than other comparably sized platforms. In contrast, the comparably sized DelFly Explorer has a wingspan of 28cm and weighs 20g including autonomy electronics and the Purdue Hummingbird, which has a wingspan of 17cm weighs 12g. The larger payload budget on Aerobat opens up the possibility of pushing the envelope for onboard perception and state estimation in flapping wing systems. Perception and state estimation in flapping wing robots is severely limited by the amount of computation possible onboard, and there are a limited number of works that have successfully demonstrated any level of onboard autonomy. <cit.>, <cit.> and <cit.> use optical flow for low-level control. Of these, only <cit.> and <cit.> perform the computations onboard. <cit.> uses a stereo rig to perform obstacle avoidance using onboard computation of disparity maps. The authors demonstrate autonomous avoidance of pillars during flight, but this method would struggle in more unstructured environments where depth information about obstacles needs to be more precise. <cit.> exploits its soft deformable wings by using them as sensors to detect wall collisions to navigate through a confined space. All of these works carry out onboard computation on small micro-controllers that can only handle basic autonomy. However, Aerobat's larger payload can support better processing, giving the opportunity to attempt more state of the art approaches. The state of the art in aerial robot perception has been established largely using multi-rotor platforms or offboard computation. Specifically, visual inertial approaches have gained much popularity due to cameras being cheap, lightweight and easily available, and complementing the noisy but high rate inertial data provided by an IMU. These have been implemented with variations in the type of visual data used (feature based <cit.> or direct <cit.>), the number of data points considered (full history, sliding window, latest only), methods for matching and estimation, back-end optimization <cit.>, loop closures, etc. and have been tested in various multi-rotor applications <cit.>. Considering the popularity and versatility of visual inertial odometry, this is chosen as the approach of choice for Aerobat. However, these algorithms typically require heavy processors to run in real time. Chapter <ref> Section <ref> provides more details about the selection of processor on Aerobat and a comparison with processors typically used in these applications, however, here it is sufficient to say that care must be taken in selecting the visual inertial algorithm to be implemented on Aerobat. <cit.> compares the performance of visual inertial odometry algorithms on various processors, noting the amount of CPU and RAM usage, processing time and accuracy. Figure <ref>, from <cit.> shows the graph of performance of different Visual Inertial Odometry (VIO) algorithms. Of the processors compared in this work, Odroid XU4 and Intel Up Board relevant for comparison with Aerobat. With 2GB and 4GB of available RAM respectfully, they are on the lower end of typically used processors in these applications. More details about the two and a comparison is presented in Section <ref>, but the results of this paper indicate that although there is a drop off in accuracy, lighter algorithms such as Multi-State Constrained Kalman Filter (MSCKF) and Semi-direct Visual-inertial Odometry + Multi-Sensor Fusion (SVOMSF) can run on limited hardware. Heavier algorithms such as SVO+GTSAM, OKVIS and ROVIO have larger processing times and consume more memory, but are also more accurate, and may potentially be implemented if the processing capacity of Aerobat is improved. This provides hope that with further optimization and tailoring of these and newer algorithms to Aerobat's specific application, it is possible to run these state of the art algorithms close to real time on limited hardware. CHAPTER: TOWARDS UNTETHERED FLIGHT Northeastern University's Aerobat is a project in development since 2016. <cit.> describe the development of the mechanical structure and actuation mechanism. <cit.> describe the development of simulation models and trajectory planning. <cit.> achieved tethered hovering flight indoors using these models on our indoor tethered test platform Aerobat Gamma. The next stage of development was focused towards developing a second version of Aerobat, called Aerobat Beta for testing untethered flight outdoors. Aerobat Beta was designed and built as part of an earlier Master's thesis presented in <cit.>. As a test platform, Aerobat Beta has lightweight laser-cut foam wings that are easily replaceable. The original goal of Aerobat Beta was to test lift-generation capabilities in isolation, and to that end, stabilizers were added to stabilize the roll and pitch axes in flight, allowing the wings to generate lift based on an open loop PWM signal sent to the motor. Stabilization was carried out using a simple PD controller that read acceleration and gyroscope values from an onboard IMU to calculate roll and pitch. PWM signal data and IMU data were relayed to a ground-station computer over Bluetooth for debugging. All this was controlled onboard by an Arduino Pico micro-controller weighing 1g, with 24kB of memory. At the start of the work presented in this thesis, Aerobat Beta was flying with intermittent success over short 3-5m distances. Testing was carried out indoors and the main focus was on increasing consistency of flight. The primary source of flight inconsistencies was found stem from the gear mechanism that keeps the two wings in sync. Additional inconsistencies came from poorly calibrated ESCs for the stabilizers and imbalanced weight. After strengthening 3D printed parts, cleaning up the gear mechanism and calibrating the ESCs, more consistent flight was observed, until finally 5-7m untethered flight was consistently achievable indoors. Figure <ref> shows one such flight. The snapshots show untethered flight before Aerobat hits the safety net, showing orientation correction in the process. The modifications that allowed this to happen served only as temporary fixes and necessitated constant maintenance of the hardware to keep Aerobat in fly-worthy condition. As an early test platform, however, this was acceptable at the time and testing was continued. With consistent flight demonstrated indoors, Aerobat was taken outdoors for longer distance flights than could be executed in the indoor space available. Outdoor tests pose an additional challenge in the form of wind. Without closed loop control and only orientation based stabilization, testing can be difficult. However, with intermittent consistency, 10m outdoor flight was demonstrated. Figure <ref> shows one such flight, again showing Aerobat correcting undesired roll to continue flying. This result sufficiently demonstrated lift generation capabilities of Aerobat Beta, and focus was shifted towards a long term fix for the gear mechanism and development of closed loop control. Chapter <ref> describes the progress made towards development of closed-loop control. The rest of this chapter, however, will be dedicated to describing the development carried out to enable the work in Chapter <ref> and beyond. § TOWARDS SAFE TESTING OF UNTETHERED FLIGHT One of the issues faced while testing Aerobat outdoors was crashes. With foam wings and no protection, each crash would lead to large reset times, allowing for only a few tests to be conducted in a given time period. As more aspects of control are developed, the ability to perform multiple repeatable tests quickly will become very important. To this end, a guard design was proposed that would protect Aerobat in the event of crashes, reduce reset times and allow a large number of tests to be carried out. Figure <ref> shows the proposed guard design with Aerobat mounted at the center. It has been named Kongming Lamp after the traditional Chinese lantern for it's distinctive shape and the safety it represents for Aerobat. Consisting of three concentric ellipses covering each of the three axes, this is designed to be a lightweight compliant addition that protects the robot in the event of a crash. Made of 11 lightweight carbon fiber rods, the structure provides strength and elasticity that would absorb impact in a crash. The rods are bound together by small snap-fit 3D printed parts that are optimized to reduce the weight to the minimum required. To test the strength of the guard, it was drop tested to see how a load at the center equivalent to the robot would survive. Figure <ref> shows the compliance of the structure absorbing the impact and protecting the representative weight. An additional modification made in the interest of testing more advanced control is shifting the stabilizers from Aerobat to the guard and providing the guard with its own IMU. Having the guard independently stabilized isolates the robot from the guard dynamics and allows it to be used as much or as little as needed. Eventually, these stabilizers and the guard itself will be phased out and Aerobat will be robust enough to fly on its own. Figure <ref> shows the full guard design with stabilizers and IMU. The guard is stabilized with the help of four BLDC motors arranged in a quad-copter-like configuration. The control algorithm for the guard runs on Aerobat's processor and uses feedback from its own IMU for independent control. Within RISE Arena, it is fitted with markers and tracked using Optitrack Motion Capture to provide pose information to the controller. Figure <ref> shows the controller logic used to stabilize the guard. For simplicity, only the roll and pitch orientations of the guard are stabilized, and velocity in only the x and y directions is considered. Altitude control will be part of future development. Stabilizing the guard is challenging due to the compliant nature of the structure. The motors and IMU are mounted on snap-fit 3D printed parts that may slide along the carbon fiber rod. The rods themselves also stretch over time and the relative positioning between the motors is not rigid. This leads to challenges in tuning the controls for the guard as it needs to be robust enough to compensate for all these inconsistencies. § ROBOTICS-INSPIRED STUDY AND EXPERIMENTATION (RISE) ARENA In order to develop controls for Aerobat, a fully controlled and repeatable environment is required where each aspect of Aerobat's dynamics may be isolated and individually studied. It needs a safe environment to test and tune controls in a rigorous and repeatable manner before it is ready to be taken outdoors for fully untethered flight. The Robotics-Inspired Study and Experimentation (RISE) Arena was created to provide this controlled test environment. Figure <ref> shows the setup of RISE Arena. At the center of it is the indoor tethered test platform Aerobat Gamma (Fig. <ref>). Aerobat Gamma is a tethered version of Aerobat with flexible electronics in its wings. It is mounted on a highly sensitive ATI 6-axis load cell (shown in Fig. <ref>). The robot and the load cell together are mounted at the end of a programmable 6 DOF manipulator. One side of RISE Arena is entirely covered by a large array of fans that can generate wind speeds of up to 2 m/s and the whole area is covered by 6 Optitrack Motion Capture Cameras. The robotic arm offers the ability to create trajectories with precise ground truth information available and do highly repeatable experiments. The arm is interfaced through Ethernet using a Python API. A wrapper was developed for the API that added new functionality, making it easier to interface with the arm, generate trajectories and execute predefined movements. Using the wrapper, keyboard teleoperation of the arm was developed, allowing a user to move the arm to any location and save the coordinates as waypoints in a trajectory. The waypoints may be saved and fed to different programs that execute different trajectories, controlling the duration and smoothness of the trajectory, and the number of loops of the trajectory to execute. It also enables setting protection zones (Fig. <ref>) to protect the arm and the robot from collisions within RISE arena, allowing safe testing of controls. From the motion capture cameras and the load cell, RISE Arena provides ground truth for flapping frequency, robot pose, lift generated, and aerodynamic forces on the robot, allowing controlled motion and pose within known stable wind conditions, making this a powerful tool for testing. RISE Arena has been used throughout this work, from validating the the aerodynamic model to testing the guard controller to calibrating sensors and testing perception. § CONCLUDING REMARKS In this chapter, the preliminary results for untethered flight was presented with successful indoor and outdoor flight tests demonstrating a proof-of-concept for untethered flight. These flights were open loop. Future development will be focused towards developing closed loop control, with initial steps for this described in Chapter <ref>. To better enable testing controls in closed loop flight, this chapter also describes the development of Kongming Lamp, a lightweight protective guard around Aerobat to save it from crashes and stabilize it while controls are being tuned. Finally, this chapter describes the development of indoor test setup RISE Arena, providing elaborate ground truth and a controlled repeatable environment for system identification and testing of controls. RISE Arena is far from a finished product, with many developments planned, including "free flight" of the robot while still attached to the manipulator using admittance control, incorporating more precise aerodynamic sensing and wind pattern detection and adding offboard processing to test more experimental and advanced algorithms. CHAPTER: AEROBAT MODELING This chapter describes the progress made towards developing a control model of Aerobat capable of executing trajectories. In order to do this, a model must be developed mapping between robot motion and control inputs to the actuators. <cit.> makes progress towards this with a description of the aerodynamic model. The dynamic modeling is derived using an unsteady aerodynamic model from the Wagner model and lifting-line theory <cit.>. Aerobat has 20 degrees of freedom, but due to the nature of the kinetic sculpture of Aerobat's mechanism, this can be reduced to just 7 degrees of freedom (6 for the body and 1 for the motor that controls the flapping) with the rest expressed as kinematic constraints. The dynamical equation of motion used in the simulation can be derived using Euler-Lagrangian dynamical formulations. Figure <ref> shows the free-body diagram of the robot, which can be presented using 5 bodies: main body, proximal and distal wings of both sides. The synchronized wing trajectory allows us to just use one side of the wing in the states. Let q = [ p^⊤, θ^⊤, q_s, q_e]^⊤ be the generalized coordinates, where p is the body center of mass inertial position, θ is the Euler angles of the body, q_s and q_e are the left wing's shoulder and elbow angles, respectively. The dynamical equation of motion of the simplified system can be defined as follows: M( q) q̈ = h( q, q̇) + u_a + u_t + J_c^⊤λ J_c q̈ = [q̈_s, q̈_e]^⊤ = y_ks, where M is the inertial matrix, h is the gravitational and Coriolis forces, u_a and u_t are the generalized aerodynamic and thruster forces, respectively. λ is the Lagrangian multiplier which enforces the constraint forces acting on q_s and q_e to track the KS flapping acceleration y_ks. λ can be solved algebraically from <ref> given the states x = [ q^⊤, q̇^⊤]^⊤ and both generalized forces u_a and u_t. These generalized forces can be derived using virtual displacement, as follows: u_a = ∑_i=1^N_b B_a,i( q) f_a,i u_t = ∑_i=1^N_t B_t,i( q) f_t,i where B matrices map the forces f ∈ℝ^3 to the generalized coordinates q, N_b is the number of blade elements, and N_b is the number of thrusters. Let the position p_k( q) be the inertial position where the force f_k defined in the inertial frame is applied. The matrix B_k for this force can be derived as follows: B_k = ( ∂ṗ_k / ∂q̇)^⊤. The aerodynamic forces generated on each blade elements and thrust forces are combined to form u_a and u_t, respectively. The aerodynamics can be derived using discrete blade elements following the derivations in <cit.>. This model uses the lifting line theory and Wagner's function to develop a model for calculating the lift coefficient. Let S be the total wingspan and y ∈ [-S/2, S/2] represents a position along the wingspan. The vortex shedding distribution can be defined as a function of truncated Fourier series of size m across the wingspan, as follows: Γ(t,y) = 1/2 a_0 c_0 U ∑^m_n=1 a_n(t) sin(n θ(y)) where a_n is the Fourier coefficients, a_0 is the slope of the angle of attack, c_0 is the chord length at wing's axis of symmetry, and U is the free stream airspeed. Let θ be the change of variable defined by y = (S/2)cos(θ) for describing a position along the wingspan y ∈ (-S/2, S/2). From Γ(t,y), we can derive the additional downwash induced by the vortices, defined as follows: w_y(t,y) = - a_0 c_0 U/4S∑^m_n=1 n a_n(t) sin(n θ)/sin(θ). Following the unsteady Kutta-Joukowski theorem, the sectional lift coefficient can be expressed as follows: C_L(t,y) = a_0 ∑^m_n=1( c_0/c(y) a_n(t) + c_0/Uȧ_n(t) ) sin(nθ), where c(y) is the chord length at the wingspan position y. The computation of the sectional lift coefficient response of an airfoil undergoing a step change in downwash Δ w(y) << U can be expressed using Wagner function Φ(t): c_L(t,y) = a_0/UΔ w(t,y) Φ(t̃) Φ(t̃) = 1 - ψ_1 e^-ϵ_1 t̃ - ψ_2 e^-ϵ_2 t̃ where t̃(t) = ∫_0^t (v_e^i/b) dt is the normalized time which is defined as the distance traveled divided by half chord length (b = c/2). Here, v_e^i is defined as the velocity of the quarter chord distance from the leading edge in the direction perpendicular to the wing sweep. For the condition where the freestream airflow dominates v_e, then we can approximate the normalized time as t̃ = Ut/b. The Wagner model in (<ref>) uses Jones' approximation <cit.>, with the following coefficients: ψ_1 = 0.165, ψ_2 = 0.335, ϵ_1 = 0.0455, and ϵ_2 = 0.3. Duhamel's principles can be used to superimpose the transient response due to a step change in downwash as defined in (<ref>). Additionally, integration by parts can be used to simplify the equation further, resulting in the following equation: C_L(t,y) = a_0/U( w(t,y) Φ(0) - ∫_0^t∂Φ(t - τ)/∂τ w(τ, y) dτ). ∂Φ(t - τ)/∂τ = -ψ_1 ϵ_1 U/b e^-ϵ_1 U/b(t-τ) -ψ_2 ϵ_2 U/b e^-ϵ_2 U/b(t-τ) Here, w(t,y) is the total downwash defined as: w(t,y) = v_n(t,y) + w_y(t,y), where v_n is the airfoil velocity normal to the wing surface which depends on the freestream velocity and the inertial dynamics. Finally, we can represent the integrals as the following states: z_1 (t,y) = ∫_0^tψ_1 ϵ_1 U/b e^-ϵ_1 U/b(t-τ) w(τ,y) dτ z_2 (t,y) = ∫_0^tψ_2 ϵ_2 U/b e^-ϵ_2 U/b(t-τ) w(τ,y) dτ. Both of these states can be expressed as an ODE by deriving the time derivatives of (<ref>). They can be derived using Leibniz integral rule, yielding the following equations: ż_1 (t,y) = ψ_1 ϵ_1 U/b( w(t,y) - ϵ_1 U/b z_1(t,y) ) ż_2 (t,y) = ψ_2 ϵ_2 U/b( w(t,y) - ϵ_2 U/b z_2(t,y) ). The sectional lift coefficient can then be defined as: c_L(t,y) = a_0/U( w(t,y) ϕ(0) + z_1(t,y) + z_2(t,y) ), and we can march the aerodynamic states z_1 and z_2 forward in time using (<ref>). Finally, we can relate the both sectional lift coefficient equations in (<ref>) and (<ref>) to solve for the Fourier coefficient rate of change, ȧ_n. The aerodynamic states are defined along the span of the wing and can be discretized into m blade elements. Therefore, we can derive the m equations relating (<ref>) and (<ref>) on each blade element to solve for the ȧ_n. Then, including z_1 and z_2 on each blade elements, we will have 3m ODE equations to solve. We can represent a_n, z_1, and z_2 of all blade elements as the vector a_n ∈ℝ^m, z_1 ∈ℝ^m, and z_2 ∈ℝ^m, respectively. This model was simulated in <cit.> (Fig. <ref>) and partially validated by the IMU data from untethered flight tests . However, to fully validate the model and close the loop, a more controlled testing setup is required. § VALIDATION OF AERODYNAMIC MODEL Using RISE Arena, the aerodynamic model presented in <cit.> was validated. Aerobat was set to flap at a fixed known frequency of about 2 Hz and load cell measurements were taken for headwind speeds of 0.5, 1.0, and 1.5 m/s. The results closely match the simulation, validating this model (Fig. <ref>) . § CONCLUDING REMARKS This chapter presented the aerodynamic model of Aerobat and the steps taken towards validating it using the newly setup RISE Arena (Section <ref>), taking Aerobat one step closer to closed-loop control. Future development will be focused towards system identification and addition of more degrees of actuation into the wings, allowing Aerobat to control roll and pitch dynamics. CHAPTER: PERCEPTION CHALLENGES AND PRELIMINARY WORKS As described in the introduction (Chap. <ref>), Aerobat needs both high level and low level control in order to execute autonomous flight. This chapter describes the progress made towards developing perception and state estimation onboard Aerobat, from selecting the electronics required to hardware and software integration and sensor calibration. § ONBOARD ELECTRONICS When selecting the onboard electronics for Aerobat, a soft payload limit of 15g was imposed on the selection for autonomy electronics. This was done to allow for additional stabilizers used for testing in the initial stages of development while the controls are still under research. §.§ Processor In order to achieve autonomy, the onboard processor must be powerful enough to interface with multiple sensors and execute control algorithms at a high enough rate. With a 15g payload limit, this did not offer a lot of options, forcing a compromise between processing power and weight. Multirotors have a larger payload capacity and can carry large processors. <cit.> use Odroid XU4 and Intel UP board onboard, both of which weigh around 40g and come with 4-core 2GHz 64-bit processors with 2/4GB of RAM. Some such as <cit.> use even larger processors such as Intel NUC (9 cores, 1.1GHz, 16GB RAM) or one of the NVIDIA Jetson series: Nano (4-core CPU, 128-core GPU, 4GB RAM), TX2 (2-core CPU, 256-core GPU, 4GB RAM), Xavier (8-core CPU, 512-core GPU, 32 GB RAM), which all weigh in the order of a few 100g. All these options are far beyond the payload capacity of Aerobat. At the other extreme, small microcontroller boards such as Arduino Nano, Arduino Pico, and Raspberry Pi Pico are becoming more capable. Arduino Pico (weighing just 1g) was used in the initial stages of Aerobat flight tests to control the actuator and stabilizer motors (Fig. <ref>). However, all of these have under 1MB of memory and are not practical for high-level computation. Arduino Portenta is a small 2-core microcontroller board that can simultaneously run an Arduino loop on one core and computer vision algorithms on the other, with support for TensorFlow Lite. Arduino Nicla Vision is another lightweight microcontroller board option that has a camera, IMU, and distance sensor embedded in the board itself and supports TinyML, OpenMV, and MicroPython. Both Arduino Portenta and Nicla come with Bluetooth and WiFi embedded. These are potentially attractive options for specialized applications. However, Portenta has just 8MB of memory (expandable up to 64MB) and Nicla is even lower at just 2MB, which is not sufficient for the level of autonomous computation targeted for Aerobat. After some consideration, the Raspberry Pi Zero 2 W (Fig. <ref>) was chosen as the ideal compromise. Weighing 11g, the Raspberry Pi Zero 2 W runs on a 4-core 1GHz 64-bit ARM processor with a Linux-based operating system. It has 512MB of RAM, Wi-Fi, and Bluetooth capability, and has an interface for a Raspberry Pi camera. This is still relatively powerful for its size, and at the time of writing to the best of my knowledge, is the most powerful processor weighing less than 15g available on the market. §.§ Sensors The choice of onboard sensors depends on the approach used for autonomy and the environment in which it is designed to operate. For example, an IR camera might be suitable for dark environments such as night flight, and simple laser rangefinders might suffice for maintaining a steady heading within a confined space such as a tunnel. Other options considered included Sonar rangefinders, optical flow sensors, and stereo cameras. At this early development stage, however, priority is given to versatility that would allow for testing under a range of environments and applications. With this in mind, a single monocular RGB camera and IMU were chosen as the onboard sensors, using a visual-inertial odometry approach for localization. §.§.§ Camera In the original configuration of Aerobat (Fig. <ref>), to get a sense of what images from an onboard camera would look like, a small FPV wireless camera was used (Fig. <ref>). It consisted of the camera itself with an attached antenna and dedicated battery mounted on Aerobat, and a radio receiver connected to a laptop offboard through USB. This streamed 640x480 resolution image at 30Hz with no noticeable lag with line of sight communication. The camera weighs 4.53g. Fig. <ref> shows an image from this camera. Without a microprocessor onboard, this camera allowed us to see the world from Aerobat's perspective while it was flying. However, this could not be a long-term solution as the only interface to this camera was through the wireless receiver. Although a few options for camera modules were compared, with the Raspberry Pi Zero 2 W selected as the onboard processor, the natural choice was to use the Raspberry Pi Camera (Fig. <ref>) as the interfaces were already in place. The Raspberry Pi Camera Model 2 weighs 3g and has an 8-megapixel sensor that offers video streaming up to 1080p at 30 fps and 720p at 60 fps. Fig. <ref> compares images from the Raspberry Pi camera and the previously used FPV camera. While the FPV camera has a wider field of view, definition in features is lost in the center of the scene when compared with the Raspberry Pi Camera. The Raspberry Pi Camera also offers a few additional perks. It allows for setting the internal sensor update rate, independent of the rate at which images are read from the camera. This allows for low motion blur in images even when reading from the camera at low frame rates. It also has an internal GPU that gives it the ability to adjust exposure, shutter speed, brightness, contrast, saturation, and rotation, taking the load off the processor. Section <ref> describes development of camera drivers. §.§.§ Inertial Measurement Unit (IMU) In this early stage of development, flight tests typically last just a few seconds at a time, removing IMU drift as a factor. However, there were two requirements for the IMU to meet: * Data rate: For good visual inertial odometry, it is ideal to have visual data at at around 20 Hz and inertial data at around 200 Hz. Therefore, the IMU must be able to provide data at 200 Hz or more. * Weight: With 14g of payload taken up by the processor (11g) and camera (3g), there is only 1g of the imposed payload limit left for the IMU. Therefore, the IMU must weigh 1g or under. Professional grade IMUs such as the VN-100 would be superfluous and expensive options at this stage when hobby-grade components are able to meet the requirements while being lighter in weight and lower in cost. Popular hobby-grade IMUs such as Adafruit's MPU6050 and ICM-20948 can comfortably meet these requirements, weighing just 1g and giving high data rates up to 400 kHz through I2C, limited only by the read/write speed of the interfaced processor. At the time of development, however, these were unavailable due to the ongoing chip shortage. Some IMUs such as Adafruit's BNO055 and WIT-motion's WT901 have similar weights and perform sensor fusion onboard to provide useful data such as absolute orientation, gravity-corrected linear acceleration, and gravity vectors. BNO055 only has a maximum rate of 100 Hz but WT901 has a maximum update rate of 200 Hz, meeting the desired data rate. Due to its availability and suitability to the requirements, this was chosen as the IMU for Aerobat (Fig. <ref>). The sensor is interfaced by I2C communication and internally updates registers containing the following information: * Linear Acceleration (x, y, z) * Angular Velocity (x, y, z) * Magnetic Field Strength (x, y, z) * Kalman Filtered Absolute Orientation in Euler angles (Roll, Pitch, Yaw) * Kalman Filtered Absolute Orientation in Quaternion (x, y, z, w) * Temperature Each value is stored in 2 Bytes of memory, bringing a total of 34 Bytes of information available to be read in each sampling of the IMU. Section <ref> describes how this data is read onboard by the IMU driver. § SENSOR INTEGRATION Results from <cit.> indicated that most standard perception approaches would struggle to run on the Raspberry Pi Zero's 512 MB of RAM, and so initial efforts were focused on testing the capabilities of the processor, the camera, and the IMU. §.§ Camera Camera drivers were implemented onboard using Python3 and various algorithms were tested for performance with greyscale images at resolutions of 640x480 and 320x240 and 30 frames per second, including SIFT feature detection, Sparse and Dense Optical Flow, and Apriltag Detection. All of these comfortably ran onboard, utilizing less than 20% of available memory. The field of view in these cases, however, appeared to be smaller than the field of view of the camera. Upon investigation, it was determined that the field of view was intentionally being clipped based on the resolution of the image and the framerate. This behaviour is described by the chart in Fig. <ref> from the Picamera documentation. To get around the clipped field of view issue, the driver was modified to read images at the full resolution (1640x922) and then resize it on the processor to the desired resolution. This gives the full field of view shown in Fig. <ref>. §.§ IMU IMU drivers were also implemented and interfaced using I2C communication and read rates of up to 1 kHz were achieved. Confident that the processor was capable of handling more, a bare-bones version of ROS Noetic was installed from source, containing only libraries required for the ROS perception stack. Continuing further testing of the processor, IMU and camera ROS drivers were implemented and tested at various publishing rates. Initial results showed that while memory usage was acceptable at around 25%, processing times were highly variable at higher rates. Figure <ref> shows the large variation in the time period of the IMU data being published over a period of 2 min of recording at a desired rate of 200 Hz. The variation in timestamps looks asymmetrical because of the way ROS handles timing. When a message takes longer than the desired rate to publish, the next message is published almost immediately with no delay. This leads to a number of messages with close to zero time difference from the previous message. This variation is further increased as the additional node to publish camera data is run alongside it <ref>. Investigation showed one of the causes to be long read times for I2C communication with the IMU. This was initially implemented as individual register reads for each of the data provided by the IMU, but optimization by reading a large contiguous block of data instead allowed reducing the number of I2C reads from 16 to 2. This considerably sped up the operation, improving performance. Further improvements to the timing were made by using timed callback functions using ros::Timers rather than rate-based sleep functions in loops to read data from the sensors and reducing the frequency from 200 Hz to 150 Hz. Figure <ref> shows the improved performance of the IMU at 150 Hz. There are still non-trivial variations in the time periods in the data. This is likely due to the number of parallel processes in operation. Running two nodes (one for camera and one for IMU) through ROS spawns over 10 threads, which, on a 4-core processor such as the Raspberry Pi Zero, would cause interruptions that increase the time between successive data reads. On a faster processor, this may not pose a challenge, with the processor able to keep up with the desired rate despite interruptions. However, this is likely a hardware limitation of the Raspberry Pi Zero. § SENSOR CALIBRATION Camera and IMU were first individually calibrated. The standard checkerboard and Matlab's camera calibration toolbox was used to calibrate the camera and 6 hours of stationary bias-corrected data was used with <cit.> to calibrate the IMU. With this calibration data, camera and IMU were calibrated together using Kalibr's <cit.> camera-imu calibration script. Using a 4x6 1cm tag size aprilgrid as the calibration target, RISE Arena's manipulator was used to move the robot around, exciting all axes of the IMU. However, despite low camera re-projection errors and estimated accelerometer and gyroscope errors in the prior, optimization fails to find a solution for this setup. This is as yet an open issue, with potential sources of error being the same timing issues still affecting the data, IMU axes not being excited enough or the camera not getting a wide enough field of view for the data. § CONCLUDING REMARKS This chapter described the challenges in selecting and integrating electronics on a tight payload budget, and the challenges associated with running the perception stack onboard with limited hardware. With a fully integrated perception system, future work will be focused on utilizing RISE Arena (Section <ref>) to test VIO algorithms onboard and evaluating their feasibility and challenges in implementing autonomous flight for Aerobat. Work will also be required to integrate the kinematics and dynamics of Aerobat into the perception algorithm for more robust state estimation. Using these, Aerobat will be flown autonomously within RISE Arena using the perception system to stay within the boundaries of the confined space while performing aerial maneuvers. CHAPTER: CONCLUSION This thesis presents the progress made towards Autonomous Untethered Flight on Northeastern University's Aerobat. This was broken down into three primary goals with the progress towards each described in their own chapter. § CHAPTER <REF> Chapter <ref> described progress made towards untethered flight. A proof-of-concept 10m outdoor untethered flight was demonstrated and two additional development was presented, towards enabling future testing for untethered flight. The first of these was the protective guard, Kongming Lamp (Sec. <ref>), which was drop tested with a representative weight at the center to demonstrate protection for Aerobat from crashes. The second development was RISE Arena (Sec. <ref>), providing elaborate ground truth information for controlled and repeatable testing and system identification. §.§ Thesis Contributions For the outdoor untethered flight demonstration, stability of flight was improved by tuning the complementary filter applied to calculate orientation from IMU for stabilization. For the design of Kongming Lamp, in addition to conceptual inputs, control code for stabilization using IMU and pose information was developed and tuned. In addition, RISE Arena was fully developed as a part of this thesis, including interfacing and control code for the manipulator, calibration of Optitrack system and integration of processing and sensing onto Aerobat-Gamma. § CHAPTER <REF> Chapter <ref> described the aerodynamic model of Aerobat and the steps taken towards validating the model. Preliminary results indicate the model is accurate, but further system identification is required to fully map out the control system and experimentally test the model in-flight under different wind conditions. Predicated on the success of this, outdoor closed-loop tests may be performed with the help of Kongming Lamp (Sec. <ref>) until Aerobat is ready to fly completely unsupported. §.§ Thesis Contributions Created the manipulator trajectories and performed the experiment using RISE Arena to collect data for validation of the aerodynamic model. § CHAPTER <REF> Chapter <ref> described the progress made towards onboard perception and state estimation. Processors and sensors were selected and integrated onto the robot (Sec. <ref>). Sensor drivers were written and iteratively optimized for timing issues and speed of processing (Sec. <ref>). ROS was installed and tested on the limited processing power available on Aerobat and preliminary data for VIO was collected with the help of RISE Arena. As an immediate next goal, this data will be run through different VIO algorithms to verify the quality of the data and benchmark the algorithms. §.§ Thesis Contributions This chapter describes research and development fully carried out as part of this thesis. § FUTURE WORK This work will be continued as part of my doctoral study, and as further progress is made on each of these goals, Aerobat will be at a mature stage where technology demonstrations may be made such as: * Controlled near ground flight akin to birds and bats demonstrating higher efficiency of near ground flight * Long distance flights demonstrating the high efficiency of flapping wing systems * Autonomous flight within a straight tunnel demonstrating precise closed loop control in confined areas * Autonomous flight within a tunnel maze demonstrating the high agility of flapping wing systems and their ability to open up previously inaccessible spaces ramezani_bat_2016, ramezani_nonlinear_nodate, ramezani_modeling_2016, hoff_synergistic_2016, hoff_reducing_2017, ramezani_describing_2017, ramezani_biomimetic_2017, syed_rousettus_2017-1, hoff_optimizing_2018, hoff_trajectory_2019, ramezani_towards_2020 sihite_mechanism_2020, sihite_computational_2020, sihite_enforcing_2020, sihite_orientation_2021, ramezani_aerobat_2022, sihite_unsteady_2022
http://arxiv.org/abs/2307.01895v2	20230704193918	First Image of the Sun with MeerKAT Solar Observations: Opening a New Frontier in Solar Physics	[ "Devojyoti Kansabanik", "Surajit Mondal", "Divya Oberoi", "James O. Chibueze", "N. E. Engelbrecht", "R. D. Strauss", "Eduard P. Kontar", "Gert J. J. Botha", "P. J. Steyn" ]	astro-ph.SR	[ "astro-ph.SR", "astro-ph.IM" ]	0000-0002-0786-7307]Devojyoti Kansabanik National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, S. P. Pune University Campus, Pune 411007, India 0000-0002-2325-5298]Surajit Mondal Center for Solar-Terrestrial Research, New Jersey Institute of Technology, 323 M L King Jr Boulevard, Newark, NJ 07102-1982, USA 0000-0002-4768-9058]Divya Oberoi National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, S. P. Pune University Campus, Pune 411007, India 0000-0002-9875-7436]James O. Chibueze Department of Mathematical Sciences, University of South Africa, Cnr Christian de Wet Rd and Pioneer Avenue, Florida Park, 1709, Roodepoort, South Africa Centre for Space Research, Physics Department, North-West University, Potchefstroom 2520, South Africa Department of Physics and Astronomy, Faculty of Physical Sciences, University of Nigeria, Carver Building, 1 University Road, Nsukka 410001, Nigeria Centre for Space Research, Physics Department, North-West University, Potchefstroom 2520, South Africa Centre for Space Research, Physics Department, North-West University, Potchefstroom 2520, South Africa 0000-0002-8078-0902]E. P. Kontar School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK 0000-0003-2099-8093]P. J. Steyn Centre for Space Research, Physics Department, North-West University, Potchefstroom 2520, South Africa, Solar radio emissions provide several unique diagnostics to estimate different physical parameters of the solar corona, which are otherwise simply inaccessible. However, imaging the highly dynamic solar coronal emissions spanning a large range of angular scales at radio wavelengths is extremely challenging. At GHz frequencies, the MeerKAT radio telescope is possibly globally the best-suited instrument at the present time and can provide high-fidelity spectroscopic snapshot solar images. Here, we present the first images of the Sun made using the observations with the MeerKAT at L-band (856 – 1711 MHz). This work demonstrates the high fidelity of the MeerKAT solar images through a comparison with simulated radio images at the MeerKAT frequencies. The observed images show extremely good mophological similarities with the simulated images. A detailed comparison between the simulated radio map and observed MeerKAT radio images demonstrates that there is significant missing flux density in MeerKAT images at the higher frequencies of the observing band, though it can potentially be estimated and corrected for. We believe once solar observations with the MeerKAT are commissioned, they will not only enable a host of novel studies but also open the door to a large unexplored phase space with significant discovery potential. § INTRODUCTION Since the discovery of solar radio emission <cit.>, the Sun has been studied in great detail in a wide range of frequencies spanning from a few kHz to several hundreds of GHz. Despite this long history of observations and studies, the Sun still harbors several mysteries. With each leap of technological advancement in building new telescopes, several of these mysteries are solved. At the same time, these new advancements probe the Sun in a very new light and hence generally open up a very rich discovery space. Interesting results coming from new instruments like the Solar orbiter <cit.>, Parker Solar Probe <cit.>, Daniel K. Inouye Solar Telescope <cit.>, Murchison Widefield Array <cit.>, LOw Frequency ARray <cit.>, Expanded Owens Valley Solar Array <cit.>, the NenuFAR <cit.>, the Owens Valley Long Wavelength Array <cit.> are testament to this. Most of these new-generation radio telescopes are not dedicated to solar observations (except EOVSA), but these are the ones that are expected to open up large expanses of pristine phase space unexplored yet. The MeerKAT <cit.> is another such new generation radio interferometric array, which can open a new frontier in solar radio physics at GHz frequencies. MeerKAT, originally known as the Karoo Array Telescope, is a new-generation radio telescope located in the MeerKAT National Park in the Northern Cape of South Africa. It consists of 64 dishes, each with a diameter of 13.5 m. Each dish is equipped with a cryogenically cooled receiver, making it extremely sensitive. At present, MeerKAT has three observing bands – UHF (544–1087 MHz), L (856–1711 MHz), and S (1750–3499 MHz) bands. The array is centrally condensed with about 39 dishes lying within 1 km and the remaining dishes distributed within a radius of ∼8 km. This provides MeerKAT with extremely good surface brightness sensitivity and also allows the generation of radio images with extremely high dynamic range and image fidelity <cit.>. The dense array layout of MeerKAT also implies that it has an excellent spectroscopic snapshot sampling in Fourier plane (uv-plane) as shown in Figure <ref>. This provides a very well-behaved point-spread-function (PSF) of the MeerKAT array and makes it well-suited for high dynamic range (DR) spectroscopic snapshot imaging. This capability is extremely useful for solar studies at radio wavelengths due to the fast dynamics seen in solar radio emissions both in the spectral and temporal domain <cit.>. There are several avenues where high dynamic range snapshot imaging can lead to extraordinary science ranging from the direct estimation of the magnetic field of the coronal mass ejections (CMEs) close to the Sun to studies of nonthermal emission from extremely weak radio transients. Here we present the first imaging observation of the Sun with the MeerKAT. Unlike standard astronomical observations, solar observations with any radio telescope pose several challenges. These challenges need to be addressed before MeerKAT can be used for solar observations. The primary reason behind this is that MeerKAT was designed for observing faint astronomical sources. To observe the Sun, the source with the highest flux density in the sky, strong attenuators need to be used to ensure that the astronomical signal stays in the linear regime of the instrument. However, these same attenuators cannot be used to observe the available calibrators as these sources are orders of magnitude weaker than the Sun. In the absence of these calibrator observations, it is hard to estimate the instrumental gains and efforts that are ongoing toward solving these issues. Here we use a different technique to observe and image the Sun. Instead of pointing at the Sun, we pointed ∼ 2.5^∘ away to keep it in the sidelobes of the primary beam to attenuate the solar emissions. The sensitivity of the MeerKAT is sufficient to image the Sun even when it is in the sidelobes of the primary beam. Availability of holographic measurements of the MeerKAT primary beam up to the second side lobe <cit.> allows us to obtain flux density calibrated solar images. We notethat there are some shortcomings of this observing strategy. Among them, the chromatic nature of the primary beam makes the sensitivity over the solar disc non-uniform. Despite these shortcomings our work substantiates the excellent imaging quality of the MeerKAT solar data and showcase its potential for enabling excellent solar science. This paper is organized as follows. Section <ref> presents the details of the observations. Section <ref> describes the data analysis procedure, including calibration, imaging, and primary beam correction. In Section <ref>, we present our results and demonstrate some early results we achieved using these data. Finally, in Section <ref>, we conclude by giving a future outlook of the MeerKAT solar observation. § OBSERVATIONS These observations were done as a Director's Discretion Time (DDT) and Science Verification (SSV) observation under project ID SSV-20200709-SA-01. Raw visibilities are already available in the public domain through SARAO data achieve[<https://archive.sarao.ac.za/>]. The observations were carried out during the 6th perihelion passage of Parker Solar Probe <cit.> from 2020 September 24 to 2020 September 30. On each day, there are about 3 hours of observations centered around 10:30 UTC. In this paper, we present results from two of these epochs – 2020, September 26, and 2020 September 27. Observations were done covering 856–1711 MHz with 32 K spectral channels and 8 s temporal resolution. This provides us with data at about 26 kHz spectral resolution. Standard MeerKAT flux density calibrator, J0408-6545, was observed at the start of observation. J0408-6545 is used for bandpass and flux density calibration (hereafter referred to as fluxcal). J1239-1023 is used as a phase calibrator (hereafter referred to as phasecal) and observed between each consecutive solar scan. Since the Sun is a non-sidereal source, its RA–DEC changes with time. Hence, the pointing center is changed every 15 minutes. For all the pointings, the Sun is kept at ∼2.5^∘ away from the pointing center. The position of the Sun on the primary beam for different scans is shown for three different frequencies in Figure <ref>. It turns out that at the lower part of the band, the Sun is in the first side lobe of the primary beam, while it lies in the second side lobes or null at the higher parts of the band. This essentially makes the observations at the lower part more sensitive than the high parts of the band. Observing the Sun keeping it at the sidelobes provide about -50 to -90 dB attenuation, depending on the frequency, to the total solar power, which is essential to keep the signal in the linear regime all through the signal chain. § DATA ANALYSIS Since the observation does not fall under the standard astronomical observation category, we did not use SARAO Science Data Processor (SDP) pipelines for the analysis. Instead, we did the analysis manually using Common Astronomy Software Application () <cit.> for flagging and calibration and <cit.> for imaging. §.§ Flagging and Calibration The flowchart of flagging and initial calibration procedures are shown in Figure <ref>. Initial flagging is performed to remove bad antennas, bad channels, and other strong radio frequency interferences (RFIs). After that, initial calibration rounds are done using the fluxcal and phasecal. A total of five rounds of initial calibration were done, each followed by post-calibration flagging steps. Detailed procedures for flagging and initial calibration are discussed in Appendix <ref> and <ref>, respectively. Once initial calibration is done, calibration solutions are applied to the solar scans, and self-calibration is performed. The Sun is present in the sidelobes of the MeerKAT primary beam. Due to the chromatic nature of the primary beam, sensitivity across the solar disc varies with frequency. On the other hand, the Sun is a non-sidereal source. So, the position of the Sun in the equatorial coordinate system changes with time. Hence, we first moved the phasecenter of the measured visibilities to the solar center and then performed self-calibration for each 20 MHz spectral and 15-minute temporal chunk separately. Improvements in DR of the images with self-calibration iterations are shown in Figure <ref>. A detailed description of the self-calibration procedure followed here is presented in Appendix <ref>. §.§ Final Imaging and Primary Beam Correction 1 Images of the Sun centered at 2020 September 27, 10:45 UTC. 1.1 Avg_image_Sep27 Avg_image_Sep27.pdf Top panel: Normalized average image over the entire MeerKAT L-band on 2020 September 27, 10:45 UTC. Bottom panels: Images at different 20 MHz spectral chunks across the L-band. The small cyan dot at the bottom left is the point-spread-function of the array. Once the self-calibration is done, we make final images of the Sun for each spectro-temporal chunk separately. For the final imaging, we used all baselines. All other imaging parameters – the number of w-layers, visibility weighting, uv-taper, multiscale parameters, and pixel size are kept the same. During the final imaging, we did not use any pre-defined mask. Instead, we use parameter available in to perform deconvolution down to 3σ, where σ is the rms calculated close to the Sun. Due to the chromatic nature of the primary beam, the low-frequency part of the band has better sensitivity compared to the high-frequency part. This is evident from the spectral images shown in the bottom panels of Figure <ref> and <ref>. At high frequencies, emissions from the active region bright points are detected with good detection significance, but the extended emission from solar limbs are not detected at all frequency chunks. To image all the structures detected across the full band, we have convolved all images at the resolution at the lowest frequency of the observing band. Then we normalized each 20 MHz spectral image with respect to the peak flux and stacked all spectral chunks for a given scan to obtain a full band image shown in the top panel of Figure <ref> and <ref>. Since the Sun is present at the sidelobes of the MeerKAT primary beam, we have to correct the primary beam response across the solar disc to obtain the correct flux density. Holographic measurements of the MeerKAT primary beam <cit.> at L-band is available[https://skaafrica.atlassian.net/wiki/spaces/ESDKB/pages/1481572357/The+MeerKAT+primary+beam#A-note-on-sidelobesMeerKAT holographic measurements of the primary beam.] over the extend of 4 degrees at an angular resolution of ∼223 arcsec. We did linear interpolation to obtain the beam values at each pixel of the image. For alt-az mount telescopes, the sky rotates with respect to the telescope beam, and this rotation is denoted by parallactic angle. If the beam of the instrument is axially symmetric, then parallactic angle correction is not important for Stokes I imaging. As evident from Figure <ref>, while the main lobe of the MeerKAT primary beam is close to axially symmetric, that is not true for its sidelobes. In the present observation, the Sun is present on the sidelobes of the primary beam. Hence, we performed parallactic angle correction to the primary beam before performing the primary beam correction. We have done image plane primary beam correction using the array-averaged response. Being at the first/second side lobe of the primary beam, flux density measurements can have errors due to primary beam measurements. Considering different kinds of errors as discussed in <cit.>, we consider a conservative 10% error on the measured flux density. § RESULTS In this section we present the results from first MeerKAT solar observations and compare it with simulated MeerKAT solar maps at frequencies spanning our L-band observations. §.§ First Solar Image using MeerKAT The first images of the Sun made using MeerKAT L-band observations are shown in Figures <ref> and <ref>. The top panel shows the average image over the entire MeerKAT L-band and the lower panels show images at individual 20 MHz bands spanning the full observing band. The entire solar disc is clearly visible once images over the full band are stacked together. We find that the solar disc is about 35 arcmin in diameter, slightly larger than the optical disc. MeerKAT images are overlaid on 193Å images from the Atmospheric Imaging Assembly (AIA) onboard Solar Dynamics Observatory <cit.> in Figure <ref>. The largest active region is co-located with the brightest radio source in the MeerKAT images. There are multiple small active region bright points visible in AIA image, which are also detected in MeerKAT images with high significance. In both of these images, the diffuse quiet Sun emission from both the limbs is also detected with good significance. Although, visually both the images show features similar to those seen in the AIA images, we go further to verify this via comparison with corresponding simulated solar radio images. §.§ Simulating Solar Radio Images and Spectra The simulated images only aim to capture the free-free thermal emission. To generate simulated images, a differential emission measure (DEM) inversion using images at different wavelengths from the AIA/SDO is performed. To reduce the computation time and imnprove the signal-to-noise of the obtained DEMs, the AIA images were smoothed to a resolution of 4.8 arcsec, before DEM inversion. Note, though that the resolution of these smoothed images is still higher than those of the MeerKAT radio images. Following <cit.>, we use the output of publicly available code[<https://github.com/ianan/demreg/tree/master/python>] to compute the expected free-free emission using the code developed by <cit.>. A uniform line-of-sight depth of 100 Mm is assumed all through the image. A chromospheric contribution has been included, assuming that it is proportional to observations at 304Å. The proportionality constant is determined assuming that the chromospheric contribution to the total brightness temperature (T_B) is 10880 K <cit.>. The left panel of Figure <ref> shows the simulated T_B map of the Sun and the right panel shows the same map convolved to the MeerKAT angular resolution. It is evident from these figures that there are emission at a range of angular scales all the way from instrumental resolution to the size of the solar disc. We note that the simulation does not incorporate any propagation effects like scattering or refraction. While their importance is well established, taking these into account appropriately is beyond the scope of this work. §.§ Comparing Simulated MeerKAT Images and Observations Radio interferometry is a Fourier imaging technique, where each baseline of the interferometer measures one Fourier component of the radio sky. Hence, the quality of the images and the scales of emission captured rely crucially on the sampling of the Fourier plane achieved by the interferometeric observations. In order to build the appropriate simulated image for comparison with the observed MeerKAT images, we first create simulated visibilities from the simulated images using MeerKAT array configuration and observing parameters used for these observations. These visibilities are then inverted to make the ideal simulated image which would have been observed by MeerKAT. A comparison between simulated MeerKAT image thus obtained and the observed MeerKAT image at the same time is shown in Figure <ref>. Left panel shows the simulated MeerKAT map and the right panel shows the observed map from MeerKAT. The similarities between the simulated and observed images are very evident. The most striking similarities are the locations and intensities of the various bright points, some of them have been marked by cyan circles in both the panels. There are also differences, the prominent ones are the presence of noise in the regions beyond the Sun. the limb being not as bright and well defined in the MeerKAT image as compared to the simulated image and the differences in details of the morphology of the brightest active region. While the first of these can be attributed to the combined effect of the thermal noise associated with the image, the imperfections in the calibration and imaging process, scattering in the solar atmosphere plays a significant role for the others. Using a simplistic description for scattering in the solar atmosphere, <cit.> found that at ∼1 GHz, one does not expect to detect sources with angular sized <10 arcsec. So one can justifiably expect the finer features approaching this angular size to be washed out in the MeerKAT map, even though they are larger than the instrumental resolution. §.§ Comparison of Observed and Simulated Spectra In this section, we compare observed spectra with the expected spectra from the simulated images. As has been mentione earlier, during these observations the Sun was in the sidelobes of the highly chromatic primary beam of the MeerKAT. At lower parts of the band (<1300 MHz), the Sun was in the first side lobe of the primary beam, while at the higher frequencies it was in the second or higher sidelobes, as evident from Figure <ref>. For further analysis we have chosen spectral points which satisfy the following two conditions: * The Sun should not lie beyond the first side lobe of the primary beam, and the value of the primary beam value towards the Sun should be > 0.001 of the peak. * The emission should be detected at a level > 5σ, where σ is the rms noise of the primary beam corrected image measured very close to the Sun. We have extracted spectra for two bright active regions present on the Sun, which are marked by red and green circles in the top panel of Figure <ref>. Corresponding spectra from this Meerkat image are shown by filled circles in the bottom panel of Figure <ref>. The conditions mentioned above are satisfied only below 1070 MHz and that limits the span of the spectra shown here. We have extracted spectra of these two regions from the corresponding simulated thermal radio maps (one example is shown in Figure <ref>), which are shown by solid lines in the left bottom panel of the same figure. It is evident that observed values shown by filled circles in the same figure are significantly different from the simulated values. We note that the simulation describes a rather ideal situation and can differ from observations due to several reasons, including the following: * Simulation assumes the thermal free-free emission from the coronal plasma to be only emission mechanism in operation. In reality, however, the emission would be a superposition of the thermal free-free emission and gyrosynchrotron/gyroresonance emission <cit.>. * The simulation ignores any propagation effects, while in reality refraction and scattering can lead to discernible effects. * Interferometers are sensitive only to variations in the brightness distributions and not to a constant background. This implies that interferometers have a tendency to not be sensitive to emissions at large angular scales. The details of the largest angular scale to which an array is sensitive depends upon the details of the array configuration and the sampling of the Fourier domain achieved by the observation under study. This can lead to a reduction in the observed flux density when compared to simulated values. It is eminently feasible to isolate the impact of the last possibility mentioned above. To do this, we sampled the simulated map of the Sun using exactly the same Fourier sampling as achieved by the MeerKAT observations and then Fourier inverted it to generate a synthetic simulated map which can be compared directly with the MeerKAT solar maps for an apples-to-apples comparison. The spectra from these synthetic simulated maps are shown by unfilled diamonds in the left panel of Figure <ref>. The spectra from synthetic simulated maps are consistent with those from the observations. This demonstrates that the large discrepancy between the observed and simulated maps is primarily due to the missing flux density in the MeerKAT maps. The ratio of the flux density measured in the simulated synthetic map to that in the simulated map is defined to be missing flux density fraction and is plotted in the right panel of Figure <ref>. The missing flux density fraction decreases with the decrease in frequency. For a given array layout, one samples increasingly shorter spacings in the uv-plane with decreasing frequency and missing flux density fraction is expected to drop. The observed variation in the missing flux density fraction show this trend and substantiates this to be cause of the observed differences between the simulated and MeerKAT solar images. As expected, the missing flux fractions for both the regions show similar spectral behavior. While the other two reasons mentioned above could also be contributing to the observed differences, their effects, however, are smaller than the uncertainty on these measurements. §.§ Comparison between observed image with that obtained using EOVSA data In this section, we compare the image produced using MeerKAT data with that obtained using data from the Expanded Owens Valley Solar Array <cit.>. EOVSA is a solar-dedicated instrument operating between 1–18 GHz. It has 13 antennas each of diameter 2.1 metres for solar observations. Due to the small number of antennas, the imaging dynamic range obtained with EOVSA is not sufficient to image the quiet sun disc with fidelity enough to make a direct comparison with the image shown in Figure <ref>. Hence we restrict ourselves to comparing only the diameter of the solar disc and the peak brightness temperature detected in the EOVSA image. Full day brightness temperature maps from EOVSA can be obtained directly from the observatory [<http://ovsa.njit.edu/browser/?suntoday_date=2020-09-26>]. The image at 1.4 GHz is shown in Figure <ref>. The peak brightness temperature reported at 1.4 GHz by the observatory is 0.58 MK. Although the EOVSA image and the MeerKAT image are not at the same time, we note that, only one C1.1 flare was reported from this active region at 23:59 UT. The radio signature of this flare, as can be seen from the EOVSA data, was extremely weak and had a very short duration as well. Additionally during the generation of the full-day image, the flare durations were flagged to ensure that the flare does not affect the image. Hence we believe that the peak brightness temperature of the two images obtained using EOVSA and MeerKAT data should be roughly comparable and we find this to be the case as well. From the EOVSA image, it is clear that even after full day synthesis, the quiet sun disc is not detected. Hence we follow a different technique to determine the diameter of the solar disc. The technique is also developed by the EOVSA team and is detailed in <http://www.ovsa.njit.edu/wiki/index.php/Full_Disk_Simulations>. The basic principle is that the solar disc can be approximated by a disc at these frequencies. Hence it is possible to determine the radius of the disc by fitting the visibilities with the fourier transform of the disc which is a Airy disc function. The functions to perform these fits are available in the https://github.com/suncasa/suncasa-srcSUNCASA package. We perform the fitting by using all baselines whose length is smaller than 800λ. Since the fitted equatorial radius and polar radius were different and we are unable to measure the radius with high accuracy along these two directions in the MeerKAT data, we compare the obtained radius from MeerKAT data with the average of the equatorial and polar radius obtained from the fit. The average diameter at 1.1 GHz turns out to be 35.8^”, which is quite comparable to the rough estimate of 35^” obtained from the produced image. We use the 1.1 GHz band of EOVSA data for comparison, as the flux density close to the solar limb in the normalised MeerKAT image is influenced mostly by the frequencies close to 929 MHz, where the limb is bright and majority of the solar disc can be seen quite nicely. §.§ Background Radio Sources for Heliospheric Observations In this section we focus on the applicability of MeerKAT for heliospheric observations. Beyond 10 R_⊙, in the outer coronal regions and heliosphere, radio emission from coronal and heliospheric plasma can not be detected directly using the ground-based radio interferometers. Although, space-based radio instruments may detect radio emissions from these heliocentric distances. Currently, no space-based radio imaging instrument is available. However, a mission comprising six CubeSats aimed at space-based interferometric imaging of solar type-II radio bursts is currently in an advanced stage of development. This mission is named the Sun Radio Interferometer Space Experiment <cit.>, and it will perform radio interferometric imaging in the frequency range 0.1 – 25 MHz. At present, beyond 10 R_⊙, in the outer coronal regions and heliosphere, several coronal and heliospheric plasma parameters can be measured using some indirect methods. These are: * Interplanetary scintillation: The plasma density fluctuations (Δ N) in the turbulent solar wind and CMEs can be measured through a phenomenon known as “Interplanetary scintillation (IPS)", which was first reported by <cit.>. In the past few decades, increasingly sophisticated algorithms have been developed for 3D reconstruction of global heliospheric parameters; density, and velocity, using observations of multiple IPS radio sources covering the entire heliosphere <cit.>. IPS observations are routinely done from SKAO-low precursor, the MWA, keeping the Sun at the null of the primary beam <cit.> and also used to study CMEs <cit.>. While the MWA observing frequency is suitable for IPS observations at larger heliocentric distances, the MeerKAT observing frequency is suitable for IPS observations at higher coronal and inner heliospheric regions. * Faraday-rotation measurements: When linearly polarized radiation passes through a magnetized plasma, its plane of polarization rotates. This phenomenon is known as Faraday rotation (FR). The amount of FR is proportional to the line-of-sight (LoS) integral of the product of electron density and LoS component of the magnetic field and also to the squared of the wavelength of observation. The wavelength-independent part of FR is expressed in terms of a quantity referred to as the Rotation Measure <cit.>. FR measurements of linearly polarized emission through CMEs <cit.> offer a promising remote-sensing tool to measure the vector magnetic field in CMEs both at the corona and the inner heliosphere <cit.>. Recent developments about FR measurements due to heliospheric plasma are discussed in <cit.>. To enable these heliospheric observations with the MeerKAT close to the Sun, the first step will be detecting numerous background galactic/extra-galactic radio sources at sufficient fidelity keeping the Sun at null/sidelobes of the primary beam. The current observation is an excellent match to test this requirement. § CONCLUSION AND FUTURE WORK The Sun is an extremely complicated radio source with emissions at angular scales ranging from few tens of arcseconds to the size of the solar disc at GHz frequencies, as is evident from simulated radio maps shown in Figure <ref>. Hence to study the solar radio emission at GHz frequencies, one requires a high DR and high-fidelity spectroscopic snapshot imaging of the Sun. Sufficiently dense spectroscopic snapshot uv-coverage of the MeerKAT allows high DR imaging of the Sun. Solar observation with the MeerKAT is not yet commissioned, and these observations were done keeping the Sun in the sidelobes of the primary beam. Here, we have presented the first images of the Sun with the MeerKAT. To the best of our knowledge, given the well-behaved spectroscopic snapshot PSF and the precise calibration, the images presented here are perhaps the highest quality spectroscopic snapshot solar images at these frequencies available to date. To demonstrate the capability of MeerKAT in producing very high-fidelity solar images, we have compared the MeerKAT images with the simulated synthetic images designed to sample exactly the same Fourier components as the MeerKAT observations. The correspondence between the observed and simulated images shown in Figure <ref> is remarkable and it is evident from that several weak solar emissions present in the synthetic image are detected with high significance in the MeerKAT image. Although the spatial structures in the observed image match well with the simulated image, from a science perspective, it is also important to test the ability of MeerKAT for determining the flux densities and spectra of solar features. As substantiated in Section <ref>, the MeerKAT spectra show evidence for missing flux at higher frequencies which drops to insignificant levels by about 900 MHz. An implication is that while the MeerKAT images in the UHF band are not expected to suffer from the missing flux density issue, one will need to be careful about the missing flux density at L band and higher. A comparison with simulated maps radio maps might provide a good way to quantify the missing flux density fraction for specific observations. While it is adequate for demonstrating the feasibility of MeerKAT for solar observations and evaluating the quality of the images it can deliver, a key limitation of the present approach arises from the issues related to imaging a source of large angular size in the chromatic primary beam sidelobes. This was however necessitated by the requirement to attenuate the solar signals to a level which would keep the signal chain downstream in its linear regime. A preferable approach for solar observing will be to keep the Sun in the main lobe of the primary beam, and adjust the gains of the appropriate elements of the signal chain to attenuate the signal to the required levels. Some members of this team are currently working with the MeerKAT engineering team to develop a calibration strategy for solar observations along these lines. Once enabled, we are convinced that, with its with its high-fidelity spectroscopic snapshot solar imaging capability, MeerKAT solar observations will open a new frontier in solar radio physics. The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation. The authors acknowledge the contribution of all those who designed and built the MeerKAT instrument. D.K. and D.O. acknowledge support of the Department of Atomic Energy, Government of India, under the project no. 12-R&D-TFR-5.02-0700. S.M. acknowledges the partial support from USA NSF grant AGS-1654382 to the New Jersey Institute of Technology. This research has also made use of NASA's Astrophysics Data System (ADS). MeerKAT <cit.>, Solar Dynamics Observatory <cit.>. astropy <cit.>, matplotlib <cit.>, Numpy <cit.>, CASA <cit.> Here we discuss in the steps followed for initial flagging, initial calibration, post-calibration flagging and self-calibration. § FLAGGING AND DATA EDITING On both the epochs, there was no dead antenna. We perform flagging on the 32 K channel data. We have visually examined the data and identify the frequency channels with persistent radio frequency interference (RFI). We first flag any data with zero amplitudes. We then flag the edge channels and all persistent RFI affects spectral channels using task in . After that we have performed an automated flagging using in its mode. Once automated flagging is done, we have extended flags for the time and frequency blocks with more than 80% data are flagged. This initial flagging gives us a comparatively clean fluxcal and phasecal dataset to proceed with the calibration. On the solar scans, we have flagged only the bad frequency channels. Since, solar flux density and spectro-temporal variation is not known a-priori, we did not perform any further automated flagging on the un-calibrated solar scans. Once initial flagging is done, we averaged eight spectral channels of total width about 200 kHz to reduce the data volume and perform the calibration. We have performed total five rounds of calibration, which are discussed in the following section. After each round of calibration, we apply the calibration solutions on the fluxcal and phasecal. Each of these calibration rounds is followed by an automated flagging using and then using mode of the . We then extended flags for time and frequencies where more than 80% data have already been flagged. We did the post-calibration flagging on the residual visibilities for the fluxcal and on the corrected visibilities for the phasecal. The post-calibration flagging allowed us to flag any low-level of RFI present in the data and make the next round of calibration solutions less affected due to outliers in the data. § INITIAL CALIBRATION We performed first two rounds of calibration and post-calibration flagging only on the flux density calibrator, J0408-6545. The model of J0408-6545 is not available in . This is brightest source in the field and contribution to the total flux density from other sources in this field is ∼1% at L-band. Hence, we use task to setup the model of J0408-6545 as described in MeerKAT calibration manual [https://skaafrica.atlassian.net/wiki/spaces/ESDKB/pages/1452146701/L-band+gain+calibratorsJ0408-6545 model]. We first perform delay calibration using the entire band using the task in . Followed by this a time dependent amplitude-phase calibration is performed on a set of good channels (channel range 1000 to 1100) after applying the delay calibration solutions. We chose a small chunk of channels to ensure that there is no significant frequency dependence while computing the time dependent gain solutions. After that we use both the delay calibration and time-dependent gain solutions, and performed a normalized bandpass calibration over the entire spectral range. Once the bandpass solution is obtained, we use the bandpass solution to correct for frequency dependence and perform a final time-dependent amplitude-phase gain calibration using the full spectral range. We used in task to make the calibration solutions robust in the presence of outliers due to RFI. We call all these steps a single calibration round. Once a single calibration round is finished, we apply delay calibration, bandpass calibration and time-dependent gain calibration solutions using the full spectral range on the calibrator corresponding scans. Each round of calibration is followed by post-calibration flagging as described in Section <ref>. After two rounds of calibration on the flux density calibrator, we apply the delay and bandpass calibration solutions on the phase calibrator. Then we perform time-dependent gain solution on the phase calibrator assuming a point source model with 1 Jy flux density at the phase center. To scale the gain amplitudes to match the flux density of the phase calibrator, we use task and obtain the scaled version of the time-dependent gain solutions. We apply these time-dependent gain solutions on the phase calibrator scans followed by post-calibration flagging. We have found that after five rounds of calibration, the residual visibilities for both flux calibrator and phase calibrator look noise like and no noticeable RFI is present on the data. We apply these final delay, bandpass and time-dependent gain solutions obtained from flux and phase calibrators of the solar scans using the task linearly interpolated across time. Since in different scans, Sun is present at different positions of the sky, we treat each 15 minutes solar scans separately for self-calibration and imaging. We also did not consider the full spectral range together, because solar flux density varies with frequency and also the sidelobe response of the primary beam varies significantly with frequency. Hence, after applying the initial calibration solutions obtained towards the phase center, we have splited each solar scans into 20 MHz spectral chunks for self-calibration and imaging. § SELF-CALIBRATION Although the Sun is at the sidelobes of the primary beam, it is still the source with the highest flux density contributing to the observed visibilities. Before primary beam correction, the brightest source in the field has flux density of 38 mJy/beam, while the peak flux density on the Sun is about 1.7 Jy/beam. Total integrated flux density of background sources is 0.5 Jy, where the integrated flux density of the Sun is about 15 Jy. Since the total contribution from background sources is about 3.3%, they will not affect the self-calibration. Since the Sun is present about 2.5^∘ away from the phase center, gain solutions towards the phase center may not valid towards the Sun. To tackle the direction-dependent effect, we shifted the phase center of the visibilities to the center of the Sun for each solar scans. We split the each solar scans into 20 MHz spectral chunks and self-calibration is performed for each spectral chunks for each scans separately. This has been done to account for the chromatic primary beam response and spectral dependence of solar structures. Another major challenge in self-calibrating the solar observation is its flux distribution with baselines. Assuming the Sun as a uniformly illuminated disc of size 32 arcmin, the first ring of the visibility amplitude distribution will lie close to 100lambda and the visibilities lie less than 100lambda increases dramatically. Hence one needs good uv-coverage at these baseline lengths to properly model the emission. But, at present, there are a limited numbers of short baselines ≤100λ at the MeerKAT, which may cause deconvolution artifacts at longer emission scales. To avoid any deconvolution artifacts due to sparse uv-coverage at ≤100λ, we use baselines >100λ during the self-calibration. We follow the following self-calibration steps: * First we make a circular mask of diameter 34 arcmin centered on the Sun. * An image is made using from the data calibrated using fluxcal and phasecal gain solutions using baselines >100λ. We use weighting[https://wsclean.readthedocs.io/en/latest/image_weighting.htmlDefinition of robustness parameter in WSClean] with robustness 0. We used a circular taper at 19kλ. * We kept w-stacking on in and numbers of w-plane is chosen automatically by . * Deconvolution is performed using the mask centered on the Sun. Average rms (σ) close to the Sun is about 0.1 Jy. Hence, we performed deconvolution down to 3σ, 0.3 Jy. We used multiscale deconvolution with Gaussian scale sizes 0, 5, 9,15, 25 and 35 times the pixel size, where one pixel is chosen to 1 arcsec. * Deconvolved model of the Sun is converted into model visibilities by and used for self-calibration. * We performed four rounds of phase-only self-calibration and five rounds of amplitude-phase self-calibration. Time-dependent gain solutions are calculated using task at 1 minute time interval using and using baselines >100λ. * Due to sidelobe response, visibility amplitudes for two parallel-hand polarizations (XX and YY) could be different. Hence, during the amplitude-phase self-calibration, we make separate sky models for XX and YY polarizations. * Since some long baseline antennas do not have sufficient signal-to-noise ratio for performing self-calibration, some of them are flagged. Hence, a time-dependent gain solutions are applied using task of in mode to keep the long baseline antenna with initial calibration solutions. We have calculated the dynamic range (DR) of the image as the ratio of peak solar flux density and the measured rms close to the Sun. Changes in DR with self-calibration iterations is shown Figure <ref>. We noticed there is jump in DR when amplitude-phase self-calibration has been initiated, but not a dramatic improvement is observed. Although, we did not see any significant change in DR with self-calibration iterations, it is evident from final images shown in Figure <ref> and <ref> that, there is no significant deconvolution artifacts, which makes these images of having high-fidelity. aasjournal
http://arxiv.org/abs/2307.02971v1	20230706131755	On the Cultural Gap in Text-to-Image Generation	[ "Bingshuai Liu", "Longyue Wang", "Chenyang Lyu", "Yong Zhang", "Jinsong Su", "Shuming Shi", "Zhaopeng Tu" ]	cs.CV	[ "cs.CV", "cs.AI", "cs.CL" ]	1 Tencent AI Lab, 2 Xiamen University, 3 Dublin City University 1{vinnylywang,norriszhang,shumingshi,zptu}@tencent.com, 2{bsliu,jssu}@ximen.com, [email protected] On the Cultural Gap in Text-to-Image Generation Bingshuai Liu,1,2Equal contribution: Bingshuai Liu and Longyue Wang. Work done while Bingshuai Liu and Chengyang Lyu were interning at Tencent AI Lab. Longyue Wang,1^* Chenyang Lyu,1,3 Yong Zhang1 Jinsong Su,2 Shuming Shi,1 Zhaopeng Tu1Zhaopeng Tu is the corresponding author. August 1, 2023 ============================================================================================================================================================================================================================================================================================================================ One challenge in text-to-image (T2I) generation is the inadvertent reflection of culture gaps present in the training data, which signifies the disparity in generated image quality when the cultural elements of the input text are rarely collected in the training set. Although various T2I models have shown impressive but arbitrary examples, there is no benchmark to systematically evaluate a T2I model's ability to generate cross-cultural images. To bridge the gap, we propose a Challenging Cross-Cultural (C^3) benchmark with comprehensive evaluation criteria, which can assess how well-suited a model is to a target culture. By analyzing the flawed images generated by the Stable Diffusion model on the C^3 benchmark, we find that the model often fails to generate certain cultural objects. Accordingly, we propose a novel multi-modal metric that considers object-text alignment to filter the fine-tuning data in the target culture, which is used to fine-tune a T2I model to improve cross-cultural generation. Experimental results show that our multi-modal metric provides stronger data selection performance on the C^3 benchmark than existing metrics, in which the object-text alignment is crucial. We release the benchmark, data, code, and generated images to facilitate future research on culturally diverse T2I generation.[<https://github.com/longyuewangdcu/C3-Bench>.] § INTRODUCTION Text-to-image (T2I) generation has emerged as a significant research area in recent years, with numerous applications spanning advertising, content creation, accessibility tools, human-computer interaction, language learning, and cross-cultural communication <cit.>. One challenge of T2I models is the inadvertent reflection or amplification of cultural gaps present in the training data, which refer to differences in norms, values, beliefs, and practices across various cultures <cit.>. The cultural gap in T2I generation signifies the disparity in image generation quality when the cultural elements of the input text are rarely collected in the training set. For example, in the LAION 400M dataset, the collected text-image pairs predominantly consist of English texts and images containing Western cultural elements. Consequently, when given a text description featuring Eastern cultural elements, the quality of the generated image is likely to be unsatisfactory. Figure <ref> shows an example. The Stable Diffusion v1-4 model that is trained on the Western cultural data fails to generate satisfying Chinese cultural elements. The lack of cultural sensitivity in the generated images can manifest in the form of images that may be inappropriate, offensive, or simply irrelevant in certain cultural contexts. Therefore, addressing these cultural gaps in AI T2I models is crucial to ensure the generation of culturally appropriate and contextually relevant images for users from diverse cultural backgrounds. However, although various T2I models have shown how the cultural gap leads to flawed images with impressive but arbitrary examples, there is no benchmark to systematically evaluate a T2I model's ability to generate cross-cultural images. To bridge the gap, we introduce a C^3 benchmark with comprehensive evaluation criteria for the target evaluation on the cross-cultural T2I generation. Given that current open-sourced T2I models are generally trained on the English data associated with Western cultural elements <cit.>, we built a evaluation set of textual prompts designed for generating images in Chinese cultural style. Specifically, we ask the powerful GPT-4 model with carefully designed context to generate the challenging prompts that can lead a T2I model to make different types of mistakes for cross-cultural generation. We also provide a set of evaluation criteria that consider characteristics (e.g. cultural appropriateness) and challenges (e.g. cross-cultural object presence and localization) of cross-cultural T2I generation. A promising way of improving cross-cultural generation is to fine-tune a T2I model on training data in target culture, which are generally in other non-English languages. Accordingly, the captions in the target-cultural data are translated to English with external translation systems, which may introduce translation mistakes that can affect the quality of the image-caption pairs. In response to this problem, we propose a novel multi-modal metric that considers both textual and visual elements to filter low-quality translated captions. In addition, analyses of generated images on the C^3 benchmark show that the object generation in target culture is one of the key challenges for cross-culture T2I generation. Accordingly, our multi-modal metric includes an explicit object-text alignment score to encourage that all necessary objects in the image are included in the translated caption. Empirical analysis shows that our metric correlates better with human judgement on assessing the quality of translated caption for T2I than existing metrics. Experimental results on the C^3 benchmark show that our multi-modal metric provides stronger data selection performance. In summary, our contributions are as follows: * We build a benchmark with comprehensive evaluation criteria for cross-cultural T2I generation, which is more challenging than the commonly-used MS-COCO benchmark with more cross-cultural objects. * We propose a multi-modal metric that considers both textual and visual elements to filter training data in the target culture, which produce better performance for fine-tuning a T2I model for cross-cultural generation. * To facilitate future research on culturally diverse T2I generation, we publicly release the resources we constructed in this paper, including the C^3 benchmark, translated dataset, the filtering scripts, and generated images. § RELATED WORK In the last several years, there has been a growing interest in T2I generation. The conventional generation models are built upon generative adversarial networks (GANs) <cit.>, which consists of a text encoder and an image generator. Recently, diffusion models have advanced state of the art in this field by improving image quality and diversity <cit.>. Previous research on text-guided image generation mainly focused on improving the understanding of complex text descriptions <cit.> or the quality of generated images <cit.>. In this work, we aim to improve the generalization of T2I models to generate images associated with cultural elements that have rarely been observed in the training data. Another thread of research turns to enhance multilingual capabilities of T2I models, which can support non-English input captions. For example, <cit.> extent the text encoder of diffusion model with a pre-trained multilingual text encoder XLM-R. <cit.> mitigated the language gap by translating English captions to other languages with neural machine translation systems. While they enhance the multilingual support for the textual caption, we focus on improving the “multilingual support” for the generated images by enabling the model to generate images of cultural content in different languages. <cit.> proposed a novel approach for benchmarking the multilingual parity of generative T2I systems by assessing the “conceptual coverage” of a model across different languages. They build an atomic benchmark that narrowly and reliably captures a specific characteristic – conceptual knowledge as reflected by a model's ability to reliably generate images of an object across languages. Similarly, we build a benchmark to capture another specific characteristic – cross-cultural generation as reflected by a model's ability to reliably generate cultural elements that are rarely collected in the training set. Closely related to this work, <cit.> introduced the PaLI model, which is trained on a large multilingual mix of pre-training tasks containing 10B images and texts in over 100 languages. This model emphasizes the importance of scale in both the visual and language parts of the model and the interplay between the two. Our work is complementary to theirs: we provide a benchmark for the target evaluation on the cross-cultural T2I generation, which can estimate how well-suited a model is to a target cultural. In addition, we propose a novel multi-modal alignment approach for fast adaption of a generation model to a target culture. Cultural Gaps One challenge of T2I models is the inadvertent reflection or amplification of cultural gaps present in the data these models are trained on or the language they are conditioned on <cit.>. These cultural gaps, which refer to differences in norms, values, beliefs, and practices across various cultures, can significantly influence the quality and diversity of the generated images, as well as the perception and interpretation of the users. When these cultural differences are not adequately represented in the training data or not properly understood by the model, it can lead to a lack of cultural sensitivity in the generated images. This insensitivity can manifest in the form of images that may be inappropriate, offensive, or simply irrelevant in certain cultural contexts. Therefore, addressing these cultural gaps in AI T2I models is crucial to ensure the generation of culturally appropriate and contextually relevant images for users from diverse cultural backgrounds. UTF8gbsn Commercial image generation systems have provided numerous examples of how these cultural gaps can lead to flawed image generation. Many of these errors are related to the misinterpretation of idioms (e.g. Chengyu) and traditional elements (e.g. dish names). In some cases, the strange images generated are a result of machine translation errors, where prompts are translated from non-English to English before generating images, rather than directly generating images from the prompts in original language. For instance, when given a Chinese transitional food “狮子头” (“a pork meatball dish”) that are often mistakenly translated to “head of a lion”, the systems may generate a misleading image of a red, flaming lion's head. This issue highlights the cultural and linguistic challenges that AI systems face, demonstrating the need for better incorporation of cultural context in their training and interpretation processes. The motivation for our paper is to address this need and contribute to the development of culturally sensitive T2I models. § CROSS-CULTURAL CHALLENGING (C^3) BENCHMARK §.§ Constructing the C^3 Benchmark with GPT-4 To generate captions for creating cross-cultural and culturally diverse images, we first asked GPT-4 to identify the types of mistakes T2I generation systems can make if they are asked to generate such cross-cultural images. We received the following insights from GPT-4, which serve as the prompt for GPT-4 to generate more challenging captions: * Language Bias: T2I systems that do not account for variations in regional dialects or Chinese script may generate text that is linguistically inaccurate or insensitive to Chinese language subtleties. * Cultural Inappropriateness: Without an accurate understanding of Chinese cultural norms and values, a T2I generation system may generate images that are seen as inappropriate or offensive. * Missed Cultural Nuances: T2I systems that lack an appreciation for the nuances of Chinese culture may generate images that are not authentic or credible. * Stereotyping and Counterfeit Representations: T2I systems that rely on popular stereotypes or inaccurate depictions of Chinese culture may generate images that perpetuate damaging myths, or counterfeit representations give mistaken impressions. * Insufficient Diversity: A T2I system that does not consider the diversity of China's 56 ethnic groups or pay attention to minority cultures' rich heritage may overgeneralize or oversimplify Chinese culture. Subsequently, we asked GPT-4 to provide five representative examples of image captions in English that could lead a T2I system, trained only on English data, to make different types of mistakes when generating images reflecting Chinese culture or elements, as listed in Table <ref>. We used the first five examples (selected and checked by humans) as seed examples to iteratively generate more diverse and different examples, which can lead to errors while generating images reflecting Chinese culture or elements. Specifically, we use the following prompt to obtain more challenging captions: 0.96 T2I systems trained only on English data can make mistakes when generating images reflecting Chinese culture/element: Language bias: T2I systems that do not account ⋯ ⋯ may overgeneralize or oversimplify Chinese culture. Can you give five representative image captions in English that could lead a T2I generation trained only on English data make different types of mistakes above when generating images reflecting Chinese culture/element based on the examples but different from the examples below: Please follow the format and only give me captions (the captions do not have to contain the word `Chinese'), no other texts: Example 1: Caption1 ⋯ Example 5: Caption5 In each iteration we randomly sample five seed examples from the generated examples as prompt examples. The collected image captions were used to construct an evaluation set for assessing the performance of T2I generation systems in generating cross-cultural and culturally diverse images. Finally, we obtain a set of 9,889 challenging captions by filtering the repetitive ones for cross-cultural T2I generation, which we name as C^3+. Since it is time-consuming and labor-intensive to manually evaluate the generated images for all the captions, we randomly sample 500 captions to form a small-scale benchmark C^3, which will serve as the testbed in the following experiments for human evaluation. The generated images for different models on the full C^3+ benchmark (without human evaluation) will also be released for future research. Figure <ref> shows the benchmark details. §.§ Evaluating Difficulty of the C^3 Benchmark To evaluate the difficulty of the C^3 benchmark, we compare with the commonly-used COCO Captions dataset <cit.>, which is extracted from the English data that is potentially similar in distribution with the training data of Stable Diffusion. Specifically, we sample 500 captions from the COCO data, and ask the Stable Diffusion v1.4 model to generate images based on the captions. Figure <ref> shows the details of the sampled COCO Caption data. Compared with C^3, the captions in COCO contain smaller sizes of words and objects, which makes it easier for T2I generation. For comparing the quality of the generated images on both benchmarks, we follow the common practices to ask human annotators to score the generated images from the perspectives of both the image-text alignment and image fidelity <cit.>. Figure <ref> lists the comparison results. Clearly, 78% of the generated images on COCO are rated above average (“≥3”), while the ratio on C^3 is 57%. Specifically, 26.2% of the generated images on C^3 is rated as the lowest 1 score, which is far larger than that on COCO. Figure <ref> shows some examples of generated images on the two benchmarks. The Stable Diffusion model successfully generates all objects in the MS-COCO captions. However, it fails to generate cultural objects (e.g. “a tea ceremony”, “a gracefully arched bridge”, and “blooming lotus flowers”) in the C^3 captions, which are rarely observed in the training data of the diffusion model. These results demonstrate that the proposed C^3 is more challenging. §.§ Human Evaluation Criteria for the C^3 Benchmark Although the metrics of image-text alignment and image fidelity are widely-used for general T2I generation, they may not be sufficient to capture the certain types of mistake in the cross-cultural scenario (e.g. cultural inappropriateness and object presence). In response to this problem, we propose a fine-grained set of criteria for the target evaluation on the cross-cultural T2I generation, which focuses on various aspects of cultural relevance and image quality: * Cultural Appropriateness that examines the extent to which the generated images reflect the cultural style and context mentioned in the caption. This criterion helps to demonstrate the model's ability to capture and generate culturally relevant visual content. * Object Presence that evaluates whether the generated images contain the essential objects mentioned in the caption. This criterion ensures that the model accurately generates the cross-cultural objects in the caption. * Object Localization that assesses the correct placement and spatial arrangement of objects within the generated images, which can be challenging for the cross-cultural objects. This criterion ensures that the model maintains the context and relationships between objects as described in the caption. * Semantic Consistency that assesses the consistency between the generated images and the translated captions, ensuring that the visual content aligns with the meaning of the text. This criterion evaluates the model's ability to generate images that accurately represent the caption. * Visual Aesthetics that evaluates the overall visual appeal and composition of the generated images. This criterion considers factors such as color harmony, contrast, and image sharpness, which contribute to the perceived quality of the generated images. * Cohesion that examines the coherence and unity of the generated images. This criterion evaluates whether all elements appear natural and well-integrated, contributing to a cohesive visual scene. As seen, in addition to generalizing the conventional image-text alignment (e.g. semantic consistency) and image fidelity (e.g. visual aesthetics and cohesion) criteria, we also propose several novel metrics that consider characteristics (e.g. cultural appropriateness) and challenges (e.g. cross-cultural object presence and localization) of cross-cultural T2I generation. We hope the fine-grained evaluation criteria can provide a comprehensive assessment of the generated images on the proposed C^3 benchmark. Table <ref> lists an example of using the criteria to evaluate the image in Figure <ref> (left panel). Table <ref> in Appendix lists the guideline of using these criteria for human evaluation. § IMPROVING CROSS-CULTURAL GENERATION A promising way of improving cross-cultural T2I generation is to fine-tune the diffusion model on the in-domain data (e.g. image-text pairs of Chinese cultural in this work). Generally, the captions of the in-domain data are translated into English, and the pairs of (translated caption, image) are used to fine-tune the diffusion model. The main challenge lies in how to filter low-quality translated captions. In this section, we first revisit existing filtering methods, which considers only either text-text alignment or image-text alignment. Inspired by recent successes on multi-modal modeling <cit.>, we propose a novel filtering approach that considers multi-modal alignment including both text-text and image-text alignment, as well as explicit object-text alignment since the objects are one of the key challenges for cross-cultural T2I generation. §.§ Revisiting Existing Methods Text-Text Alignment Since there is no reference translation for captions of in-domain data, conventional metrics such as BLEU and Meteor that rely on the reference are unsuitable for evaluating the quality of the translated captions. Accordingly, researchers turn to reference-free metric such as BertScore <cit.>, which computes a similarity score for two sentences in the same language by leveraging the pre-trained contextual embeddings from BERT. Along this direction, <cit.> propose a multilingual version – LaBSE, which can compute a similarity score for two sentences in different languages. Image-Text Alignment Another thread of research uses multi-modal pre-trained vision-language models to measure the alignment between caption and images. One representative work is CLIP <cit.>, which computes a similarity score for a sentence and image with a pre-trained model on a dataset of 400 million (image, text) pairs. While prior studies use only either text-text alignment or image-text alignment for filtering the in-domain data, they miss the useful information from the other alignment. In response to this problem, we propose a multi-modal alignment approach to better measure the quality of the (image, translated caption) pair. §.§ Our Approach – Multi-Modal Alignment As shown in Figure <ref>, our filtering metric consists of three types of alignment scores: * Text-Text Alignment A_S-T between the original and the translated captions; * Image-Text Alignment A_I-T between the image and the translated caption; * Object-Text Alignment A_O-T between the detected objects in the image and the translated caption. Formally, let S={x_1, ⋯, x_M} be the original non-English caption associated with the image I, T={y_1, ⋯, y_N} be the translated caption in English, and O={o_1, ⋯, o_K} be the list of the objects (listed in natural language) detected in the image I. We first encode the captions and objects with a multilingual BERT ℰ∈ℝ^h <cit.> to the corresponding representations: H_S = ℰ(S) ∈ℝ^M × h H_T = ℰ(T) ∈ℝ^N × h H_O = ℰ(O) ∈ℝ^K × h We encode the image I with a Vision Transformer 𝒱∈ℝ^h <cit.> into a representation vector: h_I = 𝒱(I) ∈ℝ^h We follow <cit.> to calculate the text-text alignment between two captions as a sum of cosine similarities between their tokens' embeddings: A_S-T = 1/M∑_ x∈ H_Smax_ y∈ H_T x^⊤ y/\|\| x\|\| \|\| y\|\| Similarly, we calculate the other two alignment scores by: A_O-T = 1/K∑_ o∈ H_Omax_ y∈ H_T o^⊤ y/\|\| o\|\| \|\| y\|\| A_I-T = max_ y∈ H_T h_I^⊤ y/\|\| h_I\|\| \|\| y\|\| The ultimate score is a combination of the above alignments: A = A_S-T + A_I-T + A_O-T The score A reflects the quality of the translated captions by considering both their textual and visual information. A higher A indicates that the translated caption has better quality with respect to the original caption, the relatedness between image and caption, and the similarity between image and caption at an object-level. Each term in A measures the translation quality from a specific aspect, thereby allowing for a faithful reflection of the overall translation quality of image captions. Practically, we followed previous work to implement the text-text alignment A_S-T with LaBSE and implement the image-text alignment A_I-T with CLIP. We use GRiT to implement A_O-T. GRiT will detect objects in the image and output corresponding categories. We detect the objects in the images using the GRiT model <cit.> with prediction probability >0.5. In summary, our proposed approach involves the following steps: * Obtaining embeddings of the original captions, translated captions and images. * Extracting objects in images and encode object labels. * Calculating text-to-text, image-to-text and object-to-text similarity scores. * Calculate the translation quality score as the combination of the three similarity scores. Our approach provides a novel method for estimating the quality of translated captions in non-English T2I datasets, and has the potential to improve the performance of image generation models by incorporating data in other languages. §.§ Experiments In this section, we first introduce the implementation details of our experiments. Then we evaluate our multi-modal alignment metric on two main experiments: * Assessing the quality of translated caption. * Effectiveness on the cross-cultural T2I generation using the proposed C^3 benchmark. Then we present the comparison of our multi-modal metric and prior studies with human evaluation on assessing the quality of translated caption. Last, we further investigate the cross-cultural performance of different filtering metrics on the C^3 benchmark with quantitative results and qualitative showcases. Experimental Setup We conduct experiments with the Stable Diffusion v1-4 model <cit.>.[<https://github.com/CompVis/stable-diffusion>.] For fine-tuning the diffusion model on the Chinese cultural data, we choose the Chinese subset (laion2b-zh) of the laion2b-multi dataset[<https://huggingface.co/datasets/laion/laion2B-multi>.], comprising a total of 143 million image-text pairs. We translate all image captions into English using an online translation system TranSmart <cit.>.[<https://transmart.qq.com>.] We filter the full laion-zh to 300K instances with different strategies, including 1) the text-text alignment score LaBSE <cit.>; 2) the image-text alignment score CLIP <cit.>; 3) our multi-modal metric. We fine-tune the diffusion model on the filtered laion-zh dataset for one epoch with a batch size of 2 on 8 A100 40G GPUs. We use the AdamW optimizer <cit.> with a learning rate of 1e-4 for all models. Assessing the Quality of Translated Caption We randomly sampled 500 instances from the translated laion2b-zh data, and ask human annotators to rate the quality of translated caption from two main perspectives: 1) textual translation quality, including adequacy, fluency and consistency; and 2) image correlation, including image relevance, context, and cultural appropriateness. Table <ref> in Appendix lists the evaluation guidelines. We then scored the translated captions with different automatic metrics (e.g. LaBSE, CLIP, and Ours), and calculate their Pearson correlation with the human judgements on the above criteria. [We list the evaluation guidelines in Table <ref> in Appendix.] Textual Translation Quality * Adequacy that checks how accurately the translated captions convey the meaning of the original captions. * Fluency that evaluates how well the translated captions are written in the target language. Consider factors such as grammar, syntax, and vocabulary. * Consistency that judges whether the translated captions are consistent in terms of language, tone, and style throughout the entire set. Image Correlation * Image Relevance that determines whether the translated captions are relevant to the image they describe. * Context that evaluates whether the translated captions provide enough context for the reader to understand the image and the situation in which it was taken. * Cultural Appropriateness that checks whether the translated captions are culturally appropriate for the audience. Table <ref> lists the results. Our proposed metric outperforms both LaBSE and CLIP in terms of correlation with human evaluation scores across all criteria. The positive correlation coefficients for our metric indicate a strong agreement between the multi-modal alignment metric and human judgments. This suggests that our metric is more effective in capturing the key aspects of T2I generation tasks than the other two metrics. The results clearly demonstrate the superiority of our metric in assessing the quality of translated captions for the T2I generation tasks. We also investigate the impact of object-text alignment score in our metric by removing it from the ultimate score (i.e. “-A_O-T”), which is one of the key challenges in cross-cultural T2I generation. The results confirm our hypothesis: removing the object-text alignment score drastically decreases the correlation with human judgement, indicating that the alignment is essential in assessing the translated caption for cross-cultural T2I generation. Performance on the C^3 Benchmark Table <ref> lists the results of different data filtering approaches on the proposed C^3 benchmark. We also list the results of randomly sampling 300K instances for reference. Clearly, all fine-tuned models achieve significantly better performance than the vanilla model that is trained only on the English-centric data, which confirms the necessity of fine-tuning on the target cultural data for cross-cultural generation. All filtering approaches with certain metrics outperform the randomly sampling strategy, demonstrating that these metrics are reasonable for filtering low-quality instances. Our metric obtains the best results under all criteria by maintaining high-quality instances for fine-tuning. Figure <ref> shows some example images generated by different models. The vanilla diffusion model fails to generate Chinese-cultural elements, which can be greatly mitigated by the fine-tuned models. While CLIP and Our models successfully generate all the objects in the captions (e.g. “tea ceremony with an expert” and “winding pathways, carefully placed rocks, and lush vegetation”), the elements in our images appear more natural and better-integrated. We attribute the strength of our approach to the explicit consideration of object-text alignment in data filtering. It is also worthy noting that the proposed C^3 benchmark can distinguish different models by identifying model-specific weaknesses. § CONCLUSION AND FUTURE WORK In this work, we build a C^3 benchmark of challenging textual prompts to generate images in Chinese cultural style for T2I models that are generally trained on the English data of Western cultural elements. We demonstrate how the benchmark can be used to assess a T2I model's ability of cross-cultural generation from different perspectives, which reveal that the object generation is one of the key challenges. Based on the observation, we propose a multi-modal approach that explicitly considers object-text alignment for filtering fine-tuning data, which can significantly improves cross-cultural generation over existing metrics. Future work include extending the C^3 benchmark to more non-English cultures (e.g. Arabic culture), validating our findings with more T2I models such as DALL-E 2 <cit.>, and designing automatic metric and toolkit to evaluate the quality of generated images on the C^3 benchmark. § APPENDIX §.§ Human Evaluation Guidelines Table <ref> provides a detailed guideline for evaluating the generated images on the C^3 benchmark. The evaluation is based on six criteria: Object Presence, Object Localization, Cultural Appropriateness, Visual Aesthetics, Semantic Consistency, and Cohesion. Each criterion is scored on a scale of 1 to 5, with 5 being the highest score. * Object Presence: This criterion assesses whether all essential objects described in the caption are present and clearly recognizable in the generated image. * Object Localization: This criterion evaluates whether the spatial arrangement of objects in the image accurately represents the arrangement described in the caption. * Cultural Appropriateness: This criterion measures whether the cultural style and context described in the caption are clearly and consistently reflected in the image. * Visual Aesthetics: This criterion assesses the visual appeal and composition of the image, including color harmony, contrast, and image sharpness. * Semantic Consistency: This criterion evaluates the consistency between the image and the caption, i.e., whether all elements in the image align with and accurately represent the text. * Cohesion: This criterion measures the coherence and unity in the image, i.e., whether all elements in the image appear natural and well-integrated, creating a seamless visual scene. Each of these criteria is crucial for evaluating the performance of T2I models, as they collectively assess the model's ability to generate images that are not only visually appealing and semantically consistent with the input text, but also culturally appropriate and coherent. §.§ Evaluation Scores for Example Images Table <ref> provides detailed evaluation scores for the example image generated by the vanilla stable diffusion model, as shown in Figure <ref>. The evaluation is based on six criteria: Cultural Appropriateness, Object Presence, Object Localization, Semantic Consistency, Visual Aesthetics, and Cohesion. Each criterion is scored on a scale of 1 to 5, with 5 being the highest score. * Cultural Appropriateness: The left image scores 3, as it contains distinguishable Chinese cultural elements and styles, but also includes some meaningless parts. The right image scores 5, as its overall style conforms to the cultural elements mentioned in the caption. * Object Presence: The left image scores 3, as some objects can be seen in the image, but it is difficult to distinguish specific elements. The right image scores 4, as the key object mentioned in the caption appears in the image. * Object Localization: The left image scores 2, as the temple elements in the image are not lined up correctly. The right image scores 4, as the different objects in the image are arranged correctly and logically. * Semantic Consistency: The left image scores 2, as the consistency between the image and the caption is poor. The right image scores 5, as the image is consistent with the meaning expressed by the caption. * Visual Aesthetics: The left image scores 1, as the overall image quality is very poor. The right image scores 5, as the pictures are visually appealing and the image quality is high. * Cohesion: The left image scores 2, as multiple elements in the image are not coherently matched. The right image scores 5, as the overall collocation of the elements appearing in the image is reasonable, without incoherence. These scores provide a detailed evaluation of the generated images based on multiple criteria, offering insights into the strengths and weaknesses of the vanilla stable diffusion model. §.§ Evaluation Guidelines for Translated Captions Table <ref> provides a detailed guideline for evaluating the translated captions associated with the images. The evaluation is based on six criteria: Adequacy, Fluency, Consistency, Relevance, Context, and Cultural Appropriateness. Each criterion is scored on a scale of 1 to 5, with 5 being the highest score. * Adequacy: This criterion assesses whether the translation accurately conveys the intended meaning of the original caption. * Fluency: This criterion evaluates the fluency of the translation, including grammar, syntax, and vocabulary. * Consistency: This criterion measures the consistency of the translations in terms of language, tone, and style. * Relevance: This criterion assesses whether the translations are relevant to the image they describe, capturing the essence of the image and all important details. * Context: This criterion evaluates whether the translations provide sufficient context for the reader to understand the image and the situation in which it was taken. * Cultural Appropriateness: This criterion measures whether the translations are appropriate for the target audience, demonstrating an understanding of the target culture and avoiding cultural references or language that could be offensive or confusing. These criteria provide a comprehensive framework for evaluating the quality of the translated captions associated with the images, offering insights into the strengths and weaknesses of the translation process.
http://arxiv.org/abs/2307.02185v1	20230705102545	Citation: A Key to Building Responsible and Accountable Large Language Models	[ "Jie Huang", "Kevin Chen-Chuan Chang" ]	cs.CL	[ "cs.CL", "cs.AI", "cs.CR" ]	Electroweak sphalerons, scalar multiplets, and symmetry breaking patterns Michael J. Ramsey-Musolf August 1, 2023 ========================================================================== Large Language Models (LLMs) bring transformative benefits alongside unique challenges, including intellectual property (IP) and ethical concerns. This position paper explores a novel angle to mitigate these risks, drawing parallels between LLMs and established web systems. We identify “citation” as a crucial yet missing component in LLMs, which could enhance content transparency and verifiability while addressing IP and ethical dilemmas. We further propose that a comprehensive citation mechanism for LLMs should account for both non-parametric and parametric content. Despite the complexity of implementing such a citation mechanism, along with the inherent potential pitfalls, we advocate for its development. Building on this foundation, we outline several research problems in this area, aiming to guide future explorations towards building more responsible and accountable LLMs. § INTRODUCTION The landscape of artificial intelligence is undergoing rapid transformation, spurred by the emergence of large language models (LLMs) such as ChatGPT/GPT-4 <cit.> and PaLM <cit.>. These models, recognized for their striking ability to generate human-like text, have shown enormous potential in various applications, from information provision to personalized assistance. Nonetheless, their capabilities bring along substantial risks, including intellectual property (IP) and ethical concerns <cit.>. Research by <cit.>, for instance, reveals that LLMs are prone to memorizing extensive segments of their training data, including sensitive information. This can result in violations of IP and ethical guidelines. Furthermore, studies by <cit.> suggest that current protective measures fail to provide a comprehensive and meaningful notion of safety for LLMs, making it seemingly impossible to develop safety-preserving, high-accuracy large language models even when trained on public corpora. While the notion of building an entirely safe LLM might appear daunting, it is crucial to acknowledge that many well-established systems, such as the Web, grapple with similar challenges and have not yet reached absolute safety. Recent legislation like the Online News Act[<https://www.canada.ca/en/canadian-heritage/services/online-news.html>], which requires online search engines to compensate Canadian online news outlets for their content, underscores the ongoing issues around content use and compensation on the Web. Furthermore, the Web continues to be a breeding ground for both sensitive information and misinformation. Hence, expecting a completely risk-free LLM may be an over-ask. Instead, our focus should be on accurately quantifying these risks and developing effective mitigation strategies. It is not about achieving absolute security, but about responsibly managing and minimizing risks in an ethically sound manner. Guided by these insights, we propose to examine the problem through a different lens: Can we draw parallels between the risks inherent to LLMs and those experienced by established systems such as search engines and the Web? Can we devise strategies to decrease these risks by aligning with the practices of these mature systems? In examining systems like the Web and search engines, we observe a common and robust practice employed to manage IP and ethical concerns: the use of “citations”. Broadly defined, a “citation” refers to the act of mentioning or referencing a source or piece of evidence. For example, search engine results also serve as a form of citation, with each entry typically consisting of a title, URL, and brief description. These components collectively cite the webpage's content, offering the user an overview and inviting them to explore the source in greater depth. Citations thus act as anchors for accountability and credit in these systems, providing traceability, preventing plagiarism, and ensuring credit is correctly attributed. They also contribute to transparency, allowing users to verify the information's source. Upon reflection, it becomes clear that LLMs lack this critical functionality. When LLMs generate content without citations, their output is perceived as independent and self-derived. This creates two significant issues: firstly, when the model produces valuable information, it fails to credit the source it relies on; secondly, when it generates harmful content, it becomes challenging to assign accountability. Incorporating the ability to cite could not only address these ethical and legal conundrums but also bolster the transparency, credibility, and overall integrity of the content generated by LLMs. However, implementing a “citation” mechanism in LLMs is not as straightforward as it might seem. Unlike the Web, which explicitly links and references sources, LLMs internalize the information and transform it into an abstract representation, making accurate citation a significant technical challenge. Although some strides have been made in this direction, as seen in systems like New Bing[<https://www.bing.com/new>], Bard[<https://bard.google.com>], and Perplexity AI[<https://www.perplexity.ai>], they fall short on several fronts. First, the citations provided in the response of existing systems are often inaccurate <cit.>. Moreover, these systems typically only cite non-parametric content, i.e., content directly retrieved from external sources such as the Web. However, they neglect parametric content, the knowledge embedded in the model parameters, which also needs appropriate credit attribution and consideration for potential harm. This position paper embarks on an exploratory journey into the potential of integrating a citation mechanism within large language models, examining its prospective benefits, the inherent technical obstacles, and foreseeable pitfalls. We delve into approaches to cite both non-parametric and parametric content, considering the unique characteristics of each type. We also identify and discuss potential setbacks, such as reduced creativity, dissemination of sensitive information, and citation bias. Building on this foundation, we lay bare the hurdles in our path, presenting them as enticing problems for future research. Through this endeavor, we aim to stimulate further discussion and research towards building responsible and accountable large language models. § “CITATION” IN LLMS Figure <ref> illustrates model generations with and without citations. In the absence of a citation, there is a potential risk of misunderstanding, leading one to believe that the claim is an opinion or statement formulated by the LLM itself. This not only fails to appropriately credit the original authors, but could also result in ethical dilemmas if the claim is inaccurate or misrepresented. On the other hand, the inclusion of citations can act as a multifaceted solution to these concerns. Primarily, it helps to mitigate intellectual property and ethical disputes by signaling that the information is not a product of the LLM's “opinion”, but a reflection of the cited source. Additionally, citations enhance the transparency and verifiability of the LLM's output. By indicating the source from which the information is derived, they provide a clear pathway for users to independently verify the validity and context of the information. § ROADMAP In this section, we embark on exploring the potential of incorporating a “citation” mechanism within LLMs. We start our exploration by defining when it would be ideal for an LLM to provide a citation, drawing insights from established practices in academia and existing systems like search engines and the Web. We then delve into discussing the possible strategies for effectively implementing citations in LLMs, confronting the methodological and technical intricacies this endeavor involves. §.§ When to Cite? In academic or professional writing, a citation is typically required when using someone else's ideas, concepts, data, or specific language. For LLMs, determining when to provide a citation is a considerably more challenging task. Given the vast and varied range of queries posed to LLMs, it is crucial to establish when a citation would be appropriate or necessary. A fundamental rule could be that any fact, idea, or concept that is not general knowledge should be cited. This mirrors the existing conventions on the Web, where sources for specific information are typically provided. For instance, widely known facts like “The Earth revolves around the Sun” would not necessitate a citation, while a less well-known fact like “The fastest spinning stars can rotate more than 600 times per second” would warrant one. Moreover, the need for a citation could also depend on the nature of the task LLMs are performing. Certain tasks may not necessitate citations, particularly if the output is a reformulation or reinterpretation of the input. For example, in summarization tasks, LLMs condense the input data without introducing new information. The resultant summary is hence an interpretation of the input, and typically, a citation may not be needed for such tasks. Similarly, translation tasks involve converting content from one language to another, without the introduction of novel information. In essence, while the need for citations in LLMs is task-dependent and context-specific, a guiding principle should be the commitment to knowledge integrity, respect for intellectual property, and adherence to ethical norms. These are similar principles that guide the management of intellectual property and ethical concerns on the Web and in search engines. §.§ How to Cite? Incorporating citations in LLMs ideally involves connecting outputs to the original source of information. However, this presents a notable technical challenge. During LLMs' training, information is internalized and transformed into an abstract representation, unlike search engines which possess indices to track and retrieve information. In the case of LLMs, this index is absent, which makes referencing the original source a daunting task. In this section, we delve into the consideration of citations for both non-parametric and parametric content (Figure <ref>). §.§.§ Citation for non-parametric content As a potential solution to prevailing challenges, one could design a hybrid system that merges large language models with information retrieval (IR) systems. In this approach, the model is trained to discern when a citation might be required. Subsequently, the IR system is utilized to retrieve relevant sources, namely, non-parametric content. The LLM can then incorporate these sources into its responses as citations. We identify two strategies for citing non-parametric content: * Pre-hoc citation: This approach involves first identifying the need for a citation in the upcoming dialogue or content generation. Once this requirement is recognized, the LLM triggers the IR system to retrieve the necessary information. The LLM then generates its response, seamlessly incorporating the retrieved non-parametric content as citations. This technique can be associated with the broader body of research that augments language models with retrieval <cit.>. * Post-hoc citation: Conversely, in this strategy, the LLM initially produces a response. An evaluation process then scrutinizes the generated content to ascertain whether a citation is necessary. If a citation is deemed necessary, the IR system is used to locate the appropriate non-parametric content, which is subsequently inserted into the existing text as a citation. Related research includes measuring or requiring attribution in LLMs <cit.>. In practical applications, a combination of both pre-hoc and post-hoc citation methods could be adopted for an optimized method. This mixed approach would employ the initial identification and retrieval of potential citations in line with the pre-hoc method, followed by a post-hoc evaluation to refine the integration of citations based on the generated content. This blend of proactive retrieval and reactive refinement could facilitate the creation of robust, accurate, and well-supported content, while also mitigating intellectual property and ethical concerns surrounding LLMs. §.§.§ Citation for parametric content In addition to the non-parametric content, i.e., content directly retrieved from external sources such as the Web, parametric content, which refers to information internalized from the training data, also needs appropriate credit attribution and consideration for potential harm. However, crafting a citation strategy for parametric content presents its own set of unique challenges. The fundamental challenge is the underlying nature of how LLMs process and internalize information. During training, LLMs assimilate vast amounts of data and transform them into an intricate, high-dimensional space that represents learned patterns and structures. The transformation process, rooted in complex mathematical operations, does not inherently retain any clear mapping back to individual data points in the training set. Consequently, generated content cannot easily be traced back to specific training data. This situation is further complicated by the fact that an output generated by LLMs is typically influenced by a multitude of training data points, rather than a single source. This is due to the multi-faceted and context-sensitive nature of language understanding and generation, where a single output can be influenced by a diverse range of linguistic patterns and structures. Thus, the task of accurately attributing a generated output to specific training data pieces is a complex and multifaceted problem that involves unpacking the high-dimensional representations in the model. Despite these challenges, potential solutions exist. A conceivable approach involves training the model with source identifiers, essentially tags that link specific pieces of information back to their original sources. During training, the model could then be encouraged to retain these identifiers. This would provide a more transparent lineage of information, thereby enhancing accountability. A relevant attempt in this direction was made by <cit.>, which used special reference tokens to wrap citations and trained models to predict these citations. However, it exhibited certain limitations, such as citation inaccuracy and confinement to academic citations. The successful execution of this method would likely call for advancements in model architecture and training techniques, thereby highlighting intriguing directions for future research. § PITFALLS OF CITATION IN LLMS While citations in LLMs can potentially mitigate risks such as IP and ethical issues, as well as improve transparency and verifiability, it is crucial to consider potential pitfalls. Over-Citation and Sensitive Information Dissemination The implementation of a citation system in LLMs poses the risk of over-citation, where the excessive use of references might expose more information than necessary. This overexposure could lead to information overload, diluting the significance of critical citations. Moreover, over-citation might inadvertently elevate the risk of disseminating sensitive information <cit.>. An ill-intentioned user could exploit these extensive citations to gather additional sensitive or private information. Inaccurate Citations Another potential pitfall of implementing citations in LLMs is the risk of inaccurate citations <cit.>. Given that LLMs may not possess a deep understanding of the content they are trained on or the sources they are citing, there is a chance that they could incorrectly attribute information to a source that does not actually contain that information. Inaccurate citations could mislead users, causing them to believe that a piece of information is verified and supported by a credible source, when in fact, it is not. Outdated Citations With the continuous expansion and evolution of knowledge, there is a risk that the sources an LLM cites may become outdated or irrelevant over time. This is particularly likely in fast-evolving fields where new discoveries or advancements quickly supersede existing knowledge. As LLMs are trained on a fixed dataset, their generated content and the sources they cite may not reflect the most current or accurate information. Therefore, there is a potential for LLMs to propagate outdated knowledge, misleading users who rely on the generated content and the cited sources for information. Propagation of Misinformation The risk of propagation of misinformation presents a significant concern in the application of LLMs <cit.>. As LLMs generate output based on the data they have been trained on, there is a chance they could inadvertently cite or echo unreliable or misleading sources, thereby spreading misinformation. This problem could potentially be amplified by the addition of a citation mechanism. A misinterpreted or incorrect citation could be perceived as an authoritative endorsement, inadvertently lending credibility to inaccurate or misleading content. Citation Bias Implementing citations in LLMs can also lead to citation bias <cit.>. Models may prefer to cite certain types of sources over others, either due to the characteristics of the training data or inherent biases in the retrieval mechanism of the IR system. This could lead to an over-reliance on certain types of information and unintentional promotion of certain viewpoints. Potential for Diminished Creativity The integration of citations could inadvertently cause a decrease in the creative outputs of the model. When prompted to generate innovative text or propose creative solutions, LLMs might become over-reliant on existing, citable information, thus stifling their novel content generation. Legal Implications The utilization of citations could also bring forth legal implications. The introduction of citations could imply a level of responsibility and accountability that the LLM, as an artificial entity, is not equipped to handle. Legal systems worldwide are yet to reach a consensus on how to treat legal issues related to artificial intelligence and its outputs, and the inclusion of citations could further complicate these discussions. § BARRIERS AND RESEARCH PROBLEMS Building on the potential solutions and pitfalls discussed above, we delve into the primary barriers and corresponding research problems that need to be addressed for successful citation implementation in large language models. §.§ Determining When to Cite Deciding when an LLM should cite its sources is a complex issue. While it may be intuitive to suggest that LLMs should always cite sources when they generate information that is not common knowledge (<ref>), defining what constitutes “common knowledge” is itself a difficult task. Furthermore, as discussed in <ref>, it is essential to consider the potential risks associated with over-citation, particularly the increased risk of sensitive information dissemination <cit.>. LLMs may inadvertently expose sensitive information or contribute to information overload if they include unnecessary or excessive citations. Balancing the need for transparency and accountability with the need to protect privacy and prevent information overload is a critical challenge that needs to be addressed. §.§ Addressing Hallucination in Citation Hallucination in large language models refers to the phenomenon where the models generate information not grounded in their training data, and that cannot be verified or is simply incorrect <cit.>. The incorporation of a citation feature can both alleviate and exacerbate this issue. On the one hand, requiring LLMs to link generated information to a tangible source can serve as a form of external verification, potentially restraining the model from generating completely baseless or hallucinated content. The requirement for a source may encourage the model to better align its output with the available data, thereby reducing the likelihood of hallucination. On the other hand, the citation mechanism itself can potentially hallucinate. If not meticulously designed and implemented, it may end up citing incorrect or non-existent sources <cit.>. This presents a twofold challenge: Not only is the generated content incorrect, but the citation misleads users into believing that the content is verified and substantiated by the cited source. This issue necessitates the development of techniques to enhance the model's ability to accurately represent the information present in the source, and equip the model to cross-check the consistency of the generated content with the content of the cited source. §.§ Maintaining Temporal Relevance of Citations In the pursuit of an effective citation mechanism within LLMs, it is essential to address the need for the model's ability to stay updated with the most recent and relevant knowledge. One potential approach towards this challenge is inspired by the operational principles of search engines. In their bid to stay relevant, search engines continuously update their indexes and ranking algorithms to reflect the latest web. A similar approach could be adopted for LLMs, where they could be designed for ongoing training on updated datasets, and their citation mechanisms could favor more recent sources. However, executing this in practice presents a significant research problem, considering the scale and complexity of continuously training LLMs and updating their citation mechanisms. Exploring efficient techniques for model training and designing citation mechanisms capable of consistently prioritizing the most recent and relevant sources will require substantial research and development. §.§ Evaluating Source Reliability Another important challenge is evaluating the reliability of sources used for training data and citations. As mentioned in <ref>, LLMs could potentially propagate misinformation if they cite unreliable or misleading sources. While search engines face similar challenges, they are equipped with advanced algorithms to evaluate the reliability and relevance of web pages <cit.>. Implementing analogous systems within the framework of LLMs presents an interesting and crucial direction for further exploration. §.§ Mitigating Citation Bias Citation bias in LLMs, as discussed in <ref>, can result in the uneven representation of information, leading to the propagation of certain viewpoints while others are neglected. Formulating strategies to curtail such tendencies is paramount. To begin with, sourcing a more balanced selection of training data can mitigate bias at the inception stage. Ensuring diversity in terms of viewpoints and topics in the training data can reduce bias to some extent. During citation retrieval, LLMs should utilize an impartial mechanism that does not favor specific types of sources. The underlying algorithms should be optimized to retrieve citations based on their relevance and credibility rather than the prominence of the source or its frequency in the training data. Finally, the development and application of effective evaluation techniques can help identify and measure any residual bias in LLM outputs. Quantifying the extent of bias enables more targeted corrective measures and provides an objective measure of their efficacy. §.§ Balancing Existing Content with Novel Content Generation Another intriguing area of research centers on striking a balance between the frequency of citing existing content and generating novel content. LLMs are admired for their capacity to generate creative and unique content <cit.>, as well as their reasoning ability <cit.>. An over-reliance on citations could potentially inhibit these attributes, reducing the model to a mere aggregator of existing knowledge rather than a generator of new ideas. Research into this would involve the development of techniques that allow for appropriate citation without hampering the model's creativity. One potential approach could be to create models that are capable of determining the novelty of their generated content and adjusting their citation behavior accordingly. For instance, if a model is generating content based heavily on its training data or the retrieved content, it should provide appropriate citations. Conversely, if the model is generating content that is significantly different from its training data and the retrieved content, it might deem citation unnecessary. Developing such capabilities would require significant advancements in understanding how LLMs generate novel content and how to quantify the `novelty' of such content. §.§ Navigating Copyright and Fair Use Laws The application of citation mechanisms in LLMs opens up a new array of legal challenges. Understanding and complying with copyright and fair use laws when citing sources is a complex issue. For instance, how much quoted material from a source would be considered fair use and under what conditions can it be used? In many jurisdictions, the law is not completely clear, especially as it applies to the use of AI technology. Thus, research in the legal aspects of using LLMs for generating text with citations is crucial to ensure legal compliance. § CONCLUSION In conclusion, the incorporation of a citation mechanism within large language models presents a promising approach to numerous challenges, including but not limited to intellectual property rights, ethical concerns, and the need for transparency and verifiability in AI outputs. By equipping LLMs with the ability to accurately attribute the origins of information, we can cultivate a climate of enhanced accountability for the content these models generate. This signifies a progressive step towards constructing a framework of ethical responsibility in AI that respects intellectual property rights and upholds information integrity. While introducing a citation mechanism in LLMs also brings about its own set of complexities and potential pitfalls, these challenges serve as fruitful avenues for future research and innovation. Through proactive problem-solving and innovation, we can navigate these hurdles and leverage the benefits of citation-capable LLMs. In doing so, we aim to foster more responsible, accountable, and reliable AI systems, ultimately contributing to a better, more trustworthy technological future. acl_natbib
http://arxiv.org/abs/2307.02912v1	20230706105350	LEA: Improving Sentence Similarity Robustness to Typos Using Lexical Attention Bias	[ "Mario Almagro", "Emilio Almazán", "Diego Ortego", "David Jiménez" ]	cs.CL	[ "cs.CL", "cs.AI" ]	LEA: Improving Sentence Similarity Robustness to Typos Using Lexical Attention Bias]LEA: Improving Sentence Similarity Robustness to Typos Using Lexical Attention Bias These authors contributed equally to this research. [email protected] NielsenIQ Innovation Madrid Spain [1] [email protected] NielsenIQ Innovation Madrid Spain [email protected] NielsenIQ Innovation Madrid Spain [1] [email protected] NielsenIQ Innovation Madrid Spain Textual noise, such as typos or abbreviations, is a well-known issue that penalizes vanilla Transformers for most downstream tasks. We show that this is also the case for sentence similarity, a fundamental task in multiple domains, matching, retrieval or paraphrasing. Sentence similarity can be approached using cross-encoders, where the two sentences are concatenated in the input allowing the model to exploit the inter-relations between them. Previous works addressing the noise issue mainly rely on data augmentation strategies, showing improved robustness when dealing with corrupted samples that are similar to the ones used for training. However, all these methods still suffer from the token distribution shift induced by typos. In this work, we propose to tackle textual noise by equipping cross-encoders with a novel LExical-aware Attention module (LEA) that incorporates lexical similarities between words in both sentences. By using raw text similarities, our approach avoids the tokenization shift problem obtaining improved robustness. We demonstrate that the attention bias introduced by LEA helps cross-encoders to tackle complex scenarios with textual noise, specially in domains with short-text descriptions and limited context. Experiments using three popular Transformer encoders in five e-commerce datasets for product matching show that LEA consistently boosts performance under the presence of noise, while remaining competitive on the original (clean) splits. We also evaluate our approach in two datasets for textual entailment and paraphrasing showing that LEA is robust to typos in domains with longer sentences and more natural context. Additionally, we thoroughly analyze several design choices in our approach, providing insights about the impact of the decisions made and fostering future research in cross-encoders dealing with typos. <ccs2012> <concept> <concept_id>10010147.10010178.10010179.10010184</concept_id> <concept_desc>Computing methodologies Lexical semantics</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010179</concept_id> <concept_desc>Computing methodologies Natural language processing</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10010405.10003550</concept_id> <concept_desc>Applied computing Electronic commerce</concept_desc> <concept_significance>100</concept_significance> </concept> </ccs2012> [500]Computing methodologies Lexical semantics [300]Computing methodologies Natural language processing [100]Applied computing Electronic commerce 02 February 2023 [ David Jiménez Gastón P. Fernández[Ph.D. student at the University of Leuven (KU Leuven), Department of Economics, Naamsestraat 69, box 3565, 3000 Leuven (e-mail: [email protected]). I deeply appreciate the invaluable guidance of my advisors Laurens Cherchye and Frederic Vermeulen. I would also like to thank Wietse Leleu and all participants at the Conference of the European Society for Population Economics (ESPE) in Belgrade, the Trans-Atlantic Doctoral Conference (TADC) in London, and the Public-Labor-Health Seminar, the Household Economics Gathering, and the ECORES Summer School in Leuven for their helpful comments. All errors are on my own.] University of Leuven (KU Leuven) =========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION The fast pace of information systems in society makes noise to be present in almost every text generated by either humans or automated processes. Medical reports, queries to databases, messages in social media, receipt transcriptions or product titles in e-commerce are a few examples where real production systems need to cope with a high presence of textual noise like typos, misspellings or custom abbreviations. Sentence similarity is of great interest to the scientific community for its variety of downstream applications ( question-answering, matching or retrieval) and the unresolved challenges that arises <cit.>. Transformer architectures dominate the state-of-the-art with two main alternatives: bi-encoders <cit.> and cross-encoders <cit.>. Bi-encoders focus on learning meaningful representations for each sentence independently using Siamese-like architectures, making them suitable for efficient retrieval <cit.>. However, these type of models rely only on the sentence embeddings comparison and lack of any interaction between the tokens/words of the two sentences to be compared. Cross-encoders <cit.>, on the other hand, tackle this limitation by concatenating the two sentences in the input. The attention heads in the Transformer learn the intra and inter-sentence interactions, which in many cases provides highly valuable information for achieving correct predictions in similarity tasks. Cross-encoders are often considered an upper-bound for textual similarity <cit.>, being its main limitation the computational cost of jointly encoding pairs of sentences. Recent works, attempt to use its potential by using late interaction modules for bi-encoders <cit.>, distillation techniques <cit.>, or using cross-encoders as re-rankers after retrieving potential candidates with bi-encoders <cit.>. These two stages resemble the typical pipeline of product matching, with an initial blocking stage to discard samples that are unlikely to be positive matches for a given product. This is followed by a more computationally intensive step that identifies the correct match among the selected candidates <cit.>. All these methods suffer when dealing with textual noise, that may appear in many forms: typos and misspellings as in queries for retrieval tasks <cit.> or custom abbreviations in certain domains <cit.>. This noise is challenging for vanilla Transformers for two main reasons: character information is not used in this type of architectures, and the shift in the token distribution caused by noise makes harder to relate tokens referring to the same concept, chocolate with chcolate or chclt. Prior works in information retrieval evidence performance issues <cit.> under these conditions and propose to address them mainly by training with similar typos as the ones in test. Although, this strategy has been proven, to some extent, effective to mitigate the issue of the token distribution shift, all these methods still have the limitations associated to the loss of character information from the tokenization process. All the evidence from previous works stress the importance of the character-level information to deal with textual noise. Following the same intuition, we propose to equip cross-encoders with a LExical Attention bias (LEA) that modifies the self-attention module in Transformers, guiding it towards lexically similar words. This helps the model to improve robustness under the presence of noise. We adopt standard data augmentation strategies to deal with typos and demonstrate large performance improvements when adding LEA to cross-encoders. Figure <ref> shows an example of the average attention across layers in a cross-encoder. When the typo appears in the word black ( black → blk), the tokenizer breaks it into two sub-words (b + ##lk), preventing the self-attention mechanism from capturing the relationship between both terms. However, when LEA is introduced, the lexical similarity between both adds a bias that helps the modified attention to maintain this relationship. Our main contributions are as follows: * We propose to add a lexical bias in the self-attention module of cross-encoders, which is designed specifically to improve textual noise scenarios. This bias provides the model with information about the lexical similarities between words in pairs of sentences, increasing the attention between those that are lexically close. * We evaluate LEA in five e-commerce datasets using three Transformer backbones and demonstrate that LEA consistently improves performance by a large margin. Concretely, we report an average improvement of around 6 absolute points of F1-score when dealing with synthetically generated typos. Results in textual entailment and paraphrasing tasks show that LEA achieves competitive performance or even surpasses the baselines. * We thoroughly analyze the impact of the different components of LEA to shed light on the design choices made. § RELATED WORK §.§ Sentence similarity Determining the degree of similarity between two sentences is a fundamental problem for matching, entailment or paraphrasing tasks and is normally tackled using two type of approaches: bi-encoders and cross-encoders. Bi-encoders are designed to process each sentence independently, obtaining an embedding for each of them. These models are typically trained with metric learning objectives that pull together the representations of positive pairs while pushing apart those of negative pairs. The authors in <cit.> propose SimCSE, which exploits dropout to generate embeddings that build positive pairs in the unsupervised setup. They also propose a supervised setting where they use textual entailment labels to construct the positive pair. Tracz et al. <cit.> adopt a triplet loss in a supervised setup for product matching. The approach described in <cit.>, extends SimCSE and propose to learn via equivariant contrastive learning, where representations have to be insensitive to dropout and sensitive to MLM-based word replacement perturbations. Supervised contrastive learning <cit.> is also adopted in <cit.> for sentence similarity in a general domain, while <cit.> apply it for product matching in e-commerce. Another popular method is SBERT <cit.>, which addresses both the problem of predicting several similarity degrees as a regression task and directly matching pairs of sentences via classification. Cross-encoders, on the other hand, jointly process a concatenated pair of sentences. These models are often considered to outperform bi-encoders <cit.>, obtaining robust results in general domains <cit.>, product matching <cit.> or retrieval tasks <cit.>. However, their main drawback is the need to recompute the encoding for each different pair of sentences. Therefore, many recent works adopt hybrid solutions to improve bi-encoders. Humeau et al. proposed Poly-encoders <cit.> that utilizes an attention mechanism to perform extra interaction after Siamese encoders. The TransEncoder method <cit.> alternates bi- and cross-encoder independent trainings, while distilling their knowledge via pseudo-labels. The resulting bi-encoder shows improved performance. Distillation is further explored in <cit.>, where knowledge transfer from the cross-attention of a light interaction module is adopted during training and removed at inference time. §.§ Dealing with typos and abbreviations Recent literature demonstrates that Transformer-based architectures <cit.> are not robust to textual noise, to misspellings or abbreviations of the input words <cit.>. Despite using sub-word tokenizers ( WordPiece) designed to deal with out-of-vocabulary words, Transformers exhibit in practice performance drops when exposed to typos. Words with noise are not likely to be present in the vocabulary and therefore, they are split into several sub-words yielding to token distribution shifts with respect to the noise-free counterpart <cit.>. Training the model with synthetically generated perturbations <cit.> is a standard practice to deal with typos. Some techniques use simple addition, deletion or character swaps, while others use more sophisticated methods such as common misspellings or keyboard and OCR errors <cit.>. Moreover, depending on the type of perturbation, the same type of noise can have a different impact, typos in various words of a sentence do not influence equally. This issue was reported in <cit.> showing that noise in relevant words yields larger performance drops. We can find approaches that complement this practice with different architectural designs or specific losses. For example, in <cit.> the authors realize that character-based Transformers provide improved robustness to typos and exploit this fact to propose a self-teaching strategy to boost performance of dense retrievers. Authors in <cit.> add a module to recognize the presence of typos in words just before the downstream classifier, which helps to align representations between noisy and clean versions. §.§ Relative attention bias Self-attention modules in Transformers receive as input the token representations coming from the previous layers and output contextual representations for each token estimated from a weighted combination of the tokens' representations. Modifications of the self-attention have been proposed to add a bias that accounts for the relative distance between words/tokens in the input sequence <cit.>. This strategy known as relative positional embeddings replaces the absolute positional embeddings, where the position was injected as part of the input of the Transformer. In <cit.> the authors follow this idea and extend it with long and short term relations. Wennberg et al. <cit.> propose a more interpretable representation for translation invariant relative positional embeddings using the Toeplitz matrix. The authors in <cit.> simplify the embeddings by just adding fixed scalars to the attention values that vary with the distance. This way of adding relative information between tokens has also been applied to information extraction from documents, where they use 2D relative distances <cit.> or tabular structural biases for table understanding <cit.>. § PROPOSED APPROACH In this work we propose to incorporate a Lexical-aware Attention module (LEA) to the self-attention mechanism of vanilla cross-encoder Transformers. This module considers inter-sentence lexical relations, which we demonstrate to be key for improving sentence similarity tasks, specially in the presence of typos. Notation We use capital letters to denote sets ( “X”), bold capital letters for matrices ( “𝐗”), bold lowercase letters for vectors ( “𝐱”) and lowercase for scalars ( “x”). For simplicity in the notation, equations only refer to a single layer and head in the Transformer architecture. §.§ Self-attention A key cornerstone in the success of Transformers is the multi-head self-attention mechanism, which learns token dependencies and encodes contextual information from the input <cit.>. In particular, a single head in this attention module receives an input sequence of n token representations coming from the previous layer X = (𝐱_1, …, 𝐱_n), where 𝐱_i ∈ℝ^d_h and computes a new sequence Z=( 𝐳_1,…, 𝐳_n ) of the same length and hidden dimension d_h. The resulting token representations are computed as follows: 𝐳_𝐢 = ∑_j=1^n a_ij( 𝐱_j ·𝐖^V ), 𝐳_𝐢∈ℝ^d_h. Therefore, each new token representation 𝐳_𝐢 is a weighted average of the linearly projected tokens representations 𝐱_j, using the value projection matrix 𝐖^V. The weight associated with each pair of tokens a_ij is computed using a softmax function: a_ij = expe_ij/∑^n_k=1expe_ik, where the scalar e_ij is computed using a compatibility function (dot product) between tokens i and j in the sentence: e_ij = (𝐱_i 𝐖^Q) (𝐱_j 𝐖^K)/√(d_h). The query, key and value projection matrices {𝐖^Q, 𝐖^K, 𝐖^V}∈ℝ^d_h × d_i are learned during training, where d_i refers to the dimension of the intermediate representations. §.§ Lexical attention bias for cross-encoders As we demonstrate in Section <ref>, in presence of textual noise, a_ij struggles to relate similar terms corrupted by noise. To address this issue, we propose to add a lexical attention bias to the self-attention module of cross-encoders. This bias term guides the attention towards tokens with high lexical similarity. We illustrate our proposed architecture in Figure <ref>. Cross-encoders for textual similarity receive as input the concatenation of the two sentence representations to be compared: 𝐗_c = 𝐗_l \| 𝐗_r, where 𝐗_l and 𝐗_r are the left and right sentences, respectively. Inspired by previous works in relative position embeddings <cit.>, we propose to modify the self-attention module described in Eq. <ref> as follows: ẽ_ij = e_ij + α ℓ_ij𝐖^L, ∀ i,j ∈𝐗_c, where the second term accounts for the lexical bias. 𝐖^L ∈ℝ^d_L× 1 is a learnable projection matrix, ℓ_ij∈ℝ^1 × d_L is the pairwise lexical attention embedding and α is a fixed scale factor that aligns the contributions of the lexical attention (ℓ_ij𝐖^L) and the scaled-dot product attention (e_ij). This factor is computed automatically once at the beginning of the training based on the magnitudes of both terms. To compute the pairwise lexical attention embedding, we first measure the similarity between words considering only inter-sentence relations, lexical similarities between words of the same sentence are set to 0: s_ij = 𝑆𝑖𝑚( w(𝐱_i), w(𝐱_j) ) , if 𝐱_i ∈𝐗_l and 𝐱_j ∈𝐗_r or 𝐱_i ∈𝐗_r and 𝐱_j ∈𝐗_l 0 , otherwise. where 𝐗_l and 𝐗_r represent the pair of input sentences to compare, w(𝐱_i) and w(𝐱_j) denote the input textual word associated to the i-th and j-th tokens, respectively, and 𝑆𝑖𝑚(·, ·) is a metric that measures the string similarity between two words. We elaborate on our choice for the similarity metric in Section <ref>. Inspired by <cit.>, we apply a sinusoidal function over s_ij to get an embedding that represents the lexical similarity: ℓ_ij^(s_ij,2p) = sin(2π· d_ij/β^2p/d_h), ℓ_ij^(s_ij,2p+1) = cos(2π· d_ij/β^2p/d_h), where β = 10^4 and p∈{ 0,…,d_L-1}. The final lexical embedding ℓ_ij is the concatenation of the two sinusoidal embeddings in Eq. <ref> and Eq. <ref>, respectively. Different from the original proposal <cit.> we scale the similarity s_ij by 2π to cover the full range of the sinusoidal functions. This results in embeddings more uniformly distributed across the output space. Note that by equipping LEA with a learnable projection matrix 𝐖^L we provide the model with the flexibility to adjust the contribution coming from the lexical term in the final attention values. The parameter overhead introduced by this term is d_L×#heads in all the layers where we use it. § EXPERIMENTAL WORK We structure the experimentation around four research questions to shed light on the design and capabilities of LEA. 𝐑𝐐_1. Does LEA improve performance in a consistent way across datasets and architectures under the presence of typos while remaining competitive on clean setups? 𝐑𝐐_2. How important is the choice of the lexical similarity metric for LEA under the presence of typos? 𝐑𝐐_3.What is the impact of applying LEA on varying locations of the architecture and the effect of sharing the parameters at different levels model, layer? 𝐑𝐐_4. Does LEA generalize to different noise strengths? The remaining of this section presents the experimental setting in Section <ref> and responds the four research questions in Sections <ref>, <ref>, <ref> and <ref>. §.§ Experimental setting The impact of textual noise in the prediction of the models depends on whether it appears on relevant words or not <cit.>. We argue that when the sentences are short the probability of the random noise appearing in relevant words is higher and, therefore, we expect a higher contribution of the lexical attention bias. Hence, the core of our experiments is conducted in five product matching datasets, where the sentences are short and normally with lack of syntax: Abt-Buy <cit.>, Amazon-Google <cit.> and WDC-Computers (small, medium and large) <cit.>. Moreover, we validate the contribution of LEA in two related tasks of natural language domain: textual entailment (RTE <cit.>) and paraphrasing (MRPC <cit.>). Details about the datasets are provided in Tables <ref> and <ref>, respectively. We artificially introduce typos on the aforementioned datasets as described in Section <ref>. §.§.§ Synthetic noise generation The original datasets mostly contain clean sentences with a low percentage of textual noise. To evaluate the models' generalization under the presence of noise, we synthetically generate test splits containing a wide range of typos. We apply the strategies followed in previous works <cit.> using the nlpaug[<https://github.com/makcedward/nlpaug>] library for data augmentation. In particular, we consider the following character operations: * Insertion. A random character is inserted in a random position within the word, screen → scree𝐭n. * Deletion. A random character from the word is removed, screen → sceen. * Substitution. A random character from the word is replaced by a random character, screen → s𝐛reen. * Swapping. A random character is swapped with one neighbor character in the word, screen → s𝐫𝐜een. * Keyboard substitution. A random character from the word is replaced by a close character in the QWERTY keyboard, screen → s𝐜teen. We modify all sentences in the test splits, where each word has a 20% chance to be augmented. Only one type of operation is applied to each word, which is chosen randomly among the five options. We limit the augmentation to words with more than 3 characters to mitigate the effect of words becoming non-recognizable from the original form ace → ate. §.§.§ Baseline models Due to the lack of prior works dealing with textual noise in cross-encoders for sentence similarity tasks, we adopt a benchmarking based on the comparison between three versions of cross-encoders: 1) vanilla, 2) trained with data augmentation (DA) and 3) trained with data augmentation and LEA. We adopt 2) as the reference baseline in the literature following the approach of related works in other domains, where they successfully applied data augmentation to deal with typos <cit.>. For data augmentation during training we apply the same configuration as in the synthetic generation of the test splits (see Section <ref>) and use a 50% chance for each sentence to be augmented. We use three popular pre-trained language models (PLMs) of varying sizes, Electra-small <cit.>, BERT-Medium <cit.> and BERT-Base <cit.>. §.§.§ Implementation details In all the experiments we fine-tune the PLMs described in Section <ref> for 30 epochs, using AdamW with a batch size of 32, an initial learning rate of 5e^-5, a weight decay of 5e^-5 and we apply a cosine annealing scheduler with a warm-up of 1.5 epochs. For LEA, α is automatically fixed in Eq. <ref> at the beginning of the training for each layer of the Transformer and leave 𝐖^L being trained independently per head. As similarity metric we use Jaccard (see Section <ref> for more details) and apply the proposed lexical attention bias to half of the last layers in all architectures (see Section <ref> for a detailed analysis). For more details we refer the reader to Appendix <ref>. We use the same training data for all methods and evaluate them on two different test splits, the original (clean) and the corrupted version with typos. We run three different seeds for the training and create three test splits randomly introducing typos, as their influence may differ depending on the words containing typos. Thus, we report the resulting mean and standard deviation over three and nine results for the clean and typo experiments, respectively. The test splits with typos, the binaries of the models and the required material to reproduce results are available in our repository[<https://github.com/m-almagro-cadiz/LEA>]. §.§ Robustness across datasets We compare in Table <ref> the F1-score of LEA with that of vanilla cross-encoders trained without and with data augmentation (+ DA) in five product matching datasets. We observe that applying data augmentation to mimic typos during training improves the robustness to them as reported by previous works in the retrieval domain <cit.>. When we apply LEA, we outperform the baseline by 5.4, 6.1 and 7.0 points on average across the five datasets for Electra-small, BERT-Medium and BERT-Base, respectively. Strategies solely based on data augmentation completely depend on the tokenized data, which may lose part of the lexical information when splitting into sub-words. In contrast, LEA exploits character-level similarity between words, an information that is not dependent on the tokenization. Moreover, in Table <ref> we analyze the impact of adding LEA to cross-encoders in the absence of typos. Here, the vanilla cross-encoders trained without data augmentation perform best on average. LEA, however clearly outperforms training with data augmentation and provides a competitive performance, achieving the best performance in some datasets. We refer the reader to Sections <ref> and <ref> for additional experiments with a larger architecture (BERT-Large), autoregressive models (GPT-2 and GPT-Neo) and larger datasets (WDC-XLarge and WDC-All). The results presented in Tables <ref> and <ref>, therefore provide a positive response to 𝐑𝐐_1: LEA improves cross-encoders performance to typos by a large margin, while achieving competitive performance in their absence. §.§.§ Performance on additional domains Previous experiments showing the improvements of LEA were conducted in the e-commerce domain, short product descriptions with little context. In Table <ref>, we further demonstrate the benefits of LEA using BERT-Medium in RTE (textual entailment) and MRPC (paraphrasing) datasets that represent a completely different domain with longer sentences. Again, typos dramatically reduce the performance of a cross-encoder trained without data augmentation. However, LEA palliates this drop and achieves best results in RTE with typos (∼ 6 absolute points gain), while having comparable performance to a vanilla cross-encoder trained with data augmentation in MRPC. In contrast, in a clean setup LEA suffers small performance drops with respect to the cross-encoder. We argue that Jaccard may reflect similarity worse in long texts than an edit distance because it is agnostic to character order, resulting in a higher probability of highlighting unrelated words. In turn, longer sentences reduce the probability of applying typos to relevant words, thus hiding the potential benefit of using LEA in real settings. Despite these limitations, we show that even in this situation LEA performs competitively and can even improve performance. §.§ Impact of the lexical similarity choice The lexical embeddings of LEA are computed with a similarity function between two strings. In Table <ref> (Lexical similarity metric), we analyze the impact of the choice of this similarity metric in the Abt-Buy dataset using BERT-Medium. We try LEA with the following string similarity metrics: Jaccard (Jac.), Smith-Waterman (Smith), Longest Common Subsequence (LCS), Levenshtein (Lev.) and Jaro–Winkler (Jaro) <cit.>. All the metrics improve the performance when evaluating with typos, thus supporting the positive contribution of LEA regardless of the lexical similarity metric adopted. In clean scenarios, the Smith-Waterman similarity does not outperform the regular cross-encoder (top row), while the remaining metrics does surpass it. Smith-Waterman is the metric that is penalized the most by typos appearing in the middle of words, and by lexical variations, as it relies on aligning common substrings. We decided to adopt the Jaccard similarity for LEA given that it consistently outperforms both the clean and the noisy scenarios for short sentences. The Jaccard coefficient applied to characters is order agnostic and therefore more robust to character swaps. Our intuition is that Jaccard provides higher separability between word pairs with and without typos, which is beneficial in short-texts. However, as the word context increases in long sentence domains, the probability of comparing words with different meaning that share characters increases, thus reducing the swap invariance advantage. We refer the reader to Appendix <ref> for further details on the design choices for the relative attention bias used in LEA. The evidences presented provide a positive answer to 𝐑𝐐_2: it is important to choose the right metric for better performance, although all of them help in preventing performance drops against typos with respect to the vanilla cross-encoder. §.§ LEA on different layers and sharing strategy Two important decisions to make when integrating LEA in a Transfomer architecture are: * Do we use the same lexical projection matrix for the entire model, one per layer, or one independent matrix per head? * Do we apply LEA in all layers across the architecture, or it is more beneficial to apply it only in certain layers? In Table <ref> we present results to answer these questions. For the first decision (𝐖^L parameter sharing) we show that using an independent projection matrix per head behaves best and observe an increasing performance tendency towards sharing less parameters, shared across all layers is the worst choice. We argue that this behaviour is reasonable given that using independent 𝐖^L matrices provides higher flexibility to learn the projection, as the addition to the standard self-attention term in Eq. <ref> might need different behaviour in different heads for better performance. We, therefore, use for the default LEA configuration this non-shared alternative. Regarding the second decision (Layers with LEA), we evaluate adding LEA to different layer subsets in BERT-Medium (8 layers in total): all layers ([0-8]), excluding the first two layers ([2-8]), second half of the layers ([4-8]) and the last two layers ([6-8]). We observe that all the choices help dealing with typos, achieving the best performance by adding LEA to the second half of layers. Similar behaviour is observed in clean scenarios, although only adding LEA to the last half and last two layers outperform the vanilla cross-encoder performance. Therefore, we use LEA in half of the deeper layers in all architectures and experiments. We argue that the character-level similarity provided by LEA can be considered as a high-level interaction information. Therefore, this complements the high-level features of deep Transformer layers. We left for future work to validate this hypothesis. The results obtained in this set of experiments address 𝐑𝐐_3: it is better to use dedicated lexical projection matrices for each attention head and to add LEA on late layers for better performance. §.§ Impact of the noise strength We analyze in Figure <ref> (top) the robustness of LEA to different noise strengths at test time. These results demonstrate a higher robustness to typos than vanilla cross-encoder baselines trained with and without data augmentation. For this experiment, models trained simulating typos use a 20% probability of introducing them in a word, while at test this probability is modified to change the noise strength. Intuitively, since the character-level similarities exploited by LEA are not learned during training, they provide the model with information that is, to some extent, less dependent on the amount of noise. Furthermore, Figure <ref> (bottom) shows an increasing gap between the performance of LEA with respect to the vanilla cross-encoder trained with data augmentation, suggesting a better generalization of LEA to different noise strengths. We, then, can respond 𝐑𝐐_4 based on these results, LEA maintains a robust performance across noise strengths, being dramatic the performance drop of a vanilla cross-encoder trained without data augmentation. §.§ Additional experiments §.§.§ Comparison with larger models In order to assess the effectiveness of LEA in a larger model, we perform experiments using BERT-Large. Additionally, we adopt auto-regressive architectures (GPT-2 and GPT-Neo) to compare them with the auto-encoder models used across this work. In Table <ref>, we show that despite following the same training procedure, the gap between the vanilla cross-encoder and LEA using BERT-Large increases to 28 absolute points. In Table <ref> we show the effectiveness of LEA for the clean versions of the datasets. For the GPT-like models, we followed the same approach as in <cit.> and fine-tuned the backbones as cross-encoders using the last token embedding for the sentence representation pooling (also suggested in <cit.>). We used the same hyper-parameters as the ones used in the experiments of our paper, number of epochs, size of the classification head, etc, and the publicly available pre-trained weights in HuggingFace <cit.>. As we observe in Table <ref> , embeddings obtained by fine tuning GPT-like architectures in our downstream tasks still suffer from the textual noise issue, experiencing drops in performance of 23 and 7 absolute points on average for GPT2-330M without and with DA, respectively. GPTNeo-125M also shows an average drop in performance of 21 and 4 absolute points when trained without and with DA, respectively. Despite these models being pre-trained in massive data and having more parameters, we show that by just using BERT-Base equipped with LEA we manage to outperform GPT-like architecture in the presence of textual noise. Note that we leave the addition of LEA to GPT-like architectures for future work. These results suggest that larger models might reduce the gap to some extent (as depicted in Table <ref>), but they strongly suffer with textual noise (as shown when comparing results between Table <ref> and Table <ref>). Overall our approach mitigates the impact of noise, while keeping comparable performance using clean text. §.§.§ Comparison with larger datasets We have conducted experiments considering WDC-Computers XLarge (68,461 data points in total for training) and WDC-All (214,661 samples for training) obtaining the results in Table <ref>. In all the experiments, we show that LEA consistently improves the baselines performance by a significant margin, confirming the effectiveness of our proposal in larger datasets. It is worth mentioning that the average results of “BERT-M + DA” for the 3 test splits slightly improves LEA, although with a high standard deviation. Nevertheless, LEA clearly outperforms the baselines in the remaining scenarios. § CONCLUSIONS This work proposes LEA, a LExical-aware relative Attention module designed to improve the performance of cross-encoder architectures in sentence similarity tasks. LEA is particularly intended for scenarios with textual noise ( typos) and short texts, where we show that vanilla Transformers drop performance due to tokenization shifts between noisy and clean data. In particular, we propose to modify the self-attention module by introducing a lexical bias. This lexical information tackles the tokenization shift by providing a raw character-level similarity that tends to be high for lexically close words, with and without typos. This similarity is independent of the tokenization and does not assume any prior knowledge on the type of noise present in the data. The results of LEA on five e-commerce datasets using several backbones of varying size demonstrate consistent improvements when dealing with typos over cross-encoder baselines. We further verify the robustness of LEA against typos in textual entailment and paraphrasing tasks and observe competitive performance despite not being strictly designed for these scenarios. Moreover, we provide insights to better understand the behaviour of LEA and explore the impact of: 1) different string similarity metrics, 2) introducing the lexical bias at varying subsets of layers and 3) sharing parameters when encoding the lexical similarities. Finally, we investigate the generalization to different noise strengths, demonstrating that LEA performs and generalizes better than the vanilla cross-encoder baselines. §.§ Limitations and future work Despite making no assumption about the type of noise, LEA assumes that lexical similarities between two sentences is a relevant bias for similarity matching. It is worth mentioning that in scenarios without typos there is a slight drop in performance (lower than 2 absolute points on average) as reported in Table <ref> when adding this bias. However, in the presence of typos LEA largely outperforms a Vanilla cross-encoder (more than 30 absolute points on average), thus demonstrating that the proposed lexical bias helps in these scenarios. LEA is designed for Transformer configurations where two or more sentences are used as part of the input to the model. While limited to this specific context, it encompasses a wide-ranging topic within the sentence similarity literature and LEA can be repurposed across different but related domains. Future work will focus on improving the use of lexical information on longer texts and better using this bias in clean scenarios. Another interesting research direction is the extension of LEA to bi-encoders with late-interaction techniques. ACM-Reference-Format § ALTERNATIVES TO THE RELATIVE ATTENTION BIAS Inspired by the relative attention bias introduced in <cit.> (“Fix sin.” as reported in Table <ref>), LEA introduces a small variation that scales the similarities to be in the range of [0, 2π] instead of [0, 1] (“Fix-scale sin.” in Table <ref>). The motivation behind this change is to increase the granularity of the lexical similarities by expanding the domain of the cosine and sine functions. This modification yields representations that are better distributed in the space. Note that we also considered an additional alternative to the sinusoidal embeddings by using learnable embeddings (“Learned” in Table <ref>). Apart from showing lower performance, the learnable embeddings add extra parameters and require the discretization of the similarities to map them into embeddings, a step that could potentially lead to information loss. As presented in Table <ref>, the use of sinusoidal functions to represent lexical similarities provides improved robustness and flexibility compared to the use of raw lexical similarity values ( jaccard distance) as the bias in the self-attention module. § EXPERIMENTAL SETUP DETAILS In this section, we show additional details about the hyper-parameters used across all experiments and backbone architectures. Unless otherwise stated we used the following configuration in all experiments and models: * Batch size: 32. * Learning rate scheduler: cosine annealing with warm-up. * Initial learning rate: 5e^-5. * Warm-up: 1.5 epochs. * Number of epochs: 30. * Weight decay: 5e^-5. * Sentence representation: mean pooling of all token embeddings excluding padding tokens. * Classification head: Multi-Layer Perceptron with a Linear layer of size 256 follow by LayerNorm, Dropout (0.1), GeLU activation and an output Linear layer of size 2. In Table <ref>, we show the differences between the parameters used for the experimental setup of the models under comparison. Note that the Vanilla cross-encoder, regardless the backbone architecture, does not use any data augmentation nor lexical attention embedding. For both the cross-encoder with DA and with LEA we use the same data augmentation described in Table <ref>.
http://arxiv.org/abs/2307.00637v1	20230702191629	On Embedding B-Splines in Recursive State Estimation	[ "Kailai Li" ]	eess.SY	[ "eess.SY", "cs.SY" ]	Emergent Spatiotemporal Organization in Stochastic Intracellular Transport Dynamics Kunaal Joshi,^1,∗ Harrison York,^2,∗ Charles S. Wright,^1,2 Rudro R. Biswas,^1 Senthil Arumugam^2,3,4,5,†, and Srividya Iyer-Biswas,^1,6,† ^1Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA ^2Monash Biomedicine Discovery Institute, Faculty of Medicine, Nursing and Health Sciences, Monash University, Clayton/Melbourne, VIC 3800, Australia ^3ARC Centre of Excellence in Advanced Molecular Imaging, Monash University, Clayton/Melbourne, VIC 3800, Australia ^4European Molecular Biological Laboratory Australia (EMBL Australia), Monash University, Clayton4/Melbourne, VIC 3800, Australia ^5Single Molecule Science, University of New South Wales, Sydney, NSW 2052, Australia ^6Santa Fe Institute, Santa Fe, NM 87501, USA ^∗These authors contributed equally to this work. ^†To whom correspondence should be addressed: [email protected] and [email protected]. August 1, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ empty empty We present a principled study on establishing a novel probabilistic framework for state estimation. B-splines are embedded in the state-space modeling as a continuous-time intermediate between the states of recurrent control points and asynchronous sensor measurements. Based thereon, the spline-embedded recursive estimation scheme is established w.r.t. common sensor fusion tasks, and the corresponding technique for modeling uncertain motion estimates is introduced. We evaluate the proposed estimation scheme using real-world-based synthesized data with a range-inertial setting. The numerical results demonstrate several advantages of spline embedding in recursive state estimation over the classic discrete-time filtering scheme. Estimation, filtering, stochastic systems, robotics § INTRODUCTION State estimation plays a critical role in ubiquitous application scenarios related to systems and control, including tasks such as object tracking, motion planning, robotic manipulation and locomotion, as well as cyber-physical security <cit.>. In most engineering practices, performing state estimation typically involves incorporating asynchronous measurements of various sensing modalities, such as wheel odometry, camera, ultra-wideband (UWB), light detection and ranging (LiDAR), and inertial measurement unit (IMU) <cit.>. The classic methods for multi-sensor fusion are rooted in the recursive Bayesian estimation scheme. In the context of estimating motions, popular choices of approaches include the extended Kalman filter (EKF) <cit.> and the particle filter <cit.>. The state-space representation is herein exploited to describe uncertain physical systems with a composition of the process and measurement models under uncertainty. State estimates are delivered recursively based thereon via prediction and update steps over time <cit.>. This probabilistic scheme also directly provides uncertainty quantification of the estimate, allowing for further tasks, such as outlier rejection, planning, decision and control, to be conducted with adequate accuracy and confidence <cit.>. To estimate the six-DoF rigid-body motions, recursive estimation approaches have also been developed nonlinear manifolds using error-state formulation and geometry-driven methodologies <cit.>. The conventional recursive estimation scheme is based on the Markov assumption, which requires sufficient knowledge of the underlying system dynamics and disregards state propagation beyond consecutive time steps. Recent advances in robotics, particularly in simultaneous localization and mapping, have introduced alternative schemes using optimization <cit.>. Optimal state estimates are obtained in the sense of nonlinear least squares (NLS), with residual terms built upon multi-sensor measurements through a graph structure over a certain time span. Compared with the recursive scheme, this paradigm shift can lead to improved estimation accuracy while maintaining tractable computational cost due to the sparse problem structure <cit.>. Based thereon, numerous state-of-the-art solutions have emerged in robotic state estimation and perception using multi-sensor setups. A variety of important techniques, including IMU preintegration and keyframe-based sliding-window optimization, have been proposed to achieve high-performance state estimation in terms of tracking accuracy, robustness and runtime efficiency <cit.>. In state estimation, the random variables are driven by the underlying stochastic dynamics that is inherently continuous over time, whereas observed at discrete time instants with noise. To properly address this discrepancy, most recursive estimation schemes rely on temporal discretizations of the continuous-time state-space models, resulting in the continuous-discrete and discrete-time variants <cit.>. For that, remedies such as oversampling of the system or approximation of transition densities have been introduced, and additional efforts are typically required for achieving viable accuracy and stability <cit.>. Given asynchronous sensor readings, delivering state estimates at the desired time instants (e.g., at a fixed frame rate) inevitably requires certain approximation, such as linear interpolation <cit.>. Besides, sensor measurements that contribute to the state propagation are often assumed to be constant over the sampling interval. These operations can introduce additional errors and often require tedious implementation, especially when combined with other procedures, e.g., the IMU preintegration <cit.>. Furthermore, enforcing kinematic relations over sequential motion estimates in conjunction with sensor fusion is hardly possible without introducing explicit constraints. Under unfavorable conditions, such as highly dynamic motions and noisy measurements with outliers, state estimates can exhibit physically infeasible transitions. There have been increasing efforts to enable continuous-time sensor fusion, and significant progress has been made in utilizing the B-spline <cit.>. Established atop a series of control points (knots), B-splines parameterize motion trajectories as polynomial functions over time, leading to a more efficient state representation in comparison with the discrete-time counterpart <cit.>. B-splines are smooth to a given order and inherently guarantee kinematic relations over temporal differentiations. Additionally, the locality of B-splines enables convenient interpolations via matrix multiplication and implicitly spreads correlations over dynamical motions through adjacent control points <cit.>. Using the cumulative form of B-splines, it is also possible to extend their utility from modeling vector-valued (Euclidean) to Lie group-valued functions. This makes them particularly appealing for continuous-time parameterization of six-DoF pose trajectories in various robotic applications <cit.>. Based on the methodology of B-spline fitting, a few systems have been recently developed for continuous-time state estimation using multi-sensor setups. These involve, in particular, the combination of the IMU with (event) camera, radar, UWB, and LiDAR, among other modalities <cit.>. In these systems, the control points of B-splines are obtained optimally via NLS, with residuals directly comparing the sensor measurement and the corresponding interpolated value. To realize high-performance sensor fusion, several important theories and approaches have been introduced, including quaternion-based B-spline modeling, efficient temporal and spatial differentiations, as well as sliding-window spline fitting <cit.>. Though ongoing progress has been made in utilizing B-splines for sensor fusion, all existing methods are optimization-based, leaving a considerable gap towards establishing a self-contained probabilistic framework. Besides, relevant works only refer to applying recursive filters to spline approximation for geometry modeling <cit.>. To the best knowledge of the author, there is currently no published work that systematically exploits B-splines in recursive state estimation of dynamical systems. §.§ Contributions In light of the current state of the art, we propose a novel recursive state estimation scheme for continuous-time sensor fusion. Cubic B-splines are exploited for state parameterization with kinematic interpolations unified as a concise matrix representation (sec:spline). Further, we introduce the so-called spline-state-space (TriS) model based on the concept of recurrent control points, endowing recursive feasibility and inferential power to cubic B-splines in state estimation (sec:stochastic). This also leads to expressive dynamical modeling, meanwhile utilizing the appealing locality and smoothness of B-splines. Afterward, the spline-embedded recursive estimation (SERE) scheme is newly established with corresponding probabilistic modeling of motion estimates (sec:filter). We further evaluate the SERE scheme on a real-world-based synthetic data set using a range-inertial setting (sec:eva). The numerical results show that the spline embedding enables considerable improvements on tracking performance compared to the conventional filtering scheme. As the first fundamental study on spline-based recursive state estimation, our work sketches out a self-contained paradigm for continuous-time probabilistic sensor fusion. § NOTATION CONVENTIONS Throughout the following content, we use underlined lowercase variables, such as ∈^d, to denote vectors, with letter d denoting the dimension of Euclidean spaces. Random variables are indicated by lowercase letters of bold font, such as ∈^d. Matrices are written as uppercase letters of bold case, such as ∈^d×d. We use _d∈^d×d and _d∈^d×d to denote d-dimensional identity and zero matrices, respectively. In the context of recursive state estimation, we exploit k∈ to denote a discrete time step, whereas the exact time instant at k is written as lowercase letter t_k. Moreover, we exploit calligraphic lowercase letters to denote functions, such as (x) and (x) for scalar- and vector-valued functions, respectively. The operator ⊗ denotes the Kronecker product, and the operator ⊕ stands for the direct sum of two matrices <cit.>. § CONTINUOUS-TIME MOTION REPRESENTATION For demonstrating our work in this paper, we express B-splines using their cubic form (fourth-order) in d-dimensional Euclidean spaces. Note that the presented techniques can be extended to B-splines of higher orders in a straightforward manner. Cubic B-splines exhibit continuous-time derivatives up to the second order, which is sufficient for fusing measurements of most sensory modalities related to motion estimation, including the accelerometer <cit.>. Given a set of control points {_i}_i=1^n⊂^d at time instants {t_i}_i=1^n⊂ of uniform temporal interval τ, a cubic B-spline can be established for trajectory representation. At an arbitrary time instant t∈( t_i,t_i+1], the trajectory point is determined by a local set of control points {_i+j-2}_j=0^3 according to _t=_i _t , with matrix _i=[ _i-2, _i-1, _i, _i+1 ]∈^d×4 concatenating the corresponding control points columnwise. The vector _t is given by _t=[ 1,u_t,u_t^2,u_t^3 ]^⊤∈^4 , which contains the powers of the normalized time instant u_t=(t-t_i)/τ . is the basis function matrix of cubic B-splines and follows =1/6 1 -3 3 -1 4 0 -6 3 1 3 3 -3 0 0 0 1 . We can derive the velocity trajectory of cubic B-splines by taking the first derivative of the polynomial function (<ref>) time. It is given by _t=_i _t ,_t=[ 0,1,2u_t,3u_t^2 ]^⊤/τ . Further, the acceleration trajectory can be derived as the following function over time _t=_i _t ,_t=[ 0,0,2,6u_t ]^⊤/τ^2 . In the following content, we use ∘ atop the spline function _t, i.e., _t, to denote trajectories of different orders at time instant t (including _t, _t, and _t). Therefore, the kinematic interpolations can be unified as _t=_i _t , with _t concretized in (<ref>), (<ref>), and (<ref>). § STOCHASTIC MODELING ON B-SPLINES In existing spline-based sensor fusion schemes, control points over a certain time span are optimally obtained in the sense of nonlinear least squares, delivering motion trajectories as continuous functions over time. The kinematic interpolations in sec:spline provide computational foundations for computing residuals using measurements at original time instants <cit.>. However, modeling the uncertainty of motion estimates is almost infeasible within an optimization-based scheme. Thus, we aim to establish a principled framework for stochastic modeling using B-splines. §.§ Vectorized Kinematic Interpolations We now express the basic kinematic interpolations introduced in sec:spline using a vector representation of the control points. According to <cit.>, we can perform vectorization on both sides of (<ref>) and obtain (_t) =(_i _t) =(( _t)^⊤⊗_d) (_i) , with t∈(t_i,t_i+1 ] . ⊗ denotes the Kronecker product as introduced in sec:notation. Note that _t is a d-dimensional vector. It then follows _t=_t _i ,_t=(_t)^⊤⊗_d∈^d×4d containing the interpolation coefficients control points that are combined in the vector _i=(_i)=[ _i-2^⊤, _i-1^⊤, _i^⊤, _i+1^⊤ ]^⊤∈^4d. §.§ Spline-State-Space (TriS) Model In the context of recursive Bayesian estimation, B-splines can be leveraged for continuous-time parameterization of dynamical motions discrete-time observations. We conceptualize this idea into the so-called spline-state-space (TriS) model as illustrated by the graph representation in fig:tris. A B-spline is embedded to a typical state-space model as an intermediate between the discrete-time states and observations. §.§.§ State vector Given the measurements {_i}_i=1^k, the cubic B-spline is maintained using a minimal set of control points {_i}_i=1^n_k at time instants {t_i}_i=1^n_k, with the latest measurement time instant t_k^∈( t_n_k-1,t_n_k]. The kinematic interpolation at time step k is then determined by the latest four control points {_n_k-i}_i=0^3, which we compose into the following state vector _k=[ _n_k-3^⊤,_n_k-2^⊤,_n_k-1^⊤,_n_k^⊤ ]^⊤∈^4d . We name this local set of control points as the recurrent control points to highlight their temporal relations to the most recent measurement _k. §.§.§ Measurement model Given the state vector in (<ref>), the measurement model at time step k can be expressed as the following general form _k=_t_k^(_k)+_k , t_k^∈( t_n_k-1,t_n_k] . We use to denote the domain of measurements. _k∈ is an additive noise assumed to be zero-mean Gaussian-distributed, i.e., _k∼(,_k), with _k being the measurement noise covariance. The observation function _t_k^^4d→ takes the recurrent control points at time step k as input and can be decomposed into two cascaded steps as follows _t_k^(_k)=(_t_k^(_k)) . The function _t_k^^4d→^d denotes the vectorized kinematic interpolation introduced in (<ref>) at time instant t_k^, generating the motion variable for sensing. Further, function :^d→ models the actual sensory modality, mapping the interpolated motion to the noise-free observation. In fig:tris, this cascaded measurement modeling is showcased with an ultra-wideband-accelerometer setup. §.§.§ Process model The process model describes the propagation of recurrent control points in the case that the incoming measurement falls outside the current spline time span. Suppose that the measurement _k+1 comes at time instant t_k+1^∈( t_n_k,t_n_k+τ ], given the spline of n_k control points. In order to accommodate the measurement model at t_k+1^ as introduced in (<ref>), the spline is extended to n_k+1=n_k+1 control points. Consequently, the state vector defined in (<ref>) is updated to _k+1 =[ _n_k+1-3^⊤,_n_k+1-2^⊤,_n_k+1-1^⊤,_n_k+1^⊤ ]^⊤ =[ _n_k-2^⊤,_n_k-1^⊤,_n_k^⊤,_n_k+1^⊤ ]^⊤ , where the first three control points overlap with the last three control points in the previous state _k in (<ref>). In general, the process model can be expressed as _k+1 = (_k)+_k+1 . The additive process noise _k+1∈^4d is assumed to follow a zero-mean Gaussian distribution (,_k+1), with _k+1 being the covariance. The transition function ^4d→^4d in (<ref>) can be established with reference to various dynamical principles. We hereby introduce a straightforward strategy with the following two steps: 1) we retain the first three control points {_n_k-i}_i=0^2 in _k+1 at their coordinates in the previous state _k, and 2) the latest control point _n_k+1 is added by preserving the velocity at the time instant t_n_k-1. For the latter step, we perform velocity interpolations at time instants t_n_k-1 and t_n_k+1 according to (<ref>), resulting in _t_n_k-1 =_n_k-1-_n_k-3/2τ _t_n_k+1 =_n_k+1-_n_k-1/2τ , respectively. Imposing identical velocities at these two time instants then yields _n_k+1=2_n_k-1-_n_k-3 . Therefore, the process model in (<ref>) can be concretized as the following linear expression _k+1= _k+_k+1 , with the transition matrix given by =_d _d _d _d _d _d _d _d _d _d _d _d -_d _d 2_d _d. § SPLINE EMBEDDING IN RECURSIVE ESTIMATION Based on the proposed spline-state-space model, the task of continuous-time motion estimation can be converted into estimating discrete-time recurrent control points. To concretize this concept, we now introduce the so-called spline-embedded recursive estimation (SERE) scheme together with a simple case study in this section. §.§ Spline-Embedded Recursive Estimation (SERE) As shown in alg:filter, the recurrent control points are estimated recursively following the concept of Kalman filtering based on the TriS model. Suppose that a measurement _k is obtained at time instant t_k^, given n_k-1 control points. The state _k-1 composes recurrent control points {_n_k-1-i}_i=0^3, and its previous posterior estimate mean _k-1\|k-1 and covariance _k-1\|k-1 are available. If the measurement time instant exceeds the current spline time span, namely, t_k^>t_n_k-1, we perform state propagation according to the system model in (<ref>), leading to n_k=n_k-1+1 control points and the predicted prior estimate (alg:filter, lines 1–4). Otherwise, the previous state estimate remains as the prior, with the number of control points being unchanged (alg:filter, lines 5–8). During the update step, a basic strategy is to linearize the observation function (<ref>) in the measurement model (<ref>) at the prior estimate mean _k\|k-1. The resulting observation matrix can be obtained by applying the chain rule as _k =𝒥_(_t_k^(_k\|k-1)) __t_k^(_k\|k-1) =_(_t_k^(_k\|k-1)) _t_k^ , with the first term being the Jacobian of the sensing function the kinematic motion interpolated at t_k^ using the prior estimate _k\|k-1. The second term in (<ref>) refers to the Jacobian of the kinematic interpolation function w.r.t. the state vector. According to the linear expression given in (<ref>), it follows __t_k^(_k\|k-1)=_t_k^ , which is constant given the measurement time instant t_k^ and the order of kinematic motion observed by the sensor (alg:filter, line 9). Based thereon, the prior estimate can be corrected by incorporating the measurement _k through a typical EKF update step, yielding the posterior estimate mean _k\|k and covariance _k\|k (alg:filter, line 10–13). §.§ Probabilistic Interpolation Embedding B-splines in recursive estimation decouples discrete-time state propagation from sensing continuous-time dynamical motions. Given the state estimate of mean _k and covariance _k, we now aim to retrieve the motion estimates {_t_i}_i=1^m at arbitrary time instants {t_i}_i=1^m⊂( t_n_k-1,t_n_k]. For that, the queried motions are stacked into a vector _t_1:m=[ _t_1^⊤,⋯,_t_m^⊤]^⊤∈^dm , to which we perform (<ref>) in a batchwise manner. The kinematic interpolations are linear recurrent control points. Thus, the mean and covariance of the motion estimates in (<ref>) follow _t_1:m=_t_1:m_k_t_1:m=_t_1:m_k _t_1:m^⊤ , respectively, where the matrix _t_1:m=[ _t_1^⊤,⋯,_t_m^⊤]^⊤∈^dm×4d denotes the combined coefficient matrices at each time instant. Thus, we essentially obtain the joint probability distribution of the dynamical motion estimates at any time instants, with different orders of temporal derivatives unified within a single computational procedure. To showcase the utility of the proposed SERE scheme, we now present a brief case study as follows. §.§ Case Study We synthesize a localization scenario using a GPS along different Lissajous curves <cit.> in a two-dimensional space (depicted by the red trajectories in fig:lissa). The GPS readings are simulated at a frequency of 100 Hz over a duration of 10 seconds and are further corrupted by an additive noise following a Gaussian distribution (_2,0.01 _2). We configure the TriS model with a state vector _k∈^8 that concatenates the recurrent control points {_n_k-i}_i=0^3⊂^2 of temporal interval τ=0.1 second. To model the GPS sensing, we concretize the generic formulation of the measurement model in (<ref>) to be _k=_t_k^_k+_k ,_t_k^=( _t_k^)^⊤⊗_2 being the coefficient matrix for position interpolation at t_k^∈( t_n_k-1,t_n_k]. u_t_k^=(t_k^-t_n_k-1)/τ denotes the normalized time instant according to (<ref>). The measurement noise is set to be zero-mean Gaussian-distributed with _k∼(_2,_k), where _k=0.01_2. Given the linear observation function in (<ref>), the observation matrix for spline-embedded recursive estimation in alg:filter follows _k=_t_k^ according to (<ref>). Furthermore, we exploit the expression of the process model in (<ref>) for state propagation. The process noise is Gaussian-distributed following (_8,), where the covariance =(0.02 _6)⊕(0.1 _2) , with ⊕ denoting the direct sum as introduced in sec:notation. As shown by the blue curves in fig:lissa, the proposed method delivers accurate tracking results along different trajectories. In fig:lissasig, we illustrate the estimated uncertainty along the sequence in fig:lissa-(D) via the probabilistic interpolation introduced in subsec:probItp in conjunction with downsampled measurements. These qualitative results are evident to show the efficacy of the proposed SERE scheme, including the continuous-time probabilistic interpretation of the motion estimates. § EVALUATION We now conduct an in-depth evaluation of the proposed SERE scheme using a multi-sensor setup for motion estimation. Comparisons with conventional recursive filtering scheme will be presented based on numerical results, followed by discussions on the configuration of the approach. §.§ Real-World-Based Synthetic Data Set We create a synthesized scenario for motion estimation in sensor networks using time-of-arrival (ToA) and acceleration measurements. The UTIL data set is exploited as a real-world reference <cit.>, where asynchronous UWB and IMU readings are recorded onboard a moving quadrotor platform. We choose a subset of UTIL that comprises all distinctive trajectories of autonomous maneuvers. This yields a total of 42 sequences under different UWB anchor constellations (const1-4) and operation modes (tdoa2 and tdoa3). We adopt the ground truth trajectories to simulate the accelerometer (the world frame) and ToA data at the original time instants of the IMU (1000 Hz) and UWB (200 - 500 Hz) readings of UTIL, respectively. In addition, we corrupt these generated observations with additive zero-mean Gaussian noises of covariances ^=0.01 _3 and R^=0.01 for the accelerometer and ToA readings, respectively. In the subsequent evaluation, we refer to this synthetic data set as Syn-UTIL. §.§ Setup Given the previously synthesized evaluation scenario, we now establish the TriS model used in the SERE scheme. We configure the temporal interval of control points to be τ=1 second. At step k, the state vector comprising the recurrent control points {_n_k-i}_i=0^3⊂^3 follows _k∈^12. The process model in (<ref>) is employed for state propagation, and the additive noise term _k+1 therein follows a zero-mean Gaussian distribution (_12,), with =(0.02×10^-5 _9)⊕(10^-2 _3) . The measurement models at time instant t_k^ are formulated according to (<ref>) as _k^ =_t_k_k+_k^ or _k^ =‖_t_k_k-_k‖+_k^ , given an accelerometer or a ToA measurement, respectively. Here, _k in the range observation denotes the position of the anchor indexed at t_k^ according to the original setting in UTIL. The noise terms _k^ and _k^ follow zero-mean Gaussian distributions of covariances ^ and R^ as described in subsec:data, respectively. Given the expressions in (<ref>), the observation matrices of the accelerometer and ToA measurement models can be derived according to (<ref>) as _k^ =_t_k _k^ =(_t_k_k-_k)^⊤_t_k/‖_t_k_k-_k‖ , respectively. We then perform the proposed spline-embedded recursive estimation on the recurrent control points according to alg:filter, and the trajectory estimates are obtained online via the kinematic interpolation given in (<ref>). For comparison with the conventional recursive estimation scheme, we implement an EKF as the baseline. It estimates a state vector comprising the position and velocity. The prediction and update steps are scheduled asynchronously according to the multi-sensor fusion scenario. Given an accelerometer reading, a prediction step is conducted under the assumption of constant acceleration since the previous measurement. Upon the arrival of a ToA measurement, we first propagate the state via the constant-acceleration model using the previous accelerometer reading. Afterward, an EKF update step is performed to fuse the range measurement and deliver the posterior estimate. §.§ Results For evaluating the estimation schemes, we compute the root mean squared error (RMSE) of the position estimates the ground truth throughout the trajectory. Overall 100 Monte Carlo runs are performed on each of the 42 sequences in Syn-UTIL, and the resulting RMSE distributions are summarized in fig:eva. Compared with the EKF of conventional design, the spline-embedded estimation scheme enables considerable improvement on tracking accuracy across all sequences of various settings. Furthermore, it only requires storage of the underlying control points occurring at a frequency of 1 Hz (given τ=1s) and delivers continuous motion trajectories over time, whereas the baseline method adopts discrete-time states at the rate of multi-modal measurements (asynchronous between 200 and 1000 Hz). In fig:util, we select the sequences that encompass all different shapes of trajectories in Syn-UTIL to demonstrate representative tracking results given by SERE. The proposed method delivers inherently smooth trajectory estimates with close alignment to the ground truth. To further highlight the strength of spline embedding in noise adaptation, we perform additional tests on trial4 and trial6 of tdoa2 in Syn-UTIL under increased noise level of ToA ranging (R^=0.1). As shown in the plots of top view in fig:comp, the proposed SERE scheme produces accurate trajectory estimates with inherent kinematic consistency thanks to the B-spline intermediate in the TriS model. In contrast, the baseline method exhibits clearly inferior tracking performance and delivers physically infeasible motion transitions due to the conventional state-space formulation. §.§ Discussion Achieving high performance in state estimation using spline embedding requires proper configurations of the system. In this regard, we provide insights on tuning two major parameters as follows. §.§.§ Temporal interval The interval τ for placing control points in the time domain is a global parameter for embedding B-splines in recursive estimation. In principle, a smaller value of τ allows for more detailed modeling of dynamical motions. However, this also requires higher rate of sensor measurements for more frequent state propagation and correspondingly more memory consumption for state storage. §.§.§ Process noise Tuning the process noise in deploying the SERE scheme follows common practice for conventional stochastic filtering. This typically refers to criteria regarding tracking accuracy, adaptation speed, noise sensitivity, etc <cit.>. According to (<ref>), the last three recurrent control points of the previous state are preserved during system propagation, with a new control point to be added for spline extension. Therefore, we typically construct with the following diagonal structure =(ω _3d)⊕(ν _d) , where usually a higher uncertainty is assigned to the newly added control point compared to the preserved ones, namely, ω<ν. During the update step, the latest recurrent control point usually obtains a higher gain compared with the other ones. In general, a recurrent control point receives less gain over time until being removed from the state vector, and its estimate is then fixed. To further justify our insights to parameter tuning provided previously, we now conduct a dedicated study using sequence const1-trial2-tdoa2 in Syn-UTIL under the noise level of R^=0.1 for the ToA data (The rest setup stays the same as introduced in subsec:data). The configuration of process noise follows the general expression in (<ref>), with ω∈{0.1,0.01,0.001} and ν=0.1 controlling the noise levels of preserving and adding recurrent control points, respectively. Meanwhile, the temporal interval of control points varies among τ∈{0.1,1,6}. fig:mat depicts the trajectory estimates given by SERE using the nine different parameter sets. When fixing the temporal interval τ, reducing the relative uncertainty of preserving and adding recurrent control points (smaller ω/ν) leads to better tracking accuracy. Given the same process noise, coarser control points in time domain (larger τ) induce less memory consumption for state storage, however, insufficiency in estimating complex trajectories. A small temporal interval τ can mitigate this issue, but it also results in overly dynamic motion estimates due to the fixed measurement rates. In practice, setting up the proposed SERE scheme often comes to trade-offs among multiple factors, including motion dynamics, sensor data rate and hardware constraints, etc. § CONCLUSION In this work, we have conducted a principled study on a novel continuous-time framework for recursive state estimation. The so-called spline-state-space (TriS) model has been established by embedding B-splines in state-space modeling as a continuous-time intermediate, which decouples propagating discrete-time states of recurrent control points from incorporating asynchronous sensor readings. Based thereon, we have established the spline-embedded recursive estimation (SERE) scheme and have introduced corresponding technique for probabilistic modeling of dynamical motion estimates. The proposed estimation scheme has been validated in real-world-based synthesized scenario for continuous-time state estimation using a ToA-accelerometer setting. Compared with conventional recursive filtering, it demonstrates several advantages, including improved tracking accuracy, kinematic feasibility, storage consumption, and deployment flexibility. There still exists much potential to exploit the SERE scheme. One promising direction for enhancing the current design is to develop dedicated methods for automatic system tuning. Another straightforward extension is to enable six-DoF motion estimation by incorporating spline modeling of spatial rotations <cit.>. Moreover, B-splines atop control points of nonuniform temporal interval should be considered to improve the efficiency of state representation. Also, applying the proposed SERE scheme to extensive real-world applications of various multi-sensor settings is appealing <cit.>. This can, in turn, provides valuable insights for characterizing and improving the overall design of spline-based estimation schemes. § REFERENCES IEEEtran.bst
http://arxiv.org/abs/2307.02952v1	20230706123851	Density dependent gauge field inducing emergent SSH physics, solitons and condensates in a discrete nonlinear Schrödinger equation	[ "William N. Faugno", "Mario Salerno", "Tomoki Ozawa" ]	quant-ph	[ "quant-ph", "cond-mat.quant-gas", "nlin.PS" ]
http://arxiv.org/abs/2307.00814v1	20230703075348	Finite Element Modeling of Power Cables using Coordinate Transformations	[ "Albert Piwonski", "Julien Dular", "Rodrigo Silva Rezende", "Rolf Schuhmann" ]	math.NA	[ "math.NA", "cs.NA" ]	COMPUGMAG 2023: OC1: Static and quasi-static fields / Wave propagation. Conference Paper Number: 258. Shell et al.: Bare Demo of IEEEtran.cls for IEEE Transactions on Magnetics Journals Power cables have complex geometries in order to reduce their ac resistance. Although there are many different cable designs, most have in common that their inner conductors' cross-section is divided into several electrically insulated conductors, which are twisted over the cable’s length (helicoidal symmetry). In previous works, we presented how to exploit this symmetry by means of dimensional reduction within the 𝐇-φ formulation of the eddy current problem. Here, the dimensional reduction is based on a coordinate transformation from the Cartesian coordinate system to a helicoidal coordinate system. This contribution focuses on how this approach can be incorporated into the magnetic vector potential based 𝐀-v formulation. Coordinate transformations, dimensional reduction, eddy currents, finite element modeling, magnetic vector potential formulation, power cables, tree-cotree gauging. Finite Element Modeling of Power Cables using Coordinate Transformations Albert Piwonski1, Julien Dular2, Rodrigo Silva Rezende1, Rolf Schuhmann1 1Theoretische Elektrotechnik, Technische Universität Berlin, Berlin, Germany 2 TE-MPE-PE, CERN, Geneva, Switzerland August 1, 2023 ================================================================================================================================================================================================== § INTRODUCTION Power cables are essential elements in the transmission chain of electric power from generator to consumer. Special inner conductor designs have been developed for ac operation to minimize the undesirable current displacement caused by the skin and proximity effect <cit.>. Although there are many different cable designs, most have in common that their inner conductors' cross-section is devided into several conductors, which are twisted and electrically insulated from each other (see Fig. <ref>). Investigating ac losses in power cables is an important but challenging task: Solving numerically eddy current boundary value problems (BVP) in 3-D, which model the cable’s electromagnetic behaviour in the magnetoquasistatic limit, leads to tremendous computational efforts, due to arising multiscale problems (e.g., thin insulation layers). However, if one models the power cable as a symmetric BVP, computational costs can be scaled down significantly by means of dimensional reduction. Due to the conductors' twist, however, no conventional translational symmetry is valid, as recently shown in a formal framework using Lie derivatives <cit.>. In particular though, when choosing proper boundary conditions, an eddy current BVP posed on a domain as in Fig. <ref> has a so-called helicoidal symmetry. In contrast to applying periodic boundary conditions, here the model can be solved in 2-D. This special symmetry was exploited before to calculate hysteresis losses in twisted superconductors <cit.>, but also for solving full Maxwell's equations in twisted waveguides <cit.>. Further, in previous works, we presented how to incorporate this symmetry property within the 𝐇-φ finite element formulation applied to a power cable problem <cit.>. This contribution focuses on an integration into the magnetic vector potential based 𝐀-v formulation: In Section <ref>, we define the coordinate transformation and its inverse that allow a bidirectional transition between Cartesian and helicoidal coords. Subsequently, Section <ref> presents how a 𝐀-v formulation in Cartesian coords can be reformulated equivalently into a 2-D formulation in helicoidal coords. Further, implementation details of the 2-D model are given. In Section <ref>, we compare the results of this contribution and of our previous work <cit.> with a 3-D reference model. § COORDINATE TRANSFORMATIONS In computational electromagnetism, symmetries occur quite differently: E.g., when computing electrical machines, often a translational symmetry is assumed. Further, azimuthal symmetries appear frequently when modeling cylindrical waveguides. The common key idea is the use of a coordinate system in which the geometries appear the same in one direction. Then, partial derivatives w.r.t. to this particular direction of electromagnetic field quantities are either vanishing or constant, leading to drastically reduced computational efforts. Therefore, for describing twisted conductors, a helicoidal coordinate sytem is the most suitable. In the following, we denote points represented in the Cartesian coordinate system (x, y, z) as 𝐩_xyz [x, y, z]^⊤, whereas points represented in the helicoidal coordinate system (u, v, w) are denoted as 𝐩_uvw [u, v, w]^⊤. The birectional change of coordinates is achieved by the map ϕ: Ω_xyz→Ω_uvw and its inverse ϕ^-1:Ω_uvw→Ω_xyz, where Ω_xyz, Ω_uvw⊂ℝ^3: ϕ(𝐩_xyz) = 𝐩_uvw = [ + x cos(zα/β) + y sin(zα/β); - x sin(zα/β) + y cos(zα/β); +z ], ϕ^-1(𝐩_uvw) = 𝐩_xyz = [ +ucos(wα/β) -vsin(wα/β); +usin(wα/β) +vcos(wα/β); +w ]. Here, parameters α, β are related to the number of turns and to the total longitudinal length of the helical object of interest, i.e., for different geometries ϕ resp. ϕ^-1 are defined differently as well. The effect of the global coordinate transformation is demonstrated in Fig. <ref>. It is important to note, that the maps (<ref>) & (<ref>) allow a unique bidirectional transition of points between both coordinate systems, s.t. the diagram in Fig. <ref> commutes. This property ensures that also the electromagnetic field quantities have a unique representation in the other coordinate system, which will be needed in the following. § TEXT FORMULATION IN DIFFERENT COORDINATES In previous works, we presented how to exploit helicoidal symmetries, in the context of eddy current problems, by means of dimensional reduction within the finite element formulation <cit.>. In this paper, on the other hand, we present how to incorporate this approach into the more widespread formulation. From a topological point of view, the latter formulation is simpler, since typically no topological pre-processing of the computational domain Ω is needed. In a nutshell, this can be explained by the fact that the magnetic flux density 𝐁 in Ω is closed (div 𝐁=0) and exact (∃ 𝐀, s.t. 𝐁 = 𝐜𝐮𝐫𝐥 𝐀) <cit.>. In contrast, within the formulation, we consider the magnetic field 𝐇 in the insulating domain Ω_i, which is also closed (𝐜𝐮𝐫𝐥 𝐇 = 0) as there are no currents, but not exact (∄ φ, s.t. 𝐠𝐫𝐚𝐝 φ = 𝐇). This means in particular, if one represents the magnetic field purely as the gradient of a scalar potential φ, Ampere's law would be violated in the presence of conductors, since then no net circulations of 𝐇 along closed curves could be generated. §.§ Weak formulation in Cartesian coordinates In the following, we present the weak formulation in a concise manner. More details can be found in <cit.>. In the Cartesian domain Ω_xyz with boundary ∂Ω_xyz, conducting subdomain Ω_xyz,c (N connected components), and in frequency domain with frequency ω/2π, the formulation consists of two sets of equations: (ν_xyz𝐜𝐮𝐫𝐥 𝐀_xyz, 𝐜𝐮𝐫𝐥 𝐀_xyz' )_Ω_xyz + (jωσ_xyz 𝐀_xyz, 𝐀_xyz' )_Ω_xyz,c - ⟨𝐇_xyz×𝐧, 𝐀_xyz'⟩_∂Ω_xyz + (σ_xyz 𝐠𝐫𝐚𝐝 v_xyz, 𝐀_xyz' )_Ω_xyz,c = 0, (jωσ_xyz 𝐀_xyz, 𝐠𝐫𝐚𝐝 v_xyz' )_Ω_xyz,c + (σ_xyz 𝐠𝐫𝐚𝐝 v_xyz, 𝐠𝐫𝐚𝐝 v_xyz' )_Ω_xyz,c= ∑_i=1^N I_i V_i', with the magnetic vector potential 𝐀_xyz and electric scalar potential v_xyz (with test functions 𝐀_xyz', resp. 𝐠𝐫𝐚𝐝 v_xyz'), magnetic reluctivity ν_xyz and electric conductivity σ_xyz. The term 𝐇_xyz×𝐧 denotes the fixed tangential magnetic field at the boundary of the domain. Further, the notations (𝐟, 𝐠)_Ω_xyz and ⟨𝐟, 𝐠⟩_∂Ω_xyz denote ∫_Ω_xyz𝐟·𝐠 dV_xyz and ∫_∂Ω_xyz𝐟·𝐠 dS_xyz. The electric scalar potential v_xyz is only defined in the conducting domain and is further used to associate global quantities to the i-th conductor, namely, its total current I_i and its voltage drop V_i <cit.>. In Ω_xyz,c, the magnetic vector potential is directly linked to the physical electric field 𝐄_xyz = -jω𝐀_xyz - 𝐠𝐫𝐚𝐝 v_xyz, s.t. here no gauging is needed (modified vector potential). However, in the insulating domain Ω_xyz,i, gauging is necessary to find a unique solution. §.§ Weak formulation in helicoidal coordinates The key idea for the dimensional reduction is the following: Instead of computing the integrals in (<ref>) & (<ref>) on the Cartesian domain Ω_xyz, we transform them on the domain Ω_uvw, i.e., we shift their computation to a w-invariant space (see Fig. <ref>). This transformation requires a reexpression of the involved electromagnetic field quantities. The one-forms 𝐀_xyz, 𝐠𝐫𝐚𝐝 v_xyz and 𝐇_xyz and the two-form 𝐜𝐮𝐫𝐥 𝐀_xyz (magnetic flux density 𝐁_xyz), transform as follows <cit.>: 𝐀_xyz(𝐩_xyz) = J_ϕ^-1^-⊤ 𝐀_uvw(ϕ(𝐩_xyz)), 𝐜𝐮𝐫𝐥 𝐀_xyz(𝐩_xyz) = J_ϕ^-1/det(J_ϕ^-1) 𝐜𝐮𝐫𝐥 𝐀_uvw(ϕ(𝐩_xyz)), where J_ϕ^-1 denotes the Jacobian of ϕ^-1 evaluated at point 𝐩_uvw. Here, it is important to mention, that the 𝐜𝐮𝐫𝐥 operator on the right hand side of eq. (<ref>) is not the actual differential operator in the helicoidal coordinate system. Rather, this operator is to be understood as blindly applied (applied as in Cartesian coords with a relabeling of variables and components) – its results are then directly mapped back into the Cartesian coordinate system as the equality sign in eq. (<ref>) holds componentwise for vector fields expressed in Cartesian coordinates. Also, the qualitatively same statement applies, when inserting 𝐠𝐫𝐚𝐝 v_xyz into formula (<ref>). Changing variables also introduces, rather conceptually than computationally, a factor det(J_ϕ^-1)= 1 in the integrals in eq. (<ref>) & (<ref>). In terms of the (u, v, w) coordinates, introducing trial function spaces A(Ω_uvw) & V(Ω_uvw) and test function spaces A_0(Ω_uvw) & V_0(Ω_uvw) which will be defined in Section <ref>, we can reformulate the weak formulation as follows: Seek 𝐀_uvw∈ A(Ω_uvw) and 𝐠𝐫𝐚𝐝 v_uvw∈ V(Ω_uvw,c), s.t. ∀ 𝐀_uvw'∈ A_0(Ω_uvw) and ∀ 𝐠𝐫𝐚𝐝 v_uvw'∈ V_0(Ω_uvw,c): (ν_uvw𝐜𝐮𝐫𝐥 𝐀_uvw, 𝐜𝐮𝐫𝐥 𝐀_uvw' )_Ω_uvw + (jωσ_uvw 𝐀_uvw, 𝐀_uvw' )_Ω_uvw,c - ⟨𝐇_xyz×𝐧, 𝐀_uvw'⟩_∂Ω_uvw + (σ_uvw 𝐠𝐫𝐚𝐝 v_uvw, 𝐀_uvw' )_Ω_uvw,c = 0, (jωσ_uvw 𝐀_uvw, 𝐠𝐫𝐚𝐝 v_uvw' )_Ω_uvw,c + (σ_uvw 𝐠𝐫𝐚𝐝 v_uvw, 𝐠𝐫𝐚𝐝 v_uvw' )_Ω_uvw,c= ∑_i=1^N I_i V_i', where the effect of the change of variables is fully contained in two anisotropic, w-invariant material parameters, written as tensors: σ_uvw(𝐩_uvw) = σ_xyz(ϕ^-1(𝐩_uvw)) J_ϕ^-1^-1J_ϕ^-1^-⊤det(J_ϕ^-1), ν_uvw(𝐩_uvw) = ν_xyz(ϕ^-1(𝐩_uvw)) J_ϕ^-1^⊤J_ϕ^-1/det(J_ϕ^-1). Now, since neither the integrands nor the computational domain Ω_uvw depend on the w-coordinate, one may solve the problem on any uv-plane for a fixed value of w. For simplicity, we choose w = 0, because then x=u and y=v, see eq. (<ref>). §.§ Space discretization and implementation details We solve the systems of equations (<ref>) & (<ref>) by using the open-source finite-element software GetDP <cit.>, which allows for flexible function space definitions, whereas the meshing process is performed by Gmsh controlled via the Julia API <cit.>. In previous works, we derived the computational domain (symmetry cell) analytically using the theory of envelopes <cit.>. Now, the cross-section is computed as the intersection of 3-D helicoidal conductors with a plane using built-in CAD routines. This will allow us to investigate elliptically shaped conductor cross-sections in future work, which can be created under mechanical stress in the cable manufacturing process <cit.>. Although we solve the problem on a 2-D domain, we still assume, that the magnetic vector potential 𝐀_uvw has full three components, which are interpolated separately: The in-plane components (A_u & A_v) are discretized using 2-D Whitney edge functions. In the domain Ω_uvw,i∖∂Ω_uvw,c, though, only degrees of freedom (DOF) associated with the cotree of the mesh are non-zero, i.e., a tree-cotree gauge is applied to achieve a unique solution <cit.>. This defines the function space A_uv(Ω_uvw). Further, the component A_w is spanned by nodal functions and is essentially fixed to zero on ∂Ω_uvw as we model the cable's shield as a perfect electrically conductive cylinder. Since we do not introduce an explicit gauge for A_w, this component is implicitly Coulomb-gauged. This constitutes the function space A_w(Ω_uvw), s.t. in total A(Ω_uvw) = A_uv(Ω_uvw) ⊕ A_w(Ω_uvw). The function 𝐠𝐫𝐚𝐝 v_uvw∈ V(Ω_uvw) consists of N w-directed constants (one constant per connected component of Ω_uvw,c). § RESULTS The results of the 2-D formulation and of our previously presented 2-D formulation <cit.> are compared with a 3-D reference cable model that is implemented using the commercial software CST Studio Suite <cit.>, referred to as CST. Each of the 13 helicoidal conductors (longitudinal length 0.2 m, cross-section radius 5 mm) carries a total current of amplitude √(2)/13 A at f = 50 Hz. In all models, we considered annealed copper for the conductors' material (conductivity ≈ 58.12 · 10^6 S/m) and further assumed a non-magnetic material in the whole domain (ν_0 = μ_0^-1 with μ_0 = 4π· 10^-7 H/m). Since the current constraint is differently implemented in the 3-D model (current ports located at the cuboid bounding box of the domain), we compare local quantities only at the cable's center (see Fig. <ref>). In the following, we compare the 2-D models discretized by 58.64k triangles with a 3-D model discretized by 1.19M tetrahedra: We used first order finite elements resulting in 98k DOF in the 2-D formulation, 57k DOF in the 2-D formulation and 1.21M DOF in the 3-D model. The DOF of the 2-D models differ so much on the same mesh, since only within the formulation we can directly incorporate and exploit the fact that the w-component of the magnetic field 𝐇_uvw is a constant in the insulating domain Ω_uvw,i <cit.>. So far, no way has been found to utilize this knowledge also in the formulation. The 2-D model outputs the finite-element approximated vector potential 𝐀_uvw and the gradient of the electric scalar potential 𝐠𝐫𝐚𝐝 v_uvw. Using transformation rules (<ref>) & (<ref>), all electromagnetic field quantities in the Cartesian coordinate system can be derived, e.g.: 𝐉_xyz = -σ_xyz J_ϕ^-1^-⊤ (jω𝐀_uvw + 𝐠𝐫𝐚𝐝 v_uvw), 𝐇_xyz = μ_xyz^-1J_ϕ^-1/det(J_ϕ^-1) 𝐜𝐮𝐫𝐥 𝐀_uvw. As a local comparison, 𝐉_xyz and 𝐇_xyz are evaluated along the x-axis. The results depicted in Fig. <ref> show an accurate agreement between all models. In the formulation, the current density 𝐉_xyz is multiplicatively linked to the linear interpolated magnetic vector potential 𝐀_uvw (and the piecewise-constant vector field 𝐠𝐫𝐚𝐝 v_uvw), see eq. (<ref>). In contrast, in the formulation, the current density is multiplicatively linked to the 𝐜𝐮𝐫𝐥 operator applied to the linear interpolated magnetic field 𝐇_uvw <cit.>. Therefore, the current density resulting from the formulation appears comparatively smooth, as shown in the zoom window in the upper plot of Fig. <ref>. Furthermore, the comparison of the ohmic losses, representing a global quantity, shows a good match: both 2-D models output a length-related power loss of 21.9 μWm^-1, whereas the 3-D model has a total loss of 4.34 μW. Scaling the length-related losses up to the cable’s longitudinal length results into a loss of 4.38 μW, which deviates 0.9% from the 3-D result. We suspect that this discrepancy is mainly due to the different excitation types. § CONCLUSION & OUTLOOK In this work, we have shown how a coordinate transformation can be used for the dimensional reduction of eddy current problems modeling helicoidal symmetric power cables. In particular, we have focused on integrating the symmetry approach into the vector potential based 𝐀-v formulation that can be solved in 2-D (Section <ref>). The results of this 2-D model are in excellent agreement with our previously presented 2-D model <cit.> and with the 3-D reference model (Section <ref>). This reinforces the confidence in the symmetry approach for reducing computational costs when analyzing power cables' electromagnetic behaviour numerically. Future works include the handling of also non-ideal symmetries, see e.g., <cit.>. § ACKNOWLEDGMENT Special thanks to Christophe Geuzaine (University of Liège) who provided helpful advice on implementations in GetDP. 1 a2 R. Suchantke, “Alternating Current Loss Measurement of Power Cable Conductors with Large Cross Sections Using Electrical Methods,” Ph.D. dissertation, Tech. Univ. Berlin, Berlin, Germany, 2018. a1 CST Studio Suite. (2021), Dassault Systèmes, Accessed: Mar. 18, 2022. [Online]. Available:https://www.3ds.com/products-services/simulia/products/cst-studio-suite/https://www.3ds.com/products-services/simulia/products/cst-studio-suite/. a222 A. Marjamäki, T. Tarhasaari and P. Rasilo, "Utilizing Helicoidal and Translational Symmetries Together in 2-D Models of Twisted Litz Wire Strand Bundles," IEEE Transactions on Magnetics, vol. 59, no. 5, pp. 1-4, May 2023, Art no. 7400504, doi: 10.1109/TMAG.2023.3237767. a4 A. Stenvall, F. Grilli and M. Lyly, “Current-Penetration Patterns in Twisted Superconductors in Self-Field,” IEEE Transactions on Applied Superconductivity, 2013, vol. 23, no. 3, pp. 8200105-8200105, Art no. 8200105, doi: 10.1109/TASC.2012.2228733. a22 J. Dular, “Standard and Mixed Finite Element Formulations for Systems with Type-II Superconductors,” Ph.D. dissertation, Univ. of Liège, Liege, Belgium, 2023. a3 A. Nicolet and F. Zolla, “Finite element analysis of helicoidal waveguides,” IET Science, Measurement & Technology, 2007, vol. 1, pp. 67–70, doi: 10.1049/iet-smt:20060042. a23 A. Piwonski, J. Dular, R. S. Rezende and R. Schuhmann, "2-D Eddy Current Boundary Value Problems for Power Cables With Helicoidal Symmetry," IEEE Transactions on Magnetics, vol. 59, no. 5, pp. 1-4, May 2023, Art no. 6300204, doi: 10.1109/TMAG.2022.3231054. lindell I.V. Lindell, “Differential forms in electromagnetics," John Wiley & Sons, 2004. d1 P. Dular, C. Geuzaine, F. Henrotte and W. Legros, “A general environment for the treatment of discrete problems and its application to the finite element method,” IEEE Transactions on Magnetics, vol. 34, no. 5, pp. 3395–3398, 1998. d2 C. Geuzaine and J. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” International Journal for Numerical Methods in Engineering, 2009, vol. 79, no. 11, pp. 1309-1331. c4 J. Bezanson, A. Edelman, S. Karpinski and V. Shah, “Julia: A Fresh Approach to Numerical Computing,” SIAM Review, 2017, vol. 59, no. 1, pp. 65-98, doi: 10.1137/141000671. c5 E. Creusé, P. Dular and S. Nicaise, “About the gauge conditions arising in finite element magnetostatic problems,” Computers & Mathematics with Applications, vol. 77, no. 6, pp. 1563-1582, 2019.
http://arxiv.org/abs/2307.03113v1	20230706163038	JSONoid: Monoid-based Enrichment for Configurable and Scalable Data-Driven Schema Discovery	[ "Michael J. Mior" ]	cs.DB	[ "cs.DB" ]	[email protected] Rochester Institute of Technology 102 Lomb Memorial Drive Rochester New York 14623–5608 plain plain Schema discovery is an important aspect to working with data in formats such as JSON. Unlike relational databases, JSON data sets often do not have associated structural information. Consumers of such datasets are often left to browse through data in an attempt to observe commonalities in structure across documents to construct suitable code for data processing. However, this process is time-consuming and error-prone. Existing distributed approaches to mining schemas present a significant usability advantage as they provide useful metadata for large data sources. However, depending on the data source, ad hoc queries for estimating other properties to help with crafting an efficient data pipeline can be expensive. We propose JSONoid, a distributed schema discovery process augmented with additional metadata in the form of monoid data structures that are easily maintainable in a distributed setting. JSONoid subsumes several existing approaches to distributed schema discovery with similar performance. Our approach also adds significant useful additional information about data values to discovered schemas with linear scalability. JSONoid: Monoid-based Enrichment for Configurable and Scalable Data-Driven Schema Discovery Michael J. Mior August 1, 2023 =========================================================================================== § INTRODUCTION Non-relational data formats such as JSON <cit.> have grown significantly in popularity in recent years. One of the main drivers of this growth is the flexibility provided by requiring little to no upfront schema design. This has the advantage of accepting a wide variety of data without advance planning. Such flexibility is useful in domains such as logging where different events with different attributes may be added regularly, or Web services using dynamic languages such as JavaScript and Python are well-suited to ad hoc data processing. However, this flexibility comes at the cost of providing minimal information to data analysts when consuming the data. When working with relational data, an analyst can look to the relational schema to provide information on the available data values and their types. With non-relational data, such a schema is often unavailable, and analysts are left to try to understand the data by manually examining either sample documents or existing source code which processes the data. In the case of JSON data, JSON Schema <cit.> provides a standard mechanism to represent the structure in a collection of JSON documents. There are many tools built around JSON Schema such as validators and code generators, making it easier to work with data which have an available schema. While JSON Schemas can be useful, schemas for collections of documents are also not commonly available. For example, many Web services making use of JSON data only provide written documentation, and many users of document databases which store JSON data do not provide schemas. Several mechanisms for the discovery of JSON Schemas have been proposed to automate the creation of a valid JSON Schema based on collections of documents, which we summarize in Section <ref>. This allows data analysts to explore schemas in a similar way to those of relational databases. Here, we focus on an approach that makes use of Apache Spark <cit.> to perform distributed schema inference. This enables schema discovery to be performed in a distributed fashion across a large collection of JSON documents. The key idea behind the approach is to construct a schema that precisely matches each individual document and then combines these schemas recursively to produce a schema that fits the whole collection. The data structures used in our schema discovery process are all structured in the form of monoids. Monoids are algebraic structures with an identity element (representing an empty collection) and an associated binary operation (merging schema information). Restricting the data structures we use to monoids means that we can efficiently maintain all necessary data structures to produce a final JSON Schema in a distributed fashion since individual schemas can be constructed in parallel before merging. A key element lacking from JSON Schema is information about data size, distributions, and relationships. In a relational database, this information can often be gleaned from running ad hoc SQL queries. However, sources of JSON data rarely provide simple facilities for issuing such queries. We seek to augment the distributed schema discovery process by providing additional useful information about the data while adding minimal overhead in order to maintain scalability. Our contributions are as follows: * A distributed mechanism for JSON schema discovery that incorporates not only structure, but also summary information from individual data values. * An analysis of schema information content and runtime that incorporates the value of schema enhancements. * A description of a number of use cases for such an enhanced schema, including exploration, constraint discovery, and outlier detection. The remainder of the paper starts by providing some background in Section <ref>. Next, we provide an overview of our schema discovery process in Section <ref>. We then describe each of our enhancement monoids in Section <ref>. We evaluate the information content of our generated schemas, as well as the runtime performance, in Section <ref>. Section <ref> outlines several new use cases enabled by the enhanced schemas created by JSONoid. We then describe some related work on which JSONoid is based in Section <ref>. Finally, we outline future work and conclude in Sections <ref> and <ref>. § BACKGROUND JavaScript Object Notation (JSON) arose from the JavaScript programming language representation of object literals. Primitive values are strings, numbers, arrays, booleans, or null. Objects and arrays can be nested to any depth. A simplified grammar of JSON is given in Figure <ref>. The simplicity and flexibility of JSON has resulted in it becoming a common data format for both Web APIs and document databases. As mentioned previously, unlike a relational database, it is generally not necessary to provide any schema information in order to store and process JSON data using commonly available document databases and JSON processing libraries. §.§ JSON Schema This lack of schema information means that developers often must resort to manually examining documents in a collection to gain an understanding of their structure. This can result in a “guess-and-check” approach to development where assumptions are made about the data, and those assumptions are either validated by correctly processing the dataset or broken when another document fails to meet these expectations. JSON Schema <cit.> provides a format for representing the structure of JSON documents, including the attributes that a document should contain and their associated types. An example of a JSON Schema and a document that is validated against this schema is given in Figure <ref>. Unfortunately, a JSON Schema describing a dataset is rarely available. Alternatively, a suitable schema can be mined from a collection of documents. While these mined schemas are useful, they often fail to provide a complete understanding of the dataset. If an attribute is optional, what fraction of documents contain this attribute? Are the values of this attribute unique across the dataset? How many distinct values exist across the collection? These questions cannot be answered with purely structural information. Instead, we must rely on analyzing the data. JSON Schema is capable of describing only limited information on the data contained in schemas such as minimum and maximum numerical values and regular expressions that strings must match. Our approach augments this information with additional insight in the generated schemas. Since the schema discovery process is data-driven, we decide to piggyback on this process to allow capturing of this additional information with minimal overhead since a separate pass over the data is not needed. There are many existing tools which make the existence of a complete and accurate JSON Schema description of a dataset extremely useful for developers. For example, libraries exist for most popular languages to generate code for parsing, validation, and data entry. Schemas can also be used to generate the skeleton of documentation which can be useful for APIs which produce or consume JSON data and need to provide more information on their usage. §.§ Apache Spark Apache Spark <cit.> is a computational framework for distributed big data processing. Spark is similar to MapReduce <cit.> in that datasets are loaded from a distributed file system and multiple partitions of a file are processed in parallel. While Spark does offer a richer computational model than MapReduce, a map-reduce style approach is sufficient to define the semantics of the operations required for JSONoid. We provide further details on our approach in Section <ref> We use a function to assign a schema to each individual document in a collection of JSON documents and to produce the final schema, we define a function which combines multiple schemas to produce a single schema. A similar approach is taken by Baazizi et al. <cit.> although their approach purely collects structural information. As we describe later, JSONoid mines much richer information which incorporates data values and not merely structure. §.§ Probabilistic Data Structures Probabilistic data structures (PDS) have become common in big data scenarios, as they allow efficient computation across large datasets without the need for significant memory usage at the expense of accuracy <cit.>. However, many probabilistic data structures have very low levels of error with very significant savings in space. For the purposes of this work, we make use of probabilistic data structures to solve two problems: approximate membership querying and cardinality estimation. We describe the specific data structures we use below, with their use detailed in Section <ref>. Approximate membership queries (AMQ) aim to indicate whether a value is a member of a large set with constant query time, while also using a constant amount of space which is expected to be small relative to the size of the set. For our purposes, this is useful for a succinct summary of the values stored at a particular key in the schema. AMQ data structures are typically defined such that they cannot return false negatives but may return false positives. That is, with low probability, a value may be reported as being a member of the set when that is not, in fact, the case. One of the most common AMQ data structures is the Bloom filter <cit.>. The simplest version of a Bloom filter uses a bit array to represent the set. When an item is added, multiple hash functions are computed on the item, and the bits in those positions are set. To test if an item is in the set, it suffices to check if the appropriate combination of bits is set in the array. One advantage of Bloom filters in our setting is that they can easily be treated as monoids, as we discuss in Section <ref>. Furthermore, we can also compare two Bloom filters to see if the set represented by one filter is likely to be a subset of the set represented by another filter. This is useful for constraint discovery, as we discuss in Section <ref>. For cardinality estimation, we make use of the HyperLogLog (HLL) <cit.> data structure. Cardinality estimation is useful to track the number of unique values that are likely to exist at a particular location within a JSON document. HLL works similarly to Bloom filters in that it uses hashing of input values to update arrays, which are used to estimate the final value. Like Bloom filters, we can efficiently merge two HLL data structures, making it easy to treat them as monoids. § SCHEMA DISCOVERY Our goal in the schema discovery is given a collection 𝕁 of JSON documents, produce a schema 𝐒 such that 𝐒 accepts all JSON documents j∈𝕁 as valid. To avoid the trivial case of a schema that accepts all documents, a good schema should also reject some documents not in j. We say that a schema is more descriptive if it is more likely to reject documents which are not in 𝕁. Descriptive schemas are useful for precisely describing a dataset. Another common use case for JSON schemas is checking whether a document is valid according to the underlying business rules described by the schema. In this case, an overly descriptive schema may reject documents that should otherwise be valid. For example, suppose that a numerical value should range from 0 to 100. A schema discovery algorithm that is highly descriptive will observe a collection of documents and produce a schema that shows the value in the range of 3 to 95 because those were the minimum and maximum values actually observed during the discovery process. Depending on the specific use case, we may want a schema to be more or less descriptive. To this end, schemas in JSONoid consist of a basic type and a configurable set of associated enhancement monoids. The basic types in JSONoid are object, array, boolean, string, number, or null. Any additional information about the type (e.g. the attributes in an object) is stored in monoids associated with the schema. These monoids along with their additional information are included based on the desired properties of the final schema. For primitive values such as strings, the schema can be constructed by building the initial value of all monoids, as discussed in the following section. For complex values (i.e. objects and arrays), schemas are constructed recursively, as shown in Figure <ref>. Once the schema for a single document has been constructed, the individual schemas for each document in the collection are merged to produce a final schema for the entire collection. When merging schemas, we evaluate whether they should be combined according to a configurable equivalence relation <cit.>. Equivalence relations are explained in further detail in Section <ref>. For now, it is sufficient to note that an equivalence relation is a choice made by the user on the desired granularity of the generated schema. There are two possible cases to consider: * The two schemas are of the same basic type and are equivalent according to the configured equivalence relation. * The two schemas are different types, or different according to the configured equivalence relation. In the first case, we merge two schemas by merging their associated monoids, as we discuss in the following section. In the second case, we create a product schema which specifies a schema that can be one of two possible types. This is equivalent to in JSON Schema. We envision two possible merge strategies of operation for our schema discovery process, which are shown in Figure <ref>. The streaming strategy is useful if documents appear one at a time in real time, as is the case with data logging. We start by producing a schema object from the first document, O_1, which forms the initial schema of the collection S_1. As new documents appear, we extract their schema and merge it with our schema S_i, which contains a schema representing all the documents observed in the stream so far, producing an incrementally more descriptive schema. Note that in this case we only need to hold a single document in memory at a time and two copies of the schema. One which is generated from the current document and the incrementally updated schema. Therefore, the memory used during the discovery process is bounded by a constant factor of the size of the largest document observed. The runtime of the streaming approach is linear in the number of documents. We are not aware of existing approaches which implement a similar streaming mode of schema discovery. Our second merging strategy is distributed. The distributed merge strategy targets large collections of JSON documents stored in distributed file systems. In distributed mode, construction of the schemas for individual documents can occur in parallel as they are independent of the schema generated for any other document. We can achieve another level of parallelism in the merge process by organizing schemas into a tree and merging recursively, for example, using in Spark. Both the memory consumption and the runtime of the distributed approach depend on the degree of parallelism. However, in general, we expect memory consumption to be linear in the degree of parallelism. The distributed approach therefore consumes more memory than the streaming approach, but is able to produce a schema in logarithmic time relative to the number of documents. This makes the distributed approach more attractive if all documents are available ahead of time and memory is not constrained. §.§ Extended Schema Vocabulary JSON Schema was primarily designed to define a required structure for JSON documents for the purpose of validation. As such, it only contains data-related constraints that are useful to validate data within a single instance (e.g. upper and lower bounds for integer values). However, since our goal is to provide useful metadata about a collection of JSON documents, we extend this vocabulary to include descriptive features of document collections as a whole. § MONOIDS As stated previously, to enhance the discovery process, we design all our enhancements in terms of monoids. In our setting, we use the monoid identity element to initialize the necessary data structures when starting to construct a schema. The binary operation is used during merging to combine the enhanced data structures of two separate schemas. We provide detailed descriptions of the various enhancement monoids used in JSONoid in the following subsections. Each monoid consists of an initial state (𝐌_0) that is constructed from a single value when a new instance of the monoid is required. Each monoid definition also requires a commutative and associative merge function (m_1⊗ m_2) that combines the data contained in two monoid instances. §.§ Structure Inference We take a similar approach to previous work and recursively construct a schema for each individual attribute of a document. The result is a schema that perfectly describes the structure of each document. Once a schema is constructed for each document, we iteratively merge these schemas to produce a final schema that represents the entire collection. Note that, for scalability, we can produce the individual schemas for each document in parallel. When merging schemas, we can do so in a tree structure so that O(log(n)) merges are required instead of O(n) (where n is the number of documents). Our approach also incorporates useful elements of past work in schema inference. Firstly, we consider parametric schema inference <cit.>. Parametric schema inferences merges objects according to a configurable equivalence relation. If two schemas are equivalent according to the equivalence relation, they are merged. Otherwise, two separate possible schemas are maintained in a product schema. This allows the algorithm to trade off between the precision and size of the generated schema. We incorporate equivalence relations into our approach when merging monoids that represent structural schema information. Since all monoids are merged (or not) according to the equivalence relation, this gives an easy method to adapt the granularity of the mined schema. Specifically, JSONoid currently supports both kind and label equivalence. Kind equivalence means that schemas which are of the same kind (e.g. both objects) will be merged while label equivalence requires the set of keys which are present in an object to be the same in order to allow merging. The kind equivalence relation minimizes schema size, while label equivalence maximizes precision. Although we have only implemented these two equivalence relations, others can be easily added without affecting the rest of the discovery process. Our approach to structure inference also works well with counting types <cit.> that associate observed counts with various elements of the schema. For example, a counting type for an object would indicate the number of times the object was observed, as well as the number of times each attribute in the object was observed. This is important because objects in JSON document collections do not need to contain the same attributes. This is helpful for augmenting the schema with additional information such as whether an attribute is expected to be required or optional. (That is, we consider an attribute as required if every observed object contains that attribute.) In addition, within objects, we maintain information about the co-occurrence of attributes. This is helpful in identifying groups of related attributes, which we describe more in Section <ref> §.§.§ ObjectTypes Our first monoid, ObjectTypes, tracks the type of value contained with each attribute. The initial value of this monoid constructed from a single object is a map from attribute keys to the schema of each attribute (discovered recursively as described above). The merge function takes the union of attribute schemas occurring in only one of the two input monoids and merges the schema of those occurring in both. 𝐎𝐛𝐣𝐞𝐜𝐭𝐓𝐲𝐩𝐞𝐬_0: obj → {types: {key: schema(value) for (key, value) in obj}} o_1 ⊗ o_2 = {key: o_1(key)for key in o_1.types.keys∖ o_2.types.keys}⋃ {key: o_2(key)for key in o_2.types.keys∖ o_1.types.keys}⋃ {key: o_1(key)⊗ o_2(key)for key in o_1.types.keys∩ o_2.types.keys} §.§.§ ArrayType The next monoid, ArrayType has the same purpose for arrays as ObjectType does for objects, to track the type of values contained within the array. Here, we consider arrays where all items have the same schema. (Note that this does not mean that all items must have the same type. The schema could be a product schema as described above, which accepts multiple types.) We construct the initial type contained in the monoid by reductive merging of all the types in the array. On merging, we merge the two schemas of each type. This allows the construction of arrays where elements may have different types. 𝐀𝐫𝐫𝐚𝐲𝐓𝐲𝐩𝐞_0: arr →{type: foldr(⊗, arr)} a_1 ⊗ a_2 = {type: a_1.arr⊗ a_2.arr} We note that there is a special case to be considered for the ArrayType schema. This is the case where all arrays at that particular path in the schema are of fixed identical length. JSON Schema documentation refers to this as tuple validation. When validating tuples, each element in the tuple may have a distinct type. This may be used, for example, when representing tabular data in JSON format. One possible representation is to have nested arrays of rows where each row represents a list of column values, each of which contains the same type. This is common when tabular data are represented in JSON format, which is common in open data settings <cit.>. The details are not included in the monoid definition above, but to support this use case, the implementation proceeds by maintaining an array of separate schemas, one for each element of the tuple. This continues on successive merge processes as long as the observed arrays are the same length. As soon as two monoids containing arrays of separate length are merged, the monoid collapses to the case of array validation as presented above. §.§ Data Sampling One useful view of data values is to provide specific examples of values that occur in the data set. Depending on the source of the JSON data, viewing sample column values is rarely as simple as in SQL, e.g. . For example, JSON documents can be streamed as log entries or provided via calls to a Web service. Furthermore, all documents may not have all possible attributes since some may be optional. Having a sample set of data values when viewing a schema is helpful to understand what values must be handled when writing data processing code. For each primitive data type, we collect a sample of the values available across all documents. A common technique for sampling large collections of elements is reservoir sampling <cit.>. The reservoir sampling process uses the probability that each element will be included in a sample to decide whether a newly observed element should be included, evicted from a previous sample, or discarded. Reservoir sampling is easily adopted in a distributed setting by tracking the total data size in each partition and combining two samples by repeated weighted sampling based on the sizes of the underlying data from which each of the two samples was drawn. Our monoid that retains a randomized list of examples is described below. 𝐄𝐱𝐚𝐦𝐩𝐥𝐞𝐬_0: value →{examples: [value], total: 1} e_1 ⊗ e_2 = {examples: sample( e_1.examples, e_1.total, e_2.examples, e_2.total), total: e_1.total + e_2.total} Initialization of the monoid takes the single input value and wraps it in an array. The merge function considers the second array to be a continuation of the examples presented to the reservoir sampling algorithm. We also record the total number of values used to generate the sample. This is used to weight the samples taken from each monoid. If monoid e_1 was constructed by observing 10 values, while monoid e_2 was constructed by observing 100 values, we want a 10× higher probability of selecting an example from e_2. §.§ Probabilistic Data Structures For numeric values, we provide a histogram that estimates the distribution of the observed values. For this we use a streaming histogram proposed by Ben-Haim and Tom-Tov <cit.>. This algorithm maintains a set of (value, count) pairs which are buckets in the histogram. New samples either increment the count of an existing bucket with a matching value or add a new bucket with a count of one. When a new sample causes a configurable maximum number of buckets to be exceeded, the buckets with the closest values are merged by adding their counts and taking a weighted average of their values. To merge histograms, they are concatenated and sorted by value and bins are merged to return to the original number of bins using the same merge process as for new samples. The histograms produced in the schema are useful to suggest to developers what range of values to expect. We also make use of HyperLogLog (HLL) <cit.> to estimate the number of distinct elements in a set. HLL maintains a set of registers which are incremented based on a hash function applied to observed values. These registers are later used to estimate the count of distinct elements. Since updates to these registers always store the maximum of the current and proposed values, two HLL structures can be merged by taking the maximum values of the registers in the two structures. The goal of the HLL monoid is to provide a better understanding of the cardinality of values across the dataset. Finally, we use Bloom filters <cit.>, which use bit arrays and a series of hash functions to support approximate membership queries. Specifically, using space much smaller than the size of the set of values, Bloom filters can report whether a set contains an item with no false negatives and a configurable probability of false positives. To combine filters, we take the bitwise OR of two filters. We construct a Bloom filter monoid for string and numeric values. Both the HLL and Bloom filter PDS monoids have the same structure for their definition as is given below. The initial value is added to a new data structure, and when merging values, we use the merge operator of the corresponding data structure (⊗). For HLL, this is taking the maximum of registers and for Bloom filters, the bitwise OR of the bit vector. As we describe in Section <ref>, these monoids are also useful for discovering constraints within the data. 𝐏𝐃𝐒_0: value →{pds: PDS(value)} v_1 ⊗ v_2 = {pds: o_1.pds ⊗ o_2.pds} §.§ Statistics For numerical measures, in addition to sampling values, it is helpful to provide statistical measures calculated on these values. Once again, we want to be able to calculate these measures in a distributed fashion, requiring the ability to merge values calculated across multiple partitions. For example, this can be done for the mean by maintaining a count and sum for each partition and summing values when combining partitions. The mean can then be calculated in the standard manner by dividing the sum by the count. 𝐌𝐞𝐚𝐧_0: value →{sum: value, count: 1} n_1 ⊗ n_2 = {sum: o_1.sum + o_2.sum total: n_1.total, n_2.total } Other statistics require more complicated calculations, but there has been significant work in calculating common statistical measures in an online setting. For further statistics, Knuth <cit.> provides an algorithm for calculating the standard deviation based on its corresponding recurrence relation. We adopt these online calculations of statistical measures as monoids, including others for skewness and kurtosis described by Cook <cit.>. This can be useful both for data understanding and when writing data processing code to select appropriate data structures and estimate their memory usage. We do not provide the full details of the monoid here, but we defer to the work referenced above. §.§ Structural Annotations An additional monoid we include applies to object types within the JSON data. We keep track of the counts each time a key is observed within an object, and these counts are recursively accumulated for each separate object observed. The total number of objects observed is also monitored. Then the percentage of objects with each given key can be presented as with prior work on counting types <cit.>. 𝐀𝐭𝐭𝐫𝐢𝐛𝐮𝐭𝐞𝐂𝐨𝐮𝐧𝐭𝐬_0: obj →{counts: {key: 1 for key in obj.keys}} o_1 ⊗ o_2 = {key: o_1(key)+o_2(key)for key in o_1.counts.keys∩ o_2.counts.keys} o_i(key) = 0 if key ∉o_i.keys This monoid gives the data consumer knowledge of how likely a particular key is to occur. This lets an analyst know whether they can generally rely on an attribute being present or not. JSON Schema does not provide such information. To produce similar information that can be used within the JSON Schema standard, we also define a monoid that populates the property. The property specifies which attributes are present in all instances of an object. 𝐑𝐞𝐪𝐮𝐢𝐫𝐞𝐝_0: obj →{keys: obj.keys} o_1 ⊗ o_2 = {key: o_1(key)∩ o_2(key)for key in o_1.keys.keys∩ o_2.keys.keys} o_i(key) = ∅ if key ∉o_i.keys Another interesting example of a useful monoid is one used to track whether an array contains unique elements. Here, we make use of the Examples monoid, which tracks unique examples for a schema. Since the schema and associated monoids are constructed bottom-up, we have available the examples of array elements at the time array monoids are constructed. We check if the number of (unique) examples is equal to the length of the array. If this is the case, each array element is unique. When merging this monoid, we consider the elements of the combined array to contain unique elements if each array being merged also contains unique elements. 𝐔𝐧𝐢𝐪𝐮𝐞_0: arr →{isUnique: \|arr.examples\| == \|arr\|} a_1 ⊗ a_2 = {isUnique: a_1.isUnique∧ a_2.isUnique} One final property we maintain for objects is dependencies. A dependency indicates that when one attribute occurs, another set of attributes also occurs. This can be helpful in identifying possible groups of related attributes. For example, attributes and can both be considered optional. But there may be a dependency such that whenever the attribute occurs, the attribute must also be present. We omit the details here, but dependencies are tracked by counting each time two attributes occur together as well as the total number of occurrences of each attribute individually. If the number of co-occurrences matches the total count of each attribute, a dependency has been found. §.§ Value Restrictions Existing schema discovery tools focus primarily on structural and type constraints. However, little attention has been given to what values within a type are acceptable. We define several monoids that are used to impose meaningful restrictions on values that are expressed within the data being mined. §.§.§ MaxMin The first of these value restrictions are minimum and maximum constraints. This applies to numeric values as well as the length of strings and arrays. These values can be easily maintained as monoids by starting with the initial value and then taking the minimum (or maximum) of the current value and the monoid being merged. The result after extracting and merging schemas from all documents is the minimum and maximum value across all documents. 𝐌𝐚𝐱𝐌𝐢𝐧_0: num →{min: num, max: num} n_1 ⊗ n_2 = { min: min(n_1.min, n_2.min), max: max(n_1.max, n_2.max)} §.§.§ Multiple There are also additional type-specific restrictions that can be mined. For example, JSON Schema supports the restriction for numerical values, which indicates that a numerical value must be a multiple of a specific constant. To track this property during schema discovery, we use Euclid's algorithm for computing the greatest common divisor (GCD). 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐞_0: num →{multiple: num} n_1 ⊗ n_2 = {num: gcd(n_1.multiple, n_2.multiple} The first numerical value encountered is treated as the GCD. Any subsequent values are compared with this GCD. Finally, the property is emitted if the GCD is greater than 1 to avoid the trivial case where any integer value is accepted. (In this case, we simply use the type instead of .) §.§.§ Pattern For string values, one useful property is whether string values match a particular regular expression. In this work, we consider the simple case where all strings have a common prefix and/or suffix. We initialize the prefix (and suffix) to the first string value encountered. On processing subsequent values, we take the longest common prefix (or suffix) of the new value and the previous prefix (or suffix). 𝐏𝐚𝐭𝐭𝐞𝐫𝐧_0: str →{prefix: str, suffix: str} s_1 ⊗ s_2 = {prefix: common_prefix(s_1.prefix, s_2.prefix)} suffix: common_suffix(s_1.suffix, s_2.suffix)} The final property is supported in JSON Schema by the property of string values. Patterns are regular expressions, so we anchor the expression at the start or end of a string or both depending on whether a non-empty prefix, suffix, or both are obtained. This can handle cases such as a URL that has a common prefix and file extension, providing additional context for anyone making use of the data. Extending this approach to more complex regular expressions is left for future work. §.§.§ Format Finally, we also detect possible formatted strings within JSON data. String formats are commonly used values such as dates (a datatype not natively supported in JSON), GUIDs, and URLs. To detect formats, we start by assuming that a string property has no defined format. A format detection function is written for each possible string format to determine if the string values being mined match the given format. We assume that each string property will only match a single given format, so we check all these functions to see if a string matches any defined formats. Assuming all values encountered for a string property match a given format, we include this format in the generated schema. 𝐅𝐨𝐫𝐦𝐚𝐭_0: str →{format: format(str)} s_1 ⊗ s_2 = { format: if s_1.format = s_2.format then s_1.format else ∅} § EVALUATION Our implementation of JSONoid consists of approximately 9,000 lines of Scala code that produces schemas targeting JSON Schema Draft 2019-09 and is available on GitHub[<https://github.com/dataunitylab/jsonoid-discovery/>]. We evaluate our implementation on two different dimensions: information content and runtime. The goal of the evaluation of information content is to identify the value of the additional information provided by our monoids. The runtime evaluation serves to show the scalability and efficiency of our monoid-based mining approach. Runtime is important when the goal is to produce a complete description of a large collection of documents that an analyst wishes to examine. While in some cases this may be done offline, this is prohibitive as the number of collections grows, as is the case in a data lake scenario. For our evaluation, we make use of the kind equivalence relation in JSONoid. §.§ Information Content Existing JSON schema inference techniques focus only on extracting the structure of a dataset, with no attention paid to the values each contains. The primary goal of our schema extraction approach is to provide additional useful information about the datasets under consideration. To measure the information content of a schema, we start with the simplest possible schema, S_min, which contains only the monoids ObjectType and ArrayType. Together these monoids provide basic structural information about a collection of documents, similar to the approach of early work by Baazizi et al. <cit.>. We then consider the effectiveness of additional monoids in describing a collection of documents. In order to do this, we require a ground truth. Specifically, we require a manually constructed schema which expresses the intent of its author to match against a collection of documents. We also require a collection of documents that were written in order to adhere to this schema. For our evaluation, we use , the metadata provided by package developers in the Node.js (JavaScript) ecosystem. We start with a manually authored JSON Schema, S_0 which expresses the structure of files which we obtained via JSON Schema Store <cit.>. To obtain documents that match this schema, we downloaded metadata for the 1,000 top packages from the npm registry. In order to ensure the integrity of our analysis, we validated these documents against the schema and discovered that 6 of these documents were not valid against the schema and thus excluded from our analysis. For correctness, we require that the schema which we generate against a collection of documents treats every document in that collection as valid. Indeed, we find that this is the case regardless of which monoids are used during schema creation. To determine the fidelity of our mined schemas to the created schema, we must first have a set of documents which are similar to real-world documents, but different enough that they are not valid according to the schema. Since our minimal schema consisting only of ObjectType and ArrayType is the simplest possible representation of the document structure, we generate random documents according to this schema. That is, we generate documents that have attributes with the same names and types as those that exist in the minimal schema. For each object, we randomly decide which properties to include and then generate values of the correct type for those properties. This continues recursively for any values which are nested objects. However, because the minimal schema does not consider constraints on the values within the document, many of these randomly generated documents will be invalid according to the actual schema, S_0. To create our final collection of documents for comparison, we generated random documents conforming to S_min as above and then filtered those documents to only those which do not conform to S_0. We considered two sources of values for these generated documents: sampled, and random. In the sampled case, the values generated for each attribute were selected from the same attribute in our real-world collection of documents. For the random case, we generated completely random values which have the same type, but may lie outside of the domain of the original documents. For example, a string may not be in the correct format, such as a URL, or a numeric value may lie outside a valid range. An example of a real document and a randomly generated document is given in Figure <ref>. In this case, the randomly generated document is invalid because the value is not a valid URL. We note that this invalid value cannot be detected by simply identifying that values of this attribute must be a string with a particular length, as is done by most existing approaches. In this case, the Format monoid is needed to identify that the value must be a valid URI. We also want to identify the possibility that our generated schemas will be overfit to a specific set of observed documents. In this case, the schema may reject documents that are very similar because specific values within the document have not been previously observed. This is undesirable in the case where we want to use the generated schema to validate documents or detect outliers. (But it may be useful to provide a precise summary of a collection.) To measure how prone our monoids are to overfitting, we split the collected data and created a schema on 90% of the original dataset followed by validation against the remaining 10%. If we avoid overfitting, we would expect the remaining 10% of the documents to also be valid since they are drawn from the same collection of documents. Our results are shown in Table <ref>. The final column in Table <ref> represents the fraction of documents that were not valid according to the schema generated from the remaining 10%. Values closer to 1 represent a higher level of overfitting. Note that due to excessive runtime, we were only able to generate the schema based on half of the training set for the approach of Spoth et al. While other approaches completed within seconds, waiting more than 24h did not produce a result. We believe that this is due to the large number of distinct attribute names present in this dataset. As such, we expect the performance of this method to be lower than if the schema were generated using the entire training set. Some monoids are more prone to overfitting than others. Specifically, the MaxMin monoid applied to string and array length is the most prone to overfitting in our setting. This is because the test set of documents contains strings of lengths outside the bounds observed when the schema was created. However, monoids such as Required are less prone to overfitting, since this will only happen when a document without has a property that should be considered optional is never observed when creating the schema. We find that, depending on the selection of monoids, our approach has higher accuracy than any existing approach, with comparable levels of overfitting. We note the relatively high accuracy and low overfitting of the approach of Spoth et al. This is mainly due to the design decision to create an “open” schema. That is, their approach generates a schema which specifies that any properties not defined by the schema are valid. This reduces accuracy, since properties that should not exist are considered valid. However, this also significantly reduces overfitting, since any properties that were not observed in the sample (which should still be considered valid) are accepted. In contrast, our approach and that of Baazizi et al. generate “closed” schemas, where any property not explicitly mentioned in the schema is considered invalid. To show the impact of this decision, we also modified the schema produced by the approach Spoth et al. to produce a closed version. As expected, the accuracy increases, but the overfitting of this schema also increases significantly. This is because their approach contains a step that attempts to identify distinct types of record in the schema. For example, if a schema contains some records with only attributes and and some records with only attributes and , these would be represented as two mutually exclusive types. However, this approach can lead to overfitting since not all combinations of attributes will be observed. It is possible that heuristics applied to the final schema could help reduce overfitting. Consider the case of maximum and minimum string lengths. If we observe many strings of length 100 and no strings which are any longer, it is very likely that 100 is the true maximum length which should be allowed for a string. However, if we see string lengths which are more uniformly distributed and the maximum length of any string is 93, it is much less likely that is is the true expected maximum length of string since constraints are more likely to fall on natural boundaries. We leave such heuristics for reducing overfitting for future work. Finally, we note that while we described many other monoids, many of these do not add information content according to the definition given above. For example, the statistical monoids such as Mean defined in Section <ref> do provide additional information to data consumers. However, this aggregate statistical information does not allow us to distinguish from documents that adhere to a schema that does not include these monoids. As discussed in Section <ref>, we believe that these statistical monoids as well as others such as the histogram monoid described in Section <ref> could be useful for outlier detection. We leave the evaluation of outlier detection using these monoids for future work. §.§ Runtime All runtime evaluation is performed across a cluster of four machines that contain an eight-core Intel Xeon Silver 4110 @ 2.10GHz with an ADATA SX930 240GB SSD and 16GB RAM. All four of these machines were used to evaluate distributed mode, while a single machine was used to evaluate streaming mode. To evaluate the performance of the distributed mode of JSONoid, we compare with the discovery approaches proposed by Spoth et al. <cit.> and Baazizi et al. <cit.> using implementations provided by the authors. All approaches were run using Apache Spark 2.4.8 on top of Hadoop 3.2 with all data stored on HDFS. For the algorithm used by Baazizi et alẇe used the kind equivalence relation, as we do with JSONoid. As noted previously, we are also able to produce a schema that contains the same information as the reduced structure identification graph of Klettke et al. <cit.>. However, since their approach cannot scale beyond main memory, we exclude it from our comparative analysis. We also evaluate with several different sets of monoids whose values can be computed by JSONoid. First, which contains only structural information. Second, which contains only those properties defined in JSON Schema which are inferred by JSONoid. Finally, we include all monoids defined in JSONoid as described in Section <ref>. The results for varying percentages of data sampled from ten different datasets are shown in Table <ref>. The first three columns in each group represent JSONoid for varying configurations of monoids. Each cell displays both the runtime and the fraction of a test set of 10% of documents that were valid against the generated schema. For more details on the datasets used, see Spoth et al. <cit.>. (Note that, while data were obtained from the same sources, the specific documents used and samples are not identical, so our results do not perfectly match prior work.) Note that when discovering a minimal set of monoids, which contains information similar to existing methods, our approach nearly always runs faster than Jxplain. and for smaller data sizes, also Baazizi et al. As expected, when discovering a larger set of monoids, the runtime performance of our method is less than other methods. However, as shown in the previous section, we are able to discover significantly more information. Furthermore, users of JSONoid can select the appropriate tradeoff for their application. In the cases where a schema must be discovered quickly, using a minimal set of monoids can provide useful structural information. If more time is available for analysis, using a larger set of monoids can provide more detailed information on the dataset. As a result, we note that a higher acceptance rate of documents from the test set is not necessarily an indication of a better quality schema. It may be desirable for some applications to construct a more rigid schema that closely fits the input dataset, while other applications may want to be more accepting. Our approach makes this choice configurable during the mining process by choosing the appropriate set of monoids. In some cases, the schema mined by JSONoid does not accept any of the training documents as valid. As discussed previously, this is the result of some monoids fitting the training data very precisely. The specific monoids used can be controlled more precisely instead of using the predefined sets in Table <ref> if more control is desired. JSONoid with the minimal set of properties has runtime similar to the approach proposed by Baazizi et al. Performing inference for all the monoids defined by JSONoid is significantly more expensive. However, we note that, while using the full set of monoids makes the inference process significantly slower than the fastest approach, we maintain linear scalability with respect to the number of documents. A user is free to decide on the tradeoff between runtime and the detail provided via the chosen set of monoids. Although not yet implemented, if detailed information is required and runtime is a concern, a sampling-based approach can be used to yield useful information from a subset of the data. As mentioned above, JSONoid is also capable of running in streaming mode where one document is processed at a time and the schema is updated continuously. To evaluate the performance of our various monoids in streaming mode, we consider feeding documents from the GitHub dataset used above to JSONoid one at a time and evaluating the rate of document processing. Since we cannot directly measure the performance of an individual monoid, we start with the Minimum set of monoids and then determine the additional time taken for further monoids which are added. We consider the overhead of each monoid added to reach the Simple set and again to reach all the monoids supported by JSONoid. We note that since it is necessary to measure the runtime of each monoid using independent trials, summing the results of all the monoids added between the Minimum and Simple sets will not yield the exact total for each set of monoids. Our results are given in Table <ref>. Streaming schema discovery using the Minimum set of monoids has very high throughput. While adding all possible monoids currently defined in JSONoid significantly reduces performance, users can pick and choose which monoids are required for their specific application. There are also plenty of opportunities to optimize the most expensive monoids. For example, the Format monoid currently serially checks multiple regular expressions against each string value. There are several optimized regular expression engines that would allow this matching to occur in parallel <cit.>. Finally, we note that while discovering all monoids across a stream has fairly low throughput, there are several additional techniques we can apply to improve performance in addition to optimizations to our particular implementation. Firstly, we can perform sampling on the stream to reduce the need for JSONoid to keep pace with all new documents as they arrive. Furthermore, a hybrid mode of discovery is possible, in which the stream is partitioned into multiple streams, which are processed in parallel with each of the separately discovered schemas periodically merged. We leave the development of this hybrid mode for future work. § USE CASES While Section <ref> provided an empirical evaluation of our mining approach, here we aim to identify several use cases of the generated schemas. §.§ Schema Exploration One use of our enhancement monoids is by allowing the exploration of the generated schemas from a number of datasets along with our prescribed enhancements. We have implemented a Web application[https://github.com/dataunitylab/jsonoid-web] that allows users to view sample documents from the collection, as well as the generated enhanced schemas and associated enhancements. Each of the enhancements provided is applied to all applicable elements of the schema. An example dataset is shown in Figure <ref> which consists of information about Amazon products <cit.>. A screenshot of the schema visualization based on these data is shown in Figure <ref>. On the left is the schema that was discovered based on the dataset. On the right are some enhancements to the schema element representing the values of the key that is nested under the object. In this case, the currently visible enhancements are a histogram of the observed values and statistics which were calculated across them. These enhancements provide a detailed summary of both the structure and the values contained within the documents used to create the schema. Without a schema, which is often unavailable for collections of JSON documents, it would be necessary for an analyst to manually browse through example documents to begin to develop an understanding of their structure and content. This may be followed by ad hoc analyses, which provide more detailed information. A detailed schema such as those constructed by JSONoid can alleviate much of this manual effort. §.§ Constraint Discovery One important element of relational database schemas is primary and foreign key constraints. Such constraints are typically not present in JSON schemas. However, we make further use of monoids to estimate and suggest possible constraints. Constraint discovery occurs as a post-processing step on the final schemas generated during the discovery process. For attributes expected to be unique across an entire collection (i.e. possible primary keys), we need to track the total number of documents (a simple counter). Furthermore, we can use HyperLogLog to estimate the total number of unique values. When the total number of documents is with the error bounds of the number of unique values estimated by the HLL data structure constructed using the corresponding monoid, we suggest a possible primary key. Considering our example data in Figure <ref>, the attribute would be suggested as a possible primary key, since the estimated count of unique values would be within the bounds of the total number of documents examined. Although there is a small probability of false positives, it is possible to suggest this primary key without the need to maintain a list of all possible values to ensure uniqueness. We also make use of heuristics <cit.> to provide a heuristic ranking of possible primary keys. Although we do not further evaluate this discovery technique here, we note that the features required to compute these proposed heuristic rankings, such as data type and value length, are already captured by the schemas generated by JSONoid. As a further proof of concept, we implemented a simplified version of the Many <cit.> inclusion dependency mining algorithm to identify possible foreign keys using the Bloom filter monoid. Bloom filters are bit vectors, and when the bits set in one Bloom filter (B_1) are a subset of the bits set in another filter (B_2), this indicates that the set represented by B_1 is likely a subset of the set represented by B_2. While the Bloom filters we use are monoids which can be maintained in a distributed fashion, we do need a final pass over the collected Bloom filters to determine possible foreign keys. We compare all pairs of Bloom filters to find possible subset relationships. Although this operation is quadratic, it only scales relative to the size of the generated schema, rather than the size of the dataset. In addition, the Many algorithm has other optimization techniques that could be incorporated in the future. In our example in Figure <ref>, the values contained in the and could be suggested as possible foreign keys to the attribute. As in the case of primary key detection, the heuristics of Papenbrock and Naumann <cit.> could prove helpful in ranking possible primary keys. We provide a further discussion of dependency discovery on JSON data in previous work <cit.>. Once these constraints are discovered, they can be used to aid in the process of normalization. Our prior work, as well as that of DiScala and Abadi <cit.> has proposed approaches for generating normalized models from nested key-value data, such as JSON documents. This can be useful for reducing redundancy in the schema and identifying possible real-world entities, making the schema more natural to work with. §.§ Outlier Detection Many of the statistical monoids presented can be useful in identifying single documents that are outliers compared to the generated schema. For example, the mean and standard deviation of collected values could determine whether particular numeric values in a document should be considered outliers. This also applies to other numeric values that can be tracked by our monoids, such as the length of string values and arrays. We can also compare whether schemas generated from two groups of documents are likely to have been drawn from the same underlying distribution. That is, we can identify whether a schema created from a group of samples should be considered divergent from a schema generated from reference data. As an example, two histograms of values can be compared using the Kolmogorov-Smirnov test <cit.>. Another type of outlier we can identify are structural outliers as identified by <cit.> et al. Structural outliers are attributes that occur very rarely or attributes that exist in most documents but are occasionally absent. We are able to identify such structural outliers using our AttributeCounts monoid. The result is that our schema contains the information present in Klettke et al.'s reduced structure identification graph. All the above examples demonstrate outlier detection for a single value within a schema. Although this may undoubtedly be useful, the question of whether documents are outliers is more complex. Should a single outlying value in a document result in that document being considered an outlier? Should values present anywhere in a document be given the same weight? While we do allow the ability to collect outlying values present in a document, we leave the question of document-level outlier detection for future work. § RELATED WORK Existing approaches to schema inference focus almost exclusively on structural and type information. For example, Baazizi et al. <cit.> produce a scalable mechanism for inferring schemas. As mentioned previously, our distributed discovery approach takes a similar approach to their first work on schema discovery, which uses Apache Spark to produce schemas for individual documents and recursively merge them <cit.>. Some of the additional information we collect includes the same information as the counting types described in later work. However, the resulting schema only provides information on the attributes that exist on an object and limited information on their associated types. This is missing a rich source of useful information for analysts such as some of the examples stated in the previous subsection. Other approaches similarly do not exploit any information within the data values themselves <cit.>. jHound <cit.> similarly provides basic structural information. It goes one step further to identify values incorrectly represented as strings (e.g. and instead of boolean values) with the goal of presenting more useful information to data consumers. This specific scenario was identified as common in the open data setting where jHound is applied, but it does not provide further information on the data values. This would be a trivial addition to JSONoid via additional patterns in the 𝐅𝐨𝐫𝐦𝐚𝐭 monoid. Klettke et al. <cit.> present an approach to schema extraction that also detects structural outliers or documents that contain attributes that are uncommon or which do not contain attributes that are common. The assumption is that these are either errors in the data which the consumer should be aware of or uncommon structures that the user should be prepared to handle. Several of the metrics used to detect outliers are similar to those collected in our approach, and suggesting possible structural outliers could be a possibility for future work. Since JSONoid is designed to process large volumes of data, we currently do not enable this use case with a single pass through the data. However, JSONoid can provide this information on outliers in a second pass through the data based on the metadata collected. There is also significant existing work on semantic schema discovery <cit.> where the goal is to identify meaningful entities in the discovered schemas and possibly align the entities with a preexisting knowledge base. We did not attempt to infer semantic meaning in this work, but this is an interesting area for future work. We believe that the information extracted from JSONoid schemas could provide useful features for semantic classification. § FUTURE WORK While we believe that the monoids that we present here are useful, there are many opportunities for further extension. Specifically, we aim to consider monoids which can represent combined information from multiple attributes such as joint value distributions. This could be helpful for tasks such as identifying possible functional dependencies. Other information that may be useful to surface in the inference process is the identification of schema variants. Gallinucci et al. <cit.> explore variants within a schema such as different metadata associated with different types of log records. Ruiz et al. <cit.> and Klettke et al. <cit.> considered variants that arise from a schema with a structure that changes over time. Joint statistics across attributes could also be helpful in identifying such variants. We leave the incorporation of variant discovery as future work. Our discussion of outlier detection in Section <ref> raised many questions about how to identify whether a specific document is an outlier. We may not want a single outlying value in a document to cause the entire document to be considered an outlier. However, some combination of outlying values or structural outliers may warrant flagging a document as an outlier. Surfacing all outlying values and structural outliers in a document and then using a supervised approach can tailor outlier detection to specific use cases <cit.>. To effectively use the generated schemas for tasks such as outlier detection and validation, it is necessary to avoid overfitting to the observed set of documents. As previously discussed, we expect that for each monoid it will be possible to develop heuristics to help tune the results of these monoids to limit overfitting where that is desired. There are several useful elements of real-world JSON schema instances which we have not touched on this work. For example, JSON Schema allows the creation of definitions which function as an avenue for avoiding duplication within a schema. We believe that our extracted schemas can be useful for identifying possible reuse and creating definitions in the generated schemas. This would reduce the size of the generated schemas, as well as improve the ease of working with the schema since repeated structures would be automatically identified. Such structures may also be useful for identifying real-world entities and normalizing denormalized collections of documents, as described previously. We note that some previous work, such as the approach of DiScala and Abadi <cit.> requires materializing the entire dataset in memory, a downside we believe can be avoided using the techniques presented here. § CONCLUSION We presented JSONoid, a monoid-based JSON schema discovery tool which is scalable to large collections of documents and provides significantly more useful information than existing tools. JSONoid allows users to select a tradeoff between runtime and the information content of generated schemas. When discovering simple information, JSONoid has performance nearly identical to those of existing approaches. Discovering additional information has an additional cost, but the penalty remains linear in the number of documents. We demonstrated several use cases for these schemas that are not enabled by existing schema discovery approaches. Schemas generated by JSONoid have significantly more utility for analysts as they provide insight into the actual values contained within the documents used to create the schema. This information enables analysts exploring the schema to make decisions such as the utility of various attributes and the sizing of data structures necessary during further processing without requiring manual browsing through documents. For future work, we intend to explore the practical usage of JSON Schema in more detail to explore automated approaches to mine other features from collections of JSON documents that are useful for data analysts. Two main areas where we expect JSONoid to be useful are outlier detection and the discovery of common structures within documents. IEEEtran
http://arxiv.org/abs/2307.02810v1	20230706070105	How to steer active colloids up a vertical wall	[ "Adérito Fins Carreira", "Adam Wysocki", "Christophe Ybert", "Mathieu Leocmach", "Heiko Rieger", "Cécile Cottin-Bizonne" ]	cond-mat.soft	[ "cond-mat.soft", "cond-mat.stat-mech" ]	These two authors contributed equally Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622, Villeurbanne, France These two authors contributed equally Department of Theoretical Physics and Center for Biophysics, Saarland University, 66123 Saarbrücken, Germany Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622, Villeurbanne, France Department of Theoretical Physics and Center for Biophysics, Saarland University, 66123 Saarbrücken, Germany Leibniz Institute for New Materials INM, Campus D2 2, 66123 Saarbrücken, Germany Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622, Villeurbanne, France An important challenge in active matter lies in harnessing useful global work from entities that produce work locally, e.g., via self-propulsion. We investigate here the active matter version of a classical capillary rise effect, by considering a non-phase separated sediment of self-propelled Janus colloids in contact with a vertical wall. We provide experimental evidence of an unexpected and dynamic adsorption layer at the wall. Additionally, we develop a complementary numerical model that recapitulates the experimental observations. We show that an adhesive and aligning wall enhances the pre-existing polarity heterogeneity within the bulk, enabling polar active particles to climb up a wall against gravity, effectively powering a global flux. Such steady-state flux has no equivalent in a passive wetting layer. How to steer active colloids up a vertical wall Cécile Cottin-Bizonne August 1, 2023 ================================================== Self-propelled agents, like bacteria, cells, micro-organisms, animals, or synthetic active particles, inject energy at small scale into their environment, driving the system out of equilibrium. Often, this energy only powers disordered agitation akin to thermal energy <cit.> and a major challenge is to better understand active energy flows to extract useful work from seemingly low-grade heat. This was achieved in a few configurations with, for instance, swimming micro-organisms driving unidirectional rotation <cit.>, or decreasing the apparent viscosity of their sheared medium <cit.>, both effects that have no equivalent in equilibrium systems. To date, such active energy harvesting did not show up in the ubiquitous configuration of self-propelled particles exposed to a constant and uniform force, e.g. gravity. There, indeed, active sedimentation properties were shown to be essentially that of a hotter suspension <cit.>. In this paper, we explore how this picture may change drastically when bringing a lateral confining wall in contact with sedimenting polar active particles, a configuration which may fall within active wetting phenomena. A straightforward mean for harvesting active self-propulsion forces is to herd the system. This can be obtained with a polarizing external field, where particles align with the field and are harnessed to produce useful work. For instance, the swimming direction of magnetotactic bacteria can be polarized by a magnetic field, producing a net flux <cit.>. Bottom-heavy algae <cit.>, colloids <cit.> or walkers <cit.> may point up, effectively acting against gravity. Yet, in the absence of external torque acting on particles, a constant and uniform force field can still promote a local averaged polarization <cit.>. However, this polarization only arises self-consistently to balance the sedimenting flux such that no net flow is created at steady-state. Confining walls are known to promote complex responses of active systems which challenge the intuition one has for the equilibrium. At odds with at-equilibrium thermal motion, the pressure exerted by self-propelled particles on walls depends on the wall stiffness <cit.> and curvature <cit.>. Alike, because of directional persistence <cit.>, self-propelled particles tend to accumulate at and to polarize towards walls, although the wall-particle interaction, in particular aligning interactions, have a major influence on the detention time of the particles <cit.>. Considering a heterogeneous active system displaying a fluid-fluid interface, the introduction of a wall extends wetting phenomena to active systems. With simple molecular liquids, surface tension that stems from the attractive interactions of the molecules with the wall and between each other <cit.> can trigger “wall-climbing”: from the formation of a wetting layer, to a meniscus, to capillary rise. Yet, the steady-state of this equilibrium system displays no net flux. Such phenomenology also extends to complex fluids such as a colloidal suspension with attractive interactions <cit.>. In this case the interfacial tension between the colloid-rich and the colloid-poor phase is ultralow leading to mesoscopic interface fluctuations. Starting from such equilibrium wetting configuration of phase separated polymer solutions near a vertical wall, a recent experimental and theoretical study <cit.> demonstrated surprising wetting-like phenomena when adding an active nematic into one of the two phases. In particular, an activity-induced transition towards full wetting was observed. Even without attractive interactions, active particles can also display a purely out-of-equilibrium separation into a dense and a dilute phase, called motility induced phase separation (MIPS), which is due to a slowdown occurring during collisions between particles <cit.>. We may thus anticipate for such repulsive active systems a possible extension of the passive scenario, although the mere definition of surface tension is difficult and may yield negative values <cit.>, which should be incompatible with a stable interface. Indeed, a recent theoretical study demonstrates such an analogue out-of-equilibrium behavior, where a repulsive active dense phase can form a meniscus on a wall or rise against gravity in a confining channel due to the slowdown that occurs during particle-wall collisions <cit.>. Here, we present experimental observations of the behavior of a non phase separating gaseous assembly of active colloids that are sedimenting in a gravitational field in the presence of a confining wall. We provide the evidence of an unexpected and dynamic adsorption layer at the wall that has no analogue in passive systems. We also offer a numerical model that recapitulates the experimental observations. It enables us to test the respective influences of adhesive and aligning interactions with the wall, parameters that are challenging to adjust experimentally. Our results demonstrate that a confining wall can act as a pump against a force parallel to it, opening the door to active microfluidic circuits where a configuration as simple as gravity and walls could play a role analog to a generator in an electric circuit. § RESULTS Experimental and numerical observation of “Wetting-like” behaviour — We study experimentally the behavior of sedimenting active colloidal particles in the presence of a vertical wall. The experimental configuration, as shown in Fig.<ref> and explained in detail in the Methods section, is composed of Janus microspheres of average radius R=0.8, sedimenting along the vertical direction z under an effective gravity g^. Those particles self-propel through phoretic effects in the presence of hydrogen peroxide <cit.>. In line with our previous works, we determine the activity of the system by analyzing the sedimentation profile, in the dilute regime, far away from the wall, assuming a Boltzmann distribution. From the characteristic sedimentation length, we extract the ratio of the effective to the room temperature, T_eff/T_0, which indicates how far the system is out-of-equilibrium <cit.>. By changing the concentration of hydrogen peroxide, this ratio can typically be tuned between 1 and 75. A wall is introduced in the system as a glass capillary oriented along z and immersed down to the colloidal sediment. To study how this wall impacts the active particles system, we measure the density field ϕ(x,z) for various T_eff/T_0. In Fig.<ref> we show images of the sediment together with a few iso-densities (see Methods) in the passive and active cases. As expected for purely repulsive particles with no wall-attraction, passive colloids at T_0 (Fig.<ref> A-B) are hardly affected by the wall <cit.> and the passive system does not adsorb at the wall. As we activate the colloids with hydrogen peroxide (T_eff/T_0>1), iso-densities remain horizontal and parallel to each other far from the wall. This is what is expected for activity resulting in a simply hotter system with a higher sedimentation length (Fig.<ref> C-D). Strikingly, this hot-colloids mapping breaks down in the vicinity of the wall, where we evidence an upturn of the iso-densities that rise with increasing activity. The wall causes particles to climb up at its contact. Indeed, we observe what seems to be a wetting or an adsorption layer, a layer of one particle thickness. Past this wall region, we note a small dip of the iso-densities before progressively returning to the far-field “flat” region. Overall, this departs strongly from the global features of classical passive wetting or of recent extension with active perturbation <cit.>, where a macroscopic upward meniscus smoothly bridges between the wall and far-field unperturbed region. In a classical picture, colloids excess at the wall comes from attractive wall-particle interactions. Indeed, as already mentioned in the introduction, wall accumulation of active particles is a generic feature for which directional persistence provides an underlying mechanism for effective adhesion <cit.>. However, in specific systems, interactions between walls and active-particles comprise a rich variety of contributions. For instance a direct wall-particle attraction may also arise due to activity through phoretic or hydrodynamic effects <cit.>. Likewise, hydrodynamic interaction of dipole micro-swimmers with surfaces can induce alignment with the wall <cit.>. To complement our experimental evidences, we also analyzed the predictions of a model of sedimenting repulsive active Brownian particles (ABPs) <cit.> in the presence of a vertical wall. As we shall discuss, this allows us to explore the importance of the detailed interaction between walls and active-particles. When including an activity-induced interaction, mimicking known diffusiophoretic adhesion <cit.> as well as a wall bipolar (also called nematic) alignment parallel to the wall <cit.> (see Methods for details), the whole experimental phenomenology is reproduced. As is shown in Fig.<ref> E, numerical simulations display both the adsorption layer at the wall in the active case and the small dip close to the wall. Note that consistently with experiments, we explored here an activity region for which no MIPS occurs, corresponding to moderate activities quantified by Pe_s≤ 17, far enough from Pe_crit≈ 26.7 <cit.> (see Methods for the definition of the normalized swim persistence length Pe_s). In the MIPS regime self-propelled particles generate a phenomenology that is reminiscent of classical wetting configuration as a recent theoretical study predicted a smooth macroscopic wetting meniscus <cit.>, in contrast with the present observations. Detention time at a wall without gravity — Before interpreting our results, we have to understand how active particles accumulate at a wall in the absence of gravity and the influence of wall-particle interactions on this accumulation. To do this, we generalize arguments that have been laid out in the case of steric <cit.> and hydrodynamic interactions <cit.>. To simplify the argumentation, we neglect the translational diffusion, i.e. Pe_s ≫ 1. In the absence of attractive interactions, an active particle is able to escape the wall as soon as the normal component of its propulsion force points away from the wall. Since its orientation evolves through rotational diffusion, a particle polarized towards the wall will have a long detention time, whereas a particle polarized tangentially to the wall have a shorter detention time. Therefore, wall-aligning interactions actually reduce detention time because they bring the particles closer to the escape angle. By contrast, attractive interactions push the escape angle away from the wall and thus increase detention time. A combination of attractive and wall-aligning interactions traps the orientation parallel to the wall while pushing the escape angle away, and thus increases detention time. Let us now explore how a force parallel to the wall affects these behaviours and how it explains our observation of an activity-induced adsorption layer. To do this, we use our ABP model to explore systematically the influence of the above-mentioned contributions, namely self-propulsion, direct wall adhesion, wall alignment and the combination of these. Adsorption layer height dependency — From the density field we obtain the density profiles ϕ_wall(z) close to the wall (within the adsorption layer) and ϕ_bulk(z) in the bulk, far away from the wall (see Methods). It is clear from Fig.<ref> F that at a given altitude, the adsorption layer has an excess density as compared to the bulk and that the altitude corresponding to a certain density is much higher in the adsorption layer than far from the wall. The height difference between the adsorption layer and the bulk gets even larger as the density decreases. We now define the adsorption layer height Δ H as the difference in altitudes corresponding to a given density of ϕ=0.08 in the adsorption layer and the bulk. Δ H is a global observable of the activity-induced phenomenon. For a passive system, wall adhesion promotes the creation of a gravity-fighting adsorption layer <cit.>, as can be seen in Fig.<ref>-A where however Δ H only reaches a few particle radii R. Now, let us consider an active system experiencing both wall accumulation due to the persistence of motion and an activity-induced direct adhesion. Indeed, for Janus colloids, the self-generated electro-chemical gradients responsible for self-propulsion also generate a wall-attraction force, thus scaling as the propulsion velocity (see Supp. Mat. for an explanation for the form of the adhesive energy, along with simple estimation on its magnitude). Accordingly, we take a wall adhesion strength as ϵ̃∝Pe_s in our ABP model. Comparing passive and active systems with same direct adhesion parameter in Fig.<ref>-A, we observe that Δ H of self-propelled particles exceeds by more than a decade the one of passive colloids. This is a further indication that this active system does not merely behave as an equilibrium system regarding adsorption properties, with self-propulsion of individual particles a key factor of the global response. In Fig.<ref>-B, we thus focus on self-propelled particles showing Δ H obtained numerically for different levels of activity as quantified by effective temperatures T_eff/T_0=3/8Pe_s^2+1 <cit.>. A neutral wall represents the benchmark situation where no direct interaction —neither attraction nor orientation— is present aside the short-range steric repulsion. In line with above discussion, adding a direct activity-dependent wall adhesion increases the adsorption layer height as compared to neutral wall. To mimic the tangential alignment of Janus colloids with a wall, we we choose an alignment strength as Γ̃∝Pe_s. This is consistent with the experimental characterization of Janus colloids as force dipole microswimmers <cit.> which strengh is proportional to the propulsion speed <cit.> (see Supp. Mat. for an explanation of the form of the alignment strength, along with simple estimation on its magnitude). Our model predicts that the influence of wall-particle interactions on the adsorption layer height is consistent with its influence on detention time without gravity: pure alignment interaction decreases Δ H as compared to the neutral case, and the combined effect of both alignment and direct adhesion increases Δ H above all other cases considered. However, we shall see in the following that a wall parallel to gravity also exerts a singular influence on polarity and fluxes. As a final note on the global adsorption layer height Δ H, let us stress that Fig.<ref>-B also incorporates experimental measurements. As we see, our ABP model recapitulates both the experimental order of magnitude and the dependence on activity. Polarization at the wall — To go beyond the global adsorption layer height, we measure numerically the local time- and ensemble-averaged polarization 𝐌(𝐫) (see Methods) and in particular its component parallel to gravity M_z (Fig.<ref>-B). As we already mentioned, a remarkable feature of sedimenting active particles is the existence, in the absence of lateral walls, of a non-vanishing local polarization M_z(z) <cit.>. In the dilute regime, the mean orientation points upward (M_z>0) to balance the downward sedimentation and to guarantee the absence of a net particle flux. By contrast, in the dense sediment at the bottom of the cell there is a downward polarization (M_z<0), as the total polarization has to vanish <cit.>. Far away from the wall, this bulk behaviour is well recovered in our system (Fig.<ref>-B) with a local polarization that remains unchanged irrespective of the wall properties details. At the wall, the same qualitative picture is retained for the averaged polarization, which reverses from an upward to a downward orientation when going from large z down into the sediment. There, however, the polarization strength becomes very dependent on the wall-particle interaction properties. For a neutral or a purely adhesive wall, the polarization at the wall is lower than in the bulk. Indeed, as recalled above, in the absence of gravity, a particle pointing towards the wall will in average remain longer at the wall than a particle nearly parallel to it. Mean polarization at the wall is thus obtained from an initial bulk distribution by adding extra weight to normal orientation at the expense of tangential ones, so that vertical non-aligning walls will act as dampers for the bulk z-polarization. On the opposite, a vertical bipolar aligning wall will boost any initial upward or downward polarization bias in the particles orientation. This is what we observe (Fig.<ref>-B) for all aligning configurations (with or without additional adhesion) where the wall acts as an enhancer of the bulk polarization. In a side experiment, we probed the Janus colloidal particles polarization at the wall (Fig.<ref>-A). The method, fully detailed in Supp. Mat., is based on the tiny shifts between a particle location in the three color channels of a camera that correspond to the colour difference between the two faces of the Janus. As chromatic aberrations are benchmark in the bulk region, only excess polarization at the wall is reported. We observe a strong upward excess polarization, which is a clear and independent indication of alignment interaction in the experiment. Fluxes, circulations, and wall dynamics — As we just pointed out, the colloid polarization at a vertical wall is an important feature of the present system. With self-propelled objects, this naturally raises the question of permanent fluxes in the system, a property forbidden in equilibrium wetting or adsorption. Therefore, we calculated the local particles flux 𝐉(𝐫) (see Methods) predicted by our ABP model and show the result in Fig.<ref>-C. We observe an upward flux at the attractive and aligning wall in the dilute regime, consistent with the enhanced upward polarization (Fig.<ref>-A and B). Unlike in the (unconfined) bulk, the balance between gravity and upward propulsion is broken at the wall. This fundamentally out-of-equilibrium response, which involves the presence of permanent fluxes in the steady-state, allows us to propose a rationale for the complex dependence of the rising height Δ H on aligning wall interactions (Fig. <ref>-B). When the alignment comes with an additional direct wall adhesion, particles are trapped at the wall <cit.> and display an extra upward polarization in the dilute region as compared to the sedimentation-canceling bulk reference. Accordingly, we expect a high rise of the particles at the wall, which is of purely dynamical origin. On the contrary, when the wall has only alignment interactions, the extra upward polarization is counteracted by a shorter detention time, which limits the possible rise. Although the final balance is difficult to anticipate, it turns out (Fig. <ref>-B) to yield a weaker rise than even a simple neutral wall. Moreover, an aligning wall also enhances the downwards polarization present in the dense part of the bulk. Consistently, it causes a strong downward flux close to the wall, resulting in a well pronounced vortex in the bulk. As pointed out, the existence of steady-state particle currents is a strong signature of the non-equilibrium nature of active adsorption <cit.>. Note that, so far, such currents are only accessible in our ABP model, since the velocity measured numerically from the flux is, at maximum, 0.1v_0, which is lower than the experimental resolution. For the neutral wall (Fig.<ref>-D), we observe a small downward current close to the wall. Such current can be explained by the polarization decrease at the wall, breaking the nearby flux balance and generating a downward current. As we just showed, dynamical aspects are the key in the out-of-equilibrium active phenomenon we report. So far, we have mostly discussed it in terms of single particle properties at the wall. However, when zooming in on this wall layer, we see it forms an assembly of 1D clusters, which we denote as trains, since they move collectively along the wall (see inset of Fig.<ref>-A and movie in Sup. Mat.). In a recent study of Janus particles orbiting along a circular post in quasi-2D <cit.>, similar train structures were observed resulting from collisions as well as hydrodynamic and osmotic interactions of 1D-moving swimmers. Although large trains tend to move slower than single particles at the wall, our trains are dynamical structures. They can merge or break apart. Particles leave the trains at their ends or are squeezed out into the second, more labile, layer. We focus in Fig.<ref>-A on the probability distribution P(n) of train size n in the experiments. We observe that the distribution decays exponentially and thus is completely dominated by monomers. To describe such behavior, we examine the possible origin of trains in our system, and the extent to which trains can appear randomly, without an underlying formation mechanism. To do so, we consider a simple site-adsorption model where the adsorption layer is considered as a 1D system with homogeneously distributed adsorption sites. These sites are randomly populated by adsorption-desorption exchanges with the nearby bulk reservoir (see Methods). The probability of having a train of size n is very well described by such a random site-adsoprtion model, as shown in Fig.<ref>-A. The dotted lines correspond to geometric laws, where the average density is the only adjustable parameter. The values obtained fall within the range of experimentally measured densities (see Sup. Mat.) Overall, the is no indication of collective dynamics on the wall layer which justifies the single particle arguments used so far, although we cannot rule out more subtle collective effects. Finally, a complete description of the active wall-climbing adsorption phenomenon requires a characterization of exchanges between particles in the bulk and in the adsorption layer. We now report the detention time τ_detention that is the average time a particle remains in the adsorption layer (Fig.<ref>-B). As monomers dominate train statistics, we focus on the detention time of a single particle. Our ABP model predicts how τ_detention depends on the particle-wall interaction. As expected, adhesion always increases detention time as compared to a neutral wall. On the other hand, particles escape a purely aligning wall sooner with increasing activity, however, the opposite is true for a wall that is both adhesive and aligning. These observations are consistent with the wall detention time arguments of <cit.> at Pe_s≫ 1, and with the adsorption layer height dependency on wall-particle interactions, and thus validate our single-particle arguments above. As shown also in (Fig.<ref>-B), we were also able to measure τ_detention in experiments. There, we observe τ_detention to decrease with activity, which is the same trend as for a purely aligning wall, however, with longer times overall. This is consistent with the presence of an alignment interaction, and compatible with an adhesion, however weaker than assumed in our ABP model. Nevertheless, for the detention time, there is not a perfect quantitative agreement between the experiments and simulations. Achieving a more accurate agreement would require fine-tuning of the adhesion and the alignment strengths, but beyond that, it would also be necessary to refine the ABP model, particularly by including hydrodynamic and diffusiophoretic interactions <cit.>. § DISCUSSION To summarize, we study experimentally and theoretically the behavior of an assembly of active colloidal particles in the presence of a vertical confining wall and gravity. We observe a dynamic adsorption layer at the wall that rises with activity, and we find that the interaction between the particles and the wall has a significant impact on this layer. A combination of effective adhesion, alignment and polarization enhancement at the wall of an already existing bulk polarization are most likely responsible for the significant increase of density at the wall and the persistent vertical pumping we observe in the system. How does this persistent pumping fit into the broader context of active matter studies? The exploration of mechanical aspects in active matter, such as the concept of pressure has triggered an abundant literature illustrating how active matter can depart from equilibrium systems <cit.>. These peculiarities of active systems become evident when they interact with walls or interfaces, at the core of striking features such as work extraction from a bath with ratchet-like rotors <cit.>. Following this appealing route, the extension of wetting phenomena to active matter was recently considered <cit.>. In MIPS systems, activity provides an effective attractive inter-particles interaction responsible for the phase separation into a dense active-liquid phase. It also provides an effective wall-particle attraction so that phase separated systems in contact with a solid wall display the global phenomenology obeyed by wettable walls in contact with a liquid: macroscopic ascending meniscus, capillary rise, etc. <cit.>. Also starting from phase separated systems, but here made of classical polymer-polymer solutions, a rich panel of phenomena were evidenced when adding activity into one of the phases <cit.>. Indeed, activity promoted the transition from a highly wetting meniscus to a full-wetting configuration reminiscent of the Landau-Levich coating transition <cit.> —although this was not discussed along this line in <cit.>. On the contrary, the present system does not phase separate and was only shown to form clusters <cit.>. The dense phase is held together mostly because of gravity, and overall no macroscopic meniscus forms at the wall whatever the conditions explored. In that respect, the wall layer that forms with activity would be closer to the adsorption of supercritical fluids at solid surfaces <cit.>. At odds though with this equilibrium analogy, this adsorption layer is associated with strong dynamical effects with steady-state fluxes across the system. Our results demonstrate that a vertical wall effectively harvests energy from the microscopic scale to produce a macroscopic work. More generally, a side wall can act as a pump against a force parallel to it, generating a net steady-state flux in the system. These results pave the way for active microfluidic systems, where even a basic configuration involving walls and gravity could play a role analogous to a generator in an electric circuit. § METHODS Experimental set-up. Gold particles of radius R=0.8 m were grafted with octadecanethiol <cit.> and half-coated with Platinum to form Janus microswimmers when immersed in hydrogen peroxyde (H2O2) <cit.>. Due to their high mass density μ≃ 11g/cm^3, the particles immediately sediment onto the flat bottom of the experimental cell, forming a bidimensional monolayer of sedimented active particles. A very low in-plane apparent gravity g⃗^ is obtained by tilting the whole set-up with a small angle θ≈0.1 in the z direction. An elongated borosilicate micropipette bent on the bottom of the observation cell and dipped into the 2D sediment acts as a lateral wall. We focus on the half-space on the right side of the wall. By tuning H_2O_2 concentration from c_0= 3.0e-4v/v up to 5 c_0, it is possible to vary the activity of the colloid. In practice, for each experiment, we characterize this activity by measuring, from the sedimentation profile in the dilute regime, the ratio T_eff/T_0, where T_eff and T_0 are the effective and the room temperature, respectively <cit.>. For each concentration in H_2O_2, 3000 images of 2048pixelx2048pixel are recorded at 5fps using a Basler camera (ac-A2040-90um) mounted on a Leica DMI 4000B microscope and a custom-made external dark field and a 20x fluotar objective. The pixel size is 0.273 m. We track the positions of the center of the particles using Trackpy toolkit on Python <cit.>. We measure, within one pixel (30% of R) the most occupied position x_m close to the glass wall. We set the origin of the x axis (x=0) at 0.5 R before x_m. We define the adsorption layer as the layer between x=0 and x=L=2 R. Note that the average equilibrium distance between particles is 2.4 R. Our observable for the analysis are the density maps ρ(x,z), that are obtained by averaging over time the number of particles for each pixel divided by its surface 7.45e-2m. The density profiles of the bulk are computed as ϕ_bulk(z)=π R^2/x_r-x_l∫_x_l^x_rρ(x,z) dx with x_l= 80 R corresponding to a value far enough from the wall, at least six times the average equilibrium distance between particles, and x_r= 340 R, corresponding to the border of the image. The density profile in the adsorption layer is obtained via ϕ_wall(z)=π R^2/L∫_0^Lρ(x,z) dx. From both density profiles, adsorption layer heights can be determined by measuring the difference of altitude using a 0.01 broad interval centered on ϕ=0.08. We define the trains by considering the set of particles in the adsorption layer whose centers are separated by a distance of less than 2.4 R. The train size distribution was analyzed for trains at altitudes higher than, z=300 m corresponding to a position at which the bulk density is approximately 0.08 for all studied activities. Polarity is measured on different experiments, on the same system, recorded in reflection with a Baumer HGX40c color camera, a 60x objective and a 1.6x zoom. The polarity of a particle is given by the shift between its position on the green channel and its position on the blue channel, corrected for chromatic aberration, as explained in Supplementary Methods. Random site-adsorption model. We look at the statistics of trains, that is, of segments of continuously populated sites. We note p the probability that a site is occupied by a particle and q=1-p the probability that a site is empty. p corresponds to the mean lineic fraction of particles along the wall. The probability that a random chosen site belongs to a train of size n is proportional to np^n(1-p)^2. From which we derive the probability of having a train of size n as P(n)=(1-p)p^n-1=q(1-q)^n-1 that abides a geometric law. Active Brownian Particle (ABP) model. We model the self-propelled particles as two-dimensional active Brownian particles swimming with a constant velocity v_0 in a container of size L_x and L_z along the x- and z-direction, respectively. The position 𝐫_i=(x_i,z_i) and the orientation 𝐞_i=(cosθ_i,sinθ_i) of the i-th particle evolve according to the overdamped Langevin equations: 𝐫̇_i = v_0𝐞_i+γ_t^-1𝐟_i-v_g𝐞_z+√(2D_t) η_i θ̇_i = γ_r^-1t_i^wall+√(2D_r) ξ_i for i=1,…, N. The Einstein relation for translation and rotation is γ_t=k_BT_0/D_t and γ_r=k_BT_0/D_r, respectively, where γ_t and γ_r are the friction coefficients, D_t and D_r the diffusion constants and k_BT_0 the thermal energy. η_i, ξ_i are zero-mean unit-variance Gaussian white noises. For a spherical Brownian particle, we have D_r = 3 D_t/(2R)^2. Due to the reduced gravity, particles sediment with velocity v_g along the negative z-direction -𝐞_z. The force on i-th particle 𝐟_i consists of a part due to particle-particle, ∑_j≠ i𝐟_ij, and particle-wall interaction, 𝐟_i^wall. The particles interact via a repulsive pair potential V(r)=k/2(2R-r)^2 if r≤2R, i.e., the inter-particle distance r is smaller than the particle diameter 2R, and V(r)=0 otherwise <cit.>. The repulsion strength k is chosen such that the particle overlap is 0.01 of the diameter 2R during a head on collision. The force on i-th particle due to j-th particle reads as 𝐟_ij=𝐟(𝐫_i-𝐫_j)=-∇_𝐫_iV(\|𝐫_i-𝐫_j\|). To account for a possible wall adhesion, we let the particles interact with walls via a Lennard-Jones potential, which for the left wall at x=0 reads as V^wall_left(x) = 4ϵ[ (R/x)^12-(R/x)^6 ] , where ϵ controls the attraction to the wall, and similarly for the right, bottom and top wall. For a neutral (purely repulsive) wall, we use a Lennard-Jones potential truncated at x=2^1/6R and set ϵ=k_BT_0/2. In order to mimic a possible bipolar (or nematic) wall alignment, the particles experience a position-dependent torque, which for the left wall (x=0) reads as t^wall_left=Γsin(2θ)(R/x)^3 , where Γ is the strength of alignment. For Γ>0 this torque orients the particles parallel to the wall. The form of t^wall is motivated by hydrodynamic interaction of force dipole microswimmers with surfaces <cit.>, as Janus colloids were experimentally characterized as pushers <cit.>. Four dimensionless numbers govern the system. A measure of activity is the normalized swim persistence length, Pe_s=v_0/(R D_r) sometimes also called swim Péclet number. The range of activities considered here was 5≤Pe_s≤ 17, which is below the critical point Pe_crit≈ 26.7 <cit.>. Similarly, one can define a gravitational Péclet number as Pe_g=v_g/(R D_r), which we kept fixed to the experimental value Pe_g=1. Furthermore, we define a dimensionless adhesion and alignment strength as ϵ̃=ϵ/k_BT_0 and Γ̃=Γ/k_BT_0, respectively. Our simulation box was of size L_x=500R and L_z=1500R and contained N=14000 particles. The simulation time was at least 750000/D_r. For comparison, the observation time in the experiment was approximately 300/D_r. Microscopic fields. The density field at position 𝐫 is given by ρ(𝐫)=⟨∑_i=1^Nδ(𝐫-𝐫_i)⟩ , where the angles denote a statistical average, δ the Dirac delta function, and 𝐫_i the position of i-th particle. The microscopic fields are coarse-grained using a Gaussian kernel instead of a delta function <cit.>. The local polarization is defined as 𝐌(𝐫)=⟨∑_i𝐞_i δ(𝐫-𝐫_i)⟩ , where 𝐞_i is the orientation of i-th particle. The particles flux 𝐉(𝐫) is obtained from the force density balance equation 𝐉=v_0𝐌+γ_t^-1𝐅-v_g𝐞_zρ-D_t∇ρ , where 𝐅 denotes the internal force density field <cit.>. Each term of Eq. <ref> can be measured easily in simulations. § ACKNOWLEDGEMENTS A.F.C. is supported by PhD scholarship from the doctoral school of Physics and Astrophysics, University of Lyon. C.C.B. and C.Y. acknowledge support from ANR-BACMAG. A.W. and H.R. acknowledge support from the German Research Foundation (DFG), project RI 580/15-1. § AUTHOR CONTRIBUTIONS A.F.C. and A.W. contributed equally to this article. H.R. and C.C.B. designed the original project. A.F.C. performed experiments and data analysis. A.W. performed numerical simulations and data analysis. M.L. performed the polarity analysis. C.Y. provided theoretical justification of the wall interaction used in simulations. A.W., C.Y., M.L., H.R. and C.C.B guided the research. All authors interpreted results and contributed to the manuscript. § COMPETING INTERESTS The authors declare no competing interests. § SUPPLEMENTARY MATERIAL §.§ Measurement of the excess polarity at the wall §.§.§ Colour shift We acquire RGB images with a Nikon water Immersion x60 objective, NA=1.2 and a x1.6 zoom lens. On an image, a pixel is 57 (≈ 29 pixels per particle diameter and we have 2048x2048pixel. Our color camera (Baumer HGX40c) is composed of a monochrome sensor equipped with colour filter following a Bayer matrix. Therefore there are twice as many green pixels as red or blue pixels. Since the green channel is the most spatially resolved, we use it to localize the particle position with subpixel accuray using the package Trackpy <cit.>. Around the position of each particle, we localise with subpixel accuracy the local maximum of the red (respectively blue) channel. For each particle, we can thus define two vectors: the `redshift' vector (respectively `blueshift' vector) is the difference between the position on the red channel (respectively blue channel) and the position on the green chanel. The results are shown on Fig. S<ref>. On the left zoom, it is obvious that the redshift vectors are mostly oriented towards the left and the blueshift vectors mostly towards the top right. However on the right zoom the redshift vectors are oriented towards the top right and the blueshift vectors are oriented towards the left. This means that we have chromatic aberration in our optical system: the different wavelength of light are not propagated in the same way, therefore the red image does not form at the same place on the camera as the green image or the blue image. The raw redshift and blueshift vectors must be corrected of this chromatic aberration before they can yield any information on particle polarity. §.§.§ Chromatic aberration In order to quantify and correct for the chromatic abberation, we divide the field of view into a 32× 32 grid and average the redshit and blueshift aberration of all the particles that are in a grid element at a given time, for all 6000 images of the video. Since the polarity of chemically bound dimers can be ill defined, we removed all pairs of particles closer than 28px from this ensemble average. Results are shown on Fig. S<ref>. We want to remove the global chromatic aberration, but not all polarity signal, especially close to the wall. Therefore, we remove from our grid all cells that can contain particles at walls, i.e. what is between the two vertical green dashed lines on Fig. S<ref>. The remaining field is interpolated using splines. Now, at each instantaneous position of a particle, we can remove from redshift or blueshift the value of the spline at this position. The remaining redshift and blueshit are shown on Fig. S<ref>, this time magnified by a factor 100. The abberation-corrected redshift and blueshift are typically 10 times smaller than the value of the chromatic aberration. Therefore, the arrows are now decorrelated from the apparent color shift on the image. Many particles have a redshift or blueshit measurement that is below the resolution we can expect from particle localisation from images (0.1 px, here 0.2 px since we subtract two measures). Therefore, we cannot measure precisely the polarity of a given particle, we have to estimate statistical distributions on large numbers of particles. In order to minimze the noise, all the following statistical distributions will be established by weighting a particle contribution by the magnitude of its aberration corrected redshift and/or blueshift. Furtheremore, by subtracting the average of the redshift or blueshift far from any wall, we removed not only the chromatic aberration, but also any non zero polarity of the bulk. If corrected redshift or blushift contain polarity information, it is the polarity near the wall with respect to the bulk polarity, and not the absolute polarity. In the following we consider only the abberation-corrected redshift and blueshift and thus drop the adjective. §.§.§ Statistical correlation between colour shift and polarity First, we can compute the probability distribution function of the angle θ^red blue_i between redshift S^red_i and blueshift S^blue_i of particle i: P^red blue(θ) = ∑_i S^red_i S^blue_i δ(θ^red blue_i - θ)/∑_i S^red_i S^blue_i, where δ is the binning function, S^red_i = S^red_i and dimers are excluded from the sums. Results are shown on Fig. S<ref> (Left). Redshift and blushift are statistically antiparallel in bulk, but the peak is quite large. At wall, they are cleany split between two populations, either antiparallel (majority) or parallel (minority). Then, we compute the probability distribution function of the angle θ^red_i(t) between redshift at time t and the displacement Δ x(t) of the particle i between t and t+δ t, where δ t = 100 is the time interval between two successive frames: P^red(θ) = ∑_t∑_i S^red_i δ(θ^red_i - θ)/∑_t∑_i S^red_i, where sums exclude dimers and particles at the wall. We define P^blue(θ) in the same way. Both distributions are shown on Fig. S<ref> (Right). As P^red(θ) is flat, the redshift is not correlated to displacement and is thus not a measure of polarity. By contrast, P^blue(θ) shows a peak around π, meaning that blue shift is mostly antiparallel to displacement. Even far from the walls, particle displacement is an indirect measure of polarity, as particles experience forces other than self-propulsion, i.e. Brownian motion, gravity and particle-particle interactions of electrostatic and hydrodynamic nature. In addition, automatic trajectory reconstruction is subject to tracking errors, which further decrease the signal to noise ratio. Despite these limitations, blueshift is a good statistical predictor of particle displacement and polarity. §.§.§ Rationale of blueshift Why is blueshift a good measure of polarity and not redshift? The platinum cap is thin, thus the light reflected by the particles is mostly yellow, that is to say a mix of red and green with much less blue. The difference between red position and green position (redshift) thus contains little information. By contrast, platinum reflects more equally the colors, therefore the excess of blue with respect to green (or red) indicates the platinum side. But why are redshift and blueshift antiparallel? The filters on a colour camera slightly overlap. In particular the green channels also captures some red on one side and some blue on the otherside, whereas there is very little cross-talk between red and blue channels. With respect to the red position, the green position is thus slightly shifted towards the blue position. This shift, is enough to cause the statistical correlation between redshift and blueshift, but the signal is so small with respect to noise that the redshift is a poor predictor of polarity and even poorer predictor of the direction of motion. §.§ Wall interaction for Janus self-phoretic particles We discuss hereafter the form and magnitude of the wall–particle interaction used in numerical simulations. §.§.§ Diffusiophoretic wall attraction Gold-Platinum Janus particles have been shown to experience an activity-induced adhesive interaction of phoretic origin <cit.>. Indeed each catalytic particle act as a chemical monopole source from the far-field creating environmental gradients to which each swimmer responds. One thus expects that such attractive interaction is experimentally related to the phoretic mechanism responsible for the self-propulsion for which the associated swim force reads F_swim = k_BT_0/D_t v_0. Setting that the diffusiophoretic adhesion with the wall therefore involves an adhesion energy ϵ_phor.∼ F_swim R, we thus expect an adhesion of dimensionless form ϵ_phor. = 3/4Pe_s, where as in the main text ϵ_phor. = ϵ_phor./k_BT_0 and Pe_s = v_0/(RD_r). Overall, this justifies the form used in numerical simulation for the wall adhesion ϵ = αPe_s, where moreover the constant α=0.5 consistent with experiments is very close with the simple guess 3/4. §.§.§ Wall aligning interaction As stated in the main text, the aligning torque was taken based on the effect of hydrodynamic interactions between the wall and a pusher swimmer <cit.>. Indeed this was argued to be the dominant contribution for phoretic Janus particles <cit.> with an associated torque magnitude t reading for small deviations from wall alignment t = k_BT_0 (R/R+h)^3 (2Δθ) ×1/8Pe_s, As compared to the form adopted in simulations this amounts to a dimensionless torque magnitude of Γ_hydro. = 1/8Pe_s, again in fair agreement with the magnitude used to compare with experiments Γ∼0.5Pe_s.
http://arxiv.org/abs/2307.01543v1	20230704075418	Degradation-aware data-enabled predictive control of energy hubs	[ "Varsha Behrunani", "Marta Zagorowska", "Mathias Hudoba de Badyn", "Francesco Ricca", "Philipp Heer", "John Lygeros" ]	eess.SY	[ "eess.SY", "cs.SY" ]	^1Automatic Control Laboratory, ETH Zurich, Switzerland ^2Urban Energy Systems Laboratory, Swiss Federal Laboratories for Materials Science and Technology (Empa), Dübendorf, Switzerland [email protected] Mitigating the energy use in buildings, together with satisfaction of comfort requirements are the main objectives of efficient building control systems. Augmenting building energy systems with batteries can improve the energy use of a building, while posing the challenge of considering battery degradation during control operation. We demonstrate the performance of a data-enabled predictive control (DeePC) approach applied to a single multi-zone building and an energy hub comprising an electric heat pump and a battery. In a comparison with a standard rule-based controller, results demonstrate that the performance of DeePC is superior in terms of satisfaction of comfort constraints without increasing grid power consumption. Moreover, DeePC achieved two-fold decrease in battery degradation over one year, as compared to a rule-based controller. § INTRODUCTION In 2021, the energy use in buildings represented 45% of the total energy demand in Switzerland <cit.>. Mitigating the energy use in buildings, together with satisfaction of comfort requirements are the main objectives of efficient building control systems. It has been shown that improving energy storage in buildings by introducing batteries helps achieving these objectives <cit.>. However, the operation of a battery is affected by time, use or operating conditions and the battery may degrade. The goal of this work is to devise an efficient control system for buildings taking into account battery degradation. It is important to ensure efficient operation of the controller that can take into account the degradation of the battery over time. In this context, Model Predictive Control (MPC) has been shown to reduce energy usage while maintaining comfort and operational constraints <cit.>. However, first principles models of buildings, and in particular of the effects of degradation, are costly to develop and difficult to maintain. Hence, to efficiently mitigate energy consumption over the whole lifetime of buildings, it is crucial to minimize degradation of the battery without an extensive modelling effort. In this work, we overcome the difficulties of first principles modelling by extending existing data-based approaches to capture the behaviour of a building affected by battery degradation. Data-Enabled Predictive Control (DeePC) is used to investigate the performance of an energy hub comprising of a battery affected by degradation, and a heat pump. In contrast to classical MPC, DeePC computes an optimal control strategy for a linear time-invariant system using sufficiently rich input-output trajectories of the system. In this work, we extend the use of DeePC to long-term operation of building climate control considering nonlinear battery degradation. The paper is structured as follows. In Section <ref>, we provide the theoretical background on DeePC, which outlines the basis for the problem formulation in Section <ref>. Models for the building, battery, and heat pump dynamics are summarized in Section <ref>, and results from the simulations are discussed in Section <ref>. § PRELIMINARIES ON DEEPC Consider a discrete LTI system at time k ∈ℕ_0: x_k+1= A x_k+B u_k y_k= C x_k+D u_k where x_k ∈ℝ^n is the state of the system, u_k ∈ℝ^m is the input vector, and y_k ∈ℝ^p is the output vector. The system matrices are A ∈ℝ^n × n, B ∈ℝ^n × m, C ∈ℝ^p × n, D ∈ℝ^p × m. Let u_d = (u_d(1),...,u_d(T_d))∈ℝ^T_dm and y_d = (y_d(1),...,y_d(T_d))∈ℝ^T_dp denote the input and output trajectory of length T_d. Let L, T_d∈ℤ_≥ 0 and T_d≥ L. The input trajectory u_d∈ℝ^T_dm is called persistently exciting of order L if the Hankel matrix ℋ_L(u_d):=[[ u_1 u_2 ⋯ u_T_d-L+1; u_2 u_3 ⋯ u_T_d-L+2; ⋮ ⋮ ⋱ ⋮; u_L u_L+1 ⋯ u_T_d ]] is full rank. Following <cit.>, we have that T_d≥ (m+1)(L + n) - 1. DeePC uses Hankel matrices constructed from persistently-exciting inputs and corresponding outputs in lieu of a model of the form (<ref>) to find optimal trajectories of the system <cit.>. We consider Hankel matrices ℋ_T_ini+T_f(u_d) and ℋ_T_ini+T_f(y_d), and a partitioning thereof, ([ U_p; U_f ]):=ℋ_T_ini+T_f(u_d), ([ Y_p; Y_f ]):=ℋ_T_ini+T_f(y_d). The Fundamental Lemma presented by <cit.> states that if the system (<ref>) is controllable and u_d is persistently exciting of order k+n, then any sequence col(u_ini,y_ini,u,y) ∈ℝ^T_ini + T_f is a trajectory of the system, if and only if there exists g∈ℝ^T_d - T_ini - T_f - n + 1 such that [ U_p^T Y_p^T U_f^T Y_f^T ]^Tg = [ u_ini^T y_ini^T u^T y^T ]^T. § DEEPC FOR ENERGY MANAGEMENT In this section the data-driven optimization problem for the optimal operation of the energy hub and the building is presented. The main objective is to minimize the energy consumption from the grid while satisfying operational constraints. We consider an energy hub comprising a heat pump and a battery that is used to supply the thermal demand of a five zone building. The measured output y_e∈ℝ^7 includes the temperatures of all zones in the building, y_s∈ℝ^5, the output power of the heat pump, y_h, and the voltage of the battery, y_b. The input vector u_e∈ℝ^22 constitutes the control inputs u_s∈ℝ^9 to the building, the power input to the heat pump u_h, the battery current u_b, and the disturbances, v_s∈ℝ^11. The inputs u^i_s, i=1,…,5 describe the input to the radiators in each of the five zones, and u^i_s, i=6,…,9 describe the blinds openings available in four rooms. The disturbances v_s in are assumed to be known exactly from an accurate forecast. The DeepC controller computes the setpoints and we assume that low level controllers ensure that the setpoints are reached. The key advantage of DeePC is that we work directly with data, thus avoiding the need to model these low-level controllers. The resulting DeePC optimization for the optimal energy hub control over a prediction horizon T_f is formulated as: min_u_e, y_e,g ∑_k=0^T_f-1( β p^k + 1/2βc^k )^2 + λ_ρρ_2^2 +λ_gg_2^2 s.t. [ U_p^T Y_p^T U_f^T Y_f^T ]^T g = [ u_e,ini^T y_e,ini^T u_e^T y_e^T ]^T u^k_s,min ⩽ u^k,i_s⩽ u^k_s,max, i=1…,5 y^k_s,min - ρ ⩽ y^k_s⩽ y^k_s,max + ρ u^k_b, min ⩽ u^k_b⩽ u^k_b, max y^k_b, min ⩽ y^k_b⩽ y^k_b, max 0 ⩽ y^k_h y^k_h = C_h· u^k_h y^k_h = ∑_i=1^5 1/α_i u^k,i_s u^k_h = p^k + 0.066 · u^k_b where p^k is the energy imported from the electricity grid at time k and c^k is the price of energy imported from the grid. The cost function (<ref>) comprises of the linear cost of the electricity over the prediction horizon T_f, in addition to the quadratic penalization on the power coming from the grid to improve numerical properties of the optimization problem with β=0.01. The cost also includes a regularization on the norm of g with the parameter λ_g=1000 to avoid over-fitting and improve robustness <cit.>. The slack ρ on the comfort constraints is also penalised quadratically by the parameter λ_ρ=10. The DeePC control strategy is incorporated in the constraint (<ref>) in order to optimize the room temperatures in the buildings. The inequality constraints (<ref>) limit the radiator and blind inputs of the building, and (<ref>) is the comfort constraint that ensures that the temperatures of the five zones remains within the time-dependent maximum and minimum temperatures, y^k_s,min and y^k_s,max. The temperature is between 10C and 40C between 23:00 and 5:00 when the building is unoccupied, and between 21C and 25C during regular hours. A slack variable ρ on the comfort constraints ensures that the problem remains feasible for all disturbances v_s^k. The constraints (<ref>) and (<ref>) limit the current u_b^k∈[-22,22] A and voltage y_b^k∈[63,68] V of the battery so that the battery is charged or discharged at a maximum C-rate of C/4 based on the capacity of the battery, and that the battery voltage operates in the nominal region. The static model of the heat pump is incorporated using (<ref>) and (<ref>) with the coefficient of performance C_h=3, and (<ref>) relates the output power from the heat pump to the heating power of the five radiators, where α_i are the coefficients corresponding to conversion factors with α_i=11.9 for i=1,2,3, α_4=27.77, α_5=7.58. Finally, (<ref>) is energy balance equation for the electricity in the hub, i.e. the power coming from the grid and the battery must be equal to the power going into the heat pump. Since both the voltage and current of the battery are decision variables, it would result in a bilinear equation that is difficult to solve. As a result, the battery voltage of 66V is used as an operating point in order to linearize this constraint. § ENERGY HUB MODELLING The performance of DeePC is tested on a simulated building and energy hub model, created using the energy hub component modelling (EHCM) toolbox <cit.> in . §.§ Building In this work, we use an office building with five rooms (zones). The building is modelled using the Building Resistance Capacitance Modeling (BRCM) Toolbox which describes the building’s thermal dynamics as a continuous time system, bilinear in the inputs: ẋ_s(t)= A_cx_s(t)+B_uu_s(t)+B_vv_s(t)+∑_i=1^9B_vu,iv_s(t)u_s(t) y_s(t)= C_cx_s(t) where the states of the systems, x_s∈ℝ^113, include the temperatures of each room and the temperatures of the layers of the building elements i.e. floors, roof, and inner/outer walls that connect the zones of the building. The input vector u_s∈ℝ^9 constitutes the control inputs, including the heating power of the radiators installed in the five zones (^2), and inputs for the position of the four blinds on each facade of the building. The disturbance vector v^k_s∈ℝ^11 comprises of the internal gains in the five rooms (^2), ambient temperature (C), ground temperature (C), and the global solar radiation on the four sides of the building (^2). The model thus considers external heat fluxes going into or coming out of the building including internal gains due to occupancy, lights and equipment, heating power from the radiators, and disturbances from ambient and ground temperature and heat gains by global solar radiation. A detailed description of the model and the matrices A_c, B_u, B_v, B_vu, C_c can be found in <cit.>. §.§ Battery A lithium-ion battery is considered for the energy hub modelled using a nonlinear Shepherd’s model, which describes how the terminal battery voltage changes with the input current <cit.>. The output of the model, y^k_b, is the battery terminal voltage computed as y^k_b= V_OC-R_0 · u^k_b, where u^k_b is the battery current [A], V_OC is the open circuit voltage [V], and R_0 is the internal resistance of the battery [Ω]. The internal resistance of the battery is affected by degradation. Battery degradation is modelled as ageing defined as the number of full cycles, i.e. the number of times the State-of-Charge (SoC) goes from zero to 100% and back to zero. The battery with parameters 12.8 V 40 Ah is implemented using the battery block in that includes the effects of cycling. §.§ Heat pump Heat pump uses electricity to generated heat to satisfy the building heating demand. The EHCM toolbox uses a static model of the heat pump where the input and output at each time step related through the Coefficient of Performance (COP) as given in (<ref>). § RESULTS The controller from Section <ref> has been implemented in the case study from Section <ref>. All simulations were performed in / 2022a with <cit.> and <cit.>. The total simulation horizon was chosen as one year, T_all=8760 h. The controller was implemented in a receding horizon fashion, with prediction horizon of T_f=24 h. The parameter T_ini=30 h was chosen to minimise the prediction error between Y_p and the true output over the prediction horizon T_f, at a fixed sampling time of T_s=1 h. The code is available in <cit.> §.§ Data collection The first step in solving the problem (<ref>) consists in collecting input and output data for the Hankel matrices in (<ref>). Measurements from the battery, the building and the heat pump were taken over T_d=4416 hours (184 days). The ageing effects on the battery were not taken into account in the data collection phase. For data collection, we used rule-based controllers (RBC) for the radiators and the blinds to ensure that the temperatures of the five zones stay within time-varying bounds: u_s^k,i=δ^k,i_s+ u^k_s,max if y_s^k≤ y^k_s,min u^k_s,min if y_s^k≥ y^k_s,max where δ^k,i_s is an auxiliary input disturbance, chosen as a pseudo-random binary signal (PRBS) with amplitude of 5kW, necessary to ensure the condition on persistence of excitation. The battery controller is based on State-of-Charge (SoC) and the time of day. From midnight to 4 a.m. we charge the battery with a 15A current up until its SoC reaches 90%. Then from 5 a.m. up until 23 p.m., the battery gets discharged. Then, when the SoC reaches 20%, we wait for the next charge during the night. To ensure the persistence of excitation, the applied current is also perturbed with a PRBS signal with amplitude of 15A. The rule-based controllers were then used to find u_ini and y_ini. §.§ Prediction The choice of parameters was validated by evaluating the absolute error between the predicted outputs y_s obtained from solving (<ref>) and the response of the building from simulation y_s, sim at a given prediction hour, k=1,…,T_f. Let j=1,…,5 correspond to the room number, then the average error for each room for each prediction hour is given by ϵ^k_j=∑_i=1^T_all\|y^k_s,i,j-y^k_s, sim,i,j\|/T_all, where T_all=8760 h is the complete simulation time of one year. Figure <ref>(a) shows the error for the five rooms as a function of the prediction time. For all the rooms the average error is below 0.5 ^∘C which is considered acceptable <cit.>. Moreover, the prediction error in the battery voltage remains below 0.5 V on average (Figure <ref>)(a) which is below 1 %. Figure <ref>(b) shows the output room temperatures and battery voltage obtained using DeepC and RBC over a selected day in January (24 h) and it shows how DeepC results in better temperature and voltage regulation. §.§ Long-term operation Figure <ref>(c) shows the performance of the battery over the entire year. The oscillatory nature of the battery current in RBC led to intensification of battery ageing process quantified as the number of cycles (top) and loss of capacity (bottom). Even though the objective function from (<ref>) was focused only on the cost of operation of the energy hub, by optimising the input current to the battery and reducing its oscillatory nature, DeePC led to two times smaller age of the battery as the number of cycles, compared to RBC. The capacity loss was also reduced (0.3% in DeePC compared to 0.8% in RBC). Furthermore, DeePC enabled enforcing constraints on the battery voltage. Conversely, RBC adjusts only the battery current, which led to oscillatory behaviour contributing to the intensified ageing of the battery. Quantitative results for constraint violation are collected in Table <ref>. RBC violates the comfort constraints on average up to 5.5% of the year whereas DeePC violates the constraints up to 2.8% of the time and has smaller violations. At the same time, the cost is comparable in both RBC and DeePC, with DeePC cost being 0.9% lower. § CONCLUSIONS In this work, we have investigated the performance of data-enabled predictive control for building energy management through a simulation that incorporates degradation processes that affect battery behaviour. A simulation setup with a single building and an energy hub comprising an electric heat pump and a battery was considered. A comparison between DeePC and the RBC showed that the battery ageing was reduced by over a factor of two under DeePC operation, as well as a reduction of constraint violations. The impact of the simplified model of battery degradation (no calendar ageing, no self-discharge, no influence of varying external conditions) requires further investigation, ideally in an experimental setup. Future studies also aim to investigate the performance of DeePC as compared to more realistic rule-based controllers and on larger energy hubs with uncertain PV generation and the influence of the battery, as well as to extend the proposed approach to multiple buildings in a district. § ACKNOWLEDGMENTS Research supported by NCCR Automation, a National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant no. 180545), and by the European Research Council (ERC) under the H2020 Advanced Grant no. 787845 (OCAL). § REFERENCES
http://arxiv.org/abs/2307.03362v1	20230707030534	Adaptation and Communication in Human-Robot Teaming to Handle Discrepancies in Agents' Beliefs about Plans	[ "Yuening Zhang", "Brian C. Williams" ]	cs.AI	[ "cs.AI", "cs.RO" ]	Federated Unlearning via Active Forgetting Jiaming Zhang Abstract 0.9 We review the modular flavor symmetric models of quarks and leptons focusing on our works. We present some flavor models of quarks and leptons by using finite modular groups and discuss the phenomenological implications. The modular flavor symmetry gives interesting phenomena at the fixed point of modulus. As a representative, we show the successful texture structure at the fixed point τ = ω. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators. =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== When agents collaborate on a task, it is important that they have some shared mental model of the task routines – the set of feasible plans towards achieving the goals. However, in reality, situations often arise that such a shared mental model cannot be guaranteed, such as in ad-hoc teams where agents may follow different conventions or when contingent constraints arise that only some agents are aware of. Previous work on human-robot teaming has assumed that the team has a set of shared routines, which breaks down in these situations. In this work, we leverage epistemic logic to enable agents to understand the discrepancy in each other's beliefs about feasible plans and dynamically plan their actions to adapt or communicate to resolve the discrepancy. We propose a formalism that extends conditional doxastic logic to describe knowledge bases in order to explicitly represent agents' nested beliefs on the feasible plans and state of execution. We provide an online execution algorithm based on Monte Carlo Tree Search for the agent to plan its action, including communication actions to explain the feasibility of plans, announce intent, and ask questions. Finally, we evaluate the success rate and scalability of the algorithm and show that our agent is better equipped to work in teams without the guarantee of a shared mental model. § INTRODUCTION When agents collaborate on a task, it is important that they have some shared mental model of the task routines – the set of feasible plans towards achieving the goals. However, in reality, situations often arise that such a shared mental model cannot be guaranteed. For example, in online multi-player games or search-and-rescue missions, people trained separately could form an ad-hoc team where they may follow different conventions. Even if the team has a set of shared routines, novel situations may still occur in which some contingent constraint that forbids certain plans to be taken becomes known only by some agents. In these situations, experienced teammates keep in mind what others know and what actions they may take, and communicate when necessary to make sure the team converges on a feasible plan of action. Previous work on human-robot teaming, Pike <cit.>, assumed that agents share common knowledge of the feasible plans for the task encoded in a knowledge base. Under an equal partner setting, each agent observes the actions taken by others and adapts their actions accordingly, only taking what is still feasible. This approach allows fluid human-robot interaction but breaks down when the common knowledge assumption no longer holds. In this work, we generalize the approach to handle situations where there may be discrepancies in agents' beliefs about plans by incorporating epistemic logic <cit.>, as it provides an explicit representation of agents' nested beliefs towards each other and a mechanism to model communication between agents. The contribution of this paper is threefold: (1) We propose the formalism of conditional doxastic logic <cit.> extended to knowledge bases in order to represent agents' nested beliefs on the set of feasible plans for the task and the state of execution. (2) We model both execution and a rich set of communication actions within the framework, including explanation, intent announcement, and question-asking actions, that allows agents to explicitly talk about the feasibility of plans and exchange their intent. (3) We provide an online execution algorithm based on Monte Carlo Tree Search (MCTS) for the agent to dynamically plan its action to adapt to others or communicate to resolve the discrepancy. Finally, we evaluate the success rate and performance of the algorithm through experiments. § MOTIVATING EXAMPLE Consider a pedagogical example where a robot (our agent) and a human collaborate to prepare a drink. The robot has a manipulator arm that can fetch a mug or a glass as the container, and the human can brew some coffee or take some orange juice from the fridge for the drink. For the task to succeed, it must satisfy that: (C1) the mug has to go with the coffee and the glass has to go with the orange juice. Under an equal partner setting, from the robot's perspective: Case1 If the human doesn't believe constraint C1 holds and thinks that any container can go with any drink, then the robot can adapt to the human by waiting for the human to take the drink first, then fetch the corresponding container. The robot can also explain to the human about constraint C1, especially if the task requires the robot to fetch the container first. The robot can also announce the intent for the human to choose coffee, in which case it can just fetch the mug. Case2 If the human has determined a choice of coffee or juice, but the robot doesn't know which one, the robot may wait for the human to pick first so that it can distinguish their intent, or it can ask the human about their intent. Case3 If the human picked up the juice but doesn't know that the robot couldn't reach the glass, the robot may explain the constraint and that the task has failed. § BACKGROUND In order to represent the agents' nested beliefs, our representation builds on top of conditional doxastic logic <cit.>, which is one variant in the broader epistemic logic literature. Compared to epistemic logic, it allows the modeling of false beliefs and belief revision by pre-encoding the conditional belief of the agents within the model. Given a set of agents Ag and a set of atomic propositions At, conditional doxastic logic ℒ(At, Ag) is defined by the following Backus-Naur Form (BNF): φ := p \|φ \| (φφ) \|B^φ_a φ, where p∈ At, a ∈ Ag. B^ψ_a φ reads as “agent a believes φ given ψ”. Denoting ⊤ as tautology, B_aφ := B^⊤_aφ means that agent a believes φ. Its semantic model is a plausibility model, which is a tuple M=⟨ W, {≤_a}_a∈ Ag, L⟩, where * W: a non-empty set of possible worlds, * ≤_a ⊆ W × W: binary relation on W imposing a relative plausibility order between any two worlds for agent a, * L: W → 2^At: valuation function mapping each world to the set of atomic propositions that hold in the world. w ≤_a v means that agent a considers w to be at least as plausible as v. <_a := ≤_a ∩≱_a denotes a strict plausibility order. ≃_a := ≤_a ∩≥_a denotes an equi-plausibility order. ∼_a := ≤_a ∪≥_a denotes epistemic indistinguishability, and cc_a(w) := {v ∈ W \| w ∼_a^* v} is the set of worlds that agent a finds (possibily more or less) plausible given world w, where ∼_a^* is the transitive closure of ∼_a. Note that the plausibility relation is reflexive, transitive, locally connected, that is, v ∈ cc_a(w) implies v ≤_a w or w ≤_a v, and well-founded, that is, min_a(S) := {w ∈ S \| ∀ v ∈ S: v ≮_a w} is well-defined, which is the subset of worlds in S that agent a finds most plausible. A pair (M, w) is a pointed plausibility model, which describes a conditional doxastic state with a pointed view at world w ∈ W, i.e. taking w as the true world. The truth of a formula φ∈ℒ(At, Ag) on (M, w), i.e. (M, w) φ, can be defined inductively as follows: * (M, w) p iff p ∈ L(w) * (M, w) φ iff M, w ⊭φ * (M, w) φψ iff (M, w) φ and (M, w) ψ * (M, w) B^ψ_a φ iff min_a([ψ]_M ∩ cc_a(w)) ⊆ [φ]_M, where [φ]_M := {w ∈ W \| M, w φ} is the set of worlds in M in which φ holds. Figure <ref> shows an example state represented by a pointed plausibility model (M, w_1) with agents a and b. The two worlds w_1 and w_2 are labeled with the atomic propositions that hold in the respective worlds. The pointed world w_1 highlighted in bold represents the true world in which p holds. The single arrow pointing from w_1 to w_2 labeled with b indicates that agent b considers w_2 to be strictly more plausible than w_1. We say (M, w_1) B_b p since min_b(cc_b(w_1)) = {w_2}⊆ [ p]_M. If it is instead a double-headed arrow, then it means that agent b considers w_1 and w_2 to be equally plausible. The lack of any arrow between w_1 and w_2 for agent a indicates that w_1 ≁_a w_2, that is, when in w_1 or w_2, agent a does not consider the other world plausible at all. Since the plausibility relation is reflexive, the self-loops indicate that whichever world it is, the agents find the world plausible. (M, w_1) B_a p since min_a(cc_a(w_1)) = {w_1}⊆ [p]_M, and (M, w_1) B_a B_b p since min_a(cc_a(w_1)) = {w_1}⊆ [B_b p ]_M = {w_1, w_2}. An action is defined by a plausibility action model A = ⟨Σ, {≤_a}_a∈ Ag , pre, post⟩, which has a similar structure except instead of a set of worlds W, it has a set of events Σ representing possible events that may occur in the action. pre and post are functions that assign to each event σ∈Σ a precondition and a postcondition in ℒ(At, Ag) respectively, where the postcondition of an event is restricted to a conjunction of literals over At or ⊤. A pointed plausibility action model is a pair (A, σ), σ∈Σ, which describes an action where σ is the true event. In general, a pointed plausibility model for state or action can point at multiple worlds, such as (M, W_d) or (A, Σ_d). W_d and Σ_d are called the designated worlds or events. For example, given a state (M, w), (M, W_d) with W_d = min_a(cc_a(w)) represents agent a's local perspective of the state, where W_d includes all the worlds that agent a finds the most plausible. (M, W_d) is a global state if \|W_d\| = 1. Additionally, (M, W_d) φ iff (M, w) φ for all w ∈ W_d. An action act updates a state s through action-priority update s ⊗ act, which we refer the readers to the details in <cit.>. § APPROACH OVERVIEW Our solution requires answering three questions: (1) what representation to use to capture the agents' nested beliefs of the set of feasible plans and the state of execution, (2) how to model execution and communication actions and how they update the state, and (3) how to strategically choose the next action. Our key insight is to extend conditional doxastic logic to describe knowledge bases, and use the knowledge bases to encode the feasible plans and state of execution, so that we can describe agents' beliefs on the plan space instead of their beliefs on state. As a result of this new logic, execution and communication actions can be defined which operate by adding or removing constraints from the knowledge bases. With the state and action models defined, we use an MCTS-based algorithm to simulate forward in the next k-step horizon to decide what is the best action to take. For example, Figure <ref> captures the agents' nested beliefs on plans from Case1. Each world in the plausibility model is now a knowledge base that contains the constraints of the task. Since H (human) finds w_2 more plausible, H believes that constraint C1 does not need to hold. An example action where the robot announces the intent of coffee is shown in Figure <ref> (left). The action has a single event whose precondition is that R (robot) must believe that adding the constraint of coffee is satisfiable given its belief of the current feasible plans. As a result of the action, all worlds now have the constraint of coffee added, including w_2 that H believes in. § REPRESENTING TEAM'S NESTED BELIEFS ON PLANS We describe our representation in two parts: (1) conditional doxastic logic on knowledge bases and its semantics, (2) our task representation and its encoding in the knowledge base. §.§ Conditional Doxastic Logic on knowledge Bases Given a finite set of atomic propositions At, and a finite set of agents Ag, conditional doxastic logic on knowledge bases ℒ_KB(At, Ag) is defined by the BNF: φ := in(c)\| entailed(c) \|φ \| (φφ) \|B^φ_a φ, in which a ∈ Ag, c ∈𝒞(At), where 𝒞(At) is the classical propositional logic c := p \| c \| (c c), p ∈ At. Note that the formulation naturally extends to constraint systems with finite-domain variables, which is what we use. We hence refer to c as a constraint. in(c) means constraint c is an explicit member of the constraints in the knowledge base, entail(c) means constraint c is entailed by the knowledge base, and we define sat(c) := entailed( c), which means constraint c is satisfiable by the knowledge base. The plausibility model for ℒ_KB(At, Ag) is a tuple M=⟨ W, {≤_a}_a∈ Ag , KB⟩, where W and ≤_a are the same as before and KB: W →𝐊𝐁_𝒞(At) is a function that maps each world to an associated knowledge base in 𝒞(At). When determining the truth of a formula φ∈ℒ_KB(At, Ag) on a pointed plausibility model (M, w), we replace the first rule on (M, w) p in the inductive rules with the following: * (M, w) in(c) iff c ∈ KB(w) * (M, w) entailed(c) iff KB(w) c We can say the following about the state in Figure <ref>: * B_R in((𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎)) (𝚐𝚕𝚊𝚜𝚜𝚓𝚞𝚒𝚌𝚎)) * B_R B_H in((𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎)) (𝚐𝚕𝚊𝚜𝚜𝚓𝚞𝚒𝚌𝚎)) * B_R entailed(𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎) * B_R sat(𝚖𝚞𝚐𝚓𝚞𝚒𝚌𝚎) B_R B_H sat(𝚖𝚞𝚐𝚓𝚞𝚒𝚌𝚎) §.§ Task Representation & Encoding The set of feasible plans towards achieving the goals of the task forms a plan library for the task. Additionally, actions may be ordered in the plan, such as requiring the container to be picked up first before the drink. Therefore, our task representation is a temporal plan library ⟨ V, E, O, C⟩, where: * V is a set of decision variables with domain(v), v∈ V. * E is a set of time points with guard condition guard(e) for each e ∈ E, a conjunction of decision variable assignments. e should be executed iff guard(e) is satisfied. * O is a set of ordering constraints o = ⟨ e_i, e_j, guard(o)⟩, o ∈ O, requiring time point e_i to precede time point e_j in execution order if its guard condition guard(o) is satisfied. We assume guard(o) guard(e_i) guard(e_j). * C is a set of constraints scoped on V. The time points represent the actual events of taking the actions. In multi-agent case, a multi-agent temporal plan library ⟨ V, E, O, C, Ag, f⟩ additionally has a set of agents Ag and a function f: E → Ag that maps each time point to an agent that it belongs to. In our formulation, the decision variables do not have ownership. This reflects our equal partner setting in which decisions do not belong to any agent and an announced intent can affect multiple agents' actions. The plan library represents a set of candidate subplans G, where a subplan g ∈ G is a full assignment to all the decision variables V. We use E_g and O_g to denote the set of time points and ordering constraints activated by g, i.e. those whose guard conditions are satisfied. A subplan induces a set of total orderings on E_g that satisfies O_g, which we denote by T_g. A subplan g is feasible iff all the constraints C are satisfied, i.e. ∀ c ∈ C, g c, and there exists a total ordering of E_g that satisfies O_g, i.e. T_g ≠∅. Execution As execution progresses, decision variables are gradually grounded either implicitly from the execution of time points or explicitly from announcement of intent. The execution state is a tuple ⟨ t, C_I ⟩, where t is an execution history, which is a total ordering of time points (e_i, e_j, ..., e_k) that have been executed, and C_I is the set of intents that have been announced during execution. An intent, in its most general form, can be an arbitrary constraint scoped on V, but is commonly an assignment to a specific decision variable. The subplans that are feasible with respect to ⟨ t, C_I ⟩ include any feasible subplan g such that there exists t_g ∈ T_g, where t is the prefix of t_g, and g satisfies C_I. Execution fails when there exists no feasible subplan with respect to ⟨ t, C_I ⟩. Execution succeeds when there exists a feasible subplan g with respect to ⟨ t, C_I ⟩ such that t ∈ T_g. Note that execution can succeed without ever converging to a unique subplan, and it is possible for further time points to be executed and move away from the success state. Encoding in Knowledge Base We encode the plan library and the execution state in the knowledge base, so that at any point during execution, the knowledge base contains all the feasible subplans with respect to ⟨ t, C_I ⟩. We ensure that the knowledge base is consistent iff execution has not failed. The variables of the encoding include (1) a discrete variable for each decision variable v_i ∈ V with the same domain domain(v_i), and (2) a boolean variable for each time point e_i ∈ E with domain {𝚃, 𝙵}, representing if the time point is executed. We add the following constraints to the knowledge base prior to execution: * The constraints C as defined in the plan library. * For each time point e_i, ((e_i = 𝚃) → guard(e_i)), i.e. if time point e_i is executed, its guard condition must hold. * Negation of nogoods <cit.> that represent any combination of choices of V that would result in an inconsistent ordering of time points. This can be computed from the ordering constraints O. During execution, we may additionally add to the KB: * Announced intents C_I. * For each execution of time point e_j, a conjunction of (1) assignment of variable e_j to 𝚃, and (2) the negation of the guard condition guard(o) for any ordering constraint o = ⟨ e_i, e_j, guard(o) ⟩ in which the predecessor e_i has not been executed by the time e_j is executed. The last rule ensures that for any ordering constraint o = ⟨ e_i, e_j, guard(o) ⟩, if the guard condition holds, then if e_j is executed, e_i must also have been executed, hence satisfying the ordering constraint. Note that we only encode the set of time points that have been executed, instead of their actual order of execution. With the above encoding, given a knowledge base KB, execution fails iff KB . Execution succeeds iff there exists subplan g, i.e. a full assignment of V, such that KBg ⊭ and ∀ e_i ∈ E_g, KB (e_i = 𝚃). We denote the success condition by suc_(V, E), and say that execution succeeds iff KB suc_(V, E). Additionally, (M, w) suc_(V, E) iff KB(w) suc_(V, E). Take Case 1 as an example, the knowledge base contains discrete variables container with domain {𝚖𝚞𝚐, 𝚐𝚕𝚊𝚜𝚜} and drink with domain {𝚌𝚘𝚏𝚏𝚎𝚎, 𝚓𝚞𝚒𝚌𝚎}, and boolean variables e_mug, e_glass, e_coffee, e_juice representing the events of picking up each item. Using 𝚖𝚞𝚐 as a shorthand for (container=𝚖𝚞𝚐) and similarly for others for the purpose of decluttering, the constraints include: * (𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎) (𝚐𝚕𝚊𝚜𝚜𝚓𝚞𝚒𝚌𝚎) * (e_mug=𝚃) →𝚖𝚞𝚐, similarly for other time points Note that we use the same shorthands throughout the rest of the paper. In this example, when the robot picks up the mug, constraint (e_mug = 𝚃) is added to the knowledge base. From 2 above, we now have KB 𝚖𝚞𝚐, and consequently from 1, we have KB 𝚌𝚘𝚏𝚏𝚎𝚎, which limits the human's choice of drink to coffee. Picking up juice is no longer feasible since KB (e_juice = 𝚃). Consider another case where the robot's action must precede the human's action, i.e. there are ordering constraints ⟨ e_mug, e_coffee, 𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎⟩, ⟨ e_glass, e_coffee, 𝚐𝚕𝚊𝚜𝚜𝚌𝚘𝚏𝚏𝚎𝚎⟩, etc. If the human picks up the coffee before the robot takes any action, then (e_coffee = 𝚃) ( 𝚖𝚞𝚐𝚌𝚘𝚏𝚏𝚎𝚎) ( 𝚐𝚕𝚊𝚜𝚜𝚌𝚘𝚏𝚏𝚎𝚎) is added, resulting in an inconsistent knowledge base. Need to prove Theorem 1, using Lemma on comment 2/3 Basically, KB includes the candidate subplans that are still feasible wrt the partial execution. Why? Intuitively, C is satisfied as explained, Ordering is satisfied as explained. partial execution <G', T'> is enforced by the event guard constraints. Execution hard failure, which is failure that cannot be fixed, can be determined by checking that KB. § DYNAMIC MODEL OF EVOLUTION In this section, we describe how the model evolves as a result of execution or communication actions. We first introduce the plausibility action model for our extended logic, then describe how to model each type of action. In this work, we assume that agents observe all actions that are taken, that is, all actions are public. §.§ Plausibility Action Model for Knowledge Bases A plausibility action model A for ℒ_KB(At, Ag) is a tuple ⟨Σ, {≤_a}_a∈ Ag , pre, post⟩, where Σ and ≤_a are the same, and pre and post are functions that map each event to a formula in ℒ_KB(At, Ag). The postcondition is restricted to a conjunction of in(c), which adds constraint c to the knowledge base, and in(c), which removes constraint c from the knowledge base if it exists, as well as ⊤, i.e. nothing changes. For this paper, we further restrict the postcondition to be either in(c) or ⊤, i.e. adding at most one constraint to the knowledge base. The action-priority update updates the knowledge bases as described accordingly. Execution Action An execution action is the action of an agent executing a time point, such as robot picking up mug. Recall that in our setting, each time point is assigned to an agent who can execute it. Given that the time point being executed is e_i ∈ E, and the agent who executes it is a = f(e_i), the simplest case of execution of time point e_i that has no potential predecessors is shown in Figure <ref> (left). We assume agents are rational and for agent a to execute time point e_i, it needs to believe that executing e_i is feasible, i.e. B_a sat(e_i = 𝚃). All agents observing the action also observe the truth of agent a having such belief. For the postcondition, as e_i is executed, the constraint (e_i =𝚃) is added to the knowledge base. When there are potential predecessors for e_i, we need to make sure the corresponding ordering constraints are satisfied. The postcondition of the event should always add (e_i = 𝚃) to the knowledge base, and for any ordering constraint o = ⟨ e_j, e_i, guard(o) ⟩, add ( guard(o)) on condition that entailed( e_j = 𝚃), that is, if e_j has not been executed. While a more succinct action model specification is possible <cit.>, we use the standard form defined above by taking the cross product of all the predecessors, and creating equi-plausible events for them as shown in Figure <ref>. Even though the size of the action model is exponential to the number of potential predecessors each time point has, because the preconditions of these events are mutually exclusive, the model size for the updated state will not increase as a result of the action update. Intent Announcement The model for an intent announcement action is shown in Figure <ref> (middle). For agent a to announce the intent, it must believe that it is satisfiable, hence the precondition B_a sat (c). The intent is added as a postcondition. Figure <ref> shows an example intent announcement. Explanation The model for an explanation action where agent a explains its belief of φ∈ℒ_KB(At, Ag) is shown in Figure <ref> (right). The precondition says that agent a has to believe φ, i.e. agents cannot lie about their belief. To the other agents, the explanation is essentially a public announcement that agent a believes φ. This means that whether a particular agent adopts the explainer's belief depends on the conditional belief pre-encoded in the initial pointed plausibility model, which specifies how an agent's belief gets revised when a new piece of evidence is received. An example explanation action where the robot explains constraint C1 is shown in Figure <ref>. Based on the pointed plausibility model in Figure <ref>, upon the announcement that the robot in fact believes C1 holds, w_2 is eliminated as it does not satisfy the precondition, and the human is left with w_1 in which C1 holds. Depending on the initial conditional belief, it is also possible to have situations where the human does not trust the robot and does not adopt its belief. In this paper, we restrict the explained formula to be of the BNF form φ := φ \|B_a φ \| in(c), where a ∈ Ag, c ∈𝒞(At). This simplifies the explanations in that (1) the explained formula cannot be arbitrarily complex such as B_a in(c) → B_b in(c), (2) the explanation must be about whether the knowledge base contains a constraint or not, instead of the satisfiability or entailment of an arbitrary constraint. This is similar in spirit to the idea of abductive explanations, where we want to give an explanation c such that together with the existing theory T, it explains an explanandum O, i.e. T ∪{c} O. In this case, what is satisfiable or entailed is often the explanandum, and what constraints should or should not be in the theory is what we explain. Question-Asking An agent can ask another agent about something that it is uncertain of. Since we assume public actions, the answer is observed by all agents. Given that agent a is asked about its belief on formula φ∈ℒ_KB(At, Ag), the pointed plausibility action model is shown in Figure <ref>. We place the same restriction on φ as in the explanation actions. Using Case 2 as an example, the robot does not know which choice of drink the human has determined on, which is represented by the pointed plausibility model in Figure <ref>. The robot can ask a question about the human's belief on in(𝚌𝚘𝚏𝚏𝚎𝚎), i.e. whether its intent is to take coffee. The resulting state would have the double-headed arrow in the middle labeled with R removed, i.e. the robot will be able to distinguish human's intent. § ONLINE EXECUTION PROBLEM We assume that execution is asynchronous, all actions are public, and communication has a cost. We also assume the discrepancies in beliefs come only from the agents' initial beliefs on constraints C, i.e. they share the belief on the rest of the plan library such as the ordering constraints and the guard for the time points. In this paper, we assume that a task involves two agents (e.g. robot and human), though there is no theoretical barrier to applying it to more agents. The online execution problem from a single agent's perspective, say agent a, involves taking the input of the following prior to execution: * A multi-agent temporal plan library ⟨ V, E, O, {}, Ag, f⟩. * A pointed plausibility model s_0 = (M, W_d) capturing agents' initial nested beliefs on constraints C from agent a's perspective, such that W_d = cc_a(w), ∀ w ∈ W_d. Note that constraints C is empty in the plan library as it is captured by the input of s_0. W_d includes any world that agent a finds plausible (not necessarily most plausible), and we assume that (M, W_d) captures the ground truth state (M, w^*) as one of its possibilities, so that the agent's belief can also be revised if needed. During execution, the agent receives the input of a stream of actions that are taken by itself or others in real-time, including execution and communication actions. Each action triggers a callback and the agent outputs an action to be taken or none. Overall Algorithm Upon receiving an action, our agent determines how it should act next – either take an execution action, communicate, or wait for others to act. It simulates forward to predict the utility of each possible action, e.g. if others may follow up with incorrect actions, or if many communication actions will be needed. The algorithm draws insight from epistemic planning for implicit coordination <cit.> and relies on the agent's ability to take others' perspectives to predict their actions. The overall algorithm is illustrated in Algorithm <ref>, which we name it Epistemic Pike (EPike) after Pike <cit.>. Prior to execution, the agent compiles the initial state s from s_0 and the plan library as described in the knowledge base encoding section (line 1). Upon receiving an action act', the updated state (line 3) is checked for several conditions. If the agent believes that execution has failed (line 4), then it explains the failure when some agent might not know (line 5 - 6). If the agent is unsure about whether execution has failed (line 7), then it sees if it can ask someone to distinguish it (line 8). If both conditions do not apply, then execution has not failed, and the agent checks if execution has succeeded (line 9). If so, then the agent explains it when some agent might not know (line 10 - 11). If execution has not succeeded, then the agent searches for the next action to take, if any, to progress toward completing the task (line 13). Note that when a question-asking action is taken, we wait until the answer is announced before encoding the answer as an explanation action that gets observed. Each online subroutine in Algorithm <ref> calls an MCTS algorithm with a different configuration. The MCTS algorithm simulates the team's possible execution in the next k-step horizon, and based on the result of the simulations, the agent decides if it should take an action now and which action to take. Our MCTS algorithm can be configured on (1) the termination conditions, including the horizon k, (2) which types of actions to consider for both the ego agent and the other agents, and (3) the penalties for communication actions, since we assume communication has a cost. We add a fifth type of action, noop, to represent agent taking no action and waiting for others to act. For SearchAction, a node terminates if its state s satisfies either s entailed(), which gives a utility of 0 (execution fails), or s suc_(V, E), which gives a utility of 1 (execution succeeds), or if simulation reaches a horizon of k, which gives a utility of 1 (execution has not failed). Note that only execution actions increment the horizon, since we care about the outcome after the next three physical actions. During search, we consider all five types of actions (including noop) from all agents, except for the intent announcement and question-asking actions from the other agents. They can be reasonably omitted to reduce the search space, since they may be unpredictable and ignoring it does not prevent the simulated execution to reach success state. For the rest of the subroutines, a node terminates if its state s satisfies s_i ∈ Ag B_i entailed() for ExplainFailure, s B_a entailed() B_a entailed() for AskIfFailure, and s_i ∈ Ag B_i suc_(V, E) for ExplainSuccess, all giving a utility of 1. For ExplainFailure and ExplainSuccess, only explanation and question-asking actions of the ego agent are considered. Asking a question may still be useful if the agent is uncertain about what others currently believe. For AskIfFailure, only question-asking actions for the ego agent are considered. In these cases, since the ego agent is just looking to inform others or ask a question, it is reasonable to ignore what other agents may do. To penalize communication, we set a penalty factor of 0.9 for explanation actions and question-asking actions, and 0.85 for intent announcement actions, though the values may change depending on applications. Penalty is a multiplication factor to the utility of the node. Execution actions and noop action are not penalized. Search Tree We describe the expansion of the search tree using SearchAction as an example, before discussing the details of MCTS. A partially expanded search tree is shown in Figure <ref>. There are four types of nodes in the tree: root decision node (bold circle), split nodes (diamonds), predict nodes (squares), and decision nodes (circles). Each node has its state s_i and has a utility score of between [0, 1]. The root decision node is only used as the root of the tree and finds the best action to take for the ego agent (including noop). Given input of s = (M, W_d) from the subroutine, the state of the root decision node is the ego agent's current belief of the state s_ego=(M, min_a(W_d)). The node branches on all the possible actions the ego agent can take based on its current belief, creating children split nodes. We discuss the generation of possible actions in the Appendix. If there exist children with positive scores, the agent chooses the action that leads to the child with the maximum score, and prefers non-noop actions when there is a tie. The split node represents the state after the application of the action, which may point at multiple worlds. The split node splits the state into a set of global states where only one unique world is pointed at, and answers the question of: of all the possible states that the action can lead to, what is the worst-case situation that can happen. Each predict node predicts what may happen from the global state. To do so, it expands into a set of decision nodes for the agents and predicts how each agent contributes to progressing the state toward success. If the parent split node results from an agent taking noop, then the predict node does not expand on the decision node for the same agent, that is, the agent has to wait for someone else to take action. Each agent's decision node expands on the set of possible actions the agent can take. Assuming the parent predict node has state s= (M, w), this will be the set of actions that the agent finds feasible from its perspective (M, min_a(cc_a(w))). For each action, we expand on it both from the agent's subjective view of the state (a thick arrow) to determine how good the action is from the agent's perspective, and from the objective view of the state (a single arrow), i.e. the same perspective as the parent predict node, to determine how good the action actually is. The root decision node can be considered as a special decision node where the subjective and objective views are the same. We assume that the agent only takes the best actions from its perspective, i.e. the ones with the highest subjective score that is greater than 0, and has a uniform probability of choosing any action from that set, with the exception that if there exist perfect execution actions with subjective scores of 1, then the agent would not consider taking noop action. For each node, we can determine what perspective the state is viewed from by traversing the thick arrows from the root, which represent perspective shifts. A node reaches termination state if it satisfies the termination condition defined earlier. Tree Policy & Simulation (Default) Policy Regarding the tree policy, for each node, we compute the UCB1 score of the children to select which one to descend down the tree. For the split node, we use the negative score of each child as the exploitation term, to prioritize simulations of the worst-case situation. For the decision node, we use the subjective score of each action as the exploitation term, to prioritize simulations of the actions that are likely taken by the agent. Once the action is selected, out of the two children split nodes from the objective view and the subjective view, we select the one that is less expanded. For the simulation policy, at each decision node, we only consider the execution actions of the agent, and an agent randomly selects an action with uniform probability if it is feasible from its perspective. The predict node goes through each agent in random order to find an action to simulate forward. If none exists, simulation ends with a score of 0. This means that in the ideal case where all agents share common knowledge of plans, simulation always returns a score of 1. Back Up We take a more customized approach to computing the utility score of each node during the back-up phase. The split node takes the minimum score of the children predict nodes since it cares about the worst-case outcome, similar to the work of <cit.>, then multiplies it by the penalty factor of the action that leads to the split node. The decision node computes the expected utility of the agent's action (including noop) towards contributing to the progression of the task from the perspective of the parent predict node, denoted by 𝔼_a for agent a. Given that the subjective (objective) score of an action act is sc_act (oc_act) and the set of best actions for agent a is Act, the probability of action act ∈ Act being taken, denoted by P_a(act), is sc_act/∑_act' ∈ Actsc_act'. The utility score of the decision node is then ∑_act ∈ Act P_a(act) · oc_act. Additionally, we set the objective score of the noop action to 0, since it does not contribute to the progression of the task. The score of the predict node is computed as: (1 - ∏_a ∈ AgP_a(noop)) ( ∑_a∈ Ag𝔼_a/∑_i∈ Ag 1-P_i(noop)), which is the probability of at least some agent will act, multiplied by the expected utility of action taken by the first agent who gets to act, since execution is asynchronous. Given that some agent will act, we assume that the probability of agent a acting first is proportional to its probability of taking a non-noop action, i.e. 1 - P_a(noop). Therefore, the expected utility is the sum of the normalized probability of each agent a acting first 1-P_a(noop)/∑_i∈ Ag 1-P_i(noop) multiplied by the expected utility of agent a taking a non-noop action 𝔼_a/1-P_a(noop). This means that if every agent prefers a noop action, then the predict node has a score of 0, i.e. execution is stuck. Note that in reality, agents may decide to act if nobody else does instead of waiting forever. We do not take into account such interactive behavior, but assume this is a reasonable way to approximate the utility of the predict node. Implicit Belief Revision We consider explanations to be an explicit way of revising others' beliefs. We consider it an implicit belief revision when an unexpected execution action or intent announcement action causes a less plausible world of an agent to be promoted to be a most plausible world. We assume agents do not wish to surprise others and penalize an action that causes implicit belief revision with a score of 0. This makes sure that our agent always explains its action before taking it if it is not expected by others. However, during execution, implicit belief revision may still occur, such as when others take an action unexpected by our agent which revises our agent's belief. Performance Optimization The performance of our algorithm largely depends on the speed of solving constraint satisfaction problems (CSPs) from the knowledge bases. To optimize the performance, we implement incremental checking and caching for CSPs, since the CSPs are largely similar throughout the process. At the decision node, we lazily expand the actions in the order of execution actions, noop action, then communication actions. For example, communication actions do not need to be expanded if higher-priority actions have a better score than the maximum possible score for a communication action due to penalty. § EXPERIMENT RESULTS We describe our experiment results on the success rate and the scalability of our algorithm EPike compared to Pike <cit.>. We implemented our own Pike as a naive version of EPike that assumes what it believes is believed by all, that is, it may falsely assume common knowledge when there is not. Note that the original Pike supports some additional functionalities not accounted for by this paper, such as scheduling. We use z3 as our CSP solver <cit.>. We use an exploration parameter of 4 for SearchAction, and √(2) for the rest. We use a horizon of k=3 for SearchAction. We limit our focus to a 2-agent team. The experiments are run by instantiating two EPike or Pike agents that execute together in a task. We measure the runtime in seconds for one agent being the ego agent, who we assume gets to be the first agent to act if it decides to after each action is taken. Success Rate Since MCTS is an anytime algorithm, we evaluate the success rate and failure rate of EPike and Pike, under different timeout in seconds for MCTS (or if it reaches 1000 iterations of simulations, whichever comes first). The experiments are run for the domains of (1) Breakfast, which includes variations of our motivating example, (2) Word Puzzles, (3) Search-and-Rescue (SAR), (4) Randomly generated sequential tasks. We run each hand-crafted test case for 20 times for both Pike and EPike with no timeout, with results shown in Figure <ref>. We generate random test cases that vary in the size of the task (number of variables V) and the number of constraints that agents differ on, ranging from [0, 3], for 10 cases per condition, and report the result after running each case for 2 times for both Pike and EPike under different timeout, as shown in Figure <ref>. Note that it is possible for execution to neither succeed nor fail, in which case execution hangs as no agents plan to act. This could be because (1) MCTS algorithm is stopped by the timeout before it finds a feasible next step, (2) EPike believes execution is bound to fail no matter its action, such as when the other agent would not trust its explanation, or (3) EPike falsely believes that the other agent will act. In practice, we can adopt mitigations such as allowing EPike to take the next best action after having waited for a long time. From Figure <ref>, we see that as timeout increases, EPike's success rate increases, especially for larger-sized tasks, and is higher than Pike's success rate given enough time. Meanwhile, its failure rate is consistently low and always lower than Pike. This shows that EPike is conservative, and when it does not succeed, it is mainly because it has not found a good action to take within the timeout, but it does not take an incorrect action rashly as Pike tends to do. This is consistent with the result of the hand-crafted test cases in Figure <ref>. Scalability To see how EPike scales, we run the MCTS algorithm for a fixed number of 500 iterations to see how long it takes to reach a certain level of certainty under different model parameters, such as the size of the initial plausibility model (embodied by the number of constraints that agents differ on, Diff shown by hue of the plot), the size of the task (the number of variables V, Num Variables), concurrency level of the actions (the number of ordering constraints O, Num Orders), and the number of constraints C in the task (Num Constraints). We measure the average runtime for each callback for Pike and EPike. As shown in Figure <ref>, runtime for EPike is heavily affected by how much agents' beliefs differ, and it also increases as the task size increases. Pike takes less time than EPike, as expected, but when under common knowledge, EPike's runtime is closer to Pike and can also finish relatively quickly. § RELATED WORK Human-Robot Teaming Our work is related to human-robot teaming as it considers the collaborative process of humans and robots working together to achieve tasks. Work in this field focuses primarily on recognizing and adapting to humans' intent, and in some cases, communicating about intent, which we inherit in our work. Pike inspired us to take a constraint-based approach for concurrent intent recognition and adaption, in which a library of precompiled plans are encoded in a knowledge base <cit.>. Pike is later extended to a probabilistic setting, called Riker <cit.>, where the robot can further ask the human about their intent <cit.>. Other than constraint-based approaches, work has been proposed using techniques from classical planning and MDP, such as PReTCIL <cit.> that uses probabilistic plan recognition and NOPA <cit.> that leverages inverse planning for goal recognition. In <cit.>, a human behavior model is learned through semi-supervised learning and incorporated into the robot's POMDP model that supports bi-directional communication on intent. However, most work assumes common knowledge of the task, as opposed to implementing an explicit Theory of Mind. Epistemic Planning In the field of epistemic planning, there are two main categories of approaches – the semantic approach based on Dynamic Epistemic Logic (DEL) <cit.> and the symbolic approach <cit.>. Our work leverages the DEL approach and carries over their insight on how to model announcement and question-asking actions. In particular, we take an implicit coordination approach, following from the work of Engesser et al., where the agent takes into account the spontaneous cooperation of other agents in achieving the goal, which requires recursive perspective-taking in order to predict their actions <cit.>. In <cit.>, the authors further discussed the impact of eager and lazy agents in the framework, and in <cit.>, an MCTS algorithm is developed that shares similar insights as our work. Compared to the work by Engesser et al., we differ in that our framework based on conditional doxastic logic allows the modeling of false beliefs and the revision of false beliefs, and our explanations refer directly to the plan space instead of states as a result of extending the logic to knowledge bases. XAIP Our work is related to Explainable AI in Planning (XAIP), especially to the work on plan explanations taking into account the differences in agents' mental models. In <cit.>, model reconciliation is proposed that allows robots to explain the model differences upon misalignment between the human's mental model of the robot and the robot's actual model. In <cit.>, a logic-based approach to model reconciliation is proposed, where the planning problem is encoded as a SAT problem using SatPlan, and the model differences are computed with respect to the human and the robot's knowledge bases. Since these approaches consider the entire PDDL planning model, plan explanations go beyond explaining about the differences in the initial states but can also be about agents' discrepancies in goal states and action models. In <cit.>, model reconciliation is balanced with explicable planning, which allows robots to find (potentially sub-optimal) plans that are expected by the human based on the human's understanding of the robot <cit.>, and in <cit.>, the two are unified in an expectation-aware planning framework with additional explanatory actions. This inspired us in thinking about how the robot can balance its adaptation and communication with the human. However, most of their work considers humans as observers without much human-robot cooperation. In <cit.>, the authors pointed out the importance of a richer mental modeling framework that allows human-robot collaboration, which we provide a viable way of filling the gap. Another line of work from Shvo et al. <cit.> provides explanations by considering agents' Theory of Mind represented using epistemic logic. In particular, to resolve the human's misconceptions about plans, a symbolic epistemic planner RP-MEP <cit.> is used for the robot to either take actions to align the true state to the human's belief or explain the true state to the human <cit.>. However, their explanations are also on states rather than plans. § CONCLUSION In this work, we combine insights from epistemic logic and knowledge-base encoding of plans to allow agents to understand discrepancies in their beliefs of feasible plans. We develop an online execution algorithm Epistemic Pike for the agent to dynamically plan its actions to adapt to others and communicate to resolve any discrepancy. We show that our agent is effective in working in teams where a shared mental model of plans cannot be guaranteed. A natural next step is to consider cases where actions are partially observable. § ACKNOWLEDGEMENTS This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0035. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Defense Advanced Research Projects Agency (DARPA).
http://arxiv.org/abs/2307.03005v1	20230706140954	Multi-scale hierarchy from multidimensional gravity	[ "Kirill A. Bronnikov", "Arkady A. Popov", "Sergey G. Rubin" ]	gr-qc	[ "gr-qc", "hep-th" ]	left=3.0cm right=2.0cm top=2cm bottom=2cm day month endday endmonth @newcommand[2] #20pt^to-1.4@-@@ #1. fnsymbol#1 #1 ∂̣ const ε sign ⟹ = #1(<ref>)Eq. Eqs. 1212 det _μν^μν_μ^ν_ν^μgRSchwarzschildde Sitteranti-de Sittergeneral relativityspherically symmetricstatic, spherically symmetricasymptotically flat#1#2[#1]e-mail: #2footnote1 #1 #1#1 #1 Multi-scale hierarchy from multidimensional gravity Kirill A. Bronnikov^a,b,c,1, Arkady A. Popov^d,2, Sergey G. Rubin^c,d,3 ^a Center fo Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya ulitsa 46, Moscow 119361, Russia ^b Institute of Gravitation and Cosmology, RUDN University, ulitsa Miklukho-Maklaya 6, Moscow 117198, Russia ^c National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, Moscow 115409, Russia ^d N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kremlyovskaya ulitsa 18, Kazan 420008, Russia ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== [email protected][email protected]@list.ru We discuss the way of solving the hierarchy problem. We show that starting at the Planck scale, the three energy scales — inflationary, electroweak and the cosmological ones can be restored. The formation of small parameters is proposed that leads to a successful solution of the problem. The tools involved in the process are f(R) gravity and inhomogeneous extra dimensions. Slow rolling of a space domain from the Planck scale down to the inflationary one gives rise to three consequences: an infinite set of causally disconnected domains (pocket universes) are nucleated; quantum fluctuations in each domain produce a variety of different fields and an extra-dimensional metric distribution; these distributions are stabilized at a sufficiently low energy scale. § INTRODUCTION Assuming that the Universe has been formed at the Planck scale, it is naturally implied that its initially formed parameters are of the order of the same scale. The essence of the Hierarchy problem is the question: Why are the observable low-energy physical parameters so small as compared to those of the Planck scale? How did Nature manages to decrease the parameter values so substantially? There are at least four important energy scales during evolution of the Universe: the Planck scale (∼ 10^19 GeV) at which our Universe cannot be described by classical laws; the inflationary scale (∼ 10^13 GeV) where our horizon has appeared, the electroweak scale (∼ 10^2 GeV), and the cosmological scale specified by the cosmological constant (∼ 10^-123 GeV^4) (CC). According to the inflationary paradigm, the physical laws are formed at high energies <cit.>, where the Lagrangian structure is yet unknown. The physics has been established at the energy scale M higher than the inflationary one, E_I ∼ 10^13 GeV, see <cit.> in this context. We study the way of substantially decreasing the physical parameters at the three scales mentioned above assuming natural values of the initial parameters of the order of M. In this paper, we invoke the idea of multidimensional gravity which is a widely used tool for obtaining new theoretical results <cit.>. The paper <cit.> uses warped geometry to solve the small cosmological constant problem. Multidimensional inflation is discussed in <cit.> where it was supposed that an extra-dimensional metric g_ is stabilized at a high-energy scale. Stabilization of extra space as a pure gravitational effect has been studied in <cit.>, see also <cit.>. The present research is also based on nonlinear f(R) gravity. The interest in f(R) theories is motivated by inflationary scenarios starting with Starobinsky's paper <cit.>. At present, f(R) gravity is widely discussed, leading to a variety of consequences, in particular, the existence of dark matter <cit.>. Including a function of the Ricci scalar, f(R), is the simplest extension of general relativity. In the framework of such an extension, many interesting results have been obtained. Some viable f(R) models in 4D space that satisfy the observational constraints are proposed in <cit.>. The idea that the Lagrangian parameters can be considered as some functions of a field has been widely used since Schwinger's paper <cit.>. Such fields can be involved in the classical equations of motion together with the “main” fields or treated as background fields. The latter were applied for fermion localization on branes <cit.>, gauge field localization <cit.>, extensions of gravity in a scalar-tensor form (with f(ϕ)R) <cit.> and so on. In this paper, we show that a self-gravitating scalar field can serve as a reason for the emergence of small parameters. As a mathematical tool, we use the Wilsonian approach technique, a well-known method for theoretical studies of the energy dependence of physical parameters <cit.>. In this approach, the physical parameters λ_i (M) of the Wilson action are fixed at a high energy scale M. The renormalization flow used to descend to low energies (the top-down approach) is discussed in <cit.>. In particular, quantum corrections to the Starobinsky model were discussed in <cit.>. In our approach, we add extra dimensions and study their role in different scales, with a hope that it should make the renormalization procedure much more efficient. We make use of the idea of flexible (inhomogeneous) extra space that has been developed in <cit.>. Our preliminary study of inhomogeneous extra metrics concerns such parameters as the cosmological constant <cit.>, those of the Starobinsky inflationary model and baryon asymmetry of the Universe <cit.>. It has been shown there that inhomogeneous metrics can be tuned to explain the smallness of the appropriate effective parameters. For example, encouraging results for explaining the smallness of the cosmological constant was obtained in <cit.>. The effect of quantum corrections in this context was discussed in <cit.>. Here we continue this research by including the Higgs sector of the Standard Model. There are three energy scales which we intend to describe — inflationary stage, the electroweak scale, and the cosmological one. Each of them is characterized by a specific small parameter. The initial parameters and the Lagrangian of our model are fixed at a sub-Planckian scale and do not vary during the Universe evolution. Special attention is paid to the emergence mechanism small values, specific for each of the three scales. § THE MODEL Consider f(R) gravity with a minimally coupled scalar field ζ in a = 4 + n-dimensional manifold M_: S = m_^-2/2∫_M_ d^ X √(\|g_\|) ( f(R) + ∂^ζ ∂_ζ -2 V(ζ) )+S_H_P , where g_≡ g_, , =1,, the n-dimensional manifold M_n is assumed to be closed, f(R) is a function of the D-dimensional Ricci scalar R, and m_ is the -dimensional Planck mass. Below, we will work in the units m_D=1. The term S_H_P denotes the Higgs action (<ref>) considered in Sec. <ref>, and it is assumed to be small as compared to the gravitational part of the action. It is also postulated that the scalar field ζ is very massive and hence unobservable. Nevertheless, this field plays a key role being responsible for the emergence of small parameter(s), see a discussion at the beginning of Sec. <ref>. Variation of the action (<ref>) with respect to the metric g^_ and the scalar field leads to the known equations -1/2f(R)δ_^ + (R_^ +∇^∇_ - δ_^_) f_R = - T_^, _ ζ + V_ζ =0, with f_R = df(R)/dR, _= ∇^∇_, and V_ζ = d V(ζ)/dζ. Equation (<ref>) is known to be a consequence of equations (<ref>). The corresponding stress-energy tensor of the scalar field ζ is T_^ = ∂ L_ matter/∂(∂_ζ)∂_ζ - δ_^/2 L_ matter = ∂^ζ ∂_ζ - δ_^/2 ∂^ζ ∂_ζ + δ_^V(ζ) . Here the Higgs field contribution is omitted. We use the conventions for the curvature tensor R_ ^ =∂_Γ_^-∂_Γ_^ +Γ_^Γ_^-Γ_^Γ_^ and the Ricci tensor R_=R^_. The metric is supposed in the form ds^2=^2 γ(u)(dt^2 -^2Ht(dx^2 +dy^2 +dz^2)) - du^2 -r(u)^2 d Ω_n-1^2, where d Ω_n-1^2 is the metric on a unit n-1-dimensional sphere. The metric ansatz used in this paper has been widely studied in the framework of linear gravity <cit.>, applying, in particular, to solving the Hierarchy problem <cit.>. The field equations for the metric interval and ζ = ζ(u) read R'^2 f_RRR +[R” + (3 γ' + (n-1) r'/r) R' ]f_RR - ( γ” +4γ'^2 + (n-1)γ' r'/r - 3H^2/^2 γ) f_R - f(R)2 = - ζ'^22 - V( ζ), ( 4γ'R' + (n-1) r'r R' ) f_RR - ( 4 γ” + 4γ'^2 + (n-1) r”r) f_R - f(R)2 = ζ'^22 - V(ζ) , R'^2 f_RRR + ( R” + 4γ' R' + (n - 2) r'r R') f_RR - (r”r + 4γ' r'/r + (n-2)r'^2r^2 - (n-2)r^2) f_R - f(R)2 = - ζ'^22 - V(ζ) , ζ” + (4 γ' + (n-1) r'r) ζ' - V_ζ =0, where the prime denotes d/du. Also, we will use the expression for the Ricci scalar R(u)= 12 H^2/^2 γ -8γ” -20γ'^2 - (n-1) ( 2 r”r + 8γ' r'/r + (n-2) (r'r)^2 - (n-2)r^2) as an additional equation and R(u) will be treated as a new unknown function to avoid 3rd and 4th order derivatives in tt–aa. It can be shown that one of the equations tt–scalar is a consequence of the others. The combination 2×(<ref>)- f_R ×(<ref>) is the constraint equation (8γ' + 2 (n-1) r'r) R' f_RR + (12 γ'^2 + (n-1) (8γ' r'r + (n-2)(r'^2-1)r^2) +R )f_R - 12 H^2/^- γ(u) f_R - f(R) = ζ'^2 - 2 V( ζ) containing only first-order derivatives. It plays the role of a restriction on the solutions of the coupled second-order differential equations tt–Ricci_n. As a result, we use three independent equations (<ref>), (<ref>), (<ref>) and the constraint (<ref>) to fix three functions r(u),γ(u), R(u) and the unknown metric parameter H. One of the possible numerical solutions to this system is shown in Fig. <ref>. We note that the warp factor e^γ(u)→ 0 at the boundaries, which are singular ends of the range of u and can be imagined as a kind of poles in a closed n-dimensional manifold since there r→ 0. The qualitative behavior of the solution shown in Fig. <ref> is quite generic. The particular form of solutions, including the field distribution and the extra metric, depends on the Lagrangian parameters postulated from the beginning. It also depends on the boundary conditions at u=0 that are necessary for solving the second-order differential equations. Unlike the Lagrangian parameters, the boundary conditions ultimately depend on random initial fluctuations within a pocket universe. Inflation produces a continuum set of such universes with different initial conditions and therefore with different metric functions and field distributions in the extra space. § MATTER LOCALIZATION AROUND A SINGULARITY In general, it is assumed here that matter is distributed throughout the extra dimensions like in the Universal Extra Dimensional approach <cit.>. At the same time, there is another direction that deserves discussion. Indeed, we see from the figures that there are two points where the metric is singular or has sharp peaks. They could indicate the formation of branes if the extra space is large enough and if matter is concentrated in a close neighborhood of these peaks (certainly assuming that the formal infinities are somehow suppressed by quantum effects). This opportunity is briefly discussed in this section. As a rough approximation, consider the motion of classical particles near such a singular point, see Fig. 1, bearing in mind the metric ds^2=^2γ(u)(dt^2 -dx^2 -dy^2 -dz^2) - du^2 -r(u)^2 (dξ^2 + sin^2 ξ d ψ^2). The geodesic equations have the form t_ss +2 t_s γ_u u_s =0, x_ss +2 x_s γ_u u_s =0, y_ss +2 y_s γ_u u_s =0, z_ss +2 z_s γ_u u_s =0, u_ss +^2 γ γ_u (t_s^2 -x_s^2 -y_s^2 -z_s^2) -r r_u ξ_s^2 -r r_u sin^2 ξ ψ_s^2 =0, ξ_ss +2 ξ_s r_u/r u_s -sinξ cosξ ψ_s^2 =0, ψ_ss +2 ψ_s r_u/r u_s +2ξ ξ_s ψ_s =0, where the index s denotes the derivative with respect to s, and the index u denotes the derivative with respect to u. These equations admit solutions when x, y, z, ξ, and ψ are constant. Let us assume, for simplicity, that ξ=π/2, then 0=t_ss +2 t_s γ_u u_s = t_ss+2 t_s γ_s ⇒ t_s =^C_1 -2γ, 0 = u_ss +^2 γ γ_u t_s^2 = u_ss + γ_u ^2C_1 -2 γ = 1/u_s(u_ss u_s + γ_s ^2C_1 -2γ) ⇒ u_s^2 = ^2C_1 -2 γ +C_2, with integration constants C_i. The normalization relation gives 1 = ^2 γt_s^2 -u_s^2 = -C_2. Then u_s^2 =^2C_1 -2 γ -1 = u_t^2 ^2C_1 -4 γ ⇒ u_t^2 = ^2 γ(1 -^2γ - 2 C_1 ). For nonrelativistic particles, only the second equation matters. It can be approximated as u_ss≃ -2e^2γγ_u. We see that the acceleration of a particle is directed to a singular point, which should ultimately lead to concentration of matter at such a point. As a result, matter is localized around both “poles,” as should be the case in a brane world (this time consisting of two branes on the two “poles”). It opens a door for developing a mechanism of strong reduction of the initial parameter values. For example, an interaction term of the form κ∫ d^D Z√(\|g_D\|)χ(z)ψ̅(z)ψ(z) contains overlapping integral I_ overlap≡∫ d^n y √(\|g_n\|)χ(y)ψ̅(y)ψ(y) over the extra dimensions which could be arbitrarily small if the fields χ(y) and ψ(y) are localized near different branes. It leads to the coupling constant renormalization κ→κ' =κ I_ overlap≪κ. We will leave this idea for future studies and return to our main discussion. § INTERMEDIATE ENERGIES. THE STAROBINSKY MODEL The second energy scale relates to the inflationary stage with the small parameter H/m_∼ 10^-6. Different inflationary models use different parameters of this order. It could be the inflaton mass in the simplest model of inflation with a quadratic potential or a constant factor of R^2 term in the Starobinsky model. Let us restore the latter. To this end, we should solve the system (<ref>)–(<ref>) and obtain the necessary values of its parameters. The scalar field ζ affects the extra-space metric through the Einstein equations, but here we are interested in small amplitude solutions of this field, i.e., ζ(X)≪ 1. Therefore, its role in the metric formation is negligible, and it can be considered as a test field acting in the background metric. This approximation makes the analysis easier but is not very significant for our reasoning. §.§.§ The emergence of small parameters A successful solution of the Hierarchy problem implies the presence of small parameters, and we have enough tools to create them. Indeed, in our picture, there is an infinite set of different universes created during inflation <cit.> which contain the independently fluctuating field ζ. These fluctuations decay with time and lead to static field distributions in each universe. There is an infinite set ℵ of such static distributions that form a continuum set. The situation is similar to the boson star formation model <cit.> where a self-gravitating scalar field forms a variety of dense stable clumps. The set ℵ contains a subset of small-amplitude distributions ζ(u) like those presented in Fig. <ref>, right panel. Their values averaged over the extra dimensions represent a set of small parameters to be widely used below. To proceed, let us restore some formulas from our previous paper <cit.> for a relation between the -dimensional Planck mass and the 4-dimensional one, which is needed to convert units m_=1 into the physical units. To this end, define R_4 ≡ 12 H^2, R_n ≡ R(u) - ^-2 γ(u) R_4, see (<ref>). Substitution of the Taylor series f(R) ≃ f(R_n) + f_R(R_n)^-2γ(u)R_4 + 1/2f_RR(R_n)^-4γ(u) R_4^2 + … into the gravitational part of the action (<ref>) leads to an effective theory after integration over the extra coordinates: S_ = m_^22∫_M_4 d^4 x √(\|g_4\|)(a_ R_4^2 + R_4 + c_). Here g_4 is the determinant of the 4D metric ds^2 = g_μν dx^μ dx^ν = dt^2 - ^2Htδ_ijdx^i dx^j , and m^2_ = 𝒱_n-1∫_u_min^u_max f_R (R_n) ^2γ r^n-1 du, a_ = 𝒱_-1/2m_^2∫_u_min^u_max f_RR(R_n) ^4γ r^n-1 du, c_ = 𝒱_n-1/m_^2∫_u_min^u_max(f(R_n) - (ζ')^2 - 2V(ζ)) ^4γ r^n-1 du . where 𝒱_n-1 = ∫ d^n-1 x √(\|g_n-1\|) = 2π^n/2Γ(n/2). The r.h.s. of (<ref>) is written in units m_D=1. This relation is used to express the D-dimensional Planck mass in terms of the 4D Planck mass m_. Here we suppose that the functions γ(u), r(u), ζ(u), R(u) form a particular solution to the system (<ref>)-(<ref>) for a specific value of H. Therefore the values of a_(H) and c_(H) are functions of the Hubble parameter. They are approximately constants during inflation and at the present times, being different in these two periods. The parameter c_(H) is fixed at the present epoch when H m_, while the parameter a_(H) is determined by the appropriate inflation rate. The parameter a_ must be approximately equal to the observable value obtained from the COBE normalization <cit.>, as a_Starob≃ 1.12 · 10^9 (N_60)^2m^-2_. For the solution shown in Fig. <ref>, a_≃ 7.2 · 10^8 m_^-2, and the Hubble parameter is H≃ 1 · 10^-6 m_. One can see that the Starobinsky inflationary model has been restored. The parameter values a=300, c=0.002 lead to the following values of the dimensionless parameters: a'=√(am_D^2)≃ 17, c'=√(c/ m_D^2)≃ 0.045 which looks natural. The other parameter, c_(H), needs a separate discussion. Equation (<ref>) for c_(H) is derived under the assumption that all functions in (<ref>)–(<ref>) are stationary, which takes place for a 4D de Sitter metric in which the Hubble parameter H=. This approximation is valid at slow-roll inflation with the small parameter \|H\|/H^2 ≪ 1. Fortunately, this inequality holds for a wide range of the parameter c_, in particular for c_=0<cit.>. However, (<ref>) has a practical meaning only for a pure de Sitter metric but not for slow rolling. This formula could be valid at the present times, for example, if we suppose that the cosmological constant Λ=-c_/2 is really constant. It is a subject of detailed discussion, see Sec. <ref>. § THE ELECTROWEAK SCALE. RESTORATION OF THE HIGGS PARAMETERS In this section, the reasoning is in the spirit of our previous paper <cit.>, but without introducing an external scalar field. §.§ Analytical formulas In the previous section, we have reproduced the Starobinsky model of inflation at the scale of 10^13 GeV. The appropriate values of the initial Lagrangian parameters a, c are fixed. These parameters must be the same at low scales where the Hubble parameter is equal to zero, H≃ 0, as compared to the Planck scale. On the contrary, the extra space metric depends on the energy scale, the Hubble parameter in our case. In this section we discuss the Hierarchy problem at the electroweak scale using the Higgs field as an example. Within the framework of our approach outlined in the Introduction, we assume that the physics of the Higgs field is formed at the Planck scale. Suppose that the form of the Higgs action at the Planck scale is the same as at the electroweak scale, S__P = 1/2∫ d^ X √(\|g_\|) (∂^H_P^†∂_ H_P + νH_P^† H_P - λ(H_P^† H_P)^2 ), where the symbol † means Hermitian conjugation, ν and λ > 0 are arbitrary numbers and H_P is a proto-Higgs field. We have managed to avoid large/small initial dimensionless parameter values a',c', see (<ref>) when describing the Starobinsky model acting at energies ∼ 10^13 GeV. Our intention is to repeat this success at the electroweak energies. To do that, we need to show that the initial parameters can be reduced by many orders of magnitude. All numerical values in the Lagrangian (<ref>) are of the order of unity in m_ units. More definitely, let us express the dimensionful parameters ν, λ in terms of the dimensionless ones ν', λ': ν→ (ν' m_D)^2, λ→ (λ'/m_D)^D-4. It is these dimensionless parameters ν' and λ' that should vary around unity. The classical equations of motion are obtained by varying the action (<ref>) with respect to H_P, which gives □_ H_P=ν H_P - 2 λ(H_P^† H_P)H_P. The proto-Higgs field can be presented as H_P = h(x) U(u) + δ H_P, δ H_P = ∑_k h_k(x) Y_k(u) where h(x) and h_k(x) are 2-component columns acting in the fundamental representation of SU(2). In what follows, we will consider the case H_P≃ h(x) U(u), δ H_P≪ h(x) U(u). The dimensionality of the proto-Higgs field is [H_P]=m_D^(D-2)/2, [h]=[h_k]=m_D, [U]=[Y_k] = m_D^n/2. Our immediate aim is to find the distribution of the field H_P over the extra coordinates governed by the scalar function U(u) by solving (<ref>), (<ref>). The inhomogeneities of the field h(x) are important at low energies, but they are exponentially stretched during the first de Sitter-like stage so that h(x)= with great accuracy. It means that h(x) = 1/√(2)[ 0; v_0+ρ(x) ]≃1/√(2)[ 0; v ]. Therefore, the approximation (<ref>) transforms (<ref>) in the following way: □_n U(u) =ν U(u) - λ v^2 U^3(u), with a yet unknown parameter v. We suppose further on that the metric functions as well as the Lagrangian parameters remains the same as those considered above with one exception: the Hubble parameter is extremely small at the present epoch as compared to the inflationary epoch, and below we put H≈ 0. The knowledge of solutions to (<ref>) permits us to integrate out the internal coordinates and to reduce the action (<ref>) to the 4D form S_ = 𝒱_-1/2∫ d^4 x √(\|g̃_4\|)∫_u_min^u_max[^-2γ(u) U^2(u) g̃^i j∂_i h^†∂_j h + (-(∂_u U)^2+ ν U^2(u)) h^† h - λ U^4(u) ( h^† h)^2 ] ^4γ(u) r^n-1(u) du after substitution of (<ref>) into (<ref>). To study this action at low energies, we choose the Minkowski metric g̃_4,ij = η_i j and define the following parameters by integration over u: K_h = 𝒱_-1∫_u_min^u_max U^2(u) ^2γ(u)r^-1(u) du, m^2_h =𝒱_-1∫_u_min^u_max(-(∂_u U)^2 +ν U^2(u))^4γ(u)r^-1(u) du, λ_h = 𝒱_-1∫_u_min^u_maxλ U^4(u) ^4γ(u)r^-1(u) du. §.§ Comparison with the Higgs parameters Recall that a natural range for the dimensionless parameters λ' and ν' is 10^-2 to 10^2. It means that acceptable ranges of the “physical” parameters are (10^-6÷ 10^6) for λ and (10^-4÷ 10^4) for ν according to the definitions (<ref>). The substitution H_0(x)=h(x)√(K_h) leads to the 4D effective Higgs Lagrangian S_ = 1/2∫ d^4 x √(\|g̃_4\|) (∂_i H_0^†∂^i H_0 + m_H^2 H_0^† H_0 - λ_H(H_0^† H_0)^2 ) , m_H^2 ≡m_h^2/K_h, λ_H ≡λ_h/K_h^2. Here H_0 is the observable Higgs field at zero energy. The experimentally measured parameters are the Higgs boson mass and its vacuum average, m_ Higgs=125 GeV, v_ Higgs=246 GeV . according to <cit.>. They are related to the parameters m_H and λ_H of the effective Higgs action (<ref>) as follows: m_H =m_ Higgs /√(2)=88.6 GeV≃ 10^-17 m_, and λ_H=(m_H/v_ Higgs)^2/2 ≃ 0.13, The vacuum energy of the Higgs field is V_min = -1/2 m_H^2v_ Higgs, so that the parameter c in the function f(R) should be corrected, c→ c+V_min. Note, however, that V_min is very small as compared to the D-dimensional Planck scale and may be neglected. The above formulas contain the function U(u), the solution to (<ref>) with a yet unknown constant v. It is of interest that the Lagrangian structure (<ref>) allows us to avoid the determination of this constant. Indeed, a solution to (<ref>) can be found for the function Ũ(u) = vU(u) because (<ref>)Ũ(u) does not contain the unknown parameter v in this case. Moreover, substitution of U=Ũ(u)/v into the expressions (<ref>), (<ref>) and (<ref>) gives K_h[U]= K_h[Ũ]/v^2 , m_h^2[U]= m_h^2[Ũ]/v^2, λ_h[U]= λ_h[Ũ]/v^4, hence the observable parameters m_H and λ_H in (<ref>) do not depend on v. This quantity also appears in the relation v ≃ v_ Higgs/√(K_h[U]), following from (<ref>) and the substitution H_0→ v_ Higgs, h→ v. Luckily, this relation does not depend on v as well. After taking into account the first equality in (<ref>), we obtain an additional restriction for the function Ũ: 1 ≃ v_ Higgs/√(K_h[Ũ]). The quantity K_h[Ũ] is calculated in m_D units. Therefore, v_ Higgs should be also expressed in m_D units. Figure <ref> presents the Higgs field distribution U in the extra dimensions. In this section, we have found the conditions at which the initial parameter values written in Fig.<ref> reproduce the Higgs Lagrangian with the observed parameters. The origin of small parameters has been discussed earlier, see the beginning of Sec. <ref>. Quantum fluctuations produce a variety of field amplitudes in the countable set of pocket universes. A small measure of them contains (extremely) small amplitudes U. The fields values of the order of Ũ∼ 10^-16 (see Fig. <ref>) are suitable for the relations (<ref>), (<ref>) and (<ref>). § LOW ENERGIES. THE COSMOLOGICAL CONSTANT Analytical formulas for 4D gravity have been obtained in Sec. <ref>. It is assumed that they are approximately valid at the inflationary scale ∼ 10^13 GeV. The parameter a_ was obtained for specific values of the initial parameters a, c, m_D. A calculation of c_ representing the effective cosmological constant (CC) is not necessary because its value could vary in a wide range without any effect on the inflationary process. This value is important at low energies where the Hubble parameter is ∼ 10^-61≈ 0 GeV^2. Therefore, the extra metric must be found by solving the Einstein equations at H=0. Luckily, the resulting metric weakly depends on the Hubble parameter if it varies within the interval 0 < H < 0.01, and Fig. <ref> gives the appropriate impression. At the low energy scale, the Hubble parameter H is small, and the curvature squared R_4^2 can be neglected in (<ref>). In this case, the 4D Einstein equations lead to a relation between the CC and the Hubble parameter, c_≡ -2Λ = -6 H^2, which means that c_ should be extremely small as well. On the other hand, the same value was found above starting from the initial D-dimensional action, see (<ref>). It can be presented in the following form (see the Appendix): c_ = - 6 H^2 + 𝒱_-1m_^-2/m_^2 2∫_u_min^u_max[ (f_RR R'- f_R γ') ^4γ r^-1]' du + O(H^6). A comparison of the expressions (<ref>) and (<ref>) derived with arbitrary initial parameters and boundary conditions indicates that the integral in (<ref>) must be zero. It makes sense to prove this statement directly. To this end, we should find the function Φ (u) ≡(f_RR R' -f_R γ' ) ^4γ r^-1 at the boundary points u_min and u_max. Numerical simulations indicate that this function tends to zero indeed, see Fig.<ref>. Unfortunately, the accuracy is unsatisfactory while approaching the boundary points. To clarify the situation, the following can be suggested: we modify the nonlinear term in the action R^2→ R^2 e^-ϵ R^2 from the beginning, ϵ 1 in m_D units, and put ϵ = 0 finally. It does not affect the equation of motion, but smooths out the singularities. In this case, Φ→ 0 at the boundary points, where r=0 by definition. Hence, the integral as a whole equals zero. In this subsection, we have proved that the well-known relation H^2=Λ/3 can be derived from D-dimennsional gravity. In the standard notations, c_=-2Λ, where Λ is the cosmological constant. In a more general case of the the function f(R), terms proportional to H^6 and other nontrivial terms can appear in the expression (<ref>). In this case, the inflationary dynamic requires a separate study. § THE ROLE OF QUANTUM FLUCTUATIONS §.§ The smallest scale of compact extra dimensions It is known that our instruments “feel” average values of a field ζ(u) calculated as ζ(u) = ζ_ classical(u) + δζ(u), where ζ_classical(u) is the classical part, and δζ(u) is a quantum correction to it. It makes sense to calculate the classical part only if ζ_ classical(u) ≫δζ(u). This inequality holds only if the action S ≫ 1 (the steepest descend method). Thus the classical approach can be applied in our approach if S = ∫ dv_D f(R) ≫ 1, or in other words S ≃δ v_D ⟨ f(R) ⟩≫ 1, where δ v_D ≃δ u^D is a small volume parametrized by the coordinate u, and ⟨ ...⟩ stands for averaging over this volume. It means that the volume δ v_D must not be smaller than ∼ 1/⟨ f(R)⟩→δ u ∼⟨ f(R)⟩^-1/D. Thus for a classical description to make sense (i.e. to approximately coincide with the averages), the averaging range must not be smaller than δ u ∼⟨ f(R)⟩^-1/D. For example, for a seven-dimensional space and f(R)∼ 10, we have δ u ∼ 10^-1/7∼ 1. It means that the size of extra dimensions should be larger than 1/m_D. Also, it is dangerous to make physical conclusions based on classical solutions in the vicinity of singular points at a distance smaller than 1/m_D. §.§ Fluctuations in the present epoch The field value H_P∝ 10^-17 is quite a small value. Let us estimate the probability of large fluctuations that could destroy the solution. The cosmological probability to find a field value χ_2 at the instant t_2 = t_1 + t in a spatial region of the horizon size H^-1 was studied in <cit.>. Based on the outcomes, it is possible to demonstrate that the first mode amplitude fluctuation probability, h_1, can be determined as dP = dP_1 = dh_1 ·√(q_1/π)exp [-q_1 h_1^2], t→∞ where q_1 = μ/σ^2, μ=m_1^2/3H, σ = H^3/2/2π. m_1 ∼ m_D, and the present-day Hubble parameter is H=1.2× 10^-61 m_. Their knowledge allows one to estimate the parameter q_1. The fluctuation h_1 should be of the order of the classical part, h_1∼U∼ 10^-17 (see Fig. <ref>) or larger to destroy it. Now we have everything to estimate the exponent, q_1 h_1^2 ∼^211(m_D/M_)^4, m_D>10^-5M_. This estimate can be substituted to (<ref>) to demonstrate how unlikely it is that even such tiny classical field (U∼ 10^-17) will be destroyed. §.§ Quantum corrections The essence of the Wilson approach is to fix a Lagrangian and its parameters at the highest scale and shift down to a low energy scale. It is achieved by sequentially integrating the Euclidean action over a small slice of the momentum interval Δ k_E. The renormalization group equations thus obtained are widely used in this concern <cit.>. The relations between low-energy parameter values and high-energy ones are discussed in <cit.>. Also, quantum fluctuations could modify the same form of a Lagrangian, <cit.>. The inclusion of a compact extra space into consideration complicates the procedure. Indeed, we cannot choose an arbitrarily small momentum interval due to the energy level discreteness. For example, if a size is quite small, Δ k_E < 1/r, r being the scale of extra dimensions, then this momentum interval does not contain energy levels at all. A possible way to overcome this difficulty is discussed in <cit.>, where truncated Green functions G_T(Z,Z')≡∑_N∈NY_N(Z)Y_N(Z')^*/λ_N were introduced. Here Y_N(Z) is a subset of n+4-dimensional eigenfunctions. The coordinates Z describe both 4D space and a compact extra space. It allows for approximately calculating the parameters at low energies. As a result, quantum corrections caused by a scalar field are proportional to its self-coupling. This means that such quantum effects cannot be responsible for reducing the parameter values by many orders of magnitude, from the Planck scale to the electroweak scale. The classical mechanism discussed in this paper was elaborated just for this aim. The procedure of quantum renormalization is a necessary and unavoidable element that leads to fine tuning of the physical parameters at low energies. § CONCLUSION In this paper, we have discussed an approach which provided the hierarchy of three energy scales, the inflationary, electroweak and cosmological ones. Necessary tools for small parameters formation and a successful solution of the problem are f(R) gravity and inhomogeneous extra dimensions. The set of small parameters is formed in the following way. Slow rolling of a spatial domain from a sub-Planckian scale down to the inflationary one gives rise to several consequences: (1) nucleation of an infinite set of causally disconnected domains (pocket universes), (2) quantum fluctuations in each domain produce a variety of fields and an extra-space metric distribution, (3) these distributions are stabilized when the energy scale is low enough. Self-gravitating (scalar) fields do not necessarily settle at states with minimum energy. On the contrary, e.g., the boson stars activity <cit.> is based on the fact that self-gravitating scalar fields can settle at a continuum set of static states. There are states with arbitrarily small amplitudes among them. These states are formed in a small but finite set of universes. As a result, a small but nonzero measure of different universes contains small effective parameters that are applied here to solve the Hierarchy problem at three energy scales. The mechanism developed should be accompanied by a renormalization group analysis aimed at correction of the initial parameter values. § ACKNOWLEDGEMENTS The work of SGR and KAB was funded by the Ministry of Science and Higher Education of the Russian Federation, Project "New Phenomena in Particle Physics and the Early Universe" FSWU-2023-0073 and the Kazan Federal University Strategic Academic Leadership Program. The work of AAP was funded by the development program of Volga Region Mathematical Center (agreement No. 075-02-2023-944). KAB also acknowledges support from Project No. FSSF-2023-0003. § APPENDIX The validity Condition of (<ref>) is not so trivial in the 4+n-dimensional case, and we will discuss it here. We can exclude the terms with a scalar field in the definition of c_; using the expression (<ref>), we obtain c_ = 𝒱_-1m_^D-2/m_^2∫_u_min^u_max(f(R_) - (ζ^')^2 - 2V( ζ)) ^4γ r^-1 du =𝒱_-1m_^D-2/m_^2∫_u_min^u_max {f(R_) +2R'^2 f_RRR(R) +2[R” +R' (3 γ' +(-1 ) r'r) ] f_RR(R) . . - 2( γ” +4γ'^2 + (-1)γ' r'/r) f_R(R) + 6H^2/^2γ(u) f_R - f(R) } ^4γ r^-1 du =0. Part of this expression can be transformed as follows: 𝒱_-1m_^D-2/m_^2∫_u_min^u_max {( 2R'^2 f_RRR +2[R” +R' (3 γ' +(-1 ) r'r) ] f_RR - 2( γ” +4γ'^2 . . . . + (-1)γ' r'/r) f_R} ^4γ r^-1 du = 𝒱_-1m_^D-2/m_^2 2 ∫[ (f_RR R' -f_R γ' ) ^4γ r^-1]' du. The remaining part of the expression (<ref>) can be rewritten in a more conventional form (by substituting the expansion (<ref>) and the definition (<ref>)): 𝒱_-1m_^D-2/m_^2∫ (f(R_) +6H^2/^2γ f_R(R) - f(R) ) ^4γ r^-1 du = 𝒱_-1m_^D-2/m_^2∫ [f(R_) +6H^2/^2γ(u)( f_R(R_n) +R_4/^2γ(u) f_RR(R_n) ) - ( f(R_n) . . +R_4/^2γ(u) f_R(R_n) + R_4^2/2^4γ(u) f_RR(R_n) ) +O( R_4^3/^6γ(u) f_RRR(R_n) ) ] ^4γ r^-1 du = 𝒱_-1m_^D-2/m_^2∫ [ (6H^2 -R_4)/^2γ f_R(R_n) +(12H^2 -R_4)R_4/2^4γ f_RR(R_n). . +O( R_4^3/^6γ(u) f_RRR(R_n) ) ] ^4γ r^-1 du. Since m_^2 is defined by the expression (<ref>) and R_4=12H^2, (<ref>) can be rewritten as -6 H^2 + O(H^6). Thus we can present c_ in the form of (<ref>) by summing the expressions (<ref>) and (<ref>). [title=References, heading=bibintoc] Рассмотрим более детально действие (<ref>) и метрику (<ref>). Предполагается, что часть метрики g_μν(x) ответственна за наблюдаемые гравитационные эффекты. При этом ее вариация δ g_μν(x) не может равняться нулю на границе дополнительного пространтсва, что является необходимым условием получения уравнения (<ref>). Рассмотрим (tt) уравнение системы (<ref>) более подробно. При варьировании исходного действия по g_μν(x) оно имело вид ∫ {(R'^2 f_RRR +[R” +R' (3 γ' +(-1 ) r'r) ] f_RR - ( γ” +4γ'^2 + (-1)γ' r'/r) f_R. . + 3H^2/^2γ(u) f_R - f(R)2 + (ζ^')^22 + V( ζ) } ^4γ r^-1 du =0. integration of equation for (tt)=(x^i x^i)–components over the compact extra space, i.e., ∫ d^ x √(\|g_\|) is restored. The first line of (<ref>) -10mm ∫ {(R'^2 f_RRR +[R” +R' (3 γ' +(-1 ) r'r) ] f_RR - ( γ” +4γ'^2 + (-1)γ' r'/r) f_R} ^4γ r^-1 du = ∫[ (f_RR R' -f_R γ' ) ^4γ r^-1]' du = 0 is the total derivative under the integral. It must be equal zero for smooth compact manifolds. It will be checked. Equation (<ref>) can be rewritten in more conventional form ∫( 3H^2/^2γ(u) f_R - f(R)2 + (ζ^')^22 + V( ζ) ) ^4γ r^-1 du =1/2m_^2/𝒱_-1 m_^D-2(c_eff +6H^2)+O(H^6)=0. This can be shown by substituting expansion (<ref>) and definition (<ref>) into the left hand side of this equation in the following way ∫( 6H^2/^2γ(u) f_R(R) - f(R) + (ζ^')^2 +2V( ζ) ) ^4γ r^-1 du = ∫( 6H^2/^2γ(u)( f_R(R_n) +R_4/^2γ(u) f_RR(R_n) + ... ) - ( f(R_n) +R_4/^2γ(u) f_R(R_n) + R_4^2/2^4γ(u) f_RR(R_n) +...) . . + (ζ^')^2 +2V( ζ) ) ^4γ r^-1 du = -m_^2 c_eff/𝒱_-1 m_^D-2 + ∫[ (6H^2 -R_4)/^2γ f_R(R_n) +(12H^2 -R_4)R_4/2 ^4γ f_RR(R_n) +... ] ^4γ r^-1 du = -m_^2 c_eff/𝒱_-1 m_^D-2 -m_^2 6 H^2/𝒱_-1 m_^D-2 + O(H^6). Thus, we have obtained well known relation between the cosmological constant and the Hubble parameter at the present time Λ≡ -1/2 c_eff =3H^2 + O(H^6). ++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ Using expression (<ref>)c_eff can be rewritten as follows -8mm c_eff = 𝒱_-1m_^D-2/m_^2∫_u_min^u_max(f(R_) - (ζ^')^2 - 2V( ζ)) ^4γ r^-1 du =𝒱_-1m_^D-2/m_^2∫_u_min^u_max {f(R_) +2R'^2 f_RRR(R) +2[R” +R' (3 γ' +(-1 ) r'r) ] f_RR(R) . . - 2( γ” +4γ'^2 + (-1)γ' r'/r) f_R(R) + 6H^2/^2γ(u) f_R - f(R) } ^4γ r^-1 du =0. Part of this expression can be transformed as follows -10mm 2∫_u_min^u_max {(R'^2 f_RRR +[R” +R' (3 γ' +(-1 ) r'r) ] f_RR - ( γ” +4γ'^2 . . . . + (-1)γ' r'/r) f_R} ^4γ r^-1 du = 2∫[ (f_RR R' -f_R γ' ) ^4γ r^-1]' du = . It must be equal zero for smooth compact manifolds. It will be checked. The remaining part of expression (<ref>) can be rewritten in more conventional form (by substituting expansion (<ref>) and definition (<ref>)) ∫ (f(R_) +6H^2/^2γ f_R(R) - f(R) ) ^4γ r^-1 du = ∫ [f(R_) +6H^2/^2γ(u)( f_R(R_n) +R_4/^2γ(u) f_RR(R_n) ) - ( f(R_n) +R_4/^2γ(u) f_R(R_n) . . . . + R_4^2/2^4γ(u) f_RR(R_n) ) +O( R_4^3/^6γ(u) f_RRR(R_n) ) ] ^4γ r^-1 du = ∫ [ (6H^2 -R_4)/^2γ f_R(R_n) +(12H^2 -R_4)R_4/2 ^4γ f_RR(R_n) +O( R_4^3/^6γ(u) f_RRR(R_n) ) ] ^4γ r^-1 du = -6 H^2m_^2/𝒱_-1 m_^D-2 + O(H^6). Thus, we have obtained well known relation between the cosmological constant and the Hubble parameter at the present time c_eff≡ -2 Λ = -6H^2 +𝒱_-1m_^D-2/m_^2 + O(H^6). It is true for the pure de Sitter stage (H=const) and approximately true for slow motion. Quantum corrections and contributions to other field could change this value. §.§ May be useful in any dimensions (copied from gr-qc/0601123) Consider f(R) gravity with the action S_ J = ∫ d^D x √(\|g\|)f(R) where f is a function of the scalar curvature R calculated for the metric g of a space-time = [g]. In accord with the weak field limit f∼ R at small R, we assume f(R) >0 and f_R ≡ df/dR >0, at least in a certain range of R including R=0, but admit f_R < 0 and maybe f < 0 in general. The conformal mapping ↦ with g = F(ψ) , F= \|f_R\|^-2/(D-2), transforms the “Jordan-frame” action (<ref>) into the Einstein-frame action S = ∫ d^D x √(\|\|) [ + (ψ̣)^2 - 2V(ψ)], where ψ = ±√(D-1/D-2)log \|f_R\|, 2V(ψ) = \|f_R\|^-D/(D-2) (R\|f_R\| - f). § LIFE IN ONE BRANE § STABILITY The Fig.<ref> represents particular metric depending on the extra coordinate u. Their stability is important question. Consider classical stability in relation to a matter fluctuations - some class of fluctuations. We have continuum set of such solutions each is fixed at u=0, i.e. maximal distance from singular points. This is direct analog of the Schwarzschild metric for which the metric contains one parameter (mass M) fixed by a distant observer. Mass can vary continuously as well. Suppose that we disturb the Schwarzschild metric so that new state is described by metric with new parameter M-δ M and a point-like object with mass δ M at a distance of the horizon. If one want to preserve spherical symmetry, the spherical shell can be considered instead of a particle. The evolution of such fluctuation is well known. Point-like object move along a geodesic to the center and finally contribute to the BH mass which restores its mass. Important fact is that the fluctuation never reaches the distant observer. Hence, the final static state will be the same as the initial. The similar process runs when solutions of the metric represented in Fig.<ref>. Each extra metric and the scalar field are characterized by conditions at u=0. Therefore, if we disturb the metric and the field ζ so that δζ (u) represent point-like defect along u-axis, it will move along the geodesic line to the singular point. The fluctuation never reach the distant observer at point u=0. Hence, metric and scalar field distribution tends to the same undisturbed state. It means that the observer measured the same conditions at u=0 at t→∞. That is, the state is stable in relation to small fluctuations of the scalar field. Stability in more details. Consider the interval and system (<ref>, <ref>) Disturb them slightly γ(u) + δγ (u,t); r(u) + δ r (u,t); R(u) + δ R (u,t); ζ(u) + δζ (u,t). ????? with δ X(0) <<< X(0) The interval of argument u is artificially extended from u_-∼ -∞→ u_max and u_+ ∼ +∞ . More definitely, u(w)=A+Btanh(w-w_i/Δ); A=B, A+B=u_max The solution is stable if this excitation tends to u_max or to +∞ The solution is unstable if this excitation tends to -∞ Equations for perturbed quantities for the case f(R)=aR^2 +R +c, V(ζ) = m_ζ^2 ζ^2 /2 ( t t ) ⇒2(2aR+1)/r ^2 γ∂^2 δ r/∂ t^2 +3(2aR+1)/^2 γ∂^2 δγ/∂ t^2 -(2aR+1) ∂^2 δγ/∂ u^2 +2a ∂^2 δ R/∂ u^2 +( 4aR' -2γ'(1 +2aR) )/r∂δ r/∂ u +3H(2aR+1)/^2 γ∂δγ/∂ t +( 6aR' -2(2aR +1) ( r'/r +4 γ' ) )∂δγ/∂ u -6aH/^2 γ∂δ R/∂ t +2a(2 r'/r +3 γ' ) ∂δ R/∂ u +ζ' ∂δζ/∂ u +2 r'/r^2( -2aR' +γ'(2aR +1) ) δ r -6H^2/^2 γ (2aR +1) δγ -[ a (2γ” +8γ'^2 +4γ' r'/r +R -6H^2/^2 γ) +1/2] δ R =0, ( x x ) ⇒ ( u t ) ⇒ -6mm (2aR +1) (2/r∂^2 δ r/∂ t ∂ u +3 ∂^2 δγ/∂ t ∂ u) +2a ∂^2 δ R/∂ t ∂ u -2γ' (2aR +1)/r∂δ r/∂ t -6mm -2aγ' ∂δ R/∂ t +ζ' ∂δζ/∂ t =0, ( u u ) ⇒ ( 6 6 ) ⇒ ds^2 = dt^2 - ^2 α (t)δ_ijdx^i dx^j - ^2 β (t)((d x^4)^2 +r^2(x^4) (d x^5)^2 + ... + r^2(x^4) ∏_k=5^-2(sin^2x^k) (d x^k+1)^2 ). The scalar field fluctuations at the de Sitter stage can break the maximally symmetrical extra space metric <cit.>, which is the reason for an inhomogeneous metric formation <cit.>. Here we consider an inhomogeneous n-dimensional extra metric ds^2 = ^2γ(u)(dt^2 - ^2Htδ_ijdx^i dx^j) - du^2 - r^2(u) dΩ_-1^2 , i,j=1,3 with the renaming of the coordinate x^4 ≡ u in (<ref>). The Hubble parameter H and the metric function r(u) are defined below. The Ricci scalar R(u)= 12 H^2 ^-2 γ(u) - 8 γ” - 20 γ'^2 - (-1 ) ( 2 r^''r + 8 γ' r^'r + (-2 ) (r^'r)^2 - (-2)r^2) does not depend on time. The notation used are ' ≡ d / du and ”≡ d^2 / du^2 respectively. Then equations (<ref>) for (tt)=...=(x^3 x^3), (uu) and (x^5 x^5)=...=(x^-1 x^-1)–components and (<ref>) become R'^2 f_RRR +(R” +3 γ' R' + (-1 ) r'/r R' )f_RR - - ( γ” +4γ'^2 + (-1 )γ' r'/r - 3H^2^-2 γ(u)) f_R - f(R)2 = - ζ'^22 - V( ζ), ( 4γ'R' + (-1 ) r'r R' ) f_RR - ( 4 γ” + 4γ'^2 + (-1 ) r”r) f_R - f(R)2 = ζ'^22 - V(ζ) , R'^2 f_RRR + ( R” + 4γ' R' + ( - 2) r'r R' ) f_RR - - (r^''r + 4γ' r'/r + (-2 )(r^'r)^2 - (-2)r^2) f_R - f(R)2 = - ζ'^22 - V(ζ) , ζ” + (4 γ' + (-1)r'r) ζ' - V^'_ζ =0. Also, we will use the definition of the Ricci scalar (<ref>) as the additional unknown function to avoid 3rd and 4th order derivatives in the equations. Note that equations of this system are not independent. As four independent equations we can take, for example, the following combinations 2·(<ref>)-(<ref>)· f_R, 8·((<ref>)-(<ref>))+(<ref>)· f_R, 8·(<ref>)-(<ref>)· f_R+2·(-1)·(<ref>) and (<ref>) for the four unknown functions γ(u), r(u), R(u) and ζ(u) correspondingly. §.§ Scalar field Let us consider scalar fields distribution within the extra dimensions. It can be scalars induced "by hands". Also, scalar perturbations of the extra dimensional metric are interpreted as scalar fields for 4D observer. Its action is chosen in simplest form S_ζ=1/2∫_M_ d^ x √(\|g_\|) (∂^ζ∂_ζ - m_ζ^2ζ^2 ) with the mass of the order of the m_ and hence its fluctuations are suppressed at low energies. Therefore, we assume from the beginning that the field ζ(x^4,x^5,...) is stationary and inhomogeneous in 3 dimensions. Its dependence on the internal coordinates is defined from the classical equations as is shown below. Pocket universes are characterized by different distributions of this scalar field and hence, different energy densities. The latter could vary in wide range (0÷ m_^4) from zero to the D-dim Planck energies in the analogy to the energy fluctuations in the framework of the chaotic inflaton. As one can see from Fig.<ref> the field ζ(u) is concentrated around two opposite points with coordinates u_max and u_min. Let us confirm it analytically. Equation (<ref>) can be solved analytically if we neglect the second and third terms keeping in mind that r(u),γ(u)=const in the interval far from both peaks, see Figs. <ref> ζ(u)≃ C_1 exp[- m_ζ (u-u_min) ] + C_2 exp[+ m_ζ (u-u_max) ] Behaviour of symmetrical solution: C_1=-1, C_2=1 - coincides with those in Figures... §.§ Stability §.§.§ Particle motion near peaks Consider a small local fluctuation of the scalar field in D-dimensional space. Its evolution consists of a blurring and motion its center of mass. The latter can be considered as a particle at a coordinate u=u_0<u_max. Its geodesic trajectory is d^2 u/ds^2 = - ^2 γ(u)dγ(u)/du for a nonrelativistic particle. The derivative γ'(u)<0 near the right peak, as it is evident from Fig. <ref>. Hence, the acceleration is positive and directed to the peak. One can conclude that a matter attracts to the peak and is concentrated there. It means that at least massive particles live in the vicinity critical points. Matter distributed around different peaks does not interact each other at the classical level. The Y(u) distribution is represented in Fig. <ref>, left panel. We can guess that such static field configurations are also stable. [title=References, heading=bibintoc] Now suppose that there is a regular center at u=0, which means that not only the scalar R but also each of the Ricci tensor components R^M_N should be finite at u=0 (otherwise the invariant R_MN R^MN, being a sum of squares, will diverge). These nonzero components are R^μ_μ = 3H^2 ^-2γ - γ” -γ' (4γ' + n r'/r), μ = 0,1,2,3; R^u_u = -4 γ” -4γ'^2 - 4 n r”/r, R^a_a = n-1/r^2 - r”/r - r'/r(4 γ' + (n-1) r'/r), a = 5,6,… (there is no summing over an underlined index). Assuming that each of the variables can be presented as a Taylor series near u=0, e.g., γ(u) = γ_0 + γ_1 u + 1/2γ_2 u^2 + … and similarly for other quantities, it is easy to verify that to keep R^A_B finite at u=0, some of the Taylor coefficients should vanish, and we actually have r(u) = u + 1/6 r_3 u^3 + 1/24 r_4 u^4 + …, γ (u) = γ_0 + 1/2γ_2 u^2 + 1/6γ_3 u^3 + …. Now, we must substitute these expansions to the field equations. Doing that, it is convenient to deal with R(u) as a separate unknown function, thus avoiding the emergence of third- and fourth-order derivatives. Summing Rmm–Raa, we obtain R(u)= 12 H^2 ^-2 γ(u) - 8 γ” - 20 γ'^2 - n ( 2 r”r+ 8 γ' r^'r + (n-1) r'^2r^2) + n(n-1)r^2 (recall that we have denoted n = -1). Near u=0 we have the expansion R(u) = R_0 + R_1 u + 1/2 R_2 u^2 + … = 12 H^2 ^-2γ_0 -8(n+1) γ_2 - n(n+1) r_3 + [ -4(n+2)γ_3 - 1/3 n(n+2) r_4] u +[ - 12 γ_2 H^2 ^-2γ_0 - 4 (γ_4 +5γ_2^2) - 4/3 n (γ_4 +2γ_2 r_3) - n(n+3)/12(r_3^2 + r_5)] u^2 + …. In the scalar field equation scalar we first suppose ζ = ζ_0 + ζ_1 u + 1/2ζ_2 u^2 + …, V = V_0 + V_1 (ζ - ζ_0) + 1/2 V_2 (ζ- ζ_0)^2 + …. Inserting this into scalar, we see that there must be ζ_1=0 to avoid a divergence due to r'/r ≈ 1/u. Then, since now ζ- ζ_0 ∼ u^2, the leading terms O(1) in scalar give (1 + n) ζ_2 = V_1, hence eitherζ_2 = V_1 =0 or there is a nonzero slope V_1 of the potential at u =0. Also, even if ζ_2 0, in tt–aa we have ζ'^2 = O(u^2) whereas V = V_0 + O(u^2), with the constant V_0 entering like a cosmological constant. Furthermore, in each of the equations tt–aa the only diverging term O(1/u) is proportional to R' f_RR r'/r, which means that we have to suppose eitherf_RR =0 at R=R_0 (thus substantially restricting the choice of f(R)) orR' =0 at u=0, hence R_1 =0 in R-center, or in terms of Taylor coefficients of γ(u) and r(u), 12 γ_3 + n r_4 =0. Now we are ready to consider tt–aa near u=0. We take the function f(R) near the point u=0, where R=R_0, in the general form f(R) = f_0 + f_1 (R-R_0) + 1/2 f_2 (R-R_0)^2 + …, f_i = , and assuming f_RR(R_0) = f_2 0, we have, as concluded above, R-R_0 ∼ u^2. Thus we obtain in the order O(1) of tt–aa: 1/2 f_0 + f_1 (3 H^2 ^-2γ_0 - n γ_2) + (n-1) f_2 R_2 + V_0 =0, 1/2 f_0 - f_1 (4 γ_2 + n r_3)+ n f_2 R_2 + V_0 =0, where R_2 is the coefficient of u^2 in R-center. Equation tt0 follows from tt, and uu0 from either uu or aa, they give the same result in O(1). We must also bear in mind sca0 and R1=0. The above consideration is useful if we wish to solve tt–scalar numerically specifying the boundary conditions at some small but nonzero value u = u_1 > 0 avoiding the immediate neighborhood of u =0 where a numerical solution causes difficulty due to r→ 0 in the denominators. So, suppose we know f(R), the potential V(ζ), and the dimension n, then at u=u_1 we need to specify the following quantities (since we deal with fourth-order equations for the metric): γ, γ', γ”', r, r', r”, r”', ϕ, ϕ'. For the unknowns, we have near u=0 the Taylor expansions r,g-0 and also ζ = ζ_0 + 1/2ζ_2 u^2. One can notice that γ_0 appears only together with H and can be put to zero by redefining H. Also, ζ_0 is insignificant as simply an arbitrary zero point of ζ. So, there are the following seven parameters determining the unknowns: γ_2, γ_3, γ_4, r_3, r_4, r_5, ζ_2 (r_5 takes part in the expression for R_2 in R-center that defines R_0, R_1, R_2). These seven parameters are constrained by the four relations sca0, R1=0, tt0, and uu0, thus three quantities can be specified “by hand” even if f(R) and V(ζ) are known. For f(R)=aR^2 +R +c, V(ζ) = m_ζ^2 ζ^2, γ(0)=0 and u=0 equations (<ref> - <ref>, <ref>) give a(-R_0^2/2 +6H^2 R_0 +2(n+1)(R_0” -γ_0” R_0)) +3H^2 -R_0/2 -(n+1)γ_0” -c/2 +m_ζ^2 ζ_0^2/2=0, a(-R_0^2/2 +2 n (R_0” -r_0”' R_0 ) -8γ_0” R_0) -R_0/2 -4γ_0” -n r_0”' -c/2 +m_ζ^2 ζ_0^2/2=0, (n+1)ζ_0” -m_ζ^2 ζ_0 =0. 12 H^2 -R_0 -8(n+1) γ_0” -n(n+1)r_0”' =0. After integration over extra coordinates[∫ d^ x √(\|g_\|)= m^-1_^β_ ∫ du ∫ d^-1 x √(\|g_-1\|) = 2π^2m^-_^β_ Γ(2)∫_u_min^u_max r^-1 (u) du ≡υ_-1 m^-_^β_∫_u_min^u_max r^-1 (u) du.] using decomposition f(R) = 1/2 f_RR(R_) R_4^2 + f_R(R_) R_4 + f(R_), action (<ref>) turns to the effective theory S^ II_eff = m_^22∫ d^4 x √(\|g_4\|)(a_effR_4^2 + R_4 + c_eff) for a specific form of the function f(R) as (<ref>). - Подавить сингулярности можно, наверное, использовав новые функции типа f(R) = ( R + (a/2)R^2)^-R^p/b+c . a,b,c,p - non negative parameters. Or f(R) = R ^-R^p/b+c . Or f(R) = a^-R^p/b+c . The field equations due to (<ref>) after this substitution turn into the field equations due to (<ref>). §.§ 4D spherical symmetry Consider f(R) theory applied to 4D space-time with the metric ds^2 = A(u) dt^2 - du^2/A(u) - r^2(u)dΩ^2, where we are using the so-called quasiglobal gauge g_00 g_11 = -1. Let us work in the Einstein frame, writing the action as S = 1/2∫√(-g) d^4 x [ R + 2 g_̣μϕ_̣νϕ - 2V(ϕ)], where R is the scalar curvature, g = (g), =+1 corresponds to a normal scalar field ϕ, and = -1 to a phantom one. The field equations can be written as follows: 2(A r^2 ϕ')' r^2 dV/dϕ, (A'r^2)' - 2r^2 V, r”/r - ϕ'^2, A (r^2)” - r^2 A” 2, -1 + A' rr' + Ar'^2 r^2 ( A ϕ'^2 -V), The scalar field equation (<ref>) follows from (<ref>)–(<ref>), which, given the potential V(ϕ), form a determined set of equations for the unknowns r(u), A(u), ϕ(u). Eq. (<ref>) (the 1 1 component of the Einstein equations), free from second-order derivatives, is a first integral of (<ref>)–(<ref>) and can be obtained from (<ref>)–(<ref>) by excluding second-order derivatives. Thus it is sufficient to consider 00–02. Let us find out under which circumstances we can have a regular center at (say) x =0. The corresponding necessary and sufficient conditions are r(x) → 0, A(x) = 1 + O(r^2), Ar'^2 = 1 + O(r^2), A(x) = 1 + 1/2 A_2 x^2 + …, r(x) = x + 1/6 r_3 x^3 + …, where A_2 and r_3 are Taylor coefficients. We also suppose ϕ = ϕ_0 + ϕ_1 x + 1/2ϕ_2 x^2 + …. Let us substitute all that to the field equations (including e-phi and 11 because the relationships between the equations can mix the orders of magnitude in Taylor expansions). 01 gives for the scalar field ϕ_1^2 = - r_302 leads to r_3 =0, which then immediately implies ϕ_1 = 0.00 gives for the potential V(ϕ)\|_x=0 = - 3/2 A_2. 11 does not give new information on the coefficients of interest. Lastly, from e-phi it follows dV/dϕ\|_x=0 = V_ϕ(0) = 6ϕ_2. We have seen that the mapping J-E transforms f(R) gravity to the Einstein-scalar system where = +1. If the conformal factor F = f_R is finite, a regular center maps to a regular center (under a suitable change of scales along the x and t axes), a horizon to a horizon, and flat infinity to flat infinity. Thus any solution with a regular center and =1 corresponds according to J-E to a similar solution of f(R) gravity, where f(R) should be found according to V-f(R). To find examples, it is reasonable to use 00–02, specifying the function r(x). Thus, 02 is integrated leading to the relation (A/r^2)' = 2(u_0 - u)/r^4, u_0 = , and its further integration gives A(x), so that the metric becomes completely known. Then, ϕ(x) is calculated using 01, and V(x) using 00. There is, in principle, an opportunity of a regular center in that maps to a singularity in , which is possible if f_R tends there to zero or infinity, but we leave this issue beyond the scope of the present consideration. §.§ Analysis of the system (<ref>) shows that for f(R→∞)→ 0 we have ζ'→ 0, V(ζ)=0 →ζ=0. Therefore, EVEN IN PRESENCE OF CURVATURE SINGULARITY, THE MATTER SINGULARITY IS ABSENT. What is going to be discussed: 2. D-dim gravity, brane structureKB: Right now nothing can be said on branes. Let us just try to understand whether any regular solutions can be obtained. ++++++++++++++++++++ ARKADY, please, insert text and figures above, AND what we have to do with text below?: ++++++++++++++++++++ That means that we have to study the system (<ref>) (instead of (<ref>)), (<ref>), (<ref>), (<ref>) plus (<ref>) to obtain functions γ(u), r(u), R(u), и число H. One of the equations follows from others. Comparison (<ref>) and (<ref>) gives c_eff = 𝒱_-1m_^-2/m_^2∫_u_min^u_max(6H^2 ^-2 γf_R(R_) - f(R_) + f(R_) ) ^4γ r^-1 (u) du, ≃𝒱_-1m_^-2/m_^2 6H^2 ∫_u_min^u_max(-^-2 γ f_R(R_)) ^4γ r^-1 (u) du, where f_R(R_(u) )=2aR_(u) + 1 and R_(u) = 12H^2 + R_(u) so that c_eff→ 0 if H→ 0, that looks reasonable for the effective action (<ref>). By expanding the integrand in (<ref>), one obtains expression c_eff= -6H^2 υ_-1^β_m_^2/m_^2∫_u_min^u_maxf_R(R_(u)) r^-1 (u) du,
http://arxiv.org/abs/2307.00582v1	20230702143241	A novel multi-step method for the partial pole assignment in symmetric quadratic pencil with time delay	[ "Qing Liu" ]	math.NA	[ "math.NA", "cs.NA", "math-ph", "math.MP" ]	mymainaddress,mysecondaryaddress]Qing Liumycorrespondingauthor [mycorrespondingauthor]Corresponding author. [email protected] [mymainaddress]School of Mathematics and Shing-Tung Yau Center, Southeast University, Nanjing 211189, P. R. China. [mysecondaryaddress]Nanjing Center for Applied Mathematics, Nanjing 211135, P. R. China. In this paper, we study the partial pole assignment problem in symmetric quadratic pencil with time delay. A novel multi-step method is proposed to solve this problem, resulting in the undesired eigenvalues being moved to desired values, and the remaining eigenvalues unchanged. By establishing a new matrix equality relation and using a multi-step method, the problem is transformed into solving linear systems with low order. Specifically, assuming that there are p undesired eigenvalues requiring reassigned, the size of the linear system we finally solved is p^2. Notably, the method demonstrates high efficiency for large systems with only a few poles requiring reassigned. Numerical examples are provided to illustrate the effectiveness of the proposed method. Inverse problem; Second-order linear system; Partial pole assignment; Multi-step method; Time delay. § INTRODUCTION In structural dynamics (e.g., bridges, buildings, airplanes and automobiles), the finite element method is used to discretize the damped linear system with n degrees of freedom, and the second-order linear differential equation with constant coefficients is generated as follows: M 1ptẍ(t) + C 1ptẋ(t) + Kx(t) = f(t-τ), where M,C,K ∈ℝ^n × n are the symmetric mass matrix, damping matrix and stiffness matrix with M positive definite, x(t) ∈ℝ^n is the displacement vector, f(t-τ) is the external force control or control vector and τ is the time delay between the measurements of the state and the actuation of the control. Let f(t-τ)=0, then (<ref>) may be solved by using the method of separation of variables. Substituting x(t) = xe^λ t, where λ∈ℂ, x ∈ℂ^n, into (<ref>) yields the quadratic eigenvalue problem of the open-loop system: P(λ )x: = (λ ^2M + λ C + K )x = 0, where P(λ ) = λ ^2M + λ C + K is called the open-loop quadratic pencil, λ is referred to as the eigenvalue or complex frequency or pole, and x is called the eigenvector or vibration mode. In practical engineering problems, only a few poles are unstable and lead to the system resonance. In order to prevent system instability or resonance, it is essential to modify the unstable poles of P(λ ) through feedback control. At the same time, it is crucial to ensure that the rest of the system's eigenstructure remains unaltered, thereby preserving the property of no spill-over. This leads to the partial quadratic pole assignment problem <cit.>. The system can be controlled by the external force f(t-τ) = Bu(t-τ), and the corresponding equation of state is described by M 1ptẍ(t) + C 1ptẋ(t) + Kx(t) =Bu(t-τ), where B ∈ℝ^n × m is the full column rank control matrix and u(t-τ)∈ℝ^ m is the feedback control vector. If m =1, the system (<ref>) is reduced into the single-input control system, and if m > 1, it is called the multi-input control system. In this paper, u(t-τ) is chosen as the displacement-velocity feedback control u(t-τ) = G^⊤ x(t-τ)+F^⊤ẋ(t-τ), where G, F ∈ℝ^n× m are the displacement feedback matrix and velocity feedback matrix, respectively. Substituting (<ref>) into (<ref>), the following second-order closed-loop system can be obtained: M 1ptẍ(t) + C 1ptẋ(t) + K 1pt x(t) = B(F^⊤ẋ(t-τ) + G^⊤x(t-τ) ) . It follows that the corresponding quadratic eigenvalue problem with time delay is P_τ(λ )x: = [λ ^2M + λ (C - BF^⊤e^-λτ) + ( K - BG^⊤e^-λτ) ]x = 0, where P_τ(λ )= λ ^2M + λ (C - BF^⊤e^-λτ) + ( K - BG^⊤e^-λτ). Suppose that {λ _i}_i = 1^p(p< 2n) are undesired poles of the second-order system (<ref>), which need to be reassigned to the desired poles {μ _i}_i = 1^p. This leads to the following problem, known as the partial quadratic pole assignment problem with time delay. Problem 1. Given the symmetric matrices of the second-order system M,C,K∈ℝ^n× n with M positive definite, the full column rank control matrix B ∈ℝ^n × m, the time delay τ> 0, the self-conjugate subset {λ _i}_i = 1^p( p < 2n) of the open-loop spectrum {λ _i}_i = 1^2n and the corresponding eigenvector set {x_i}_i = 1^p, and the desired self-conjugate set {μ_i}_i = 1^p. Find the state feedback matrices F,G ∈ℝ^n × m such that the closed-loop delayed pencil P_τ(λ ) has the desired eigenvalues {μ_i}_i = 1^p and the eigenpairs {( λ _i,x_i)}_i = p + 1^2n. If the system (<ref>) is partially controllable <cit.> with respect to the eigenvalues λ=λ_i (i=1,2,…,p), then rank{λ ^2M + λ C + K,B} = n. In this paper, we assume that {λ _i}_i = 1^p and {μ _i}_i = 1^p are all distinct, the system (<ref>) is partially controllable with respect to {λ _i}_i = 1^p, and {μ_i }_i=1^p ∩{λ_i }_i=1^2n = ∅, {λ_i }_i = 1^p ∩{λ_i }_i=p+1^2n = ∅. Let Λ =diag(λ_1,λ_2, … ,λ_2n), X = [x_1,x_2, … ,x_2n], Λ_1 = diag(λ_1,λ_2, …, λ_p ), Λ_2 = diag(λ_p+1,λ_p+2, …,λ_2n), X_1 = [x_1, x_2, … , x_p], X_2 = [x_p+1,x_p+2, … ,x_2n]. Datta et al <cit.> derived the following three orthogonality relations Λ X^⊤ MX Λ -X^⊤ KX=D_1, Λ X^⊤ CX Λ + Λ X^⊤ KX + X^⊤ KX Λ =D_2, Λ X^⊤ MX + X^⊤ MX Λ +X^⊤ CX =D_3, where D_1,D_2,D_3 ∈ℂ^2n × 2n are diagonal matrices. Via orthogonality relation (<ref>), Datta et al <cit.> presented the explicit solution of feedback control matrices F and G for the partial quadratic pole assignment problem for the single-input system. Ram and Elhay <cit.> generalized it to the multi-input system such that a multi-step method for the partial quadratic pole assignment problem is developed. However, this method cannot reassign zero or near zero poles and needs to solve (m· p) n-order linear systems. Based on the receptance method <cit.>, Ram et al <cit.> proposed an algorithm for the partial quadratic pole assignment problem with time delay for the single-input system. Bai et al <cit.>, Ram and Mottershead <cit.>, Chen and Xie <cit.>, and Richiedei et al <cit.> developed the receptance methods for the partial quadratic pole assignment problems for the multi-input system. However, the receptance matrix needs to be measured by modal tests, and the receptance matrix is sensitive to the test noise <cit.>. Mao <cit.> provided a method for solving the partial quadratic pole assignment problem without using the receptance matrix. However, this method needs to solve the Sylvester equation. Liu and Yuan <cit.> developed a multi-step method for solving the partial quadratic pole assignment problem with time delay. However, this method needs to solve (m· p) n-order linear systems, which the computational cost is relatively expensive for the large-scale system. In this paper, we study an efficient numerical method for the partial pole assignment in symmetric quadratic pencil with time delay. The main innovations presented in this paper can be summarized as follows. Firstly, we provide an explicit solution to Problem 1 for the single-input system, utilizing the third orthogonality relation (<ref>) and the inverse of the Cauchy matrix. This method involves only the multiplication of matrices and vectors. Secondly, for the multi-input system, we establish a new connection between system matrices, undesired eigenvalues and corresponding eigenvectors, as well as desired eigenvalues and control vectors. This enables us to develop a novel multi-step method that transforms Problem 1 into solving p^2-order linear systems, and does not involve solving n-order linear systems. This approach proves to be highly efficient for large-scale systems, as it only requires reassigning a small number of poles, i.e. it is particularly suitable for p≪ n. The remaining of this paper is organized as follows. In section <ref>, we present an explicit solution to Problem 1 for the single-input system. In section <ref>, a novel multi-step method is proposed to solve Problem 1 for the multi-input system. Numerical examples are presented in section <ref>. Finally, we draw some conclusions in section <ref>. § SINGLE-INPUT CONTROL In this section, we consider the single-input control system, which means m=1, B = b ∈ℝ^n, F = f ∈ℝ^n, G = g ∈ℝ^n. To begin with, we provide the following three lemmas. <cit.> The single-input control system (<ref>) is partially controllable with respect to the eigenvalues λ _i if and only if b^⊤ x_i ≠ 0 (i=1,2, …, p). According to the orthogonality relation (<ref>), we can obtain Λ_1 X_1^⊤ MX_2 + X_1^⊤ MX_2 Λ_2 +X_1^⊤ CX_2 =0. Therefore, the following lemma can be verified. <cit.> For arbitrary β∈ℂ^p, define f = MX_1β, g = (MX_1Λ_1 + CX_1)β , then the closed-loop delayed pencil P_τ(λ ) has the eigenpairs {( λ _i,x_i)}_i = p + 1^2n, that means MX_2Λ_2^2 + CX_2Λ_2- bf^⊤X_2Λ_2e^-τΛ_2 + KX_2 - bg^⊤X_2e^-τΛ_2 = 0. Lemma <ref> indicates that the feedback vectors f and g defined by (<ref>) can make the closed-loop delayed pencil P_τ(λ )= λ ^2M + λ(C - bf^⊤e^-λτ ) + ( K - bg^⊤e^-λτ) keep the remaining eigenstructure of the original system unchanged, i.e., the no spill-over property is preserved. Assume that {(μ_i,y_i) }_i=1^p are the eigenpairs of the closed-loop delayed pencil P_τ(λ ), define Σ_1 = diag(μ_1, μ_2, …, μ_p), Y_1 = [y_1, y_2, …, y_p], H = (h_ij) =Λ_1X_1^⊤ MY_1+X_1^⊤ MY_1Σ_1+X_1^⊤ CY_1. <cit.> If the single-input control system (<ref>) is partially controllable with respect to the eigenvalues {λ_i }_i=1^p, {λ _i}_i = 1^p and {μ _i}_i = 1^p are all distinct and satisfy (<ref>), and {y_i }_i=1^p are the solutions of the linear equations (μ_i^2M+μ_i C+K )y_i=b, i=1,2, …, p , then H is nonsingular, and H = H̃Ĥ, where H̃ = diag(b^⊤ x_1, b^⊤x_2, …, b^⊤ x_p), Ĥ = ( [ 1/μ_1 - λ_1 ⋯ 1/μ_p - λ_1; ⋯ ⋯ ⋯; 1/μ_1 - λ_p ⋯ 1/μ_p - λ_p ]). From Lemma <ref>, solving Problem 1 is transformed into determining vector β in (<ref>) such that MY_1Σ_1^2 + CY_1Σ_1 - bf^⊤ Y_1Σ_1e^-τΣ_1 + KY_1 - bg^⊤ Y_1e^-τΣ_1 = 0. Substituting f and g defined in (<ref>) into (<ref>), and according to the definition of H in (<ref>), we can derive MY_1Σ_1^2 + CY_1Σ_1 + KY_1 = bβ^⊤(Λ_1X_1^⊤MY_1 + X_1^⊤MY_1Σ_1 + X_1^⊤CY_1)e^-τΣ_1 = bβ^⊤H e^-τΣ_1 : = bγ, where γ = β^⊤ H e^-τΣ_1. Moreover, the vector γ is related to the eigenvector Y_1 of the closed-loop delayed pencil P_τ(λ ). In order to obtain Y_1, we can calculate the eigenvectors {y_i }_i=1^p from the linear equations (μ_i^2M+μ_i C+K )y_i=b, i=1,2, …, p, which corresponds to the selection γ=[1,1,…,1] ∈ℝ^p. Therefore, parameter vector β satisfies β^⊤ H = [1,1,…,1]e^τΣ_1. According to Lemma <ref>, the linear system (<ref>) has a unique solution, then we can obtain the following result. If the single-input control system (<ref>) is partially controllable with respect to the eigenvalues {λ_i }_i=1^p, {λ _i}_i = 1^p and {μ _i}_i = 1^p are all distinct and satisfy (<ref>), and {y_i }_i=1^p are the solutions of (<ref>), then Problem 1 has a unique solution, which can be expressed by (<ref>), where β is the solution of the linear system (<ref>). In fact, we need not to solve p linear systems (<ref>). Since Ĥ defined in (<ref>) is a Cauchy matrix, define T = (t_ij)_p × p = Ĥ^-1 and β = [β_1, β_2, …, β_p]^⊤, based on the inverse of the Cauchy matrix <cit.>, we can derive t_ij = ∏_k=1^p (λ_j - μ_k) ∏_k=1^p (μ_i - λ_k)/(λ_j - μ_i)∏_k=1,k i^p (μ_i - μ_k)∏_k=1,k j^p (λ_j - λ_k). Therefore, the explicit expression of β in (<ref>) is given by β_j = 1/b^⊤ x_j∑_i=1^p t_ij e^τμ_i. Substituting t_ij defined in (<ref>) into (<ref>), we can obtain β_j = 1/b^⊤ x_j∏_k=1^p (λ_j - μ_k)/∏_k=1,k j^p (λ_j - λ_k)∑_i=1^p ∏_k=1^p (μ_i - λ_k)/(λ_j - μ_i)∏_k=1,k i^p (μ_i - μ_k)e^τμ_i. The explicit expression of β in (<ref>) is easy to implement without solving any linear system. If time delay τ = 0, (<ref>) can be further simplified. Datta et al <cit.> prove that ∑_i=1^p ∏_k=1,k j^p (μ_i - λ_k)/∏_k=1,k i^p (μ_i - μ_k) = 1, then, β_j = 1/b^Tx_j(μ_j - λ_j)∏_k=1,k j^p λ_j - μ_k/λ_j - λ_k. It is easy to prove that if the complex eigenvalues of Λ_1 and Σ_1 appear as conjugate pairs, the velocity feedback vector f and displacement feedback vector g calculated by this method are both real. Based on Theorem <ref>, we can summarize the following algorithm. Algorithm 1 A direct method for Problem 1 by single-input control. Input: The symmetric matrices M,C∈ℝ^n× n with M positive definite; The control vector b ∈ℝ^m and the time delay τ > 0; The self-conjugate eigenpairs {(λ_i,x_i)}_i=1^p and a self-conjugate set {μ_i}_i = 1^p. Output: State feedback real vectors f and g. 1: Form Λ_1 = diag(λ_1,λ_2, … ,λ_p) and X_1 = [x_1,x_2, … ,x_p]; 2: For j = 1,2, …, p, calculate β_j according to (<ref>); 3: Form β = [β_1,β_2, … ,β_p]^⊤; 4: Compute f = MX_1β, g = (MX_1Λ_1 + CX_1)β. § MULTI-INPUT CONTROL In this section, we consider Problem 1 for the multi-input control system, which means m>1. The closed-loop system (<ref>) can be written as M ẍ(t) + C ẋ(t) + Kx(t) = ∑_k=1^m b_k (f_k^⊤ẋ(t-τ) + g_k^⊤ x(t-τ)), where b_k, f_k, and g_k are the k-th columns of B, F and G, respectively. The corresponding closed-loop delayed pencil is given by P_τ(λ) = λ^2M + λ(C - ∑_k=1^m b_kf_k^⊤ e^-λτ) + K - ∑_k=1^m b_kg_k^⊤ e^-λτ. For j = 1, 2, … ,p and k = 1, 2, … ,m, define ξ_jk = η_jk + k/m(μ_j - η_jk), ξ_jm = μ_j. Let C_1 = C, C_k = C - ∑_i=1^k-1 b_if_i^⊤ e^-λτ, K_1 = K, K_k = K - ∑_i=1^k-1 b_ig_i^⊤ e^-λτ , where k = 2,3, …, m, then Problem 1 can be transformed into partial pole assignment problems for m single-input second-order linear systems. For k = 1, f_1 and g_1 can be obtained by Algorithm 1 such that the eigenvalues {λ _i}_i = 1^p of the open-loop pencil P(λ ) are reassigned to {ξ_j1}_j=1 ^p. However, the system matrices C_2 and K_2 are no longer symmetric after the first-step assignment. Therefore, Algorithm 1 cannot be applied for subsequent assignment. For multi-input control systems, we propose the following multi-step method to solve Problem 1. Suppose that after (k-1)-step assignment, where 2≤ k<m, the system M ẍ(t) + C_k ẋ(t) + K_k x(t) = 0 has the eigenvalues {ξ_j,k-1}_j=1^p and the eigenpairs {( λ _i,x_i)}_i = p + 1^2n. For the k-step, we need to find the feedback vectors f_k, g_k ∈ℝ^n, by single-input feedback control M ẍ(t) + C_kẋ(t) + K_kx(t) = b_k(f_k^⊤ẋ(t)e^-λτ + g_k^⊤ x(t)e^-λτ), such that the eigenvalues {ξ_j,k-1}_j=1^p of the system (<ref>) are reassigned to {ξ_j,k}_j=1^p and the remaining eigenstructure keeps the no spill-over property. After m-step assignment, the feedback matrices F and G can be obtained. Then finding the feedback vectors f_k, g_k ∈ℝ^n can be expressed as the following problem. Problem 2. Given M, C, K, B, Λ_1, X_1, τ and Σ_1. For k = 1, 2,…, m, let {ξ_jk}_j=1^p and {C_k,K_k } be defined by (<ref>) and (<ref>) respectively. Find the two feedback vectors f_k and g_k in sequence such that the single-input closed-loop delayed pencil P_τ k(λ) = λ^2M + λ (C_k - b_kf_k^⊤ e^-λτ) + K_k - b_kg_k^⊤ e^-λτ has the desired eigenvalues {ξ_jk}_j=1^p and the eigenpairs {(λ_i, x_i) }_i=p+1^2n. <cit.> For k = 1, 2,…, m, and arbitrary β_k∈ℂ^p, define f_k = MX_1β_k, g_k = (MX_1Λ_1 + CX_1 )β_k, then the single-input closed-loop delayed pencil P_τ k(λ) has the desired eigenpairs {(λ_i, x_i) }_i=p+1^2n, i.e., the no spill-over property is preserved. Let {(ξ_jk,y_jk)}_j=1^p be the eigenpairs of the P_τ k(λ), and define D_k = diag(ξ_1k,ξ_2k, … ,ξ_pk), Y_k = [y_1k,y_2k, … ,y_pk], H_k =(h_ls^(k))= X_1^⊤ MY_kD_k + Λ_1X_1^⊤ MY_k + X_1^⊤ CY_k. From Lemma <ref>, solving Problem 2 is transformed into determining vector β_k such that MY_kD_k^2 + C_kY_kD_k- b_kf_k^⊤Y_kD_ke^-τ D_k + K_kY_k - bg_k^⊤Y_ke^-τ D_k = 0. Substituting f_k, g_k defined in (<ref>) into (<ref>), and according to the definition of H_k in (<ref>), we have MY_kD_k^2 + C_kY_kD_k + K_kY_k = b_kβ_k^⊤(X_1^⊤ MY_kD_k + Λ_1X_1^⊤ MY_k + X_1^⊤ CY_k)e^-τ D_k = b_kβ_k^⊤ H_k e^-τ D_k:= b_kγ_k, where γ_k = β_k^⊤ H_k e^-τ D_k. Similarly, choose γ_k=[1,1,…,1] ∈ℝ^p, then β_k^⊤ H_k = [1,1,…,1]e^τ D_k. From (<ref>), we can solve the linear systems (ξ_jk^2M + ξ_jkC_k + K_k)y_jk = b_k, j = 1,2, … ,p to get Y_k by choosing the appropriate parameters {η_jk} such that the coefficient matrix of linear systems (<ref>) are nonsingular. Then the matrix H_k can be formed from (<ref>), and β_k in (<ref>) can be obtained by solving the linear system (<ref>). Clearly, the method mentioned earlier necessitates the solution of at least ((m-1)· p) n-order linear systems. When dealing with a second-order system with a large degree of freedom, the computational expenses can be quite substantial. However, in practical engineering scenarios, the requirement to reassign poles is usually limited, with p≪ n as a general rule. To minimize computational costs, we establish a new connection among system matrices, undesired eigenvalues and their corresponding eigenvectors, desired eigenvalues, and control vectors. Subsequently, we develop an efficient algorithm for forming matrix H_k without the need to calculate Y_k. Based on the definition of H_k in (<ref>), we can derive h_ls^(k) = x_l^⊤[ξ_skM+λ_lM+C]y_sk = x_l^⊤/ξ_sk-λ_l[ξ_sk^2M+ξ_skC-(λ_l^2M+λ_lC)]y_sk = x_l^⊤/ξ_sk-λ_l(ξ_sk^2M+ξ_sk(C-∑_i=1^k-1 b_if_i^⊤e^-τD_k ) +(K-∑_i=1^k-1 b_ig_i^⊤e^-τD_k ) -(λ_l^2M+λ_lC+K) +ξ_sk∑_i=1^k-1b_if_i^⊤e^-τD_k +∑_i=1^k-1b_ig_i^⊤e^-τD_k )y_sk =x_l^⊤/ξ_sk-λ_l((ξ_sk^2M+ξ_skC_k+K_k)-(λ_l^2M+λ_lC + K) +∑_i=1^k-1b_i(ξ_skf_i^⊤+g_i^⊤)e^-τD_k )y_sk. According to (λ_l^2M+λ_lC + K)x_l=0 and (<ref>), we have h_ls^(k) = x_l^⊤ b_k/ξ_sk-λ_l + e^-τξ_skx_l^⊤(∑_i=1^k-1 b_i(ξ_skf_i^⊤+g_i^⊤) )y_sk/ξ_sk-λ_l. Let G_k = X_1^⊤∑_i=1^k-1 b_iβ_i^⊤, W_k =(w_ls^(k))= X_1^⊤∑_i=1^k-1 b_iβ_i^⊤ H_k=G_kH_k. Substituting H_k define in (<ref>) into W_k, we can obtain w_ls^(k) = x_l^⊤(∑_i=1^k-1b_iβ_i^⊤(ξ_skX_1^⊤ M + Λ_1X_1^⊤ M + X_1^⊤ C) )y_sk = x_l^⊤(∑_i=1^k-1 b_i(ξ_skf_i^⊤ + g_i^⊤) )y_sk. From (<ref>) and (<ref>), we find that each element of W_k has a strong correlation with the corresponding element of H_k. We can use this relationship to derive a low order matrix equation about H_k. Let V_k = ( [ 1/ξ_1k - λ_1 … 1/ξ_pk - λ_1; ⋮ ⋱ ⋮; 1/ξ_1k - λ_p … 1/ξ_pk - λ_p ]), A_k=diag(x_1^⊤ b_k, …, x_p^⊤ b_k), T_k = diag(e^-τξ_1k, …,e^-τξ_pk),R_k = V_kT_k. By (<ref>) and (<ref>), H_k satisfies the matrix equation described by H_k = U_k + (V_k T_k) ∗ (G_kH_k)=U_k + R_k ∗ W_k, where U_k =(u_ls^(k))= (x_l^Tb_k/ξ_sk-λ_l)=A_k V_k and ∗ represents the Hadamard product. Let A= [a_1, a_2, …, a_n ] be an m × n matrix, a_i denote the i-th column of the matrix A, the mn dimension vector vec(A)=[a_1^⊤, a_2^⊤, …, a_n^⊤]^⊤ be called the column straightening of matrix A, A ⊗ B represent the Kronecker product of matrix A and B, and I_n represent the n-order identity matrix. Then the column straightening of the matrix with the Hadamard product and Kronecker product of the matrix has the following properties. <cit.> Let A be an m × n matrix. (1) If B is an m × n matrix, then vec(A ∗ B)=diag(vec(A))vec (B); (2) If B and C are n × p and p × q matrix, respectively, then vec(ABC)=(C^⊤⊗ A)vec(B). From Lemma <ref>, the matrix equation (<ref>) can be transformed into the following p^2-order linear system [I_p^2-diag(vec(R_k))(I_p ⊗ G_k)]vec(H_k)=vec(U_k). Let Z_k=I_p^2-diag(vec(R_k))(I_p ⊗ G_k), because R_k, G_k and U_k are all known, the matrix H_k can be obtained by solving the p^2-order linear system (<ref>), and the β_k can be obtained by solving the p-order linear system (<ref>). In summary, we can derive the following theorem. For k=1,2,… m, if the single-input control system ( M, C_k,K_k , b_k) is partially controllable with respect to the {ξ_j,k}_j=1^p, {λ _i}_i = 1^p and {ξ_jk}_j=1^p are all distinct, {ξ_jk}_j=1^p ∩{λ_i }_i=1^2n = ∅, {λ_i }_i = 1^p ∩{λ_i }_i=p+1^2n = ∅, and {y_jk}_j=1^p are the solutions of (<ref>), then Problem 2 has a solution, which can be expressed by (<ref>), where β_k is the solution of linear system (<ref>). Based on the Theorem <ref> and Theorem <ref>, the calculation process of solving the multi-input partial quadratic pole assignment problem with time delay can be summarized as the following algorithm. Algorithm 2 A multi-step method for Problem 1 by multi-input control. Input: The symmetric matrices M,C∈ℝ^n× n with M positive definite; The full column rank control matrix B ∈ℝ^n× m and the time delay τ > 0; The self-conjugate eigenpairs {(λ_i,x_i)}_i=1^p and a self-conjugate set {μ_i}_i = 1^p. Output: State feedback real matrices F and G. 1: Form Λ_1 = diag(λ_1,λ_2, … ,λ_p) and X_1 = [x_1,x_2, … ,x_p]; 2: Choose η_jk and compute ξ_jk = η_jk + k/m(μ_j - η_jk), j = 1,2, … ,p, k = 1,2, … ,m; 3: Compute f_1 = MX_1β_1, g_1 = (MX_1Λ_1 + CX_1)β_1 by Algorithm 1; 4: For k = 2,3,… , m, do 4.1: Compute R_k,G_k,U_k. If cond(Z_k) is large, take another η_jk; 4.2: Compute H_k by solving the linear system (<ref>); 4.3: Compute β_k by solving the linear system (<ref>); 4.4: Compute f_k = MX_1β_k, g_k = (MX_1Λ_1 + CX_1)β_k; 5: Form F = [f_1,f_2, … ,f_m], G = [g_1,g_2, … ,g_m]. § NUMERICAL EXAMPLES In this section, we provide three examples to illustrate the effectiveness of Algorithm 1 and Algorithm 2. All the numerical examples are implemented in MATLAB R2020a and run on the PC with 3.80 GHz Inter Core i7, 10700K CPU. We use Error_1=MY_1Σ_1^2 + CY_1Σ_1 - BF^⊤ Y_1Σ_1e^-τΣ_1 + KY_1 - BG^⊤ Y_1e^-τΣ_1_F and Error_2=MX_2Λ_2^2 + CX_2Λ_2- BF^⊤X_2Λ_2e^-τΛ_2 + KX_2 - BG^⊤X_2e^-τΛ_2_F to measure the residuals of the closed-loop system. Example 1 and Example 2 show the correctness of Algorithm 2, and Example 3 shows Algorithm 2 is more efficient than the traditional multi-step method in <cit.> in the case p≪ n. Example 1. <cit.> Consider the second-order control system M=I_3, C = ([ 2.5 2 0; 2 1.7 0.4; 0 0.4 2.5 ]),K = ( [ 16 12 0; 12 13 4; 0 4 29 ]). Time delay τ = 0.1 and B = b = ( [ 1 3 3 ])^⊤. The corresponding open-loop pencil has 6 eigenvalues: λ_1,2 = - 0.0129 ± 1.4389i, λ_3,4 = - 1.3342 ± 5.2311i, λ_5,6 = - 2.0030 ± 4.7437i. In order to make the system more stable, the first two eigenvalues λ_1,2 = - 0.0129 ± 1.4389i are reassigned to μ_1 =- 0.2 and μ_2 =- 0.3, and the remaining eigenvalues of the original system are kept unchanged. Using Algorithm 1, we can obtain β_1 = ( - 0.3023 + 0.4678i , - 0.3023 - 0.4678i)^⊤, the feedback vectors F = f = ( [ 0.1428; -0.1541; 0.0215 ]),G = g = ( [ -0.9698; 1.2224; -0.1852 ]), and Error_1 = 6.0497e-15, Error_2 = 1.9486e-13. Example 2. Consider the same system matrices in Example 1 with time delay τ = 0.1 and control matrix B = ( [ 1 2; 3 2; 3 4 ]). The first two eigenvalues λ_1,2 =-0.0129 ± 1.4389i are reassigned to μ_1 =-0.2 and μ_2 =- 0.3, and the remaining eigenvalues of the original system are kept unchanged. Let η_jk = λ_j, j=1,2,…,p. Applying Algorithm 2, we can obtain β_1 = ( - 0.3779 + 0.4117i , - 0.3779 - 0.4117i)^⊤, β_2 = ( - 0.4108 - 0.3047i , - 0.4108 + 0.3047i)^⊤, the feedback matrices F = ( [ 0.0220 - 0.6561; - 0.0131 0.7658; 0.0005 - 0.1141 ]), G = ( [ - 1.0119 - 0.2284; 1.2347 0.0669; -0.1844 0.0047 ]), and Error_1 = 1.5638e-12, Error_2 = 2.0668e-13. Example 3. Consider a mass-damp-spring system with n degrees of freedom shown as in Figure <ref>. The system matrices are M =diag(m_1,m_2,…,m_n), C = ( [ c _1 + c _2 - c _2 ; - c _2 c _2 + c _3 - c _3 ; ⋱ ⋱ ; - c _n c _n ]), K = ( [ k _1 + k _2 - k _2 ; - k _2 k _2 + k _3 - k _3 ; ⋱ ⋱ ; - k _n k _n ]). Let M=I_n, c_1=k_1=0, c_i=8, k_i=150, i = 2,3, … ,n, time delay τ = 0.1 and control matrix B = [e_1,e_2], where e_1 and e_2 represent the first and second columns of matrix I_n, respectively. Let the degrees of freedom of the system be 500, 1000, 2000, 3000, 4000 and 5000. In each of the six cases, only one eigenvalue is unstable, and they are λ_969=7.8049e-14, λ_1993=6.4293e-13, λ_3997=1.3911e-12, λ_5999=1.5965e-12, λ_7999=1.9989e-07 and λ_9999=1.6973e-07, respectively. These eigenvalues are all reassigned to μ_1 = -0.2. Let η_jk = λ_j, j=1,2,…,p, the numerical results are shown in Table <ref>. Algorithm 2 shows that the calculations related to the degree of freedom n are all the multiplication of matrices and vectors, and do not involve solving n-order linear systems. Specifically, we only need to solve the p-order linear system (<ref>) and the p^2-order linear system (<ref>). Therefore, the time consumption of Algorithm 2 is much less than that of Algorithm in <cit.>, and the results in Table <ref> are consistent with the theoretical prediction. Moreover, as the degree of freedom of the system increases, the time consumption of Algorithm <cit.> increases markedly, while the time consumption of Algorithm 2 does not increase much. § CONCLUSION In this paper, the partial pole assignment problem in symmetric quadratic pencil with time delay is considered. We present a simple explicit solution for this problem for the single-input system, which involves only the multiplication of matrices and vectors. For the multi-input system, by establishing a new matrix equality relation, we develop a novel multi-step method to transform this problem into solving linear systems with low order. Numerical examples show that the proposed method is effective, and both the computational time and cost of the method proposed in this paper are markedly reduced than the traditional multi-step method in the case p≪ n. Our method is particularly well-suited for the large-scale system where only a small part of the spectrum needs to be reassigned. § DECLARATION OF COMPETING INTEREST The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. § DATA AVAILABILITY Data will be made available on request. § ACKNOWLEDGMENTS The author sincerely thanks Professor Hua Dai for the careful guidance on the partial pole assignment in symmetric quadratic pencil. § REFERENCES 100 url<#>1urlprefixURL href#1#2#2 #1#1 1 B.N. Datta, D.R. Sarkissian, Theory and computations of some inverse eigenvalue problems for the quadratic pencil, Contemp. Math. 280 (2001) 221-240. 2 W.W. Lin, J.N. Wang, Partial pole assignment for the quadratic pencil by output feedback control with feedback designs, Numer. Linear Algebra Appl. 12 (2005) 967-979. 3 S. Xu, J. 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http://arxiv.org/abs/2307.00451v2	20230702015920	Dissipative Preparation of Many-Body Spin Steady States Using Trapped Ultracold Atoms	[ "Roland Cristopher F. Caballar" ]	cond-mat.quant-gas	[ "cond-mat.quant-gas" ]	APS/123-QED [email protected] Department of Physics and Mathematics, College of Science Research Center for the Natural and Applied Sciences University of Santo Tomas, España Blvd., Sampaloc, Manila, Philippines This article presents a dissipative method of creating a spin steady state, or a state whose spin expectation values approaches a fixed value over time, using a trapped gas of ultracold atoms coupled to a background BEC. The ultracold atoms are trapped in a double potential well embedded in a wide harmonic trap, which has a higher energy level than the double wells. The trapped atoms are then excited out of the double well trap into the harmonic trap using Raman lasers. Due to the coupling of the system to the background BEC, the atoms are then able to return to the double potential well by emitting an excitation into the background BEC, which serves as a reservoir of these excitations. By repeatedly coupling and uncoupling the trapped ultracold atoms and the background BEC over fixed intervals of time, the expectation value of the total spin of these atoms will, over time, reach a steady - state value. Dissipative Preparation of Many - Body Spin Steady States Using Trapped Ultracold Atoms Roland Cristopher F. Caballar August 1, 2023 ======================================================================================= § INTRODUCTION The role of dissipation in quantum dynamics has been and continues to be an active area of research <cit.>. In particular, quantum dissipation has been used as a resource to prepare quantum states that are used in both quantum computing and quantum information <cit.>. One advantage of the use of dissipative methods in quantum state preparation is that by interacting with an environment with a much larger number of degrees of freedom, a quantum system will, over time, eventually attain a steady state with regards to some physical property, thus allowing for a minimal amount of control on the part of the experimenter. One particular dissipative quantum state preparation system involves the use of single trapped atoms which are coupled to a reservoir and whose ground states are coupled to their excited states via Raman lasers with a given detuning and Rabi frequency. Examples of these dissipative quantums state preparation schemes are described in Refs. <cit.>, wherein individual atoms are trapped in optical fields. The atoms are excited from their ground states to one or more of their excited states, and they decay back to their ground states via spontaneous emission of photons into the optical trap, which act as a reservoir of these photons. Through this driven - dissipative mechanism, the atom then evolves over time towards a steady state, with steady state to which it evolves to dependent on the type of atom that is trapped, as well as the trap configuration (e. g. optical lattice, optical cavity or optical tweezers). The resulting states prepared are of interest in quantum computation and quantum information. However, it is also possible to use many - body systems such as trapped bosonic or fermionic atoms or Bose - Einstein Condensates (BECs) for dissipative quantum state preparation schemes, as shown in Refs. <cit.>. The dissipative quantum state preparation schemes described in Refs. <cit.>, in particular, are of interest because instead of using optical fields, they make use of superfluids or BECs as the bath or reservoir of excitations. This quantum state preparation scheme has the advantage of being able to prepare many - body quantum states which are of interest not just in quantum information and quantum computation, but also for more general purposes such as those produced in Refs. <cit.>, wherein the resulting steady - state of the initial many - body system consisting of spin - 1/2 particles is a BEC, and that produced in Ref. <cit.>, which is a p - wave superconductor prepared using a finite number of one - dimensional fermions. Finally, it should be noted that it is also possible, as demonstrated in Ref. <cit.>, to formulate a dissipative quantum state preparation scheme wherein macroscopic systems such as mechanical resonators serve as reservoirs of excitations for a quantum system, in this case an ensemble of microwave photons, which results in the photons behaving coherently. The dissipative quantum state preparation schemes mentioned above are but a sampling of many others that have been proposed and implemented over the years. However, focusing on the dissipative quantum state preparation schemes formulated in Refs. <cit.>, we see that it is possible to induce collective quantum behavior in the form of Bose - Einstein condensation or superconductivity via dissipative mechanisms. For BEC preparation, in particular, these results are significant, considering that the standard method of preparing BECs via optical trapping and laser cooling <cit.> requires that the gas of atoms be isolated from the surrounding environment to reduce the risk of thermal losses. This in turn requires a high degree of control over the BEC preparation process. However, by introducing dissipation as a dynamical resource, this significantly reduces the degree of control required of the experimenter during the process, since the dissipative dynamics can be used to drive the time evolution of the gas of atoms towards a BEC. Hence, motivated by these dissipative BEC preparation mechanisms, this paper proposes a dissipative preparation mechanism using a gas of trapped ultracold atoms coupled to a background BEC which will serve as a reservoir of excitations. The mechanism proposed in this paper is based on that formulated in Ref. <cit.>, which in turn is a proposed physical realization of the theoretical mechanism proposed in Ref. <cit.> for the preparation of number - and phase - squeezed states. However, instead of producing squeezed states, the dissipative preparation mechanism described in this paper will produce spin steady states, that is, states whose expectation value of their total spin remains constant over time. It is to be noted that this is not the first dissipative quantum state mechanism which will affect the spin of a many - body quantum system. Dissipative quantum state preparation systems that are capable of controlling the spin of many - body quantum systems have been proposed in Refs. <cit.>. However, the significance of this many - body dissipative quantum preparation scheme is its use of a background BEC as the environment to which the many - body ultracold atom system is coupled, with interatomic interactions between the atoms comprising the trapped ultracold atom gas and the background BEC, instead of quantum electrodynamic interactions between these trapped atoms and an optical cavity, being the mechanism that enables the dissipative dynamics of the system. The rest of this paper proceeds as follows. In section II, we describe the components for the dissipative quantum state preparation scheme, specifying in particular the form of the system and reservoir Hamiltonians, the trapping potential to be used for the ultracold atom gas, and the interaction Hamiltonian that will describe the coupling between the trapped ultracold atom gases and the background BEC. In section III, we examine the dissipative dynamics of the system as a result of its coupling with the reservoir, deriving in particular the quantum master equation that will describe the time evolution of the trapped ultracold atom gas as it interacts with the background BEC. Section IV presents the numerical and graphical results from the implementation of this dissipative state preparation scheme, demonstrating in particular that the resulting state prepared using this mechanism will have total spin expectation values that will evolve over time to a steady state. Section V summarizes the results obtained in this paper and outlines further applications and prospects for future work. § COMPONENTS OF THE DISSIPATIVE QUANTUM STATE PREPARATION SCHEME §.§ System and Reservoir Hamiltonians The physical system to be used for the dissipative BEC preparation scheme is a gas of ultracold bosonic atoms, trapped in a modified double - well potential, similar to the system used in Ref. <cit.>, with the schematic diagram and its corresponding energy levels shown in figure <ref>. As shown in the figure, the narrow double wells in the potential are centered at x=± x_0, which contain the degenerate ground states ϕ_g,±(x) corresponding to the energy ϵ_g. The double wells, in turn, are embedded in a wide harmonic potential, which contains the excited state ϕ_e,0(x) corresponding to the energy ϵ_e. The Hamiltonian for the trapped ultracold atom system can then be written in its second quantized form as Ĥ_S=ϵ_g(â^†_g,+â_g,++â^†_g,-â^†_g,-)+ϵ_e â^†_e,0â_e,0, where the operators â_g,± and â^†_g,± are the annihilation and creation operators, respectively, corresponding to the degenerate ground state energy level ϵ_g at the location of the double wells x=± x_0, and the operators â_e,0 and â^†_e,0 are the annihilation and creation operators, respectively, corresponding to the non - degenerate excited energy level ϵ_e located at x=0. We note that the Hilbert space in which the trapped ultracold atom is described has the form ℋ_S=ℋ_S,E⊗ℋ_S,p, where the subscripts E and p denote Hilbert subspaces whose basis vectors are given in terms of the eigenstates of the number and position operators N̂ and x̂, respectively. In particular, we can write the system operators â_g,±, â^†_g,± and â_e,0 in the following form: â_g,±=â_g ⊗\|± x_0 ><± x_0 \|, â_e,0=â_e ⊗\|0><0\| Here, â_g and â_e are the annihilation operators for the ground and excited energy states, with â_g \|N_g>=√(N_g)\|N_g -1> and â_e \|N_e>=√(N_e)\|N_e -1>, where \|N_g> and \|N_e> are particle number states corresponding to the ground state and excited state of the system. On the other hand, the operators \|± x_0 ><± x_0 \| and \|0 ><0 \| are position operators corresponding to the locations of the double wells x=± x_0 and x=0 in the potential diagrammed in Fig. <ref>. On the other hand, the Hamiltonian for the background BEC in which the trapped ultracold atoms are immersed is given by Ĥ_B=∑_𝐤 E_𝐤b̂^†_𝐤b̂_𝐤 In this Hamiltonian, b̂_𝐤 and b̂^†_𝐤 are the annihilation and creation operators for excitations in the BEC, with those excitations having energy E_𝐤=√(ϵ_k^0(ϵ_k^0 +2ρ_b U_B)), where ϵ_k^0 = ħ^2 k^2/2m_B is the energy of the excitations when they behave as free particles, and U_B=√(4πħ^2 a_BB/m_B) is the interaction potential, where m_B is the mass of the atoms in the background BEC, a_BB is the scattering length of the atoms in the condensate, and ρ_B is the density of atoms in the condensate. §.§ Interaction Hamiltonian As shown in figure <ref>, the trapped bosonic atoms interact with the BEC in which it is immersed by emitting excitations into it in order for them to return from the excited state ϵ_e in the harmonic potential to one of the degenerate ground states ϵ_g contained in the double well potential. This is done after the atoms, which are initially in the double well, are excited by Rabi lasers with identical Raman frequencies Ω and detuning Δ to the excited state in the harmonic potential. This interaction between the trapped atoms and the background BEC is described using the interaction Hamiltonian Ĥ_SB=4π a_SB/2μ∫ d^3 rψ̂^†_Sψ̂_Sψ̂^†_Bψ̂_B In this expression, a_SB is the inter-atomic species scattering length between the trapped ultracold atoms in the system (S) and the atoms comprising the background BEC (B), while μ is the reduced mass of the system and BEC atoms. In constructing the interaction Hamiltonian, we need the explicit form of the field operators for both the system of trapped ultracold atoms and the background BEC in which they are immersed. For the trapped ultracold atom system, the corresponding field operator has the following form: ψ̂_S = [â_g,-ϕ_g,-(x)+â_g,+ϕ_g,+(x)+â_e,0ϕ_e,0(x)] × w_y (y)w_z (z) In this field operator, ϕ_g,±(x)w_y(y)w_z(z) and ϕ_e,0(x)w_y(y)w_z(z) are the eigenfunctions corresponding to the ground state energy ϵ_g and the excited state energy ϵ_e in position representation of the trapped ultracold atom system's Hamiltonian. The field operator for the background BEC, on the other hand, can be written as ψ̂_B = √(ρ_B)+δψ̂_B where the excitation term δψ̂_B has the explicit form δψ̂_B = 1/√(V)∑_𝐤(u_𝐤e^i𝐤·𝐫b̂_𝐤+v_𝐤e^-i𝐤·𝐫b̂_𝐤), with V being the total volume of the background BEC, u_𝐤=(1-L^2_𝐤)^-1/2, v_𝐤=L_𝐤(1-L^2_𝐤)^-1/2, L_𝐤=(E_𝐤-(k^2/2m_B)-m_Bc^2)/m_Bc^2, and E_𝐤 is the excitation energy given by Eq. <ref>. In carrying out the integration necessary to obtain the interaction Hamiltonian for this system, we will be operating under the assumption that the excitations emitted by the ultracold atoms into the background BEC are sound - like, i. e. E_𝐤≈ ck, where c=√(ρ_B U_B/m_B). Also, we will assume that the x - components of the ground state and the excited state wavefunctions for the trapped ultracold atom systems will have the following forms: ϕ_g,±(x)=(m_Sω_g,x/πħ)^1/4exp(-m_Sω_g,x/2ħ(x± x_0)^2) ϕ_e,0(x)=√(2)/π^1/4(m_Sω_e,x/ħ)^3/4xexp(-m_Sω_e,x/2ħx^2) In both equations, ω_g,x and ω_e,x are the frequencies corresponding to the ground state and excited state energies ϵ_g and ϵ_e of the system, respectively, and m_S is the mass of the atoms comprising the system. Also, we assume that the transverse components w_y(y) and w_z(z) of the ground state and excited state wavefunctions have the same form as the x - component for the ground state wavefunction, given by Eq. <ref>, with ω_y and ω_z being the frequencies corresponding to these wavefunctions, and y and z replacing the x coordinate, with y_0 = z_0 = 0. Let us now proceed to the derivation of the interaction Hamiltonian itself. Substituting the explicit forms of the field operators given by equations <ref> and <ref> into equation <ref>, we obtain Ĥ_SB=2π a_SB/μ∫ d^3x w_y^2(y)w_z^2(z){ϕ_g,+^2(x)â_g,+^†â_g,+. +ϕ_g,-^2(x)â_g,-^†â_g,-+ϕ_e,0^2(x)â_e,0^†â_e,0 +ϕ_g,+(x)ϕ_e,0(x)(â_g,+^†â_e,0+â_e,0^†â_g,+) .+ϕ_g,-(x)ϕ_e,0(x)(â_g,-^†â_e,0+â_e,0^†â_g,-)} ×{ρ_B+√(ρ_B/V)∑_k(u_k +v_k )(e^i𝐤·𝐫b̂_k +e^-i𝐤·𝐫b̂^†_k)} In carrying out the substitution, we keep only the terms that are linear in the condensate field operators b̂_k and b̂^† _k. For the succeeding steps, we will work in Cartesian coordinates to facilitate ease of calculation of the interaction Hamiltonian. Using now our earlier assumption that w_y(y) and w_z(z) have the same form as ϕ_g,±(x), we can carry out the integration over the transverse variables as follows: ∫ w^2_u(u)e^± ik_u udu=√(m_S ω_g,u/πħ)∫ du e^-m_S ω_g,u/ħu^2e^± ik_u u In evaluating this integral, we can treat the ground state transverse wavefunction w_u^2 (u) as a Dirac delta function by making the assumption that ω_g,u→∞. Under this assumption, we then obtain the following: ∫ w^2_u(u)e^± ik_u udu →∫ du δ(u)e^± ik_u u=1 Also, by making the assumption that ω_g,x→∞ for the ground state wavefunctions along the x - axis, we can simplify the task of evaluating the overlap integrals ∫ dx ϕ_g,±(x)ϕ_e,0(x)e^± ik_x x in Eq. <ref> in the following manner: ∫ dx ϕ_g,±(x)ϕ_e,0(x)e^± ik_x x=2√(m_S ω_e,x/ħ)(ω_e,x/ω_g,x)^1/4 ×√(m_S ω_g,x/2πħ)∫ dx e^-m_S ω_g,x/2ħ(x± x_0)^2e^-m_S ω_e,x/2ħx^2e^± ik_x xx =± 2√(m_S ω_e,x/ħ)(ω_e,x/ω_g,x)^1/4e^-m_S ω_e,x/2ħx_0^2e^ik_x x_0x_0 ≈± 2√(m_S ω_e,x/ħ)x_0 The same approximation can be made for the overlap integral ∫ dx ϕ_g,±(x)ϕ_e,0(x)e^∓ ik_x x. For both of these overlap integrals, we assume that ω_e,x≈ω_g,x, k_x x_0<<1, and m_S ω_e,x/ħx_0^2<<1 in order to make the approximations given in Eqs. <ref>. Finally, using the same assumptions as those given in the previous paragraphs regarding ω_g,x and ω_e,x, the overlap integrals ∫ dx \|ϕ_g,±(x)\|^2 e^± ik_x x and ∫ dx \|ϕ_e,0(x)\|^2 e^± ik_x x can be evaluated as follows: ∫ dx \|ϕ_g,±(x)\|^2 e^± ik_x x→∫ dx δ(x± x_0)e^± ik_x x =e^± ik_x x_0≈ 1 ∫ dx \|ϕ_e,0(x)\|^2 e^± ik_x x→ 2(m_S ω_e,x/ħ) ∫ dx δ(x)x^2 e^± ik_x x =0 Combining all of these results, we obtain the following form of the interaction Hamiltonian: Ĥ_SB=2π a_SB/μ(ρ_B(â_g,+^†â_g,++â_g,-^†â_g,-+â_e,0^†â_e,0). +√(ρ_B/V)∑_k(u_k +v_k )(â_g,+^†â_g,++â_g,-^†â_g,-)(b̂_k +b̂_k^† ) +2√(m_S ω_e,xρ_B/Vħ)x_0∑_k(u_k +v_k ) .×((â^†_g,+-â^†_g,-)â_e,0+(â_g,+-â_g,-)â^†_e,0)(b̂_k +b̂_k^†)) §.§ Simplification of the Interaction Hamiltonian Now let us simplify this interaction Hamiltonian by first making use of the explicit form of the coefficients u_k and v_k. We will be working in the phonon limit, i. e. in the limit where the excitations have energy E_k = ck, where, again, c=√(ρ_B U_B/m_B). In the phonon limit, k is very small, and in particular, k<<m_B c so that L_𝐤=(E_𝐤-(k^2/2m_B)-m_Bc^2)/m_Bc^2≈k/m_B c-1 As such, evaluating the sum of u_k and v_k, we obtain u_k +v_k = 1+L_k/√(1-L^2_k)=√(1+L_k/1-L_k)=√(k/2m_B c/2-k/2m_B c) 1/2√(k/m_B c)(1-k/4m_B c)^-1/2≈1/2√(k/m_B c) Substituting this in Eq. <ref>, we then obtain Ĥ_SB=2π a_SB/μ(ρ_B(â_g,+^†â_g,++â_g,-^†â_g,-+â_e,0^†â_e,0). +1/2√(ρ_B/Vm_B c)∑_k√(k)(â_g,+^†â_g,++â_g,-^†â_g,-)(b̂_k +b̂_k^† ) +√(m_S ω_e,xρ_B/Vħ m_B c)x_0∑_k√(k)((â^†_g,+-â^†_g,-)â_e,0. ..+(â_g,+-â_g,-)â^†_e,0(b̂_k +b̂_k^†))) § DYNAMICS OF THE DRIVEN - DISSIPATIVE BEC PREPARATION SCHEME §.§ Time Evolved Interaction Hamiltonian The interaction Hamiltonian is evolved over time by applying the Baker - Campbell - Hausdorff (BCH) identity. In doing so, the time - evolved system and background BEC annihilation operators will then have the following form: â_j,n(t)=e^it/ħĤ_Sâ_j,ne^-it/ħĤ_S=e^-it/ħϵ_gâ_j,n, b̂_k(t)=e^it/ħĤ_Bb̂_ke^-it/ħĤ_B=e^-it/ħE_kb̂_k In obtaining these time - evolved annihilation operators, we make use of the following commutator identities, in conjunction with the BCH identity: [â_j,n,â_j',n'^†]=δ_j,j'δ_n,n', [â_j,n,â_j',n']=[â^†_j,n,â^†_j',n']=0 [b̂_k,b̂_k'^†]=δ_k,k', [b̂_k,b̂_k']=[b̂_k^†,b̂_k'^†]=0 In the first two identities, j=g,e and n=+,0,-. The time - evolved system and background BEC creation operators, on the other hand, are obtained by taking the Hermitian conjugate of the time - evolved system and background BEC annihilation operators. By evolving the interaction Hamiltonian in the manner described above, we then obtain the following explicit form of the time - evolved interaction Hamiltonian: Ĥ_SB(t)=π a_SB/μ√(ρ_B/Vm_B c)∑_k√(k)((â_g,+^†â_g,++â_g,-^†â_g,-). ×(e^-it/ħE_kb̂_k +e^it/ħE_kb̂_k^†) +2√(m_S ω_e,x/ħ)x_0(e^-it/ħ(E_k +(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂_k. +e^it/ħ(E_k -(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂^†_k +e^-it/ħ(E_k -(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂_k ..+e^it/ħ(E_k +(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂^†_k)) §.§ Derivation of the Master Equation Having obtained the time - evolved interaction Hamiltonian, we now proceed to derive the master equation which governs the time evolution of the system. To do so, we make use of the Born - Markov approximation, under which the master equation has the form dρ̂/dt=ℒρ̂=-∫_0^∞dt' Tr_R̂[Ĥ_SB(t),[Ĥ_SB(t-t'),ρ̂⊗R̂]] In this equation, R̂ is the density matrix corresponding to the background BEC, and ρ̂ is the density matrix corresponding to the trapped ultracold atom system. Let us now substitute Eq. <ref> into this equation, and evaluate the commutators and the integral accordingly. In doing so, we obtain terms proportional to Tr(b̂_k b̂_k' R̂) and Tr(b̂_k^†b̂_k'^†R̂), both of which are equal to zero, following Refs. <cit.> and <cit.>. At the same time, we will also be obtaining terms proportional to Tr(b̂_k^†b̂_k' R̂) and Tr(b̂_k b̂_k'^†R̂). Following Ref. <cit.>, under the assumption that the background BEC has a temperature T≈ 0, Tr(b̂_k^†b̂_k' R̂)≈ 0 and Tr(b̂_k b̂_k'^†R̂)≈δ_k,k', so that we will only consider terms in the master equation that are proportional to the latter expression. The explicit forms of these expressions are given by Eqs. <ref> to <ref> in the Appendices of this paper. Next, as per Eq. <ref>, we integrate Eqs. <ref> to <ref> over time and sum them over k. In doing so, we will have to evaluate integrals of the form ∑_kk∫_0^∞dt' e^±it'/ħE_k and ∑_k∫_0^∞dt e^±it'/ħ(E_k±(ϵ_e -ϵ_g)). To evaluate these terms, let us first recall that the excitations emitted by the trapped ultracold atom gas into the background BEC are phonons with energies E_k = ck. At the same time, we can replace the summation over k with an integration over the same variable, treating it as continuous instead of discrete. Finally, considering that the integrals over t' are oscillatory integrals, we can treat them as Dirac delta distributions over k. Taking these all together, we obtain the following expressions: ∑_kk∫_0^∞dt' e^±it'/ħE_k=∫ dk k δ(c/ħk)=0 ∑_kk∫_0^∞dt e^±it'/ħ(E_k±(ϵ_e -ϵ_g))=∓ħ/c(ϵ_e -ϵ_g) Substituting these terms into Eqs. <ref> to <ref> will result in the elimination of terms in the master equation which are proportional to the oscillatory integral ∫_0^∞dt' e^±it'/ħck. This leaves us with terms which are proportional to the oscillatory integral ∫_0^∞dt e^±it'/ħ(ck±(ϵ_e -ϵ_g)). Further simplification of the master equation can be made by making the assumption that 4m_S ω_e,x/ħ x^2_0 >>2√(m_S ω_e,x/ħ)x_0. Finally, we perform an adiabatic elimination of the excited states in order to express them as a linear combination of the ground states trapped in the two wells in Fig. <ref>, which can be done by ensuring that the Raman lasers coupling the ground states in the double wells to the excited state in the harmonic trap are both weak and far detuned. In so doing, and by noting that both Raman lasers have the same frequency Ω, we obtain the following expression for the excited state annihilation operator â_e,0: â_e,0≈Ω/√(2)Δ(â_g,++â_g,-) We then substitute this expression for â_e,0, together with its Hermitian conjugate, into the master equation, and group together like terms in the equation. In doing so, the master equation will have the following form: dρ̂/dt≈γ((2(ĉρ̂ĉ^†-{ρ̂,ĉ^†ĉ})-(2ĉ^†ρ̂ĉ-{ρ̂,ĉĉ^†}). .+e^-2it/ħ(ϵ_e -ϵ_g)[ρ̂,ĉĉ]-e^2it/ħ(ϵ_e -ϵ_g)[ρ̂,ĉ^†ĉ^†]) In this equation, the jump operator ĉ has the explicit form ĉ=(â^†_g,+-â^†_g,-)(â_g,++â_g,-) and the coefficient γ, also known as the coupling coefficient since it describes the strength of coupling or interaction between the system and the environment, has the form γ=4N_B Ωm_S ω_e,x/√(2)Δ(π a_SB/μ Vcx_0)^2 (ϵ_e -ϵ_g) § NUMERICAL RESULTS §.§ Time Evolution of the System Let us now carry out the time evolution of the system using the master equation given by Eq. <ref>. For our initial state, it is given by the density matrix ρ̂_0=\|ψ_0><ψ_0\|, where the initial state ket \|ψ_0 > will have the form \|ψ_0>=1/√(u)∑_j=1^u\|N_j,g>⊗\|x_0>+1/√(v)∑_j=1^v\|N_j,g>⊗\|-x_0> Here, \|N_j,g,±>=\|N_j,g>⊗\|± x_0> are eigenstates of the operator N̂_g,±=N̂_g⊗\|± x_0⟩⟨± x_0\|, where N̂_g is the particle number operator for the ground state. N̂_g,± represents a measurement of the number of particles in one of the double wells located at x=± x_0. These eigenstates have corresponding eigenvalues N_j,g,pm, and u,v≤ N is the number of particle number states corresponding to the double well located at x=± x_0, respectively in the initial state. We then substitute the initial state given by Eq. <ref> into the master equation given by Eq. <ref>, in order to evolve the state over time. §.§ Spin Steady State Formation As the trapped ultracold atom gases evolve over time, we calculate the expectation values of the SU(2) generators which describe the spin-x, spin-y and spin-z of these states. These SU(2) generators have the following form, as specified in Ref. <cit.> and <cit.>, have the form Ŝ_x = 1/2(â^†_g,+â_g,-+â^†_g,-â_g,+) Ŝ_y = -i/2(â^†_g,+â_g,--â^†_g,-â_g,+) Ŝ_z = 1/2(â^†_g,+â_g,+-â^†_g,-â_g,-) The resulting expectation values of these SU(2) generators will have the following explicit form: ⟨Ŝ_j(t)⟩ = ⟨ψ(t)\|Ŝ_j\|ψ(t)⟩, j=x,y,z. We consider a system where N=200, ϵ_e - ϵ_g = 1.0× 10^-1 and γ=1.0× 10^-5, with all quantities in natural units and where we set ħ=c=1, with the system evolved from t=0 to t=80, and the time increment being δ t=1.0× 10^-3. The resulting expectation values for the SU(2) generators Ŝ_j are shown in figure <ref>. As can be seen in the figure, both the expectation values of Ŝ_y and Ŝ_z exhibit very small variations over time compared to the variations shown by the expectation value of Ŝ_x. Furthermore, the expectation value of Ŝ_x oscillates over time, with an amplitude much larger than the expectation values of Ŝ_y and Ŝ_z. As such, for the trapped ultracold atom gases to be steady states with respect to Ŝ_j, there is a need to evolve the system in such a way that Ŝ_x will approach a fixed value as the system evolves over time. One way to achieve this is by noting that for a fixed value of δϵ = ϵ_e - ϵ_g, the amplitude of oscillations for Ŝ_x decrease as the coupling coefficient γ, whose explicit form is given by Eq. <ref>, decreases in magnitude, as shown in fig. <ref>. However, it should be noted that decreasing the value of γ alone while keeping it constant throughout the time evolution of the system will not cause the system to evolve towards a steady - state value of Ŝ_x, since the oscillatory behavior will still remain, albeit with a reduced amplitude. Instead, what can be done is to evolve the trapped ultracold atom gas in a stroboscopic manner, similar to the dissipative quantum state preparation scheme described in Refs. <cit.>. To do this, we first evolve the trapped ultracold atom gas over a time interval Δ t_1=t_1 for an initial value of the coupling constant γ=γ_1, then turn off the coupling between the system and the environment (i. e. set γ=0) for an interval of time τ_1<<t_1. We then turn the coupling between the system and the environment back on again as the system evolves over a time interval Δ t_2=t_2-(t_1+τ_1). However, unlike what was proposed in Refs. <cit.>, instead of the coupling constant remaining constant, for this time interval, the coupling constant is now reduced from γ=γ_1 to γ_2=γ_1 Δγ, where Δγ=γ_n+1/γ_n <1 for n≥ 1. We then repeat the process multiple times until the amplitude of the oscillations have been reduced significantly such that the expectation value of Ŝ_x for the time - evolved state is almost constant. The result is a state whose expectation value of Ŝ_x evolves towards a steady state, with the oscillations of this expectation value being dampened as a result of the successive decoupling and coupling (with ever - decreasing strength) of the system with the environment. To illustrate this, we compare the time evolution of the system with and without stroboscopic coupling and decoupling with the environment. As shown in figure <ref>, we can see that using the stroboscopic method of time evolution outlined above, the oscillations of ⟨Ŝ_x (t)⟩ are continuously dampened over each interval of time Δ t_n during which the system is evolved while coupled to the BEC excitation reservoir, with ⟨Ŝ_x (t)⟩ approaching a steady - state value as t increases. Contrast this with the case wherein the coupling constant of the system with the environment remains constant. If the system is evolved over time in this manner, then ⟨Ŝ_x (t)⟩ will not approach a steady - state value due to its oscillatory behavior. Now by varying the magnitude of the decrease Δγ in the coupling constant γ_n, we can control the steady - state value which ⟨Ŝ_x (t)⟩ approaches over time, as well as the interval of time that elapses before the amplitude of the oscillations have decreased sufficiently such that ⟨Ŝ_x (t)⟩ can be definitively said to be approaching its steady state value. This is shown in figure <ref>. As Δγ decreases, the steady - state value that ⟨Ŝ_x (t)⟩ approaches decreases, but so too does the interval of time over which the amplitude of oscillations of ⟨Ŝ_x (t)⟩ have decreased sufficiently so that it begins to approach its steady - state value. At this point, we have shown that evolving the trapped ultracold atom gas stroboscopically, with the coupling constant γ being decreased each time the coupling is re-established, would result in the suppression of the oscillatory behavior of ⟨Ŝ_x (t)⟩ and allow it to approach a steady - state value over time. However, there is a question of how one can achieve this stroboscopic time evolution of the trapped atom gas. To answer this question, let us recall the explicit form of the coupling constant γ, given by Eq. <ref>. It can be seen from this expression that γ is directly proportional to the Rabi frequency Ω of the laser that couples the ground state in the double wells to the excited state in the harmonic potential in which the double wells are embedded. As such, it is possible for us to reduce the magnitude of the coupling constant γ by reducing the laser's Rabi frequency until the desired value is attained. We note that while the stroboscopic method of evolving the system suppresses the oscillations of the expectation value of Ŝ_x (t), it also has a similar effect for both Ŝ_y (t) and Ŝ_z (t), as shown in figure <ref>. In particular, comparing this with figure <ref>, we find that the oscillations in both Ŝ_x (t) and Ŝ_y (t) are damped, allowing the expectation values of both observables to evolve towards a steady - state value. On the other hand, instead of increasing over time, Ŝ_z (t) is now evolving in a manner similar to both Ŝ_x (t) and Ŝ_y (t), approaching a steady - state value as it continues to evolve over time. Therefore, we can say that coupling a trapped ultracold atom gas with a background BEC that acts as a reservoir of excitations and turning this coupling on and off with decreasing strength as the system evolves over time will result in the trapped ultracold atom gas becoming a system whose spin approaches a steady - state value. § CONCLUSION We have shown, in this paper, that a gas of ultracold bosonic atoms trapped in a double - well potential embedded in a wide harmonic potential which in turn is coupled to a background BEC that acts as a reservoir of excitations will evolve towards a total spin steady state, as evidenced by the expectation value of these two observables for this trapped ultracold bosonic atom gas approaching a constant value over time. However, this steady state can only be achieved by stroboscopic coupling between the trapped ultracold atom gas and the background BEC, wherein the coupling between the system and the environment is turned on and off over fixed time intervals, with the coupling strength decreasing each time it is turned on. The resulting steady state from this dissipative quantum state preparation mechanism is of significance not just in many - body algorithms for quantum computing that can be used to simulate quantum chemistry processes <cit.>, but also for simulation of quantum many - body systems <cit.> such as the Hubbard model and spin models, for which definite values of the particle number and total spin of the system are necessary. At the same time, considering that the steady states resulting from this dissipative quantum state mechanism have definite spin, one can use this for many - body spintronic applications, such as those described in Refs.<cit.>. However, for this to be experimentally feasible, the features shown in this paper must also be seen if the number of bosonic atoms in the trapped ultracold atom gas are increased by at least one order of magnitude, which corresponds to the standard number of atoms that are present in a BEC. The author would like to acknowledge support from the Department of Mathematics and Physics of the College of Science, and the Research Center for Natural and Applied Sciences, of the University of Santo Tomas. § EVALUATION OF THE COMMUTATORS AND TRACING OUT THE BACKGROUND BEC OBSERVABLES OF THE TIME - EVOLVED INTERACTION HAMILTONIAN IN THE MASTER EQUATION Let us evaluate the explicit form of the commutators appearing in the master equation given by Eq. <ref>, afterwhich we trace out the background BEC observables, specifically the BEC creation and annihilation operators. In evaluating these commutators and tracing out these observables, we make use of the approximations Tr(b̂_k b̂_k' R̂)=Tr(b̂_k^†b̂_k'^†R̂)=0, Tr(b̂_k^†b̂_k' R̂)=0 and Tr(b̂_k b̂_k'^†R̂)≈δ_k,k'. We start with the double commutators involving both of the first terms of the time - evolved interaction Hamiltonian: Tr_R̂∑_k,k'√(kk')[(â_g,+^†â_g,++â_g,-^†â_g,-)(e^-it/ħE_kb̂_k +e^it/ħE_kb̂_k^†),[(â_g,+^†â_g,++â_g,-^†â_g,-)(e^-i(t-t')/ħE_k'b̂_k' +e^i(t-t')/ħE_k'b̂_k'^†),ρ̂⊗R̂]] =∑_kk e^-it'/ħE_k((â_g,+^†â_g,++â_g,-^†â_g,-)(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)) -∑_kk e^it'/ħE_k((â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)-ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)(â_g,+^†â_g,++â_g,-^†â_g,-)) Next, we evaluate the double commutators in the master equation involving the first and the second terms of the time - evolved interaction Hamiltonian, obtaining the following expression: 2√(m_S ω_e,x/ħ)x_0 Tr_R̂∑_k,k'√(kk')[(â_g,+^†â_g,++â_g,-^†â_g,-)(e^-it/ħE_kb̂_k +e^it/ħE_kb̂_k^†),[e^-i(t-t')/ħ(E_k' +(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂_k'.. +e^i(t-t')/ħ(E_k' -(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂^†_k'+e^-i(t-t')/ħ(E_k' -(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂_k' ..+e^i(t-t')/ħ(E_k' +(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂^†_k',ρ̂⊗R̂]] =2√(m_S ω_e,x/ħ)x_0 ∑_kk(e^-it/ħ(ϵ_e -ϵ_g)e^-it'/ħ(E_k-(ϵ_e -ϵ_g))((â_g,+^†â_g,++â_g,-^†â_g,-)(â^†_g,+-â^†_g,-)â_e,0ρ̂. -(â^†_g,+-â^†_g,-)â_e,0ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-))+e^it/ħ(ϵ_e -ϵ_g)e^-it'/ħ(E_k+(ϵ_e -ϵ_g))((â_g,+^†â_g,++â_g,-^†â_g,-)(â_g,+-â_g,-)â^†_e,0ρ̂ .-(â_g,+-â_g,-)â^†_e,0ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-))) -2√(m_S ω_e,x/ħ)x_0 ∑_kk(e^-it/ħ(ϵ_e -ϵ_g)e^it'/ħ(E_k+(ϵ_e -ϵ_g))((â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â^†_g,+-â^†_g,-)â_e,0. -ρ̂(â^†_g,+-â^†_g,-)â_e,0(â_g,+^†â_g,++â_g,-^†â_g,-))+e^it/ħ(ϵ_e -ϵ_g)e^it'/ħ(E_k-(ϵ_e -ϵ_g))((â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â_g,+-â_g,-)â^†_e,0 .-ρ̂(â_g,+-â_g,-)â^†_e,0(â_g,+^†â_g,++â_g,-^†â_g,-))) Evaluating the double commutators in the master equation involving the second and the first terms of the time - evolved interaction Hamiltonian, we obtain the following expression: 2√(m_S ω_e,x/ħ)x_0 Tr_R̂∑_k,k'√(kk')[e^-it/ħ(E_k +(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂_k+e^it/ħ(E_k -(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂^†_k. +e^-it/ħ(E_k -(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂_k+e^it/ħ(E_k +(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂^†_k', .[(â_g,+^†â_g,++â_g,-^†â_g,-)(e^-i(t-t')/ħE_k'b̂_k'+e^i(t-t')/ħE_k'b̂_k'^†),ρ̂⊗R̂]] =2√(m_S ω_e,x/ħ)x_0 ∑_kk(e^-it'/ħE_ke^-it/ħ(ϵ_e -ϵ_g)((â^†_g,+-â^†_g,-)â_e,0(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂. -(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â^†_g,+-â^†_g,-)â_e,0)+e^-it'/ħE_ke^it/ħ(ϵ_e -ϵ_g)((â_g,+-â_g,-)â^†_e,0(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂ .-(â_g,+^†â_g,++â_g,-^†â_g,-)ρ̂(â_g,+-â_g,-)â^†_e,0)) -2√(m_S ω_e,x/ħ)x_0 ∑_kk(e^it'/ħE_ke^-it/ħ(ϵ_e -ϵ_g)((â^†_g,+-â^†_g,-)â_e,0ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-). -ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)(â^†_g,+-â^†_g,-)â_e,0)+e^it'/ħE_ke^it/ħ(ϵ_e -ϵ_g)((â_g,+-â_g,-)â^†_e,0ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-) .-ρ̂(â_g,+^†â_g,++â_g,-^†â_g,-)(â_g,+-â_g,-)â^†_e,0)) Finally, evaluating the double commutators in the master equation involving both of the last terms in the time - evolved interaction Hamiltonian, we obtain the following expression: 4m_S ω_e,x/ħ x^2_0∑_k,k'√(kk')Tr_R̂[e^-it/ħ(E_k +(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂_k+e^it/ħ(E_k -(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂^†_k. +e^-it/ħ(E_k -(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂_k+e^it/ħ(E_k +(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂^†_k,[e^-i(t-t')/ħ(E_k' +(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂_k' +e^i(t-t')/ħ(E_k'-(ϵ_e -ϵ_g))(â^†_g,+-â^†_g,-)â_e,0b̂^†_k'+e^-i(t-t')/ħ(E_k' -(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂_k' .+e^i(t-t')/ħ(E_k'+(ϵ_e -ϵ_g))(â_g,+-â_g,-)â^†_e,0b̂^†_k',ρ̂⊗R̂]] =4m_S ω_e,x/ħ x^2_0∑_kk(e^-2it/ħ(ϵ_e -ϵ_g)e^-it'/ħ(E_k -(ϵ_e -ϵ_g))((â^†_g,+-â^†_g,-)â_e,0(â^†_g,+-â^†_g,-)â_e,0ρ̂-(â^†_g,+-â^†_g,-)â_e,0ρ̂(â^†_g,+-â^†_g,-)â_e,0). +e^-it'/ħ(E_k+(ϵ_e -ϵ_g))((â^†_g,+-â^†_g,-)â_e,0(â_g,+-â_g,-)â^†_e,0ρ̂-(â_g,+-â_g,-)â^†_e,0ρ̂(â^†_g,+-â^†_g,-)â_e,0) +e^-it'/ħ(E_k-(ϵ_e -ϵ_g))((â_g,+-â_g,-)â^†_e,0(â^†_g,+-â^†_g,-)â_e,0ρ̂-(â^†_g,+-â^†_g,-)â_e,0ρ̂(â_g,+-â_g,-)â^†_e,0) +e^2it/ħ(ϵ_e -ϵ_g)e^-it'/ħ(E_k+(ϵ_e -ϵ_g))((â_g,+-â_g,-)â^†_e,0(â_g,+-â_g,-)â^†_e,0ρ̂-(â_g,+-â_g,-)â^†_e,0ρ̂(â_g,+-â_g,-)â^†_e,0).) -4m_S ω_e,x/ħ x^2_0∑_kk(e^-2it/ħ(ϵ_e -ϵ_g)e^it'/ħ(E_k+(ϵ_e -ϵ_g))((â^†_g,+-â^†_g,-)â_e,0ρ̂(â^†_g,+-â^†_g,-)â_e,0-(â^†_g,+-ρ̂â^†_g,-)â_e,0(â^†_g,+-â^†_g,-)â_e,0). +e^it'/ħ(E_k-(ϵ_e -ϵ_g))((â^†_g,+-â^†_g,-)â_e,0ρ̂(â_g,+-â_g,-)â^†_e,0-ρ̂(â_g,+-â_g,-)â^†_e,0(â^†_g,+-â^†_g,-)â_e,0) +e^it'/ħ(E_k+(ϵ_e -ϵ_g))((â_g,+-â_g,-)â^†_e,0ρ̂(â^†_g,+-â^†_g,-)â_e,0-ρ̂(â^†_g,+-â^†_g,-)â_e,0(â_g,+-â_g,-)â^†_e,0) +e^2it/ħ(ϵ_e -ϵ_g)e^it'/ħ(E_k-(ϵ_e -ϵ_g))((â_g,+-â_g,-)â^†_e,0ρ̂(â_g,+-â_g,-)â^†_e,0-ρ̂(â_g,+-â_g,-)â^†_e,0(â_g,+-â_g,-)â^†_e,0).)
http://arxiv.org/abs/2307.03159v1	20230706173859	On the Linear Stability of the Lamb-Chaplygin Dipole	[ "Bartosz Protas" ]	physics.flu-dyn	[ "physics.flu-dyn" ]	LaTeX2e ε ø Δ ∂ ℐ 𝒥 𝒞 𝒜 ℬ Ø𝒪 𝒩 𝒫 𝒬 ℛ ℰ ℱ 𝒦 Łℒ 𝒟 𝒢 ℋ̋ 𝒯 𝒳 𝔹 𝕂 ℙ ℝ ℂ 𝕊 𝕋 ℤ A C D E G H F U S P R B I J K M Z ã r̃ ω λ ψ K_max E_max DK Π_ess τ_max 𝐚 𝐞 𝐠 𝐦 𝐧 x 𝐤̨ 𝐪 ŭ 𝐝̣ 𝐭 𝐜̧ 𝐬 𝐱 𝐲 𝐳 𝐟 𝐛̱ 𝐮̆ 𝐯̌ 𝐕 0 ξ ω ψ ∇ Δ Φ #1 #1 #1#2 ∂#1 ∂#2 #1#2#3 ∂^2 #1 ∂#2 ∂#3 #1#2#3 ∂^#3 #1 ∂#2^#3 #1#2 <#1,#2> #1 #1 #1 [c]#1 #1 [t]#1 #1 [b]#1 u lemmaLemma corollaryCorollary assumptionAssumption On the Linear Stability of the Lamb-Chaplygin Dipole Bartosz ProtasEmail address for correspondence: [email protected] Department of Mathematics and Statistics, McMaster University Hamilton, Ontario, L8S 4K1, Canada August 1, 2023 ================================================================================================================================================================================= The Lamb-Chaplygin dipole <cit.> is one of the few closed-form relative equilibrium solutions of the 2D Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time-integration of the 2D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions. Keywords: Vortex instability, Computational methods § INTRODUCTION The Lamb-Chaplygin dipole is a relative equilibrium solution of the two-dimensional (2D) Euler equations in an unbounded domain ^2 that was independently obtained by <cit.> and <cit.>; the history of this problem was surveyed by <cit.>. The importance of the Lamb-Chaplygin dipole stems from the fact that this is a simple exact solution with a continuous vorticity distribution which represents a steadily translating vortex pair <cit.>. Such objects are commonly used as models in geophysical fluid dynamics where they are referred to as “modons” <cit.>. Interestingly, despite the popularity of this model, the stability properties of the Lamb-Chaplygin dipole are still not well understood and the goal of the present investigation is to shed some new light on this question. We consider an unbounded flow domain Ω := ^2 (“:=” means “equal to by definition”). Flows of incompressible inviscid fluids are described by the 2D Euler equation which can be written in the vorticity form as ωt + (·̆) ω = 0 in Ω, where t ∈ (0,T] is the time with T>0 denoting the length of the interval considered, ω : (0,T] ×Ω→ is the vorticity component perpendicular to the plane of motion and =̆ [u_1, u_2]^T : (0,T] ×Ω→^2 is a divergence-free velocity field (i.e., ·=̆ 0). The space coordinate will be denoted = [x_1, x_2]^T. Introducing the streamfunction ψ : (0,T] ×Ω→, the relation between the velocity and vorticity can be expressed as =̆^⊥ψ, where ^⊥ := [x_2,-x_1]^T and Δψ = - ω. System (<ref>)–(<ref>) needs to be complemented with suitable initial and boundary conditions, and they will be specified below. In the frame of reference translating with the velocity -U _1, where U>0 and _i, i=1,2, is the unit vector associated with the ith axis of the Cartesian coordinate system, equilibrium solutions of system (<ref>)–(<ref>) satisfy the boundary-value problem <cit.> Δψ = F(ψ), in Ω, ψ →ψ_∞:= U x_2, for \|\| →∞, where the “vorticity function” F : → need not be continuous. Clearly, the form of the equilibrium solution is determined by the properties of the function F(ψ). Assuming without loss of generality that it has unit radius (a = 1), the Lamb-Chaplygin dipole is obtained by taking F(ψ) = -b^2 (ψ - η), ψ > η 0, otherwise, where b ≈ 3.8317059702075123156 is the first root of the Bessel function of the first kind of order one, J_1(b) = 0, and η∈ (-∞,∞) is a parameter characterizing the asymmetry of the dipole (in the symmetric case η = 0). The solution of (<ref>)–(<ref>) then has the form of a circular vortex core of unit radius embedded in a potential flow. The vorticity and streamfunction are given by the following expressions stated in the cylindrical coordinate system (r,θ) (hereafter we will adopt the convention that the subscript “0” refers to an equilibrium solution) * inside the vortex core (0 < r ≤ 1, 0 < θ≤ 2π): ω_0(r,θ) = 2 U b/J_0(b)[J_1(br) sinθ - η b/2 U J_0(br)], ψ_0(r,θ) = 2 U/b J_0(b) J_1(br) sinθ + η[ 1 - J_0(br)/J_0(b)], * outside the vortex core (r > 1, 0 < θ≤ 2π): ω_0(r,θ) = 0, ψ_0(r,θ) = U (1 - 1/r) sinθ. The vortical core region will be denoted A_0 := {∈^2 : ≤ 1 } and ∂ A_0 will denote its boundary. The streamline pattern inside A_0 in the symmetric (η = 0) and asymmetric (η > 0) case is shown in figures <ref>a and <ref>b, respectively. Various properties of the Lamb-Chaplygin dipole are discussed by <cit.>. In particular, it is shown that regardless of the value of η the total circulation of the dipole vanishes, i.e., Γ_0 := ∫_A_0ω_0 dA = 0. We note that in the limit η→±∞ the dipole approaches a state consisting of a monopolar vortex with a vortex sheet of opposite sign coinciding with the part of the boundary ∂ A_0 above or below the flow centerline, respectively, for positive and negative η. Generalizations of the Lamb-Chaplygin dipole corresponding to differentiable vorticity functions F(ψ) were obtained numerically by <cit.>, whereas multipolar generalizations were considered by <cit.>. Most investigations of the stability of the Lamb-Chaplygin dipole were carried out in the context of viscous flows governed by the Navier-Stokes system. While relations (<ref>)–(<ref>) do not represent an exact steady-state solution of the Navier-Stokes system, this approximate approach was justified by the assumption that viscous effects occur on time scales much longer than the time scales characterizing the growth of perturbations. A first study of this type was conducted by <cit.> who considered perturbations with dependence on the axial wavenumber and found several unstable eigenmodes together with their growth rates by directly integrating the three-dimensional (3D) linearized Navier-Stokes equations in time. Additional unstable eigenmodes were found in the 2D limit corresponding to small axial wavenumbers by <cit.>. The transient growth due to the non-normality of the linearized Navier-Stokes operator was investigated in the related case of a vortex pair consisting of two Lamb-Oseen vortices by <cit.> and <cit.>, whereas <cit.> studied Widnall-type instabilities of such vortex pairs. The effect of stratification on the evolution of a perturbed Lamb-Chaplygin dipole in 3D was considered by <cit.>. The history of the studies concerning the stability of vortices in ideal fluids was recently surveyed by <cit.>. The only stability analysis of the Lamb-Chaplygin dipole in the inviscid setting we are aware of is due to <cit.> who employed methods based on imperfect velocity-impulse diagrams applied to an approximation of the dipole in terms of a piecewise-constant vorticity distribution and concluded that this configuration is stable. Finally, there is a recent mathematically rigorous result by <cit.> who established orbital stability of the Lamb-Chaplygin dipole (orbital stability implies that flows corresponding to “small” perturbations of the dipole remain “close” in a certain norm to the translating dipole; hence, this is a rather weak notion of stability). As noted by several authors <cit.>, the stability properties of the Lamb-Chaplygin dipole are still to be fully understood despite the fact that it was introduced more than a century ago. The purpose of this work is to shed some new light on this question. We demonstrate that the Lamb-Chaplygin dipole is in fact linearly unstable, but the nature of this instability is quite subtle and cannot be understood without referring to the infinite-dimensional nature of the linearized governing equations. More specifically, both the asymptotic and numerical solution of an eigenvalue problem for the 2D linearized Euler operator suitably localized to the vortex core A_0 confirm the existence of an essential spectrum with the corresponding approximate eigenfunctions in the form of short-wavelength oscillations localized near the vortex boundary ∂ A_0. However, the time-integration of the 2D Euler system reveals the presence of a single exponentially growing eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also found not to be a smooth function and exhibits short-wavelength oscillations. These findings are consistent with the general mathematical results known about the stability of equilibria in 2D Euler flows <cit.> and have been verified by performing computations with different numerical resolutions and, in the case of the eigenvalue problem, with different arithmetic precisions. The structure of the paper is as follows: in the next section we review some basic facts about the spectra of the 2D linearized Euler equation and transform this system to a form in which its spectrum can be conveniently studied with asymptotic methods and numerically; a number of interesting properties of the resulting eigenvalue problem is also discussed, an approximate asymptotic solution of this eigenvalue problem is presented in <ref>, the numerical approaches used to solve the eigenvalue problem and the initial-value problem (<ref>)–(<ref>) are introduced in <ref>, whereas the obtained computational results are presented in <ref> and <ref>, respectively; discussion and final conclusions are deferred to <ref>; some more technical material is collected in three appendices. § 2D LINEARIZED EULER EQUATIONS The Euler system (<ref>)–(<ref>) formulated in the moving frame of reference and linearized around an equilibrium solution {ψ_0,ω_0} has the following form, where ψ', ω' : (0,T] ×Ω→ are the perturbation variables (also defined in the moving frame of reference) ω't = - (^⊥ψ_0 - U_1 )·ω' - ^⊥ψ' ·ω_0 = - (^⊥ψ_0 - U_1 )·ω' + ω_0 ·(^⊥Δ^-1)ω' =: Łω', in Ω, Δψ' = - ω', in Ω, ψ' →0, for \|\| →∞, ω'(0) = w', in Ω, in which Δ^-1 is the inverse Laplacian corresponding to the far-field boundary condition (<ref>) and w' is an appropriate initial condition assumed to have zero circulation, i.e., ∫_Ω w' dA = 0. Unlike for problems in finite dimensions where, by virtue of the Hartman-Grobman theorem, instability of the linearized system implies the instability of the original nonlinear system, for infinite-dimensional problems this need not, in general, be the case. However, for 2D Euler flows it was proved by <cit.> that the presence of an unstable eigenvalue in the spectrum of the linearized operator does indeed imply the instability of the original nonlinear problem. Arnold's theory <cit.> predicts that equilibria satisfying system (<ref>) are nonlinearly stable if F'(ψ) ≥ 0, which however is not the case for the Lamb-Chaplygin dipole, since using (<ref>) we have F'(ψ_0) = -b^2 < 0 for ψ_0 ≥η. Thus, Arnold's criterion is inapplicable in this case (<cit.> refer to this condition, but there seems to be confusion as regards signs in their analysis leading the authors to an incorrect conclusion about the stability of the dipole). §.§ Spectra of Linear Operators When studying spectra of linear operators, there is fundamental difference between the finite- and infinite-dimensional cases. To elucidate this difference and its consequences, we briefly consider an abstract evolution problem du/dt = u on a Banach space (in general, infinite-dimensional) with the state u(t) ∈ and a linear operator : →. Solution of this problem can be formally written as u(t) = e^ t u_0, where u_0 ∈ is the initial condition and e^ t the semigroup generated by <cit.>. While in finite dimensions linear operators can be represented as matrices which can only have point spectrum Π_0(), in infinite dimensions the situation is more nuanced since the spectrum Λ() of the linear operator may in general consist of two parts, namely, the approximate point spectrum Π() (which is a set of numbers λ∈ such that ( - λ) is not bounded from below) and the compression spectrum Ξ() (which is a set of numbers λ∈ such that the closure of the range of ( - λ) does not coincide with ). We thus have Λ() = Π() ∪Ξ() and the two types of spectra may overlap, i.e., Π() ∩Ξ() ≠∅ <cit.>. A number λ∈ belongs to the approximate point spectrum Π() if and only if there exists a sequence of unit vectors {f_n}, referred to as approximate eigenvectors, such that ( - λ) f_n _→ 0 as n →∞. If for some λ∈Π() there exists a unit element f such that f = λ f, then λ and f are an eigenvalue and an eigenvector of . The set of all eigenvalues λ forms the point spectrum Π_0() which is contained in the approximate point spectrum, Π_0() ⊂Π(). If λ∈Π() does not belong to the point spectrum, then the sequence {f_n} is weakly null convergent and consists of functions characterized by increasingly rapid oscillations as n becomes large. The set of such numbers λ∈ is referred to as the essential spectrum Π_ess() := Π() \Π_0(), a term reflecting the fact that this part of the spectrum is normally independent of boundary conditions in eigenvalue problems involving differential equations. It is, however, possible for “true” eigenvalues to be embedded in the essential spectrum. When studying the semigroup e^ t one is usually interested in understanding the relation between its growth abscissa γ() := lim_t →∞ t^-1ln e^ t_ and the spectrum Λ() of . While in finite dimensions γ() is determined by the eigenvalues of with the largest real part, in infinite dimensions the situation is more nuanced since there are examples in which sup_z ∈Λ()(z) < γ(), e.g., Zabczyk's problem <cit.> also discussed by <cit.>; some problems in hydrodynamic stability where such behavior was identified are analyzed by <cit.>. In regard to the 2D linearized Euler operator Ł, cf. (<ref>), it was shown by <cit.> that its essential spectrum is a vertical band in the complex plane symmetric with respect to the imaginary axis. Its width is proportional to the largest Lyapunov exponent λ_max in the flow field and to the index m ∈ of the Sobolev space H^m(Ω) in which the evolution problem is formulated (i.e., = H^m(Ω) above). The norm in the Sobolev space H^m(Ω) is defined as u _H^m := [ ∫_Ω∑_\|α\| ≤ m(∂^\|α\| u/∂^α_1 x_1 ∂^α_2 x_2)^2 dA]^1/2, where α_1,α_2 ∈ with \|α\| := α_1+α_2 <cit.>. More specifically, we have <cit.> (Ł) = { z ∈, -\|m\| λ_max≤(z) ≤ \|m\| λ_max}. In 2D flows Lyapunov exponents are determined by the properties of the velocity gradient (̆) at hyperbolic stagnation points _0. More precisely, λ_max is given by the largest eigenvalue of (̆) computed over all stagnation points. As regards the Lamb-Chaplygin dipole, it is evident from figures <ref>a and <ref>b that in both the symmetric and asymmetric case it has two stagnation points _a and _b located at the fore and aft extremities of the vortex core. Inspection of the velocity field ^⊥ψ_0 defined in (<ref>) shows that the largest eigenvalues of (̆) evaluated at these stagnation points, and hence the Lyapunov exponents, are λ_max = 2 regardless of the value of η. While characterization of the essential spectrum of the 2D linearized Euler operator Ł is rather complete, the existence of a point spectrum remains in general an open problem. Results concerning the point spectrum are available in a few cases only, usually for shear flows where the problem can be reduced to one dimension <cit.> or the cellular cat's eyes flows <cit.>. In these examples unstable eigenvalues are outside the essential spectrum (if one exists). On the other hand, it was shown by <cit.> that when an unstable eigenvalue is embedded in the essential spectrum, then the corresponding eigenfunctions need not be smooth. One of the goals of the present study is to consider this issue for the Lamb-Chaplygin dipole. §.§ Linearization Around the Lamb-Chaplygin Dipole The linear system (<ref>) is defined on the entire plane ^2, however, in the Lamb-Chaplygin dipole the vorticity ω_0 is supported within the vortex core A_0 only, cf. (<ref>). This will allow us to simplify system (<ref>) so that it will involve relations defined only within A_0, which will facilitate both the asymptotic analysis and numerical solution of the corresponding eigenvalue problem, cf. <ref> and <ref>. If the initial data w' in (<ref>) is also supported in A_0, then the initial-value problem (<ref>) can be regarded as a free-boundary problem describing the evolution of the boundary ∂ A(t) of the vortex core (we have A(0) = A_0 and ∂ A(t) = ∂ A_0). However, as explained below, the evolution of this boundary can be deduced from the evolution of the perturbation streamfunction ψ'(t,), hence need not be tracked independently. Thus, the present problem is different from, e.g., the vortex-patch problem where the vorticity distribution is fixed (piecewise constant in space) and in the stability analysis the boundary is explicitly perturbed <cit.>. Denoting ψ_1' : (0,T] × A_0 → and ψ_2' : (0,T] ×^2\A_0 → the perturbation streamfunction in the vortex core and in its complement, system (<ref>) can be recast as ω't = - (^⊥ψ_0 - U_1 )·ω' - ^⊥ψ_1' ·ω_0, in A_0, Δψ_1' = - ω', in A_0, Δψ_2' = 0, in ^2\A_0, ψ_1' = ψ_2' = f', on ∂A_0, ψ_1'n = ψ_2'n, on ∂A_0, ψ_2' →0 , for \|\| →∞, ω'(0) = w', in A_0, where is the unit vector normal to the boundary ∂ A_0 pointing outside and conditions (<ref>)–(<ref>) represent the continuity of the normal and tangential perturbation velocity components across the boundary ∂ A_0 with f' : ∂ A_0 → denoting the unknown value of the perturbation streamfunction at that boundary. The velocity normal to the vortex boundary ∂ A(t) is u_n := ·̆ = ∂ψ_1 / ∂ s = ∂ψ_2 / ∂ s, where s is the arc-length coordinate along ∂ A(t), cf. (<ref>). While this quantity identically vanishes in the equilibrium state (<ref>)–(<ref>), cf (<ref>), in general it will be nonzero resulting in a deformation of the boundary ∂ A(t). This deformation can be deduced from the solution of system (<ref>) as follows. Given a point ∈∂ A(t), the deformation of the boundary is described by d / dt = u_n\|_∂ A(t). Integrating this expression with respect to time yields (τ) = (0) + ∫_0^τ u_n\|_∂ A_τ dτ' = (0) + τ u_n\|_∂ A_0 + 𝒪(τ^2), where (0) ∈∂ A_0 and 0 < τ≪ 1 is the time over which the deformation is considered. Thus, the normal deformation of the boundary can be defined as ρ(τ) := ·[ (τ) - (0)] ≈ u_n\|_∂ A_0τ. We also note that at the leading order the area of the vortex core A(t) is preserved by the considered perturbations ∮_∂ A_0ρ(τ) ds = τ∮_∂ A_0ψs ds = τ∮_∂ A_0 dψ = 0 ⟹ \|A(t)\| ≈ \|A_0\|. We notice that in the exterior domain ^2\A_0 the problem is governed by Laplace's equation (<ref>) subject to boundary conditions (<ref>)–(<ref>). Therefore, this subproblem can be eliminated by introducing the corresponding Dirichlet-to-Neumann (D2N) map M : ψ_2'\|_∂ A_0→ψ_2'n\|_∂ A_0 which is constructed in an explicit form in Appendix <ref>. Thus, equation (<ref>) with boundary conditions (<ref>)–(<ref>) can be replaced with a single relation ψ_1'n = Mψ_1' holding on ∂ A_0 such that the resulting system is defined in the vortex core A_0 and on its boundary only. We therefore conclude that while the vortex boundary ∂ A(t) may deform in the course of the linear evolution, this deformation can be described based solely on quantities defined within A_0 and on ∂ A_0 using relation (<ref>). In particular, the transport of vorticity out of the vortex core A_0 into the potential flow is described by the last term on the right-hand side (RHS) in (<ref>) evaluated on the boundary ∂ A_0. Noting that the base state satisfies the equation Δψ_0 = - b^2 (ψ_0 - η) in A_0, cf. (<ref>)–(<ref>), and using the identity (^⊥ψ_1')·ψ_0 = - (ψ_1')·^⊥ψ_0, the vorticity equation (<ref>) can be transformed to the following simpler form Δψ_1't = - (^⊥ψ_0) ·(Δψ_1' + b^2 ψ_1') in A_0, where we also used (<ref>) to eliminate ω' in favor of ψ_1'. Supposing the existence of an eigenvalue λ∈ and an eigenfunction : A_0 →, we make the following ansatz for the perturbation streamfunction ψ_1'(t,) = () e^λ t which leads to the eigenvalue problem λ = Δ_M^-1 [ (^⊥ψ_0) ·(Δ+ b^2 ) ] in A_0, Δr = 0, at r=0, where Δ_M^-1 is the inverse Laplacian subject to the boundary condition ∂ / ∂ n - M = 0 imposed on ∂ A_0 and the additional boundary condition (<ref>) ensures the perturbation vorticity is differentiable at the origin (such condition is necessary since the differential operator on the RHS in (<ref>) is of order three). Depending on whether or not the different differential operators appearing in it are inverted, eigenvalue problem (<ref>) can be rewritten in a number of different, yet mathematically equivalent, forms. However, all these alternative formulations have the form of generalized eigenvalue problems and are therefore more difficult to handle in numerical computations. Thus, formulation (<ref>) is preferred and we will focus on it hereafter. We note that the proposed formulation ensures that the eigenfunctions have zero circulation, as required Γ' := ∫_A_0ω' dA = - ∫_A_0Δψ_1' dA = - ∮_∂ A_0ψ_1'n ds = - ∮_∂ A_0ψ_2'n ds = - ∫_^2 \A_0Δψ_2' dA = 0, where we used the divergence theorem, equations (<ref>)–(<ref>) and the boundary conditions (<ref>)–(<ref>). Since it will be needed for the numerical discretization described in <ref>, we now rewrite the eigenvalue problem (<ref>) explicitly in the polar coordinate system λ = Δ_M^-1[ ( u_0^r r + u_0^θ/r θ ) (Δ+ b^2 ) ] =: for 0 < r ≤1, 0 ≤θ≤2π, Δr = 0, at r=0, where Δ = r2 + 1/rr + 1/r^2θ2 and the velocity components obtained as [ u_0^r, u_0^θ] := ^⊥ψ_0 = [ 1/rθ, -r] ψ_0 are u_0^r = 2U J_1(br)cosθ/b J_0(b) r, u_0^θ = - 2U [ J_0(b r) - J_1(b r)/b r]sinθ + η b J_1(b r)/J_0(b). They have the following behavior on the boundary ∂ A_0 u_0^r(1,θ) = 0, u_0^θ(1,θ) = 2U sinθ. Since ψ_L^2∼Δψ_H^-2 = ω_H^-2, where “∼” means the norms on the left and on the right are equivalent (in the precise sense of norm equivalence), the essential spectrum (<ref>) of the operator $̋ will havem = - 2, so that()̋is a vertical band in the complex plane with\|(z)\| ≤4,z ∈(sinceλ_max = 2). Operator$̋, cf. (<ref>) has a non-trivial null space ()̋. To see this, we consider the “outer” subproblem ϕ := ( u_0^r r + u_0^θ/r θ ) ϕ= 0 for 0 < r ≤1, 0 ≤θ≤2π, ϕr = 0, at r=0, whose solutions are ϕ(r,θ) = ϕ_C(r,θ) := B [J_1(br) sinθ]^C, B ∈, C = 2,3,… (see Appendix <ref> for derivation details). Then, the eigenfunctions _C spanning the null space of operator $̋ are obtained as solutions of the family of “inner” subproblems ( r2 + 1/rr + 1/r^2θ2 + b^2 ) _C = ϕ_C for 0 < r ≤ 1, 0 ≤θ≤ 2π, _Cr + M _C = 0, at r=0, whereC = 2,3,…. Some of these eigenfunctions are shown in figures <ref>a–d, where distinct patters are evident for even and odd values ofC. § ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEM (<REF>) A number of interesting insights about certain properties of solutions of eigenvalue problem (<ref>) can be deduced by performing a simple asymptotic analysis corresponding to the short-wavelength limit. We focus here on the case of the symmetric dipole (η= 0) and begin by introducing the ansatz (r,θ) = ∑_m=0^∞ f_m(r) cos(mθ) + g_m(r)sin(mθ), wheref_m, g_m : [0,1] →,m=1,2,…, are functions to be determined. Substituting this ansatz in (<ref>) with the Laplacian moved back to the left-hand side (LHS) and applying well-known trigonometric identities leads after some algebra to the following system of coupled third-order ordinary differential equations (ODEs) for the functionsf_m(r),m=1,2,…, λ_m f_m = 1/2P(r) d/dr( _m-1f_m-1 + b^2 f_m-1 + _m+1f_m+1 + b^2 f_m+1) 1/2Q(r) m/r( _m-1f_m-1 + b^2 f_m-1 - _m+1f_m+1 - b^2 f_m+1), r ∈ (0,1), f_m bounded at r = 0, d/dr f_m = - m f_m at r = 1, d/dr_m f_m = 0 at r = 0, where the Bessel operator_mis defined via_m f := d^2/dr^2 f + 1/r d/dr f - m^2/r f, whereas the coefficient functions have the form, cf. (<ref>), P(r) := 2U J_1(br)/b J_0(b) r, Q(r) := - 2U [ J_0(b r) - J_1(b r)/b r]/J_0(b). The functionsg_m(r),m=1,2,…, satisfy a system identical to (<ref>) which shows that the eigenfunctions(r,θ)are either even or odd functions ofθ(i.e., they are either symmetric or antisymmetric with respect to the flow centerline). Moreover, the coupled form of system (<ref>) implies that the eigenvectors(r,θ)are not separable as functions ofrandθ. Motivated by our discussion in <ref> about the properties of approximate eigenfunctions of the 2D linearized Euler operator, we will construct approximate solutions of system (<ref>) in the short-wavelength limitm →∞. We thus consider the asymptotic expansions λ = λ^0 + 1/mλ^1 + Ø(1/m^2), f_m(r) = f_m^0(r) + 1/m f_m^1(r) + Ø(1/m^2), whereλ^0, λ^1 ∈andf_m^0, f_m^1 : [0,1] →. Plugging these expansions into system (<ref>) and collecting terms proportional to the highest powers ofmwe obtain Ø(m^3): f_m-1^0 - f_m+1^0 = 0, Ø(m^2): 1/2Q(r)/r^3( f_m-1^1 - f_m+1^1 ) = 1/2 P(r) d/dr[ 1/r^2( f_m-1^0 + f_m+1^0 ) ] = + Q(r)/r^3( f_m-1^0 + f_m+1^0 ) + λ^0/r^2 f_m^0. It follows immediately from (<ref>) thatf_m-1^0 = f_m+1^0. Since this analysis does not distinguish between even and odd values ofm, we will assume thatf_m^0 = f_m-1^0 = f_m+1^0. Furthermore, we will also assume thatf_m-1^1 = f_m+1^1(as will be evident below, these assumptions do lead to consistent asymptotic solutions of system (<ref>); however, it is possible that they do not exhaust all possibilities and that solutions with other properties can also be obtained if these assumptions are not made). With these assumptions, the LHS in (<ref>) vanishes and the RHS takes the form P(r) d/dr(1/r^2 f_m^0 ) - 2 Q(r)/r^3 f_m^0 - λ^0/r^2 f_m^0 = 0, r ∈ (0,1), which is a first-order differential equation defining the leading-order termf_m^0(r)in (<ref>). Without loss of generality the boundary condition (<ref>) can be replaced withf(0) = 1. In this analysis the eigenvalueλ^0can be treated as a free parameter such that equation (<ref>), which is separable, can be solved via quadrature to give f_m^0(r) = exp[ i ∫_0^r I_i(r') dr' ] exp[ ∫_0^r I_r(r') dr' ], r ∈ [0,1], where I_r(r) := (λ^0) b J_0(b) r^2 - 4 U b J_0(br) r + 8 U J_1 (br)/2 U J_1(b r) r , I_i(r) := (λ^0) b J_0(b) r/2 U J_1(b r). The limiting (asr →1) behavior of functions (<ref>)–(<ref>) exhibits an interesting dependence onλ^0, namely, lim_r → 1 I_r(r) = + ∞, (λ^0) < 4 + 0, (λ^0) = 4 - ∞, (λ^0) > 4 , lim_r → 1 I_i(r) = - [(λ^0)] ∞. In particular, the limiting value ofI_r(r)asr →1changes when(λ^0) = 4, which defines the right boundary of the essential spectrum in the present problem, cf. (<ref>). BothI_r(r)andI_i(r)diverge asØ(1/(1 - r))whenr →1which means that the integrals under the exponentials in (<ref>), and hence the entire formula, are not defined atr = 1. While the factor involvingI_i(r)is responsible for the oscillation of the functionf_m^0(r), the factor depending onI_r(r)determines its growth asr →1: we see that\|f_m^0(r)\|becomes unbounded in this limit when(λ^0) < 4and approaches zero otherwise. The real and imaginary parts off_m^0(r)obtained for different eigenvaluesλ^0are shown in figures <ref>a,b, where it is evident that both the unbounded growth and the oscillations off_m^0(r)are localized in the neighbourhood of the endpointr = 1. Given the singular nature of the solutions obtained at the leading order, the correction termf_m^1(r)is rather difficult to compute and we do not attempt this here. We thus conclude that when(λ^0) < 4, the solutions of eigenvalue problem (<ref>) constructed in the form (<ref>) are dominated by short-wavelength oscillations whose asymptotic, asm →∞, structure involves oscillations in both the radial and azimuthal directions and are localized near the boundary∂A_0. We remark that the asymptotic solutions constructed above do not satisfy the boundary conditions (<ref>)-(<ref>), which is consistent with the fact that they represent approximate eigenfunctions associated with the essential spectrum()̋of the 2D linearized Euler operator. In order to find solutions of eigenvalue problem (<ref>) which do satisfy all the boundary conditions we have to solve this problem numerically which is done next. § NUMERICAL APPROACHES In this section we first describe the numerical approximation of eigenvalue problem (<ref>)–(<ref>) and then the time integration of the 2D Euler system (<ref>)–(<ref>) with the initial condition in the form of the Lamb-Chaplygin dipole perturbed with some approximate eigenfunctions obtained by solving eigenvalue problem (<ref>)–(<ref>). These computations will offer insights about the instability of the dipole complementary to the results of the asymptotic analysis presented in <ref>. §.§ Discretization of Eigenvalue Problem (<ref>)–(<ref>) Eigenvalue problem (<ref>)–(<ref>) is solved using the spectral collocation method proposed by <cit.>, see also the discussion in <cit.>, which is based on a tensor grid in(r,θ). The discretization inθinvolves trigonometric (Fourier) interpolation, whereas that inris based on Chebyshev interpolation where we taker ∈[-1,1]which allows us to avoid collocating (<ref>) at the origin when the number of grid points is even. Since then the mapping between(r,θ)and(x_1,x_2)is 2-to-1, the solution must be constrained to satisfy the condition (r,θ) = (-r,(θ+π)(mod 2π)), r ∈ [-1,1], θ∈ [0,2π] which is fairly straightforward to implement <cit.>. In contrast to (<ref>), the boundary condition (<ref>) does need to be evaluated at the origin which necessities modification of the differentiation matrix (since our Chebyshev grid does not include a grid point at the origin). The numbers of grid points discretizing the coordinatesr ∈[-1,1]andθ∈[0,2π]are linked and both given byNwhich is an even integer. The resulting algebraic eigenvalue problem then has the form λ = , where∈^N^2is the vector of approximate nodal values of the eigenfunction and∈^N^2 ×N^2the matrix discretizing the operator$̋, cf. (<ref>), obtained as described above. Since the operator $̋ and hence also the matrixare singular, conditioning of problem (<ref>) is improved by eliminating a part of its null space by performing projections on a certain numberN_Cof eigenfunctions associated with the eigenvalueλ= 0. They are obtained by solving problem (<ref>) with different source termsϕ_C,C = 2,3,…,N_C+1, cf. (<ref>). Problem (<ref>) is implemented in MATLAB and solved using the function eig. In addition to examining convergence of the results with respect to grid refinement (by increasing the resolutionNas discussed in <ref>), we have also checked the effect of arithmetic precision using the toolbox <cit.>. However, increasing the arithmetic precision up toØ(10^2)significant digits was not found to have a significant effect on the results obtained with small and medium resolutionsN ≤100(at higher resolutions the cost of such computations becomes prohibitive). In the light of the discussion in <ref>– <ref>, we know the spectrum of the operator$̋ includes essential spectrum in the form of a vertical band in the complex plane \|(z)\| ≤ 4, z ∈. Available literature on the topic of numerical approximation of infinite-dimensional non-self-adjoint eigenvalue problems, especially ones featuring essential spectrum, is very scarce. However, since the discretized problem (<ref>) is finite-dimensional and therefore can only have point spectrum, it is expected that at least some of the eigenvalues of the discrete problem will be approximations of the approximate eigenvalues in the essential spectrum ()̋, whereas the corresponding eigenvectors will approximate the approximate eigenfunctions (we note that the term “approximate” is used here with two distinct meanings: its first appearance refers to the numerical approximation and the second to the fact that these functions are defined as only “close” to being true eigenfunctions, cf. <ref>). As suggested by the asymptotic analysis presented in <ref>, these approximate eigenfunctions are expected to be dominated by short-wavelength oscillations which cannot be properly resolved using any finite resolution N. Thus, since these eigenfunctions are not smooth, we do not expect our numerical approach to yield an exponential convergence of the approximation error. To better understand the properties of these eigenfunction, we also solve a regularized version of problem (<ref>) in which is replaced with _δ := _δ^-1, where _δ := ( - δ^2 Δ), δ > 0 is a regularization parameter and the inverse of _δ is defined with the homogeneous Neumann boundary conditions. The regularized version of the discrete problem (<ref>) then takes the form λ_δ = _δ _δ^-1 =: _δ , where the subscript δ denotes regularized quantities and _δ is the discretization of the regularizing operator _δ. Since the operator _δ^-1 can be interpreted as a low-pass filter with the cut-off length given by δ, the effect of this regularization is to smoothen the eigenvectors by filtering out components with wavelengths less than δ. Clearly, in the limit when δ→ 0 the original problem (<ref>) is recovered. An analogous strategy was successfully employed by <cit.> in their study of the stability of Hill's vortex where the eigenfunctions also turned out to be singular distributions. §.§ Solution of the Time-Dependent Problem (<ref>)–(<ref>) The 2D Euler system (<ref>)–(<ref>) is solved in the frame of reference moving with the velocity -U _1 and with the vorticity equation (<ref>) rewritten in terms of the difference with respect to the equilibrium solution, i.e., for ω_1(t,) := ω(t,) - ω_0(). Since the resulting system is solved using a standard Fourier pseudospectral method <cit.>, we assume that the flow domain in a 2D periodic box Ω = ^2 instead of the 2D plane ^2. We note that a similar approximation was also used in earlier studies by <cit.>. Since the instability has the form of localized short-wavelength oscillations, interaction of the perturbed dipole with its periodic images does not have a significant effect. The exponential filter proposed by <cit.> is used in lieu of dealiasing and the discretized problem is integrated in time using the RK4 method. We use a massively-parallel implementation based on MPI with Fourier transforms computed using the FFTW library <cit.>. Convergence of the results with refinement of the resolution M, representing the number of grid points in each direction, and of the time step Δ t was carefully checked. § SOLUTION OF THE EIGENVALUE PROBLEM In this section we describe solutions of the discrete eigenvalue problem (<ref>) and its regularized version (<ref>). We mainly focus on the symmetric dipole with η = 0, cf. figure <ref>a. In order to study dependence of the solutions on the numerical resolution, problems (<ref>)–(<ref>) were solved with N ranging from 20 to 260, where the largest resolution was limited by the amount of RAM memory available on a single node of the computer cluster we had access to. The discrete spectra of problem (<ref>) obtained with N = 40, 80, 160, 260 are shown in figures <ref>a–d. We see that for all resolutions N the spectrum consists of purely imaginary eigenvalues densely packed on the vertical axis and a “cloud” of complex eigenvalues clustered around the origin (for each N is there is also a pair of purely real spurious eigenvalues increasing as \|λ\| = Ø(N) when the resolution is refined; they are not shown in figures <ref>a–d). We see that as N increases the cloud formed by the complex eigenvalues remains restricted to the band -2 ⪅(λ) ⪅ 2, but expands in the vertical (imaginary) direction. The spectrum is symmetric with respect to the imaginary axis as is expected for a Hamiltonian system. The eigenvalues fill the inner part of the band ever more densely as N increases and in order to quantify this effect in figures <ref>a–d we show the eigenvalue density defined as the number of eigenvalues in a small rectangular region of the complex plane, i.e., μ(z) := number of eigenvalues λ∈{ζ∈ : \|(ζ -z)\| ≤Δλ_r, \|(ζ -z)\| ≤Δλ_i }/4 Δλ_r Δλ_i, where Δλ_r, Δλ_i ∈ are half-sizes of a cell used to count the eigenvalues with Δλ_i ≈ 500 Δλ_r reflecting the fact that the plots are stretched in the vertical direction. We see that as the resolution N is refined the eigenvalue density μ(z) increases near the origin. However, with the exception of the eigenvalue λ_0, we did not find evidence for individual eigenvalues to converge to well-defined limits as the resolution N increases. As discussed in <ref>, a key question concerning the linear stability of 2D Euler flows is the existence of point spectrum Π_0(Ł) of the linear operator Ł, cf. (<ref>). Anticipating the discussion in <ref>, for each resolution N we have identified a conjugate pair of eigenvalues λ_0 associated with the linearly unstable mode discussed in that section. These eigenvalues are given in Table <ref> and are marked (together with their counterparts with negative real parts) in figures <ref>a–d. As we see from Table <ref>, the differences between the real parts of the eigenvalue λ_0 computed with different resolutions N are very small and just over 1%, although the variation of the imaginary part is larger. We now take a closer look at the purely imaginary eigenvalues which are plotted for different resolutions N in figure <ref>. It is known that these approximate eigenvalues are related to the periods of Lagrangian orbits associated with closed streamlines in the base flow <cit.>. In particular, if the maximum period is bounded < ∞, this implies the presence of a horizontal gaps in the essential spectrum. However, as shown in Appendix <ref>, the Lamb-Chaplygin dipole does involve Lagrangian orbits with arbitrarily long periods, such that the essential spectrum (Ł) includes the entire imaginary axis i. The results shown in figure <ref> are consistent with this property since the gap evident in the spectra shrinks, albeit very slowly, as the numerical resolution N is refined. The reason why these gaps are present is that the orbits sampled with the discretization described in <ref> have only finite maximum periods which however become longer as the discretization is refined. Finally, we analyze eigenvectors of problem (<ref>) and choose to present them in terms of vorticity, i.e., we show _i = - Δ_i, i=0,1,2. To fix attention in figures <ref>a,c,e we show the real parts of these eigenvectors associated with the following eigenvalues: the complex eigenvalue λ_0 corresponding to the exponentially growing mode, cf. Table <ref>, a purely real eigenvalue λ_1 and a purely imaginary eigenvalue λ_2. It is clear that all these eigenvectors are dominated by short-wavelength oscillations mostly localized near the boundary ∂ A_0 of the vortex core, a feature that was predicted by the asymptotic solution constructed in <ref>, cf. figures <ref>a,b. However, in the eigenvector _0 associated with the eigenvalue λ_0 these oscillations are mostly concentrated near the azimuthal angles θ = ±π / 4, ± 3π /4, cf. figure <ref>a. In the other eigenvectors _1 and _2 the oscillations are mostly concentrated near the stagnation points _a and _b. In addition, while the eigenvector _0 is symmetric with respect to the flow centerline, the eigenvectors _1 and _2 are antisymmetric. The eigenvectors associated with all other eigenvalues are also dominated by short-wavelength oscillations localized near different parts of the boundary ∂ A_0. Since due to their highly oscillatory nature the eigenvectors shown in figures <ref>a,c,e are not fully resolved, in figures <ref>b,d,f we show the corresponding eigenvectors of the regularized eigenvalue problem (<ref>) where we set δ = 0.05. The eigenvalues λ_δ,i, i=0,1,2, these regularized eigenvectors correspond to are slightly shifted with respect to the original eigenvalues λ_i, i=0,1,2, since regularization affects some fine details of the spectrum, although its key properties remain unchanged. We see that in the regularized eigenvectors oscillations are shifted to the interior of the domain A_0 and their typical wavelengths are much larger. Solution of the discrete eigenvalue problem (<ref>) for asymmetric dipoles with η > 0 leads to eigenvalue spectra and eigenvectors qualitatively very similar to those shown in figures <ref>a–d and <ref>a,c,e, hence for brevity they are not shown here. The only noticeable difference is that the eigenvectors are no longer symmetric or antisymmetric with respect to the flow centerline. § SOLUTION OF THE EVOLUTION PROBLEM As in <ref>, we focus on the symmetric case with η = 0. The 2D Euler system (<ref>)–(<ref>) is solved numerically as described in <ref> with the initial condition for the perturbation vorticity ω_1(t,) given in terms of the eigenvectors shown in figures <ref>a–f, i.e., ω_1(0,) = ε_i_L^2(Ω)/ω_0 _L^2(Ω)_i() or ω_1(0,) = ε_δ,i_L^2(Ω)/ω_0 _L^2(Ω)_δ,i(), i=0,1,2. Unless indicated otherwise, the numerical resolution is M = 512 grid points in each direction. By taking ε = 10^-4 and T = 40 we ensure that the evolution of the perturbation vorticity is effectively linear and to characterize its growth we define the perturbation enstrophy as (t) := ∫_Ωω_1(t,)^2 d. The time evolution of this quantity is shown in figure <ref>a for the six considered initial conditions. In all cases we see that after a transient period the perturbation enstrophy starts to grow exponentially as exp( t ), where the growth rate ≈ 0.127 is very close to the real part of the eigenvalue λ_0, cf. Table <ref>. The duration of the transient, which involves an initial decrease of the perturbation enstrophy, is different in different cases and is shortest when the eigenfunctions _0 and _δ,0 are used as the initial conditions in (<ref>) (in fact, in the latter case the transient is barely present). Hereafter we will focus on the flow obtained with the initial condition (<ref>) given in terms of the eigenfunction _0, cf. figure <ref>a. The effect of the numerical resolution N used in the discrete eigenvalue problem (<ref>) is analyzed in figure <ref>b, where we show the perturbation enstrophy (<ref>) in the flows with the eigenvector _0 used in the initial conditions (<ref>) computed with different N. We see that refined resolution leads to a longer transient period while the rate of the exponential growth is unchanged. The enstrophy spectrum of the initial condition (<ref>) and of the perturbation vorticity ω_1(t,) at different times t ∈ (0,60] is shown in figure <ref> as a function of the wavenumber k := \|\|̨. It is defined as e(t,k) := ∫_S_k\| [ω_1(t)]_\|^2 dσ, where [ω_1(t)]_, ∈̨^2, are the Fourier coefficients of the perturbation vorticity ω_1(t,), σ is the angle in the wavenumber space and S_k denotes the circle of radius k in this space (with some abuse of notation justified by simplicity, here we have treated the wavevector $̨ as a continuous rather than discrete variable). Since its spectrum is essentially independent of the wavenumberk, the eigenvector_0in the initial condition (<ref>) turns out to be a distribution rather than a smooth function. The enstrophy spectra of the perturbation vorticityω_1(t,)during the flow evolution show a rapid decay at high wavenumbers which is the effect of the applied filter, cf. <ref>. However, after the transient, i.e., for20 ⪅t ≤60, the enstrophy spectra have very similar forms, except for a vertical shift which increases with timet. This confirms that the time evolution is dominated by linear effects as there is little energy transfer to higher (unresolved) modes. This is also attested to by the fact that for all the cases considered in figure <ref>a the relative change of the total enstrophy∫_Ω ω(t,) ^2 d, which is a conserved quantity, does not exceed0.1%. We now go on to discuss the time evolution of the perturbation vorticity in the physical space and in figures <ref>a and <ref>b we showω_1(t,)at the timest = 4andt = 21, respectively, which correspond to the transient regime and to the subsequent period of an exponential growth. During that period, i.e., for20 ⪅t ≤60, the structure of the perturbation vorticity field does not change much. We see that as the perturbation evolves a number of thin vorticity filaments is ejected from the vortex coreA_0into the potential flow with the principal ones emerging at the azimuthal anglesθ≈±π/ 4, ±3π/ 4, i.e., in the regions of the vortex boundary where most of the short-wavelength oscillations evident in the eigenvector_0are localized, cf. figure <ref>a. The perturbation remains symmetric with respect to the flow centerline for all times and since the vorticityω_0of the base flow is antisymmetric, the resulting total flowω(t,)does not possess any symmetries. The perturbation vorticityω_1(t,)realizing the exponential growth in the flows corresponding to the initial condition involving the eigenvectors_1and_2(and their regularized versions_1,δand_2,δ) is essentially identical to the perturbation vorticity shown in figure <ref>b, although its form during the transient regime can be quite different. In particular, the perturbation eventually becomes symmetric with respect to the flow centerline even if the initial condition (<ref>) is antisymmetric. The same is true for flows obtained with initial condition corresponding to all approximate eigenvalues other thanλ_0,λ_1andλ_2. We did not attempt to study the time evolution of asymmetric dipoles withη> 0in (<ref>), since their vorticity distributions are discontinuous making computation of such flows using the pseudospectral method described in <ref> problematic. § DISCUSSION AND FINAL CONCLUSIONS In this study we have considered an open problem concerning the linear stability of the Lamb-Chaplygin dipole which is a classical equilibrium solution of the 2D Euler equation in an unbounded domain. We have considered its stability with respect to 2D circulation-preserving perturbations and while our main focus was on the symmetric configuration withη= 0, cf. figure <ref>a, we also investigated some aspects of asymmetric configurations withη> 0. Since the stability of the problem posed on a unbounded domain is difficult to study both with asymptotic methods and numerically, we have introduced an equivalent formulation with all relations defined entirely within the compact vortex coreA_0, which was accomplished with the help of a suitable D2N map accounting for the potential flow outside the core, cf. Appendix <ref>. The initial-value problem for the 2D Euler equation with a compactly supported initial condition is of a free-boundary type since the time evolution of the vortex boundary∂A(t)is a priori unknown and must be determined as a part of the solution of the problem. This important aspect is accounted for in our formulation of the linearized problem, cf. relation (<ref>). The operator representing the 2D Euler equation linearized around the Lamb-Chaplygin dipole has been shown to have an infinite-dimensional null space(Ł)and the eigenfunctions_C,C=2,3,…, spanning this null space, cf. figures <ref>a–d, can potentially be used to search for nearby equilibrium solutions. An approximate solution of eigenvalue problem (<ref>) obtained in <ref> using an asymptotic technique reveals the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the vortex boundary∂A_0. Remarkably, eigenfunctions with such properties exist when(λ^0) < 4, i.e., whenλ^0is in the essential spectrum()̋of the 2D linearized Euler operator and it is interesting that the asymptotic solution has been able to capture this value exactly. We remark that with exponential terms involving divergent expressions as arguments, cf. (<ref>), this approach has the flavour of the WKB analysis. We note however thatλ^0serves as a parameter, rather than an eigenvalue, in this approach. Moreover, since the obtained approximate solution represents only the asymptotic (in the short-wavelength limitm →∞) structure of the eigenfunctions, it does not satisfy the boundary conditions (<ref>)-(<ref>). To account for these limitations, complementary insights have been obtained by solving eigenvalue problem (<ref>) numerically as described in <ref>. Our numerical solution of eigenvalue problem (<ref>) obtained in <ref> using different resolutionsNyields results consistent with the general mathematical facts known about the spectra of the 2D linearized Euler operator, cf. <ref>. In particular, these results feature eigenvalues of the discrete problem (<ref>) filling ever more densely a region around the origin which is bounded in the horizontal (real) direction and expands in the vertical (imaginary) direction as the resolutionNis increased, which is consistent with the existence of an essential spectrum()̋in the form of a vertical band with the width determined by the largest Lyapunov exponent of the flow, cf. (<ref>). The corresponding eigenvectors are dominated by short-wavelength oscillations localized near the vortex boundary∂A_0, a feature that was predicted by the asymptotic solution constructed in <ref>. However, solutions of the evolution problem for the perturbation vorticity with the initial condition (<ref>) corresponding to different eigenvectors obtained from the discrete problems (<ref>)–(<ref>) reveal thatλ_0(and its complex conjugateλ_0^) are the only eigenvalues associated with an exponentially growing mode with a growth rate effectively equal to the real part of the eigenvalue, i.e., for which≈(λ_0). When eigenvectors associated with eigenvalues other thanλ_0orλ_0^are used in the initial condition (<ref>), the perturbation enstrophy (<ref>) reveals transients of various duration followed by exponential growth with the growth rate again given by(λ_0). This demonstrates that±λ_0and±λ_0^are the only “true” eigenvalues and form the point spectrumΠ_0()̋of the operator associated with the 2D Euler equation linearized around the Lamb-Chaplygin dipole. On the other hand, all other eigenvalues of the discrete problems (<ref>)–(<ref>) can be interpreted as numerical approximations to approximate eigenvalues belonging to the essential spectrum()̋. More precisely, for each resolutionNthe eigenvalues of the discrete problems other than±λ_0and±λ_0^approximate a different subset of approximate eigenvalues in the essential spectrum()̋and the corresponding eigenvectors are approximations to the associated approximate eigenvectors. This interpretation is confirmed by the eigenvalue density plots shown in figures <ref>a–d and is consistent with what is known in general about the spectra of the 2D linearized Euler operator, cf. <ref>. In figure <ref>a we noted that when the initial condition (<ref>) is given in terms of the eigenvector_0, the perturbation enstrophy(t)also exhibits a short transient before attaining exponential growth with the rate≈(λ_0). The reason for this transient is that, being non-smooth, the eigenvector_0is not fully resolved, which is borne out in figure <ref> (in fact, due to the distributional nature of this and other eigenvectors, they cannot be accurately resolved with any finite resolution). Thus, this transient period is needed for some underresolved features of the perturbation vorticity to emerge, cf. figure <ref>a vs. figure <ref>b. However, we note that in the flow evolution originating from the eigenvector_0the transient is actually much shorter than when other eigenvectors are used as the initial condition (<ref>), and is nearly absent in the case of the regularized eigenvector_0,δ. We emphasize that non-smoothness of eigenvectors associated with eigenvalues embedded in the essential spectrum is consistent with the known mathematical results <cit.>. Interestingly, the eigenfunctions_C,C=2,3,…, associated with the zero eigenvalueλ= 0are smooth, cf. figures <ref>a–d. We also add that there are analogies between our findings and the results of the linear stability analysis of Hill's vortex with respect to axisymmetric perturbations where the presence of both the continuous and point spectrum was revealed, the latter also associated with non-smooth eigenvectors <cit.>. There is a potentially intriguing connection with the so-called “tygers” which are short-wavelength oscillations arising when a truncated inviscid system begins to thermalize. They have been observed in 1D inviscid Burgers and 3D Euler flows <cit.>. In the course of the linear evolution of the instability the vortex regionA(t)changes shape as a result of the ejection of thin vorticity filaments from the vortex coreA_0, cf. figures <ref>a,b. However, both the area\|A(t)\|of the vortex and its total circulationΓare conserved at the leading order, cf. (<ref>) and (<ref>). We reiterate that the perturbation vorticity fields shown in figures <ref>a,b were obtained with underresolved computations and increasing the resolutionMwould result in the appearance of even finer filaments such that in the continuous limit (M →∞) some of the filaments would be infinitely thin. In this study we have considered the linear stability of the Lamb-Chaplygin dipole with respect to 2D perturbations. It is an interesting open question how the picture presented here would be affected by inclusion of 3D effects. We are also exploring related questions in the context of the stability of other equilibria in 2D Euler flows, including various cellular flows. § ACKNOWLEDGMENTS The author wishes to thank Roman Shvydkoy for bringing the mathematical results concerning the stability of equilibria in 2D Euler flows to his attention. The author is also thankful to Xinyu Zhao for her help with the solution of the time-dependent problem and to the Matthew Colbrook for discussions about numerical solution of eigenvalue problems for non-self-adjoint infinite-dimensional operators. Miguel Bustamante is acknowledged for pointing out the potential connection with tygers. Partial support for this research was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant. The author would also like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Mathematical aspects of turbulence: where do we stand?” where a part of this study was conducted. This work was supported by EPSRC grant number EP/R014604/1. Computational resources were provided by the Digital Research Alliance of Canada (DRAC) under its Resource Allocation Competition. § CONSTRUCTION OF THE DIRICHLET-TO-NEUMANN MAP We consider the Laplace subproblem consisting of (<ref>)–(<ref>) and (<ref>) whose solution has the general form ψ_2'(r,θ) = ∑_k=1^∞α_k cos(k θ) + β_k sin(k θ)/r^k, r ≥ 1, 0 ≤θ≤ 2π, whereα_k,β_k ∈,k = 1,2,…, are expansion coefficients to be determined and the constant term is omitted since we adopt the normalization∮_∂A_0 f'(s) ds = 0. The boundary valuef'of the perturbation streamfunction on∂A_0serves as the argument of the D2N operator, cf. (<ref>). Expanding it in a Fourier series gives f'(θ) = ∑_k=1^∞f_k^c cos(k θ) + f_k^s sin(k θ), wheref_k^c, f_k^s ∈,k = 1,2,…, are known coefficients. Then, using the boundary conditionψ_2'(1,θ) = f'(θ),θ∈[0,2π], cf. (<ref>), the corresponding Neumann data can be computed as [M f'](θ) := ψ_2'n\|_∂ A_0 = ψ_2'r\|_r=1 = - ∑_k=1^∞ k [f_k^c cos(k θ) + f_k^s sin(k θ) ] which expresses the action of the D2N operatorMonf'. In order to make this expression explicitly dependent onf', rather than on its Fourier coefficients as in (<ref>) , we use the formulas for these coefficients together with their approximations based on the trapezoidal quadrature (which are spectrally accurate when applied to smooth periodic functions <cit.>) f_k^c = 1/π∫_0^2π f'(θ') cos(k θ') dθ' ≈ 2/N∑_l=1^N f'(θ_l) cos(k θ_l), f_k^s = 1/π∫_0^2π f'(θ') sin(k θ') dθ' ≈ 2/N∑_l=1^N f'(θ_l) sin(k θ_l), where{θ_l }_l=1^Nare grid points uniformly discretizing the interval[0,2π]. Using these relations, the D2N map (<ref>) truncated atN/2Fourier modes and evaluated at the grid pointθ_jcan be written as [M f'](θ_j ) ≈∑_l=1^N M_jl f'(θ_l), j=1,…,N, where M_jl := - 2/N∑_k=1^N/2 k [ cos(k θ_j) cos(k θ_l) + sin(k θ_j) sin(k θ_l) ] are entries of a symmetric matrix∈^N ×Napproximating the D2N operator. § SOLUTION OF OUTER PROBLEM (<REF>) Assuming separability, we use the ansatzϕ(r,θ) = R(r) T(θ), whereR : [0,1] →andT : [0,2π] →. Plugging this ansatz into (<ref>), we obtain the relationu_0^r T(θ) (dR/dr) = - (u_0^θ/ r) R(r) (dT/dθ), which using expressions (<ref>)–(<ref>) for the velocity components can be rewritten as r J_1(b r)/ b r J_0(b r) - J_1(b r)1/R(r)dR/dr = tan(θ)/(θ)dT/dθ = C with some real constantC ≠0. The azimuthal partdT/dθ- C (θ) T(θ) = 0can be integrated using the periodic boundary conditions to give T(θ) = A sin^C(θ), A ∈. The radial part of (<ref>) isdR/dr - C [ b r J_0(b r) - J_1(b r) ] / [ r J_1(b r) ] R(r) = 0, which upon integration gives R(r) = B [ J_1(b r) ]^C, B ∈. Imposing the boundary condition (<ref>) and requiring the solution to be real-valued while noting thatJ_1(0) = 0and(d/dr) J_1(b r)\|_r=0 ≠0, restricts the values ofCto integers larger than 1. Thus, combining (<ref>) and (<ref>) finally gives ϕ(r,θ) = ϕ_C(r,θ) := B [J_1(br) sinθ]^C, C = 2,3,…. § MAXIMUM PERIODS OF LAGRANGIAN ORBITS In this appendix we estimate the maximum periodof Lagrangian orbits in the flow field of the Lamb-Chaplygin dipole where we focus on the symmetric case withη= 0in (<ref>). We consider the heteroclinic trajectory connecting the two hyperbolic stagnation points_aand_b, cf. figure <ref>a, which coincides with a part of the boundary∂A_0. Lets = s(t)denote the arc-length coordinate of a material point on this orbit. Then, assuming the dipole has unit radiusa = 1, we have, cf. 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http://arxiv.org/abs/2307.00748v2	20230703044910	Decoherence Limits the Cost to Simulate an Anharmonic Oscillator	[ "Tzula B. Propp", "Sayonee Ray", "John B. DeBrota", "Tameem Albash", "Ivan Deutsch" ]	quant-ph	[ "quant-ph" ]	#1#2=0pt#1#2 #1#2=0pt#1#2 theoremTheorem 0.0cm 0.0cm 0.cm 0.0cm corollaryCorollary[theorem]
http://arxiv.org/abs/2307.02006v1	20230705033112	PULSAR at MEDIQA-Sum 2023: Large Language Models Augmented by Synthetic Dialogue Convert Patient Dialogues to Medical Records	[ "Viktor Schlegel", "Hao Li", "Yuping Wu", "Anand Subramanian", "Thanh-Tung Nguyen", "Abhinav Ramesh Kashyap", "Daniel Beck", "Xiaojun Zeng", "Riza Theresa Batista-Navarro", "Stefan Winkler", "Goran Nenadic" ]	cs.CL	[ "cs.CL" ]	2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CLEF 2023: Conference and Labs of the Evaluation Forum, September 18–21, 2023, Thessaloniki, Greece 1,2]Viktor Schlegel 2]Hao Li[[email protected]] 2]Yuping Wu[[email protected]] 1,3]Anand Subramanian 1]Thanh-Tung Nguyen 1]Abhinav Ramesh Kashyap 4]Daniel Beck 2]Xiaojun Zeng 2]Riza Theresa Batista-Navarro 1,3]Stefan Winkler 2]Goran Nenadic [1]ASUS Intelligent Cloud Services (AICS), Singapore. [2]Dept. of Computer Science, University of Manchester, United Kingdom. [3]Dept. of Computer Science, National University of Singapore, Singapore. [4]School of Computing and Information Systems, University of Melbourne, Australia. This paper describes PULSAR, our system submission at the ImageClef 2023 MediQA-Sum task on summarising patient-doctor dialogues into clinical records. The proposed framework relies on domain-specific pre-training, to produce a specialised language model which is trained on task-specific natural data augmented by synthetic data generated by a black-box LLM. We find limited evidence towards the efficacy of domain-specific pre-training and data augmentation, while scaling up the language model yields the best performance gains. Our approach was ranked second and third among 13 submissions on task B of the challenge. Our code is available at <https://github.com/yuping-wu/PULSAR>. Abstractive Summarisation AI for Healthcare Dialogue Summarisation Natural Language Processing PULSAR at MEDIQA-Sum 2023: Large Language Models Augmented by Synthetic Dialogue Convert Patient Dialogues to Medical Records [ August 1, 2023 ============================================================================================================================= § INTRODUCTION With the recent successes of generative large language models (LLMs) on a variety of tasks <cit.> and domains <cit.>, even in the face of data scarcity <cit.>, there is vivid interest in identifying potential application scenarios that could benefit from the power of LLMs. One of the promising domains is healthcare <cit.> as many administrative tasks involve the transformation of textual data. LLM-based approaches that assist hospital staff in repetitive administrative tasks have the potential to improve operational efficiency and documentation quality, optimise revenue streams, reduce cognitive load on healthcare experts, and ultimately result in better and more effective patient care <cit.>. A range of different scenarios have been investigated for the suitability of LLM-based assistance, such as summarising patient progress notes as discharge summaries <cit.> or identifying problems that need treatment during a patient's hospital course <cit.>. One of the potential tasks is summarising doctor-patient dialogue as medical records <cit.>. Dialogue summarisation, an established task in the Natural Language Processing (NLP) community, aims to identify salient topics in a multi-turn dialogue <cit.>. State-of-the-art approaches typically formulate the problem as abstractive summarisation, making the task a prime candidate for further investigation of the potential of LLMs in clinical settings. In this scenario, conversations between patients and doctors need to be transformed into (excerpts of) clinical documentation. For example, if a 27 year old female patient mentions that they are experiencing “Sore throat, runny nose, dry cough and fever 37.5 ^∘C”, the corresponding entry can be the “Subjective” section of a medical record excerpt, e.g., “Patient is a 27 year old female who presents with sore throat, runny nose dry cough and a fever of 37.5 ^∘C.” This documentation is typically performed by the consulting doctor or an attending nurse. Despite bearing potential impact for automation, with clinical staff spending at least 35 minutes of their time every other day on writing such clinical notes <cit.>, this task was underexplored by the NLP community, compared to other hospital-related tasks, such as clinical coding <cit.>, or generating radiology reports <cit.>. More recently, the task has received more attention <cit.>, however studies thus far have either focussed on narrow department selections <cit.>, did not focus on medical documentation generation <cit.>, or have not released their data publicly <cit.>. To that end, the ImageClef 2023 MediSum shared task released a collection of dialogues and corresponding clinical notes in an effort to spark interest and advance the state of the art in dialogue as clinical note summarisation <cit.>. The task revolves around three core sub-tasks: (A) identifying the topic of a conversation from a selection of possible medical note sections (i.e., “Subjective” in the previous example), (B) summarising conversation snippets to appropriate sections in medical records, and, finally, (C) summarising full conversations to full medical records. While conversations are synthetic, the corresponding clinical notes are real, doctor-written documentation. Our guiding objective to participate in this task was to investigate, how well a recently proposed LLM training framework can generalise to new tasks with as little adaptation as possible <cit.>. At its core, the framework (i) fine-tunes a LLM with a pre-training objective that learns to reconstruct a pseudo-summary consisting of automatically extracted medical terms and (ii) employs data augmentation (DA) by instructing Black-Box LLMs to obtain task-specific training data. As such, the DA framework supports any LLM, such as Bloom <cit.>, GPT-3 <cit.> or GPT-3.5 <cit.>. Our submission for task B was ranked second best overall among all participants. Although we have not actively sought to compete in Task C, we observed that our data augmentation technique could improve the performance, particularly when the training data is scarce. These findings underline the potential of LLMs in various settings as well as the generalisability of our proposed approach. § TASK DEFINITION In this section, we describe and formalise the three tasks of the ImageClef 2023 MediSum challenge. Task A – Dialogue2Topic Classification In this task, participants need to identify the topic of a conversation. The list of possible topics corresponds to the 20 different fine-grained sections that can be part of a medical record, such as “Subjective”, i.e., the subjective description of symptoms by the patient. Task B – Dialogue2Note Summarization Here, participating systems need to convert a conversation on a specific topic into a corresponding section in the medical record. This task can be regarded as conditional generation, sequence-to-sequence translation or abstractive summarisation. Approaches are evaluated on multiple natural language generation metrics, both based on n-gram overlap, i.e., ROUGE <cit.>, as well as semantic similarity <cit.>. 1201 training and 100 validation examples are provided. 200 examples form the test set. Task C – Full-Encounter Dialogue2Note Summarization This task is formulated similarly to Task B, however here, the inputs are full notes and the evaluated systems need to generate medical record outputs for the four general sections “Subjective”, “Objective Exam”, “Objective Results” and “Assessment and Plan”. This task features only 67 training and 20 validation examples, with 40 examples reserved for testing. The systems are evaluated based on their output for each of the sections using the ROUGE metrics from Task B; the results are averaged across all sections. An alternative mode of evaluation combines all outputs into one single record and measures the n-gram overlap by means of the ROUGE score. The tasks appear to be arranged as a progression, where, given a dialogue, a segmentation and classification model could segment the topics of the conversation (Task A) to be used as input for a Dialogue Snippet Summarisation Model (Task B), the output of which can be arranged as a full medical record (Task C). However, as our goal was to evaluate how well the proposed framework generalises to the tasks with as little adaptation as possible, we decide not to make any task-specific adaptations even if they could provide beneficial given the particular arrangement of the tasks. Thus, we do not rely on any additional information, treat tasks B and C in isolation, and disregard task A for not being a generative task. § METHODOLOGY §.§ Language model Pre-training Motivated by the success of predicting masked words <cit.> and contiguous spans <cit.> as self-supervised training objectives, we customised the pre-training objectives for the medical domain generation task to concatenate “gap text spans (sentences)” into a pseudo-summary. Each masked span is a medical term from the input text identified by the QuickUMLS <cit.> or a NER model fine-tuned on a N2C2 dataset (i2b2-2010 challenge <cit.>). Specifically, as shown in Figure <ref>, pre-training consisted of three different policies: first, when both QuickUMLS and N2C2 NER models identified entities, the QuickUMLS results were used in 70% of cases and the results of the N2C2 NER model were used in 30%. Second, when only one of them predicted any output, that output was used for masking. Third, when neither had any output, then 15% of the sentences were masked at random. These text spans were replaced with “sentinel” mask tokens <extra id i> to inform the model that input was masked. In order to provide the model with sufficient medical knowledge, we used the MIMIC-III <cit.>, a pre-trained corpus of 2 million data, which consists of a large number of clinical records, such as admission notes, discharge summaries or lab results. §.§ Data Augmentation (DA) Both tasks suffer from scarcity of training data, specifically Task C, which requires generating comprehensive clinical notes based on lengthy patient-doctor conversations based on only 67 training examples. These may be insufficient to train a model capable of performing well on the task. To address this issue, we adopt data augmentation to generate additional examples for training, as this has been shown to improve performance in data-scarce scenarios <cit.>. Prompting Strategy We observed that Large Language Models (LLMs) such as ChatGPT are proficient in understanding clinical context and manipulating clinical data. Therefore, we utilise a pre-existing LLM to generate data for the model's training. Ideally, the data generation approach would involve providing conversations and requesting the LLM to produce the corresponding medical note. However, we are limited by the fact that we only have 67 full-length conversations in our dataset. Nonetheless, we have access to a significantly larger number of medical notes. Hence, we invert the task by prompting the LLM with a medical note (or its snippet) and ask it to generate a hypothetical conversation between the doctor and the patient. We then use the generated conversations as input to train our model to produce the corresponding clinical note. We employ the OpenAI ChatGPT API (gpt-35-turbo) for data augmentation, utilising a two-stage prompting strategy to generate data effectively. In the first stage, we use in-context learning with one-shot prompting to prompt the LLM to generate a fictitious conversation between the doctor and patient based on the medical note, while adhering to important guidelines. We provide only one example picked from the training set as we are limited by token context windows for the API. In the second stage (only performed for task C), we prompt the model to include conversational fillers such as “ums”, “uh”, and “hmm” to the generated conversation from the first stage, as we noticed that the model did not include these fillers despite our instructions in the first stage. Dataset Utilised For task B, we extract matching subsection headings from the MIMIC-III database <cit.>, adapting the pre-processing method from <cit.> to identify section headers. We rank the generations based on their average Rouge similarity to all training instances and pick the top-scoring n conversations. For task C, we utilise a corpus of freely available medical notes scraped from MTSamples, which is available on Kaggle[<https://mtsamples.com/> and <https://www.kaggle.com/datasets/tboyle10/medicaltranscriptions>, respectively]. Since the dataset contains medical transcriptions of notes from various medical specialities, we devise a method to pick samples from the dataset that are the closest to the medical notes in our training set. To do this, we identify and curate a list of the section headers in the training set through a heuristic approach by exploiting the fact that section headers are usually written in all capital letters. We split the document by newline and extract the lines which are fully upper-cased and add these contents to our list of section headers. We then score the medical notes in MTSamples based on the number of headers that each document has based on the curated list from the previous step and pick the top n documents from MTSamples with the highest scores to use as input for DA. We end up with a corpus of 746 data samples due to the fact that some inputs were flagged as offensive by OpenAI's content moderating policy. § EMPIRICAL EVALUATION §.§ Experiments set-up We aim to empirically evaluate, how well our framework can solve the problem of converting patient dialogues to medical records. We pursue the following questions: * How well can our proposed approach convert doctor-patient dialogues to Medical Records? * Does the domain-specific pre-training objective improve performance? * What is the impact of model scale on the performance? * Does synthetic data augmentation improve performance on the tasks? To answer question (i) we empirically evaluate our proposed framework on the task B and C test sets of the ImageClef Challenge. For evidence towards question (ii), we compare the performance of to that of equally-sized models. Regarding question (iii), we compare the performance of variously sized models of the same architecture and for question (iv), we compare the performance of models trained on available data only to those fine-tuned on synthetically generated conversation data. §.§ Implementation Details Pre-training is initialised with weights from the corresponding models <cit.> and pre-trained with four NVIDIA Tesla A100 80GB GPUs for 1 epoch on all MIMIC-III notes. Huggingface Accelerate is used to optimise GPU memory usage with the Fully Sharded Data Parallel (FSDP) paradigm. We set the training batch size per GPU device as 4 and the gradient accumulation step as 8 to accelerate the training process. Fine-tuning We fine-tune all models for 3 epochs. We experiment with encoder-decoder , and <cit.> models, with the configurations (0.9B Parameters), and . Unless stated otherwise, the models are trained on two A100 80GB GPUs with cumulative batch size of 8 and the learning rate of 3^-5. For the largest of them, i.e., ( and ), we use FSDP with CPU offloading. We also experiment with a decoder-only model, , freezing and quantising the base model in eight bit <cit.> and using the parameter-efficient LoRA <cit.> method. More details on hyper-parameter choice are reported in the appendix. §.§ Results and analysis At a glance, Table <ref> shows the results of our empirical study and Table <ref> shows the final ranking of all participating systems according to the official evaluation by task organisers In the following, we discuss our findings in context of the questions outlined in the motivation of this empirical study. Our approach generalises well to the dialogue summarisation task. Overall, our approach generalises well to Task B, with our best model (Table <ref>, 11B1) surpassing the 50 Rouge-1 scores mark, which means that on average, half of the prediction tokens are found in reference and vice versa. The high Rouge-L score of 44 suggests that most of these overlapping tokens indeed form a sequence. However, these scores may be “boosted” by the presence of many short target sequences in the evaluation data, such as “Noncontributory.” or “No known allergies.”, when a dialogue revolves around a topic that does not contribute to the patients' hospital visit. We find that the utilising the outputs of task A—the section headers—does not contribute to improving the overall performance, compare Table <ref>, L2 and L4. We observed the same trend across all model sizes (not reported here for brevity). In the absence of established baselines, we interpret the official rankings of the shared task in Table <ref> as additional evidence towards the success of our approach. There is no conclusive evidence that domain-specific pre-training is beneficial. Comparing 11B1 and 11B2, and 3B1 and 3B2 in Table <ref>, respectively, we observe that domain-specific pre-training by learning to predict missing medical terms in MIMIC-III notes appears not beneficial, with the gap being smaller for bigger models. One possible reason for this is the domain mismatch between pre-training and application data. MIMIC-III is dominated by inpatient progress notes which track the patients' status along the hospital stay and contain abbreviations, repetitions, incomplete sentences and medical jargon. Conversely, the medical records in the challenge are well-written and stem most likely from admission notes or outpatient encounters, where most of the initial documentation about a new patients' particulars, such as their chief complaint, medical history and drug allergies happens. Additionally, input dialogues have a colloquial tone, further adding to the domain mismatch between pre-training and fine-tuning. Model scale yields the biggest performance improvements. Comparing L, 3B and 11B* results in Table <ref>, we can see a clear trend where larger models of the same family consistently perform better. The biggest hike in performance is observed between the 3B and 11B models. This observation is in line with most literature on model scale as driver of performance and the reason for emergent abilities in LLMs <cit.>. We also find that the model trained with adapters can learn to perform on the task successfully, despite the relatively small (around 1.1% of the full 7B model) number of trainable parameters. However, our results suggest that updating all models' parameters is more effective, as even smaller models outperform the 7B adapter model (Table <ref>, L2, 3B* compared to 7B1). Data Augmentation can be helpful if training data is extremely scarce. Larger models obtain enough signal from the training data of Task B, as there is no clear improvement in scores for the 3B models (Table <ref>, 3B1 vs. 3B3 and 3B2 vs. 3B4). Meanwhile, data augmentation can lead to consistent, albeit minor, improvements for smaller models (Table <ref>, L2 vs. L3). When training data is scarce (i.e., Task C) data augmentation helps with the performance. Subjectively, models exhibit typical generation errors such as hallucination and input copying, (see Figure <ref> in Appendix) and data augmentation seems to alleviate this issue. Quantitatively, data augmentation improves performance across all metrics (27.64 vs 29.41 R1, 9.79 vs 11.60 R2, 16.24 vs 19.18 RL and 23.63 vs 26.08 RLSum without and with DA, respectively). We find the results promising, as the optimised model seems to perform well without any task-specific adaptation. Ultimately, however, this simple approach does not compete with other, potentially task specific information exploiting submissions, with the best of them scoring almost 20 Rouge-1 points higher (20.32 R2, 24.30 RL and 45.06 RLSum). § CONCLUSION In this work, we present an LLM framework and adapt it to the task of dialogue note summarisation. While we find that the approach generalises well to this new task, there is mixed evidence of the efficacy of both domain-specific pre-training and data augmentation. Our experiments seem to align with the “bitter lesson of AI”[<http://www.incompleteideas.net/IncIdeas/BitterLesson.html>], in that model scale seems to trump domain-specific adaptations. This, in turn, supports the narrative of the transformative potential of LLMs in healthcare <cit.>, as larger LLMs become more readily available. Our findings suggest further avenues for future work: We argued that the pre-training objective may suffer from domain mismatch. As such, experimenting with other domain-specific objectives might improve the performance of the downstream tasks. Furthermore, it is unclear how the choice of hyper-parameters for both training and inference stages (i.e., decoding arguments) impacts the overall performance. Finally, we have left it for future work to investigate, whether data augmentation could provide beneficial with a more advanced filtering strategy, for example by only augmenting examples with certain length or specific section headers. As such, we will expand the work reported in this paper by experimenting with different pre-training objectives, performing a more rigorous hyper-parameter optimisation and investigating the impact of data augmentation more closely. § LIMITATIONS The results described in this paper should be interpreted within the following context: * The language of the conversations is English. Due to the dominance of English data during pre-training, it is expected that all LLMs that we inspected perform better on English. It is unclear how well the approach will transfer to other languages. * The conversations are synthetic in that they have been written based on existing medical notes, rather than transcribed from real patient-doctor dialogues. While the quality has been evaluated by medical professionals, it is unclear how well the performance would translate to real-world scenarios. * The obtained results should be regarded as preliminary, as robust empirical results such as hyper-parameter optimisation for fine-tuning, pre-training policy selection, exhaustive search for best-performing prompts for data augmentation and strategies for data selection are often impossible given the time constraints of academic challenges and shared tasks. § HYPER-PARAMETERS FOR TRAINING AND INFERENCE We initialise LoRA with r=16, α=16 on the query, key, value and output projection weights of all layers of the base model (Q, K, V and O, respectively). The model is trained on a single A100 80GB GPU with a learning rate of 3^-4 for the adapter weights. For both encoder-decoder and decoder only settings, during training, we optimise the parameters of the language models to minimise the cross-entropy loss between each token of the prediction and the corresponding token of the ground truth answer sequence using teacher forcing. For encoder-decoder models, we limit the length of input dialogues to at most 496 and the length of output notes to at most 214 tokens, respectively (95th percentile). For the decoder model, we limit the length of input and output combined to at most 696 tokens. During inference, we set no limits to input and output sequence lengths and decode the prediction using beam search with 6 (4 for LLaMa), temperature of 1.0, top k of 50 (40 for LLaMa) and top p of 1.0 (0.7 for LLaMa). For task C, we use the same arguments as for task C, with the exception of limiting the input length to 2048 and output length to 990 during training, in order to fit the GPU during training. § QUALITATIVE EXAMPLE Figure <ref> shows qualitative examples generated by our models trained on task C training data with and without data augmentation, respectively.
http://arxiv.org/abs/2307.00960v1	20230703122509	Neural Architecture Transfer 2: A Paradigm for Improving Efficiency in Multi-Objective Neural Architecture Search	[ "Simone Sarti", "Eugenio Lomurno", "Matteo Matteucci" ]	cs.LG	[ "cs.LG", "cs.AI", "cs.CV" ]	Challenges in Domain-Specific Abstractive Summarization and How to Overcome them Anum Afzalsup1, Juraj Vladikasup1, Daniel Braunsup2, Florian Matthessup1 sup1Department of Computer Science, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching bei Muenchen, Germany sup2Department of High-tech Business and Entrepreneurship, University of Twente, Hallenweg 17, 7522NH Enschede, The Netherlands {anum.afzal, juraj.vladika, matthes}@tum.de, [email protected] Received date ; Accepted date =========================================================================================================================================================================================================================================================================================================================================================================================================== Deep learning is increasingly impacting various aspects of contemporary society. Artificial neural networks have emerged as the dominant models for solving an expanding range of tasks. The introduction of Neural Architecture Search (NAS) techniques, which enable the automatic design of task-optimal networks, has led to remarkable advances. However, the NAS process is typically associated with long execution times and significant computational resource requirements. Once-For-All (OFA) and its successor, Once-For-All-2 (OFAv2), have been developed to mitigate these challenges. While maintaining exceptional performance and eliminating the need for retraining, they aim to build a single super-network model capable of directly extracting sub-networks satisfying different constraints. Neural Architecture Transfer (NAT) was developed to maximise the effectiveness of extracting sub-networks from a super-network. In this paper, we present NATv2, an extension of NAT that improves multi-objective search algorithms applied to dynamic super-network architectures. NATv2 achieves qualitative improvements in the extractable sub-networks by exploiting the improved super-networks generated by OFAv2 and incorporating new policies for initialisation, pre-processing and updating its networks archive. In addition, a post-processing pipeline based on fine-tuning is introduced. Experimental results show that NATv2 successfully improves NAT and is highly recommended for investigating high-performance architectures with a minimal number of parameters. § INTRODUCTION Deep learning has emerged as a significant revolution in recent years, significantly impacting various aspects of modern society. It has notably transformed numerous activities by leveraging artificial neural networks. These networks possess remarkable capabilities, outperforming conventional approaches in multiple tasks. One notable advantage is their ability to eliminate the requirement for manual feature engineering, as they autonomously discern meaningful patterns from the provided data. The effectiveness of deep learning networks stems from their meticulously designed layered architecture, enabling proficient feature extraction. While human research endeavors have achieved notable advancements in performance, the resulting models have exhibited a trend towards increased size <cit.>. Consequently, their production necessitates not only specialized expertise but also hardware, energy, and production times that have become progressively unattainable <cit.>. Neural Architecture Search (NAS) emerged to address the need for innovative neural architectures that are universally applicable and don't require extensive expertise. Its primary goal is to automatically discover the optimal configuration for a given dataset and task <cit.>. Over time, NAS has also incorporated considerations for computational and temporal constraints. These techniques aim to improve the performance of found models trading-off with the whole search process complexity. This includes minimizing time, energy consumption, and CO_2 emissions, as well as achieving a favorable trade-off between model performance and complexity in terms of parameters and operations. Additionally, NAS must account for scenarios where devices running these models have limited memory or constraints. Therefore, it is crucial to broaden the scope of NAS beyond benchmark rankings and consider it as a means to address such limitations <cit.>. The work known as Once-For-All (OFA) represents a significant milestone in this direction. As the name suggests, the objective of this approach is to perform massive computations in a single instance by constructing a super-network, from which sub-networks satisfying different constraints can be readily extracted, while maintaining excellent performance <cit.>. This work has been successfully expanded through the technique known as Once-For-All-2 (OFAv2). The authors maintained the same underlying training principle as the original algorithm but adapted it to a network search space which was extended with proven techniques from the field of artificial neural network design, thereby elevating the obtained super-network to higher levels <cit.>. The extraction of sub-networks represents the final and crucial step. While OFA already proposed a potential solution in this regard, the Neural Architecture Transfer (NAT) algorithm was specifically developed to maximize the effectiveness of this step. NAT seeks to generate neural architectures that exhibit strong performance across diverse objectives by leveraging knowledge transfer and adaptation from pre-trained super-network models. It employs a combination of transfer learning and many-objective evolutionary search steps. Specifically, it adapts only the portions of the super-network corresponding to sub-networks discovered along the trade-off front by the search algorithm <cit.>. This paper introduces NATv2, an extension of NAT that enhances the capabilities of multi-objective search algorithms on dynamic super-network architectures. NATv2 replaces the original super-network, OFAMobileNetV3, used in NAT and pre-trained with OFA's Progressive Shrinking algorithm, with super-networks generated by OFAv2. Consequently, significant qualitative improvements are achieved in the extractable sub-networks' topology, allowing for the inclusion of parallel blocks, dense skip connections, and early exits. To enhance the NATv2 archive, new policies are implemented for initialization, pre-processing, and updates. Moreover, a novel encoding type is proposed to accommodate these improvements, while the pipeline's predictors are upgraded to higher-performance techniques. Additionally, a post-processing pipeline based on fine-tuning is introduced, which further enhances model performance at a marginal increase in parameters and MACs. By integrating all these advancements, NATv2 demonstrates the ability to generate image classification networks that surpass the accuracy achieved by NAT. Furthermore, NATv2 achieves this improvement with a reduced number of parameters and MACs. An overview of the proposed technique is depicted in Figure <ref>. The rest of the paper is divided into the following sections. Section <ref> provides an introduction to NAS, the main works in the field of image classification, and the rationale behind their design choices. Section <ref> describes the NATv2 method in detail, reporting on its workflow and paying special attention to the additions and improvements introduced by the new version. The experiments performed, the configurations used and the qualitative and quantitative comparisons are described in Section <ref>. Finally, Section <ref> summarises the contributions of this work and concludes the manuscript. § RELATED WORKS Neural Architecture Search (NAS) is an evolving research area within the deep learning community that combines various techniques from machine learning and optimization domains. The primary goal of NAS is to automatically design complex neural network architectures in an end-to-end process without human intervention. Despite its popularity in the AI community, NAS lacks standardised approaches due to the variety of techniques involved. Howerver, Elsken et al. proposed a widely accepted classification of NAS algorithms based on three key characteristics <cit.>: * The search space, referred to as the set of all possible architectures that can be found by the algorithm. * The search strategy, which defines how the algorithm explores the search space to find optimal architectures for the given task. * The performance evaluation strategy, which determines how to efficiently evaluate the quality of the architectures during the search process. Early NAS research achieved remarkable model quality, but required significant computational resources. For example, NASNet, a pioneering work in the cell-based NAS approach, emerged as a competitive solution for image classification, rivalling state-of-the-art human-designed neural networks <cit.>. Inspired by highly successful models such as ResNet <cit.> and InceptionNet <cit.>, which featured the sequential repetition of convolutional modules, NASNet aimed to identify the most effective set of layers and connections for the given task and encapsulate them in a computational macro unit called a cell. This cell could then be stacked multiple times according to the desired depth of the final network. Unfortunately, this early work remained a challenge constrained within the confines of large computing centres, limiting its accessibility. However, a significant democratising advance came with the introduction of the PNAS algorithm. PNAS introduced a sequential model-based optimisation technique into the NAS context to relax the computational and time constraints. The key idea is that the search process could start with thinner models and progressively add new parallel and sequential layers based on guidance from a surrogate model called predictor. The predictor is a machine learning model that estimates the potential accuracy of each candidate architecture and dynamically adapts to the training results of previously sampled networks <cit.>. In further advancements, the POPNAS series of algorithms were developed to improve the efficiency of the cell-based approach. These algorithms extended the use of predictors to estimate the training times of the architectures to be searched. This allowed a transition to multi-objective optimisation, which considered both minimisation of search times and quality of architectures by explicitly training networks on the Pareto front. This improvement significantly increased the search efficiency without compromising the quality of the best architectures obtained <cit.>. An alternative approach is taken by works such as AmoebaNet <cit.> and NSGANet <cit.>, which use evolutionary algorithms for architecture search and employ gradient descent techniques to optimise the weights of the discovered architectures. In evolutionary algorithms, the search space represents the phenotype, while the architectures being searched are encoded as genotypes. At each iteration, a population of architectures is maintained, and their genotypes are modified by mutation and crossover operations to produce offspring. Mutations in this context involve the random swapping of bits in the encoding, often resulting in the addition or removal of a layer, or the establishment or removal of a connection between two layers. DARTS improves search efficiency by introducing a relaxation to the search space, making it continuous. This allows the entire search space to be represented as a single large network, known as a super-network, where each edge between layers is parameterised and part of the optimisation process. This modification allows both model weights and structure to be optimised using gradient descent via two specialised training steps, one for the whole super-network and one for the sub-networks within it. In this way, the final architecture is nothing more than a sub-graph extracted from the super-network itself <cit.>. The efficiency of DARTS has contributed to the emergence of a new class of methods known as one-shot architecture searches. During the search process, each sampled architecture can be viewed as a composition of paths within the super-network. The weights of these paths are inherited from the super-network and fine-tuned to quickly assess the accuracy of the network <cit.>. However, training super-networks is challenging as they are more sensitive to hyperparameter changes and weight co-adaptation. Specialised training techniques are required to avoid favouring only certain subsets of architectures in the final estimation phase. §.§ Once-For-All Once-For-All (OFA) is a pivotal work in the realm of super-network-based NAS techniques, designed to maximize search efficiency. As a prominent component of hardware-aware NAS techniques, OFA has gained significant recognition and widespread adoption due to its Progressive Shrinking (PS) optimization strategy <cit.>. This strategy not only enables the acquisition of excellent starting points for sub-network extraction but also concentrates the computational load into a single end-to-end training process. The PS algorithm is organized into four elastic steps, each comprising multiple phases. The first step, Elastic Resolution, involves randomly varying the size of input images. The second step, Elastic Kernel Size, gradually reduces the maximum kernel size for convolutional operators across the entire network. The third step, Elastic Depth, progressively decreases the minimum depth achievable for sub-networks. Finally, the fourth step, Elastic Width, aims to reduce the number of filters available for each convolutional layer. The algorithm begins by defining the maximal network, which includes all PS parameters set to their maximum values. Subsequently, the PS training steps and phases are executed sequentially. The values unlocked by each phase for a specific elastic step remain available for selection in all subsequent training phases, enabling the addition of smaller networks to the search space. During each batch of images, a certain number of sub-networks are sampled from the current sample space and activated within the super-network. Their gradients are accumulated, and a single weight update step is performed. Throughout the training process, the maximal sub-network serves as the teacher network for Knowledge Distillation <cit.>. KD involves transferring knowledge from a large pre-trained network (referred to as the teacher network, in this case, the maximal sub-network) to a smaller network (referred to as the student network, in this case, an active sub-network). This is achieved by using the output of the teacher network as a soft target for the student network during training. By learning from the predictions of the teacher network, the student network can achieve similar performance to the larger teacher network while being smaller in size. Once the PS algorithm is concluded, it is possible to sample sub-networks to extract their configuration via encoding, train surrogate models also known as performance predictors, and finally exploit them to identify the most suitable and performing sub-network according to different hardware constraints. The results presented show how OFA effectively addresses the multi-model forgetting problem, which refers to the performance degradation caused by weight sharing when sequentially training multiple neural architectures within a super-network <cit.>. Among the notable macro-architectures introduced by the authors, the main one, called OFAMobileNetV3, is based on the Inverted Residual Bottleneck (IRB), i.e. an highly efficient type of block, based on depthwise separable convolutions, originally introduced in MobileNetV2 <cit.> and further refined in MobileNetV3 <cit.>. By default, the network consists of five stages, each consisting of four blocks. In cases where the number or size of the feature maps from the input to the output of a block is changed, the residual connection cannot be used. Therefore, the IRB block is replaced by its sequential counterpart known as an Inverted Bottleneck (IB). §.§ Neural Architecture Transfer Neural Architecture Transfer (NAT) is a recent advancement in NAS that builds on the OFA framework. It serves as an adaptive post-processing step that replaces the simple sub-networks extraction in the original OFA algorithm. NAT aims to progressively transform a pre-trained generic super-network into a task-specific super-network. The goal is to directly search and extract sub-networks that achieve the best trade-off across a range of objectives from the task-specific super-network, without the need for re-training <cit.>. To optimise the efficiency of the super-network adaptation process, NAT selectively fine-tunes only those parts of the super-network that correspond to sub-networks whose structures can be directly sampled from the current trade-off front distribution. This approach saves computation by focusing on the parts that contribute to improvements in the current task. The multi-objective evolutionary search in NAT is guided and accelerated by a performance prediction model that is updated online using only the best sub-networks configurations and their scores. This approach helps maintain a high-performance prediction model despite using a relatively small number of sub-networks as training samples. By relying on the predictor, NAT can save time by avoiding evaluating each member of the intermediate populations generated during the evolutionary search. Throughout the process, the best architectures found are gradually added to an archive. In the end, NAT returns three main outputs: the set of non-dominated sub-networks from the archive, which represent the best trade-offs across objectives; the set of high trade-off sub-networks, which are solutions where choosing any neighbouring solution would result in a significant loss in some objectives compared to a unitary gain in others; and the resulting task-specific super-network, which can be reused as a starting point for a new NAT process in different deployment scenarios. §.§ Once-For-All-2 Once-For-All-2 (OFAv2) represents a significant evolution from its original version, aiming at the same goal of a one-shot NAS algorithm to construct a super-network from which models suitable for different devices can be extracted. This new version introduces significant improvements, particularly in the search space <cit.>. In particular, the original OFAMobileNetV3 macro-architecture has been enhanced by the authors through the incorporation of parallel blocks, dense skip connections, and early exits, the latter taking advantage of the Anticipate Ensemble and Prune (AEP) technique <cit.>. These additions increase the flexibility and performance of the super-network. To accommodate the aforementioned architectural changes, the PS training algorithm has been extended to the Extended Progressive Shrinking (EPS) algorithm, incorporating two new elastic steps: Elastic Level and Elastic Exit. Elastic Level supports parallel networks, while Elastic Exit is applied in the presence of early exits. In addition, OFAv2 introduces a novel teacher network extraction strategy. This strategy dynamically updates the teacher network at the end of each EPS step, ensuring the transfer of relevant knowledge for subsequent training steps. § METHOD This section presents the second version of the Neural Architecture Transfer (NATv2) algorithm, which builds on the Once-For-All-2 (OFAv2) technique used to generate the the super-networks used as staring points. The focus is on the modifications made to the original algorithm to accommodate any architecture generated by OFAv2. In addition, changes to the sub-network sampling method and performance predictor are presented. NATv2 introduces two new steps in its pipeline: a pre-processing step, which incorporates the new archive initialisation method, and a two-stage post-processing method. The pre-processing step is designed to ensure that the archive is appropriately set up to start with a set of performing architectures from the beginning. The post-processing step is applied after the conclusion of the algorithm to fine-tune and refine the selected architectures, ultimately improving their performance. Being an upgraded version, it is assumed that all steps and algorithms not explicitly mentioned have remained unchanged from the original approach. §.§ Expanded Search Space NATv2 requires a new encoding paradigm to enable evolutionary search on OFAv2 super-networks. The existing encoding method used by NAT to represent sub-network structures in OFAMobileNetV3 is insufficient to capture the full range of architectural permutations. However, it serves as a starting point and undergoes precise modifications to meet the new set of constraints. For the OFAMobileNetV3 architecture, the baseline NAT representation utilizes integer-encoded strings consisting of 22 values, as depicted in Figure <ref> under the label “Baseline". Each compressed representation contains specific information. The first value encodes the resolution R of the input image for the network. The second value represents the active width multiplier W. Width multipliers serve as scaling factors for the number of filters during the execution of the OFA algorithm. In the vanilla configuration, as in NATv2, two distinct super-networks are constructed, with W values of 1.0 and 1.2, respectively. These two initial encoding rules are retained in the NATv2 encoding scheme. In NAT, the set of 20 encoded values represents the combinations of kernel size K and expansion ratio E for each of the 20 internal IRB or IB blocks. To allow the inclusion of parallel blocks in the super-networks, as illustrated in Figure <ref>, these pairs are expanded into triplets by introducing the additional term A. The value of A ranges from 1 to 7, covering all possible permutations of parallel blocks activation states. Including the special value 0, which represents the ith block or level being excluded from the network, thus reducing the stage depth, each of the 20 P_i values can take up to 64 different values. This expansion of the search space allows greater exploration without changing the encoding size, as shown in the second row of Figure <ref> labelled “Parallel". In order to incorporate early exits into super-networks, and to capture sub-networks capable of making inferences at intermediate stages, it is necessary to encode this information appropriately. This ensures that the extracted sub-networks have the potential to make inferences at the end of each stage of the super-network, as illustrated in Figure <ref>. To allow for this, a new variable called X is introduced in the third position of the encoded strings, as shown in the third row of the Figure <ref> labelled “Early Exits". The value of X corresponds directly to the index of the selected exit, providing information about the selected exit if early exits are present in the super-network architecture. The final encoding used in NATv2 is shown in the fourth line of Figure <ref> and is named “Early Exits + Parallel". It encompasses the combined modifications necessary to incorporate parallel blocks and early exits. Compared to the version employed in NAT, the search space has been significantly increased at the minimal cost of one additional character in the encoding string. §.§ Archive Initialization and Update NATv2 takes a different approach to managing the archive of optimal sub-networks. The changes primarily affect two key stages of the NAT algorithm: the archive initialisation and archive growth steps. With respect to the archive initialisation step, NAT directly samples architectures from the search space, thus evenly distributing possible values within the encoding. As a result, the initial archive has a strong bias towards networks with a maximum stage depth of 4. This bias arises from the fact that skippable IRB blocks (specifically, the 3rd or 4th block within a stage) can be encoded with values ranging from 0 to 9, but only assigning a value of 0 will cause these blocks to be skipped. Essentially, due to uniform sampling, all values have an equal chance of being selected, making stages with a depth of 3 uncommon and stages with a depth of 2 quite rare. This problem becomes more pronounced when parallel blocks are supported, as the number of possible encodings for each network level is expanded up to 64 values. To address this issue, NATv2 approaches the archive initialisation phase by sampling subnets in a way that ensures uniformity not within the search space domain, but rather within the depth (for both stages and networks) and in the block configurations (for both parallel and non-parallel levels). This adaptation encourages greater heterogeneity in the NATv2 initialisation process, while improving the generalisation of the predictor through a more diverse training set. In contrast to the archive growth step in NAT, NATv2 introduces a sub-network replacement step. Instead of starting with a limited set of architectures and allowing the archive to grow iteratively by adding newly discovered sub-networks, NATv2 directly populates the archive with the maximum number of architectures right from the start. During each iteration, the weakest architectures within the archive are replaced with those obtained through the evolutionary search. Consequently, the size of the archive remains constant throughout the process. This approach aims to enhance the average quality of the architectures within the archive, leading to improved performance of the predictor model. Notably, this improvement is particularly evident in the early iterations due to the larger pool of input data available for the predictor to learn from. The overall effect is an increased capability of the performance predictor in gauging the quality of sub-networks in NATv2. In addition, NATv2 includes a pre-processing step within the archive initialization phase. This pre-processing entails sampling a significantly larger number of architectures, specifically ten times larger than the desired archive size, denoted as A_s. The sampled sub-networks are evaluated and compared, and only the top A_s-2 networks, along with the maximal and minimal networks, are selected to form the initial archive. Introducing a set of high-quality architectures at the beginning of the algorithm can contribute to obtain better performing sub-networks throughout the process. It is worth noting that the inclusion of the pre-processing step comes at the cost of increased execution time, but this is only a one-time cost per execution. The extent of this impact depends on the size of the initially sampled architecture set. In NATv2, the archive size A_s has been defined as 300. §.§ Performance Predictor The execution of the evolutionary search process in NATv2 generates a significant number of sub-networks. However, evaluating the performance of each of these sub-networks individually would render the algorithm computationally impractical, even with the use of weight sharing. To overcome this challenge, NATv2, like its predecessor, relies on a performance predictor model. This predictor performs a regression task and is trained online. At the start of each NAT iteration, the predictor model is fitted by taking as input the set of encodings corresponding to the sub-networks currently present in the archive, with their corresponding top-1 accuracy serving as the target. By leveraging this dataset, the predictor model is trained to predict the performance of architectural encodings that it has not encountered before. This approach allows NATv2 to efficiently estimate the performance of numerous sub-networks without having to evaluate each one individually, making the algorithm computationally tractable. In NATv2, a significant expansion of candidate predictor models has been undertaken. Initially, the following machine learning models were considered: * Gaussian Process (GP) <cit.>, * Radial Basis Function (RBF) <cit.>, * Multilayer Perceptron (MLP) <cit.>, * Classification and Regression Tree (CART) <cit.>, * Radial Basis Function Ensemble (RBFE) <cit.>. In order to thoroughly investigate the effectiveness of the predictor, various regression mechanisms have been explored, leading to the inclusion of the following models * Support Vector Regressor (SVR) <cit.>, * Ridge Regressor <cit.>, * K-Nearest Neighbours Regressor (KNN) <cit.>, * Bayesian Ridge Regressor <cit.>. In addition, given their claimed success in the literature, the following models have been added to the candidate predictor list: * End-to-End Random Forest-based Performance Predictor (E2EPP) <cit.>, * Light Gradient Boosting Machine (LGBM) <cit.>, * Catboost <cit.>. This extensive selection of machine learning models allows a comprehensive exploration of potential predictors in NATv2, highlighting the importance of the role of the predictor in the algorithm. §.§ Training Networks with Early Exits Special attention was paid to the training algorithm applied to super-networks with early exits, enriching it with new techniques introduced and proven effective in both the AEP and OFAv2 works. In particular, the AEP training method is relied upon; given a network with multiple exits, the network undergoes a form of joint training by creating a weighted ensemble of its exits. Different weighting strategies can be used to adjust the contribution of each exit, but in general the benefits of this training method were clearly demonstrated in the original study. Another important technique is ENS-KD, a new knowledge distillation technique introduced in OFAv2, which is also based on the AEP approach. As shown in Figure <ref>, given a teacher network with multiple exits, the knowledge transferred to the student isn't limited to that available at the last layer of the teacher network, but rather information from all its exits is weighted and combined according to the AEP method to distill more significant information, resulting in improved performance of the student networks. Returning to the training of the NATv2 super-networks with early exits, during the warm-up phases, the maximal networks extracted from the OFAv2 super-network under experimentation, i.e. those corresponding to width multipliers 1.0 and 1.2, are trained using the AEP training method, following the DESC weighting strategy. The use of the AEP joint training method is possible because the maximal network within an OFAv2 super-network with early exits retains all the exits present in the super-network. For the NATv2 adaptation phase, in which the super-network is fine-tuned by sequentially activating sub-networks within it, a training algorithm corresponding to the last phase of the EPS algorithm is used. This is because NAT immediately makes all the possible values of each elastic parameter available for sampling. On the contrary, the steps and phases of EPS progressively reveal new elastic parameters and their values. It is only in the last phase of EPS that all elastic parameter values are available for sampling. NAT uses Knowledge Distillation to improve sub-networks during the super-network adaptation phase. In NATv2, the standard Knowledge Distillation technique is replaced by the ENS-KD technique when performing the adaptation step on early exit super-networks. To be consistent with the EPS training that the super-networks underwent in OFAv2, the sub-networks activated during the super-network adaptation step in NATv2 are single exit networks. §.§ Post-Processing The multi-objective optimization nature of the search process in NATv2 may result in the best sub-networks achieving high performance across multiple objectives but still not reaching their maximum classification potential. To address this, two distinct fine-tuning post-processing methods are introduced to maximize the accuracy of these networks. The first method is applicable to sub-networks derived from any super-network, while the second method is specifically designed for sub-networks extracted from super-networks with early exits. Both methods consist of two sequential phases. In the first phase, the optimal number of training epochs, denoted as e, is determined for the given sub-network through fine-tuning on the target dataset. During this phase, the performance of the sub-network is continuously evaluated on the corresponding validation set. In the second phase, the sub-network is fine-tuned for e epochs using the combination of the training and validation sets. Finally, the test classification performance is computed and returned. The difference between the two post-processing methods lies in the fine-tuning algorithm used. The first method directly fine-tunes the networks returned by NATv2 as they are, i.e. single exit networks. On the other hand, the second post-processing method utilizes the Anticipate, Ensemble and Prune (AEP) technique <cit.>. In this method, all exits above the one selected by NATv2 for the sub-network are extracted from the super-network and reattached to the sub-network. A joint fine-tuning of the exits is then performed. The second post-processing method allows for greater gains in accuracy compared to the traditional fine-tuning method. However, it comes at the cost of a slightly increased number of parameters and MACs. § RESULTS AND DISCUSSION This section outlines the experiments conducted to evaluate the performance of NATv2 and compare it to NAT, the method on which our approach is based. First, the experiments are described in detail, including the experimental setup, implementation specifics, and hardware used to ensure reproducibility. Then, three ablation studies are presented to provide insight into the intermediate steps and to evaluate the decision-making process that led to the final results. The first study focuses on identifying the optimal performance predictor to use during the search. The second examines the effectiveness of the new super-networks and their encodings compared to those used in NAT. The last explores different post-processing optimisation strategies to determine the most effective approach. Finally, we present the results of our study by comparing the performance of NAT and NATv2, with and without post-processing, on the proposed datasets and configurations. The aim is to provide compelling evidence to support our claims and convince the reader of the superiority of NATv2. §.§ Experiments Configuration The experiments conducted for NATv2 used the super-network obtained with OFAv2 using the Extended Progressive Shrinking (EPS) technique. In contrast, the NAT experiments used the baseline OFAMobileNetV3 super-network obtained by OFA. Two versions of the same pre-trained super-networks were required to allow for the warm-up steps in both algorithms. The width multiplier configurations used were W=1.0 and W=1.2, maintaining consistency with those used in the NAT paper. While the OFA networks were trained on ImageNet, the OFAv2 networks were trained on the Tiny ImageNet dataset instead. To maintain consistency, and due to resource constraints, the Tiny ImageNet dataset was also used instead of ImageNet in both the NAT and NATv2 configurations. The experiments were performed on three widely used image classification datasets: CIFAR-10, CIFAR-100 and Tiny ImageNet. Further details on these datasets are given in Table <ref>. To ensure comparable results, the same set of hyperparameters was used for all experiments. The NATv2 models were trained using the SGD optimiser with a momentum of 0.9 and a weight decay parameter set to 3·10^-4. The learning rate was initially set to 2.5·10^-3 and adjusted using a cosine annealing scheduler. A batch size of 256 was used, and during each super-network adaptation epoch, four sub-networks per batch were sampled. The same hyperparameters were used for the warm-up phases, with the exception of the initial learning rate, which was set to 7.5·10^-3. In order to maximise the effectiveness of the post-processing step, numerous combinations of optimisers and learning rates were tested. These experiments aimed to determine the optimal configurations and were conducted on sub-networks derived from the initial NATv2 experiments using the CIFAR-100 dataset, with NATv2 run with the objectives “Accuracy & Params" and “Accuracy & MACs". To explore different post-processing combinations for single exit networks and early exit networks, the following optimisers were tested: SGD, AdamW <cit.>, and Ranger (the combination of the LookAhead <cit.> and RAdam <cit.> optimisers). For each optimiser, two initial learning rates were tuned, i.e. 10^-4 and 10^-5. In the case of AEP-based post-processing, which is applicable to sub-networks derived from early exit super-networks, experiments were conducted for all four AEP exit weighting strategies. When the SGD optimiser was used for post-processing, the previously reported values for momentum and weight decay were used, while no additional hyperparameters were specified for the other two optimisers. The batch size for these experiments was set to 64, and the networks were trained for a maximum of 150 epochs using a cosine annealing learning rate scheduler. Early termination was used, with a patience value of 30 epochs based on validation loss. All models were implemented using PyTorch 1.12.1 and experiments were run on an NVIDIA Quadro RTX 6000 GPU. The evolutionary algorithms used for the NAT and NATv2 search steps, specifically NSGA3 <cit.>, were obtained from the pymoo library <cit.>. §.§ Performance Predictors Analysis In the first ablation study, the predictor models were compared by keeping the training set size fixed at 300 and using the encoding methods proposed in Section <ref> to generate input features. The performance of the surrogate models were evaluated using correlation values as the reference metric. These correlation values were obtained by analysing sub-networks extracted from the available configurations of the OFAv2 algorithm, which also includes sub-networks from OFA. The aim of this study was to identify the most accurate predictor of sub-network accuracy that could be used in both NAT and NATv2 evaluations. To obtain the most accurate estimate, all models were evaluated using the 10-fold cross-validation technique. After calculating the validation performance by averaging, the 10 models trained on different data splits were aggregated by ensemble to form a single macro-model. The best macro-model among these ensembles was selected as the predictor. Referring to the candidate models presented in Section <ref>, Figure <ref> shows their performance, measured as the Spearman correlation between the predicted accuracy and the actual accuracy, referred to as the “rho correlation". The best performing models were CatBoost and LGBM, both above 0.9, followed by CARTS and E2EPP. CatBoost and LGBM also consistently produced the smallest error standard deviation intervals. This result has two positive implications. Firstly, it demonstrates the correlation between the proposed encoding and the accuracy of the corresponding model. Secondly, it means that the encoding successfully captures the large heterogeneity of possible networks, resulting in excellent error standard deviation intervals for the metric under consideration. Figure <ref> illustrates the time required by the different models to complete the entire training process. Given the reasonable size of the dataset, it is not surprising that the average times are generally positive for most models. As expected, the CART, RIDGE and BAYESIAN models are the fastest due to their simplicity. However, as their performance was not so promising, these models were not considered. When choosing a performance predictor, maximising the regression metric (in this case the rho correlation) is a crucial consideration. However, it is not the only factor to consider. The time taken to fit each model is also important. Although CatBoost performed slightly better than LGBM, it was significantly slower to fit. As a result, LGBM was ultimately selected as the predictor model. Once the predictor model was fixed, the correlation values were evaluated for different training set sizes, using both integer and one-hot encodings to generate the input features. The results shown in Figure <ref> indicate that for the given configuration, integer encoding outperforms one-hot encoding in terms of performance and stability. Furthermore, it appears that the number of samples reaches an optimal point, as the improvements in rho correlation seem to plateau in both encoding configurations. Finally, considering the performance variance achieved with smaller datasets, it can be concluded that the decision to replace the small but growing archive of NAT with a larger fixed-size archive in NATv2 is advantageous and improves the accuracy of the best estimated sub-networks from the beginning of the search process. §.§ OFAv2 Super-Networks Analysis The second ablation study focuses on evaluating the effectiveness of the NAT algorithm enhanced with the new encoding methods described in Section <ref> and the new performance predictor. This study serves as a preliminary evaluation to determine the improvements achieved by our approach compared to the initial model at this stage. The aim of this analysis is to evaluate the variation in performance resulting from the modification of the initial super-network. Specifically, two variations are considered: the initial super-network obtained via OFA and the starting super-networks obtained via OFAv2. The results of this study are summarised in Figure <ref>. Each row corresponds to a different dataset and each column represents a different optimisation objective. Starting from the first row, which corresponds to the experiments carried out on CIFAR10, it can be seen that changing the super-network leads to an overall improvement in all the configurations analysed. When optimising for accuracy only, the best model obtained from OFAv2 shows an improvement in accuracy of about 2% compared to the model obtained from OFA. Moving on to the results of the multi-objective searches, it can be observed that the models obtained not only have higher accuracy, but also have significantly fewer parameters and MACs on average. For example, when comparing architectures with fewer parameters, in addition to a 1% accuracy advantage for the architecture found by OFAv2, the number of parameters is approximately five times lower compared to the best architecture found by OFA. Looking at the results obtained on CIFAR100 and Tiny ImageNet, which are more complex datasets compared to CIFAR10, the previous observations are even more significant. In particular, when focusing only on accuracy optimisations, there is an improvement of about 5% and 9% respectively in favour of the models obtained from OFAv2. The performance of the solutions found in multi-objective optimisations further supports the claim that the proposed approach becomes more effective as the complexity of the problem increases. In addition, these results highlight the success and effectiveness of the proposed encoding method in this context. The encoding demonstrates its ability to capture the complexity of the problem and contribute to the improved performance of the models. §.§ Post-Processing Optimisation The third and final ablation study focuses on fine-tuning the parameters of the post-processing strategy once the optimal sub-networks have been found by NATv2. The goal of this study is to determine the optimal combination of optimiser, initial learning rate, and weight assignment for networks with early exits. In the case of networks with early exits, the choice of weights applied to the outputs is also part of the tuning process. The results presented in this study are based on the Tiny ImageNet dataset, with NATv2 run using the “Accuracy & Params" and “Accuracy & MACs" objectives. We found that for each optimiser, regardless of the type of post-processing, the average performance obtained by setting the initial learning rate to 10^-4 was better than that obtained by setting it to 10^-5. This finding is consistent with the expectation that starting with a low learning rate may result in too much dilution of learning due to the cosine annealing schedule. Regarding the choice of optimizer, single-exit architectures tend to benefit from using SGD, while almost all multiple-exit architectures show better performance with AdamW and Ranger optimizers, with a slightly stronger preference for AdamW. In terms of the AEP strategy, it was found that using a uniform weight distribution for networks with early exits yields better results on average. Quantitatively, as demonstrated by the final results of this work, presented below in Table <ref>, the inclusion of post-processing provides significant benefits to model accuracy, with an average improvement of 2.63% on Tiny ImageNet. However, it is important to consider the trade-off in terms of increased parameters (21.84% increase) and MACs (7.63% increase) when early exits are incorporated. Nevertheless, the architectures discovered by NATv2 often exhibit high computational efficiency, and the increase in complexity can be justified by the significant performance gains. Additionally, it is worth noting that the post-processing step is optional and that incurs a minimal time cost of a few minutes, which is negligible compared to the overall research process. §.§ Final Results The final set of experiments aims to compare the performance of the NAT reference model and its enriched version, NATv2, with and without the post-processing step. The results of these experiments are presented in Table <ref>, which encompasses the evaluation on CIFAR10, CIFAR100, and Tiny ImageNet datasets. The results are reported in terms of top-1 accuracy, number of parameters, and number of multiply-accumulate (MAC) operations, both measured in millions. For CIFAR10, the NATv2 + PP model obtaines the highest accuracy of 93.17%, which is better than the other models. However, it also has the highest number of parameters (12.42M) and MAC (77.82M). Considering the trade-off between accuracy and number of parameters, the NATv2 model obtained by optimising “Accuracy Params” with only 0.27M parameters achieved an accuracy of 91.46%, which is higher than any model obtained using NAT. Similarly, considering the trade-off between accuracy and MACs, the NATv2 model obtained by the “Accuracy MACs” optimisation achieves an accuracy of 89.67% with a minimum number of MACs equal to 6.35M and a very small number of parameters equal to 0.19M. This is particularly suitable for devices with very strict constraints. Generally speaking, it is fair to say that for each optimisation scenario considered, NATv2 significantly outperforms NAT for each of the research objectives in most cases. On average, without exceeding either accuracy or parameter gains, the application of post-processing is beneficial. If we turn to the CIFAR100 dataset, we observe similar trends in terms of the performance of the models and the optimisation targets. The NATv2 + PP model found in the single-target search emerges as the best performer overall, achieving an accuracy of 73.39%. This result demonstrates the model's ability to cope with the increased complexity of the CIFAR100 dataset, far exceeding the baseline accuracy of 66.93%. Considering the trade-off between accuracy and parameter count, NATv2 successfully optimises the search, finding a model with 0.86M parameters and an accuracy of 69.50%. Compared to the NAT models identified by the same optimisation, this is a significant improvement. It is also interesting to note that this model found with NATv2, although in a multi-objective optimisation, performs better in terms of accuracy than even the best NAT model optimised without taking parameters and MACs into account. It improves its accuracy by 2.57% and reduces parameters and MACs by 86.26% and 38.80% respectively. The NATv2 model stands out again when considering the trade-off between accuracy and MAC. The model with the lowest number of MACs (8.14M) consists of a very small number of parameters, 0.21M, with an accuracy of 66.29%, which is higher or comparable to any model found by NAT in any optimisation configuration. In addition to losing 7.68 percentage points of accuracy, NAT's lightest model requires more than twelve times as many parameters and twice as many MACs. Let us now turn our attention to the results obtained for the Tiny ImageNet data set, which represents the most challenging problem. Considering accuracy as the only optimisation objective, the NATv2 + PP model achieves the highest accuracy of 54.82%, which exceeds the baseline accuracy of 43.06%. This is a further evidence of the effectiveness of the NATv2 + PP model in the improvement of classification performance. In the analysis of the trade-off between accuracy and number of parameters, NATv2 results in a model with only 0.10 million parameters, but still achieves an accuracy of 39.92 By applying post-processing to this model, which is obviously an architecture without the possibility of inserting early exits and therefore extremely flat, the accuracy rises to 45.03% without increasing either the parameters or the MACs. This further demonstrates the usefulness of this fine-tuning step. The model thus obtained turns out to be better than any model found by NAT, this time with a truly minimal number of parameters. Overall, the experiments show that NATv2 + PP consistently achieves the highest level of accuracy across all three data sets. However, by achieving competitive accuracy with significantly fewer parameters, NATv2 demonstrates its strength in terms of parameter efficiency. Furthermore, by achieving reasonable accuracy and minimising computational requirements, NATv2 also demonstrates its efficiency in terms of MAC. In general, it can be said that NATv2 is able to achieve significantly better trade-offs than NAT, in some cases decimating the number of parameters, and is therefore particularly suitable for model searches for devices with a small amount of memory. On the other hand, if secondary objectives are to be sacrificed for the sake of accuracy, the use of post-processing has proven to be a beneficial step in the vast majority of cases, as it allows for architectures that are still considerably lighter and faster than those realised by NAT, while at the same time achieving much better performance in terms of accuracy. § CONCLUSION In this paper we have presented Neural Architecture Transfer 2 (NATv2), the extension of the Neural Architecture Transfer (NAT) technique by the implementation of two recent algorithms, namely Once-For-All-2 (OFAv2) and Anticipate Ensemble and Prune (AEP). In particular, we have shown that NATv2 can find networks that are significantly smaller in terms of parameters and operations, and more accurate than those found by applying NAT, by modifying the architectural design of the super-networks as well as the algorithm used. The greatest improvements were obtained by applying NATv2 exactly to these modified super-networks. Among the modifications, the one that contributed to the greatest improvements was the introduction of early exits in the architectures. Among the most important algorithmic improvement, the introduction of the post-processing phase, which allows further refinement of the returned sub-networks, proved to be an extremely effective addition, virtuously increasing the performance at a negligible cost in terms of parameters and operations. The results suggest that NATv2 is a successful extension of NAT, which was already an excellent tool for the realisation of deep learning models by satisfying very strict constraints. In particular, NATv2 is highly recommended for exploring high-performance architectures with an extremely small number of parameters. § ACKNOWLEDGMENT This project has been supported by AI-SPRINT: AI in Secure Privacy-pReserving computINg conTinuum (European Union H2020 grant agreement No. 101016577) and FAIR: Future Artificial Intelligence Research (NextGenerationEU, PNRR-PE-AI scheme, M4C2, investment 1.3, line on Artificial Intelligence). unsrt
http://arxiv.org/abs/2307.01451v1	20230704025629	Electrical operation of planar Ge hole spin qubits in an in-plane magnetic field	[ "Abhikbrata Sarkar", "Zhanning Wang", "Mathew Rendell", "Nico W. Hendrickx", "Menno Veldhorst", "Giordano Scappucci", "Mohammad Khalifa", "Joe Salfi", "Andre Saraiva", "A. S. Dzurak", "A. R. Hamilton", "Dimitrie Culcer" ]	cond-mat.mes-hall	[ "cond-mat.mes-hall" ]	School of Physics, The University of New South Wales, Sydney 2052, Australia School of Physics, The University of New South Wales, Sydney 2052, Australia School of Physics, The University of New South Wales, Sydney 2052, Australia QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands Department of Electrical and Computer Engineering. University of British Columbia, Vancouver, B.C., Canada. Quantum Matter Institute, University of British Columbia, Vancouver, B.C., Canada. Department of Electrical and Computer Engineering. University of British Columbia, Vancouver, B.C., Canada. Quantum Matter Institute, University of British Columbia, Vancouver, B.C., Canada. School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney 2052, Australia School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney 2052, Australia School of Physics, The University of New South Wales, Sydney 2052, Australia School of Physics, The University of New South Wales, Sydney 2052, Australia Hole spin qubits in group-IV semiconductors, especially Ge and Si, are actively investigated as platforms for ultrafast electrical spin manipulation thanks to their strong spin-orbit coupling. Nevertheless, the theoretical understanding of spin dynamics in these systems is in the early stage of development, particularly for in-plane magnetic fields as used in the vast majority of experiments. In this work we present a comprehensive theory of spin physics in planar Ge hole quantum dots in an in-plane magnetic field, where the orbital terms play a dominant role in qubit physics, and provide a brief comparison with experimental measurements of the angular dependence of electrically driven spin resonance. We focus the theoretical analysis on electrical spin operation, phonon-induced relaxation, and the existence of coherence sweet spots. We find that the choice of magnetic field orientation makes a substantial difference for the properties of hole spin qubits. Furthermore, although the Schrieffer-Wolff approximation can describe electron dipole spin resonance (EDSR), it does not capture the fundamental spin dynamics underlying qubit coherence. Specifically, we find that: (i) EDSR for in-plane magnetic fields varies non-linearly with the field strength and weaker than for perpendicular magnetic fields; (ii) The EDSR Rabi frequency is maximized when the a.c. electric field is aligned parallel to the magnetic field, and vanishes when the two are perpendicular; (iii) The orbital magnetic field terms make the in-plane g-factor strongly anisotropic in a squeezed dot, in excellent agreement with experimental measurements; (iv) Focusing on random telegraph noise, we show that coherence sweet spots do not exist in an in-plane magnetic field, as the orbital magnetic field terms expose the qubit to all components of the defect electric field. These findings will provide a guideline for experiments to design ultrafast, highly coherent hole spin qubits in Ge. Electrical operation of planar Ge hole spin qubits in an in-plane magnetic field Dimitrie Culcer August 1, 2023 ================================================================================ § INTRODUCTION Solid state spin qubits are prime candidates for scalable, highly coherent quantum computing platforms. <cit.> Among these group IV materials such as Ge and Si stand out thanks to the absence of piezo-electric interaction with phonons <cit.> and the possibility of isotopic purification, which eliminates the contact hyperfine coupling to the nuclear field,<cit.> with the maturity of semiconductor microfabrication as an added advantage. Recent years have witnessed a concerted push towards all-electrical qubit operation, since electric fields are easier to apply and localise than magnetic fields, and electrically operated qubit gates offer significant improvements in speed and power consumption as compared to magnetic gates. A series of theoretical predictions<cit.> as well as experimental leaps in growth techniques and sample quality <cit.> have led to a surge in interest in spin-3/2 hole systems in group IV materials. The strong and multifaceted spin-orbit coupling experienced by holes, <cit.> their anisotropic and tunable g-tensor, <cit.> and the absence of a valley degree of freedom makes them ideal for electrical spin manipulation, with Ge offering the additional advantages of a small effective mass <cit.> and ease of ohmic contact formation. Compared to electrons, the weaker hyperfine coupling for holes<cit.> due to the absence of the contact interaction significantly reduces the nuclear field contribution to spin decoherence, while the hole spin-3/2 is responsible for physics with no counterpart in electron systems, <cit.> which may offer flexibility in future design strategies – for example magic angles have been predicted for acceptor qubits, <cit.> at which dipole-dipole entanglement can be switched off without switching off the electric dipole moments of single qubits. Remarkable progress on hole spin qubits in several architectures has spanned more than a decade, with an overwhelming focus on Ge and Si.<cit.> Initial work focused on measuring hole spin states, <cit.> relaxation and dephasing times, <cit.> single spin electrical control, <cit.> readout and control of the g-tensor <cit.> and of spin states in multiple dots, <cit.> and achieving single-spin spin qubits. <cit.> In recent years the development of strained germanium in SiGe heterostructures <cit.> provided a low-disorder environment, which supported the development of single-hole qubits, <cit.> singlet-triplet qubits, <cit.> universal quantum logic, <cit.> and a four-qubit germanium quantum processor.<cit.> Experiments have demonstrated ultrafast spin manipulation using the spin-orbit interaction <cit.> and EDSR Rabi oscillations as fast as 540 MHz, <cit.> electrical control of the underlying spin-orbit coupling <cit.> and charge sensing using a superconducting resonator, <cit.> while relaxation times of up to 32 ms have been measured in Ge dots. <cit.> Hole spins in Ge have been used as quantum simulators <cit.> and control of an array of 16 Ge hole dots has been demonstrated. <cit.> The development of hybrid structures offers another path towards entanglement, with the demonstrations of superconductivity in planar Ge, <cit.> hole coupling to a superconducting resonator, <cit.> dipole coupling to a microwave resonator, <cit.> charge sensing using a superconducting resonator <cit.> and devices such as transistors and interferometers. <cit.> Theoretically, the interplay of spin-orbit coupling and superconductivity in hybrid semiconductor-superconductor structures is only now being studied in the context of quantum computing. <cit.> Concomitantly, Si qubits have also registered remarkable recent progress, with coherence times of up to 10ms in Si:B acceptors <cit.> and the detection of sweet spots as a function of the top gate field in quantum dots, <cit.> sweet spots having been the subject of a number of theoretical studies. <cit.> Anisotropic exchange was also used to entangle two hole qubits, <cit.> hole coupling to a superconducting resonator has been demonstrated, <cit.> and progress has been made towards higher-temperature operation with the observation of Coulomb diamonds up to 25K <cit.> and single-qubit operation above 4K. <cit.> Despite experimental advances on many fronts, constructing an all-encompassing theory to describe hole physics in group-IV semiconductor quantum dots is challenging. In particular, owing to different effective masses and intrinsic spin-orbit gap in the valence band of these materials, the wide range of QD sizes as well as spin manipulation timescales exhibited in experiments render it difficult to provide a theory of spin qubits common to Si and Ge. In this work we will focus on Ge, which is somewhat more tractable analytically than Si, and describe electrical qubit operation in an in-plane magnetic field. This choice is motivated by the observation that experiments has overwhelmingly favoured in-plane magnetic fields, <cit.> partly because a transverse magnetic field makes it easier to suppress hyperfine and cyclotron quantization effects, and partly in order to avoid the strong orbital coupling of the perpendicular field, which can cause a large diamagnetic shift and also affect tunnel rates. For spin-3/2 holes, in-plane magnetic fields are highly non-trivial, because the spin and orbital degrees of freedom are intertwined. The in-plane g-factor is very small, and in Ge most of it comes from the octupole interaction with the magnetic field.<cit.> Whereas in-plane magnetic fields have been considered for realistic devices in recent theoretical studies, <cit.> this has been largely from an engineering point of view, and the orbital effect of an in-plane magnetic field in planar quantum dots has largely been neglected, though they have been shown to play an important role in nanowires. <cit.> It has thus not been possible to date to construct a full picture of spin dynamics and electrical spin operation in planar Ge hole dots in an in-plane magnetic field. This has left several outstanding questions unanswered: What determines the speed of EDSR, as well as the relaxation time T_1? Is there an optimal magnetic field orientation for driving a spin-orbit qubit? Do coherence sweet spots exist when the magnetic field is in the plane? In this work we seek to answer these questions. Our focus will be on gate-defined Ge quantum dots, with the parent 2DHG exhibiting very high mobility<cit.>, low percolation density,<cit.>, and a low effective mass of m_h= 0.05 m_e;<cit.>, which all aid the formation of quantum dots. One reason for our choice of Ge is its band alignment, which makes it the only group IV material suitable for growing quantum wells. Another reason is pragmatic – it is easier to describe theoretically. This is because the spin-orbit splitting in Ge (Δ_SO=325 meV) is stronger than that in Si (Δ_SO=44 meV), resulting in a large separation of the spin-orbit/split-off (SO) band from the heavy- and light-hole subspaces. This ensures that the 4×4 Luttinger Hamiltonian formalism for spin-3/2 is adequate for the topmost Ge valence band, as opposed to Si where the Rashba mediated electrical control features both heavy hole-light hole (HH-LH) coupling contributions and heavy hole-split off (HH-SO) coupling contributions. Ge has a noticeable cubic-symmetry contribution to the Luttinger Hamiltonian. It is strong enough to enable electrical spin manipulation in planar dots, <cit.> making Ge ideal for electrical spin operation, but not as strong as in Si, such that it can be treated perturbatively. Figure. <ref> provides a schematic of the device architecture for studying a planar germanium hole quantum dot qubit. In this paper we concentrate on developing the key formalisms for describing the spin physics of hole spin qubits in an in-plane magnetic field. To avoid unneccessary complexity, and to keep our results generally applicable, we avoid making the analysis overly device specific; therefore we do not consider effects of gate electrode induced non-uniform strain which can lead to a significant modification of the spin-orbit interaction;<cit.> or Fowler-Norheim tunnelling of hole states through the SiGe barrier leading to charge accumulation at the interface between the semiconductor and the gate dielectric; or light-hole penetration through the SiGe barrier. Addressing these would require detailed finite element numerical simulation techniques on top of the theory developed here. For an applied in-plane magnetic field operation of planar Ge hole QD, in the presence of uniaxial strain but neglecting shear strain, we show that: (i) EDSR is linear in the magnetic field with nonlinear corrections emerging at larger fields, and is driven by Rashba spin-orbit coupling rather than by the orbital magnetic field terms; the picture that emerges is that the orbital B terms give rise to the finite Zeeman splitting between the qubit energy levels, while the Rashba spin-orbit coupling gives rise to transitions between them. The EDSR Rabi frequency is a maximum when the electric field is parallel to the magnetic field, and vanishes when the two are perpendicular; it can be sizable despite the smallness of the in-plane g-factor, and has a non-monotonic dependence on B; (ii) The relaxation rate is due to bulk acoustic phonons and is proportional to B^3 in the leading power; <cit.> (iii) For a squeezed (elliptical) dot with an aspect ratio L_y/L_x= 2 the spin-flip frequency is an order of magnitude faster than for a circular dot allowing Rabi time ∼10 ns with ∼10^5 operations within time T_1; (iv) For a squeezed dot, due to the effect of the magnetic field vector potential terms, the in-plane g-factor exhibits a strong anisotropy resulting in oscillatory behavior as the magnetic field is rotated in the plane of the dot, an observation supported by new experimental measurements shown in Sec. V of this paper; (v) Although extrema in the qubit Zeeman splitting as a function of the top gate voltage exist in the same way as for a perpendicular magnetic field,<cit.> coherence sweet spots do not exist for in-plane magnetic fields for noise induced by charge defects in the plane of the quantum well. The physics underlying this process is associated with the orbital magnetic field terms and is not captured by the Schrieffer-Wolff approximation (although electrical qubit operation including EDSR can be described in a simplified Schrieffer-Wolff picture). The outline of this paper is as follows. Section <ref> lays down the foundation of the semi-analytical model of a Ge hole quantum dot qubit withing the framework of effective mass theory. Next we discuss the properties of circularly symmetric dots in Sec. <ref>, including the qubit Zeeman splitting, EDSR, relaxation and dephasing. In Sec. <ref> we focus on elliptical dots and study their EDSR and coherence properties, while in Sec. <ref> we discuss the consequences of g-factor anisotropy and compare our predictions with recent experimental results. We end with a summary and outlook in Sec. <ref>. § HAMILTONIAN AND MODEL The topmost valence band in Ge has orbital angular momentum l=1. When the hole spin s=1/2 is taken into account the resultant states at the Γ-point are eigenstates of the total angular momentum 𝐉=(𝐋+𝐒). The four-fold degenerate j=3/2 states are separated by the spin-orbit gap Δ_0 from the j=1/2 two-fold degenerate split-off states. For Ge the spin-orbit gap Δ_0=325 meV, so the split-off band is safely disregarded in describing hole motion in the topmost valence bands. The \|3/2⟩ and \|-3/2⟩ states constitute the heavy-hole (HH) manifold while the \|1/2⟩ and \|-1/2⟩ states represent the light-hole (LH) manifold. The Luttinger Hamiltonian describes the hole motion in the topmost valence bands and has the following form in the j=3/2 basis {\|3/2⟩,\|-3/2⟩,\|1/2⟩,\|-1/2⟩}: H_LK(k)=[ P'+Q' 0 L' M'; 0 P'+Q' M'^* -L'^; L'^ M' P'-Q' 0; M'^* -L' 0 P'-Q'; ] where the matrix elements of the Luttinger Hamiltonian comprise the k-dependent part and strain-induced perturbations: P'=P(𝐤)+P_ε, Q'=Q(𝐤)+Q_ε, L'=L(𝐤)+L_ε, M'=M(𝐤)+M_ε. The kinetic energy terms are P =ħ^2γ_1/2m_0(k_x^2+k_y^2+k_z^2), Q =ħ^2γ_2/2m_0(k_x^2+k_y^2-2k_z^2), L =-√(3)ħ^2γ_3/m_0({k_x,k_z}-i {k_y,k_z}) and M =√(3)ħ^2/2m_0{-γ_2(k_x^2-k_y^2)+2iγ_3{k_x,k_y}}. The strain terms are: P_ε=-a(ε_xx+ε_yy+ε_zz), Q_ε=-b(ε_xx+ε_yy-2ε_zz), L_ε=d(ε_xz-i ε_yz) and M_ε=√(3)/2b (ε_xx-ε_yy)-d ε_xy. Here m_0 is the bare electron mass while γ_1,γ_2 and γ_3 are Luttinger parameters. The constant a=2 eV is the hydrostatic deformation potential, b=-2.16 eV is the uniaxial deformation potential, and d=-6.06 eV accounts for the shear deformation potential. In most Ge/GeSi samples there is considerable strain in the quantum well, which significantly increases the splitting between light and heavy holes compared to silicon - here we take the compressive strain to be 0.6%.<cit.> The strain tensor components in the plane are: ε_xx=ε_yy=-0.006. The in-plane compressive strain elongates the out-of-plane lattice constant via ε_zz=-2C_12/C_11ε_xx=0.004; with C_12=44 GPa, C_11=126 GPa.<cit.> We assume the off-diagonal shear elements of the strain tensor to be ε_ij\|_i≠ j=0. The out-of-plane confinement is described by a one-dimensional infinite square well potential V(z) = ∞ z∈{-L_z/2,L_z/2} 0 otherwise . The coupling to the top-gate electric field, denoted by F_z, gives an additional term eF_zz in the Hamiltonian. The in-plane confinement is modelled by a parabolic potential V_x,y=1/2(λ_x^2x^2+λ_y^2y^2), where λ_x, λ_y are determined by the dot dimensions L_x, L_y in the plane. The effective hole QD Hamiltonian is given by: H_0D=H_LK(𝐤)+eF_zz+V_conf, where V_conf=V_x,y+V(z) is the total confinement potential. The Zeeman interaction is given by, H_Z=-2κμ_B𝐁·𝐉-2qμ_B𝐁·𝒥, where 𝐉={J_x, J_y, J_z}, 𝒥={J_x^3,J_y^3,J_z^3}; and J_x, J_y, J_z are the 4×4 Pauli matrices. The 𝒥-terms are vital in order to obtain the correct in-plane g-factor ≈ 0.25. In the presence of a magnetic field, the canonical momentum of holes in topmost valence band becomes 𝐤→(𝐤+e𝐀/ħ). The hole spin in a planar Ge quantum dot is then described by H_QD=H_LK(𝐤+e𝐀/ħ)+eF_zz+V_conf+H_Z. The resultant modifications in H_LK(𝐤+e𝐀/ħ) lead to non-trivial contributions to the effective spin-orbit interaction in the HH and LH manifolds, as well as between them. To check the gauge invariance of our theoretical framework, we consistently diagonalise the effective quantum dot Hamiltonian using two different gauges: * 𝐀=-1/2B_z yê_𝐱+1/2B_z xê_𝐲+(B_x y-B_y x)ê_𝐳. * The symmetric gauge: 𝐀 = 1/2 𝐁×𝐫 = 1/2(B_yz-B_zy) ê_𝐱+1/2(B_zx-B_xz) ê_𝐲+1/2(B_x y-B_y x) ê_𝐳. All calculations have been performed in both gauges, yielding consistent results. The eigenstates of the hole QD can be expressed as linear combinations of states belonging to a basis in which the bare QD Hamiltonian is diagonal, i.e. \|Ψ_QD⟩=∑_i c_iψ_i(x,y,z)\|ϕ_j⟩, where the bare Hamiltonian refers to H_QD of Eq. <ref> with its off-diagonal elements set to zero, following the practice of k· p theory, as well as the external magnetic field set to zero. We choose the spatial wave functions as ψ(x,y,z)=ψ_n(x) ψ_m(y) ψ_l(z), where the in-plane basis states are 1-D Harmonic oscillator states for x and y and the out-of-plane basis states are given by solutions of the infinite potential well: ψ_n(x) = 1/√(2^n n!)1/√(L_x√(π))e^-x^2/2L_x^2H_n(x/L_x), x∈{-∞,∞} ψ_m(y) = 1/√(2^m m!)1/√(L_y√(π))e^-y^2/2L_y^2H_m(y/L_y), y∈{-∞,∞} ψ_l(z) = cos((l+1)π z/L_z) l is even sin((l+1)π z/L_z) l is odd ,z∈{-L_z/2,L_z/2} The indices n,m,l in Eqn. <ref> can take integer values 0,1,2,3.. etc. The hole spinors represent the j=3/2 spin states: \|ϕ_j⟩∈{\|3/2⟩, \|-3/2⟩, \|1/2⟩, \|-1/2⟩}. When operated at low in-plane 𝐁 we are in the ω_c≪ω_0 limit, with ω_c the cyclotron frequency, so any effect of the in-plane magnetic field on the dot size are generally irrelevant (they are taken into account in our formalism). This means the Fock-Darwin solutions have a one-on-one analogy to the 2D Harmonic oscillator solutions in Eqn. <ref>. The in-plane basis states Ψ_n(x)Ψ_m(y) are ordered according to their energy ∝(n+m+1). We find converging solutions to Eqn. <ref> by considering 55 in-plane basis states, i.e. (n+m)∈[0,9]; and 15 out-of-plane Ψ_l(z) basis states, i.e. l∈[0,14]. We note that the (n+m)=0 level has no degeneracy; but considering the degeneracies of (n+m)=1,2....9; the simulation spans 55 in-plane levels. The numerical diagonalisation of the resultant 3300×3300 Hamiltonian yields the energy levels of the hole quantum dot: H_QD\|Ψ_QD⟩=λ_E\|Ψ_QD⟩. In the Schrieffer-Wolff approximation, which assumes the longitudinal confinement is much stronger than the in plane confinement, theoretical models show that the in-plane magnetic field gives rise to an effective spin-orbit coupling whose magnitude is proportional to B.<cit.> Recent experimental works have shown unambiguously that the g-factor of a 2D hole gas is a strong function of density,<cit.>, which can be captured by effective 2D theoretical models<cit.>. Moreover, the effective spin-orbit interaction due to the orbital magnetic field terms has a highly non-trivial interplay with the Rashba spin-orbit interaction stemming from the top-gate potential.<cit.> For a hole QD in an in-plane B, as considered here, the orbital magnetic field terms also contribute to the spin dynamics. Nevertheless, this contribution, as the following sections make clear, cannot be captured by a naive Schrieffer-Wolff transformation, because the orbital magnetic field terms couple the in-plane and out-of-plane dynamics in a way that makes them inseparable: if one first reduces the 3D Hamiltonian to an effective 2 × 2 Hamiltonian for a 2D hole gas, and then attempts to understand QD dynamics based on this effective 2D Hamiltonian (in analogy with electron systems) all the physics of the orbital magnetic field terms is lost. Hence a full 3D theoretical model is essential to understand the full spin dynamics of a hole quantum dot in an in-plane magnetic field. The model we present in this work can treat arbitrary quantum dot sizes, magnetic field strengths and orientations. We comment briefly on the choice of spatial Ge basis functions. Variational analyses of the z-wave function incorporating the top gate potential have successfully described Ge hole QDs in Refs. <cit.>, but the variational model is hard to extend to a full 3D numerical analysis due to the complicated form of the variational excited states. For example, the Airy function<cit.> provides the exact solution if the z confinement is modelled as a triangular potential well but can yield a residual Rashba spin-orbit interaction at non-zero top gate potential (F_z>0), requiring a careful choice and implementation of boundary conditions. In the present paper we describe the z confinement using an infinite square well augmented by a linear electrostatic potential that accounts for the top gate, and consider top gate fields up to 50MV/m (although values up to 100 MV/m can also be studied with this method). Ref. wang2022modelling used a sophisticated model that incorporates Fowler-Nordheim tunnelling, which is beyond the scope of the present study, as explained below. We stress that the range of F_z is at the lower end of what we consider here (up to 2.5 MV/m), and thus our studies can be regarded as complementary. § CIRCULAR QUANTUM DOT Qubit Zeeman splitting. We solve the full 3D Hamiltonian in an external magnetic field along x for a QD with following dimensions: L_x=50 nm,L_y=50 nm,L_z=11 nm. The ground state is given by \|Ψ_QD⟩=∑_i c_iψ_i(x,y,z)\|ϕ_j⟩. We evaluate the QD ground state and first excited state, labelling them as state \|0⟩ with eigenenergy 𝔼_0 and \|1⟩ with eigenenergy 𝔼_1 repectively. The ground state and the first excited state are of heavy hole-type with admixtures due to heavy hole-light hole coupling, and split by an amount Δ𝔼=𝔼_1-𝔼_0, which we refer to as the qubit Larmor frequency, while continuing to express it in units of energy. The HH-LH splitting Δ_LH∼ 60 meV is governed by the z-energy bands in the quasi 2D dot limit , so that many in-plane (quantum dot) levels are contained between any two out-of-plane (quantum well) levels. This allows us to write \|0⟩=(c_1\|0,0,0,3/2⟩+admixtures) and \|1⟩=(c_1'\|0,0,0,-3/2⟩+admixtures), where the four indices denote n, m, l, J_z. When plotted against the top-gate voltage, the qubit Larmor frequency exhibits an extremum as a function of F_z.<cit.> The in-plane magnetic field gives rise to a finite Zeeman splitting, which shows a linear trend with B_x (figure <ref>a). The extrema in the qubit Larmor frequency as a function of F_z in an in-plane 𝐁 are explained by the same mechanism as for out-of-plane magnetic field operation (Figure <ref>b). At small values of F_z the matrix elements connecting the HH and LH states, which give rise to Rashba spin-orbit coupling, increase linearly with the gate field, while the change in the HH-LH splitting is negligible. At large values of F_z the increase in the HH-LH splitting outweighs all other effects and the Rashba spin-orbit coupling decreases as a function of F_z. These competing effects give rise to an extremum in the qubit Larmor frequency at a certain value of the top gate electric field, where the qubit is insensitive to z-electric field fluctuations. The qubit Zeeman splitting is linear in 𝐁, Δ𝔼=g_∥μ_BB_x, where the effective in-plane g-factor ranges between 0.215-0.219, expected to be much smaller than out-of-plane g-factor; g_∥≪ g_⊥. EDSR. An alternating electric field 𝐄̃(t) can induce spin-flip transition between the qubit states \|0⟩ and \|1⟩ via electron dipole spin resonance (EDSR) when the frequency of the ac electric field matches the Zeeman splitting of the hole spin qubit, Δ𝔼=hν. The EDSR Rabi frequency is calculated as the matrix element of the ac field between the qubit ground state (GS) and excited state (ES): f_EDSR = ⟨0\|e𝐄̃(t)·𝐫\|1⟩ Here we will focus on the scenario in which the alternating electric field is in the plane. In the 3D model, the effective interaction between the qubit GS and ES wavefunctions lies in the LH admixture to \|±3/2⟩ HH states. The admixture is the results of the two primary spin-orbit interactions in the system: structure inversion asymmetry (SIA) due to the top-gate potential F_z gives rise to the first Rashba term, which stems primarily from HH-LH coupling. The second contribution to the spin-orbit interaction comes from the orbital terms due to the in-plane magnetic field 𝐁. As 𝐁 is ramped up, the magnitude of this latter contribution increases. For an applied oscillatory electric field of strength E_0=10 kV/m, figure <ref>a presents the spin-flip Rabi frequency variation with the top gate field and applied B_x, with the key features of the EDSR frequency exhibiting a maximum at a certain value of F_z, as well as a nonlinear dependence of f_EDSR on B_x. In figure <ref>b, f_EDSR shows a non-linear variation as a function of the in-plane magnetic field. The best fit corresponds to f_EDSR=a_fB_x+b_fB_x^2+c_fB_x^3 at a constant top gate field. At B_x=0.1 T the Rabi frequency increases slowly with the top gate field. On the other hand at (B_x=2 T) the Rabi frequency increases more rapidly with F_z, and the maximum shifts towards lower values of F_z (Fig. <ref>c). We find the EDSR rate is a maximum when the electric field is parallel to the magnetic field, and vanishes when the two are perpendicular. The nonlinearity in F_z in figure <ref>c reflects the quadrupole Rashba spin-orbit coupling unique to spin-3/2 systems,<cit.> having no counterpart in electron spin qubits. Hole qubits in Ge quantum dots can exhibit fast EDSR of the order of ∼100 ns, despite the fact that the in-plane g-factor is two orders of magnitude smaller than the out-of-plane g-factor. The linear term in B_x in the Rabi frequency (figure <ref>b) stems from the Zeeman terms cubic in angular momentum ∝ q in Eq. <ref>. To see this one can write an effective 2×2 qubit Hamiltonian using the Schrieffer-Wolff (SW) transformation up to third order in small quantities: H_eff^2×2=H_0+H_SO+g_∥μ_BB_x^3σ_x+V(x,y)+eE_x(t)x Here H_0(𝐤→(𝐤+e𝐀/ħ))=[𝒜k^2-ℬk^4-D(k_+^2-k_-^2)^2]I_2×2 is the kinetic energy including a correction ∝ k^4,<cit.> and the D term represents warping of the energy contours.<cit.> The spin-orbit Hamiltonian H_SO comprises two important k-cubic Rashba terms induced by the top gate field, a spherical term ∝α_R2 and a cubic-symmetric correction ∝α_R3. The next three terms represent the Zeeman splitting, confinement energies and the ac electric potential along the x-direction, respectively. Starting from this effective Hamiltonian and projecting it onto the in-plane states, one may apply the SW approximation again to obtain an effective 2 × 2 qubit Hamiltonian, in which the EDSR driving term appears on the off-diagonal. Reading off the EDSR Rabi frequency we find f_EDSR∝B_x to leading order, while the next order in the expansion gives f_EDSR∝ B_x^3 (Appendix. <ref>). The significant quadratic B_x^2 dependence of the EDSR Rabi frequency at large F_z (Fig. <ref>) is a signature of the orbital magnetic field terms. One can determine possible paths connecting the lowest energy \|0⟩↔\|1⟩ HH-qubit states, mainly composed of \|0,0,0,3/2⟩ and \|0,0,0,-3/2⟩ with admixtures, respectively. Choosing the symmetric gauge for 𝐁=(B_x,0,0), the magnetic field induces the following transitions in the Luttinger-Kohn picture: {\|3/2⟩\|1/2⟩, \|-1/2⟩\|-3/2⟩}. The gate potential gives {n_zn_z±1} which is spin conserving. The Luttinger terms L(𝐤), M(𝐤) couple HH and LH states e.g. {\|n_x,n_y,n_z,3/2⟩\|n_x±1,n_y,n_z±1,3/2⟩}. We consider the complete SO picture in the Ge hole QD following eqn. <ref> to sketch a few example paths as follows: * \|0,0,0,3/2⟩\|0,0,0,1/2⟩\|0,1,1,3/2⟩ \|1,2,1,-1/2⟩\|1,2,0,-1/2⟩\|0,2,0,-1/2⟩ \|0,0,0,-1/2⟩\|0,0,0,-3/2⟩. * \|0,0,0,3/2⟩\|0,1,1,1/2⟩\|1,1,1,1/2⟩ \|1,0,2,1/2⟩\|1,0,0,1/2⟩\|0,1,0,-3/2⟩ \|0,0,1,-3/2⟩\|0,0,0,-3/2⟩. These paths explain the quadratic B_x^2 dependence at large F_z in the presence of the ac electric field. Phonon mediated Relaxation. The relaxation mechanism in Ge hole QD qubits is well explained by acoustic phonon coupling to the hole spins through the valence band deformation potential 𝒟_i,j of Ge. There are no piezo-electric phonons in Ge, but the hole spin interacts with the thermal bath of the bulk phonons via the hole-phonon Hamiltonian H_h-ph, where α∈{l,t,w} denotes the polarization directions of the phonon and 𝐪 is the phonon wave vector. We write H_h-ph=∑_i,j𝒟_i,jε^α_i,j(r) with ε^α_i,j(r)= ∑_qi/2√(ħ/2ρ NVω_𝐪,α)(q_iĉ_j+q_jĉ_i) e^i q·r√(N_q^α+1). The relaxation rate 1/T_1 is written as: 1/T_1=2π/ħ∑_α,𝐪\|⟨0\|H_h-ph\|1⟩_α\|^2δ(Δ𝔼-ħω_α,𝐪) We calculate the relaxation rate using our semi-analytical 3D QD model by computing in real space the overlap integral ⟨0\|H_h-ph\|1⟩_α due to position-dependent local strain, followed by the scattering integral in the phonon wave vector q-space for a specific polarization direction. The full analytical integrations and dipole approximation calculations are detailed in Appendix. <ref>. Figure <ref>a shows the nonlinear variation of the relaxation rate T_1^-1 w.r.t. the external magnetic field B_x, with B^3, B^4 and B^5 terms present in the fitting obtained from the full 3D numerical model. While the B^3 and B^5 dependence are explained from the first two terms in the dipole approximation, the B^4 term is understood from the orbital B-admixture. A minimum relaxation time T_1∼ 80 ms at an in-plane magnetic field B_x=0.67 T is obtained. This result from the theory compares well with a single hole relaxation time measurement in Ge of over 30 ms and a five-hole relaxation time of approximately 1 ms by Lawrie et al.<cit.> The magnetic field in the experimental setup<cit.> is B=0.67 T, as also used in Fig. <ref>a. There exists a minimum in T_1 in the range of F_z considered here for B_x=0.67 T (figure <ref>b). At this minimum the Rabi ratio T_1/T_π≈ 2× 10^5, where T_π is the time required for an EDSR π-rotation, demonstrating that fast Rabi oscillations can be achieved without sacrificing T_1. Random Telegraph Noise (RTN) Dephasing. The large spin-orbit coupling exposes the hole spin qubit to charge noise. The dephasing time T^_2,RTN is evaluated from the fluctuation in the qubit energy gap, denoted by δω, caused by the screened potential U_s( r) of a nearby single charge defect in the 2DHG.<cit.> The mathematical formulation of U_s( r) is given in Appendix. <ref>. The matrix elements ⟨ n,m,l\|U_s(r)\|n',m',l'⟩ are added to the full Hamiltonian, and the 3300× 3300 matrix is diagonalised to evaluate the qubit energy splitting in the presence of the charge defect as Δ𝔼+δω. The dephasing rate is: (T^_2,RTN)^-1=(δω)^2τ/2ħ^2 where the defect switching time is taken as τ=10^3 t_Rabi. This picture assumes the most significant contribution to RTN comes from charge defects away from the top gate, close to the qubit plane; hence we consider a single charge defect in the qubit plane situated 200 nm away from the center of the qubit. Fluctuating single charge defects right above the qubit will be screened by the presence of the top-gate, where the image charge changes the interaction to a much weaker dipole interaction. In contrast fluctuating charges in the plane of the quantum well are less effectively screened by surface gates, and may be the dominant source of charge noise. <cit.> It is evident that for hole qubit in an in-plane magnetic field, the dephasing time T_2,RTN^* actually decreases as a function of the top gate electric field F_z, and reaches a minimum at a certain value of this field (Fig. <ref>a), in other words, a coherence hot spot. The location of this dephasing time hotspot is closely related to the extremum in the qubit Zeeman splitting (Fig. <ref>b). This behaviour is in sharp contrast to hole spin qubits in a perpendicular magnetic field, where the qubit exhibits a sweet spot at a certain value of the top gate field (Fig. <ref>a), at which its sensitivity to noise vanishes to leading order in the noise strength, and dephasing time T_2,RTN^* reaches a maximum. The location of the sweet spot in F_z for out-of-plane qubit operation is closely related to an extremum in the qubit Zeeman splitting (Fig. <ref>b). B_⊥ vs. B_∥ coherent qubit operation. In the context of qubit coherence one must distinguish between extrema in the qubit Zeeman splitting and actual sweet spots in the coherence time. It is important to recall that in a spin qubit the dephasing time T_2^ depends on the magnitude of the magnetic field. This follows from time-reversal symmetry considerations, since the combination of charge noise and spin-orbit coupling cannot give rise to an energy difference between qubit states that form a Kramers doublet. The magnetic field dependence involves both the Zeeman terms and the orbital vector potential terms, a fact that is responsible for the main difference between in-plane and out-of-plane magnetic fields with regard to qubit dynamics: the make up of the ground and first excited states is very different when the magnetic field is in the plane and when it is out of the plane. For an out of plane magnetic field the hole g-factor is large, having a textbook value of 20.4 for Ge.<cit.> With the magnetic field out of the plane one can understand the physics qualitatively by considering an approximate decomposition of in-plane and out-of-plane dynamics by means of a Schrieffer-Wolff transformation <cit.> The picture that emerges is that the top gate electric field primarily affects spin dynamics in the plane by enabling a Rashba term. In a quantum dot this Rashba term is responsible for a renormalization of the g-factor. In other words, one can think of the magnetic field terms as providing the qubit Zeeman splitting, and the Rashba spin-orbit terms as renormalizing this Zeeman splitting. Background charge fluctuations generating an electric field perpendicular to the plane are the biggest danger for this qubit, because they directly affect the Rashba interaction and through it the g-factor, generating pure dephasing. A more detailed analysis of hole spin qubit in B_⊥ <cit.> reveals that in-plane charge fluctuations do not produce pure dephasing to leading order. Hence for B_⊥ operation, when the qubit Zeeman splitting is at an extremum with respect to the top gate electric field (Fig. <ref>b), the qubit is protected against noise and one also observes a sweet spot in T_2^* in the vicinity of this point (Fig. <ref>a). On the other hand, for hole spin qubit in B_∥, we recall that to a first approximation the in-plane g-factor is zero, hence the entire qubit Zeeman splitting is given by coupling to the excited states. This coupling involves Luttinger spin-orbit terms, the orbital magnetic field terms, the top gate electric field, and any other electric fields present in the system. The orbital terms due to the magnetic field mix the in-plane and out-of-plane coordinates regardless of the gauge choice. There is no clear separation between in-plane and out-of-plane dynamics, and no suitable Schrieffer-Wolff transformation from the 3D picture to the asymptotic 2D limit. One may at best envisage a combined Rashba-Zeeman interaction with contributions from all the components of the electric field, not just the top gate. The qubit states contain a strong admixture of all the higher orbital excited states in all three directions, which exposes the qubit to all components of the electric field of the defect. Thus, even though one can still identify an extremum in the qubit Zeeman splitting as a function of the top gate field (Fig. <ref>b), this does not offer protection against noise and does not constitute a sweet spot (Fig. <ref>a). It only protects against noise fields perpendicular to the plane, without offering any protection against the in-plane electric field of a defect. We check explicitly that for a defect that produces only an out-of-plane electric field at the qubit location the sensitivity to this out-of-plane noise is minimised at the extremum in the Zeeman splitting. We have also checked that the qubit is not shielded from the in-plane electric field of the defect at this extremum: there is nothing special about the extremum from this perspective. We note that in an experimental sample exposed to an ensemble of defects it is possible for the net in-plane electric field to cancel out, or nearly cancel out. Hence, to achieve a more complete understanding of coherence, it is vital to consider a realistic configuration leading to 1/f noise. In light of this, and of additional complexities identified recently in modelling hole spin coherence,<cit.> we defer the full theory of hole spin coherence in the presence of 1/f noise to a future publication. We note that our findings appear to agree with recent experimental work reporting sweet spot operation of a Ge hole spin qubit <cit.> as well as strong anisotropy in the noise sensitivity. Sensitivity to charge noise is found to increase significantly when the qubit is operated in an in-plane magnetic field. This is in agreement with the finding of the present paper that in-plane magnetic fields expose the qubit to noise much more strongly than out-of-plane magnetic fields, leading to the coherence hot spot seen in Fig. <ref>a. Remarkably, the dominant source of noise in Ref. Nico2023SweetSpot is believed to lie directly above the qubit, implying charge fluctuations predominantly in the perpendicular electric field component, and suggesting the qubit was not operated in the sweet spot for out-of-plane charge fluctuations. Nevertheless, a full comparison between theory and experiment is premature at this stage, given that tilting of the g-tensor and local strain have not been considered in the present work. § ELLIPTICAL QUANTUM DOT Introducing asymmetry into the planar confinement, i.e. having one lateral confinement potential stronger than the other will bring in additional sources of structure inversion asymmetry (SIA). For such elliptical hole QDs, the resultant Rashba spin-orbit interaction is stronger, bridging the gap between planar QD and nanowires <cit.> in terms of fast gate operations. A theoretical understanding of QD ellipticity for holes is thus important. Insight into our numerical results can be obtained from the effective 2×2 spin-orbit Hamiltonian in Eqn. <ref>, which we rewrite as: H_SO = α_R2[k_+^3σ_--k_-^3σ_+]+α_R3[{k_+^2,k_-}σ_+-{k_+,k_-^2}σ_-] where, α_R2=eħ^4/m_0^2γ_3γS_hh; α_R3=eħ^4/m_0^2γ_3δ S_hh; with the spherical Luttinger parameter γ=(γ_2+γ_3)/2, the cubic-symmetry parameter δ=(γ_3-γ_2)/2, and S_hh the sub-band interaction in 3^rd order perturbation theory (Appendix. <ref>). In a circular dot the cubic-symmetry correction α_R3 is responsible for EDSR, while the spherical term α_R2 does not contribute. In contrast, in an elliptical dot the α_R2 term is nonzero. From Eqn. <ref>, α_R3=(δ/γ) α_R2; using the lattice parameters of Ge we evaluate α_2≈10 α_3. We present the results for a dot size of L_z=11 nm, L_x=50 nm, and varying L_y. Fig. <ref>a shows the variation of the EDSR Rabi frequency with the aspect ratio L_x/L_y, showing that an increase in the aspect ratio results in a larger Rabi frequency. The qubit Zeeman splitting is linear in the applied in-plane magnetic field, similar to the circular case. The relaxation rate varies as B^3. The EDSR Rabi frequency is linear in B (Fig. <ref>b), which is reminiscent of out-of-plane B-field operation in presence of strong SIA; and the Rabi frequency exhibits a maximum as a function of F_z (Fig. <ref>c). § G-FACTOR ANISOTROPY OF ELLIPTICAL QD AND COMPARISON WITH EXPERIMENT The in-plane g-factor of an elliptical dot is strongly anisotropic and exhibits an oscillatory behaviour as the magnetic field is rotated in the plane. We compare the predicted variation of the g-factor with experimentally measured values from EDSR in a planar germanium hole qubit. The qubit sample is a gate-defined double quantum dot in a Ge/Si_0.2Ge_0.8 heterostructure. (See Ref. for further sample info). The dots are assumed to be elliptical, with a slight misalignment of their semi-major axes. Fig. <ref>a shows a false colour SEM of the gate design. Plunger gates (purple) are used to define the two dots, while the barrier gates (green) are used to control coupling to the leads and between the dots. Metal ohmic contacts (yellow) act as a reservoir for holes. By negatively biasing the barrier and plunger gates, the sample can be tuned to the few hole regime. The relative angle between the applied magnetic field direction in the plane and the double dot transport direction is denoted by θ. Bias triangles of the double dot measured via transport for positive and negative bias are shown in Fig. <ref>b and Fig. <ref>c. A region of Pauli spin blockade is visible at the base of one charge transition as indicated by a yellow circle. By applying an external magnetic field and a microwave tone to the P2 gate, we are able to drive spin rotations via EDSR when the microwave frequency matches the Larmor frequency (ħ f=gμ_B B). These spin rotations lift the Pauli spin blockade, causing a change in the current through the double dot. Using a lock-in amplifier, we measure the difference in current through the double dot when the microwave is on vs off. Fig. <ref>d shows the change in the leakage current for the double quantum dot with the external magnetic field applied in the direction indicated in Fig. <ref>a. Clear EDSR lines are visible for both dots, and both single and multi photon lines can be seen. From the slope of these resonance lines the g-factor can be calculated for each dot. Using this technique, we measure the g-factor as a function of field angle by rotating a magnetic field of B=0.7 T in the 2D plane. Fig. <ref>e shows the results of this measurement for both quantum dots, revealing an oscillatory variation in the g-factor as a function of in-plane magnetic field angle. The direction of θ is shown in Fig. <ref>a. Using the model developed in Sec. <ref>, we fit to the experimental data. For both dots we use the same size, shape and strain. We are able to account for the difference in g-factor between the dots by considering only a rotation of the dot axes in-plane and a change in the magnitude of the vertical electric field. A full list of fitting parameters is given in Table <ref>. The maximum value of the g-factor is not aligned with the external magnetic field or sample axes, and is also different for the left and right dots. To account for this, we introduce a phase shift angle (θ_ps) which effectively rotates the axes of the quantum dots. Here θ_ps,l=3π/4 and θ_ps,r=5π/8 The magnitude of the g-factor is also different for each dot. This is accounted for by changing the vertical electric field applied to each dot. Here we use 10MV/m for the left dot and 45MV/m for the right dot. The results of the fits for both dots are shown by the solid lines in Fig <ref> e). The theoretical fit in Fig. <ref>e shows good agreement between the phase shift angle θ_ps parameter choices and B-field angular orientations where the experimental g-factors are maximum. In other words, the largest value of the g-factor occurs when the magnetic field is parallel to the semi-major axis of the elliptical hole QD. This behaviour is consistent with the effective in-plane g-factor being primarily the result of coupling to higher excited states brought about by the orbital magnetic field terms. We note that inhomogeneous strain in the sample, or the Ge/SiGe heterostructure interface induced roughness and disorder, or a misalignment of the sample with respect to the in-plane B-field (since g_⊥≫ g_∥) could potentially lead to significant modulation of the g-tensor. We can rule out the latter, since there is a different phase shift for the left and right dots in Fig. <ref>e. The effects of strain and inhomogeneities on g-factor anisotropy will be considered in a future publication. § CONCLUSIONS AND OUTLOOK We have presented a generalised, semi-analytical model that fully describes the electrical operation of a planar germanium hole qubit in presence of an in-plane magnetic field. A comprehensive theory for spin manipulation via electron dipole spin resonance (EDSR) is given: surface inversion asymmetry (SIA) mediated fast EDSR is a result of the 𝐤·𝐩 coupling of the heavy hole ground state to higher energy light hole bands. The EDSR rate is linear in B with important nonlinear corrections due to orbital mixing. Qubit relaxation is induced by acoustic phonons and the relaxation rate 1/T_1 has terms with B^3, B^4, B^5 dependence, again reflecting the importance of the orbital mixing. In-plane operation demonstrates an excellent trade-off between relaxation and EDSR. The in-plane g-factor is strongly anisotropic and oscillates as the magnetic field is rotated in the plane. Random telegraph noise from charges in the plane of the quantum well results in decoherence, with an optimal top gate potential where it is insensitive to Δ F_z; although the in-plane magnetic field exposes the qubit to x-y electric field of the fluctuator. Hence, in contrast to the case of out-of-plane magnetic fields, coherence sweet spots cannot be identified in an in-plane B for a qubit exposed to electric field fluctuations in all spatial directions. For an elliptical QD of aspect ratio L_y/L_x=2, EDSR is shown to be faster by an order of magnitude compared to a circular dot of L_x=50 nm radius; and the non-linear correction to EDSR is suppressed as rotational asymmetry induces more SIA Rashba. Acknowledgments. 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Loss, @noop journal journal Physical Review B volume 104, pages 115425 (year 2021b)NoStop [Rendell(2021)]Matthew author author M. Rendell, title Spin dynamics of holes in GaAs and Gesemiconductor nanostructures, @noop Ph.D. thesis, school UNSW, Sydney, address School of Physics UNSW, Sydney (year 2021)NoStop § GAUGE INVARIANCE The hole motion in the topmost valence band is described by the 4×4 Luttinger-Kohn (LK) Hamiltonian, which in the general operator form is given by: H_LK = ħ^2/2m_0[(γ_1+5γ_2/2)k^2-2 γ_2(k_x^2J_x^2+k_y^2J_y^2+k_z^2J_z^2)-4γ_3({k_x,k_y}{J_x,J_y}+c.p.)] Expanding the anti-commutators, LK Hamiltonian has the form: H_LK = ħ^2/2m_0[(γ_1+5γ_2/2)(k_x^2+k_y^2+k_z^2)-2 γ_2(k_x^2J_x^2+k_y^2J_y^2+k_z^2J_z^2)-4γ_3((k_xk_y+k_yk_x/2)(J_xJ_y+J_yJ_x/2).. +(k_yk_z+k_zk_y/2)(J_yJ_z+J_zJ_y/2)..+(k_xk_z+k_zk_x/2)(J_xJ_z+J_zJ_x/2))] with m_0 notifying bare electron mass, γ_1=13.38, γ_2=4.24 and γ_3=5.69 are Luttinger parameters. We have tried two different gauges: 1/2𝐁×𝐫 and gauge choice from Loss et al.<cit.> §.§ Gauge 1: In presence of magnetic field, the momentum correction would be: 𝐤→(𝐤+e𝐀/ħ). We use the gauge (symmetric in z, Landau in x-y) 𝐀=-1/2B_z yê_𝐱+1/2B_z xê_𝐲+(B_x y-B_y x)ê_𝐳. Momentum Correction: * The components of corrected momentum: k_x → (k_x-eB_z/2ħy) , k_y→(k_y+eB_z/2ħx) , k_z →(k_z+eB_x/ħy-eB_y/ħx) where, 𝐤=-ι∂. For B_y=0, B_z=0; we write: k_x → k_x , k_y→ k_y , k_z →(k_z+eB_x/ħy) * We evaluate k_i^2 terms below: k_x^2 ⇒ k_x^2 , k_y^2⇒ k_y^2 , k_z^2⇒(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2) and the cross-terms k_ik_j are: k_xk_y⇒ k_xk_y; k_yk_z⇒(k_yk_z+eB_x/ħ{y,k_y}); k_xk_z⇒(k_xk_z+eB_xyk_x/ħ) * Using these corrections, the LK Hamiltonian is: H_LK=[ γ_1+5γ_2/2(k_x^2+k_y^2+k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)-2γ_2(k_x^2J_x^2+k_y^2J_y^2+(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)J_z^2) -4γ_3(k_xk_y {J_x,J_y}+(k_yk_z+eB_x/ħ{y,k_y}){J_y,J_z}+(k_xk_z+e B_x y k_x/ħ){J_z,J_x})]ħ^2/2m_0 where, H_LK^11/22 = ħ^2/2m_0[(γ_1-2γ_2)(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)+(γ_1+γ_2)(k_x^2+k_y^2)] H_LK^33/44 = ħ^2/2m_0[(γ_1+2γ_2)(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)+(γ_1-γ_2)(k_x^2+k_y^2)] H_LK^13=L = -√(3)ħ^2γ_3/m_0[(k_xk_z+eB_xyk_x/ħ)-i(k_yk_z+eB_x/ħ{y,k_y})] H_LK^14=M = √(3)ħ^2/2m_0[-γ_2 (k_x^2-k_y^2)+2iγ_3 k_xk_y] §.§ Gauge 2: In this section, we use the symmetric gauge (1/2𝐁×𝐫): 𝐀=1/2(B_yz-B_zy)ê_𝐱+1/2(B_zx-B_xz)ê_𝐲+1/2(B_x y-B_y x)ê_𝐳. Momentum Correction: * The components of corrected momentum with B_y=0 ,B_z=0: k_x → k_x , k_y→(k_y-eB_x/2ħz) , k_z →(k_z+eB_x/2ħy) where, 𝐤=-ι∂. * We evaluate k_i^2 terms below: k_x^2 ⇒ k_x^2 , k_y^2⇒(k_y^2-eB_xzk_y/ħ+e^2B_x^2z^2/4ħ^2) , k_z^2⇒(k_z^2+eB_xyk_z/ħ+e^2B_x^2y^2/4ħ^2) and the cross-terms k_ik_j are: k_xk_y ⇒ (k_xk_y-eB_xzk_x/2ħ);k_yk_z⇒(k_yk_z+eB_x/2ħ({y,k_y}-{z,k_z})-e^2B_x^2/4ħ^2yz);k_xk_z⇒(k_xk_z+eB_xyk_x/2ħ) * Using these corrections, the LK Hamiltonian terms become: H_LK=ħ^2/2m_0[(γ_1+5γ_2/2)(k_x^2+k_y^2-eB_xzk_y/ħ+e^2B_x^2z^2/4ħ^2+k_z^2+eB_xyk_z/ħ+e^2B_x^2y^2/4ħ^2) -2γ_2(k_x^2J_x^2+(k_y^2-eB_xzk_y/ħ+e^2B_x^2z^2/4ħ^2)J_y^2+(k_z^2+eB_xyk_z/ħ+e^2B_x^2y^2/4ħ^2)J_z^2)-4γ_3((k_xk_y-eB_xzk_x/2ħ) {J_x,J_y}. .+(k_yk_z+eB_x/2ħ{y,k_y}-eB_x/2ħ{z,k_z}-e^2B_x^2/4ħ^2yz){J_y,J_z}+(k_xk_z+eB_xyk_x/2ħ){J_z,J_x})] where, H_LK^11/22 = ħ^2/2m_0[(γ_1-2γ_2)(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)+(γ_1+γ_2)(k_x^2+k_y^2-eB_xzk_y/ħ+e^2B_x^2z^2/4ħ^2)] H_LK^33/44 = ħ^2/2m_0[(γ_1+2γ_2)(k_z^2+2eB_xyk_z/ħ+e^2B_x^2y^2/ħ^2)+(γ_1-γ_2)(k_x^2+k_y^2-eB_xzk_y/ħ+e^2B_x^2z^2/4ħ^2)] H_LK^13=L= -√(3)ħ^2γ_3/m_0[(k_xk_z+eB_xyk_x/2ħ)-i(k_yk_z+eB_x/2ħ{y,k_y}-eB_x/2ħ{z,k_z}-e^2B_x^2/4ħ^2yz)] H_LK^14=M= √(3)ħ^2/2m_0[-γ_2 (k_x^2-k_y^2+eB_xzk_y/ħ-e^2B_x^2z^2/4ħ^2)+2iγ_3 (k_xk_y-eB_xzk_x/2ħ)] § SCHREIFFER-WOLFF TRANSFORMATION In the effective 2×2 system Hamiltonian given by Eqn. <ref>, H_0+V(x,y) produces the quantized in-plane energies. We consider a basis with following in-plane states: {e^-(x^2/2L_x^2+y^2/2L_y^2)/√(π L_x L_y), √(2)y e^-(x^2/2L_x^2+y^2/2L_y^2)/√(π L_y^3 L_x), √(2)x e^-(x^2/2L_x^2+y^2/2L_y^2)/√(π L_x^3 L_y)}⊗{↑, ↓} where the spatial states are the first three harmonic oscillator product states: (n,m)={(0,0),(0,1),(1,0)}, we denote the orbital energies of these states as E_0, E_1, E_1. The spinors are the effective spin-up and spin-down states of the 2D hole qubit, and gives ± g_∥μ_BB_x Zeeman energies for the up and down spin states. The spin-orbit interactions can be listed as: H_SO=iα_R1(k_-σ_--k_+σ_+)+iα_R2(k_+^3σ_–k_-^3σ_+)+iα_R3({k_+,k_-^2}σ_--{k_+^2,k_-}σ_+)+iα_R4({k_z^2,k_-}σ_--{k_z^2,k_+}σ_+) The k-linear Rashba terms come from the coupling of the bonding VB p-orbitals to the antibonding CB p-orbitals, which is very small. In the Luttinger formalism thus the α_R1, α_R4 terms vanish. The nonzero contributions are: H_SO=iα_R2(k_+^3σ_–k_-^3σ_+)+iα_R3({k_+,k_-^2}σ_--{k_+^2,k_-}σ_+) the spin-orbit matrix elements are calculated below: ⟨(0,0)↑\|H_SO\|(0,1)↓⟩ = -iα_R2⟨(0,0)↑\|k_-^3σ_+\|(0,1)↓⟩+iα_R3⟨(0,0)↑\|k_+k_-k_+σ_+\|(0,1)↓⟩ = -iα_R2⟨(0,0)\|k_-^3\|(0,1)⟩+iα_R3⟨(0,0)\|k_+k_-k_+\|(0,1)⟩ = -3iα_R2/2√(2)L_y^3(1-L_y^2/L_x^2)+iα_R3/2√(2)L_y^3(1+3L_y^2/L_x^2)=R_c ⟨(0,0)↑\|H_SO\|(1,0)↓⟩ = -iα_R2⟨(0,0)↑\|k_-^3σ_+\|(1,0)↓⟩+iα_R3⟨(0,0)↑\|k_+k_-k_+σ_+\|(1,0)↓⟩ = -iα_R2⟨(0,0)\|k_-^3\|(0,1)⟩+iα_R3⟨(0,0)\|k_+k_-k_+\|(0,1)⟩ = -3α_R2/2√(2)L_xL_y^2(1-L_y^2/L_x^2)+α_R3/2√(2)L_xL_y^2(1+3L_y^2/L_x^2)=R_r The ac electric field eE_x(t)x is spin-conserving and connects the states (0,0) and (1,0). Putting all of it together, the 6×6 Hamiltonian would be: H'=[ E_0-1/2g_∥μ_BB_x 0 0 R_c eE_xa R_r; 0 E_0+1/2g_∥μ_BB_x R_c^* 0 -R_r eE_xa; 0 R_c E_1-1/2g_∥μ_BB_x 0 0 0; R_c^* 0 0 E_1+1/2g_∥μ_BB_x 0 0; eE_xa -R_r 0 0 E_1-1/2g_∥μ_BB_x 0; R_r eE_xa 0 0 0 E_1+1/2g_∥μ_BB_x ] 2^nd order perturbation theory gives the EDSR matrix element as: H̃_12 = ∑_j=3^61/2H'_1jH'_j2(1/ϵ_1-ϵ_j+1/ϵ_2-ϵ_j)=1/2H'_15H'_52(1/ϵ_1-ϵ_5+1/ϵ_2-ϵ_5)+1/2H'_16H'_62(1/ϵ_1-ϵ_6+1/ϵ_2-ϵ_6) = -1/2eE_xa R_r(1/-Δ_01+1/-Δ_01+Z)+1/2eE_xa R_r(1/-Δ_01-Z+1/-Δ_01) = eE_xa R_r(1/Δ_01(1-Z/Δ_01)-1/Δ_01(1+Z/Δ_01))=(eE_xa R_r Z)/Δ_01^2 Where we have used Z=g_∥μ_BB_x, Δ_01=E_1-E_0. This gives H̃_12∝B_x, explaining the linear magnetic field dependence of EDSR. From Eqn. <ref>, <ref>, the α_2 term vanishes when L_x=L_y i.e. for a symmetric dot, while it is strongly nonzero for an elliptical dot; making the linear B_x-dependence stronger. § BULK PHONON MEDIATED RELAXATION RATE ANALYTICS The total strain in the 4×4 Luttinger-Kohn-Bir-Pikus formalism is given as: H^LKBP_strain( r)=[ P_ε( r)+Q_ε( r) 0 L_ε( r) M_ε( r); 0 P_ε( r)+Q_ε( r) M_ε( r)^* -L_ε( r)^; L_ε( r)^ M_ε( r) P_ε( r)-Q_ε( r) 0; M_ε( r)^* -L_ε( r) 0 P_ε( r)-Q_ε( r); ] where, P_ε( r)=-a(ε_xx( r)+ε_yy( r)+ε_zz( r)), Q_ε( r)=-b/2(ε_xx( r)+ε_yy( r)-2ε_zz( r)), L_ε( r)=d(ε_xz( r)-i ε_yz( r)), M_ε( r)=(√(3)/2b(ε_xx( r)-ε_yy( r))-idε_xy( r)). The non-zero static component of the strain tensor is given by ε_xx=ε_yy=-0.006, and ε_zz=-C_12/C_11ε_xx with C_12=44 GPa, C_11=126 GPa. Considering the lattice deformation potential 𝒟( r), the 'local' strain is given by: ε_i,j^α( r)=1/2(∂ u_i( r)/∂ r_j+∂ u_j( r)/∂ r_i), i,j∈{x,y,z} Here u vector designates the deformation field at the position r. For a phonon traveling with wave vector 𝐪 in the polarized state α, the strain tensor is given by, ε_i,j^α( r)=i/2√(ħ/2V_cρω_𝐪,α)q(ĉ_i^αq̂_j+ĉ_j^αq̂_i)(e^-i𝐪·𝐫a_𝐪,α+e^i𝐪·𝐫a^†_𝐪,α) where ĉ is the polarisation unit vector. We consider three polarisations with con-ordinate systems understood as l:(r,θ,ϕ); t:(r,θ+π/2,ϕ); w:(r,π/2,ϕ+π/2). Using ω_𝐪,α= v_α q, ε_i,j^α( r)=i√(ħ/2V_cρ v_α)√(q)𝒜_ε,ij^α e^i𝐪·𝐫(a_-𝐪,α+a^†_𝐪,α) where v_α are the acoustic phonon velocities, and we assumed 1/2(ĉ_i^αq̂_j+ĉ_j^αq̂_i)=𝒜_ε,ij^α. The matrix elements of 𝒜_ε,ij^α are sketched out below: 𝒜_ε^α=1/2[ 2ĉ_x^αq̂_x ĉ_x^αq̂_y+ĉ_y^αq̂_x ĉ_x^αq̂_z+ĉ_z^αq̂_x; ĉ_y^αq̂_x+ĉ_x^αq̂_y 2ĉ_y^αq̂_y ĉ_y^αq̂_z+ĉ_z^αq̂_y; ĉ_z^αq̂_x+ĉ_x^αq̂_z ĉ_z^αq̂_y+ĉ_y^αq̂_z 2ĉ_z^αq̂_z ] The phonon wave vector has the components q→{q sinθcosϕ,q sinθ sinϕ,q cosθ}; so the unit vectors for q are given by: q̂→{sinθcosϕ, sinθ sinϕ, cosθ}. The polarization wave vectors are as follows: * ĉ^l→ q^-1{q_x,q_y,q_z}={sinθcosϕ, sinθcosϕ, cosθ} * ĉ^t→ q^-1(q_x^2+q_y^2)^-1/2{q_xq_z,q_yq_z,-(q_x^2+q_y^2)}={cosθcosϕ, cosθsinϕ,-sinθ} * ĉ^w→ (q_x^2+q_y^2)^-1/2{q_y,-q_x,0}={-sinϕ, cosϕ, 0} We can write the matrix elements of 𝒜_ε,ij^α for the three polarisations using the decompositions above: 𝒜_ε^l = 1/2[ 2 q_x^2/q^2 2 q_xq_y/q^2 2 q_xq_z/q^2; 2 q_yq_x/q^2 2 q_y^2/q^2 2 2 q_yq_z/q^2; 2 q_zq_x/q^2 2 q_zq_y/q^2 2 q_z^2/q^2 ]=1/q^2[ q_x^2 q_xq_y q_xq_z; q_yq_x q_y^2 q_yq_z; q_zq_x q_zq_y q_z^2; ] 𝒜_ε^t = 1/2[ 2 q_z/qq_x/√(q_x^2+q_y^2)q_x/q 2 q_z/qq_x/√(q_x^2+q_y^2)q_y/q q_xq_z/q√(q_x^2+q_y^2)q_z/q-√(q_x^2+q_y^2)/qq_x/q; 2 q_z/qq_x/√(q_x^2+q_y^2)q_y/q 2 q_z/qq_y/√(q_x^2+q_y^2)q_y/q q_yq_z/q√(q_x^2+q_y^2)q_z/q-√(q_x^2+q_y^2)/qq_y/q; q_xq_z/q√(q_x^2+q_y^2)q_z/q-√(q_x^2+q_y^2)/qq_x/q q_z^2q_y-q_x^2q_y-q_y^3/q^2√(q_x^2+q_y^2) -2q_z/q√(q_x^2+q_y^2)/q ] 𝒜_ε^w = 1/2[ -2q_y/√(q_x^2+q_y^2)q_x/q -q_y/√(q_x^2+q_y^2)q_y/q+q_x/√(q_x^2+q_y^2)q_x/q -q_y/√(q_x^2+q_y^2)q_z/q; q_x^2-q_y^2/q√(q_x^2+q_y^2) 2q_xq_y/q√(q_x^2+q_y^2) q_xq_z/q√(q_x^2+q_y^2); -q_yq_z/q√(q_x^2+q_y^2) q_xq_z/q√(q_x^2+q_y^2) 0 ] We wish to evaluate the angular integrals, so we write the matrices in terms of θ and ϕ: 𝒜_ε^l=[ sin^2θcos^2ϕ sin^2θsinϕcosϕ sinθcosθcosϕ; sin^2θsinϕcosϕ sin^2θsin^2ϕ sinθcosθsinϕ; sinθcosθcosϕ sinθcosθsinϕ cos^2θ ] 𝒜_ε^t = 1/2[ 2cosθcosϕsinθcosϕ 2cosθcosϕsinθsinϕ cosθcosϕcosθ-sin^2θcosϕ; 2cosθcosϕsinθsinϕ 2 cosθsinϕsinθsinϕ cosθsinϕcosθ-sin^2θsinϕ; cosθcosϕcosθ-sin^2θcosϕ cosθsinϕcosθ-sin^2θsinϕ -2sinθcosθ ] = [ 1/2sin2θcos^2ϕ 1/4sin2θsin2ϕ 1/2cos2θcosϕ; 1/4sin2θsin2ϕ 1/2sin2θsin^2ϕ 1/2cos2θsinϕ; 1/2cos2θcosϕ 1/2cos2θsinϕ -1/2sin2θ ] 𝒜_ε^w = 1/2[ -2sinϕsinθcosϕ -sinϕsinθsinϕ+cosϕsinθcosϕ -sinϕcosθ; -sinϕsinθsinϕ+cosϕsinθcosϕ 2cosϕsinθsinϕ cosϕcosθ; -sinϕcosθ cosϕcosθ 0 ] = 1/2[ -sin2ϕsinθ cos2ϕsinθ -sinϕcosθ; cos2ϕsinθ sin2ϕsinθ cosϕcosθ; -sinϕcosθ cosϕcosθ 0 ] The total hole-phonon Hamiltonian can be added as per the following equation: H_h-ph^α=𝒟_11^α𝒜_ε,11^α+𝒟_12^α𝒜_ε,12^α+𝒟_13^α𝒜_ε,13^α+𝒟_22^α𝒜_ε,22^α+𝒟_23^α𝒜_ε,23^α+𝒟_33^α𝒜_ε,33^α where α is the polarisation index, and 𝒟 are the 4×4 LK deformation potential matrices. l-polarisation: Putting in the local strain terms, we can write: H_h-ph^l=i√(q)√(ħ/2V_cρ v_l)[ P_ε^l+Q_ε^l 0 L_ε^l M_ε^l; 0 P_ε^l+Q_ε^l M_ε^l^* -L_ε^l^; L_ε^l^ M_ε^l P_ε^l-Q_ε^l 0; M_ε^l^* -L_ε^l 0 P_ε^l-Q_ε^l ]e^i𝐪·𝐫(a_-𝐪,l+a^†_𝐪,l) where, P_ε^l = -a(sin^2θcos^2ϕ+sin^2θsin^2ϕ+cos^2θ)=-a Q_ε^l = -b/2(sin^2θcos^2ϕ+sin^2θsin^2ϕ- 2cos^2θ)=-b/2(1-3cos^2θ) L_ε^l = d(sinθcosθcosϕ-isinθcosθsinϕ)=d/2sin2θ e^-iϕ M_ε^l = √(3)b/2(sin^2θcos^2ϕ-sin^2θsin^2ϕ)-idsinϕcosϕsin^2θ=√(3)b/2sin^2θcos2ϕ-id/2sin2ϕsin^2θ t-polarisation:The hole-phonon Hamiltonian in this case takes the following form in angular co-ordinates: H_h-ph^t=i√(q)√(ħ/2V_cρ v_t)[ P_ε^t+Q_ε^t 0 L_ε^t M_ε^t; 0 P_ε^t+Q_ε^t M_ε^t^* -L_ε^t^; L_ε^t^ M_ε^t P_ε^t-Q_ε^t 0; M_ε^t^* -L_ε^t 0 P_ε^t-Q_ε^t ]e^i𝐪·𝐫(a_-𝐪,t+a^†_𝐪,t) where, P_ε^t = -a(1/2sin2θcos^2ϕ+1/2sin2θsin^2ϕ-1/2sin2θ)=0 Q_ε^t = -b/2(1/2sin2θcos^2ϕ+1/2sin2θsin^2ϕ+sin2θ)=-3b/4sin2θ L_ε^t = d(1/2cos2θcosϕ-i/2cos2θsinϕ)=d/2cos2θ e^-iϕ M_ε^t = √(3)b/2(1/2sin2θcos^2ϕ-1/2sin2θsin^2ϕ)-id/4sin2θsin2ϕ=√(3)b/4sin2θcos2ϕ-id/4sin2θsin2ϕ w-polarisation:For the 3^rd polarisation direction: H_h-ph^w=i√(q)√(ħ/2V_cρ v_w)[ P_ε^w+Q_ε^w 0 L_ε^w M_ε^w; 0 P_ε^w+Q_ε^w M_ε^w^* -L_ε^w^; L_ε^w^ M_ε^w P_ε^w-Q_ε^w 0; M_ε^w^* -L_ε^w 0 P_ε^w-Q_ε^w ]e^i𝐪·𝐫(a_-𝐪,w+a^†_𝐪,w) where, P_ε^w = -a(-1/2sin2ϕsinθ+1/2sin2ϕsinθ)=0 Q_ε^w = -b/2(-1/2sin2ϕsinθ+1/2sin2ϕsinθ)=0 L_ε^w = d(-1/2sinϕcosθ-i/2cosϕcosθ)=-id/2cosθ e^-iϕ M_ε^w = √(3)b/2(-1/2sin2ϕsinθ-1/2sin2ϕsinθ)-id/2cos2ϕsinθ=-√(3)b/2sin2ϕsinθ-id/2cos2ϕsinθ The relaxation rate is characterized by the spontaneous and stimulated phonon scattering, hence: Γ_1=1/T_1=∑_α(2π/ħ∑_𝐪\|⟨0\|H_h-ph\|1⟩_α\|^2δ(Δ𝔼-ħω_α,𝐪)) The summation over the wave vectors can be changed to continuous integral, and the creation-annihilation operators can be approximated to produce a factor of N_q, which denotes the number of acoustic phonons with q momentum: Γ_1=1/T_1 = ∑_α(2π/ħV/(2π)3∫_V_q d^3𝐪 q ħ/2V_cρ v_α\|⟨0\|e^i𝐪·𝐫H^LKBP_strain,αN_q\|1⟩_α\|^2δ(Δ𝔼-ħ v_α q)) = ∑_α(1/8π^2∫_V_q q^3 dq sinθ dθ dϕ1/ρ v_α\|⟨0\|e^i𝐪·𝐫H^LKBP_strain,α\|1⟩_α\|^2 1/ħ v_αδ(Δ𝔼/ħ v_α-q)) = ∑_α(1/8π^2ħρ v_α^2∫_0^2πdϕ ∫_0^πdθsinθ ∫_0^∞q^3dq \|⟨0\|e^i𝐪·𝐫H^LKBP_strain,α\|1⟩_α\|^2 δ(Δ𝔼/ħ v_α-q)) The qubit ground state \|0⟩ and excited state \|1⟩ are 1×4 spinors with each component multiplied to spatial functions. ⇒ ⟨0\|e^i𝐪·𝐫H^LKBP_strain,α\|1⟩=[ g_1^* g_2^* g_3^* g_4^* ]e^i𝐪·𝐫[ P_ε^α+Q_ε^α 0 L_ε^α M_ε^α; 0 P_ε^α+Q_ε^α M_ε^α^* -L_ε^α^; L_ε^α^ M_ε^α P_ε^α-Q_ε^α 0; M_ε^α^* -L_ε^α 0 P_ε^α-Q_ε^α; ][ e_1; e_2; e_3; e_4 ] = [ g_1^* g_2^* g_3^* g_4^* ]e^i𝐪·𝐫[ (P_ε^α+Q_ε^α)e_1+L_ε^α e_3+M_ε^α e_4; (P_ε^α+Q_ε^α)e_2+M_ε^α^* e_3-L_ε^α^* e_4; L_ε^α^* e_1+M_ε^α e_2+(P_ε^α-Q_ε^α)e_3; M_ε^α^* e_1-L_ε^α e_2+(P_ε^α-Q_ε^α)e_4; ] = [{(P_ε^α+Q_ε^α)e^i𝐪·𝐫g_1^e_1+L_ε^α e^i𝐪·𝐫g_1^e_3+M_ε^α e^i𝐪·𝐫g_1^e_4}+{(P_ε^α+Q_ε^α)e^i𝐪·𝐫g_2^e_2+M_ε^α^* e^i𝐪·𝐫g_2^e_3-L_ε^α^ e^i𝐪·𝐫g_2^e_4}. + .{L_ε^α^e^i𝐪·𝐫g_3^e_1+M_ε^α e^i𝐪·𝐫g_3^e_2+(P_ε^α-Q_ε^α) e^i𝐪·𝐫g_3^e_3}+{M_ε^α^e^i𝐪·𝐫g_4^e_1-L_ε^α e^i𝐪·𝐫g_4^e_2+(P_ε^α-Q_ε^α) e^i𝐪·𝐫g_4^e_4}] According to our model g_i^=∑_{m,n,l,i'}c_ii'^g^ψ_n(x)ψ_m(y)ψ_l(z), and e_j^=∑_{m',n',l',j'}c^e_jj'ψ_n'(x)ψ_m'(y)ψ_l'(z); implies that the terms in Eqn. <ref> have the form e^i𝐪·𝐫g_i^e_j⇒(∑_i',j' c^g^_ii'c^e_jj')(∑_n,n'∑_m,m'∑_l,l')∫_-∞^∞dx e^iq_xxψ_n(x)ψ_n'(x) ∫_-∞^∞dy e^iq_yyψ_m(y)ψ_m'(y) ∫_-∞^∞dz e^iq_zzψ_l(z)ψ_l'(z) To our advantage, the inversion-symmetric basis wave-functions we use to describe our hole QD, i.e. an infinite barrier in z and harmonic potential in x-y, have closed form solutions of the e^i𝐪·𝐫 integrals. This allows us to evaluate the Relaxation rate Γ_1 analytically. We also show the dipole approximation to agree with the analytical results for T_1; thirdly, a numerical pathway is sketched as an alternative. Analytical: in-plane integrals. The matrix elements of e^iq_xx between two x(or y) wavefunctions are given by, ⟨ n\|e^iq_xx\|n'⟩=∫_-∞^∞dx e^iq_xxψ_n(x)ψ_n'(x) e^iq_xx can be written as an infinite expansion in Hermite polynomial basis: e^iq_xx=e^iq_xL_xx/L_x=e^(-iq_xL_x)^2/4∑_r=0^∞(iq_xL_x)^r/2^r r!H_r(x/L_x) ⇒⟨ n\|e^iq_xx\|n'⟩=1/√(2^n n! L_x√(π))1/√(2^n' n'! L_x√(π))e^-q_x^2L_x^2/4∑_r=0^∞(iq_xL_x)^r/2^r r!𝕀(r,n,n') with 𝕀(r,n,n')=∫_-∞^∞ dx e^-x^2/L_x^2H_r(x/L_x)H_n(x/L_x)H_n'(x/L_x)=L_x∫_-∞^∞ d(x/L_x) e^-x^2/L_x^2H_r(x/L_x)H_n(x/L_x)H_n'(x/L_x). Substituting x/L_x→ x, we write: ⇒⟨ n\|e^iq_xx\|n'⟩ = (2^n+n'n!n'!π)^-1/2e^-q_x^2L_x^2/4∑_r=0^∞(iq_xL_x)^r/2^r r!∫_-∞^∞ dx e^-x^2H_r(x)H_n(x)H_n'(x) The product of two Hermite polynomials can be expanded in the Hermite polynomial basis: H_n(x)H_n'(x)=2^nn!n'!∑_k=0^nH_2k+n'-n(x)/2^kk!(k+n'-n)!(n-k)! ⇒⟨ n\|e^iq_xx\|n'⟩ = e^-q_x^2L_x^2/4/√(2^n+n'n!n'!π)∑_r=0^∞(iq_xL_x)^r/2^r r!2^nn!n'!∑_k=0^n1/2^kk!(k+n'-n)!(n-k)!∫_-∞^∞ dx e^-x^2H_r(x)H_2k+n'-n(x) = e^-q_x^2L_x^2/4/√(2^n+n'n!n'!π)∑_r=0^∞(iq_xL_x)^r/2^r r!2^nn!n'!∑_k=0^n2^2k+n'-n(2k+n'-n)!√(π)δ_r,2k+n'-n/2^kk!(k+n'-n)!(n-k)! where we have used the orthonormality relation:∫_-∞^∞ e^-x^2H_n(x)H_n'(x) dx=2^nn!√(π)δ_n,n'. The δ-function boils the r-sum down to only one term, such that: ⇒⟨ n\|e^iq_xx\|n'⟩=√(n'!n!/2^n'-n)e^-q_x^2L_x^2/4∑_k=0^n (iq_xL_x)^2k+n'-n/2^2k+n'-n(2k+n'-n)!2^2k+n'-n(2k+n'-n)!/2^kk!(k+n'-n)!(n-k)! = √(n!/2^n'-nn'!)e^-q_x^2L_x^2/4∑_k=0^nn'!(iq_xL_x)^n'-n(iq_xL_x)^2k/2^kk!(k+n'-n)!(n-k)!=√(n!/2^n'-nn'!)e^-q_x^2L_x^2/4(iq_xL_x)^n'-n∑_k=0^nn'!((iq_xL_x)^2)^k/2^kk!(k+n'-n)!(n-k)! = √(n!/2^n'-nn'!)e^-q_x^2L_x^2/4(iq_xL_x)^n'-n∑_k=0^n(-1)^k(n+n'-n)!/k!(k+n'-n)!(n-k)!(q_x^2L_x^2/2)^k Using the formula for associated Laguerre polynomial ℒ_n^a(x)=∑_k=0^n(-1)^k(n+a)!/k!(k+a)!(n-k)!x^k, the matrix element of e^iq_xx can be analytically evaluated as: ⇒⟨ n\|e^iq_xx\|n'⟩=√(1/2^n'-nn!/n'!)e^-q_x^2L_x^2/4(iq_xL_x)^n'-nℒ_n^n'-n(q_x^2L_x^2/2) ⇒⟨ m\|e^iq_yy\|m'⟩=√(1/2^m'-mm!/m'!)e^-q_y^2L_y^2/4(iq_yL_y)^m'-mℒ_m^m'-m(q_y^2L_y^2/2) The z-basis wavefunctions are: ψ_l(z)= cos((l+1)π z/L_z) l=0,2,4,.. sin((l+1)π z/L_z) l=1,3,5.. ,z∈{-L_z/2,L_z/2} The matrix element ⟨ l\|e^iq_zz\|l'⟩ is evauated as: ∫_-L_z/2^L_z/2dz e^iq_zzψ_l(z)ψ_l'(z)=∫_-L_z/2^L_z/2dz (cos q_zz+isin q_zz)ψ_r(z)ψ_r'(z) where r=l+1 is the new index for z-integral (for simpler mathematical expressions). r odd, r' even. For the case of evaluating the matrix element between even and odd z-basis functions is calculated as = ∫_-L_z/2^L_z/2dz(cos q_zz+isin q_zz)√(2/L_z)cos(rπ z/L_z)√(2/L_z)sin(r'π z/L_z) = 2/L_z∫_-L_z/2^L_z/2dz(cos q_zz+isin q_zz)1/2(sin(r+r'π z/L_z)-sin(r-r'π z/L_z)) = 1/L_z∫_-L_z/2^L_z/2dz(cos q_zzsin(r+r'π z/L_z)-cos q_zzsin(r-r'π z/L_z)+isin q_zzsin(r+r'π z/L_z)-isin q_zzsin(r-r'π z/L_z)) = 1/2L_z∫_-L_z/2^L_z/2dz(sin( q_zz+r+r'π z/L_z)-sin(q_zz-r+r'π z/L_z)-sin(q_zz+r-r'π z/L_z)+sin(q_zz-r-r'π z/L_z)) + i/2L_z∫_-L_z/2^L_z/2dz(cos( q_zz+r+r'π z/L_z)+cos(q_zz-r+r'π z/L_z)-cos(q_zz+r-r'π z/L_z)-sin(q_zz-r-r'π z/L_z)) The sin and cos terms can be now evaluated straight forwardly to give us the final simplified expression: =i/2[sin(q_zL_z/2+(r+r')π/2)/q_zL_z/2+(r+r')π/2-sin(q_zL_z/2-(r+r')π/2)/q_zL_z/2-(r+r')π/2-sin(q_zL_z/2+(r-r')π/2)/q_zL_z/2+(r-r')π/2+sin(q_zL_z/2-(r-r')π/2)/q_zL_z/2-(r-r')π/2] r,r' both odd/even. For the case of evaluating the matrix element between both even(both odd) z-basis functions is calculated as = ∫_-L_z/2^L_z/2dz(cos q_zz+isin q_zz)√(2/L_z)sin(rπ z/L_z)√(2/L_z)sin(r'π z/L_z) = -1/2[sin(q_zL_z/2+(r+r')π/2)/q_zL_z/2+(r+r')π/2+sin(q_zL_z/2-(r+r')π/2)/q_zL_z/2-(r+r')π/2+sin(q_zL_z/2+(r-r')π/2)/q_zL_z/2+(r-r')π/2+sin(q_zL_z/2-(r-r')π/2)/q_zL_z/2-(r-r')π/2] Finally we put the results from Eqns. <ref>, <ref> into Eqn. <ref> to evaluate: e^i𝐪·𝐫g_i^e_j ⇒ (∑_i',j' c^g^_ii'c^e_jj')(∑_n,n'∑_m,m'∑_l,l')√(1/2^n'-nn!/n'!)e^-q_x^2L_x^2/4(iq_xL_x)^n'-nℒ_n^n'-n(q_x^2L_x^2/2) × √(1/2^m'-mm!/m'!)e^-q_y^2L_y^2/4(iq_yL_y)^m'-mℒ_m^m'-m(q_y^2L_y^2/2) × f(l,l',q_zL_z) where, f(l,l',q_zL_z)=i/2[sin(q_zL_z/2+(r+r')π/2)/q_zL_z/2+(r+r')π/2-sin(q_zL_z/2-(r+r')π/2)/q_zL_z/2-(r+r')π/2-sin(q_zL_z/2+(r-r')π/2)/q_zL_z/2+(r-r')π/2+sin(q_zL_z/2-(r-r')π/2)/q_zL_z/2-(r-r')π/2] l± l'=odd -1/2[sin(q_zL_z/2+(r+r')π/2)/q_zL_z/2+(r+r')π/2+sin(q_zL_z/2-(r+r')π/2)/q_zL_z/2-(r+r')π/2+sin(q_zL_z/2+(r-r')π/2)/q_zL_z/2+(r-r')π/2+sin(q_zL_z/2-(r-r')π/2)/q_zL_z/2-(r-r')π/2] l± l'=even Next we make the substitution q_x→ qsinθcosϕ, q_y→ qsinθsinϕ, q_z→ qcosθ to write e^i𝐪·𝐫g_i^e_j ⇒ (∑_i',j' c^g^_ii'c^e_jj')(∑_n,n'∑_m,m'∑_l,l')√(1/2^n'-nn!/n'!)e^-q^2L_x^2sin^2θcos^2ϕ/4(iqL_xsinθcosϕ)^n'-nℒ_n^n'-n(q^2L_x^2sin^2θcos^2ϕ/2) × √(1/2^m'-mm!/m'!)e^-q^2L_y^2sin^2θsin^2ϕ/4(iqL_ysinθsinϕ)^m'-mℒ_m^m'-m(q^2L_y^2sin^2θsin^2ϕ/2) × f(l,l',qL_zcosθ) These set of equations allow us to calculate the terms in Eqn. <ref> to express \|⟨0\|e^i𝐪·𝐫H^LKBP_strain,α\|1⟩\|^2 as I_α(q,θ,ϕ). From Eqn. <ref>, Γ_1=1/T_1 = ∑_α(1/8π^2ħρ v_α^2∫_0^2πdϕ ∫_0^πdθsinθ ∫_0^∞q^3dq I_α(q,θ,ϕ) δ(Δ𝔼/ħ v_α-q)) = ∑_α(Δ𝔼^3/8π^2ħ^4ρ v_α^5∫_0^2πdϕ ∫_0^πdθsinθ I_α(q→Δ𝔼/ħ v_α,θ,ϕ))=∑_α(Δ𝔼^3/8π^2ħ^4ρ v_α^5 Ω_α(Δ𝔼/ħ v_α)) where Ω denotes the angular integration. Dipole approximation. Alternatively, one can expand e^i𝐪·𝐫≈1(+i𝐪·𝐫-\|𝐪·𝐫\|^2+..) which simplifies the product \|⟨0\|e^i𝐪·𝐫H^LKBP_strain,α\|1⟩\|^2≈\|⟨0\|H^LKBP_strain,α\|1⟩+i⟨0\|𝐪·𝐫H^LKBP_strain,α\|1⟩-⟨0\|\|𝐪·𝐫\|^2H^LKBP_strain,α\|1⟩+..\|^2. In sharp contrast to electron spin-1/2 qubits, where the local strain is a diagonal tensor and the leading zeroth order term ⟨0\|H_ε\|1⟩ vanishes; for the spin-3/2 holes the leading term is the zeroth order, resulting in a α_r B^3+β_r B^4 + γ_r B^5 relaxation rate variation, compared to B^5 variation in electron spin-1/2 qubits. While the B^3 and B^5 dependence are explained by the first two terms in dipole approximation, the orbital B-terms in the qubit admixture give rise to the B^4 dependence. § RANDOM TELEGRAPH NOISE(RTN) DEPHASING: SCREENED POTENTIAL OF A CHARGE DEFECT Taking into account the screening effect of the 2DHG being formed in Ge, the Fourier q-space form of the potential of single defect with charge e is given by: U_scr(q)=e^2/2 ϵ_0 ϵ_r e^-q dΘ(2 k_F-q)/q+q_T F. where U_scr(q) is known as the Thomas-Fermi screened potential, q is the Fourier space variable, q_TF=0.49 nm^-1 is the Thomas-Fermi wave vector in germanium independent of the density of holes. k_F is the Fermi wave vector which is estimated to be 0.1 nm^-1 in our calculations, and Θ is the Heaviside Theta step function. Considering constant screening, which only holds for q<2k_F, the screened potential in real space under the limit d ≪r is approximated to be: U_s(r)=e^2/4 πϵ_0 ϵ_r1/q_TF^2( 1/\|𝐫-𝐫_D\|^3+ dq_TF/\|𝐫-𝐫_D\|^3 ) The relative electrical permeability of germanium is ϵ_r = 15.8; ϵ_0 being the vacuum electrical permeability. 𝐫_D={80 nm,80 nm,0} is the vector denoting the distance of the in-plane charge defect from the center of the QD. § G-FACTOR ANISOTROPY AND FITTING PARAMETERS In the main text, Fig.<ref>, we mentioned that the fitting paramters are not unique. Here we suggest some other possible configurations to fit the parameters. The main fitting parameters are dot size (L_x, L_y, L_z), the electric field can be tuned via gate to modulate the g-factors. * L_x=40 nm, L_y=60 nm L_z = 10 nm * L_x=30 nm, L_y=50 nm L_z = 10.5 nm * L_x=44 nm, L_y=60 nm L_z = 9.5 nm Note that there are many other possible combinations of parameters if the strains (both uniaxial strain and shear strain) are included.
http://arxiv.org/abs/2307.02583v1	20230705183011	User Perspectives on Branching in Computer-Aided Design	[ "Kathy Cheng", "Phil Cuvin", "Alison Olechowski", "Shurui Zhou" ]	cs.HC	[ "cs.HC" ]	User Perspectives on Branching in Computer-Aided Design]User Perspectives on Branching in Computer-Aided Design University of Toronto 27 King's College Circle Toronto Canada [email protected] University of Toronto Toronto Canada University of Toronto Toronto Canada University of Toronto Toronto Canada Branching is a feature of distributed version control systems that facilitates the “divide and conquer” strategy present in complex and collaborative work domains. Branching has revolutionized modern software development and has the potential to similarly transform hardware product development via CAD (computer-aided design). Yet, contrasting with its status in software, branching as a feature of commercial CAD systems is in its infancy, and little research exists to investigate its use in the digital design and development of physical products. To address this knowledge gap, in this paper, we mine and analyze 719 user-generated posts from online CAD forums to qualitatively study designers’ intentions for and preliminary use of branching in CAD. Our work contributes a taxonomy of CAD branching use cases, an identification of deficiencies of existing branching capabilities in CAD, and a discussion of the untapped potential of CAD branching to support a new paradigm of collaborative mechanical design. The insights gained from this study may help CAD tool developers address design shortcomings in CAD branching tools and assist CAD practitioners by raising their awareness of CAD branching to improve design efficiency and collaborative workflows in hardware development teams. <ccs2012> <concept> <concept_id>10010405.10010497.10010500</concept_id> <concept_desc>Applied computing Document management</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10010405.10010432.10010439.10010440</concept_id> <concept_desc>Applied computing Computer-aided design</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10003120.10003130.10011762</concept_id> <concept_desc>Human-centered computing Empirical studies in collaborative and social computing</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> [500]Human-centered computing Empirical studies in collaborative and social computing [500]Applied computing Computer-aided design [300]Applied computing Document management [ Shurui Zhou August 1, 2023 ================== § INTRODUCTION Hardware design and development – the creation of physical products – is an indispensable part of human history and essential to technological innovation. Contemporary hardware development is practically impossible without the use of CAD (computer-aided design) software to facilitate all aspects and stages of the development process, from initial conceptual design to manufacturing of the final product <cit.>. The field of hardware development is long-standing, as are the tools and systems involved in the hardware design process. Traditional CAD ecosystems such as PLM (Product Lifecycle Management), PDM (Product Data Management), and centralized version control that first appeared in the 1980s remain the standard collaboration systems in place at most hardware organizations today <cit.>. As design practices are becoming increasingly globalized, distributed, and demanding <cit.>, these traditional ecosystems are unable to keep up with the changing nature of hardware design and development work <cit.>. Recent advances in new generation collaboration and version control tools – such as cloud-based CAD, fully-synchronous CAD, and distributed version control (i.e., branching and merging) – have the potential to support today's complex collaborative workflows. Similar tools have become second nature to the related field of software development, fundamentally changing the way in which software developers collaborate. Branching and merging is a feature from distributed version control systems (e.g., Git[https://git-scm.com/] <cit.>, Mercurial[https://www.mercurial-scm.org/]) popularized by cloud-based code hosting platforms (e.g., GitHub, GitLab, Bitbucket), that is essential to modern distributed version control systems (DVCS) <cit.>. Branching makes it possible for developers to split a codeline away from the production code version to make contributions without affecting the existing version or disturbing collaborators unnecessarily <cit.>, while merging allows changes made in “source branches” to be integrated into a single “target branch” or “main branch” <cit.>. Though branching and merging are related operations, in this work, we focus on branching practices due to the limited abilities of current CAD merging functionalities (as we discuss later in this paper) and the fact that merging is a consequence of branching; thus, it is in our best interest to first study users' intentions for branching. Related work in software literature likewise prioritizes thorough understanding of branching practices before further investigation of merging <cit.>. Branching and merging functionality has revolutionized modern software development; collaboration has not only become more efficient and effective, but a new paradigm of open-source and crowdsourced software development work has been facilitated with branching tools <cit.>. Although hardware and software developers create different products (physical vs. digital), both fields are complex, large-scale, and inherently collaborative work domains which require the collective and joint contribution of many individuals <cit.>. Both fields are challenged by collaborative distance, overhead of synchronization, and lack of group awareness <cit.>, which necessitates version control systems to manage file documentation, track version changes, and enable access to shared files <cit.>. While there are parallels between software and hardware development, artifacts in CAD are 3D models that represent geometric and topological data (e.g., STL, STEP files), which are fundamentally different from artifacts in software, which consist primarily of text-based code and documentation <cit.>. Therefore, while lessons from software development are informative to CAD researchers and users, their fundamental differences motivate a dedicated study for hardware branching. To date, little research has investigated CAD branching for the design and development of hardware products. Existing work has focused on proposing computational or data management methods for accomplishing branch and merge operations (e.g., hash algorithms <cit.>, “diff and merge” using persistent identifiers (POID) <cit.>, and processing-oriented modeling <cit.>), but few studies have focused on CAD and CAD-related systems from the user perspective <cit.>. Furthermore, support for branching and merging in commercial CAD platforms is a relatively recent development. Onshape[https://www.onshape.com/en/] was the first to bring this capability to CAD artifacts in 2015, followed by Fusion360[https://forums.autodesk.com/] in 2017, and SolidWorks PDM[https://www.solidworks.com/product/solidworks-pdm] in 2018 <cit.>. However, the implementation of these tools has not always been successful, for example, commercial releases from Fusion360 were discontinued soon after launch <cit.>. As branching technology is still relatively new to the CAD community, this exploratory study is needed to uncover how branching is currently used and adopted among CAD designers, what the pain points are of current tools, and how to improve branching functionalities to fulfill the unmet design needs of hardware designers. Our study aims to close this knowledge gap through an empirical investigation of designers’ current intentions and expectations for the use of branching in CAD. The following two questions guided our research: RQ1: How do CAD designers use branching functionality in the hardware design process? RQ2: What are the design shortcomings and gaps of existing CAD branching tools? To answer these research questions, we mine and qualitatively analyze 719 user-generated posts from online CAD forums, inclusive of CAD-platform-specific forums (i.e., Fusion360[https://forums.autodesk.com/], Onshape[https://www.onshape.com/en/], and SolidWorks PDM[https://www.solidworks.com/product/solidworks-pdm]), and platform-agnostic forums (i.e., CAD Forum[https://cadforum.net/] and Eng-Tips[https://www.eng-tips.com/]). We chose to mine posts from online CAD forums for our exploratory study because searching in online forums allows us to reach more CAD users and obtain more discussions of branching than we would with other qualitative research methods, such as interviews. Furthermore, online communities which communicate in user forums provide a rich and comprehensive dataset that can help researchers understand real user requirements and behaviours; other CSCW (Computer-Supported Cooperative Work) studies have similarly mined online forums to understand the privacy concerns of software developers <cit.>, misinformation in social media platforms <cit.>, and codes of conduct in collaborative software projects <cit.>. We therefore contribute the following insights to the growing CAD community within CSCW: * An empirical investigation of 719 user-generated online posts from five forum sites to identify shortcomings of existing branching capabilities in commercial CAD platforms, providing tool builders with actionable improvements for CAD branching tools. * A taxonomy of CAD branching use cases, comprised of three high-level groupings, Product Line Management, Risk Isolation, and Designer Support. * Evidence that CAD designers use branches for several use cases within each grouping, most frequently for production version maintenance, creating new product variants, and presenting the design to project stakeholders. Our findings further revealed that not all identified use cases are intentional functions of branching tools; in other words, branching in CAD is often used as a workaround to perform other desired design tasks – potentially due to lack of technical support in these areas. * A discussion of the potential of branching to not only support a new paradigm of collaborative mechanical design, but improved coordination, cooperation, and communication for CSCW domains as a whole. § BACKGROUND & RELATED WORK §.§ Branching and Merging for Software Development In software development, branching is a mechanism supported by distributed version control systems (e.g., Git, Mercurial) that allows for a code file to be cloned and edited independently of the main codeline, and to subsequently be merged to the original file to reconcile the changes with the main codeline <cit.>. Branching and merging is currently the most widely adopted method for software version control <cit.>. Branching functionality has been present in software version control tools since the 1980s <cit.>, but modern branch-based distributed version control only entered wider use with the introduction of Git (an open-source version control tool which provided lightweight and robust branching and merging workflows) in 2005, and with the introduction of GitHub in 2008, which leveraged branching and merging functionality derived from Git to enable large-scale social collaboration on software projects and to provide a platform for open-source development <cit.>. Branching enables parallel and simultaneous collaboration on the same code file and offers developers the ability to isolate the risk associated with changing code (and potentially breaking the program) from the main branch (also known as the “mainline” or the “root” branch) <cit.>. Therefore, one might branch for bug fixes, new feature development, experimental code creation, and maintenance of build scripts <cit.>. Branching is also used to support organizational and non-technical workflow requirements, such as file sharing or dividing work between programming teams <cit.>. §.§.§ Purposes of Branching in Software Development Software version control branches often fulfill more than a single purpose, as a branch can accomplish multiple development objectives at the same time (e.g., a branch can simultaneously isolate a bug fix from the root branch and assign the bug fix task to a separate team) <cit.>. As such, a use case, as we will present, represents a purpose or intention of branching that does not overlap with any other. Given this definition of a branching use case, we synthesized prior literature that explored how software developers use branching in practice, in order to provide an overview of known software branching use cases. Previous studies on software branching have demonstrated that developers primarily use branches for version control purposes (e.g., tracing design history, tracking version changes, managing product evolution) and this is consistent with how branches were intended to be used, as described in foundational handbooks and guides <cit.>. The most popular use cases discussed in literature are branching to separate product releases for different devices or versions of operating systems (i.e., Parallel Version branching), and branching to build new features that are intended to be integrated into the main branch and put into production (i.e., New Feature branching) <cit.>. Creating a branch per feature is a development strategy that presents the opportunity for Modularization branching, where branches separate code into modules that can be combined to result in different product configurations <cit.>. Prior work has also shown that developers use branches to build faster, better, and more optimized versions of the mainline code, without altering the function of the code, known as Refactorization branching <cit.>. An important theme that emerges from these studies is that the aforementioned use cases serve the development, iteration, and evolution of the mainline branch, or production code version. In practice however, developers not only use branching to serve the mainline branch, but also to isolate the risk of unverified and untested changes from the mainline; such branches are described as “small-scope, short-lived, and used for single, targeted purposes” <cit.> and “more temporary and for minor changes” <cit.>. For example, bug fixing commonly requires developers to create these “ad-hoc” branches, where an error in the code must be repaired and merged back <cit.>. Developers may branch to experiment with building new code that is not yet confirmed as a feature (i.e., Experimental/Prototype branching) <cit.>, or branch to create an isolated environment for testing code changes prior to integration with the mainline (i.e., Testing branching) <cit.>. Branches are further used to maintain and store process-specific code which are not directly related to the main codebase, but are necessary for building, testing, or deploying software (e.g., build scripts, macros, regression test suites) outside of the main development branch (i.e., Tool/Utility branching) <cit.>. Software literature has also identified that branching can be used to serve developers' organizational or non-technical needs <cit.>. A frequently discussed purpose of branching is to support collaboration by dividing coding work between different individuals or teams (i.e., Work Division branching) <cit.>, or allowing shared files to be easily accessed by multiple developers (i.e., File Sharing branching) <cit.>. It has been shown that the parallel workflow made possible by branching can improve working efficiency in collaborative software development projects <cit.>. Overall, prior work has identified numerous branching use cases for software development, summarized in Table <ref>. These studies provide an important foundation for our knowledge of software branching and have high potential to inform our understanding of how branches may be used in other contexts, like the development of hardware products. Nonetheless, compared to software branching, relatively little is known about CAD branching purposes in the hardware development world, therefore motivating the present study. §.§ Branching and Merging for CAD Branching and merging as a method for CAD version control is not a new concept in theory; literature as early as the 1980s has discussed distributed version control systems for CAD <cit.>. However, the practical branching of CAD artifacts in commercial platforms is a relatively recent development – to our knowledge, support for branching and merging of parametric CAD files has emerged in only three CAD platforms: PTC's Onshape in 2015 <cit.>, Autodesk's Fusion360 in 2017 (soon after removed as functionality, also in 2017) <cit.>, and Dassault Systèmes's SolidWorks PDM in 2018 <cit.>. While branches for CAD artifacts (e.g., 3D models, CAD drawings) have been supported in commercial CAD tools since 2015 (by Onshape), the investigation of practical usage of branching and merging in CAD literature remains scarce. Richter and Beucke explored version control for distributed civil engineering design and proposed “diff and merge” solutions inspired by the software development field to compare and merge versions of CAD drawings <cit.>. They identified that the “parallel and consistent collaboration” that is enabled by branching can be a superior workflow compared to traditional centralized document management systems. Le performed a SWOT (strength, weakness, opportunity, threat) analysis to compare the CAD platforms, Onshape and SolidWorks and concluded that one of the main reasons why users preferred Onshape is the flexibility of its branch and merge tool that allows users to model concurrently as well as experiment with multiple design variations <cit.>. Stirling et al. explored the possibility of adapting DevOps workflows – widely utilized in software development – to facilitate open collaborative development of hardware products, creating “HardOps” <cit.>. In open-source hardware projects, it is essential that multiple designers can simultaneously contribute to a design and for all contributors to have unfettered access to all product data, and this cannot be achieved with the centralized version control systems in place at most design organizations <cit.>. With distributed (i.e., branch-based) version control, contributors to open hardware projects can work independently, test and refine their changes, and ensure that their changes are carefully reviewed before being integrated into the main project. Before Git-style branching was introduced in commercial CAD software, researchers have considered branching's suitability for CAD models through examining the commonalities and differences between hardware and software development processes <cit.>. Bricogne et al. and Fresemann et al. compared version control and change management in PDM (product data management) for the CAD design process to SCM (software configuration management) in the software development process. The major differences between hardware and software data management identified were: (1) software product versions are modified at the same time without affecting one another, but mechanical product versions are developed sequentially <cit.>; (2) mechanical product data is seldom modified after the product is released, whereas a released software product will likely change <cit.>; and (3) several developers modify a code version in parallel with the branch/merge approach, whereas only one CAD designer can develop a hardware product version with the check-in/check-out approach used in PDM (centralized version control) <cit.>. Bricogne et al. suggest that these differences between hardware and software data management contribute to the limited implementation of branch and merge in CAD <cit.>, and Fresemann et al. consider these characteristics in their design of a comparable branch-based PDM prototype inspired by SCM <cit.>. Together these studies highlight the motivation for Git-style branching and merging mechanisms to support hardware development with CAD. Existing work has discussed potential usages and implementations of CAD branching but has not investigated practical use of commercial CAD branching tools by designers. Therefore, our study aims to determine the real user requirements of branching with commercial CAD platforms, the limitations of current branching functionality, and how to improve CAD branching tools to fulfill these user requirements. § METHODS: MINING ONLINE CAD FORUMS The goal of this study is to establish CAD branching use cases, and to identify the deficiencies of current CAD branching tools. We do this by mining user-generated posts from online CAD forums that discuss branching and merging and related activities for CAD, to build an understanding of what kinds of tasks CAD designers use branching for and what the limitations are to such tools. Through mining online forums, we analyze data from a large set of CAD users, and thus we can access a greater variety of discussions of branching use cases and shortcomings. Online CAD forums can be considered as a type of community of practice – a “group of people informally bound together by shared expertise and passion for a joint enterprise” that facilitates learning, problem-solving, information exchange, and the sharing of best practices <cit.> – in this case, offering a place for CAD users to help each other solve design and development-related problems, share CAD knowledge, and discuss various CAD topics (e.g., drafting, modelling, simulations). Previous work by Kiani et al. has shown that online forums are the most frequently used resource among design professionals who use feature-rich CAD software, and the third most frequently used resource for CAD newcomers <cit.>. A survey study by Robertson et al. similarly found that “experienced, constant users are more likely to visit online CAD forums”, stating that these users “were keen to pass on the lessons they had learned from their years of experience” <cit.>. Matejka et al. likewise found that CAD professionals are frequent online forum-goers – nearly 25% of their study participants visit forums daily <cit.>. Evidently, online CAD forums are important sites for knowledge sharing, knowledge producing, and learning for novice and experts alike, and researchers can glean real user discussions, complaints, and requests that occur in such forums. In this section, we first provide a brief overview of the studied forum sites. We then review the data collection and data processing steps, followed by the qualitative coding analysis used. Figure <ref> summarizes our research methodology. §.§ Selection of Online Forums The five forum sites that we studied are described in Table <ref>. It should be noted that these sites were not chosen at random; we purposefully selected online forums targeted towards CAD professionals or CAD enthusiasts who have a personal interest or motivation for discussing CAD-related topics. The web forums examined in this study include those officially affiliated with CAD programs (i.e., Onshape[https://www.onshape.com/en/], SolidWorks PDM[https://www.solidworks.com/product/solidworks-pdm], Autodesk Fusion360[https://forums.autodesk.com/]), as well as general, non-platform-specific forums (i.e., Eng-Tips[https://www.eng-tips.com/] and CAD Forum[https://cadforum.net/]). We chose Onshape, SolidWorks PDM, and Autodesk Fusion360 forums, specifically, because these CAD programs are the only ones that currently support or previously supported branching and merging <cit.>. When seeking online help, CAD users tend to visit product-owned forums over general forums because the terminology, construction workflows, and features can differ greatly between different CAD platforms, thus, product-owned forums offer more specific help <cit.>. That said, non-platform-specific forums were also included in our study to capture general discussions of CAD version control that are not limited to a single platform. The platform-specific forums are moderated by employees of the company, whereas non-platform-specific forums are moderated by volunteers, who are typically professional CAD users or CAD enthusiasts. All five sites allow public visitors to view posts, but in order to be able to further interact within the forums (e.g., create new posts, “like” comments, reply to others, and participate in a discussion), users must register as a member of that forum – which involves creating an account that is verified with an email address – to help ensure the quality and legitimacy of contributors. With our selection of forums, we aim to provide a representative sample of CAD designers who use or are familiar with branching. §.§ Data Collection In the online forums, participants discuss a wide range of topics, such as reporting bugs or requesting troubleshooting help from others. To narrow our search before data scraping, we searched for the following keyword terms (stemmed): branch, merg, copy, clon, fork, git, github, version. We included the terms copy and clone because they are frequently used interchangeably with branching <cit.>. The term fork was included because of its close relationship to branching in the software field, and the terms git, github, and version to capture discussions of version control. For some forums, we could only search by one keyword at a time (e.g., a search for branch, then a search for copy), which resulted in duplicate posts. We built a custom web-scraper using Python (with packages Beautiful Soup[https://www.crummy.com/software/BeautifulSoup/] and Selenium[https://selenium-python.readthedocs.io/]) to extract the data from the forum posts which range from April 2005 to April 2022. Although commercial CAD platforms have only begun supporting branching since 2015 (by Onshape), we chose to include posts starting from April 2005 in our search criteria because this is when branching was introduced to the software development field by Git <cit.>; we found numerous posts discussing the potential of “Git for CAD”, suggesting that CAD users have been asking for the ability to branch and merge long before its implementation in commercial CAD platforms. The data we collected include the title and content of the post, the initial posting date, the username of the post author, an ordered list of comments and corresponding usernames of commenters. Here we define post as the initial posting and thread as the entire discussion, which includes the initial post and any comments. The dataset was stored in a local Excel file for analysis. In total, our search across the five forum sites returned 14,448 unique threads. Aggregate statistics for each forum are provided in Table <ref>. The average length of a thread was 792 words (SD = 1,037). Over the 14,448 threads, there were 8,528 unique authors and the average number of posts per author was 1.7 (SD = 9.9). On average, each post had 8.6 comments (SD = 11.7) and 1,247 posts had no comments. The number of posts grew rapidly from 2005 to 2017 and the forum activity remained relatively steady from 2015 to 2022; the number of posts peak in 2017, which may be the result of Autodesk's launch and subsequent retraction of branching and merging in 2017 <cit.>. Note that data collection in this work was done in April 2022, thus only four months of data was available for that year. Figure <ref> shows some characteristics of our analyzed dataset. §.§ Data Processing After data collection, the dataset was cleaned, and duplicate threads were removed. As a first step in preparing the data, a random sample of 1,000 threads were manually parsed to identify keywords present in threads that were irrelevant to our research questions. Such keywords are presented in Table <ref> and represent discussions about CAD software performance issues and specifications of computer hardware. Unlike other online review mining studies that included references to bug fixes, runtime errors, or glitches in their analysis, our study is interested broadly in how current branching features are used in CAD design and how to improve usability. Thus, we omitted threads that described unexpected bugs and glitches as their main topic, as these problems will likely be fixed by the developers in subsequent versions of the CAD software. The exception to this filter criteria were threads containing the word “branch” – these threads were deemed as relevant even if they contained keywords like “error”. We prioritized filtering out irrelevant posts from the list of forum posts, knowing that this would result in a larger but more inclusive dataset, rather than to first filter by relevant posts and overlook potentially pertinent posts that did not fit our keyword search criteria. In this sense, if relevant posts are considered positive, we aimed to minimize false negatives and favour false positives. This was done because in the data analysis stage, manual coding and tagging was used, where the researchers could then determine the relevancy of posts with the help of additional semantic context. To give a confidence level of 95% and confidence interval 5%, we tested our keyword filter on a random sample of 400 threads, out of the total 14,448. Of the 400 threads, we found 0% false negatives, 13% positives (true and false, to be determined in the manual coding stage) and 87% true negatives. After sample testing our keyword filter, the remaining threads were filtered for relevance to the research questions which returned 1,761 threads (around 12.2%). §.§ Data Analysis Once all 1,761 potentially relevant threads were extracted, the new dataset was manually coded by two researchers who have qualitative research experience and familiarity with CAD and CAD version control <cit.>. A hybrid coding approach was used, whereby we started with the following broad categories for coding the data: (1) use cases: how branches are currently being used; and (2) status of feature: whether the branching use case is supported by existing functionalities. Within the use cases category, we started with the high-level groupings of branching use cases found in software development literature (as presented in Sec. <ref>) to guide our initial three subcategories of product line management, risk isolation, and designer support – though more subcategories could be added if they were deemed necessary. Codes within these three subcategories were then developed using an inductive approach, meaning that codes were added as new branching use cases were identified from the threads. A forum thread could be assigned more than one code if there was discussion of multiple use cases. Threads that did not discuss a branching use case were tagged by the researchers as irrelevant and were discarded. The first two authors used the 80/20 approach to code the data; first, 20% of the 1,761 threads (353 threads) were open coded collaboratively, then the remaining 80% of threads were divided equally between the two researchers. According to the Institute of Educational Sciences (IES) guidelines, a minimum acceptable percentage agreement of 80% must be achieved from 20% of the total dataset <cit.> for the coding scheme to be deemed repeatable and consistent across all coders involved. Out of the 353 threads that were coded by both researchers, 83% displayed complete agreement, 6% displayed partial agreement, and 11% displayed no agreement on any codes. Overall, the intercoder reliability percentage agreement was 88% <cit.>. For threads with no agreement on any codes, one researcher deemed the thread irrelevant to branching whereas the other researcher considered the thread relevant and assigned the thread a branching use case. In this situation, the researchers revisited the thread and discussed until a consensus was reached. Of the 1,761 threads that were coded, 719 (40.8%) relevant threads were found. Aggregate statistics for each forum are provided in Table <ref>. § RESULTS In this section, we first provide a detailed overview of our taxonomy of CAD branching activities, followed by an analysis of the shortcomings of branching tools. Where excerpts from the forum threads are quoted, we include the name of the forum site and the year it was posted in brackets; for example, a quote from an Onshape thread posted in 2021 would be followed by “(OS21)” – “OS” for Onshape and “21” for 2021[Abbreviations for the five forum sites are: Autodesk (AD), CAD Forum (CF), Eng-Tips (ET), Onshape (OS), and SolidWorks (SW).]. §.§ Taxonomy of CAD Branching Use Cases (RQ1) To answer RQ1: Why is branching used for CAD?, user-generated posts from online CAD forums were manually coded to identify and categorize branching use cases. Mimicking the three groupings of software branching use cases (as described in Section <ref>), we found the similar three categories for CAD branching use cases: (1) Product Line Management encompassing branching to manage the files and version of the mainline design, (2) Risk Isolation used to experiment with changes without affecting the stability of the mainline design, and (3) Designer Support to serve non-technical or organizational needs. The name of Designer Support was adapted from Developer Support to describe CAD users more accurately. Figure <ref> shows the final taxonomy. To provide additional insight on designers' use of branching for CAD, we summarize the prevalence of use cases mentioned in the online forums, displayed in Figure <ref>. The identified CAD branching use cases are discussed in further detail below, and organized by the groupings of Product Line Management, Risk Isolation, and Designer Support. §.§.§ Product Line Management Product Line Management was identified as a top-level grouping of branching use cases in CAD, comprising all use cases that directly serve the mainline, release, or production version of the CAD file. In general, Product Line Management branching facilitates the organization of a product's design process and evolution throughout the development pipeline, and thus, is highly related to the coordination aspect of collaboration <cit.>. The branching use cases that were found for Product Line Management are: production version maintenance, configuration, modularization, and new product variant. Software and hardware development share the common use case of Modularization branching, but other use cases are unique to CAD. The Product Line Management category is by far the most mentioned in the online forums (67.5% of threads), and this is reflective of branching’s primary use in CAD as a version control tool. Production Version Maintenance Production Version Maintenance branching encompasses any instance of branching used to separate the production version of the CAD design from any dependent files and to maintain the design’s currency and availability to designers. The production design version is the main branch that is intended to be revised, released, and then produced. As explained by a forum participant, “The main branch is where you put all the important changes to your model, and you only make changes to your main branch that are complete. The idea is that each version of the main branch is fully manufacturable, with no errors” (OS16). Production Version Maintenance additionally encompasses using branching for version control, such as rolling back the design version to a previous iteration, branching from an older design, or storing design history. Regarding the use of branching for version control, one user explained, “I use merging/branching to save a version of my base part on the main branch, then create branches for additional features that I want to add to them. I try to make my branches precise and clean so that when they merge back in there are no issues” (OS15). When describing how branching is adopted in their workflow, another user wrote, “the easiest way to get some benefit from the branching system is to use it as a fancy undo, redo system. Whenever you are particularly happy with the state of a document-in-progress, save a new version. If you hit a dead-end in your design, open up the version control and branch off the last saved version that has the state you want to retry from. Rinse and repeat” (OS16). Of the 11 use cases, branching for Production Version Maintenance was the most frequently discussed, mentioned in 28.4% of forum threads. Configuration Configurations are used to allow design parameters (e.g., component dimensions, quantity, and orientation of components) to be modified within a single design file to manage families of models, inclusive of parts and assemblies. Creating and managing configurations is commonly supported by CAD platforms and is achievable without the use of branching <cit.>. Yet, Configuration branching was frequently mentioned in the forum threads as a workaround that CAD users have developed in place of using the existing configuration functionality present in CAD platforms. Designers use Configuration branches for many instances, such as to create two configuration branches of a design for two separate clients, or create a configuration to reflect each step in the manufacturing and machining process (e.g., investment casting configuration, then post-machined configuration). When choosing between these two methods of configuration control, there are trade-offs; as written in a forum thread, “The question is how many variations you have. If you only have a few, then branching/merging is a decent workflow. If you have 100’s of variations, then a table-driven approach seems more appropriate. In SolidWorks, this is called ‘table driven configurations”' (OS16). Using table-driven automatically generated configurations is convenient when creating many permutations with few parameter changes. On the other hand, branching is effective for fewer, more exploratory design configurations, where many design parameters and features must be modified. Configuration branching was mentioned in 12.5% of forum threads. Every major CAD platform supports configuration management without using branching; nevertheless, the frequency of threads discussing branching to serve the purpose of configurations suggests that for certain applications, designers consider branching to be more effective or efficient for creating configurations. Modularization Modularization branching encompasses using branches to save different sub-systems or modules that can be combined through merging or assemblies to form a variety of different completed designs. Modularization branching is frequently used for large and complex designs – especially to coordinate interdisciplinary (typically, mechanical and electrical) designs that involve multiple interacting subassemblies with a high number of variable parts – to allow for variability in module and component choice. One forum thread that described projects using Modularization branching explained the scale that would warrant such practice: “My first couple of efforts were extremely complicated designs. One of them has some 50 components, another design has about two dozen components, but average about 20 to 30 bodies each” (AD20). However, smaller, and simpler designs (comprised of few parts) rely less on Modularization, because the simplicity of the design does not warrant dividing into modules – Configuration would be more suitable to allow for parameter variability. In the relevant forum posts, branching for Modularization (6.7%) was mentioned nearly half as often as Configuration (12.5%), despite their similar objectives. However, considering that modularization is intended for models with high complexity and degree of variability between variations, the results suggest that simpler models are more common, or that branching tools are less frequently applied to highly complex and variable models. New Product Variant New Product Variant encompasses using branching to build the basis for a new/separate design based on the original design. The high count (20.3%) of posts relating to New Product Variant is reflective of the popularity of the practice of reusing designs (which saves design time, allows for component interchangeability, and ensures a product's quality and reliability), and suggests that mechanical design best practices are actively employed by designers that use branching <cit.>. Creating a new product variant is analogous to “hard forking” in software development, whereby developers copy a repository with the intent to split off a new independent design that will not be later merged back to the original repository <cit.>. In CAD, designers use branching functionality to create new independent projects, even though hard forking and branching fulfill different development objectives. A further understanding of New Product Variant branching in family-based hardware development may be informed by software literature regarding software product-line engineering <cit.>. §.§.§ Risk Isolation Risk Isolation was adapted as a top-level grouping of branching use cases from software to hardware, encompassing all instances when branches are created to isolate unapproved or unverified file changes from the main branch of the design. Risk Isolation additionally includes the use of branching to iterate new features and prepare CAD files for testing, as these are processes performed on files outside of the main branch. Risk Isolation branching can be thought of as a cooperation tool because it enables designers to collaborate on shared tasks through a separate, yet linked workspace. Our taxonomy contains three use cases for Risk Isolation: Error Repair, Testing/Simulation, and Experimentation. Comparing Risk Isolation branches in software and hardware, Error Repair is analogous to Bug Fix, Testing/Simulation resembles Testing branching, and Experimentation serves a similar but nonequivalent purpose as Experimental/Prototype branching. Of the three groupings, Risk Isolation use cases are the least frequently discussed, present in 15.3% of all relevant posts. Error Repair Branching for Error Repair in CAD is the process of creating a branch to fix a design error in a CAD file, with the intention of merging the file to the main design branch once the fix is complete, analogous to Bug Fix branching in software design. Branching for error repair is performed to allow other design operations to take place on a CAD file while errors are being fixed, allowing the design workflow to continue uninterrupted during error repair. Error repair branching is typically performed at the project/assembly level, where the entire file is branched to fix an error. Regarding how to implement a branching approach to solve design errors, one forum user wrote: “if you make a change that breaks [the model], [...] find a spot on the history tree where it was working, version and branch from here and take a look at what's different” (OS16). Despite being one of the most frequent uses of branching in software, branching to isolate error repairs from the main branch is uncommon in hardware design, mentioned in only 1.0% of threads. This may be due to the difficulty or impossibility of selectively merging changes and error fixes to the main branch in CAD platforms, greatly reducing the utility of branching for error fixes. We further discuss this discrepancy between software and hardware branching for Error Repair in Section <ref>. Testing/Simulation Branches for Testing/Simulation are created to run simulations, motion studies, and other CAE (computer-aided engineering) tests on a CAD model. Testing branches are used in two ways: (1) to run many different tests on the same design, or (2) to run the same test on multiple variations of a design and compare the results. Using branches to run different tests on the same model can be superior to other existing practices (e.g., creating many clones of the design or running the tests sequentially and resetting the model between tests) because branches can contain all of the tests within a single document and as put by one of the post commenters, “the branching method is much less `weight' than the full document copy method” (OS19). One post author expressed a need to run an identical heat transfer simulation on an assembly with five different solid materials specified for one of the parts, while all other parts, parameters, and boundary conditions are held constant. By creating a branch for each of the five assembly variations, the simulations can be run simultaneously, and the designer can easily switch between branches to view and compare the results of each simulation. Branching for Testing/Simulation was mentioned in 1.9% of threads and is thus one of the least mentioned use cases. Experimentation Experimentation branching involves the creation of a branch to serve as a “sandbox” for a designer to explore an alternative design, design and iterate new features, improve the design, or compare and select new features for inclusion in the production design. In the product design and development process, several candidate models are created and explored concurrently to allow for comparison and enable the selection of a single optimal design. Regardless of whether branching tools are available on the CAD platform, hardware designers invariably create and compare several candidate designs, but branching additionally allows for the light-weight creation of each candidate design in a separate branch, and for streamlined comparison and integration of the selected design to the main branch. One forum author illustrates: “being able to share variations of a design and have them available if you need to change directions [...] beats manual copies and renaming or playing with references. Not all users understand file management well enough to keep it straight, but branch and merge is more controlled and doesn't require that level of know how” (SW22). Discussion of Experimentation branching is present in 12.4% of relevant forum posts. Experimentation is identified by CAD literature and CAD branching guides as one of the expected use cases of CAD branching; branching functionality in contemporary CAD platforms is explicitly designed to support branching for iteration and experimentation <cit.>. §.§.§ Designer Support Designer Support use cases are intended to serve organizational or non-technical needs, and were found to predominately facilitate communication between designers and other stakeholders. Branching for Designer Support was frequently discussed in the forums, mentioned in 38.6% of relevant posts. Through mining online forums, we have found four branching use cases for Designer Support: Work Division, Demonstration, Documentation, and Collaborative Troubleshooting. Software and hardware share Work Division branching as a common use case, as well as File Sharing, which is similar to Demonstration in hardware design. Work Division Branches for Work Division are used to divide modelling work between teams or individuals, so multiple designers can work on the same design concurrently. Branches provide each designer with their own individual workspace so that they can work without being disturbed by potentially distracting actions from collaborators, like hiding/showing parts, suppressing/un-suppressing features, or moving parts out of view. Using branches, collaborators can also access a CAD file without needing for the file to be “checked-in” to a central version control system. In addition to facilitating concurrent design, Work Division branching is also used to coordinate a design's workflow – for example, if the components of a product must be developed by multiple teams, branches are used to split and route design work between the teams. An example of this is provided in a forum thread: “User 1 creates a branch and adds a hole. User 2 creates a branch and modifies the size of a cutout. Each branch gets merged back into the main workspace and updates the model at different times” (CF21). In this regard, Work Division branching is frequently used in combination with other branching uses, like Experimentation and Modularization, whereby each collaborator can branch to work on a different part or sub-assembly of the model or create their own alternative design. Work Division branching was the third most mentioned use case (16.1% of threads). A specific use case within Work Division branching is to support crowdsourced mechanical design. Crowdsource branches are specifically used to allow a large group of, often geographically distributed, participants to contribute to the same design through the internet. Crowdsourced CAD projects typically rely on freelance designers, hobbyists, or general CAD enthusiasts to develop and maintain <cit.>. We elaborate on the use of branching in crowdsource mechanical design in Section <ref>. Demonstration Demonstration branching is used to share and present a design to stakeholders. While related to Work Division branching, the purpose of Demonstration branching is to support design communication, whereas Work Division branching is for coordinating design tasks and enabling parallel work. Throughout the development process, CAD models must frequently be accessed by many stakeholders, such as vendors, clients, external collaborators, manufacturers, or suppliers, for a variety of reasons, such as to provide feedback or approval, estimate a price quote, or receive design updates. Branching allows the designer to exact fine control over how the model is demonstrated; for a vendor, a designer may only want to share the parts of an assembly that they would like to be quoted and hide all other parts; for a client, protecting the intellectual property of a design is crucial, so a designer may create a branch to provide a “clean” version of the design that hides the design history (and parametric data necessary to recreate the model). In any case, branching also allows the stakeholder to view, manipulate, or annotate the design without disturbing the designer working on the main design branch. Using branching can also be preferred over other methods of sharing CAD designs (e.g., email) because it preserves “an audit trail of what model(s) were sent to the customer, at what Revision/Version” (SW19). Demonstration is a frequently mentioned use case, appearing in 15.7% of relevant forum posts even though it was not a branching use case that appeared in software literature. So, the use of branching in this context implies that branching permits, at least for some designers, better or easier model demonstration to stakeholders than existing tools in CAD. Documentation Branching for Documentation involves attaching and storing design documentation to a CAD file within a branch to improve the traceability of related files. From the forum threads, it was found that Documentation branches are used to house a variety of files, “from quotes to drawings to engineering documents, pictures, test results, user manuals, purchases, invoices, etc.” (SW16). Documentation branching is a workaround (similar to Configuration and New Product Variant branching), as using branches to store design documentation is not an intended use case of branching and merging mechanisms, as far as we know. Documentation branching was not a frequently mentioned use case, present in 2.4% of threads. The low discussion of Documentation could be attributed to the fact that branching tools were not designed to be used for this purpose <cit.>, thus designers are not aware of this workaround. Collaborative Troubleshooting Collaborative Troubleshooting entails creating a branch with the intention of helping to improve or troubleshoot another person’s design. From the forums, we found that posts created to request assistance (e.g., deciding which type of mate to use, creating a new feature, or solving a design error) would often contain a shared link to the cloud-based model, in order to facilitate the communication of the design problem or error. With access to the post author’s design, commenters were able to view the complete model in its full context, as well as branch the design to propose direct and traceable modifications to the CAD model, without having to recreate the design problem on their personal workstation. The post's original author can then decide whether to merge or discard the changes. Naturally, finding this use case could be an implication of our methodology of mining and analyzing online forum posts, where some users go to seek help. However, branching to share knowledge has practical applications in industry hardware design teams. For example, a senior designer in a design firm can use branching to help troubleshoot a junior designer’s CAD model. Collaborative Troubleshooting branching was present in 4.4% of forum threads. In all instances, the use case was mentioned in the comments of the thread, where a commenter branched the post author's design in response to their request for help. An example of such a response is: “Can you please either make your document public and post a link here, so we can take a look?” (OS16). §.§ Shortcomings of Current CAD Branching Tools (RQ2) The data mined from online forums provide important insights on why designers use branches, but also contain rich user discussions, in the form of criticisms and complaints, that are pertinent to improving future generations of CAD branching tools. In this section, we explore such user feedback and functional shortcomings, with the aim of answering RQ2 and proposing improvements to branching and merging functionality to better support collaborative hardware design. Of the total 719 relevant forum threads, we found 221 threads that mentioned a shortcoming of current branching functionalities. The prevalence of these shortcomings in the forums is shown in Figure <ref>. It should be noted that we only present the most commonly mentioned shortcomings; shortcomings that were uncommon or only described minor issues were tagged as “Other”. §.§.§ Product Line Management Across the analyzed forum threads, three main shortcomings of CAD branching functionality for Product Line Management were identified: (1) poor visualization of branch history, (2) lack of support for New Product Variant branching, and (3) lack of ability to clean design history. The most frequently mentioned shortcoming was that among those limited CAD platforms with existing branching functionality, there exists poor support of visualization and navigation of a model’s branching history, which can make it difficult for a designer to access and edit previous versions of a model. In software development branching tools (e.g., Git), the branching history of a codeline is available in the form of a commit history visualization, displaying branches and merges over time relative to a selected branch, shown in Figure <ref>a <cit.>. CAD users recognize the value of this functionality; several forum posts directly cite Git-style branching and commit history as a feature that would improve the utility of branching in CAD software. Although Onshape and briefly, Fusion360, have implemented a similar graphical user interface for designers to visualize a design’s branching history and activity (Figure <ref>b), navigating the graph is cumbersome and does not scale when the number of branches becomes large, and the commit history grows longer. The comment below illustrates: I’ve always had some sort of mental block with the Versions And History tree. The animation displayed for the SCROLLING VERSION GRAPH above is like a personal nightmare, complete with a sensation of vertigo while viewing. Usually I just avoid the tree and pretend it doesn’t exist, surviving on aptly-named tabs and documents. But that scrolling tree animation can’t be ignored: It jumps off the screen almost. I hope to comprehend this tree and its long vine-like branches eventually (OS16). As shown in Figure <ref>, this challenge of poor visualization and navigation of commit history is present in both CAD and software development, where it is difficult for contributors to maintain an overview of forking/branching activity, and this lack of awareness problem has not yet been solved in either field <cit.>. The second major improvement request related to Product Line Management is increased support for New Product Variant branching. CAD designers commonly branch out an existing design to build the basis of a new, separate product variant, with no intention of merging back the variant. To support the creation of new product variants, designers need the ability to specify design aspects they would like to be carried over or discarded, such as comments, notes, or design history (versions and branches). Presently, branching to create a New Product Variant only preserves the entire geometry of the model, and purges comments, notes, and design history. Finally, CAD users have expressed a desire for a feature to prune the design history (versions or branches) of a CAD file in order to maintain a “clean” design, similar to the git-rebase[https://git-scm.com/docs/git-rebase] feature. Predominantly, designers wish to use this feature to suppress Risk Isolation branches, since these branches are often created for ad hoc reasons (e.g., fixing a design error or experimenting with a design change). As written in a forum thread: The notion of cleaning up a timeline reminds me very much of preparing a patch series in Git, in software development. I will often commit away until I’m happy with my changes, then rewrite the changes in a clean branch, creating a more logical commit series. That helps reviewers read the commits in order, which are now more like a story. Once they have made suggestions, I can often rebase them into that logical commit series and maintain the order for future readers (AD16). §.§.§ Risk Isolation A frequent complaint of existing CAD branching tools is the lack of granularity in branching. Current tools only support branching of an entire project file, but CAD users expressed the need to branch just a subset of the project, like a single part file or subassembly, similar to the Git shallow clone[https://github.blog/2020-12-21-get-up-to-speed-with-partial-clone-and-shallow-clone/] feature. The need for finer granularity is particularly relevant for Risk Isolation tasks because designers often want to experiment with a change or error repair of a specific component of the larger design and would like to do so without branching the entire project unnecessarily. Although current tools lack granularity in branching, selective merging (or “cherry-picking”[https://git-scm.com/docs/git-cherry-pick]) at the part level has been introduced to CAD in 2022 <cit.>, whereby a designer can choose to include or exclude changes from the source branch for each part in an assembly. Still, many forum posts discussed the need for even finer branching and merging granularity (e.g., at the feature level), such as for Experimentation branching, where several candidate designs are being developed in parallel and designers want to merge a specific set of features of one candidate design (e.g., design A) to another (e.g., design B), while still maintaining other features of candidate design B. §.§.§ Designer Support In terms of the deficiencies of branching for Designer Support, CAD users emphasized the need for more specific access permissions, which pertains to both Demonstration and Work Division branching. Since designs are shared for many reasons to many different stakeholders, it is necessary for designers to have tight control over the access permissions of shared designs. Forum participants discussed the ability to configure a variety of access levels, such as view only, copy only, edit, ownership, access to linked documents, and even automatic expiration dates for access privileges. Similarly, CAD users require the ability to assign branch- or version-specific permissions to stakeholders to protect intellectual property or prevent unwanted/unauthorized model edits. Currently, access permissions can only be set at the project level, and customized version control is needed to support CAD collaboration. § DISCUSSION §.§ Hardware Versus Software Development Branching We begin our discussion with a comparison of hardware and software development, and how the respective characteristics of these related yet distinct fields impact the design and use of branching technology. §.§.§ Alternative Designs Our study found evidence that CAD designers frequently discuss using branches to create and develop several alternative designs in parallel, with the intention of selecting one to put into production and discarding the rest, which we call Experimentation branching. Typically in software development, the developed code is continually refactored, enhanced, and improved, but in the hardware development process, generating alternative designs to ultimately choose one is standard practice <cit.>. A possible explanation for this is that developing traditional software products is a fundamentally different process from developing hardware products. For physical products, 40% of the development cost is attributed to physical prototyping and manufacturing <cit.>. Due to the significance of these two phases, a great deal of development effort is spent ensuring that the optimal design is created before physical production, as it is much harder to change a physical product after it is released <cit.>. On the contrary, improving a software product after release is comparatively easier and faster, since a developer can still branch the codebase to deliver patches to the end user in order to fix bugs or add/remove a feature with the support of continuous integration, continuous delivery, and continuous deployment (known as CI/CD <cit.>), thus it is less vital that the initial released design is the best design. While it is common for software developers to iteratively improve a single design, other programming tasks (e.g., data science, building machine-learning models) involve experimentation with a variety of alternative designs, which is the practice of “exploratory programming” <cit.>. To support exploratory programming for data scientists, researchers have proposed different file structures to effectively express, compare, and visualize design alternatives, such as a side-by-side layout of alternative call paths within a single computational notebook <cit.>. Although the new structure was helpful for design comparison, Weinman et al. also highlighted that more work is needed to make the interface more sophisticated and suitable for complex, real-world applications that may involve many more alternatives. This view is shared by Kery et al., who writes that there is currently a lack of tool support for experimentation, including a lack of support for recording and sensemaking of exploration history and a lack of support collaborative exploration <cit.>. Another study by Kery et al. found that branching is seldom used for alternative design exploration in software contexts <cit.>, but as our findings have shown, experimentation branching is common among CAD designers. Evidently, tooling support for comparing alternative designs is an evolving challenge in both software and hardware development, and future studies should investigate how best to support design exploration. §.§.§ Branching for Risk Isolation An unanticipated finding of this study was that branching to fix design errors is not frequently performed or desired by CAD designers, with discussion of Error Repair branching present in only 1.0% of the relevant forum threads. This is contrary to studies of software development branching, where bug fixing is found to be one of the most common uses of branching <cit.>. There are several possible explanations for this result. For one, it is possible that errors are fixed directly in the production design version, as they are of sufficiently high priority to take over the entire design evolution – in which case, this would apply to Production Version Maintenance branching. Another potential cause is the multiple levels of extensive review that engineering firms require for CAD models before their inclusion into the main assembly file, meaning that design errors are identified and rectified in the initial component design and testing phases <cit.>. Finally, the infrequent usage of branching to fix design errors could be a consequence of CAD software’s lack of finer branching granularity (discussed in Section <ref>) which may prevent designers from being able to seamlessly branch and fix an error within an individual component, then merge the fixed component back to the main assembly. One of the new branching use cases found in our study is Testing/Simulation branching, which has not been previously mentioned in CAD literature. Although Testing is a known use of branching in software development <cit.>, testing hardware products is a more complex undertaking, as hardware products must be tested for multiple dimensions (e.g., thermal, fluid flow, materials, finite element analysis) that are not present in software design. A fundamental distinction is that in the software field, the developed code is the final released product. For hardware development, however, the CAD model is a representation of the product during development, and not the end product itself. Thus, running simulations and testing on a CAD model is an estimation of how the physical product will perform in real life, but the only way to definitively evaluate a product’s performance is to physically build and test it – which is a significantly costly and time-consuming process <cit.>. Given the importance of testing designs before fabrication and the utility of branching for this purpose, CAD users may find Testing/Simulation branching to be a useful introduction of branching to their design practice. §.§ Branching as a Tool for Collaboration In this work, we introduce a taxonomy that categorizes CAD branching use cases into the high-level groups of Product Line Management, Risk Isolation, and Designer Support. Through studying these groupings and their use cases for branching, we are advancing a fundamental understanding of CAD collaboration, as defined as coordination, cooperation, and communication (i.e., the 3C model of collaboration <cit.>). In CSCW literature, coordination is defined as “the management of people, their activities and resources” to facilitate cooperation and communication <cit.>. The need to coordinate is extremely critical to the CAD design process; many individuals and teams share potentially 1000s of CAD files, and all prior file versions must be carefully tracked and managed <cit.>. With the ability to branch, CAD users can improve coordination, via: organizing a design's evolution (Production Version Maintenance branching); coordinating shared and individual tasks (Work Division branching); splitting a large task into smaller tasks (Modularization); and initiating a new goal to work towards (New Product Variant). Given that branching has the potential for improving coordination in CAD collaboration, we envision that branching is an important feature that should be brought to the forefront for not only CAD designers, but for other stakeholders, such as project managers. Managers may find that a branch-based workflow can help organize personnel, timelines, resources, and tasks for product development projects. Similarly, branches can enable cooperation (defined as “the joint production of members of a group within a shared space” <cit.>) in CAD work. With branching, designers can work in separate branches or “copies” of a CAD file concurrently, which allows tasks like Error Repair, Testing/Simulation, and Experimentation to occur without impeding others who are also working on the same model in a parallel branch. Finally, our findings support that branching can help CAD users with the third aspect to collaboration, communication (defined as “the exchange of messages and information among people” <cit.>). For example, creating a branch for Demonstration allows a CAD model to act as a shared artifact between individuals that can transfer a variety of different information about the design (e.g., geometry, material, size, construction sequence). CAD models are considered as virtual prototypes that have the ability to represent designs in varying levels of fidelity, and they are essential to effective communication throughout the product design stages <cit.>. Collaborative Troubleshooting branching helps to make a design error common among individuals, which allows them to achieve a shared understanding of the problem <cit.>. To facilitate asynchronous communication in the CAD workflow, Documentation branching is a way to preserve design-relevant information, such that it is accessible to other stakeholders in later stages of the development lifecycle. Our results provide a framework and foundation for future work to advance theory and practice in branching of CAD artifacts. However, our work also has practical implications for the broader CSCW community. The mechanisms of branching and merging are not only used by software and hardware developers; we conjecture that the lessons learned in our study about the use cases for branching are highly applicable to other fields, particularly in graphical design, such as UX (user experience) design. Virtually any collaborative work field requires the management of prior versions of artifacts, co-development of intellectual goods, and the “divide and conquer” strategy enabled by branching and merging <cit.>. Similar to our study, researchers in the CSCW community recognize the collaboration implications of branching and have made recent contributions regarding the application of branch-based version control to Scientific Workflow Management Systems (SWfMSs) <cit.>, open collaborative writing <cit.>, and creative practice <cit.>. With these and future insights, we hope to broaden the applicability and understanding of version control branching to improve collaboration efficiency across all CSCW domains. §.§ Supporting Crowdsourced and Open-Source Hardware Development Our study found that CAD users are interested in using branching technology to support crowdsourced mechanical design. The term “Crowdsourcing” is defined as “the act of a company or institution taking a function once performed by employees and outsourcing it to an undefined (and generally large) network of people in the form of an open call” <cit.>. Crowdsourcing has become a popular workflow in many design-centric CSCW domains, such as collaborative writing <cit.>, web design <cit.>, graphic design <cit.>, and software development <cit.>. In software development, crowdsource branching is used as a way to easily delegate work amongst distributed collaborators which can reduce a product’s time to market, reduce the cost of development, leverage experts in the field, democratize and lower the barriers to participation, and offer invaluable learning opportunities for contributors <cit.>. The world of crowdsourced hardware development is much less mature than crowdsourced software development <cit.>; yet our findings confirm that there is a growing interest in this area. In the forums, designers discussed recreational crowd design projects, such as a public CAD model of a virtual chess game that could be played when players create branches to move the chess pieces. Other, more formal products designed through crowdsourcing are intended to be put to production to solve a legitimate need. An example is Enabling the Future, an online global community of volunteers that uses crowdsourcing with Fusion360 to design and develop low-cost 3D-printed prosthetic hands for underserved populations <cit.>. When the scope of crowdsourcing is broadened, this is called “open-source”. Open-source is a decentralized and collaborative method of development that not only allows for open contribution from an undefined and large network of people, but also gives the entire open-source community the freedom to study, use, modify, and share the product, maximizing the project’s beneficiaries <cit.>. The field of software development has been revolutionized by open-source (also known as open-source software or OSS) – creating world-leading products like Linux and Mozilla FireFox – and this was largely made possible by the introduction of branching and merging functionality to distributed version control systems (DVCS) <cit.>. The concept of open-source has also been applied to hardware development (OSHW), whereby the developed product is a tangible artifact that is free for the public to: (1) study (i.e., the right to access sufficient information to understand how the product works and was designed), (2) modify (i.e., the right to edit the product documentation and to create variants of the product), (3) make (i.e., the right to physically fabricate and manufacture the product), and (4) distribute (i.e., the right to share or sell the physical product or associated documentation) <cit.>. OSHW is rising in popularity – attracting the attention of researchers, educators, and practitioners alike <cit.>. However, the general sentiment towards OSHW is that while promising, the practical applications are presently limited <cit.>. The organizational challenges for adopting open-source in a hardware context are well-known, particularly with regards to protecting the intellectual property of designs <cit.>, and establishing legal and regulatory frameworks to support open use, modification, and distribution of designs (which is more complicated than OSS due to the complex nature of hardware artifacts that may need to be described through 2D or 3D schematics, CAD files, bill of materials, or assembly instructions, instead of solely source code text) <cit.>. However, even beyond organizational challenges, the expansion of OSHW is further limited by the shortcomings of current branching and merging tools (e.g., the lack of branching overview, lack of specific support for hard forking, inability to cherry-pick merges, and poor selection of access permissions, as we presented in Section <ref>) and the lack of public, online platforms with sophisticated DVCS to support open-source CAD to the same extent as OSS <cit.>. In recent years, platforms for OSHW have emerged, such as Thingiverse[https://www.thingiverse.com/] and GrabCAD[https://grabcad.com/library] which play a significant role in “Maker culture.” This culture is an extension of Do-It-Yourself culture: a community of people interested in designing and creating physical objects. Makers often use CAD software and 3D printing <cit.>, as it allows them to access and use CAD software and CAD models for free and to modify and share designs with the community <cit.>. Yet, while these platforms are sometimes viewed as the open-source version of CAD, they lack advanced tracking of file versions and dependencies (e.g., assembly relationships, hard forks) <cit.>, leading to poor traceability, confusion caused by multiple versions of the same design, and low reuse of models <cit.>. In a previous study by Bonoisin et al. that mined online hardware repositories, it was found that within a distributed hardware design project, there was significantly less collaboration on CAD files compared to other design-related files (e.g., documentation, images), suggesting that existing OSHW features are not sufficient to support the nuanced version control needed for open CAD collaboration <cit.>. Although explicit discussion of branching for OSHW was not found in the forums, we conjecture that the growing interest in crowdsourcing hardware development is a promising step towards the possibility of OSHW. However, to accelerate a widespread and large-scale OSHW movement, the development of better branching functionalities and open-source CAD infrastructure is necessary. §.§ Limitations Here, we discuss the limitations of our study, first with respect to external validity, followed by internal validity. External Validity The first limitation to our work is that we collected data from English-language forum sites, so our findings of branching use cases and design shortcomings may not be generalizable to the global CAD user community. This lack of representation could have an impact on our findings of branching shortcomings (Section <ref>), especially those that pertain to user experience and user interface; research has shown that cultural compatibility is an important factor that influences a user's design preferences and interaction with a software system <cit.>. Furthermore, we recognize that there may be a degree of bias in results derived from CAD platform-specific forums (in our case, Autodesk, Onshape, and SolidWorks), given that these sites are moderated by company employees who have their own incentives and biases. To address this, we also included non-platform specific forums (CAD Forum and Eng-Tips) in our data collection, where user feedback is more likely to be impartial. Lastly, there may be other valid use cases of CAD branching that were not found by us in the forums, either due to a limitation of our keyword search criteria, or simply the absence of public discussion of these use cases. Thus, it is in the interest of future studies to investigate branching practices in real-life design projects, as has been done in software research <cit.>. The findings from our work serve as a promising first step in identifying and categorizing CAD branching activities. We also recognize that in this work, we do not claim that the identified use cases are best practices in CAD version control. However, with our developed taxonomy, a natural progression of this work would be to identify and develop best practices for branching in CAD, as has been similarly done in collaborative software development research <cit.>. Internal Validity Qualitative studies that rely on manual coding are susceptible to the researchers' bias. To mitigate this, two researchers coded the data with an 80/20 approach to reduce the effects of bias on our results from a single researcher. Additionally, a coding scheme was developed with detailed descriptions and specific examples for each code and sub-code to maintain consistency between the two coders. §.§ Future Work: Merging in CAD This paper aimed to explore branching intentions and purposes in hardware development. With this foundation, future work should study the related operation of merging, to advance our overall understanding of distributed version control in CAD. As previously discussed, merging was not the focus of this study because of the limitations of current CAD merging features (Sec. <ref>) and the dependence of merging on branching. Moreover, after our analysis, we found that the forum threads only minimally mentioned merging, and did not provide a comprehensive view of CAD merging practices (such as how often merges happen, how many branches are merged versus discarded, what prompts a designer to merge or discard, etc.). Nonetheless, given the importance of both branching and merging operations to software configuration management (SCM) and the need for better merging tools in CAD software, we believe that merging is a necessary area for future work. In SCM, the version control system easily highlights changes between two branches or versions of code, identifies potential merge conflicts, and then merges branches by updating the root branch with the new lines of code made in the source branch. This process for merging is possible because software artifacts consist of lines of code expressed as text – but CAD artifacts are different in structure <cit.>. Artifacts in CAD represent a physical object in 3D space using either a 3D point cloud (e.g., STEP files) or an ordered series of parametric operations that define the geometry of the object (e.g., SLDPRT files). As such, merging in CAD involves replacing an entire 3D point cloud with the new version of the point cloud (because selective merging at the feature level is not currently supported by any CAD software), or updating operations in a parametric model to implement the changed geometric features in the branch <cit.>. Merging a 3D point cloud file without the ability to selectively merge features is redundant, as the part file is effectively being overwritten by the new version. In the case of a parametric CAD file, changes made in a branch might include those that remove geometries referenced by other elements. Consequently, a merge requires rolling back all changes to the point of divergence of the branch to allow the merge to be incorporated without conflicts, otherwise elements in the root component would reference geometries that have been changed or no longer exist <cit.>. The latter situation can create broken or impossible geometries that can render the entire file un-editable, and necessitates rolling back to a file version before the merge. Compared to the relatively straightforward operation of merging changed lines of code back into the root in software development, merging in CAD is complex, error-prone, and poorly supported by existing tools. As Frazelle writes in the article, A New Era for Mechanical CAD, “today there is no way to push a CAD file to a git repo, have several people modify the file, and resolve merge conflicts. (Well, maybe it could be done, but it would be the opposite of fun.)” <cit.>. Thus, without improved functionality of the merging operation, the utility and appeal of branching will be severely limited for hardware designers. To fully realize the capabilities of CAD branching, future development needs to target merging functionality in CAD. § CONCLUSION In this work, we conducted an empirical study of practical use of branch-based version control for hardware development with CAD. Through mining data from online CAD forums – a rich and expansive source of user feedback – we gathered over 14,000 user-generated posts across five forum sites, representative of over 8,000 unique CAD users. In the end, we analyzed 719 of these posts to build an understanding of envisioned and enacted branching practices in hardware design with CAD. Our work contributes the first taxonomy of CAD branching use cases, as well as identification of actionable areas for improvement of current branching technology to better support modern product design and development. The taxonomy encompasses three high-level groupings of CAD branching use cases, Product Line Management, Risk Isolation, and Designer Support, that parallel the groupings of branching use cases in software development, the field from which branching and merging originates. Our analysis of forum posts revealed that CAD designers not only use branching to fulfill technical design requirements, but branching is additionally used to support collaborative workflows and non-technical needs, such as designer communication, cooperation, and coordination. Moreover, branching was found to serve as a workaround for design tasks like creating design configurations, modularizing complex designs, and storing design documentation. We further identified and uncovered deficiencies of existing branching tools related to poor design history navigation, branching granularity, and configuration of specific access permissions, that may contribute to the slow adoption of this new functionality. Addressing these and other branching shortcomings is a fruitful area for future work and development. Future work will continue the investigation of CAD branching and merging in the context of real-life design projects, in order to confirm the branching use cases identified in this paper as well as expand our understanding of branching application in hardware development. Our work demonstrates that there is a real user need for robust Git-style branching and merging mechanisms within the collaborative CAD field, and the availability of efficient and effective branching and merging functionality in CAD could broaden the collaborative scale of hardware product design. ACM-Reference-Format January 2023 [revised]April 2023 [accepted]July 2023
http://arxiv.org/abs/2307.01623v1	20230704101945	A controller-stopper-game with hidden controller type	[ "Andi Bodnariu", "Kristoffer Lindensjö" ]	math.PR	[ "math.PR", "math.OC" ]	Photonic bound states in the continuum governed by heating I. V. Timofeev August 1, 2023 ========================================================== Abstract: We consider a continuous time stochastic dynamic game between a stopper (Player 1, the owner of an asset yielding an income) and a controller (Player 2, the manager of the asset), where the manager is either effective or non-effective. An effective manager can choose to exert low or high effort which corresponds to a high or a low positive drift for the accumulated income of the owner with random noise in terms of Brownian motion; where high effort comes at a cost for the manager. The manager earns a salary until the game is stopped by the owner, after which also no income is earned. A non-effective manager cannot act but still receives a salary. For this game we study (Nash) equilibria using stochastic filtering methods; in particular, in equilibrium the manager controls the learning rate (regarding the manager type) of the owner. First, we consider a strong formulation of the game which requires restrictive assumptions for the admissible controls, and find an equilibrium of (double) threshold type. Second, we consider a weak formulation, where a general set of admissible controls is considered. We show that the threshold equilibrium of the strong formulation is also an equilibrium in the weak formulation. § INTRODUCTION We consider a continuous time two player stochastic game between a stopper (Player 1) and a controller (Player 2). The controlled process (X_t) is given by X_t=∫_0^t(θλ_s-c)ds+W_t, where (λ_t) is a process chosen by the controller, (W_t) is a Brownian motion, θ is an independent Bernoulli random variable with ℙ(θ=1) = 1-ℙ(θ=0) =p ∈ (0,1) indicating whether the controller is effective (or active) or not, and c > 0 is a constant. On the other, based on the observations of (X_t), the stopper selects a stopping time τ at which the game ends. For a given stopping-control strategy pair (τ,(λ_t)) the reward of the stopper is 𝒥^1(τ,(λ_t),p)=[∫_0^τ e^-rs(θλ_s-c)ds] and the reward of the controller is 𝒥^2(τ,(λ_t),p)=[∫_0^τ e^-rs(c-(λ_s-λ)^2)ds\|θ=1], where r> is a constant (discount rate). The control process values are restricted to take one of two constants {λ̅,λ} at each time t, where we assume that λ̅>λ > c > 0. The model is further specified in Sections <ref> and <ref> where we also define notions of (Nash) equilibria corresponding to both players wishing to maximize their respective rewards. The interpretation is that (X_t) is the accumulated income of Player 1 who wants to maximize the discounted accumulated income by selecting a time τ at which the game ends. This income has a positive drift if θ(ω)=1 and a negative drift if θ(ω)=0; however, the outcome of θ cannot be observed by Player 1 who must make the stopping decision based only on observations of (X_t). The stopping decision will in equilibrium, as we will show, be made based on the probability that Player 1 assigns to the event {θ=1}, which is dynamically updated based on the observations of (X_t). On the other hand, an active Player 2, i.e., in case θ(ω)=1, can affect the drift of the income (X_t) by dynamically selecting the effort level, i.e., (λ_t), and thereby, as we shall see, affect the probability that Player 1 assigns to the event {θ=1}. The accumulated income of an active Player 2 is based on the constant income rate c minus the cost rate (λ_t-λ)^2 which is zero when effort is low and positive otherwise; cf. (<ref>). Moreover, an active Player 2 wants to dynamically select the effort level (λ_t) in order to maximize the discounted accumulated income of Player 2 until Player 1 ends the game. Player 2 must therefore consider the trade-off between exerting a large effort, which is costly, and a small effort, which implies no cost but decreases the probability that Player 1 assigns to {θ=1} compared to the large effort. An inactive Player 2 cannot act at all. In line with the interpretation above, our ansatz to this problem is to use stochastic filtering methods to search for an equilibrium which depends on the conditional probability of the event {θ=1} based on the observations of (X_t), which corresponds to Player 1's continuously updated belief about Player 2 being active. Indeed, we find such an equilibrium of (double) threshold type meaning that we find two thresholds 0<b_1^<b_2^<1 such that an equilibrium is that Player 2 exerts the smaller effort λ when the conditional probability of {θ=1} is above b_2^* and the larger effort λ̅ when the conditional probability is below b_2^, and Player 1 stops the game whenever the conditional probability of {θ=1} falls below b_1^; see Remark <ref> for details. We study this game in an increasing order of generality regarding the set of admissible control strategies. First using a strong formulation and second using a weak formulation of the game. The same threshold equilibrium is obtained in both formulations. * Strong formulation: In the strong formulation we admit control strategies only of Markovian type in the sense that λ_t = λ(P_t), where λ:[0,1]→{λ̅,λ} (a deterministic function) and (P_t) is defined as a process which in equilibrium coincides with the conditional probability that the stopper assigns to {θ=1}. The process (P_t) is here the strong solution to a particular stochastic differential equation; see the beginning of Section <ref> for details. In this formulation the main results are: (i) we provide a verification theorem for a double threshold equilibrium, and (ii) we prove that a double threshold equilibrium exists under certain parameter restrictions. * Weak formulation: In the weak formulation, admissible control strategies correspond to a general set of stochastic processes adapted to a filtration generated by (X_t) taking values in {λ,λ̅}. Here, however, we start by defining (X_t) as a Brownian motion and we achieve a controlled process analogous to the one in (<ref>) by means of a measure change, with which we define the reward functions and a corresponding equilibrium; see Section <ref> for details. The main result is that the double threshold equilibrium found in the strong formulation is also an equilibrium in the weak formulation, i.e., when allowing a larger set of admissible control strategies. In Section <ref> we survey related previous literature and clarify the contribution of the present paper. In Section <ref> we present stochastic filtering arguments which are relevant to the subsequent sections. The strong formulation of our game is studied in Section <ref>. In particular, the beginning of Section <ref> specifies the strong formulation further, Section <ref> contains a heuristic derivation of an equilibrium candidate, Section <ref> reports the verification result, and Section <ref> reports the equilibrium existence result. The weak formulation is studied in Section <ref>. §.§ Previous literature and contribution The problem studied in the present paper belongs to a new class of dynamic stochastic control and stopping games with the key feature being that the player's may be ghosts (cf. <cit.>) in the sense that a player does not necessarily exist, or equivalently is not activity, or not effective. This ghost feature was first studied in <cit.>, where a two-player stopping game is studied and the term ghost was introduced. In <cit.>, a controller-stopper-game where the stopper faces unknown competition in the form of a ghost controller is studied, in the context of a fraud detection application. In <cit.>, a de Finetti controller-stopper-game (of resource extraction) where the controller faces unknown competition in the form of a stopper ghost with the option to extract all the remaining resources instantaneously is studied. From a game theoretic interpretation our main contribution is that our game is a non-zero-sum game where the player objectives agree in the sense that both players would benefit if the hidden controller were revealed (in the case of an active Player 2). In this sense, both players are not exactly competing against each other, but rather aiming on finding an agreement that would benefit both. Typically, such situations are complicated since it makes existence of (non-trivial) Nash equilibria sensitive to the specific player payoffs. This stands in contrast to previously studied games of these type, see e.g., <cit.> where the profit of one player is an immediate loss for the other, which results in opposite player objectives in the sense that the controller aims at staying hidden, which is the opposite to our situation. From a technical view-point, our main contribution is twofold. First, we constrain the control process to take values in a finite set, i.e., {λ̅,λ}. This means that we can interpret the problem of the controller as an optimal switching problem without a cost for switching, implying that switching (between the two control values {λ̅,λ}) may occur infinitely often, which stands in contrast to the usual formulation of optimal switching problems; see e.g., <cit.> and the references therein. Second, we consider a weak formulation for these types of games, based on defining the state process (X_t) as a Brownian motion, and the reward functions in terms of measure changes. Then we show that the Nash equilibrium in Markovian strategies (i.e., in the strong formulation), is also a Nash equilibrium in the weak formulation. This weak approach is inspired by <cit.>, which formulates a weak approach in the study of a sequential estimation problem, where the optimizer can choose a bounded control representing the rate at which the information is received and a stopping time at which the experiment ends, in particular, <cit.> considers an optimization problem and not a game. Weak solution approaches to dynamic stochastic games have previously been considered in a variety of recent papers; see e.g., <cit.>, and <cit.> which contain surveys of the related literature. In a broader context, the problem studied in the present paper can be regarded as a controller-stopper-game under incomplete information. Controller-stopper-games were first studied for zero-sum games. In <cit.> a zero-sum game between a controller and a stopper is studied for a one-dimensional diffusion, whereas <cit.> considers the game in a multidimensional setting. In <cit.> a zero-sum game between a stopper and a controller choosing a probability measure is studied. Singular controls for zero-sum controller-stopper-games were studied in <cit.> for a one-dimensional diffusion and in <cit.> for the multidimensional setting. In <cit.> zero-sum controller-stopper-games with singular control are studied for a spectrally one-sided Lévy process. A zero-sum game between a stopper and a player controlling the jumps of the state process is studied in <cit.>. Stochastic games under asymmetric information were first considered in <cit.>, which considers a zero-sum stochastic differential game between two controllers. In <cit.> path-wise non-adaptive controls are studied for a zero-sum game between two controllers. An asymmetric information Dynkin game with a random expiry time observed by one of the players is studied in <cit.>. In <cit.> a two-player zero-sum game under asymmetric information is considered where only one player can observe the underlying Brownian motion, while the second the player only observes the strategy chosen by the first player. A zero-sum game where both players observe different processes is studied in <cit.>. Non-Markovian zero-sum games under partial information are also considered; see e.g., <cit.>. For a background regarding the interpretation of our game as a dynamic signaling game between an owner (Player 1, the stopper) and a manager (Player 2, the controller) see <cit.> and the references therein. §.§ The underlying stochastic filtering theory arguments The present section contains a brief account of the stochastic filtering arguments that underlie the analysis of the present paper. The section is included as an informal and heuristic precursor for content of the subsequent sections. A formal result in the direction of this section is Proposition <ref>. Let us first consider the perspective of the stopper. Assuming that the controller uses a control strategy (λ^_t) we obtain—using standard filtering theory; see e.g., <cit.>—that the innovations process defined by Ŵ_t=X_t+ct-∫_0^t[λ^_sθ\|ℱ^X_s]ds is a Brownian motion with respect to ((ℱ^X_t),ℙ), where (ℱ^X_t) is defined as the smallest right-continuous filtration to which (X_t) is adapted. Relying again on basic filtering theory, and arguments similar to those in <cit.>, we find that if the strategy (λ^_t) is (ℱ^X_t)-adapted—we shall later see that an equilibrium with this property can indeed be found—then the conditional probability (process) that the stopper assigns to the controller being active, i.e., ℙ(θ=1\|ℱ^X_t)=𝔼[θ\|ℱ^X_t], t ≥ 0 is given by the stochastic differential equation (SDE) dP_t=λ^_tP_t(1-P_t)dŴ_t, P_0=p. Note that the observations above rely implicitly on the assumption that the control strategy (λ^_t) is fixed in the sense that the stopper knows which process (λ^_t) that the controller uses. However, in order to verify that a candidate equilibrium strategy (λ^_t) is indeed an equilibrium strategy (cf. Definition <ref> below) we must be able to analyze what happens to an equilibrium stopping strategy—which as we shall see will be determined as a threshold time in terms of the conditional probability process—when the controller deviates from the candidate equilibrium strategy. To this end observe that if we consider an (ℱ^X_t)-adapted candidate equilibrium strategy (λ^_t) and an arbitrary admissible deviation (control) strategy (λ_t), and now define a process (P_t) to be given by dP_t =λ^_tP_t(1-P_t)(θλ_t-λ^_tP_t)dt+λ^_tP_t(1-P_t)dW_t, P_0 = p, then (P_t) depends, of course, on the equilibrium candidate (λ^_t) as well as the deviation strategy (λ_t). However, using the observations above it is also directly verified that P_t = ℙ(θ=1\|ℱ^X_t), t ≥ 0 in the special case of no deviation (i.e., with (λ^_t)=(λ_t)). In other words, (P_t) defined as in (<ref>) coincides with the conditional probability process in the case of no deviation, but it also tells us how the controller affects (P_t) in the case of deviation, and we may therefore, as we will see, use this definition of (P_t) to find an equilibrium. § STRONG FORMULATION Let (Ω,ℱ,ℙ) be a probability space supporting the standard one-dimensional Brownian motion (W_t) and the independent Bernoulli random variable θ, where we recall that ℙ(θ=1) = 1- ℙ(θ=1)=p ∈ (0,1). Observe that if both the candidate equilibrium strategy and the deviation strategy in (<ref>) are of Markov control type, specifically in the sense that λ^_t=λ^(P_t) and λ_t=λ(P_t) where λ^,λ:[0,1]→{λ̅,λ}, then P_t = P^λ,λ^_t in (<ref>) will be given by the SDE dP_t=λ^(P_t)P_t(1-P_t)(θλ(P_t)-λ^(P_t)P_t)dt+λ^(P_t)P_t(1-P_t)dW_t, P_0=p. (Depending on the context we will, to ease notation, sometimes write P_t and sometimes write P_t^λ,λ^.) In the present section we will restrict the set of admissible control strategies to be of Markov control type. Recall that Section <ref> contains a weak formulation of our game where we relax the notion of admissible strategies to be a set of general stochastic processes (taking values in {λ̅,λ}). By restricting to Markov controls we ensure that (P_t) is obtained as the strong solution to (<ref>); see Proposition <ref> in Appendix <ref>. Furthermore, using the definition of (X_t) in (<ref>) as well as (<ref>) we note that the dynamics of (P_t) can be written as dP_t =λ^(P_t)P_t(1-P_t)(c-λ^(P_t)P_t)dt+λ^(P_t)P_t(1-P_t)dX_t, P_0=p, and that (P_t) is (ℱ^X_t)-adapted. Formally, we restrict the set of admissible control strategies to be of Markov control type by identifying an admissible control strategy (λ_t) with a deterministic function λ:[0,1]→{λ̅,λ} according to λ_t=λ(P_t), where (P_t) is given by (<ref>), and where λ satisfies the conditions of Definition <ref> (which also defines the set of admissible stopping strategies). * A Markov control (deterministic function) λ:[0,1]→{λ,λ̅} is said to be an admissible control strategy if it is RCLL (right-continuous with left hand limits). The set of admissible control strategies is denoted by 𝕃. * A stopping time τ is said to be an admissible stopping strategy if it is adapted to (ℱ^X_t). The set of admissible stopping strategies is denoted by 𝕋. To clarify, a control process (λ_t) is obtained by λ_t =λ(P^λ,λ^_t), t ≥ 0, where P^λ,λ^_t = P_t, with (λ,λ^)∈𝕃^2, is given by (<ref>); i.e., a control process (λ_t) depends generally on a pair of admissible strategies (λ^,λ)∈𝕃^2 which represents the candidate equilibrium strategy λ^* and the deviation strategy λ, respectively. In line with Section <ref>, both players want to maximize their respective rewards and we define our Nash equilibrium accordingly. A pair of admissible strategies (τ^,λ^) ∈𝕋×𝕃 is a said to be a Nash equilibrium if the corresponding rewards, (<ref>)–(<ref>), satisfy 𝒥^1(τ^,(λ^(P^λ^,λ^_t)),p)≥𝒥^1(τ,(λ^(P_t^λ^,λ^)),p), 𝒥^2(τ^,(λ^(P_t^λ^,λ^)),p)≥𝒥^2(τ^,(λ(P_t^λ,λ^)),p), for any pair of deviation strategies (τ,λ) ∈𝕋×𝕃. In line with the usual interpretation of a Nash equilibrium we note that the first condition in (<ref>) implies that deviating from the equilibrium is sub-optimal for the stopper, and that the second condition implies the same for the controller. Note also that the appearance of the equilibrium control λ^ in the right hand side of the second condition in (<ref>) is due to the role that it plays for the determination of P_t=P^λ,λ^_t also when the controller deviates from the equilibrium, cf. (<ref>). A connection between our equilibrium definition and a fixed-point in a suitable best response mapping can be established. In fact we will use this connection when proving the equilibrium existence result Theorem <ref>. Let (τ^,λ^) be any given admissible strategy pair. Then we may, in line our equilibrium definition, define the (point-to-set) best response mapping of the stopper as λ^ ∈𝕃↦_τ∈𝕋𝒥^1(τ,λ^(P^λ^,λ^),p), while the (point-to-set) best response mapping of the controller is given by (τ^,λ^) ∈𝕋×𝕃↦_λ∈𝕃𝒥^2(τ^,λ(P^λ,λ^),p). It is then immediately clear that our equilibrium definition corresponds to a fixed-point in the best response mapping (τ^,λ^) ∈𝕋×𝕃↦( _τ∈𝕋𝒥^1(τ,λ(P^λ^,λ^),p), _λ∈𝕃𝒥^2(τ^,λ(P^λ,λ^),p) ). In the following result we conclude this section by establishing that (P_t) does indeed correspond to the conditional probability of an active controller, i.e., {θ=1}, in case the controller does not deviate from an equilibrium (candidate). Let λ^ ∈𝕃 be an arbitrary admissible control. Suppose λ^* = λ in (<ref>) and consider a constant 0 < T<∞. Then the solution P_t = P_t^λ^,λ^ to (<ref>) satisfies a.s., for each 0≤ t≤ T, P_t=[θ\|ℱ_t^X] and dP_t=λ^_t(P_t)P_t(1-P_t)dŴ_t, P_0=p, where Ŵ_t=X_t+ct-∫_0^tλ^(P_s)P_sds, is a Brownian motion w.r.t. ((ℱ^X_t),ℙ). This proof is similar to that of <cit.>. Define Π_t=𝔼[θ\|ℱ_t^X]. Then 𝔼[θλ^(P_t)\|ℱ_t^X]= λ^(P_t)Π_t, since (P_t) is (ℱ^X_t)-adapted. Relying on standard filtering theory (see e.g., <cit.> and arguments similar to those in the proof of <cit.>), it can now be seen, for 0≤ t≤ T, that dΠ_t=λ^(P_t)Π_t(1-Π_t)dW̅_t, where W̅_t:=X_t+ct-∫_0^tλ^(P_s)Π_sds is a Brownian motion with respect to ((ℱ^X_t),ℙ). Hence, by the definition of (X_t) in (<ref>) it is directly seen that (Π_t) satisfies the SDE dΠ_t=λ^(P_t)Π_t(1-Π_t)(θλ^(P_t)-λ^(P_t)Π_t)dt+λ^(P_t)Π_t(1-Π_t)dW_t, Π_0 = p. Recalling the definition of (P_t) in (<ref>), we observe that (P_t) and (Π_t) are both strong solutions to the same SDE in case λ^* = λ. The results follow. §.§ Searching for a threshold equilibrium The aim of the present section is to search for an equilibrium of threshold type in the sense that the equilibrium strategy pair satisfies (τ^,λ^)=(τ_b_1^,λ_b_2^) where τ_b_1^: =inf{t ≥ 0:P_t≤ b_1^}, p →λ_b_2^(p): =λ+(λ̅-λ)I_{p< b^_2}, with 0<b_1^<b_2^<1. The (double) threshold strategy pair defined by (<ref>)–(<ref>) corresponds to (i) stopping the first time that (P_t)—whose dynamics is in this case given by (<ref>) with λ(P_t) = λ^(P_t)=λ_b_2^(P_t)—falls below b^_1, and (ii) the controller using the control process (λ_b_2^(P_t)), which is equal to the small controller rate λ when P_t≥ b_2^* and the large controller rate λ̅ when P_t< b_2^. We remark that the content of this section is mainly of motivational value and that a corresponding formal result is the verification theorem reported in Section <ref>, below. §.§.§ The perspective of the controller Given a candidate equilibrium strategy λ^∈𝕃 and supposing that the stopper uses a candidate equilibrium threshold strategy of the kind (<ref>) where b_1^∈ (0,1), the controller faces the optimal control problem v(p,b_1^):= sup_λ∈𝕃𝒥^2(τ_b_1^,λ(P^λ,λ^_t),p), where we recall that (P_t) = (P_t^λ,λ^) is given by (<ref>); however, due to the conditioning on θ = 1 in the controller reward 𝒥^2 (see (<ref>)) we may here set θ = 1 in (<ref>). Indeed writing v(p)=v(p,b_1^) and relying on (<ref>) with the underlying process (P_t) in the representation (<ref>) with θ=1, we expect, using the usual dynamic programming arguments, that the optimal value v(p) satisfies λ^(p)^2p^2(1-p)^2/2v_pp(p) +(λ^(p)p(1-p)λ -λ^(p)^2p^2(1-p)) v_p(p) -r v(p)+ c-(λ-λ)^2≤ 0, for all λ∈{λ̅,λ} and p ∈ (b_1^,1), while equality should hold in case λ^(p)=λ, i.e., λ^(p)^2p^2(1-p)^2/2v_pp(p) +(λ^(p)p(1-p)λ^(p) -λ^(p)^2p^2(1-p)) v_p(p) -r v(p) + c-(λ(p)^-λ)^2 = 0. We will from now on ease the presentation by sometimes writing e.g., λ^* instead of λ^(p). By subtracting one of the two equations above from the other we obtain (λ^)^2p(1-p)v_p-(λ^-λ)^2-λ^p(1-p)λ v_p+(λ-λ)^2 ≥ 0, which is equivalent to (λ^-λ)λ^p(1-p)v_p≥ (λ^-λ)^2-(λ-λ)^2. We conclude that if (τ_b_1^, λ^) is an equilibrium then λ^= λ^(p) must satisfy (<ref>) for λ∈{λ̅,λ} and all p ∈ (b_1^,1). We now first consider the case λ^(p)=λ with the deviation λ=λ̅ (if λ=λ, then (<ref>) trivially holds). In this case (<ref>) becomes (λ-λ̅)λp(1-p)v_p≥-(λ̅-λ)^2, which is equivalent to p(1-p)v_p≤λ̅-λ/λ=:(A). Supposing that p(p-1)v_p is decreasing (this is under additional assumptions on the model parameters verified in Proposition <ref>, below) we see, for any given equilibrium strategy λ^, that if we can find a value for p that gives equality in (<ref>), then it is a lower threshold for the set of points p where λ^(p)=λ is possible; i.e., for any p smaller than this threshold we must have λ^(p)=λ̅. The interpretation is that if the stopper assigns a small probability to an active controller then the controller will control with the large rate λ̅. We now consider the case λ^(p)=λ̅ and obtain, similarly to the above, the condition (λ̅-λ)λ̅p(1-p)v_p≥(λ̅-λ)^2, which in turns gives the condition p(1-p)v_p≥λ̅-λ/λ̅=:(B). Similarly to the analysis of (A) above, this gives us an upper threshold for p where λ^(p)=λ̅ is possible; i.e., for any p exceeding this threshold we need λ^(p) = λ. In order for (<ref>) and (<ref>) to be feasible conditions we need that (A) minus (B) is non-negative, which is is directly verified. Hence, with the observations above as a motivation we will search for an equilibrium strategy λ^ of the threshold type (<ref>), where the threshold switching point b_2^* is a such that λ̅-λ/λ̅≤ b_2^(1-b_2^)v_p(b_2^)≤λ̅-λ/λ. Note that (<ref>) indicates that there may be multiple Nash equilibria, since every b_2^ satisfying (<ref>) results in an equilibrium candidate strategy for the controller. As our equilibrium controller candidate we will, however, consider a switching point b_2^* that corresponds to equality in the right hand side inequality in (<ref>). More precisely, we will search for an equilibrium controller strategy given by (<ref>), with b_2^* ∈(b_1^,1) satisfying b_2^(1-b_2^)v_p(b_2^)= λ̅-λ/λ. Let us lastly note that if the players use a threshold strategy pair (b^_1,b_2^), defined as in (<ref>)–(<ref>), with 0<b_1^<b_2^<1, then it can be shown that the corresponding value for the controller, i.e., 𝒥^2(τ_b_1^,λ_b_2^(P_t),p) = [∫_0^τ_b_1^* e^-rs(c-(λ_b_2^(P_t)-λ)^2)ds\|θ=1], where (P_t)=(P^λ_b_2^,λ_b_2^_t) is given by (<ref>) with λ(P_t)=λ^(P_t)=λ_b^_2(P_t), coincides with v:=v(p,b_1^,b_2^) defined as the solution to λ̅^2p^2(1-p)^2/2v_pp(p)+λ̅^2(1-p)^2pv_p(p)-rv(p)+c-(λ̅-λ)^2 = 0, p ∈ (b_1^,b_2^), λ^2p^2(1-p)^2/2v_pp(p)+λ^2(1-p)^2pv_p(p)-rv(p)+c = 0, p ∈ (b_2^,b_1^), v(p) =0, p ∈ [0, b_1^], v(1) =c/r, v∈𝒞(0,1) ∩𝒞^1(b_1^,1)∩ 𝒞^2((b_1^,b_2^)∪(b_2^,1)). Indeed we will in the subsequent analysis show that we may choose a stopper-controller threshold pair (b_1^,b_2^) which is an equilibrium with a controller value given by (<ref>), under certain parameter assumptions; see Theorems <ref> and <ref>. Note that (<ref>) is a boundary value problem on (b_1^,1), whose solution v has been extended to be equal to zero on [0,b_1^). The boundary conditions of (<ref>) follow immediately from the boundary cases p ≤ b_1^* and p=1, which result in immediate stopping (corresponding to no income for the controller) and never stopping (corresponding to the income rate c earned forever), respectively. §.§.§ The perspective of the stopper If the players use a threshold strategy pair (b^_1,b_2^), defined as in (<ref>)–(<ref>), with 0<b_1^<b_2^<1, then the corresponding value for the stopper is 𝒥^1(τ_b_1^,λ_b_2^(P_t),p) = [∫_0^τ_b_1^* e^-rs(θλ_b_2^(P_t)-c)ds], where (P_t)=(P^λ_b_2^,λ_b_2^_t) is given by (<ref>) with λ(P_t)=λ^(P_t)=λ_b^_2(P_t). However, since λ=λ^ we may equivalently consider the dynamics of (P_t) in the representation (<ref>); cf. Proposition <ref>. Relying again on Proposition <ref> we may moreover use that P_t=[θ\|ℱ_t^X] and iterated expectation to replace θ in the stopper reward with P_t; in other words we have the representation 𝒥^1(τ_b_1^,λ_b_2^(P_t),p) = [∫_0^τ_b_1^* e^-rs(P_t λ_b_2^(P_t)-c)ds], where (P_t) is given by (<ref>). Based on this it can be shown that the stopper reward coincides with u=u(p,b_1^,b_2^) defined as the solution to λ̅^2p^2(1-p)^2/2u_pp(p)-ru(p)+pλ̅-c =0, p ∈ (b^_1,b^_2), λ^2p^2(1-p)^2/2u_pp(p)-ru(p)+pλ-c =0, p ∈ (b^_2,1), u(p) =0, p ∈ [0,b^_1], u(1) =λ-c/r, u ∈𝒞(0,1) ∩𝒞^1(b^_1,1)∩ 𝒞^2((b^_1,b_2^)∪(b_2^,1)). Note that (<ref>) is also a boundary value problem on (b^_1,1) whose solution u has been extended to be equal to zero on [0,b^_1). The boundary conditions of (<ref>) can be interpreted using arguments similar to those in Remark <ref>. Lastly note that if (b_1^,b^_2) corresponds to an equilibrium, then it should hold that u_p(b_1^,b_1^,b_2^)=0, by the smooth fit principle of optimal stopping theory, which motivates condition (<ref>) in Theorem <ref> below. §.§ A threshold equilibrium verification theorem Here we present our first main result, which is a verification theorem based on the equilibrium conditions that were informally derived in Section <ref>. Let b^_1,b_2^∈ (0,1) satisfy b_1^<b_2^. Let u(p)=u(p,b_1^,b_2^) and v(p)=v(p,b_1^,b_2^) be solutions to the boundary value problems (<ref>) and (<ref>). Suppose that u(p) ≥ 0, p∈ [0,1],I u_p(b_1^) =0,II d/dp(p(1-p)v_p(p)) <0, p∈ (b^_1,b_2^)∪(b_2^,1),III b_2^(1-b_2^)v_p(b_2^) =λ̅-λ/λ.IV Then the stopper-controller strategy pair (τ_b^_1,λ_b_2^)∈𝕋×𝕃 corresponding to τ_b^_1=inf{t ≥ :P_t≤ b^_1} and p →λ_b_2^(p):=λ+(λ̅-λ)I_{p< b_2^} is a Nash equilibrium (Definition <ref>). Moreover, u and v correspond to the equilibrium values for the stopper and the controller respectively, i.e., u(p)=𝒥^1(τ_b^_1,λ_b_2^(P_t),p), v(p)=𝒥^2(τ_b^_1,λ_b_2^(P_t),p). (i) Recall that (<ref>) corresponds to the control process being λ_b_2^(P_t) where (P_t) is given by (<ref>) with λ(P_t)=λ^(P_t)=λ_b^_2(P_t). (ii) If a pair (b_1^,b_2^) corresponds to an equilibrium as in Theorem <ref> then the equilibrium values u(p)=u(p,b_1^,b_2^) and v(p)=v(p,b_1^,b_2^) can be determined explicitly by solving (<ref>) and (<ref>); cf. Section <ref>. (of Theorem <ref>.) For ease of exposition we write in this proof λ^=λ_b^_2 and τ_b_1^=τ_b^. Optimality of τ_b^. Note that (<ref>) and (<ref>), together with the boundary condition u(b_1^)=0, imply that u_pp(b_1^+)≥ 0. Thus, using the ODE in (<ref>), we obtain b_1^≤c/λ̅. Let n be fixed number. Relying on Proposition <ref> which implies that (P_t) solves (<ref>), as well as (<ref>) and Itô's formula we obtain for an arbitrary stopping time τ that e^-r(τ∧ n)u(P_τ∧ n) =u(p) +∫_0^τ∧ ne^-rt((λ^(P_t)^2P_t^2(1-P_t)^2/2u_pp(P_t)-ru(P_t) )1_{P_t ∉{b^_1,b^_2}}dt +∫_0^τ∧ ne^-rtλ^(P_t)P_t(1-P_t)u_p(P_t)dŴ_t, where the Itô integral is a martingale since the integrand is bounded. Now use (<ref>) and (<ref>) to see that -((λ^)^2p^2(1-p)^2/2u_pp-ru) ≥λ^p-c. Using the above together with Proposition <ref> and iterated expectation, and (<ref>), we find that u(p) = [e^-r(τ∧ n)u(P_τ∧ n)-∫_0^τ∧ ne^-rt((λ^(P_t)^2P_t^2(1-P_t)^2/2u_pp(P_t)-ru(P_t) ) 1_{P_t ∉{b^_1,b^_2}}dt] ≥[e^-r(τ∧ n)u(P_τ∧ n)+∫_0^τ∧ ne^-rt(λ^(P_t) P_t -c)dt] ≥[∫_0^τ∧ ne^-rt(λ^(P_t)𝔼[θ\|ℱ^X_t] -c)dt] =[∫_0^τ∧ ne^-rt(θλ^(P_t)-c)dt]. By sending n→∞ and relying on dominated convergence we thus obtain u(p)≥[∫_0^τe^-rt(θλ^(P_t)-c)dt]=𝒥^2(τ,λ^,p). Using similar arguments as above with τ = τ_b^* we find, by observing that we have equality in (<ref>) for p∈ (b^_1,1), that u(p) =[e^-r(τ_b^∧ n)u(P_τ_b^∧ n)] + [∫_0^τ_b^∧ ne^-rt(θλ^(P_t) -c)dt]. (Note that the equality above is trivial when p ≤ b^_1, since u(0)=0 for p ≤ b^_1). Using that u is bounded together with u(b^_1)=0 we find using dominated convergence that the first expectation above converges to zero as →∞. Hence, using dominated convergence again, we find that u(p) =[∫_0^τ_b^e^-rt(θλ^(P_t) -c)dt]= 𝒥^1(τ_b^, λ^,p). We conclude that u(p) = 𝒥^1(τ_b^, λ^,p) = sup_τ∈𝕋𝒥^1(τ, λ^,p). Optimality of λ^. The controller reward (<ref>) is conditioned on θ= 1. Hence, in order to find the optimal strategy for the controller, we consider the process (P_t) defined by (<ref>) with θ=1; in particular, if the controller selects an admissible control λ, then (P_t) is given by dP_t=λ^(P_t)P_t(1-P_t)(λ(P_t)-λ^(P_t)P_t)dt+λ^(P_t)P_t(1-P_t)dW_t. We now define the process (N_t) given by N_t=e^-r(t∧τ_b^)v(P_t∧τ_b^)+∫_0^t∧τ_b^e^-rs(c-(λ(P_s)-λ)^2)ds. Consider now an arbitrary admissible control strategy λ∈𝕃. Using Itô's formula we obtain for t≤τ_b^* that dN_t =e^-rt(1/2(λ^(P_t))^2P_t^2(1-P_t)^2v_pp(P_t)-rv(P_t))1_{P_t ≠ b^_2}dt +e^-rtλ^(P_t)P_t(1-P_t)(λ(P_t)-λ^(P_t)P_t)v_p(P_t) 1_{P_t ≠ b^_2}dt +e^-rt(c-(λ(P_t)-λ)^2) 1_{P_t ≠ b^_2}dt +e^-rtλ^(P_t)P_t(1-P_t)v_p(P_t)dW_t. Hence, (N_t) is an Itô process with a drift coefficient given, for p∈(b^_1,b^_2) ∪ (b^_2,1), by e^-rt( 1/2(λ^(p))^2p^2(1-p)^2v_pp(p) -rv(p) + λ^(p)p(1-p)(λ(p)-λ^(p)p)v_p(p) + c-(λ(p)-λ)^2). Note that it also holds, for p∈(b^_1,b^_2) ∪ (b^_2,1), that 1/2(λ^(p))^2p^2(1-p)^2v_pp(p)-rv(p) +λ^(p)p(1-p)(λ^(p)-λ^(p)p)v_p(p) + c-(λ^(p)-λ)^2 =0. To see this use (<ref>) and that λ^=λ_b_2* is given in (<ref>). Multiplying the equation above by e^-rt and subtracting the resulting left hand side (which is zero) from the drift coefficient of (N_t) yields that the drift coefficient of (N_t) can, for p∈(b^_1,b^_2) ∪ (b^_2,1), be written as e^-rt( (λ(p)-λ^(p))λ^(p)p(1-p)v_p(p) + (λ^(p)-λ)^2 - (λ(p)-λ)^2 ). With arguments similar to those in Section <ref> we find that conditions (<ref>) and (<ref>) imply that the expression above is non-positive (compare the above expression with (<ref>)), i.e., the drift of (N_t) is non-positive (regardless of the choice of λ∈𝕃). We conclude that (N_t) is a bounded process with non-positive drift. Using optional sampling we find v(p)=N_0≥𝔼[N_τ_b^∧ n\|θ=1] for any n∈. Using dominated convergence and lim_n→∞e^-r(τ_b^∧ n)v(P_τ_b^∧ n)=0 a.s. we find v(p)=N_0≥𝔼[lim_n→∞N_τ_b^∧ n\|θ=1] =[∫_0^τ_b^e^-rt(c-(λ(P_t)-λ)^2)dt\|θ=1]= 𝒥^2(τ_b^,λ,p). Repeating the same arguments with λ=λ^* we obtain that the drift of (N_t) vanishes and that v(p)=N_0=[lim_n→∞N_τ_b^∧ n\|θ=1]]=[∫_0^τ_b^e^-rt(c-(λ^(P_t)-λ))^2dt\|θ=1]= 𝒥^2(τ_b^,λ^,p). We conclude that v(p)=𝒥^2(τ_b^, λ^,p)=sup_λ∈𝕃𝒥^2(τ_b^,λ,p). §.§ Equilibrium existence The main result of this section is Theorem <ref> which reports conditions on the primitives of the model that guarantee the existence of a threshold equilibrium. The proof of this result, which is reported in Section <ref>, relies on the Poincaré-Miranda theorem and is in this sense a fixed-point type proof. In particular, the Poincaré-Miranda theorem follows from the Brouwer fixed-point theorem; cf. e.g., <cit.>. The following notation will be used throughout this section α_1(λ):=1/2 + √(8r/λ^2+1)/2, α_2(λ):=1/2-√(8r/λ^2+1)/2. In particular, we will use these to express solutions to the ODEs in (<ref>) and (<ref>). Suppose the model parameters λ,λ̅,c and r are such that -α_2(λ̅)((λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ)<λ̅-λ/λ<c/r and (λ̅-λ)^2≤ c ≤(1-α_1(λ))λ̅+α_1(λ)λ. Then there exists constants 0<b^_1<b^_2<1 such that the strategy pair (τ_b^_1,λ_b^_2) given by (<ref>) is a Nash equilibrium. Figure <ref> contains a numerical example. (i) The conditions (<ref>)–(<ref>) of Theorem <ref> can be directly examined for any given parameter specification. (ii) If we set λ̅= λ + h, then we can write these conditions as -α_2(λ̅)/rh^2+α_2(λ̅)/α_1(λ)λh <h/λ<c/r and h^2 ≤ c ≤(1-α_1(λ))h + λ. Using this observation it is easily verified that there exists, for fixed c and r, a constant h̅∈(0,∞) such that these conditions are satisfied for each h≤h̅. In other words, the conditions of Theorem <ref> hold, i.e., an equilibrium exists, whenever λ and λ̅ are sufficiently close to each other. §.§ The proof of Theorem <ref> The proof of Theorem <ref> is found in Section <ref>. It relies on the content of Sections <ref>–<ref>. §.§.§ Observations regarding Equation (<ref>) Let 0<b_1^<b_2^<1 be arbitrary constants. It can be verified that the solution v(p)=v(p,b_1^,b_2^) to (<ref>) is v(p,b_1^,b_2^) = 0, p≤ b_1^, k_1(1-p/p)^α_1(λ̅)+k_2(1-p/p)^α_2(λ̅) + c-(λ̅-λ)^2/r, b^_1 < p< b^_2, k_3(1-p/p)^α_1(λ)+k_4(1-p/p)^α_2(λ) + c/r, p≥ b^_2, where the constants k_i,i=1,..,4 can be determined by the boundary and smoothness conditions in (<ref>). (Recall that α_i(λ),i=1,2 are defined in (<ref>).) However, instead of directly determining k_i,i=1,..,4 to attain these conditions we will determine these constants in order to attain only the boundary conditions and the continuity in (<ref>) as well as the condition v_p(b^_2+,b_1^,b_2^)=λ̅-λ/b^_2(1-b^_2)λ. (The interpretation of (<ref>) is that condition (<ref>) in Theorem <ref> holds from the right.) After this we will show that b^_2 can be chosen so that (<ref>) also holds from the left (i.e., so that v satisfies all conditions of (<ref>) as well as (<ref>)); see Lemma <ref> below. First use that v(1,b_1^,b_2^)=c/r implies that k_4=0. Note also that (<ref>) implies that k_3=-λ̅-λ/α_1(λ)λ(1-b_2^/b_2^)^-α_1(λ). Using these constants we obtain from (<ref>) that v(b^_2+,b_1^,b_2^)=c/r-λ̅-λ/α_1(λ)λ. Using the condition v(b_1^,b_1^,b_2^)=0 we obtain k_2=(λ̅-λ)^2-c/r(1-b_1^/b_1^)^-α_2(λ̅)-k_1(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅) and hence, using also continuity v(b^_2-,b_1^,b_2^)=v(b^_2+,b_1^,b_2^), we obtain k_1=(λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ+c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅)/(1-b^_2/b^_2)^α_1(λ̅)-(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅). We need the following technical result in the proof of Theorem <ref> (in Section <ref>). The proof can be found in Appendix <ref>. Suppose (<ref>) holds. (i) Let v(p,b_1^,b_2^) be given by (<ref>) with the constants k_i,i=1,…,4 determined above. Then lim_b^_2 ↗ 1b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^)= -α_2(λ̅)((λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ) and lim_b^_2↘ b_1^b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^)=∞. In particular, lim_b^_2↗ 1b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^)<λ̅-λ/λ<lim_b^_2↘ b^_1 b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^). (ii) For any fixed b_1^∈ (0,1) there exists a b^_2∈(b_1^,1) such that the solution v(p) to (<ref>) satisfies (<ref>). §.§.§ Observations regarding Equation (<ref>) Let 0<b_1^<b_2^<1 be arbitrary constants. It can be verified that the solution u(p)=u(p,b_1^,b_2^) to (<ref>) is u(p)= 0, p≤ b_1^, p(c_1(1-p/p)^α_1(λ̅)+c_2(1-p/p)^α_2(λ̅))+pλ̅-c/r, b^_1 < p< b^_2, p(c_3(1-p/p)^α_1(λ)+c_4(1-p/p)^α_2(λ))+pλ-c/r, p≥ b^_2, where the constants c_i are determined by the boundary and smoothness conditions in (<ref>). The boundary condition u(1,b_1^,b_2^)=λ-c/r gives us c_4=0, while u(b_1^,b_1^,b_2^)=0 gives us c_2=c-b_1^λ̅/b_1^r(1-b_1^/b_1^)^-α_2(λ̅)-c_1(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅). Finally, the remaining two conditions give us c_3=(c_1(1-b^_2/b^_2)^α_1(λ̅)+c_2(1-b^_2/b^_2)^α_2(λ̅)+λ̅-λ/r) (1-b^_2/b^_2)^-α_1(λ) and c_1=α_1(λ)(λ̅-λ)/r(1-b^_2/b^_2)^-α_2(λ̅)+(α_2(λ̅)-α_1(λ))b_1^λ̅-c/b_1^r(1-b_1^/b_1^)^-α_2(λ̅)/(α_1(λ̅)-α_1(λ))(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)-(α_2(λ̅)-α_1(λ))(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅). We will make use of the following technical result in the proof of Theorem <ref>. The proof can be found in Appendix <ref>. Suppose (<ref>) holds. Then, for the solution to (<ref>) it holds that lim_b_1^↘ 0sup_b^_2∈(b_1^,1)u_p(b_1^+,b_1^,b_2^) = -∞, and lim_b_1^↗ 1inf_b^_2∈(b_1^,1)u_p(b_1^+,b_1^,b_2^) =∞. §.§.§ The proof (Of Theorem <ref>.) The idea of the proof is to establish existence of a threshold strategy pair (b_1^,b_2^) satisfying the conditions of Theorem <ref>. The proof consists of several parts. Here we establish existence of a pair (b_1^,b_2^) with 0<b_1^<b_2^<1 and corresponding functions u(p)=u(p,b_1^,b_2^) and v(p)=v(p,b_1^,b_2^) such that (<ref>) and (<ref>), as well as (<ref>) and (<ref>) hold. The remaining conditions are established in Appendix <ref>. In particular, (<ref>) follows from Proposition <ref> and (<ref>) follows from Proposition <ref>. Consider a pair (b_1^,b^_2) with 0<b_1^<b^_2<1 and let u(p,b_1^,b^_2) and u(p,b_1^,b^_2) be given by (<ref>) and (<ref>) with the constants c_i,k_i,i=1,...,4 determined as in Sections <ref> and <ref>. Then all we have left to do to is to show that the pair (b_1^,b_2^) can be chosen so that u_p(b^_1+,b^_1,b^_2) =0, b^_2(1-b^_2)v_p(b^_2-,b^_1,b^_2))-λ̅-λ/λ =0. To this end we introduce the notation f(b_1,b_2) = u_p (b_1+,b_1,b_2), g(b_1,b_2) = arctan(b_2(1-b_2)v_p(b_2-,b_1,b_2)-λ̅-λ/λ), for an arbitrary threshold strategy pair (b_1,b_2)∈ A:= {(x,y)∈^2:0<x<y<1}. Using Lemma <ref>, it is easy to see that there exist constants 0<b_1<b̅_1<1 such that: (i) f(b_1,b_2)<0 for all b_2>b_1, and f(b̅_1,b_2)>0 for all b_2>b̅_1, and (ii) f is continuous on the set Ã:=A∩ ([b_1,b̅_1]× [0,1]). Fix two such values b_1 and b̅_1 (arbitrarily). We can now find a continuous extension of f on the whole rectangle [b_1,b̅_1]× [0,1] by f̃(b_1,b_2)= f(b_1,b_2), (b_1,b_2)∈Ã, f(b_1,b_1), (b_1,b_2)∉Ã. We conclude that f̃ is continuous on [b_1,b̅_1]× [0,1] with the properties that f(b_1,b_2)<0 for all b_2 ∈ [0,1] and f(b̅_1,b_2)>0 for all b_2 ∈ [0,1]. Using Equation (<ref>) and (<ref>) we find a continuous extension of g on the whole rectangle [b_1,b̅_1]× [0,1] by g̃(b_1,b_2)= g(b_1,b_2), b_1<b_2, π/2, b_1≥ b_2. Based on Lemma <ref> (in particular the left hand side inequality of (<ref>)) we may now conclude that: g̃(b_1,1)<0, for any b_1 ∈ [b_1,b̅_1] and g̃(b_1,0)= π/2 >0, for any b_1 ∈ [b_1,b̅_1]. The conclusions noted for f̃ and g̃ imply that we may use the Poincaré-Miranda theorem (cf. <cit.>). In particular, it implies that there exists a pair (b^_1,b^_2)∈ [b_1,b̅_1]× [0,1] such that f̃(b^_1,b^_2)=g̃(b^_1,b^_2)=0. Moreover, using (<ref>) and g̃(b_1,1)<0 (cf. above) we obtain g̃(b_1,b_2) ≠ 0 for (b_1,b_2) ∉Ã, and hence (b^_1,b^_2)∈Ã⊂ A, i.e., 0<b^_1<b^_2<1. It is now directly seen by the definitions of f,f̃, g and g̃ that the pair (b^_1,b^_2) satisfying (<ref>) is such that also (<ref>)–(<ref>) hold and we are done. § WEAK FORMULATION The purpose of this section is to consider a more general class of admissible control strategies compared to that of the strong formulation in Section <ref>. To this end we consider here a weak formulation of our game based measure changes and Girsanov's theorem. We remark that this formulation is closely related to <cit.>, where a similar weak solution approach is used for an optimal control problem with discretionary stopping. The main finding of the present section is that the double threshold equilibrium of Theorem <ref> is a Nash equilibrium also in the weak formulation. Let (Ω, 𝒜,ℙ) be a probability space supporting a one-dimensional Brownian motion (X_t) and a Bernoulli random variable θ with ℙ(θ=1)=p∈ (0,1). Denote by (ℱ_t^X) the smallest right continuous filtration to which (X_t) is adapted. Define the terminal filtration according to ℱ^X_∞:=σ(⋃_0≤ t≤∞ℱ_t^X). Define (ℱ_t^X,θ) and ℱ_∞^X,θ analogously. * A process (λ_t) is said to be an admissible control process if it has RCLL paths, is adapted to (ℱ_t^X,θ), and takes values in {λ̅,λ}. The set of admissible control processes is denoted by 𝕃̃. * A stopping time τ is said to be an admissible stopping strategy if it is adapted to (ℱ^X_t). The set of admissible stopping strategies is denoted by 𝕋. The set of admissible stopping strategies in the weak formulation is analogous to set of admissible stopping strategies in the strong formulation. The main difference is instead that we define (X_t) as a Brownian motion in the weak formulation, whereas (X_t) is given by (<ref>) in the strong formulation. Now for any given control process (λ_t)∈𝕃̃ we define the process (W^λ_t) according to X_t = ∫_0^t(θλ_s-c)ds+W^λ_t. By Girsanov's theorem (<cit.>) there exists a measure ℙ_t^λ∼ℙ on (Ω,(ℱ_t^X,θ)), given by dℙ_t^λ/dℙ\|_ℱ_t^X,θ= exp(∫_0^t(θλ_t-c)dX_t-1/2∫_0^t(θλ_t-c)^2dt ):=Λ^λ_t, such that {W_t^λ,ℱ_t^X,θ;0≤ t≤ T} is a Brownian motion on (Ω, ℱ^X,θ_T,ℙ_T^λ) for each fixed T∈ [0,∞). Moreover, we note that (Λ^λ_t) is a martingale by Novikov's condition. Thus, the theory of the Föllmer measure gives us the existence of a measure ℙ^λ on ℱ_∞^X,θ, which satisfies ℙ^λ(A)=ℙ_t^λ(A) for every t∈[0,∞) and A∈ℱ^X,θ_t; see <cit.>, and also <cit.> and <cit.>. This allows us to give definitions of the reward functions based on measure changes. Given a strategy pair (τ,(λ_t))∈𝕋×𝕃̃ we define the payoff of the stopper as 𝒥̃^1(τ,(λ_t),p)=^ℙ^λ[∫_0^τ e^-rs(θλ_s-c)ds], and the payoff of the controller as 𝒥̃^2(τ,(λ_t),p)=^ℙ^λ[∫_0^τ e^-rs(c-(λ_s-λ)^2)ds\|θ=1]. Let us motivate Definition <ref> further. In the strong formulation (Section <ref>) we consider a fixed probability measure and define the controlled process (X_t) in terms of a control process (λ_t) and a given Brownian motion (W_t); cf. (<ref>). In the present weak formulation we instead define (X_t) as a Brownian motion, and let the control process (λ_t) imply a measure change ℙ^λ, such that W^λ defined by (<ref>) is a Brownian motion under this measure. By comparing the resulting weak formulation equation for (X_t) (i.e., (<ref>)) and the equation for (X_t) in the strong formulation (in (<ref>)) the connection between the formulations becomes clear. The Nash equilibrium is now defined in the usual way: A pair of admissible strategies (τ^,(λ^_t)) ∈𝕋×𝕃̃ is a said to be a Nash equilibrium if 𝒥̃^1(τ^,(λ^_t),p)≥𝒥̃^1(τ,(λ^_t),p), 𝒥̃^2(τ^,(λ^_t),p)≥𝒥̃^2(τ^,(λ_t),p), for any pair of deviation strategies (τ,(λ_t)) ∈𝕋×𝕃̃. In line with the strong formulation solution approach (cf. (<ref>) and (<ref>)) we define a double threshold strategy pair (τ_b_1^,λ_b_2^), where 0<b_1^<b_2^<1, by τ_b_1^:=inf{t≥ 0:P_t≤ b_1^} λ_b_2^(P_t)=λ+(λ̅-λ)I_{P_t<b_2^}, where (P_t) is (in analogy with (<ref>)) given by the SDE dP_t=λ^(P_t)P_t(1-P_t)(c-λ^(P_t)P_t)dt+λ^(P_t)P_t(1-P_t)dX_t, P_0=p. (Recalling that (X_t) is a Brownian motion we find that (<ref>) has a strong solution using analogues arguments as in the strong solution formulation; cf. Proposition <ref>). The main result of the present section is that the double threshold equilibrium investigated in the strong formulation is also an equilibrium in the weak formulation. Note that this implies that equilibrium existence in the weak formulation is guaranteed by the same parameter conditions as in Theorem <ref>. Suppose (b^_1,b^_2) with 0<b^_1<b^_2<1 are such that the conditions of Theorem <ref> hold (implying that they correspond to a double threshold equilibrium (<ref>)–(<ref>), in the strong formulation). Then, (τ_b^_1,λ_b^_2) given by (<ref>), (<ref>) and (<ref>) is an equilibrium in the weak formulation (Definition <ref>). In this proof we write λ^=λ_b_2^. For any admissible deviation strategy (λ_t) it follows from (<ref>) and (<ref>) that (P_t) has the representation dP_t=λ^(P_t)P_t(1-P_t)(θλ_t-λ^(P_t)P_t)dt+λ^(P_t)P_t(1-P_t)dW^λ_t, where we recall that (W^λ_t) is a Brownian motion under the measure ℙ^λ. We remark that (P_t) depends in this sense on both λ^* and (λ_t) when the controller deviates from the equilibrium. Note that the representation of (P_t)=(P_t^λ^,λ^) in (<ref>) is analogous to (<ref>) in the strong formulation. Moreover, it is directly seen that the value functions are the same for both formulations in the case of no deviation, i.e., 𝒥^1(τ^,λ^(P^λ^,λ^),p)=𝒥̃^1(τ^,λ^,p), 𝒥^2(τ^,λ^(P^λ^,λ^),p)=𝒥̃^2(τ^,λ^,p). Hence, u(p)=𝒥̃^1(τ^,λ^,p) and v(p)=𝒥̃^2(τ^,λ^,p), with u(p) and v(p) as in Theorem <ref>. Using this it is directly checked that the proof of Theorem <ref> can be adjusted so that it shows that (b_1^,b_2^) corresponds to an equilibrium also in the present weak formulation. Indeed, this requires only minor adjustments including that (P_t) is here given by (<ref>), and that the deviation strategies are allowed to be processes in 𝕃̃. Particularly, note that Proposition <ref> holds also in this case. Acknowledgment The authors are grateful to Erik Ekström at Uppsala University for discussions regarding games of the kind studied in the present paper, and suggestions that lead to improvements of this manuscript. § PROPERTIES OF (P_T) IN THE STRONG FORMULATION The SDE (<ref>) has a strong solution (P_t) = (P^λ,λ^_t) for any admissible pair of Markov strategies (λ^,λ) ∈𝕃^2. Consider the interval I=[ϵ,1-ϵ], for a small arbitrary constant ϵ>0. Then the diffusion coefficient is uniformly bounded away from zero in I. Thus we obtain for both cases {θ=0} and {θ=1} that: (i) a weak solution (P_t) to (<ref>) exists (cf. e.g., <cit.>), and (ii) a solution to (<ref>) is pathwise unique in I (see <cit.>). By Lemma <ref>, we obtain that (P_t) cannot reach 0 or 1 in finite time. Hence, (<ref>) admits a strong solution (P_t) by <cit.>. For any pair (λ^,λ)∈𝕃^2 it holds for (P_t) = (P^λ,λ^_t) given by (<ref>) that τ_0 :=inf{t≥ 0:P_t=0}=∞, τ_1 :=inf{t≥ 0:P_t=1}=∞. In order to prove (<ref>), it is suffices to show that τ̃_0 :=inf{t≥ 0:P̃_t=0} = ∞, where P̃_t solves the SDE dP̃_t=-λ̅^2P̃^2_t(1-P̃_t)dt+λ^(P̃_t)P̃_t(1-P̃_t)dW_t, with P̃_̃0̃=P_0=p; indeed, it follows by comparison (see <cit.>) that τ̃_0≤τ_0 a.s. (The existence of a strong solution to (<ref>) is given by arguments similar to those in the proof of Proposition <ref>.) Since λ^ is RCLL in [0,1] (and is piece-wise constant), there exists a z∈ (0,1) such that λ^(p)=λ^(z) for p∈ (0,z]. With some calculations we now obtain that the scale function of (<ref>), is for a ≤ z given by s'(a) =exp(-2∫_z^a-λ̅^2p^2(1-p)/(λ^(p))^2p^2(1-p)^2dp) = exp(-2(λ̅/λ^(z))^2∫^z_a1/(1-p)dp) =C(z)1/(1-a)^2(λ̅/λ^(z))^2, where C(z)>0, and density of the speed measure for p≤ z is given by m(p)=2/λ^(z)^2p^2(1-p)^2s'(p). Using that s'(x) is increasing for x∈(0,1) we have ∫_0^zs'(a)(∫_a^zm(p)dp)da ≥ C(z) 2/λ^(z)^2(1-z)^2s'(z)∫_0^z(∫_a^z1/p^2dp)da = ∞. Hence, τ̃_0=∞ follows from Feller's test for explosion, and (<ref>) follows. For reasons similar to the proof of the previous statement it is sufficient to prove that τ̂_1:=inf{t≥ 0:P̂_t=1} = ∞ where dP̂_t=λ̅^2P̂_t(1-P̂_t)dt+λ^(P̂_t)P̂_t(1-P̂_t)dW_t. We fix a z∈ (0,1) such that λ^(p)=λ^(z) for p ∈ [z,1). For (<ref>) and a ∈ (z,1), we have s'(a) =exp(-2∫_z^aλ̅^2p(1-p)/(λ^(p))^2p^2(1-p)^2dp) =D_1(z)(1-a/a)^2(λ̅/λ^(z))^2, where D_1(z)>0, and for p ∈ (z,1), we have m(p)=2/λ^(z)^2p^2(1-p)^2s'(p). Hence, for some positive constants D_2(z),D_3(z),D_4(z), we have ∫^1_zs'(a)(∫_z^am(p)dp)da ≥ D_2(z)∫_z^1(1-a)^2(λ̅/λ^(z))^2(∫^a_z1/(1-p)^2(λ̅/λ^(z))^2+2dp)da =D_3(z)∫_z^11/1-ada-D_4(z)=∞. Hence, τ̂_1=∞ follows by Feller's test for explosion, and (<ref>) follows. § PROOFS OF LEMMAS <REF> AND <REF> (of Lemma <ref>.) Observe that b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^) =-k_1(α_1(λ̅)(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅)) +α_2(λ̅)c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅). First we consider the limit b^_2→ 1. For the first part we have that k_1α_1(λ̅)(1-b^_2/b^_2)^α_1(λ̅) =((λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ)(1-b^_2/b^_2)^-α_2(λ̅)+c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)/(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)-(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)α_1(λ̅)(1-b^_2/b^_2)^α_1(λ̅)→ 0, as b^_2↗ 1, since α_1(λ̅),-α_2(λ̅)>0. We consider the remaining term. We obtain k_1α_2(λ̅)(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅)+α_2(λ̅)c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅) =α_2(λ̅)(λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ/(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)(1-b_1^/b_1^)^α_2(λ̅)-α_1(λ̅)-1 +α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅)( c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)/(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)(1-b_1^/b_1^)^α_2(λ̅)-α_1(λ̅)-1+c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)) = (A1)+(B1). For (A1) we obtain using α_1(λ̅)-α_2(λ̅)>0, that (A1)=α_2(λ̅)(λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ/(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)(1-b_1^/b_1^)^α_2(λ̅)-α_1(λ̅)-1→ -α_2(λ̅)((λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ), as b^_2 ↗ 1. For (B1) we have (B1) =α_2(λ̅)c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)( (1-b^_2/b^_2)^α_1(λ̅)(1-b_1^/b_1^)^α_2(λ̅)-α_1(λ̅)/(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)(1-b_1^/b_1^)^α_2(λ̅)-α_1(λ̅)-1) → 0, as b^_2 ↗ 1. Adding the limits gives us (<ref>), and using (<ref>) we thus obtain the first part of (<ref>). For the second limit we find lim_b^_2→ b_1^b^_2(1-b^_2)v_p(b^_2-,b_1^,b_2^) =α_2(λ̅)c-(λ̅-λ)^2/r+(1-b_1^/b_1^)^α_1(λ̅)(α_1(λ̅)-α_2(λ̅)) lim_b^_2→ b_1^-k_1. We note that α_1(λ̅)-α_2(λ̅)>0 and further investigate the limit by considering the denominator and numerator of k_1 separately. For the denominator of k_1 we have that (1-b^_2/b^_2)^α_1(λ̅)-(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅) =(1-b^_2/b^_2)^α_1(λ̅)(1-(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅)-α_1(λ̅))↗ 0, as b^_2↘ b_1^; to see this use e.g., that x ↦1-x/x is decreasing for x>0. For the numerator of k_1 we use (<ref>) to find (λ̅-λ)^2/r-λ̅-λ/α_1(λ)λ+c-(λ̅-λ)^2/r(1-b_1^/b_1^)^-α_2(λ̅)(1-b^_2/b^_2)^α_2(λ̅) →c/r-λ̅-λ/α_1(λ)λ>0. It follows that lim_b^_2↘ b_1^k_1=-∞ and by (<ref>) we obtain (<ref>) (from which the second part of (<ref>) follows). Hence, statement (i) has been proved. Relying on the continuity of v_p(b_2^-,b_1^,b_2^) for b_2^∈ (b_1^,1) it follows immediately from (<ref>) and the intermediate value theorem that we can choose b^_2 so that v_p(b^_2-,b_1^,b_2^)=λ̅-λ/b^_2(1-b^_2)λ. Hence, if we choose b_2^* in this way then p ↦ v(p,b_1^,b_2^) satisfies (<ref>) as well as (<ref>) and hence statement (ii) holds. We need the following technical result in the proof of Lemma <ref>. Let c_1 be given by (<ref>), then we have * c_1 >-(1-b_1^/b_1^)^-α_1(λ̅)b_1^λ̅-c/b_1^r, for b_1^≤ c/λ̅, c_1 < (α_1(λ)(λ̅-λ)/r(α_1(λ̅)-α_2(λ̅))-b_1^λ̅-c/b_1^r)(1-b_1^/b_1^)^-α_1(λ̅), for b_1^* ≥ c/λ̅. Let us prove (a) by showing that showing that c_1 is strictly decreasing in b^_2 (recall that b^_2∈ (b_1^,1)); the result then follows by taking b^_2=1 in c_1. It holds that the denominator of c_1 is positive and strictly increasing in b^_2. To see this use e.g., that α_1(λ̅)-α_1(λ)<0 and that 1-x/x is strictly decreasing for x>0, which implies that (α_1(λ̅)-α_1(λ))(1-b^_2/b^_2)^α_1(λ̅)-α_2(λ̅)-(α_2(λ̅)-α_1(λ))(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅) ≥(1-b_1^/b_1^)^α_1(λ̅)-α_2(λ̅)(α_1(λ̅)-α_2(λ̅))>0. Additionally, -α_2(λ̅)>0 and b_1^≤ c/λ̅ implies that the numerator is positive and decreasing in b^_2. We conclude that c_1 is strictly decreasing on b_2^. In order to prove (b) we write c_1=α_1(λ)(λ̅-λ)/r(1-b_2^/b_2^)^-α_2(λ̅)/D(b_1^,b_2^)+(α_2(λ̅)-α_1(λ))b_1^λ̅-c/b_1^r(1-b_1^/b_1^)^-α_2(λ̅)/D(b_1^,b_2^) where D(b_1^,b_2^) denotes the denominator of c_1. Note that b_1^* ≥ c/λ̅ implies that the second expression is non-positive. Thus, by similar arguments to (a) the result follows by taking b_2^=b_1^ in the first and b_2^=1 in the second expression. (of Lemma <ref>.) We find with some work (use e.g.,(<ref>)) that u_p(b_1^+,b_1^,b_2^) =c/b_1^r-1/1-b_1^(α_1(λ̅)c_1(1-b_1^/b_1^)^α_1(λ̅)+c_2α_2(λ̅)(1-b_1^/b_1^)^α_2(λ̅)) =c/b_1^r+α_2(λ̅)(b_1^λ̅-c)/(1-b_1^)b_1^r-α_1(λ̅)-α_2(λ̅)/1-b_1^c_1(1-b_1^/b_1^)^α_1(λ̅). Suppose b_1^≤ c/λ̅. Thus, Lemma <ref>(a) implies u_p(b_1^+,b_1^,b_2^) <c/b_1^r+α_2(λ̅)(b_1^λ̅-c)/(1-b_1^)b_1^r+(α_1(λ̅)-α_2(λ̅))(b_1^λ̅-c)/(1-b_1^)b_1^r =(1-b_1^)c-α_1(λ̅)c/(1-b_1^)b_1^r+α_1(λ̅)λ̅/(1-b_1^)r→ -∞, as b_1^↘ 0, since α_1(λ̅)>1, and the first result follows. Now suppose b_1^ ≥ c/λ̅. Then Lemma <ref>(b) implies u_p(b_1^+,b_1^,b_2^) >c/b_1^r+α_2(λ̅)(b_1^λ̅-c)/b_1^(1-b_1^)r-α_1(λ)(λ̅-λ)/(1-b_1^)r+(α_1(λ̅)-α_2(λ̅))(b_1^λ̅-c)/(1-b_1^)b_1^r =c/b_1^r+(α_1(λ̅)(λ̅-c/b_1^)-α_1(λ)(λ̅-λ))/(1-b_1^)r. Fix a sufficiently small ϵ>0. Then, α_1(λ̅)>1 implies that there exists a b̅∈ (0,1) such that α_1(λ̅)(λ̅-c/b_1^)>λ̅-c+ϵ for any b_1^∈(b̅,1). Thus, (<ref>) implies that for b_1^∈(b̅,1), we have that u_p(b_1^+,b_1^,b_2^) >c/b_1^r+(λ̅-c+ϵ-α_1(λ)(λ̅-λ))/(1-b_1^)r ≥c/b_1^r+ϵ/(1-b_1^)r→∞, for b_1^↗ 1. Hence, the second result follows. § RESULTS FOR THE PROOF OF THEOREM <REF> Throughout this section we consider the setting of the proof of Theorem <ref>. Particularly, we here consider a pair (b_1^,b_2^) such that (<ref>) and (<ref>) hold. We also rely on condition (<ref>). It holds that v(p)<c/r for p∈ (b^_1,1). Let us first prove v(p)≤ c/r by contradiction. Assume there exists a p̃∈ (b^_1,1) such that v(p̃)>c/r. Then v∈𝒞^1(b_1^,1) and v(1)=c/r implies that there exists p̂∈ (p̃,1) with v(p̂)>c/r such that v attains a local maximum at p̂. Using the ODE in (<ref>) we find v_pp(p̂-)≥ v_pp(p̂+)>0, which is a contradiction. We continue to prove v(p)< c/r by contradiction. For this purpose, assume that v(p̃)= c/r for some p̃∈(b^_1,1). Then by v≤ c/r we have that v_p(p̃)=0. We have two cases: If p̃∈(b_1^,b_2^] then the ODE (<ref>) implies that v_pp(p̃-)>0, which contradicts v≤ c/r. * If p̃∈ (b_2^,1), using the ODE we find v(p)=c/r for p∈ [b^_2,p̃] (cf. also (<ref>)). Since v∈𝒞^1(b_1^,1), we obtain v_pp(b^_2-)>0 as above, which again contradicts v≤ c/r. It holds that (a) v_p(b^_1+)>0, and (b) v_p(b^_2)>0. We will only prove the first statement, since the second statement follows using analogues arguments. Suppose that v_p(b^_1+)≤ 0. We will show that this implies that v has a local minimum below zero, i.e., there exists a point p̃∈ (b^_1,1) such that v_p(p̃)=0, v_pp(p̃+)≥ 0 and v(p̃)< 0. This contradicts the ODE in (<ref>) since c-(λ^(p̃+)-λ)^2≥ 0 (by (<ref>)). We have three cases: If v_p(b^_1+)<0, then v(1)=c/r and continuity immediately imply that v has a local minimum below zero. If v_p(b^_1+)=0,c-(λ̅-λ)^2>0, then the ODE in (<ref>) implies that v_pp(b^_1+)<0. Analogously to the first case this implies that v has a local minimum below zero. * If v_p(b^_1+)=0,c-(λ̅-λ)^2=0, then we find using the ODE that v(p)=0 for p∈ [b^_1,b^_2). Using also v∈𝒞^1(b^_1,1) and the ODE we conclude that v(b^_2)=v_p(b^_2)=0 and v_pp(b^_2+)<0. With v(1)=c/r this implies that v has a local minimum below zero. It holds that v_p(p)>0 for p∈ (b^_1,1). We show that v_p(p)>0 for p∈ (b^_1,b^_2) by contradiction. The remaining case can be proved using analogues methods. To this end, assume that p_1∈ (b^_1,b^_2) is the smallest point such that v_p(p_1)=0. We consider three cases: * If v_pp(p_1)=0, then the ODE in (<ref>) implies that v is constant on (p_1,b^_2), which is a contradiction to v∈𝒞^1(b^_1,1) and Lemma <ref>(b). * If v_pp(p_1)>0, then p_1 is a local minimum and Lemma <ref>(a) implies that p_1 cannot be the first point with v_p=0. * Consider the case v_pp(p_1)<0. Then since v_p(b^_2)>0 and v∈𝒞^2(b^_1,b^_2), we see that there must exist a second p_2∈(p_1,b^_2) such that v_p(p_2)=0 and v(p_2) is a local minimum. Let p_2 be the first such a point. Then it is easy to see, that v_pp(p_2)≥ 0 and v(p_2)<v(p_1). However, using v_p(p_1)=v_p(p_2)=0, v_pp(p_1)<0 ≤ v_pp(p_2) and the ODE we find the contradiction v(p_1)<c-(λ̅-λ)^2/r≤ v(p_2). Let f(p):=p(1-p)v_p(p). Then * f'(p_1)<0 for any p_1∈ (b^_1,b^_2)∪ (b^_2,1) satisfying rv(p_1)≤ c-(λ^(p_1)-λ)^2, * f'(b_2^-)<0, f”(p_2)>0 for any p_2∈ (b^_1,b^_2) satisfying f'(p_2)=0. Let us prove the statement in (a). With the help of the ODE, we observe that f'(p) =v_ppp(1-p)+(1-p)v_p-pv_p =2(rv-c+(λ^(p)-λ)^2)/p(1-p)(λ^)^2(p)-v_p. Hence, (a) follows from Lemma <ref>. Let us prove (b). Using, (<ref>), (<ref>), (<ref>) and v∈𝒞^1(b_1^,1), we find f'(b_2^-) =1/b_2^(1-b_2^)(-2r(λ̅-λ)/α_1(λ)λ̅^2λ+2(λ̅-λ)^2/λ̅^2-λ̅-λ/λ) ¨ <λ̅-λ/b_2^(1-b_2^)(2λ(λ̅-λ)-λ̅^̅2̅/λ̅^2λ)<0. We continue to prove (c). Using (<ref>), we find f”(p) =2rv_p/p(1-p)λ̅^2-2(rv-c+(λ̅-λ)^2)(1-2p)/p^2(1-p)^2λ̅^2-v_pp =2rv_p/p(1-p)λ̅^2-2(rv-c+(λ̅-λ)^2)(1-2p)/p^2(1-p)^2λ̅^2+2v_p/p-2(rv-c+(λ̅-λ)^2)/p^2(1-p)^2λ̅^2 =2rv_p/p(1-p)λ̅^2-4(rv-c+(λ̅-λ)^2)/p^2(1-p)λ̅^2+2v_p/p =2rv_p/p(1-p)λ̅^2-2f'(p)/p. Using f'(p_2)=0 and Lemma <ref> we thus obtain f”(p_2)=2rv_p(p_2)/p_2(1-p_2)λ̅^2>0. It holds that f(p)=p(1-p)v_p(p) is strictly decreasing for p∈ (b^_1,1). By Lemma <ref>(a), the statement holds for p∈ (b^_2,1), since v< c/r (Lemma <ref>) and λ^-λ=0 on (b^_2,1). We prove f'(p)<0 for p∈(b_1^,b^_2) by contradiction. Note that f'(b_1^+)<0, by 0=r v(b^_1)≤ c-(λ̅-λ)^2 and Lemma <ref>(a). For this purpose, let p̃∈ (b^_1,b^_2) be a point such that f'(p̃)=0, which is then a local minimum (by Lemma <ref>(c)). Hence, we obtain f'(p)≥ 0 for p∈(p̃,b_2^) (cf. Lemma <ref>(c)). This is a contradiction to Lemma <ref>(b). It holds that b^_1<c/λ̅. We prove the statement by contradiction. To this end assume b^_1 ≥ c/λ̅. Then we can use the calculations in the proof of Lemma <ref> to arrive at (<ref>), which with (<ref>) gives us 0=u_p(b^_1) >c/b_1^r+b_1^(α_1(λ̅)(λ̅-c/b_1^)-α_1(λ)(λ̅-λ))/b_1^(1-b_1^)r =(1-α_1(λ̅))c+b_1^(α_1(λ̅)λ̅-α_1(λ)(λ̅-λ)-c)/b_1^(1-b_1^)r. Related to the numerator we introduce the function f(b)=(1-α_1(λ̅))c+b(α_1(λ̅)λ̅-α_1(λ)(λ̅-λ)-c). It is easily verified that f is increasing and that f(c/λ̅)≥ 0 (use e.g., (<ref>)). This is a contradiction to u_p(b^_1)=0 and the statement follows. It holds that u(p)≥ 0 for p∈ [0,1]. We prove that u(p)> 0 for p∈ (b^_1,1] by contradiction. By definition, u(p)=0 for p≤ b^_1. Lemma <ref> establishes b^_1< c/λ̅. Thus the ODE (<ref>) and u_p(b^_1+)=0 imply that u(p) is strictly increasing and convex on {p∈(b^_1,1):ru(p̂)+c-p̂λ^(p̂)>0, ∀p̂∈ (b^_1,p) }, which is non-empty. Suppose, in order to obtain a contradiction, that p̃∈ (b^_1,1] is the smallest point such that u(p̃)=0. It can then be verified that p̃ > c/λ̅. Let us consider the case p̃<b^_2. Recall that (<ref>) holds for any c/λ̅≤ b_1^<b_2^<1. Hence, using p̃ instead of b^_1 in (<ref>) and the same reasoning that lead to (<ref>) gives u_p(p̃+; p̃, b_2^) > (1-α_1(λ̅))c+p̃(α_1(λ̅)λ̅-α_1(λ)(λ̅-λ)-c)/p̃(1-p̃)r. Note that that we have u_p(p̃+; p̃, b_2^)=u_p(p̃; b_1^, b_2^):=u_p(p̃), since both functions u(·; p̃, b_2^) and u(·; b_1^, b_2^) satisfy the ODE in (<ref>) on (p̃,1) with the same boundary conditions, in particular u(p̃; p̃, b_2^)=0 (by (<ref>)) and u(p̃; b_1^, b_2^)=0 (by the contradiction assumption). Hence, using p̃ > c/λ̅ and arguments analogous to those after (<ref>) we find that u_p(p̃)> 0. However, u_p(p̃)> 0 is a contradiction to the definition of p̃ being the smallest point in (b^_1,1] where u(p̃)=0. Let us consider the case p̃≥ b^*_2. Then the contradiction u_p(p̃)> 0 is obtained in a similar way. More precisely, using the ODE (cf. (<ref>)) with the boundary conditions u(p̃)=0 and u(1)=λ-c/r, we obtain u(p)=c_3p(1-p/p)^α_1(λ)+pλ-c/r, for p>p̃, where c_3=-p̃λ-c/p̃r(1-p̃/p̃)^-α_1(λ). Using also p̃ > c/λ̅, (<ref>), and arguments analogous to those after (<ref>), we find with some work that u_p(p̃)=c/p̃r+α_1(λ)(p̃λ-c)/p̃(1-p̃)r>0. abbrv
http://arxiv.org/abs/2307.01126v3	20230703155626	Minimal Supergeometric Quantum Field Theories	[ "Viola Gattus", "Apostolos Pilaftsis" ]	hep-th	[ "hep-th", "hep-ph" ]	July 2023 Minimal⁠[4] Supergeometric⁠[4] Quantum⁠[4] Field⁠[4] Theories Viola Gattus[E-mail address: [email protected]] and Apostolos Pilaftsis[E-mail address: [email protected]] Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom ABSTRACT We formulate minimal SuperGeometric Quantum Field Theories (SG-QFTs) that allow for scalar-fermion field transformations in a manifestly reparameterisation covariant manner. First, we discuss the issue of uniqueness in defining the field-space supermetric of the underlying supermanifold, and clarify the fact that different supermetric definitions can lead to distinct theories in the off-shell kinematic region. By adopting natural choices for the field-space supermetric, we then show that scalar fields alone cannot induce a non-trivial field-space Riemannian curvature in the fermionic sector, beyond the one originating from the scalar part of the theory. We present for the first time minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions. Physical applications of SG-QFTs are discussed. Keywords: Supergeometry, Quantum Field Theory empty § INTRODUCTION For more than half a century, covariant and differential geometric methods still continue to play a central role in the development of Quantum Field Theory (QFT) <cit.>. Besides aspects of gauge covariance in effective actions <cit.>, these methods were used to compute transition amplitudes of chiral loops in a reparameterisation invariant manner <cit.>. They have also been applied within the context of non-linearly realised supersymmetric theories <cit.>. Beyond the classical approximation, these covariant and differential geometric methods have been put on a more rigorous footing by Vilkovisky and DeWitt (VDW) <cit.> to address the issue of gauge-fixing parameter independence in gauge and quantum gravity theories. This VDW framework was developed further by several other authors <cit.>. More recently, a related differential geometric formalism was utilised to resolve the so-called quantum frame problem in cosmological single-field and multi-field inflation <cit.>, along with the issue of uniqueness of the path-integral measure of the VDW effective action <cit.> beyond the Born approximation. On the other hand, similar geometric techniques were employed to analyse new-physics phenomena within the framework of Effective Field Theories (EFTs) beyond the Standard Model (SM) <cit.>, also known as SMEFT (for a recent review, see <cit.>). In spite of the enormous progress made in field-theoretic differential geometric techniques for bosonic theories, the inclusion of fermions in the VDW formalism encountered a number of theoretical difficulties and limitations. Specifically, earlier attempts to include fermions in a reparameterisation and frame-invariant manner were either related to the geometry of bosons through supergravity <cit.> or dependent on a specific EFT-operator <cit.>. Unlike the commuting boson fields, fermion fields are anti-commuting Grassmannian variables, so considering them consistently as independent chart variables in the path-integral configuration space requires the consideration of differential supergeometry (SG) on supermanifolds <cit.>. Recently, a manifestly reparameterisation-invariant formulation of scalar-fermion theories was put forward <cit.>, where the field-space metric was defined from the action. The formalism enabled to obtain earlier known results of the effective action at the one-loop level, but also a new expression for the SG effective action at the two-loop order. However, definite models with non-zero fermionic curvature have not been presented yet in the existing literature. In this paper we present for the first time novel minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions. In formulating these minimal models, we discuss the issue of uniqueness in defining from the action the field-space metric of the underlying supermanifold, _α G_β, which is also termed supermetric. We explain how different definitions of the supermetric _α G_β will usually lead to distinct theories in the off-shell kinematic region. However, some natural choices can be made that rely on a model-function which appears in the kinetic term of the fermions. As noted in <cit.>, the derived metric of the field-space supermanifold should be supersymmetric, which means that _α G_β should be invariant under the operation of supertransposition (sT) to be defined in Section 2. Moreover, we show that scalar fields alone cannot induce a non-trivial field-space Riemannian curvature in the fermionic sector, beyond the one that has its origin in the scalar part of the theory. The paper is organised as follows. After this brief introductory section, we discuss in Section <ref> the basic covariant structure of scalar-fermion SG-QFTs including their key model functions, _α k_β and ζ^μ_α. Here we ignore the effect of gravity and gauge interactions, and leave their study in future works. In the same section, we also present our approach to deriving the supermetric from the classical action of an SG-QFT. In Section <ref> we first show a no-go theorem for the generation of a non-zero super-Riemannian curvature in a bilinear kinetic fermionic sector from the existence of scalar fields only in SG-QFTs. Then, we present two minimal models that realise non-zero fermionic curvature when the model function ζ^μ_α contains non-linear fermionic terms. Section <ref> presents the scalar-fermion superpropagators, as well as the three- and four-point supervertices, thereby generalising earlier results that were derived in pure bosonic theories. Finally, Section <ref> summarises the main results of our study and provides an outlook for physical applications within this novel SG-QFT framework. § SUPERGEOMETRY ON THE SCALAR-FERMION FIELD SPACE Let us briefly review some basic aspects of differential supergeometry on the scalar-fermion field space <cit.> that are relevant to the formulation of SG-QFTs. To start with, we note that a set of N real scalar fields and M Dirac fermions describe a field-space supermanifold of dimension (N\|8M) in four spacetime dimensions (4D). A chart of this supermanifold may be denoted by Φ ≡ {Φ^α} = (ϕ^A , ψ^X , ψ^Y, )^ . Here and in what follows, a Greek index like =1,2,…, N+M labels all fields. Otherwise, we will use Latin letters from the beginning of the alphabet to denote individual bosonic degrees of freedom and letters from the end to denote fermionic ones. In analogy to the standard theory of manifolds, general field reparameterisations of the form, Φ^α → Φ^α = Φ^α(Φ) , become now diffeomorphisms on the supermanifold. Notice that the class of transformations in (<ref>) cover any ultralocal redefinitions of the scalar and fermions fields without introducing extra spacetime derivatives of fields like ∂_μΦ^α, along the lines of the VDW formalism. Up to second order in ∂_μΦ^α, the Lagrangian for a general scalar-fermion theory, which is invariant under field-space diffeomorphisms, can be written in terms of three model functions: (i) a rank-2 field-space tensor _α k_β(Φ), (ii) a mixed spacetime and field-space vector ζ_α^μ(Φ), and (iii) a zero-grading scalar U(Φ) describing the potential and Yukawa interactions. Such a diffeomorphically- or frame-invariant Lagrangian reads <cit.> ℒ = 1/2 g^μν∂_μΦ^α _α k_β(Φ) ∂_νΦ^β + i/2 ζ_α^μ(Φ) ∂_μΦ^α - U(Φ). In (<ref>), _α k_β vanishes when the indices α or β are fermionic, i.e. _X k_A = _A k_Y = _X k_Y = 0. Note that _α k_β plays the role of the field-space metric <cit.> for a purely bosonic theory. Therefore, the function ζ_α^μ is introduced to describe the fermionic sector. Notice that ζ_α^μ may also be used to include chiral fermions by decomposing each Dirac fermion into pairs of Majorana fermions. The model functions, _α k_β and ζ_α^μ, can unambiguously be extracted from the Lagrangian according to the following prescription <cit.>: _α k_β = g_μν/D∂/∂(∂_μΦ^α)ℒ∂/∂(∂_νΦ^β) , ζ_α^μ =2/i(ℒ-1/2 g^μν∂_μΦ^γ_γ k_δ ∂_νΦ^δ) ∂/∂(∂_μΦ^α) . To equip the supermanifold with a metric, we need to construct a pure field-space covector ζ_α from ζ_α^μ. As it is evident from (<ref>) and explained in <cit.>, the Lorentz index μ in ζ_α^μ can only arise from the presence of a γ^μ-matrix, or a σ^μ = (σ^0 ,σ) matrix in the chiral basis, where σ = (σ^1 , σ^2 , σ^3) are the three Pauli matrices. To find the metric of the field-space supermanifold, it proves useful to distinguish two categories of SG-QFTs depending on the actual structure of the model function ζ_α^μ. In the first category, the fermionic components of ζ_α^μ may be expressed in a factorisable form as ζ^μ_ =ζ_ ^(Γ^μ)_ , where Γ^μ =([ γ^μ 0; 0 (γ^μ)^ ]). The second category of SG-QFTs does not possess the factorisation property (<ref>). As we will see in Section <ref>, the distinction between factorisable and non-factorisable ζ_α^μ affects the geometric properties of the field-space supermanifold. For the first category of SG-QFTs, it is straightforward to project a proper field-space covector ζ_ from ζ_α^μ given in (<ref>). The simplest way would be to introduce a differentiation with respect to the γ^μ matrix as done in <cit.>, i.e. ζ_α = 1/D δζ_α^μ/δγ^μ , where D is the number of space-time dimensions. But for the second category of SG-QFTs for which ζ^μ_ does not obey (<ref>), one may alternatively use the more natural projection operation, ζ_β^μ ^(Σ_μ)_α = ζ_α , where Σ_μ =1/D([ ∂/∂γ^μ 0; 0 Γ_μ ]) . In the above, the differentiation acting on the fermionic components of ζ_α^μ is replaced by contraction with γ^μ matrices. In this way, the spin-3/2 degrees of freedom (dofs) contained in ζ^μ_a are projected onto spin-1/2 dofs in ζ_a. In this study we adopt the projection method (<ref>) which can be applied to both categories of SG-QFTs. An important geometric property of SG-QFTs as described by the Lagrangian ℒ in (<ref>) is that ℒ is a scalar in the field-space supermanifold. In other words, ℒ remains invariant under the field redefinitions in (<ref>), provided all model functions and the field-space tangent vectors are appropriately transformed. In this SG framework, we have ∂_μΦ^(Φ) = ∂_μΦ^(̱Φ) _J^(Φ) , where _J^=̱ _,Φ^$̱ is the Jacobian of the transformation and the subscriptαbefore the comma on the left side ofΦ^$̱ denotes ordinary left-to-right differentiation with respect to the field Φ^. A field-space supermanifold of interest to us must be endowed with a rank-2 field-space tensor _ G_$̱, which is supersymmetric, i.e. _α G_β = (_α G_β)^ sT = (-1)^α +β +αβ_β G_α , and non-singular. Such a supermanifold is called Riemannian <cit.> and the rank-2 field-space tensor_G_$̱ is known as the supermetric. Its inverse ^ G^$̱, deduced from the identity: _G_G^̧̱= _δ^, satisfies ^ G^ = G^ = (-1)^ G^ . In the above, we have employed the compact index calculus and conventions by DeWitt in <cit.>, so that the exponents of(-1)determine the grading of the respective quantities and take the values0or1for commuting or anticommuting fields, respectively. According to DeWitt's conventions, the usual tensor contraction between indices can only be performed if the two indices to be summed over are adjacent. Otherwise, extra factors of(-1)must be introduced whenever two indices are swapped. Given the supermetric_αG_β, the Christoffel symbolsΓ^_ ̧̱can be evaluated as in <cit.>, from which the super-Riemann tensor is obtained R^_ ̧̱δ = - Γ^_ ̧̱,δ + (-1)^δ̧ Γ^_ δ̱, + (-1)^(̧σ+)̱ Γ_ σ^Γ_ δ̱^σ - (-1)^δ(σ++̱)̧ Γ_ σδ^Γ_ ̧̱^σ . The super-Ricci tensor is obtained by contracting the first and third indices of the super-Riemann tensor <cit.>, R_ = (-1)^(̧ + 1)R^_ ̧̱ . Further contraction of the remaining two indices ofR_yields the super-Ricci scalar, R = R_ G^ . Note that the super-Ricci tensor is supersymmetric, i.e.R_ = (R_)^sT = (-1)^R_. To determine the supermetric_G_$̱ of the scalar-fermion field space, we follow the procedure presented in <cit.>. After calculating the projected model function ζ_ as stated in (<ref>), we may now construct the rank-2 field-space anti-supersymmetric tensor _αλ_β = 1/2 (_,ζ_β - (-1)^α+β+αβ _,̱ζ_α) . Exactly as happens for the anti-symmetric field strength tensor F_μν in QED in curved spacetime, the derivatives appearing in (<ref>) are ordinary derivatives and not covariant ones, since the Christoffel symbols drop out for such constructions of anti-supersymmetric rank-2 tensors like _λ_$̱. The so-constructed _α λ_βturns out to be singular in the presence of scalar fields, and so the scalar contribution_α k_βhas to be added which results in the new rank-2 field-space tensor, _αΛ_β = _α k_β + _αλ_β . However,_α Λ_βcannot act as a supermetric, since it is not supersymmetric. To find a suitable rank-2 tensor that satisfies the latter property, we make use of the vielbein formalism <cit.> which allows to compute the field-space vielbeins_e^a, if the form of_α Λ_βis known in the local field-space frame. For the latter, we demand that the Lagrangian (<ref>) assumes the canonical Euclidean form in this local frame. In this way, we may compute the field-space supermetric as <cit.> _ G _= _ e^a _a H_b ^be^_, where _a H_b≡([ 1_N 0 0; 0 0 1_4M; 0 - 1_4M 0 ]) is the local field-space metric in 4D. Finally, according to VDW formalism <cit.>, we need to promote the field space to a configuration space, so as to take into account the spacetime dependence of the fields. In this configuration space, the coordinate charts are extended as Φ^α̂ ≡ Φ^α(x_α), wherex_is the spacetime coordinate of a generic fieldΦ^. Likewise, the supermetric gets generalised as _α̂ G_β̂ = _α G_β δ(x_α - x_β) , whereδ(x_α - x_β)is theD-dimensionalδ-function. This generalisation affects the Christoffel symbols and the Riemann tensors, as given in more detail in <cit.>. § MINIMAL MODELS In this section we first show a no-go theorem that no non-zero field-space curvature can be generated in the fermionic sector from a single scalar field and multiple fermion species, as long as the model functionζ^μ_αonly contains linear terms in the fermionic fields. We then present two minimal SG-QFT models with non-zero fermionic curvature which is induced by the addition of non-linear terms in fermion fields toζ^μ_α. §.§ No-Go Theorem on Fermionic Field-Space Curvature The simplest case with one bosonic fieldϕand one Dirac fermionψwas analysed in <cit.>. This case reduces to a flat field space, and as such, we will not repeat it here. Instead, we consider a more general scenario with a single bosonϕand a multipletψ ={ ψ^X} of Dirac fermion fields (withX=1,2,…M). In 4D, such a scenario has(1\|8M)field-space coordinates. Up to second order in spacetime derivatives, the Lagrangian for such a system is given by ℒ = 1/2 k(ϕ) (∂^μϕ) (∂_μϕ) - 1/2 h_X Y(ϕ) ψ^Xγ^μψ^Y (∂_μϕ) + i/2 g_X Y(ϕ)[ψ^Xγ^μ (∂_μψ^Y) - (∂_μψ^X) γ^μψ^Y] . Evidently, the single fieldϕcannot induce by itself a non-zero Riemannian curvature in the scalar sector. Hence, if there is a non-trivial field-space curvature, this can only come from the fermionic sector of the Lagrangian (<ref>). If the field space is flat, one should be able to find a suitable reparameterisation of the fields to bring the Lagrangian (<ref>) into a canonical Cartesian form. To this end, let us consider the following redefinition of fermionic fields: ψ ⟶ ψ = K(ϕ)^-1 ψ , whereKis a4M×4M-dimensional matrix that only depends on the scalar fieldϕ. The field reparameterisation (<ref>) modifies the fermionic part of the Lagrangian (<ref>) as follows: ℒ_f = -1/2[h_X Y K_W^† X K_Z^Y-i g_X Y( K_W^† X∂ K_Z^Y/∂ϕ-∂ K_W^† X/∂ϕ K_Z^Y)] ψ^Wγ^μψ^Z (∂_μϕ) +i/2 g_X Y K_W^† X K_Z^Y[ψ^Wγ^μ (∂_μψ^Z) - (∂_μψ^W) γ^μψ^Z] . To put Lagrangian (<ref>) into a canonical form, it suffices to eliminate the first term on the RHS of (<ref>). We therefore require the vanishing of the matrix expression, i K^†hK + K^†g∂K/∂ϕ - ∂K^†/∂ϕgK = 0 , withg ={ g_XY}andh ={ h_XY}. Even though it is straightforward to find a solution to (<ref>) for the case of a single fermion field <cit.>, it becomes non-trivial in the presence of many fermions. However, after some familiarisation with the analytic expression on the LHS of (<ref>), a simple and intuitive solution can be obtained, given by K(ϕ) = exp(- i/2∫_0^ϕg^-1h d ϕ) . Consequently, the field-space supermanifold for this theory is flat. It is worth noting here that one can confirm the field-space flatness of the theory within our formalism of an SG-QFT as well. In detail, the supermetric derived from (<ref>) reads <cit.> _α G_β = ([ k-1/2ψ(g^'- i h)g^-1(g^' + i h)ψ -1/2ψ(g^'- i h) 1/2ψ^ (g^' + i h^); 1/2(g^' - i h^) ψ^ 0 g^ 1_4; -1/2(g^'+ i h) ψ -g 1_4 0 ]) . However, as opposed to what was conjectured in <cit.>, we find that the super-Riemann tensor computed from the field-space supermetric (<ref>) vanishes identically, thus implying that the field-space supermanifold is flat. This exercise shows that the scalar fieldϕacts only as an external parameter in the fermionic sector. If we add more scalar fields, this last result will not alter as long as the model function_αk_β≡k_ABdoes not introduce any curvature in the scalar sector, in which casek_ABcan then be brought into a canonical (Cartesian) form, i.e.k_AB = δ_AB. The rest of the proof goes along the lines discussed above for the single scalar case. In addition, one would need to consider an extra indexA = 1,2,…Ncounting theNscalarsϕ^A. As a consequence, one would have to solve a set of independentNmatrix equations like (<ref>), resulting inNdifferent matricesK_A(ϕ^A). This concludes our proof of the no-go theorem. For a pure fermionic theory, non-zero fermionic curvature effects can only then be generated if the model functionζ^μ_αdepends non-linearly on the fermion fields. In the next two subsections, we present two SG-QFT models that realise non-zero fermionic curvature. §.§ Non-zero Fermionic Field-Space Curvature: Model I Let us first consider a 2D SG-QFT model, also called Model I, which includes one scalar fieldϕand one Dirac fermion represented asψ^= (ψ_1 , ψ_2). The Lagrangian of this simple model is given by ℒ_I = 1/2 k (∂_μϕ) (∂^μϕ) + i/2(g_0 +g_1ψψ)[ψγ^μ (∂_μψ) - (∂_μψ) γ^μψ] + Y ψψ - V , whereγ^μ= (σ^1 , -iσ^2). Here, all the model functionsk,g_0,g_1,YandVdepend on the scalar fieldϕ. Note that the model functionζ^μ_αderived from (<ref>) takes on the factorisable form of (<ref>), with ζ_ = {0 , (g_0 + g_1ψψ) ψ , (g_0 + g_1ψψ) ψ^} . Using the method of <cit.> briefly outlined in Section 2, we may derive the field-space supermetricG = { _αG_β}in the superspaceΦ^= (ϕ , ψ^ , ψ), G = ([ k + b^ (d^-1)^ a^ - a d^-1 b -a b^; a^ 0 d^; -b -d 0 ]), where a = 1/2 ψ(g_0^' + g_1^'ψψ) , b = 1/2(g_0^' + g_1^'ψψ)ψ , d = (g_0+ g_1ψψ) 1_2 + g_1ψψ , and a prime(^')on the model functionsg_0,1denotes differentiation with respect toϕ. Note thatGis supersymmetric, sinceb^(d^-1)^a^= -a d^-1 b. Given the supermetricG, we may now compute the non-zero components of the Riemann tensor. For instance, ifg_0=g_1=1, these components are found to be R^ψ_1_ ψ_1ψ_1ψ_2 =-R^ψ_2_ ψ_2ψ_1ψ_2= ψ_1ψ_2-1 , R^ψ_1_ ψ_1ψ_2ψ_2 =R^ψ_1_ ψ_2ψ_1ψ_2=-R^ψ_1_ ψ_1ψ_2ψ_2= -ψ_1ψ_1 , R^ψ_2_ ψ_1ψ_1ψ_2 = R^ψ_2_ ψ_2ψ_1ψ_1=-R^ψ_2_ ψ_2ψ_1ψ_1 =ψ_2ψ_2 , R^ψ_2_ ψ_1ψ_2ψ_2 =R^ψ_1_ ψ_1ψ_2ψ_1=-R^ψ_2_ ψ_2ψ_2ψ_1=-R^ψ_1_ ψ_2ψ_1ψ_2= 1-ψ_2ψ_1 . Hence, the minimal SG-QFT model of (<ref>) exhibits a non-zero fermionic field-space curvature. Allowing forϕ-dependent model functionsg_0,1, the super-Ricci scalar evaluates to[In our study, we did not pay attention to the dimensionality of the model parameters, k, g_0, g_1, and their derivatives g'_0, g'_1, that enter the supermetric G from which R is evaluated. Their dimensionality can be restored following the approach in <cit.> to ensuring uniqueness of the path-integral measure. In detail, the energy (E) dimension of R is the same as the inverse of the squared field-space line element: dΣ^2 = dΦ^_ G_ dΦ^$̱, i.e.[R] = [dΣ^2]^-1in anyDdimensions. Like in <cit.>, we takedΣ^2to be dimensionless, whilst considering the dimensions of the fields:[ϕ ] = E^(D-2)/2,[ψ ] = E^(D-1)/2. To do so, we rescale the model parametersk,g_0,1as:k̃ = ℓ^D-2 k,g̃_0,1 = ℓ^D-1 g_0,1, whereℓ = ℓ(Φ)plays the role of an effective Planck length, with[ℓ ] = E^-1. In addition,g̃_1should be divided by the energy cut-offΛ^D-1that normalises the fermionic bilinearψψ, in the context of an effective field theory. Hence, inDdimensions we have:[k̃] = E^-(D-2),[g̃_0] = E^-(D-1),[g̃_1] = E^-2(D-1),[g̃'_0] = E^2 -3D/2and[g̃'_1] = E^3 -5D/2. One may then verify that with the rescaled parametersk̃,g̃_0,1andg̃'_0,1, the so-redefined super-Ricci scalarRis dimensionless in (<ref>) and (<ref>), i.e.[R̃] = E^0. R = 4g_1/g_0^2 + (2 g_1 g_0^' g_1^'/g_0^3 k-2 g_1^2 g_0^' 2/g_0^4 k-g_1^' 2/2 g_0^2 k)(ψψ)^2 . Observe thatRis a Lorentz scalar, but not a real-valued expression due to the appearance of the fermionic bilinear term(ψψ)^2. Forg_0=g_1=1, the super-Ricci scalar simplifies to R = 4 . It is important to remark here that the same result (<ref>) would have been obtained in the absence of the bosonic fieldϕ. Consequently, the non-vanishing field-space curvature arises from the non-linear terms in the fermion fields inζ_αthrough the model functiong_1in (<ref>). The above consideration can be easily extended to a 4D version of the SG-QFT Model I considered in (<ref>). In this case,γ^μstand for the usual 4D Dirac matrices, and the Dirac fermion has four components:ψ^ = (ψ_1 , ψ_2 , ψ_3 , ψ_4). The 4D SG-QFT model has(1\|8)dimensions giving rise to rather lengthy expressions for the super-Riemann tensor, which we will not present here. Instead, we give the field-space super-Ricci scalar, R = 24 g_1/g_0^2 - 24g_1^2/g_0^3(ψψ) + (2 g_1 g_0^' g_1^'/g_0^3 k-2 g_1^2 g_0^' 2/g_0^4 k-g_1^' 2/2 g_0^2 k-4 g_1^3/g_0^4) (ψψ)^2 +(-16 g_1^2 g_0^' g_1^'/g_0^4 k+16 g_1^3 g_0^' 2/g_0^5 k+4 g_1 g_1^' 2/g_0^3 k+40 g_1^4/g_0^5) (ψψ)^3 +(80 g_1^3 g_0^' g_1^'/g_0^5 k-80 g_1^4 g_0^' 2/g_0^6 k-20 g_1^2 g_1^' 2/g_0^4 k+20 g_1^5/g_0^6) (ψψ)^4 . Forg_0=g_1=1, the field-space Ricci scalar takes on the simpler form, R = 24 - 24 (ψψ) - 4 (ψψ)^2 + 40 (ψψ)^3 + 20 (ψψ)^4 . We note that (<ref>) becomes identical to the result one would obtain in a system with two fermions in 2D. This should be expected, since the number of degrees of freedom and the structure of the Lagrangian (<ref>) are exactly the same for the two cases. Let us now discuss an important feature of the geometric construction of Lagrangian (<ref>), and SG-QFTs in general. Specifically, one may notice that under a naive non-linear reparameterisation of the fermion fields, ψ = ψ √(1+ψψ) , ψ = √(1+ψψ) ψ , one can turn a standard (canonical) Dirac Lagrangian, ℒ_D = i/2 [ψ(∂ψ) - (∂ψ)ψ] , into the Lagrangian (<ref>), in whichg_0 = g_1 = 1and all remaining model functions are set to zero,k = Y = V = 0. This would seem to suggest that a curved field-space theory can be obtained from a flat one by means of a non-linear reparameterisation like (<ref>), and vice-versa. However, within our SG-QFT framework, such a transformation is not possible. More explicitly, in an SG-QFT, the standard Dirac Lagrangian must be recast into the covariant form, ℒ_D = i/2 ζ_α ∂Ψ^α , withΨ^ = {ψ , ψ^}andζ_ = {ψ , ψ^}. Any change of the fermionic field chart,Ψ^a →Ψ^a, must be done according to the transformations, ∂_μΨ^ = ∂_μΨ^ _J̱^ , ζ_ = ζ_ ^(̱J^-1)^_ . However, the Jacobian transformations (<ref>) do not alter the form ofℒ_Din (<ref>). Therefore, different analytic forms of the model functionζ_α^μgive rise to distinct supergeometric constructions of Lagrangians, involving different supermetricsG. The superdeterminant of the latter usually affect the path-integral measure and so the effective action beyond the classical approximation <cit.>. Finally, we should comment on the flavour covariance of an SG-QFT with many species of fermions. Indeed, an equivalent class of Lagrangians can be consistently constructed through flavour field redefinitions,ψ→ψ = Uψ, whereUis a unitary flavour-rotation matrix that may only depend on the scalar fields. The new supermetric in the flavour-transformed basis is derived from the usual rank-2 covariance relation, _G_=̱ _ (J^-1)^_Ģ_δ ^δ (J^-1)^_ . The so-derived supermetric in (<ref>) can be shown to be equivalent to the supermetric that would be obtained by extracting the new model functions from a flavour-transformed Lagrangian, e.g.ℒ. This last property provides further support of the mathematical and physical consistency of the SG-QFT framework under study. §.§ Non-zero Fermionic Field-Space Curvature: Model II We now turn our attention to the second category of SG-QFTs, for which the model functionζ^μ_cannot be written in the factorisable form of (<ref>). For brevity, we call this scenario Model II, in order to distinguish from Model I discussed in the previous subsection. To showcase the rich geometric structure of this new class of SG-QFTs, we ignore all scalar fields and only consider one Dirac fermionψin 4D. A minimal SG-QFT Model II is described by the Lagrangian, ℒ_II = i/2[ψγ^μ(∂_μψ)-(∂_μψ)γ^μψ] + i/2ψγ^μψ[ψ(∂_μψ)-(∂_μψ)ψ] . As outlined in Section <ref>, we employ (<ref>) to calculate the model functionζ^μ_, ζ^μ_ ={0 , ψγ^μ+(ψγ^μψ) ψ , ψ^γ^μ +(ψγ^μψ) ψ^} . To extract the covectorζ_from this latter expression, we make use of the projection method given in (<ref>). Following the approach of <cit.>, we construct the anti-supersymmetric rank-2 field-space tensor_λ_$̱, which in turn was used to determine the vielbeins _ e^a. With the help of _ e^a, the following field-space supermetric is derived: G ≡ { _ G_}̱ = ([ 0 d^; -d 0 ]), where d is a 4× 4-dimensional matrix in the Dirac spinor space given by d = 1_4 + 1/4(ψγ^μψ) γ_μ + 1/4γ^μ ψψ γ_μ . Knowing the analytic form of the supermetric G, we may now compute the super-Ricci scalar of this theory, R = -8+2 (ψψ) +23/8(ψψ)^2+9/8(ψγ_5ψ)^2 +5/4(ψγ_μψ)(ψγ^μψ) -29/12(ψψ)^3 +7/16(ψψ)^4 . Observe that the super-Ricci scalar R is both parity-preserving and Lorentz invariant, so it shares the same properties like the original Lagrangian (<ref>) from which it was obtained. Interestingly enough, the expression for R of Model II has a much richer expansion than that found in Model I [cf. (<ref>)]. In addition to (ψψ)^2 terms, R now contains new Lorentz-invariant four-fermion operators, such as (ψγ_5ψ)^2 and (ψγ_μψ)(ψγ^μψ). We should remark here that had we used the method of <cit.> given in (<ref>) to deduce the covector ζ_, we would then have obtained a different supermetric leading to an expression for R similar to Model I as powers of the fermionic bilinears ψψ but with different coefficients. Nevertheless, we find that the projection method introduced in (<ref>) is more appropriate, since it reflects more accurately the geometric structure of the SG-QFT models in the second category. Furthermore, it should be noted that as opposed to Model I, the SG-QFT Model II cannot be brought into the canonical form of (<ref>) by naive redefinitions of the fermion fields: ψ = ψ f(ψψ ) and ψ = f^*(ψψ ) ψ, where f is some judicious function like (<ref>). As we will see in the next section, field-space geometry governs the Feynman rules of an SG-QFT through superpropagators and supervertices. § SUPERPROPAGATORS AND SUPERVERTICES In this section, we present analytical results of the superpropagator and the three- and four-point supervertices related to the fermionic part of SG-QFTs, where the model function _ k_$̱ was set to zero. However, we allow for the possible presence of background scalar fieldsϕ^A. In this simplified SG-QFT setting, we first give the equation of motion of the fields, S_; = S_, = i(-1)^ _λ^μ_ρ ∂_μΦ^ρ - U_, . Here and in the following, a semicolon (;) stands for covariant configuration-space differentiation andSis the classical action pertinent to the Lagrangian (<ref>), with_ k_=̱ 0. In addition, we introduced in (<ref>) a modified version of the tensor_αλ_βof (<ref>) defined as _αλ^μ_β ≡ 1/2 (_,ζ^μ_β - (-1)^α+β+αβ _,̱ζ^μ_α) , which will appear in our expressions for the superpropagator and the supervertices given below. Notice that_αλ^μ_βis a proper spacetime vector and rank-2 field-space tensor as it is derived from functional differentiation ofζ^μ_α. From (<ref>), the covariant inverse superpropagatorS_;may be evaluated as follows: S_; = i(-1)^( _λ^μ_ρ ∂_μΦ^ρ_ ; + (-1)^ρ_λ^μ_ρ; ∂_μΦ^ρ) - U_; . In the spacetime homogeneous limit of the theory in which∂_μΦ→ 0, expression (<ref>) becomes in momentum space, S_;\|_∂_μΦ=0 = ((-1)^ _λ^μ_p^_̱μ - U_;) δ(p^+p^)̱ . Our next step is to calculate the covariant three-supervertexS_;. As before, we start evaluating this in the coordinate space, S_;=i (-1)^( _λ^μ_ρ ∂_μΦ^ρ_ ;̧̱ + (-1)^(̧ρ+)̱_λ^μ_ρ; ∂_μΦ^ρ_ ; +(-1)^ρ_λ^μ_ρ; ∂_μΦ^ρ_ ; + (-1)^ρ(+̱)̧_λ^μ_ρ;̧̱∂_μΦ^ρ) - U_;̧̱ . In the momentum space and homogeneous limit∂_μΦ→ 0of the theory, the covariant three-supervertex reads S_;\|_∂_μΦ=0 = ( (-1)^_λ^μ_;̱ p^_̱μ + (-1)^+̧̱_λ^μ_;̧ p^_̧μ - U_;̧̱) δ (p^+p^+̱p^)̧ . Notice that unlike a pure bosonic theory, the covariant three-supervertexS_;does not vanish in fermionic SG-QFTs in the absence of a potential termU. In a similar fashion, we can compute the four-supervertex in the configuration space as S_; = i(-1)^( _λ^μ_ρ ∂_μΦ^ρ_ ;̧̱δ + (-1)^δ(+̱+̧ρ)_λ^μ_ρ;δ ∂_μΦ^ρ_ ;̧̱ +(-1)^(̧+̱ρ)_λ^μ_ρ; ∂_μΦ^ρ_ ;δ̱ +(-1)^ρ_λ^μ_ρ; ∂_μΦ^ρ_ ;δ̧ + (-1)^(+̧δ)(ρ+)̱ _λ^μ_ρ;δ̧ ∂_μΦ^ρ_ ; + (-1)^ρ(+̱δ)+δ _λ^μ_ρ;δ̱ ∂_μΦ^ρ_ ; +(-1)^ρ(+̱)̧_λ^μ_ρ;̧̱ ∂_μΦ^ρ_ ;δ + (-1)^ρ(+̱+̧δ) _λ^μ_ρ;̧̱δ ∂_μΦ^ρ) - U_;̧̱δ . In the momentum space and the homogeneous limit, the latter expression simplifies to . S_;\|_∂_μΦ=0= ( (-1)^_λ^μ_ρ R^ρ_ ̧̱δ p^δ_μ + (-1)^_λ^μ_;̱δ̧ p^_̱μ + (-1)^+̧̱_λ^μ_;̧δ̱ p^_̧μ + (-1)^+δ(+̱)̧_λ^μ_δ;̧̱ p^δ_μ -U_;̧̱δ) δ (p^ + p^+̱p^+̧ p^δ) . One can now make explicit the field-space metric dependence on the supervertices by writing_λ^μ_$̱ as: _λ^μ_=̱_(λ^μ)^ρ _ρ G_$̱. When computing covariant derivatives of_λ^μ_$̱ through the contraction _(λ^μ)^ρ _ρ G_$̱, there will be a vanishing contribution arising from covariant derivatives of the supermetric_ G_$̱. One should bear in mind that _ G_$̱ satisfies the metric compatibility condition: _ G_;̱ ≡ _ G_,̱ - _ G_ρ Γ^ρ_ ̧̱ - (-1)^+ρ+(̱ρ+) _ρ G_ Γ^ρ_ = 0 , and so non-zero contributions can only come from covariant differentiations of_(λ^μ)^$̱. Hence, it is this misalignment between _λ^μ_$̱ and_ G_$̱ that yields a non-vanishing three-supervertex S_; in (<ref>), even in the absence of potential terms. This is in contrast to the bosonic case as shown in <cit.>. We may now verify that the four-supervertices given in (<ref>) satisfy two essential super-Ricci identities involving the supercommutator of covariant derivatives. First, we remind the reader that the supercommutator of two covariant derivatives is defined as <cit.>: S_;[α̂,β̂] ≡ S[∇_α̂ ,∇_β̂] = S_;α̂β̂ - (-1)^α̂β̂ S_;β̂α̂ . Then, one can show that the following super-Ricci identities are satisfied in the homogeneous limit: .S_;α̂[β̂,γ̂]δ̂\|_∂_μΦ=0 = (-1)^δ̂(ρ̂+α̂+β̂+γ̂) .S_;ρ̂δ̂\|_∂_μΦ=0 R^ρ̂_ α̂β̂γ̂ , .S_;α̂β̂[γ̂,δ̂]\|_∂_μΦ=0 = .S_;α̂ρ̂\|_∂_μΦ=0 R^ρ̂_ β̂γ̂δ̂ + (-1)^β̂(ρ̂+α̂) .S_;ρ̂β̂\|_∂_μΦ=0 R^ρ̂_ α̂γ̂δ̂ . These identities turn out to be rather useful in simplifying the process of supersymmetrisation of higher-point supervertices. We conclude this section by noting that pure scalar contributions to the superpropagators and supervertices <cit.> can also be included in the above expressions. Unlike the fermionic contributions which depend linearly on particle momenta, bosonic effects are quadratic in the momenta and so they enter additively to (<ref>), (<ref>) and (<ref>). § SUMMARY AND OUTLOOK We have studied in detail the frame-covariant formalism presented earlier in <cit.> on scalar-fermion theories. The scalar and fermion fields define a coordinate system or a chart which describe a supermanifold in the configuration space of the respective QFTs. We discussed the issue of uniqueness of the supermetric and clarified that different choices of the latter lead to distinct Supergeometric QFTs in the off-shell kinematic region, as well as beyond the classical approximation. Adopting a natural and self-consistent choice for the supermetric, we have shown that scalar fields alone do not provide a new source of curvature in the fermionic sector of the theory beyond the one that originates from the model function _α k_β. In particular, we have explicitly demonstrated that non-linear powers of fermionic fields in the model function ζ^μ_α can give rise to non-zero fermionic curvature, as expressed by a non-zero super-Riemann tensor. Hence, we have presented for the first time novel minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions up to second order in spacetime derivatives. It should be emphasised here that the resulting super-Riemann tensor and super-Ricci scalar may contain fermionic bilinears which are no proper real numbers. This should be contrasted with Supergravity theories <cit.> where the curvature is a real-valued expression dictated by the scalar part of the Kaehler manifold, on which the fermions were treated as tangent vectors. In addition, we have derived new generalised expressions for the scalar-fermion inverse superpropagator, and the three- and four-supervertices. As opposed to pure bosonic theories, we have found that the three-supervertices are non-zero in fermionic theories in the absence of a zero-grading scalar potential U [cf. (<ref>)]. These ingredients are all necessary for future considerations in evaluating amplitudes and higher-loop effective actions in SG-QFTs. Furthermore, one may wish to include further gauge and gravitational symmetries in SG-QFTs which will act as isometries <cit.> on the supermanifold. We expect that SG-QFTs will lead to a complete geometrisation of realistic theories of micro-cosmos, such as the SM and its gravitational sector. We may even envisage that SG-QFTs will provide a new portal to the dark sector, where dark-sector fermionic fields may modify the dispersion properties of weakly interacting particles, like SM neutrinos and axions. We plan to investigate the above issues in future works. §.§ Acknowledgements The authors thank Alejo Rossia and Thomas McKelvey for discussions. The work of AP is supported in part by the STFC Research Grant ST/T001038/1. VG acknowledges support by the University of Manchester through the President's Doctoral Scholar Award. 99DeWitt:1967ub B. S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, https://doi.org/10.1103/PhysRev.162.1195Phys. Rev. 162 (1967) pg. 1195–1239b Author, Title, arxiv:1234.5678. c Author, Title, Publisher (year). ]
http://arxiv.org/abs/2307.02871v1	20230706091538	Contrastive Label Disambiguation for Self-Supervised Terrain Traversability Learning in Off-Road Environments	[ "Hanzhang Xue", "Xiaochang Hu", "Rui Xie", "Hao Fu", "Liang Xiao", "Yiming Nie", "Bin Dai" ]	cs.RO	[ "cs.RO" ]	[ Spin and orbital Edelstein effect in a bilayer system with Rashba interaction Annika Johansson August 1, 2023 ============================================================================= ] *This work was supported by the National Natural Science Foundation of China under No.61790565 and No.61803380. ^1Hanzhang Xue, Xiaochang Hu, and Hao Fu are with the College of Intelligence Science and Technology, National University of Defense Technology, Changsha, 410073, China [email protected], [email protected], [email protected]. ^2Hanzhang Xue, Liang Xiao, Yiming Nie, and Bin Dai are with the Unmanned Systems Technology Research Center, Defense Innovation Institute, Beijing, 100071, China [email protected], [email protected], [email protected]. ^3Rui Xie is with the Key Laboratory of Machine Perception (MOE), Peking University, Beijing, 100871, China [email protected]. Discriminating the traversability of terrains is a crucial task for autonomous driving in off-road environments. However, it is challenging due to the diverse, ambiguous, and platform-specific nature of off-road traversability. In this paper, we propose a novel self-supervised terrain traversability learning framework, utilizing a contrastive label disambiguation mechanism. Firstly, weakly labeled training samples with pseudo labels are automatically generated by projecting actual driving experiences onto the terrain models constructed in real time. Subsequently, a prototype-based contrastive representation learning method is designed to learn distinguishable embeddings, facilitating the self-supervised updating of those pseudo labels. As the iterative interaction between representation learning and pseudo label updating, the ambiguities in those pseudo labels are gradually eliminated, enabling the learning of platform-specific and task-specific traversability without any human-provided annotations. Experimental results on the RELLIS-3D dataset and our Gobi Desert driving dataset demonstrate the effectiveness of the proposed method. § INTRODUCTION For autonomous driving, understanding the traversability of surrounding environments is one of the most fundamental and critical tasks. The majority of existing related works primarily concentrates on structured environments where the traversable regions are explicitly defined, and treat the traversability analysis as a binary classification task. Closely related tasks include road detection <cit.>, ground segmentation <cit.>, or free-space detection <cit.>. However, most of these approaches may not work well in complex off-road environments. There are two main reasons: Firstly, there is a high degree of similarity between traversable and non-traversable regions in some off-road environments; Secondly, countless terrain types and irregular terrain shapes in off-road environments present intricate possibilities for traversability, and it is challenging to analyze them with a unified rule. Recently, several researchers <cit.> have attempted to employ supervised semantic segmentation approaches to obtain semantic information for each region in off-road environments, and to analyze traversability by establishing mapping relationships between different semantic categories and traversability. Although these methods achieved impressive results on some specific datasets, they are difficult to adapt to previously unseen environments since the inherent ambiguity of the traversability in off-road environments. On the one hand, it is challenging to define semantic categories and their corresponding traversability in diverse off-road environments without ambiguity. On the other hand, traversability itself is also platform-related and task-related, different platforms or autonomous tasks may yield different traversability results. Furthermore, these supervised learning approaches also require exhausting human labor for manual annotation of training samples. It is unaffordable and unsustainable for re-annotating a tremendous amount of data each time a new environment is encountered. Bearing the purpose of rapidly learning platform-specific and task-specific traversability in new off-road environments without any human-provided annotations, we shift from directly define traversability or semantic categories in off-road environments to understand traversability by learning from demonstration. When an unmanned ground vehicle (UGV) encounters an unknown off-road environment or a new autonomous task, a large amount of weakly labeled data can be automatically generated by simply driving the UGV for a short distance with the assistance of a human driver (shown as the middle figure in Fig. <ref>). As derived from actual driving experiences, these weakly labeled data are platform-specific and task-specific. They consist of scarce positive samples (actually traversed regions) and numerous unlabeled samples, which can be employed as training data in the problem of self-supervised traversability learning. Recently, although a few approaches such as positive-unlabeled learning <cit.> or anomaly detection <cit.> have been applied to address this problem, they possess limited ability to discriminate traversability in off-road environments with high similarity. Furthermore, common forms of input data for this problem include images <cit.> or single-frame LiDAR scan <cit.>. Images are sensitive to illumination changes, and single-frame LiDAR scan is sparse and prone to noises. Consequently, neither can provide a stable and robust representation of off-road environments, directly affecting the stability of traversability learning. To address these challenges, we propose a novel self-supervised terrain traversability learning framework. After generating stable, complete, and accurate terrain models in real-time using our previous work <cit.>, automatic data annotation is conducted in those constructed terrain models based on actual driving experiences. Those actually traversed regions are assigned determined positive labels, while the remaining regions are assigned candidate pseudo labels. Inspired by the impressive progress of partial label learning (PLL) <cit.>, a prototype-based contrastive representation learning method with the aid of a local window based transformer encoder is designed to learn distinguishable embeddings for updating those candidate pseudo labels, and the refined pseudo labels in turn facilitate representation learning. As the iterative interaction between representation learning and label updating, the ambiguities associated with those pseudo labels are gradually eliminated, enabling the learning of specific traversability in off-road environments. This learning process can be referred to as contrastive label disambiguation. To demonstrate the effectiveness of the proposed method, we conduct experiments on both the publicly available RELLIS-3D dataset <cit.> and a Gobi Desert driving dataset collected by our own UGV. Experimental results show that the proposed method can learn specific traversability from human-selected driving routes in a self-supervised manner. The rest of this paper is organized as follows. Section <ref> discusses some related works. Section <ref> provides detailed information about the proposed method. Experimental results on both the RELLIS-3D dataset and our Gobi Desert driving dataset are presented in Section <ref>. Finally, Section <ref> summarizes the conclusions. § RELATED WORK Traversability analysis plays a crucial role in autonomous driving and has garnered significant attention in recent years. In the existing literature, most approaches treat traversability analysis as a binary classification task. One common method projects point cloud or RGB images onto a 2D Bird's Eye View (BEV) grid map and extracts geometric features <cit.> or appearance features <cit.> for traversability classification. Another kind of approaches determines traversability by estimating terrain models of local environments, such as Gaussian process regression <cit.>, Bayesian generalized kernel inference <cit.>, and B-spline surface <cit.>. With the rise of deep learning, convolutional neural networks (CNNs) have also been utilized for end-to-end traversable region detection by using RGB images <cit.>, point cloud <cit.>, or a combination of both <cit.> as input. Although these binary classification approaches work well in structured environments, their suitability for complex off-road environments is limited. It is more suitable to adopt semantic mapping for traversability analysis in off-road environments, which allows for distinction of semantic information among different regions. Some methods <cit.> perform fine-grained semantic mapping to assign a detailed semantic label (such as dirt roads, grass, bushes, etc.) to each region by using semantic segmentation networks. The traversability can be further analyzed through mapping relationships between semantic categories and traversability. Some other works <cit.> argue that fine-grained semantic information is not necessary for autonomous navigation and propose solutions based on coarse-grained semantic mapping. In these approaches, different regions are segmented by traversability levels. Although these semantic mapping approaches achieve good results in some specific off-road environments, they face some intractable challenges. Firstly, it is difficult to define uniform and unambiguous semantic categories or semantic-traversability mapping relationships that are suitable for all off-road environments. Additionally, these methods heavily rely on supervised labels and the burden of manually re-annotating pixel-level or grid-level semantic labels each time a new environment is encountered proves to be impractical, thus limiting the practical application of these approaches. Recently, there has been a growing interest in self-supervised learning methods for traversability analysis. The physical experiences of the UGV, rather than human-provided annotations, are used to automatically label the training data. These methods can be divided into two categories. The first type utilizes on-board proprioceptive sensors to measure signals that directly reflect information about terrain traversability. Commonly used sensors include the Inertial Measurement Unit (IMU) <cit.>, force-torque sensors <cit.>, or acoustic sensors <cit.>. Then, traversability-related signals are used to annotate data from exteroceptive sensors (such as images or point cloud), and the generated weakly labeled data is employed as training samples for self-supervised learning traversability classification <cit.> or regression <cit.>. The second category of methods obtain labeled data directly from the driving experiences of the UGV. Vehicle trajectories are projected into image space or point cloud space, successful and failed traverses provide positive and negative traversability labels, respectively. Some works <cit.> utilize only positive samples annotated by the footprints of the UGV for learning traversability in a positive-unlabeled learning manner. Part of works try to enhance performance by incorporating negative samples, for example, using LiDAR-based obstacle detection algorithms to annotate obstacle regions as negative samples <cit.>. However, these approaches can not distinguish those non-traversable regions that are not obstacles. Additionally, Chavez-Garcia et al. <cit.> employ a simulation system for self-learning traversability estimation, labeling regions where the UGV gets stuck as negative samples. Bae et al. <cit.> introduce a small amount of manually labeled support data to provide negative samples, resulting in better performance with reduced labor costs. § THE PROPOSED APPROACH In this paper, a self-supervised traversability learning framework is proposed with treating this problem as a contrastive label disambiguation task. The pipeline of the proposed framework is illustrated in Fig. <ref>, consisting of four core modules: a spatial-temporal 3D terrain modeling module, an automated label generation module, a local window based transformer encoder, and a prototype-based contrastive representation learning module. §.§ Spatial-temporal 3D Terrain Modeling To provide a stable, complete, and objective representation of off-road environments and overcome the limitations of sparse and noise-prone point cloud data, a spatial-temporal terrain modeling approach proposed in our previous work <cit.> is applied to generate dense 3D terrain models in real-time. In this approach, a normal distributions transform (NDT) mapping technique is first utilized to recursively fuse information from consecutive LiDAR scans into a global grid map. The elevation of each observed grid cell is modeled as a normal distribution 𝒩 (μ̂, Σ̂). Subsequently, a bilateral filtering-aided Bayesian generalized kernel (BGK) inference approach is employed to infer a predicted elevation distribution 𝒩(μ, Σ) for each grid cell, thus producing a dense and stable elevation map. Furthermore, various terrain features can be calculated by considering the geometric connectivity properties between adjacent grid cells. After constructing the 3D terrain models, a multi-channel terrain feature map F can be generated by projecting various terrain features onto the 2D grid map (as shown in Fig. <ref>). The features contained within each grid cell include: (1) The mean μ̂ and variance Σ̂ of the observed elevation distribution; (2) The mean μ and variance Σ of the predicted elevation distribution; (3) The maximum-minimum observed elevation difference δ_z; (4) The normal angle θ_n; (5) The average concavity angle θ̅_c <cit.> between the 4-neighboring grid cells. §.§ Automated Label Generation To provide training samples without any human-provided annotations, vehicle trajectories are utilized to annotate terrain patches in the proposed method. The process of the automated label generation is illustrated in Fig. <ref>. A vehicle trajectory is defined as positions of four contact points P = {P_lf, P_rf, P_lr, P_rr} between four vehicle wheels and the ground plane. The past or future vehicle trajectory at timestamp τ can be transformed into the local body coordinate system {B} at current timestamp t by: P^B_τ, t = (T^WB_t)^-1·T^WB_τ·P^B , where P^B_τ, t represents the transformed vehicle trajectory. P^B denotes the local position of the contact points P, which can be measured by a simple calibration process. T^WB_τ denotes the transformation matrix from the global coordinate system {W} into the local body coordinate system {B} at timestamp τ, and it is estimated by using an online pose estimation module proposed in our previous work <cit.>. The transformed vehicle trajectory set {P^B_τ, t}_τ∈[t_p, t_f] ([t_p, t_f] denotes valid time interval) is projected onto the terrain feature map F_t generated at current timestamp t. Four projected points from each vehicle trajectory are connected to form a quadrilateral in F_t. As shown in Fig. <ref>, those grid cells lying within the quadrilateral are considered as actually traversed regions and are annotated as positive cells G_p, while the remaining grid cells are unlabeled cells G_u. Then, weakly annotated terrain patches can be extracted. Each terrain patch x_i consists of M × M grid cells (M is an odd number), and its pseudo label y_i is determined by the annotation of its central grid cell G_i,c. y_i is represented as a one-hot encoded form: y_i = {[1 0 0 ⋯ 0 ] (G_i,c = G_p ) , [1 1 1 ⋯ 1 ] (G_i,c = G_u ) , . 1 1 1 ⋯ 1_K where K is the total number of traversability categories. §.§ Local Window Based Transformer Encoder For traversability analysis, it is helpful to effectively utilize terrain information from spatially adjacent terrain patches since traversability itself is a spatially-related concept. However, it is challenging to determine the optimal range of supported spatial neighborhoods. To address this issue and extract more representative embeddings, a local window based transformer encoder f inspired by the Swin Transformer <cit.> is introduced in this subsection. An illustration of the proposed encoder f is shown in Fig. <ref>. The input terrain feature map F is partitioned into a sequence of non-overlapping local windows, and each local window is further split into W × W terrain patches. All terrain patches are automatically labeled using the approach introduced in Section <ref>. A local window is treated as a whole and fed into the f. Each flatten terrain patch contained in the local window is treated as a token. In f, a linear embedding layer is applied to the input flatten tokens to project them to a D-dimensional embedding E∈ℝ^W^2 × D. E is then fed into eight transformer blocks. The shape of the output embedding for each transformer block remains unchanged. Each transformer block consists of a local window based multi-head self-attention (LW-MSA) module, followed by a 2-layer MLP module. A Layer Normalization (LN) layer is applied before each LW-MSA and MLP module, and a residual connection is applied after each LW-MSA and MLP module. The whole process of the proposed encoder f can be formulated as: Z^0 = E , Ẑ^l = LWMSA(LN(Z^l-1)) + Z^l-1 , Z^l = MLP(LN(Ẑ^l)) + Ẑ^l , where Ẑ^l ∈ℝ^W^2 × D and Z^l ∈ℝ^W^2 × D represent the output embedding vectors of the LW-MSA module and the MLP module in the l-th transformer block, respectively. The final extracted embedding for each terrain patch x_i is z^8_i ∈ℝ^D , which is simply denoted as z_i in the subsequent contents. §.§ Prototype-based Contrastive Representation Learning Inspired by <cit.>, a prototype-based contrastive representation learning approach is proposed to learn discriminative embeddings for self-supervised traversability learning. The overall process of this approach is illustrated in Fig. <ref>. In this approach, a given local window is first processed by separate query encoder f_q and key encoder f_k, respectively, generating a query embedding z_q and a key embedding z_k for each token. Only the parameters θ_q of f_q are updated by back-propagation, while the parameters θ_k of f_k are momentum updated by θ_q: θ_k = m_θ·θ_k + (1-m_θ) ·θ_q , where m_θ is a momentum coefficient for updating encoder. The query embedding z_q is then fed into an MLP-based classifier f_c. By combining the output of f_c with the pseudo label y of the token corresponding to z_q, a masked predicted label ỹ_q can be generated by: ỹ_q = max_j ∈[1, K][f_c^j(z_q) ·y] , where f_c^j(z_q) denotes the j-th component of the output vector f_c(z_q) ∈ℝ^K. An embedding queue Q_e and a predicted label queue Q_l are maintained to store the recently encoded key embeddings and their corresponding predicted labels. The embeddings and labels of the latest tokens are enqueued, and the same number of the oldest embeddings and labels are dequeued to ensure a fixed queue size. For a token x with a predicted label ỹ_q, its positive embeddings can be selected from Q_e. Specifically, any embedding z' in Q_e with the same predicted label as ỹ_q is selected as a positive embedding, while the remaining embeddings are considered as negative embeddings. After the positive/negative embeddings selection, the per-token contrastive loss ℒ_cont(x) can be defined as: ℒ_cont(x) = -1/\|A(x)\|∑_z^+∈A(x)logexp(z_q^T ·z^+ / τ)/∑_z_j ∈Q_eexp(z_q^T ·z_j/τ) , where τ is a temperature hyper-parameter, and \|A(x)\| denotes the total number of positive samples in the positive embedding set A(x). To ensure the generation of discriminative embeddings, a high-quality classifier is required for accurate positive/negative embeddings selection. However, improving the performance of the classifier solely through the contrastive loss is challenging due to the inherent ambiguity of pseudo labels. To alleviate this problem, K prototype vectors Ψ = {ψ_c}_c=1:K are created for incremental updating of the pseudo labels. Each prototype serves as a representative embedding for a group of similar embeddings. During training, ψ_c is momentum updated by those query embeddings z_q whose predicted labels ỹ_q belong to class c, and the update process can be expressed as: ψ_c = m_p ·ψ_c + (1-m_p) ·z_q/‖ m_p ·ψ_c + (1-m_p) ·z_q‖_2 , where m_p is a momentum coefficient for updating the prototypes, ‖·‖_2 denotes L2-norm of a vector. After the prototype updating, the pseudo label updating process is performed. An initial normalized vector y_n = 1/∑_i=1^Kyy is assigned to each token based on its pseudo label y in the first batch. Then, an indicator vector ξ∈ℝ^K is computed by comparing the similarity between z_q and Ψ, and y_n is momentum updated by: y_n = m_l ·y_n + (1 - m_l) ·ξ , ξ^c = { 1 if c = max_j ∈[1, K](z_q^T ·ψ_j) 0 else. , where m_l is a momentum coefficient for updating pseudo labels, and ξ^c denotes the c-th component of ξ. y_n is considered as the refined pseudo label, and is utilized for calculating the per-token cross-entropy loss ℒ_cls(x) as: ℒ_cls(x) = ∑_j=1^K -y_n^j ·log(f_c^j (z_q)) . In the training process, the MLP-based classifier and the query encoder are jointly trained, and the overall loss function is: ℒ_sum = ℒ_cls + λℒ_cont , where λ is a weight used for balancing ℒ_cls and ℒ_cont. In summary, the proposed prototype-based contrastive representation learning approach consists of two components that mutually reinforce each other. The discriminative embeddings learned from contrastive learning enhance the quality of positive/negative embeddings selection, while the refined pseudo labels in turn improve the performance of contrastive representation learning. As the iterative interaction of prototype updating and pseudo label updating, the ambiguities associated with those pseudo labels are gradually eliminated, leading to the understanding of the specific traversability. § EXPERIMENTAL RESULTS §.§ Experimental Datasets To evaluate the proposed method, experiments are conducted on two off-road datasets: the publicly available RELLIS-3D <cit.> dataset and a Gobi Desert driving dataset collected by our UGV. The data collection platforms and some typical scenes of both datasets are shown in Fig. <ref>. The RELLIS-3D dataset consists of five sequences of LiDAR frames collected in a rugged off-road environment using a Warthog all-terrain UGV. The UGV is equipped with an Ouster OS1 LiDAR and a Vectornav VN-300 inertial navigation system. Each LiDAR frame is point-wise annotated with 20 different semantic classes (such as grass, fence, tree, barrier, etc.). Additionally, ground-truth pose for each frame is provided by a high-precision Simultaneous Localization and Mapping (SLAM) system. For our experiments, we select 50 key-frames from sequence 01 for training, 200 random frames from the remaining frames of sequence 01 for validation, and all 2059 frames from sequence 04 for quantitative and qualitative testing. In our Gobi Desert driving dataset, LiDAR frames were collected in a Gobi desert scene. Our UGV is equipped with a Robosense RS-Ruby128 LiDAR and a StarNeto XW-GI7660 GNSS/INS system. High-frequency 6-degree of freedom (DoF) poses with centimeter-level accuracy can be obtained by using an online pose estimation module proposed in our previous work <cit.>. For our experiments, we select 100 key-frames for training, and 1900 frames for qualitative testing. §.§ Evaluation Metrics To quantitatively evaluate the performance of the proposed method, we utilize the annotations from the RELLIS-3D dataset to generate ground-truth traversability maps. The semantic categories are grouped into three traversability levels (traversable, non-traversable, and risky) based on their travel costs. In the process of ground-truth generation, several annotated LiDAR frames are first assembled by using the provided ground-truth poses. The merged dense point cloud is then projected onto a 2D grid map, and the traversability of each grid cell is determined by the semantic labels of the projected points. If all the projected points within a grid cell have labels such as “grass", “puddle", “asphalt", or “concrete", it is considered as a traversable cell; if the labels of all projected points are “bush" or “fence", it is considered as a risky cell; otherwise, it is considered as a non-traversable cell. Given the ground-truth traversability maps, we evaluate the traversability analysis results by two performance metrics widely used for semantic segmentation: Pixel Accuracy (PA) and mean Intersection over Union (mIoU). PA measures the proportion of correctly classified grid cells in the prediction results, and mIoU calculates the degree of overlap between the ground-truth and prediction results. Both metrics provide a quantitative measure of the grid-level prediction accuracy. §.§ Implementation Details In our experiments, we set the resolution of each grid cell to 0.2m× 0.2m, and the map size is set to 40m× 40m. Each terrain patch consists of 11 × 11 (M = 11) grid cells, and each local window comprises 10 × 10 (W = 10) terrain patches. The dimensionality D of the embedding is set to 32. The lengths of the embedding queue Q_e and the predicted label queue Q_l are kept as 81920. The momentum coefficients m_θ and m_p are set to 0.999 and 0.99, respectively. The initial momentum coefficient m_l is set to 0.99, and its value decays polynomially after the initial 10 epochs. For hyper-parameters, we set τ to 0.07, and λ to 0.5. During training, we use Stochastic Gradient Descent (SGD) as the optimizer, with a weight decay of 1e^-5, a momentum of 0.9, and an initial learning rate of 0.02. The network is trained for 50 epochs on a NVIDIA RTX A6000 GPU, with an exponentially decayed learning rate. §.§ Ablation Studies §.§.§ Prototype Num To evaluate how different number of prototypes K affects the performance of the proposed method, an ablation study is conducted with varying number of K. The quantitative experimental results are presented in Fig <ref>. It can be found that an increasing number of K boosts the model's performance until when K = 4, and then the model's performance decreases and tends to converge when K > 4. Therefore, we choose K = 4 as the optimal number of prototypes for our subsequent experiments. Furthermore, we also conduct visualization to gain insights into the generated prototypes and their semantic meaning. First, we visualize the traversability classification results (Fig.<ref>(b)). Notably, the proposed method automatically divides the “bushes" category into “tall bushes" and “low bushes", resulting a finer semantic categories compared to the original annotations (Fig.<ref>(a)). Subsequently, we employ t-SNE <cit.> visualization to explore the embedding space (Fig.<ref>(c)). We observe that well-separated clusters are generated in the embedding space. Each cluster represents a specific semantic category, and can be represented by a prototype. Based on these visualizations, we can interpret the semantic meaning of each prototype. In the subsequent traversability analysis, the grass category corresponds to traversable regions, the tree category corresponds to non-traversable regions, and both low bushes and high bushes are considered as risk regions. §.§.§ Input data To verify the validity of the input terrain feature map F in the proposed method, we conduct an ablation study by varying the forms of input data. In LiDAR-based traversability analysis approaches, a common input data format is the BEV grid map <cit.>. In this ablation study, we consider two common variations of BEV grid maps: the single LiDAR scan BEV (S-BEV) and the multiple LiDAR scans BEV (M-BEV). The S-BEV is generated from a single LiDAR scan, while the M-BEV is formed by fusing multiple LiDAR scans. The S-BEV and M-BEV are applied as two forms of input data in the proposed framework for comparative analysis. The quantitative experimental results are shown in Table <ref>. The results indicate that the model achieves the worst performance when using S-BEV as the input data. This can be attributed to the sparse nature of a single LiDAR scan, which may fail to provide stable and complete representations of the local environment. Although the model's performance improves significantly when using M-BEV as the input data, there still exists a performance gap compared to using F as the input data. The reason is that F contains richer information compared to M-BEV. §.§.§ Encoder Network To evaluate the validity of the proposed local window based transformer (LW-Transformer), two commonly used backbone networks (AlexNet and ResNet-18) are employed as encoders in the proposed framework for comparative analysis. The results in Table <ref> show that model's performance decreases when using AlexNet or ResNet-18 as encoders. The primary reason is that traversability is a spatially-related concept, the traversability of a terrain patch not only depends on the patch itself, but also on those neighboring terrain patches within a certain range. The self-attention mechanism incorporated in the LW-Transformer enables it to capture the implicit spatial dependencies between adjacent terrain patches. This capability is crucial for accurate traversability analysis. In contrast, CNN-based encoder networks lack the modeling of spatial dependencies, which results in performance degradation. §.§.§ Loss Function To validate the impact of the loss functions ℒ_cont and ℒ_cls in the prototype-based contrastive representation learning, we conduct an ablation study by considering each loss function individually. The results presented in Table <ref> clearly indicate that the model's performance decreases significantly when utilizing only ℒ_cont or ℒ_cls as the loss function. This finding validates the necessity of using a joint loss function that combines ℒ_cont and ℒ_cls as Eq. (<ref>) in the proposed method. §.§ Comparative Experiments We compare the proposed method with two recent LiDAR-based traversability analysis approaches. The first one <cit.> is a rule-based approach that estimates the travel cost for each region by using the constructed 3D terrain models. It determines traversability based on some cost thresholds derived from vehicle trajectories. The second approach is a self-supervised learning based off-road drivable area extraction network (ORDAE-Net) <cit.>. ORDAE-Net segments the environments into obstacle regions, traversable regions, and grey regions using vehicle paths and auto-generated obstacle labels. Fig. <ref> shows some qualitative comparison results, and the quantitative results are presented in Table <ref>. The results in Fig. <ref> show that the rule-based approach can roughly distinguish the overall shape of regions with different traversability, but the results often contain a significant amount of noise. The ORDAE-Net excels in detecting non-traversable regions but struggles to distinguish traversable regions from those similar risky regions. In contrast, the proposed method demonstrates superior capability in distinguishing regions with varying traversability. The qualitative results shown in Table <ref> further supports the superiority of the proposed method. It significantly surpasses the other two approaches in terms of both PA and mIoU. §.§ Qualitative Results on Gobi Desert Driving Dataset We further conduct experiments on our Gobi Desert driving dataset to evaluate whether the proposed method can be adapted to different off-road environments. Some qualitative results are presented in Fig. <ref>. It can be observed that the proposed method works well in the Gobi environment which has a high degree of terrain similarity, and accurately identifies those traversable regions. However, as shown in the bottom figure of Fig. <ref>, some gullies are misclassified as traversable regions since the terrain features of these regions closely resemble those of traversable regions, leading to this incorrect classification. § CONCLUDING REMARKS In this paper, we present a novel terrain traversability learning method that leverages a contrastive label disambiguation strategy to learn platform-specific and task-specific traversability in a self-supervised manner, without any human-provided annotations. To achieve this, a prototype-based contrastive representation learning approach is designed to learn discriminative embeddings by using weakly labeled terrain patches obtained from actual driving experiences. 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http://arxiv.org/abs/2307.02273v2	20230705131714	Joint Hierarchical Priors and Adaptive Spatial Resolution for Efficient Neural Image Compression	[ "Ahmed Ghorbel", "Wassim Hamidouche", "Luce Morin" ]	cs.CV	[ "cs.CV", "eess.IV" ]	IEEE Transactions on Circuits and Systems for Video Technology, Under Review, July 2023 Shell et al.: A Sample Article Using IEEEtran.cls for IEEE Journals Joint Hierarchical Priors and Adaptive Spatial Resolution for Efficient Neural Image Compression Ahmed Ghorbel Wassim Hamidouche Luce Morin Ahmed Ghorbel and Luce Morin were with Univ. Rennes, INSA Rennes, CNRS, IETR - UMR 6164, Rennes, France, e-mail: (Ahmed.Ghorbel; Luce.Morin)@insa-rennes.fr. Wassim Hamidouche was with Technology Innovation Institute, Masdar City, P.O Box 9639, Abu Dhabi, UAE, e-mail: [email protected]. August 1, 2023 =================================================================================================================================================================================================================================================================================================================================================================== Recently, the performance of nic has steadily improved thanks to the last line of study, reaching or outperforming state-of-the-art conventional codecs. Despite significant progress, current nic methods still rely on ConvNet-based entropy coding, limited in modeling long-range dependencies due to their local connectivity and the increasing number of architectural biases and priors, resulting in complex underperforming models with high decoding latency. Motivated by the efficiency investigation of the Tranformer-based transform coding framework, namely SwinT-ChARM, we propose to enhance the latter, as first, with a more straightforward yet effective Tranformer-based channel-wise auto-regressive prior model, resulting in an absolute ict. Through the proposed ict, we can capture both global and local contexts from the latent representations and better parameterize the distribution of the quantized latents. Further, we leverage a learnable scaling module with a sandwich ConvNeXt-based pre-/post-processor to accurately extract more compact latent codes while reconstructing higher-quality images. Extensive experimental results on benchmark datasets showed that the proposed framework significantly improves the trade-off between coding efficiency and decoder complexity over the vvc reference encoder (VTM-18.0) and the neural codec SwinT-ChARM. Moreover, we provide model scaling studies to verify the computational efficiency of our approach and conduct several objective and subjective analyses to bring to the fore the performance gap between the aict and the neural codec SwinT-ChARM. All materials, including the source code of aict, will be made publicly accessible upon acceptance for reproducible research. § INTRODUCTION Visual information is crucial in human development, communication, and engagement, and its compression is necessary for effective storage and transmission over constrained wireless and wireline channels. Thus, thinking about new enhanced lossy image compression solutions is a goldmine for scientific research. The goal is to reduce an image file size by permanently removing redundant data and less critical information, particularly high frequencies, to obtain the most compact bit-stream representation while preserving a certain level of visual fidelity. Despite this, optimizing the rate-distortion tradeoff involves a fundamental objective for achieving a high compression ratio and low distortion. Conventional image and video compression standards, including JPEG <cit.>, JPEG2000 <cit.>, H.265/hevc <cit.>, and H.266/vvc <cit.>, rely on hand-crafted creativity within a block-based encoder/decoder diagram. In addition, these codecs employ intra-prediction, fixed transform matrices, quantization, context-adaptive arithmetic encoders, and various in-loop filters to reduce spatial and statistical redundancies and alleviate coding artifacts. However, it has taken several years to standardize a conventional codec. Moreover, existing image compression standards are not anticipated to be an ideal and global solution for all types of image content due to the rapid development of new image formats and the growth of high-resolution mobile devices. On the other hand, with recent advancements in machine learning and artificial intelligence, new nic schemes have emerged as a promising alternative to traditional compression methods. nic consists of three modular parts: transform, quantization, and entropy coding. Each of these components can be represented as follows: i) autoencoders as flexible nonlinear transforms where the encoder (i.e., analysis transform) extracts a latent representation from an input image and the decoder (i.e., synthesis transform) reconstructs the image from the decoded latent, ii) differentiable quantization that quantizes the encoded latent iii) deep generative entropy model estimating the conditional probability distribution of the quantized latent to reduce the rate. Further, these three components are jointly optimized in end-to-end training by minimizing the distortion loss between the original image and its reconstruction, and the rate needed to transmit the bit-stream of latent representation. Recently, we have seen a significant surge of deep learning-based lines of study exploring the potential of ann to develop various nic frameworks, reaching or even outperforming state-of-the-art conventional codecs. Some of these previous works leverage hyperprior-related side information <cit.> to capture short-range spatial dependencies or additional context model <cit.>, and others use non-local mechanism <cit.> to model long-range spatial dependencies. For example, Mentzer <cit.> proposed a generative compression method achieving high-quality reconstructions. In contrast, Minnen <cit.> introduced channel-conditioning and latent residual prediction taking advantage of an entropy-constrained model that uses both forward and backward adaptations. Current research trends has focused on attention-guided compressive transform, as Zhu <cit.> replaced the ConvNet-based transform coding in the Minnen <cit.> architecture with a Transformer-based nonlinear transform. Later, Zou et <cit.> combined the local-aware attention mechanism with the global-related feature learning and proposed a window-based attention module. An additional series of efforts have addressed new entropy coding methods, as Zhu <cit.> proposed a probabilistic vector quantization with cascaded estimation to estimate pairs of mean and covariance under a multi-codebooks structure. Further, Kim <cit.> exploited the joint global and local hyperpriors information in a content-dependent manner using an attention mechanism, He <cit.> adopted stacked residual blocks as nonlinear transform and multi-dimension entropy estimation model. More recently, El-Nouby <cit.> replaced the vanilla vector quantizer with pq <cit.> in a compression system derived from vqvae <cit.> offering a large set of rate-distortion points and then introduced a novel mim conditional entropy model that improves entropy coding by modeling the co-dependencies of the quantized latent codes. Further, Muckley <cit.> introduced a new adversarial discriminator based on vqvae that optimizes likelihood functions in the neighborhood of local images under the mean-scale hyperprior Minnen <cit.> architecture. Other interesting attempts <cit.>, known as coordinate-based or implicit neural representations, have shown good ability to represent, generate, and manipulate various data types, particularly in nic by training image-specific networks that map image coordinates to RGB values, and compressing the image-specific parameters. Through these numerous pioneering works, we can estimate the importance of nic in the research field and the industry. Thus, identifying the main open challenges in this area is crucial. The first one is to discern the most relevant information necessary for the reconstruction, knowing that information overlooked during encoding is usually lost and unrecoverable for decoding. The second challenge is to enhance the trade-off between coding efficiency and decoding latency. While the existing approaches improve the transform and entropy coding accuracy, they still need to improve the decoding latency and reduce the model complexity, leading to an ineffective real-world deployment. To tackle those challenges, we propose a nonlinear transform coding and channel-wise auto-regressive entropy coding built on Swin Transformer <cit.> blocks and paired with a neural scaling network, namely aict. Figure <ref> portrays a high-level diagram to provide a more comprehensive overview of the proposed framework. The contributions of this paper are summarized as follows: * We propose the ict, a nonlinear transform coding and spatio-channel auto-regressive entropy coding. These modules are based on Swin Transformer blocks for effective latent decorrelation and a more flexible receptive field to adapt to contexts requiring short/long-range information. * We propose the aict model that adopts a scale adaptation module as a sandwich processor to enhance compression efficiency. This module consists of a neural scaling network, and ConvNeXt-based <cit.> pre-/post-processor to optimize differentiable resizing layers jointly with a content-dependent resize factor estimator. * We conduct extensive experiments on four widely-used benchmark datasets to explore possible coding gain sources and demonstrate the effectiveness of aict. In addition, we carried out a model scaling analysis and an ablation study to substantiate our architectural decisions. The experimental results reveal the impact of the spatio-channel entropy coding, the sandwich scale adaptation component, and the joint global structure and local texture learned by the attention units through the nonlinear transform coding. These experiments show that the proposed ict and aict achieve respectively -4.65% and -5.11% BD-rate (psnr) reduction over VTM-18.0 while considerably reducing the decoding latency, outperforming conventional and neural codecs in the trade-off between coding efficiency and decoding complexity. The rest of this paper is organized as follows. First, Section <ref> briefly describes the background and related works. Then, Section <ref> presents our overall framework along with a detailed description of the proposed architecture. Further, we devote Section <ref> to present and analyze the experimental results, then, finally, Section <ref> concludes the paper. § BACKGROUND AND RELATED WORKS Over the past years, research has renewed interest in modeling image compression as a learning problem, giving a series of pioneering works <cit.> that have contributed to a universal fashion effect, and have achieved great success, augmented by the efficient connection to variational learning <cit.>. In the early stage, some of these methods adopted ConvNets and activation layers coupled with gdn layers to perform non-linear transform coding over a vae architecture. This framework creates a compact representation of the image by encoding them to a latent representation. The compressive transform squeezes out the redundancy in the image with dimensional reduction and entropy constraints. Following that, some studies focus on developing network architectures that extract compact and efficient latent representations while providing higher-quality image reconstruction. This section reviews relevant nic techniques, including works related to our research, while focusing on the following aspects. First, we briefly present the auto-regressive context related works. Then, we describe the end-to-end nic methods that have recently emerged, including attention-guided and Transformer-based coding. Finally, we introduce adaptive downsampling within the context of neural coding. §.§ Auto-Regressive Context Following the success of autoregressive priors in probabilistic generative models, Minnen <cit.> was the first to introduce autoregressive and hierarchical priors within the variational image compression framework, featuring a mean-scale hyperprior. An additional context model is added to boost the rate-distortion performance. Although the combined model demonstrated superior rate-distortion performance compared to neural codecs, it came with a notable computational cost. Later, Cheng <cit.> proposed the first model achieving competitive coding performance with vvc, using a context model in an auto-regressive manner. They improved the entropy model by using a discretized K-component gmm. In addition, Minnen <cit.> estimated the latent distribution's mean and standard deviation in a channel-wise manner and incorporated an auto-regressive context model to condition the already-decoded latent slices and the latent rounding residual on the hyperprior to further reduce the spatial redundancy between adjacent pixels. Finally, He <cit.> proposed a parallelizable spatial context model based on the checkerboard-shaped convolution that allows parallel-friendly decoding implementation, thus increasing the decoding speed. §.§ Auto-Regressive Context In the mean-scale hyperprior framework <cit.>, an additional context model is added to boost the rate-distortion performance. Considering the entire autoencoder structure (g_a, g_s), hyper autoencoder (h_a, h_s), context model g_cm, and a parameter inference network g_ep which estimates, from the latent ŷ, the location and scale parameters Φ=(μ, σ) of the entropy model. Let h_s(ẑ) denotes the hyperprior feature and g_cm(ŷ_<i) denotes the context feature. The parameter prediction for i-th representation ŷ_i is expressed as follows: Φ_i=g_ep(h_s(ẑ), g_cm(ŷ_<i)), where Φ_i=(μ_i, σ_i) is used to jointly predict entropy parameters, and ŷ_<i ={ŷ_1, …, ŷ_i-1} is the observable neighbors of each symbol vector ŷ_i at the i-th location. Cheng <cit.> proposed the first model achieving competitive coding performance with vvc, using a context model in an auto-regressive manner. They improved the entropy model by using a discretized K-component gmm: p_ŷ\|z(ŷ_i\|ẑ)=∑_0<k<Kπ_i^k[𝒩_(μ_i^k, σ_i^2k) * 𝒰_(-1/2, 1/2)](ŷ_i), where K groups of entropy parameters (π^k, μ^k, σ^k) are calculated by g_ep, 𝒩_ (μ, σ^2) represents the mean and scale Gaussian distribution, and 𝒰_(-1/2, 1/2) denotes the uniform noise. In addition, Minnen <cit.> estimated the latent distribution's mean and standard deviation in a channel-wise manner and incorporated an auto-regressive context model to condition the already-decoded latent slices and the latent rounding residual on the hyperprior to further reduce the spatial redundancy between adjacent pixels. Finally, He <cit.> proposed a parallelizable spatial context model based on the checkerboard-shaped convolution that allows parallel-friendly decoding implementation, thus increasing the decoding speed. §.§ Attention-Guided Coding Attention mechanism was popularized in nlp <cit.>. It can be described as a mapping strategy that queries a set of key-value pairs to an output. For example, Vaswani <cit.> have proposed mha methods in which machine translation is frequently used. For low-level vision tasks <cit.>, spatially adaptive feature activation is made possible by the attention mechanism, focusing on more complex areas, like rich textures, saliency, etc. In image compression, quantized attention masks are used for adaptive bit allocation, e.g., Li <cit.> used a trimmed convolutional network to predict the conditional probability of quantized codes, Mentzer <cit.> relied on a 3D-cnn-based context model to learn a conditional probability model of the latent distribution. Later, Cheng <cit.> inserted a simplified attention module (without the non-local block) into the analysis and synthesis transforms to pay more attention to complex regions. More recently, Zou <cit.> combined the local-aware attention mechanism with the global-related feature learning within an effective window-based local attention block, which can be used as a specific component to enhance ConvNet and Transformer models. §.§ Transformer-based Coding Recently, Transformers have been increasingly used in neural codecs. They exempt convolution operators entirely and rely on attention mechanisms to capture the interactions between inputs, regardless of their relative position, thus allowing the network to focus more on pertinent input data elements. Qian <cit.> replaced the auto-regressive hyperprior <cit.> with a self-attention stack and introduced a novel Transformer-based entropy model, where the Transformer's self-attention is used to relate different positions of a single latent for computing the latent representation. Zhu <cit.> replaced all convolutions in the standard approach <cit.> with Swin Transformer <cit.> blocks, leading to a more flexible receptive field to adapt tasks requiring both short/long-range information, and better progressive decoding of latent. Apart from their effective window-based local attention block, Zou <cit.> proposed a novel symmetrical Transformer (STF) framework with absolute Transformer blocks for transform coding combined with a charm prior. Inspired by the adaptive characteristics of the Transformers, Koyuncu <cit.> proposed a Transformer-based context model, which generalizes the de facto standard attention mechanism to spatio-channel attention. §.§ Adaptive Downsampling Learning sampling techniques were first developed for image classification to improve image-level prediction while minimizing computation costs. stn <cit.> introduced a layer that estimates a parametrized affine, projective, and splines transformation from an input image to recover data distortions and thereby improve image classification accuracy. Recasens <cit.> suggested that when downsampling an input image for classification, salient regions should be "zoomed-in" to learn a saliency-based network jointly. Talebi <cit.> jointly optimize pixel value interpolated at each fixed downsampling location for classification. Marin <cit.> recently argued that a better downsampling scheme should sample pixels more densely near object boundaries and introduced a strategy that adapts the sampling locations based on the output of a separate edge-detection model. Further, Jin <cit.> introduced a deformation module and a learnable downsampling operation, which can be optimized with the given segmentation model in an end-to-end fashion. In the context of nic, Chen <cit.> proposed a straightforward learned downsampling module that can be jointly optimized with any nic kernels in an end-to-end fashion. Based on the stn <cit.>, a learned resize parameter is used in a bilinear warping layer to generate a sampling grid, where the input should be sampled to produce the resampled output. They also include an additional warping layer necessary for an inverse transformation to maintain the same resolution as the input image. § PROPOSED AICT FRAMEWORK In this section, we first formulate the nic problem. Next, we introduce the design methodology for the overall aict architecture, followed by the description of each component individually. §.§ Problem Formulation The objective of nic is to minimize the distortion between the original image and its reconstruction under a specific distortion-controlling hyperparameter. For an input image x, the analysis transform g_a, with parameter ϕ_g, removes the image spatial redundancies and generates the latent representation y. Then, this latent is quantized to the discrete code ŷ using the quantization operator ⌈.⌋, from which a synthesis transform g_s, with parameter θ_g, reconstructs the image denoted by x̂. The overall process can be formulated as follows: y = g_a( x\|ϕ_g), ŷ = ⌈y⌋, x̂ = g_s(ŷ\|θ_g). A hyperprior model composed of a hyper-analysis and hyper-synthesis transforms (h_a, h_s) with parameters (ϕ_h, θ_h) is usually used to reduce the statistical redundancy among latent variables. In particular, this hyperprior model assigns a few extra bits as side information to transmit some spatial structure information and helps to learn an accurate entropy model. The generated hyper-latent representation z is quantized to the discrete code ẑ using the quantization operator ⌈.⌋. The hyperprior generation can be summarized as follows: z = h_a(y\|ϕ_h), ẑ = ⌈z⌋, p_ŷ\|ẑ(ŷ\|ẑ) ← h_s(ẑ\|θ_h). Further, considering a context model g_cm with parameter ψ_cm, and a parameter inference network g_ep with parameter ψ_ep which estimates, from the latent ŷ, the location and scale parameters Φ=(μ, σ) of the entropy model. The parameter prediction for i-th representation ŷ_i is expressed as follows: Φ_i=g_ep(h_s(ẑ), g_cm(ŷ_<i\|ψ_cm) \|ψ_ep), where Φ_i=(μ_i, σ_i) is used to jointly predict entropy parameters, and ŷ_<i ={ŷ_1, …, ŷ_i-1} is the observable neighbors of each symbol vector ŷ_i at the i-th location. p_ŷ_i\|ẑ(ŷ_i\|ẑ)=∑_0<k<Kπ_i^k[𝒩_(μ_i^k, σ_i^2k) * 𝒰_(-1/2, 1/2)](ŷ_i), where K groups of entropy parameters (π^k, μ^k, σ^k) are calculated by g_ep, 𝒩_ (μ, σ^2) represents the mean and scale Gaussian distribution, and 𝒰_(-1/2, 1/2) denotes the uniform noise. Both transform and quantization introduce distortion D = MSE(x, x̂) for mse optimization that measures the reconstruction quality with an estimated bitrate R, corresponding to the expected rate of the quantized latent and hyper-latent, as described below: R = 𝔼 [-log _2(p_ŷ\|ẑ(ŷ\|ẑ)) -log _2(p_ẑ(ẑ)) ]. In the case of adaptive resolution (i.e., aict), we consider the rpn, the downscale, and the upscale modules as (r_s, a_d, a_u) with parameters (ω_r, ω_d, ω_u), respectively. The generation process of x_d and x̂ is described as follows: s = r_s(x\|ω_r), x_d = a_d(x, s\|ω_d), x̂ = a_u(x̂_d, s\|ω_u). Representing (g_a,g_s), (h_a,h_s), (g_cm,g_ep), and (r_s, a_d, a_u) by dnn enables jointly optimizing the end-to-end model by minimizing the rate-distortion trade-off ℒ, giving a rate-controlling hyperparameter λ. This optimization problem can be expressed as follows: ϕ_g^⋆, ϕ_h^⋆, θ_g^⋆, θ_h^⋆, ψ_cm^⋆, ψ_ep^⋆, ω_r^⋆, ω_d^⋆, ω_u^⋆ = ϕ_g, ϕ_h, θ_g, θ_h, ψ_cm, ψ_ep, ω_r, ω_d, ω_uminℒ (x, x̂) = ϕ_g, ϕ_h, θ_g, θ_h, ψ_cm, ψ_ep, ω_r, ω_d, ω_uminD (x, x̂)+λ R, = ϕ_g, ϕ_h, θ_g, θ_h, ψ_cm, ψ_ep, ω_r, ω_d, ω_umin\|\|x- x̂\|\|^2_2 + λ ( ℍ(ŷ) + ℍ (ẑ) _R ), minℒ (x, x̂) = minD (x, x̂)+λ R, = min\|\|x- x̂\|\|^2_2 + λ ( ℍ(ŷ) + ℍ (ẑ) _R ), where ℍ stands for the cross entropy. Finally, we recall that training the model with the gradient descent method requires substituting the quantization with additive uniform noise <cit.>, preventing the gradient from vanishing at the quantization. We follow this method in this paper, where the noisy representations of the latent are used to compute the rate during the training phase. §.§ Overall Architecture The overall pipeline of the proposed solution is illustrated in Figure <ref>. The framework includes three modular parts. First, the scale adaptation module, composed of a tiny rpn <cit.>, a ConvNeXt-based pre-/post-processor, and a bicubic interpolation filter. Second, the analysis/synthesis transform (g_a,g_s) of our design consists of a combination of patch merging/expanding layers and Swin Transformer <cit.> blocks. The architectures of hyper-transforms (h_a,h_s) are similar to (g_a,g_s) with different stages and configurations. Then, a Transformer-based slice transform inside a charm is used to estimate the distribution parameters of the quantized latent. Finally, the resulting discrete-valued data (ŷ, ẑ) are encoded into bit-streams with an arithmetic encoder. §.§ Scale Adaptation Module Given a source image x∈ℝ^H × W × C, we first determine an adaptive spatial resize factor s ∈ℝ∩ [0, 1] estimated by the rpn module, which consists of three stages of resblock. Indeed, the estimated resize parameter s is used to create a sampling grid τ_M following the convention stn, and used to adaptively down-scale x into x_d∈ℝ^H' × W' × C through the bicubic interpolation, with H'=s H and W'= s W. The latter (i.e., x_d) is then encoded and decoded with the proposed ict. Finally, the decoded image x̂_d∈ℝ^H' × W' × C is up-scaled to the original resolution x̂∈ℝ^H × W × C using the same, initially estimated, resize parameter s. The parameterization of each layer is detailed in the rpn and resblock diagrams of Figure <ref> (a) and (b), respectively. In addition, a learnable depth-wise pre-/post-processor is placed before/after the bicubic sampler to mitigate the information loss introduced by down/up-scaling, allowing the retention of information. This neural pre-/post-processing method consists of concatenation between the input and the output of three successive ConvNeXt <cit.> blocks, using depth-wise convolutions with large kernel sizes to obtain efficient receptive fields. Globally, the ConvNeXt block incorporates a series of architectural choices from a Swin Transformer while maintaining the network's simplicity as a standard ConvNet without introducing any attention-based module. These design decisions can be summarized as follows: macro design, ResNeXt's grouped convolution, inverted bottleneck, large kernel size, and various layer-wise micro designs <cit.>. In Figure <ref> (c), we illustrate the ConvNeXt block, where the DConv2D(.) refers to the depthwise 2D convolution, LayerNorm for the layer normalization, Dense(.) for the densely-connected neural network layer, and gelu for the activation function. Finally, it is essential to note that we propose to skip the scale adaptation module for a better complexity-efficient design when the predicted scale does not change the input resolution, i.e., s ≅ 1. The overhead to store and transmit the scale parameter s can be ignored, given the large bitstream size of the image. §.§ Transformer-based Analysis/Synthesis Transform The analysis transform g_a contains four stages of patch merging layer and Swin Transformer block to obtain a more compact low-dimensional latent representation y. In order to consciously and subtly balance the importance of feature compression through the end-to-end learning framework, we used two additional stages of patch merging layer and Swin Transformer block in the hyper-analysis transform to produce the hyperprior latent representation z. During training, both latents y and z are quantized using a rounding function to produce ŷ and ẑ, respectively. During inference, both latents y and z are first quantized using the same rounding function as training and then compressed using probability tables. The quantized latent variables ŷ and ẑ are then entropy coded regarding an indexed entropy model for a location-scale family of random variables parameterized by the output of the charm, and a batched entropy model for continuous random variables, respectively, to obtain the bit-streams. Finally, quantized latents ŷ and ẑ feed the synthesis and hyper-synthesis transforms, respectively, to generate the reconstructed image. The decoder schemes are symmetric to those of the encoder, with patch-merging layers replaced by patch-expanding layers. The Swin Transformer block architecture, depicted in Figure <ref> (d), is a variant of the vit that has recently gained attention due to its superior performance on a range of computer vision tasks. Therefore, it is essential to highlight the unique features and advantages to motivate the choice of the Swin Transformer over other vit variants. One key advantage of the Swin Transformer is its hierarchical design, which enables it to process images of various resolutions efficiently. Unlike other vit variants, Swin Transformer divides the image into smaller patches at multiple scales, allowing it to capture both local and global information. This hierarchical design has been shown to be particularly effective for large-scale vision tasks. Another advantage of the Swin Transformer is its ability to incorporate spatial information into its attention mechanism. Swin Transformer introduces a novel shifted window attention mechanism, which aggregates information from neighboring patches in a structured way, allowing it to capture spatial relationships between image features, leading to linear complexity w.r.t. the input resolution. This attention mechanism has been shown to outperform the standard vit attention mechanism, whose complexity is quadratic, on a range of benchmarks. Overall, Swin Transformer's efficiency and superior performance make it a promising architecture for nic. In addition, its ability to capture both global and local features efficiently, and its adaptability to different image resolutions, make it a strong contender among other transformer-based architectures. §.§ Transformer-based Slice Transform Although there are strong correlations among different channels in latent space, the strongest correlations may come from the spatio-channel dependencies. Thus, to better parameterize the distribution of the quantized latents with a more accurate and flexible entropy model without increasing the compression rate, we propose a Transformer-based slice transform inside the charm. Unlike previous works, our model considers the spatio-channel latent correlations for entropy modeling in an auto-regressive manner. As a side effect, it also leads to faster decoding speed thanks to the Transformer's parallelization abilities on gpu. In contrast to ConvNets that depend on matrix multiplications, and are susceptible to communication overhead while parallelizing across multiple gpu, Transformers exhibit better parallelizability on gpu. This is attributed to their independent output computation mechanism using self-attention, and a structured architecture that facilitates efficient parallelization. The slice transform consists of two successive Swin Transformer blocks with an additional learnable linear projection layer, used to get a representative latent slices concatenation. This charm estimates the distribution p_ŷ (ŷ \| ẑ) with both the mean and standard deviation of each latent slice and incorporates an auto-regressive context model to condition the already-decoded latent slices and further reduce the spatial redundancy between adjacent pixels. § EXPERIMENTAL RESULTS In this section, we first describe the experimental setup, including the used datasets, the baselines against which we compared, and the implementation details. Then, we assess the compression efficiency of our method with a rate-distortion comparison and compute the average bitrate savings on four commonly-used evaluation datasets. We further elaborate a model scaling study to consistently examine the effectiveness of our proposed method against pioneering ones. Additionally, we perform a resize parameter analysis to show the variations of the predicted parameter s. Finally, we conduct a latent analysis, an ablation study, and a qualitative analysis to highlight the impact of our architectural choices. §.§ Experimental Setup Datasets. The training set of the CLIC2020 dataset is used to train the proposed models. This dataset contains professional and user-generated content images in RGB color and grayscale formats. We evaluate image compression models on four datasets, including Kodak <cit.>, Tecnick <cit.>, JPEG-AI <cit.>, and the testing set of CLIC21 <cit.>. Figure <ref> gives the number of images by pixel count for the four test datasets. Finally, for a fair comparison, all images are cropped to the highest possible multiples of 256 to avoid padding for neural codecs. Baselines. We compare our approach with the state-of-art neural compression method SwinT-ChARM proposed by Zhu <cit.>, and the Conv-ChARM proposed by Minnen <cit.>, and non-neural compression methods, including bpg(4:4:4), and the up-to-date vvc official Test Model VTM-18.0 in All-Intra profile configuration. Table <ref> gives the configuration of each of the considered image codec baselines with B1 and B2 referring to Conv-ChARM and SwinT-ChARM, respectively, and O1 and O2 refer to our proposed approaches ict and aict, respectively. C_i and d_i are the hyperparameters defined in Figure <ref>. We intensively compare our solutions with Conv-ChARM <cit.> and SwinT-ChARM <cit.> from the state-of-the-art models <cit.>, under the same training and testing conditions. Nevertheless, Figure <ref> compares our models with additional state-of-the-art solutions. Implementation details. We implemented all models in TensorFlow using tfc library <cit.>, and the experimental study was carried out on an RTX 5000 Ti gpu. All models were trained on the same CLIC2020 training set with 2M steps using the ADAM optimizer with parameters β_1=0.9 and β_2=0.999. The initial learning rate is set to 10^-4 and drops to 10^-5 for the last 200k iterations. The loss function, expression in Equation (<ref>), is a weighted combination of bitrate R and distortion D, with λ being the Lagrangian multiplier steering rate-distortion trade-off. mse is used as the distortion metric in RGB color space. Each training batch contains eight random crops x^j ∈ R^256 × 256 × 3 from the CLIC2020 training set. To cover a wide range of rate and distortion points, for our proposed method and respective ablation models, we trained four models with λ∈{1000, 200, 20, 3}× 10^-5. The inference time experiments on the cpu are performed on an Intel(R) Xeon(R) W-2145 processor running at 3.70 GHz. §.§ Rate-Distortion Performance To demonstrate the compression efficiency of our proposed solutions, we plot the rate-distortion curves of ict, aict, and the baselines on benchmark datasets. Figure <ref> (a) (1^st row) gives the psnr versus the bitrate for our solutions and baselines on the Kodak dataset. The latter figure shows that aict and ict equally outperform the neural approaches Conv-ChARM and SwinT-ChARM, as well as bpg(4:4:4) and VTM-18.0 traditional codecs, achieving a higher psnr values for the different bitrate ranges. Moreover, we introduce Figure <ref> (a) (2^nd row), showing the rate savings over the VTM-18.0 on the Kodak dataset. ict and aict achieve significant rate savings compared to the baselines, demonstrating their ability to compress images more efficiently. More specifically, aict, including the adaptive resolution module, achieves the highest bitrate gain at a low bitrate/quality range, where it is more beneficial to reduce the spatial resolution. To further generalize the effectiveness of our solutions, we extend the evaluation to three high resolutions datasets (Tecnick, JPEG-AI, and CLIC21), as shown in Figure <ref>. The figure illustrates the psnr versus bitrate (1^st row), rate savings (2^nd row) on the considered datasets. aict and ict consistently achieve better rate-distortion performance and considerable rate savings compared to the existing traditional codecs and the neural codecs Conv-ChARM and SwinT-ChARM, demonstrating their efficiency across different high-resolution images and datasets. Furthermore, we assessed the effectiveness of our methods using the perceptual quality metric ms-ssim on the four benchmark datasets. Figure <ref> gives the ms-ssim scores expressed versus the bitrate for the four test datasets. As illustrated in Figure <ref>, our methods yield better coding performance than the current neural baselines in terms of ms-ssim. Note that we haven't optimized our approaches and baselines using ms-ssim, as a differentiable distortion measure in the loss function, during the training process. Thus, optimizing the solutions with ms-ssim will further improve the performance regarding this metric. Besides the rate-distortion rate savings curves, we also evaluate different models using Bjontegaard's metric <cit.>, which computes the average bitrate savings (%) between two rate-distortion curves. In Table <ref>, we summarize the BD-rate (psnr) of image codecs across all four datasets, compared to the VTM-18.0 as the anchor. On average, ict and aict are able to respectively achieve -4.65% and -5.11% rate reductions compared to VTM-18.0 and -3.47% and -3.93% relative gain from SwinT-ChARM. In addition, Table <ref> presents the BD-rate (psnr/ms-ssim) of SwinT-ChARM, ict, and aict across the considered datasets, compared with the anchor Conv-ChARM. Once again, ict and aict are able to outperform the neural approach Conv-ChARM, with average rate reductions (psnr) of -8.79% and -9.13% and average rate reductions (ms-ssim) of -8.18% and -8.36%, respectively, outperforming SwinT-Charm solution on the four benchmark datasets and the two image quality metrics. Overall, the proposed ict and aict have demonstrated strong rate-distortion performance on various benchmark datasets. This indicates that our approaches can better preserve image quality at lower bitrates, highlighting its potential for practical applications in image compression. §.§ Models Scaling Study We evaluated the decoding complexity of the proposed and baseline neural codecs by averaging decoding time across 7000 images at 256×256 resolution, encoded at 0.8 bpp. Table <ref> gives the image codec complexity features, including the decoding time on gpu and cpu, flops, and the total model parameters. Finally, we recall that the models run with Tensorflow 2.8 on a workstation with one RTX 5000 Ti gpu. Compared to the neural baselines, ict can achieve faster decoding speed on gpu but not on cpu, which proves the parallel processing ability to speed up compression on gpu and the well-engineered designs of both transform and entropy coding. This is potentially helpful for conducting high-quality real-time visual data streaming. Our aict is on par with ict in terms of the number of parameters, flops, and latency, indicating the lightweight nature of the scale adaptation module with consistent coding gains over four datasets and two quality metrics. Figure <ref> gives the BD-rate (with VTM-18.0 as an anchor) performance versus the Kflops per pixel of the ict, aict, SwinT-ChARM and Conv-ChARM on the Kodak dataset. We can notice that ict and aict are in an interesting area, achieving a good trade-off between BD-rate score on Kodak, total model parameters, and Kflops per pixel, reflecting an efficient and hardware-friendly compression model. Finally, Figure <ref> shows the BD-rate (with VTM-18.0 as an anchor) versus the decoding time of various codecs on the Kodak dataset. It can be seen from the figure that our ict and aict achieve a good trade-off between BD-rate performance and decoding time, outperforming state-of-the-art learning and conventional solutions. Furthermore, the symmetrical architecture of the proposed solutions allows similar complexity at both the encoder and decoder. This feature can be an advantage of neural codecs, since the best conventional codecs like vvc exhibit more complex encoding than decoding. §.§ Resize Parameter Analysis We conduct a resize parameter analysis through the benchmark datasets, including the images of the four datasets with various resolutions, as illustrated in Figure <ref>. Figure <ref> shows how the parameter s varies according to the weighting parameter λ (i.e., bitrate) for the four datasets. First, we can notice that the estimated resize parameter s depends on the bitrate and the spatial characteristics of the image content. Resizing the input image to a lower resolution is frequently observed at a low bitrate, where the compression removes the image details. In contrast, the down-sampling is not performed at a high bitrate to reach high image quality, significantly when the up-sampling module cannot recover the image details at the decoder. Nevertheless, even at a high bitrate, a few samples are down-sampled to a lower resolution, especially images with low spatial information that the up-sampling module can easily recover on the decoder side. This also can explain the higher coding gain brought by the adaptive sampling module of aict on datasets, including more high-resolution images such as JPEG-AI and CLIC21 (see Figure <ref>). In addition, skipping the resize modules for a predicted scale close to 1 s ≅ 1 contribute to reducing encoding and decoding complexity. §.§ Latent Analysis Transform coding is motivated by the idea that coding is more effective in the transform domain than in the original signal space. A desirable transform would decorrelate the source signal so that a simple scalar quantization and factorized entropy model can be applied without constraining coding performance. Furthermore, an appropriate prior model would provide context adaptivity and utilize distant spatial relations in the latent tensor. The effectiveness of the analysis transforms g_a can then be evaluated by measuring the level of correlation in the latent signal ŷ. We are particularly interested in measuring the correlation between nearby spatial positions, which are heavily correlated in the source domain for natural images. In Figure <ref>, we visualize the normalized spatial correlation of ŷ averaged over all latent channels and compare Conv-ChARM and SwinT-ChARM with the proposed ict at λ=0.002. We can observe that while both lead to small cross-correlations, ict can decorate the latent with a slight improvement when compared to SwinT-ChARM and a much considerable improvement when compared to Conv-ChARM. This suggests that Transformer-based transforms with Transformer-based entropy modeling incur less redundancy across different spatial latent locations than convolutional ones, leading to an overall better rate-distortion trade-off. §.§ Ablation Study To investigate the impact of the proposed ict and aict, we conduct an ablation study according to the reported BD-rate↓ w.r.t. vvc and Conv-ChARM in Table <ref> and Table <ref>, respectively. Image compression performance increases from Conv-ChARM to SwinT-ChARM on the considered datasets due to the inter-layer feature propagation across non-overlapping windows (local information) and self-attention mechanism (long-range dependencies) in the Swin Transformer. With the proposed spatio-channel entropy model, ict is able to achieve on average -3.47% (psnr) and -1.39% (ms-ssim) rate reductions compared to SwinT-ChARM. Moreover, aict is enhancing ict on average by -0.46% (psnr) and -0.18% (ms-ssim) rate reductions with consistent gain over the four datasets. This indicates that introducing a scale adaptation module can further reduce spatial redundancies and alleviate coding artifacts, especially at low bitrate for higher compression efficiency. More importantly, the adaptive resolution may also reduce the complexity of the encoder and decoder regarding the number of operations per pixel since fewer pixels are processed on average by the codec when the input image is downscaled to a lower resolution, i.e., s < 1. §.§ Qualitative Analysis To assess the perceptual quality of the decoded images, we visualize two reconstructed samples with the proposed ict and aict methods, along with Conv-ChARM and SwinT-ChARM, all trained at the same low bitrate configuration (λ=0.002). Figure <ref> presents the visualization of the reconstructed kodim14 and kodim04 images from the Kodak dataset. Under a similar rate budget, ict and aict perform better in maintaining texture details and clean edges while suppressing visual artifacts of the decoded images compared to Conv-ChARM and SwinT-ChARM neural approaches. Additionally, the self-attention mechanism focuses more on high-contrast image regions and consequently achieves more coding efficiency on such content. § CONCLUSION In this work, we have proposed an up-and-coming neural codec aict, achieving compelling rate-distortion performance while significantly reducing the latency, which is potentially helpful to conduct, with further optimizations, high-quality real-time visual data transmission. We inherited the advantages of self-attention units from Transformers to approximate the mean and standard deviation for efficient entropy modeling and combine global and local texture to capture correlations among spatially neighboring elements for non-linear transform coding. Furthermore, we have presented a lightweight scale adaptation module to enhance compression ability, especially at low bitrates. The experimental results, conducted on four datasets, showed that ict and aict approaches outperform the state-of-the-art conventional codec vvc, achieving respectively -4.65% and -5.11% BD-rate reductions compared to the VTM-18.0 by averaging over the benchmark datasets. With the development of gpu chip technology and further engineering optimization, neural codecs will be the future of coding, achieving better compression efficiency when compared with traditional codecs and aiming to bridge the gap to a real-time operation. IEEEtran
http://arxiv.org/abs/2307.00882v1	20230703092348	Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures	[ "David Machado", "Roberto Mulet", "Federico Ricci-Tersenghi" ]	cond-mat.stat-mech	[ "cond-mat.stat-mech" ]	Group of Complex Systems and Statistical Physics, Department of Theoretical Physics, University of Havana, Cuba Group of Complex Systems and Statistical Physics, Department of Theoretical Physics, University of Havana, Cuba Dipartimento di Fisica, Sapienza Università di Roma, and CNR-Nanotec, Rome unit and INFN, Sezione di Roma1, 00185 Rome, Italy We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, that translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes very well the out-of-equilibrium dynamics at high temperatures, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms, not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of the glassy dynamics), but with a relaxation timescale too short. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations. Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures Federico Ricci-Tersenghi August 1, 2023 ========================================================================================================================== § INTRODUCTION A myriad of problems from condensed matter physics <cit.>, combinatorial optimization <cit.>, neuroscience <cit.> and machine learning <cit.> are formulated via the extremization of some function of N variables. The optimization landscape is sometimes very complex, and the solution becomes difficult, or impossible, because of the presence of local attractors that slow down the exploration of the configuration space. The system is called frustrated <cit.>. The relevance of the subject is widely recognized, and some theoretical tools for the study of systems' equilibrium properties are now becoming well-established science, after years of applications in diverse contexts. The replica symmetry breaking <cit.>, the cavity method <cit.> and the Thouless-Anderson-Palmer approach <cit.> allowed to tackle numerous problems. On the other hand, the progress in out-of-equilibrium situations has been considerably slower. The field lacks a general understanding of the problems, and their treatments are highly specific depending on the type of variables (continuous <cit.> or discrete <cit.>), their interactions topology (fully-connected <cit.>, random <cit.> or latticed <cit.>), or the nature of time itself (also continuous <cit.> or discrete <cit.>). If we complicate the scenario by looking at problems with a known spin-glass phase in equilibrium, the available results are even more scarce. The pioneering work of Sompolinsky and Zippelius <cit.> studied the continuous-time dynamics of a soft-spin version of the Sherrington-Kirkpatrick (SK), where the variables are continuous. This case is usually modelled with a Langevin formalism as it is possible to write a differential equation directly for the soft-spin variables. Later, Sompolinsky <cit.> attempted the construction of a consistent mean-field theory that takes account of both equilibrium and non-equilibrium spin-glass behavior. Motivated by the dynamical properties of spin-glasses, like the aging regime <cit.>, a series of works <cit.> pointed some inconsistencies in Sompolinsky's approach and exploited the picture of weak ergodicity breaking <cit.> to re-visit the problem. Besides the study of the SK model <cit.>, they included a simpler but illustrative model: the spherical spin-glass model with p-spin interactions <cit.>. The theory reproduced aging and allowed for the existence of multiple time-scales. However, recently, this theory has been questioned for more complicated spherical models <cit.> and for Ising models on sparse random graphs <cit.>. While the spherical p-spin is a long-range model (fully-connected) with continuous variables, we focus our attention on the diluted ferromagnetic p-spin model, defined for discrete spin variables on sparse random graphs. Although in our case there is no quenched disorder in the interactions, and despite all the differences mentioned before, there is a common phenomenology. At low temperatures, both models exhibit multiple relaxation processes within the dynamic evolution, one of them with a very large characteristic time scale. This relation between traditional mean-field spin-glass models and more realistic structural glass models has been discussed in various occasions <cit.>. Our work exploits shared aspects between them, like the existence of multiple time scales, to give a simple mean-field theory to describe the spin glass dynamics of discrete variables on random graphs. Section <ref> introduces a new closure for the master equation governing the system dynamics, written for a single instance of the interactions graph. For simplicity, we applied them to the p-spin ferromagnet on random regular hypergraphs, where everything is reduced to one average case equation. In Section <ref> we generalize the work of Montanari and Semerjian <cit.> on dynamical phase transitions to the out-of-equilibrium scenario. The latter allows us to re-interpret the results of the dynamical theory presented in Section <ref> by providing a way to compute a new time scale for our calculations. The results are compared with Monte Carlo simulations in Section <ref>. § CONDITIONED DYNAMIC APPROXIMATION In its simplest form, the ferromagnetic p-spin model is defined by the Hamiltonian H = -∑_i_1, i_2, …, i_pσ_i_1σ_i_2…σ_i_p, where we have N discrete-spin variables σ_i=±1. The interaction is structured in groups, called factor nodes or plaquettes, of exactly p variables. The continuous time dynamics of the probabilities P^t(σ⃗) of having some configuration σ⃗={σ_1, σ_2, …, σ_N } is governed by the Master Equation <cit.>: d P(σ⃗)/dt = -∑_k=1^N r_k(σ⃗) P(σ⃗) + ∑_k=1^N r_k(F_k[σ⃗]) P(F_i[σ⃗]) where r_k(σ⃗) is the probability per time unit that the spin σ_k changes to -σ_k, when the system's configuration is σ⃗. These are called dynamic transition rates. The operator F_k[·] takes any configuration σ⃗ and flips the k-th spin to get: σ⃗' = {σ_1, σ_2, …, -σ_k, …, σ_N }. When the transition rates r_k depends only on the variables that directly interact with the k-th node, we can choose some site i and sum (<ref>) over all the configurations that keep the spin σ_i fixed, the result is the local equation: d P(σ_i)/dt = - ∑_σ_∂ i r_i(σ_i, σ_∂ i) P(σ_i, σ_∂ i) + ∑_σ_∂ i r_i(-σ_i, σ_∂ i) P(-σ_i, σ_∂ i) The symbol ∂ i represents the set of nodes that interact with i according to the model's Hamiltonian. Of course, we have an equation like (<ref>) for all spins in the system. But these are not the only equations we can get. We could in principle marginalize (<ref>) keeping fixed a plaquette of p interacting variables and thus obtain a differential equation for the probability of a plaquette's configuration, or we could fix a variable σ_i and all its neighborhood σ_∂ i to obtain a differential equation for P(σ_i, σ_∂ i): d/dt P(σ_a) = -∑_i ∈ a∑_σ_∂ i ∖ a[ r_i(σ_i, σ_∂ i) P(σ_∂ i ∖ a, σ_a) - r_i(-σ_i, σ_∂ i) P(σ_∂ i ∖ a, F_i[σ_a]) ] d/dt P(σ_i, σ_∂ i) = -r_i(σ_i, σ_∂ i) P(σ_i, σ_∂ i) + r_i(-σ_i, σ_∂ i) P(-σ_i, σ_∂ i) - - ∑_b ⊂∂ i∑_j ∈ b ∖ i∑_σ_∂ j ∖ b[ r_j(σ_j, σ_∂ j) P(σ_∂ j ∖ b, σ_∂ i, σ_i) - r_j(-σ_j, σ_∂ j) P(σ_∂ j ∖ b, F_j[σ_∂ i], σ_i) ] Here, the indexes i and j represent variable nodes, while the indexes a, b denote plaquettes, and σ_a is the configuration of the variables inside the plaquette a. By choosing each time a larger group of spins we can construct a hierarchical system of differential equations that we should truncate at some point. The simpler approximation that one can make is to neglect all connected correlations C_ij=⟨σ_i σ_j ⟩ - ⟨σ_i ⟩ ⟨σ_j ⟩, closing the system at the level of the equation (<ref>). In practice, to substitute P(σ_∂ j ∖ b, σ_∂ i, σ_i) by P(σ_i) ∏_k ∈∂ i ∖ j P(σ_k) In the sake of simplicity and concretness, let us consider a random regular hypergraph, where all variables σ_i participate in the same number c of plaquettes (c is the node's connectivity), and all plaquettes contain exactly p=3 variables. With homogeneous initial conditions, the system can be characterized by just one differential equation. d ϕ/dt = -∑_u=0^ccu r(u) [f_1(ϕ)]^u [f_2(ϕ)]^c-u ϕ + ∑_u=0^ccu r(u) [f_2(ϕ)]^u [f_1(ϕ)]^c-u (1 - ϕ) where ϕ(t) is the probability that a spin points up, f_1(ϕ) = 2 ϕ (1-ϕ) and f_2(ϕ)=ϕ^2 + (1-ϕ)^2. Within this approximation, starting the relaxation at low temperatures from the initial condition ϕ(0)=1/2, the equation (<ref>) gives ϕ̇(0) = 0 and ϕ(t)=1/2 at all times. The energy density is then easily computed as e(t) = 0 for all times, which obviously is very different from the real dynamics of the model. A less trivial approximation, similar to the one employed in <cit.> is the following: P(σ_a ∖ i , σ_i, σ_b ∖ i) = P(σ_b ∖ i\|σ_i, σ_a ∖ i) P(σ_a) ≈ P(σ_b ∖ i\|σ_i) P(σ_a) P(σ_a ∖ i , σ_i, σ_b ∖ i) ≈ P(σ_a) P(σ_b)/P(σ_i) In the first line of (<ref>) we approximated the conditional probability P(σ_b ∖ i\|σ_i, σ_a ∖ i) by P(σ_b ∖ i\|σ_i), thus neglecting the connected correlations C_a ∖ i, b ∖ i = ⟨σ_a ∖ iσ_b ∖ i⟩ - ⟨σ_a ∖ i⟩⟨σ_b ∖ i⟩. By applying (<ref>) to all sites i=1, …, N, we obtain that all connected correlations of variables at distance d=2 in the graph are zero. Now the values of C_ij=⟨σ_i σ_j ⟩ - ⟨σ_i ⟩ ⟨σ_j ⟩, where i and j are at distance d=1, are the only ones which can be non-zero. In this case, the same initial condition P(σ_i)=1/2 for all i in a random regular hypergraphs leads to a single equation for P(σ_a), where we can drop the index a and define Φ≡∑_σ_a P(σ_a) δ( ∏_k ∈ aσ_k, -1) The corresponding differential equation is: d Φ/dt = -∑_u=0^c - 1c-1u r(u + 1) Φ^u+1 (1 - Φ)^c - 1 - u + ∑_u=0^c - 1c-1u r(u) Φ^u (1 - Φ)^c - u In what follows we call (<ref>) the Conditional Dynamic Approximation of the first order (CDA-1), for reasons that will be clearer latter. This equation can be numerically integrated in time to get a non-trivial relaxation of the energy density e(t). The results for the Glauber dynamics of this model, where we make a specific choice for the rates r_i(σ_i, σ_∂_i) = 1/2( 1 - σ_i tanh[β J ∑_k ∈∂ iσ_k] ), are compared with Monte Carlo simulations in Fig. <ref>. We represent the CDA-1 with dashed lines. For high temperatures, see Fig. <ref>, the approximation works reasonably well for the transient regime and provides the right stationary value for the energy. However, below the dynamical spin glass transition, which occurs at temperature T_d ≈ 0.51, the CDA-1 approximation is very poor and returns a relaxation very far from the behavior measured in the simulations (see Fig. <ref>). We are interested precisely in this two-step relaxation, which is typical of glassy dynamics at low temperatures. Given that we cannot obtain a non-zero correlation between neighboring plaquettes from the CDA-1, we will take another step. Analogously to (<ref>), to close equation (<ref>) we can write: P(σ_∂ j ∖ b, σ_i, σ_∂ i) = P(σ_∂ j ∖ b\|σ_i, σ_∂ i) P(σ_i, σ_∂ i) ≈ P(σ_∂ j ∖ b\|σ_j, σ_b ∖ j) P(σ_i, σ_∂ i) P(σ_∂ j ∖ b, σ_i, σ_∂ i) ≈ P(σ_j, σ_∂ j)/P(σ_i, σ_b ∖ i) P(σ_i, σ_∂ i) = P(σ_j, σ_∂ j)/∑_σ_∂ j ∖ bP(σ_j, σ_∂ j) P(σ_i, σ_∂ i) Fig. <ref> illustrates the meaning of equation (<ref>), which we call Conditioned Dynamic Approximation of the second order (CDA-2) in what follows. The target is the conditional probability P(σ_∂ j ∖ b\|σ_i, σ_∂ i), which involves the configuration of the nodes in the neighborhoods of i and j. By approximating this probability by P(σ_∂ j ∖ b\|σ_j, σ_b ∖ j) we are neglecting the effect of the nodes which are at distance d=3 from ∂ j ∖ b (see Fig. <ref>), and thus not considering connected correlations between spins which are at a distance greater than d=2 from each other. Then we say that the closure is of the second order in space. It is straightforward to generalize (<ref>) and then write other levels of approximation: CDA-3, CDA-4, etcetera. After putting Eqs. (<ref>) and (<ref>) together we obtain a closed system of differential equations for P(σ_i, σ_∂ i). d/dt P(σ_i, σ_∂ i) = -r_i(σ_i, σ_∂ i) P(σ_i, σ_∂ i) + r_i(-σ_i, σ_∂ i) P(-σ_i, σ_∂ i) - - ∑_b ⊂∂ i∑_j ∈ b ∖ i∑_σ_∂ j ∖ b r_j(σ_j, σ_∂ j) P(σ_j, σ_∂ j)/∑_σ_∂ j ∖ bP(σ_j, σ_∂ j) P(σ_i, σ_∂ i) + + ∑_b ⊂∂ i∑_j ∈ b ∖ i∑_σ_∂ j ∖ b r_j(-σ_j, σ_∂ j) P(σ_j, σ_∂ j)/∑_σ_∂ j ∖ bP(-σ_j, σ_∂ j) P(σ_i, F_j[σ_∂ i]) The equations (<ref>) do not contain any further assumption about the actual structure of the interactions. Actually, it is not even necessary to have exactly p spins in each plaquette for these equations to be valid. The dynamical rates r_i(σ_i, σ_∂ i) can also take any the form of any function of the spin on site i and its neighborhood. Thus, in the way we presented it, the CDA-2 is of general purpose. If we consider again a random regular hypergraph with the initial condition P(σ_i) = 1/2, ∀ i=1, …, N, it is possible to write an average case version of the equations that simplifies the numerical computations. The Appendix <ref> contains the derivation of these equations. We concentrate here on their solution. Fig. <ref> illustrates the behaviour of e(t) for c=3, computing results for three specific temperatures. Together with Monte Carlo simulations (points) and the previous approximation (CDA-1, with dashed lines), we represent the results of the CDA-2 with continuous lines. When the temperature decreases, the theoretical technique becomes slightly less accurate to describe the transient regime but it keeps predicting the steady state of the system very accurately. In all cases, the transient regime obtained from the CDA-2 is closer to the simulations than the results of the CDA-1. More importantly, Fig. <ref> shows the behaviour of the model below T_d≈0.51. In this case, the energy relaxation obtained from the CDA-2 approximation takes place through two different relaxation processes. This important feature of the glassy dynamics has been obtained thanks to the fact that the CDA-2 approximation takes into account the correlation between nearest neighbour energy defects. Such a correlation starts playing a key role exactly at the energy value where the first plateau develops (let us call e_p this energy value). The plateau at e_p becomes much longer (mind the log scale) when the temperature is decreased. Notwithstanding the important result about the two-steps relaxation (not captured by simpler closure schemes), the CDA-2 approximation shows a too-short time scale to leave the plateau and an asymptotic energy value which depends on the temperature and is different from the energy reached in Monte Carlo numerical simulations. We believe these two failures can be ascribed to the following approximation underlying any mean-field closure scheme: at every time during the dynamics, the average is taken over a measure assuming that all relevant configurations are easily accessible by the dynamics itself. This is in general true at high enough energies, where correlations are weak. However, for low enough energies correlations become very strong (e.g. the correlation between energy defects below e_p) and make the microscopic time scale to sample the measure larger. When this microscopic timescale grows, the curves like e(t) should be “stretched” and eventually, if the divergence of such a time scale takes place, the relaxation would reach a stop. It is clear that at the plateau energy e_p some relevant barrier must appear: indeed, the T=0 dynamics is not able to relax below e_p both in Monte Carlo simulations and in the mean-field equations. These barriers are likely to have an entropic origin as discussed in detail in Ref. <cit.>. Thus we can safely assume that no barrier is present for e>e_p (at least along the typical trajectories followed by the relaxation dynamics) and the microscopic time scale can be set to the (conventional) unit value in such a regime. On the contrary, barriers are present below e_p. But these are non-extensive barriers, that remain finite in the large N limit, as witnessed by the observation that the dynamics with any T>0 can relax below e_p. Fig. <ref> also shows that Monte Carlo simulations relax to an asymptotic energy e_d≈ -0.96 strictly larger than -1. This is the so-called dynamical threshold energy. At this energy value, barriers become extensive, i.e. diverge in the large N limit, and the relaxation gets stuck. In the next Section, we are going to compute the dynamical threshold energy, extending the classical computation of the point-to-set correlation function to the out-of-equilibrium regime. § NON-EQUILIBRIUM DYNAMICAL TRANSITION Models on random graphs that manifest frustration have been shown to exhibit purely dynamical phase transitions <cit.>, where the ergodicity is broken but the free energy remains analytic. In such a complex situation, the work of Montanari and Semerjian <cit.> gives a simple method to detect the occurrence of the transition. In short, the method reduces to look at the relation of a variable σ_0 in the system, with the variables {σ_l } at a distance l of σ_0. For a given configuration of {σ_l }, it is possible to use a message passing technique <cit.> to compute the expected value m_0^l of the spin σ_0. As we explain in the Appendix <ref>, a set of self-consistent equations allows to obtain the probability distribution of expected values, Q^l(m_0^l), as a function of the distance l. Then the magnitudes to measure are the point-to-set correlation 𝒞^l = ∫ dm Q^l (m) m and the corresponding length l(ϵ) = min{ l: 𝒞^l < ϵ}. The point-to-set correlation length l(ϵ) diverges exactly at T_d, regardless of the value chosen for ϵ. Thus, the problem of detecting the presence of a dynamic phase transition is simplified to the determination of the divergence of a single magnitude <cit.>. Here, we extend the calculation to consider local measures that are completely out of equilibrium. Let us assume that from a dynamical computation we are able to obtain the local probabilities: P(S_a ∖ i, S_b ∖ i\|σ_i) = ∑_σ_a ∖ i∑_σ_a ∖ i P(σ_a ∖ i, σ_b ∖ i\|σ_i) δ(S_a ∖ i , ∏_j ∈∂ a ∖ iσ_j) δ(S_b ∖ i, ∏_k ∈∂ b ∖ iσ_k) where δ(x, y) stands for the Kronecker delta on x and y. With the information contained in (<ref>) we can write the set of self-consistent equations for the cavity probability distributions Q^l(μ) and Q^l(μ̂) (see Appendix <ref>): Q^l(μ) = ∑_S_1, …, S_c-1π̂(S_1, …, S_c-1) ∫[∏_i=1^c-1 dμ̂_i Q̂^l, S_i(μ̂_i) ] δ( ℱ_1[μ, {μ̂_i }_i=1^c-1] ) Q̂^l, S(μ̂) = 2^-p+2∑_σ_1, …, σ_p-1δ(S, ∏_j=1^p-1σ_j) ∫[∏_j=1^p-1 dμ_j Q^l-1(σ_j μ_j) ] δ(ℱ_2^S[ μ̂, {μ_j }_j=1^p-1] ) where we considered again the case where P(σ_i) = 1/2 for all i=1, …, N and have defined π̂(S_1, …, S_c-1) ≡ P(S_1, …, S_c-1\|σ = 1). The Dirac delta functions in these equations enforce the relations: ℱ_1[μ, {μ̂_i }_i=1^c-1] = 1 + μ/2 - ∏_i=1^c-1(1 + μ̂_i )/∑_σ[∏_i=1^c-1(1 + σμ̂_i ) ] ℱ_2^S[ μ̂, {μ_j }_j=1^p-1] = μ̂ - S ⟨ S ⟩∏_j=1^p-1μ_j where ⟨ S ⟩ = ∑_S S π̂(S) Now we can explore the solutions to Eqs. (<ref>) and (<ref>) to locate the divergence of the corresponding dynamic point-to-set correlation. For the sake of simplicity, we write the approximation π̂(S_1, …, S_c-1) ≈∏_i=1^c-2π̂(S_i, S_i+1) / ∏_i=2^c-2π̂(S_i) for any local probability π̂. The “pair” probabilities π̂(S_i, S_i+1) can be exactly written in terms of two parameters: the energy density e and the correlation between neighboring plaquettes C. Fig. <ref> shows the position in the plane (e,C) of the divergence of the point-to-set correlation, i.e. the physical time scale, obtained from Eqs. (<ref>) and (<ref>). In Fig. <ref> we report the same data shown in Section <ref>, but as a parametric plot of (e(t),C(t)). In this representation the absolute time becomes irrelevant and we observe a good similarity between the Monte Carlo data (points) and the analytical results based on the CDA-2 approximation (lines). This is a strong indication that a proper redefinition of the microscopic time scale in the dynamical mean-field equations could eventually provide a solution very close to the physical one. The inset in Fig. <ref> is a zoom on the region reached at very long times. The black line with points marks the place where the point-to-set correlation length diverges. As Ref. <cit.> points out, this is to be associated with the divergence of the physical time scale. Indeed the Monte Carlo simulation is not able to go to the left of this line, into the non-ergodic region (see Fig. <ref>), and it seems to converge very closely to the dynamical critical line at large times. On the contrary, the CDA-2 approximation returns trajectories that enter into the non-ergodic region, evidently ignoring the dynamical phase transition. § SETTING THE PROPER TIME SCALE IN THE DYNAMICAL MEAN-FIELD EQUATIONS As shown in Fig. <ref>, at low temperatures, the CDA-2 approximation shows up two failures compared to the physical dynamics obtained through Monte Carlo simulations: (i) a too-fast relaxation below the plateau energy e_p and (ii) a convergence to energy values below the dynamical threshold energy e_d (see the inset of Fig. <ref>), in the non-ergodic region, which should not be accessible on finite time scales. Nonetheless, in the (e,C) plane, the CDA-2 approximated trajectories follow very closely the true dynamics, showing non-equilibrium correlations (i.e. C≠ 0). So, we foresee the possibility of achieving a very good matching between the true and the CDA-2 approximated dynamics by just redefining the microscopic timescale in the dynamical mean-field equations. The reason beyond this sort of time-reparametrization comes from the following argument: within the mean-field approximation, at every time, one takes the average over a particular probability distribution that should correspond to the configurations that the system is likely to visit at that time. As long as this probability distribution is well ergodic, the hypothesis that the required average can be taken in a short time is reasonable. However, when barriers come into play and the probability distribution to sample is no longer well ergodic, this could, in turn, correspond to a longer effective time scale. Only if this longer time scale is used in the solution to the mean-field dynamical equations, then there is a chance of describing the actual dynamics. The same argument can be brought to the extreme consequences when the CDA-2 approximated trajectory approaches the dynamical critical line, where the relaxation time scale diverges because in the probability measure the ergodicity breaks down. As discussed above, the microscopic time scale does not need to be renormalized for energies large enough (e>e_p) because in this region there are no barriers. However, below e_p barriers arise and we assume that the exploration of the accessible configurations is slowed down by a factor exp(Δ_S). The entropic barrier Δ_S is constant with respect to time and does not depend on the temperature. Finally, approaching the dynamical phase transition, the microscopic time scale must diverge and we assume a simple power law divergence (e-e_d)^-γ. So, the simpler ansatz for the effective time scale is the following τ̂(e) = {[ 1 e_p < e; e^Δ_S((e_p-e_d) / (e - e_d))^γ e_d < e < e_p; ∞ e < e_d ]. This ansatz depends on two important energy values, e_p and e_d, that we now discuss in detail, and two parameters, Δ_S and γ, that will be adjusted to match the actual physical dynamics. On the one hand, e_p is the energy density of the intermediate plateau, which does not depend on the temperature, and thus can be computed by solving the CDA-2 equations at T=0. On the other hand, e_d marks the frontier between ergodic and non-ergodic configurations as computed in the Section <ref> (see Fig. <ref>). As already discussed, e_d marks the place where the physical time scale diverges. Below e_d the system relaxation would proceed by activation processes, which are ignored in the present approach. The expression in Eq. (<ref>) for the effective time scale depends on four parameters: e_p, e_d, Δ_S and γ. Two of them, e_p and e_d, can be computed analytically as explained above. The remaining two, Δ_S and γ, will be fixed by best fitting the data from Monte Carlo simulations. We stress, however, that their values do not depend on the temperature and so we have to fix just two parameters to describe the relaxation in the entire low temperature phase. Fig. <ref> shows a comparison between Monte Carlo simulations and the dynamics derived from the CDA-2 equations when the proper time scales τ̂(e) is taken into account. We observe a very good agreement between the actual physical dynamics and the predictions from the CDA-2 approximation. The results are presented for two different connectivities (c=3 and c=5) in order to show their robustness. The best-fitting parameters are reported in the caption. While Δ_S seems to change a little bit with c, the γ parameter is very stable (γ=2 fits perfectly the data). This γ value lies within the bounds derived in Ref. <cit.>. In particular, one could be tempted to compare it to the results of the Monte Carlo simulations reported in Ref. <cit.> (γ_ MC≈ 3.2). However, the key observation is that the computation of Ref. <cit.> is made on the equilibrium measure, while here we are studying an out-of-equilibrium process. The latter is likely to make smart choices to relax towards the lowest energy e_d and it is not surprising that along these smart relaxation paths, the divergence of the time scale approaching the ergodicity-breaking transition is less severe. § CONCLUSIONS While the glassy dynamics of continuous and unbounded variables interacting through a fully connected topology has been solved a long time ago <cit.> (although maybe not in all the aspects <cit.>) the glassy dynamics of other types of models is very hard to approximate at the analytical level. In this work, we have made an important step forward in the study of Ising models defined on sparse random graphs. By using a continuous time description, we have been able to reproduce faithfully the relaxation of the energy in the low-temperature regime of the diluted Ising p-spin model, where such a relaxation takes place following a two steps process (see Fig. <ref>). Our solution takes into account two crucial aspects of the dynamics, which were not fully considered in previous works. On the one hand, we derive the Conditioned Dynamical Approximation, which is a closure scheme for the master equation that keeps track of the correlation between nearest neighbour energy defects. This correlation is related to the presence of local barriers that need to be crossed in order to proceed with the relaxation to lower energies. On the other hand, we realized that the mean-field equations are valid only under the assumption that all configurations with given mean-field parameters are well sampled: this can happen only on a time scale that grows approaching the ergodicity breaking transition. Following this idea, we have computed the point-to-set correlation <cit.> on the out-of-equilibrium measure predicted by the mean-field approximation, and defined a proper effective time scale τ̂(e) for the mean-field evolution, see Eq.(<ref>). The ideas above provide a very satisfying result, with the solution to the CDA-2 dynamical mean-field equations matching perfectly the complex energy relaxation measured in Monte Carlo simulations. Nonetheless, there are many possible extensions of our approach which is worth pursuing in the near future. For example, observables depending on two times may show aging, even when one-time observables (like the energy) have reached their stationary value <cit.>. Checking the quality of the mean-field approximation presented in this work for those observables is certainly very useful. Moreover, we foresee the possibility to extend the present approach to other interaction topologies, like non-regular random graphs, or even graphs with short loops. The extension to other models (e.g. vector spin models) is very open and requires some analytical work in addition. A quite different, but equally interesting, developing direction is to consider dynamics different from the Glauber one, eventually dynamics not satisfying detailed balance. This very broad class of processes are fundamental, for example, in the description of smart search algorithms to solve combinatorial optimization problems. Our methodology could be used to predict possible regions of divergence of the dynamical time scale and to estimate the relaxation near the algorithmic thresholds. unsrt § DYNAMIC INDEPENDENT NEIGHBORS APPROXIMATION In this section we start from CDA and re-derive a result by Semerjian and Weigt <cit.>, written in an average case. For the sake of simplicity, we will re-write the Conditioned Dynamic Approximation in the case of pairwise interactions: d/dt P(σ_i, σ_∂ i) = -r_i(σ_i, σ_∂ i) P(σ_i, σ_∂ i) + r_i(-σ_i, σ_∂ i) P(-σ_i, σ_∂ i) - - ∑_j ∈∂ i∑_σ_∂ j ∖ i r_j(σ_j, σ_∂ j) P(σ_j, σ_∂ j)/∑_σ_∂ j ∖ iP(σ_j, σ_∂ j) P(σ_i, σ_∂ i) + + ∑_j ∈∂ i∑_σ_∂ j ∖ i r_j(-σ_j, σ_∂ j) P(σ_j, σ_∂ j)/∑_σ_∂ j ∖ iP(-σ_j, σ_∂ j) P(σ_i, F_j[σ_∂ i]) } The equation (<ref>) provides a new single-instance closure to the Master Equation that will be Conditioned Dynamic Approximation of the second order (CDA-2) in what follows. It contains already the main approximation needed to derive the Dynamic Independent Neighbors Approximation (DINA) as presented in <cit.>. To do so, it is necessary to assume that our system of spins is defined over a regular graph, where every node has the same number c of neighbors. In that case, if the initial conditions for the probabilities are independent of the site, all probabilities will be governed by identical equations. The only important parameters are then the central spin σ and the number u of unsatisfied interactions with its neighbors. Equations can be re-written as: d/dt P(σ, u) = -r(u) P(σ, u) + r(c - u) P(-σ, c - u) - - u [∑_û=0^c - 1c - 1û P(-σ, û + 1) ]^-1∑_û=0^c - 1c - 1û r(û + 1) P(-σ, û + 1) P(σ, u) + u [∑_û=0^c - 1c - 1û P(σ, û) ]^-1∑_û=0^c - 1c - 1û r(û) P(σ, û) P(σ, u - 1) - (c - u) [∑_û=0^c - 1c - 1û P(σ, û) ]^-1∑_û=0^c - 1c - 1û r(û) P(σ, û) P(σ, u) + (c - u) [∑_û=0^c - 1c - 1û P(-σ, û + 1) ]^-1∑_û=0^c - 1c - 1û r(û + 1) P(-σ, û + 1) P(σ, u + 1) In <cit.>, the equations are written for the probabilities: P̂(σ, u) = cu P(σ, u) The equation (<ref>) can be expressed in terms of this P̂ probabilities making some small re-arrangements. The sums over the variable û, that go from û=0 to û = c - 1, should be modified like in the following example: ∑_û=0^c - 1c - 1û P(-σ, û + 1) = ∑_û=0^c - 1(c - 1)!/û! (c - 1 - û)! P(-σ, û + 1) ∑_û=0^c - 1c - 1û P(-σ, û + 1) = ∑_û=0^c - 1(c - 1)!/û! (c - 1 - û)! (û + 1)! (c - û - 1)!/c!P̂(-σ, û + 1) ∑_û=0^c - 1c - 1û P(-σ, û + 1) = 1/c∑_û=0^c - 1 (û + 1) P̂(-σ, û + 1) = 1/c∑_u=0^c u P̂(-σ, u) ∑_û=0^c - 1c - 1û P(-σ, û + 1) = 1/c ⟨ u ⟩_-σ The equation (<ref>) was used in the second line of (<ref>) to write P(-σ, û + 1) in terms of P̂(-σ, û + 1). In the last line, the notation ⟨·⟩_σ≡∑_u=0^c [ · ] P(σ, u) was introduced. The following identities can be derived similarly as in (<ref>): ∑_û=0^c - 1c - 1û P(σ, û) = 1/c ⟨ c - u ⟩_σ ∑_û=0^c - 1c - 1û r(û + 1) P(-σ, û + 1) = 1/c ⟨ u r(u) ⟩_-σ ∑_û=0^c - 1c - 1û r(û) P(σ, û) = 1/c ⟨ (c - u) r(u) ⟩_σ Putting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), the differential equation becomes: d/dt P(σ, u) = -r(u) P(σ, u) + r(c - u) P(-σ, c - u) - - ⟨ u r(u) ⟩_-σ/⟨ u ⟩_-σ [ u P(σ, u) - (c - u) P(σ, u + 1) ] - ⟨ (c - u) r(u) ⟩_σ/⟨ c - u ⟩_σ [ (c - u) P(σ, u) - u P(σ, u - 1) ] Finally, the equation presented in <cit.> is obtained multiplying (<ref>) by cu and making use of (<ref>) and the relations: cu (c - u) P(σ, u + 1) = (u + 1) P̂(σ, u + 1) cu u P(σ, u - 1) = (c - u + 1) P̂(σ, u - 1) The result is: d/dtP̂(σ, u) = -r(u) P̂(σ, u) + r(c - u) P̂(-σ, c - u) - - ⟨ u r(u) ⟩_-σ/⟨ u ⟩_-σ [ u P̂(σ, u) - (u+1) P̂(σ, u + 1) ] - ⟨ (c - u) r(u) ⟩_σ/⟨ c - u ⟩_σ [ (c - u) P̂(σ, u) - (c - u + 1) P̂(σ, u - 1) ] These are the equations as presented in <cit.> § BROADCASTING PROCESS Let us start in a system with a single spin variable σ_0, called root, and perform the following broadcasting process: * Set some positive integer l * Create c groups of p - 1 variable nodes. * Select the values of the p-1 variables in each group according certain conditional probability π(σ_1, …, σ_p-1\|σ_0). * Connect each group to σ_0, thus forming c factor nodes, each one of order p. * For the newly created variable nodes, repeat the steps 2, 3 and 4, but instead of creating c factor nodes, create c-1. * Repeat 5 until the last created nodes are at distance d=l of the central node with value σ_0 At the end, we obtain a graph of connected nodes: the root σ_0, and l generations of nodes at distances d=1, 2, 3, …, l from the root. The last generation is known as the border of the graph. Now we can ask ourselves: If we repeat this process many times for a given l and keep track of the configurations of the border, can we recover the value of σ_0? The quantity we would like to study is the expected value m_0^l of the central variable for a given configuration of the border at distance l. Then, one possible way of answering the previous question is to define some self-consistent equations for the probability distribution Q_σ_0, π^l(m_0^l) of having the expected value m_0^l according to the border, if the border itself was generated following a broadcasting process done with root value σ_0 and using some measure π(σ_1, …, σ_p-1\|σ_0). This is actually a somewhat involved object, and we can understand it by imagining the following stochastic process: from a given σ_0 and using some π, generate a border at distance l from σ_0, and using only the information from that border, compute somehow the expected value m_0^l of the central variable. Repeat this to get a collection of values of m_0^l and then define Q_σ_0, π^l(m_0^l) as the probability of obtaining a given m_0^l when doing many realizations of the stochastic process. In order to be able of recovering the original value of the central variable, Q_σ_0, π^l(m_0^l) must have some non-trivial shape. Now, what is a non-trivial shape? One has to define some quantity to measure, and that is the expected point-to-set correlation (following the definition in <cit.>): C_PS^l, π = ⟨ σ_0 m_0^l ⟩_l, π - ⟨ σ_0 ⟩_l, π⟨ m_0^l ⟩_l, π C_PS^l, π = ∫ dm_0^l∑_σ_0 P(σ_0) Q_σ_0, π^l(m_0^l) σ_0 m_0^l - ( ∑_σ_0 P(σ_0) σ_0 ) ( ∫ dm_0^l∑_σ_0 Q_σ_0, π^l(m_0^l) m_0^l) If we draw σ_0=± 1 from the uniform distribution P(σ_0) = 1 / 2, the second term in (<ref>) vanishes and it is not difficult to see that: C_PS^l, π = ∫ dm_0^l Q_+, π^l (m_0^l) m_0^l where Q_+, π^l(m_0^l)≡ Q_σ_0=1, π^l(m_0^l), Q_-, π^l(m_0^l)≡ Q_σ_0=-1, π^l(m_0^l) and it is necessary to use the fact that when P(σ_0) is uniform the symmetry of the problem leads to the relation Q_+, π^l(m_0^l)≡ Q_-, π^l(-m_0^l). An important point to solve the problem is to introduce two new parameters. They will be defined in broadcasting processes that are slightly different from the original one. The first parameter will be μ_0^l, which is the expected value of the root defined in a broadcasting process where also at the first step one generates c-1 new factor nodes, instead of c. We will denote by Q_σ_0, π^l(μ_0^l) the probability density of having some value of μ_0^l. The second parameter is the expected value μ̂_0^l taken in a broadcasting process where at the first step one generates only one factor node, and the corresponding probability density will be denoted as Q̂_σ_0, π^l(μ̂_0^l). For those distribution the relations Q_+, π^l(μ_0^l) ≡ Q_-, π^l(-μ_0^l) and Q̂_+, π^l(μ̂_0^l) ≡Q̂_-, π^l(-μ̂_0^l) must also hold. One has then two different distributions Q_+, π^l(μ_0^l) and Q̂_+, π^l(μ̂_0^l), that in what follows will be denoted simply as Q_π^l(μ) and Q̂_π^l(μ̂). We will only need then the probability π(σ_1, …, σ_p-1\|σ_0=1), which will also be re-denoted as π(σ_1, …, σ_p-1). The advantage of introducing μ and μ̂ is that one can compute the corresponding distributions using the following iterative equations: Q_π^l(μ) = ∫[∏_i=1^c-1 dμ̂_i Q̂_π^l(μ̂_i) ] δ( ℱ_1[μ, {μ̂_i }_i=1^c-1] ) Q̂_π^l(μ̂) = ∑_σ_1, …, σ_p-1π(σ_1, …, σ_p-1) ∫[∏_j=1^p-1 dm_j Q_π^l-1(σ_j μ_j) ] δ(ℱ_2[ μ̂, {μ_j }_j=1^p-1] ) with the relations ℱ_1[μ, {μ̂_i }_i=1^c-1] = 1 + μ/2 - ∏_i=1^c-1(1 + μ̂_i )/∑_σ[∏_i=1^c-1(1 + σμ̂_i ) ] ℱ_2[ μ̂, {μ_j }_j=1^p-1] = μ̂ - S ⟨ S ⟩∏_j=1^p-1μ_j where S=∏_i=1^p-1σ_i and ⟨ S⟩ = ∑_σ_i, …, σ_p-1( ∏_i=1^p-1σ_i ) π(σ_i, …, σ_p-1) When π(σ_1, …, σ_p-1) is taken according the Boltzmann distribution, if a non-trivial fixed point of (<ref>) and (<ref>) exists, it corresponds to a non-trivial fixed point of the cavity method at the 1-RSB level with parameter x=1 <cit.>. In such non-trivial fixed points, C_PS^l, π↛0 when l →∞. We can define a correlation length as the minimum distance l at which the value of C_PS^l, π is smaller that certain parameter: l^∗(ϵ) = min{ l: C_PS^l, π < ϵ} The point-to-set correlation length l^∗(ϵ) must then diverge at T_d the temperature of the dynamic spin-glass transition. To compute C_PS^l, π we could use first a populations dynamics algorithm to obtain the distributions Q_π^l(μ) and Q̂_π^l(μ̂). With the latter it is possible to find the probability density Q_π^l(m) of the original broadcasting problem in a random regular hypergraph with: Q_π^l(m) = ∫[∏_i=1^c dμ̂_i Q̂_π^l(μ̂_i) ] δ( ℱ_3[m, {μ̂_i }_i=1^c] ) ℱ_3[m, {μ̂_i }_i=1^c] = 1 + m/2 - ∏_i=1^c(1 + μ̂_i )/∑_σ[∏_i=1^c(1 + σμ̂_i ) ] And then, with Q_π^l(m) one can directly compute (<ref>). However, if we just want to know where the point-to-set correlation length diverges, it is enough to define the cavity point-to-set correlation C̃_PS^l, π = ∫ dμ Q_π^l (μ) μ and the corresponding length l̃^∗(ϵ) = min{ l: C̃_PS^l, π < ϵ}. Fig. <ref>a shows that near T_d the value of l̃^∗(ϵ) becomes larger, and Fig. <ref>b shows a non-linear fit of these values according to the law C / (T - T_d)^1/2, which gives T_d ≈ 0.51. This is the value that has been reported in the literature for the transition temperature of the p-spin ferromagnet defined over random regular hypergraphs with c=3 and p=3
http://arxiv.org/abs/2307.01346v1	20230703203948	Patch-CNN: Training data-efficient deep learning for high-fidelity diffusion tensor estimation from minimal diffusion protocols	[ "Tobias Goodwin-Allcock", "Ting Gong", "Robert Gray", "Parashkev Nachev", "Hui Zhang" ]	cs.CV	[ "cs.CV", "cs.LG", "eess.IV" ]	Patch-CNN: Training data-efficient DL for diffusion tensor estimation T. Goodwin-Allcock et al. Department of Computer Science and Centre for Medical Image Computing, University College London, London, United Kingdom Department of Brain Repair & Rehabilitation, Institute of Neurology, UCL, London, United Kingdom. Patch-CNN: Training data-efficient deep learning for high-fidelity diffusion tensor estimation from minimal diffusion protocols Tobias Goodwin-Allcock1 Ting Gong1 Robert Gray2 Parashkev Nachev2 Hui Zhang1 This version: August 1, 2023 =============================================================================================================================== We propose a new method, Patch-CNN, for diffusion tensor (DT) estimation from only six-direction diffusion weighted images (DWI). Deep learning-based methods have been recently proposed for dMRI parameter estimation, using either voxel-wise fully-connected neural networks (FCN) or image-wise convolutional neural networks (CNN). In the acute clinical context—where pressure of time limits the number of imaged directions to a minimum—existing approaches either require an infeasible number of training images volumes (image-wise CNNs), or do not estimate the fibre orientations (voxel-wise FCNs) required for tractogram estimation. To overcome these limitations, we propose Patch-CNN, a neural network with a minimal (non-voxel-wise) convolutional kernel (3×3×3). Compared with voxel-wise FCNs, this has the advantage of allowing the network to leverage local anatomical information. Compared with image-wise CNNs, the minimal kernel vastly reduces training data demand. Evaluated against both conventional model fitting and a voxel-wise FCN, Patch-CNN, trained with a single subject is shown to improve the estimation of both scalar dMRI parameters and fibre orientation from six-direction DWIs. The improved fibre orientation estimation is shown to produce improved tractogram. § INTRODUCTION Diffusion tensor (DT) magnetic resonance imaging (MRI) allows for non-invasive quantification of white matter (WM) microstructure and connectivity, providing measures shown to be sensitive to disease <cit.>. However, accurate estimation of tissue microstructure—under conventional estimation routines—requires at least 30 diffusion weighted images (DWI) <cit.>. This requirement is incompatible with current acute clinical practice, where commonly only six-direction DWIs can be acquired within the feasible envelope of available time and patient tolerability. Recently, deep learning (DL) has been proposed to reduce the number of DWIs required for accurate estimation of tissue microstructure parameters <cit.>. Early works aimed to directly estimate scalar parameters from a reduced number of DWIs and treated each voxel independently. This was first achieved in q-Space deep learning <cit.>; this method estimated scalar diffusion kurtosis imaging (DKI) <cit.> and neurite orientation dispersion and density imaging (NODDI) <cit.> parameters—such as radial kurtosis and neurite orientation dispersion index—accurately with 12× acceleration. This method was then adapted to DiffNet <cit.> which accurately estimates scalar diffusion tensor (DT) measures, fractional anisotropy (FA) and mean diffusivity (MD), directly from as little as three DWIs. However, none of these methods estimate the primary diffusion direction required for tractography. Direction has previously been accurately estimated from six-directional DWIs. This was achieved by DeepDTI <cit.>, but, this network uses a more complex—image-wise—convolutional neural network (CNN) architecture. This architecture requires a training dataset that contains a representative distribution of the macro-scale brain structure, therefore an order of magnitude more training subjects is required. Additionally, these methods may not be robust to abnormal brain structure e.g. pathology, and may not easily learn pathology given the wide space of possible pathological characteristics. Here we aim to estimate high-fidelity DTs from only six DWIs, like DeepDTI, but having a minimal training-data requirement, like DiffNet. To achieve that, we adopt a patch-wise CNN architecture. This approach has been shown to significantly improve estimation accuracy of scalar measures from DT and DKI models compared to FCNs when only a small number of DWIs are available <cit.>. This shows the benefit of incorporating local anatomical information in a judicious manner. Additionally, patch-wise CNNs have been shown to improve fibre orientation estimation <cit.>. In this work, the full fibre orientation distributions were estimated accurately from a patch-wise input, however, this input used a large number (>6) of diffusion-encoding directions. We therefore propose to adopt a patch-wise architecture to estimate DTs with only six-directional DWIs as input. We organise the rest of the paper as follows: Section <ref> describes the Patch-CNN method and how we assess it; Section <ref> shows the results which are then discussed in Section <ref>. § METHODS §.§ Proposed deep learning technique Here, we propose Patch-CNN, an adaptation H-CNN <cit.>, for DT estimation from only six DWIs. To minimise training data requirements, we must avoid learning macro-scale brain anatomy, for an inadequately learnt global pattern may be misleading. Therefore, the network's input size needs to be minimal. The minimum input size is a voxel-wise network; however, we will show later that voxel-wise networks do not improve estimation for the directional parameters. Increasing the input window to a minimal (non-voxel-wise) 3D patch (3×3×3) has been shown to increase estimation accuracy <cit.> (H-CNN) for scalar parameter estimation. This is attributed to combining local neighbourhood information. Additionally, H-CNN does not rely upon macro-scale brain anatomy. The proposed method, Patch-CNN, is based on the previous methods architecture except it is adapted for estimation of the DT by estimating the six values of the DT. As the hierarchical aspect is no longer required we estimate all six values from the final layer of the network. To ensure semi-positive definiteness of the estimated DT's, the network is trained to estimate the matrix log of the diffusion tensor. In summary, Patch-CNN consists of a convolutional layer with a 3x3x3 kernal and a ReLu activation function followed by 2 fully-connected hidden layers each with 150 units and ReLu activation functions finally outputting to the 6 independent parameters of the matrix log of the diffusion tensor. §.§ Evaluation method §.§.§ Baseline techniques To assess the proposed method we compare against conventional model fitting (Conventional) and voxel-wise machine learning (Voxel-NN) at DT estimation from six DWIs. Voxel-NN uses the same architecture as Patch-CNN except for the kernel size of the input layer which is reduced to 1×1×1. This is equivalent to a voxel-wise fully-connected layer of 150 units. This architecture is similar to other voxel-wise networks <cit.>. For comparison against conventional model fitting we apply linear least squares using FSL <cit.>. §.§.§ Tractography For all methods, tractograms are derived from estimated DTs using the FACT algorithm <cit.>. The FACT algorithm generates streamlines in an iterative process starting at seed points in the white matter (WM) and follows the primary fibre orientation until it terminates; we used the FACT algorithm implemented in DTITK <cit.>. WM masks are defined as being voxels with estimated linearity <cit.> > 0.6. The tractogram is separated into tract bundles by including all streamlines which pass through specific regions of interest (ROI) <cit.>. The corticospinal tract (CST) and corpus callosum (CC) bundles are chosen for assessment as they are well known large structures whose fibres are tangential to each other. §.§.§ Dataset For this evaluation we require a dataset for training and testing; this dataset must contain a large set of DT-compatible measurements to provide 1) a high-quality ‘ground-truth’ (GT) DT and tractogram using conventional fitting 2) DWI subsets with fewer measurements that mimic six directional DWI gradient schemes. These specifications are satisfied by the Human Connectome Project (HCP) <cit.> dataset due to it's DTI-compatible measurements that include b=0’s (18) and b=1000’s (90). All of these DWIs are used to estimate the GT diffusion tensors; estimation is performed using conventional linear least squares fitting. To imitate a clinical scan consisting of a single b=0 image and an optimal subset of six b=1000’s we chose to subsample the gradient scheme to be maximally similar to Skare’s <cit.> six DWI gradient scheme. This scheme minimises the condition number of the bvec matrix and is used in DeepDTI <cit.>. §.§.§ Training Both the proposed and baseline ML method are trained using the same training settings. To mimic minimal training data requirements, data from a single HCP subject was used; this data consists of the optimal six DWI subset, for the input, and the matrix-logarithm of the ‘ground-truth’ DTs, for the output. Training settings follow the implementation of H-CNN: a batch size of 256; optimisation uses the ADAM optimiser <cit.>; the learning rate is set to 0.001 initially and subsequently reduced by 50% at each plateau. Plateaus in training are defined by no improvement in training loss over 10 consecutive epochs. To reduce overfitting, early stopping is applied with a random choice of 20% of the brain voxels as validation, with the remaining 80% for training. §.§.§ Testing To test all of the methods we estimate DTI’s for 12 subjects—unseen during training—using each method and measure the error. §.§ Evaluation Metrics Estimation error is computed on the estimated DT’s as well as the derived scalar parameters, directional parameters and tractograms. To assess the ability to estimate the tensors and scalar parameters, the errors between estimated and GT measurements are calculated over the whole brain. The metrics used are the Frobenius norm for tensors and absolute difference for FA and MD. To assess its ability to estimate the directional measures—required to reconstruct WM bundles—the primary fibre orientation estimation error is calculated. For the primary fibre orientation the angular errors between the estimated primary fibre orientation and the ones from GT tensors are computed over the WM, where white matter is determined by voxels with a ground truth linearity coefficient <cit.> > 0.6. The accuracy of generated tract bundles is assessed on a tract bundle level. For each bundle the accuracy is calculated by the similarity between the estimated tract bundles and the ground truth tract bundle. This is computed by the dice overlap of the voxels containing at least 25 streamlines. § RESULTS Figure <ref> shows the error of the estimated tensors—measured by the Frobenius norm—over an example axial slice for each of the different estimation techniques. On six directional data, the estimation technique with the lowest error is Patch-CNN; this method performs better than both Voxel-NN and conventional fitting. Additionally, the error map for Patch-CNN is comparable with 30 directional conventional fitting. These qualitative results are affirmed by the boxplots in Figure <ref>a. Here, each boxplot corresponds to median of the Frobenius norm for each of the 12 subjects for each of the methods. Not only is Patch-CNN shown to outperform all methods estimating from the same number of directions, it also outperforms conventional fitting from twice as many DWIs. Figure <ref> shows a coronal slice of the FA weighted colour encoded primary fibre orientation for the ground truth and estimations across multiple fitting methods. Qualitatively, the Patch-CNN estimate is far more accurate than both of its six directional counterparts as it is both more visually similar to the ground truth and less noisy. To see the effect in more detail we enlarge a small region of the motor tract and corpus callosum, shown in the yellow boxes. In this enlarged view the primary fibre orientations, visualised as sticks, estimated by Patch-CNN are coherent like the ground truth and 30 directional conventional estimation. In contrast both of the other estimations from six directions have very incoherent fibres. Figure <ref>b confirms this quantitatively over the 12 testing subjects and again shows that Patch-CNN outperforms conventional fitting with twice as many DWIs. The estimation performance on scalar measures derived from the diffusion tensor is shown in Figure <ref>. Axial slices of the estimated FA and MD are shown above their error maps. Qualitatively, Patch-CNN performs the best of the six directional estimations because it is the most similar to the ground truth image as well as having the least amount of noise. For FA estimation, conventional fitting largely overestimates the FA whereas Voxel-NN largely underestimates the FA. Only Patch-CNN estimates FA with far greater accuracy. Figures <ref>c and <ref>d further affirms that Patch-CNN outperforms conventional fitting with twice as many DWIs. Figure <ref>, shows the tract bundles of the corpus callosum and corticospinal tract. Qualitative assessment shows that Patch-CNN performed the best, the CC bundle is far denser and the left CST has streamlines terminating in two regions just like the ground truth. Figure <ref> quantitatively supports this as Patch-CNN has the highest average dice score of the three methods on all of the tract bundles. § DISCUSSION We have developed a data-efficient method of estimating the diffusion tensor with high accuracy, leading to improved performance in all derived parameters both scalar and directional, as well as greatly improved estimates of the tractograms, in comparison with rival methods. Patch-CNN outperformed conventional fitting as well as Voxel-NN at estimating the corpus callosum and corticospinal tract. The most improvement was shown in the estimation of the corticospinal tract. Although the corticospinal tract was the tract estimation most improved by Patch-CNN it was still also worse than the CC estimation. We believe that both of these factors are due to the length of the corticospinal streamlines being longer than that of the CC streamlines; longer tracts travel through a larger number of voxels and so premature termination is much more likely. Tensors estimated from Voxel-NN have a lower average Frobenius norm on the error compared to conventional fitting (0.22, 0.28) this is reflected in the accuracy of the scalar measures MD and FA and is consistent with the literature. However, Voxel-NN's estimation of the primary fibre orientation is not as accurate as conventional fitting. It is no surprise that the tractograms—determined by the primary fibre orientation—also shows a decrease in performance when using Voxel-NN. Therefore, we see that the increase in estimation accuracy for primary fibre orientations and the resultant tractograms from Patch-CNN is due to including local neighbourhood information into the input. Although we have only shown the accuracy of MD and FA scalar measures, as the whole diffusion tensor is estimated, any scalar measure which can be derived from the diffusion tensor—such as radial diffusivity, linearity, planarity and sphericality—can be estimated with no new training regime required. Patch-CNN is more clinically viable than image-wise ML models due to Patch-CNN’s modest training data requirements. Patch-CNN only requires a single subject for training because it’s small input field size means that each subject’s data contains hundreds of thousands of training examples. For an ML model whose input field size is the entire image, a single subject’s data is a single training example. Therefore, a much larger number of subjects must be recruited to train image-wise models. This is especially important as each model is tied to the acquisition settings, e.g. gradient scheme, of the data it was trained on. For every new set of acquisition settings more training data must be collected. As Patch-CNN only requires one subject to train, it is still clinically feasible. A limitation of the proposed method, along with Voxel-NN, is a tendency to underestimate the high FA regions. A possible cause of this is class imbalance as high FA voxels are vastly underrepresented in the training data. Class imbalance could be combated with data balancing of the training data. Robustness to lesion presence was not shown and is left as future work. § CONCLUSION Patch-CNN has been demonstrated to outperform both conventional fitting and voxel-wise machine learning for estimating the DT and the clinically useful measures derived from it, matching the performance of conventional fitting with twice the number of DWIs. Tractograms derived from Patch-CNN’s tensor estimates have higher fidelity than from conventionally estimated tensors. Patch-CNN achieves this performance with only one training subject which is a vast improvement from the image-wise CNNs such as DeepDTI which uses around 40 training subjects as well as both T1w and T2w anatomical images. Patch-CNN allows for robust estimation of tracts from only six DWI's in a clinically feasible manner. § ACKNOWLEDGEMENTS This work is supported by the EPSRC-funded UCL Centre for Doctoral Training in Medical Imaging (EP/L016478/1), the Department of Health’s NIHR-funded Biomedical Research Centre at UCLH and the Wellcome Trust. splncs04
http://arxiv.org/abs/2307.00179v1	20230701001140	Unsupervised Coordinate-Based Video Denoising	[ "Mary Damilola Aiyetigbo", "Dineshchandar Ravichandran", "Reda Chalhoub", "Peter Kalivas", "Nianyi Li" ]	cs.CV	[ "cs.CV", "eess.IV" ]	Tuning a magnetic energy scale with pressure in UTe_2 Johnpierre Paglione August 1, 2023 ===================================================== In this paper, we introduce a novel unsupervised video denoising deep learning approach that can help to mitigate data scarcity issues and shows robustness against different noise patterns, enhancing its broad applicability. Our method comprises three modules: a Feature generator creating features maps, a Denoise-Net generating denoised but slightly blurry reference frames, and a Refine-Net re-introducing high-frequency details. By leveraging the coordinate-based network, we can greatly simplify the network structure while preserving high-frequency details in the denoised video frames. Extensive experiments on both simulated and real-captured demonstrate that our method can effectively denoise real-world calcium imaging video sequences without prior knowledge of noise models and data augmentation during training. § INTRODUCTION Video denoising is a critical task in computer vision, with various applications such as video surveillance, medical imaging, and autonomous driving. The goal is to remove the noise present in videos, which can be caused by various factors such as sensor noise, compression, and low-light conditions. Supervised Convolutional Neural Networks (CNNs) have been successful in solving this problem by learning the mapping between noisy and clean images from large datasets <cit.>. However, obtaining such datasets can be challenging and time-consuming, especially when the noise is complex and diverse. To address this challenge, unsupervised video denoising methods have emerged in recent years, which do not require labeled data for training. One such approach is based on the "blind-spot" technique, where the CNN is trained to estimate each noisy pixel from its surrounding spatial neighborhood without considering the pixel itself <cit.>. However, most of these self-supervised methods require either training on synthetic datasets by adding random noise to the clean image during training, or need long video sequences to learn the implicit noise model <cit.>. In applications like microscopy, where video sequences might be short, obtaining noiseless ground truth videos and estimating the underlying tractable model of the noise pose significant challenges. In such cases, the performance of denoising methods based on "blind-spot" CNN can significantly degrade <cit.>. Therefore, developing an unsupervised video denoising method that can work directly on raw noisy videos is critical to improve the accuracy and robustness of video denoising in challenging scenarios. The architecture of our network is shown in Fig. <ref>, comprises three key components, each designed to effectively denoise video data. First, the feature generator ℱ renders feature maps that align with the coordinates of the input frames. Second, the Denoise-Net 𝒟 utilizes these feature maps to produce denoised yet slightly blurry reference frames. Finally, the Refine-Net ℛ restores high-frequency details to the denoised frames, enhancing overall image clarity. The network is trained by minimizing the discrepancy between the generated denoised frame, the input noisy frame, and the refined final output frame, ensuring both efficient denoising and preservation of the original video data's integrity and quality. To streamline the network architecture and enhance training efficiency, we incorporate coordinate-based networks, as referenced in <cit.>, into both 𝒟 and ℛ. We conduct comprehensive experiments on a diverse range of noisy videos, including both simulated and real-captured footage. In comparison to state-of-the-art method, our approach demonstrates superior performance in effectively correcting noise, highlighting its efficacy and potential for widespread application. § RELATED WORKS Supervised Video Denoising. Supervised CNN approaches to denoising, such as those presented in <cit.>, have achieved state-of-the-art results in both single image and video denoising, requiring clean images for model training. Several techniques, like residual learning <cit.> and batch normalization <cit.> in DnCNN <cit.>, or the use of dilated convolution <cit.> for a faster model, have been developed. The introduction of downsampling and upsampling layers in FFDNet <cit.> aimed to increase the receptive field. Two-stage denoising without explicit motion estimation was explored by DVDNet <cit.>, ViDeNN <cit.>, and FastDVDnet <cit.>. However, these models' reliance on unrealistic noisy/clean pairs is a drawback, especially in medical imaging where clean ground truth images are seldom available. Unsupervised Video Denoising. Unsupervised denoising algorithms like Noise2Noise (N2N) <cit.>, Frame2Frame (F2F) <cit.>, and Multi Frame2Frame (MF2F) <cit.> leverage noisy images as both inputs and targets, or consecutive frames as noisy targets for denoising. Techniques like the blind spot technique in Noise2Void (N2V) <cit.> and the convolution blind-spot network architecture <cit.> have been developed to estimate the underlying clean signal. While UDVD <cit.> achieved state-of-the-art results, their methodology necessitates noise addition at every model iteration and substantial data augmentation. Furthermore, the deep interpolation algorithm by Lecoq et al. <cit.> required over 200,000 data samples and showed limitations in generalizability. In contrast to these methods, our approach can effectively denoise short video sequences and generalize to different types of noisy videos without requiring excessive data augmentation or iterative noise addition. Implicit Neural Representation. Implicit neural representations, also known as coordinate-based representations, utilize fully connected neural networks to associate input coordinates with their corresponding signal values. They have shown remarkable utility across a range of tasks, including view synthesis <cit.>, image representation <cit.>, and 3D shape representation <cit.>. One significant development in this domain is the Sinusoidal Representation Networks (SIREN) <cit.>, which leverage periodic activation functions to encode positional information and model complex natural signals with high precision. This approach allows for rapid training convergence, which significantly improves efficiency, especially in resource-intensive applications. In our framework, we employ coordinate-based networks in the feature generator ℱ and Refine-Net ℛ to enhance the denoising performance. § UNSUPERVISED COORDINATE-BASED VIDEO DENOISING Given a sequence of noisy video frames {I_t \| t = 1, 2, …, N} and their corresponding coordinates {G_t \| t = 1, 2, …, N}, where G_t(p_t ) represents the coordinates of pixel p_t = (x, y,t) in the noisy frame I_t, our objective is to recover the noise-free video frames {J_t \| t = 1, 2, …, N}. Our approach consists of three primary components: a feature generator ℱ_θ, a Denoise-Net 𝒟_ϕ, and a Refine-Net ℛ_η, as illustrated in Fig. <ref>. §.§ Feature Generator ℱ_θ Our Feature Generator processes a batch of uniformly sampled coordinate grids {G_t \| t = 1,...,B}, where each G_t ∈ℝ^H × W × 3, to generate a corresponding batch of feature maps {F_t \| t = 1,...,B}, each being F_t ∈ℝ^H × W × C. Here, B is the batch size, while C is the number of feature channels. Before the coordinate grid is fed into ℱ_θ, each coordinate p_t=(x,y,t) is subject to positional encoding <cit.>. This encoding step transforms low-dimensional input coordinates into a higher-dimensional space, enabling the model to better learn and represent high-frequency details inherent in the image data. The positional encoding function we adopt is given as: γ(𝐩_t) = [sin(2^0 π𝐩_t), cos(2^0 π𝐩_t), ..., sin(2^L-1π𝐩_t), cos(2^L-1π𝐩_t)] In this equation, L serves as a hyperparameter controlling the level of detail or high-frequency information in the output. By selecting a smaller value for L, we can effectively reduce the level of high-frequency noise in the image data, as noise often manifests as high-frequency information. For our experiments, we set L=30. The input coordinates, normalized to the range [-1, 1] using a mesh grid, are passed through the encoding function γ(.). The resulting high-dimensional output γ(G_t)∈ℝ^H × W × 6L is subsequently fed into the feature generator ℱ_θ: F_t = ℱ_θ(γ(G_t)) where F_t∈ℝ^H × W × C is the feature maps corresponding to each noisy frame, and C denotes the channel size of the output features. Our feature generator is composed of 6 convolution layers, each featuring a kernel size of 3 and identical padding to maintain the size throughout the model layers. Each layer has 256 feature channels with batch normalization (BN) applied solely to the first two layers. ReLU activation is employed for all layers, except for the last one. §.§ Denoiser 𝒟_ϕ The denoiser network takes the concatenated feature maps output from the feature generator as input, and generates a denoised central frame Î_B: Î_B = 𝒟_ϕ([F_1,F_2,...F_B]) where the concatenation is applied along the feature channel dimension. This allows 𝒟_ϕ to learn spatial-temporal patterns along the neighboring time frames eliminate the noise. Note that the output of 𝒟_ϕ may be somewhat blurred, because we've set a low L in the feature generator. The architecture of 𝒟_ϕ includes 6 convolutional layers. ReLU activation is used in the first five layers, and a sigmoid activation function is used in the last layer. Unlike in the feature generator, batch normalization is not applied in this stage. Each layer uses 256 filters with a kernel size of 3, except for the last two layers which have 96 and the number of color channels filters, respectively, with kernel sizes of 1. §.§ Refine-Net ℛ_η The refine-net is built upon the backbone of the Sinusoidal Representation Networks (SIREN) <cit.>, which is a type of coordinate-based network that uses periodic activation functions, particularly sine functions, in place of traditional activation functions like ReLU. SIREN's unique characteristic lies in its ability to naturally model the high-frequency details of complex patterns by leveraging its intrinsic periodic activation functions. In our context, ℛ_η uses the SIREN network to take the coordinates grid of the central frame as input and generate the refined image Î_R. Î_R = ℛ_η(G_t^c) where G_t^c is the coordinates of the central frames in the input batch. Note that the SIREN-based refine-net is particularly beneficial in our case, as it helps to further refine the denoised output from 𝒟_ϕ by enhancing the finer details and correcting any blurring introduced in the denoising process. The output from ℛ_η represents the final denoised and refined video frames. § NETWORK OPTIMIZATION Given the unsupervised nature of our architecture, the network optimization problem is highly non-convex with a vast parameter search space. To navigate this challenge, we propose a two-step network optimization strategy that exploits the structural similarity between neighboring frames to reconstruct the central frame. §.§ First Stage: Joint Training of the Feature Generator ℱ_θ and Denoiser 𝒟_ϕ In the first stage, we jointly train the feature generator ℱ_θ and the denoiser 𝒟_ϕ in an end-to-end fashion. The objective function for this stage is composed of two parts. The first part is the l_1 loss, which measures the difference between the denoised central frame Î_B and the central frame I^c of the input batch of noisy frames {I_t \| t = 1, 2, …, B}: ℒ_𝒟 = \|\|Î_B - I^c\|\| The second part of the objective function ensures that the feature maps generated by ℱ_θ capture the image information. This is achieved by enforcing similarity between the central channels of each feature map and the corresponding noisy frame: ℒ_ℱ = 1/B∑_t=1^B\|\| F_t^c - I_t\|\| where F_t^c is the central channels of each feature map ℱ_θ(γ(G_t)) and I_t is the corresponding noisy frame, as shown in Fig. <ref> (a). The final loss function for the first stage is the sum of these two losses: ℒ_1 = ℒ_𝒟 + λ_1 ℒ_ℱ where λ_1 is a weight parameter that controls the trade-off between the two terms. To ensure our model doesn't simply learn to reproduce the noise present in the input, we train the network for approximately 2000 epochs. §.§ Second Stage: Training the Refine-Net ℛ_η In the second stage, we focus on training the Refine-Net ℛ_η. Unlike in the first stage, where ℱ_θ and 𝒟_ϕ were trained together, here we fix ℱ_θ and 𝒟_ϕ and solely train ℛ_η. We use the coordinates of the central frame G_t^c as input to ℛ_η, which outputs a refined denoised image Î_R. As this network is designed to further improve the quality of the denoised frames, the loss function for this stage is defined to measure the difference between the output of ℛ_η and both the noisy central frame I_t^c and the denoised central frame Î_B from the first stage. Specifically, the loss function is defined as: L_2 = λ_2\|\|ℛη(G_t^c) - I^c\|\| + λ_3\|\|ℛη(G_t^c) - Î_B\|\|, where λ_2 and λ_3 are weight parameters controlling the contribution of each term. These parameters help balance the network's objectives of reducing noise (by making the output similar to I_B^c) and preserving details (by making the output similar to I_t^c). This strategy allows the Refine-Net to leverage the advantages of both the noisy and denoised frames, by enhancing details and suppressing noise. The training of ℛη is performed until satisfactory results are obtained, typically for about 2000 epochs. It's important to note that the second stage training does not affect the training of ℱ_θ and 𝒟_ϕ, which is crucial for preserving the generalization capability of the whole framework. Our two-stage optimization can effectively remove the noise in the video frames. The first image on the left represents the initial state, which is typically a noisy video frame. The joint training of the feature generator ℱ_θ and the denoiser 𝒟_ϕ shows a noticeable reduction in noise (middle frame), but some blurriness might still be present due to the low L we used in our feature generator. The right image showcases the result of the refining stage ℛ_η. At this stage, the high-frequency details that might have been lost in the denoising process are recovered, resulting in a crisp, clean frame that retains the original structure and details of the scene, demonstrating the step-by-step improvement of our method. § EXPERIMENTS In this section, we evaluate the performance of our proposed method through extensive experiments. Our method is tested on a variety of video sequences with different types of noise, and the results are compared with state-of-the-art denoising methods to demonstrate its effectiveness. §.§ Datasets and Setup Our approach is tested on a variety of datasets, encompassing both synthetic and real-world scenarios, to provide a comprehensive evaluation of its performance. Synthetic data are derived from established benchmarks and intentionally corrupted with diverse types of noise, while real-world data are sourced from calcium imaging experiments. Moreover, we outline the specifics of our computational setup and training parameters, detailing the choices that were made to optimize our model's performance. This thorough experimental setup is aimed at providing a robust assessment of our proposed video denoising method and its potential applicability to different types of video data. Synthetic. We employed a variety of benchmark datasets, including the DAVIS dataset <cit.>, and SET8<cit.> videos s captured with GoPro camera, to provide a comprehensive evaluation of our algorithm. These datasets include a diverse array of video sequences, each with unique content and characteristics, thus allowing us to test our method under numerous conditions. To challenge our algorithm's robustness, we deliberately introduced various types of noise to the clean video sequences. Gaussian noise was added as per the formula: N(c\|μ,σ^2) = 1/√(2πσ^2)e^-(c-μ)^2/2σ^2 where c is a pixel value, and μ and σ^2 are the mean and variance of the Gaussian distribution, respectively <cit.>. In our experiment, we set μ=0. Poisson noise was added according to the equation: P(c\|λ) = λ^c e^-λ/c!, where λ is the expected number of occurrences <cit.>. Salt-and-pepper noise, also known as impulse noise, was added by randomly selecting pixels and setting their values to the minimum and maximum values representing 'salt' and 'pepper' respectively <cit.>. Real-World. In addition to the synthetic datasets, we also applied our algorithm to real-world, highly noisy calcium imaging data, as shown in Fig. <ref>. These were locally sourced recordings from freely behaving transgenic mice engaged in cocaine/sucrose self-administration experiments. The recordings were captured using single-channel epifluorescent miniscopes and were subsequently processed using a motion correction algorithm to adjust for translational motion artifacts. This dataset represents the practical complexity and noise levels that are often present in real-world scenarios, further challenging our algorithm's ability to effectively denoise video sequences. Our proposed method was implemented using the PyTorch framework and trained on an NVIDIA A100 GPU. We employed the Adam optimizer during the training process, with an initial learning rate of 1e-4 set for the first stage and 1e-5 for the second stage. Both learning rates were reduced by a factor of 10 every 100 epochs. In our loss function, we set λ_1=0.1 and λ_2=1.0 as the balancing factors for our dual-term loss. §.§ Quantitative Evaluation To facilitate a quantitative evaluation of our approach, we employ two widely-accepted metrics in the image and video processing community: the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM). These measures allow us to objectively compare our results with those of the current state-of-the-art unsupervised video denoising algorithm, UDVD<cit.>. The outcomes, as displayed in Table <ref>, highlight that our method consistently outperforms UDVD across all tested datasets and noise types. This superior performance is primarily attributed to our approach's effective utilization of spatial-temporal information from the video sequence. Our algorithm leverages this information to efficiently eliminate noise while simultaneously preserving high-frequency details within the frames. When examining the visual comparison results, as shown in Fig. <ref>, it becomes clear that UDVD often struggles to effectively remove noise when dealing with short input sequences. This challenge is particularly evident when confronted with Poisson and Impulse noise types, where UDVD tends to produce noticeable artifacts. Conversely, our method shows remarkable resilience even in these demanding situations. We tested our approach on ten short video sequences, each consisting of ten frames, and the results consistently demonstrated our method's superior noise removal capability and robustness. Overall, the quantitative and qualitative evaluations underscore the potential of our proposed video denoising method as a robust and effective solution, particularly for applications dealing with short, noisy video sequences. §.§ Qualitative Evaluation A visual comparison between our method and UDVD on the highly noisy calcium imaging sequences further underscores our superior performance, as shown in Fig. <ref>. In the noisy frames, it can be challenging to distinguish individual cells due to high levels of noise. UDVD, while effective in reducing some noise, often blurs the intricate cellular structures, making it difficult to identify individual cells. In contrast, our approach not only removes the noise effectively but also preserves the intricate cellular structures, allowing for better visualization and identification of individual cells. This difference is particularly notable in regions with a high density of cells, where our method is able to maintain the distinct boundaries between cells, whereas UDVD tends to blur them together. This visual comparison highlights our method's ability to handle real-world data with significant noise, offering promising potential for applications in biological and medical imaging. §.§ Ablation Study We conduct an ablation study to understand the contribution of each component in our method. Notably, when we omit the refining stage ℛ_η, the denoised frames tend to be slightly blurry due to the low L in the feature generator ℱ_θ. However, with the incorporation of the refining stage, our method is able to effectively recover high-frequency details, thereby underscoring its crucial role in enhancing the overall quality of the denoised frames. We show the visual comparison between each of network variants in Fig. <ref>. A detailed presentation of the ablation study results is provided in Table <ref>. § DISCUSSION In this study, we proposed an innovative unsupervised video denoising framework that leverages a two-stage optimization strategy and a novel refine-net that employs the SIREN architecture as its backbone. Our model has been demonstrated to be highly effective across a range of video datasets, from natural videos to challenging fluorescence microscopy and calcium imaging recordings. One key strength of our approach is its adaptability to various noise types and levels without needing prior knowledge about the noise characteristics. This flexibility is primarily due to our unsupervised training strategy which enables the model to learn directly from the noisy frames and adapt to the specific noise present in the video. Moreover, the use of SIREN in our refine-net is another significant advantage, as it excels in representing complex patterns and structures in the data, making it particularly suitable for refining the denoised output. However, like all models, ours also has certain limitations. The denoising quality could be affected by the value of L in the feature generator. Setting a lower L might result in blurry output frames, whereas a higher L could potentially lead to overfitting. Therefore, the selection of L requires careful tuning based on the specific characteristics of the dataset. Another challenge arises when dealing with extremely noisy video sequences. Our model might struggle with recovering high-quality frames from these videos as the noise might overwhelm the actual content in the frames. Future research could focus on developing more robust denoising models that can handle higher noise levels. In conclusion, our proposed video denoising framework presents a promising approach to tackle the video denoising problem, and we hope that our work will inspire future research in this direction. plain § NETWORK ARCHITECTURE AND TRAINING §.§ Feature Generator The feature generator ℱ_θ takes in a batch of sampled coordinated grids {G_t \| t = 1,...,B} corresponding to a batch of noisy frames {I_t \| t = 1, 2, …, B}, where B is the batch size. In our experiment, we set B = 5, and we used 2 consecutive frame coordinate grids before and after the central frame, resulting in 5 frame coordinate grids including the central frame itself. Our goal is to generate a batch of feature map, denoted as {F_t \| t = 1,...,B}, corresponding to each of the input frame. Initially, a positional encoding γ(.) is applied to these batched coordinate grids before they are processed by the feature generator. γ(.) expands the low-dimensional coordinates into a high-dimension feature space using the following equation: γ(𝐩_t) = [sin(2^0 π𝐩_t), cos(2^0 π𝐩_t), ..., sin(2^L-1π𝐩_t), cos(2^L-1π𝐩_t)] where L signifies the frequency level of output feature map. The output size of γ(.) depends on the size of input coordinate C_coord and L. Next, the feature generator ℱ_θ takes positional encoded grid γ(G_t) as input, and output the feature map F_t∈ℝ^H × W × C_feat. §.§ Denoisier The network architecture for the Denoiser 𝒟_ϕ is described in Table <ref>. The batched feature map output from ℱ_θ is concatenated to form an input dimension of 1 × B*C_feat× H × W, and then fed into 𝒟_ϕ. The output of the Denoiser network is C_in, where C_in =3 for color images and C_in=1 for grayscale images. §.§ Refine-Net The network architecture of the Refine-Net ℱ_θ is depicted in Table <ref>. The backbone of this network is a Sinusoidal Representation Networks (SIREN) as described in <cit.>. This network takes as its input the coordinates of the denoised output from the denoise-net and produces a refined denoised central frame where each pixel coordinate is denoted by p_t = (x,y,t). The Refine-Net is composed of fully connected layers, which include 4 hidden layers, with the activation function at each layer being a sine function. Each hidden layer contains 256 units, and the output layer is composed of C_in nodes, where C_in=3 for RGB images and C_in=1 for grayscale images. The refined denoised output is generated by processing the denoised image's coordinates through this network, resulting in a more precise final image. §.§ Training Details Our proposed network is trained using a two-stage process to effectively leverage the functionalities of each individual component - the feature generator, denoiser, and refiner. In the first stage, we train the feature generator and denoiser together as a combined model. Here, we optimize for the reduction of noise in the video frames. The aim at this stage is to extract relevant spatio-temporal features from the noisy inputs and then utilize those features to output an initial, denoised estimate of the central frame of the batch. In the second stage of training, we keep the weights of the feature generator and denoiser fixed, and focus on training the refiner. The refiner uses the denoised output from the previous stage and enhances it further, particularly focusing on the high-frequency details. This two-stage training approach ensures an effective noise reduction while preserving and enhancing the structural details in the video sequences. It is important to note that each stage of training involves its own data augmentation strategy and the whole network is trained end-to-end, with the weights of each component optimized for their specific tasks. By doing so, our network provides an overall pipeline that delivers high-quality denoised video frames. Optimization Details: The training of our network leverages the Adam optimizer <cit.>. Initially, the feature generator and denoise-net are jointly trained, minimizing l_1 loss between the central channel of the feature map and corresponding noisy frames, along with the l_1 loss between the denoised central frame and its corresponding original frame. The balance between these two loss terms is controlled by λ_1 in <ref>, which is set to 1.0. ℒ_1 = \|\|Î_B - I^c\|\|_1 + λ_1(1/B∑_t=1^B\|\| F_t^c - I_t)\|\|_1 Subsequently, we refine the denoised output using the refine-net, minimizing the l_1 loss between the refine-net's output and both the denoise-net's output and the corresponding noisy frame. The regularization terms λ_2 and λ_3, set to 0.1 and 1.0 respectively, help to strike a balance between detail preservation and noise reduction. L_2 = λ_2\|\|ℛ_η(G_t^c) - I^c\|\|_1 + λ_3\|\|ℛ_η(G_t^c) - Î_B\|\|_1 The initial learning rates are set to 0.0001 for the joint training stage and 0.00001 for the refine-net training. Both rates are reduced by a factor of 10 after every 1000 epochs. During the refine-net training stage, the weights of the feature generator and denoise-net are frozen. § MORE VISUAL COMPARISON OF DENOISING ALGORITHMS In this section, we present a comprehensive visual comparison of our denoising method against the UDVD <cit.>, utilizing several test datasets. Our experiments use the DAVIS and Set8 datasets, each corrupted with various noise types: Gaussian noise with σ = 30 and 50, Poisson noise with λ=30 and 50, and Impulse noise with α = 0.2. The denoising performance is evaluated and visually demonstrated on several distinct scenarios, including 'bus', 'hypersmooth', and 'snowboard'. § ABLATION STUDY Positional encoding. We provide ablation studies on the positional encoding hyperparameters in <ref> using video sequences in DAVIS dataset. We experimented with different frequency levels in <ref> to encode the input coordinates. Table <ref> describes the average PSNR and SSIM value using different L:
http://arxiv.org/abs/2307.01745v1	20230704142845	Holographic baryons, dense matter and neutron star mergers	[ "Matti Jarvinen" ]	hep-ph	[ "hep-ph", "nucl-th" ]
http://arxiv.org/abs/2307.02077v1	20230705073907	A new pulsar candidate in 47 Tucanae discovered with MeerKAT imaging	[ "Ian Heywood" ]	astro-ph.HE	[ "astro-ph.HE", "astro-ph.IM" ]	firstpage–lastpage Graph Contrastive Topic Model Sophia Ananiadou August 1, 2023 ============================= MeerKAT imaging of the globular cluster 47 Tucanae (47 Tuc) reveals 1.28 GHz continuum emission at the locations of 20 known millisecond pulsars (MSPs). We use time series and spectral imaging to investigate the image-domain characteristics of the MSPs, and search for previously unknown sources of interest. The MSPs exhibit a range of differences in their temporal and spectral properties compared the general background radio source population. Temporal variability differs strongly from pulsar to pulsar, some appearing to vary randomly on 15 min timescales, others varying coherently by factors of >10 on timescales of hours. The error in the typical power law fit to the spectrum emerges as a powerful parameter for indentifying the MSPs. This behaviour is likely due to differing diffractive scintillation conditions along the sight lines to the MSPs. One MSP exhibits tentative periodic variations that are consistent with modulation due the orbit of an eclipsing binary system. One radio source has spectro-temporal properites closely resembling those of the MSP population in the cluster, and we report its position as a candidate new MSP, or alternatively an interferometric localisation of one of six MSPs which do not yet have an accurate position from the timing solutions. globular clusters: individual: 47 Tucanae – pulsars: general – radio continuum: general § INTRODUCTION Globular clusters (GCs) contain high numbers of millisecond pulsars (MSPs) which are spun-up via the high interaction rates with other stars in these extremely dense environments <cit.>. The recent deployment of next-generation radio interferometers such as the sensitive and wide-field Square Kilometre Array precursor instruments ASKAP <cit.> and MeerKAT <cit.> has led to renewed interest in the prospects of using image-domain data to discover new pulsars, and to expand the parameter space for the discovery of variables and transienst in general <cit.>. Indeed, the first millisecond pulsar was discovered following an investigation of an unusual feature in a radio continuum image which was arising due to interplanetary scitillation <cit.>, and at the other end of the periodicity scale, recent results have demonstrated the potential for using image-domain observations to detect potential neutron star systems that are emitting radio pulses with periodicities that are beyond the death lines in the period–period-derivative (P–Ṗ) parameter space <cit.>. Motivated by this, we present an analysis of archival MeerKAT imaging observations of 47 Tucanae (47 Tuc), and present the first MeerKAT image of this GC. 47 Tuc is located at a distance of 4.52 ± 0.03 kpc <cit.> and contains 29 known MSPs[<https://www3.mpifr-bonn.mpg.de/staff/pfreire/GCpsr.html>], of which 23 have published positions <cit.>. Due to this high number of known MSPs we can investigate their characteristics in the image domain data, in both the time and frequency integrated `continuum' image, as well as by generating sub-images of the data along those dimensions. From there we can attempt to search for previously uncharacterised sources that share these properties, as well as search for other sources of interest that exhibit unusual properties when compared to the general (predominantly extragalactic) compact source population. Note that throughout this letter we refer to the pulsars in 47 Tuc using their alphabetical designation, and omit the J0023-7204 or 47 Tuc prefix. § OBSERVATIONS AND DATA PRODUCTS We make use of publicly available data from the MeerKAT radio telescope, taken during its science verification (SV) phase, see Table <ref> for details. The calibration strategy during SV was conservative, with frequent scans of calibrator sources. The secondary calibrator was observed for 1.73 min for every 14.8 min target scan, and the primary calibrator was visited approximately once per hour, and observed for 9.7 min. The data were recorded in full polarisation mode, and scans of a primary polarisation calibrator were included, however we did not perform full-Stokes imaging for this study. The data were retrived from the archive in Measurement Set (MS) format using the KAT Data Access Library[katdal; <https://github.com/ska-sa/katdal>], and averaged by a factor of 4 in frequency, resulting in 1024 channels. The MS was then processed using oxkat[v0.3; <https://github.com/IanHeywood/oxkat>] <cit.>, a set of scripts for automatically processing MeerKAT continuum data. The process involves cross-calibration using casa <cit.>, automatic removal of radio frequency interference using casa and tricolour <cit.>, self-calibration using cubical <cit.>, direction dependent calibration using killms <cit.>, and imaging and deconvolution performed using either wsclean <cit.> or ddfacet <cit.>. Primary beam correction is applied using an azimuthally-averaged model of the Stokes I beam produced using the katbeam package[<https://github.com/ska-sa/katbeam>]. Diagnostic plots are provided by ragavi[<https://github.com/ratt-ru/ragavi>] and shadems <cit.>, and all packages are containerised using singularity <cit.>. Full details and validation of the data processing workflow are provided by <cit.>. A time-integrated, multi-frequency synthesis (MFS) image of 47 Tuc is presented in Figure <ref>, with details of the annotations provided in the figure caption. Continuum emission is seen at the position of 20 of the 23 pulsars in 47 Tuc, with non-detections of W, Y, and Z, although the latter two have hints of associated emission. Additionally, at this angular resolution we are unable to separate the emission for the groupings G/I and also F/S, with T also partially overlapping with G/I. The components associated with ab/R and O/L are also partially blended. Thirteen MSPs have clear, isolated radio continuum detections. Figure <ref> shows an intriguing feature, in that the pulsars C and J have not deconvolved properly. Calibration deficiencies, in particular those due to direction dependent effects <cit.>, often result in residual PSF-like structures around the brighter sources. The dominant DDE for MeerKAT L-band obsevations is related to the antenna primary beam pattern coupled with pointing errors. These effects do not have a strong manifestation close to the pointing centre, and sources of comparable brightness to MSPs C and J do not suffer similar deficiencies. One could thus make the prediction at this stage that these sources are strongly time-variable over the course of the observation. Intrinsic variability of a particular radio source during the observation is analogous to a direction-dependent amplitude error in the eyes of the calibration and imaging process. The presence of persitent, unexpected PSF-like patterns around a source is a good indicator that its radio emission may be variable, either intrinsically or due to propagation effects <cit.>. We restrict the analyses that follow to the inner 0.625 × 0.625 deg^2 of the image, covering up to 7 times the half mass radius from the cluster centre. The primary beam gain in this region is no less than 0.75. The main lobe of the MeerKAT Stokes I beam is not azimuthally symmetric, and our selection limits any instrumental source variability induced by rotation of the beam pattern with respect to the sky to <1 per cent. To extract a `control' sample of sources – a representative sample of the general source population, predominantly background extra galactic sources, but that may also include new pulsar candidates – we used the pybdsf source finder <cit.> with its default settings. In the absence of confusion or significant nebular emission, a pulsar will be detected as true point source. The control sample was therefore filtered to exclude extended sources by enforcing a peak brightness to integrated flux density ratio 0.9 < S_peak / S_int < 1.0, and selecting only single point / Gaussian components with S_peak > 10 μJy beam^-1. Following this the control sample contains 927 sources. Detections of known pulsars were flagged, and any known pulsar that was not picked up by pybdsf was fitted using the casa imfit task. In order to study the temporal properties of the pulsars and other sources in the field in the image domain, we used ddfacet to produce deconvolved, full-band MFS images with direction-dependent corrections applied for each of the 21 × 14.8 min scans. We extract light curves from this sequence of images for both MSPs and control sources by force-fitting a Gaussian with the shape of the epoch-specific restoring beam at the position of each source using the casa imfit task. Spectral properties of the sources are examined by means of a spectral index[We adopt that convention that the power law S ∝ ν^α relates the spectral index α to the flux density (or peak brightness) S.] map and an accompanying error map. These are made by imaging and deconvolving the data in 8 sub-bands. The angular resolutions of the resulting stack of images are homogenised by convolving the images to a marginally coarser resolution than that of the lowest sub-band (85). We then apply primary beam correction appropriate for each frequency. Following this, we fit for the linear gradient (α) of each sight-line through the resulting cube in log-ν / log-S space. These values are then recorded as a spectral index and spectral index error (σ_α) map. For each source we weight the spectral index and spectral index error map with a circular 85 Gaussian centred on its position, and the brightness-weighted mean values of α and σ_α are then associated with each component in the catalogue for further analysis. § DISCUSSION §.§ Temporal variability statistics The light curve for each object can be boiled down to three statistics, which when plotted as an ensemble can reveal interesting behaviour via the identification of outliers <cit.>. For a light curve consisting of N brightness measurements, S_i, the first statistic we make use of is V, which is simply the standard deviation of the light curve divided by the mean: V = σ_S / S. The V statistic thus provides some measure of how spread out the values of S_i are. The second statistic η is based on the reduced χ^2 statistic: η = N/N-1 ∑_i=1^N (S_i - S)^2/σ^2_i where σ_i is the standard deviation noise measurement of S_i. A constant source should therefore have a value of η close to unity. The third statistic ξ_max is based on the median absolute deviation (MAD). For each element in the light curve, the residual of the data and the median (S̃) is expressed as a fraction of the MAD: ξ_i = S_i - S̃/median(\|S_i-S̃\|) and ξ_max is the maximum value from the resulting N-length set. The ξ_max statistic thus has a higher value for light curves that contain short, burst-like features. These three statistics are computed for the objects in the control sample, as well as for the known pulsars. The resulting values of V, η, and ξ_max are plotted against the median peak brightness for each object in log-log space on panels a, b and c of Figure <ref>. The control sample is plotted in grey, with known MSPs plotted as blue circles. Note that the parent catalogues for these figures are drawn from the full time-integrated MFS images described in Section <ref>. Assuming constant sensitivity of the telescope during a given observation, the per-scan snapshot images used to derive the light curves will have noise levels that are 4.6 times higher than the fully integrated image. We will therefore be performing photometric extraction via the fitting of epoch-specific Gaussians at the positions of some sources that are dominated by the thermal noise in the time series imaging. However, this is not an issue in terms of confusing plots of the the variability statistics introduced above, as they are all formed relative to a mean or median baseline. Indeed the correlation between the signal to noise ratio (SNR) of the source brightness and the V parameter is evident in panel a of Figure <ref> (and self-evident in Equation <ref>). Similarly, some of the pulsars with fainter continuum detections in the integrated images may also be dominated by noise in the time series imaging. Again we apply force-fitting to extract their light curves to search for brief brightness changes that may not be evident in the time and frequency integrated images. §.§ Spectral index properties As can be be seen in panels d and e of Figure <ref>, spectral index measurements exhibit increasing amounts of scatter as the SNR of the source decreases. Mean α measurements for ensembles of objects as a function of their mean brightness eventually become entirely unreliable due to the imposition of strong selection effects that are coupled to the observational frequency coverage and the intrinsic spectral indices of the source population <cit.>. The tail of steep negative spectral indices visible in panel d of Figure <ref> is almost certainly an artefact of these effects. We thus restrict interpretation of the spectral index measurements to the region where their distribution remains roughly symmetric at log_10 S_peak > 1.5. Pulsars are known to typically exhibit steep spectra <cit.>, and therefore may appear as outliers from the general population of compact radio sources, which is dominated by extragalactic synchrotron sources with a typical spectral index of -0.7. The error in the spectral index fit σ_α, i.e. a measure of how well the spectrum is captured by a simple power law, is shown in panel e. §.§ The MSP population and other outlier sources The temporal and spectral properties of the continuum counterparts of the known MSP population are plotted using blue markers on Figure <ref>. Deviations from the control sample are seen most readily in panels a (V), b (η), and e (σ_α). A linear fit to the distribution of these three parameters is shown by the solid line. The dashed lines are ±2σ, where σ is the mean of the standard deviations of the distribution computed in eight logarithmically-spaced peak brightness bins. These lines crudely delineates regions of the parameter space containing outliers. Plots of both the MSP and control sample lightcurves can be found online[<https://github.com/IanHeywood/globular>]. Temporal variability in the MSP continuum emission could be explained by either differing diffractive interstellar scintillation conditions along the sightlines to the pulsars, intrinsic variability of the pulsars themselves, modulations due to binary orbits, or some combination of these. Approximately 20 per cent of the known MSPs could be identified as outliers in the V and η plots. The strong variability of MSPs C and J that was predicted via their associated imaging artefacts is confirmed, with these two sources being the most apparent. The distribution of the ξ_max statistic shows that at the 15 minute imaging cadence none of the MSPs appear as outliers, and thus do not have single epochs that deviate significantly from their general behavior. Investigation of the six components with log_10 ξ_max > 1 in panel c of Figure <ref> revealed that they are all pairs of sources that are close in projection one another. Scans that are at lower elevations exhibit larger, more elongated synthesised beams, and in those instances the emission from the sources encroaches on the position of the fainter one, causing the high ξ_max value when the photometry of the fainter source is extracted. Such instances could be automatically identified by considering the angular resolution of each individual image and the angular separations between a source and its neighbours, but we made no attempt to implement such a scheme here. The outlier source visible to the lower right of 47 Tuc C in the V plot is also the partially blended source with the highest ξ_max value. A far more effective way to identify the outlying MSPs is to examine the σ_α distribution, in which approximately half of the MSPs can be distinguished. This suggests that scintillation is the dominant source of the variability, as the interference pattern induced by the ionized material along the lines of sight to each pulsar introduces random fluctuations into the radio emission in frequency as well as time. This can also explain the spectral index values that are extremely steep (e.g. 47 Tuc H) and inverted (e.g. 47 Tuc J), as seen in panel d. §.§ J002402.7-720539.4: A candidate millisecond pulsar The source from the control sample that is highlighted by the pink box in Figure <ref> occupies the same regions of parameter space as the known MSP population. We identify this as a candidate new MSP, or perhaps the localised radio continuum counterpart of one of the six MSPs in 47 Tuc that do not yet have good positions available from the timing solutions. These are P and V <cit.>, ac and ad <cit.>, and ae and af[<http://trapum.org/discoveries/>]. The radio position of the candidate is 1” offset from the position of the X-ray source W52 <cit.>, which has an intrinsic X-ray luminosity in the 0.5–2.5 keV of 2.5 × 10^30 erg s^-1, consistent with the X-ray luminosities of the general MSP population in 47 Tuc. § SUMMARY We have investigated the image domain properties of the MSP population in 47 Tuc by exploiting the time and frequency dimensions of an archival MeerKAT observation. Approximately 20 per cent of the MSPs could be discovered using the temporal variability statistics that are generally employed in image domain searches for transients and variables. This fraction increases to approximately 50 per cent when the spectral domain is considered. We have identified one source as a candidate previously unknown MSP, or otherwise provide an interferometric localisation of one of the known MSPs in 47 Tuc for which no position is yet available through timing solutions. Scintillation is the likely cause of the MSPs appearing as outliers in the various metrics we have explored, with differing sightlines to the MSPs in the cluster resulting in a range of differing light curves. Modulation of the brightness due to binary orbits is hinted at in the light curve of at least one MSP (47 Tuc O), which is known to have a 3.12 hour orbital period that is coincident with peaks in the lightcurve <cit.>. Additional monitoring (possibly using MeerKAT's S-band to lessen the effects of scintillation) required to investigate this. S-band imaging would also potentially allow some of the blended pairs of MSPs to be resolved separately. The approach used in this paper is computationally cheaper than the true variance imaging method discussed by <cit.>, but is demonstrated to be a fruitful method for the detection of MSPs and other scintillating sources. We have made no attempt to optimise the time/frequency imaging intervals to match the scintillation scales in those domains. This may result in higher detection fractions, and will be investigated in a future work, which we will extend to include imaging of additional GCs and full Stokes imaging where possible. This work demonstrates that commensal imaging of fields that are being targeted for beamformer pulsar searches offers a potentially valuable way to localise pulsars immediately, as well as the potential for discovering new pulsars in image domain programs that fully exploit the time and frequency dimensions of the data. § ACKNOWLEDGEMENTS The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation. This research made use of Astropy,[<http://www.astropy.org>] a community-developed core Python package for Astronomy <cit.>. This work has made use of the Cube Analysis and Rendering Tool for Astronomy <cit.>. This research has made use of NASA's Astrophysics Data System. I acknowledge support from the Breakthrough Listen initiative, the UK Science and Technology Facilities Council, and from the South African Radio Astronomy Observatory. Breakthrough Listen is managed by the Breakthrough Initiatives, sponsored by the Breakthrough Prize Foundation. I thank my colleagues in the Oxford pulsar and relativistic accretion groups for useful discussions. I thank Paulo Freire for maintaining a comprehensive web resource on pulsars in globular clusters which was useful for this work. Finally, I thank the anonymous referee for their useful comments. § DATA AVAILABILITY The visibility data are available from the SARAO archive under the capture block ID listed in Table <ref>. Tabular data and light curve images are available at < https://github.com/IanHeywood/globular>. Synthesised images are available from the author. 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http://arxiv.org/abs/2307.00367v1	20230701153447	Dark Dust III: The high-quality single-cloud reddening curve sample. Scrutinizing extinction curves in the Milky Way	[ "R. Siebenmorgen", "J. Smoker", "J. Krełowski", "Karl Gordon", "Rolf Chini" ]	astro-ph.GA	[ "astro-ph.GA" ]	Dark Dust III: Scrutinizing extinction curves in the Milky Way. R. Siebenmorgen et al. Reddening in the diffuse ISM European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany email: [email protected] European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK Materials Spectroscopy Laboratory, University of Rzeszów, Pigonia 1 Street, 35-310, Rzeszów, Poland Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), 44780 Bochum, Germany Universidad Católica del Norte, Instituto de Astronomía, Avenida Angamos 0610, Antofagasta, Chile Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland The nature of dust in the diffuse interstellar medium can be best investigated by means of reddening curves where only a single interstellar cloud lies between the observer and the background source. Published reddening curves often suffer from various systematic uncertainties. We merge a sample of 895 reddening curves of stars for which both FORS2 polarisation spectra and UVES high-resolution spectra are available. The resulting 111 sightlines toward OB-type stars have 175 reddening curves. For these stars, we derive their spectral type from the UVES high-resolution spectroscopy. To obtain high-quality reddening curves we exclude stars with composite spectra in the IUE/FUSE data due to multiple stellar systems. Likewise, we omit stars that have uncertain spectral type designations or stars with photometric variability. We neglect stars that show inconsistent parallaxes when comparing DR2 and DR3 from GAIA. Finally, we identify stars that show differences in the space and ground-based derived reddening curves between 0.28 μm and the U-band or in R_V. In total, we find 53 stars with one or more reddening curves passing the rejection criteria. This provides the highest quality Milky Way reddening curve sample available today. Averaging the curves from our high-quality sample, we find R_V = 3.1 ± 0.4, confirming previous estimates. A future paper in this series will use the current sample of precise reddening curves and combine them with polarisation data to study the properties of dark dust. The high-quality single-cloud reddening curve sample R. Siebenmorgen1, J. Smoker2,3, J. Krełowski4, Karl Gordon5, and Rolf Chini6,7,8 Received: August 1, 2022/ Accepted: July 1, 2023 ========================================================================================= § DUST IN THE INTERSTELLAR MEDIUM The disc of our Galaxy, as well as of other spiral galaxies, is filled with the interstellar medium (ISM), consisting of gas, molecules, and dust grains. The ISM is clumpy <cit.>, and covers a wide range of temperatures and densities <cit.>; based on the latter parameter one may distinguish three categories – diffuse, translucent, and dense clouds. Dense clouds are typically star-forming regions, where the high extinction (A_V > 5 mag) generally impedes optical observations of embedded stars. Therefore, this component of the ISM is best analysed at infrared and microwave wavelengths e.g. <cit.>. Translucent clouds, in contrast, offer the possibility to study the ISM optically via photometric and spectroscopic observations of stars, shining through the material at moderate density of A_V < 3 mag <cit.>. The disadvantage here is that in the majority of cases several clouds are present along a single-sightline – an effect that increases with stellar distance. Likewise, there is the diffuse medium which contributes to the extinction along the sightline <cit.>. Furthermore, translucent clouds show striking differences in their chemical composition and physical parameters as witnessed by varying intensity ratios of spectral lines or bands. The study of the ISM has two origins: Firstly, interstellar gas and dust were discovered by chance during photometric and spectroscopic observations of stars <cit.>, revealing that the precise knowledge of both the interstellar lines and the amount of dust along the sightline were crucial for the interpretation of stellar spectra and photometric data. Secondly, <cit.> recognized the fundamental role of the ISM in the process of star and planet formation. After the detection of interstellar dust nearly 100 years ago <cit.>, it was soon learned to compare the spectral energy distribution of unreddened stars with those of reddened stars (see Sect. <ref>) to quantify the wavelength-dependant reddening by dust grains – leading to the famous extinction curve. Its normalisation R_V = A_V / E(B-V) – the ratio of total-to-selective extinction – was further treated like a constant of nature with a value of R_ V ∼ 3.1. Therefore, early model fits of the extinction curve led to fairly simple dust models with a power-law grain-size distribution of silicate and graphite grains. Only photometric studies, some decades later, of stars in dense, star-forming clouds revealed R_V-values of up to five, suggesting substantial grain growths – either by coagulation and/or by the formation of mantles <cit.>. Furthermore, PAHs were detected by IR spectroscopy in the ISM <cit.>. Nevertheless, the R_V-value kept its importance and was suggested to be the only parameter that determines the extinction law, e.g. <cit.>. In the present work, we readdress the issue of reddening curves by analyzing and completing published data. The ideal set of data required to determine a reddening curve would be a spectroscopically accurately classified star, with data from the UV to IR, at a reliable GAIA <cit.> distance and with a single translucent cloud along the sightline. Unfortunately, such cases are rare, calling into question many published reddening curves. For example, due to their large distances, light from OB-type stars typically passes through several intervening clouds <cit.>. Therefore, the observed extinction curves for different sightlines are usually quite similar to each other, simulating a similar R_V-value. However, they are ill-defined averages in the sense that measurements of single-cloud sightlines with similar chemical compositions should preferentially be used when studying the physical properties of dust or the magnetic field direction when extracted from the optical polarisation angle. To improve the sample from which precise reddening curves can be derived, we selected sightlines where spectra show interstellar atomic lines or features of simple radicals, dominated by only one Doppler component. Such single-cloud sightlines can be interpreted in terms of better defined physical conditions. Furthermore, we focus on OB-type stars with known trigonometric parallaxes from GAIA, as this may allow to estimate both column densities and also local (volume) densities. For future observations, it seems important to select and scrutinize all cases from the extensive catalogues by <cit.>. This paper is the third in a series concerned with dark dust, which is composed of very large and very cold grains in the ISM. The first paper <cit.> presented initial results derived from distance unification and the second <cit.> presented a dust model for the general ISM, which was tested against the contemporary set of observational constraints <cit.>. The current paper presents a high-quality sample of reddening curves obtained from the literature as needed for detailed modelling of particular sightlines. It is laid out as follows: Sect. <ref> presents the sample of sightlines, many of them dominated by single interstellar lines. Sect. <ref> describes how interstellar reddening curves are calculated and discusses the impact of binarity and uncertainties in the spectral type (SpT) on the derived curves. The method used to ascribe a SpT to our sample stars and how uncertainties in the SpT of sample and comparion stars affect the reddening curve is described in Sect. <ref>. Systematic issues affecting the quality of the reddening curves in the literature are discussed in Sect <ref>. The high-quality sample of reddening curves is presented in Sect. <ref>, followed by a discussion of the systematic scatter in published reddening curves and R_V of the same star. A mean Milky Way reddening curve is derived in Sect. <ref> and the main findings are summarized in Sect. <ref>. § THE SAMPLE Whenever sightlines are observed that intersect different components of the ISM, the relationship between the physical parameters of the dust and the observing characteristics provided by the extinction and polarisation data is lost. Hence studying variations of dust properties in translucient clouds requires a sample of single-cloud sightlines for which the wavelength dependence of the reddening and polarisation are available <cit.>. Reddening curves have been measured in the near-infrared (J H K) using the Two Micron All Sky Survey (2MASS, <cit.>), in the optical (U B V) utilizing ground-based facilities <cit.>, and at shorter wavelengths from space. In particular, the International Ultraviolet Explorer (IUE) and the Far Ultraviolet Spectroscopic Explorer (FUSE) provide (far) UV spectra below 0.3 μm down to the Lyman limit. Reddening curves have been derived from the IUE for 422 stars by <cit.>, 351 by <cit.>, including sightlines observed with HST/STIS by <cit.>, and for 75 stars with FUSE by <cit.>, who adjusted the FUSE spectra to the larger IUE aperture. In total, 895 reddening curves towards 568 early types (OB) stars are available from the references described above. In order to obtain a sample of sightlines with few interstellar components, we examined 186 stars mainly observed with the Ultraviolet and Visual Echelle Spectrograph (UVES; <cit.> of the ESO Very Large Telescope <cit.>. The term single-cloud sightline was introduced when the interstellar line profiles show one dominant Doppler component at a spectral resolving power (R) of λ/Δλ∼ 75,000 (full width at half maximum, FWHM ∼ 4 km/s) and accounting for more than half of the observed column density. In total, 65 single-cloud sightlines were detected predominantly by inspecting the K i line at 7698 Å <cit.>. Single-cloud sightlines may include two or more fine-structure components in the line profiles with slightly different radial velocities especially when observed at an even higher resolution <cit.>. The detection of 65 single-cloud sightlines is a substantial increase as so far only eight single-cloud sightlines were available for a detailed analysis. Complementary linear polarisation data of 215 stars were taken from the Large Interstellar Polarisation Survey <cit.>. Simultaneous modelling of reddening and polarisation data provides important constraints of the dust <cit.>. Merging the data of 186 OB stars with UVES spectra, 568 OB stars with 895 reddening curves, and 215 OB stars with polarisation spectra yields our sample of 111 sightlines. For this sample, 175 reddening curves are published: 18 by <cit.>, 70 by <cit.>, and 87 by <cit.>. They are displayed in Fig. <ref>. § INTERSTELLAR REDDENING AND EXTINCTION Generally, interstellar reddening and extinction curves are derived by measuring the flux ratio of a reddened and unreddened star with the same SpT and luminosity class (LC), the so-called standard pair-method <cit.>. This method includes uncertainties in the calibraton of the observed spectra, in the SpT and LC estimates, and in the need of a close-match in SpT and LC between the reddened and unredded star, respectively. An alternative to the pair-method is to use stellar atmosphere models <cit.>. This method avoids the comparison star but relies on the accuracy of stellar models. In the current paper we examine the accessible databases of reddening curves by <cit.> and <cit.> for our sample and apply the following notation: The observed flux of a star is derived from the spectral luminosity L(λ), the distance D, and the extinction optical depth τ(λ), which is due to the absorption and scattering of photons along the sightline. The observed flux of the reddened star is given by F(λ) = L(λ) / 4 π D^2 e^-τ(λ) . We follow <cit.> and denote the flux of the unreddened (τ_0=0) comparison star at distance D_0 by F_0. The apparent magnitude is related to the flux through m(λ) = 2.5 log _10( w_λ/F(λ) ), where w_λ is the zero point of the photometric system. The difference in magnitudes between the reddened and the unreddened star is Δ m(λ) = 1.086 ×( τ(λ) + 2 log _10(D/D_0) ). The accuracy of the dust extinction derived by this method depends critically on the match of both the SpT and LC, and on how well the distances to both stars are known. Unfortunately, distances to hot, early-type stars, commonly used to measure interstellar lines, are subject to large errors <cit.>. Hence one relies on relative measurements of two wavelengths and defines the color excess E(λ - λ')=Δ m(λ)-Δ m(λ'). The common notations are e.g. for the B and V-band E(B-V) = (B-V) - (B-V)_0. The reddening curve E(λ) is traditionally represented by a colour excess that is related to the V-band and employs a normalisation to avoid the distance uncertainties, vis: E(λ) = E(λ - V)/E(B-V) = A_λ - A_V/A_B - A_V = τ_λ - τ_V/τ_B - τ_V By definition E(V) = 0 and E(B) = 1. The extinction in magnitudes at wavelength λ is denoted by A(λ). An extrapolated estimate of the visual extinction A_V is obtained from photometry. This requires measuring E(B-V) and extrapolating the reddening curve to an infinite wavelength E(∞). In practice, E(λ-V) is observed at the longest wavelength which is not contaminated by either dust or any other emission components of early type stars <cit.>. From this wavelength, e.g. the K-band, one extrapolates to infinite wavelength assuming some a priori shape of E(λ) and hence estimating E(∞ -V). By introducing the ratio of total-to-selective extinction R_V = A_V / E(B-V) a simple relation of the reddening to the extinction curve is τ(λ)/τ_V = E(λ)/R_V + 1 where obviously E(∞) = - R_V. The total-to-selective extinction of the dust is R_V = τ_V/τ_B - τ_V = κ_V/κ_B - κ_V where κ = κ_abs + κ_sca is the extinction cross-section which is the sum of the absorption and scattering cross-section of the dust model. In the ISM of the Milky Way the total-to-selective extinction scatters between 2 < R_V 4.1; A_V∼ 3.1 E(B-V) is given as a mean value <cit.>. Other approximate formulae using near IR colours may also be applied, e.g. <cit.> proposed R_V ∼ 1.1 E(V-K) / E(B-V). We note that a large R_V-value (e.g. 5) does not necessarily exclude a low reddening (e.g. E(B-V) 0.3). Indeed, at first sight four stars in our sample show low values of E(B-V) combined with high R_V. However, a more detailed investigation indicates that each of these stars has some issue which could impact the derivation of R_V and the reddening curve. In particular, HD 037022 <cit.> includes in the IUE apertures multiple equally bright objects that contaminate the observed spectra. The second star HD 037041 shows photometric instabilities with time variations in the GAIA G-band of 0.07 mag, which is significant, considering that E(B-V) = 0.2 <cit.>. Finally, for HD 104705 and HD 164073 varying estimates of R_V have been derived by different authors, placing doubt on the “true" value of R_V. As we will show in Sect. <ref>, there is a large systematic error associated with published R_V estimates of the same star. Hence, whenever possible, we try to avoid the R_V parameter and thus prefer to discuss reddening instead of extinction curves. § STELLAR CLASSIFICATION In this section we describe how we determined the spectral classes and luminosity types of our sample in a uniform manner. Accurate stellar classification is of utmost importance for deriving the reddening. A Simbad search of the MK classification in our sample (Sect. <ref>) shows a large spread in the SpT and the LC. In order to reduce systematics we therefore reclassified the spectral types of our stars in the MK system using UVES spectra that were fitted to standard stars. For the standard star spectra we used the library “libr18” by <cit.>. The library includes spectra at wavelengths between 380 - 462 nm. The reduction and analysis of the UVES spectra are explained in paper I <cit.> and were complemented with spectra available in the ESO Science Archive Facility under ESO programme IDs listed in Appendix <ref>. The UVES spectra are at a resolving power of λ /Δλ∼ 75,000 and high signal-to-noise (≥ 200). The different settings of the UVES spectra were rectified, merged, shifted in wavelength to match the 410.2 nm feature, and smoothed by a Gaussian kernel to equal the spectral resolution of the spectra in the library. SpT and LC were determined by the best-fitting element of the library to the spectrum of the target star. The best fit was identified using a minimum χ^2 condition. The precision in the classification of O-type stars was estimated by comparing the <cit.> standards to the SpT derived in the Galactic O-star survey by <cit.>. There are 34 O-stars common to both catalogues. For 32 stars the SpT agrees to better than one subclass and for two stars the SpT differs by more than that. These are HD 093129 and HD 303308; both are O3 standards by <cit.> whereas <cit.> classifies them in agreement to our fitting procedure as O5 and O4.5, respectively. Additionally, our SpT and LC estimates were compared with classifications of early-type standards by <cit.>. These authors provide 38 O-type and 37 B-type standard stars with different LCs. The <cit.> library includes 24 spectra for stars between O4 - O9 and 37 spectra for stars between B0 - B9. Our SpT agrees to better than one subclass for 36 stars. A larger spread is only found for two stars; HD 037043 is classified as an O9V and HD 163758 as an O6.5Ia standard by <cit.>; in agreement with <cit.> we assign spectral types of O7V and O5Ib, respectively. Most of the 37 B stars are earlier than B3 with only one B5 and one B8 star. Our SpT estimates match to within better than one subclass for 35 of these stars. Only HD 51309 and HD 53138, both classified as B3Iab <cit.>, yield B5Iab in our fitting procedure. The goodness of the <cit.> best-fit to the 75 OB standard star spectra by <cit.> is at minimum χ^2 = 0.5 ± 0.6 < 1.45 and the ratio of both spectra varies by 2.4 ± 0.4 < 3.7 (%). The SpT estimates of the stars agree to within ΔSpT = 0.6 ± 0.3. <cit.> distinguish luminosity classes I, III, and V. The LC determination of our procedure agrees with <cit.> to ΔLC = 0.5 ± 0.8. The accuracy of the classification on the MK system by our fitting procedure is the same as reached by <cit.>. These authors report a precision comparable to human classifiers of ΔSpT = 0.6 of a spectral subclass and ΔLC = 0.5 of a luminosity class, respectively. <cit.> classified stellar spectra of the LAMOST survey using <cit.> and confirm the quoted accuracy. A similar precision in stellar classification is also reached by <cit.>, again indicating the accuracy of our classification is on a par with other works in the literature. The SpT procedure was also applied to the unreddened comparison stars by <cit.> and <cit.>. The ELODIE[http://atlas.obs-hp.fr/elodi] and ESO archives[http://archive.eso.org] were inspected for available data from high resolution spectrographs ELODIE <cit.>, ESPRESSO <cit.>, FEROS <cit.>, HARPS <cit.>, XSHOOTER <cit.>, and UVES. High resolution spectra of 21 standards were found and the various SpT estimates agree to previous estimates within one subclass (Table <ref>). § SCRUTINIZING REDDENING CURVES Reddening curves offer the possibility of deriving fundamental characteristics of dust such as particle sizes and abundances. Systematic issues that affect reddening curves must be minimized for dust modelling work. To this aim, the trustworthiness of the reddening curves towards the 111 stars presented in Sect. <ref> have been inspected to establish a high-quality reddening curve sample with predominantly single-cloud sightlines. According to Eq. <ref> reddening curves have a similar shape making it extremely difficult to exclude less good cases by simple inspection (Fig. <ref>). Even if there exist comparable results for the same star, there might still be errors whenever the conditions for the derivation of the reddening are not fulfilled. The quality of the reddening curve depends critically on the precision of the SpT and LC estimates, the photometric and spectral stability of the reddened and unreddened star, the de-reddening of the comparison star, and the photospheric model when applied. In general, reddening curves are derived assuming single stellar systems. In the following subsections, we discuss the systematic effects that affect the reddening curves of our sample. Uncertain cases and those that are not qualified for detailed dust modelling are rejected and listed in Table <ref>, while the sample of high-quality reddening curves are given in Table <ref>. §.§ Parametrization of reddening curves Reddening curves in the UV range between 3.3 μm^-1λ^-1 11 μm^-1 are represented by a spline fit the a Drude profile for the 2175 Å extinction bump, and a polynomial for the UV rise E(λ - V)/E(B-V) = c_1 + c_2 x + c_3 D(x,γ, x_0) + c_4 F(x) where x=λ^-1. The Drude profile is given by D(x,γ, x_0) = x^2/(x^2-x_0^2)^2 + (x γ)^2 with damping constant γ and central wavelength x_0^-1. The non-linear increase of the reddening in the far UV is described by F(x). <cit.> and <cit.> applied a form that is given by <cit.> F(x) = 0.5392 (x - 5.9)^2+ 0.05644 (x - 5.9)^3 : x ≥ 5.9 μm^-1 while <cit.> used F(x) = (x - c_5)^2 : x ≥ c_5 . At longer wavelengths F(x) = 0. Following <cit.>, we reduce c_4 by 7.5 % when data from the IUE alone were used to estimate the reddening curve, and extrapolate the reddening to x = 11 μm^-1 when necessary. Reddening in spectral regions close to wind lines at 6.5 and 7.1 μm^-1 and Ly-α at 8 μm^-1≤ x ≤ 8.45 μm^-1, or with apparent instrumental noise at x 3.6 μm^-1 is ignored. Relations between the reddening curve parameters c_i and R_V (Eq. <ref>) and the dust model parameters are given in Table 5 by <cit.>. For example, an uncertainty of 10% in c_1 implies an uncertainty of 5% in the the abundance ratio of the large silicate to carbon particles; a 10% error in R_V translates into a variation of the exponent q of the dust size power-law distribution of 10%. Eventually, a variation of 10% in c_4 results in an uncertainty in the abundance ratio of very small to large grains of 25%. §.§ Composite FUSE and IUE spectra Reddening curves derived from a composite spectrum that includes the program star and other bright objects are invalid. Reddening curves of our sample are flagged when there are multiple objects in the IUE (separation ≤ 10) or in the FUSE aperture (separation ≤ 15) that contribute by more than 10 % to the flux of the program star (Δ V 2.5 mag). The Hipparcos (ASCC-2.5 <cit.>) and Simbad databases were also used to detect potential contaminating objects. For 12 stars the label M is assigned in Table <ref> indicating that the observed spectrum is a composite of multiple bright objects in the IUE aperture. No companions were found in the FUSE aperture. §.§ Multiple star systems The multiplicity of stellar systems is typically investigated through imaging and interferometry <cit.> and/or spectroscopy <cit.>. The latter method is biased towards finding close companions down to a few AU separation by providing a measure of time variable radial velocities. Differences in the line profiles are still visible in inclined systems for which the brightness difference between primary and companion may become marginal. Such a high resolution radial velocity survey with spectra taken at multiple epochs (2-12) of about 800 OB stars is presented by <cit.>. Companions in that survey are detected down to a brightness difference of Δ V ∼ 2 mag. This translates to detectable companions of an O5 star range between O5 - B2, and those of a B9 star from B9 - A7. The results of the surveys listed above indicate that nearly 100 % of O-type stars have one or more companions within 1 mas to 8” separation and at a contrast down to Δ H = 8 mag <cit.>, falling to 80 % for early-B and to 20 % for late B-types <cit.>. The HIPPARCOS Tycho photometric catalogue <cit.> provides various labels indicating the duplicity or variability status of the stars. Unfortunately, there are no striking features in the reddening curves noticeable when inspecting sub-samples with variability/binary flag set or unset. §.§ GAIA The GAIA space observatory launched by the European Space Agency (ESA) in 2013 measures positions, parallaxes, motions, and photometry of stars with unprecedented precision <cit.>. The GAIA data release 2, DR2 <cit.> was based on observations made between July 2014 and May 2016 and was followed by data release 3, DR3 <cit.> which includes observations until May 2017. Stars that have inconsistent GAIA parallaxes π_DR2 versus π_DR3 are suspicious. Their parallax measurements at higher than 3 σ confidence remain unconfirmed either due to instrumental artefacts, which we doubt, bright companions or stellar activity. In our sample, there are 102 stars with parallax measurements in DR2 at a typical S/N ratio of 13; in DR3 there are 106 of our stars with a typical S/N ratio of 25. From these stars, 95 have a ratio π_DR2/ π_DR3 higher than 3σ confidence with a mean ratio of 0.94 ± 0.16. The 6 % deviation from unity is driven by unsecured DR2 measurements at large distances. By considering a sub-sample with π_DR2^-1 2 kpc there are 72 stars with a mean ratio of π_DR2/ π_DR3∼ 0.98 ± 0.14 which is consistent with being unity. Four stars are identified by 3σ clipping as outliers and are indicated with the flag π in Table <ref>. The inverse parallax (pc) of π^-1_DR3 versus π^-1_DR2 is shown with outliers marked in magenta in Fig. <ref>. Note the deviation from the identity curve at π^-1_DR2 2 kpc. The photometric stability of the stars of our sample was verified by comparing both GAIA data releases. In Fig. <ref> the differences in G-band (330 - 1050 nm) photometry between DR3 and DR2, Δ G = G_DR3 - G_DR2, is shown for our sample of 111 stars. In that sample, the mean and 1σ scatter of Δ G = 14 ± 11 mmag. Nine stars show variability in the photometry with Δ G outside of the range 14 ± 33 mmag. They are marked in Fig. <ref> in magenta outside the ± 3 σ curves shown as dashed lines. Their reddening curves are classified as uncertain. Six of these sightlines were not yet rejected and they are labelled Δ G in Table <ref>. We note that seven stars from our sample would be rejected due to colour variability in B - G, whereas stars outside of the range (B-G)_DR3 - (B-G)_DR2 of 31± 66 mmag would be excluded using 3σ clipping. These stars were already rejected using the previous criteria. §.§ Photometric variability Stellar variability will impact the results for reddening. Early-type stars may vary due to multiplicity or due to winds. About 50% of OB stars reveal extra-photospheric infrared excess emission that is likely caused by winds <cit.>, and strikingly, as discussed in Sect. <ref>, the great majority of O and early-type B stars form close binary systems. The reddening curves that we obtained from the literature made use of the Johnson UBV system <cit.>. Ground-based (GB) photometry of our sample is given by <cit.> and refers to observations between 1950 - 2000. Stellar photometry from the Hipparcos satellite between 1989 - 1993 is also available. <cit.> merged Hipparcos, Tycho, PPM, and CMC11 observations of 2.5 million stars and transformed V and B magnitudes to the Johnson system. This resulted in a colour-dependent correction of 20 - 40 mmag, and a typical error below 10 mmag of the ASCC-2.5 catalogue <cit.>. The photometric stability of the stars was verified by comparing ground-based V_GB <cit.> and Hipparcos V_Hip <cit.> photometry (Fig. <ref>). There are 73 stars for which photometry in both catalogues is available and which are not rejected by the previous confidence criteria (Sect. <ref> - <ref>). In that sample the mean and 1σ scatter of Δ V = V_GB - V_Hip = 5 ± 31 mmag. The small offset is within the photometric error. Seven stars are identified by 3σ clipping as outliers showing significant variability in the V-band. Their red- dening curves are classified as uncertain and labelled Δ V in Table <ref>. These stars are marked in magenta in Fig. <ref> and are outside the ± 3 σ variation shown as dashed lines. HD 037023 was identified as a composite star in Sect. <ref> and fails with Δ V = 1.6 mag. The photometric variable star HD 091983 was used by <cit.> to derive the reddening curves towards HD 122879 (Table <ref>) and HD 168941. The same procedure of outlier rejection was repeated by comparing the B-band photometry and the (B - V) colour provided from the ground by <cit.> and by Hipparcos <cit.>. The mean and 1σ scatter in the colours is Δ_BV = (B-V)_GB - (B-V)_Hip = -28 ± 28 mmag. The two stars HD 147701 and HD 169454 are identified by 3σ clipping as outliers. Their reddening curves are classified as uncertain and labelled Δ_BV in Table <ref>. No additional stars were rejected due to photometric variability in the B-band. §.§ Unfeasible stellar classification O- and B-type stars are often fast rotators; the peak in the rotational velocity probability distribution for B-type stars is around 300 km s^-1 <cit.>. As a consequence their SpT and LC determination is highly uncertain because: (1) most useful diagnostic lines such as Mg ii or He i are blended and (2) unless the spectra have a very high S/N ratio, many stellar absorption lines merge into the continuum. Several of the stars have also a bright companion making the stellar classifcation unfeasible. For nine stars the stellar classification procedure of Sect. <ref> is uncertain at χ^2 > 2. These sightlines are rejected for the high-quality reddening curve determination with label χ^2 in Table <ref>. §.§ Inaccurate stellar classification A spectral type and luminosity miss-match of the target or comparison star can give a large variation in E(B-V), in the infrared extinction, differences in position and width of the extinction bump, and in the far UV rise of the reddening curve. As shown by <cit.>, photometric and systematic errors of the extinction curves scale as 1/E(B-V). A miss-classification in the LC or SpT of more than one subclass in either of the reddend or the unreddened star may introduce large (∼ 20 %) systematic errors in E(B-V) and a significant difference in the far UV rise <cit.>. They also show that such a change in the far UV rise expressed in parameter c4 (Eq. <ref>) may vary by a factor ∼ 1.5. Whenever the comparison star is hotter than the reddened star the far UV rise will be overestimated. In the UV, the IUE and FUSE spectrographs were used to estimate SpT and LC. Besides the low resolving power of these UV spectrographs, there are major diagnostic lines in the optical but not in the UV. <cit.> show that the UV spectral diagnostics indicate often earlier SpT than obtained from optical spectra. They considered uncertainties of one luminosity class and one spectral sub-type, which increases to up to two sub-types for mid/late B stars because of fewer spectral diagnostics in that range. <cit.> list the SpT of the reddened and comparison stars. <cit.> apply comparison stars selected from <cit.>, who provide those for types earlier than B3. For later stars, the choice of the comparison star was not detailed; same holds for stars earlier than O7. We note that the stellar atmosphere model-based method, as used by <cit.>, does not need to apply a comparison star. For some stars we find differences in the SpT and LC of more than one subclass when derived from observations in the UV <cit.> and in the optical (Sect. <ref>). This introduces systematic errors in the reddening curves as mentioned above. This affects fifteen curves by <cit.>, two by <cit.>, and six by <cit.>. For the latter there is an extra difficulty that the temperature of the <cit.> model atmospheres needs to be related to the MK system by means of a stellar temperature scale <cit.>. In that scheme we add an extra uncertainty of 1,000 K in favour of non-rejection. In 40 out of 54 cases, the SpT of the reddened and the comparison stars agree within 1.5 subclasses. These stars are indicated below the line in Table <ref> and are kept in the high-quality sample, whereas the cases above that line show larger deviations. Their reddening curves are considerered to be of lower quality. For example, the spectral type of HD 093843 is uncertain; we find O4 Ib, <cit.> O5 III, and <cit.> O6 III. For deriving the reddening curve <cit.> used an O7 V comparison star, which deviates by more than two types; therefore, we reject the reddening. HD 046660 whose spectral type is also uncertain, when using the fitting procedure (Sect. <ref>), a similar χ^2 minimum is found for either B0 III or O7 V. We adopt the latter type as the star displays He ii. It has been classified as O9 V by <cit.> implying a temperature 30,000 K, in agreement with 31,067 K derived <cit.>. However, <cit.> derived the reddening with a best fitting model atmosphere at 26,138 K, hence later than B1 <cit.>. Additionally, <cit.> derived the reddening with a B1.5 V comparison star. Due to the discrepancy between these results and our derived spectral type, we also reject this reddening curve. As detailed in Table <ref>, the reddening curve of HD 168076 (O4 III/O5 V) was derived by <cit.> using an O5 V type star, while <cit.> used HD 091824 (O7 V) and is therefore removed. For HD 167771 the reddening curve by <cit.> was derived with a comparison star that differs by two subclasses and is removed, while the <cit.> derived curve with a smaller SpT difference is included (Table <ref>). The reddening curve towards HD 108927 (B5 V) was derived by <cit.> using a B3 V comparison star. Reddening curves derived with comparison stars with unconfirmed SpT in Simbad are kept. For example, <cit.> used as comparison stars HD 051013 (B3V?) for deriving the reddening of HD 027778 (B3 V) and HD 062542 (B5 V). Both reddening curves show (Fig. 6) excellent agreement with those derived by <cit.>, respectively. Further comments on individual stars are given in Appendix <ref>. In total, 15 sightlines have uncertain reddening curves because of a SpT mismatch (Table <ref>). They are flagged in Table <ref> by attaching the SpT and LC to that star. §.§ Comparison of reddening derived from the IUE and Ground-based observations After applying the confidence criteria, we discovered that some of the remaining reddening curves display a jump when going from the longest wavelength of the IUE spacecraft, that is not contaminated by instrumental features at 0.28 μm, to the shortest wavelength accessible from the ground, the U-band. The reddening between 0.28 μm and the U-band for these 96 curves has a mean and 1 σ scatter of E(0.28μm - U)/E(B-V) = 1.3 ± 0.3 mag and are shown in Fig. <ref>. Peculiar reddening curves that show an offset in E(0.28μm - U)/E(B-V) were identified by means of 3 σ clipping. These include eight reddening curves which are derived by <cit.>. Two of these, HD 108927 and HD 122879, were rejected above. For HD 079286, HD 089137, and HD 149038 no other reddening curve is available. For the remaining three sightlines – HD 091824, HD 156247, and HD 180968 – only the reddening curves derived by <cit.> show this striking jump whereas this feature is not present in the reddening curves derived by <cit.> for the same stars (Fig. <ref>). These six reddening curves are marked as peculiar in Table <ref>. § THE HIGH QUALITY SAMPLE OF REDDENING CURVES IN THE MILKY WAY §.§ The sample Our high-quality sample contains only reddening curves that respect the following six confidence criteria derived in Sect. <ref>: a) The IUE spectra include only a single star that dominates the observed spectrum. b) The GAIA parallaxes π_DR3 and π_DR2 are consistent within their 3σ error estimates. c) The GAIA photometric variability between DR3 and DR2 is within -19 Δ G 47 (mmag). d) The variability of the star observed from the ground and by Hipparcos in the V-band is within -100 Δ V 88 (mmag) and in the B-V colour within -55 Δ_BV 112 (mmag), respectively. e) SpT and LC are derived as in Sect. <ref> at high confidence (χ^2 < 2). f) The stellar classification of program and comparison star agrees within one subtype. With these confidence criteria, the high-quality sample is given in Table <ref>. It comprises 80 reddening curves for 53 sightlines of which 35 are the rare cases of single-cloud dominated sightlines. Six reddening curves show a peculiar jump between 0.28μm and the U-band, i.e. E(0.28μm -U)/E(B-V) > 2.2 mag. We give preference to reddening curves by <cit.> as they include besides IUE also FUSE data. We also prefer reddening curves by <cit.> over <cit.> because comparison star observations that were used in the latter work are not needed in the former derivation procedure. Our choice of the reference reddening curve E^Ref for each sightline is given in column 3; when available, other accepted reddening curves E^i with i ∈{FM07, V04} are listed in column 4. None of the stars of Table <ref> is associated with reflection nebulosity within 5” <cit.> as otherwise, the curves would not represent the reddening in the translucient cloud. All reddening curves of both the high-quality sample (Table <ref>) and those of the stars with uncertainties in the reddening curve are shown in Fig. <ref> and Fig. <ref>, respectively. A nearly perfect match for stars with accepted curves by <cit.>, <cit.>, and <cit.> are visible in Fig. <ref> and for about half of the stars with high-quality reddening curves by <cit.> and <cit.> in Fig. <ref>. We note that the consistency between various reddening curves of a star cannot be taken as full proof of the correctness of the derivation. In the far UV a steeper rise in the reddening is present for six curves derived by <cit.> and for three stars by <cit.> when compared to any other of the available reddening curves for these stars. In the near IR differences in the reddening between <cit.> and <cit.> are visible for three sightlines (Fig. <ref>). §.§ Intrinsic errors in the high-quality sample The intrinsic error of the reddening curves in the high-quality sample (Sect. <ref>) was estimated by measuring the variance between individual reddening curves towards the same star. For each star of Table <ref>, which is not flaged as peculiar, the ratio E_λ^ i/E_λ^Ref of the two reddening curves was computed, where the reference for the reddening curve E_λ^Ref is given in column 3 and the reference for the reddening curve E^i with i ∈{FM07, V04} in column 4 of Table <ref>. For example, for HD 027778 the ratio of the reddening curves E_λ^FM07/E_λ^G09 were determined, for HD 030470 there exists no such ratio, and for HD 037903 there are even two ratios available – E_λ^FM07/E_λ^G09 and E_λ^V04/E_λ^G09. In total there are 27 (E^ i/ E^Ref) ratios of reddening curves that all pass the confidence criteria of Sect. <ref>. The mean and 1 σ error of these (E^ i/ E^Ref) ratios are computed by omitting data at 1/λ 7.5 μ m^-1 when not observed by FUSE. In the V-band, to which the curves are normalized, the scatter of these ratios is σ(E^ i/E^Ref) = 0 by definition and remains naturally small for wavelengths close to it. In the IUE range σ(E^ i/E^Ref) ∼ 10 % and grows to ∼ 15 % in the far UV at x ∼ 11 μm^-1. Similarly σ(E^ i/E^Ref) 7 % for λ∼ 1 μm and increases to 11% at longer wavelengths. The typical error in the high-quality sample when averaged over wavelengths is σ(E^ i/E^Ref) ∼ 9 % and stays below 16 %. Repeating the same exercise for the 58 stars with uncertain reddening curves gives 23 such ratios and shows a larger scatter of σ(E^ i/E^Ref) ∼ 15 %. The ratios of uncertain reddening curves of the same star varies by up to 39 %, which is about a factor of two larger spread than for the high-quality sample. §.§ Uncertainties in R_V The shape of the extinction curves depends on the total-to-selective extinction R_V <cit.>. This parameter is estimated by extrapolating a derived reddening, e.g. in the near or mid infrared, to infinite wavelengths; thus R_V is not an observable. It successfully describes averaged properties with clear deviations in the reddening for specific sightlines <cit.>. The systematic uncertainties of R_V = - E(∞) (Eq. <ref>) are hence of interest. The differences Δ R_V = E^ref(∞) - E^ i(∞) of the available estimates of R_V for the same star are computed. They provide an estimate of the systematic errors and are shown in Fig. <ref> as histograms for the sample of uncertain and confident reddening curves. The references to the reddening curves E^ref and E^ i are listed for the high-quality sample in colums 3 and 4 of Table <ref>. For the unsecure sample we use – whenever available – the estimate of E^ref(∞) by <cit.> and otherwise <cit.>. Three stars HD 104705, HD 164073 and HD 147933 deviate with Δ R_V > 1.3 and are therefore omitted. For example, HD 164073 has R_V = 2.96 <cit.> or R_V = 5.18 <cit.>. The distributions in Δ R_V appear similar in both samples and are flatter than Gaussian, which indicates that systematic errors dominate. The high-quality sample has a peak-to-peak scatter, mean, and 1σ error of -0.33 Δ R_V = 0.09 ± 0.21 0.67. The reference reddening curves of the high-quality sample vary between 2.01 R_V = 3.08 ± 0.44 4.34 (Sect. <ref>). We find that the derivation of R_V for the same star by the various authors agrees to better than σ(Δ R_V) / mean(R_V)∼ 7%. §.§ Reddening in the Milky Way Galactic extinction studies derive the typical wavelength dependence of interstellar reddening. Such curves are commonly used, in particular in extra-galactic research, as the standard for dereddening the observed flux of objects for which there is no specific knowledge about the dust. Large variations from cloud-to-cloud of the grain characteristics and the physical parameters of the dust are reported by <cit.>. Although averaging a sufficiently large number of clouds and sightlines leads to similar mean parameters, they likely do not reflect the true nature of the dust. The degree to which mean Galactic extinction curves can be taken as typical shall always be put in question. The shape of the average Galactic extinction for a sample of 243 stars with 2.4 ≤ R_V ≤ 3.6 at 1/λ≤ 8.6 μ m^-1 has been presented by <cit.>. Their mean curve has R_V = 3.1 as derived earlier <cit.> and in recent studies <cit.>; <cit.> find a similar value of R_ V = 3.16. Here we compute average Milky Way reddening curves using the high-quality sample (Table <ref>, col. 3). As in the previous sections a 3σ rejection criterium is applied, so that stars have A_V 2.4 and R_V 4.4. HD 093222 and HD 168076 violate this criterium and are not included in the analysis. We distinguish sightlines of translucent clouds at 1 < τ_V 2.2 and single-cloud sightlines of the diffuse ISM at τ_V 1. The average curve of the 33 translucent clouds in our sample has a peak-to-peak scatter, mean, and 1σ error of 2.7 R_V = 3.3 ± 0.4 4.1 at median of R_V = 3.16. That of the 15 single-cloud diffuse ISM sightlines has 2.3 R_V = 3.0 ± 0.3 3.6 at median of R_V = 3.1, respectively. These curves are shown in Fig. <ref> with the average Galactic reddening by <cit.> and <cit.>. Driven by the same R_V of these curves a nearly perfect match from optical to longer wavelengths is found; for translucent clouds this incldues the IUE range and for the diffuse ISM the reddening is smaller at λ < 0.2μm. Noticeable is the large diversity of the reddening curves (Fig. <ref>) and agreement of the various mean Milky Way reddening curvesl within σ(R_ V) = 0.4. § SUMMARY Individual clouds in the ISM can be drastically different from each other. Therefore, the framework of single-cloud sightlines was introduced as they provide an unambiguous view of physical relations between dust properties and observables such as extinction and polarisation <cit.>. Reddening curves allow to derive fundamental characteristics of dust. However, before extracting dust properties from a physical model one must ensure that the observational basis is solid and the assumptions in the derivation of the reddening are met to an acceptable level. We discussed the current database of available reddening curves. The initial sample of 895 reddening curves towards 568 OB stars, which cover the spectral range from the near IR to the Lyman limit was merged with 186 OB stars with high-resolution UVES spectra and with polarisation spectra towards 215 OB stars from the Large Interstellar Polarisation survey. This yields a sample of 111 sightlines for which the reddening curves were scrutinized against systematic errors by the following means: a) Whenever IUE/FUSE spectra were identified as a composite of multiple sources the corresponding reddening curves were rejected. Stars with assigned binary information did not provide a direct link or a striking feature in the appearance of the reddening curves for qualifying a rejection. b) The stellar classification of the stars was derived at high confidence. Objects whose spectral types do not match within one subtype the comparison star were removed from the sample. c) Stars with detected variability between 1950 - 2017 from ground-based and Hipparcos V-band photometry and B-V colurs of about 0.1 mag were excluded. The same holds for GAIA G-band photometric variations of more than 47 mmag. d) Reddening curves of stars showing inconsistencies in the GAIA parallaxes between DR2 and DR3 were declared as spurious. e) The reddening curves with different estimates of R_V of the same star exceeding 50% were also rejected. In total, we find 53 stars with one or more reddening curves passing the rejection criteria of which 35 are the rare cases of single-cloud sightlines. This provides the highest quality Milky Way reddening curve sample available today. The average Milky Way reddening curve is determined for translucent clouds and the diffuse ISM to be R_V = 3.1 ± 0.4, confirming earlier estimates. The high-quality reddening curve sample together with polarisation properties will be subject to dust modelling in a future paper in this series of the Dark Dust project. JK acknowledge the financial support of the Polish National Science Centre, Poland (2017/25/B/ST9/01524) for the period 2018 - 2023. This research has made use of the services of the ESO Science Archive Facility and the SIMBAD database operated at CDS, Strasbourg, France (Wanger et al. 2000) and partially based on observations collected at the European Southern Observatory under ESO programmes (Sect. <ref>). § COMMENTS ON INDIVIUDAL STARS Reddening curves with a miss-match of more than one type between our estimates (Sect. <ref>) and the SpT used in the derivation are rejected by the following reason: HD 054306: we classify it as B3V whereas <cit.> finds it at 22,409 K so earlier than B1.5 I and <cit.> used a B1 V comparison star (HD 031726). HD 072648: we classify it as B5 Ib whereas <cit.> finds it at 21,035 K so earlier than B1.5 I and <cit.> used a B1.5 III comparison star. HD 091983: we classify this photometric variable star (Fig. <ref>) as O9 V, <cit.> as O9 IV, and <cit.> finds it as O9.5 Ib. HD 096675: we classify it as B7 V, <cit.> as B6 IV and <cit.> used a B5 V comparison star. HD 103779: we classify it as B2 Ib whereas <cit.> used a B0.5 Ib comparison star. HD 134591: we classify it as B8 III whereas <cit.> used a B4 V comparison star. HD 151804, HD 153919, HD 162978: we classify these stars as O 8Ia, O4 Ib, O8 II and <cit.> as O8 Iab, O6 V, O8 III; whereas <cit.> used a O 9.5Ia comparison star. HD 164536: we classify it, in agreement to <cit.>, as O7 V whereas <cit.> finds it at 33,500 K, close to O9 V <cit.>. HD 164816: we classify it as O8 V, <cit.> as O9.5 V+B0 V binary, <cit.> as B0 V, and <cit.> finds it at 31,427 K, close to O9.5 V <cit.>, <cit.> used HD 097471 a B0 V comparison star. HD 168076: we classify it as O5 V at χ^2=0.31 and as O4 III at χ^2=0.17 in agreement to <cit.>. HD 185859: we classify it as B2 Ib whereas <cit.> used a B0.5 Ib comparison star. HD 203532: we classify it as B5 III, <cit.> finds it at 17,785 K so earlier than that <cit.> and <cit.> used a B3 IV comparison star. HD 315033: we classify it as B3 V whereas <cit.> finds it at 25,609 K so earlier than B1.5 V and <cit.> used a B1.5 III comparison star. § ESO PROGRAMMES Observations collected at the European Southern Observatory are based under ESO programmes: 060.A-9036(A), 067.C-0281(A), 071.C-0367(A), 072.D-0196(A), 073.C-0337(A), 073.C-0337(A), 073.D-0609(A), 074.D-0300(A), 075.D-0061(A), 075.D-0369(A), 075.D-0369(A), 076.C-0164(A), 076.C-0431(A), 076.C-0431(B), 079.D-0564(A), 081.C-0475(A), 081.D-2008(A), 082.C-0566(A), 083.D-0589(A), 083.D-0589(A), 086.D-0997(B), 091.D-0221(A), 092.C-0019(A), 102.C-0040(B), 102.C-0699(A), 187.D-0917(A), 194.C-0833(A), 194.C-0833(B), 194.C-0833(D), 194.C-0833(E), 194.C-0833(F), 194.C-0833(H), 0102.C-0040(B), 072.A-0100(A), 072.B-0123(D), 072.B-0218(A), 072.C-0488(E), 076.B-0055(A), 077.C-0547(A), 078.D-0245(C), 079.A-9008(A), 079.C-0170(A), 083.A-0733(A), 088.A-9003(A), 089.D-0975(A), 091.C-0713(A), 091.D-0221(A), 092.C-0218(A), 096.D-0008(A), 097.D-0150(A), 106.20WN.001, 1102.A-0852(C), 165.N-0276(A), 194.C-0833(A), 194.C-0833(B), 194.C-0833(C), 266.D-5655(A), 65.H-0375(A), 65.N-0378(A), 65.N-0577(B), 70.D-0191(A). aa
http://arxiv.org/abs/2307.01820v1	20230704164949	Failure of the curvature-dimension condition in sub-Finsler manifolds	[ "Mattia Magnabosco", "Tommaso Rossi" ]	math.MG	[ "math.MG", "math.DG" ]	cell-space-limits = 1pt showonlyrefs § 0.5em §.§ 0.4em remark 10pt 10pt . .4em proof 10pt 10pt . .4em definition 10pt 10pt . .4em theorem 10pt 10pt . .4em theorem theoremTheorem[section] prop[theorem]Proposition corollary[theorem]Corollary lemma[theorem]Lemma conj[theorem]Conjecture definition definition[theorem]Definition remark remark[theorem]Remark notation[theorem]Notation proof proProof pr @←→ Failure of the curvature-dimension condition in manifolds Mattia Magnabosco[Institut für Angewandte Mathematik, Universität Bonn. E-mail: mailto:[email protected]@iam.uni-bonn.de] and Tommaso Rossi[Institut für Angewandte Mathematik, Universität Bonn. E-mail: mailto:[email protected]@iam.uni-bonn.de] August 1, 2023 ===================================================================================================================================================================================================================================================================================== The Lott–Sturm–Villani curvature-dimension condition (K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It has been recently proved that this condition does not hold in geometry for every choice of the parameters K and N. In this paper, we extend this result to the context geometry, showing that the (K,N) condition is not well-suited to characterize curvature in this setting. Firstly, we show that this condition fails in (strict) manifolds equipped with a smooth strictly convex norm and with a positive smooth measure. Secondly, we focus on the Heisenberg group, proving that curvature-dimension bounds can not hold also when the reference norm is less regular, in particular when it is of class C^1,1. The strategy for proving these results is a non-trivial adaptation of the work of Juillet <cit.>, and it requires the introduction of new tools and ideas of independent interest. Finally, we demonstrate the failure of the (weaker) measure contraction property (K,N) in the Heisenberg group, equipped with a singular strictly convex norm and with a positive smooth measure. This result contrasts with what happens in the Heisenberg group, which instead satisfies (0,5). § INTRODUCTION In the present paper, we address the validity of the Lott–Sturm–Villani curvature-dimension (in short (K,N)) condition in the setting of geometry. In particular, we prove that this condition can not hold in a large class of manifolds. Thus, on the one hand, this work shows that the (K,N) condition is not well-suited to characterize curvature in geometry. On the other hand, we discuss how our results could provide remarkable insights about the geometry of (K,N) spaces. §.§ Curvature-dimension conditions In their groundbreaking works, Sturm <cit.> and Lott–Villani <cit.> introduced independently a synthetic notion of curvature-dimension bounds for non-smooth spaces, using Optimal Transport. Their theory stems from the crucial observation that, in the Riemannian setting, having a uniform lower bound on the Ricci curvature and an upper bound on the dimension, can be equivalently characterized in terms of a convexity property of suitable entropy functionals in the Wasserstein space. In particular, it was already observed in <cit.> that the Ricci bound ≥ K· g holds if and only if the Boltzmann–Shannon entropy functional is K-convex in the Wasserstein space. More generally, let (M,g) be a complete Riemannian manifold, equipped with a measure of the form =e^-V vol_g, where vol_g is the Riemannian volume and V∈ C^2(M). Given K∈ and N∈ (n,+∞], Sturm <cit.> proved that the (generalized) Ricci lower bound _N,V := + ∇^2 V - ∇ V ⊗∇ V/N-n≥ K · g, holds if and only if a (K,N)-convexity inequality holds for Rényi entropy functionals, defined with respect to the reference measure . While (<ref>) involves a differential object, the Ricci tensor, entropy convexity can be formulated relying solely upon a reference distance and a reference measure, without the need of the underlying smooth structure of the Riemannian manifold. Therefore, it can be introduced in the non-smooth setting of metric measure spaces and taken as definition of curvature-dimension bound. This condition is called (K,N) and represents a synthetic lower bound on the (Ricci) curvature by K∈ and a synthetic upper bound on the dimension by N∈ (1,∞], see Definition <ref>. In this sense, according to the discussion above, the (K,N) condition is coherent with the Riemannian setting. Moreover, it was proved by Ohta <cit.> that the relation between curvature and (K,N) condition holds also in the context of Finsler manifolds. Remarkably, (K,N) spaces (i.e. spaces satisfying the (K,N) condition) enjoy several geometric properties which hold in the smooth setting. Some of them are expected (and in a way necessary) for a reasonable curvature-dimension bound, such as the scaling <cit.>, tensorization <cit.> and globalization <cit.> properties or the monotonicity with respect to the parameters <cit.>, i.e. (K',N') (K,N) if K'≥ K and N'≤ N. Others are completely non-trivial and highlight some notable geometric features. Among them, we mention the Bonnet–Myers diameter bound and the Bishop–Gromov inequality, that provides an estimate on the volume growth of concentric balls. Particularly interesting in the context of this work is the Brunn–Minkowski inequality (K,N), which, given two sets A and B in the reference metric measure space (,,), provides a lower estimate on the measure of the set of t-midpoints M_t(A,B)={ x ∈ : (a,x)=t (a,b), (x,b)=(1-t) (a,b) for some a∈ A,b∈ B}, in terms of (A) and (B), for every t∈[0,1], cf. (<ref>). The notable feature of the (K,N) inequality is that its formulation does not invoke optimal transport, or Wasserstein interpolation, and because of that, it is easier to handle than the (K,N) condition. Nonetheless, it contains a strong information about the curvature of the underlying space, to the extent that it is equivalent to the (K,N) condition in the Riemannian setting, cf. <cit.>. In particular, in the proof of Theorem <ref> and Theorem <ref>, we show the failure of the (K,N) condition by contradicting the Brunn–Minkowski inequality (K,N). Finally, another fundamental property of the (K,N) condition is its stability with respect to the (pointed) measured Gromov–Hausdorff convergence <cit.>. This notion of convergence for metric measure spaces essentially combines the Hausdorff convergence for the metric side and the weak convergence for the reference measures. As in a metric measure space, the tangent spaces at a point are identified with a measured Gromov–Hausdorff limit procedure of suitably rescalings of the original space, the stability of the curvature-dimension condition implies that the metric measure tangents of a (K,N) space is a (0,N) space. In the setting of metric measure spaces, it is possible to define other curvature-dimension bounds, such as the so-called measure contraction property (in short (K,N)), introduced by Ohta in <cit.>. In broad terms, the (K,N) condition can be interpreted as the Brunn–Minkowski inequality where one of the two sets degenerates to a point. In particular, it is implied by (and strictly weaker than) the (K,N) inequality, and therefore it is also a consequence of the (K,N) condition. §.§ The curvature-dimension condition in geometry While the (K,N) condition is equivalent to having bounded geometry in the Riemannian setting, a similar result does not hold in the setting. Sub-Riemannian geometry is a broad generalization of Riemannian geometry where, given a smooth manifold M, we define a smoothly varying scalar product only on a subset of horizontal directions _p⊂ T_pM (called distribution) at each point p∈ M. Under the so-called Hörmander condition, M is horizontally-path connected, and the usual length-minimization procedure yields a well-defined distance _SR. In particular, differently from what happens in Riemannian geometry, the rank of the distribution r(p):=_p may be strictly less than the dimension of the manifold and may vary with the point. This may influence the behavior of geodesics, emphasizing singularities of the distance _SR. For this reason, we can not expect the (K,N) condition to hold for truly manifolds. This statement is confirmed by a series of papers, most notably <cit.>, that contributed to the proof of the following result. Let M be a complete manifold, equipped with a positive smooth measure . Then, the metric measure space (M,_SR,) does not satisfy the (K,N) condition, for any K∈ and N∈(1,∞). In <cit.>, Juillet proved Theorem <ref> for manifolds where the rank of the distribution r(p) is strictly smaller than the topological dimension n:= M, for every p∈ M. His strategy relies on the construction of two Borel subsets for which the Brunn–Minkowski inequality (K,N) does not hold. Namely, for all R,ε>0, one can find A,B⊂ M such that diam(A∪ B)<R, (A)≈(B), and such that there exists t∈ (0,1) for which (M_t(A,B))≤1/2^𝒩-n (B)(1+ε), where 𝒩 is the so-called geodesic dimension of M, see <cit.> for a precise definition. The sets A and B are metric balls of small radius, centered at the endpoints of a short segment of an ample geodesic, see <cit.> for details. The inequality (<ref>) allows to contradict the Brunn–Minkowski inequality (K,N) if and only if the geodesic dimension 𝒩 is strictly greater than n, which is the case if r(p)<n, for every p∈ M. While Julliet's result is quite general, it does not include geometry. Roughly speaking, an manifold is a manifold where the rank of the distribution coincides with the dimension of M, at almost every point. In <cit.>, we addressed this issue, proposing a new strategy for proving Theorem <ref> in this setting. Our idea is to exploit the following one-dimensional characterization of the (K,N) condition: (K,N) ⇒ ^1(K,N), proved by Cavalletti and Mondino in <cit.>, and contradict the ^1(K,N) condition. On a metric measure space (,,), given a 1-Lipschitz function u∈(), it is possible to partition in one-dimensional transport rays, associated with u, and disintegrate the measure accordingly. Then, the ^1(K,N) condition asks for the validity of the (K,N) condition along the transport rays of the disintegration associated with u, for any choice of u∈(). In <cit.>, when M is either strongly regular or M=2, we are able to explicitly build a 1-Lipschitz function, and compute the associated disintegration, showing that the (K,N) condition along the rays does not hold for any K∈ and N∈ (1,∞). Most recently, Rizzi and Stefani <cit.> proposed yet another strategy to prove Theorem <ref>. Differently from the strategies presented above, they pursue the “Eulerian” approach to curvature-dimension bounds, based on a suitable Gamma calculus, see <cit.> for details. This approach can be adopted for metric measure spaces that satisfy the infinitesimal Hibertian condition (cf. <cit.>) which forces the space to be Riemannian-like and ensures the linearity of the heat flow. According to <cit.>, an infinitesimally Hilbertian (K,N) space supports the so-called Bakry–Émery inequality (K,∞), which, in the setting reads as 1/2Δ(∇ f^2) ≥ g (∇ f, ∇Δ f)+K∇ f^2, ∀ f∈ C^∞_c(M), where ∇ is the horizontal gradient and Δ is the sub-Laplacian. In <cit.>, the authors show that (<ref>) implies the existence of enough isometries on the metric tangent to force it to be Euclidean at each point, proving Theorem <ref> (including also the case N=∞). §.§ Other curvature-dimension bounds in geometry Given that the (K,N) condition does not hold in geometry, considerable efforts have been undertaken to explore potential curvature-dimension bounds that may hold in this class. A first observation in this direction is that the weaker (K,N) condition does hold in many examples of manifolds. In particular, it was proved by Juillet <cit.> that the Heisenberg group satisfies the (0,5) condition, where the curvature-dimension parameters can not be improved. Moreover, in <cit.> it was observed that the optimal dimensional parameter for the measure contraction property coincides with the geodesic dimension of the Heisenberg group (i.e. 𝒩=5). This result has been subsequently extended to a large class of manifolds, including ideal Carnot groups <cit.>, corank-1 Carnot groups <cit.>, generalised H-type Carnot groups <cit.> and two-step analytic structures <cit.>. In all these cases, the (0,N) condition holds with the dimensional parameter N greater than or equal to the geodesic dimension 𝒩. Another attempt is due to Milman <cit.>, who introduced the quasi curvature-dimension condition, inspired by the interpolation inequalities along Wasserstein geodesics in ideal sub-Riemannian manifolds, proved by Barilari and Rizzi <cit.>. Finally, these efforts culminated in the recent work by Barilari, Mondino and Rizzi <cit.>, where the authors propose a unification of Riemannian and sub-Riemannian geometries in a comprehensive theory of synthetic Ricci curvature lower bounds. In the setting of gauge metric measure spaces, they introduce the (β,n) condition, encoding in the distortion coefficient β finer geometrical information of the underlying structure. Moreover they prove that the (β,n) condition holds for compact fat manifolds, thus substantiating the definition. §.§ Sub-Finsler manifolds and Carnot groups In the present paper, we focus on manifolds, which widely generalize both and Finsler geometry. Indeed, in this setting, given a smooth manifold M, we prescribe a smoothly varying norm (which needs not be induced by a scalar product) on the distribution _p⊂ T_pM, at each point p∈ M. As in the setting, must satisfy the Hörmander condition, and consequently the length-minimization procedure among admissible curves gives a well-defined distance _SF. Note that, on the one hand, if the _p=T_pM for every p∈ M, we recover the classical Finsler geometry. On the other hand, if the norm on _p is induced by a scalar product for every p∈ M, we fall back into geometry. Replacing the scalar product with a (possibly singular) norm is not merely a technical choice, as the metric structure of a manifold reflects the singularities of the reference norm. Indeed, even though manifolds can still be investigated by means of classical control theory <cit.>, deducing finer geometrical properties is more delicate compared to what happens in the setting, as the Hamiltonian function has a low regularity, cf. Section <ref>. In this regard, manifolds provide an interesting example of smooth structures which present both the typical and Finsler singular behavior. A particularly relevant class of manifolds is the one of Carnot groups. A Carnot group is a connected, simply connected Lie group G with nilpotent Lie algebra 𝔤, admitting a stratification 𝔤= 𝔤_1 ⊕⋯⊕𝔤_k, where 𝔤_i+1= [𝔤_1,𝔤_i], for every i=1,…, k-1, and [𝔤_1,𝔤_k]={0}. Given a Carnot group G, if we equip the first layer 𝔤_1 of its Lie algebra with a norm, we naturally obtain a left-invariant structure on G. We refer to the resulting manifold as a Carnot group. Motivated from the results presented in the previous section, cf. Theorem <ref>, and especially from the ones obtained in the present work (see Section <ref>), we formulate the following conjecture. Let G be a Carnot group, endowed with a positive smooth measure . Then, the metric measure space (G,_SF,) does not satisfy the (K,N) condition for any K∈ and N∈(1,∞). Our interest in Carnot groups stems from the fact that they are the only metric spaces that are locally compact, geodesic, isometrically homogeneous and self-similar (i.e. admitting a dilation) <cit.>. According to this property, sub-Finsler Carnot groups naturally arise as metric tangents of metric measure spaces. Let (,,) be a geodesic metric measure space, equipped with a doubling measure . Assume that, for -almost every x∈, the set Tan(, x) of all metric tangent spaces at x contains only one element. Then, for -almost every x ∈, the element in Tan(, x) is a Carnot group G. In particular, this result applies to (K,N) spaces, where the validity of the doubling property is guaranteed by the Bishop–Gromov inequality. Moreover, as already mentioned, the metric measure tangents of a (K,N) space are (0,N). Therefore, the study of the (K,N) condition in Carnot groups, and especially the validity of Conjecture <ref>, has the potential to provide deep insights on the structure of tangents of (K,N) spaces. This could be of significant interest, particularly in connection with Bate's recent work <cit.>, which establishes a criterion for rectifiability in metric measure spaces, based on the structure of metric tangents. §.§ Main results The aim of this paper is to show the failure of the (K,N) condition in the setting, with a particular attention to Conjecture <ref>. Our results offer an advance into two different directions: on the one hand we deal with general structures, where the norm is smooth, cf. Theorem <ref> and Theorem <ref>, and, on the other hand, we deal with the Heisenberg group, equipped with more general norms, cf. Theorem <ref> and Theorem <ref>. In order to extend the result of Theorem <ref> to the setting, one can attempt to adapt the strategies discussed in Section <ref>, however this can present major difficulties. Specifically, the argument developed in <cit.> has little hope to be generalized, because the infinitesimal Hilbertianity assumption does not hold in Finsler-like spaces, see <cit.>. It is important to note that this is not solely a “regularity" issue, in the sense that it also occurs when the norm generating the structure is smooth, but not induced by a scalar product. Instead, the approach proposed in <cit.> could potentially be applied to manifolds as it relies on tools developed in the non-smooth setting, see (<ref>). However, adapting the computations that led to a contradiction of the ^1(K,N) condition seems non-trivial already when the reference norm is smooth. Finally, the strategy illustrated in <cit.> hinges upon geometrical constructions and seems to be well-suited to generalizations to the setting. In this paper, we build upon this observation and adapt the latter strategy to prove our main theorems. Our first result is about the failure of the (K,N) condition in smooth manifolds, cf. Theorem <ref>. Let M be a complete manifold with r(p)<n:= M for every p∈ M, equipped with a smooth, strictly convex norm and with a positive smooth measure . Then, the metric measure space (M,_SF,) does not satisfy the (K,N) condition, for any K∈ and N∈ (1,∞). This result is the analogue of <cit.>. Although the strategy of its proof follows the blueprint <cit.>, the adaptation to our setting is non-trivial and requires many intermediate results of independent interest. First of all, we establish the existence of geodesics without abnormal sub-segments, cf. Theorem <ref>, proposing a construction that is new even in the framework and relies on the regularity properties of the distance function from the boundary of an open set. Note that, while these properties are well-known in the context (cf. <cit.>), inferring them in the setting becomes more challenging due to the low regularity of the Hamiltonian, which affects the regularity of the normal exponential map. Nonetheless, we settle a weaker regularity result that is enough for our purposes, cf. Theorem <ref>. Second of all, we prove an analogue of the theorem, indicating that the volume contraction along ample geodesics is governed by the geodesic dimension, see <cit.>. Indeed, in a smooth manifold, we establish that the volume contraction rate along geodesics without abnormal sub-segments is bigger than M+1, cf. Theorem <ref>. Finally, we mention that these technical challenges lead us to a simplification of Juillet's argument (cf. Theorem <ref>), which revealed itself to be useful also in the proof of Theorem <ref>. Observe that, since sub-Finsler Carnot groups are equiregular (and thus r(p)<n, for every p∈ G) and complete, we immediately obtain the following consequence of Theorem <ref>, which constitutes a significant step forward towards the proof of Conjecture <ref>. Let G be a Carnot group, equipped with a smooth, strictly convex norm and with a positive smooth measure . Then, the metric measure space (G,_SF,) does not satisfy the (K,N) condition, for any K∈ and N∈ (1,∞). In the proof of Theorem <ref>, the smoothness of the norm plays a pivotal role in establishing the correct volume contraction rate along geodesics. When the norm is less regular, it is not clear how to achieve an analogue behavior in full generality. Nonetheless, we are able to recover such a result in the context of the Heisenberg group , equipped with a possibly singular norm (see Section <ref>). Working in this setting is advantageous since, assuming strict convexity of the norm, the geodesics and the cut locus are completely described <cit.> and there exists an explicit expression for them in terms of convex trigonometric functions <cit.> (see also <cit.> for an example of the non-strictly convex case). For the Heisenberg group, we prove two different results, with the first addressing the case of C^1,1 reference norms and thus substantially relaxing the smoothness assumption of Theorem <ref>, cf. Theorem <ref>. Let be the Heisenberg group, equipped with a strictly convex and C^1,1 norm and with a positive smooth measure . Then, the metric measure space (,_SF,) does not satisfy the (K,N), for any K∈ and N∈ (1,∞). The proof of this statement follows the same lines of <cit.>. However, the low regularity of the norm, and thus of geodesics, prevent us to exploit the same differential tools developed for Theorem <ref>. Nonetheless, using the explicit expression of geodesics and of the exponential map, we can still recover an analogue result. In particular, guided by the intuition that a contraction rate along geodesics, similar to the one appearing in the smooth case, should still hold, we thoroughly study the Jacobian determinant of the exponential map. Building upon a fine analysis of convex trigonometric functions, cf. Section <ref> and Proposition <ref>, we obtain an estimate on the contraction rate of the Jacobian determinant of the exponential map, but only for a large (in a measure-theoretic sense) set of covectors in the cotangent space. This poses additional challenges that we are able to overcome with a delicate density-type argument, together with an extensive use of the left-translations of the group, cf. Theorem <ref> and also Remark <ref>. Remarkably, for every C^1,1 reference norm, we obtain the exact same contraction rate, equal to the geodesic dimension 𝒩=5, that characterizes the Heisenberg group. Our second result in the Heisenberg group deals with the case of singular (i.e. non-C^1) reference norms, cf. Theorem <ref>. Let be the Heisenberg group, equipped with a strictly convex norm which is not C^1, and let be a positive smooth measure on . Then, the metric measure space (, _SF, ) does not satisfy the measure contraction property (K,N) for any K∈ and N∈ (1,∞). Observe that this theorem also shows the failure of the (K,N) condition, which is stronger than the measure contraction property (K,N). However, Theorem <ref> has an interest that goes beyond this consequence, as it reveals a phenomenon that stands in contrast to what typically happens in the setting. In fact, as already mentioned in section <ref>, the (K,N) condition holds in many manifolds, and, in the particular case of the Heisenberg group, holds with parameters K=0 and N=5. Therefore, Theorem <ref> shows that a singularity of the reference norm can cause the failure of the measure contraction property (K,N). A similar phenomenon is highlighted in the recent paper by Borza and Tashiro <cit.>, where the authors prove that the Heisenberg group equipped with the l^p-norm cannot satisfy the (K,N) condition if p>2. Our strategy to show Theorem <ref> consists in finding a set A⊂, having positive -measure, such that the set of t-midpoints M_t({},A) (where denotes the identity in ) is -null for every t sufficiently small. This construction is based on a remarkable geometric property of the space (, _SF, ), where geodesics can branch, even though they are unique. This has independent interest, as examples of branching spaces usually occur when geodesics are not unique. We conclude this section highlighting that the combination of Theorem <ref> and Theorem <ref> proves Conjecture <ref> for a large class of Heisenberg groups. This is particularly interesting as the Heisenberg groups are the unique Carnot groups with Hausdorff dimension less than 5 (or with topological dimension less than or equal to 3), up to isometries. §.§ Structure of the paper In Section <ref> we introduce all the necessary preliminaries. In particular, we present the precise definition of the (K,N) condition with some of its consequences, and we introduce the notion of structure on a manifold. Section <ref> is devoted to the study of the geometry of manifolds. For the sake of completeness, we include generalizations of various results, especially regarding the characterizations of normal and abnormal extremals and the exponential map. In Section <ref>, we present the proof of Theorem <ref>. We start by developing the building blocks for it, namely the existence of a geodesic without abnormal sub-segments and the regularity of the distance function. Then, we estimate the volume contraction rate, along the previously selected geodesic. Finally, in Section <ref>, we adapt Juillet's strategy to obtain our first main theorem. Section <ref> collects our results about the failure of the (K,N) condition in the Heisenberg group. After having introduced the convex trigonometric functions in Section <ref>, we use them to provide the explicit explicit expression of geodesics, cf. Section <ref>. We conclude by proving Theorem <ref> in Section <ref> and Theorem <ref> in Section <ref>. §.§ Acknowledgments T.R. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the collaborative research centre “The mathematics of emerging effects” (CRC 1060, Project-ID 211504053). The autors wish to thank Luca Rizzi for stimulating discussions regarding the regularity of the distance function in manifolds and Lorenzo Portinale for his careful reading of the introduction. § PRELIMINARIES §.§ The CD(K,N) condition A metric measure space is a triple (,,) where (,) is a complete and separable metric space and is a locally finite Borel measure on it. In the following C([0, 1], ) will stand for the space of continuous curves from [0, 1] to . A curve γ∈ C([0, 1], ) is called geodesic if (γ(s), γ(t)) = \|t-s\| ·(γ(0), γ(1)) for every s,t∈[0,1], and we denote by () the space of geodesics on . The metric space (,) is said to be geodesic if every pair of points x,y ∈ can be connected with a curve γ∈(). For any t ∈ [0, 1] we define the evaluation map e_t C([0, 1], ) → by setting e_t(γ) := γ(t). We denote by () the set of Borel probability measures on and by _2() ⊂() the set of those having finite second moment. We endow the space _2() with the Wasserstein distance W_2, defined by W_2^2(μ_0, μ_1) := inf_π∈𝖠𝖽𝗆(μ_0,μ_1)∫^2(x, y) π(x, y), where 𝖠𝖽𝗆(μ_0, μ_1) is the set of all the admissible transport plans between μ_0 and μ_1, namely all the measures in (×) such that (_1)_♯π = μ_0 and (_2)_♯π = μ_1. The metric space (_2(),W_2) is itself complete and separable, moreover, if (,) is geodesic, then (_2(),W_2) is geodesic as well. In particular, every geodesic (μ_t)_t∈ [0,1] in (_2(),W_2) can be represented with a measure η∈(()), meaning that μ_t = (e_t)_#η. We are now ready to introduce the (K,N) condition, pioneered by Sturm and Lott–Villani <cit.>. As already mentioned, this condition aims to generalize, to the context metric measure spaces, the notion of having Ricci curvature bounded from below by K∈ and dimension bounded above by N>1. In order to define the (K,N) condition, let us introduce the following distortion coefficients: for every K ∈ and N∈ (1,∞), τ_K,N^(t)(θ):=t^1/N[σ_K, N-1^(t)(θ)]^1-1/N, where σ_K,N^(t)(θ):= sin(tθ√(K/N))/sin(θ√(K/N)) if Nπ^2 > Kθ^2 > 0, t if K =0, sinh(tθ√(-K/N))/sinh(θ√(-K/N)) if K < 0. Observe that for every K∈, N∈ (1,∞) and t∈ [0,1] we have lim_θ→ 0σ_K,N^(t) (θ) = t and lim_θ→ 0τ_K,N^(t) (θ) = t. A metric measure space (,,) is said to be a (K,N) space (or to satisfy the (K,N) condition) if for every pair of measures μ_0=ρ_0,μ_1= ρ_1 ∈_2(), absolutely continuous with respect to , there exists a W_2-geodesic (μ_t)_t∈ [0,1] connecting them and induced by η∈(()), such that for every t∈ [0,1], μ_t =ρ_t ≪ and the following inequality holds for every N'≥ N and every t ∈ [0,1] ∫_ρ_t^1-1/N'≥∫_×[ τ^(1-t)_K,N'((x,y) ) ρ_0(x)^-1/N' + τ^(t)_K,N'((x,y) ) ρ_1(y)^-1/N'] π( x,y), where π= (e_0,e_1)_#η. One of the most important merits of the (K,N) condition is that it is sufficient to deduce geometric and functional inequalities that hold in the smooth setting. An example which is particularly relevant for this work is the so-called Brunn–Minkowski inequality, whose definition in the metric measure setting requires the following notion. Let (,) be a metric space and let A,B ⊂ be two Borel subsets. Then for t∈ (0,1), we defined the set of t-midpoints between A and B as M_t(A,B) := { x ∈ : x = γ(t) , γ∈() , γ(0) ∈ A , and γ(1) ∈ B } . We can now introduce the metric measure version of the Brunn–Minkowski inequality, whose formulation is stated in terms of the distortion coefficients (<ref>). Given K ∈ and N∈ (1,∞), we say that a metric measure space (,, ) satisfies the Brunn–Minkowski inequality (K,N) if, for every nonempty A,B ⊂spt() Borel subsets, t ∈ (0,1), we have (M_t(A,B)) )^ 1/N≥τ_K,N^(1-t) (Θ(A,B)) ·(A)^ 1/N + τ_K,N^(t) (Θ(A,B)) ·(B)^ 1/N , where Θ(A,B):= {[ inf_x ∈ A, y ∈ B(x, y) if K ≥ 0 ,; sup _x ∈ A, y ∈ B(x, y) if K<0 . ]. As already mentioned, the Brunn–Minkowski inequality is a consequence of the (K,N) condition, in particular we have that (K,N) (K,N), for every K∈ and every N∈ (1,∞). In Sections <ref> and <ref>, we are going to disprove the (K,N) condition for every choice of the parameters K∈ and N∈ (1,∞), by contradicting the Brunn–Minkowski inequality (K,N). A priori, this is a stronger result than the ones stated in Theorem <ref> and Theorem <ref>, since the Brunn–Minkowski inequality is (in principle) weaker than the (K,N) condition. However, recent developments (cf. <cit.>) suggest that the Brunn–Minkowski (K,N) could be equivalent to the (K,N) condition in a wide class of metric measure spaces. Another curvature-dimension bound, which can be defined for metric measure spaces, is the so-called measure contraction property (in short (K,N)), that was introduced by Ohta in <cit.>. The idea behind it is basically to require the (K,N) condition to hold when the first marginal degenerates to δ_x, a delta-measure at x∈spt(), and the second marginal is \|_A/(A), for some Borel set A⊂ with 0<(A)<∞. Given K∈ and N∈ (1,∞), a metric measure space (,,) is said to satisfy the measure contraction property 𝖬𝖢𝖯(K,N) if for every x∈spt() and a Borel set A⊂ with 0<(A)<∞, there exists a Wasserstein geodesic induced by η∈(()) connecting δ_x and \|_A/(A) such that, for every t∈[0,1], 1/(A)≥(e_t)_#(τ_K,N^(t)((γ(0),γ(1)))^Nη(dγ)). For our purposes, we will use an equivalent formulation of the inequality (<ref>), which holds whenever geodesics are unique, cf. <cit.> for further details. More precisely, let x∈spt() and a Borel set A⊂ with 0<(A)<∞. Assume that for every y∈ A, there exists a unique geodesic γ_x,y:[0,1]→ joining x and y. Then, (<ref>) is verified for the measures δ_x and \|_A/(A) if and only if (M_t({x},A')))≥∫_ A'τ_K,N^(t)((x,y))^N (y), for any Borel A'⊂ A. The (K,N) condition is weaker than the (K,N) one, i.e. (K,N) (K,N), for every K∈ and every N∈ (1,∞). In Theorem <ref> (for the case of non-ample geodesics) and Theorem <ref> (cf. Theorem <ref>) in the Heisenberg group, equipped with singular norms, we contradict the (K,N) condition. More precisely, we find a counterexample to (<ref>). §.§ Sub-Finsler structures Let M be a smooth manifold of dimension n and let k∈. A structure on M is a couple (ξ,) where : ^k →_+ is a strictly convex norm on ^k and ξ: M×^k→ TM is a morphism of vector bundles such that: each fiber of the (trivial) bundle M×^k is equipped with the norm ; * The set of horizontal vector fields, defined as := {ξ∘σ : σ∈Γ(M×^k) }⊂Γ(TM), is a bracket-generating family of vector fields (or it satisfies the Hörmander condtion), namely setting Lie_q():={X(q) : X∈span{[X_1,…,[X_j-1,X_j]] : X_i∈,j∈}}, ∀ q∈ M, we assume that Lie_q()=T_qM, for every q∈ M. We say that M is a smooth manifold, if the norm of the structure (ξ,) is smooth, namely ∈ C^∞(^k∖{0}). Although this definition is not completely general in context, since it does not allow the norm to vary on the fiber of M×^k, it includes geometry (where is induced by a scalar product), as every structure is equivalent to a free one, cf. <cit.>. At every point q∈ M we define the distribution at q as _q := {ξ(q,w) : w ∈^k }={X(q) : X∈}⊂ T_qM. This is a vector subspace of T_qM whose dimension is called rank (of the distribution) and denoted by r(q):=_q≤ n. Moreover, the distribution is described by a family of horizontal vector fields. Indeed, letting {e_i}_i=1, …, k be the standard basis of ^k, the generating frame is the family {X_i}_i=1, …, k, where X_i(q) := ξ(q, e_i) ∀ q∈ M, for i= 1, … ,k. Then, according to (<ref>), _q= span{X_1(q), …, X_k(q)}. On the distribution we define the induced norm as v_q := inf{w : v = ξ(q,w) } for every v∈_q. Since the infimum is actually a minimum, the function _q is a norm on _q, so that (_q, _q) is a normed space. Moreover, the norm depends smoothly on the base point q∈ M. A curve γ: [0,1]→ M is admissible if its velocity γ̇(t) exists almost everywhere and there exists a function u=(u_1,…,u_k)∈ such that γ̇(t)=∑_i=1^k u_i(t)X_i(γ(t)), for a.e. t∈ [0,1]. The function u is called control. Furthermore, given an admissible curve γ, there exists u̅=(u̅_1,…, u̅_k): [0,1]→^k such that γ̇(t) = ∑_i=1^k u̅_i(t) X_i(γ(t)), and γ̇(t)_γ(t) = u̅(t), for a.e. t∈[0,1]. The function u̅ is called minimal control, and it belongs to , cf. <cit.>. We define the length of an admissible curve: ℓ(γ):=∫_0^1 γ̇(t)_γ(t) t∈[0,∞). We can rewrite the length of a curve as the L^1-norm of the associated minimal control, indeed by (<ref>), ℓ(γ)=∫_0^1u̅(t) t=u̅_L^1([0,1];(^k,·)). For every couple of points q_0,q_1∈ M, define the distance between them as _SF (q_0,q_1)= inf{ℓ(γ) : γ admissible, γ(0)=q_0 and γ(1)=q_1}. Since every norm on ^k is equivalent to the standard scalar product on ^k, it follows that the structure on M given by (ξ,··) induces an equivalent distance. Namely, denoting by _SR the induced distance, there exist constants C>c>0 such that c _SR≤_SF≤ C_SR, on M× M. Thus, as a consequence of the classical Chow–Rashevskii Theorem in geometry, we obtain the following. Let M be a manifold. The distance is finite, continuous on M× M and the induced topology is the manifold one. From this proposition, we get that (M,_SF) is a locally compact metric space. The local existence of minimizers of the length functional can be obtained as in the setting, in particular, one can repeat the proof of <cit.>. Finally, if (M,_SF) is complete, then it is also a geodesic metric space. § THE GEOMETRY OF SMOOTH MANIFOLDS §.§ The energy functional and the optimal control problem Let γ :[0,1]→ M be an admissible curve. Then, we define the energy of γ as J(γ)=1/2∫_0^1 γ̇(t)^2_γ(t) t. By definition of admissible curve, J(γ)<+∞. In addition, a standard argument shows that γ:[0,1]→ M is a minimum for the energy functional if and only if it is a minimum of the length functional with constant speed. The minimum of J is not invariant under reparametrization of γ, so one needs to fix the interval where the curve is defined. Here and below, we choose [0,1]. The problem of finding geodesics between two points q_0,q_1∈ M can be formulated using the energy functional as the following costrained minimization problem: Pγ:[0,1]→ M, admissible, γ(0)=q_0 and γ(1)=q_1, J(γ) = 1/2∫_0^1 γ̇(t)^2_γ(t) t →min. The problem (<ref>) can be recasted as an optimal control problem. First of all, a curve is admissible if and only if there exists a control in satisfying (<ref>). Second of all, we can consider the energy as a functional on the space of controls. Indeed, as in (<ref>), given an admissible curve γ:[0,1]→ M, we let u̅∈ be its minimal control, as in (<ref>). Then, we have J(γ)=1/2∫_0^1u̅(t)^2 t = 1/2u̅^2_L^2([0,1];(^k,·)). Hence, we regard the energy as a functional on , namely J:→_+; J(u):=1/2u^2_L^2([0,1];(^k,·)), and we look for a constrained minimum of it. Thus, the problem (<ref>) becomes: P'γ̇(t) = ∑_i=1^k u_i(t) X_i(γ(t)), γ(0)=q_0 and γ(1)=q_1, J(u) = 1/2∫_0^1 u(t)^2 t →min. Note that, by (<ref>), the solutions of (<ref>) and (<ref>) coincide. An application of Pontryagin Maximum Principle (see <cit.>) yields necessary conditions for optimality. For every u∈^k and ν∈, introduce the following Hamiltonian: h_u^ν (λ) := λξ(π(λ),u)+ ν/2u^2, ∀λ∈ T^M. Recall that for h∈ C^1(T^M), its Hamiltonian vector field h⃗∈ Vec(T^M) is defined as the unique vector field in T^M satisfying d_λ h = σ(·, h⃗(λ)), ∀ λ∈ T^M, where σ is the canonical symplectic form on T^M. Let M be a manifold and let (γ,u̅) be a solution of (<ref>). Then, there exists (ν,λ_t) 0, where ν∈ and λ_t∈ T^_γ(t)M for every t∈ [0,1], such that Hλ̇_t = h⃗_u̅(t)^ν (λ_t) for a.e. t∈ [0,1], h_u̅(t)^ν (λ_t)= max_v∈^k h_v^ν (λ_t) for a.e. t∈ [0,1], ν≤ 0. If ν <0 in (<ref>), (λ_t)_t∈ [0,1] is called normal extremal. If ν =0 in (<ref>), (λ_t)_t∈ [0,1] is called abnormal extremal. By homogeneity of the Hamiltonian system, if ν 0 in (<ref>) we can fix ν=-1. §.§ Characterization of extremals and the exponential map In this section, we recall some characterizations of normal and abnormal extremal, which are well-known in geometry. We include the proofs in our case, for the sake of completeness. Recall that the annihilator ()⊂ T^M is defined by ()_q := {λ∈ T^_qM : λw=0, ∀ w∈_q}, ∀ q∈ M. Let M be a manifold. Let (γ,u̅) be a non-trivial solution to (<ref>) and let (λ_t)_t∈ [0,1] be its lift. Then, (λ_t)_t∈ [0,1] is an abnormal extremal if and only if λ_t≠ 0 and λ_t∈()_γ(t) for every t∈ [0,1], where γ(t):=π(λ_t). The claim is an easy consequence of the maximization property of the Hamiltonian along the dynamic. More precisely, by (<ref>), we have h_u̅(t)^ν(λ_t)=max_v∈^kh_v^ν(λ_t), for a.e. t∈ [0,1], where the function h_u^ν is defined in (<ref>). Assume that ν=0, then (<ref>) reads as λ_tξ(γ(t),u̅(t))=max_v∈^k λ_tξ(γ(t),v), for a.e. t∈ [0,1], with λ_t≠0 for every t∈ [0,1]. Now, since ξ is linear in the controls, the right-hand side is +∞ unless λ_t∈()_γ(t) for a.e. t∈ [0,1]. By continuity of t↦λ_t, this is true for every t∈[0,1]. Conversely, assume that λ_t∈()_γ(t), then the maximization condition (<ref>) becomes ν/2u̅(t)^2=max_v∈^kν/2v^2. Since ν≤ 0, we may distinguish two cases, either ν=0 and the extremal is abnormal, or ν=-1 and the extremal is normal. In the second case, the optimal control must be 0, so that λ̇_t=0 and the extremal is constant and constantly equal to λ_0. Since we are assuming (λ_t)_t∈[0,1] to be non-constant the latter can not happen. The (or maximized) Hamiltonian is defined as H(λ) := max_u∈^k h_u^-1 (λ_t) = max_u∈^k(∑_i=1^k λ u_i X_i(π(λ)) - u^2/2) The Hamiltonian can be explicitly characterized in terms of the norm _, which denotes the dual norm of . To this aim, we prove the following lemma describing the dual element of v∈(^k,), when is a C^1 norm. Recall that its dual vector v^∈(^k,)^ is uniquely characterized by v^_= v and v^v= v^2, where ·· is the dual coupling. Let (^k,) be a normed space and assume :^k →_+ is a strictly convex C^1 norm, i.e. ∈ C^1(^k∖{0}). Then, for every non-zero vector v∈^k, it holds that v^= v· d_v . Set λ:= d_v ∈ (^k,)^, where we recall that d_v(u):=lim_t→0v+tu-u/t, ∀ u,v∈^k. Then, on the one hand, it holds that λv=d_v(v)=lim_t→0v+tv-v/t= v. On the other hand, we have λ_ = sup_u ∈ B_1λu = sup_u ∈ B_1d_v(u) = sup_u ∈ B_1lim_t→01/t( v+tu-v) ≤sup_u ∈ B_1u = 1, where B_1:=B_1^·(0)⊂^k is the ball of radius 1 and centered at 0, with respect to the norm . The converse inequality can be obtained by taking u=v/v. Finally, the conclusion follows by homogeneity of the dual norm. Let M be a smooth manifold. Given λ∈ T^M, define λ̂=(λ̂_i)_i=1,…,k where λ̂_i:=λX_i(π(λ)) for every i=1,…,k. Then, H^-1(0)=() and H(λ)=1/2λ̂^2_ , ∀λ∈ T^M∖(). where _ is the dual norm to in ^k. Moreover, H∈ C^∞(T^M∖())∩ C^1(T^M). Let q∈ M. Assume that λ∈ T_q^M∖()_q and set F(u):= λ̂u - u^2/2, ∀ u∈^k. Since is smooth, its square is a C^1-function, thus F∈ C^1(^k). Moreover, by homogeneity of the norm, F(u)→-∞ as u→∞, hence F admits a maximum. We compute its differential: d_uF=λ̂- u· d_u= λ̂- u^, according to Lemma <ref>. Therefore, F has a unique critical point (which is also the unique point of maximum) given by u=u^*=λ̂^. Finally, using also (<ref>), this implies that H(λ)=max_u∈^k F(u) = F(λ̂^)=λ̂λ̂^-1/2λ̂_^2=1/2λ̂_^2. To conclude, observe that if λ∈()_q, then λ̂=0 and H(λ)=max_u∈^k(-u^2/2)=0. Conversely, if H(λ)=0 we must have λ∈(). Indeed, if this is not the case, λ̂≠ 0 and hence λ̂_≠0, giving a contradiction. This proves that H^-1(0)=(). Finally, we prove the regularity of H. Note that _ is a smooth norm itself. Indeed, as is smooth and strictly convex, the dual map of Lemma <ref>, which is v^=v d_v=1/2 d_v(^2)=:N(v), is smooth on v≠ 0, invertible and with invertible differential on v≠ 0. Thus, by the inverse function theorem, N^-1∈ C^∞(^k∖{0}). But now the dual norm satisfies (<ref>), hence λ̂_=N^-1(λ̂), and the claim follows. Therefore, we deduce that H∈ C^∞(T^M∖ H^-1(0))∩ C^1(T^M) = C^∞(T^M∖())∩ C^1(T^M). Let (λ_t)_t∈[0,1] be a normal extremal for the problem (<ref>), then the associated control is given by u̅(t)=λ̂_t^, for a.e. t∈ [0,1]. This is again a consequence of the maximality condition in (<ref>), together with the characterization of the Hamiltonian. In particular, we must have h^-1_u̅(t)(λ_t)=max_u∈^k h^-1_u(λ_t)=H(λ_t), ∀ t∈ [0,1], From this and (<ref>), we deduce that the control associated with (λ_t)_t∈[0,1] satisfies the identity (<ref>). The next result relates the system (<ref>) with the Hamiltonian system associated with the Hamiltonian (<ref>). A similar statement can be found in <cit.>. The main difference is the regularity of the Hamiltonian function, which in the classical statement is assumed to be smooth outside the zero section. Let M be a smooth manifold. Let H∈ C^∞(T^ M ∖())∩ C^1(T^M) be the Hamiltonian defined in (<ref>). If (λ_t)_t∈[0,1] is a normal extremal, and λ_0∈(), then λ_t≡λ_0. If λ_0∈ T^ M ∖(), then λ̇_t = H⃗(λ_t). Conversely, if (λ_t)_t∈[0,1] is a solution of (<ref>) with initial condition λ_0∈ T^* M ∖(), then there exists u̅∈ such that (λ_t)_t∈[0,1] is a normal extremal with control u̅ (i.e. the pair (-1,(λ_t)) is a solution of (<ref>)). If (λ_t)_t∈ [0,1] is a normal extremal, there exists an optimal control u̅ such that the pair (-1,(λ_t)) is a solution to (<ref>). Now, if the initial covector λ_0∈(), then λ_t∈() for all t∈ [0,1], as the Hamiltonian is constant along the motion and H(λ_0)=0. Using Corollary <ref>, this implies that the control u̅≡ 0 and that λ_t≡λ_0 as claimed. If λ_0∈ T^M∖(), we follow the blueprint of <cit.>. By the definition of Hamiltonian (<ref>), we have H(λ)≥ h_u̅(t)(λ), ∀ λ∈ T^M, t∈ [0,1], with equality along the dynamic t↦λ_t. This means that the function T^M∋λ↦ H(λ)-h_u̅(t)(λ) has a maximum at λ_t. Therefore, using that H∈ C^1(T^M), we deduce that d_λ_tH=d_λ_th_u̅(t), ∀ t∈ [0,1]. Such an equality immediately implies that the Hamiltonian vector fields are equal along the dynamic, namely H⃗(λ_t)=h⃗_u̅(t)(λ_t), ∀ t∈[0,1]. For the converse implication, recall that the Hamiltonian is constant along the motion. So, if (λ_t)_t∈[0,1] is a solution to (<ref>) with initial condition λ_0∈ T^M∖(), then H(λ_0)=H(λ_t) and λ_t∈ T^M∖(), for every t∈ [0,1]. Since H is smooth outside the annihilator bundle of , we deduce that (λ_t)_t∈ [0,1] is uniquely determined by λ_0 and, repeating verbatim the argument of <cit.>, we conclude the proof. Fix t∈ and consider the (reduced) flow of H⃗ on T^M∖(): e_r^tH⃗:𝒜_t→ T^M∖(), where 𝒜_t⊂ T^M∖() is the set of covectors such that the associated maximal solution (λ_s)_s∈ I, with I⊂ such that 0∈ I, is defined up to time t. Under the assumption of completeness of (M,_SF), H⃗ is complete as a vector field on T^M∖() (and thus 𝒜_t=T^M∖()). We state below this result without proof as the latter is analogous to the classical proof, cf. <cit.>, in view of Lemma <ref> and Lemma <ref>. Let M be a smooth complete manifold. Then any normal extremal t↦λ_t=e^tH⃗_r(λ_0), with λ_0∈ T^M∖(), is extendable to . Let (M,_SF) be a complete smooth manifold and let q∈ M. Then, the exponential map at q is defined as exp_q(λ):= π∘ e^H⃗_ r(λ) if λ∈ T^_qM∖()_q, q if λ∈()_q. The exponential map is smooth in T_q^M∖()_q and, by homogeneity, we also have exp_q∈ C^1(T^_qM). However, note that the we do not have spatial regularity. More precisely, setting ℰ: T^M→ M to be ℰ(q,λ)= exp_q(λ), then ℰ∈ C(T^M)∩ C^∞(T^M∖()), and we can not expect a better spatial regularity as the vector field H⃗ is only continuous on the annihilator bundle. §.§ The end-point map Let M be a manifold, consider u∈ and fix t_0∈ [0,1]. Define the non-autonomous vector field ξ_u(t)(q):=ξ(q,u(t))=∑_i=1^k u_i(t) X_i(q), ∀ q∈ M, t∈[0,1], and denote by P^u_t_0,t: M→ M its flow. This means that, for q_0 ∈ M, the curve t↦ P^u_t_0,t(q_0) is the unique maximal solution to the Cauchy problem γ̇(t)=∑_i=1^m u_i(t) X_i(γ(t)), γ(t_0)=q_0. Moreover, we denote by γ_u: I→ M, the trajectory starting at q_0 and corresponding to u, namely γ_u(t):=P_0,t^u(q_0), for every t∈ I, which is the maximal interval of definition of γ_u. Let M be a manifold and let q_0∈ M. We call 𝒰_q_0⊂ the open set of controls for which the corresponding trajectory γ_u is defined on the interval [0,1]. We define the end-point map based at q_0 as E_q_0 : 𝒰_q_0→ M, E_q_0(u)= γ_u(1). Let M be a manifold. The end-point map is smooth on 𝒰_q_0. Moreover, for every u∈𝒰_q_0 the differential d_u E_q_0: → T_E_q_0(u) M has the expression d_u E_q_0(v)=∫_0^1(P_t, 1^u)_ξ_v(t)(E_q_0(u)) t, for every v ∈. Using the explicit expression for the differential of the end-point map we deduce the following characterization for normal and abnormal extremals in geometry, see <cit.> for the analogous result. Let M be a smooth manifold and let (γ,u) be a non-trivial solution to (<ref>). Then, there exists λ_1∈ T_q_1^M, where q_1= E_q_0(u), such that the curve (λ_t)_t∈[0,1], with λ_t:=(P_t,1^u)^* λ_1∈ T^_γ(t)M, ∀ t∈[0,1], is a solution to (<ref>). Moreover, one of the following conditions is satisfied: (i)(λ_t)_t∈[0,1] is a normal extremal if and only if u satisfies λ_1d_u E_q_0(v) = u^v for every v ∈; (ii)(λ_t)_t∈[0,1] is an abnormal extremal if and only if u satisfies λ_1d_u E_q_0(v) = 0 for every v ∈. Firstly, (<ref>) is a well-known consequence of the Pontryagin maximum principle. We prove (i), the proof of (ii) is analogous. For every v ∈, using Lemma <ref> we deduce that λ_1d_u E_q_0(v) = ∫_0^1 λ_1(P^u_t,1)_* ξ_v(t)(E_q_0(u)) t =∫_0^1 (P^u_t,1)^λ_1ξ_v(t)(γ (t)) t = ∫_0^1 λ_tξ_v(t)(γ (t)) t = ∫_0^1 ∑_i=1^k λ_tξ_i(γ (t)) v_i(t) t. Assume (λ_t)_t∈ [0,1] is a normal extremal, then, by Corollary <ref>, the associated optimal control u satisfies λ_tξ_i(γ(t))= u^_i(t) for i=1,…,k. Therefore, using (<ref>) we deduce that for every v∈ it holds λ_1 d_u E_q_0(v) = ∫_0^1 ∑_i=1^k λ_tξ_i(γ (t)) v_i(t) t = ∫_0^1 u^(t)v(t) t = u^v. This proves (<ref>). Conversely, assume that the control u satisfies (<ref>). We are going to prove that (λ_t)_t∈[0,1] is a normal extremal. Using (<ref>) and (<ref>) we deduce that for every v∈ u^v_L^2 = λ_1 d_u E_q_0(v) = ∫_0^1 ∑_i=1^k λ_tξ_i(γ (t)) v_i(t) t, As a consequence, since v is arbitrary, we conclude that λ_tξ_i(γ(t))= u^_i(t) for i=1,…,k. § FAILURE OF THE CD(K,N) CONDITION IN SMOOTH MANIFOLDS In this section, we prove our main result regarding smooth manifolds, cf. Theorem <ref>. One of the crucial ingredients for the proof is the construction of a geodesic, enjoying good regularity properties, cf. Theorem <ref>. §.§ Construction of a geodesic without abnormal sub-segments This section is devoted to the construction of a geodesic without abnormal sub-segments, in smooth manifolds. The main idea is to choose a short segment of a normal geodesic that minimizes the distance from a hypersurface without characteristic points. We recall the definition of strongly normal geodesic and of geodesic without abnormal sub-segments. Let M be a manifold and let γ : [0, 1] → M be a normal geodesic. Then, we say that γ is (i) left strongly normal, if for all s ∈ [0, 1], the restriction γ\|_[0,s] is not abnormal; (ii) right strongly normal, if for all s ∈ [0, 1], the restriction γ\|_[s,1] is not abnormal; (iii) strongly normal, if γ is left and right strongly normal. Finally, we say that γ does not admit abnormal sub-segments if any restriction of γ is strongly normal. Let Σ⊂ M be a hypersurface and let γ[0,T]→ M be a horizontal curve, parameterized with constant speed, such that γ(0)∈Σ, γ(T) = p ∈ M∖Σ. Assume γ is a minimizer for _SF(·,Σ), that is ℓ(γ)=_SF(p,Σ). Then, γ is a geodesic and any corresponding normal or abnormal lift, say λ :[0,T]→ T^M, must satisfy the transversality conditions, cf. <cit.>, ⟨λ_0, w⟩=0, ∀ w∈ T_γ(0)Σ. Equivalently, the initial covector λ_0 must belong to the annihilator bundle (Σ) of Σ with fiber (Σ)_q= {λ∈ T_q^M \|⟨λ, T_q Σ⟩ = 0}, for any q∈Σ. In the setting, the normal exponential map E, defined as the restriction of the exponential map to the annihilator bundle of Σ, allows to build (locally) a smooth tubular neighborhood around non-characteristic points, cf. <cit.>. This may fail in the setting as E is not regular at Σ, cf. Remark <ref>. Nonetheless, we are able to deduce a weaker result that is enough for our construction, see Theorem <ref>. Recall that q∈Σ is a characteristic point, and we write q∈ C(Σ), if _q⊂ T_qΣ. As it happens in the case, also in the setting, minimizers of _SF(·,Σ) whose initial point is a non-characteristic point, can not be abnormal geodesics. Let M be a smooth manifold. Let p∈ M∖Σ and let γ:[0,1]→ M be a horizontal curve such that γ(0)∈Σ, γ(1)=p and ℓ(γ)=_SF(p,Σ). Then, γ(0)∈ C(Σ) if and only if γ is an abnormal geodesic. The proof is a straightforward adaptation of the analogous result in the setting. We sketch here the argument for completeness. By the Pontryagin maximum principle, cf. Theorem <ref>, there exists a lift λ:[0,1]→ T^M verifying the system (<ref>) with the additional condition (<ref>). Using the characterization of Lemma <ref>, λ is an abnormal lift if and only if λ_0∈()_γ(0). The latter, combined with the transversality condition concludes the proof. From now on, we assume that Σ is the boundary of an open set Ω⊂ M. As our results are local in nature, this assumption is not necessary, however it makes the presentation easier. Let Ω⊂ M be a non-characteristic domain in M, so that ∂Ω is compact and without characteristic points. Then, there exists a never-vanishing smooth section of (∂Ω), i.e. a smooth map λ^+:∂Ω→(∂Ω) such that λ^+(q)∈_q(∂Ω) and 2H(λ^+)=1, which is uniquely determined, up to a sign. Define the normal exponential map as the restriction of the exponential map to the annihilator bundle, namely E:D→ M, E(q,λ)=exp_q(λ), where D⊂(∂Ω) is the largest open sub-bundle where E is defined. Furthermore, we define the distance function from ∂Ω as δ: M→ [0,∞), δ(p):=_SF(p,∂Ω). Let M be a smooth manifold. There exists ϵ>0 such that on the sub-bundle D_ϵ:={(q,λ)∈(∂Ω): E(q,λ)∈Ω and 0<√(2H(λ))<ϵ}⊂ D the map E\|_D_ϵ is injective and E(D_ϵ)={0<δ<ϵ}∩Ω. Without loss of generality, we assume that M is complete, so that D=(∂Ω). We may proceed by contradiction and assume that there does not exist a choice of ϵ>0 so that E\|_D_ϵ is injective. Hence, we can find sequences {(q_n,λ_n)},{(q_n',λ_n')}⊂(∂Ω) such that (q_n,λ_n)≠ (q_n',λ_n') E(q_n,λ_n)=E(q_n',λ_n'), and H(λ_n), H(λ_n')→0. Note that, as ∂Ω has no characteristic points, the Hamiltonian is a norm on the fibers of (∂Ω). Therefore, by compactness, (q_n,λ_n)→ (q,0) and (q_n',λ_n')→(q',0), up to subsequences. Thus, recalling that E is continuous on D, passing to the limit in (<ref>), we get that E(q,0)=E(q',0), meaning that q=q'. As a consequence λ_n,λ_n'∈(∂Ω)_q so they are multiple of the section defined in (<ref>), namely λ_n=t_nλ^+(q), λ_n'=t_n'λ^+(q), where t_n,t_n'→ 0 and their signs agree. Finally, recall that the length of the normal curve [0,1]∋ t ↦ E(q,tλ) is exactly √(2H(λ)). This forces t_n=t_n' which is a contradiction with (<ref>). We are left to prove the last part of the statement. Fix ϵ>0 so that E\|_D_ϵ is injective, then E(D_ϵ)⊂{0<δ<ϵ}∩Ω. For the converse inclusion, pick p∈{0<δ<ϵ}∩Ω and let γ:[0,1]→ M be a geodesic joining γ(0)∈∂Ω and p=γ(1) such that ℓ(γ)=δ(p). Then, γ(0) is a non-characteristic point, therefore γ is a normal geodesic, whose lift satisfies (<ref>), according to Lemma <ref>. Hence, there exists 0≠λ∈(∂Ω)_γ(0) such that γ(t)=E(γ(0),tλ) and ℓ(γ)=√(2 H(λ))<ϵ. Thus, (q,λ)∈ D_ϵ concluding the proof. We state here a useful lemma regarding the regularity of the distance function from a boundary. Recall that a function f:M→ is said to be locally semiconcave if, for every p∈ M, there exist a coordinate chart φ: U⊂ M→^n, with p∈ U, and a constant C∈ such that F:^n→; F(x):=f∘φ^-1(x) - C\|x\|^2/2, is concave, where \|·\| denotes the Euclidean norm. Let M be a smooth manifold. Let Ω⊂ M be an open and bounded subset. Assume that ∂Ω is smooth and without characteristic points. Then, the distance function from ∂Ω, δ is locally semiconcave in Ω. We do not report here a complete proof, since it follows the same arguments of <cit.>, with the obvious modification for the case. In particular, applying Lemma <ref>, we deduce there are no abnormal geodesic joining points of Ω to its boundary and realizing δ. Thus, the proof of <cit.> shows that δ is locally Lipschitz in coordinates, meaning that the function δ written in coordinates is Lipschitz with respect to the Euclidean distance. Then, using <cit.> implication (3)⇒ (2), we conclude. Since δ is locally semiconcave, Alexandrov's theorem ensures that δ is differentiable two times ^n-a.e. (in coordinates) and, letting 𝒰⊂Ω be the set where δ is differentiable, the function dδ:𝒰→ T^M is differentiable ^n-a.e., cf. <cit.> for the precise statement of Alexandrov's theorem. This observation, combined with Lemma <ref> below, gives us an alternative description of geodesics joining ∂Ω and differentiablity points of δ in {0<δ<ϵ}∩Ω. Let M be a smooth manifold. Let p,q∈ M be distinct points and assume there is a function ϕ : M→ differentiable at p and such that ϕ(p) = 1/2^2_SF(p, q) and 1/2^2_SF(z, q) ≥ϕ(z), ∀ z ∈ M. Then, the geodesic joining p and q is unique, has a normal lift and is given by γ : [0, 1] → M; γ(t) = exp_p(-td_pϕ). This is a well-known result in geometry, cf. <cit.>. The same proof can be carried out without substantial modifications in the setting of manifolds, in light of Proposition <ref>. Let M be a smooth manifold and let Ω⊂ M be an open and bounded subset. Assume that ∂Ω is smooth and without characteristic points. Let p∈{0<δ<ϵ}∩Ω be a differentiability point of δ. Then, the unique geodesic γ:[0,1]→ M joining p and ∂Ω and such that δ(p)=ℓ(γ) is defined by γ(t)=exp_p(-t/2d_pδ^2). Since p∈{0<δ<ϵ}∩Ω, from Lemma <ref> we know that the geodesic joining p and ∂Ω, and realizing δ is normal and unique. Let q∈∂Ω be its endpoint and define ϕ:M→; ϕ(z):=1/2δ^2(z). Note that ϕ is differentiable at the point p, ϕ(p)=1/2ℓ(γ)^2=1/2_SF^2(p,q) and, since q∈∂Ω, it also satisfies the inequality in (<ref>). Thus, we may apply Lemma <ref> and conclude the proof. Collecting all the previous results, we are in position to prove the following theorem concerning the regularity of the normal exponential map. Let M be a smooth manifold. The restriction of the normal exponential map to D_ϵ, namely E\|_D_ϵ:D_ϵ→{0<δ<ϵ}∩Ω, defines a diffeomorphism on an open and dense subset 𝒪⊂ D_ϵ. Moreover, δ is smooth on E(𝒪)⊂{0<δ<ϵ}∩Ω, which is open and with full-measure. We are going to show that d_(q,λ)E is invertible for every (q,λ) in a suitable subset of D_ϵ. By Corollary <ref>, letting U⊂{0<δ<ϵ}∩Ω be the set where δ is twice-differentiable, the map Φ: U→(∂Ω); Φ(p)=e^-H⃗(d_pδ) is a right-inverse for the normal exponential map, namely E∘Φ=𝕀_U. Note that Φ(U)⊂(∂Ω) by Corollary <ref>, in combination with the transversality condition (<ref>). Moreover, recalling that the Hamiltonian is constant along the motion, we also have: √(2H(Φ(p)))=ℓ(γ)=δ(p)∈ (0,ϵ), so that Φ(U)⊂ D_ϵ. But now by the choice of the set U, δ is twice-differentiable on this set and it has a Taylor expansion up to order 2. Thus, expanding the identity E∘Φ=𝕀_U at a point p=E(q,λ), we deduce that d_(q,λ)E must be invertible for every (q,λ)∈Φ(U)⊂ D_ϵ, and thus E is a local diffeomorphism around every point in Φ(U). Furthermore, observing that Φ(U) is dense in D_ϵ, we see that E is a local diffeomorphism everywhere on a open and dense subset 𝒪⊂ D_ϵ, containing Φ(U). Hence, we conclude that E\|_𝒪 is a diffeormorphism onto its image, being a local diffeomorphism that is also invertible, thanks to Lemma <ref>. Finally, in order to prove that δ is smooth on E(𝒪)⊂{0<δ<ϵ}∩Ω, it is enough to observe that, by construction, δ(E(q,λ))=√(2H(λ)), ∀ (q,λ)∈ D_ϵ. On D_ϵ, H is smooth, hence we conclude that δ is smooth on E(𝒪). Now U⊂ E(𝒪) so that E(𝒪) is open, dense and has full-measure in {0<δ<ϵ}∩Ω. An immediate consequence of the previous theorem is the existence of many geodesics that are strongly normal in the sense of Definition <ref>. Let M be a smooth manifold and let Ω⊂ M be an open and bounded subset. Assume that ∂Ω is smooth and without characteristic points. Let p∈ E(𝒪)⊂{0<δ<ϵ}∩Ω and let γ:[0,1]→ M be the unique geodesic joining p and ∂Ω and realizing δ. Then, γ is strongly normal. Let q:=γ(0)∈∂Ω the endpoint of γ on the boundary of Ω. Then, since q is non-characteristic point, Lemma <ref> ensures that γ\|_[0,s] can not have an abnormal lift. Hence, γ is left strongly normal. In order to prove that γ is also right strongly normal, we reason in a similar way but with {δ=δ(p)} in place of ∂Ω. Indeed, since δ is smooth on the open set E(𝒪) by Theorem <ref> and d_p̅δ is not vanishing for every p̅∈ E(𝒪) as a consequence of Corollary <ref>, the set Σ:={δ=δ(p)} defines a smooth hypersurface in a neighborhood of the point p. In addition, δ_Σ(q):=_SF(q,Σ)=δ(p) and γ is the unique geodesic realizing δ_Σ. Finally, applying once again Lemma <ref>, we also deduce that p∉ C(Σ), so repeating the argument we did before, we conclude that γ must be also right strongly normal. This concludes the proof. Since E(𝒪) has full measure in {0<δ<ϵ}∩Ω, we can find (q,λ)∈𝒪 such that, denoting by γ:[0,1]→ M the corresponding geodesic minimizing δ, we have that γ(t)∈ E(𝒪) for ^1-a.e. t∈ [0,1]. This means that ^1-almost every level set defines locally a hypersurface and, recalling that restrictions of abnormal geodesics are still abnormal, the proof of Corollary <ref> can be repeated to show that the curve γ does not contain abnormal sub-segments. Let M be a smooth manifold. Then, there exists a strongly normal geodesic γ:[0,1]→ M, which does not contain abnormal sub-segments. Note that Theorem <ref> was stated for a hypersurface that is the boundary of non-characteristic domain Ω. However, without substantial modifications, one can prove that an analogous result holds locally around a non-characteristic point of a given smooth hypersurface Σ⊂ M. In particular, letting q∈Σ∖ C(Σ), there exists V_q⊂Σ open neighborhood of q, and ϵ>0 such that, denoting by D̃_ϵ:={(q̅,λ):q̅∈ V_q, 0<√(2H(λ))<ϵ}, the map E\|_D̃_ϵ:D̃_ϵ→ E(D̃_ϵ)⊂{0<δ_Σ<ϵ} is a diffeomorphism on an open and dense subset 𝒪⊂D̃_ϵ and δ_Σ is smooth on E(𝒪). Now, Corollary <ref> shows that there exists a point p∈ E(𝒪) such that the unique geodesic γ:[0,1]→ M minimizing δ is strongly normal and, also according to Remark <ref>, it does not contain abnormal sub-segments. In order to conclude, we need to show that there exists a hypersurface Σ with Σ∖ C(Σ)≠∅. But this is a consequence of the Hörmander condition, indeed if _q⊂Σ_q for every q∈Σ, then Frobenius' theorem would ensure be involutive and thus it would not be bracket-generating. §.§ Regularity of the distance function We state below the definition of conjugate and cut loci in a manifold, following the blueprint of the setting, cf. <cit.> or <cit.>. Let M be a smooth manifold and let γ : [0, 1] → M be a normal geodesic with initial covector λ∈ T^_pM, that is γ(t)=exp_p(tλ). We say that q = exp_p(t̅λ) is a conjugate point to p along γ if t̅λ is a critical point for exp_p. Let M be a smooth manifold and let p∈ M. We say that q∈ M is a smooth point with respect to p, and write q∈Σ_p, if there exists a unique geodesic γ:[0,1]→ M joining p and q, which is not abnormal and such that q is not conjugate along p. Define the cut locus of p∈ M as (p):=M∖Σ_p. Finally, the cut locus of M is the set (M):={(p,q)∈ M× M: q∈(p)}⊂ M× M. In the setting, according to <cit.>, the set of smooth points with respect to p is open and dense. However, it is an open question to understand whether its complement, that is the cut locus, is negligible. Outside the cut locus of a manifold, we can define the t-midpoint map, for t∈ [0,1], as the map ϕ_t:M× M ∖(M)→ M assigning to (p,q) the t-midpoint of the (unique) geodesic γ_p,q joining p and q. More precisely, for every (p,q)∈ M× M ∖(M), ϕ_t(p,q):=e_t(γ_p,q)=exp_q((t-1)λ_p), where λ_p ∈ T^_q M such that p=exp_q(-λ_p). Note that, by definition of cut locus, the t-midpoint map is well-defined since the geodesic joining p and q for q∉(p) is unique and strictly normal, i.e. without abnormal lifts. We report a useful result relating the regularity of the squared distance function on a manifold M with the cut locus. Such result can be proved repeating verbatim the proof of <cit.>, in light of Proposition <ref> and Lemma <ref>. For every p∈ M, let 𝔣_p:=1/2_SF^2(·,p). Let M be a smooth manifold and let p,q∈ M. Assume there exists an open neighborhood 𝒪_q⊂ M of q such that 𝔣_p is smooth. Then, 𝒪_q⊂Σ_p and ϕ_t(p,z)=exp_z((t-1)d_z𝔣_p), ∀ z∈𝒪_q. Thanks to Proposition <ref>, the regularity of the squared distance ensures uniqueness of geodesics and smoothness of the t-midpoint map. Thus, it is desirable to understand where the squared distance is smooth. In this regard, <cit.> proves the regularity of the squared distance function along left strongly normal geodesics. We refer to <cit.> for further details. Let M be a smooth manifold and let γ : [0, 1] → M be a left strongly normal geodesic. Then there exists ϵ > 0 and an open neighborhood U ⊂ M × M such that: (i) (γ(0), γ(t)) ∈ U for all t ∈ (0, ϵ); (ii) For any (p, q) ∈ U there exists a unique (normal) geodesic joining p and q, shorter than ϵ; (iii) The squared distance function (p,q)↦_SF^2(p, q) is smooth on U. The regularity of the squared distance function can be “propagated” along geodesics that do not admit abnormal sub-segments, applying the previous theorem for every sub-segment. Let M be a smooth manifold and let γ:[0,1]→ M be a geodesic that does not admit abnormal sub-segments. Then, for every s∈ [0,1], there exists ϵ > 0 and an open neighborhood U ⊂ M × M such that: (i) (γ(s), γ(t)) ∈ U for all t ∈ [0,1] such that 0<\|t-s\|<ϵ; (ii) For any (p, q) ∈ U there exists a unique (normal) geodesic joining p and q, shorter than ϵ; (iii) The squared distance function (p,q)↦_SF^2(p, q) is smooth on U. §.§ Volume contraction rate along geodesics Our goal is to quantify the contraction rate of small volumes along geodesics. To do this, we combine the smoothness of the t-midpoint map, with a lower bound on the so-called geodesic dimension. The latter has been introduced in <cit.> for manifolds and in <cit.> for general metric measure spaces. We recall below the definition. Let M be a smooth manifold. Given a point p∈ M and a Borel set Ω⊂ M ∖(p), we define the geodesic homothety of Ω with center p and ratio t ∈ [0, 1] as Ω_t^p:={ϕ_t(p,q): q∈Ω}. In the sequel, we say that is a smooth measure if, in coordinates, is absolutely continuous with respect the Lebesgue measure of the chart with a smooth and positive density. We will consider the metric measure space (M,_SF,). Let M be a smooth manifold, equipped with a smooth measure . For any p∈ M and s > 0, define C_s(p) := sup{lim sup_t→ 01/t^s(Ω_t^p)/(Ω): Ω⊂ M ∖(p) Borel, bounded and (Ω)∈(0,+∞)}, We define the geodesic dimension of (M, _SF, ) at p∈ M as the non-negative real number 𝒩(p) := inf{s > 0 : C_s(p) = +∞} = sup{s > 0 : C_s(p) = 0}, with the conventions inf∅ = +∞ and sup∅ = 0. In <cit.>, the definition of geodesic dimension is given for metric measure spaces with negligible cut loci. In this work, we adapted the definition by taking the supremum (<ref>) over sets Ω which are outside the cut locus (p). We now prove a fundamental theorem which relates the geodesic and topological dimensions of a sub-Finsler manifold M. This result is a suitable adaptation of <cit.> to our setting. Let M be a smooth manifold, equipped with a smooth measure . Assume that r(p)<n:= M for every p∈ M. Then, 𝒩 (p) ≥ n+1, ∀ p∈ M. Let _SR be a distance on the manifold M, equivalent to _SF (see (<ref>)). The Ball-Box theorem, cf. <cit.>, ensures that for every p ∈ M there exist n_p≥ n+1 and a positive constant C_p such that (B_r^SR(p)) ≤ C_p · r^n_p for r sufficiently small. Since _SF and _SR are equivalent, up to changing the constant, the same estimate holds for balls, in particular lim sup_r → 0(B_r^SF(p))/r^k=0 for every k<n+1. Take any Ω⊂ M ∖(p) Borel, bounded and with (Ω)∈ (0,+∞) and consider R>0 such that Ω⊂ B_R^SF(p). Note that Ω_t^p ⊂ B_tR^SF(p) and thus for every k<n+1 we have that lim sup_t → 0(Ω_t^p)/t^k (Ω)≤lim sup_t → 0(B_tR^SF(p))/t^k (Ω) = lim sup_t → 0(B_tR^SF(p))/(tR)^k·R^k/(Ω)=0, where we used (<ref>) for the last equality. Since Ω was arbitrary, we deduce that C_k(p)=0 for every k<n+1 and then 𝒩(p)≥ n+1. For an equiregular manifold, with the same proof, it is possible to improve the estimate of Proposition <ref>. In fact, in this case the Ball-Box theorem provides the estimate (<ref>) with n_p equal to the Hausdorff dimension _H(M), for every p, and consequently 𝒩(p)≥_H(M), cf. <cit.>. By construction, the geodesic dimension controls the contraction rate of volumes along geodesics. This information can be transferred to the t-midpoint map, provided that is smooth. By invoking Theorem <ref>, we can always guarantee the smoothness of the t-midpoint map for a sufficiently short segment of a geodesic without abnormal sub-segments. Let M be a smooth manifold equipped with a smooth measure and such that r(p)<n:= M for every p∈ M. Let γ:[0,1]→ M be a geodesic that does not admit abnormal sub-segments, with endpoints p and q. Assume that (p,q) belongs to the open set U, found in Theorem <ref>. Then, either \|(d_qϕ_t(p,·))\| has infinite order at t=0 or \|(d_qϕ_t(p,·))\| ∼ t^m_p, as t→ 0 for some integer m_p≥𝒩(p)≥ n+1. Since, by assumption (p,q)∈ U, we can apply item (iii) of Theorem <ref>, deducing the regularity of the distance function. Combining this with Proposition <ref> and the homogeneity of the Hamiltonian flow, there exists an open neighborhood V⊂ M of q, such that the function [0,1)× V∋(t,z) ↦ d_zϕ_t(p,·) = d_z(exp_z( (t-1) d_z𝔣_p)) is smooth. Thus, we can compute the Taylor expansion of its determinant in the t-variable at order N:= ⌈𝒩 (p)⌉-1 < 𝒩(p), obtaining: (d_zϕ_t(p,·))= ∑_i=0^N a_i(z)t^i+t^N+1R_N(t,z), ∀ z∈ V, where the functions a_i and R_N are smooth. Arguing by contradiction, we assume that there exists j≤ N such that a_j(q) 0 and define m:= min{ i≤ N : ∃ z∈ V such that a_i(z)≠ 0}. Note that m≤ j since a_j(q)≠ 0 and thus m≤ N. Without loss of generality, we can assume that V and p are contained in the same coordinate chart and that a_m>0 on an open subset Ṽ⊂ V with positive measure. Then, in charts, it holds that ^n (Ṽ_t^p)=∫_Ṽ\|(d_zϕ_t(p,·))\| z = ∫_Ṽ a_m(z) z · t^m + o(t^m) as t → 0. Therefore, recalling that is a smooth measure, there exists a constant a>0 such that (Ṽ_t^p) ≥ a · t^m, for every t sufficiently small. As a consequence, taking any s∈ (N, 𝒩(p)) we have that lim sup_t→ 01/t^s(Ṽ_t^p)/(Ṽ)≥lim sup_t→ 01/(Ṽ)a · t^m/t^s = +∞, and therefore we deduce C_s(p)=+∞, which in turn implies 𝒩 (p)≤ s, giving a contradiction. Theorem <ref> motivates the following definition. Let M be a smooth manifold and let γ:[0,1]→ M be a strictly normal geodesic not admitting abnormal sub-segments. We say that γ is ample if, for every couple of distinct points p,q∈γ([0,1]), \|(d_qϕ_t(p,·))\| exists and has finite order in t=0. The concept of ample geodesic in the setting has been introduced in <cit.> and it differs from Definition <ref>. However, we remark that, in manifolds, for ample geodesics in the sense of <cit.>, \|(d_qϕ_t(p,·))\| has finite order equal to the geodesic dimension at p, cf. <cit.>. Thus, our definition is weaker, but enough for our purposes. §.§ Proof of Theorem <ref> Let M be a smooth manifold and let ϕ_t the t-midpoint map, defined as in (<ref>). For ease of notation, set (p,q) := ϕ_1/2(p,q), ∀ (p,q)∈ M× M∖(M), be the 1/2-midpoint map or simply midpoint map. Reasoning as in <cit.>, we obtain the following result as a consequence of Corollary <ref> and Theorem <ref>. This argument hinges upon Theorem <ref>, which establishes the existence of a geodesic without abnormal sub-segments in a manifold. Let M be a smooth manifold equipped with a smooth measure and such that r(p)<n:= M for every p∈ M. Let γ:[0,1]→ M be the geodesic identified in Theorem <ref> and let ε>0. Then, there exist 0≤ a<b≤ 1 such that, letting p̅:=γ(a), q̅:=γ(b), the following statements hold: (i) p̅∉(q̅), q̅∉(p̅) and, for every t∈ (a,b), we have p̅,q̅∉(γ(t)). Moreover, for every t∈ (a,b), 𝔣_γ(t) is smooth in a neighborhood of p̅ and in a neighborhood q̅. (ii) If, in addition, γ is ample, the midpoint map satisfies \| d_q̅(p̅,·)\|≤ (1+ε)2^-m_p̅, \| d_p̅(·,q̅)\|≤ (1+ε)2^-m_q̅ where m_p̅ and m_q̅ are defined by (<ref>) and m_p̅, m_q̅≥ n+1. Given z∈ M, define the inverse geodesic map _z: M ∖(z) → M as _z(p) = exp_z(- λ) where λ∈ T^_z M such that p=exp_z( λ). We may interpret this map as the one associating to p the point _z(p) such that z is the midpoint of x and _z(p). We prove now the main theorem of this section, which also implies Theorem <ref>. Our strategy is an adaptation to the setting of the one proposed in <cit.>. Let M be a complete smooth manifold equipped with a smooth measure and such that r(p)<n:= M for every p∈ M. Then, the metric measure space (M,_SF,) does not satisfy the Brunn–Minkowski inequality (K,N), for every K∈ and N∈ (1,∞). Fix ε>0, K∈ and N∈(1,∞). Let γ:[0,1]→ M be the geodesic identified by Theorem <ref> and assume it is contained in a coordinate chart with (sub-Finsler) diameter D>0. Up to restricting the domain of the chart and the geodesic, we can also assume that (1-ε)^n ≤≤ (1+ε)^n and τ_K,N^(1/2) (θ) ≥1/2 - ε, ∀θ≤ D, where the second inequality can be fulfilled, according to Remark <ref>. Moreover, let 0≤ a < b ≤ 1 be as in Proposition <ref>. We proceed by contradiction and assume that (M,_SF,) satisfies the (K,N). First of all, suppose that γ is not ample. According to <cit.>, the Brunn–Minkowski inequality (K,N) implies the (K,N) condition[In that paper, the proposition is proved for essentially non-branching metric measure spaces. However, what is needed is a measurable selection of geodesics which we have locally around the curve γ.]. Therefore, (M,_SF,) satisfies the (K,N) condition and, for the moment, assume K=0. Set p̅:=γ(a) and q̅:=γ(b) and let Ω_ϱ:=B_ϱ(q̅) for ϱ>0. From the (0,N) condition we get (Ω_ϱ,t^p̅)≥ t^N(Ω_ϱ), ∀ t∈[0,1], ϱ>0. If ϱ is sufficiently small, then Ω_ϱ,t^p̅=ϕ_t(p̅,Ω_ϱ) for t∈ [0,1), therefore, employing the first estimate in (<ref>), the inequality (<ref>) can be reformulated as follows: 1+ε/1-ε_Ω_ϱ\|(d_zϕ_t(p̅,·))\| z≥(Ω_ϱ,t^p̅)/(Ω_ϱ)≥ t^N, ∀ t∈[0,1), ϱ>0. Taking the limit as ϱ→0, and then the limit as t→ 0, we find that the order of \|(d_q̅ϕ_t(p̅,·))\| should be smaller than or equal to N, giving a contradiction. Finally, if K≠ 0, observe that the behavior of the distortion coefficients, as t→ 0, is comparable with t, namely there exists a constant C=C(K,N,(p̅,q̅))>0 such that τ_K,N^(t)(θ)≥ C t, as t→ 0, ∀ θ∈ ((p̅,q̅)-ϱ,(p̅,q̅)+ϱ). Therefore, repeating the same argument that we did for the case K=0, we obtain the sought contradiction. Suppose instead that the geodesic γ is ample and let m be the unique midpoint between p̅=γ(a) and q̅=γ(b). According to item (i) of Proposition <ref>, the map _m is well-defined and smooth in a neighborhood of p̅ and q̅, moreover by definition _m(q̅)=p̅ and _m(p̅)=q̅. Note that _m ∘_m = 𝕀 (where defined), thus \|(d_p̅_m)\| · \|(d_q̅_m)\| = \|(d_q̅ (_m ∘_m) )\| =1. Therefore, at least one between \|(d_q̅_m)\| and \|(d_p̅_m)\| is greater than or equal to 1, without loss of generality we assume \|(d_q̅_m)\| ≥ 1. Let B_ϱ:=B^eu_ϱ(q̅) the (Euclidean) ball of radius ϱ>0 centered in q̅. Introduce the function F: B_ϱ× B_ϱ→ M, defined as B_ϱ× B_ϱ∋ (x,y) ↦ F(x,y):= (_m(x),y). Observe that, for ϱ small enough, F is well-defined and by construction F(x,x)=m for every x∈ B_ϱ. Therefore, we deduce that for every vector v∈ T_q̅M≅^n, the following holds: 0 = d_(q̅,q̅) F (v,v) = (d_p̅ (·,q̅) ∘ d_q̅_m) v + d_q̅(p̅,·) v. Since the former identity is true for every vector v∈^n, we can conclude that d_p̅(·,q̅)∘ d_q̅_m + d_q̅(p̅,·)=0, and consequently, for every v,w∈^n, we have d_(q̅,q̅) F (v,w) = (d_p̅ (·,q̅)∘ d_q̅_m) v + d_q̅ (p̅,·) w = d_q̅ (p̅,·) (w-v). In particular, we obtain a Taylor expansion of the function F at the point (q̅,q̅) that in coordinates takes the form: F(q̅+v,q̅+w) - m - d_q̅ (p̅,·) (w-v)_eu = o (v_eu+w_eu), as v,w→ 0. Then, as v and w vary in B^eu_ϱ(0), v-w varies in B^eu_2ϱ(0), and we obtain that F(B_ϱ,B_ϱ) ⊆ m + d_q̅ (p̅,·) (B^eu_2ϱ(0)) + B^eu_ω(ϱ)(0), where ω:_+→_+ is such that ω(r)=o(r) when r→ 0^+. Now, consider A_ϱ:= _m(B_ϱ) and note that by definition M_1/2(A_ϱ,B_ϱ) = F(B_ϱ,B_ϱ), then using (<ref>) we conclude that, as ϱ→ 0, ^n (M_1/2(A_ϱ,B_ϱ))=^n(F(B_ϱ,B_ϱ)) ≤^n( d_q̅ (p,·) (B^eu_2ϱ(0))) + o (ϱ^n) = \|(d_q̅ (p,·))\| ·ω_n 2^nϱ^n + o(ϱ^n) ≤ (1+ε)2^n-m_qω_n ϱ^n + o(ϱ^n) ≤1/2 (1+ε) ω_n ϱ^n + o(ϱ^n) where ω_n=^n(B^eu_1(0)) and the two last inequalities follow from (<ref>) and m_q̅≥ n+1. On the other hand, it holds that ^n (B_ϱ)= ω_n ϱ^n and, as ϱ→0, ^n (A_ϱ)= ^n( _m(B_ϱ)) = ( \|(d_q̅_m)\| + O (ϱ)) ^n (B_ϱ)≥ω_n ϱ^n +o(ϱ^n). Taking into account the first estimate of (<ref>), we deduce the following inequalities for the measure , as ϱ→ 0, (M_1/2(A_ϱ,B_ϱ)) ≤1/2 (1+ε)^2 ω_n ϱ^n + o(ϱ^n), (A_ϱ) ≥ (1-ε)ω_n ϱ^n + o(ϱ^n) and (B_ϱ) ≥ (1-ε) ω_n ϱ^n. Finally, if ε is small enough we can find ϱ sufficiently small such that (M_1/2(A_ϱ,B_ϱ)) ^1/N < (1/2 - ε) (A_ϱ)^1/N + (1/2 - ε) (B_ϱ)^1/N ≤τ_K,N^(1/2)(Θ(A_ϱ,B_ϱ)) (A_ϱ)^1/N + τ_K,N^(1/2)(Θ(A_ϱ,B_ϱ)) (B_ϱ)^1/N, which contradicts the Brunn–Minkowski inequality (K,N). Observe that the argument presented in this section is local, around the geodesic without abnormal sub-segments. Thus, repeating the same proof, we can extend Theorem <ref> if the assumption on the rank holds on an open set V⊂ M, namely r(p) <n for every p∈ V. § FAILURE OF THE CD(K,N) CONDITION IN THE SUB-FINSLER HEISENBERG GROUP In this section, we disprove the curvature-dimension condition in the Heisenberg group, cf. Theorem <ref>. Our strategy relies on the explicit expression of geodesics in terms of convex trigonometric functions, found in <cit.>. §.§ Convex trigonometry In this section, we recall the definition and main properties of the convex trigonometric functions, firstly introduced in <cit.>. Let Ω⊂^2 be a convex, compact set, such that O:=(0,0)∈Int (Ω) and denote by its surface area. Let θ∈ denote a generalized angle. If 0≤θ < 2 define P_θ as the point on the boundary of Ω, such that the area of the sector of Ω between the rays Ox and OP_θ is 1/2θ (see Figure <ref>). Moreover, define (θ) and (θ) as the coordinates of the point P_θ, i.e. P_θ = ( (θ), (θ) ). Finally, extend these trigonometric functions outside the interval [0,2) by periodicity (of period 2), so that for every k∈ℤ. (θ)= (θ+2k ), (θ)= (θ+2k ) and P_θ = P_θ +2k. Observe that by definition (0)=0 and that when Ω is the Euclidean unit ball we recover the classical trigonometric functions. Consider now the polar set: Ω^∘ := {(p,q)∈^2 : px+qy≤ 1 for every (x,y)∈Ω}, which is itself a convex, compact set such that O∈Int(Ω^∘). Therefore, we can consider the trigonometric functions and . Observe that, by definition of polar set, it holds that (θ) (ψ) + (θ)(ψ)≤ 1, for every θ,ψ∈. We say that two angles θ,ψ∈ correspond to each other and write θΩψ if the vector Q_ψ:= ((ψ),(ψ)) determines a half-plane containing Ω (see Figure <ref>). By the bipolar theorem <cit.>, it holds that Ω^∘∘=Ω and this allow to prove the following symmetry property for the correspondence just defined. Let Ω⊂^2 be a convex and compact set, with O∈Int (Ω). Given two angles θ,ψ∈, θΩψ if and only if ψΩ^∘θ. Moreover, the following analogous of the Pythagorean equality holds: θΩψ if and only if (θ) (ψ) + (θ)(ψ)= 1. The correspondence θΩψ is not one-to-one in general, in fact if the boundary of Ω has a corner at the point P_θ, the angle θ corresponds to an interval of angles (in every period). Nonetheless, we can define a monotone multi-valued map C^∘ that maps an angle θ to the maximal closed interval containing angles corresponding to θ. This function has the following periodicity property: C^∘(θ+2 k)= C^∘(θ) +2^∘ k for every k∈ℤ, where ^∘ denotes the surface area of Ω^∘. If Ω is strictly convex, then the map C^∘ is strictly monotone, while if the boundary of Ω is C^1, then C^∘ is a (single-valued) map from to and it is continuous. Analogously, we can define the map C_∘ associated to the correspondence ψΩ^∘θ. Proposition <ref> guarantees that C_∘∘ C^∘ = C^∘∘ C_∘= id. Let Ω⊂^2 as above. The trigonometric functions and are Lipschitz and therefore differentiable almost everywhere. At every differentiability point θ of both functions, there exists a unique angle ψ corresponding to θ and it holds that '(θ)= (ψ) and '(θ)= - (ψ). Naturally, the analogous result holds for the trigonometric functions and . As a corollary of the previous proposition, we obtain the following convexity properties for the trigonometric functions. The functions and are concave in every interval in which they are non-negative and convex in every interval in which they are non-positive. This convexity properties of the trigonometric functions will play a small but fundamental role in Section <ref> in the form of the following corollaries. Given a non-null constant k∈ and every angle θ, consider the function g:→; g(t):=(θ)(θ+ k t) - (θ) (θ+ k t). If k>0 this function is convex for positive values of t and concave for negative values of t, locally around 0. Vice versa, If k<0 it is concave for positive values of t and convex for negative values of t, locally around 0. The function g(t) can be seen as a scalar product of two vectors in ^2, therefore it is invariant by rotations. In particular, we consider the rotation that sends θ to 0: this maps P_θ to the the positive x-axis and the set Ω to a convex, compact set Ω̃⊂^2. Then, g(t) in (<ref>) is equal to the function t↦ - cos_Ω̃(0) sin_Ω̃( k t). The conclusion immediately follows from Corollary <ref>. Given a non-null constant k∈ and every angle ψ, the function h:→; h(t):= 1 - (ψ)((ψ+ k t)_∘) - (ψ) ((ψ+ k t)_∘). is non-decreasing for positive values of t and non-increasing for negative values of t, locally around 0. Note that h is the derivative of the function ∋ t↦ kt +(ψ)(ψ+ k t) - (ψ) (ψ+ k t), divided by k. The thesis follows from Corollary <ref>, since the function (<ref>) is the sum of a linear function and of a function of the type (<ref>). In the following we are going to consider the trigonometric functions associated to the unit ball of a strictly convex norm on ^2, i.e. Ω:= B^·_1(0). In this case, the polar set Ω^∘ is the unit ball B^·__1(0) of the dual norm _. Moreover, according to the Pythagorean identity (<ref>), if θΩψ then Q_ψ is a dual vector of P_θ. In particular, if is a C^1 norm, Lemma <ref> ensures that ((ψ),(ψ))=Q_ψ = d_P_θ= d_((θ),(θ)). We conclude this section by recalling a well-known result on the relation between a norm and its dual _. This will be employed in the subsequent sections, as the geodesics of the Heisenberg group, equipped with the norm , follow the shape of the boundary of B^·__1(0), cf. Theorem <ref>. Let be a norm on ^2, and let _ be its dual norm, then: (i) _* is a strictly convex norm if and only if is a C^1 norm; (ii) _* is a strongly convex norm if and only if is a C^1,1 norm. §.§ Geodesics in the Heisenberg group We present here the Heisenberg group and study its geodesics. Let us consider the Lie group M=^3, equipped with the non-commutative group law, defined by (x, y, z) ⋆ (x', y', z') = (x+x',y+y',z+z'+1/2(xy' - x'y)), ∀ (x, y, z), (x', y', z')∈^3, with identity element =(0,0,0). In the notation of Section <ref>, we define the following morphism of bundles ξ: M×^2→ TM, ξ(x,y,z;u_1,u_2)=(x,y,z;u_1,u_2,1/2(u_2 x - u_1 y)). The associated distribution of rank 2 is spanned by the following left-invariant vector fields: X_1=∂_x-y/2∂_z, X_2=∂_y+x/2∂_z, namely =span{X_1,X_2}. It can be easily seen that is bracket-generating. Then, letting :^2→_+ be a norm, the Heisenberg group is the Lie group M equipped with the structure (ξ,). By construction, also the resulting norm on the distribution is left-invariant, so that the left-translations defined by L_p:→; L_p(q):=p⋆ q, are isometries for every p∈. In this setting, the geodesics were originally studied in <cit.> and <cit.> for the three-dimensional case and in <cit.> for general left-invariant structures on higher-dimensional Heisenberg groups. We recall below the main results of <cit.> for strictly convex norms. Let be the Heisenberg group, equipped with a strictly convex norm. Then, the following statements hold: (i) for any q∈∖{x=y=0}, there exists a unique geodesic γ:[0,1]→ joining the origin and q. (ii) γ:[0,T]→ is a geodesic starting at the origin if and only if it satisfies the Pontryagin's maximum principle for the time-optimal control problem: γ̇(t) = u_1(t) X_1(γ(t))+u_2(t) X_2(γ(t)), u(t)∈ B^·_1(0), γ(0)=q_0, and γ(T)=q_1, T→min. Note that the geodesics in <cit.> are found solving the Pontryagin maximum principle of Theorem <ref> for the time-optimal problem. The latter is an equivalent formulation of (<ref>), however it produces arc-length parameterized geodesics. The next step is to compute explicitly the exponential map. In <cit.>, the author provides an explicit expression for geodesics starting at the origin, using the convex trigonometric functions functions presented in Section <ref>. Since therein the author solves the time-optimal problem, we prefer to solve explicitly the Hamiltonian system (<ref>), in the case of the Heisenberg group. Let be the Heisenberg group, equipped with a strictly convex norm. Let γ : [0, 1] → be the projection of a (non-trivial) normal extremal (λ_t)_t∈ [0,1] starting at the origin, then γ(t) = (x(t), y(t), z(t)), with x(t) = r/w((ϕ+ω t ) - (ϕ)), y(t) = -r/w((ϕ+ω t ) - (ϕ)), z(t) = r^2/2ω^2(ω t + (ϕ+ω t ) (ϕ) - (ϕ+ω t ) (ϕ)), for some ϕ∈ [0, 2𝕊^∘), ω∈∖{0} and r > 0. If ω = 0, then x(t) = (r(ϕ^∘)) t, y(t) = (r(ϕ^∘)) t, z(t) = 0. Firstly, we characterize the Hamiltonian in the Heisenberg group. Note that, without assuming additional regularity on , we can not apply directly Lemma <ref>. Nevertheless, we can still obtain an analogous result by means of convex trigonometry. Indeed, let h_i(λ):=⟨λ,X_i(π(λ))⟩ for i=1,2, then H(λ):=max_u∈^2(∑_i=1^2 u_ih_i(λ)-u/2), ∀ λ∈ T^. We introduce polar coordinates on ^2 associated with and its dual norm _, namely (u_1,u_2)↦ (ρ,θ) and (h_1,h_2)↦(ζ,ψ) where u_1 =ρ(θ), u_2 =ρ(θ), and h_1 =ζ(ψ), h_2 =ζ(ψ). Hence, the Hamiltonian becomes H(λ) =max_u∈^2(∑_i=1^2 u_ih_i(λ)-u/2) =max_θ∈[0,2) ρ>0(ρζ((θ)(ψ)+(θ)(ψ))-ρ^2/2) ≤max_ρ>0(ρζ-ρ^2/2) = ζ^2/2, where the last inequality is a consequence of (<ref>). Moreover, we attain the equality in (<ref>) if and only if ρ=ζ and ψ=C^∘ (θ). Therefore, since ζ=λ̂_* with λ̂=(h_1(λ),h_2(λ)), we conclude that H(λ)=1/2λ̂^2_, ∀ λ∈ T^∖(), and the maximum is attained at the control u=λ̂^. Furthermore, H∈ C^1(T^M) by strict convexity of , cf. Proposition <ref>. We write the system (<ref>) in coordinates (x,y,z;h_1,h_2,h_3) for the cotangent bundle, where h_3(λ):=⟨λ,∂_z⟩. The vertical part of (<ref>) becomes ḣ_1(t) = λ̂_t_d_λ̂_t_·(0,-h_3(t)), ḣ_2(t) = λ̂_t_d_λ̂_t_·( h_3(t),0), ḣ_3(t) = 0. Let (λ_t)_t∈ [0,1] be a normal extremal with associated maximal control given by t↦ u(t), then we use Lemma <ref> to deduce that λ̂_t_d_λ̂_t_=λ̂_t^= u(t). Therefore, letting h_3(t)≡ω∈, we may rewrite (<ref>) as ḣ_1(t) = - ω u_2(t), ḣ_2(t) = ω u_1(t). To solve this system, we use the polar coordinates (<ref>): letting t↦ (ρ(t),ψ(t)) be the curve representing λ̂_t=(h_1(t),h_2(t)), we deduce that ρ(t) and ψ(t) are absolutely continuous and satisfy ρ(t)=λ̂_t_, ψ̇(t)=h_1(t)ḣ_2(t)-ḣ_1(t) h_2(t)/ρ^2(t). We may compute explicitly ρ̇(t) and ψ̇(t), using once again Lemma <ref>, the system (<ref>) and identity (<ref>): ρ̇(t) = d_λ̂_t_·(ḣ_1(t),ḣ_2(t))= ω/λ̂_t_u(t)· (-u_2(t),u_1(t))=0, ψ̇(t) = ω. Thus, integrating the above identities, we obtain ρ(t)≡ r and ψ(t)=ω t +ϕ for some r>0 and ϕ∈ [0,2𝕊^∘). Finally, we find an explicit expression for the maximal control: u(t)=(r(C_∘ (ϕ+ω t )),r(C_∘ (ϕ+ω t ))). From this, we may explicitly integrate the horizontal part of the Hamiltonian system, obtaining the desired expression. In particular, if ω=0 we immediately obtain (<ref>). If ω≠ 0, we may employ Proposition <ref> to conclude. As (,_SF) is complete, normal extremals can be extended to , according to Proposition <ref>. Thus, we may define the (extended) exponential map at the origin on the whole T_0^×: G: ([0,2𝕊^∘)×× [0,∞))× ⟶, (ϕ,ω,r;t) ⟼(x(ϕ,ω,r;t), y(ϕ,ω,r;t), z(ϕ,ω,r;t)), where (x(ϕ,ω,r;t), y(ϕ,ω,r;t), z(ϕ,ω,r;t)) correspond to the curve (x(t),y(t),z(t)) defined by Proposition <ref> with initial datum (ϕ,ω,r) and with the understanding that G(ϕ,ω,0;t)≡ 0. By the properties of the convex trigonometric functions, G is a C^1 map for ω≠ 0. Moreover, thanks to Theorem <ref>, for every initial datum (ϕ,ω,r), the curve t↦ G(ϕ,ω,r;t) is a geodesic between its endpoints for sufficiently small times. More precisely, it is minimal for \|t\|<t^=t^(ϕ,ω,r), where t^>0 is the first positive time such that G(ϕ,ω,r;t^) lies on the z-axis. In particular, a direct computation shows that t^= 2𝕊^∘/\|ω\|, if ω≠ 0, ∞, if ω =0. We conclude this section by highlighting a property of geodesics in the Heisenberg group that will be relevant in our analysis. For the sake of notation, denote by Ω^∘_(ϕ,ω,r) the following transformation of Ω^∘=B^·__1(0): Ω^∘_(ϕ,ω,r):= R_-π/2[r/ω(Ω^∘-((ϕ),(ϕ)))], where R_-π/2 is counter-clockwise rotation in the plane of angle -π/2. Let be the Heisenberg group, equipped with a strictly convex norm and let γ:[0,1]→ be a geodesic starting at the origin, with γ(t)=(x(t),y(t),z(t)). Then, the curve t↦ (x(t),y(t)) is either a straight line or belongs to the boundary of Ω^∘_(ϕ,ω,r). Moreover, for every t∈[0,1], z(t) equals the oriented area that is swept by the vector joining (0,0) with (x(s),y(s)), for s∈ [0,t]. §.§ Failure of the CD(K,N) condition for C1,1-norms In this section we contradict the validity of the (K,N) condition in the Heisenberg group, equipped with a strictly convex and C^1,1 norm and with a smooth measure. The strategy follows the blueprint of the one presented in Section <ref>. The main issue we have to address here is the low regularity (cf. Remark <ref>) of the midpoint and inverse geodesic maps of (<ref>) and (<ref>). Nevertheless, using the explicit expression of geodesics presented in Proposition <ref>, we successfully overcome these challenges through a series of technical lemmas, culminating in Corollary <ref>, Proposition <ref> and Theorem <ref>. Let be the Heisenberg group, equipped with a C^1,1 and strictly convex norm . According to Proposition <ref>, the dual norm _ is C^1 and strongly convex. Thus, in the notations of Section <ref>, the correspondences C^∘ and C_∘ are continuous functions. In order to ease the notation, in this section we sometimes use the shorthands: θ^∘ = C^∘(θ) and ψ_∘ = C_∘(ψ), ∀ θ, ψ∈. Alexandrov's theorem ensures that the dual norm _* has a second derivative and a second-order Taylor expansion almost everywhere, we call D_⊂^2 the set of twice differentiability of it. Let ψ∈ [0,2^∘) be an angle such that Q_ψ∈ D_, then the function C_∘ is differentiable at ψ with positive derivative. Consider a vector v∈^2 orthogonal to d_Q_ψ_* such that v_eu=1. Then, since Q_ψ∈ D_, there exists a constant C∈ such that Q_ψ+ s v_ = 1 + C s^2 + o(s^2), as s → 0. Observe that, since the norm _* is strongly convex, the constant C is strictly positive. Consider the curve s ↦ x(s):= Q_ψ+ s v/Q_ψ+ s v_, which by definition is a parametrization of an arc of the unit sphere S^·__1(0)=∂Ω^∘. Call A(s) the signed area of the sector of Ω^∘ between the rays O Q_ψ and O x(s) (see Figure <ref>). As a consequence of (<ref>), we deduce that A(s)= 1/2 k s + o(s^2), as s → 0, where k is the scalar product between Q_ψ and v^⊥, that is the vector obtained by rotating v with an angle of -π/2. In fact, the first-order term 1/2 k s is the area of the triangle of vertices O, Q_ψ and Q_ψ+sv, while the error term is controlled by the area of the triangle of vertices x(s), Q_ψ and Q_ψ+sv. The latter is an o(s^2) as s→ 0, thanks to (<ref>). In particular, letting ψ(s) be the angle such that x(s)= Q_ψ(s), by definition of generalized angles, it holds that ψ(s)-ψ= 2 A(s)= k s + o(s^2), as s → 0. Up to substituting the vector v with -v, we can assume k>0. Then, in order to conclude, it is enough to prove that the function s ↦ C_∘ (ψ(s)) is differentiable in s=0 with positive derivative. First of all, by our choice of k>0, s ↦ C_∘ (ψ(s)) is monotone non-decreasing close to s=0, being a composition of monotone non-decreasing functions. Second of all, we can show that it has a first-order expansion. To this aim, note that the curve s ↦ y(s):=d_x(s)_* is a parametrization of an arc of the sphere S^·_1(0)= ∂Ω (cf. Lemma <ref>). Moreover, recalling that Q_ψ∈ D_* and using the homogeneity of the norm, we have that y(s) = d_Q_ψ+sv_* = d_Q_ψ_* + a s + o (s), as s → 0, where a:= Hess_Q_ψ(_)(v). Observe that a≠ 0 because _ is strongly convex and Q_ψ_=1. Then, call B(s) the (signed) area of the sector of Ω between the rays O y(0) and O y(s) (see Figure <ref>). Reasoning as we did for A(s), from (<ref>) we deduce that B(s) = 1/2y(0)a^⊥ s + o(s^2), as s → 0. On the other hand, by definition C_∘ (ψ(s)) - C_∘(ψ(0)) = 2 B(s)=y(0)a^⊥ s + o(s^2), as s → 0. This shows that the function s ↦ C_∘ (ψ(s)) is differentiable in s=0 with derivative y(0)a^⊥. In addition, since C_∘∘ψ is non-decreasing close to s=0, (<ref>) also implies that y(0)a^⊥≥ 0. We are left to show that y(0)a^⊥ is strictly positive. If y(0)a^⊥=0 then a is parallel to y(0), however, according to (<ref>), the vector a is tangent to the sphere S^·_1(0) at y(0) and therefore we obtain a contradiction. Since the norm is invariant by homotheties, then also D_ is so, thus the set of angles ψ such that Q_ψ∉D_* has null ^1-measure. In particular, the function C_∘ is differentiable with positive derivative ^1-almost everywhere, as a consequence of Proposition <ref>. As already mentioned, the strategy to prove the main theorem of this section is the same of Section <ref>. In particular, it is fundamental to prove estimates on the volume contraction along geodesic homotheties. To this aim, we consider the Jacobian determinant of the exponential map (<ref>): J(ϕ,ω,r;t):= \| ∂ x/∂ r ∂ x/∂ϕ ∂ x/∂ω ∂ y/∂ r ∂ y/∂ϕ ∂ y/∂ω ∂ z/∂ r ∂ z/∂ϕ ∂ z/∂ω (ϕ,ω,r;t) \| where we recall x(ϕ,ω,r;t), y(ϕ,ω,r;t), z(ϕ,ω,r;t) are defined in Proposition <ref>. In order to study this, we will use the following formulation: J(ϕ,ω,r;t) = \| ∂ z/∂ω(ϕ,ω,r;t) (M_1) - ∂ z/∂ϕ(ϕ,ω,r;t) (M_2) + ∂ z/∂ r (ϕ,ω,r;t)(M_3) \| where M_1:= ∂ x/∂ r ∂ x/∂ϕ ∂ y/∂ r ∂ y/∂ϕ(ϕ,ω,r;t), M_2:= ∂ x/∂ r ∂ x/∂ω ∂ y/∂ r ∂ y/∂ω(ϕ,ω,r;t), M_3:= ∂ x/∂ϕ ∂ x/∂ω ∂ y/∂ϕ ∂ y/∂ω(ϕ,ω,r;t). We are particularly interested in studying the behaviour of J(ϕ,ω,r;t) as t→0. In the following lemmas we estimate the behaviour of every term in (<ref>) as t→0. Let I⊂ be an interval containing 0. Given a function f:I → and n∈, we write f(t) ∼ t^n, as t → 0, if there exists a constant C≠ 0 such that f(t)= C t^n + o (t^n), as t→ 0. Let ϕ∈[0,2^∘) be a differentiability point for the map C_∘, r>0 and ω≠ 0, then ( M_1 (ϕ,ω,r;t) ) ∼ t^2, as t → 0, while ( M_2 (ϕ,ω,r;t) ), ( M_3 (ϕ,ω,r;t) ) = O(t^3), as t → 0. Let us begin by proving (<ref>). Firstly, since the function C_∘ is differentiable at ϕ, we can compute the following Taylor expansions as t→0, using Proposition <ref>: ( (ϕ+ω t)_∘) = ( ϕ_∘) - t ω C'_∘ (ϕ) (ϕ) + o(t), ( (ϕ+ω t)_∘) = ( ϕ_∘) + t ω C'_∘ (ϕ) (ϕ) + o(t). Therefore, we may expand the entries of M_1 as t→ 0: ∂ x/∂ r (ϕ,ω,r;t) = 1/ω((ϕ+ω t ) - (ϕ))=(ϕ_∘) t+o(t), ∂ y/∂ r (ϕ,ω,r;t) = -1/ω((ϕ+ω t ) - (ϕ))=(ϕ_∘) t+o(t), ∂ x/∂ϕ (ϕ,ω,r;t) = r/ω( ( (ϕ+ω t )_∘) - ( ϕ_∘) )= -t rC'_∘ (ϕ) (ϕ) + o(t), ∂ y/∂ϕ (ϕ,ω,r;t) = r/ω( ( (ϕ+ω t )_∘) - ( ϕ_∘) ) = tr C'_∘ (ϕ) (ϕ) + o(t) , where we used once again Proposition <ref>. Finally, the determinant has the following Taylor expansion as t→0: (M_1) = ∂ x/∂ r∂ y/∂ϕ - ∂ y/∂ r∂ x/∂ϕ = t^2 r C'_∘ (ϕ) ( (ϕ) ( ϕ_∘) + (ϕ)( ϕ_∘) )+o(t^2) = t^2 r C'_∘ (ϕ)+o(t^2), where, in the last equality, we used Proposition <ref>. This proves (<ref>), keeping in mind Proposition <ref>, which guarantees that C_∘'(ϕ)>0. Now we prove (<ref>) for (M_2), the proof for (M_3) is analogous. As a first step, reasoning as before, we can Taylor expand at second-order the following quantities, as t→0: (ϕ+ω t) = ( ϕ) - ω t (ϕ_∘) - 1/2 (ω t)^2 C_∘'(ϕ)( ϕ) + o (t^2), (ϕ+ω t) = ( ϕ) + ω t (ϕ_∘) - 1/2 (ω t)^2 C_∘'(ϕ)( ϕ) + o (t^2) . Hence, we deduce the expansion for the derivative of x in the ω direction, as t→ 0: ∂ x/∂ω (ϕ,ω,r;t) = -r/ω^2( (ϕ+ω t) - ( ϕ) ) + rt/ω( (ϕ+ ω t)_∘) =-r/ω( t (ϕ_∘) - 1/2ω t^2 C_∘'(ϕ)( ϕ))+rt/ω(( ϕ_∘) - t ω C'_∘ (ϕ) (ϕ))+ o (t^2) =-1/2 r t^2 C_∘'(ϕ)( ϕ)+ o (t^2). An analogous computation shows that the derivative of y in ω has the ensuing expansion as t→ 0: ∂ y/∂ω (ϕ,ω,r;t) =1/2 r t^2 C_∘'(ϕ)( ϕ)+ o (t^2). Note that, on the one hand, (<ref>) and (<ref>) imply that ∂ x/∂ω=O(t^2) and ∂ y/∂ω=O(t^2), as t→ 0. On the other hand, by (<ref>), we can deduce the following behavior, as t→0: ∂ x/∂ r=O(t) and ∂ y/∂ r=O(t). Thus, (<ref>) and (<ref>) prove the claimed behavior of (M_2) as t→ 0, since (M_2) = ∂ x/∂ r∂ y/∂ω- ∂ y/∂ r∂ x/∂ω. In the next lemmas, we study the derivatives of z. These are the most delicate to estimate, since the second-order Taylor polynomial of z is zero and higher-order derivatives may not exist. Let g:→ be a function. We write g(t)= C(1+O(ε)) f(t), ∀ t∈ [-ρ,ρ]. if there exists a constant K>0 and a function f:→ such that, for every ε>0, there exist positive constants C=C(ε),ρ=ρ(ε)>0 for which the following holds C(1-Kε) f(t)<g(t)< C(1+Kε) f(t), ∀ t∈ [-ρ,ρ]. Given ε>0 sufficiently small, for ^1-almost every ϕ∈ [0,2^∘), every r>0 and ω≠ 0, there exist two positive constants k=k(r) and ρ=ρ (ϕ,ω,r) such that ∂ z/∂ω(ϕ,ω,r;t)=(1+O(ε)) k t^3, ∀ t∈ [-ρ,ρ]. First of all, we compute that ∂ z/∂ω(ϕ,ω,r;t) = r^2t/2ω^2 ( 1 - (ϕ)((ϕ+ω t)_∘) - (ϕ) ((ϕ+ω t)_∘) ) - r^2/ω^3 ( ω t + (ϕ)(ϕ+ω t) - (ϕ) (ϕ+ω t) ). In order to evaluate this quantity, fix an angle ψ∈[0,2^∘), for which Proposition <ref> holds, and consider the function f_ψ, defined as s ↦ f_ψ(s) := 1 - (ψ+s)(ψ_∘) - (ψ+s) (ψ_∘). Notice that (<ref>) ensures that f_ψ(0)=0, moreover direct computations show that f'_ψ(0)=0 and f”_ψ(0) = C'_∘ (ψ) >0. Consequently, it holds that f_ψ(s)= C'_∘ (ψ) · s^2 + o(s^2), as s → 0. For every n∈ℤ and m∈ define the set of angles E_n,m := {ψ∈ [0,2^∘) : (1+ε)^n-1 s^2 < f_ψ (s) < (1+ε)^n+1 s^2, for every s∈[-1/m,1/m]}. Observe that, by (<ref>), we have that E:=⋃_n∈ℤ,m∈ E_n,m covers all differentiability points of C_∘, in particular E has full ^1-measure, cf. Remark <ref>. Now, fix ω> 0, r>0 and take ϕ∈ [0,2^∘) to be a density point[We say that r∈ is a density point for a measurable set J⊂ if lim_s→ 0^+^1(J ∩ [r-s,r+s])/2s = 1. ] for the set E_n,m, for some n∈ℤ, m∈. We are going to prove the statement (<ref>) for our choice of parameters and for positive times. The cases with ω<0 and negative times are completely analogous. Let 0<ρ(ϕ,ω,r)< 1/2 ω m be sufficiently small such that for every t∈(0,ρ] ^1(E_n,m∩ [ϕ-2ω t,ϕ+2ω t])> 4 ω t (1-ε/4). Introduce the set F_n,m:= {s ∈ : ϕ+ω s ∈ E_n,m} . Observe that, from (<ref>), we can deduce that for every t∈(0,ρ], ^1(F_n,m∩ [-2 t,2 t])> 4 t (1-ε/4). Now, given every t∈(0,ρ], (<ref>) ensures that there exists s̅∈ [t(1-ε),t] such that s̅∈ F_n,m. Then, thanks to Corollary <ref>, we obtain that 1 - (ϕ) ((ϕ+ω t)_∘)- (ϕ) ((ϕ+ω t)_∘) ≥ 1 - (ϕ)((ϕ+ωs̅)_∘)- (ϕ) ((ϕ+ωs̅)_∘) = f_ϕ+ωs̅(-ωs̅) ≥ (1+ε)^n-1 (ωs̅)^2 ≥ (1-ε)^2 (1+ε)^n-1 (ω t)^2, where the second to last inequality holds by our choice of the parameter ρ and because ϕ+ωs̅∈ E_n,m. With an analogous argument, we can find an element in [t,t(1+ε)]∩ F_n,m and deduce the estimate: 1 - (ϕ)((ϕ+ω t)_∘)- (ϕ) ((ϕ+ω t)_∘) ≤ (1+ε)^2 (1+ε)^n+1 (ω t)^2. Combining (<ref>) and (<ref>), we conclude that, on (0,ρ], the following holds 1 - (ϕ)((ϕ+ω t)_∘)- (ϕ) ((ϕ+ω t)_∘) = (1+ O(ε))(1+ε)^n (ω t)^2, in the Notation <ref>. Consequently, we deduce: r^2t/2ω^2 ( 1 - (ϕ)((ϕ+ω t)_∘) - (ϕ) ((ϕ+ω t)_∘) ) = (1+ O(ε))(1+ε)^nr^2 t^3/2, for t∈ (0,ρ]. To estimate the second term in (<ref>), observe that ∂/∂ s ( ω s + (ϕ) (ϕ+ω s) - (ϕ) (ϕ+ω s) ) = ω ( 1 - (ϕ)((ϕ+ω s)_∘) - (ϕ) ((ϕ+ω s)_∘) ). In particular, since in s=0 this quantity is equal to 0, we have that for t∈ (0,ρ] ω t + (ϕ)(ϕ+ω t) - (ϕ) (ϕ+ω t) =ω∫_0^t ( 1 - (ϕ)((ϕ+ω s)_∘) - (ϕ) ((ϕ+ω s)_∘) ) s = ω∫_0^t (1+ O(ε))(1+ε)^n(ω t)^2 s = (1+ O(ε))(1+ε)^n(ω t)^3 /3, where the second equality follows from (<ref>). Then, we obtain that: r^2/ω^3 ( ω t + (ϕ)(ϕ+ω t) - (ϕ) (ϕ+ω t) )= (1+ O(ε))(1+ε)^nr^2t^3/3. Finally, putting together (<ref>) and (<ref>), we conclude that ∂ z/∂ω(ϕ,ω,r;t) = (1+ O(ε))(1+ε)^nr^2t^3/6, ∀ t∈ (0,ρ], that is (<ref>) with k=k(r) :=(1+ε)^nr^2/6. To conclude, observe that we proved the statement for (every r>0, ω≠ 0 and) every ϕ∈ [0,2^∘) which is a density point of some E_n,m and the set of such angles has full ^1-measure in [0,2^∘). Indeed, E=⋃_n∈ℤ,m∈ E_n,m has full ^1-measure in [0,2^∘) and almost every point of a measurable set is a density point. Let ϕ∈ [0,2^∘) be a differentiability point for the map C_∘, r>0 and ω≠ 0, then ∂ z/∂ r(ϕ,ω,r;t), ∂ z/∂ϕ (ϕ,ω,r;t) = o (t^2), as t→ 0. We start by proving the statement for ∂ z/∂ r. We have that ∂ z/∂ r (ϕ,ω,r;t) = r/ω^2(ω t + (ϕ+ω t ) (ϕ) - (ϕ+ω t ) (ϕ)). Direct computations show that, on the one hand, ∂/∂ t\|_t=0(ω t + (ϕ+ω t ) (ϕ) - (ϕ+ω t ) (ϕ)) = ω - ω(ϕ_∘)(ϕ) - ω(ϕ_∘) (ϕ) =0, where we applied Proposition <ref>, and, on the other hand, ∂^2/∂ t^2\|_t=0(ω t + (ϕ+ω t ) (ϕ)- (ϕ+ω t ) (ϕ)) =- ω^2(ϕ)(ϕ) C'_∘(ϕ) + ω^2 (ϕ)(ϕ) C'_∘(ϕ) =0. Consequently, we conclude the proof of the first part of the statement: ∂ z/∂ r (ϕ,ω,r;t) = r/ω^2· o(t^2) = o(t^2), as t→ 0. In order to prove the statement for ∂ z/∂ω, we use a geometric argument based on Proposition <ref>. First of all, recall that d_Q_ϕ_* identifies a half-plane tangent at Q_ϕ and containing Ω^∘. Thus, we can find a rigid transformation R:^2→^2, such that R(Q_ϕ)= (0,0) and R(Ω^∘) is contained in {y≥0}⊂^2, see Figure <ref>. Then, as _* is C^1,1, the image of the unit sphere R(∂Ω^∘), can be described (locally around O) as the graph of a non-negative function f∈ C^1,1() with f(0)=0. In addition, by our choice of ϕ∈[0,2^∘), f is twice differentiable in 0 with strictly positive second derivative f”(0):=c>0. Now consider the function p defined in a neighborhood of ϕ as p(ψ):= _x ( R(Q_ψ)), where _x:^2→ denotes the projection on the x-axis, i.e. _x(a,b)=a. Second of all, for s_1,s_2∈, call F(s_1,s_2) the signed area between the segment connecting (s_1,f(s_1)) and (s_2,f(s_2)) and the graph of f (intended positive if s_1< s_2 and negative if s_1>s_2), see Figure <ref>. Proposition <ref> ensures that for ψ in a neighborhood of ϕ it holds that z (ψ,ω,r;t) = r^2/ω^2F (p(ψ), p(ψ+ω t)). In particular, we obtain that ω^2/r^2 ∂ z/∂ϕ (ϕ,ω,r;t) = ∂/∂ s_1 F(0, p(ϕ+ω t))· p'(ϕ) + ∂/∂ s_2 F(0,p(ϕ+ω t))· p'(ϕ+ ω t). We now proceed to compute the terms in the last formula, starting from the ones involving p'. To this aim, consider the point (x_0,y_0):= R(O) and, for every q in a neighborhood of 0, call A(q) the signed area inside R(∂Ω^∘) between the segments (x_0,y_0)O and (x_0,y_0)(q,f(q)). Observe that A'(q)= 1/2(1,f'(q))(y_0-f(q), q-x_0) = 1/2 y_0 + O(q), as q→ 0. Note that, in the last equality, we have used that f(0)=f'(0)=0 and f∈ C^1,1(). Consequently, since A(0)=0, we have that A(q) = 1/2 y_0 q + O(q^2), as q→ 0. On the other hand, by the definition of angle it holds that 2A(p(ϕ+ϑ))=ϑ for every ϑ sufficiently small and therefore, invoking (<ref>) and observing that p∈ C^1, we obtain that p(ϕ+ ϑ) = 1/y_0ϑ + o(ϑ) and p'(ϕ+ ϑ) = 1/y_0 + o(1), as ϑ→ 0. Now we compute the partial derivatives of the function F. Observe that F can be calculated in the following way F(s_1,s_2) = 1/2 (f(s_1)+f(s_2)) (s_2-s_1) - ∫_s_1^s_2 f(x) x. As a consequence, we compute that ∂/∂ s_1 F(s_1,s_2) = 1/2 f'(s_1) (s_2-s_1) + 1/2( f(s_1) - f (s_2)) ∂/∂ s_2 F(s_1,s_2) = 1/2 f'(s_2) (s_2-s_1) + 1/2( f(s_1) - f (s_2)). Combining these two relations with (<ref>), we conclude that ω^2/r^2∂ z/∂ϕ (ϕ,ω,r;t) = - 1/2 f (p(ϕ+ω t)) · p'(ϕ), + 1/2 [ f'(p(ϕ+ω t)) p(ϕ+ω t) - f (p(ϕ+ω t))]· p'(ϕ+ ω t). Now, recall that f is twice differentiable in 0 with positive second derivative c, therefore we have that f(x)= 1/2 c x^2 + o(x^2) and f'(x)= c x + o(x). Using these relations, together with (<ref>), we can conclude that ω^2/r^2 ∂ z/∂ϕ (ϕ,ω,r;t) = 1/2 f'(p(ϕ+ω t)) p(ϕ+ω t) p'(ϕ+ ω t)- 1/2 f (p(ϕ+ω t)) [p'(ϕ) + p'(ϕ+ω t)] = 1/2 [c p(ϕ+ω t) + o (p(ϕ+ω t)) ] p(ϕ+ω t) p'(ϕ+ ω t) - 1/4 [c p(ϕ+ω t)^2 + o (p(ϕ+ω t)^2)] [p'(ϕ) + p'(ϕ+ω t)] =c/2 y_0^3 (ω t)^2 - c/2y_0^3 (ω t)^2 + o(t^2)= o(t^2). This concludes the proof. As a consequence of the these lemmas, we obtain the following estimate of the quantity J(ϕ,ω,r;t), as t→ 0. Given ε>0 sufficiently small, for ^1-almost every ϕ∈ [0,2^∘), every r>0 and ω≠ 0, there exist two positive constants C=C(ϕ,ω,r) and ρ=ρ(ϕ,ω,r) such that J(ϕ,ω,r;t) = C(1 + O(ε)) \|t\|^5, ∀ t∈[-ρ,ρ], in the Notation <ref>. Let ϕ∈ [0,2^∘) be a differentiability point for the map C_∘ and such that the conclusion of Lemma <ref> holds, and fix r>0 and ω≠ 0. Observe that, on the one hand, as a consequence of Lemma <ref> and Lemma <ref>, we have that \|∂ z/∂ϕ(M_2) \|(ϕ,ω,r;t), \| ∂ z/∂ r(M_3)\|(ϕ,ω,r;t) = o(t^5), as t → 0. On the other hand, Lemma <ref> and Lemma <ref> ensure that there exist positive constants C=C(ϕ,ω,r),ρ=ρ(ϕ,ω,r)>0 such that \|∂ z/∂ω(M_1) \|(ϕ,ω,r;t)= C(1 + O(ε)) \|t\|^5, ∀ t∈[-ρ,ρ] where, in particular, ρ has to be smaller than the constant identified by Lemma <ref>. Up to taking a smaller ρ and keeping in mind (<ref>), we may conclude that J(ϕ,ω,r;t) = C(1 + O(ε)) \|t\|^5, ∀ t∈[-ρ,ρ]. Note that, in the Heisenberg group, the contraction rate of volumes along geodesic is exactly t^5, cf. <cit.>. In our setting, we are able to highlight the same behavior for the Jacobian determinant of the exponential map J(ϕ,ω,r;t), as t→0. Now that we know the behaviour of J(ϕ,ω,r;t) as t→ 0, in the next proposition, we obtain a statement similar to Proposition <ref>, which will allow us to disprove the (K,N) condition in the Heisenberg group. In particular, the proof of the following proposition uses Corollary <ref> and some ideas developed in <cit.>. In our setting, we define the midpoint map as: (p,q):=e_1/2(γ_pq), if p⋆ q^-1∉{x=y=0}, where γ_pq:[0,1]→ is the unique geodesic joining p and q, given by Theorem <ref>. Similarly, we define the inverse geodesic map I_m (with respect to m∈) as: I_m(q)=p, if there exists x∈ such that (p,q)=m. Recall the definition of midpoint map in (<ref>) and inverse geodesic map in (<ref>). Both maps were defined using the differential structure of a smooth manifold, however they are characterized by the metric structure of the space. In particular, if the norm is sufficently regular, they coincide with (<ref>) and (<ref>). Let be the Heisenberg group, equipped with a strictly convex and C^1,1 norm. For ^1-almost every ϕ∈[0,2^∘), every r>0 and ω≠ 0, there exists a positive constant ρ=ρ(ϕ,ω,r) such that for every t∈[-ρ,ρ]: (i) the inverse geodesic map I_ is well-defined and C^1 in a neighborhood of G(ϕ,ω,r;t); (ii) the midpoint map is well-defined and C^1 in a neighborhood of (,G(ϕ,ω,r;t)), moreover \| d_G(ϕ,ω,r;t)(,·) \| ≤1/2^4. Take ε sufficiently small, let ϕ be an angle for which the conclusion of Corollary <ref> holds. Fix r>0 and ω≠ 0, and let ρ=ρ(ϕ,ω,r) be the (positive) constant identified by Corollary <ref>. Let t∈ [-ρ,ρ] and consider the map E_t:T^_→ defined as E_t(ϕ,ω,r) := G(ϕ,ω,r;t)= (x(ϕ,ω,r;t), y(ϕ,ω,r;t), z(ϕ,ω,r;t)), where G is defined in (<ref>). Note that J(ϕ,ω,r;t) is the Jacobian of E_t(ϕ,ω,r) and in particular, since t∈ [-ρ,ρ], Corollary <ref> ensures that J(ϕ,ω,r;t)>0. Then, from the inverse function theorem, we deduce that E_t is locally invertible in a neighborhood B_t⊂ of E_t(ϕ,ω,r) with C^1 inverse E_t^-1:B_t → T^_. Then, according to Theorem <ref> and Proposition <ref>, the curve [-t,t] ∋ s ↦ G(ϕ,ω,r;s) is the unique geodesic connecting G(ϕ,ω,r;-t) and G(ϕ,ω,r;-t), and such that G(ϕ,ω,r;0)=, provided that ρ is sufficiently small. Hence, we can write the map I_:B_t →^3 as I_(q)= E_-t(E_t^-1(q)), ∀ q∈ B_t. Therefore, the map I_ is C^1 on B_t, being a composition of C^1 functions, proving item (i). With an analogous argument, the midpoint map (with first entry ), _(·):=(,·):B_t →^3, can be written as _(q)= E_t/2(E_t^-1(q)), ∀ q∈ B_t. As before, we deduce this map is well-defined and C^1. To infer regularity of the midpoint map in a neighborhood of (,G(ϕ,ω,r;t)), we take advantage of the underline group structure, in particular of the left-translations (<ref>), which are isometries. Indeed, note that (p,q)= L_p(_( L_p^-1 (q))), ∀ p,q ∈, and, for every (p,q) in a suitable neighborhood of (,G(ϕ,ω,r;t)), we have L_p^-1 (q)∈ B_t, therefore is well-defined and C^1. Finally, keeping in mind (<ref>) and applying Corollary <ref>, we deduce that \| d_G(ϕ,ω,r;t)_(·) \| = \| d_(ϕ,ω,r) E_t/2\| ·\| d_(ϕ,ω,r) E_t \|^-1 = J(ϕ,ω,r;t/2) · J(ϕ,ω,r;t)^-1 = C(1 + O(ε)) \|t/2\|^5/C(1 + O(ε)) \|t\|^5= 1/2^5 (1 + O(ε)) ≤1/2^4, where the last inequality is true for ε sufficiently small. This concludes the proof of item (ii). Let be the Heisenberg group, equipped with a strictly convex and C^1,1 norm and with a smooth measure . Then, the metric measure space (,_SF,) does not satisfy the Brunn–Minkowski inequality (K,N), for every K∈ and N∈ (1,∞). Take an angle ϕ for which the conclusion of Proposition <ref> holds, fix r>0, ω≠ 0 and call γ the curve ∋ s ↦γ(s):= G(ϕ,ω,r;s). Fix t∈ (0,ρ], where ρ=ρ(ϕ,ω,r) is the positive constant identified by Proposition <ref>. Recall the map E_t (see (<ref>)) from the proof of Proposition <ref>. E_t is invertible, with C^1 inverse, in a neighborhood B_t⊂ of E_t(ϕ,ω,r)= γ(t). Consider the function s ↦Φ(s) := _1 [ E_t^-1(L_γ(s)^-1( γ(t+s)))], where _1 denotes the projection onto the first coordinate. Observe that, for s sufficiently small, L_γ(s)^-1( γ(t+s)) ∈ B_t, thus, Φ is well-defined and C^1 (being composition of C^1 functions) in an open interval I⊂ containing 0. Moreover, note that Φ(s) is the initial angle for the geodesic joining and γ(s)^-1⋆γ(t+s). Now, we want to prove that there exists an interval Ĩ⊂ I such that, for ^1-almost every s∈Ĩ, Φ(s) is an angle for which the conclusion of Proposition <ref> holds. We have two cases, either Φ' ≡ 0 in I or there is s̅∈ I such that Φ'(s̅) ≠ 0. In the first case, since by definition Φ(0)=ϕ, we deduce that Φ(s)≡ϕ, thus the claim is true. In the second case, since Φ is C^1, we can find an interval Ĩ⊂ I such that Φ'(s) ≠ 0 for every s∈Ĩ. Then, consider J:={ψ∈Φ(Ĩ): ψ is an angle for which Proposition <ref> holds}⊂Φ(Ĩ) and observe that J has full ^1-measure in Φ(Ĩ). Therefore, the set J̃:=Φ^-1(J)⊂Ĩ has full ^1-measure in Ĩ, it being the image of J through a C^1 function with non-null derivative. Thus the claim is true also in this second case. At this point, let s̅∈Ĩ such that Φ(s̅) is an angle for which the conclusion of Proposition <ref> holds and consider ρ̅:=ρ( E_t^-1(L_γ(s̅)^-1( γ(t+s̅))))>0. For every s ∈ [-ρ̅,ρ̅] ∖{0}, from Proposition <ref>, we deduce that the inverse geodesic map I_ and the midpoint map are well-defined and C^1 in a neighborhood of G(E_t^-1(L_γ(s̅)^-1( γ(t+s̅)));s) and (, G(E_t^-1(L_γ(s̅)^-1( γ(t+s̅)));s)), respectively. Moreover, we have that \| d_G(E_t^-1 (L_γ(s̅)^-1( γ(t+s̅)));s)(,·) \| ≤1/2^4. Observe that, since the left-translations are smooth isometries, the inverse geodesic map I_γ(s̅) is well-defined and C^1 in a neighborhood of γ(s̅ + s), in fact it can be written as I_γ(s̅) (p) = L_γ(s̅)[I_(L_γ(s̅)^-1 (p))], and L_γ(s̅)^-1(γ(s̅ + s))=G(E_t^-1(L_γ(s̅)^-1( γ(t+s̅)));s). Similarly, we can prove that the midpoint map is well-defined and C^1 in a neighborhood of (γ(s̅),γ(s̅ + s)), with \| d_γ(s̅ + s)(γ(s̅),·) \| ≤1/2^4. In conclusion, up to restriction and reparametrization, we can find a geodesic η:[0,1]→ with the property that, for ^1-almost every s̅∈ [0,1], there exists λ(s̅)>0 such that, for every s∈ [s̅ -λ(s̅), s̅ + λ(s̅)]∩ [0,1]∖{s̅}, the inverse geodesic map I_η(s̅) and the midpoint map are well-defined and C^1 in a neighborhood of η (s) and (η(s̅),η(s)) respectively, and in addition \| d_η( s)(η(s̅),·) \| ≤1/2^4. Set λ(s)=0 on the (null) set where this property is not satisfied and consider the set T:={(s,t)∈ [0,1]^2 : t∈ [s-λ(s),s + λ(s)]}. Observe that, introducing for every ϵ>0 the set D_ϵ:= {(s,t)∈ [0,1]^2 : \|t-s\|<ϵ}, we have that ^2(T ∩ D_ϵ)/^2( D_ϵ) =^2(T ∩ D_ϵ)/2ϵ-ϵ^2→ 1, as ϵ→ 0. On the other hand, we can find δ>0 such that the set Λ_δ :={s∈ [0,1] : λ(s)>δ} satisfies ^1(Λ_δ) > 3/4. In particular, for every ϵ<δ sufficiently small we have that ^2({(s,t)∈ [0,1]^2 : s+t/2∉Λ_δ}∩ D_ϵ) < 1/2ϵ. Therefore, putting together (<ref>) and (<ref>), we can find ϵ<δ sufficiently small such that ^2(T ∩ D_ϵ∩{(s,t)∈ [0,1]^2 : s+t/2∈Λ_δ}) > 1/2^2( D_ϵ). Then, since the set D_ϵ is symmetric with respect to the diagonal {s=t}, we can find s̅≠t̅ such that (s̅,t̅),(t̅,s̅) ∈ T ∩ D_ϵ∩{(s,t)∈ [0,1]^2 : s+t/2∈Λ_δ} . In particular, this tells us that: (i) t̅∈ [s̅-λ(s̅),s̅ + λ(s̅)] and s̅∈ [t̅-λ(t̅),t̅ + λ(t̅)]; (ii) \|t̅-s̅\|< ϵ <δ; (iii) s̅+t̅/2∈Λ_δ. Now, on the one hand, (i) ensures that the midpoint map is well-defined and C^1 in a neighborhood of (η(s̅), η(t̅)) with \| d_η( t̅)(η(s̅),·) \| ≤1/2^4 and \| d_η( s̅)(·, η(t̅)) \| ≤1/2^4. While, on the other hand, the combination of (ii) and (iii) guarantees that the inverse geodesic map I_η(s̅+t̅/2) is well-defined and C^1 in a neighborhood of η(s̅) and in a neighborhood of η(t̅) respectively. Indeed, we have: s̅,t̅∈[s̅+t̅/2- δ,s̅+t̅/2+ δ] ⊂[s̅+t̅/2- λ(s̅+t̅/2),s̅+t̅/2+ λ(s̅+t̅/2)], and, by the very definition of λ(·), we obtain the claimed regularity of the inverse geodesic map. Once we have these properties, we can repeat the same strategy used in the second part of the proof of Theorem <ref> and contradict the Brunn–Minkowski inequality (K,N) for every K∈ and every N∈ (1,∞). If we want to replicate the strategy of Theorem <ref>, we ought to find a short geodesic γ:[0,1]→ such that (i) the midpoint map is C^1 around (γ(0),γ(1)) and satisfies a Jacobian estimates at γ(1) of the type (<ref>); (ii) the midpoint map satisfies a Jacobian estimates at γ(0) of the type (<ref>); (iii) the inverse geodesic map _γ(1/2), with respect to γ(1/2), is C^1 around γ(0) and γ(1). Proposition <ref> guarantees the existence of a large set 𝒜⊂ T_γ(0)^* of initial covectors for which the corresponding geodesic γ satisfies (i). The problem arises as the set 𝒜 of “good” covectors depends on the base point and is large only in a measure-theoretic sense. A simple “shortening” argument, mimicking the strategy of the smooth case, is sufficient to address (ii) . However, once the geodesic is fixed, we have no way of ensuring that (iii) is satisfied. In particular, it may happen that the map _γ(1/2) does not fit within the framework of Proposition <ref> item (i), as the corresponding initial covector may fall outside the hypothesis. To overcome such a difficulty, we use a density-type argument to choose simultaneously an initial point and an initial covector in such a way that (i)–(iii) are satisfied. §.§ Failure of the MCP(K,N) condition for singular norms In this section we prove Theorem <ref>, showing that the measure contraction property (see Definition <ref>) can not hold in a Heisenberg group, equipped with a strictly convex, singular norm. Our strategy is based on the observation that, in this setting, geodesics exhibit a branching behavior, despite being unique (at least for small times). Let be the Heisenberg group, equipped with a strictly convex norm which is not C^1, and let be a smooth measure on . Then, the metric measure space (, _SF, ) does not satisfy the measure contraction property (K,N) for every K∈ and N∈ (1,∞). For simplicity, we assume =^3. As it is apparent from the proof, the same argument can be carried out in the general case. According to Proposition <ref>, since is not C^1, its dual norm _* is not strictly convex. In particular, there exists a straight segment contained in the sphere S^·__1(0)=∂Ω^∘. Since the differential structure of the Heisenberg group is invariant under rotations around the z-axis, we can assume without losing generality that this segment is vertical in ^2∩{x>0}, i.e. there exists x̅∈ and an interval I:=[y_0,y_1]⊂ such that {x̅}× I ⊂∂Ω^∘. Moreover, we can take the interval I to be maximal, namely for every y∉I we have (x̅,y)∉Ω (see Figure <ref>). Let ψ_0 ∈ [0,2^∘) be such that Q_ψ_0=(x̅,y_0), then it holds that (x̅, y)= Q_ψ_0 + (y- y_0)x̅, for every y∈ I. As a consequence, we have that (ψ_0 + (y-y_0)x̅)= x̅ and (ψ_0 + (y-y_0)x̅)= y, for y∈ I. Let y_2= 1/2 (y_0+y_1) and ϕ_0= ψ_0 + 1/2(y_1-y_0) x̅, so that (x̅, y_2)= Q_ϕ_0 by (<ref>). Moreover, take ϕ_1>ψ_1:= ψ_0+(y_1-y_0)x̅ sufficiently close to ψ_1 (so that Q_ϕ_1 is not in the flat part of ∂Ω^∘) and call r̅= ϕ_1-ϕ_0>0. We are now going to prove that there exists a suitably small neighborhood 𝒜⊂ T_0^≅ [0,2^∘)×× [0,∞) of the point (ϕ_0,r̅,r̅)[ Here the angle ϕ_0 has to be intended modulo 2^∘.] such that ^3( G(𝒜;1)) >0. For proving this claim, one could argue directly by computing the Jacobian of the map G(·,1) at the point (ϕ_0,r̅,r̅), however the computations are rather involved and do not display the geometrical features of the space. Thus, we instead prefer to present a different strategy, which highlights the interesting behaviour of geodesics. Consider the map F(ϕ,ω,r):= (x(ϕ,ω,r;1),y(ϕ,ω,r;1)), where x(ϕ,ω,r;t),y(ϕ,ω,r;t) are defined as in (<ref>), and observe that F(ϕ_0,r̅,r̅)= ((ϕ_1) - (ϕ_0), (ϕ_0) - (ϕ_1))= ((ϕ_1) - y_2, x̅ -(ϕ_1)). Proceeding with hindsight, let ε>0 such that ε<min{1/2(ϕ_1-ψ_1),1/4 (ψ_1-ϕ_0)} and consider the intervals I_ϕ=[ϕ_0-ε,ϕ_0+ε] and I_r=[r̅-ε,r̅ + ε], then the set F(I_ϕ× I_r× I_r) is a neighborhood of F(ϕ_0,r̅,r̅). Indeed, due to our choice of ϕ_1 the set {F(ϕ_0,r,r) : r ∈ [r̅ - ε/2, r̅ + ε/2] }⊂^2 is a curve that is not parallel to the x-axis. Moreover, for every small δ such that \|δ\| < ψ_1-ϕ_0 = ϕ_0-ψ_0 and every r∈ [r̅ - ε/2, r̅ + ε/2], the equalities in (<ref>) imply the following relation: F(ϕ_0+ δ,r-δ, r-δ) = ((ϕ_0+ r) - (ϕ_0+δ), (ϕ_0+δ) - (ϕ_0+r)) = ((ϕ_0+ r) - (ϕ_0)- δ/x̅, x̅ - (ϕ_0+r) ) = (-δ/x̅,0) + F(ϕ_0, r, r). This shows that F(I_ϕ× I_r× I_r) contains all the sufficiently small horizontal translation of the set in (<ref>) (see Figure <ref>), so it is a neighborhood of F(ϕ_0,r̅,r̅). In particular ^2(F(I_ϕ× I_r× I_r))>0. Now we claim that, for every point (x̃,ỹ,z̃)= G(ψ̃,ω̃,r̃;1) with ψ̃∈ I_ϕ, ω̃∈ I_r and r̃∈ I_r, there exists an interval J_z∋z̃ (depending on x̃ and ỹ) such that {(x̃, ỹ, z) : z∈ J_z}⊂ G([ψ̃- ε, ψ̃+ ε], [ω̃- ε, ω̃+ ε], [r̃ - ε, r̃ + ε];1). This is enough to prove (<ref>), indeed, on the one hand, (<ref>) implies that {(x̃, ỹ, z) : z∈ J_z}⊂ G(I'_ψ× I'_r× I'_r;1), where I'_ψ=[ ϕ_0 - 2ε, ϕ_0 + 2ε], and I'_r=[r̅ - 2ε, r̅ + 2ε]. On the other hand, since (<ref>) holds for every point (x̃, ỹ)∈ F(I_ϕ× I_r× I_r), we deduce that ^3(G(I'_ψ× I'_r× I'_r;1))≥∫_F(I_ϕ× I_r× I_r)^1(J_z(x̃,ỹ))x̃ỹ >0. which implies (<ref>) with 𝒜=I'_ψ× I'_r× I'_r. We proceed to the proof of claim (<ref>): let (x̃,ỹ,z̃)=G(ψ̃,ω̃,r̃;1) with ψ̃∈ I_ϕ, ω̃∈ I_r and r̃∈ I_r and consider the family of parallel lines { l(s)={y=s + kx} : s ∈} in ^2, following the direction identified by the vector (x̃,ỹ), see Figure <ref>. Call S'⊂^2 the sphere ∂Ω^∘ dilated by r̃/w̃ and rotated by -π/2. Then, there exists s̅∈ such that l(s̅) intersects S' in the points r̃/w̃((ψ̃), -(ψ̃)) and r̃/w̃((ψ̃+ r̃), -(ψ̃+r̃)). Let a(s) be the function that associates to s the area inside S' and below l(s) and let d(s) be the function that associates to s the (Euclidean) distance between the two intersections of l(s) with S' (see Figure <ref>). In particular, by our choice of s̅, we have d(s̅)=(x̃, ỹ)_eu and, according to Proposition <ref>, a(s̅)=z̃. Moreover, note that, by Lemma <ref>, the function s ↦a(s)/d(s)^2 is strictly increasing. Now, for every s close enough to s̅, the line l(s) intersects S' in the points r̃/ω̃(( ψ(s)), -(ψ(s))) and r̃/ω̃(( ψ(s)+ r(s)), -( ψ(s)+r(s))), with ψ(s)∈ [ψ̃- ε, ψ̃+ ε] and r(s)∈ [r̃ - ε/2, r̃ + ε/2]. By Proposition <ref> and our choice of parallel lines in (<ref>), we deduce that G(ψ(s), r(s), r(s) (x̃, ỹ)_eu/d(s);1)= (x̃, ỹ, (x̃, ỹ)_eu^2/d(s)^2· a(s)). Observe that, since d is a continuous function and d(s̅)= (x̃, ỹ)_eu, for every s sufficiently close to s̅ we have r(s) ∈ [r̃-ε,r̃+ε]⊂ I'_r and r(s) d(s)/(x̃, ỹ)_eu∈ [r̃ - ε, r̃ + ε] ⊂ I'_r. Then, (<ref>) is sufficient to conclude the existence of an interval J_z⊂ as in (<ref>). This concludes the proof of claim (<ref>) with the choice 𝒜=I'_ψ× I'_r× I'_r. Finally, we are ready to disprove the measure contraction property (K,N), taking as marginals μ_0:=δ_ and μ_1:= 1/^3(G(𝒜;1)) ^3\|_G(𝒜;1). Note that, thanks to our construction of the set 𝒜, the curve t↦ G(λ;t), with λ∈𝒜, is the unique geodesic joining the origin and G(λ;1) (cf. Theorem <ref>). Therefore, according to Remark <ref>, it is enough to contradict (<ref>) with A'=A= G(𝒜;1). In particular, we prove that there exists t_0∈(0,1) such that M_t({},A) ⊂{y=0,z=0}, ∀ t<t_0. To this aim, fix any (ϕ,ω,r)∈𝒜 and note that, for every t<ψ_1-ϕ/ω, (<ref>) implies that (ϕ+ω t ) = x̅ and (ϕ+ω t )= (ϕ) + ω t/x̅. From these relations, it follows immediately that y(ϕ,ω,r;t)=0 and z(ϕ,ω,r;t) = 0, for every t<ψ_1-ϕ/ω. Observe that, by our choice of ε small enough, ψ_1-ϕ/ω is bounded from below by a positive constant uniformly as ϕ∈ I'_ϕ and ω∈ I'_r, thus ensuring the existence of a constant t_0∈ (0,1) for which (<ref>) holds. In the last step of the proof of the preceding theorem, we established the existence of a family of branching geodesics: namely those corresponding to a flat part of ∂Ω^∘. In particular, when is equipped with a strictly convex and singular norm, geodesics can branch, although they are unique. This is remarkable as examples of branching spaces usually occur when geodesics are not unique. Let f:→ be a concave and C^1 function. Assume that there exist α_0<β_0 such that f(α_0)=f(β_0)=0 and f>0 on (α_0,β_0). For every s∈ [0,max f), define α(s)<β(s) such that {y=s}∩ Graph(f)={(α(s),s);(β(s),s)}. Denote by a(s) the area enclosed by the line {y=s} and the graph of f, and by d(s):=β(s)-α(s) (see Figure <ref>). Then, [0,max f)∋ s↦a(s)/d^2(s) is strictly decreasing. Fix 0≤ s_1< s_2 < max f, then it is sufficient to prove that A_1:= a(s_1) > d^2(s_1)/d^2(s_2) a (s_2) =: A_2. Observe that, by definition, a(s)=∫_α(s)^β(s)(f(t)-s) t, therefore: A_1= ∫_α(s_1)^β(s_1)(f(t) - s_1 ) t, A_2=d(s_1)/d(s_2)∫_α(s_1)^β(s_1)[f(α(s_2)+d(s_2)/d(s_1)(t-α(s_1)))-s_2] t, where, for A_2, we used the change of variables t↦α(s_1)+d(s_1)/d(s_2)(t-α(s_2)). For ease of notation, set g to be the integrand of A_2, namely: g(t):=d(s_1)/d(s_2)[f(α(s_2)+d(s_2)/d(s_1)(t-α(s_1)))-s_2], ∀ t∈. Now, let t̃∈ [α(s_1),β(s_1)] be such that t̃ = α(s_2)+d(s_2)/d(s_1)(t̃-α(s_1)). Note that, by linearity, t≤t̃ if and only if t≤α(s_2)+d(s_2)/d(s_1)(t-α(s_1)). Thus, for every t≤t̃, the concavity of f yields that g'(t) = f'(α(s_2)+d(s_2)/d(s_1)(t-α(s_1))) ≤ f'(t). Therefore, observing that g(α(s_1))=0, we deduce that, for every t≤t̃, g(t) = ∫_α(s_1)^t g'(r) r ≤∫_α(s_1)^t f'(r) r = f(t) - s_1. The same inequality can be proved for every t≥t̃, proceeding in a symmetric way. Thus, integrating both sides of (<ref>), we obtain A_1≥ A_2. Finally, observe that if A_1=A_2 then also (<ref>) is an equality, for every t< t̃. By concavity, this implies that f'(t)≡ c_1 for every t< t̃. Analogously, f'(t)≡ c_2 for every t > t̃ and (<ref>) implies that we must have c_1≠ c_2. Thus, f is linear on (α(s_1)),t̃) and (t̃,β(s_1))) and not differentiable at t̃. But, this contradicts that f∈ C^1(), proving claim (<ref>). alpha
http://arxiv.org/abs/2307.03082v1	20230706155027	A two-sample comparison of mean survival times of uncured sub-populations	[ "Dennis Dobler", "Eni Musta" ]	stat.ME	[ "stat.ME", "math.ST", "stat.TH" ]	Mean survival of uncured patients in two samples – Part I Vrije Universiteit Amsterdam The Netherlands e2 University of Amsterdam The Netherlands e1 D. Dobler and E. Musta ^: The authors contributed equally and are given in alphabetical order. Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary measure. In the presence of subjects that are immune to the event of interest (`cured'), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured sub-populations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups and develop permutation tests for inference. In this first out of two connected papers, we focus on nonparametric approaches. The methods are illustrated with medical data of leukemia patients. asymptotic statistics cure models inference random permutation right-censoring § INTRODUCTION In many applications, it is of interest to compare survival probabilities among two different samples, e.g., two treatment arms. One common approach is to test for the equality of the survival functions although it does not provide information on the size of the difference. Alternatively, as a graphical tool, one could plot the difference between the two estimated survival curves together with confidence bands. However, in practice it is preferred to have a summary measure of such difference. This facilitates the understanding and interpretation of study results even though it provides limited information since no single metric can capture the entire profile of the difference between two survival curves. The hazard ratio (HR) is commonly used to quantify this difference under the assumption that the ratio of the two hazard functions remains constant over time; for example, the proportionality of hazard rates is central to the famous semiparametric model by Cox <cit.>. However, such assumption is often not satisfied in practice and the use of the HR would be problematic. One alternative approach is given by the restricted mean survival time (RMST). The difference between restricted mean survival times for different groups has been advocated as a useful summary measure that offers clinically meaningful interpretation <cit.>. The RMST is defined as the the expected lifetime truncated at a clinically relevant time point τ. The restriction to τ is used to accommodate the limited study duration, as a result of which the upper tail of the survival function cannot be estimated, unless one is willing to assume a specific parametric model for extrapolation beyond the range of the observed data. Recently, <cit.> and <cit.> investigated a random permutation method for inference on the difference in restricted mean (net) survival times. While their test is finitely exact under exchangeable data, <cit.> stated for the case of non-exchangeable data that “Further research to develop methods for constructing confidence intervals for RMST difference with a small sample data is warranted. It is quite challenging to construct an exact confidence interval for the difference in RMST.” <cit.> continued in the direction of this remark and analyzed a studentized permutation version of the just-mentioned approach. Their resulting hypothesis test is exact under exchangeability and it even controls the type-I error probability asymptotically under non-exchangeability. Because of this additional feature of exactness under exchangeability, permutation tests also enjoy great popularity in survival analytic applications beyond the RMST: for instance, <cit.> and <cit.> researched permutation-based weighted log-rank tests. In this paper, we will consider the not unusual case that some of the subjects are immune to the event of interest (`cured') instead of the classical survival problem. The challenge arises because, as a result of censoring, the cured subjects (for which the event never takes place) cannot be distinguished from the susceptible ones. Cure rate models, which account for the presence of a cured sub-population have become increasingly popular particularly for analyzing cancer survival and evaluating the curative effect of treatments. More in general, they have found applications in many other domains including fertility studies, credit scoring, demographic and social studies analyzing among other things time until marriage, time until rearrest of released prisoners, time until one starts smoking. For a complete review on cure models methodology and applications, we refer the reader to <cit.>. In presence of immune subjects, comparing survival between two samples becomes more complicated than in the standard setting since one can compare overall survivals, cure chances and survival probabilities for the uncured subjects. Several papers have focused on testing for differences in the cure rates in a nonparametric setting. <cit.> On the other hand, different methods have been proposed to test for equality of the survival functions among the uncured sub-populations. <cit.> However, as in the standard survival analysis setting, just testing for equality of the two distributions would not be sufficient for many practical applications. Thus, apart from comparing the cure probabilities, it would be meaningful to compare relevant statistical summaries for the sub-population of uncured subjects. To this end, we propose to analyze mean survival times of the uncured. This has the advantage of being an easy-to-interpret extension of the RMST in the present context, whereas we will not impose a time restriction. Hence, we will use the abbreviation MST for our purposes. To the best of our knowledge, this has rarely been investigated in the literature so far. One notable exception is the recent paper <cit.> where a semiparametric proportional model for the mean (residual) life of the uncured was proposed and analyzed in a one sample context. Their approach will be further discussed in Part II of the present two companion papers. To illustrate the above-mentioned concepts, Figure <ref> visualizes different survival models with cure fractions. These may be summarized with the following three important numbers: the cure fraction p, i.e., the height of the plateau for late time points; the time point τ_0 when the plateau is reached; the mean survival time of the uncured patients. It is evident that that mean is potentially completely unrelated to the cure fraction. On the other hand, the mean is of course affected by τ_0. In each panel of Figure <ref> except for the top-right one, the two survival curves reach their plateaus at different time points. The top-left panel exhibits the same cure fraction in both populations but clearly different mean survival times (of the uncured). In all other panels, the cure fractions differ. In the top-right and the bottom-left panels, the mean survival times of the uncured populations are equal. This can be seen by affine-linearly transforming the vertical axis such that the thus transformed survival curves cover the whole range from 1 to 0; then, the areas under the transformed curves are the same in each of both just-mentioned panels. Lastly, the bottom-right panel shows two survival curves for which all parameters are different. Measures other than the first moment could obviously also be used to summarize the survival curves of the uncured patients, e.g., the median or other quantiles of these proper survival functions, or other moments. In our opinion, however, the mean offers the easiest interpretation: how much (in absolute numbers) of the wholly available area of τ_0 is below the survival curve? The more, the better. This offers another means of comparing the usefulness of two (or more) treatments: which treatment prolongs the mean survival times most effectively, next to comparing the cure fractions? Let us now discuss the agenda for the present paper and the benefits of the proposed procedure: We will propose a method for comparing the lifetimes of the uncured subject via mean survival times. * This will allow for comparisons in two sample problems. * Restrictions of time will not be necessary. * Only weak assumption, e.g., for the sake of identifiability will be made. * In particular, hazard rates are generally allowed to be discontinuous. * Inference will be based on the random permutation method which gives rise to finitely exact hypothesis tests in the case of exchangeable samples and, otherwise, good small sample properties. In this Part I of our two companion papers, we will consider nonparametric models with a constant cure rate. In Part II, we will extend the method to a semiparametric mixture cure model that allows for expressions of mean survival times conditionally on covariates. For the latter, we will assume a logistic-Cox model since it is the most widely used in practice. This article is organized as follows. The model and notation are introduced in Section <ref>. In subsections therein, we also offer a toy example for a discussion about comparisons of survival in the two-sample problem and the formal definition of mean survival times of the uncured. In Section <ref>, we propose a Kaplan-Meier estimator-based estimation procedure, introduce the random permutation scheme, and present large sample properties which are crucial for inference. Section <ref> contains a description of an extensive simulation study as well as the numerical results. Data from a study on leukemia are illustrated and analyzed in the light of the proposed method in Section <ref>. We conclude this Part I with a discussion in Section <ref> where we also offer an interim conclusion in preparation of Part II. All proofs are contained in Appendix <ref>. The R code with an implementation of our methods is available in the GitHub repository <https://github.com/eni-musta/MST_uncured>. § MODEL AND NOTATION We consider i.i.d. survival times T_11, …, T_1n_1 and T_21, …, T_2n_2 from two independent groups (i=1,2) that consist of a mixture of cured and uncured individuals, meaning that a fraction of the study population in each group (the cured ones) would not experience the event of interest. We denote the event time of the cured individuals by ∞ and assume that, for the uncured ones, the event can happen on the interval [0,τ_0,i], i=1,2, respectively for each group. We do not assume the cure threshold τ_0,i<∞ to be known in advance but depending on the application at hand one might have some information about it, for example in oncology based on the medical knowledge τ_0,i is expected to be somewhere between 5 or 10 years depending on the cancer type. For the remainder of this paper, we denote by (Ω,𝒜,ℙ) the underlying probability space, 𝔼 denotes expectation, d→ denotes convergence in distribution, and d= denotes equality in distribution. Let us denote by F_i and S_i=1-F_i, i=1,2, the improper cumulative distribution function and the improper survival function of the time-to-event variables in the two treatment groups. Let F_u,i and S_u,i=1-F_u,i be the proper cumulative distribution function and survival function for the uncured individuals in the two treatment groups. Let p_i=(T_i1>τ_0,i)=S_i(τ_0,i)∈(0,1) denote the cure fractions in both groups. In particular, we have F_i(t)=(1-p_i)F_u,i(t), S_i(t)=p_i+(1-p_i)S_u,i(t) In the presence of right censoring, instead of the survival times, we observe the follow-up times Y_i1,…,Y_in_i and the censoring indicators Δ_i1,…,Δ_in_i, where Y_ij=min{T_ij,C_ij}, Δ_ij=_{T_ij≤ C_ij} and C_ij are the censoring times. We assume that censoring is independent of the survival times and has bounded support [0,τ_i] in each group. In particular, because of the finite censoring times, all the cured individuals will be observed as censored. In order to be able to identify the cure fraction, we need τ_0,i≤τ_i and F_i continuous at τ_i in case τ_0,i=τ_i, which is known as the sufficient follow-up assumption <cit.>. The idea is that, since the cure status is not observed and F_u,i(·) is left unspecified, if τ_i<τ_0,i then the events {T_i1∈(τ_i,τ_0,i]} and {T_i1=∞} would be indistinguishable. As a result, the cure rate could not be identified. A statistical test for this assumption is proposed in <cit.> but its practical behavior is not very satisfactory given the unstable behaviour of the Kaplan Meier estimator in the tail region. In practice, a long plateau of the Kaplan-Meier estimator, containing many censored observations, is considered to be an indication of sufficient follow-up. §.§ Comparison of overall survival We first illustrate why comparing overall survival functions is not appropriate in the presence of a cure fraction. The difference in overall survival combines together the difference in cure fractions and in the survival times of the uncured in a way that it is difficult to interpret. For example, if group one has a higher cure fraction but lower survival times for the uncured, the overall survival functions might cross and the difference between them would be a weighted combination of the two effects S_1(t)-S_2(t)=(p_1-p_2){1-S_u,2(t)}+(1-p_1){S_u,1(t)-S_u,2(t)}. On the other hand, if the two groups have the same cure fraction p, then S_1(t)-S_2(t)=(1-p){S_u,1(t)-S_u,2(t)} This means that, particularly for a large cure fraction, the observed difference in overall survival functions is much smaller than the actual difference of the survival functions for the uncured. Using a one number summary of the difference in overall survival is even more problematic in the presence of a cure fraction. First, the proportional hazard assumption is clearly violated on the level of the whole population and, as a result, the hazard ratio cannot be used. For the Mann-Whitney effect, what counts are the chances of having longer survival times for one group compared to the other, but the actual difference between these times does not matter. So one cannot distinguish between having a larger cure fraction or just slightly longer survival. Consider for example the following hypothetical scenario: patients receiving treatment A have a 20% chance of being cured while with treatment B there is no cure chance; a random person receiving treatment B lives several months longer compared to an uncured patient who received treatment A. Let T_1 and T_2 represent the random lifetimes of patients receiving treatments A and B, respectively. According to the above description, the Mann-Whitney effect would then be (T_1>T_2)+1/2(T_1=T_2)=0.2<0.5, leading to the conclusion that treatment B should be preferred. This is counter-intuitive because, given the small difference in survival times of the uncured, in practice one would probably prefer the treatment that offers some chance of getting cured. On the other hand, if one would use the restricted mean survival time as a summary measure, the actual survival times matter. However, because of the restriction to a specified point τ (duration of the study), there would still be no distinction between the cured individuals and those who survive more than τ RMST^(i)_τ=[min(T_i,τ)]=(1-p_i)[min(T_i,τ)\| T_i < τ]+τ p_i, i=1,2. To illustrate this, consider the following example: patients receiving treatment A have 20% chance of being cured, while with treatment B there is no cure chance; a random person receiving treatment B lives on average 60 months, while an uncured patient who received treatment A lives on average 24 months. Let us assume that τ_0,1=τ_0,2=120 months. If the duration of the study was also 120 months (sufficient follow-up), we would obtain RMST^(1)_120-RMST^(2)_120=0.8·[T_1\|T_1 < 120]+120·0.2-[T_2]=-16.8, leading to the conclusion that treatment B should be preferred. However, if the study had continued for longer, e.g., 240 months, we would obtain RMST^(1)_240-RMST^(2)_240=0.8·[T_1\|T_1<240]+240·0.2-[T_2]=7.2, suggesting that treatment A is better, which contradicts the previous conclusion. For these reasons, we think that in the presence of a cure fraction, it is more informative to compare separately the cure fractions and the survival functions of the uncured. In practice, one can then make a personalized decision by choosing to put more weight to one component compared to the other, based on the life expectancy if uncured and the risks one is willing to take. For example, for children there is an essential difference between cure and 10 year survival, while such difference might be less significant for elderly patients. §.§ Mean survival time of the uncured The problem of comparing cure fractions has already been considered in the literature. Here, we focus on comparing the survival times of the uncured. In particular, we propose the mean survival time as a summary measure. We are interested in the difference of mean survival times of the uncured individuals among the two groups MST_u,1-MST_u,2=[T_11\| T_11 < ∞]-[T_21\| T_21 < ∞]. In combination with the cure fractions, such mean survival times provide useful summaries of the improper survival curves. Using the relations in (<ref>), we obtain the following expression for the mean survival times MST_u,i= ∫_0^τ_0,iS_u,i(u) u=∫_0^τ_0,iS_i(u)-p_i/1-p_i u, i=1,2. § ESTIMATION, ASYMPTOTICS, AND RANDOM PERMUTATION §.§ Estimators and their large sample properties Estimation of the cure rate and of the nonparametric survival function in this setting has been considered in <cit.> and is based on the Kaplan-Meier (KM) estimator. In particular, we can estimate S_i and p_i by Ŝ_i(t)=∏_u∈(0,t](1- N_i(u)/R_i(u)) and p̂_i=Ŝ_i(Y_i,(m_i)), respectively, where Y_i,(m_i) is the largest observed event time in group i, N_i(u)=∑_j=1^n_i_{Y_ij≤ u,Δ_ij=1} counts the number of observed events up to time u and R_i(u)=∑_j=1^n_i_{Y_ij≥ u} counts the numbers of individuals at risk at time u. By a plug-in method, we estimate the mean survival time of the uncured by MST_u,i=∫_0^Y_i,(m_i)Ŝ_i(u)-p̂_i/1-p̂_i u Let m̂= MST_u,1- MST_u,2 be an estimator of m=MST_u,1-MST_u,2. Using the asymptotic properties of the KM estimator, we obtain the following result for the mean survival times, for which we define v_i(t)=∫_0^t F_i(u)/S_i(u){1-H_i(u-)}, H_i(t)=(Y_i1≤ t)={1-F_i(t)}{1-G_i(t)}, and G_i(t)=(C_i1≤ t) denotes the distribution of the censoring times. For i=1,2, assume p_i∈(0,1) and that one of the following conditions holds: a) τ_0,i<τ_i b) τ_0,i=τ_i, F_i is continuous and ∫_0^τ_i F_i(t)/1-G_i(t-)<∞; c) τ_0,i=τ_i, F_i is continuous at τ_i, lim_t↑τ_i{F_i(τ_i)-F_i(t)}^2v_i(t)=0, and lim_t↑τ_i∫_t^τ_i_{0≤ G_i(u-)<1}S_i(u)/{1-G_i(u-)}S_i(u-) F_i(u)=0. Then the variable √(n_i)(MST_u,i-MST_u,i) is asymptotically normally distributed, i.e. √(n_i)(MST_u,i-MST_u,i)N∼𝒩(0,σ_i^2) as n→∞. The limit variance σ_i^2 is defined in (<ref>) in the appendix. The extra technical conditions in b) and c) of the previous theorem are the conditions needed to obtain weak convergence of the normalized Kaplan-Meier estimator to a Gaussian process <cit.>. Note that in the particular case τ_0,i < τ_i, the conditions unrelated to the continuity of F_i are automatically satisfied, leading to no extra requirements for case a). Continuity of F_i is nowhere needed in case a). From the independence assumption between the two groups and Theorem <ref>, we obtain the following result for which we define a_n=√(n_1n_2/(n_1+n_2)). Assume that n_1/(n_1+n_2)→κ∈(0,1) as min(n_1, n_2) →∞. Under any of the two conditions in Theorem 1, we have that a_n(m̂-m) is asymptotically normally distributed with mean zero and variance σ^2=(1-κ)σ_1^2+κσ_2^2. The canonical plug-in estimator of σ^2, say σ̂^2 = n_2/n_1 + n_2σ̂_1^2 + n_1/n_1+n_2σ̂_2^2 is obviously consistent. Hence, the combination of Theorem <ref> and Corollary <ref> could be used to justify inference methods for MST_u,1 - MST_u,2 based on the asymptotic normal approximation. However, such inference procedures can usually be made more reliable by means of resampling methods. §.§ Inference via random permutation across samples We propose random permutation to construct inference methods for m. To introduce this procedure, let π=(π_1, …, π_n_1+n_2) be any permutation of (1,2, …, n_1+n_2). When applied to the pooled sample, say, (Y_1, Δ_1), …, (Y_n_1+n_2, Δ_n_1+n_2), this permutation leads to the permuted samples (Y_π_1, Δ_π_1), …, (Y_π_n_1, Δ_π_n_1) and (Y_π_n_1+1, Δ_π_n_1+1), …, (Y_π_n_1+n_2, Δ_π_n_1+n_2). In the special case of exchangeability, i.e. (Y_1j, Δ_1j) d= (Y_2j, Δ_2j), m̂ = MST_u,1 - MST_u,2 would have the same distribution as m̂^π = MST_u,1^π - MST_u,2^π. Here, MST_u,i^π are the estimators of the mean survival times, just based on the i-th permuted sample. So, under a sharp null hypothesis of exchangeability, a test for the equality of mean survival times would reject the null hypothesis if m̂ belongs to the α× 100% most extreme values of m̂^π across all (n_1+n_2)! permutations. However, under the weak null hypothesis of equal mean survival times, H: m=0, the samples are in general not exchangeable. As a consequence, the asymptotic variances of m̂ and m̂^π cannot be assumed equal and hence they must be studentized. We will thus focus on m̂^π/σ̂^π as the permutation version of m̂/σ̂, where σ̂^π2 = n_2/n_1 + n_2σ̂_1^π2 + n_1/n_1+n_2σ̂_2^π2, and σ_i^π2 is the plug-in variance estimator based on the i-th permuted sample. Consequently, our aim is to compare m̂/σ̂ to the conditional distribution of m̂^π/σ̂^π given the data to reach a test conclusion. Because it is computationally infeasible to realize m̂^π/σ̂^π for all (n_1+n_2)! permutations, we will realize a relatively large number B of random permutations π and approximate the conditional distribution by the collection of the realized m̂^π/σ̂^π of size B. In the following, we will discuss the asymptotic behaviour of m̂^π/σ̂^π to justify the validity of the resulting inference procedures. From now on, we understand the weak convergence of conditional distributions (in probability) as the convergence of these distributions to another with respect to e.g. the bounded Lipschitz metric (in probability); see e.g. Theorem 1.12.4 in <cit.>. Assume that τ_0,i<τ_i and p_i∈(0,1) for i=1,2. Then, as n_1, n_2→∞ with n_1/(n_1+n_2) →κ∈ (0,1), the conditional distribution of a_nm̂^π given the data converges weakly in probability to the zero-mean normal distribution with variance given in (<ref>) in the appendix. Under continuity assumptions similar to those in Theorem <ref> and additional assumptions on the censoring distributions, we conjecture that a similar weak convergence result holds for the case of τ_0,i=τ_i. This could potentially be shown by extending the results of <cit.> to the random permutation method instead of the classical bootstrap. Deriving such results, however, is beyond the scope of the present paper. The structure of asymptotic variance in the previous theorem motivates a canonical permutation-type variance estimator σ̂^π2, that is, the plug-in estimator based on the pooled sample. Due to the obvious consistency of this estimator, we arrive at the following main result on the permuted studentized mean survival time: Assume that τ_0,i<τ_i and p_i∈(0,1) and F_i is continuous for i=1,2. Then, as n_1, n_2→∞ with 0 < lim inf n_1/(n_1+n_2) ≤lim sup n_1/(n_1+n_2) < 1, the (conditional) distributions of a_n m̂^π/σ̂^π (given the data) and a_n (m̂ - m)/σ̂ converge weakly (in probability) to the same limit distribution which is standard normal. We conclude this section with a remark on the inference procedures deduced from the random permutation approach. Corollary <ref> gives rise to asymptotically exact 1- and 2-sided tests for the null hypotheses H_0^(1): m ≤ 0, H_0^(2): m ≥ 0, or H_0^(3): m≠ 0 against the respective complementary alternative hypotheses: comparing m̂/σ̂ with data-dependent critical value(s) obtained from the collection of realized m̂^π/σ̂^π (for fixed data) allows for controlling the chosen significance level α∈ (0,1) as n_1, n_2 →∞ under the assumptions made above. A similar remark holds true for more general null hypotheses in which m is compared to some hypothetical value m_0 ∈. In addition, as a well-known property of permutation tests based on studentized test statistics, the just-mentioned tests are exact in the special case of exchangeability between both sample groups for which m_0=0 is automatically fulfilled. Similarly, by inverting hypothesis tests into confidence intervals, asymptotically exact confidence intervals for m can be constructed; see the subsequent section for details. § SIMULATION STUDY In this section, we study the finite sample performance of both the asymptotic and the permutation approach when constructing confidence intervals for m and testing the one sided hypothesis H_0:m≤ 0 versus H_1: m>0. In order to cover a wide range of scenarios, we consider nine settings as described below. Settings 1-4 correspond to having two samples with the same mean survival time for the uncured (m=0) but possibly different distributions, while settings 5-9 correspond to having two samples with different mean survival times for the uncured (m≠ 0) with different magnitudes and signs for m. The cure and censoring rates also vary across the settings. In all of the following settings, the survival times for the uncured are truncated at τ_0,i equal to the 99% quantile of their distribution in order to satisfy the assumption of compact support. The censoring times are generated independently from an exponential distribution with parameter λ_C,i and are truncated at τ_i=τ_0,i+2. The truncation of the censoring times is done only to reflect the bounded follow-up but τ_i does not play any role apart from the fact that τ_i>τ_0,i. Note also that the reported censoring rate includes the cured subjects, which are always observed as censored, hence it is larger than the cure rate. §.§ Simulation settings Setting 1. The two samples are exchangeable (m=0): cure rate 40%, Weibull distribution for the uncured with shape and scale parameters 0.75 and 1.5 respectively, censoring rate around 50% (λ_C,i=0.3, i=1,2). Setting 2. The uncured have the same Weibull distribution in both samples (m=0) with shape and scale parameters 0.75 and 1.5 respectively. The cure rate is 20% in sample 1 and 60% in sample 2, the censoring rate is around 30% and 70% in sample 1 and 2 respectively (λ_C,1=0.25, λ_C,2=0.5). Setting 3. The uncured have the same Weibull distribution in both samples (m=0) with shape and scale parameters 0.75 and 1.5 respectively. The cure rate is samples 1 and 2 is 60% and 20% respectively; censoring rate is around 65% and 25% respectively (λ_C,1=0.3, λ_C,2=0.1). Setting 4. The uncured in sample 1 follow a Weibull distribution with shape and scale parameters 0.75 and 1 respectively, while the uncured in sample 2 follow a Gompertz distribution with scale parameter 1 and shape parameter 0.327. The parameters are chosen such that the two groups have the same mean survival time (so m=0). The cure rate is 40% in both samples; censoring rate is around 50% in both samples (λ_C,1=0.2, λ_C,2=0.15). Setting 5. The uncured in the two samples have different Gompertz distributions with the same scale parameter 1 and shape parameters 0.1 and 0.5 respectively. The difference of mean survival times is m=1.09. For this choice of parameters, the supports of the event times in the two samples are [0,3.8] and [0,2.3] respectively. The cure rate is 40% in both samples; censoring rate is around 65% in sample 1 and 45% in sample 2 (λ_C,1=0.3, λ_C,2=0.1). Setting 6. This is the same as Setting 5 but the two groups are exchanged, i.e. m=-1.09. Setting 7. The uncured in the two samples have different Gompertz distributions with the same scale parameter 1 and shape parameters 0.08 and 0.1 respectively. The difference of mean survival times is m=0.18 For this choice of parameters, the supports of the event times in the two samples are [0,4.1] and [0,3.9] respectively. The cure rate is 60% in sample 1 and 20% in sample 2; censoring rate is around 70% in sample 1 and 40% in sample 2 (λ_C,1=0.2, λ_C,2=0.15). Setting 8. The survival distributions of the uncured are as in Setting 7, i.e. m=0.18. The cure rate is samples 1 and 2 is 30% and 20% respectively; censoring rate is around 44% in sample 1 and 34% in sample 2 (λ_C,1=0.1, λ_C,2=0.1). Setting 9. The event times of the uncured in the sample 1 follow a Gompertz distribution with scale parameter 1 and shape parameter 0.08, while in sample 2 they follow a Weibull distribution with shape ans scale parameters 2 and 0.28 respectively. Both distributions have support [0,4.1] but different mean survival times (m=0.52). The cure rate is 40% in both samples; censoring rate is around 50% in both samples (λ_C,1=0.1, λ_C,2=0.1). §.§ Simulation results First, considering different sample sizes n_1=2n_2∈{50,200} or n_1=n_2∈{100,200}, 95% confidence intervals for m are constructed based on both the asymptotic approximation and the permutation approach: I_n=[m̂∓ q_1-α/2σ̂/a_n], I^π_n=[m̂- q^π_1-α/2σ̂,m̂- q^π_α/2σ̂], where σ̂ is given in (<ref>), q_1-α denotes the 100(1-α)%-quantile of the standard normal distribution, α=0.05 and q^π_1-α denotes the 100(1-α)%-quantile of the conditional distribution of m̂^π/σ̂^π for the permutation approach. We take B=500 random permutations, which seemed to be sufficient since increasing B to 1000 did not have much effect in the results. Average length and coverage probabilities over 1,000 repetitions are reported in Table <ref>. The coverage rates closest to 95% among both types of confidence intervals is printed in bold-type. We observe that the confidence intervals based on the permutation approach are in general slightly wider and have better coverage, particularly for small sample sizes. As the sample sizes increase, the two approaches give more comparable results. For some settings, much larger sample sizes are needed to have coverage close to the nominal level but, for most of them, coverage is close to 95%. When the sample sizes are the same, settings 2 and 3 are almost the same, with setting 3 having less censoring, leading to shorter confidence intervals and better coverage. When n_1=2n_2, setting 2 is more difficult because the smaller sample has a very large cure and censoring rate, leading to worse coverage probabilities. Similarly, when the sample sizes are the same, setting 5 and 6 are the same, leading to same length confidence intervals and approximately same coverage (due to sampling variation). When n_1=2n_2, setting 6 is more difficult because has higher censoring rate in the smaller sample. As a result, we observe longer confidence intervals and worse coverage probabilities. Setting 8 is similar to setting 7 but the first sample has lower cure rate, leading to shorter confidence intervals. When the two samples are not comparable in terms of cure and censoring rate, increasing the sample size of the sample in which it is easier to estimate MST_u,i, does not usually lead to better coverage (compare settings 2 and 3, 5 and 6). On the other hand, increasing the sample size of the sample in which estimation of MST_u,i is more difficult usually leads to better coverage. We further investigate settings 2, 3, 4 which exhibit the worst performance in terms of coverage probabilities. In setting 2, as estimation in the second sample is more difficult (because of higher cure and censoring rates), the performance of both the asymptotic and permutation approach in worse when the size of sample 1 is larger than the size of sample 2. In setting 3, estimation of the first sample is more challenging and we observed that the coverage is better when the size of sample 1 is larger. In setting 4, both samples have the same censoring and cure rate but the coverage seems to be worse when the sample sizes are the same. Results for larger sample sizes under the most difficult scenarios for each of these three settings are reported in Table <ref>. They show that, as expected, the coverage probabilities for both approaches converge to the nominal level. In addition, we selected 2 of the settings (setting 2 and 9) and further investigated the effect of the censoring and cure rates for sample sizes 100-100 and 200-100. First, we keep the cure rate fixed at 40% (moderate) and consider 3 censoring levels: 45% (low), 50% (moderate) and 60% (high). Secondly, we vary the cure rate: 20% (low), 40% (moderate) and 60% (high), while maintaining the same moderate censoring level equal to the cure rate plus 10%. Average length and coverage probabilities over 1,000 repetitions are reported in Table <ref>. As expected, we observe that, as the censoring or cure rate increases, the length of the confidence intervals increases. In setting 4 the coverage deteriorates significantly for a high censoring rate, while in setting 9 the coverage remains stable and close to the nominal value throughout all scenarios. Next, we consider a one-sided hypothesis test for H_0: m≤ 0 versus H_1: m> 0 at level 5%. The rejection rates for the test are reported in Table <ref>. Looking at settings 1-4 and 6 for which H_0 is true (with m=0 for settings 1-4), we observe that most of the time the rejection rate is lower or close to 5% for both methods. Setting 2 is again the most problematic one with rejection rate higher than the level of the test. This might be because in setting 2 the second sample has a high cure (and censoring) rate, which might lead to underestimation of the mean survival times for the uncured in sample 2 and as a result an overestimation of m. For settings 2, 3, and 4 we also considered larger sample sizes. The results are reported in Table <ref>. In particular, we observe that the rejection rates in setting 2 decrease and approaches the significance level as the sample size increases. In terms of power, as m or the sample size increase, the power increases. Both methods are comparable, with the permutation approach usually leading to slightly lower rejection rate under both hypothesis. Again, in settings 4 and 9 we also investigate the effect of the censoring and cure rate as above. Results are given in Table <ref>. As the censoring or cure rate increases, the power of the test decreases, while the rejection rate in setting 4 (H_0 is true) remains below the 5% level throughout all scenarios. Finally, to acknowledge that the case of unbalanced sample sizes where the smaller sample meets the higher censoring rate, we would like to point to <cit.>. For very small sample sizes, their studentized permutation test about the RMST also exhibited the worst control of the type-I error rate in this challenging context; see Table 1 therein, and also Tables S.1 and S.2 in the supplementary material accompanying that paper. Of note, their proposed permutation test is still quite accurate with a size not exceeding 6.8% even in the most challenging setting. § APPLICATION In this section, we apply the developed methods to a real data set from research on leukemia <cit.>; the study ran from March 1982 to May 1987. 91 patients were treated with high-dose chemoradiotherapy, followed by a bone marrow transplant. n_1=46 patients received allogeneic marrow from a matched donor and n_2=45 patients without a matched donor received autologous marrow, i.e. their own. They were followed for 1.4 to 5 years. For other details such as additional patient characteristics and the frequency of the graft-versus-host disease among allogeneically transplanted patients we refer to the original study <cit.>. In our analysis, we are going to re-analyze relapse-free survival. The data sets are available in the monograph <cit.>. They contain the (potentially right-censored) times to relapse or death (in days), together with the censoring status. It was argued in both <cit.> and <cit.> that a cure model is appropriate for the data. The authors of the just-mentioned book first fit parametric accelerated failure time mixture cure models (see Section 2.6) and did not find a significant difference in either cure rates or survival times for the uncured. Additionally, in Section 3.6 of <cit.> they fit semiparametric logistic-Cox and logistic-AFT models. Under the logistic-Cox model they did find a significant effect for the survival times of the uncured between both treatment groups. In particular, they concluded that the Autologous group has significantly higher hazard than the Allogeneic group with an estimated hazard ratio of 1.88, p-value 0.04 and 95% confidence interval (1.03,3.45). On the other hand, with the semiparametric logistic-AFT model, no statistically significant difference is detected between the two groups. Let us briefly summarize the data sets. The censoring percentages amounted to 28% and 20% in the allogeneic and autologous groups, respectively, i.e. 13 and 9 patients in absolute numbers. The percentages of data points in plateaus were 15% and 16% and the estimated cure fractions p̂_i = Ŝ_i(τ_0,i) were 26% and 19%, respectively. The test of <cit.> for the equality of cure fractions resulted in a non-significant p-value of 0.453. Permutation with 500 iterations as. CI: [3,255] perm. CI: [3,260] as. p-value: 0.045 perm. p-value: 0.058 Figure <ref> shows an illustration of the Kaplan-Meier curves. It shows that the curves are crossing and very close to each other in during the first weeks. After that, the curve for the autologous group clearly stays below the one for the allogeneic group. However, that discrepancy melts down as time progresses, hence the non-significant p-value for the equality of cure fractions. On the other hand, the difference in estimated mean survival times for the uncured patients amounts to m=129. The 95% confidence intervals for those mean survival differences were [3, 255] (asymptotic) and [1, 255] (permutation). The two-sided hypothesis tests for equal mean survival times of the uncured resulted in the p-values 0.045 (asymptotic) and 0.046 (permutation), i.e. just significant at the significance level α=5%. The random permutation-based inference methods have been run with 5,000 iterations. The asymptotic and the permutation-based inference methods thus agreed on their outcomes, despite the rather small sample sizes. In view of the rather low censoring rates and the simulation results for moderate censoring settings presented in the previous section, we deem all applied inference methods reliable. § DISCUSSION AND INTERIM CONCLUSION In this article, we considered a two-sample comparison of survival data in the presence of a cure fraction. In such situations, instead of just looking at the overall survival function, it is more informative to compare the cured fractions and the survival of the uncured sub-populations. We propose the use of the mean survival time as a summary measure of the survival curve for the uncured subjects since it is a model-free parameter and easy to interpret. We introduced a nonparametric estimator of the mean survival time for the uncured and developed both an asymptotic and a permutation-based method for inference on the difference between the MST_u. Based on our simulation results, both methods were quite reliable, with the permutation approach being recommended particularly for small sample sizes. However, more caution is required when applying the methods in the presence of a high censoring or cure rate, for which larger sample sizes are needed to obtain reliable results. The MST_u is useful in assessing whether there is a difference in the survival times of the uncured among the two groups. However, we would like to point out that, even in the context of a randomized clinical trial, MST_u does not necessarily have a causal interpretation as a direct effect of the treatment on survival of the uncured. This is because of the conditioning on being uncured; the uncured sub-populations between the two treatment arms might, in general, fail to be comparable. For example, it might be that treatment A is more beneficial in curing patients compared to treatment B and those who do not get cured with treatment A are the patients with worse condition. As a result, the survival of the uncured for treatment A might be worse compared to treatment B, but that does not mean that treatment A shortens the survival of the uncured. However, the fact the MST_u does not have an interpretation as a direct causal effect on the uncured is not a problem when the goal is to choose which treatment should be preferred. Randomized clinical trial data allow us to understand the causal effect of the treatment on the joint distribution of _T=∞ and T_T<∞. One can then in practice define a utility function, for example w_11_T=∞+w_2T1_T<∞ for certain weights w_1,w_2 that represent whether the curative effect or the life prolonging one is more important. Maximizing the expected utility reduces to choosing the treatment that maximizes w_1(T=∞)+w_2(T<∞)MST_u. On the other hand, if one is interested only in the causal effect of the treatment on the uncured subpopulation, determining the relevant quantity is not straightforward and extra caution is required since we are conditioning on a post-treatment variable, which cannot be directly intervened upon. If all the possible variables that can affect both the cure status and the survival time of the uncured were observed, one could condition on those to estimate the effect of treatment on the uncured but this is a quite unrealistic scenario. However, if the treatment does not have any effect on the cure fraction, conditioning on being uncured is not problematic and we can still give a causal interpretation to the effect on the MST_u. I have thought again about the previous statement. I believe, there could still be causal problems even in the case of equal cure rates. Because, even for a RCT, it could still be that under Treatment A some other subpopulation is cured than under Treatment B. Should we perhaps simply delete the previous sentence in blue and then also the following blue sentence? (Because I feel that we might be vulnerable in this point when we go under review.) This is probably the most interesting case in practice since one would usually give priority to treatments that cure more patients and, only when cure fractions are comparable, one would be more interested in comparing the effect of treatment on the uncured. We would also like to point out that, instead of comparing the MST_u's over some time horizon [0, τ_0], trivial adjustments of the methods and proofs would also allow for comparing the mean residual survival times. Similarly as in <cit.>, these are defined as MRST_u,t = [T - t \| t < T < ∞ ]. In addition to the theorems presented in the current paper, we conjecture that a variant of Theorem <ref> still holds true in the case of τ_0,i = τ_i. However, the extra assumptions given in Theorem <ref> and the asymptotic results in <cit.> about bootstrapping the unrestricted Kaplan-Meier estimator suggest that stronger assumptions on the censoring distributions will be necessary. Clearly, it is beyond the scope of the present paper to explore this case further. Despite the advantage of not requiring any model assumptions, in several situations, one might want to account for covariate effects when comparing the survival times of the uncured among two groups. This will be the focus of our study in Part II, where the results of this article will be further extended to the conditional MST_u estimated under the assumption that each of the groups follows a semiparametric logistic-Cox mixture cure model. § PROOFS Since the KM estimator remains constant after the last observed event time, we have p̂_i=Ŝ_i(τ_0,i) and MST_u,i=∫_0^τ_0,iŜ_i(u)-Ŝ_i(τ_0,i)/1-Ŝ_i(τ_0,i) u. Hence MST_u,i-MST_u,i=ψ(Ŝ_i)-ψ(S_i), where ψ: D̃[0,τ_0,i]→ ψ(θ)=∫_0^τ_0,iθ(u)-θ(τ_0,i)/1-θ(τ_0,i) u and D̃[0,τ_0,i]={f∈D[0,τ_0,i]\|sup_t∈[0,τ_0,i]\|f(t)\|<1}. We will derive the asymptotic distribution using the functional delta method based on standard limit results for the KM estimator. First, we consider for simplicity the case of a continuous distribution F_i and assume that either condition a) or b) of the theorem holds. From results of <cit.>, we have that the stochastic process √(n_i){Ŝ_i(t)-S_i(t)} converges weakly in D[0,τ_0,i] to the process S_i(t)· B(v_i(t)) where B is a standard Brownian motion and v_i is defined as in (<ref>). Returning to the delta method, the function ψ is Hadamard-differentiable tangentially to C[0,τ_0,i] with derivative ψ_θ given by ψ_θ· h=h(τ_0,i)∫_0^τ_0,iθ(u)-θ(τ_0,i)/{1-θ(τ_0,i)}^2 u+∫_0^τ_0,ih(u)-h(τ_0,i)/1-θ(τ_0,i) u By Theorem 3.9.4 in <cit.>, we conclude that √(n_i){MST_u,i-MST_u,i} converges weakly to N = S_i(τ_0,i)B(v_i(τ_0,i))∫_0^τ_0,iS_i(u)-S_i(τ_0,i)/{1-S_i(τ_0,i)}^2 u +∫_0^τ_0,iS_i(u)B(v_i(u))-S_i(τ_0,i)B(v(τ_0,i))/1-S_i(τ_0,i) u =p_iB(v_i(τ_0,i))∫_0^τ_0,iS_i(u)-p_i/(1-p_i)^2 u+∫_0^τ_0,iS_i(u)B(v(u))-p_iB(v_i(τ_0,i))/1-p_i u The variable N is normally distributed with mean zero and variance σ_i^2 =∫_0^τ_0,i∫_0^τ_0,iS_i(u)S_i(t)/(1-p_i)^2v_i(u∧ t) u t+p_i^2/(1-p_i)^2(MST_u,i-τ_0,i)^2v_i(τ_0,i) +2p_i/(1-p_i)^2(MST_u,i-τ_0,i)∫_0^τ_0,iS_i(u)v_i(u) u. If F_i is not continuous and either condition a) or c) of the theorem is satisfied, we only have weak convergence in D[0,τ_i] of the stopped process L̂_i(t) =√(n_i)S_i(t)/S_i(t∧ Y_i,(n)){Ŝ_i(t∧ Y_i,(n))-S_i(t∧ Y_i,(n))} =√(n_i){S_i(t)Ŝ_i(t∧ Y_i,(n))/S_i(t∧ Y_i,(n))-S_i(t)} where Y_i,(n) denotes the largest observation in group i (see for example Theorem 3.14 in <cit.>). If τ_0,i<τ_i, then Y_i,(n)>τ_0,i with probability converging to one, which leads to the uniform convergence of √(n_i){Ŝ_i(t)-S_i(t)} on D[0,τ_0,i] as in part a). Otherwise, if τ_0,i=τ_i, applying the Delta method as before, we would obtain the limit distribution of √(n_i){ψ(Q̂_i)-ψ(S_i)}, Q(t)=S_i(t)Ŝ_i(t∧ Y_i,(m_i))/S_i(t∧ Y_i,(m_i)) It then remains to deal with the difference √(n_i){ψ(Ŝ_i)-ψ(Q̂_i)} and show that it converges to zero. For this one can use Lemma 3 in <cit.>, which does not require continuity of F_i. We begin by analyzing the asymptotic behaviour of (W_1^π, W_2^π) = √(n_1n_2/n_1+n_2) (Ŝ_1^π - Ŝ , Ŝ_2^π - Ŝ) as a random element of (D[0,τ_0])^2, where τ_0 = max(τ_0,1, τ_0,2) and Ŝ denotes the Kaplan-Meier estimator based on the pooled sample. We refer to Lemma 2 in the online supplementary material to <cit.> for the result that, as min(n_1, n_2)→∞, the conditional distribution of (W_1^π, W_2^π) converges weakly on (D[0,τ_0])^2 in probability to the distribution of the Gaussian process ((1-κ) S(·) B(v(·)), -κ S(·) B(v(·))). Here, B again denotes a standard Brownian motion, S(t) = exp(- ∫_0^t κ (1-G_1(u-)) F_1(u) + (1-κ) (1-G_2(u-)) F_2(u)/κ (1-G_1(u-)) S_1(u-) + (1-κ) (1-G_2(u-)) S_2(u-)) is the limit of the pooled Kaplan-Meier estimator, and v(t) = ∫_0^t κ (1-G_1(u-)) F_1(u) + (1-κ) (1-G_2(u-)) F_2(u)/{κ (1-G_1(u-)) S_1(u-) + (1-κ) (1-G_2(u-)) S_2(u-)}^2. Now, write MST_u,i^π = ψ(Ŝ^π_i) and MST_u = ψ(Ŝ) for the estimated mean survival time based on the pooled sample. We pointed out the Hadamard-differentiability of ψ in the proof of Theorem <ref> above. Thus, Theorem 3.9.11 in <cit.> applies and it follows that, conditionally on the data, √(n_1 n_2/(n_1 + n_2))(MST_u,1^π-MST_u, MST_u,2^π-MST_u) converges weakly to the following two-dimensional Gaussian random vector, say, (N_1, N_2) in probability: [ S(τ_0)B(v(τ_0))∫_0^τ_0S(u)-S(τ_0)/{1-S(τ_0)}^2 u +∫_0^τ_0S(u)B(v(u))-S(τ_0)B(v(τ_0))/1-S(τ_0) u ] · (1-κ, -κ) = [ p B(v(τ_0))∫_0^τ_0S(u)-p/{1-p}^2 u +∫_0^τ_0S(u)B(v(u))-pB(v(τ_0))/1-p u ] · (1-κ, -κ), where p = S(τ_0). Finally, we take the difference of both entries of the pair to conclude that the weak limit of the conditional distribution of √(n_1 n_2/(n_1 + n_2))(MST_u,1^π-MST_u,2^π) is normal with mean zero and variance σ^π2 =∫_0^τ_0∫_0^τ_0S(u)S(t)/(1-p)^2v(u∧ t) u t+p^2/(1-p)^2(MST_u-τ_0)^2v(τ_0) +2p/(1-p)^2(MST_u-τ_0)∫_0^τ_0S(u)v(u) u. As is apparent from a comparison of (<ref>) and (<ref>), both limit variances coincide up to sample size-related factors if both samples are exchangeable. [Acknowledgments] The authors would like to thank Joris Mooij for insightful comments regarding the causal interpretation and the decision theoretic perspective on choosing between two treatments. imsart-number Mean survival of uncured patients in two samples – Part II Vrije Universiteit Amsterdam The Netherlands e2 University of Amsterdam The Netherlands e1 D. Dobler and E. Musta ^: The authors contributed equally and are given in alphabetical order. The restricted mean survival time has been recommended as a useful summary measure to compare lifetimes between two groups, avoiding the assumption of proportional hazards between the groups. In the presence of cure chances, meaning that some of the subjects will never experience the event of interest, it has been illustrated in Part I that it is important to separately compare the cure chances and the survival of the uncured. In this study, we adjust the mean survival time of the uncured (MST_u) for potential confounders, which is crucial in observational settings. For each group, we employ the widely used logistic-Cox mixture cure model and estimate the MST_u conditionally on a given covariate value. An asymptotic and a permutation-based approach have been developed for making inference on the difference of conditional MST_u's between two groups. Contrarily to available results in the literature, in the simulation study we do not observe a clear advantage of the permutation method over the asymptotic one to justify its increased computational cost. The methods are illustrated through a practical application to breast cancer data. asymptotic statistics cure models random permutation inference right-censoring logistic-Cox mixture model § INTRODUCTION Comparing lifetimes between two groups is a common problem in survival analysis. The survival function carries all the necessary information for making such comparison. In practice, however, to facilitate interpretations and decision making, it is important to have a single (or at least some few) metric summary measure(s) that quantifies the difference between the two groups. Moving away from the routine use of hazard ratios, the restricted mean survival time (RMST) has been recently advocated as a better and safer alternative, which does not rely on the proportional hazards assumption <cit.>. In observational studies, however, it is important to also account for possible confounders. One can then adjust for imbalances in the baseline covariates between the two groups by using regression-based methods to estimate the RMST; see <cit.> and references therein. In the presence of a cure fraction, some of the subjects do not experience the event of interest and are referred to as `cured'. In this context, it was illustrated in Part I that it is more appropriate and informative to separately compare the cure fraction and the survival times of the uncured subpopulations, for example by means of the expected survival times, MST_u. Despite the clear advantage of using a nonparametric, model-free approach, in the presence of covariate information semiparametric models are usually preferred in terms of interpretability and balance between complexity and flexibility. Several semiparametric mixture cure models have been introduced in the literature to account for the possibility of cure; see, e.g., <cit.>. Among them, the most commonly used in practice is the logistic-Cox mixture model, which considers a logistic model for the cure probability given the covariates and a Cox proportional hazards (PH) model for the survival times of the uncured subjects. For this reason, we will employ a logistic-Cox model for both groups in this paper. Note, however, that the baseline hazards for the uncured subpopulations in the two groups are in general different, leading to non-proportional hazards. Based on the maximum likelihood estimates of the components of a logistic-Cox model (computed using the package in R), we propose an estimator for the conditional mean survival time of the uncured given the covariates. Despite this model choice, the estimation procedure and the results could be similarly extended to other semiparametric mixture cure models. Recently, the problem of estimating the conditional mean survival time for the uncured in a one-sample context has been considered in <cit.>. They assume a semiparametric proportional model for the mean (residual) life of the uncured and propose an estimation method via inverse-probability-of-censoring weighting and estimating equations. As a consequence, their approach requires also estimation of the censoring distribution. In addition to the estimation of the mean survival time for the uncured, we also analyze both the asymptotic and the permutation-based approach for inference on the difference between the conditional MST_u of the two groups, for a fixed covariate value. In the nonparametric setting, it has been observed that the permutation approach improves upon the asymptotic method for small sample sizes, while maintaining good behavior asymptotically <cit.>. To the best of our knowledge, the permutation approach has not been used before for semiparametric models in a similar context of maximum likelihood-based estimators that we will pursue. One notable permutation-based inference approach in the survival literature concerns the weighted logrank test <cit.>; there, the semiparametric model arises from the form of the null hypothesis which is a cone or subspace of hazard derivatives. Given also the complexity of our model, several challenges arise from both the computational and theoretical point of view. In order to obtain results on the asymptotic validity of the permutation approach, we first derive a general Donsker-type theorem for permutation based Z-estimators. Secondly, when fitting the model to the permuted sample, there is an issue of model mispecification. This leads to the convergence of the estimators to a minimizer of the Kullback-Leibler divergence. Thirdly, since the variance estimators need to be computed via a bootstrap procedure, combination of bootstrap and permutation becomes computationally expensive. Hence, it is of interest to investigate whether the gain in accuracy is sufficient to compensate for the increased computation cost compared to the asymptotic approach. The article is organized as follows. The model and notation are introduced in Section <ref>. In Section <ref>, we propose a semi-parametric estimator for the difference in conditional mean survival time of the uncured and derive its asymptotic distribution. Additionally, we also introduce a random permutation approach for inference and justify its asymptotic validity. The behavior of the asymptotic and permutation based methods for finite sample sizes is evaluated through a simulation study in Section <ref>. Practical application of the methods is illustrated through a study of breast cancer in Section <ref>. Finally, we conclude with a discussion in Section <ref>. All proofs are contained in Appendix <ref>, while the general Donsker-type theorem for permutation based Z-estimators is given in Appendix <ref>. The R code with an implementation of our methods is available in the GitHub repository <https://github.com/eni-musta/MST_uncured>. Appendix <ref> contains a few lines of R code for accessing the breast cancer data analyzed in this paper. § MODEL AND NOTATION We consider i.i.d. survival times T_11, …, T_1n_1 and T_21, …, T_2n_2 from two independent groups (i=1,2), each comprising a mixture of cured (immune to the event of interest) and uncured subjects. Note that the distributions of the survival times are allowed to differ between groups. For mathematical convenience, the event time of the cured subjects is set to ∞, signifying that the event never actually occurs. Conversely, a finite survival time implies that subjects are susceptible and will experience the event at some point. However, because of censoring, the cure status is only partially known. Specifically, instead of the actual survival times, we observe the follow-up times Y_i1,…,Y_in_i and the censoring indicators Δ_i1,…,Δ_in_i, where Y_ij=min{T_ij,C_ij}, Δ_ij=_{T_ij≤ C_ij} and C_ij are the censoring times. Assume that, for each individual in both groups, we observe two covariate vectors X_ij∈^p and Z_ij∈^q, i=1,2, j=1,…, n_i, representing the variables that affect the probability of being susceptible (incidence) and the survival of the uncured (latency). In this way, we allow for these two components of the model to be affected by different variables. However, we do not exclude situations in which the two vectors X and Z are exactly the same or share some components. Using the framework of mixture cure models, the relations in (<ref>) of Part I now hold conditionally on the covariates. In particular, we have that the survival function of T_i1 given X_i1 and Z_i1 is given by S_i(t\|x,z)=(T_i1>t\|X_i1=x, Z_i1=z)=p_i(x)+(1-p_i(x))S_u,i(t\|z), where S_u,i(t\|z)=(T_i1>t \| Z_i1=z, T_i1<∞) is the conditional survival function of the susceptibles and p_i(x)=(T_i1=∞ \| X_i1=x) denotes the conditional cure probability in group i. Instead of independent censoring, now we assume that censoring is independent of the survival times conditionally on the covariates: T⊥ C\| (X,Z). Among various modeling approaches for the incidence and the latency, the most common choice in practice is a parametric model, such as logistic regression, for the incidence and a semiparametric model, such as Cox proportional hazards, for the latency <cit.>. The popularity of such choice is primarily due to simplicity and interpretability, particularly when dealing with multiple covariates. We focus on this type of models and assume that 1-p_i(x)=ϕ(γ_i^Tx), where ϕ: →[0,1] is a known function, γ_i∈^p+1 and γ^T_i denotes the transpose of the vector γ_i. Here, the first component of x is taken to be equal to one and the first component of γ_i corresponds to the intercept. In particular, for the logistic model, we have ϕ(u)=e^u/1+e^u. One can in principle allow also for a different function ϕ in the two groups but for simplicity we assume that to be the same. For the latency, we assume a semiparametric model S_u,i(t\|z) =S_u,i(t\|z;β_i,Λ_i) depending on a finite-dimensional parameter β_i∈^q, and a function Λ_i. For example, for the Cox proportional hazards model, we have S_u,i(t\|z) = exp{-Λ_i(t)exp(β^T_iz)}, where Λ_i is the baseline cumulative hazard in group i. One challenge with mixture cure models is model identifiability, i.e., ensuring that different parameter values lead to different distributions of the observed variables. General identifiability conditions for semiparametric mixture cure models were derived by <cit.>. In the particular case of the logistic-Cox model the conditions are: (I1) for all x, 0 <ϕ(γ_i^Tx) < 1, (I2) the function S_u,i has support [0; τ_0,i] for some τ_0,i<∞, (I3) P(C_i1 >τ_0,i\|X_i1;Z_i1) > 0 for almost all X_i1 and Z_i,1, (I4) the matrices Var(X_i1) and Var(Z_i1) are positive definite, Condition I3 corresponds again to the assumption of sufficient follow-up. In practice, this can be evaluated based on the plateau of the Kaplan-Meier estimator and the expert (medical) knowledge. In the presence of covariate information, we are now interested in the difference of mean survival times of the uncured individuals among the two groups conditional on the covatiates: MST_u,1,z-MST_u,2,z=[T_11\| T_11 < ∞,Z_11=z]-[T_21\| T_21 < ∞,Z_21=z]. Note that we use only the covariate Z because that affects the survival of the uncured individuals. In combination with the conditional cure probabilities p_i(x), such conditional mean survival times provide useful summaries of the conditional survival curves. § ESTIMATION, ASYMPTOTICS, AND RANDOM PERMUTATION The conditional mean survival time can be written as MST_u,i,z= ∫_0^τ_0,iS_u,i(u\|z) u i=1,2. This leads to the following estimator MST_u,i,z= ∫_0^Y_i,(m_i)Ŝ_u,i(u\|z) u, where Ŝ_u,i(·\|z) is an estimate of the conditional survival function for the uncured and Y_i,(m_i) is the largest observed event time in group i. Next we focus on the logistic-Cox mixture model, given by (<ref>)-(<ref>), and consider the plug-in estimate Ŝ_u,i(t\|z)=exp(-Λ̂_i(t)e^β̂^T_iz), where Λ̂_i, β̂_i are the maximum likelihood estimates of Λ_i and β_i respectively. Maximum likelihood estimation in the logistic-Cox model was initially proposed by <cit.> and is carried out via the EM algorithm. The procedure is implemented in the R package <cit.>. In practice, the survival Ŝ_u,i(t\|z) is forced to be equal to zero beyond the last event Y_i,(m_i), meaning that the observations in the plateau are considered as cured. This is known as the zero-tail constraint as suggested in <cit.> and is reasonable under the assumption of sufficient follow-up: τ_0,i<τ_i, which follows from (I3). The asymptotic properties of the maximum likelihood estimates Λ̂_i, β̂_i, γ̂_i were derived in <cit.> under the following assumptions: (A1) The function Λ_i(t) is strictly increasing, continuously differentiable on [0,τ_0,i) and Λ_i(τ_0,i):=lim_t→τ_0,iΛ_i(t)<∞. (A2) γ_i,β_i lie in the interiors of compact sets and the covariate vectors Z_ij and X_ij have compact support: there exist m_i >0 such that: (‖ Z_ij‖ < m_i and ‖ X_ij‖ < m_i) = 1. (A3) There exists a constant ϵ>0 such that (T_i1 =τ_0,i\| T_i1<∞,Z_i1)>ϵ with probability one. (A4) (Y_i1≥ t\| Z_i1,X_i1) is continuous in t ≤τ_0,i. Assumptions (A1),(A3) are formulated slightly in a different way in <cit.> but, given the identifiability constraints (I1)-(I4), they reduce to the ones stated above. In particular, assuming that the survival distribution for the uncured has a positive mass at the end point of the support (A3) is a technical condition needed to guaranty that Λ_i stays bounded on [0,τ_0,i] while ensuring the identifiability of the model. In the Cox model without cure fraction, one does not encounter this problem because the support of the event times is larger than the follow-up of the study. Even though (A3) might seem not realistic, one can think of such assumption being satisfied with a very small ϵ. In such case it is unlikely to observe events at τ_0,i as we see in real-life scenarios. If instead of the maximum likelihood estimation, one considers estimation via presmoothing as proposed in <cit.>, this condition can be avoided at the price of additional technicalities. This is because the conditional probability of {T=∞} is identified beforehand by means of a nonparametric smooth estimator. As a result, when estimating Λ_i in the second step, one could restrict to a smaller interval [0,τ^]⊂[0,τ_0,i]; see the discussion in Section 5.1 of <cit.>. Let m̂_z= MST_u,1,z-MST_u,2,z be an estimator of m_z=MST_u,1,z-MST_u,2,z. Using the large sample properties of the estimators Λ̂_i and β̂_i from <cit.>, we first derive the limit distribution of the process √(n_i){Ŝ_u,i(·\|z)-S_u,i(·\|z)} and then obtain the following result for the conditional mean survival time. Assume that the identifiability conditions (I1)-(I4) and the assumptions (A1)-(A4) are satisfied. Then, for any z∈𝒵, the variable √(n_i)(Ê_i,z-E_i,z) is asymptotically normally distributed, i.e., √(n_i)(MST_u,i,z-MST_u,i,z)N∼𝒩(0,σ_i,z^2) as n_i→∞. The limit variance σ_i,z^2 is defined in (<ref>) in the Appendix <ref>. From the independence assumption between the two groups and Theorem <ref>, we obtain the following result for which we define a_n=√(n_1n_2/(n_1+n_2)). Assume that n_1/(n_1+n_2)→κ∈(0,1) as min(n_1,n_2) →∞. Then a_n(m̂_z-m_z) is asymptotically normally distributed with mean zero and variance σ^2_z=(1-κ)σ_1,z^2+κσ_2,z^2. We restrict for simplicity to the logistic-Cox model and the maximum likelihood estimation method but the previous results can be generalized to other estimation methods or other semiparametric mixture cure models. For example, if the presmoothing approach introduced in <cit.> is used instead of the MLE, then the asymptotic properties could be derived in the same way using Theorem 4 in <cit.>. We also note that because of their complicated expressions, the variances of the estimators in the semiparametric mixture cure model are estimated via a bootstrap procedure <cit.>. As a result, we will also use the bootstrap to estimate σ_z. As in Part I, we would like to investigate whether the asymptotic inference can be made more reliable by means of a permutation approach. Again π=(π_1, …, π_n_1+n_2) denotes any permutation of (1,2, …, n_1+n_2). Write (Y_j, Δ_j,X_j,Z_j), j=1,…, n_1+n_2 for the pooled sample which consists of the data points of the first group (j ≤ n_1) and those of the second group (j> n_1). Applying π to the pooled sample leads to the permuted samples (Y_π_1, Δ_π_1,X_π_1,Z_π_1), …, (Y_π_n_1, Δ_π_n_1,X_π_n_1,Z_π_n_1) and (Y_π_n_1+1, Δ_π_n_1+1,X_π_n_1+1,Z_π_n_1+1), …, (Y_π_n_1+n_2, Δ_π_n_1+n_2,X_π_n_1+n_2,Z_π_n_1+n_2). In the special case of exchangeability, for any z, m̂_z would have the same distribution as m̂^π_z = MST_u,1,z^π - MST_u,2,z^π. Here, MST_u,i,z^π are the estimators of the conditional mean survival times, just based on the i-th permuted sample. Since the samples are in general not exchangeable and, as a consequence, the asymptotic variances of m̂_z and m̂^π_z cannot be assumed equal, we use their studentized version. We will thus focus on m̂^π_z/σ̂^π_z as the permutation version of m̂_z/σ̂_z, where σ̂_z^π2 is the estimated variance of m̂^π_z, estimated via bootstrap. Because it is computationally infeasible to realize m̂^π_z/σ̂^π_z for all (n_1+n_2)! permutations, we will realize a relatively large number B of random permutations π and approximate the conditional distribution by the collection of the realized m̂^π_z/σ̂^π_z of size B. In the following, we will discuss the asymptotic behaviour of m̂^π_z/σ̂^π_z to justify the validity of the resulting inference procedures. One challenge that arises in this setting is that, since we are assuming a semiparametric model, the permuted samples will in general not follow the same model. Hence, when we fit the logistic-Cox model to obtain the estimates in the permuted samples, the model is misspecified. Hence, we first show in a series of lemmas in Appendix <ref> that the maximum likelihood estimators converge to the parameters of a logistic-Cox likelihood that minimize the Kullback-Leibler divergence from the true distribution of the pooled data. We can indeed argue that such minimizer exists and we assume that it is unique. In case of non-uniqueness, we expect that the results can be extended and, depending on the starting point of the algorithm, the estimates would converge to one of such minimizers. However, such extension is beyond the scope of the current paper. In practice, we observed that the EM algorithm converges and the limit was stable with respect to the initial point, which might indicate that the minimizer was indeed unique. Secondly, to obtain the asymptotic distribution of the permuted estimators, we first obtain a general Donsker-type theorem for permutation based Z-estimators (see Appendix <ref>). That result holds in a great generality, so it would also apply to countless other two sample problems. Thus, it is of interest of its own. But let us first return to the main result about the permuted estimators in the present context: Assume that the identifiability conditions (I1)-(I4) and (A1)-(A4) hold. Assume also that the minimizer of the KL divergence definied in (<ref>) in Appendix <ref> is unique. Then, for any z∈𝒵, as min(n_1, n_2) →∞ with n_1/(n_1 + n_2) →κ∈(0, 1), the conditional distribution of a_nm̂_z^π given the data converges weekly in probability to the zero-mean normal distribution with variance σ^π2_z given in (<ref>) in Appendix <ref>. Under the assumptions of the previous theorem, for any z∈𝒵, as min(n_1, n_2)→∞ with 0 < lim inf n_1/(n_1+n_2) ≤lim sup n_1/(n_1+n_2) < 1, the (conditional) distributions of a_nm̂_z^π/σ̂^π_z (given the data) and a_n(m̂_z - m_z)/σ̂_z converge weakly (in probability) to the same limit distribution which is standard normal. § SIMULATION STUDY In this section, we investigate the practical performance of the permutation approach and of the asymptotic method when comparing mean survival times for the uncured sub-population in a two-sample problem. We consider two samples of size 200 and 100, respectively, from logistic-Cox mixture cure models. Note that, in practice, semiparametric cure models are usually not used for sample sizes much smaller than these because of their complexity (more parameters need to be estimated compared to standard Cox model for example) and the need to observe a long plateau with a considerable amount of censored observations (as a confirmation of the sufficiently long follow-up assumption). Since the permutation approach is computationally intensive and asymptotically we expect the behavior of the two methods to be more similar, we also did not consider larger sample sizes. Instead, we focus on three different scenarios as described below by varying the distributions of the uncured subjects, the cure proportions and the censoring rates. For simplicity, we also consider the same covariates in the incidence and latency components, i.e., X=Z. § SIMULATION SETTINGS Setting 1. Both samples are generated from the logistic-Cox mixture cure model with Weibull baseline distribution with shape parameter 0.75 and scale parameters 1.5 and 2, respectively. The survival times of the uncured subjects are truncated at τ_0,i, i=1,2 equal to the 99% quantile of the corresponding baseline Weibull distribution in order to have finite supports. We consider two independent covariates Z_1 and Z_2, which affect both the cure probability and the survival of the uncured. In the first sample, Z_1∼ N(0,1), Z_2∼ Bern(0.4) while in the second sample Z_1∼ N(1,1), Z_2∼ Bernoulli(0.6). The regression coefficients are γ_1=(0,0.5,0.8), β_1=(0.3,0.5), γ_2=(0.1,1,0.6), β_2=(0.3+log(0.75),0.5). This corresponds to having around 43% and 24% cured subjects in each sample. The censoring times are generated independently of the other variables from exponential distributions with parameters 0.4 and 0.2, respectively. They are truncated at τ_i=τ_0,i+2, i=1,2 to reflect the limited length of studies in practice. The censoring rate in sample 1 is 52%, while in sample 2 it is 28%. In both cases, we have around 15% of the observations in the plateau. In this setting, the covariate distributions, the cure and censoring rates, and the survival distributions of the uncured are different among the two samples. However, depending on the covariate values, the conditional mean survival times of the uncured can be the same, i.e., m_z=0. We consider a range of possible covariates, see Table <ref>, including some extreme and unlikely values in order to get more different mean survival times between the two groups. Setting 2. Both samples are generated from logistic-Cox mixture cure models with Gompertz baseline distribution with shape parameter 1 and rate parameters 0.1, 0.3, respectively. The survival times of the uncured subjects are truncated at τ_0,i, i=1,2, equal to the 99% quantiles of the corresponding baseline distributions in order to have finite supports. We consider two independent covariates, Z_1∼ N(0,1) and Z_2∼ Unif(-1,1) with the same distribution in both samples. The regression coefficients are γ_1=γ_2=(0.8,-1,1), β_1=(-0.6,0.5), γ_2=(0.1,1,0.6), β_2=(-0.05,0.4). This corresponds to having around 35% cured subjects in each sample. The censoring times are generated independently of the other variables from exponential distributions with parameters, 0.1 and 0.2, respectively. They are truncated at τ_i=τ_0,i+2, i=1,2, to reflect the limited length of studies in practice. The censoring rate in sample 1 is 46%, while in sample 2 it is 48%. In both cases, we have around 20% of the observations in the plateau. In this setting, the covariate distributions, the cure rates, and the censoring rates are the same for both samples. The survival distributions of the uncured are different but again, for certain values of the covariates, the conditional mean survival times of the uncured are the same. We consider different covariate values as in Table <ref>. In particular, z_8 is a very extreme and unlikely value but it was considered in order to have a case with larger negative value for m_z. Setting 3. Both samples are generated from the same distribution as for sample 1 in Setting 1. This means that the two samples are exchangeable and for any covariate value we have m_z=0. § SIMULATION RESULTS For each setting and covariate value, we construct 1-α=95% confidence intervals for m_z based on the asymptotic and the permutation approach I_z^=[m̂_z∓ q_1-α/2σ̂_z/a_n], I_z^π=[m̂_z-q^π_1-α/2,m̂_z- q^π_α/2σ̂_z] where m̂_z is computed as in Section <ref>, σ̂_z^2 is the variance of a_nm̂_z estimated via the bootstrap, q_1-α/2 denotes the quantile of the standard normal distribution and q^π_1-α/2 is the quantile of the distribution of m̂_z^π/σ̂_z^π. Due to the computational cost, we use 100 bootstrap samples combined with 500 random permutation samples. This procedure was repeated 1,000 times. The lengths and coverage probabilities of the confidence intervals are given in Tables <ref>-<ref>. The coverage rates closest to 95% among both types of confidence intervals is printed in bold-type. For the exchangeable Setting 3, since the permutation confidence intervals are exact, we only provide the results of the asymptotic approach. We observe that the coverage of both confidence intervals is very low for some covariate values. That happens mainly when m_z is large in absolute value (either positive or negative depending on the setting). This seems to be because, for certain covariate values, the errors that we make in the estimation of the coefficients and baseline survival get amplified when computing the survival function conditional on z, resulting in a biased estimate for m_z. Much larger sample sizes would be needed to get a good estimate of m_z for such covariates z. In the other cases, the coverage probabilities are close to the nominal value and the two methods are comparable. For some z, the permutation approach does slightly better than the asymptotic one, but vice versa for other choices of z. Next, we consider testing the hypothesis H_0: m_z=0 against H_1: m_z≠ 0 at level α=5%. Again the variance of m̂_z is estimated via the bootstrap with 100 bootstrap samples and the quantiles of m̂_z/σ̂_z are estimated via 500 permutation samples. The procedure was repeated 1,000 times. In Tables <ref>-<ref>, we report the percentages of the cases in which H_0 was rejected. In Settings 1 and 2, the levels of the test seem to be close to the nominal level and the power is larger when \|m_z\| is larger, even though it does not only depend on \|m_z\| but also on the sign of m_z (deviations in conditional mean survival times might be easier to detect in one direction compared to the other). In Setting 1, the asymptotic method has more power when m>0 (m_1, m_2,m_7,m_8), while the permutation approach has more power when m<0 (m_5, m_6). In Setting 2, we observe that the results for both methods are comparable when m>0 but the permutation approach has more power when m<0. For the exchangeable setting, the level of the asymptotic test is larger than the nominal value for some of the covariate values (the ones for which the coverage probabilities were anti-conservative; see above). Overall, we conclude that, unless the two samples are exchangeable, there is no clear advantage of using the permutation approach to justify its much higher computational cost. This is different from what is observed previously in the literature and it might be related to the fact that the logistic-Cox model is misspecified in the permutation samples. Computationally, the EM algorithm still converges and is stable with respect to the initial estimates. This suggest that the problem should not be about the existence of a unique maximizer of the likelihood for the misspecified model. § APPLICATION In this section, we illustrate the methods developed in this paper by analyzing a data set about breast cancer. In Appendix <ref>, we provide the R code for accessing this freely available data set. The data come from an observational study that included 286 lymph-node-negative breast cancer patients collected between 1980 and 1995. Thereof, 209 patients were oestrogen-receptor-positive (ER+) and 77 were ER-negative (ER-). These two will later form the subgroups to be analyzed in a two-sample inference problem. As additional covariates, we consider the patients' age (ranging from 26 to 83 with a median of 52 years) and a tumour size score which is an integer number between 1 and 4. We refer to <cit.> for a more complete description of the study and other specifics of the dataset. Additionally, <cit.> compared this dataset in the light of two competing models and corresponding statistical methods: a Cox-logistic cure model versus a Single-Index/Cox model. We, on the other hand, do not model the ER-status semiparametrically but nonparametrically by means of two subgroups. Our aim is to conduct a regression analysis to investigate differences in disease progression expectations for ER+/- patients while taking the covariates tumour size (ordinal) and age into account. The outcomes of this two-sample analysis could be used to justify why the two groups should not be pooled, and how or how not to model the ER-status semiparametrically. These questions are relevant if one wishes to make predictions, e.g., for the remaining expected lifetime of a patient. From a technical point of view, we consider the composite endpoint of relapse-free survival (measured in months), which here means that deaths and the occurrence of distant metastases are combined into one event of interest. We excluded those 8 patients from our analysis who had a tumour size exceeding 2. This results in two samples of sizes n_1=203 (ER+) and n_2= 75 (ER-). Let us briefly summarize the data: in the ER+ subgroup, about 55% have a tumour size score of 1, as opposed to 47% in the ER- subgroup. The age distributions in the four subgroups (ER+/-, tumour size score 1/2) are generally similar (rather symmetric, no outliers), although the patients with ER- and smaller tumour sizes exhibit a smaller dispersion in age; see Figure <ref>. For both groups, the latest uncensored events were observed after 80 and 48 months, respectively. The censoring rates amount to 62% and 64%, respectively, and the majority of censorings occurred in the plateau. Thus, there is sufficient follow-up. Figure <ref> shows the nonparametric Kaplan-Meier estimates for relapse-free survival for the subsamples of ER+ and ER- patients. These two curves are crossing twice: once, but insignificantly, soon after the time origin, and once again after the last observed event in the ER- subgroup. These crossings underline that the classical proportional hazards model <cit.> might not be appropriate for a combined modeling of all these data within a single, extended Cox-logistic cure model: such a model would contradict crossing Kaplan-Meier curves as seen in Figure <ref> (after rescaling both curves to exhibit the same cure rate). We begin our inferential data analysis by observing a non-significant difference in cure rates between both groups (ER+/-): the nonparametric test of <cit.> resulted in a p-value of p=0.659. This motivated us to check for differences in the residual mean survival times for uncured patients. Let us now turn to the hypothesis tests for equal differences in mean survival times, for H_0: m=0 versus H_a: m≠ 0, at level α and (1-α)-confidence intervals for the difference m. Throughout, we have chosen α=5%, 500 permutations, and 100 bootstrap iterations. Starting with our nonparametric test of equal residual mean survival times for the uncured, we first of all obtained a point estimate of m̂=16. This resulted in p-values of p=0 for both the asymptotic and the permutation approach. The corresponding 95%-confidence intervals are [10,22] and [9,22], respectively. Our present approach is to compare the outcomes of two independently fitted Cox-logistic models for both sample groups ER+/-. We have first checked the proportional hazards assumption for the latency parts of the model by means of the test proposed in <cit.>. The resulting p-values are 1 and 0.96 respectively for the covariates age and tumour size in sample 1; 1 and 0.81 respectively for the covariates age and tumour size in sample 2. Thus, there is no reason to reject the proportional hazards assumption. We would also like to point out that we do not rely on the proportional hazards assumption for ER status, which is nonparametrically modeled in terms of two separate subgroups. Thus continuing with the two independently fitted Cox-logistic models, Table <ref> contains all point estimates of the parametric model components. Note that none of the covariates was found to have a significant influence on any of the two models. From the point estimates, we also see that for ER+ patients age seems protective for both, incidence (γ_1 <0) and latency (β_1<0), and generally harmful for ER- (γ_1, β_1> 0). Also, a bigger tumour size is generally harmful for ER+ patients (γ_2, β_2>0) but, for ER- patients, it has a protective influence on incidence (γ_2 <0) although a harmful influence on the latency (β_2>0). Testing whether there is a significant difference of any of these parameters from 0 could be easily achieved by studentizing the parameter point estimates and comparing the results with quantiles from the standard normal distribution or those from a corresponding permutation version of the studentized parameter estimates. Since this was not our main focus, these results are not shown. Just like for the nonparametric test, the results from the semiparametric model were unambiguous (see Table <ref>): for all considered covariate combinations, uncured patients with ER+ have, at level α=5%, a significantly larger expected mean survival time than uncured patients with ER-. This difference seems to grow with progressing age, especially for patients with tumour size 1. This is in line with the parameter estimates related to age and latency. The influence of the tumour size on the mean survival difference is not that obvious. All in all, both the asymptotic and the permutation-based method resulted in very similar outcomes. § DISCUSSION AND FUTURE RESEARCH In this article, we focused on the comparison of survival times for the uncured subpopulations in a two-sample problem by means of the mean survival time adjusted for adjusted for potential confounders. We proposed a semiparametric estimation method for the MST_u conditionally on a covariate value and developed an asymptotic and a permutation-based approach for inference. We encountered several theoretical and computational challenges related to the permutation approach in a semi-parametric model. Moreover, based on our simulation study, we did not observe a clear advantage of the permutation method over the asymptotic one, contrarily to existing findings in the literature for the nonparametric setting. We employed a logistic-Cox mixture cure model because it is the most widely used semiparametric cure model. However, the logistic model for the incidence could be replaced by any other parametric model that might be more suitable in specific applications. Also, the Cox model for the latency component could for example be replaced by the accelerated failure time model. In such cases, both the estimators and the theory would need some adjustments but we expect that the same challenges would arise and the practical performance would be similar to the current model. Another possibility would be to allow for covariates without imposing a specific model on the latency. For example, <cit.> focused on estimating the cure rate, while leaving the distribution of the uncured unspecified, and they obtained a nonparametric estimator for the conditional survival function of the uncured. Another nonparametric estimator was proposed in <cit.>, relying on the Beran estimator for the conditional survival function. It would be of interest to further extend our method to these nonparametric settings. One important extension of the present methodology would include a rectification for causal interpretations, as motivated in the discussion section of Part I. For example, the combination of an inverse-probability-of-treatment weighted version of the MST_u,i,z's with the cure rates would allow for causal insights and a more complete picture in comparisons of two groups with cure fractions. If the data did not originate from randomized clinical trials in the first place, also the estimator for the cure rate would require an adjustment for enabling causal interpretations. These considerations are clearly beyond the scope of the present paper and are thus left for future research. Apart from the permutation approach for inference, one could also consider a pooled bootstrap approach; cf. Section 3.7.2 of <cit.>. However, even with the pooled bootstrap we would encounter the same theoretical and computational problems and we expect that the practical behavior would be similar to the permutation approach. Also, one would lose the benefit of the finite sample exactness of the permutation-based inference procedures under exchangeability. Finally, to avoid the model misspecification issue for the permutation approach in the semiparametric setting, one could estimate a mixed model instead of a logistic-Cox model in the permutation samples, i.e., fitting the correct model of the pooled data, which is a linear combination of two logistic-Cox models. Afterwards, we could estimate the MST as usual. Then one would still need to develop the asymptotic results theoretically for the resulting estimator; these asymptotics will not be the same as in the standard logistic-Cox. In practice, we do not expect this approach to behave better since more parameters need to be estimated for each permuted sample. § PROOFS Let θ_i=(γ_i,β_i) denote the vector of parameters of the semiparametric mixture cure model. Consider the space ℋ_m={h=(h_1,h_2)∈ BV[0,τ_0,i]×^p+q: ‖ h_1‖_v+‖ h_2‖_1≤ m}, where m<∞ and ‖ h_1‖_v is the absolute value of h_1(0) plus the total variation of h_1 on the interval [0, τ_0,i ]. From Theorem 3 in <cit.> we have that the process ⟨ n_i^1/2(Λ̂_i-Λ_i),n_i^1/2(θ̂_i-θ_i)⟩(h)=n_i^1/2∫_0^τ_0,ih_1(s) (Λ̂_i-Λ_i)(s)+n_i^1/2h^T_2(θ̂_i-θ_i) indexed by h∈ℋ_m converges weakly in l^∞(ℋ_m) to a tight Gaussian process G_i in l^∞(ℋ_m) with mean zero and covariance process Cov(G_i(h),G_i(h^))=∫_0^τ_0,ih_1(s)σ_(1),i^-1(h^)(s)Λ_i(s)+h^T_2σ_(2),i^-1(h^) where σ_(1),i(h)(s)=[_{Y_i1≥ s}V_i(s;θ_i,Λ_i)(h)g_i(s;θ_i,Λ_i)e^β^T_iZ_i1] -[∫_s^τ_0,i_{Y_i1≥ u}V_i(s;θ_i,Λ_i)(h)g_i(u;θ_i,Λ_i){1-g_i(u;θ_i,Λ_i)}e^2β^T_iZ_i1Λ_i(u)], σ_(2),i(h)=[∫_0^τ_0,i_{Y_i1≥ s}W_i(s;θ_i,Λ_i)V_i(s;θ_i,Λ_i)(h)g_i(s;θ_i,Λ_i)e^β^T_iZ_i1Λ_i(s)] and g_i(s;θ,Λ)=ϕ(γ^TX_i1)exp(-Λ(s) exp(β^T Z_i1))/1-ϕ(γ^TX_i1)+ϕ(γ^TX_i1)exp(-Λ(s) exp(β^T Z_i1)), V_i(s;θ_i,Λ_i)(h)=h_1(s)-{1-g_i(s;θ_i,Λ_i)}e^β^T_iZ_i1∫_0^sh_1(u)Λ_i(u)+h^T_2W_i(s;θ_i,Λ_i) W_i(s;θ_i,Λ_i)=({1-g_i(s;θ_i,Λ_i)}X^T_i1,[1-{1-g_i(s;θ_i,Λ_i)}e^β^T_iZ_i1Λ_i(s)]Z^T_i1)^T. Using this result we first obtain weak convergence of the process √(n){Ŝ_u,i(t\|z)-S_u,i(t\|z)} for fixed z∈𝒵. By definition and a series of Taylor expansions we have √(n_i){Ŝ_u,i(t\|z)-S_u,i(t\|z) =√(n_i){exp(-Λ̂_i(t)e^β̂^T_iz)-exp(-Λ_i(t)e^β^T_iz)} =-√(n_i){Λ̂_i(t)e^β̂^T_iz-Λ_i(t)e^β^T_iz}exp(-Λ_i(t)e^β^T_iz)+R_1 =-√(n_i){Λ̂_i(t)-Λ_i(t)}e^β^T_izexp(-Λ_i(t)e^β^T_iz) - √(n_i){e^β̂^T_iz-e^β^T_iz}Λ_i(t)exp(-Λ_i(t)e^β^T_iz)+R_1+R_2 =-√(n_i){Λ̂_i(t)-Λ_i(t)}e^β^T_izexp(-Λ_i(t)e^β^T_iz) - √(n_i){β̂_i-β_i}^Tze^β^T_izΛ_i(t)exp(-Λ_i(t)e^β^T_iz)+R_1+R_2+R_3 where the remainder terms R_1,R_2,R_3 converge uniformly to zero in probability because of the boundedness of Λ_i, β_i and Theorems 2-3 in <cit.>. Note that, as in assumption (A1), we denote by S_u,i(τ_0,i\|z) the left limit S_u,i(τ_0,i-\|z) so that S_u,i(·\|z) is a continuous function, bounded from below away from zero. This modification does not influence the value of MST_u,i,z. Consider the functions h∈ℋ_m of the form h_t,z =(h_1;t,z,h_2;t,z) =(_[0,t](·)e^β^T_izexp(-Λ_i(t)e^β^T_iz),(0_p,ze^β^T_izΛ_i(t)exp(-Λ_i(t)e^β^T_iz))), where 0_p denotes a zero vector in ^p since we are not interested in the γ component. For an appropriate choice of m and any t∈[0,τ_0,i], such functions belong to ℋ_m because of assumptions (A1)-(A2). For these functions we have √(n_i){Ŝ_u,i(t\|z)-S_u,i(t\|z)}=⟨ n_i^1/2(Λ̂_i-Λ_i),n^1/2(θ̂_i-θ_i)⟩(h_t,z)+o_P(1) from which we conclude that the process √(n){Ŝ_u,i(t\|z)-S_u,i(t\|z) converges weakly in D^∞([0,τ_0,i]) to a mean zero Gaussian process G_i,z^* with covariance ρ_i,z(s,t) =Cov(G_i,z^(t),G_i,z^(t^)) =∫_0^τ_0,ih_1;t,z(s)σ_(1),i^-1(h_t^,z)(s)Λ_i(s)+h^T_2;t,zσ_(2),i^-1(h_t^,z), where h_t,z is as in (<ref>) and σ_(1),i,σ_(2),i as in (<ref>),(<ref>). Next we use the delta method to obtain the asymptotic distribution of the conditional mean survival time. We have √(n_i)(MST_u,i,z-MST_u,i,z) =√(n_i)∫_0^τ_0,i{Ŝ_u,i(t\|z)-S_u,i(t\|z)} u -√(n_i){τ_0,i-Y_i,(m_1)}Ŝ_u,i(τ_0,i). The first term in the previous equation is equal to √(n_i){ψ(Ŝ_u,i)-ψ(S_u,i)} where ψ:D[0,τ_0,i]→ ψ(ξ)=∫_0^τ_0,iξ(u) u The function ψ is Hadamard-differentiable with derivative ψ_ξ· h=∫_0^τ_0,ih(u) u. By Theorem 3.9.4 in <cit.> it follows that √(n_i){ψ(Ŝ_u,i)-ψ(S_u,i)} converges weakly to N=∫_0^τ_0,iG_i,z^(u) u which is normal distributed with mean zero and variance σ^2_i,z=∫_0^τ_0,i∫_0^τ_0,iρ_i,z(s,t) s t, where ρ_i,z is defined in (<ref>). Next we show that the second term on the right hand side of (<ref>) converges to zero in probability, from which we can conclude that the asymptotic distribution of √(n_i)(MST_u,i,z-MST_u,i,z) is determined by that of the first term. Since Ŝ_u,i(τ_0,i) converges to S_u,i(τ_0,i), it is sufficient to show that √(n_i){τ_0,i-Y_i,(m_1)}=o_P(1). For any δ>0 we have (√(n_i){τ_0,i-Y_i,(m_1)}>δ) =(Y_i,(m_1)<τ_0,i-δ/√(n_i)) =(Δ_i1Y_i1<τ_0,i-δ/√(n_i))^n =[1-(Δ_i1Y_i1≥τ_0,i-δ/√(n_i))]^n =[1-∫_𝒵×𝒳∫_τ_0,i-δ/√(n_i)^τ_0,i(C_i1≥ t\|x,z) F_T_i1\|X_i1,Z_i1(t\|x,z) F_(Z_i1,X_i1)(z,x)]^n_i ≤[1-∫_𝒵×𝒳(C_i1> τ_0\|x,z)ϕ(γ^T_ix)(T_i1=τ_0,i\|T_i1<∞, z) F_(Z_i1,X_i1)(z,x)]^n_i. Because of assumptions (I1),(I3) and (A3), for some K>0 we have (√(n_i){τ_0,i-Y_i,(m_1)}>δ) ≤[1-Kϵ]^n_i→ 0. This concludes the proof. In order to prove our main Theorem <ref>, we first need some preliminary results, which are provided in the following lemmas. In what follows, we assume that conditions (I1)-(I4) and (A1)-(A4) are satisfied. Denote by Λ̂^π_n_i,i and θ̂_n_i,i^π the estimators of Λ and θ=(γ,β) obtained by fitting a logistic-Cox model to the i-th permuted sample, which will be denoted for notational convenience as (Δ_i1^π,Y_i1^π,X_i1^π,Z_i1^π),…, (Δ_in_i^π,Y_in_i^π,X_in_i^π,Z_in_i^π). Let Λ̅_n_1+n_2 and θ̅_n_1+n_2 denote the estimators of Λ and θ based on the pooled sample (Δ_1,Y_1,X_1,Z_1),…,(Δ_n_1+n_2,Y_n_1+n_2,X_n_1+n_2,Z_n_1+n_2). Note that the true distribution of the pooled sample is =κ_1+(1-κ)_2, where _i denotes the distribution of the ith sample. In particular, does not correspond to a logistic-Cox model. Let Q_Λ,θ be the corresponding distribution of a logistic-Cox model with parameters (Λ,θ) and corresponding log-likelihood l(δ,y,x,z;Λ,θ) =δ{logϕ(γ^Tx)+log f_u(y\|z;Λ,β) } +(1-δ)log{1-ϕ(γ^Tx)+ϕ(γ^Tx)S_u(y\|z;Λ,β) } By assumption (I2), the event times on the pooled sample happen on [0,τ̅_0], where τ̅_0=max{τ_0,1,τ_0,2}. Hence S_u(t\|z;Λ,β)=0 for t≥τ̅_0 which corresponds to Λ being defined on [0,τ̅_0). In addition, (Δ=1, Y=τ̅_0)>0 by assumptions (A3) and (I3). Hence, we can restrict on distributions Q_Λ,θ that have a positive mass at τ̅_0, meaning that lim_t→τ̅_0Λ(t)<+∞ and we can denote the limit by Λ(τ̅_0). To reflect the existence of the jump in the likelihood, for the terms with Δ=1 and Y=τ̅ we have f_u(τ̅\|z;Λ,β)=S_u(τ̅-\|z;Λ,β)=exp(-Λ(τ̅_0)e^β^Tz), instead of the usual expression f_u(t\|z;Λ,β)=λ_u(t\|z;β)S_u(τ̅-\|z;Λ,β). Here λ_u denotes the hazard function corresponding to S_u and the baseline hazard function corresponding to Λ will be denoted by λ. Define (Λ̅,θ̅)=_Λ,θ_[l(Δ,Y,X,Z;Λ,θ)]=_Λ,θKL(\|Q_Λ,θ) where KL(·\|·) denotes the Kullback-Leibler divergence between two distributions. The argmax defined in (<ref>) exists. We show that the argmax can be restricted on a bounded set, from which the existence follows because of continuity. In the three steps below we deal consequently with β, Λ and γ. Step 1. First we show that for any K>0 there exists c̅>0 such that for any c≥c̅ we have inf_β̃∈ S^q-1(c \|β̃^TZ\|>K)>0, where S^q-1 is the unit circle in ^q. Suppose by contradiction that there exists K such that for any c we have inf_β̃∈ S^q-1(c \|β̃^TZ\|>K)=0. Note that the infimum is actually a minimum because S^q-1 is compact and the function is continuous. Hence, it means that for any c there exists β̃∈ S^q-1 for which (c \|β̃^TZ\|≤ K)=1. Equivalently, for any ϵ>0, there exists β̃∈ S^q-1 for which (\|β̃^TZ\|≤ϵ)=1. The closed subsets of S^q-1 defined by B_m={β̃∈ S^q-1 \| (\|β̃^TZ\|≤1/m)=1} are non-empty for all m and B_m↓ B=∩_m B_m. B cannot be empty because then (B_m^c)_m form an open covering of the compact S^q-1 and there would exists a finite sub-covering, which is impossible since all B_m are non-empty. It follows that B is not empty, which is equivalent to saying that there exists β̃∈ S^q-1 for which (\|β̃^TZ\|=0)=1. This contradicts the assumption that Var(Z) has full rank. Next, let η=1/2Λ(τ̅_0)inf_z(Y=τ̅_0\|Δ=1,Z=z) and choose K such that x≤η e^x for all x≥ K. Let β=cβ̃ with β̃∈ S^q-1, c>c̅. We will show that, as c increases, the expectation in (<ref>) becomes arbitrarily small. For fixed γ and Λ, we can write _[l(Δ,Y,X,Z;Λ,θ)] =_[Δβ^TZ_{Y<τ̅_0}-ΔΛ(Y) e^β^TZ]+R_1 =_[{Δcβ̃^TZ_{Y<τ̅_0}-ΔΛ(Y) e^cβ̃^TZ}_{β̃^TZ>0}_{c \|β̃^TZ\|>K}] +_[{Δcβ̃^TZ_{Y<τ̅_0}-ΔΛ(Y) e^cβ̃^TZ}_{β̃^TZ<0}_{c \|β̃^TZ\|>K}]+R_2, where R_1 and R_2 denote terms that are bounded in absolute value. Using _[Λ(Y)\|Z,Δ=1]>Λ(τ̅_0)inf_z(Y=τ̅_0\|Δ=1,Z=z)=2η>0, we obtain the following bound _[l(Δ,Y,X,Z;Λ,θ)] ≤_[{Δcβ̃^TZ-2Δη e^cβ̃^TZ}_{β̃^TZ>0}_{c \|β̃^TZ\|>K}] +c_[Δβ̃^TZ_{Y<τ̅_0}_{β̃^TZ<0}_{c \|β̃^TZ\|>K}]+R_2 ≤ -η_[Δ e^cβ̃^TZ_{β̃^TZ>0}_{c \|β̃^TZ\|>K}] -c_[Δ\|β̃^TZ\|_{Y<τ̅_0}_{β̃^TZ<0}_{c \|β̃^TZ\|>K}]+R_2 ≤ -η_[Δ e^cβ̃^TZ_{β̃^TZ>0}_{c̅ \|β̃^TZ\|>K}] -c_[Δ\|β̃^TZ\|_{Y<τ̅_0}_{β̃^TZ<0}_{c̅ \|β̃^TZ\|>K}]+R_2 This further leads to _[l(Δ,Y,X,Z;Λ,θ)] ≤ -η e^cK/c̅(Δ=1, β̃^TZ>0,c̅ \|β̃^TZ\|>K) -cK/c̅(Δ=1,Y<τ̅_0,β̃^TZ<0,c̅ \|β̃^TZ\|>K)+R_2 ≤ -cK/c̅(Δ=1,Y<τ̅_0,c̅ \|β̃^TZ\|>K)+R_2 ≤ -cK/c̅inf_z(Δ=1,Y<τ̅_0\|Z=z)inf_β̃∈ S^q-1(c̅ \|β̃^TZ\|>K)+R_2 Since both infimums are strictly positive, _[l(Δ,Y,X,Z;Λ,θ)] can be made arbitrarily small for c sufficiently large (and how large c should be does not depend on β̃). Hence, we can restrict the argmax on a bounded set for β. Step 2. Next we show that, there exists M>0 such that it suffices to search for the maximizer among Λ that are bounded by M. Let Λ be such that Λ(τ̅_0)>M. We can construct Λ̃(t)=cΛ(t) with c=M/Λ(τ̅_0)∈(0,1). We have Λ̃(τ̅_0)=M and λ̃=cλ. We show that _[l(Δ,Y,X,Z;Λ,θ)]< _[l(Δ,Y,X,Z;Λ̃,θ)]. Indeed we have _[l(Δ,Y,X,Z;Λ,θ)]- _[l(Δ,Y,X,Z;Λ̃,θ)] =_[-Δlog c_{Y<τ̅_0} -ΔΛ(Y)e^β^TZ+ΔcΛ(Y)e^β^TZ. .+(1-Δ)log1-ϕ(γ^TX)+ϕ(γ^TX)S_u(Y\|Z;Λ,β)/1-ϕ(γ^TX)+ϕ(γ^TX)S_u(Y\|Z;Λ̃,β)] Since S_u(Y\|Z;Λ̃,β)>S_u(Y\|Z;Λ,β) the ratio is smaller than 1 and as a result the (1-Δ) term in the expectation is negative. Hence _[l(Δ,Y,X,Z;Λ,θ)]- _[l(Δ,Y,X,Z;Λ̃,θ)] <_[-Δlog c_{Y<τ̅_0} -ΔΛ(Y)e^β^TZ+ΔcΛ(Y)e^β^TZ] =-(Δ=1,Y<τ̅_0)log c-(1-c)_[ΔΛ(Y)e^β^TZ] ≤-(Δ=1,Y<τ̅_0) log c-(1-c)_[ΔΛ(Y)e^β^TZ_{Y=τ̅_0}] =- (Δ=1,Y<τ̅_0)log c-(1-c)Λ(τ̅_0)_[Δe^β^TZ_{Y=τ̅_0}]. Since we are restricting β on a compact and Z is assumed to have bounded support, there exist c_2>0 such that e^β^TZ>c_2 a.s.. It follows that _[l(Δ,Y,X,Z;Λ,θ)]- _[l(Δ,Y,X,Z;Λ̃,θ)] <-(Δ=1,Y<τ̅_0)log c-(1-c)Λ(τ̅_0)c_2(Δ=1,Y=τ̅_0) =(Δ=1,Y<τ̅_0)(logΛ(τ̅_0)-log M) -(Λ(τ̅_0)-M)c_2(Δ=1,Y=τ̅_0) <(Δ=1,Y<τ̅_0)(Λ(τ̅_0)-M)1/M -(Λ(τ̅_0)-M)c_2(Δ=1,Y=τ̅_0) =(Λ(τ̅_0)-M){1/M(Δ=1,Y<τ̅_0) -c_2(Δ=1,Y=τ̅_0)}<0 for large enough M since (Δ=1,Y=τ̅)>0 by assumption. Hence we conclude that, there exists M such that it is sufficient to search for the maximizer among Λ's bounded by M. Step 3. We can also restrict the argmax on a bounded set for γ because as ‖γ‖→∞, for fixed values of β and Λ, the expectation converges to -∞. Indeed we have _[l(Δ,Y,X,Z;Λ,θ)] =_[Δlogϕ(γ^TX)_{γ^TX>0}]+_[Δlogϕ(γ^TX)_{γ^TX≤0}]+R_3, where R_3 denotes terms bounded in absolute value. The first term is bounded and using the same reasoning as with β it can be shown that the second term converges to -∞. We conclude that we can restrict the argmax on a bounded set, from which the existance of the argmax follows as the criteria is continuous with respect to the parameters. In what follows, we assume that the maximizer (Λ̅,θ̅) is unique. It will also be useful to characterize it as the solution of the score equation defined similarly to <cit.>. As in the proof of Theorem <ref>, consider ℋ_m={h=(h_1,h_2)∈ BV[0,τ̅_0]×^p+q: ‖ h_1‖_v+‖ h_2‖_1≤ m}, where m<∞ and ‖ h_1‖_v is the absolute value of h_1(0) plus the total variation of h_1 on the interval [0, τ̅_0 ]. Define the functions ψ_(Λ,θ),h(δ,y,x,z) =δ[h_1(y)+h_21^Tx+h^T_22z]-{ϕ(γ^Tx)-(1-δ)g(y,Λ,θ)}h_21^Tx - {δ+(1-δ)g(y,Λ,θ)}{e^β^Tz∫_0^yh_1(s)Λ(s)+e^β^TzΛ(y)h^T_22z }, where g(t,Λ,θ)=ϕ(γ^Tx)exp(-Λ(t) exp(β^T z))/1-ϕ(γ^Tx)+ϕ(γ^Tx)exp(-Λ(t) exp(β^Tz)). We will denote by ^π_n_i,iψ_(Λ,θ),h the score function for the i-th permuted sample ^π_n_i,iψ_(Λ,θ),h =1/n_i∑_j=1^n_iΔ^π_ij[h_1(Y_j^π)+h_21^TX_ij^π+h^T_22Z_ij^π] -1/n_i∑_j=1^n_i{ϕ(γ^TX_ij^π)-(1-Δ_ij^π)g_ij^π(Y_ij^π,Λ,θ)}h_21^TX_ij^π -1/n_i∑_j=1^n_i{Δ_ij^π+(1-Δ_ij^π)g_ij^π(Y_ij^π,Λ,θ)} ×{e^β^TZ_ij^π∫_0^Y_ij^πh_1(s)Λ(s)+e^β^TZ_ij^πΛ(Y_ij^π)h^T_22Z_ij^π}, where h=(h_1,h_2)=(h_1,h_21,h_22)∈ℋ_m and g_ij^π(t,Λ,θ)=ϕ(γ^TX_ij^π)exp(-Λ(t) exp(β^T Z_ij^π))/1-ϕ(γ^TX_ij^π)+ϕ(γ^TX_ij^π)exp(-Λ(t) exp(β^TZ_ij^π)). Similarly, _n_1+n_2ψ_(Λ,θ),h and ψ_(Λ,θ),h are defined using the empirical distribution of the pooled sample or the true distribution of the pooled sample =κ_1+(1-κ)_2 respectively. By definition (Λ̅,θ̅) is the solution of ψ_(Λ,θ),h!=0. Assume the maximizer (Λ̅,θ̅) defined in (<ref>) is unique. The pooled maximum likelihood estimator (Λ̅_n_1+n_2,θ̅_n_1+n_2) is a (weakly) consistent estimator of (Λ̅,θ̅). We will pursue similar ideas as <cit.> in the proofs of his Lemma 2 and Theorem 2. Comparing the arguments in <cit.> that lead to the maximum likelihood estimator in Display (12) of that paper, it is evident that the pooled estimator Λ̅_n_1+n_2 must exhibit a similar structure. In particular, Λ̅_n_1+n_2(t) = ∫_0^t d N (s)/∑_j=1^n_1+n_2 R_j(s)exp(β̅_n_1+n_2^T Z_j) {Δ_j + (1- Δ_j) g̅_j( Y_j; θ̅_n_1+n_2, Λ̅_n_1+n_2)}, where N = N_1 + N_2 is the pooled counting process, R_j(s) = _{ Y_j ≥ s} denotes the at-risk process of the j-th pooled individual and g̅_i(t; Λ, θ) = ϕ( γ^T X_i) exp(- Λ(t) exp(β^T Z_i))/1 - ϕ( γ^T X_i) + ϕ( γ^T X_i) exp(- Λ(t) exp(β^T Z_i)). Similarly, we define Λ̃_n_1+n_2(t) = ∫_0^t d N (s)/∑_j=1^n_1+n_2 R_j(s)exp(β̅^T Z_j) {Δ_j + (1- Δ_j) g̅_j( Y_j; θ̅, Λ̅)}. We have already noticed that sup_n_1,n_2Λ̃_n_1+n_2(τ) < ∞ a.s.; also, following the lines of Lemma 2 (ii) in <cit.>, there exists a non-negative and integrable function η: [0, τ̅_0] → (0, ∞) bounded away from 0 such that, for each ω∈Ω, there exists a subsequence (n_1,k_1(ω)+n_2,k_2(ω)) such that sup_t ∈ (0,τ̅_0] \| Λ̅_n_1,k_1+n_2,k_2/Λ̃_n_1,k_1+n_2,k_2(t) - η(t) \| → 0. For notational convenience, we will write n̅_k = n_1,k_1+n_2,k_2 from now on. Arguing for a fixed ω and along subsequences had similarly been done by <cit.> and <cit.> based on Helly's theorem. To prove the desired consistency of the pooled estimators, we will show that the difference of the log-likelihoods, say ℓ̅_n̅_k (Λ̅_n̅_k, θ̅_n̅_k) and ℓ̅_n̅_k(Λ̃_n̅_k, θ̅) converges to zero. Clearly, 0 ≤ ℓ̅_n̅_k (Λ̅_n̅_k, θ̅_n̅_k) - ℓ̅_n̅_k(Λ̃_n̅_k, θ̅) = 1/n̅_k∑_i=1^n̅_k[ Δ_ilogg̅_i( Y_i; Λ̅_n̅_k, θ̅_n̅_k)/g̅_i( Y_i, Λ̃_n̅_k, θ̅) + Δ_i logΔΛ̅_n̅_k( Y_i)/ΔΛ̃_n̅_k( Y_i) + Δ_i(β̅_n_k - β̅)^T Z_i +logS̅_i( Y_i; Λ̅_n̅_k, θ̅_n̅_k)/S̅_i( Y_i; Λ̃_n̅_k, θ̅)] = 1/n̅_k∑_i=1^n̅_k[ Δ_ilogϕ(γ̅^T_n_k X_i)/ϕ(γ̅^T X_i) - Δ_i Λ̅_n_k ( Y_i)exp(β̅^T_n_k X_i) + Δ_i Λ̃_n_k ( Y_i)exp(β̅^T X_i) + Δ_i logΔΛ̅_n̅_k( Y_i)/ΔΛ̃_n̅_k( Y_i) + Δ_i(β̅_n_k - β̅)^T Z_i + (1- Δ_i)logS̅_i( Y_i; Λ̅_n̅_k, θ̅_n̅_k)/S̅_i( Y_i; Λ̃_n̅_k, θ̅)] where and S̅_i(t; Λ, θ) = 1 - ϕ( γ^T X_i) + ϕ( γ^T X_i) exp(- Λ(t) exp(β^T Z_i)). The space of bounded, increasing functions with discontinuities only at τ_0,1 and τ_0,2 is separable with respect to the supremum norm. Also, Euclidean spaces are separable. Denote by (Λ_l,θ_l)_l∈ℕ a countable subset that is dense in the product of the just-described spaces. For each l ∈ℕ, by the strong law of large numbers, 1/n̅_k∑_i=1^n̅_k [ Δ_ilogϕ( γ_l^T X_i)- Δ_iΛ_l(t)exp(β^T_l Z_i)+ Δ_i logΛ_l( Y_i)+ Δ_iβ_l^T Z_i +(1- Δ_i)logS̅_i( Y_i; Λ_l, θ_l) ] converges a.s. to its expectation, i.e., for all ω∈Ω_l with ℙ̅(Ω_l)=1. From now on, we restrict ω to be in the intersection ⋂_l ∈ℕΩ_l which also has probability 1. Consequently, also due to the continuity of the likelihoods in Λ and θ, 0 ≤ ℓ̅_n̅_k (Λ̅_n̅_k, θ̅_n̅_k) - ℓ̅_n̅_k(Λ̃_n̅_k, θ̅) = 𝔼_ℙ̅[ Δ_ilogϕ( γ^^T X_i)/ϕ( γ̅^T X_i)-{Λ^(t)exp(β^^T Z_i)-Λ̅(t)exp(β̅^T Z_i)} + Δ_i logΛ^( Y_i)/Λ̅( Y_i) + Δ_i(β^* - β̅)^T Z_i + logS̅_i( Y_i; Λ^, θ^)/S̅_i( Y_i; Λ̅, θ̅)]+o(1) where the expectation is taken with respect to X,Y,Z and Λ^, β^, γ^* are fixed (depending on ω). For a.e. ω, the conditional expectation in the previous display represents a negative KL-divergence of the logistic-Cox model specified by (Λ̅, θ̅) from the model specified by ( Λ^(ω), θ^(ω)). As a consequence, it must be 0, i.e., ℓ̅(Λ^, θ^) = ℓ̅(Λ̅, θ̅) ℙ̅-a.e.. We use this fact to identify all model components, one by one; every equality below is to be understood ℙ̅-a.s.. We first consider Δ=0 and Y≥τ̅_0, for which S̅( Y; Λ^, θ^) =ϕ( γ^T X) and S̅( Y; Λ̅, θ̅)=ϕ(γ̅^T X). From this we can identify γ^=γ̅ a.s. for the logistic model. Next, for Δ=0 and Y < τ̅_0, we obtain S̅( Y; Λ^, θ^) = S̅( Y; Λ̅, θ̅), hence exp (-Λ^( Y) exp( β^T Z)) = exp (-Λ̅( Y) exp(β̅^T Z)) Upon inserting different combinations of Y and Z, we conclude that β^* = β̅ and Λ^* = Λ̅ a.s.. The following lemma establishes the consistency of randomly permuted Z-estimators. Since the proof does not make use of the specific underlying model structure, it is clear that this result holds more generally, i.e., also beyond logistic-Cox cure models. Assume the maximizer (Λ̅,θ̅) defined in (<ref>) is unique. The permutation estimators (Λ̂_n_1,1^π, θ̂_n_1,1^π) and (Λ̂_n_2,2^π, θ̂_n_2,2^π) converge in probability to (Λ̅,θ̅). First, we would like to point out that conditional convergence in probability (given a σ-algebra) is equivalent to the unconditional convergence in probability; a variant of Fact 1 in the Supporting Information of <cit.> similarly holds for the present setting. That is why we do not distinguish between conditional and unconditional consistency. To prove the consistency of the permuted estimators, we are going to employ the permutation version of the score equations, i.e., ℙ_n_i,i^πψ_(Λ, θ),h!=0 for all indexing h, i=1,2. So far, we know by definition that ℙ_n_i,i^πψ_(Λ̂^π_n_i,i, θ̂^π_n_i,i),h = 0 and that ℙ_n_1+n_2ψ_(Λ̅_n_1+n_2, θ̅_n_1+n_2),h = 0. Also, since n_1ℙ_n_1,1^πψ_( Λ, θ),h + n_2ℙ_n_2,2^πψ_( Λ, θ),h = (n_1+n_2)ℙ_n_1+n_2ψ_( Λ, θ),h, we have that n_1ℙ_n_1,1^πψ_(Λ̅_n_1+n_2, θ̅_n_1+n_2),h = - n_2ℙ_n_2,2^πψ_(Λ̅_n_1+n_2, θ̅_n_1+n_2),h. Thus, because both of these permuted expressions are connected, we will only focus on the index i=1 from now on. Furthermore, upon integrating out all permutations, it is easy to see that the (conditional) expectation is [ℙ_n_1,1^πψ_(Λ̅_n_1+n_2, θ̅_n_1+n_2),h \| Y_i, Δ_i, X_i, Z_i : i = 1, …, n_1+n_2] = 0. Additionally, straightforward and standard algebra for permuted linear statistics for the conditional variance leads to Var[ℙ_n_1,1^πψ_(Λ̅_n_1+n_2, θ̅_n_1+n_2),h \| Y_i, Δ_i, X_i, Z_i : i = 1, …, n_1+n_2] = O_p((n_1+n_2)^-1). Consequently, Chebychev's inequality (applied to the conditional distribution) verifies that the permutation-based score equations evaluated at the point (Λ̅_n_1+n_2, θ̅_n_1+n_2) all converge to 0 in probability. Similar convergences in probability (not necessarily to zero) also hold for other evaluation points. Hence, the pooled estimator is asymptotically a solution to the permutation-based score equations. Now, since (Λ̂^π_n_1,1, θ̂^π_n_1,1) is another (finite sample) solution and the “true” solution (Λ̅, θ̅) is assumed to be unique, the permuted estimator must approach the pooled estimator in probability as the sample size goes in infinity. Anything else would contradict the continuity of the map (Λ, θ) ↦𝔼̅_ℙ̅( ℓ̅(Λ, θ)). We will show this last step by reductio ad absurdum. Thus, we assume for a moment that consistency of the permuted statistic does not hold which entails that there exists a subsequence (n_1',n_2')⊂ (n_1, n_2) for which every further subsequence (n”_1,n”_2) ⊂ (n_1',n_2') results in ℙ(lim sup_n_1”, n_2”→∞Λ̂^π_n_1”,1 - Λ̅_∞∨θ̂^π_n_1”,1 - θ̅ > 0 ) >0, also similarly with (Λ̅, θ̅) replaced by (Λ̅_n”_1+n”_2, θ̅_n”_1+n”_2), and lim sup_n_1”, n_2”→∞\|ℙ_n”_1,1^πψ_(Λ̂^π_n_1”,1, θ̂^π_n_1”,1),h - ℙ_n”_1,1^πψ_(Λ̅_n_1”+n”_2, θ̅_n_1”+n”_2),h\| >0 for some index h with positive probability along that subsequence. But the first term in the difference in (<ref>) is 0 by definition, and the second converges to zero almost surely along subsequence. This yields a contradiction to the inequality in (<ref>). Assume the maximizer (Λ̅,θ̅) defined in (<ref>) is unique. Conditionally on the observations, the process ⟨ n_i^1/2(Λ̂^π_n_i,i-Λ̅_n_1+n_2),n_i^1/2(θ̂_n_i,i^π-θ̅_n_1+n_2)⟩, i=1,2 defined as in (<ref>) and indexed by h∈ℋ_m converges weakly in l^∞(ℋ_m) to a tight Gaussian process G^_i in l^∞(ℋ_m), in outer probability. We will apply Theorem <ref>. We need to show that the sample specific estimators (Λ̂^π_n_i,i,θ̂^π_n_i,i) satisfy the conditions of Theorem 3.3.1 in <cit.>. Consistency of the estimators was shown in Lemmas <ref> and <ref> above. Verification of the other conditions can be done as in <cit.>. We omit the other details here since the proof goes along the same lines as points a)-c) below. a) To verify condition (<ref>) of Theorem <ref>, it suffices to show that for any sequence ϵ_n_i→ 0, sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i, ‖β-β̅_n_1+n_2‖≤ϵ_n_i, ‖γ-γ̅_n_1+n_2‖≤ϵ_n_i\|(^π_n_i,i-_n_1+n_2)ψ_(Λ,θ),h-(^π_n_i,i-_n_1+n_2)ψ_(Λ̅_n_1+n_2,θ̅_n_1+n_2),h\|/n_i^-1/2∨‖β-β̅_n_1+n_2‖∨‖γ-γ̅_n_1+n_2‖∨‖Λ-Λ̅_n_1+n_2‖_∞ converges to zero in probability given the data. For simplicity, we can write (^π_n_i,i-_n_1+n_2)ψ_(Λ,θ),h-(^π_n_i,i-_n_1+n_2)ψ_(Λ̅_n_1+n_2,θ̅_n_1+n_2),h =∑_j=1^6(^π_n_i,i-_n_1+n_2)a_j,h where a_1,h(δ,y,x,z) =-h_21^Tx{ϕ(γ^Tx)-ϕ(γ̅^T_n_1+n_2x)} a_2,h(δ,y,x,z) = (1-δ) h_21^Tx{g(y,Λ,θ)-g(y,Λ̅_n_1+n_2,θ̅_n_1+n_2)} a_3,h(δ,y,x,z) = δ{e^β^Tz∫_0^yh_1(s)Λ(s)-e^β̅^T_n_1+n_2z∫_0^yh_1(s)Λ̅_n_1+n_2(s)} a_4,h(δ,y,x,z) = δ h^T_22z {e^β^TzΛ(y)-e^β̅^T_n_1+n_2zΛ̅_n_1+n_2(y)} a_5,h(δ,y,x,z) = (1-δ) {g(y,Λ,θ)e^β^Tz∫_0^yh_1(s)Λ(s). -.g(y,Λ̅_n_1+n_2,θ̅_n_1+n_2)e^β̅^T_n_1+n_2z∫_0^yh_1(s)Λ̅_n_1+n_2(s)} a_6,h(δ,y,x,z) = (1-δ) h^T_22z {g(y,Λ,θ)e^β^TzΛ(y). -.g(y,Λ̅_n_1+n_2,θ̅_n_1+n_2)e^β̅^T_n_1+n_2zΛ̅_n_1+n_2(y)}. Next we consider the third term. The other terms can be handled similarly. First note that, by Lemma <ref>, Λ̅_n_1+n_2 and θ̅_n_1+n_2 are consistent estimates of Λ̅ and θ̅, and as a result they are bounded on a set of probability converging to one. From a Taylor expansion we have (^π_n_i,i-_n_1+n_2)a_3,h =(β-β̅_n_1+n_2)^T∫ zδ e^β̅^T_n_1+n_2z∫_0^yh_1(s)Λ(s) (^π_n_i,i-_n_1+n_2)(δ,y,x,z) +∫δ e^β̅^T_n_1+n_2z∫_0^yh_1(s)(Λ-Λ̅_n_1+n_2)(s) (^π_n_i,i-_n_1+n_2)(δ,y,x,z) +o^_P(∫δ e^β̅^T_n_1+n_2z∫_0^yh_1(s)(Λ-Λ̅_n_1+n_2)(s) (^π_n_i,i-_n_1+n_2)(δ,y,x,z)) +o^_P(‖β-β̅_n_1+n_2‖) For the first term, since the class of functions that we are integrating is Donsker and uniformly bounded, by Theorem 3.7.2 in <cit.> it follows that, conditionally on the observations sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i∫ zδ e^β̅^Tz∫_0^yh_1(s)Λ(s) (^π_n_i,i-_n_1+n_2)(δ,y,x,z)=o^_P(1). The second term can be rewritten as ∫_0^τ̅_0 D_n(s)h_1(s)(Λ-Λ̅_n_1+n_2)(s) where D_n(s)=∫δ_{y>s}e^β̅^Tz(^π_n_i,i-_n_1+n_2)(δ,y,x,z). By integration by parts and the chain rule we have ∫_0^τ̅_0 D_n(s)h_1(s)(Λ-Λ̅_n_1+n_2)(s) = D_n(τ̅_0)h_1(τ̅_0)(Λ-Λ̅_n_1+n_2)(τ̅_0)- ∫_0^τ̅_0 (Λ-Λ̅_n_1+n_2)[D_n(s)h_1(s)] =D_n(τ̅_0)h_1(τ̅_0)(Λ-Λ̅_n_1+n_2)(τ̅_0)-∫_0^τ̅_0 (Λ-Λ̅_n_1+n_2)(s) D_n(s) h_1(s) +∫δ (Λ-Λ̅)(y)h_1(y) e^β̅^Tz (^π_n_i,i-_n_1+n_2)(δ,y,x,z) + ∫δ (Λ̅-Λ̅_n_1+n_2)(y)h_1(y) e^β̅^Tz (^π_n_i,i-_n_1+n_2)(δ,y,x,z) Again, by Theorem 3.7.2 in <cit.>, it follows that, conditionally on the observations D_n=o^_P(1). Since h_1 is bounded, it follows sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i\|D_n(τ̅_0)h_1(τ̅_0)(Λ-Λ̅_n_1+n_2)(τ̅_0)\|/‖Λ-Λ̅_n_1+n_2‖_∞=o^_P(1). In addition we also have that conditionally on the observations sup_s∈[0,τ_0]\|D_n(s)\|=o^_P(1) and since h_1 is of bounded variation sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i\|∫_0^τ̅_0(Λ-Λ̅_n_1+n_2)(s) D_n(s) h_1(s)\|/‖Λ-Λ̅_n_1+n_2‖_∞ ≤sup_t∈[0,τ̅_0]\|D_n(s)\| ∫_0^τ̅_0 \| h(s)\|=o_P^(1). Since ‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i implies ‖Λ-Λ̅‖_∞≤ϵ̃_n_i for some ϵ̃_n_i→ 0, the class {g_Λ(y,δ,z)=δ(Λ-Λ̅)(y)h_1(y)e^β̅^Tz: ‖Λ-Λ̅‖_∞≤ϵ̃_n_i} is a Donsker class (product of bounded variation functions, uniformly bounded) and _[Δ(Λ-Λ̅)(Y)^2h_1(Y)^2 e^2β̅^TZ]=O(ϵ̃_n^2)=o(1) , we have that, conditionally on the data, sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i√(n)∫δ (Λ-Λ̅)(y)h_1(y) e^β̅^Tz (^π_n_i,i-_n_1+n_2)(δ,y,x,z)=o^_P(1). Finally, since ‖Λ̅_n_1+n_2-Λ̅‖_∞→ 0 a.s., by Proposition A.5.3 in <cit.> it follows that, conditionally on the data, ∫δ (Λ̅-Λ̅_n_1+n_2)(y)h_1(y) e^β̅^Tz (^π_n_i,i-_n_1+n_2)(δ,y,x,z)=o^_P(1). Combining all the results we obtain that sup_‖Λ-Λ̅_n_1+n_2‖_∞≤ϵ_n_i, ‖β-β̅_n_1+n_2‖≤ϵ_n_i, ‖γ-γ̅_n_1+n_2‖≤ϵ_n_i\|(^π_n_i,i-_n_1+n_2)a_3,h\|/n_i^-1/2∨‖β-β̅_n_1+n_2‖∨‖γ-γ̅_n_1+n_2‖∨‖Λ-Λ̅_n_1+n_2‖_∞ converges to zero in probability, given the data. The terms related to the other a_j can be treated similarly. b) Next we check condition (<ref>). From (<ref>) it follows in particular that, conditionally on the data, √(n_i)(^π_n_i,i-_n_1+n_2)ψ_(Λ̅,θ̅),h-√(n_i)(^π_n_i,i-_n_1+n_2)ψ_(Λ̅_n_1+n_2,θ̅_n_1+n_2),h=o_P^(1) almost surely. Hence, it is sufficient to show that √(n_i)(^π_n_i,i-_n_1+n_2)ψ_(Λ̅,θ̅),h⇝ Z_1 on l^∞(ℋ_m) in outer probability, where Z_1 is a tight random element (actually a Gaussian process). This follows from Theorem 3.7.1. in <cit.> since the class of functions {ψ_(Λ̅,θ̅),h: h∈ℋ_m} is Donsker and bounded. This is already shown in step 1 of the proof of Theorem 3 in <cit.> (the class of functions is the same, just evaluated at a different point (Λ̅,θ̅)). c) Since the functions ψ_(Λ,θ),h, h∈ℋ_m are the same as in <cit.>, it can be proved in the same way that ψ_(Λ̅,θ̅),h is Fréchet-differentiable at (Λ̅,θ̅) and the derivative is given by (ψ̇_(Λ̅,θ̅))((Λ,θ)-(Λ̅,θ̅))(h)=∫_0^τ̅_0σ̅_(1)(h) (Λ-Λ̅)(t)+(θ-θ̅)^Tσ̅_(2)(h) where σ̅_(1), σ̅_(2) are defined as in (<ref>)-(<ref>) respectively, with __i replaced by _ and evaluated at (Λ̅,θ̅) instead of (Λ_i,θ_i). Also the proof that the derivative is continuously invertible remains the same as in <cit.>. d) For condition (<ref>), consider a sequence (Λ_n_1+n_2,θ_n_1+n_2) converging to (Λ̅,θ̅) as n_1+n_2→∞. We have (ℙ_n_1+n_2ψ_θ_n_1+n_2,h - ℙ_n_1+n_2ψ_θ̅,h) - (ℙ̅ψ_θ_n_1+n_2,h - ℙ̅ψ_θ̅,h) =(ℙ_n_1+n_2-ℙ̅) ψ_θ_n_1+n_2,h - (ℙ_n_1+n_2-ℙ̅) ψ_θ̅,h=∑_j=1^6(ℙ_n_1+n_2-ℙ̅) a̅_j,h where a̅_j,h are defined as in (<ref>) replacing (Λ,θ) and (Λ̅_n_1+n_2,θ̅_n_1+n_2) by (Λ_n_1+n_2,θ_n_1+n_2) and (Λ̅,θ̅) respectively. We can deal with this similarly to what we did to show (<ref>). The difference is that now (Λ_n_1+n_2,θ_n_1+n_2) and (Λ̅,θ̅) are fixed and we consider the class of functions with respect to h∈ℋ_m. For example, for the term corresponding to a̅_3,h we have (ℙ_n_1+n_2-ℙ̅) a̅_3,h =(β_n_1+n_2-β̅)^T∫ zδ e^β̅^Tz∫_0^yh_1(s)Λ̅(s) (ℙ_n_1+n_2-ℙ̅)(δ,y,x,z) +∫δ e^β̅^Tz∫_0^yh_1(s)(Λ_n_1+n_2-Λ̅)(s) (ℙ_n_1+n_2-ℙ̅)(δ,y,x,z) +o^_P(∫δ e^β̅^Tz∫_0^yh_1(s)(Λ_n_1+n_2-Λ̅)(s) (ℙ_n_1+n_2-ℙ̅)(δ,y,x,z)) +o^_P(‖β_n_1+n_2-β̅‖). The integral in the first term converges to zero since the class of functions that we are integrating is Donsker and uniformly bounded. The second term can be rewritten as ∫_0^τ̅_0D̅_n_1+n_2(s)h_1(s)(Λ_n_1+n_2-Λ̅)(s) where D̅_n_1+n_2(s)=∫δ_{y>s}e^β̅^Tz(ℙ_n_1+n_2-ℙ̅)(δ,y,x,z). By integration by parts and the chain rule we again have ∫_0^τ̅_0D̅_n_1+n_2(s)h_1(s)(Λ_n_1+n_2-Λ̅)(s) =D̅_n_1+n_2(τ̅_0)h_1(τ̅_0)(Λ_n_1+n_2-Λ̅)(τ̅_0)-∫_0^τ̅_0 (Λ_n_1+n_2-Λ̅)(s) D̅_n_1+n_2(s) h_1(s) +∫δ (Λ_n_1+n_2-Λ̅)(y)h_1(y) e^β̅^Tz (ℙ_n_1+n_2-ℙ̅)(δ,y,x,z). By the Glivenko-Cantelli theorem sup_s∈[0,τ̅_0]D̅_n_1+n_2(s)=o_P(1). Since h_1∈ℋ_m are uniformly bounded, it follows that sup_h_1∈ℋ_m\|D̅_n_1+n_2(τ̅_0)h_1(τ̅_0)(Λ_n_1+n_2-Λ̅)(τ̅_0)\|/‖Λ_n_1+n_2-Λ̅‖_∞=o_P(1). In addition, since h_1 are functions of bounded variation and uniformly bounded norm, sup_h_1∈ℋ_m\|∫_0^τ̅_0 (Λ_n_1+n_2-Λ̅)(s) D̅_n_1+n_2(s) h_1(s)\|/‖Λ_n_1+n_2-Λ̅‖_∞ ≤sup_s∈[0,τ̅_0]\|D̅_n_1+n_2(s)\| ∫_0^τ̅_0 \| h_1(s)\|=o_P(1). Finally, from Theorem 2.11.23 in <cit.> it follows that sup_h_1∈ℋ_m√(n_1+n_2)∫δ (Λ_n_1+n_2-Λ̅)(y)h_1(y) e^β̅^Tz (ℙ_n_1+n_2-ℙ̅)(δ,y,x,z)/‖Λ_n_1+n_2-Λ̅‖_∞ is bounded in probability. Combining all the results we obtain sup_h_1∈ℋ_m(ℙ_n_1+n_2-ℙ̅)a_3,h=o_P( ‖θ_n_1+n_2-θ̅‖∨‖Λ_n_1+n_2-Λ̅‖_∞). The other terms can be handled similarly obtaining ‖(ℙ_n_1+n_2ψ_θ_n_1+n_2,h - ℙ_n_1+n_2ψ_θ̅,h) - (ℙ̅ψ_θ_n_1+n_2,h - ℙ̅ψ_θ̅,h)‖ =o_P(‖θ_n_1+n_2-θ̅‖∨‖Λ_n_1+n_2-Λ̅‖_∞). We proceed similarly to the proof of Theorem <ref>. From Lemma <ref>, it follows that the process ⟨ n_i^1/2(Λ̂^π_n_i,i-Λ̅_n_1+n_2),n_i^1/2(θ̂^π_n_i,i-θ̅_n_1+n_2)⟩_i=1,2(h) converges to a Gaussian process G^=(G_1^,G_2^) with G_2^=-√(κ/(1-κ))G_1^. We start by deriving the weak convergence of the process √(a_n)(Ŝ^π_n_1,1-S̅_n_1+n_2,Ŝ^π_n_2,2-S̅_n_1+n_2), where a_n=√(n_1n_2/(n_1+n_2)). By a series of Taylor expansions we can write √(a_n){Ŝ^π_n_1,1(t\|z)-S̅_n_1+n_2(t\|z)} =√(a_n){exp(-Λ̂^π_n_1,1(t)e^β̂^π^T_n_1,1z)-exp(-Λ̅_n_1+n_2(t)e^β̅^T_n_1+n_2z)} =-√(a_n){Λ̂^π_n_1,1(t)e^β̂^π^T_n_1,1z-Λ̅_n_1+n_2(t)e^β̅^T_n_1+n_2z}exp(-Λ̅_n_1+n_2(t)e^β̅^T_n_1+n_2z)+R_1 =-√(a_n){Λ̂^π_n_1,1(t)-Λ̅_n_1+n_2(t)}e^β̅^T_n_1+n_2zexp(-Λ̅_n_1+n_2(t)e^β̅^T_n_1+n_2z) - √(a_n){e^β̂^π^T_n_1,1z-e^β̅^T_n_1+n_2z}Λ̅_n_1+n_2(t)exp(-Λ̅_n_1+n_2(t)e^β̅^T_n_1+n_2z)+R_1+R_2 =-√(a_n){Λ̂^π_n_1,1(t)-Λ̅_n_1+n_2(t)}e^β̅^Tzexp(-Λ̅(t)e^β̅^Tz) - √(a_n){β̂^π_n_1,1-β̅_n_1+n_2}^Tze^β̅^TzΛ̅(t)exp(-Λ̅(t)e^β̅^Tz)+R_1+R_2+R_3 where the remainder terms converge to zero in probability. Considering functions h∈ℋ_m of the form h_t,z =(h_1;t,z,h_2;t,z) =(_[0,t](·)e^β̅^Tzexp(-Λ̅(t)e^β̅^Tz),(0_p,ze^β̅^TzΛ̅(t)exp(-Λ̅(t)e^β̅^Tz))), where 0_p denotes a zero vector in ^p, we have √(a_n){Ŝ^π_n_1,1(t\|z)-S̅_n_1+n_2(t\|z)} =√(1-κ)⟨ n_1^1/2(Λ̂^π_n_1,1-Λ̅_n_1+n_2),n_1^1/2(θ̂^π_n_1,1-θ̅_n_1+n_2)⟩(h_t,z)+o_P^(1) from which we conclude that, given the data, the process √(a_n){Ŝ^π_n_1,1(t\|z)-S̅_n_1+n_2(t\|z)} converges weakly in D^∞([0,τ̅_0]) to a mean zero Gaussian process G̅_1,z with covariance ρ_1,z(s,t) =Cov(G̅_1,z(s),G̅_1,z(t)) =(1-κ){∫_0^τ_0,1h_1;s,z(u)σ̅_(1)^-1(h_t,z)(u)Λ̅(u)+h^T_2;s,zσ̅_(2)^-1(h_t,z)}, where σ̅_(1), σ̅_(2) are defined as in (<ref>)-(<ref>) respectively, with __i replaced by _ and evaluated at (Λ̅,θ̅) instead of (Λ_i,θ_i) Defining MST_u,z as the conditional expected lifetime of the uncured in the pooled sample, we have √(a_n)(MST^π_u,1,z-MST_u,z) =√(a_n)∫_0^τ̅_0{Ŝ^π_n_1,1(t\|z)-S̅_n_1+n_2(t\|z)} t -√(a_n){τ̅_0-Y^π_1,(m_1)}Ŝ^π_n_1,1(τ̅_0)-√(a_n){τ̅_0-Y_(m)}S̅_n_1+n_2(τ̅_0), where Y^π_1,(m_1) and Y_(m) denote the largest uncensored observation in the first permuted sample and the pooled sample respectively. As in the proof of Theorem <ref>, the second term and third term in the right hand side of the previous equation can be shown to converge to zero in probability. Considering the map ψ:D[0,τ̅_0]→ ψ(ξ)=∫_0^τ̅_0ξ(u) u which is Hadamard-differentiable, it follows that, given the data, √(a_n)(MST^π_u,1,z-MST_u,z,MST^π_u,2,z-MST_u,z) converges weakly to a two dimensional Gaussian random vector (N_1,N_2)=(∫_0^τ̅_0G̅_1,z(u) u,∫_0^τ̅_0G̅_2,z(u) u)=(N_1,-κ/1-κN_1). Taking the difference of both entries of the pair, we conclude that, given the data, √(a_n)(MST^π_u,1,z-MST^π_u,2,z) converges weakly (in probability) to a mean-zero Gaussian random variable with variance σ^π2_z=1/(1-κ)^2∫_0^τ̅_0∫_0^τ̅_0ρ_1,z(s,t) s t, where ρ_1,z is defined in (<ref>). This concludes the proof. § PERMUTATION OF Z-ESTIMATORS IN A TWO-SAMPLE SET-UP In this appendix, we discuss the asymptotic properties of randomly permuted Z-estimators. For this, we consider a two independent samples set-up with n_1 and n_2 i.i.d. random vectors W_11, …, W_1n_1∼_1 and W_21, …, W_2n_2∼_2, respectively. Let Θ be a subset of a Banach space, Ψ_n_1,1, Ψ_n_2,2 : Θ→𝕃 be random maps, and Ψ: Θ→𝕃 be a deterministic map. Solutions (or approximate solutions) θ̂_n_i,i to the equations Ψ_n_i,i(θ)!=0 will be called Z-estimators. Due to the i.i.d. set-up, we assume the structure Ψ_n_i,i(θ)h = ℙ_n_i,iψ_θ,h, for given measurable functions ψ_θ,h indexed by Θ and h ∈ℋ for some index set ℋ, where ℙ_n_i,i denotes the i-th empirical process. Thus, we understand the equation system in the space 𝕃 = ℓ^∞(ℋ). For the random permutation approach, we randomly re-assign the n_1+n_2 observations of the pooled sample (W_11,…, W_1n_1, W_21, …, W_2 n_2)=:(W_1, …, W_n_1+n_2) to the groups 1 and 2 without changing the original sample sizes. For a random permutation π of the numbers 1, …, n_1+n_2, the permuted samples can be expressed as W_π(1), …, W_π(n_1) and W_π(n_1+1), …, W_π(n_1+n_2). For notational convenience, we denote the permuted samples by W^π_i1, …, W^π_in_i, for sample group i=1,2, and the corresponding i-th permutation empirical process by ℙ_n_i,i^π. Let Ψ_n_i,i^π(θ)h = ℙ_n_i,i^πψ_θ,h!=0 for all h∈ℋ be the estimating equation corresponding to Ψ_n_i,i(θ)!=0, just based on the i-th permuted sample. We denote the (approximate) solution to the i-th permuted estimating equation by θ^π_n_i,i. For future uses, let 𝔾^π_n_i,i = √(n_i)(ℙ_n_i,i^π - ℙ_n_1+n_2) be the i-th normalized permutation empirical process, where ℙ_n_1+n_2 denotes the empirical process of the pooled sample. The centering at ℙ_n_1+n_2 seems reasonable, as this has an interpretation as a conditional expectation: E(ℙ_n_i,i^πψ_θ,h \| W_ij: i=1,2; j=1,…, n_i) = ℙ_n_1+n_2ψ_θ,h . Let θ̅_n_1+n_2 be the (approximate) solution to ℙ_n_1+n_2ψ_θ,h!=0 for all h ∈ℋ. The following theorem represents a version of Theorem 3.3.1 of <cit.> for the random permutation-based estimators. Assume that n_1/n_1+n_2→λ∈ (0,1) as n_1+n_2→∞, define = λ_1 + (1-λ) _2, and assume that Theorem 3.3.1 holds for each sample-specific Z-estimator θ̂_n_1,1 and θ̂_n_2,2. Let the criterion functions ψ_·, h be such that 𝔾_n_i,i^π (ψ_θ_n_i,i^π, h - ψ_θ̅_n_1+n_2, h) _ℋ = o_P^(1 + √(n_i)θ^π_n_i,i - θ̅_n_1+n_2) . Conditionally on W_11, W_21, W_12, W_22, …, assume that (√(n_i)(^π_n_i,i - _n_1+n_2) ψ_θ̅_n_1+n_2,h)_i=1^2 ⇝ (Z_1, Z_2) on (ℓ^∞(ℋ))^2 in outer probability, where (Z_1, Z_2) is a tight random element. We assume that θ↦ψ_θ, h is Fréchet-differentiable in ℓ^∞(ℋ) at θ̅ with a continuously invertible derivative ψ̇_θ̅, h, and that, for any sequence (θ_n_1+n_2)_n_1,n_2 converging to θ̅, (ℙ_n_1+n_2ψ_θ_n_1+n_2,h - ℙ_n_1+n_2ψ_θ̅,h) - (ℙ̅ψ_θ_n_1+n_2,h - ℙ̅ψ_θ̅,h) _ℋ = o_P^(θ_n_1+n_2 - θ̅) as n_1+n_2 →∞. If θ^π_n_i,i and θ̅_n_1+n_2 satisfy ^π_n_i,iψ_θ^π_n_i,i,h_ℋ= o_P^(n^-1/2), i=1,2, and _n_1+n_2ψ_θ̅_n_1+n_2,h_ℋ= o_P^(n^-1/2), respectively, and if all three estimators converge in outer probability to θ̅, then √(n_i) (ψ̇_θ̅, h) (θ_n_i,i^π - θ̅_n_1+n_2) = - √(n_i)(_n_i,i^π - _n_1+n_2) ψ_θ̅_n_1+n_2,h + o_P^(1) ⇝ -(Z_1,Z_2) as n_1+n_2 →∞ conditionally on W_11, W_21, W_12, W_22, …, in outer probability. Finally, ( √(n_1) (θ^π_n_1,1 - θ̅_n_1+n_2) , √(n_2) (θ^π_n_2,2 - θ̅_n_1+n_2) ) ⇝ -((ψ̇_θ̅, h)^-1 Z_1, (ψ̇_θ̅, h)^-1 Z_2) conditionally on W_11, W_21, W_12, W_22, …, in outer probability. Note that, due to the equality ^π_n_2,2 - _n_1+n_2 = - n_1/n_2 (^π_n_1,1 - _n_1+n_2), Z_1 and Z_2 are perfectly negatively linearly correlated: Z_2 = - √(λ/1-λ) Z_1. The essential steps of this proof are similar to those in the proof of Theorem 3.3.1 of <cit.>. But for the sake of completeness, we shall present the whole proof. The assumed consistencies of the pooled and the permuted estimators and then assumption (<ref>) entail that √(n_i)(_n_1+n_2ψ_θ^π_n_i,i,h - _n_1+n_2ψ_θ̅_n_1+n_2,h) = √(n_i)(_n_1+n_2ψ_θ^π_n_i,i,h - _n_i,i^πψ_θ_n_i,i^π,h) + o_P^(1) = - 𝔾_n_i,i^π ( ψ_θ^π_n_i,i,h - ψ_θ_n_i,i^π,h) + o_P^(1) = - √(n_i)(_n_i,i^π - _n_1+n_2) ψ_θ̅_n_1+n_2,h + o_P^(1 + √(n_i)θ_n_i,i^π - θ̅_n_1+n_2). The consistencies of θ^π_n_i,i and θ̅_n_1+n_2 for θ̅ in combination with the approximation (<ref>) applied twice imply that the norm of the left-hand side of (<ref>) equals √(n_i)( ℙ̅ψ_θ^π_n_i,i,h - ℙ̅ψ_θ̅_n_1+n_2,h_ℋ + o_P^(θ^π_n_i,i - θ̅_n_1+n_2 ) ). Additionally, the Fréchet-differentiability of θ↦ψ_θ, h at θ̅ and the continuous invertibility of the derivative ψ̇_θ̅, h respectively imply that ψ_θ, h - ψ_θ̅, h_ℋ = (ψ̇_θ̅, h)(θ-θ̅) _ℋ + o(θ-θ̅) and that the right-hand side in the previous display is bounded below by c θ-θ̅ + o(θ-θ̅) for some positive constant c. Combine this with (<ref>) and (<ref>) to see that √(n_i)θ^π_n_i,i - θ̅_n_1+n_2 (c + o_P(1)) ≤ O_P(1) + o_P(1 + √(n_i)θ_n_i,i^π - θ̅_n_1+n_2), conditionally on W_11, W_21, W_12, W_22, … in probability. Thus, in the same manner, θ^π_n_i,i is √(n_i)-consistent for θ̅_n_1+n_2 in norm. Next, apply the approximation in (<ref>) to the left-hand side of (<ref>) and use the Fréchet-differentiability of θ↦ψ_θ, h at θ̅ to find that (<ref>) equals √(n_i) (ψ̇_θ̅, h) (θ_n_i,i^π - θ̅_n_1+n_2) + o_P^(√(n_i)θ_n_i,i^π - θ̅_n_1+n_2). Since o_P^(√(n_i)θ_n_i,i^π - θ̅_n_1+n_2) and also o_P^*(1+√(n_i)θ_n_i,i^π - θ̅_n_1+n_2) in the right-hand side of (<ref>) are both o_P(1), the assertion given in (<ref>) follows from assumption (<ref>). The continuity of (ψ̇_θ̅, h)^-1 together with the continuous mapping theorem and the just established conditional weak convergence (<ref>) in outer probability imply the corresponding conditional weak convergence ( √(n_1) (θ^π_n_1,1 - θ̅_n_1+n_2) , √(n_2) (θ^π_n_2,2 - θ̅_n_1+n_2) ) ⇝ -((ψ̇_θ̅, h)^-1 Z_1, (ψ̇_θ̅, h)^-1 Z_2) in outer probability. § R CODE FOR ACCESSING THE BREAST CANCER DATA SET [Acknowledgments] The authors would like to thank Sarah Friedrich for useful comments regarding the data set. imsart-number
http://arxiv.org/abs/2307.00515v1	20230702084307	Beam current from downramp injection in electron-driven plasma wakefields	[ "Céline Hue", "Anton Golovanov", "Sheroy Tata", "Sébastien Corde", "Victor Malka" ]	physics.plasm-ph	[ "physics.plasm-ph", "physics.acc-ph" ]	A re-examination to the SCoTLASS problems for SPCA and two projection-based methods for themt1 [ ============================================================================================== We study the stability of plasma wake wave and the properties of density-downramp injection in an electron-driven plasma accelerator. In this accelerator type, a short high-current electron bunch (generated by a conventional accelerator or a laser-wakefield acceleration stage) drives a strongly nonlinear plasma wake wave (blowout), and accelerated electrons are injected into it using a sharp density transition which leads to the elongation of the wake. The accelerating structure remains highly stable until the moment some electrons of the driver reach almost zero energy, which corresponds to the best interaction length for optimal driver-to-plasma energy transfer efficiency. For a particular driver, this efficiency can be optimized by choosing appropriate plasma density. Studying the dependence of the current of the injected bunch on driver and plasma parameters, we show that it does not depend on the density downramp length as long as the condition for trapping is satisfied. Most importantly, we find that the current of the injected bunch primarily depends on just one parameter which combines both the properties of the driver (its current and duration) and the plasma density. § INTRODUCTION Plasma accelerators that rely on high-amplitude plasma wakefields are promising several orders of magnitude higher acceleration gradients than in conventional radio-frequency accelerators <cit.>. The two main types of plasma accelerators are laser–wakefield accelerators (LWFAs) based on driving a plasma wake with a short intense laser pulse <cit.> and plasma–wakefield accelerators (PWFAs) based on using a short high-current particle bunch as a driver <cit.>. LWFAs have demonstrated rapid growth both in terms of the energy of accelerated electrons (reaching the energies of 8 GeV at the distance of 20 cm <cit.>) and stability of beam properties <cit.>. For PWFA, experimental studies have demonstrated high-gradient acceleration <cit.> and efficient energy transfer from the driver to the witness <cit.>. Despite some advantages such as the long dephasing length and a more stable wakefield structure, PWFAs saw comparatively less development than LWFAs because the sources of short (comparable to plasma wavelength) electron bunches with a high enough current to excite a nonlinear wake were not widely available. Recently, the concept of hybrid LWFA–PWFA multi-staged plasma accelerators <cit.> based on the idea of using electron bunches from the first LWFA stage to drive a second PWFA stage has started gaining popularity, broadening the possibilities for experimental PWFA studies. LWFA-produced electron bunches are very short and high-current, so they naturally have excellent parameters to drive a highly-nonlinear (blowout) plasma wake. The density-downramp injection technique also proved to be an effective way of injecting electrons in the PWFA stage <cit.>. Even though the electron-driven second stage cannot significantly surpass the performance of the first LWFA stage in terms of the total energy of the accelerated electrons, due to its stable nature it can serve as a “quality booster” by generating bunches with improved properties <cit.>. However, although relatively high energy transfer efficiency is achieved and high-quality beams with hundreds of MeVs are predicted, the energy stability for such accelerator in previous research is still comparable to the a single-stage LWFA accelerators <cit.>, and more studies are required to understand the parameter dependence and physics behind this stage. For PWFA to be efficient, one of the important steps is to optimize the driver-to-witness energy transfer efficiency. The beam currents of both the driver and the witness beams play decisive roles in optimizing the energy transfer efficiency. Few studies so far have systematically focused on controlling the beam currents produced in plasma-based accelerators. The production in LWFA of the current profile finely tuned for the use in the second stage was studied in <cit.>, but studies for PWFAs are lacking. In this paper, we focus on performance of PWFA based on density-downramp injection of electrons. One important step in calculating and optimising the energy transfer efficiency is to understand the beam dynamics. Previous research described such important phenomena as the hosing instability experienced at the tail of the bunch <cit.>, beam head erosion <cit.>, energy depletion <cit.>, and the transformer ratio <cit.>. Yet, driver parameters used in these studies corresponded mostly to electron beams produced by conventional linear accelerators, which are very different from beams produced in LWFA stage for hybrid accelerators. In this article, the dynamics of flattop–current electron drivers with a total beam charge of 100–500 pC, 100–300 MeV energy typical for the LWFA-produced bunches <cit.> is studied. In Section <ref>, the stability of the wakefield and driver-to-plasma energy transfer efficiency are discussed. In Section <ref>, we numerically study density-downramp injection in the PWFA stage and the dependence of the current of the injected electron bunch on the parameters of the driver, the plasma, and the downramp. Despite having such a multidimensional parameter space, we demonstrate that the injected current is determined mostly by one parameter, the effective current J_ = J_b (k_ξ_b)^2/3, which combines both the properties of the driver (its current J_b and length ξ_b) and the plasma (wavenumber k_). The dependence of the witness current on this parameter is linear. We also show the limitation of this scaling for longer driver bunches which cannot efficiently excite a nonlinear wake. § DRIVER STABILITY AND DRIVER-TO-PLASMA EFFICIENCY In this section, we study the evolution and propagation of an electron beam in PWFA assuming fully pre-ionized plasma. For sufficiently short, tightly-focussed and high-current beams, the wakefield is excited in the strongly nonlinear (“bubble” or “blowout”) regime <cit.>, when a cavity (a bubble) devoid of plasma electrons is formed behind the driver. The excitation of the bubble by an electron beam can be self-consistently described by a model by <cit.> which can be used to calculate the shape of the bubble as well as the distribution of all the fields in it based solely on the charge density distribution of the driver. As this model is based on the relativistic limit of the theory by <cit.>, it is strictly valid only in the case when the transverse size of the bubble R_ is large in terms of plasma units, k_ R_≫ 1, which is not necessarily the case for the parameters considered in the paper. A more accurate model for comparatively small bubbles (k_ R_∼ 1) was recently proposed in <cit.>, but it lacks simple analytical scalings. Theoretical models also cannot fully describe the driver evolution and self-injection, and thus cannot completely replace numerical simulations. Still, we can use the predictions of the models to compare to the simulation results. To study the evolution of the driver in PWFA, we perform numerical simulations of the beam–plasma interaction using the 3D quasistatic particle-in-cell (PIC) code QuickPIC <cit.>. The beam has the mean energy of 250 and the charge of 137pC (corresponding to the total energy of 17mJ) with a flat-top longitudinal current profile with the length ξ_b = 13.4 (or the duration 45fs corresponding to the current of 3.1kA) and a Gaussian transverse profile exp(-r^2/2σ_r^2) with the radius σ_r = 0.52. The peak number density of the beam n_b = 3.75e19^-3. The chosen parameters correspond to typical electron beams generated by the density-downramp injection in the first LWFA stage and are taken from simulations in <cit.>. In fact, as will be briefly explained in the next section, the tunability of the LWFA-produced driver parameters is fairly limited. The plasma density n_0 is chosen to be much lower than the density of the beam n_b, so that the driver excites a highly non-linear plasma wakefield, or a bubble (see Fig. <ref>). The transverse emittance of the beam is chosen to be close to matched to the plasma density to prevent significant changes of the transverse size. In this wakefield, the driver generally experiences two forces: the focussing force from the ion column (linear in the distance r from the axis inside the bubble and leading to betatron oscillations of the electrons of the driver) and the decelerating longitudinal force. We observe that the accelerating structure remains highly stable before rapidly collapsing when electrons of the driver start dephasing due to deceleration to very low energies. This inherent stability owes to the fact that the transverse distribution of the matched driver does not influence the structure of the bubble, while the longitudinal velocity of the ultrarelativistic particles of the bunch stays almost equal to the speed of light until some of the electrons are decelerated to sub-relativistic energies, corresponding to the moment of collapse (compare the first three distances in Fig. <ref> before the collapse to the fourth one after the collapse began). The propagation length corresponding to this collapse is much shorter than the distance of particle loss due to head erosion or beam hosing instability, so it effectively determines the PWFA stage length. Head erosion happens due to the initial emittance of the driver in the region at the head of the bunch where the wakefield providing the focusing force is not high enough amplitude yet <cit.>. The Coulomb self-force is proportional to γ^-2 and can be usually ignored for ultrarelativistic bunches. In the blowout regime of PWFA, head erosion affects only a small portion of the bunch at the head, as the wakefield amplitude quickly becomes sufficient to maintain focusing of almost the entire bunch. As the head erodes, it still continues to drive a lower-amplitude wake behind it, which makes the spread of this erosion to other parts of the bunch very slow. As can be seen from Fig. <ref>, it leads only to a small shift in the phase of the nonlinear wake over the propagation distance and cannot significantly affect the acceleration process. Another effect which is believed to limit the length of PWFAs is the hosing instability which should lead to the exponential growth in oscillations of the beam centroid and the corresponding growth of oscillations of the bubble <cit.>. However, recent research suggests that energy depletion of the driver as well as its energy spread or energy chirp can efficiently suppress and saturate this instability due to the dephasing of betatron oscillations <cit.>. In our simulations, hosing is not taken into account as the initial charge density distribution is ideally symmetric. So, energy depletion of the driver is the main mechanism which determines the acceleration length. It leads to electrons reaching sub-relativistic energies, after which significant portion of the electrons of the driver are quickly lost and the accelerating structure collapses (see Fig. <ref> at 14.4 mm). For a bunch with no energy chirp, this process happens at the point of the peak decelerating electric field the value of which can be estimated from the solution based on the model in <cit.> (shown with dotted lines in Fig. <ref> at 0 mm). The length at which the accelerating structure collapse happens can be thus estimated as the length at which an electron with the kinetic energy K ≈γ m c^2 is decelerated to zero energy in the field E_, L_≈γ m c^2/e E_, where γ is the Lorentz factor, m is the electron mass, e > 0 is the elementary charge, and c is the speed of light. The comparison to values observed in the simulations (see Table <ref>) provides a fairly good estimate for the collapse length. When the collapse of the accelerating structure begins, the driver still has part of its energy left (as decelerating field is non-uniform, and different parts of the driver experience different field values), which limits driver-to-plasma energy transfer efficiency defined as the percentage of the bunch initial energy spent on generating the wakefield by the moment the collapse begins, η = γ_0 - γ_/γ_0, where averaging is performed over the bunch particles. Assuming that the bunch is monoenergetic, it can also be estimated as η≈E_z/E_, so the efficiency mostly reflects how uniform the distribution of the decelerating electric field inside the driver is. The comparison between the actual efficiency observed in simulations to the estimated efficiency based on the field distribution E_z according to the model by <cit.> is shown in Table <ref>. The efficiency changes depending on the plasma density, so it is required to carefully choose the plasma density for the second PWFA stage in order to increase the energy transfer efficiency. For a very high plasma density, the driving bunch can become too long compared to the plasma wavelength (determined by the dimensionless value k_ξ_b in Table <ref>), leading to its tail being accelerated in the created bubble and a significant drop in the driver-to-plasma energy transfer efficiency due to non-uniformity of the field distribution. The efficiency can thus be increased by lowering the plasma density. In very low-density plasmas, the efficiency also starts to slightly drop due to the non-uniformity of the field at the front of a short driver. In addition to that, lower densities can be less desirable due to lower acceleration gradients, which increase the size of the accelerator. Therefore, there is an optimal range of plasma densities at which the beam-to-plasma efficiency is high. As results of numerical simulations show (see Table <ref>), for the considered electron driver, the optimal plasma density of the second PWFA stage is around e18^-3, corresponding to the length of the driver in plasma units k_ξ_b ∼ 3. § DOWNRAMP INJECTION In this section, we study the dependence of the current profile of the witness electron beam produced by density-downramp injection on the driver and plasma parameters. A recent study on downramp injection for LWFA <cit.> reports that the injected beam current is strongly influenced by the laser intensity, and, consecutively, the wakefield strength. Unlike for the laser driven counterpart, the electron driver can be considered non-evolving during the beam injection process in the PWFA stage since the beam evolution scale is the period of betatron oscillations √(2 γ)λ_, usually of millimeter to centimeter scale, much longer than the downramp length, usually in the range of hundreds of micrometers or less. As a result, the location of the downramp does not play such a crucial role as in the LWFA case. §.§ The influence of downramp steepness Now we consider the influence of downramp steepness on the parameters of the injected beam. As quasistatic codes like QuickPIC used in Sec. <ref> cannot self-consistently describe the injection of particles into the wakefield, we perform PIC simulations with FBPIC <cit.>. The downramp is modelled as a linear change of plasma density from n_0 to n_0/2 of length L. For a given downramp steepness, particles are injected only when their energy is high enough, the required energy grows with the decrease of the downramp steepness <cit.>. The energy of the particles depends only on the nonlinearity of the created bubble (which will be quantified later) irrespective of other driver properties. Four group of simulations are presented in Fig. <ref> that explain the role that the non-linearity of the bubble (which depends on the density ratio n_b/n_0 between the driver and the plasma for a fixed driver shape and size) and L play on the witness injection. The injected beam currents are plotted and the corresponding wakefield structures are illustrated in the inset of subfigures (c) and (g). As subfigure (b) demonstrates, for a fixed downramp length L, the injection occurs only when the bubble non-linearity is strong enough. Also, once the condition for injection is reached, the witness beam current is barely influenced by the downramp steepness for the same bubble strength (the same driver), as shown in subfigure (c). Similar to the results in <cit.>, a current peak is observed at the head of the witness bunch. This happens due to the non-linear phase mixing caused by a sharp plasma density transition when the downramp begins, and the peak can be mitigated with a smooth density transition at the beginning of the downramp, usually found in experimental cases. Behind the head of the bunch with respect to the co-moving coordinate ξ = z - ct, the injected current stabilises at a constant value. In the following considerations, we will determine the value of the injected witness current J_w as the current of the constant part of the profile. As subfigures (c, g) and corresponding inset figures show, the length and thus the charge of the injected beam also does not depend much on the transition length L. It is mostly determined by the ratio between the initial and the final plasma density (fixed at 2 in our case), as this ratio determines the change of the length of the bubble during the transition. As shown by Fig. <ref>(a) and (b), the higher value of ratio n_b/n_0 (and thus the wakefield non-linearity) leads to the higher current J_w. From the comparison of subfigures (a) and (b), one can see that shorter and steeper downramp enables injection at lower driver charge and lower non-linearity. No injection at all is observed for the case where n_b/n_0=10 shown in sub-figure (b), and the higher the wakefield non-linearity n_b/n_0 is, the stronger the injected witness current becomes. §.§ Injected beam current scaling As shown in the previous section, as long as the density downramp provides stable injection, its properties do not significantly affect the injected current J_w, so it should mostly be determined by the parameters of the driver and their relation to the plasma density. It is suggested by Fig. <ref> that the higher values of n_b/n_0 which should correspond to stronger nonlinearity lead to a stronger injected beam current. As we want to investigate the dependence of J_w on the nonlinearity of the bubble, we need to introduce a measure of it first. In the previous section, we used the ratio n_b/n_0 as this measure, which is only suitable for a fixed shape and size of the driver. A more general measure requires introducing the energy properties of the bubble. Following <cit.>, we introduce the quantity Ψ(ξ) = ∫ (c W - S_z) d^2r_⊥ which depends on the comoving coordinate ξ = ct - z and contains the integral over the transverse plane of the energy density W and the longitudinal energy flux S_z of both the EM field and the plasma particles. The value of Ψ is also equal to the total energy flux in the comoving window <cit.>. As it has the dimension of power, we will refer to Ψ as “the power of the bubble”. In the absence of energy exchange with bunches, Ψ is a conserved property in any wake; drivers lead to the increase of Ψ, while accelerated witnesses decrease it. As a property describing the energetic properties of the bubble, it is equal to the total power of deceleration felt by the driving bunch and is also equal to the maximum achievable power of acceleration for the witness bunch. For the blowout regime of plasma wakefield, Ψ is fully determined by the size of the bubble R_ and grows with it; in the limit of a large bubble size (k_ R_≫ 1), Ψ∝ (k_ R_)^4 (see <cit.>). The power of the bubble Ψ serves as quantification of nonlinearity of the wake and it is the most important property of the bubble. Regardless of the shape the driver which creates a bubble with a certain value of Ψ, the effect of the bubbles with the same power on acceleration of particles and on downramp injection will be mostly the same. Therefore, we can expect that injection only depends on the value of Ψ. According to <cit.>, for a blow-out wakefield excited by a sufficiently tighly focussed (k_ r_b ≪ 1) driver with the charge Q and a flattop current profile of length ξ_b, the power of the bubble in the large-bubble limit (k_ R_≫ 1) is calculated using the following formula: Ψ≈4π m^2 c^6 ε_0/e^2 J_ k_ Q [√(2 c Q/J_ξ_b) - k_ξ_b/8], where J_ = 4 πε_0 m c^3 / e ≈17kA is the Alfvén current, ε_0 is the vacuum permittivity. It can be rewritten as Ψ≈√(2) m c^2 J_/e(J_/J_)^3/2[1 - (k_ξ_b)^4/3/√(128 J_ / J_)], where we introduce the quantity J_ which we call “the effective current” of the driver: J_ = c Q/ξ_b (k_ξ_b)^2/3 = J_b (k_ξ_b)^2/3. For high-current (J_∼ J_) sufficiently short electron bunches, the last factor in Eq. (<ref>) is close to 1, and Ψ∝ J_^3/2. In general, this factor describes the weakening of the bubble when the bunch becomes long enough compared to the plasma wavelength. In the limiting case when the back of the bunch is already located in the accelerating phase, the bubble effectively transfers the energy from one part of the driver to another, limiting the possible efficiency of accelerating the witness, which corresponds to this factor tending to 0. When it becomes negative, this solution for Ψ is formally incorrect, which corresponds to the situation when the driver cannot fit inside the bubble. Since J_ fully determines Ψ and thus the properties of the bubble for sufficiently short bunches, it should be the only combination of the driver and plasma parameters affecting the injection process. Although the model in <cit.> is not necessarily strictly valid in the case of smaller-size bubbles, because this dependence holds in the important limit of large bubbles, we make a conjecture that J_ should be the most important combination of the driver's parameters. To explore the influence of J_ on the injected witness current J_w, we perform numerical PIC simulations with FBPIC <cit.> for two types of drivers: 3D Gaussian beams which are usually used to model the driver beam produced by a conventional accelerator <cit.>, and flattop longitudinal bunch which models the LWFA produced driver beam <cit.>. To prevent any effects due to the evolution of the driver beam, we artificially freeze it by setting its initial particle energy to 50GeV. The ratio between the plasma density before and after the downramp is fixed to 2 (going from n_0 to n_0/2) and the downramp length is between 20 and 160. For Gaussian drivers with the density profile defined as n_b exp[-r^2 / 2 σ_r^2 - (ξ-ξ_0)^2 / 2 σ_z^2], the beam transverse size σ_r = 0.5. Flattop drivers correspond to the cylinder of constant density n_b, length ξ_b, and fixed radius r_b = 0.6. As the effective current J_ defined by Eq. (<ref>) depends on the current of the driver J_b and its length ξ_b, we study the relationship J_w and J_ by varying these two parameters. Of course, the value of J_ also depends on the plasma density, so we use the plasma density n_0 before the downramp to calculate the plasma wavenumber k_ used in Eq. (<ref>). Because definition Eq. (<ref>) is written for flattop profiles, for Gaussian beams we use ξ_b = √(2)σ_z and the peak value of J_b. This combination was shown to provide a good approximation for Ψ in simulations (not presented in the paper). The results for varying the driver beam current J_b at fixed length are shown in Fig. <ref>, and the results for varying the driver bunch length while the current is fixed are presented in Fig. <ref> for different plasma densities. For the fixed beam length (Fig. <ref>), we observe a linear dependence of the injected current on the effective current, which seems to indicate that J_w ∝Ψ^2/3∝ J_. The slope of the linear dependence depends only on the shape of the driver and remains the same for different plasma densities and lengths of the driver. We observe that the witness current becomes slightly lower for denser plasmas (thus larger values of k_), which corresponds to the lowering efficiency due to the driver length in plasma units, as predicted by the length factor in Eq. (<ref>). At very low J_, the injected current goes to 0, which corresponds to the injection threshold discussed in Sec. <ref>. To study the dependence of the witness current J_w on the driver length ξ_b in more details, we also plot the dependence of J_w on the effective current J_ for a fixed current of the driver, which means that the increase in J_ corresponds to the increase in its length (Fig. <ref>). For smaller J_ and thus shorter driver lengths, the already found linear trend with the same slope is again observed for both types of drivers. But for higher J_ and longer driver length, we see the deviation of J_w from the linear trend and the decline in it. This behavior is qualitatively consistent with Eq. (<ref>) which predicts that the elongation of the driver lowers the efficiency of exciting a bubble, so we can expect that the witness current J_w ∝Ψ^2/3 behaves like J_w = c_1 J_[1 - c_2 (k_ξ_b)^4/3/√(128 J_/J_)]^2/3. By choosing the value of c_2, the qualitative behavior of J_ can be recreated using this formula (see dotted lines in Fig. <ref>). However, this formula cannot show the full picture, because downramp injection for longer drivers is more complex than for short ones. According to Eq. (<ref>), the power of the bubble changes as the driver propagates through the density downramp which changes the plasma wavenumber k_. For short drivers, Ψ∝ J_^3/2∝ n_^1/2 scales exactly the same for all drivers with the same J_, and thus the injected current is still fully determined by J_. However, longer drivers which have a lowered efficiency in dense plasma begin exciting the bubble more effectively when transitioning into lower density plasma at the downramp, as their length in plasma units becomes small. An example of such a driver is shown in Fig. <ref>: in the initial larger density it is too long to fit inside the bubble, so during the density downramp it passes through the point of having almost zero efficiency of exciting a bubble (Ψ≈ 0), and then Ψ starts growing again as the driver becomes shorter than the excited bubble. In addition to the complex behaviour of the nonlinearity of the bubble, the corresponding velocity of the back of the bubble which determines the injection threshold will also significantly depend on the length of the driver. The dynamics of downramp injection for such cases cannot be reduced to a simple formula such as linear dependence on J_ or even corrected Eq. (<ref>) and requires the full description of the evolution of the bubble in the downramp. However, as explained in Sec. <ref>, these cases are suboptimal for efficient utilisation of the driver's energy and should be avoided. In more optimal cases, when the driver is comparatively short, the linear dependence of J_w and J_ holds. § CONCLUSION We studied a plasma-wakefield accelerator driven by a short high-current electron bunch. The accelerating structure in this case remains stable until the moment when some of the electrons of the driver lose all their energy, which is the condition which determines the acceleration length optimizing the driver-to-plasma energy transfer efficiency. The dependence of the witness bunch generated by density-downramp injection on the parameters of the driver was studied. We showed that a steeper downramp enables injection for a weaker driver, but the current of the injected bunch does not significantly depend on the downramp length as long as the criterion for the injection is met. The witness current of the injected bunch mostly depends on the effective current of the driver J_ which combines both the driver's current and its length in plasma units. The dependence of the injected witness current on the effective current is linear for reasonably short drivers. This work was supported by the Fondation Jacques Toledano, the Schwartz Reisman Center for Intense Laser Physics, and by ERC PoC Vherapy grant. jpp
http://arxiv.org/abs/2307.02682v2	20230705230126	Zero-Shot Dense Video Captioning by Jointly Optimizing Text and Moment	[ "Yongrae Jo", "Seongyun Lee", "Aiden SJ Lee", "Hyunji Lee", "Hanseok Oh", "Minjoon Seo" ]	cs.CV	[ "cs.CV", "cs.CL" ]	The Kibble-Zurek Scenario and Coarsening Across Nonequilibrium Phase Transitions in Driven Vortices and Skyrmions C. Reichhardt and C. J. O. Reichhardt August 1, 2023 =================================================================================================================== Dense video captioning, a task of localizing meaningful moments and generating relevant captions for videos, often requires a large, expensive corpus of annotated video segments paired with text. In an effort to minimize the annotation cost, we propose , a novel method for dense video captioning in a zero-shot manner. Our method does not require any videos or annotations for training; instead, it localizes and describes events within each input video at test time by optimizing solely on the input. This is accomplished by introducing a soft moment mask that represents a temporal segment in the video and jointly optimizing it with the prefix parameters of a language model. This joint optimization aligns a frozen language generation model (i.e., GPT-2) with a frozen vision-language contrastive model (i.e., CLIP) by maximizing the matching score between the generated text and a moment within the video. We also introduce a pairwise temporal IoU loss to let a set of soft moment masks capture multiple distinct events within the video. Our method effectively discovers diverse significant events within the video, with the resulting captions appropriately describing these events. The empirical results demonstrate that surpasses zero-shot baselines and even outperforms the state-of-the-art few-shot method on the widely-used benchmark ActivityNet Captions. Moreover, our method shows greater robustness compared to supervised methods when evaluated in out-of-domain scenarios. This research provides insight into the potential of aligning widely-used models, such as language generation models and vision-language models, to unlock a new capability—understanding temporal aspects of videos. § INTRODUCTION Dense video captioning is a task that temporally localizes multiple meaningful events (or moments) within a video and provides captions for each event <cit.>. As wild videos are often untrimmed and contain multiple events within a single video, this task is particularly useful in real-world scenarios. Dense video captioning requires a deep understanding and accurate representation of temporal information present in the video. As a result, it typically requires a substantial collection of annotations for temporal segments within videos, each paired with corresponding captions, which is often prohibitively costly. For this reason, performing dense video captioning without access to language captions or annotated temporal segments is especially valuable, but the literature lacks previous work on a zero-shot setup. In this paper, we propose (), which tackles the problem of zero-shot dense video captioning by jointly optimizing text generation and moment localization for a single video at test time in an end-to-end manner. This joint optimization ensures that the generated text aligns with the discovered temporal moment and, simultaneously, that the discovered temporal moment accurately corresponds to the generated text. Our model comprises two modules: the text generation module and the moment localization module (Figure <ref>). The design of the text generation module is inspired by <cit.>, where they address image and video captioning tasks without training data. Likewise, we leverage a frozen language generation model (i.e., GPT-2 <cit.>) and a frozen vision-language model (i.e., CLIP <cit.>), and align GPT2 with CLIP using a small number of learnable prefix parameters as in prefix-tuning <cit.>. Although CLIP is pretrained on image-text pairs with contrastive learning and GPT-2 is pretrained on text-only data without video knowledge, can effectively localize and generate captions for different moments in a video. This work hints at how we can align models such as a language generation model and a vision-language contrastive model, to build a compositional model that is capable of temporal understanding. For the design of the moment localization module, we propose one new masking mechanism and one new loss term: soft moment masking and pairwise temporal IoU loss. The soft moment masking ensures the text generation focuses solely on the corresponding video moment by introducing a differentiable temporal mask onto video frames. Pairwise temporal Intersection over Union (IoU) loss ensures that our approach generates multiple captions from distinct time segments within a video, thus enhancing the richness of the dense captions. This loss is calculated on a group of moments that are jointly optimized for a given video. We validate the effectiveness of our approach in accurately identifying and describing significant moments in a given video. Our zero-shot method surpasses various zero-shot baselines and outperforms the state-of-the-art few-shot method pretrained on a billion-scale video-text data <cit.> on the widely-used ActivityNet Captions benchmark. Furthermore, we demonstrate the robustness of our method in out-of-domain scenarios when compared to supervised models. When assessed on a dataset distinct from the one used for model training, supervised models struggle to adapt to the new dataset. Conversely, our zero-shot approach exhibits better resilience in this situation. The out-of-domain setup is especially valuable when seeking to make use of real-world videos, which are characterized by distinctly different domains. To summarize, we provide the following key contributions: * We propose (), a pioneering zero-shot dense video captioning method by aligning pretrained models to unlock a new capability of temporal understanding. * We propose soft moment masking for end-to-end optimization of temporal localization and pairwise temporal IoU loss for the diversity of localized moments. * Our method surpasses various zero-shot baselines and even outperforms the state-of-the-art few-shot method on the ActivityNet Captions benchmark. Also, our method is more robust in out-of-domain scenarios than supervised models. § RELATED WORK Dense video captioning Dense video captioning (also called dense event captioning <cit.>) extends the task of video captioning <cit.> by incorporating fine-grained temporal localization and generate multiple captions per video. Due to the complexity of the task, most existing methods <cit.> require a strong supervision with a large amount of video-text-timestamp data. To mitigate annotation costs, existing attempts <cit.> have focused on addressing dense video captioning tasks with lower levels of supervision. Specifically, <cit.> introduced a weakly supervised methodology for dense video captioning, utilizing video data paired with captions without time intervals annotation during the training process. However, these approaches still rely on video paired with text corpus and make a somewhat unrealistic assumption of a one-to-one correspondence between video segments and their respective captions. In contrast, we present a zero-supervision paradigm that eliminates the need for a video or text corpus for training. <cit.> recently introduced a few-shot dense video captioning setup, which involves first pretraining a model on narrative videos and then fine-tuning it with a small portion of the downstream training data. Our approach extends this few-shot setting further by introducing zero-shot dense video captioning. Also, our method does not need pretraining on video data. Vision-language alignment Our approach is related to the models that bridge between visual and textual modalities. CLIP <cit.> is one such model that has gained noteworthy recognition. Recent works <cit.> show that pretrained image and text models can be tuned together to be applied to various vision-language tasks. In particular, <cit.> showed visual representations from frozen vision models can be projected onto frozen language models with a single linear layer. Similarly, <cit.> connected image features into the word embedding space using a trainable projection matrix. Our method follows a similar approach and incorporates projected visual embeddings as a prefix into a frozen language model. <cit.> combine a visual-semantic model with a language model, leveraging knowledge from both models to generate descriptive text given an image or a video, respectively. Inspired by these works, we take a step further to apply this approach to solve zero-shot dense video captioning tasks for the first time. Notably, The task of dense video captioning requires a temporal understanding of a video, which an image-text visual-semantic model has never been trained on. Moment localization Moment localization is the task of identifying specific moments from a video that are relevant to a given natural language query <cit.>. Since obtaining annotations for moment localization can be costly, several studies have explored ways to lessen the need for supervision. As part of these efforts, the weakly supervised setup for moment localization has been proposed <cit.>. Although these methods reduce the costs related to temporal annotations, the remaining cost associated with the creation of natural language queries continues to be significant. A few works explored zero-shot setup for moment localization <cit.>. <cit.> extract nouns and verbs from moment proposals by object detection and simple language modeling, then use them as pseudo-queries to train a moment localization model. While this method produces simplified sentences resembling dense video captions during the procedure, the constructed queries are mere lists of nouns and verbs that lack natural language properties. As such, they are not designed to address the dense video captioning task. Similarly, <cit.> takes a simpler approach to zero-shot moment localization by utilizing CLIP, but it does not generate discrete captions in natural language. § METHOD Dense video captioning aims to describe with natural language events within a given untrimmed video, while also temporally localizing them with start and end time stamps (Figure <ref>). Formally, the task of dense video captioning can be described as follows: Given a video 𝐕 of L frames, the objective is to determine a function F: 𝐕→{(s_k, m_k)}^N_k=1 where s_k represents the caption, m_k denotes the corresponding moment, and N is the number of moments. Each caption s_k is a sequence of tokens, and each moment m_k is a consecutive subset of video frames. A moment signifies a meaningful temporal segment of the video. In this work, we treat N as a hyperparameter that is predetermined before the input is given. In zero-shot dense video captioning, the model does not have access to language captions or annotated time stamps for training. Therefore, the challenges are two-fold. First, the model needs to accurately identify significant moments within a long video without annotated captions. Second, it must generate natural language captions for each of these identified moments, without annotated moments. To tackle these two challenges at the same time, we design a training-free method, (). As in Figure <ref>, is composed of two modules. The first is the text generation module (left of Figure <ref>) that utilizes a frozen language model, which is conditioned on a learnable prefix context. The prefix context and vision loss (L_vision) are designed to produce text that aligns with the visual content corresponding to a specific moment, as detailed in Section <ref>. The second module, referred to as the moment localization module (right part of Figure <ref>), is responsible for learning the parameters that specify a moment in a video while ensuring the diversity of moments, as presented in Section <ref>. Finally, we combine the losses for both modules and optimize the model in an end-to-end manner, as described in Section <ref>. §.§ Text generation The text generation module uses a pretrained language model (i.e., GPT-2) to infer the next word from a prefix context. The language model parameters are fixed, and only the prefix context parameters (Section <ref>) are optimized during the test time to align the generated text to the corresponding moment. The optimization takes place during auto-regression and is iterated for each generation step. Taking inspiration from using a vision-language alignment model and a language model for image and video captioning <cit.>, we adopt two losses during the optimization process for the text generation module. The first loss, the vision loss (Section <ref>), aims to enhance the similarity between the generated text and the corresponding moment. The second loss, the language loss (Section <ref>), focuses on preserving the naturalness of the generated text. §.§.§ Prefix context The prefix context has three parts: soft prompt, projected video embedding, and hard prompt. These three parts are concatenated and used by the language model as a prefix for language generation. The first part of the prefix context is the tunable soft prompt. Similar to the prefix-tuning <cit.>, the transformer blocks within the length of the soft prompt have their key and value embeddings learned during the optimization process. The frozen language model then attends to this soft prompt, providing guidance during the generation process. The second part of the prefix context is the projected video embedding. To obtain these embeddings, we first extract the image features from video frames using a pretrained image encoder (CLIP image encoder), aggregate the features with a soft moment mask (Section <ref>), and then apply a simple trainable projection layer (W) to the aggregated video feature embedding. The trainable projection layer is a single linear layer used to project video feature embedding to language model token embedding space <cit.>. By the projection, the dimensionality of video feature embedding matches that of the language model token embedding. The third part of the prefix context is the hard prompt. These are prefix tokens such as 'Video showing,' 'Video of,' etc. We randomly sample a hard prompt from a list of prefix tokens. The list of the prefix tokens we used in experiments is in the Appendix Section <ref>. §.§.§ Vision loss To steer the language model toward a specific visual direction at each generation step, we incorporate vision loss. This loss is obtained through a vision-language alignment model (CLIP). CLIP scores the relevance between the generated tokens up to the current step and a video moment, which we call the alignment score (Eq. <ref>). For i-th candidate token t^i_k,l at generation step l of caption s_k, we form the associated candidate sentence s^i_k,l by concatenating the candidate token with previously generated tokens s^i_k,l = {t_k,1, …, t_k,l-1, t^i_k,l} and calculate alignment score for each candidate sentence[For efficiency, we compute the scores only for the top 512 candidate tokens.]. The alignment score (a^i_k,l) of the i-th candidate token at generation step l of caption s_k is computed as a^i_k,l∝exp(cos(E_Text(s^i_k,l), E_Image(m_k)) / τ) where cos denotes the cosine similarity, and E_Text and E_Image represent the textual and image encoder of the vision-language alignment model (CLIP). This measures the similarity between the textual embedding of candidate sentence s^i_k,l and the image embedding of the moment m_k. τ > 0 is a temperature hyperparameter. The vision loss is defined as the average cross-entropy loss (CE) between the alignment score distribution (a_k,l) and the probability distribution of the candidate tokens (q_k,l) obtained by the language model: L_vision = 1/N∑_k CE(a_k,l, q_k,l) This loss stimulates token generation towards higher text-visual matching scores between the generated text and visual information from the moment. §.§.§ Language loss In order to preserve the natural language quality of the generated text while aligning it with the visual content, we employ a regularization term, which we call language loss. This loss quantifies the average cross-entropy (CE) between the probability distribution of words from the language model with the prefix context (q_k,l) and without the prefix context (q'_k,l). By minimizing this loss, we ensure that the probability distribution of words with the prefix context closely matches that of the original language model without the prefix context. This regularization step helps maintain the overall language model coherence while incorporating visual alignment <cit.>. L_language = 1/N∑_k CE(q_k,l, q'_k,l) §.§ Moment localization Similar to how the text generation module aligns generated text with a video moment, the moment localization module is responsible for aligning the video moment with the generated text. Previous works performed the selection of temporal moments through a separate module, relying solely on visual feature similarity <cit.>. However, such an approach is sub-optimal as the moments are selected without considering the corresponding captions. To remedy this, we introduce soft moment masking (Section <ref>). Dense video captioning requires identifying multiple temporal moments from a given video. To accomplish this, instead of generating a single moment-text pair, we optimize a group of moments simultaneously. In order to enhance the diversity among temporal moments and ensure that each moment captures distinct meaningful segments of a video, we introduce the pairwise temporal IoU loss (Section <ref>). §.§.§ Soft moment masking A soft moment mask specifies a moment m_k with two parameters: center c_k and width w_k. These two parameters are randomly initialized and tuned during end-to-end optimization. To construct a soft mask that spans the length of the video using the two parameters, we employ the following steps: * Apply the sigmoid function to c_k and w_k to convert it to normalized values 0 ≤c̃_k ≤ 1 and 0 ≤w̃_k ≤ 1 that indicate their relative positions to the length of the video; 0 represents the start of the video, and 1 represents the end of the video. * Let p_j ∈{p_1, …, p_L} denote the normalized frame position value between 0 and 1, representing the relative position of a frame to the length of the video. * Calculate the L1 distance between each frame position p_j and c̃_k. * Subtract an offset of half the normalized width (w̃/2) from the distance, multiply it by the sharpness hyperparameter γ, and then apply the sigmoid function on it. We can summarize the above procedure with the following formula. The value of jth position in the mask for the moment m, mask_j, is mask_j = sigmoid(γ * (\|p_j - c̃_k\| - w̃_k / 2)) where c̃_k = sigmoid(c_k) and w̃_k = sigmoid(w_k) The resulting values become close to 1 when the frame is near the center of the moment, approximately 0.5 when it is at the start or end of the moment, and towards 0 as the frame is further away from the moment. The sharpness hyperparameter promotes a sharp contrast between the values inside and outside the moment. The value of the sharpness can be progressively increased with each iteration, enhancing the contrast over the course of the optimization (Figure <ref>). Since the soft moment mask is differentiable, it can be optimized in an end-to-end manner alongside the text generation module. By introducing merely two parameters per moment, the optimization process of the temporal moment mask is both highly stable and efficient. Moreover, our parameterization of a moment using center and width parameters provides straightforward interpretability and applicability. §.§.§ Pairwise temporal IoU loss To discover multiple temporal segments at the same time we optimize a group of moments for a video simultaneously, each with separate soft moment masking and prefix context. To encourage the model to capture different moments of distinct regions, we introduce pairwise temporal IoU loss between different moments. Pairwise temporal IoU loss between N moments is calculated by the following equation: L_ptIoU = 1/N2∑_k=1^N-1∑_l=k+1^NIoU(m_k, m_l) In this expression, N2 represents the total number of possible pairwise combinations between N moments. IoU(m_k, m_l) calculates the temporal Intersection over Union between the two moments m_k and m_l. §.§ Joint optimization The total loss of our method is the weighted sum of vision loss, language loss, and pairwise temporal IoU loss. The model is optimized in an end-to-end manner. L_total = λ_1 · L_vision + λ_2 · L_language + λ_3 · L_ptIoU λ_1, λ_2, and λ_3 are hyperparameters that represent the weights assigned to each loss term. § EXPERIMENTS This section demonstrates the effectiveness of our proposed model by comparing it to baselines and the state of the art. We begin by providing an overview of our experimental setup in Section <ref>. We then present quantitative analysis in Section <ref>. Note that we add qualitative results in the Appendix Section <ref>. §.§ Experimental setup §.§.§ Datasets For zero-shot dense video captioning, we use two datasets for evaluation: ActivityNet Captions <cit.> and YouCook2 <cit.>. Adhering to a zero-shot setup, we refrained from using any caption or temporal annotations in training data. ActivityNet Captions includes 20K untrimmed videos showcasing various human activities. Each video in this dataset lasts around 120 seconds on average and is annotated with an average of 3.7 temporally-localized captions. YouCook2 comprises 2K untrimmed cooking procedure videos, with an average duration of 320 seconds per video. Each video in the dataset is annotated with an average of 7.7 temporally-localized sentences. §.§.§ Implementation Details We uniformly sample one frame per second from a given video. The visual feature extraction and text-image similarity calculation are done using the pre-trained CLIP ViT-L/14. We use the pretrained GPT-2 medium for the language model. In the case of ActivityNet Captions, the number of moments k for a video is set to 4. For the YouCook2 dataset, the number of moments k for a video is set to 8. The initialization of the center and width parameters is based on the respective dataset distributions. We set the vision loss weight to λ_1 = 1, the language loss weight to λ_2 = 0.8, and the pairwise temporal IoU loss weight to λ_3 = 10. The sharpness hyperparameter γ is linearly increased starting from 10 and incremented by 1 after each generation iteration. The temperature hyperparameter τ is set to 1.0. Throughout the experiments, we employ 12 generation iterations. For the further implementation details, refer to <ref> §.§.§ Evaluation metrics For dense video captioning, we adopt three widely used metrics: CIDEr <cit.> (C), METEOR <cit.> (M), and SODA_c <cit.> (S). Both CIDEr and METEOR initially determine the matched pairs between the predicted moments and the ground truth annotations across IoU (Intersection over Union) thresholds of 0.3, 0.5, 0.7, and 0.9. The captioning metrics are then calculated based on these matched pairs. SODA_c, on the other hand, addresses the limitations of traditional captioning metrics in the context of dense video captioning and considers the overarching narrative of the video. §.§.§ Baselines Since this work is the first attempt at zero-shot dense video captioning, there is no prior work directly addressing this task. Therefore, we evaluate our method by comparing it against several straightforward baseline approaches: 1) Scene detection using PySceneDetect[https://scenedetect.com] followed by image captioning with BLIP <cit.> (PySceneDetect+BLIP). PySceneDetect is a widely used scene detector for splitting a video into separate clips. We extract the center frame from each detected clip and use BLIP to generate corresponding captions. 2) Scene detection using PySceneDetect followed by a video captioner (PySceneDetect+TimeSformer+GPT-2). This is the same as the one with BLIP but uses an open-source pretrained video captioning model based on TimeSformer <cit.> and GPT2[https://huggingface.co/Neleac/timesformer-gpt2-video-captioning]. 3) Video captioning with TimeSformer+GPT2 model followed by frame matching with CLIP (TimeSformer+GPT-2+CLIP). This baseline first generates multiple captions using beam search with a video captioner and matches the most similar frame with each caption using CLIP. Then, the frame that best matches each caption is regarded as the center of the moment, with a fixed width applied across all moments. We add more implementation details of the baselines in the Appendix Section <ref>. §.§ Results In this section, we evaluate and analyze the performance of our model in comparison to baselines and the current state-of-the-art models. Table <ref> presents a performance comparison between our model, zero-shot baselines and methods that have stronger supervision. Table <ref> shows a performance comparison in out-of-domain settings. We add more detailed ablation studies in the Appendix Section <ref>. Joint optimization is more effective than two-stage methods In dense video captioning, our model outperforms various zero-shot baselines on both ActivityNet Captions and YouCook2 datasets. These baselines utilize two-stage approaches with a segmenting component and captioning component to tackle dense caption generation. Despite the fact that the image captioning and the video captioning components of these baselines are trained directly using additional captioning data and captioning loss, there remains a noticeable gap in performance when compared to our approach. This observation highlights the critical role of joint training in text generation and moment localization, enabling effective dense caption generation even in the absence of data. outperforms a state-of-the-art few-shot model Compared to models that have stronger supervision than ours, we observe that surpasses the performance of few-shot Vid2Seq, a model with pretraining. It is worth noting that Vid2Seq is pretrained on the YT-Temporal-1B dataset, which consists of 18 million narrated videos spanning 1 billion frames paired with transcribed speech sentences. Remarkably, despite our model never having access to video data or temporal annotations, we achieved better performance than Vid2Seq fine-tuned with 1% of the training data. Text space of the target task and that of CLIP need to match YouCook2 shows a different trend compared to ActivityNet Captions. Here, our method underperforms the few-shot Vid2Seq. This divergence can be attributed to the distinct style of language annotation inherent to the dataset. ActivityNet Captions typically contain conventional captions briefly describing the visual content, such as "Cheerleaders are standing on the side of the road.". In contrast, YouCook2 is characterized by task-oriented, instructional textual annotations like "place a slice of cheese on the bread." Since our model relies on CLIP, which is pretrained with conventional image captions, the generated text resembles these captions. This style of resulting captions conflicts with YouCook2's ground truth captions, thus degrading performance in metrics. See Section <ref> for more discussion. is robust in out-of-domain setups Our method demonstrates greater robustness in out-of-domain setup, surpassing fully-trained state-of-the-art models. Unlike fine-tuned models, which are optimized for a target domain and thus struggle to adapt to new ones, our zero-shot approach maintains the performance across different domains. Its inherent domain-agnostic nature allows for flexibility, avoiding the overfitting pitfalls of specialized models. § LIMITATION AND DISCUSSION Our zero-shot method, by design, doesn't encounter any text or temporal annotation associated with the dataset. Consequently, it doesn't have the opportunity to learn the particular style of the output text and moments of the dataset. While this limitation could potentially be addressed by extending the method in various ways, including few-shot learning, we reserve this for future work. § CONCLUSION In this work, we present a novel zero-shot method for dense video captioning, , which utilizes soft moment masking and pairwise temporal IoU loss for end-to-end temporal localization. Our method, despite not requiring any video or annotations for training, not only surpasses various zero-shot baselines but also outperforms the state-of-the-art few-shot method on the widely-used benchmark, ActivityNet Captions. Moreover, it demonstrates superior robustness in out-of-domain scenarios compared to fully-supervised models, thereby showcasing its adaptability to diverse and previously unseen video data. This research not only presents a pioneering approach to zero-shot dense video captioning but also sheds light on the potential of aligning language and vision models. By combining the power of pretrained models of different modality, we can unlock new capabilities such as understanding temporal aspects in videos. These contributions advance the field of dense video captioning and offer valuable insights for future research in zero-shot alignment of language and vision models. plainnat § EXPERIMENTAL SETUP In this section, we complement the description of our experimental setup outlined in Section <ref>. We provide the implementation details (Section <ref>) and also give additional information about baselines (Section <ref>). §.§ Implementation details We uniformly sample one frame per second from a given video. The visual features of each frame are extracted and the similarity between text and frames is measured using the pre-trained CLIP ViT-L/14. We use the pretrained GPT-2 medium for the language model. For prefix context, we employ soft prompt of length 5. Also, we use the projected video embedding of length 20, i.e., we project the averaged frame CLIP embeddings to 20 token embeddings of GPT-2. The initialization of the center and width parameters is based on the respective dataset distributions. In the case of ActivityNet Captions, the number of moments for a video k is set to 4. The center parameters are initialized in a way that the sigmoid of their values uniformly transition from the start to the end of the video. The width parameter of each moment is initialized to -0.8472, resulting in a sigmoid value of 0.3. For the YouCook2 dataset, the number of moments for a video k is set to 8. The center parameters are initialized in a way that the sigmoid of their values uniformly transition from 0.1 to 0.9 of the video duration. This initialization aims to exclude irrelevant start and end frames, which usually contain intro and outro scenes. The width paramter of each moment is initialized to -2.1972, yielding a sigmoid value of 0.1. Additionally, a maximum width parameter value of -0.8472 is applied, which corresponds to 0.3 of the video duration. We set the vision loss weight to λ_1 = 1, the language loss weight to λ_2 = 0.8, and the pairwise temporal IoU loss weight to λ_3 = 10. The temperature hyperparameter τ is set to 1.0. Throughout the experiments, we employ 12 generation iterations. The sharpness hyperparameter γ is linearly increased starting from 10 and incremented by 1 after each generation iteration. For the generation of each new sentence, a hard prompt is randomly selected from the set of {"Video showing", "Video shows", "Video of", "Photo showing", "Photo shows", "Photo of", "Picture showing", "Picture shows", "Picture of", "Image showing", "Image shows", "Image of"} The AdamW optimizer is employed with β = (0.9, 0.999) and weight decay = 0.0018. We use a learning rate of 6e^-3 with a cosine annealing learning rate scheduler. The experiments are conducted on NVIDIA A100 GPUs. §.§ Baselines PySceneDetect+BLIP. We split a given video using the default adaptive content detector method that analyzes the changes in average frame intensity/brightness using PySceneDetect. For image captioning, we use the BLIP Base image captioning model based on ViT-B/32. During decoding, we employ beam search with a beam size of 5 for caption generation. PySceneDetect+TimeSformer+GPT-2. Here the configuration of PySceneDetect is the same as PySceneDetect+BLIP. For video captioning, we employ an open-source pretrained video captioner based on TimeSformer and GPT2. We use beam search with a beam size of 8 for decoding. TimeSformer+GPT-2+CLIP. In this baseline, instead of initially splitting the video, we first perform the captioning process, following which each caption is matched to a specific moment within the video. To generate multiple captions from a video, we use a beam search with a beam size of 8. We employ the same video captioner as PySceneDetect+VideoCaptioner. Subsequently, we compute the CLIP scores to measure the similarity between each generated caption and all the frames within the video. We use CLIP ViT-B/32 for this process. The frame with the highest CLIP score is considered the central frame for the moment associated with that particular caption. Finally, we apply a fixed width of 0.3 of the total duration to each moment. § QUALITATIVE RESULTS Figure <ref> presents the qualitative results of dense event captioning obtained by our model. Here, We show additional results attained from the ActivityNet Captions and YouCook2 datasets in Figures <ref>, <ref>, and <ref>. Figure <ref> shows that can capture meaningful moments and generate corresponding captions, even without any training data. In Figure <ref>, we observe that although the style of the caption may differ from the ground truth (as discussed in Section <ref>), still manages to generate meaningful dense captions and identify moment boundaries. Figure <ref> illustrates failure cases, such as (1) generating descriptions that lack visual grounding (i.e., hallucination) and (2) failing to capture all significant moments due to the fixed number of moments per video. § ABLATION STUDIES In this section, we provide ablation studies that complement the results presented in Section <ref>. We use the same default hyperparameters, evaluation metrics, and downstream datasets for these experiments. Vision-language similarity model In Table <ref>, we analyze the benefits of scaling up the size of the pretrained CLIP model. We find that scaling up the CLIP size from ViT-B/32 to ViT-L/14 brings considerable performance improvements. These results suggest that further performance improvements could potentially be achieved by scaling up CLIP to even larger models. Due to computational constraints, we did not conduct experiments with CLIP models larger than ViT-L/14, leaving this as an area for future exploration. Language model In Table <ref>, we evaluate the effect of scaling up the size of the pretrained GPT-2 language model. We find that scaling up the language model size also increases the overall performance of the model. Note that, due to computational constraints, our default setting across all other experiments is GPT-2 medium. Projected video embedding Table <ref> presents an ablation study on the projected video embedding part of the prefix context presented in Section <ref>. By default, the process projects the averaged video frame embeddings into 20 token embeddings. We find that incorporating the projected video embeddings in the prefix context results in improved performance compared to the model without it. Additionally, projecting video embeddings into a greater number of tokens is beneficial. Sharpness of soft moment mask In the default settings, we increment the sharpness hyperparameter γ by 1 after each generation iteration, starting from a base value of 10. In Table <ref>, we ablate the scheduling of the sharpness hyperparameter. Our findings suggest that starting with a lower sharpness value and gradually increasing it can lead to better performance than constant scheduling schemes.
http://arxiv.org/abs/2307.02445v1	20230705171511	Quantifying Poynting flux in the Quiet Sun Photosphere	[ "Dennis Tilipman", "Maria Kazachenko", "Benoit Tremblay", "Ivan Milic", "Valentin Martinez Pillet", "Matthias Rempel" ]	astro-ph.SR	[ "astro-ph.SR" ]	Dennis Tilipman [email protected] 0000-0001-9361-6629]Dennis Tilipman National Solar Observatory, University of Colorado Boulder, Boulder, CO, USA Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, Boulder, CO, USA 0000-0001-8975-7605]Maria Kazachenko National Solar Observatory, University of Colorado Boulder, Boulder, CO, USA Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, Boulder, CO, USA 0000-0002-5181-7913]Benoit Tremblay High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO, USA 0000-0002-0189-5550]Ivan Milić Leibniz Institute for Solar Physics (KIS), Freiburg, Germany Faculty of Mathematics, University of Belgrade, Belgrade, Serbia 0000-0001-7764-6895]Valentin Martínez Pillet National Solar Observatory, University of Colorado Boulder, Boulder, CO, USA 0000-0001-5850-3119]Matthias Rempel High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO, USA Poynting flux is the flux of magnetic energy, which is responsible for chromospheric and coronal heating in the solar atmosphere. It is defined as a cross product of electric and magnetic fields, and in ideal MHD conditions it can be expressed in terms of magnetic field and plasma velocity. Poynting flux has been computed for active regions and plages, but estimating it in the quiet Sun (QS) remains challenging due to resolution effects and polarimetric noise. However, with upcoming DKIST capabilities, these estimates will become more feasible than ever before. Here, we study QS Poynting flux in Sunrise/IMaX observations and MURaM simulations. We explore two methods for inferring transverse velocities from observations – FLCT and a neural network based method DeepVel – and show DeepVel to be the more suitable method in the context of small-scale QS flows. We investigate the effect of azimuthal ambiguity on Poynting flux estimates, and we describe a new method for azimuth disambiguation. Finally, we use two methods for obtaining the electric field. The first method relies on idealized Ohm's law, whereas the second is a state-of-the-art inductive electric field inversion method PDFI_SS. We compare the resulting Poynting flux values with theoretical estimates for chromospheric and coronal energy losses and find that some of Poynting flux estimates are sufficient to match the losses. Using MURaM simulations, we show that photospheric Poynting fluxes vary significantly with optical depth, and that there is an observational bias that results in underestimated Poynting fluxes due to unaccounted shear term contribution. § INTRODUCTION Quantitative estimates of vertical energy transport in solar photosphere have been limited, yet they are explicitly relevant to many observed phenomena on the Sun, including flux emergence <cit.>, chromospheric and coronal heating <cit.>, and solar flares and coronal mass ejections <cit.>. The flux of magnetic energy, i.e. Poynting flux or Poynting vector, defined as the cross product of electric and magnetic fields, has long been considered a primary mechanism for the energy transport from the photosphere to the overlaying atmosphere, but specific magnetically-driven processes and their relative importance have remained somewhat elusive <cit.>. Typically, the flux of magnetic energy is divided into emergence and shear terms. The emergence term arises from advection of magnetic field lines by upward plasma flows, and the shear term (also called wave term) is associated with twisting of the field lines by horizontal flows. Quantitative investigations of photospheric Poynting flux are a relatively recent development, owing to the fact that the intermediate quantities needed to compute it – full electric and magnetic field vectors – are difficult to obtain even from modern state-of-the-art observations. Significant strides have been made in both magnetic field inversions from observed Stokes profiles <cit.>, and electric field inversions <cit.>. However, most of the quantitative studies of Poynting flux have been constrained to either simulated data <cit.>, or active regions and plages <cit.>, since in these settings one deals with relatively high polarimetric signal-to-noise ratios (SNR). In particular, <cit.> used high-fidelity simulations of the quiet-Sun (QS) photosphere to explain heating in a coronal loop, while <cit.> computed Poynting flux from the active region AR 11158 and found it to be sufficient to explain the heating of chromosphere and corona, according to theoretical estimates in <cit.>. However, an analogous, observation-based study into Poynting flux in QS has not been conducted. <cit.> and <cit.> put constraints on the coronal energy associated with motions of photospheric footpoints and plage, but they used ideal MHD formulation of Ohm's law and they assumed zero upward advective motion, thereby neglecting the emergence term of Poynting flux. More recently, <cit.> produced quantitative estimates of QS Poynting flux, but their focus was mostly on the horizontal flux and their method also included several simplifications, such as the idealized Ohm's law and reliance on apparent motions of magnetic field concentrations to obtain velocities transverse to the line-of-sight (i.e. parallel to plane of sky). The studies of magnetic features in the QS have been few and far between due to both the noisiness of observations and systematic issues. The Sunrise/IMaX balloon-borne probe provides some of the best currently available QS polarimetry <cit.>, yet even in this data sample, strong linearly polarized light constitutes only about 10% of the field of view <cit.>. Furthermore, there is the outstanding problem of magnetic field 180° azimuthal ambiguity, wherein spectropolarimetric inversions of Stokes profiles return two mathematically valid configurations of transverse magnetic field. While many methods of disambiguation have been proposed <cit.>, none of them have been validated on QS magnetograms. Since full magnetic vector is necessary to compute Poynting flux, the task of disambiguation is necessary. As a result of these observational and methodological challenges, quantitative investigations into Poynting flux in QS have been limited. At the same time, there will soon be unprecedented observations of QS from the Daniel K. Inouye Solar Telescope (DKIST), which will allow us to improve significantly on spatial resolution, cadence, and/or polarimetric sensitivity <cit.>. There are also sophisticated methods of computing Poynting flux, which have not yet been tested on QS data. This presents a gap in the current state of this discipline, which this paper seeks to fill. Since QS constitutes the majority of observed photosphere area-wise, it is imperative that we understand the energy flux from it. The goal of this paper is to compute Poynting flux in the QS photosphere. To this end, we use several methods and we apply them to both observational and simulated data, with a focus on the former. The remainder of the paper is structured as follows: in <ref> we describe the observational and simulated data we used in this work, in <ref> we explain how we obtain Poynting flux and the necessary intermediate quantities – full velocity, magnetic field, and electric field vectors. In <ref> we describe Poynting flux estimates from the various employed methods, and in <ref> we discuss them. Finally, in <ref> we summarize our findings and outline some of the possible future work. § DATA §.§ Observational Data: IMaX We perform our analysis on spectropolarimetric observations from the Imaging Magnetograph eXperiment (IMaX) instrument on board the SUNRISE balloon-borne observatory <cit.>. We use one continuous IMaX/SUNRISE time series taken on June 9th, 2009 between 01:30:54–02:02:29 UT. This data set covers a 40×40 Mm region of QS at the disk center and includes a slowly evolving region of relatively high (>200 G, for filling factor unity) magnetic field concentration seen at the bottom of the V Stokes vector map in panels e–h of Fig. <ref>. The photon SNR of 1000, cadence of 33.25 s, and sampling resolution of 0.0545”/px make the IMaX data set the best available source for the purposes of studying Poynting flux in QS. With this combination of cadence and spatial resolution, a typical flux element moving at a moderate speed of 3 km s^-1 in the plane of sky <cit.> would traverse two pixels. IMaX provides high-quality, diffraction-limited polarimetric observations of QS in the Fe I 5250.2 Å line, which is sensitive to photospheric magnetic fields. The observations include full Stokes vector (I,Q,U,V) sampled in five wavelength positions: ±40 and ±80 mÅ on either side of the Fe I 5250.2 Å line, and at +227 Å in the continuum, with spectral resolution of 65 mÅ (85 mÅ Gaussian). The IMaX Fabry-Perot sensor introduces a systematic blue shift which grows as a function of distance from the center of field of view (FOV). We apply a correction in the form of distance-to-center-dependent red shift to account for this effect on LOS velocity. The level 0 data had also been corrected to minimize instrumental effects, such as dark and flat-fielding and removal of dust-induced effects, resulting in non-reconstructed (NR) data <cit.>. The Q and U noise levels in the NR data set were estimated to be 8.3×10^-4 I_c and 1.1×10^-3 I_c, respectively <cit.>. In addition, the IMaX point-spread function (PSF) was used to apply a phase diversity reconstruction (PDR) to NR data, thereby increasing spatial resolution to 0.15” at the expense of increasing Q and U noise levels to 2.6×10^-3 I_c and 3.6×10^-3 I_c, respectively <cit.>. We used the NR data, with their lower polarimetric noise, for magnetic field inversions, and the PDR data, with their higher spatial resolution, for velocity inversions. §.§ Simulation Data: STAGGER STAGGER <cit.> is a 3-D radiative magneto-hydrodynamic (MHD) code, which solves for conservation of mass, energy, and momentum equations. Those equations are coupled with radiative transfer equations in local thermodynamic equilibrium (LTE) non-grey atmosphere on a 48-km grid size. The simulation cadence is 60 s. We use continuum intensities and transverse velocities from STAGGER simulations of QS to validate velocities obtained with FLCT and the neural network based method DeepVel, which isdiscussed further in Section <ref>. §.§ Simulation Data: MURaM MURaM <cit.> is a state the of art radiative MHD code used to model a variety of features in the solar atmosphere and below. The MURaM code solves for mass and energy transfers between subsurface convective zone and the photosphere, chromosphere, and corona. The simulation we analyze here is based on the case `O16bM' from <cit.> and was extended in the vertical direction by about 500 km. The simulation solves for all the main MHD quantities (magnetic and velocity vectors, temperature, pressure, heat and energy fluxes) in a domain with the physical extent of 24.576× 24.576× 8.192 Mm^3, with an isotropic grid spacing of 16 km, resulting in a 1536×1536×512 grid. It spans optical depths between approximately 5× 10^-8< τ <10^9, i.e. from the convection zone to the upper chromosphere and transition region. The location τ=1 is found about 2 Mm beneath the top boundary. The relevant simulation quantities from a QS MURaM simulation (LOS velocity, \|B\|, S_z, and S_h) are shown in Figure <ref>. § METHODOLOGY Recall that Poynting flux is defined as S = 1/4πE×B, where B and E are magnetic and electric field vectors, respectively. In <ref>, we first describe how we use the polarimetric observations to infer the magnetic field B in the quiet Sun. In <ref>, we summarize the three methods we use to disambiguate the azimuth of the horizontal magnetic field: ME0 <cit.>, random azimuth and Poynting-flux optimization methods. Finally, in <ref>, we overview the two approaches we use to derive the electric field E: the PDFI_SS electric field inversion method, that solves the Faraday's induction equation -∇×E = ∂B/∂ t, and the simplified electric field inversion method that strictly imposes the idealized Ohm's law E = -v×B, where v is the velocity vector. In the simplified formulation of Poynting flux, where idealized Ohm's law is imposed strictly, we can express vertical Poynting flux (S_z) as follows: S_z = 1/4π[v_zB_h^2 - (v_h ·B_h) B_z], where the z and h subscripts denote vertical and transverse variables, respectively. In this expression, the first term is the emergence term and the second is the wave, or shear, term. In <ref>, we describe the two transverse velocity reconstruction methods, DeepVel and FLCT, since transverse velocities are a required intermediate quantity for using either of the electric field inversions. For brevity, we refer to the simplified approach as “ideal-MHD” method and to the PDFI_SS method as “inductive method”. We emphasize, however, that both approaches could enforce the ideal MHD condition, but to different extents: the “ideal-MHD” method does so strictly but the “inductive method” does not, but could enforce it via ideal non-inductive contribution (see Section 2.4 in <cit.>) §.§ Magnetic Field inversions To obtain the magnetic field configuration and LOS velocity from level-1 IMaX polarimetry, we apply the Milne-Eddington (ME) inversion code pyMilne <cit.> to the NR IMaX data set. We chose this method for its relative computational efficiency – the assumptions of Milne-Eddington atmosphere simplify the inversion scheme while adequately capturing the physics of the photospheric Fe I 5250.2 Å line formation. b For each IMaX frame, the inversion code uses several seeds ( parameter) to prevent the scheme from converging to local minima, and several Levenberg-Marquardt iterations () per seed. It should be noted that pyMilne assumes magnetic filling factor of unity, which may introduce bias in transverse magnetic field inversions <cit.>. We show an example of level 1 Stokes data in panels a–d in Figure <ref> and the corresponding outputs of Milne-Eddington inversions in panels e–h. Clearly visible is the high-V signal region at the bottom of FOV. It corresponds to a strong B-field region that persists and slowly evolves throughout the observation window. We designate it as the region of interest (ROI) and denote it by a green rectangle in the four right panels. The ROI is not associated specifically with either upflows or downflows, as seen from the dopplergram. We also show the resulting distributions of LOS and transverse magnetic field components in Figure <ref>. The negative polarity in ROI is clearly seen in the skewed shape of B_z histogram. As seen from the bottom-left panel, other regions with strong polarization signal are much smaller in extent. They are also more transient, highlighting the difficulties of QS polarimetric observations. As can be seen in Figure <ref>, the polarimetric signal, particularly in Q and U (panels b and c), is quite weak in our data (<200 G in most of the FOV). This is of course to be expected in the quiet Sun regime, where magnetic fields are only strong enough to produce distinct linear polarization features in 3–16 % of pixels in the FOV of SUNRISE/IMaX <cit.>. In parts of the analysis that follows, we only consider regions of the FOV with signal strengths above a certain threshold. We chose the threshold of 50 G for masking out pixels with insufficiently strong magnetic fields, for the following reasons: 1) 50 G is approximately equal to 3 σ in magnetic field strength distribution (bottom right panel of Figure <ref>), 2) the subset of pixels in FOV with B>50G closely (within 5 G) corresponds to the pixels where at least one of Q, U, or V spectra exhibits strong enough (>3σ) deviations from continua, 3) this threshold is consistent with the minimum horizontal field strength described in <cit.>, where the strength of linear polarization features in IMaX magnetograms was found to be in the range 50–500 G. §.§.§ Azimuth Disambiguation Azimuthal 180^∘ ambiguity is a well-known problem, wherein spectropolarimetric inversions based on Zeeman effect produce two solutions for B-field azimuth, and the two solutions are mathematically equally valid. Several solutions to this problem have been proposed. Those include global optimization mechanisms, such as ME0, where the preferred magnetic field configuration results in a globally minimized magnetic energy <cit.>. Other methods select the orientation of magnetic fields that results in the highest B_z, if the magnetogram is taken off disk center, or they look for opposite polarities and select an orientation that would close field lines between the polarities <cit.>. It should be noted that none of these methods have been rigorously tested in the QS regime, as linear polarization strength is usually too low to adequately employ these methods. In this work, we attempt to use three (and end up using two) methods to disambiguate azimuths: ME0, randomization, and Poynting flux optimization. We first use ME0, as it is the most physically rigorous of the three methods and it has been extensively used, including, for example, in Hinode and SDO/HMI data processing pipelines <cit.>. ME0, or the minimum energy method, is an optimization algorithm that minimizes the global quantity λ \|J_z\| + \|∇·B\|, where J_z is vertical current and λ is a modifiable scalar parameter that determines the relative importance of the two terms. As mentioned, ME0 has not been tested on QS data, so, to that end, we tested ME0 on synthetic magnetograms obtained from the 3D MHD STAGGER code (see <ref>). Unfortunately, ME0 performed poorly on QS magnetograms produced by STAGGER, likely due to different physical assumptions under which STAGGER and ME0 operate. The issue with ME0 validation warrants a more detailed investigation, but we leave it for future work, as it is not the focus of the present paper. While we cannot use full ME0 capabilities, the code is capable of performing a potential field acute angle disambiguation. We use this method to disambiguate azimuths in the first frame, and for each subsequent frame, we resolve the ambiguity using acute angle with respect to the previous frame. Another approach is to use the regular ME0 disambiguation while setting the λ weighting factor for \|J_z\| to 0. The minimized quantity is then simply the divergence of magnetic field. Like in the potential field disambiguation, we only apply this method to the first frame and then select the azimuths resulting in an acute angle with respect to the previous frame. In both cases this is done in order to minimize temporal discontinuities. While not strictly physical, this approach has been taken before, e.g. in <cit.>. In the absence of a validated physical disambiguation method, we asked two questions: how sensitive is Poynting flux to the orientation of transverse magnetic fields (in other words, how much does the "choice" of azimuth affect our computed quantities of Poynting flux), and what is the maximum Poynting flux that can be obtained from any given magnetogram that is yet to be disambiguated? These two questions lead us, respectively, to two other disambiguation methods: azimuth randomization and Poynting flux optimization. Azimuth randomization can be thought of as an absolutely imperfect disambiguation, wherein we randomly add either 0^∘ or 180^∘ to the azimuth value of each pixel in a magnetogram. The random assignment for each pixel is performed independently of its neighboring pixels or earlier azimuth values in that pixel. Thus, this method yields a disambiguated magnetogram that almost certainly has spatial and temporal discontinuities in transverse field orientations. The Poynting flux optimization disambiguation method consists of two steps: in the first magnetogram (t=0), we disambiguate azimuths in each pixel by selecting the one that results in higher value of S_z as computed using the ideal-MHD method, i.e. using Equation <ref>. Then, for each consecutive magnetogram, we select for each pixel the azimuth value that is closer to the azimuth value of that pixel in the previous frame. In contrast with the randomization method, where every pixel is completely independent from both its surrounding pixels and that pixel in adjacent magnetograms in the time series, the Poynting flux optimization method results in some degree of spatial and temporal azimuth continuity, while also providing us a physical ceiling (i.e. upper boundary) for the Poynting flux. We stress, however, that this disambiguation method is only physically meaningful insofar as it provides the ceiling for Poynting flux. §.§ Electric Field Inversion Methods To find the electric field needed to estimate the Poynting flux (Equation <ref>), we use two approaches. The first “ideal-MHD” approach strictly enforces the ideal MHD condition (Equation <ref>). We then use Doppler measurements to derive the vertical velocity component and two reconstruction methods, FLCT and DeepVel, to invert the transverse velocity component (see <ref> below). In the second “inductive” approach we use the PDFI_SS method to derive the electric field directly by inverting Faraday's law without necessarily enforcing the ideal MHD condition (see <ref> below). §.§.§ Transverse Velocity Inversion Methods As shown in Section <ref>, the full plasma velocity vector (or alternatively, the horizontal electric field) is required to compute Poynting flux. Unlike the LOS velocity, which could be recovered from Doppler data (e.g. ), transverse velocities cannot be directly inferred from observables. The two velocity retrieval methods we use in this work are Fourier Local Correlation Tracking <cit.> and a convolutional neural network (CNN) DeepVel <cit.>. FLCT <cit.> is a plasma flow tracking method that takes two consecutive magnetograms or intensitygrams and, using a finite sliding window, infers the plane-of-sky displacement needed to produce the second map from the first one. It has been used as the flow inversion method for PDFI_SS <cit.> and for tracking flows in various environments <cit.>, but it has some constraints. The main constraint of the FLCT approach is that it assumes that any change in continuum or magnetic field intensity is due to advective motion without obeying the induction equation (i.e. FLCT measures an optical flow). Secondly, FLCT has been applied to data with either relatively strong magnetic fields <cit.>, where tracking is made possible by relatively large S-N ratios, or to low-resolution and large FOV images, where the objective was to track meso- and super-granular motions <cit.>. Neither of these contexts applies to our QS case: magnetic concentrations are, for the most part, transient and limited in spatial extent and strength, making it necessary to rely on continuum images, and the relevant scales of plasma motions are well below even meso-granular scales. To validate FLCT plasma flow inferences in a setting more closely resembling the IMaX observations, we apply the FLCT to continuum images from STAGGER simulations. We find the correlation between FLCT and reference flows to be low, with the Pearson correlation coefficient of at most r<0.45. The correlation is even weaker if the σ parameter, which defines the width of sliding Gaussian window, is lower than 10 pixels or higher than 15. In our work, we pick σ=10, as it produces the strongest correlation. Following analyses in <cit.> and <cit.>, we consider three other correlation metrics between reference STAGGER velocities and velocities obtained from inversions: the spatially averaged relative error E_rel[v_inv,v_ref] ≡<√((v_ref-v_inv)·(v_ref-v_inv)/v_ref·v_ref)>, the vector correlation coefficient C[v_inv,v_ref] ≡<v_inv·v_ref>/√(<v_ref·v_ref>·<v_inv·v_inv>), and the cosine similarity index, which measures the global spatial distribution of velocity vector orientations A[v_inv,v_ref] ≡<v_inv·v_ref/v_invv_ref>, where the <·> operation denotes spatial averaging.The C coefficient is defined so that it is 0 when the velocity vectors are perpendicular everywhere and 1 when parallel everywhere. Likewise, the A coefficient is -1 when the vectors are anti-parallel and 1 when identical. Thus, the agreement between two vectors is the better the closer both C and A are to unity. For mathematical expressions of these metrics, see equations 3–5 in <cit.>. For FLCT, the values of these metrics are (E_rel=1.09, C=0.35, A=0.21). FLCT and reference STAGGER flows are also qualitatively different (see Figure <ref>, left and right panels). STAGGER velocities show a clear pattern of divergence in granules and convergent flows with vortices in intergranular lanes (IGLs), whereas FLCT velocity fields are much more laminar (and smaller in magnitude) on average. We find that FLCT velocity inferences in QS can be improved by averaging instantaneous velocities over 30-minute time windows <cit.>. The correlation coefficient between FLCT and STAGGER velocities then improves to r=0.75, but, considering the photospheric timescales are on the order of five minutes, such improvement comes at a cost of losing time-dependent information. We therefore conclude that the FLCT method is an inadequate velocity inversion for our purposes, where instantaneous or near-instantaneous (<2.5 minutes) velocities are to have high fidelity. In addition to FLCT, we use Deepvel – a convolutional neural network that has been previously used to infer velocities on granular scales, including in the quiet Sun <cit.>. DeepVel is trained using simulation data, for which all flow components are known, to map a pair of input images (e.g., continuum intensity images at two timesteps) to the transverse flows at a given optical depth or geometrical height. This approach is known as supervised learning. In other words, the output velocities approximate what the flows in the training simulation would be if we assume that the input data provided to the neural network was generated by the training simulation (i.e., there is a model dependency). For this work, we train DeepVel on a set of STAGGER data frames. To test the trained network, we run it on a STAGGER intensity map that is outside of the training set. The test map has the same properties as those described in Section <ref>. Similarly to the previous works, we find that DeepVel instantaneous velocities are highly (r=0.91) correlated with simulated velocities (Figure <ref>). We find that correlation metrics values are significantly higher for DeepVel (E_rel=0.74, C=0.91, A=0.87) than for FLCT. These are stark improvements over FLCT in terms of the accuracy achieved without losing temporal resolution via time averaging. Even though DeepVel is limited in ways that may have implications for our results <cit.>, we choose to apply DeepVel to IMaX intensitygrams to retrieve transverse velocities in our analysis. Hereafter, all retrieved transverse velocities are obtained with DeepVel rather than FLCT. We note, however, that DeepVel is not without its limitations. Even though we get really good agreement (r≈0.9) between simulation and DeepVel velocities and divergences, DeepVel is not as reliable at reproducing vorticities (see Figure <ref>, panel d). §.§.§ PDFI Electric Field Inversion Method To find Poynting flux without assuming ideal MHD conditions, we use the PDFI_SS method <cit.>. Briefly, the magnetic field in the PDFI_SS method is expressed as a sum of poloidal and toroidal components. This decomposition allows to derive the inductive component of the electric field from observed quantities by uncurling the Faraday law (Equation <ref>). The gradient of a scalar part of the electric field that appears due to uncurling of the Faraday's law is called “non-inductive” and could be computed from additional constraints, including ideal MHD constraint E·B=0 <cit.>. PDFI_SS has been used to describe the evolution of Poynting flux and magnetic helicity in multiple works, but notably, these were all concerned with either observed or simulated active regions <cit.> or regions of flux emergence <cit.>. To our knowledge, PDFI_SS has not been applied to QS magnetic fields. Apart from the general challenge of studying QS magnetism, the reliance of PDFI_SS on the ∂B/∂ t term in Faraday's law makes it especially susceptible to noise. To mitigate the influence of noise, we set the parameter, which masks pixels with lower magnetic field strength <cit.>, to 50 G – the same threshold we choose in <ref>. PDFI_SS also requires high cadence observations, so as to not miss the transient magnetic concentrations that are ubiquitous in QS <cit.>. § RESULTS We compute Poynting fluxes using two approaches: from velocity fields together with the ideal MHD assumption, and the PDFI_SS electric fields, where time derivatives of magnetic fields are used as a source term. Within each approach, we use randomly disambiguated azimuths and azimuths obtained via the optimization procedure (see Section <ref>). We show temporal evolution of Poynting fluxes in both settings in Figure <ref>. As discussed in <ref>, azimuthal orientation of vector magnetic fields can affect Poynting flux magnitudes. Since one of our two principal methods of azimuthal disambiguation relies on randomizing azimuths on a pixel-by-pixel basis, we investigate the resulting uncertainty in Poynting fluxes in ideal-MHD setting by repeating the randomization for each magnetogram 5000 times. We find average Poynting flux estimates in each frame to be highly robust to different realizations of azimuth randomization, with both signed (net) and unsigned (absolute values) fluxes tightly clustered (see Figure <ref>). We also find that in the ideal-MHD setting, the emergence term v_z B_h^2 dominates both signed and unsigned fluxes over the shear, or wave, term (v_h ·B_h) B_z (see Equation <ref>), which accounts for less than 1% of the total vertical Poynting flux. The left panel of Figure <ref> shows the Poynting flux evolution for the ideal-MHD case. Different plot colors correspond to spatially averaged Poynting flux (S_z) in all pixels as well as only in pixels where the magnetic field strength (\|B\|) exceeds 50 G and 100 G thresholds. The first thing to note here is that the choice of azimuth disambiguation method plays a negligible role on S_z values. The largest difference is in the first frame, where, in the optimization procedure, we explicitly optimize for largest S_z value. In just over one minute this difference disappears, and S_z values stabilize within the range 6.0± 0.56 × 10^5 erg cm^-2 s^-1 and 1.1±0.087 × 10^7 erg cm^-2 s^-1 for all pixels and pixels with high B-fields, respectively. This is consistent with our analysis of the randomization procedure shown in Figure <ref>, where we find very little variation in FOV-integrated S_z across different azimuth realizations. We also observe that selecting pixels with relatively strong \|B\| increases the average S_z by an order of magnitude, but there isn't much variation between 50 G and 100 G thresholds (or even 150 G threshold and above, which aren't shown here). Increasing the threshold to 100 G, however, reveals quasi-periodic oscillations that could conceivably be linked to 5-minute photospheric oscillations <cit.>. Figure <ref> (right panel) shows Poynting fluxes derived from the PDFI_SS method. Recall that, since we set the parameter to 50 G, all pixels with magnetic fields below that threshold are set to zero and are not considered in the following analysis. We find that these estimates are very different from the ideal-MHD estimates shown on the left panel. Firstly, the optimized (randomized) S̅_̅z̅ is -2.1± 13 × 10^5 (-1.3± 9.4 × 10^5) erg cm^-2 s^-1 – significantly lower, even in terms of absolute values, than S̅_̅z̅ in the ideal-MHD case with pixels with weak magnetic fields counted. Secondly, in both cases (randomized and optimized azimuths), S_z oscillates around zero and is sometimes well below it, meaning that magnetic energy is transported downwards instead of upwards. Thirdly and less surprisingly, S_z values obtained from the randomization and S_z optimization disambiguation methods are much more different from one another than in the ideal-MHD case. This is due to the fact that PDFI_SS uses spatial and temporal derivatives of the B-fields to compute S_z, and both are affected in the randomization procedure which produces highly discontinuous magnetic field configurations, more so than in the case of optimized azimuths. However, the two other azimuthal ambiguity resolutions – from potential field and from \|∇·B\| = 0 – also produce Poynting fluxes that oscillate frequently around zero and almost never exceed 2 × 10^6 erg cm^-2 s^-1. This most likely indicates that significant spatial and temporal discontinuities are present in IMaX QS magnetograms regardless of the azimuthal disambiguation method, as can be seen in Figure <ref>. To evaluate how Poynting flux and its components vary in height, we use the outputs of MURaM simulations, since IMaX data set only includes data from one optical surface. In Figure <ref>, we compare Poynting fluxes derived directly from MURaM simulations. We find that MURaM averaged vertical Poynting flux reverses sign very close to the τ=1 surface, and that it is exceeded by \|S_h\| from the convection zone until well above τ=0.1. We find that at τ=1, S_z = 4.38 ×10^6 erg cm^-2 s^-1, and it rises to 2.28 ×10^7 erg cm^-2 s^-1 at τ=0.1. § DISCUSSION In their seminal paper, <cit.> derived a threshold of upward energy flux from the photosphere that would be necessary to explain chromospheric and coronal heating in the quiet Sun – S_z,thr=4.3×10^6 erg cm^-2 s^-1.In MURaM simulations, vertical Poynting flux at τ=1 is just above the S_z,thr value from <cit.>. This is consistent with existing MURaM simulations, where hot corona is maintained by photospheric magnetoconvection <cit.>. However, we find that in IMaX observations, there is not enough Poynting flux whether we use the ideal-MHD method or PDFI_SS, unless we consider only strong B-field pixels in the ideal-MHD case (Figure <ref>). Furthermore, in the ideal-MHD case, vertical Poynting flux is lower than the minimum value required for heating by an order of magnitude. On the other hand, PDFI_SS values are closer to 4.3×10^6 erg cm^-2 s^-1 in magnitude, but are frequently negative, indicating downward energy flux. What are the possible causes of discrepancies between different Poynting flux estimates, and why is Poynting flux negative in some of them? We propose several physical and methodological explanations. A non-trivial methodological issue with our analysis is the weak signal strength in the quiet Sun. This is particularly severe when it comes to Q and U Stokes vector signal strength, which adversely affects our inversions of B_h. Even in the ROI, where magnetic field strength exceeds 200 G – a relatively high value for our data set – there are significant discontinuities in the spatial distribution of horizontal magnetic field (right panel of Figure <ref>). These gaps affect both ideal-MHD and PDFI_SS Poynting flux inversion methods. In ideal-MHD, as can be seen from Equation <ref>, Poynting flux is highly sensitive to B_h as it appears in both terms of the expression. In PDFI_SS, B_h uncertainties affect both the v×B term as above and the spatial and temporal derivatives of the magnetic field. Uncertainties in transverse magnetic field inversions B_h propagate into issues with azimuth disambiguation. However, we see that with the IMaX signal strength, they do not meaningfully affect Poynting flux estimates, especially in the ideal-MHD scenario (Figure <ref>). Instead, the emergence term v_z B_h^2 is responsible for virtually all signed Poynting flux and 99% of unsigned Poynting flux. This is qualitatively consistent with some of the existing literature <cit.>, but this fraction is much higher than in previous works. This is likely due to the weak magnetic field signal in our observational sample: For the shear term to be present, both linear and circular polarization signatures must be strong in the same pixel. Transverse velocity inversions are another potential source of errors in Poynting flux inversions, though in the present work it is also likely to be a secondary order error. As discussed in <ref>, vorticity values inferred with DeepVel may be unreliable (Figure <ref>). Further, we do not have access to “ground truth" when it comes to transverse velocity on the real Sun, and a neural network trained using supervised learning generates predictions that are only as good as the simulations they were trained on (from the relationship between continuum intensity and transverse flows, to the topology and magnitude of the flows). In MHD simulations, including STAGGER and MURaM, vortices are mostly concentrated in IGLs and have been shown to be spatially correlated with vertical Poynting flux in MURaM simulations <cit.>. This, then, presents a clear avenue for improvement, particularly when DKIST observations with higher spatial resolution <cit.> become available, since features in IGLs are especially vulnerable to resolution effects. Another way to improve the neural network approach is to train it to match coherence spectra, i.e. to match velocities at different frequencies in the Fourier space, as was done in <cit.>. Poynting flux inversions themselves can still be improved. We already explained how, unlike PDFI_SS, the ideal-MHD method does not account for Poynting flux derived from ∂B/∂ t, but PDFI_SS also has limitations. It has not been tested in the QS regime, particularly when only one polarity (negative in our case) of B_z is present in the FOV. Another explanation for negative vertical Poynting flux coulde be due to physical reasons. There are several pieces of evidence in favor of that possibility. First, we are studying Poynting flux at the boundary layer between convection-dominated layers below the photosphere and radiation-dominated atmosphere. In such environment, it is reasonable to expect all energy fluxes (e.g. mass flux, convective flux) averaged over a representative FOV to become dominated by their horizontal components, which are mostly self-canceling, while vertical components approach near-zero values <cit.>. This is indeed the case in IMaX observations. Assuming ideal MHD conditions, we find that the horizontal Poynting flux (\|S_h\|) exceeds \|S_z\| by a factor of ≈ 3. <cit.> analyzed the same IMaX data set we use in our work and reported an even higher ratio of horizontal-to-vertical Poynting fluxes, likely due to higher velocity values in their inversions. In MURaM simulations, we see that Poynting fluxes are principally concentrated in IGLs. An important corollary from this observation is that the emergence term of Poynting flux, which can only be negative provided v_z<0, is primarily negative in photospheric and chromospheric heights, which is indeed what we find (see Figure <ref>). Therefore, on average, the wave term of Poynting flux is larger in magnitude than the emergence term, in stark contrast with our findings in IMaX observations. As discussed above, this is likely due to a strong observational bias, wherein both IGL structure and magnetic concentrations that are not trivially oriented (neither completely parallel nor perpendicular to LOS) are subject to instrumental limitations. Observed ideal-MHD Poynting fluxes in IMaX, which are dominated by the emergence term yet positive, likely arise from magnetic concentrations located in granule interiors, such as those inside the ROI (see Figure <ref>). Analogs of such a structure can be seen in MURaM simulations as well, e.g. in Figures <ref> and <ref> at [x,y]=[3.5 Mm,3.5 Mm] (just left and below of the center of FOV). There are of course caveats when it comes to optical depth. First, it is unclear what optical depth corresponds to the formation of Fe I 5250.2 Å line, which we used for spectropolarimetric inversions. It is evident that the line forms somewhere in the photosphere, but we are not aware of existing studies that looked at its response function. We find that this could be important, since in MURaM simulations average Poynting flux values are sensitive to changes in optical depth: from τ=1.1 to τ=0.63, S_z increases by a factor of 2.7 (see Figure <ref>). To see where the Fe I 5250.2 line could form, we calculate its response function using a MURaM atmospheric profile <cit.>. We use one profile from granule and one from IGL, and in both cases (the latter is shown in Figure <ref>) we find that the line forms in the photosphere (300–400 km), but that it has a broad formation height range. The second caveat related to optical depth is that a constant τ surface is very different from a constant height surface, and Poynting fluxes computed on optical surfaces deviate significantly from those computed on geometrical surfaces with comparable optical depth averaged over the FOV (see Figures <ref> and <ref>). In particular, while the average Poynting flux computed on the geometrical surface with τ=1 may be sufficient to match energy losses in the chromosphere and corona, the average Poynting flux computed on the geometrical surface τ=1 is far from it, as it is -3.11×10^7 erg cm^-2 s^-1 (Figure <ref>). Despite these differences, we treat all of our vector quantities' components as being either parallel or perpendicular to plane-of-sky (and line-of-sight). This can lead to unphysical results, since we are essentially dealing with vector projections and not true vectors. It is especially so in regions where optical surfaces are least aligned with geometrical ones, such as on the boundaries between granules and intergranular lanes. Incidentally, this is where 1) MURaM vertical Poynting flux is primarily concentrated, 2) transverse flows have the most shear and vorticity, and 3) resolution constraints have the highest detrimental effect <cit.>. § CONCLUSIONS In this work we used two approaches – the ideal-MHD and the PDFI_SS methods – to compute average photospheric Poynting fluxes from IMaX polarimetric observations. We tested several methods for deriving intermediate quantities required for computing Poynting flux. Principally, such quantities include the magnetic field azimuth, transverse velocities, and electric fields. We also looked at the outputs of 3-D radiative MHD code MURaM between τ=10^9 and τ=-5×10^-8 to glean insights from simulated photospheric data. Our quantitative estimates of Poynting flux do not reveal a consistent picture with respect to whether photospheric Poynting flux is sufficient to explain chromospheric and coronal heating (Table <ref>). However, we can outline several important findings: * The ideal-MHD approach yields ambiguous estimates of S_z, but this could be explained by the quality of available data. When considering only pixels with relatively high magnetic field values (\|B\|>50 G), the resulting average Poynting flux suffices to explain chromospheric heating, but S_z averaged over all pixels does not (Figure <ref>). It is possible that, due to instrumental limitations, we miss on many of the small and/or transient magnetic concentrations; * The 180^∘ azimuthal ambiguity barely affects the estimates of the ideal-MHD approach (Figure <ref>). This is because Poynting fluxes derived via the ideal-MHD method are dominated by the emergence term v_z B_h^2. This could also explain the lack of Poynting flux when averaged over all pixels. The importance of the emergence term has been reported before, but it is likely exaggerated in our results, since pixels with both vertical and transverse magnetic field (which are both necessary to produce the wave term (v_h ·B_h) B_z of Poynting flux) are difficult to detect in QS magnetograms. Indeed, in MURaM simulations, the wave term is on average positive and larger in magnitude than the emergence term, which is concentrated in IGLs and, consequently, is negative on average. When more advanced observations are available, such that the bias against the wave term is diminished and resolving azimuthal ambiguity becomes relevant, we point to our Poynting flux optimization method as a way to disambiguate azimuths while meaningfully constraining S_z; * Poynting flux obtained with the PDFI_SS method is highly time-dependent, insufficient for chromospheric and coronal heating, and is negative in many of the frames in our time series (Figure <ref>). It is also sensitive to azimuth disambiguation. The variability and sensitivity to magnetic field azimuthal orientation can be caused by the reliance of PDFI_SS on spatial and temporal derivatives, combined with the noisy data sample. The closeness of it to zero can be attributed to the photosphere being a boundary layer between convection-dominated subsurface and radiation-dominated lower atmosphere; * MURaM simulations also display vertical Poynting flux that flips sign around τ=1 and is dominated by (unsigned) horizontal Poynting flux, supporting the boundary layer explanation (Figure <ref>). S_z in MURaM simulations is frequently negative around the τ=1 surface, particularly in IGLs (see Figures <ref> and <ref>). At the same time, the upward Poynting flux is more than sufficient to explain chromospheric and coronal heating. While it may look like Poynting flux is close to the heating threshold around τ=1 (Figure <ref>), this region is in the deep photosphere, where the sign of Poynting flux flips and below the formation height of most observable spectral lines. It should be noted that MURaM simulations that extend into the corona <cit.> produce a self-maintained QS corona (about 1.5 Million K) with sufficient Poynting flux. However, those simulations have lower resolution and the Poynting flux comes more from braiding of QS network field that is mostly absent in our simulation set. MURaM simulations of coronal loops also show that photospheric energy output is sufficient to maintain hot corona <cit.>. The main question – whether observed QS photosphere produces enough magnetic energy in the form of Poynting flux to heat the chromosphere and corona – remains open. There are, however, promising signs that this uncertainty will be cleared up in the future. DKIST observations, particularly with VBI, DL-NIRSP, and VTF instruments, can be used to observe photospheric magnetic fields with unprecedented polarization sensitivity, resolution, and cadence <cit.>. Repeating this analysis with DKIST data is one of the most obvious avenues for future work. We can also improve our methodology moving forward, particularly as it pertains to transverse magnetic field inversions, including azimuth disambiguation, and transverse velocity inversions. For the former, a physics-based approach such as ME0 is preferable to the more stochastic or optimizing approaches used in this work. We can also use acute angle disambiguation, provided we have QS observations sufficiently far from the disk center. We may also achiever higher fidelity in transverse magnetic field inversions by using an inversion scheme that solves for magnetic filling factor <cit.>. For transverse velocity inversions, modifying DeepVel so that it is trained to match vorticity as well as velocity can be useful, since Poynting flux is associated with shear flows and vortices in IGLs. Finally, numerical MHD simulations present a convenient avenue of exploring relationships between time- and height-dependent upward flux of magnetic energy and different structures in the quiet Sun. This area has remained largely unexplored, due to a lack of observational counterparts with which to verify potential findings, but it is more relevant now, in the era of DKIST. For an investigation of Poynting flux that is more directly comparable to observations, observables such as Stokes vectors must be computed using forward models and then inverted. It should be noted that it is unclear whether such an approach will result in physical values, since inversions produce quantities on optical surfaces where vector cross products are not meaningful. However, an approach involving forward modeling can be used to assess the model fidelity and, by extension, whether it can be used to make useful Poynting flux predictions. Detailed and focused studies of numerical simulations are therefore necessary. § ACKNOWLEDGEMENTS This work is supported by NASA FINESST award 20-HELIO20-0004. This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977. The authors thank Anna Malanushenko for general comments on the paper and K.D. Leka for her valuable advice on azimuth disambiguation. The authors also thank the anonymous referee for useful suggestions, particularly in regards to azimuth disambiguation. SUNRISE (IMaX). https://github.com/jaimedelacruz/pyMilnepyMilne <cit.>, https://www.cora.nwra.com/AMBIG/ME0.htmlME0 <cit.>, https://pyflct.readthedocs.io/en/latest/index.htmlFLCT <cit.>, https://github.com/aasensio/deepvelDeepVel <cit.>, http://cgem.ssl.berkeley.edu/cgi-bin/cgem/PDFI_SS/indexPDFI_SS <cit.>, MURaM <cit.>. aasjournal
http://arxiv.org/abs/2307.00914v2	20230703101932	Revisiting equilibrium condensation and rocky planet compositions: Introducing the ECCOplanets code	[ "Anina Timmermann", "Yutong Shan", "Ansgar Reiners", "Andreas Pack" ]	astro-ph.EP	[ "astro-ph.EP", "astro-ph.IM" ]	Introducing the ECCOplanets code Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany [email protected] Geowissenschaftliches Zentrum Abteilung Geochemie und Isotopengeologie Goldschmidtstraße 1, 37077 Göttingen, Germany Centre for Earth Evolution and Dynamics, University of Oslo, Sem Salands vei 2A ZEB-bygget 0371 Oslo, Norway The bulk composition of exoplanets cannot yet be directly observed. Equilibrium condensation simulations help us better understand the composition of the planets' building blocks and their relation to the composition of their host star. We introduce ECCOplanets, an open-source Python code that simulates condensation in the protoplanetary disk. Our aim is to analyse how well a simplistic model can reproduce the main characteristics of rocky planet formation. For this purpose, we revisited condensation temperatures (T_c) as a means to study disk chemistry, and explored their sensitivity to variations in pressure (p) and elemental abundance pattern. We also examined the bulk compositions of rocky planets around chemically diverse stars. Our T-p-dependent chemical equilibrium model is based on a Gibbs free energy minimisation. We derived condensation temperatures for Solar System parameters with a simulation limited to the most common chemical species. We assessed their change (Δ T_c) as a result of p-variation between 10^-6 and 0.1. To analyse the influence of the abundance pattern, key element ratios were varied, and the results were validated using solar neighbourhood stars. To derive the bulk compositions of planets, we explored three different planetary feeding-zone (FZ) models and compared their output to an external n-body simulation. Our model reproduces the external results well in all tests. For common planet-building elements, we derive a T_c that is within ±5 of literature values, taking a wider spectrum of components into account. The T_c is sensitive to variations in p and the abundance pattern. For most elements, it rises with p and metallicity. The tested pressure range (10^-6 – 0.1) corresponds to Δ T_c ≈ +350, and for -0.3 ≤[M/H]≤0.4 we find Δ T_c ≈ +100. An increase in C/O from 0.1 to 0.7 results in a decrease of Δ T_c ≈ -100. Other element ratios are less influential. Dynamic planetary accretion can be emulated well with any FZ model. Their width can be adapted to reproduce gradual changes in planetary composition. Revisiting equilibrium condensation and rocky planet compositions Anina Timmermann1 Yutong Shan3 Ansgar Reiners1 Andreas Pack2 Received XX XX XXXX / Accepted XX XX XXXX =================================================================== § INTRODUCTION The chemical composition of rocky planets is, among other factors such as the surface temperature or the presence of a magnetic field, an important parameter with respect to the habitability and the potential existence of extraterrestrial life. The bulk composition of a planet can be roughly estimated using density measurements from observational data of radial velocity (planet mass) and planetary transits <cit.>. Transit spectroscopy is beginning to provide answers regarding the existence of specific molecules in planetary atmospheres <cit.>. However, the observational answer to the pivotal question of the composition of rocky planets, in particular the makeup of their interior structure, still remains largely elusive. Currently, the only technique that allows glimpses into solids compositions is the spectroscopic analysis of polluted white dwarfs <cit.>. This lack of observational data on planetary compositions is currently bridged by simulations of planet formation. To estimate the elemental composition of a planet, it is common to look to its stellar host. The star's chemical composition is assumed to approximately represent that of its protoplanetary disk due to the fact that stars and their disks form from the same molecular cloud <cit.>. Out of such a disk, planetesimals and planet embryos form from locally condensed solid materials, whose atomic and mineralogic makeups depend not only on the bulk proportions of elements available, but also on the local temperature and pressure, as governed by the principles of chemical equilibration. In other words, to the first order, the types of building blocks that comprise a planet are determined by the condensation sequence for a given stellar composition and formation location. Due to the assumption of both (1) a chemical equivalence of the molecular cloud and the protoplanetary disk and (2) a chemical equilibrium within the disk, it is immaterial if planetary building blocks truly develop in situ or are inherited from molecular clouds. The validity of this paradigm has been well tested within our Solar System <cit.>. Information about the composition of the protoplanetary disk of the Solar System has traditionally been inferred from the analysis of meteorites. The carbonaceous chondrites, in particular of the Ivuna-type (CI chondrites), have been found to reflect the elemental abundance of the Sun's photosphere to a very high degree <cit.> – if one allows for some depletion of volatile elements, especially a deficiency in H, C, N, O, and noble gases <cit.>. Regarding the rocky planets of our Solar System, it has been found that the composition of the Earth conforms very well to expectations <cit.>: relative to the composition of the Sun (measured by spectroscopy and approximated by the CI chondrites), the bulk Earth is depleted in moderately volatile elements, that is, elements that condense into mineral dust grains at lower temperatures <cit.>, but reflects the Sun's elemental abundance pattern for refractory elements. Much less is known about the composition of the other rocky planets of the Solar System. There are several processes that can modify the bulk composition of a planet. For instance, the planetary material may be taken out of the chemical equilibrium of the disk at some threshold temperature (`incomplete condensation'), as was likely the case for Earth <cit.>. Another example is Mercury, with its high density. Mercury has lost about 80 percent of its rocky (low density) mantle via collisional erosion <cit.>. In contrast to Mercury, the bulk density of Mars is lower than expected when assuming a CI chondritic bulk composition <cit.>. These findings show the limits of predicting planetary bulk compositions based on the element abundance of their host star alone. Complex processes, such as dynamical interactions, migration, radial mixing, discontinuous distribution of solids, or even giant impact events can offset a planet's abundance pattern <cit.>. The identification of such anomalies, however, requires knowledge of the expected bulk density, which can only be obtained from the bulk chemistry and size of a planet. It is now becoming possible to investigate how well exoplanet systems conform to this picture, at least indirectly. <cit.> explored the statistical likelihood of rocky planets having the same composition as their host stars, by comparing their expected core mass fraction to the core mass fraction that could be expected given the elemental abundance of the star. Of their sample of eleven planets, only two were found to be incompatible with the null-hypothesis assumption of the planet reflecting the composition of its host star at the 1σ level. Similarly, <cit.> analysed an ensemble of planets and stars and find that the predicted composition of the population of planets spans an overlapping, yet wider, range with respect to the corresponding host stars, both in terms of the Fe/Si distribution and the core mass fraction <cit.>. Given these findings, we can take advantage of the possibility to deduce a star's composition from its spectrum and assume the same elemental ratios for the protoplanetary disk. The advances in modern spectrographs, coupled with an improved understanding of spectral line characteristics, allows the derivation of precise abundances of major rock-building elements, at least for F, G, and K stars <cit.>. Despite all the advances in this field, it should be kept in mind that deducing concrete stellar elemental abundances from the depth and width of absorption features in a spectrogram is far from straightforward. As compiled and analysed by <cit.>, there are a multitude of methods that obtain quite different elemental abundances with different error margins, even from the same spectra <cit.>. Using simulations to find the composition of exoplanets depending on the composition of their host star has become a fairly common practice. Apart from trying to recreate the planets of the Solar System in order to test our understanding of planet formation <cit.>, there have been great efforts to explore the compositional diversity of exoplanets <cit.>, and especially the influence of stellar elemental abundance patterns that deviate significantly from that of our Sun <cit.>. Due to its provision of an extensive thermochemical database and its widely applicable general chemical equilibrium computations, the powerful commercial software suite HSC Chemistry[http://www.hsc-chemistry.nethttp://www.hsc-chemistry.net] has been the backbone of many recent studies of exoplanet compositions <cit.>. There are also several general equilibrium condensation codes that were written specifically for applications in planetary science but which are generally not publicly available, such as the Condor code, developed by and described in <cit.>, and the PHEQ code, developed by and described in <cit.>. A freely available Fortran code is GGchem <cit.>. This code simulates the equilibrium chemistry in the protoplanetary disk down to 100 and includes an extensive thermochemical database. Another open-source program is the TEA code by <cit.>, which, however, is limited to gas-phase simulations. Originally intended to study geochemical processes, but also usable for planetary simulation, is the SUPCRTBL software package by <cit.>, with its extensive thermochemical database SUPCRT92. Building on the foundation of results from general equilibrium calculations, additional effects can be included to capture the complexity of the disk evolution process. Examples include: combining a thermochemical equilibrium simulation with a dynamical simulation of disk development <cit.>; including dust enrichment in the composition of the protoplanetary disk to account for deviations of planet compositions from stellar elemental ratios <cit.>; adding the notion of the isolation of a fraction of the condensed solids from the chemical equilibrium <cit.>; and looking into non-ideal solid solutions <cit.>. Once the most likely composition of a rocky planet has been found, further simulations can follow to estimate the internal structure of the planet <cit.>, its geological evolution <cit.>, the development of an atmosphere <cit.>, and even the formation and composition of clouds <cit.>. With our ECCOplanets[Equilibrium Condensation & Composition Of planets, or in Italian roughly `There you have it: planets'. Available at https://github.com/AninaTimmermann/ECCOplanetshttps://github.com/AninaTimmermann/ECCOplanets.] code, we provide a general equilibrium condensation code as a simplified, open-source Python alternative to use for simulations. The main focus of our code is its ease of use, which allows it to be tailored to specific research questions and extended. As a first application of our code, we show its projection for the composition of exoplanets in different stellar systems and the condensation temperatures of common planet-building molecules and elements. We also study the sensitivity of condensation temperatures to variations in disk pressure and elemental abundance patterns within the protoplanetary disk. With this analysis, we want to highlight the limitations of the application of element volatility, as determined from Solar System parameters, in general theories of planet formation. In Sect. <ref> we describe the underlying thermochemical principles of our simulation. Section <ref> deals with the mathematical properties of the thermochemical equations. Our data sources and processing are presented in Sect. <ref>. The basics of our code are shown in Sect. <ref>. Finally, we present our simulation results regarding condensation temperatures of certain species and their variability (Sect. <ref>) and the composition of exoplanets (Sect. <ref>). Apart from showing our own results, these simulations are used as a benchmark test for our code. § THERMOCHEMICAL BASIS A protoplanetary disk, at least during the stage of solid condensation, can be approximated as a closed system, that is, closed with respect to matter but open with respect to heat exchange. The timescale on which the disk cools is generally large compared to the timescale of condensation <cit.>. This is, however, only true for the formation of condensates from the gas phase, not for the rearrangement of the condensates into their thermochemically favoured phases <cit.>. Thus, assuming that the disk evolves through a sequence of equilibrium conditions is, to some degree, a simplification, especially for large bodies at low temperatures. §.§ Chemical equilibrium and Gibbs free energy minimisation A system is in thermochemical equilibrium when its Gibbs free energy is minimised <cit.>. Accordingly, we can compute the disk's equilibrium composition at each temperature by minimising its Gibbs free energy. The Gibbs free energy is an extensive property, that is, the total Gibbs energy of a multi-component system is given by the sum of the Gibbs energies of its constituents <cit.>: G_tot(T) = ∑_i G_i(T). The Gibbs energy of a substance i is given by its molar amount x_i and its chemical potential μ_i(T), also referred to as its molar Gibbs free energy <cit.>: G_i(T) = x_i μ_i(T). The chemical potential at a temperature T depends on the chemical potential at standard state, μ_i^∘(T), and the natural logarithm of the chemical activity, a_i, of the component <cit.> μ_i(T) = μ_i^∘(T) + R Tlna_i, where R is the ideal gas constant. Regarding the first term of Eq. (<ref>), we use the relation of the standard Gibbs free energy, G^∘, to the standard enthalpy, H^∘, and the standard entropy, S^∘: G^∘ = H^∘ - T S^∘. If we use these variables as molar quantities, that is, enthalpy and entropy per mole of a substance, this equation holds for the standard chemical potential, μ^∘(T) <cit.>. The dependence of the enthalpy and entropy on changes in temperature is defined by the heat capacity, C_p^∘ <cit.>: H^∘ = C_p^∘(T) T S^∘ = C_p^∘(T)/T T. The heat capacity, C_p^∘, is well approximated by the Shomate polynomial <cit.>, allowing us to integrate the differentials analytically. With τ = T ×10^-3 it takes the form C_p^∘(T) = A + B τ + C τ^2 + D τ^3 + E τ^-2. The Shomate equation is only valid for temperatures larger than T=298.15. This is also the reference temperature of the standard state of all thermochemical data used in our code. Therefore, it constitutes the lower limit of the integration. The upper limit is given by any temperature, T: H^∘(T) - H^∘(298.15) = ∫_298.15^T C_p^∘(T) T = A τ + 1/2 B τ^2 + 1/3 C τ^3 + 1/4 D τ^4 - E τ^-1 + F S^∘(T) - S^∘(298.15) = ∫_298.15^T C_p^∘(T)/T T = A ln(τ) + B τ + C τ^2/2 + D τ^3/3 - E/2 τ^2 + G , where F and G denote the negative value of the integrated polynomials evaluated at T=298.15. We can rearrange the equations and add the constants H^∘(298.15) and S^∘(298.15) to the constants F and G, respectively, on the right-hand side. The new constants are denoted with a tilde sign.[This is a slight deviation from the definition of the constants in the NIST-JANAF web-book <cit.>.] Combining the resulting polynomials for H^∘(T) and S^∘(T) gives us an equation for the standard chemical potential of each species defined by the Shomate parameters: μ_i^∘(T) =H_i^∘(T) - T S_i^∘(T) = 10^3 [A τ (1-lnτ) - B τ^2/2 - C τ^3/6 - D τ^4/12 - E/2 τ + F - G τ]. The usage of τ = T ×10^-3 entails that for consistent constants A to G, the enthalpy, H^∘, is given in units of [], whereas the heat capacity, C_p^∘, and the entropy, S^∘, are given in []. The chemical potential, μ^∘(T), is given in []. In this version of the code, we treat the gas phase as ideal gas and only consider pure solid phases, that is, no solid solutions. We can therefore use a common approximation for the activity of a substance <cit.>: For solids, the activity is unity; for components in the gas phase, the activity is assumed to equal the partial pressure of this component. This also entails that we do not differentiate between stable and unstable condensed phases on the basis of the activity. The presence of a phase is solely determined by the product of its molar amount and chemical potential at standard state. The gas-phase approximation is valid, as long as deviations from an ideal gas are negligible. This deviation can be quantified with the fugacity coefficient, which depends on the gas in question, the pressure, and the temperature. As a rule of thumb, this approximation of an ideal gas is better the lower the pressure and the higher the temperature <cit.>, we do not expect significant influences on our result for our parameter range where T > 300 and p < 1. The partial pressure of a component, p_i, can be expressed as the product of the total pressure with the fraction of the molar amount of the species in question, x_i, of the total molar amount of all gaseous species, X. In our case the total pressure is that of the protoplanetary disk p_disk = p_tot := p. The activity can be summed up as a_i = p_i = x_i/X p gas-phase species, 1 solid-phase species. By combining Eqs. (<ref>) to (<ref>), and assembling all molar amounts x_i in the vector 𝐱, we get the Gibbs free energy function of the protoplanetary disk that needs to be minimised in order to find the equilibrium composition <cit.>: G_sys(𝐱,T,p) = ∑_i x_i [μ_i^∘ + R T lna_i] = ∑_gas,i x_i[μ_i^∘ + R T (lnp + lnx_i/X)] + ∑_solid,i x_i μ_i^∘. This equation is a function of the molar amounts x_i of all chemical species in the system. The temperature T and disk pressure p can be specified or sequentially varied, all other factors are constants. Accordingly, minimising Eq. (<ref>) means finding the molar amounts x_i of all chemical species in chemical equilibrium. §.§ Constraints There are two constraints in the minimisation of the Gibbs free energy. The most obvious constraint to our problem is non-negativity, which means that no molar amount can take values below zero: x_i ≥ 0 ∀ x_i ∈System. The second constraint, mass balance, is imposed by the elemental composition of the protoplanetary disk. The amount of any element bound in molecular species cannot exceed the initial molar amount of that element. If we include all possible forms in which an element can occur, we can formulate this requirement as an equality constraint: ∑_i α_ij x_i = β_j, where β_j is the initial molar amount of element j and α_ij is the stoichiometric number of element j in species i (see the example in Appendix <ref>). It should be noted that our code only considers relative amounts of species, based on relative initial abundance patterns. We specify all molar amounts relative to an initial abundance of 10^6 Si atoms. § MATHEMATICAL ANALYSIS We characterise the problem as a constrained, non-linear optimisation of the general form <cit.>: minimise f(x) subject to g(x) ≥ 0 h(x) = 0, where the target function f(x) is the function of the Gibbs free energy of the system, g(x) is the non-negativity constraint and h(x) is the mass balance constraint. The target function f(x) is mathematically the most complex of the three functions. As shown in the previous section, for n gas-phase species and m solid-phase species, it can be written as f:ℝ^n+m→ℝ, x→ G_sys(𝐱,T,P) = c^T x_linear + d ∑_i=1^n x_i lnx_i/X_transcendental, with c, x∈ℝ^n+m and d ∈ℝ, c = [ μ_1^∘ + R T lnp; ⋮; μ_n^∘ + R T lnp; μ_n+1^∘; ⋮; μ_n+m^∘ ] d = R T. As denoted, the function is the sum of a linear and a transcendental function. For the further characterisation of the transcendental term, we use the fact that it is the sum of terms that are identical in mathematical form, f_trans, i(x_i) = d x_i lnx_i/X. The molar amount x_i of any species i has to be between 0 and the total molar amount of all species X. For computational purposes, we use the limit and define lim_x → 0 x lnx = 0 define⟶ 0 ln0 := 0. With this definition, f_trans, i(x_i), x_i ∈ [0,X], is U-shaped and has zero intercepts at zero and X only. It is convex over its domain. The total transcendental term is the sum of convex functions, and thus convex itself <cit.>. Adding the linear term has no influence on this characterisation. So, the target function is convex. The main implication of the convexity of a function is that finding a local minimum is always equivalent to finding a global minimum <cit.>, that is, our minimisation solution is unique. The non-negativity constraint can be phrased as an inequality constraint: 𝐱≥0or g(𝐱) = 𝐱≥ 0, with 𝐱 as defined above. The equality constraint can be written as 𝐀 𝐱 = 𝐛or h(𝐱) = 𝐀 𝐱 - 𝐛 = 0, where 𝐀∈ℕ^o× p is the stoichiometry-matrix for the number balance of a system composed of initially o elements resulting in a final composition with p = m+n molecular species (see the example in Appendix <ref>), and 𝐛∈ℝ^o is the vector containing the total abundances of the elemental components. Typically, p > o because the most common molecular species are only made up of a handful of different elements. The gradient of the target function is given by ∂ f/∂ x_i = c_i + d lnx_i/X i ≤ n, x_i ≠ 0 c_i i > n or x_i = 0. The Hesse matrix of the target function is given by ∂^2 f/∂ x_i ∂ x_j = -d/X i,j ≤ n, i ≠ j or x_i = 0, d/x_i - d/X i,j ≤ n, i = j, x_i ≠ 0, 0 else. In summary, our problem can be characterised as a convex minimisation problem subject to two linear constraints. For the numerical minimisation we can make use of the gradient and Hesse matrix of the target function, and can be sure of the uniqueness of a found solution due to the convexity of the target function. § DATA §.§ Data types and sources There are different types of data used in this code. Regarding their function within the code, we can distinguish the thermochemical data of molecules (this term includes minerals; melts were not considered), on the one hand, and the stellar elemental abundance data, on the other hand. In terms of their mathematical usage, the former plays the main role in the target function of the minimisation procedure, whereas the latter is used in the number balance constraint. The thermochemical data describes laboratory-measured properties of molecules and is constant for all simulations; the stellar abundance data were derived from astronomical observations of particular stars and can be varied as an input parameter between simulations. We use ancillary atomic weight data to express the atomic composition of planets in terms of wt-%. Our thermochemical database is limited to the most common species expected to form in a protoplanetary disk, and does not contain any charged species at the moment. It is likely that the lack of certain molecules and ions increases the expected error of the computed condensation temperatures, especially for high-T condensates containing Mg and Na. For the sake of formal comparability between the thermochemical data of different species (i.e. identical derivation, processing, and presentation of data), we used as few different data sources as possible. Most data, especially the gas-phase data, were taken from the comprehensive NIST-JANAF Thermochemical Tables.[https://janaf.nist.gov/https://janaf.nist.gov/] Most of the mineral data were taken from three bulletins of the U.S. Geological Survey <cit.>. The data were extracted from these sources in their given tabulated form. An overview of the included data is shown in Appendix <ref>. The stellar elemental abundance data are given as the absolute number of atoms of each element, normalised to N_Si = 10^6. This is an arbitrary scaling commonly used in cosmochemistry <cit.>. The exact normalisation is inessential for the code, as we only consider the element ratios in the disk. We included the data of 1617 F, G, and K stars from the <cit.> database in the code. Further data can easily be added to all databases if considered useful for a simulation. §.§ Data processing, uncertainties, and extrapolation The tabulated data are only available at discrete temperatures, with intervals of 100. To obtain continuous thermochemical data for any temperature, we used the tabulated enthalpies to fit Shomate parameters A, B, C, D, E, and F, via the respective Shomate equation (cf. Eq. (<ref>)). Subsequently, we use the found values (A to E), the tabulated entropies and the respective Shomate equation (Eq. (<ref>)) to find the last parameter G. As an example, we show the thermochemical data of Al2O3(S) in Fig. <ref>. The top and middle panel show the entropy and enthalpy values of the species, respectively. We see that our Shomate fit (black solid line) retraces the tabulated data (blue diamonds) well. The bottom panel shows the Gibbs free energy derived from the other two properties. In cases where the tabulated data contains discontinuities in the form of jumps at certain temperatures (commonly in reference-state data including multiple phases of a species), we fitted separate Shomate parameters for either side of the discontinuity. If the tabulated data do not span our entire temperature range of 300 to 6000, we applied a linear extrapolation for the Shomate enthalpy equation (parameters A and F) using the last three tabulated enthalpy values. The G parameter was fitted subsequently. The uncertainty in thermochemical data can be very large, especially for solids. The reason for this is a combination of the limited precision of experimental measurements and errors introduced in the subsequent data processing, for instance by extrapolating measured data beyond the temperature range of the experiment. Regarding the measurement uncertainty, our data sources state an uncertainty of the order of 10% for the entropy at reference state for many minerals; for gases the uncertainty is generally much lower, often below 1% (for the exact values, we refer to the data sources themselves, Sect. <ref>). This uncertainty is propagated to the other values in the tables. Regarding the data processing uncertainty, <cit.> and <cit.> studied the deviations between the thermochemical equilibrium constants of the species (k_p) as stated in different data collections as a function of temperature. They found that only for 65% of the species the agreement between different sources is good over the entire temperature range, that is, better than 0.1dex at high temperatures and better than 0.4dex at low temperatures. We did not take the uncertainty in the thermochemical data into account in the assembling of our thermochemical database, and it is not considered in any way in our simulations. It should, therefore, be kept in mind for the interpretation of our simulation results. There are also uncertainties in the stellar elemental abundance data. For most rock-building elements, the estimated error of the given abundances is smaller than ±0.03dex for the stars in our database <cit.>. We discuss the implications of these uncertainties in Sect. <ref>. § COMPUTATIONAL SOLUTION Our computational solution to the chemical equilibrium problem is provided as an open-source software on GitHub[https://github.com/AninaTimmermann/ECCOplanetshttps://github.com/AninaTimmermann/ECCOplanets]. It is written in Python and only relies on the standard Python libraries , and .[We acknowledge the fact that Python is not a particularly efficient coding language (see our performance test in Table <ref> and compare to e.g. <cit.>, who use Fortran-90 in their GGchem-code), and understand the growing concern about the ecological impact of computational astrophysics <cit.>. Our choice is motivated by the conjecture that Python is most commonly taught and used in physics, astronomy, and geosciences, and our aim to also make our code accessible to an audience with limited coding experience.] Both the databases and minimisation code can be easily expanded to include more species or integrate a more sophisticated scientific approach, for instance by including an isolation fraction for the simulated solids or a thermochemical activity model. §.§ Scope of our simulation We limited the scope of our simulation in several areas. Most importantly, it is a purely thermochemical simulation. We do not consider disk profiles, disk dynamics, dynamical planet formation models or planet migration. The temporal development of the disk is only considered indirectly, in that we simulate a decrease in temperature, but do not set an absolute timescale for its evolution. The thermochemical approach is limited to equilibrium condensation, that is, we assume that all components of the system stay in chemical equilibrium for the whole temperature range of 6000 to 300 , or, in any case, down to the temperature at which the planet is extracted from the disk. This is a twofold approximation: firstly, we assume that the cooling timescale of the disk is large compared to the thermochemical equilibration timescale, and secondly, we assume that no part of the condensates becomes isolated from the equilibration process, for instance, by being integrated into larger bodies. Furthermore, our model only includes ideal gases and solids. We do not include the condensation of trace elements. Our scientific objective is the analysis of the composition of rocky planets; thus, it is limited to the solid materials found after condensation. While we do simulate the evolution of gas-phase species in the disk, they are not considered to be part of a forming planet. We only look at relative amounts of species and make no assumptions regarding absolute planet sizes. §.§ Condensation simulation The code's main function is to compute the temperature-dependent equilibrium composition of a protoplanetary disk and to allow for the subsequent analysis of the results. This is realised by performing a sequence of Gibbs Free Energy minimisations at decreasing temperatures. As a result, the minimisation routine returns the relative amounts of each species – gaseous and solid – at each temperature step. This result matrix allows the inference of the temperature ranges in which a species is stable, and, therefore, expected to be present in the protoplanetary disk. The code requires the specification of a start and end temperature, as well as the temperature increment, the disk pressure, the elemental abundance pattern in the disk, and the list of molecular species to be considered. The start and end temperature, as well as the temperature increment have no physical meaning; their computational impact is discussed in Appendix <ref>. The disk pressure is kept constant for the entire simulation, which means it is not automatically adapted to a decrease of gaseous material or temperature. Its value should be chosen in accordance with disk profiles. The elemental abundance pattern needs to be specified in terms of the normalised absolute number of atoms per element (see Sect. <ref>). §.§.§ Molecule selection A meaningful selection of the species to be included in the simulation is the most important, albeit most complex, aspect of the simulation. Our ideal is to include as few species as possible, in order to make the minimisation problem as numerically easy to solve as possible, thereby reducing the expected computation time and the likelihood of numerical errors. On the other hand, one wishes to find a stable mineral assemblage and hence would like to include all phases where thermodynamic data are available. Currently, there are >5000 minerals known to occur in nature; including all would likely lead to computational problems. The difficulty is determining the set of crucial species, that is, those that have a notable influence on the thermochemistry of the disk, for instance, by controlling the total pressure or otherwise determining condensation temperatures. The total pressure is generally controlled by He and H2. On top of that, the simulation results will only be reliable if we include the most stable species at any sampled (T,p)-tuple for each element contained in the simulation. The most stable species carrying a particular element at a given (T,p)-value depends on the overall elemental abundance pattern; thus, compiling the list of species is typically an iterative process. The starting point of this process can be based on the extensive simulations of the Solar System, for instance by <cit.>, <cit.>, and <cit.>. §.§.§ Minimisation procedure The minimisation at each temperature step is done using a trust region method, due to its known stability when solving bounded and constrained non-linear problems <cit.>. The Gibbs function G(x,T, p), as defined in Eq. (<ref>), is passed to the minimisation function as the target function, in combination with the non-negativity and number-balance constraints, the gradient (Eq. (<ref>)) and Hessian matrix (Eq. (<ref>)) in their respective capacity. We derive the initial starting point x_0 for the minimisation at the highest temperature of the simulation by solving the linear part of the Gibbs equation (Eqs. (<ref>), (<ref>)) with a simplex method. We ignore the transcendental part of the function, but respect non-negativity and number-balance. The vector c, as defined in Eq. (<ref>), is constructed from the Shomate parameters of all included species, the specified temperature, and disk pressure. In all subsequent temperature steps, the solution at the previous temperature step is used as the initial guess input of the minimisation procedure. §.§ Simulation analysis and definition of the condensation temperature We provide several functions to analyse the results of the condensation simulation. These analysis functions roughly fall into three categories. First, the basic parameters of the simulation (temperature range, included molecules, abundance pattern, and disk pressure) can be retrieved. Second, temperature progression curves of species can be plotted to analyse their amounts as a function of temperature and, in the case of solid species, their condensation behaviour. And third, the projected composition of a rocky planet as a function of its formation temperature can be studied. Temperature progression curves are a powerful tool for analysing different aspects of planet formation. As an example, we show a plot of the relative molar amounts (mol-%)[The specification `mol' is used to distinguish it from the later used weight percentage (wt-%).] of all the solid species included in a simulation as a function of decreasing temperature in Fig. <ref>. The amounts are relative to the total molar content of the disk, that is, both gas- and solid-phase species (n_speciesn_solids + n_gases in -%). From this figure, we can gather various information. For instance, the temperature of the first onset of condensation, the approximate total proportion of solids in the disk as a function of temperature, and the dominant contributors to a planet's bulk composition at any given temperature, especially the distribution between typical planetary metal-core and silicate mantle components. We implemented a computation of the condensation temperature of elements, as a common parameter to assess planet compositions. This is defined as the temperature at which 50% of the total amount of an element is bound in solid-phase species. As an example, we show the condensation of Ca in Fig. <ref>. The dark blue line denotes the fraction of Ca-atoms bound in gas-phase species, (n_Ca(g)/n_Ca(tot)), in %, the light blue line the fraction bound in solid-phase species, (n_Ca(s)/n_Ca(tot)). The curves intersect at T = 1512, signalling the 50% condensation temperature of Ca. Additionally, we define a `condensation temperature' of specific solid-phase molecules (synonymously used for minerals) in order to analytically assess the appearance of condensates and sequences of phases containing specific elements. This value is, however, far less distinct than the condensation temperature of an element. There are two reasons for this: firstly, the maximum amount of a particular phase is not known a priori and depends on the temperature range considered; secondly, phases often disintegrate at lower temperatures in favour of more stable phases. This behaviour is exemplified in the left panel of Fig. <ref> for perovskite (CaTiO3(s)), which is superseded by Ti4O7(s). Some species show even more complex condensation behaviours, involving successions of increases and decreases in their relative amounts. This is demonstrated in the right panel of Fig. <ref> for forsterite (Mg2SiO4(s)), which is part of the intricate Mg-Si-chemistry in the protoplanetary disk. Irrespective of the precise curve shape, we only compute one condensation temperature per species, based on the 50% level of the local maximum closest to the first onset of condensation. If a species is present in such small quantities that it cannot be distinguished from numerical noise, it is assumed to not have condensed in the simulation, and therefore no condensation temperature is reported. For the analysis of the composition of an exoplanet itself, we use the code's output of relative molar amounts of the chemical species that are stable in chemical equilibrium as a function of temperature. We convert this information into the relative amounts of the individual elements bound in solid species, using the structural formulae of the species. This procedure results in a temperature-series of the elemental composition of solid material in the disk. As default, we give the resulting composition in units of wt-%, meaning the total mass of one element bound in solid species relative to the total mass of all elements in solid species, using the atomic weights data specified in Sect. <ref>. To get the composition of an individual planet out of the temperature series, we have to specify its `formation temperature'. This temperature defines the point, at which the planetary material is taken out of the chemical equilibrium of the disk. This does not necessarily mean that the planet is fully formed at this temperature, nor that it corresponds to the final disk temperature at the location of the planet. It is sufficient that the planetesimals have become so large that their interiors are effectively shielded from the disk chemistry. For instance, the depletion pattern of volatile elements in the Earth suggests a formation temperature between 1100 and 1400 <cit.>. We discuss our different models connecting the formation temperature to the planet's composition in Sect. <ref>. § CONDENSATION TEMPERATURES AND THEIR DEPENDENCE ON PRESSURE AND ELEMENT ABUNDANCE We used our code to look at condensation temperatures and their variability. The condensation temperatures of rocky species give a first idea of the building blocks of a planet, because only material that has condensed at the formation temperature of a planet can be accreted from the protoplanetary disk. The condensation temperature of elements takes this idea to a slightly more abstract level, as we are not looking at the specific material that can be accreted onto a planet anymore, but rather think about the final elemental composition of the planet, even after the original planetesimals have undergone chemical changes in the formation and consolidation of the planet, for instance due to thermal processes. This relies on the assumption that while the specific molecules that have been accreted might not be found in the final planet in their original form, their elemental proportions will be retained. Finally, the variability of the condensation temperatures gives an indication as to how applicable our understanding of the connection between the formation and composition of the Earth is to different planet formation regions in the disk and to exoplanetary systems. To validate the results of our code externally, we compared them against benchmark results from the literature. In Sect. <ref> we show our computed condensation temperatures of common rocky species (Sect. <ref>) and of elements (Sect. <ref>) for a Solar System elemental abundance pattern at a constant pressure, and compare them against the results by <cit.> and <cit.>. In Sect. <ref> we explore the variability of the condensation temperatures of elements as a function of the disk pressure and the stellar elemental abundance pattern. §.§ Condensation temperatures for a solar elemental abundance pattern and constant pressure First, we analyse the condensation temperatures of species and elements that can be derived for the solar elemental abundance pattern at the disk pressure associated with the formation of Earth. In order to also use our results as a benchmark test for our code, we used the same system parameters that were used in the studies we compare our results against, even if these values are not necessarily in line with the currently most accepted ones. Namely, we used the Solar System elemental abundances pattern as reported by <cit.> and a disk pressure of 1e-4, which was found to represent the total pressure in the solar nebula near 1AU by <cit.>, and which was also used for the simulations of <cit.> and <cit.>. Our simulation covered a temperature range from 2000 to 300, with a resolution of 1. The number of species included in our simulation (47 gases + 25 solids) was much smaller than in the comparison studies. The simulation parameters are summarised in Table <ref> in the Appendix. We estimate very conservatively that different codes should return condensation temperatures of molecules within 100 of each other and condensation temperatures of elements within 20 for a given disk pressure and abundance pattern. This presumes that the most common elements and the majority of the most stable molecules are included in the simulation. This estimate is based on our experience regarding the response of our own code to variations in the molecule selection, the thermochemical data, and the definition of the condensation temperatures, in the case of molecular species. §.§.§ Condensation temperatures of rocky species For the validation of our simulated condensation temperatures of common rocky species, we used the benchmark results of the seminal work by <cit.>. Their results have been computed with the Condor-code, which is not publicly available. Their thermochemical database contains 2000 gas-phase species and 1600 solids, which are all considered for the simulation. In their code, the chemical equilibration is based on equilibrium constants of formation, rather than Gibbs free energy, and the reported condensation temperatures pertain to the point at which the computed activity of a solid species reaches unity <cit.>. Visually, this point would typically correspond to the onset of condensation, that is, the initial sharp change in gradient seen in our condensation curves of molecules (see Fig. <ref> as an example). Depending on the slope of the curve, this temperature might easily be 20 higher than our 50% condensation temperature, as defined in Sect. <ref>. Keeping this in mind, we found a high degree of agreement between the two sets of values, as shown in Table <ref>. Our condensation temperatures are mostly within ±50 of the literature values. Our values are on average lower than those of <cit.>, confirming our expectation based on the different definitions of the condensation temperature. Regarding condensation sequences, we found that the order in which the molecules are expected to condense has been reproduced well with our code. The slight observed differences, as well as our failure to condense grossite (CaAl4O7), are likely due to the fact that our code does not include solid solutions. The solid solutions of the olivine and pyroxene mineral groups especially will lead to a shift in the Mg budget, potentially affecting the condensation of most of the shown species. In conclusion, the found agreement between the two codes is much better than our conservative estimate. The combination of vagueness of the definition of the condensation temperature of a molecule and the large uncertainty in the thermochemical data itself (see Sect. <ref>) suggests that one should not attach great meaning to their exact simulated values. §.§.§ Condensation temperatures of elements In contrast to the condensation of a rocky species, the 50% condensation temperature of the elements (the temperature at which 50% of the element is bound in solid species) is very sharply defined, since its maximum amount is known a priori from the given elemental abundance (cf. Sect. <ref>). Additionally, the selection of species is less influential in the equilibrium computation, since the elements tend to only be exchanged between different solid-phase species after the initial onset of condensation, but do not to return to gas-phase species. Accordingly, not including a particular species does not affect the amount of the element being bound in solid-phase species, and consequently the 50% condensation temperature. Furthermore, the condensation temperatures of elements enable us to easily estimate the composition of a rocky planet. Most elements condense (10% to 90%) within a T-interval smaller than 100. This implies that there is hardly any of the element in solid form at temperatures above the 50% condensation temperature, whereas all of it is in solid form at temperatures below. Hence, if the formation temperature of a planet is above the 50% condensation temperature of an element, it will not contain significant amounts of this element. Otherwise, the element will have a similar relative abundance in the planet as in its host star. If the formation temperature of the planet is close to the condensation temperature of an element, this element will likely be depleted to some degree in the planet. For the comparison, we again used <cit.>, as well as the more recent study of <cit.>. The latter applied the PHEQ code, developed by and described in <cit.>, which is very similar in its thermochemical approach to our code. The agreement with <cit.> and <cit.> for the condensation temperatures of major rock-forming elements are excellent, as shown in Table <ref>. The average discrepancy from the mean condensation temperature of each element is below 5 and the largest deviation is below 15 (cf. Fig. <ref>). We conclude that the 50% condensation temperatures of the most common planet-building elements are not sensitive to details of the used condensation algorithm. In other words, even our very simplistic approach with only a few included species, will return results with a projected error below ±10, for a given elemental abundance pattern and disk pressure. §.§ Dependence of the condensation temperatures of elements on system parameters We explored the variability of the condensation temperature of elements as a function of disk pressure and elemental abundance pattern. While it has been widely acknowledged that the chemical processes in the protoplanetary disk are controlled by its elemental composition and especially its C/O and Mg/Si ratio <cit.>, the condensation temperatures of the elements are sometimes implicitly treated almost as material constants, at least within certain limits of system parameters <cit.>. In Sect. <ref> we explore the influence of the disk pressure, and in Sect. <ref> we analyse the effect of a variation in the elemental abundance pattern. In Sect. <ref> we briefly discuss implication of our findings. §.§.§ Dependence on disk pressure The condensation temperatures of elements are often reported for a total disk pressure of 1e-4. In general, however, the disk pressure depends both on the radial distance from the central star, the vertical distance from the mid-plane, and the total material within the system <cit.>. We varied the disk pressure logarithmically between 1e-6 and 1e-1, while keeping all other parameters of the simulation constant. Depending on the disk model, this range of pressure values might correspond to a radial distance range from 0.1AU to 5AU in a Solar-System-like disk <cit.>, that is, distances within the water snow line, where we expect to find rocky planets. As shown in the top panel of Fig. <ref>, we found an overall trend of a higher disk pressure corresponding to a higher condensation temperature of the elements. Raising the pressure in a system where nothing else is changed, is equivalent to increasing the particle concentration. This implies an increase in the reaction rates. Since the pressure raises the effective concentration of all species equally, we found a quantitatively very similar relation between the disk pressure and the condensation temperature for all analysed elements, except for S. This can be seen particularly clearly in the bottom panel of Fig. <ref>, where we show the deviation of the condensation temperature of an element at a given disk pressure from the element's mean condensation temperature within the analysed pressure range. For the analysed pressure range all species have their mean condensation temperature at roughly the same pressure (between 4e-4 and 5e-4). Also, the deviation from this mean is similar for all elements at each disk pressure. On average, the condensation temperature of all elements except for S increases by 357±57 over the analysed five orders of magnitude in disk pressure. S behaves differently, its condensation temperature does not change at all over the pressure range.[The deviation is, however, not relevant for our argument, because the condensation temperature of S is much lower than those of the other main rock-forming elements.] The consistency in pressure versus condensation temperature relations between the different elements has implications for the analysis of planet formation at different radial locations within the protoplanetary disk. In disk models, both the midplane temperature and the disk pressure decrease with increasing distance from the central star. Generally, when simulating planet formation, both pressure and temperature are considered in combination. However, since the disk pressure affects the condensation temperatures of all elements very similarly, the variation in disk pressure does not change the equilibrium chemistry and equilibrium composition as a function of temperature qualitatively, but only shifts the equilibrium composition to higher temperatures (for an increased pressure) or to lower temperatures (for a decreased pressure). The small variations in the pressure response of the elements' condensation temperature will likely be rendered insignificant by the fact that a planet does not comprise material of one specific (T-p)-equilibrium condition but rather a mixture over a range of conditions. As shown in Fig. <ref>, the greater the change in pressure, the larger the difference in T_c between the elements. Our argument is therefore only valid for small pressure ranges. §.§.§ Dependence on the elemental abundance pattern There have been many studies assessing the diversity of exoplanetary compositions as a result of the changed equilibrium chemistry in a protoplanetary disk, due to variations in its elemental abundance pattern <cit.>. Certain element ratios control which molecular species will form out of the available elements. Different molecules can have vastly different condensation temperatures even if they consist of similar elements. The species in which an element is predominantly bound, generally determines its 50% condensation temperature. To systematically analyse the influence of the elemental abundance pattern on the condensation temperatures of the elements, we ran condensation simulations for synthetic abundance patterns, only varying one key element ratio at a time. We explored the role of the overall metallicity and the element ratios C/O, Mg/Si, Fe/O, and Al/Ca. We specify metallicities logarithmically and normalised to solar values: [M/H] = log(N_M/N_H)_star - log(N_M/N_H)_sun, where N_M is the sum of the relative number of atoms in the system of all elements larger than He, and N_H the relative number of H atoms. All other element ratios are given as non-normalised number ratios, for instance, `C/O' means C/O = (N_C/N_O)_star. As a basis for our analysis, we used the <cit.> catalogue of 1617 F, G, and K stars. To ensure the reliability of the abundance data, we only took stars into account whose spectra have a signal-to-noise ratio (S/N) larger than 100. Furthermore, to avoid giant stars, we excluded all stars with log g ≤ 3.5 <cit.>. We used the remaining 964 stars to (1) generate a representative abundance pattern as a starting point for the element ratio variations, (2) determine the parameter ranges of the element ratios we were interested in, and (3) pick roughly 100 stars, covering the whole parameter space, to verify that any trends found in the analysis of the synthetic data are also followed by the real stellar data. Figure <ref> shows the parameter ranges (T_eff versus metallicity, C/O versus Al/Ca, and Mg/Si versus Fe/O) covered by the <cit.> stars, and the distribution of the sample of comparison stars. For most variations of the element ratios, we kept the abundance of one element constant in our representative abundance pattern, and only varied the other. For the overall metallicity, only the H abundance was changed. For the Al/Ca, Mg/Si, and Fe/O ratios, Al, Mg, and Fe were varied, respectively. The C/O ratio was treated differently. In the studied stellar sample, there is a strong correlation between the C/O ratio and the ratio of O to the sum of other abundant elements, such as Mg, Si, and Fe. In order to avoid a distortion of the analysis due to an unrealistic abundance pattern of the synthetic data, we approximated this correlation with a parabola, as shown in Fig. <ref>, and adapted both the C and O abundance accordingly. Our tests show that an element ratio can affect the condensation temperatures in different ways. We can differentiate the effects in terms of the number of affected elements, and the curve shape of the correlation between the ratio and condensation temperature. Table <ref> shows the summary of our findings. The overall metallicity and C/O ratio have the most profound impact on the condensation temperatures. These two variations stand out, because (1) they affect a large number of elements, (2) the magnitude of change in condensation temperatures is high, and (3) the correlation between the ratio and the condensation temperature of all affected elements is systematic. We therefore limit our discussion to these two parameters and only cover the others in a cursory fashion. In Fig. <ref>, we show the influence of the overall metallicity of the system on the condensation temperatures of a selection of common elements. The coloured foreground markers represent the simulation result for the synthetically varied metallicity, the grey background markers show the simulation results of the random sample of comparison stars. The figure clearly demonstrates a linear correlation between the logarithmic metallicity and condensation temperature for all elements. We found an increase in condensation temperature between 52 (Fe) and 117 (Al) over the covered metallicity range. Despite the fact that the abundance patterns of the comparison stars are quite diverse (cf. Fig. <ref>), the log-linear correlation between metallicity and condensation temperatures can also be found there.[The deviation of the Fe condensation temperatures at low metallicities is caused by the superposition of the effect of disproportionately low Fe-abundances in the stellar sample <cit.>.] The median deviation of the condensation temperatures from the interpolation curve of the synthetic simulation results is below 20 for all the elements. The effect of the overall metallicity is reminiscent of the effect of disk pressure variations, seen in Sect. <ref>. This is unsurprising as both variations effectively change the relative number of rock-building particles per volume, that is, the chemical reaction rates. We have found a similar log-linear correlation for the Fe/O ratio, which does, however, only affect the condensation temperature of Fe itself. Again, we suspect this to be explainable by the fact that we increased the partial pressure of Fe. Since its dominant solid species in our simulation is pure solid iron, this increased partial pressure does not shift the balance of any other chemical reaction. The result would apply analogously to similar condensation patterns, where the dominant solid species do not include any other elements, such as the Ni condensation. In Fig. <ref>, we show the influence of the C/O ratio on the condensation temperature of some common elements. Again, the synthetic simulation results are depicted with coloured foreground markers, the comparison stars with grey background markers. It is important to note that in this figure the effect of the metallicity, as described above, has already been removed from the results. The order of magnitude of the change in condensation temperature caused by the variation of the C/O ratio is similar to that of the metallicity. There are, however, also several qualitative differences. The most obvious difference is that an increase in the C/O ratio causes a decrease in condensation temperatures, in contrast to the increase caused by a higher metallicity. Also, while there certainly seems to be a systematic effect of the C/O ratio on all condensation temperatures, the correlation is not log-linear. Finally, while the metallicity affected all condensation temperatures, the C/O ratio only affects elements whose dominant species contain O, for instance, the condensation of Fe and Ni are unaffected. The expected correlation mapped out by the synthetic data is followed exceptionally well by the real data simulations, especially for the range 0.3 ≤C/O≤ 0.7. For the whole parameter range, the median deviation from the expected curve is below 10 for all elements. For C/O values below 0.3, the real data condensation temperatures of Al and Ca do not follow the synthetic data well. This is caused by the superposition of the influence of the Al/Ca ratio. The diverging systems coincidentally all feature a particularly low Al/Ca ratio. Our tests with synthetically varied Al/Ca ratios have shown that it causes a roughly log-linear increase of the Al condensation temperatures and a step-function increase for Ca (see Fig. <ref>, left panel). This explains why the condensation temperatures of Al shown in Fig. <ref> gradually taper off from the expectation curve, whereas the Ca condensation temperatures suddenly jump to values more than 100 below the expectation. These differences between the effect of the variations in metallicity and in the C/O ratio allude to different underlying mechanisms. We have argued that an increase in metallicity implies increasing the number of all reactants per volume, thereby increasing all reaction rates. In contrast, changing the C/O ratio tilts the reaction balances in the formation of many major species, by only changing the availability of one of the reactants or by changing them to different degrees. The effect of a changed reaction balance is far more difficult to predict than the effect of globally increased reaction rates. Reaction balances are particularly strongly affected when the involved element ratios in a system are typically close to unity, because then a change in the ratio can imply that the availability one of the reactants is exhausted before the other, inhibiting this reaction. This is the case for the C/O, Mg/Si, and Al/Ca ratios. §.§.§ Implications of the variability in condensation temperatures Our findings regarding the variation of the condensation temperatures of elements have several implications. Most importantly, a combination of variations in pressure and elemental abundance pattern, even over a moderate parameter range, can easily change the condensation temperatures of elements by more than 100. This needs to be taken into account, when they are used to estimate planet compositions in other stellar system, for instance when applying the elemental devolatilization pattern of the Earth to exoplanets <cit.> or in the context of white dwarf pollution <cit.>. We have, however, seen that the overall metallicity and the disk pressure affect all elements very similarly. Neglecting those will likely not be of great consequence to any derived exoplanetary compositions. That is to say, the computed element ratios would agree well with a model taking these parameters into account over the whole simulation range, but these ratios would be predicted for shifted radial distances. Other variations in the elemental abundance pattern, however, cause more unpredictable changes to the condensation temperatures of some elements. As a result, both the sequence in which the elements condense, as well as the difference in their condensation temperatures can be significantly altered. These changes entail substantial qualitative deviations in the most likely composition of a planet expected to form in a given system compared to its Solar System analogue. We explore this point further in the next section (Sect. <ref>). Furthermore, our findings give us an idea of the potential influence of the uncertainty of stellar abundance measurements. While the uncertainties of the abundances of most planet-building elements might be lower than ±0.03dex for many well-studied F, G, and K stars <cit.>, and even an uncertainty at the ±0.01dex level seems feasible for these stars <cit.>, the situation is generally much worse. For M-dwarfs, where abundance measurements are in their infancy, typical errors exceed ±0.1dex <cit.>. As we see in Fig. <ref>, a difference of ±0.1dex in the C/O ratio can signify a difference of some tens of Kelvins in the condensation temperature of certain elements, at least at the upper end of our tested range, that is, C/O≥ 0.5. An in-depth analysis of the impact of uncertainty is available in <cit.>. § EXOPLANET COMPOSITIONS We now look at the bulk composition of rocky planets around chemically different stars. To emulate the dynamical formation of planets we compare different methods of assembling the solid material from our equilibrium condensation simulation. To externally validate our results and qualitatively assess the merits of our composition models, we compare them against the n-body simulations of <cit.> (in this Sect. abbreviated as B10). §.§ Derivation of planet compositions Our underlying disk model is vastly simplified. The only parameter changing within our disk is the temperature, which is a proxy for distance from the central star. For our study, we are not interested in an exact temperature-distance relation but only in qualitative tendencies. We keep the pressure constant at a value of 1e-4. This pressure value is often assumed for the formation of Earth <cit.>. In this context, however, the choice was arbitrary. As shown above in Sect. <ref>, variations in disk pressure affect the condensation of all species very similarly, especially if the expected variation in pressure is small.[Note that, in contrast to our simplified model, realistic disk models have a two-dimensional structure, show pressure gradients and pressure bumps; they likely have inhomogeneous element distributions, and they evolve in time.] To derive the bulk composition of a planet for any given formation temperature (see above, Sect. <ref>), we use three different methods of assembling planetary material to emulate planet formation via accretion. While our models are loosely rooted in the idea of planetary accretion from within the planet's Hill sphere <cit.>, we only use them to bypass computationally expensive n-body simulations. That means there is no correspondence between the expected physical size of the planet and the diversity of the material assembled in our code; we instead made the latter match the results of the n-body simulation. As illustrated in Fig. <ref>, we compare two differently shaped planetary feeding zones (FZs) to a model without a FZ. In the simplest approach, the planet is only made up of the solid material that is stable at the planet's formation temperature, with the relative amounts dictated by the thermochemical equilibrium at that temperature. This approach corresponds to taking an infinitesimally thin section of the elements-temperature-progression described in Sect. <ref>. It is illustrated in the bottom panel of Fig. <ref>, where the x-axis represents the temperature decreasing with distance from the central star, all material except that at T=T_central is discarded. The first FZ is an equal weights temperature band, illustrated in the middle panel of Fig. <ref>, later referred to as `boxcar FZ'. We specify the width of the temperature band, add up the elemental equilibrium compositions within the temperature range, and normalise the result. This normalised result is taken to be the planetary composition at the central temperature of the band. Since a lower temperature generally entails a higher total amount of solids in the equilibrium, it follows that the lower-temperature edge of the band effectively has a stronger influence on the resulting planetary composition than the higher-temperature edge. The second type of FZ is a Gaussian profile, as illustrated in the top panel of Fig. <ref>. Here, the total material at each temperature is first multiplied by a normal distribution with a specified standard deviation, σ, and subsequently added up. The argument regarding more solid material being present at lower temperatures also applies to this FZ. The width of the FZ is given by 2σ. The location of the peak of the normal distribution gives the planetary formation temperature. For both of these types of FZ, the effect of a ring geometry on the amount of available material is not taken into consideration for the final planetary make-up. That means we neglected the fact that a lower temperature corresponds to a greater distance from the star. A larger radius of the ring implies more material with that particular composition being available for accretion onto the planet. To quantify this geometric effect, we would have to connect our temperature profile to specific distances, which is not part of our simplified model. §.§ Application and comparison study Following the approach of B10, we analyse the predicted planetary compositions with our simplified disk model for chemically diverse stars, delineated primarily by differences in their C/O ratios. In particular, we explore the simulated compositions of a rocky planet formed around a low-carbon star (HD27442), around a medium-carbon star (HD17051), and around a high-carbon star (HD19994), using the three different planetary FZ models described above. The physical properties of the stars are listed in Table <ref>. We use the elemental abundances of these stars as reported by B10, in order to facilitate the comparison of the results. It should be noted, though, that these abundances have later been found to be inaccurate, especially the C/O ratios are vastly overestimated <cit.>. Based on more recent studies of stars in the solar neighbourhood <cit.>, all of the C/O ratios analysed in this section would be classified as moderately high or high. We compare our results against the results of B10. They simulated the composition of rocky planets by combining a chemical equilibrium condensation with a dynamical accretion simulation. The chemical equilibrium condensation was done with the HSC suite. The T-p input parameters were based on the <cit.> temperature-pressure-profile for the midplane of the protoplanetary disk.[The <cit.> disk profile is not considered state-of-the-art anymore, as it is a purely diffusive model. It has been shown that introducing radiative transfer to the model inverses the vertical temperature profile of the disk (i.e. T increases for increasing z) compared to a purely diffusive model in which the temperature decreases with distance from the midplane, and a shadowing effect results in an overall cooler midplane <cit.>. However, for the purpose of our analysis, the only essential aspect of the disk-profile is that the midplane temperature decreases with distance from the star.] The solids formed in equilibrium at the specified temperature and pressure constitute the planetary building blocks for the dynamical n-body simulation, which was done using the SyMBA integrator <cit.>. They ran four accretion simulations for each of the studied stars, slightly varying the initial distribution of planetesimals, and recorded the composition of the final planets as the sum of all accreted material. Each simulation run returned between zero and three planets per star. For each star, we show the planet compositions in order of formation distance across the whole set of the B10 simulations. This illustrates representative compositional patterns as a function of distance, and captures their variability due to dynamics. §.§ Results for chemically diverse systems Figures <ref> to <ref> show the comparisons among our results of planet compositions for the three stars and to the B10 results. We describe them in more detail in the following subsections. Each figure contains four panels, showing the bulk composition of a planet (in wt-%) simulated to form around the respective star. The top panel shows the discrete results of the B10 simulation as a function of distance, the second panel from the top shows our composition for a Gaussian FZ, the third panel shows our simulation for a boxcar FZ, and the bottom panel shows the composition without any FZ. Where possible, we group the B10 planets with similar composition and indicate roughly which section of our simulation best corresponds to them with arrows between the top most and second panel. §.§.§ Low carbon abundance In Fig. <ref> we show the system of HD27442, which has the lowest carbon abundance of the three analysed systems (C/O=0.61). The elemental abundance pattern of this star is similar to the solar values. This implies that the simulated planets can be expected to also resemble the inner planets of the Solar System in bulk composition. No S abundance is reported for HD27442. Since its C/O ratio is far below 1, S species are not expected to play a major role in the equilibrium chemistry (see footnote <ref> and Sect. <ref>). We therefore excluded all species containing S from the simulation of this system. Starting closest to the central star, two planets from the B10 simulations formed at 0.33 and 0.35AU with similar compositions of mostly O and Al, significant amounts of Mg, Ca, and Si. We find a very similar composition in our simulation without a FZ at or slightly below 1400. The boxcar FZ, with a width of 100, produces a slightly different composition for this formation temperature, with a reduced Mg-content. This is caused by mixing in material from the higher-temperature regions, where only Al-Ca-O species have condensed. We cannot reproduce the composition of the two innermost planets with our Gaussian profile, because it does not create a region that contains Mg but no Fe. At intermediate distances, we find a group of five planets in the B10 simulation with similar O and Si contents as the innermost planets, but with ever increasing Fe amounts. These planets formed between 0.36AU and 0.52AU, which seems to correspond to the location Fe snow line in the B10 simulation. Due to the very abrupt condensation of Fe at T≈1360, we cannot reproduce this planetary composition without resorting to a FZ. The gradual change in planet composition can be reproduced with both FZ models. The boxcar model would profit from a larger FZ width than the one used here, though. At greater distances, we find a group of three planets in the B10 simulation that can be characterised by their large content of Fe, Mg, O, and Si. These planets formed between 0.52AU and 0.77AU. As shown in our continuous simulations, the composition of the solids in the disk converges to these specific ratios, which are in accordance with the stellar elemental abundance ratios. This is due to the fact that we now look at planetary formation temperatures below the condensation temperatures of the main planetary components. Once we enter this region, the FZ model becomes obsolete, as the material to either side of the central temperature is identical. There is one final planet left in the B10 simulation that has no correspondence with our continuous simulation. It formed at the greatest distance from the star, but its composition rather resembles the second group of planets. The formation of this planet requires substantial dynamical processes, likely in the form of planet migration, which cannot be emulated by our simple FZ model. §.§.§ Medium carbon abundance In Fig. <ref> we show the planets simulated to form around the star HD17051, which has an intermediate carbon abundance among the analysed systems (C/O=0.87). Closest to the central star, we see five B10 planets with almost identical compositions. They formed between approximately 0.3AU and 0.4AU. They contain a large fraction of O, similar amounts of Al and Ca, and a small amount of Si. We find the same composition in our simulation for all three types of FZs between roughly 1500 and 1400. For the Gaussian FZ, though, the region corresponding to the composition of the B10 planets is very narrow, which is difficult to reconcile with the consistent compositions returned by the n-body simulation. The model without a FZ has a broad plateau of the same composition as the B10 planets. The boxcar FZ does not have such a broad plateau, but shows a section with a sufficiently constant composition to be compatible with the formation of similar planets over an extended range of distances. At greater distances, we identify two B10 planets with very similar composition that are vastly different from the first group. These planets are dominated by their high Fe content, exceeding 50% of the total weight of the planet, and suggesting a very extensive planetary core. The remaining composition is made up of O, Mg, and Si, with only small contributions of Ni, Ca, and Al. In contrast to the HD27442 system, this group of planets has not formed in region of the convergence composition of the disk. We can clearly see in our simulations that the composition changes significantly all the way down to approximately 600, when S condenses. Both the high similarity of composition over a fairly large distance range, as well as the deviation from it in the form of a slight increase in Mg, O, and Si can be seen in our three continuous models in the temperature range from approximately 1300 to 1200. Both the Gaussian and the boxcar profile reproduce the gradual changes in the composition of the B10 planets. The composition without a FZ compares less favourably, because at the onset of the Mg and Ni condensation, it changes are very rapidly and strongly, and at lower temperatures, it stays constant. §.§.§ High carbon abundance Finally, we show the planets simulated to form around the high-carbon star HD199944 in Fig. <ref>. Based on the elemental abundance data we use for this simulation, the star has the exceptional C/O ratio of 1.26. We expect a completely altered disk chemistry for systems with C/O ratios exceeding unity. All O-atoms are bound to C-atoms to form highly stable CO gas molecules <cit.>. Accordingly, O is no longer available for the solid-phase chemistry, inhibiting the condensation of some of the most common species in planet formation, such as Al2O3, CaAl12O19, and MgSiO3. Because all O is bound to CO, no O is available to bind with H2 to form H2O and the system becomes highly reducing. This means that all Fe is in reduced state and that some Si occurs in metal instead of silicates. A C/O ratio exceeding 1 also means that free C is available to form exotic phases like SiC or free C in form of, for example, graphite. In high C/O systems, S replaces O as anion, which leads to a much higher condensation temperature of S as it condenses with Ca into refractory phases like oldhamite (CaS). Although the solar C/O is approximately 0.5, some portions of the early Solar System apparently had C/O ratios close to 1 as it is evident from the presence of CaS in reduced enstatite chondrites or exotic elemental ratio patterns of the rare Earth elements in ordinary chondrite chondrules <cit.>. A planet with a bulk C/O > 1 would certainly not allow the presence of liquid water and thus would likely be hostile for life. From a practical, computational point of view, it should be noted that not many S species are taken into account in condensation simulations, due to their limited importance in solar-like systems. For instance, B10 only considers the solid S species FeS, MgS, and CaS; we only added Al2S3 to this selection.[The GGchem code <cit.>, on the other hand, contains thermochemical data of 12 different solid S species.] This means that the simulations likely do not reflect the true disk chemistry of a C-rich system. As expected, the composition of the simulated planets is completely different from the planets discussed so far. Starting again closest to the central star, we find two B10 planets only containing C and Si. They formed at 0.31 and 0.33AU. In all our models, the same composition can be found for a large range of temperatures. The condensation of C and SiC in this system occurs at a much higher temperature than any of the other species, and in combination with the suppression of the Al-Ca-O species, this C-Si composition of solids is very stable in the disk for an extended temperature range. This implies that the FZ type has hardly any influence on the predicted composition of the innermost planets in the system. At intermediate distances, we find two B10 planets with a small fraction of Fe. They formed at 0.35 and 0.37AU. We cannot reproduce this composition without using a FZ. The very rapid condensation of Fe means that the disk composition changes from no Fe in solid form to all Fe in solid form within a few kelvins. This makes it difficult to form a planet with a small amount of Fe. The next two B10 planets in the sequence, formed at 0.45 and 0.46AU, show increasing amounts of Fe and traces of other elements. Despite their almost identical distance from the central star, the ratios of these elements are substantially different. While we can identify a section in our FZ models, in which the Fe fraction increases rapidly, we do not find these exact compositions in any of our models. Especially the Al traces cannot be reproduced, as we found Al to condense at a temperature that is too low to allow for mixing of the material into the region in which Fe has not yet fully condensed. One reason for this deviation might be the geometric effect we described in Sect. <ref>. This would increase the relative amount of the lower-temperature material, making it available for a redistribution to the higher-temperature regions. It is also possible that this composition requires a more dynamical accretion of different types of materials than we can emulate with our FZ models. The most distant B10 planet, at 0.7AU, has a much more diverse composition. This planet formed at a temperature at which CO starts to lose its role as the dominant C gas phase and is replaced by CH4, removing C from the solid phase and freeing O for the condensation of more common rocky species. Accordingly, the relative C and Si contents are significantly reduced, but there is also a large fraction of the typical rock components of O and Mg. Additionally, S becomes a more abundant trace element. Qualitatively, we find this composition in all our models, the only difference seems to be that our models predict a much lower relative S abundance at the location at which there is still a significant amount of C in solids. §.§ Implication for simplified planet formation models We learned several things in the comparison between combined thermochemical-dynamic model of B10 and our simplified continuous planet composition models, where the only free parameter was the disk temperature. Firstly, since the analysed B10 planets were confined to a radial distance between approximately 0.3AU and 0.8AU from their central star, the variations in disk pressure in these simulations is only about one order of magnitude. As we show in Sect. <ref>, the condensation temperatures of the elements do not change significantly within one order of magnitude in pressure. This makes it unsurprising that we can recreate the B10 results so easily without a variable pressure input. Regarding the emulation of dynamical planet formation, a FZ is generally able to reproduce the results of an n-body simulation. The continuum compositions of all three systems show that we can distinguish sections in which the element ratios are fairly constant over a large temperature range, and section with rapid changes. The greatest variability in composition occurs in the vicinity of the condensation temperatures of the major planet-building elements Mg, Si, and Fe. At these temperatures, using a FZ is crucial to reproduce the gradual variations in planet composition found in n-body simulations. In regions where the element ratios are constant over a large temperature range, using a FZ is less relevant, or, in the case of the convergence composition, completely obsolete. The exact shape of the FZ does not seem to be particularly significant, as their width can be adapted to generate the required effect on the final composition. For instance, <cit.> has shown that the measured elemental depletion pattern of Earth compared to the Sun can be achieved by using a Gaussian FZ with a standard deviation of approximately σ≈216, whereas that of Vesta, with its mass of 4e-5M_⊕, requires a standard deviation of σ≈57. There are, however, some arguments in favour of the boxcar model. On the one hand, it seems to be better at reproducing the composition of the innermost planets formed in the n-body simulation. At the onset of condensation, when there is no solid material at higher temperatures, a Gaussian profile results in a very asymmetric assemblage of material that is skewed towards low-temperature material. On the other hand, a boxcar profile seems to be more compatible with the physical concept of accretion from a region within the gravitational influence of the forming planet. This could also be achieved by cutting off the wings of the Gaussian profile, for example at 2σ or 3σ. We have, however, seen some deviations from our continuum composition in the B10 planets, which we could not reproduce with any of our FZ models, and which must therefore be a result of the dynamical accretion simulation. This shows the limitation of our simplistic model. N-body simulations can help us explore the extent to which processes that entail large displacements of planetary building blocks from their formation region might affect the final composition of a planet. Taking this idea even further, these simulations would also allow us to study the composition of planets that are partly formed by accreting material from remote reservoirs <cit.>. § SUMMARY AND CONCLUSIONS ECCOplanets is a simple, accessible, and versatile Python code that can be used to simulate the equilibrium condensation of the main building blocks of rocky planets in the protoplanetary disk of stars, as a function of the elemental abundance pattern and disk pressure, based on a Gibbs free energy minimisation. The performance of our code is stable and robust for a variety of starting conditions. The software package, which we make publicly available, includes a limited built-in (and extendable) library of thermochemical data representative of common problems in exoplanet formation. In this paper we have used our code for two typical applications in planetary science: finding the condensation temperature of elements and condensates, and deriving the composition of rocky planets as a function of the stellar abundance pattern. Both these analyses were also used as a benchmark test for the results of our code against literature values. The computed condensation temperature of a condensate is very sensitive to its exact definition and to the selection of molecules included in the simulation. In combination with the uncertainty in thermochemical data, this suggests that the exact value of simulated molecular condensation temperatures is not very meaningful. Nevertheless, under reasonably simple assumptions, we have shown that our code outputs condensation temperatures within 50 of accepted literature values for most tested species. The derived 50% condensation temperatures of elements are a far more robust measure of disk chemistry. They are unambiguously defined and less sensitive to the selection of molecules. Here, the agreement between our results and the literature values is of the order of 5. The condensation temperatures of elements are highly sensitive to physical variations in the system, that is, the disk pressure and elemental abundance pattern. The disk pressure affects the condensation temperature of all elements in a similar way, with higher pressures corresponding to higher condensation temperatures. Over the analysed range 10^-6 to 10^-1 , we find an average increase in condensation temperatures of 357±57 for the studied elements. To understand the influence of variations in the elemental abundance pattern, we performed simulations with synthetically altered key element ratios and compared them to a representative selection of stars. We identified different groups of systematic variations to the condensation temperatures, which hint at different underlying chemical processes. Regarding the number of affected elements and the magnitude of the change in condensation temperature, the metallicity and C/O ratio have the greatest impact. An increase in metallicity results in a log-linear increase in elemental condensation temperatures; in contrast, an increase in C/O lowers the condensation temperature exponentially. While not all elements are affected to the same degree, the condensation temperatures can easily vary by more than 100 for the sampled parameter ranges of 4×10^-4 to 2×10^-3 in metallicity and 0.1 to 0.7 in C/O. We conclude that the combined effect of the pressure and elemental abundance pattern on the condensation temperature of elements limits the applicability of the values derived in the context of the formation of the Earth to other planet formation locations within the Solar System, and especially other stellar systems. Finally, we studied the composition of rocky planets forming around three exemplary stars, delineated by their C/O ratio. To explore the effects of profoundly limited model assumptions, we used a one-parameter (T) disk model and only emulated planetary accretion with FZ models. We compared our results against a study using a (T-p) disk model in a combined thermochemical and n-body simulation. Our simple model was able to reproduce almost all compositions of the combined thermochemical-dynamical simulation. This serves as a further confirmation that the disk pressure has an almost uniform influence on the whole condensation regime, and that neglecting it does not affect the results qualitatively for small pressure ranges. It also provides insights into the effects of dynamical accretion. Dynamical accretion leads to gradual changes in the planetary composition as a function of distance from the star. As most elements condense abruptly, these gradual changes require the mixing of condensates from the equilibrium conditions of a large temperature range, that is, a FZ. The shape of the FZ appears to be insignificant, as any FZ can be tailored to achieve the required degree of redistribution of material by adjusting its width. We conclude that the most likely main characteristics of rocky planet compositions can be determined with very simplified model assumptions. Adding further model parameters can give us invaluable insights into the variability and deviations from equilibrium conditions to be expected in a real exoplanet population. aa § EXAMPLE OF A STOICHIOMETRY MATRIX We consider a system only containing the elements H, O, C. The initial amount of each element i is denoted as b_i. We assume that these elements can only form the molecules H2, O2, H2O, C, CO2, and CH4. The amount of each of the molecules i is denoted as x_i. Then, the number balance of the system requires b_O = 2 x_O2 + x_H2O + 2 x_CO2 b_H = 2 x_H2 + 2 x_H2O + 4 x_CH4 b_C = x_C + x_CO2 + x_CH4. Alternatively, this equation can be written in vector notation as [ b_O; b_H; b_C ] = [ 2 1 2 0 0 0; 0 2 0 2 4 0; 0 0 1 0 1 1 ]·[ x_O2; x_H2O; x_CO2; x_H2; x_CH4; x_C; ] 𝐛 = 𝐀·𝐱. § SIMULATION PARAMETERS § STABILITY AND PERFORMANCE TESTS We performed several simulations to assess the robustness and internal consistency of our results. In particular, we investigated in the influence of (1) the starting temperature, T_start, (2) the temperature resolution, Δ T, and (3) he scaling of the elemental abundance pattern of the system. The less noticeable the influence of these variations on the simulation result, the more reliable we judge them to be. In general, the quality of a simulation can be assessed by comparing the computed condensation temperatures against literature values, checking the smoothness of the condensation curves, that is, the lack of numerical errors. We tested the starting temperatures T_start = [6000, 4000, 2000, 1700], that is, only temperature above the expected onset of condensation at T ≈1670 for this simulation. The temperature resolutions covered the values Δ T = [0.1, 1, 5, 10]. For the abundance scaling, we normalised all elements to the abundance of Si, with n_Si = [10^5, 10^6, 10^7]. If not otherwise specified, the simulations were run with a resolution of Δ T = 1 and a start temperature of T_start = 4000. Our test problem includes 33 common species, divided into 15 solid-phase species and 18 gas-phase species, made out of the elements Fe, O, Al, Ca, Si, Ti, Mg, C, and H. These elements, except for C and H, are the major constituents of the rocky planets in the Solar System. For the sake of simplicity, Ni, which behaves similar as metallic Fe and S, which mainly occurs in the outer core, have been neglected here. We use both the Solar System relative elemental ratios and the presumed disk pressure at 1AU of 1e-4 reported by <cit.>.[We ran many simulations with larger sets of species, containing more elements, without encountering any major computational problems, but did not perform any systematic performance tests.] The simulation parameters are summarised in Table <ref>. We used two types of simulation results to quantitatively assess the robustness of the simulation: (1) the 50% condensation temperature of elements and (2) the elemental composition of the solids in the system at three different temperatures (T = [1600, 1400, 1200]). In Table <ref>, we summarise the result of our tests with regard to the condensation temperatures of elements. It is clear that neither the starting temperature, T_start, nor the temperature resolution, Δ T, have an effect on the simulation. The only deviations we observe are due to the reduced/increased temperature sampling that is entailed by changing the temperature resolution of the simulation. In Table <ref> we summarise the second type of results: the relative elemental composition of solids at sample temperatures. The results were identical irrespective of the simulation parameters. We did observe some qualitative differences when examining the temperatures progressions of the test simulations, though. The chosen temperature resolution Δ T was most influential in this regard. The higher the resolution, the more stable the simulation's response to abrupt changes in gradients of the curves. Our simulations with low resolutions (Δ T > 1) showed a tendency to overshoot significantly at these locations. Regarding the starting temperature, we found that irrespective of T_start, the simulations usually need a few temperature steps as a `burn in' period, in which the simulation results are unreliable. This suggests that a simulation should always be started at a temperature at least 20 above the temperature range of interest. There were almost no differences in the simulation results as a function of the abundance scaling. Only for the normalisation n_Si = 10^7 we found some isolated numerical errors (overshoots at sharp gradient changes), which we cannot explain. Additionally, we perform a pure gas-phase benchmark test against the open-source condensation code GGchem by <cit.>, which itself is benchmarked against the open-source gas-phase code Tea by <cit.>. In this test, we run both codes `as is', that is, we do not try to make the simulations as similar as possible, but use them in their default configuration, with regards to solar abundance pattern[GGchem uses the solar abundance pattern as reported by <cit.>.] and included number of species. We set the disk pressure to the same value in the two simulations and restrict both to the same set of elements.[The simulation details can be found in the git repository of ECCOplanets.] The gas-phase benchmark test against GGchem shows an overall good agreement (see Fig <ref>). The curve shapes are identical for all species included in both simulations, despite the large difference in the total number of included molecules. Most deviations in the relative amounts are explained by differences in the assumed elemental abundance pattern. We consider the found accuracy sufficient for all intended uses of our code. While the results of the stability tests are all in good agreement, the simulation parameters obviously affect the computation time of the simulation. As shown in Table <ref>, the computation time per temperature step is very similar for all test except the lowest resolution of Δ T = 10. Disregarding this simulation, we find a computation time per temperature step of t=0.52 ± 0.01. Accordingly, the total runtime scales linearly with the number of temperature steps to be calculated. § INCLUDED MOLECULE DATA lll Included gas-phase molecule data. formula common name data source continued. formula common name data source Al(g) Aluminium NIST-JANAF SiN(g) Silicon nitride NIST-JANAF SiH(g) Silylidyne NIST-JANAF SiC(g) Silicon carbide NIST-JANAF Si(g) Silicon NIST-JANAF S2(g) Sulfur NIST-JANAF S(g) Sulfur NIST-JANAF PS(g) Phosphorus sulfide NIST-JANAF PO(g) Phosphorus oxide NIST-JANAF PN(g) Phosphorus nitride NIST-JANAF PH(g) Phosphinidene NIST-JANAF P(g) Phosphorus NIST-JANAF O2(g) Oxygen NIST-JANAF O(g) Oxygen NIST-JANAF NS(g) Nitrogen sulfide NIST-JANAF NO(g) Nitrogen oxide NIST-JANAF NiS(g) Nickel sulfide NIST-JANAF Ni(g) Nickel NIST-JANAF NH3(g) Ammonia NIST-JANAF NaOH(g) Sodium hydroxide NIST-JANAF NaO(g) Sodium oxide NIST-JANAF Na2(g) Sodium NIST-JANAF Na(g) Sodium NIST-JANAF N2(g) Nitrogen NIST-JANAF N(g) Nitrogen NIST-JANAF MgS(g) Magnesium sulfide NIST-JANAF MgOH(g) Magnesium hydroxide NIST-JANAF MgO(g) Magnesium oxide NIST-JANAF SiO(g) Silicon oxide NIST-JANAF MgN(g) Magnesium nitride NIST-JANAF SiS(g) Silicon sulfide NIST-JANAF SO2(g) Sulfur dioxide NIST-JANAF Mn(g) Manganese NIST-JANAF CrO2(g) Chromium Oxide NIST-JANAF Ca2(g) Calcium NIST-JANAF COS(g) Carbon Oxide Sulfide NIST-JANAF PO2(g) Phosphorus Oxide NIST-JANAF PH3(g) Phosphine NIST-JANAF PH2(g) Phosphino NIST-JANAF P4O6(g) Phosphorus Oxide NIST-JANAF P2(g) Phosphorus NIST-JANAF HAlO(g) Aluminium Hydride Oxide NIST-JANAF Al2O2(g) Aluminium Oxide NIST-JANAF NaCN(g) Sodium Cyanide NIST-JANAF Na2O2H2(g) Sodium Hydroxide NIST-JANAF AlO2H(g) Aluminium Hydroxide NIST-JANAF CaO2H2(g) Calcium Hydroxide NIST-JANAF MgO2H2(g) Magnesium Hydroxide NIST-JANAF FeO2H2(g) Iron Hydroxide NIST-JANAF OH(g) Hydroxyl NIST-JANAF NH2(g) Amidogen NIST-JANAF NH(g) Imidogen NIST-JANAF CH2(g) Methylene NIST-JANAF PCH(g) Methinophosphide NIST-JANAF SiO2(g) Silicon Oxide NIST-JANAF SiH4(g) Silane NIST-JANAF TiO2(g) Titanium dioxide NIST-JANAF TiO(g) Titanium oxide NIST-JANAF Ti(g) Titanium NIST-JANAF SO(g) Sulfur oxide NIST-JANAF MgH(g) Magnesium hydride NIST-JANAF NaH(g) Sodium hydride NIST-JANAF CP(g) Carbon phosphide NIST-JANAF Al2O(g) Aluminium(I) oxide NIST-JANAF AlH(g) Aluminium hydride NIST-JANAF Fe(g) Iron NIST-JANAF CS(g) Carbon sulfide NIST-JANAF CrO(g) Chromium oxide NIST-JANAF CrN(g) Chromium nitride NIST-JANAF AlO(g) Aluminium(II) oxide NIST-JANAF AlOH(g) Aluminium hydroxide NIST-JANAF FeO(g) Iron oxide NIST-JANAF AlS(g) Aluminium sulfide NIST-JANAF Cr(g) Chromium NIST-JANAF CO2(g) Carbon dioxide NIST-JANAF CO(g) Carbon monoxide NIST-JANAF CN(g) Cyanogen NIST-JANAF Ca(g) Calcium NIST-JANAF CH4(g) Methane NIST-JANAF CaS(g) Calcium sulfide NIST-JANAF CaOH(g) Calcium hydroxide NIST-JANAF C(g) Carbon NIST-JANAF FeS(g) Iron sulfide NIST-JANAF CaO(g) Calcium oxide NIST-JANAF HS(g) Mercapto NIST-JANAF Mg(g) Magnesium NIST-JANAF He(g) Helium NIST-JANAF H(g) Hydrogen NIST-JANAF HCO(g) Formyl NIST-JANAF H2(g) Hydrogen NIST-JANAF HCN(g) Hydrogen cyanide NIST-JANAF H2S(g) Hydrogen sulfide NIST-JANAF H2O(g) Water NIST-JANAF
http://arxiv.org/abs/2307.01267v1	20230703180032	Sequential Quantum Circuits as Maps between Gapped Phases	[ "Xie Chen", "Arpit Dua", "Michael Hermele", "David T. Stephen", "Nathanan Tantivasadakarn", "Robijn Vanhove", "Jing-Yu Zhao" ]	cond-mat.str-el	[ "cond-mat.str-el", "quant-ph" ]	Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, CA, USA =-1 Department of Physics, Caltech, Pasadena, CA, USA Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA =-1 Department of Physics, Caltech, Pasadena, CA, USA Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA =-1 Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA Department of Physics, Caltech, Pasadena, CA, USA Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA =-1 Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, CA, USA =-1 Department of Physics, Caltech, Pasadena, CA, USA Department of Physics, Caltech, Pasadena, CA, USA Institute for Advanced Study, Tsinghua University, Beijing 100084, China =-1 Finite-depth quantum circuits preserve the long-range entanglement structure in quantum states and map between states within a gapped phase. To map between states of different gapped phases, we can use Sequential Quantum Circuits which apply unitary transformations to local patches, strips, or other sub-regions of a system in a sequential way. The sequential structure of the circuit on the one hand preserves entanglement area law and hence the gapped-ness of the quantum states. On the other hand, the circuit has generically a linear depth, hence it is capable of changing the long-range correlation and entanglement of quantum states and the phase they belong to. In this paper, we discuss systematically the definition, basic properties, and prototypical examples of sequential quantum circuits that map product states to GHZ states, symmetry-protected topological states, intrinsic topological states, and fracton states. We discuss the physical interpretation of the power of the circuits through connection to condensation, Kramers-Wannier duality, and the notion of foliation for fracton phases. Sequential Quantum Circuits as Maps between Gapped Phases Jing-Yu Zhao August 1, 2023 ========================================================= Understanding the nature of entanglement in ground state wave functions has led to important developments in the theoretical understanding and classification of quantum phases of matter. In particular, for zero-temperature gapped phases, it was understood that gapped ground states connected by a finite-depth local unitary quantum circuit – quantum circuits composed of a finite number of layers of non-overlapping local unitaries – have the same `long-range entanglement' structure and are hence in the same phase <cit.>. This understanding has been helpful in the classification or systematic construction of gapped phases in various dimensions. To map between ground states of different gapped phases, what kind of quantum circuits do we need? It has been proven in Ref. Bravyi2006 that, to map from a product state (ground state of a trivial phase) to either the GHZ state (ground state of a symmetry breaking phase) or a topological state, a quantum circuit of at least linear depth is needed. The intuition is simply that we need an `effort' or more precisely time that grows linearly with the total system size in order to establish the long-range correlation in the GHZ state or the long-range entanglement in the topological states. On the other hand, however, applying a generic linear depth circuit to a product state could easily lead to too much correlation and entanglement that cannot be accommodated in gapped ground states. Therefore, the question becomes: which subset of linear (or higher) depth circuits can map between gapped phases and not beyond? The Sequential Quantum Circuit <cit.> (SQC) is such a subset. As the name suggests, sequential quantum circuits are local unitary quantum circuits that act in a sequential way. As shown in Fig. <ref>, a sequence of local unitary quantum gates or finite depth quantum circuits can be applied to local patches, strips, annuli, or other lower dimensional sub-regions in the system one at a time. To cover the whole system, the circuit depth scales as the number of sub-regions, which can be constant, linear, or higher in the linear size of the system. As the action of each layer is restricted to one sub-region, it ensures that each local region in the system gets acted upon by only a finite number of local unitaries in the whole circuit. We use this as the defining feature of a sequential quantum circuit. That is, A quantum circuit (composed of local unitary gates) is called a Sequential Quantum Circuit if each local degree of freedom is only acted upon by a finite number of gates in the circuit. Finite depth circuits are hence examples of sequential quantum circuits, although only a small subset. Sequential quantum circuits have been used to propose efficient generation methods for matrix product states and a large class of tensor product states and extensively discussed <cit.>. They also play an important role in various quantum protocols. See for example Ref. Lin2021, Lamata2008,Saberi2011. A direct consequence of the sequential structure is that, if the initial quantum state satisfies the entanglement area law, the final state also satisfies the entanglement area law. Therefore, gapped systems remain gapped under the action of SQC. (When we say a state is gapped, it means the state is the gapped ground state of a local Hamiltonian.) On the other hand, the linear depth gives SQC the power to generate or change the long-range correlation and long-range entanglement in the quantum state, hence mapping between different quantum phases. In this paper, we discuss the sequential quantum circuit that generates various gapped quantum phases, including symmetry breaking phases (section <ref>), symmetry protected topological phases (section <ref>), 2+1D topological phases (section <ref>), 3+1D topological phases and Walker Wang models (section <ref>), and fracton phases (section <ref>). In section <ref>, we show that all locality preserving unitary operators, i.e. quantum cellular automata, can be realized as sequential quantum circuits. We discuss circuits acting in different dimensions and in the presence of different symmetries. We note that some sequential circuits in the context of generating gapped phases of matter have been discussed in previous literature <cit.>. We interpret the power of the SQC by connecting the circuit action to Kramers-Wannier duality between symmetric and symmetry-breaking phases, condensation on gapped boundaries of topological / Walker Wang models, the foliation structure of fracton models, etc. Note that to demonstrate that a sequential quantum circuit can connect a generic state in one gapped phase to a generic state in another gapped phase, we only need to show how to map between particular chosen states (usually fixed point states) in the two phases. The mapping from a generic state to the fixed point state in the same phase can be accomplished by an additional finite depth circuit. § MAP TO SYMMETRY BREAKING PHASES In this section, we consider the mapping between symmetric and symmetry-breaking phases. We will first focus on the prototypical example of the 1+1D transverse field Ising model, before generalizing it to all dimensions and all finite symmetry groups. We show that the circuit preserves the global Z_2 symmetry of the 1+1D transverse field Ising model, but does not preserve locality. The transverse field Ising model in 1+1D, H = -J∑_i Z_iZ_i+1 -B∑_i X_i, has a symmetric phase (B>J>0) and a symmetry breaking phase (J>B>0) with respect to the global Z_2 symmetry ∏_i X_i. The fixed point wave functions of the symmetry preserving phase and the symmetry breaking phase are, \|ψ_SP⟩ = \|++...+⟩, \|ψ_SB⟩ = 1/√(2)\|00...0⟩ + 1/√(2)\|11...1⟩, where \|+⟩ = 1/√(2)(\|0⟩ + \|1⟩) and we have chosen the symmetrized GHZ state to represent the symmetry breaking ground space. To motivate the sequential circuit, it is insightful to map the above Hamiltonian to fermions using the Jordan-Wigner transformation, Z_i Z_i+1 → i γ̃_i γ_i+1, X_i → i γ_i γ̃_i, where γ_i = c_i+c^†_i and γ̃_i = -i(c_i-c^†_i). Under such a mapping, the symmetric and symmetry-breaking phases map to the trivial and nontrivial Kitaev chain phases respectively. It is now clear that the sequential circuit that maps between these two phases is composed of Majorana swaps <cit.>, 𝒰_F = e^π/4γ̃_N γ_1∏_i=N-1^1 e^π/4γ_i+1γ̃_i+1 e^π/4γ̃_i γ_i+1. Here, the ordering is chosen such that from right to left the product goes from 1 to N-1. Mapping back to the spin operators, and defining, R(𝒪) ≡ e^-iπ/4𝒪, the desired circuit is therefore, 𝒰 = R(Z_1 Z_N ) ∏_i=N^1 U_i,i+1, where, U_i,i+1 = R(X_i+1) R( Z_i Z_i+1), as shown in Fig. <ref>. We note that a convenient property for R(𝒪) is that for Pauli operators P and Q, R(Q) P R(Q)^† = P ; [P,Q]=0 iPQ; {P,Q }=0 . Using the above property, one can immediately see that conjugation by U_i,i+1 sends X_i+1 to Z_i Z_i+1. The complete mapping of operators using the unitary (<ref>) is [ X_1 → X_1X_2...X_N-1X_N · Z_1Z_N; X_i → Z_i-1Z_i, i=2,...,N; X_1X_2...X_N → X_1X_2...X_N; Z_1 → Z_1; Z_i → X_iX_i+1...X_N · Z_1, i = 2...N; Z_iZ_i+1 → X_i, i =2...N; Z_1Z_2 → X_2...X_N ] Note that in the ∏_i X_i=1 subspace, the mapping sends X_i → Z_i-1Z_i and Z_i Z_i+1→ X_i as desired. A similar circuit has been discussed in Ref. Ho2019. We want to point out some important features of this circuit. * \|ψ_SP⟩ maps to \|ψ_SB⟩ and vice versa. * In the bulk of system, the transverse field term maps to the Ising term and vice versa. Near the two end points, the correspondence is lost and some terms map to nonlocal terms. * Applying the circuit twice generates translation by one site in the bulk of the system. * The circuit is of linear depth, which saturates the lower bound required to generate long-range correlation in the GHZ state from a product state. * The circuit is symmetric. That is, the circuit is composed of local gates (gate sets in the blue boxes) that commute with the global Z_2 symmetry ∏ X_i. * The circuit is not locality-preserving, as for example the operator X_1 maps to Z_1Z_N· X_1X_2...X_N. This is also necessary because it is known that locality-preserving unitaries in one spatial dimension are either finite depth circuits or translation, neither is able to generate long-range correlation and map from symmetric to symmetry-breaking phases.[If we perform the Jordan Wigner transformation and map the transverse field Ising model to the Majorana chain, the mapping between the two phases can be realized by translation by a single Majorana mode.] In appendix <ref>, we generalize the circuit to all finite groups. Higher dimensional versions of the circuit can be built starting from the 1+1D version. For example, in 2+1D, to map from the symmetric state to the symmetry breaking state with Z_2 symmetry, we can use the circuit shown in Fig. <ref>. § MAP TO SYMMETRY-PROTECTED TOPOLOGICAL PHASES In this section, we construct symmetric, linear-depth SQCs which generate fixed points of symmetry-protected topological (SPT) phases from symmetric product states. We focus on the bosonic SPT phases described by group cohomology <cit.>. We begin with 1D SPT phases, which are described by an element [ω] of the second cohomology class H^2(G,U(1)) of the protecting symmetry G <cit.>. We can construct fixed-point states realizing such phases by picking a representative cocycle ω(g,h) and constructing a d-dimensional projective representation V(g) such that V(g)V(h) = ω(g,h)V(gh) <cit.>. Define the d^2-dimensional linear representation U(g)=V(g)⊗ V(g)^* where * is complex conjugation. Trivial and non-trivial representative fixed-point states with this symmetry can be defined on a chain of N sites, where the Hilbert space of each site has dimension d^2 and consists of two d-dimensional particles. In the trivial state, the two particles within each site are in the maximally entangled state \|Ω⟩ =∑_i=0^d-1\|ii⟩, so the state is a product state. In the non-trivial state, the right particle of one site is maximally entangled with the left particle of the next site. This is pictured in Fig. <ref>(a). These states are both symmetric under the symmetry U(g)^⊗ N. It was shown in Ref. Huang2015 that these two states cannot be related by a finite-depth symmetric circuit, but that they can be related by a linear-depth symmetric circuit of gates, as pictured in Fig. <ref>(a). Each gate commutes with the symmetry U(g)^⊗ N, as only particles which transform under the representation V(g)^* are swapped, so this is a linear-depth symmetric SQC mapping a trivial SPT state to a non-trivial SPT state. It is interesting to consider the effect of truncating this SQC. Namely, if we decide to stop applying gates at some point in the circuit, what state is left over? This is illustrated in Fig. <ref>(b), which shows that, after applying some of the gates in Fig. <ref>(a), the resulting state is a non-trivial 1D SPT on periodic boundary conditions (PBC). This is necessary in order to ensure that the symmetry is preserved after each step. Each gate in the SQC serves to extend the region of the lattice occupied by the non-trivial SPT, and the circuit ends when this region is the entire lattice. This will contrast with the circuits we construct for topological phases in section <ref>, in which truncating the SQC can result in a droplet of topological order with open boundaries and a particular gapped boundary to vacuum. Now we consider 2D SPT phases, focusing on one example and giving the general case in Appendix <ref>. We consider the CZX model, which is a fixed-point representative of a 2D SPT phase with ℤ_2 symmetry <cit.> and is defined on a square lattice with four qubits per site, as pictured in Fig. <ref>(c). Similar to the 1D fixed-point states, the qubits are divided into a tensor product of four-qubit entangled states 1/√(2)(\|0000⟩ + \|1111⟩) which are contained within one site in the trivial SPT fixed-point and shared between four sites in the non-trivial SPT fixed-point, see Fig. <ref>(c). The symmetry acts on each site as U_CZX = CZ_12CZ_23CZ_34CZ_41 X_1X_2X_3X_4 where the four qubits within the site are numbered clockwise. It is straightforward to check that U_CZX^2=1, and that the trivial and non-trivial states are both symmetric under U_CZX^⊗ N where the tensor product is over all N sites. Therefore, U_CZX defines a ℤ_2 global symmetry of the models. As these two states are in different phases with respect to the ℤ_2 symmetry <cit.>, there is no finite-depth symmetric circuit that relates them. To construct a linear-depth SQC, we might first try to mimic the 1D case and use gates to shift the four-qubit entangled states from within the sites to between the sites. This will map between the two states. However, in this case, the gates do not commute with the Z_2 symmetry since they drag around the CZ gates involved in U_CZX. To remedy this, we can dress the gates with additional operations that restore the symmetry without affecting the action of the gate on the fixed-point states. Define the dressed gate acting on two adjacent sites as ^CCZ = _26_37 × CCZ_265CCZ_261CCZ_378CCZ_374 where the qubits are numbered as in Fig. <ref>(c) and CCZ is the controlled-controlled-Z gate which acts on three qubits as CCZ\|ijk⟩=(-1)^ijk\|ijk⟩. These CCZ are placed such that the additional phase factors created by commuting the 's in ^CCZ past the CZ's in U_CZX are canceled by phase factors created by commuting the CCZ's in ^CCZ past the X's in U_CZX. One can indeed check that ^CCZ acting on any neighbouring pair of sites commutes with U_CZX, as is shown for a more general case in Appendix <ref>. Now we can use these symmetric ^CCZ gates to map between the two states. Beginning with the trivial state, we first apply the gates sequentially between each pair of adjacent columns, noting that all gates within a given column can be performed in parallel such that the depth of this step is linear. Then we apply 90^∘-rotated versions of the gates sequentially to each row, which also has linear depth. This results in the non-trivial SPT states. It is important that, at each step, the extra CCZ gates included in ^CCZ cancel out pairwise when acting on the state, such that they do not affect the final state. This procedure is pictured in Fig. <ref>(d). The general construction of dressing operators with additional phases to make symmetric gates is described in Appendix <ref> for arbitrary 2D SPT phases. From that construction, and our general understanding of fixed-point SPT states in all dimensions <cit.>, it is clear that we should be able to construct similar circuits for SPT phases within the group cohomology classification in all dimensions. Indeed, the construction of symmetric SQCs for Quantum Cellular Automata (QCA), described in Sec. <ref>, can be applied to obtain symmetric SQCs for all SPT states that can be created via a (non-symmetric) FDQC, which includes all group cohomology SPTs <cit.> and some beyond cohomology SPTs <cit.>. § MAP TO 2+1D TOPOLOGICAL PHASES In this section, we discuss the mapping from product states to 2+1D topological states with gappable boundary – the string-net states <cit.>. We discuss first the circuit for generating Toric Code (TC) ground states and then generalize to all string-nets. A key feature of our general construction is that we can write down SQCs which generate string net models with arbitrary gapped boundaries to vacuum, as well as models with periodic boundary conditions. §.§ 2+1D Toric Code We start with the ground state of the 2+1D Toric Code <cit.>. The model has qubit DOFs on edges of a square lattice, and the Hamiltonian is, H_2dTC = -∑_vertex∏_e ∋vertex Z_e - ∑_plaquette∏_e∈plaquette X_e. We discuss two distinct SQCs in detail. It is convenient to define the following notation for a unitary gate, < g r a p h i c s > where the arrows represent CNOT gates acting on the qubits on the edges of the lattice, with the arrow pointing from the control qubit to the target qubit. This gate will be generalized for string-net models in the next section. The first circuit closely follows the construction given in Ref. Liu2022, and is pictured in Fig. <ref>(a). The circuit consists of two distinct parts. First, we apply parallel gates to rows of the lattice sequentially. Each gate generates one plaquette term of H_2dTC. After each row of gates is applied, we are left with a TC state on a cylinder with periodic boundaries in the x direction and smooth <cit.> boundaries to vacuum (i.e. the qubits that are still in a product state) on the top and bottom edges. In other words, each row of gates pushes the gapped boundary one row further into the vacuum, expanding the region spanned by the topological order. The second part of the circuit involves the gates acting on the final row. We act on each plaquette in the final row (except the last) sequentially from left to right. This has the effect of `zipping' the smooth boundaries on the top and bottom of the cylinder, resulting in a TC ground state on the torus. The specific ground state which is obtained is the +1 eigenstate of the Z-type Wilson loops winding in both directions. The overall depth of the SQC scales like ∼ L_x+L_y where L_x, L_y are the lengths in the x and y directions. We can use a similar circuit on the dual lattice to prepare the eigenstate X-type Wilson loops in both directions, such that truncating the circuit results in a rough <cit.> boundary to vacuum. The second circuit is pictured in Fig. <ref>(b). The key difference compared to the first circuit is that, after each step in the circuit, we are left with a TC state with periodic boundaries in both directions. That is, there is never a gapped boundary to vacuum. Rather, the first two steps of the circuit prepare a TC state on a thin torus, and each subsequent row of gates serves to extend the region over which the periodic Toric Code state is defined, similar to the circuits for SPT states defined in the previous section. Interestingly, because the final `zipping' step is not required (since the periodic boundaries in the vertical direction are present from the start), the depth of this circuit scales only as ∼ L_y. Accordingly, the ground state of the TC which this circuit prepares is the +1 eigenstate of both the X and Z-type Wilson loops that wrap around the torus in the y direction. We remark that the circuit in Fig. <ref>(b) can also be run such that all blue CNOTs are first applied in sequence, and then the plaquette operators in Eq. <ref> are applied in sequence. The first step prepares a stack of 1D GHZ states, and the second step merges them into the TC ground state, with each step having depth ∼ L_y. §.§ Levin-Wen string-net models In this section, we show how the construction in the previous section for the 2D TC can be generalized to arbitrary Levin-Wen string-net states <cit.>, writing down a sequential circuit that generates the bulk string-net ground state on a cylinder and on the torus. In the first case, the state is created row-by-row from a product state, creating and moving arbitrary gapped boundaries in the process like in the previous section. We generalize the unitary circuit that generates the `smooth' boundary at the ends of the cylinder constructed in Ref. Liu2022 to arbitrary gapped boundaries. We show the general framework here and present the technical details in appendix <ref>. The general construction follows from Kitaev and Kong <cit.> and bears many resemblances to the (finite-depth) quantum circuit written down in Ref. lootens2022mapping, mapping between different Morita equivalent string-nets. The Hilbert space of string-net models consists of configurations on a hexagonal lattice, with the edges labeled by simple objects a,b,... of a unitary fusion category 𝒞. The simple objects obey fusion constraints: a × b = ∑_c N_ab^c c, which are assumed to be multiplicity free here (N_ab^c ∈0,1). There exists a unit object 1, such that ∀ a, 1× a = a. The string-net Hamiltonian is a sum of commuting projectors, generalizing Eq. <ref>: H_SN = -∑_v A_v - ∑_p B_p. In the ground state, the vertex terms (A_v) enforce the fusion rules at every vertex and the plaquette terms (B_p) give dynamics to the string-net by projecting every plaquette on the trivial anyon sector B_p = ∑_s d_s/D B_p^s, (B_p)^2 = B_p, with d_s the quantum dimensions of the simple objects s and D = ∑_s (d_s)^2 the total quantum dimension. The action of B_p can be schematically drawn as B_p < g r a p h i c s > = < g r a p h i c s > . After the weighted sum of s-loops (<ref>) is created in the middle of the plaquette, the loops are fused in the lattice and every vertex is recoupled until the hexagonal basis states are recovered (Eq. <ref>). An explicit ground state on the torus can be obtained by initializing all the edges on the trivial object, a state which trivially satisfies the vertex constraints, and then applying the plaquette operator on every plaquette. We remark that these plaquette operators are not unitary, so this does not directly lead to a circuit to prepare the ground state. Similarly, to define our circuit, we start with an initial state with all edges fixed on the trivial object. We leave the lattice unoriented here for simplicity and choose an explicit orientation in appendix <ref>. Just like for the Toric Code, the circuit is constructed such that a control edge can be assigned for every individual plaquette operator. This edge is not yet entangled in the lattice and is still in the trivial state \|1⟩. To merge the edge into the latttice, it is first mapped to \|1⟩→∑_sd_s/D\|s⟩, after which it is used as the control to draw the loops around the added plaquette. The controlled-B_p (C-B_p) action is similar as the action of B_p^s, but with the control edge treated as in the trivial state <cit.>: < g r a p h i c s > = ∑_sd_s/D (C-B_p) < g r a p h i c s > = ∑_sd_s/DB_p^s < g r a p h i c s > = ∑_sd_s/D < g r a p h i c s > . The C-B_p operator now acts as a unitary with the same action as B_p^s, provided at least one edge is in the trivial state initially (grey). This is shown in Ref. Liu2022, wang2022renormalization. The proof relies on the unitarity of the F-symbols of the fusion category 𝒞 (<ref>). The full circuit is shown in Fig. <ref>. A cylinder is generated with boundaries labeled by 𝒞. This boundary is the generalization of the `smooth' boundary in the Toric Code for general string-nets with Hamiltonian Eq. <ref>, where both the bulk- and boundary edges are labeled by objects in 𝒞. The effect of applying operators in a new row is to push the boundary down and enlarge the bulk string-net ground state. Following Kitaev and Kong, the string-net boundaries with edges labeled by objects in 𝒞 are not the only possible ground state solution of Eq. <ref>. More generally, the edges can carry labels in a 𝒞-module category ℳ (indicated by blue lines in the diagrams). Given the input category 𝒞 and the choice of boundary ℳ, it is always possible to define the `Morita dual' category 𝒞_ℳ^* and consider ℳ as a (𝒞_ℳ^, 𝒞)-bimodule category <cit.>. This is the structure we require to define our general quantum circuit, the details of which are explained in appendix <ref>. The opposite category ℳ is a (𝒞, 𝒞_ℳ^)-bimodule category, with objects that are in one-to-one correspondence with those of ℳ, but with their orientation reversed. One can now construct a weighted sum of simple objects in ℳ (B_p^ℳ): B_p^ℳ < g r a p h i c s > = < g r a p h i c s > , B_p^ℳ = ∑_S d_S/D B_p^S,ℳ, with d_S the quantum dimensions of the objects in ℳ. When applied to all plaquettes, B_p^S,ℳ maps the ground state of a string-net with input category 𝒞_ℳ^* to that of a string-net with input category 𝒞 (the operator B_p^ℳ does the reverse). The initial trivial state, on which our general sequential circuit will act, is interpreted as a state with all edges fixed on the trivial object in the Morita dual category 𝒞_ℳ^. The objects in ℳ have an action on the objects in 𝒞_ℳ^ (𝒞_ℳ^×ℳ→ℳ), such that on the boundary, the loops are fused with the trivial edges (∈𝒞_ℳ^) to obtain boundaries with ℳ labels. In the bulk, loops are fused pairwise on every edge ℳ×ℳ→𝒞 (by virtue of <ref>), such that the desired bulk string-net 𝒞 is recovered. The general sequential circuit is shown in Fig. <ref>. It is the generalization of the one shown in Fig. <ref> for arbitrary gapped boundaries labeled by ℳ. The circuit is slightly more complicated as we need different plaquette operators on different sublattices for the first two rows. The general picture, however, is very similar. The same isometry argument can be invoked here for the general plaquette operator, the proof of which we omit, but is a generalization of the one given in Ref. Liu2022, wang2022renormalization, and relies on the unitarity of all the F-symbols involved (see Appendix <ref>). Note that 𝒞 is always a valid choice for the module category over itself (ℳ = 𝒞), in which case the Morita dual is simply 𝒞 itself. This case corresponds to the original circuit and the `smooth' boundary is recovered. The Toric Code is recovered from this general picture by choosing 𝒞 = Vec_ℤ_2. The `smooth' and `rough' boundaries are produced by the sequential circuit by choosing ℳ = 𝒞 = Vec_ℤ_2 and ℳ = Vec (the category of vector spaces, with only one trivial object) respectively <cit.>. We finish this section by showing how the circuit needs to be adapted for generating the string-net ground state on the torus. We can try to close the cylinder in Fig. <ref> by applying a new row with periodic boundary conditions, using the side edges as the control edges this time (because the bottom edges of the new row are already occupied after the circuit for the first row). However, we cannot completely close the cylinder, as there are no unoccupied edges left to use as the control on the very last plaquette. The solution is explained in Ref. wang2022renormalization and we use a generalization of it here to construct the torus ground state in different representations labeled by ℳ. The string-net ground state is created on a minimal torus first (a torus with two vertices, one plaquette and three edges), by initializing the edges on the trivial object in 𝒞 and applying the plaquette operator in the reverse direction of Eq. <ref> on the one plaquette (Eq. <ref>). The result is a minimal torus ground state of a string-net with input category 𝒞_ℳ^. After this initial step, the original sequential circuit is now used to add plaquettes to obtain the ground state for a string-net with input category 𝒞 in the bulk. For the last row, the circuit needs to be altered, using side edges as control qudits similar to the Toric Code circuit. The plaquette operator can now be applied on the last plaquette since we are guaranteed to be in the torus ground state already. The procedure is shown in Fig. <ref>. The general circuit can be used to generate a Projected Entangled Pair State (PEPS) representation of the string-net ground state, in which case the representation from Ref. lootens2021matrix is recovered, generalizing the original representation from Ref. buerschaper2009explicit,gu2009tensor. Note that in the abelian group case, the minimal torus is trivial, and for ℳ=Vec_ℤ_2 we recover the Toric Code sequential circuit (on a hexagonal lattice). Just like for the Toric Code in Fig. <ref>(b), we can change the string-net circuit such that after each step, we are left with a string-net with periodic boundary conditions in both directions (without any open boundaries). The periodic boundary condition in the vertical direction can be enforced in the circuit, without applying a last row that zips the boundaries together, by first initializing a one-row torus groundstate and enlarging the torus in every step. § MAP TO 3+1D TOPOLOGICAL PHASES AND WALKER-WANG MODELS We now describe SQCs for preparing topological phases in 3+1D. For the 3+1D Toric Code, there are two ways to view the bulk wavefunction: either as a condensate (equal-weighted superposition) of membrane-like excitations or as a condensate of loop-like excitations<cit.>. Correspondingly, there are two types of gapped boundary: a rough boundary that condenses the charges, and a smooth boundary that condenses the loop-like fluxes. In this section, we discuss two sequential quantum circuits for generating the 3+1D Toric Code from product states. The first one generates a membrane condensate in the bulk with a rough boundary while the second one generates a loop condensate in the bulk with a smooth boundary. The first type of circuit can be generalized to other 3+1D Dijkgraaf-Witten gauge theories while the second type of circuit can be generalized to other Walker-Wang models, as we discuss later in this section. §.§ Point Charge Condensed Boundary Consider the 3+1D Toric Code defined on a cubic lattice. In the membrane condensate picture, the Z_2 DOFs are on each plaquette. The Hamiltonian contains two types of terms, one associated with each cube, and one associated with each edge. H_3dTC = -∑_cube∏_p∈cube X_p - ∑_edge∏_p ∋edge Z_p Starting from a product state Hamiltonian H=-∑_p Z_p, the Toric Code can be generated with a sequential circuit as shown in Fig. <ref>. Within each cube (Fig. <ref>(a)), the bottom plaquette is first transformed by a Hadamard gate that exchanges X_p and Z_p. It is then used as the control qubit for a set of controlled-Not operations targeting all the other plaquettes in the same cube. After these operations, the Z_p term on the bottom plaquette gets mapped to the cube term ∏_p∈cubeX_P in the Toric Code Hamiltonian. The sequential circuit acts by applying this transformation first to all the cubes (at the same time) in layer 1, then to all the cubes in layer 2 underneath layer 1, and so on. This circuit is a direct generalization of the circuit generating 2D Toric Code discussed in section <ref>. Applying the circuit up to layer n generates a gapped boundary between the 3+1D Toric Code and the vacuum state. It is easy to see that this is the `rough' boundary of the Toric Code (`rough' in the dual lattice) where the gauge charge excitation condenses. The sequential circuit described in Fig. <ref> can be generalized to all Dijkgraaf-Witten (DW) gauge theories in 3+1D. DW gauge theories are membrane condensates in 3+1D where different membrane configurations can come with different phase factors (for Toric Code the phase factors are all 1). To generate the membrane condensate, we can generalize the sequential circuit in Fig. <ref> from Z_2 DOF to group G labeled DOF and then supplement each elementary closed membrane generation operation in (a) with the appropriate phase factor. Similar to the string-net circuit discussed in section <ref>, the elementary closed membrane generation operations in the same layer can be shown to commute with each other. Therefore, the DW states can be generated layer by layer with a linear depth circuit. If the circuit is terminated at a certain layer, the gapped boundary is a condensate of the bosonic gauge charges, just like in the Toric Code case. §.§ Flux Loop Condensed Boundary A different sequential circuit can give rise to the flux loop condensed boundary for the 3+1D Toric Code. The circuit generates a loop condensate in the bulk. To describe this circuit, we take the dual lattice of that in Fig. <ref> so that DOF are associated with each edge. The Toric Code Hamiltonian is then expressed as H_3dTC = -∑_vertex∏_e ∋vertex X_e - ∑_plaquette∏_e∈plaquette Z_e We start with a trivial product state with Hamiltonian H=-∑ X_e. The circuit to obtain the Toric Code is shown in Fig. <ref>. Within each plaquette (Fig. <ref>(a)), a Hadamard gate is first applied to a chosen edge (the thick black one) which is then used as target for a set of controlled-Not gates with all the other edges in the plaquette as the control. After these operations, the X_e term on the target edge gets mapped to the plaquette term ∏_e∈plaquette Z_e in the Toric Code Hamiltonian. The sequential circuit acts by applying this transformation first to all the plaquettes in the first row on the bottom surface with the black edges as the target, then to all the plaquettes in the second row with the grey edges as the target, and so on for all N rows in the bottom surface. Then to move up, apply the transformation in (a) to all the vertical plaquettes immediately above the bottom surface with the orange edges as the target in step N+1. This generates the plaquette term for all the involved vertical plaquettes. After this step, the plaquette terms on the orange plane are automatically satisfied because the product of the six plaquette terms around a cube is the identity. Therefore, repeating step N+1 allows us to expand the topological region and push the boundary upward. It can be checked that, if the sequential circuit is terminated at step N+m, the gapped boundary at the mth plane is the `smooth' boundary type where the flux loop condenses. §.§ Walker-Wang with Loop Condensed Boundary The circuit in Fig. <ref> can be generalized to construct all Walker-Wang (WW) models in 3+1D <cit.>. The ground state of any WW model is a loop condensate, although the loops are more complicated than that in the Toric Code. To construct the ground state wavefunction from a product state, we can choose a target edge in each plaquette and use it to draw loops in each plaquette in a way similar to that described in section <ref> for the string-nets. The only difference is that, in the WW wavefunctions, when we draw loops, there can be extra phase factors due to edges lying over or under the plaquettes. Applying these operations in each plaquette in the sequence shown in Fig. <ref>(b) generates the WW wavefunction. Note that, the plaquette terms in the WW model satisfy the same local constraint as that in the Toric Code model: the product of the six plaquette terms around a cube is identity. For Toric Code, this constraint is satisfied in the whole Hilbert space while for a generic WW model, this constraint is only satisfied in the closed loop subspace. To see why this constraint holds, notice that the product of plaquette terms in the WW model in a closed surface measures the strings that go through the surface. For a topologically trivial closed surface like the surface of a cube, in the closed loop subspace, all strings going into the surface must come out. Therefore, the net flux going through a cube is zero in the closed loop subspace – the product of all the plaquette terms must be identity. Therefore, generating the plaquette terms in the bottom and side faces of a cube automatically gives rise to the plaquette term on the top face. The circuit depicted in Fig. <ref> hence proceeds in the same way as for Toric Code. The gapped boundary created with a circuit terminating at step N+m has the corresponding 2+1D topological order of the WW model. § MAP TO FRACTON PHASES In this section, we construct sequential circuits for the preparation of the X-cube model, which is a well-known example of fracton topological order <cit.>. The X-cube model is a commuting projector model where the projectors are Pauli operators, as shown in Fig. <ref>. We use two different ways to construct the circuit: the first one is associated with the `foliation' structure of the X-cube model <cit.> while the second one is related to the `p-string condensation' picture <cit.>. §.§ Foliation circuit for X-cube model We now state the `foliation' circuit for the X-cube model. We write the circuit to prepare an X-cube model of height L_z cubes with smooth boundaries at the top and bottom and periodic boundary conditions on the sides; however, it can be generalized to other choices of boundary conditions. We start with L_z layers of Toric Code with periodic boundary conditions which were prepared using the sequential circuit shown in Sec. <ref> and a layer of trivial qubits in state \|0⟩. We then combine the Toric Code layers and the layer of trivial qubits into an X-cube model with smooth boundaries using a sequential circuit as shown in Fig. <ref>. The CNOT gates are first applied in the top layer of vertical plaquettes, all in parallel, and then in the next layer, and so on. The overall circuit depth for preparing the X-cube model with smooth boundaries is linear. We apply a circuit of depth 𝒪(L_x+L_y) for the Toric Code layers and then we need a sequential circuit of depth 𝒪(L_z) to combine them into an X-cube model of height 𝒪(L_z) cubes and with smooth boundaries (Fig. <ref>). The same circuit can be used to prepare the X-cube model with periodic boundary condition in the vertical direction with fewer starting trivial qubits. To be specific, under periodic boundary conditions, the layer of trivial qubits shown in purple is not needed along the vertical direction and the orange layer at the bottom will connect to the top orange layer. The entangling gates as shown in Fig. <ref> will then prepare the X-cube model on a 3-torus. However, if we first prepare the height L_z X-cube model with smooth boundaries and then convert it to the X-cube model on a 3-torus with height L_z+1 by doing local gates at the top layer, we need a sequential circuit as shown in Fig. <ref>. We add a layer of trivial qubits on vertical edges above the top smooth boundary in state \|+⟩; these vertical edges now connect to the bottom smooth boundary due to periodic boundary conditions. Using these vertical qubits as control qubits, we do CNOT gates sequentially to prepare the layer of cube stabilizers. The CNOT gates that are applied in parallel are specified by the choice of the control qubits shown on the vertical edges with the same color and along the diagonal (left of Fig. <ref>) and by the gates shown (right of Fig. <ref>). Thus, to convert the X-cube model with smooth boundaries into a 3-torus, we need a sequential circuit of depth 𝒪(L_x+L_y) (Fig. <ref>). A generalization of the X-cube model is the Ising cage-net model which is obtained from a coupled layer construction of double-Ising string-net models instead of Toric Code layers <cit.>. In Ref. wang2022renormalization, a generalized notion of foliation and a sequential circuit for preparation of the Ising Cage-net model is presented. For the X-cube model, the resource Toric Code layers can be prepared in parallel before being inserted one by one into the bulk to increase the size of the X-cube model. For Ising Cage-net, however, this is not possible. Using the scheme presented in Ref. wang2022renormalization, a thin slab (of small L_z) of Ising Cage-net can first be prepared with 𝒪(L_x+L_y) steps. Then the height of the slab L_z can be increased one at a time, each time requiring a circuit of depth 𝒪(L_x+L_y). Therefore, overall a quadratic (𝒪(L^2)) depth circuit is needed to generate the Ising Cage-net model. Whether one can do better than quadratic circuit depth for the Ising cage-net model is left as an open question. It will also be interesting to construct sequential circuit maps for more general fracton models such as the type-2 fracton models which have fractal-shaped logical operators <cit.>. §.§ p-string condensation circuit for X-cube model We now construct a linear-depth sequential circuit for the preparation of the X-cube model using p-string condensation <cit.>. We first consider three decoupled stacks of Toric Codes in xy, yz, and zx planes such that a part of this stack z>L has undergone p-string condensation. That is, Z-Z stabilizers have been added to the double edges of the 3-foliated stack lattice for z>L, such that only the product of Toric Code X plaquette stabilizers around a cube which share the support of Z-Z double edge stabilizers survive as stabilizers. In the part z≤ L, we do not add these Z-Z stabilizers and hence, we still have Toric Code vertex and plaquette stabilizers. Thus, the part z>L is like the X-cube model, the z<L part is like the decoupled Toric Code stack, and there is a gapped boundary between the two at z=L. We assume periodic boundary conditions in the x- and y-directions, while the boundary conditions in the z-direction do not play a role, since we will only consider moving the z=L interface by one lattice constant. See Fig. <ref>(a) for this starting configuration of stabilizers. In the X-cube-like part, the stabilizers are the 24-body X stabilizer terms around a cube, Z-Z stabilizers on every composite edge, and the original Z vertex stabilizers of the Toric Code. At the z=L plane interface between the X-cube-like part and the decoupled Toric Code part, the X-stabilizers supported on the cube are a 20-body X term on the cube above z=L and the Toric Code X-plaquette term in the z=L plane. Our goal is to grow the X-cube portion of the model from z>L to z>L-1, i.e. to push the gapped boundary between the two models, similar to how our previously defined circuits push a gapped boundary to vacuum. In Fig. <ref>(b), we write the finite-depth circuit that acts on the layer of cubes right below z=L to achieve this. The Z-Z stabilizers on the vertical bonds right below z=L follow from the relation among the Toric Code Z vertex stabilizer terms around a “vertex” of the 3d stack lattice and the Z-Z double edge stabilizers terms supported on the same qubits as those vertex stabilizers. In this manner, the interface which was at z=L has moved to z=L-1 after the circuit is applied on the layer of cubes right below z=L. Starting from Toric Code layers, the circuit depth of the p-string condensation circuit is linear i.e., it scales as 𝒪(L_z) since it takes a finite-depth circuit to grow the X-cube part by one unit length in L_z. Since the Toric Code layers in xy planes can be prepared in parallel in circuit depth 𝒪(L_x+L_y), the overall circuit depth scales linearly in the system sizes as 𝒪(L_x+L_y+L_z). § QUANTUM CELLULAR AUTOMATA In this section, we show that arbitrary quantum cellular automata (QCA) can be realized as sequential circuits. A QCA is a unitary operator Q that preserves locality, meaning that, for any operator O_i supported on lattice site i, QO_iQ^† is supported on sites that are at most a distance c away from i for some constant c <cit.>. We focus on implementing translationally invariant QCA on periodic boundary conditions. Any finite-depth quantum circuit (FDQC) gives an example of a QCA, but there are also QCA that cannot be realized as FDQC, such as the 1D shift operator S which acts on operators on a 1D lattice as SO_iS^† = O_i+1. However, it is easy to show that S can be realized as a sequential circuit of SWAP gates <cit.>. Now, we will show that any QCA in any spatial dimension can be realized as a sequential circuit. Additionally, if the QCA Q commutes with some global on-site symmetry U_g=u_g^⊗ N, the gates in the corresponding sequential circuits will also commute with U(g). Therefore, symmetric QCA are a strict subset of symmetric SQCs. On the other hand, SQC are not always QCAs as they do not always preserve locality. The construction we use is closely related to the construction used in Ref. Stephen2023 to realize QCA as finite-depth circuits with geometrically non-local long-range gates. Let us first consider 1D QCA, and then later generalize to higher dimensions. The starting point is the standard form of a 1D QCA. After blocking sufficiently many sites, any QCA will have unit range, meaning that QO_iQ^† is supported only on sites i-1,i,i+1. Then, it was shown in Ref. Schumacher2004 (see also Ref. Cirac2017) that the QCA on a ring of N sites can be expressed in the following way, Q^(N) = ( ∏_i=1^N/2 v_2i-1,2i)( ∏_i=1 ^N/2 u_2i,2i+1). where we take periodic boundaries. Therein, u is a unitary map between a d× d-dimensional Hilbert space to an ℓ× r-dimensional Hilbert space where ℓ r=d^2, and similarly v maps from and r×ℓ-dimensional space to a d× d-dimensional space. We emphasize that these internal dimensions ℓ and r do not represent any physical degrees of freedom, nor do they appear in the sequential circuits we construct; they are simply a convenient technical tool. Indeed, since the input and output Hilbert spaces differ, u and v do not represent physical operations on their own, but they can be combined to define physical operations such Q^(N). As such, Eq. <ref> is not an FDQC, so it can also represent QCA that are not FDQCs such as the shift QCA. To realize this QCA as a SQC, we follow closely the idea used for SPT states in Sec. <ref>. The fundamental unitary gates in the SQC are the QCA on a ring of length 4, Q^(4)_i,j,k,l =v_i,jv_k,lu_j,ku_l,i and also the gate, w_i,j = u^-1_j,iv^-1_i,j. Note that, unlike u and v, the operators Q^(4) and w are genuine physical unitary operators acting on 4 and 2 sites, respectively. Using these gates, we realize the QCA on a chain of even length N>4 as an SQC in the following way, Q^(N) ≈ Q^(4)_1,2,3,4 ×∏_i=1^N-4/2SWAP_2i+2,2i+4w_2i+1,2i+4Q^(4)_2i+1,2i+2,2i+3,2i+4 This equality follows from commuting all SWAP gates to one side of the equation, as is shown graphically in Fig. <ref>. Therein, the `≈' sign means equivalence up to the SWAP gates which can easily be undone in a sequential manner. Now we show how to apply the same construction to arbitrary spatial dimension D. We assume the topology of a D-dimensional hypercubic lattice with periodic boundary conditions. Once again, we can block sites such that Q spreads any operator by at most one site in all directions. Now, suppose we compactify all dimensions except one by viewing the D-dimensional system as a ring of supersites, each consisting of (D-1)-dimensional tori. Then Q can be viewed as a 1D QCA acting on this compactified system. Therefore, we may use the above result to express it as a SQC in one direction, see Fig. <ref>(a). According to the Eq. <ref>, the gates in this SQC are Q^(4) and w and SWAP. The SWAP gate acts on the compactified systems by swapping corresponding sites between the two tori, so each SWAP of tori is local and can be done in depth 1. Since Q is locality preserving in all dimensions, Q^(4) can be viewed as a D-1-dimensional QCA. Likewise, w, which essentially acts as the QCA on two supersites, is also a (D-1)-dimensional QCA, as was proven explicitly in Ref. Stephen2023. Now, we use an inductive argument. We have already shown how to realize any 1D QCA as an SQC. Now, assume we can realize a (D-1)-dimensional QCA as an SQC. According to the above discussion, any D-dimensional QCA Q can be written as an SQC consisting of (D-1)-dimensional QCA. By the inductive assumption, each of these QCA is in turn an SQC, such that Q is itself an SQC. When constructed in this way, the depth of the SQC realizing Q grows like 𝒪(N) where N is the number of sites. This inductive procedure is illustrated in Fig. <ref> for D=2. It turns out that the SQCs constructed above are also composed of symmetric gates, so they are symmetric SQCs. That is, suppose that Q^(N) commutes with some global on-site unitary symmetry u^⊗ N for all N. It is clear that the SWAP gates commute with the global symmetry, as does Q^(4) by assumption. It was shown in Ref. Stephen2023 that w also commutes with the symmetry. Therefore, each of the local gates in the SQC (Eq. <ref>) is symmetric, so it is a symmetric SQC. One application of this comes from applying the construction to the FDQCs which create SPT ground states from symmetric product states <cit.>. These FDQCs commute with the global symmetry protecting the SPT as a whole, but the individual gates do not commute with the symmetry. Applying our construction gives an SQC consisting of symmetric gates that realizes the same unitary. This gives an alternative, but similar, construction of SQCs for SPT states compared to those given in Sec. <ref>. It is interesting to consider truncating the SQCs we have constructed. Consider the case of D=3, where an important class of QCA is given by those which disentangle ground states of Walker-Wang models <cit.>. It has been argued that these QCA are non-trivial, meaning that they cannot be written as a product of FDQCs and translations. The argument is rooted in the fact that, if the QCA acts as an FDQC followed by a translation, then it can be easily truncated to act only in a finite region of space. These truncated circuits could then be used to generate an isolated boundary of the Walker-Wang models that host surface topological order, which is conjectured to be impossible in some cases. Thus, the non-trivial nature of the QCA is tied to the inability to straightforwardly truncate its action to a finite region of space. Since we have constructed SQCs realizing QCA, and these SQCs can be truncated, one may worry that there is a contradiction. However, this is not the case, as truncating the SQC simply implements the QCA exactly on a smaller periodic system, without introducing any boundaries. Therefore, the SQCs cannot be used to isolate boundaries of the Walker-Wang models and the non-triviality of the QCA, as described above, is compatible with the existence of a SQC realizing the QCA. The sequential circuit shown in Section <ref> for constructing the Walker-Wang wavefunction is not mapped from a QCA and does leave open boundaries if truncated. But as it always generates two surfaces (top and bottom) at the same time, there is no contradiction with the non-triviality of the QCA either. § SUMMARY AND OUTLOOK In this paper, we discussed the generation of nontrivial gapped quantum states, starting from product states, using Sequential Quantum Circuits. In particular, we discussed how the circuit that generates the (symmetrized) symmetry-breaking state and the SPT states preserves global symmetry but not necessarily locality; how the circuit that generates fractional topological states is associated with gapped boundaries to the vacuum with either charge or loop condensation; and how the circuit that generates fracton states are related to the foliation structure or p-string condensation in the fracton states. One major class of states that are not covered is the invertible chiral topological states like the Integer Quantum Hall. It is not clear what kind of circuit (beyond finite depth circuit) is needed to generate such states and this will be an interesting question to answer. We can compare four types of many-body unitaries: the finite depth quantum circuit (FDQC), the quantum cellular automata (QCA), the sequential quantum circuit (SQC) and the linear depth quantum circuit (LDQC). Table <ref> summarizes the similarities and differences between them. Based on what we know, we see that they have a strict containment relation FDQC ⊂ QCA ⊂ SQC⊂LDQC Table <ref> summarizes the similarities and differences between the three sets in terms of their effect on global properties of gapped many-body states. Linear depth circuit is the most powerful which can map gapped states to generic entanglement volume law states. The other three types of unitary all preserve entanglement area law, hence mapping gapped states to gapped states. SQC can change short-range correlation into long-range correlation, short-range entanglement into long-range entanglement. It can also change the locality of their parent Hamiltonian and its ground state degeneracy, hence capable of mapping between different gapped phases. They can also generate a non-zero finite correlation length from a state with zero correlation length [For example, SQCs can generate arbitrary matrix product states on open boundary conditions, which can have finite, non-zero correlation length <cit.>]. FDQC and QCA, on the other hand, preserve the locality of operator, correlation function and entanglement and cannot change ground state degeneracy of gapped Hamiltonian. The Multi-scale Entanglement Renormalization procedure (MERA)<cit.> is another way to map nontrivial entangled states to product states. It takes a many-body entangled wave function, applies one layer of finite depth circuit which disentangles local degrees of freedom (DOF) and maps the wave function back to its original form but with doubled unit cell. Such a step is then repeated at the renormalized length scale until the wave function is completely disentangled after log L steps. It is different from the sequential circuit in that in later steps of the procedure the degrees of freedoms are very far away from each other and the unitary gates applied are not local any more. This non-locality in unitary gates is also the reason why the MERA procedure does not violate the linear lower bound in circuit depth to generate GHZ or topological states<cit.>. MERA is more powerful than a sequential circuit as it can map product states to gapless states<cit.>. We are indebted to inspiring discussions with Wenjie Ji, Laurens Lootens, Bram Vancraeynest-De Cuiper, Xiao-Gang Wen, and Cenke Xu. X.C. and A.D. are supported by the National Science Foundation under award number DMR-1654340, the Simons Investigator Award (award ID 828078) and the Institute for Quantum Information and Matter at Caltech. X.C. and N.T. are supported by the Walter Burke Institute for Theoretical Physics at Caltech. R.V. is supported by the Belgian American Educational Foundation. The work of MH on fracton systems was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under Award number DE-SC0014415. The work of D.T.S., work of M.H. on aspects other than fractons, and (in part) the work of X.C. and A.D. was supported by the Simons Collaboration on Ultra-Quantum Matter, which is funded by grants from the Simons Foundation (651440, DTS and MH; 651438, XC and AD). § MAP TO SYMMETRY BREAKING PHASES OF GENERAL FINITE GROUPS In this section, we show how the construction in section <ref> can be generalized from the Z_2 group to arbitrary finite groups. We start by generalizing the Hilbert space from the Z_2 case {\|0⟩,\|1⟩} to a finite group G {\|z⟩:z∈ G} and replacing the X and (1± Z)/2 operators with X^g\|z⟩ = \|gz⟩, g,z∈ G T^h \|z⟩ = δ_h,z\|z⟩, h,z ∈ G, where (X^g)^† = X^g^-1 is unitary and T^h is a projection onto the state \|h⟩. These operators satisfy the commutation relation X^gT^h=T^ghX^g. Using these new operators, we can rewrite the Z_2 invariant transverse field Ising model (<ref>) as H = -J∑_i∑_h∈ GT_i^hT_i+1^h - B1/\|G\|∑_i∑_g∈ G X_i^g, which is now invariant under a global symmetry G, i.e. (∏_i X_i^g) H(∏_i X_i^g)^†=H. Similar to the transverse field Ising model, the Hamiltonian above has a symmetric phase (B≫ J>0) and a symmetry breaking phase (J≫ B>0). The corresponding fixed-point wave functions are given by \|ψ_SY⟩ = \|++⋯+⟩ \|ψ_SB⟩ = 1/√(\|G\|)∑_g\|gg⋯ g⟩, where \|+⟩ = 1/√(\|G\|)∑_g\|g⟩. To construct a quantum circuit that maps \|ψ_SY⟩ to \|ψ_SB⟩, we define R_ij(𝒲) ≡∑_h∈ GT_i^hX^h_j𝒲_j X^h^-1_j, where 𝒲 is a unitary operator so that R_ij(𝒲) is unitary in each subspace projected by T_i^h and, consequently, unitary in the full Hilbert space. One can also check that R_ij(𝒲) is invariant under the symmetry operation (∏_kX^g_k)R_ij(𝒲)(∏_kX^g_k)^† = ∑_h∈ G T_i^ghX_j^gh𝒲_jX_j^(gh)^-1 = R_ij(𝒲). Any circuit made up of R_ij(𝒲) gates is hence symmetric. In analogy to Eqs. (<ref>) and (<ref>), consider a circuit of the following form 𝒰 = R_N1(S) ∏_i=N^1 U_i,i+1, where U_i,i+1= R_i,i+1(S^†HadS^† ) R_i,i+1(S). The matrices R_i,i+1(S^†HadS^†) and R_i,i+1(S) in (<ref>) are chosen such that they will reduce to R(X_i+1) and R(Z_i,Z_i+1) in (<ref>) for the G=Z_2 case, respectively. To map \|ψ_SY⟩ to \|ψ_SB⟩, we need to find unitaries Had and S that map Had\|+⟩ = \|e⟩, S^† \|z⟩ = e^-iθ(z)\|z⟩, such that U_i,i+1\|+,+⟩ = 1/√(\|G\|)∑_hX_i+1^h S^†_i+1Had_i+1\|h,+⟩ = e^-iθ(e)/√(\|G\|)∑_h X^h_i+1\|h,e⟩ = e^-iθ(e)/√(\|G\|)∑_h\|h,h⟩. According to the mapping (<ref>), the first row of Had is restricted to be 1/√(\|G\|)(1,⋯,1), and S must be diagonal. In addition to these two constraints, the Had and S matrices can be selected freely as long as Had is unitary and S is diagonal. A particular choice of Had can be taken as Had = ∑_p,μ,ν∑_z√(ρ_p/\|G\|)A^p_μν(z)\|p,μ,ν⟩⟨ z\|. Here p labels inequivalent irreducible representations of the group G, μ and ν are the row and column indices of the representing matrix A^p, and ρ_p is the dimension of the representation space. Had is a mapping from group elements to entries in the irreducible representation matrices and in particular it should map the identity element to the trivial representation of G in order to ensure the mapping (<ref>). We can also check the mappings of operators under the circuit (<ref>) and discuss whether these mappings hold for a general group G. First, the symmetry operator remains invariant under the circuit: ∏ _i=1^N X_i^g →∏ _i=1^N X_i^g. as our unitary circuit converses the symmetry. And the transverse magnetic field term is always mapped to the ferromagnetic term: 1/\|G\|∑_g∈ GX^g_i→∑_h T^h_i-1T^h_i, i=2,⋯,N, which can be proved by the following steps U_i-1,i(1/\|G\|∑_g∈ GX^g_i)U^†_i-1,i = ∑_hT^h_i-1 X_i^h S^†_iHad_i \|+⟩⟨ +\| Had^†_iS_iX_i^h^-1 = ∑_h T^h_i-1T^h_i, and U_i,i+1(∑_h T^h_i-1T^h_i)U^†_i,i+1 = ∑_h T^h_i-1T^h_i. Unfortunately, the other mappings in the Z_2 case in (<ref>) do not hold for a general finite group G. For instance, the ferromagnetic term ∑_h T^h_iT^h_i+1 generally does not map to the transverse field term 1/\|G\|∑_g∈ GX^g_i+1. Things become easier in the case where G is abelian, as the Had matrix in (<ref>) reduces to conventional Fourier transformations of the abelian group G. Without loss of generality, we consider the example of a cyclic group Z_M, where we can choose Had = ∑_α,z=0^M-11/√(M)exp(2π i/Mα z)\|α⟩⟨ z\|, S = ∑_z=0^M-1exp(π i/Mz^2)\|z⟩⟨ z\|. In the simplest case where G=Z_2, the Had and S matrices will reduce to the conventional Hadamard gate and phase gate, respectively. It can also be verified that R_ij(S) = exp(-iπ/4(Z_iZ_j-1)) and R_ij(S^†HadS^†)=exp(-iπ/4X_j), which implies that the generalized circuit (<ref>) will reduce to (<ref>) in the Z_2 case up to a global phase. For the abelian groups, all the operator mappings for the Z_2 case are preserved as long as we choose the gate as in (<ref>). The mappings in (<ref>) will become 1/\|G\|∑_gX^g_i →∑_hT_i-1^hT_i^h, i = 2,⋯, N, 1/\|G\|∑_gX^g_1 →∑_g 1/\|G\|∑_h,h'e^-2π i/\|G\|(h'-h)gT_1^hT_N^h'∏_i=1^NX_i^g, ∑_hT^h_iT^h_i+1 →1/\|G\|∑_g X^g_i, i = 2,⋯,N ∑_hT^h_1T^h_2 →1/\|G\|∑_g X^g_1∏_i=1^NX_i^g. In the bulk of the system, the transverse field term maps to the Ising term and vice versa. Near the left endpoint, the correspondence is preserved only under the symmetric subspace ∏_i=1^NX_i^g=1. Generalization to higher dimensional versions shows no difference with the Z_2 case, as shown in Fig. <ref>. § MAP TO STRING-NET STATES WITH GENERAL GAPPED BOUNDARIES AND THE TORUS This section contains some of the necessary notions from category theory to define the general sequential circuit in section <ref>. Following Kitaev and Kong, gapped boundaries of string-net models are classified by 𝒞-module categories ℳ, where 𝒞 is the (unitary) fusion category of the string-net model. The boundary can be interpreted as a gapped domain wall between the string-net 𝒞 and the vacuum, given by the category of vector spaces Vec, in which case ℳ can be viewed as a (𝒞, Vec)-bimodule category. At the same time, given ℳ and 𝒞, we can always find the (unique) Morita dual category of 𝒞, 𝒟 = 𝒞_ℳ^, in which case ℳ is an invertible (𝒞,𝒟)-bimodule category. This is the structure we will require to define our quantum circuit. We refer the reader to Ref. lootens2021matrix, etingof2016tensor for more details about the underlying mathematics and will present only some key ingredients here in order to define the sequential circuit. Given the resemblance of the operators needed to define our sequential circuit and the (finite-depth) circuit from Ref lootens2022mapping, we follow the same conventions here. The Hilbert space of string-net models consists of configurations on a hexagonal lattice, with the edges labeled by simple objects a,b,... of a unitary fusion category 𝒞. The simple objects obey fusion constraints: a × b = ∑_c N_ab^c c, which are assumed to be multiplicity free (N_ab^c ∈0,1) from now on. There exists a unit object 1, such that ∀ a, 1× a = a. Each simple object has a quantum dimension d_a associated to it. We can diagrammatically write down the resolution of the identity: < g r a p h i c s > . The fusion category has an associator or 𝒞-symbol -we use the 𝒞-superscript to distinguish the symbol from other associators that will be defined below- expressed in terms of the simple objects as: < g r a p h i c s > . This 𝒞-symbol is simply the usual F-symbol for fusion categories and obeys the well-known pentagon equation. We also require the “bubble pop" identity: < g r a p h i c s > . A (right) 𝒞-module category ℳ, has simple objects that will be denoted by capital Roman letters (A,B,C,...) and a right-action of 𝒞 on ℳ, ◃: ℳ×𝒞→ℳ. The action is strict: A ◃1 = A and generally we can write A ◃ a = ∑_B∈ℳ N_Aa^B B. Note that these new N_Aa^B are distinct from the original fusion rules of 𝒞 (N_ab^c) and may carry multiplicities (N_Aa^B > 1) even if the N_ab^c don't. ℳ has an associator ◃ that implements the following recoupling at the boundary of the string-net: < g r a p h i c s > , where the boundary is now blue, labeled by objects in the module category ℳ. A suitable gauge choice can be made such that: (^◃F^A1b_B )^B,m_b,1k = (^◃F^Aa1_B )^A,m_a,k1 = δ_k^m. The inverse symbol is denoted by ◃. We now have a second unitary fusion category 𝒟 with simple objects labeled by Greek letters (α, β, δ,...). In our case, this category will be chosen to be the unique Morita dual of 𝒞, given ℳ: 𝒟 = 𝒞_ℳ^. The F-move for 𝒟 is defined in the following way: < g r a p h i c s > . Just like for ℳ, it is not guaranteed that 𝒟 is multiplicity-free, even if 𝒞 is. We will now take ℳ to be a left 𝒟-module category, with an associator ▹. The bulk string-net 𝒟 is now on the left of the boundary: < g r a p h i c s > . A similar gauge choice can be made such that: (^▹F^1β A_B )^B,m1_β ,1k = (^▹F^α1A_B )^A,1m_a,1k = δ_k^m. The inverse symbol is denoted by ▹. For ℳ to be a (𝒟,𝒞)-bimodule category we require one last additional associator : < g r a p h i c s > . with a similar gauge choice leading to: (^F^1Ab_B )^b,m1_A,1k = (^F^α A1_B )^A,1m_B,k1 = δ_k^m. The inverse symbol is denoted by . There are in total 6 coupled pentagon equations that the set of F-symbols should satisfy. These are given in Ref. lootens2021matrix. One final ingredient we need is a resolution of the identity that allows for fusion of two ℳ lines in terms of a 𝒟- and a 𝒞 line. Although ℳ has no intrinsic duality, we can define the `opposite' bimodule category ℳ, a (𝒞,𝒟)-bimodule category, which contains the dual objects of ℳ. The rigorous mathematical picture behind this is called a Morita context <cit.>. This allows us to define: < g r a p h i c s > , and the opposite resolution in terms of a 𝒟 line: < g r a p h i c s > , where d_A,d_B are quantum dimensions for the objects in ℳ. Note that the invertibility of the bimodule ℳ guarantees <ref> and <ref>. We are now ready to write down the precise action of the general sequential circuit in figure <ref> (cylinder case) and <ref> (torus case) in terms of the individual plaquette actions for the 7 sublattices L_1,...,L_7. The original `smooth' boundary case in figure <ref> is recovered by choosing ℳ = 𝒞. In this case, the Morita dual is simply 𝒞 itself (𝒟 = 𝒞) and all F-symbols defined above reduce to 𝒞 (up to some suitable permutations of the labels). For simplicity, we will only treat the general case. We will follow the orientation convention as in Ref. lootens2022mapping. The control edges in the sequential circuit are colored red in Eqs. <ref>-<ref>. In the full sequential circuit, the control edge has not been acted on by previous operations and is still labeled by the trivial object in 𝒟. For the purpose of generality, we assume it to be arbitrary in Eqs. <ref>-<ref>. < g r a p h i c s > = ∑_{Ẽ_i,h_i} [^1B_p^S,ℳ]_{ϵ_i,k_i}^{Ẽ_i,h_i} < g r a p h i c s > , [^1B_p^S,ℳ]_{ϵ_i,k_i}^{Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_6d_ϵ_5d_ϵ_3d_ϵ_2d_Ẽ_4d_Ẽ_1/d_α_5d_α_2d_ϵ_4d_ϵ_1d_S^2) ( ▹^α_1ϵ_2S_Ẽ_1)^Ẽ_2,j_2h_1_ϵ_1,k_1 n_1 ( ▹^ϵ_2ϵ_3Ẽ_3_Ẽ_2)^S,n_3n_2_α_2,k_2h_2 ( ▹^ϵ_3α_3Ẽ_4_S)^Ẽ_3,h_3n_3_ϵ_4,k_3n_4 ( ▹^ϵ_5α_4Ẽ_4_S)^Ẽ_5,h_4n_5_ϵ_4,k_4n_4 ( ▹^ϵ_6ϵ_5Ẽ_5_Ẽ_6)^S,n_6n_5_α_5,k_5h_5 ( ▹^α_6ϵ_6S_Ẽ_1)^Ẽ_6,n_6h_6_ϵ_1,k_6n_1. The case for the original plaquette operator B_p^s (<ref>) is recovered by choosing ℳ = 𝒞 = 𝒟 in <ref>. The matrix [B_p^s] we recover in this case is different than the one in the original Levin-Wen string-net paper, because of the different orientation convention and the fact that no tetrahedral symmetry of the 𝒞-symbols is assumed <cit.>. < g r a p h i c s > = ∑_{ẽ_iẼ_i,h_i} [^2B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} < g r a p h i c s > , [^2B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_2d_ϵ_3d_ϵ_4d_Ẽ_1d_Ẽ_3d_ẽ_1d_ẽ_2/d_ϵ_1d_A_2d_A_3d_Ẽ_2d_Ẽ_4d_S^2) ( ▹^α_1ϵ_2S_Ẽ_1)^Ẽ_2,n_2h_1_ϵ_1,k_1n_1 ( ^ϵ_2Sẽ_1_A_1)^E_1,n_3k_2_Ẽ_2,n_2h_2 ( ^ϵ_3Ẽ_3ẽ_1_E_1)^A_2,h_3k_3_S,n_4n_3 ( ^ϵ_3Ẽ_3ẽ_2_E_2)^A_3,h_4k_4_S,n_4n_5 ( ^ϵ_4Sẽ_2_A_4)^E_2,n_5k_5_Ẽ_4,n_6h_5 ( ▹^α_2ϵ_4S_Ẽ_1)^Ẽ_4,n_6h_6_e_1,k_6n_1. < g r a p h i c s > = ∑_{ẽ_iẼ_i,h_i} [^3B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} < g r a p h i c s > , [^3B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_2d_ϵ_3d_A_2d_Ẽ_1d_ẽ_1d_ẽ_3/d_ϵ_1d_E_3d_α_2d_Ẽ_2d_Ẽ_3d_ẽ_2) ( ▹^α_1ϵ_2S_Ẽ_1)^Ẽ_2,n_2h_1_ϵ_1,k_1n_1 ( ^ϵ_2Sẽ_1_A_1)^E_1,n_3k_2_Ẽ_2,n_2h_2 ( ◃^Sẽ_1a_2_E_2)^ẽ_2,n_4_E_1,n_3k_3 ( ◃^Sẽ_3a_3_E_2)^ẽ_2,n_4_E_3,n_5k_4 ( ^ϵ_3Sẽ_3_A_2)^E_3,n_5k_5_Ẽ_3,n_6h_3 ( ▹^α_4ϵ_3S_E_1)^Ẽ_3,n_6h_4_e_1,k_6n_1. < g r a p h i c s > = ∑_{ẽ_iẼ_i,h_i} [^4B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} < g r a p h i c s > , [^4B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_1d_E_3d_E_1d_ẽ_1d_ẽ_2d_ẽ_4d_Ẽ_2/d_A_1d_E_4d_A_4d_Ẽ_2d_ẽ_3d_S^2) ( ^ϵ_1E_1ẽ_1_Ẽ_1)^S,n_1n_2_A_1,k_1h_1 ( ^ϵ_1Sẽ_2_A_2)^E_2,n_3k_2_Ẽ_1,n_2h_2 ( ◃^Sẽ_2a_1_E_3)^ẽ_3,n_4_E_2,n_3k_3 ( ◃^Sẽ_4a_2_E_3)^ẽ_3,n_4_E_4,n_5k_4 ( ^ϵ_2Sẽ_4_A_3)^E_4,n_5k_5_Ẽ_2,n_6h_3 ( ^ϵ_2E_1ẽ_1_Ẽ_2)^S,n_1n_6_A_4,k_6h_4. < g r a p h i c s > = ∑_{ẽ_iẼ_i,h_i} [^5B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} < g r a p h i c s > , [^5B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_1d_E_1d_E_3d_ẽ_2d_ẽ_4d_ẽ_5/d_A_2d_E_2d_a_3d_ẽ_1d_S^2) ( ◃^E_1a_1ẽ_2_S)^ẽ_1,n_1_E_2,k_1n_2 ( ^ϵ_1A_1ẽ_2_S)^E_1,h_1n_3_E_2,k_2n_2 ( ^ϵ_1Ẽ_1ẽ_3_E_3)^A_2,h_2k_3_S,n_3n_4 ( ◃^Sẽ_4a_2_E_3)^ẽ_3,n_4_E_4,n_5k_4 ( ◃^E_5ẽ_5ẽ_4_E_4)^a_3,k_5_S,n_6n_5 ( ◃^E_1a_4ẽ_5_S)^ẽ_1,n_1_E_5,k_6n_6. < g r a p h i c s > = ∑_{ẽ_iẼ_i,h_i} [^6B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} < g r a p h i c s > , [^6B_p^S,ℳ]_{ϵ_i,E_i,k_i}^{ẽ_i,Ẽ_i,h_i} = ∑_{n_i}√(d_ϵ_1d_E_3d_E_1d_ẽ_2d_ẽ_4d_ẽ_5/d_A_1d_E_2d_a_3d_ẽ_3d_S^2) ( ^ϵ_1E_1ẽ_1_Ẽ_1)^S,n_1n_2_A_1,k_1h_2 ( ^ϵ_1Sẽ_2_A_2)^E_2,n_3k_2_Ẽ_1,n_2h_2 ( ◃^Sẽ_2a_1_E_3)^ẽ_3,n_4_E_2,n_3k_3 ( ◃^Sẽ_4a_2_E_3)^ẽ_3,n_4_E_4,n_5k_4 ( ◃^E_5ẽ_5ẽ_4_E_4)^a_3,k_5_S,n_6n_5 ( ◃^E_1a_4ẽ_5_S)^ẽ_1,n_1_E_5,k_6n_5. < g r a p h i c s > = ∑_{ẽ_i} [^7B_p^S,ℳ]_{E_i,k_i}^{ẽ_i} < g r a p h i c s > , [^7B_p^S,ℳ]_{E_i,k_i}^{ẽ_i} = ∑_{n_i}√(d_E_1d_E_4d_ẽ_2d_ẽ_3d_ẽ_5d_ẽ_6/d_a_2d_a_5d_ẽ_1d_ẽ_4d_S^2) ( ◃^E_1a_1ẽ_2_S)^ẽ_1,n_1_E_2,k_1n_2 ( ◃^E_2ẽ_2ẽ_3_E_3)^a_2,k_2_S,n_2n_3 ( ◃^Sẽ_3a_3_E_4)^ẽ_4,n_4_E_3,n_3k_3 ( ◃^Sẽ_5a_4_E_4)^ẽ_4,n_4_E_5,n_5k_4 ( ◃^E_6ẽ_6ẽ_5_E_5)^a_5,k_5_S,n_6n_5 ( ◃^E_1a_6ẽ_6_S)^ẽ_1,n_1_E_6,k_6n_6. Finally, we show the action of growing an S-loop on the minimal torus with trivial edges: < g r a p h i c s > = ∑_α,β,γ{h_i}√(d_αd_γ/d_βd_S^2) ( ▹^αγ S_S)^S,h_3h_1_β,h_5h_2( ▹^γα S_S)^S,h_1h_3_β,h_4h_2 < g r a p h i c s > . This action is the first step in the sequential circuit to construct the string-net ground state on the torus (Fig. <ref>). It guarantees that the state is already in the ground state before the last plaquette operator (<ref>) is performed. Therefore, <ref> is unitary, even though no edge is left to serve as a valid control. § MAP TO SPTS IN TWO DIMENSIONS AND HIGHER The sequential circuit in section <ref> for generating the CZX state (the 2D ℤ_2 SPT) from the trivial state can be generalized to group cohomology SPT states with any symmetry and in any dimension <cit.>. Following the notation in section <ref>, consider fixed point wavefunctions as shown in Fig. <ref>(c). There are four spin DOFs in each site with basis states \|g⟩, g∈ G. The symmetry acts on each site as U_g\|g_1, g_2, g_3, g_4⟩→α(g_2,g_1) α(g_1,g_4)/α(g_3,g_4) α(g_2,g_3)\|gg_1, gg_2, gg_3, gg_4⟩ where α(g_i,g_j) = ν_3(g_i,g_j,g^-1g^,g^). Four spins connected by a black square are in a local entangled state 1/√(\|G\|)∑_g \|gggg⟩. When the entangled states are each contained within a lattice site, the state is a trivial SPT. When the entangled states are between lattice sites, the state is a nontrivial SPT. See Fig. <ref>(c). The circuit that maps between the two proceeds with a sequence of SWAP gates, as shown in Fig. <ref>(d). The sequence of SWAP gates maps between the two wavefunctions. However, after each SWAP gate, the symmetry action changes and needs to be restored. Following the notation in Fig. <ref>(c), after the step of SWAP, Spin 1 and 5 are exchanged and spin 4 and 8 are exchanged. Therefore, the phase difference that needs to be corrected to restore the symmetry action is α(g_2,g_1) α(g_1,g_4)/α(g_3,g_4) α(g_2,g_3)α(g_6,g_5) α(g_5,g_8)/α(g_7,g_8) α(g_6,g_7)×α(g_3,g_8) α(g_2,g_3)/α(g_2,g_5) α(g_5,g_8)α(g_7,g_4) α(g_6,g_7)/α(g_6,g_1) α(g_1,g_4) which can be separated into two parts for the upper and lower half of the squares α(g_2,g_1)α(g_6,g_5)/α(g_2,g_5)α(g_6,g_1), α(g_3,g_8)α(g_7,g_4)/α(g_3,g_4)α(g_7,g_8) The two parts can be corrected separately with phase factors acting on 1,2,5,6 and 4,3,8,7 respectively. We will focus on the upper part. Using cocycle calculation, we see that [ α(g_2,g_1)α(g_6,g_5)/α(g_2,g_5)α(g_6,g_1) = ν_3(g_2,g_1,g^-1g^,g^)ν_3(g_6,g_5,g^-1g^,g^)/ν_3(g_2,g_5,g^-1g^,g^)ν_3(g_6,g_1,g^-1g^,g^); = ν_3(g_2,g_1,g^-1g^,g^)ν_3(g_6,g_2,g^-1g^,g^)/ν_3(g_6,g_1,g^-1g^,g^)×ν_3(g_6,g_5,g^-1g^,g^)/ν_3(g_2,g_5,g^-1g^,g^)ν_3(g_6,g_2,g^-1g^,g^); = ν_3(g_6,g_2,g_1,g^)/ν_3(g_6,g_2,g_1,g^-1g^)×ν_3(g_6,g_2,g_5,g^-1g^)/ν_3(g_6,g_2,g_5,g^); = ν_3(g_6,g_2,g_1,g^)/ν_3(gg_6,gg_2,gg_1,g^)×ν_3(gg_6,gg_2,gg_5,g^)/ν_3(g_6,g_2,g_5,g^) ] which can be achieved by conjugating the symmetry operation by a phase factor of ν_3(g_6,g_2,g_1,g^)/ν_3(g_6,g_2,g_5,g^*). Similar calculation can be done for the lower part of the squares and for vertical swaps. Therefore, the SWAP gate can be dressed by phase factors to make sure that symmetry operators remain invariant throughout the transformation. Moreover, the phase correction coming from the upper half and the lower half cancel each other when acting on the ground state wavefunction, therefore, the added phases do not change the ground state, so the ground state transforms as we wanted under the SWAP gates. A similar construction applies to group cohomology SPT states in higher dimensions as well.
http://arxiv.org/abs/2307.01561v1	20230704083209	Microlocal categories over the Novikov ring I: cotangent bundles	[ "Yuichi Ike", "Tatsuki Kuwagaki" ]	math.SG	[ "math.SG" ]	Approximate information for efficient exploration-exploitation strategies Jean-Baptiste Masson August 1, 2023 ========================================================================= This is the first part of a series of papers studying categories defined over the Novikov ring arising from microlocal sheaf theory. In this paper, we define a family of categories associated with each cotangent bundle, which is an enhanced version of the category first introduced by Tamarkin. Using our categories, for any (possibly non-exact immersed) Lagrangian brane, we develop a theory of sheaf quantization generalizing those of Tamarkin, Guillermou, Jin–Treumann, and others. In particular, our theory involves the notion of a sheaf-theoretic bounding cochain, which is a counterpart of the theory of Fukaya–Oh–Ohta–Ono. We also study several structures and properties known in the classical Tamarkin category; separation theorem, intersection points estimates, interleaving distances, energy stability with respect to Guillermou–Kashiwara–Schapira autoequivalence, and the completeness of the distance. We conjecture that our category is equivalent to a Fukaya category defined over the Novikov ring. § INTRODUCTION §.§ Background: Sheaf quantization and Fukaya category Sheaf-theoretic study of symplectic geometry has evolved since the work of Nadler–Zaslow <cit.> (appeared on arXiv in 2006) and Tamarkin <cit.> (appeared on arXiv in 2008). The underlying key idea of both studies is the microlocal sheaf theory due to Kashiwara and Schapira <cit.>. For a sheaf on a manifold M, the theory associates a subset called microsupport of the cotangent bundle T^M. This is the topological counterpart of the notion of characteristic variety in the theory of -modules. From the viewpoint of deformation quantization, a module should be viewed as a quantization of its characteristic variety. Hence we can view a sheaf as a “topological" quantization of its microsupport. The key philosophy is that these sheaf-theoretic quantizations form a category closely related to Fukaya categories of Lagrangian submanifolds. Nadler–Zaslow's work suggests that the association of microsupport can be upgraded to an equivalence between the category of constructible sheaves over M and a version of Fukaya category of T^M. Roughly speaking, for a Lagrangian submanifold (more precisely, brane) L, the corresponding constructible sheaf has its microsupport in the conical limit of L. Nadler–Zaslow's work motivates many recent studies. One of the most notable ones is Ganatra–Pardon–Shende's work <cit.> that proves equivalences between wrapped Fukaya categories and microsheaf categories for a large class of non-compact symplectic manifolds, which enables us to compute many Fukaya categories. One disadvantage of Nadler–Zaslow's work is that it is designed to be Hamiltonian-invariant from the beginning, and hence the geometric shape of a Lagrangian submanifold is somehow lost (through taking the conical limit) in the formulation. This is related to the fact that microsupport is conic by definition. In other words, one can sheaf-quantize only conic objects. This makes quantitative study or geometric study impossible/difficult. There is a known prescription for the above problem in the context of deformation quantization (e.g., the work of Polesello–Schapira <cit.>); it is to introduce one-additional parameter ħ. Then conic objects in 2n+2-dimension can express non-conic objects in 2n-dimension. Tamarkin <cit.> adopted this trick in sheaf theory. We would like to describe it more precisely. Let M be a manifold. We consider sheaves on M×_t having the 1-dimensional additional parameter, rather than considering sheaves on M. The Tamarkin category is then defined as a quotient category of the category of sheaves on M×_t. An object of this category has its well-defined positive part of microsupport (i.e., τ>0) in T^M× T^_t≅ T^M×_t×_τ. We consider the twisted projection ρ T^M×_t×{τ>0}→ T^M; (p, t, τ)↦ p/τ. Then we consider a sheaf quantization of a subset L of T^M as a positively-microsupported sheaf with ρ(()∩{τ>0})=L. In the past several years, this notion has been proven to be quite useful: Tamarkin proved the non-displaceability theorem, Guillermou–Kashiwara–Schapira made it precise through quantization of Hamiltonian isotopies, Guillermou and Jin–Treumann constructed sheaf quantization of exact Lagrangian branes, Asano–Ike introduced pseudo-metric and studied its properties, etc. A partial review is given in <cit.>. One disadvantage of the Tamarkin category is that it works quite well for exact Lagrangian submanifolds, but it does not work well for non-exact Lagrangian submanifolds. The reason is that ρ^-1(L) is huge and redundant, and we have to take a Lagrangian leaf of it, which is possible only for the exact case. On the other hand, Nadler–Zaslow–Ganatra–Pardon–Shende's approach can treat non-exact (unobstructed) Lagrangian branes if one works over the Novikov field, as pointed out by Shende <cit.>. In <cit.>, for WKB-theoretic construction of sheaf quantization, the second-named author enhanced Tamarkin's approach to treat possibly non-exact Lagrangian submanifolds. The key idea is to consider the equivariant sheaf version of the Tamarkin category. Then the equivariancy cancels the redundancy of ρ^-1(L). Moreover, the resulting category is automatically enhanced by the Novikov ring. In this paper, based on <cit.>, we would like to start a systematic sheaf-theoretic study of non-exact Lagrangian geometry, which can be considered as a step to unify the two approaches by Nadler–Zaslow and Tamarkin. §.§ Our results We start with defining our working category, which is a slight generalization of that of <cit.>. Let M be a real manifold, _t (resp. _u) be the real line with the standard coordinate t (resp. u). For a positive real number c, we denote by _u<c the subset of _u defined by the inequality u<c. We regard _u as the case c=∞. We also fix a field and a subgroup of the additive group . Then we consider the derived category of sheaves of -vector spaces over M ×_u<c×_t, and denote it by ^(M ×_u<c×_t). We quotient out the non-positively microsupported sheaves and obtain _τ>0^(M ×_u<c×_t). If is the trivial group, this category is nothing but the category introduced by Tamarkin <cit.>. We explain the role of t and u. Let us start with t. As explained above, t is introduced to treat non-conic Lagrangians. Namely, by using this additional variable, one can lift any non-exact Lagrangian in T^M to a conic Lagrangian in T^M× T^_t. As clarified in <cit.>, this variable encodes energy information. By introducing the group action, <cit.> observed the category _τ>0^(M ×_u ×_t) is enriched over the Novikov ring associated with : Λ_0^_∋ c→∞[∩_≥ 0]/T^c[∩_≥ 0], where T^c is the indeterminate corresponding to c∈∩_≥ 0. This is analogous to the fact that Fukaya categories are defined over the Novikov ring <cit.> that encodes energy information in Floer theory. On the other hand, the variable u is first used by Guillermou <cit.> for an intermediate step to produce sheaf quantizations through “doubling movies”. It is further used by Guillermou <cit.>, Asano–Ike <cit.>, and Nadler–Shende <cit.> to produce “obstructed" sheaf quantizations. We view such sheaf quantizations as an object of the category _τ>0^(M ×_u<c×_t). In Floer theory, the counterpart is obstructed Lagrangian branes whose Floer cohomology is defined modulo T^c. Since objects of _τ>0^(M ×_u<c×_t) do not always have the shape of “doubling movies”, we restrict ourselves to the subcategory of such objects. We denote the subcategory by μ^(T^M;u<c). When is trivial or , the theory of sheaf quantizations is developed in various literature, including <cit.>. Here we develop the analogous theory for non-exact Lagrangian submanifolds inside the category μ^(T^M;u<c). Now we state our first theorem. Let L be a Lagrangian brane, which is a tuple of a Lagrangian immersion L↬ T^M, a grading α, a relative Pin structure b, and a local system . Associated with (L, α, b), we have a functor assigning a local system on L to an object with L=()ρ(()∩τ>0), where is the microsupport[We need a mild modification of ρ in the presence of u-variable. See the body of the paper.]. We say that an object is a sheaf quantization of L if the associated local system is . Let L be an end-conic Lagrangian brane in T^M whose projection to the base is compact. Then there exists a positive number c such that there exists a sheaf quantization of L in μ^_L(T^M;u<c), where _L is the period group of L. This is a sheaf-theoretic counterpart of the Gromov compactness: For any Lagrangian brane, there exists a positive number c such that an unobstructed Floer complex is defined over Λ_0/T^cΛ_0. In the following, we set to be and omit from the notation. In general, the Floer complex over Λ_0 is not a complex (i.e., d^2≠ 0). To get a genuine complex, in the book <cit.>, Fukaya–Oh–Ohta–Ono developed the notion of bounding cochain, which deforms the Floer differential to be non-curved over Λ_0/T^c'Λ_0 for a larger c'. Analogously, to extend our sheaf quantization to a sheaf quantization in μ(T^M;u<∞), we need the notion of sheaf-theoretic bounding cochain. Our theorem is the following: Let L be a Lagrangian brane. We fix a sufficiently fine discrete monoid σ. Then, there exists a curved dga CF_SQ(L, L,σ) such that any Maurer–Cartan element of CF_SQ(L, L,σ) ( sheaf theoretic bounding cochain) gives a sheaf quantization of L. Conversely, any sheaf quantization of L gives a sheaf-theoretic bounding cochain. See <ref> for a precise statement. We conjecture that CF_SQ(L, L,σ) is quasi-isomorphic to the curved Floer complex. From <ref> to the end of the paper, we give a generalization of structural properties of the Tamarkin category in our setting. The first such one is the separation theorem, first proved by Tamarkin. We adopt it in our setting and remove the compactness assumption. Suppose , ∈μ(T^M;u<c) satisfy the following: * () and () are end-conic, * π(()) ∩π(()) is compact, where π T^M→ M is the projection, and () ∩() = ∅. Then, one has _μ(T^M;u<c)(,)=0. The second such theorem is the intersection point estimate, which is proved by <cit.> in the compact exact setting. Our generalization here is twofold: (1) We remove the compactness assumption, and (2) we identify what is in the clean case. As a result, we obtain the following (see the body of the paper for a more precise statement): Let L_1,L_2 be end-conic Lagrangian branes and _1, _2 be their sheaf quantizations, respectively. Assume that π(L_1) ∩π(L_2) is compact and L_1 cleanly intersects with L_2. Then _Λ_0/T^c Λ_0^∙_μ(T^M;u<c)(_1, _2) = ∑_k ∈_ H^k(L_1∩ L_2; _L_1, L_2), where _L_1, L_2 is the sheaf-theoretic Maslov index local system, which is defined only with information of the brane structures. In particular, in the case c=∞, ∑_k ∈_ H^k(L_1∩ L_2; _L_1, L_2) ≥_Λ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ. If L_1 transversely intersects with L_2 and _1, _2 are simple, then the left-hand side is #(L_1∩ L_2). The third is about the metric aspect. Motivated by sheaf-theoretic study of persistence distance <cit.>, Asano–Ike <cit.> introduced a pseudo-distance on the sheaf category _τ>0(M×_t). On the other hand, Guillermou–Kashiwara–Schapira <cit.> introduced sheaf quantization of Hamiltonian isotopy by sophisticating <cit.>. For the pseudo-distance on the sheaf category _τ>0(M×_t), <cit.> proved the stability property of GKS-autoequivalence. Asano–Ike <cit.> and Guillermou–Viterbo <cit.> also proved the completeness of the metric. In our setup, the metric is defined with the Λ_0-linear structure. Then we can prove the completeness and the Hamiltonian stability: There exists a pseudo-metric d_I on μ(T^M; u<c) such that the following holds: * For , ∈μ(T^M; u<c), if d_I(, )<∞ , then ≅ over the Novikov field. The pseudo-metric d_I is complete. * For any compactly supported Hamiltonian diffeomorphism φ, there exists a GKS-autoequivalence Φ on μ(T^M; u<c) that satisfies (Φ()) = φ(()) for any ∈μ(T^M;u<c). Moreover, for any ∈μ(T^M;u<c), one has d_I(,Φ()) ≤φ_H, where φ_H is the Hofer norm of φ. Although d_I is a degenerate metric on μ(T^M; u<c), it is natural to expect it to be non-degenerate on limits of sheaf quantization. Such a kind of result is proved by Guillermou–Viterbo <cit.> in the compact exact setting. In our setup, we prove the following. Let L be a Lagrangian brane with a (sheaf-theoretic) bounding cochain b. We assume that the corresponding sheaf quantization has rank 1 endomorphism space. We denote by (L, b) the set of Lagrangian branes with bounding cochains that are isotopic to (L, b) by compactly-supported Hamiltonian isotopy. This set is equipped with the Hofer norm. We denote the completion by (L, b). There exists a canonical functor (L, b)→μ(T^M;u<∞) extending the sheaf quantization. More precisely, if two Cauchy sequences in (L, b) define the same limiting object, the corresponding limits of sheaf quantizations are also isomorphic. As we mentioned in the abstract, this is the first paper of a series of papers we plan to write. In the forthcoming papers, we will address the problems (1) how to generalize the story to Liouville manifolds <cit.>, (2) how to give an alternative model of our categories that is closer to Fukaya categories <cit.>, and (3) how to prove the conjecture comparing our categories with Fukaya categories <cit.>. §.§ Organization of this paper Here is the organization of the paper. In <ref>, we define our category, which is a combination of the setup in <cit.> and <cit.>. In <ref>, we develop sheaf quantization formalism for possibly non-exact Lagrangian branes with the notion of sheaf-theoretic bounding cochains. <ref>, we study the structural properties of our microlocal category, which are known for the classical Tamarkin category. §.§ Acknowledgment Y. I. is supported by JSPS KAKENHI Grant Numbers 21K13801 and 22H05107. T. K. is supported by JSPS KAKENHI Grant Numbers 22K13912 and 20H01794. Y. I. thanks Tomohiro Asano for the helpful discussions. T. K. thanks Kaoru Ono and Hiroshi Ohta for having helpful discussions on the ideas of bounding cochain in various occasions. §.§ Notation Throughout this paper, we let be a field. Let X be a manifold and let π T^X → X denote its cotangent bundle. We write T^_XX for the zero-section of T^X. We set _X to be the constant sheaf on X with stalk . We also let (X) denote the derived dg-category of the abelian category of _X-modules. For an object ∈(X), we write () for the microsupport of (see <cit.> for the definition), which is a closed conic subset of T^X. We also let ∂ T^X denote the contact boundary defined as (T^X ∖ T^_XX)/_>0. § EQUIVARIANT SHEAVES AND THE NOVIKOV RING Some of the following results have already appeared in <cit.>, but we recall them systematically. Until the end of <ref>, all the statements without proofs are proved in <cit.> (in updated version). §.§ Equivariant sheaves and the Novikov action Let Q be a manifold and _t be the real line with the standard coordinate t. On the product space Q ×_t, we consider the action of a subgroup of the additive group via the addition on the right factor; T_c Q ×_t → Q ×_t; (x,t) ↦ (x,t+c) for c ∈. We consider this action as a continuous action with the discrete[If one equips with the usual Euclidean topology, then the resulting equivariant geometry is the same as M.] topology on . We denote the trivial subgroup of by . Let _(Q ×_t) denote the abelian category of sheaves on Q ×_t. An equivariant sheaf on Q ×_t with the action of is given by the following data: * A sheaf on Q ×_t, i.e., an object ∈_(Q ×_t). * A collection of isomorphisms e_a → T_a (a ∈) satisfying e_a+b=T_a e_b ∘ e_a. As usual, these objects form an abelian category. Let ^_(Q ×_t) be the abelian category of equivariant sheaves. This is a Grothendieck abelian category <cit.>. Note that (Q ×_t)= ^(Q ×_t) for the non-equivariant category. For an inclusion of subgroups ⊂⊂, we denote the functor forgetting equivariant structures by _^(Q ×_t)→^(Q ×_t). We also use the notation _. For an object of ^(Q ×_t), we define its microsupport by ()(()), where on the right-hand side is the usual microsupport. We write τ for the cotangent coordinate of t and write {τ > 0} for the subset of T^Q× T^_t defined by τ >0. For ∈^(Q ×_t), we also define _τ>0() () ∩{τ >0}. We write {τ≤ 0} for the subset of T^Q× T^_t defined by τ≤ 0. We consider the full subcategory ^_τ≤ 0∈^(Q×_t)()⊂τ≤ 0 and set _τ>0^(Q ×_t)^(Q ×_t)/^_τ≤ 0. * When = the trivial group, we denote the category ^_τ>0(Q ×_t) by _τ>0(Q ×_t), which is the (usual) non-equivariant version of the Tamarkin category <cit.>. * When is a non-trivial discrete group in with the Euclidean topology, it is isomorphic to as a subgroup of . Hence the category is ^_τ>0(Q ×_t) is equivalent to _τ>0(Q × S^1), which was used in <cit.>. * When =, the category is the one introduced in <cit.>. The functor _ induces a functor ^_τ>0(Q ×_t)→^_τ>0(Q ×_t), which will also be denoted by _. We note that _τ>0() = ()∩{τ>0} is well-defined for objects in _τ>0^(Q ×_t). §.§ Equivariant version of Tamarkin projector For , ∈_(Q ×_t), we set ⋆ m_! δ_Q ^-1(⊠), where δ_Q Q ×^2 → Q^2 ×^2 is the diagonal embedding on the first factor and m Q ×^2 → Q ×_t is the addition on the second factor. Suppose that is equipped with an equivariant structure. Then each structure morphism e_a induces an isomorphism e_a⋆𝕀⋆→ (T_a)⋆=T_a(⋆) for ∈_(Q ×_t), which gives an equivariant structure on ⋆. As a result, we get a functor (-)⋆_G ^_(Q ×_t)→^_(Q ×_t) for each ∈_(Q ×_t). By deriving it, we get an endofunctor of ^(Q ×_t). We denote it by the same symbol (-)⋆_G. The following is obvious. One has ((-)⋆_G)=(-)⋆. In particular, consider the case when _t≥ 0, which is the constant sheaf supported on (x, t) t≥ 0. By combining the above lemma with techniques developed by Tamarkin <cit.> and Guillermou–Schapira <cit.>, one can deduce ⋆_G _≥ 0≅ 0 for ∈^_τ≤ 0. Hence we get the induced functor (-)⋆_G _≥ 0_τ>0^(Q ×_t)→^(Q ×_t). The following is an analogue of Tamarkin's result. The functor (-)⋆_G_≥ 0_τ>0^(Q ×_t)→^(Q ×_t) is fully faithful left adjoint of the quotient functor ^(Q ×_t)→^_τ>0(Q ×_t). The essential image coincides with the left orthogonal ^⊥_τ≤ 0^. §.§ Monoidal operations We consider equivariant operations, which is an adaptation of the materials in <cit.>. Here, we briefly recall them. We first recall some basic operations for equivariant sheaves. For details, we refer to <cit.>. Let G be a group and X_1, X_2 be G-spaces. Let f X_1→ X_2 be a G-map. Then we have functors f_, f_! ^G(X_1)→^G(X_2), f^-1, f^! ^G(X_2)→^G(X_1). For these functors, usual adjunctions hold. We can also define tensor and internal hom. Let ϕ G→ H be a surjective group homomorphism and Y be an H-space. Then G acts on H through ϕ. By setting K to be the kernel of ϕ, we have the invariant functor (-)^K^G(Y)→^H(Y) and the coinvariant functor (-)_K^G(Y)→^H(Y). If one has an H-equivariant sheaf on Y, it can also be considered as a G-equivariant sheaf, which gives a functor ι^ϕ^H(Y)→^G(Y). The functor ι^ϕ is the right adjoint of (-)_K and the left adjoint of (-)^K. Let us go back to our particular situations. We consider Q ×^2 on which ^2 acts by the addition on each component. Through the addition m ^2 →, the group ^2 also acts on Q ×_t. The kernel of the map m ^2 → is the anti-diagonal Δ_a{(t, -t) ∈×}. We also consider the addition map m Q ×^2 → Q ×_t on _t-factors. We then have a functor m_!^^2(Q ×^2)→^^2(Q ×_t). By combining it with the coinvariant functor (-)_Δ_a^^2(Q ×_t)→^(Q ×_t), we define m_!^Δ_a (-)_Δ_a∘ m_!^^2(Q ×^2)→^(Q ×_t). We have the right adjoint of m_!^Δ_a: m^!_Δ_a m^!∘ι^m^(Q ×_t) →^^2(Q ×^2). For i=1,2, let p_i Q ×^2 → Q ×_t be the i-th projection. We also have the corresponding projection q_i^2 →. We then have p_i^ q_i (-)^ q_i∘ p_i^^2(Q ×^2) →^(Q ×_t) and p_i^-1 p_i^-1∘ι^q_i^(Q ×_t) →^^2(Q ×^2). Combining the above statements, we deduce: The functor p_i^-1 is the left adjoint of p_i^ q_i. For objects , ∈^(Q ×_t), we set, ⋆_ m_!^Δ_a(p_1^-1⊗ p_2^-1). For , ∈^(Q ×_t), we also set ^⋆_(,) p_1^ q_1_* _Q ×^2(p_2^-1, m^!_Δ_a). This is the right adjoint of ⋆_. The functors ⋆_ and ^⋆_ descend to functors on _>0^(Q ×_t). We also introduce the relative operation. Suppose ⊂. For ∈^(Q×_t) and ∈^(Q×_t), we can consider ⋆_. For the same reason as in defining ⋆_G, this object can be viewed as an object in ^(Q×_t). Hence we get a functor, (-)⋆_ (-)^(Q×_t)×^(Q×_t)→^(Q×_t), (-)⋆_ (-)^(Q×_t)×^(Q×_t)→^(Q×_t). In particular, we have ⋆_=⋆_G and ⋆_=⋆_. §.§ Equivariant and non-equivariant We have introduced the forgetful functor _^_τ>0(Q ×_t)→_τ>0^(Q ×_t). The left adjoint of this functor is given by ^L_ (-)⋆_⊕_c∈_t≥ c_τ>0^(Q ×_t)→^_τ>0(Q ×_t). For any 0<a∈ and ∈_τ>0(Q ×_t), we have a morphism T^a→ T_a. We have an analogous morphism T^a→ T_a for any a>0 and ∈^_τ>0(Q ×_t). For these morphisms, we have the following compatibility. One has _^L(T^a)=T^a. We name a particularly important object: 1_μ⊕_c∈_t≥ c =_^L(⊕_c∈_t≥ c) ∈^_τ>0(Q ×_t). With this notation, we have _^L = (-) ⋆_ 1_μ. §.§ Novikov ring action In this section, we see that the Novikov ring acts on _τ>0^(Q ×_t). First, we mention the following lemma, which is easily proved by an argument in <cit.>. The endofunctor (-)⋆_ 1_μ on ^_τ>0(Q ×_t) is naturally isomorphic to 𝕀. We denote the submonoid of the non-negative elements of by _≥ 0. We set Λ_0^_c →∞[_≥ 0]/T^c[_≥ 0]. This is the Novikov ring Λ_0 introduced by Fukaya–Oh–Ohta–Ono <cit.> when =. The ring Λ_0^ for a general can be regarded as the counterpart of -gappedness in <cit.>. The indeterminate corresponding to a∈_≥ 0 will be denoted by T^a. Let Λ_0^+ denote the unique maximal ideal of the Novikov ring Λ_0^. The quotient Λ_0^/Λ_0^+ is isomorphic to . If Q is a singleton, we have an isomorphism (⊕_c∈_≥ c)≅Λ_0^. For general Q, we have a morphism Λ^_0 →((-)⋆_1_μ) (see <cit.>). Hence we get the following homomorphism: Λ_0^→((-)⋆_1_μ)→(⋆_1_μ , ⋆_ 1_μ)=(, ), where the last equality follows from <ref>. Hence we have: The category _τ>0^(Q ×_t) is Λ_0^-linear. § MICROLOCAL CATEGORY In this section, we introduce our microlocal category over the Novikov ring. §.§ Doubling variable The doubling variable was originally used by Guillermou <cit.> as an auxiliary variable to construct sheaf quantization. Later, Guillermou <cit.> and Asano–Ike <cit.> used it crucially to consider sheaf quantization of obstructed Lagrangians. In Nadler–Shende <cit.>, it is also essential to have relative doubling. In what follows, until the end of the paper, let M be a manifold. Moreover, we let c be a real positive number or ∞ and denote the open interval (-∞, c) with the standard coordinate u by _u<c. The cotangent coordinate of T^_u<c will be denoted by υ. Let A be a subset of T^M× T^_τ>0_t invariant under the -translation along _t. The doubling AA of A is defined as AA AA_h∪ AA_t⊂ T^M × T^_u<c× T^_τ>0_t, where AA_h (p,u,0,t,τ) (p, t, τ) ∈ A, u ≥ 0 ⊂ T^M × T^_u<c× T^_τ >0_t, AA_t (p, u, υ, t, τ) (p, t, τ) ∈ A, u ≥ 0, υ=-τ⊂ T^M × T^_u<c× T^_τ>0_t. We call AA_h (resp. AA_t) the horizontal (resp. tilted) component of the doubling. A subset B of T^M × T^_u<c× T^_τ>0_t is said to be a doubling if B=AA for some A ⊂ T^M × T^_τ>0_t that is invariant under the translation. A subset B of T^M × T^_u<c× T^_τ>0_t is said to be a weak doubling if B⊂ AA for some A ⊂ T^M × T^_τ>0_t that is invariant under the -translation. §.§ Microlocal category over the Novikov ring We start with the notion of doubling movies. * An object of _τ>0^(M×_u<c×_t) is said to be a doubling movie if \|_u≤ 0=0 and _τ>0(\|_u>0) is a doubling. * An object of _τ>0^(M×_u<c×_t) is said to be a weak doubling movie if \|_u≤ 0=0 and _τ>0(\|_u>0) is a weak doubling. Now we define our main microlocal category and non-conic microsupport of an object of the category. The subcategory of _τ>0^(M×_u<c×_t) spanned by the weak doubling movies is denoted by μ^(T^M;u<c). We call it the microlocal category of T^M. For an object of μ^(T^M;u<c), we set ()⋃_u_0∈_u<cρ(_τ>0(\|_u_0)) ⊂ T^M, where ρ T^M× T^_τ >0_t→ T^M;(x, ξ, t, τ)↦ (x, ξ/τ). Let A' be a closed subset of T^M. Then ()⊂ A' if and only if _τ>0(\|_u>0)⊂ρ^-1(A')ρ^-1(A') for ∈μ^(T^M; u<c). Suppose _τ>0(\|_u>0)⊂ρ^-1(A')ρ^-1(A'). By the definition of the doubling, the inclusion of u=u_0 is non-characteristic for ρ^-1(A')ρ^-1(A'). Hence we have ρ(_τ>0(\|_u_0))⊂ A', which implies ()⊂ A'. Conversely, suppose ()⊂ A'. By the definition of , we have ρ(_τ>0(\|_u_0))⊂ A'. Since _τ>0(\|_u>0) is a weak doubling, we have _τ>0(\|_u>0)⊂ρ^-1(A')ρ^-1(A'). The category μ^(T^M;u<c) is a triangulated category with arbitrary coproducts. By the definition and the microsupport triangle inequality, we can see that μ^(T^M;u<c) is a triangulated category. The cocompleteness is also clear. For 0<c<c', we have the canonical functor r_c'cμ^(T^M;u<c')→μ^(T^M;u<c). We sometimes denote it by r_c for simplicity. For 0<c<c', we denote the inclusion M×_u<c×_t↪ M×_u<c'×_t by ι_cc'. Let (μ^(T^M;u<c)) be the category of modules over μ^(T^M;u<c). For an object ∈μ^(T^M;u<c), we get an object of (μ^(T^M;u<c')) as __τ>0^(M×_u×_t)(-, ι_cc')μ^(T^M;u<c')→(Λ_0). This structure seems to be important, but we never use it in this paper. Our category μ(T^M, u<c) does not have an obvious convolution unit; ^L∘ C_u(1_μ) is not a unit, where ^L∘ C_u will be defined later in <ref>. §.§ Without doubling variable It is sometimes convenient to consider the situation where u does not exist. In particular, for many purposes (for example, the situation where one focuses on unobstructed branes), we can forget u. For this purpose, we prepare some notation for the category without u. We define μ^(T^M)^_τ>0(M×_t). For an object ∈μ^(T^M), we also define ()ρ(_τ>0()) ⊂ T^M. For each 0<u∈_u<c, we set i_u M×_t ↪ M ×_u<c×_t; (x, t) ↦ (x,u,t). Then we have the specialization functor _u i_u^-1μ^(T^M;u<c)→μ^(T^M), which is Λ_0-linear. When c<∞, we also consider the inclusion j_c M ×_u< c×_t → M ×_u≤ c×_t. With these maps, we also set _c i_c^-1j_cμ^(T^M;u<c)→μ^(T^M), which is a kind of nearby cycle. When c=∞, we consider the map j_∞ M ×_u ×_t ↪ M × (-∞, ∞] ×_t associated with the one-point partial compactification _u ↪ (-∞, ∞]. With this map, we set _∞ i_∞^-1j_∞μ^(T^M;u<∞)→μ^(T^M), where i_∞ M ×_t ↪ M × (-∞,∞] ×_t; (x,t) ↦ (x,∞,t). Let A' be a closed subset of T^M. For ∈μ^(T^M; u<c) with (\|_u>0)⊂ρ^-1(A')ρ^-1(A'), we have (_u())⊂ A' for any u≤ c. It follows from the standard microsupport estimate (see <cit.>). §.§ /N-grading setup As we will see in the next section, to have an object corresponding to a Lagrangian submanifold, one needs the existence of a brane structure. One can actually relax the existence of a brane structure by considering the orbit category, which we will define below. Let N ∈_≥ 2. We define the category μ^, /N(T^M;u<c) as follows: * The set of objects is the same as μ^(T^M;u<c). For objects , ∈μ^, /N(T^M;u<c) and i=1,…,N, we set ^i_μ^,/N(T^M;u<c)(, )⊕_j∈(, [i-jN])e^-j, where e is an indeterminate. Then μ^, /N(T^M;u<c) is a Λ_0[e^±]-enriched category. We can define μ^(T^M) in a similar manner. We regard μ^(T^M;u<c) and μ^(T^M) introduced in the previous sections as the case N=∞. §.§ Over the Novikov field Since Λ^_0 is an integral domain, we can take its fraction field. We denote it by Λ^, which is called the (universal) Novikov field. Given a Λ_0^-linear dg-category , we can define the base-changed category ⊗_Λ^_0Λ^ as follows: * The set of objects (⊗_Λ_0^Λ^) is (). * For , ∈(⊗_Λ_0^Λ^), we set _⊗_Λ_0^Λ^(, )_(, )⊗_Λ_0^Λ^. The differentials and compositions are naturally induced. The resulting category is a “filtered" Λ^-linear dg-category: We set F^ϵ_⊗_Λ_0^Λ^(, ) to be the image of T^ϵ_(, ) under the natural morphism _(, )→_⊗_Λ_0^Λ^(, ). § SHEAF QUANTIZATIONS §.§ Microsheaves along Lagrangian In this subsection, we explain microsheaves along a Lagrangian immersion. First, we recall general notation. Let X be a manifold. For an open subset of U of T^X, we set (_X;U) (_X)/∈(_X)()⊂ T^X ∖ U. This forms a prestack and the stackification is called the Kashiwara–Schapira stack. We can consider the subsheaf spanned by the objects supported on Y⊂ T^X. It is a subsheaf supported on Y, which we denote by _Y. Let λ be the Liouville form of T^M. Let i_L L ↬ T^M be a Lagrangian immersion. We set i_L(L) for the simplification. We can locally take a primitive f U∩→ of λ\|_U, where U is an open subset of T^M. We set U_f (x, ξ, t, τ)∈ T^M× T^_τ>0_t (x, ξ/τ)∈, t=-f(x, ξ/τ). Then we can define _U_f by the argument above. Note that U_f is conic with respect to the scaling, and we have an isomorphism U_f/_>0≅ U∩. Under this isomorphism, _U_f descends to a sheaf on _U∩ and does not depend on f, but locally depends only on U∩. Then we can glue them over . We denote it by _. The definition of our _ is different from that of Jin's _L <cit.> if L^ has self-intersection. As an effect, <ref> below is stated in a form slightly different from <cit.>. We introduce some notation following Tamarkin. For a closed subset A'⊂ T^M, we set μ_A'(T^M;u<c) ∈μ(T^M;u<c) ()⊂ A' and μ_A'(T^M) ∈μ(T^M) ()⊂ A'. See also <ref>. In <cit.>, the author constructed the microlocalization functor μ_μ^_(T^M)→_(), by mimicking the known constructions in the exact case. We explain how to construct the microlocalization functor in a general -equivariant case. We first consider the restriction functor: μ^_(T^M)→_ρ^-1()(ρ^-1()). By restricting it to a local conic lift U_f⊂ρ^-1(), we have _ρ^-1()(ρ^-1())→_ρ^-1()\|_U_f(U_f). In <cit.>, it is proved that the image of this functor is in _U_f(U_f). Hence we further have μ^_(T^M)→∏_c∈/_U_f+c(U_f+c)→_U_f(U_f), where the last functor is taken by the direct product using the canonical equivalences along _U_f+c(U_f+c)→_U_f(U_f). Using the equivariant structure, we get a functor μ_μ^_(T^M) →_L^(L^). This is the desired microlocalization functor. As an analogue, we can construct a functor μ^_μ_(T^M;u<c) →_() as follows: For some u> 0, we take a contractible neighborhood U of u in (0, ∞). The microlocalization along AA_h\|_U as in loc. cit. gives an object of _× U(× U) along × U. Since U is contractible, it equivalently gives a functor to _(). Let L_1 and L_2 be two Lagrangian immersions and let _i∈μ^__i(T^M). Then for i=1,2, we have μ__i(_i)∈__i(_i). Since __i(_i)⊂__1∪_2(_1∪_2), we can consider μ__i(_i) as an object of the latter category. We then set ^(_1, _2)___1∪_2(μ__1(_1), μ__2(_2)), which is a sheaf on L_1∩ L_2. We can similarly construct /N-version of _, which will be denoted by ^/N_. We can also construct μ_^/Nμ_^/N(T^M;u<c) →^/N_() in a similar manner. We regard the usual case as N=∞. In the following, we mostly set to be and omit from the notation. This is only for the simplicity of notation, and all the results hold for other 's. In <ref>, we use different 's explicitly, where we would like to compare our sheaf quantization with the previously existing studies. §.§ Lagrangian branes In this subsection, we recall the notion of a Lagrangian brane. Let A' be a closed subset of T^M. We say that A' is end-conic if there exists a disk bundle D^M in T^M cut out by some fiber metric such that A' ∖ D^M is conic, i.e., stable under the scaling action of _≥ 1. For an end-conic subset A', by taking a disk bundle D^M so that A' ∖ D^M is conic, we set ∂ A' (_>0· (A' ∖ D^M))/_>0⊂∂ T^M, the boundary of A'. There exists an isomorphism S^M→∂ T^M induced by the projection, where S^M is the boundary of D^M. By this identification, we sometimes consider ∂ A' as a subset of T^M. We make the following assumption for Lagrangian immersion hereafter in this paper. Any Lagrangian immersion i_L L↬ T^M in this paper is assumed to satisfy the following. (1) The image is end-conic. (2) The image π() under the projection π T^M→ M is compact. (3) The immersion i_L has clean self-intersection of ≥ 1. That is, the following hold. The fiber product L ×_T^M L is a smooth submanifold of L × L with ≤ n-1 except for the diagonal. For (p_1, p_2) ∈ L ×_T^M L, we have T_(p_1,p_2)(L ×_T^M L) = (V,W) ∈ T_p_1L × T_p_2L (d_p_1i_L)(V) = (d_p_2i_L)(W) . The assumption of clean self-intersection of ≥ 1 is not essential. If i_L has codimension 0 clean self-intersection, we can treat it as an embedded Lagrangian brane with higher rank local systems. For example, let L be an embedded Lagrangian and (α_i, b_i, _i) (i=1,2) be brane structures on L, which will be defined below. Then the immersed Lagrangian brane given by L⊔ L→ T^M with the brane structure _i=1,2(α_i, b_i, _i) can be regarded as an embedded Lagrangian brane (L, α_1, b_1, (_1⊕_2⊗ (b_2-b_1)[α_2-α_1]), where (b_2-b_1) is the principal (1)-bundle and α_2-α_1∈. Since TT^M is a symplectic vector bundle, we have the associated Lagrangian Grassmannian bundle LGr(T^M). The Lagrangian distribution given by the fiber directions gives a section s_f of LGr(T^M)→ T^M. By using the image of the section s_f as the base point set, we can take the fiberwise N-fold covering (N∈_≥ 2∪{∞}) of LGr(T^M). We denote it by LGr^N(T^M). Let i_L L↬ T^M be a Lagrangian immersion. A /N-grading of L is a lift of the Gauss map L→ i_L^LGr(T^M) to α L→ i_L^* LGr^N(T^M). A relative Pin structure on a Lagrangian immersion i_L L↬ T^M is a null-homotopy of i_L^* w_2(TT^M)-w_2(L). Fix N ∈_≥ 2∪{∞}. An (immersed) Lagrangian brane is a tuple L=(L, α , b, ), where i_L L↬ T^M is a Lagrangian immersion, α is a /N-grading of L, * b is a relative Pin structure of L, and * is a (shifted) local system over L. A Lagrangian brane L is said to be a simple if is a rank 1 local system. When is a trivial rank 1 local system, we simply say L=(L, α, b) is a Lagrangian brane. For a Lagrangian brane L=(L, α, b, ), we denote by L^0 the associated simple Lagrangian brane (L, α, b) with the trivial rank 1 local system. Lagrangian brane data can be defined for more general coefficients (i.e., symmetric monoidal presentable stable ∞-category). See <cit.> for further information. The definition above is for the case is the derived category of -modules. Given a Lagrangian immersion with a /N-grading α and a relative Pin structure b, there exists a faithful embedding ^/N(L)↪^/N_(), where the left-hand side is the category of derived /N-graded local systems. The image of the embedding is denoted by _L^/N(L). If L is an embedded Lagrangian, we have _L^/N(L)≅_^/N(). For a Lagrangian immersion i_L L↬ T^M, we set μ^/N_L(T^M; u<c) (μ_^/N)^-1(_L^/N(L)), where μ_^/Nμ_^/N(T^M;u<c) →^/N_() is the microlocalization defined in <ref>. If L is embedded, it is compatible with the definition of μ_L(T^M; u<c) introduced in the previous section. Given a simple Lagrangian brane L=(L, α, b), by composing the functor given above with μ_L^Im, we obtain a functor μ_Lμ^/N_L(T^M;u<c) →^/N(L). If L is further equipped with a rank 1 local system (i.e., L (L,α,b,) is a simple Lagrangian brane), we set μ_Lμ_L^0⊗^-1μ_L^/N(T^M;u<c) →^/N(L). We can similarly define μ_Lμ_L^/N(T^M)→^/N(L). An object of μ^/N(T^M;u<c) is said to be smooth if () is an embedded Lagrangian and μ_(L, α,b)() is of finite rank for some (α,b). An object of μ^/N(T^M;u<c) is said to be holonomic if () is possibly singular Lagrangian and μ_(L^sm, α,b)() is of finite rank for some (α,b) on the smooth part L^sm. The subcategory of μ^/N(T^M;u<c) spanned by the holonomic objects (resp. smooth objects) is denoted by μ^/N, hol(T^M;u<c) (resp. μ^/N, sm(T^M;u<c)). We call it the holonomic (resp. smooth holonomic) microlocal category of T^M. The inclusion μ^/N, sm(T^M;u<c)↪μ^/N, hol(T^M;u<c) induces an equivalence μ^/N, sm(T^M;u<c)⊗_Λ_0Λμ^/N, hol(T^M;u<c)⊗_Λ_0Λ over the Novikov field. There exists an equivalence of categories between μ^/N, sm(T^M) and the infinitesimally wrapped /N-graded Fukaya category of end-conic immersed Lagrangians over Λ_0. §.§ Sheaf quantization and brane structure In the following, we fix N ∈_≥ 2∪{∞}, and we suppress the notation /N. We would like to describe sheaf quantization more precisely. Let L=(L,α, b, ) be a Lagrangian brane. An object ∈μ^_L(T^M;u<c) is said to be a sheaf quantization of a Lagrangian immersion L over Λ_0^/T^cΛ_0^ if μ_(L, α,b)() is of finite rank for some (α,b). More precisely, we define: An object ∈μ^(T^M;u<c) (resp. ∈μ^(T^M)) is said to be a sheaf quantization of L over Λ_0^/T^cΛ_0^ if (1) ∈μ^_L(T^M;u<c) (resp. ∈μ^_L(T^M)) and (2) μ_L^0() ≅. We say that a sheaf quantization is simple if is of rank 1. We remark that for 0<c<c', if ∈μ^(T^M;u<c') is a sheaf quantization of L (resp. L), the restriction r_c'c() is a sheaf quantization of L (resp. L) in μ^(T^M;u<c). We also give the relationship between sheaf quantizations over Λ_0^ and Λ_0^. Let be a sheaf quantization of L over Λ_0^. Then _^L() is a sheaf quantization of L over Λ_0^ for ⊂. We shall see the following relationship between sheaf quantization with/without the u-variable. Let p_u M ×_u ×_t → M×_t be the projection forgetting _u. We consider the sheaf _t≥ u≥ 0∈(M×_u×_t). We have the map _t≥ 0→_t≥ u≥ 0 that corresponds to 𝕀 under the adjunction isomorphism (_t≥ 0, _t≥ u≥ 0)≅(_t≥ u≥ 0, _t≥ u≥ 0). Then we have the induced map p_u^-1=p_u^-1⋆_t≥ 0→ p_u^-1⋆_t≥ u≥ 0. We denote the fiber (i.e., [-1]-shifted cone) of the above map by C_u(). Let ∈μ^(T^M) be a sheaf quantization of L. Then the object ^L(C_u())∈μ(T^M;u<∞) is a sheaf quantization of L. Moreover, ^L∘ C_u gives a faithful embedding μ_L^(T^M)↪μ^_L(T^M;u<∞). The first part is easy. The composition _∞∘ (^L∘ C_u) is the identity of μ_L^(T^M). Hence the faithfulness follows. In several cases, the existence of sheaf quantizations is known, e.g., Guillermou <cit.>, Jin–Treumann <cit.> for a large class of exact Lagrangian submanifolds, Asano–Ike <cit.> for strongly rational Lagrangian immersions. From their results and <ref>, we can deduce the existence of sheaf quantizations in our sense. In the next section, we will (partly) generalize their results. §.§ Existence of low-energy sheaf quantization of Lagrangian immersions Let L=(L, α, b, ) be a Lagrangian brane, where the underlying Lagrangian immersion i_L L→ T^M satisfies <ref>. Firstly, we would like to discuss the existence of small sheaf quantizations. In this subsection, we write M ×_t ×_u instead of M ×_u ×_t for notational convenience. Consider the pull-back of the Liouville form i_L^λ. We can take a primitive f of i_L^λ as a multi-valued function on L. We denote the domain of f as a single-valued function by L, which is a covering of L. Hence we have the associated immersive map i_LL→ T^M. We set L_f (x, ξ, t, τ)∈ T^M× T_τ>0^_t, ∃p̃∈L s.t. -f(p̃)=t and i_L(p̃)=(x, ξ/τ). We denote the pull-back of ∂ L under the projection T^M T^_MM→∂ T^M by _>0·∂ L. We then set L_f L_f∪_>0·∂ L× T^__t_t ⊂ T^M × T^_t. The cone closure of the doubling L_f L_f can be perturbed into the cusp doubling L_f^≺, which was introduced in <cit.>. The advantage of the cusp doubling is that it is finite over the base if L is so. We define the period group _L by _L i_L^λ(H_1(L, ))⊂. Note that L_f carries a canonical _L-action. If the closure of _L is not in the Euclidean topology, we set c_L,mininf c∈_>0 c∈_L. We denote the sheaf of category of sheaves on M×_t×_u whose microsupport included in L_f^≺ by _L_f^≺. The substack whose objects are 0 on _u<0 is denoted by _L_f^≺;0. We denote the projection T^M× T^_t× T^_u→ M×_t×_u by π. For every end-conic Lagrangian brane L, there exists a sheaf quantization of _L∈μ^_L(T^M; u<c) of L for a sufficiently small c. We can take c as c_L, min if it is defined. The case when _L={0}, the argument is essentially known <cit.>. See also the last part of the proof. Hence we assume ⊂_L below. The category ^_L_τ>0(M×_t×_u) is equivalent to ^_L/_τ>0(M× S^1_t×_u). We denote the projection along M×_t×_u→ M× S^1_t×_u of L_f (resp. L_f^≺) by L_f^S^1 (resp. L_f^S^1^≺), which are closed by the end-conic assumption. One can perturb the Legendrian boundary of L_f^S^1 in ∂ T^(M× S^1_t) into a finite position. By the usual argument using <cit.> and the microlocal cut-off result <cit.>, we have an equivalence of sheaves of categories _L_f^S^1^≺;0→π__L_f^S^1^≺ along M× S^1_t×{u=0}. Note also that the natural _L/-actions on both sides are compatible under this equivalence. A Lagrangian brane structure of L gives an _L-invariant global section of the right-hand side. Hence it gives a _L-invariant global section s of the left-hand side over M × S^1_t ×{u=0}. Next, we would like to lift the section of _L_f^S^1^≺;0(M ×_t ×{ u=0}) to a section over M ×_t × (-c,c), which gives an object of μ(T^M; u<c). Here we can adapt an argument similar to <cit.>. For any point q ∈π() × S^1 ×{u=0} we can take an open neighborhood V'_q and a section s'_q on V'_q such that s'\|_V'_q ∩ M × S^1 ×{0} = s'_q\|_V'_q ∩ M × S^1 ×{0}. Since π() is compact, we can take a finite subset J of π() × S^1 ×{u=0} such that {V'_j}_j ∈ J is an open covering. For any q ∈π() × S^1 ×{u=0}, the set J(q)={j ∈ J \| q ∈ V'_j } is finite. Hence, for any q ∈π() × S^1 ×{u=0}, we can find an open neighborhood W'_q = W”_q × (-c_q,c_q) such that s'_j\|_W'_q=s'_k\|_W'_q for any j,k ∈ J(q). Then { W'_q }_q ∈π() × S^1 is an open covering of π() × S^1 ×{u=0} and s'_j\|_W'_q=s'_k\|_W'_q for any j,k ∈ J(q). Again by the compactness of π(), we can take a finite subset J' of π() × S^1 ×{u=0} such that {W'_j }_j ∈ J' is an open covering of π() × S^1 ×{u=0}. If we set c=min_j ∈ J' c_j and U'_j=V”_j × (-c,c), then {U'_j}_j ∈ J is an open covering and s'_i\|_U'_i ∩ U'_j = s'_j\|_U'_i ∩ U'_j. Hence we get a section over [0, c) for some c. Incorporating the equivariant structure, this is a desired sheaf quantization. If c_L, min>0, we can go further. Namely, for 0<c<c_L,min, L_f∪ T_cL_f can be considered as an image under the contact translation flow of L_f∪ T_c'L_f for 0<c'<c. Hence the usual argument (cf. <cit.>) works and gives a sheaf quantization provided that there exists a non-trivial object in _L_f(L_f). Such a non-trivial object is given by the brane structure. This completes the proof. Suppose that L is an exact Lagrangian submanifold. Then _L= and c_L, min=∞. Hence we get a sheaf quantization in _τ>0(M ×_t ×_u). Specializing at ∞, we get a sheaf quantization in _τ>0(M×_t). The resulting object is nothing but the one constructed in Guillermou <cit.> and Jin–Treumann <cit.>. * Suppose that L is an strongly rational immersed Lagrangian in the sense of <cit.>. Then we find that _L ≅, and we can take c_L, min as the period in the sense of <cit.>. Then we get a sheaf quantization in ^(M ×_t ×_u<c_L, min)≅(M × S^1 ×_u<c_L, min), which recovers the result in <cit.>. §.§ u-translation and stupid extensions We would like to glue objects of μ(T^M;u<c) for different c's. For this purpose, we introduce some notation. For a positive real number c', we consider the translation map S_c' (-∞, c)→ (-∞, c+c'); a↦ a+c'. By pushing forward, we obtain the functor S_c'μ(T^M;u<c)→^_τ>0(M ×_u<c+c'×_t). We also define stupid extension functors. We consider the following map l_c+c', c_u < c+c'→_u≤ c; a↦ a if a<c c otherwise. Let j_c M ×_u<c×_t ↪ M ×_u≤ c×_t be the inclusion. We define a functor s_c' as s_c'μ(T^M;u<c) →^_τ>0(M ×_u<c+c'×_t); ↦ l_c+c', c^-1j_c. §.§ Abstract intersection Maslov indices In this subsection, we introduce the notion of abstract intersection Maslov indices and investigate their relationship to sheaf quantizations. We first recall the construction of the positive definite path. Let V be a symplectic vector space and ℓ be a Lagrangian subspace of V. Then the tangent space of the Lagrangian Grassmannian LGr(V) at ℓ is canonically identified with the space of quadratic forms on ℓ. The tangent cone spanned by the non-degenerate positive forms is denoted by C_0(ℓ). [Positive definite path, transverse case] Let ℓ_1, ℓ_2 be two transverse Lagrangians in a symplectic vector space V. A path c [0,1]→ LGr(V) from ℓ_1 to ℓ_2 is said to be positive definite if it satisfies the following: * c(0)=ℓ_1 and c(1)=ℓ_2, * c(t) is transverse with ℓ_1 for t>0, and * ċ(0)∈ C_0(ℓ_1). Such a path exists and is unique up to homotopy relative to ends. Let ℓ_1, ℓ_2 be two Lagrangians in a symplectic vector space V. Then (ℓ_1+ℓ_2)/ℓ_1∩ℓ_2 is naturally a symplectic vector space, and a Lagrangian ℓ containing ℓ_1∩ℓ_2 defines a Lagrangian in (ℓ_1+ℓ_2)/ℓ_1∩ℓ_2. [Positive definite path, clean case, cf. <cit.>] Let ℓ_1, ℓ_2 be two Lagrangians in a symplectic vector space V. A path c [0,1]→ LGr(V) from ℓ_1 to ℓ_2 is said to be positive definite if it satisfies the following: * c(0)=ℓ_1 and c(1)=ℓ_2, * c(t) contains ℓ_1∩ℓ_2 for every t, and * c(t)/ℓ_1∩ℓ_2 is a positive definite path in LGr((ℓ_1+ℓ_2)/ℓ_1∩ℓ_2) from ℓ_1/ℓ_1∩ℓ_2 to ℓ_2/ℓ_1∩ℓ_2 in the sense of <Ref>. Such a path exists is unique up to homotopy relative to ends. Now we recall the definition of the intersection Maslov index. Fix a base point * in LGr(V). Then the universal covering space LGr(V) is given by the based path space of LGr(V). Suppose that ℓ_1, ℓ_2∈LGr(V) are points over ℓ_1, ℓ_2∈ LGr(V), respectively. By connecting ℓ_1^-1 and ℓ_2 at , we get a path from ℓ_1 to ℓ_2. Moreover, by connecting it to a positive definite path from ℓ_2 to ℓ_1, we get a loop in LGr(V). By evaluating it by the Maslov cycle, we get a number, called the intersection Maslov index. Let L be a Lagrangian submanifold of T^M. Let be the coefficient category (cf. <ref>). We denote the classifying map of the Kashiwara–Schapira stack by c_L L→ U/O B(), where KS is the universal Kashiwara–Schapira map. A -brane data on L is a null-homotopy of the map c_L. We now give the definition of the abstract intersection Maslov index. We first consider the embedded case. Let L_1, L_2 be Lagrangian submanifolds intersecting cleanly in X with -brane data. On the intersection locus L_1∩ L_2, we have two maps c_L_i\|_L_1∩ L_2 (i=1,2) with null homotopies. On the intersection locus, we also have a (family of) positive definite path connecting TL_i\|_L_1∩ L_2 L_1∩ L_2→ U/O (i=1,2). By composing it with KS, we get a homotopy connecting c_L_i\|_L_1∩ L_2 (i=1,2). Moreover, by connecting the null homotopies of c_L_1\|_L_1 ∩ L_2 and c_L_2\|_L_1 ∩ L_2, we obtain a family of loops in B() parameterized by L_1∩ L_2: (L_1∩ L_2) × S^1→ B(). By the adjunction, we get a map m_L_1, L_2 L_1∩ L_2→(). We call the map m_L_1, L_2 the abstract intersection Maslov index. In the case =(), -brane data is what we call brane data with trivial local systems, and we have ()≅× B/2. We denote the grading of L_i by α_i and the relative Pin structure by b_i. By the construction, the part of m_L_1, L_2 going to is the usual intersection Maslov index, denoted by α_2-α_1. The second part going to B/2 gives a principal O(1)-bundle on L_1∩ L_2. We denote it by b_2-b_1. Let L_i (L_i, α_i, b_i, _i) (i=1,2) be Lagrangian branes intersecting cleanly. The sheaf-theoretic Maslov index local system _L_1, L_2 is defined as (_1,_2)⊗_(b_2-b_1)[α_2-α_1], which is a (shifted) local system on L_1∩ L_2. The following is a generalization of what is known in the transverse case <cit.>: Let L_i=(L_i, α_i, b_i, _i) (i=1,2) be embedded Lagrangian branes intersecting cleanly. Then each brane gives an object _L_i of _L_1∪ L_2(L_1∪ L_2). We have the following isomorphism ^(_L_1, _L_2)__L_1∪ L_2(L_1∪ L_2)(_L_1, _L_2)≅_L_1, L_2 in (L_1∩ L_2). We set L_12 L_1∩ L_2. Let L'_1 be a small positive definite perturbation of L_1 such that L_1∩ L_1'=L_12. Then one can import the brane structure of L_1 to L_1'. Then the abstract intersection Maslov index from L_1 to L_1' is trivial, and it is easy to see that the corresponding is the rank 1 constant sheaf. Hence we have done for the case of L_2=L_1'. Let us consider the general case. Topologically, L_1∪ L_1'≅ L_1∪ L_2 around L_12. Moreover, _L_1∪ L_1'≅_L_1∪ L_2 locally around L_12. Under this equivalence, we import the microsheaf _L_2∈_L_1∪ L_2(L_1∪ L_2) to _L_2∈_L_1'(L_1')⊂_L_1∪ L_1'(L_1∪ L_1'). Then we have μ_L_1'(_L_2)≅π_L_1'^-1_L_1, L_2, where π_L_1' L_1'→ L_1'∩ L_1 is the projection. By combining this with the first half of the proof, we get the conclusion. We now consider the immersed case. Let L_1, L_2 be cleanly intersecting immersed Lagrangian branes. Each connected component of L_1×_T^ML_2 can be considered as a clean intersection of embedded Lagrangian branes. Hence we get an object _L_1, L_2 in (L_1×_T^M L_2). We denote the canonical map L_1×_T^ML_2→ T^M by i_12. Let L_i=(L_i, α_i, b_i, _i) (i=1,2) be Lagrangian branes intersecting cleanly. Then each brane gives an object _L_i of _L^Im_1∪ L^Im_2(L_1^Im∪ L_2^Im). We have the following isomorphism ^(_L_1, _L_2)__L^Im_1∪ L^Im_2(L_1^Im∪ L_2^Im)(_L_1, _L_2)≅ i_12_L_1, L_2. This is an isomorphism of sheaves on L_1^Im∩ L_2^Im. For the case of self-intersecting Lagrangian L, we get _L, L on the self-intersection locus L_SI L×_TML. We conjecture that _SI_L, L is equal to Θ^- in immersed Floer theory <cit.>. In the situation of <cit.>, it is easy to see that this conjecture is true. §.§ Microlocal extension classes Let L=(L, α, b, ) be a Lagrangian brane and ∈μ(T^M; u<c) be a simple sheaf quantization of L. For any 0<u<c, the microlocalization of _u() along L is ⊕[-1]. We set A=ρ^-1(L). Perturbing AA slightly by a Hamiltonian isotopy, we can make the configuration at u=0 be of the cusp type as in Nadler–Shende <cit.>. We put the brane structure (α, b, ) on AA_h and extend it to AA_t. When crossing the cusp, the Maslov grading changes by 1. This completes the proof. For c_1<c, _1 r_cc_1() and _2 S_-c_1(s_c-c_1r_cc_1()→)[-1] are sheaf quantizations of L such that μ_L(_i)≅_L. Then ∈μ(T^M;u<c) is an extension of s_c-c_1_1 by S_c_1_2 such that μ_L()≅_L. For c>c_2>c_1, we have the following exact triangle: _c_2-c_1(_2)→_c_2()→_c_1(_1)→. Consider the microlocalization along L in μ(T^M). Then we have the associated exact triangle: μ_L(_c_2-c_1(_2))→μ_L(_c_2())→μ_L(_c_1(_1))→, which is isomorphic to ⊕[-1]→⊕[-1]→⊕[-1]→ in (L) by <ref>. The space of extensions is described as H^1^∙__L^(L^)(μ_L(_c_1(_1)), μ_L(_c_2-c_1(_2)))≅ ^1(, [-1])⊕ H^1(L_SI;_SI[-1]) ⊕^1(, ) ⊕ H^1(L_SI;_SI) ⊕^1([-1], [-1]) ⊕ H^1(L_SI;_SI) ⊕^1([-1], ) ⊕ H^1(L_SI;_SI[1]). Note that the leftmost [-1] in (<ref>) comes from the tilted component of the doubling. The right ⊕[-1] and the leftmost comes from the horizontal component. Hence, for the third and fourth lines of the left-hand side of (<ref>), the microlocalization of the extension class does not have a non-trivial component there. Hence the object μ_L(_c()) is classified by an element of the first and second lines of the right-hand side of (<ref>). We call this element the microlocal extension class of at c_1. The microlocal extension class can be normalized into the form (x=1,y=0,z,w) ∈ ^1(, [-1])⊕ H^1(L_SI;_SI[-1]) ⊕^1(, ) ⊕ H^1(L_SI;_SI). We view (x,y) as a morphism →. If (x,y) is not an isomorphism, the corresponding extension is not ⊕[-1], which is a contradiction. We can normalize it as (x,y)=𝕀=(1,0). By this lemma, in the following, we regard the microlocal extension class as an element of ^1(, ) ⊕ H^1(L_SI;_SI)=H^1^∙__L^(L^)(μ_L(), μ_L()). We say that is the standard extension at c_1 if the microlocal extension class is zero. Namely, z=w=0 in the expression of <ref>. For any sheaf quantizations _1, _2∈μ(T^M;u<c) of a given brane structure L for some c∈_>0∪∞, there exists c' such that r_cc'(_1)≅ r_cc'(_2). Since they are isomorphic on M×_t×{u=0} as global sections of sheaves of (<ref>), we can lift the isomorphism for small c'. This completes the proof. §.§ Standard sheaf quantization of non-exact Lagrangians Now we shall discuss “standard" sheaf quantizations. For this purpose, we will inspect the above construction of sheaf quantizations more. Let L=(L, α, b, ) be a non-exact(=not necessarily exact) immersed Lagrangian brane in T^M. We first perturb L to be finite over M. Fix a sheaf quantization ∈μ_L(T^M;u<c_0) of L whose existence is proved in <ref>. Now we also construct local sheaf quantizations. Let ={ U_i }_i be a finite open covering of M such that L_iπ^-1(U_i)∩ L is component-wise contractible for every U_i∈. In particular, L_i is exact. Then we can take a single-valued primitive f_i of λ on L_i. Then we have the lift L_f_i (see (<ref>)). By applying <ref>, we get a sheaf quantization for small c_i, which we denote by ^0_L_i. Then _L_i⊕_c∈T_c_L_i^0 is a sheaf quantization of L_i in μ_L_i(T^U_i; u<c_i). By <ref>, there exists c_i' such that r_c_ic_i'(_L_i)≅ r_c_ic_i'(\|_U_i). We set c_Lmin_ic_i'. Then we obtain r_c_L() by gluing r_c_L(_L_i)'s together. In the following, we apply r_c_L to all the sheaves, and remove the symbol r_c_L. We denote by _L_ij (resp. ^0_L_ij) the restriction of _L_i (resp. ^0_L_i) to U_i∩ U_j. A gluing morphism of is represented as an automorphism of _L_ij. Here we have _μ(T^U_i; u<c_L)(_L_ij, _L_ij) =__τ>0(M ×_u<c_L×_t)(^0_L_ij, ⊕_c'∈T_c'^0_L_ij) ≅__τ>0(M ×_u<c_L×_t)(^0_L_ij, ⊕_0≤ c' <c_LT_c'^0_L_ij). Since the support of ^0_L_ij is proper in the direction t, the last space is isomorphic to ⊕_0≤ c' <c_L__τ>0(M ×_u<c_L×_t)(^0_L_ij, T_c'^0_L_ij). Now we have the set of gluing morphisms g_ij= {g_ij^c'}_c'∈⊕_0≤ c' <c_L__τ>0(M ×_u<c_L×_t)(^0_L_ij, T_c'^0_L_ij) associated with the sheaf quantization . The family {g_ij}_i,j is said to be the Čech data of . Let us consider the Kashiwara–Schapira stack _L(-, Λ_0/T^cΛ_0) over Λ_0/T^cΛ_0. The above construction says that there exists a sufficiently small c such that μ_L(T^M; u<c) is equivalent to _L(L, Λ_0/T^cΛ_0) for a Lagrangian submanifold L. We now define c-standard sheaf quantizations: For a Lagrangian brane L and c∈_>0, an object ∈μ(T^M;u<c) is said to be a c-standard sheaf quantization if it satisfies the following: (1) is a sheaf quantization of L and (2) g_ij^c=0 for c>0. Note that the Čech cohomology class of g_ij^0 is uniquely determined by the brane data. There exists a c-standard sheaf quantization of a given Lagrangian brane L for sufficiently small c. In the above discussion, we can take 0<c<min c'{ g^c'_ij}_i,j≠ 0. For every sheaf quantization ∈μ(T^M;u<c) of a given brane structure for some c∈_>0∪∞, there exists c' such that r_cc'() is c-standard. It follows from <ref>. From the viewpoint of <ref>, we can regard the c-standard sheaf quantization as follows: The obvious ring inclusion ↪Λ_0/T^cΛ_0 induces a morphism _L(L, )→_L(L, Λ_0/T^cΛ_0). Then the brane structure _L(L, )≅(L) gives a canonical object in _L(L, Λ_0/T^cΛ_0) as the image of the rank 1 constant sheaf under the composition (L)≅_L(L, )→_L(L, Λ_0/T^cΛ_0) of the functors defined above. This canonical object is nothing but the c-standard sheaf quantization. §.§ Twisted complex presentations We first recall the notion of a twisted complex. Let be a Λ_0-linear dg-category. A (simple one-sided) twisted complex is a tuple ({V_i}_i∈,{f_ij}_i > j), where * V_i is an object of and * f_ij is a morphism V_i→ V_j of degree i-j+1 such that df_ij+∑_kf_kj∘ f_ik=0 holds for any i>j. There exists a more general version of the notion of a twisted complex. See <cit.>. The twisted complexes form a dg-category . The following is fundamental: Suppose is pretriangulated. Then there exists a fully faithful functor →. By this proposition, we view a twisted complex as an object of . Let L=(L,α, b, ) be a non-exact(=not necessarily exact) immersed Lagrangian brane in T^M. Let ϵ be a positive real number such that there exists a ϵ-standard sheaf quantization. Fix real numbers σ c_0 0<c_1<c_2<⋯ <c_n<c=c_n+1 with c_i+1-c_i≤ϵ for any i. Then we have (c_i-c_i-1)-standard sheaf quantizations ^i_L. We further set ^i S_c_i-1s_c-c_i^i_L[i-1]. Let ∈μ(T^M; u<c) be a sheaf quantization of L. A twisted complex presentation (TCP) of gapped by σ for L is a set of morphisms f_ij∈^i-j+1_^_τ>0(M ×_u<c×_t)(^i, ^j)_i>j satisfying (1) the tuple ({^i}_i,{f_ij}_i>j) defines a twisted complex, i.e., the twisted complex Maurer–Cartan equation df_ij+∑_kf_kj∘ f_ik=0 holds for any i,j and (2) the resulting twisted complex is isomorphic to . Note that ^i-j+1_^_τ>0(M ×_u<c×_t)(^i, ^j) ≅^1_^_τ>0(M ×_u<c×_t)(S_c_i-1s_c-c_i^i_L, S_c_j-1s_c-c_j^j_L). One has f_ij=f_i'j' if c_i-c_j=c_i'-c_j'. Without loss of generality, we may assume that c_j<c_j'. After some refinements, the action of T^c_j'-c_j=T^c_i'-c_i is induced by morphisms ^j'→^j and ^i'→^i. By the compatibility with twisting morphisms, the result follows. Since ^i_L are standard, we have an isomorphism μ_L(^i_L)≅⊕[-1]. We then have μ_L(f_ij)∈^0(, )⊕^1(, )=^∙__L^(L^)(μ_L(), μ_L()) by the observation given in the paragraph below (<ref>). Moreover, by <ref>, the first component is an isomorphism if the resulting twisted complex is a sheaf quantization. Hence we can normalize it to 𝕀. Under this normalization, we view μ_L(f_ij) as a morphism in ^1(, ). We denote by _L^c the set of isomorphism classes of sheaf quantizations of L in μ(T^M;u<c). For ∈_L^c, we can decompose into sheaf quantizations _i∈μ(T^M; u<c_i+1-c_i) with c_i+1-c_i≤min(ϵ,c_L) and ∑ c_i+1=c as s_c-c_1_0 →→_0', s_c-c_2_1 → S_-c_1_0'→ S_-c_1_1', ⋮ s_c-c_n-1_n-2 → S_-c_n-2_n-3'→ S_-c_n-2_n-2', s_c-c_c_n_n-1 → S_-c_n-1_n-2'→ S_-c_n-1_n. We can and do refine c_i's so that all the _i's are standard. Then we can rephrase the situations as follows: is obtained as an iterated extension of _i S_c_is_c-c_i+1^i in ^_τ>0(M ×_u<c×_t). In other words, we get a twisted complex, TCP. §.§ Curved dga and A_∞-algebra We briefly recall the notion of a curved dga and an A_∞-algebra. A curved dga is a pair (A, d), where * A is a graded Λ_0-algebra and * d is Λ_0-linear morphism of degree 1 such that A is free as Λ_0-module and d^2⊗_Λ_0Λ_0/Λ_0^+=0. A filtered A_∞-algebra is a generalization of curved dga. A morphism of curved dga is a morphism of graded Λ_0-algebras compatible with d. A morphism of curved dga is said to be a quasi-isomorphism if it induces a quasi-isomorphism over Λ_0/Λ_0^+. Suppose A, B are countably generated filtered A_∞-algebras. A quasi-isomorphism has a homotopy inverse. Let A be a curved dga/A_∞-algebra. A Maurer–Cartan element is an element b of A such that it satisfies the Maurer–Cartan equation ∑_i≥ 0m_i(b^⊗ i)=0, where m_i is the i-th operation of the curved A_∞-structure of A. Let A_1, A_2 be a curved dga and f A_1→ A_2 be a morphism between them. Let b a Maurer–Cartan element of A. Then f(b) is a Maurer–Cartan element of b. Let I be a filtered poset and A_∙ be a curved dga parameterized by I; for each i∈ I, we have a curved dga A_i, and a morphism ρ_ij A_i→ A_j for any i<j. Then one can define the colimit curved dga A as the quotient of the direct sum ⊕_i∈ I A_i by the relations induced by ρ_ij's. Then a Maurer–Cartan element of A_i for some i∈ I defines a Maurer–Cartan element of A. §.§ Curved complex and bounding cochain in Floer theory We recall the notion of bounding cochain by <cit.> briefly. For a Lagrangian brane L, there exists a curved A_∞-algebra CF(L,L) defined over Λ_0 such that (1) the cohomology of CF(L,L)⊗_Λ_0Λ_0/Λ_0^+ is isomorphic to the “cohomology ring”[If it is immersed, one has to take care of the definition. See <cit.> and <ref>.] of L, and (2) there exists an inclusion map C^1(L;)⊗_Λ_0↪ CF(L, L), where C^1(L;) is a model of singular chain complex of L. Of course, the statement here is far from the complete statement. We state this theorem in this form to compare with our results below. A bounding cochain of L is an element b of CF^1(L, L) such that it is a Maurer–Cartan element of the curved A_∞-algebra. If b is a bounding cochain, one can deform CF^1(L, L) and get a genuine A_∞-algebra. §.§ Curved complex and bounding cochain in sheaf theory We now would like to give analogous statements for our setup. Let L be a Lagrangian brane. Let σ c_0=0, c_1, c_2, … be a discrete submonoid of . In the terminology of <cit.>, we will consider the setup gapped by σ. By taking σ sufficiently fine, we obtain standard sheaf quantizations ^i_L and ^i defined as in (<ref>). Then we have a morphism f_ij^st^i→^j[i-j] corresponding to the standard extension. This defines a degree 1 endomorphism f of the dga ⊕_i≥ 1^i[i-1]. In general, the dga ⊕_i≥ 1^i[1-i] is not well-behaved in the following construction. The reason is that one should rather consider the notion of derived dga (or E_1-algebra), since we are working over a ring. Instead, we take a resolution of ⊕_i≥ 1^i[i-1] on which Λ_0 acts freely. Such a resolution can be easily found through a variant of the Godement resolution. We fix such a model. The dga (⊕_i≥ 1^i[i-1]⊗_Λ_0/T^cΛ_0Λ_0/Λ_0^+, (d+f)⊗_Λ_0Λ_0/Λ_0^+) is isomorphic to the cohomology dga __L^Im(L^Im)(,). In particular, the pair (⊕_i≥ 1^i[i-1], d+f) defines a curved dga. Here d denotes the differential of ⊕_i≥ 1^i[i-1]. We denote the resulting curved dga by CF_SQ(L, L,σ). Note that <ref> below gives an isomorphism ⊕_i≥ 1^i[i-1]⊗_Λ_0Λ_0/Λ_0^+≅__L^Im(L^Im)(⊕_iμ_L(^i), ⊕_iμ_L(^i)). Then f is microlocalized and gives a twisted complex in _L^Im(L^Im). Each f_ij is microlocalized to an isomorphism between . Hence the resulting twisted complex is isomorphic to . Thus, the endoalgebra is isomorphic to __L^Im(L^Im)(μ_L(^i), μ_L(^i)) __L^Im(L^Im)(, ) for some (then any) i. Let σ' be a refinement of σ. Then there exists a morphism induced by a quasi-isomorphism given by the refinement between CF_SQ(L, L,σ) and CF_SQ(L, L,σ'). Then it induces a quasi-isomorphism over Λ_0/Λ_0^+. Hence we obtain a quasi-isomorphism from CF_SQ(L, L,σ) to CF_SQ(L, L,σ'). Let Σ be the set of discrete monoids of , which forms a filtered poset. We take a cofinal subset Σ' such that CF_SQ(L, L,σ) is defined for any σ. We denote the colimit curved dga along Σ' by CF_SQ(L, L). Note that it does not depend on the choice of Σ'. There exists a quasi-isomorphism CF_SQ(L, L)≃ CF(L, L) of curved A_∞-algebras. A sheaf-theoretic bounding cochain of L gapped by σ is an element b of CF^1_SQ(L, L, σ) such that it satisfies the Maurer–Cartan equation (d+b)^2=0. For σ' refining σ, a sheaf-theoretic bounding cochain gapped by σ' is induced by a sheaf-theoretic bounding cochain gapped by σ. Hence it also induces a Maurer–Cartan element of CF^1_SQ(L, L). A bounding cochain of L gapped by σ and a TCP gapped by σ are the same notions. This is a direct comparison between the Maurer–Cartan equation of the twisted complex and the Maurer–Cartan equation of sheaf-theoretic bounding cochain. Note the following two things: * Let b be a sheaf-theoretic bounding cochain gapped by σ. Let σ' be a refinement of σ and let b' be the sheaf-theoretic bounding cochain gapped by σ' that is induced by b. Then the two TCPs corresponding to b and b' give the same sheaf quantization. * The space ^i-j+1_^_τ>0(M ×_u<c×_t)(^i, ^j) is isomorphic to ^1__L^(L^)(, ) in the limit as c_i-c_i-1→ 0. Hence through <ref>, we can package a bounding cochain as an element of C^1_SQ(L)⊗_Λ_0b∈^1__L^(L^)(, )⊗_Λ_0b=∑_i≥ 1, b_i≠ 0b_iT^c_i'. We denote the subset of bounding cochains of C^1_SQ(L)⊗_Λ_0 by _L. We would like to summarize this section as follows: Let L be a Lagrangian brane. We denote the set of sheaf quantizations of L by _L. There exists a curved dga CF_SQ(L, L) and a subset _L of C^1_SQ(L)⊗_Λ_0 such that (1) H^(CF_SQ(L, L)⊗_Λ_0/T^cΛ_0Λ_0/Λ_0^+)≅ H^(L;)⊕ H^(L_SI;_SI), (2) any element b associates a non-curved dga deformed from CF_SQ(L, L), and (3) there exist maps bc_L→_L, real_L→_L such that real∘bc =𝕀. § SEPARATION THEOREM In this section, we prove the following separation theorem for our microlocal category, which is an analogue of the separation theorem for the original Tamarkin category <cit.>. Let A' and B' be end-conic closed subsets T^M. Suppose that π(A') ∩π(B') is compact and A' ∩ B' = ∅. Then for any ∈μ_A'(T^M;u<c) and any ∈μ_B'(T^M;u<c), one has _μ(T^M;u<c)(,)=0. We prepare some lemmas in <ref> and give a proof in <ref>. §.§ Cut-off result In this subsection, we prove the following: For ∈μ^(T^M;u<c), () ⊂ T^M controls the whole microsupport () ⊂ T^(M ×_u<c×_t), where is regarded as an object of (M ×_u<c×_t) through the projector (-)⋆_≥ 0 in <ref>. First, we give a cut-off lemma in the special case. Let V be a finite-dimensional real vector space and γ be a closed convex proper cone in V with 0 ∈γ and γ≠{0}. We also set U_γ T^M × V ×γ^∘ and Z_γ T^M × T^V ∖ U_γ as in <cit.>, where γ^∘ denotes the polar cone of γ: {θ∈ V^ \|⟨θ,v ⟩≥ 0 for any v ∈γ}. We recall known cut-off results from <cit.>, which we will use in the following proofs. For ∈(_V), we have (⋆_γ)⊂ (()∩ U_γ)∪ V×∂γ^∘. In particular, (⋆_γ)∩ U_γ⊂ (() ∩ U_γ). We slightly generalize this and <cit.> as follows. Let ∈^⊥_Z_γ(_V) and assume that there exists a closed cone C^* ⊂ V^* such that * C^* ⊂γ^∘ and * () ∩ U_ γ⊂ V × C^. Then one has () ⊂ (() ∩ U_ γ) ∪ (V × (C^ ∩∂γ^∘)). We mimic the proof of <cit.>. We first claim the following. For any proper closed convex cone β of V, we have () ∩ U_β⊂ (() ∩ U_γ) ∪ V × ((C^* ∩β^∘) ∩∂γ^∘) ∩ U_β. Let us consider the sheaf = ⋆_β. By <ref>, we have () ∩ U_β = () ∩ U_β, which implies () ∩ U_γ∩ U_β⊂ V × C^* ∩ U_β by the condition (2). Set λ (C^* ∩β^∘)^∘⊂ V. We shall show (⋆_λ) ∩ U_β⊂ (() ∩ U_γ) ∪ V × ((C^* ∩β^∘) ∩∂γ^∘) ∩ U_β and ≅⋆_λ. Since ∈^⊥_Z_γ(_V), by <ref>, we have ()=(⋆_γ) ⊂ V ×γ^∘ and () ∩ U_β = (⋆_γ) ∩ U_β ⊂ ((() ∩ U_γ) ∪ (V ×∂γ^∘)) ∩ U_β ⊂ V × ((λ^∘∪∂γ^∘) ∩β^∘ ). In particular, we have () ⊂ V × (λ^∘∪∂γ^∘∪∂β^∘ ) by <ref>. Since ⋆_λ≅ (⋆_γ) ⋆_λ, by the microsupport estimate for ⋆, we have (⋆_λ) ∩ U_β ⊂ V × (λ^∘∩ (λ^∘∪∂γ^∘∪∂β^∘ ) ∩β^∘ ) ⊂ V × (λ^∘∪ (λ^∘∩∂γ^∘) ∩β^∘) ⊂ (U_γ∪ (V × ((C^* ∩β^∘) ∩∂γ^∘))) ∩ U_β, which proves (<ref>). Moreover, since (⋆_λ∖γ) ∩ U_β ⊂ V × ((γ^∘∖λ^∘) ∩ (λ^∘∪∂γ^∘∪∂β^∘) ∩β^∘) ⊂ V× (∂γ^∘∩β^∘)⊂ Z_γ∩ U_β. From the exact triangle ⋆_λ∖γ→⋆_λ→⋆_γ, we find that ⋆_λ→⋆_γ is an isomorphism in (_V;U_γ∩ U_β)=(_V;U_(γ^∘∩β^∘)^∘) by the above microsupport estimate. Since = ⋆_γ⋆_β, we have ()⊂ V× (γ^∘∩β^∘). Hence we have ⋆_(γ^∘∩β^∘)^∘≅. Through the equivalence (-) ⋆_(γ^∘∩β^∘)^∘(_V;U_(γ^∘∩β^∘)^∘)→^⊥_Z_(γ^∘∩β^∘)^∘(_V), we have an isomorphism ⋆_λ→ in (_V). Hence, we obtain (<ref>), which proves the claim. Let us return to the proof of <ref>. If we set β={0} in the claim, we get () ⊂ (() ∩ U_γ) ∪ V × ((C^) ∩∂γ^∘). Suppose (x,ξ) ∈() ∩ V × (((C^) ∖ C^) ∩∂γ^∘). Then, we can find a proper closed convex cone β such that ξ∈β^∘ and C^ ∩β^∘={0}. By applying the claim, we get (x,ξ) ∈() ∩ V ×β^∘⊂ (() ∩ U_γ) ∪ V × 0 ∩ V ×β^∘, which is a contradiction. This proves <ref>. In the above lemma, if C^* is a strict γ-cone in the sense of <cit.>, then C^* ∩∂γ^∘={0}. Hence, the lemma generalizes <cit.>. Let M be an open subset of E=^d and ∈^⊥_Z_γ(_M × V). Assume that there exists a closed cone C^* ⊂ E^* × V^* such that * C^* ⊂ E^* ×γ^∘ and * () ∩ U_ γ⊂ (M × V) × C^. Then one has () ⊂ (() ∩ U_ γ) ∪ (M × V) × (C^ ∩ (E^* ×∂γ^∘)). The proof is similar to <cit.>. We replace <cit.> with <ref>. The statement is local on M. Let x_0 ∈ M and K be a compact neighborhood of x_0. We choose an open neighborhood W of K and an diffeomorphism W ≅^d=E satisfying (-1,1)^d ⊂ K. We take a diffeomorphism φ (-1,1) such that dφ(t) ≥ 1 for any t ∈ (-1,1). Define Φ U (-1,1)^d × V E × V by Φ(x'_1,…,x'_d, x”) (φ(x'_1),…,φ(x'_d),x”). Then, we obtain Φ_πΦ_d^-1(U × C^) ⊂ E × C^ (see <cit.>). Hence, we can apply <ref> to the sheaf Φ_(\|_U) on the vector space E × V with the cone {0}×γ. For an end-conic closed subset A' of T^M, we set ∂ A' A' ∩∂ T^M ⊂∂ T^M. Let A' be an end-conic closed subset of T^M and set A=ρ^-1(A'). Then for any ∈μ^_A'(T^M;u<c), one has (\|_u>0) ⊂ AA ∪{ (x,aξ,u,0,t,0) \| (x,ξ) ∈∂ A',a>0 }∪ T^_M ×_u<c×_t(M ×_u<c×_t) Fix (x,u)∈ M×_u<c and take a coordinate compact neighborhood K. Then the cotangent bundle is trivialized over K, namely, T^(M×_u<c)\|_K ≅ K × V^* with V=^n+1. Set A=ρ^-1(A'). For each (x',u') ∈ K, we set C_(x',u')^* T^_(x',u')(M×_u<c) ×_τ∩ AA⊂ V^ ×_τ. We also define C_K^* ⋃_(x',u')∈ KC_(x',u')^* ⊂ V^* ×_τ, C^∞,_(x',u') _> 0· (∂ A'∩∂ T^_x' M) ×{υ=0}×{τ=0}∪{(0,0)}⊂ V^* ×_τ. Since A' is end-conic, we find that C_K^* ∩{τ=0 }⊂⋃_(x',u') ∈ KC^∞,_(x',u'). We take an open subset U ⊂ K containing (x,u) and set _U\|_U ×_t. Then we have (_U) ∩{τ>0}⊂ (U ×_t) × C_K^. Hence, by applying <ref> to _U with V=_t and γ=_≥ 0, we obtain (_U) ⊂ ((_U) ∩ U_γ) ∪ ((U×_t) × (C_K^* ∩{τ =0})). Here, (_U) ∩ U_γ⊂ AA and C_K^* ∩{τ=0}⊂⋃_(x',u') ∈ KC^∞,_(x',u'). In particular, (\|_{(x,u) }×_t) ⊂ AA ∪ (U×_t) ×⋃_(x',u') ∈ KC^∞,_(x',u') for any coordinate compact neighborhood K of (x,u). By taking the intersection over the neighborhoods of x, we get the desired estimate over x. This completes the proof. §.§ Proof of separation theorem We first prove the separation theorem in the non-equivariant case. Let A' and B' be end-conic closed subsets T^M. Suppose that π(A') ∩π(B') is compact and A' ∩ B' = ∅. Then for any ∈μ^_A'(T^M;u<c) and any ∈μ^_B'(T^M;u<c), one has _μ^(T^M;u<c)(,)=0. We will prove that _(M ×_u<c×_t)(_M × (-∞,a) ×_t,)=0 for a>0. If the claim holds, since _a ↗ c_M× (-∞, a)×_t=_M×_u<c×_t, we have _(M ×_u<c×_t)(_M ×_u<c×_t,)≅lim_a ↗ c_(M ×_u<c×_t)(_M × (-∞,a) ×_t,)≅ 0. Hence, we will check the claim below. We argue similarly to <cit.> to estimate (_M × (-∞,a) ×_t). Since (\|_u>0) ∩{τ=0}⊂{υ=0} and N^(M × (0,a) ×_t) ⊂{ξ=0, τ =0 }, we find that () ∩ N^(M × (0,a) ×_t) ⊂ T^_M ×_u<c×_t(M ×_u<c×_t). Noticing that \|_u ≤ 0≅ 0 and applying <cit.>, we get (_M × (-∞,a) ×_t) ⊂ N^(M × (0,a) ×_t)^a + (). By <ref>, we have (_M × (-∞,a) ×_t) ∩{ u=a}⊂ { (x,ξ,a,υ,t,τ) \| (x,ξ,t-a,τ) ∈ A, υ≥ -τ} ∪{ (x,ξ,a,υ,t,0) \| (x,ξ) ∈∂ A', υ≥ 0 }, where A=ρ^-1(A'). Since _M × (-∞,a) ×_t≅_M ×_u<c× [0,∞)⋆_M × (-∞,a) ×_t, we have isomorphisms _(M ×_u<c×_t)(_M × (-∞,a) ×_t, ) ≅ _(M ×_u<c×_t)(_M ×_u<c× [0,∞)⋆_M × (-∞,a) ×_t, ) ≅ _(M ×_u<c×_t)(_M ×_u<c× [0,∞), ^⋆(_M × (-∞,a) ×_t, )). Here, ^⋆ is the right adjoint of ⋆ defined as in (<ref>). By <cit.>, ^⋆ can be also written as ^⋆(,) ≅ m_* (p_2^-1i^-1, p_1^!), where p_i M ×_u<c×^2 → M ×_u<c×_t is the i-th projection and i M ×_u<c×_t → M ×_u<c×_t is the involution (x,u,t) ↦ (x,u,-t). By adjunction, we find that q_* ^⋆(_M × (-∞,a) ×_t, ) ∈_{τ≤ 0}(_t)^⊥, where q M ×_u<c×_t→_t is the projection. Since (i^-1_M × (-∞,a) ×_t) ∩() ⊂ T^_M ×_u<c×_t(M ×_u<c×_t), by the microsupport estimate, we have (^⋆(_M × (-∞,a) ×_t, )) ⊂ { (x,ξ,u,υ,t,τ) \| there exist (ξ_1,υ_1), (ξ_2,υ_2) ∈ T^_(x,u)(M ×_u<c) and t_1, t_2 ∈_t with t=t_1+t_2 such that (x,ξ_1,u,υ_1,t_1,τ) ∈(_M × (-∞,a) ×_t), (x,ξ_2,u,υ_1,t_2,τ) ∈(), ξ = -ξ_1+ξ_2, and υ=-υ_1+υ_2 }. Again by <ref>, we obtain (^⋆(_M × (-∞,a) ×_t, )) ∩ (T^_M ×_u<c(M ×_u<c) × T^_t) ⊂ T^_M ×_u<c×_t(M ×_u<c×_t). Since the map q is proper on (^⋆(_M × (-∞,a) ×_t, )), we get (q_ ^⋆(_M × (-∞,a) ×_t, )) ⊂ 0__t. By combining this estimate with the fact that it is in _{τ≤ 0}(_t)^⊥, we find that q_* ^⋆(_M × (-∞,a) ×_t, ) ≅ 0. This implies _(M ×_u<c×_t)(_M × (-∞,a) ×_t, ) ≅ _(_t)(_[0,∞), q_* ^⋆(_M × (-∞,a) ×_t, )) ≅ 0 as desired. Finally, we prove the separation theorem for our equivariant microlocal category. We have _μ(T^M;u<c)(, ) ≅ _μ(T^M;u<c)(⊕_c∈_M ×_u<c× [c,∞)⋆_, ) ≅ _μ(T^M;u<c)(⊕_c∈_M ×_u<c× [c,∞) ,^⋆_(, ) ≅ _(M×_u<c×_t)(_M ×_u<c× [c,∞) ,(^⋆_(, )) ≅ _^()(M×_u<c×_t)(_M ×_u<c× [c,∞) , p_1_ _M ×^2(p_2^-1, m^!)). Here ^()(M×_u<c×_t) is the derived category of equivariant sheaves with respect to the trivial -action. The last space is -invariant of the -representation _(M×_u<c×_t)(_M ×_u<c× [c,∞) , p_1_* _M ×^2(p_2^-1, m^!)). Hence the vanishing follows from that of the non-equivariant version (<ref>). § INTERSECTION POINTS ESTIMATE In this section, we prove that the rank of the hom-space between simple sheaf quantizations associated with two Lagrangians gives a lower bound of the number of intersection points. To prove the bound, we mimic the proof in <cit.>. One advantage of the present setup is that one can state the intersection equality neatly by using the language of the Novikov ring. Throughout this section, let L_i (L_i, α_i, b_i, _i) be an end-conic Lagrangian brane and _i∈μ(T^M;u<c) be a sheaf quantization of L_i for i=1,2. §.§ Statements First, we note the notion of rank of Λ_0/T^cΛ_0-modules. Let N be a Λ_0/T^cΛ_0-module. We define the derived rank of N by _Λ_0/T^cΛ_0 N∑_i_ H^i(N⊗^_Λ_0/T^cΛ_0Λ_0/Λ_0^+). In the following, we omit from the notation of the derived tensor product. We note that _Λ N⊗_Λ_0Λ≤_Λ_0N. when c=∞. Assume that π(_1) ∩π(_2) is compact. Then one has _Λ_0/T^cΛ_0^∙_μ(T^M;u<c)(_1, _2) = ∑_k ∈_ H^k(L_1^Im∩ L_2^Im; ^(_1,_2)). In particular, in the case c=∞, ∑_k ∈_ H^k(L_1^Im∩ L_2^Im; ^(_1,_2)) ≥_Λ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ. When L_1 and L_2 intersect cleanly, we obtain the following. Assume that π(_1) ∩π(_2) is compact and L_1 cleanly intersects with L_2. One has _Λ_0/T^cΛ_0^∙_μ(T^M;u<c)(_1, _2) = ∑_k ∈_ H^k(L_1×_T^M L_2; _L_1, L_2), where _L_1, L_2 is the sheaf-theoretic Maslov index local system (see <ref>). In particular, in the case c=∞, ∑_k ∈_ H^k(L_1×_T^M L_2; _L_1, L_2) ≥_Λ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ. * Moreover, we assume that (1) L_1 and L_2 are embedded, (2) L_1 and L_2 intersect transversely, and (3) _1 and _2 are simple sheaf quantization. Then _Λ_0/T^cΛ_0^∙_μ(T^M;u<c)(_1, _2) = # (L_1∩ L_2). In particular, in the case c=∞, # (L_1∩ L_2)≥_Λ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ. This follows from the computation in <ref> and <ref>. §.§ Proof of <ref> We first reduce the problem to the case when c is finite. We have ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ_0/T^cΛ_0≅^∙_μ(T^M;u<∞)(_1, _2). We first have ^∙_μ(T^M;u<∞)(_1, _2)⊗_Λ_0Λ_0/T^cΛ_0 ≅^∙_μ(T^M;u<∞)((T^c_1→_1), _2) ≅^∙_μ(T^M;u<∞)((_1 → S_c_1), _2). Then by the construction of sheaf quantization, the cone of the canonical morphism (_1→ S_c_1) is the stupid extension of _1\|_<c, namely, l_c^_1\|_<c≅(_1→ S_c_1). By the homotopy invariance, we have ^∙_μ(T^M;u<∞)((_1 → S_c_1), _2) ≅^∙_μ(T^M;u<∞)(l_c^_1\|_<c, _2) ≅^∙_μ(T^M;u<c)(l_c^_1\|_<c,_2\|_<c). This completes the proof. By the above lemma, we only have to consider the case when c<∞. In the following, we assume that c is finite. Now we have the following chain of isomorphisms: ^∙_μ(T^M;u<c)(_1, _2) ≅^∙_^_τ>0(M×_u<c×_t)(_1⋆_⊕_c∈_t≥ c, _2) ≅^∙_^_τ>0(M×_u<c×_t)(⊕_c∈_t≥ c, ^⋆_(_1, _2)) ≅^∙__τ>0(M×_u<c×_t)(_t≥ 0, ^⋆_(_1, _2)) ≅Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2)), where q M ×_u <c×_t →_t is the projection. The explanations of the isomorphisms above are now in order: The first isomorphism is by the definition and the unit property. The second one is the adjoint property. The third one follows from the description of the adjoint of ^L given in <ref>. The fourth one follows from the definition of local cohomology. Moreover, we have the following exact triangle: Γ(_t; (q_* ^⋆_(_1, _2))_(0, ∞)) → Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))) → (Γ_[0, ∞)q_* ^⋆_(_1, _2))_0 →. Next, we compute the leftmost object of the above exact triangle in the following lemma. We have Γ(_t; (q_* ^⋆_(_1, _2))_(0, ∞)) ≅Γ(_t;Γ_[0, ∞)(q_* ^⋆_(_1, _2)))⊗_Λ_0/T^cΛ_0Λ_0^+/T^cΛ_0, and the first arrow of the above exact triangle (<ref>) is given by the canonical morphism Γ(_t;Γ_[0, ∞)(q_* ^⋆_(_1, _2))⊗_Λ_0/T^cΛ_0Λ_0^+/T^cΛ_0^+ →Γ(_t;Γ_[0, ∞)(q_* ^⋆_(_1, _2))). We first note that the action of T^c∈Λ_0 on Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))) is induced by the morphism Γ_[c, ∞)→Γ_[0, ∞) through Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))) ≅Γ(_t; Γ_[c, ∞)(q_* ^⋆_(_1, _2))) →Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))), where the first isomorphism comes from the equivariant structure. Hence, for c>0, it factors as Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))) ≅Γ(_t; Γ_[c, ∞)(q_* ^⋆_(_1, _2))) →Γ(q_* ^⋆_(_1, _2))_(0,∞)) →Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2))). Hence we get a morphism Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2)))⊗_Λ_0/T^cΛ_0Λ_0^+/T^cΛ_0→Γ(q_* ^⋆_(_1, _2))_(0,∞)). Here, any element of Γ(, q_* ^⋆_(_1, _2))_(0,∞)) is represented by a section of Γ(_t; Γ_[c, ∞)(q_* ^⋆_(_1, _2))) for some c>0, since of the positively-microsupported condition of q_^⋆_(_1, _2). Using the equivariancy, such a section can be identified with a section s of Γ(_t; Γ_[0, ∞)(q_ ^⋆_(_1, _2))). Hence we can denote a section of Γ(_t; Γ_[c, ∞)(q_* ^⋆_(_1, _2))) as T^c s. Then this expression precisely gives the corresponding element in the left-hand side of (<ref>). Hence the morphism in (<ref>) is an isomorphism. This completes the proof. By taking the cone of the first morphism of (<ref>), we get Γ(_t; Γ_[0, ∞)(q_* ^⋆_(_1, _2)))⊗_Λ_0/T^cΛ_0Λ_0/Λ_0^+ ≅ (Γ_[0, ∞)q_* ^⋆_(_1, _2))_0. Combining with (<ref>), we get ^∙_μ(T^M; u<c)(_1, _2)⊗_Λ_0/T^cΛ_0Λ_0/Λ_0^+ ≅Γ_[0, ∞)(q_ ^⋆_(_1, _2))_0. Now we shall describe the right-hand side by ^. Assume that π(_1) ∩π(_2) is compact. We have Γ_[0, ∞)(q_* ^⋆_(_1, _2))_0 ≅Γ(L_1^Im∩ L_2^Im, ^(_1, _2)). Combining with (<ref>), we have ^∙_μ(T^M; u<c)(_1, _2)⊗_Λ_0/T^cΛ_0Λ_0/Λ_0^+ ≅Γ(L_1^Im∩ L_2^Im, ^(_1, _2))). We mimic the the argument of <cit.> (see also <cit.>). We first prove Γ_[0, ∞)(q_ ^⋆_(_1, _2))_0 ≅Γ({τ > 0}; (π'_* (_1,_2))^ m\|_{τ>0}), where π' T^(M ×_u<c×_t) → T^_0_t is the projection. We will prove the isomorphism in the following steps: Γ_[0, ∞)(q_* ^⋆_(_1, _2))_0 ≅Γ({τ>0 };μ_{0}(q_^⋆_(_1, _2))\|_{τ>0 }) ≅Γ({τ>0 }; q'_ μ_M ×_u<c×{0}(^⋆_(_1, _2))) ≅Γ({τ>0 }; q'_* (π_m_* μ_M ×_u<c× m^-1(0)(p_2^-1i^-1'_1, p_1^!_2))^ m) ≅Γ({τ>0 }; q'_* (π_m_* π_M ×_u<c_* ι_('_1,_2))^ m) ≅Γ({τ>0 }; q'_ (π_m_* π_M ×_u<c_* ι_(_1,_2))^ m) ≅Γ({τ > 0}; (π'_ (_1,_2))^ m\|_{τ>0}), where q', π_m, i, _1', and ι will be defined in the course of the proof. The first step (<ref>) is obvious from the definition of μ. Next, we prove (<ref>). By the adjunction, we have ^⋆(_1,_2) ≅ j^-1^⋆(j_! _1, j_! _2), where j M ×_u<c×_t ↪ M ×_u ×_t is the inclusion and ^⋆ in the right-hand side is the operation on M ×_u ×_t. Note that j_!_1 and j_!_2 are supported in M × [0,c] ×_t. By abuse of notation, we also let q denote the projection M ×_u ×_t →_t. By the assumption that π(_1) ∩π(_2) is compact, q is proper on (^⋆_(j_!_1, j_!_2)). Hence by <cit.> we get μ_{0}(q_^⋆_(_1, _2)) ≅μ_{0}(q_ j^-1^⋆_(j_!_1, j_!_2)) ≅μ_{0}q_* (^⋆_(j_!_1, j_!_2)\|_M ×_u < c× T^_0) ≅ q'_ (μ_M ×_u ×{0}(^⋆_(j_! _1, j_!_2))\|_M ×_u < c× T^_0) ≅ q'_ μ_M ×_u<c×{0}(^⋆_(_1, _2)), where q' M ×_u<c× T^_0_t → T^_0_t is the projection. This completes (<ref>). Next, we prove (<ref>). Here we note that μ_{0}(q_^⋆_(_1, _2)) ≅μ_{0}(q_^⋆_(_1, _2)\|_(-1,1)). Moreover, we can also rewrite ^⋆(,) as ^⋆(,) ≅ m_^ m(p_2^-1i^-1, p_1^!), where i M ×_u<c×_t → M ×_u<c×_t; (x,u,t) ↦ (x,u,-t) is the involution. By the end-conic assumption of L_1 and L_2, there exists a sufficiently large B>0 such that (_i\|_M ×_u<c× ((-∞, -B+2) ⊔ (B-2,∞))) ⊂{τ=0 } for i=1,2. Then we have _i ≅^-_i ⊠_(-∞,-B+2)⊕^+_i ⊠_(B-2,∞) on M ×_u<c× ((-∞, -B+2) ⊔ (B-2,∞)). By setting U M ×_u<c× (-1,1), we find that p_1^! _2 ≅^-_2 ⊠_(-∞,-B+1) ×[1] ⊕^+_2 ⊠_(B-1, ∞) ×[1] on m^-1(U) ∩ (M ×_u<c×_t × ((-∞,-B+1) ⊔ (B-1,∞)). Hence, we have (p_2^-1i^-1 (_1)_M ×_u<c× ((-∞,-B] ⊔ [B,∞)), p_1^!_2)\|_m^-1(U) ≅ (^+_1 ⊠_× (-∞,-B]⊕^-_1 ⊠_× [B,∞), ^-_2 ⊠_(-∞,-B+1) ×[1] ⊕^+_2 ⊠_(B-1, ∞) ×[1])\|_m^-1(U) ≅ ((^+_1, ^-_2) ⊠(_× (-∞,-B], _(-∞,-B+1) ×[1]))\|_m^-1(U) ⊕ ((^-_1, ^+_2) ⊠(_× [B,∞), _(B-1, ∞) ×[1]))\|_m^-1(U). By applying m_ and taking sections, we get m_* (p_2^-1i^-1 (_1)_M ×_u<c× ((-∞,-B] ⊔ [B,∞)), p_1^!_2)\|_m^-1(U)≅ 0 by the Künneth theorem. By the exact triangle '_1 →_1 → (_1)_M ×_u<c× ((-∞,-B] ⊔ [B,∞))→ with '_1 being supported in M ×_u<c× [-B,B], we obtain an isomorphism (m_* (p_2^-1i^-1_1, p_1^!_2)\|_(-1,1)≅ (m_* (p_2^-1i^-1'_1, p_1^!_2)\|_(-1,1) and m is proper on ((p_2^-1i^-1'_1, p_1^!_2)). Hence, we can apply <cit.> again to get μ_M ×_u<c×{0}(^⋆_(_1, _2)) ≅ (π_m_* μ_M ×_u<c× m^-1(0)(p_2^-1i^-1'_1, p_1^!_2))^ m, where π_m M ×_u<c× T^_m^-1(0)^2 → M ×_u<c× T^_0_t is the canonical map associated with m. This completes (<ref>). Next, we prove (<ref>). Let p̃_i (M ×_u<c×)^2 → M ×_u<c× denote the i-th projection for i=1,2. We first note that the diagonal map δ M ×_u<c×^2 → (M ×_u<c)^2 ×^2 is non-characteristic for ((p̃_2^-1i^-1'_1, p̃_1^!_2)) on the subset T^_M×_u<c(M×_u<c) × T^_τ>0_t ⊂ T^_M×_u<c(M×_u<c)× T^_t ≅ T^_M×_u<c× m^-1(0)(M×_u<c×^2) (see <cit.> for the definition). One can see this as follows: Let us fix a fiber metric of the T^M. For any R∈_>0, consider the codisk D^_R M bundle of radius R. We set _R(_i) (_i)∩ρ^-1(D^_RM). Then we have ((p̃_2^-1i^-1'_1, p̃_1^!_1))⊂⋃_R p̃_2^-1i^-1_R(_1)^a+p̃_1^-1_R(_2)∪τ_1=τ_2=0. For any R, the map δ is non-characteristic for p̃_2^-1i^-1_R(_1)^a+p̃_1^-1_R(_2) by <ref>. Hence δ^#_∞(p̃_2^-1i^-1_R(_1)^a+p̃_1^-1_R(_2))=∅ for any R by <cit.>. Hence δ^♯_∞(((p̃_2^-1i^-1'_1, p̃_1^!_1))) is in {τ_1=τ_2=0}. This confirms the desired non-characteristicity. Moreover, the restriction δ\|_M ×_u<c× m^-1(0) is a submersion onto its image. Hence, by <cit.>, we find that μ_M ×_u<c× m^-1(0)(p_2^-1i^-1'_1, p_1^!_2)\|_{τ >0 } ≅ μ_M ×_u<c× m^-1(0)δ^! (p̃_2^-1i^-1'_1, p̃_1^!_2)\|_{τ >0 } ≅ π_M ×_u<c_* μ_Δ_M ×_u<c× m^-1(0)(p̃_2^-1i^-1'_1, p̃_1^!_2)\|_{τ >0 } ≅ π_M ×_u<c_* ι_('_1,_2)\|_{τ >0 }, where π_M ×_u<c T^(M ×_u<c×_t) → M ×_u<c× T^_t is the projection and ι T^_Δ^2 T^_m^-1(0)^2; (t,t,τ,-τ) ↦ (t,-t,τ,τ). Moreover, we used the identification T^(M ×_u<c) ≅ T^_Δ_M ×_u<c(M ×_u<c)^2 and T^_t ≅ T^_Δ^2 by the first projections. The only nontrivial part is the second isomorphism where we used <cit.>. This completes (<ref>). Next, we prove (<ref>). Since (((_1)_M × ((-∞,-B] ⊔ [B,∞)),_2)) ⊂{τ=0}, by the exact triangle (<ref>), we have an isomorphism ('_1,_2)\|_{τ>0}≅(_1,_2)\|_{τ >0 }, which proves (<ref>). Finally, since π' = q' ∘π_m ∘π_M ×_u<c∘ι, we obtain (<ref>), which completes the proof of (<ref>). Next, we prove Γ({τ>0};(π'_ (_1,_2))^ m\|_{τ>0})≅Γ(L_1^Im∩ L_2^Im; ^(_1, _2)). By taking c sufficiently small, we can represent _1, _2 by the Čech data: Let be an open covering of M representing the Čech data of M. On each U∈, we have _i=⊕_c∈T_c^0_i for some _i^0 i=1,2. Then, on T^U, we have Γ({τ>0};(π'_ (⊕_c∈T_c^0_1,⊕_c∈T_c^0_2))^ m\|_{τ>0}) ≅Γ({τ>0};(π'_* (^0_1,⊕_c∈T_c^0_2))\|_{τ>0}). Moreover, by the definition of ^, we have an isomorphism (^0_i,⊕_c∈T_c^0_i))\|_{τ>0}≅ρ^-1^(_1, _2)\|_T^U, where ρ is defined as follows: ρ (x,ξ,u,0,t,τ) τ>0, (x,ξ/τ)∈ L_1^Im∩ L_2^Im∩ T^U, t=-f(x, ξ/τ) ∪ (x,ξ,u,-τ,t+u,τ) τ>0, (x,ξ/τ)∈ L_1^Im∩ L_2^Im∩ T^U, t=-f(x, ξ/τ) ∪ (x,ξ,0,-υ,t,τ) τ > 0, (x,ξ/τ) ∈ L_1^Im∩ L_2^Im∩ T^U, -τ≤υ≤ 0,t=-f(x, ξ/τ) → L_1^Im∩ L_2^Im∩ T^U; (x,ξ,u,υ,t,τ) ↦ (x, ξ/τ), where f is a primitive of λ\|_L_1^∩ L_2^∩ T^U. By combining the above results, we get the desired isomorphism (<ref>). This completes the proof. We now complete the proof of the main theorem in this section. The statement follows from a direct combination of <ref> and the definition of rank. § METRIC ON THE OBJECTS We discuss the metric on the set of objects following <cit.>. Our treatment is closer to that of <cit.>. Then the completeness with respect to the interleaving distance studied in Asano–Ike <cit.> and Guillermou–Viterbo <cit.> will be discussed in our setup. In this section, we take = (if not, it will produce a non-interesting theory). §.§ Λ_0-linear category Let be a triangulated Λ_0-linear category. Note that a Λ_0/T^cΛ_0-linear category can be viewed as a Λ_0-linear category. The following three definitions are analogues of the notion in <cit.>. Let ∈ be an object. Then is said to be an a-torsion, if T^a𝕀∈() is zero. Let , ∈ and c∈_≥ 0. The objects and are said to be c-isomorphic if there exist morphisms α→ and β→ such that (1) the composite is equal to T^c𝕀_ and (2) the composite is equal to T^c𝕀_. In this case, the pair of morphisms (α,β) is called a c-isomorphism. In this situation, we also say that α is c-invertible and β is a c-inverse of α. Following <cit.>, one can also define (a,b)-isomorphism as follows. Let , ∈ and a, b∈_≥ 0. The pair (, ) is said to be weakly (a,b)-isomorphic if there exist morphisms α, δ→ and β, γ→ such that (1) the composite is equal to T^a+b𝕀_, (2) the composite is equal to T^a+b𝕀_, and (3) T^aα=T^aδ and T^bβ=T^bγ. If (,) is (a,b)-isomorphic, then and are 2(a+b)-isomorphic. The proofs of the following can be easily read off from the corresponding proofs of <cit.>, hence omitted. For <ref>, see also <ref>. Let ∈ and a∈_≥ 0, consider the exact triangle → C→. Then C is 2a-torsion. Let →→→ is an exact triangle in and assume is c-torsion. Then and are 2c-isomorphic. Assume that (, ) is c-isomorphic. Let (α, β) be a c-isomorphism on (, ). Then the cone of α is 3c-torsion. §.§ Interleaving distance We interpret the interleaving distance of <cit.> in terms of the Novikov ring. Let be a triangulated Λ_0-linear category. For objects c_1, c_2 of , the interleaving distance d_I(c_1, c_2) between them is the infimum of ϵ≥ 0 satisfying the following: There exist morphisms α∈_(c_1, c_2) and β∈_(c_2, c_1) such that β∘α=T^ϵ𝕀_c_1, α∘β=T^ϵ𝕀_c_2. It is easy to see that d_I is a pseudo-distance on . The following is also a straightforward generalization of the result in <cit.>. Let be a triangulated Λ_0-linear category with arbitrary coproduct. Then d_I is a complete pseudo-metric. Hence, as a corollary of this theorem and <ref>, we obtain: The category μ(T^M;u<c) is complete with respect to d_I. We denote the completion of μ^sm(T^M;u<c) in μ(T^M;u<c) with respect to d_I by μ^lsm(T^M;u<c). We can also discuss the Gromov–Hausdorff distance in the sense of Fukaya <cit.> in this setting (up to slight modification). An advantage of sheaf theory (already evident in <cit.> and <cit.>) is that the sheaf category already contains limits, and hence the limiting category is more concrete than the approach in <cit.>. §.§ Interleaving distance on the non-equivariant category To relate our interleaving distance with that of <cit.>, we recall the formulation in loc. cit. Let , ∈_τ>0(M×_u<c×_t) and c∈_≥ 0. The pair (, ) is said to be (a,b)-isomorphic if there exist morphisms α→ T_a and β→ T_b such that (1) the composite T_aT_a+b is equal to T^a+b and (2) the composite T_bT_a+b is equal to T^a+b. We also set d(, )inf a+bthe pair (,) is (a,b)-isomorphic. For , ∈_τ>0(M×_u<c×_t), we have d_I(^L(), ^L())≤ d(, ). We prove the lemma in a more general form in <ref>. The following is also easy: For , ∈μ(T^M;u<c), we have d_I(, )≥ d((), ()). §.§ Fukaya's Hofer distance We can consider a slightly weaker version of the distance. Let be a triangulated Λ_0-linear category. For objects c_1, c_2 of , the interleaving distance d_H(c_1, c_2) between them is the infimum of ϵ≥ 0 satisfying the following: There exist morphisms α∈ F^-ϵ_1_⊗_Λ_0Λ(c_1, c_2) and β∈ F^-ϵ_2_⊗_Λ_0Λ(c_2, c_1) such that β∘α=𝕀_c_1, α∘β=𝕀_c_2. and ϵ_1+ϵ_2≤ϵ. It is easy to see that d_H is a pseudo-distance on . The definition is motivated by the Hamiltonian invariance of the Floer cohomology over the Novikov field. The following is easy. For , ∈, we have d_H(, )≤ d_I(, ). § HAMILTONIAN AUTOMORPHISM In this section, we explain that the energy stability result <cit.> also holds in our microlocal category. §.§ Hamiltonian automorphism and energy stability Let I be an open interval containing the closed interval [0,1]. For a time-dependent compactly supported smooth function H T^M × I →, we let ϕ^H=(ϕ^H_s)_s ∈ I T^M × I → T^M denote the associated Hamiltonian isotopy. In Guillermou–Kashiwara–Schapira <cit.>, they proved the existence and the uniqueness of sheaf quantization K^H ∈((M ×)^2 × I) of a Hamiltonian isotopy ϕ^H associated with a compactly supported function H. One can prove that the restriction K^H_1 K^H\|_(M ×)^2 ×{1}∈((M ×)^2) depends only on the time-1 map ϕ^H_1 (see Asano–Ike <cit.>, for example). Hence, we can define the sheaf quantization of a compactly supported Hamiltonian diffeomorphism φ of T^M by setting K^φ K^H_1 with φ=ϕ^H_1. Let φ be a compactly supported Hamiltonian diffeomorphism of T^M and K^φ∈((M ×)^2) be the associated sheaf quantization. We define ^φ (q_* K^ϕ) ⋆_≥ 0∈(M^2 ×_t), where q is the base-change of the map ^2 →_t; (t_1,t_2) ↦ t_1-t_2. The following result is proved by Asano–Ike <cit.> (see also <cit.> and <cit.>). For a compactly supported Hamiltonian diffeomorphism φ of T^M, one has d(^𝕀,^φ) ≤φ_H, where the right-hand side is the Hofer norm of φ defined by φ_H inf∫_0^1 max_p ∈ T^M H_s(p) -min_p ∈ T^M H_s(p) ds H T^M × I → T^M is compactly supported and φ = ϕ^H_1. To consider the action of Hamiltonian diffeomorphisms on the microlocal category, we introduce a general operation that combines composition and convolution. For ∈(X_1 × X_2 ×_t) and ∈(X_2 × X_3 ×_t), we define ∙_X_2π_13_!(q̃_12^-1⋆q̃_23^-1) ∈(X_1 × X_3 ×_t), where q̃_12 X_1 × X_2 × X_3 ×→ X_1 × X_2 ×; (x_1,x_2,x_3,t) ↦ (x_1,x_2,t), q̃_23 X_1 × X_2 × X_3 ×→ X_2 × X_3 ×; (x_1,x_2,x_3,t) ↦ (x_2,x_3,t), π_13 X_1 × X_2 × X_3 ×→ X_1 × X_3 ×; (x_1,x_2,x_3,t) ↦ (x_1,x_3,t). If there is no risk of confusion, we simply write ∙ instead of ∙_X_2. We also have the -equivariant version: For ∈(X_1 × X_2 ×_t) and ∈^(X_2 × X_3 ×_t), we define ∙_R π_13_!(q̃_12^-1⋆_R q̃_23^-1) ∈^(X_1 × X_3 ×_t). Obviously, ( ∙_R ) = ⋆(). For a compactly supported Hamiltonian diffeomorphism φ of T^M, the functor ^φ∙_R(-) induces an endofunctor ^φ^φ∙_R (-) μ(T^M;u<c) →μ(T^M;u<c), which satisfies (^φ()) = φ(()) for any ∈μ(T^M;u<c). Moreover, for any ∈μ(T^M;u<c), one has d_I(,^φ∙_R ) ≤φ_H. We find that ^φ∙ (-) induces an autoequivalence of _τ≤ 0(M ×_u<c×_t). Hence ^φ∙_R (-) induces an autoequivalence of ^_τ≤ 0(M ×_u<c×_t), since (^φ∙_R )=((^φ∙_R ))=(^φ∙())⊂{τ≤ 0} for ∈^_τ≤ 0(M ×_u<c×_t). This implies that ^φ∙_R (-) induces an autoequivalence of ^⊥^_τ≤ 0(M ×_u<c×_t). Since ^φ∙_R is also a weak doubling movie for ∈μ(T^M;u<c), the functor induces an autoequivalence of μ(T^M;u<c). Furthermore, by <cit.> (see also <cit.>), we find that (^φ∙_R ) ⊂φ(()). By replacing φ with φ^-1 and by ^φ∙_R, we obtain () ⊂φ^-1((^φ∙_R )), which implies that (^φ∙) = φ(()). To prove the second assertion, we prove the following lemma. Let _0, _1 ∈(M^2 ×_t) and ∈μ(T^M;u<c). Then d_I(_0 ∙_R ,_1 ∙_R ) ≤ d(_0,_1). Let α' _0 → T_a _1 and β' _1 → T_b _0 be a pair of (a,b)-isomorphism. These morphisms induce morphisms α_0 ∙_R →_1 ∙_R and β_1 ∙_R →_0 ∙_R, which satisfy β∘α=T^a+b𝕀__0 ∙_R and α∘β=T^a+b𝕀__1 ∙_R. Since ^𝕀∙_R (-) is the identity functor, the result follows from the above lemma and <ref>. For any object ∈μ(T^M;u<c) and any compactly supported Hamiltonian diffeomorphism φ, we have ≅ K^φ∙_R in μ(T^M;u<c)⊗_Λ_0Λ. Note that this corollary is only meaningful when c=∞. Otherwise, the claimed equality means 0≅ 0. <ref> says that there exists α→ K^φ∙_R and β K^φ∙_R → such that α∘β=T^ϵ, β∘α=T^ϵ for some ϵ≥ 0. After the base change to Λ, we can rewrite the equalities as (T^ϵα)∘β=𝕀, β∘ (T^-ϵα)=𝕀. This completes the proof. §.§ Hamiltonian automorphisms on the completed category In the following, we consider an action of the group of Hamiltonian homeomorphisms on μ^lsm(T^M;u<c). Let I be an open interval containing the closed interval [0,1]. An isotopy of homeomorphisms ϕ=(ϕ_s)_s T^M × I → T^M of T^M is said to be a continuous Hamiltonian isotopy if there exist a compact subset C ⊂ T^M and a sequence of smooth Hamiltonian functions (H_n T^M × I →)_n ∈ supported in C satisfying the following two conditions. (1) The sequence of flows (ϕ^H_n)_n ∈ C^0-converges to ϕ, uniformly in s ∈ I. (2) The sequence (H_n)_n ∈ converges uniformly to a continuous function H T^M × I →. That is, H_n-H_∞sup_p ∈ T^M\|H_n(p)-H(p)\| → 0. A homeomorphism of T^M is called a Hamiltonian homeomorphism if it is the time-1 map of a continuous Hamiltonian isotopy. The group of Hamiltonian homeomorphism is denoted by (T^M). The group (T^M) acts on the category μ^lsm(T^M;u<c). The sheaf quantization ^φ_∞ of a Hamiltonian homeomorphism φ_∞ is constructed in <cit.>. In fact, for a sequence (H_n)_n satisfying the above conditions, the sequence (ϕ^H_n_1)_n forms a Cauchy sequence with respect to the Hofer norm ∙_H and we can take a limit of (^H_n_1)_n by using the energy stability (<ref>). Then by <ref>, <ref>, and <ref>, the functor ^φ_∞∙_R (-) preserves μ^lsm(T^M;u<c). § SHEAF QUANTIZATION OF LIMITS OF LAGRANGIAN BRANES In this section, we discuss sheaf quantization of limits of Lagrangian branes, which is previously discussed in <cit.>. To simplify the following discussion, we introduce some terminologies. A morphism f→ in μ(T^M;u<∞) is said to be an isomorphism over Λ if it induces an isomorphism in μ(T^M;u<∞)⊗_Λ_0Λ. Note that a c-invertible morphism for some c≥ 0 is an isomorphism over Λ. Let L (resp. L_i) be a Lagrangian brane with a bounding cochain b (resp. b_i). We assume that the corresponding sheaf quantization satisfies ^0(_L)=Λ_0𝕀. * If (L_1, b_1) and (L_2, b_2) are Hamiltonian isotopic by a compactly supported Hamiltonian diffeomorphism φ, we denote it by (L_1, b_1) ∼_φ (L_2, b_2). * We denote by (L, b) the set of Lagrangian branes with bounding cochains which are isotopic to (L, b) by a compactly supported Hamiltonian diffeomorphism. * We set d((L_1, b_1),(L_2, b_2))infφ_H (L_1, b_1) ∼_φ (L_2, b_2), where φ_H is the Hofer norm defined as in (<ref>). We denote by (L, b) the metric completion of (L, b). Now we can state our theorem in this section: There exists a canonical functor (L, b)→μ^lsm(T^M) extending the sheaf quantization. More precisely, if two Cauchy sequences in (L, b) define the same limit object, the corresponding limits of sheaf quantizations are also isomorphic. Let ((L_i, b_i))_i and ((L'_i, b'_i))_i be Cauchy sequences in (L, b) that define the same limit. We let and ' be the corresponding sheaf quantizations, respectively. Then it follows that d_I(, ')=0. We clarify the situation a little more. For each i, there exist compactly supported Hamiltonian diffeomorphisms φ_i and φ'_i such that (1) (L_i, b_i) ∼_φ_i (L_i+1, b_i+1), (L'_i, b'_i) ∼_φ'_i (L'_i+1, b'_i+1) and (2) φ_i_H, φ_i'_H → 0 as i→∞. We fix such choices of Hamiltonians. Associated with the choice, we obtain a sequence of sheaf quantizations _i (resp. '_i) of (L_i, b_i) (resp. (L'_i, b'_i)), * sequences of morphisms f_i, i+1∈(_i, _i+1) and g_i, i+1∈(_i+1, _i) (resp. f'_i, i+1∈('_i, '_i+1) and g'_i, i+1∈('_i+1, '_i)) such that f_i, i+1 and g_i, i+1 (resp. f'_i, i+1 and g'_i, i+1) are mutually ϵ_i-inverse (resp. ϵ'_i-inverse) with ϵ_i, ϵ'_i→ 0, and * the colimit of (_i, f_i, i+1) (resp. ('_i, f_i, i+1')) is isomorphic to (resp. '). We denote the associated morphism _i→ (resp. '_i →') by f_i (resp. f_i'). By taking a subsequence if necessary, we can further assume: (iv) there exists a morphism g_i (resp. g_i') which is η_i-inverse (resp. η_i'-inverse) of f_i (resp. f_i') with η_i, η_i'→ 0. Let us now construct morphisms between and '. Since d_I(,')=0, we have α_1→' and β_1'→ that are mutually ρ_1-inverse. Take a positive number ρ_2 less than ρ_1/2. We also have α_2→' and β_2'→ that are mutually ρ_2-inverse as well. We also note that α_2 is a scalar multiple of α_1 by the construction over the Novikov field, since ^0_Λ(, ')≅Λ by the assumption of ^0(_i)=Λ_0𝕀. Repeating the arguments, we obtain the following: There exists a sequence of morphisms α_j →' and β_j '→ in μ(T^*M, u<∞) such that β_j ∘α_j =T^ρ_j, α_j ∘β_j = T^ρ_j with ρ_j → 0 as j →∞ and the subspace of _Λ(, ') spanned by α_j does not depend on j. For each j, we set θ_α^j maxθ T^-θα_j ∈ F^0(, ') θ_β^j maxθ T^-θβ_j ∈ F^0(', ). We have θ_α^j+θ_β^j=ρ_j. This follows from the one-dimensionality of the hom-spaces. If necessary, we take a subsequence of α_j so that θ_α^j→ 0 as j→∞. We again denote the subsequence by {α_j}_j. We similarly take such a subsequence for β_j. Note that α_j=T^θ_α^j-θ_α^j'α_j' if j'>j. We set α T^-θ_α^jα_j ∈ F^0_Λ(, ') and β T^-θ_β^jβ_j ∈ F^0_Λ(', ) (which does not depend on j). Then we find that α∘β=𝕀 and β∘α=𝕀. There exist α∈_Λ_0(, ') and β∈_Λ_0(', ) that lift α and β, respectively, and satisfy T^θ_α^jα=α_j, T^θ_β^jβ=β_j for any j. We have _Λ_0(, ')≅_(M×_t)(_[c,∞), ^⋆(, ')) for any c∈. Then the action of T^c on _Λ_0(, ') is given by _(M×_t)(_[c,∞), ^⋆(, '))→_(M×_t)(_[0,∞), ^⋆(, ')) induced by _[0, ∞)→_[c, ∞). We regard α_j as α_j∈_(M×_t)(_[-θ_α^j,∞), ^⋆(, ')). For j'>j, we have a morphism _(M×_t)(_[-θ_α^j',∞), ^⋆(, ')) →_(M×_t)(_[-θ_α^j,∞), ^⋆(, ')), which maps α_j' to α_j. Hence {α_j}_j forms a direct system of elements in the direct system {_(M×_t)(_[-θ_α^j,∞), ^⋆(, '))}_j. Hence it defines an element α=_j →∞α_j ∈_j →∞_(M×_t)(_[-θ_α^j,∞), ^⋆(, ')) ≅_(M×_t)(_j →∞_[-θ_α^j,∞), ^⋆(, ')) ≅_(M×_t)(_[0,∞), ^⋆(, ')), which is our desired lift. Indeed, we have _(M×_t)(_[0,∞), ^⋆(, ')) →_(M×_t)(_[-θ_α^j,∞), ^⋆(, ')); α ↦ T^θ_α^jα=α_j by the definition of the limit. We can prove similarly for β. By the construction, we have T^ϵα∘β=T^ϵ𝕀, T^ϵβ∘α=T^ϵ𝕀 for any ϵ>0. By running the argument of <cit.>, we complete the proof of <ref>. One can carry out the argument of <cit.> in this setup, by replacing the γ-metric with the Hofer metric. Then one can conclude that the γ-support of an element of (L, b) is the closure of of the corresponding sheaf quantization. Yuichi Ike: Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka-shi, Fukuoka 819-0395, Japan. E-mail address: , Tatsuki Kuwagaki: Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan. E-mail address:
http://arxiv.org/abs/2307.00897v1	20230703095008	Fixing confirmation bias in feature attribution methods via semantic match	[ "Giovanni Cinà", "Daniel Fernandez-Llaneza", "Nishant Mishra", "Tabea E. Röber", "Sandro Pezzelle", "Iacer Calixto", "Rob Goedhart", "Ş. İlker Birbil" ]	cs.LG	[ "cs.LG", "cs.AI" ]	Periodicity of general multidimensional continued fractions using repetend matrix form Vítězslav Kala Electronic address: ====================================================================================== Feature attribution methods have become a staple method to disentangle the complex behavior of black box models. Despite their success, some scholars have argued that such methods suffer from a serious flaw: they do not allow a reliable interpretation in terms of human concepts. Simply put, visualizing an array of feature contributions is not enough for humans to conclude something about a model's internal representations, and confirmation bias can trick users into false beliefs about model behavior. We argue that a structured approach is required to test whether our hypotheses on the model are confirmed by the feature attributions. This is what we call the “semantic match” between human concepts and (sub-symbolic) explanations. Building on the conceptual framework put forward in <cit.>, we propose a structured approach to evaluate semantic match in practice. We showcase the procedure in a suite of experiments spanning tabular and image data, and show how the assessment of semantic match can give insight into both desirable (e.g., focusing on an object relevant for prediction) and undesirable model behaviors (e.g., focusing on a spurious correlation). We couple our experimental results with an analysis on the metrics to measure semantic match, and argue that this approach constitutes the first step towards resolving the issue of confirmation bias in XAI. § INTRODUCTION The success of machine learning techniques in solving a variety of tasks, along with a parallel surge in model complexity, has rekindled interest in the interface between humans and machines. The field of Explainable AI (XAI henceforth) is concerned with unpacking the complex behavior of machine learning models in a way that is digestible by humans <cit.>. Among several proposed solutions, one approach has risen to prominence in the last half decade, namely what is known as feature attribution or feature importance. Loosely speaking, feature attribution methods explain machine behavior by indicating the extend to which different parts of the input contribute to the model's output. It is hard to overstate how widespread such methods are: they are currently employed in a plethora of scenarios, including virtually all data modalities, and deployed in production in low- as well as high-risk environments <cit.>. Yet, such techniques are not free from criticism. Beside doubts about consistency between explanations and faithfulness to the model, scholars have argued that feature attribution techniques expose the users of machine learning applications to confirmation bias, namely the reasoning pitfall that leads us to believe an explanation just because it aligns with our expectations <cit.>. For instance, a clinician using AI to diagnose metabolic disorders from images – after inspecting some explanations highlighting build-ups of fat in the liver – might be prone to believe that the model has learned to pay attention to fatty liver. As fatty liver is a known metabolic condition, the clinician will recognize it and possibly assume the machine recognizes it too. This may influence the level of trust the clinician has in the model, affecting the way care is delivered. But how can we be sure the model has learned this? More generally, due to the sub-symbolic nature of feature attributions (i.e., the fact that they are just strings or matrices of numbers), we currently have no systematic way to ascertain whether explanations capture a concept we are interested in. Some authors advocate checking explanations against human intuition <cit.>, but this exercise must be structured in a way that allows us to measure alignment between human concepts and explanations, lest we fall back into the problem of confirmation bias. In this article, we build on the framework of semantic match proposed by <cit.> and formalize a procedure that allows us (1) to formulate a hypothesis with the form “the model behaves in this way”, and (2) to obtain a score representing the extent to which the model's explanations confirm or reject this hypothesis. Such procedure is general and can in principle be applied to any model and to any local feature attribution method. In this article, we focus on SHapley Additive exPlanations (SHAP) <cit.> due to their widespread use in practice. Such a procedure is paired with a discussion on what metrics are appropriate to measure semantic match. We display the procedure in two sets of experiments on tabular and image data. We investigate different kinds of hypotheses about model behavior, showing that the procedure can give insight both into desirable behaviors – what we hope the model is doing well – as well as undesirable behaviors. All experiments use publicly available data and are fully reproducible. § RELATED WORK Feature attribution methods in XAI. The majority of attribution-based methods provide local explanations, i.e., they aim to explain the prediction for an individual instance rather than the model as a whole and henceforth we focus our work on attribution-based methods (agnostic or specific) for local explainability. We can further break down attribution-based methods into gradient-based and perturbation-based methods. The former category includes, for instance, DeConvNet <cit.>, guided backpropagation (GBP) <cit.>, Grad-CAM <cit.>, and integrated gradients <cit.>. Methods that fall in the second category include Occlusion sensitivity maps <cit.>, LIME <cit.> and SHAP <cit.>. We refer to review papers such as those of <cit.> and <cit.> for a more detailed account of the different methods. Criticism of feature attribution methods. Although feature attribution methods are the most studied and deployed in practice <cit.>, they are facing several criticisms. First, feature attribution methods do not produce stable results. They have been shown to be sensitive to adversarial perturbations that are perceptively indistinguishable and to produce drastically different results for similar inputs <cit.>. For perturbation-based methods, due to sampling, two independent runs can result in different attributions <cit.>. These concerns have been studied in the literature and different approaches to tackle robustness and reliability have been proposed <cit.>. Second, the methods tend to be sensitive to the choice of baseline, i.e., the reference values that feature importance scores are compared to <cit.>. Third, the features deemed most important differ between methods for the same input. For example, <cit.> compare the results obtained from logistic regression, random forest, and LIME <cit.> applied to both models and observe that different features are detected with these methods. <cit.> compare the rank correlation between feature attribution methods and attention-based methods and find that there is no strong correlation between those methods. In general, there have been concerns about the performance of attribution-based methods, especially due to a lack of some ground truth for (quantitative) evaluation <cit.>. Confirmation bias in XAI. Confirmation bias is a well-known concept from psychology, first described by <cit.>, and followed by plenty of empirical studies to investigate this phenomenon <cit.>. The American Psychological Association defines confirmation bias as “the tendency to gather evidence that confirms preexisting expectations, typically by emphasizing or pursuing supporting evidence while dismissing or failing to seek contradictory evidence” <cit.>. Even though the literature on human cognition including confirmation bias is rich and the problem of this type of cognitive bias has been acknowledged in the (X)AI literature <cit.>, the empirical research on confirmation bias in XAI is scarce. <cit.> propose a conceptual framework for building human-centered and decision-theory-driven XAI, in which they consider human decision making and the role of confirmation bias in relation to XAI. <cit.> conducted a field experiment in which the study subjects were tasked with performing risk assessments aided by a predictive model, while <cit.> conducted two studies in the real estate industry investigating how humans shift their mental models. Both results find that confirmation bias is present in human-XAI interaction. The risk of falling prey to confirmation bias is especially present if we use feature attribution methods on high-level features <cit.>. Especially known from the field of computer vision and deep learning, high-level features refer to patterns in groups of features, while low-level features are the entries of the input vector <cit.>. <cit.> argue that the meaning of feature attributions for low-level features is intuitive, if the low-level features have a predefined semantic translation. This is the case in most tabular data structures, where every features has a concrete meaning. In image data, however, individual pixels do not carry any semantic meaning and hence the use of feature attribution methods is not sensible unless we know whether the high-level features match our semantic representation, i.e., if we have semantic match <cit.>. Related approaches. To our knowledge, there has been little work to check whether (something like) semantic match is present. In natural language processing an approach that is similar in spirit is probing classifiers, which is a way of understanding whether a language model's internal representation is encoding some linguistic property <cit.>. Despite the shared intention to unpack sub-symbolic representations, model embeddings are not explanations and probing does not appeal to intuitions in the same way as feature attribution methods do. In image classification, a typical approach to explain the classification is using prototypes. Essentially, the explanation relies on `this looks like that' reasoning and provides prototypical images from the training data to explain some classification <cit.>. In <cit.>, this idea is extended by explaining in what visual aspects, such as color hue, saturation, shape, texture, and contrast, the test image is similar to the prototype. They quantify the influence of these aspects in a prototype and by that clarify the classification of the test image. Another approach for interpretable image classification is concept-based models or concept bottleneck models (CBMs) <cit.>. The core idea of such approaches is to map inputs onto some user-defined concepts, which are then used to predict the outcome class. In both CBMs and prototype explanations, the intention is to ground explanations by latching them to concepts or prototypes for which semantic match is given. Contributions. We propose an approach to test directly whether semantic match is present for the hypotheses that we are interested in, without the need of pre-defined concepts or prototypes. Our metrics for semantic match can also support a quantitative analysis of model behavior, and when semantic match is not achieved, we can side-step our intuition and thus avoid confirmation bias. § METHODOLOGY In this section we describe the main methodology in full generality and elaborate on the metrics to assess semantic match. We also outline the setup of the experiments on tabular and image data. §.§ Main procedure Consider a dataset with input vectors X = {x_1, x_2, …, x_n} and the corresponding output labels Y = {y_1, y_2, …, y_n}. Thus, the data point i is denoted by the pair (x_i, y_i) ∈ X × Y. We assume that a machine learning (ML) model f has been trained on this dataset and a local feature attribution method M is specified. The term M(f, x_i, y_i) = e_i defines the explanation e_i obtained by M for data point i that is classified with model f. When evaluating semantic match, we consider a specific sample x_c ∈ X and a corresponding explanation e_c. We are interested in testing whether we have semantic match with the explanation e_c, or in other words, whether what we `see' in the explanation is indeed what the explanation is capturing. At an intuitive level, what we want to ascertain is that an explanation matches our translation of it. This is encoded in the commutation of the semantic diagram from <cit.>, namely that all the data points giving rise to a certain explanation are also complying with our translation hypothesis, which we will indicate with θ, and vice versa. Mathematically this can be written as {x ∈ X \| M(f, x, y) ≈ e_c } = {x∈ X \| (x, y) θ}, where (x, y) θ denotes that the corresponding data point complies with the hypothesis θ. Note that we expect our reference point x_c to comply with θ, since it is the explanation e_c that elicited it. We propose a procedure to test how much such equality holds in a practical case. The procedure requires a distance metric between explanations, which we will denote as d, a maximum distance allowed ϵ, a number of samples n, and a method to test a hypothesis θ on data points. In case of explanations as vectors or matrices, there are standard notions of distance to employ; see Section <ref> for several examples. The choices of ϵ and n depend on how thorough an inspection one wants to conduct. We thus rewrite the test as X^ϵ{x ∈ X \| d(M(f, x, y), e_c) ≤ϵ} = X^θ{x ∈ X \| (x, y) θ} where the left-hand side is a set containing all data points generating an explanation close to e_c (subset X^ϵ), and the right-hand side is a set containing all data points satisfying θ (subset X^θ). To exemplify the procedure, suppose one has developed an algorithm to classify pictures of animals. Presented with a picture of a dog and an explanation e_c, one may formulate the translation hypothesis θ that the explanation highlights the tail of the dog. Following the procedure, one would first obtain a dataset with input images and explanations, and then identify the images with explanations sufficiently similar to e_c (i.e., subset X^ϵ), as well as the images that contain tails (i.e., subset X^θ). It is then possible to evaluate the overlap between these two sets. Note that, while our algorithm uses a single dataset for both X^ϵ and X^θ, it would also be possible to obtain separate samples for these two sets. For example, one can consider the use of generative models to generate samples satisfying θ, and evaluating the the similarity of the explanations obtained for this set compared to e_c. §.§ Metrics for semantic match We now turn our attention to quantifying the semantic match. Since we have defined distances in terms of a reference point, each of the metrics defined in this section will have to be re-computed by sampling data points within X_θ to understand how much the results depend on the choice of data point. Our two tests boil down to two questions: (1) how necessary is θ for an explanation similar to e_c, and (2) how sufficient is θ for an explanation similar to e_c? In proportion/probability notation, for some random input x_i ∈ X, we define q_1 = P(x_i ∈ X^θ \| x_i ∈ X^ϵ) and q_2 = P(x_i ∈ X^ϵ \| x_i ∈ X^θ). These two metrics effectively offer two perspectives on the overlap between the area defined by θ and the area defined by setting a threshold ϵ on the distances.[This formulation in terms of necessity and sufficiency might seem reminiscent of <cit.> but here the key difference is that we are not considering whether an explanation is sufficient for a certain prediction, we are instead considering how it matches with respect to θ.] One drawback of these quantities is that they are threshold-dependent, and the choice of threshold is somewhat arbitrary. An alternative way to test for semantic match is to think about the distances of the explanations from e_c as constituting a ranking of the data points. This ranking can then be used to `predict' which data points satisfy θ. Semantic match as classification. Using the training dataset, we construct a new dataset by relabeling the samples using the hypothesis θ. That is, we obtain new labels with θ by defining y̅_i = 1( (x_i, y_i) θ), i=1, …, n. Here, 1 stands for the indicator function. In other words, a sample is relabeled as one if it complies with the hypothesis; otherwise, its label is zero. Given the machine learning model f, the local feature attribution method M and explanation e_c, we can also construct `predictions' as follows: h_c(x_i) = 1( d(M(f, x_i, y_i), e_c) ≤ϵ), In light of this construction, the metrics in Eq. (<ref>) are the precision and recall values, respectively. When considering the distances as a ranking, we need to flip the sign since in our case smaller distances are supposed to indicate higher chance of satisfying θ, while in standard classification problems larger values are supposed to indicate the positive class.[One can think of the flipped distances as a proximity score, with higher values for explanation closer to e_c.] With this approach, we can resort to well-known metrics to measure the discrimination of rankings, such as the area under the ROC curve (shortened with AUC): for every threshold on the distance we can obtain a value of true positive rate and true negative rate, and vary the threshold to obtain the standard AUC plot. Coherence of explanations. Obtaining a high AUC for semantic match indicates that explanations allow us to separate the points satisfying θ from those that do not. However, AUC is invariant to monotone transformations and does not indicate how coherent explanations are with each other. In case of global behavior of the model (e.g., is the model placing attention on the object to classify), we are also interested in measuring the consistency between θ and explanations (for this dataset, these explanations and this model), not just on discrimination. In order to measure the coherence among explanations, we resort to the median distance of all explanations from the reference point e_c, to understand how much explanations cluster close to the θ region. §.§ Experimental setup For the sake of simplicity, we limit our experiments to a single explanatory technique and opted for SHAP values <cit.>, since it is a widely used technique that is easily applicable across data modalities. Experiments on tabular data. To illustrate the procedure to test semantic match, we first design a controlled experiment on synthetic data. We want to control the data generating process and have clarity on what is the high-level feature the model may be picking up, so that we can have clear expectations on whether semantic match should work or not. We generated a tabular dataset consisting of two continuous features normally distributed, x_1 and x_2, and one binary feature x_3. We proceeded to define a binary outcome by passing the function x_1x_3 – (1-x_3)x_1 + x_2 through a sigmoid and a 0.5 threshold. In this way, we incorporate a feature interaction between features x_1 and x_3 into the outcome; this will be the high-level feature of interest. We then trained a random forest on the dataset in order to predict the outcome from the three features, and generated explanations using SHAP. In principle, the random forest algorithm should be able to pick up on such feature interaction. Next, we wanted to use explanations to understand whether the model had learned about the feature interaction. We picked a data point x_c with negative x_1 value and x_3 = 0, whose explanation e_c gave positive contribution for both these features. We formulated the following hypothesis θ: “the model has learned that x_3 = 0 flips the effect of x_1 and thus increase the probability of the outcome when x_1 is negative and x_3 = 0”. We operationalized θ by considering the subset of the data where x_1<0 and x_3 = 0: this is the subset of data points for which – if semantic match is achieved – we expect explanations to be close to e_c. Finally, we define a notion of distance between explanations. We opted for Euclidean distance between vectors of SHAP values and deemed two explanations to be 'similar' whose distance was below a threshold ϵ, which we tested at different values. With these ingredients we are then able to test for semantic match. Experiments on images. We further experimented on a computer vision task from the literature with the goal of assessing semantic match for vision-related hypotheses. We employed data from MALeViC <cit.>, a dataset of synthetically-generated images depicting four to nine colored geometric objects with varying areas. The objects are generated at random locations in the images. Since MALeViC was originally introduced to study an object's contextually-defined size, each shape has a corresponding binary label – big or small – which stands for its size in the context of the whole image, i.e., whether it counts as big or small given the surrounding objects. These are based on an underlying threshold function considering the area occupied by the objects. For each image, the threshold T is computed as follows: T(I) = Max – k(Max-Min), where I is the image, k is randomly sampled from the normal distribution of values centered on 0.29 (μ = 0.29, σ = 0.066), and Max and Min are the areas, in pixels, of the biggest and smallest objects in I, respectively. During the construction of the dataset, an object is deemed big if its area exceeds T; otherwise, small. To solve this task, a model will need to construct high-level features capturing the role of the target object and the relationship with the other shapes. Here, we focus on the partition of the MALeViC dataset where all the objects in an image are either squares or rectangles. This choice has a practical motivation, namely to have a direct mapping between objects and their bounding boxes. Furthermore, we select images that contain one single red object. The resulting dataset is balanced in terms of objects' sizes. We split our dataset into training, validation and test sets (80:10:10). To augment our training data, we flip each image horizontally and vertically. Thus, we end up with 4800 images in the training set and 200 images in the test set. We trained a model to predict whether red objects are big or small. The model is a convolutional neural network (CNN) that takes the three-channel input images and outputs a probability over the two classes (small or big). Further details on the implementation are specified in Appendix <ref>. We generate explanations by obtaining pixel-level SHAP values. It is difficult to compare heatmaps directly because the shapes spawn at random locations in the images. The MALeVIC dataset comes equipped with metadata on the location of the bounding box for each shape in an image. After segmenting the whole image with Segment Anything Model (SAM) <cit.>, we are able to identify the relevant masks by matching their coordinates to the coordinates of the relevant bounding boxes using the metadata. This allowed us to consider only the SHAP values of the pixels inside a certain shape, as depicted in Figure <ref>. All the hypotheses considered concern the contribution placed on the target red object. Thus, we calculate the amount of SHAP values placed on the relevant shape using the bounding box as a mask on the SHAP heatmap. We add up the absolute value of the contribution of the pixels within the bounding box and normalize this quantity by the total contribution in the whole image. This gives us the proportion of the attention placed on the target object in the reference image x_c. Given any other image, we will calculate the proportion of contribution of the red shape with the same method, and measure the distance as the absolute difference in these proportions. Therefore, if our reference explanation places 40% of the contribution on the red object and in another image the contribution of the red object is 10%, the distance between the two explanations is 30% or 0.3. In this way, we have a notion of distance that is only considering the relevant parts of the images and is not affected by the fact that shapes can be located in different places. § RESULTS We summarize our results in the two sets of experiments. §.§ Synthetic experiments on tabular data On this simple task, the model has an AUC of 1.00, thus it must have learned about the feature interaction. Our experiment was geared towards testing whether the explanations matched our hypothesis “the model has learned that x_3 = 0 flips the effect of x_1 and thus increase the probability of the outcome when x_1 is negative and x_3 = 0”. By choosing a first data point as a point of reference, we obtained a semantic match AUC of 0.99 and a median distance of 0.20. This indicates that the distances allow us to distinguish data points where the first feature is negative and the third is zero, but explanations are not coherent. Inspection of the histogram of distances of the explanations (Figure <ref>, blue histogram) reveals a group of data points for which the distance is rather large. What is at play here is that x_2 is confounding our notion of distance: while x_2 is irrelevant for the hypothesis we formulated, our naive notion of distance does take into account the distance on that dimension too. This simple example brings into light the fact that hypotheses may be local. Revising our notion of distance to only consider dimension x_1 and x_3, we see a drop in median distance reaching 0.09, while semantic match AUC remains high at 0.92. We also observe in the orange histogram of Figure <ref> a shape that complies with expectations: explanations cluster close to e, with fewer and fewer examples as we allow for more distance. These observations allow us to conclude that we have a reasonable level of semantic match, and thus we are confident that the explanations reveal what our hypothesis has described. Note that in this process we have not dissected the model itself, which in principle has remained a black box. We elaborate on the robustness of these results in Appendix <ref>. §.§ Experiments on images On the MALeViC dataset, the model reached an accuracy of 91.5% on the classification task. We were thus interested in confirming the model has learned which shapes it is supposed to consider, a theory that is also suggested by visual inspection of some explanations. We formulated various hypotheses regarding the contribution placed on specific objects in the image. Notably, the distance between reference explanations (e_c) and all explanations expresses a difference in percentage of attention placed on the objects of interest. First, we studied the amount of contribution placed on the target object and the correctness of the model's predictions by formulating the following hypotheses: * θ_1: `≥ 10% of the attention is placed on the target object' * θ_2: `≥ 10% of the attention is placed on the target object and the prediction is correct' * θ_3: `< 5% of the attention is placed on the target object and the prediction is correct' * θ_4: `< 5% of the attention is placed on the target object and the prediction is not correct' The results are summarized in Figure <ref>. Hypotheses θ_1, θ_2, θ_3, θ_4 all obtain high AUC, suggesting the explanations clearly separate the data points complying to the hypotheses from the rest. Overall, the median distances are relatively small (i.e., the median of median distances stands close to 6%). This suggests that, for all explanations, variability is relatively low compared to the reference points. In particular, since the reference points for hypotheses θ_3, θ_4 have less than 5% of contribution placed on the target objects, this entails that all explanations put little attention on the target object. We can thus conclude that semantic match for those hypotheses is high and therefore the model does not behave as desired, i.e., focus on the target object. For further insights into the semantic match on the hypotheses considered, we refer the reader to Appendix <ref>. If the focus on the target object does not explain the high performance of the model, perhaps the model is using the smallest and largest object to perform the classification (see Section <ref> for explanations on the data generation). Therefore we formulated a similar set of hypotheses, expanding their scope to encompass the contribution placed on the triple of the target, biggest and smallest objects in the image: * θ_5: `≥ 30% of the attention is placed on the target, biggest and smallest objects' * θ_6: `≥ 30% of the attention is placed on the target, biggest and smallest objects and the prediction is correct' * θ_7: `< 15% of the attention is placed on the target, biggest and smallest objects and the prediction is correct' * θ_8: `< 15% of the attention is placed on the target, biggest and smallest object and the prediction is not correct' The results are summarized in Figure <ref>. Notably, the trends are similar to the ones observed in the previous set of hypothesis. Hypotheses θ_5, θ_6, θ_7, θ_8 all also obtain high AUC (albeit the first two with high variability), suggesting that explanations complying with the hypothesis and those which do not can be separated easily. The AUC distributions for this set of hypotheses is broadly comparable to the ones observed in hypotheses θ_1, θ_2, θ_3, θ_4, respectively. Overall, the median distances are small (i.e., roughly between 10% and 20%), although slightly larger than in the previous set of hypotheses. Overall, the budget of contribution devoted to the biggest, smallest and target object tends to be less than half. This indicates that there are more factors playing a role for the model to come up with a decision. Since the target, smallest and largest objects should be the only objects affecting the classification, this results strongly suggests that the model is utilizing some spurious correlation. Further insights into semantic match on the hypotheses considered can be found in the Appendix <ref>. § DISCUSSION In the previous sections, we laid out a procedure to investigate the semantic match between human-understandable concepts and attribution-based explanations. The procedure begins in a way that is akin to how such explanations are commonly used: by examining a data point with its explanation, and formulating a hypothesis on what the model has learned. We showed how this first step can continue by making the hypothesis more precise, and by defining a notion of distance between explanations that is hypothesis-driven. We then proposed some diagnostic tools to measure the level of semantic match between said hypothesis and the explanations. Such numerical analysis of the explanations can ground our intuitions and prevent confirmation bias. This framework was put to the test on synthetic tabular data and on a computer vision task on the MALeViC dataset. We observed how, without any prior knowledge about the model, the semantic match framework allows us to draw conclusions about model behavior. In the computer vision task, we started by investigating a desirable behavior, and concluded that the model was behaving in an undesirable way (i.e., not placing enough attention to the relevant shapes). Such experiments, while revealing the complexity of the problem, showcase that this framework can elicit useful information on model behavior and prevent confirmation bias. When it comes to limitations of this approach, it should be noted that this whole endeavor is predicated on the assumption that explanations have some degree of faithfulness to the model <cit.>. If explanations misrepresent the model, a semantic match between our ideas and the explanations is not going to give us information about the model. Moreover, we only experimented with SHAP, which is but one of the many feature attribution techniques available; it remains to be shown that this framework also generalizes to other explainability techniques. Furthermore, the experiments revealed several interesting aspects of semantic match. First, hypotheses are often local, in the sense that they pertain to a part of the input data, and it may not be straightforward to define a relevant notion of distance between explanations (see for example the bounding box problem in Section <ref>). Second, perhaps unsurprisingly, the results we obtained were sensitive to the specification of the hypothesis, highlighting the importance of formalizing hypotheses precisely and testing different specifications. Third, we find that sharpening the hypothesis does not necessarily lead to crisper results, see for instance θ_1 and θ_2 from the set of image experiment in Figure <ref>. More generally, the role of the logical structure of the hypotheses remains to be investigated; the divergent AUCs values for θ_3 and θ_4 from the set of image experiments in Figure <ref> might be related to fact that θ_4 contains a negation of a term of θ_3. In future work, we plan on expanding the suite of experiments further, tackling more real-world tasks and datasets, as well as expand to other feature attribution methods. We also intend to formalize more precisely a language for posing hypotheses, in the vein of a query language, so that the process of testing semantic match can be further refined and automatized. § EXPANDED RESULTS ON SYNTHETIC TABULAR DATA The simulation reported in <ref> was repeated adding a noise factor k to the outcome, to assess robustness of semantic match in light of noisy outcomes. More specifically, the baseline function that was passed through a sigmoid was changed to x_1x_3 – (1-x_3)x_1 + x_2 to x_1x_3 – (1-x_3)x_1 + x_2 + kx_4, where k is a chosen constant and x_4 is a standard normal variable (such as x_1 and x_2). Thus, a value of k=0.5 leads to a noise impact on the model of around half the impact of x_2. Simulations with different combinations of ϵ and k – with the same θ and sample size N=1000 – give the expected results (where q_1r and q_2r are the metrics calculated with the revised distance including only features x_1 and x_3); we display here some examples: * ϵ=0.05, k=0: q_1 = 1.0, q_1r = 0.84, q_2 = 0.025, q_2r = 0.1338. * ϵ=0.05, k=0.5: q_1 = 0.5833, q_1r = 0.5745, q_2 = 0.02229, q_2r = 0.0860. * ϵ=0.2, k=0: q_1 = 0.6633, q_1r = 0.5299, q_2 = 0.4204, q_2r = 0.7898. * ϵ=0.2, k=0.5: q_1 = 0.5735, q_1r = 0.5009, q_2 = 0.5096, q_2r = 0.8694. As the distance constitutes a reverse ranking (smaller distances should mean higher chance to fulfill the hypothesis), larger threshold ϵ means lower proportion q_1 (precision) and higher proportion q_2 (recall). In other words, the larger ϵ, the less similar explanations become, and P(x_i ∈ X^e) converges to 1. On the other hand, the larger the threshold the more q_1 (precision) tends towards P(X^θ). This is displayed clearly in plots such as Figure <ref> and <ref>. The noise appears to worsen semantic match – and performance on the downstream task – but these trends remain. § EXPANDED RESULTS ON THE MALEVIC DATA The input for the convolutional neural network is a 3-channel image containing squares or rectangles. The model used consists of three convolutional layers with 3,16 and 32 filters respectively. The dimension of the filters is 3x3. Each convolutional layer is followed by max pooling (2x2 filter) and a ReLU activation function. The output is flattened and passed through a dropout layer (25% rate), two fully connected layers and a sigmoid to output probabilities. The overall architecture is shown in Figure <ref>. The network was trained for 20 epochs using a batch size of 128. The chosen optimization algorithm was Adam with a learning rate set to 0.001 using the binary cross entropy loss. The random seed is set to 42. During training, the model with lowest validation loss was selected for inference. We report in Figures <ref> the histograms of the distances and in Figure <ref> the precision-recall curves for the four hypotheses. § EXPANDED RESULTS ON THE SST2 DATA §.§ Experiment Setup We trained a random forest classifier on the SST2 dataset. The input provided to the model was a bag of words-based binary vector representations of the sentences. In the first iteration, we created a vocabulary of words that occur in at least 25 instances in the training data. This led to a feature space of 2826 words, which was computationally cumbersome, both for training and subsequent analysis. Using the random forest model trained with this dataset, we selected the 50 most important features. The feature importance for the filtering was computed as the mean and standard deviation of accumulation of the impurity decrease within each random forest tree. Once we had the filtered vocabulary of 50 features, we trained a final random forest classifier for the final semantic match analysis and generated explanations using SHAP values. We verified that the trigger words of interest for our hypotheses i.e. {'no',' n't', 'not'} were part of the final curtailed vocabulary. The random forest model trained was a standard implementation from the Scikit learn library and was an ensemble of 100 trees, each tree using its own bootstrapped samples. It used the Gini index as the measure for selecting features for split and required a minimum of two samples per node for splitting. The SHAP algorithm used for this model was TreeSHAP. Note that this pruning of features, even though it had a negative impact on the downstream task of sentiment analysis, in no way impacted our major goals of verifying and quantifying semantic match, infact it help by reducing redundant complexity and made analysis easier within a smaller feature space. §.§ Visualizations We report in Figures <ref> the histograms of the distances and in Figure <ref> the precision-recall curves for the four hypotheses.
http://arxiv.org/abs/2307.07521v1	20230706225710	Artistic Strategies to Guide Neural Networks	[ "Varvara Guljajeva", "Mar Canet Sola", "Isaac Joseph Clarke" ]	cs.HC	[ "cs.HC", "cs.CY", "I.2.0; I.2.m" ]	Encoder-Decoder Networks for Self-Supervised Pretraining and Downstream Signal Bandwidth Regression on Digital Antenna Arrays This work was funded under US Defense Advanced Research Projects Agency agreement HR00112190100. Rajib Bhattacharjea DeepSig, Inc. Atlanta, Georgia, USA [email protected] Nathan West DeepSig, Inc. Rosslyn, Virginia, USA [email protected] ================================================================================================================================================================================================================================ Artificial Intelligence is present in the generation and distribution of culture. How do artists exploit neural networks? What impact do these algorithms have on artistic practice? Through a practice-based research methodology, this paper explores the potentials and limits of current AI technology, more precisely deep neural networks, in the context of image, text, form and translation of semiotic spaces. In a relatively short time, the generation of high-resolution images and 3D objects has been achieved. There are models, like CLIP and text2mesh, that do not need the same kind of media input as the output; we call them translation models. Such a twist contributes toward creativity arousal, which manifests itself in art practice and feeds back to the developers’ pipeline. Yet again, we see how artworks act as catalysts for technology development. Those creative scenarios and processes are enabled not solely by AI models, but by the hard work behind implementing these new technologies. AI does not create a ‘push-a-button’ masterpiece but requires a deep understanding of the technology behind it, and a creative and critical mindset. Thus, AI opens new avenues for inspiration and offers novel tool sets, and yet again the question of authorship is asked. § INTRODUCTION It is claimed that recent advancements in AI, such as CLIP-based products Midjourney and DALL-E, are supposed to augment our creativity. For the first time, it does not sound so absurd that artists can find themselves out of jobs nicholas2017these. Not that artists would have ever had a secure and stable job, but deep learning (DL) tools might eventually lead to losing some commercial commissions. Such thinking relies on a modern art approach where skills are in the centre of attention and not the conceptual idea. Quoting Lev Manovich: “Since 1970 the contemporary art world has become conceptual, ie focused on ideas. It is no longer about visual skills but semantic skills.” manovich2022ai As these new tools advance, the interfaces and techniques become more complex and sophisticated as our eyes are becoming more accustomed to not being easily surprised. Echoing Aaron Hertzmann, once painters were in a similar situation when photography was invented and took over the niche of portrait-making. Then visual artists had to re-invent themselves and re-think the meaning of painting. Photography had to wait another 40 years until it got recognized as an artistic medium hertzmann2018can. So-called AI artists have faced similar challenges in gaining acceptance within the art world and even inside the digital art niche roose2022ai. Computer art emerged with the invention of the computer. Artists, such as Vera Molnar and Manfred Mohr, created their first computer-generated artworks in the 1960s using scientific lab computers at night when they were not used by scientists. Early computer artists were re-purposing a machine for artistic use and writing code to make art on it. Since the creation process was mediated by a computer, it may seem to the general audience that the artists were simply pressing a button and the computer doing art for them. Hence, the question of authorship emerged: is the artist a machine or human? Today, with the appearance of neural networks (NN) and their creative applications, the same question re-appears. Hertzmann has written several articles arguing that people do art and not computers hertzmann2018can,hertzmann2020computers. Manovich also describes how AI-generated images that imitate realist and modernist paintings are claimed to be art manovich2022ai. At the same time, experimental art forms, like installation, interactive format, performance and sound art, are often overlooked unless they are promoted by a large corporation. Instead of re-telling a short but very dense history of DL technology development, in the next section, we focus on the appearance of neural network tools that raised interest amongst artists and led to meaningful artwork production. § HISTORICAL OVERVIEW OF DL DEVELOPMENT DL is a subset of machine learning (ML) using Deep Neural Networks (DNN) to learn underlying patterns and structures in large datasets. In 2012, a DNN designed by Alex Krizhevsky outperformed other computer vision algorithms to achieve the new state of the art in the ImageNet Large Scale Visual Recognition Challenge heravi2016classification. This model, AlexNet, signalled the start of a new DL era. As AI technology has developed and become more prevalent in real-world systems, artists have been exploring its limits and potentials, adapting these models to their own practices. As the number of scientific publications on AI grows exponentially it is useful to map out the influential papers, and related applications, to help track the evolution of the AI-Art space in relation to the technological advances krenn2022predicting. Figure <ref> shows a timeline of the development of generative models for images and text. Using this diagram we can make a few observations on the past ten years: the dominance of GANs for image generation, the influence of the Transformer on Large Language Models (LLM), and the growing interest in multi-modal approaches and translation models. The starting period of image generation using DNNs can be traced back to the creation of the Variational Auto-Encoder (VAE) in 2013, and the Generative Adversarial Network (GAN) in 2014 kingma2013auto, goodfellow2020generative. These models showed different ways in which a NN can be trained on a large dataset, and then used to generate outputs that resemble but do not copy the original dataset. For much of the past decade, GAN art has been a dominant and defining element of AI Art. GANs are trained using a competitive lying game, played by two players: the Generator and the Discriminator. The Generator wins by making an image that the Discriminator thinks is from the original dataset. The Discriminator wins by successfully identifying which images the Generator has made. By playing this game repeatedly, both sides slowly learn when they have been fooled and remember information so they don’t fall for the same tricks again. The Generator gets better at making images, and the Discriminator gets better at detecting these fakes. At the end of the game we are left with a Generator that is very good at generating new images, with the qualities and style of our original inputs. After the original GAN paper, there was a rush of exploration of this new technique for generating images. Alongside general improvements to the models architecture and stability, new ways of guiding the outputs and applying GANs to specific problems were also explored radford2015unsupervised,arjovsky2017wasserstein. Image-to-Image Translation with Conditional Adversarial Nets (2016), also known as pix2pix, showed a process of converting one type of image into another type isola2017image. Mario Klingemann’s work Alternative Face[<https://underdestruction.com/2017/02/04/alternative-face/>] used the pix2pix model with a dataset of biometric face markers and the music videos of the singer François Hardy. This allowed him to control the movement of the face with this form of digital puppetry, which he then demonstrated by transferring the facial expressions of the political consultant Kellyanne Conway onto Hardy’s face as she talks about “alternative facts”. In 2015, on the Google research blog, the post Inceptionism: Going Deeper into NNs described a tool that attempted to understand how image features are understood in the hidden layers of the NN mordvintsev2015inceptionism. Alongside this post they released a tool called DeepDream. This model enhances an image with the NN's attempts to find the features of the dataset it was trained on. The creative use of DeepDream was proposed by the authors in the original article “It also makes us wonder whether neural networks could become a tool for artists—a new way to remix visual the creative process in general”.[13] DeepDream’s psychedelic imagery quickly caught the attention of the internet and of artists around the world, resonating with those interested in understanding the cross-over between biological and neurological construction of images. Memo Atken’s work All Watched Over By Machines Of Loving Grace[<https://www.memo.tv/works/all-watched-over-by-machines-of-loving-grace-deepdream-edition/>]: Deepdream edition, hallucinated over an aerial photograph of the GCHQ headquarters. This work raises questions around the motivations of the organisations funding the development of AI, and in doing so make the dreamlike qualities a little more nightmarish. In the same year, the paper A Neural Algorithm of Artistic Style introduced a DNN “to separate and recombine content and style of arbitrary images, providing a neural algorithm for the creation of artistic image” gatys2015neural. Neural Style Transfer (later known simply as StyleTransfer) takes two inputs, a style image and a content image, it extracts textural information from the style image and compositional information from the content image, then generates an image with minimal distance between the two. The paper demonstrates this with images of a DeepDream’s psychedelic imagery quickly caught the attention of the internet and of artists around the world, resonating with those interested in understanding the cross-over between biological and neurological construction of images. Memo Atken’s work All Watched Over By Machines Of Loving Grace: Deepdream edition, hallucinated over an aerial photograph of the GCHQ headquarters. This work raises questions around the motivations of the organisations funding the development of AI, and in doing so make the dreamlike qualities a little more nightmarish. In the same year, the paper A Neural Algorithm of Artistic Style introduced a DNN “to separate and recombine content and style of arbitrary images, providing a neural algorithm for the creation of artistic image” gatys2015neural. Neural Style Transfer (later known simply as StyleTransfer) takes two inputs, a style image and a content image, it extracts textural information from the style image and compositional information from the content image, then generates an image with minimal distance between the two. The paper demonstrates this with images of a photograph represented in various styles of famous paintings, such as Van Gogh’s The Starry Night. In 2017, CycleGAN continued with the problem of image-to-image generation shown in pix2pix, but removed the requirement of aligned image pairs being needed for training zhu2017unpaired. Instead a set of source images and a set of target images that are not directly related can be used. The advantage of this is it is simpler to scale to larger datasets, making the process more accessible for artists. Helena Sarin has been using CycleGAN for a number of years, and recently in Leaves of Manifold[<https://www.nvidia.com/en-us/research/ai-art-gallery/artists/helena-sarin/>][<https://twitter.com/NeuralBricolage/status/954027624728354821>] she collected and photographed thousands of leaves to build her own training dataset, and then implemented a custom pipeline with changes that improve results when working with smaller datasets. This personalised approach in crafting the models resonates with the hand-made, collaged aesthetic of the images generated. Other notable developments to GANs brought improvements to image quality and resolution karras2017progressive,wang2018esrgan. In late 2018, the release of StyleGAN, a model built on a combination of ideas from Style Transfer and PGGAN, demonstrated very convincing images of human faces karras2019style. In his article “How to recognize fake AI-generated Images”, the artist Kyle McDonald investigated the images generated by StyleGAN, and highlighted the visual artefacts he found mcdonald2018recognize. At a glance these images look like photographs, but on closer inspection irregularities such as patches of straight hair, misaligned eyelines, or mismatched earrings reveal the difficulties GANs have in managing “long-distance dependencies” in images. In 2017 the paper Attention Is All You Need proposed a new network architecture called the Transformer vaswani2017attention. This model addressed the long-distance dependency issue in RNNs and CNNs by rethinking how we could handle sequences. Rather than looking at a sentence word by word, the Transformer observes the relationship between all elements of the sequence simultaneously. Being able to better handle long distance dependencies meant the Transformer was appropriate for natural language generation. Artists have explored the use of VAEs for short text generation, but with the emergence of LLM passages of long, coherent, texts could be generated brown2020language. As dataset sizes increased, along with hardware costs for training these large models, they have become harder for individuals to train themselves, and the mode of interaction has shifted from curated datasets and homemade scripts, to web APIs and third party services. While it is more difficult to participate in the training process, the availability of services and interfaces provides new ways of working with these models that can produce less technical and more playful approaches. For example, Hito Steyerl used GPT-3 to create Twenty-One Art Worlds: A Game Map and described the process as “fooling around” with GPT-3 to write descriptions of different Art Worlds steyerl2022twentyone. In the resulting text it is difficult to distinguish which words may have been written by Steyerl and which were written by GPT-3. The learnings from LLM for text generation were soon applied to image generation (Image GPT, Vision Transformer), and the simultaneous release of CLIP and DALL-E in January 2021 signalled the start of a new era of image generation chen2020generative,dosovitskiy2020image. Although the DALL-E model was not released, CLIP was made available to the public, and the model was quickly adopted by AI artists who applied the idea of CLIP guidance to various image generation techniques. Ryan Murdock produced the colab notebooks DeepDaze[<https://github.com/lucidrains/deep-daze>] (combining CLIP and SIREN) and BigSleep[<https://github.com/lucidrains/big-sleep>] (CLIP and BIGGAN), which were subsequently adapted by Katherine Crowson in the widely distributed VQGAN+CLIP[<https://github.com/EleutherAI/vqgan-clip>] notebook. The paper Denoising Diffusion Probabilistic Models introduced a different method for creating generative models ho2020denoising. This technique trains a model by adding increasing amounts of noise to an image and then having the model remove the noise, resulting in a model that can generate images from only noise. Diffusion models, when combined with CLIP or other conditioning processes, enable much faster text-to-image processing. The popularity and accessibility of these techniques was further raised by the release of DALL-E 2 and Midjourney in 2022. Midjourney became so popular it is now the largest Discord server with over 5 million members. Following the releases of these products, open source models such as Stable Diffusion have also been developed. There are many benefits of using free and open source models for artists. Being able to modify code and develop on your own software allows the artist to pursue their own experimental approaches, not restricted to the interface designed by a service provider. The artist's involvement in generating new images with these models is vastly different to working with GANs. Rather than building custom datasets and training models, instead the focus has shifted to writing prompts that can generate the images the artist wants to find, and designing interfaces for exploring these prompts and their translations. The artist Johannez coined the term Promptism for describing his art practice, and wrote a humorous Prompist manifesto using GPT-3. Against a backdrop of models trained on hundreds of millions of images scraped from the internet, including many artists’ portfolios, the manifesto asserts “The prompt must always be yours” johannezz2022promptist. § ARTIST-GUIDED NEURAL NETWORKS Many papers discuss AI from the point of view of creativity taking mostly one position of two: either AI as an amazing tool for artists and creativity, or AI is seen as something negative in art. It is easy to see that the people from industry advocate for the first position, and theory scholars for the second one. But, how do practitioners see contemporary AI technology themselves? And in which ways AI is deployed in art practice? Hence, it is not the focus of this paper to discuss whether AI can make art, but rather how AI can be useful for artists and what new ideas it can offer. By using practice-based research methodology, we decode the role of AI tools in artistic practice and trace the evolution of such artistic work. In this paper, the practice of artist duo Varvara & Mar was used as a case study, which provided us with the insides in this research. We divide the case studies into four categories based on medium: synthetic image, synthetic text, synthetic form, and translation models. From the view of the practitioner, the limitations, new possibilities, and change in production processes are discussed. §.§ Synthetic Image Our DL exploration began in 2017 with Google DeepDream, focusing on image generation (Fig.<ref>). The concept behind Neuronal Landscapes[<https://var-mar.info/neuronal-landscapes/>] project was to imagine how Estonian landscape will look like in 100 years time (commission work for the Estonian History Museum). Through synthetic vistas created by machines, the artwork offers a glimpse into the environment from a machine's perspective, immersing viewers in a hallucinated neural net simulacrum. To depict the evolution of Estonian society over time, from forests and farmlands to urbanization and digitalization, a 360º VR video was created. Filmed with drone-mounted two 360º cameras, the footage was edited and processed using DeepDream. The rendering process spanned 30 days on powerful machines with Nvidia TitanX GPUs. While some customization was possible, the algorithm's aesthetic footprint remained prominent. In the next art project, ProGAN was deployed. For the first time we worked with datasets and training GAN models. Plasticland[<https://var-mar.info/plasticland/>] (2019) talks about plastic waste and ecological problems this material causes (Fig.<ref>). We composed four different datasets of images of layered plastics in our planet: landfills, plastic on top of water, plastic underwater, and plastiglomerates. The ProGAN model was trained on a local machine using pyTorch and took a week to train, and the artist used a selection of generated images to create a video composition. A metal totem displaying those synthetic, as plastic is, layers, we draw attention not only to the problem of waste but also question whether AI has some similarity with this material. Since the invention of plastic, this material was applied almost everywhere because of its perfect qualities, until we realised that it is not sustainable and ecology-friendly. Will a similar story happen with AI? From the practice-based research perspective, this work shows artists’ desire to move from a still to moving image and towards sculptural form that is held back by the early stage of machine learning technology: low resolution images jumping from one frame to another. The next artworks POSTcard Landscapes from Lanzarote I (00:18:37) and II (00:18:40)[<https://var-mar.info/postcard-landscapes-from-lanzarote/>] in 2021 demonstrate the artist's ability to create video works with StyleGAN2 (Fig.<ref>). The hypnotic appearance of these works, where one frame morphs naturally into another, shows the artists' ability in guiding the outputs of the neural network. Vector curation and composition of a journey through the latent space, created by training the model on specific datasets of 2000+ images, were crucial and integral parts of the artistic process. The artwork talks about critical tourism and how circulation of images representing touristic gaze overpower the nature of seeing. In the words of Jonas Larsen “‘reality’ becomes touristic, and item for visual consumption” larsen2006geographies. Hence, we scraped, where licence allowed, the location-tagged images from Flickr and composed two datasets of photos categorised as tourism or landscape. As we have written earlier: “The two videos are random walks in the latent space of the Stylegan2 trained models, creating a cinematic synthetic space. The audiovisual piece shows an animated image through the melted liquid trip of learning acquired from the dataset composed of static images. The video flows from point to point, generating new views and meaning spaces through the latent space’s movement. The audio was created after the video was generated in response to the visual material to complete the art piece.” guljajeva2022postcard. The sound for local or landscape view was created by a sound artist from Lanzarote, Adrian Rodd, who aimed to give a socio-political voice to the piece. In contrast, the sound design created by Taavi Varm is a soundscape replying to touristic gaze. The artists aimed to initiate collaborations with others but also to experiment with human-AI co-creation. In a similar vein is the artwork Phantom Landscapes of Buenos Aires (00:20:00, 2021), with sound work by Cecilia Castro. Our last experiment with GAN models Synthetic-scapes of Tartu (00:10:00, 2022), demonstrates a different approach. Taking a dataset composed from our own video footage (flaneur walks), we first produced the sound (a composition by Taavi Varm, Ville MJ Hyvönen with piano by J. Kujanpää) and used this to inform the direction of the video . The result was a sound-guided AI-generated visual output. §.§ Synthetic Text In this section, we focus on artwork incorporating AI text generation as part of the artistic concept. Our journey to text generation started with the online participative theatre project ENA[<https://var-mar.info/ena/>] and ended with a hand-bound publication (Fig.<ref>). During the first lockdown in May 2020, together with theatre maker Roger Bernat, we created an online participative theatre piece ENA on the website of Theater Lliure in Barcelona. ENA is a generative chatbot that talks to its audience, and together (AI and audience), they make theatre. As we have described before: “Although in the description of the project it was stated explicitly that people were talking to a machine, multiple participants were convinced that on the other side of the screen another human was replying to them—more precisely the theatre director himself, or at least an actor.” guljajeva2021ena. Analysing synthetic books, Varvara Guljajeva has stressed the importance of human input in the AI text-generation systems guljajeva2021synthetic. In addition, one also needs to guide the audience participation and interaction with the chatbot. For this purpose, we have adopted the traditional theatre method for guiding actors, as a way to guide the audience, and thus, the bot, too. Stage directions were used as a guiding method, which triggered thematic conversation and offered meaningful dialogue between humans and the AI system. We found the conversations so meaningful that we decided to publish a book that contains all the conversations with ENA. With this project, we learned that it is essential to guide neural networks via audience interaction. In order to do this, it is also necessary to guide the audience. Without audience interaction guidance, it is nearly impossible to achieve meaningful navigation of neural networks. §.§ Translation models This category focuses on translation models that enable interactive and installation-based formats. Translation refers to the conversion of mediums, or as we put it, translation of semiotic spaces. To illustrate this, we introduce Dream Painter[<https://var-mar.info/dream-painter/>] an art installation that translates audience’s spoken dreams to a line-drawing produced by a robot (Fig.<ref>). As described earlier: “Dream Painter is an interactive robotic art installation that explores the creative potential of speech-to-AI-drawing transformation, which is a translation of different semiotic spaces performed by a robot. We extended the AI model CLIPdraw which use CLIP encoder and the differential rasterizer diffvg for transforming the spoken dreams into a robot-drawn image.” canet2022dream. “Design- and technology-wise, the installation is composed of four larger parts: audience interaction via spoken word, AI-driven multi-colored drawing software, control of an industrial robot arm, and kinetic mechanism, which makes paper progression after each painting has been completed. All these interconnected parts are orchestrated into an interactive and autonomous system in a form of an art installation […].” guljajeva2022dream. Out of all the projects discussed, this was the most difficult to realise. This is because of the large scale of the artwork, and multiple parts of software and hardware that need to run automatically and synchronously. In this project we investigated how guidance of neural networks could be interactive and real-time instead of non-interactive and pre-determined, as shown in previous examples of our work. It is important to notice that methods, such as dataset composition and output curation were not used in this case. In fact, visual output curation is totally missing. The artists created an interactive system to be experienced and discovered by the audience. This means the audience determines the output. Instead of curating a dataset, a CLIP model is used that can produce nearly real-time output guided by a text prompt. As we have written earlier: “Translation of semiotic spaces, such as spoken dreams to AI-generated robot-drawn painting, allowed us to deviate from image-to-image or text-to-text creation, and thus, imagine different scenarios for interaction and participation.” guljajeva2022dream. This project indicates our search for transformative outputs of AI technology, and thus, shows the evolution in practice. By extending available DL tools and combining with other technology, for example, text-to-speech models, real-time industrial robot control, and physical computing, it offered an interactive robotic and kinetic experience of neural network latent space navigation. This contributes towards the explainability of AI because the audience could experience how the words affected the drawing, and which concept triggered which outcome. Being inspired by Sigmund Freud's work on the interpretation of the human mind while unconscious, we speculatively ask if AI is powerful enough to understand our dreamworld. Through practice we question the capacities of neural networks and investigate how far we can push this technology in the art context. This artwork allows the audience to experience the limits of concept-based navigation with AI. The system is unable to interpret and can only illustrate our dreams. It cannot understand the prompt semantically and only gets the concepts. §.§ Synthetic Form In this section, we ask how artists can guide neural networks when creating volumetric forms, and what happens when AI meets materiality. After working for a while with DL tools that produce 2D outputs, it is an obvious step to explore possibilities to produce 3D results. To our surprise, it was not an easy task to find the solution (Oct 2021). Psychedelic Forms is a series of sculptures produced in ceramics and recycled plastic through which we investigated the possibilities of AI in producing physical sculptures. The project re-interprets antique culture in the contemporary language and tools guljajeva2023ai. Following the same paradigm shift as in the previous section, text2mesh is a CLIP-based model that does not require a dataset, but a 3D object and text prompt as input michel2022text2mesh. Hence, the model actually does not create a 3D model but stylises the inserted one, guided by inputted text. We decided to go back to the origins, in terms of ancient sculptures and material selection. Although it was said that there was no dataset, we still had a collection of 3D models of ancient sculptures because, by far, not all produced a desirable output. In this sense, there was definitely an output curation present in the process. The criteria for selection were the following: first, the form had to be intriguing, and second, it should be possible to produce it in material afterwards. It was clear that we had to modify each model because the physical world has gravity, and the DL model does not take this into account. Some generated models were discarded because they were seen as not-fixable, although interesting in their shape. The process demonstrated here is quite an unusual way to create an object. After extensive experimentation with the tool, we learned how certain words triggered certain shapes and colours. This knowledge gave us a chance to treat text prompts as poetic input. Thus, we created short poems to guide NN. The best ones survived as titles and are reflected in the forms. The artists did not strictly follow the original model but took the creative liberty to modify the shape and determine the colour by manually glazing the sculptures. The dripping technique was used for colouring the sculptures. This served as a metaphor for liquid latent space and the psychedelic production process (this was the artists’ inner feeling about the creative process because they did not know what results would be achieved in the end). Sometimes, AI-generated vertex colouring was taken as inspiration, sometimes totally ignored. Nevertheless, digital sculptures were exhibited alongside the physical ones to underline the transformation and human role in the creative process. Although ceramic sculptures were 3D printed in clay, the fabrication process had to follow the traditional way of producing pottery (Fig.<ref>). Since the artists had never engaged in ceramics before, the whole production process felt psychedelic: unexpected neural network processes led to transformation by numerical, physical, and chemical processes, all guided by both the artists and chance. Hence, the art project highlights the relationship between different agencies. In the end, we can say that AI is not prepared for the physical world. It created nice images, but when one wants to materialise the output, it requires considerable additional work. However, those extra processes were very rewarding and creative in our case. In this project, AI served as an inspiration or a departing point more than anything else. In other words, the experimental phase of technology is necessary for experimental practices, and this can lead to the creation of a new production pipeline. The fine line between control and chance when guiding the neural networks and related processes is likely the main creative drive for the artists. § DISCUSSION According to the media hype around AI, this technology is intelligent enough to create art autonomously perez2018microsoft,vallance2022art. However, the reality is different. A computer scientist and a co-inventor of Siri Luc Julia, AI does not exist. He advocates for machines’ multiple intelligences that often outperform humans. However, machine intelligence is limited and discontinuous compared to human intelligence julia2020there. Therefore, it is vital to have artistic practices around this technology, as a counterbalance to the AI fantasies served by the industry and mass media. We see AI as a creative tool with its own possibilities and limitations, which can stimulate artists’ creativity through unexpected outputs. Research has shown that tool-making expands human cognitive level and constitutes evolution in culture stout2011stone,stout2016tales. Similarly, as a new tool, generative AI could potentially enrich creativity by allowing new production pipelines that can create unique results. Coming back to the synthetic images, we can say that all machine-created synthetic image-based works discussed here have particular aesthetics: both with DeepDream and GAN. Unlike the output of GANs, DeepDream has a more recognizable style and can be seen more as a filter that transforms every inputted image instead of learning from the given dataset. Regarding GAN aesthetics, such visual appearance is inherited from two entities to a large extent: the dataset and the model itself. GANs have a particular footprint, as seen in all works produced with this model. The visual palette comes from the used datasets. For example, if a dataset is homogeneous (only landscape images), then we will easily recognize landscapes in the generated output. However, if images in the dataset have a lot of visual variation, the output is rather abstract. POSTcard Landscapes from Lanzarote II illustrates this well. Also, when photos in the dataset look similar, the output will also be similar, as was the case with the Synthetic-scapes of Tartu video work where frames from recorded flaneur walks in a city were extracted. When we talk about video works generated with the neural net, then manual guidance of latent space offered more variations than an audio-led approach. Synthetic image works have encouraged us to work with formats like images and videos that we did not engage in before in our art practice, but we found it exciting working with AI and video. For example, AI video generation has some affordances, like starting and ending can be done in a perfect loop since images are synthetically generated. However, creating real-time AI work is much more complex because some models are too slow. It might take a few minutes to render a single image. The limitations inspire us to devise new solutions and work in new mediums. Moreover, the limitations of the medium has always been a good challenge for our creativity. Working with GANs or other image-generation tools has become much easier in recent years, although it used to be quite difficult. We must note that for practitioners, easy-to-use tools, such as DALL-E and Midjourney, offer little creative freedom, and thus, are less attractive to the artists. Those products tend to instrumentalize the user rather than the other way around. At the same time, open source models offer more creative freedom and enable broader use of artistic ideas. The work with generated text demonstrates that AI is not context-aware but maps concepts automatically without understanding semantics. More importantly, as shown in the ENA project the audience must also be guided alongside the AI. In the case of ENA, stage directions were used, and in the Dream Painter project, the concept of dream telling was applied to guide the participants who in turn guided the neural net through their interaction, creating a chain reaction. Navigating concepts in latent space is artistically interesting and inspiring, this was especially evident when working with form. The artists went beyond semantics and learned how to guide neural networks with a text prompt and 3D object. The presented practice represents a paradigm shift in machine learning, moving away from composing datasets for GANs and toward translating semiotic spaces enabled by diffusion models. The evolution in practice shows how artists discover and learn to work with the DL toolset, embracing its possibilities and limitations. In the case of practice-based research, practice can be seen as a lab for testing artistic ideas with technology through chance until control is encountered. § CONCLUSION In this article, we have summarised DL development from the perspective of artists’ interests concentrating on the image, video, text, 3D object generation, and translation models. We applied practice-based research methodology to investigate the role and possibilities of recent co-creative AI tools in artistic practice. It is difficult to keep pace with AI development. In less than a decade, we have gone from blurry black-and-white faces to impressive high-resolution images guided by text prompts. The user level has gone from difficult to easy, which on one side, broadens possibilities for creation, but on another, it diminishes experimentation and creativity, since AI outputs seem ready-made. This is also demonstrated by the explorative nature of the body of work presented here. Furthermore, it was noticed that creative AI, especially GAN models, have recognizable aesthetics, which in the long run, become repetitive. This led to the change of tools by the artists. The curation of datasets, models, and outputs, along with neural network guidance, have become the toolset of an artist working with AI. Finally, these models can generate multitudes of outputs, but the art is giving the right input to guide the desired output and selecting the results that best serve the concept As Andy Warhol had envisioned in 1963, eventually, art production will become mechanised and automated 10.1093/oxartj/kcy001. In his own words: “I want to be a machine” bergin1967andy, which was also a reflection on that time's vast industrialization process. Resonating with today’s deep learning age: I want my machine to do art. § AUTHOR CONTRIBUTIONS, ACKNOWLEDGMENTS AND FUNDING MSC is supported as a CUDAN research fellow and ERA Chair for Cultural Data Analytics, funded through the European Union’s Horizon 2020 research and innovation program (Grant No.810961).
http://arxiv.org/abs/2307.02228v1	20230705121008	Local flow measurements around flexible filaments under rotating magnetic field	[ "Andris P. Stikuts", "Abdelqader Zaben", "Ivars Driķis", "Māra Šmite", "Rūdolfs Livanovičs", "Andrejs Cēbers", "Guntars Kitenbergs" ]	physics.flu-dyn	[ "physics.flu-dyn", "cond-mat.soft" ]	Local flow measurements around flexible filaments under rotating magnetic field]Local flow measurements around flexible filaments under rotating magnetic field MMML lab, Department of Physics, University of Latvia, Riga, Jelgavas 3, LV1004, Latvia [email protected] Effective mixing of fluids at the microfluidic scale is important for future applications in biology, medicine, and chemistry. A promising type of micromixers are magnetic filaments, which can be activated by an external magnetic field. However, there is a lack of research that combines experiments and numerical modelling of hydrodynamics around such filaments. Here we use micro-particle image velocimetry to measure flow fields around rotating flexible ferromagnetic filaments and compare them to numerical data from elastic rod model. We measure that rotating filaments hover above the surface, whereas the resulting fluid velocities are highly dependent on the hovering distance. We also find that the rotating filament causes a 3D flow coming from the rotational plane and being extracted along the axis of rotation. These findings will help develop better micromixers. § INTRODUCTION Mixing in a low Reynolds number regime is an increasingly active topic of research because of its importance in microfluidics and development of lab-on-a-chip devices. Mixing is required for various applications, particularly in the fields of biology, medicine, and chemistry <cit.>. A promising approach is magnetic mixing or magnetic stirring, where mixing is done with magnetic materials and an external magnetic field <cit.>. Although it is an active mixing method, application of an external field is easy to implement, and there is a wide selection of magnetic materials. An interesting active micromixer that shows promising efficiencies as presented by D.Owen et al. <cit.>, is realised by means of actuating magnetic beads under a rotating magnetic field. Here, we focus on micrometre-sized ferromagnetic beads linked to form a flexible filament and driven by an external rotating magnetic field for stirring. Their advantages include commercial availability and relatively low cost for filament synthesis <cit.>. Besides the on-demand control of mixing, magnetic filaments have been shown to be a promising candidate for cargo delivery <cit.>, sensing <cit.>, and microrheology <cit.>. The dynamics of flexible ferromagnetic filaments under the action of a rotating magnetic field has been investigated by our group both numerically <cit.> and experimentally <cit.>. These studies allow us to estimate the physical properties and deformation of the filament shape under the action of rotating magnetic field. Moreover, it was found that ferromagnetic filaments take a 3D shape when operating at higher frequencies <cit.>. This is different for flexible paramagnetic filaments where back-and-forth motion is observed for higher frequencies <cit.>. This also limits the mixing rate for paramagnetic chains, where a rapid drop in efficiency occurs after an optimal frequency <cit.>. The difference in behaviour gives us the motivation to further investigate the use of ferromagnetic filaments as mixers. For a better understanding of the physics related to micromixing, it is esential to characterise the hydrodynamics around such filaments. Experimentally, this includes measurements of flow fields, accessible by the micro Particle Image Velocimetry (μPIV) technique. Although challenging due to resolution limits, this approach has been used to study hydrodynamics at low Reynolds number for artificial micron sized swimmers and microorganisms, including L-shaped swimmer, driven by a rotating magnetic field <cit.>, flow measurements around microalgae <cit.> and 3D flow measurements for magnetically actuated artificial cilia using stereoscopic μPIV <cit.>. However, coupling of experimental measurements around a simpler model system flexible filament with the corresponding numerical simulations has not been done. We aim to solve this problem with this paper. First, the experimental system is described, including the steps to improve the resolution. Second, the numerical model is introduced, and the important aspect of the filament's height above the surface is discussed. Finally, we compare the experimental and numerical results, noticing and characterising a 3D flow, induced by the filament rotation. § EXPERIMENTAL SYSTEM §.§ Filament Synthesis For filament synthesis we follow the method proposed by Dreyfus et al. <cit.>. It uses the formation of a streptavidin-biotin bond between DNA fragments and magnetic particles, aligned by a static magnetic field. In detail, this method has been described by K.Ērglis <cit.>, and was further adopted for our earlier studies on filament dynamics under a rotating field <cit.>. The ferromagnetic particles (Spherotech, 1 %w/v) are made of polystyrene and are covered with a layer of chromium oxide, functionalised with streptavidin, and have an average diameter of 4.26 μm. DNA fragments (Latvian Biomedical Research and Study Centre), biotinylated at the 5' ends, have a length of 1000bp and a concentration of 192 μg/ml. Samples are made with the following procedure. 40μl of 0.53 μm tracer Nile red fluorescent particles (Spherotech, 1 %w/v), for particle displacement measurements, is mixed with 0.5 ml of 10% TE buffer solution (η = 0.001 Pa, pH = 7.5), resulting in a concentration of 0.08 %(w/v). 10 μl of 0.01% Sodium Dodecyl Sulphate (SDS) is then added and the sample is sonicated for 30 min, to reduce particle aggregations. 10 μl DNA (6.2 nM) and 2 μl of magnetic particles (0.004% w/v) are then added to the sample, and placed for two minutes between two Neodymium magnets, creating a homogeneous field of ≈ 50 mT. §.§ Experimental setup The experimental setup consists of a micro Particle Image Velocimetry (μPIV) system combined with a coil system for magnetic field generation. The μPIV setup (Dantec Dynamics) consists of an inverted optical microscope (Leica DM-ILM with a Y3 filter cube, 40x objective), a HiSense MkII camera for image acquisition, which has a CCD sensor with double frame mode (maximum frame rate 6.1 Hz, minimum interframe time 200 ns, double pulsed YAG laser (λ = 532 nm) which is used as a light source for exciting the florescent particles and has a maximum frequency of 50 Hz. The camera and the laser are synchronised through a timer box (NI PCI-6602) and controlled by DynamicStudio software. Magnetic field generation is done by an in-house built coil system, which has 3 pairs of coils to generate magnetic field in the three dimensions. The coils are connected to 3 AC power supplies (Kepco BOP 20-10M), giving a maximum current of 7 A, which correspond to fields up to 12 mT. The desired field profile is generated by a Labview code. For a rotating field profile, two sine signals with a phase shift of 90 degrees are generated and sent through National Instruments data acquisition card (NI PCI-6229) that is connected to the power supplies. §.§ Experimental procedure For observation we use fluidic cells, which are made from two glass slides separated by a 211 μm thick double-sided adhesive tape. Experiments were conducted as follows. 20 μl of the prepared filament sample is pipetted in the fluidic cell and placed under the microscope. The acquisition mode is set to double frame mode while the inter-frame time and the acquisition frequency is chosen based on the field frequency. These parameters are given later. For each experiment, double frames are acquired for filaments that complete 9 to 25 rotations. The laser power was set to 100% and a maximum attenuator level of 25%. For higher levels, the filament was found to break. Due to gravity, the filaments were found to sediment to the bottom of the fluidic cell, with free particles around, which may have detached during the transfer process. A rotating field (B=1.72 mT, f=1 Hz) is applied first for 10 to 15 minutes to allow the free particles to connect to the filament. During the rotations, filaments were also found to have a slight shift, likely due to interactions with the wall, which differs for particles with different sizes. An example of an acquired image is given in figure <ref>, where (a) shows the rotating filament with fluorescent tracer particles around, while (b) and (c) visualise the first and second frames of the double frame mode on a small zoom-in region. For better visability, the contrast of these images is increased. §.§ Particle Image Velocimetry and data processing For processing, we export the experimental images from Dynamic Studio and do initial image processing in Matlab. We first mask the filament to eliminate its signal in the cross-correlation calculations, and we detect only the flow profile around the filament. The filament was first segmented, based on intensity thresholding, then dilated to increase the mask size to remove any tracer particles that may got attached to the filament during the experiment. An example of the masked region is indicated in figure <ref> (d) with a green border. Patricle image velocimetry (PIV) analysis was performed using PIVlab, an open source software developed by W. Thielicke and E.J. Stamhuis <cit.>. The PIV algorithm used here is based on Fourier transform cross-correlation, and the analysis was made by defining the interrogation area as 48×48 pixels with a PIV step size of 16 pixels. As the PIV measurement takes place close to the resolution limit, the resulting velocity values are noisy. An example of flow field obtained from a single double frame image is given by the red vectors in figure <ref> (a)&(d). A qulitative character of the flow field can be obtained, but the pecularities are lost in noise. Such noise is usually eliminated by time averaging. In our experiments, the filament rotates with its centre of rotation moving, so summing the velocity fields simply frame by frame is not an option. We use Python for further experimental image and data processing and solve this problem as follows. The first step is a geometric preprocessing. We find the filament's rotation angle in each frame, by selecting the data sequence where the rotation rate is steady. Due to the chosen camera framerate and field frequency, there are only a small number (1, 2, 3 or 6) of filament rotation angles against a fixed direction.. We find the precise angle values by cross-correlating images generated by the forward Radon transform, obtained using functions from scikit-image library <cit.>. To find the filament displacements during the experiment, we first find the displacements for filaments with the same angle of rotation by cross-correlation. As a result, filaments with the same angle of rotation are aligned with each other. Finally, a common centre of mass is found by the least-squares method. The quality of this can be assessed in figure <ref> (e), which shows a superimposed image of all filament images for the particular filament, field and frequency. Once the offsets of the centre are found, we calculate the coordinates for the centre of rotation for each of the frames above. In the second step, we average the velocity fields. First, we shift and rotate each PIV velocity field so that the centre of rotation is at the origin and filament's centre line is parallel to the x axis. It is important to note that the grids of points where the velocities are determined by the PIV do not coincide. To overcome this, we create a new grid of points where we want to find the velocity field values with a chosen gridsize, here fixed at 16 px. For each of the grid points we calculate the velocity values as the average of the PIV velocity measurement points that fall within a chosen radius around this point, here chosen as 9 px. The same approach is used when calculating velocities along other paths, e.g. lines or circles, used for characterising flows. Several examples of experimentally measured and averaged flow fields are given in figure <ref>, while the averaged flowfield corresponding to the figure <ref> (a) is given in the middle bottom graph of figure <ref>. § NUMERICAL MODEL The filament is numerically modelled using the Kirchhoff theory of elastic rods <cit.>, where the effect of the magnetic field is added <cit.>. The filament is described by its centre line y(l), which is parameterized by its arc length l∈[-L/2,L/2], where L is the length of the filament. The cross-section of the filament experiences a force F = -A ∂^3 y/∂ l^3 - μ_0 M H + Λ∂ y/∂ l, where A is the bending modulus, μ_0 is the vacuum permeability, H is the external magnetic field intensity and M is the filament's linear magnetization (magnetic moment per unit length), and Λ is the tension force to ensure the inextensibility of the filament. The velocity of the filament is determined using the resistive force theory, which states that the velocity is proportional to the linear force density v = f_∥/ζ_∥ + f_⊥/ζ_⊥, where f = ∂ F / ∂ l is the linear force density, ∥ denotes the projection along the tangent of the filament, ⊥ denotes the projection normal to the filament, and ζ_∥ and ζ_⊥ are the drag coefficients. We enforce free end boundary conditions (zero force and torque at l=± L/2), which give F\|_l=± L/2=0 and ∂^2 y / ∂ l^2 \|_l=± L/2=0. The drag coefficients are affected by the presence of the no-slip wall below the filament. For a filament oriented and moving parallel to the wall, the drag coefficients can be approximated as <cit.> ζ_∥ = 4πη/ln( L^2/4 h^2)-1-E_1+E_2+E_3+2α, ζ_⊥ = 8πη/ln( L^2/4 h^2)+1-E_1-2E_2+2α, where η is the viscosity of the surrounding fluid and E_1 = 2 L/4 h, E_2 = 1/2√(1+16h^2/L^2), E_3 = 1/2( 1+16h^2/L^2)^3/2, α = ln( h + √(h^2-a^2)/a) , where is the inverse of the hyperbolic sine, h is the height above the bottom wall measured from the centerline of the filament, and a is the filament's cross-section radius. Far away from the wall h/L→∞, the drag coefficients take the familiar <cit.> values of ζ_∥ = 2πη/ln( L/a) - 1/2, ζ_⊥ = 4πη/ln( L/a) + 1/2. However, they diverge to infinity as h→ a. The filament is discretized into N nearly cylindrical segments, and the velocity of each segment is determined by v_1 = 1/Δ l( F_1∥/ζ_∥ + F_1⊥/ζ_⊥), v_i = f_i∥/ζ_∥ + f_i⊥/ζ_⊥, i=2… N-1 v_N = -1/Δ l( F_N∥/ζ_∥ + F_N⊥/ζ_⊥), where the numerical index denotes the segment number, Δ l is the length of the cylindrical segment. Since the magnetic term in eq. (<ref>) is constant, only the first and the last segment directly experience the magnetic force. The filament is assumed to be inextensible; therefore, the calculated velocities are projected on the space of inextensible motion using the procedure described in ref. <cit.>. This permits us to bypass the direct tension force calculation. The derivatives with respect to l are calculated using finite differences. The segments are moved with the calculated velocity using an automatically selected stiff ODE solver from the package <cit.> in Julia language. N=80 segments were used in the calculations. The relative error in the equilibrium shape (characterised by the maximum curvature) resulting from this choice was estimated to be 3·10^-3. During the calculations in this work, the relative length of the filament changed less than 10^-5. The flow that is produced around the rotating filament is calculated using an integral of near-wall Stokeslets and source doublets distributed along the center-line y(l) of the filament u_i( x) = 1/8πη∫_-L/2^L/2( S_ij( x, y (l)) + a^2/2 D_ij( x, y (l)) ) f^tot_j(l) dl, where f^tot is the total force density (including tension) that is acting on the filament's cross-section. For each cylindrical segment it is calculated as f^tot = ζ_∥ v^inext_∥ + ζ_⊥ v^inext_⊥, where v^inext is the velocity that was obtained after the projection on the space of inextensible motion <cit.>. The Stokes flow singularities are given by <cit.> S_ij = ( δ_ij/r + r_i r_j/r^3) - ( δ_ij/R + R_i R_j/R^3) +2h(δ_jαδ_α k - δ_j3δ_3k)∂/∂ R_k[ h R_i/R^3 - ( δ_i3/R + R_i R_3/R^3) ], D_ijf_j= f_j [ ( δ_ij/r^3 - 3 r_i r_j/r^5) - ( δ_ij/R^3 - 3 R_i R_j/R^5) ] -f_3 δ_i α6 R_α R_3/R^5 -f_αδ_i36 R_α R_3/R^5 + 2f_3 δ_iα( -3R_α x_3/R^5 + 15 R_3^2 R_α x_3/R^7) +2 f_3δ_i3( -9R_3 x_3/R^5 + 15 R_3^3 x_3 /R^7) -2f_αδ_iβ( -3R_3 δ_αβ x_3/R^5 + 15 R_3 R_α R_β x_3 /R^7) -2f_αδ_i3( -3R_α x_3/R^5 + 15 R_3^2 R_α x_3 /R^7) , where here δ_ij is the Kronecker delta, r = x - y, R = x - y^, x = {x_1,x_2,x_3}, y = {y_1,y_2,h}, y^ = {y_1,y_2,-h}. The no-slip wall is located at x_3=0, and the filament is hovering at a distance h above it. Repeated Latin indices imply summation ∑_j=1^3, whereas repeated Greek indices imply summation ∑_α=1^2. The Stokeslet term S_ij in the velocity integral (<ref>) describes the flow that is produced by the force density f^tot applied along the center-line of the filament. Whereas, the addition of the source doublet term D_ij ensures that the velocity is constant across a given cross-section of the filament <cit.>, which gives an improved velocity correction near the filament, but is negligible at large distances. § VELOCITY FIELD AROUND A STRAIGHT ROTATING ROD, AND DETERMINATION OF HEIGHT ABOVE SURFACE It has been reported previously that slender rotating bodies raise vertically above the bottom wall <cit.>. Accounting for this is important since the no-slip condition on the wall significantly affects the velocity field around the filament (Figure <ref>). Furthermore, height affects the parallel and perpendicular drag coefficients, as shown in eqs. (<ref>), which in turn impact the calculated shape of the filament. Assuming that the filament is instead a straight rotating rod, it is possible to calculate the velocity fields around it analytically. Let the rod lie momentarily along the x axis y(l)={l,0,h}. The force density exerted by the rod rotating with the angular velocity ω is f^rod = -ζ_⊥ω x e_y, where e_y is the unit vector along y axis. Then the velocity field is given by the integral u_i^rod( x) = -ζ_⊥ω/8πη∫_-L/2^L/2( S_ij( x, y (ξ)) + a^2/2 D_ij( x, y (ξ)) )δ_j2ξ dξ where when expanded, the integrand contains terms of the form A ξ^n/(B+(C-ξ)^2)^m/2, n=1,2,3 m=3,5,7 , which can be readily integrated. The integration is simple, but cumbersome, therefore it was done using Wolfram Mathematica. The notebook with the calculated velocity field is given in the supplementary material. Finally we obtain the expression for the velocity field u^rod(x,y,z,h,ω,a,L), which has been plotted for several values of h in figure <ref>. For a straight rod, the velocity field is symmetric with regards to point inversion through the origin. Furthermore, with regards to the reflection around either x or y axis, the velocity field is "antisymmetric" - it only acquires a "-" sign in front. For a flexible ferromagnetic filament the tips bend in the rotation direction, and the velocity field is no longer "antisymmetric" around the x an y axis. Nonetheless, we try to determine the height h of the filament by comparing its velocity field to the field produced by a rotating straight rod of the same size. We plug into u^rod the ω,a,L values from the experiment, and set z=h. To avoid the influence of the bent tips, but to still have a sizable velocity magnitude, we observe the velocity component in the y direction across the filament at x=± L/4 (figure <ref>). We then determine the height h and its error using the function in Wolfram Mathematica. The fitted heights are shown in figure <ref>. There is a tendency that increasing the frequency makes the filament rotate higher up in the sample cell. For filament 0 and 2, for large frequencies the velocity field is such that the fitted height h is several meters, and the error of the fit is several times lager than the fitted h. We interpret this result as follows: the velocity field around the filament in these cases is indistinguishable from that of a filament in an unbounded fluid, and this way of determining the height of the rotating filament is no longer accurate. § COMPARISON OF SIMULATED AND EXPERIMENTAL VELOCITY FIELDS Having determined the height h of the rotating filaments, it is possible to calculate their shape and the velocity field around them. Other parameters used in the simulations are outlined in table <ref>. The magnetization per unit length M was taken from magnetization measurements <cit.>. The precise experimental determination of the bending modulus A is quite intricate. A method that has been used in the past was based on the observation of the rate at which a bent filament straightens once the magnetic field is turned off <cit.>. This rate, however, is dependent on the height h above the surface. The value of A in this work was determined using a height independent method, which will be outlined in more detail in a future paper. For the viscosity η we used the viscosity of water. The calculated velocity field around the filament is sampled at the same points as in the experiment. If the sample point is inside the filament, the filament's velocity at that point is used. A representative comparison between experimental and simulated velocities around the filament is shown in figure <ref>. The comparison for other frequencies and filaments is given in the supplementary material. There is good agreement both for the filament shapes and the distant velocity field. The velocity close to the filament is larger in the simulations than in the experiment. This can be attributed to the experimental underestimate of the velocity next to the filament, as masking step during image processing excludes tracer particle signal close to the filament, where the highest velocities are expected. Moreover, some of the tracer particles close to the filament move out of the image plane in order to go around it, while PIV can still detect the fluorescence signal. This results in an undetectable velocity component in z direction and a smaller in-plane velocity. § CHARACTERIZING THE FLOW AROUND THE FILAMENT To more quantitatively compare the experimental and simulated velocity fields, we calculate the flux and the circulation of the velocity field u around circular contours centered on the filament's center of rotation. In particular, we calculate for different distances r from the origin the integrals ∮ u_n ds, ∮ u_t ds, where u_n= u · e_r is the component of the velocity field along the radial direction, u_t= u · e_ϕ is the component along the azimuthal direction, and ds is the differential of arc length. e_r and e_ϕ are the unit vectors in the radial and azimuthal direction, respectively. Note that for simulations, to avoid integrating over velocity singularities (eq. (<ref>)), when the integral passes over the filament, we instead use the velocity of the filament at that point. The values of eqs. (<ref>) for filament 0 are shown in figure <ref>. The plots for other filaments can be found in the supplementary material. The integral of the azimuthal velocity shows a reasonably good quantitative agreement between simulated and measured velocity fields. The magnitude of circulation increases as the radius r of the contour increases up until r≈ L/2, after which it decays. The experimental circulation maximum is systematically more shifted towards larger r, which might be explained by the fact that experimentally the beads may be connected with small gaps between them and thus the total length of the filament is larger. A more curious situation can be observed in the graph of the radial velocity integral (right side of figure <ref>). There is a velocity flux in the plane of rotation of the filament directed towards the center of rotation. This flux is most pronounced for the radius of the contour r≈ L/2. Experimentally the flux is larger, but the qualitative behavior is captured also by the simulations. Since the surrounding fluid is incompressible this implies that the rotating filament moves it towards the center of the rotation and pushes it out perpendicular to the plane of rotation. To illustrate this effect more clearly, we have plotted the trajectories of simulated tracer particles around a straight rotating rod and a bent rotating filament (figure <ref>). Around the rod the trajectory oscillates as the filament passes beneath it, but overall remains in the same height and undergoes roughly a circular motion. Whereas for a bent filament the particle starts off nearly at the same height as the filament, but over several rotations it gets sucked towards the center and pushed out away from the filament. It reaches a height of approximately L/2 after 30 filament rotations. Why does a bent rotating filament eject the fluid perpendicular to the plane of rotation and not a straight one? We propose the following mechanism (figure <ref>). The velocity v at a particular point on the rotating filament is perpendicular to the radius vector from the center of rotation. However, due to the anisotropy of drag ζ_∥ < ζ_⊥, the force density f that the filament exerts on the fluid at this point can have a component along the radius vector. In particular if the tips are bent in the direction of rotation (as it is the case for magnetic filaments), there is a force component towards the center of rotation. This forces the fluid towards the center and due to incompressibility it is ejected perpendicular to the plane of rotation. In an infinite fluid it is ejected both up and down by the same amount, however, if the filament is rotating near a wall, fluid is mostly ejected in the direction away from it. This effect has implications for microfluidic mixing, where we propose that, all else being equal, magnetic filaments will outperform straight rigid rod mixers by transporting the fluid also perpendicular to the rotation plane. § CONCLUSIONS In summary, we have shown that micro-particle image velocimetry can be successfully used to measure the flow around microscopic filaments. However, to increase the precision of the method, averaging of the velocity field is needed, which can be readily done for symmetric, time-periodic motion such as rotating filaments. Our measurements indicate that rotating filaments hover above the bottom and the hovering height increases with the increase of the rotation frequency. This height greatly influences the velocity field around the filament. Fitting the numerically calculated velocity field of a straight rotating rod to the experimentally measured velocity fields allow to determine this height. From physics point of view, the height above the bottom influences the drag force that the filament experiences. This in turn has an effect on the filament's shape. Decreasing the height increases the drag, which increases the bending of the tips. This has the same effect as increasing the frequency of the field. If the height above the bottom is taken into the account, the elastic rod model can reasonably accurately calculate both the shapes and the velocity fields around rotating ferromagnetic filaments. Our experimental and numerical results show a good agreement. By this we discover that rotating flexible ferromagnetic filaments, as opposed to rigid straight rods, suck in the fluid in the rotation plane and push it out along the axis of rotation. This effect is the result of the bent tips of the filament. The induced 3D flow gives an additional applicability in micro-mixing. § DATA AVAILABILITY STATEMENT Data will be made available in Zenodo repository, following the OpenData principles, as a part of community "Magnetics and Microhydrodynamics: From guided transport to delivery". § ACKNOWLEDGMENT A.P.S., I.D., M.Š. and G.K. acknowledge the funding by the Latvian Council of Science, project A4Mswim, project No. lzp-2021/1-0470. A.Z. acknowledges the supprot from the European Union's Horizon 2020 research and innovation programme project MAMI under grant agreement No.766007. A.C. & R.L. acknowledge the funding by the Latvian Council of Science, project BIMs, project No. lzp-2020/1-0149 and the financial support from the M-era.net project FMF No.1.1.1.5./ERANET/18/04.
http://arxiv.org/abs/2307.02718v1	20230706015519	A composition law and refined notions of convergence for periodic continued fractions	[ "Bradley W. Brock", "Bruce W. Jordan", "Lawren Smithline" ]	math.NT	[ "math.NT", "math.CO", "math.GR", "11J70 (Primary) 11G30 (Secondary)" ]	[2020]Primary 11J70; Secondary 11G30 B. W. Brock]Bradley W. Brock Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA [email protected] Department of Mathematics, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY 10010-5526, USA [email protected] L. Smithline]Lawren Smithline Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA [email protected] We define an equivalence relation on periodic continued fractions with partial quotients in a ring ⊆, a group law on these equivalence classes, and a map from these equivalence classes to matrices in () with determinant ±1. We prove this group of equivalence classes is isomorphic to /2∗ and study certain of its one- and two-dimensional representations. For a periodic continued fraction with period k, we give a refined description of the limits of the k different k-decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all k of these limits exist in ^1() and a strict majority of them are equal. A composition law and refined notions of convergence for periodic continued fractions [ ===================================================================================== § INTRODUCTION Continued fractions with positive integer partial quotients are a staple of classical number theory, with an intimate connection to Euclid's algorithm. Here, we consider the more general case of continued fractions with partial quotients in a subring ∋ 1 of the complex numbers , as in <cit.>. We define an equivalence relation on the set () of periodic continued fractions over and a group law on the resulting equivalence classes (). The theory of periodic continued fractions, unlike continued fractions in general, has connections to both arithmetic groups and diophantine equations. The work <cit.> relates the theory of () to the group ^±(), the subgroup of () consisting of matrices with determinant ±1, and to diophantine problems on associated varieties. In this paper we further explore the relationship between () and the group ^±(). A major influence in our study is the 1966 paper <cit.> by Cohn. Section <ref> sets out preliminaries related to actions of 2×2 matrices. We consider the action of () by linear fractional transformations on the projective line ^1(). We also name subgroups of () determined by common eigenspaces. In Section <ref>, we introduce the concatentation binary operation ⋆ on the semigroup () of finite continued fractions over . We develop an equivalence relation on () and show that the equivalence classes () form a group under ⋆ . We exhibit a map M: () →^±() in Proposition <ref> arising from the algorithm for computing the value of a finite continued fraction. We show that M is well defined on equivalence classes, giving a homomophism M:()→^±(). We review results of Cohn <cit.> that give information on (M) for certain rings . Using this, we show how the standard amalgam presentation ()≅/4∗_/2/6 arises from presenting () as an explicit quotient of (). Section <ref> extends the structure on finite continued fractions to periodic continued fractions (PCFs). In Theorem <ref>, we show that this yields a natural equivalence relation on periodic continued fractions yielding a group of equivalence classes () isomorphic to (). Section <ref> relates the group () to quantities defined in <cit.>. We review the fundamental matrix E(P)∈() associated to a periodic continued fraction P and apply the isomorphism of Theorem <ref> to show that E is well-defined on (), giving a homomophism E:()→^±(). For A∈() we define subgroups _(A) of () by requiring E(P) to be a linear combination of A and the identity matrix I. We exhibit characters corresponding to each eigenvalue of A. These characters have values that are units in a quadratic extension of . In Section <ref>, we give examples where is a number ring. In Section <ref>, we reformulate the PCF-convergence criteria of <cit.> in parallel with the cases of convergence in Section <ref>. We show that when E(P) has eigenvalues of different magnitudes (which implies ), the k different k-decimations of its sequence of convergents all converge in ^1(), and a strict majority agree. Unanimity is equivalent to convergence. In Section <ref>, we demonstrate the sharpness of the result of Section <ref>. We give examples showing that the proportion of agreement of the k different k-decimations of the sequence of convergents of a quasiconvergent periodic continued fraction can be arbitrarily close to 1/2 or arbitrarily close to and different from 1. We show that the equivalence relation ∼ on () respects quasiconvergence but does not respect convergence. § ACTIONS OF 2×2 MATRICES Let ⊆ be a ring with 1. A finite continued fraction with partial quotients in is an iterated quotient. It can be interpreted in terms of the action of ()≤() on the projective line ^1() by linear fractional transformations which we review in this section. We also name certain subgroups of () determined by common eigenspaces. §.§ Linear fractional transformations Suppose M=[ m_11 m_12; m_21 m_22 ]∈() and β∈^1(). The matrix M acts on β by linear fractional transformation Mβ = m_11β + m_12/m_21β + m_22, where we interpret 1/0 = ∞∈^1(). For β∈^1(), let v(β)=β1 if β∞ and v(∞)=10. The vector v(β) is an eigenvector of M if and only if Mβ = β. For nonzero polynomial Q=aX^2+bX+c∈[X], let (Q) be the multiset of zeros of Q[x] in ^1(), with the convention that if (Q)=1, then ∞ is a simple root of Q; and if (Q)=0, then 0≠ Q has a double root at ∞. When Q = 0, say (Q) = ^1(). For a 2× 2 matrix M as above, set (M)=m_21X^2+ (m_22-m_11)X- m_12. We use (M) to denote ((M)). When β∈(M), v(β) is an eigenvector of M with eigenvalue λ(β) m_21β +m_22=m_11+m_12/β. In particular, λ(∞)=m_11 and λ(0)=m_22. Let A, B be 2× 2 matrices over , I be the 2× 2 identity matrix. Let κ, λ∈ be not both zero. There exists μ∈ such that κ B = λ A + μ I if and only if κ(B)=λ(A). Suppose κ, λ, μ∈ with (κ,λ)≠ (0,0) such that κ B = λ A + μ I. The polynomial (λ A+μ I) is independent of μ. We have κ(B) = (κ B) = (λ A + μ I) = (λ A) = λ(A). Conversely, suppose κ(B)=λ(A) for κ,λ∈, (κ,λ)≠ (0,0). Since is linear map, (κ B - λ A) = 0. Thus, there is μ∈ such that κ B-λ A = μ I. For a matrix M, (M) and the eigenvalues of M govern the convergence behavior of the sequence M^nβ, n ≥ 0, for β∈^1(). For any x, y, let δ(x,y) denote the diagonal matrix δ(x,y) = [ x 0; 0 y ]. For M∈(), one of four mutually exclusive possibilities holds: * (M) has one root β̂ of multiplicity and lim_n→∞M^nβ=β̂ for all β∈^1(). * (M) = 0. For all β∈^1(), Mβ = β and so lim_n →∞M^nβ = β. * (M) has distinct roots β_+, β_-, and M has corresponding eigenvalues λ_+=λ(β_+), λ_-=λ(β_-) as in (<ref>), with \|λ_+\| > \|λ_-\|. For all ββ_-∈^1(), lim_n →∞M^nβ = β_+. For β = β_-, lim_n →∞M^nβ = β_-. * (M) has distinct roots with M having distinct eigenvalues of the same magnitude. For β∈(M), lim_n →∞M^nβ = β. Otherwise, the sequence M^n β diverges. Every M is conjugate to its Jordan normal form, so the result follows if it is so for M in Jordan normal form. Let β∈^1(). The design of cases (<ref>)–(<ref>) makes evident that exactly one of the following holds, where λ_1, λ_2 are the eigenvalues of M. In cases (a) and (b) below there is one eigenvalue λ_1 of multiplicity 2. * M is defective, M-λ_1I is nilpotent, and lim_n→∞[ λ_1 1; 0 λ_1 ]^nβ =∞; * M = λ_1 I, and lim_n→∞δ(λ_1, λ_1)^nβ =β; * M = δ(λ_1, λ_2), \|λ_1\| > \|λ_2\|, and lim_n→∞δ(λ_1, λ_2)^nβ = ∞, β0, 0, β=0; * M = δ(λ_1, λ_2), \|λ_1\| = \|λ_2\|, λ_1 ≠λ_2, and lim_n→∞δ(λ_1, λ_2)^nβ =β, for β∈{0, ∞}, does not exist for β∉{0,∞}. §.§ Subgroups of GL(C) determined by eigenspaces The subsets of () with particular eigenvectors form subgroups of (). Let M ∈(). Every β∈^1() satisfies Mβ=β if and only if M is a scalar multiple of the identity matrix I. Write M=[m_ij]_1≤ i,j≤ 2 as in (<ref>). If M = m_11I, then every β satisfies Mβ = β. Conversely, if every β satisfies Mβ = β, then v(0) and v(∞) are eigenvectors of M and M is diagonal. Since v(1) is also an eigenvector, M must be a scalar multiple of the identity. Let 0≠ L(X)∈[X], (L)≤ 1, with root β∈^1(). Set T(β) = {M ∈() \| Mβ = β}. * The set T(β) is a subgroup of () conjugate to the Borel subgroup of () consisting of upper triangular matrices. * For every M∈ T(β), L(X) divides (M). If L(X)=0, L(X)\|(M) means (M)≤ 1. * The group T(β) has a multiplicative character λ_β mapping M to its v(β)-eigenvalue. (<ref>): The stabilizer T(∞) of ∞∈^1() consists of the upper triangular matrices. Since () acts transitively on ^1(), T(β) is a subgroup of () conjugate to T(∞) for every β∈^1(). (<ref>): Suppose M∈ T(β). Then β∈(M) and L(X) divides (M). (<ref>): Suppose M, M'∈ T(β). Then MM'∈ T(β) and λ_β(MM')v(β)=MM'v(β)=Mλ_β(M')v(β)= λ_β(M)λ_β(M')v(β), so λ_β is a multiplicative character on T(β). Let 0≠ Q∈[X] with (Q)≤ 2. The set G(Q) = {M ∈() \|∃λ∈ such that (M) = λ Q } is a group. When Q is not a square, G(Q) = T(β) ∩ T(β^) with {β, β^ }= (Q), β≠β^∗, and G(Q) is conjugate to the group of diagonal matrices in (). When Q is a square in [X], G(Q) is conjugate to the the subgroup of upper triangular matrices in () with equal diagonal entries. When Q is not a square, (Q) = {β, β^}, β≠β^. In this case, M∈ G(Q) is equivalent to M having eigenvectors v(β) and v(β^). Hence, G(Q) = T(β) ∩ T(β^). A change of basis mapping β to ∞ and β^* to 0 conjugates G(Q) to G(x), the group of diagonal matrices. When Q is a square, (Q) has one element β of multiplicity 2.. Every M∈ G(Q) is either a scalar multiple of I or defective. A change of basis mapping β to ∞ conjugates G(Q) to G(1). A matrix in G(1) is either a scalar multiple of I or upper triangular with equal diagonal entries and not diagonal. § FINITE CONTINUED FRACTIONS Let ⊆ be a ring containing 1. We define the set () of finite continued fractions with partial quotients in , and an operation ⋆ making () a semigroup. We give an equivalence relation on () and show that the equivalence classes () form a group. Let c_i∈, 1≤ i≤ n. A (FCF) F= [c_1,c_2, …, c_n] is the formal expression c_1 + 1c_2+1c_3+1c_4+1-1.2em-2.4em. We say that the FCF F=[c_1,…, c_n] has (F)=n. The elements c_i are called the . For k ≤ n, the _k(F) of F is the evaluation of the finite continued fraction [c_1,c_2,…, c_k], where we interpret 1/0 as ∞∈^1(). We distinguish between the formal object F = [c_1, c_2, … c_k] and its value F̂_k(F) ∈^1( ). There is a simple rule to iteratively compute _k(F) as a ratio p_k/q_k. Let D(x)=[ x 1; 1 0 ]. Let be the 2× 2 identity matrix and F=[c_1, …, c_n] be a finite continued fraction as in (<ref>). Define p_0, q_0, p_-1, q_-1 by the matrix equation [ p_0 p_-1; q_0 q_-1 ] = . Recursively define p_k=p_k(F), q_k=q_k(F) for n≥ k ≥ 1 by [ p_k p_k-1; q_k q_k-1 ] = [ p_k-1 p_k-2; q_k-1 q_k-2 ] D(c_k)=D(c_1)⋯ D(c_k). The numerators p_k and denominators q_k with _k(F)=p_k(F)/q_k(F) can be written in terms of continuant polynomials (see, e.g., <cit.>). An -FCF is a FCF [c_1,…, c_n] having all partial quotients c_i∈𝒪. §.§ An equivalence relation on finite continued fractions The semigroup () is the set {F\| F is an -FCF} with the concatenation operation ⋆: [c_1,… , c_n]⋆[c'_1,… , c'_n']= [c_1,… , c_n][c'_1,… , c'_n'] [c_1,…, c_n, c_1', …, c_n']. Equivalently, () is the semigroup of the set of words in formal symbols (x), for x ∈, with the two descriptions identified by [x_1,…, x_n]↔(x_1)⋯(x_n). Let () be the set () modulo the equivalence relation ∼ generated by the relations (x)(0)(y) = (x+y), x, y ∈. For F=[c_1,…, c_n]∈() denote by F= c_1,…, c_n∈() the equivalence class containing F. For a word 𝐰 in the 𝐃(x)'s denote by 𝐰 the equivalence class containing 𝐰. * If F_1,G_1, F_2, G_2 ∈() with F_1=F_2 and G_1=G_2, then F_1⋆ G_1=F_2⋆ G_2∈(). * The binary operation ⋆ on () induces a well-defined binary operation ⋆ on (), making () a semigroup. * The semigroup () is a group with identity 0,0↔(0)^2. * The element (x)∈() has inverse (x)^-1=(0)(-x)(0). (<ref>) is immediate, and (<ref>) is implied by (<ref>). (<ref>): Substituting y = 0 into relation (<ref>) shows that the class of (0)^2 is the identity in (). The equality (x)(0)(-x)(0) = (0)^2, x ∈, exhibits the inverse (x)^-1= (0)(-x)(0). Define ,(x),(x)∈() for x∈ by (0), (x)(x)(0), and (x)(0)(x). We consider the group /2∗, where is viewed as an additive group and j∈/2 is a generator. Let :()→/2∗ be the surjective map of semigroups :(x)↦ xj∈/2∗. In particular, note that ()=j∈/2, ((x))=x∈, and ((x)))=jxj. The morphism of semigroups :()→/2∗ of Definition induces an isomorphism of groups :()≃⟶/2∗. Verify that ((x)(0)(y))=((x+y)) for x,y∈. Hence induces a well-defined map on equivalence classes ()=()/∼ by the Definition <ref> of ∼. To see that is an isomorphism, verify that its inverse is given by ^ -1:/2∗⟶() with ^ -1(j)= and ^ -1(x)=(x) for x∈. For F=[c_1,…, c_n]∈(), set F^∗ [0,-c_n,…, -c_1,0]. Observe that F^∗= F^ -1∈() c_1,…, c_n⋆ 0, -c_n, …, -c_1,0 = c_1,…, c_n 0, -c_n, …, -c_1,0 = 0,0 . This formula for F^ -1 also follows from Proposition <ref>(<ref>). The finite continued fraction F=[c_1,… ,c_n] is if it has no interior zeros: c_i 0 for 2≤ i≤ n-1. For F ∈(), there is a unique reduced F_red∈() such that F_red∼ F. Given F ∈(), iteratively apply relation (<ref>) to remove interior zeros to produce a reduced F_red∼ F. On the other hand, suppose F_red, F'_red∈() are both reduced and F_red=F'_red. Then (F_red)= (F'_red)∈/2∗. Since they are reduced, both F_red and F'_red map under to words in () ≅ / 2∗, each literally expressed without spurious insertions of the identity, and they are equal. Since / 2∗ is a free product, these words are the same and F'_red = F_red. For any F∈(), the of F is the unique reduced representative F_red∈() of the equivalence class F. Two elements F, F'∈() are equal if and only if they have the same normal forms in (): F=F'⟺ F_red=F'_red∈(). In practice one computes F_red for F∈() by the algorithm of repeatedly applying relation (<ref>) until all interior zeros are eliminated. For example, consider F= [0,-2,0,2,0,3,0,5]∈(). We have F=[0,-2,0,2,0,3,0,5]∼ [0,0,0,3,0,5]∼[0,3,0,5]∼[0,8], and hence F_red=[0,8]∈() with F= 0,-2,0,2,0,3,0,5 = 0,8 =F_red∈(). §.§ Mapping finite continued fractions to matrices Let () = {g∈()\|(g)=± 1}. The kernel of the determinant map, (), is normal of index 2 in (). The map :()→() given by ([c_1,… ,c_n])=D(c_1)⋯ D(c_n) so that ((x)) = D(x) induces a well-defined homomorphism of groups :()→(). We therefore have a homomorphism of groups M∘^ -1:/2∗→(). We have specified the map on (). We must show that respects equivalence under (<ref>). By direct computation, ((x)(0)(y)) = D(x)D(0)D(y) = D(x+y) = ((x+y)). Suppose F_1, F_2∈() and F_1 ∼ F_2. * The values of F_1 and F_2 are equal: F̂_1=F̂_2. * We have (F_1)≡(F_2) 2, so there is a well-defined function :()→/2 given by (F)=(F) 2. (<ref>): Since F_1∼ F_2, (F_1)=(F_2)∈() by Proposition <ref> We have that F̂_1=F̂_2, because F̂_1 =(F_1)_11/(F_1)_21 and likewise for F̂_2 by (<ref>). (<ref>): Reformation of a word by relation (<ref>) maintains the parity of the word length. Let χ:()→⟨± 1⟩ be ∘ M. Let χ̃:/2∗→⟨± 1⟩ be the group homomorpism to ⟨± 1⟩ that maps j to -1 and is trivial on the second factor. For w∈(), we have χ(w) = (-1)^(w) and χ̃∘=χ. The maps χ and χ̃∘ are determined by their action on the generators (x) of (), namely χ((x)) = -1 and χ̃∘((x))=χ̃(xj)=-1. Hence χ=χ̃∘ and for w ∈(), χ(w) = (-1)^(w)= (-1)^(w). Set ^ +()=()◃(). Let χ̃:/2∗→⟨± 1 ⟩ be as in Proposition . There is an isomorphism ψ:∗→χ̃⊆/2∗ given by sending x∈ in the first factor to x∈/2∗ and y∈ in the second factor to jyj∈/2∗. Hence maps ^ +() isomorphically onto ∗≃(χ̃). The definition of the free product /2∗ implies that ψ is injective. A word in /2∗ with an even number of j's is of the form x_1jy_1jx_2jy_2j⋯ x_njy_nj with possibly x_1, y_n, or both 0, which is in the image of ψ. Set W_^+ψ^-1∘ W_\|_^+(): ^+()⟶∗. * The map W_^+ of Definition induces an isomorphism W_^+:^ +()≃⟶∗. * Let , (x), (x)∈() for x∈ be as in Definition . We have W_^+((x))=x in the first factor of ∗ and W_^+((y))=y in the second factor. In particular {(x),(y)\| x,y∈} generates ^ +(). The fact that the equivalence relation does not change the parity of the number of 's implies (<ref>). (<ref>) follows from Proposition <ref> after unwinding the definitions. The map of Proposition <ref> on () takes to J D(0) = [[ 0 1; 1 0 ]]. For x∈, (𝐃(x))=D(x)[ x 1; 1 0 ], ((x)) = U(x)[ 1 x; 0 1 ], ((x))= L(x) [ 1 0; x 1 ]. * The in () are {L(c), U(c)\| c∈}. * The in () are {L(c), U(c), J\| c∈}. * The group () is the subgroup of () generated by elementary matrices in (). * The group () is the subgroup of ^±() generated by elementary matrices in ^±(). * Set M^+=M\|_^ +():^ +()⟶(). The image of :()→() is () and (^ +())=(). By Remark <ref>, maps generators of () to generators of () and generators of ^ +() to generators of (). We have ()=⟨ D(x) : x∈⟩ and ()∩()=(). Since () is generated by (x), the equality ()=⟨ D(x) : x∈⟩ is an immeditate consequence of Proposition <ref>. The kernel of the determinant map applied to () is ()∩(). The group ^ +() is the kernel of determinant composed with by Proposition <ref> and Definition <ref>. Thus, the equality ()∩()=() follows. Note that ()=() if and only if ()=(). For a number field K with integers _K, it is known that (_K)=(_K) unless K=𝐐(√(-D)) where D 1,2,3,7,11 and is squarefree. Vaseršteĭn <cit.> proved that (_K)=(_K) if K is not imaginary quadratic (see also <cit.>), and Cohn <cit.> had previously settled the imaginary quadratic case. Nica <cit.> provides a short constructive proof that for _K the ring of integers of an imaginary quadratic field K, either (_K)=(_K) or (_K) is an infinite index non-normal subgroup of (_K). §.§ The kernel of M and presentations of E(O) The map :()→() of Proposition <ref> has image () by Proposition <ref>. It is a difficult problem to find the kernel of . We begin with the observation that is not injective. Let x, y, z ∈ with w = -x - z - xyz. Suppose x, y, z ≠ 0, and there exists a, b ∈ such that xy = aw, yz = bw. Note is an integral domain since ⊂, so x,y0 implies w0. We also have y+a+b+awb=0 because w(y+a+b+awb)=wy+xy+yz+xy^2z=y(w+x+z+xyz)=0. This is sufficient to compute that M( x,y,z, a, w, b) = ( D(x)D(y)D(z))( D(a)D(w)D(b)) = [ -w aw+1; bw+1 y ][ -y aw+1; bw+1 w ]=1∈(). Hence, for example we have the following nontrivial elements in the kernel of : x, -x^-1, x ,x^-1, -x, x^-1, x, -4x^-1, x , -2x^-1, 2x, -2x^-1, x, -3x^-1, x , -3x^-1, x, -3x^-1, x, α, -α^-1 ,xα^2, α^-1,-α, provided 1/x,2/x,3/x,1/α∈, respectively. A ring ⊂ has a norm \|·\| given by the usual absolute value. A ring ⊂ is <cit.> when every unit x ∈^× has norm 1, and every nonzero, nonunit x ∈ has norm at least 2. An imaginary quadratic number ring is discretely normed unless it has discriminant -D ∈{-3,-4,-7,-8,-11,-12}. The exceptions arise from elements x such that \|x\|=√(2) when -D=-4,-7,-8 or \|x\|=√(3) when -D=-3,-8,-11,-12, namely x=1+i, 1+√(-7)/2, √(-2), and x=3+√(-3)/2, 1+√(-2), 1+√(-11)/2, √(-3) up to signs. These all give nontrivial kernel elements when substituted into expressions (<ref>) and (<ref>), respectively, such that all partial quotients are nonzero nonunits. If is any discretely normed ring and c_1, c_2, …, c_n∈, then some c_i is a unit or zero as shown by a modification of <cit.>. In parallel with <cit.>, we define the following subgroup of (). * For a ∈^×, let (a) = a, -a^-1, a ∈(). * Let () be the normal closure in () of the group generated by (1)^2, (a)(b)(a^-1b^-1)(1), a,b∈^×, (x)(a)(0)(a^2x)(0)(-a), a∈^×, x∈. Calculating that ((a)) = δ(a, -a^-1) in the notation of Definition <ref>, it is routinely verified that (<ref>) – (<ref>) map to I∈() under and hence () is a normal subgroup of . Relations (<ref>) and (<ref>) give ⟨(a) : a ∈^×⟩ mod () ⊂()/() the same commutative group structure as ⟨((a)) ⟩⊂^±(). In particular, (1)(-1) mod ()=(-1)(1) mod () has order 2 and maps by to -I. Relation (<ref>) implies (-a)(x)(a)=(-a^2x)(), from which we see that (1)(-1)() is central in ()/(). Borrowing the terminology of <cit.>, we say a ring is for ^± when () / () ≅^±(), in other words, if M=(). Note that taking a=-1, b=1 in (<ref>) gives (-1)(1)(-1)(1)∈() for any , implying the same for its conjugate (1)(-1)(1)(-1). Set 𝐊'()⟨(1)(-1)(1)(-1), x, 1,-1,x,1,-1, x, -1,1,x,-1,1, x∈⟩ , the normal closure being taken in (). Note that (1)^2, (-1)^2∈'() by taking x=1, x=-1 in (<ref>). From (<ref>), (<ref>) we see that '() (). When ^× =⟨± 1⟩, we have '()=(). Let ”()=⟨(1)^2, (-1)^2,(1)(-1)(1)(-1)⟩, the normal closure being taken in (). Then ()='()=”() and () is a normal subgroup of ^ +(), equal to the normal closure of ⟨(1)^2, (-1)^2,(1)(-1)(1)(-1)⟩ in ^ +(). To show that ”()='()=(), it suffices to show that ⟨ x, 1,-1,x,1,-1, x, -1,1,x,-1,1, x∈⟩⊆”() in light of Remark <ref>. Verify that (1)^2 x^-1 x-1,1,-1,x-1,1,-1 x = x^-1 x,1,-1,x,1,-1 x for x∈, and hence x-1,1,-1,x-1,1,-1∈”()⟺ x,1,-1,x,1,-1∈”(). Since (-1)^2= -1,1,-1, -1,1,-1∈”(), it follows by induction that x,1,-1, x,1,-1∈”() for all x∈. Likewise the identity (-1)^2 x^-1 x+1,-1,1,x+1,-1,1 x = x^-1 x,-1,1,x,-1,1 x implies that x+1,-1,1,x+1,-1,1∈”()⟺ x,-1,1,x,-1,1∈”(). Since 1,-1,1,1,-1,1 =(1)^2∈”(), induction shows that x,-1,1,x,-1,1∈”() for all x∈, establishing (<ref>). We have ()≤^ +() since each of its generators in (<ref>) has even length, satisfying Definition <ref>. Since ^ +() is a normal subgroup of index 2 with the nontrivial coset of ()/^ +() containing 0 = 0^-1, to show that the normal closure of ⟨(1)^2,(-1)^2,(1)(-1)(1)(-1)⟩ in () is equal to its normal closure in ^ +(), it suffices to observe that 0(-1)^2 0 = (1)^-2 0(1)^2 0 = (-1)^-2 0(1)(-1)(1)(-1) 0 =[(1)(-1)(1)(-1)]^-1. The map M:()→^±() =() is surjective with kernel (). Likewise M^ +:^ +()→() is surjective with kernel (). Cohn <cit.> shows that discretely normed rings are universal for ^± as in Definition <ref>, and is discretely normed. The statement for M implies that for M^ +. Of course, it follows from the Euclidean algorithm that ^±()= (); see, for example,<cit.>. Knowing the kernel of M^ +:^+()≅∗→()≤() gives a presentation of (), and a presentation of () if M^ + is surjective. More generally, information on (M^ +) gives information on the generation of () if ()=(). It is an interesting question whether information on (M^ +), such as knowing ()≤(M^ +), gives results on bounded generation in case ^× is infinite as in <cit.> and <cit.>. We show that the presentation of () given by Theorem <ref> gives the familiar amalgamated product presentation. We have ()≅/6∗_/2/4. Note that by Remark <ref>, (-1)^-1= 0,1,-1,1,0. Let α= 1,-1∈^+() and β= 1,-1,1,0=(1) 0∈^+(), so that α^3=(1)(-1), β^2=(1)(-1)^-1, M(α)=[ 0 1; -1 1 ], and M(β)=[ 0 1; -1 0 ]. We now consider the group /6∗_/2/4. A presentation of this group is the free group on two generators α and β modulo the obvious relations: /6∗_/2/4≅⟨α⟩∗⟨β⟩/⟨α^6, α^3β^2, β^4⟩, where ⟨α^6, α^3β^2, β^4⟩ is the normal closure of ⟨α^6, α^3β^2, β^4⟩ in ⟨α⟩∗⟨β⟩. We now state and prove a series of claims: Claim 1. Set ”()⟨α^6, α^3β^2,β^4⟩, the normal closure of ⟨α^6, α^3β^2,β^4⟩ in ^ +(). Then ”() is the normal closure of ⟨(1)^2, (-1)^2, (1)(-1)(1)(-1) ⟩ in ^ +(). So by Proposition , ”()=()='(). Proof. ”()⊆(): Using (<ref>), verify that M(α^6)= M(α^3β^2)=M(β^4)=1∈(). Hence ”()⊆ (M) =(M^ +)=() by Theorem <ref>. ()⊆”(): It suffices to note the following: (1)(-1)(1)(-1) = α^6, (-1)^2 =β^-2α^3=(α^3β^2)^-1α^6, (1)^2 =βα^-3β =β^-1[β^4(α^3β^2)^-1]β∈⟨α^6,α^3β^2,β^4⟩”(), proving Claim 1. Claim 2. The elements α and β generate ^ +(). Proof. By Proposition <ref>, ^ +() is a free group on the two generators (-1)=(1)^-1 and (-1)=(1)^-1. But (-1) = -1,0 = 0,0,-1,1,-1,0 1,-1 = β^-1α (-1) = 0,-1 = 0,0,-1,1,-1,0 1,-1 1,-1 =β^-1α^2, proving Claim 2. We can now conclude the proof of Corollary <ref>. By Theorem <ref> we have () ≅^ +()/()≅⟨α⟩∗⟨β⟩/⟨α^6, α^3β^2, β^4⟩ by Claims 1 and 2 ≅/6∗_/2/4 by (<ref>). When is a discretely normed quadratic imaginary number ring, Cohn shows ()=M and, in <cit.>, ()⊂() is a proper containment. § PERIODIC CONTINUED FRACTIONS An infinite continued fraction C is a formal expression [c_1,c_2,…] where the sequence of partial quotients does not terminate. The infinite continued fraction C=[c_1, c_2, …] may also be expressed as a nonterminating version of (<ref>). As in the finite case, the convergent _k(C) is the evaluation of [c_1, c_2, …, c_k]. For as in (<ref>), set M_k(C)([c_1, c_2, …, c_k]) with M_0(C) =I so that _k(C)=M_k(C)_11/M_k(C)_21. The (or ) of C is Ĉ=lim_k→∞_k(C) when the limit exists in ^1(), in which case C . A (PCF) P is an infinite continued fraction [c_1,c_2,…] together with a (N,k) with N≥ 0, k≥ 1 such that c_n+k=c_n for n>N. We denote the sequence of partial quotients of the PCF P by P=[b_1, …, b_N, a_1, a_2, …, a_k ]. The natural number k is the of P; the of P in (<ref>) is the FCF (P)=[b_1,…, b_N]; and the of P is the FCF (P)=[a_1, …, a_k]. The PCF P as in (<ref>) has Galois dual P^∗ [b_1, …, b_N,0,-a_k,…, -a_1 ]; see, e.g., <cit.>. For a ring ⊆ containing 1, say P= [b_1,…, b_N,a_1, a_2, …, a_k ] is an - if b_i, a_j∈ for 1≤ i≤ N, 1≤ j≤ k. Let () denote the set of all -PCFs. If P∈(), then P^∗∈(). A PCF with type (0,k) is purely periodic. For brevity, we abbreviate purely periodic continued fraction as RCF, after the phrase . The set of RCFs which are -PCFs is denoted (). An (UCF) U is an infinite continued fraction which is periodic of some type (N,k). Notice that the type of a UCF is not uniquely determined: A UCF U with type (N,k) also has type (N',mk) for every N'≥ N and every multiple mk of k. There are three equality relationships between a PCF P of type (N,k) and a PCF P' of type (N',k'). * P and P' are (written =_CF) when P and P' are the same as UCFs. * P and P' are (written =_k) when P and P' are the same as UCFs and k=k'. * P and P' are (written =) when P and P' are the same as PCFs, that is, P=_CFP' and (N,k)=(N',k'). For example, [1,2]=_CF [1,2,2], but [1,2 ]≠_k [1,2,2 ]. And [1,2,1 ]=_k [1,2,1,2 ], but [1,2,1 ]≠ [1,2,1,2 ]. §.§ An equivalence relation and group law on PCFs We give an equivalence relation on () and a group law on the equivalence classes by means of the group law on (). The map : ()→/2∗ maps the PCF P with initial part (P) and repeating part (P) to (P) = ((P))((P))((P))^-1= ((P))((P))((P)^∗), where is as in Definition <ref> and (P)^∗ is as in Remark <ref>. The type (N,k) is essential data attached to P for computing the map . We also have maps :()→() and :()→()⊆() defined by ([b_1,…, b_N,a_1,… , a_k ]) = [b_1,…, b_N,a_1, …, a_k,0,-b_N, …, -b_1,0], N > 0 [a_1,… , a_k], N=0. ([c_1,…, c_n]) = [ c_1,…, c_n ]. Note that ∘ is the identity on () and the map factors as =∘. The maps and \|_() give inverse bijections: \|_():()≃⟶() and :()≃⟶(). The equivalence relation ∼ on () defined in Section <ref> induces an equivalence relation, also denoted ∼. on (). For P, P'∈(), say P∼ P' ⟺(P)∼(P')⟺((P))=((P')) ⟺(P)=(P')∈/2∗ . For F, F'∈() we have F∼ F'⟺(F)∼(F') since ((F))=F. If P, P'∈() and P=_kP' as in Definition , then P∼ P'. First consider the case that P is of type (N,k) and P' is of type (N+1,k) with P=_kP'. Then we must have P=[b_1,…, b_N,a_1,…, a_k] and P'=[b_1,…, b_N, a_1,a_2, …, a_k, a_1]. We have (P) =[b_1,…, b_N, a_1, …, a_k,0,-b_N,…, -b_1,0] and (P') =[b_1,…, b_N,a_1,a_2,… , a_k, a_1,0,-a_1,-b_N,…, -b_1,0] with as in (<ref>). But then (P)∼(P') by applying (<ref>). The proposition then follows from this by iteration. An RCF R=[ a_1,… a_n ]∈() is when (R)∈() is reduced as in Definition <ref>, i.e., a_i≠ 0 for 2≤ i≤ n-1. Note that F∈() is reduced if and only if (F)∈() is reduced since ∘(F)=F. For P∈(), let (P)_red∈() be as in Proposition . Then the element P_red = ((P)_red) is the unique reduced element of () such that P_red∼ P. For P∈() we have that P_red = ((P)_red)∈() is reduced, as stated in Remark <ref>. We claim that P∼ P_red. Note that (P)∼(P)_red implies that ((P))∼((P)_red) as in Remark <ref>. By Definition <ref>, (P)=(((P))) implies P∼((P)). Thus, P∼((P)_red). Suppose R, R'∈() are both reduced and R∼ R'. In this case, (R),(R')∈() are both reduced and (R)∼(R') by Definition <ref>. Proposition <ref> shows that (R)=(R')∈() and hence R=R' by (<ref>). Write ()()/∼. For P=[b_1,…, b_N,a_1,… , a_k ]∈(), let P= b_1,…, b_N,a_1, … , a_k ∈() be the equivalence class containing P. By Definition <ref> and Remark <ref>, the maps and induce inverse bijections :()≃⟶() and :() ≃⟶(). Via the bijections , the binary operation ⋆ of Proposition <ref> on () induces a binary operation, also denoted ⋆, on (): for P, P'∈() we have (P⋆P')= (P)⋆(P') and P⋆P'=((P)⋆(P')). The semigroup () with its binary operation ⋆ is a group isomorphic to ()≅/2∗. The map :()→/2∗ of Definition induces an isomorphism of groups :()≃⟶/2∗. For the convenience of the reader we give in Figure 1 the maps defined thus far and the relations between them. If P∈() with P∈(), then Proposition <ref> shows that the equivalence class P contains a unique reduced RCF P_red=P_red. We call the P_red=P_red the of P. Two elements P, P'∈() are equal if and only if they have the same normal forms: P=P'⟺P_red= P'_red∈(). For , there are PCFs P with not all partial quotients in with normal form P_red∈(); most simply, [ c,0,-c ] ∼ [ 0 ] for any c ∈. We test the equivalence of P, P'∈() via their normal forms P_red, P'_red∈(). It is never necessary to make an excursion outside to determine equivalence of PCFs with partial quotients in . We give an example of how to compute P_red using Proposition <ref> and Remark <ref>. Suppose P=[3,0,-3,4,0,1,0,-5 ]∈(). Then (P)=[3,0,-3,4,0,1,0,-5,0,-4,3,0,-3,0]∈(). We have (P)∼[0,4,0,1,0,-5,0,-4,3,0,-3,0]∼ [0,5,0,-9,0,0]∼ [0,-4,0,0]∼ [0,-4]. Hence (P)_red=[0,-4]∈() and P_red=([0,-4])=[ 0,-4 ]∈() with 3,0,-3,4,0,1,0,-5 = P=P_red= 0,-4 ∈(). In practice, P⋆P'∈() is computed using the normal forms P_red,P'_red: if P_red=[ c_1, …, c_n ]∈() and P'_red=[ c'_1,…, c'_n' ]∈(), then P⋆P'= c_1, …, c_n, c'_1, … , c'_n' ∈(). Suppose P=[ b_1,a_1,…, a_k ], P'=[ b_1',a_1',…, a_k'' ] ∈() are of types (1,k) and (1,k') respectively. Then the product takes a particularly simple form with a representative of type (1,k+k'): P⋆P' = b_1, a_1,…, a_k,0,-b_1,0,b_1', a_1',… a_k'', 0, -b_1',0 by (<ref>) = b_1, a_1,…, a_k+b_1'-b_1, a_1', …, a_k''-b_1',0 using (<ref>) = b_1,a_1,…, a_k-1, a_k+b'_1-b_1, a'_1, …, a'_k'-1, a'_k'-b_1',0,b_1 = b_1, a_1,…, a_k-1, a_k+b_1'-b_1, a_1', …, a'_k'-1, a'_k'-b'_1+b_1 . A similar exercise shows that the product is simple for P,P'∈() with (P)=(P'). Suppose P=[b_1,…, b_N,a_1,…, a_k ] and P'=[b_1,…, b_N,a_1', … , a_k'' ]. Then P⋆P'= b_1,…, b_N,a_1,…, a_k,a_1',…, a_k'' . * The set () with binary operation ⋆ forms a group with identity e 0,0. * For P∈(), the inverse P^-1∈() is P^∗ with the Galois dual P^∗ as in (<ref>). * The map of (<ref>) induces an isomorphism W_𝖯:()→()=/2∗, where is viewed as an additive group. (<ref>): The maps and transfer the group structure on () defined in Theorem <ref> to (). Hence () with induced binary operation ⋆ is a group with identity e=(e), where e is the identity in (). But by Theorem <ref>(<ref>), e= 0,0, and hence e= 0,0. (<ref>): For P=[b_1,…, b_N,a_1,…, a_k ]∈(), we have (P) =[b_1,…, b_N,a_1,… ,a_k,0,-b_N,…, -b_1,0] by (<ref>), (P)^∗ = [0,0,b_1,…, b_N,0,-a_k,…, -a_1,-b_N,…, -b_1,0] as in Theorem <ref>(<ref>), P^∗ =[b_1,…, b_N,0,-a_k,…, -a_1 ] by (<ref>), (P^∗) =[b_1, …, b_N,0,-a_k,…, -a_1,0,0,-b_N,…, -b_1,0] by (<ref>). Note that (P)^∗∼(P^∗) using the equivalence of Proposition <ref>(<ref>). By Theorem <ref>(<ref>) we have (P^-1)= ( P)^-1=((P))^∗= (P^∗). Hence P^-1=P^∗. (<ref>):The map induces an isomorphism of groups :()≃⟶() and the map W_𝖥 induces an isomorphism of groups W_𝖥:()≃⟶() by Theorem <ref>(<ref>). Since W_𝖯=W_𝖥∘ by (<ref>), W_𝖯:()→()=/2∗ is an isomorphism. For every PCF of the form P=[b_1,…, b_N,0 ], (P) ∈/2∗ is an involution, but, in general, P is not equivalent to [ 0 ]. Every PCF of the form [b_1,…, b_N,0,0 ] is in the class of the identity in (), that is, P ∼ [ 0,0 ]. § THE SUBGROUP PCF(A)PCF(O) AND ITS REPRESENTATIONS For P∈(), <cit.> defines a matrix E(P) which plays a large role in the study of P. We review the definition, using the notation of this paper. Let E:()→() be the map E=∘ with as in (<ref>) and as in (<ref>). We denote the quadratic polynomial (E(P)) as in (<ref>) by (P)∈[x] and the multiset of its roots (E(P)) by (P). The map E induces a homomorphism E:() →() with image ()≤() as in Section . From Definition <ref> and Proposition <ref> we see that E is well-defined on equivalence classes in (). By Proposition <ref>, the image of E is (). Let β∈^1(). Set _(β) = {P∈() \|E(P)β = β}. Recall from Proposition <ref> that T(β) is the subgroup {M∈()\| Mβ=β} and T(β) has a multiplicative character λ_β mapping M to its v(β)-eigenvalue. The preimage of T(β) ∩() under E is _(β), so _(β) is a subgroup of (). The multiplicative character λ_β∘E maps an element P∈_(β) to the v(β)-eigenvalue of E(P). * For Q∈[X] with Q ≤ 2, set _(Q)={P∈()\|E(P)∈ G(Q)}, where G(Q) is the subgroup of () defined in Proposition <ref>. * For A∈(), set _(A) = _((A)). * Recall from Proposition <ref> that E:()→() factors through (): E:()⟶()E⟶(). For P ∈(), set _(P) = _(P)=_(E(P))=_ (E(P)). For Q∈[X] with Q ≤ 2, the preimage of G(Q) ∩() under E is _(Q), so _(Q) is a group. For β∈(Q), the group _(Q) has a multiplicative character λ_β∘E inherited from the containment _(Q) ⊂_(β). Let A ∈(). * If (A) = {β, β^}, β≠β^, and A has corresponding eigenvectors v(β), v(β^), then _(A) = _(β) ∩_(β^), and the values of the characters λ_β∘E and λ_β^∘ E on _(A) are quadratic integral over and in [β]^×. The product of these characters is ∘ E. If (A) has one element β of multiplicity 2, _(A) has a character λ_β∘ E whose square is ∘ E. * Otherwise, A is a scalar multiple of the identity and _(A) has a character λ∘ E equal to λ_β∘E for every β. Its square is ∘ E. Let P∈(A) and E = E(P) ∈(). (<ref>): If (A) is not zero and not a square, then (A) has distinct elements β, β^* and E has corresponding eigenvalues λ_β(E), λ_β^(E) that solve the monic quadratic equation (E-λ I). It follows from Propostion <ref> that _(A) = _(β) ∩_(β^). Relation (<ref>) shows the eigenvalue λ_β(E) ∈[β]. The sum λ_β(E)+λ_β^(E) is in , so λ_β^(E)∈[β]. The product λ_β(E)λ_β^(E) is E = ±1, so both λ_β(E) and λ_β^(E) are units. (<ref>): When (A) is a nonzero square, A has a single eigenvector v(β). For E as above, the eigenvalue λ_β(E) solves a monic linear equation over and λ_β(E)^2 = ±1. (<ref>): When (A) is zero, every E ∈_(A) is μ_E I, a scalar multiple of I. In this case, λ_β(E) = μ_E for every β. Let P=[b_1,a_1,…,a_k] be a PCF of type (1,k). The convergent 𝒞_k(P) relates (P) and the eigenvalues of E = ((P)). When P ∼ [b_1, a_1, …, a_k-1, a_k-b_1,0], we have E(P)=([b_1,a_1, …, a_k-1, a_k-b_1,0]). For M=M([b_1,a_1,…,a_k-1])=[[ m_11 m_12; m_21 m_22 ]], we have E=[ E_11 E_12; E_21 E_22 ]= [ m_11 m_12; m_21 m_22 ][ 1 a_k-b_1; 0 1 ]= [ m_11 m_11(a_k-b_1)+m_12; m_21 m_21(a_k - b_1) + m_22 ]. Thus, E_11/E_21 is the kth convergent 𝒞_k(P). If (P) has two distinct elements β, β^* and E has associated eigenvalues λ, λ^, λ=E_11-E_21β^∗. If (P) has one element β of multiplicity two, E_11-E_21β is the eigenvalue of E. If E is a scalar multiple of I, its eigenvalue is E_11. The maps defined in this section form a commutative diagram as given in Figure 2. § EXAMPLES OF THEOREM <REF> ARISING FROM NUMBER RINGS §.§ P=[1,2], P̂=2 The -PCF P_1 is of type (N,k)=(1,1) and E(P_1)=D(1)D(2)D(1)^-1=[ 1 2; 1 1 ], so (P_1)=x^2-2, (P_1)={√(2),-√(2)}, and the first convergent 𝒞_1(P_1) =1/1. We have P_1∈_(P_1), with _(P_1) as in Definition <ref>(<ref>). The characters λλ_√(2)∘E, λ^λ_-√(2)∘E of _(P_1) map P_1 to the eigenvalues of E(P_1): λ(P_1)=1+√(2) and λ^∗(P_1)=1-√(2), which are units in [√(2)], and λ(P_1)λ^∗(P_1) =(E(P_1))=(-1)^k=-1. §.§ P=[1,2,2,2], P̂=2 The -PCF P_2 has type (N,k)=(1,3), and E(P_2) =D(1)D(2)^3D(1)^-1=[ 7 10; 5 7 ]= E(P_1)^3=[ 1 2; 1 1 ]^3, so (P_2)=5x^2-10 = 5(P_1) and (P_2)=(P_1)= {√(2), -√(2)}. By (<ref>), the equivalence class P_2 satisfies P_2 = P_1⋆P_1 ⋆P_1 ∈_(P_2)=_(P_1). The convergent 𝒞_3(P_2) is 7/5. The characters λ, λ^∗ of _(P_2)=_(P_1) in (<ref>) map P_2 to the eigenvalues of E(P_2): λ(P_2)=7+5√(2) and λ^∗(P_2)=7-5√(2), which are units in [√(2)], and λ(P_2)λ^∗(P_2) =(E(P_2))=(-1)^3=-1. Consequent to λ being a homomorphism, λ(P_2)=λ(P_1⋆P_1⋆P_1)=5√(2)+7= (1+√(2))^3=λ(P_1)^3 using (<ref>). §.§ P=[2,-2,4], P̂=2 The -PCF P_3 is of type (N,k)=(1,2) and E(P_3) = D(2)D(-2)D(4)D(2)^-1=[ -3 -4; -2 -3 ], so (P_3)=-2x^2+4 = -2(P_1) and (P_3)=(P_1). The convergent 𝒞_2(P_3) is 3/2. The class P_3 is in _(P_1)=_(P_3). For the characters λ, λ^∗ as in (<ref>), we have λ(P_3)=-3-2√(2) and λ^∗(P_3)=-3+2√(2), with which are units in [√(2)], and λ(P_3)λ^∗(P_3)= (E(P_3))=(-1)^2=1. §.§ P=[2,3,-2,3], P̂=2 The -PCF P_4 is of type (N,k)=(1,3) and E(P_4) = D(1)D(3)D(-2)D(3)D(1)^-1=[ -7 -10; -5 -7 ], so (P_4)=-5x^2+10=-5(P_1) and (P_4)=(P_1). The convergent 𝒞_3(P_4) is 7/5. We have P_4∈_Z(P_4)=_(P_1) and, by (<ref>), P_4= 1, 2-1+2, -2, 4-2+1 = 1, 2 2, -2,4 = P_1⋆P_3. The values of the characters λ, λ^∗ in (<ref>) on P_4 are λ(P_4)=-7-5√(2) and λ^∗(P_4)=-7+5√(2). We have λ(P_4)λ^∗(P_4)= (E(P_4))=(-1)^3=-1. From Example <ref> λ(P_1)=1+√(2) and from Example <ref> λ(P_3)=-2√(2)-3. We verify using (<ref>) and (<ref>) λ(P_4)=λ(P_1⋆P_3) =-5√(2)-7=(1+√(2))(-2√(2)-3) =λ(P_1)λ(P_3). §.§ P=[442+3122,-298532+2110942,884+6242], P̂=(2+2) The [√(2)]-PCF P_5 from <cit.> is of type (N,k)=(1,2). We have E=E(P_5) = D(442+312√(2))D(-298532+211094√(2)) D(884+624√(2))D(442+312√(2))^-1 = [ -228487+161564√(2) -174876+123656√(2); -298532+211094√(2) -228487+161564√(2) ], so (P_5) =(-298532+211094√(2))x^2+174876-123656√(2) =(-298532+211094√(2))(x^2-2-√(2)), (P_5)={β√(2 + √(2)), -β}, and the convergent 𝒞_2(P_5)=(11502+8105√(2))/52≈ 441.6, which is a horrible approximation to β in keeping with <cit.>, which remarks on and quantifies the extremely slow convergence of P_5. The values of the characters μλ_β∘E, ,μ^∗λ_-β∘E :_[√(2)](P_5)→[√(2)][β]^× on the class P_5∈_[√(2)](P_5) are μ(P_5) =(-228487+161564√(2))+(-298532+211094√(2))β and μ^∗(P_5) =(-228487+161564√(2))-(-298532+211094√(2)) β, which are both units in the ring [√(2)][β]=[β]. In fact _(β)/(√(2))μ(P_5)=_(β)/(√(2))μ^∗(P_5)= μ(P_5)μ^∗(P_5)= (E(P_5))=1. § REFINED NOTIONS OF CONVERGENCE FOR PERIODIC CONTINUED FRACTIONS For a PCF P, the multiset (P) contains the limit P̂ of P when that limit exists; see, e.g., <cit.>. Let P=[b_1,…, b_N,a_1,…, a_k] and P'=[b'_1,…, b'_N',a'_1,… , a'_k'] be PCFs of types (N,k) and (N',k') that are -equal. * If k'=k so that P and P' are -equal, then E(P)=E(P'), (P)=(P'), and (P)=(P'). * If k' k, then E(P)^k'=E(P')^k, and (P) and (P') are linearly dependent. If (P) and (P') are both nonzero, then (P)=(P'). * The PCF P converges if and only if P' converges and in this case their limits are equal: P̂=P̂'∈(P)=(P'). (<ref>): Suppose k=k'. If N=N', then P equals P'. In the alternative N ≠ N', we may assume N'>N with N'=N+j. In this case, P'=[b_1,…, b_N,b_N+1,…,b_N+j,a_j+1,… , a_j+k], where we take the subscripts of the a's k to put them in the range 1 to k and b_i=a_i-N for i>N. So E(P') =M([b_1,…, b_N,b_N+1,…,b_N+j,a_j+1,… , a_j+k,0, -b_N+j,…,-b_1,0]) =M([b_1,…, b_N,a_1,a_2,…, a_j+k,0, -a_j,…,-a_1,-b_N,…,-b_1,0]) =M([b_1,…, b_N,a_1,a_2,…, a_j+k-1,0, -a_j-1,…,-a_1,-b_N,…,-b_1,0]) ⋮ =M([b_1,…, b_N,a_1,…, a_k,0,-b_N,…,-b_1,0])=E(P) using the fact that M([a,0,-a])=M([0]) and a_i=a_i+k j times. The rest of (<ref>) follows trivially. (<ref>): Since the choice of N does not affect E, we may assume N=N'. The equality E(P)^k'=E(P')^k follows from the observation that the concatenation of k' copies of [a_1,…, a_k] equals the concatenation of k copies of [a'_1,… , a'_k']. The Cayley–Hamilton theorem implies that any power of a 2×2 matrix M is a linear combination of M and I. This implies E(P), E(P'), and are linearly dependent, which implies (P) and (P') are linearly dependent, which implies they have the same roots if both are nonzero. (<ref>): This is trivial, since P and P' are -equal. For the PCF P=[b_1,…, b_N,a_1,…, a_k] of type (N,k), we introduce notation for the matrix M_N+j(P)^-1M_N+j+k(P) and its entries: R_j(P) = M_N+j(P)^-1M_N+j+k(P) = M([a_j+1,…, a_k, a_1,…, a_j])= [ r_j(P) s_j(P); t_j(P) u_j(P) ], and for the limits, if they exist, P̂_j = lim_n→∞_N+j+nk(P). The matrix R_j(P) is when t_j(P) = 0 and \|u_j(P)\| > 1. Rephrased with this terminology, the condition <cit.> says that R_j(P) is heavy for some 0≤ j≤ k-1. The matrix R_0(P) is M((P)). The matrices R_j(P) are all conjugate to E(P). From the definition, we see P̂_j = P̂_j+k. Let P be a PCF. Suppose R_j(P) is heavy. * The limit P̂_j exists and v(P̂_j) is an r_j(P)-eigenvector of E(P). * R_j+1(P) is not heavy. (<ref>): The heavy condition implies R_j(P) has an eigenvector 10 with eigenvector r_j(P). Therefore, E=M_N+jR_jM_N+j(P)^-1 has eigenvector M_N+j(P)10. We compute the limit P̂_j=lim_n→∞M_N+j+nk(P)∞ =lim_n→∞M_N+j(P)R_j^n∞=M_N+j∞. We are now done because v(P̂_j) and M_N+j(P)10 are both elements of the equivalence class M_N+j(P)∞. (<ref>): The calculation, with argument P suppressed and a = a_j+1-nk, for the value of nk such that 1 ≤ j+1-nk ≤ k, [ r_j+1 s_j+1; t_j+1 u_j+1 ] = [ 0 1; 1 -a ][ r_j s_j; 0 u_j ][ a 1; 1 0 ] = [ u_j 0; (r_j-u_j)a+s_j r_j ] shows that R_j(P) and R_j+1(P) cannot both be heavy. If they are, u_j+1 = r_j, yielding \|u_j+1\| < 1, a contradiction. We recall a theorem proved in the appendix of <cit.>, with the cases aligned to match Proposition <ref>. Let P be a PCF, let λ_± be the eigenvalues of E = E(P) chosen so that \|λ_+\| ≥ 1≥\|λ_-\|, and let v(β_±) be the corresponding eigenvectors. If P converges, its limit P̂=β_+. Exactly one of the following holds. * (P) has one root β_+=β_- of multiplicity 2, and P̂=β_+=β_-. * (P) = 0, E=λ_+=λ_-, and P diverges. * (P) has roots β_+, β_- with \|λ_+\| > \|λ_-\|. * For some j ≥ 0, R_j(P) is heavy, P̂_j = β_-, P̂_j+1=β_+, and P diverges. * For all j≥ 0, R_j(P) is not heavy, and P converges to P̂ = β_+. * (P) has distinct roots β_+, β_- with \|λ_+\|=\|λ_-\|, and P diverges. In fact, P̂_j does not exist for some j. In case (<ref>), by Proposition <ref>(<ref>), for all j, P̂_j= β_+, so P converges to β_+. In case (<ref>), R_j(P) = E for all j, so P̂_j= _N+j(P). No two consecutive ones are equal: P̂_jP̂_j+1 by the more general rule that no two consecutive convergents can be equal, so P diverges. Now assume that cases (<ref>) and (<ref>) do not hold. (P) has two distinct roots, β_-, β_+. The matrix E has corresponding distinct eigenvalues λ_-, λ_+. In case (<ref>), \|λ_+\| > \|λ_-\|, and Proposition <ref>(<ref>) shows all β_j exist. Suppose, as in the first subcase, there exists j such that R_j(P) is heavy, Proposition <ref> shows P̂_j= β_- and R_j+1 is not heavy. Proposition <ref>(<ref>) shows P̂_j+1=β_+, so P diverges. In the alternative subcase of (<ref>), where there is no j such that R_j is heavy, we have P̂_j= β_+ for all j, and P converges to β_+. In case (<ref>) of divergence, we have \|λ_+\|=\|λ_-\| and λ_+λ_-. By Proposition <ref>(<ref>), the limit P̂_j exists if and only if t_j=t_j(P)=0. We claim there exists j such that t_j(P)0. Assume, to the contrary, that t_j = 0 for all j. To deduce a contradiction, we show t_j=t_j+1=t_j+2=0 implies a_j+2=0. From the equality R_j+1[ a_j+2 1; 1 0 ] =[ a_j+2 1; 1 0 ]R_j+2, we see that t_j+1=t_j+2=0 implies s_j+2=0. Similarly, t_j=t_j+1=0 implies s_j+1=0, so R_j+1 and R_j+2 are diagonal with distinct diagonal entries λ_±. Equation (<ref>) implies a_j+2=0. Since t_j = 0 for all j, we have that a_j = 0 for all j. If k is odd then t_j=1 for every j, a contradiction. If k is even then we are in case (<ref>), also a contradiction, thus proving the claim. Hence, for some j, P̂_j does not exist, and P diverges. We name the convergence behaviors of a PCF P identified in Theorem <ref>. In cases <ref>(<ref>) and <ref>(<ref>), P is . In cases <ref>(<ref>) and <ref>(<ref>), P is . In the divergent subcase of <ref>(<ref>), P is . Divergent means strictly divergent or strictly quasiconvergent. Quasiconvergent means convergent or strictly quasiconvergent. Let PCFs P and Q be equivalent. P is quasiconvergent if and only if Q is quasiconvergent. If P and Q both converge, then P̂ =Q̂. PCFs P and Q are quasiconvergent together, because E(P) = E(Q). For the same reason, if P and Q both converge, they converge to the same value. The following lemma and theorem characterize strict quasiconvergence. Let P=[b_1,…, b_N, a_1,…, a_k] be a PCF. If R_j(P) and R_j+2(P) are both heavy, then a_j+2-n'k=0 for the value of n' such that 1 ≤ j+2-n'k ≤ k. Inspect the relation, with argument P suppressed, a = a_j+1-nk, and a' = a_j+2-n'k: [ r_j+2 s_j+2; t_j+2 u_j+2 ] = [ 1 -a; -a' aa'+1 ][ r_j s_j; t_j u_j ][ aa'+1 a; a' 1 ]. Observe that t_j=0 implies r_j+2 = r_j + aa'r_j -aa'u_j + a's_j, t_j+2 = a'(u_j-r_j+2). If t_j+2 = 0 and a' 0, then r_j+2 = u_j, contradicting \|u_j\|>1>\|r_j+2\|. Hence, t_j+2=0 implies a' = a_j+2-n'k = 0. The PCF P is strictly quasiconvergent if and only if for each j, 1≤ j ≤ k, P̂_j exists, a strict majority, but not all, of which are the same. Let the eigenvalues of E = E(P) be λ_+ and λ_-, with \|λ_+\| ≥ 1. If both eigenvalues have magnitude 1, arbitrarily assign one to be λ_+. Define β_+ and β_- such that v(β_+), v(β_-) are eigenvectors of E with corresponding eigenvalues λ_+ and λ_-. If P is strictly quasiconvergent, then there exists j such that R_j(P) is heavy and λ_+ = u_j. Proposition <ref> shows that for every j such that R_j(P) is heavy, P̂_j = β_- and R_j+1(P) is not heavy. For those j such that R_j(P) is not heavy, P̂_j = β_+. If R_j(P) is heavy for every even j or every odd j, then either k is even and, by Lemma <ref>, every second partial quotient of (P) is zero, or k is odd, and all partial quotients of (P) are zero. In the even case, E is conjugate to an elementary matrix and P satisfies <ref>(<ref>) or <ref>(<ref>), a contradiction. In the odd case, E is conjugate to the matrix J of Remark <ref> and P satisfies <ref>(<ref>). Hence, for only a minority of j, 1 ≤ j ≤ k, can R_j(P) be heavy and P̂_j= β_-. Conversely, suppose for each j, 1≤ j ≤ k, P̂_j exists, a strict majority, but not all, of which are the same. If P satisfies Theorem <ref>(<ref>), then P̂_j P̂_j+1. So a majority of the P̂_j, 1≤ j≤ k, are not equal to a particular value. The PCF P satisfies Theorem <ref>(<ref>) if and only if E has eigenvalues λ_+λ_- with \|λ_+\|=\|λ_-\|=1. We claim that for 1≤ j≤ k, either P̂_j does not exist, or P̂_j exists and _N+j+nk(P) is a constant independent of n. We shall prove this for j=k. The result follows for all j by rotating the periodic part of P. Write v(P̂_k) = c v(β_+) + d v(β_-), and observe E^n v(P̂_k) = c λ_+^n v(β_+) + d λ_-^n v(β_-). If P̂_k exists, then v(P̂_k) is an eigenvector of E, so either c=0 or d=0. This proves the claim. By assumption that the limits P̂_j exist and the fact that consecutive convergents cannot be equal, we cannot have a strict majority of P̂_j agree and P satisfy Theorem <ref>(<ref>) The only possibility left is that P satisfies Theorem <ref>(<ref>), so P is strictly quasiconvergent. § EXAMPLES FOR QUASICONVERGENCE §.§ Strict majority in Theorem <ref> is necessary The strict majority hypothesis in the sufficiency part of the proof of Theorem <ref> is essential. For example, P = [a,b,0,0] has limits P̂_1=a and P̂_2=a+1/b. The split is equal and P is not quasiconvergent. It is strictly divergent because it satisfies Theorem <ref>(<ref>). §.§ The fraction of (P)= can be made arbitrarily close to ½ We can make the fraction of P̂_j= β_+ arbitrarily close to 1/2. Let c 0 and P=[a,b,c,-1/c,c,0,…,0 ], have odd length period k≥ 3. We have P̂_2j=a and P̂_2j+1=a+1/b, 1≤ j < k/2. If c^2=-1 we are in case (<ref>) of Theorem <ref>, and P̂_1=a+1/(b-c). If \|c\|=1, c^2 -1, we are in case (<ref>), and P̂_1 does not exist. If \|c\|1 we are in case (<ref>) and P̂_1=β_+=a+1/b if \|c\|>1, a if \|c\|<1, so β_+ has a slim (k+1)/2 majority. A similar construction can be made for even periods by taking P'=[a,b,c,-1/c,c,0,…,0,d,-1/d,d,0,…,0 ] where the length of the strings of 0's have the same parity. §.§ The fraction of (P)= can be made arbitrarily close to 1 Likewise we can make the fraction of P̂_j=β_+ arbitrarily close to 1. Let {F_n}_n=1^∞ =1,1,2,3,5,… be the . Let b≠ 0 and P Q_k=[a,b,1, …, 1, -F_k-2/F_k-1 ], where there are (k-1) 1's. The continued fraction P for k≥4 provides an example where exactly k-1 of the k P̂_j's are β_+. (If k=3, P̂_2 doesn't exist, and if k=2, β_+=β_-=P̂.) If k≥ 4 and b0, then P̂_j=β_+(Q_k)=a+F_k-1/bF_k-1+F_k-2 for 1≤ j≤ k-1, but P̂_k=β_-(Q_k)=a+1/b. We compute M=M(k) M([1,1,…, 1, -F_k-2/F_k-1]): M =[ 1 1; 1 0 ]^k-1[ -F_k-2/F_k-1 1; 1 0 ] = [ F_k F_k-1; F_k-1 F_k-2 ][ -F_k-2/F_k-1 1; 1 0 ] = [ (-1)^k/F_k-1 F_k; 0 F_k-1 ], using Cassini's identity F_k-1^2-F_kF_k-2=(-1)^k (<cit.>). The next step is to compute the P̂_j, 1≤ j≤ k, as in (<ref>). We begin with P̂_k. Set A=D(a)D(b)= [ a 1; 1 0 ][ b 1; 1 0 ]= [ ab + 1 a; b 1 ]. Using (<ref>), we have M(n,k) M_2+nk(Q_k) = AM^n = [ ab + 1 a; b 1 ][ (-1)^k/F_k-1 F_k; 0 F_k-1 ]^n = [ ab+1 a; b 1 ][ (-1)^kn/F_k-1^n ∗; 0 F_k-1^n ] =[ (ab+1)(-1)^kn/F_k-1^n ∗; b (-1)^kn/F_k-1^n ∗ ]. Hence 𝒞_N+nk=M(n,k)_11/M(n,k)_21=ab+1/b=a + 1/b for all n≥ 1 and P̂_k=lim_n→∞𝒞_N+nk=a + 1/b. For the other P̂_j, we need to calculate M_j D(1)^-jM D(1)^j for 1≤ j ≤ k-1, so that we can apply Proposition <ref>. Firstly note that D(1)^j = [ F_j+1 F_j; F_j F_j-1 ], and so D(1)^-j=(-1)^j[ F_j-1 -F_j; -F_j F_j+1 ]. Hence from (<ref>) and (<ref>) we get M_j =(-1)^j[ F_j-1 -F_j; -F_j F_j+1 ][ (-1)^k/F_k-1 F_k; 0 F_k-1 ][ F_j+1 F_j; F_j F_j-1 ] = (-1)^j[ ∗ ∗; F_j((-1)^k+1F_j+1/F_k-1+F_j+1F_k-1-F_jF_k) ∗ ]. If (M_j)_21=0, then we would have F_k-1\|F_j+1 from (<ref>). But this is not possible for j≤ k-1 unless j=k-2: F_k-1\|̸ F_k takes care of j=k-1≥ 3 and if j≤ k-3, then F_k-1>F_j+1 since k≥ 4. Hence (M_j)_21≠ 0 if 1≤ j≤ k-1, j≠ k-2, so in this case P̂_j=β_+(Q_k) by Proposition <ref>(<ref>), with the “M” there equal to AMA^-1 and the “β” there equal to AD(1)^j∞. For the exceptional case j=k-2, explicit computation of M_k-2 using the Cassini/Vajda identities for Fibonacci numbers gives M_k-2=[ F_k-1 F_kF_k-3/F_k-1; 0 (-1)^k/F_k-1 ]. Hence by (<ref>) (M_k-2)_21=0 and \|(M_k-2)_11\|>\|(M_k-2)_22\| since k≥ 4. Hence P̂_k-2=β_+(Q_k) again by Proposition <ref>(<ref>), with the “M” there equal to AMA^-1 and the “β” there equal to AD(1)^k-2∞. Lastly we show the computation giving β_-(Q_k) and β_+(Q_k): β_-(Q_k)=a+ 1/b and β_+(Q_k)= a+F_k-1/bF_k-1+F_k-2. Calculating β_-(Q_k) and β_+(Q_k) entails finding the eigenvectors of E(Q_k)=AMA^-1 with A as in (<ref>) and M as in (<ref>). Let v_-[ 1; 0 ], λ_-(-1)^k/F_k-1, and v_+[ F_k-1; F_k-2 ], λ_+ F_k-1. Observe that \|λ_+\|>\|λ_-\| since k≥ 4. Using the Cassini identity yet again, verify that Mv_-=λ_-v_- and Mv_+=λ_+v_+. So E(Q_k)=AMA^-1 will have eigenvectors Av_-, Av_+ with eigenvalues λ_-, λ_+, respectively. We have Av_- =[ ab+1; b ]=b[ a+1/b; 1 ]=bv(β_-) and Av_+ =[ (ab+1)F_k-1+aF_k-2; bF_k-1+F_k-2 ]= (bF_k-1+F_k-2)[ a+F_k-1/bF_k-1+F_k-2; 1 ] =(bF_k-1+F_k-2)v(β_+), establishing the formulas (<ref>) for β_-(Q_k) and β_+(Q_k). §.§ The equivalence ~ on PCF(O) does not respect convergence Theorem <ref> shows quasiconvergence is a property of PCF equivalence class. It is possible for P∼ Q, both quasiconvergent, with P strictly quasiconvergent (and hence divergent) and Q convergent. For example, take a 0 and let P=[a,0,-1/a], Q =[a-1/a]. Let the convergents of P be 𝒞_i(P), i≥ 1. If \|a\|<1, then P satisfies Theorem <ref>(<ref>), and hence is strictly quasiconvergent. The limits P̂_j for 1≤ j≤ 3 are [ P̂_1 = lim_i→∞𝒞_1+3i(P) = a,; P̂_2 = lim_i→∞𝒞_2+3i(P) = -1/a,; P̂_3 = lim_i→∞𝒞_3+3i(P) = -1/a. ] The PCF Q converges to -1/a, since the convergents are just 𝒞_i(Q)=𝒞_3i(P). alpha
http://arxiv.org/abs/2307.02108v1	20230705083454	Proportional Response: Contextual Bandits for Simple and Cumulative Regret Minimization	[ "Sanath Kumar Krishnamurthy", "Ruohan Zhan", "Susan Athey", "Emma Brunskill" ]	cs.LG	[ "cs.LG", "stat.ML" ]	[ Stéphane Moreau August 1, 2023 =================== Simple regret minimization is a critical problem in learning optimal treatment assignment policies across various domains, including healthcare and e-commerce. However, it remains understudied in the contextual bandit setting. We propose a new family of computationally efficient bandit algorithms for the stochastic contextual bandit settings, with the flexibility to be adapted for cumulative regret minimization (with near-optimal minimax guarantees) and simple regret minimization (with SOTA guarantees). Furthermore, our algorithms adapt to model misspecification and extend to the continuous arm settings. These advantages come from constructing and relying on “conformal arm sets" (CASs), which provide a set of arms at every context that encompass the context-specific optimal arm with some probability across the context distribution. Our positive results on simple and cumulative regret guarantees are contrasted by a negative result, which shows that an algorithm can't achieve instance-dependent simple regret guarantees while simultaneously achieving minimax optimal cumulative regret guarantees. § INTRODUCTION Personalized treatment assignment policy learning is crucial across domains such as healthcare and e-commerce <cit.>. Traditional randomized control trials (RCTs), while foundational <cit.>, can be inefficient and costly <cit.>. Bandit algorithms, as a modern, adaptive sequential experimentation approach, can improve policy learning (simple regret minimization) and participant experience (cumulative regret minimization), making them particularly useful in various applications including adaptive clinical trials, where the goal is to optimize treatment while improving patient health <cit.>. In many practical settings, we are interested in balancing gathering information useful for decision making beyond the current study with minimizing cumulative regret. Yet there has been relatively little work so far into algorithms that can balance multiple objectives, like cumulative regret and simple regret (though see <cit.> for studies that address this empirically or juxtapose minimizing cumulative regret with estimating treatment effects or arm parameters). This work presents a new computationally efficient contextual bandit algorithm that can balance cumulative and simple regret minimization. It is known that minimax simple regret guarantees can be obtained using a randomized controlled trial (RCT) <cit.>, and one of our key objectives was to create an algorithm that outperforms minimax optimal RCTs for simple regret minimization across a broad range of contextual bandit instances, while also enabling good cumulative regret guarantees. To accomplish this, we introduce the notion of a "Conformal Arm Set" (CAS) at each context. These CASs encompass the context-specific optimal arm with some probability across the context distribution. Our algorithm maintains CASs via offline regression oracles, allowing us to make exponentially fewer oracle calls compared to previous methods that only relied on the more computationally expensive cost-sensitive classification oracle <cit.>. Furthermore, with CASs and an additional misspecification test, we can leverage regression approaches for very general reward regression models without relying on realizability assumptions (unlike prior work on upper confidence bound approaches, e.g. <cit.>). We use these with a new covering objective to balance simple and cumulative regret. Finally, our results also extend to the continuous arm setting. Our algorithm operates with a tuning parameter ω. With ω=1, we achieve near minimax optimal cumulative regret. While with ω>1, we attain improved instance-dependent simple regret guarantees over minimax optimal RCTs at the cost of increasing worst-case cumulative regret by up to a factor of √(ω). These upper bounds suggest a tension between the two objectives, and we present a lower bound showing that this tradeoff is unavoidable. Our result complements recent work that highlights the impossibility of achieving both (0,δ)-PAC instance-optimal sample efficiency and minimax optimal cumulative regret <cit.>. Note the focus in <cit.> was on creating an instance-optimal PAC algorithm.[This tradeoff is mitigated in simple MAB settings by arm elimination-based algorithms <cit.>.]. Our key contributions include the following. (i) A surrogate objective, optimal cover, linking data collection with simple regret minimization. (ii) Statistically and computationally efficient uncertainty quantification with conformal arm sets (CASs), with the flexibility of handling general reward model class for both finite and continuous arm settings. (iii) A versatile algorithm capable of trading-off worst-case cumulative regret guarantees with instance-dependent simple regret guarantees. (iv) A hardness result of the inherent tension in simultaneously improving simple and cumulative regrets. (v) A succinct simulation demonstration of the computational tractability of our approach. Related Work. Contextual bandit algorithms broadly fall into two categories: regression-free and regression-based. Regression-free algorithms create an explicit policy distribution, randomly choosing a policy for decision-making at any time-step <cit.>. While these algorithms provide worst-case cumulative regret <cit.> or instance-dependent PAC <cit.> guarantees without additional assumptions, they can be computationally intensive <cit.>: they require solving and storing the output of Ω(poly(T)) NP-hard cost-sensitive classification problems <cit.> at every epoch (or update step). In contrast, regression-based algorithms <cit.> construct a conditional arm distribution using regression estimates of the expected reward, allowing for methods that need only (1) oracle calls at every epoch. Traditionally, these algorithms relied on realizability assumptions for optimal regret guarantees, but recent advances allow for misspecified reward model classes <cit.>. We develop regression-based algorithms. Our work also recovers some cumulative regret guarantees for the continuous arm case <cit.>, with new guarantees on simple regret and robustness to misspecification. Note that our restriction to “slightly randomized" policies for the continuous arm case results in regret bounds with respect to a “slightly randomized" (smooth) benchmark <cit.>. Our work also connects to the literature on pure exploration, extensively studied in MAB settings (see overview in <cit.>). <cit.> study elimination-based algorithms for fixed confidence best-arm identification (BAI). <cit.> study variants of Thompson Sampling with optimal asymptotic designs for BAI. <cit.> propose sequential halving for fixed budget BAI. Our algorithm provides fixed confidence simple regret guarantees and can be seen as a generalization of successive elimination <cit.> to the contextual bandit setting. The key technical difference being it often isn't possible to construct sub-gaussian confidence intervals on conditional expected rewards. The uncertainty quantification we use is similar to the notion of conformal prediction (see <cit.> for a detailed exposition). Until recently, pure exploration had been nearly unstudied in contextual bandits. <cit.> provide a static exploration algorithm that achieves the minimax lower bound on sample complexity for linear contextual bandits. <cit.> then provided the first algorithm with instance-dependent (ϵ,δ)-PAC guarantees for contextual bandits. This algorithm is regression-free (adapts techniques from <cit.>) and requires a sufficiently large dataset of offline contexts as input. Hence, unfortunately, it inherits high memory and runtime requirements <cit.>. However, these costs come with the benefit that their notion of instance dependence leverages structure not only in the true conditional expected reward (as in <Ref>) but also leverages structure in the policy class (similar to policy disagreement coefficient <cit.>). As discussed earlier, they also prove a negative result, showing that it is not possible for an algorithm to have instance-dependent (0,δ)-PAC guarantees and achieve minimax optimal cumulative regret guarantees. Our hardness result is similar but complementary to their result, for we show a similar result for simple regret (rather than their (0,δ)-PAC sample complexity).[In (ϵ,δ) PAC sample complexity results, given an input (ϵ,δ), the objective is to minimize the number of samples needed in order to output an ϵ-optimal policy with probability at least 1-δ (a "fix accuracy, compute budget" setting). In contrast, in our simple regret case, we consider how to minimize the error ϵ as the number of samples increases.] § PRELIMINARIES §.§ Stochastic Contextual Bandits We consider the stochastic contextual bandit setting, with context space , (compact) arm space , and a fixed but unknown distribution D over contexts and arm rewards. D_ refers to the marginal distribution over contexts, and T signifies the number of rounds or sample size. At each time t ∈ [T][For any n∈ℕ^+, we use notation [n] to denote the set {1,...,n }], the environment draws a context x_t and a reward vector r_t ∈ [0,1]^ from D; the learner chooses an arm a_t and observes a reward r_t(a_t). To streamline notation for discrete and continuous arm spaces, we consider a finite measure space (,Σ,μ) over the set of arms, with K shorthand for μ()[Here Σ is a σ-algebra over and μ is a bounded set function from Σ to the real line.]. For ease of exposition, we focus on the finite/discrete arm setting. Here =[K] and μ is the count measure, and μ(S)=\|S\| for any S⊆. An action selection kernel p: ×→ [0,1] is a probability kernel that describes a distribution p(·\|x) over arms at every context x. Let D(p) be the induced distribution over ×× [0,1], where sampling (x, a, r(a)) ∼ D(p) is equivalent to sampling (x, r) ∼ D and then sampling a ∼ p(·\|x). A policy π maps contexts to singleton arm sets Σ_1:=a\|a∈[The introduction of Σ_1 is to allow for easy generalization to the continuous arm setting.] . With some abuse of notation, we also let π refer to the kernel given by π(a\|x)=I(a∈π(x)). A reward model f maps × to [0,1], with f^(x,a):=_D[r_t(a)\|x_t=x] denoting the true conditional expected reward model. Policy value and regret. For a given model f and action selection kernel p, we denote the expected instantaneous reward of p with f as R_f(p). We write R_f^(p) as R(p) to simplify notation when no confusion arises. The optimal policy associated with reward function f is defined as π_f[subject to any tie-breaking rule.]. R_f(p) = _x ∼ D__a∼ p(·\|x)[f(x, a)], and π_f ∈max_π R_f(π). For any S⊆, with some abuse of notation, we let f(x,S)=∫_a f(x,a)dμ(a)/μ(S). Note that π_f(x)∈max_S∈Σ_1 f(x,S) for all x. The “regret” of a policy π with respect to f is the difference between the optimal value and the actual value of π, denoted as _f(π) = R_f(π_f) - R_f(π). Objectives. In the experiment, arm selection utilizes estimated models of expected reward from a reward model class , allowing for violation of realizability (or model misspecification). The policy π_f induced by f∈ is assumed to be within policy class Π without loss of generality [Note that Π may contain policies that are not induced by models in the class .]. Here, π^* denotes an optimal policy in Π with maximum R(π). The experimenter's two primary goals are: Simple regret minimization. After the experiment, the experimenter uses the gathered data to infer a policy π̂∈Π aiming to maximize policy value. The simple regret _Π(π) quantifies how the inferred policy π̂ falls short compared to the optimal policy π^[ Note that _Π(π) should not be confused with _f^(π).], i.e., _Π(π) := R(π^) - R(π). Hence, the objective here is to minimize this simple regret. Cumulative regret minimization. The experimenter also seeks to maximize the cumulative expected reward during the experiment, equivalent to minimizing the cumulative regret against π_f^. Given action sampling kernels {p_t}_t∈[T] used by the CB algorithm over T rounds, the cumulative regret (_T) is _T:= ∑_t=1^T _f^(p_t). Cover. We now define an important quantity called the cover, which guides the learning procedure of our algorithm and is a measure of the overlap between the policy π and the action selection kernel p. Given a kernel p and a policy π, we define the cover of policy π under the kernel p to be, V(p,π):=_x∼ D_, a∼π(·\|x)[π(a\|x)/p(a\|x)]. Additionally, for any pair of kernels (p,q), we let V(p,q):=_x∼ D_, a∼ q(·\|x)[q(a\|x)/p(a\|x)]. Finally, we use the term optimal cover for kernel p to refer to V(p,π^). The cover measures the quality of data collected under the action selection kernel p for evaluating a given policy π and bounds the variance of commonly used unbiased estimators for policy value <cit.>. In particular, the cover under optimal policy 1/T∑_t=1^T V(p_t,π^) can be treated as a surrogate objective for simple regret minimization (proven in Appendix), which is particularly instructional in designing our algorithm to minimize simple regret. Extending notation to continuous arms. In the continuous arm setting, evaluating arbitrary deterministic policies can be infeasible without extra assumptions <cit.>. Thus, we focus on “slightly randomized” policies by generalizing Σ_1 to be the arm sets with measure one (Σ_1:={S∈Σ\|μ(S)=1}).[Note that our restriction to “slightly randomized" policies for the continuous arm case results in regret bounds with respect to a “slightly randomized" (smooth) benchmark. Hence for the continuous arm case, our cumulative regret bounds translate to smooth regret bounds from <cit.> with K=1/h. Where h is the measure of smoothness in smooth regret (a leading objective for this setting). ] The granularity of these sets can be adjusted by scaling the finite measure μ, which also affects the value of K=μ(). We then continue letting policies (π) be maps from to Σ_1 and Π is a class of such policies. With some abuse of notation, the induced kernel is still given by π(a\|x)=I(a∈π(x)). This is a valid kernel since ∫_aI(a∈π(x))dμ(a)=μ(π(x))=1. All the remaining definitions, including R_f(π),π_f,π^ and V(p,π), relied on these induced kernels and continue to hold. While there are some measure theoretic issues that remain to be discussed, we push these details to the appendix. Uniform sampling. Our algorithm frequently selects an arm uniformly from a constructed set of arms. In the context of a set S⊆, uniform sampling refers to selecting an arm from the distribution q(a):=I(a∈ S)/μ(S). This constitutes a probability measure since its integral over equals 1. In the discrete arm setting, uniform sampling from a set S⊆ implies selecting an arm according to the distribution I(a∈ S)/\|S\|. §.§ Oracle Assumptions Our algorithm relies on two sub-routines. For generality, we abstract away these sub-routines by stating them as oracle assumptions, for which we describe two oracles, and in Assumptions <ref> and <ref> respectively. The sub-routine is for estimating conditional expected reward models (<Ref>), and the sub-routine is for estimating policy values (<Ref>) according to the true and estimated reward models. These sub-routine tasks are supervised learning problems. Hence, the average errors for the corresponding tasks can be bounded in terms of the number of samples (n) and a confidence parameter (δ'). The oracle assumptions describe these estimation rates. We let ξ:×[0,1]→[0,1] denote the estimation rate for these oracles; for simplicity, we assume that they share the same rate and that ξ(n,δ') scales polynomially in 1/n and log(1/δ'). In order to simplify the analysis, we also require ξ(n/3,δ'/n^3) be non-increasing in n.[This ensures that ξ_m defined in <Ref> is non-increasing in m for any epoch schedule with increasing epoch lengths.] We now formally describe these oracle assumptions, starting with . [Estimation Oracle] We assume access to a reward model estimation oracle () that takes as input an action selection kernel p, and n independently and identically drawn samples from the distribution D(p). The oracle then outputs an estimated model ∈ such that for any δ'∈(0,1), the following holds with probability at least 1-δ': _x∼ D__a∼ p(·\|x)[ ((x, a) - f^(x,a))^2 ] ≤ B + ξ(n,δ') Where B≥ 0 is a fixed but unknown constant that may depend on the model class and distribution D, but is independent of the action selection kernel p. In <Ref>, the parameter B measures the bias of model class ; under realizability, B equals 0. The function ξ characterizes the estimation variance, which decreases with increasing sample size. As long as the variance term (which shrinks as we gather more data) is larger than the fixed unknown bias (B), we have from <Ref> that the expected squared error for the estimated reward model is bounded by 2ξ. We use this bound to further bound how accurately the estimated reward model evaluates policies in the class Π. However, since B is unknown, we need to validate/test if these bounds hold. Therefore, our algorithm relies on which provides consistent independent policy value estimates and helps compare them with policy value estimates with respect to the estimated reward model. [Evaluation Oracle] We assume access to an oracle () that takes as input an action selection kernel p, n independently and identically drawn samples from the distribution D(p), a set of m models {g_i\|i∈[m]}⊆, and another action selection kernel q. The oracle then outputs a policy evaluation estimator , and a set of m policy evaluation estimator {_g_i\|i∈[m] }} that estimate policy value with respect to the models g_1,g_2,…,g_m respectively. Such that for any δ'∈(0,1), the following conditions simultaneously hold with probability at least 1-(m+1)δ': \|(π)-R(π)\| ≤√(2V(p,π)ξ(n,δ')) + 2ξ(n,δ')/(min_(x,a)∈×p(a\|x)) for all π∈Π∪{q}. * \|_f(π)-R_f(π)\| ≤√(2ξ(n,δ')) for all π∈Π∪{q} and for all f∈{g_i\|i∈[m]}. When and Π are finite, one can construct oracles such that Assumptions <ref> and <ref> hold with ξ(n,δ')=(log(max(\|\|,\|Π\|)/δ')/n). One example of such a construction is given by using empirical squared loss minimization for , using inverse propensity scores (IPS) for estimating R(π) in , and using the empirical average for estimating R_f(π) in . The guarantees of these assumptions can be derived using Bernstein's inequality and union bounding. When has pseudo-dimension <cit.> bounded by d and Π has the Natarajan-dimension bounded by d <cit.>, one can construct oracles such that Assumptions <ref> and <ref> hold with ξ(n,δ')=(dlog(nK/δ')/n). § ALGORITHM At a high level, our algorithm operates in two modes, indicated by a Boolean variable “". During mode one (=), where estimated reward models are sufficiently accurate at evaluating policies in the class Π,[Where the estimated reward models pass the misspecification test.] we use our estimated models to update our action selection kernel used during exploration. During mode two (=), where the condition for mode one no longer holds, we stop updating the action selection kernel used for exploration. Operationally our algorithm runs in epochs/batches indexed by m. Epoch m begins at round t=τ_m-1+1 and ends at t=τ_m, with m(t) denoting the epoch index containing round t. We let denote the critical epoch, at the end of which our algorithm changes mode (with “" being updated from “" to “"), and refer to this epoch as the algorithmic safe epoch. For all rounds in epoch m≤, our algorithm samples action using the action selection kernel p_m defined later in (<ref>). For m>, our algorithm samples action using the action selection kernel p_ – which is the action selection kernel used in the algorithmic safe epoch m̂. We will now describe the critical components of our algorithm. These include (i) data splitting and using oracle sub-routines; (ii) misspecification tests, which we use to identify the -mode switching epoch ; and (iii) conformal arm sets, which presents a new form of uncertainty quantification that is critical in constructing p_m+1 at the end of each epoch m∈[]. Finally, we use these components to describe our final algorithm. Data Splitting and Oracle Sub-routines. Consider any epoch m∈[]. Let S_m denote the set of samples collected in this epoch, that is S_m={(x_t,a_t,r_t(a_t))\|t∈[τ_m-1,τ_m]}. Our algorithm starts by splitting the sample set S_m into three equally sized subsets S_m,1,S_m,2 and S_m,3. As described in <Ref>, we use these sets of samples and the oracles described in <Ref> to estimate reward models and policy evaluators. Based on Assumptions <ref> and <ref>, we bound the errors for these estimates in terms of ξ_m+1=2ξ((τ_m-τ_m-1)/3,δ/(16m^3)), where δ is a pre-specified confidence parameter. As we will see in the following sections, our algorithm relies on these bounds to test for misspecification and construct action selection kernels for exploration. Misspecification Test. We now discuss the need for our misspecification test and state it later. Recall that <Ref> is flexible and allows our reward model class to be misspecified. In particular, the squared error of our reward model estimate may depend on an unknown bias term B. To account for this unknown bias term, we will find it particularly useful to center our analysis around the safe epoch :={m≥ 1\|ξ_m+1≥ 2B}; is the last epoch where variance dominates bias. We show that for any epoch m∈[], the estimated reward model _m+1 is “sufficiently accurate" at evaluating the expected reward of any policy in Π∪{p_m+1}. This property is critical in ensuring that the constructed action selection kernel p_m+1 has low exploration regret _f^(p_m+1) and a small optimal cover (V(p_m+1,π^)). Since B and are unknown, we need to test and ensure that the estimated reward model is sufficiently accurate at evaluating these policies. When the test fails, the algorithm sets the variable “safe” to and stops updating the action selection kernel used for exploration. The core idea for this test comes from <cit.> although its application to simple regret minimization is new, and the form of our test differs a bit. At the end of each epoch m, the misspecification test is passed if the following inequality holds. max_π∈Π∪{p_m+1}\|_m+1,_m+1(π)-_m+1(π)\| - √(α_mξ_m+1)∑_∈[m]_m+1,_(π__)-_m+1,_(π)/40^2 √(α_-1ξ_) ≤ 2.05√(α_mξ_m+1) + 1.1√(ξ_m+1), where α_ empirically bounds V(p_,π^), the optimal cover for the action selection kernel used in epoch . The first term in (<ref>) measures how well the estimated reward model _m+1 evaluates the policy π, and the second term helps normalize for under-explored policies. Conformal Arm Sets. We proceed to introduce the notion of conformal arm sets (CASs), based on which we construct the action selection kernels employed by our algorithms. At the beginning of each epoch m, we construct conformal arm sets, denoted as {C_m(x,ζ)\|x∈,ζ∈[0,1]}; here ζ controls the probability with which the set C_m contains the optimal arm. The construction of these sets rely on the models (_1,…,_m) estimated from data up to epoch m-1, as defined below. Consider ζ∈(0,1). At epoch m, for context x, the arm set C_m(x, ζ) is given by (<ref>). C_m(x, ζ) := π__m(x) ⋃C̅_m(x, ζ), C̅_m(x, ζ) := ⋂_∈[m]C_(x, ζ/2^2), C_(x, ζ') := {a: _(x, π__(x)) - _(x,a)≤20√(α_-1ξ_)/ζ'} ∀∈[m], ζ'∈(0,1). Similar to conformal prediction (CP) <cit.>, CASs have marginal coverage guarantees. We show that with high probability, we have π^(x) lies in C_m(x,ζ) with probability at least 1-ζ under the context distribution. That is, _x∼ D_(π^(x)∈ C_m(x,ζ)) ≥ 1 - ζ with high-probability. However, there is also a key technical difference. While CP provides coverage guarantees for the conditional random outcome, CASs provide coverage guarantees for π^(x) – which is not a random variable given the context x. Hence, intervals estimated by CP need to be wide enough to account for conditional outcome noise, whereas CASs do not. Moreover, CASs have several advantages compared to pointwise confidence intervals used in UCB algorithms. First, CASs are computationally easier to construct. Second, CAS widths have a polynomial dependency on model class complexity, whereas pointwise intervals may have an exponential dependence for some function classes <cit.>. Third, pointwise intervals require realizability, whereas the guarantees of CASs hold even without realizability (as long as the misspecification test in (<ref>) holds). However, it's important to note that these benefits of CASs come with the risk of marginal coverage guarantees. Risk Adjusted Proportional Response Algorithm. We now describe the design of our algorithm, which is summarized in <Ref>. The algorithm depends on the following input parameters: ω∈[1,K] which controls the trade-off between simple and cumulative regret, the proportional response threshold ∈ (0,1/2], and confidence parameter δ. The algorithm also computes η_m+1 (risk adjustment parameter for p_m+1), α_m+1 (empirical bound on optimal cover for p_m+1), and λ_m+1(·) (empirical bound on average CAS size). At the end of every epoch m∈[], we construct the action selection kernel p_m+1 given by (<ref>). p_m+1(a\|x) = (1-β_max)I[a∈ C_m+1(x,β_max/η_m+1)]/μ(C_m+1(x,β_max/η_m+1)) + ∫_0^I[a∈ C_m+1(x,β/η_m+1) ]/μ(C_m+1(x,β/η_m+1))β. At any context x, sampling arm a from p_m+1(·\|x) is equivalent to the following. Sample β uniformly from [0,1], then sample arm a uniformly from the set C_m+1(x,min(,β)/ η_m+1). A small β results in a larger CAS and a higher probability of containing the optimal arm for the sampled context. However, uniformly sampling an arm from a larger CAS also implies a lower probability on every arm in the set. Sampling β uniformly allows us to respond proportionately to the risk of not sampling the optimal arm while enjoying the benefits of smaller CASs. We refer to this as the proportional response principle. Similarly, a larger risk-adjustment parameter η_m+1 encourages reliance on less risky albeit larger CASs. Let λ_m+1(η) be a high-probability empirical upper bound on E_x∼ D_[μ(C_m+1(x,/η))]. We then choose η_m+1∈[1,√(ω K/α_m)] in order to balance the risk of a small η with the benefits of a small λ_m+1(η). Our analysis shows that the choice of η_m+1 in (<ref>) helps minimize our upper bound on the optimal cover for epoch m+1 subject to constraints bounding regret during exploration. λ_m+1(η) := min(1+1/\|S_m,2\|∑_t∈ S_m,2μ(C̅_m+1(x, β_max/η)) + √(K^2ln(8\|S_m,2\|(m+1)^2/δ)/2\|S_m,2\|),K), η_m+1←max{η_m, max{η=\|S_m,2\|/n\|n ∈ [\|S_m,2\|], η≤√(ω K/α_m),λ_m+1(η) ≤K/η}}. With η_m+1 chosen, the action selection kernel p_m+1 is completely described. We now let α_m+1 be a high-probability empirical upper bound on V(p_m+1,π^). We then use α_m+1 at the end of epoch m+1: to construct CASs, to compute the risk-adjustment parameter, and to test for misspecification. α_m+1←3K/η_m+1. Computation. We have (log(T)) epochs. At the end of any epoch m∈[], we solve three optimization problems. The first is for estimating _m+1, which often reduces to empirical squared loss minimization and is computationally tractable for several function classes . The second is for computing the risk-adjustment parameter in (<ref>) which can be solved via binary search. The third is for the misspecification test in (<ref>), which can be solved via two calls to a cost-sensitive classification (CSC) solver (don't need this when assuming realizability, further if we only care about cumulative regret, sufficient to use the simpler test in <cit.>). Finally, to learn a policy π̂ at the end of T rounds, we need to solve (<ref>) using a CSC solver (under realizability we can set π̂=π__m(T)-1). Hence, overall, ω-RAPR makes exponentially fewer calls to solvers compared to regression-free algorithms like <cit.>. § MAIN RESULTS We first characterize the cumulative regret guarantees provided by RAPR. As mentioned earlier, our algorithm/analysis/results hold for both the discrete and continuous arm cases. theoremthmRAPRCumulativeRegret Suppose Assumptions <ref> and <ref> hold. Then with probability 1-δ, ω-RAPR attains the following cumulative regret guarantee. _T ≤( ∑_t=τ_1+1^T√(K/α_m(t)α_m(t)-1/α_m(t))(√(KB)+√(Kξ_m(t))) ) ≤(√(ω KB)T + ∑_t=τ_1+1^T√(ω Kξ_m(t))). Where ξ_m+1 = ξ((τ_m-τ_m-1)/3,δ/(16m^3); and we use to hide terms logarithmic in T,K,ξ(T,δ). We start with discussing (<ref>). The first part √(ω KB)T comes from the bias of the regression oracle with model class and will vanish under the realizability assumption. The second part ∑_t=τ_1+1^T√(ω Kξ_m(t)), when setting ω=1, recovers near-optimal (upto logarithmic factors) minimax cumulative regret guarantees for common model classes, as demonstrated by the following examples. We consider ω-RAPR with appropriate oracles in the following cases and let B denote the corresponding bias terms. When and Π are finite, _T≤(√(ω KB)T + √(ω K T log(max(\|\|,\|Π\|)/δ))) with probability at least 1-δ. When has a finite pseudo dimension d, Π has a finite Natarajan dimension d, and is finite, _T≤(√(ω KB)T + √(ω K T dlog(TK/δ))) with probability at least 1-δ. Note that under realizability (B=0), 1-RAPR achieves near-optimal minimax cumulative guarantees. In (<ref>), we observe that the multiplicative √(ω) cost to cumulative regret is only incurred if the empirical bound on optimal cover (α_m∈[1,3K]) can get small. That is, our cumulative regret bounds degrades only if our algorithm better bounds the optimal cover and thus ensures better simple regret guarantees. [For large t, once our bounds on optimal cover can't be significantly improved, we have α_m(t)/α_m(t)-1= O(1). Hence for large T, our cumulative regret is a factor of √(K/α_m(T)+1) larger than the near optimal minimax guarantees. As we will see in <Ref>, this multiplicative factor is unavoidable.] We now provide instance dependent simple regret guarantees for our algorithm. theoremthmRAPRSimpleRegret Suppose Assumptions <ref> and <ref> hold. For some environment parameters λ∈ [0,1], Δ>0, and A∈[1,K], consider an instance where the following (<ref>) holds. ℙ_x∼ D_(μ({a∈: f^(x,π_f^(x)) - f^(x,a) ≤Δ}) ≤ A ) ≥ 1- λ. Let m'=min(,m(T))-1 and let the learned policy π̂ be given by (<ref>). π̂∈max_π∈Π_m(T)(π)- 1/2√(α_m'ξ_m(T))∑_∈[m']_m(T),_(π__)-_m(T),_(π)/40^2 √(α_-1ξ_) . Then with probability 1-δ, we have the following simple regret bound when T samples are collected by ω-RAPR. _Π(π̂)≤(√(α_m'ξ_m(T))) ≤(√(ξ_m(T)min(K, A+Kλ+K/ω + K^3/2ω^1/2/Δ√(ξ_min(,m(T)-1)-⌈log_2log_2(K)⌉))) ) ). Under (<ref>), we can only argue that the expected (over context distribution) measure of Δ optimal arms is at most (1-λ)A+Kλ = O(A+Kλ). Hence for large T, the best we can hope for is instance-dependant simple regret guarantees that shrink/improve over minimax guarantees by a factor of (√((A+Kλ)/K)). We show that this is guaranteed by <Ref>. Suppose ω=K, has a finite pseudo dimension bounded by d, and Π has a finite Natarajan dimension bounded by d. The simple regret guarantee of <Ref> reduces to (min((√(Kd/T), √((A+Kλ)d/T)+(K/√(Δ))√(d/T)√(B+d/T))). When the reward model estimation bias B is small enough, the term (K/√(Δ))√(d/T)√(B+d/T)) is dominated by the remaining terms for large T. Hence, in this case, we get a simple regret bound of (√((A+Kλ)d/T)) for large T. As promised, this improves upon the minimax guarantees by a factor of (√((A+Kλ)/K)). Note that <Ref> guarantees are better for ω closer to 1 whereas <Ref> guarantees are better for ω closer to K. Hence these theorems show a tradeoff between the cumulative and simple regret guarantees for ω-RAPR. <Ref> shows that improving upon minimax simple regret guarantees for instances satisfying (<ref>) must come at the cost of worse cumulative regret guarantees. This finding substantiates the inescapable trade-off observed in the simple/cumulative regret bounds for ω-RAPR. theoremthmLowerBound Given parameters K,F,T∈ and ϕ∈ [1,∞). There exists a context space and a function class ⊆ (×→ [0,1]) with K actions such that \|\|≤ F and the following lower bound on cumulative regret holds: inf_𝐀∈Ψ_ϕsup_D∈𝒟 _D[∑_t=1^T (r_t(π^(x_t)) - r_t(a_t) ) ]≥Ω( √(K/ϕ)√(KTlog F)) Here (a_1,… a_T) denotes the actions selected by an algorithm A. 𝒟 denotes the set of environments such that f^∈ and (<ref>) hold with (A,λ,Δ)=(1,0,0.24). Π denotes policies induced by . Ψ_ϕ denotes the set of CB algorithms that run for T rounds and output a learned policy with a simple regret guarantee of √(ϕlog F/T) for any instance in 𝒟 with confidence at least 0.95, i.e., Ψ_ϕ:={A: ℙ((π̂_𝐀)≤√(ϕlog F/T))≥ 0.95 for any instance in 𝒟}. Finally, Ω(·) hides factors logarithmic in K and T. Note that the environments constructed in <Ref> satisfy f^∈ with max(\|\|,\|Π\|)≤ F. Hence, with appropriate oracles, Assumptions <ref> and <ref> are satisfied with B=0 (i.e. =∞) and ξ(n,δ')=(log(F/δ')/n). Further, these environments also satisfy (<ref>) with (A,λ,Δ)=(1,0,0.24). Hence for large enough T, ω-RAPR achieves a simple regret bound of (√((K/ω)log F/T)) with probability at least 0.95. Therefore, ω-RAPR is a member of Ψ_ϕ for some ϕ=(K/ω). <Ref> lower bounds the cumulative regret of such algorithms by Ω( √(K/ϕ)√(KTlog F) )=Ω( √(ω)√(KTlog F) ). Up to logarithmic factors, this matches the cumulative regret upper bound for ω-RAPR. Re-emphasizing that the trade-off obsererced in Theorems <ref> and <ref> is unavoidable. Simulation. We ran a uniform RCT and a version of K-RAPR with a linear function class, =1/2, and δ=0.05 on synthetic data. The immediate goal is to illustrate the tractability of our algorithm (and our ongoing work is expanding these simulations). With =[4], =[-1,1]^2, and T=1000, contexts are uniformly sampled from four regions in , where each region is associated with a unique optimal arm. The underlying expected outcomes are a sum of linear functions (in arm and context) and an indicator function denoting whether the context-arm pair corresponds to the same region. A simulation run takes about a minute on a laptop with 16GB RAM and an Apple M1 Pro chip. Averaging over 1000 runs, our algorithms improved simple regret over the RCT by 25%. § CONCLUSION We develop Risk Adjusted Proportional Response (RAPR), a computationally efficient regression-based contextual bandit algorithm. It is the first contextual bandit algorithm capable of trading-off worst-case cumulative regret guarantees with instance-dependent simple regret guarantees. The versatility of our algorithm allows for general reward models, handles misspecification, extends to finite and continuous arm settings, and allows us to choose the trade-off between simple and cumulative regret guarantees. The key ideas underlying RAPR are conformal arm sets (CASs) to quantify uncertainty, proportional response principle for cumulative regret minimization, optimal cover as a surrogate for simple regret, and risk adjustment for better bounds on the optimal cover. Future directions. The proportional response principle is related to IGW at a high level. For example, <Ref> shows that the p_m(a\|x) is lower bounded by a quantity that depends on the inverse estimated gap for arm a from the estimated optimal arm at context x. This suggests that the Inverse Gap Weighting algorithm <cit.>, which does not appear to rely on any explicit uncertainty quantification, could be interpreted to rely on a version of CASs. Understanding this connection better could help in the development of better algorithms and could be an exciting direction for future work. Our algorithm also generalizes successive elimination <cit.> to the contextual bandit setting.[Consider the non-contextual finite arm bandit setting (\|\|=1). Here each policy is defined by a single arm. Let a^ be the optimal arm/policy. Since only one context exists, we do not need to parameterize CASs with the context. That is, C_m(x,ζ)≡ C_m(ζ). Note that, for any ζ∈(0,1), _x∼ D_(a^* ∈ C_m(ζ)) ≥ 1-ζ > 0 is equivalent to a^* ∈ C_m(ζ). Hence, ζ doesn't influence the conformal arm set guarantee. Therefore, the construction of CASs need not depend on ζ. We can let CASs be defined by a single set that contains all potentially optimal arms. That is, C_m(ζ)≡ C_m. At every round in epoch m, the risk adjusted proportional response strategy selects the set C_m and samples an arm uniformly from this set. Therefore, this strategy is equivalent to successive elimination for the non-contextual setting.] Therefore, another interesting direction for future work is to extend other elimination algorithms to the contextual bandit setting. Limitations. A limitation of our approach is that we do not utilize the structure of the policy class being explored. Further refining CASs with other forms of uncertainty quantification that leverage such structure can lead to significant improvements. Another limitation is our focus on rates; the constants used in <Ref> should be optimized. §.§.§ Acknowledgements S.A. and S.K.K. are grateful for the generous support provided by Golub Capital Social Impact Lab and the Office of Naval Research grant N00014-19-1-2468. abbrvnat § EXPANDED NOTATIONS We start with expanding our notation from <Ref> to include notation helpful for our proofs and expand to the continuous arm setting. Measure over arms. To recap, our algorithm and analysis adapt to both discrete and continuous arm spaces, where we consider a finite measure space (,Σ,μ) over the set of arms (i.e. μ() is finite) to unify the notation.[Here Σ is a σ-algebra over and μ is a bounded set function from Σ to the real line.] As short hand, we use K in lieu of μ(). We let Σ_1 be a set of arms in Σ with measure one, i.e. Σ_1:={S∈Σ\| μ(S)=1 }. Policies. Let Π denote the universal set of policies. That is, Π is the set of all functions from to Σ_1. The policy class Π is a subset of Π. We use π(x) to denote the set of arms given x∈ and use p_π(a\|x)=𝕀(a∈π(x)) to denote the induced probability measure over arms at x.[Note that for any π∈Π, we have ∫_a∈ p_π(a\|x) dμ = μ(π(x)) = 1 at any x∈.] With some abuse of notation, we use the notation π(a\|x) in lieu of p_π(a\|x). Below is the elaboration of our notation to both discrete and continuous arm spaces. * Discrete arm space. We choose μ to be the count-measure, where μ(S)=\|S\| for any S⊆ and μ()=K. In this case, Σ_1 contains singleton arm sets, and Π denotes deterministic policies from to where each policy maps a context to an action. * Continuous arm space. We choose μ to any finite measure, where μ(S) =∫_ Sdμ(a) for any S⊆, and in particular μ()=K. In this case, Σ_1 contains arm sets that may have an infinite number of arms but with total measure be 1 with respect to μ. Space of action selection kernels. In this paper, we will always define our action selection kernels with respect to the reference measure μ, that is p(S\|x)=∫_a∈ S p(a\|x)dμ for any S∈Σ and x∈. Based on the notation in <Ref>, for any kernel p, we let _f(p) = R_f(π_f) - R_f(p). Now let denote the set of action selection kernels such that p(a\|x)≤ 1 for all (x,a)∈×, and in particular, we have the policy class Π⊂. We note that all action selection kernels (p) considered in this paper belong to the set , allowing our analysis to rely on the fact that p(·\|·)≤ 1. Note that, _f(p) is non-negative for any p∈. To see this, consider any context x. Recall that π_f(x)∈max_S∈Σ_1 f(x,S) for all x. Since p(·\|·)≤ 1 for any p∈, we have ∫_a∈p(a\|x)f(x,a) dμ is maximized when p(a\|x)=1 for all a∈π_f(x). That is, f(x,π_f(x))=max_p∈_a∼ p(·\|x)[f^(x,a)]. Hence, max_p∈𝒫R_f(p) = R_f(π_f), so _f(p) is non-negative for any p∈. Connection to smooth regret <cit.>. Recall that we define cumulative regret as _T:= ∑_t=1^T _f^(p_t), which measures regret w.r.t the benchmark R_f^(π_f^) = max_p∈𝒫R_f^(p). As discussed earlier, <cit.> shows that smooth regret bounds are stronger than several other definitions of cumulative regret in the continuous arm setting <cit.>. Hence to show that our bounds are comparable/competitive for the continuous arm setting, we argue that our definition of cumulative regret (_T) is equivalent to the definition of smooth regret in <cit.>. Let the loss vectors l_t in <cit.> be given by -r_t. Let the smoothness parameter h in <cit.> be given by 1/K. And, let the base probability measure in <cit.> be given by μ/K. Then, our benchmark (max_p∈𝒫R_f^(p)) is equal to the smooth benchmark ([Smooth_h(x)]) considered in <cit.>. Hence, our definition of cumulative regret (_T) is equal to smooth regret (_CB,h(T)) when the loss, smoothness parameter, and base probability measure are given as above. This shows the equivalence in our definitions. Hence our near-optimal cumulative regret bounds (with ω=1) recover several existing results for the stochastic contextual bandit setting up to logarithmic factors using only offline regression oracles. Our algorithm also handles reward model misspecification and does not assume realizability. We also provide instance-dependent simple regret bounds (for larger choices of ω). The parameter ω allows us to trade-off between simple and cumulative regret bounds. Measure theoretic issues with continuous arms. To avoid measure-theoretic issues, we require that for all models f∈∪{f^}, all contexts x∈, and all real numbers z∈ℝ, we have the level set of arms {a\| f(x,a)≤ z } must lie in Σ. That is the reward models f∈∪{f^} are measurable at every context x with the Lebesgue measure on the range of f(x,·) and the measure (,Σ,μ) on the domain of f(x,·). We note that this isn't a strong condition and usually trivially holds. Moreover, we require an additional condition as follows to simplify our arguments and allow for easy construction of our uncertainty sets (see <Ref>). We require that for all models f∈ and all contexts x∈, we have f(x,π_f(x)) is equal to max_a∈ f(x,a). This condition trivially holds for the finite-arm setting with μ as a count measure. For the continuous arm setting, this condition follows from requiring _a∈ f(x,a) lies in Σ and has measure of at least one. Additional notation. For notational convenience, we let U_m=20√(α_m-1ξ_m) for any epoch m. Note that by construction (see (<ref>)) α_m is non-increasing in m. Further, from the conditions in <Ref>, we have ξ_m+1=2ξ((τ_m-τ_m-1)/3,δ/(16m^3)) is non-increasing in m. Hence U_m is also non-increasing in m. We also let α_0:=α_1=3K, and let α_m:=α_ for any epoch m≥. Similarly, we let η_0:=η_1=1, and let η_m:=η_ for any epoch m≥. Sometimes, we use use C_m(x,β,η) in lieu of C_m(x,β/η). § BOUNDING CUMULATIVE REGRET This section derives the cumulative regret bounds for ω-RAPR. We start with analyzing the output of oracles described in Assumptions <ref> and <ref>. Note that we do not make the “realizability" assumption in this work – i.e., we do not assume that f^* lies in . Hence, as in <Ref>, the expected squared error of our estimated models need not go to zero (even with infinite data) and may contain an unknown non-zero irreducible error term (B) that captures the bias of the model class . It is useful to split our analysis into two regimes to handle this unknown term B, similar to the approach in <cit.>. In particular, we separately analyze oracle outputs for epochs before and after a so-called “safe epoch". Where we define the safe epoch as the epoch where the variance of estimating from the model class (ξ_m) is dominated by the bias of estimating from the class (B). That is, :={m≥ 1\|ξ_m+1≥ 2B}. §.§ High Probability Events We start with defining high-probability events under which our key theoretical guarantees hold. The first high probability event characterizes the accuracy of estimated reward models and the policy evaluation estimators, the tail bound of which can be obtained by taking a union bound of each epoch-specific event that happens with probability 1-δ/4m^2 under assumptions in <Ref>. Suppose <Ref> and <Ref> hold. The following event holds with probability 1-δ/2, := { ∀ m , ∀π∈Π∪{p_m+1}, ∀ f∈{_1,…,_m+1}, _x∼ D__a∼ p_m(·\|x)[ (_m+1(x, a) - f^(x,a))^2 ] ≤ B + ξ_m+1/2 \|_m+1,f(π)-R_f(π)\| ≤√(ξ_m+1), \|_m+1(π)-R(π)\| ≤√(V(p_m,π)ξ_m+1) + ξ_m+1/(min_(x,a)∈×p_m(a\|x)). ) }. Where ξ_m+1=2ξ((τ_m-τ_m-1)/3,δ/(16m^3)).[Our epoch schedules will always be increasing in epoch length. Under such conditions, we have ξ_m is non-increasing in m.] The second high probability event characterizes the measure of conformal arm sets, which directly follows from Hoeffding's inequality and union bound. The following event holds with probability 1-δ/2, := { ∀ m, ∀η∈{\|S_m,2\|/n\|n ∈ [\|S_m,2\|] }, \| _x∼ D_[μ( C̅_m+1(x, /η))] - 1/\|S_m,2\|∑_t∈ S_m,2μ(C̅_m+1(x_t,/η)) \| ≤√(K^2ln(8\|S_m,2\|m^2/δ)/2\|S_m,2\|)}. Together both and hold with probability 1-δ. The rest of our analysis works under these events. §.§ Analyzing the Cover In this sub-section, we upper bound the cover (V(p_m,·)) for the action selection kernel used in epoch m.[Recall that V(p,q):=_x∼ D_, a∼ q(·\|x)[q(a\|x)/p(a\|x)].] To upper bound V(p_m,q), we first lower bound p_m(·\|·). Recall that in <Ref>, we define U_m=20√(α_m-1ξ_m) for any epoch m. Starting from here, our lemmas and proofs will use U_m and 20√(α_m-1ξ_m) interchangeably. For any epoch m, we have (<ref>) holds. p_m(a\|x) ≥1-/μ(C_m(x, , η_m)) + /μ(), if a∈ C_m(x, , η_m) η_m/μ()min_∈[m]2^2U_/_(x,π__(x))-_(x,a), if a∉ C_m(x, , η_m) ≥1/μ(), if a∈ C_m(x, , η_m) η_mmin_∈[m] 2^2U_/μ(), if a∉ C_m(x, , η_m) Recall that p_m given by (<ref>). p_m(a\|x) = (1-β_max)I[a∈ C_m(x,,η_m)]/μ(C_m(x,β_max,η_m)) + ∫_0^I[a∈ C_m(x,β,η_m) ]/μ(C_m(x,β,η_m))β. We divide our analysis into two cases based on whether a lies in C_m(x, , η_m), and lower bound p_m(a\|x) in each case. Case 1 (a∈ C_m(x, , η_m)). Note that C_m(x, , η_m)⊆ C_m(x, β, η_m) ⊆ for all β∈[0,]. Hence, a∈ C_m(x, β, η_m) and μ(C_m(x, β, η_m))≤μ() for all β∈[0,]. Therefore, in this case, p_m(a\|x)≥(1-β_max)/μ(C_m(x,β_max,η_m))+/μ()≥1/μ(). Case 2 (a∉ C_m(x, , η_m)). For this case, the proof follows from (<ref>). p_m(a\|x) ≥ ∫_0^I[a∈ C_m(x,β,η_m) ]/μ(C_m(x,β,η_m))β (i)≥ 1/μ()∫_0^I[a∈ C_m(x,β,η_m) ]β (ii)≥ I(a∉ C_m(x,,η_m))/μ()∫_0^I[a∈ C_m(x,β,η_m) ]β (iii)= I(a∉ C_m(x,,η_m))/μ()∫_0^1I[a∈ C_m(x,β,η_m) ]β (iv)= I(a∉ C_m(x,,η_m))/μ()∫_0^1∏_∈[m] I[_(x,π__(x))-_(x,a)≤2^2η_mU_/β]β = I(a∉ C_m(x,,η_m))/μ()∫_0^1∏_∈[m] I[β≤2^2η_mU_/_(x,π__(x))-_(x,a)]β = η_mI(a∉ C_m(x,,η_m))/μ()min_∈[m]2^2U_/_(x,π__(x))-_(x,a) (v)≥ η_mI(a∉ C_m(x,,η_m))/μ()min_∈[m] 2^2U_ where (i) is because the measure of the conformal set C_m can be no larger than the measure of the action space ; (ii) follows from I(a∉ C_m(x,,η_m))≤ 1; (iii) follows from the fact that if a∉ C_m(x,,η_m) then a∉ C_m(x,β,η_m) for all β≥; (iv) follows from <Ref>; and (v) follows from .[Note that if a∈π__m(x) then I(a∉ C_m(x,,η_m)) = 0.] Using <Ref>, we get an upper bound on V(p_m,q) in terms of [μ(C_m(x, , η_m))], K/η_m, and expected regret with respect to the models _1,…,_m. lemmalemBoundVCont For any epoch m and any action selection kernel q∈, we have (<ref>) holds. V(p_m, q)≤[μ(C_m(x, , η_m))]/1- + K/η_m∑_∈[m]__(q)/2^2U_. From <Ref>, we have (<ref>) holds. [Note that we require f(x,π_f(x))≥ f(x,a), for all x∈, a∈, and f∈. ] I(a∈ C_m(x, , η_m))/p_m(a\|x)≤μ(C_m(x,,η_m))/1-, I(a∉ C_m(x, , η_m))/p_m(a\|x)≤μ()/η_mmax_∈[m]_(x,π__(x))-_(x,a)/2^2U_. We now bound the cover V(p_m,q) as follows, V(p_m,q)=_x∼ D_, a∼ q(·\|x)[q(a\|x)/p_m(a\|x)] (i)≤ _x∼ D_, a∼ q(·\|x)[1/p_m(a\|x)] = _x∼ D_, a∼ q(·\|x)[I[a∈ C_m(x,, η_m)] + I[a∉ C_m(x,, η_m)]/p_m(a\|x)] (ii)≤ _x∼ D_, a∼ q(·\|x)[μ(C_m(x,,η_m))/1-] + _x∼ D_, a∼ q(·\|x)[μ()/η_mmax_∈[m]_(x,π__(x))-_(x,a)/2^2U_] ≤ [μ(C_m(x, , η_m))]/1- + μ()/η_m∑_∈[m]__(q)/2^2U_. Here (i) follows from the fact that q∈ and (ii) follows from (<ref>). Having bounded the cover for the kernel p_m in terms of [μ(C_m(x, , η_m))] and K/η_m. We now bound these terms with α_m. lemmalemConditionAlpham Suppose holds. Then for any epoch m, we have (<ref>) holds. [μ( C_m(x, , η_m))]/1- + K/η_m≤α_m≤3K/η_m. Since μ( C_m(x, , η_m))≤ K and = 1/2, the bound trivially holds if η_m=1. Suppose η_m>1. Note that η_ is non-decreasing in by construction. Let m' be the smallest epoch index such that η_m'=η_m. We now have the following holds. [μ( C_m(x, , η_m))]/1- + K/η_m (i)≤min{1+[μ( C̅_m(x, , η_m))],K}/1- + K/η_m (ii)≤min{1+[μ( C̅_m'(x, , η_m))],K}/1- + K/η_m (iii)≤λ_m'(η_m)/1-+K/η_m (iv)≤ 2λ_m'(η_m)+K/η_m (v)≤3K/η_m Here (i) follows from the definition of CASs. (ii) follows from C̅_m⊆C̅_m' which follows from the fact that C̅_m=∩_∈[m]C̃_ and m'≤ m. (iii) follows from and the definition of λ_m' in (<ref>). (iv) follows from the fact that = 1/2. (v) follows from (<ref>) and η_m'-1<η_m'=η_m – note that η_m'-1≠η_m' gives us that η_m' was set using the constrained maximization procedure in (<ref>), hence the constraint λ_m'(η)≤ K/η is satisfied at η=η_m'=η_m. §.§ Evaluation Guarantees Under Safe Epoch This sub-section provides guarantees on how accurate R__m+1 is at evaluating policies when we are within the safe epoch. lemmalemBoundSafeError Suppose holds. For all epochs m∈ [], for any q∈, we have, \|R_f̂_m+1(q)-R(q)\|≤√(V(p_m, q)ξ_m+1). Consider any epoch m∈[] and policy q∈. We then have, \|R_f̂_m+1(q)-R(q)\| =\| _x∼ D_, a∼ q[_m+1(x,a)-f^(x,a)]\| (i)=\| _x∼ D_, a∼ p_m[q(a\|x)/p_m(a\|x)(_m+1(x,a)-f^(x,a))]\| ≤_x∼ D_, a∼ p_m[q(a\|x)/p_m(a\|x)\|_m+1(x,a)-f^(x,a)\|] = _x∼ D_, a∼ p_m[√((q(a\|x)/p_m(a\|x))^2\|_m+1(x,a)-f^(x,a)\|^2)] (ii)≤√(_x∼ D_, a∼ p_m[( q(a\|x)/p_m(a\|x))^2])√(_x∼ D_, a∼ p_m[ (_m+1(x,a)-f^(x,a))^2] ) (iii)=√(_x∼ D_, a∼ q[ q(a\|x)/p_m(a\|x)])√(_x∼ D_, a∼ p_m[ (_m+1(x,a)-f^(x,a))^2] ) (iv)≤√(V(p_m, q)ξ_m+1) , where (i) and (iii) follow from change of measure arguments, (ii) follows from Cauchy-Schwartz inequality, and (iv) follows from . By combining the guarantees of <Ref> and <Ref>, we get <Ref>. lemmalemRefineSafeErrorBound Suppose and hold. Then for any action selection kernel q∈, we have: \|R_f̂_m+1(q)-R(q)\| ≤√(α_mξ_m+1) + 1/2√(α_mξ_m+1)∑_∈[m]__(q)/2^2U_. From <Ref>, we have (<ref>) holds for any q∈. V(p_m, q) ≤[μ( C_m(x, , η_m))]/1- + K/η_m∑_∈[m]__(q)/2^2U_ ≤([μ( C_m(x, , η_m))]/1- + K/η_m) + ([μ( C_m(x, , η_m))]/1- + K/η_m)∑_∈[m]__(q)/2^2U_ = α_m + α_m ∑_∈[m]__(q)/2^2U_ Combining (<ref>) with <Ref> we have: \|R_f̂_m+1(q)-R(q)\| ≤ √(V(p_m,q)ξ_m+1) (i)≤ 1/2√(α_mξ_m+1) + 1/2√(ξ_m+1/α_m)V(p_m,q) (i)≤ √(α_mξ_m+1) + 1/2√(α_mξ_m+1)∑_∈[m]__(q)/2^2U_. Where (i) follows from AM-GM inequality and (ii) follows from (<ref>). §.§ Testing Safety The misspecification test (<ref>) is designed to test if we are within the safe epoch. In principle, it works by comparing the accuracy of R__m+1 (<Ref>) and _m+1 (<Ref>). Formally, <Ref> shows that the misspecification test in (<ref>) fails only after . Hence, ≥+1. <Ref> then describes the implication of (<ref>) continuing to hold. In what follows, we let _m+1,_(π):=_m+1,_(π__)-_m+1,_(π). We start with <Ref> which provides accuracy guarantees for _m+1 in any epoch. lemmalemBoundIPSError Suppose and hold. Then for any epoch m and all π∈Π∪{p_m+1}, we have, \|_m+1(π)-R(π)\| ≤√(α_mξ_m+1) + 1/2√(α_mξ_m+1)∑_∈[m]__(π)/2^2U_ + Kξ_m+1/η_m min_∈[m]U_. From <Ref>, we have (<ref>) holds, which provides a worst-case lower bound on p_m. min_(x,a)∈×p_m(a\|x)≥min(1/K,η_m min_∈[m] (2^2)U_/K) ≥η_m min_∈[m] U_/K Where the last inequality follows from U_m≤ 1. Now from , we have, \|_m+1(π)-R(π)\| ()≤√(V(p_m,π)ξ_m+1) + Kξ_m+1/η_m min_∈[m] U_ (i)≤ 1/2√(α_mξ_m+1) + 1/2√(ξ_m+1/α_m)V(p_m,π) + Kξ_m+1/η_m min_∈[m] U_ (ii)≤ √(α_mξ_m+1) + 1/2√(α_mξ_m+1)∑_∈[m]__(π)/2^2U_ + Kξ_m+1/η_m min_∈[m] U_. Where (i) follows from AM-GM inequality, and (ii) follows from (<ref>) in the proof of <Ref>. Lemmas <ref> and <ref> provide useful inequalities that help construct the misspecification test (<ref>). lemmalemTriangleIneq For any epoch m, policy π∈, and model f∈{_1, _2, …,_m+1}. We have, \|\|R_f(π)-R(π)\| - \|_m+1,f(π)-_m+1(π)\| \| ≤ \|_m+1(π)-R(π)\|+\|R_f(π)-_m+1,f(π)\|. The proof follows from noting that, \|R_f(π)-R(π)\| = \|_m+1(π)-R(π)+R_f(π)-_m+1,f(π)+_m+1,f(π)-_m+1(π)\| ≤ \|_m+1(π)-R(π)\|+\|R_f(π)-_m+1,f(π)\|+\|_m+1,f(π)-_m+1(π)\|. and from noting that, \|_m+1(π)-_m+1,f(π)\| = \|_m+1(π)-R(π)+R_f(π)-_m+1,f(π)+R(π)-R_f(π)\| ≤ \|_m+1(π)-R(π)\|+\|R_f(π)-_m+1,f(π)\|+\|R_f(π)-R(π)\|. lemmalemBoundModelRegret Suppose and hold. Then for any epoch m, any model f∈{_i\|i∈[m+1] }, and any policy π∈Π∪{p_m+1}, we have, \|_f(π)-_m+1,f(π)\| ≤ 2√(ξ_m+1) Follows from triangle inequality and , \|_f(π)-_m+1,f(π)\| ≤ \|R_f(π_f)-_m+1,f(π_f)\| + \|R_f(π)-_m+1,f(π)\| ≤ 2√(ξ_m+1) As discussed earlier, <Ref> shows that the misspecification test in (<ref>) fails only after . Hence, ≥+1. lemmalemMisspecificationTest Suppose and hold. Now for any epoch m∈[] we have that, max_π∈Π∪{p_m+1}\|_m+1,_m+1(π)-_m+1(π)\| - √(α_mξ_m+1)∑_∈[m]_m+1,_(π)/2^2U_ ≤ 2.05√(α_mξ_m+1) + 1.1√(ξ_m+1). For any epoch m∈[] and for any π∈Π∪{p_m+1}, we have, \|__m+1(π)-_m+1(π)\| (i)≤ \|_m+1(π)-R(π)\| + \|R_f̂_m+1(π)-R(π)\| +\|R_f̂_m+1(π)-__m+1(π)\| (ii)≤ 2√(α_mξ_m+1) + √(α_mξ_m+1)∑_∈[m]__(π)/2^2U_ + Kξ_m+1/η_m min_∈[m] U_ + √(ξ_m+1) (iii)≤ 2√(α_mξ_m+1) + √(α_mξ_m+1)∑_∈[m]__(π)/2^2U_ + α_mξ_m+1/U_m+ √(ξ_m+1) (iv)≤ 2√(α_mξ_m+1) + √(α_mξ_m+1)∑_∈[m]_m+1,_(π)/2^2U_ + 2√(α_m)ξ_m+1/U_m + α_mξ_m+1/U_m + √(ξ_m+1) (v)≤ 2.05√(α_mξ_m+1) + √(α_mξ_m+1)∑_∈[m]_m+1,_(π)/2^2U_ + 1.1√(ξ_m+1) Where (i) follows from <Ref>. (ii) follows from <Ref>, <Ref>, and . (iii) follows from <Ref> and the fact that U_ is non-increasing in (giving us min_∈[m] U_=U_m). (iv) follows from <Ref>, the fact that U_ is non-increasing in , and the fact that ∑_=1^∞1/(2^2)≤ 1. (v) follows from U_m=20√(α_m-1ξ_m), α_m≤α_m-1, and ξ_m+1≤ξ_m. <Ref> now describes the implication of (<ref>) continuing to hold. lemmalemTestImplication Suppose and hold. Now for any epoch m∈[-1] and any policy π∈Π∪{p_m+1}, we then have that, \|R__m+1(π)-R(π)\|≤ 2.2√(ξ_m+1) + 3.1√(α_mξ_m+1) + 3/2√(α_mξ_m+1)∑_∈[m]__(π)/2^2U_. \|R__m+1(π)-R(π)\| (i)≤ \|_m+1(π)-R(π)\| + \|__m+1(π)-_m+1(π)\| +\|R_f̂_m+1(π)-__m+1(π)\| (ii)≤ 3.1√(α_mξ_m+1) + 3/2√(α_mξ_m+1)∑_∈[m]__(π)/2^2U_ + 2.2√(ξ_m+1) Where (i) follows from <Ref>. And (ii) follows from <Ref>, <Ref>, <Ref>, and . §.§ Inductive Argument This sub-section leverages the guarantee of <Ref> and applies it inductively to derive <Ref>. This lemma bounds _Π(π) in terms of __m+1(π) and vice-versa for any policy π∈Π. The proof of <Ref>, relies on the following helpful lemma. lemmalemInductiveArg Consider any class of policies Π' ⊇Π and consider any fixed constants l_1,l_2,l_3,C'>0. At any epoch m, suppose the policy evaluation guarantee of <Ref> holds. ∀π∈Π', \|R__m+1(π)-R(π)\| ≤ l_1√(ξ_m+1) + l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]z_,m+1__(π)/2^2 Now consider fixed constants C_1,C_2≥ 0. As an inductive hypothesis, suppose <Ref> holds. ∀∈[m],∀π∈Π', __(π) ≤4/3_Π(π) + C_1√(ξ_)+ C_2√(α_-1ξ_). We then have that <Ref> holds. ∀π∈Π', _Π(π) ≤6/5__m+1(π) + 12/5(l_1 + l_3C_1/C')√(ξ_m+1) + 12/5(l_2 + l_3C_2/C')√(α_mξ_m+1). Now consider C_3≥ 0 and further suppose <Ref> holds. ∀∈[m], __(π__m+1)≤ C_3√(α_-1ξ_). We then also have that <Ref> holds, ∀π∈Π', __m+1(π) ≤7/6_Π(π) + (2l_1 + l_3C_1/C')√(ξ_m+1) + (2l_2 + l_3(C_2+C_3)/C')√(α_mξ_m+1). Where C'≥ 8l_3, z_,m+1:=√(α_mξ_m+1/α_-1ξ_)≤ 1, and α_m≤α_-1 for all ∈[m]. Consider any policy π∈Π'. Suppose (<ref>) and (<ref>) hold. We first show (<ref>). _Π(π) - __m+1(π) = R(π^) - R(π) - R__m+1(π__m+1) + R__m+1(π) ≤ R(π^) - R__m+1(π^) + (R__m+1(π) - R(π) ) (i)≤ 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]z_,m+1/2^2(__(π)+__(π^)) (ii)≤ 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]z_,m+1/2^2( 4/3_Π(π)+ 2C_1√(ξ_) + 2C_2√(α_-1ξ_)) = 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]1/2^2( 4z_,m+1/3_Π(π)+ 2C_1√(ξ_m+1)/√(α_-1/α_m) + 2C_2√(α_mξ_m+1)) (iii)≤(2l_1 + 2l_3C_1/C')√(ξ_m+1) + (2l_2 + 2l_3C_2/C')√(α_mξ_m+1) + 4/3l_3/C'_Π(π) , Where (i) follows from (<ref>), (ii) follows from (<ref>) and from _Π(π^)=0, and finally (iii) follows from z_,m+1≤ 1, α_m≤α_-1, and ∑_∈[m]1/(2^2)≤ 1. Now (<ref>) immediately implies (<ref>). (1-4l_3/3C')_Π(π) ≤__m+1(π) + (2l_1 + 2l_3C_1/C')√(ξ_m+1) + (2l_2 + 2l_3C_2/C')√(α_mξ_m+1) (i) _Π(π) ≤6/5__m+1(π) + 12/5(l_1 + l_3C_1/C')√(ξ_m+1) + 12/5(l_2 + l_3C_2/C')√(α_mξ_m+1) Where (i) follows from the fact that C'≥ 8l_3. Similar to (<ref>), we will now show (<ref>). __m+1(π) - _Π(π) = R__m+1(π__m+1) - R__m+1(π) - (R(π^)-R(π)) ≤(R__m+1(π__m+1) -R(π__m+1)) +( R(π)-R__m+1(π)) (i)≤ 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]z_,m+1/2^2(__(π__m+1)+__(π)) (ii)≤ 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]z_,m+1/2^2( 4/3_Π(π)+ C_1√(ξ_) + (C_2+C_3)√(α_-1ξ_)) = 2l_1√(ξ_m+1) + 2l_2√(α_mξ_m+1) + l_3/C'∑_∈[m]1/2^2( 4z_,m+1/3_Π(π)+ C_1√(ξ_m+1)/√(α_-1/α_m) + (C_2+C_3)√(α_mξ_m+1)) (iii)≤(2l_1 + l_3C_1/C')√(ξ_m+1) + (2l_2 + l_3(C_2+C_3)/C')√(α_mξ_m+1) + 4l_3/3C'_Π(π) Where (i) follows from (<ref>), (ii) follows from (<ref>), (<ref>), and (iii) follows from z_,m+1≤ 1, α_m≤α_-1, and ∑_∈[m]1/(2^2)≤ 1. Now (<ref>) immediately implies (<ref>). __m+1(π) ≤(1+4l_3/3C')_Π(π) + (2l_1 + l_3C_1/C')√(ξ_m+1) + (2l_2 + l_3(C_2+C_3)/C')√(α_mξ_m+1) (i)__m+1(π) ≤7/6_Π(π) + (2l_1 + l_3C_1/C')√(ξ_m+1) + (2l_2 + l_3(C_2+C_4)/C')√(α_mξ_m+1). Where (i) follows from the fact that C'≥ 8l_3. lemmalemRegret Suppose and hold. Now for any epoch m∈[-1], we then have that (<ref>) holds. ∀π∈Π, _Π(π)≤4/3__m+1(π) + 6.5√(ξ_m+1) + 12√(α_mξ_m+1), __m+1(π)≤4/3_Π(π) + 6.5√(ξ_m+1) + 12√(α_mξ_m+1). Moreover when m∈[], we have (<ref>) holds for all policies π∈. Note that (<ref>) trivially holds for m=0. We will now use an inductive argument. Consider any epoch m∈[]. As an inductive hypothesis, let us assume (<ref>) holds. (i.e. (<ref>) holds for epoch m-1.) ∀π∈Π, ∈[m], _Π(π)≤4/3__(π) + 6.5√(ξ_) + 12√(α_-1ξ_), __(π)≤4/3_Π(π) + 6.5√(ξ_) + 12√(α_-1ξ_). Hence from (<ref>), we have (<ref>) holds with C_1=6.5 and C_2=12. Since m∈[], from <Ref>, we have (<ref>) holds. ∀π∈Π∪{p_m+1}, \|R__m+1(π)-R(π)\| ≤22/10√(ξ_m+1) + 31/10√(α_mξ_m+1) + 3/40∑_∈[m]z_,m+1__(π)/2^2 Hence from (<ref>), we have (<ref>) holds with l_1=2.2, l_2=3.1, l_3=1.5, C' = 20, and Π=Π. Hence from <Ref>, we have (<ref>) holds. ∀π∈Π, _Π(π) ≤6/5__m+1(π) + 12/5(22/10 + 1.56.5/20)√(ξ_m+1) + 12/5(31/10 + 1.512/20)√(α_mξ_m+1) = 6/5__m+1(π) + 6.45√(ξ_m+1) + 9.6√(α_mξ_m+1) ≤4/3__m+1(π) + 6.5√(ξ_m+1) + 12√(α_mξ_m+1) Now from (<ref>) and (<ref>), we have (<ref>) holds. ∀∈[m], __(π__m+1) ≤4/3_Π(π__m+1) + 6.5√(ξ_) + 12√(α_-1ξ_) ≤4/3(0 + 6.5√(ξ_m+1) + 12√(α_mξ_m+1)) + 6.5√(ξ_) + 12√(α_-1ξ_) ≤ 43.2√(α_-1ξ_) Hence from (<ref>), we have (<ref>) holds with C_3 = 43.2. Therefore from <Ref> we have (<ref>). ∀π∈Π, __m+1(π) ≤7/6_Π(π) + (2l_1 + l_3C_1/C')√(ξ_m+1) + (2l_2 + l_3(C_2+C_3)/C')√(α_mξ_m+1) = 7/6_Π(π) + (22.2 + 1.56.5/20)√(ξ_m+1) + (23.1 + 1.5(12+43.2)/20)√(α_mξ_m+1) = 7/6_Π(π) + 4.8875√(ξ_m+1) + 10.34√(α_mξ_m+1) ≤4/3_Π(π) + 6.5√(ξ_m+1) + 12√(α_mξ_m+1) From (<ref>) and (<ref>), we have (<ref>) holds for epoch m. This completes our inductive argument. An immediate implication of <Ref> is that we have __m(π^)≤ U_m for all m∈ []. Hence, from <Ref> and <Ref>, we have (<ref>) holds. V(p_m,π^) ≤[μ( C_m(x, , η_))]/1- + K/η_m≤α_m, ∀ m∈[]. §.§ Bounding Exploration and Cumulative Regret This sub-section leverages the structure of the kernel p_m+1, and the guarantees in Lemmas <ref> and <ref> to bound the expected regret during exploration (<Ref>). Then, summing up these exploration regret bounds, we get our cumulative regret bound in <Ref>. We start with <Ref> which leverages structure in p_m to bound __(p_m) for any ∈[m]. lemmalemExplorationModelRegret For any pair of epochs m∈[+1] and ∈[m], we have that (<ref>) holds. __(p_m)≤ 15.2√(ξ_)+28√(α_-1ξ_)+2^2η_mU_(1/+ ln/2^2η_m U_) We first make the following observation. _x∼ D_x_a∼ p_m(a\|x)[I(a∉π__m(x))·(_(x,π__(x))-_(x,a))] =_x∼ D_x[∫_a∈∖π__m(x)(_(x,π__(x))-_(x,a))p_m(a\|x) dμ(a)] (i)=_x∼ D_x[∫_a∈∖π__m(x)∫_β∈[0,1](_(x,π__(x))-_(x,a))I[a∈ C_m(x,min(β,)/η_m) ]/μ(C_m(x,min(β,)/η_m))βμ(a)] (ii)≤_x∼ D_x[∫_β∈[0,1]∫_a∈∖π__m(x)min(1, 2^2η_mU_/min(β,))I[a∈ C_m(x,min(β,)/η_m) ]/μ(C_m(x,min(β,)/η_m))μ(a)β] ≤∫_β∈[0,1]min(1, 2^2η_mU_/min(β,))β≤ (1-)2^2η_m U_/ +∫_0^min(1, 2^2η_mU_/β)β where (i) follows from the definition of p_m given in (<ref>). (ii) follows from the fact that for any ζ∈(0,1) and a∈ C_m(x,ζ)∖π__m(x) we have _(x,π__(x))-_(x,a)≤min(1,2^2/ζ), since C_m(x,ζ)∖π__m(x) ⊆C̅_m(x,ζ) ⊆C̃_(x,ζ/(2^2)) by <Ref>, . We now bound __(p_m). __(p_m) = _x∼ D_x_a∼ p_m(a\|x)[_(x,π__(x))-_(x,a)] = _x∼ D_x_a∼ p_m(a\|x)[(I(a∈π__m(x))+I(a∉π__m(x)))·(_(x,π__(x))-_(x,a))] (i)≤__(π__m)+( (1-)2^2η_m U_/ +∫_0^min(1, 2^2η_mU_/β)β) ≤__(π__m)+( 2^2 η_m U_(1-)/ + 2^2η_mU_ + 2^2η_mU_∫_2^2η_mU_^1/ββ) = __(π__m)+2^2η_mU_(1/+ ln/2^2η_m U_) (ii)≤4/3_Π(π__m)+6.5√(ξ_)+12√(α_-1ξ_)+2^2η_mU_(1/+ ln/2^2η_m U_) (iii)≤4/3(6.5√(ξ_m)+12√(α_m-1ξ_m))+6.5√(ξ_)+12√(α_-1ξ_) +2^2η_mU_(1/+ ln/2^2η_m U_) (iv)≤ 15.2√(ξ_)+28√(α_-1ξ_)+2^2η_mU_(1/+ ln/2^2η_m U_) where (i) follows from (<ref>), (ii) follows from <Ref>, (iii) follows from <Ref> and __m(π__m)=0, and (iv) follows from ≤ m. Now from the guarantees in Lemmas <ref>, <ref>, and <ref> we get the following bound on _Π(p_m+1). lemmalemExplorationRegret Suppose and hold. Now for any epoch m∈[-1], we have that (<ref>) holds. _Π(p_m+1)≤ 100(m+1)^2η_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) Since m∈[], from <Ref>, we have (<ref>) holds. ∀π∈Π∪{p_m+1}, \|R__m+1(π)-R(π)\| ≤22/10√(ξ_m+1) + 31/10√(α_mξ_m+1) + 3/40∑_∈[m]z_,m+1__(π)/2^2 We will now bound _Π(p_m+1) in terms of __(p_m+1) for ∈[m+1]. _Π(p_m+1) - __m+1(p_m+1) = R(π^) - R(p_m+1) - R__m+1(π__m+1) + R__m+1(p_m+1) ≤ R(π^) - R__m+1(π^) + (R__m+1(p_m+1) - R(p_m+1) ) (i)≤44/10√(ξ_m+1) + 62/10√(α_mξ_m+1) + 3/40∑_∈[m]z_,m+1/2^2(__(p_m+1)+__(π^)) (ii)≤44/10√(ξ_m+1) + 62/10√(α_mξ_m+1) + 3/40∑_∈[m]z_,m+1/2^2(__(p_m+1) + 6.5√(ξ_)+12√(α_-1ξ_)) (iii)≤44/10√(ξ_m+1) + 62/10√(α_mξ_m+1) + 3/40∑_∈[m]1/2^2(z_,m+1__(p_m+1) + 6.5√(ξ_m+1)+12√(α_mξ_m+1)) (iv)≤ 4.9√(ξ_m+1) + 7.1√(α_mξ_m+1) + 3/40∑_∈[m]z_,m+1/2^2__(p_m+1). Where (i) follows from (<ref>), (ii) follows from <Ref> and from _Π(π^)=0, (iii) follows from z_,m+1:=√(α_mξ_m+1/α_-1ξ_) and α_m≤α_-1, finally (iv) follows from ∑_∈[m]1/(2^2)≤ 1. We now simplify the last term in the upper bound of (<ref>). ∑_∈[m]z_,m+1/2^2__(p_m+1) (i)≤∑_∈[m]z_,m+1/2^2(15.2√(ξ_)+28√(α_-1ξ_)+2^2η_m+1U_(1/+ ln/2^2η_m+1 U_) ) (ii)≤∑_∈[m]1/2^2(15.2√(ξ_m+1)+28√(α_mξ_m+1) +40^2η_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) ) (iii)≤ 15.2√(ξ_m+1)+28√(α_mξ_m+1)+20mη_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) Where (i) follows from <Ref>, (ii) follows from z_,m+1:=√(α_mξ_m+1/α_-1ξ_), choice of U_m, and α_m≤α_-1, finally (iii) follows from ∑_∈[m]1/(2^2)≤ 1. By combining (<ref>), (<ref>), and <Ref>, we get our final result. _Π(p_m+1) (i)≤__m+1(p_m+1) + 6.04√(ξ_m+1) + 9.2√(α_mξ_m+1) + 1.5mη_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) (ii)≤ 21.3√(ξ_m+1)+37.2√(α_mξ_m+1) +41.5(m+1)^2η_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) Where (i) follows from (<ref>) and (<ref>), and (ii) follows from <Ref>. The earlier bound on _Π(p_m+1) now immediately gives us the following bound on _f^(p_m+1). lemmalemExplorationRegretWithBenchmark Suppose and hold. Now for any epoch m∈[-1], we have that (<ref>) _f^(p_m+1)≤ 2√(KB) + 100(m+1)^2η_m+1√(α_mξ_m+1)(1/+ ln/40η_m+1√(α_mξ_m+1)) From <Ref> (properties of ), we know the bias of the model class is bounded by B. In particular, we know there exists g∈ such that _x∼ D_, a∼()[ (g(x,a)-f^(x,a))^2]≤ B. Hence, we have, the following. _f^(π^) (i)≤_f^(π_g) = R(π_f^) - R(π_g) = (R(π_f^)-R_g(π_f^)) -_g(π_f^) + (R_g(π_g)-R(π_g)) (ii)≤ \|R(π_f^)-R_g(π_f^)\| + \|R_g(π_g)-R(π_g)\| (iii)≤(√(_x∼ D_, a∼π_f^[ π_f^(a\|x)/1/K]) + √(_x∼ D_, a∼π_g[ π_g(a\|x)/1/K])) ·√(_x∼ D_, a∼()[ (g(x,a)-f^(x,a))^2]) (iv)≤ 2√(KB). Here (i) follows from the fact that π_g∈Π since g∈. (ii) follows from triangle inequality and the fact that _g(π_f^)≥ 0. (iii) follows from the proof of <Ref>. And (iv) follows from _x∼ D_, a∼()[ (g(x,a)-f^(x,a))^2]≤ B and π_f(a\|x)=I(a∈π_f(x)). Since _f^(p_m+1) = R(π_f^) - R(π^) + R(π^) - R(p_m+1) = _f^(π^) + _Π(p_m+1), the result follows from combining the above with <Ref>. We now get our final cumulative regret bound by summing up the exploration regret bounds in <Ref>. * From <Ref>, both and hold with probability 1-δ. We prove our cumulative regret bounds under these events. Under , from <Ref>, we have ≥+1. Further, from conditions in <Ref>, we have ξ_m is non-increasing in m. Since ξ(n,δ') scales polynomially in 1/n and log(1/δ'), there exists a constant Q_0> 1 such that the doubling epoch structure ensures ξ_m≤ Q_0ξ_m+1 for all m. Hence ξ_≤ Q_0ξ_+1≤ Q_0ξ_+2≤ 2Q_0B. Let m'(t)=min(m(t),). Hence, ξ_m'(t)≤max(ξ_m(t),ξ_) ≤ 2Q_0B+ξ_m(t). Therefore, by summing up the bounds in <Ref>, we have the following cumulative regret bound. _T ≤∑_t=1^T _f^(p_m'(t)) ≤τ_1+∑_t=τ_1+1^T (2√(KB) + 100(m'(t))^2η_m'(t)√(α_m'(t)-1ξ_m'(t))(1/+ ln/40η_m'(t)√(α_m'(t)-1ξ_m'(t))) ) ≤(∑_t=τ_1+1^T (η_m'(t)√(α_m'(t)-1ξ_m'(t)))) = (∑_t=τ_1+1^T η_m(t)√(α_m(t)-1/K)(√(Kξ_m'(t)))) ≤(∑_t=τ_1+1^T η_m(t)√(α_m(t)-1/K)(√(KB)+√(Kξ_m(t)))) Now the theorem follows from the fact that we have: η_m√(α_m-1/K)<Ref>≤ 3√(K/α_mα_m-1/α_m)<Ref>≤ 3η_m√(α_m-1/K)(<ref>)≤ 3√(ω). § BOUNDING SIMPLE REGRET In this section, we prove our simple regret bound (<Ref>). Our analysis starts with <Ref>, which provides instance dependent bounds on _x∼ D_[μ(C_m(x,β,η)) ]. We will later use <Ref> to derive instance-dependent bounds on α_m. This bound then helps us derive instance-dependant bounds on simple regret. lemmalemBoundConformalSetSize For some environment parameters λ∈ (0,1), Δ>0, and A∈[1,K], consider an instance where (<ref>) holds. ℙ_x∼ D_(μ({a∈: f^(x,π_f^(x)) - f^(x,a) ≤Δ}) ≤ A ) ≥ 1- λ. Suppose and hold. For all epochs m, suppose the action selection kernel is given by <ref>, and suppose (<ref>) holds for all ∈[m]. Then for any epoch m∈[], we have (<ref>) holds. _x∼ D_[μ(C_m(x,β,η)) ] ≤(1+A+Kλ)+25K/Δη/β√(α_m-1ξ_m). For any β∈(0,1/2] and η∈[1,K]. Consider any epoch m∈[]. In this proof, for short-hand, let C := [μ(C_m(x,β,η))]. We then have, C = [μ(C_m(x,β,η))] ≤ (A+1) P(μ(C_m(x,β,η))≤ A+1) + K P(μ(C_m(x,β,η))> A+1) ≤ A+1 + K P(μ(C_m(x,β,η))> A+1) ≤ A+1 + K - K P(μ(C_m(x,β,η))≤ A+1) The above immediately implies (<ref>). P(μ(C_m(x,β,η))≤ A+1) ≤A+1+K-C/K. Let π_0∈Π be defined by (<ref>). ∀ x∈, π_0(x) ∈_S∈Σ_1\| S⊆ C_m(x,β,η) f^(x,S). Since π_0 only selects arms in C_m(x,β,η), from <Ref>, we have (<ref>). __m(π_0)≤η/βU_m. We can lower bound the regret of π_0 as follows, _f^(π_0) ≥ P( f^(x,π_f^(x)) - f^(x, π_0(x))> Δ)·Δ (i)= P(∃ S∈Σ_1\| S⊆ C_m(x,β,η), f^(x,π_f^(x)) - f^(x, S)> Δ)·Δ (ii)≥ P(μ(C_m(x,β,η))≥ A+1 μ({a :(f^(x,π_f^(x)) - f^(x, a)> Δ}) ≥ K-A)·Δ = P(μ(C_m(x,β,η))≥ A+1 μ({a :(f^(x,π_f^(x)) - f^(x, a)≤Δ}) ≤ A)·Δ = (1-P(μ(C_m(x,β,η))< A+1 μ({a :(f^(x,π_f^(x)) - f^(x, a)≤Δ}) > A) ) ·Δ (iii)≥ (1-P(μ(C_m(x,β,η))< A+1) -P(μ({a :(f^(x,π_f^(x)) - f^(x, a)≤Δ}) > A) ) ·Δ = (P(μ({a :(f^(x,π_f^(x)) - f^(x, a)≤Δ}) ≤ A) -P(μ(C_m(x,β,η))< A+1) ) ·Δ (iv)≥ (1-λ- A+1+K-C/K)Δ = (C-A-1/K - λ)Δ. where (i) is because by construction π_0(x)⊆ C_m(x,β,η) for all x, (ii) is by the fact that μ is a finite measure with μ()=:K, (iii) follows from union bound, and (iv) follows from (<ref>) and (<ref>). We will now work towards upper bounding _f^(π_0), and use this bound in conjunction with (<ref>) to obtain our desired bound on C. To upper bound _f^(π_0) using <Ref>, we will upper bound __m-1(π_0) and __m-1(π_f^). __m-1(π_0) (i)≤4/3_Π(π_0) + 12√(α_m-2ξ_m-1) + 6.5 √(ξ_m-1) (ii)≤4/3(4/3__m(π_0) + 12√(α_m-1ξ_m) + 6.5 √(ξ_m)) + 12 √(α_m-2ξ_m-1) + 6.5 √(ξ_m-1) (iii)≤16/9__m(π_0) + 28√(α_m-2ξ_m-1) + 91/6√(ξ_m-1) (iv)≤16/9__m(π_0) + 259/6√(α_m-2ξ_m-1). Where (i) and (ii) follow from <Ref>, (iii) follows from z_m-1=√(α_m-1ξ_m/α_m-2ξ_m-1)≤ 1, and (iv) follows from α_m-2≥ 1. __m-1(π_f^) (i)≤ 12√(α_m-2ξ_m-1) + 6.5√(ξ_m-1) (ii)≤37/2√(α_m-2ξ_m-1). Where (i) follows from <Ref>, (ii) follows from α_m-2≥ 1. _f^(π_0) = R(π_f^) - R(π_0) = (R(π_f^) - R__m(π_f^) ) - (R(π_0) - R__m(π_0) ) + (R__m(π_f^) - R__m(π_0) ) (i)≤ 2√(α_m-1ξ_m) + 1/2√(α_m-1ξ_m)∑_m̅∈[m]1/2m̅^2U_m̅(__m-1(π_f^)+__m-1(π_0)) + __m(π_0) (ii)≤ 2√(α_m-1ξ_m) + 1/40∑_m̅∈[m]z_m̅,m-1/2m̅^2(16/9__m(π_0)+(37+18.54/3)√(α_m-2ξ_m-1)) + __m(π_0) (iii)≤ 3.6√(α_m-1ξ_m)+ 47/45__m(π_0) (iv)≤√(α_m-1ξ_m)(3.6+47/45* 20 η/β) ≤ 25η/β√(α_m-1ξ_m) Where (i) follows from <Ref>. (ii) follows from (<ref>), (<ref>), and U_m-1=20√(α_m-2ξ_m-1), (iii) follows from z_m-1≤ 1, and (iv) follows from (<ref>) and U_m=20√(α_m-1ξ_m). Finally, combining (<ref>) and (<ref>), we have, (C-A-1/K - λ)Δ≤_f^(π_0) ≤ 25η/β√(α_m-1ξ_m) C ≤ A+1+Kλ+25K/Δη/β√(α_m-1ξ_m). In <Ref>, we use the bound from <Ref> to derive instance-dependent bounds on α_m. <Ref> is an immediate implication of <Ref>, and provides a bound on α_m that doesn't depend on α_m-1. Finally, <Ref> is used to derive our instance-dependant bound on simple regret. lemmalemboundalphamtheoretically For some environment parameters λ∈ (0,1), Δ>0, and A∈[1,K], consider an instance where (<ref>) holds. Suppose and hold, and η_m is chosen using (<ref>). For all epochs m, suppose the action selection kernel is given by <ref>, suppose <ref> holds, and suppose (<ref>) holds for all ∈[m]. Then for any epoch m∈[], we have (<ref>) holds. α_m ≤(max(√(Kα_m-1/ω), A+Kλ + √(K^3ωξ_m)/Δ ) ) Suppose η_m≤√(Kω/α_m-1)-1/\|S_m-1,2\|, we then have, K/η_m(i)≤\|S_m-1,2\|+1/\|S_m-1,2\|K/η_m+1/\|S_m-1,2\| (ii)≤\|S_m-1,2\|+1/\|S_m-1,2\|λ_m(η_m+1/\|S_m-1,2\|) (iii)≤\|S_m-1,2\|+1/\|S_m-1,2\|(1+[μ(C_m(x_t,β_max,η_m+1/\|S_m-1,2\|))] + √(2K^2ln(8\|S_m-1,2\|m^2/δ)/\|S_m-1,2\|)) (iv)≤\|S_m-1,2\|+1/\|S_m-1,2\|((2+A+Kλ) + 25K/Δη_m+1/\|S_m-1,2\|/β_max√(α_m-1ξ_m) + √(2K^2ln(8\|S_m-1,2\|m^2/δ)/\|S_m-1,2\|)) (v)≤\|S_m-1,2\|+1/\|S_m-1,2\|((1+A+Kλ) + 50/Δ√(K^3ωξ_m) + √(2K^2ln(8\|S_m-1,2\|m^2/δ)/\|S_m-1,2\|)) Where (i) follows from η_m≥ 1, (ii) follows from (<ref>), (iii) follows from , (iv) follows from <Ref>, and (v) follows from (<ref>) and the fact that = 0.5. Finally, the result now follows from <Ref>. For some environment parameters λ∈ (0,1), Δ>0, and A∈[1,K], consider an instance where (<ref>) holds. Suppose and hold. For all epochs m, suppose the action selection kernel is given by <ref>, suppose <ref> holds, and suppose suppose (<ref>) holds for all ∈[m]. Then for any epoch m∈[], we have (<ref>) holds. α_m ≤(K/ω+ A+Kλ + √(K^3ωξ_m-⌈log_2log_2(K)⌉)/Δ) Where for notational convenience, we let ξ_i=1 for i≤ 0. By repeatedly applying <Ref>, we have: α_m ≤(max((K/ω)^1/2+1/4+⋯+1/2^⌈log_2log_2(K)⌉K^0.5^⌈log_2log_2(K)⌉, A+Kλ + √(K^3ωξ_m-⌈log_2log_2(K)⌉)/Δ ) ) (i)≤(max((K/ω)K^0.5^⌈log_2log_2(K)⌉, A+Kλ + √(K^3ωξ_m-⌈log_2log_2(K)⌉)/Δ ) ) (ii)≤(max((K/ω), A+Kλ + √(K^3ωξ_m-⌈log_2log_2(K)⌉)/Δ ) ) ≤(K/ω+ A+Kλ + √(K^3ωξ_m-⌈log_2log_2(K)⌉)/Δ) where (i) follows from ∑_i=1^∞1/2^i=1, and (ii) follows from K^1/2^⌈log_2log_2(K)⌉≤ K^1/2^log_2log_2(K)= K^1/log_2K = K^log_K 2=2. We now re-state and prove <Ref>. As discussed earlier, this result relies on the bound in <Ref>. From <Ref>, both and hold with probability 1-δ. We prove our simple regret bounds under these events. Let m=m(T), we then have the following bound. R(π̂) (i)≥_m(π̂) - √(α_m'ξ_m) - 1/2√(α_m'ξ_m)∑_∈[m']__(π̂)/2^2U_ - Kξ_m/η_m'min_∈[m']U_ (ii)≥_m(π̂) - √(α_m'ξ_m) - 1/2√(α_m'ξ_m)∑_∈[m']_m,_(π̂)/2^2U_ - 2√(α_m')ξ_m/U_m' - α_m'ξ_m/U_m' (iii)≥_m(π^) - √(α_m'ξ_m) - 1/2√(α_m'ξ_m)∑_∈[m']_m,_(π^)/2^2U_ - 2√(α_m')ξ_m/U_m' - α_m'ξ_m/U_m' (iv)≥_m(π^) - √(α_m'ξ_m) - 1/2√(α_m'ξ_m)∑_∈[m']__(π^)/2^2U_ - 4√(α_m')ξ_m/U_m' - α_m'ξ_m/U_m' (v)≥ R(π^) - 2√(α_m'ξ_m) - √(α_m'ξ_m)∑_∈[m']__(π^)/2^2U_ - 4√(α_m')ξ_m/U_m' - 2α_m'ξ_m/U_m' (vi)≥ R(π^) - 3.3√(α_m'ξ_m) . Here (i) follows from <Ref>. (ii) follows from <Ref>, <Ref>, and the fact that U_m'≤ U_ for any ∈[m']. (iii) follows from (<ref>). (iv) follows from <Ref>. (v) follows from <Ref>, <Ref>, and the fact that U_m'≤ U_ for any ∈[m']. Finally, (vi) follows from <Ref> and U_m'≤ 20√(α_m'ξ_m). Hence _Π(π̂)≤(√(α_m'ξ_m)). Now the final bound follows from the fact that α_m'≤α_1=3K, α_m'≤α_min(,m(T)-1), and <Ref>. § OLD LOWER BOUND Consider a policy class Π={π:𝒳→ [K]} of K actions. A set {x^(1),… x^(m)}⊂𝒳 is shattered by Π if there exist two functions f_0, f_1:𝒳→ [K] such that For every i∈[m], f_0(x^(i))≠ f_1(x^(i)). * For every ν=(ν_1,…,ν_m)∈{±1}^m, there exists a policy π∈Π such that for every i∈[m], we have π(x^(i)) = {[ f_1(x^(i)) ν_i=1;; f_-1(x^(i)) ν_i=-1.; ]. The size of the largest set shattered by Π is defined to be its Natarajan dimension (Π). We show the lower bounds of cumulative regret and simple regret by following similar approaches used in Lemma D.5 <cit.> and Theorem 1 <cit.>. For notation convenience, we use _T and _Π(π) to denote cumulative regret and simple regret of policy π respectively. Let a policy class Π be given. Denote Δ∈(0,1/8) as the uniform gap between the best and the second-best arms. Then there exist a Bayesian-reward environment with a context sampling distribution D_X and reward function sampling from unifom(ℱ), such that for any bandit algorithm, the following properties hold: * (Cumulative regret) if 𝔼_[_T]≤Δ T/8, then the incurred cumulative regret must have 𝔼_[_T]≥(Π)/64Δ. * (Simple regret) if at each time t, the probability of sampling from suboptimal arms is upper bounded by g_t, then any policy π̂ learned from realized data must have 𝔼_[_Π(π̂)]≥Δ/4exp(- 32Δ^2/(Π)∑_t=1^T g_t ). Particularly, if Δ =√((Π)/32∑_t=1^T g_t ), we have 𝔼_[_T]≥Ω( √((Π) ∑_t=1^T g_t)) 𝔼_[_Π(π̂)]≥Ω( √((Π)/∑_t=1^T g_t)). Theorem <ref> indicates a trade-off between cumulative regret and simple regret. In this environment, if the bandit does a good job such that the probability of sampling from sub-optimal arms is low (i.e., ∑_t g_t is small), then we have low cumulative regret; but this will create challenge for later offline policy learning algorithm to distinguish between different arms, and thus we will have large simple regret. Let {x^(1),… x^(m)}⊂𝒳 witness the Natarajan dimension (Π), for which we abbreviate as d. We choose D_X to select x uniformly from {x^(1),… x^(m)}. For each ν∈{± 1}^d, define π_ν∈Π as follows π_ν(x^(i)) = {[ f_1(x^(i)) ν_i=1;; f_-1(x^(i)) ν_i=-1. ]. By definition of Natarajan dimension, we can always construct {π_ν} as above. Now we define . For each ν∈{± 1}^d, define f_ν(x^(i) as follows: f_ν(x^(i),a) = {[ 1/2+Δ a = f_1(x^(i));; 1/2+2Δ a = f_-1(x^(i)) ν_i=-1;; 1/2, ]. It immediately yields that π_ν = π_f_ν. Part I: cumulative regret. Denote p_t as the probability kernel used at time t, and denote the average kernel as p̅=1/T∑_t=1^T p_t. For each ν, denote _ν({(x_1,a_1,r_1),…, (x_T,a_T,r_T)}) as the law with x_t∼𝒟, a_t∼(p_t(·,x_t)) and r_t ∼(f_ν(x_t,a_t)) and _P_ν as the expectation under this distribution. Given a ν, we have _T ≥∑_t=1^T∑_i=1^d P_D_X(x_t=x^(i)) ∑_a p_t(a;x^(i))(f_ν(x^(i), π_ν(x^(i))) - f_ν(x^(i), a)) = 1/d∑_t=1^T∑_i=1^d∑_a p_t(a;x^(i))(f_ν(x^(i), π_ν(x^(i))) - f_ν(x^(i), a)) ≥Δ/d∑_t=1^T∑_i=1^d(1 - p_t(π_ν(x^(i));x^(i))) = Δ T/d∑_i=1^d(1 - p̅(π_ν(x^(i));x^(i))) ≥Δ T/2d∑_i=1^d {p̅(π_ν(x^(i));x^(i)) < 1/2}. Define _+i = 1/2^d-1∑_ν:ν_i=1_ν and _-i = 1/2^d-1∑_ν:ν_i=-1_ν. For a ν∈{± 1}^d, define M_i(ν)∈{± 1}^d be the vector that differs from ν only in element i: [M_i(ν)]_i=-ν_i and [M_i(ν)]_j=ν_j for any j≠ i. The expectation of (<ref>) gives 𝔼_[_T] = 1/2^d∑_ν∈{±1}^d _P_ν [_T] ≥1/2^d∑_ν∈{±1}^d Δ T/2d∑_i=1^d ℙ_ν(p̅(π_ν(x^(i));x^(i)) < 1/2 ) = Δ T/d2^d+1∑_i=1^d ∑_ν:ν_i=1( ℙ_ν(p̅(f_1(x^(i));x^(i)) < 1/2 ) + ℙ_M_i(ν)(p̅(f_-1(x^(i));x^(i)) < 1/2 ) ) ≥Δ T/d2^d+1∑_i=1^d ∑_ν:ν_i=1( ℙ_ν(p̅(f_1(x^(i));x^(i)) < 1/2 ) + ℙ_M_i(ν)(p̅(f_1(x^(i));x^(i)) ≥ 1/2 ) ) ≥Δ T/d2^d+1∑_i=1^d ∑_ν:ν_i=1( ℙ_ν(p̅(f_1(x^(i));x^(i)) < 1/2 ) + 1 - ℙ_M_i(ν)(p̅(f_1(x^(i));x^(i)) < 1/2 ) ) ≥Δ T/4d∑_i=1^d ( ℙ_+i(p̅(f_1(x^(i));x^(i)) < 1/2 ) + 1 - ℙ_-i(p̅(f_1(x^(i));x^(i)) < 1/2 ) ) ≥Δ T/4(1 -1/d∑_i=1^d ℙ_+i- ℙ_-i_TV) We continue lower bound (<ref>): 1/d∑_i=1^d ℙ_+i- ℙ_-i_TV(i)≤( 1/d∑_i=1^d ℙ_+i- ℙ_-i_TV^2)^1/2 = ( 1/d∑_d=1^d 1/2^d-1∑_ν:ν_i=1(ℙ_ν- ℙ_M(ν))_TV^2)^1/2 (ii)≤ ( 1/d2^d-1∑_i=1^d ∑_ν:ν_i=1ℙ_ν- ℙ_M(ν)_TV^2)^1/2(iii)≤( 1/d2^d∑_i=1^d ∑_ν:ν_i=1D_KL(ℙ_ν, ℙ_M(ν)))^1/2 where (i) and (ii) are by Cauchy-Schwartz inequality, and (iii) is by Pinsker's inequality. Combining (<ref>) and (<ref>), together with condition that 𝔼_[_T]≤Δ T/8, we have 1/2^d-1∑_i=1^d ∑_ν:ν_i=1D_KL(ℙ_ν, ℙ_M(ν))≥d/2. On the other hand, for any fixed ν∈{± 1}^d with ν_i=1, we have D_KL(ℙ_ν, ℙ_M(ν)) ≤_P_ν[\|{t:x_t=x^(i), a_t= f_-1(x^(i))}\|] D_KL((1/2), (1/2 + 2Δ) ) ≤_P_ν[\|{t:x_t=x^(i), a_t≠ f_1(x^(i))}\|] 8Δ^2/1-16Δ^2 (i)≤_P_ν[\|{t:x_t=x^(i), a_t≠ f_1(x^(i))}\|] · 16Δ^2, where in (i) we use Δ<1/8. Consequently, we have 1/2^d-1∑_i=1^d ∑_ν:ν_i=1D_KL(ℙ_ν, ℙ_M(ν))≤1/2^d-1∑_i=1^d ∑_ν:ν_i=1_P_ν[\|{t:x_t=x^(i), a_t≠ f_1(x^(i))}\|] · 16Δ^2. Combining (<ref>) and (<ref>), we have 1/2^d-1∑_i=1^d ∑_ν:ν_i=1_P_ν[\|{t:x_t=x^(i), a_t≠ f_1(x^(i))}\|]≥d/32Δ^2. Now we lower bound the cumulative regret [_T] = 1/2^d∑_ν∈{±1}^d_P_ν[_T]≥1/2^d∑_i=1^d∑_ν:ν_1=1_P_ν[_T] ≥1/2^d∑_i=1^d∑_ν:ν_1=1_P_ν[\|{t:x_t=x^(i), a_t≠ f_1(x^(i))}\|]Δ ≥d/64Δ. Part II: simple regret. Denote π̂ as the policy learned from logged bandit data. For each ν, we have _Π(π̂) ≥∑_i=1^d P_(x=x^(i))∑_a π̂(a;x^(i))(f_ν(x^(i), π_ν(x^(i))) - f_ν(x^(i), a) ) =1/d∑_i=1^d ∑_a π̂(a;x^(i))(f_ν(x^(i), π_ν(x^(i))) - f_ν(x^(i), a) ) ≥Δ/d∑_i=1^d (1 -π̂(π_ν(x^(i)); x^(i)) ≥Δ/2d∑_i=1^d {π̂(π_ν(x^(i)); x^(i))<1/2}. We therefore have _[_Π(π̂)] ≥1/2^d∑_ν∈{± 1}^d_Q_ν[ Δ/2d∑_i=1^d {π̂(π_ν(x^(i)); x^(i)))<1/2}] ≥1/2^d∑_ν∈{±1}^d Δ/2d∑_i=1^d P_ν(π̂(π_ν(x^(i));x^(i)) < 1/2 ) = Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1( P_ν(π̂(f_1(x^(i));x^(i)) < 1/2 ) + P_M_i(ν)(π̂(f_-1(x^(i));x^(i)) < 1/2 ) ) ≥Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1( P_ν(π̂(f_1(x^(i));x^(i)) < 1/2 ) + P_M_i(ν)(π̂(f_1(x^(i));x^(i)) ≥ 1/2 ) ) ≥Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1( P_ν(π̂(f_1(x^(i));x^(i)) < 1/2 ) + 1 - P_M_i(ν)(π̂(f_1(x^(i));x^(i)) < 1/2 ) ) ≥Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1( 1 - P_ν- P_M_i(ν)_TV) (i)≥Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1exp(-D_KL(P_ν, P_M_i(ν))), where (i) is a result of Lemma <ref>. Let P and Q be any two probability measures on the same measurable space. Then 1 - P-Q_TV≥1/2exp(-D_KL(P,Q)). For any ν with ν_i=1, we have D_KL(P_ν, P_M(ν)) ≤_P_ν[\|{t:x_t=x^(i), a_t= f_-1(x^(i))}\|]D_KL((1/2), (1/2+2Δ)) ≤_P_ν[\|{t:x_t=x^(i), a_t= f_-1(x^(i))}\|]D_KL((1/2), (1/2+2Δ)) · 16Δ^2, =16Δ^2/d_P_ν[∑_t=1^T p_t(f_-1(x^(i) ); x^(i))] Thus, [_S] ≥Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1exp(- 16Δ^2/d_P_ν[∑_t=1^T p_t(f_-1(x^(i) ); x^(i))] ) = Δ/d2^d+1∑_i=1^d ∑_ν:ν_i=1exp(- 16Δ^2/d_P_ν[∑_t=1^T p_t(π_M(ν)(x^(i) ); x^(i))] ) (i)≥Δ/4d∑_i=1^d exp(- 1/2^d-1∑_ν:ν_i=116Δ^2/d_P_ν[∑_t=1^T p_t(π_M(ν)(x^(i) ); x^(i))] ) ≥Δ/4d∑_i=1^d exp(- 1/2^d-1∑_ν16Δ^2/d_P_ν[∑_t=1^T p_t(π_M(ν)(x^(i) ); x^(i))] ) = Δ/4exp(- 32Δ^2/d𝔼_[∑_t=1^T p_t(π_M(ν)(x^(i) ); x^(i))] ), where (i) is by Jensen inequality. If at each time t, the probability of sampling from suboptimal arms is upper bounded by g_t, then Δ/4exp(- 32Δ^2/d𝔼_[∑_t=1^T p_t(π_M(ν)(x^(i) ); x^(i))] )≥Δ/4exp(- 32Δ^2/d∑_t=1^T g_t ), which completes the proof. § LOWER BOUND * We prove <ref> in the following sub-sections. §.§ Basic Technical Results The following result is established in <cit.>, with this version taken from the proof of Lemma D.2 in <cit.>. Let = (x_1,a_1,r_1(a_1)), …, (x_T,a_T,r_T(a_T)), and let {ℙi}_i∈[M] be a collection of measures over , where M≥ 2. Let be any reference measure over , and let ℙ be the law of (m^,) under the following process: Sample m^* uniformly from [M]. * Sample ∼ℙm^. Then for any function (), if ℙ( = ) ≥ 1-δ, then (1-1/M)log(1/δ) - log2 ≤1/M∑_i=1^M (Q\|\|ℙi). §.§ Construction If K≤ 10 or T≤ 152^2Klog F or ϕ≥ K, our lower bound directly follows from the cumulative regret lower bound in <cit.>. Hence, without loss of generality, we can assume K≥ 10, T≥ 152^2Klog F, and ϕ≤ K. The following construction closely follows lower bound arguments in <cit.>. Let = {a1, a2, …, aK} be an arbitrary set of discrete actions. Let k = ⌊ 1/ϵ⌋, and d be parameters that will be fixed later. With ϵ∈(0,1), note that 1/(2ϵ) ≤ k ≤ 1/ϵ. We will now define the context set as the union of d disjoint partitions 1, 2, …, d, where i = {xi,0, xi,1,…, xi,k} for all i∈[d]. Hence, we have = ∪i and \|\|=d(k+1). For each partition index i∈[d], we construct a policy class Πi⊆ (i→) as follows. First we let πi,0:i→ be the policy that always selects arm a1, and let πi,l,b:i→ be defined as follows for all l∈ [k] and b∈_0:=∖{a1}, ∀xi,j∈i, πi,l,b(xi,j)= a1, if j ≠ l, b, if j=l. Construct Πi:={πi,l,b\|l∈ [k] and b∈_0 }∪{πi,0}.[Here [k]={1,2,…,k}] Finally, we let Π := Π1×Π2×…×Πd. We will now construct a reward model class that induces Π. Let Δ:=1/4. For each partition index i∈[d], we construct a reward model class i⊆ (i×→ [0,1]) as follows. First we let fi,0:i×→ [0,1] be defined as follows, ∀ (xi,j,a)∈i×, fi,0(xi,j,a) = 1/2+Δ, if a=a0, 1/2, if a∈_0. For all l∈ [k] and b∈_0, we define fi,l,b:i×→ [0,1] as follows, ∀ (xi,j,a)∈i×, fi,l,b(xi,j,a) = 1/2+Δ, if a=a0 1/2+2Δ, if j=l and a=b, 1/2, otherwise. Note that fi,l,b differs from fi,0 only at context (xi,l,b). Construct i:={fi,l,b\|l∈ [k] and b∈_0 }∪{fi,0}. Finally, we let := 1×2×…×d. Hence, we have, \|\| = \|i\|^d ≤ (k · K)^d d ≥log\|\|/log(K· k)≥log\|\|/log(K/ϵ). We choose d to be the largest value such that F≥ (k· K)^d. Hence we choose d = ⌊log F/log (K· k) ⌋≥log F/(2log (K· k)). To use <ref>, we will describe a collection of environments that share a common distribution over contexts and only differ in the reward distribution. The context distribution D_ is given by D_:=1/d∑_i D_i, where D_i is a distribution over i, with ϵ probability of sampling each context in i∖{xi,0}, and 1-kϵ probability of sampling the context xi,0. For each block i, we let ℙi,0 denote the law given by the reward distribution r(a)∼(fi,0(x,a)) for all x∈i. Further, for any l∈[k] and b∈_0, we let ℙi,l,b denote the law given by the reward distribution r(a)∼(fi,l,b(x,a)) for all x∈i. For any policy π∈Πi, we let Ri,l,b(π)=_ℙi,l,b[r(π(x))] denote expected reward under ℙi,l,b, and let i,l,b(π)=Ri,l,b(πi,l,b)-Ri,l,b(π) denote expected simple regret under ℙi,l,b. We use ρ to index environments. Here ρ=(ρ_1,…,ρ_d), where ρ_i=(l_i,b_i) for l_i∈{0,1,…,k} and b_i∈_0. We let ℙ_ρ denote an environment with the law ℙi,l_i,b_i for contexts in i.[Here ℙi,0≡ℙi,0,b for all b∈_0.] Finally let π_ρ denote the optimal policy under ℙ_ρ, and let π_ρi denote its restriction to i. Let _ρ[·] denote the expectation under ℙ_ρ. Let R_ρ(π)=_ρ[r(π(x))] denote the expected reward of π under ℙ_ρ, and let _ρ(π)=R_ρ(π_ρ)-R_ρ(π) denote the simple regret of π under ℙ_ρ. §.§ Lower bound argument We sample ρ from a distribution ν defined as follows. For each i∈[d], set l_i=0 with probability 0.5, otherwise l_i is selected uniformly from [k]. Select b_i uniformly from _0. Note that when l_i=0, we disregard the value of b_i. We let π̂_𝐀∈Π denotes the policy recommended by the contextual bandit algorithm 𝐀 at the end of T rounds, and let π̂_𝐀i∈Πi be the restriction of π̂_𝐀 to block i. Note that the policy recommended by 𝐀 will depend on the environment ρ. Let :={i∈[d]\| i,l_i,b_i(π_𝐀i) ≤ 19√(ϕlog F/T)}. Since we only consider algorithms that guarantee the following with probability at least 19/20, 1/d∑_i=1^d i,l_i,b_i(π_𝐀i) = _ρ(π̂_𝐀) ≤√(ϕlog F/T). Under this event, we have that at most d/19 block indices satisfy i,l_i,b_i(π_𝐀i) > 19√(ϕlog F/T). Therefore, we have \|\|≥ 18d/19. Define event M_i = {i∈}. We have ∑_i=1^d P(M_i) = ∑_i=1^d [1{i∈}}]≥19/20[\|\|\|_ρ(π̂_𝐀) ≤√(ϕlog F/T)]≥9d/10. Consider any fixed index i, under the event M_i, we have the following. First observe for any (l,b)≠ (l',b'), we have i,l,b(πi,l',b')≥ϵΔ. Let (l̂_i,b̂_i) be indices such that π_𝐀i=πi,l̂_i,b̂_i. We now choose ϵ such that, ϵ = 38/Δ√(ϕlog F/T)ϵΔ/2 = 19√(ϕlog F/T). Hence from definition of , we have i,l_i,b_i(π_𝐀i) ≤ϵΔ/2. Further since i,l_i,b_i(π)≥ϵΔ for all π∈Πi∖{πi,l_i,b_i}, we have (l̂_i,b̂_i) = (l^_i,b^_i). Restating the above result, we have the following. For any i∈, with probability 1-1/16, we have (l̂_i,b̂_i) = (l^_i,b^_i). Hence from <ref>, we have, (1 - 1/(K-1)k)log(1/P(M_1)) - log2 ≤1/(K-1)k∑_l=1^k∑_b∈_0(ℙi,0\|\|ℙi,l,b) (i)=1/(K-1)k∑_l=1^k∑_b∈_0((1/2)\|\|(1/2+2Δ))_ℙi,0[\|{t\|x_t=xi,l,a_t=b }\|] (ii)≤1/(K-1)k∑_l=1^k∑_b∈_0 4Δ^2_ℙi,0[\|{t\|x_t=xi,l,a_t=b }\|] = 4Δ^2/(K-1)k_ℙi,0[\|{t\|x_t∈i∖{xi,0},a_t∈_0 }\|]. Where (i) follows from the fact that ℙi,0 and ℙi,l,b are identical unless x_t=xi,l and a_t=b, and (ii) follows from Δ≤ 1/4. Clearly we have: _ρ∼ν_ρ[∑_t=1^T (r_t(π^(x_t)) - r_t(a_t) ) ] ≥Δ_ρ∼ν_ρ[∑_t=1^T ∑_i=1^d ({x_t∈i∖{xi,0}, a_t∈_0, l_i=0 }) ] (i)≥Δ/2∑_i=1^d_ℙi,0[ \|{t\| x_t∈i∖{xi,0}, a_t∈_0 } \| ] (ii)≥Δ/2∑_i=1^d (-(1 - 1/k(K-1))log(P(M_i) ) - log2) ·(K-1)k/4Δ^2 =∑_i=1^d (-(1 - 1/k(K-1))log(1-P(M_i) ) - log2) ·(K-1)k/8Δ (iii)≥∑_i=1^d ((1 - 1/k(K-1))P(M_i) - log2) ·(K-1)k/8Δ (iv)≥{(1 - 1/k(K-1))9/10 - log 2}·(K-1)kd/8Δ (v)≥Kkd/100Δ(vi)≥dK/200Δϵ(vii)=1/7600√(T d^2 K^2/ϕlog F)(viii)≥1/15200√(K^2 T log F/ϕlog^2 (K· k)) (ix)≥1/15200√(K^2 T log F/ϕlog^2 (K· T)). Where (i) follows from the fact that ν(l_i=0)=1/2, (ii) follows from (<ref>) and that \|\|≥ d/2, (iii) uses log(1+x)≤ x, for x>-1, , (iv) uses (<ref>), (v) follows from k≥ 1 and K≥ 10, (vi) follows from k≥ 1/(2ϵ), (vii) follows from choice of ϵ, (viii) follows from (<ref>), and (ix) since k≤ 1/ϵ<ref>=1/152√(T/ϕlog F)≤ T. This completes the proof of <ref>. § ADDITIONAL DETAILS §.§ Conformal Arm Sets The below lemma shows that, for any given policy π, the conformal arm sets given in <ref> can be probabilistically relied on (over the distribution of contexts) to contain arms recommended by π, with low regret under the models estimated up to epoch m. Recall we earlier define U_m=20√(α_m-1ξ_m). [Conformal Uncertainty]lemmalemConformalUncertainty For any policy π and epoch m, we have: _x∼ D_,a∼π(·\|x) (a∈ C_m(x,ζ)) ≥ 1 - ζ∑_∈ [m]__(π)/ (2^2)U_ For any policy π, we have (<ref>) holds. _x∼ D_,a∼π(·\|x) (a∉ C_m(x,ζ)) ≤_x∼ D_,a∼π(·\|x) (⋃_∈[m]{a∉C_(x,ζ/(2^2)) }) (i)≤∑_∈[m]_x∼ D_,a∼π(·\|x) ( _(x, π__(x)) - _(x,a)> (2^2)U_/ζ) (ii)≤∑_∈[m]__(π)/(2^2)U_/ζ. Where (i) follows from union bound and (ii) follows from Markov's inequality. Recall that right after <Ref>, we show that __≤ U_ with high-probability for any ∈[]. Hence, <Ref> gives us that with high-probability we have _x∼ D_,a∼π^(·\|x) (a∈ C_m(x,ζ))≥ 1-ζ. While we don't directly use <Ref>, this lemma helps demonstrate the utility of CASs. §.§ Argument for Surrogate Objective <Ref> is a self-contained result proving that guarantying tighter bounds on the optimal cover leads to tighter simple regret bounds for any contextual bandit algorithm. Hence the optimal cover is a valid surrogate objective for simple regret. This lemma is not directly used in the analysis of ω-RAPR, however similar results (see <Ref>) were proved and used. Note that the parameters below (including α) are not directly related to parameters maintained by ω-RAPR. [Valid Surrogate Objective]lemmalemPolicyLearning Suppose Π is a finite class and suppose a contextual bandit algorithm collects T samples using kernels (p_t)_t∈[T] such that p_t(·\|·)≥√(ln(4\|Π\|/δ)/α T). Further suppose that the following condition holds with some α∈[1,∞): 1/T∑_t=1^T V(p_t,π^) ≤α Then we can estimate a policy π̂∈Π such that with probability at least 1-δ, we have: \|R(π^)-R(π̂)\| ≤(√(αln(4\|Π\|/δ)/T)). WOLG we assume T≥ln(4\|Π\|/δ), since otherwise the result trivially holds. Now consider any policy π. Let y_t := r_t(π(x_t)=a_t)/p_t(π(x_t)\|x_t). Now note that: Var_t[y_t] ≤_D(p_t)[y_t^2] = _(x_t,a_t,r_t)∼ D(p_t)[r_t^2(π(x_t)=a_t)/p^2_t(π(x_t)\|x_t)] ≤_x∼ D_[1/p_t(π(x_t)\|x_t)]=V(p_t,π). Then from from a Freedman-style inequality <cit.>, we have with probability at least 1-δ/(2\|Π\|) that the following holds: \| ∑_t=1^T (y_t - R(π)) \| ≤ 2max{√(∑_t=1^TVar(y_t)ln(4\|Π\|/δ)), ln(4\|Π\|/δ)/√(ln(4\|Π\|/δ)/α T)} (i) \| 1/T∑_t=1^T r_t(π(x_t)=a_t)/p_t(π(x_t)\|x_t) - R(π) \| ≤ 2√(ln(4\|Π\|/δ)/Tmax{1/T∑_t=1^T V(p_t,π), α}) Here (i) follows from (<ref>). Similarly with probability at least 1-δ/(2\|Π\|) the following holds: \| 1/T∑_t=1^T 1/p_t(π(x_t)\|x_t) - 1/T∑_t=1^T V(p_t,π) \| (i)≤2/Tmax{√(∑_t=1^TVar(1/p_t(π(x_t)\|x_t))ln(4\|Π\|/δ)), ln(4\|Π\|/δ)/√(ln(4\|Π\|/δ)/α T)} (ii)≤2/Tmax{√(Tα T/ln(4\|Π\|/δ)ln(4\|Π\|/δ)), √(α Tln(4\|Π\|/δ))}(iii)≤ 2√(α)(iv)≤ 2α. Here (i) follows from Freedman's inequality, (ii) follows from the lower bound on p_t, (iii) follows from T≥ln(4\|Π\|/δ), and (iv) follows from α≥ 1. Hence the above events hold with probability at least 1-δ for all policies π∈Π. Now let π̂ be given as follows. π̂∈_π∈Π1/T∑_t=1^T r_t(π(x_t)=a_t)/p_t(π(x_t)\|x_t) - 2√(ln(4\|Π\|/δ)/T( 2α + 1/T∑_t=1^T 1/p_t(π(x_t)\|x_t))) We then have the following lower bound on R(π̂) using the definition of π̂ and the above to high-probability events. R(π̂)(i)≥1/T∑_t=1^T r_t(π̂(x_t)=a_t)/p_t(π̂(x_t)\|x_t) - 2√(ln(4\|Π\|/δ)/Tmax{1/T∑_t=1^T V(p_t,π̂), α}) (ii)≥1/T∑_t=1^T r_t(π̂(x_t)=a_t)/p_t(π̂(x_t)\|x_t) - 2√(ln(4\|Π\|/δ)/T( 2α + 1/T∑_t=1^T 1/p_t(π̂(x_t)\|x_t))) (iii)≥1/T∑_t=1^T r_t(π^(x_t)=a_t)/p_t(π^(x_t)\|x_t) - 2√(ln(4\|Π\|/δ)/T( 2α + 1/T∑_t=1^T 1/p_t(π^(x_t)\|x_t))) (iv)≥ R(π^) - 4√(ln(4\|Π\|/δ)/T( 4α + 1/T∑_t=1^T V(p_t,π^)))≥ R(π^) - 4√(5αln(4\|Π\|/δ)/T) Here (i) follows from (<ref>), (ii) follows from (<ref>), (iii) follows from (<ref>), and (iv) follows from (<ref>) and (<ref>). This completes the proof. §.§ Testing Misspecification via CSC We restate the misspecification test that is used at the end of epoch m and argue how this test can be solved via two calls to a cost sensitive classification solver. First, let us restate the test in (<ref>). max_π∈Π∪{p_m+1}\|_m+1,_m+1(π)-_m+1(π)\| - √(α_mξ_m+1)∑_∈[m]_m+1,_(π__)-_m+1,_(π)/40^2 √(α_-1ξ_) ≤ 2.05√(α_mξ_m+1) + 1.1√(ξ_m+1), We are interested in calculating the value of the maximization problem in (<ref>). To calculate this maximum, we need to fix our estimators. Let _m+1,f(π):=1/\|S_m,3\|∑_t∈ S_m,3f(x_t,π(x_t))=1/\|S_m,3\|∑_t∈ S_m,3_a∼π(·\|x_t) f(x_t,a) for any policy π and reward model f, which is the only obvious estimator we could think off for R_f(π). Also let us use IPS estimaton for policy evaluation (the same argument works for DR), _m+1(π):=1/\|S_m,3\|∑_t∈ S_m,3π(a_t\|x_t)r_t(a_t)/p_m(a_t\|x_t). [When evaluating a general kernel q, we use the natural extension of these estimators of policy value. In particular, simply replace π(·\|x) with q(·\|x) in their formulas.] [Up to constant factors, these estimators give us the best rates in <Ref> with finite classes. These estimators are also used in several contextual bandit papers <cit.>.] Note that the value of the maximization problem in (<ref>) is equal to max(L_1,L_2,L_3), where {L_i\|i∈[3]} are defined as follows. L_1:=max_π∈Π_m+1,_m+1(π)-_m+1(π) - √(α_mξ_m+1)∑_∈[m]_m+1,_(π__)-_m+1,_(π)/40^2 √(α_-1ξ_) L_2:=max_π∈Π_m+1(π)-_m+1,_m+1(π) - √(α_mξ_m+1)∑_∈[m]_m+1,_(π__)-_m+1,_(π)/40^2 √(α_-1ξ_) L_3:=\|_m+1,_m+1(p_m+1)-_m+1(p_m+1)\| - √(α_mξ_m+1)∑_∈[m]_m+1,_(π__)-_m+1,_(p_m+1)/40^2 √(α_-1ξ_) Note that L_3 doesn't involve any optimization and can be easily calculated. Substituting value of these estimators for L_1 and L_2, we get. L_1=max_π∈Π∑_t∈ S_m,31/\|S_m,3\|(_m+1(x_t,π(x_t))-π(a_t\|x_t)r_t(a_t)/p_m(a_t\|x_t) - √(α_mξ_m+1)∑_∈[m]_(x_t,π__(x_t))-_(x_t,π(x_t))/40^2 √(α_-1ξ_)) L_1=max_π∈Π∑_t∈ S_m,31/\|S_m,3\|(π(a_t\|x_t)r_t(a_t)/p_m(a_t\|x_t)-_m+1(x_t,π(x_t)) - √(α_mξ_m+1)∑_∈[m]_(x_t,π__(x_t))-_(x_t,π(x_t))/40^2 √(α_-1ξ_)) Clearly, both L_1 and L_2 are cost-sensitive classification problems <cit.>.In both, we need to find a policy (classifier) that maps contexts to arms (classes), incurring a score (cost) for each decision such that the total score (cost) is maximized (minimized). Hence the misspecification test we use only requires two calls to CSC solvers. §.§ Simulation Details Data generating process. We consider four arms, i.e., =[4]. The context x=(x_1,x_2) is uniformly sampled from four regions on the two-dimensional unit ball; and in specific, x is generated via the following distribution: x̃_1∼(0.8, 1.0) * x̃_2=√(1-x̃_1^2)· z, where z∼{-1,1}. * Sample region index r∼{0,1,2,3}: * if r=0: {x_1,x_2}={x̃_1,x̃_2}, arm 1 is the best. * if r=1: {x_1,x_2}={x̃_2,x̃_1}, arm 2 is the best. * if r=2: {x_1,x_2}={-x̃_1,-x̃_2}, arm 3 is the best. * if r=3: {x_1,x_2}={-x̃_1,-x̃_2}, arm 4 is the best. Given context x=(x_1,x_2), we have the reward when pulling arm a∈[4] be generated as 1000(x_1·𝕀[a=1] + x_2·𝕀[a=2]-x_1·𝕀[a=3]-x_2·𝕀[a=4]+ϵ), where ϵ∼[-2√(3),2√(3)]. In this way, the context space is divided into four equal regions, and each region is associated with a unique arm that separates from the others by 1000(0.8-√(1-0.8^2)) regarding the expected outcome.
http://arxiv.org/abs/2307.00688v1	20230702235443	Orientational dynamics of anisotropic colloidal particles in a planar extensional flow	[ "Dinesh Kumar" ]	cond-mat.soft	[ "cond-mat.soft" ]	[email protected] Department of Chemical and Biomolecular Engineering University of Illinois at Urbana-Champaign, Urbana, IL, 61801 Suspensions of anisotropic particles are commonly encountered in a wide spectrum of applications, including industrial and architectural coatings, targeted drug delivery and manufacturing of fiber-reinforced composites. A grand challenge in the field of chemical and material processing is robust production of strongly aligned fibers at the microscopic level, as this is routinely linked with enhanced mechanical properties at the macroscopic level. While the investigation of the microstructure of anisotropic colloids in shear flows has garnered a lot of theoretical and experimental attention, the case of extensional flow remains poorly understood due to several experimental challenges. In this article, we present a theoretical framework for predicting the steady and transient orientations of anisotropic particles in a flowing liquid undergoing precisely defined steady and time-dependent planar extensional flow at the stagnation point of a Stokes trap device. In particular, we analytically solve the Fokker-Planck equation for estimating the probability distribution function describing the orientation dynamics of rod-like objects as a function of flow strength (Peclet number, Pe) and probing frequency (Deborah number, De). The theoretical results are compared with recent experiments and reasonable agreement is found. We also discuss the challenges involved in obtaining a full closed-form solution for the transient dynamics of anisotropic particles in oscillatory time-dependent extensional flow. Overall, our theoretical framework provides a way to compare the orientation dynamics of rod-like particles with experiments that have been performed using a new experimental technique involving the Stokes trap and precise flow-control over the orientation of particles. Orientational dynamics of anisotropic colloidal particles in a planar extensional flow Dinesh Kumar August 1, 2023 ====================================================================================== § INTRODUCTION There are numerous applications in science and engineering where it is useful to orient fibers, nanowires and nanotubes in a certain fashion to control the overall rheology and hydrodynamic characteristics of the suspension <cit.>. For instance, obtaining desired material properties of fiber-reinforced composites involves specific alignment of the particles to improve the overall mechanical, optical, electrical and magnetic properties <cit.>. Such composites are routinely processed in complex flows through a combination of shearing and elongation. A first step towards understanding the processing of fiber composites is to study their dynamics in simple flows, such as shear and extensional flows, as they are relatively easy to model and can be replicated in laboratory experiments. The ability of extensional flows to strongly align the fibers along the stretching direction also makes them an attractive model flow-field as the composite is stiffer and stronger in this direction compared with any other direction. Hence, characterizing the flow-induced orientation dynamics of anisotropic particles through experiments and theory is of paramount importance in order to tailor the microstrucutre and desired properties with minimum cost and maximum reliability. Recently, the experimental results on transient orientation dynamics of a single rod-like particle was investigated <cit.> using the Stokes trap <cit.>. Unfortunately, predicting the orientation dynamics of anisotropic particles in extensional flow has been challenging due to the complex interplay between the hydrodynamic forces exerted by the flow, and random thermal fluctuations in the suspending fluid causing Brownian rotational diffusion. Generally, the Fokker-Planck equation (FPE) accurately describes the time evolution of the particle orientation distribution function,ψ(ϕ,t), (or its moments) at an orientation, ϕ, at time, t : ∂ψ ( ϕ, t )/∂ t=D∂^2 ψ ( ϕ, t ) /∂ϕ^2-∂ [ω ( ϕ ) ψ ( ϕ, t ) ]/∂ϕ where D is the rotational diffusivity and ω is the rotational velocity of the anisotropic particle. The time evolution of ψ(ϕ,t) is of considerable interest to the fluid dynamics community as it determines the start up rheology of suspensions of anisotropic particles. However, it is well known that while the stationary solution at long times of the FPE can be given in closed form, full time-dependent solutions for an arbitrary ω(ϕ) are extremely rare and the analytical solution exists for only a limited number of cases. The orientation dynamics and distributions of rod-like particles has been extensively studied in shear flow through experiments and theory. Jeffery <cit.> investigated the suspension of non Brownian spheroidal particles in simple shear flow and found that there exists no steady state distribution as they periodically rotate in an unsteady motion known as Jeffery orbits. Hinch and Leal <cit.> added the effect of rotational diffusion which led the distribution ψ(ϕ,t) to approach steady state and they made scaling arguments to qualitatively describe the evolution of ψ(ϕ,t) at high and low Pe, where Pe=ϵ̇/D, and ϵ̇ is the imposed strain-rate by the shear flow. Recently, Leahy et al. <cit.> obtained an exact solution to the full Fokker-Planck equation (1) in orientation-space by an appropriate coordinate transformation to the phase-angle space under steady and time-dependent shear flows. An analytical solution of the orientation dynamics provided direct insights into the enhanced rotational diffusion due to shear. Leahy and co-workers published another paper <cit.> in 2017 where they used this analytical solution to describe the behavior of a dilute suspension of rod-like colloids under an arbitrary periodic, high-frequency shear flow. They presented optimized periodic waveforms to control the particle alignment and hence, the suspension rheology at an arbitrary moment in the flow-cycle. While the dynamics of rod-like particles has been extensively studied in shear-flows, it has not received much attention in the case of elongational flows. Another interest in the community is to compare the analytical approach of probability distribution functions with experiments under similar conditions. Since the demonstration of anisotropic colloids over long observation times in extensional flows has been challenging as the fluid elements separate exponentially in time <cit.>, the theoretical formulation of orientation dynamics in steady and time-dependent flows has not been attempted. Motivated by recent experimental work which demonstrated arbitrary control over both the center of mass position and orientation of rod-like colloids using a Stokes trap <cit.>, we aim to develop a theoretical framework for predicting the steady and transient orientations of anisotropic particles in a flowing liquid undergoing precisely defined steady and time-dependent planar extensional flow and to check whether these theoretical predictions agree with the experimental results. § QUANTITATIVE MODEL AND GOVERNING EQUATIONS Let us consider a freely-suspended anisotropic particle placed at the center of a four-channel microfluidic cross-slot device having two alternating fluid inlets and outlets as shown in Fig 1a The slow, viscous and incompressible flow produced in the slot is well described in the Hele-Shaw regime by: ẋ = 1/π H∑_i=1^4(𝐱-𝐑_i)/ (𝐱-𝐑_i) ^2q_i where 𝐱 is an arbitrary position, q_i is a vector containing the flow rate through each channel and 𝐑_i=(R_ix,R_iy) is the position vector of i^th channel and H is the depth of the device. We model the anisotropic particle as two beads connected by a massless rod, where the lower bead is at the center of mass x_c=(x_c,y_c) of the rod, and the upper bead is at the extreme end, x_t of the major axis of the particle, being separated by a distance L. We can write the rotational velocity of the rod as: ϕ̇=ω(ϕ)=1/π H∑_i=1^4sin ( 2ϕ )𝐱_c-𝐫_i^2-2cos ( 2ϕ) ( x_c-R_ix ) ( y_c-R_iy )/ (𝐱-𝐫_i) ^4q_i where ϕ is the angle made by the particle's major axis with the flow axis. We further assume that the center of mass position of the particle is kept fixed at origin of the device. This further simplifies equation (3) to: ϕ̇=-ϵ̇sin (2ϕ) where ϵ̇=2q_1/πHR_cell^2 is the strain-rate imposed by the flow and R_cell is the half-width of the channel.The corresponding probability distribution function (PDF) ψ(ϕ,t) of finding a rod of negligible cross-section at orientation ϕ and at time t is given by the Fokker-Planck equation: ∂ψ ( ϕ, t )/∂ t=∂^2 ψ ( ϕ, t ) /∂ϕ^2+Pe∂ [ sin ( 2ϕ ) ψ ( ϕ, t ) ]/∂ϕ Since ψ(ϕ,t) is periodic with a period of π as the rods are indistinguishable when oriented at ϕ or ϕ+π, the boundary conditions and initial condition become: ψ ( 0,t )=ψ ( π,t )=0 ; ψ' ( 0,t )=ψ' ( π,t ) =0 ∫_0^πψ ( ϕ,t )dϕ=1 ; ψ ( ϕ,0 )=δ ( ϕ-ϕ_0 ) where ϕ_0 is the initial orientation angle of the particle. § RESULTS In this section, we present the analytical solution of equation (5) under certain cases: §.§ Steady state dynamics in continuous extension: At long times, the left hand side of equation (5), namely ∂ψ ( ϕ, t ), equals zero and a closed form solution of the FPE is obtained as: ψ ( ϕ )=e^-Pecos2ϕ/∫_0^πe^-Pecos2ϕ The results we find for the long-time distribution function are displayed in Fig. 2a for small and large values of Peclet number ranging from 0.01 to 10. As the Peclet number Pe increases, the orientation PDF becomes sharply peaked along the axis of extension ϕ=π/2. Without going into the details, we also infer that the nature of differential equation in (5) changes for Pe greater than 1000 as it becomes a singular perturbation problem in Pe^-1. Hence, our analytical solution in (8) cannot predict the distribution function for Pe>1000. To characterize the degree of alignment along the extension axis, we also define a 2D order parameter S=2<𝐩𝐩>-δ, where 𝐩 is the unit vector along the major axis of the particle such that 𝐩 = [cos(ϕ),sin(ϕ)]^T. In the limit of Pe→ 0, the order parameter S is zero and the distribution is termed isotropic. Also, at high Peclet number, the order parameter S approaches unity and the distribution is strongly aligned along the extensional axis. From Eq. (3), we can get the time evolution of ϕ(t) which is further used to predict the time taken by particle to change its orientation from an initial angle ϕ_0 at time, t=0 to a final orientation ϕ_f at time, t as a function of strain-rate, ϵ̇. In the case where particle center of mass is not allowed to change, (3) reduces to (4) which is solved by integration through a simple separation of variable as the strain rate ϵ̇ is uniform at the center of cross-slot device. Finally, this yields: t=-2ϵ̇logϕ_f/ϕ_0 Also, the strain-rate ϵ̇ is linked with the pressure-drop ΔP across inlet and outlet of the cross-slot device through ϵ̇=2q_1/πHR_cell^2 and ΔP=q_1/R, where R is the hydrodynamic resistance of the channel. Fig. 4 depicts the time-prediction of change in the particle orientation from ϕ_0=89^∘ to ϕ_f=1^∘ as given in Eq. (9). To go further, we compared the recent experimental curve <cit.> to Eq.(9). We see a reasonable agreement between theory and experiments even though the particles experience perturbations when they reorient in the experiments which could change the strain-rate by a small magnitude. It is of note that the experimental curve describes the time-profile for an orientation change from ϕ_0=90^∘ to ϕ_f=0^∘ but slightly different values, ϕ_0=89^∘ and ϕ_0=1^∘ were chosen in theory because the trigonometric function ϕ diverges near ϕ=0^∘. §.§ Transient dynamics in continuous extension : As mentioned earlier, a full closed-form solution to Eq. (5) does not exist. However, we can determine the transient dynamics by exactly solving (5) under the small angle approximation, sin2ϕ≈ 2ϕ. This limits the choice of initial condition in Eq. (6) to \|δϕ \|⩽ 15^∘ around the extensional axis ϕ=π/2. Experimentally, this can be achieved by controlling the particle orientation to within ± 15^∘ along the extension axis at zero flow using the method described in <cit.> and gradually, increasing the applied flow. This can provide direct comparison with the transient dynamics obtained from theory, which we describe below. We solve Eq. (5) using a Fourier transform method and obtain the solution as: ψ ( ϕ,t )=√(%s/%s)Peπ ( 1-e^-4ϵ̇ ̇ṫ )exp (-Pe ( ϕ-ϕ_0e^-2ϵ̇ ̇ṫ )^2/1-e^-4ϵ̇ ̇ṫ ) At long times t→∞, Eq. (10) approaches steady state and becomes similar to Eq. (8). For illustration, we plot the time-evolution of solution ψ(ϕ,t) and their corresponding order parameter S as a function of time for two values of the Peclet number, Pe=10.28 , 102.8 in Figs. 4 and 5. The particle reaches steady-state corresponding to the onset of plateau in the plot of Order parameter versus time. As observed, the transient probability distribution function typically reaches steady state over a time scale of ϵ̇^-1. §.§ Transient dynamics in oscillatory extension In this subsection, we aim to characterize the transient orientation distribution function ψ(ϕ,t) in large amplitude oscillatory extension (LAOE) over a wide range of Peclet number Pe and Deborah number De where De=1/DT and T is the cycle period of flow <cit.>. As shown in Fig.6, we impose a sinusoidal strain-rate input ϵ̇=ϵ̇_̇0̇sin ( 2π/T ), where ϵ̇_̇0̇ is the maximum strain rate . The microdynamical equation: tan (ϕ_2 )=tan (ϕ_1 )exp ( -Pe/π De ( cos(2π DDet) ) -1 ) also allows us to plot the evolution of orientation ϕ as a function of sinusoidal strain in Fig.6. As expected, peak is observed for ϕ at t=T/2. Next, we solve the FPE in Eq. (5) under the small amplitude approximation, and the result can be written as: ψ ( ϕ,t )=1/√(2π A(t))exp ( -(ϕ-B(t))^2/2A(t) ) where B(t)=ϕ_0exp ( --Tϵ̇/πcos ( 2π/T ) ) A(t)=2D∫_0^texp ( --Tϵ̇/πcos ( 2π/T ) )dt The solution of Eq. (11) is obtained through numerical integration by integrating over at-least five periodic cycles so that effect of initial flow transience have completely decayed out. The solution is then used to express the orientation dynamics in the context of two-dimensional (De,Pe) space. It is noteworthy that our solution in Eq. (12) is only valid for a set of (Pe,De) at which the PDF ψ(ϕ,t) decays to zero beyond ±15^∘ around the axis of extension ϕ=π/2. Representative plots of order parameter in a full LAOE cycle are shown in Fig.7 as a function of Pe for oscillatory extensional flow De>0. It is observed that the order parameter in LAOE appears to be less on increasing De (steady extension has higher order parameter compared to LAOE at the same imposed flow-strength) and reaches steady state at a higher Pe for larger De. The complex interplay between Pe and De causes a decrease in order parameter due to presence of flow-oscillations at higher De. Following the logic in <cit.>, we next aim to construct a master curve for Fig.7. Our analytical solution is only valid in a small orientation space [π/2-δϕ, π/2+δϕ] where δϕ≈15^circ and it cannot be used to define the average order parameter over the entire orientation space. However, it can be used to define a critical flow-strength at which transition from isotropic S<1 to aligned state S=1 happens since the transition occurs around a small angle around π/2. In plotting the master curve and hence, the dynamic oscillatory-flow based phase diagram, a key question is: What defines the critical Peclet number, Pe at the boundary between isotropic and aligned probability distribution function? While examining the steady-state distributions Fig.2 at De=0, we observed that order parameter S saturates at a value of S≈0.996. Using this observation, we define the transition from non-aligned to highly-aligned behavior to occur at a critical Peclet number Pe_crit at which the order parameter S=0.996. Fig.8 shows a preliminary sketch of particle dynamic orientation distribution behavior in oscillatory extensional flow in the context of (De,Pe) space. As discussed in <cit.>, we classify four different regimes: (1) Regime I: Isotropic, quasi-steady state for De<1 and Pe<Pe_crit. Here, the period T is large enough so that rods have sufficient time to relax through diffusion and achieve steady state. (2) Regime II: Isotropic, unsteady state for De>1 and Pe<Pe_cric in which the rods do not get sufficient time in a cycle period to relax. (3) Regime III: Aligned unsteady state for De>1 and Pe>Pe_crit. (4) Regime IV: Aligned steady state for De<1 and Pe>Pe_crit. This phase diagram is derived from simulations and it can be experimentall verified by using the Stokes trap with large amplitude oscillatory extensional flow (LAOE) <cit.>. Since the analytical solution is based on an assumption that particles are found only in a small orientations-space around the axis of extension throughout the LAOE cycle, it will be interesting to look at the comparisons with the experiments. In the future, the validity of this assumption will be verified. In this respect, the results in this article provide an important first step in confirming that oscillatory extension reduces the alignment of anisotropic particles along extensional axis. §.§ Future experiments and simulations The results presented above provide a direct way to compare the stationary and dynamic orientation distribution function of rod-like particles with the recent experiments <cit.> that have been performed using a new experimental technique involving the Stokes trap and precise flow-control over the orientation of particles. The detailed predictions in this paper could be tested experimentally by precisely controlling the center of mass position of a single anisotropic particle using Stokes trap and directly observing the dynamics in steady flow. Also, our results on transient dynamics for a single rod-like particle in oscillatory extension can be matched by performing experiments in large amplitude oscillatory extension (LAOE) using Stokes trap, similar to <cit.>. In terms of modeling, the future work will involve completely solving the Fokker-Planck equation in Eq.5 by an appropriate series solution or perturbation method to establish the transient dynamics in complete orientation space. The effect of LAOE on the suspension stress and effective viscosity will be also be probed in the future through theory and experiments. § CONCLUSION In this work, we investigated the stationary and transient dynamics of a single anisotropic particle in steady planar extensional and time-dependent oscillatory flow by analytically solving the full Fokker-Planck equation under the small angle approximation. Good agreement is found between the theoretical results and experiments for the case of dynamic evolution of the particle orientation as a function of strain-rate. Generally, steady-state distributions of particle orientations strongly peak along the extensional axis with increasing Pe. In oscillatory flow, the elongational axis rotates by an angle π/2 in each half-cycle and hence, the orientation of rod periodically changes to align with the stretching direction. We further characterized the average transient dynamics in LAOE in a two-dimensional (De,Pe) space where the dynamic behavior was classified into four regimes based on values of Pe and De to predict the flow-based phase boundary for the transition from isotropic to an aligned state. Overall, our theoretical results establish new knowledge into our understanding of the non-equilibrium dynamics of single anisotropic particles in time-dependent flows and calls for controlled experimental studies of rod-like particles in oscillatory extensional flows. unsrt
http://arxiv.org/abs/2307.01784v1	20230704154437	The Inner Sentiments of a Thought	[ "Chris Gagne", "Peter Dayan" ]	cs.CL	[ "cs.CL", "cs.AI" ]	Constraining the binarity of black hole candidates: a proof-of-concept study of Gaia BH1 and Gaia BH2 [ August 1, 2023 ========================================================================================================= § INTRODUCTION How do sentences do affective work? That is, how do sentences, with their complex syntax and semantics, toy with our expectations in order to pack an emotional punch? How could we generate new sentences that are finely pitched at a particular strength of sentiment? Equally, what does someone's choice of sentences reveal about their psychological state? If even the idiosyncratic choice of innocuous function words (e.g., pronouns) can be used to predict affective states, from the momentary to the more long lasting, such as depression <cit.>, how much more might be encoded in the detailed affective topography of someone's utterances? Until recently, we have lacked access to the inner semantic and syntactic content of sentences required to perform fine-grained, automatic analyses of their emotional dynamics, and thus answer these questions. However, the advent of large-scale pretrained language models (LLMs)<cit.> has dramatically altered the scene. The hidden activities in these models as they `read' text have been used as representations to predict, with incredible accuracy, a wide range of linguistic features, from the parts-of-speech of individual words to logical entailment of adjacent utterances. They have been used to perform various tasks like question-answering, text-retrieval, sentiment analysis, and document clustering <cit.>. And of course, they have been used to generate uncannily human-like sentences, paragraphs, (bad) limericks, and even articles. This has led to a flurry of papers trying to understand how LLMs work, the linguistic information encoded their hidden activities (or attention weights) <cit.>, and whether they `understand' language <cit.> (or even reason)<cit.> like we do. Comparatively less emphasis has been placed on the other direction – using LLMs to understand how text, on a micro-level, can be used to convey information like emotional sentiment. Here we take initial steps in this direction. We use LLMs to provide an embedding space for the state within an utterance, and train models to predict from this the quantiles of the distribution of the end ratings of emotionally relevant sentiments. The evolution of the resulting quantile predictions in new sentences tells us how the sentence is moving in high and low-dimensional emotion space – showing clearly, for instance, how the conjunction “but” can suddenly reverse the sentiment or cue the imminent rise of more subtle emotional tones. Equipped with predictive distributions of emotion, we then demonstrate how to generate text that targets particular affective quantiles, in a method that relates to long-run risk-sensitive choice. Although many recent methods (finetuning, prompting, etc.) have been developed to alter the writing style of LLMs <cit.>, our method uniquely provides fine-grained control over particular quantiles for a wide range of emotional sentiments. § RESULTS §.§ Predicting distributions of sentiment Our testbed is the abundant, and often emotionally-charged, discourses of people on Reddit <cit.>. We first extract the hidden states of a large-scale language model (we chose the comparatively venerable GPT-2<cit.> for availability and computational practicality) applied to 2 million sentences. We then train a set of smaller models to predict the quantiles of the distributions of the end ratings for these sentences along several different emotionally-relevant dimensions: positive and negative valence, determination, admiration, annoyance, and anxiety. A separate model was trained for each of these dimensions. We chose these dimensions to highlight the applicability of our method to both low-dimensional (i.e., valence and arousal) and high-dimensional models of emotion. The end ratings were obtained automatically using other language models <cit.>, which had been fine-tuned to predict human emotional judgements of short whole utterances. As depicted in Figure <ref>, the quantile models are provided with the hidden states from GPT-2 for each token, and are trained using Monte Carlo methods and quantile regression<cit.> to predict the distribution of end scores. Scores are only provided at the end of the sentence, and therefore the quantile model must predict how the sentence might end for each prefix (i.e., at each token position). For some common prefixes, such as “I think this is …”, the model encounters an empirical distribution of different possible end scores in the dataset (Figure <ref>; top right) and can learn to approximate it. For other prefixes, the model may only see a single example and its score, and therefore must learn to generalize across utterances with similar emotional potentials. The model generates a trajectory of predicted quantiles across the successive tokens (Figure <ref>; top left). Observe how the width of the predicted valence distribution varies throughout the sentence, reaching a maximal width at the token “such”, and then collapses around a single (correct) predicted score by the end as the final sentiment is certain. As seen in the figure, the quantile model can capture bi- and multi-modal predictions. §.§.§ Model calibration To evaluate the accuracy of the predicted quantiles, we use 1M validation sentences that were unseen during training. We first assess global (marginal) calibration by calculating the percentages of end scores that fall below each of the predicted quantiles, aggregated across all sentences. For a perfectly calibrated model, the end scores would fall below the α-quantile exactly α-% of the time, regardless of the fact that the quantile predictions correspond to different sentences and different prefixes within a sentence. In Supplemental Figure <ref>, we show that the models are highly accurate, with calibration curves barely deviating from the identity line and with a maximum absolute deviation less than 1% for valence and between 2-7% for the four emotions, which have much sparser empirical distributions of end scores (Supplemental Figure <ref>). The models are well-calibrated even at the start of the sentence, meaning that they can accurately predict the distribution a dozen or more words ahead (since the median sentence length is 15). As an alternative to Monte Carlo methods, we also investigated temporal difference (TD) learning for training the quantile models<cit.>; however, the resulting models were more poorly calibrated, with a maximum absolute deviation of 9% for valence (see Methods for more details). We also compared the models’ predicted quantiles to the empirical distributions of end scores for frequently occurring prefixes. Figure <ref>a contains several representative examples, which are organized according to the mean (columns) and variance (rows) of the predicted distribution. The model predictions for these prefixes closely match the quantiles of the empirical distributions of end valence scores, despite these distributions differing considerably from the marginal distribution (shown in Figure <ref> and Supplemental Figure <ref>). This demonstrates the model's ability to capture distributions of non-trivial shapes, from those that are highly skewed (e.g., “Thank you …”) to those that are more bimodal (e.g., “This is the …”). Furthermore, note that although these examples were hand-selected, their estimation error (average maximum deviation between quantile sets) does not differ to that of the top 1000 most frequently occurring prefixes. In Figure <ref>b, we plot examples for each emotion, showing large positive shifts in the distribution for phrases such as “I will …” (for determination), “I appreciate …” (for admiration), “The problem is …” (for annoyance), and “I'm having …” (for anxiety). Note that the marginal distributions, shown in lighter color beneath each example, are much more skewed for these four emotions than for valence – and yet the models are still able to make accurate predictions. This means that quantiles could also be estimated for other semantic characteristics that have sparse distributions, such as toxicity or conversational topic<cit.>. §.§ The evolution of emotional sentiment across a sentence Next, we explore how the predicted distributions evolve throughout a sentence. As paradigmatic examples, we look at the case of the conjunction “but” and common intensifiers, such as “really” and “extremely”. §.§.§ Conjunctions that reverse expectations The typical function of the word “but” is to present contrast or exception. For the case of emotions, we are also all too familiar with its role in suddenly reversing our expectations, as in the example, “I'd love to chat, but …”. In Figure <ref>a, we show this effect brought to life, for four different sentences. In two cases (top row), the predicted valence rises (or dips) to an extreme value before swinging to the opposite valence at the word “but”; with the outermost quantile (5th or 95th), however, anticipating this potential turnabout. For the other two cases, the valence starts at nearly the same neutral level prior to the word “but”, but then rises or falls depending on the preceding context. To show this effect on aggregate, we plot the median predicted sentiment at the word “but” against the preceding sentiment, for the validation sentences (Figure <ref>b; the four red encircled points are the examples from panel a). A significant deviation away from the identity line shows how often these sentiment reversals occur. These data also reveal that many occurrences of the word “but” are characterized by a transition from neutral to either positive or negative valence (like the bottom two examples in panel a). To show that these model predictions are also accurate, we performed a separate calibration analysis for the quantiles of “but”. This had a maximum absolute error of 1.8% across quantiles, confirming the accuracy of these predictions. The word “but” can also signal more subtle shifts in the expression of other emotions. We demonstrate this again by way of example, in Figure <ref>c. Here, the word “but” signals rising determination for a sentence starting with hardship (e.g., “Some days are really hard, but …”) and rising admiration when the preceding text is socially critical (e.g., “Sometimes he isn’t very good, but …”). The reverse predictions, however, are not made, pointing to the specificity of the models, even for two related and positive emotional sentiments. §.§.§ Words that widen the distribution A benefit of predicting the whole distribution of emotional sentiments, rather than simply their expected value, is that one can analyze differences in the level of uncertainty about how a sentence might end. Thus, we next turn to identifying locations in a sentence at which the variance peaks – that is, places where the sentiment is poised to become more extreme, but the direction of change is not yet known. We used the absolute value of the difference between the 25th and 75th quantiles as a measure of variance, and scanned the validation sentences for local maximums. Three examples are shown in Figure <ref>a, with the peaks in variance and the phrases that precede them highlighted in red. For each of these examples (“It makes me …”, “I am so …”, and “it was extremely …”), it is clear that a judgment is pending, but what that may be is completely opaque. After the variance and the envelope of quantiles expands, it typically collapses, and the predicted valence drops or rises as the ambiguity around sentiment is resolved. Context-sensitive predictions for variance could also be used enhance traditional lexicon-based analyses for the use of extreme language, which might simply count the occurrences of common intensifiers, such as the words “really” or “extremely”. Tokens associated with peak variance in our analysis (defined as the top 1% of maximum variances across sentences), indeed include these words and many others (Figure <ref>b). However, this set also includes more colloquial intensifiers, such as “honestly”, “pretty”, and “seriously”, whose role as such will likely strongly depend on context. In Supplemental Table <ref>, we list common English intensifiers from Wikipedia<cit.> and statistics associated their predicted variances in the validation set. Roughly half of these words occur at peaks in variance (i.e., top 1% of variances), and the other half have either only slightly lower maximal variance or occur very infrequently in our data. §.§ Generating text from the tails of a distribution Having demonstrated the predictive capabilities of the quantile models, we next turn to whether these can be used steer a language model to write in a way that is more positive, negative, or emotionally colored. Generating text from a large language model operates, in its most basic form, by mapping partially completed text to a probability distribution over possible next tokens and then sampling one of these tokens. We intervene on this process by adjusting these probabilities, so that tokens that predict (and engender) certain emotions are more likely to be selected. This means that if we want, for instance, to generate text from the lower α-quantile (which we denote by α^-) of a prompt's distribution, we would increase the probability for tokens that have a more negative set of predicted quantiles and decrease the probability for tokens that have a more positive set. More specifically, we re-weight the next-token probabilities proportional to the amount of each next token's predicted distribution (i.e., the number of predicted quantiles) that falls below the target α-quantile. Sampling from these re-weighted probabilities then provides a principled way of generating sentences that reside within the lower α-tail of the prompt (for more details, see Methods). To generate sentences above the (1-α)-quantile (which we denote by α^+), we reverse the values of the quantiles. We provide a schematic example of this procedure in Figure <ref>. Here, the initial α^-, for the prompt “I think this is”, is set to 0.05 (to sample from the extreme lower tail). The next possible tokens include “the”, “going”, “just”, “wrong”, “too”, and “great”, and each is assigned a probability by the language model. To re-weight these probabilities, we look at the estimated quantiles for each next token, and upweight/downweight those tokens depending on if they have more/fewer of their quantiles (i.e., more/less probability mass) below the α^--quantile of the prompt's distribution. “Wrong”, “just” and “too” are upweighted because they have more negative valence distributions, while “great” and “there” are downweighted because they have more positive distributions. The next token is then sampled according to the re-weighted probabilities and the process is repeated. To demonstrate the effectiveness of this method, we use the trained quantile models along with GPT-2 to complete five prompts (“I think this is”, “This week is really busy,”, “I”, “Yesterday I went to school and”, “My kids”) with varying degrees of negativity or emotional tones. In Figure <ref>b, we show how lowering α^- generates sentences from increasingly negative lower tails of the valence distribution. For values of α^- less than 1, the majority of the generated distribution lies below the target quantile (vertical black dashed line) with only a small percentage of sentences with scores above that value. In Figure <ref>c, we set α^+=0.05 (for the upper tail) and sample according to each of the emotional models; here, we can see that the distributions are strongly shifted to the right relative to the distribution of sentences generated from an unbiased process (shown inverted and underneath; α=1.0); however, sampling sentences from an extreme quantile for these sparse emotion distributions proved to be more difficult than for valence, with the quantile targeting being imperfect. In Figure <ref>, we show example sentences from each of the models (with α^- or α^+ set to 0.05) and color the text to show the re-weighted probabilities for each token. As expected, the model chooses to upweight words that have a clear valence or emotional tone, such as “sorry” and “mistake”, and at the opposite end of the spectrum “exciting” and “amazed”. However, because the model is predictive, it also upweights words and phrases that are less obviously valenced, but which make the target sentiment or emotion more likely to occur. Some of these are predictive because of their semantics, such as “test results” for anxiety, “sky” for admiration or “gym” for determination, and others for their grammatical role, such as “not”, “and”, and “very”. Indeed, the word “but” plays a critical role in the generation of the positive emotions. Finally, there are also phrases, such as “this is my day” for determination or “this is the point” for annoyance, which convey emotions idiomatically (and subtly), but are nevertheless able to be exploited by the model. More examples are shown in Supplemental Figures <ref>-<ref>. § DISCUSSION In this paper, we demonstrated how large language models can provide a novel lens onto the inner structure and emotional dynamics of sentences. We developed a method for training a model to predict, word-by-word, the quantiles of a distribution of the final emotional sentiments of text, and validated these predictions on unseen sentences. We then showed how the trajectories of the predicted quantiles capture the intuitive emotional effects of conjunctions (i.e., “but”) and intensifiers (e.g., “really” and “so”). Finally, we used the quantile models to encourage GPT-2 to write more emotionally colorful sentences from specific quantiles of a distribution. Training quantile models and using them to generate text is related to a number of recent trends in machine learning and natural language processing. Predicting a distribution rather than a single value for sentiment is related to distributional reinforcement learning, which estimates distributions of future rewards in addition to the expected value and then optimizes for some aspect of that distribution; indeed, using quantile regression to predict end scores can be viewed as a Monte Carlo (also known as TD(1) in the reinforcement learning literature<cit.>) version of the learning algorithm used in previous work<cit.> and sampling tokens from the re-weighted probabilities can be seen as a basic form of risk-“aware” control. For quantile estimation, we found that Monte-Carlo methods produced better calibrated models than temporal difference methods such as TD(0). We speculate that this may be due to the need of TD(0) to pass predictive information backwards through every token, which could be susceptible to tokens with lower quality hidden state representations. For control, we showed how a language model can be steered to generate from either the lower or upper tail of a distribution by reweighting next-token probabilities according their predicted quantiles. However, it is also possible to optimize for other aspects of the predicted distributions, for instance, generating in a way that avoids (rather than targets) the lower tail. This could be useful for reducing the amount of toxic content, when paired with a toxic comment classifier<cit.>. A simple approach for this would be to re-weight next token probabilities according to 1-α instead of α, though future work would needed to explore this and other risk-sensitive approaches to text generation more fully. Our method for text generation is also related to several recent methods from NLP that re-weight the next-token probabilities to encourage language models to generate text with particular characteristics. For example, <cit.> use lightweight discriminative classifiers to heuristically bias the next-token logits. Other work <cit.> estimates Q-value functions, which like our quantile models are forward-looking, to steer text generation (again via probability re-weighting). However, none of this prior work has tried to estimate distributions of future characteristics (i.e., sentiment) and target particular quantiles (which is why we do not compare directly with these methods); moreover, many of the probability re-weighting techniques are heuristic, as opposed to the principled re-weighting scheme we develop. This also contrasts with prompting, which while often effective at generating text with certain characteristics, is largely conducted on a trial-and-error basis. Other methods for controlled text generation involve re-training large language models. One well-known approach is to use control codes during pretraining<cit.> (or finetuning <cit.>) to write in different styles or about different topics. There is also a recent trend of reinforcement learning with human feedback (RLHF)<cit.>, which uses a model trained on human feedback to provide `end' scores, and using finetuning via policy gradient to optimize for these scores. Predicting end scores via a separate (quantile) model and probability re-weighting can be seen as a model-based variant of this approach. Finally, the most closely related work is <cit.>, conducted contemporaneously, which uses finetuning and conditioning on a special tokens to generate from one of five quantiles. An interesting future path would be to explore the differences between conditional pretraining or finetuning and online probability-reweighting methods, such as ours. The latter involve more computation at inference but avoid the need to re-train large language models, which is becoming increasingly difficult to do as the scale of these models increases. Using smaller models to guide larger ones also allows for easier mixing and matching between control tasks and LLMs, for instance using BERT (110M parameters) to predict the accuracy of BlenderBot (2.7B parameters) and reduce the number of over-confidently expressed falsehoods<cit.>. More work is needed, however, to know the domains in which smaller models can be used to guide larger ones effectively (and the domains in which richer representations of larger models are truly needed). Generation aside, predictive quantile models may be useful for analyzing text in a wide variety of ways, complementing the long tradition of lexical analysis, use of word-embeddings, and more recently the use of contextualized embeddings. Our analyses of conjunctions (i.e., “but”) and intensifiers (e.g., “really”, “so”) are only a starting point. One could search for collections of phrases with unique emotional dynamics using unsupervised methods, such as time-series motif analyses or clustering. Features of the emotional quantile trajectories, such as variance, could also be used to augment more traditional lexicon-based analyses<cit.>, in, for instance, quantifying extreme language. They could also provide insights into the workings of humor. Word-by-word level features, like variance, and aggregate features, like the α-quantiles that individuals favor, can also be used to compare writing from different sources. In the Supplemental Information, we provide an example the latter by inverting our text-generation process to infer the most likely alpha value for each sentence. Averaging alpha values across sentences, we observe a slight pessimism bias in posts made by Reddit users who self-report depression relative to those from healthy control users. We also use this method to score excerpts from well-known speeches and books, highlighting the dominant emotion in each text (e.g., determination in the "We shall fight on the beaches" speech delivered by Winston Churchill, Figure <ref>). Analyses like these, as well as those based on other aggregate features, including the posterior distributions over the values of alpha, could be used for the sake of understanding the style of individuals (famous writers, or newspaper outlets), analyzing historical trends <cit.>, or identifying the characteristics of effective therapeutic language <cit.>. Moreover, predictive models could be estimated for other semantic characteristics, such as a forward-looking version of the topic classifier used to analyze the narrative arc of short dialogs or scripts<cit.>. Note that forward-looking models (like the ones we explore here) differ from classification methods (which view the whole text), and could be an alternative to feature importance methods such as attention-weight analysis, SHAP<cit.> or LIME<cit.> for highlighting important words; moreover, forward looking models reveal how predictions for end-scores change as information is added, mirroring the dynamics of human expectations as we read. A relevant (albeit difficult) next direction would be to develop methods for predicting the emotional arc of text at longer time-horizons – conducting the same sort of analyses and generations at the scale of paragraphs, sections, chapters and beyond. Relatedly, one could attempt to predict the longer-run destination of a person's thought (or at least the sentiment thereof) based on the words so far said. Although this would be undoubtedly difficult for most situations, there may be contexts in which repeatable patterns might be expected, e.g. in the context of psychotherapy. If successful, this could be used to flag cognitive biases <cit.> (e.g., use of absolute language like “always”, etc.), which although they are relatively neutral in themselves, can nevertheless lead perniciously to more extreme (and often) negative thoughts. This could be done real-time during telehealth therapy sessions or retrospectively as patients and therapists review previous conversations. In recent work<cit.>, we also explored a connection between worry and risk aversion in an abstract (and very simplified) computational model, which presented worry as a form of model-based (offline) planning aimed at mitigating (bad) outcomes from the lower α-quantile of a distribution. In this model, the process of worrying involves upweighting the probabilities (in an internal cognitive model) for the lower α-tail of the possible outcomes and the states that precede them, and then designing contingency plans to avoid these scenarios. Since worry is often thought to be mediated by verbal thought (i.e., language), it would be useful to develop a verbal version of this model, using a large language model and next-token probability weighting, to model phenomena such as the forms of catastrophizing evident in 'what if' chains-of-thought<cit.>. One major limitation of our work is that we use a state-of-yesterday's-art language model (GPT-2 large), which pales in comparison to the recent and much larger, but unfortunately inaccessable, models such as Chinchilla<cit.>, Lambda<cit.>, and ChatGPT<cit.>. Indeed, generation from GPT-2 (even without re-weighting) tended to produce run-on sentences and flaunt standard grammatical convention (e.g., using comma splices). The quantile models were also sometimes `jumpy' for rare or unusual tokens or text, despite the very good marginal calibration for common prefixes. A second limitation is that the quantile models rely on the underlying scoring algorithms applied to the dataset employed. For some of the emotions that we did not report here, the algorithm was rather shallow, overly emphasizing a small subset of words. Furthermore, the Reddit dataset appears to be a poor source for capturing the range of certain emotions. Finally, both the end sentiment model and emotion classifier were trained to predict the probability of the text containing that sentiment or emotion (as judged by a human rater) and therefore do not directly capture the intensity of emotional expression. Nevertheless, our method and demonstrations stand as a proof-of-principle, and we expect only improvements with models of increased scale and predictive capabilities. § METHODS §.§ Datasets and text preprocessing Text used for training the quantile models was combined from several sources: SMHD <cit.>, Social IQ <cit.>, Positive Psychology Frames <cit.>, and sentences generated from GPT-2 <cit.>. The largest source was the SMHD dataset, which contains posts to Reddit between January 2006 and December 2017 (inclusive). A random subset of posts from the full dataset was sampled, split into sentences, and then grouped into a training set (2M sentences) and a validation set (1M sentences). The SMHD also includes labels for each Reddit user for whether they self-reported a psychiatric diagnosis (e.g., depression) or whether they were recruited as a healhy control. We also included 30k sentences from the Social IQ dataset, 13k sentences from the Positive Reframes dataset, and 60k sentences generated from GPT-2 in response to short prompts, such as “I woke up early.”, “I made dinner” etc. These additional sentences were added out of convenience and originally intended for analyses unrelated to the current work. They were also divided into train and validation sets. Sentences were extracted from longer segments of text using Spacy’s NLP “sentencizer”<cit.>. Sentences that were fewer than 5 words or those contained special punctuation characters, such as pipes (“\|”), new line characters (“\r”, “\n”), colons, parentheses, and brackets were excluded. Sentences were therefore only punctuated by a period, a question mark or an exclamation mark. This was done to limit the number of run-on utterances in the dataset. Prior to training the quantile models, the data was tokenized using the GPT-2 tokenizer implemented by Huggingface and sentences were truncated to 32 tokens for computational reasons; this resulted in only about 7% of the sentences being truncated. Again for computational efficiency, the token embeddings for each sentence were extracted from the final layer of GPT-2 large and saved to disk prior to fitting the quantile models. §.§ Pretrained models The large version of GPT-2<cit.> (with 774M parameters), implemented using code from Huggingface, was used to extract text embeddings and to generate text. Positive and negative valence scores were provided by a version of RoBERTa finetuned on tweets from the TweetEval sentiment dataset <cit.>. This model outputs separate probabilities for negative and positive valence, however we combined these by multiplying by -1 and 1 (respectively) and summing to form a single scale, which varied continuously between -1 to 1. Scores for the other emotions were provided by a version of BERT that was finetuned on short English comments (3-30 tokens long) posted to Reddit and rated by human annotators for emotional content. The emotion model, provided by Hume AI, is an improved version of that used in previous work <cit.>, retrained using approximately 5x more data and roughly double the number of emotional categories (53 instead of 27). Scores ranged between 0 to 1 for each emotion and represent the probability that a human rater would label a piece of text with that emotion. The top two principal components of these emotional scores correspond approximately to the dimensions of ‘valence’ and ‘arousal’ that are central to the circumplex model of affect <cit.> (Supplemental Figure <ref>). For our analyses, we focused on two positive emotions (determination and admiration) and two negative emotions (anxiety and annoyance). These were selected based on the linguistic diversity of the sentences receiving high scores for that emotion. §.§ Quantile models Separate quantile models were used to predict the distributions of end scores for valence, determination, admiration, annoyance, and anxiety. For each token x_t in a sentence, a quantile model outputs ten equally-spaced quantiles between α=0.05 and α=0.95. The model's predictions, denoted by q_α(x_t\|x_<t; θ), are also conditioned on the preceding text x_<t through the hidden states of the large language model, which are provided as input. Each quantile model consisted of single hidden layer with 100 hidden units (whose parameters are denoted by θ). The input dimension was 1280, corresponding to GPT-2 large’s hidden state size for a single token, and the output dimension was 10, corresponding to the number of quantiles used (5%, 15%, …, 95%). Each network had 130k parameters, which is substantially smaller than GPT-2 (at 774M parameters). A tanh or sigmoid transformation was applied to the output depending on if the model was being trained on positive and negative valence ([-1,1]) or emotions ([0,1]). Given a sentence x and an end score y sampled from the training dataset, the parameters θ for a quantile model were updated by performing gradient descent on the following Huber quantile loss <cit.>: L(θ) = ∑_α∑_t ρ^H_α( y - q_α(x_t\|x_<t; θ) ), ρ^H_α(u) = \|1_u<0 - α \| H(u) Where ρ^H_α is an asymmetric weighting function that penalizes underestimation errors (y>q_α) with a weight of α and overestimation error (y < q_α) with a weight of 1-α, and 1 is an indicator function which takes on the value 1 if the condition in the subscript is true and the value 0 otherwise. To make the loss smooth at zero, the function H is applied to the estimation errors, where H(x) is 0.5x^2 if \|x\| ≤ k and (\|x\|-0.5k)k otherwise. We set k=0.001. Batches of 20 sentences were used to train the model, with the order of the sentences randomized at the start of each epoch. The models were trained for 25 epochs and evaluated periodically on a 100k-sentence subset of the validation dataset. Adam<cit.> was used with an initial learning rate of 1e-4; the learning rate was also decayed by half each time the validation loss plateaued for two consecutive epochs. The quantile models at the end of training were used for all analyses. The training and validation curves are plotted in Supplemental Figure <ref>. §.§ Temporal difference methods As an alternative, we also experimented with using temporal difference methods, specifically TD(0), to train the quantile models <cit.>. Instead of using the end scores as the target for the quantile predictions of each token, the predicted distributions for the next token were used. This was accomplished by substituting q_α^'(x_t+1\|x_<t+1; θ) with y in the equation for the quantile loss and additionally summing over all pairwise combinations of α^' and α (see <cit.> for more details). Note that using quantile regression directly on the end scores y can be viewed as a Monte Carlo (TD(1)) version of distributional temporal difference methods <cit.>. However, using this method resulted in quantile models that were more poorly calibrated, with an maximum absolute error of 9% for valence. §.§ Model calibration 1M validation sentences were used for global calibration. For each sentence, the end score (valence or emotion) was compared to the predicted quantiles at each token position. If the end score fell below a predicted quantile, a 1 was stored, otherwise a 0 was stored. Averaging across all sentences gives the empirical probability that the end score falls below the quantile of a certain level. An accurately estimated α-quantile would mean that the end scores fell below that quantile α-% of the time. Quantiles were calibrated separately for each token position; the number of sentences used for each position differed, because sentences shorter than the token position analyzed were naturally excluded. The results of the calibration are shown in Supplemental Figure <ref>. §.§ Generating text for specific α-quantiles Text generation in many large language models, like GPT-2, is autoregressive. Given a partially completed sentence x_< t (and prompt x_p), the LLM outputs a probability distribution over possible next tokens p(x_t\|x_< t, x_p). A single token x_t is sampled from this distribution and is then fed back into the model as input for the next step. To generate more positively or negatively valenced text, or text with a specific emotional tone, we reweight these next-token probabilities using the (valence or emotion) quantiles predicted for each token. More specifically, our goal is to generate sentences from below a prespecified α-quantile of a prompt x_p's distribution. To do so, we adjust the next-token probabilities to be: p^'(x_t\|x_< t, x_p) ∝α_x_tp(x_t\|x_< t, x_p) Where the weight α_x_t corresponds to the proportion of each next token's predicted distribution that resides below the α-quantile of prompt's distribution q_α(x_p). These weights are estimated by linearly interpolating the value for α_x_t such that its corresponding quantile equals the target quantile, i.e. such that q_α_x_t(x_t \| x_< t, x_p) = q_α(x_p) for each next token. Normalization is also applied to ensure that the reweighted probabilities sum to one. For intuition, consider a next token whose predicted distribution is entirely above the target quantile. This would be assigned a weight of 0, because it does not contribute to the lower α-tail of the prompt's distribution, and selecting it would make generating sentences from that lower tail impossible (given perfectly accurate predictions). Conversely, a token whose predicted distribution is entirely below the target quantile (i.e., within the target area) would be assigned a weight of 1 (and its probability would be unaltered). To generate sentences from above the (1-α)-quantile, the same procedure is applied to quantiles that are multiplied by -1. We denote the lower tail using α^- and the upper tail using α^+. After the next token is sampled from the reweighted probabilities, the process is repeated by comparing the distributions for the next set of possible tokens to the target quantile of the prompt distribution. Probability re-weighting is done after truncating the distribution using top-p (with p=0.95) and top-k (k=50) sampling. This was done to reduce the probability that extremely rare or odd tokens are selected and to increase computational speed. The temperature parameter was set to 1. §.§.§ Further details for text generation Text was generated from the model until a period token was chosen or a maximum sequence length of 40 tokens was reached. For both the unbiased and biased models many generation runs failed to self-terminate before the maximum sequence length. For the unbiased GPT-2 model (α=1.0), 79/500 utterances did not terminate. For the negative-valence model, this number was 77/500 (for α^-=0.5), 113/500 (for α^-=0.25), and 111/500 (for α^-=0.05). For the positive-valence model it was 79/500 (for α^+=0.5), 82/500 (for α^+=0.25), and 74/500 (for α^+=0.05). For the emotion models at α^+=0.05, it was 67/500 (determination), 79/500 (admiration), 91/500 (anxiety), and 130/500 (annoyance). In general, GPT-2 size language models are known to have issues with repetition and run-on sentences <cit.>. This issue is likely also exacerbated by using very simple prompts, which do not indicate how long the sentence should be. However, given that the quantile models were trained on short sentences (<32 tokens), we opted for shorter prompts to keep the quantile models in-distribution. Aside from using a larger language model, this problem could also potentially be mitigated by longer prompts or adjusting the temperature parameter. As the current work aims merely to demonstrate the validity of this approach, we will leave these explorations for future work. § DATA AVAILABILITY The primary dataset we used to train our models, SMHD <cit.>, was obtained from Georgetown University with permission to use but not to share, in order to protect user privacy. We therefore cannot make this dataset available. However, scripts used to process the data and train the models will be included in the project’s Github repository. § ACKNOWLEDGEMENTS CG and PD are funded by the Max Planck Society. PD is also funded by the Alexander von Humboldt Foundation. The authors would like to thank Alan Cowen and Hume AI for kindly providing access to the language model that was used to emotionally score text. We would also like to thank Christian Adam, Y-Lan Boureau, Marc Bellemare, and Marcel Binz for helpful comments on an earlier version of this manuscript. § AUTHOR CONTRIBUTIONS STATEMENT C.G. and P.D. conceived the experiments and analyses; C.G. conducted the experiments; C.G. and P.D. wrote and reviewed the manuscript. § ADDITIONAL INFORMATION Competing interests: The author(s) declare no competing interests. § SUPPLEMENTAL INFORMATION §.§ Inferring the α for a given text For generating text, we specified a particular α-quantile below (or above) which to sample sentences. In this section, we explore a method for inverting this process and inferring the the most likely α that would have produced a given sentence or a set of sentences written by a single source. §.§.§ Estimating α for single sentences To estimate α, we follow the same procedure as generation, except that we yoke the next tokens to those in a sentence. As the model processes that sentence, we separately sum the tokens’ log-probabilities under the language model and the reweighted log-probabilities under different values of α^- and α^+. The alpha value with the highest summed log-probability is considered the best fitting for that sentence (including α=1.0, which corresponds to the baseline language model). Under different alphas, token probabilities are reweighted by different amounts, with valenced words being more strongly up- (or down-) weighted by the lowest values of alpha. Therefore, highly valenced sentences will be most likely under the lowest values of alpha, and ambivalent sentences will be best fit by a value of alpha closer to 1. In Figure <ref>a, we show two example sentences, both with negative valence. For the first sentence, negative words are primarily upweighted, and the rest untouched, leading to a best fitting α^-=0.1. For the second, which is more ambivalent, the positive part of the sentence is too unlikely under the extreme α^-'s (<=0.1), leading to a best fitting α^-=0.25. The final valence of the two sentences is -0.97 and -0.66, respectively, in an order that conforms with the best fitting values of alpha. However, the posterior distributions over α^- (or, here, the likelihood curves, assuming a flat prior) provides a finer-scale depiction of the emotional workings of the sentence, showing, for instance, how very unlikely it is that the second sentence was associated with a really low value of α^- (presumably because of the change in sentiment of the end clause), whereas the first sentence has a far flatter curve. §.§.§ Generation and recovery analysis We also performed a generation-recovery analysis to assess the accuracy with which we can infer the alpha value for a single source (e.g., an individual or another LLM). To have a ground truth against which to validate, we generated 500 hundred sentences for each α^- and α^+ value from a larger set of possible values (i.e., [0.05, 0.1, 0.25, 0.5, 0.75]) and treated each set of sentences as a single source. Averaging the best-fitting α^- or α^+ across sentences produced an almost entirely correct re-estimation of the alpha value (i.e., the one used to generate that set of sentences; Pearson correlation > 0.99; Figure <ref>b). Note that before averaging, the α^+ values were converted to [1.25, 1.5, 1.75, 1.9, 1.95], so that they could be differentiated from the α^- values. We then repeated this analysis for different sample sizes from 1 sentence to 500 sentences per source (with 50 simulations per sample size) and show that even with 50 sentences, the ground truth α can be recovered 77% of the time (compared to 90% for 500 sentences; top left inset in Figure <ref>b). This demonstrates that around fifty sentences suffices to estimate the statistical tendency to generate sentences from below/above a particular quantile of the distribution. §.§.§ Analyzing text written by Reddit users As a test of external validity, we identified Reddit users who had written at least fifty sentences from our validation set and split them into two groups: those with self-reported diagnoses of depression and those who were designated as health controls. These labels were derived from the SMHD dataset<cit.>, the largest component of our training and validation sets. To balance the two groups, 192 participants were randomly selected from larger group (depression) and fifty sentences were randomly selected from each user. Estimating the best fitting α per user revealed a significant difference between the two groups (mean α^-=0.92 for depression; mean α^-=0.97 for control; t= -2.57, p=0.01, df=382; where α^-<1.0 corresponds to pessimistic sampling and α=1.0 corresponds to unbiased sampling). Moreover, 69% of the users who self-reported a diagnosis of depression had a best fitting α^- below 1 compared to 57% for control users. This is notable because the SMHD<cit.> excluded posts with explicit mentions of mental health diagnoses or those posted to mental health subreddits, and instead contains posts on a variety of topics, such as sports and politics. §.§.§ Analyzing real world examples Finally, we show the same procedure applied to each emotion model and applied to real-world examples, to highlight the applicability of the method to analyzing different sorts of writing. We selected excerpts from well-known authors that we thought strongly conveyed each one of the four emotions (see below for the full excerpts). α^+ was estimated first per sentence and then averaged across all sentences for each of the four texts. The estimated α^+'s for each example was low specifically for one emotion, matching our intuitions (Figure <ref>c). For instance, Oprah Winfrey's eulogy commemorating Rosa Parks has an mean α^+ of 0.4 for admiration, and the excerpt from Winston Churchill's famous speech, “We shall fight on the beaches”, has a mean α^+ of 0.31 for determination. These values of α^+ indicate the upper-tail quantile, with lower values signifying more extreme emotional scores. For determination, we used the following speech from Winston Churchill delivered on June 4, 1940: I have, myself, full confidence that if all do their duty, if nothing is neglected, and if the best arrangements are made, as they are being made, we shall prove ourselves once again able to defend our Island home, to ride out the storm of war, and to outlive the menace of tyranny, if necessary for years, if necessary alone. At any rate, that is what we are going to try to do. That is the resolve of His Majesty’s Government-every man of them. That is the will of Parliament and the nation. The British Empire and the French Republic, linked together in their cause and in their need, will defend to the death their native soil, aiding each other like good comrades to the utmost of their strength. Even though large tracts of Europe and many old and famous States have fallen or may fall into the grip of the Gestapo and all the odious apparatus of Nazi rule, we shall not flag or fail. We shall go on to the end, we shall fight in France, we shall fight on the seas and oceans, we shall fight with growing confidence and growing strength in the air, we shall defend our Island, whatever the cost may be, we shall fight on the beaches, we shall fight on the landing grounds, we shall fight in the fields and in the streets, we shall fight in the hills. We shall never surrender, and even if, which I do not for a moment believe, this Island or a large part of it were subjugated and starving, then our Empire beyond the seas, armed and guarded by the British Fleet, would carry on the struggle, until, in God’s good time, the New World, with all its power and might, steps forth to the rescue and the liberation of the old. For admiration, we used the following section of the eulogy for Rosa Parks, delivered by Oprah Winfrey in October 2005: To Reverend Braxton, family, friends, admirers, and this amazing choir: I – I feel it an honor to be here to come and say a final goodbye. I grew up in the South, and Rosa Parks was a hero to me long before I recognized and understood the power and impact that her life embodied. I remember my father telling me about this colored woman who had refused to give up her seat. And in my child's mind, I thought, “She must be really big.” I thought she must be at least a hundred feet tall. I imagined her being stalwart and strong and carrying a shield to hold back the white folks. And then I grew up and had the esteemed honor of meeting her. And wasn't that a surprise. Here was this petite, almost delicate lady who was the personification of grace and goodness. And I thanked her then. I said, “Thank you,” for myself and for every colored girl, every colored boy, who didn't have heroes who were celebrated. For anxiety, we used the following passage from the book “Kiterunner” by Khaled Hosseini: Panic. You open your mouth. Open it so wide your jaws creak. You order your lungs to draw air, NOW, you need air, need it NOW. But your airways ignore you. They collapse, tighten, squeeze, and suddenly you’re breathing through a drinking straw. Your mouth closes and your lips purse and all you can manage is a croak. Your hands wriggle and shake. Somewhere a dam has cracked open and a flood of cold sweat spills, drenches your body. You want to scream. You would if you could. But you have to breathe to scream. Panic. And for annoyance, we used the following passage from the commencement speech delivered by David Foster Wallace at Kenyon College in 2005 titled “This Is Water: Some Thoughts, Delivered on a Significant Occasion, about Living a Compassionate Life”: … And many more dreary, annoying, seemingly meaningless routines besides. But that is not the point. The point is that petty, frustrating crap like this is exactly where the work of choosing is gonna come in. Because the traffic jams and crowded aisles and long checkout lines give me time to think, and if I don’t make a conscious decision about how to think and what to pay attention to, I’m gonna be pissed and miserable every time I have to shop. Because my natural default setting is the certainty that situations like this are really all about me. About MY hungriness and MY fatigue and MY desire to just get home, and it’s going to seem for all the world like everybody else is just in my way. And who are all these people in my way? And look at how repulsive most of them are, and how stupid and cow-like and dead-eyed and nonhuman they seem in the checkout line, or at how annoying and rude it is that people are talking loudly on cell phones in the middle of the line. And look at how deeply and personally unfair this is. § SUPPLEMENTAL TABLES AND FIGURES
http://arxiv.org/abs/2307.02657v2	20230705211604	DiskMINT: A Tool to Estimate Disk Masses with CO Isotopologues	[ "Dingshan Deng", "Maxime Ruaud", "Uma Gorti", "Ilaria Pascucci" ]	astro-ph.EP	[ "astro-ph.EP", "astro-ph.IM", "astro-ph.SR" ]	0000-0003-0777-7392]Dingshan Deng Lunar and Planetary Laboratory, the University of Arizona, Tucson, AZ 85721, USA Dingshan Deng [email protected] 0000-0003-0522-5789]Maxime Ruaud NASA Ames Research Center, Moffett Field, CA 94035, USA Carl Sagan Center, SETI Institute, Mountain View, CA 94043, USA 0000-0002-3311-5918]Uma Gorti NASA Ames Research Center, Moffett Field, CA 94035, USA Carl Sagan Center, SETI Institute, Mountain View, CA 94043, USA 0000-0001-7962-1683]Ilaria Pascucci Lunar and Planetary Laboratory, the University of Arizona, Tucson, AZ 85721, USA CO is one of the most abundant molecules in protoplanetary disks, and optically thin emission from its isotopologues has been detected in many of them. However, several past works have argued that reproducing the relatively low emission of CO isotopologues requires a very low disk mass or significant CO depletion. Here, we present a code, , which includes gas density and temperature structures that are both consistent with the thermal pressure gradient, isotope-selective chemistry, and conversion of CO into CO_2 ice on grain-surfaces. The code generates a self-consistent disk structure, where the gas disk distribution is obtained from a Spectral Energy Distribution (SED)-derived dust disk structure with multiple grain sizes. We use to study the disk of RU Lup, a high-accreting star whose disk was previously inferred to have a gas mass of only ∼ 1.5×10^-3 M_⊙ and gas-to-dust mass ratio of ∼ 4. Our best-fit model to the long-wavelength continuum emission can explain the total C^18O luminosity as well as the C^18O velocity and radial intensity profiles, and obtains a gas mass of ∼ 1.2×10^-2 M_⊙, an order of magnitude higher than previous results. A disk model with parametric Gaussian vertical distribution that better matches the IR-SED can also explain the observables above with a similarly high gas mass ∼ 2.1×10^-2 M_⊙. We confirm the conclusions of <cit.> that optically thin rotational lines provide reasonable estimates of the disk mass and can therefore be used as gas disk tracers. § INTRODUCTION Disks of gas and dust around young stars (hereafter, protoplanetary disks) are the sites of planet formation, and their mass is fundamental to understanding when and how planets and small bodies form. While the gas content sets limits on the potential masses of forming giant planets, the dust mass constrains the masses and formation times for the cores of gaseous planets and terrestrial planets. The gas-to-dust mass ratio (Δ_𝐠𝐝), moreover, is an indicator of the relative rates of planet formation and gas disk dispersal and indicates the stage of disk evolution and planet formation (e.g., for a recent review). Ideally, independent and reliable dust and gas mass estimations are needed to infer the disk physics and evolution, but measuring both masses is complicated and challenging. Dust masses (M_dust) are estimated by the dust thermal emission at (sub)millimeter wavelengths, which is sensitive to particles with sizes 1 cm and is mostly optically thin (e.g., ). However, M_dust estimates rely on the dust opacity κ_ν which depends on the composition of dust grains and their size distribution. Therefore, estimates of M_dust from a single flux measurement strongly depend on the assumptions made on the dust properties <cit.>. Improved estimates of M_dust can be made by fitting the spectral energy distribution (SED) at long wavelengths (100) where the emission is typically optically thin <cit.>. Gas masses (M_gas) are more difficult to estimate since there are very few optically thin gas emission lines that may trace the disk mass reservoir. H_2 is the most abundant molecule in the gas phase in the disk, but its emission is faint. This is because H_2 is a light, homonuclear molecule with no permanent dipole moment and hence has only transitions at high energy levels (E_u ∼ few 100-1000K), while the majority of the gas in the disk around T-Tauri stars is far colder (∼ 30 K). The less abundant isotopologue HD is favored to measure M_gas, although it also traces relatively warm gas (needed to excite the first rotational level of HD at E_u ∼ 128 K), and therefore has some limitations on its suitability as a mass tracer <cit.>. Carbon monoxide (CO) is the most abundant molecule after H_2 and is co-spatially distributed with H_2 at the disk surface. In the disk mid-plane, CO freezes out on the dust grain surface (when T_dust≲ 20 K) where it can be processed into more refractory ices. With its high detectability at (sub)millimeter wavelengths in disks, CO and its isotopologues have long been considered among the best tracers of gas disk mass. However, recent Atacama Large Millimeter/submillimeter Array (ALMA) observations of Class-II disks have cast doubts about its ability as a mass tracer because model-predicted line emissions of CO and its isotopologues are higher than observed even after accounting for the fact that CO freezes-out in the mid-plane <cit.>. This raises questions as to whether CO chemical abundances in disks differ from that in the interstellar medium (ISM) or whether the disk gas masses are low. Furthermore, the CO-based M_gas were smaller by ∼ 1 -2 orders of magnitude compared with the HD-based values for the few disks where HD has also been detected <cit.>. Thus, some works have argued for higher gas masses but large-scale depletion of CO due to dynamical processes that sequester CO into forming planetesimals and proto-planets <cit.>. A different solution was proposed recently by <cit.> (hereafter RGH22), who argued that by including (a) the density distribution given by self-consistent vertical hydrostatic pressure equilibrium, (b) isotopologue-selective chemistry, and (c) grain-surface chemistry where CO to CO_2 conversion is a key reaction, the apparent discrepancy between the HD and C^18O derived masses can be resolved. They concluded that CO chemistry in disks is in fact similar to that in the ISM and that the optically thin lines from C^18O can be used as a gas mass tracer. Although they could retrieve typical C^18O fluxes observed for the Lupus sample, they did not consider individual disks in detail or compare the profile and radial distribution of the line emission. In this work, we develop a tool to estimate the disk mass: (Disk Model for INdividual Targets). It uses the dust temperature-based approach suggested in RGH22: generating a self-consistent gas disk structure on top of a SED-derived dust disk. It also uses a reduced chemical network that properly captures the conversion of CO into CO_2 ice. The tool is tested in considerable detail for the Class II source RU Lup. We select RU Lup because this disk has been previously inferred <cit.> to have a low gas mass of M_gas∼ 1.5×10^-3 M_⊙ with Δ_gd∼ 4 which is at odds with the large mass accretion rate onto the star <cit.> and the large disk size <cit.>. The paper is organized as follows. First, we describe the modeling procedure in Section <ref>. Then, we summarize the stellar parameters, observational data, and model setup for RU Lup in Section <ref>, followed by the results and discussion in Section <ref>. We present our summary and outlook in Section <ref>. § MODEL DESCRIPTION is a dust temperature-based disk model, and uses the recommendations made by RGH22. From their analysis using a full thermo-chemical model that includes isotope-selective photodissociation and 3-phase grain-surface chemistry, RGH22 identified two main components that can be used to construct a simplified model to accurately simulate C^18O emission. The two components are: (a) a self-consistent disk physical structure, based on the dust temperature T_d and imposing vertical hydrostatic pressure equilibrium (hereafter VHSE) to calculate densities consistent with this vertical temperature; and (b) a reduced chemical network that includes isotope-selective photodissociation and grain-surface chemistry that accounts for conversion of CO into CO_2 ice (see Appendix A of RGH22). Since C^18O traces the vertical layer where gas temperature T_g is still very similar to T_d, RGH22 found that a simplified dust disk structure model (which does not consider the self-consistent gas temperature T_g computed from full thermal equilibrium) can be used to estimate the C^18O emission. As such, we build based on this simplified model. The overall method adopted in our analysis is summarized in the flow chart shown in Figure <ref>. Two main steps are involved in obtaining a self-consistent disk model that fits the C^18O line and continuum data. The goal of Step 1 is to find a density structure — based on the dust temperature profile — that is self-consistent with pressure equilibrium and fits the SED. This is achieved by iteration: starting from an arbitrary initial density, computing the dust temperature using <cit.>, determining the resulting gas temperature, solving for vertical hydrostatic pressure equilibrium, and subsequently updating the density and temperatures in iterations until convergence. Step 2 computes the C^18O abundance distribution via the reduced chemical network that includes isotopologue-selective dissociation and CO to CO_2 ice conversion on grains. It then computes the C^18O line emission using the radiative transfer tool (Line Modelling Engine Version 1.9.5, ) and compares it to the observed line emission profiles. If the agreement is poor, then the initial parameters (e.g., surface density distribution Σ, gas-to-dust mass ratio Δ_gd) are modified to repeat the entire modeling procedure from Step 1 until a satisfactory match with both SED and line emission is obtained. Details about the two steps are provided in the following subsections. §.§ Model Step 1: Finding a Self-consistent Disk Structure that Fits the SED The main input parameters for this step (apart from the stellar parameters) are the surface density distribution, dust size distribution and opacity, and the disk gas-to-dust ratio. Surface Density The surface density distribution is assumed to be that of a viscously evolving disk <cit.> and is specified as: Σ(r) = Σ_1 (r/1 AU)^-pexp[- (r/r_tap)^2-γ]; (r_in < r< r_out) where r is the radial distance from the star, Σ_1 is the surface density at 1 AU that is scaled according to the chosen disk mass, γ is the tapering-off exponent and p is the power-law index. We further assume that γ = p, which represents the self-similar viscous solution. The inner radius cut-off, r_in, is assumed to be the dust sublimation radius while the outer radius r_out is chosen to be much larger than r_tap to ensure that all the mass is included. Dust Properties As discussed later, the gas temperature is computed assuming an equilibrium value from collisional heating/cooling by dust grains. To determine the gas temperature accurately, we use multiple dust sizes and calculate the size-dependent dust temperature. The dust species are divided into multiple grain-size bins equally distributed in log-space. The dust number density follows a power-law distribution: n(a) ∝ a^-q with a∈[a_min, a_max] where a is the dust grain size and q is the exponent describing the size distribution. We adopt a dust composition consisting of 64% astronomical silicates and 36% graphite by volume fraction, which is representative of ISM dust with the ratio of visual extinction to reddening R_V = 5.5 <cit.>. A similar composition has been adopted in many previous disk models <cit.>. The package from <cit.> is used to compute the wavelength dependence of dust opacity, and the optical constants are those of astrosilicate from WD01 and graphite from <cit.>. Gas-to-dust ratio In order to determine the vertical hydrostatic pressure equilibrium solution, the gas pressure gradient and hence the gas density are needed. In , the surface density distribution of gas and dust can in principle be specified separately as a function of radius, and this determines the local gas-to-dust ratio Δ_gd(r). However, for our modeling of RU Lup, we assumed a constant value throughout the disk for simplicity, which, as we show later, can already match the C^18O data. The vertical dust density distribution, ρ_ d(r, z), is initially set as an arbitrary Gaussian profile. This is then distributed according to the mass fraction in each grain size bin to obtain ρ_ d(r, z, a). <cit.> is used to compute the dust temperature T_d(r, z, a) for each grain size bin. We first determine the gas temperature T_g(r, z) balancing collisional energy exchange with dust grains; this contribution is denoted as T_g,d. Since RU Lup is a high accretor, the near and mid-infrared SED can be affected by viscous heating <cit.>. does not currently include this viscous heating term, and we hence add this as a separate contribution to the gas (T_g,v), and the dust as described later below. T_g,d is estimated from the following equation balancing dust heating and cooling [ ∑_T_d(a) > T_g,d A_H n_d(a) π a^2 n_Hv̅_H 2k_B [T_d(a) - T_g,d]; = ∑_T_d(a) < T_g,d A_H n_d(a) π a^2 n_Hv̅_H 2k_B [T_g,d - T_d(a)], ] where T_d(a) is the dust temperature at the grain size a, A_H is the mean accommodation coefficient, n_d(a) is the dust number density distribution, n_H is the gas number density, v̅_H is the gas thermal velocity, and k_B is the Boltzmann constant. This thermal balance equation simplifies to [ ∑_T_d(a) > T_g,d n_d(a) a^2 [T_d(a) - T_g,d]; = ∑_T_d(a) < T_g,d n_d(a) a^2 [T_g,d - T_d(a)], ] and only the terms related to dust size remain. The gas temperature contributed by dust grain collisions (T_g,d) is thus a cross-section weighted mean value between the hot (small) and cold (large) dust grain temperatures, and therefore the number of grain size bins (N) used could potentially affect the accuracy of the gas temperature evaluation. We adopt N=20 as we find that this results in gas temperature deviations (caused by N) to be less than 5%. We next estimate the temperature due to a balance between accretion heating and radiative cooling <cit.>. Viscous heating is given by (9/4)νΣΩ_k^2 where ν is the kinematic viscosity, and Ω_K is the Keplerian angular frequency. Cooling is given by 2σ_SB T^4, and for a disk accreting in steady state the accretion rate (Ṁ_acc∼ 3πνΣ) we have T_g,v = [ 3 G M_⋆Ṁ_acc/8 πσ_ SB r^3× ( 1 - √(r_⋆/r)) ]^1/4, where G is the gravitational constant and σ_SB is the Stefan–Boltzmann constant. The resulting gas temperature is determined by adding the two temperatures in quadrature and is therefore given by T_g(r, z) = [T_g,d^4(r, z) + T_g,v^4(r, z)]^1/4. Viscous heating dominates only at the mid-plane in the inner disk (10 AU) for typical disk densities <cit.>. Once the gas temperature is computed, the new density structure is calculated from the pressure gradient by solving dP(r, z)/dz = -ρ_gas(r, z) Ω^2 z, where P(r, z), ρ_gas and Ω are the gas pressure, gas density (assumed to be the total dust density times a constant Δ_gd) and Keplerian frequency, respectively. For the next iteration, the dust density profile with z is rescaled with this vertical gas density profile, and re-normalized to the surface density at this radius. The dust temperatures are re-calculated with the new dust density distribution using . The steps above are recomputed until convergence is achieved at the iteration m: \| [ρ_gas, m(r, z) - ρ_gas, m-1(r, z) ] /ρ_gas, m-1(r, z) \| < 5% for regions with ρ_gas > 10^-20 g cm^-3 (corresponding to n_ H 10^3 cm^-3). The error tolerance was chosen as a reasonable compromise between accuracy and speed of computation (∼ 4 hours to achieve convergence when running with 24 threads with 2.10 GHz CPUs). Lower tolerances did not significantly change the results. The above procedure results in a dust and gas density and temperature distribution which are all self-consistent with the local vertical pressure gradient. We described viscous heating for gas above, but this term is also relevant for heating dust grains. Since this is difficult to incorporate into the code, we include this effect by adding it to the dust grains before computing the SED. This is done by considering the gas as a thermal reservoir that equilibrates the dust temperature in regions where dust and gas are highly coupled. In practice, we estimate the extent of this mid-plane region as the region where the temperature differences between the hottest/smallest grain and coldest/largest grain are small enough as \|(T_d(a_min) - T_d(a_max))/T_d(a_max)\| < 10%. T_d(a) = T_g is set for all grain sizes in this coupled region. We then run to compute the SED and compare it with the observed SED. We vary the disk dust parameters until a satisfactory match to the SED is obtained. The dust opacity κ_ν and the dust mass M_dust are two main parameters affecting the synthetic SED: Changing κ_ν alters the slope of the long-wavelength portion of the SED and M_dust moves the flux density up and down. In practice, we find the best fit κ_ν by comparing the slope of the dust opacity β_abs = - d log(κ_abs)/d logλ with the slope of the SED at long wavelength α_SED = - d logF_ν/d logλ (λ≫ 100) based on the relation between the two slopes β_abs = α_SED - 2. When the dust composition is fixed, we first vary the maximum particle size a_max and keep the slope of the number density distribution with size fixed to q = 3.5, which is the value expected in collisional equilibrium <cit.>. If the upper limit of a_max = 1 cm is reached while varying a_max, then q is varied to find the best match of the slope. After the best-fit κ_ν is found, the M_dust is derived by matching the absolute value of the flux density at long wavelengths. §.§ Model Step 2: Computing the Line Emission and Profile The next step in our modeling approach is to run the reduced chemical network described in RGH22 to obtain the C^18O abundance with (r,z). The photodissociation rates (for our application target RU Lup) are computed from the UV HST/COS median-resolution spectrum obtained by <cit.> (see also Figure <ref> for average photometric values from this spectrum). We assume all gas is molecular in the disk structure calculation but explicitly solve for the chemistry by specifying the corresponding H nuclei density (n_H) for the chemical network. This means that all molecular abundances in the chemical network are defined by their density ratio compared to the density of H nuclei. Finally, the gaseous abundances of C^18O and the disk structure are inputs to <cit.> to compute the non-LTE(local thermal equilibrium) synthetic C^18O (2-1) and (3-2) emission. The model parameters are varied until the synthesized SED and C^18O line emission match the observations. We fix κ_ν and M_dust to the values determined in the SED fitting, and explore a range of gas-to-dust ratios Δ_gd = 5, 10, 50, 100 (which covers the low disk Δ_gd reported in the literature up to the ISM value) to generate a grid of M_gas. Since the self-consistent VHSE solution depends on the gas mass (which varies with Δ_gd in the gas mass grid), the vertical density structure of each of these models slightly differ. However, the SED at long wavelengths traces the optically thin thermal emission from the large grains and remains the same as it is not sensitive to the vertical dust density distribution. The derived M_dust therefore remains unaltered even as Δ_gd is varied. We start from the beginning for each grid point and find that we do not need to re-fit the SED, hence we calculate the dust thermal structure with , solve the VHSE through iterations, and then derive the abundance by the reduced chemical network. Next, the C^18O line luminosity (L_C^18O) is computed to compile a L_C^18O vs. M_gas relation. The best-fit M_gas is then determined as the value where the modeling relation (L_C^18O vs. M_gas) intersects the luminosity inferred from the observations. Finally, we run the model with best-fit M_gas again also from the beginning to verify the estimate found above. In this work, we not only compare total line luminosities as in RGH22 but also match the velocity profile and radial distribution of the C^18O (2-1) line. These are generated by the package <cit.> from the simulated image and follow the same procedure used on observational data. The slope of the surface density distribution and the gas-to-dust ratio as a function of radius are parameters that can be changed to improve the fit on the line profile, if necessary. § APPLICATION TO RU LUP §.§ The Highly Accreting RU Lup Star and Its Dust and Gas disk RU Lup (Sz 83, 2MASS J15564230-3749154) is a K7-type star located at a distance of 158.9 pc <cit.> and a member of the Lupus II star-forming region <cit.>. RU Lup has the highest mass accretion rate (∼ 10^-7 M_⊙/yr, ) and is one of the most active stars in the region with large irregular variations in both spectroscopy and photometry from ultraviolet (UV) to infrared (IR) wavelengths <cit.>. The stellar mass estimates range from 0.2 to 1.2 M_⊙ <cit.>. Here, we adopt the value of 0.7 M_⊙ from more recent evolutionary models <cit.> over the dynamical mass of 0.2 M_⊙. This is because the disk of RU Lup is close to face-on which introduces a large uncertainty in the dynamical mass <cit.>. As one of the most extensively observed Class II objects in Lupus, photometry and spectra are available from the UV to radio wavelengths resulting in the multi-wavelength spectral energy distribution (SED) shown in Figure <ref>, where average photometry is reported for multi-epoch observations. A large-scale, complex proto-planetary disk has also been recently revealed by ALMA. The millimeter dust disk appears symmetric with multiple annular gaps and rings and extends out to a radius of ∼ 63 AU <cit.>. In contrast, CO emission has a more asymmetric morphology. <cit.> identified a Keplerian disk with a radius of ∼ 120 AU, similar in size to that inferred via scattered light <cit.>, surrounded by an envelope extending out to ∼ 260 AU with spiral arms and clumps. However, the C^18O emission, which we focus on and aim to model in this work, is less complex. The C^18O emission is symmetric, only traces the Keplerian disk, and has a radius of ≲ 100 AU. The lower panels of Figure <ref> show the (2-1) line profile and radial intensity cut from publicly available datacubes <cit.> generated using . We choose the same aperture and wavelength range used in <cit.>, 1.5∼ 240 AU and 1.75-7.25 km/s, as the maximum extent to include all the emitting areas and channels when computing the line profile and radial profile. In the line profile, there is clear dark cloud contamination at 5 km/s (dashed line). Linear interpolation (grey point and line) is utilized to recover the disk emission in this channel, which brings the integrated total flux of (2-1) from 0.34 ± 0.03 Jy km/s to 0.37 ± 0.03 Jy km/s. For the radial profile, deprojection is applied using the disk position angle PA = 121 and inclination i = 18.8 <cit.>. The dust and gas mass of RU Lup, hence the gas-to-dust mass ratio Δ_gd, have been previously estimated using continuum millimeter emission and CO isotopologue emission. <cit.> measured the ^13CO(3-2) and C^18O(3-2) line fluxes and compared them to a grid of simple disk models by <cit.>: They inferred a gas disk mass of ∼ 2.7_-1.7^+7.3×10^-3 M_⊙ and a gas-to-dust mass ratio Δ_gd∼ 8.9_-5.7^+24.4. <cit.> included isotope-selective dissociation in the thermo-chemical physical code <cit.> and used the same line luminosities to infer an even lower gas disk mass (∼ 1.5_-1.0^+2.5×10^-3 M_⊙) and gas-to-dust ratio (Δ_gd∼ 3.8_-2.6^+6.2). Clearly, the low inferred disk mass and gas-to-dust mass ratio are hard to reconcile with the large dust and gas disk of RU Lup and the high accretion rate onto the star; we therefore re-examine the dust and gas mass constraints using the modeling approach. §.§ Specific Models Two models are considered in this work with different vertical density distributions: (a) the VHSE model uses a self-consistent vertical hydrostatic pressure equilibrium solution; (b) the Gaussian model uses a parameterized Gaussian vertical structure. Both models share the same surface density distribution and use the same dust grains (same κ_ν and M_dust) determined by fitting the long wavelength portion of the SED (λ 100). The Gaussian model additionally fits the IR wavelengths (10 λ 100) by assuming the pressure gradient to be a free parameter and thus varying pressure scale height in the Gaussian structure: H_p = H_p, 100(r/100 AU)^α with free characteristic height H_p, 100 and flaring index α. This is the approach taken in a few recent studies to estimate disk masses and Δ_gd <cit.>. The model input parameters are presented in Table <ref>. Stellar mass M_⋆, radius r_⋆ as well as mass accretion rate Ṁ_acc are fixed and taken from the literature (see Section <ref>). The inner radius r_in is fixed at the dust sublimation radius, and the tapering-off radius is set as the dust outer radius r_tap∼ r_dust given in <cit.>. The dust opacity κ_ν is computed by : It uses a dust composition described in Section <ref>, and has fixed volume fraction from a_ min∼ 1.0×10^-6 cm through to a_ max, in which a_ max and the power law index q are free parameters. The other two free parameters are the dust disk mass M_dust, and the gas-to-dust mass ratio Δ_gd. The synthetic imaging setup for the models is obtained from observations (summarized in Section <ref>). The output synthetic image is created with a pixel size of 0.04, and with source distance, i and PA. The image has 151 × 151 pixels to include all disk emission within 3.0. The dust continuum emission is also included in the synthetic image, and then the continuum is subtracted in the final line imaging datacube. Then, the output image is convolved with a beam of 0.32 × 0.32 to get the final synthesized image. lcccc Model Parameters 0.9 Parameter 2cSymbol 2cValue Dust Properties Volume fraction 64% Silicate 36% Graphite minimum size 2ca_min 2c1×10^-6 cm maximum size 2ca_max 2cfree parameter exponential slope 2cq 2cfree parameter Radial Structure inner radius of the disk 2cr_in 2c0.035 AU tapering-off radius 2cr_tap 2c63 AU surface density slope 2cp 2c1 Vertical Structure VHSE Gaussian Characteristic Scale Height 2cH_p, 100 solved free parameter Flaring Index 2cα solved free parameter In principle, all parameters in this table could be varied to fit the observations. However, only a_max and q are changed here for the VHSE model as the default settings for other parameters could already give a good fit. H_p, 100 and α are also set free for the Gaussian model while the vertical structure for the VHSE model is solved self-consistently from pressure equilibrium. The best-fit free parameters are summarized in Table <ref>. § RESULTS AND DISCUSSION The inferred dust parameters, dust and gas masses are summarized in Table <ref>. One of the main results of this work is that our model can explain RU Lup's long-wavelength (λ≳100) SED, the C^18O (2-1) and (3-2) line luminosities, and the velocity and radial profiles, with a higher M_gas and thus higher Δ_gd than previously inferred. We present details on these models in Section <ref>. Effects of CO↔CO_2 conversion on grain-surface and differences between the Gaussian and VHSE models are discussed in Section <ref>. Our VHSE model under-estimates the strong IR excess of RU Lup by a factor of ≲ 3, and we discuss possible reconciliations in Section <ref>. lccccccc Model Main Results 0.45 Model H_p, 100 α a_max q M_dust M_gas Δ_gd (AU) (cm) (M_⊙) (M_⊙) VHSE - - 0.3 3.5 4.0× 10^-4 1.2× 10^-2 30 Gaussian 30 1.1 0.3 3.5 4.0× 10^-4 2.1× 10^-2 52 §.§ Emission Indicates a Relatively High Gas Disk Mass for RU Lup In , the disk density structure is based on the dust temperature profile, and the dust disk is constructed by fitting the SED (See Section <ref>). The SED fits of the two models (VHSE and Gaussian) introduced in Section <ref> are shown in the top panel of Figure <ref>. Both models share the same dust grain properties described in Table <ref> and the best-fit free parameters are reported in Table <ref>. The best-fit maximum grain size a_max, q parameters and dust disk mass M_dust are the same for both models: a_max = 0.3 cm, q = 3.5 and M_dust∼ 4.0×10^-4 M_⊙. Since the pressure scale height is determined using free parameters to match the SED in the Gaussian model, this model provides a better fit to the IR SED. To find the best parameters, we start from the best-fit pressure scale height with H_p, 100 = 20.9324 AU and α = 1.1301 reported in <cit.> for the disk of RU Lup, and generate a grid of H_p, 100 = 15, 20, 25, 30 AU and α = 1.05, 1.10, 1.15, 1.20. Although we use a different dust composition and updated parameters for the central star, we find a relatively close pressure scale height with H_p, 100 = 30 AU and α = 1.10 and a very similar synthetic SED for the Gaussian model. For the VHSE models, the procedure of iteration to determine the vertical density structure to be consistent with the temperature profile sets the pressure scale height; there is no simple power law to describe the scale height thus the parameters H_p, 100 and α are not valid. We then run the reduced chemical network and to obtain the synthetic luminosity, which is compared with the observation to obtain the best-fit Δ_gd and hence the M_gas (see Section <ref>). The synthetic (2-1) and (3-2) luminosities vs. M_gas using different Δ_gd are presented in Figure <ref>. There are four data points on each modeling line representing Δ_gd = 5, 10, 50, 100 (points in Figure <ref>), and one additional best-fit model (`×' in Figure <ref>), which is obtained at the cross point with the observation of C^18O (2-1) luminosity at left panels. The best-fit gas masses for both models are within a factor of two: M_gas∼ 1.2×10^-2 M_⊙ and ∼ 2.1×10^-2 M_⊙ for the VHSE and Gaussian models, respectively. In addition to matching the luminosity, the line spectrum and radial distribution from the synthetic line images are also compared with the observations. The lower panels of Figure <ref> present the C^18O (2-1) spectra and radial distribution for different models generated from simulated datacubes with and the same setup (see Section <ref>) as for the observational data. These panels demonstrate that the VHSE model with default input parameters (Table <ref>) also matches the C^18O (2-1) line profile and radial cut. The Gaussian model can reproduce the line luminosity and matches the (2-1) line velocity profile relatively well, but its emission is more compact than the VHSE model with the intensity peaking closer to the host star. We note that the models also fit the (3-2) luminosity <cit.>, as shown in the right panel of Figure <ref>. The (3-2) line emission has a similar velocity profile and radial cut, but it is a factor of ∼ 5 more luminous than the (2-1) line. For both models, even better fits may be achieved by changing the surface density distribution and by including a radial-dependent gas-to-dust ratio, but we did not consider these modifications necessary for RU Lup. The Gaussian model has an emission profile that is less radially extended compared with the observations. This is because it has a very puffed-up density distribution which appears necessary to fit the IR SED: H_p, 100 = 30 AU by this work (also H_p, 100∼ 21 AU reported in <cit.> which is a better match to the SED). Since the scale height is parameterized as a power-law, this implies that the flaring index in the outer emitting regions of the model disk is also higher. The increased flaring moves the emitting layer closer to the star and higher. It is nearly a factor of ∼ 3-4 higher than the VHSE disk at r ∼ 50 AU where most of the emission comes from (Figure <ref>). Although it is hard to obtain the height of the emitting layer for the RU Lup disk due to its small inclination angle, this unrealistically puffed-up disk scenario – with the emitting layer as high as z/r ∼ 0.8 at r ∼ 50 AU – is at odds with recent observations which instead find the emitting layer of Class II disks to be at z/r ∼ 0.1 for r < 100 AU <cit.>. We note that, in principle, if the height of the emitting layer could be measured as it has been in some disks, then this information could be used to fit the radial and velocity profiles for the Gaussian model. We also find that if we assume the scale height obtained from the VHSE model and repeat the Gaussian modeling for RU Lup, it results in a combination of (M_gas, L_C^18O) similar to the best-fit VHSE model, although the synthetic IR SED is no longer an improved match to the data. While it may be possible to fit all of the observational data using a Gaussian disk model, determining the emission scale height requires very high spatial resolution observations and only works for disks with favorable inclination angles. In their absence, the disk structure parameterization can deviate substantially from reality as we show for RU Lup. On the other hand, the VHSE model is physically motivated, determines the scale height at each radius via coupling of the disk density and temperature structure, and can simultaneously fit the radial and velocity distribution of flux. Hence, we believe it is a more reliable indicator of conditions in the disk. In summary, our VHSE model fits the SED, total line emission, velocity, and radial profiles from recent observations <cit.> with relatively high M_gas and Δ_gd (∼ 30) in comparison with the previously inferred Δ_gd of ∼ 4. Using a Gaussian vertical distribution, our model also derives a similarly high M_gas within a factor of ∼ 2 of the one obtained from the VHSE model. Thus, we conclude that the RU Lup disk is not significantly low in its gas mass and nor has it undergone any substantial change in CO chemistry due to changes in C/H and O/H caused by planet formation processes. We confirm the conclusions of RGH22, and find that optically thin lines provide reasonable estimates of the disk mass. We also note that the RGH22 models compare favorably not only with the fluxes, but also with the ^13CO, CO, and atomic carbon forbidden line [CI] fluxes for a sample of large disks (R≥200 AU) <cit.>, and cold water emission as well (Ruaud & Gorti, submitted). §.§ Comparisons with Literature Values Our work is the first to focus on specifically modeling CO isotopologue emission from RU Lup, and matches the SED, line luminosity, spectrum and radial profile. In this section, we compare our source-specific model with the grids generated in previous works and discuss possible explanations for the different results in gas masses and gas-to-dust ratios. First, we comment on the differences between the VHSE model presented here and those in RGH22. Here our dust-temperature based model gives a factor of ∼ 2 larger L_C^18O compared with the full VHSE thermo-chemical model by RGH22 (all possible results are shown in Figure <ref> magenta regions). This is approximately consistent with the differences found by RGH22 for the dust and gas temperature based modeling, and a similar result of a factor of ∼ 2 difference was found at these M_gas values. We also note a few additional differences. We use similar grain-surface chemistry (as the reduced network was adopted from tests conducted in RGH22), but use the dust temperature to set our gas temperature whereas RGH22 computed the gas temperature. We also do not include settling, while in RGH22 most of the a≳ 100 settles and plays a negligible role in the thermal balance, because the balance is dominated by the small grains that has higher density (Equation <ref>). Another important difference is the dust composition used in our models vs. RGH22. In this work, we adopt a combination of astrosilicate and graphite based on WD01 – that is similar to the dust composition used in <cit.> – while RGH22 used a mix of olivine (76% by volume) and amorphous carbon (24% by volume); more importantly, we construct the dust disk by fitting the SED of RU Lup. How different dust compositions affect the disk structure, temperature and grain-surface chemistry, and how they could be better constrained are out of the scope of this paper and will be the subject of future work. We find similar differences in the models for RU Lup from <cit.>, although the dust composition used in our model is similar to theirs. Their M_dust estimation was derived from the flux at mm-wavelength and not by fitting the SED, but the dust mass estimation of the two models converge to the same M_dust∼ 4×10^-4 M_⊙. However, their best-fit value of M_gas is a factor of ∼ 8 smaller than the VHSE result and ∼ 14 smaller than our Gaussian gas disk model estimate. This can be partially attributed to the fact that the grid of Gaussian disk models used in <cit.> are not tailored to RU Lup. For example, as noted earlier, the scale height parameters adopted impact the inferred line luminosity and therefore the mass estimate. For the range of scale height parameters (together with other free parameters) used in <cit.>, the mass estimates in fact range from 4×10^-3 M_⊙ to 4.8×10^-4 M_⊙. Another contributor is the CO↔CO_2 grain-surface chemistry conversion which is not accounted for in <cit.>; this could bring a discrepancy of a factor ∼ 2-3 as noted by <cit.> and RGH22. We would like to note that there could be other processes at work that may deplete gas-phase CO at the surface, e.g., vertical diffusion of gas into the icy midplane where it may freeze out, although the extent to which this occurs will also depend on the ability of small grains to form and transport ices back into the surface layers <cit.>. However, to correctly consider those processes require a full 2D transport model including the particle dynamics. Such simulations are not suitable for detailed modeling of observational data on individual targets, as they require knowledge of the disk's history; in fact, modeling presented here may help decipher disk conditions at different evolutionary stages from observations and inform the development of theoretical transport models. In summary, the derived M_gas for RU Lup in this work lies between the model grids from RGH22 and <cit.>, see Figure <ref>. Our model is the first one that is specifically built for RU Lup. We also fit the SED, line spectrum, and radial distribution, while both previous models only matched the luminosity using a grid of models which resulted in larger uncertainties on the derived parameters. We thus demonstrate that is a promising tool for modeling individual disks and deriving more robust disk mass estimates. §.§ The Missing IR Emission in VHSE Models As discussed so far, the VHSE model successfully reproduces the observations including the line velocity profile and radial distribution. While the VHSE model presented in this work is capable of fitting the entire SED of the average ∼ 1-3 Myr-old disk <cit.>, and can also match all available continuum photometry of RU Lup beyond 100, it underestimates the infrared emission from the disk of RU Lup by a factor of ∼ 2 between ∼ 2 - 60 (Figure <ref> upper panel). We first check and confirm that this IR continuum underestimation does not affect the gas mass determination from the C^18O (2-1) line. simulations show that the IR continuum emission comes from within a radial distance of 10 AU (see the cumulative dust emission in Figure <ref> lower panel), but the C^18O line emission mostly arises from the outer disk radius (Figure <ref> lower right panel). There is therefore a deficit of dust emission from within 10 AU, indicating a possible missing physical process in our simple disk models. This lack of strong IR emission in VHSE models has also been noted previously, e.g., <cit.> for T-Tauri stars and <cit.> for Herbig Ae/Be stars. Moreover, RU Lup has one of the strongest IR excesses, a factor of ∼ 2 higher than the upper boundary of the Taurus median SED (Figure <ref>), a region of similar age to Lupus <cit.>. One obvious shortcoming of our VHSE models is that we ignore gas thermal processes that are important, especially at the surface of the disk at small radii. Here other heating processes – notable stellar high energy X-ray and UV photons – will heat the gas to higher temperatures. When densities are high, gas and dust are better coupled which leads to more small dust at higher elevations, increasing the IR excess. Another intriguing possibility is that small dust grains (∼ size) are uplifted by a wind in the inner part of the disk ( for a recent review on disk winds). This would lead to hotter dust at a higher scale height and thus increase the IR emission <cit.>. A parametric wind and disk model has previously been used to fit the strong IR excess from an Herbig disk <cit.>. Interestingly, RU Lup has a well-known inner wind detected via optical forbidden lines <cit.>. It is quite likely that the wind (if dense, i.e., n ≳ 10^6-7 cm^-3) can loft small amounts of dust to greater heights and can explain the factor of ∼ 2 deficit in the IR excess we find with the VHSE models. The hypothesis of a wind lifting small dust and increasing the IR excess warrants further exploration. § SUMMARY AND OUTLOOK We developed a dust temperature-based self-consistent vertical hydrostatic pressure equilibrium disk model, , to compute gas disk masses. is a code built on and for the continuum and gas line radiative transfer, respectively; and it includes a reduced chemical network suggested in RGH22 to determine the distribution. With , we introduce a target-based approach to estimate the disk mass in considerable details, where we fit the SED and also the line emission. We further test it on RU Lup, whose disk was previously inferred to have just over a Jupiter-mass gas disk (M_gas∼ 1.5×10^-3 M_⊙) and a gas-to-dust mass ratio Δ_gd of only ∼ 4. We show that our model can match the long wavelength portion of the SED, the total (2-1) and (3-2) line luminosity as well as the (2-1) velocity and radial profiles with an order of magnitude higher mass (M_gas∼ 1.2×10^-2 M_⊙) and gas-to-dust ratio (Δ_gd∼ 30). We also test a Gaussian vertical density distribution that fits the SED better from IR- to millimeter-wavelengths and considers CO↔CO_2 conversion. We find this Gaussian model that can match the line luminosity with even higher gas mass (M_gas∼ 2.1×10^-2 M_⊙) and gas-to-dust ratio (Δ_gd∼ 52) but consider its large vertical height unrealistic. We also find that the VHSE model underestimates the IR SED (λ∼ 2 - 60) by a factor of ∼ 2, which may indicate the need for considering more detailed gas thermal balance in the inner disk and/or an inner dusty wind from RU Lup. With our target-based approach, the RU Lup's estimated disk mass is better-constrained, and it is larger than the Minimum Mass Solar Nebula <cit.>. The larger mass is more in agreement with the young-age, high accretion rate, large disk size, and a lack of strong radial substructures in the disk of RU Lup. Our derived Δ_gd is just a factor of a few lower than the ISM value of ∼ 100. This may indicate the disk has lost some of its gas within 1 Myr, or alternately, that CO is depleted by a factor of few. If the CO is depleted however by a factor of ∼ 10 for RU Lup as suggested by <cit.> for disks in Lupus star forming region – based on the data by and attributed by them to the coupling of physical and chemical processes – then the true M_gas would be as high as ∼ 0.1-0.2 M_⊙. Given that the stellar mass of RU Lup is ∼ 0.2-1.2 M_⊙, such a massive disk would be gravitationally unstable, and we, therefore, consider large depletion factors unlikely. In summary, a better understanding of disk physics and evolution could be achieved by modeling target-by-target and obtaining better-constrained disk masses for more disks of different ages. The procedure of fitting the long-wavelength portion of the SED in combination with the C^18O line emission demonstrated in this work could be easily implemented on other targets with sufficient photometric data. The code is also released <cit.> and available in the public repository[https://github.com/DingshanDeng/DiskMINT], so that the community can extend this approach to other disks. Acknowledgments The authors thank C.P. Dullemond for helpful discussions and assistance on building our wrapper based on , thank J. Barnes, A. Youdin and the anonymous referee for helpful suggestions and comments. DD, IP and UG acknowledge support from the NASA/XRP research grant 80NSSC20K0273 which made this work possible. Support for MR's research was provided by NASA’s Planetary Science Division Research Program, through ISFM work package `The Production of Astrobiologically Important Organics during Early Planetary System Formation and Evolution' at NASA Ames Research Center. aasjournal
http://arxiv.org/abs/2307.01090v1	20230703150910	Streamlined Lensed Quasar Identification in Multiband Images via Ensemble Networks	[ "Irham Taufik Andika", "Sherry H. Suyu", "Raoul Cañameras", "Alejandra Melo", "Stefan Schuldt", "Yiping Shu", "Anna-Christina Eilers", "Anton Timur Jaelani", "Minghao Yue" ]	astro-ph.GA	[ "astro-ph.GA", "astro-ph.CO", "astro-ph.IM", "cs.CV", "cs.LG" ]	Technical University of Munich, TUM School of Natural Sciences, Department of Physics, James-Franck-Str. 1, D-85748 Garching, Germany [email protected] Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), 11F of ASMAB, No.1, Section 4, Roosevelt Road, Taipei 10617, Taiwan Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy Purple Mountain Observatory, No. 10 Yuanhua Road, Nanjing, Jiangsu, 210033, People's Republic of China MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA Astronomy Research Group and Bosscha Observatory, FMIPA, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia U-CoE AI-VLB, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia Quasars experiencing strong lensing offer unique viewpoints on subjects like the cosmic expansion rate, the dark matter profile within the foreground deflectors, and the quasar host galaxies. Unfortunately, identifying them in astronomical images is challenging since they are overwhelmed by the abundance of non-lenses. To address this, we have developed a novel approach by ensembling cutting-edge convolutional networks (CNNs) – i.e., ResNet, Inception, NASNet, MobileNet, EfficientNet, and RegNet – along with vision transformers (ViTs) trained on realistic galaxy-quasar lens simulations based on the Hyper Suprime-Cam (HSC) multiband images. While the individual model exhibits remarkable performance when evaluated against the test dataset, achieving an area under the receiver operating characteristic curve of >97.4% and a median false positive rate of 3.1%, it struggles to generalize in real data, indicated by numerous spurious sources picked by each classifier. A significant improvement is achieved by averaging these CNNs and ViTs, resulting in the impurities being downsized by factors up to 40. Subsequently, combining the HSC images with the UKIRT, VISTA, and unWISE data, we retrieve approximately 60 million sources as parent samples and reduce this to 892,609 after employing a photometry preselection to discover z>1.5 lensed quasars with Einstein radii of θ_E<5. Afterward, the ensemble classifier indicates 3991 sources with a high probability of being lenses, for which we visually inspect, yielding 161 prevailing candidates awaiting spectroscopic confirmation. These outcomes suggest that automated deep learning pipelines hold great potential in effectively detecting strong lenses in vast datasets with minimal manual visual inspection involved. Streamlined Lensed Quasar Identification via Ensemble Networks Andika et al. Streamlined Lensed Quasar Identification in Multiband Images via Ensemble Networks Irham Taufik Andika <ref>, <ref> Sherry H. Suyu <ref>, <ref>, <ref> Raoul Cañameras <ref>, <ref> Alejandra Melo <ref>, <ref> Stefan Schuldt <ref> Yiping Shu <ref> Anna-Christina Eilers <ref>Pappalardo Fellow Anton Timur Jaelani <ref>, <ref> Minghao Yue <ref> ================================================================================================================================================================================================================================================================================= § INTRODUCTION Quasars are fueled by matter accretion onto supermassive black holes (SMBHs) and are among the most luminous objects in the universe, emitting enormous amounts of energy. This attribute makes them ideal probes for studying the distant universe and the physical processes that govern the emergence of the SMBHs and their host galaxies across cosmic time <cit.>. In rare occurrences, the presence of a nearby galaxy in the observer's line of sight can distort the light originating from a distant quasar in the background, resulting in so-called gravitational lensing <cit.>. In the event of strong lensing, where highly magnified and multiple images of the quasar are produced, the mass distribution of the deflectors can be examined by analyzing the observed lens configuration, providing insights into the dark matter profile and the processes that drive the mass assembly of galaxies and clusters <cit.>. Lensed quasars also serve as crucial tracers for understanding the fundamental physics of our universe. For example, the cosmic expansion rate, age, and critical density are related to the Hubble constant (H_0), which can be inferred via lens mass distribution modeling and time delay analysis of lensed quasar images <cit.>. Considering the current tension of H_0 values inferred from the different late and early universe probes, independent methods using the lensed quasars analysis are critically important <cit.>, and more precise measurement is expected with higher number statistics. In addition, lensing could provide flux magnification and increase the effective spatial resolution of the target of interest that would otherwise be too faint (compact) to be detected (resolved). This effect enables us to study quasars with intrinsically lower luminosity along with their host galaxies in unprecedented detail <cit.>. At the time of writing, around 300 lensed quasars have been discovered through various observational techniques, including locating multiple point sources with quasar-like colors or selecting objects with unusual shapes consistent with lensing configurations. In the early days, for example, the Cosmic Lens All-Sky Survey identified many multiply-imaged flat-spectrum radio sources, which are then confirmed as radio-loud lensed quasars <cit.>. Shortly after, in the optical wavelength, the Sloan Digital Sky Survey Quasar Lens Search confirmed a few tens of lenses based on the morphological analysis and color selection of spectroscopically classified quasars <cit.>. Over time, modern wide-field sky surveys can reach deeper limiting magnitudes and deliver great data quality, making it feasible to select more lensed quasar candidates via imaging data alone without the need for spectroscopic preselection. For instance, data mining on photometric catalogs to detect multiple quasar sources allows for the discovery of lensed quasars in numerous projects, such as the Dark Energy Survey <cit.>, the Kilo-Degree Survey <cit.>, and the Dark Energy Spectroscopic Instrument Legacy Imaging Surveys <cit.>. Besides, complementing optical data with infrared photometry and adding astrometric measurements could further reduce the number of false detections <cit.>. Note that despite the high success rates of the previous lensed-finding approaches, they still face a substantial challenge: the inevitability of human involvement in time-consuming and exhaustive visual inspection stages to acquire a final list of candidates with high purity <cit.>. Over the past few years, automated lens-finding methods, either using conventional point sources and lens arcs finder or state-of-the-art machine learning algorithms – e.g., convolutional neural network (CNN), variational autoencoder (VAE), and vision transformer (ViT) – are being explored to reduce human interventions further <cit.>. However, more optimizations are still required since applying these classifiers to real survey data frequently yields samples dominated by false detections. Often, manual visual inspection still needs to be done at the final stage on more than ten thousand candidates recommended by automated classifiers to select high-grade lenses and remove the contaminants <cit.>. This hassle might be caused by the need for more realistic training datasets to improve the classifier performance, coupled with complications caused by the very low fraction (≲10^-3) of strong lenses in all galaxies per sky area <cit.>. Although machine learnings are now widely used for finding galaxy-galaxy strong lenses and performing lens modeling to those systems <cit.>, their use case for lenses containing galaxy-quasar pairs is still limited and has not been explored much. They might have worked well for lensed galaxies since these sources display extended lens arcs that can be distinguished from other astronomical sources. On the other hand, lensed quasars often show only two or more point sources that outshine the light from the lensing galaxy. They are also overwhelmed by visually identical impurities such as binary stars or quasar-star projections. As mentioned earlier, previous lensed quasar searches are proven to be effective, but they might not be efficient enough and not scalable in larger datasets. Specifically, current estimations for the upcoming data from the next-generation surveys such as Euclid <cit.> and Vera C. Rubin Observatory's Legacy Survey of Space and Time <cit.> expect that these projects would expand the number of candidates for strong lenses by at least a few orders of magnitude <cit.>. Therefore, developing a highly efficient, automated lensed quasar selection algorithm is very much indispensable. Here, we develop a novel lens finder using the ensemble of state-of-the-art convolutional and transformer-based neural networks <cit.>. Our classifier is particularly optimized to detect lensed quasars in multiband images of the Hyper Suprime-Cam Subaru Strategic Program <cit.>, extending the selection space to higher redshift ranges that might be missed by previous surveys. Complementing the primary optical data with infrared photometry, we further apply spectral color modeling to obtain lensed quasar candidates with minimal contaminants. This paper is presented as follows. Section <ref> begins with a description of data collection and target preselection using photometric color cuts. Section <ref> explains the simulation for understanding the color and morphology of the galaxy–quasar lens systems. Section <ref> then goes through the specifics of lens detection using automated classifiers, including the datasets utilized for training and evaluating the neural networks. Section <ref> then discusses the classification outputs and the lensed quasar candidates. Section <ref> completes with a summary and our conclusions. Throughout this paper, we employ the ΛCDM cosmological model where Ω_Λ = 1 - Ω_m = 0.685 to simulate lensed images in the training set <cit.>. Note that the resulting lensed images do not depend on the exact value of H_0. In addition, the written magnitudes are reported using the AB system. § DATASET AND PRESELECTION Our lensed quasar hunt comprises two steps: (1) selecting candidates based on their photometric color using multiband data, and (2) calculating the relative likelihoods of the candidates being a lens or contaminant utilizing a machine learning classifier. In this first step, we want to increase the purity of the candidates by restricting the search to objects that we suspect, based on catalog-level photometry, are more likely to be lenses. This approach offers a strategy for efficiently distinguishing the candidates from the majority of the contaminants while requiring the least amount of computer resources. The following section will describe the first part of our search method in more detail. §.§ Primary Optical Photometric Data As the primary catalog in the optical regime, we make use of the wide-layer data of HSC-SSP Public Data Release 3 <cit.>. The observations are conducted by utilizing the Hyper Suprime-Cam mounted on the Subaru 8.2m telescope <cit.>, capturing the sky image of 670^2 wide in five bands (grizy) at the full depth of ∼26 mag (5σ for point sources), a pixel scale of 0168, and seeing of 06–08. Note that if we also account for the partially observed areas, the current data release covers up to approximately 1300 ^2 instead. This larger HSC footprint will be used to construct the parent sample of our lensed quasar candidates selection. To begin the initial selection, we pick all sources detected in the i, z, and y bandpasses with signal-to-noise ratio (S/N) values of more than 3, 5, and 8, respectively. These sources should also have g and r images in the HSC data, but we do not impose any S/N cuts for these bands. As a note, these S/N cut values are derived based on our lens simulation, which will be discussed in the later section. Also, in this case, we adopt the flux and magnitude measurements within the 2 (≈12 pixels) aperture diameter from the HSC [<https://hsc-release.mtk.nao.ac.jp/schema/#pdr3.pdr3_wide.summary>] catalog entries. This table incorporates forced photometry on stacked images containing frequently-selected columns of primary objects. Furthermore, we apply the flags: (1) , and (2) , to retrieve only the sources with reliable photometry. The science images with a size of 72×72 pixels and their corresponding point spread function (PSF) cutouts are subsequently downloaded using the HSC data access tools[<https://hsc-gitlab.mtk.nao.ac.jp/ssp-software/data-access-tools/>]. At this point, 57,464,157 unique sources – i.e., defined as our parent sample – pass our preliminary S/N cuts and flag criteria, implying that a lot of computing power is required to process them all. As additional information, the summary of our selection criteria will be reported in Table <ref>. After that, since most kpc-scale quasar pairs and lensed quasars have separations of ≲3 <cit.>, we try to narrow the selection to sources that show the presence of nearby companions within a 2 radius. We are aware that this choice might be too strict. As an illustration, out of 22 optically bright lensed quasars with spectroscopic confirmation in the HSC catalog <cit.>, we only recover 15 of them via the above preselection – i.e., a recovery rate of 68%. Seven lensed quasars are missed due to the absence of neighboring sources in their vicinity, probably because they are too faint or some failures in the HSC object deblending process. Nevertheless, this method managed to reduce the number of selected objects significantly while keeping many known lenses to be recovered with minimal contaminations and required computational power later. We then crossmatch our parent sample with catalogs of known quasars <cit.>, galaxies/stars <cit.>, strong lenses[ The list of previously published lens systems is compiled from the Master Lens Database (hereafter MLD; <https://test.masterlens.org/>) and the Gravitationally Lensed Quasar Database (dubbed as GLQD; <https://research.ast.cam.ac.uk/lensedquasars/>) ], and brown dwarfs[The brown dwarf catalogs consist of late-M stars plus L and T dwarfs.] <cit.> to identify the spectroscopic classification of these sources when available. In the end, after selecting only sources that: (1) have at least one neighboring source within a 2 radius or (2) have spectroscopic classifications, we managed to reduce the number of objects to only 4,854,831. §.§ Infrared Photometry from Public Surveys The near-infrared (NIR) data is then acquired from the catalogs of the UKIRT Infrared Deep Sky Survey <cit.>, the UKIRT Hemisphere Survey <cit.>, the VISTA Hemisphere Survey <cit.>, and the VISTA Kilo-degree Infrared Galaxy (VIKING) Survey <cit.>. We use here the photometry in the J, H, and K (or K_s) bands, when available. As a note, most of the southern hemisphere is covered by VHS and VIKING, while UKIDSS and UHS capture a large sky area in the north. When we began the candidate selection, UKIDSS and VIKING had completed their observations. However, UHS had only published its J-band photometry, while VHS had only provided its J and K_s-band photometry for most sky regions. As a consequence, the precise photometry accessible for each source is determined by its location in the sky. We also exploit mid-infrared (MIR) observations from the unWISE catalog <cit.>, which contains about two billion objects identified by the Wide-field Infrared Survey Explorer <cit.> throughout the whole sky. With its ≈0.7 magnitudes of deeper imaging data and better source extraction in crowded sky areas, unWISE data surpasses the quality of the predecessor WISE catalog. To combine the HSC data with the compiled infrared catalogs, we use a crossmatching radius of 2 between the sources. Together with the NIR photometry, the MIR W1 (3.4 μm) and W2 (4.6 μm) bands from unWISE are highly valuable for determining if the sources are quasars, stars, or brown dwarfs <cit.>. This crossmatching technique also works for removing unwanted sources, such as cosmic rays or moving sources that are present in one survey but not in others <cit.>. Subsequently, we retrieve the candidates with fluxes of respectively at least 5σ and 3σ in the W1 and W2 bands, as well as having the color of 0.1 < y-W1 < 3.6 and -0.7 < W1-W2 < 0.7, resulting in a remaining 911,263 sources. The NIR color cut is then conducted by keeping the sources with J-band S/N > 3, leaving us only 621,713 candidates. Further cut is made by taking only sources with -0.8 < z-y < 3.9 and -0.2 < y-J < 2.8, which yields 601,277 objects. Note that the criteria we employed so far are derived empirically and managed to preserve 68% previously discovered lensed quasars and 57% known unlensed quasars within the HSC footprint while removing 93% and 28% of contaminating stars and galaxies. Next, we will focus on sources that present the existence of nearby companions within a 2 radius, reducing the number of candidates further to 389,263. Since a lens candidate could have one, two, or more detected companions, we also need to take account of the neighbors around the primary targets, so the total number of sources that will be analyzed at the next stage is 892,609. At the end of this preselection, we are still able to recover all of the 15 known lensed quasars mentioned before. As a reminder, Table <ref> contains an overview of all the selection steps employed up to this point. It is also worth mentioning that all photometric measurements have been corrected from Galactic reddening employing the dust map from <cit.> and with the updated bandpass corrections from <cit.> and <cit.> extinction relation, implemented via the library of <cit.>. § SIMULATING THE LENSES To find lensed quasar candidates based on their multiband images, we need to understand their spectral energy distribution (SED) and morphology – i.e., composed by the addition of lights between the galaxy in the foreground and the quasar in the background. Following some lensing configurations, we can ray trace the lights and produce highly-realistic mock lens images by overlaying the lights of deflected point sources, representing the background quasar emission, over the real deflector images. These mock images will serve as input for the training dataset for building our neural networks model at the later selection step. As a brief illustration, the outline of our simulation workflow is presented in Figure <ref>. More details on (1) the deflector galaxies data retrieval, (2) the mock quasar spectra generation, and (3) the galaxy-quasar lens image production are explained in this part of the paper. §.§ Assembling the Deflector Galaxy Samples We first need to look for a sample of spectroscopically verified galaxies in the Sloan Digital Sky Survey Data Release 18 catalog <cit.>, accessible through the [<https://skyserver.sdss.org/CasJobs/>] website, to assemble the deflectors for our lens simulation. Since the velocity dispersion (σ_v) is a critical metric for computing the lensing effect later, we pick all pipeline-classified “GALAXY” sources and narrow our search to those with the ratio of velocity dispersion to its error of σ_v/σ_v, err > 5 to retrieve samples with accurate measurements. We also exclude galaxies with σ_v ≤ 50 km s^-1 to discard lenses with too small or potentially inaccurate mass. Furthermore, because the bulk of the lensing optical depth for high-z sources originates from the early-type lens galaxies at a redshift of z∼1 <cit.>, we limit our selection to deflectors at z=0.05 to 4 (see Figure <ref> as a reference). The resulting samples are then matched to the HSC catalog, with a radius of search of 1, to get their associated magnitudes and image cutouts when present. As a result, we acquire a sample of 78,619 deflectors dominated by the luminous red galaxies (LRGs) population, peaked at z∼0.5 and σ_v∼250 km s^-1, extending out to z≲1.5. §.§ Generating the Quasar Spectral Colors We proceed now to create simulated quasar emissions by generating a thousand quasar spectra, distributed uniformly at redshifts of 1.5 ≤ z ≤ 7.2 and absolute magnitudes of -30 ≤ M_1450≤ -20 at the rest-frame wavelength of 1450 Å. The simulation is done using the [<https://simqso.readthedocs.io/en/latest/>] module <cit.>, following the prescription of <cit.>. This kind of simulation has been proven to mimic the SDSS quasar colors in high accuracy while also frequently used to assess the completeness of various quasar surveys <cit.>. As a quick summary, the foundation of our quasar spectral model consists of continuum emission, represented by a broken power-law function. The slopes of the continuum (α_ν) follow a Gaussian distribution with mean values of -1.5 and -0.5 for the wavelengths at ≤1215 Å and >1215 Å, respectively, while each of their dispersions is fixed to 0.3. Afterward, the series of iron emissions at the rest wavelengths of <2200 Å, 2200–3500 Å, and 3500–7500 Å are consecutively appended to the model following the templates from <cit.>, <cit.>, <cit.>. The broad and narrow lines are then added to the spectra, complying with the ratio and width distributions of SDSS quasars <cit.>. Furthermore, the mock spectra incorporate the intergalactic medium (IGM) absorption by the Lyα forest in the sightline <cit.>. On top of that, for z≳5.5 quasars, we apply the Lyα damping wing effect based on the theoretical approximation proposed by <cit.>, with a fixed proximity zone size of 3 Mpc and a randomly assigned neutral hydrogen fractions of 0–10% <cit.>. Finally, using the <cit.> model and randomly picked E(B-V) values of -0.02 to 0.14, the internal reddening effect from the dust is applied to the spectra. Note that the negative reddening parameters are for creating quasar models with bluer continua than the original templates can accommodate. The photometry is then estimated from the mock spectra, and the associated errors are calculated using the magnitude–error relations of each survey <cit.>. §.§ Producing the Multiband Images of Lensed Quasars As the next step, we adopt a singular isothermal ellipsoid <cit.> model to characterize the lens mass profile, which is specified by the Einstein radius (θ_E), axis ratio translated into a complex ellipticity, position angle, and image centroid <cit.>. Subsequently, the Einstein radius can be calculated from σ_v enclosed by the gravitational potential using: θ_E = 4πσ_v^2/c^2D_ds/D_s, where the light speed is c, while the angular diameter distances of the lens to source is D_ds and the observer to source is D_s. Given the ratio of distances in Equation <ref>, θ_E is independent of H_0. Nonetheless, for computing each of the distances, a value of H_0 = 67.4 km s^-1 Mpc^-1 is used <cit.>. The SIE axis ratio, centroid, and position angle are then estimated directly by fitting the light distribution of each deflector on its HSC i-band image. Here, we perform the light profile fitting using the combination of elliptical Sérsic and exponential functions implemented in the [<https://pyautogalaxy.readthedocs.io/en/latest/>], an open-source library for investigating the galaxy morphologies and structures in multiwavelength data <cit.>. The external shears are then added at random following a Gaussian distribution with a mean strength of 0 and a standard deviation of 0.058 <cit.>, while the corresponding position angles are selected randomly in the range of 0 to 180 . Next, the simulated lens images are generated by coupling each real galaxy with a mock quasar taken at random. The quasar is then randomly positioned behind the lens within 001 ≤β≤θ_E, where β is the true angular position of the respective source. After that, the source image is projected onto the lens plane, while the magnification and deflection angle are traced based on the lensing structure using the [<https://pyautolens.readthedocs.io/en/latest/>] code <cit.>. We also convolve the deflected quasar lights with the associated HSC PSF model before overlaying them to the original HSC galaxy images. The quasar pairing and placement can be repeated up to 500 times to find a suitable lens configuration. Otherwise, we drop the current deflector and move to the next one. We finally take the mock lens if: (1) it has a strong lensing effect with a magnification factor of μ > 5, (2) the lensed quasar y-band peak flux is detected at ≥5σ against the mean background noise, and (3) its y-band magnitude is >15 mag to exclude unusually bright objects or saturated images. Throughout this simulation, we also exclude systems with θ_E≥5 since the largest Einstein radius detected so far in the cases of galaxy-scale lensing corresponds to that limit <cit.>. At last, we acquire 72,626 surviving lens configurations that fit our criteria from the initial 78,619 deflectors and 1000 mock quasars. Figure <ref> depicts the distribution of the lens galaxy redshifts, velocity dispersions, Einstein radii, and i-band magnitudes adopted in our simulation. We also refer to Figure <ref> for the resulting grz-band color images of the previously created mock lens systems. It is apparent that the redshifts of our deflectors peak at z≈0.5 and extend out to z≲1.5, as mentioned before. Concerning the i-band magnitudes, we witness a spike in the deflector galaxy numbers up to i_HSC≈19.5, followed by an abrupt decrease near the faint end. As a result, our training dataset is weighted toward brighter and larger lens galaxies. This occurrence is mostly produced by how SDSS picks its target galaxies for spectroscopy, which fulfills the guidelines outlined by <cit.> and <cit.> to investigate the universe's large-scale structure. The majority of the targets are luminous elliptical galaxies at z<1, which are ideal tracers for studying the baryon acoustic oscillation signal and, hence, the expansion of our cosmos <cit.>. The magnitude boundaries for galaxies selected for spectroscopic surveys are i = 19.9 for the SDSS III and i = 21.8 for the SDSS IV projects <cit.>. § LENS FINDING WITH DEEP NEURAL NETWORKS The second step of our lensed quasar search strategy involves supervised, deep neural network classification, requiring realistic training datasets as inputs to function. CNNs, for example, have been demonstrated to be successful in pattern recognition, such as discovering gravitational lenses in enormous sets of data <cit.>. While the exact design of the CNNs is generally determined by the challenge at hand, it typically consists of images as data inputs, which are subsequently processed by a sequence of convolutional, fully connected, and output layers. In this part, we describe our automated classifier that has been trained to distinguish lensed quasars from non-lensed sources. The section that follows will discuss the principles of our approach. §.§ Preparing the Input Data The inputs used to train our classifier will be divided into four categories: (1) the mock lensed quasars created in the previous section, (2) the real HSC galaxies that are not picked for the lensing simulation <cit.>, (3) previously discovered quasars from the local universe up to z∼7 <cit.>, and (4) a sample of stars and brown dwarfs <cit.>. Here, the distribution is balanced so that each class contains around 60,000 objects, and in sum, we use approximately 240,000 sources. Note that the images utilized for the training inputs have been built based on the grizy bands of HSC cutouts with a size of 72 pixels on a side, which is comparable to an angular dimension of ≈12. Next, the images are min-max adjusted so that the fluxes vary from zero to one and are square-root stretched to boost features with low fluxes and enhance the visual appearance. The relative pixel brightnesses across bandpasses are maintained, and therefore the colors of the associated sources are retained. The images are subsequently augmented with random ± π/2 rotations, 7-pixel translations, and horizontal or vertical flips on the fly each time they are called for training. This strategy will expand the quantity of training data while increasing the possibility that the network will properly categorize several perspectives of the same image. §.§ Network Architectures Ensemble networks, which combine the predictions of multiple classifiers (e.g., CNNs or ViTs), have been shown to outperform individual models in various machine-learning tasks <cit.>. There are several reasons why this approach is often superior. First, we can leverage the diversity of individual models. Each classifier in the ensemble is trained with a different initialization, architecture variation, or data augmentation scheme, leading to diverse learned representations. By combining these diverse models, the grouped networks can capture a broader range of patterns and variations in the data, improving overall generalization performance. Second, ensemble networks can reduce overfitting, mitigate the impact of individual model biases or errors, and offer improved performance stability. By averaging the predictions of multiple classifiers, they compensate for these biases and reduce the impact of individual errors, resulting in more reliable and robust predictions. Therefore, several network models will be discussed in this section to assemble our ensemble network architecture. §.§.§ Baseline Convolutional Network We start with creating a simple CNN as a baseline, dubbed as BaseNet, following the same model presented by <cit.> and motivated by other classical network designs <cit.>. BaseNet has three convolutional layers containing kernels with sizes of 3×3× C, with C=32, 64, and 64 for the first, second, and third layers, respectively, along with a stride of 1×1 and the same padding[ Same padding refers to the padding of additional rows and columns of zeros around the input image in such a way that the output feature map has the same spatial dimensions as the input. ]. Each convolutional layer is then followed by a max pooling with the stride of 2×2, the size of 2×2, and the same padding. At the end of this sequence, a fully connected layer of 128 neurons is appended, and the final output layer is attached to retrieve four outputs – i.e., the chances of a target being a lensed quasar, a galaxy, an unlensed quasar, and a star. The dropout regularizations are utilized everywhere, where the drop rates are set to 0.2 and 0.5 for convolutional and fully connected layers, respectively. At the start, the learning rate is adjusted to 10^-4 while the bias and weight of each neuron are set randomly and subsequently updated during the training. The activation functions based on the Rectified Linear Unit (ReLU) are utilized throughout the networks, except for the output layer, which employs the softmax activation. The [<https://www.tensorflow.org/>] deep learning platform is used to carry out all training procedures and CNN modeling <cit.>. After establishing the baseline network, additional classifiers will be added to the ensemble model, and an overview of each network architecture will be discussed. However, here we only provide a concise, high-level understanding and comparisons of the network architectures while acknowledging that in-depth information, technical specifications, and implementation details can be found in the respective cited references <cit.>. §.§.§ Residual Learning Network Residual Network, which is often abbreviated as ResNet, is a deep CNN architecture familiarized by <cit.>. The core idea of ResNet is based on the observation that deeper networks could suffer from diminishing performance or even degradation due to vanishing/exploding gradients. The introduction of skip connections bypasses certain layers, allowing the network to learn residual functions or the difference between the input and the desired output. This strategy enables the model to focus on learning the residual information, which is often easier to optimize. Further progress is made with the introduction of ResNetV2, which follows the original ResNet designs but integrates the bottleneck residual blocks <cit.>. It also presents the concept of identity shortcuts to handle the skip connections, enhancing the overall network performance. Another variant of of this family is ResNetRS, which incorporates the Squeeze-and-Excitation (SE) modules into the residual blocks, aiming to capture channel-wise dependencies and recalibrate the feature maps adaptively <cit.>. By selectively amplifying informative features, these modules enhance the representation capacity of the network and improve its discriminative ability. Here, ResNet50V2 and ResNetRS50, which have 50 layers, will be picked as our choices for building the ensemble model components. This starting point is also chosen since deeper ResNet with 101 or 152 layers did not improve classification performance for the tiny image cutouts we studied. §.§.§ Inception Network Inception is an architecture that is developed by <cit.> to handle the challenges of effectively capturing multi-scale features and reducing computational complexity in deep neural networks. The central notion behind the Inception model is to employ parallel convolutional filters of different sizes within a single layer, allowing the network to grasp features at different spatial scales. These filters are usually composed of 1×1, 3×3, or 5×5 convolutions, along with a 3×3 max pooling operation. By combining these different-sized filters and pooling operations, Inception enables the network to apprehend both fine-grained details and broader contextual information simultaneously. The Inception family has undergone several improvements over time, leading to subsequent versions such as InceptionV3, Xception, and InceptionResNetV2 <cit.>. These variants, which we will utilize in this work, incorporated additional design elements, including factorized convolutions, depthwise separable convolutions, batch normalization, or residual connections, to further enhance the gradient flow and training stability. §.§.§ Neural Search Architecture Network Neural Architecture Search Network, or NASNet, is a category of CNNs that revolutionizes network modeling, developed by <cit.>. Its ability to automatically discover high-performing architectures has remarkably reduced the need for human expertise and computational resources in the design process. The idea behind NASNet is to utilize a reinforcement learning-based search algorithm to explore a vast search space of potential models. NASNet is composed of diverse essential elements. The main building block is a “cell” structure, represented by a directed acyclic graph, which captures the connectivity pattern of the neural network. The search algorithm learns to discover the optimal cell structure, which is then repeated multiple times to construct the complete network. A notable characteristic of NASNet is the incorporation of skip connections, or residual connections, within its cell structure. These connections allow for seamless information flow and facilitate gradient propagation during training, leading to more effective learning. Additionally, NASNet introduces reduction cells that downsample feature maps spatially, enabling the network to handle larger input images and capture high-level spatial information. Here, we consider a smaller variant of the NASNet family called NASNetMobile. §.§.§ MobileNet MobileNet is a class of lightweight CNNs designed for mobile and embedded devices. MobileNetV3, developed by <cit.>, brings several critical improvements over its predecessors – i.e., MobileNetV1 <cit.> and MobileNetV2 <cit.> – to deliver enhanced performance while maintaining efficiency, catering to the limitations of resource-constrained devices. The architecture of MobileNetV3 incorporates an efficient backbone structure consisting of various lightweight layers, including depthwise separable convolutions, pointwise convolutions, and linear bottleneck layers. SE blocks are also employed to recalibrate channel-wise features adaptively, emphasizing their importance and improving the model capacity. In addition, the use of hard-swish activation functions introduces non-linearity while keeping the computational cost low. By leveraging these components, MobileNetV3 reduces complexity while retaining the ability to capture crucial visual features. Additionally, the adoption of neural architecture search (NAS) allows it to automatically discover the optimal architecture through an algorithmic exploration of a vast search space. MobileNetV3 is available in different versions, and in this case, we employ the MobileNetV3Large, aiming to achieve higher accuracy with a sacrifice of slightly larger model size and calculation requirements compared to its smaller variant. §.§.§ EfficientNet EfficientNet is a family of CNN architectures presented by <cit.> that has demonstrated outstanding performance across various computer vision tasks, including image classification, object detection, and semantic segmentation. The EfficientNet models are designed to attain cutting-edge performance while being computationally efficient, requiring fewer parameters and computations compared to other architectures. The pivotal innovation behind EfficientNet is the concept of compound scaling, which uniformly scales the network's depth, width, and resolution, allowing for an optimal trade-off between model size and performance. The EfficientNet models also employ other techniques to enhance performance, such as the use of mobile inverted bottleneck convolutional (MBConv) layers and a compound coefficient for controlling the number of channels in each layer. Further improvement is then proposed by <cit.> by introducing a new convolutional operation called “Fused-MBConv”, combining depthwise separable convolutions with inverted bottleneck residual connections. These approaches additionally improve the efficiency and effectiveness of the models. Here, we adopt the simplest form of EfficientNet, represented by EfficientNetB0 and EfficientNetV2B0. Note that other variants of this family (B1–B7, S–XL, etc.) introduce progressive scaling to increase the model sizes and complexity. §.§.§ Regularized Network Regularized Network, hereafter RegNet, is a family of CNNs introduced by <cit.> to address the challenge of network scaling by promoting a design principle that improves both accuracy and efficiency, utilizing adaptive regularization of weights and adjustment of scaling coefficients. This approach applies regularization proportional to the magnitude of network weights, which helps prevent overfitting and improves generalization performance <cit.>. RegNet architectures also comprise a channel-wise group convolution technique, effectively reducing the computational cost without significantly sacrificing accuracy. We will implement here two of the smallest RegNet variants. The first one is RegNetX002, which emphasizes the model depth as the primary scaling factor, has deeper layers, and aims to capture more complex patterns in the data. The second one is RegNetY002, which prioritizes the model width as the scaling factor, has a wider network and seeks to capture more fine-grained details in the data with increasing feature diversity. §.§.§ Vision Transformer Vision Transformer, shortened as ViT, is a state-of-the-art deep learning architecture introduced by <cit.>. It brings the powerful transformer-based architecture, originally designed for natural language processing, to the field of computer vision. It also represents a significant departure from traditional CNNs by relying solely on self-attention mechanisms without any convolutional layers. The principal concept of ViT is to treat an image as a sequence of patches, where each patch is regarded as a token. These image patches are flattened and fed into a transformer encoder consisting of multiple stacked self-attention layers and feed-forward neural networks. The self-attention mechanism allows the model to capture global dependencies and relationships between different patches in the image, enabling it to learn contextual information and high-level representations. Since the transformer architecture does not inherently encode spatial information, positional encoding is introduced to provide the model with the relative positions of the image patches. This information helps the model understand the spatial structure of the image and retain spatial relationships between different patches during the self-attention process. However, ViT has shown to be more data-hungry, meaning it typically requires larger amounts of labeled training data to achieve competitive performance compared to CNNs. This phenomenon is partly due to its reliance on self-attention mechanisms and the challenges in capturing fine-grained spatial details. The combination of ViT with Shifted Patch Tokenization (SPT) and Locality Self-Attention (LSA) is an alternative modification that aims to improve the efficiency and effectiveness of ViT models <cit.>. SPT shift the image patches by half of their size horizontally and vertically. This method improves the alignment between the patches and the objects in the image, enhancing the model's ability to capture accurate spatial information. On the other hand, LSA restricts the network attention to a local neighborhood of tokens instead of attending to all of them, significantly reducing the computational cost while still capturing relevant contextual information. Hence, we will use the original ViT model (dubbed as ViT-Vanilla) and ViT with the implementation of SPT and LSA (named as ViT-Lite) for constructing our ensemble network. §.§ Training the Network Models The CNN and ViT models discussed before need to be separately trained first. Once the optimum parameters for each classifier are obtained, we will combine them to construct the ensemble networks. This strategy aims to improve predictive power by averaging the forecasts of multiple instances into a unified and robust decision-making framework. One shortcoming of this technique is that each model contributes the same proportion to the ensemble forecast, irrespective of how well the network performs. A weighted average ensemble is a version of this strategy that weights the role of every ensemble member by its performance on the test dataset. This scheme allows high-performing classifiers to contribute more while low-performing models influence less. This technique may be further generalized by substituting the linear weighted sum model used to integrate the sub-model predictions with any learning algorithm and correspondingly known as layered generalization or stacking. A stacked generalization ensemble, as compared to a weighted average ensemble, can utilize the set of forecasts as a context and dynamically select how to weigh the input predictions, possibly leading to higher performance <cit.>. However, for simplicity and considering that: (1) manually assigning each of the model contributions in the weighted average ensemble is not straightforward and (2) the stacked generalization ensemble usually needs more independent datasets for its training to prevent overfitting, we will use the model averaging ensemble without the weighing approach instead, as shown in Figure <ref>. Training the network models involves several steps, including data splitting, batch subdivision, and iterative forward and backward propagation. In this case, the input data is split into training, validation, and test datasets using a ratio of 70:20:10, allowing for proper evaluation and testing of the trained models. The dataset is further subdivided into smaller batches consisting of 128 samples to facilitate efficient computation by processing a subset of the data at a time. During the training process, the models undergo iterative forward and backward propagation. Forward propagation involves passing the input data through the network and generating predictions. These calculations are then compared to the ground-truth labels using the sparse categorical cross-entropy[<https://www.tensorflow.org/api_docs/python/tf/keras/losses/SparseCategoricalCrossentropy>], a loss function that is often used for multilabel classification tasks. The backpropagation step estimates the gradients of the loss function, which are then utilized to update the corresponding weights and biases employing a stochastic gradient descent technique <cit.>. This optimization process allows the models to iteratively adjust their parameters, improving their predictions and overall performance. The training and validation losses need to be monitored to check whether the model is able to learn or if overfitting occurs – i.e., where the network becomes too specialized to the training data and fails to generalize well to new, unseen data. In an attempt to prevent overfitting and achieve more accurate predictions, we randomly shuffle the training and validation data after each epoch. At the start, the learning rate is adjusted to 10^-4 while the bias and weight of each neuron are set randomly and subsequently updated during the training. When learning becomes stagnant, the network models often benefit from reducing the learning rate by a factor of 2–10. Due to this reason, we apply a callback function, which lowers the model learning rate by a factor of ten if the loss curve shows a plateau for five consecutive epochs. After numerous epochs, we stop the training if the lowest average validation loss over multiple runs is reached, or to put it another way, early stopping is applied if the loss difference fails to decrease below 10^-4 over ten consecutive epochs. Typically, the training cycle could reach 50 to 100 epochs before the optimization is converged, and the classifier performance can not be improved further (see Figure <ref> as reference). The best model is then stored, which corresponds to the combination of the weights and other parameters that generate the lowest cross-entropy loss tested in the validation dataset. §.§ Classifier Performance Evaluation Each of our classifiers will return the probability estimates for an individual tensor that is passed into the networks, indicating whether the respective images contain a lensed quasar, an unlensed galaxy, an ordinary quasar, or a star – i.e., P_lens, P_galaxy, P_quasar, and P_star – where the sum of this probabilities equals to unity. The predicted category is then allocated by selecting the class with the highest likelihood score. It is worth noting that P_lens = 1 indicates that there is a strong probability that the categorized images include a lensed quasar. P_lens = 0, on the other hand, indicates that the cutouts do not comprise a lensed quasar and are more likely to contain contaminating sources. As commonly perceived, the excellent accuracy achieved in the training process might have been attributed to overfitting. Naturally, we then examine the accuracy-loss learning curves obtained by assessing network predictions on validation and training datasets (see Figure <ref> for an example). The training and validation losses are settling and maintaining the same trends after declining for numerous epochs. The absence of any overfitting signals – i.e., the training loss stays to decline while the validation loss starts rising after several epochs – gives us confidence that our classifier can generalize and learn well. In order to evaluate the overall performance of the trained models further, another commonly used metric is the receiver operating characteristic (ROC) curve. The area under the ROC curve (AUROC) provides insights into how effectively a binary classifier distinguishes between two classes as the decision threshold is adjusted. Therefore, to utilize the ROC curve and calculate the AUROC, we mark lenses as the positive (P) and non-lenses or contaminating sources as the negative (N) cases. True positives (TP) are instances in which the model properly predicts the lenses, distinct from true negatives (TN), which are accurate identifications of non-lenses. False positives (FP) emerge when the classifier wrongly labels contaminants as lenses. Finally, false negatives (FN) are occasions in which the model incorrectly rejects lenses. The ROC curve compares the false-positive rate (FPR) to the true-positive rate (TPR) for the unseen test dataset, where: TPR = TP/P = TP/TP + FN; FPR = FP/N = FP/FP + TN. The ROC curve is then made by gradually raising the probability cutoff from 0 to 1. This result in AUROC = 1 for a flawless classifier, while AUROC = 0.5 for a classifier that only predicts randomly. Since we have four categories for classifying the candidates (i.e., a multilabel classification), we have to binarize each network's prediction using the so-called “one versus all” framework. As a result, we generate four ROC curves constructed based on the evaluation of the classifier using a previously unseen test dataset and display them in Figure <ref>, which encompasses the following scenarios: (1) distinguishing lensed quasars from galaxies and other point-source contaminants, represented by the solid blue line; (2) discriminating galaxies from lens systems and other contaminating sources, symbolized by the dashed magenta line; (3) separating stars from other sources, illustrated by the dashed yellow line; and (4) classifying quasars with respect to lenses, galaxies, and other sources, expressed by the dashed cyan line. Notably, these curves exhibit high AUROC values, indicating the exceptional performance of the classifier in these scenarios. We then employ the geometric mean or G-mean[The definition is G-mean = √(TPR× (1-FPR))] metric to find a balance of the TPR and FPR ratios based on the ROC curves. The highest G-mean score shows an ideal P_lens threshold for maximizing TPR while reducing FPR. In this situation, the reasonable P_lens and the resulting FPR and TPR for each network model and the combined classifiers are reported in Table <ref>. It is noteworthy to mention that below the previously mentioned P_lens limit, the amount of candidates increases rapidly while their quality declines, which means that the visual assessment required at the next stage will be more tedious and less practical. Concerning the compromise involving completeness and purity, we have to strike a balance in which the quantity of candidates is reasonable for follow-up observations at the next step. § RESULTS AND DISCUSSION §.§ Final List of Lens Candidates After the catalog-level preselection explained before and employing our ensemble network classification, we obtained 3991 surviving targets with P_lens > 0.3. To further remove the contaminating sources from this list, we consider the relevant astrometric excess noise <cit.> and proper motion significance <cit.> parameters when the astrometry of these lens candidates is available in the Gaia Data Release 3 catalog <cit.>. Note that only about 30% of our candidates have this astrometric information. A high value of AEN (≳10 mas) could point to a potential star-forming galaxy, while a significant PMSIG number (≳10σ strongly implies that the system includes a star <cit.>. These additional criteria manage to select 3423 sources (2308 unique systems/groups), which are then visually inspected, yielding 161 candidates with high lens probabilities. We further split these candidates into A and B grades. Grade A is assigned when the cutout exhibits a definite configuration of strong lensing, even without the assistance of a higher-resolution image. This case implies the presence of multiple-imaged sources or the indication of a counter-image, along with the existence of a possible lens galaxy. On the other hand, grade B means that the candidates display a potentially lensing-like configuration, although visual identification of multiple images is not feasible. This category includes scenarios where multiple objects or a single arc-like object are positioned on one side of the central galaxy without a clear counter-image visible on the opposite. Note that our compilation of candidates encompasses not only lensed quasars but also galaxy-galaxy lenses. These strongly lensed galaxies are not included explicitly in the training dataset. However, the lensed point-source lights from the multiply-imaged quasars sometimes could mimic the extended arcs of the galaxy-galaxy lens systems and confuse our ensemble classifier. Therefore, more information beyond the optical images is required to discriminate between these systems, and we will attempt to check their SEDs later (see Appendix <ref>). In addition, we provide the complete list of our lens candidates in Table <ref> of Appendix <ref>. Based on the evaluation applied to the test dataset and reported in Table <ref>, our ensemble classifier appears to have an FPR as low as 1.3% for recognizing lensed quasars at z = 1.5–7.2. Yet, how many lensed quasars that we expect to discover? Using the latest estimate from <cit.>, assuming an i-band 5σ depth of ≈26 mag, and sky coverage of around 1300^2, we expect to find about 153 lensed quasars, including 13 quadruply-imaged sources, within the HSC wide-layer footprint. Approximately 80% (50%) of these systems have a separation greater than 05 (1). We remark that this number is about two to three times lower compared to the earlier model from <cit.>, primarily caused by the discrepancy in the details of the simulated quasars. adopt a steeper faint-end slope of the quasar luminosity function, do not impose an absolute magnitude cut, and disregard redshift evolution of the deflector velocity dispersion function <cit.>. On the other hand, an improvement from is made by considering quasars with i-band absolute magnitude of M_i<-20 and employing a new VDF that decreases with redshift <cit.>. While our current strategy appears to yield reasonable candidate samples, we believe there is an opportunity for optimization. The occurrence of FP-classified sources is, at present, not zero. Fortunately, trained astronomers can swiftly rule out the spurious sources in this candidate list, as presented in Figure <ref>. Based on our visual inspection, they are highly improbable to be lenses or have no apparent indicators of strong lensing characteristics. Moreover, they have a variety of visible forms, such as irregular galaxies, spiral arms, or groups of multiple sources that imitate lensing arcs. Aside from that, sources with unusual morphologies that do not belong to either category in the data used for the training can get unexpected network-based classification scores. Due to these reasons, our network models have been trained iteratively by including the sample of identified FPs to keep improving the classifier performances. §.§ Selection of High-redshift Lenses Lensed quasars at z≳6 is another intriguing case we want to explore because so far, only 2 lenses have been found at this distance among ≈300 known quasars throughout the whole sky <cit.>. Subsequently, we expect to discover at least 2 lensed quasars at z≳6 in the current dataset if we consider the high-z lens fraction of ≈1% <cit.> among ∼150 quasars that have been found within the HSC footprint <cit.>. However, this estimate could be much larger, depending on the chosen model. For example, <cit.> suggest a lens fraction of >4%, which results in the expected number of lenses of more than 6. This tension might be the result of unaccounted-for biases that have not been thoroughly examined, in particular, the disparity between the adopted quasar luminosity function and deflector VDF used by and . In retrospect, the reasons why many lenses are left undiscovered might be very evident. Most methods to select quasar candidates have included extra magnitude cuts or full “dropout” criteria at all bandpasses bluer than the Lyα emission <cit.>. This approach makes sense because the light emitted by z≳6 quasars at wavelengths shorter of Lyα is severely absorbed by the foreground IGM, forming a prominent break in the spectrum and thus a primary marker for the quasar preselection. To put it another way, we are not anticipating any substantial flux emanating from the g or r bands for quasars at these redshifts. This characteristic, however, is not true if the lens galaxies exist at 0.1 ≲ z ≲ 1.5, which might produce substantial emission at the observed wavelengths of λ_obs≲ 8000. Testing our methodology against prior quasar selection approaches validates the expanded selection space. If we imposed an extra cut of S/N(g, r) < 5, or the dropout criterion, none of the mock lenses generated in Section <ref> would survive. In simpler terms, we would overlook all lens systems featuring luminous galaxies as deflectors. Hence, to include prospective lens systems, we have eliminated such dropout requirements in our photometric preselection. Instead, we exploit the whole spatial and color information by using the associated multiband images and process them with our deep learning classifier. The intersection with earlier search methods occurs only when the deflectors are dim and, as a result, less massive. These configurations produce compact lenses with small θ_E and image separations. Therefore, these systems are expected to have lights dominated by quasar emission, with a slightly extended shape, at least in ground-based images. When they are below a certain lensing mass, standard search techniques will pick them up, albeit their lensing natures will be challenging to show. Obviously, executing this diagnostic using higher-resolution, space-based imaging would significantly expand the accessible parameter space. Accordingly, as an effort to: (1) detect compact, faint lenses with the light dominated by the bright z≳6 quasars, (2) identify quasars in the cases that they are well separated (≳05) from the associated foreground deflectors, and (3) recognize unwanted sources (e.g., binary stars and quasar-star pairs) solely using their catalog-level photometric information, we also implement a conventional object classification using the SED fitting method. The goal is to pinpoint candidates of quasars, galaxies, and stars based on their multiwavelength data, estimate the associated photometric redshifts, and reassess our final list of targets. We refer to Appendix <ref> for a detailed explanation of this procedure. However, within the current dataset and using the current selection method, we could not find any new z≳6 quasars, and all of our lens candidates are located at lower redshifts. In the future work, we want to relax the present photometric preselection to discover high-z lenses. §.§ Quasar Selection Completeness As previously stated, the parent population of quasars in our simulation follows a uniform distribution within the redshifts of 1.5 ≤ z ≤ 7.2 and absolute magnitudes of -30 ≤ M_1450≤ -20. Without the contribution of strong gravitational lensing, our ensemble network classifier can only discover quasars with M_1450≲-22 at z≳6. Fortunately, the lensing event could shift this boundary towards a lower luminosity territory, depending on the factor of magnification values. To examine this in more depth, we initially define our selection function (or completeness) as the proportion of simulated quasars with specific M_1450, z, and intrinsic SEDs that are successfully identified by our selection criteria. The outcomes are displayed in Figure <ref>. Quasars with inherently low brightness can only be detected if they experience a substantial magnification boost from gravitational lensing, and such occasions are relatively rare. Therefore, as the quasars possess inherently lower luminosity, our completeness rate diminishes at a given redshift. As an extra note, since in Section <ref> we exclude sources with y-band magnitude >15 mag to discard unusually bright objects or saturated images, we missed all intrinsically luminous quasars with M_1450≲ -27 at low redshifts (z≲3), which might not even exist in the real universe. Then, looking at Figure <ref>, our classifier could recover most of the lensing configurations without a significant bias. A slight decrease in recovery rate is apparent in the cases of systems with deflector redshifts of z_gal≳1 and magnitudes of i≳21, which might be caused by farther lens galaxies having fainter apparent fluxes and more challenging to be detected. Though these sources could still be identified if the lensed quasars are well separated, these phenomena can be exacerbated for the lens systems that are too compact due to small lens masses. For example, we notice a slight drop in lens recovery rate in Figure <ref> for systems with σ_v ≲ 100 km s^-1 and θ_E≲ 05. §.§ Evaluation with Independent Datasets We also conduct a supplementary test with an alternative dataset assembled using the list of known quasars compiled in the GLQD <cit.>. In this case, HSC images are available for about 22 of the 220 lenses in the database (see Figure <ref>). To evaluate the completeness and purity of our ensemble classifier, we first combine these known lenses with a sample of contaminants compiled in Section <ref>, which includes galaxies, stars, and quasars, with the non-lensed sources expected to outweigh the lens population by a factor of a few thousand. As an outcome, our model correctly identifies 15 known lensed quasars, resulting in a TPR (or completeness) of 68.2% and an FPR of 1.3%. In addition, we discover that the classifier has a purity[ Assuming that the fraction of lenses among all sources in the universe is in the order of S=10^-3, or corresponds to 1 lens per 1000 objects, we then define the purity as AP = TPR × S / (TPR × S + FPR × (1-S)). ] of 6.9% in identifying the lens candidates. Note that other individual model explained in Section <ref> demonstrates impressive performance when assessed against the test dataset, with an AUROC exceeding 97.4% and a median false positive rate as low as 3.1%. However, they still encounter difficulties in generalizing to real-world data. This thing is evident from the presence of numerous spurious sources identified by each classifier. For example, ResNetRS50 and InceptionV3 are best-performing classifiers, recovering 19 and 20 known lensed quasars, respectively – i.e., a completeness of 86.4–90.9%. Yet, these networks give high scores to more than a hundred thousand sources, making the required visual inspection for the resulting lens candidates at the later stage extremely time-consuming. Better selection purity is achieved by ViT-Vanilla and ViT-Lite, which have a completeness of 68.2–72.7% while keeping the number of lens candidates in the order of ten to twenty thousand. Then, a much better improvement is uncovered by ensembling the predictions of all of the CNNs and ViTs via model averaging, leading to a reduction in impurities by up to a factor of ≈40. This ensemble network manages to reduce the number of candidates to just a few thousand while maintaining the completeness of 68.2%. When evaluated against real lens systems, the model seems to perform worse than when tested against simulated lenses, which is indicated by the declining TPR from 97% to 68% (see Table <ref>). This lower performance is somewhat expected and might be attributed to the uniqueness of some of the lens systems that are not taken into account by our simulation. Some missed lenses might contain arcs or counter images that are too dim to be recognized, contamination of the bright deflector lights, compact lenses, saturated images, or other factors. Still, our classifier can generalize and obtain a high enough accuracy for our objectives, where the purpose of our network-based classification is to reduce the false candidates as much as possible before proceeding with the visual inspection and compiling the final list of targets. In addition, to assess the flexibility of our ensemble model when applied to the next-generation ground-based survey data, we perform one more test using a mock dataset that closely resembles the LSST photometry <cit.>. This dataset, kindly provided by <cit.>, encompasses 3628 strongly-lensed quasars at z≳5. Employing our ensemble model directly, we successfully identified 3095 systems of the parent sample, yielding a completeness of 85.3%. We expect that applying transfer learning by retraining our classifier on a subset of this dataset could increase the network performance further <cit.>. This outcome illustrates the adaptability and robustness of our ensemble model, showcasing its ability to excel in the upcoming LSST data. § SUMMARY AND CONCLUSION In this paper, we conduct a systematic hunt for lensed quasars at 1.5 ≤ z ≤ 7.2 by exploiting the HSC, UKIRT, VISTA, unWISE, and Gaia data. Our approach is divided into two key stages. First, we use catalog-level information to preselect the candidates based on their photometric color, decreasing the number of sources from ∼60 million to only 892,609. Second, we use an ensemble of CNN and ViT classifiers to assess the relative likelihood of each source being a lens or contaminant, yielding 3991 prevailing candidates. It is worth noting that the training input is created by overlaying deflected point-source lights on the images of real HSC galaxies. This strategy allows us to generate realistic strong-lens simulations and concentrate on identifying systems within the Einstein radii of θ_E<5. We then obtain 161 newly found lens candidates after inspecting their astrometric data when available and visually evaluating the objects with the lens probability of P_lens > 0.3. These findings indicate that automated neural network-based classifiers, with minimal human involvement, are promising for spotting lensed quasars in big datasets. The technique presented in this paper is readily applicable to seeking out galaxy-quasar lenses across a wide range of redshifts. It also appears to be suitable for next-generation surveys such as Euclid <cit.>, which will provide high-resolution NIR imagery over a large portion of the extragalactic sky, and Rubin Observatory Legacy Survey of Space and Time <cit.>, which will have extensive optical multiband data. In this case, modifications to the bandpass profiles, seeing values, and image scale will be essential to obtain optimal results. In addition, adopting more complex galaxy mass profiles beyond the SIE model might also help to enhance the classifier performance. To fully exploit the scientific potential of our catalog of lenses, it is essential to conduct spectroscopic observations to confirm the redshifts of the deflectors and sources, along with high-resolution imaging to perform accurate lens modeling. The authors thank James Chan for his valuable contributions and constructive discussions, which have significantly improved the quality of this manuscript. This research is supported in part by the Excellence Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2094 – 390783311. SHS thanks the Max Planck Society for support through the Max Planck Fellowship. SS acknowledges financial support through grants PRIN-MIUR 2017WSCC32 and 2020SKSTHZ. YS acknowledges the support from the China Manned Spaced Project (No. CMS-CSST-2021-A07 and CMS-CSST-2021-A12). ATJ is supported by the Program Riset ITB 2023. This work is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, operated by the Subaru Telescope and Astronomy Data Center at the National Astronomical Observatory of Japan. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan, Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), the Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at <http://dm.lsst.org>. This project has included data from the Sloan Digital Sky Survey (SDSS). Funding for SDSS-IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is <https://www.sdss.org/>. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration. The unWISE catalog utilized in this paper is based on data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. We acknowledge the use of the VHS, VIKING, UKIDSS, and UHS data. This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. Facilities. ESO:VISTA (VIRCAM), Gaia, Sloan (eBOSS/BOSS), Subaru (HSC), UKIRT (WFCAM), WISE. Software. Astropy <cit.>, EAZY <cit.>, Matplotlib <cit.>, NumPy <cit.>, Pandas <cit.>, PyAutoLens <cit.>, SIMQSO <cit.>, TensorFlow <cit.>. aa § ELIMINATING SPURIOUS SOURCES WITH SPECTRAL MODELING We implement SED fitting as an additional practice to discover unresolved lens systems with the light dominated by the background quasar and to remove spurious sources in our lens search. In principle, we want to separate the candidates of quasars, galaxies, and stars based on their multiwavelength data and estimate the associated photometric redshifts. Therefore, the [<https://github.com/gbrammer/eazy-py>] module, a Pythonic photometric redshift tool based on <cit.>, will be used to implement the SED modeling. The way it works is by going across a grid of spectral templates, matching them to the photometry of the targets, and trying to discover the best model. Here, we select the best models with the lowest reduced chi-square (χ^2_red) as solutions. The candidates with a high likelihood of being quasars are chosen based on the derived χ^2_red of the quasar (χ^2_red,Q), galaxy (χ^2_red, G), and star (χ^2_red, S) templates along with their associated ratios. Hence, the sources which are best fitted with models of stars or brown dwarfs will be removed from our list of lens candidates. To establish the templates, we first compile the brown dwarf spectra from the SpeX Prism Library[<http://pono.ucsd.edu/ adam/browndwarfs/spexprism/library.html>] <cit.>. This database contains 360 spectra of M5–M9, L0–L9, and T0–T8 stars with wavelength spanning from 0.625 μm to 2.55 μm. Following the prescription of <cit.>, we then extend these templates into the wavelength covered by unWISE bands – i.e., W1 (3.4 μm) and W2 (4.6 μm). We also add the stellar models[<http://www.eso.org/sci/facilities/paranal/decommissioned/isaac/tools/lib.html>] provided by <cit.>, which contains 131 spectra in the range of 1150–25,000 Å to include the SED of main sequence stars. After that, we utilize the latest version of the XMM-COSMOS galaxy and active galactic nucleus (AGN) SEDs <cit.>. These templates are discussed in detail by <cit.> as part of their work on estimating the redshift of X-ray AGNs. Similar to what has been done by <cit.>, we then apply the dust reddening by employing the attenuation levels of 0 ≤ A_V ≤ 2, following the <cit.> extinction law. While the original template list covers a wide range of galaxy spectral types, we only pick the SEDs of luminous quasars dominated by the continuum and broad-line emissions for our purpose. Next, to ascertain that our targets do not resemble unlensed galaxies, we fit them with the set of SEDs established from the Flexible Stellar Population Synthesis code <cit.>. These templates are composed of a combination of stellar lights, nebular lines, and MIR dust-reprocessed emissions. They also encompass ultraviolet to infrared wavelengths and contain information about stellar population properties, such as ages, metallicities, and initial mass functions. A grid of SED sets is then assembled by distributing the quasar and galaxy templates in the redshifts of 0 ≤ z ≤ 8 with a step size of Δ z = 0.005. We have to note that in these SED models, we incorporate the attenuation produced by the H i in the IGM using the analytical approximation from <cit.>. While the star, brown dwarf, and quasar templates are fitted utilizing the single template mode in , the galaxy models are matched to each source's photometry with non-negative linear combinations, allowing each component to contribute to the fit. An example of our SED fitting result is displayed in Figure <ref>. § COMPLETE LIST OF LENS CANDIDATES We present here the complete list of our lens candidates selected in the main text. The photometry measured within a 2 aperture diameter of each system is reported in Table <ref>, while the associated color images are displayed in Figure <ref>. cccccccccccc List of rediscovered strong lenses and newly found lensed quasar candidates. ID Name g r i z y J W1 P_lens Grade Reference continued. ID Name g r i z y J W1 P_lens Grade Reference 2074 J000.67138+02.81479 22.51 ± 0.01 20.84 ± 0.01 19.93 ± 0.00 19.55 ± 0.00 19.42 ± 0.01 19.48 ± 0.07 16.68 ± 0.01 0.43 A [1] 10677 J003.99017-00.17165 23.67 ± 0.02 23.23 ± 0.02 22.90 ± 0.03 22.62 ± 0.03 22.56 ± 0.05 0.35 A 23509 J015.65965+01.98248 24.19 ± 0.03 22.74 ± 0.01 21.39 ± 0.01 20.72 ± 0.00 20.43 ± 0.01 19.59 ± 0.07 17.90 ± 0.01 0.62 A [1] 26276 J016.54800+00.00320 23.77 ± 0.03 22.35 ± 0.01 21.15 ± 0.00 20.86 ± 0.01 20.48 ± 0.01 19.80 ± 0.10 18.94 ± 0.03 0.75 A 28188 J017.13767+00.54788 24.35 ± 0.04 23.49 ± 0.02 23.28 ± 0.03 23.12 ± 0.05 23.06 ± 0.10 20.09 ± 0.11 18.76 ± 0.03 0.56 A 31098 J018.11680+01.28198 22.35 ± 0.01 20.64 ± 0.00 19.78 ± 0.00 19.54 ± 0.00 19.24 ± 0.00 18.55 ± 0.03 17.44 ± 0.01 0.47 A [2] 37338 J020.07557+00.19049 23.30 ± 0.02 21.87 ± 0.01 20.76 ± 0.00 20.51 ± 0.00 20.17 ± 0.01 19.85 ± 0.14 18.46 ± 0.02 0.70 A [3] 52971 J029.38124-03.51603 22.71 ± 0.01 21.58 ± 0.01 20.60 ± 0.00 20.20 ± 0.00 19.98 ± 0.01 19.35 ± 0.04 18.41 ± 0.02 0.63 A [4] 53914 J029.51814-00.48833 23.05 ± 0.01 22.44 ± 0.01 22.12 ± 0.01 21.92 ± 0.02 21.76 ± 0.02 18.60 ± 0.03 17.96 ± 0.01 0.34 A [5] 58909 J030.19785-03.73786 23.62 ± 0.02 23.14 ± 0.02 22.54 ± 0.01 22.21 ± 0.02 21.91 ± 0.03 19.51 ± 0.07 18.39 ± 0.02 0.59 A [4] 66838 J031.26894-01.05441 22.74 ± 0.01 22.12 ± 0.01 21.76 ± 0.01 21.58 ± 0.01 21.50 ± 0.02 18.30 ± 0.03 17.93 ± 0.01 0.53 A 67286 J031.32065-01.38927 23.00 ± 0.01 22.51 ± 0.01 22.09 ± 0.01 21.81 ± 0.01 21.58 ± 0.02 19.20 ± 0.03 18.20 ± 0.02 0.90 A [4] 72413 J032.04445-02.33796 26.49 ± 0.29 23.62 ± 0.03 22.89 ± 0.02 22.47 ± 0.03 22.20 ± 0.03 19.41 ± 0.06 19.07 ± 0.03 0.35 A [6] 73180 J032.14510-00.29402 22.87 ± 0.01 21.33 ± 0.00 20.39 ± 0.00 20.10 ± 0.00 19.76 ± 0.00 19.20 ± 0.05 18.25 ± 0.02 0.44 A 90052 J034.35567-01.98047 26.41 ± 0.29 23.55 ± 0.03 22.13 ± 0.01 21.53 ± 0.01 21.31 ± 0.01 19.55 ± 0.08 18.04 ± 0.01 0.43 A 90377 J034.40516-05.22500 22.58 ± 0.01 22.42 ± 0.01 22.10 ± 0.01 21.92 ± 0.01 21.76 ± 0.02 0.37 A [4] 97746 J035.41724-02.17223 23.72 ± 0.02 22.30 ± 0.01 21.10 ± 0.00 20.63 ± 0.00 20.37 ± 0.01 19.46 ± 0.06 18.25 ± 0.02 0.37 A [4] 104517 J036.23320+01.45236 23.22 ± 0.02 22.85 ± 0.02 22.25 ± 0.01 22.07 ± 0.01 21.66 ± 0.03 18.77 ± 0.05 17.60 ± 0.01 0.82 A [1] 124577 J038.94314-02.32526 21.59 ± 0.00 21.18 ± 0.00 21.13 ± 0.01 21.10 ± 0.01 21.11 ± 0.01 19.51 ± 0.05 18.81 ± 0.02 0.93 A 126115 J039.15545-03.53893 21.09 ± 0.00 19.60 ± 0.00 19.10 ± 0.00 18.82 ± 0.00 18.56 ± 0.00 17.87 ± 0.02 17.58 ± 0.01 0.71 A [4] 151222 J132.11497+00.81026 20.33 ± 0.00 20.01 ± 0.00 19.73 ± 0.00 19.79 ± 0.00 19.69 ± 0.00 19.42 ± 0.05 18.56 ± 0.02 0.42 A 170245 J135.64663+03.75000 22.07 ± 0.00 20.75 ± 0.00 19.93 ± 0.00 19.73 ± 0.00 19.65 ± 0.00 18.60 ± 0.04 18.41 ± 0.02 0.51 A 173697 J136.03305-00.99807 23.42 ± 0.01 22.14 ± 0.01 20.96 ± 0.00 20.55 ± 0.00 20.27 ± 0.01 19.73 ± 0.11 18.54 ± 0.02 0.47 A [4] 185334 J137.41113+00.47872 25.17 ± 0.08 25.07 ± 0.09 24.18 ± 0.06 23.84 ± 0.07 23.46 ± 0.09 0.45 A [6] 192262 J138.37916+00.65203 23.10 ± 0.01 21.80 ± 0.01 20.80 ± 0.00 20.05 ± 0.00 19.84 ± 0.00 18.08 ± 0.01 16.89 ± 0.01 0.51 A [7] 206063 J140.33649+04.74183 22.70 ± 0.02 21.50 ± 0.01 20.33 ± 0.00 20.03 ± 0.00 19.77 ± 0.01 19.37 ± 0.06 18.21 ± 0.02 0.41 A [3] 231613 J145.88780-00.72746 23.24 ± 0.02 23.10 ± 0.02 22.58 ± 0.01 22.51 ± 0.02 22.24 ± 0.04 19.87 ± 0.11 18.71 ± 0.02 0.39 A 244751 J148.68976+03.39569 22.76 ± 0.01 21.61 ± 0.00 20.71 ± 0.00 20.33 ± 0.00 20.08 ± 0.01 19.52 ± 0.05 18.29 ± 0.02 0.96 A 256581 J151.21543-00.52913 23.17 ± 0.02 21.74 ± 0.01 20.59 ± 0.00 20.25 ± 0.00 20.14 ± 0.01 19.27 ± 0.04 17.00 ± 0.01 0.40 A [2] 306065 J160.59709+00.25609 23.10 ± 0.01 22.69 ± 0.01 22.28 ± 0.01 21.98 ± 0.02 21.62 ± 0.02 19.21 ± 0.06 18.09 ± 0.01 0.79 A [8] 379069 J171.99042+04.40394 21.86 ± 0.01 21.64 ± 0.00 21.47 ± 0.00 21.30 ± 0.01 20.95 ± 0.01 19.41 ± 0.06 18.33 ± 0.02 0.47 A [9] 380704 J172.25016-01.70394 22.61 ± 0.01 21.15 ± 0.00 19.94 ± 0.00 19.58 ± 0.00 19.40 ± 0.00 18.81 ± 0.04 17.50 ± 0.01 0.36 A [10] 398135 J174.62114+03.96712 22.86 ± 0.01 22.36 ± 0.01 22.05 ± 0.01 21.83 ± 0.02 21.68 ± 0.02 18.68 ± 0.03 17.69 ± 0.01 0.53 A 419901 J177.23536+02.21434 24.11 ± 0.03 23.80 ± 0.03 23.23 ± 0.02 22.92 ± 0.04 22.69 ± 0.06 19.32 ± 0.05 18.03 ± 0.01 0.84 A 420960 J177.36290+02.40679 24.54 ± 0.04 23.32 ± 0.02 22.76 ± 0.01 22.39 ± 0.03 22.28 ± 0.04 19.63 ± 0.06 18.51 ± 0.02 0.34 A 426556 J178.05915+00.52403 21.66 ± 0.00 20.47 ± 0.00 19.73 ± 0.00 19.41 ± 0.00 19.17 ± 0.00 18.61 ± 0.03 17.54 ± 0.01 0.41 A [4] 445898 J180.32619-00.21206 24.97 ± 0.07 22.75 ± 0.01 21.32 ± 0.00 20.78 ± 0.00 20.51 ± 0.01 19.94 ± 0.12 18.61 ± 0.02 0.65 A [6] 447014 J180.45139+01.97095 23.80 ± 0.02 23.39 ± 0.02 22.96 ± 0.02 22.83 ± 0.03 22.72 ± 0.06 19.25 ± 0.05 18.67 ± 0.02 0.69 A 449138 J180.73692+00.65845 23.54 ± 0.02 21.66 ± 0.00 20.38 ± 0.00 19.90 ± 0.00 19.62 ± 0.00 18.80 ± 0.03 17.43 ± 0.01 0.37 A [11] 456623 J181.74491-00.11098 22.69 ± 0.01 20.91 ± 0.00 20.02 ± 0.00 19.65 ± 0.00 19.48 ± 0.00 18.14 ± 0.01 0.37 A 479732 J185.05065+04.11383 22.67 ± 0.02 22.12 ± 0.01 21.76 ± 0.01 21.61 ± 0.01 21.43 ± 0.02 18.84 ± 0.03 18.02 ± 0.01 0.53 A [12] 479945 J185.07896+01.21497 22.00 ± 0.00 21.05 ± 0.00 20.24 ± 0.00 19.92 ± 0.00 19.89 ± 0.00 19.31 ± 0.06 18.19 ± 0.02 0.92 A [3] 481128 J185.25948+00.31493 24.56 ± 0.04 22.71 ± 0.01 21.31 ± 0.00 20.55 ± 0.00 20.43 ± 0.01 19.69 ± 0.08 18.14 ± 0.01 0.39 A [3] 486405 J186.05202+01.28664 23.80 ± 0.02 23.51 ± 0.03 23.01 ± 0.02 24.19 ± 0.12 23.91 ± 0.13 19.55 ± 0.06 18.05 ± 0.01 0.70 A [4] 495027 J187.45272+01.61460 23.45 ± 0.02 21.55 ± 0.00 20.17 ± 0.00 19.67 ± 0.00 19.49 ± 0.00 18.90 ± 0.04 17.48 ± 0.01 0.68 A [2] 516365 J190.34810+00.26748 23.85 ± 0.03 23.56 ± 0.03 22.86 ± 0.03 22.46 ± 0.03 22.23 ± 0.04 19.57 ± 0.11 18.28 ± 0.02 0.36 A [2] 559372 J199.90890-00.69446 23.22 ± 0.02 22.18 ± 0.01 21.12 ± 0.00 20.63 ± 0.00 20.37 ± 0.01 19.27 ± 0.05 17.59 ± 0.01 0.70 A 598787 J209.24615-00.89224 22.28 ± 0.01 21.25 ± 0.00 20.76 ± 0.00 20.46 ± 0.00 20.19 ± 0.01 19.31 ± 0.04 0.47 A 598833 J209.25645+01.07771 24.91 ± 0.08 22.80 ± 0.01 21.57 ± 0.01 20.74 ± 0.01 20.43 ± 0.01 19.88 ± 0.08 18.14 ± 0.01 0.60 A 603402 J210.21613-00.66769 23.18 ± 0.01 21.64 ± 0.00 20.95 ± 0.00 20.62 ± 0.00 20.50 ± 0.01 19.63 ± 0.04 0.39 A [2] 619815 J213.25027-01.43561 23.37 ± 0.02 22.13 ± 0.01 20.86 ± 0.00 20.47 ± 0.00 20.10 ± 0.01 19.71 ± 0.09 18.37 ± 0.02 0.67 A [3] 631388 J214.87622+43.69132 21.40 ± 0.00 21.37 ± 0.00 21.34 ± 0.00 21.47 ± 0.01 21.43 ± 0.02 19.78 ± 0.11 18.66 ± 0.02 0.34 A [3] 633575 J215.20170+00.12608 23.81 ± 0.02 22.18 ± 0.01 21.15 ± 0.00 20.75 ± 0.00 20.62 ± 0.01 19.35 ± 0.09 18.20 ± 0.02 0.40 A [3] 633707 J215.22372+00.93918 26.61 ± nan 26.22 ± nan 23.89 ± 0.03 23.45 ± 0.05 23.14 ± 0.07 19.50 ± 0.03 0.42 A [7] 635069 J215.41200+44.43782 22.38 ± 0.01 21.39 ± 0.01 20.97 ± 0.01 20.81 ± 0.01 20.52 ± 0.02 19.20 ± 0.07 18.23 ± 0.01 0.73 A [13] 641575 J216.36693-01.25125 22.88 ± 0.01 21.20 ± 0.00 20.22 ± 0.00 19.86 ± 0.00 19.63 ± 0.00 19.11 ± 0.04 18.19 ± 0.01 0.87 A [3] 670584 J220.25645+01.56248 23.57 ± 0.02 23.36 ± 0.03 22.78 ± 0.01 22.44 ± 0.03 22.21 ± 0.04 0.36 A [10] 676870 J220.97906-00.12570 23.18 ± 0.01 22.71 ± 0.01 22.51 ± 0.01 22.41 ± 0.02 22.26 ± 0.03 20.10 ± 0.14 18.89 ± 0.03 0.76 A [11] 696744 J223.50993+43.76861 23.02 ± 0.02 22.79 ± 0.02 22.54 ± 0.01 22.34 ± 0.02 22.58 ± 0.05 19.70 ± 0.09 18.38 ± 0.01 0.45 A [2] 703049 J224.38524-01.98824 23.42 ± 0.02 22.99 ± 0.02 22.74 ± 0.02 22.51 ± 0.03 22.28 ± 0.05 19.02 ± 0.02 18.10 ± 0.01 0.52 A [4] 734136 J238.82391+41.86073 22.38 ± 0.02 21.18 ± 0.01 20.15 ± 0.00 19.90 ± 0.00 19.52 ± 0.01 18.94 ± 0.06 17.85 ± 0.01 0.57 A [3] 734908 J239.61106+43.47521 21.93 ± 0.00 20.33 ± 0.00 19.45 ± 0.00 19.11 ± 0.00 18.94 ± 0.00 18.46 ± 0.03 17.88 ± 0.01 0.40 A [4] 736419 J240.78050+43.23936 22.64 ± 0.01 21.07 ± 0.00 19.91 ± 0.00 19.46 ± 0.00 19.28 ± 0.00 18.84 ± 0.05 17.61 ± 0.01 0.45 A [3] 747537 J247.92887+43.66653 23.63 ± 0.01 22.15 ± 0.01 21.60 ± 0.00 21.35 ± 0.01 21.25 ± 0.01 19.37 ± 0.09 18.57 ± 0.01 0.34 A 771737 J333.82346+04.74438 22.29 ± 0.01 20.75 ± 0.00 20.05 ± 0.00 19.79 ± 0.00 19.62 ± 0.00 19.19 ± 0.07 18.79 ± 0.02 0.43 A 786779 J336.48464+04.16030 20.98 ± 0.00 20.82 ± 0.00 20.41 ± 0.00 20.21 ± 0.00 20.11 ± 0.01 19.60 ± 0.09 18.27 ± 0.02 0.42 A [9] 793290 J337.67317-00.01680 23.84 ± 0.03 22.25 ± 0.01 21.46 ± 0.01 21.08 ± 0.01 20.88 ± 0.01 18.63 ± 0.03 17.68 ± 0.01 0.43 A [6] 796622 J338.28654+00.49941 23.72 ± 0.02 23.64 ± 0.03 23.36 ± 0.02 23.08 ± 0.04 23.35 ± 0.12 19.90 ± 0.11 18.69 ± 0.02 0.47 A [6] 799332 J338.82260+02.37483 26.22 ± 0.26 23.58 ± 0.03 22.97 ± 0.02 22.73 ± 0.03 22.57 ± 0.06 18.66 ± 0.03 17.81 ± 0.01 0.89 A 809323 J340.47762+00.05874 22.19 ± 0.01 20.97 ± 0.00 20.19 ± 0.00 19.78 ± 0.00 19.71 ± 0.00 19.09 ± 0.06 18.32 ± 0.02 0.93 A [4] 837427 J346.33984-00.03664 22.61 ± 0.01 22.40 ± 0.01 22.16 ± 0.01 22.00 ± 0.01 21.86 ± 0.02 18.70 ± 0.03 17.87 ± 0.01 0.90 A [14] 857582 J350.94141-00.51090 24.97 ± 0.09 24.00 ± 0.04 23.60 ± 0.04 23.79 ± 0.08 22.86 ± 0.05 19.78 ± 0.11 18.50 ± 0.02 0.48 A [4] 859449 J351.28928+00.85322 22.74 ± 0.01 21.30 ± 0.00 20.41 ± 0.00 20.13 ± 0.00 19.99 ± 0.00 19.39 ± 0.09 18.62 ± 0.02 0.67 A [3] 860199 J351.44998+00.62763 21.89 ± 0.00 20.35 ± 0.00 19.48 ± 0.00 19.26 ± 0.00 19.07 ± 0.00 18.43 ± 0.03 17.66 ± 0.01 0.83 A [3] 867305 J352.87690+00.62623 23.74 ± 0.03 23.05 ± 0.02 22.31 ± 0.01 21.94 ± 0.01 21.77 ± 0.02 19.31 ± 0.08 17.91 ± 0.01 0.98 A [4] 867677 J352.94338+01.64599 22.49 ± 0.01 20.80 ± 0.00 19.90 ± 0.00 19.66 ± 0.00 19.44 ± 0.00 18.98 ± 0.04 17.90 ± 0.01 0.44 A [3] 877180 J354.92148+00.01208 21.12 ± 0.00 20.82 ± 0.00 20.58 ± 0.00 20.49 ± 0.00 20.19 ± 0.01 19.44 ± 0.06 18.48 ± 0.02 0.99 A [12] 889892 J359.05923+02.52107 21.72 ± 0.00 20.11 ± 0.00 19.39 ± 0.00 19.10 ± 0.00 18.95 ± 0.00 18.51 ± 0.03 17.90 ± 0.01 0.65 A 891809 J359.72159+01.40142 23.98 ± 0.03 23.29 ± 0.02 22.76 ± 0.01 22.51 ± 0.03 22.05 ± 0.03 19.30 ± 0.06 18.12 ± 0.01 0.39 A [15] 50799 J029.07523-01.12969 22.88 ± 0.01 22.29 ± 0.01 21.72 ± 0.01 21.29 ± 0.01 21.06 ± 0.02 19.19 ± 0.07 18.05 ± 0.01 0.50 B [3] 51372 J029.16338-02.04496 23.24 ± 0.02 21.53 ± 0.01 20.73 ± 0.00 20.39 ± 0.00 20.17 ± 0.01 19.50 ± 0.07 18.96 ± 0.03 0.47 B 52499 J029.32110-00.42419 22.57 ± 0.01 20.77 ± 0.00 20.14 ± 0.00 19.91 ± 0.00 19.61 ± 0.00 18.74 ± 0.02 18.28 ± 0.02 0.34 B 66605 J031.23188-02.14772 20.39 ± 0.00 19.84 ± 0.00 19.40 ± 0.00 19.30 ± 0.00 19.17 ± 0.00 0.64 B 68389 J031.48689-02.14248 23.86 ± 0.03 23.46 ± 0.03 22.94 ± 0.02 22.57 ± 0.03 22.28 ± 0.04 18.87 ± 0.03 17.69 ± 0.01 0.35 B [3] 75188 J032.43229+00.62493 21.45 ± 0.00 20.84 ± 0.00 20.55 ± 0.00 20.45 ± 0.00 20.37 ± 0.01 18.79 ± 0.05 18.12 ± 0.01 0.50 B 77117 J032.66791+02.19009 20.54 ± 0.00 19.86 ± 0.00 19.24 ± 0.00 18.96 ± 0.00 18.88 ± 0.00 18.00 ± 0.02 17.70 ± 0.01 0.40 B 77381 J032.70028+00.09824 24.19 ± 0.03 22.63 ± 0.01 21.52 ± 0.01 20.82 ± 0.01 20.37 ± 0.01 18.46 ± 0.02 17.42 ± 0.01 0.49 B 84319 J033.59273-02.91557 23.32 ± 0.02 22.02 ± 0.01 21.18 ± 0.00 20.78 ± 0.01 20.70 ± 0.01 20.11 ± 0.14 19.49 ± 0.04 0.35 B 86073 J033.83070+01.46474 21.90 ± 0.01 21.60 ± 0.01 21.50 ± 0.01 21.48 ± 0.01 21.45 ± 0.03 19.81 ± 0.10 19.01 ± 0.03 0.38 B 93880 J034.90323-00.91156 23.29 ± 0.01 21.63 ± 0.01 20.51 ± 0.00 20.11 ± 0.00 19.86 ± 0.00 19.45 ± 0.09 18.24 ± 0.02 0.54 B 95632 J035.12890+00.73894 24.76 ± 0.05 24.21 ± 0.04 22.47 ± 0.01 21.75 ± 0.01 21.55 ± 0.02 0.34 B 103355 J036.07595+02.78420 23.42 ± 0.03 21.71 ± 0.01 20.64 ± 0.01 20.28 ± 0.01 19.97 ± 0.01 18.87 ± 0.05 17.86 ± 0.01 0.72 B [9] 126604 J039.22622-03.93275 23.34 ± 0.01 21.82 ± 0.01 20.77 ± 0.00 20.37 ± 0.00 20.13 ± 0.01 19.48 ± 0.07 18.45 ± 0.02 0.52 B 157310 J133.42566+01.38783 22.02 ± 0.00 21.66 ± 0.01 21.48 ± 0.01 21.33 ± 0.01 21.23 ± 0.01 18.82 ± 0.05 18.35 ± 0.02 0.65 B 163347 J134.62093-00.60642 23.50 ± 0.01 22.36 ± 0.01 21.49 ± 0.00 20.98 ± 0.01 20.87 ± 0.01 20.11 ± 0.05 19.12 ± 0.03 0.35 B 163604 J134.67324+03.89196 22.40 ± 0.01 20.88 ± 0.00 20.15 ± 0.00 19.80 ± 0.00 19.74 ± 0.00 19.10 ± 0.04 0.66 B 168255 J135.38959+04.60414 22.24 ± 0.01 21.06 ± 0.00 20.44 ± 0.00 20.05 ± 0.00 20.01 ± 0.01 18.41 ± 0.03 18.25 ± 0.02 0.77 B 170439 J135.66838+04.82183 20.83 ± 0.00 19.70 ± 0.00 18.76 ± 0.00 18.28 ± 0.00 18.20 ± 0.00 0.39 B 170929 J135.73529+04.22977 22.54 ± 0.01 21.26 ± 0.00 20.40 ± 0.00 20.18 ± 0.00 20.12 ± 0.00 19.20 ± 0.05 18.29 ± 0.02 0.45 B 182764 J137.09298-01.13119 22.63 ± 0.01 22.10 ± 0.01 21.12 ± 0.00 20.82 ± 0.00 20.52 ± 0.01 20.45 ± 0.17 19.01 ± 0.03 0.61 B [6] 183185 J137.14126+00.13506 22.71 ± 0.01 21.69 ± 0.00 20.58 ± 0.00 20.03 ± 0.00 19.80 ± 0.00 19.23 ± 0.04 17.79 ± 0.01 0.44 B [4] 184295 J137.27788+03.39111 22.29 ± 0.01 21.20 ± 0.00 20.75 ± 0.00 20.49 ± 0.00 20.29 ± 0.01 0.81 B 209801 J140.92640-00.92415 22.57 ± 0.01 21.68 ± 0.01 21.14 ± 0.00 21.01 ± 0.01 20.73 ± 0.01 19.40 ± 0.03 0.45 B 211435 J141.21699-00.09576 23.05 ± 0.01 22.02 ± 0.01 20.95 ± 0.00 20.71 ± 0.00 20.36 ± 0.01 19.77 ± 0.06 18.49 ± 0.02 0.36 B 216863 J142.17544+01.32812 22.44 ± 0.01 21.02 ± 0.00 20.05 ± 0.00 19.74 ± 0.00 19.41 ± 0.00 18.99 ± 0.03 18.12 ± 0.01 0.78 B [3] 235814 J146.80458+02.79517 21.08 ± 0.00 20.63 ± 0.00 20.36 ± 0.00 20.31 ± 0.00 20.26 ± 0.01 19.13 ± 0.04 0.35 B [9] 238736 J147.43035+00.09320 20.60 ± 0.00 20.32 ± 0.00 20.14 ± 0.00 20.16 ± 0.00 20.10 ± 0.01 18.18 ± 0.02 16.97 ± 0.01 0.44 B [9] 248367 J149.45874+01.75211 22.85 ± 0.01 21.57 ± 0.00 20.49 ± 0.00 20.17 ± 0.00 19.95 ± 0.00 19.54 ± 0.07 18.42 ± 0.02 0.35 B [4] 254398 J150.70561+00.51154 23.70 ± 0.02 22.22 ± 0.01 21.20 ± 0.00 20.85 ± 0.00 20.65 ± 0.01 19.02 ± 0.03 17.68 ± 0.01 0.60 B 297613 J159.35616+02.52577 22.65 ± 0.01 21.79 ± 0.01 21.06 ± 0.00 20.87 ± 0.01 20.61 ± 0.01 20.23 ± 0.12 19.14 ± 0.03 0.43 B 335597 J165.10725+00.33480 23.06 ± 0.01 21.43 ± 0.00 20.53 ± 0.00 20.11 ± 0.00 20.02 ± 0.01 19.58 ± 0.08 18.84 ± 0.03 0.74 B 359977 J169.10389+02.86011 23.06 ± 0.01 21.40 ± 0.00 20.36 ± 0.00 19.94 ± 0.00 19.73 ± 0.00 19.86 ± 0.09 17.45 ± 0.01 0.39 B 376283 J171.57689+04.10215 21.35 ± 0.01 20.17 ± 0.00 19.59 ± 0.00 19.13 ± 0.00 19.10 ± 0.00 18.64 ± 0.03 17.41 ± 0.01 0.37 B 396574 J174.43810-00.97871 23.60 ± 0.02 23.50 ± 0.03 22.98 ± 0.02 22.78 ± 0.04 22.29 ± 0.05 20.09 ± 0.05 19.20 ± 0.04 0.55 B 398472 J174.66084-00.90392 24.38 ± 0.04 24.57 ± 0.07 23.76 ± 0.04 23.28 ± 0.05 23.09 ± 0.07 19.91 ± 0.04 19.08 ± 0.03 0.38 B [2] 403630 J175.25814+00.71887 23.45 ± 0.02 21.93 ± 0.01 21.06 ± 0.00 20.70 ± 0.00 20.54 ± 0.01 20.02 ± 0.09 19.11 ± 0.03 0.71 B 408025 J175.79693-01.65983 22.39 ± 0.01 21.48 ± 0.01 20.34 ± 0.00 19.94 ± 0.00 19.78 ± 0.00 18.95 ± 0.04 17.73 ± 0.01 0.83 B [10] 421588 J177.44370-00.36609 22.83 ± 0.01 21.64 ± 0.01 20.75 ± 0.00 20.40 ± 0.00 20.26 ± 0.01 19.74 ± 0.11 18.84 ± 0.03 0.36 B [3] 428218 J178.26375+03.81240 23.18 ± 0.01 22.88 ± 0.01 22.56 ± 0.01 22.36 ± 0.02 22.18 ± 0.03 19.22 ± 0.02 18.03 ± 0.01 0.44 B 429919 J178.46578-00.88741 23.97 ± 0.03 23.01 ± 0.02 22.27 ± 0.01 21.86 ± 0.01 21.68 ± 0.03 19.14 ± 0.04 17.91 ± 0.01 0.41 B [3] 439412 J179.58976+01.17015 25.14 ± 0.07 23.11 ± 0.02 21.84 ± 0.01 21.40 ± 0.01 21.14 ± 0.01 19.34 ± 0.04 18.09 ± 0.01 0.43 B 447096 J180.46200+04.89641 22.62 ± 0.01 21.08 ± 0.00 20.24 ± 0.00 19.94 ± 0.00 19.67 ± 0.00 19.18 ± 0.04 18.54 ± 0.02 0.76 B 457430 J181.85322-00.56724 22.20 ± 0.01 20.74 ± 0.00 20.14 ± 0.00 19.86 ± 0.00 19.67 ± 0.00 19.20 ± 0.05 18.39 ± 0.02 0.42 B 472144 J183.88950-00.97856 22.18 ± 0.01 20.38 ± 0.00 19.48 ± 0.00 19.11 ± 0.00 19.07 ± 0.00 18.26 ± 0.02 17.64 ± 0.01 0.49 B 485479 J185.90854+00.81986 21.34 ± 0.00 20.28 ± 0.00 19.85 ± 0.00 19.47 ± 0.00 19.61 ± 0.00 18.89 ± 0.03 18.14 ± 0.01 0.35 B 504748 J188.80300-00.06022 23.21 ± 0.02 21.81 ± 0.01 20.75 ± 0.00 20.44 ± 0.00 20.30 ± 0.00 19.01 ± 0.09 0.36 B 509467 J189.45326+04.13141 21.85 ± 0.02 21.05 ± 0.00 20.55 ± 0.00 20.29 ± 0.01 20.15 ± 0.01 19.68 ± 0.07 18.24 ± 0.02 0.39 B 510782 J189.64236-00.72435 23.99 ± 0.03 22.36 ± 0.01 21.65 ± 0.01 21.30 ± 0.01 21.12 ± 0.01 18.53 ± 0.02 17.64 ± 0.01 0.73 B [10] 525892 J191.69633+02.21569 23.03 ± 0.02 21.45 ± 0.00 20.45 ± 0.00 20.02 ± 0.01 19.88 ± 0.00 18.89 ± 0.02 17.92 ± 0.01 0.36 B 564583 J201.38247-01.35862 20.48 ± 0.00 20.02 ± 0.00 19.58 ± 0.00 19.43 ± 0.00 19.31 ± 0.00 0.91 B 566973 J202.04765+00.26065 20.04 ± 0.00 19.88 ± 0.00 19.75 ± 0.00 19.65 ± 0.00 19.67 ± 0.00 18.13 ± 0.02 0.42 B 572002 J203.34213+01.71111 22.85 ± 0.02 21.01 ± 0.00 19.93 ± 0.00 19.57 ± 0.00 19.33 ± 0.00 18.94 ± 0.05 17.39 ± 0.01 0.34 B 574483 J203.97402+00.94847 23.16 ± 0.02 22.23 ± 0.01 21.15 ± 0.00 20.69 ± 0.00 20.38 ± 0.01 19.80 ± 0.08 18.67 ± 0.02 0.66 B 616231 J212.70374-01.16278 22.64 ± 0.01 22.30 ± 0.01 22.13 ± 0.01 22.06 ± 0.01 21.88 ± 0.02 19.84 ± 0.09 18.50 ± 0.02 0.51 B [11] 639109 J216.01461+00.13081 23.32 ± 0.01 21.68 ± 0.01 20.73 ± 0.00 20.34 ± 0.00 20.22 ± 0.00 19.92 ± 0.10 18.45 ± 0.02 0.57 B 640138 J216.18209-00.97517 21.53 ± 0.00 20.75 ± 0.00 20.51 ± 0.00 20.28 ± 0.00 20.31 ± 0.01 18.84 ± 0.02 19.16 ± 0.03 0.86 B 640692 J216.24943+00.91866 24.44 ± 0.04 23.61 ± 0.02 23.17 ± 0.02 22.86 ± 0.03 22.70 ± 0.05 19.25 ± 0.02 18.26 ± 0.02 0.51 B [2] 644700 J216.81339-00.06825 22.61 ± 0.01 21.66 ± 0.01 21.27 ± 0.00 21.10 ± 0.01 20.69 ± 0.01 19.62 ± 0.06 18.93 ± 0.03 0.53 B 646668 J217.10931+01.25926 22.18 ± 0.01 21.49 ± 0.01 21.32 ± 0.00 21.27 ± 0.01 21.03 ± 0.01 19.27 ± 0.05 18.79 ± 0.02 0.60 B 652665 J218.06127+01.59046 23.62 ± 0.03 22.06 ± 0.01 20.78 ± 0.00 20.25 ± 0.00 20.04 ± 0.01 19.46 ± 0.07 18.05 ± 0.01 0.36 B 653035 J218.11986-00.20185 21.26 ± 0.00 20.80 ± 0.00 20.80 ± 0.00 20.12 ± 0.00 20.60 ± 0.01 19.83 ± 0.10 19.25 ± 0.03 0.58 B 655412 J218.44448+43.81522 22.35 ± 0.01 21.22 ± 0.00 20.65 ± 0.00 20.44 ± 0.00 20.43 ± 0.01 18.98 ± 0.05 18.26 ± 0.01 0.51 B 666651 J219.82191-00.65496 24.18 ± 0.03 22.22 ± 0.01 20.80 ± 0.00 20.35 ± 0.00 20.08 ± 0.00 19.56 ± 0.07 18.31 ± 0.02 0.59 B [6] 673740 J220.62905-00.39813 22.30 ± 0.01 20.53 ± 0.00 19.71 ± 0.00 19.30 ± 0.00 19.14 ± 0.00 18.43 ± 0.03 17.63 ± 0.01 0.54 B [3] 691834 J222.88068-01.67836 21.83 ± 0.01 20.47 ± 0.00 19.75 ± 0.00 19.47 ± 0.00 19.35 ± 0.00 18.81 ± 0.04 18.54 ± 0.02 0.36 B 703092 J224.39202+43.67890 23.01 ± 0.01 22.61 ± 0.01 22.02 ± 0.01 21.83 ± 0.01 21.72 ± 0.02 19.16 ± 0.05 18.14 ± 0.01 0.62 B [3] 720080 J229.29518+44.12815 23.04 ± 0.01 22.62 ± 0.01 22.31 ± 0.01 22.10 ± 0.02 22.08 ± 0.04 20.43 ± 0.13 19.05 ± 0.02 0.66 B [3] 725694 J233.42897+42.83963 23.50 ± 0.02 22.09 ± 0.01 20.76 ± 0.00 20.23 ± 0.00 20.03 ± 0.01 19.36 ± 0.06 17.86 ± 0.01 0.64 B [4] 736755 J241.05542+43.37910 22.70 ± 0.01 21.90 ± 0.00 21.21 ± 0.00 20.93 ± 0.01 20.81 ± 0.01 20.40 ± 0.20 19.67 ± 0.04 0.46 B 751225 J249.97872+43.62234 21.46 ± 0.00 20.12 ± 0.00 19.54 ± 0.00 19.20 ± 0.00 19.27 ± 0.00 18.69 ± 0.04 18.22 ± 0.01 0.49 B 764917 J332.57047+02.12108 22.56 ± 0.01 20.86 ± 0.00 19.93 ± 0.00 19.53 ± 0.00 19.38 ± 0.00 18.03 ± 0.02 17.07 ± 0.01 0.34 B 768996 J333.27851-00.51019 24.17 ± 0.05 22.71 ± 0.03 21.52 ± 0.01 21.08 ± 0.01 20.86 ± 0.02 19.05 ± 0.06 17.20 ± 0.01 0.63 B [4] 793076 J337.63423-00.12474 21.31 ± 0.00 19.96 ± 0.00 19.31 ± 0.00 19.05 ± 0.00 18.78 ± 0.00 18.26 ± 0.02 17.87 ± 0.01 0.38 B 795264 J338.04846+01.99870 23.30 ± 0.02 21.65 ± 0.00 20.50 ± 0.00 20.03 ± 0.00 19.83 ± 0.00 19.35 ± 0.06 17.82 ± 0.01 0.62 B [3] 806087 J339.96430+02.52779 21.29 ± 0.00 19.80 ± 0.00 19.23 ± 0.00 18.89 ± 0.00 18.77 ± 0.00 17.82 ± 0.02 17.60 ± 0.01 0.41 B 816392 J341.56034+05.97432 25.44 ± 0.30 24.63 ± 0.24 24.04 ± 0.18 22.09 ± 0.02 21.88 ± 0.05 0.79 B 862254 J351.88233+00.14406 22.12 ± 0.01 20.94 ± 0.00 20.34 ± 0.00 20.12 ± 0.00 20.04 ± 0.00 18.68 ± 0.04 18.48 ± 0.02 0.43 B 865485 J352.52481-01.05834 23.08 ± 0.02 22.33 ± 0.01 21.99 ± 0.02 21.78 ± 0.02 21.52 ± 0.02 19.23 ± 0.06 18.22 ± 0.02 0.37 B 868470 J353.06212-00.77878 21.69 ± 0.01 20.13 ± 0.00 19.44 ± 0.00 19.08 ± 0.00 19.08 ± 0.00 18.45 ± 0.03 17.90 ± 0.01 0.38 B 872424 J353.80073+02.02693 21.60 ± 0.01 20.57 ± 0.00 20.23 ± 0.00 20.15 ± 0.00 19.33 ± 0.00 19.45 ± 0.11 18.35 ± 0.02 0.78 B 874767 J354.30696+00.93632 20.03 ± 0.00 19.73 ± 0.00 19.19 ± 0.00 19.22 ± 0.00 19.13 ± 0.00 18.40 ± 0.04 16.90 ± 0.01 0.42 B Column (1): identification number for each candidate. Column (2): name of the source. Columns (3)–(7): HSC grizy-band magnitudes and the corresponding 1σ errors. Column (8): NIR J-band magnitude. Column (9): unWISE W1-band magnitude. Column (10): lens probability based on the ensemble network classification. Column (11): grade after visual inspection. Column (12): references for the list of lenses or candidates published by earlier works. The magnitudes are in AB values, corrected for Galactic extinction based on the dust map of <cit.> and considering the <cit.> reddening equation. Depending on the visual inspection grades, best and good lens candidates are marked with A and B, respectively. To name the candidates, we follow the “JRRR.rrrrr+DD.ddddd” convention, where RRR.rrrrr and +DD.ddddd are, respectively, the R.A. and decl. in decimal degrees (J2000). [1] <cit.>, [2] <cit.>, [3] <cit.>, [4] <cit.>, [5] <cit.>, [6] <cit.>, [7] <cit.>, [8] <cit.>, [9] <cit.>, [10] <cit.>, [11] <cit.>, [12] <cit.>, [13] <cit.>, [14] <cit.>, [15] <cit.>
http://arxiv.org/abs/2307.02486v2	20230705175938	LongNet: Scaling Transformers to 1,000,000,000 Tokens	[ "Jiayu Ding", "Shuming Ma", "Li Dong", "Xingxing Zhang", "Shaohan Huang", "Wenhui Wang", "Nanning Zheng", "Furu Wei" ]	cs.CL	[ "cs.CL", "cs.LG" ]	Phenomenology of bond and flux orders in kagome metals Mark H. Fischer August 1, 2023 ====================================================== Scaling sequence length has become a critical demand in the era of large language models. However, existing methods struggle with either computational complexity or model expressivity, rendering the maximum sequence length restricted. To address this issue, we introduce , a Transformer variant that can scale sequence length to more than 1 billion tokens, without sacrificing the performance on shorter sequences. Specifically, we propose dilated attention, which expands the attentive field exponentially as the distance grows. has significant advantages: 1) it has a linear computation complexity and a logarithm dependency between any two tokens in a sequence; 2) it can be served as a distributed trainer for extremely long sequences; 3) its dilated attention is a drop-in replacement for standard attention, which can be seamlessly integrated with the existing Transformer-based optimization. Experiments results demonstrate that yields strong performance on both long-sequence modeling and general language tasks. Our work opens up new possibilities for modeling very long sequences, e.g., treating a whole corpus or even the entire Internet as a sequence. Code is available at <https://aka.ms/LongNet>. § INTRODUCTION Recent years have witnessed a trend toward scaling neural networks <cit.>. The depth is primarily scaled up for exponential expressivity, producing many powerful deep networks <cit.>. Then, the sparse MoE models <cit.> and model parallelism approaches <cit.> efficiently enlarge the hidden dimension. Sequence length, as the last atomic dimension of the neural network, is desirable to be unlimited. Breaking the limitation of sequence length introduces significant advantages. First, it provides large memory and receptive field for models, which is practical for them to interact with human and the world. Second, a longer context contains more complex causality and reasoning paths that models can exploit in training data. In contrast, short dependency has more spurious correlations, which is harmful to generalization. Third, it enables to explore the limits of in-context learning, which has the potential to be a paradigm shift for many-shot learning, as an extremely long context may help the models alleviate catastrophic forgetting. The major challenge of scaling up sequence length is striking the right balance between the computational complexity and the model expressivity. RNN-style models are primarily implemented to increase the length. However, its sequential nature limits the parallelization during training, which is essential in long-sequence modeling. More recently, state space models <cit.> are appealing to sequence modeling. It can operate as a CNN during training, and transform to an efficient RNN at test time. While they perform well at long-range benchmarks <cit.>, their performance on regular lengths is not as good as Transformers, limited mainly by the model expressivity <cit.>. Another strand of scaling the sequence length is to decrease the complexity of Transformers, i.e., the quadratic complexity of self-attention. Implementing sliding windows or convolution modules over the attention is a straightforward way to make the complexity nearly linear. Nevertheless, this sacrifices the ability to recall the early tokens, forgetting the prompts at the very beginning of the sequence. Sparse attention reduces the computation by sparsifying the attention matrix, preserving the possibility of recalling long-distant information. For example, <cit.> obtains 𝒪(N√(N)d) time complexity with a fixed sparse pattern. Besides the heuristic patterns <cit.>, the learnable patterns prove to be useful for sparse attention <cit.>. There are also some other efficient Transformer-based variants, including low-rank attention <cit.>, kernel-based methods <cit.>, downsampling approaches <cit.>, recurrent models <cit.>, and retrieval-based methods <cit.>. Yet, none has been scaled to 1 billion tokens (see <ref>). In this work, we successfully scale the sequence length to 1 billion tokens. Our solution is , which replaces the attention of vanilla Transformers with a novel component named dilated attention. The general design principle is - attention allocation decreases exponentially as the distance between tokens grows. We prove that it obtains a linear computation complexity and a logarithm dependency between tokens. This deals with the contradiction between limited attention resources and the accessibility to every token. In the implementation, can be transformed into a dense Transformer, which seamlessly supports the off-the-shelf optimization for Transformers (e.g., kernel fusion, quantization, and distributed training). Taking advantage of the linear complexity, can parallelize the training across nodes, breaking the constraint of both computation and memory with a distributed algorithm. This allows us to efficiently scale up the sequence length to 1B tokens with nearly constant runtime (see <ref>), while vanilla Transformer suffers from quadratic complexity. § §.§ Preliminary The core of Transformers <cit.> is self-attention, which maps a query and a set of keys and values to output. Given the inputs Q,K,V ∈ℝ^N × d, it computes the outputs O with O = softmax(QK^T)V Self-attention struggles with long sequences, due to its quadratic dependency on the sequence length. One query would attend to all keys and values, leading to computational inefficiencies. Sparse attention alleviates this issue by restricting the query's access to a subset of keys and values. The key of sparse attention is the sparse attention pattern S ∈{0, 1}^N × N, which determines specific keys and values that the query Q can attend to. O = softmax(QK^T⊙1_S)V For example, the fixed pattern of sparse Transformer <cit.> is composed of a local pattern and a strided pattern. The sequence is divided into blocks of length l. The local pattern allows one query to attend to tokens within the same block, while strided pattern allows one query to attend to the last c tokens of each block. Formally, the local pattern S_i^(1) = { j \|⌊ j/l ⌋ = ⌊ i/l ⌋}, and the strided pattern S_i^(2) = { j \| j mod l ∈{t, t+1,...,l}}. §.§ Dilated Attention <ref> illustrates the overview of dilated attention. Dilated attention splits the input (Q, K, V) into segments {(Q_i,K_i,V_i)}^N/w equally with a segment length w. Each segment is then sparsified along the sequence dimension by selecting the rows with an interval r. The computation can be written as: Q_i = [Q_iw, Q_iw+r, Q_iw+2r, ..., Q_(i+1)w-1] K_i = [K_iw, K_iw+r, K_iw+2r, ..., K_(i+1)w-1] V_i = [V_iw, V_iw+r, V_iw+2r, ..., V_(i+1)w-1] The sparsified segments {(Q_i,K_i,V_i)}^N/w are fed into the attention in parallel, after which are scattered and concatenated as the output O: O_i = softmax(Q_iK_i^T)V_i Ô_i = {O_i,j \| j mod r=0; 0 \| j mod r ≠ 0} O = [Ô_0, Ô_1, ..., Ô_N/w-1] In the implementation, the dilated attention can be transformed into dense attention between a gathering operation over the input (Q, K, V) and a scattering operation over the output O_i, so it can directly reuse any optimization for vanilla attention (e.g., flash attention <cit.>). Dilated attention can significantly reduce the computation cost by a factor of N/wr^2 over the vanilla attention. In practice, the segment size w trades the globality of attention for efficiency, while the dilation with a size r reduces the computation cost by approximating the attention matrix. To capture both long-range and short-range information efficiently, we implement a mixture of dilated attentions with different segment sizes and dilation rates {r_i, w_i}^k: O = ∑_i=1^kα_i O\|_r_i,w_i α_i = s_i/∑_j s_j where s_i denotes the denominator of the attention softmax for O\|_r_i,w_i. Note that the computations for {O\|_r_i,w_i}^k are in parallel because there is no computation dependency among them. Experiments show that dynamic weights calculated by the denominator of the attention softmax are better than learnable fixed weights. For a query attends to keys in different dilated attentions, our method to mix dilated attentions is equivalent to gather keys in different parts and calculate softmax together. Intuitively, the local attention should be precisely computed, while the global attention can be approximate. Therefore, we set a larger w_i with a bigger r_i. Moreover, we gradually increase the w_i for each attention until it reaches the maximum length N or the number of attention patterns k: w={w_0, w_1, w_2, ..., N}^k (w_i<w_i+1<N) r={1, r_1, r_2, ..., r_k}^k (1<r_i<r_i+1) In practice, we set w and r to geometric sequences for an exponential attentive field. §.§ Multi-Head Dilated Attention As shown in <ref>, we differ in the computation among different heads by sparsifying different parts of the query-key-value pairs. Specifically, for the j-th head, we have an offset s_j = j mod r when selecting the (Q, K, V): Q_i = [Q_iw+s_j, Q_iw+s_j+r, Q_iw+s_j+2r, ..., Q_(i+1)w+s_j-1] K_i = [K_iw+s_j, K_iw+s_j+r, K_iw+s_j+2r, ..., K_(i+1)w+s_j-1] V_i = [V_iw+s_j, V_iw+s_j+r, V_iw+s_j+2r, ..., V_(i+1)w+s_j-1] Following the vanilla multi-head attention, the outputs of different heads are concatenated into a final output. The rest of the computation remains the same as the single-head counterpart in <ref>. §.§ Computational Complexity and Token Dependency Given dilated attention with a segment size and dilation rate of (r, w), each query-key-value pair is sparsified from (Q,K,V) ∈ℝ^N × d to (Q,K,V) ∈ℝ^w/r× d, so the flops of the attention computation are estimated as: FLOPs=2N/w(w/r)^2d=2Nwd/r^2 We further extend it to dilated attention with multiple segment sizes and dilation rates. The flops can be written as: FLOPs=2Nd ∑_i=1^kw_i/r_i^2 With the segment sizes and dilation rates in <ref> and <ref>, the flops are given by FLOPs=2w_0Nd ∑_i=0^k-11/α^i≤2α/α - 1 w_0Nd (α > 1) where w_0 is a predefined constant and α is the common ratio for geometric sequences w and r. Therefore, the computation complexity of dilated attention is approximate to 𝒪(Nd). Moreover, the information of each tokens can be propagated to a maximum distance of D: D=∑_i=0^l-1w_i=w_0∑_i=0^l-1α^i ≈w_0/α-1α^l where l is the length of the propagated path. Therefore, the maximum path length of a sequence with N tokens can be estimated as: L ≈log_αN(α-1)/w_0 (α > 1) This proves that the token dependency is approximate to 𝒪(log N). § AS A DISTRIBUTED TRAINER: SCALING UP TO 1B TOKENS Although the computation complexity of dilated attention has been greatly reduced to 𝒪(Nd), it is infeasible to scale the sequence length to the million level on a single GPU device due to the computation and memory constraints. There are some distributed training algorithms for large-scale model training, such as model parallelism <cit.>, sequence parallelism <cit.>, and pipeline parallelism <cit.>. However, they are insufficient for especially when the sequence dimension is extremely large. §.§ Distributed Algorithm We take advantage of the linear computation complexity of for the distributed training of sequence dimension. Without loss of generality, <ref> presents our distributed algorithm on two GPUs, which can be further scaled to an arbitrary number of devices. We start by splitting the input sequence along the sequence dimension. Each sequence is put on one device separately: X = [X_1, X_2] Then, they are projected into queries, keys, and values on the two devices: [Q_1, K_1, V_1] = [W_Q, W_K, W_V] X_1, [Q_2, K_2, V_2] = [W_Q, W_K, W_V] X_2 For the segment length w_i ≤ l (where l is the sequence length on the local device), we compute the attention locally with <ref> to <ref>. For the segment length w_i > l, the keys and values are distributed across different devices. Therefore, we collect the key-value pairs before computing the attention. We use <ref> to <ref> to sparsify the {Q, K, V} into {Q, K, V}. An all-gather operation is implemented to collect the key-value pairs: K = [K_1, K_2], V = [V_1, V_2] Note that the all-gather operation in the backward becomes a reduce-scatter operation. Different from vanilla attention, both sizes of K_i and V_i are independent of the sequence length N, making the communication cost constant. Finally, we compute the cross-attention with the local queries Q_i and the global key-value pairs {K, V}. The formulation is written as: O_1 = softmax(Q_1K^T)V, O_2 = softmax(Q_2K^T)V The concatenation of the outputs across different devices becomes the final attention output: O = [O_1, O_2] The distributed algorithm described above is orthogonal to other parallelisms, including data parallelism which partitions the batch dimension, model parallelism which partitions the hidden dimension, and pipeline parallelism which partitions the layers. §.§ Scaling up to 1B Tokens We verify the feasibility of scaling to 1B tokens with the modern distributed systems. Starting from 8K, we gradually scale the sequence length until the limit of GPU memory. We reduce the batch size accordingly to keep the number of tokens per batch at 1 billion. Each model of different sequence lengths has up to 3 segment lengths, which are 2,048, the number of tokens per device, and the sequence length. We compute the average speed in the forward propagation for 10 different runs. <ref> reports the runtime of vanilla attention and our dilated attention. Both of them are implemented with FlashAttention Kernel for saving memory and improving speed. It shows that dilated attention can successfully scale up the sequence length with almost constant latency. By partitioning the sequence dimension, it can leverage the distributed systems to scale the sequence length to 1 billion tokens. In contrast, vanilla attention suffers from the quadratic dependency on the sequence length. Its latency dramatically increases as the length grows. Moreover, there is no distributed algorithm for vanilla attention to break sequence length limitation. This proves the advantage of the linear complexity as well as the distributed algorithm for . § EXPERIMENTS ON LANGUAGE MODELING §.§ Setup We implement on language modeling. The backbone architecture is Magneto <cit.> with xPos <cit.> relative position encoding, except that we replace the standard attention with our dilated attention. We use the base-size configuration of Magneto, which has a hidden dimension of 768, 12 attention heads, and 12 decoder layers. We pre-train the model with The Stack dataset <cit.>, a source code collection in over 300 programming languages. The data is preprocessed with the tiktoken tokenizer[<https://github.com/openai/tiktoken>] with encoding. The models are trained with a batch size of 0.5M tokens for 300K steps. More details regarding the hyperparameters can be found in the appendix. All experiments are conducted based on the torchscale <cit.> codebase. §.§ Results We compare with both vanilla Transformer and sparse Transformers. The differences among the architectures are the attention layers, while the others remain the same. We scale the sequence length of these models from 2K to 32K, while reducing the batch size to keep the number of tokens per batch constant. For , we use segment lengths of w={2048, 4096, 8192, 16384, 32768}, and the dilated ratios are r={1,2,4,6,12}. We implement the fixed pattern for sparse attention as in <cit.> with multiple heads attending to distinct subblocks. The block size is set to 2048. We adjust their sparse ratios to match the computation flops with so that the comparison is fair. The attention layers in vanilla Transformers are dense and fully connected, so the computation cost is much higher. Due to the computation constraints, we only scale it up to 32K sequence length. All of our implementations of attention variants are based on FlashAttention[<https://github.com/HazyResearch/flash-attention/tree/main>] for training efficiency. We customize the flash attention kernels for both sparse attention and dilated attention. <ref> summarizes the results of these models on the Stack dataset. We use perplexity as the evaluation metric. The models are tested with different sequence lengths, ranging from 2K to 32K. When the input is longer than the maximum length that the models support, we implement blockwise causal attention (BCA) <cit.>, a state-of-the-art extrapolation method for language model inference. Besides, we remove the absolute position encoding. Primarily, the results demonstrate that increasing the sequence length during training generally leads to a better language model. Secondly, the extrapolation of sequence length in inference does not apply to the case when the length is much larger than the model supports. Finally, consistently outperforms the baseline models, proving its effectiveness in language modeling. §.§ Scaling Curves of Sequence Length Previous work <cit.> has shown that language models follow some scaling laws by increasing parameters or training tokens. We are interested in the performance of language models when the context length is scaled up during training. We test the losses with inputs of a mixture of different lengths, from 1K to 32K. We use blockwise causal attention during inference to improve the generalization of sequence lengths. <ref> plots the scaling curves of sequence length for both vanilla Transformers and . We estimate the amount of compute by calculating the total flops of matrix multiplication. The results show that both vanilla Transformers and benefit from a larger context length during training. However, can scale up the context length more efficiently, achieving a lower test loss with a smaller amount of computing. This demonstrates the advantage of longer training input over extrapolation. In conclusion, our experiments show that is a more efficient way to scale up the context length in language models. This is because can learn longer-range dependencies more effectively. §.§ Scaling up Model Size An important property of large language models is that the loss scales as a power law with compute. To verify whether still follows the similar scaling law, we train a series of models with different model sizes, from 125 million to 2.7 billion parameters. The 2.7B model is trained with 300B tokens, while the rest digest about 40B tokens. <ref> plots the scaling curve of regarding the compute. We compute the perplexity on the same test set. The amount of compute is estimated by calculating the total flops of matrix multiplication during training. It proves that can still follow the power law. This implies that the dense Transformer is not a prerequisite for scaling the language models. Additionally, the scalability and the efficiency are both obtained by . §.§ Long Context Prompting Prompting is an essential method to guide and provide additional information to the language models. We conduct experiments to verify whether can benefit from a longer context window for prompting. Specifically, we reserve a piece of prefixes as the prompt and test the perplexity of its suffixes. We gradually scale the length of the prompt from 2K to 32K. For a fair comparison, we keep the suffixes the same, while increasing the length of the prefixes to the maximum lengths of the models. The results on the test set are reported in <ref>. It shows that the test loss of gradually decreases as the context window grows. This demonstrates the superiority of in fully leveraging the long context to improve the language model. § CONCLUSION AND FUTURE WORK We present , a Transformer variant that can scale the sequence length to 1 billion tokens and beyond, with no loss in shorter sequences. The core of is dilated attention, which reduces the computation complexity from quadratic to linear. can be served as a distributed trainer that parallelizes the training of a sequence across multiple GPU devices. Experiments show that has superior performance over the strong baselines on modeling both long and short sequences. In the future, we will extend to support more tasks, e.g., multimodal large language modeling <cit.>, BEiT pretraining <cit.>, and genomic data modeling. Acknowledgement We would like to acknowledge Yuqing Xia and Jilong Xue for the early exploration of the flash attention kernel. logsparse alpha § HYPERPARAMETERS
http://arxiv.org/abs/2307.01380v2	20230703221728	Information Synergy Maximizes the Growth Rate of Heterogeneous Groups	[ "Jordan T Kemp", "Adam G Kline", "Luís MA Bettencourt" ]	physics.soc-ph	[ "physics.soc-ph", "nlin.AO" ]	APS/123-QED ^1Department of Physics, University of Chicago, Chicago, Illinois 60637, USA ^2Department of Ecology and Evolution, University of Chicago, Chicago, Illinois 60637, USA ^3Mansueto Institute for Urban Innovation, University of Chicago, Chicago, Illinois 60637, USA Collective action and group formation are fundamental behaviors among both organisms cooperating to maximize their fitness, and people forming socioeconomic organizations. Researchers have extensively explored social interaction structures via game theory and homophilic linkages, such as in kin selection and scalar stress, to understand emergent cooperation in complex systems. However, we still lack a general theory capable of predicting how agents benefit from heterogeneous preferences, joint information, or skill complementarities in statistical environments. Here, we derive a general statistical dynamics for the origin of cooperation based on the management of resources and pooled information. Specifically, we show how groups that optimally combine complementary agent knowledge about resources in statistical environments maximize their growth rate. We show that these advantages are quantified by the information synergy embedded in the conditional probability of environmental states given agents' signals, such that groups with greater diversity of signals maximize their collective information. It follows that, when there are constraints placed on group formation, agents must intelligently select with who they cooperate with to maximize the synergy available to their own signal. Our results show how the general properties of information underlie the optimal collective formation and dynamics of groups of heterogeneous agents across social and biological phenomena. Information Synergy Maximizes the Growth Rate of Heterogeneous Groups August 1, 2023 ===================================================================== § INTRODUCTION Collective behavior is a general feature of biological and social systems. It mediates the survival and evolution of populations under resource constraints, competition, or predation in natural systems <cit.> and the formation and persistence of social organizations in human societies <cit.>. Much past work has modeled collective dynamics using homogeneous interaction rules, common to all agents, that are often also phenomenological. While these models have produced diverse insights, they typically lack a theoretical foundation to explain how specific social behavior emerges among individual agents with heterogeneous information and behavior. Thus, there remain significant knowledge gaps in most realistic situations, where agents with distinct but potentially complementary traits act collectively to maximize their joint growth (fitness, wealth) in knowable, noisy environments. Some examples help illustrate the present situation. Game theorists and ecologists have considered many different cooperative interaction schemes <cit.> and explored evolutionary stable behavior <cit.>, particularly on networks <cit.>, where optimal behavior is identifiable under given interaction rules. Elaborating these schemes by introducing higher order interactions has broadened our understanding of more complex social networks <cit.>, and their dynamical phase-stability under varying interaction strengths <cit.>. Researchers have also studied, both theoretically and in the laboratory, how memory of previous interactions influences agents' preferences for future encounters <cit.>, the spread of social crises across distance <cit.>, and the formation and scaling properties of social collectives <cit.>, such as cities <cit.>. In addition to interaction rules and associated payoffs, collective dynamics is predicated on maximum principles, which specify agents' preferences in view of a goal and thus render their behavior intelligent (optimal). For example, inclusive fitness theory, which assumes a reproductive benefit to cooperation because of shared genes <cit.> has been studied in mixing populations and over networks <cit.> where it predicts population benefits to cooperation through several forms of reciprocity <cit.>. More recently, researchers have studied resource pooling in models of growth as a means to minimize environmental uncertainty and associated loss of fitness among agents experiencing independent fluctuations with shared statistics <cit.>. Such approaches remain limited by the association between collective behavior and (genetic) homophily but they can help explain the existence of phase transitions in cooperation networks <cit.>, and specify agents' plausible behavioral patterns <cit.>, even if doubts remain about inclusive fitness's predictive power <cit.>. Generally, however, most current quantitative frameworks fail to address collective dynamics when agents remain heterogeneous across skills, knowledge, and behavior <cit.>. Developing more general approaches to collective behavior that include adaptation and learning along with heterogeneity, is a crucial step towards understanding how agents self-organize in more complex and dynamical environments, where specialization and the division of labor and knowledge become key. Adaptive behavior requires agents to acquire and process information over time <cit.> in response to their environments and to each other. In any realistic situation, limited experience, specialization costs and physical limitations of effort, energy and time, all prevent agents from perfecting their knowledge of complex environments <cit.>. A natural way to mitigate these individual limitations is to pool knowledge across agents leading to the formation of social organizations <cit.>, and the division and coordination of labor in terms of their behavior <cit.>. This is widely observed in human organizations, but also in animal social behavior starting with the division of labor by age and sex. By working jointly to predict characteristics of their environment <cit.> and gather resources, groups of agents can maximize their collective fitness even when each individual has very limited knowledge. In a setting where there are resource returns to successful prediction and behavior, information of the state of a statistical environment determines the fitness of the population <cit.>, though there are questions about how such benefits emerge quantitatively <cit.>. Here we formalize the calculation of these social benefits in terms of the properties of information and show how maximizing knowledge complementarities (synergy) maximizes the long-term growth rate of collectives. Specifically, we derive an expression for the additional payoff to cooperative behavior in terms of the joint information synergy about the agents' dynamical environment. These results lead us to introduce the principle of maximum synergy, which maps the maximization of collective resource growth rates into optimal social interaction structures. This work adds new dimensions to the study of collective dynamics by connecting the structure of groups to that of information in complex environments mediated by agents' diverse subjective characteristics, such as their present knowledge and their life histories. § THEORY OF COLLECTIVE GROWTH We start by demonstrating how the benefits of collective action emerge from pooling information in synergistic situations. Synergy means the combination of behavior, knowledge, and skills that complement each other towards a goal. This concept is necessary for creating effective organizations that embody complex information <cit.>, but it is often not sufficiently formalized in common language, such as in discussions of innovation <cit.> or firm structure. Here, we will refer to synergy as an explicit information theoretic quantity that measures the additional predictive power that a group has upon pooling its agent's information together, relative to the knowledge of each individual separately. This quantity has been introduced sometime ago in the context of studying circuits in information processing systems <cit.>, and has provided a framework for studying higher-order neuron interactions in the brain <cit.>, and causality and information in complex systems <cit.>. As we will show, synergy results formally from the conditional dependence between the probability of predictive signals distributed in a population and events in a shared environment. The gain in predictive power from agents pooling information as collectives allows them to obtain additional resources from a knowable environment beyond what agents alone can do, thus boosting their relative fitness or productivity. It follows that collectives that seek to maximize their resources over long times must combine the information from their agents' individual models of the world in a way that accesses the most synergy. Groups that do not know a priori how to realize their synergies must adjust their collective knowledge and interaction structure by observing outcomes of their environment in an iterative learning process. After developing the general framework for group formation and collective growth across group sizes, we demonstrate a model environment that exhibits synergy using logic gates that take signals as inputs, and output probabilistic events. We will demonstrate how synergy scales with the number of unique signals in a collective, and how specific combinations of signals affect the average growth of resources for the group. §.§ Collective Growth in Synergistic Environments We consider a population of N agents, each with initial resources r_i that can be (re)invested into the set of outcomes of their environment to generate returns. Each agents has access to a private signal, s_j∈ S_j, which is used to predict the state of the environment and makes resource allocations to possible outcomes e∈ E. This signal may represent a number of different processes such as sensory input, or a lead retrieved from memory. With accurate parameterization of a model of the environment, P(E\|S_j), an agent's optimal investment strategy leads to a resource growth rate γ=I(E;S_j) <cit.>. Agents with better models (and better statistical estimations) experience higher average growth rates given by the information rate of the agent's signal for the environment. We now define our environment by a set of l signals with unique statistics, S≡{S_1,… S_l} as P(E\| S), with marginals of events P(E) and signals P( S). The joint information that S has on E is at least equal to S_j, that is I( S;E)≥ I(S_j;E). Generally, this inequality is strict if the conditional information I(S\|E)>I( S) <cit.>. We compute the total information by summing over the mutual information between the signals independently, subtracted by an interaction term across signals, I(E; S)=∑_jI(E;S_j)-R_P. The quantity, R_P, denoted the coefficient of redundancy, measures the strength of this conditional dependence across larger sets of signal (two, three, etc way). It is defined in App. <ref> R_P = ∑_j>k=1^lR(E;S_j;S_k)+∑_j>k>m=1^lR(E;S_j;S_k;S_m) +…+R(E;S_1;…;S_l). The coefficient of redundancy can have a positive or negative value, indicating different conditional relationships between the signals and environmental states. When R_P>0, there is information across signals irrespective of environmental events. This means that signals are redundant, and consequently there are diminished returns to pooling information as I(E; S)<∑_jI(E;S_j). Conversely, when R(E; S)=0, the signals are statistically independent, and the benefits of pooling information increase linearly with the information of each signal on the environment but there is no synergy. Finally, when R(E; S)<0, there is conditional dependence of the signals on the environment. This is called synergy and corresponds to a superlinear benefit to pooling information in the number of agents, above and beyond the information contributed from each signal individually. §.§.§ Group formation and collective decision making We have now defined individual resource growth as a quantity of information and discussed how information can be aggregated across signals to express their synergy relative to states of the environment. Now we can explore how agents with different signals can pool information together as coordinated groups, and access the synergy in their environment through collective decision-making. Consider the undirected hypergraph H=(A,G) of vertices, A, and hyperedges G. We consider a discrete number of vertices, A={a_1,a_2,…,a_N}, where a_i identifies agent i. The set of hyperedges, g∈ G={1,2,…}, called groups, defines the number of cooperating collectives. A hyperedge connects 1≤ N_g≤ N agents, and we assume that agents can only belong to a single group. Therefore, by construction, ∑_g N_g=N and the sum over all nodes of every hyperedge yields the number of agents in the population. There exist two extremes of cooperation. First, when a single hyperedge spans every node, meaning all agents pool information in one group. In the limit of no cooperation, N_g=1 for all g, and no agents pool information. In this case, the dynamics of the model are similar to previous work <cit.>. Let S_g be the set of unique signals held by the agents of a group g to be pooled, such that S_g⊆ S. The number of cooperants is defined by the number of unique signals, \| S_g\|=k_g, and is bounded by 1≤ k_g≤ l. When k_g=l and the group has a complete signal, the collective can make maximally informed decisions. Conversely, when k_g<l, the signal is considered incomplete, and the collective can only interpret and act on a subset of signals. As we will see, the number of unique signals a collective can observe determines the amount of information they can access. Now that we have defined how agents organize into groups of various sizes, we can discuss how agents pool their information to make collective decisions and grow their resources in dynamic environments. At every time step, a collective with access to all signal types observes a unique private signal s={s_1,…,s_l}∈ S. Each agent then allocates its resources r_i on events according to collective g's allocation matrix B(E\| s). As the event e is observed, the agent is rewarded with returns w_e to the fraction of resources invested in e, B(e\| s). In the limit of many sequential investments n, the average growth rate of resources converges to γ=1/nlogr_n/r_i≈∑_e, sP(e, s)log[B(e\| s)w_e], The optimal investment in the large n limit is the conditional probability of the event given the signals, B(e\| s)=P(e\| s). When the rewards are “fair", and w_e=1/P(e), the optimal growth rate is given by the mutual information <cit.> defined in equation <ref>, γ=I(E; S). The typical collective may not have a complete signal, and instead may only observe and interpret a subset of all unique signals S_g. Their optimal allocation, given by P(E\| S_g), then has mutual information I(E; S_g)≤ I(E; S), with equality only if the omitted signals are completely redundant with present signals. Unless there are redundant signals, an incomplete group is guaranteed to have suboptimal information and growth rate. Furthermore, agents may also not start out with perfect knowledge and must invest using their best estimate of the true environmental probability, X(E\| S_g)≠ P(E\| S_g). In this case, the collective's average growth will be submaximal by the number of signals and lack of information on signals, and is described by γ_g=I(E; S_g)-E_ s_g(D_KL[P(E\| s_g)\|\|X(E\| s_g)]), where E_ s_g is the expectation value over the states of the group's signals, and D_KL[P(E; s_g)\|\|X(E; s_g)]=∑_eP(e\| s_g)log(P(e\| s_g)/X(e\| s_g))≥0 is the Kullback-Leibler divergence, an information measure expressing how similar the distributions are in their inputs. This result shows that collectives with both a better model as reflected by the first term, a better characterization of the model and its various synergies by the second, and a more complete signal will experience higher growth rates. Furthermore, γ_g<γ unless g is the full set of signals, so it is typically valuable to add more signals to the group. This setup is illustrated in Figure <ref>. §.§ Maximum synergy principle and optimal growth These results introduce important considerations for how collective innovation and growth determine strategies for group formation. In theories of cooperation such as kin selection <cit.> and scalar stress <cit.>, group formation is advantaged by member relatedness and disadvantaged by unfamiliarity. This is intuitive in many situations, as agents are more likely to cooperate when they are more certain others will reciprocate <cit.>, and cooperating with similar agents minimizes this uncertainty. Equation <ref> counters this intuition by defining an explicit benefit to cooperating with dissimilar agents across heterogeneous, complementary skills and information. Specifically, a group with more synergistic signals, as defined through the conditional dependence of their decisions on states of the environment, will experience higher growth. So, even if there are additional coordination costs for more heterogeneous agents, there is now a possiblity that cooperation will emerge as there are also greater informational benefits, formalizing intuitive ideas about the value of diversity <cit.>. The beneficial contribution of synergy to the growth rate of resources provides an important input to models of random multiplicative growth, such as those commonly used to study wealth dynamics and mathematical finance. In its simplest form, the stochastic growth rate in such models is characterized by its first two temporal moments. The average over time, η, and the resource temporal standard deviation (volatility), σ, combine under Itô integration to give actual growth rate γ=η-σ^2/2. Maximizing this growth rate (as a positive quantity) entails maximizing η and minimizing σ, which at the individual agent level can be achieved by (Bayesian) learning over time <cit.>. At the population level, it has been proposed that pooling resources in groups would naturally emerge as a means to reduce σ, when growth rate fluctuations are independent across agents, and thus maximize γ <cit.>. Our results introduce a different possibility of cooperation, through pooling information in structured groups, that maximizes η (and γ) through synergy effects. Thus, to maximize γ, agents should pool information with a most diverse set of collaborators possible to access the most mutual synergy viz. the environment. This maximum synergy principle defines the benefit of intelligent collective behavior in complex environments where there are agent level limitations to knowing the environment fully and where mechanisms of the division of labor and knowledge are favored. This principle is general and applies across levels of cooperation, whether it be individuals matching skills to form groups, or specialized groups organizing into more complex collectives <cit.>, all the way to large scale societies. Generally, these two strategies– information synergy versus resource pooling under independence– are distinct modes of cooperation over which groups can maximize γ, as demonstrated in Figure <ref>. As we will see later, the decision of who to cooperate with is not trivial, as different combinations of signals may yield varying synergies. This means that under constraints of group size, such as from cooperation costs per connection, groups satisfying the maximum synergy principle must intelligently select which signals and agents to integrate, and which to exclude as redundant. Furthermore, collectives may not a priori know the optimal allocation strategy that leverages the synergy available to their signals, meaning that intelligent collective behavior must itself be learned over time and by exploring the best possible matchings. We will now develop the dynamics of how a group can organize itself optimally so as to maximize its synergy. §.§.§ Synergy maximization through Bayesian inference Bayesian learning is the optimal strategy to incorporate new information from observed events into the estimate of conditional probabilities, such as those of environmental states given agents' signals <cit.>. Agents can also learn the synergy embedded in their environment in groups by collectively weighing their conditional observations across their individual signals. A group wanting to maximize their synergy must then update their conditional relationship through a Bayesian inference process X_n(e\| s)=AP( s_n\|e_n)X(e_n)=[∏_i=1^nP( s_i\|e_i)/P( s_i)]X(e), where the normalization A=(∫ de_nP( s_n\|e_n)X(e_n))^-1. We take the prior probability, X(e_1) = X(e), because we are assuming that the environment is stationary or at least slowly changing relative to groups’ learning rates. Bayesian inference converges X(E\| S)→ P(E\| S) over time, decreasing the information divergence, and maximizing synergy and average growth. For groups with incomplete signals, the information acquired through learning is still bounded by what is available in the incomplete signal space. We have thus far defined collective growth in terms of information synergy, and shown how agents can learn as a collective to increase their growth rate over time. We will now illustrate these general results using a model based on logic circuits. § MODELING SYNERGY WITH LOGIC CIRCUITS Logic circuits have been used extensively as models for synergistic interactions <cit.>. This is because their outputs are predicted by combinations of inputs, much like events are predicted by combinations of signals. Among other logic circuits (like AND or OR), the XOR gate is unique in that information between inputs and outputs only exists as synergy across all inputs <cit.>; no individual input has mutual information with the output. In the following, we will show how modifying the XOR gate relaxes this condition, such that information exists for any input and scales on average with the number of cooperating signals. Similar to <cit.>, while this model will be used to study synergy in a simplified setting, the theory is defined for general dynamical environments. §.§ The Uniform XOR Gate Consider the space of statistically independent binary signals s_j∈ 0,1, such that a sample set s has uniform probability P( s)=2^-l. We assign each input s a binary event, e∈ 0,1, using the generalized XOR rule, e=M_2( s)≡[∑_j=1^ls_j ](mod 2) with binomial probability p_s. From the sets of sampled signals, s, and binomial coefficients p={p_s}, we can define this generalized XOR circuit as a joint distribution on signals and events as P(E, S\|p)≡ f( p,l)=1/2^l∏_ s(p_ s)^M_2( s)(1-p_ s)^1-M_2( s). This distribution is called the uniform XOR (UXOR). It performs a unique, l dimensional XOR gate on each input s with probability p_ s. In the case where p_ s=1 for all input permutations, this circuit behaves deterministically like an XOR gate, and the complete group has 1 bit of information. In the limit of p_ s=.5, this no longer models a logic gate as the output is uncorrelated to the inputs. The truth table of this circuit is shown in Figure <ref>A for an environment with two signals. §.§.§ Information scaling in the UXOR environment With this explicit choice of distribution, we can explore quantities of information that will define a group's growth process. For simplicity, we choose a uniform prior for the distribution of p, but in principle any prior distribution is admissible. The information available in the environment measures the maximum average growth rate a group with a complete signal can experience. When averaged over all configurations of p, the information is given by I(E;S)=log2-1/2≈ .28 bits (App. <ref>). For groups with incomplete signals (when k_g<l), we compute the information by marginalizing equation <ref> over the λ_g=l-k_g signals unavailable to the group. The procedure for marginalization is defined in App. <ref>, but in general, marginalization of one signal halves the size of the parameter space p that describes the distribution. The average information for an incomplete signal is approximately (App. <ref>) I(E; S_g\| p)≈ 2^-λ(log2 - 1/2). Average information increases exponentially, ∼ 2^k, as more signals are included. The mutual information of the complete signal is independent of the number of signals, so the information of a single signal must converge to zero in the limit of large l. The exponential scaling of the information with the number of cooperants is demonstrated in Figure <ref>B, as lines on a logarithmic scale for environments of increasing l. The curves are computed by Monte Carlo sampling circuits for l signals by measuring the information after λ=l-k marginalizations. §.§.§ Growth and group learning Until now we have explored the mean behavior of this environment subject to a uniform prior. In general, collectives do not have perfect information on a single prior. In this case, their inaccurate guess for the set of binomial coefficients is parameterized by x_g≡{x_s_g}, indexed by the signals available to the group s_g∈ S_g, and the collective's likelihood model becomes X(e\| s_g)=f( x_g,k_g). The information divergence term of equation <ref> becomes the divergence between f( x_g,k_g) and f( p_g,k_g), where p has been projected into the subspace spanned by S_g, averaged over all signals E_ s_g[D_KL]=⟨ p_ s_glog(p_ s_g/x_ s_g)+(1-p_ s_g)log[(1-p_ s_g/(1-x_ s_g)]⟩ here angle brackets denote sample averages over the binomial values. Subtracting the mutual information by this term yields the growth rate under imperfect, incomplete group information. γ_g=⟨ p_ s_glog x_ s_g+(1-p_ s_g)log(1-x_ s_g)⟩+log2, We have so far described growth rate dynamics under a stationary x_g. To illustrate growth dynamics under group learning, we turn to the Latent Dirichlet Allocation (LDA) model. Through a categorical description of pairs of events and signals, agents experience average dynamics to x_g in the limit of high sampling rate ω=n/t≫1 x_g(t)= p_ gt/2κ+ x_g/1+t/2κ, where κ defines the Bayesian update time. The details of LDA are given in Ref. <cit.> and provide parametric dynamics that converge to full information as a power law in time, in stationary environments. To study the dynamics of resources in the UXOR environment, we simulated agent investments in a Monte Carlo sampled environment. We randomly assigned N=2000 agents signals in an l=4 environment, then randomly assigned them to groups sized l≤ N_g≤ 11. This results in an ensemble of groups with cooperants 1≤ k_g≤ 4. We reveal Bernoulli-sampled signals to the groups, whose agents make collective decisions on which events to allocate resources. For each group, we track the resources of a representative agent, informed by the group, investing their individual resources through time. Figure <ref> illustrates the results of this simulation. In subfigures A and B, the Monte Carlo simulated means are shown as solid lines, with 95% Confidence Interval (CI) shaded regions. Theoretical means are computed from the initial population configuration using equation <ref>, plotted as dashed lines, with hash-filled uncertainty regions. The more unique signals a group can access increases, the more they can learn, and the more resources they acquire over time. A high signal-to-noise ratio when k_g=1,2 causes growth rates lower than the theoretical mean, and cumulatively fewer resources over time. §.§.§ Constrained intelligent group formation For the groups with k_g<4 (incomplete signals) there is significantly higher variance in both information and resources compared to k_g=4. This is attributed to differences in synergy between groups with different combinations of signals of order k. This illustrates a general feature of the maximal synergy principal; that signal combinations with higher conditional dependence on the environment will have higher synergy and experience higher growth rates than other combinations. Figure <ref> demonstrates the synergy effects across different combinations of signals. For each group of size k, the left, smaller dot indicates the amount of information each signal has averaged over the signals present. The right, larger dot indicates the total information the combination of signals has when pooled. The difference between the two dots gives the amount of synergy. We see, for example, that even though signals 0 and 3 have less information than signal 2, both signals have higher synergy effects when pooled with 1 individually, as indicated by their crossover with the 1, 2 line. For a group aggregator, not only does this mean that signal choice is nontrivial, but also that individual information is not generally a good indicator of synergy benefits that can be realized when pooled. This result reinforces the complexity that fulfilling the maximum synergy principle entails, as one must understand signal complementarities for a given model of the environment to all orders, a likely costly process. As demonstrated by the bottom plots in Figure 4C, through a smart selection of p, we can also design special environments such as where either no synergy is present, or where there are uniform benefits of synergy across combinations of signals. The procedure for constructing environments with specific synergy profiles will developed in future work. § DISCUSSION In this paper, we developed a novel mechanism of cooperation among heterogeneous agents that use shared information to grow resources in noisy environments. We derived the benefits of cooperation in terms of synergy gained by pooling information across agents' unique signals. This motivates the principle of maximum synergy, whereby a group's aggregate growth is optimized when that group maximizes the synergy of its members relative to a statistical environment. We proposed this principle as a complementary avenue to cooperation compared to the reduction of volatility through resource pooling in multiplicative growth models. We then showed that a group with no a priori knowledge of its potential synergy can learn it through Bayesian inference. We illustrated these principles using a model of a high-dimensional probabilistic logic gate and showed that, on average, group synergy scales superlinearly (exponentially) with the number of unique signals in the group. We also illustrated the challenge faced by groups under constraints to size to pick not just unique signals but also admit new group members as additional signals that maximize their potential collective synergy. These results formalize several insights into the causes and benefits of cooperation. First, the properties of information allow us to consider how the limits to human effort and ability motivate group formation. Specialization through learning or adaptation is costly in terms of time and resources, motivating a division of labor to fully learn and maximize productivity across disparate but synergistic agents <cit.>. Second, these results motivate analyses of how information and resource pooling strategies affect different levels of selection within an organizational hierarchy. Effective resource pooling relies on uncorrelated fluctuations across participants, which is not possible when agents are making coordinated decisions across correlated signals. We therefore expect information and resource pooling strategies to create tradeoffs in group formation, and apply to different environmental features and levels of selection. Groups lacking informational complementarities (because they are homogeneous) operating in variable environments should pool resources. This may apply to people in insurance pools, or independent economic sectors within a common population, such as a city or nation. Conversely, groups in complex environments made up of agents with complementary knowledge, such as within a firm or innovation ecology, should engage in information pooling and skills specialization to maximize their production whenever the variability of the environment and costs of cooperation are sufficiently low. Parsing out these modes of cooperation becomes more important when considering how groups respond to changing environmental or social conditions. As new environmental conditions emerge, such as new industries, the distribution of synergy across different group configurations will also change, selecting for different group compositions and skills combinations. This has the interesting implication that new knowledge (science, technology, institutional change) should be disruptive of established social and economic structures because it enables new synergies. This also has implications for natural ecosystems <cit.> where changing environmental conditions and variability, such as via climate change, may alters their structures. Third, the framework developed here describes a general approach to describing interaction dynamics in many fields. The conditional probabilities P(e\|s) capture the general structure of information between populations and their environment. Through synergy, that information becomes encoded in how groups form and are structured, and which sets of coordinated behaviors produce beneficial or detrimental behaviors across agents. By tracing over states of E, averaging over (stationary) environments, we can produce a set of rules for (average) rewards associated with agents' perceptions and actions. This shows how general conditional probabilities of choices and behaviors in given environments may underlie particular games and other phenomenological agent interaction rules <cit.>. Furthermore, because conditional distributions are general and multi-dimensional they also provide natural models of higher order interactions expressing large groups' synergy, such as reciprocal cooperation and the emergence of culture as shared knowledge and behavior <cit.>. In summary, the formal properties of information, made explicit over group structures and time, provide the theoretical basis for a broad class of agent interaction models found throughout the social and ecological sciences. This includes the formation of complex societies made up of diverse cooperating agents in situations where large scale synergy becomes possible. This work is supported by the Mansueto Institute for Urban Innovation and the Department of Physics at the University of Chicago and by a Na- tional Science Foundation Graduate Research Fellowship (Grant No. DGE 1746045 to JTK), and by the National Science Foundation, through the Center for the Physics of Biological Function (PHY-1734030), as well as the National Institutes of Health BRAIN initiaitive (R01EB026943) to AGK. § APPENDIX §.§ Information Aggregation Consider a target statistical variable E (environment), that we wish to predict using l other variables (signals) S={S_1,…,S_l}. The mutual information between each signal S_i separately and E is given by <cit.> I(E;S_i)=H(E)-H(E\|S_i)=-Δ H(E)/Δ S_i. where H(E) is the Shannon entropy of E, and the variation measures the difference in entropy of the event when conditioned on the signal. From the rules of information aggregation, this expression generalizes to information across every added signal <cit.>. The mutual information between the event and the set of several signals is given by I(E; S)= -∑_i=1^lΔ H(E)/Δ S_i -∑_i>j=1^lΔ^2H(E)/Δ S_iΔ S_j -… -Δ^lH(E)/Δ S_1…Δ S_l. The first term of this expansion is just a sum over the mutual information of each individual signal and the environment. The goal of this section is to show that the inclusion of each new signal introduces a coefficient of redundancy of progressively higher order. The first term is Δ ^2 H(E)/Δ S_iΔ S_j= H(S_i)-H(S_i\|E)+H(S_j)-H(S_j\|E) -H(S_i,S_j)+H(S_i,S_j\|E) = H(S_i)-H(S_i\|E)-H(S_i\|S_2)+H(S_i\|S_j,E) = I(S_i;S_j)-I(S_i;S_j\|E)≡ R(E;S_i;S_j), where we used the identity H(A,B)=H(A\|B)+H(B). We denote R as the coefficient of redundancy, which measures the difference in mutual information between the variables, I(S)≡ I(S_1;…,S_k), and the mutual information of the variables conditioned on E, I(S\|E). When I(S_i;S_j)<I(S_i;S_j\|R), the signals contain less mutual information in the absence of the event (we gain information by considering the event), and R(S_i;S_j;X)<0. In this case agents experience a positive benefit from pooling information, which we call synergy. To demonstrate this effect to higher orders in goups of signals, we perform a similar calculation for a three-signal interaction. Δ ^3 H(E)/Δ S_iΔ S_jΔ S_k= H(E)-H(E\|{S_i,S_j,S_k}) = H(S_i)+H(S_j)+H(S_k)-H(S_i\|E) -H(S_j\|E)-H(S_k\|E)-H(S_i,S_j)-H(S_j,S_k) -H(S_k,S_i)+H(S_i,S_j\|E) +H(S_j,S_k\|E)+H(S_k,S_i\|E) +H(S_i,S_j,S_k)-H(S_i,S_j,S_k\|E) = H(S_i,S_j,S_k)-H(S_i\|S_j)-H(S_j\|S_k) -H(S_k\|S_i)+H(S_i\|S_j,E)+H(S_j\|S_k,E) +H(S_k\|S_i,E)-H(S_i,S_j,S_k\|E) = I(S_i;S_j;S_k)-I(S_i;S_j;S_k\|E)≡ R(E;S_i,S_j,S_k). We see that an analogous redundancy coefficient arises in three dimensions. This can generally be retrieved for arbitrary number of dimensions through a similar iterative procedure. We refer to the sum of these moments collectively as the redundancy of the joint distribution, denoted R_P <cit.>, R_P≡ -∑_i>j=1^lΔ^2H(E)/Δ S_iΔ S_j-… -Δ^lH(E)/Δ S_1…Δ S_l = ∑_i>j=0^l[I(S_i;S_j)-I(S_i;S_j\|E)]+… +I(S_1;…;S_l)-I(S_1,…,S_l\|E) . Note that redundancies of lower order than cardinality of the signal space must be computed over every combination of signals. For example, when l=3, there are three second order redundancy terms. This expansion generally defines the benefits to cooperation over increasingly higher orders of cooperation (number of signals). This expression can be used to compute the relative strengths of the various orders of interaction for any set of signals and environmental variables, given their conditional distributions. §.§ Kelly Growth rate Consider an environment with events conditionally dependent on signals characterized by a joint distribution P(E,S) for event E and l signals S. Consider a cooperative Kelly investment scheme whereby each participant, agent i, witnesses signal s_i∈ S_i, and informs the collective how to invest their shared resources r. The mechanics of pooling resources and collectively investing will be discussed below. Kelly's formalism can be adapted by expanding the environmental probability to contain l signals, P(E,S)→ P(E,S), as can the betting matrix X(E\|S)→ X(E\|S), where S={S_1,…,S_l}. When odds are fair, the Kelly growth rate is given by the returns to each investment, averaged over the probability of that signal, event pair G=∑_e,s^E,Sp(e,s)logx(e\|s)/p(e). We expand this equation by inserting p(e,s) into the numerator and denominator of the log G=∑_e,s^E,sp(e,s) [ logp(e,s)/p(s)p(e) -logp(e\|s)/x(e\|s)]. These two terms can be simply expressed as G=I(E;S)-E_s[D_KL(P(E\| s)\|\|X(E\| s))], similar to previous work, but we can decomposes this equation in terms of redundant information across the signals using equations <ref> and <ref>. §.§ Information for UXOR circuits Here we compute I(E;S) for the UXOR logic circuit. This represents the information that a group of agents with l distinct signals have about the output E of the probabilistic gate, averaged over all configurations of the gate for a uniformly distributed prior. Because the signals s_i are independent Bernoulli trials with probability 1/2, I(E; S) = ∑_e, s P(e, s) logP(e, s)/P(e)P( s) = 1/2^l∑_e, s P(e\| s) logP(e\| s)/P(e) . Then, using the fact that the output E is also a Bernoulli variable, P(e=1\| s) = 1-P(e=0\| s), and I_l = I(E; S) = 1/2^l∑_ s g(P(e=0\| s)) - g(P(e=0)) , where g is a function representing application of the UXOR gate and is defined over binomial parameters x as g(x) = x log x - (1-x) log (1-x) . The number of terms in (<ref>) grows exponentially with l and quickly becomes large. When it is sufficiently large, the sum can be approximated by an average. In particular, for uniformly distributed P(e=0\|s), 1/2^l∑_ sg(P(e=0\| s)) ≈⟨ g(x) ⟩_x∼ U(0,1). This expectation can be analytically evaluated. With this, (<ref>) gives I_l = ⟨ g(x)⟩_x∼ U(0,1) - g(1/2) = log 2 - 1/2 . §.§ Mutual Information for incomplete signal sets Here, we demonstrate that the information the collective has about the gate output scales exponentially with respect to the number of cooperants, k, as is depicted in Figure <ref>. Following the previous section, the introduction of the function g(x) = xlog x + (1-x)log(1-x) simplifies the expression for mutual information I_k = I(E; S) = 1/2^k∑_ s g(P(e=0\| s)) - g(P(e=0)) . As before, this sum can be interpreted as an average over the uniform distribution when the number of terms is large. Here, the removal of parts of the signal changes the distribution of parameters, so the measure that approximates this sum also changes. We call this new measure P_k. Furthermore, whereas in the main text, the subscript of S_g denoted the signals of group g, here the subscript of S_k will denote the signal set of cardinality k to be marginalized. With this new notation, the first term in (<ref>) may be written approximately as: 1/2^k∑_ s_kg(P(e=0\| s_k)) ≈⟨ g(x) ⟩_x∼ P_k. To compute P_k for k<l, we need to calculate how the probability of E conditional on the remaining signals s_k changes under the removal of the k^th signal. For this model, P(e\| s_k-1) = 1/2(P(e\| s_k-1,s_k=0) + P(e\| s_k-1,s_k=1)). By iterating this sum, we reduce the number of parameters required to describe P_k(k), which in the main text are given by the set of binomial parameters p. Additionally, the distribution P_k(x) becomes increasingly narrow, centered around 1/2, which is the mean of all probabilities P(e\| s). Parameterizing P_k(x) by its moments allows us to directly compute the mutual information. The moment expansion of the distribution is given by I_k ≈⟨ g(x) ⟩_P_k - f(1/2) = ∑_a=0^∞1/a!f^(a)(x_0)⟨ (x-x_0)^a ⟩_P_k - f(1/2). Using standard arguments, which we provide in the following section, these moments approximately scale like m_k-1^(a) = ⟨(x - 1/2)^a⟩_P_k-1≈m_k^(a)/2^a/2 , m_l^(a) = 1/(a+1) 2^a , which is related to the onset of central limit theorem behavior. This provides us with a heuristic explanation for why I_k scales as 1/2^λ. After only a few cooperants are removed, higher order terms in the expansion (with order denoted by a) quickly die away, leaving only the second-order term I_k ≈∑_n=0^∞f^(a)(1/2)/a! m_k^(a)→ 2 m_k^(2) + O((m_k^(2))^2) . Once this occurs, we can see plainly that between each marginalization the mutual information reduces by half, I_k/I_k+1→1/2. While this explanation gives approximately the correct scaling behavior, it does not admit a good estimate of I_k near full cooperation, since there the higher-order terms in the expansion are not small. To explicitly include these terms, we need all derivatives of f, evaluated at x=1/2 f^(a)(1/2) = (-1)^a2^a a!/a(a-1). Inserting these derivatives and the approximation (<ref>) for moments of P_k gives an approximation of I_k as a series. Then, evaluating this series analytically yields a closed form expression. I_k = ∑_a=0f^(a)(1/2)/a! m_k^(a) - g(1/2) ≈∑_a=1^∞(2a + 1)!/(2a-1)!1/2^λ a = 1/2[(2^-λ/2 + 2^λ/2) arctanh(2^-λ/2) + log(1-2^-λ) - 1] , λ > 0 This expression gives a good approximation for I_k in the small λ regime and also captures the scaling in the intermediate regime. For λ→∞, an estimate of the asymptotic behavior is given by setting z = 2^-λ/2 and Taylor expanding around 0. 1/2[(z + z^-1) arctanh(z) + log(1-z^2) - 1] →1/2[(z + z^-1)(z + 1/3z^3 + O(z^4)) - z^2 + O(z^4) - 1] =1/6 z^2 + O(z^4) Because z=2^-λ/2, the quadratic leading term agrees with the observed exponential scaling I_k ∝ 2^-λ. Although the scaling prediction is correct, there is not a regime where this high-λ expression consistently estimates I_k in its actual value. The essential reason is that as k decreases, the number of terms in the sum over remaining signal states decreases, and the approximation of that sum as an average over P_k(x) begins to break down. This manifests as an error in the overall scale of the estimate, but not in the exponential dependence on k. Instead, using the fact that I_l is known to a very good approximation and further that information is approximately exponential in k, a good estimate for I_k at large to intermediate k is given by I_k = 2^-λI_l = 2^-λ( log 2 - 1/2) . This is the quantity quoted in (<ref>) in the main text. §.§.§ Moment scaling with k The following argument justifies (<ref>) and is standard. We produce it here for completeness. Upon moving from k to k-1 cooperants, the new conditional distribution is given by (<ref>). This means that P_k-1 is given by a convolution of P_k with itself. P_k-1(x) = ∫_ℝ^2 dy dz δ(x - 1/2(y + z))P_k(y)P_k(z) = 2 ∫_ℝ dy P_k(2x - y)P_k(y) The characteristic function of a distribution over a continuous variable is its Fourier transform. Since P_k-1 is a convolution, its characteristic function is the product of characteristic function of P_k with itself. A slightly more convenient object to work with is therefore the logarithm of the characteristic function: φ_k(z) = log∫_ℝ dx P_k(x) e^izx The sum rule above gives a recursion relation for φ_k φ_k-1(z) = 2φ_k(z/2) + log 2 . Now, the cumulants of P_k can be calculated from φ_k c^(a)_k = 1/i^ad^a/dz^aφ_k(z)\|_z=0 , Meaning there are also recursion relations for the cumulants: c^(a)_k-1 = 2^1-ac^(a)_k This leads directly to the central limit theorem, since when the second cumulant is rescaled to remain constant with respect to n, all higher-order cumulants scale to zero. Here, by using the fact that the n^th moment m^(a)_k can be expressed in terms of cumulants c^(b)_k for b≤ a, we can see that the second-order cumulant dominates all of these expressions once k is sufficiently small. For example, the fourth moment quickly scales like (m^(2))^2 as k is decreased from l because the second cumulant begins much larger than the fourth cumulant and remains dominant. c^(2)_k = 2^-λ c^(2)_l = 1/12·2^λ c^(4)_k = 2^-3λ c^(4)_l = -1/120 · 2^3λ m^(4)_k = c^(4)_k + 3 (c^(2)_k)^2 ≈ 3 (c^(2)_k)^2 ∼ 2^-2λ Hence, due to (<ref>), m^(4)_k-1≈ m^(4)_k/4, which agrees with (<ref>). ieeetr
http://arxiv.org/abs/2307.00558v1	20230702125241	Conditionally Invariant Representation Learning for Disentangling Cellular Heterogeneity	[ "Hananeh Aliee", "Ferdinand Kapl", "Soroor Hediyeh-Zadeh", "Fabian J. Theis" ]	cs.LG	[ "cs.LG", "q-bio.QM" ]	Entropy Accumulation under Post-Quantum Cryptographic Assumptions Rotem Arnon-Friedman August 1, 2023 ================================================================= This paper presents a novel approach that leverages domain variability to learn representations that are conditionally invariant to unwanted variability or distractors. Our approach identifies both spurious and invariant latent features necessary for achieving accurate reconstruction by placing distinct conditional priors on latent features. The invariant signals are disentangled from noise by enforcing independence which facilitates the construction of an interpretable model with a causal semantic. By exploiting the interplay between data domains and labels, our method simultaneously identifies invariant features and builds invariant predictors. We apply our method to grand biological challenges, such as data integration in single-cell genomics with the aim of capturing biological variations across datasets with many samples, obtained from different conditions or multiple laboratories. Our approach allows for the incorporation of specific biological mechanisms, including gene programs, disease states, or treatment conditions into the data integration process, bridging the gap between the theoretical assumptions and real biological applications. Specifically, the proposed approach helps to disentangle biological signals from data biases that are unrelated to the target task or the causal explanation of interest. Through extensive benchmarking using large-scale human hematopoiesis and human lung cancer data, we validate the superiority of our approach over existing methods and demonstrate that it can empower deeper insights into cellular heterogeneity and the identification of disease cell states. § INTRODUCTION Learning high-level, latent variables from multi-domain[In this paper, multi-domain data refers to different datasets that may have different characteristics, such as different distributions or different sources of biases.] data that explain the variation of data within each domain as well as similarities across domains is an important goal in machine learning <cit.>. The task involves identifying features of the training data which exhibit domain-varying spurious correlations that are unwanted and features which capture the true correlations of interest with labels that remain invariant across domains <cit.>. It has been shown that the data representations or predictors that are based on the true correlations improve out-of-distribution generalization <cit.>. The problem of learning invariant representations by exploiting the varying degrees of spurious correlations naturally present in training data has been addressed in several works <cit.>. The general formulation of invariant representation learning is a challenging bi-level optimization problem, and existing theoretical guarantees often require linear constraints on data representations and/or classifiers <cit.>. The conditions and limitations of these approaches are extensively studied here <cit.>. In nonlinear settings, the work in <cit.> proposes a variational autoencoder (VAE) framework that uses a flexible conditionally non-factorized prior to capture complicated dependencies between the latent variables. It is a two-stage method that conducts the PC algorithm to discover the invariant latent variables. Despite the promising theory, the applications of invariant models for solving real-world problems, particularly in biology, is underexplored. Understanding the underlying factors and their dependencies within complex biological systems is a challenge in biology <cit.>. In this paper, we focus on data integration and classification of multi-domain single-cell genomics data. Single-cell genomics allows us to study individual cells and their genetic makeup, providing a comprehensive view of the heterogeneity across cells. However, datasets are often generated in different labs and under various experimental conditions where cells could have been exposed to chemical or genetic perturbations or sampled from individuals with diseases. The exceptional complexity in such data makes the process of disentangling technical artifacts or distractors from relevant biological signals, referred to as data integration, non-trivial. Data integration in single-cell genomics is essential for capturing the cellular and molecular landscapes across domains. Deep generative models have shown great potentials in analyzing biological data <cit.>, but existing integration methods often struggle to distinguish relevant biological signals from noise, leading to over-correction and loss of biological variations. To address this challenge, we propose a conditionally invariant deep generative model that effectively integrates single-cell genomics data while preserving biological variations across datasets. Our model incorporates specific biological mechanisms, such as gene programs, disease states, or treatment conditions to capture the biological context of the data and facilitates deeper insights into cellular heterogeneity. Our main contributions are as follows: * We revisit the fundamental assumptions in invariant representation learning and argue that in complex biological processes, the assumptions of independent and invariant causal mechanisms might not be sufficient. * We propose an invariant representation learning method that identifies both spurious and invariant latent variables. * We prove that our proposed method is identifiable up to a simple transformation and a permutation of the latent variables. * We test our method using two large-scale datasets including human hematopoiesis <cit.> and human lung cancer single-cell RNA-seq data <cit.> with 49 samples, spanning two lung cancer types and healthy individuals. * We intensively benchmark several invariant and identifiable deep generative models and demonstrate the superiority of our method for single-cell data integration, cell state identification, and cell type annotation. Related work. Invariant representation learning <cit.> leverages grouped data from multiple environments for inferring the underlying invariant causal relationships, as learning the true causal structures between variables is not possible from i.i.d data from a single domain. In traditional representation learning, independent component analysis (ICA)<cit.> and non-linear ICA have been used to find independent latent features that generate the observations. However, identifying the true latent factors in the general nonlinear case of unsupervised representation learning is impossible <cit.>. A recent line of pioneering works provide identifiability results for the nonlinear representation learning case under additional assumptions, such as weakly- or self-supervised approaches which leverage additional information in the form of multiple views <cit.>, auxiliary variables <cit.> and temporal structure <cit.>. To allow learning latent variables that may not be independent but causally related and motivated by <cit.>, there has been a shift towards causal representation learning <cit.>. Similar to works by <cit.>, the present work is based on identifiable VAEs. However, unlike <cit.>, our work disentangles latent representations that contain invariant information from the part that corresponds to changes across domains. The current work is akin to recent works on causal disentanglement of style and content features in images with identifiability guarantees <cit.>, which partition the latent space into an invariant and a changing part. However, it differs from these works in the assumptions made on the data generation process, the causal graph and/or the requirement for having access to paired or multi-view observations <cit.>. Related work on the state-of-the-art generative models for single cell data is given in Appendix <ref>. § SETUP AND BACKGROUND Consider an observed data variable x^u∈ℝ^n from the domain u∈𝒰 with corresponding label y^u∈𝒴 that is generated using a latent random vector z^u∈ℝ^m (lower dimensional, m≤ n). We are interested in learning a data representation Φ∈ℋ_Φ: 𝒳→𝒵, where z^u=(z_I,z_S)_z_I,z_S ∈ℋ_Φ and z_I is invariant across domains 𝒰. We can further use the latent representation z_I for training an invariant predictor ω∘Φ such that ω∈ℋ_ω:𝒵_ℐ→𝒴 performs well across all domains, that is: ω∈argmin_ω̅∈ℋ_ωℒ^u(ω̅∘Φ), ∀u∈𝒰 where ∘ refers to function composition and ℒ^u is the prediction loss in domain u. Besides identifying the latent variables that are stable across environments, this paper is concerned with providing a causal semantic to those latent variables. Here, we briefly discuss some of the prominent related works. §.§ Identifiable VAEs The lack of identifiability in variational autoencoders (VAEs) <cit.> often leads to their failure in accurately approximating the true joint distribution of observed and latent variables. In the work presented in <cit.>, the authors address this problem by extending nonlinear independent component analysis (ICA) <cit.> to a wide range of deep latent-variable models. They demonstrate that by employing a conditionally factorized prior distribution over observed and latent variables, denoted as p_θ( z\|u), where u represents an additional observed variable such as a class label y, identifiability can be ensured up to simple transformations. In other words, disregarding transformation for simplicity, if two different choices of model parameters θ and θ' yield the same marginal distribution p_θ(x)=p_θ'(x), then it implies that θ = θ', resulting in similar joint distributions p_θ(x,z) = p_θ'(x,z), and therefore similar posteriors p_θ(z\|x) = p_θ'(z\|x). Let θ = (f, T, λ) represent the parameters of the conditional generative model as follows: p_θ(x, z\|u) = p_f(x\|z)p_T,λ(z\|u) Here, p_f(x\|z) = p_ϵ(x - f(z)) where f:ℝ^m→ℝ^n is a non-linear injective function, and ϵ is an independent noise variable with a probability density function p_ϵ(ϵ). The identifiable VAE (iVAE) <cit.> assumes that the prior distribution p_T,λ(z\|u) is conditionally factorial. In this assumption, each element z_i ∈z follows a univariate exponential family distribution characterized by the parameters λ and the sufficient statistics T. The identifiability result of iVAE is expanded upon in the work of non-factorized iVAE (NF-iVAE) by Lu et al. <cit.>. This extension relaxes the assumption of a factorized prior distribution for the latent representation, allowing for a more general exponential family distribution. By adopting a non-factorized prior, the model gains the ability to capture complex dependencies between latent variables, which is often observed in real-world problems. However, the proposed model does not explicitly disentangle the latent variables that contain invariant information from the domain-specific variables that remain relevant for the prediction task. §.§ Invariant risk minimization The concept of invariant risk minimization (IRM) <cit.> involves the development of an optimal and invariant predictor, denoted as ω, that demonstrates strong performance across all environments, collectively referred to as ℰ_all. The underlying assumption in IRM is that spurious correlations lack stability when observed across different environments <cit.>. This assumption becomes particularly crucial when the training and testing datasets possess dissimilar distributions. IRM utilizes training data gathered from distinct environments, represented as ℰ_tr to learn a data representation Φ that enables the optimal classifier ω to consistently achieve satisfactory performance across all environments. Mathematically, this concept can be formulated as follows: min_Φ:𝒳→ℋ ω:ℋ→𝒴∑_e∈ℰ_tr R^e(ω∘Φ) The authors in <cit.> show that if both Φ and ω come from the class of linear models, under certain conditions, the predictor ω∘Φ remains invariant across ℰ_all. § PROBLEM SPECIFICATION Two commonly-made assumptions in causal representation learning are independent and invariant causal mechanisms which are generally seen as simplifying assumptions that may not hold across complex biological data. The independent causal mechanisms (ICM) principle <cit.> states that the generative process consists of independent modules that do not inform or influence each other. These modules are considered as conditional distributions of each variable in the system given its direct causes. Biological processes are, however, interconnected and can affect each other in complex ways. For example, genes do not work in isolation but instead interact with each other in the form of gene programs (modules) or pathways that carry out specific biological functions. Gene programs often share some genes and therefore they correlate. Additionally, there is a cross-talk between biological pathways, meaning that changes in one pathway or process can have downstream effects on other pathways and processes <cit.>. There are, therefore, complicated dependencies between mechanisms in biological data where the ICM principle may not hold. The invariant causal mechanisms assumption is often made to enable generalization across different settings <cit.>. However, in biology invariance might not be an appropriate assumption since biological systems and processes can be affected by a variety of factors which can result in the causal relationships between variables to change over time or across different experiments. For example, the causal relationship between a gene and a phenotype may depend on background genetic variation (e.g. mutations, cancer subtype, cell-line, treatment history in cancer patients, etc) that can simply vary across datasets generated from various patients under different conditions. In this case, an invariant causal learning model is very likely to leave out on important biological mechanisms that would be deterministic of patient outcome. To ensure generality in our model, we assume that latent factors can be dependent and refrain from interpreting inferred mechanisms as causally invariant unless there is sufficient evidence to support such an interpretation. Our focus instead is on separating noise and spurious correlations from biologically-interpretable factors within the model, with the aim of learning a structured representation that remains invariant to noise. §.§ Motivating application in single-cell genomics Single-cell genomics allows us to study individual cells and their genetic makeup, providing a comprehensive view of the heterogeneity across cells, cellular and molecular processes underlying biological systems <cit.>. However, analyzing single-cell data can be complex and challenging, as the datasets may be generated using different experimental techniques or platforms which add technical artifacts, called batch effects, to the observations <cit.>. The datasets may also measure cells exposed to chemical or genetic perturbations <cit.>, or in diseases, where the presence of biological variations further complicates the analysis <cit.>. To address these challenges, data integration methods aim to infer a representation of the datasets that captures the relevant biological signals while minimizing noise and technical artifacts. These methods help to identify the cell states that may be missed in individual datasets, helping us to build a comprehensive reference map, or atlas, of the cellular and molecular landscapes. This reference map can then be used to compare and analyze new single-cell data <cit.>. In the field of single-cell genomics, various methods have been proposed for data integration (see Appendix <ref>). However, these methods often encounter challenges such as over-correction for batch effects and the removal of biological variations across similar cell states. Existing integration methods based on VAEs <cit.> also lack identifiability. To address these limitations, our work introduces an identifiable deep generative model for effective data integration in single-cell genomics. This model disentangles representations that remain invariant across different environments from those that vary with the environment, such as batch effects. This disentanglement is crucial for preserving biological variations across datasets and surpassing the limitations of existing methods. Moreover, our model can incorporate specific biological mechanisms, such as gene programs, disease states, or treatment conditions, allowing for their integration into the data integration process. This incorporation of biological factors enables the model to capture the biological context of the data and provide deeper insights into cellular heterogeneity. §.§ Assumptions on generative process r0.45 < g r a p h i c s > The data generating process. Our aim is to learn two sets of latent variables z_I and z_S such that z=(z_I,z_S). Latent variables z_I ∈ℝ^i represent inherent correlations of interest that remain consistent across different environments. In contrast, latent variables z_S ∈ℝ^s (where i+s = m) correspond to spurious correlations that tend to fluctuate across varying environments. We assume that the spurious correlations stemming from data biases are unrelated to the causal explanation of interest, although they are required for a perfect reconstruction. We also substitute u in iVAE as (d,e) where d refers to the specialized knowledge about the samples such as cell annotations, disease state, time, transcription factor, or gene program activity, etc. The variable e also refers to the environment and encodes potential biases in the data. We assume that the causal graph underlying data generated under various conditions and environments satisfies the following assumptions: (a) For any component i in the latent space, z_i depends either on d or e <ref>. (b) The variables z_I and z_S are conditionally independent. Therefore, z_I is invariant to noise. (c) The variable z depends on both d and e and p(x\|z) is not necessarily invariant across the environments. (d) The label y is independent of e given z_I: ye\|z_I. Therefore, we assume p(y\|z_I) is invariant across all environments. The NF-iVAE model, as outlined in <cit.>, operates under the assumption that the variable x is conditionally independent of e and d given z: xe,d\|z. This condition of independence suggests that the probability distribution p(x\|z) remains consistent and invariant across all environments. However, our findings, discussed in Section <ref>, reveal that in practical applications, environmental noise can still impact the latent representation z of NF-iVAE, which challenges this assumption. To tackle this issue, our proposed model disentangles spurious correlations z_S and assumes that the resulting probability distribution p(y\|z_I) is independent of e and, consequently, remains invariant across all environments. We also assume that z_I and z_S are conditionally independent meaning that the invariant features are not biased and only capture the stable information across the environments: p(z_I,z_S\|u) = p(z_I\|u)p(z_S\|u). §.§ Assumptions on priors In accordance with Assumption <ref>, we make the following assumptions regarding the conditional priors to attain identifiability: (a) The conditional prior on the invariant data representation p(z_I\|d) belongs to a general exponential family distribution that is not necessarily factorized. (b) Prior distribution p(z_S\|e) is conditionally factorial. Assumption <ref> (a) follows <cit.> and allows for a more versatile and flexible prior given the domain specific knowledge that can effectively account for complex interdependencies between the invariant latent variables. The conditional prior on the spurious latent variables z_S can be either factorized or non-factorized depending on the application. In this paper, we assume that the spurious variables are factorized and independent of the invariant ones (Assumption <ref> (b)). The density function of the prior for the invariant latent variables z_I ∈ℝ^i is given by: p_T,λ(z_I\|d) = 𝒬(z_I)/𝒵(d)exp[ T(z_I)^Tλ(d)] where 𝒬 is the base measure and 𝒵 is the normalizing constant (more details in Appendix <ref>). The sufficient statistics T(z_I) = [T_f(z_I)^T,T_NN(z_I)^T]^T are the concatenation of the sufficient statistics T(z_I) = [T_1(z_I_1)^T,..,T_n(z_I_n)^T] of a factorized exponential family, where the dimension of T_i(z_I_i) is greater or equal to two. T_NN(z_I) is also the output of a neural network that allows the prior to model and capture arbitrary dependencies between the latent variables. The density function of the prior for the spurious features z_S ∈ℝ^s is: p_T,λ(z_S\|e) = Π_i=1^s𝒬_i(z_S_i)/𝒵_i(e)exp[ ∑_j=1^kT_i,j(z_S_i)λ_i,j(e)] where z_S are assumed to follow a factorized but unknown distribution p(z_S) = Π_i=1^s p_i(z_S_i) <cit.>. 𝒬_i is the base measure, z_S_i is the i-th dimension of z_S, 𝒵_i(e) the normalizing constant, T_i = (T_i,1,..,T_i,k) the sufficient statistics, λ_i(e) = (λ_i,1,..,λ_i,k) the corresponding natural parameters depending on e, and k the fixed dimension of the sufficient statistics. § PROPOSED METHOD Our model, referred to as the invariant VAE (inVAE), is depicted in Figure <ref> and incorporates several key features. These features include: (i) disentangled spurious and invariant latent representations, (ii) partially independent latent variables, (iii) invariant representation using z_I, (iv) identifiable data representation up to simple transformations. The observed variables (x,d,e) are encoded into two separate latent variables, namely z_I and z_S. The approximated posterior distribution, denoted as q_φ(z\|x,d,e), represents the conditional distribution of the latent variables z given the observed variables (x,d,e). The variable e is encoded using a one-hot encoder. This component encodes the biases in the data. Prior knowledge, such as known biological mechanisms underlying the samples, is encoded through 𝒟_ψ. Two encoders infer the distribution parameters of the priors, denoted as p_Θ(z_I\|d) and p_Θ(z_S\|e), where Θ=(T,λ) for exponential distributions. Finally, the likelihood p_ϕ(x\|z_I,z_s) evaluates the reconstruction quality of the generated data. In our VAE model, the decoder component decodes the latent variables z into the parameters of a negative binomial distribution. The negative binomial distribution is commonly used to model count data, which is often encountered in genomics applications. §.§ Loss function Our loss function contains three main parts: the evidence lower bound (ELBO) (ℒ_ELBO), the score matching (SM) loss (ℒ_SM), and the total correlation between the distributions of z_I and z_S (ℒ_TC). The ELBO for the proposed model is as follows (proof in Appendix <ref>): ℒ_ELBO(φ,ϕ,T,λ):= 𝔼_p_D[ 𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z)] - D_KL (q_φ(z\|x,u)\|\|p_T,λ(z\|u))] = 𝔼_p_D[𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z) - log q_φ(z\|x,u) + logp_T,λ(z_I\|d) + log p_T,λ(z_S\|e) ] ] To model the factorized prior in Eq. <ref>, we use Gaussian distribution in practice, so we can calculate the corresponding conditional probability by directly learning the mean and the variance of the distribution and optimize the ELBO in Eq. <ref>, correspondingly. However, the direct optimization of the non-factorized prior in Eq. <ref> is impossible as the normalization constant 𝒵(d) for a general multivariate exponential family distribution is unknown. In this work, we use score matching <cit.> for optimizing unnormalized probabilistic models, similar to the work in <cit.>. The score matching loss is as follows: ℒ_SM (T, λ) : = -𝔼_p_D[𝔼_q_φ(z_I\|x,u)[ ∇_zlog q_φ (z_I\|x,u) - ∇_zlogp̃_T, λ (z_I\|d)^2 ]] = -𝔼_p_D[𝔼_q_φ(z_I\|x,u)[ ∑_j=1^i [ ∂^2 logp̃_T, λ (z_I\|d)/∂z_j^2 + 1/2( ∂logp̃_T, λ (z_I\|d)/∂z_j)^2]]] +cst. In practice to solve Eq.<ref>, we use partial integration to simplify the evaluation shown in Eq.<ref> where p̃_T, λ is the non-normalized density. A summary of the score matching technique is provided in Appendix <ref>. In this work, we aim at disentangling the noise and the invariant factors within the data. Previous studies, as suggested by Chen et al. <cit.> and Higgins et al. <cit.>, have highlighted the importance of two key aspects for achieving a disentangled representation: (i) maximizing the mutual information between the latent variables and the data variables, and (ii) promoting independence among the variables. While the invariant variables may exhibit correlation in real-world applications, we only encourage independence between z_I and z_S by minimizing their total correlation. The total correlation is a measure of dependence between two random variables, and penalizing this correlation encourages the model to learn statistically independent factors <cit.>. To assess the total correlation between the distributions of z_I and z_S, we employ the Kullback–Leibler (KL) divergence as follows: ℒ_TC : = D_KL(q_φ(z\|u)\|\|q_φ(z_I\|u)q_φ(z_S\|u)) Note that q_φ(z\|u) requires the evaluation of the density 𝔼_p(n\|u)[ q_φ(z\|u,x_n)], where p(n\|u) is the probability of observing the data point x_n in the domain u. We refer to q_φ(z\|u) as the conditional aggregated posterior, following <cit.>. To measure the above density in practice, we extend the minibatch-weighted sampling proposed in <cit.> as follows: 𝔼_q_φ(z\|u)[ log q_φ(z\|u)]≈1/b∑_i=1^b[log 1/nb∑_j=1^bq_φ(z(x_i)\|x_j,u)] where (x_1, ..., x_b) is a minibatch of samples and z(x_i) is a sample from q_φ(z\|u,x_i). Both x_i and x_j are drawn from the same domain under the assumption of i.i.d data (proof in Appendix <ref>). We can similarly estimate q_φ(z_I\|u) and q_φ(z_S\|u) in Eq. <ref>. The overall loss is then: ℒ(φ,ϕ,Θ, T, λ) = ℒ_ELBO + ℒ_SM + β·ℒ_TC = 𝔼_p_D[𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z)] - D_KL(q_φ(z\|x,u)\|\|p_Θ(z_I\|d)p_Θ(z_S\|e))] - 𝔼_p_D[𝔼_q_φ(z\|x,u)[ ∑_j=1^n [ ∂^2 logp̃_T, λ (z\|u)/∂z_j^2 + 1/2( ∂logp̃_T, λ (z\|u)/∂z_j)^2]]] + β· D_KL(q_φ(z\|u)\|\|q_φ(z_I\|u)q_φ(z_S\|u)) ℒ(φ,ϕ, T, λ) = ℒ_ELBO + ℒ_SM + β·ℒ_TC = 𝔼_p_D[𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z) - log q_φ(z\|x,u) + logp̃_T̂,λ̂(z_I\|d) + log p_T,λ(z_S\|e) ] ] - 𝔼_p_D[𝔼_q_φ̂(z_I\|x,u)[ ∑_j=1^i [ ∂^2 logp̃_T, λ (z_I\|d)/∂z_j^2 + 1/2( ∂logp̃_T, λ (z_I\|d)/∂z_j)^2]]] + β· D_KL(q_φ(z\|u)\|\|q_φ(z_I\|u)q_φ(z_S\|u)) where β is a tunable parameter and φ̂,T̂,λ̂ are copies of φ,T,λ, and are treated as constant during the training. The parameters of the encoder q_φ(z\|x,u), decoder p_ϕ(x\|z), and the spurious prior p_T,λ(z_S\|e) are optimized using the ℒ_ELBO, while the invariant prior p̃_T,λ(z_I\|d) remains fixed. Additionally, the ℒ_SM is employed solely for training the non-factorized prior with φ being fixed. Further implementation details are provided in Appendix <ref>. The identifiability proofs of each of the invariant and spurious representations as well as the joint representation are presented in <ref>. §.§ Extension to discrete data with negative binomial distribution Negative Binomial is commonly used for modeling single-cell gene expressions <cit.>. Let ρ_cg be the expected frequency of expression of gene g in cell c and θ_g be the inverse-dispersion of gene g. Denote l_c as the library size (i.e. total read counts) in cell c. We use a non-linear transformation f_ϕ(z) to generate the distributional parameters ρ and θ of each gene in each cell. The gene expression x_cg is then generated as x_cg∼NegativeBinomial(l_cρ_cg, θ_g) <cit.>. § RESULTS In this section, we present empirical results on two extensive datasets: human hematopoiesis <cit.> and human lung cancer single-cell RNA-seq data <cit.>. We evaluate various deep generative models, including scVI <cit.>, scANVI <cit.>, iVAE <cit.>, NF-iVAE <cit.>, principal component analysis (PCA), and our own inVAE. We examine their impact on data integration, cell state identification, cell annotation, and generalization to unseen data—key challenges in single-cell genomics. We assess these methods using 7 different metrics outlined in <cit.>. Appendix <ref> provides an explanation of the metrics, and the architectures are discussed in Appendix <ref>. §.§ Invariant representation of single-cell hematopoiesis data To explore invariant representation learning, we initially focus on the human hematopoiesis data from <cit.>. For training, we utilize three datasets consisting of 15,496 cells from two distinct donors, generated at two different sites, with 21 cell types (classes) (additional details in Appendix <ref>). We anticipate that datasets with similar donors but different sites would exhibit biological similarities, making the variability across those datasets undesirable. Conversely, we assume that datasets from different donors should contain interesting biological variations. To visualize the data and compare the latent representations of different methods, we utilize UMAP visualizations <cit.> colored with three batches, as illustrated in Figure <ref>. Without any correction using the raw data (top left), the cells are clustered based on their batch, making it challenging to interpret the heterogeneity across cells. For downstream tasks, we employ the invariant latent variables of inVAE and refer to it as inVAE-bio. While all other methods successfully cluster similar cells (Figure <ref> bottom right and Figure <ref>), only inVAE-bio effectively separates batch s1d2 (donor 2) from s1d1 and s2d1 (donor 1) while simultaneously eliminating technical noise from samples with similar donors. These observations are further evaluated in Figure <ref> using scIB metrics <cit.>. scVI and scANVI aggressively mask batch effects, whereas inVAE-bio achieves higher biological conservation and interpretability by disentangling noise from biological signals. On the other hand, inVAE using both invariant and spurious latent variables, as well as iVAE and NF-iVAE, perform poorly, as expected due to the presence of spurious correlations in their representations. §.§ Predicting cell types using invariant predictors Classifying cell types (also called cell annotation) in new datasets poses a challenge in single-cell genomics due to technical variations and biases across datasets generated in different environments. We examine the effectiveness of various deep generative models for cell type classification. The prediction results are shown in Table <ref>. The results indicate that inVAE-bio surpasses the others and achieves the highest performance on two unseen datasets with different donors and sites (additional details in Appendix <ref>). The superior performance of inVAE, compared to inVAE-bio, on training datasets can be attributed to the presence of spurious correlations in its latent representation, leading to overfitting. §.§ Invariant representation with interventions Lastly, we examine the integration of human lung cancer data from <cit.>. The dataset comprises 155,098 cells from 21 Small Cell Lung Cancer (SCLC) clinical samples obtained from 19 patients, along with 24 Lung Adenocarcinoma (LUAD) and 4 tumor-adjacent normal lung samples serving as controls (details in <ref>). According to literature <cit.>, we anticipate variability among epithelial cells in SCLC samples across different patients, while expecting minimal biological variations among other cells. UMAP visualizations of control and cancer cells using inVAE-bio are presented in Figure <ref> and <ref>. Our method effectively captures patient-level variability (Figure<ref>, middle) and removes spurious correlations in other cells (Figure <ref>, left). However, scVI (and scANVI) over-corrects the batch effect and fails to capture relevant biological variations across samples (Figure <ref>). In <cit.>, a small subset of epithelial cells in SCLC samples is annotated as recurrent, containing samples from all batches. Our results in Figure <ref> (right) demonstrate that our method successfully captures the similarity among these cells and clusters them appropriately. § CONCLUSIONS In this study, we introduce inVAE, a deep generative model that utilizes identifiable variational autoencoders. Our model infers a two-partition latent space, where one part captures spurious correlations or biases that vary across domains, while the other remains invariant to unwanted variability. Importantly, the latent features in these partitions are independent, promoting the disentanglement of the invariant representation from the varying representation. We apply our approach to large and heterogeneous single-cell gene expression data for cell state detection and cell type classification. Our results demonstrate that inVAE addresses a main limitation of current deep generative models in single-cell biology, namely their inability to disentangle biological signals from unrelated data biases during data integration. We envision that our model will find applications in more complex scenarios, such as cell state detection in multi-scale biological data (e.g., genes, proteins, chromatin accessibility, spatial gene expression), as well as in computer vision for tasks like style-content isolation in images. It is important to take caution when making consequential decisions, particularly in healthcare, based on the structures learned from observational data. § ACKNOWLEDGMENT We express our gratitude to Stefan Bauer for engaging discussions on causal models and generative processes, as well as his feedback on the paper. We would also like to acknowledge Sergei Rybakov for reviewing the equations and providing valuable feedback. We extend our appreciation to Jason Hartford and Sebastien Lachapelle for fruitful discussions on identifiability during the initial phases of this project. Finally, we are thankful to Julien Gagneur and his administrative team, particularly Florian Hölzlwimmer, for their generous and invaluable support in providing computational resources. unsrt § RELATED WORK: DATA INTEGRATION Deep generative models are common frameworks for integration of single cell RNA sequencing data. Single cell Variational Inference (scVI) <cit.> appends one-hot encoding of the variables denoting the environments (e.g. experimental batches, sequencing protocols, site, etc) to genes which are then used as inputs to a VAE. The one-hot encoding is appended to the first layer of both the encoder and the decoder and has proven to be highly effective in removal of technical noise, also called batch effect, in certain datasets. In this case scVI is akin to conditional VAEs (cVAEs) <cit.>. While scVI is completely unsupervised, scanVI <cit.>, an extension of scVI, is a semi-supervised deep generative model based on M1 + M2 model in <cit.> which additionally uses cell annotations (cell type labels) to integrate cells across environments. Both scVI and scANVI are widely recognized as state-of-the-art integration models in the single-cell community. The latent representations produced by these models primarily capture variations corresponding to cell types, which represent the primary axis of variability in single-cell data, disregarding technical noise. However, other sources of biological variation, including inter-patient differences and treatment effects (especially when the treatment's impact is subtle), tend to become entangled with batch effects and may be masked as a result. Consequently, these methods are susceptible to potential over-correction issues. § ELBO FOR INVAE log p_θ(x\|u) = log p_θ(x\|u)∫ q_φ(z\|x,u) dz = ∫log p_θ(x\|u) q_φ(z\|x,u) dz = 𝔼_q_φ(z\|x,u)[log p_θ(x\|u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/p_θ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)q_φ(z\|x,u)/p_θ(z\|x,u)q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/q_φ(z\|x,u)]+ D_KL( q_φ(z\|x,u)\|\|p_θ(z\|x,u)) ≥ 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_ϕ(x\|z)p_Θ(z\|u)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_ϕ(x\|z)p_Θ(z_I,z_S\|d,e)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_ϕ(x\|z)p_Θ(z_I\|d)p_Θ(z_S\|e)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z) + logp_Θ(z_I\|d)p_Θ(z_S\|e)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z)] - D_KL (q_φ(z\|x,u)\|\|p_Θ(z_I\|d)p_Θ(z_S\|e)) Given ∫ q_φ(z\|x,u) dz = 1. § ELBO FOR IVAE ELBO for iVAE: log p_θ(x\|u) = log p_θ(x\|u)∫ q_φ(z\|x,u) dz = ∫log p_θ(x\|u) q_φ(z\|x,u) dz = 𝔼_q_φ(z\|x,u)[log p_θ(x\|u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/p_θ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)q_φ(z\|x,u)/p_θ(z\|x,u)q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/q_φ(z\|x,u) + KL( q_φ(z\|x,u)\|\|p_θ(z\|x,u))] ≥ 𝔼_q_φ(z\|x,u)[logp_θ(x,z\|u)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[logp_ϕ(x\|z)p_Θ(z\|u)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z) + logp_Θ(z\|u)/q_φ(z\|x,u)] = 𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z)] - D_KL (q_φ(z\|x,u)\|\|p_Θ(z\|u)) Given ∫ q_φ(z\|x,u) dz = 1. § SCORE MATCHING We provide here a short summary of the Score Matching (SM) method by <cit.>. Let us assume that we observe data x∼ p_x(·) with x∈^n where we want to approximate the observed density p_x (·) by some parametric distribution with density p(· ; θ) with θ∈^t. We assume that we can only calculate the density up to a multiplicative constant, i.e. p(ξ; θ) = 1/𝒵(θ)p̃ (ξ; θ) where we know how to calculate the non-normalized density p̃(ξ; θ) (e.g. by an analytical form) but can not calculate the normalization constant 𝒵(θ), for example in the case where the integral ∫_ξ∈^np̃(ξ; θ) d ξ is intractable. Estimation by score matching now works by first defining the score function (in our context) to be the gradient of the log density with respect to the data vector: ψ (ξ; θ) = [ ∂log p(ξ; θ)/∂ξ_1; ⋮; ∂log p(ξ; θ)/∂ξ_n ] = [ ψ_1 (ξ; θ); ⋮; ψ_n (ξ; θ) ] = ∇_ξlog p(ξ;θ). Here, it is important to note that the score function of our model (the parametric form of the density we chose) does not depend on 𝒵(θ), i.e. ψ (ξ; θ) = ∇_ξlogp̃(ξ; θ). To go on, <cit.> now define, together with the score function of the data ψ_x (·) = ∇_ξlog p_x(·), the score matching estimator of θ by J (θ) = 1/2∫_ξ∈^n p_x (ξ) ψ(ξ;θ) - ψ_x (ξ)^2 d ξ and the corresponding estimate of θ is θ̂ = _θ J(θ). This theoretic estimator, that works by minimizing the expected squared distance between the model and the data score function, is justified by the following two points: * Although calculating the score function of the data ψ_x (·) is again a hard non-parametric estimation problem, we can use the score matching estimator with a different, equivalent minimization objective avoiding this computation (which is one of the main results in <cit.>) to estimate the parameters θ. * Even more important, under some conditions, minimizing the score matching estimator J(θ) corresponds to finding the true parameter θ^* such that p_x (·) = p(·; θ^*). For point 1 above, it is shown in <cit.> that the score matching estimator (<ref>), assuming ψ (ξ; θ) is differentiable and some weak regularity conditions are fulfilled, can be also written as J(θ) = ∫_ξ∈^n p_x (ξ) ∑_i=1^n [ ∂^2 logp̃(ξ; θ)/∂ξ_i^2 + 1/2( ∂logp̃(ξ; θ)/∂ξ_i)^2 ] d ξ + const. where the constant does not depend on θ. In practice, we can sample this estimator J via T observations of the random variable x, i.e. (x^(1), …, x^(T)), by J̃ (θ) = 1/T∑_t=1^T∑_i=1^n[ ∂^2 logp̃(x^(t); θ)/∂ (x^(t)_i)^2 + 1/2( ∂logp̃(x^(t); θ)/∂x^(t)_i)^2 ] where we drop the constant for the purpose of optimizing J̃ (θ) over θ. Furthermore, many extensions of the score matching method exist. One of these extensions, that we call regularized score matching in the following, is given by <cit.>. They argue that the conditions necessary for the above derivations to hold, are often violated in practice, especially the assumption of a smooth enough data density. Therefore, they derive the regularized score matching estimator by an approximation to the true SM estimator for input data with Gaussian noise as J̃ (θ) = 1/T∑_t=1^T∑_i=1^n[ ∂^2 logp̃(x^(t); θ)/∂ (x^(t)_i)^2 + 1/2( ∂logp̃(x^(t); θ)/∂x^(t)_i)^2 + λ_regSM( ∂^2 logp̃(x^(t); θ)/∂ (x^(t)_i)^2)^2 ], where λ_regSM is a hyperparameter controlling the strength of the regularization. This hyperparameter can be thought of as smoothing the optimization landscape for score matching. In our experiments, we randomly sampled λ_regSM as either zero, i.e. regular SM, or as a random non-zero value. § MINIBATCH-WEIGHTED SAMPLING Here, we extend the proof in Appendix C of <cit.>. Let ℬ^u = {x_1,…,x_b}^u be a minibatch of b indices from domain u where each element is sampled i.i.d from p_x\|u. So, for any sampled batch instance ℬ^u, p(ℬ^u)=(1/n^u)^b, where n^u is the total number of samples from domain u. 𝔼_q(z\|u)[log q(z\|u)] = 𝔼_q(z,x_i\|u)[log 𝔼_x_j∼ p_x\|u[ q(z\|x_j,u)]] = 𝔼_q(z,x_i\|u)[log 𝔼_p(ℬ^u)[ 1/b∑_j=1^bq(z\|x_j,u)]] ≥𝔼_q(z,x_i\|u)[log 𝔼_p(ℬ^u\|x_i)[ p(ℬ^u)/p(ℬ^u\|x_i)1/b∑^j=1_bq(z\|x_j,u)]] = 𝔼_q(z,x_i\|u)[log 𝔼_p(ℬ^u\|x_i)[ 1/n^ub∑_i=1^bq(z\|x_j,u)]] This can be estimated as follows (more discussions in <cit.>): 𝔼_q(z\|u)[log q(z\|u)] ≈1/b∑_i=1^b[log 1/n^ub∑_j=1^bq(z(x_i)\|x_j,u)] where p(ℬ^u\|x_i) is the probability of the sampled minibatch where one of its element is fixed to be x_i and the rest are sampled i.i.d from p_x\|u. This can be intuitively interpreted as having a mixture distribution over each domain u, where the data index j indicates a mixture component. § ON INVARIANT REPRESENTATION LEARNING Consider the following structural equation model <cit.>: X_1 ← μ_1 + Guassian(0,σ_1(e)) Y ← X_1 + Guassian(0,σ_2(e)) X_2 ← Y + Guassian(0,σ_3(e)) where the environment e influences σ_1, σ_2, and σ_3. To predict Y from X_1 and X_2, we use a linear least-square predictor as follows: Ŷ^e = α_1 X_1^e + α_2 X_2^e The only predictor that generalizes well to unseen environment is Ŷ = 1 . X_1 + 0 . X_2. Nevertheless, this remains valid only if we have knowledge of the X_1 and X_2 variables and can employ invariant risk minimization to deduce the causal connections between them. In practical scenarios, the (hidden) variables and their precise quantities are often unknown. When the model deduces the obscure variables of variation from the data, such as in iVAE and NF-iVAE, the previously mentioned structural equation model may appear as: X_1 ← μ_1 + Guassian(0,σ_1(e)) X_3 ← Guassian(0,σ_2(e)) Y ← X_1 + X_3 X_2 ← Y + Guassian(0,σ_3(e)) In this scenario, the variables X_1, X_2, and X_3 are inferred from the data. The new invariant predictor can be denoted by Ŷ = 1 . X_1 + 0 . X_2 + 1 . X_3. However, X_3 is a spurious feature that has been erroneously considered in the invariant predictor. To address this issue, we distinguish the correlations arising from spurious features and those intrinsic to the data, and eliminate the spurious correlations. This enables us to identify the causal graph that is invariant to the presence of the spurious feature X_3, as well as the invariant predictor Ŷ = 1 . X_1 + 0 . X_2. § THE TRAINING LOSS IN PRACTICE In practice, we follow <cit.> and model only the log of the unnormalized probability of our invariant prior, i.e. the part inside the exponential function in Eq. (<ref>). This is done in the following way logp̃_T,λ(z_I\|d) = < T_NN (z_I), λ_NN (d) > + <T_f(z_I), λ_f (d) > = <NN(z_I;param1), NN(d; param2) > + <concat(z_I,z^2_I), NN(d;param3) > where < ·,·> is the dot product of two vectors and concat(·,·) denotes the concatenation of two vectors. Here, every part except T_f(z_I) is implemented through their own respective neural network with corresponding input variables and correct output dimension. For the output dimension we only need to make sure that the respective outputs of the dot product have the same dimensions, but for example T_NN (z_I) and T_f (z_I) do not need to have the same size. The first dot product allows the prior to capture, as previously mentioned, arbitrary dependencies between the latent variables since NN(z_I;param1) can model complicated nonlinear transformations of the latents. Going on, T_f (z_I) is the concatenation of the latent variables and their square values and therefore fulfills the theoretic requirement of the definition <ref> of the prior that each T_i (z_i) has at least 2 dimensions. Finally, the second dot product (without the first dot product) corresponds to a factorized exponential family. ℒ(φ,ϕ,Θ, T, λ) = ℒ_ELBO + ℒ_SM + β·ℒ_TC = 𝔼_p_D[𝔼_q_φ(z\|x,u)[log p_ϕ(x\|z) - log q_φ(z\|x,u) + logp̃_Θ(z_I\|d) + log p_Θ(z_S\|e) ] ] - 𝔼_p_D[𝔼_q_φ(z_I\|x,u)[ ∑_j=1^i [ ∂^2 logp̃_T, λ (z_I\|d)/∂z_j^2 + 1/2( ∂logp̃_T, λ (z_I\|d)/∂z_j)^2]]] + β· D_KL(q_φ(z\|u)\|\|q_φ(z_I\|u)q_φ(z_S\|u)) , Concretely, the training loss in Eq. <ref>, is optimized as follows. The ℒ_ELBO part of the loss is responsible for optimizing the parameters of the encoder q_φ(z\|x,u), decoder p_ϕ(x\|z) and spurious prior p_Θ(z_S\|e) while keeping the invariant prior p̃_Θ(z_I\|d) as fixed. In practice, this is done via the Pytorch framework by setting requires_grad to False for the respective operations of the invariant prior for the calculation of ℒ_ELBO such that its parameters are not updated through that part. Similarly, the score matching part in the loss ℒ_SM only updates the parameters of the invariant prior p̃_Θ(z_I\|d) by keeping the rest of the model as fixed. This is achieved by, first, calling detach on the latent space variable z, so that the parameters of the encoder are not updated by the score matching loss. Secondly, also keeping the parameters of the spurious prior fixed, by either setting requires_grad to False or not calculating the gradients in the score matching loss with respect to the indices that belong to the spurious prior. § IDENTIFIABILITY PROOF Identifiability of the invariant partition of the latent representation We place a non-factorized prior p_T,λ(z_I\|d) from the exponential family distribution <cit.> on the invariant partition z_I, where z_I ∈^i, is shown to be identifiable under certain conditions based on the results in <cit.>. The identifiability of latent representations belonging to this partition imply that the ∼_A- and ∼_P-equivalence relations (see Definition 8 and 9 in <cit.>), that is identifiability up to simple transformations and permutations respectively, hold. Therefore: ∃ A_I, c_I s.t. T(f^-1(z_I)) = A_I T'(f'^-1(z_I)) + c_I, where A_I ∈^i × i is an invertible matrix and c_I ∈^i is a constant, and T' and f' are different parameterization of the model. Similarly, there exists a block permutation matrix P∈^i × i such that: ∃ P_I, c_I s.t. T(f^-1(z_I)) = P_I T'(f'^-1(z_I)) + c_I, Identifiability of the spurious partition of the latent representation The prior on the spurious partition, p_T,λ(z_S\|e), is designed to belong to a factorized exponential family distribution <cit.>, where z_S ∈^s, is proved to be identifiable under certain conditions as described in <cit.>. Therefore ∼_A- and ∼_P-equivalence relations also hold here and we have: ∃ A_S, c_S s.t. T(f^-1(z_S)) = A_S T'(f'^-1(z_S)) + c_S, where A_S ∈^s × s is an invertible matrix and c_S ∈^s is constant. Similarly, there exists a block permutation matrix P ∈^s × s such that: ∃ P_S, c_S s.t. T(f^-1(z_S)) = P_S T'(f'^-1(z_S)) + c_S, Identifiability of joint latent representation To prove identifiability of the joint latent representation, we first demonstrate that the two priors can be viewed as a joint prior that belongs to a general exponential family distribution. We then use the assumptions and results from NF-iVAE in <cit.> to complete the proof. Assumption 1 in <cit.> on causal graph is satisfied for our method as discussed in Section <ref>. We assume that data is sampled from a deep generative model with parameters θ= (f, T, λ ) defined according to: p_θ (x,z \| d,e) = p_f (x\|z) p_T,λ (z\|d,e) p_f (x\|z) = p_ϵ(x - f(z)) which means that the value of x can be decomposed as x= f(z) + ϵ, where ϵ is an independent noise variable with density function p_ϵ(ϵ), and is independent of z or f (we show later that this assumption might not hold in real applications). The prior on the latent variables is represented as: p_T,λ (z\|d,e) = p_T,λ(z_I, z_S \| d,e) = p_T,λ(z_I\|d)p_T,λ(z_S\|e) = 𝒬(z_I)/𝒵(d)exp[ T(z_I)^Tλ(d)] Π_i=1^s𝒬_i(z_S_i)/𝒵_i(e)exp[ ∑_j=1^kT_i,j(z_S_i)λ_i,j(e)] = 𝒬(z)/𝒵(d,e)exp[<T_f(z_I),λ_f(d)> + <T_NN(z_I),λ_NN(d)>_non-factorized prior + <T_f(z_S), λ_f(e)>_factorized prior] If we concatenate z_I and z_S and their sufficient statistics T, the expression above can be written as: p_T,λ (z\|d,e) = 𝒬(z)/𝒵(d,e)exp[<T_f(z),λ_f(d,e)> + <T_NN(z),λ_NN(d,e)>] = 𝒬(z)/𝒵(d,e)exp[T(z)^Tλ(d,e)] The resulting density belongs to a general exponential family with parameter vector given by arbitrary function λ(d,e) and sufficient statistics T(z) = [T_f(z)^T, T_NN(z)^T]^T given by concatenation of sufficient statistics of a factorised exponential family T_f(z) where all T_i(z_i) have dimension larger or equal to 2, and T_NN(z) which is an output of a neural network with ReLU activations, as required by Assumption 2 in <cit.>. We can intuitively view T_NN(z) as T_NN((Z_I,0)). Additionally, the results in (<ref>) and (<ref>) imply that there exists a block-diagonal, invertible matrix A ∈^m × m, with m = i + s, such that the parameter set (f, T, λ ) is ∼_A-identifiable: ∃ A, c s.t. T(f^-1(z)) = A T'(f'^-1(z)) + c, where c ∈^m is a constant and the blocks in A correspond to A_I and A_S. Since both A_I and A_S are invertible, the resulting block-diagonal matrix A is also invertible. Thus, the ∼_A equivalence relation still holds for θ= (f, T, λ) given the results from iVAE <cit.> and NF-iVAE <cit.> and independence of z_I and z_S. A similar reasoning can be argued to demonstrate ∼_P equivalence relation under certain conditions on parameters holds for a block permutation matrix P, m × m, using the results in (<ref>) and (<ref>) and independence of z_I and z_S. Hence, the parameter set (f, T, λ) is also ∼_P-identifiable: ∃ P, c s.t. T(f^-1(z)) = P T'(f'^-1(z)) + c, where c ∈^m is constant and T' and f' are different parameterization of model parameters T and f. The parameters θ are therefore identifiable up to a simple transformation and a permutation of the latent variables z. That is, for all parameter sets θ = (f, T, λ ) and θ' = (f', T', λ'), we have: ∀(θ,θ'): p_x(θ) = p_x(θ') ⇒θ∼θ' Notes on identifiability assumptions in single-cell models This work and similar works in the field <cit.> assume that x can be decomposed as x = f(z) + ϵ, where f is a differentiable bijection with a differentiable inverse. This is in general a very strong assumption in single-cell RNAseq, where the generative models estimate the parameters of the data distribution from f(z) to model gene expression count data x. So, the assumptions made on the decoder f in the previous works where theoretical proofs of identifiabilty are presented may not hold in discrete gene expression count data, as raised and discussed in <cit.>. However, empirical observations in <cit.> demonstrate improved performance for data with discrete features. Additionally, there are concerns regarding the size effect of d on the variability captured by the latent features z_I, that is how the conditional distribution p(z_I\|d) sufficiently vary with d <cit.>. We leave closer investigations of the validity of the above assumption on the decoder as well as the evaluation of the sufficient variability as future directions. § SPARSE MECHANISM SHIFT To represent cellular perturbations as interventions on latent variables, related work assumes that the underlying mechanisms undergo sparse shifts when subjected to interventions <cit.>. In the previous method, called sVAE+ <cit.>, an extension of sVAE <cit.> to single-cell data, each perturbation is considered a stochastic intervention that targets an unknown, yet sparse, subset of latent variables. Specifically, only a few latent components z_i are influenced by perturbation p, while the other components remain invariant. This can be expressed as follows: z_i\|p∼γ_i^pN(f(p),1) + (1-γ_i^p) N(0,1) Here, γ_i^p is a learnable binary variable assumed to be sparse, and N(f(p),1) represents a normal distribution with a mean determined by the function f(p), which depends on the perturbation p. The extension of this work to non-factorized priors, where the latent components can be dependent, is an interesting follow-up research direction that we defer to future work. § ARCHITECTURES All the neural networks used in this work are fully connected, feed-forward neural networks. For our experiments, we sampled randomly from reasonable ranges for the relevant hyperparameters and chose the setting with the best ELBO on the validation set. In general, we found that usual settings for VAEs, for example encoder and decoder as a neural network with 2 hidden layers and 128 neurons with ReLU activation functions, work well for our models. Furthermore, for a fair comparison, all the methods in Figure <ref> including iVAE and NF-iVAE are extended to use Negative-Binomial distribution for modeling the likelihood of the gene expression counts (i.e. for the decoder). In practice, for inVAE we use the following distributions: Gaussian with diagonal covariance for the encoder, Negative-Binomial for the decoder, the non-factorized general exponential family distribution for the invariant prior and a Gaussian distribution with diagonal covariance for the spurious prior. For inVAE, as well as iVAE and NF-iVAE, the gene expression counts are fed to the encoder by taking the log (x + 1) for numerical stability, same as per default for scVI. Additionally, we use Batch-Norm layers in the encoder and decoder for the same reasons. All models are optimized using Adam <cit.> with an initial learning rate of 0.01. We use PyTorch's default weight initialization scheme for the weights. All models can be trained entirely on CPUs on consumer grade Laptop machines within minutes or hours (see table <ref>). For training human hematopoiesis data, "site" is considered as a technical covariate and "donor" and "cell type" as biological covariates. For the lung cancer data, "assay" and "treatment" are used as technical covariates and "disease" and "cell type" as biological covariates. § EVALUATION METRICS We quantified the quality of the data integration using the following metrics from <cit.>, which are designated metrics for the evaluation of integration of single cell data, implemented in the scIB package. We provide a short explanation of each metric, for more details we suggest to read the original publication <cit.>. Cell type ASW, isolated label F1, isolated label silhouette, NMI and ARI were used as biological conservation metrics. To quantify batch mixing we used principal component regression, graph connectivity and batch ASW. NMI: NMI (Normalized Mutual Information) measures the overlap between two different clusterings. This score is used between the cell type labels and the clusters obtained via unsupervised clustering of the dataset after integration. The score is bounded between 0 and 1, with 1 denoting a perfect match between the two clusterings and 0 the absence of any overlap. ARI: ARI stands for Adjusted Rand Index. The raw Rand Index scores the similarity between two clusterings, it considers both correct overlaps and correct disagreements in the computation. This score is computed between the cell type labels and the clusters obtained on the integrated dataset. The score spans from 0 to 1, with 1 representing a perfect score. KBET: The kBET algorithm <cit.> determines whether the label composition of a k nearest neighborhood of a cell is similar to the expected (global) label composition. The test is repeated for a random subset of cells, and the results are summarized as a rejection rate over all tested neighborhoods. kBET works on a kNN graph. Cell type ASW: ASW (Average Silhouette Width) is a measure of the relationship between the within-cluster distances of a cell and the between-cluster distances of the same cell to the closest cluster. This metric can range between -1 and 1. Values of -1 indicate total misclassification, values of 0 imply overlap between clusters and values of 1 occur when clusters are well separated. We use two versions of this scores, one computed on cell type labels, and a modified version to quantify batch mixing (see ASW batch below). The score is scaled to have values between 0 and 1 using the following equation: cell type ASW = ASW_C + 1/2, where C represents the set of all cell type labels. Batch ASW: The batch ASW quantifies the quality of batch mixing in the integrated object. We obtain it by computing the ASW but on batch labels instead of cell type labels. Scores of 0 are indicative of good batch mixing, while any deviation from this score is the result of batch effects. In order to have a metric bound between 0 and 1, the following transformation is applied: batch ASW_j = 1/C_j∑_i∈ C_j 1 - s_batch(i) Here C_j is the set of cells with label j and \|C_j\| is the support of the set. The final score is obtained by averaging the batch ASW values obtained for each label: batch ASW = 1/\|M\|∑_j ∈ M batch ASW_j with M being the set of unique cell labels. Isolated label F1: Isolated labels are defined as cell type labels which are found in the smallest number of batches. We aim to determine how well these cell types are separated from the rest in the integrated data. To do so, we find the cluster containing the highest amount of cells from such isolated labels, and we then compute the F_1 score of the isolated label against all other labels within the cluster. We use the standard F_1 formulation: F_1 = 2 precision · recall/precision + recall Once again, scores of 1 represent the desirable outcome in which all cells from the isolated labels are grouped in one cluster. Isolated label silhouette: This is a cell type ASW score but computed only on the isolated labels. Graph connectivity This metric measures whether a kNN graph (G) computed on the integrated object connects cells that fall within the same cell type. It is bound between 0 and 1, with 0 indicating a graph where all cells are unconnected and, and 1 occurring when all cells of the same cell type are connected in the integrated output. For each cell type label a subset graph G(N_c, E_c) is computed, the final score is obtained using the following formulation: GC = 1/\|C\|∑_c ∈ C\|LCC (G(N_c, E_c))\|/\|N_c\| where C is the set of unique cell type labels, \|LCC()\| is the number of nodes in the largest connected component in the kNN graph and \|N_c\| is the number of nodes in the graph. § SINGLE-CELL DATASETS §.§ Human hematopoiesis This dataset contains 120,000 single cells from the human bone marrow of 5 diverse donors measured with two multi-modal technologies capturing gene expression profiles. This dataset is multi-site. Donors are all healthy non-smokers without recent medical treatment. Donors varied by age (22 - 40), sex, and ethnicity. We were interested in preserving between-donor variation. We therefore use site as a distractor for the spurious prior. Raw gene counts were downloaded as per author instructions <cit.>. For data processing, we follow the Best Practices guidelines for single cell genomics data analysis <cit.>. We select 2000 highly variable genes (HVGs) using the scanpy framework <cit.> for processing of single cell data. We obtain UMAP visualisations using the scanpy pipeline The batch correction metrics often evaluate how well the cells from different batches are mixed within a neighborhood of cells. Therefore, the results in Figure <ref> are shown for the two training datasets with similar donors but different sites where we expect similar cells from different batches to cluster together. §.§ Human lung cancer This dataset contains 155,098 cells from 21 fresh Small Cell Lung Cancer (SCLC) clinical samples obtained from 19 patients, as well as 24 Lung Adenocarcinoma (LUAD) and 4 tumor-adjacent normal lung samples as controls. The SCLC and LUAD cohorts include treated and untreated patients. Samples were obtained from primary tumors, regional lymph node metastases, and distant metastases (liver, adrenal gland, axilla, and pleural effusion). The data was downloaded from the Human Tumor Atlas portal, <data.humantumoratlas.org>. For data processing, we follow the Best Practices guidelines for single cell genomics data analysis <cit.>. § EXTENDED RESULTS §.§ Human hematopoiesis data The latent representations in Figure <ref> for scANVI, NF-iVAE, and iVAE colored by cell types are presented in Figure <ref>. §.§ Lung cancer data The data integration results using inVAE and scVI are presented in Figures <ref> and <ref>.
http://arxiv.org/abs/2307.03228v1	20230706180003	TuRMoiL of Survival: A Unified Survival Criterion for Cloud-Wind Interactions	[ "Matthew W. Abruzzo", "Drummond B. Fielding", "Greg L. Bryan" ]	astro-ph.GA	[ "astro-ph.GA" ]	0000-0002-7918-3086]Matthew W. Abruzzo Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA 0000-0003-3806-8548]Drummond B. Fielding Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA 0000-0003-2630-9228]Greg L. Bryan Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Cloud-wind interactions play an important role in long-lived multiphase flows in many astrophysical contexts. When this interaction is primarily mediated by hydrodynamics and radiative cooling, the survival of clouds can be phrased in terms of the comparison between a timescale that dictates the evolution of the cloud-wind interaction, (the dynamical time-scale ) and the relevant cooling timescale . Previously proposed survival criteria, which can disagree by large factors about the size of the smallest surviving clouds, differ in both their choice of and (to a lesser extent) . Here we present a new criterion which agrees with a previously proposed empirical formulae but is based on simple physical principles. The key insight is that clouds can grow if they are able to mix and cool gas from the hot wind faster than it advects by the cloud. Whereas prior criteria associate with the cloud crushing timescale, our new criterion links it to the characteristic cloud-crossing timescale of a hot-phase fluid element, making it more physically consistent with shear-layer studies. We develop this insight into a predictive expression and validate it with hydrodynamic simulations of ∼10^4 K, pressure-confined clouds in hot supersonic winds, exploring, in particular, high wind/cloud density contrasts, where disagreements are most pronounced. Finally, we illustrate how discrepancies among previous criteria primarily emerged due to different choices of simulation conditions and cooling properties, and discuss how they can be reconciled. § INTRODUCTION A variety of long-lived multiphase flows appearing in galaxy-related contexts involve or derive from cloud-wind interactions, in which a body of cooler gas (the cloud) moves with respect to a hotter volume-filling flow (the wind). Examples of such flows that shape galaxy evolution include multiphase galactic outflows <cit.>, the intermediate and high velocity clouds in a galactic fountain <cit.>, or the filamentary accretion of cold intergalactic material <cit.>. Other flows provide valuable information about their progenitor, such as the Magellanic Stream <cit.> or the tails of jellyfish galaxies <cit.>. In the absence of cooling the cloud-wind velocity differential drives turbulent mixing that destroys the cloud (i.e. homogenizes it with the other phase) faster than ram pressure removes the differential by accelerating and entraining the cloud. This is fundamentally at odds with the observed long lifetimes of the cool phase of galactic winds <cit.>. However, cooling is observed to be present in many cloud-wind interactions and can allow clouds to survive for long times <cit.>. The key question is what sets the criteria for when cooling outweighs the disruptive effects of mixing, enabling cloud survival? Cloud survival has received significant attention in the context of galactic winds, in which hot supersonic winds produced by supernovae are expected to accelerate and entrain cool interstellar clouds. Previous work has explored various physical mechanisms that extend the cloud's lifetime long enough for it to be entrained <cit.> or alter the acceleration mechanism <cit.>. In this letter we consider the regime in which rapid cooling facilitates the long-term survival of clouds <cit.>. We refer to this process as turbulent radiative mixing layer (TRML; which we pronounce as “turmoil”) entrainment. This process transfers mass and momentum from the wind to the cloud, which ultimately leads to cloud growth <cit.>. While the precise details have been a topic of vigorous debate, clouds undergo TRML entrainment when they incorporate new hot gas at a rate faster than they lose cold gas. This is generally phrased as the competition between some timescale that dictates the evolution of the cloud-wind interaction, which we generically term the dynamical time-scale, with the relevant cooling timescale . Most wind-tunnel studies <cit.> have adopted for the timescale on which the cloud is dispersed in the absence of cooling, which is associated with the cloud crushing timescale, although we note that studies of individual shear layers <cit.> typically find that is more closely related to the shear time, which has a different dependence on the cloud/wind density contrast. There has also been significant disagreement over the appropriate definition for largely because the cooling time is phase-dependent and it is not obvious which phase to use when computing the cooling time. Here we propose a new survival criterion that aims to reproduce the accuracy of previous empirically calibrated criteria with a simple physically motivated argument, thereby resolving many of the disagreements from prior works. In <ref>, we describe the physical conditions under which we compare survival criteria; we present the various criteria in <ref> and compare them in <ref>. Finally, we summarize and discuss our results in <ref>. § SCOPE OF THIS WORK To maximize our results' generality, we focus on TRML entrainment of a pressure-confined cloud in idealized conditions. We omit all physics beyond non-dissipative hydrodynamics and optically-thin radiative cooling (at uniform solar metallicity). While a cooling wind can affect an interaction's dynamics <cit.>, we consider it a distinct effect from TRML entrainment and take steps to omit it (see below). We describe regimes where omitted effects are relevant in <ref>. We restrict this work to supersonic winds (=/>1), density contrasts (χ=/) of ≥100, and a uniform initial thermal pressure of p/k_B = 10^3 K cm^-3. We primarily consider clouds with a physically motivated initial temperature of ≈5×10^3 K, which is approximately equal to the thermal equilibrium temperature for gas at this pressure exposed to the z=0 metagalactic UV background. We also consider several cases with ≈4×10^4 K for comparisons with <cit.>. In both cases, we shut cooling off below ; the modified curves are shown in <ref>. This cooling floor has minimal impact on (T) for the ≈5×10^3 K case. All simulations have a resolution of 16 cells per cloud radius and use identical setups and numerical schemes to <cit.>. They were run with [<http://enzo-e.readthedocs.io>], a rewrite of <cit.> built on the AMR framework <cit.>. We modeled radiative cooling with the pre-tabulated solver from the library[<https://grackle.readthedocs.io/>] <cit.> and assume a z=0 <cit.> UV background. We present ≈5×10^3 K simulations taken directly from <cit.>, supplemented by additional χ=100 runs with =3 and =6. All ≈4×10^4 K simulations were run for this work. To avoid cooling of the wind, cooling is artificially turned off above ∼0.6 in all cases other than the runs with χ=10^4,≈5×10^3 K and χ=10^3,≈4×10^4 K.[ These are a subset of our runs where is large compared to . More generally, previous works illustrate that turning cooling of the wind off in such cases has negligible impact on cloud survival <cit.>. ] § SURVIVAL CRITERIA §.§ New Criterion We now introduce the physical picture underlying our cloud survival criterion. To begin with, there is a large imbalance in the specific thermal energy <cit.> and momentum content of the cloud and wind material. Turbulent mixing redistributes these imbalances between interacting fluid elements. Thus, as hot phase fluid elements travel along the length of the cloud, they mix with cooler fluid elements. This mixing produces intermediate-temperature gas at the interface of the phases, which is highly susceptible to radiative cooling. The key physical insight is that clouds will not grow (and therefore will eventually be destroyed) if an incoming parcel of hot gas is not able to mix and cool before it is advected past the cloud. In other words, if the characteristic cooling timescale, , is short enough then mixed gas will be added to the cool-phase enabling the cloud to grow and ultimately survive. The window of opportunity for a cloud to capture a fluid element from the wind closes once the fluid element reaches the end of the cloud. The cartoon in <ref> illustrates that this corresponds to a timescale on the order of (but slightly longer than) , where =/. The cloud, therefore, survives when α where α is a constant on the order of a few to ∼ 10 that accounts for the fact that the relative velocity is somewhat less than due to momentum mixing, and that the length the element can traverse is somewhat longer than . We emphasize that this choice for differs from the usual adoption of the cloud-crushing timescale, which remains the time required for the cloud to be completely destroyed; however, the key point is that clouds need to incorporate fresh hot gas in order to grow, and this process – and its timescale – differs from the process of overall cloud destruction and its timescale. Next, we shift focus to the cooling timescale , which likely involves a weighted average over the cooling function <cit.>. Like prior works, we approximate it with computed at a representative intermediate temperature or specific internal energy, e = p / ((γ-1) ρ).[ Throughout this letter, we frame our discussion in terms of specific internal energy, rather than temperature, T. This simplifies equations when the mean molecular weight varies with T; for a varying μ, <χ while =χ. The reader can replace all occurrences of e with T/μ multiplied by some constants. ] <cit.> make a compelling argument for using the cooling timescale of the gas in the mixing layer = (,p). The definition of derives from the cloud-wind interaction's quasi-isobaric nature and an estimate for the specific internal energy of the mixed gas, = √(), motivated by analytic arguments from <cit.>. While we certainly agree with this overall philosophy, we argue that defined in this way does not always capture the sensitivity of to alterations in the e-dependence of <cit.> – in particular, we note that both the gas distribution across phases and the appropriate phase-weighting must depend on the precise form of the cooling function. A fully self-consistent weighting must await future work so here we adopt a subtle but important refinement to this original estimate in which ≈=(√(χ),p) <cit.>, where coincides with the minimum for ≤ e≤.[For the cooling function used in our ≈ 4×10^4 K runs (as was used in the original <cit.> paper), = since , the e that coincides with the minimum over ≤ e≤, matches .] Now, we plug our choice for into <ref>. Our new criterion predicts cloud survival for α = (√(χ), p). We find that α∼7 provides a good fit to the data and is consistent with our physical picture. This predicts that clouds survive when their radius exceeds the critical value = √(χ) (√(χ), p)/α, where =√(γ(γ - 1)) is the initial sound speed within the cloud. §.§ Existing Criteria To facilitate comparisons, we now briefly review some previously proposed survival criteria. The existing criteria are conceptually different from our new proposed criterion. As opposed to comparing cooling to advection, they argue that a cloud survives when cooling is more rapid than the rate at which it is destroyed. The cloud destruction timescale, is generally linked to =√(χ). Realistic galactic environments commonly have 10^2 χ 10^4. <cit.> proposed the first survival criterion, >, and <cit.> later proposed an improved refinement, >, that we term the criterion. These criteria specify critical cloud radii of ∼ (√(χ), p) ∼ (√(χ), p). <cit.> proposed a separate criterion f>=(,p), where f(, R_ cl, ) is an empirical parameterization of cloud lifetime described by (f/9±1)^10/3 = η 2 /1 pc /10^-2 cm^-3 ^2/10^4 km^2 s^-2, when η=1. We actually focus on a variant proposed by <cit.> that improves accuracy when >1, by using η = 2 (/1.5)^-2.5. This criterion, gives a critical cloud radius of = 72.2 pc (γ/5/3 0.6/μ_ w p/k_B/10^3 K cm^-3)^-3/13 (^2.9(χ, p)/ Gyr/10 km s^-1)^10/13 . § RESULTS AND RECONCILLATION We now turn to the key question: how well does this criterion work in simulations over a wide parameter range? Fig. <ref> and <ref> illustrate the robustness of our new survival criterion, <ref>. Simulations with various parameter choices are shown as symbols coded by their survability, while the lines show the predicted minimum cloud radius for survival. In these plots, rather than a binary categorization of the clouds' fates, we show the minimum cool phase mass, before the cloud starts growing.[There are two cases where the cool phase mass remains positive, but never show growth. However, in both cases the minimum mass is under 1% of the initial value.] As in prior works <cit.>, we define the cool phase as all gas with ρ> = /√(χ). We consider clouds to be destroyed when the cool-phase mass drops below some threshold fraction of its initial value. Our new criterion clearly matches the simulation results for sensible choices of the threshold (i.e., values ranging from 1% to 10%). In particular, it reproduces the steep χ dependence on R_ crit that is seen in the numerical simulations and is also consistent with the results when varying Mach number. One natural question is how this criterion compares to those that have been previously suggested and how to reconcile any differences. One possibility is that the interpretation of the simulation results might differ. Indeed, the exact definition of cloud destruction in a simulation is a thorny topic[For example, it may be important to differentiate between the case where a cloud survives and the case where cloud material is converted to hotter gas that seeds prompt cool-phase precipitation at a downstream location. This distinction has important implications for the transport of dust in galactic winds. However, such an endeavor is well beyond the scope of this letter.] without an obvious definition; there are almost as many definitions as there are survival criteria. These differing definitions have sometimes been cited as potential explanations for discrepant predictions made by survival criteria <cit.>; however, as we will demonstrate below, we generally find agreement when the criteria are appropriately compared, and these definitional differences are of secondary importance. To understand the relationships between survival criteria, it is insightful to compare the dependence on parameters describing the cloud-wind interaction, rather than the exact normalization. For interactions with ≈5×10^3 K and 100≤χ≤10^4, we find[Under these conditions, is constant and ∝ e^2.4 p^-1 for the relevant values of √() and .] ∝ χ^1.2 p^-1 ∝ χ^1.85 ^2.23 p^-1 R_ crit,new ∝ χ^1.7 p^-1, where R_ crit,new corresponds to our new scaling relation, <ref>. Comparisons of χ dependence are of central importance. We start by comparing the χ dependence of the various criteria to the results of our simulations. <ref>a and <ref> illustrate that the steeper χ dependence of our new criterion and the criterion both better match the observed high-χ cloud survival than the criterion. However, although both criteria predict the same χ scaling, the physical picture behind the strong χ scaling in our criterion is quite different from that of the criterion. The criterion essentially argues that should heavily weight high-T cooling (i.e. cooling in the hot wind itself). While ≈ may over-emphasize the low-T cooling in some cases, given the fact that many of our simulations shut off high-T cooling by hand and still find the same scaling, it is clear that ≈ incorrectly highlights the importance of cooling at high-T. Interestingly, since it matches the simulation results, we suggest that the accurate predictions of the criterion arise because of a surprising cancellation: the adoption of ≈ compensates for the incorrect χ-dependence of ∼. We now compare the results of simulations to the various predictions for the scaling of the critical cloud radius. <ref>b illustrates that there is some dependence on the definition of cloud destruction, with more relaxed definitions (i.e. allowing small survival fractions to count as survived) favoring our criterion's relatively shallow scaling, and stricter definitions (i.e. requiring higher survival fractions) favoring an intermediate dependence. While the criterion favors a steeper dependence, the supersonic simulations from <cit.> are consistent with our result. The consistency of their ≥1.5 runs with an intermediate dependence becomes apparent when the clouds' fates, specified by their fairly strict definition of cloud destruction, are plotted as a function of and . Their simulations only favor a steeper dependence when their =0.5 runs are also considered. This is not entirely surprising given the evidence that the interaction changes for subsonic winds <cit.>. We expect followup work to show that has steeper scaling for subsonic winds. Finally, we explain how criteria with weaker χ-dependence emerged. <cit.> focused on simulations with ≈4× 10^4 K and 100 ≤χ≤ 1000, and, as illustrated in <ref>, reached the very reasonable conclusion that < describes cloud survival. At this (essentially at the peak of the cooling curve), = and the only difference with our criterion is the use of in place of . Due to the exploratory nature of their work, <cit.> varied cloud radius by factors of ≈10 at each considered χ; while this allowed them to sample a large region of parameter space, it was too sparsely sampled to capture the √(χ) difference in scaling between and . When <cit.> determined that should replace , they used runs that varied for fixed χ. Because χ was fixed, = and = are equally consistent with their results. There have been several works that highlighted differences between the <cit.> and criteria <cit.>, however these came before the refinement introduced by <cit.>. Reconciliation efforts were stymied by the fact that only the combined differences in and are sufficient to explain the <cit.> criterion's weaker χ-dependence for ≈10^4 K (i.e. the commonly considered ). Further challenges likely arose from the effectiveness of ∼ for ≈ 4× 10^4 K <cit.> and association of with by both criteria. § CONCLUSION In this letter we propose a new criterion for predicting the survival of cool (∼10^4 K) clouds that encounter hot supersonic flows, based on the ratio between the timescale for hot-phase fluid elements to cross the length of the cloud, α, and , an estimate for the weighted average of the cooling timescale. Our new criterion shares the accuracy of the more empirically calibrated criteria from <cit.> and <cit.>, particularly in the high-χ limit, while employing the better-physically motivated choice of ∼, as in the <cit.> criterion. Additionally, discrepancies among criteria primarily emerged/persisted due to choices of validation simulation conditions (,,χ), while differences in the definition of cloud destruction within the simulations turn out to be less important. While previous survival criteria compare against , we are encouraged that our choice to compare against α produces more consistent scaling compared to the eddy turnover timescale, which is the relevant dynamical timescale on the micro-scale <cit.>. We are also encouraged that tall-box simulations of more realistic cloud-wind interactions find cloud-size distributions consistent with our new criterion <cit.>. We predict cloud survival when α, for α∼7. We tested our criterion for 5×10^3≲ (/ K) ≲ 4× 10^4 and supersonic winds, under the assumption that cooling in the wind is unimportant. Our criterion provides a minimum survival radius of = 271 pcℳ_ wχ_3^1.5 c_ s,cold,1^3/p̂_3 Λ_,-21.4 T_ min,cool/10^4.25 K 0.71/μ(T_ min,cool,p) 7/α ≈ 618 pcℳ_ wχ_3^1.5 T_ cl,4^1.5/p̂_3 Λ_,-21.4 T_ min,cool/10^4.25 K, where Λ_,-21.4≡Λ(T_)/(10^-21.4 erg cm^3 s^-1), p̂_3≡ p k_B^-1/(10^3 K cm^-3), χ_3≡χ/10^3, c_ s,cold,1≡/(10 km s^-1), T_ cl,4≡/(10^4 K), and T_ min,cool is the temperature where is minimized (between and ). In calculations where the temperature dependence of μ is modeled, the first line of <ref> is accurate, while approximations in the second line reduce accuracy to within ≈30% for 5×10^3 K and our fiducial pressure p̂_3 = 1.[ This error primarily arises from the omission of the ^-1.5 dependence. Additional error can arise at other pressures from the omitted μ_ min,cool^-1 dependence. When <10^4 K, larger errors can arise when the value of T_, used to compute Λ(T_), is estimated while ignoring μ dependence. ] Our criterion's predictions are most meaningful for clouds in thermal equilibrium (for reasonable equilibration timescales). For our fiducial pressure, p̂_3 = 1, T_ min,cool=10^4.25 K, the equilibrium is ≈5× 10^3 K, and ≈ 8.5 km s^-1. With that said, our criterion can still be applied at other choices of under the assumption that cooling is shut off below that temperature.[ We expect our criterion to be accurate for any spatially uniform (T) that satisfies three properties on the interval < T<: (i) it has no local maximum, (ii) there is up to 1 local minimum, which lies between and =√(χ)/, and (iii) cooling in the wind is relatively slow.] Under such contrived conditions, with p̂_3 = 1 and >5× 10^3 K, T_ min,cool=max(, 10^4.25 K). Given that lower pressure environments have larger equilibrium , larger T_ min,cool and smaller magnitudes of Λ, our criterion suggests that clouds likely require unreasonably large sizes to survive in such cases. In contrast, our new survival criteria is likely relevant for colder clouds, like those expected in the higher pressure environments (the equilibrium temperature significantly drops for p/k_B≳ 10^4 K cm^-3) where starburst galaxies launch their outflows <cit.>. Based on the recent study of colder (10^3 K) gas clouds in <cit.>, we expect that our criterion specifies conditions for the long-term survival of a cool (∼10^4 K) phase and suspect that ψ t_ cool,max may specify conditions for the colder phase's survival; ψ is a normalization constant and t_ cool,max is the maximum of between and T_ min,cool. Because <cit.> considered runs with fixed χ, their results fit these criteria equally as well as their own criteria (the primary difference is they use instead of ). It is also noteworthy that the criterion does not generalize well to these conditions <cit.>. Followup work is clearly required. Next, we highlight limitations of our idealized setup. The shaded regions in Fig. <ref> and <ref> illustrate where other effects are relevant in more realistic simulations. The light gray region's lower bound highlights where a cooling wind may become dynamically important , and the dark gray region highlights where pressure confinement of the cloud fails <cit.>. The cyan region highlights Jeans-unstable clouds that will likely collapse <cit.>. Self-shielding against ionizing radiation may also be dynamically relevant at most radii. In other words, simulations of cool (∼10^4 K) clouds that undergo TRML entrainment in a supersonic flow only make robust predictions for a narrow range of conditions, while neglecting self-gravity, cooling in the wind, and self-shielding. However, these assumptions may be robust for subsonic flows, if the survival criterion has steeper dependence in those cases. This regime is of particular interest in the context of starburst galaxy outflows because most cloud acceleration occurs when the outflow is subsonic <cit.>. Future work may also want to consider the impact of other physical effects. For example, there is evidence that magnetic fields with intermediate to strong field-strengths may make it easier for clouds to survive <cit.>. While the effects of conduction <cit.> could have large impacts, there's reason to believe they won't strongly influence the survival criterion <cit.>. There are other context-specific effects that may also be relevant, such as external sources of turbulence <cit.>, large-scale external gravitational fields <cit.>, cloud geometry <cit.>, or ensembles of clouds <cit.>. Finally, we consider convergence of the cool phase mass, M_> mix, evolution. While our resolution, 16 cells per cloud radius, is sufficient for convergence in most cases, convergence is more difficult when =6 <cit.> or χ=10^4 <cit.>. For =6, we are encouraged by convergence in the M_> mix evolution (at low resolutions) of the case taken from <cit.> and the monotonic scaling of the minimum M_> mix with (see <ref>b). Even if the χ=10^4 runs are not converged, they are not of central importance to our conclusions. We are grateful to R. Farber, M. Gronke, P. Oh, and B. Tan for useful conversations about survival criteria. We are also thankful for the efforts of J. Bordner, M. Norman and the efforts of the other developers. GLB acknowledges support from the NSF (AST-2108470, XSEDE grant MCA06N030), NASA TCAN award 80NSSC21K1053, and the Simons Foundation (grant 822237) and the Simons Collaboration on Learning the Universe. numpy <cit.>, matplotlib <cit.>, yt <cit.>, scipy <cit.>, pandas <cit.>, aasjournal
http://arxiv.org/abs/2307.01987v1	20230705023051	Tetrahedron genuine entanglement measure of four-qubit systems	[ "Meng-Li Guo", "Zhi-Xiang Jin", "Bo Li", "Shao-Ming Fei" ]	quant-ph	[ "quant-ph" ]	[email protected] School of Mathematical Sciences, Capital Normal University, Beijing 100048, China School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, China School of Computer and Computing Science, Hangzhou City University, Hangzhou 310015, China [email protected] School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples. Tetrahedron genuine entanglement measure of four-qubit systems Shao-Ming Fei ============================================================== Keywords: four-qubit, genuine entanglement measure, concurrence, tetrahedron § INTRODUCTION Quantum entanglement is an essential feature of quantum mechanics that distinguishes the quantum from the classical world <cit.>. It has potential applications in quantum information processing such as quantum cryptography <cit.>, quantum dense coding <cit.>, quantum secret sharing <cit.>, quantum teleportation <cit.> and measurement-based quantum computing <cit.>. To use the entanglement as a resource is not only a matter of how to detect it, but also how to quantify it. The entanglement measures for bipartite systems have been well studied <cit.>, such as the concurrence <cit.>, entanglement of formation <cit.> and negativity <cit.>. For a bipartite pure state \|ψ⟩_AB in a finite-dimensional Hilbert space ℋ_A⊗ℋ_B=ℂ^d_1⊗ℂ^d_2 the concurrence is defined as <cit.> C( \|ψ⟩_AB)=√(2[1-Tr(ρ^2_A)]), where ρ_A is the reduced density matrix by tracing over the subsystem B, ρ_A=Tr_B(\|ψ⟩_AB⟨ψ\|). The concurrence is extended to mixed states ρ=∑_i p_i \| ψ_i ⟩⟨ψ_i \|, 0 ≤ p_i ≤ 1, ∑_i p_i =1, by the convex roof extension C(ρ_AB)=min_{p_i, \| ψ_i ⟩}∑_i p_i C(\| ψ_i ⟩ ), where the minimum is taken over all possible pure state decompositions of ρ_AB. For entanglement in multipartite systems, although the experimental observation has been successfully implemented <cit.>, the quantification of multipartite entanglement is still far from being satisfied. In <cit.> Ma et al. introduced the genuine multipartite entanglement measure. A measure of genuine multipartite entanglement (GME) E(ρ) of a state ρ should at least satisfy: (a) the measure must be zero for all product and biseparable states; (b) the measure must be positive for all non-biseparable states (e.g., the Greenberg-Horn-Selinger (GHZ) state and W state for the three-qubit case). In addition, besides conditions (a) and (b), a well defined GME measure also needs to satisfy the following condition: (c) the measure should be nonincreasing under local operations and classical communications. Recently, Xie et al. <cit.> proposed a new triangle measure specifically for tripartite systems, which has a simple form and an elegant geometric interpretation, but does not satisfy the GME requirement (c). In <cit.> the authors presented an improved GME measure that satisfies all the requirements. Further related researches have also been presented very recently <cit.>. In this paper, we analyze in detail the multipartite entanglement measures for four-qubit systems and show how the concurrence tetrahedron constitute a well defined GME measure. The rest of this article is organized as follows. In Sec. 2, we analyze and calculate the concurrences of all nine-type four-qubit states. It is found that except for the product states and biseparable states, the non-biseparable states form a concurrence tetrahedron, which shows that the concurrence tetrahedron is a new bona fide entanglement measure. In addition, we also compare the concurrence tetrahedron with genuinely multipartite concurrence (GMC). It is found that GMC and the concurrence tetrahedron are not equivalent. For four-qubit systems the concurrence tetrahedron has more advantages than the generalized method of moments. Finally we discuss and summarize the results in Sec.3. § TETRAHEDRAL ENTANGLEMENT MEASUREMENT OF FOUR-QUBIT STATES Two quantum states that can be converted to each other with limited success probability under local operation and classical communication (LOCC) or stochastic local quantum operations assisted by classical communication (SLOCC) belong to the same entanglement class <cit.>. For three-qubit systems, there are two types of entangled states with completely different properties: the GHZ states and W states. Westerlet et al. show that there are nine different types of entanglements for four-qubit systems <cit.> under SLOCC operations. For an arbitrary N-qubit pure state \|Ψ⟩, denote E_j=E_j\|1...k≠ j...N the entanglement under the bipartition of the j-th qubit and the rest ones. It is shown that <cit.>, E_j≤∑_k≠ jE_k, which is valid for a generic entanglement measure E such as the von Neumann entropy <cit.>, concurrence <cit.> and negativity <cit.>. Consider the concurrence of general four-qubit pure states \|ψ_ABCD⟩. Denote C_i=C_i\|jkl and C_ij=C_ij\|kl the concurrences under bipartitions i\|jkl and ij\|kl, respectively, where i≠ j≠ k≠ l∈{1,2,3,4}. We have C_i\|jkl≤ C_j\|kli+C_k\|ijl+C_l\|ijk and C_ij\|kl≤ C_ik\|lj+C_il\|jk, where the equality C_ij\|kl= C_ik\|lj+C_il\|jk holds when the state is biseparable under the partition <cit.>. To have an explicit geometric picture, consider the value of each concurrences C_i and C_ij as the length of lines. For non-biseparable states, according to inequality (<ref>), C_ij, C_ik and C_il form a triangle. Moreover, let u,v,w be the smallest three of {C_i, C_j, C_k, C_l} with 0≤ u≤ v ≤ w. Then u,v,w and C_ij, C_ik, C_il form a tetrahedron, as shown in Fig. <ref>. We call it concurrence tetrahedron. We first give a lemma that will be used in deriving our main results. Let ABC be a triangle with three vertices A, B and C. Denote 3R the sum of the distances from the center O of the circumscribed circle of the triangle to the three vertices. Let D is a point which is not coplanar with the triangle ABC, and H is the sum of the distances from D to the three vertices of ABC. If H-3R>0, then A, B, C and D can form a tetrahedron. Proof Since the sum of the distances from the center of the circumscribed circle of the triangle to the three vertices of the triangle is the longest, if H(\|ψ_ABCD⟩) ≤ 3R, then D is inside the triangle ABC, and the four points A,B,C,D cannot form a tetrahedron. Particularly, when H(\|ψ_ABCD⟩)=3R, D is the center of the circumscribed circle of the triangle. Therefore, H(\|ψ_ABCD⟩)>3R implies that D is outside the BCD, and the A,B,C,D can form a tetrahedron. The volume of the concurrence tetrahedron is another interesting quantity. For both product and biseparate states, the lengths of the tetrahedron is zero, i.e., the volume is zero and thus the condition (a) of GME is satisfied. In addition, the volume of the concurrence tetrahedron also satisfies the condition (b). In Fig. <ref> we list in table the relations between the entanglement of four-qubit states and the corresponding concurrence tetrahedron. For any four-qubit pure state \|ψ_ABCD⟩, the volume of the concurrence tetrahedron defines a bona fide GME measure, 𝒱_1234(\|ψ_ABCD⟩)= 1/12√(4 u^2 v^2 w^2-u^2 D^2- v^2 E^2-w^2 F^2+DEF), where D=v^2+w^2-C_ij^2, E=u^2+w^2-C_ik^2, F=u^2+v^2-C_il^2, u,v,w and C_ij, C_ik, C_il are the edges of concurrence tetrahedron. Proof We first prove that \|ψ_ABCD⟩ is GME iff 𝒱_1234(\|ψ_ABCD⟩)>0. On one hand, if 𝒱_1234(\|ψ_ABCD⟩)>0, then each edge of the concurrence tetrahedron, u, v, w, C_ij, C_ik and C_il are all greater than zero. Hence \|ψ_ABCD⟩ is a genuine multipartite entangled state. On the other hand, if 𝒱_1234(\|ψ_ABCD⟩)=0, at least one edge of the concurrence tetrahedron is zero, namely, either ψ_ABCD=ρ_A⊗ρ_BCD or ψ_ABCD=ρ_B⊗ρ_ACD or ψ_ABCD=ρ_C⊗ρ_ABD or ψ_ABCD=ρ_D⊗ρ_ABC. Therefore, ψ_ABCD is not a genuine multipartite entangled state. We next prove that 𝒱_1234(\|ψ_ABCD⟩) cannot increase under LOCC, i.e., 𝒱_1234(\|ψ_ABCD⟩)≥𝒱_1234(Λ(\|ψ_ABCD⟩)) for any LOCC map Λ. As concurrence C does not increase under LOCCs, the edges of the tetrahedron do not increase under LOCCs. Hence, we only need to prove that the volume 𝒱_1234 is an increasing function of its edges u, v, w, C_ij, C_ik and C_il. In the concurrence tetrahedron as shown in Fig. <ref>, with respect to Lemma <ref> we set H(\|ψ⟩)=u+v+w and denote R the radius of the circumscribed circle of the triangle on the bottom, G(\|ψ⟩)=H(\|ψ⟩)-3R, where R=C_ij C_ik C_il/4 √(p (p-C_ij) (p- C_ik) (p-C_il)) with p=1/2 (C_ij+C_ik+C_il). In the following prove the conclusion one by one according to all the types of four-qubit states under LOCC. (C_1) The representative state of the L_0_3⊕1̅0_3⊕1̅ family is L_0_3⊕1̅0_3⊕1̅=\|0000⟩+\|0111⟩ <cit.>. Taking \|ψ_1 ⟩= 1/√(2)L_0_3⊕1̅0_3⊕1̅, we have C_1 (\|ψ_1 ⟩)=0, C_2 (\|ψ_1 ⟩) =C_3 (\|ψ_1 ⟩)=C_4 (\|ψ_1 ⟩) =1. That is to say, one of {u, v, w} is 0. Hence, \|ψ_1 ⟩ cannot gives rise to a tetrahedron. It is a product state of the one-qubit state \|0⟩ and the three-qubit \|GHZ⟩ state. (C_2) The representative state of the L_0_7⊕1̅ family is L_0_7⊕1̅=\|0000⟩ +\|1011⟩ +\|1101⟩+\|1110⟩. Taking \|ψ_2 ⟩= 1/2L_0_7⊕1̅, we have C_1 (\|ψ_2 ⟩) =√(3)/2, C_2 (\|ψ_2 ⟩) =C_3 (\|ψ_2 ⟩)=C_4 (\|ψ_2 ⟩) =1, C_12 (\|ψ_2 ⟩)=C_13 (\|ψ_2 ⟩)=C_14 (\|ψ_2 ⟩)=√(5)/2. Therefore, u=√(3)/2, v=w=1. Then G(\|ψ_2 ⟩)=H(\|ψ⟩)-3R=0.92953>0. \|ψ_2 ⟩ can form a tetrahedrons and \|ψ_2 ⟩ is genuine multipartite entangled state. We need to consider the derivatives of 𝒱_1234(\|ψ_2⟩) with respect to C, ∂𝒱_1234(\|ψ_2⟩)/∂ u=0.1049>0, ∂𝒱_1234(\|ψ_2⟩)/∂ v=0.0692>0, ∂𝒱_1234(\|ψ_2⟩)/∂ w=0.0692>0, ∂𝒱_1234(\|ψ_2⟩)/∂ C_12(\|ψ_2⟩)=0.0389>0, ∂𝒱_1234(\|ψ_2⟩)/∂ C_13(\|ψ_2⟩)=0.0542>0, ∂𝒱_1234(\|ψ_2⟩)/∂ C_14(\|ψ_2⟩)=0.0387>0. Thus the monotonicity of 𝒱_1234(\|ψ_2⟩) holds and hence 𝒱_1234(\|ψ_2⟩) is non-increasing under LOCC. (C_3) The representative state of the L_0_5⊕3̅ family is L_0_5⊕3̅=\|0000⟩ +\|0101⟩ +\|1000⟩+\|1110⟩. Let \|ψ_3 ⟩=1/2L_0_5⊕3̅. We have C_1 (\|ψ_3 ⟩) =C_2 (\|ψ_3 ⟩)=C_4 (\|ψ_3 ⟩) =√(3)/2, C_3 (\|ψ_3 ⟩) =1, C_12 (\|ψ_3 ⟩)=√(5)/2, C_13 (\|ψ_3 ⟩)=C_14 (\|ψ_3 ⟩)=1. We have u=v=w=√(3)/2 and G(\|ψ_3 ⟩)=0.92298>0. Therefore, \|ψ_3 ⟩ can form a tetrahedrons and \|ψ_3 ⟩ is genuine multipartite entangled state. The derivatives of 𝒱_1234(\|ψ_3⟩) with respect to C are given by ∂𝒱_1234(\|ψ_3⟩)/∂ u=0.0587>0, ∂𝒱_1234(\|ψ_3⟩)/∂ v=0.0783>0, ∂𝒱_1234(\|ψ_3⟩)/∂ w=0.0783>0, ∂𝒱_1234(\|ψ_3⟩)/∂ C_12(\|ψ_3⟩)=0.0101>0, ∂𝒱_1234(\|ψ_3⟩)/∂ C_13(\|ψ_3⟩)=0.0452>0, ∂𝒱_1234(\|ψ_3⟩)/∂ C_14(\|ψ_3⟩)=0.0452>0. Thus the monotonicity of 𝒱_1234(\|ψ_3⟩) holds and hence 𝒱_1234(\|ψ_3⟩) is non-increasing under LOCC. (C_4) The representative state of the L_a_20_3⊕ 1 family is L_a_20_3⊕1̅=a(\|0000⟩+\|1111⟩)+(\|0011⟩+\|0101⟩+\|0110⟩). Set \|ψ_4 ⟩=1/√(2a^2+3)L_a_20_3⊕ 1. We have C_1 (\|ψ_4 ⟩) =√(4 a^2(a^2+3))/2 a^2+3, C_2 (\|ψ_4⟩)=C_3(\|ψ_4⟩)=C_4(\|ψ_4⟩)=2 √(a^4+3 a^2+2)/2 a^2+3, C_12 (\|ψ_4 ⟩)=C_13 (\|ψ_4 ⟩)=C_14 (\|ψ_4 ⟩)=2 √(a^4+4 a^2+2)/2 a^2+3. If a=0, one has C_1 (\|ψ_4 ⟩)=0 and C_2 (\|ψ_4⟩)=C_3(\|ψ_4⟩)=C_4(\|ψ_4⟩)=2 √(2)/3. Then u=0 and \|ψ_4 ⟩ cannot form a tetrahedron. It is a product state of the one-qubit state \|0⟩ and the three-qubit \|W⟩ state. When a ≠ 0, C_1(\|ψ_4 ⟩)>0, C_2 (\|ψ_4⟩)=C_3(\|ψ_4⟩)=C_4(\|ψ_4⟩)>0 and √(4 a^2(a^2+3))/2 a^2+3<2 √(a^4+3 a^2+2)/2 a^2+3. So, u=√(4 a^2(a^2+3))/2 a^2+3 and v=w=2 √(a^4+3 a^2+2)/2 a^2+3. Then we have H(\|ψ_4 ⟩) =u+v+w=2 √(a^2 (a^2+3)/(2 a^2+3)^2)+4 √(a^4+3 a^2+2/(2 a^2+3)^2), G(\|ψ_4 ⟩) =-2 √(3)(a^4+4 a^2+2/(2 a^2+3)^2)^3/2/√((a^4+4 a^2+2)^2/(2 a^2+3)^4)+2 √(a^2 (a^2+3)/(2 a^2+3)^2)+4 √(a^4+3 a^2+2/(2 a^2+3)^2). We have G(\|ψ_4 ⟩)>0, see Fig. <ref> (a). Hence, if a ≠ 0 \|ψ_4 ⟩ can form a tetrahedrons and thus \|ψ_4 ⟩ is genuine multipartite entangled state. Next, we can get the derivatives of 𝒱_1234(\|ψ_4⟩) with respect to C, the expression of ∂𝒱_1234(\|ψ_4 ⟩)/∂ C(\|ψ⟩) is given in Appendix A. So the monotonicity of 𝒱_1234(\|ψ_4⟩) holds and hence 𝒱_1234(\|ψ_4⟩) is non-increasing under LOCC. (C_5) The representative state of the L_a_4 family is La_4=a(\|0000⟩ +\|0101⟩ +\|1010⟩+\|1111⟩)+(i\|0001⟩ +\|0110⟩ -i\|1011⟩). Let \|ψ_5 ⟩=1/√(4a^2+3)L_a_4. We have C_1 (\|ψ_5 ⟩) =C_2 (\|ψ_5 ⟩) =√(8 (2 a^4+7 a^2+2))/4 a^2+3, C_3 (\|ψ_5 ⟩) =C_4 (\|ψ_5 ⟩)=√(8 (2 a^4+7 a^2+1))/4 a^2+3, C_12 (\|ψ_5 ⟩)=C_14 (\|ψ_5 ⟩)=√(4 (6 a^4+14 a^2+3))/4 a^2+3, C_13 (\|ψ_5 ⟩)=√(8 (6 a^2+1))/4 a^2+3. Since C_1 (\|ψ_5 ⟩)>C_3 (\|ψ_5 ⟩), u=v=√(8 (2 a^4+7 a^2+1))/4 a^2+3 and w=√(8 (2 a^4+7 a^2+2))/4 a^2+3, we have H(\|ψ_5 ⟩) =u+v+w=2 √(2)(2 √(2 a^4+7 a^2+1/(4 a^2+3)^2)+√(2 a^4+7 a^2+2/(4 a^2+3)^2)), G(\|ψ_5 ⟩) =√(2)(-3 (6 a^4+14 a^2+3) √(6 a^2+1/(4 a^2+3)^2)/(4 a^2+3)^2 √(72 (a^2+2) a^4+52 a^2+5/(4 a^2+3)^4)+4 √(2 a^4+7 a^2+1/(4 a^2+3)^2)+2 √(2 a^4+7 a^2+2/(4 a^2+3)^2)). As shown in Fig. <ref> (b), G(\|ψ_5 ⟩)>0. Hence \|ψ_5 ⟩ can form a tetrahedrons and \|ψ_5⟩ is genuine multipartite entangled state. Next, we can get the derivatives of 𝒱_1234(\|ψ_5⟩) with respect to C, the expression of ∂𝒱_1234(\|ψ_5 ⟩)/∂ C(\|ψ⟩) is given in Appendix B. So the monotonicity of 𝒱_1234(\|ψ_5⟩) holds and hence 𝒱_1234(\|ψ_5⟩) is non-increasing under LOCC. (C_6) The representative state of the L_a_2b_2 family is L_a_2b_2=a(\|0000⟩ +\|1111⟩)+b(\|0101⟩+\|1010⟩)+\|0110⟩ +\|0011⟩. Setting \|ψ_6 ⟩=1/√(2(1+a^2+b^2))L_a_2b_2, we have C_1(\|ψ_6⟩)=C_2(\|ψ_6⟩)=√(2 a^2 (b^2+1)+a^4+b^2 (b^2+2))/a^2+b^2+1, C_3 (\|ψ_6 ⟩) =C_4 (\|ψ_6 ⟩)=1, C_12(\|ψ_6⟩)=C_14(\|ψ_6⟩)=√(a^2(4b^2+2)+a^4+(b^2+1)^2)/a^2+b^2+1, C_13 (\|ψ_6 ⟩)=√(-2 a^2 (b^2-2)+a^4+b^2 (b^2+4))/a^2+b^2+1. When a=b=0, we have C_1(\|ψ_6⟩)=C_2(\|ψ_6⟩)=0 and u=v=0. Hence, \|ψ_6 ⟩ cannot form a tetrahedron, which is a product state: 1/√(2)(\|01⟩ _13⊗ (\|01⟩ +\|10⟩ )_24). Consider the case that a and b are not both 0. Since 0<C_1(\|ψ_6⟩)<C_3 (\|ψ_6 ⟩), we have u=v=√(2 a^2 (b^2+1)+a^4+b^2 (b^2+2))/a^2+b^2+1 and w=1. We get H(\|ψ_6 ⟩) =u+v+w=2 √(1-1/(a^2+b^2+1)^2)+1 , G(\|ψ_6 ⟩) =1-3 (a^2 (4 b^2+2)+a^4+(b^2+1)^2) √(-2 a^2 (b^2-2)+a^4+b^4+4 b^2/(a^2+b^2+1)^2)/(a^2+b^2+1)^2 √((-2 a^2 (b^2-2)+a^4+b^4+4 b^2) (2 a^2 (9 b^2+2)+3 a^4+3 b^4+4 b^2+4)/(a^2+b^2+1)^4)+2 √(1-1/(a^2+b^2+1)^2). The G(\|ψ_6 ⟩) is shown in Fig. <ref> (c). Obviously, G(\|ψ_6 ⟩)>0. Thus \|ψ_6 ⟩ can form a tetrahedron and \|ψ_6 ⟩ is genuine multipartite entangled state. Next, we can get the derivatives of 𝒱_1234(\|ψ_6⟩) with respect to C are shown in Fig. <ref> (a), and the expression of ∂𝒱_1234(\|ψ_6 ⟩)/∂ C(\|ψ⟩) is given in Appendix C. Obviously, when a and b are not all 0 at the same time, ∂𝒱_1234(\|ψ_6⟩)/∂ C(\|ψ_6⟩)≥0. Thus the monotonicity of 𝒱_1234(\|ψ_6⟩) holds and hence 𝒱_1234(\|ψ_6⟩) is non-increasing under LOCC. (C_7) The representative state of L_ab_3 is L_ab_3= a(\|0000⟩+\|1111⟩)+a+b/2(\|0101⟩+\|1010⟩) +a-b/2(\|0110⟩ +\|1001⟩) +i/√(2)(\|0001⟩ +\|0010⟩+\|0111⟩+\|1011⟩). With normalization we set \|ψ_7 ⟩= 1/√(3 a^2+b^2+2)L_ab_3. We have C_1 (\|ψ_7 ⟩) = C_2 (\|ψ_7 ⟩)=C_3 (\|ψ_7 ⟩) =C_4 (\|ψ_7 ⟩) =√(a^2 (6 b^2+44)+9 a^4+b^4+12 b^2+3)/3 a^2+b^2+2, C_12(\|ψ_7⟩)= √(4 (a^2 (3 b^2+10)+3 a^4+2 b^2+1))/3 a^2+b^2+2, C_13(\|ψ_7⟩)=√(M_11-24 a^3 b+16 a b)/√(2)(3 a^2+b^2+2), C_14(\|ψ_7⟩)= √(M_11+24 a^3 b-16 a b)/√(2)(3 a^2+b^2+2), and H(\|ψ_7 ⟩) =u+v+w=3 √(a^2 (6 b^2+44)+9 a^4+b^4+12 b^2+3/(3 a^2+b^2+2)^2) , G(\|ψ_7 ⟩) =12X_11√(8 a^2-17/(3 a^2+b^2+2)^2+8/3 a^2+b^2+2+1)-3 √(a^2 (6 b^2+20)+6 a^4+4 b^2+2/(3 a^2+b^2+2)^2)√(N_11-8 a (2-3 a^2) b/(3 a^2+b^2+2)^2)√(8 a (2-3 a^2) b+N_11/(3 a^2+b^2+2)^2)/4X_11, where M_11=a^2 (6 b^2+88)+15 a^4+3 b^4+24 b^2+8, N_11=6 (a^2+4) b^2+15 a^4+88 a^2+3 b^4+8, X_11=√(a^6 (294-45 b^2)+a^4 (9 (b^2+42) b^2+707)+a^2 (9 b^6+90 b^4+322 b^2+128)+27 a^8+6 b^6+43 b^4+32 b^2+6/(3 a^2+b^2+2)^4). According to Fig. <ref> (d), we have G(\|ψ_7 ⟩)>0. So, \|ψ_7 ⟩ can form a triangular pyramid and is genuine multipartite entangled state. The derivatives of 𝒱_1234(\|ψ_7⟩) with respect to C are shown in Fig. <ref> (b), and the expression of ∂𝒱_1234(\|ψ_7 ⟩)/∂ C(\|ψ⟩) is given in Appendix D. As shown in Fig. <ref> (b), all the derivatives of 𝒱_1234(\|ψ_7⟩) with respect to C satisfy ∂𝒱_1234(\|ψ_7⟩)/∂ C(\|ψ_7⟩)>0. Thus the monotonicity of 𝒱_1234(\|ψ_7⟩) holds and hence 𝒱_1234(\|ψ_7⟩) is non-increasing under LOCC. (C_8) The representative state of L_abc_2 is L_abc_2= a+b/2(\|0000⟩+\|1111⟩)+a-b/2(\|0011⟩+\|1100⟩) +c(\|0101⟩ +\|1010⟩)+\|0110⟩. From the normalized state of L_abc_2, \|ψ_8 ⟩= 1/√(1+a^2+b^2+2c^2)L_abc_2, we have C_1 (\|ψ_8 ⟩) = C_2 (\|ψ_8 ⟩)=C_3 (\|ψ_8 ⟩) =C_4 (\|ψ_8 ⟩) = √(2 a^2 M_12+2 b^2( c^2+N_12) +4 c^2 N_12 +L_12)/a^2+b^2+2c^2+1, C_12(\|ψ_8⟩) = √(4(a^2 M_12+b^2 ( c^2+N_12) +c^4))/a^2+b^2+2c^2+1, C_13 (\|ψ_8 ⟩) = √(O_12+8 a b (c^2+ N_12)+3 L_12)/√(2)( a^2+b^2+2 c^2+1), C_14 (\|ψ_8 ⟩) = √(O_12-8 a b (c^2+ N_12)+3 L_12)/√(2)(a^2+b^2+2 c^2+1), where M_12=b^2+2c^2+1, N_12=c^2+1, L_12=a^4+b^4, O_12=2 a^2 ( M_12+1)+4 b^2 N_12+8 c^2( N_12+1). When a=b=c=0, u=v=w=0, \|ψ_8 ⟩ cannot form a tetrahedron. It is a full separable state. In <cit.> the authors divide the family L_abc_2 into three subfamilies, the subfamily L_abc_2 with c=0, with abc≠ 0 and with c≠ 0 and ab=0. Substituting these three subfamilies into C_i(\|ψ_8 ⟩) and C_ij(\|ψ_8 ⟩) respectively, we all have C_i(\|ψ_8 ⟩)>0 and G(\|ψ_8 ⟩) >0. When c=0, see Fig. <ref> (e), \|ψ_8 ⟩ can form a triangular pyramid and is genuine multipartite entangled state. Consider the derivatives of 𝒱_1234(\|ψ_8⟩) with respect to C for c=0, and the expression of ∂𝒱_1234(\|ψ_8 ⟩)/∂ C(\|ψ⟩) is given in Appendix E. As shown in Fig. <ref> (c), we have ∂𝒱_1234(\|ψ_8⟩)/∂ C(\|ψ_8⟩)>0. Similar to the case of c=0, one easily obtains ∂𝒱_1234(\|ψ_8⟩)/∂ C(\|ψ_8⟩)>0 for abc≠ 0 or c≠ 0, ab=0. Thus the monotonicity of 𝒱_1234(\|ψ_8⟩) holds and hence 𝒱_1234(\|ψ_8⟩) is non-increasing under LOCC. (C_9) The representative state of G_abcd is G_abcd= a+d/2(\|0000⟩+\|1111⟩)+a-d/2(\|0011⟩+\|1100⟩)+b+c/2(\|0101⟩+\|1010⟩)+b-c/2(\|0110⟩ +\|1001⟩). Set \|ψ_9 ⟩=1/√(a^2+ab+3/2b^2+c^2-ad+d^2/2)G_abcd. We have C_1 (\| ψ_9 ⟩) = C_2 (\|ψ_9 ⟩)=C_3 (\|ψ_9 ⟩) =C_4 (\|ψ_9 ⟩) = √(2 (M_22+N_22+4 a b L_22) )/2 a^2+2 a b-2ad+3 b^2+2 c^2+d^2, C_12(\|ψ_9 ⟩) = √(2 (M_32+N_32+4 a^2 (L_22+b^2)))/2 a^2+2 a b-2ad+3 b^2+2 c^2+d^2, C_13 (\|ψ_9 ⟩) = √(2 (M_42-N_42+L_42-24 a b c d) )/2 a^2+2 a b-2ad+3 b^2+2 c^2+d^2, C_14 (\|ψ_9 ⟩) = √(2 (M_42-N_42+L_42+24 a b c d) )/2 a^2+2 a b-2ad+3 b^2+2 c^2+d^2, where M_22= 8 a^3 b+2 a^4+4 ad^2+8 b^2 c^2+7 b^4+2 c^4+2 b^2 d^2-d^4, N_22= 4 a^2 (-2 a d+3 b^2+c^2)-4 a d (3 b^2+2 c^2+d^2), L_22= -2a d+3 b^2+2 c^2+d^2, M_32= 8 a^3 b+4 ad^2+12 b^2 c^2+6 b^2 d^2+5 b^4+4 c^2 d^2-3 d^4, N_32= 4 a b (-2 ad+3 b^2+2 c^2+d^2)-4 ad(3 b^2+2 c^2+d^2), M_42= 8 a^3 b+3 a^4+4 ad^2+6 b^2 c^2+8 b^4-2 c^2 d^2+3 c^4, N_42= 2 a^2 (4 ad-5 b^2-c^2+d^2)+4 ad(3 b^2+2 c^2+d^2), L_42= 4 a b (-2 ad+3 b^2+2 c^2+d^2). For the cases a=b=c=d; x=y=z=0 and u≠ 0; x=y=z=-u; x=y=-z=-u, where x, y, z, u are distinct and x,y,z,u∈{a,b,c,d}, \|ψ_9 ⟩ cannot form a tetrahedron if u=v=w=0, which is a product state of two EPR pairs. In addition to the above special cases, the authors in <cit.> divide the family G_abcd into four subfamilies: 1) x=y=0 and zu≠ 0, where x ≠ y ≠ u≠ v∈{a, b, c, d}; 2) x=± y≠ 0 and u=± v≠ 0, where x ≠ y ≠ u≠ v∈{a, b, c, d}; 3) a=± d≠ 0 and b≠± c, or b=± c≠ 0 and a≠± d; 4) x≠± y, or x≠± y but only one r=s, where x, y∈{a, b, c, d}, r∈{± a, ± d} and s∈{± b, ± c}. By straightforward calculation we have C_i(\|ψ_9 ⟩)>0 and G(\|ψ_9 ⟩) >0 for all these subcases. Fig. <ref> (f) shows that case of x=± y≠ 0, u=± v≠ 0 in a=-d and b=c. Hence, \|ψ_9 ⟩ can form a triangular pyramid and is genuine multipartite entangled state. Let x=± y≠ 0, u=± v≠ 0 in a=-d and b=c, we consider the derivatives of 𝒱_1234(\|ψ_9⟩) with respect to C, the expression of ∂𝒱_1234(\|ψ_9 ⟩)/∂ C(\|ψ⟩) is given in Appendix F. We easily get that, see Fig. <ref> (d), ∂𝒱_1234(\|ψ_9⟩)/∂ C(\|ψ_9⟩)≥0. Similarly, we can get that for the remaining subfamilies of G_abcd, ∂𝒱_1234(\|ψ_9⟩)/∂ C(\|ψ_8⟩)≥ 0 also holds. Thus the monotonicity of 𝒱_1234(\|ψ_9⟩) holds and hence 𝒱_1234(\|ψ_9⟩) is non-increasing under LOCC. The above a,b,c,d are all the unique eigenvalues of a 2n× 2n complex symmetric matrix P with non-negative real part. The indices L_αβ⋯ are representative for the Jordan block structure of P (e.g. L_a_20_3⊕1̅ means that the eigenstructure of P consists of two 2× 2 Jordan blocks with eigenvalues a and -a, and a degenerated pair of dimension 3 and 1 respectively). From the above analysis on the concurrence tetrahedron with respect to the nine different types of entanglements for four-qubit systems under LOCC, the concurrence tetrahedron is a well-defined measure of genuine multipartite entanglement. Next, we compare the tetrahedron concurrence with GMC introduced in <cit.>. For n-partite pure states \|Ψ⟩∈ℋ_1⊗ℋ_2⊗⋯⊗ℋ_n with dim(ℋ_i)=d_i, i=1,2, ⋯ ,n, the GMC is given by C_GME(\|Ψ⟩):=min_γ_i ∈γ√(2(1- (ρ^2_A_γ_i))), where γ={γ_i} represents the set of all possible bipartitions {A_i\|B_i} of {1,2,…,n}. Consider the states \|ψ_A⟩∈ C_2 and \|ψ_B⟩∈ C_3, \|ψ_A ⟩ = 1/2(\|0000⟩ +\|1011⟩ +\|1101⟩+\|1110⟩), \|ψ_B ⟩ = 1/2(\|0000⟩ +\|0101⟩ +\|1000⟩+\|1110⟩). Direct calculation shows that C_GME(\|ψ_A⟩)=C_GME(\|ψ_B⟩)=0.8660, while 𝒱_1234(\|ψ_A⟩)=0.1254, 𝒱_1234(\|ψ_B⟩)=0.0960. To further illustrate the advantage of the tetrahedron concurrence, let us choose the states \|ψ_C⟩∈ C_4 and \|ψ_D⟩∈ C_5, \|ψ_C ⟩ = 1/√(5)(\|0000⟩+\|1111⟩ +\|0011⟩+\|0101⟩+\|0110⟩), \|ψ_D ⟩ = 1/√(4 (5 √(113)/32+51/32)+3)(√(5 √(113)/32+51/32)(\|0000⟩ +\|0101⟩ +\|1010⟩+\|1111⟩) +(i\|0001⟩ +\|0110⟩ -i\|1011⟩)). We have C_GME(\|ψ_C⟩)=C_GME(\|ψ_D⟩)=0.8000, 𝒱_1234(\|ψ_C⟩)=0.1084 and 𝒱_1234(\|ψ_D⟩)=0.1624. Clearly, in both cases, the GMC cannot tell the difference between the entanglements of \|ψ_A⟩ and \|ψ_B⟩, as well as \|ψ_C⟩ and \|ψ_D⟩, since C_GME only depends on the length of the shortest edge which is the same for both states. However, our concurrence tetrahedron takes into account all the information about the length of the edges. The entanglement of \|ψ_A⟩ detected by the concurrence tetrahedron is larger than the entanglement of \|ψ_B⟩, and that of \|ψ_C⟩ is less than that of \|ψ_D⟩. In this sense, GMC and concurrence tetrahedron are two inequivalent measures. The concurrence tetrahedron shows more advantages in characterizing the entanglement of four-qubit systems. § CONCLUSION Quantum entanglement plays an crucial role in quantum information theory. Proper GME measures to quantify the genuine four-qubit entanglement faithfully are of great significance. We have constructed a concurrence tetrahedron based on the concurrence of nine different types of four-qubit states under LOCC, and presented a new measure of genuine multipartite entanglement for four-qubit systems. Furthermore, if a pure state is not GME, from our measure one can certify which part is separated from the rest. A specific example shows that compared with the GMC, our measure can characterize the genuine four-qubit entanglement in a finer way. Our approach may highlight investigations on quantification of genuine entanglement for multipartite systems with high dimensions. § ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (NSFC) under Grants 12075159, 12171044 and 12175147; Beijing Natural Science Foundation (Grant No. Z190005); the Academician Innovation Platform of Hainan Province. § APPENDIX §.§ The expression of ∂𝒱_1234(\|ψ_4 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_4⟩)/∂ u=√(a^2 (a^2+3)/(2 a^2+3)^2)(a^4+4 a^2+2) (a^4+4 a^2+6)/3 (2 a^2+3)^4 √(a^2 (a^4+4 a^2+2) (2 a^6+13 a^4+26 a^2+18)/(2 a^2+3)^6)≥0, ∂𝒱_1234(\|ψ_4⟩)/∂ v=∂𝒱_1234(\|ψ_4⟩)/∂ w=a^2 (a^2+4) (a^4+4 a^2+2) √(1/4-1/4 (2 a^2+3)^2)/3 (2 a^2+3)^4 √(a^2 (a^4+4 a^2+2) (2 a^6+13 a^4+26 a^2+18)/(2 a^2+3)^6)≥0, ∂𝒱_1234(\|ψ_4⟩)/∂ C_12(\|ψ_4⟩)=a^2 √(a^4+4 a^2+2/(2 a^2+3)^2)(a^6+6 a^4+12 a^2+12)/3 (2 a^2+3)^4 √(a^2 (a^4+4 a^2+2) (2 a^6+13 a^4+26 a^2+18)/(2 a^2+3)^6)≥0, ∂𝒱_1234(\|ψ_4⟩)/∂ C_13(\|ψ_4⟩)=∂𝒱_1234(\|ψ_4⟩)/∂ C_14(\|ψ_4⟩)=a^2 (a^2+2) (a^4+4 a^2+2/(2 a^2+3)^2)^3/2/3 (2 a^2+3)^2 √(a^2 (a^4+4 a^2+2) (2 a^6+13 a^4+26 a^2+18)/(2 a^2+3)^6)≥0, §.§ The expression of ∂𝒱_1234(\|ψ_5 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_5⟩)/∂ v=8 a^2 √(2 a^4+7 a^2+2/(4 a^2+3)^2)(18 a^4+27 a^2+4)/3 (4 a^2+3)^4 √((6 a^2+1) (60 a^8+344 a^6+496 a^4+176 a^2+15)/(4 a^2+3)^6)>0, ∂𝒱_1234(\|ψ_5⟩)/∂ u=∂𝒱_1234(\|ψ_5⟩)/∂ w=2 √(2 a^4+7 a^2+1/(4 a^2+3)^2)(36 a^6+90 a^4+44 a^2+5)/3 (4 a^2+3)^4 √((6 a^2+1) (60 a^8+344 a^6+496 a^4+176 a^2+15)/(4 a^2+3)^6)>0, ∂𝒱_1234(\|ψ_5⟩)/∂ C_12(\|ψ_5⟩)=∂𝒱_1234(\|ψ_5⟩)/∂ C_14(\|ψ_5⟩)=√(2)(6 a^2+1) √(6 a^4+14 a^2+3/(4 a^2+3)^2)(2 (a^2+7) a^2+3)/3 (4 a^2+3)^4 √((6 a^2+1) (60 a^8+344 a^6+496 a^4+176 a^2+15)/(4 a^2+3)^6)>0, ∂𝒱_1234(\|ψ_5⟩)/∂ C_13(\|ψ_5⟩)=√(6 a^2+1/(4 a^2+3)^2)(4 (15 a^6+74 a^4+80 a^2+25) a^2+7)/3 (4 a^2+3)^4 √((6 a^2+1) (60 a^8+344 a^6+496 a^4+176 a^2+15)/(4 a^2+3)^6)>0, §.§ The expression of ∂𝒱_1234(\|ψ_6 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_6⟩)/∂ u=n_11√(1-1/(a^2+b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6), ∂𝒱_1234(\|ψ_6⟩)/∂ v=(10 a^2 b^2+a^4+b^4+3) (-2 a^2 (b^2-2)+a^4+b^4+4 b^2) √(1-1/(a^2+b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6) , ∂𝒱_1234(\|ψ_6⟩)/∂ w=(-2 a^2 (b^2-2)+a^4+b^4+4 b^2-2) (a^2 (4 b^2+2)+a^4+(b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6), ∂𝒱_1234(\|ψ_6⟩)/∂ C_12(\|ψ_6⟩)=r_11√(a^2 (4 b^2+2)+a^4+(b^2+1)^2/(a^2+b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6), ∂𝒱_1234(\|ψ_6⟩)/∂ C_13(\|ψ_6⟩)=s_11√(-2 a^2 (b^2-2)+a^4+b^4+4 b^2/(a^2+b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6), ∂𝒱_1234(\|ψ_6⟩)/∂ C_14(\|ψ_6⟩)=t_11√(a^2 (4 b^2+2)+a^4+(b^2+1)^2/(a^2+b^2+1)^2)/12 (a^2+b^2+1)^4 √(m_11/(a^2+b^2+1)^6). where m_11 = 2 a^10(6 b^2+7)+a^8 (-6 b^4+94 b^2+31)+2 a^6 (-8 b^6+66 b^4+79 b^2+17), +a^4 (-6 b^8+132 b^6+262 b^4+70 b^2+23)+2 a^2 (6 b^10+47 b^8+79 b^6+35 b^4+22 b^2-1), +2 a^12+b^2 (b^2+2) (b^2 (b^2+4) (2 (b^4+b^2)+3)-1)-1, n_11 = 2 a^6 (b^2+3)+2 a^4 (-3 b^4+9 b^2+5)+2 a^2 (b^6+9 b^4+12 b^2+2)+a^8+b^8+6 b^6+10 b^4+4 b^2+2, r_11 = -2 a^6 (b^2-3)+2 a^4 (b^4+b^2+4)+2 a^2 (-b^6+b^4+11 b^2-1)+a^8+b^8+6 b^6+8 b^4-2 b^2+1, s_11 = 16 a^6 b^2+a^4 (26 b^4+24 b^2-1)+2 a^2 (8 b^6+12 b^4-2 b^2+3)+a^8+b^8-b^4+6 b^2, t_11 = -2 a^6 (b^2-3)+a^4 (2 (b^4+b^2)+7)-2 a^2 (b^6-b^4-6 b^2+1)+a^8+b^8+6 b^6+7 b^4-2 b^2-2. §.§ The expression of ∂𝒱_1234(\|ψ_7 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_7⟩)/∂ u=n_22√(8 a^2-17/(3 a^2+b^2+2)^2+8/3 a^2+b^2+2+1)/3 (3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6), ∂𝒱_1234(\|ψ_7⟩)/∂ v=r_22√(8 a^2-17/(3 a^2+b^2+2)^2+8/3 a^2+b^2+2+1)/6 (3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6), ∂𝒱_1234(\|ψ_7⟩)/∂ w=s_22√(8 a^2-17/(3 a^2+b^2+2)^2+8/3 a^2+b^2+2+1)/6 (3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6), ∂𝒱_1234(\|ψ_7⟩)/∂ C_12(\|ψ_7⟩) =t_22√(a^2 (3 b^2+10)+3 a^4+2 b^2+1/(3 a^2+b^2+2)^2)/24 (3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6), ∂𝒱_1234(\|ψ_7⟩)/∂ C_13(\|ψ_7⟩)=x_22√(6 (a^2+4) b^2+8 (2-3 a^2) a b+15 a^4+88 a^2+3 b^4+8/(3 a^2+b^2+2)^2)/6 √(2)(3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6), ∂𝒱_1234(\|ψ_7⟩)/∂ C_14(\|ψ_7⟩)=y_22√(6 (a^2+4) b^2+8 (3 a^2-2) a b+15 a^4+88 a^2+3 b^4+8/(3 a^2+b^2+2)^2)/6 √(2)(3 a^2+b^2+2)^4 √(m_22/(3 a^2+b^2+2)^6). where m_22 = 9 a^10(2278-159 b^2)+27 a^8 (-18 b^4+594 b^2+3869)+2 a^6 (117 b^6+7638 b^4+75426 b^2+89216) +a^4 (369 b^8+8316 b^6+64954 b^4+134208 b^2+40200)+a^2 (45 b^10+1278 b^8+10628 b^6+34432 b^4 +20656 b^2+3136)+1269 a^12+(2 b^2+1) (15 b^8+304 b^6+1416 b^4+672 b^2+80), n_22 = (a^2 (3 b^2+10)+3 a^4+2 b^2+1) (3 (-2 a^2 (b^2-8)+a^4+b^4)+16 b^2+4) , r_22 = (6 (a^2+4) b^2+8 (2-3 a^2) a b+15 a^4+88 a^2+3 b^4+8) (a^2 (3 b^2+10)+6 a^3 b+3 a^4-4 a b+2 b^2+1), s_22 = (6 (a^2+4) b^2+8 (3 a^2-2) a b+15 a^4+88 a^2+3 b^4+8) (a^2 (3 b^2+10)-6 a^3 b+3 a^4+4 a b+2 b^2+1), t_22 = 36 a^6 (3 b^2+52)-2 a^4 (87 b^4+456 b^2-4636)+4 a^2 (15 b^6+204 b^4+1556 b^2+288) -9 a^8+15 b^8+272 b^6+1016 b^4+384 b^2+32, x_22 = 3 a^6 (39 b^2+158)+72 a^5 b (b^2+10)+a^4 (57 b^4+558 b^2+913)+8 a^3 b (3 b^2 (b^2+8)-62) +a^2 (3 b^6+118 b^4+446 b^2+128)+144 a^7 b+63 a^8-16 a b (b^4+10 b^2+2)+(2 b^2+1) (b^4+24 b^2+4), y_22 = 3 a^6 (39 b^2+158)-72 a^5 b (b^2+10)+a^4 (57 b^4+558 b^2+913)-8 a^3 b (3 b^2 (b^2+8)-62) +a^2 (3 b^6+118 b^4+446 b^2+128)-144 a^7 b+63 a^8+16 a b (b^4+10 b^2+2)+(2 b^2+1) (b^4+24 b^2+4). §.§ The expression of ∂𝒱_1234(\|ψ_8 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_8⟩)/∂ u=(-2 a^2 b^2+3 a^4+3 b^4) (a^2 (b^2+1)+b^2) √(1-1/(a^2+b^2+1)^2)/3 (a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6) , ∂𝒱_1234(\|ψ_8⟩)/∂ v=(a^2 (b^2+1)-2 a b+b^2) √(1-1/(a^2+b^2+1)^2)(2 a^2 (b^2+2)+3 a^4+8 a b+b^2 (3 b^2+4))/6 (a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6) , ∂𝒱_1234(\|ψ_8⟩)/∂ w=(a^2 (b^2+1)+2 a b+b^2) √(1-1/(a^2+b^2+1)^2)(2 a^2 (b^2+2)+3 a^4-8 a b+b^2 (3 b^2+4))/6 (a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6) ∂𝒱_1234(\|ψ_8⟩)/∂ C_12(\|ψ_8⟩) = n_33√(a^2 (b^2+1)+b^2/(a^2+b^2+1)^2)/24 (a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6), ∂𝒱_1234(\|ψ_8⟩)/∂ C_13(\|ψ_8⟩)=(r_33+a^6 (b^2+1)+a^4 (6 b^4+11 b^2+4)) √(2 a^2 (b^2+2)+3 a^4+8 a b+b^2 (3 b^2+4)/(a^2+b^2+1)^2)/6 √(2)(a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6) , ∂𝒱_1234(\|ψ_8⟩)/∂ C_14(\|ψ_8⟩)=(s_33+a^6 (b^2+1)+a^4 (6 b^4+11 b^2+4)) √(2 a^2 (b^2+2)+3 a^4-8 a b+b^2 (3 b^2+4)/(a^2+b^2+1)^2)/6 √(2)(a^2+b^2+1)^4 √(m_33/(a^2+b^2+1)^6) . where m_33 = 15 a^10(b^2+1)+a^8 (36 b^4+91 b^2+40)+2 a^6 (13 b^6+59 b^4+24 b^2+8)+2 a^4 b^2 (18 b^6 +59 b^4+40 b^2-8)+a^2 b^4 (15 b^6+91 b^4+48 b^2-16)+b^6 (15 b^4+40 b^2+16) , n_33 = 4 a^6 (5 b^2+6)-2 a^4 (3 (b^2+4) b^2+8)+4 a^2 b^2 (5 b^4-6 b^2+8)+15 a^8+b^4 (15 b^4+24 b^2-16), r_33 = -8 a^3 (b^3+b)+a^2 b^2 (b^4+11 b^2+8)-8 a^5 b-8 a (b^5+b^3)+b^4 (b^2+4) , s_33 = 8 a^3 (b^3+b)+a^2 b^2 (b^4+11 b^2+8)+8 a^5 b+8 a (b^5+b^3)+b^4 (b^2+4) , §.§ The expression of ∂𝒱_1234(\|ψ_9 ⟩)/∂ C(\|ψ⟩) ∂𝒱_1234(\|ψ_9⟩)/∂ u=∂𝒱_1234(\|ψ_9⟩)/∂ v=n_44√(38 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4/(5 c^2-2 c d+5 d^2)^2)/6 √(2)(5 c^2-2 c d+5 d^2)^4 √(m_44/(5 c^2-2 c d+5 d^2)^6) , ∂𝒱_1234(\|ψ_9⟩)/∂ w=r_44√(38 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4/(5 c^2-2 c d+5 d^2)^2)/6 √(2)(5 c^2-2 c d+5 d^2)^4 √(m_44/(5 c^2-2 c d+5 d^2)^6) ∂𝒱_1234(\|ψ_9⟩)/∂ C_12(\|ψ_9⟩) =∂𝒱_1234(\|ψ_9⟩)/∂ C_13(\|ψ_9⟩)=s_44√(54 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4/(5 c^2-2 c d+5 d^2)^2)/6 √(2)(5 c^2-2 c d+5 d^2)^4 √(m_44/(5 c^2-2 c d+5 d^2)^6) , ∂𝒱_1234(\|ψ_9⟩)/∂ C_14(\|ψ_9⟩)=t_44√(6 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4/(5 c^2-2 c d+5 d^2)^2)/6 √(2)(5 c^2-2 c d+5 d^2)^4 √(m_44/(5 c^2-2 c d+5 d^2)^6). where m_44 = (6 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) (2236 c^6 d^2-8 (355 c+17) c^4 d^3+2 (1887 c+80) c^3 d^4 -8 (335 c+54) c^2 d^5-680 c^7 d+289 c^8-104 (5 c+1) d^7+4 (c (451 c+40)+4) d^6+169 d^8) , n_44 = (6 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) (54 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) , r_44 = (6 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) (102 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4), s_44 = (6 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) (22 c^2 d^2-20 c^3 d+17 c^4-4 (5 c+1) d^3+13 d^4) , t_44 = 3324 c^6 d^2-8 (515 c+17) c^4 d^3+2 (3039 c+80) c^3 d^4-8 (495 c+86) c^2 d^5-680 c^7 d+289 c^8 -104 (5 c+1) d^7+4 (c (659 c+40)+4) d^6+169 d^8. 00 op1 Mintert F, Kus M, Buchleitner A 2004 Concurrence of Mixed Bipartite Quantum 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http://arxiv.org/abs/2307.02627v1	20230705195906	Proxy Selection in Transitive Proxy Voting	[ "Jacqueline Harding" ]	cs.GT	[ "cs.GT", "cs.MA", "econ.GN", "q-fin.EC" ]	Jacqueline Harding Stanford University [email protected] Proxy Selection in Transitive Proxy Voting Jacqueline Harding ========================================== Transitive proxy voting (or `liquid democracy') is a novel form of collective decision making, often framed as an attractive hybrid of direct and representative democracy. Although the ideas behind liquid democracy have garnered widespread support, there have been relatively few attempts to model it formally. This paper makes three main contributions. First, it proposes a new social choice-theoretic model of liquid democracy, which is distinguished by taking a richer formal perspective on the process by which a voter chooses a proxy. Second, it examines the model from an axiomatic perspective, proving (a) a proxy vote analogue of May's Theorem and (b) an impossibility result concerning monotonicity properties in a proxy vote setting. Third, it explores the topic of manipulation in transitive proxy votes. Two forms of manipulation specific to the proxy vote setting are defined, and it is shown that manipulation occurs in strictly more cases in proxy votes than in classical votes. § INTRODUCTION Transitive proxy voting (or `liquid democracy') is a novel form of collective decision making. It is often framed as an attractive hybrid of direct and representative democracy, purporting to balance pragmatic factors with the ability to represent a population. Recently, it has been used by the German branch of the Pirate Party to aid intra-party decisions Litvinenko2012. Although the ideas behind liquid democracy have garnered widespread support, there has been little rigorous examination of the arguments offered on its behalf. In particular, there have been relatively few attempts to model liquid democracy formally (discussed in Section <ref>). A formal model has the potential to serve as a testing ground for the conceptual and empirical claims put forward by supporters of liquid democracy. This paper attempts to fill this gap, presenting a new model of liquid democracy. The model is distinguished by the fact that it takes a richer formal perspective on proxy selection, the process by which a voter chooses a proxy; it is argued that this allows it better to capture features relevant to claims made about liquid democracy. The model is examined from an axiomatic perspective, then put to work exploring the hitherto undeveloped topic of manipulation in a proxy vote setting. (, p.165) characterise liquid democracy as the conjunction of four principles. A voter can (a) vote directly, (b) delegate her vote to a representative to vote on her behalf (this representative is called her `proxy'), (c) delegate those votes she has received via delegation to another representative and; (d) terminate the delegation of her vote at any time. It is argued that the flexibility of liquid democracy confers it certain benefits which classical voting — where voters vote directly — lacks. Let us recall some of these benefits here. Balancing Practicality and Democracy. Direct democracy, where citizens vote directly on issues through frequent referenda, is seen as `strongly democratic but highly impractical' (<cit.>, p.190), whilst representative democracy, where citizens elect representatives to make decisions on their behalf, is `practical but democratic to a lesser degree' (<cit.>, p.190). Under proxy voting, no such trade-off need occur. If people want their particular views to be represented in a vote, they can ensure this by voting directly. If they are undecided on an issue (or practical factors prevent them from becoming sufficiently informed, or even from casting their vote directly), they can choose to delegate their vote to someone they perceive as competent or trustworthy. Increasing Voter Turnout. There are at least three arguments for the claim that proxy voting increases voter turnout. Firstly, Miller1969 argues that a major barrier to voters' participation in elections is simply the opportunity cost of voting directly. Secondly, both Miller1969 and Alger2006 identify apathy towards political representatives as a reason for poor voter turnout. If a voter can be represented by someone whom she trusts, they argue, she will be more likely to vote. Since proxy voting (at least as normally construed) allows voters to delegate their votes to any other voter, it seems more likely that voters will be represented by someone they approve of. Thirdly, voters are often deterred from voting by the fact that they haven't made their mind up about all the alternatives being considered in the election (even if they have some sense of what they think). By choosing their proxy carefully, they can vote on some alternatives but not others. Behrens2017 argues that the `transitivity' of liquid democracy (where proxies can delegate the votes they have been given) accentuates this benefit, since it can only increase the number of potential representatives for a voter. Increasing Competence of Voters. A voter might delegate her vote when she believes another voter to be better informed than her. Assuming this perception of competence is truth-tracking, Green-Armytage2015 argues that this implies that proxy voting leads to votes being cast by voters who are (on average) better informed. Increasing Diversity of Viewpoints. Alger2006 observes that proxy voting might lead to greater diversity in the viewpoints expressed by voters. In a representative democracy, only a very small proportion of the population is a potential representative; this means that such representatives tend to be pushed towards viewpoints with more broad appeal. By increasing the number of potential representatives, proxy voting could allow voters to express more idiosyncratic viewpoints. The ability to represent the benefits outlined above is a desideratum of a formal model of liquid democracy. In Section <ref>, existing models of liquid democracy from within the social choice literature are surveyed. It is argued that none of these models properly represents the means by which a voter chooses a proxy. This motivates the model of transitive proxy voting proposed in Section <ref>, which is distinguished by its ability to include preference information in proxy selection. Formally, the model presented augments a classical vote (N,A,f) (where N is a set of voters, A is a set of alternatives and f is a resolute social choice function) with a novel function g, called a `proxy mechanism'. In Section <ref>, novel properties of proxy mechanisms g are explored, and a natural proxy mechanism (the mechanism) is characterised using some of these properties. In Section <ref>, properties of pairs (f,g) are defined. A proxy vote analogue of May's Theorem (which characterises the majority rule when \|A\|=2) is proved and an impossibility result in a proxy vote setting is presented and proved, showing that certain desirable properties of pairs (f,g) are incompatible with natural properties of their individual components f and g. In Section <ref>, manipulation in a proxy vote setting is examined. A novel form of manipulation (`proxy choice manipulability') is defined and connections between this form of manipulation and classical manipulation are explored. The effect of single peakedness on manipulation in a proxy vote setting is also examined; it is demonstrated that strategyproofness is strictly harder to come by in proxy elections. § RELATED WORK This section outlines existing social-choice theoretic models of transitive proxy voting. Discussion of these models motivates a model which takes a more robust perspective on proxy selection; such a model is introduced in the next section. §.§ Pairwise Delegation Brill2018PairwiseLD observe that we can view ordinal preference rankings as collections of pairwise comparisons (or `edges') between alternatives. When voters provide linear orders ≻ over some set of alternatives A, they are effectively choosing whether a ≻ b or b ≻ a for each a,b∈ A. In the ordinal preference setting, then, this allows us to model voters with partial opinions as having fixed some edges but not others. Similarly, Christoff17 have voters vote or delegate on interdependent binary issues. This model generalises that of Brill2018PairwiseLD; once we translate ordinal rankings into a binary aggregation setting, pairwise comparisons become binary issues. In both 's and 's models, for each pair of alternatives (a,b) (or each binary issue p) that a voter has not decided between, she chooses some delegate from amongst the other voters to decide on her behalf whether a ≻ b or b≻ a. So her delegations are `pairwise'; a voter might have a different delegate for each edge she is undecided on, meaning she ends up submitting an intransitive preference order (or, in the more general aggregation setting, a ballot violating some rationality requirement). This means that we must either modify the social choice function to accommodate intransitivity or provide a systematic way of forming preference profiles from the outputs of delegations. Of course, the issue with intransitivity arises only when we allow pairwise delegations. The model I propose in this paper will restrict delegations such that each voter picks at most one proxy. This loss of generality is motivated not merely from a desire to circumvent issues of intransitivity but also by consideration of proxy selection, the process by which a voter chooses a proxy. For example, according to , delegation is done on the basis of the perception of competence. But if we accept that it is irrational to hold intransitive preferences oneself (an assumption which I won't challenge here), then it is unclear why we should think it rational to accept an intransitive preference resulting from delegation. Surely the conclusion a voter ought to draw when her delegates present her with an intransitive ballot is that she was mistaken in her initial assessment of the competence of her delegates? It appears that — by allowing pairwise delegations in the absence of a well-developed account of proxy selection — we are condoning irrationality at a distance. §.§ Proxies Represent Voters' Interests In the model proposed by Bloembergen2018OnRD, voters in a network choose between two alternatives. For each voter, one alternative is better, but voters are not aware which alternative is better for any of the voters, including themselves. If a voter votes directly, there is a publicly-known probability that she will vote for the alternative which is worse for her. Each voter either votes directly or delegates her vote to one of her neighbours in the network (so delegations are transitive). Note that since voters are aware of neither their interests nor other voters', it's possible for a voter to delegate to a neighbour whose interests aren't aligned with hers. The utility a voter gains from voting is proportional to the probability that the voter who ends up casting her vote — her terminal delegate, or `guru' — votes for the alternative which is better for her. In the model proposed by Abramowitz2018, an electorate votes on a set of binary issues. The electorate is composed of voters and delegates; voter preferences over the issues are private, but delegate preferences are public. Voters then express preferences over delegates; a voter's attitude towards a delegate is assumed to correspond solely with the degree of agreement between the voter's and delegate's preferences over the issues. In 's model, proxy selection depends on a perception of correspondence between a voter's views and those of her proxy (this could also be said of 's model; in their case, a perception of correspondence can be mistaken); this seems to me an essential component of any account of proxy selection, and my model will incorporate it. The models lack, though, a representation of all of the other factors that can make a voter choose a proxy (knowledge, charisma, etc); we saw above that these factors are important in motivating transitive proxy voting. §.§ Preferences over Delegates In the model proposed by escoffier_convergence_iterative, each voter i ∈ N (where N is the set of voters) submits a preference ordering ≻_i over N∪{0}, interpreted as representing who i would prefer to end up casting her vote (i.e. a preference relation over her potential gurus), with `0' representing the possibility of abstention. Similarly, Kotsialou2018 propose a model where voters submit preferences over the set of alternatives A or preferences over N (the latter interpreted as a preference over their immediate proxy, rather than their guru). In the first case, voters are taken to cast their own vote. In the second case, voters are taken to delegate their vote, with the delegate decided on by a central mechanism. By having voters rank other voters, both models neatly represent the idea that several factors can inform a voter's choice of proxy. What is missing from the models, though, is an important component of proxy selection we discussed above, namely that a voter's choice of proxy ought to depend on correspondence between her views and the proxy's. In 's model, for example, there is no actual election in which voters are participating, meaning there is no way to model such correspondence. Similarly, 's model takes an all-or-nothing approach to delegation. Voters are immediately categorised into direct voters or delegators, regardless of the actual content of the preferences they submit (the existence of a preference order of either sort is sufficient to determine this categorisation). So the model is unable to allow voters to express preferences on some issues but not others, and relate proxy selection to these partial preferences. §.§ Ground Truth The models I've discussed above take a voting procedure to aggregate the preferences of an electorate; the model I propose in this paper is of this sort. A very different sort of model measures the accuracy of a voting procedure relative to an underlying ground truth. For example, Cohensius2017 consider an infinite population N of voters distributed on a real interval [a,b]; a distinguished finite N'⊆ N are allowed to cast their votes directly (a vote consists of reporting their position on the interval), whilst each other voter delegates her vote to the closest voter in N' (so delegation is non-transitive). The authors find that proxy voting is always more accurate than direct voting when the ground truth is taken to be the taken to be the median of the voters' positions, and generally (through simulation) more accurate when the ground truth is the mean of the voters' positions. Similarly, Kang2018 consider a vote on a binary issue, for which it is assumed there is a ground truth. N voters are arranged in a social network. Similarly to Bloembergen2018OnRD, each voter i ∈ N has a public competence level p_i, interpreted as the probability she would vote correctly if she voted directly. Voters can either vote directly or delegate their vote to a neighbour in the network whose competence level is strictly higher than their own (note that this eliminates delegation cycles). For each voter i who decides to delegate, a `local delegation mechanism' takes in the competence levels of the voter's neighbours and returns a probability distribution over the delegations available to i. Delegations carry weight according to this probability distribution, and the collective decision is made by the majority rule. prove a negative result: there is no local delegation mechanism which is strictly more accurate than allowing voters to vote directly. §.§ Structure of the Delegation Graph The models discussed in this section typically divide transitive proxy voting into three stages. In the first stage, a `delegation graph' (a graph representing delegations between voters) is formed from voters' preferences over alternatives and/or delegates. In the second stage, a preference profile is formed from this graph. In the third stage, this profile is used as an input to an aggregator. The model I propose in this paper works in the same way. It is possible, though, to consider questions regarding the three stages independently. Golz2018 focus on the first stage, the formation of the delegation graph from information about voters' delegation preferences. In the model they propose, each voter specifies a subset of the other voters who they would be happy to delegate their vote to. They consider the problem of assigning each voter a delegate from within this subset so as to minimise the number of voters delegating their vote to a single proxy (so as to minimise the maximum voting power of a voter who votes directly). Boldi focus on the second stage; specifically, they consider the problem of forming a profile from the delegation graph in a way which minimises the maximum power an individual voter can accrue. Given a set of voters arranged in a network, their innovation is a `viscous' delegation factor α∈ (0,1), representing the extent to which a delegation between neighbours in the network preserves voting power. The smaller α is, the more weight is lost every time a vote is transferred, so fine-tuning α could affect the feasible length of delegation chains. They discuss the impact of the structure of an underlying social network on the number of possible winners. § INTRODUCING THE MODEL §.§ Proxy Selection It's not my aim to give a full account of the factors that go into a voter's choice of proxy. That said, given the discussion in the previous section, I will take it that (1) and (2) are plausible starting points for an account of proxy selection: (1) There is a large range of factors which inform a voter's choice of proxy (for example, a perception of competence, intelligence, honesty, etc). (2) Voters pick proxies whom they think will represent their interests. Example of Proxy Selection. Suppose I am asked to give an ordinal ranking over the available options for the UK's future relationship with the European Union. I know that I prefer remaining in the EU to the other options, but I am unsure how to compare various intermediate levels of integration. If given the option of choosing a trusted delegate to submit an opinion on my behalf, I will do so. I know that my friend Alice is exceptionally well informed about the intricacies of the EU. Ceteris paribus, then, she would be an excellent candidate to be my delegate (this is (1)). I learn, though, that Alice prefers leaving the EU without a deal to remaining in the EU. The fact that Alice prefers a no-deal Brexit to remaining in the EU doesn't make me think that she's any less informed, or trustworthy, and so on, but it is sufficient to ensure that I won't pick her as my proxy. Since she disagrees with me so strongly on the issues on which I have made up my mind, I don't think she will represent my interests if she votes on my behalf (this is (2)). I use this example to show that a model of proxy selection should have at least two interacting components. First and foremost, voters will only consider delegates who represent their interests (this is (2)). Beyond this, it is futile to attempt to place restrictions on their choice amongst potential delegates who they feel represent their interests, since many factors are relevant to this decision (this is (1)). §.§ Notation For a finite set X, let 𝒫(X) denote the set of all binary relations on X which are irreflexive, anti-symmetric and transitive. I will call P∈𝒫(X) a `partial order', to emphasise that P need not be total. Technically, of course, the relation usually called a `partial order' is reflexive rather than irreflexive. The reader should be mindful of this terminological idiosyncrasy, but it makes no difference to the content of this paper. Following Brill2018PairwiseLD, I represent a partial order as a set of strict pairwise comparisons. This affects the notation I use. Suppose X={a,b,c}. Then, using my terminology, the following are all examples of partial orders on X: * P=∅ * P'={a ≻ b} * Q = {a ≻ b, a ≻ c} but the following would not be a partial order, since it is not closed under transitivity (since it doesn't contain a ≻ c): * Q' = {a ≻ b, b ≻ c} I will also speak of specific pairwise comparisons (or `edges') being members of a partial order. For example, I will say that P' contains the edge a≻ b. Formally, I will write that a≻ b ∈ P' (or, equivalently, that {a≻ b}⊆ P'), but a≻ b ∉ P ({a≻ b}⊈ P). Let ℒ(X) denote the set of all binary relations on X which are irreflexive, anti-symmetric, transitive and also complete. Then I call L∈ℒ(X) a `linear order'. Note that, by definition, ℒ(X)⊆𝒫(X). Throughout the paper, I will speak of profiles of partial (linear) orders. We can think of a profile of partial orders as a list of partial orders, one for each voter. So if N={1,...,n} is the set of voters, and A is the set of alternatives, then P=(P_1,...,P_n)∈𝒫(A)^n is a list of partial orders (note I use the bold type face for lists of orders). By P_i, I designate the partial order submitted by voter i. Fix some P=(P_1,...,P_n). Then, as is standard, we can also write P=(P_i,P_-i) or P=(P_i, P_j, P_-i,j), for some i,j∈ N. I will write (P'_i,P_-i) to designate the profile that is an `i-variant' of (P_i,P_-i). The same notational conventions apply to profiles of linear orders. §.§ Properties of Social Choice Functions I will use the phrase `social choice function' to refer to a resolute social choice function f:ℒ(A)^n A (so the reader can assume that the functions I consider have some sort of tie-breaking system built in). There are various familiar properties of social choice functions which will be relevant (there are many references for these standard axioms; see, for example, <cit.>). Recall that N is the set of voters i,j,k, ... (with \|N\|=n), and A is the set of alternatives a,b,c, ... (with \|A\|=m). A social choice function f is anonymous if, for any bijection ψ:N N and profile L=(L_1,...,L_n)∈ℒ(A)^n, we have that f(L_1,...,L_n) = f(L_ψ(1),...,L_ψ(n)) Let ψ:A A be a bijection. Let P∈𝒫(A). By ψ(P), I denote the alternative-wise application of the bijection. So if P={a≻ b}, ψ(a)=b and ψ(b)=a, then ψ(P)={b≻ a}. A social choice function f is neutral if, for any bijection ψ:A A and profile L=(L_1,...,L_n)∈ℒ(A)^n, we have that ψ(f(L_1,...,L_n)) = f(ψ(L_1),...,ψ(L_n)) A social choice function f is weakly monotonic if the following holds for every L∈ℒ(A)^n. Suppose f(L)=a, for some a ∈ A. Let L'=(L'_i,L_-i) be an i-variant of L, where L'_i = L_i\{b ≻ a}∪{a ≻ b} for some b ∈ A (in other words, voter i moves alternative a up at most one place in her ordering). Then we have that f(L')=a. I will also define a novel property of social choice functions, which I will make use of in Section <ref>. Let L+ be the profile we get when we augment L with \|A\|! new voters, one holding each possible ranking in ℒ(A) (if f is not anonymous, we must assume some ordering on the rankings in ℒ(A)). A social choice function f is Uniform Voter Addition Invariant (UVAI) iff f(L+)=f(L) for every L∈ℒ(A)^n. §.§ Proxy Votes My model extends a classical vote (N,A,f) with a proxy mechanism, g. Recall that 𝒫(A) denotes the set of all partial orders over A. By 𝒫(N), I designate the powerset of N. A function g: 𝒫(A)^n× N →𝒫(N) is a proxy mechanism iff, for every P=(P_1,...,P_n)∈𝒫(A)^n, for every i ∈ N: 1. If P_i=∅, then g(P,i)=N\{i}. 2. If P_i∈ℒ(A), then g(P, i)={i}. 3. If P_i∉ℒ(A), then i∉ g(P,i). Intuitively, a proxy mechanism takes in a profile of partial orders and assigns to each voter a set of `permitted proxies', the voters whom they are allowed to choose as their delegate. The idea is that this set of permitted proxies constitutes the delegates who represent the voter's interests. So the proxy mechanism is designed to represent the features of proxy selection described above. Firstly, if voter i submits an empty order, we require (in 1.) that she can choose any other voter as her proxy (every other voter is in her set of permitted proxies). This is because she has no preferences over the alternatives, implying that there is no way for a potential delegate to fail to represent her interests. Secondly, if voter i submits a linear order, we require (in 2.) that she casts her own vote (she is the only voter in her set of permitted proxies). If she has already made her mind up about the alternatives, there is no need for her to delegate her vote to another voter. Finally, if voter i submits a partial order which is not a linear order, then she is not allowed (by 3.) to cast her own vote (she does not appear in her set of permitted proxies). This is because the social choice function takes profiles of linear orders as inputs; the model I propose modifies the method of collecting preferences, not the method of aggregating preferences. So a proxy vote is a tuple (N,A,f,g). Each voter i ∈ N submits a triple (P_i,S_i,D_i), where: * P_i∈𝒫(A) is a partial order over the alternatives. So the model allows voters to have made their mind up about some pairwise comparisons but not others. * S_i∈ℒ(N) is a linear order over the voters. Intuitively, this order corresponds to a ranking over potential proxies (capturing all the reasons that i might have to prefer a delegate as her proxy independently of the delegate's ability to represent her views). * D_i∈ℒ(A) is a linear order over the set of alternatives, with P_i⊆ D_i. D_i is a `default vote'. In the situation where i has no permitted proxies (so g(P,i)=∅), i is required to vote directly, submitting this default vote. When each voter submits a triple, we have a proxy vote profile (P,S,D), where P is a (partial) preference profile, S is a proxy choice profile and D is a default vote profile. Each voter i then receives g(P,i), a set of permitted proxies, given the preference profile. If g(P,i)=∅, then i must submit her default vote D_i∈ℒ(A). If g(P,i)≠∅, then i must pick some j ∈ g(P,i) to cast her vote on her behalf. Let N'⊆ N. Then by S_i\|_N' I denote the restriction of S_i to N'. Voter i will pick the potential proxy who is ranked highest when we consider S_i\|_g(P,i) (in other words, the most preferred delegate from amongst her permitted proxies). Suppose that this is j. Then I will abuse notation by writing that S_i\|_g(P,i)=j. For the sake of convenience, I will write S_i\|_{i}=i and S_i\|_∅=i, since i casts her own vote if g(P,i)={i} or if g(P,i)=∅. So, given a voting profile P=(P_1,...,P_n) and proxy choice profile S=(S_1,...,S_n), each i ∈ N has a proxy. So we have a delegation graph (N, R) where iRj iff j = S_i\|_g(P,i) Note that, where it does not have a negative impact on accuracy, I will speak of `i choosing j to be her proxy' as expressing this formal condition. Let R^* be the transitive closure of R. For each i, let Π_i = {j ∈ N \| iR^j and jRj} If Π_i is non-empty, it is easy to see that it will be a singleton {π_i}. Call π_i voter i's guru. Note that if i casts her own vote, then we have π_i=i (so i will be her own guru). We can then define a guru voting profile P_π, S,D=(P_π_1,S,D,...,P_π_n,S,D) Where P_π_i,S,D is the preference order submitted by voter i's guru π_i, generated according to (P,S,D). I use the notation P_π, S,D to emphasise that this profile results from the proxy vote profile (P,S,D). The use of π is supposed to remind the reader that the votes are actually submitted by the gurus π_1,...,π_n. Note that, by construction, P_π_i,S,D∈ℒ(A) for every i ∈ N, since each guru must cast her own vote. So we can use P_π, S,D as the input to a social choice function. The outcome of the proxy vote is given by f(P_π, S,D).[One might worry that there is a prohibitively large effort involved in submitting S_i and D_i. To defend the cost of D_i, note that the default vote can be thought of as little more than a placeholder, a device which serves some practical purpose but has little ideological significance (e.g. each voter extends P_i at random, or chooses the lexicographically earliest extension of P_i). To defend S_i, note that any real world version of transitive proxy voting will be situated within a dynamic environment. If we assume that the model is a static representation of a process which is inherently dynamic, then we might interpret the situation described by the model as follows. A voter i submits P_i. She then calculates g(P,i), the set of proxies she feels represent her interests, using the preferences submitted by the other voters. If g(P,i)=∅, she submits D_i⊃ P_i, her default preference. If g(P,i)≠∅, she picks some j ∈ g(P,i) to be her proxy. So, rather than S_i, the voter i is really only required to specify the name of a proxy in g(P,i). It's true that the calculation of g(P,i) will require some computation, but this shouldn't surprise us: the process of choosing a proxy does take some effort from the voter!] §.§ Examples of Proxy Mechanisms It is worth giving some examples of simple proxy mechanisms. By definition, for every proxy mechanism g we have that g(P,i)=N\{i} if P_i=∅, and g(P,i)={i} if P_i∈ℒ(A). So it suffices to define the mechanism for the case where P_i≠∅ and P_i∉ℒ(A). () (P,i) = ∅ if P_i≠∅ and P_i∉ℒ(A) If g=, then every voter will cast her own vote, unless she has no preferences at all over the alternatives (in which case she will delegate). So we are close to a classical vote; proxy selection plays little role here. () (P,i) = N\{i} if P_i≠∅ and P_i∉ℒ(A) If g=, then each voter who has not made her mind up fully can delegate her vote to any other voter, regardless of what she thinks on the issues she has made her mind up on. In effect, this is the formal set up of many of the models we discussed in the previous section; the strictly partial components of the preference profile P are irrelevant to proxy selection. () (P,i) = {j ∈ N\{i} \| P_i⊆ P_j} if P_i≠∅ and P_i∉ℒ(A) If g=, each voter who has not made up her mind fully can delegate to those voters whose preferences include her own as a subset. () For each i∈ N, fix some j_i∈ N\{i}. Then (P,i) = {j_i} if P_i≠∅ and P_i∉ℒ(A) If g=, then each voter i has a unique dictator j_i; when i submits some but not all pairwise comparisons, she must delegate her vote to j_i. won't be used in the remainder of the paper, but I define it to remind the reader that a proxy mechanism can act very differently for each voter it acts upon. §.§ Representational Power I want to make three points about the representational power of the model I have presented. Firstly, a proxy vote (N,A,f,g) is a generalisation of a classical vote (N,A,f). In the case where P_i∈ℒ(A) for every i ∈ N, every voter casts her vote directly; we have a classical vote. In particular, this implies that any impossibility result concerning social choice functions f will carry over into this setting. So, for example, the Gibbard-Sattherthwaite Theorem (<cit.>, <cit.>) holds in this novel setting. Secondly, the model permits an easy resolution to delegation cycles. Following Christoff17, I have had voters submit a default vote which extends their existing vote. One could simply specify that this default vote is submitted directly by any voter who features in a delegation cycle. Thus delegation cycles require that the voters involved submit more pairwise comparisons in their votes, but do not prevent them from voting. Thirdly, the model permits at least two ways of constraining delegations through arranging voters in a network (something explored by Boldi, Golz2018 and Bloembergen2018OnRD). Fix some social network (N,T⊆ N× N). Then we can build the social network into the range of the proxy mechanism g, requiring that g(P,i)⊆ T[i] for all i∈ N, so that a voter can only delegate to her neighbours in the network. Another option is to place a requirement on S, the proxy choice profile. For example, we could stipulate that (j,k)∈ S_i iff (j∈ T[i] and k ∉ T[i]) for every i ∈ N. What this says is that i will always delegate to one of her neighbours if they are in her set of permitted proxies, but is able to delegate further afield if none of her neighbours represents her sufficiently. § PROPERTIES OF PROXY MECHANISMS In this section, some natural properties of proxy mechanisms are defined. Some of these properties are then used to characterise the mechanism. §.§ Defining Properties of Proxy Mechanisms Recall that a proxy mechanism is a function g: 𝒫(A)^n× N →𝒫(N). Let ψ:N N be a bijection. For N'⊆ N, I write ψ(N') to denote the image of N' under ψ. Let P∈𝒫(A)^n be a partial preference profile. Abusing notation, I write ψ(P) = ψ(P_1,...,P_n) = (P_ψ(1),..., P_ψ(n)) (Proxy Mechanism Anonymity) A proxy mechanism g is anonymous iff for every preference profile P∈𝒫(A)^n and every bijection ψ:N N, we have that ψ(g(P,i)) = g(ψ(P),ψ(i)) Proxy Mechanism Anonymity says that if we rename the voters, then a renamed voter's set of permitted proxies will just be the original voter's set of permitted proxies renamed. In other words, the proxy mechanism is blind to the identity of the individual voters. Let ψ:A A be a bijection. For P∈𝒫(A), I write ψ(P) to denote the alternative-wise application of the bijection. So if P={a≻ b}, ψ(a)=b and ψ(b)=a, then ψ(P)={b≻ a}. Let P∈𝒫(A)^n be a partial preference profile. Abusing notation, I write ψ(P) = ψ(P_1,...,P_n) = (ψ(P_1),...,ψ(P_n)) (Proxy Mechanism Neutrality) A proxy mechanism g is neutral iff for every preference profile P∈𝒫(A)^n and every bijection ψ:A A, we have that g(P,i) = g(ψ(P),i) Proxy Mechanism Neutrality says that we can rename the alternatives without affecting each voter's set of permitted proxies. (Proxy Availability (PA)) g satisfies PA iff for every P_i∈𝒫(A), for every i ∈ N, there is some P_-i∈𝒫(A)^n-1 such that g((P_i, P_-i),i)≠∅ Proxy Availability (PA) says that every voter should be able to find potential proxies for her votes, regardless of what views she holds, in at least some profile. In other words, every voter is capable of being represented, regardless of her views. (Independence of Irrelevant Proxies (IIP)) g satisfies IIP iff for every P,P'∈𝒫(A)^n, for every i,j∈ N, if P_i=P'_i and P_j=P'_j, then j ∈ (P,i) iff j ∈ g(P',i) Independence of Irrelevant Proxies (IIP) says that whether j is a permitted proxy for i should depend only on i's and j's preferences, not on the preferences of the other voters. It should be clear that IIP is motivated by the conception of proxy selection that I have argued for above. (Zero Regret (ZR)) g satisfies ZR iff there is no triple (P,S,D) (where P∈𝒫(A)^n, S∈ℒ(N)^n and D∈ℒ(A)^n) such that, for some i ∈ N: P_i⊈ P_π_i,S,D Zero Regret (ZR) says that a proxy mechanism guarantees that every voter's vote ends up being cast by someone who agrees with them completely (i.e. that they have no regrets about the vote submitted by their guru). Let P_i,Q_i∈𝒫(A). Then I write Agree(P_i,Q_i) = {a≻ b ∈ P_i \| a≻ b ∈ Q_i} and Disagree(P_i,Q_i) = {a≻ b ∈ P_i \| b≻ a ∈ Q_i}. (Preference Monotonicity (PM)) g satisfies PM iff the following condition holds for every P∈𝒫(A)^n, for every i∈ N. Suppose j∈ g(P,i) and j≠ i. Then for every k∈ N\{i}, if Agree(P_i,P_j)⊆ Agree(P_i, P_k) and Disagree(P_i,P_k)⊆ Disagree(P_i, P_j) then k ∈ g(P,i). Preference Monotonicity (PM) says that if j is a permitted proxy for i and k agrees with i on at least the same things as j whilst disagreeing with i on at most the same things as j, then k should also be a permitted proxy for i. It should be clear that PM is motivated by the account of proxy selection I have presented. §.§ Characterising Recall that was defined as follows: (P,i) = N\{i} if P_i=∅ {i} if P_i∈ℒ(A) {j ∈ N\{i} \| P_i⊆ P_j} otherwise is the unique proxy mechanism satisfying Proxy Availability, Independence of Irrelevant Proxies, Zero Regret and Preference Monotonicity. Clearly, satisfies PA, IIP and ZR. To see that satisfies PM, suppose that j∈(P,i) and j≠ i, for some P∈𝒫(A)^n, i,j ∈ N. So P_i⊆ P_j. Suppose that there is k ∈ N such that Agree(P_i,P_j)⊆ Agree(P_i,P_k) and Disagree(P_i,P_k)⊆ Disagree(P_i,P_j). Then we must have P_i⊆ P_k, since Agree(P_i,P_j)=P_i, since P_i⊆ P_j. So k ∈(P,i), as required. For the other direction (i.e. to show uniqueness), I prove the contrapositive. Suppose g≠ is a proxy mechanism, and suppose g satisfies PA, IIP and PM. I will show that g does not satisfy ZR. It will help to prove the following intermediate claim. Let P∈𝒫(A)^n and i,j ∈ N, such that P_i∉ℒ(A). Then if P_i⊆ P_j, and g satisfies PA, IIP and PM, we have j ∈ g(P,i). Since g satisfies PA, there must be some P'∈𝒫(A)^n such that P'_i=P_i and g(P',i)≠∅. Suppose k ∈ g(P',i), for some k ∈ N. Then we can construct a new profile P” where P”_i = P'_i=P_i P”_j = P_j P”_k = P'_k By IIP, we must have k ∈ g(P”,i), since P”_i=P'_i and P”_k=P'_k. But then by PM, we must have j ∈ (P”,i), since P”_i=P_i⊆ P_j = P”_j implying that j must agree at least as much with i as k in profile P”. But then by another application of IIP, we must have j ∈ g(P,i), since P”_i=P_i and P”_j=P_j. So Lemma <ref> holds. We are now ready to prove the uniqueness of . Since g≠, there must be some P∈𝒫(A)^n and i,j∈ N with P_i∉ℒ(A) such that either P_i⊈ P_j and j ∈ g(P,i) or P_i⊆ P_j and j ∉ g(P,i) But note that Lemma <ref> rules out this latter case. So we only need to consider the case where P_i⊈ P_j and j ∈ g(P,i). Since P_i⊈ P_j, there must be some a≻ b∈ P_i such that a≻ b∉ P_j. But now consider a profile P' where, for some k ∈ N: P'_i =P_i P'_j =P_j P'_j∪{b≻ a} ⊆ P'_k, and P'_k∈ℒ(A) Note that this profile is well defined; since a≻ b∉ P_j=P'_j, we must have that P'_j∪{b≻ a} is still anti-symmetric, and thus can be extended to a linear order P'_k. Since P'_i=P_i and P'_j=P_j, we must have j∈ g(P',i), by IIP. We must also have k∈ g(P',j) by Lemma <ref>, since P'_j⊆ P'_k. So then if i picks j as her proxy and j picks k as her proxy, then k will be i's guru. But P'_i⊈ P'_k, since a≻ b∈ P_i=P'_i and b≻ a∈ P'_k. So g is not ZR. In fact, PA, IIP, ZR and PM are all individually necessary for characterising . (defined in Section <ref>) is a proxy mechanism which does not satisfy PA. But note that does satisfy IIP, ZR and PM. (defined in Section <ref>) is a proxy mechanism which does not satisfy ZR. But note that does satisfy PA, IIP and PM. Consider g defined as follows: g(P,i) = N\{i} if P_i=∅ {i} if P_i∈ℒ(A) {j ∈ N \| P_j∈ℒ(A) and P_i⊂ P_j} otherwise g is a proxy mechanism which does not satisfy PM (just consider some P_j∉ℒ(A) such that P_i⊆ P_j). But note that g does satisfy PA, IIP and ZR. Consider g defined as follows: g(P,i) = N\{i} if P_i=∅ {i} if P_i∈ℒ(A) {j ∈ N \| P_i⊆ P_j} if P_i∉ℒ(A) and (∀ j∈ N\{i} such that P_j∈ℒ(A)) we have P_i⊆ P_j ∅ otherwise g is a proxy mechanism which does not satisfy IIP. But note that g does satisfy PA, PM and ZR. § PROPERTIES OF PROXY VOTES In this section, properties of pairs (f,g) are examined. §.§ Defining Properties of Proxy Votes Suppose ψ:N N is a bijection. Suppose P∈𝒫(A)^n is a preference profile. Then it will be convenient to write ψ(P)=(P_ψ(1),...,P_ψ(n)) to denote the voter-wise application of the bijection ψ. Likewise with a default vote profile D∈ℒ(A)^n. Suppose S∈ℒ(N). Then I write ψ(S) to denote the voter-wise permutation of S. For example, if S={i≻ j}, then ψ(S)={ψ(i)≻ψ(j)}. Suppose S∈ℒ(N)^n is a proxy choice profile. Then, abusing notation, I write ψ(S)=(ψ(S)_ψ(1),...,ψ(S)_ψ(n)) to denote the voter-wise application of the bijection ψ to both the profile and the content of each voter's proxy choice. (Proxy Vote Anonymity) A pair (f,g), where f is a social choice function and g is a proxy mechanism, satisfies Proxy Vote Anonymity iff for every P∈𝒫(A)^n, for every S∈ℒ(N)^n for every D∈ℒ(A)^n, and for every bijection ψ:N N: f(P_π,S,D) = f(ψ(P)_π,ψ(S),ψ(D)) Proxy Vote Anonymity says that renaming the voters does not affect the result of the proxy vote. Suppose ψ:A A is a bijection. Let P∈𝒫(A). Then I write ψ(P) to denote the alternative-wise permutation of P. For example, if P={a≻ b}, then ψ(P)={ψ(a)≻ψ(b)}. Suppose P∈𝒫(A)^n is a (partial) preference profile. Then, abusing notation, I write ψ(P)=(ψ(P_1),...,ψ(P_n)) to denote the voter-wise application of the bijection ψ. Likewise with a default vote profile D∈ℒ(A)^n. (Proxy Vote Neutrality) A pair (f,g), where f is a social choice function and g is a proxy mechanism, satisfies Proxy Vote Neutrality iff for every P∈𝒫(A)^n, for every S∈ℒ(N)^n, for every D∈ℒ(A)^n and for every bijection ψ:A A: ψ(f(P_π,S,D)) = f(ψ(P)_π,S,ψ(D)) Proxy Vote Neutrality says that renaming the alternatives just renames the outcome of the proxy vote. In a proxy vote setting, voters submit partial orders over alternatives. This means there are two ways they can increase their support for an alternative a∈ A. They can either add an edge a ≻ b, or remove an edge b ≻ a. With this in mind, we can distinguish between two notions of monotonicity in a proxy vote setting: `addition monotonicity' and `deletion monotonicity'. As an anonymous reviewer observes, the proxy vote analogue of the classical monotonicity property can be analysed as a conjunction of these two notions. (Proxy Vote Addition Monotonicity (PVAM)) A pair (f,g), where f is a social choice function and g is a proxy mechanism, satisfies PVAM iff the following holds for every P∈𝒫(A)^n, every S∈ℒ(N)^n and every D∈ℒ(A)^n. Suppose f(P_π,S,D)=a for some a ∈ A. Consider an i-variant of P, P'=(P'_i,P_-i), where P'_i=P_i∪{a≻ b}, for some b ∈ A. Then f(P'_π,S,D)=a. Proxy Vote Addition Monotonicity (PVAM) says that if the winner under some proxy vote profile (P,S,D) is a, and we modify P by having some voter add a pairwise comparison to favour a, then the winner should remain a. (Proxy Vote Deletion Monotonicity (PVDM)) A pair (f,g), where f is a social choice function and g is a proxy mechanism, satisfies PVDM iff the following holds for every P∈𝒫(A)^n, every S∈ℒ(N)^n and every D∈ℒ(A)^n. Suppose f(P_π,S,D)=a, for some a ∈ A. Consider an i-variant of P, P'=(P'_i,P_-i), where P'_i=P_i\{b≻ a}, for some b ∈ A. Then f(P'_π,S,D)=a. Proxy Vote Deletion Monotonicity (PVDM) says that if the winner under some proxy vote profile (P,S,D) is a, and we modify P by having some voter delete a pairwise comparison which favours some other alternative over a, then the winner should remain a. §.§ Interaction with Properties of f and g Having defined properties of pairs (f,g), it's interesting to explore how they relate to properties of the individual components f and g. The first two results follow immediately from the relevant definitions; simply consider preference profiles where voters submit linear preferences. If (f,g) satisfies proxy vote anonymity, then f satisfies anonymity. If (f,g) satisfies proxy vote neutrality, then f satisfies neutrality. If (f,g) satisfies proxy vote addition monotonicity and proxy vote deletion monotonicity, then f satisfies weak monotonicity. By contraposition. Suppose f is not weakly monotonic. We will show that either (f,g) fails to satisfy PVAM or (f,g) fails to satisfy PVDM. Since f is not weakly monotonic, there must be P,P'∈ℒ(A)^n, where P'=(P'_i,P_-i) and P'_i = P_i\{b ≻ a}∪{a ≻ b} such that f(P)=a and f(P')≠ a, for some a,b∈ A. Define P”_i=P_i\{b≻ a}. By definition, P'_i=P”_i∪{a≻ b}. Define P”=(P”_i,P_-i). Fix some arbitrary proxy choice profile S and default vote profile D. Then P_π,S,D=P and P'_π,S,D=P', by construction. So f(P_π,S,D)=a and f(P'_π,S,D)≠ a. If f(P”_π,S,D)=a, then (f,g) fails to satisfy PVAM (since adding the edge a≻ b changes the winner from a). If f(P”_π,S,D)≠ a, then (f,g) fails to satisfy PVDM (since removing the edge b≻ a has changed the winner from a). If f is anonymous and g is proxy mechanism anonymous, then (f,g) is proxy vote anonymous. Let a proxy vote profile (P,S,D) be arbitrary. Pick some bijection ψ:N N. Then we must have f(ψ(P)_π,ψ(S),ψ(D)) = f(ψ(P_π,S,D)) (since g is anonymous) = f(P_π,S,D) (since f is anonymous) If f is neutral and g is proxy mechanism neutral, then (f,g) is proxy vote neutral. Let a proxy vote profile (P,S,D) be arbitrary. Pick some bijection ψ:A A. Then we must have f(ψ(P)_π,S,ψ(D)) = f(ψ(P_π,S,D)) (since g is neutral) = ψ(f(P_π,S,D)) (since f is neutral) Note that, in general, the other direction of Lemmas <ref> and <ref> won't hold. §.§ A Proxy Vote Analogue of May's Theorem When \|A\|=2, the majority rule selects the alternative which receives the most first choice votes. When \|N\| is odd (meaning the majority rule is resolute), May (<cit.>) shows that we can characterise the majority rule as the unique rule satisfying anonymity, neutrality and weak monotonicity. We can use the proxy vote analogues of these properties to achieve the same characterisation result. The key point is that setting \|A\|=2 fully specifies the proxy mechanism g, by the definition of a proxy mechanism (since either a voter submits an empty order, meaning she is allowed to delegate to any other voter, or she submits a linear order, meaning she casts her vote directly). In effect, we are close to a classical vote; the only voters who delegate their votes are the voters who submit empty orders. So it is unsurprising that the following result holds irrespective of the choice of g. Suppose \|A\|=2 and \|N\| is odd. Then a pair (f,g) satisfies Proxy Vote Anonymity * Proxy Vote Neutrality * Proxy Vote Addition Monotonicity (PVAM), and * Proxy Vote Deletion Monotonicity (PVDM) iff f is the majority rule. The left to right direction follows from Propositions <ref>, <ref>, <ref> and May's Theorem. For the other direction, suppose that f is the majority rule. So f is anonymous and neutral. When \|A\|=2, note there is only a single proxy mechanism g, by the definition of a proxy mechanism. So g will be anonymous and neutral. By Lemmas <ref> and <ref>, this implies that (f,g) is anonymous and neutral. It remains only to show that (f,g) satisfies PVAM and PVDM. I will write A={a,b}. Suppose that for some proxy vote profile (P,S,D), we have f(P_π,S,D) =a Fix some i ∈ N. Then there are two cases to consider. To see that (f,g) satisfies PVAM, suppose that P_i=∅, then consider the case where P'_i={a≻ b}. So i casts her own vote, meaning P'_π_i,S,D={a≻ b}. Note that if j picked i as her proxy when P_i=∅, then j must still pick i as her proxy (since this implies P_j=∅, since \|A\|=2). And we know that P_π_i,S,D was either {b≻ a} or {a≻ b}. To see that (f,g) satisfies PVDM, suppose that P_i={b≻ a}, then consider the case where P'_i=∅. So i delegates her vote, meaning P'_π_i,S,D is either {a≻ b} or {b ≻ a}. Note that if j didn't pick i as her proxy when P_i={b≻ a}, then j must still not pick i as her proxy (since this implies either that P_j∈ℒ(A), or that P_j=∅ and j prefers some other voter in N\{i}). And we know that P_π_i,S,D was {b≻ a}, since i cast her own vote. In either case, changing from P_i to P'_i can only decrease the number of {b≻ a} edges submitted in the profile (P'_i,P_-i)_π,S,D from in the profile P_π,S,D and increase the number of {a≻ b} edges submitted in the profile (P'_i,P_-i)_π,S,D from in the profile P_π,S,D. Since f is weakly monotonic, this implies that f((P'_i,P_-i)_π,S,D)=a. So (f,g) satisies PVAM and PVDM. §.§ Proxy Vote Monotonicity: An Impossibility Result Given some plausible restrictions on g, the monotonicity property of g turns out to be incompatible with monotonicity properties of the pair (f,g) for a large class of social choice functions. A social choice function f is a scoring rule if it can be expressed as a vector (s_1,...,s_m) with s_1≥ ... ≥ s_m≥ 0 and s_1>s_m. Each a∈ A receives s_p points for each voter putting it in the pth position in her ballot, and the outcome is the alternative a with the most points s_a. Suppose \|A\|=3. Then, for \|N\|≥14, there is no pair (f,g), where f is a scoring rule and g is a proxy mechanism, such that: * (f,g) satisfies proxy vote addition monotonicity (PVAM) and proxy vote deletion monotonicity (PVDM) * g satisfies preference monotonicity (PM) and independence of irrelevant proxies (IIP) I write \|A\|={a,b,c}. Let f be a scoring rule, and assume g satisfies PM and IIP. I show that (f,g) must either fail to satisfy PVAM or fail to satisfy PVDM. The following lemma is at the core of the proof. It says that for any proxy mechanism satisfying PM and IIP there must exist profiles where adding an edge {a ≻ b} or removing an edge {b ≻ a } switches some voter i's guru's vote from {b ≻ a ≻ c} to {c ≻ a ≻ b} (for some a,b,c∈ A). I will show that this implies that at least one of the monotonicity properties fails for scoring rules, completing the proof of Theorem <ref>. Note, though, that Lemma <ref> is a result about proxy mechanisms g; it is entirely independent of social choice functions f. Let \|N\|≥ 3 and \|A\|=3. If g satisfies PM and IIP, then, for some i∈ N, for some a,b,c∈ A, there exist P_i,P'_i∈𝒫(A), P_-i∈𝒫(A)^n-1, S∈ℒ(N)^n, D∈ℒ(A)^n such that: * Either P'_i=P_i∪{a≻ b} or P'_i=P_i\{b ≻ a} * (P_i,P_-i)_π_i,S,D = {b≻ a ≻ c} * (P'_i,P_-i)_π_i,S,D = {c≻ a ≻ b} In other words, we can construct a profile where the vote cast by i's guru changes from {b≻ a ≻ c} to {c≻ a ≻ b} when i's vote changes from P_i to P'_i. There are three collectively exhaustive cases: Case 1: For some i∈ N, for some a,b∈ A, for some P_-i∈𝒫(A)^n-1: P'_i = {a ≻ b} and g((P'_i,P_-i),i) ≠ N\{i}. Proof of Case 1: Let P_i=∅. So P'_i=P_i∪{a ≻ b}. We construct a profile where the vote cast by i's guru is {b≻ a ≻ c} when she submits P_i and {c≻ a ≻ b} when she submits P'_i. By assumption, g((P'_i,P_-i),i) ≠ N\{i}. If g((P'_i,P_-i),i)=∅, then let D_i={c≻ a≻ b} (in this case, i would be her own guru when she casts the vote P'_i; so her guru would vote for {c≻ a≻ b}, as required). If g((P'_i,P_-i),i)≠∅, then pick some j ∈ g((P'_i,P_-i),i). Let P'_j={b≻ a ≻ c}, and let S_i\|_N\{i}=j. Since g satisfies IIP and g((P'_i,P_-i),i) ≠ N\{i}, we must have g((P'_i,P'_j, P_-i,j),i) ≠ N\{i}. To see this, consider some voter in N\{j} who was not a permitted proxy for i before j changed her vote; since neither this voter nor i have changed their votes, it follows from the fact that g is IIP that this voter must not be a permitted proxy for i after j changes her vote. Since g satisfies PM, this implies that we must have j ∉ g((P'_i,P'_j, P_-i,j),i). To see this, note that P'_j has the minimum number of agreements and maximum number of disagreements possible with P'_i. So if j were a permitted proxy for i, then every other voter would have to be – regardless of what she voted – since g is PM, contradicting our earlier reasoning. Pick some k ∈ N\{i,j} and set P'_k={c ≻ a ≻ b}. Since g satisfies IIP, we must have j ∉ g((P'_i,P'_j,P'_k, P_-i,j,k),i). If g((P'_i,P'_j,P'_k, P_-i,j,k),i)=∅, set D_i={c≻ a≻ b} (as above, i would here be her own guru; so her guru would vote for {c≻ a≻ b}, as required). If g((P'_i,P'_j,P'_k, P_-i,j,k),i)≠∅, then we must have k ∈ g((P'_i,P'_j,P'_k, P_-i,j,k),i), since g satisfies PM. Set S_i\|_g((P'_i,P'_j,P'_k, P_-i,j,k),i)=k. So i chooses k as her proxy when she votes for P'_i. Note that, since P_i=∅, we have g((P_i,P'_j,P'_k, P_-i,j,k),i)=N\{i} by definition. It follows that we must have (P_i,P'_j,P'_k, P_-i,j,k)_π_i,S,D = {b≻ a ≻ c}, since i will pick j as her proxy in this case (since we have specified that S_i\|_N\{i} = j). By construction, we must have (P'_i,P'_j,P'_k, P_-i,j,k)_π_i,S,D = {c≻ a ≻ b}, since i will either pick k as her proxy in this case or submit her default vote. Case 2: For every i ∈ N, a,b∈ A, Q_-i∈𝒫(A)^n-1, if P_i={a≻ b}, we have g((P_i,Q_-i), i) = N\{i} (i.e. Case 1 doesn't hold). For some i∈ N, for some a,b,c∈ A, for some P_-i∈𝒫(A)^n-1: P'_i = {a ≻ b, c ≻ b} and g((P'_i,P_-i),i) ≠ N\{i}. Proof of Case 2: Let P_i={c ≻ b}. So P'_i=P_i∪{a≻ b}. By assumption, g((P'_i,P_-i),i)≠ N\{i}. If g((P'_i,P_-i),i)=∅, then let D_i={c≻ a≻ b} (as in the proof of the previous case, i is her own guru in this situation). Otherwise, let P'_j={b ≻ a ≻ c} for some j ∈ g((P'_i,P_-i),i) and P'_k={c ≻ a ≻ b} for some k ∈ N\{i,j}. Set S_i\|_N\{i}=j. If g((P'_i,P'_j,P'_k, P_-i,j,k),i) = ∅, set D_i={c≻ a≻ b} (as above, i is her own guru in this situation). Otherwise, set S_i\|_g((P'_i,P'_j,P'_k, P_-i,j,k),i)=k. Since g satisfies IIP and PM, identical reasoning to that in Case 1 shows that j ∉ g((P'_i,P'_j,P'_k, P_-i,j,k),i) and k ∈ g((P'_i,P'_j,P'_k, P_-i,j,k),i). Note that g((P_i,P'_j,P'_k, P_-i,j,k),i)=N\{i}, by the assumption that Case 1 is false. It follows that we must have (P_i,P'_j,P'_k, P_-i,j,k)_π_i,S,D = {b≻ a ≻ c}, since i will pick j as her proxy in this case. By construction, we must have (P'_i,P'_j,P'_k, P_-i,j,k)_π_i,S,D = {c≻ a ≻ b}, since i will either pick k as her proxy in this case or submit her default vote. Case 3: For every i ∈ N, a,b∈ A, Q_-i∈𝒫(A)^n-1, if P_i={a≻ b}, we have g((P_i,Q_-i), i) = N\{i} (i.e. Case 1 doesn't hold). For every i ∈ N, a,b,c∈ A, Q_-i∈𝒫(A)^n-1, if P_i={a≻ c, b ≻ c}, we have g((P_i,Q_-i), i) = N\{i} (i.e. Case 2 doesn't hold). Proof of Case 3: Let P_i={b ≻ a ≻ c} and P'_i={a ≻ c, b≻ c}. Then P'_i=P_i\{b≻ a}. Let P_k={c≻ a ≻ b} for some k ∈ N\{i}. Let S_i\|_N\{i}=k, and fix some P_-i,k∈𝒫(A)^n-2. By construction, (P_i,P_k, P_-i,k)_π_i,S_i,D_i = {b≻ a ≻ c}, since i casts her own vote (by the definition of a proxy mechanism, since P_i is a linear order). By the assumption that Case 2 doesn't hold, g((P'_i,P_k, P_-i,k),i)=N\{i}. So k ∈ g((P'_i,P_k, P_-i,k),i). So (P'_i,P_k, P_-i,k)_π_i,S,D = {c≻ a ≻ b}, since i delegates her vote to k in this situation. Since the three cases are collectively exhaustive, Lemma <ref> holds. What we have shown, then, is that there must exist profiles where adding an edge {a ≻ b} or removing an edge {b ≻ a } switches a voter i's guru's vote from {b ≻ a ≻ c} to {c ≻ a ≻ b}. In these profiles, we require that P_j={b≻ a ≻ c} and P_k={c≻ a ≻ b} for some j,k∈ N\{i}. Crucially, though, since g is IIP, we are free to vary the votes of the voters in N\{i,j,k} as we wish whilst ensuring that i's final vote will still change in the constructed way. To complete the proof, all we need do is set the votes submitted by the voters in the set N\{i,j,k} to construct profiles where a wins when i's final vote is {b≻ a≻ c}, and c wins when i's final vote is {c≻ a ≻ b}. We do this as follows. Note that since we are dealing with resolute scoring rules, we require slightly different solutions depending on whether the tie break contains a ≻ c or c ≻ a.[It should be clear that we could modify the proof to accommodate irresolute rules.] For a tie break containing a ≻ c, the following solution works for even n. Have two voters vote for b ≻ c ≻ a, one voter vote for a ≻ c ≻ b and the remaining n-6 voters divide their vote evenly between c ≻ a ≻ b and a ≻ c ≻ b. If i's guru votes for b ≻ a ≻ c, then: s_a = (n/2-2)s_1 + (n/2)s_2 + 2s_3 s_b = 4s_1 + (n - 4)s_3 s_c = (n/2-2)s_1 + (n/2)s_2 + 2s_3 For \|N\|≥ 14, we must have that s_a,s_c> s_b, since s_1>s_3 and s_2≥ s_3 by definition. So a wins the election. If i's guru votes for c ≻ a ≻ b, then: s_a = (n/2-2)s_1 + (n/2)s_2 + 2s_3 s_b = 3s_1 + (n - 3)s_3 s_c = (n/2-1)s_1 + (n/2)s_2 + s_3 So s_c>s_a, meaning c wins the election. For a tie break containing a ≻ c and odd n, the exact solution depends on the scoring rule. If s_1>s_2=s_3, then add one voter with b≻ a ≻ c to the solution for even n. If s_1>s_2>s_3 or s_1=s_2>s_3, add one voter with a ≻ c ≻ b to the solution for even n. This exhausts the possible scoring rules. For a tie break containing c ≻ a, the following solution works for odd n. Have two voters vote for b ≻ c ≻ a, one voter vote for a ≻ c ≻ b, one voter vote for a ≻ b ≻ c and the remaining n-7 voters divide their vote evenly between c ≻ a ≻ b and a ≻ c ≻ b. It is easily verified that s_a>s_c when i's guru votes for b ≻ a ≻ c (meaning a wins), and s_a=s_c when i's guru votes for c ≻ a ≻ b (meaning c wins). For a tie break containing c ≻ a and even n, the exact solution depends on the scoring rule. If s_1>s_2=s_3 or s_1>s_2> s_3, add one voter with b ≻ c ≻ a to the solution for odd n. If s_1=s_2>s_3, add one voter with a ≻ c ≻ b to the solution for odd n. This exhausts the possible scoring rules. Any tie break must contain either a≻ c or c≻ a. It follows that (f,g) must either fail to satisfy PVAM or fail to satisfy PVDM. § MANIPULATION I turn now to the topic of manipulation. Manipulation has hitherto received little attention in the literature on liquid democracy. () mention manipulation as a potential issue with democratic systems, but the notion of manipulation they have in mind is that by the agenda-setter (often called `control' in the social choice literature), rather than by voters themselves. () notes that introducing delegation permits voters to manipulate the outcomes of votes in novel ways, but does not develop a formal model to support this claim. The most developed formal model of manipulation in a proxy vote setting comes from (), discussed in Section <ref>. Recall that in their model, voters submit preferences over potential gurus, but do not also submit preferences over some set of alternatives (that is, there is no background election against which the delegation takes place). The authors consider ways in which voters might misrepresent their preferences so as to obtain more preferred gurus. So the notion of manipulation they discuss is not manipulation of the outcome of an election, but rather of the endpoints of individual delegation chains. In particular, this implies that – unlike the notions of manipulation defined in this section – there is no relationship between their notion of manipulation and standard notions of manipulation. In this section, a novel form of manipulation (`proxy choice manipulation') is defined, which is shown to occur roughly as often as classical manipulation. Classical manipulation is then generalised to the proxy vote setting (`preference misrepresentation manipulation'), and it is shown that manipulation occurs strictly more often in proxy votes. §.§ Proxy Choice Manipulation In a classical vote (N,A,f), voters can manipulate by misrepresenting their preferences to achieve a better outcome. In a proxy vote (N,A,f,g), there is an additional option for manipulation. Voters can manipulate by misrepresenting their choice of proxy (i.e. by picking one proxy over another for strategic reasons). I call this sort of manipulation `proxy choice manipulation'. Note that in a proxy vote setting, manipulability is no longer a property of a social choice function f alone, but rather of a pair (f,g). A pair (f,g) is proxy choice manipulable (PC-manipulable) iff there exists i ∈ N, P∈𝒫(A)^n, S∈ℒ(N)^n, D∈ℒ(A)^n such that: f(P_π,S,D) ≺ f(P_π,(S'_i,S_-i),D)∈ P_i for some S_i,S'_i∈ℒ(N). Intuitively, a pair (f,g) is PC-manipulable if there is a profile where a voter would prefer one of her potential proxies over another for purely strategic reasons. A natural question to investigate is how PC-manipulability relates to the standard notion of manipulability, which I'll call `Gibbard-Satterthwaite Manipulability' (GS-manipulability). A social choice function f is Gibbard-Satterthwaite manipulable (GS-manipulable) iff there exists i ∈ N, P_-i∈ℒ(A)^n-1 such that: f((P_i,P_-i))≺ f((P'_i,P_-i))∈ P_i for some P_i,P'_i∈ℒ(A). One way of investigating the connection between PC-manipulability and GS-manipulability is to fix a particular proxy mechanism g. If (f,) is PC-manipulable for n voters and m alternatives, then f is GS-manipulable for n voters and m alternatives. Suppose (f,) is PC-manipulable for n voters and m alternatives. Then there is some preference profile P, default profile D and proxy choices of the voters in N\{i}, S_-i, where i strictly prefers the outcome of the vote when she submits (P_i, S'_i, D_i) to the outcome when she submits (P_i, S_i, D_i). Let S=(S_i,S_-i) and S'=(S'_i,S_-i). Let Proxy_i = {j ∈ N \| P_π_j,S,D≠ P_π_j,S',D} be the set of voters whose guru's vote changes when i changes her choice of proxy (i.e. the set of voters whose vote `flows through' i; note that i ∈ Proxy_i, by assumption). Without loss of generality, suppose f(P_π,S,D) = b and f(P_π,S',D)=a. So a≻ b∈ P_i. Since we are using the mechanism, this implies that a≻ b ∈ P_π_j,S,D for every j∈ Proxy_i. Suppose now that we move from the profile P_π,S,D towards the profile P_π,S',D by changing, for each j ∈ Proxy_i, P_π_j,S,D to P_π_j,S',D. We know that when we start, the outcome is b. We know that when we have made all the changes, the outcome is a. If the outcome changes directly from b to a at some stage in the process, then we have a profile with respect to which f is GS-manipulable (since a≻ b ∈ P_π_j,S,D for every j∈ Proxy_i). If the social outcome first changes to some c≠ b ≠ a, then there are two cases. If c≻ b ∈ P_π_i,S,D, then the same reasoning shows that f is GS-manipulable. If c≺ b ∈ P_π_i,S,D, then we can just carry on making the changes until the social outcome changes to a, then apply the same reasoning as above. It follows that f is GS-manipulable. So we have shown that PC-manipulability implies GS-manipulability, assuming g is the mechanism. In general, the converse won't hold (just consider the case where \|N\|=2). But we can say something about the converse relationship, using a stronger form of GS-manipulability. A social choice function f is IIA- manipulable if there is some L∈ℒ(A) such that, for some i∈ N, L'_i≠ L_i: * f(L'_i,L_-i) ≻ f(L)∈ L_i * f(L'_i,L_-i) ≻ f(L)∈ L'_i Intuitively, f is IIA-manipulable if a voter can reverse the social ranking of two alternatives whilst maintaining her personal ranking of the alternatives.[IIA-manipulability can be thought of as a much weaker condition than `one-way monotonicity' (<cit.>), which features in the preference reversal paradox (<cit.>). In effect, one-way monotonicity says that every example of GS-manipulability is an example of IIA-manipulability.] If f is: * IIA-manipulable over n voters and m alternatives. * Invariant to Uniform Voter Additions then (f,) is PC-manipulable over n+m! voters, and m alternatives. Suppose f is IIA-manipulable over \|N\|=n voters and \|A\|=m alternatives. Note that we must have m>2, by the definition of IIA-manipulability. Since f is IIA-manipulable, there must be some profile L∈ℒ(A)^n and some i ∈ N such that, for some L'_i≠ L_i, we have f(L'_i,L_-i)≻_L_i f(L) For the sake of readability, let f(L)=b and f(L'_i,L_-i)=a. Since f is IIA-manipulable, we can assume that both a ≻ b ∈ L_i and a ≻ b ∈ L'_i without loss of generality. Let us now consider L+ and L'+, the uniform-voter augmentations of L and (L'_i,L_-i) respectively. Since f is IUVA, it follows that f(L+) = b and f(L'+) = a. Since L+ contains, for every linear order over A, at least one voter who submits that order, we must have voters j and k such that L+_j=L_i and L+_k=L'_i. Define P_i = a ≻ b. Note that both P_i⊂ L+_j and P_i⊂ L+_k. Since we are using the proxy mechanism, this implies that both j and k are permitted proxies for i in the preference profile (P_i,L+_-i). If i picks j as her proxy, then the guru profile for (P_i,L+_-i), written as (P_i,L+_-i)_π,S,D, is simply L+. So we must have f((P_i,L+_-i)_π,S,D)=b. If i picks k as her proxy, then the guru profile for (P_i,L+_-i), written as (P_i,L+_-i)_π,S',D, is simply L'+. So we must have f((P_i,L+_-i)_π,S',D)=a. Since a≻ b ∈ P_i, it follows that we have a situation where i would strictly prefer picking k over j as her proxy. So f is PC-manipulable on a profile of n+m! voters. §.§ Preference Misrepresentation Manipulation It is also natural to generalise GS-manipulation in the proxy vote setting. A pair (f,g) is preference misrepresentation manipulable (PM-manipulable) iff there exists i ∈ N, P∈𝒫(A)^n, S∈ℒ(N)^n such that: f(P_π,S,D) ≺ f((P'_i,P_-i)_π,S,D)∈ P_i for some P_i,P'_i∈𝒫(A). Given PM-manipulability is just the generalisation of GS-manipulability to the proxy vote setting, one might wonder whether domain restrictions which result in GS-strategyproofness also result in PM-strategyproofness. The following result shows that this does not hold. There are social choice functions f such that (f,) is PM-manipulable on the domain of single-peaked preference profiles but f is not GS-manipulable on the domain of single-peaked preference profiles. When \|A\|= 3 and \|N\|≥ 2, there is no social choice function f which is non-dictatorial, surjective, and such that (f,) is PM-strategyproof, even when we restrict the domain to include only single-peaked preference profiles. We know that if a social choice function f is GS-manipulable, the pair (f,) is PM-manipulable. Moulin characterises the class of surjective, non-dictatorial and GS-strategyproof social choice functions on the domain of single-peaked preferences as the class of generalised median voter rules (<cit.>, <cit.>). To prove the theorem at hand, then, it suffices to show that for a generalised median voter rule f, (f,) is PM-manipulable on the domain of single-peaked preferences when \|A\|= 3. I write A={a,b,c}. Let f be an arbitrary generalised median voter rule with n-1 phantoms. Without loss of generality, suppose that at least one phantom has peak a. We construct the profile P=(P_1,...,P_n), where P_j = {c ≻ b} ∀ j ∈ N\{i} P_i = {b ≻ c ≻ a} Suppose also that D_j={a ≻ c ≻ b} for every j ∈ N\{i}, and that S_j\|_N\{j}=i (i.e. that every j would pick i as her proxy if permitted). Note that P is single-peaked along the dimension a,c,b. As it stands, we have that g(P,j)=N\{i,j} for every j ∈ N\{i} (since g is the mechanism). It follows that each of these voters will enter a delegation cycle, casting her default vote {a ≻ c ≻ b}. So the peak of the median voter will be a, since n - 1 voters and at least one phantom have peak a. So the winner will be a. Now suppose i switches from {b≻ c≻ a} to {c≻ a ≻ b}. Note that the profile is still single-peaked along the dimension a,c,b. Now we have that g(P,j)=N\{j} for every j ∈ N\{i}. By construction, it follows that each of the voters has i as her guru. So the peak of all n voters will be c, implying that the peak of the median voter will be c (even if none of the phantoms has peak c). So the winner will be c. Since c≻ a ∈ P_i, it follows that i has an incentive to change her preference from {b≻ c≻ a} to {c≻ a ≻ b}. § CONCLUSION This paper introduced a novel model of transitive proxy voting, which paid more attention to `proxy selection', the process by which voters select delegates. The properties of the model were explored from an axiomatic perspective; it was shown that (given plausible assumptions) we cannot expect proxy votes to satisfy intuitively desirable monotonicity properties. The model was also put to work in analysing manipulation in a proxy vote setting. It was shown not only that novel forms of manipulation arise in a proxy vote setting, but also that there are strictly more situations in which manipulation is available to voters in a proxy vote than in a classical vote. A natural question concerns the relation between the main results in this paper and the informal arguments surrounding transitive proxy voting of the sort discussed in Section <ref>. Do these formal results have implications for real world discussion of liquid democracy? Underpinning the model I've introduced is the claim that voters will only delegate their votes to proxies who represent their interests. I introduced the claim normatively (akin to a constraint on voters' rationality); in order for the claim to be testable empirically, we need to operationalise the notion of a proxy representing a voter's interests. Proxy mechanisms provide us with useful tools with which to do this – different formal properties of proxy mechanisms will lend themselves to developing and evaluating competing notions of what it means for a proxy to represent a voter's interests. In particular, Theorem <ref> (which characterises the mechanism) can be seen as providing a formal argument for the plausible empirical claim that a voter will only delegate to a proxy who agrees with her on all the issues on which she's already made up her mind. When there are only two alternatives, it's natural to think that the degree to which a proxy vote resembles a classical vote depends entirely on the number of voters who choose to delegate rather than vote directly. Since it supports this intuition, Theorem <ref> (the proxy vote analogue of May's Theorem) is best viewed as justification for the formal model introduced, rather than as a result with practical implications of its own. Given plausible assumptions on the process by which voters select proxies, Theorem <ref> shows that proxy votes will always fail to satisfy monotonicity properties for a large class of monotonic social choice functions. In effect, it serves to highlight the instability inherent to votes involving delegations; small changes in an individual voter's behaviour can change the outcomes of votes in counterintuitive ways. My sense is that Theorem <ref> ought to be taken seriously by opponents of liquid democratic systems. Recall that a key motivation for transitive proxy voting is that it is more `democratic', in that it better represents the views of the whole electorate (<cit.>). But a failure of monotonicity is precisely a situation in which an aggregation procedure has represented the views of an electorate poorly. Similarly, it is often claimed that proxy voting increases participation (<cit.>, <cit.>). One might think that the sort of reasoning employed in the proof of Theorem <ref> suggests that the model proposed in this paper is well positioned to challenge this participation claim formally (future work could do just this). As noted in Section <ref>, it has been claimed informally that transitive proxy voting equips voters with the ability to manipulate in ways unavailable to them in classical votes (<cit.>, <cit.>). The model proposed in this paper confirms such claims, and provides a formal framework with which to examine the material consequences of novel forms of manipulation. Some of the results in this section will reassure proponents of liquid democracy; for example, Theorems <ref> and <ref> challenge the idea that a voter's novel ability to misrepresent her choice of proxy gives her any additional power to manipulate the outcome of the election. Another result, though, lends support to arguments advanced against liquid democracy; Theorem <ref> shows that outcomes of proxy votes are strictly more vulnerable than those of classical votes to manipulation by an individual voter who misrepresents her preferences. Furthermore, the proof of Theorem <ref> exploits the fact that voting power in a transitive proxy voting system can concentrate in the hands of individual `super-voters', a common worry raised against liquid democratic systems (<cit.>). Future work could apply the model in this paper to a wider range of manipulation and control problems. My aim in this paper has not been to provide full formal coverage of topics relevant to transitive proxy voting, but rather to showcase interesting features of the model I've introduced. My hope is that the reader thinks my model sufficiently rich to enable non-trivial formal discussion of the arguments surrounding transitive proxy voting. I'd like especially to thank Ulle Endriss, who provided whole hosts of fruitful suggestions at every stage in the writing of this paper. Thanks also to Davide Grossi and Ronald de Haan for generous feedback on an earlier draft.
http://arxiv.org/abs/2307.01246v1	20230703180000	Bootstrapping Pions at Large $N$. Part II: Background Gauge Fields and the Chiral Anomaly	[ "Jan Albert", "Leonardo Rastelli" ]	hep-th	[ "hep-th", "hep-ph" ]	=1 OT1pzc OT1pzcmit<-> s * [1.10] pzcmi7t OT1pzcmit #1 (#1) #1 (#1) e_1 e_2 e_3 e_4 n n' e_1·e_2
http://arxiv.org/abs/2307.01051v1	20230703142933	On the reach of isometric embeddings into Wasserstein type spaces	[ "Javier Casado", "Manuel Cuerno", "Jaime Santos-Rodríguez" ]	math.MG	[ "math.MG", "49Q20, 28A33, 30L15, 49Q22, 53C21, 55N31" ]	W R]On the reach of isometric embeddings into Wasserstein type spaces J. Casado]Javier Casado^∗ M. Cuerno]Manuel Cuerno^∗∗ J. Santos-Rodríguez]Jaime Santos-Rodríguez^∗∗∗ ^Supported in part by the FPU Graduate Research Grant FPU20/01444, and by research grants MTM2017-‐85934-‐C3-‐2-‐P and PID2021-124195NB-C32 from the Ministerio de Economía y Competitividad de Españã (MINECO) ^∗∗Supported in part by the FPI Graduate Research Grant PRE2018-084109, and by research grants MTM2017-‐85934-‐C3-‐2-‐P and PID2021-124195NB-C32 from the Ministerio de Economía y Competitividad de Españã (MINECO) ^∗∗∗ Supported in part by a Margarita Salas Fellowship CA1/RSUE/2021-00625, and by research grants MTM2017-‐85934-‐C3-‐2-‐P, PID2021-124195NB-C32 from the Ministerio de Economía y Competitividad de Españã (MINECO) [J. Casado]Department of Mathematics, Universidad Autónoma de Madrid and ICMAT CSIC-UAM-UC3M, Spain [email protected] [M. Cuerno]Department of Mathematics, Universidad Autónoma de Madrid and CSIC-UAM-UC3M, Spain and Department of Mathematical Sciences, Durham University, UK [email protected], [email protected] [J. Santos-Rodríguez]Department of Mathematics, Universidad Autónoma de Madrid, Spain and Department of Mathematical Sciences, Durham University, UK [email protected], [email protected] [2020]49Q20, 28A33, 30L15, 49Q22, 53C21, 55N31 We study the reach (in the sense of Federer) of the natural isometric embedding X↪ W_p(X) of X inside its p-Wasserstein space, where (X,) is a geodesic metric space. We prove that if a point x∈ X can be joined to another point y∈ X by two minimizing geodesics, then (x, X⊂ W_p(X)) = 0. This includes the cases where X is a compact manifold or a non-simply connected one. On the other hand, we show that (X⊂ W_p(X)) = ∞ when X is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding X↪ W_ϑ(X) into an Orlicz–Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of X into _∞, the space of persistence diagrams equipped with the bottleneck distance. [ [ August 1, 2023 ================== § INTRODUCTION The concept of the reach of a subset in Euclidean space was first introduced by Federer in <cit.>. It is used as a way to measure how much the subset folds in on itself (i.e. how far apart two pieces of the set are in the ambient space despite them being far in the intrinsic metric of the set). Loosely speaking (see Definition <ref>) a subset A⊂ X has positive reach if there is a neighbourhood of any point on A such that every point in this neighbourhood has a unique metric projection into A. That is, every x∈ X inside that neighbourhood is sent by the projection to its unique nearest point in A. The reach of a subset has been of interest not only for its geometric and topological properties (see for example <cit.>) but also for its application as a useful parameter for manifold learning and topological data analysis (see <cit.> and references therein). In the survey <cit.>, the interested reader can also find a summary of some results for sets of positive reach. Given a geodesic metric space (X, ), one can equip the space of probability measures supported on X with a distance induced by the solutions to an optimal transport problem. Usually the cost comes from taking the p-power of the distance function, the so called p-Wasserstein spaces. One advantage of considering these ambient spaces is that they share many geometrical properties with the base space X such as non-branching of geodesics, compactness, and lower sectional curvature bounds amongst others. In this article we focus on determining the reach of the image of the natural isometric embedding, given by mapping each point x ∈ X to the corresponding Dirac delta δ_x∈ W_p(X). We denote this by (X⊂ W_p(X)), where W_p(X) is the p–Wasserstein space of X (see Section <ref>). Our first result shows that the cost considered affects the reach of the embedding significantly: <ref> Let (X,) be a metric space, and consider its 1–Wasserstein space, W_1(X). Then, for every accumulation point x∈ X, (x, X⊂ W_1(X)) = 0. In particular, if X is not discrete, (X⊂ W_1(X)) = 0. theorem-1 Geometric features of X also play an important role. In the presence of multiple geodesics joining the same pair of points we obtain the following: <ref> Let X be a geodesic metric space, and x∈ X a point such that there exists another y∈ X with the property that there exist at least two different minimising geodesics from x to y. Then, for every p>1, (x, X⊂ W_p(X))=0. In particular, if there exists a point x∈ X satisfying that property, (X⊂ W_p(X))=0 for every p>1. theorem-1 This theorem leads us to obtain two interesting corollaries related to two important classes of manifolds: <ref> If M is a compact manifold, then (x,M⊂ W_p(M)) =0 for every x∈ M and every p>1. coro-1 <ref> If M is a not simply connected complete manifold, then (x,M⊂ W_p(M)) =0 for every x∈ M and p>1. coro-1 In <cit.>, Kell studied several convexity conditions, such as (resp. strictly, uniformly) p–convexity or Busemann, on the distance of a geodesic metric space and some other more general conditions about metric spaces, such as reflexivity (see also Definitions <ref> and <ref>) obtaining existence and uniqueness of barycenters, i.e., certain points on the metric space that minimise the distance to a given measure/density. In order to formalise that concept, we define a barycenter as a point in X that minimises the distance between a given element from a Wasserstein type space and some isometric embedding of the metric space inside that space. Kell's conditions allow us to determine that the reach is infinite for a broad class of spaces: <ref> Let (X,) be a reflexive metric space, Then the following assertions hold: If X is strictly p–convex for p∈[1,∞) and uniformly ∞–convex if p=∞, then (X⊂ W_r(X))=∞ for r>1. * If X is Busemann, strictly p–convex for some p∈[1,∞], and uniformly q–convex for some q∈[1,∞], then (X⊂ W_r(X))=∞ for r>1. theorem-1 We also study properties of the projection map projection map, i.e., _2: W_2(X) → X μ ↦ r_μ, that sends each measure to its 2-barycenter (i.e. the barycenter on the 2–Wasserstein space), and showing that this map is in fact a submetry for a certain class of spaces. <ref> Let (^n,) the Euclidean space with the canonical distance, then _2 is a submetry. theorem-1 Our next results focus on the embedding into Wasserstein type spaces with more general metrics, such as the Orlicz-Wasserstein spaces defined by Sturm in <cit.> and the space of persistence diagrams, the key tool in Topological Data Analysis <cit.>. For the Orlicz-Wasserstein spaces we require reasonable assumptions on the cost (stated in the theorem) in order to ensure that the natural embedding using Dirac deltas is indeed isometric. In a similar fashion to the case of p-Wasserstein spaces, we obtain: <ref> Let X be a geodesic metric space, and x∈ X a point such that there exists another y∈ X with the property that there exists at least two different minimising geodesics from x to y. Suppose X is isometrically embedded into an Orlicz-Wasserstein space W_ϑ(X). Then, for every φ (as explained in Subsection <ref>) such that φ(t_0) ≠ t_0 for some t_0>1, (x, X⊂ W_ϑ(X))=0. In particular, if there exists a point x∈ X satisfying that property, (X⊂ W_ϑ(X))=0 for every p>1. theorem-1 The last case of isometric embedding into a Wasserstein type space is the one into , the space of persistence diagrams. We can equip with Wasserstein type distances, involving a minimisation process as in an optimal transport problem. The bottleneck distance, w_∞, is one of the most used distances in . Bubenik and Wagner proved in <cit.> the existence of an isometric embedding of separable and bounded metric spaces into (_∞,w_∞). We have studied the reach of these embeddings: <ref> Let (X,) be a separable, bounded metric space and (_∞,w_∞) the space of persistence diagrams with the bottleneck distance. If x∈ X is an accumulation point, then (x, X⊂_∞)=0. In particular, if X is not discrete, (X⊂_∞)=0. theorem-1 The paper is organized as follows: In Section <ref>, we state the necessary technical definitions that we need. Section <ref> is devoted to present the Wasserstein type spaces we use and some of their properties. Sections <ref>, <ref> and <ref> contain the results about the reach of spaces embedded into their p-Wasserstein space, their Orlicz–Wassertein space and the persistence diagram space respectively. The authors would like to express their sincere gratitude to Professor Luis Guijarro for his invaluable comments and insights during the elaboration of this paper. They would also like to extend their appreciation to Professors Fernando Galaz-García and David González for their enlightening discussions and contributions to the final manuscript. Additionally, the authors wish to acknowledge the Department of Mathematical Sciences at Durham University for their warm hospitality and the excellent working conditions provided during the final months of preparing this paper. § PRELIMINARIES §.§ Reach First we recall the definition of the reach of a subset of a metric space. Let (X, ) be a metric space and A⊂ X a subset. We define the set of points having a unique metric projection in A as (A) = {x ∈ X : there exists a unique a such that (x,A) = (x,a)}. For a∈ A, we define the reach of A at a, denoted by (a, A), as (a, A) = sup{ r≥ 0 : B_r(a) ⊂(A)}. Finally, we define the global reach by (A) = inf_a∈ A(a, A). The intuitive idea is that (A)=0 if and only if we do not have an ε–neighbourhood of A admitting a metric projection into A. Conversely, (A)=∞ will occur if and only if the entirety of X admits a metric projection into A. §.§ CAT(0) spaces We recall the definition of a CAT(0) metric space. A complete metric space (X,) is CAT(0) if for all z, y ∈ X there exists m∈ X such that for all x∈ X, (x,m)^2≤(x,y)^2 + (x,z)^2 2 - (y,z)^24. This is a generalization of the concept of nonpositive curvature for Riemannian manifolds to metric spaces. So, in particular, the Euclidean space or the hyperbolic space are examples of CAT(0) spaces. A few basic properties of these spaces are: * For any two points in X, there exists a unique geodesic segment between them. * X is simply connected. §.§ General metric definitions Most of the definitions in this section appear in <cit.>. First, we recall the well known definition of existence of midpoints: We say that (X,) admits midpoints if , for every x,y∈ X, there is m(x,y)∈ X such that (x,m(x,y))=(y,m(x,y))=1/2(x,y). This technical detail allows us to present the following definitions: Let (X,) be a metric space that admits midpoints. * X is p–convex for some p∈[1,∞] if, for each triple x,y,z∈ X and each midpoint m(x,y) of x and y, (m(x,y),z)≤(1/2(x,z)^p+1/2(y,z)^p)^1/p. The space X is called strictly p–convex for p∈(1,∞] if the inequality is strict for x≠ y and strictly 1–convex if the inequality is strict whenever (x,y)>\|(x,z)-(y,z)\|. * X satisfies the p–Busemann curvature condition if, for all x_0,x_1,y_0,y_1∈ X with midpoints m_x=m(x_0,x_1) and m_y=m(y_0,y_1), (m_x,m_y)≤(1/2(x_0,y_0)^p+1/2(x_1,y_1)^p)^1/p for some p∈[1,∞]. If X satisfies the p–Busemann condition, we say that (X,) is p–Busemann. In particular, if p=1, we say that (X,) is Busemann. It turns out that (X,) is a Busemann space if and only if (m(x,z),m(x,y))≤1/2(z,y). * X is uniformly p–convex for some p∈[1,∞] if, for all ϵ>0, there exists ρ_p(ϵ)∈(0,1) such that, for every x,y,z∈ X satisfying (x,y)>ϵ(1/2(x,z)^p+1/2(y,z)^p)^1/p, for some p>1, or (x,y)>\|(x,z)-(y,z)\|+ϵ(1/2(x,z)+1/2(y,z)), for p=1, the following inequality holds: (m(x,y),z)≤(1-ρ_p(ϵ))(1/2(x,z)^p+(y,z)^p)^1/p. For example, every CAT(0)–space is uniformly 2–convex. By <cit.>, the following assertions hold: * A uniformly p–convex metric space is uniformly p'–convex for all p'≥ p. * Assume (X,) is Busemann. Then (X,) is strictly (resp. uniformly) p–convex for some p∈[1,∞] if and only if it is strictly (resp. uniformly) p–convex for all p∈[1,∞]. * Any CAT(0)–space is both Busemann and uniformly 2–convex, thus uniformly p–convex for every p∈[1,∞]. In order to apply some of Kell's results, we introduce the notion of reflexivity on metric spaces. Let I be a directed set. A metric space (X,) is reflexive if, for every non–increasing family {C_i}_i∈ I⊂ X of non–empty bounded closed convex subsets (i.e. C_i⊂ C_j whenever i≥ j), we have ⋂_i∈ IC_i≠∅. § WASSERSTEIN TYPE SPACES AND DISTANCES In this section we will recall the standard notions of optimal transport and Wasserstein distance. Then we will provide an introduction to the Orlicz–Wasserstein spaces initially proposed by Sturm in <cit.>. Finally, we present our last Wasserstein-type space, the one formed by persistence diagrams, the key element in the field of Topological Data Analysis <cit.>. §.§ Wasserstein space From now on, X will be a metric space with distance function . Denote by 𝒫(X) the set of probability measures on X and by 𝒫 _p (X) the probability measures with finite p-moment, i.e. 𝒫_p(X) := {μ∈𝒫(X) : ∫_X (x, x_0)^p d μ(x) < ∞ for some x_0∈ X}. A transference plan between two positive measures μ, ν∈𝒫(X) is a finite positive measure π∈𝒫 (X × X) which satisfies that, for all Borel subsets A, B of X, π(A× X) = μ(A), and π(X × B) = ν(B). Note that we require 1=\|μ\| = \| ν\| = π( X × X), so we are not considering all measures of the product space. We denote by Γ(μ, ν) the set of transference plans between the measures μ and ν. Then, we define the p–Wasserstein distance for p≥ 1 between two probability measures as W_p( μ, ν) := ( min_π∈Γ(μ, ν)∫_X × X (x,y)^p dπ (x, y) ) ^1/p . The metric space (𝒫_p (X) , W_p) is denoted as the p–Wasserstein space of X. It is easy to see that, for x, y ∈ X, W_p(δ_x, δ_y) = (x,y). Therefore the inclusion x ↦δ_x is an isometric embedding of X inside W_p(X). When we are calculating W_p(δ_x, μ), there exists only one pairing π = δ_x ⊗μ∈Γ(δ_x, μ) between a delta and a general probability measure. Therefore the Wasserstein distance can be easily computed by W_p^p(δ_x, μ) = ∫_X (x, y) ^p dμ(y). In particular, fixing a, x, y∈ X, and 0 ≤λ≤ 1, we have W_p^p(δ_a, λδ_x + (1-λ) δ_y) = λ(a, x)^p + (1-λ) (a, y) ^p. §.§ Orlicz–Wasserstein space Let ϑ: ^+ →^+ be a strictly increasing, continuous function. Assume ϑ admits a representation ϑ = φ∘ψ as a composition of a convex and a concave function φ and ψ, respectively. This includes all 𝒞^2 functions <cit.>. Let (X, ) be a complete separable metric space. The L^ϑ–Wasserstein space 𝒫_ϑ (X) is defined by all probability measures μ in X such that ∫_X φ(1/tψ((x,y))) d μ(x) < ∞. The L^ϑ–Wasserstein distance of two probability measures μ, ν∈𝒫_ϑ (X) is defined as W_ϑ (μ, ν) = inf{ t>0 : inf_π∈Γ(μ, ν) ∫_X× Xφ( 1/tψ((x,y))) dπ(x,y) ≤ 1 }. The function W_ϑ is a complete metric on 𝒫_ϑ(X) (see <cit.>, Proposition 3.2). The metric space (𝒫_ϑ (X), W_ϑ) is known as the ϑ-Orlicz–Wasserstein space of X. Notice that for every x∈ X, the probability measure δ_x belongs to 𝒫_ϑ(X). Therefore, we can embed the metric space X inside its Orlicz–Wasserstein space by mapping x ↦δ_x. In addition, this map is an isometric embedding if and only if ψ≡Id and φ(1) = 1. §.§ Space of persistence diagrams and bottleneck distance We will now define (_p,w_p) the space of persistence diagrams with a Wasserstein metric. For that purpose we being with the basic notion of the elements of our metric space: A persistence diagram is a function from a countable set I to ^2_<, i.e. D I→^2_<, where ^2_<={(x,y)∈^2 x<y}. In this definition, all the points have multiplicity one. Other authors suggest considering persistence diagrams as multisets of points, i.e. sets of points where we can repeat points (see <cit.>). This consideration is closer to the performance of the persistence diagrams in the TDA setting as various homological features can have the same birth and death. Also, in <cit.>, the authors extend the notion of persistence diagrams beyond the Euclidean setting and present a general definition for points in metric spaces. Once we have the points of our metric space, we want to define distance functions on it. For that purpose, first, we present two preliminary definitions. Let D_1 I_1→^2_< and D_2:I_2→_<^2 be persistence diagrams. A partial matching between them is a triple (I_1',I_2',f) such that I_1'⊆ I_1, I_2'⊆ I_2, and f I_1'→ I_2' is a bijection. In the same spirit of the original p–Wasserstein distance, we want to define a new one between persistence diagrams D_1 and D_2 as the minimal cost of a partial matching between them. In particular, the cost of a partial matching will be the ℓ^p norm of distances between matched pairs and the distances between unmatched pairs and Δ, as Bubenik stated on <cit.> , where Δ denotes the diagonal in ^2. Let D_1 I_1→^2_< and D_2 I_2→^2_< be persistence diagrams and (I_1',I_2',f) a partial matching between them. We endow ^2 with the infinity metric _∞(a,b)=a-b_∞=max(\|a_x-b_x\|,\|a_y-b_y\|). Observe that, for a∈^2_<, we have that _∞(a,Δ)=inf_t∈Δ_∞(a,t)=(a_y-a_x)/2. We denote by _p(f) the p–cost of f, defined as follows. For p<∞, let _p(f)=(∑_i∈ I_1'_∞(D_1(i),D_2(f(i)))^p+∑_i∈ I_1\ I_1'_∞(D_1(i),Δ)^p+∑_I_2\ I_2'_∞(D_2(i),Δ)^p)^1/p, and for p=∞, let _∞(f)=max{sup_i∈ I_1'_∞(D_1(i),D_2(f(i))),sup_i∈ I_1\ I_1'_∞(D_1(i),Δ),sup_i∈ I_2\ I_2'_∞(D_2(i),Δ)}. If any of the terms in either expression is unbounded, we declare the cost to be infinity. Now we can define the distance functions and the metric space of persistence diagrams: Let 1≤ p≤∞ and D_1, D_2 persistence diagrams. Define w̃_p(D_1,D_2)=inf{_p(f) f is a partial matching between D_1 and D_2}. Let (_p,w_p) denote the metric space of persistence diagrams D such that w̃_p(D,∅)<∞ with the relation D_1∼ D_2 if w̃_p(D_1,D_2)=0, where ∅ is the unique persistence diagram with empty indexing set. The metric w_p is called the p–Wasserstein distance and w_∞ is called the bottleneck distance. § REACH OF THE WASSERSTEIN SPACE The p–Wasserstein space of a metric space (X,) gives us an example of a well-known isometric embedding between a total space and an infinite dimensional space (in this sense, there also exists the Kuratowski embedding between a compact metric space (Y,_Y) and the space of functions L^∞(Y)). We first recall the definition of the set of unique points and reach: Let (X, ) be a metric space and A⊂ X a subset. We define the set of points having a unique metric projection in A as (A) = {x ∈ X : there exists a unique a such that (x,A) = (x,a)}. For a∈ A, we define the reach of A at a, denoted by (a, A), as (a, A) = sup{ r≥ 0 : B_r(a) ⊂(A)}. Finally, we define the global reach by (A) = inf_a∈ A(a, A). The set of unique points of the isometric embedding of a metric space into its p–Wasserstein space is dense into the total space: Let (X,) be a non-branching metric space and W_p(X) with p>1 its p–Wasserstein space. Then the set of unique points (X⊂ W_p(X)) is dense in W_p(X). Let μ∈ W_p(X) be a measure with x ∈ X a barycenter. Take ν inside a geodesic between δ_0 and μ. Suppose that there exists some other point z∈ X that is as barycenter for ν. This implies that W_p(ν, δ_z) ≤ W_p(ν, δ_x) and with this we get W_p(μ,δ_z)≤ W_p(μ,ν)+W_p(ν,δ_z)≤ W_p(μ,ν)+W_p(ν,δ_x)= W_p(μ,δ_x). So then z is also a barycenter for μ. Furthermore we notice that there is a branching geodesic joining μ with δ_z. This gives us the contradiction as W_p(X) is non-branching. Then ν is a measure in (X⊂ W_p(X)) which can be taken arbitrarily close to μ. This density fact motivates the question of the existence of metric spaces with positive reach into its p–Wasserstein space. §.§ Null reach The first result of this paper is that the reach of a metric space inside its 1–Wasserstein space is always 0. The proof follows the idea of the proof of Theorem 1.6. of <cit.>. Let (X,) be a metric space, and consider its 1–Wasserstein space, W_1(X). Then, for every accumulation point x∈ X, (x, X⊂ W_1(X)) = 0. In particular, if X is not discrete, (X⊂ W_1(X)) = 0. Following <cit.>, let ϵ>0. We will show that inside B_ϵ(x)⊂ W_1(X) there exists at least one measure μ∉(X). By hypothesis, there exists y∈ X, y≠ x, such that d(x,y)<ϵ. Then μ:=1/2δ_x+1/2δ_y. First, notice that μ≠δ_z for any z∈ X because the support of μ is different from the support of any of the δ_z∈ X. In addition, due to the triangle inequality, W_1(δ_a,μ)=1/2(a,x)+1/2(a,y)≥1/2(x,y). By inequality (<ref>) above, we can clearly see that μ∈ B_ϵ(x), because W_1(δ_x,μ)=1/2(x,y)<ϵ. Finally, we observe that both a=x and a=y minimize the distance to μ. Therefore, μ∉ (X) and (x, X⊂ W_1(X)) = 0. Note that the hypothesis of the point being an accumulation point is necessary, because, if x_0∈ X is an isolated point, then the quantity ℓ = inf_x∈ X(x, x_0) is strictly positive, and B_ℓ/2(x) admits a unique metric projection to X. An interesting observation is that, combining the same argument in the proof of Theorem <ref> with the previous remark, if X is a discrete metric space isometrically embedded into another metric space Y, then (X ⊂ Y) = inf_x_1≠ x_2(x_1, x_2)/2>0. Now we will provide results about the reach of a geodesic metric space inside its p–Wasserstein space with p>1. We have found that these results are closely related to the uniqueness of the geodesics. This next proposition has important consequences about the reach inside a Wasserstein space, as it constructs measures with possibly several projections in X. Let (X,) be a geodesic metric space, and x, y∈ X two points with x≠ y. Consider the probability measure μ= λδ_x + (1-λ) δ_y, for 0<λ<1. Then μ minimizes its p–Wasserstein distance to X exactly once for every minimizing geodesic between x and y. The proof is structured in the following way: First, we choose a candidate for the distance–minimizer of μ, supposing it lies inside a minimizing geodesic. Then, we show that the global minimum distance can only be achieved inside a minimizing geodesic. Choose γ(t) [0,1] → X a minimizing geodesic from x to y. We can compute the cost W_p^p( δ_γ(t), μ) and then minimize in t. Indeed, W_p^p(δ_γ(t), μ) = λ(γ(t), x)^p + (1-λ) (γ(t), y) ^p = (λ t^p + (1-λ) (1-t)^p )( x, y)^p. The minimum will be achieved at the parameter t_0 which verifies ddt\|_t=t_0 W_p^p (δ_γ(t), μ) = 0. We know this because that derivative is negative for t=0, and positive for t=1, and vanishes at only one point t=t_0. An easy computation shows us that the only solution in our interval is t_0 = (1-λ)^p-1/λ^p-1+(1-λ)^p-1. Thus, the Wasserstein distance between μ and this geodesic minimum is W_p^p(δ_γ(t_0), μ ) = λ(1-λ)^(p-1)p + (1-λ) λ^(p-1)p/(λ^p-1+ (1-λ)^p-1)^p·^p(x,y). Observe that this value is independent from the minimizing geodesic γ of our choice. Finally, we only have to prove that the minimum can only be achieved inside a minimizing geodesic. For that purpose, we will choose any a ∈ X, and we will construct another point a' inside a minimizing geodesic segment γ verifying W_p^p(δ_a, μ) ≥ W_p^p(δ_a', μ). The case (a, y) ≥(x, y) is straightforward, as choosing a' = x we have W_p^p (δ_a, μ) = λ(a, x) ^p + (1-λ) (a, y) ^p ≥ (1-λ) (a, y) ^p ≥ (1-λ) (x, y) ^p = W_p^p (δ_x, μ). Now, if (a, y) < (x, y), we can pick a' inside γ at distance (a, y) to y. Observe that (a, x) ≥(a', x) or γ would not be minimizing. Then W_p^p (δ_a, μ) = λ(a, x) ^p + (1-λ) (a, y) ^p = λ(a, x) ^p + (1-λ) (a', y) ^p ≥λ(a', x) ^p + (1-λ) (a', y) ^p = W_p^p (δ_a', μ). Therefore, the minimum can only be achieved inside minimizing geodesics between x and y and our proof is complete. Now, we will apply the preceding proposition to construct measures with multiple projections close to any point in X. We will use this to derive sufficient conditions for attaining (p,X)=0 for all p∈ X. Let X be a geodesic metric space, and x∈ X a point such that there exists another y∈ X with the property that there exist at least two different minimising geodesics from x to y. Then, for every p>1, (x, X⊂ W_p(X))=0. In particular, if there exists a point x∈ X satisfying that property, (X⊂ W_p(X))=0 for every p>1. The probability measure μ_λ = λδ_x + (1-λ )δ_y will have at least two different points minimizing its distance to X by proposition <ref>. Now simply observe that W_p^p(μ_λ, δ_x) = (1-λ) (x, y)^p, which decreases to 0 when λ→ 1. Hence (x, X) = 0 for every x∈ X satisfying that property, and therefore (X⊂ W_p(X))=0. When X is a Riemannian manifold, some common hypothesis will grant us reach 0. For example, a classic result by Berger (see for example <cit.>) proves that our theorem can be applied when X is compact. In this case, for any p∈ X, there always exists another q∈ X such that there exist two minimizing geodesics starting at p to q. More precisely, for every p∈ X we can choose a maximum q of the function (p, ·) and there will be at least two minimal geodesics from p to q. There is a similar result in <cit.>, where it is shown that for every p, there exists q∈ X such that p and q are joined by several minimizing geodesics. If M is a compact Riemannian manifold, then (x, M⊂ W_p(M)) =0 for every p>1 and x∈ M. Also, we can apply our Theorem <ref> to the non simply connected case: If M is a complete Riemannian manifolds with non–trivial fundamental group (i.e. not simply connected), then (x, M⊂ W_p(M)) =0 for every p>1 and x∈ M. Consider the universal cover πM̃→ M. Let x∈ M, and let x̃ be a point with π(x̃) = x. Denote by G the fundamental group of M. We know that G acts on M̃ by isometries and that Gx̃ is a discrete, locally finite set. Then, we may take x̃'∈ Gx̃ at minimal distance from x̃. Then we can take a minimizing geodesic γ̃: [0, ℓ] →M̃ from x̃ to x̃', and the projection γ= π∘γ̃ will be a geodesic loop such that γ(0)= γ(ℓ) = x, and γ is globally minimizing on [0, ℓ/2] and [ℓ/2, ℓ]. Otherwise, by taking a shorter curve to the midpoint γ(ℓ/2) and lifting it we could construct a shorter geodesic from x̃ to another point in Gx̃ and our two points would not be at minimal distance. §.§ Infinite reach For this subsection, we will use results obtained by Kell <cit.>, employing the metric definitions presented in Section <ref>. The combination of these elements yields results that imply infinite reach for certain metric spaces. Let (X,) be a reflexive metric space. Then the following assertions hold: * If X is strictly p–convex for p∈[1,∞) or uniformly ∞–convex if p=∞, then (X⊂ W_r(X))=∞, for r>1. * If X is Busemann, strictly p–convex for some p∈[1,∞] and uniformly q–convex for some q∈[1,∞], then (X⊂ W_r(X))=∞, for r>1. In <cit.>, Kell establishes that any p-convex and reflexive metric space possesses p-barycenters, as he defined them in <cit.>. His Theorem 4.4. establishes the existence of such barycenters but not uniqueness. To establish it, we require the conditions we stated in both cases of our theorem. Now we present how these restrictions give us infinity reach. * Following <cit.>, the spaces (X,) which satisfy the hypotheses in item (1) of the theorem have unique r–barycenters for r>1. In other words, every μ∈ W_r(X) has a unique barycenter. This finishes the proof of the first assertion of the theorem. * Following <cit.>, if (X,) is strictly (resp. uniformly) p–convex for some p, then it is strictly (resp. uniformly) p–convex for all p. Hence, we are in the case (1). As we pointed in Section <ref>, CAT(0)–spaces are a well–known example of metric spaces satisfying some of the hypotheses in Theorem <ref>. In that sense, there is a straightforward corollary to our Theorem <ref> in terms of CAT(0)–spaces: Let (X,) be a reflexive CAT(0)–space, then (X⊂ W_p(X))=∞, for p>1. As Kell stated in <cit.>, CAT(0)-spaces are both Busemann spaces and uniformly p–convex for every p∈[1,∞]. Moreover, from the definition of CAT(k)-spaces, with k=0, we have that (m(x,y),z) ≤_𝔼^n(m(x',y'),z') 1/2(_𝔼^n(x',z')+_𝔼^n(y',z'))=1/2((x,z)+(y,z)). Hence, CAT(0)–spaces are strictly 1–convex and, by <cit.> they are strictly p–convex for all p. The conclusion now follows from item (2) in Theorem <ref>. It is easy to check, from the definition, that CAT(0)–spaces are contractible, and, therefore, simply connected. This is a necessary condition for Theorem <ref>, because if this were not the case, we would have a closed geodesic and Proposition <ref> would give us zero reach for the points inside that geodesic. As particular cases of CAT(0)–spaces, we have Hadamard manifolds (complete, simply connected Riemannian manifolds with non-positive sectional curvature everywhere) and, in particular, Euclidean n–space. So, as a corollary, we obtain the following: * Let (M^n,g) be a Hadamard manifold. Then (M^n⊂ W_p(M^n))=∞, for p>1. * Let 𝔼^n be the Euclidean n–space. Then (𝔼^n⊂ W_p(𝔼^n))=∞, for p>1. Other authors have considered the existence of barycenters in the CAT(κ)–space context, specifically κ=0. In <cit.>, Sturm proved the existence and uniqueness of barycenters for CAT(0)–spaces only for the 2–Wasserstein space. In <cit.>, Yokota stated a condition on CAT(κ)–spaces, with κ>0, to have unique barycenters. This condition is related to the size of the diameter of the CAT(κ)–space, which needs to be small in order to have unique barycenters. §.§ Projection map The infinity of the reach leads to a natural question about the regularity of the projection map, i.e., _p: W_p(X) → X μ ↦ r_μ, where r_μ∈ X denotes the barycenter of the measure μ, that is, the only point in X that minimizes the distance to μ. Let (X,) be a metric space for which Theorem <ref> holds. Then X has infinity reach; in other words, every measure has a unique barycenter and _p is well–defined. Moreover, the fibres of the map are convex. Let μ, ν∈{σ∈ W_p(X) r_σ=a}, λ∈(0,1) and b∈ X. Then W_p^p(λμ+(1-λ)ν,δ_b)=λ W_p^p(μ,δ_b)+(1-λ)W_p^p(ν,δ_b)≥λ W_p^p(μ,δ_a)+(1-λ)W_p^p(ν,δ_a), since μ, ν∈{σ∈ W_p(X) r_σ=a}. A submetry between two metric spaces X, Y, is a map f Y → X such that, for every a∈ Y and every r≥ 0, we have f(B_Y(a, r)) = B_X(f(a), r). For more information about this type of maps, we refer the reader <cit.>. We briefly recall Kuwae's property B, (see Section 4.3 in <cit.> and references therein). Take two geodesics γ, η such that they intersect at an unique point p_0. Assume that for all points z ∈γ [0,1] the minimum of the map t ↦z-η_t is achieved only by the point p_0. Then for every point w ∈η[0,1] the minimum of the map t ↦w-γ_t is achieved only by p_0. Let (X,·) be a reflexive Banach space equipped with a strictly convex norm and satisfying property B. Then proj_2 is a submetry. First let us make a couple observations. From the strict convexity of the norm it follows that between any two points x,y ∈ X there is a unique geodesic joining them, more precisely it is the curve [0,1]∋ t ↦ (1-t)x+ty. In particular this tells us that m(x,y)= 1/2x+1/2y. Let p>1 and x,y,z∈ X then m(x,y)-z^p = 1/2x+1/2y-z^p < 2^p-1(1/2(x-z)^p+1/2(y-z)^p) = 2^p-1(1/2^px-z^p+1/2^py-z^p) = 1/2x-z^p+1/2y-z^p Hence (X,·) is strictly p-convex and so it satisfies the conditions of Theorem <ref>, with this barycenters exist and are unique. Therefore the projection map _2 is well defined. Now notice that m(x,y)-m(y,z) = 1/2(x+z)-1/2(y+z) = 1/2x-y. Which implies that for p>1 m(x,y)-m(y,z)^p < 1/2x-y^p, i.e. it is p-Busemann. Then the 2-Jensen inequality (see Section 4.3 in <cit.>) holds and so in addition we have that by Proposition 4.8 in <cit.> _2 is 1-Lipschitz. Let B_r(μ) be a ball in the Wasserstein space. We just proved that _2(B_r(μ) )⊂ B_r(_2(μ)). Then, it suffices to see that every point in B_r(_2(μ)) is the image of a point (the barycenter of a measure) in B_r(μ). Fix μ, r≥ 0 and let b∈ B_r(_2(μ)). Let T be the translation from _2(μ) to b. Let us show that T_#μ has b as a barycenter. For any a∈ X, W_2^2(T_#μ, δ_T(a)) = ∫_Xx-T(a)^2 d(T_#μ )(x) = ∫_XT(x)-T(a)^2 dμ (x) = ∫_Xx-a^2 dμ(x) = W_2^2(μ, δ_a). Hence, if a=_2(μ), then a minimizes the distance from X to μ, and then T(a)=b minimizes the distance to T_#μ. It remains to see that T_#μ is contained in B_r(μ). Choosing (Id, T)_#μ as a transport plan in Π(μ, T_#μ), W_2^2(μ, T_#μ) = inf_π∈Π(μ, T_#μ)∫_X × Xx-y^2 dπ(x,y) ≤∫_Xx-T(x)^2 dμ(x) = _2(μ)-b^2 < r^2. Therefore, T_#μ∈ B_r(μ). Examples of spaces satisfying the assumptions of Theorem include Hilbert spaces and L^p spaces (see Examples 4.5, and 4.6 in <cit.>). § REACH OF THE ORLICZ–WASSERSTEIN SPACE An introduction to Orlicz–Wasserstein spaces can be found in subsection <ref>. More information about this type of spaces along with a proof of their completeness can be found in <cit.>. §.§ Null reach We start this section with a simple remark. Let φ≡ Id. Observe that ψ∘ is a distance when ψ is a positive concave function with ψ(0)=0. Then W_ϑ is a 1-Wasserstein distance for the metric space (X, ψ∘). Therefore, (x, X⊂ W_ϑ(X))=0 whenever x∈ X is an accumulation point, by Theorem <ref>. We can replicate Proposition <ref> for the case where X is isometrically embedded into an Orlicz–Wasserstein space using a more delicate argument. Let X be a geodesic metric space, and let x, y∈ X be two points with x≠ y. Consider the probability measure μ= λδ_x + (1-λ) δ_y, for 0<λ<1. Then, the following assertions hold: * μ can only minimize its ϑ–Wasserstein distance to X inside a minimizing geodesic between x and y. * If λ is close to one, and there exists a constant c>1 such that φ^-1(t)< t for every t>c, then the minimum will be attained inside the interior of each geodesic. First we will see that the minimum can only be attained inside a geodesic. For that purpose, we will replicate the argument in the proof of Proposition <ref>. That is, given a∈ X, we construct a'∈γ([0, ℓ]), where γ is a minimizing geodesic, with W_ϑ(δ_a, μ) > W_ϑ(δ_a', μ) . Again, it suffices to consider the case (a, y) ≤(x, y). We can pick a' ∈γ([0,ℓ]) such that (a', y) = (a, y). Then, (a', x) < (a, x) or a is also inside a minimizing geodesic. Let S={ t> 0: λφ( 1/t(a, x) ) + (1-λ) φ( 1/t(a, y) ) ≤ 1 }. As we have only one transport plan π = δ_a ⊗μ, we can write W_ϑ(δ_a, μ) = inf S. Thus, it is enough to see that, if t_0 verifies the inequality inside that infimum for a, then it will verify it for a'. Indeed, 1 ≥λφ( 1/t_0(a, x) ) + (1-λ) φ( 1/t_0(a, y) ) = λφ( 1/t_0(a, x) ) + (1-λ) φ( 1/t_0(a', y) ) > λφ( 1/t_0(a', x) ) + (1-λ) φ( 1/t_0(a', y) ). The last inequality comes from the monotonicity of φ, and the assumption (a', x) < (a,x). Observe that, because the previous inequality is strict, we will have a strict inequality in W_ϑ(δ_a, μ) > W_ϑ(δ_a', μ). Now we will prove the second part of our proposition. Assuming λ close to 1, and that φ differs from the identity for big enough values, we will see that there are points a ∈γ((0, ℓ) ) with W_ϑ(δ_a, μ) ≤min{W_ϑ(δ_x, μ),W_ϑ(δ_y, μ) }. First, we observe that the right hand side in inequality <ref> above is easy to compute. Using that φ^-1 is an increasing function, W_ϑ(δ_x, μ) = inf{ t>0 : (1-λ) φ( 1/t(x, y) ) ≤ 1 } =inf{ t>0 : φ( 1/t(x, y) ) ≤1/1-λ} =inf{ t>0 : 1/t≤φ^-1( 1/1-λ)/(x,y)} = inf{ t>0 : (x,y)/φ^-1( 1/1-λ)≤ t } = (x,y)/φ^-1( 1/1-λ). Similarly, W_ϑ(δ_y, μ) =(x,y)φ^-1( 1/λ). If we want λ close to one, we can suppose λ > 1- λ. Therefore, 1/(1-λ) > 1/λ, and because φ^-1 is increasing, φ^-1(1/(1-λ))> φ^-1(1/λ). Thus, we know that t_0 := min{W_ϑ(δ_x, μ),W_ϑ(δ_y, μ) } = (x,y)/φ^-1( 1/1-λ). Now, we will show that we can find a point inside the geodesic a = γ(s), s∈ (0, ℓ) verifying inequality (<ref>). It suffices to see that t_0 ∈ S, because W_ϑ(δ_a, μ) is the infimum of S and by definition will be smaller. First, observe that, by monotonicity of φ^-1 the inequality defining S is equivalent to φ^-1( λφ( 1/t(a, x) ) + (1-λ) φ( 1/t(a, y) ) ) ≤φ^-1 (1) = 1. By concavity of φ^-1, it is enough to have λ1/t(a, x) + (1-λ) 1/t(a, y) ≤ 1. We will evaluate t=t_0 and look for a condition in s so the preceding inequality is verified. Observe that (a, x) = s, (a, y) = ℓ - s and (x,y) = ℓ. Then λ1/t_0(a, x) + (1-λ) 1/t_0(a, y) ≤ 1 λs/ℓ·φ^-1 (1/(1-λ)) + (1-λ) ℓ-s/ℓ·φ^-1 (1/(1-λ)) ≤ 1 λ s/ℓ + ℓ-s/ℓ - λ·ℓ-s/ℓ≤1/φ^-1 (1/(1-λ)) s · ( 2λ -1) ≤ℓ( 1/φ^-1 (1/(1-λ)) + 1-λ) s ≤ℓ( 1/φ^-1 (1/(1-λ)) -(1-λ) ) · (2λ-1) ^-1. If we show that our bound for s is strictly positive, the minimum will be attained inside the geodesic and we will finish the proof. Choosing λ close enough to 1, we have (2λ-1) >0 and 1/(1-λ) > c. Therefore, φ^-1( 1/(1-λ)) - 1/(1-λ) < 0 and, because the function t ↦ 1/t is decreasing, 1φ^-1( 1/(1-λ)) - (1-λ) > 0 and we have finished our proof. An immediate consequence from our proposition is the following theorem, providing us with examples of manifolds with zero reach inside their Orlicz–Wasserstein space: Let X be a geodesic metric space, and x∈ X a point such that there exists another y∈ X with the property that there exists at least two different minimising geodesics from x to y. Suppose X is isometrically embedded into an Orlicz-Wasserstein space W_ϑ(X). Then, for every φ such that φ(t_0) ≠ t_0 for some t_0>1, (x, X⊂ W_ϑ(X))=0. In particular, if there exists a point x∈ X satisfying that property, (X⊂ W_ϑ(X))=0 for every p>1. Also, in compact manifolds and non-simply connected manifolds, (x, X⊂ W_ϑ(X)) = 0 for every x∈ X. The proof is identical to the one from Theorem <ref>. It remains to see that φ(t_0) ≠ t_0 implies the condition we ask for in Proposition <ref>. Indeed, the convexity and φ(1) = 1 imply φ(t) > t for every t>t_0. And, because φ^-1 is increasing, we also have t > φ^-1(t) for every t>t_0, which is what we need to apply Proposition <ref>. §.§ Positive Reach Similarly to the p–Wasserstein case, several results by Kell <cit.> imply that reflexive CAT(0)–spaces inside some Orlicz–Wasserstein spaces have infinite reach. Let (X,) be a reflexive CAT(0)–space. Suppose φ is a convex function which can be expressed as φ(r) = ψ(r^p), where ψ is another convex function and p>1. Then (X⊂ W_ϑ(X))=∞, where ψ≡ and φ(1)=1. As we pointed in the proof of Corollary <ref>, CAT(0) spaces are strictly p-convex, so by <cit.> they are strictly Orlicz φ-convex. Thus, the result is derived directly from <cit.> which confirms the existence of unique barycenters for every μ∈ W_ϑ(X). All proper metric spaces (i. e., those where every bounded closed set is compact) is reflexive <cit.>. As Caprace pointed out in <cit.>, symmetric spaces of non–compact type (i.e. with non-positive sectional curvature and no non-trivial Euclidean factor) and Euclidean buildings are proper CAT(0) spaces and are examples for which Theorem <ref> holds. § REACH OF THE PERSISTENCE DIAGRAM SPACE In <cit.>, Bubenik and Wagner construct an explicit isometric embedding of bounded separable metric spaces into (_∞,w_∞). φ:(X,) →(_∞,w_∞) x ↦{(2c(k-1),2ck+(x,x_k))}_k=1^∞, where c>(X)=sup{(x,y) x,y∈ X} and {x_k}_k=1^∞ is a countable, dense subset of (X,). The authors stated that this embedding can be thought of as a shifted version of the Kuratowski embedding (for more information about this embedding see <cit.>). Let (X,) be a separable, bounded metric space and (_∞,w_∞) the space of persistence diagrams with the bottleneck distance. If x∈ X is an accumulation point, then (x, X⊂_∞)=0. In particular, if X is not discrete, (X⊂_∞)=0. For every two points x, y∈ X, we can construct a persistence diagram P with at least those two points minimizing the bottleneck distance from the diagram P to the embedded space φ(X). That P will be a midpoint between φ(x) and φ(y), so by choosing y arbitrarily close to x, we will have a diagram with several barycenters (x and y) that is also arbitrarily close to x. Therefore, (x,X⊂_∞)=0 for every accumulation point x∈ X, and, thus, (X⊂_∞)=0. Then, it suffices to prove our first claim. For x, y ∈ X, choose the diagram P = {(2c(k-1), 2ck + (x, x_k) + (y, x_k)/2)}_k=1^∞. Now, observe that w_∞(φ(x), P) = sup_k∈ℕ\| (x, x_k) - (y, x_k) + (x, x_k)/2\| =sup_k∈ℕ\|(x, x_k) - (y, x_k)\|/2 = 1/2 w_∞(φ(x), φ(y)) = (x,y)/2. And, by a symmetric argument, w_∞(φ(y), P)=(x,y)/2. Note that, similarly to the end of the proof of <cit.>, any other pairing between points of the diagrams would pair two points from different vertical lines. Those points would be at distance at least 2c. On the other hand, any possibly unpaired points are at distance at least c from the diagonal. So those pairings would have a cost bigger than c>(x,y)/2, and therefore we always pair points in the same vertical lines. Now, if z∈ X, we will see that P is at distance at least 1/2(x,y) from z. Indeed, we can give a lower bound for the distance simply by ommiting the supremum: w_∞(φ(z), P) = sup_k∈ℕ\|(z, x_k) - (x, x_k) + (y, x_k)/2\| ≥\|(z, x_k) - (x, x_k) + (y, x_k)/2\|. Looking at x_k arbitrarily close to z, we get that w_∞(φ(z), P) ≥\| (x, z) + (y, z)/2\| ≥(x,y)/2. This proves that P is not in the image of φ, and that φ(x), φ(y) both minimize the distance from P to φ(X), as we wanted to see. *
http://arxiv.org/abs/2307.01409v1	20230703235923	Winds versus jets: a comparison between black hole feedback modes in simulations of idealized galaxy groups and clusters	[ "Filip Huško", "Cedric G. Lacey", "Joop Schaye", "Folkert S. J. Nobels", "Matthieu Schaller" ]	astro-ph.GA	[ "astro-ph.GA" ]	firstpage–lastpage Sliding suffix trees simplified Laurentius Leonard10000-0001-8477-7033 Shunsuke Inenaga20000-0002-1833-010X Hideo Bannai30000-0002-6856-5185 Takuya Mieno40000-0003-2922-9434 ============================================================================================================================================================== Using the SWIFT simulation code we study different forms of active galactic nuclei (AGN) feedback in idealized galaxy groups and clusters. We first present a physically motivated model of black hole (BH) spin evolution and a numerical implementation of thermal isotropic feedback (representing the effects of energy-driven winds) and collimated kinetic jets that they launch at different accretion rates. We find that kinetic jet feedback is more efficient at quenching star formation in the brightest cluster galaxies (BCGs) than thermal isotropic feedback, while simultaneously yielding cooler cores in the intracluster medium (ICM). A hybrid model with both types of AGN feedback yields moderate star formation rates, while having the coolest cores. We then consider a simplified implementation of AGN feedback by fixing the feedback efficiencies and the jet direction, finding that the same general conclusions hold. We vary the feedback energetics (the kick velocity and the heating temperature), the fixed efficiencies and the type of energy (kinetic versus thermal) in both the isotropic and the jet case. The isotropic case is largely insensitive to these variations. In particular, we highlight that kinetic isotropic feedback (used e.g. in IllustrisTNG) is similar in its effects to its thermal counterpart (used e.g. in EAGLE). On the other hand, jet feedback must be kinetic in order to be efficient at quenching. We also find that it is much more sensitive to the choice of energy per feedback event (the jet velocity), as well as the efficiency. The former indicates that jet velocities need to be carefully chosen in cosmological simulations, while the latter motivates the use of BH spin evolution models. galaxies: jets – galaxies: evolution – galaxies: clusters: intracluster medium § INTRODUCTION Supermassive black holes (BHs), situated in the central regions of their host galaxies, are often observed to be releasing significant amounts of energy into their environment. In this role they are referred to as active galactic nuclei (AGN). Many AGN have been observed through the radiation they release (often in the form of very luminous quasars), from infrared to X-ray frequencies, from very early in the Universe's history all the way to the present day (e.g. ). AGN appear to be affecting their environment on kiloparsec scales through the inflation of lobes of relativistic gas that are visible at radio frequencies, again from the cosmic dawn to today (e.g. ). This injection of energy by AGN, in various forms, is thought to affect their host galaxies and larger-scale environment – a process referred to as AGN feedback. Most importantly, AGN feedback appears to be responsible for the quenching of star formation in massive galaxies and could thus explain their current state as ‘red and dead' (e.g. , , , ). BHs can grow through two processes; accretion (of gas, as well as stars and dark matter, albeit the last two are usually ignored when modeling BH growth) and BH-BH mergers. As a consequence of the accreting gas having net angular momentum, an accretion disc is typically formed around an accreting BH. Depending on the accretion rate, different types of discs can form. The classical <cit.> solution (and its general-relativistic counterpart; ) describes a geometrically thin and optically thick accretion disc in which gas orbits are almost circular (Keplerian at large distances). As the matter slowly funnels downwards towards the BH, it is heated up by viscous stresses. Around 10 per cent of the total mass-energy of the matter in this type of accretion disc is radiated outwards through this process, leading to the observed quasars. An alternative solution found by <cit.> (see for the general-relativistic version) instead describes a geometrically thick and optically thin, advection-dominated accretion flow (ADAF). In this solution, radial gas motions are dominated by advection, and magnetic fields are also thought to be advected inwards (although this is thought to also occur, to a smaller degree, in the thin disc). In combination with the dynamo effect, this leads to the buildup of strong magnetic fields near the BH. These magnetic fields then facilitate the launching of relativistic jets through the <cit.> process, in which energy is extracted from the rotation of the BH. Radiation from AGN is thought to couple to gas, leading to the launching of winds. These winds are thought to be launched mainly due to radiation or thermal pressure (e.g. ). While observations of bright AGN are extremely numerous (e.g. ), and winds appearing to emanate from them have also been frequently observed (e.g. , , ), direct observational evidence of negative feedback on their host galaxies is not conclusive. Observations have found both enhanced and suppressed star formation rates (SFRs) in galaxies hosting AGN (see and references therein). As <cit.> show using simulations (see also ), this may be explained by highly-accreting BHs (visible as bright AGN) being triggered by large amounts of cold gas and star formation. These simulated galaxies appear quenched only once they are devoid of large amounts of cold gas and star formation, which necessarily means that BHs are accreting at lower rates and the AGN are faint by that point. Direct evidence of negative AGN feedback is more easily found in galaxy groups and clusters (see reviews by and , respectively). X-ray observations of the circumgalactic/intracluster medium (CGM/ICM) surrounding the central galaxies of these systems (‘brightest group/cluster galaxies', which we refer to simply as BCGs hereafter) have revealed evidence of AGN feedback in the form of cavities in the X-ray emitting gas (, , , , ). These cavities are often coincident with synchrotron-emitting plasma taking the form of two-sided lobes (, , ). This plasma originates from jets of relativistic particles launched from close to the BHs (, ). The power of these jets, inferred from the power required to inflate the cavities, suggests that they inject sufficient energy to shut off the cooling flows that would otherwise develop in the centres of the CGM/ICM (, , , , ). This feedback mechanism is often referred to as ‘mechanical', ‘maintenance' or ‘radio' mode feedback. We refer to it simply as jet feedback throughout the rest of this paper. Semi-analytical models of galaxy formation typically employ N-body simulations of cosmic structure formation to populate dark matter haloes with galaxies in post-processing (e.g. , , ). Early versions of such models were successful at reproducing the numbers of massive galaxies, but only if AGN feedback is included (e.g. , , ). Hydrodynamical cosmological simulations of galaxy formation and evolution also invariably find that AGN feedback is necessary in order to quench star formation in massive galaxies. Most such simulations have implemented AGN feedback as isotropic heating of gas (thermal isotropic feedback), usually intended to represent the effects of radiatively-driven winds from quasars[While this feedback mode is in principle similar in different simulations, the practical aspects of how it is implemented can lead to significant differences. Most significantly, if the feedback energy is injected in all particles/cells around the BH equally at every time-step (‘thermal dump'), the heated gas can be prone to numerical overcooling, and the feedback is thus not very effective. In contrast, if the feedback energy is held in a reservoir until a sufficient amount of it has been accumulated to heat particles near the BH by some chosen heating temperature Δ T, these problems can be avoided (). The feedback is most effective if only a single gas resolution element receives all of the accumulated energy.]. Examples of such simulations include Magneticum (), EAGLE (), MassiveBlack-II (), Romulus () and ASTRID (), among others. Other simulations have employed somewhat more complicated AGN feedback prescriptions, using different mechanisms of energy injection at low BH accretion rates (alongside thermal isotropic feedback also being used at high accretion rates in all cases). In Illustris (), pairs of thermal bubbles were injected at large distances in haloes. This feedback mode represents the late-time effects of relativistic jets that inflate lobes. However, the inflation process itself (which includes strong shocks that may be critical for heating the ICM) was not included, with the bubbles placed ‘by hand', already inflated. IllustrisTNG () instead uses kinetic isotropic feedback at low accretion rates, representing the effects of winds that may be active alongside the jets (e.g. ). For this feedback channel, the critical accretion rate below which it is used is highly dependent on BH mass, leading to effectively no kinetic feedback for low-mass BHs and little thermal feedback for high-mass ones. The SIMBA simulations () use kinetic jets at low accretion rates (and high BH masses, M_BH≥10^7.5 M_⊙, similar to IllustrisTNG) that are launched in the direction of the angular momentum of the gas surrounding the BH, alongside an additional X-ray feedback mechanism (implemented isotropically, as a mixture of heating and kicking particles), representing the equivalent of the kinetic wind used in IllustrisTNG at low accretion rates. In Horizon-AGN (), a similar prescription is used for the jets as in SIMBA (in that the jets are launched in the direction of the angular momentum of the gas close to the BH). Its successor New-Horizon () uses a more sophisticated prescription, wherein the BH spin is evolved for all BHs using accretion disc models, and the jets are launched along the direction of the BH spin vectors, with spin-dependent efficiencies. In addition to being more realistic, this approach has the benefit (from a numerical perspective) that the BH spin vector is not as prone to perturbations and changes from time-step to time-step, as compared to the gas angular momentum, since the BH spin is a quantity that is integrated over the history of each BH. The radiative efficiency of AGN at high accretion rates also depends on BH spin in this model. In this work we will focus on modifications to the AGN feedback prescription of the EAGLE galaxy formation model (, ), which is based on the <cit.> AGN feedback scheme developed for the OWLS simulations (). The EAGLE simulations used a fairly simple AGN feedback prescription – despite this, the model correctly predicts the number of galaxies as a function of mass (as measured through the stellar mass function or the stellar mass-halo mass relation; ) and redshift () as well as many galaxy properties (e.g. the metallicities and sizes; , molecular gas content; , and colours; ). The Hydrangea simulations used the EAGLE model to evolve a sample of galaxy clusters (). Despite the EAGLE model working well for the overall population of galaxies, these simulations found that BCGs were too massive, from about a factor of two for low-mass clusters (halo masses of order 10^14 M_⊙) to a factor of nearly ten for high-mass clusters (halo masses of order 10^15 M_⊙). The same galaxies were also found to be too highly star-forming compared to observations. This problem possibly originates from overly strong cooling flows in the simulations, which could be a consequence of insufficient heating by thermal isotropic AGN feedback at large radii (e.g. >100 kpc). The C-EAGLE project () also used the EAGLE model to simulate a broadened sample (relative to Hydrangea) of galaxy clusters. Mock X-ray observations () showed that these clusters appear to have central entropies of the ICM that are too high (a problem confirmed by on a separate sample of galaxy groups and clusters, using an updated version of the EAGLE model). This is also true for the temperature, and the reverse is true for the density. A related problem is in the cool-core (CC) versus non-cool-core (NCC) dichotomy of clusters (e.g. ): simulated clusters are likely too often NCC as compared to observed ones (the fraction of CC clusters is too low), although firm conclusions on this are complicated by varying definitions in the literature of what is a CC versus a NCC cluster (). <cit.> studied AGN feedback using an updated version of the EAGLE model in idealized galaxy groups and clusters, and found that thermal isotropic feedback can quench star formation in central galaxies for long times (many Gyr) only in galaxy groups, while in galaxy clusters, the BCGs have recurrent cooling flows. They found that clusters initialized as CC largely remain CC. This indicates that the potential lack of CC clusters, as measured through the central entropy, in realistic, cosmologically simulated samples of clusters (C-EAGLE) may be unrelated to AGN feedback, and could instead be a result of other physical processes in the evolution of these clusters. Alternatively, <cit.> found significant differences in entropy profiles between their clusters and those in the C-EAGLE sample. Their cosmological zoom-in simulations of groups and clusters used a slightly updated EAGLE model with, most significantly, a new hydrodynamics scheme (). They also found substantial differences when turning off artificial conduction in the hydrodynamics solver. These results indicate that the differences between observed and simulated clusters may be partly or wholly due to numerical issues. If the differences between the observed clusters and ones simulated with EAGLE are not entirely due to numerics, including a more realistic feedback mechanism (representing the effects of relativistic jets) may be helpful, presumably by allowing more effective coupling of the feedback energy to larger radii instead of only to the core of the ICM. A similar modification may be beneficial in the IllustrisTNG model (see e.g. the results of the MillenniumTNG simulations, ). This is despite that model using kinetic feedback at low accretion rates (alongside thermal isotropic feedback at high accretion rates), and the reason may be that the feedback mechanism is also isotropic. As we will show in this paper, kinetic isotropic and thermal isotropic feedback are fairly similar in their effects, at least in the context of idealized cluster simulations. The potential problems we have discussed may be present even for the SIMBA simulations (), which also show somewhat too high entropies, albeit at intermediate radii rather than in the core of the ICM (). While SIMBA includes AGN jets, they are launched in the direction of the angular momentum of the gas near the BH, which may not be very stable (especially in clusters and at low resolutions). As we will show in this work, the jet direction needs to be relatively stable for the jets to lead to significant differences compared to isotropic feedback. Performing idealized simulations of AGN jets (on kpc scales) is important in order to further our understanding of the effects of these jets, and their precise mechanisms of action, in a more controlled environment than in cosmological simulations (see review by ). In <cit.> we simulated constant-power jet episodes in an idealized ICM: these simulations were the first smoothed particle hydrodynamics (SPH) simulations of their type, i.e. of idealized episodes of AGN jets. We performed them mainly to validate the numerical method for the jet launching and to confirm that the hydrodynamics of these jets, as well as the lobes they inflate and their interaction with the ambient medium, are correctly simulated with our chosen SPH method (). We found good agreement with theoretical predictions. Surprisingly, jet episodes represented with only ≈500 particles per jet were found to be sufficiently resolved in terms of basic properties (e.g. the sizes of the inflated lobes). In a subsequent paper () we studied the evolution of jet-inflated bubbles in an idealized ICM over relatively long time-scales (≈Gyr). We found that heating of the ICM dominates early on (while the jets are active and shortly afterwards), but AGN feedback is done mostly through gas uplift and the reduction of its central density at late times (once buoyancy starts to act on the bubbles). In both papers we found that the jet velocity parameter[The jet velocity in cosmological simulations, as well as idealized ones such as in this paper, is both a physical and numerical parameter. Typically, real AGN jets are highly relativistic and the lobes they inflate are very light. The mass resolution of the simulations limits the sampling of particles being launched into the jets, and effectively provides a lower limit to the mass of jets and lobes. The spatial resolution of the simulations provides a lower limit to the injection scale on which the AGN jets are launched. For these reasons, jet velocities in these simulations need to be subrelativistic, of order 10^4 km s^-1.] plays a very important role, and thus needs to be carefully chosen. In our latest paper (, hereafter Paper I), we studied self-consistent BH accretion and feedback using BH spin evolution and the EAGLE model in simulations of idealized galaxy groups and clusters. We found that jets were successful at preventing cooling flows and quenching star formation in this setting. Here we will broaden this analysis and consider isotropic feedback as well – the AGN feedback mode used in EAGLE and all other large, cosmological hydrodynamical simulations (at least at high BH accretion rates). Our main goal here is to compare these two feedback modes in terms of their impact on the BCGs, their BHs and the ICM. Some previous works focusing on feedback in idealized galaxy clusters have studied different feedback implementations, with many of them comparing thermal and kinetic feedback (e.g. , , , ). Most of these works employed thermal feedback as a ‘thermal dump' (see footnote <ref>), meaning that it will likely have been prone to numerical overcooling, unlike the kinetic variety. The work we are presenting here builds on previous studies by broadening the comparison between AGN feedback modes to include both realistic feedback (with BH spin evolution) and a more simplified implementation (with fixed efficiencies and jet directions). In addition, we compare these feedback implementations for different halo masses, ranging from the galaxy group (M_200=10^13 M_⊙) to the high-mass galaxy cluster (M_200=10^15 M_⊙) scales. In our simplified feedback scenario, we systematically vary relevant parameters such as the heating temperatures and kick velocities, as well as feedback efficiencies. Furthermore, for both isotropic and jet feedback, we vary the type of energy being injected (thermal versus kinetic). In terms of results, we focus mostly on SFRs and entropy profiles, as stellar masses of BCGs and entropy profiles of the ICM appear to show the largest or most easily observable discrepancies between observed clusters and those simulated with EAGLE. In <ref> we present our BH spin evolution model and the feedback efficiencies used in the simulations. We focus on the thin, radiatively-efficient disc, with the thick, advection-dominated disc having been presented in Paper I. In <ref> we discuss the code and galaxy evolution model that we use, alongside the physical set-up. We also list all of the simulations we have performed and discuss how their parameters were chosen. <ref> contains our results using the BH spin evolution model, whereas <ref> contains the ones using simpler feedback without BH spin dependencies. In <ref> we summarise and conclude. In the Appendices <ref>, <ref>-<ref> and <ref> we discuss, in turn: 1) the role of redirection and precession in the jet case, 2) some additional quantities related to our BH spin evolution simulations and 3) the origin of the periodicity in the cooling flows that will be apparent. § BLACK HOLE SPIN EVOLUTION AND FEEDBACK The dimensionless BH spin, a, is a proxy for the angular momentum of the BH, J_BH, through its definition a=J_BHc/M_BH^2G, where M_BH is the mass of the BH, and c and G the speed of light and Newton's constant, respectively. In order to avoid naked singularities, the BH spin is expected to be no larger than 1 in magnitude (). We actually limit the upper end of this range to 0.998 (see )[The emission of radiation by accreting gas and its swallowing by the BH causes a counteracting torque that acts against spinup of the BH, and which is important for a>0.99. The difference between a maximal BH spin of 0.998 and 1 may seem negligible, but the radiative efficiency of a thin disc is substantially lower for the former (32 per cent) than for the latter (42 per cent); see <ref>.]. The inner accretion disc is expected to be in the equatorial plane due to the effects of <cit.> torques (see <ref>). In this region, the gas may be corotating with the BH (prograde accretion), in which case a>0, or it may be counter-rotating (retrograde accretion), with a<0. For the purpose of evolving the BH spin, we assume that BHs can be in one of two different accretion states depending on the accretion rate (more precisely, the Eddington ratio – see the next subsection): 1) the geometrically thick, advection-dominated disc (i.e. ADAF; advection-dominated accretion flow, ) at low accretion rates and 2) the geometrically thin, radiatively-efficient disc () at high accretion rates. We refer the reader to these papers for details on the properties of the discs. For our purpose, it is most important that the thick disc features a turbulent dynamo effect and strong advection – this includes the advection of magnetic fields, which then build up near the event horizon and lead to the launching of strong jets. The thin disc, on the other hand, releases most of the gravitational binding energy of the gas (as it flows inwards) as radiation, which results in winds that may act as a feedback mechanism on the galaxy scale. In Paper I we assumed the first of these two accretion states to evolve the BH spin and launch jets. However, for simplicity, we made the unrealistic assumption that the equations describing this accretion flow are valid at all accretion rates, and that the jet efficiencies are also high at all accretion rates. Here we will present an accretion model that includes both accretion states, in which the BH is assumed to be in one of the two modes depending on its current accretion rate. In the simulations we may instead assume that one or the other accretion state is active at all accretion rates, in order to compare simulations with simpler BH spin evolution/feedback prescriptions. Below we give a summary of our method and the assumptions we make for the modelling of the thin disc state. We also refer the reader to Paper I, where we describe the BH spin evolution model in more detail. While that model was presented for the thick disc, a very similar one is used for the thin disc, but with some different assumptions for disc structure. We also modify the thick disc model slightly (as compared to Paper I) by updating the model for the spinup/spindown rates in this accretion state. §.§ Deciding the nature of the accretion state The state of the accretion flow is thought to depend on the dimensionless accretion rate (also often referred to as the Eddington ratio), defined as ṁ=Ṁ_BH/Ṁ_Edd, where the Eddington accretion rate is Ṁ_Edd=L_Edd/ϵ_r,0c^2=4πG M_BHm_p/ϵ_r,0σ_Tc. Here, L_Edd is the Eddington luminosity, m_p is the proton mass, σ_T the Thomson cross-section and ϵ_r,0=0.1 is a nominal radiative efficiency used only for the definition of ṁ in this paper (the actual radiative efficiency is allowed to depend on BH spin, see <ref>). According to numerical calculations by <cit.> done soon after the discovery of the the thick disc solution, the thick disc is not always stable and it should transition to being thin once ṁ≳α^2. Here, α is a numerical parameter that is related to the kinematic viscosity ν through ν=α c_sH, where c_s and H are the sound speed and height of the disc at a given radius, respectively. The factor α is used to encapsulate our ignorance of the detailed behaviour and origin of the kinematic viscosity of accretion discs. It is usually taken to be constant with radius, for simplicity, although it very likely varies with radius and possibly with accretion state. More recent and detailed calculations suggest that this picture (of a transition between accretion solutions at ṁ≈α^2) is somewhat too simple (see the review by ). In particular, the properties of the thick disc already begin to change at ṁ=0.2α^2, and the transition appears to be complete by ṁ=0.7α. Between these two values, the disc takes on a transition state whose properties are not well understood. For conceivable values of α, which may be as low as 0.05 based on simulations () and as high as 0.1-0.4 based on observations (), the transition state may occupy the range 0.001-0.3 in ṁ. Observations of both X-ray binary spectra () and AGN spectra () find this transition to occupy a narrower range of ṁ=0.01-0.03. <cit.> analysed the radiative and mechanical powers of AGN and found the transition to span the same range. We assume the lower end of this range to be the critical transition rate at which the two accretion states interchange; ṁ_crit=0.01. Given this choice, we can set a value for the viscosity parameter α, which appears in many of the equations describing accretion disc structure that we will discuss. For this purpose we use the finding of numerical calculations that the transition spans the range between 0.2α^2 and 0.7α in ṁ. We assume that the geometric mean of these two boundaries corresponds to ṁ_crit=0.01, which is true for α≈0.2, so we set α=0.2 for the remainder of this paper. §.§ Feedback efficiencies For the purpose of the simulations presented in this paper, the feedback power P is the end-product of interest of any BH spin evolution model (with the jet direction also being of interest). For this reason, we specify here the feedback efficiencies ϵ used in our two accretion states, before elaborating on how we evolve the BH spin. We define the feedback efficiency (both the radiative and jet efficiency) using the relation P=ϵṀ_BHc^2. Thick discs are thought to have low radiative efficiencies and thin discs low jet efficiencies. We therefore assume, for simplicity and as a first-order approximation, that no jets are launched at high accretion rates (ϵ_j=0 for ṁ>ṁ_crit=0.01) and no radiation is emitted at low accretion rates (ϵ_r=0 for ṁ<ṁ_crit=0.01). Given some assumed radiative and jet efficiencies, the BH grows at the rate Ṁ_BH=(1-ϵ_r-ϵ_j)Ṁ_BH,0, where Ṁ_BH,0 is the rest-frame large-scale accretion rate (before radiative or jet losses). The radiative efficiency of the thin disc, ϵ_r, is taken from the general-relativistic solution presented by <cit.>. It is assumed that the radiative efficiency is related to the binding energy of the gas at the innermost stable circular orbit (ISCO). Within this radius, R_ISCO, orbits are assumed to decay quickly, carrying all of the gas energy into the BH before it can be radiated away. From infinity to the ISCO, a parcel of gas of mass Δ M loses a fraction ϵ_r,ISCO of its total rest-frame mass-energy Δ Mc^2 to radiation, while a fraction e_ISCO = 1 - ϵ_r,ISCO of it is kept. This fraction is the (dimensionless) binding energy. Using the known analytical expression for e as a function of radius from <cit.>, in combination with an analytical expression for the dimensionless radius r_ISCO=R_ISCO/R_G (see e.g. Online Appendix A of Paper I), where R_G=M_BHG/c^2 is the gravitational radius of the BH, we can express the radiative efficiency of the thin disc as ϵ_r(a) = 1-e_ISCO(a)=1-√(1-2/3r_ISCO(a)), This formula yields an efficiency that grows monotonically as the BH spin is increased from a=-1 to a=1, due to the ISCO approaching the event horizon with increasing a. It grows slowly from 4.5 per cent at a=-1 to 15 per cent at a=0.9. Beyond this, the efficiency grows very steeply to reach a value of 42 per cent at a=1 (or 30 per cent at a=0.998, which is our actual cap). For the jet efficiency, ϵ_j, we take the same approach as in Paper I (Section 2.2), to which we refer the reader for more details. Here we provide a condensed version. We assume that the jets are powered by the <cit.> (BZ) process, i.e. they are launched by means of the extraction of energy from the rotational ergosphere of the BH. Whilst analytical expressions for jet powers exist for BZ jets, these rely on assuming classical accretion disc solutions and their magnetic fields (e.g. ). The magnetic fields are highly uncertain in these solutions. We instead use jet efficiency formulas inferred from general-relativistic magnetohydrodynamical (GRMHD) simulations that have converged onto very similar jet powers (e.g. , , , ). These simulations find that magnetic fields are dynamically important in the inner regions of the disc, where they ‘choke' the accretion flow. In this self-regulated and quasi-periodic state (the magnetically arrested disc, i.e. MAD, see ), the magnetic field saturates at some value that depends on the accretion rate and the BH spin. We take the jet efficiency formula presented in <cit.>, which we reproduce in Paper I. The main features of this formula are as follows: 1) at low BH spin, it leads to ϵ_j∝ a^2, in agreement with the classical BZ analysis, while at higher BH spin (a>0.9) the dependence is steeper (ϵ_j∝ a^4 or even ϵ_j∝ a^6), 2) the normalization for the thick disc is overall much higher than in the classical accretion disc solution, to the point that jet spindown becomes very important and (the efficiency may even be larger than 100%, so the BH loses mass as it accretes and launches jets; see Eqn. <ref>), and 3) the efficiency is higher for prograde accretion (a>0) than for retrograde accretion (a<0). §.§ Evolving the magnitude of the black hole spin The evolution of the magnitude of the BH spin can be described by da/dM_BH,0/M_BH=ℓ_in-2a e_in - s_j, where dM_BH,0 is an increment of mass being accreted at large radii (i.e. before radiative or jet losses) and ℓ_in=cL_in/GM_BH is the dimensionless specific angular momentum, where L_in is the specific angular momentum at some inner radius R_in, at which orbits are unstable and at which gas begins to quickly plunge into the BH. The first term in Eqn. (<ref>) is due to gas accretion onto the BH, the second one originates from the definition of the BH spin a through the presence of the BH mass as a factor, while the last term encapsulates spindown from jets. The second term includes the specific binding energy e_in at R_in. For the thin disc, R_in corresponds to the radius of the ISCO. We use an analytical expression for ℓ_in=ℓ_ISCO, which is given in the Online Appendix A of Paper I. The binding energy e_in=e_ISCO can be read off from Eqn. (<ref>). Since we assume that no jets are launched from the thin disc, we also set s_j=0[We should, in principle, add a term representing radiation () in the thin disc regime. This term causes spindown and is relevant for a>0.99. If a>0.998, the spindown from this term is stronger than the spinup from accretion, and vice-versa if a<0.998. For simplicity we neglect this term and instead simply cap the BH spin to a value of 0.998.]. For the thick disc, we replace the entire right-hand side of Eqn. (<ref>) with a fitting formula for the spinup/spindown rates provided by <cit.>, who confirmed the results obtained by <cit.>, and many authors since, on the jet production mechanisms and its dependence on BH spin in GRMHD simulations. Note that this is different from Paper I, where we used a mixture of numerical and analytical expressions that were not motivated by these simulations. The fitting formula used in the present paper is given by da/dM_BH,0/M_BH=0.45 - 12.53a - 7.8a^2 +9.44a^3 + 5.71a^4 -4.03a^5. Since we use the jet powers from GRMHD simulations (see <ref>), using the spinup(down) rates from the same simulations is more consistent. The right-hand-side of Eqn. (<ref>) is positive for a<0.05, leading to spinup, while it is negative for a>0.05, leading to spindown. Thus, a_eq≈0.05 is an equilibrium BH spin value at which accretion and jet launching are exactly balanced in terms of angular momentum flux into/out of the BH (a result recently also confirmed by with even more sophisticated simulations, albeit with a slightly different value a_eq≈0.07). This GRMHD-derived value of the equilibrium spin is significantly lower than a_eq≈0.25, the value we obtained by using our analytical prescription in Paper I. The spindown is so much stronger (for positive spins) when using the results from the GRMHD simulations because of two separate reasons: 1) accretion provides even less angular momentum than is typically assumed from analytical calculations and 2) jets tap energy from the rotation of the BH even more efficiently. §.§ Deciding the sign and direction of the black hole spin Eqn. (<ref>) for the evolution of the BH spin depends only on how much matter is being accreted and the current BH spin, including its sign (direction). The sign of the BH spin encapsulates whether gas accretion is prograde or retrograde relative to the BH spin vector. In the inner accretion disc, the BH's angular momentum always dominates and it becomes either aligned or counter-aligned with the BH's spin vector through <cit.> torques. In the latter case, we consider accretion to be retrograde and the BH spin negative. The thin disc develops a warp due to <cit.> torques and is aligned or counteraligned with the BH within a warp radius R_warp, which is the radius out to which the ‘communication' of the BH and the disc is effective (), in terms of torques. Outside this radius, the accretion disc is undisturbed and aligned with the large-scale accretion flow (see and for a detailed discussion of the structure of the disc in this case). For the thick disc, the assumption of exact (counter-)alignment is invalid. Instead, the disc precesses about the BH spin vector. This precession occurs on very short time-scales, much shorter than the ones we are simulating. For this reason we may also assume (counter-)alignment of the thick disc, in a time-averaged sense. Thus, in our model, the two accretion states are treated equally in this regard (but with different assumptions about the properties and structure of the accretion disc, which affects the size of the aligned or precessing region). The sign of the BH spin (i.e. whether the disc aligns or counteraligns) is decided based on the <cit.> criterion (see Paper I for a detailed discussion). In this prescription, the BH and the inner accretion disc are assumed to come into (counter-)alignment in such a way that the magnitude of the BH spin does not change, and that the total angular momentum (of the BH + inner accretion disc) is conserved. The condition for counteralignment (and for spin to be negative) in this approach can be stated as follows: cosθ<-J_warp/2 J_BH, where cosθ=Ĵ_BH·Ĵ_d is the initial misalignment between the BH and the (large-scale) angular momentum of the disc, whose direction is Ĵ_d. J_warp is the total angular momentum of the inner accretion disc out to R_warp. We describe how both are calculated in <ref>. Eqn. (<ref>) implies that if cosθ>0 (i.e. if the angle θ between the BH spin vector and the angular momentum of the outer accretion disc is smaller than 90), the inner accretion disc is always aligned with the BH spin vector. On the other hand, if cosθ<0 (the BH spin vector and the angular momentum of the outer accretion disc are misaligned by more than 90), the warp angular momentum has to be at most similar in magnitude to J_BH for counteralignment to be possible. If J_warp is much larger than J_BH, counteralignment cannot occur (even in the case of complete misalignment between the BH spin vector and the angular momentum of the outer accretion disc), which can be understood as a consequence of <cit.> torques being incapable of overpowering the large amount of angular momentum in the inner accretion disc compared to that of the BH. From a numerical standpoint, the direction of the BH spin is evolved in the following way. For each increment of mass M_warp consumed by the BH, the BH-inner accretion disc system is assumed to come into equilibrium (with the inner accretion disc aligned or counter-aligned with the BH), so that the direction of the angular momentum of both the BH and the inner accretion disc is parallel with the direction of the total angular momentum J_tot=J_BH+J_warp. Here, 𝐉_𝐰𝐚𝐫𝐩=J_warpĴ_𝐝 is the angular momentum of a single warp increment, which is assumed to be directed along the angular momentum of the outer accretion disc (i.e. the large-scale accretion flow, which we calculate directly from the simulation). For a more detailed description of this process and the motivation for this implementation, we again refer the reader to Paper I. §.§ The structure of the accretion disc In order to calculate the warp angular momentum, J_warp, we have to: 1) know R_warp, 2) assume some accretion disc solution, which yields a surface density profile Σ(r) and 3) assume the specific angular momentum as a function of radius, L(R). For the thick disc, we refer the reader to Paper I for all three of these. For the thin disc, the radius R_warp, which separates the inner and outer accretion disc, can be calculated by equating the Lense-Thirring precession time-scale (t_p=2π/Ω_p, with Ω_p=2GJ_BH/c^2R^3 the precession rate) and the vertical warp propagation time-scale (t_warp=R^2/ν_2, with ν_2 the kinematic viscosity in the vertical direction) (e.g. , , ). The vertical kinematic viscosity ν_2 can be related to the horizontal one, ν_1, by ν_2=ξν_1, with ξ a numerical factor (e.g. ). We use the relation Ṁ=3πν_1 Σ (for R≫ R_ISCO, e.g. ) to calculate ν_1, and therefore ν_2. The warp radius depends on which regime of the thin disc we assume, with each having its own expression for Σ. The <cit.> solution of the thin disc describes three regions: a) an inner one where radiation pressure dominates, which is often unstable and usually does not extend far out, b) a middle one where gas dominates the pressure and electron-electron scattering dominate the opacity and c) an outer one where gas also dominates the pressure, but the opacity is dominated by free-free absorption. We ignore region a) (because the mass and angular momentum associated with that region is relatively small for our purpose) and assume, for simplicity, that the entire accretion disc, at least out to R_warp, can be described by either region b) or c). We have tested both assumptions and they appear to have little effect. However, we keep both choices as options in our model and specify them both here for clarity and completeness. For the remainder of the paper, we assume the disc to be described by region b). In region b), the surface density can be expressed as Σ_TD,b=6.84 × 10^5 g cm^-2α^-4 / 5ṁ^3 / 5(M_BH/10^8 M_⊙)^1 / 8(R/R_S)^-3 / 5, () whereas in region c) Σ_TD,c=3.41 × 10^4 g cm^-2α^-4 / 5ṁ^7/10(M_BH/10^8 M_⊙)^1 / 20(R/R_S)^-3 / 4 (see appendix in ). Here, R_S=2R_G is the Schwarzschild radius. Using these surface densities, the warp radii can be calculated as R_warp,TD,b=3410 R_S a^5 / 8ξ^-5/8α^-1 / 2ṁ^-1 / 4(M_BH/10^8 M_⊙)^1 / 8 for region b) () and R_warp,TD,c=2629R_Sa^4/7ξ^-4/7α^-16/35ṁ^-6/35(M_BH/10^8M_⊙)^4/35, for region c). The latter is equivalent to equation A8 from <cit.> (but with a different definition of ξ; we use ξ=ν_2/ν_1, whereas they use ξ=(ν_2/ν_1)2α^2). The ratio of the vertical and horizontal viscosity, ξ, is a constant parameter, often also expressed in the form α_2/α. Early theoretical calculations predicted α_2=1/2α for small α (), which has also been confirmed by simulations (). Later simulations have found that higher-order corrections to this prediction may need to be included for realistic values of α (e.g. ), such as α=0.2, as assumed in this paper. These numerical results agree with the theoretical prediction by <cit.>, which we assume here: ξ=ν_2/ν_1=α_2/α=2/α^21+7α^2/4+α^2, which reduces to 1/2α^2 for small α. For our assumed value of α (α=0.2) we obtain ξ=15.84 using the full expression, as opposed to ξ=12.5 when using the approximation. We use the former value. We are finally able to calculate the warp angular momentum by using the expression J_warp(R_warp)=2π∫_0^R_warpL(R)Σ(R)RdR, where L(R) is the specific angular momentum at a distance R from the BH. A similar integral (without the L(R) factor) is used to calculate the warp mass M_warp. For the thin disc, we assume Keplerian orbital velocities, i.e. L(R)=√(M_BHGR), and the surface densities are given by equations (<ref>) and (<ref>) for the two cases. Thin accretion discs can extend to large enough distances that they are prone to the effects of self-gravity (see for a review). At large enough radii, the gravity due to the disc locally becomes comparable to that due to the BH. Here we assume that the disc extends out to a radius R_sg where the Toomre instability parameter, Q=Ω c_s /π G Σ, is equal to the critical value of 1. This equation, the Toomre instability criterion, be solved to obtain R_sg,TD,b=6460 R_Sα^28/51ṁ^-18/51(M_BH/10^8 M_⊙)^-49/51 for region b) and R_sg,TD,c=2456 R_Sα^28/45ṁ^-22/45(M_BH/10^8 M_⊙)^-52/45 for region c) (). For Q<1 (i.e. R>R_sg), the disc is prone to local gravitational instabilities and it likely undergoes gravitational collapse/fragmentation and star formation. In the case that R_sg<R_warp, we simply assume that the entire accretion disc is (counter-)aligned and use R_sg instead of R_warp in all equations where R_warp makes an appearance. § SIMULATIONS, METHODS AND SET-UP In this section we will describe the code, subgrid galaxy evolution model and physical set-up used to perform the simulations presented in this paper, the details of which we will also describe here. The simulations of idealized galaxy groups and clusters discussed in this paper are the same in substance as the ones presented in Paper I. For this reason, we will provide only a summary of the methods we use. For an even more detailed description than in Paper I, we refer the reader to <cit.>, where the physical set-up of these idealized galaxy groups and clusters is discussed in great detail. §.§ Numerical code and subgrid physics model In this paper we use the SWIFT[https://swiftsim.comhttps://swiftsim.com] hydrodynamics and gravity code () and its SPH method SPHENIX (). SWIFT includes various subgrid physical processes, including our BH spin evolution model presented in <ref> for the thin, radiatively efficient disc. Additionally, it includes the BH spin evolution model for the thick, advection-dominated disc described in Paper I. In a future paper, such a model will also be presented for the slim, super-Eddington disc (e.g. , ). SWIFT includes a thermal isotropic AGN feedback mode that we use in the thin and slim disc, as well as kinetic AGN jets that we use in the thick and slim disc. We describe these feedback modes in <ref> (the kinetic jet mode is described in more detail in Paper I), alongside other feedback variations that we test. In addition to AGN feedback, we include subgrid physics in the form of radiative gas cooling, an entropy floor and star formation. We do not include stellar feedback (nor stellar enrichment) in order to simplify the interpretation of the results. We use essentially the same subgrid model for these additional processes as in the EAGLE galaxy formation model (). We again refer the reader to that paper, <cit.> or Paper I for details. We use a slightly updated version of the EAGLE model with new cooling tables (). The large-scale accretion rate Ṁ_BH,0 is set equal to the Bondi-Hoyle-Lyttleton rate (), which also differs slightly from the EAGLE model, where that rate was suppressed by an additional factor related to the angular momentum of the gas near the BH. We do not suppress the Bondi-Hoyle-Lyttleton rate for the effects of gas turbulence nor vorticity (unlike in , where the suppression due to both effects was accounted for). We also do not boost it, which would account for unresolved high gas densities that the BH would sometimes be embedded in (). Alongside being used to calculate the accretion rate, we also use the gas in the BH smoothing kernel to calculate the direction of its angular momentum. We then assume that this determines the direction of the angular momentum of the outer regions of the subgrid accretion disc Ĵ_d ( <ref>). This is a strong assumption: the direction of the angular momentum of the gas may change significantly as the gas moves down from the scales we are simulating , ∼100-1000 pc, to the scales of the accretion disc, <1 pc (see section 2.6 of Paper I for a detailed discussion of this assumption). §.§ Implementation of AGN feedback In this paper we compare two different forms of AGN feedback in terms of how it is directed: isotropic and jet feedback (the former of which is done as in , at least for the thermal case). In the isotropic case, energy is imparted to the closest particle in the BH smoothing kernel. This implementation is not precisely isotropic, since isotropic feedback would entail choosing random angles and imparting energy to the particles closest to those chosen angles. <cit.> compared different numerical implementations of kinetic feedback (albeit stellar, but this makes no difference for the following argument), including ‘Min distance' and ‘Isotropic'. In the former, the closest particles to the BH are heated (corresponding to what we do here), while in the latter, particles were chosen in pairs along rays (that do not generally pass through the central star or BH that is injecting the energy) to ensure conservation of not only energy, but also linear and angular momentum. They found the results to be very similar in the two kicking schemes. Throughout the rest of this paper, for simplicity we refer to the scheme we use (‘Min distance' from ) as ‘isotropic', since we use it to represent the effects of isotropic winds, and since it is much different from jet feedback regardless of the details of how it is implemented. In the jet case, energy is always imparted to two particles instead of one, and the same criterion is used to choose the particles as in our isotropic feedback (the ones closest to the BH; see Paper I for other choices and their effects on jet feedback). In order to find a pair of particles to kick in roughly opposite directions, we define two hemispheres within the BH smoothing kernel. The equatorial plane separating them is perpendicular to the vector that defines the launching direction of the jets (the z-axis or the BH spin vector). Several parameters can be tuned to affect the behaviour of these feedback mechanisms. The first of these is the feedback efficiency ϵ, which controls how much feedback energy is injected given some amount of BH accretion. The feedback power is given by P=ϵṀ_Bc^2, where Ṁ_B is the Bondi accretion rate of the BH. We use variable feedback efficiencies in the case where the BH spin and its evolution are used ( <ref>), but also values fixed throughout the duration of a given simulation in a simplified model ( <ref>). The feedback power is funneled to a reservoir of energy. Once the reservoir exceeds some threshold value Δ E, a feedback event occurs (either one particle receiving energy in the isotropic case, or a pair in the jet case). The energy Δ E is imparted in either thermal, kinetic or mixed form (in the latter case, half of the energy is injected as thermal and half as kinetic). Thus, there are three choices to make in both the isotropic and jet case: 1) the feedback efficiency ϵ, 2) the type of energy being received and 3) the energy threshold Δ E. In all of our isotropic cases, we use large enough values of Δ E that the feedback is energy-dominated. We expect no additional radiative cooling (of a physical or numerical nature) in the regions immediately ahead or behind the outflows associated with feedback, as seen for momentum-driven outflows that appear if low velocities are used for kinetic feedback (e.g. ). In the thin, radiatively efficient disc (used at high accretion rates), we use the thermal isotropic variant of AGN feedback to represent the effects of radiation-driven winds[In the thermal isotropic case, we generally refer to the total feedback efficiency ϵ, which is different from the radiative efficiency ϵ_r for the following reason. The BH radiates at a rate ϵ_rṀ_Bc^2, but only a fraction ϵ_f (the coupling efficiency) of that actually couples with the gas in the simulation. The total feedback efficiency is therefore ϵ=ϵ_fϵ_r. This distinction has a small effect in the simulations in that the BH accretes only (1-ϵ_r) of the total accretion rate, rather than a fraction (1-ϵ) of it. We fix ϵ_f=0.1 and vary ϵ_r in our simulations. For the jets we assume ϵ_f=1 and drop the factor hereafter.]. This assumption is valid if the radiatively-driven winds shock and deposit their energy on small scales (e.g. 1-100 pc) that we do not resolve in these simulations, leading to hot gas that expands on account of thermal pressure (e.g. ). For the thick, advection-dominated disc, we use kinetic jets to represent the effects of relativistic jets launched in this accretion regime. In both cases, our BH spin evolution model is used to evolve the radiative and jet feedback efficiencies ( <ref> for the former, and Paper I for the latter), when we allow them to vary. For the jet case, this also results in a variation of the jet direction, which is assumed to be directed along the BH spin axis. In the case that particles are being isotropically heated, we refer to (and vary) the heating temperature Δ T instead of Δ E; the two are related by Δ E = (3m_g/2μm_p)k_BΔ T, where m_g is the gas particle mass, μ=0.62 the mean molecular weight of ionized gas and k_B the Boltzmann constant. In the kinetic jet case, we express the energy being received by the particles through the jet velocity v_j as Δ E=2× m_gv_j^2/2, where the multiplication by two is present since we always kick in pairs. We do not kick particles perfectly along the jet direction, but instead implement a finite half-opening angle of θ_j=10. This is accomplished by assigning a new kick direction every time a kick event occurs; this direction is given by a unit vector 𝐧_j that is drawn randomly and uniformly in solid angle within a cone with a half-opening angle θ_j directed along the chosen jet direction (either aligned with the BH spin vector or the z-axis). Since we always kick in pairs, the above procedure is done for one particle in the ‘positive' direction (along the jet direction) and for another particle in the ‘negative' direction (counteraligned with the jet direction). We kick particles by increasing their velocity by Δ𝐯 = Δ v 𝐧_j. The magnitude of the velocity increase Δ v is chosen in such a way that the kinetic energy of each particle increases exactly by Δ E/2. Conservation of kinetic energy gives 1/2m_g(𝐯_i+Δ𝐯)^2 - 1/2m_g𝐯_i^2 = Δ E/2, where 𝐯_i is the initial velocity. This equation can be solved for the magnitude of the velocity increase Δ v, yielding Δ v = √(v_i,j^2 + v_j^2) - v_i,j, where v_i,j=𝐯_i ·𝐧_j is the initial velocity projected onto the kick direction. This equation implies that the change in the particle velocity is always smaller than the target velocity, i.e. Δ v < v_j, if the initial velocity is non-zero. However, we use fairly large values of v_j that are at least a factor of 10 larger than the initial particle velocities, so in practice Δ v ≈ v_j. §.§ Physical set-up In order to test the different implementations of AGN feedback introduced above, we simulate idealized galaxy groups and clusters. The initial set-up of these systems follows <cit.>. We focus on three halo masses, which correspond to a galaxy group (M_200=10^13 M_⊙), a low-mass galaxy cluster (M_200=10^14 M_⊙) and a high-mass galaxy cluster (M_200=10^15 M_⊙). Here, the halo masses are defined as the masses within the radius R_200, the radius of a sphere within which the mean density is 200 times the critical density at z=0. We use an <cit.> (NFW) gravitational potential to represent the dark matter. The value of the concentration parameter, c_NFW, is chosen for each halo to be in line with the mass-concentration relation found by <cit.>. A <cit.> profile is used to represent the stellar population of the BCG (for which we use live particles), given some total stellar mass M_* and a stellar half-light radius a_* (i.e. the scale length of the profile). Using the NFW and Hernquist potentials, a gaseous halo representing the ICM is generated in such a way that it is in hydrostatic equilibrium, and that the baryonic mass fraction (ratio of enclosed baryonic and total masses) within radius R_500 (defined in a similar way as R_200, but using an overdensity factor of 500) is equal to some value f_b,500. These values are calibrated on the BAHAMAS simulations (). In the central regions of the ICM, the gaseous halo is modified such that its temperature approaches some value T_0, which controls how cool-core (CC) or non-cool-core (NCC) the halo is. At the centre of the halo we place a BH, which is fixed there throughout the simulation. We assume some initial BH mass and spin, the latter of which is directed along the z-axis. All of the above parameters vary with halo mass. For the following parameters we assume values based on general expectations and scaling relations between these quantities and halo masses: halo concentration, baryonic fraction, stellar mass and half-light radius, and BH mass. Our assumed values for each halo mass are listed in Table <ref>. These parameters do not vary in any of our simulations, other than with halo mass, as shown in the table. The initial central temperature of the gas T_0 has a strong impact on the simulations (see for the thermal isotropic feedback case, and for kinetic jet case). For this study we choose relatively low values that lead to significant cooling and feedback on the Gyr time-scales of the simulations we are performing here. In other words, these choices of T_0 correspond to a relatively CC set-up, rather than NCC (the majority of observed groups and clusters do not have appreciable amounts of ongoing cooling or AGN feedback). While this choice may not be representative of the entire population of galaxy clusters, we make it for a few reasons: 1) it leads to more AGN activity, allowing us to compare different AGN feedback schemes more easily, 2) the cooling flows are stronger, so the potential of various AGN feedback schemes to shut them off is tested to a stronger degree, 3) the BH accretion rates are higher, leading to the accretion regime more often corresponding to the thin, radiatively-efficient disc (which we wish to test in detail in this setting). The BH spins we choose are relatively low; in galaxy groups and clusters we do not expect fully spun-up BHs due to spindown from jets and BH-BH merger activity. Other choices also have to be made in setting up the ICM, although they are independent of halo mass (for our study). We assume a constant gas metallicity (as found in at least some observations, e.g. and ) of 0.3Z_⊙ (with the solar metallicity chosen as Z_⊙=0.0134; ). We also assume rotation of the ICM about the z-axis (see for details on how this is set up) with a spin parameter of λ=0.05 (), which is slightly larger than that of the DM (). [table]skip=0pt We assume that the ICM extends out to 3R_200. In the central 500 kpc of the ICM, we use a fixed gas particle mass resolution m_g,0, while outside 500 kpc, the particle masses (in the initial conditions) increase as m_g,0(r/500kpc)^2. This drop in resolution allows us to perform relatively higher-resolution simulations (in terms of how well the central regions of the halo are resolved). The central mass resolution m_g,0 is chosen to depend on the halo mass, since more massive objects require more computational resources to be simulated. The values we have chosen are also given in Table <ref>, alongside the gravitational softening lengths ϵ_g. We run all of our simulations for a duration of 8 Gyr. §.§ Simulations We perform a total of nine simulations using the BH spin evolution model presented in Paper I and <ref>; three for each halo mass. The three for each case use different variations of BH spin evolution and feedback: 1) one simulation using the thin, radiatively efficient disc and thermal isotropic feedback, 2) one using the thick, advection-dominated disc with kinetic jets and 3) one with hybrid accretion and feedback modes, with the thin disc mode used at high accretion rates (ṁ>ṁ_crit) and the thick disc one at low accretion rates (ṁ<ṁ_crit). The details of these simulations are given in Table <ref>. This last model represents the most realistic one and should thus replicate the behaviour of BHs in the real Universe most closely. In this work we use heating temperatures, Δ T, of order 10^9 K as motivated by many previous works (e.g. ). For jet velocities, v_j, we choose values of order 10^4 km s^-1, instead of relativistic ones, mainly due to limitations related to resolution (see footnote <ref>). We increase the heating temperatures and jet velocities with halo mass, in order to sample feedback at a similar level (using the same values would result in the sampling of feedback being significantly better as halo mass is increased, which might thus lead to artificial numerical differences between the three simulated haloes). The increase in jet velocity with halo mass is also motivated by previous simulations we have done (e.g. and ), where we found that jets need to be highly supersonic relative to the external medium (by a factor M=v_j/c_s,ICM≥10) in order to inflate lobes. As the ICM temperature increases with increasing halo mass, this implies that an increase in jet velocity is well-motivated. [table]skip=0pt We also perform simulations with simplified feedback prescriptions. For these we fix the feedback efficiencies to constant values, as well as fixing the jet directions to be along the z-axis. The details of these simulations are given in Table <ref>. We perform these simulations only for the galaxy clusters (M_200=10^14 M_⊙ and M_200=10^15 M_⊙) since these simulations show more interesting (or variable) behaviour than the galaxy group ones. The motivation for these simulations is to provide a comparison of different feedback modes by removing any differences due to variations in the feedback efficiency. To this end we include runs where we fix the efficiency to ϵ=0.01 in both the thermal isotropic and kinetic jet cases. For the kinetic jet case we test two options: 1) using jet velocities that are ≥10 times higher than the sound speed of the ICM and 2) using lower velocities (by a factor ≈3 relative to option 1) that, however, lead to the energy per feedback event Δ E being the same as in the equivalent thermal isotropic simulations. We consider option 1) our fiducial choice, for the reasons laid out in the paragraph above. For the low-mass galaxy cluster (M_200=10^14 M_⊙) case, we also perform a series of simulations whose parameters are specified in the last two rows Table <ref>. The purpose of these simulations is to vary all parameters of interest: the feedback efficiency, the energy per feedback event and the type of energy being injected. These variations were done for both the isotropic and jet cases. For the jet case, we also tested the importance of the jet direction by manually redirecting the jets in random directions with a given periodicity, and also by precessing them with varying opening angles and periods. These simulations, and their results, are discussed in detail in Appendix <ref>. We found the jet direction to be largely unimportant for the type of simulations being performed here. [table]skip=0pt §.§ Observational sample of entropy profiles In this work, we mainly focus on the gas entropy when discussing the impact of feedback on the ICM. For this purpose we define the entropy as K=k_BT/n_e^2/3, where k_B is the Boltzmann constant and T and n_e are the gas temperature and electron number density, respectively. We will compare our simulated entropy profiles of the ICM (as a function of radius) to observed ones inferred from X-ray observations. For high-mass clusters there are plentiful such samples due to the hot ICM gas falling well into the range observable by X-ray observatories such as Chandra, and since these clusters are easier to observe due to a larger intrinsic brightness. We compare the simulated high-mass galaxy cluster (M_200=10^15 M_⊙) with the observed ones from <cit.>, who studied 31 nearby clusters using XMM-Newton, as well as those from <cit.> using the same telescope, but with a different sample of 12 galaxy clusters. We also compare with Chandra observations by <cit.>, who provide entropy profiles for a large sample of 239 high-mass galaxy clusters (M_500≈10^15 M_⊙, where M_500 is the halo mass using a virial overdensity of factor 500 relative to the critical density). They also split their sample into CC and NCC clusters based on whether the central entropy is below or above 50 keVcm^2. For galaxy groups and low-mass clusters (M_500≤10^14 M_⊙), such observations are inherently difficult (e.g. ,, , ). The sample sizes tend to be small and/or they span a large range in halo mass. The halo masses of these galaxies cannot currently be measured through X-ray observations, since their ICM/CGM may not be in hydrostatic equilibrium, nor is the X-ray emission typically measured up to the virial radius (or an appreciable fraction of it so that one may extrapolate the pressure profile). The samples may also be biased towards CC (low-entropy) ones since such X-ray atmospheres are more likely to be bright and therefore observed. Finally, it is also likely that many of these observed X-ray atmospheres surround satellite galaxies rather than being the central ones of primary haloes. Tidal stripping may be affecting many such galaxies, or it may also be biasing the samples towards the X-ray bright ones. Notwithstanding those currently unavoidable shortcomings, we compare the entropy profiles of our galaxy group (M_200=10^13 M_⊙) and low-mass cluster (M_200=10^14 M_⊙) simulations with a set of different observational papers. We use the data based on 28 and 43 observed galaxy groups and clusters by <cit.> (using XMM-Newton) and <cit.> (using Chandra), which provide useful constraints on the entropy profiles between roughly r=30 kpc and r=1 Mpc. At relatively small radii (r<100 kpc) we compare with data from <cit.>, who compiled observed profiles of 40 galaxies/groups and 110 galaxy clusters, all observed with Chandra. For all these systems, <cit.> find a universal median entropy profile, which they fit with K∝ r^2/3 at small radii and K∝ r^1.1 at large ones. Finally, we compare with <cit.>, who presented entropy profiles of 49 bright elliptical galaxies observed with Chandra. These data are largely consistent with the <cit.> ones, although they tend to follow a single slope with radius. § RESULTS I: FEEDBACK WITH BLACK HOLE SPIN EVOLUTION We first consider the results of using the BH spin evolution model for all three of the halo masses, from the galaxy group (M_200=10^13 M_⊙) to the high-mass cluster (M_200=10^15 M_⊙) scale. For each of the halo masses, we performed three simulations: 1) using the thin, radiatively-efficient disc and thermal isotropic feedback, 2) using the thick, advection-dominated disc and kinetic jet feedback and 3) a hybrid case where the two accretion and feedback modes interchange at ṁ=ṁ_crit=0.01. The details of these simulations are given in <ref> and Table <ref> (in terms of physical set-up and halo mass) as well as <ref> and Table <ref> (in terms of feedback implementation). In Fig. <ref> we show visualizations of the gas temperature in our hybrid simulation of the high-mass cluster. These show the qualitative behaviour of the feedback and cooling cycle, which we consider to be representative of all our simulations. These visualizations highlight the rich variety of structures we find in our simulations, with many of them similar to features observed in the ICM of real galaxy clusters. The bulk of the ICM on the spatial scales shown in Fig. <ref> has a temperature of order T≈10^7.5-10^8 K (light-blue to dark-purple colours), varying with radius. Black colours indicate gas that is slightly hotter (mostly due to shock waves), while orange-to-white colours indicate gas that is a factor of several times hotter than the ambient medium (the gas launched as part of feedback or entrained in the same process). These visualizations also show gas that is strongly cooling (white colours). The two left-hand panels show simulation times when the kinetic jet activity is peaking, while the two right-hand panels show the same for thermal isotropic feedback. From the two left-hand panels, we see that jet feedback can lead to asymmetrical large-scale outflows, as a result of several processes, some of which are: 1) jet redirection and/or precession, 2) variability in the jet power and 3) the complex structure of the ICM in the jets' path (including uplifted low-entropy gas due to previous feedback episodes; we discuss this below). From the right-hand panels, we see that thermal isotropic feedback generally does not lead to isotropic outflows. This is partly a result of how it is implemented: gas is heated to large temperatures (Δ T=10^9.5 K in this case). This hot gas tends to not expand isotropically, but rather in the 'path of least resistance' away from the BCG. The first few heating events in a given feedback episode create a channel that represents the preferred direction in which the subsequently heated gas will expand. For both thermal isotropic and kinetic jet feedback, we see that the typical temperature of the hot gas outflows and bubbles is not similar to the temperature associated with the launching events (Δ T=10^9.5 K and Δ T_j≈10^10 K[This temperature represents the typical temperature of hot gas making up the jet-inflated lobes if one assumes that all of the kinetic energy of a single jet kicking event, with a velocity of v_j=3×10^4 km s^-1 in this case, is transformed to thermal energy through shocks, as well as that none of it is transferred to the ambient medium through the shocks, and that no ambient ICM is entrained. This typical temperature, obtained through (3/2)k_BΔ T_j=(1/2)μ m_pv_j^2, is expected to be an overestimate for the aforementioned reasons, but it is a useful order-of-magnitude estimate, especially when comparing to thermal feedback.], respectively). It is instead a factor of 10 or so lower in temperature, which is likely on account of several processes, including the transferal of energy from the outflows to the ambient medium (through shocks or other processes), as well as adiabatic expansion and entrainment of ambient gas. From the bottom two panels we see that both kinetic jets and thermal isotropic feedback lead to the generation of roughly spherical shock waves, which is one of the ways in which AGN feedback can heat the ambient medium (e.g. , see also review by ). From all four panels we see that the ICM has a generally very complex structure, with actively cooling gas draping and trailing the outflows and bubbles associated with feedback (to distances as large as 300 kpc in this case). Particularly noticeable are filamentary structures that arise from the feedback-induced uplift of the low-entropy ICM from the core of the ICM to larger radii (see discussion in Paper I and for a detailed study of AGN feedback-induced uplift). The process of gas uplifting is one of the ways feedback is done, by reducing the central gas density and therefore delaying radiative cooling. However, the uplifted gas rises to some radius where the thermal pressure is lower, so its thermal pressure also reduces. The gas cools adiabatically and it thus may be more prone to further radiative cooling. It is thus possible that a positive AGN feedback loop exists, at least to some degree (not necessarily dominant over the negative feedback), in the systems we are simulating. §.§ Feedback powers We begin our quantitative comparison of the different simulations with BH spin evolution by considering the variation of feedback powers with time. In Paper I we showed the power to be high when the central regions of these simulated haloes are strongly cooling, i.e. undergoing a cooling flow that leads to significant amounts of cool gas (which we define to be T<2×10^4 K for the purposes of this paper) and a non-zero SFR (see for a discussion of the small delay between cooling and feedback). On the other hand, if the central regions of these haloes have been sufficiently heated by feedback, or if gas has been transported outwards through feedback-induced uplift, the feedback powers are low since the BHs are accreting directly from the hot halo, rather than the cold gas. As a result, the feedback power serves as a good tracer of the overall behaviour of the cooling and feedback cycle of these haloes. In the top two rows of Fig. <ref> we show the feedback power as a function of time in simulations with different feedback prescriptions, for all three of our studied halo masses. The top two left-hand panels show the feedback powers for the galaxy group. In all cases there is an initial feedback episode, after which the feedback power settles down to much lower, roughly constant values for the rest of the simulations. This constant value is around 5 times lower for the thermal isotropic case (bottom panel) than the kinetic jet case. The difference can be explained by considering the feedback efficiencies in these simulations, which are set by the spins of the BHs (see <ref> and Fig. <ref> for a detailed discussion of the evolution of the BHs in these simulations). In the jet case, the feedback efficiency is ϵ_j≈0.025, whereas the radiative efficiency in the thermal case is ϵ_r≈0.06. The thermal isotropic feedback power is 10 times lower than that due to a coupling efficiency factor of ϵ_f=0.1, so the total thermal isotropic efficiency is ϵ=ϵ_fϵ_r≈0.006. This value is around 5 times lower than the jet one, ϵ_j≈0.025, leading to a 5 times lower feedback power at late times. The thermal isotropic power is also (on average) lower at all except very early times (t>100 Myr). This indicates that these haloes go through a very similar thermodynamic state as the feedback is in the process of quenching them. In other words, the system is not self-regulated. Instead, any feedback mechanism is sufficient to quench the cooling flow in the centre very quickly, and any residual feedback is merely an ‘after-effect'. While this is not easily visible from the plot, the thermal power is higher than the jet power at very early times. This could be either due to the lower feedback efficiency in that case or due to thermal isotropic feedback generally being less effective at quenching cooling flows than kinetic jets even with the same efficiency (as we show in <ref>), so a stronger initial cooling flow develops. As a consequence, there is more cold gas in the centre of the halo (visible in the third row) at these times, feeding the BH more strongly. The feedback power is also more variable in that case as compared to the jet case. This difference is a result of isotropic feedback regularly blowing away clumps of cool gas from the centre of the halo, which eventually fall back and periodically feed the BH. In the same two panels (top two left-hand ones) we show the feedback powers in a simulation with hybrid feedback and interchanging accretion modes. We find that there is only a small amount of thermal feedback in the very beginning in this simulation, with jets dominating at all other times (because the accretion rate in terms of the Eddington ratio is generally ṁ<ṁ_crit=0.01; see Fig. <ref>). The jet power in this case is very similar to the jet-only case, although it appears to be more variable, possibly as a result of more cold gas being present (see third row). We now turn to the more massive, galaxy cluster-size haloes (M_200≥10^14 M_⊙), which quickly become self-regulated. We begin with the low-mass cluster (M_200=10^14 M_⊙), the results for which are shown in the top two middle panels of Fig. <ref>. In all cases we see multiple cycles of cooling and feedback. The peaks in the feedback powers do not occur at the same times for the different simulations, for two reasons: 1) the feedback implementation is inherently different and 2) these simulations are chaotic (see Appendix A of ). The feedback powers averaged over the 8 Gyr simulation run times, shown with arrows on the plot, are P≈10^44 erg s^-1 in both the kinetic-only and thermal-only cases (slightly above that value for the former and slightly below it for the latter). The hybrid case again shows very little thermal feedback, except some activity at t=2-3 Gyr. The mean jet power in this simulation is, however, roughly a factor of two lower than in the kinetic-only one. Despite the overall similarity in the mean feedback powers between these three simulations, there are differences in their variability. The kinetic-only case has 3-7 distinct episodes of feedback (depending on how one counts them) with some activity at all times except at the end of the simulation. The thermal-only one has 3-4 episodes (depending on whether the first bout of activity, between t=0 Gyr and t=2 Gyr, is considered as one or two episodes) with very clear quiescent periods at t≈4 Gyr and t≈7 Gyr. This difference is likely a result of jet feedback being able to react more quickly to the formation of a cooling flow, possibly due to the higher feedback efficiency (see <ref> and Fig. <ref>), which allows a cooling flow to be shut off before it becomes overly strong. In the hybrid case, jet feedback appears yet more variable. Instead of multiple coherent episodes being discernible in the variability of the jet power, we see relatively frequent variations around a jet power of P≈3×10^43 erg s^-1. This difference is likely caused by the higher jet efficiency in this case, since in the jet-only mode the BH can be and does become spun down to very low BH spins (see <ref>, <ref> and Fig. <ref>). We now move to our most massive galaxy cluster, with M_200=10^15 M_⊙, the results for which are shown in the top two right-hand panels of Fig. <ref>. Similar to the low-mass cluster, the feedback powers show multiple cycles of activity, with the thermal-only case this time showing significant variability, while the jet-only case has a few distinct episodes of activity. From the hybrid case we see that that thermal feedback is often active. While it may appear that thermal isotropic and kinetic jet feedback are often active at the same time, this is merely a consequence of the the feedback modes interchanging more frequently than our sampling of the feedback powers (which are in this case plotted using adaptive bin widths containing 10 feedback events), as well as other quantities (e.g. Eddington ratios shown in Fig. <ref>) Comparing the jet powers in the jet-only and hybrid case, we find that they are overall similar (even in the positions of the peaks), but there is a difference towards the end of the simulations. The jet-only one has a jet power that increases towards P_j≈10^46 erg s^-1 by the end – this is a result of jet-induced spindown leading to a very low BH spin and therefore low jet efficiency, which in turn leads to a very high (unrealistically so) BH mass (see <ref> and Fig. <ref>). With such a high mass, the BH is able to launch strong jets by accreting from the hot gas halo, leaving the system fully quenched (see bottom panels). From the thermal power, we see that thermal feedback is active more often in this case than in the low-mass galaxy cluster. This is a result of the massive galaxy cluster having significant amounts of gas cooling and star formation, which is connected to the accretion rate of the BH, which is also higher (similar as in Paper I; see bottom panels and Fig. <ref>). We can also compare the mean feedback powers in all of the simulations we have discussed thus far (see arrows in Fig. <ref>). We find that the kinetic jet power is higher than the thermal power in all cases. We interpret this as a result of a larger fraction of the energy related to jet feedback reaching larger radii (regions that do not ‘need' to be heated, since they already have long cooling times) than in the thermal isotropic case, which generally has more central heating. A larger fraction of feedback energy coupling to larger radii thus leads to overall more energy needing to be injected to shut off the central cooling flows. §.§ Impact of feedback on galaxy growth We will now discuss quantities related to the BCGs and their growth in our simulations with BH spin evolution, which are shown in the bottom two panels of Fig. <ref>. These are the cold gas masses (defined as the total masses of all gas with T<2× 10^4 K) and SFRs. We consider galaxies as quenched if their specific SFR (sSFR), i.e. the SFR divided by M_, is below 0.01 Gyr^-1 (e.g. ). We find that our results are largely insensitive to this exact choice. We again begin with the left-hand panels, showing the results for the lowest-mass simulations (M_200=10^13 M_⊙). In the kinetic-only case, there is barely any cold gas and star formation, and then only at the very beginning (hardly discernible in the plot). The thermal-only and hybrid cases show a similar amount of cold gas (M_cold≈3×10^7 M_⊙) and star formation (SFR=0.1 M_⊙ yr^-1) at the peak, although the hybrid case more quickly reaches a state of no cold gas being present and therefore no star formation. In all three cases, the system is considered quenched at all times. We now move to the low-mass galaxy cluster case with M_200=10^14 M_⊙ (middle panels). The cold gas and SFR exhibit multiple episodes that generally coincide with the peaks in the feedback powers (see top rows). The cold gas mass has peak values close to M_cold=10^9 M_⊙, with the peaks being slightly lower in the cases with jet feedback (see also the mean values, indicated on the plot with arrows). The SFR peaks at ≈10 M_⊙ yr^-1, which is sufficient to consider the galaxies non-quenched at these rare times. The kinetic jet case exhibits very little cold gas or star formation at early times (before t=3.5 Gyr). This indicates that hot halo accretion is sufficient to keep the halo quenched with this feedback mode for quite a long time. By t=3.5 Gyr, a strong cooling flow develops, and it lasts ≈2 Gyr. During this time, the BH experiences a significant amount of growth. Since it was spun down to a very low value of the BH spin even earlier (see <ref> and Fig. <ref>), it means that the BH cannot quickly react to the development of a cooling flow. As a result, a strong cooling flow develops, to the degree that it results in feedback strong enough to heat the ICM at large radii, thus preventing any cooling flows from occurring in at least the next 2.5 Gyr (until the end of the simulation). The thermal isotropic case has the largest amounts of cold gas and star formation, and its first cooling flow develops in the very beginning of the simulation (whereas jet feedback, in both the jet-only and hybrid simulations, is able to delay the initial cooling flow). The hybrid case has a moderate amount of cool gas and star formation. The shape of each peak is similar to the thermal-only case. Whereas the jet-only case has sharp declines in the cold gas mass and SFR after every peak, these two cases have gradual declines that can last up to 2 Gyr. We interpret this as possibly being due to thermal feedback blowing away clumps of cold gas, which thus take a longer time to be consumed through SF, and in the meantime they are not feeding the BH and producing feedback. Finally, we discuss the massive galaxy cluster case (M_200=10^15 M_⊙). The cold gas mass reaches peaks of up to 10^10 M_⊙ in all three cases, with the SFR reaching several hundred M_⊙ yr^-1. The hybrid case has only a mildly lower mean cold gas mass and SFR than the thermal-only case, since the operating feedback mode is quite often thermal (Fig. <ref>). The jet-only simulation has an appreciably lower cold gas masses and SFR, and is fully quenched at around t=4.5 Gyr. In Appendix <ref> we discuss whether BH growth and feedback interferes with star formation directly or indirectly. We probe this by considering the ratio M_i/M_,formed as a function of time, where M_i is the total mass accreted, launched into the jets or heated by the BH, and M_,formed is the total mass of all stars formed. We find that this ratio is often comparable or larger than unity, suggesting that BH growth and feedback do indeed directly interfere with star formation in our simulations, by depriving it of its fuel (cold gas) through direct processes (algorithmically choosing it to be heated or kicked), rather than, for example, through entrainment. The implications of this finding for realistic, cosmological simulations may not be problematic for BH accretion, as long as we assume that the amount of BH growth is realistic in these simulations. However, this finding for the mass flux of particles associated with feedback may be more problematic, especially since these fluxes are also typically higher than those associated with BH accretion. The rate at which the BH is heating or kicking gas particles depends not only on the feedback powers, but also on the heating temperature Δ T and jet velocity v_j. Both of these parameters are at least partially numerical in nature. Decreasing their values (at a fixed feedback power) increases the mass flux of particles being heated/kicked. If too low values are chosen, the mass flux of particles associated with feedback may be close in magnitude to the SFR, which we sometimes find to be the case in our simulations. One would ideally want to avoid this situation, and ensure that the mass flux of particles being heated/kicked is always much smaller than the rate at which the gas is being converted into stars. In practice, this limit may be hard to avoid, at least at low resolutions, since decreasing the mass flux of the particles being heated/kicked also decreases how well sampled the feedback is, which then means that feedback is resolved more poorly. We do not propose a particular solution here, but merely point out that the mass flux in question is probably quite large (close to the SFR) in most implementations of AGN feedback in cosmological simulations. §.§ Evolution of black hole properties In Fig. <ref> we show the evolution of various BH-related properties, including from top to bottom the BH mass, Eddington-normalised mass accretion rate, the BH spin magnitude, the angle between the BH spin vector and the z-axis, and finally the jet and radiative efficiencies. We discuss each of these quantities in turn. The BH mass (first row of Fig. <ref>) remains unchanged in the galaxy group case, whereas in the galaxy cluster cases, there is always some appreciable growth. The low-mass cluster exhibits BH growth by more than a factor of ten in the kinetic-only case, partly as a result of low efficiencies (due to jet-induced spindown). The thermal-only and hybrid cases show much less growth – about a factor of two for both cases. The results are similar for the high-mass galaxy cluster, but these simulations show even more growth. The kinetic-only one shows BH growth beyond M_BH=10^11 M_⊙ by t=2.5 Gyr (the final BH mass in this simulation is ≈5×10^11 M_⊙, which highlights the unrealistic nature of using the jet mode in isolation, at least with the strong jet spindown rates we have assumed). The other two cases both show growth by a factor of 5-10, with the hybrid one, interestingly, showing the least amount of growth. The Eddington-normalized accretion rates (second row of Fig. <ref>) peak near ṁ_crit=0.01 for all three galaxy group simulations. They also all reach ṁ=10^-5 by the end, although the thermal-only case takes the longest time to reach that value. The two higher-mass cases show much more variability in the accretion rate, with it often peaking above ṁ_crit (which is why the hybrid cases have the feedback modes often interchanging). Interestingly, the hybrid simulations do not feature high values of the accretion rate (e.g. ṁ=0.1 or approaching ṁ=1) as often as the two other simulations. From the evolution of the Eddington ratios, it is clear that BHs sometimes have an accretion rate high enough for the accretion mode to correspond to the thin disc, instead of the thick disc, and feedback to be thermal isotropic (ṁ>ṁ_crit=0.01), at least in the galaxy cluster cases (M_200≥10^14 M_⊙). However, it is not clear how much growth actually occurs in which accretion regime. While the Eddington ratio appears to be ṁ<0.01 most of the time for all 9 simulations shown in Fig. <ref>, it is possible for most of the growth to occur at ṁ>0.01 due to the accretion rates being higher. In Appendix <ref> we discuss the cumulative mass fractions of growth that occur at low versus high Eddington ratios. We find that neither regime is negligible in terms of growth. Perhaps surprisingly, we find that most growth occurs when ṁ>0.01 in the galaxy cluster cases, despite the accretion rate satisfying ṁ<0.01 most of the time. This implies that radiatively-efficient accretion and its associated ‘quasar feedback' should not be ignored for galaxy clusters, despite its rarity. This finding is likely a consequence of cooling flows becoming progressively stronger for more massive clusters. The picture of ‘maintenance-mode' feedback (by relativistic jets) that keeps BCGs quenched is thus probably an oversimplification for relatively CC clusters, such as the ones we are simulating here. This is in agreement with the analysis of a wide range of observations done by <cit.>, who found that the systems with the largest cooling flows (and star formation rates) tend to have the highest star formation efficiencies, which could be explained by the central BH more often being in the radiative versus mechanical feedback mode (the former of which is less efficient as a feedback mechanism, as we have found already in this section, and which we also confirm more robustly in <ref>). The BH spin (third row of Fig. <ref>) exhibits very little evolution in the galaxy group case (except a small amount of spindown at the very beginning), which is a direct result of little-to-no BH mass growth. In the other two cases there is significant BH spin evolution. The low-mass cluster shows spindown in the kinetic-only simulation (down to values a<0.05, as a result of using a GRMHD spindown formula, see <ref>), as well as in the hybrid one, where larger values of the BH spin are reached (although still very low ones, a≈0.05-0.1). The thermal-only case instead shows occasional spin-up to values around a=0.4. In the massive galaxy cluster case, all three simulations have a median BH spin that is below the initial value (a_0=0.4). The kinetic-only one behaves similar to the low-mass cluster case, although the mean BH spin is even lower. The hybrid one has the BH spin varying between a=0 and a=0.3 – higher values are achieved than in the low-mass cluster case due to more spinup, as the BH spends more time in the thin disc regime due to high accretion rates. The thermal-only case, on the other hand, shows a lower mean BH spin than in the low-mass cluster simulation. This could be due to the cold gas being more chaotic in terms of its angular momentum (or due to the high-mass simulation having poorer resolution), which would lead to more frequent retrograde accretion of the BH, and therefore more frequent spindown. The angle between the BH spin vector and the z-axis (fourth row of Fig. <ref>) contains information on how much redirection the BH spin vector has experienced. In the galaxy group case, there is some redirection that occurs in the very beginning of both the thermal-only and hybrid simulations. In the two higher-mass cases, there is much more redirection – these plots show how chaotic the angular momentum of the accreting gas is in these simulations. These results are in qualitative agreement with the ‘chaotic cold accretion' (CCA) scenario presented by <cit.>. This is despite the fact that we use Bondi accretion, which is often portrayed as being mutually exclusive with CCA. We argue instead that a version of CCA naturally emerges if cold gas is included in the Bondi formula (instead of restricting it to hot, X-ray emitting gas). This mixed approach can reproduce the main features of CCA, including the chaotic nature of the cold gas that is accreting onto the BH and the boosting of the BH accretion rate (relative to the Bondi rate inferred from hot gas). In the final two rows of Fig. <ref> we show the feedback efficiencies in these simulations. These are calculated as moving averages over 5-Myr wide bins, but we only include times when the BH is in the appropriate accretion state (the thick disc for the jet efficiency and the thin disc for the radiative efficiency), and they are also weighted by the accretion rate at every time-step. In the galaxy group case, the efficiencies show some variability – this can occur despite the magnitude of the BH spin being constant because the feedback efficiencies also depend on the sign of the BH spin, which itself depends on the angular momentum direction of the gas in the BH smoothing kernel. The jet efficiency quickly drops to per cent-level values for the kinetic-only case in both galaxy cluster simulations. In the hybrid cases, the jet efficiencies depend highly on the current BH spin; in the low-mass case it is below 2 per cent, while in the high-mass case it sometimes grows to several per cent. The radiative efficiency in the thermal-only simulations is in all cases between 4 and 8 per cent. This lack of strong variability is a result of the radiative efficiency being weakly dependent on BH spin, except at a>0.9. §.§ Entropy profiles We now turn to the entropy profiles in these simulations, which are shown in Fig. <ref>. These profiles are compared to the observed ones, which are described in <ref>. We do not expect the simulated profiles to agree perfectly with the observed ones, for several reasons. Firstly, our simulations represent only single realizations in terms of how CC they are, i.e. we could vary the initial central ICM temperature parameter T_0 to obtain different profiles. We chose low values (relatively CC systems) for reasons laid out in <ref>. Secondly, real clusters undergo merging activity, which is not included in our idealized simulations. Thirdly, observed profiles are deprojected given some assumed model. Fourthly, the profiles are not volume-weighted, but X-ray emission-weighted. Finally, for the galaxy group and low-mass cluster simulations, the observed profiles we are comparing these simulations with span the mass range M_200=10^13-10^14 M_⊙, i.e. they are centred on a median halo mass of M_200≈10^13.5 M_⊙. Additional shortcomings of these observations (in the context of applying them here for purposes of comparison) are discussed in <ref>. For these reasons, care should be taken when comparing the galaxy group (M_200≈10^13 M_⊙) and low-mass cluster (M_200≈10^14 M_⊙) simulations with these observations (the observed profiles shown in the left-hand and middle panels of Fig. <ref> are the same). Furthermore, this sample of observed galaxies does not only include centrals, but also satellites. Given these considerations, we use the observed profiles as a baseline to compare the shapes of the profiles (and their differences between models), rather than seeking full agreement. Before discussing the cases individually (and comparing with the observed profiles), we first comment on some common features seen in all three cases. From Fig. <ref> we see that the hybrid simulations have the lowest central entropy, even lower than the kinetic-only simulations. This is potentially caused by a combination of two effects whose impacts on the central entropy are opposite. Firstly, jets are able to heat the halo at larger distances than thermal isotropic feedback, and they deposit less of their energy in the very central regions (see also Fig. <ref>). This means that the cores in simulations with jets should be cooler. Secondly, the haloes are more effectively quenched by jets than by thermal isotropic feedback, which means that the central ICM undergoes strong cooling flows less frequently, or they are weaker and/or shorter-lived. This in turn means that the central entropy should, on average, be higher if jets are used. When comparing our kinetic-only and thermal-only simulations, it appears that the first of these two effects dominates, at least for the two lower-mass cases. Allowing the two feedback modes to interchange, as in the hybrid simulations, leads to the lowest central entropies for all three halo masses since these simulations exhibit both strong cooling flows and less central heating. Another common feature between all three feedback cases is the difference in scatter. The thermal-only simulations show the largest scatter because they have both central ICM heating and strong cooling flows, whereas the opposite is true for the kinetic-only simulations. The hybrid ones have an intermediate amount of scatter. For the galaxy group case, shown in the left-hand panel of Fig. <ref>, all three simulations (with differing feedback implementations) fail to reproduce the shape (slope) of the observed entropy profiles within 10 kpc, but agree with them at larger distances (by construction). Within 10 kpc, the observed profiles behave as K∝ r^2/3, whereas our simulated entropy profiles all have cores. This is unlikely to be affected strongly by the T_0 parameter. Instead we interpret this disagreement as showing that lower jet velocities may need to be used (the velocity used for these simulations was v_j=5×10^3 km s^-1). As we show in Fig. <ref>, lower velocities lead to lower central entropies and more sloped profiles. Alternatively, it is possible that the observations used for comparison here are biased towards brighter groups that therefore have lower central entropies. If the satellites were removed from the observational samples, it is likely that the disagreement would be worse, since satellites are less likely to be undergoing cooling flows, due to stripping of their CGM. For the low-mass cluster simulations (middle panel of Fig. <ref>), we again find agreement with the observed profiles at large distances (in this case r>50 kpc). In the central regions, the thermal-only case appears to have the correct slope at small distances (r<10 kpc), but it has a flat section extending from r=10 kpc to r=30 kpc – a feature not visible in the observed entropy profiles. Our kinetic-only and hybrid simulations show similar slopes as the observed profiles, with perhaps a slightly too shallow slope in the very centre. This could be mitigated by a different choice of T_0 or a slightly lower jet velocity. Finally, we move to the high-mass galaxy cluster case, shown in the right-hand panel of Fig. <ref>. In the inner regions, all of our entropy profiles are lower than those in observations of <cit.>, <cit.> and <cit.> (although this could have been prevented by choosing a higher T_0, but we instead chose a highly CC setup to maximize differences between the AGN feedback implementations). They also show a central entropy core, but signs of such a core appear to be present in observations as well. All three of the simulations are consistent with being CC at most times (in agreement with the CC sample of , as well as with the lower end of the scatter from and ). Out of the three simulations, only the thermal feedback case sometimes has a central entropy approaching the median entropy of the NCC sample from <cit.>. However, NCC clusters may also be explained as a result of mergers (e.g. , ). We note that changing the implementation of AGN feedback is not the only way of affecting simulated entropy profiles (e.g. ). The details of the hydrodynamics scheme appear to be at least as important (e.g. , ). In particular, turning off artificial conduction in the SPH solver appears to lead to significantly more sloped entropy profiles. Entropy profiles are often compared in a rescaled form, such that instead of plotting K versus r, one plots K/K_500 versus r/r_500, where K_500 is a typical entropy that depends only on the halo mass. We discuss such profiles in Appendix <ref>. We find that they are fairly similar between the different simulated haloes, and all of them lie below the median observed entropies, likely because we simulate relatively CC systems. § RESULTS II: SIMPLIFIED FEEDBACK In the previous section we presented results of our model with BH spin evolution, for both thermal isotropic and kinetic jet feedback. Here we will simplify our implementation by instead fixing the efficiencies, as well as the direction for the jets (along the z-axis). In Appendix <ref> we show that the latter is justified as long as jet redirection occurs less frequently than ≈1000 Myr, and if jets precess with small opening angles (≤15) and long time-scales (Δ t≥20 Myr). The typical redirection time-scale, if redirection is allowed, is indeed typically longer than this (see the hybrid feedback cases in Fig. <ref>), since it occurs only a few times during an 8 Gyr long simulation (if we define ‘redirection' as a change in direction that is larger than a few dozen degrees). These simplifications are motivated by a desire to isolate the effects of varying efficiencies, as well as to make the simulations with different feedback implementations as similar as possible. To this end we fix the efficiencies to a value ϵ=0.01 for both the thermal isotropic and kinetic jet feedback. We do not test hybrid cases in these simplified simulations, and instead assume the feedback to be either thermal isotropic or kinetic jets regardless of accretion rate. We test the case where the feedback energies per heating/kicking event are the same for thermal isotropic and kinetic jet feedback, but this is not our fiducial choice. We instead typically use much higher jet velocities, since they are required in order to lead to inflation of lobes that turn into bubbles and create cavities in X-ray emitting gas, as seen in observations. We present results for the low- and high-mass galaxy clusters here (M_200=10^14 M_⊙ and M_200=10^15 M_⊙, respectively). We then vary the feedback efficiency, heating/kicking energy and energy type (thermal versus kinetic, as well as mixed) for both isotropic and jet feedback. These variations are intended to show the effects of choosing a particular implementation of feedback. For simplicity we vary these only for the low-mass galaxy cluster. §.§ General results In Fig. <ref> we show the feedback powers and SFRs in the galaxy cluster simulations with the simplified feedback prescriptions. We find that the thermal isotropic simulations are quite similar to those presented in <ref>, i.e. with BH spin evolution. This is likely due to the radiative efficiencies being near-constant in the case with BH spin evolution (Fig. <ref>). The kinetic jet simulations (with fiducial jet velocities, 1.5×10^4 km s^-1 and 3×10^4 km s^-1 for the two halo masses) are somewhat different from the BH spin evolution case. This is largely due to the jet efficiency not dropping below 1 per cent (unlike in the BH spin evolution case; Fig. <ref>), which means that very strong cooling and BH growth are prevented. As in Paper I, we find that fixing the jet direction to be along the z-axis does not prevent efficient feedback. Comparing the thermal isotropic and kinetic jet simulations, we find that the former reach lower minima of the feedback power, despite the fact that the same constant efficiency is used. This means that the BH reaches lower accretion rates in the thermal isotropic case (same as in the cases with BH spin evolution, see Fig. <ref>). This is caused by the thermal feedback simulations often featuring a significant presence of hot gas near the BH (originating from the feedback itself), which reduces its accretion rate. We find that using a constant efficiency leads to periodicity between cooling flow episodes, which seems more pronounced in the high-velocity jet cases. In these cases, we see periods of ≈1.5 Gyr in the low-mass cluster and ≈2 Gyr in the high-mass cluster. This periodicity occurs because AGN feedback effectively heats all gas out to a radius at which the ratio of the cooling time and the dynamical time is ≈10. The period between cooling flows is then roughly equal to the cooling time at that radius. These findings are illustrated in Appendix <ref> and are supported by previous works (e.g. , see also discussion therein). In Fig. <ref> we also show the results of using low jet velocities (6.5×10^3 km s^-1 and 1.15×10^4 km s^-1 for the low-mass and high-mass cluster, respectively), which are supersonic by only a factor of a few relative to the ICM. These velocities are chosen such that the energy per kicking event is equal to the heating energies used in the corresponding thermal isotropic simulations (Δ T=10^9 K and Δ T=10^9.5 K for the low-mass and high-mass cluster, respectively). We find that such low velocities lead to the period between cooling flow episodes increasing (to the point that the high-mass cluster shows no periodicity in this case, at least within 8 Gyr), and the SFR reaching smaller peaks, as well as being lower on average. In Fig. <ref> we show radial profiles of the ICM density, temperature and entropy for these same simulations. From the top panels we see that using jets leads to higher central densities (within 20-40 kpc), by a factor of a few. There is only a small difference between the fiducial and low-velocity jet cases. In the middle panels we compare the temperatures. The jet cases have lower central temperatures (within the same radii as for the densities) than the thermal isotropic ones, by up to a factor of two. At intermediate radii (up to r=100 kpc), the jet cases have higher temperatures, indicating that more of the feedback energy couples to larger radii in the jet cases. In the bottom panels we compare the entropy profiles. Due to a combination of higher central densities and lower central temperatures, the central entropies in the jet cases are lower by a factor of ≈5 and ≈2 for the two halo masses, respectively. In the low-mass case, the low-velocity simulation appears to have the same slope as the observed profiles, which are also shown in the figure. This potentially indicates that lower velocities should be used (rather than highly supersonic ones with Mach numbers ≥10, which we find to be required for the inflation of hot lobes and for X-ray cavities to be present), at least in these lower-mass systems. For the high-mass case, we again find that using thermal isotropic or kinetic jet feedback leads to similarly-shaped profiles as the observed ones (the same conclusion as found from Fig. <ref>, showing the BH spin evolution case). The entropies are visibly higher for jet feedback at r=30-100 kpc and r=40-300 kpc in the two mass cases, respectively. This supports the interpretation that kinetic jets are able to heat at larger radii than thermal isotropic feedback. In all of the presented profiles we see less scatter in the kinetic jet cases than with thermal isotropic feedback – this is a result of fewer or weaker cooling flows, and less violent central heating. The thermal isotropic form of feedback leads to very similar results in terms of the entropy profiles as the cases with BH spin evolution (Fig. <ref>). This is likely due to very similar feedback efficiencies (Fig. <ref>), although as we show in the next section, the entropy profiles are also largely insensitive to a much larger variation of the efficiency. §.§ Varying the implementation of feedback §.§.§ Visual differences We now turn to variations on the cases presented above. We vary the efficiencies of both types of feedback (isotropic and jet), energy per each feedback event and the type of energy used for feedback – thermal[In the thermal jet variant, particles are preferentially heated along a particular direction (the z-axis in this case). No kinetic energy is imparted to the gas, but it can still form outflows in the form of jets.], mixed or kinetic. We performed all of these for the low-mass galaxy cluster (M_200=10^14 M_⊙). In Fig. <ref> we show visualizations of some of these simulations. In particular, we show jets with different energy types and velocities (top row, left- and right-hand sides, respectively), and the same for isotropic feedback (bottom row, with the latter variation corresponding to the heating temperature). These are shown on the same spatial scales for purposes of comparison, but we find that isotropic feedback is generally more confined to the central regions than jet feedback. We also note that these visualizations are generally not shown for the same simulation time. Doing so would result in very little visible activity in some of the cases, since all of these simulations peak in feedback activity at different times. We have therefore attempted to show these visualizations at representative times for each of the cases. We begin our comparison of different types of feedback with variations of energy type for jet feedback (left-hand side of the top row in Fig. <ref>). Kinetic jets inflate well-defined ellipsoidal lobes, and they also create strong bow shocks. Using mixed jets also leads to fairly symmetrical lobes that create bow shocks, although they appear to be weaker (judging by the typical temperature in the shock fronts). Thermal jets do lead to biconical outflows, but these are asymmetrical since they are much more susceptible to perturbations. Relatively weak shocks are visible in this case. In the right-hand side of the top row of Fig. <ref> we show variations of the jet velocity in the kinetic jet case. The lowest-velocity case (v_j≈8500 km s^-1) does not appear to feature hot, ellipsoidal lobes. Instead, the outflows resemble <cit.> type I (conical) jets. Increasing the jet velocity leads to the inflation of lobes and stronger generation of spherical shocks, and this activity tends to be concentrated to smaller radii. The highest-velocity case shows lobes that appear similar to observed X-ray cavities, although we caution that this may be merely a consequence of low resolution (increasing the jet velocity at fixed power decreases the number of jet particles inside the lobes/bubbles, making them more spherical). In the left-hand side of the bottom row of Fig. <ref> we show results of varying the type of energy in the isotropic case. Using less kinetic energy leads to weaker spherical shocks, but typically hotter outflows. In the last row of Fig. <ref> we show the results of varying the heating temperature in the thermal isotropic case. These simulations all appear similar, and we do not find that increasing the heating temperature leads to hotter outflows, as one might have expected. From the visualizations shown here, it is also apparent that thermal isotropic feedback can sometimes lead to the emergence of biconical outflows – this is typically a result of a cold gas disc forming in the centre and feeding the BH. The feedback then results in the launching of biconical outflows that are perpendicular to the disc (see also ), since the heated gas tends to expand along the ‘path of least resistance'. §.§.§ Differences in feedback powers and SFRs In Fig. <ref> we show the feedback power and SFR for all of the cases discussed above, as well as cases with varying feedback efficiency. We begin by discussing the jet cases (left-hand column), and then the isotropic ones (right-hand column). We find that varying the type of jet energy (top-left panel) does not lead to very large differences in the jet powers. The mean jet power does increase slightly, however, by making it more kinetic rather than thermal. In addition, thermal jets lead to lower minima in the jet power, similar as in the thermal isotropic case, due to the gas near the BH often being hotter (which leads to lower accretion rates). From the SFR plot we see that kinetic jets are the most efficient at quenching, with the purely thermal ones quite similar to isotropic feedback (discussed below), and the mixed ones somewhere in between. In the middle-left panel of Fig. <ref> we show results of varying the jet velocity of kinetic jets. We already showed a variation of this kind in Figs. <ref> and <ref>, although we do it here more systemically. We find that using higher jet velocities results in more episodic feedback cycles, with higher peaks in the jet power and lower minima. The former is a result of more cold gas feeding stronger feedback, while the latter is a result of stronger shocking or shocking at smaller distances, which leads to more hot gas feeding the BH and reducing its accretion rate. Decreasing the velocity leads to a decrease in the SFR. Note, however, that decreasing the jet velocity at fixed resolution worsens the sampling of feedback (leading to fewer particles making up the jets and lobes), an effect that might be the main cause of these differences. In the bottom-left panel of Fig. <ref> we show results of varying the feedback efficiency of the kinetic jets. As we can see, the differences are significant. Increasing the feedback efficiency results in fewer and fewer feedback episodes. With an efficiency of 100 per cent, there is only one initial episode and effectively no star formation. Using an intermediate efficiency leads to two feedback and SFR episodes. It should be noted, however, that all three cases are quenched and thus show negligible star formation as compared to star-forming galaxies. Interestingly, all three simulations show the same minimum in the jet power (P_j≈3×10^42 erg s^-1). This minimum corresponds to hot halo accretion. In the right-hand panels of Fig. <ref> we show the corresponding variations of isotropic feedback. We find that all of the simulations are fairly similar, especially when compared with the variations in the jet case. It should be kept in mind that these simulations are chaotic in nature, so differences in the timing of peaks in the SFR and feedback power may not be very significant. With this in mind, we find that changing the energy type (top right panel) is the variation that has the most significant impact, in the form of changing the periodicity of the feedback events – the purely thermal case appears to have the longest period between feedback events. It also reaches the lowest value in the feedback power and SFR at t≈6 Gyr. Regardless of these small differences, the typical powers and SFRs are still similar. The similarity of thermal and kinetic isotropic feedback, as implemented here, has bearing for cosmological simulations. In particular, the EAGLE simulations () used a thermal isotropic AGN feedback implementation, whereas in IllustrisTNG (), feedback by AGN is mostly done through kinetic isotropic winds. The results shown here imply that these two feedback implementations are quite similar in their effects (and both of them quite different from kinetic jets). A caveat to this is that our simulations are of idealized clusters. In reality, feedback is expected to occur in various contexts, such as during and after galaxy mergers (see e.g. for observational evidence or for indications of the same in cosmological simulations) or triggered by disc instabilities (e.g. ). In these situations the effects of AGN feedback might be more sensitive to the various parameters and choices we have discussed. In the middle panel of Fig. <ref> we show the results of varying the heating temperature in the thermal isotropic case. The results are again very similar, although the two higher-temperature cases (Δ T ≥10^9 K) show a somewhat lower recurrent peak in the feedback power and SFR at t≈6-7 Gyr. This result may be due to stochastic noise, rather than an indication of an actual trend. In the bottom right-hand panel, we see the results of varying the feedback efficiency. The highest-efficiency case has both lower maxima and higher minima in the power and SFR (more easily visible in the latter), which is likely due to the BH reacting more quickly to the development of a cooling flow: the maxima reached are lower because the feedback can shut off the cooling flow before too much cooling occurs, while the minima are higher because the feedback is then not as explosive. §.§.§ Entropy profiles Finally, in Fig. <ref> we show the entropy profiles for the variations discussed above. From the top-left panel we see that increasing the fraction of kinetic energy in the jets leads to steeper inner entropy profiles, which are in closer agreement to observed ones (in terms of the slope). From the middle-left panel we see that decreasing the jet velocity can also bring the entropy profiles into closer agreement with observations. This may appear counterintuitive considering that real AGN jets are relativistic, and thus high-velocity (see e.g. review by ). However, one should keep in mind that jets are not necessarily relativistic on all scales; they are often transrelativistic[Velocities at which relativistic effects begin to become important, v_j≈0.1-0.5c.] (e.g. and ) or subrelativistic (e.g. ) on kpc scales, the ones we are simulating in this paper. The subrelativistic launching velocities (v_j<0.05c) favoured by these simulations may be indicative of observed jets experiencing significant amounts of entrainment on subgrid scales relative to what we are resolving here (i.e. below ≈300 pc). We find that the two lower-velocity cases shown in the panel have an almost identical entropy profile, indicating that the profiles converge to the same one as the velocity is decreased. From the two higher-velocity cases, we see that increasing the velocity leads to differences in the profiles: the central entropies are higher, and the slope is changed. In addition, the scatter between the different snapshots is increased. Overall, these results indicate that increasing the velocity leads to entropy profiles that are progressively more similar to those found with thermal isotropic feedback, likely due to shock heating (thermalisation) of the jets and inflation of lobes/bubbles at smaller radii. In the bottom-left panel we show variations of the feedback efficiency. These results indicate that higher efficiencies lead to entropies that are too flat in the centre. These results indicate that the CC/NCC dichotomy could partially or wholly be a result of the BH population differing in BH spin – the low spin ones having lower feedback efficiencies and therefore lower central entropies, whereas the higher-spin ones would be the opposite in this picture. In the right-hand panels of Fig. <ref> we show the same variations for the isotropic case. Overall these are very similar to each other, with the energy type variations (top right-hand panel) being the only ones that show appreciable differences. In particular, the mixed or purely kinetic isotropic wind cases have lower entropies than the purely thermal one, by roughly a factor of two. However, the overall shape of the entropy profile is still the same, and it still disagrees with the observed profiles in terms of the slope. §.§.§ Comparison with previous simulations We will compare our variations of the feedback implementation with previous work, mostly on idealized galaxy clusters and mainly for the low-mass cluster case (M_200=10^14 M_⊙), since other studies have largely focused on such haloes. <cit.> performed SPH simulations and compared several implementations of kinetic feedback, as well as one thermal variation. Their feedback implementation is intermediate to our isotropic and jet feedback, since it is bipolar in nature, but with a large opening angle (45). They found that using kinetic feedback leads to less star formation than if thermal feedback is used, in agreement with our findings. However, their entropy profiles with thermal feedback are lower than the ones with kinetic feedback, a conclusion opposite to ours (both for the isotropic and jet cases). This is likely a result of their thermal feedback being implemented as a ‘thermal dump' (see footnote <ref>), which likely resulted in numerical overcooling. They find that lower-velocity feedback leads to higher central entropies, again in disagreement with our finding. These difference could be due to differences in the hydrodynamics schemes (GADGET-3 vs. SPHENIX). <cit.> compared kinetic jets with the kinetic wind implemented in IllustrisTNG; they found that jets are slightly more efficient at quenching star formation, in agreement with our results. They also found that the feedback powers are less time-variable in the jet case than in the kinetic wind case, which is again in agreement with our results. Their interpretation of this is that jets act more on the actively cooling (but not yet star-forming gas), while the wind acts on the star-forming ISM, including in the vicinity of the BH. The remaining simulations we compare with were performed using grid-based codes. <cit.> compared mechanical (kinetic) jet feedback and thermal isotropic feedback across a range of halo masses (10^13-10^15 M_⊙), finding that the former leads to cooler cores, in agreement with our results. However, they implemented kinetic jets as self-regulated (with the accretion rate determined from the properties of gas), while their thermal feedback was implemented as a blast with a fixed power (heating all gas near the BH), which is not a fair comparison. <cit.> compared different feedback models in a massive halo (M_200≈10^15 M_⊙). They found that purely thermal jets are less efficient at preventing cooling flows from developing than either mixed or purely kinetic ones, in agreement with our findings. However, in disagreement with our results, they find lower central entropies with thermal feedback, similar to <cit.> (this is, again, probably a result of using low heating temperatures as part of a ‘thermal dump' that likely led to too much numerical overcooling). <cit.> compared dense (i.e. low-velocity) and light (high-velocity) jets, finding that the results are relatively similar. However, it should be pointed out that the majority of these papers, including the last one, perform their simulations for a relatively short time (usually 1-2 Gyr or less). This is of order the length of the typical cycle of activity (cooling and feedback) we find in our simulations. Thus, most of these papers may be biasing their results to the first episode of high-activity. Finally, while we found that the choice of heating temperature used for thermal isotropic feedback has little effect on our results, especially for the entropy profiles, this is in disagreement with previous work in a cosmological context (e.g. , ). Those studies found that the choice of heating temperature affects both the total mass of the ICM (the gas fraction) as well as its distribution and properties (the thermodynamical profiles). This difference between our results and cosmological studies is likely due to our simulations focusing on isolated and self-regulated systems (assumptions that break down for realistic haloes). § SUMMARY AND CONCLUSIONS Using the SWIFT simulation code () and the SPHENIX SPH implementation (), we have compared different prescriptions of AGN feedback. For this purpose we used a well-tested set-up of idealized galaxy groups and clusters () with virial masses M_200=10^13, 10^14 and 10^15 M_⊙. We focused on comparing thermal isotropic () and kinetic jet feedback () – the former representing the effects of radiatively-driven wind (i.e. quasar) feedback, and the latter the effects of feedback by relativistic jets. We first tested these AGN feedback implementations in unison with a BH spin evolution model based on equations describing subgrid accretion discs. This model gives variable feedback efficiencies, as well as variable jet directions. We assumed that thermal isotropic feedback occurs at high normalized accretion rates (Eddington ratios ṁ>0.01), when the disc is thin and radiatively efficient, whereas kinetic jets are assumed to be launched at low Eddington ratios (ṁ<0.01), when the disc is thick and advection-dominated. We compared this hybrid model with one where the disc is always thin and launching isotropic winds, as well one where the disc is always thick and launching jets. We then simplified the set-up by assuming constant feedback efficiencies and fixing the jet direction. In this simplified set-up, we further varied the feedback efficiency, the energy per feedback event, as well as the type of energy being used for feedback (thermal vs. kinetic) for both the isotropic and jet cases. From the simulations performed and the analysis presented in this paper, we find the following: Kinetic jet feedback leads to more efficient quenching of star formation in the central galaxies than thermal isotropic (wind) feedback. This applies across the whole mass scale range we have tested. It is true in simulations using detailed models of BH spin evolution (resulting in variable feedback efficiencies/jet directions), as well as ones without (using constant feedback efficiencies/jet directions). A larger fraction of the feedback energy couples to large radii in the jet case, resulting in overall more energy being injected in that case in order to quench cooling flows. * Due to a smaller fraction of the feedback energy coupling to the intracluster gas at smaller radii, and a larger fraction at larger radii, the central gas entropies are significantly lower with kinetic jet feedback than with thermal isotropic feedback. They are also in closer agreement with observations in terms of the inner slope. In addition to the median central entropies being lower, median central densities are higher and median central temperatures lower, despite cooling flows being weaker and/or shorter-lived. * We find that isotropic feedback is largely insensitive to the choice of feedback efficiency and energy per feedback event. By varying the type of energy being injected (kinetic, mixed and thermal), we find that the thermal isotropic case has a somewhat higher central entropy and a feedback cycle with the longest periodicity. However, all of these isotropic feedback implementations are still more similar to each other than any of them is to kinetic jet feedback. This may indicate that the isotropic kinetic feedback employed in some cosmological simulations (e.g. IllustrisTNG) is quite similar in its effects to the isotropic thermal feedback employed in other simulations (e.g. EAGLE). However, all of our isotropic feedback is energy-dominated, so the conclusions may change somewhat for momentum-dominated winds. * Jet feedback is sensitive to all of the choices mentioned in the previous point. High feedback efficiencies can prevent any cooling flows from developing, leading to higher central entropies. Increasing the jet velocity leads to more frequent cooling flows (and more star formation), but it also leads to higher mean central entropies with shallower slopes, due to strong shocks and heating at small radii. In other words, kinetic jet feedback is progressively more similar to thermal isotropic feedback as the jet velocity is increased. Jet feedback is most efficient if it is kinetic, rather than thermal or mixed. The jet direction is unimportant, as long as it does not change more frequently than every ≈1 Gyr, which it is unlikely to do in galaxy clusters with realistic BH spin evolution. Constant jet efficiencies lead to highly periodic cooling flows, unlike in the variable-efficiency cases. * In order to recover the observed entropy profiles across a large range of masses (galaxy group to rich cluster scales), it may be necessary to choose jet velocities carefully. In particular, low velocities may be required in galaxy groups/low-mass clusters in order to yield steep entropy profiles, while high jet velocities may be required to reproduce cored entropy profiles and X-ray cavities in rich galaxy clusters. Alternatively, variable jet efficiencies from a BH spin evolution model, in conjunction with different accretion/merger histories, might naturally lead to some of these differences. We find that a hybrid model with both thermal isotropic and kinetic jet feedback (depending on the BH accretion rate) has the lowest central entropies, and may thus be the most promising. On the other hand, our jet-only model is disfavoured on account of excessive BH mass growth. This growth is due to strong jet-induced spindown of BHs, leading to very low BH spins (and therefore jet efficiencies of order 0.1 per cent). * The differences between simulated entropy profiles with varying AGN feedback implementations are similar in magnitude to differences that arise if the numerical details are varied (e.g. the artificial conductivity and viscosity of the hydrodynamics code). This means that these physical and numerical variations are somewhat degenerate with regards to entropy profiles. Bringing different numerical codes in agreement would thus significantly improve the potential of simulations to discriminate between different AGN feedback implementations. We caution that these conclusions may only be valid for isolated systems such as the ones studied in this paper. Thus, some of them may not fully apply in the context of cosmological simulations. Despite this caveat, the results presented in this paper should be valuable when considering different implementations of AGN feedback in cosmological simulations of galaxy formation and evolution. This is particularly true as the AGN feedback implementations are becoming more complicated and realistic. § ACKNOWLEDGEMENTS The research in this paper made use of the SWIFT open-source simulation code (<http://www.swiftsim.com>, ) version 0.9.0. The swiftsimio Python library was used to analyze and visualize the data from the simulations (, ). The work has been performed under the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the Horizon 2020 Framework Programme of the European Union (H2020); in particular, FH gratefully acknowledges the support of the Leiden Observatory and the computer resources and technical support provided by SURFsara, the Dutch national HPC facility. This project was also funded by the AHEAD2020 Programme under Grant Agreement 871158. This project has received funding from the Netherlands Organization for Scientific Research (NWO) through research programme Athena 184.034.002. FH would like to acknowledge support from the Science Technology Facilities Council through a CDT studentship (ST/P006744/1), and CGL acknowledges support from STFC consolidated grants ST/T000244/1 and ST/X001075/1. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. § DATA AVAILABILITY The data underlying this article will be provided upon reasonable request to the corresponding author. The code and initial conditions used to generate the data are available online: <https://gitlab.cosma.dur.ac.uk/swift/swiftsim>. mnras § EFFECTS OF JET REDIRECTION AND PRECESSION For the purposes of the main results in this paper, we fixed the jets to be along the z-axis when considering simplified feedback without BH spin evolution. This immediately leads to the following questions: how justified is this assumption, and how important is the change of the jet direction for the effects of feedback? We ran some additional simulations in order to answer these questions. These simulations employed either manually redirecting or precessing jets. There are many ways in which both of these processes could be implemented. We used a fairly simple implementation, since these results are meant to be illustrative. We tested these cases in our fiducial high-mass galaxy cluster set-up (M_200=10^15 M_⊙), since we found redirection to be more likely for this halo mass (if BH spin evolution is used). In the redirecting case, the jets were initially directed along the z-axis. With a period of Δ t, they were then instantaneously redirected to another, randomly chosen axis. We tested three periods: Δ t=1000 Myr, Δ t=200 Myr, and Δ t=40 Myr. These are compared with the fixed-direction case in Fig. <ref>, alongside a case that has spin-driven jet redirection, but a constant jet efficiency (ϵ_j=0.01, as in all of these simulations). The spin-redirecting case appears to show similar behaviour as the fixed-axis case, despite the jets being redirected during each of the cooling episodes (for a total of 4 times, i.e. once per each cooling episode, although this is not shown here). The case with manual redirection every Δ t=1000 Myr is again very similar to the fixed case, and therefore to the spin-redirecting case. However, if redirection is done more often (Δ t ≤200Myr), the jet powers are more variable and the SFRs are more similar to the thermal isotropic case. We interpret this to be a result of the redirection time-scale being similar to (of the same order of magnitude as) the typical duration of a jet episode, which can be up to 100 Myr. We speculate that this may be due to jets often being redirected while they are in the process of inflating a pair of lobes, or otherwise moving to large radii (where effective heating seems to be necessary in order for cooling flows to be shut off effectively). In the precessing cases, we manually precessed the jets with a period of Δ t about the z-axis, with a precession angle of θ_prec. We did not nutate the jets as well, i.e. they did not ‘cover' the region between the z-axis and the circle on which they were precessing. Note that the effects of jet precession are probably quite similar to the effects of using a larger opening angle. We tested three values of Δ t: Δ t=100 Myr, Δ t=20 Myr, and Δ t=4 Myr. These are relatively shorter than in the redirecting case, because we expect that the BH spin vector can change in direction by small values (e.g. 15) with very little mass accretion, which is not true for full redirection. For each of the precession time-scales, we tested three precession angles: θ_prec=15, θ_prec=30 and θ_prec=45. The results of these tests are shown in Fig. <ref>. It appears that jet precession leads to significant differences in all cases shown here. The only combination(s) that result in fairly low SFRs are those with θ_prec=15 and Δ t≥20 Myr. However, even these cases show higher SFRs than the fixed-axis case. Cases with larger precession angles appear quite similar in their effects to thermal isotropic feedback. The precession time-scale does not appear to have a large impact. § MASS FLUX ASSOCIATED WITH ACCRETION AND FEEDBACK In Fig. <ref> we show some additional quantities from the same simulations as in Fig. <ref>, namely: the ratio of total mass accreted, launched into the jets and heated by the BH, to the total stellar mass formed. These ratios are plotted for our simulations with spin evolution spanning the galaxy group (M_200=10^13 M_⊙) to high-mass cluster scale (M_200=10^15 M_⊙). We plot these quantities in order to glean information on whether BH accretion and feedback are directly interfering with star formation (by depriving it of cool gas either by accreting, kicking or heating it), or indirectly by e.g. causing outflows of the same gas or shutting off cooling flows that supply this gas. These ratios should be treated as meaningful only once the amount of stars that have formed is appreciable, and not very low due to feedback being effective. For this reason, the results from the left-hand column should not be considered too meaningful (since very little star formation occurs in the galaxy group case), while for the galaxy cluster cases, they become meaningful only at t=2-4 Gyr, depending on the case. From the galaxy cluster cases we see that the amount of mass accreted by the BH is significant in all cases, with the mass flux associated with feedback even more significant (for at least one of the feedback channels in the given simulation). This is true even for the high-mass cluster, which is the most star-forming of the systems we study, and where we find that the combined mass of all the heated and kicked particles in the hybrid case (as an example) to be roughly as large as the total mass of all stars formed (≈3×10^11 M_⊙). Overall, these plots indicate that feedback mechanisms in simulations may directly interfere with the formation of stars (by depriving it of cold gas), even when the rate of star formation is relatively high. This effect may, however, still be subdominant to the indirect effects of feedback. § FRACTION OF BLACK HOLE GROWTH AT LOW VS. HIGH EDDINGTON RATIOS In Fig. <ref> we show the cumulative fraction of mass accreted when ṁ<ṁ_crit=0.01 (corresponding to the thick disc in the hybrid and kinetic-only case) as a function of time, for all 9 simulations discussed in <ref>. This figure is an extension of Fig. <ref>. We see that in the galaxy group case, most of the growth is at low Eddington ratios, except at the very beginning. However, this reflects the fact that there is an initial burst of high accretion rate growth at the beginning of the simulation, after which the system is fully quenched. In the galaxy cluster cases, we see that most of the BH growth occurs when ṁ>0.01, despite the fact that this condition is not fulfilled most of the time. The growth at low Eddington ratios, however, is by no means negligible. § DIMENSIONLESS ENTROPY PROFILES In Fig. <ref> we show the dimensionless entropy profiles K/K_500 as a function of the scaled radius r/r_500, for the 9 simulations with BH spin evolution, discussed in <ref> (this plot may be considered an alternative way of showing the data in Fig. <ref>). We define the entropy K_500 as k_BT_500/n_e,500^2/3, where T_500=GM_500μ m_p/2r_500 and n_e,500=500f_b,0ρ_c/μ_e m_p. Here, ρ_c is the critical density, μ_e=1.14 the mean molecular weight per free electron and f_b,0≈0.16 the cosmic baryon fraction (e.g. ). Overall, from Fig. <ref> we see that all of the simulated dimensionless profiles are similar, although there is some disagreement in normalisation at large radii. This can be avoided (and the profiles made even more similar) if the cosmic baryon fraction f_b,0 in the definition of K_500 is replaced by the actual baryon fraction of each of the haloes, f_b,500, although we do not show those profiles here. We find that the low-mass clusters (M_200=10^14 M_⊙) with jets show the lowest dimensionless entropy. This result may not be significant, however, given that these are single realizations of idealized and isolated clusters. Most of our profiles lie below the observations shown in the figure, although this is by construction (we simulate relatively CC systems). § PERIODICITY BETWEEN JET EPISODES In Fig. <ref> we show the approximate periodicity in relevant quantities related to the feedback cycle for our fiducial simulation (with a fixed feedback efficiency ϵ_j=0.01 and the jets directed along the z-axis) of the low-mass galaxy cluster (M_200=10^14 M_⊙). The top left-hand panel shows the jet power: it has 5 peaks that appear to be roughly equally separated in time, while the top right-hand panel shows the same for the star formation rate. The peaks in jet power and SFR roughly coincide. The bottom row shows the cooling time and the cooling time to dynamical time ratio ratio at several radii from r=3.16 kpc to r=316 kpc. According to the hypothesis of <cit.> (and references therein), all gas with t_cool/t_dyn<10 will cool effectively and contribute to the cooling flow. From the right-hand panel we see that the gas at r=31.6 kpc has a roughly constant value of the ratio, t_cool/t_dyn≈10. If we then look at the left-hand panel, that same gas has a roughly constant cooling time of ≈1.5 Gyr. This is also roughly the period between the cooling flow episodes. Our results are thus in agreement with the above-mentioned hypothesis. While we have shown only this one case, we find that the same holds true across all our simulations.
http://arxiv.org/abs/2307.00535v1	20230702102732	Goal-oriented Tensor: Beyond Age of Information Towards Semantics-Empowered Goal-Oriented Communications	[ "Aimin Li", "Shaohua Wu", "Sumei Sun", "Jie Cao" ]	cs.IT	[ "cs.IT", "eess.SP", "math.IT" ]	Goal-oriented Tensor: Beyond Age of Information Towards Semantics-Empowered Goal-Oriented Communications Aimin Li, Graduate Student Member, IEEE, Shaohua Wu, Member, IEEE, Sumei Sun, Fellow, IEEE, and Jie Cao, Member, IEEE August 1, 2023 ============================================================================================================================================= Optimizations premised on open-loop metrics such as Age of Information (AoI) indirectly enhance the system's decision-making utility. We therefore propose a novel closed-loop metric named Goal-oriented Tensor (GoT) to directly quantify the impact of semantic mismatches on goal-oriented decision-making utility. Leveraging the GoT, we consider a sampler & decision-maker pair that works collaboratively and distributively to achieve a shared goal of communications. We formulate a two-agent infinite-horizon Decentralized Partially Observable Markov Decision Process (Dec-POMDP) to conjointly deduce the optimal deterministic sampling policy and decision-making policy. To circumvent the curse of dimensionality in obtaining an optimal deterministic joint policy through Brute-Force-Search, a sub-optimal yet computationally efficient algorithm is developed. This algorithm is predicated on the search for a Nash Equilibrium between the sampler and the decision-maker. Simulation results reveal that the proposed sampler & decision-maker co-design surpasses the current literature on AoI and its variants in terms of both goal achievement utility and sparse sampling rate, signifying progress in the semantics-conscious, goal-driven sparse sampling design. Goal-oriented communications, Goal-oriented Tensor, Status updates, Age of Information, Age of Incorrect Information, Value of Information, Semantics-aware sampling. § INTRODUCTION The recent advancement of the emerging 5G and beyond has spawned the proliferation of both theoretical development and application instances for Internet of Things (IoT) networks. In such networks, timely status updates are becoming increasingly crucial for enabling real-time monitoring and actuation across a plethora of applications. To address the inadequacies of traditional throughput and delay metrics in such contexts, the Age of Information (AoI) has emerged as an innovative metric to capture the data freshness perceived by the receiver <cit.>, defined as AoI(t)=t-U(t), where U(t) denotes the generation time of the latest received packet before time t. Since its inception, AoI has garnered significant research attention and has been extensively analyzed in the queuing systems <cit.>, physical-layer communications <cit.>, MAC-layer communications <cit.>, industrial IoT <cit.>, energy harvesting systems<cit.>, and etc. (see <cit.> and the references therein for more comprehensive review). These research efforts are driven by the consensus that a freshly received message typically contains more valuable information, thereby enhancing the utility of decision-making processes. Though AoI has been proven efficient in many freshness-critical applications, it exhibits several critical shortcomings. Specifically, (a) AoI does not provide a direct measure of information value; (b) AoI does not consider the content dynamics of source data and ignores the effect of End-to-End (E2E) information mismatch on the decision-making process. To address shortcoming (a), a typical approach is to impose a non-linear penalty on AoI<cit.>. In <cit.>, the authors map the AoI to a non-linear and non-decreasing function f(AoI(t)) to evaluate the degree of “discontent” resulting from stale information. Subsequently, the optimal sampling policy is deduced for an arbitrary non-decreasing penalty function. The authors in <cit.> introduce two penalty functions, namely the exponential penalty function a^AoI(t)-1 and the logarithmic penalty function log_a(AoI(t+1)), for evaluating the Cost of Update Delay (CoUD). In <cit.> and <cit.>, the binary indicator function 1_{AoI(t)>d} is applied to evaluate whether the most recently received message is up-to-date. Specifically, the penalty assumes a value of 1 when the instantaneous AoI surpasses a predetermined threshold d; otherwise, the penalty reverts to 0. The freshness of web crawling is evaluated through this AoI-threshold binary indicator function. In <cit.>, an analogous binary indicator approach is implemented in caching systems to evaluate the freshness of information. The above works tend to tailor a particular non-linear penalty function to evaluate the information value. However, the selection of penalty functions in the above research relies exclusively on empirical configurations, devoid of rigorous derivations. To this end, several information-theoretic techniques strive to explicitly derive the non-linear penalty function in terms of AoI <cit.>. One such quintessential work is the auto-correlation function 𝔼[X_t^X_t-AoI(t)], which proves to be a non-linear function of AoI when the source is stationary <cit.>. Another methodology worth noting is the mutual information metric between the present source state X_t and the vector consisting of an ensemble of successfully received updates 𝐖_t <cit.>. In the context of a Markovian source, this metric can be reduced to I(X_t;X_t-AoI(t)), which demonstrates a non-linear dependency on AoI under both the Gaussian Markov source and the Binary Markov source <cit.>. In <cit.>, an analogous approach is utilized to characterize the value of information (VoI) for the Ornstein-Uhlenbeck (OU) process, which likewise demonstrates a non-linear dependency on AoI. In <cit.>, the conditional entropy H(X_t\|𝐖_t) is further investigated to measure the uncertainty of the source for a remote estimator given the history received updates 𝐖_t. When applied to a Markov Source, this conditional entropy simplifies to H(X_t\|X_t-AoI(t)), thus exemplifying a non-linear penalty associated with AoI. To address shortcoming (b), substantial research efforts have been invested in the optimization of the Mean Squared Error (MSE), denoted by (X_t-X̂_t)^2, with an ultimate objective of constructing a real-time reconstruction remote estimation system <cit.>. In <cit.>, a metric termed effective age is proposed to minimize the MSE for the remote estimation of a Markov source. In <cit.> and <cit.>, two Markov sources of interest, the Wiener process and the OU process are investigated to deduce the MSE-optimal sampling policy. Intriguingly, these policies were found to be threshold-based, activating sampling only when the instantaneous MSE exceeds a predefined threshold, otherwise maintaining a state of idleness. The authors in <cit.> explored the trade-off between MSE and quantization over a noisy channel, and derived the MSE-optimal sampling strategy for the OU process. Complementary to the above MSE-centered research, variants of AoI that address shortcoming (b) have also been conceptualized <cit.>. In <cit.>, Age of Changed Information (AoCI) is proposed to address the ignorance of content dynamics of AoI. In this regard, unchanged statuses do not necessarily provide new information and thus are not prioritized for transmission. In <cit.>, the authors introduce the context-aware weighting coefficient to propose the Urgency of Information (UoI), a metric capable of measuring the weighted MSE in diverse urgency contexts. In <cit.>, the authors propose a novel age penalty named Age of Synchronization (AoS), a novel metric quantifying the time since the most recent end-to-end synchronization. Moreover, considering that an E2E status mismatch may exert a detrimental effect on the overall system's performance over time, the authors of <cit.> propose a novel metric called Age of Incorrect Information (AoII). This metric quantifies the adverse effect stemming from the duration of the E2E mismatch, revealing that both the degree and duration of E2E semantic mismatches lead to a utility reduction for subsequent decision-making. The above studies focused on the open-loop generation-to-reception process within a transmitter-receiver pair, neglecting the closed-loop perception-actuation timeliness. A notable development addressing this issue is the extension from Up/Down-Link (UL/DL) AoI to a closed-loop AoI metric, referred to as the Age of Loop (AoL) <cit.>. Unlike the traditional open-loop AoI, which diminishes upon successful reception of a unidirectional update, the AoL decreases solely when both the UL status and the DL command are successfully received. Another advanced metric in <cit.>, called Age of Actuation (AoA), also encapsulates the actuation timeliness, which proves pertinent when the receiver employs the received update to execute timely actuation. Notwithstanding the above advancements, the question on how the E2E mismatch affects the closed-loop utility of decision-making has yet to be addressed. To address this issue, <cit.> introduce a metric termed Cost of Actuation Error to delve deeper into the cost resulting from the error actuation due to imprecise real-time estimations. Specifically, the Cost of Actuation Error is denoted by an asymmetric zero diagonal matrix 𝐂, with each value C_X_t,X̂_t representing the instant cost under the E2E mismatch status (X_t,X̂_t)_X_tX̂_t. In this regard, the authors reveal that the utility of decision-making bears a close relation to the E2E semantic mismatch category, as opposed to the mismatch duration (AoII) or mismatch degree (MSE). For example, an E2E semantic mismatch category that “Fire” is estimated as “No Fire” will result in high cost; while in the opposite scenario, the cost is low. Nonetheless, we notice that i) the method to obtain a Cost of Actuation Error remains unclear, which implicitly necessitates a pre-established decision-making policy; ii) Cost of Actuation Error does not consider the context-varying factors, which may also affect the decision-making utility; iii) the zero diagonal property of the matrix implies the supposition that if X_t=X̂_t, then C_X_t,X̂_t=0, thereby signifying that error-less actuation necessitates no energy expenditure. Fig. <ref> provides an overview of the existing metrics. Against this background, the present authors have recently proposed a new metric referred to as GoT in <cit.>, which, compared to Cost of Actuation Error, introduces new dimensions of the context Φ_t and the decision-making policy π_A to describe the true utility of decision-making. Remarkably, we find that GoT offers a versatile degeneration to established metrics such as AoI, MSE, UoI, AoII, and Cost of Actuation Error. Additionally, it provides a balanced evaluation of the cost trade-off between the sampling and decision-making. The contributions of this work could be summarized as follows: ∙ We focus on the decision utility issue directly by employing the GoT. A controlled Markov source is observed, wherein the transition of the source is dependent on both the decision-making at the receiver and the contextual situation it is situated. In this case, the decision-making will lead to three aspects in utility: i) the future evolution of the source; ii) the instant cost at the source; iii) the energy and resources consumed by actuation. ∙ We accomplish the goal-oriented sampler & decision-maker co-design, which, to the best of our knowledge, represents the first work that addresses the trade-off between semantics-aware sampling and goal-oriented decision-making. Specifically, we formulate this problem as a two-agent infinite-horizon Decentralized Partially Observable Markov Decision Process (Dec-POMDP) problem, with one agent embodying the semantics and context-aware sampler, and the other representing the goal-oriented decision-maker. Note that the optimal solution of even a finite-horizon Dec-POMDP is known to be NEXP-hard <cit.>, we develop the RVI-Brute-Force-Search Algorithm. This algorithm seeks to derive optimal deterministic joint policies for both sampling and decision-making. A thorough discussion on the optimality of our algorithm is also presented. ∙ To further mitigate the “curse of dimensionality” intricately linked with the execution of the optimal RVI-Brute-Force-Search, we introduce a low-complexity yet efficient algorithm to solve the problem. The algorithm is designed by decoupling the problem into two distinct components: a Markov Decision Process (MDP) problem and a Partially Observable Markov Decision Process (POMDP) issue. Following this separation, the algorithm endeavors to search for the joint Nash Equilibrium between the sampler and the decision-maker, providing a sub-optimal solution to this goal-oriented communication-decision-making co-design. § GOAL-ORIENTED TENSOR §.§ Specific Examples of Goal-Oriented Communications Consider a time-slotted communication system. Let X_t∈𝒮 represent the perceived semantic status at time slot t at the source, and X̂_t∈𝒮 denote the constructed estimated semantic status at time slot t. Real-time reconstruction-oriented communications is a special type of goal-oriented communications, whose goal is achieving real-time accurate reconstruction: minlimsup_T →∞1/T∑_t=0^T-1(X_t-X̂_t)^2. Although real-time reconstruction is important, it does not represent the ultimate goals of communications, such as implementing accurate actuation (as opposed to merely real-time reconstruction) and minimizing the system's long-term Cost of Actuation Error, as in <cit.>: minlimsup_T →∞1/T∑_t=0^T-1C_X_t,X̂_t, where C_X_t,X̂_t represents the instantaneous system cost of the system at time slot t. This cost is derived from the erroneous actuation stemming from the status mismatch between transceivers. Following the avenue of Cost of Actuation Error, we notice that the matrix-based metric could be further augmented to tensors to realize more flexible goal characterizations. For example, drawing parallels with the concept of Urgency of Information <cit.>, we can introduce a context element Φ_t to incorporate context-aware attributes into this metric.[It is important to note that the GoT could be expanded into higher dimensions by integrating additional components, including actuation policies, task-specific attributes, and other factors.] Accordingly, we can define a three-dimensional GoT through a specified mapping: (X_t,Φ_t,X̂_t)∈𝒮×𝒱×𝒮ℒ→GoT(t)∈ℝ, which could be visualized by Fig. <ref>. In this regard, the GoT, denoted by ℒ(X_t,Φ_t,X̂_t) or GoT(t), indicates the instantaneous cost of the system at time slot t, with the knowledge of (X_t,Φ_t,X̂_t). Consequently, the overarching goal of this system could be succinctly expressed as: minlimsup_T →∞1/T∑_t=0^T-1GoT(t). §.§ Degeneration to Existing Metrics In this subsection, we demonstrate, through visualized examples and mathematical formulations, that a three-dimension GoT can degenerate to existing metrics. Fig. <ref> showcases a variety of instances of the GoT metric. ∙ Degeneration to AoI: AoI is generally defined as AoI(t) ≜ t-max{G_i:D_i<t}, where G_i is the generated time stamp of the i-th status update, D_i represents the corresponding deliver time slot. Since AoI is known to be semantics-agnostic <cit.>, the values in the tensor only depend on the freshness context Φ_t≜AoI(t). In this case, each tensor value given a determined context status Φ(t) is a constant, and the GoT is reduced to GoT(t)=ℒ(X_t,Φ_t,X̂_t)(a)=Φ(t)=AoI(t). where (a) indicates that AoI is semantics-agnostic. In this case, AoI refers to the context exactly. The process of reducing GoT to AoI is visually depicted in Fig. <ref>(a). ∙ Degeneration to AoII: AoII is defined as AoII(t) ≜ f(AoS(t))· g(X_t,X̂_t), where AoS(t) ≜ t- max{τ:t≤τ, X_t=X̂_t}. AoII embraces the semantics-aware characteristics and is hence regarded as an enabler of semantic communications <cit.>. The inherent multiplicative characteristic of AoII guarantees the existence of a base layer of the tensor representation, from which other layers are derived by multiplying this base layer, as depicted in Fig. <ref>(b). Let Φ_t≜ f(AoS(t)), the GoT is reduced to GoT(t)=ℒ(X_t,Φ_t,X̂_t)(a)=Φ_t· g(X_t,X̂_t)=f(AoS(t))· g(X_t,X̂_t)= AoII(t), where g(X_t,X̂_t), typically characterized by 1_{X_tX̂_t}, represents the base layer in the tensor, and (a) indicates the inherent multiplicative characteristic of AoII. The visual representation of the GoT reduction to AoII is depicted in Fig. <ref>(b). ∙ Degeneration to MSE: MSE is defined as MSE(t) ≜(X_t-X̂_t)^2. MSE is intuitively context-agnostic. In the scenario where g(X_t,X̂_t)=(X_t-X̂_t)^2, the GoT collapses to the MSE metric: GoT(t)=ℒ(X_t,Φ_t,X̂_t)(a)=g(X_t,X̂_t) =(X_t-X̂_t)^2=MSE(t), where (a) establishes due to the context-agnostic nature of MSE. The visualization of the GoT reduction to MSE is shown in Fig. <ref>(c). ∙ Degeneration to UoI: UoI is defined by UoI(t) ≜Φ_t·(X_t-X̂_t)^2, where the context-aware weighting coefficient Φ_t is further introduced <cit.>. When g(X_t,X̂_t)=(X_t-X̂_t)^2, the GoT could be transformed into the UoI by GoT(t)=ℒ(X_t,Φ_t,X̂_t)(a)=Φ_t· g(X_t,X̂_t) =Φ_t·(X_t-X̂_t)^2=UoI(t), where (a) indicates the inherent multiplicative characteristic of UoI. The visualization of the GoT reduction to UoI is shown in Fig. <ref>(d). ∙ Degeneration to Cost of Actuation Error: Cost of Actuation Error is defined by C_X_t,X̂_t, which indicates the instantaneous system cost if the source status is X_t and the estimated one X̂_t mismatch <cit.>. Let g(X_t,X̂_t)=C_X_t,X̂_t, the GoT collapses to Cost of Actuation Error: GoT(t)=ℒ(X_t,Φ_t,X̂_t)(a)= g(X_t,X̂_t)=C_X_t,X̂_t, where (a) establishes due to the context-agnostic nature of Cost of Actuation Error. The visualization of the GoT reduction to Cost of Actuation Error is shown in Fig. <ref>(e). §.§ Goal Characterization Through GoT As shown in Fig. <ref>(f), a more general GoT exhibits neither symmetry, zero diagonal property, nor a base layer, offering considerable versatility contingent upon the specific goal. Here we propose a method that constructs a specific GoT, there are four steps: ∙ Step 1: Clarify the scenario and the communication goal. For instance, in the wireless accident monitoring and rescue systems, the goal is to minimize the cumulative average cost associated with accidents and their corresponding rescue interventions over the long term. ∙ Step 2: Define the sets of semantic status, 𝒮, and contextual status, 𝒱. These sets can be modeled as collections of discrete components. For instance, the set 𝒮 might encompass the gravity of industrial mishaps in intelligent factories, whereas the set 𝒱 could encompass the meteorological circumstances, which potentially influence the source dynamics and the costs. ∙ Step 3: The GoT could be decoupled by three factors: [The definitions of costs are diverse, covering aspects such as financial loss, resource usage, or a normalized metric derived through scaling, depending on the specific objective they are designed to address.] i) The status inherent cost C_1(X_t,Φ_t). It quantifies the cost associated with different status pairs (X_t, Φ_t) in the absence of external influences; ii) The actuation gain C_2(π_A(X̂_t)), where π_A is a deterministic decision policy contingent upon X̂_t. This cost quantifies the positive utility resulting from the actuation π_A(X̂(t)); iii) The actuation resource expenditure C_3(π_A(X̂_t)), which reflects the resources consumed by a particular actuation π_A(X̂(t)). ∙ Step 4: Constructing the GoT. The GoT, given a specific triple-tuple (X_t, X̂_t,Φ_t) and a certain decision strategy π_A, is calculated by GoT^π_A(t)=ℒ(X_t,Φ_t,X̂_t,π_A)=[C_1(X_t,Φ_t) - a C_2(π_A(X̂_t))]^+ + bC_3(π_A(X̂_t)). The ramp function [·]^+ ensures that any actuation π_A(X̂_t) reduces the cost to a maximum of 0. A visualization of a specific GoT construction is shown in Fig. <ref>. The GoT in Fig. <ref> is obtained through the following definition: [ C_1(X_t,Φ_t)= ([ 0 1 2; 0 0 1 3; 1 0 2 5 ]), π_A(X̂_t)=([ 0 1 2; a_0 a_1 a_2; ]),; C_2(a_A(t))=([ a_0 a_1 a_2; 0 2 4; ]), C_3(a_A(t)) =([ a_0 a_1 a_2; 0 1 2; ]). ] § SYSTEM MODEL We consider a time-slotted perception-actuation loop where both the perceived semantics X_t∈𝒮={s_1,⋯,s_\|𝒮\|} and context Φ_t∈𝒱={v_1,⋯,v_\|𝒱\|} are input into a semantic sampler, tasked with determining the significance of the present status X_t and subsequently deciding if it warrants transmission via an unreliable channel. The semantics and context are extracted and assumed to perfectly describe the status of the observed process. The binary indicator, a_S(t)=π_S(X_t,Φ_t,X̂_t) ∈{0, 1}, signifies the sampling and transmission action at time slot t, with the value 1 representing the execution of sampling and transmission, and the value 0 indicating the idleness of the sampler. π_S here represents the sampling policy. We consider a perfect and delay-free feedback channel <cit.>, with ACK representing a successful transmission and NACK representing the otherwise. The decision-maker at the receiver will make decisions a_A(t)∈𝒜_A={a_1,⋯,a_\|𝒜_A\|} based on the estimate X̂_t[We consider a general abstract decision-making set 𝒜_A that exhibits adaptability to diverse applications. Notably, this decision-making set can be customized and tailored to suit specific requirements.], which will ultimately affect the utility of the system. An illustration of our considered model is shown in Fig. <ref>. §.§ Semantics and Context Dynamics Thus far, a plethora of studies have delved into the analysis of various discrete Markov sources, encompassing Birth-Death Markov Processes elucidated in <cit.>, binary Markov sources elucidated in <cit.>, and etc. In real-world situations, the context and the actuation also affect the source's evolution. Consequently, we consider a context-dependent controlled Discrete Markov source: ( X_t+1=s_u\| X_t=s_i, a_A(t)=a_m, Φ_t=v_k.)=p_i,u^(k,m), where the dynamics of the source is dependent on both the decision-making a_A(t) and context Φ_t. Furthermore, we take into account the variations in context Φ_t, characterized by: ( Φ_t+1=v_r\| Φ_t=v_k.)=p_k,r. Note that the semantic status X_t and context status Φ_t could be tailored according to the specific application scenario. In general, these two processes are independent with each other. §.§ Unreliable Channel and Estimate Transition We assume that the channel realizations exhibit independence and identical distribution (i.i.d.) across time slots, following a Bernoulli distribution. Particularly, the channel realization h_t assumes a value of 1 in the event of successful transmission, and 0 otherwise. Accordingly, we define the probability of successful transmission as (h_t=1)=p_S and the failure probability as (h_t=0)=1-p_S. To characterize the dynamic process of X̂_t, we consider two cases as described below: ∙ a_S(t)=0. In this case, the sampler and transmitter remain idle, manifesting that there is no new knowledge given to the receiver, i.e., X̂_t+1=X̂_t. As such, we have: (X̂_t+1=x\|X̂_t=s_j,a_S(t)=0.)=1_{x=s_j}. ∙ a_S(t)=1. In this case, the sampler and transmitter transmit the current semantic status X_t through an unreliable channel. As the channel is unreliable, we differentiate between two distinct situations: h_t=1 and h_t=0: (a) h_t=1. In this case, the transmission is successful. As such, the estimate at the receiver X̂_t+1 is nothing but X(t), and the transition probability is (X̂_t+1=x\|X̂_t=s_j,X_t=s_i,a_S(t)=1,h_t=1.)=1_{x=s_i}. (b) h_t=0. In this case, the transmission is not successfully decoded by the receiver. As such, the estimate at the receiver X̂_t+1 remains X̂(t). In this way, the transition probability is (X̂_t+1=x\|X̂_t=s_j,X_t=s_i,a_S(t)=1,h_t=0.)=1_{x=s_j}. As the channel realization h_t is independent with the process of X_t, X̂_t, and a_S(t), we have that (X̂_t+1=x\|X̂_t=s_j,X_t=s_i,a_S(t)=1.) =∑_h_tp(h_t)(X̂_t+1=x\|X̂_t=s_j,X_t=s_i,a_S(t)=1,h_t.) =p_S·1_{x=s_i}+(1-p_S)·1_{x=s_j}. Combing (<ref>) with (<ref>) yields the dynamics of the estimate. §.§ Goal-oriented decision-making and Actuating We note that the previous works primarily focused on minimizing the open-loop freshness-related or error-related penalty for a transmitter-receiver system. Nevertheless, irrespective of the fresh delivery or accurate end-to-end timely reconstruction, the ultimate goal of such optimization efforts is to ensure precise and effective decision-making. To this end, we broaden the open-loop transmitter-receiver information flow to include a perception-actuation closed-loop utility flow by incorporating the decision-making and actuation processes. As a result, decision-making and actuation enable the conversion of status updates into ultimate effectiveness. Here the decision-making at time slot t follows that a_A(t)=π_A(X̂_t), with π_A representing the deterministic decision-making policy. § PROBLEM FORMULATION AND SOLUTION Traditionally, the development of sampling strategies has been designed separately from the decision-making process. An archetypal illustration of this two-stage methodology involves first determining the optimal sampling policy based on AoI, MSE, and their derivatives, such as AoII, and subsequently accomplishing goal-oriented decision-making. This two-stage separate design arises from the inherent limitation of existing metrics that they fail to capture the closed-loop decision utility. Nevertheless, the metric GoT empowers us to undertake a co-design of sampling and decision-making. We adopt the team decision theory, wherein two agents, one embodying the sampler and the other the decision-maker, collaborate to achieve a shared goal. We aim at determining a joint deterministic policy π_C=(π_S,π_A) that minimizes the long-term average cost of the system. It is considered that the sampling and transmission of an update also consumes energy, incurring a C_S cost. In this case, the instant cost of the system could be clarified by GoT^π_A(t)+C_S· a_S(t), and the problem is characterized as: [ P 1: min_π_C∈Υlimsup_T →∞1/T𝔼^π_C( ∑_t = 0^T - 1GoT^π_A(t)+C_S· a_S(t)); ], where π_C=(π_S,π_A) denotes the joint sampling and decision-making policy, comprising π_S=(a_S(0),a_S(1),⋯) and π_A=(a_A(0),a_A(1),⋯), which correspond to the sampling action sequence and actuation sequence, respectively. Note that GoT^π_A(t) is characterized by (<ref>). §.§ Dec-POMDP Formulation The problem in (<ref>) aims to find the optimal decentralized policy π_C so that the long-term average cost of the system is minimized. To solve the problem 𝒫1, we ought to formulate a DEC-POMDP problem, which is initially introduced in <cit.> to solve the cooperative sequential decision issues for distributed multi-agents. Within a Dec-POMDP framework, a team of agents cooperates to achieve a shared goal, relying solely on their localized knowledge. A typical Dec-POMDP is denoted by a tuple ℳ_DEC-POMDP≜⟨ n, ℐ, 𝒜, 𝒯, Ω, 𝒪, ℛ⟩: ∙ n denotes the number of agents. In this instance, we have n=2, signifying the presence of two agents within this context: one agent 𝒜gent_S embodies the semantics-context-aware sampler and transmitter, while the other represents the X̂_t-dependent decision-maker, denoted by 𝒜gent_A. ∙ ℐ is the finite set of the global system status, characterized by (X_t, X̂_t,Φ_t)∈𝒮×𝒮×𝒱. For the sake of brevity, we henceforth denote 𝐖_t=(X_t, X̂_t,Φ_t) in the squeal. ∙ 𝒯 is the transition function defined by 𝒯(𝐰,𝐚,𝐰')≜(𝐖_t+1=𝐰'\|𝐖_t=𝐰,𝐚_t=𝐚), which is defined by the transition probability from global status 𝐖_t=𝐰 to status 𝐖_t+1=𝐰', after the agents in the system taking a joint action 𝐚_t=𝐚=(a_S(t),a_A(t)). For the sake of concise notation, we let p(𝐰'\|𝐰,𝐚) symbolize 𝒯(𝐰,𝐚,𝐰') in the subsequent discourse. Then, by taking into account the conditional independence among X_t+1, Φ_t+1, and X̂_t+1, given (X_t,Φ_t,X̂_t) and 𝐚(t), the transition functions can be calculated in lemma <ref>. The transition functions of the Dec-POMDP: p((s_u,x,v_r)\|(s_i,s_j,v_k),(1,a_m).)= p_i,u^(k,m)· p_k,r·(p_S·1_{x=s_i}+(1-p_S)·1_{x=s_j}), p((s_u,x,v_r)\|(s_i,s_j,v_k),(0,a_m).)= p_i,u^(k,m)· p_k,r·1_{x=s_j}, for any x∈𝒮 and indexes i, j, u∈{1,2,⋯,\|𝒮\|}, k, r∈{1,2,⋯,\|𝒱\|}, and m∈{1,2,⋯,\|𝒜_A\|}. The transition function can be derived by incorporating the dynamics in equations (<ref>), (<ref>), (<ref>), and (<ref>). A more comprehensive proof is presented in Appendix <ref>. ∙ 𝒜=𝒜_S×𝒜_A, with 𝒜_S≜{0,1} representing the set of binary sampling actions executed by the sampler, and 𝒜_A≜{a_0,⋯,a_M-1} representing the set of decision actions undertaken by the actuator. ∙ Ω=Ω_S×Ω_A constitutes a finite set of joint observations. Generally, the observation made by a single agent regarding the system status is partially observable. Ω_S signifies the sampler's observation domain. In this instance, the sampler 𝒜gent_S is entirely observable, with Ω_S encompassing the comprehensive system state o_S^(t)=𝐖_t. Ω_A signifies the actuator's observation domain. In this case, the actuator (or decision-maker) 𝒜gent_A is partially observable, with Ω_A comprising o_A^(t)=X̂(t). The joint observation at time instant t is denoted by 𝐨_t=(o_S^(t),o_A^(t)). ∙ 𝒪=𝒪_S×𝒪_A represents the observation function, where 𝒪_S and 𝒪_A denotes the observation function of the sampler 𝒜gent_S and the actuator 𝒜gent_A, respectively, defined as: 𝒪(𝐰, 𝐨)≜(𝐨_t=𝐨\|𝐖_t=𝐰), 𝒪_S(𝐰, o_S)≜(o_S^(t)=o_S\|𝐖_t=𝐰), 𝒪_A(𝐰, o_A)≜(o_A^(t)=o_A\|𝐖_t=𝐰). The observation function of an agent 𝒜gent_i signifies the conditional probability of agent 𝒜gent_i perceiving o_i, contingent upon the prevailing global system state as 𝐖_t=𝐰. For the sake of brevity, we henceforth let p_A(o_A\|𝐰) represent 𝒪_A(𝐰, o_A) and p_S(o_S\|𝐰) represent 𝒪_S(𝐰, o_A) in the subsequent discourse. In our considered model, the observation functions are deterministic, characterized by lemma <ref>. The observation functions of the Dec-POMDP: p_S((s_u,s_r,v_q)\|(s_i,s_j,v_k)) =1_{(s_u,s_r,v_q)=(s_i,s_j,v_k)} p_A(s_z\|(s_i,s_j,v_k)) =1_{s_z=s_j}· for indexes z, i, j, u, r∈{1,2,⋯ \|𝒮\|}, and k, q∈{1,2,⋯ \|𝒱\|}. ∙ ℛ is the reward function, characterized as a mapping ℐ×𝒜→ℝ. In the long-term average reward maximizing setup, resolving a Dec-POMDP is equivalent to addressing the following problem min_π_C∈Υlimsup_T →∞1/T𝔼^π_C( -∑_t = 0^T - 1 r(t) ). Subsequently, to establish congruence with the problem in (<ref>), the reward function is defined as: r(t)=ℛ^π_A(𝐰,a_S)=-GoT^π_A(t)-C_S· a_S(t). §.§ Solutions to the Infinite-Horizon Dec-POMDP In general, solving a Dec-POMDP is known to be NEXP-complete for the finite-horizon setup <cit.>, signifying that it necessitates formulating a conjecture about the solution non-deterministically, while each validation of a conjecture demands exponential time. For an infinite-horizon Dec-POMDP problem, finding an optimal policy for a Dec-POMDP problem is known to be undecidable. Nevertheless, within our considered model, both the sampling and decision-making processes are deterministic, given as a_S(t)=π_S(𝐰) and a_A(t)=π_A(o_A). In such a circumstance, it is feasible to determine a joint optimal deterministic policy via Brute-Search-across the decision-making policy space. §.§.§ Optimal Solution The idea is based on the finding that, given a deterministic decision-making policy π_A, the sampling problem can be formulated as a standard fully observed MDP problem denoted by ℳ^π_A_MDP≜⟨ℐ,𝒯^π_A,𝒜_S,ℛ⟩. Given a deterministic decision-making policy π_A, the optimal sampling problem could be formulated by a typical fully observed MDP problem ℳ^π_A_MDP≜⟨ℐ,𝒜_S,𝒯_MDP^π_A,ℛ⟩, where the elements are given as follows: ℐ: the same as the pre-defined Dec-POMDP tuple. * 𝒜_S={0,1}: the sampling and transmission action set. * 𝒯^π_A: the transition function given a deterministic decision-making policy π_A, which is 𝒯^π_A(𝐰,a_S,𝐰')=p^π_A(𝐰'\|𝐰,a_S)=∑_o_A∈𝒪_Ap(𝐰'\|𝐰,(a_S,π_A(o_A)))p_A(o_A\|𝐰) , where p(𝐰'\|𝐰,(a_S,π_A(o_A))) could be obtained by Lemma <ref> and p(o_A\|𝐰) could be obtained by Lemma <ref>. * ℛ: the same as the pre-defined Dec-POMDP tuple. We now proceed to solve the MDP problem ℳ^π_A_MDP, which is characterized by a tuple ⟨ℐ,𝒯^π_A,𝒜_S,ℛ⟩. In order to deduce the optimal sampling policy under a deterministic decision-making policy π_A, it is imperative to resolve the Bellman equations <cit.>: θ^_π_A+V_π_A(𝐰)=max_a_S∈𝒜_A{ℛ^π_A(𝐰,a_S)+∑_𝐰'∈ℐp(𝐰'\|𝐰,a_S)V_π_A(𝐰')}, where V^π_A(𝐰) is the value function and θ^_π_A is the optimal long-term average reward given the decision-making policy π_A. We apply the relative value iteration (RVI) algorithm to solve this problem. The details are shown in Algorithm <ref>: With Definition <ref> and Algorithm <ref> in hand, we could then perform a Brute-Force-Search across the decision-making policy space Υ_A, thereby acquiring the joint sampling-decision-making policy. The algorithm is called RVI-Brute-Force-Search Algorithm, which is elaborated in Algorithm 2. In the following theorem, we discuss the optimality of the RVI-Brute-Force-Search Algorithm. The RVI-Brute-Force-Search Algorithm (Algorithm <ref>) could achieve the optimal joint deterministic policies (π_S^,π_A^), given that the transition function 𝒯^π_A follows a unichan. Proof. If the the transition function 𝒯^π_A follows a unichan, we obtain from <cit.> that for any π_A, we could obtain the optimal deterministic policy π_S^* such that θ^(π_A,π_S^)≤θ^(π_A,π_S). Also, Algorithm 2 assures that for any π_A, θ^(π^_A,π_S^)≤θ^(π_A,π_S^). This leads to the conclusion that for any π_C=(π_S,π_A)∈Υ, we have that θ^(π^_A,π_S^)≤θ^(π_A,π_S^)≤θ^(π_A,π_S). Nonetheless, the Brute-Force-Search across the decision-making policy space remains computationally expensive, as the size of the decision-making policy space Υ_A amounts to \|Υ_A\|=𝒜_A^𝒪_A. This implies that executing the RVI algorithm 𝒜_A^𝒪_A times is necessary to attain the optimal policy. Consequently, although proven to be optimal, such an algorithm is ill-suited for scenarios where 𝒪_A and 𝒜_A are considerably large. To ameliorate this challenge, we propose a sub-optimal, yet computation-efficient alternative in the subsequent section. §.§.§ A Sub-optimal Solution The method here is to instead find a locally optimal algorithm to circumvent the high complexity of the Brute-Force-Search-based approach. We apply the Joint Equilibrium-Based Search for Policies (JESP) for Nash equilibrium solutions <cit.>. Within this framework, the sampling policy is optimally responsive to the decision-making policy and vice versa, i.e., ∀π_S, π_A,θ(π_S^,π_A^)≤θ(π_S,π_A^),θ(π_S^,π_A^)≤θ(π_S^,π_A). To search for the Nash equilibrium, we first search for the optimal sampling policy prescribed a decision-making policy. This problem can be formulated as a standard fully observed MDP problem denoted by ℳ^π_A_MDP≜⟨ℐ,𝒜_S,𝒯_MDP^π_A,ℛ⟩ (see Definition <ref>). Next, we alternatively fix the sampling policy π_S and solve for the optimal decision-making policy π_A. This problem can be modeled as a memoryless partially observable Markov decision process (POMDP), denoted by ℳ^π_S_POMDP≜⟨ℐ,𝒜_A,Ω_A,𝒪_A,𝒯^π_S_POMDP,ℛ⟩ (see Definition <ref>). Then, by alternatively iterating between 𝒜gent_S and 𝒜gent_A, we could obtain the Nash equilibrium between the two agents. Given a deterministic sampling policy π_S, the optimal sampling problem could be formulated as a memoryless POMDP problem ℳ^π_S_POMDP≜⟨ℐ,𝒜_A,Ω_A,𝒪_A,𝒯^π_S_POMDP,ℛ⟩, where the elements are given as follows: * ℐ, Ω_A, 𝒜_A, and 𝒪_A: the same as the pre-defined Dec-POMDP tuple. * 𝒯^π_S_POMDP: the transition function given a deterministic sampling policy π_S, which is 𝒯^π_S_POMDP(𝐰,a_A,𝐰')=p^π_S(𝐰'\|𝐰,a_A)=p(𝐰'\|𝐰,(π_S(𝐰),a_A)) , where p(𝐰'\|𝐰,(π_S(𝐰),a_A)) could be obtained by Lemma <ref>. * ℛ: the reward function is denoted as ℛ^π_S(𝐰,a_A), which could be obtained by (<ref>). We then proceed to solve the memoryless POMDP problem discussed in Definition <ref> to obtain the deterministic decision-making policy. Denote p_π_A^π_S(𝐰'\|𝐰) as the transition probability {𝐖_t+1=𝐰'\|𝐖_t=𝐰} under the sampling policy π_A and π_S, we then have that p_π_A^π_S(𝐰'\|𝐰)=∑_o_A∈𝒪_Ap(o_A\|𝐰)∑_a_A∈𝒜_Ap^π_S(𝐰'\|𝐰,a_A)π_A(a_A\|o_A), where p(o_A\|𝐰) could be obtained by Lemma <ref> and p^π_S(𝐰'\|𝐰,π_A(o_A)) is obtained by (<ref>). By assuming the ergodicity of the p^π_S_π_A(𝐰'\|𝐰) and rewrite it as a matrix 𝐏_π_A^π_S, we could then solve out the stationary distribution of the system status under the policies π_S and π_A, denoted as μ_π_A^π_S, by solving the balance equations: μ_π_A^π_S𝐏_π_A^π_S=μ_π_A^π_S , μ_π_A^π_S𝐞=1, where 𝐞 is the all one vector [1,⋯,1]_\|ℐ\|×1, μ_π_A^π_S could be solved out by Cramer's rules. Denote μ_π_A^π_S(𝐰) as the stationary distribution of 𝐰. Also, we denote r_π_A^π_S(𝐰) as the expectation reward of global system status 𝐰 under policies π_A and π_S. It could be calculated as: r_π_A^π_S(𝐰)=∑_o_A∈𝒪_Ap(o_A\|𝐰)ℛ^π_S(𝐰,π_A(o_A)). The performance measure is the long-term average reward: η_π_A^π_S= limsup_T →∞1/T𝔼^(π_S,π_A)(∑_t = 0^T - 1 r(t) )=∑_𝐰∈ℐμ_π_A^π_S(𝐰)· r_π_A^π_S(𝐰) With η_π_A^π_S in hand, we then introduce the relative reward g_π_A^π_S(𝐰), defined by g_π_A^π_S(𝐰)≜limsup_T →∞1/T𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|𝐖_0=𝐰. ], which satisfies the Poisson equations <cit.>: η_π_A^π_S+g_π_A^π_S(𝐰)=r_π_A^π_S(𝐰)+∑_𝐰'∈ℐp_π_A^π_S(𝐰'\|𝐰)g_π_A^π_S(𝐰'). Denote 𝐠^π_S_π_A as the vector consisting of g_π_A^π_S(𝐰), 𝐫^π_S_π_A as the vector consisting of r^π_S_π_A(𝐰),𝐰∈ℐ. 𝐠^π_S_π_A could be solved by utilizing <cit.>: 𝐠^π_S_π_A=[(I-𝐏^π_S_π_A+𝐞μ^π_S_π_A)^-1-𝐞μ^π_S_π_A]𝐫^π_S_π_A With the relative reward g_π_A^π_S in hand, we then introduce Q_π_A^π_S(𝐰,a_A) and Q_π_A^π_S(o_A,a_A) as follows: Q_π_A^π_S(𝐰,a_A) and Q_π_A^π_S(o_A,a_A) are defined and calculated as: Q_π_A^π_S(𝐰,a_A)≜limsup_T →∞1/T𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|𝐖_0=𝐰,a_A(0)=a_A. ] =ℛ^π_S(𝐰,a_A)-η_π_A^π_S+∑_𝐰'∈ℐp^π_S(𝐰'\|𝐰,a_A)g_π_A^π_S(𝐰'), Q_π_A^π_S(o_A,a_A)≜limsup_T →∞1/T𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|o_A^(0)=o_A,a_A(0)=a_A. ] =∑_𝐰∈ℐp_π_A^π_S(𝐰\|o_A)Q_π_A^π_S(𝐰,a_A), where p_π_A^π_S(𝐰'\|𝐰) can be obtained by (<ref>) and p_π_A^π_S(𝐰\|o_A) can be obtained by the Bayesian formula: p_π_A^π_S(𝐰\|o_A)=μ_π_A^π_S(𝐰)p(o_A\|𝐰)/∑_𝐰∈ℐμ_π_A^π_S(𝐰)p(o_A\|𝐰). Please refer to Appendix <ref>. With Q_π_A^π_S(o_A,a_A) in hand, it is then easy to conduct the Policy Iteration (PI) Algorithm with Step Sizes <cit.> to iteratively improve the deterministic memoryless decision-making policy π_A. The detailed steps are shown in Algorithm <ref>. Thus far, we have solved two problems: i) by capitalizing on Definition <ref> and Algorithm <ref>, we have ascertained an optimal sampling strategy π_S^* contingent upon the decision-making policy π_A; ii) by harnessing Definition <ref> and Algorithm <ref>, we have determined an optimal actuation strategy π_S^* predicated on the decision-making policy π_A. Consequently, we could iteratively employ Algorithm 1 and Algorithm <ref> in an alternating fashion, whereby Algorithm 1 yields the optimal sampling strategy π_S^(k)(π_A^(k-1)), subsequently serving as an input for Algorithm 3 to derive the decision-making policy π_A^(k)(π_S^(k)). The procedure shall persist until the average reward θ^(π_S^(k),π_A^(k)) reaches convergence, indicating that the solution achieves a Nash equilibrium between the sampler and the actuator.The intricacies of the procedure are delineated in Algorithm <ref>. Generally, the JESP algorithm should restart the algorithm by randomly choosing the initial decision-making policy π_A^(1) to ensure a good solution, as the initialization of decision-making policy π_A^(1) may often lead to poor local optima. We here investigate a heuristic initialization to find the solution quickly and reliably. Specifically, we assume that the decision-maker is fully observable and solve a MDP problem: ℳ_MDP≜⟨ℐ,𝒜_A,𝒯_MDP,ℛ⟩, where the elements are given as follows: * ℐ: the set of (X_t,Φ_t)∈𝒮×𝒱. * 𝒜_A={a_0,a_M-1}: the decision-making set. * 𝒯: the transition function, given as 𝒯((X_t,Φ_t),a_A,(X_t+1,Φ_t+1)) =p(X_t+1\|X_t,a_A,Φ_t)· p(Φ_t+1\|Φ_t), where p(X_t+1\|X_t,a_A,Φ_t) and p(Φ_t+1\|Φ_t) could be obtained by (<ref>) and (<ref>), respectively. * ℛ: the same as the pre-defined Dec-POMDP tuple. Through solving the above MDP problem, we could explicitly obtain the Q function Q(X_t,Φ_t,a_A), define Q(X_t,a_A) as 𝔼_Φ_t[Q(X_t,Φ_t,a_A)], the initial decision-making policy π_A^(1) is given as π_A^(1)(X̂_t)=min_a_AQ(X̂_t,a_A). § SIMULATION RESULTS Tradition metrics such as Age of Information have been developed under the assumption that a fresher packet or more accurate packet, capable of aiding in source reconstruction, holds a higher value for the receiver, thus promoting goal-oriented decision-making. Nevertheless, the manner in which a packet update impacts the system's utility via decision-making remains an unexplored domain. Through the simulations, we endeavor to elucidate the following observations of interest: ∙ GoT-optimal vs. State-of-the-art. In contrast with the state-of-the-art sampling policies, the proposed goal-oriented sampler & decision-maker co-design is capable of concurrently maximizing goal attainment and conserving communication resources, accomplishing a closed-loop utility optimization via sparse sampling. (See Fig. <ref> and <ref>) ∙ Separate Design vs. Co-Design. Compared to the two-stage sampling-decision-making separate framework, the co-design of sampling and decision-making not only achieves superior goal achievement but also alleviates resource expenditure engendered by communication and actuation implementation. (See Fig. <ref> and <ref>) ∙ Optimal Brute-Force-Search vs. Sub-optimal JESP. Under different successful transmission probability p_S and sampling cost C_S, the sub-optimal yet computation-efficient JESP algorithm will converge to near-optimal solutions. (See Fig. <ref>) ∙ Trade-off: Transmission vs. Actuation. There is a trade-off between transmission and actuation in terms of resource expenditure: under reliable channel conditions, it is apt to increase communication overhead to ensure effective decision-making; conversely, under poor channel conditions, it is advisable to curtail communication expenses and augment actuation resources to attain maximal system utility. (See Fig. <ref>) §.§ Comparing Benchmarks Fig. <ref> illustrates the simulation results, which characterizes the utility by the average cost composed by status inherent cost C_1(X_t,Φ_t), actuation gain cost C_2(π_A(X̂_t)), actuation inherent cost C_3(π_A(X̂_t)), and sampling cost C_S. For the simulation setup, we set 𝒜_A={a_0,⋯,a_10}, 𝒮={s_0,s_1,s_2}, 𝒱={v_0,v_1} and the corresponding costs are given as: 𝐂_1^\|𝒱\|×\|𝒮\|=[ 0 20 50; 0 10 20 ], C_2(π_A(X̂_t)) and C_3(π_A(X̂_t)) are both linear to the actuation with C_2(π_A(X̂_t)=C_g·π_A(X̂_t) and C_3(π_A(X̂_t))=C_I·π_A(X̂_t). The following comparing benchmarks of interest are considered: ∙ Uniform. Sampling is triggered periodically in this policy. In this case, a_S(t)=1_{t=K*Δ}, where K=0,1,2,⋯ and Δ∈ℕ^+. For each Δ, we could calculate the sampling rate as 1/Δ and explicitly obtain the long-term average cost through Markov chain simulations under pre-defined decision-making policy π_A=[a_0,a_3,a_7]. The setup of Δ represents a trade-off between utility and sampling frequency, as depicted in Fig. <ref>: If C_S is minimal, sampling and transmission will contribute positively to the utility; If the sampling action is expansive, sampling may not yield adequate utility; If a single sampling consumes moderate resource, the utility will exhibit a U-shaped pattern in terms of the sampling rate. ∙ Age-aware. Sampling is executed when the AoI attains a predetermined threshold, a principle that has been established as a threshold-based result for AoI-optimal sampling <cit.>. In this case, a_S(t)=1_{AoI(t)>δ}, where the AoI-optimal threshold δ can be ascertained using the Bisection method delineated in Algorithm 1 of <cit.>. In this context, rather than determining a fixed threshold that minimized AoI, we dynamically shift the threshold to explore the balance between sampling and utility. As evidenced in Fig. <ref>, the utility derived from this sampling policy consistently surpasses that of the uniform sampling policy. ∙ Change-aware. Sampling is triggered whenever the source status changes. In such a case, a_S(t)=1_{X_t X_t-1}. The performance of this policy is dependent on the dynamics of the system, i.e., if the semantics of the sources transfers frequently, then the sampling rate will be higher. The utility of this policy may be arbitrarily detrimental owing to its semantics-unaware nature. In our considered model, the Change-aware policy will turn out 89.5% to have the unsynchronized status X_t=s_0 while X̂_t=s_3. In this case, the actuator will implement the actuation a_A=a_7 according to the estimate X̂_t=s_3, which will in turn make the status X_t converges to s_0. ∙ Optimal MSE. This is a type of E2E goal-oriented sampling policy if the goal is determined as achieving real-time reconstruction. Nevertheless, this policy disregards the semantics conveyed by the packet and the ensuing actuation updating precipitated by semantics updates. The problem could be formulated as a standard MDP formulation and solved out through RVI Algorithm. The sampling rate and average cost are obtained given the MSE-optimal sampling policy and the pre-defined decision-making policy π_A=[a_0,a_3,a_7]. ∙ Optimal AoII (also optimal AoCI). From <cit.>, it has been proven that the AoII-optimal sampling policy turns out to be a_S(t)=1_{X_tX̂_t }. From <cit.>, the AoCI-optimal sampling policy is a_S(t)=1_{X_tX_t-AoI(t)}. Note that X̂_t=X_t-AoI(t), these two sampling policies are equivalent. The sampling rate and average cost are obtained given this sampling policy and the greedy-based decision-making policy. π_A=[a_0,a_3,a_7]. §.§ Separate Design vs. Co-Design Conventionally, the sampling and actuation policies are designed in a two-stage manner: they first emphasize open-loop performance metrics such as average mean squared error (MSE) or average Age of Information (AoI), we then focus on the decision-making policy π_A design. Specifically, we consider that π_A is predetermined using a greedy methodology: π_A(X̂_t)=min_a_A∈𝒮_A𝔼_Φ_t{[C_1(X̂_t,Φ_t) - C_2(π_A(X̂_t))]^+ C_3(π_A(X̂_t))}. This greedy approach entails selecting the actuation that minimizes cost in the current step, given that the estimate X̂_t is perfect. By calculating (<ref>), we obtain π_A = [a_0,a_3,a_7]. However, we notice that sampling and actuation are closely intertwined, highlighting the potential for further co-design. In this paper, we have proposed the RVI-Brute-Force-Search and the Improved JESP algorithms for such optimal co-design. As shown in Fig. <ref>, the sampler & decision-maker co-design achieves the optimal utility through sparse sampling. Specifically, only semantically important information is sampled and transmitted, while non-essential data is excluded. This goal-oriented, semantic-aware, and sparse sampling design represents a significant advancement in sampling policy design. By incorporating a best-matching decision-making policy, the sparse sampling achieves superior performance compared to existing methods. Fig. <ref> presents a comparative analysis between the AoII (or AoCI)-optimal sampling policy, the MSE-optimal sampling policy, and our proposed GoT-optimal sampler & decision-maker co-design. It is verified that under different p_S and C_S, the proposed GoT-optimal sampler & decision-maker co-design achieves the optimal goal-oriented utility. Importantly, under the condition of an extremely unreliable channel p_S=0.2 and high sampling cost C_S=10, the proposed co-design facilitates a significant reduction in long-term average cost, exceeding 60%. This underscores the superiority of the GoT-optimal sampler & decision-maker co-design. §.§ Optimal vs. Sub-Optimal Fig. <ref> presents a comparative visualization between optimal and sub-optimal solutions over a wide range of C_S and p_S values. The negligible zero-approaching value in Fig. <ref> implies a trivial deviation between the optimal and sub-optimal solutions, suggesting the latter's potential for convergence towards near-optimal outcomes. The minimal variance testifies to the sub-optimal algorithm's consistent ability to approximate solutions with high proximity to the optimal. This critical observation underscores the practical advantages of employing sub-optimal improved JESP Algorithm, especially in scenarios with extensive 𝒜_A and 𝒪_A. §.§ Trade-off: Transmission vs. Actuation Fig. <ref> exemplifies the resource allocation trade-off between transmission and actuation when the long-term average cost is minimized. When the probability of p_S remains low (signifying an unreliable channel) or C_S is high (indicating expensive sampling), it becomes prudent to decrease sampling and transmission, while concurrently augmenting actuation resources for optimal system utility. In contrast, when the channel is reliable, sampling and transmission resource can be harmonized with actuation resources to achieve the goal better. This indicates that through the investigation of the optimal co-design of the sampler & decision-maker paradigm, a trade-off between transmission and actuation resources can be achieved. § CONCLUSION In this paper, we have investigated the GoT metric to directly describe the goal-oriented system decision-making utility. Employing the proposed GoT, we have formulated an infinite horizon Dec-POMDP problem to accomplish the co-design of sampling and actuating. To address this problem, we have developed two algorithms: the computationally intensive RVI-Brute-Force-Search, which is proven to be optimal, and the more efficient, albeit suboptimal algorithm, named JESP Algorithm. Comparative analyses have substantiated that the proposed GoT-optimal sampler & decision-maker pair can achieve sparse sampling and meanwhile maximize the utility, signifying the initial realization of a sparse, goal-oriented, and semantics-aware sampler design. IEEEtran § THE PROOF OF LEMMA <REF> By taking into account the conditional independence among X_t+1, Φ_t+1, and X_t+1, given (X_t,Φ_t,X_t) and 𝐚(t), we we can express the following: {𝐖_t+1=(s_u,x,v_r)\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).} ={X_t+1=s_u\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}× {X̂_t+1=x\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}× {Φ_t+1=v_r\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}, wherein the first, second, and third terms can be derived through conditional independence, resulting in simplified expressions of (<ref>), (<ref>), and (<ref>), respectively: {X_t+1=s_u\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}=p_i,u^(k,m), {X̂_t+1=x\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}=p_S·1_{x=s_i}+(1-p_S)·1_{x=s_j}, {Φ_t+1=v_r\|𝐖_t=(s_i,s_j,v_k),𝐚(t)=(1,a_m).}=p_k,r, Substituting (<ref>), (<ref>), and (<ref>) into (<ref>) yields the (<ref>) in Lemma <ref>. In the case where a_S(t)=0, we can obtain a similar expression by replacing 𝐚(t)=(1,m) with 𝐚(t)=(0,m). Substituting (<ref>), (<ref>), and (<ref>) into this new expression results in the proof of (<ref>) in Lemma <ref>. § PROOF OF LEMMA 3 Q_π_A^π_S(𝐰,a_A) could be simplified as follows: limsup_T →∞1/T𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|𝐖_0=𝐰,a_A(0)=a_A. ] =ℛ^π_S(𝐰,a_A)-η_π_A^π_S+∑_𝐰'∈ℐp^π_S(𝐰'\|𝐰,a_A)·limsup_T →∞1/T-1𝔼^(π_S,π_A)[∑_t = 1^T - 1(r(t)-η_π_A^π_S) \|𝐖_1=𝐰'. ]_g_π_A^π_S(𝐰') =ℛ^π_S(𝐰,a_A)-η_π_A^π_S+∑_𝐰'∈ℐp^π_S(𝐰'\|𝐰,a_A)g_π_A^π_S(𝐰'), Q_π_A^π_S(o_A,a_A) could be solved as: limsup_T →∞1/T𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|o_A^(0)=o_A,a_A(0)=a_A. ] =∑_𝐰∈ℐp_π_A^π_S(𝐰\|o_A,a_A)·limsup_T →∞1/T-1𝔼^(π_S,π_A)[∑_t = 0^T - 1(r(t)-η_π_A^π_S) \|𝐖_0=𝐰,a_A(0)=a_A. ]_Q_π_A^π_S(𝐰,a_A) =∑_𝐰∈ℐp_π_A^π_S(𝐰\|o_A,a_A)Q_π_A^π_S(𝐰,a_A). where p_π_A^π_S(𝐰\|o_A,a_A) is the posterior conditional probability. Note that the state 𝐰 is independent of a_A when o_A is known, we have that p_π_A^π_S(𝐰\|o_A,a_A)=p_π_A^π_S(𝐰\|o_A)=p_π_A^π_S(𝐰,o_A)/p_π_A^π_S(o_A)=μ_π_A^π_S(𝐰)p(o_A\|𝐰)/∑_𝐰∈ℐμ_π_A^π_S(𝐰)p(o_A\|𝐰).
http://arxiv.org/abs/2307.02220v1	20230705115331	Spherical Basis Functions in Hardy Spaces with Localization Constraints	[ "Christian Gerhards", "Xinpeng Huang" ]	math.NA	[ "math.NA", "cs.NA" ]	1]C. [email protected] 1]X. [email protected] [1] TU Bergakademie Freiberg, Institute of Geophysics and Geoinformatics, Gustav-Zeuner-Str. 12, 09599 Freiberg, Germany 1.1 equationsection thmTheorem[section] lem[thm]Lemma prop[thm]Proposition cor[thm]Corollary definition settSetting defi[thm]Definition expl[thm]Example rem[thm]Remark abst [c]0.9 Abstract. key[c]0.9 Keywords. amsclass[c]0.9 AMS Subject Classification. Spherical Basis Functions in Hardy Spaces with Localization Constraints [ ======================================================================= Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces H_+() and H_-(), respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration. § INTRODUCTION It is well-known that the space of square-integrable vector fields on the sphere can be decomposed as follows: L^2(,^3)=H_+()⊕ H_-() ⊕ D(), where H_+() and H_-() denote the Hardy spaces of inner and outer harmonic gradients, respectively, and D() the space of tangential divergence-free vector fields (see, e.g., <cit.>, with roots in terms of vector spherical harmonic representations going back to <cit.>). Applications to inverse magnetization problems and the separation of magnetic fields into contributions of internal and external origin are various and can be found, e.g., in <cit.>. Concerning the inverse magnetization problem, knowing the magnetic field in the exterior of a magnetized sphere, only the H_+()-component of the underlying magnetization can be reconstructed uniquely. However, if it is known a priori that the magnetization is locally supported in some proper subdomain of the sphere, also the H_-()-component is determined uniquely (which may help for the reconstruction of certain properties of the magnetization, e.g., the direction of an underlying inducing field or the susceptibility in the case of a known direction of the inducing field, e.g., <cit.>). Therefore, it is of interest to have a suitable setup that allows the computation of the H_-()-component based on knowledge of the H_+()-component under the assumption that the underlying vector field is locally supported. A first step is to investigate appropriate approximation methods in H_+() and H_-(), respectively, under such localization constraints. The latter is the main motivation and content of the paper at hand. More precisely, let L^2(Σ^c,^3) be the space of square-integrable vector fields that are locally supported in a subdomain Σ^c⊂, and let P_+(L^2(Σ^c,^3)) and P_-(L^2(Σ^c,^3)) be the subspaces obtained by orthogonal projection onto the corresponding Hardy space H_+() and H_-(), respectively. Here, we will investigate approximation in these projected subspaces. The specific vectorial Slepian functions from, e.g., <cit.> can address this to a certain extent. However, their construction requires joint optimization of the H_+()-, H_-()-, and D()-contributions and often focuses on a spectral representation. Furthermore, the computations have to be redone for every new subdomain of the sphere. Spherical basis functions, on the other hand, have the advantage that only their centers need to be adapted if the subdomain under consideration changes. Additionally, the construction as presented here allows for a separate treatment of the H_+()- and H_-()-contributions, respectively. The course of this paper will be as follows: It is known from <cit.> that the orthogonal projection of L^2(Σ^c,^3) onto the Hardy space H_+() produces the subspace P_+(L^2(Σ^c,^3))=B_+(D_+,Σ), where B_+ denotes a particular vectorial linear operator that is expressible via layer potentials (cf. Section <ref> for details and notations) and D_+,Σ⊂ L^2() denotes the scalar function space D_,+=L^2(^c)+(K+12I)V_ (cf. Section <ref> for details and notations; for now, we let K denote the double layer potential and V_Σ the space of functions whose single layer potential is harmonic in Σ). Thus, since the operator B_+ is bounded and invertible, the problem of finding adequate vectorial spherical basis functions for the approximation of the H_+()-component of locally supported vector fields reduces to finding adequate (scalar) spherical basis functions for approximation in the subspace D_+,Σ. The latter is, in fact, what we focus on in this paper (a similar statement holds true for the H_-()-component and a related subspace D_-,Σ). The structure of D_+,Σ according to (<ref>) indicates that it suffices to investigate approximation in L^2(Σ^c) and approximation in V_Σ. The former is discussed in Section <ref>, the latter in Section <ref>. An approximation result for locally supported spherical basis functions similar to the one in Section <ref> has already been obtained, e.g., in <cit.>. However, in their case, the support of the used spherical basis functions does not necessarily lie within Σ^c, while we want to enforce this support condition in order to guarantee that the functions used for approximation are members of L^2(Σ^c) themselves. In a Euclidean setup, the latter has been achieved in <cit.>. We simply transfer their result to the sphere by applying the spherical techniques used, e.g., in <cit.> and the stereographic projection. For the study of approximation in V_Σ we rely on regularized fundamental solutions of the Laplace-Beltrami operator, as described, e.g., in <cit.>. These spherical basis functions have the advantage that they are locally harmonic and naturally lead to a function system that is contained in V_Σ itself. Both results are then joined in Section <ref> to obtain the desired approximation property in D_+,Σ, which is stated in Theorem <ref>. As indicated before, the new aspect here is not the approximation property in general but that the investigated spherical basis functions lead to vectorial functions that are themselves members of the projected subspace P_+(L^2(Σ^c,^3)) of H_+(). Sections <ref> and <ref> additionally provide some considerations on locally supported vector fields of minimum norm and their representation in the given context. Sections <ref> and <ref> gather the required notations and background material, some numerical illustration is provided in Section <ref>. § GENERAL PRELIMINARIES We define some necessary notations required throughout the course of this paper and gather some known results on spherical basis function interpolation, cubature on the sphere, and the fundamental solution for the Laplace-Beltrami operator on the sphere. §.§ Sobolev spaces on the sphere Let ={x∈^3:\|x\|=1} denote the unit sphere. We restrict ourselves to the three-dimensional case to simplify some considerations, especially in Sections <ref> and <ref>, although they should also hold in higher dimensional setups. Let {Y_n,k}_n∈ℕ,k=-n,…,n denote an orthonormal set of real spherical harmonics of order n and degree k (each spherical harmonic of degree n and order k is the restriction of a homogeneous, harmonic polynomial of degree n to the sphere; details can be found, e.g., in <cit.>). For an infinitely often differentiable function f:→ of class C^∞(), one can write f=∑_n=0^∞∑_k=-n^nf̂_n,kY_n,k with Fourier coefficients f̂_n,k=∫_ f(y)Y_n,k(y)dω(y), where ω denotes the surface measure on the sphere. When dealing with such Fourier coefficients, we often say that we are arguing in spectral domain. The Sobolev space H^s() of smoothness s∈ is defined as the closure of C^∞() with respect to the norm ·_H^s() induced by the inner product ⟨ f,g⟩_H^s()=∑_n=0^∞∑_k=-n^n(n+1/2)^2sf̂_n,k ĝ_n,k, i.e., H^s()=C^∞()^·_H^s(). In particular, it holds that H^0()=L^2(), the space of square-integrable functions, and that C^0()⊂ H^s() for s>1, where C^0() denotes the set of continuous functions on the sphere. By ·_C^0() we denote the supremum norm. At some occasions, we use the notation L^2()/⟨1⟩ to indicate the space of functions f in L^2() with vanishing mean ⟨ f,1⟩_L^2()=0. We denote by the (surface) gradient on the sphere. For f in H^1(), its gradient f is well-defined and lies in the space L^2(,^3) of square-integrable vector fields, which is equipped with the inner product ⟨,⟩_L^2()=∫_(y)·(y) dω(y) and the corresponding norm ·_L^2(,^3). Typically, we denote vector fields ,:→^3 by bold-face letters. The (surface) divergence operator · is defined as the negative adjoint -^* (with respect to the L^2-inner product) and we define the Laplace-Beltrami operator via Δ_=·=-^. If Σ⊂ is a Lipschitz domain with connected boundary, then the space of square-integrable locally supported vector fields on Σ is denoted by L^2(Σ,^3), naturally embedded in L^2(,^3) via extension by zero. Furthermore, we define H^s(Σ) as the closure of C_0^∞(Σ)={f∈ C^∞():(f)⊂Σ} with respect to the ·_H^s()-norm. Thus, H^s(Σ) is naturally embedded in H^s(). For f in H^2() it holds that (Δ_ f)^∧_n,k=-n(n+1)f̂_n,k and, consequently, that ((-Δ_+1/4) f)^∧_n,k=(n+1/2)^2f̂_n,k. Thus, one can (formally) see from (<ref>) that a function f is in H^s() if and only if (-Δ_+1/4)^s/2f is in . A function f:→ is called 𝐩-zonal (for some fixed 𝐩∈) if there exists a function F:[-1,1]→ such that f(x)=F(x·𝐩) for all x∈. In that case, we define _xf(y)=f(z), for z∈ chosen such that x· y=z·𝐩. This allows the definition of the convolution fg of two functions f and g via f∗ g (x)= ∫__xf(y) g(y)dω(y), x∈. For F:[-1,1]→, the notation Fg is meant in the sense F∗ g (x)= ∫_ F(x· y) g(y)dω(y), naturally relating to the definition (<ref>). If f is a 𝐩-zonal function in s and g is in k, for some s,k∈, then f∗ g_s+k+1/2≤ C f_s g_k, where C>0 is a constant depending on s and k. A proof of the proposition above is provided in Appendix <ref>. Furthermore, the following estimate will be helpful. <cit.> Let s>1. Then s is a Banach algebra, i.e., there exists a constant C>0, depending on s, such that for any f,g in s, the following inequality holds fg_s≤ Cf_sg_s. To conclude this section, we want to mention that occasionally it is convenient to rely on known results in the Euclidean setup by using the stereographic projection. Given a fixed pole 𝐩∈, let :∖{𝐩}→^2 be the stereographic projection as defined, e.g., in <cit.>. Typically we do not indicate the pole explicitly in the notation of the stereographic projection. If it should be required for clarity, we write instead of . The stereographic projection of a set Σ⊂∖{𝐩} is given by (Σ)={(x):x∈Σ}. Furthermore, the stereographic projection of a function f:∖{𝐩}→ is defined via f:^2 →, y ↦ f(^-1(y)). The following are some useful estimates involving the stereographic projection. Let 𝐩∈ be fixed and Σ⊂ be a Lipschitz domain with connected boundary whose closure satisfies Σ⊂∖{𝐩}. Then there exist constants C>c>0 such that, for any x,y∈Σ, c\|x-y\| ≤ \|(x)-(y)\|≤ C \|x-y\|, c ω(Σ) ≤μ((Σ))≤ C ω(Σ), where μ denotes the Lebesgue measure in ^2 and ω the surface measure on the sphere . Let s≥ 0, then there exist constants C≥ c>0 (possibly different from above), depending on s, such that for any f∈ H^s(Σ) the following inequalities hold true: cf_H^s()≤ f_H^s(^2)≤ Cf_H^s(). Throughout the course of this article, we allow all appearing constants to depend on the subdomain Σ⊂ without explicitly mentioning this. In cases where the constant does not depend on the subdomain, this should become clear from the context. §.§ Cubature rules on the sphere Given a set of points X={x_1,…,x_N}⊂ and weights w_1,…,w_N∈, then the corresponding cubature rule Q_X takes the form Q_Xf=∑_i=1^N w_i f(x_i). We say that Q_X has polynomial precision of degree L if Q_Xf=∫_ f(y)dω(y) for all spherical polynomials of degree at most L, i.e., for all functions of the form f=∑_n=0^L∑_k=-n^nf̂_n,kY_n,k. If all weights are positive, i.e., if w_i>0 for all i=1,,…,N, then we call Q_X a cubature rule with positive weights. The spatial distribution of the points in X is paramount for the existence of cubature rules with a prescribed polynomial precision and positive weights. An important parameter that provides information on the spatial distribution is the mesh width of X relative to some subset Σ⊂, which is defined by h_X, Σ=sup_y∈Σmin_x_i∈ X∩Σy-x_i. If Σ=, we typically abbreviate h_X, Σ simply by h_X. The separation distance is defined by q_X,Σ=1/2min_x_i,x_j∈ X∩Σ x_i≠x_jx_i-x_j, again, we drop the index Σ if Σ=. (Existence of cubature rules <cit.>) There exists a constant C>0 such that following holds true: For every pointset X⊂ and for every L∈ℕ satisfying h_X≤C/L, there exist positive weights w_1, … ,w_N>0 such that the corresponding cubature rule Q_X has polynomial precision of degree L. The existence of cubature rules with polynomial precision of degree L and positive weights then guarantees the following error estimate for our later considerations in Section <ref>. (Error estimate for cubature rules <cit.>) Let s>1 and X⊂. Then there exist a constant C>0 that depends on s such that, for every L∈ℕ and for every cubature rule Q_X with polynomial precision of degree L and positive weights, the worst-case error in H^s() can be estimated by sup_f∈s,f_s≤ 1\|Q_Xf-∫_ f(y)dω(y)\|≤C/L^s. §.§ Spherical basis functions and interpolation In this section, we briefly recapitulate some definitions and results for spherical basis functions derived from classical radial basis functions, as indicated, e.g., in <cit.>. Let ψ:[0,∞)→ be compactly supported with (ψ)⊂[0,1] such that ψ(\|·\|):^3→ is a positive definite radial basis function. Then we define Ψ:×→ via Ψ(x,y)=ψ(\|x-y\|) and introduce scaled versions, for δ>0, via Ψ_δ(x,y)=1/δ^2Ψ(x/δ,y/δ), x,y∈. We call these restrictions to the sphere spherical basis functions. Often, we use the notation Ψ_x and Ψ_δ,x to more concisely express the functions Ψ(x,·) and Ψ_δ(x,·), respectively. Further, observing that Ψ(x,y)=∑_n=0^∞∑_k=-n^nΨ̂_n Y_n,k(x)Y_n,k(y) for adequate coefficients Ψ̂_n∈[0,∞), we define a norm ·_Ψ via f_Ψ^2 = ∑_n=0^∞∑_k=-n^nf̂_n,k^2 /Ψ̂_n, f∈ C^∞(). The native space corresponding to Ψ is then given as the closure of C^∞() with respect to this norm ·_Ψ. Analogous definitions hold for Ψ_δ, where the involved coefficients are denoted by (Ψ_δ)^∧_n. A popular choice for the underlying ψ in Definition <ref> that we will use in our numerical examples is, e.g., the Wendland function (cf. <cit.>) ψ(r) =(1-r)_+^4(4r+1), which corresponds to the native space H^5/2(), although various other choices are equally valid. A closed-form representation of its Legendre coefficients is provided in <cit.>. <cit.> Let Ψ_δ be given as in Definition <ref>, and the underlying ψ be given as in (<ref>). Then, Ψ_δ has the following spherical harmonic expansion Ψ_δ(x,y)=∑_n=0^∞∑_k=-n^nΨ̂^δ_n (x)(y)=∑_n=0^∞2n+1/4πΨ̂^δ_n P_n(x· y), with coefficients Ψ̂^δ_n=π/7 _3F_2(-n,n+1,5/2;4,9/2;δ^2/4), where _3F_2 denotes the general hypergeometric function. Next, we state some known properties for interpolation via spherical basis functions. <cit.> Let X⊂ have meshwidth h_X, and let s>1. Then there exists a constant C>0, depending on s, such that, for all f in s with f\|_X=0, it holds f_L^2()≤ C h_X^sf_s. We do not actually need the spherical estimate from Proposition <ref> in the later course of this paper, it is stated mainly for illustration purposes. For the proof of Theorem <ref>, we will infact use the stereogrpahic projection in order to retreat to the Euclidean setup and rely on related results from <cit.>. Let Ψ be as in Definition <ref> with native space H^s(), for some s>1. Moreover, let X⊂ and denote by I_X,Ψ:H^s()→span{Ψ_x: x∈ X} the interpolation operator that is characterized by I_X,Ψ(f)\|_X=f\|_X. Then, it holds f_Ψ^2 =f-I_X,Ψ(f)_Ψ^2+I_X,Ψ(f)_Ψ^2. Some useful results on the spectral behaviour of the scaled spherical basis functions and on the boundedness of the corresponding interpolation operator are the following. <cit.> Let Ψ be as in Definition <ref> where ψ(\|·\|) is a radial basis funtion in ^3 with native space H^s+1/2(^3) for some s>1, then Ψ is a spherical basis function with native space H^s(). Moreover, there exist constants C≥ c>0, depending on s, such that c(1+δ n)^-2s≤(Ψ_δ)^∧_n ≤ C(1+δ n)^-2s. As a result, there exists constants C̅≥c̅>0, depending on s, such that, for any f in H^s(), c̅f_Ψ_δ≤f_s≤C̅δ^-sf_Ψ_δ. Let Ψ satisfies the condition in Proposition <ref>. Given X⊂, let c>1 be such that h_X< c q_X, and let the scaling factor δ satisfy ν q_X<δ< cν q_X for some ν≥ 1. Then there exists a constant C>0, depending on c, ν, and s, such that it holds for any f in H^s() that I_X, Ψ_δ(f) _L^∞()≤ C f _L^∞(). Proposition <ref> is a simplified spherical version of <cit.>. A proof can be found the Appendix <ref>. §.§ Regularized fundamental solution The function G(;·,·):{(x,y)∈×:x≠y}→ given by G(;x,y)=1/4πln(1-x· y)+1/4π(1-ln2). is called fundamental solution with respect to and is twice continuously differentiable on its domain. It satisfies y G(;x,y) = -1/4π, x≠y, and it has the Fourier expansion, for x≠y, G(;x,y)=∑_n=1^∞∑_k=-n^n1/-n(n+1)Y_n,k(x)Y_n,k(y)=1/4π∑_n=1^∞2n+1/- n(n+1)P_n(x· y), where P_n denotes the Legendre polynomial of degree n (see, e.g., <cit.>). Furthermore, Green's third formula yields for f in H^3+κ(), with κ>0, that f(x)=1/4π∫_f(y)dω(y) +∫_ G(;x,y) yf(y)dω(y), x∈. From (<ref>), we can easily check, for fixed x∈, that G(;x,·) lies in the Sobolev space H^s() for any s<1 but not for s≥ 1. Regularization of the fundamental solution around its singularity leads to higher smoothness and is considered here for later use. A once continuously differentiable regularization will suffice for our purposes and a particular choice is presented in Definition <ref>, although various other choices are possible as well. The concept is described and applied in more detail, e.g., in <cit.>. For fixed ρ∈(0,2), we define the once continuously differentiable regularized fundamental solution G^ρ(;·,·):×→ via G^ρ(;x,y)= 1/4πln(1-x· y)+1/4π(1-ln 2), x· y≤1-ρ, 1-x· y/4πρ +1/4π(lnρ-ln 2), x· y>1-ρ . A later on useful notation will be the following, with x,x̅∈ fixed, G^ρ_x,x̅=G^ρ(;x,·)-G^ρ(;x̅,·). It is directly obvious that G^ρ(:x,·) and G(:x,·) only differ on the spherical cap 𝒞_ρ(x)={y∈:x· y> 1-ρ} with center x and polar radius ρ∈(0,2). Thus, G^ρ_x,x̅ is locally supported in 𝒞_ρ(x)∪𝒞_ρ(x̅). One should note that the spherical basis functions in Section <ref> as well the mesh width h_X in Section <ref> are defined with respect to the Euclidean distance, i.e., they relate to the spherical ball ℬ_r(x)∩={y∈:\|x-y\|< r} with center x and (Euclidean) radius r>0. The distinction between these two notions of spherical caps and spherical balls is solely made to pay tribute to the origins of the two different setups, not because they are conceptually different. In order for 𝒞_ρ(x) and ℬ_r(x)∩ to cover the same area, one needs to choose the radii to be related via r=(2ρ)^1/2. As already mentioned, the regularization (<ref>) is only one of many possible choices. It has been chosen solely so that we can rely on some already known explict computations that lead to the following properties. Let the regularized fundamental solution G^ρ(;·,·) be given as in Definition <ref>. It holds that sup_x∈ G^ρ(:x,·)_2∼sup_x∈ G^ρ(:x,·)_L^2()∼ρ^-12, where the notation f_ρ∼ g_ρ is meant in the sense that there exist constants C≥ c>0 such that c g_ρ≤ f_ρ≤ C g_ρ for all sufficiently small ρ>0. * For f of class C^0(), it holds that (cf. <cit.>) sup_x∈\| ∫_(G^ρ(:x,y) -G(;x,y)) f(y)dω(y)\| ≤ Cρln(ρ)f_C^0(), sup_x∈\| ∫_(x G^ρ(:x,y)-x G(;x,y)) f(y)dω(y)\| ≤ Cρ^1/2f_C^0(), with some constant C>0. * G^ρ(;·,·) has the following spherical harmonic expansion (cf. <cit.>) G^ρ(;x,y)=∑_n=0^∞∑_k=-n^nĜ^ρ_n (x)(y)=∑_n=0^∞2n+1/4πĜ^ρ_n P_n(x· y), with coefficients Ĝ^ρ_0 =1/4ρ, Ĝ^ρ_1 =-1/2+1/4ρ -1/24ρ^2, Ĝ^ρ_n =P_n+1(1-ρ)-P_n-1(1-ρ)/2n(n+1)(2n+1)-2-ρ/2n(n+1)P_n(1-ρ) +1/2ρP_n(1-ρ)-P_n+2(1-ρ)/(2n+1)(2n+3)-1/2ρP_n-2(1-ρ)-P_n(1-ρ)/(2n+1)(2n-1), n≥2. § PRELIMINARIES ON HARDY SPACES AND LOCALIZATION CONSTRAINTS In order to state the desired relations between H_+() and H_-() under localization constraints, we very briefly recapitulate classical layer potentials and the definition of Hardy spaces. Subsequently, we state the subspaces of H_+() and H_-() in which we want to perform approximation. This forms the foundation for most considerations in the later course of this paper. §.§ Layer potentials and Hardy spaces For f ∈ we define the (boundary) single layer potential S → H^1() via Sf(x) =-1/4π∫_f(y)/\|x-y\| d ω(y), x∈. S is bounded and invertible. Furthermore, we define the operators K - 12𝕀 / ⟨ 1 ⟩→ / ⟨ 1 ⟩ and K + 12𝕀→, where 𝕀 is the identity operator and K is the (boundary) double layer potential Kf(x)= -p.v. 1/4π ∫_(y)· (x-y)/\|x-y\|^3 f(y) d ω(y), x∈, with (y)=y simply denoting the unit normal vector to the sphere at the location y∈. Both operators K + 12𝕀 and K - 12𝕀 are invertible and self-adjoint. The operator K + 12𝕀 preserves constant functions while the operator K - 12I annihilates them. Latter is the reason why we formally define it only on the space / ⟨ 1⟩, although we will not require this particular distinction throughout most parts of the paper. For more information on (boundary) layer potentials we refer the reader, e.g., to <cit.> and references therein. Here, we merely use them to define the operators B_+ / ⟨ 1 ⟩→ L^2(,^3) and B_-→ L^2(,^3) given by B_+= (K-12𝕀)+ S, B_-= (K+12 I)+ S. The Hardy spaces are then defined as the ranges of the corresponding operators B_+ and B_-, i.e., H_+()=B_+( / ⟨ 1 ⟩), H_-()=B_-( ). This notation will be useful for the purposes of the paper at hand; in particular, it allows to reduce considerations in the vectorial Hardy spaces to considerations in the scalar space. For the statement that these definitions are equivalent to the more classical definition of Hardy spaces via non-tangential limits of harmonic gradients, we refer the reader to <cit.>. There, one can also find the statement of the Hardy-Hodge decomposition L^2(,^3)=H_+()⊕ H_-() ⊕ D(), where D()={∈ L^2(,^3): is tangential and ( S)^=0} denotes the space of tangential, divergence-free vector fields on the sphere. Note again that B_+ can also be considered as acting on the full L^2() where it annihilates constant functions. Furthermore, it should be mentioned that the operators B_+, B_- closely resemble the operators õ^(1), õ^(2) in <cit.> (however, the notation of B_+ and B_- more naturally allows considerations on Lipschitz surfaces, e.g., in <cit.>, while the notation of õ^(1) and õ^(2) is rather specific to the sphere). Some basic calculations show that, for f∈ L^2(), it holds (Sf)^∧_n,k=-1/2n+1f̂_n,k (see, e.g., <cit.>). Analogously, it holds (Kf)^∧_n,k=1/4n+2f̂_n,k. Thus, the application of the single and double layer potential can be easily computed in spectral domain. §.§ Localization constraints For the remainder of this paper, we assume that Σ⊂, with Σ≠, is a Lipschitz domain with connected boundary. Let Σ^c=∖Σ be the open complement of Σ, which again is a Lipschitz domain with connected boundary. By P_+:L^2(,^3)→ H_+() and P_-:L^2(,^3)→ H_-() we denote the orthogonal projections onto H_+() and H_-(), respectively. For any vector field in L^2(,^3), we abbreviate its Hardy components by _+=P_+() and _-=P_-(). Additionally, P_Σ:L^2()→ L^2(Σ) denotes the projection onto L^2(Σ), and we define the auxiliary space V_Σ={f∈ L^2():·( Sf)\|_Σ=0 as distribution on Σ} of functions whose single layer potential is harmonic in Σ. In order to comply with the notation in <cit.>, we will implicitly assume that (𝐟)⊂Σ^c if we say that a vector field is locally supported, i.e., we assume that is in L^2(Σ^c,^3). The upcoming theorem (which is a collection of results from <cit.>) states the desired unique relation between the H_+()-contribution _+ and the H_-()-contribution _- of such a locally supported vector field . In particular, it characterizes the relevant subspaces P_+(L^2(Σ^c,^3))⊂ H_+() and P_-(L^2(Σ^c,^3))⊂ H_-(). It holds that P_+(L^2(Σ^c,^3))=B_+(D_+,Σ) and P_-(L^2(Σ^c,^3))=B_-(D_-,Σ), with D_+,Σ =L^2(Σ^c)+(K+1/2I)V_Σ, D_-,Σ =L^2(Σ^c)+(K-1/2I)V_Σ. Both D_+,Σ and D_-,Σ are dense but not closed in L^2() and, consequently, B_+(D_+,Σ) and B_-(D_-,Σ) are dense but not closed in H_+() and H_-(), respectively. Furthermore, we define the linear mappings τ_ptm:D_+,Σ→ D_-,Σ and :D_-,Σ→ D_+,Σ/⟨1⟩ by =[P_Σ(K+1/2I)]^-1P_Σ-I =-[P_Σ(K-1/2I)]^-1P_Σ-I+⟨·,1⟩_L^2(). Both operators are surjective and unbounded; they are inverse to each other in the sense ∘=I\|_D_-,Σ and ∘=I\|_D_+,Σ/⟨1⟩. It holds that for f∈ D_+,Σ, the function g=τ_ptm(f) is the unique function in L^2() such that B_+f+B_-g differs from a locally supported vector field in L^2(Σ^c,^3) only by a tangential divergence-free vector field from D(). Vice versa, if is in L^2(Σ^c,^3) and _+=B_+f with f∈ D_+,Σ, then _-=B_-τ_ptm(f). Observing that the operators B_+ and B_- are bounded and invertible, Theorem <ref> implies that, for the construction of vectorial spherical basis functions in the Hardy subspaces of interest, it suffices to find adequate spherical basis functions in the scalar-valued spaces D_+, and D_-,. Therefore, for the remainder of the paper, in order to simplify notations, we focus on the scalar setup of the spaces D_+,, D_-,, and the operators , . The corresponding results for the vectorial setup are obtained immediately by application of the operators B_+ and B_-. §.§ Norm minimizing localized vector fields Remembering that the original motivation to study localization constraints came from the study of non-uniqueness properties for inverse magnetization problems, we may ask the following question: given a magnetic field in the exterior of the sphere , what is the magnetization of minimal norm that is localized within a given source region and that produces the given magnetic field? In terms of the paper at hand, the corresponding question would be: given the H_+()-contribution of some vector field in L^2(Σ^c,^3), what is the vector field in L^2(Σ^c,^3) of minimal norm such that f_+=f̃_+? This question is answered in the next proposition. Let f be in D_+,Σ and set =B_+f+B_-(f)+_df with _df={[ -B_+f-B_-(f), on Σ,; -∇_ h, on Σ^c, ]. and h∈ L^2(Σ^c) such that Δ_ h=0 on Σ^c and ν·∇_ h=ν·(B_+f+B_-(f)) on ∂Σ (the latter two properties are meant in the distributional sense; and ν denotes the unit vector that is tangential to , normal to the boundary ∂Σ, and points into the exterior of Σ, i.e., into Σ^c). Then it holds that is in L^2(Σ^c,^3), _df is in H_df(), and 0.95_L^2(,^3)=min{_L^2(,^3):=B_+f++𝐝∈ L^2(Σ^c,^3), ∈ H_-(), 𝐝∈ H_df()}. The statement that _df of the form (<ref>) exists and is in H_df() has been shown in <cit.>. By construction it is then clear that vanishes on Σ, i.e., is in L^2(Σ^c,^3). It remains to show (<ref>). From the previous section it is known that =B_- (f) is necessary in order to enable B_+f++𝐝 to be in L^2(Σ^c,^3), so that only the choice of $̣ needs to be investigated. Due to the orthogonality ofH_+(),H_-(), andH_df(), we know B_+f+B_-(f)+𝐝_L^2(,^3)^2=B_+f+B_-(f)_L^2(,^3)^2+𝐝_L^2(,^3)^2. Furthermore, any𝐝that guaranteesB_+f+B_-(f)+𝐝to vanish onΣmust coincide with_dfonΣ. Thus, we obtain=̣_df+, for some∈H_df()with()⊂Σ^c. We can compute further, 𝐝_L^2(,^3)^2 =_df_L^2(,^3)^2+𝐝̃_L^2(,^3)^2+2⟨_df,⟩_L^2(,^3) =_df_L^2(,^3)^2+𝐝̃_L^2(,^3)^2. The last summand in the first row vanishes due to the local support condition onand the fact that_dfis expressible as a gradient onΣ^c. The minimization of the norm of$̣ therefore leads to =0, which concludes the proof. § SPHERICAL BASIS FUNCTIONS RELATED TO HARDY SPACES We now consider spherical basis functions contained in and suited for approximation in D_+, and D_-,, respectively. Remembering the characterizations (<ref>) and (<ref>), we first deal with approximation in the space L^2(^c), then with approximation in V_, and, subsequently, in Section <ref> we join both considerations to deal with approximation in the actual spaces D_+, and D_-,. From Remark <ref> we know that this suffices for approximation in the corresponding subspaces of H_+() and H_-(). Again, throughout, we assume that Σ⊂ is a Lipschitz domain with connected boundary. §.§ Approximation in L^2(Σ^c) Simply for notational convenience, we drop the index "c" in this subsection and consider approximation in L^2() instead of L^2(^c). Obviously, all results equally hold true on Σ^c just by taking the complement. We prove the upcoming theorem following step by step the argumentation in the proof of the main result in <cit.> for the Euclidean setup, solely adapting it to the spherical setup where necessary. The main framework for the spherical setup has already been described in Section <ref>, based on, e.g., <cit.> and the stereographic projection. Let X_n⊂ be pointsets with associated mesh widths h_n=h_X_n,Σ and separation widths q_n=q_X_n,Σ that satisfy c_1γ h_n≤ h_n+1≤γ h_n and q_n≤ h_n≤ c_2 q_n, for some fixed constants c_1,γ∈(0,1) and c_2≥ 1. Furthermore, let Ψ be as in Proposition <ref> with native space H^s(), for some s>1, and choose scaling factors δ_n=ν h_n, for some fixed ν>1. Eventually, set X̃_n={x∈ X_n:ℬ_δ_n(x)∩⊂Σ}. Then, for sufficiently large ν and sufficiently small γ,h_1, the following holds true: For any f in H^s(Σ), there exists a f_n in span{Ψ_δ_i,x_i:x_i∈X̃_i, i=1,…,n}⊂ H^s(Σ) with f-f_n_L^2()≤ C η^nf_H^s(), where the constants C>0 and η∈(0,1) depend on ν, γ, c_2, h_1, and s. Before we proceed to the proof, we mention the following proposition that states some technical properties of Lipschitz domains with a tube around the boundary removed. <cit.> Let Ω⊂^2 be a bounded Lipschitz domain that satisfies the interior cone condition with parameters r and θ (reflecting the radius and the opening angle of the cone). Let c>max{2,π/θsin(π/8)} be fixed. Then there exist constants C,δ_0>0 so that, for any δ∈(0,δ_0), there exists a Lipschitz domain K_δ⊂Ω satisfying Ω_cδ⊂ K_δ⊂Ω_δ, where Ω_δ={x∈Ω:inf_y∈∂Ωx-y>δ}, * μ(Ω∖ K_δ) ≤ C cδ, where μ denotes the Lebesgue measure, * K_δ satisfies the interior cone condition with parameters θ̃ =min{π/5,θ/2} and r̃=3rsinθ̃/8(1+1sinθ̃). Theorem <ref> We choose f_n to be the function obtained by a multiscale interpolation procedure as indicated, e.g., in <cit.>. Namely, set e_0=f, iterate by setting, for i=1,…, n, s_i=I_X̃_i, Ψ_δ_i(e_i-1), e_i=e_i-1-s_i, and eventually choose f_n=∑_i=1^n s_i=f-e_n. By I_X̃_i, Ψ_δ_i we mean the interpolation operator with respect to the set X̃_i and the scaled spherical basis function Ψ_δ_i, as indicated in Section <ref>. We now follow the steps from <cit.> in order to estimate e_n_L^2() in our spherical setup. First, we observe the following chain of inequalities: e_i+1_Ψ_δ_i+1≤ Ce_i+1_s≤ Cδ_i+1^-se_i+1_Ψ_δ_i+1≤ Cδ_i+1^-se_i_Ψ_δ_i+1, where the first two inequalities follow form Proposition <ref>, and the last one follows form Proposition <ref> since e_i+1=e_i-I_X̃_i+1,Ψ_δ_i+1(e_i). The constant C>0 depends on s (throughout the course of this proof, we consider it a generic constant that may change; and only if its dependencies change, we will mention this explicitly). We can split e_i_Ψ_δ_i+1 into e_i_Ψ_δ_i+1^2 = ∑_n=0^∞∑_k=-n^n\|(e_i)^∧_n,k\|^2/(Ψ_δ_i+1)^∧_n ≤ c^-1(∑_n≤δ_i+1^-1∑_k=-n^n\|(e_i)^∧_n,k\|^2 (1+δ_i+1n)^2s_=I_1 +∑_n> δ_i+1^-1∑_k=-n^n\|(e_i)^∧_n,k\|^2 (1+δ_i+1n)^2s_=I_2), where the constant c>0 ist the lower bound in (<ref>) of Proposition <ref> that depends on s. For the low frequency part I_1, we get I_1 ≤ 2^2se_i_L^2()^2 =2^2se_i_L^2(Σ)^2 ≤ C 2^2s e_i_L^2((Σ))^2. The second equality follows from the observation that e_i is locally supported in Σ and the last inequality from Proposition <ref>. Setting Ω=(Σ) and letting K_C̃(δ_i+h_i) be the set from Proposition <ref> (where C̃>0 denotes the constant from the upper bound in (<ref>) of Proposition <ref>), we can now further estimate I_1 ≤C 2^2s e_i_L^2(K_C̃(δ_i+h_i))^2_=I_1,1 + C 2^2s e_i_L^2(Ω∖ K_C̃(δ_i+h_i))^2_=I_1,2 Using the notation from Proposition <ref> i), we get K_C̃(δ_i+h_i) ⊂Ω_C̃(δ_i+h_i)⊂⋃_x∈X̃_i(ℬ_h_i(x)∩)⊂⋃_x∈X̃_iℬ_C̃h_i((x)). The above implies h_(X̃_i),K_C̃(δ_i+h_i)≤C̃h_i for the mesh width. Additionally, we have by construction that e_i\|_(X̃_i)=0. Therefore, we can use <cit.> to obtain the estimate e_i_L^2(K_C̃(δ_i+h_i))≤Ĉ(C̃h_i)^s e_i_H^s(K_C̃(δ_i+h_i)), where the constant Ĉ>0 depends on s and Ω but indendent of i, due to the interior cone property from Proposition <ref> iii) and the discussion in <cit.>. Thus, we can continue to estimate I_1,1 ≤ C h_i^2s e_i^2_H^s(K_C̃(δ_i+h_i))≤ C h_i^2s e_i^2_H^s(Ω) ≤ C h_i^2se_i^2_H^s()≤ C (h_i/δ_i)^2se_i_Ψ_δ_i^2 ≤ C (h_i/δ_i)^2se_i-1_Ψ_δ_j^2= C ν^-2se_i-1_Ψ_δ_j^2, where the third inequality follows from Proposition <ref>, the fourth from Proposition <ref>, and the fifth from Proposition <ref>. I_1,2 can be controlled using Propositions <ref> ii), <ref>, and <ref>: I_1,2 ≤ C μ(Ω∖ K_C̃(δ_i+h_i)) e_i_L^∞(Ω)^2 ≤ C (δ_i+h_i)e_i_L^∞()^2 ≤ C(δ_i+h_i) (1+C̅)^2ie_0_L^∞()^2= Cδ_i(1+ν^-1) (1+C̅)^2if_L^∞()^2 ≤ Cδ_1/γ(1+ν^-1) (√(γ)(1+C̅))^2if_H^s()^2=Cδ_1/γ(1+ν^-1) η^2if_H^s()^2, where we have set η=√(γ)(1+C̅). For the third inequality one needs to observe that e_i=e_i-1-I_X̃_i,Ψ̃_δ_i(e_i-1) before applying Proposition <ref> (the constant C̅> depends on c_2, ν, and s). For the last inequality we have used that H^s() is compactly embedded in C^0(), for s>1. For the high frequency contribution I_2 we observe that, for n>δ_i+1^-1, (1+δ_i+1 n)^2s≤ (2 δ_i+1 n)^2s≤ 2^2s(δ_i+1/δ_i)^2s(1+δ_in)^2s. Thus, it holds I_2 ≤ 2^2s(δ_i+1/δ_i)^2s∑_n=0^∞∑_k=-n^n(e_i)^∧_n,k\|^2(1+δ_i n)^2s ≤ C(δ_i+1/δ_i)^2s∑_n=0^∞∑_k=-n^n\|(e_i)^∧_n,k\|^2/(Ψ_δ_i)^∧_n ≤ Cγ^2se_i_Ψ_δ_i^2 ≤ Cγ^2se_i-1_Ψ_δ_i^2 For the second inequality we have used Proposition <ref> and for the last one Proposition <ref>. Eventually, combining (<ref>), (<ref>), (<ref>) with (<ref>), we obtain e_i_Ψ_δ_i+1^2 ≤ C(ν^-2s + γ^2s) e_i-1_Ψ_δ_i^2 + Cδ_1(1+ν^-1)/γη^2if_H^s()^2. For brevity, we set α^2=C(ν^-2s + γ^2s) and C'=Cδ_1(1+ν^-1)/γ. Iterating the estimate with respect to i until i=0 is reached, this leads to e_i_Ψ_δ_i+1^2 ≤α^2(i-1)f_Ψ_δ_1^2+ C'f_H^s()^2 ∑_k=1^iη^2kα^2(i-k) ≤ C α^2(i-1)f_H^s()^2+ C'η^2(η^2i-α^2i)/η^2-α^2f_H^s()^2, where Proposition <ref> is used for the second inequality. Remembering that η=√(γ)(1+C̅), one can choose ν sufficiently large and γ sufficiently small such that α<η and η<1 (note that C̅ depends on ν, so that one would first choose ν and subsequently adapt γ). Possibly increasing the constants C and C' appropriately, one can now cancel the contributions in (<ref>) that contain a factor α^2i, and using the fact that ·_Ψ_δ_i uniformly controls ·_L^2() regardless of varying i by (<ref>), we end up with the desired result f-f_n_L^2()=e_n_L^2()≤ Cη^nf_H^s() The final constant C as well as the previously fixed η∈(0,1) depend on ν, γ, s, c_2, and h_1(since δ_1=ν h_1). §.§ Approximation in We consider approximation in V_ based on regularized fundamental solutions as introduced in Section <ref>. First, a general approximation property of the regularized fundamental solution in Sobolev spaces on the sphere is shown. Afterwards, we proceed to locally harmonic functions on Lipschitz domains Σ⊂. Let L∈ℕ and X⊂ be such that condition (<ref>) is satisfied. Furthermore, let κ∈(0,1/2] and x̅∈ be fixed. Then, for ρ∈(0,2), the following holds true: For every f in 3+κ, there exists a f_ρ,L in span{1_,G^ρ_x,x̅(;·): x∈ X} such that f-f_ρ,L_1≤ C(ρ^1/2+ρ ^-1/2L^-(1+κ)) f_1+κ, with a constant C>0 depending on κ. We can assume ⟨ f,1⟩_L^2()=0 for the remainder of the proof since constant functions are obviously contained in span{1_,G^ρ_x,x̅(;·): x∈ X}. For ease of notation, we write X={x_1,…,x_N}, and Q_X be a cubature rule with corresponding (not yet specified) weights w_1,…, w_N. Setting F= f, we get F∈1+κ and by Green's third theorem, we have f=∫_G(; y,·)F(y)dω(y). In partiuclar, F is continuous and allows pointwise evaluations. Finally, observing that ∫_G^ρ(; x̅,·)F(y)dω(y)=0, we can write f-∑_x_i∈ XF(x_i)w_i G^ρ_x_i,x̅_1 = ∫_G(; y,·)F(y)dω(y)-∑_x_i∈ XF(x_i)w_i G^ρ_x_i,x̅(;·)_1 ≤∫_G(; y,·)F(y)dω(y)-∫_G^ρ(: y,·)F(y)dω(y)_1_=I_1 +∫_G^ρ_y,x̅(: ·)F(y)dω(y)-∑_x_i∈ XF(x_i)w_i G^ρ_x_i,x̅(: ·)_1_=I_2. The I_1 contribution can be estimated using Proposition <ref> ii): I_1^2 ≤∫_G(; y,·)F(y)dω(y)-∫_G^ρ(: y,·)F(y)dω(y)_L^2()^2 +∫_G(; y,·)F(y)dω(y)-∫_G^ρ(: y,·)F(y)dω(y)_L^2(,^3)^2 ≤ C^2F_C^0()^2 (ρ^2 \|ln(ρ)\|^2+ρ)≤ C^2F_C^0()^2ρ≤ C^2F_H^1+κ()^2ρ, where the last estimate follows from the compact embedding of H^s() in C^0() for s>1. Throughout the course of the proof, C>0 is a generic constant that may change. In order to estimate I_2, we first observe that, since condition (<ref>) is satisfied by assumption, Theorem <ref> guarantees the existence of a cubature rule Q_X with polynomial precision of degree L and positive weights w_1,…, w_N>0. From now on we fix the weights appearing in our computations to those corresponding to such a particular cubature rule. We can estimate the 1 norm using the principle of duality I_2 =∫_G^ρ_y,x̅F(y)dω(y)-∑_x_i∈ XF(x_i)w_i G^ρ_x_i,x̅(;·)_1 = sup_g∈ C^∞(),g_-1≤ 1\|∫_ g(z)(∫_G^ρ_y,x̅(; z)F(y)dω(y)-∑_x_i∈ XF(x_i)w_i G^ρ_x_i,x̅(;z)) dω(z) \| = sup_g∈ C^∞(),g_-1≤ 1\|∫_(∫_ g(z)G^ρ_y,x̅(; z)dω(z))F(y)dω(y)-∑_x_i∈ XF(x_i)w_i ∫_ g(z)G^ρ_x_i,x̅(; z)dω(z) \| = sup_g∈ C^∞(),g_-1≤ 1\|∫_H_g^ρ(y)F(y)dω(y)-∑_x_i∈ XF(x_i)w_i H_g^ρ(x_i) \|, where we have used the abbreviation H_g^ρ=∫_ g(z)G^ρ_·,x̅(; z)dω(z). The last expression in (<ref>) is precisely the cubature error for the integrand H_g^ρ F. Thus we can use Theorem <ref> to estimate I_2≤C/L^1+κH_g^ρ F_1+κ The constant C>0 appearing here depends on the choice of κ. Applying Proposition <ref> and <ref>, we see that H_g^ρ F lies in 1+κ since we have assumed κ≤1/2. In particular, together with Hölder's inequality, we get H^ρ_g F _1+κ ≤ C H^ρ_g_1+κF_1+κ ≤ C H^ρ_g_1+1/2F_1+κ ≤ C (G^ρ(;x̅,·)_2+G^ρ(;x̅,·)_1) g_-1F_1+κ ≤ Cρ^-12F_1+κ. The last inequality follows from Proposition <ref> i) and from the condition in (<ref>) that g_-1≤ 1. Therefore, combining the above with (<ref>), (<ref>), and (<ref>) yields the desired estimate (<ref>). Based on the above, we can verify the following approximation property in S, the space of functions that are locally harmonic in a subdomain Σ. Note that for Proposition <ref> we would not need to use G^ρ_x,x̅(;·), with the auxiliary fixed center x̅∈, to obtain the desired estimate; solely approximating with G^ρ(;x,·) would yield the same estimate. However, when interested in locally harmonic functions, using G^ρ_x,x̅(;·) is more appropriate because it is locally harmonic itself, namely, y G^ρ_x,x̅(;y)=0 for all y∉𝒞_ρ(x)∪𝒞_ρ(x̅). Let L∈ℕ and X⊂ be such that the condition (<ref>) is satisfied, and x̅∈ be fixed. Furthermore, let κ∈ (0,1/2]. Then, for ρ∈(0,2), the following holds true: For every f in S∩3+κ, there exist a f_ρ,L in span{1_,G^ρ_n_x,x̅: x∈^c∩ X} such that f-f_ρ,L_1≤ C(ρ^1/2+ρ ^-1/2L^-(1+κ)) f_1+κ, with a constant C>0 depending on κ. We have F(x)= f(x)=0 for all x∈ due to our assumption that f is in S∩3+κ. Thus, observing (<ref>) in the proof of Proposition <ref>, no G^ρ_x_i,x̅ with x_i∈∩ X is required to achieve the estimate from Proposition <ref>, which already yields the result desired here. Before proceeding to the main assertion, we consider the following proposition that helps to guarantee a favorable behaviour of pointsets near the boundary of Lipschitz domains. Let Σ⊂, with Σ≠, be a Lipschitz domain with connected boundary. Then there exist constants C,r_0>0 such that, for any r<r_0 and x∈Σ, there exists a x'∈Σ satisfying ℬ_r(x')∩⊂Σ and \|x-x'\|≤ Cr. Let R and θ be the radius and angle parameters of the interior cone condition satisfied by (Σ)⊂^2. Then, for any h∈(0,(1+sinθ)^-1R) and any x∈Σ, there exists y∈(Σ) with ℬ_h sinθ(y)⊂(Σ) and y-(x)=h (see, e.g., <cit.>). Now, let r=C̅^-1hsinθ (where C̅>0 is the constant from the upper bound in (<ref>) of Proposition <ref>) and take x'=^-1(y). Then it holds that ℬ_r(x')∩⊂^-1ℬ_h sinθ(y)⊂Σ and x'-x≤c̅^-1y-(x)≤c̅^-1C̅(sinθ)^-1 r (where c̅>0 is the constant from the lower bound in (<ref>) of Proposition <ref>). Choosing C=c̅^-1C̅(sinθ)^-1 and r_0=C̅^-1sinθ(1+sinθ)^-1R finishes the proof. Let X_n⊂ and L_n∈ℕ (with L_n→∞ monotonically) be such that condition (<ref>) is satisfied and, additionally, such that there exists a constant c̅>0 with h_X_n>c̅L_n^-1 for all n∈ℕ. Furthermore, we define ρ_n=L_n^-2 and X̃_n={x∈ X_n:𝒞_ρ_n(x)⊂Σ^c}, and we assume x̅∈Σ^c with 𝒞_ρ_1(x̅)⊂Σ^c and κ∈(0,1/2] to be fixed. If ρ_1 is sufficiently small, then the following holds true: For every f in H^2+κ()∩ V_Σ, there exists a f_n in span{1_,S^-1G^ρ_n_x,x̅:x∈X̃_n}⊂ V_Σ with f-f_n_L^2()≤ C h_X_n^κf_H^2+κ(), where the constant C>0 depends on κ and c̅. Due to the definition of X̃_n and the choice of x̅, the properties of G^ρ_n_x,x̅ yield that S^-1G^ρ_n_x,x̅ is contained in V_. If ρ_1 is sufficiently small, it holds that √(2ρ_n)+h_X_n<r_0, where r_0 is the constant from Proposition <ref>. Thus, for every x∈{y∈Σ^c:𝒞_ρ_n(y)⊄Σ^c}, there exists a x'∈Σ^c with ℬ_√(2ρ_n)+h_X_n(x')∩⊂Σ^c and \|x-x'\|≤ C'(√(2ρ_n)+h_X_n) (where C'>0 is the constant from Proposition <ref>). For this particular x', by construction, there must exist a x̃∈X̃_n with \|x-x̃\|<h_X_n. In consequence, this means that, for every x∈Σ^c, there must exist some x̃∈X̃_n with \|x-x̃\|≤ 2h_X_n+C'(√(2ρ_n)+h_X_n)≤ (2+C'+√(2)C'c̅^-1)h_X_n. This leads us to h_X̃_n,Σ^c≤C̃ h_X_n, where C̃>0 depends on c̅. Applying a similar argument to Σ instead of Σ^c, we obtain for the overall set X_n'={x∈ X_n: x∈Σ or x∈X̃_n} that h_X'_n≤C̃ h_X_n. From Corollary <ref> and the bounded invertibility of S:L^2()→1, we now get the existence of a f_n in span{1_,S^-1G^ρ_n_x,x̅:x∈X̃_n}⊂ V_Σ such that f-f_n_L^2() ≤ CSf-Sf_n_1 ≤ C(ρ_n^1/2+ρ_n ^-1/2L_n^-(1+κ)) Sf_1+κ =C(ρ_n^1/2+ρ_n ^-1/2ρ_n^1+κ/2) Sf_1+κ ≤ Cρ_n^κ/2f_2+κ, where C>0 is a generic constant, which may change in every step of estimation but which eventually depends on κ and c̅. The desired assertion (<ref>) follows from the observation that ρ_n=L_n^-2≤c̅^-2h_X_n^2. §.§ Approximation in the subspaces D_±,Σ Due to the structure of D_+,Σ and D_-,Σ, the considerations in the two previous Sections <ref> and <ref> directly provide the following theorem. Let X'_n⊂ and δ_n>0 satisfy the conditions of Theorem <ref> (with Σ substituted by Σ^c). Furthermore, let X_n⊂ and ρ_n>0 satisfy the conditions of Theorem <ref>, and x̅∈Σ^c with 𝒞_ρ_1(x̅)⊂Σ^c be fixed. We define the sets X̃'_n={x'∈ X'_n:ℬ_δ_n(x')∩⊂Σ^c} and X̃_n={x∈ X_n:𝒞_ρ_n(x)⊂Σ^c}. Additionally, let Ψ be as in Definition <ref> with native space H^s(), for s=2+κ with a fixed κ∈(0,1/2]. Then the following holds true: For every f in D_+,Σ/⟨1⟩, there exists a f_n in span{(K+1/2I)S^-1G^ρ_n_x,x̅, Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}⊂ D_+,Σ with f-f_n_L^2()≤ C (η^n+h_X_n^κ)(f_s+ (f)_s). Analogously, for every g in D_-,Σ, there exists a g_n in span{(K-1/2I)S^-1G^ρ_n_x,x̅, Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}⊂ D_-,Σ with g-g_n_L^2()≤ C (η^n+h_X_n^κ)( (g)_s+g_s). The constants C>0 and η∈(0,1) depend on the parameters listed in Theorems <ref> and <ref>. According to Theorem <ref> we can write f=h+(K+1/2I)k with h∈ L^2(Σ^c) and k∈ V_Σ. Applying τ_ptm yields τ_ptm(f)=-h+k-(K+1/2I)k=k-f. Thus, we obtain k=f+ (f) h=f-(K+1/2I)k=-(K-1/2I)f-(K+1/2I) (f) Since K±1/2I:H^s()→ H^s() is bounded, we have k_s≤ C (f_s+ f_s) and h_s≤ C(f_s+ f_s), for some constant C>0. Now, let h_n∈span⋃_i=1^n{Ψ_δ_i,x':x'∈X̃'_i} be an approximation of h according to Theorem <ref> and k_n∈span{S^-1G^ρ_n_x,x̅:x∈X̃_n} an approximation to k according to Theorem <ref>, then f_n=h_n+(K+1/2I)k_n is in D_+,Σ and satisfies the estimate (<ref>). The estimate (<ref>) follows along the same lines. It can easily be seen from (<ref>) that τ_ptm((K+12I)S^-1G^ρ_n_x,x̅)=-(K-12I)S^-1G^ρ_n_x,x̅, τ_ptm(Ψ_δ_n,x')=-Ψ_δ_n,x', under the assumptions of Theorem <ref>. Therefore, for any f in the finite-dimensional subspace span{(K+1/2I)S^-1G^ρ_n_x,x̅, Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}⊂ D_+,Σ, one directly obtains the corresponding (f) in D_-,Σ without much additional computational effort. Yet, one should keep in mind the unboundedness of the operator when mapping between the infinite-dimensional spaces D_+,Σ and D_-,Σ. §.§ Norm minimization and vectorial approximation In Section <ref> we have briefly discussed norm minimizing vector fields with localization constraints. A corresponding statement with respect to the spherical basis functions considered above is presented next. For a given function f in D_+,Σ that is expanded in terms of those spherical basis functions, the norm minimizing vector field that is supported in Σ^c and that satisfies f̃_+=f can be expressed fairly explicitly. Let ρ∈(0,2) and x,x̅∈ be fixed. Then we denote by N^ρ_x,x̅:Σ^c→ the solution in L^2(Σ^c)/⟨1⟩ to the following Neumann boundary value problem: Δ_ N^ρ_x,x̅=0 in Σ^c and ν·∇_ N^ρ_x,x̅=ν·∇_ G^ρ_x,x̅ on ∂Σ. If x,x̅∈Σ, then clearly N^ρ_x,x̅=G^ρ_x,x̅. Otherwise, if x,x̅∈Σ^c, which is the relevant case for our setup, an explicit expression for N^ρ_x,x̅ is more tedious to obtain. However, for the special case of Σ^c being a spherical cap, which is sufficient for many considerations, an explicit expression of N^ρ_x,x̅ is known: namely, N^ρ_x,x̅(y)=Φ^(N)(x,y)-Φ^(N)(x̅,y), with Φ^(N) as derived in <cit.> for the construction of a Neumann Green function on spherical caps. Let the general setup be as in Theorem <ref>. Furthermore, let f be in span{(K+1/2I)S^-1G^ρ_n_x,x̅, Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}⊂ D_+,Σ, i.e., f=∑_ℓ=1^M c_ℓ(K+1/2I)S^-1G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,kΨ_δ_i,x_i,k', with c_ℓ,c_i,k∈ and M,K_1,…,K_n∈ℕ. For =B_+f+B_-f̃_-+_df with f̃_-=-∑_ℓ=1^M c_ℓ(K-1/2I)S^-1G^ρ_n_x_ℓ,x̅-∑_i=1^n∑_k=1^K_i c_i,kΨ_δ_i,x_j,k and _df={[ -∑_ℓ=1^M c_ℓ∇_ G^ρ_n_x_ℓ,x̅, on Σ,; -∑_ℓ=1^M c_ℓ∇_ N^ρ_n_x_ℓ,x̅, on Σ^c, ]. the following holds true: is in L^2(Σ^c,^3), _df is in H_df(), and 0.95_L^2(,^3)=min{_L^2(,^3):=B_+f++𝐝∈ L^2(Σ^c,^3), ∈ H_-(), 𝐝∈ H_df()}. We already know that f̃_-=(f) is necessary in order to enable to be in L^2(Σ^c,^3). The representation (<ref>) then directly follows from Remark <ref>. Furthermore, we can compute 0.95B_+f+B_-f̃_-= 0.95∑_ℓ=1^M c_ℓη(K-1/2I)(K+1/2I)S^-1G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,kη(K-1/2I)Ψ_δ_i,x_i,k' 0.95-∑_ℓ=1^M c_ℓη(K+1/2I)(K-1/2I)S^-1G^ρ_n_x_ℓ,x̅-∑_i=1^n∑_k=1^K_i c_i,kη(K+1/2I)Ψ_δ_i,x_i,k' +0.95∑_ℓ=1^M c_ℓ∇_(K+1/2I)G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,k∇_ SΨ_δ_i,x_i,k' 0.95-∑_ℓ=1^M c_ℓ∇_(K-1/2I)G^ρ_n_x_ℓ,x̅-∑_i=1^n∑_k=1^K_i c_i,k∇_ SΨ_δ_i,x_i,k' 0.95= 0.95∑_ℓ=1^M c_ℓ∇_ G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,kη Ψ_δ_i,x_i,k'. Due to the properties of the point sets X̃_i' in Theorem <ref> and the assumption x_i,k'∈X̃_i', the functions Ψ_δ_i,x_i,k' vanish on Σ. Therefore, the expression above implies B_+f+B_-f̃_-=∑_ℓ=1^M c_ℓ∇_ G^ρ_n_x_ℓ,x̅ on Σ. Furthermore, on the boundary ∂Σ, it holds that ν·(B_+f+B_-f̃_-)=∑_ℓ=1^M c_ℓν·∇_ G^ρ_n_x_ℓ,x̅, since ν is orthogonal to η by definition. The choice h=∑_ℓ=1^M c_ℓ N^ρ_n_x_ℓ,x̅ then satisfies h=0 on Σ^c and ν·∇_ h= ν·(B_+f+B_-f̃_-) on ∂Σ. The assertion of Proposition <ref> is now a direct consequence of Proposition <ref>. Furthermore, the above computations also guide us to an error estimate for the joint approximation in D_+,Σ and D_-,Σ. In fact, this yields a slightly different proof of Theorem <ref> and it provides a slightly more general statement, since it guarantees the existence of a sequence f_n such that both f_n and (f_n) satisfy the desired error estimate. Let the general setup be as in Theorem <ref> and let be in L^2(Σ^c,^3). Then there exists a _n in span{∇_ G^ρ_n_x,x̅, η Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}, i.e., _n=∑_ℓ=1^M c_ℓ∇_ G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,kη Ψ_δ_i,x_i,k' for some c_ℓ,c_i,k∈ and M,K_1,…,K_n∈ℕ, such that (_++_-)-_n_L^2(,^3)≤ C (η^n+h_X_n^κ)(f_+_s+f_-_s), where =_++_-+_df=B_+f_++B_-f_-+_df reflects the Hardy-Hodge decomposition of . The constants C>0 and η∈(0,1) depend on the parameters listed in Theorems <ref> and <ref>. Furthermore, the functions f_+^n =∑_ℓ=1^M c_ℓ(K+1/2I)S^-1G^ρ_n_x_ℓ,x̅+∑_i=1^n∑_k=1^K_i c_i,kΨ_δ_i,x_i,k', f_-^n =(f_+^n)=-∑_ℓ=1^M c_ℓ(K-1/2I)S^-1G^ρ_n_x_ℓ,x̅-∑_i=1^n∑_k=1^K_i c_i,kΨ_δ_i,x_i,k', with c_ℓ,c_i,k∈ and M,K_1,…,K_n∈ℕ as above, lie in D_+,Σ and D_-,Σ, respectively, and satisfy f_+-f^n_+_L^2() ≤C(η^n+h_X_n^κ)(f_+_s+f_-_s), f_–f^n_-_L^2() ≤C(η^n+h_X_n^κ)(f_+_s+f_-_s). It is well-known known that any in L^2(,^3) can be decomposed into the Helmholtz decomposition =η f_η+∇_ S f_cf+_df, with f_η,f_cf in L^2() and _df in H_df() (see, e.g., <cit.>). Since the statement of the theorem assumes that is in L^2(Σ^c,^3), we get that f_η must be in L^2(Σ^c) and f_cf must be in V_Σ. Theorems <ref> and <ref> then guarantee the existence of an _n^1 in span{η Ψ_δ_i,x_i': x_i'∈X̃'_i, i=1,…,n} and an _n^2 in span{∇_ G^ρ_n_x,x̅: x∈X̃_n} such that η f_η-_n^1_L^2(,^3) ≤ C η^nf_η_s, ∇_ S f_cf-_n^2_L^2(,^3) ≤ C h_X_n^κf_cf_s. Combining the two estimates above and observing that _++_-=η f_η+∇_ S f_cf as well as that B_+ and B_- are bounded and invertible, we directly obtain (<ref>), with a possibly modified constant C>0. The desired estimates (<ref>) and (<ref>) follow from (<ref>) when reading the equations in (<ref>) backwards and when observing the mutual orthogonality of the Hardy spaces H_+() and H_-(). § NUMERICAL EXAMPLE We provide a brief numerical example that illustrates the suitability of the constructed spherical basis functions for approximation in D_+,Σ and that reproduces the derived convergence rates. For that purpose, we choose the test vector field (x)=Q(x)(3(x·)̣x-)̣ with =̣(0,0.6,0.8)^T and Q(x)=Q(t,φ)={[ (t-a)^3(t-1)^2(φ-2π)^3 sin(2φ)φ^3, if t>a,; 0, else, ]. noticing that x=(√(1-t^2)cos(φ),√(1-t^2)sin(φ),t)^T∈ for t∈[-1,1] and φ∈[0,2π). Clearly, is supported in the spherical cap 𝒞_1-a(𝐞_3) with polar radius 1-a∈(0,2) and center 𝐞_3=(0,0,1)^T. In the upcoming example we choose a=0.9, i.e., is supported in 𝒞_0.1(𝐞_3) (the function as well as its H_+()-contribution _+=B_+f are illustrated in Figure <ref>). As a consequence, the corresponding f lies in D_+,Σ, for Σ^c=𝒞_0.1(𝐞_3) but also for Σ^c=𝒞_0.2(𝐞_3), 𝒞_1(𝐞_3). In the following we want to approximate this f by solving f_n=argmin{f-g_L^2()^2+λ^2 g_H^s()^2:g∈𝒟_n}, with s=9/4, a not yet specified regularization parameter λ, and 𝒟_n⊂ D_+,Σ the finite dimensional subspace 𝒟_n=span{(K+1/2I)S^-1G^ρ_n_x,x̅, Ψ_δ_i,x_i': x∈X̃_n, x_i'∈X̃'_i, i=1,…,n}. The pointsets X̃_n, X̃'_i and the parameters ρ_n, δ_i are chosen to satisfy the conditions from Theorem <ref> (cf. Figure <ref> for more details on 𝒟_n and the used pointsets). We run tests for levels n=1,2, …,6, and for the three choices of Σ indicated above, i.e., for Σ^c=𝒞_0.1(𝐞_3),𝒞_0.2(𝐞_3), 𝒞_1(𝐞_3), while we keep the underlying f the same. Note that X̃_n, X̃'_i depend on Σ according to their definition in Theorem <ref>. Finally, for the evaluation of (<ref>), we assume to know f only via its spherical harmonic coefficients up to a maximal degree 100 (an assumption that is reasonable for typical geomagnetic applications, and which also reflects some noise in the data by neglecting higher degree information). Figure <ref> indicates the relative errors f_n-f_L^2()/(f_H^s()+τ_ptm(f)_H^s()) obtained for the solutions f_n of (<ref>) within the setup described above (only the outcome for the best choice of λ, among a tested range on regularization parameters, is plotted). One can see that the evolution of the error with respect to n generally follows the predicted rate from Theorem <ref>. Additionally one can see that the error is smaller for larger Σ^c, which is probably due to the increased number of available basis functions contained in 𝒟_n. The observation in the left plot of Figure <ref> that the error reaches a plateau at n=5 is likely due to the restriction of information on f to spherical harmonic degree 100. If we increase the information on f up to degree 200, the error decreases further as n grows, although it still does not reach the predicted error bound (cf. right plot in Figure <ref>). § CONCLUSION We have investigated a set of spherical basis functions suitable for the approximation in subspaces of the Hardy space H_+() that are obtained by orthogonal projection of locally supported vector fields. The new aspect has been that the considered spherical basis functions lead to vectorial functions that are themselves members of these subspaces. In this sense they are related to certain vector spherical Slepian functions but may have some advantages since they do not require simultaneous computations in H_+(), H_-(), H_df() and only their centers need to be adapted for new domains of support. However, they do not feature an additional optimization in spectral domain that Slepian functions typically have. The obtained theoretical approximation results have been illustrated by a numerical example in Section <ref>. Remark <ref> shows that the derived spherical basis functions further allow a simple mapping into H_-(), given the underlying localization constraints, which can be of interest for various inverse magnetization problems. The latter, however, requires further studies on possible regularization strategies that go beyond the scope of the paper at hand (a first naïve estimate is provided in the Appendix <ref>). plain § APPENDIX §.§ Proof of Proposition <ref> If f is a 𝐩-zonal function, then f(x)=∑_n=0^∞f̂_n∑_k=-n^n(p)(x), for coefficients f̂_n∈. From the addition theorem for spherical harmonics, we get f_s^2 =∑_n=0^∞(n+12)^2sf̂_n^2∑_k=-n^n(p)^2 =∑_n=0^∞(n+12)^2sf̂_n^2 2n+1/4π. Thus, it must hold \|f̂_n\|≤√(2π)f_s (n+12)^-s-1/2. Let g∈k with spherical harmonic coefficient ĝ_n,k. The convolution defined in (<ref>) then has the spectral expression (f∗ g)^∧_n,k=f̂_n ĝ_n,k Hence we get by use of (<ref>) that f∗ g_s+k+1/2^2 =∑_n=0^∞∑_k=-n^nf̂_n^2ĝ_n,k^2(n+12)^2s+2k+1 ≤∑_n=0^∞∑_k=-n^n2πf_H^s()^2/(n+1/2)^2s+1ĝ_n,k^2(n+12)^2s+2k+1 =2πf_H^s()^2∑_n=0^∞∑_k=-n^nĝ_n,k^2(n+12)^2k =2π f_s^2 g_k^2, which concludes the proof. §.§ Proof of Proposition <ref> Consider a set X̃⊂ℬ_r={x∈^3:\|x\|<r}, with r=1+cν q_X, such that X̃\|_=X, q_X̃=q_X, dist(X̃∖ X,)>cν q_X, and h_X̃≤ 4cν q_X. For f in H^s(), we denote by f̃ an extension to H^s(ℬ_r) with f̃\|_=f and f̃_L^∞(ℬ_r)≤ 2f_L^∞(). Finally, let Ψ̃_δ be the same as Ψ_δ, simply extended to act on ℬ_r×ℬ_r instead of × (which is unproblematic since Ψ relies on a Euclidean radial basis function ψ as indicated in Definition <ref>). With this setup, <cit.> states that there exists a C>0, depending on c, ν, and s, such that for every scale ν q_X<δ< cν q_X, it holds I_X̃, Ψ̃_δ(f̃) _L^∞(ℬ_r)≤ C f̃_L^∞(ℬ_r). The assumptions dist(X̃∖ X,)>cν q_X and δ<cν q_X yield that I_X̃, Ψ̃_δ(f̃)\|_=I_X, Ψ_δ(f) and subsequently, since by construction I_X, Ψ_δ(f) _L^∞()≤I_X̃, Ψ̃_δ(f̃) _L^∞(ℬ_r)≤ C f̃_L^∞(ℬ_r)≤ 2C f _L^∞(), that (<ref>) holds true. §.§ A Bounded Extremal Problem for Approximation in D_±,Σ We are interested in the following bounded extremal problem: given f_e=f_++e in L^2() and c> 0, find f_+^n,e in 𝒟_n such that f_+^n,e=argmin{f_e-g_L^2():g∈𝒟_n, τ_ptm(g)_L^2()≤ c}.BEP The finite-dimensional subspace 𝒟_n⊂ D_+,Σ be defined as in (<ref>). In order to quantify the weak convergence of τ_ptm(f_+^n,e) to τ_ptm(f_+), we further consider the following adjoint bounded extremal problem: For some y∈ L^2() and some bound t>0, find h_t∈dom(τ_ptm^) such that h_t=argmin{y-h_L^2():h∈dom(τ_ptm^), τ_ptm^(h)_L^2()≤ t}. For convenience, we denote J_y(t)=y-h_t_L^2(). The existence of a solution h_t and the guarantee that J_y(t) tends to zero as t→∞ follows along the same lines as for the original problem (e.g., <cit.>). With this notation, we can now deduce the following estimate. Let f_e=f_++e be in L^2(), with f_+ in D_+,Σ and e_L^2()<. Let all further setup be as in Theorem <ref> and set δ_n=C (η^n+h_X_n^κ)(f_+_s+ (f_+)_s), with constants C>0 and η∈(0,1) as indicated in Theorem <ref>. If f_+^n,e is the solution to (<ref>), then, for any y∈ L^2() and c> τ_ptm(f_+)_L^2(), we have that f_+^n,e-f_+_L^2()≤ 2+δ_n and \|⟨(f_+^n,e), y⟩_L^2()-⟨(f_+), y⟩_L^2()\| ≤inf_t>0{2c J_y(t) + (2+δ_n) }. Without loss of generality we assume c≥(f)+δ_n. Theorem <ref> then guarantee the existence of an 𝔣_n in 𝒟_n such that f_+-𝔣_n≤δ_n and (𝔣_n)≤ c, i.e., 𝔣_n lies in the feasible set of (<ref>). Thus, f_+^n,e-f_+_L^2() ≤f_+^n,e-f_e_L^2()+≤𝔣_n-f_e_L^2()+ ≤𝔣_n-f_+_L^2()+f_+-f_e_L^2()+≤δ_n+2. We then get that, for any t>0 and any y in L^2(), \|⟨(f_+^n,e), ⟩_L^2()-⟨(f_+), ⟩_L^2()\| ≤\|⟨(f_+^n,e)- (f_+), - h_t⟩_L^2()\|+\|⟨(f_+^n,e-f_+), h_t⟩_L^2()\| ≤((f_+^n,e)_L^2()+(f_+)_L^2()) J_() + \|⟨ f_+^n,e-f_+, ^(h_t)⟩_L^2()\| ≤ 2cJ_()+(2+δ_n)t. Taking the infimum over t leads to the desired assertion. The proposition above guarantees that f_+^n,e converges to f_+ and that τ_ptm(f_+^n,e) weakly converges to τ_ptm(f_+) as n→∞ and →0 (assuming that both f_+ and τ_ptm(f_+) are in H^s()). The rate of the weak convergence clearly depends on the behaviour of J_y(t) with respect to t. A more precise study of this, however, is beyond the scope of the paper at hand.
http://arxiv.org/abs/2307.01963v2	20230705001829	Disorder-free localisation in permutation symmetric fermionic quantum walks	[ "A. P. Balachandran", "Anjali Kundalpady", "Pramod Padmanabhan", "Akash Sinha" ]	quant-ph	[ "quant-ph", "cond-mat.str-el", "hep-th" ]	=1 calc,positioning patterns,arrows,decorations.pathreplacing >=stealth 22.5cm16.8cm 0pt0pt-1cm 1.3 1ex =.5ex addtoresetequationsection
http://arxiv.org/abs/2307.03246v2	20230706183250	Semantic-Aware Image Compressed Sensing	[ "Bowen Zhang", "Zhijin Qin", "Geoffrey Ye Li" ]	eess.IV	[ "eess.IV" ]	This work was conducted at the Department of Mechanical Engineering, Stanford University, Stanford, CA 94305 USA. ======================================================================================================================== Deep learning based image compressed sensing (CS) has achieved great success. However, existing CS systems mainly adopt a fixed measurement matrix to images, ignoring the fact the optimal measurement numbers and bases are different for different images. To further improve the sensing efficiency, we propose a novel semantic-aware image CS system. In our system, the encoder first uses a fixed number of base CS measurements to sense different images. According to the base CS results, the encoder then employs a policy network to analyze the semantic information in images and determines the measurement matrix for different image areas. At the decoder side, a semantic-aware initial reconstruction network is developed to deal with the changes of measurement matrices used at the encoder. A rate-distortion training loss is further introduced to dynamically adjust the average compression ratio for the semantic-aware CS system and the policy network is trained jointly with the encoder and the decoder in an end-to-end manner by using some proxy functions. Numerical results show that the proposed semantic-aware image CS system is superior to the traditional ones with fixed measurement matrices. Compressed sensing, semantic sensing, deep learning, image reconstruction § INTRODUCTION The traditional image acquisition systems based on the Nyquist-Shannon sampling theorem require the sampling ratio of image sensors to be no less than twice the bandwidth of the original signal <cit.>, which is unfriendly to the applications where inexpensive sensors shall be used or oversampling may be harmful to the object being sensed (e.g. medical imaging). Also, as many sensed images will be compressed for storage or transmission purposes, the sensing costs for pixels that will be discarded in the compressing process are higher than needed in traditional sensors. Based on these considerations, image compressive sensing (CS) that jointly implements the sampling and compression processes has been proposed as a new paradigm for image acquisition and reconstruction <cit.>. The CS theory <cit.> also shows the number of measurements required for image CS is much fewer than suggested by the Nyquist-Shannon sampling rate as images can be well sparsely represented. Recently, deep learning based image CS methods have been developed to improve the sensing efficiency and reconstruction accuracy in image CS problem. For example, based on the block-based image CS architecture <cit.>, Shi et al. <cit.> propose a convolutional neural network (CNN)-based image CS network architecture, CSNet, where sensing matrices and reconstruction network are jointly optimized. Motivated by the iterative algorithm, deep unfolding networks, such as ADMM-Net <cit.> and AMP-Net <cit.>, are introduced as reconstruction networks for image CS, which balances reconstruction speed and network interpretation. To address the problem of CNN-based networks in modelling long distance relationships, a cascaded visual transformer (ViT) architecture is developed in <cit.>. The information bottleneck measurement in <cit.> can enhance the training process of sensing network by explicitly modelling the importance level of different measurements. Despite the fast development of image CS methods, existing methods mainly use a fixed sensing measurement matrix for different images. Recent research on semantic communications has, however, demonstrated that data transmission efficiency can be increased if the communication policy is modified in accordance with the semantic information in the data <cit.>. This inspires us to think if data acquisition process can also be improved in a semantic-aware manner. In fact, instead of using a fixed sensing matrix, images with varying types of semantic information shall be sensed and compressed by different measurement matrices, including different numbers of measurements and different measurement bases[Each row of the measurement matrix is called as a measurement base in this paper]. It is well-known that different semantic information will have different sparsity levels when they are represented sparsely under a sparse transformation matrix. From CS theory, more measurements shall be used for signals less sparse to satisfy the restricted isometric property (RIP) requirement <cit.>. Therefore, the sparse signal cannot be well recovered if fewer samples than required are collected; but the sensing costs are higher than needed if more samples are used. This inspires us to adjust the number of measurements according to the semantic information type. Furthermore, it is also helpful to adjust the measurement bases for different semantic information for image CS problem. Specifically, the sparse representations of different semantic information may have different support sets (or sparsity patterns)[considering the case where different semantic information has different frequency components and a discrete Fourier transform (DFT) matrix is used as the sparse transformation matrix.]. Without these support information, one need to ensure that the correlations between all pairs of columns of measurement matrix are small enough so that the sparse recovery methods, such as orthogonal matching pursuit <cit.>, can operate successfully. By contrast, if the support set information can be roughly estimated by analysing the semantic information type and is available before CS process, we can enhance the sensing and sparse signal recovery process by explicitly reducing the correlations of the columns belonging to the support set and reducing the search space in recovery process. There are two challenges for this semantic-aware CS process: 1) how to estimate the semantic information of an area and use it to adjust the sensing process before sensing it; 2) how to dynamically adjust the measurement matrix according to the semantic information type without storing a matrix per type. To address the first challenge, we divide the CS process into two steps. In the first step, we use a fixed measurement matrix for all areas and estimate the semantic information from these observations for each area. The estimated semantic information is then used to decide the measurement matrix for different areas in the second step. To deal with the second challenge, we learn a relatively large measurement matrix and dynamically select rows from this large matrix to construct the semantic-aware measurement matrix for each individual area. The selection process is done by a policy network, which is trained jointly with the measurement matrix and the reconstruction network via some proxy functions in an end-to-end manner. Note that the whole network follows the designs in block-based image CS problem (BCS)<cit.>. The most related work to ours is the content-aware scalable network (CASNet) proposed in <cit.>. Our work differs from <cit.> in the following three aspects: 1) Our method adjusts both the number of measurements and the measurement bases; 2) Instead of using the same compression ratio from different images, we adjust the compression ratio for different images under the constraint that the average compression ratio over the the training/validation data-set meets the requirement. 3) To reduce memory and computational costs for sensor, our policy network works on the measurement space, which is far smaller than the image signal space. § PROBLEM FORMULATION AND SYSTEM MODEL In this section, we will introduce the problem formulation and system models for existing network-based BCS and the proposed semantic-aware BCS. §.§ Block-based image compressive sensing Given an image I∈ℛ^H× W×3, BCS first divides the image into non-overlapping blocks of size B× B×3 and reshapes the blocks into vectors. Then, each block is sensed by a learned measurement matrix ϕ of size n× 3B^2. This process can be represented as, y_i,j=ϕx_i,j, for i=1,2,⋯,H/B, j=1,2,⋯,W/B, where x_i,j and y_i,j are the (i,j)-th block in 2D image space and corresponding measurements. In this process, the compression ratios for each block and the whole image are the same, i. e., n/3B^2. In this work, we set H=W=224 and B=32, resulting in 7× 7 blocks. After sensing, the goal of BCS is to reconstruct the original image from these CS measurements. In this work, we mainly focus on network-based BCS methods <cit.>. In particular, after obtaining CS measurements, network-based BCS first obtains an initial reconstructed image via a trainable matrix θ of size 3B^2× n <cit.>. Given CS measurement y_i,j of the (i,j)-th block, its initial reconstruction result x̂_i,j can be represented as, x̂_i,j=θy_i,j. To this end, the initial reconstruction result for each block is still a vector. Network-based BCS methods will further reshape and concatenate these reconstructed vectors to get an initial reconstructed image Î <cit.>. After initial reconstruction, a deep reconstruction network D(·) is utilized to refine the initial reconstruction result, Ĩ=D(Î), where Ĩ denotes the final reconstructed images. Depending on the network architecture, the deep reconstruction network can be categorized as model-driven networks <cit.>, data-driven networks <cit.>, and hybrid networks <cit.>. §.§ Semantic-aware block-based image compressive sensing Based on the BCS methods, we now give the pipeline of the proposed semantic-aware BCS methods, which is also shown in Fig. <ref>. As aforementioned, the semantic-aware BCS is divided into two steps. In the first step, a learned base measurement matrix ϕ_b of size n_b× 3B^2 is utilized to sense each block as follows, y_i,j^b=ϕ_bx_i,j, where y_i,j^b is the CS measurements under base measurement matrix for the (i,j)-th block. After obtaining these base CS measurements, a policy network P(·) will take these measurements as inputs, analyse its semantic information type, and generate 0-1 row-selection vectors for each block, which can be represented as, m_1,1,m_1,2,⋯,m_⌈H/B⌉,⌈W/B⌉ =P(y_1,1^b,y_1,2^b,⋯, y_⌈H/B⌉, ⌈W/B⌉^b), where m_i,j∈{0,1}^n_max and n_max is the number of rows of a shared large measurement matrix ϕ_f∈ℛ^n_max× 3B^2. The sensor then constructs semantic-aware measurement matrix ϕ_i,j^s∈ℛ^n_i,j^s× 3B^2 by selecting rows from ϕ_f according to the locations of values 1 in m_i,j for each block, where n_i,j^s is the number of values 1 in m_i,j. Usually, n_i,j^s will have a higher value for blocks where the sparse representations are less sparse. Next, each ϕ_i,j^s is utilized to sense the corresponding area in the second step. This process can be represented as, y_i,j^s=ϕ_i,j^sx_i,j, where y_i,j^s is the CS measurements under the learned semantic-aware measurement matrix for the (i,j)-th block. After these two steps, there are n_b+n_i,j^s measurements for the (i,j)-th block. The average compression ratio r_avg for an image dataset with N images can be calculated as r_avg=n_avg/3B^2, where n_avg=1/N⌈HW/B^2⌉∑_k=1^N∑_i=1^⌈H/B^2⌉∑_j=1^⌈W/B^2⌉ (n_i,j^s,k+n_b) denotes the average number of measurements per block, where n_i,j^s,k is the number of measurements for the (i,j)-th block in image k at the second step. Following BCS, semantic-aware BCS also has initial reconstruction and deep reconstruction stages. However, different from BCS, semantic-aware BCS needs to tackle the changes of measurement matrices used in the sensing stage. Specifically, since y_i,j^s is generated through different ϕ_i,j^s in each block, it is hard to reconstruct initial reconstruction result x̂_i,j through a shared matrix θ for all blocks. Therefore, in the initial reconstruction stage, we first generate a block-wise matrix θ_i,j^s for each block using another weight-generation network A(·), which takes base measurements obtained from ϕ_b as inputs, θ_1,1^s,θ_1,2^s,⋯, θ_⌈H/B⌉, ⌈W/B⌉^s=A(y_1,1^b,y_1,2^b,⋯, y_⌈H/B^2⌉, ⌈W/B^2⌉^b), where θ_i,j^s∈ℛ^3B^2× n_i,j^s is the generated initial reconstruction matrix for block (i,j). After that, x̂_i,j can be represented as, x̂_i,j=[ θ_b, θ_i,j^s][ [ y_i,j^b; y_i,j^s ]], where θ_b∈ℛ^3B^2× n_b is the initial reconstruction matrix for base measurements. In this work, we do not directly learn the θ_i,j^s due to the large matrix size. We decompose θ_i,j^s into a large matrix θ_s∈ℛ^3B^2× n_max, which is shared among blocks, and a small matrix θ̃_i,j^s∈ℛ^n_max× n_i,j^s, which is actually learned for each block. Here, n_max(≪ 3B^2) is a pre-defined value representing the maximum number of measurements for each block in step two. With this decomposition, Eq. (<ref>) is re-written as, x̂_i,j=[ θ_b, θ_s][ [ I_n_b× n_b 0; 0 θ̃_i,j^s ]][ [ y_i,j^b; y_i,j^s ]], The deep reconstruction network should also be designed in a way adaptive to the changes of ϕ_i,j^s in different blocks, which will be considered in our future work. In this work, we use a memory-friendly deep reconstruction network for simplicity. More details will be given hereafter. If we substitute semantic-aware block-wise matrices ϕ_i,j^s in Eq.(<ref>) with a fixed matrix ϕ_bf of size (n_avg-n_b)× 3B^2 and θ_i,j^s in Eq.(<ref>) with a fixed matrix θ_bf of size 3B^2×(n_avg-n_b) for different blocks in the above architecture, the semantic-aware BCS can easily degrade to the traditional BCS with the same average compression ratio, enabling a fair comparison between semantic-aware BCS and traditional BCS. § SEMANTIC-AWARE IMAGE BCS NETWORK In this section, we will introduce the network architectures of the proposed semantic-aware image BCS network and the training details. §.§ Network architecture As shown in Fig. <ref>, the architecture of the proposed coding method is composed of an encoder and a decoder. We first introduce the encoder. Given an image I of size H× W × 3, we first apply a B× B convolution layer (Conv) with a stride size of B and n_b output channels to I, generating features C ∈ℛ^H/B×W/B× n_b. This process corresponds to Eq.(<ref>)[More details for this step are explained in <cit.>] and c_ij=C[i,j,:]∈ℛ^n_b, for i=1,2,⋯,H/B and j=1,2,⋯,W/B, denotes the base CS measurements for the image area I[(i-1)B:iB,(j-1)B:jB,:], which is also called as (i,j)-th block in previous section. Next, a policy network, denoted as P-net, takes C as inputs and generates an intermediate feature G∈ℛ^H/B×W/B× n_max, which is then quantified into a 0-1 mask matrix M by a binarizer. The detailed architecture of P-net is shown in Fig. <ref>, where FEN denotes a feature extraction network consisting of three 3× 3 Convs with stride 1 and 256 output channels, and sigm denotes sigmoid activation layer. The binarizer is used to conduct binary quantization to G. It outputs 1 if the input is over 0.5; otherwise it outputs 0. Due to binarizer, the backward gradient is zero almost everywhere, restricting the parameter update of the P-net. To solve this non-differentiate issue, we use a straight-through estimator of the gradient <cit.> which directly uses the gradients to M as the gradients to G. The generation of m_i,j=M[i,j,:]∈{0,1}^n_max corresponds to Eq.(<ref>). After that, we apply a B× B Conv layer with stride size B and n_max output channels to I and obtain features D ∈ℛ^H/B×W/B× n_max. This process is equivalent to sense each block using the shared large measurement matrix ϕ_f mentioned before. We then multiple D and 0-1 mask matrix M element-wisely to get E. e_i,j=E[i,j,:]∈ℛ^n_max can be regarded as a zero-padded version of y_i,j^s in Eq.(<ref>), where the values of unimportant measurements indicating by the values 0 in m_i,j are set as 0. Note that in the training process, we first sense each block with the maximum number of measurements and then set unimportant ones to 0. In this way, the number of measurements generated by each block is the same, making it easier to implement the batch training method. However, during the test phase, we first need to construct ϕ_i,j^s by considering m_i,j and ϕ_f stored in the Conv layer and then sense each block with ϕ_i,j^s to get y_i,j^s. Only in this way can we reduce the actual sensing costs. The differences between the training phase and the testing phase are shown in Fig. <ref>. At the decoder side, base CS measurements C are first fed into the P-net, which shares the same architecture and parameters as the one in the encoder. And then, the outputs of FEN in the P-net are used as the inputs of a weight-generation network, A-net, which generates weights W ∈ℛ^H/B×W/B× (n_max× n_max). Here, w_i,j=W[i,j,:]∈ℛ^n_max× n_max is similar to the θ̃_i,j^s in Eq.(<ref>). The only difference is that more columns are generated in w_i,j than θ̃_i,j^s for zero-padded values in e_i,j. The redundant columns will not affect the final results. In the training phase, for block (i,j), we conduct matrix multiplication between w_ij and e_ij, and repeat this process for all blocks. This process corresponds to operation θ̃_isy_is in Eq.(<ref>). The results are features F∈ℛ^H/B×W/B× n_max. After obtaining F, we concatenate F and C along the channel dimension, which equals forming the [Iy_ib;θ̃_isy_is] in Eq.(<ref>). Following that, we input them into a 1× 1 Conv with stride 1 and 3B^2 output channels. The convolution operation represents multiplying [θ_b, θ_s] by the left in Eq.(<ref>). In the testing phase, operations are slightly different. We first select columns from W under the guidance of M and obtain θ̃_i,j^s, which is then used to multiply with y_i,j^s. At last, a D-net is used for deep reconstruction. We show the architecture of the D-net in Table <ref>, where Depth-to-Space layer is used to rearrange features from the channel dimension into spatial dimension and Resblock layer is a cascaded of Conv-relu-Conv with skip connections. The parameters of the Convs used in Resblock are shown in Table <ref>. Note that as the main goal of this work is to verify the semantic-aware operations introduced in the encoder and the initial reconstruction process in the decoder, we do not spend much effort in designing the deep reconstruction network, whose designs, however, are important for comparison with state-of-the-arts BCS works and will be left for our future work. In this work, the adopted D-net has comparable performance with the one used in CSNet <cit.>. Different from CSNet whose D-net operates on image signal space and thus has a high memory and computational costs, the D-net shown in Table <ref> extracts features mainly in feature space with low spatial dimension and has faster speed and lower memory consumption. §.§ Rate-distortion trade-off Here, we will describe the training process of the whole network. In this semantic-aware BCS, we hope the number of CS measurements to be small while at the same time the peak signal-to-noise ratio (PSNR) to be high. Under this design goal, we can formulate the training loss as the well-known rate-distortion trade-off <cit.>, which can be defined as follows, ℒ=∑_I∈ℐℒ_2(I,Î)+γℒ_R(I), where ℐ denotes the training dataset, ℒ_2=\|\|I-Î\|\|_2^2 denotes the distortion, and ℒ_R(I)=∑_i=1^H/B∑_j=1^W/B∑_k=1^n_n_max G(I)[i,j,k] denotes the rate loss, and G(I) is the outputs of P-net when I is the network input. As G(I) determines the number of values 1 in M, minimizing G(I) equals minimizing the number of measurements. Besides, γ is an introduced trade-off parameter between rate loss and reconstruction accuracy. Increasing the value of γ will penalize more on the number of measurements and reduce the average compression ratio of ℐ sets. § EXPERIMENTS In this section, we compare the proposed semantic-aware image BCS methods with fixed-ratio image BCS methods under the same number of average compression ratio. We name our method as SemBCS. The fixed-ratio version of SemBCS, FixBCS, can be obtained by using a B× B Conv with strides B and n_avg output channels as the encoder and a 1× 1 Conv with stride 1 and 3B^2 output channels as the decoder, followed by the same D-net used in the SemBCS. §.§ Dataset We use two different datasets in our experiments. The first one, the MS-COCO 2014 <cit.> dataset, is composed of all kinds of images in human life and contains rich semantic information. The second one, the MPI-INF-3DHP <cit.> dataset, is widely used for human mesh recovery task and contains the video sequences where some human objects are doing some specific actions in an indoor environement with a green screen background. Therefore, the semantic information in this dataset is quite limited. As discussed above, all images are scaled to the size of 224× 224 × 3 for experiments. For the MS-COCO 2014 dataset, we use 82,783 training samples, 2,000 validation samples, and 2,000 testing samples. For the MPI-INF-3DHP dataset, we first extract images from the training video sequences and then randomly choose 5% for validation, 5% for test, and the rest for training. §.§ Experimental settings The values of n_b and n_max are different in the two datasets. For MS-COCO 2014 dataset, we choose n_b from {150,250,350,450,550}, set n_max=200, and ensure n_avg≈ n_b+100 by tuning the value of γ. For MPI-INF-3DHP dataset, we set n_b=20, n_max=200 and ensure that n_avg changes from around 50 to 140 by changing γ. The main reason for these setting difference is that the growth speed of PSNR value alongwith the increasement of compression ratio in MPI-INF-3DHP is much higher than MS-COCO 2014 dataset. In both experiments, we train the networks until the PSNR value in validation datasets stops increasing under the specific n_avg values. §.§ MPI-INF-3DHP experiment We show the PSNR versus the average compression ratio r_avg of different BCS methods in the MPI-INF-3DHP dataset in Fig. <ref>. From the figure, the SemBCS works significantly better than the FixBCS. For example, the SemBCS uses 23% fewer samples than the FixBCS when the targeted PSNR value is 27 dB. This experiment shows the superiority of the proposed semantic-aware BCS system over traditional BCS systems. To further understand the P-net learned in the SemBCS, we show some examples of the learned number of measurements at stage 2 and the underlying sparsity levels for each block in Fig. <ref>. To estimate the sparsity level of each block, we solve a sparse linear inverse problem, y=Ax, where y∈ℛ^3B^2 is the vectorized version of the pixels in each block, A∈ℛ^3B^2× 12B^2 is a predefined overcomplete discrete cosine transform (DCT) dictionary, x is the underlying sparse representation. We calculate the number of elements in x whose absolute value is larger than 10 and use it as the sparsity level of each block. Note that as the D-Net is used to fuse the information from different blocks in the deep reconstruction stage, the number of measurements for one block will be affected by the neighbouring blocks. From Fig. <ref>, more measurements are generally used for blocks containing richer semantic information, such as human areas, light areas, and green screens with folds. Also, more measurements are used for blocks that are less sparse, which means the P-net is able to estimate the sparsity level and assign the measurement matrix accordingly by analysing the semantic information contained in the base CS measurements. This experiment is a very good verification of our design concept. §.§ MS-COCO2014 experiment We also show the PSNR versus the average compression ratio r_avg of different BCS methods in the MS-COCO2014 datatset in Fig. <ref>. From the figure, the SemBCS still has a steady performance gain over the FixBCS, indicating the generality of the proposed semnatic-aware operations on different datasets. However, we find the performance gain is not as large as the previous experiment. This is because the P-net in SemBCS only has five Convs and is not deep enough to conduct semantic reasoning for datasets with rich semantic contents. Some examples of the learned number of measurements at stage 2 and the underlying sparsity levels for each block are shown in Fig. <ref>. We can see that more measurements are allocated to the human areas while fewer to the floor and the wall. However, we also notice the learned number of measurements is not strictly allocated alongwith the amount of semantic information and the sparsity levels for some blocks, which means this version of SemBCS can be further improved for datasets containing rich semantic information. § CONCLUSION In this work, we have proposed a novel semantic-aware image compressive sensing system, where the best measurement matrices for different images are decided by the images' semantic information. We also verify the effectiveness of the proposed method in MPI-INF-3DHP and MS-COCO2014 datasets. Improving the architecture of the policy network and the deep reconstruction network will be left as our future work. IEEEbib
http://arxiv.org/abs/2307.01796v1	20230704160428	Gravitational collapse of a spherical scalar field	[ "Roberto Giambò" ]	gr-qc	[ "gr-qc" ]	Roberto Giambò () University of Camerino (Italy), Scuola di Scienze e Tecnologie, Mathematics Division. [email protected] INAF, Sezione di Roma INdAM – GNAMPA INFN, Sezione di Perugia Gravitational collapse of a spherical scalar field Roberto Giambò August 1, 2023 ================================================== Examining the relativistic collapse of a spherical spacetime where gravity is coupled with a scalar field, this review provides a thorough analysis of some of the most relevant studies from both analytical and numerical perspectives. The discussion includes achievements made in this field, with a focus on those related to cosmic censorship, as well as recent perspectives on the topic. § INTRODUCTION AND HISTORICAL CONTEXT From the second half of last century's 60s, the so called ”singularity theorems” of Stephen Hawking and Roger Penrose, together with the first observational evidence of black holes candidates, shifted attention to the formation of singularities as an outcome of a dynamical process. Actually, despite the official reasons that were given half a century later for awarding the Nobel Prize to Roger Penrose, it is always worth remembering that the theorems of Hawking and Penrose are rather results that assert the geodesic incompleteness of a Lorentzian manifold. It was Hawking himself that supported (see e.g. <cit.>) the notion of causal geodesic incompleteness as a mathematically viable definition of spacetime singularity that, however, is not always completely satisfactory under the physical point of view <cit.>. In fact, one can conceive solutions where initial data on a would–be Cauchy hypersurface evolve into an incomplete spacetime that terminates at a Cauchy horizon, i.e. a boundary of the maximal future development – here, solution can be possibly extended but not in an unique way. This surprising feature is observed even in seemingly simple examples as the Kerr spacetime, indicating that the formation of high curvature regions evolving into a trapped surface from regular data is far from being solved by simply resorting to geodesic incompleteness. For a comprehensive account of the history, stemming from the Penrose and Hawking theorems, let me recommend the beautiful review <cit.>. As the interest in the dynamical features of spacetimes grew, it became clear that Schwarzschild (or Kerr) spacetime could only be seen as an asymptotic state of more realistic configurations that could potentially develop a singularity. Therefore, it was necessary to consider solutions with richer features in order to gain insight into the problem of relativistic collapse. However, the nonlinear complexity of the Einstein Field Equations suggested the need for a trade-off between mathematical and physical reasons. As a result, researchers turned their attention to the collapse of spherically symmetric models, with a particular focus on the study of collapse in dust clouds, radiation, and scalar fields. This review will specifically examine the latter model. Studies on scalar fields also date back to the late 1960s, with the introduction of the concept of spin-zero particles described by a scalar wave, which led to the development of the so-called boson star – for a comprehensive review, see <cit.>. From a mathematical standpoint, as we will see, the problem has been modelled in many different ways, using various gauges and assumptions. However, almost every approach shares some common features: * a metric g on the 4–dimensional manifold M=𝒬× S^2, that is given by the product metric of a Lorentzian metric on the 2-dimensional spacetime 𝒬 and a conformal metric to the induced Riemannian metric on S^2↪ℝ^3, with conformal factor depending on 𝒬; * an energy momentum tensor arising from a Lagrangian density L=1/2 g(∇ϕ,∇ϕ)+1/2 m^2ϕ^2+V(ϕ), where ϕ:𝒬→ℝ is the scalar field, ∇ is the gradient induced by g, m is the constant mass of the scalar field (m=0 is the so–called massless case) and a potential function V:ℝ→ℝ that can be possibly set to zero (free case) is also introduced. The Lagrangian may contain other terms when the scalar field is coupled with other energy sources, for example an electromagnetic field (see Section <ref>). Therefore, in its more general extent, in the present paper (M,g) will be such that R_μν-1/2 g_μνR=8π T_μν, holds and, using an effective potential V(ϕ) possibly embodying the mass contribution, (<ref>) gives T_μν=ϕ,_μϕ,_ν-(1/2 g^αβϕ,_αϕ,_β+V(ϕ)) g_μν. It is straightforward to calculate the conservation law T^μ_ν;μ=ϕ,_ν( ϕ-V'(ϕ))=0, where ϕ=g^μνϕ_,μ;ν is the curved wave operator induced by g. The equation ϕ=V'(ϕ) is often referred to as Klein–Gordon equation. To trace the beginning of research on scalar field collapse, we must start with the earlier works of Demetrios Christodoulou. As he recounts in his monograph <cit.>, the problem of the evolutionary character of trapped surfaces was posed to him by his PhD mentor at Princeton, John A. Wheeler. After working on stars filled with dust matter, where gravitational attraction is counterbalanced by outward pushing forces due to internal nuclear reactions, he turned his attention to the study of the spherical scalar field. As explained in <cit.>, one reason why he chose to study this model was because the dynamics governing its evolution are precisely described by the wave equation. Therefore, one could view this model as a perturbation of some background spacetime. Christodoulou's first work on the subject, published in <cit.>, focused on the massless free case (V=0 in (<ref>)), as did his subsequent works. The main result of this paper is a theorem that shows that if one considers initial data that are sufficiently small, they evolve into a globally unique solution that is null geodesically complete and asymptotically equivalent to flat spacetime. This result had a significant impact on the field of research, as it showed that despite the absence of forces counterbalancing gravitational attraction, as in the case of a dust star, the collapse of a scalar field – at least in the massless free case – does not necessarily lead to the formation of spacetime singularities. At the time, Christodoulou was not the only one working on scalar field collapse. Matthew W. Choptuik had already addressed the problem using numerical methods in his PhD thesis at British Columbia, under William Unruh supervision, and had published a couple of papers in which the presumed inconsistency of some numerical codes useful for numerically solving the Cauchy problem for some relativistic models was discussed and fixed. Choptuik got in touch with Christodoulou at the end of the 80s, when the latter was at Syracuse. On one hand, a picture was emerging that suggested the existence of a sort of phase transition that governed initial data of solutions, with some evolving into a singularity and others regularly dispersing in the infinite future. However, it was still unclear at the time whether one could fine–tune the data to develop a horizon and obtain black holes of arbitrarily small mass, and whether one could fully capture the causal behaviour of this hypothetical phase transition between black hole formation and dispersion. Indeed, Choptuik's numerical analysis published in <cit.> confirmed the existence of what would have been subsequently known as the black hole threshold, dividing the two classes of initial data. In other words, considering various 1–parameters family of initial data, there exists a critical value of the parameter separating the two possible evolutions – i.e. trapped surface formation vs dispersion. This behaviour was found later to be common to many collapsing models, and for a complete account on the subject the main reference is the Living Review by Carsten Gundlach and José M. Martín–García <cit.>. The solution corresponding to the critical value of the parameter can be shown to exhibit surprising features <cit.>. To begin, it is a discretely self–similar solution. Moreover, it contains a point singularity that can communicate with the infinite future boundary – it is, therefore, an example of a naked singularity, thereby violating the cosmic censorship conjecture made by Roger Penrose <cit.>. By the way, it was Choptuik's discovery that in 1997 put an end to the first bet on naked singularity between Hawking, who was unfavourable to their existence, and John Preskill and Kip Thorne, who supported the possibility of naked singularities. Numerical analysis were suggesting that the initial data leading to naked singularities might not be generic, and this led to a new version of the bet. Furthermore, it boosted research towards an analytical approach to the analysis of naked singularities. And this bring us back to Christodoulou's work. After some follow-ups to <cit.>, in the attempt to prove Penrose conjecture, he had published in 1991 the first of four papers that are still nowadays considered cornerstones of the subject. The result <cit.> expresses sufficient condition on the initial data of the problem leading to trapped surface formation, hence to a geodesically incomplete spacetime. That is a key result for the topic under exam here and for all following research (see discussion after Theorem <ref> below). The solutions considered so far were sufficiently regular, i.e. at least differentiable. The second paper <cit.> of the series improves previous results leading to dispersing behaviour but mostly takes into account bounded variation solutions, that prove to be extremely important for the other two papers of the series. Indeed, in <cit.> an analytical example of a scalar field solution collapsing to a singularity without horizon formation, hence naked, is found out. In the same work it is announced, for “a subsequent paper”, that the space of initial data leading to solutions with this feature is unstable in the larger space of bounded variation solutions considered in <cit.>. Curiously enough this last paper was published only five years later <cit.> – although received by the journal a couple of years before publication – and it has since often been popularized as the proof of Penrose conjecture. In Section <ref> we will get back to the influence of this work in relation to cosmic censorship. The methods pioneered by Christodoulou were refined for a number of more recent results, especially concerning the Einstein–Maxwell–scalar field model, see Section <ref> below. These techniques, sharing a common approach involving the analysis of an initial value problem for the PDE governing system's evolution, led to a number of considerable result and advances in the study of problems in Mathematical Relativity, under very general assumptions. Some literature can be traced back from <cit.> On the other side, since the dynamical system picture was not fully resolved by the theorems in <cit.>–<cit.>, a significant effort was devoted to many analytical and/or numerical studies of the 'critical' case found out by Choptuik. The aim was to determine the extent to which Christodoulou's non-genericity result was essentially describing the same situation as the Choptuik black hole threshold. Despite some initial enthusiasm, it must now be concluded that the link between the two pictures, while displaying some overlapping features, has not been fully resolved to date. On quite a different pathway, a few years later, stimulated by connections with extended gravity theories and string theory, a growing interest arose about models self interacting with a potential. Dealing with a cosmological setting, in this case one usually restricts to homogeneous models where the background metric interacting with the scalar field is a Robertson–Walker (RW) spacetime – this gives the obvious advantage to reduce the model's equations to an ODEs system. A huge variety of potentials V(ϕ) has been discussed within this framework, finding conditions under which the universe collapse to a future singularity, and global models without the formation of an apparent horizon has been discussed also in this situation. Of course, this study interacted with those aiming to find quantum mechanisms to avoid recollapsing universe, see e.g. <cit.> – a quite recent and complete review on this subject is <cit.>. The review is organized as follows. The basic results from Christodoulou and PDE approach are described and commented in Section <ref>. The following section <ref> is devoted to the advances in the study of the black hole threshold and the critical case, and the homogeneous case is described in Section <ref>. One last section <ref> is dedicated to some hints and studies that are not completely related to the topics covered in the previous section, and to sketch conclusions. § SCALAR FIELD COLLAPSE AS A PDE INITIAL VALUE PROBLEM In this section we review among the most influential and important results regarding the scalar field collapse, when the evolutionary PDE approach is embraced. As said in the Introduction, this field can be traced back to the earlier works of Christodoulou on the subject, but it must be also said that these were pioneering works, with a notation often invented specifically to solve the problem at hand, and therefore not yet consolidated in the literature at the time and sometimes not fully optimized, so the less accustomed reader may find it challenging at first glance. For example, in <cit.> a geometric Bondi coordinate system is used to describe the metric, ds^2=-a(u,s) du^2-2 du ds+r^2(u,s) γ_𝕊^2 (we refer to γ_𝕊^2 as the Euclidean metric induced by 𝕊^2↪ℝ^3) but many of the calculations in the same work are done in a double null frame. The use of double null coordinates is rather effective in this PDE approach: ds^2=-Ω^2(u,v) du dv+r^2(u,v) γ_𝕊^2. Coherently with <cit.> and other works in this thread, we normalize the scalar field ϕ→1/√(4π)ϕ and the potential V→√(2π) V to get rid of a 4π factor in the coupling constant, thereby obtaining from (<ref>) r,_uv =-1/r r,_u r,_v-1/4rΩ^2+1/2 rΩ^2 V(ϕ), (logΩ),_uv =(Ω/2r)^2+r,_ur,_v/r^2-ϕ,_uϕ,_v ϕ,_uv =-ϕ,_ur,_v+ϕ,_vr,_u/r-1/4Ω^2 V'(ϕ), as a system of independent PDEs. Moreover, the two relations (r,_u/Ω^2),_u =-r(ϕ,_u/Ω)^2, (r,_v/Ω^2),_v =-r(ϕ,_v/Ω)^2 are consequence of (<ref>)–(<ref>). It will also be useful to consider the Misner–Sharp mass[often referred to as Hawking mass too, see e.g. <cit.>.] m(u,v) defined by the relation 1-2m/r=g(∇ r,∇ r), that in view of (<ref>) becomes m(u,v)=r/2(1+4r,_ur,_v/Ω^2) §.§ Trapped region formation in the free massless case As already mentioned, Christodoulou considers the free massless case, V(ϕ)=0, that implies (see (<ref>)) than the scalar field satisfies the homogeneous wave equation. The equations (<ref>)–(<ref>) can be transformed in a first order (overdetermined) system with the positions ν=r,_u, λ=r,_v, ζ=rϕ,_u, θ=rϕ,_v in the unknown functions (r,λ,ν,m,ζ,θ) as follows (see e.g. <cit.>) r,_u =ν, r,_v =λ, λ,_u =λ(2m ν/r^2-2m r), ν,_v =ν(2m λ/r^2-2m r), m,_u =(r-2m)ζ^2/2r ν, m,_v =(r-2m)θ^2/2r λ, θ,_u =-ζλ/r, ζ,_v =-θν/r. The first paper on the subject, as already recalled, is <cit.>, where a condition on the initial data is derived to have global forward–in–time solutions. This result was recently refined in <cit.>, where it is proven that initial data can be chosen such that their L^1 norm is infinite but they evolve again to a forward–in–time global solution. The construction can be extended in order to prescribe initial data at past null infinity. Moreover these refinements contain estimates useful for proving recent stability results of dispersive spherical solutions with respect to non–spherical perturbations <cit.>. On the other side, on <cit.> a sufficient condition for trapped surface formation starting from collapsing initial data is formulated. The theorem is stated – and re–proved – in terms of the double null framework sketched above in the recent paper <cit.>, that here we follow. First of all, notice that spherical symmetry allows to consider a timelike curve Γ that will play the role of the symmetry centre. Characteristic initial data along u=u_0 and v=v_1 are prescribed (see Figure <ref>), and v_2>v_1 is taken to define the quantities η_0:=m(u_0,v_2)-m(u_0,v_1)/r(u_0,v_2), δ_0:=2 r(u_0,v_2)-r(u_0,v_1)/r(u_0,v_2), Now, a collapsing situation amounts to assume r,_u<0 and then, recalling (<ref>), trapped surface forms when also the other partial derivative becomes negative somewhere, r,_v<0. There exists a constant c_1≥ 1 such that if δ_0≤1/e and η_0>c_1 δ_0log(1/δ_0) then a trapped surface forms in the region [u_0,u_]×[v_1,v_2], where u_ is such that r(u_,v_2)=3δ_01+δ_0. It is worth mentioning here that the techniques pioneered in <cit.> to prove the above Theorem lie at the foundations of many other important results in the gravitational collapse mathematical theory, let me cite the celebrated <cit.> proof of trapped surface development from a packet of waves emitted at past infinity. There, the technique is refined to assume that the incoming radiation per unit solid angle is uniformly bounded from below. The original idea is contained in the above Theorem: the conditions on η_0 and δ_0 in Theorem <ref> statement [In the original formulation of <cit.>, the conditions were stated as δ_0<12 and η_0>E(δ_0), with E(x)=x/(x+1)^2(5-x-log(2x)). The formulation shown here is introduced by the Author since <cit.>, where the result is used for the proof of naked singularity instability (see below). ] are roughly equivalent to assuming that a sufficient amount of energy is emitted by the scalar field, at a sufficiently short amount of time. Actually, <cit.> contains other important results for the present context, that haven't been reported in the above statement of Theorem <ref>. In particular, see Figure <ref>, the existence of a (non-central) strictly spacelike singular future boundary is proven. This boundary terminates at the boundary of the centre of symmetry where also the apparent horizon 𝒜 arrives. In other words, the centre of symmetry becomes trapped exactly when it becomes singular[This is a common feature shared by other models of gravitational collapse with matter, for instance collapsing stars with dust <cit.> or anisotropic <cit.> matter. See also <cit.> and references therein.]. In a more recent paper <cit.> Mihalis Dafermos will show that these solutions can be regularly extended backwards up to past null infinity. The paper <cit.> study solutions with bounded variation (BV) initial data. With this notation it is means that the functions ϕ, r,_u, r,_v, (rϕ),_u, (rϕ),_v are BV with respect to one null variable, uniformly in the other, and vice versa; plus some technical regularity conditions on the centre of symmetry Γ. The paper considers both regular solutions and those developing singularities. A regularity condition is stated in terms on the behaviour at the centre (see Fig. <ref>): if v_>0 is such that, ∀ v<v_, a solution exists in 𝒟(0,v), the domain of dependence of (0,v), then ∃ϵ>0 such that the condition lim_u→ v_sup_𝒟(u,v_)2m/r<ϵ implies the regular extension of the solution on 𝒟(0,v_1) for some v_1>v_. In other words, in order to extend the regular centre at some point, 2m/r must tend to zero as this point is approached from every direction coming from 𝒟(u,v_). If this condition isn't satisfied then a central singularity develops, and the existence of a non-central boundary, preceded by a trapped region as in Fig. <ref>, is proved. §.§ Non-genericity of naked singularity As said in the introduction, the solution exhibiting a naked singularity is built in the subsequent paper <cit.>. It is better to recall here that the naked singularity here corresponds to a situation where horizon does not exist at all[In this sense this situation is very different from the naked singularity developing in the collapse of some matter models, see previous footnote. In those contexts there may be situations where outgoing null geodesics are “emitted” from the central singularity and detected by a faraway observer, even when a horizon forms.]. We will discuss this example in next Section, since it is in some way connected to the critical solution numerically found by Choptuik <cit.>. Here we only stress that this example exhibits a pointwise singularity on the centre of symmetry. This fact is relevant because, in <cit.>, it is shown that if one has bounded variation initial data on u=0 developing such a solution, or more in general a solution that is neither future regular nor develops an event horizon, then a suitable perturbation of this initial data restores the apparent horizon and, hence, trapped surface formation. More specifically, assigned BV initial data such that a naked pointwise singularity develops at the centre, then two BV functions f_1(v), f_2(v) exist, such that any (non-trivial) perturbation of the initial data with a linear combination of f_1(v) and f_2(v) results in a trapped surface formation. Let us discuss in brief this important result. First of all, it must be said that the proof in <cit.> is carried out again in a Bondi coordinate setting similar to (<ref>). However, in recent times a new proof appeared <cit.> where the argument is adapted in double null coordinates. The rough idea of the argument is sketched in Fig. <ref>. Coordinates are shifted in such a way that (0,0) is an isolated singularity in the (u,v)–plane as a development of some particular initial data α_0(v)=rϕ,_v(u_0,v). The idea follows these steps: finding proper estimates of the quantities involved (not satisfied with the initial data α_0(v)), in order to determine a sequence (u_n,δ_n) approaching (0,0), in such a way that one can apply Theorem <ref> with v_1=0 and v_2=δ_n, and determine trapped surfaces existence. At every step of the sequence one is able to determine a piece of the apparent horizon. Since the sequence approaches the singular point, so does the apparent horizon, and the singularity is no more naked. * finding conditions on r ϕ,_v(0,v) such that estimates cited on step 1 hold. This is done by considering two separate cases, see <cit.>, corresponding to <cit.>. * determining – see <cit.> and, in its new version, <cit.> – two suitable functions f_1(v) and f_2(v) and proving that the initial data α_0(v)+λ_1 f_1(v)+λ_2 f_2(v) (∀λ_1,λ_2∈ℝ) evolves in such a way that r ϕ,_v(0,v) necessarily satisfies the conditions stated on step 2. The instability result in <cit.> marks an important point in the history of Penrose cosmic censorship conjecture, and many still regard that paper as “the proof” of this conjecture, or at least of a re–designed version of it, allowing for naked singularities at most for an exceptional set of zero measure in the set of initial data for the evolution problem. The result shows that the codimension of the space of initial data is at least 2, being at least two “directions” where an initial data developing a point singularity can be perturbed, to restore trapped surface formation. This and other results in this context seems to highlight a non-genericity feature of pointlike singularities developing without an horizon, but a central question may be the choice of the suitable – under a physical point of view – regularity requested for the initial data, because demanding a mild regularity property – as bounded variation solution may be, as commented in <cit.>– can result in a too large space of admissible initial data. In this sense, if we embrace the evolutionary PDE setting that is the object of the present section, and therefore identify the double null as the natural coordinate setting to work with, taking initial data on a null hypersurface, it must be said, regarding the instability theorem proved in <cit.> or in its new version of <cit.>, that at least the function f_2(v) is absolutely continuous in [0,+∞[ (whereas f_1 has a jump discontinuity). This means that non-genericity (with codimension at least 1) is granted when one restricts to absolutely continuous functions. Of course, the scalar field model treated so far is not the most general one (recall that potential is been set to zero so far), and then in the following subsection we wish to explore some more recent result that aim to extend the knowledge on the subject through analytical studies. §.§ Cosmological constant and potential Theorem <ref> has been proved in other contexts. For example, in <cit.> the proof is made where the addition of a positive cosmological constant is considered, thereby obtaining the modified system with respect to (<ref>) R_μν-1/2 g_μνR+Λ g_μν=8π T_μν. This amounts to work with system given by (<ref>), where the potential V(ϕ) is assumed to be constant and positive. Therefore, with the same normalization of the scalar field considered above (see right after (<ref>)) we arrive to the system (<ref>)–(<ref>) with V(ϕ)=Λ/2. The conditions are similar to those appearing in <ref>, paying attention to that the introduction of Λ affects the definition of mass (<ref>) that in this context must be normalized as ϖ=m-Λ6 r^3. The case with non-zero potential has in general been less covered in the PDE approach. Introducing a potential in principle adds a source term to Klein–Gordon equation and may significantly complicate the evolution in that, just to mention, scale invariance does not hold anymore. However some results have been proved all the same. Among these it must be certainly mentioned one of Dafermos paper on the subject, i.e. <cit.>. At the time, paper <cit.> appeared that – heuristically trying to approximate the model with the anti–deSitter homogeneous spacetime, corresponding to the asymptotic state of a negative minima for the potential V(ϕ), was conjecturing the appearance of naked singularities generically arising but not from the centre of symmetry of the system. The paper <cit.> shows that this situation cannot happen for a collapsing scalar field with bounded from below potential, regardless of the sign of V(ϕ) at the minima. In particular, the situation described in <cit.> is the following. We consider the future evolution 𝒬^+ from a spacelike Cauchy hypersurface S under the collapsing assumption r,_u<0 in 𝒬^+. The notion of first singularity is introduced as follows: A boundary point p∈∂𝒬^+ is a first singularity if * J^+(p)∩𝒬^+ is eventually compactly generated (i.e. ∃ X⊂𝒬^+ compact such that J^-(p)⊂ D^+(X)∪ J^-(X)); * if Y⊊ J^-(p)∩𝒬^+ is eventually compactly generated ⇒∃ q∈𝒬^+ : Y=J^-(q). The result proved in <cit.> is that (see Fig. <ref>) non central first singularities are such that J^-(p)∩𝒬^+∩ D^+(X)∩𝒯∅, where 𝒯 is the trapped surface defined by r,_v<0. This means that the horizon must form prior to the appearance of a non central singularity. The argument lies in careful estimates on the quantities involved in the “diamond” J^-(p)∩ D^+(X), and that is why non centrality of p is crucial for this proof. The paper <cit.> of the same period states general sufficient conditions on energy momentum tensor for the future null infinity I^+ to be future complete when 𝒬^+∖ J (I^+)∅ (i.e., when a trapped region arises) and in the scalar field collapse the result can be applied when V(ϕ)≥ 0. Indeed, one of the sufficient conditions stated in <cit.> is the dominant energy condition that in this case requires V(ϕ)≥ 0 to hold. In the evolutionary PDE approach, this corresponds to a reformulation of weak cosmic censorship given in <cit.>, see also <cit.>. It is also worth mentioning that the results proved in <cit.> have been extended to higher dimensional spacetimes in <cit.>. §.§ Interaction with an electromagnetic field Recalling the dualism between the (spherically symmetric) Reissner–Nordström spacetime and the (rotating) Kerr spacetime, in an effort to step out of the spherically symmetric case, the first attempt is to consider a spherical situation when an electromagnetic field is added to the system. This leads to modify the Lagrangian (<ref>) as happens in the Einstein–Maxwell–scalar model, where the energy momentum tensor T=T^SF+T^EM is made by the sum of two components T^SF given by (<ref>) and an electromagnetic field T^EM_μν= F_μλF_νρg^λρ-1/4 g_μνF_λρF_στg^λσg^ρτ. In this situations the scalar field and the electromagnetic field are uncoupled, so that each component of T separately satisfies Bianchi equations, i.e. the divergence of each T^EM and T^SF separately vanishes. Consequently, the Klein–Gordon equation still holds, together with Maxwell equations F^μν_ ;ν=0, F_[μν,ρ]=0, that in view of the sphericity of the metric, gives a quite simplified form for T^EM: (T^EM)^μ_ν=q^2/2r^4·diag{-1,-1,1,1}, where the constant q is the charge of the EM field. The mass m given in (<ref>) is correspondingly renormalized as ϖ=m+q^2/2r^2. The addition of this Maxwell field in the model, as studied by Dafermos in a series of his earlier papers <cit.>, results in a quite different evolution of the collapse. Indeed, the future maximal development of the Cauchy problems ends at least partially in a Cauchy horizon that is not necessarily a singularity in the classical sense[Interestingly enough, adding a charge to other matter models also results in weak singularity formation such as the shell crossing developing from the collapse of charged anisotropic models studied in <cit.>]. Near this region the function r is positive and bounded away from zero <cit.>, but the renormalized mass ϖ blows up the at the Cauchy horizon <cit.>. These facts imply <cit.> that the metric can be (not uniquely) C^0 extended across the Cauchy horizon. In the same paper <cit.> a sufficient condition, based on the growth estimate of ϕ,_v along the event horizon, is shown to imply that C^1 extension is not allowed. Actually Jonathan Luk and Sung–Jin Oh observe in papers <cit.> that this pointwise estimate condition is difficult to be verified; in the same works, the C^2 inextendibility across the Cauchy horizon is proved, supporting a C^2–version of strong cosmic censorship. These results and the techniques employed have been constantly refined in order to tackle the stability problem of the Cauchy horizon in Reissner–Nordstrom and later in Kerr spacetimes. This long programme has produced a huge literature in this sense and fostered a great interest on the study of stability properties of these features even outside the realm of spherical symmetry, see <cit.> and more recently <cit.> and again <cit.> with therein references. As we have already said, Theorem <ref> has been proved again in the paper <cit.>, to carry on the argument in double null coordinates that are useful to study a generalization of the system that we have considered so far, the so called Einstein–Maxwell–charged scalar field. In this situation the scalar field is coupled to the electromagnetic field, which results in a modification of Maxwell equations because Bianchi identity does not hold anymore separately for the two components of T. We refer the reader to <cit.> of Maxime Van de Moortel for some mathematical and physical aspects of this model. In the same paper, the existence of a Cauchy horizon is proven, together with C^0–extendibility across this horizon. In <cit.>, with suitable modifications of the sufficient conditions from the original theorem in <cit.>, the existence of trapped surface is proved for charged scalar field, confirming previous numerical insights on this model <cit.>. The above cited paper <cit.> actually also tackles the massive case m 0 (recall (<ref>)), where Cauchy horizon existence is proven also where mass of the scalar field is taken into account. In a later work <cit.> it is proven that the C^2–extension is not possible since Ricci curvature diverges. Results about continuous extendibility have recently been found too <cit.>. § SCALAR FIELD COLLAPSE AND THE NUMERICAL APPROACH We have already commented in the Introduction that the “numerical counterpart” to Christodoulou for the scalar field collapse is represented by Choptuik and mostly by his celebrated [ Almost 1000 citations received at the time the present work was written, Scopus data.] paper <cit.> (but see also <cit.> for some more details on the numerical scheme). His work is important because it represents a crucial advance to understand the scalar field's dynamics, but also because it revealed a behaviour that was surprisingly found to be satisfied by many other relativistic models along the years, as explained in the reviews <cit.>. Gundlach and his collaborators gave a huge contribution in understanding this surprising behaviour – most of the material in the present section is taken from their works. Scalar field spherical model for numerical analysis purposes is usually cast into a Schwarzschild gauge: -α^2(t,r) dt^2+a^2(t,r) dr^2+ r^2 (dθ^2+sin^2θ dφ^2). Einstein equations (<ref>) in the free massless case become[It should be noted that there is no longer the normalization usually done in analytical studies to eliminate π from Einstein field equations.] a,_r/a+a^2-1/2r =2π r(Π^2+Φ^2), α,_r/α-a,_r/a-a^2-1/r =0, a,_t/α =4π rΦΠ, where Φ=ϕ,_r, Π=a/αϕ,_t. One advantage of using this gauge is that the first two equations contains only r–derivatives of the metric coefficients, and in principle they can be integrated in r once the dynamics of the scalar field is known. The latter is captured by the wave equation ϕ=0 which takes the form Φ,_t =(α/aΠ),_r, Π,_t =1/r^2(r^2α/aΦ),_r. Also notice that the mass in this formulation obey the relation m,_r=2π(r/a)^2(Φ^2+Π^2), which allows quite simply to calculate the total mass of the spacetime at each time just by integration, M(t)=2π∫_0^∞(r/a)^2(Φ^2+Π^2) dr. The numerical evaluation is performed on a typical finite difference grid. Φ and Π are specified at some initial time t=0, and this completely defines α(0,r) and a(0,r). The numerical scheme ignores (<ref>) and uses (<ref>)–(<ref>) together with (<ref>)–(<ref>) to determine completely the evolution of the system from the initial data – for more details reader is referred to <cit.>, especially concerning how the discretization scale varies both in space and in time to keep the local truncation error below a fixed limit. Noticeably enough, the system is invariant under the rescaling (t,r)→ (kt, kr), ∀ k>0; we'll get back on the symmetry features of this model in a while. §.§ The black hole threshold To describe the nature of the critical behaviour found out by Choptuik, let us alternatively assign ϕ_0(r)=ϕ(0,r) instead of Φ(0,r) and Π(0,r), for instance ϕ_0(r)=ϕ_0 r^3 e^-(r-r_0/δ)^q, together with a condition that the scalar field is purely ingoing at initial time. Recalling the discussion in Section <ref>, if the mass 2M≪ L, where L is the pulse thickness (controlled in (<ref>) by δ) then the wave packet disperses, whereas when 2M≈ L then the strong gravity regime allows for trapped surfaces formation. What Choptuik did was to fix all the parameters of the initial data (<ref>) but one, call it p, and vary it in order to obtain the critical value p^* by a simple bisection technique – that is, find two parameter values p^HI and p^LO such that p>p^HI certainly implies black hole and p<p^LO certainly implies dispersion, and then proceed by subsequent bisections of the interval [p^LO,p^HI] to approximately determine p^. The study of this 1-parameter family of numerical solutions revealed that the mass of the black hole was obeying a scaling form depending on the free parameter p when p→ (p^)^+: M≃ C\|p-p^\|^γ, where C and, of course p^, depend on the parameter chosen to describe the family, but γ is a universal constant (γ≈ 0.37) at least for this matter model. Moreover, the critical solution is discretely self–similar, that is there exists a conformal isometry of the spacetime with conformal factor equal to e^-2Δ for some Δ. This scale–periodicity (Christodoulou's terminology) or “echoing” (Choptuik's terminology) behaviour can be formalized as follows. Let us introduce the variable change τ=logL/t_-t, x=logr/t_- t, then for values t_* and L depending on the model, the unknown functions α, a, ϕ as a function of (τ,x) are periodic with respect to τ (for instance ϕ(τ,x)=ϕ(τ+Δ,x)). For massless scalar field the period Δ is approximatively equal to 3.447. Therefore, representing the space of initial data as the phase space associated to the dynamics of scalar field collapse, the situation is roughly explained in Fig. <ref>. The stable manifold for the critical solution is the so–called black hole threshold, and separates data evolving to a blackhole to those that dispersing to future infinity. This is a rough description for many reasons: one of these is that the unstable mode for the black hole is not unique, as pointed out in <cit.>. Moreover, it's the numerical analysis made by considering several 1-parameter families of initial data, all exhibiting the same behaviour, that supports this universality, towards the conclusion that the black hole threshold is a codimension one submanifold. It is to remember that the scalar field case is the first example exhibiting features of universality, – solutions' endstates depend on the positions of their data with respect to a critical submanifold only – black hole mass scaling like (<ref>), and critical solution's scale–invariance of some kind. Attempts to study the critical behaviour have been made <cit.> with both quantitative and qualitative results not completely overlapping with those from numerical studies. All these hints are probably the signature of profound phenomena that have yet to be entirely understood in their full generality. What's for sure, is that numerical analysis has confirmed critical behaviour existence in many cases. In the realm of scalar field collapse, when considering a massive scalar field <cit.>, in the non-minimally coupled case with a potential <cit.>, and in other gravity theories like Brans–Dicke theory <cit.>. The latter case is relevant since in this context a proof of the existence of naked singularity, extending to this case the example of Christodoulou <cit.> that we have already cited and that we will discuss in next Subsection too, is given <cit.>. Further examples of critical behaviour in other matter models are given for instance in the case of the Einstein cluster studied in <cit.> For a much broader overview on critical behaviour we refer the reader to <cit.> once again and to the more recent review <cit.> also. §.§ Naked singularity in the critical case The picture sketched in the above subsection shows that there is a so–called critical solution that lives in the threshold black hole and is neither regular nor contains a black hole covered by a horizon. In Fig. <ref> this solution is represented as a limit cycle in the phase space since, as we have observed, it has periodicity features. This critical solution, as mentioned in the previous sections, is important also because it possesses a singularity that yet is not covered by a event horizon, and is visible a future null infinity. It is therefore a naked singularity. The global structure of this spacetime has been the object of several works <cit.>. However it must be remembered that its existence basically comes from numerical inference, although quite recently a computer–assisted proof appeared <cit.> that, starting from and approximate solution shows the existence of a real analytic true solution, precisely interpreted as the Choptuik critical case. The picture emerging from the analysis of Gundlach and Martín–García (see in particular <cit.>) shows that it is a spacetime that can be uniquely continued up to the future outgoing null cone of the pointlike singularity, that therefore is a Cauchy horizon. This analysis seems confirmed also using other numerical approaches, see e.g. <cit.>. The solution can be extended in a non unique way beyond the Cauchy horizon, but the only discrete self–similar possible extensions are either with a regular centre or with a timelike central singularity (see Figure <ref>), excluding null singularity followed by a spacelike dynamical singularity, similar to other collapsing matter model such as spherical dust. The past light cone is a self similar horizon – a null surface invariant by the self–similarity. Without considering the possible extension across the Cauchy horizon, the model has then a pointlike singularity, in that resembling Christodoulou counterexample built in <cit.>. The two solutions anyway cannot coincide, because the latter is actually a continuous self–similar solution, with metric g=-e^2ν du^2-2e^λ+ν du dr+r^2γ_𝕊^2, where λ and ν are function of x=-r/u In this way, defined the vector field S=u∂_u+r∂_r, one has ℒ_S g=2g, Sr=r. Moreover the vector field acts on the scalar field ϕ so that Sϕ=-k∈ℝ. Self–similarity reduces Einstein equations to an ODE system, and in <cit.> it is shown that a solution without a horizon exists in the half–plane u<0, such that the scalar field along two null directions terminating in (u=0,r=0) diverges like log r, and this makes (0,0) a singular point not covered by a horizon. In this example the solution is extended to the half plane u>0 isometrically, so that one can obtain a solution that is regular everywhere except one single point. At this stage, one may argue that Christodoulou analysis described in Section <ref> and the picture emerging from critical behaviour don't overlap, since one may apply the instability argument described in Section <ref> above to an initial data in the black hole threshold, obtaining a 2-parameter family of initial data always leading to black hole formation. As pointed out in <cit.>, one possible explanation for this apparent contradiction lies in that the picture emerging from numerical evidence concerns solutions starting from smooth functions, unlike the argument presented in Subsection <ref>. Indeed (see Fig. <ref>) let us consider initial data α_0 corresponding to a solution in the black hole threshold. In view of the above discussion we can suppose this data at past null infinity I^-, in order to be consistent with the setup of Section <ref>. The perturbative argument of Christodoulou considers two functions f_1(v) and f_2(v) (see step 3. in Section <ref>). These functions will be set to zero for v<0, so that the evolution is left unchanged in the interior of the past null cone of ℬ_0. Then they are defined for v>0 in order to obtain an apparent horizon extendable until ℬ_0, as explained in Section <ref>. This results in that f_1(v) has a jump discontinuity at v=0, while f_2(v) is absolutely continuous on the real line, but not smooth. Therefore, although the whole 2-parameters family of initial data ℱ(p_1,p_2)={α_0(v)+p_1 f_1(v)+p_2 f_2(v) : p_1,p_2∈ℛ} evolves to a black hole, this is not a family of smooth initial data, and then lies outside the range covered by numerical studies. § HOMOGENEOUS SCALAR FIELDS The scalar field collapsing model that we have discussed so far has no constraint imposed on the evolution of the scalar field, excluding the initial assumption that we have made at the very beginning that ϕ is defined on the quotient space 𝒬 for symmetry reasons. In a cosmological setting, however, isotropy arguments impose the further constraint that ϕ is a function on the cosmological time only. Of course this considerably reduce the mathematical complexity of the model, that will be now ruled by a system of ODEs. On the other side, this allows to include without significant drawbacks the self–interaction potential V(ϕ), and to study solutions' evolution by means of dynamical systems techniques <cit.>, retaining information on the qualitative behaviour of the scalar field depending on assumptions made on V(ϕ). The physical significance of scalar fields with potential in a cosmological setting is related to the interpretation of inflationary expansion phase of the universe <cit.>. But we find them also in string theory or in higher dimensional gravity, using Kaluza–Klein reduction <cit.>. Moreover, they arise also in nonlinear theories of gravitation <cit.> such as f(R) theories, since as is well known they can be reduced, up to a conformal transformation, to the ordinary gravity theory where the energy momentum tensor has a scalar field component, where V(ϕ) depends on the form of f(R). Scalar field and spacetime singularities are also connected of course within the study of primordial gravitation, for instance to suggest alternative model to the standard Λ CDM paradigm <cit.> or within loop quantum cosmology <cit.>. Perhaps the best known example of homogeneous collapsing solutions is the Oppenheimer–Snyder <cit.> homogeneous dust ball collapsing to a black hole. Similarly, one may consider a scalar field – possibly coupled with other matter types – as interior of a star, that must be matched with a reasonable exterior to get a global model where one can, once again, study the collapse evolution and see whether singularities form with or without a horizon. In this context one can build homogeneous “scalar field stars” matching a FR interior with a – possibly non homogeneous – exterior solution. All in all, there is a huge variety of situations that have been studied, and a wide account on the existing literature on the subject can be found in the two companion reviews <cit.>. Einstein's field equations for the RW model ds^2=-dt^2 + a(t)^2[1/1-k r^2 dr^2+r^2(dθ^2+sin^2θ dφ^2)], with the stress–energy tensor (<ref>) are (G^0_0=8π T^0_0): -3(k+ȧ^2)/a^2=-(1/2ϕ̇^2+V(ϕ)), (G^1_1=8π T^1_1): -(k+ȧ^2)+2aä/a^2=(1/2ϕ̇^2-V(ϕ)). in the unknown functions a(t),ϕ(t). [As in the previous situations, the scalar field and the potential are normalized in order to get rid of the π factor in Einstein field equations. Unfortunately, this normalization changes from paper to paper in literature – here we will follow the normalization made (for the flat k=0 case) in <cit.>, that will possibly differ from those used in other papers that we will refer to.] These equations imply as usual Bianchi identity that in this context becomes T^μ_ 0;μ=-ϕ̇(ϕ̈+V'(ϕ)+3ȧ/aϕ̇)=0. The above second order system can be translated into a first order system introducing the Hubble function h=ȧ/a, and v=ϕ̇. We first briefly consider solutions such that ϕ(t)=ϕ_0 constant on some interval so (<ref>) holds, but V'(ϕ_0) 0 (so the term in round bracket in (<ref>) is non zero). These evolutions are obtained by (<ref>): ȧ^2=V(ϕ_0)/3 a^2-k, corresponding to a (anti)deSitter solution (the regular value of the potential in ϕ_0 plays the role of the cosmological constant). Excluding this case we have that (ϕ,v,h) solves the regular system ϕ̇ =v, v̇ =-V'(ϕ)-3h v, ḣ =-h^2-13(v^2-V(ϕ)). Observe that the function W(ϕ,v,h)=h^2-1/3(1/2v^2+V(ϕ)) satisfies by (<ref>) the identity W(t)=-k/a(t)^2, where k=-sgn(W(0)). The sign of W is invariant by the flow (Ẇ=-2hW) – in other words the system (<ref>)–(<ref>) represents at once all RW cosmologies (<ref>). The equilibrium points are given by (ϕ_,0,h_) such that V'(ϕ_)=0, h_^2=1/3 V(ϕ_) (notice that W(ϕ_,0,h_)=0), that physically correspond to (anti)deSitter universe. The sign of the Hubble function h tells us whether the solution is expanding (h>0) or collapsing (h<0). §.§ (Re)collapsing solutions and nature of the singularity Let us first briefly discuss the evolution from expanding initial conditions, i.e. h(0)>0 <cit.>. Fig. <ref> sketches some of the integral curves of system (<ref>)–(<ref>) in this case. A minimum of the potential V(ϕ) with positive critical value is a stable equilibrium point, and nearby cosmologies approach the corresponding deSitter state asymptotically: no collapse will take place here. These are the curves on the right of Fig. <ref>. For the open topologies k≤ 0, corresponding to the portion of the phace space W≥ 0, the upper branch of W=0 near this local minimum acts like a constraint forcing the solutions to approach the minimum. The deSitter space is attractive also for the portion (W<0) of the phase space between the two branches of W=0, corresponding to k=1 cosmologies, at least when the minimum of V(ϕ) is strictly positive. When V(ϕ_0)=0 then recollapse is possible in k=1 case. On the left of Fig. <ref> we have a situation near a local minimum for V(ϕ) with negative critical value. In this case the two branches of W>0 intersect somewhere at h=0, determining a sort of “tunnel” crossed by solutions in the open topology k=-1, that in this way reach the h<0 portion of the space and then recollapse. A fortiori, that happens also for solutions with k=0 and k=1. Once we have established that homogeneous scalar fields may collapse, even if they start expanding, we can ask whether the evolution ends into a singularity and if a horizon forms. Since the dependence on h in the right-hand side of (<ref>) is quadratic, a singularity will likely form in a finite amount of cosmological time. In <cit.> the flat case k=0 is approached by using the scale factor a(t) as a sort of time, and expressing the energy density ρ_SF of the model, given by (minus) the right-hand side in (<ref>), as ρ_SF(t):=1/2ϕ̇(t)+V(ϕ(t))=3(ψ(a(t))/a(t))^2. Using (<ref>) we obtain ȧ=-ψ(a), and then the singularity form in a finite amount of cosmological time if and only if <cit.> ∫_0^a_01/ψ(a) da<+∞ for some a_0>0 corresponding to the initial data at t=0. Moreover, all the dynamical quantities involved can be expressed as a function of a, for example using (<ref>)–(<ref>) we have (dϕ/da)^2 =2/a^2(1-a ψ'(a)/ψ(a)), V(a) =2ψ^2(a)/a^2(1+a ψ'(a)/2ψ(a)), whereas the mass in (<ref>) takes the form m=r^3/2 a(t)ȧ^2(t)=r^3/2 a ψ(a)^2, which allows to write the trapped region R<2m as ψ(a)^2 r^2>1, and in this way one can find out sufficient conditions on the function ψ(a) to develop a horizon or a naked singularity <cit.>. The idea sketched above extends the approach used by Rituparno Goswami and Pankaj S. Joshi <cit.>, where ρ_SF=ρ_0 a^-ν with ν>0 is considered. In the language of (<ref>) that amounts to postulate ψ(a)= ψ_0 a^1-ν/2, and in view of (<ref>) a singularity always forms in a finite amount of time if ν>0. On the other side, if the inequality r<1/ψ_0a^ν2-1 is satisfied until a=0, then (<ref>) cannot hold and then an apparent horizon with a trapped region cannot form. Then, if ν<2, the right-hand side of (<ref>) diverges in the approach to the singularity, and then ∀ r>0 there is a region near the singular boundary that is untrapped, see Figure <ref>. To build a global model one can consider a RW metric for r≤ r_b for some r_b, and get initial data a(t)=a_0 such that r_b<1ψ_0 a_0^ν2-1. In this way the solution will live in the untrapped region ∀ r∈[0,r_b], ∀ t : a(t)∈]0,a_0]. Then a matching at r=r_b using Israel–Darmois junction conditions can be performed with an exterior spacetime. This is made in <cit.> where generalized Vaidya spacetimes are considered, while in <cit.> an anisotropic generalization of deSitter – that is observed to be a subclass of the former – is used. The choice of the external solution is important also because it allows to evaluate the strength of the singularity. Many definitions in this sense exists in literature, and we refer to <cit.> for a discussion about it, where a sufficient condition for the singularity discussed here to be strong is given. As a consequence, in the example above sketched, there exists timelike geodesics terminating into a strong naked singularity. §.§ Instability of the naked singularity The example discussed above has been further developed in <cit.> where is shown that loop quantum gravity effects prevent the formation of a singularity. But yet at a classical level, the fact that they develop a naked singularity for values of the parameter ν in an open interval of the real line, and therefore are stable with respect to that parameter's perturbation, could in principle be interpreted as a non-generic violation of cosmic censorship for these models. In the more recent <cit.> the problem is cast into a classical setting but adding a cosmological constant to (<ref>), and finding once again naked singularities for a non-generic choice of a free parameter. However, as already remarked, the genericity question crucially relies on what we set as fixed and what we consider as “variable”. For instance, in the above example, the free parameter is ν, expressing the dependence of the energy density ρ_SF from the scale factor a. But this does not keep the potential V(ϕ) fixed. Indeed notice that, from (<ref>)–(<ref>) and knowing the profile ψ(a) one can find that ϕ diverges like -√(ν)log a and then reconstruct back the original potential V(ϕ), that is given by V(ϕ)=V_0(3-ν/2)e^√(ν)ϕ. Things radically change if we fix V(ϕ) and study the end state of homogeneous collapsing scalar field as a function of the initial data for t=0. Let us examine again the flat case k=0 and start by observing that the choice of (ϕ_0,v_0)=(ϕ(0),ϕ̇(0)) fully determines the data at t=0, because in system (<ref>)–(<ref>) the initial condition h_0=h(0) is determined by the constraint W(ϕ_0,v_0,h_0)=0 and the fact that we consider a collapse scenario. In the paper <cit.> a class of C^2 potentials V(ϕ) is considered, that includes quite general polynomial potentials such as those asymptotically diverging like ϕ^2n as a subclass. In particular, V(ϕ) is supposed to satisfy (see <cit.>) the following properties: (i) it must be bounded from below, (ii) its critical points must be isolated and possible local maxima must be non-degenerate, (iii) a suitable compact sublevel set {V≤ V_} must exist and (iv) a growth condition on the first and second derivative of log V(ϕ) at ±∞ is assumed to hold. Other class of potentials were considered in <cit.>. Under the above assumptions is proved <cit.> that – except at most for an exceptional zero–measured set of initial data (ϕ_0,v_0) – the scalar field diverges together with its derivative and develops a singularity at a finite amount of cosmological time. Moreover <cit.> the scale factor a(t) vanishes in the approach to the singularity but generically ȧ(t)→ -∞. This means, recalling (<ref>), that (<ref>) is always satisfied in a right neighbourhood of a=0, ∀ r>0, and then an horizon is developing before the singularity. We stress that this happens up to a non-generic choice of initial data, once V(ϕ) is fixed. The proof of the above results is carried on by observing first that, introduced the function y:=V(ϕ)/12ϕ̇^2, then the scalar field energy ρ_SF and the scale factor a(t) satisfy the differential relations ρ̇_SF(t) =2√(3)ρ_SF^3/2(t)1/1+y(t), ä(t) =-ȧ^2(t)/a(t)(2-y(t)/1+y(t)) Now the proof determines the asymptotic behaviour of y(t), as follows: * first, it is shown that the scalar field ϕ(t) with collapsing initial data diverges and its derivative ϕ̇(t) is eventually non-zero; then we can express – at least eventually the dynamical quantities with respect to ϕ; * the function y, seen as a function of ϕ, satisfies a differential relation that is studied using the growth assumption on V(ϕ), finding that, up a non-generic choice of initial data, y(ϕ) 0; * using steps 1. and 2. in (<ref>)–(<ref>), one finds that, for these non-generic data, ρ_SF(t) diverges in a finite amount of cosmological time, and then a singularity develops, but ȧ(t) is unbounded, which as observed above results generically in the formation of the horizon. The function y(t) is basically a modified way to “normalize” the dynamical system, that is a classical approach to the system under suitable condition on the potential V(ϕ), see <cit.> where, instead of considering the unknown functions and (<ref>)–(<ref>), takes their ratios to the Hubble function h, that in the flat k=0 case is connected, by (<ref>), to the energy density ρ_SF. It follows that, the cases depicted in <cit.> leading to a naked singularity, necessarily correspond to the non-generic case excluded by the analysis of step 2 above, see <cit.>). Same argument applies to the examples shown in <cit.>. In this case the potential considered is V(ϕ)=λϕ^2(ϕ^2-83), and it is prescribed ρ_SF(a)=64λ(k-log a)^2 for k>0. Using the above relations it can be actually proved that the spacetime is regular for all t>0 – although in <cit.> it is observed that an ultra high density region is reached in a finite amount of time. However, once again, it can be seen that this situation corresponds to the non-generic choice of initial data excluded by step 2 described above. In fact, as observed in <cit.>, if a potential V(ϕ) in this class is fixed, then initial data generically lead to black hole formation, and prescribing the dependence of ρ(a) as done in <cit.> precisely corresponds to the non-generic data (ϕ(0),ϕ̇(0)) that possibly may not lead to the formation of an horizon. In conclusion, once a potential V(ϕ) obeying suitable – yet rather inclusive – conditions is prescribed, the collapse may end into a naked singularity only for a non-generic choice of the initial data on the scalar field. Nevertheless, the non-generic naked singularity can be gravitationally strong, which interestingly enough is the same conclusion reached in <cit.> for the critical collapse described before in Subsection <ref>. § OTHER DEVELOPMENTS AND CONCLUSIONS The literature discussed in this review, although covering some of the undoubtedly recognized as cornerstones in the analytical and numerical studies on scalar field collapse, is necessarily a selection in the huge amount of contributions on this subject. Concerning the analytical studies, we have already cited some recent developments aiming to prove stability of spherically results with respect to general – hence possibly nonspherical – perturbations <cit.> in the free massless case. Just to name a few other analytical results, when a potential is added, collapse in cylindrical symmetry under self–similarity assumptions has been discussed in <cit.>. With aim to study the big bang singularity, backwards evolution of a free massless scalar field is considered in <cit.>, where the initial data are close to the homogeneous case and given on a 3-dimensional torus 𝕋^3. In <cit.>, a scalar field self–interacting with a negative exponential potential is Kaluza–Klein lifted to a vacuum solution in ℝ^1+3× S^2, to find singular solutions without trapped surface formation developing from regular data. Notice that the family of 4-dimensional scalar field solutions considered contains an example that extends to a case with potential the self–similar naked singularity of <cit.>. The stability of this important example, as we have said, remains a problem with some issues to be fully understood, although <cit.> represent of course a crucial step. When confining to the self–similar collapse, the situation has been studied e.g. in <cit.>. It is worth citing also <cit.> that attempts a first important generalization of the naked singularity instability result when no symmetry on the spacetime geometry is assumed. The PDE approach that has been presented here is not the only one used, even in an analytical context. Some results based on Fuchsian techniques can be found in <cit.>, while an interesting approach involving inverse reconstruction methods recently appeared <cit.> where no particular geometric symmetries are required. Also numerical studies have tried to cover solutions not necessarily spherically symmetric, in the attempt to reproduce the critical behaviour also in more general cases, see e.g. <cit.>. As already said, the study of scalar field cosmologies via dynamical system techniques also are a huge part of this subject, although many are understandably focused on expanding solutions. Having recalled once again reviews <cit.> for extensive treatises in this context, let us cite the paper <cit.> where back–reaction terms are considered to find collapsing solutions that do not develop singularities. A great deal of attention has been also paid to (re)collapsing models where a matter contribution, typically a perfect fluid – either coupled or uncoupled to the scalar field – is added, mostly arising from higher order gravity theory <cit.>. The problem has been studied in <cit.> for the case of positive exponential potentials and in <cit.> for the same class of potentials studied in <cit.>, where again black hole formation is shown to generically form. As one may expect, big crunch singularities naturally arise also when one deals with negative potentials <cit.>. Coupling effect may be mimicked also using deformation of the phase space, producing corrections to the dynamical system also in absence of a perfect fluid, as studied in <cit.>, where anyway the deformation is shown to prevent, in some cases, the formation of singularities from gravitational collapse. There are other approached to the problem worth to be cited, for instance <cit.> where the scalar field is assumed to be timelike, in such a way that comoving coordinates can be built considering a possibly inhomogeneous spacetime, although ϕ=ϕ(t) only. In summary, the collapse of a scalar field has been a valuable tool for investigating various aspects of gravitational physics. The scalar field's multiplicity and rich physical interpretations, both alone and when coupled with other matter models, have contributed to its continued relevance. 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http://arxiv.org/abs/2307.02818v2	20230706072306	Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph $\boldsymbolβ$-Model	[ "Sagnik Nandy", "Bhaswar B. Bhattacharya" ]	math.ST	[ "math.ST", "cs.IT", "cs.SI", "math.IT", "stat.ML", "stat.TH" ]	Inference in the Hypergraph β-Model]Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph β-Model S. Nandy and B.B. Bhattacharya]Sagnik Nandy and Bhaswar B. Bhattacharya Department of Statistics and Data Science University of Pennsylvania Philadelphia PA 19104 United States [email protected] Department of Statistics and Data Science University of Pennsylvania Philadelphia PA 19104 United States [email protected] The β-model for random graphs is commonly used for representing pairwise interactions in a network with degree heterogeneity. Going beyond pairwise interactions, <cit.> introduced the hypergraph β-model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph β-model with multiple layers, which allows for hyperedges of different sizes across the layers. To begin with, we derive the rates of convergence of the maximum likelihood (ML) estimate and establish their minimax rate optimality. We also derive the limiting distribution of the ML estimate and construct asymptotically valid confidence intervals for the model parameters. Next, we consider the goodness-of-fit problem in the hypergraph β-model. Specifically, we establish the asymptotic normality of the likelihood ratio (LR) test under the null hypothesis, derive its detection threshold, and also its limiting power at the threshold. Interestingly, the detection threshold of the LR test turns out to be minimax optimal, that is, all tests are asymptotically powerless below this threshold. The theoretical results are further validated in numerical experiments. In addition to developing the theoretical framework for estimation and inference for hypergraph β-models, the above results fill a number of gaps in the graph β-model literature, such as the minimax optimality of the ML estimates and the non-null properties of the LR test, which, to the best of our knowledge, have not been studied before. [ [ August 1, 2023 ================== § INTRODUCTION The β-model is an exponential family distribution on graphs with the degree sequence as the sufficient statistic. Specifically, in the β-model with vertex set [n] := {1, 2, …, n }, the edge (i, j) is present independently with probability p_ij := e^β_i + β_j/1+ e^β_i + β_j, for 1 ≤ i < j ≤ n and β = (β_1, β_2, …, β_n) ∈^n. This model was first considered by <cit.> and can also be viewed as the undirected version of the p_1-model that appear in the earlier work of <cit.>. Thereafter, the β-model has been widely used for capturing degree heterogeneity in networks (see <cit.>, among several others). The term β-model can be attributed to the seminal paper of <cit.>, which provides the theoretical foundations for parameter estimation in this model. While random graph models, such as the β-model, are important tools for understanding binary (pairwise) relational data, in many modern applications interactions occur not just between pairs, but among groups of agents. Examples include folksonomy <cit.>, collaboration networks <cit.>, complex ecosystems <cit.>, biological networks <cit.>, circuit design <cit.>, computer vision <cit.>, among others. Hypergraphs provide the natural mathematical framework for modeling such higher-order interactions <cit.>. Formally, a hypergraph H is denoted by H = (V(H), E(H)), where V(H) is the vertex set (the agents in the network) and E(H) is a collection of non-empty subsets of V(H). The elements in E(H), referred to as hyperedges, represent the interactions among groups of agents. Motivated by the emergence of complex relational data with higher-order structures, there has been a slew of recent results on modeling random hypergraphs, community detection, recovery, clustering, and motif analysis, among others (see <cit.> and the references therein). In this paper we study the hypergraph β-model, introduced by <cit.>, that allows one to incorporate degree heterogeneity in higher-order networks. Like the graph β-model (<ref>), this is an exponential family on hypergraphs where the (hypergraph) degrees are the sufficient statistics. In its general form it allows for layered hypergraphs with hyperedges of different sizes across the layers. To describe the model formally we need a few notations: For r ≥ 2, denote by [n] r the collection of all r-element subsets of [n]: ={1, 2, …, n}. A hypergraph H = (V(H), E(H)) is said to be r-uniform if every element in E(H) has cardinality r. (Clearly, 2-uniform hypergraphs are simple graphs.) We will denote by _n, r the collection of all r-uniform hypergraphs with vertex set [n] and _n, [r] := ⋃_s=2^r_n, s, the collection of all hypergraphs with vertex set [n] where every hyperedge has size at most r. Then the r-layered hypergraph β-model is a probability distribution on _n, [r] defined as follows: <cit.> Fix r ≥ 2 and parameters B := (β_2, …,β_r), where β_s:=(β_s, v)_v ∈ [n]∈^n. The r-layered hypergraph β-model is a random hypergraph in _n, [r], denoted by 𝖧_[r](n, B), where, for every 2 ≤ s ≤ r, the hyperedge {v_1, v_2, …, v_s}∈[n] s is present independently with probability: p_v_1, v_2, …, v_s:= e^β_s, v_1+…+β_s, v_s/1 + e^β_s, v_1+…+β_s, v_s. This model can be expressed as an exponential family on _n, [r] with the hypergraph degrees up to order r as the sufficient statistics (see (<ref>)). Specifically, the parameter β_s, u encodes the popularity of the node u ∈ [n] to form groups of size s, for 2 ≤ s ≤ r. Consequently, β_s, u controls the local density of hyperedges of size s around the around node u. The model (<ref>) includes as a special case the classical graph β-model (when r=2) and also the r-uniform hypergraph β-model, where only the hyperedges of size r are present. More formally, given parameters β= (β_1, β_2, …, β_n) ∈^n, the r-uniform hypergraph β-model is a random hypergraph in _n, r, denoted by 𝖧_r(n, β), where each r-element hyperedge {v_1, v_2, …, v_r}∈[n] r is present independently with probability: p_v_1, v_2, …, v_r:= e^β_v_1+…+β_v_r/1 + e^β_v_1+…+β_v_r. It is worth noting that, since the degrees are the sufficient statistics in the aforementioned models, it is enough to observe only the degree sequences (not the entire network) for inference regarding the model parameters. This feature makes the β-model particularly attractive because collecting information about the entire network can often be difficult for cost or privacy reasons. For example, <cit.> (see also <cit.>) studied social networks between a group of Swiss students before and during COVID-19 lockdown, where, for privacy reasons, only the total number of connections of each student in the network (that is, the degrees of the vertices) were released. The β-model is also relevant in the analysis of aggregated relational data, where instead of asking about connections between groups of individuals directly, one collects data on the number of connections of an individual with a given feature (see, for example, <cit.> and the references therein). <cit.> proposed two algorithms for computing the maximum likelihood (ML) estimates for the hypergraph β models described above and reported their empirical performance. However, the statistical properties of the ML estimates in these models have remained unexplored. §.§ Summary of Results In this paper we develop a framework for estimation and inference in the hypergraph β-model. Along the way, we obtain a number of new results on the graph β-model as well. The following is a summary of the results: * Estimation: In Section <ref> we derive the rates of convergence of the ML estimates in r-layered hypergraph β-model (<ref>), both in the L_∞ and the L_2 norms. Specifically, we show that given a sample H_n ∼𝖧_[r](n, B) from the r-layered hypergraph β-model, the ML estimate B̂ = (β̂_2,…, β̂_r) of B satisfies: β̂_s-β_s_2 ≲_s, M √(1/ n^s-2) and β̂_s-β_s_∞≲_s, M √(log n/ n^s-1) , for 2 ≤ s ≤ r, with probability going to 1 (see Theorem <ref>). These extend the results of <cit.> on the graph β-model, where the rate of convergence of the ML estimate was derived only in the L_∞ norm, to the hypergraph case. Next, in Theorem <ref> we show that both the rates in (<ref>) are, in fact, minimax rate optimal (up to a √(log n) factor for the L_∞ norm). To the best of our knowledge, these are the first results showing the statistical optimality of the ML estimates in the β-model even for the graph case. * Inference: In Section <ref> we derive the asymptotic distribution of the ML estimate B̂. In particular, we prove that the finite dimensional distributions of the ML estimate converges to a multivariate Gaussian distribution (see Theorem <ref>). Moreover, the covariance matrix of the Gaussian distribution can be estimated consistently, using which we can construct asymptotically valid confidence sets for the model parameters (see Theorem <ref>). * Testing: In Section <ref> we study the goodness-of-fit problem for the hypergraph β-model, that is, given γ∈^n we wish to distinguish: H_0:β_s=γ H_1:β_s ≠γ. We show that the likelihood ratio (LR) statistic for this problem (centered and scaled appropriately) is asymptotically normal under H_0 (see Theorem <ref> for details). Using this result we construct an asymptotically level α test for (<ref>). Next, we study the power properties of this test. In particular, we show that the detection threshold for the LR test in the L_2 norm is n^-2s-3/4, that is, the LR test is asymptotically powerful/powerless in detecting γ' ∈^n depending on whether γ'-γ_2 is asymptotically greater/smaller than n^-2s-3/4, respectively. We also derive the limiting power function of the LR test at the threshold γ'-γ_2 = Θ(n^-2s-3/4) (see Theorem <ref>). Further, in Theorem <ref> we show that this detection threshold is, in fact, minimax optimal, that is all tests are asymptotically powerless when γ'-γ_2 is asymptotically smaller than n^-2s-3/4. In Section <ref> we also obtain the detection threshold of the LR test in the L_∞ norm and establish its optimality. Again, these appear to be the first results on the non-null properties of the LR test and its optimality in the β-model for the graph case itself. In Section <ref> we illustrate the finite-sample performances of the proposed methods in simulations. §.§ Related Work on the Graph β-Model As mentioned before, <cit.> initiated the rigorous study of estimation in the graph β-model. They derived, among others things, the convergence rate of the ML estimate in the L_∞ norm. Thereafter, <cit.> derived necessary and sufficient conditions for the existence of the ML estimate in terms of the polytope of the degree sequences. <cit.> proved the asymptotic normality of ML estimate and later, <cit.> derived the properties of a moment based estimator. <cit.> studied the problem of estimation in the β-model under privacy constraints. In the context of hypothesis testing, <cit.> considered the problem of sparse signal detection in the β-model, that is, testing whether all the node parameters are zero versus whether a (possibly) sparse subset of them are non-zero. Recently, <cit.> derived the asymptotic properties of the LR test for the goodness-of-fit problem in the graph β-model, under the null hypothesis. The graph β-model has also been generalized to incorporate additional information, such as covariates, directionality, sparsity, and weights (see <cit.> and the references therein). For other exponential random graph models with functions of the degrees as sufficient statistics, see <cit.>. §.§ Asymptotic Notation For positive sequences {a_n}_n≥ 1 and {b_n}_n≥ 1, a_n = O(b_n) means a_n ≤ C_1 b_n and a_n =Θ(b_n) (and equivalently, a_n ≍ b_n) means C_2 b_n ≤ a_n ≤ C_1 b_n, for all n large enough and positive constants C_1, C_2. Similarly, for positive sequences {a_n}_n≥ 1 and {b_n}_n≥ 1, a_n ≲ b_n means a_n ≤ C_1 b_n and a_n ≳ b_n means a_n ≥ C_2 b_n for all n large enough and positive constants C_1, C_2. Moreover, subscripts in the above notation, for example O_□, ≲_□, ≳_□, and Θ_□, denote that the hidden constants may depend on the subscripted parameters. Also, for positive sequences {a_n}_n≥ 1 and {b_n}_n≥ 1, a_n ≪ b_n means a_n/b_n → 0 and a_n ≫ b_n means a_n/b_n →∞, as n →∞. § MAXIMUM LIKELIHOOD ESTIMATION IN HYPERGRAPH Β-MODELS In this section we consider the problem of parameter estimation in the hypergraph β-model using the ML method. In Section <ref> we derived the rates of the consistency of the ML estimate. The central limit theorem of the ML estimate and the construction of confidence intervals for the model parameters are presented in Section <ref>. §.§ Rates of Convergence Given a sample H_n ∼𝖧_n, [r](n, B) from the r-layered hypergraph β-model, the likelihood function can be written as follows: L_n( B) = ∏_2 ≤ s ≤ r∏_{v_1, v_2, …, v_s}∈[n] s e^β_s, v_1+…+β_s, v_s/1 + e^β_s, v_1+…+β_s, v_s. Therefore, the negative log-likelihood is given by ℓ_n( B) := - log L_n( B) = - ∑_s=2^r{∑_v=1^nβ_s, v d_s(v)-∑_{v_1, v_2, …, v_s}∈[n] slog(1+exp(β_s, v_1+…+β_s, v_s))} , where d_s(v) := ∑_ e ∈ E(H_n): \| e \| = s 1{v ∈ e}, is the s-degree of the vertex v ∈ [n], that is, the number of hyperedges of size s in H_n passing through v. The negative log-likelihood in (<ref>) can be re-written as: ℓ_n( B)= ∑_s=2^rℓ_n, s (β_s) , where ℓ_n, s (β) := ∑_{v_1, v_2, …, v_s}∈[n] slog(1+exp(β_s, v_1+…+β_s, v_s)) - ∑_v=1^nβ_s, v d_s(v) . Note that (<ref>) is separable in β_2,…,β_r, hence, the ML estimate of B = (β_2, …,β_r) is given by B̂=(β̂_2,…,β̂_r), where β̂_s := _βℓ_n, s(β). This implies that the ML estimate β̂_s satisfies the following set of gradient equations: For all v ∈ [n] and 2 ≤ s ≤ r, d_s(v)=∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β̂_s, v+β̂_s, v_2+…+β̂_s, v_s/1+ e^β̂_s, v+β̂_s, v_2+…+β̂_s, v_s, where [n] \{v} s-1 denotes the collection of all (s-1)-element subsets of [n] \{v}. <cit.> presented two algorithms for computing the ML estimate, an iterative proportional scaling algorithm and a fixed point algorithm, and showed that both algorithms converge if the ML estimate exists. In this paper we study the asymptotic properties of the ML estimates. In the following theorem we show that the likelihood equations (<ref>) have a unique solution with high-probability and derive its rate of convergence. Hereafter, we denote by x_∞ and x _2, the L_∞ and the L_2 norms of a vector x, respectively. Also, denote _M = { x : x _∞≤ M }, the L_∞ the ball of radius M. Throughout we will assume β_s ∈_M, for all 2 ≤ s ≤ r, for some constant M > 0. Suppose H_n ∼𝖧_n, [r](n, B) is a sample from the r-layered hypergraph β-model as defined in (<ref>). Then with probability 1-o(1) the likelihood equations (<ref>) have a unique solution B̂ = ( β̂_2, …, β̂_r), that satisfies: β̂_s-β_s_2 ≲_s, M √(1/ n^s-2) and β̂_s-β_s_∞≲_s, M √(log n/ n^s-1) , for 2 ≤ s ≤ r. Theorem <ref> provides the rates for the ML estimate both in the L_2 and L_∞ norms for the parameters in a r-layered hypergraph β-model. To interpret the rates in (<ref>) note that s-degree of a vertex (recall (<ref>)) in the r-layered model 𝖧_n, [r](n, B) is O(n^s-1) with high probability. This means there are essentially O(n^s-1) independent hyperedges containing information about each parameter in the s-th layer. Hence, each parameter in the s-th layer can be estimated at the rate 1/√(n^s-1). Aggregating this over the n coordinates gives the rates in (<ref>) for the vector of parameters β_s in the s-th layer. The proof of Theorem <ref> is given in Appendix <ref>. The following discussion provides a high-level outline of the proof. * For the rate in the L_2 norm we first upper bound the gradient of the log-likelihood at the true parameter value. Specifically, we show that ∇ℓ_n, s(β_s)_2^2 = O(n^s) with high probability (see Lemma <ref> for details). Next, we show that the smallest eigenvalue of the Hessian matrix ∇^2 ℓ_n, s(β_s) is bounded below by n^s-1 (up to constants) in a neighborhood of the true parameter (see Lemma <ref>). Then a Taylor expansion of the log-likelihood around the true parameter, combined with the above estimates, imply the rate in the L_2 norm in (<ref>) (see Appendix <ref> for details). * The proof of the rate in the L_∞ norm is more involved. For the graph case, <cit.> analyzed the fixed point algorithm for solving the ML equations and developed a stability version of the Erdős-Gallai condition (which provides a necessary and sufficient condition for a sequence of numbers to be the degree sequence of a graph) to derive the rate of ML estimate in the L_∞ norm. One of the technical challenges in dealing with the hypergraph case is the absence of Erdős-Gallai-type characterizations of the degree sequence. To circumvent this issue, we take a more analytic approach based on the `leave-one-out' technique, that appear in the analysis of ranking models <cit.>. Here the idea is to decompose, for each u ∈ [n], the log-likelihood function of the s-th layer ℓ_n, s (recall (<ref>)) into two parts: one depending on β_s, u and the other not depending on it. Using the part of the log-likelihood not depending on β_s, u we first analyze the properties of the constrained leave-one-out ML estimate, which is the maximizer of the part of the log-likelihood not depending on β_s, u in a neighborhood of the leave-one-out true parameter. Then from the part of the log-likelihood depending on β_s, u we obtain, by a Taylor expansion around the true parameter value β_s, u, the L_∞ rate in (<ref>) with an extra additive error term which depends on the constrained leave-one-out ML estimate. Using the bound on the latter obtained earlier we show this error term is negligible compared to the L_∞ rate in (<ref>). The following corollary about the r-uniform model is an immediate consequence of Theorem <ref>. We record it separately for ease of referencing. Suppose H_n ∼𝖧_n, r(n, β) is a sample from the r-uniform hypergraph β-model as defined in (<ref>). Then with probability 1-o(1) the ML estimate β̂ is unique and β̂-β_2 ≲_r,M√(1/ n^r-2) and β̂-β_∞≲_r,M√(log n/ n^r-1) . §.§ Minimax Rates In the following theorem we establish the tightness of the rates of ML estimate obtained in the previous section by proving matching lower bounds. Suppose H_n ∼𝖧_n, [r](n, B), with B = (β_2, …, β_r), such that β_s ∈(M), for 2 ≤ s ≤ r. Given δ∈ (0, 1) there exists a constant C (depending on M, r, and δ) such that the following holds for estimation in the L_2 norm: min_β̂max_β_s ∈(M)ℙ(β̂-β_s _2 ≥ C √(1/n^s-2)) ≥ 1 - δ . Moreover, for estimation in the L_∞ norm the following holds: min_β̂max_β_s ∈(M)ℙ(β̂-β_s _∞≥ C √(1/n^s-1)) ≥ 1 - δ . This result shows that the ML estimate is minimax rate optimal in the L_2 metric and (up to a √(log n) factor) in the L_∞ metric. The proof of Theorem <ref> is given in Appendix <ref>. The bound in (<ref>) is proved using Fano's lemma. For this we construct 2^Θ(n) well-separated points in the parameter space which have `small' average Kulbeck-Leibler (KL) divergence with the origin (see Appendix <ref>). The bound in (<ref>) follows by a direct application of Le Cam's 2-point method (see Appendix <ref>). §.§ Central Limit Theorems and Confidence Intervals The results obtained in the previous section show that the vector ML estimates are consistent in the L_∞-norm. However, for conducting asymptotically precise inference on the individual model parameters, we need to understand the limiting distribution of the ML estimates. In Theorem <ref> below we show that the finite dimensional distributions of the ML estimates (appropriately scaled) converge to a multivariate Gaussian distribution. Using this result in Theorem <ref> we construct joint confidence sets for any finite collection of parameters. Towards this, for H_n ∼𝖧_n, [r](n, B) denote the variance of the s-degree of the node v ∈ [n] as: σ_s(v)^2 := [d_s(v)] = ∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β_s, v+β_s, v_2+…+β_s, v_s/(1+ e^β_s, v+β_s, v_2+…+β_s, v_s)^2 . Then we have the following result: Suppose H_n ∼𝖧_n, [r](n, B) is a sample from the r-layered hypergraph β-model as defined in (<ref>). For each 2 ≤ s ≤ r fix a collection of a_s ≥ 1 indices J_s:={v_s, 1, ⋯, v_s, a_s}∈[n] a_s. Then as n →∞, [ [ D_2 (β̂_2 - β_2)]_J_2; [ D_3 (β̂_3 - β_3)]_J_3; ⋮; [ D_r (β̂_r - β_r)]_J_r ]D→_∑_s=2^r a_s( 0, I ) , where D_s = diag (σ_s(v))_v ∈ [n], for 2 ≤ s ≤ r and for any vector x ∈^n, [ x]_J_s = (x_v)_v ∈ [J_s]^⊤. The proof of Theorem <ref> is given in Appendix <ref>. The idea of the proof is to linearize β̂_s, v - β_s, v in terms of the s-degrees of the node v ∈ [n]. Since the s-degree of a node is the sum of independent random variables, applying Lindeberg's CLT gives the result in (<ref>). In the special case of the r-uniform model, Theorem <ref> can be written in the following simpler form: Suppose H_n ∼𝖧_n, r(n, β) is a sample from the r-uniform hypergraph β-model as defined in (<ref>). For all v ∈ [n], let σ(v)^2 := ∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β_v+β_v_2+…+β_v_s/1+ e^β_v+β_v_2+…+β_v_s . Then for any collection of a ≥ 1 indices J:={v_1, ⋯, v_a}∈[n] a, as n →∞, [ D]_J([β̂]_J-[β]_J) _a( 0, I ) , where D = diag (σ(v))_v ∈ [n], [ D]_J = diag (σ(v))_v ∈ J, [β̂]_J = (β_v)_v ∈ [J]^⊤, and [β_s]_J = (β_s, v)_v ∈ [J]^⊤. To use the above results to construct confidence sets for the parameters, we need to consistently estimate the elements of the matrix D_s. Note that the natural plug-in estimate of σ_s(v) is σ̂_s(v)^2 := ∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β̂_s, v+ β̂_s, v_2+…+ β̂_s, v_s/(1+ e^β̂_s, v+ β̂_s, v_2+…+ β̂_s, v_s)^2 . This estimate turns out to be consistent for σ_s(v), leading to the following result (see Appendix <ref> for the proof): Suppose H_n ∼𝖧_n, [r](n, B) is a sample from the r-layered hypergraph β-model as defined in (<ref>). For each 2 ≤ s ≤ r fix a collection of a_s ≥ 1 indices J_s:={v_s, 1, ⋯, v_s, a_s}∈[n] a_s. Then lim_n →∞ℙ( {∑_s=2^r ([(β̂_s - β_s)]_J_s)^⊤ [ D̂_s^2]_J_s ([(β̂_s - β_s)]_J_s) ≤χ^2_∑_s=2^r a_s, 1 - α}) = 1-α , where D̂_s^2 = diag(σ̂_s(v)^2)_v ∈ [n], [ D̂_s^2]_J_s = diag ( σ̂_s(v)^2)_v ∈ J_s, for 2 ≤ s ≤ r, and for a ≥ 1, χ^2_a, 1 - α is the (1-α)-th quantile of the chi-squared distribution with a degrees of freedom. § GOODNESS-OF-FIT: ASYMPTOTICS OF THE LIKELIHOOD RATIO TEST AND MINIMAX DETECTION RATES In this section we consider the problem of testing for goodness-of-fit in the hypergraph β-model. In particular, given γ∈^n and a sample H_n ∼𝖧_n, [r](n, B), with B = (β_2, …, β_r), we consider the following hypothesis testing problem: For 2 ≤ s ≤ r, H_0:β_s=γ H_1:β_s ≠γ. This section is organized as follows: In Section <ref> we derive the asymptotic distribution and detection rates of the likelihood ratio (LR) test for the problem (<ref>). In Section <ref> we show that the detection rate of the LR test is minimax optimal for testing in L_2 norm. Rates for testing in L_∞ norm are derived in Section <ref>. §.§ Asymptotics of the Likelihood Ratio Test Consider the LR statistic for the testing problem (<ref>): logΛ_n, s = ℓ_n, s(γ)-ℓ_n, s(β̂_s) , where ℓ_n, s is the log-likelihood function (<ref>) and β̂_s is the ML estimate (<ref>). The following theorem proves the limiting distribution of the LR statistic (<ref>) under H_0. Suppose γ∈(M). Then under H_0, λ_n, s := 2logΛ_n, s - n/√(2n)(0,1) , for logΛ_n, s as defined in (<ref>). The proof of Theorem <ref> is given in Appendix <ref>. To prove the result we first expand logΛ_n, s around the null parameter γ and derive an asymptotic expansion of λ_n, s in terms of the sum of squares of the s-degree sequence (d_s(1), d_s(2), …, d_s(n))^⊤ (see (<ref>)). Since the degrees are asymptotically independent (recall Theorem <ref>), we can show that the sum of squares of the degrees (appropriately centered and scaled) is asymptotically normal (see Proposition <ref>), establishing the result in (<ref>). Theorem <ref> shows that the LR test ϕ_n, s := 1 { \|λ_n, s\| >z_α/2} , where z_α/2 is the (1-α/2)-th quantile of the standard normal distribution, has asymptotic level α. To study the power of this test consider the following testing problem: H_0:β_s=γ H_1:β_s =γ' , where γ' ≠γ is such that γ - γ' _2=O(1). Recall that d_s = (d_s(1), d_s(2), …, d_s(n))^⊤ is the vector of s-degrees. Also, _γ[ d_s] will denote the covariance matrix of the vector of s-degrees (see (<ref>)). Suppose (<ref>) holds and γ' as in (<ref>). Then the asymptotic power of the test ϕ_n, s defined in (<ref>) satisfies: lim_n →∞_γ'[ϕ_n, s] = α 1 Moreover, if n^2s-3/4γ'-γ_2 →τ∈ (0, ∞), then there exists η∈ (0, ∞) depending on τ such that η = lim_n →∞(γ'-γ)^⊤_γ[ d_s] (γ'-γ)/√(n), where the limit always exists along a subsequence, and lim_n →∞_γ'[ϕ_n, s] = (\| (-η√(2), 1)\| > z_α/2) . The proof of Theorem <ref> is given in Appendix <ref>. It entails analyzing the asymptotic distribution of the scaled LR statistic λ_n, s under H_1 as in (<ref>). Specifically, we show that when γ'-γ_2 ≪ n^-2s-3/4, then λ_n, s(0, 1), hence the LR test (<ref>) is asymptotically powerless in detecting H_1. On the other hand, if γ'-γ_2 ≫ n^-2s-3/4, then the λ_n, s diverges to infinity, hence the LR test is asymptotically powerful. In other words, n^-2s-3/4 is the detection threshold in the L_2 norm of the LR test. We also derive the limiting power function of the LR test at the threshold n^2s-3/4γ'-γ_2 →τ∈ (0, ∞). In this case, λ_n, s(-η/√(2), 1), where `effective signal strength' η is the limit of the scaled Mahalanobis distance between γ and γ', where the dispersion matrix is the covariance matrix of the degrees. In the next section we will show that this detection rate is, in fact, minimax optimal. §.§ Minimax Detection Rate in the L_2 Norm In this section we will show that the detection threshold of the LR test obtained in Theorem <ref> is information-theoretically tight. To formalize this consider the testing problem: For ε > 0 and γ∈(M), H_0:β_s=γ H_1: β_s -γ_2 ≥ε . The worst-case risk of a test function ψ_n for the testing problem (<ref>) is defined as: (ψ_n, γ ) = _H_0( ψ_n=1) + sup_γ' ∈(M) : γ' - γ_2 ≥ε_γ'( ψ_n=0), which is the sum of the Type I error and the maximum possible Type II error of the test function ψ_n. Given H_n ∼𝖧_n, s(n, β_s), for some β_s ∈(M), and ε = ε_n (depending on n), a sequence of test functions ψ_n is said to be asymptotically powerful for (<ref>), if for all γ∈(M) lim_n →∞( ψ_n, γ)=0. On the other hand, a sequence of test functions ψ_n is said to be asymptotically powerless for (<ref>), if there exists γ∈(M) such that lim_n →∞( ψ_n, γ )=1. Given H_n ∼𝖧_n, s(n, β_s) and γ∈(M), consider the testing problem (<ref>). Then the following hold: (a) The LR test (<ref>) is asymptotically powerful for (<ref>), when ε≫ n^-2s-3/4. (b) On the other hand, all tests are asymptotically powerless for (<ref>), when ε≪ n^-2s-3/4. The result in Theorem <ref> (a) is a direct consequence of Theorem <ref>. The proof of Theorem <ref> (b) is given in Appendix <ref>. For this we chose γ = 0 ∈^n and randomly perturb (that is, randomly add or subtract ε/√(n)) the coordinates of γ to construct β_s ∈(M) satisfying β_s -γ_2 ≥ε. Then a second-moment calculation of the likelihood ratio shows that detecting these two models is impossible for ε≪ n^-2s-3/4. These results combined show that n^-2s-3/4 is the minimax detection rate for the testing problem (<ref>) and the LR test attain the minimax rate. (Comparison between testing and estimation rates.) Recall from (<ref>) and (<ref>) that the minimax rate of estimating β̂_s in the L_2 norm is n^-s-2/2. On the other hand, Theorem <ref> shows that the minimax rate of testing in the L_2 norm is n^-2s-3/4≪ n^-s-2/2. For example, in the graph case (where s=2), the estimation rate is Θ(1) whereas the rate of testing is n^-1/4. This is an instance of the well-known phenomenon that high-dimensional estimation is, in general, harder that testing in the squared-error loss. §.§ Testing in the L_∞ Norm In this section we consider the goodness-of-fit problem when separation is measured in the L_∞ norm. This complements our results on estimation in L_∞ norm in Theorem <ref>. Towards this, as in (<ref>), consider the testing problem: For ε > 0 and γ∈(M), H_0:β_s=γ H_1: β_s -γ_∞≥ε . In this case the minimax risk of a test function is defined as in (<ref>) with the L_2 norm γ' - γ_2 replaced by the L_∞ norm γ' - γ_∞. Then consider the test: ϕ_n, s^max := 1{β̂_s - γ_∞≥ 2C √(log n/n^s-1)}, where C: = C(s, M) > 0 is chosen according to (<ref>) such that _κ( β̂_s - κ_∞≤ C √(log n/n^s-1)) → 1, for all κ∈(M). This implies, _γ[ϕ_n, s^max] → 0. Also, for γ' ∈(M) such that γ - γ' _∞≥ε, _γ'[ϕ_n, s^max] = _γ'( β̂_s - γ_∞≥ 2C √(log n/n^s-1)) ≥_γ'( β̂_s - γ' _∞≤ C √(log n/n^s-1)) → 1 , whenever ε≫√(log n/n^s-1). This is because β̂_s - γ' _∞≤ C √(log n/n^s-1) implies, β̂_s - γ_∞≥γ - γ' _∞ - β̂_s - γ' _∞≥ε - C √(log n/n^s-1)≥ 2C √(log n/n^s-1), whenever ε≫√(log n/n^s-1). This implies that the test ϕ_n, s^max in (<ref>) is asymptotically powerful for (<ref>) whenever ε≫√(log n/n^s-1). The following result shows that this rate is optimal (up to a factor of √(log n)) for testing in the L_∞ norm. Given H_n ∼𝖧_n, s(n, β_s) and γ, β_s ∈(M), consider the testing problem (<ref>). Then the following hold: (a) The test ϕ_n, s^max in (<ref>) is asymptotically powerful for (<ref>), when ε≫√(log n/n^s-1). (b) On the other hand, all tests are asymptotically powerless for (<ref>), when ε≪√(1/n^s-1). The proof of Theorem <ref> (b) is given in Appendix <ref>. Note that in this case minimax rates of estimation and testing are the same, since the effect of high-dimensional aggregation does not arise when separation is measured in the L_∞ norm. § NUMERICAL EXPERIMENTS In this section we study the performance of the ML estimates and the LR tests discussed above in simulations. To begin with we simulate a 3-uniform hypergraph β-model 𝖧_3(n, β), with n=400 vertices and β = 0 ∈^n. Figure <ref>(a) shows the quantile-quantile (QQ) plot (over 200 iterations) of the first coordinate of the ML estimate [ D]_1 ([β̂ - β]_1) (where β̂ is computed using the fixed point algorithm described in <cit.> and D is as defined in Corollary <ref>). We observe that the empirical quantiles closely follow the quantiles of the standard normal distribution, validating the result in Corollary <ref>. In the same setup as above, Figure <ref>(b) shows the 95% confidence interval for [β]_1 over 50 iterations. Specifically, we plot the intervals [ [β̂]_1 - 1.96/[D̂]_1, [β̂]_1 + 1.96/[D̂]_1], where D̂ is the estimate of D as defined in Theorem <ref>. This figure shows that 47 out of 50 of intervals cover the true parameter, which gives an empirical coverage of 47/50= 0.94. Next, we consider the goodness of fit problem in s-uniform hypergraph β-model: H_0:β=0 H_1:β≠ 0, for s=2,3. For this we simulate H_n ∼𝖧_3(n, γ), with n=250 and γ = α· u, where u is chosen uniformly at random from the n-dimensional unit sphere and α∈ [0, 1]. Figure <ref>(c) shows the empirical power of the LR test (<ref>) (over 50 iterations) as α varies over a grid of 25 uniformly spaced values in [0,1], for s=2, 3. In both cases, as expected, the power increases with α, which, in this case, determines the signal strength. Also, the LR test is more powerful in the 3-uniform case compared to the 2-uniform case. This aligns with conclusions of Theorem <ref>, which shows that the detection threshold of the LR test in the 3-uniform case is n^-3/4, while for 2-uniform case it is n^-1/4. Hence, one expects to see more power at lower signal strengths (smaller α) for s=3 compared to s=2. §.§ Acknowledgements B. B. Bhattacharya was supported by NSF CAREER grant DMS 2046393, NSF grant DMS 2113771, and a Sloan Research Fellowship. abbrvnat § PROOF OF THEOREM <REF> §.§ Convergence Rate in the L_2 Norm As mentioned in the Introduction, the proof of Theorem <ref> involves showing the following: (1) a concentration bound on the gradient of negative log-likelihood ℓ_n, s (recall (<ref>)) at the true parameter value B = (β_1, β_2, …, β_r), and (2) the strong convexity of ℓ_n, s in a neighborhood of the true parameter. We begin with the concentration of the gradient ∇ℓ_n, s in both the L_2 and the L_∞ norms: Suppose the assumptions of Theorem <ref> hold. Then for each 2 ≤ s ≤ r, there exists a constant C>0 (depending on r and M) such that the following hold: ∇ℓ_n, s(β_s)_2^2 ≤ C n^s and ∇ℓ_n, s(β_s)^2_∞≤ C n^s-1log n , with probability 1-O( 1/n^2). The next step is to establish the strong convexity of ℓ_n, s. Towards this we need to show that the smallest eigenvalue λ_min(∇^2 ℓ_n, s) of the Hessian matrix ∇^2 ℓ_n, s (appropriately scaled) is bounded away from zero in a neighborhood of the true value β_s. This is the content of the following lemma, which also establishes a matching upper bound on the largest eigenvalue λ_max(∇^2 ℓ_n, s) of the Hessian matrix ∇^2 ℓ_n, s. Suppose the assumptions of Theorem <ref> hold. Fix 2 ≤ s ≤ r and a constant K > 0. Then there exists a constants C_1', C_2' >0 (depending on r and M) such that the following hold: C_1' e^-s β - β_s _2 n^s-1≤λ_min(∇^2ℓ_n, s(β)) ≤λ_max(∇^2ℓ_n, s(β)) ≤C_2' n^s-1. As a consequence, there exists a constants C_1, C_2 >0 (depending on r, K, and M) such that the following hold: C_1 n^s-1≤inf_β: β - β_s _2 ≤ Kλ_min(∇^2ℓ_n, s(β)) ≤sup_β: β - β_s _2 ≤ Kλ_max(∇^2ℓ_n, s(β)) ≤C_2 n^s-1. The proofs of Lemma <ref> and Lemma <ref> are given in Appendix <ref> and Appendix <ref>, respectively. We first apply these results to prove the rate of convergence in the L_2 norm in Theorem <ref>. §.§.§ Deriving the L_2 Norm Bound in (<ref>) To begin with suppose the ML equations (<ref>) have a solution B̂=(β̂_2, …, β̂_r). This implies, ℓ_n, s(β̂_s) = 0, for 2 ≤ s ≤ r, where ℓ_n, s is as defined in (<ref>). For 2 ≤ s ≤ r and 0 ≤ t ≤ 1, define β_s(t) := t β̂_s + (1- t) β_s, and g_s(t) := (β̂_s - β_s)^⊤ℓ_n, s(β_s(t)). Note that ℓ_n, s(β_s(1)) = ℓ_n, s(β̂_s) = 0. Hence, by the Cauchy-Schwarz inequality, \|g_s(1) - g_s(0)\| = \|(β̂_s - β_s)^⊤ℓ_n, s(β_s)\| ≤β̂_s - β_s _2 ·ℓ_n, s(β_s) _2 . Also, g_s'(t) = (β̂_s - β_s)^⊤^2 ℓ_n, s(β_s(t) ) (β̂_s - β_s) ≥λ_min(^2 ℓ_n, s(β_s(t) )) β̂_s - β_s _2^2. We now consider two cases: To begin with assume s ≥ 3. By Lemma <ref>, given a constant K > 0 there exists a constant C_1 > 0 (depending on r, K, M) such that inf_β: β - β_s _2 ≤ Kλ_min(^2 ℓ_n, s(β)) ≥ C_1 n^s-1. Note that β_s(t) - β_s _2 = \|t\| β̂_s - β_s _2. Then \| g_s(1) - g_s(0)\| ≥ g_s(1) - g_s(0) = ∫_0^1 g_s'(t) dt ≥∫_0^min{1, K/β̂_s - β_s _2} g_s'(t) dt ≥ C_1 n^s-1β̂_s - β_s _2^2 min{1, K/β̂_s - β_s _2} , where the last step follows from (<ref>) and (<ref>). Therefore, by (<ref>) and Lemma <ref>, with probability 1-O(1/n^2), min{β̂_s - β_s _2, K }≲_r, K, M1/n^s-1·ℓ_n, s(β_s) _2 ≲_r, K, M√(1/n^s-2) . Since K > 0 is fixed and the RHS of (<ref>) converges to zero for s ≥ 3, the L_2 norm bound in (<ref>) follows, under the assumption that ML equations (<ref>) have a solution. Next, suppose s=2. Since β_2(t) - β_2 _2 = \|t\| β̂_2 - β_2 _2. By Lemma <ref>, λ_min(^2 ℓ_n, 2(β_2(t) )) ≥ C_1' e^-2 \|t\| β̂_2 - β_2 _2 n , for some constant C_1' > 0 depending on M. Then \| g_2(1) - g_2(0)\| ≥ g_2(1) - g_2(0) = ∫_0^1 g_2'(t) dt ≥ C_1' n β̂_2 - β_2 _2^2 ∫_0^1 e^-2 \|t\| β̂_2 - β_2 _2 d t (by (<ref>) and (<ref>)) ≥C_1'/2 n β̂_2 - β_2 _2 ( 1- e^ - 2 β̂_2 - β_2 _2 ) . Therefore, by (<ref>) and Lemma <ref>, with probability 1-o(1), β̂_2 - β_2 _2 ( 1- e^- 2 β̂_2 - β_2 _2 ) ≤2/C_1' n ·ℓ_n, 2(β_2) _2 ≤ C' , for some constant C' > 0 depending on M. Note that if β̂_2 - β_2 _2 ≥ 1, then (1- e^-2 β̂_2 - β_2 _2 ) ≥ 1-e^-2. Hence, (<ref>) implies, β̂_2 - β_2 _2 ≤ C, for some constant C > 0 depending on M. Therefore, β̂_2 - β_2 _2 ≤max{1, C }. This implies the L_2 norm bound in (<ref>), under the assumption that the ML equations (<ref>) have a solution, for s=2. To complete the proof we need to show that bounded solution to equation (<ref>) exists. To this end, for 2 ≤ s ≤ r, denote by 𝒟_s, the set of all possible degree sequences in an s-uniform hypergraph on n vertices. Moreover, let ℛ_s be the set of all expected degree sequences in a hypergraph on n vertices sampled from the s-uniform model (<ref>). The following result shows that any convex combination of s-degree sequences in 𝒟_s can be reached by the limit of expected degree sequences of the s-uniform hypergraph β-model. This was proved in the graph case (s=2) by Chatterjee et al. <cit.>. Here, we show that the same holds for all 2 ≤ s ≤ r. Fix 2 ≤ s ≤ r and let 𝒟_s and ℛ_s be as defined above. Then conv (𝒟_s)=ℛ̅_s, where conv (𝒟_s) denotes the convex hull of 𝒟_s and ℛ_s is the closure of ℛ_s. The proof of the above result is given in Appendix <ref>. Using this proposition we now show the existence of bounded solutions of the ML equations (<ref>). Note that by Proposition <ref>, given H_n ∼𝖧_n, [r](n, B) the s-degree sequence (d_s(1),…, d_s(n)) ∈_s ⊆ℛ̅_s. This implies, there exists a sequence { x_t}_t ≥ 0∈_s satisfying lim_t →∞ x_t = (d_s(1), …, d_s(n)) . Since x_t ∈_s, there exists {β̂_1^(t), …, β̂_r^(t)} such that x_t = ∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β^(t)_s, v+β^(t)_s, v_2+…+β^(t)_s, v_s/1+ e^β^(t)_s, v+β^(t)_s, v_2+…+β^(t)_s, v_s , for 2 ≤ s ≤ r. In other words, for each t ≥ 0, {β̂_1^(t), …, β̂_r^(t)} is a solution of the ML equations (<ref>) with (d_s(1), d_s(2), …, d_s(n)) replaced by x_t. By the previous argument, there exists C > 0 (not depending on t) such that with probability 1-o(1), max_2 ≤ s ≤ rβ̂^(t)_s_∞≤ C, for all t ≥ 0. Therefore, the sequence {(β̂^(t)_1, β̂^(t)_2, …, β̂^(t)_r) }_≥ 0 has a limit point. This limit point is a solution to (<ref>) (by taking limit as t →∞ in (<ref>)) and is bounded. Finally, since ℓ_n,s is strongly convex for β∈ℬ(M) (see (<ref>)), if the gradient equations have a bounded solution, it is the unique minimizer. Therefore, there exists a unique bounded solution to (<ref>) which is the minimizer of ℓ_n,s. §.§.§ Proof of Lemma <ref> Recalling (<ref>) note that, for v ∈ [n], v-th coordinate of the gradient of ℓ_n, s is given by: ℓ_n, s(β_s)_v = [d_s(v)] - d_s(v) where [d_s(v)] = ∑_{v_2, …, v_s}∈ [n] \{v} s-1 e^β_s, v+β_s, v_2+…+β_s, v_s/1+ e^β_s, v+β_s, v_2+…+β_s, v_s. Since d_s(v) is the sum of O(n^s-1) independent random variables, by Hoeffding's inequality and the union bound, ( ℓ_n, s(β_s) _∞^2 ≥ 4 C_s,M n^s-1log n ) ≤1/n^2, for some constant C_s,M>0 (depending on s and M). This establishes the second bound in (<ref>). Next, we prove the first bound in (<ref>). Denote by 𝔹^n := { x ∈^n : x _2 ≤ 1} the unit ball in ℝ^n. By <cit.>, we can construct an 1/2-net 𝒱 of 𝔹^n satisfying log\|𝒱\| ≤ C_1 n for some constant C_1 >0. Now, for any unit vector a = (a_1, a_2, …, a_n)^⊤∈^n and the corresponding point b = (b_1, b_2, …, b_n)^⊤∈𝒱, recalling (<ref>) gives, ∑_v=1^n a_v ℓ_n, s(β_s)_v = ∑_v=1^n a_v ([d_s(v)] - d_s(v)) = ∑_v=1^n b_v ([d_s(v)] - d_s(v)) + Δ, where Δ := ∑_v=1^n (a_v - b_v)([d_s(v)] - d_s(v)) ≤√(∑_v=1^n (a_v - b_v)^2 ∑_v=1^n ([d_s(v)] - d_s(v))^2 ) ≤1/2√(∑_v=1^n ([d_s(v)] - d_s(v))^2 ) = 1/2ℓ_n, s(β_s) _2 , by the Cauchy-Schwarz inequality and the fact that a - b ≤1/2. Using the above in (<ref>) gives, ∑_v=1^n a_v ℓ_n, s(β_s)_v ≤∑_v=1^n b_v ([d_s(v)] - d_s(v)) + 1/2ℓ_n, s(β_s) _2 . Maximizing over a ∈^n and b ∈ on both sides of (<ref>) and rearranging the terms gives, ℓ_n, s(β_s) _2 ≤ 2 max_ b ∈𝒱∑_v=1^n b_v ([d_s(v)] - d_s(v)) . For e = (u_1, u_2, …, u_s) ∈ [n]^s denote β_s, e = (β_s, u_1, β_s, u_2, …, β_s, u_s)^⊤. Hence, by (<ref>), Hoeffding's inequality, and union bound, ( ℓ_n, s(β_s) _2^2 > 4 C^2 n^s ) ≤∑_ b ∈ ( ∑_v=1^n b_v ([d_s(v)] - d_s(v)) > 2 C n^s/2) = ∑_ b ∈( ∑_v=1^n∑_ e ∈[n] s : v ∈ e b_v {e^β_s, e^⊤ 1/1+e^β_s, e^⊤ 1 - 1 { e ∈ E(H_n)}} > 2 C n^s/2) ≤∑_ b ∈ e^- 2 C^2 n/∑_v=1^n b_v^2≤ 2^C_1 n e^-2 C^2 n → 0 , by choosing C > C_1 to be large enough. This proves the first inequality in (<ref>). §.§.§ Proof of Lemma <ref> For e = (u_1, u_2, …, u_s) ∈[n] s and β = (β_1, β_2, …, β_n) ∈^n, denote β_ e = (β_u_1, β_u_2, …, β_u_s)^⊤. Recalling (<ref>) note that, the Hessian matrix ^2 ℓ_n, s can be expressed as: ^2 ℓ_n, s(β) = ∑_u, v ∈ [n]∑_ e ∈[n] s e^β^⊤_ e 1/(1+ e^β^⊤_ e 1)^2 η_u η_v^⊤ 1{ u, v ∈ e } , where η_u is the u-th basis vector in ^n, for 1 ≤ u ≤ n. Note that for β∈^n and β_s ∈(M), \| 1^⊤β_ e \| ≤ s β_∞≤ s β_s _∞ + s β_s - β_∞≤ s M + s β_s - β_2. Hence, 1/4e^-s (M + β_s - β_2 ) ≤e^ 1^⊤β_ e/(1+e^ 1^⊤β_ e)^2≤ 1 . For x ∈^n, consider x^⊤^2 ℓ_n, s(β) x = ∑_u, v ∈ [n]∑_ e ∈[n] s e^β^⊤_ e 1/(1+ e^β^⊤_ e 1)^2 x_u x_v 1{ u, v ∈ e } = ∑_ e ∈[n] s e^β^⊤_ e 1/(1+ e^β^⊤_ e 1)^2 ( ∑_u, v ∈ [n] x_ux_v 1{ u, v ∈ e }) = ∑_ e ∈[n] s e^β^⊤_ e 1/(1+ e^β^⊤_ e 1)^2 (∑_ u ∈ [n] x_u 1{ u ∈ e })^2 ≥1/4e^-s (M + β_s - β_2 ) ∑_ e ∈[n] s(∑_ u ∈ [n] x_u 1{ u ∈ e })^2 , where the last step uses (<ref>). Observe that for any x ∈^n ∑_ e ∈[n] s(∑_ u ∈ [n] x_u 1{ u ∈ e })^2 = x^⊤ L x, where L := ∑_u, v ∈ [n]∑_ e ∈[n] sη_u η_v^⊤ 1{ u, v ∈ e } = ( n-1 s-1 - n-2 s-2) I _n+ n-2 s-2 1 1^⊤ , where I_n is the n × n identity matrix and 1 = (1, 1, …, 1)^⊤. Similarly, we can show from (<ref>) that for any x ∈^n x^⊤^2 ℓ_n, s(β) x ≤ x^⊤ L x. Thus, for β∈^n 1/4e^-s (M + β_s - β_2 ) λ_min( L ) ≤λ_min(∇^2 ℓ_n, s(β)) ≤λ_max(∇^2 ℓ_n, s(β)) ≤λ_max( L ) . Note that L is a circulant matrix with 2 non-zero eigenvalues: n-1 s-1 and n-1 s-1-n-2 s-2. Hence, there exists constants C_1”, C_2”>0 (depending on r), such that n-1 s-1≤ C_1” n^s-1 n-1 s-1-n-2 s-2≥ C_2” n^s-1 . This implies, from (<ref>), that there exists constants C_1', C_2'>0 (depending on r and M) such that (<ref>) hold. The result in (<ref>) from hold from (<ref>) directly. §.§ Convergence Rate in the L_∞ Norm Suppose H_n ∼𝖧_n, [r](n, B) as in the statement of Theorem <ref>. From the arguments in Appendix <ref> we know that, with probability 1-o(1), the ML equations (<ref>) have a bounded solution B̂=(β̂_1, β̂_2, …, β̂_r), that is, ℓ_n, s (β̂_s) = 0, for 2 ≤ s ≤ r, and max_2 ≤ s ≤ rβ̂_s _∞ = O(1). To establish the rate in L_∞ norm we decompose the likelihood for the s-th layer as follows. ℓ_n, s(β) = ∑_{v_1, v_2, …, v_s}∈[n] slog(1+ e^β_v_1+…+β_v_s) - ∑_v=1^nβ_v d_s(v) = ∑_ e ∈[n] s{log(1+ e^β^⊤_e 1 ) - 1 { e ∈ E(H_n)}β^⊤_e 1 } = ℓ^+_n, s(β_u\|β_u̅) + ℓ^-_n, s(β_u̅) , where β_u̅ = (β_1, …, β_u-1, β_u+1, …, β_n), ℓ^+_n, s(β_u\|β_u̅) := ∑_ e ∈[n] s : u ∈ e{log(1+ e^β^⊤_e 1 ) - 1 { e ∈ E(H_n)}β^⊤_e 1 } ℓ^-_n, s(β_u̅) := ∑_ e ∈[n] s : u ∉ e{log(1+ e^β^⊤_e 1 ) - 1 { e ∈ E(H_n)}β^⊤_e 1 } . Fix a constant K > 0 and define β̂^∘_s,u̅ = argmin_β_u̅: β_u̅-β_s,u̅_2 ≤ Kℓ^-_n, s(β_u̅) , where β_s, u̅ = (β_s, 1, …, β_s, u-1, β_s, u+1, …, β_s, n). This is the leave-one-out ML estimate on the constrained set β_u̅-β_s,u̅_2 ≤ K. First we bound the difference (in L_2 norm) of constrained leave-one-out ML estimate defined above and the leave-one-out true parameter β_s, u̅. Let β̂^∘_s,u̅ and β_s,u̅ be as defined above. Then, for u ∈ [n], with probability 1 - o(1), max_u ∈ [n] β̂^∘_s,u̅-β_s,u̅_2^2 ≲ _s,M,K1/n^s-2 . To begin with, observe that ℓ^-_n, s(β_s,u̅) ≥ℓ^-_n, s(β̂^∘_s,u̅) = ℓ^-_n, s(β_s,u̅) + (β̂^∘_s,u̅-β_s,u̅)^⊤∇ℓ^-_n, s(β_s,u̅) + 1/2(β̂^∘_s,u̅-β_s,u̅)^⊤∇^2 ℓ^-_n, s(β̃)(β̂^∘_s,u̅-β_s,u̅), where β̃ - β_s,u̅_2 ≤β̂^∘_s,u̅-β_s,u̅_2 ≤ K. This implies, β̂^∘_s,u̅-β_s,u̅_2 ·∇ℓ^-_n, s(β_s,u̅)_2 ≥ - (β̂^∘_s,u̅-β_s,u̅)^⊤∇ℓ^-_n, s(β_s,u̅) ≥1/2(β̂^∘_s,u̅-β_s,u̅)^⊤∇^2 ℓ^-_n, s(β̃)(β̂^∘_s,u̅-β_s,u̅) . By Lemma <ref>, (β̂^∘_s,u̅-β_s,u̅)^⊤∇^2 ℓ^-_n, s(β̃)(β̂^∘_s,u̅-β_s,u̅) ≳_ s,M,K β̂^∘_s,u̅-β_s,u̅^2 n^s-1 . Also, by Lemma <ref>, ∇ℓ^-_n, s(β_s,u̅)^2_2 ≲ _ s,M,K n^s with probability 1 - O(1/n^2). Plugging in the above inequalities in (<ref>), and using the union bound we get (<ref>). Next, we bound the difference between the constrained leave-one-out ML estimate β̂^∘_s,u̅ and the (unconstrained) leave-one-out ML estimate β̂_s, u̅ = (β̂_s, 1, …, β̂_s, u-1, β̂_s, u+1, …, β̂_s, n). Let β̂^∘_s,u̅ and β̂_s,u̅ be as defined above. Then, with probability 1-o(1), max_u ∈ [n]β̂^∘_s,u̅-β̂_s, u̅^2_2 ≲ _ s,M,K 1/n^s-1 + β̂_s-β_s^2_∞/n^s-1 , where X_ e = 1{ e ∈ E(H_n)}, ψ(x)=e^x/1+e^x, and β_s, e = (β_s, u_1, β_s, u_2, …, β_s, u_s)^⊤, for e = (u_1, u_2, …, u_s) ∈ [n]^s. By the definition of β̂^∘_s,u̅ (recall (<ref>)) ℓ^-_n, s(β̂_s,u̅) ≥ℓ^-_n, s(β̂^∘_s,u̅) = ℓ^-_n, s(β̂_s, u̅) + (β̂^∘_s,u̅-β̂_s, u̅)^⊤∇ℓ^-_n, s(β̂_s,u̅) + (β̂^∘_s,u̅-β̂_s, u̅)^⊤∇ℓ^-_n, s(β̅)(β̂^∘_s,u̅-β̂_s,u̅), where β̅-β̂_s,u̅_2 ≤β̂^∘_s,u̅-β̂_s,u̅_2. Note that β̂^∘_s,u̅-β̂_s,u̅_2 = O(1), since β̂_s _∞ = O(1) and β̂^∘_s,u̅ = O(1). Then by Lemma <ref>, β̂^∘_s, u̅-β̂_s, u̅^2_2 ≲ _s,M,K∇ℓ^-_n, s(β̂_s,u̅)^2_2 /n^2(s-1). Since ∇ℓ_n, s(β̂_s)=0, that is, ∂/∂β_vℓ_n,s(β̂_s)=0, for v ∈ [n]. Hence, we have from (<ref>), ∂/∂β_vℓ^-_n, s(β̂_s, u̅) = - ∂/∂β_vℓ^+_n, s(β̂_s, u\|β̂_s,u̅) = - ∑_ e ∈[n] s: {u, v}∈ e{ X_ e - ψ( 1^⊤β̂_s, e)} , where ψ(x):= e^x/1+e^x. This implies, ∇ℓ^-_n, s(β̂_s,u̅)^2_2 = ∑_v ∈ [n]\{ u }(∑_ e ∈[n] s: {u, v}∈ e{X_ e - ψ(β̂^⊤_s, e1)})^2 ≲∑_v ∈ [n]\{ u }[ (∑_ e ∈[n] s: {u, v}∈ e{X_ e - ψ( 1^⊤β_s, e)})^2 + (∑_ e ∈[n] s: {u, v}∈ e{ψ( 1^⊤β̂_s, e) - ψ( 1^⊤β_s, e)})^2 ] ≲_r∑_v ∈ [n]\{ u }(∑_ e ∈[n] s: {u, v}∈ e{X_ e - ψ(1^⊤β_s, e)})^2 + n^s-1β̂_s-β_s^2_∞ , using \|ψ( 1^⊤β̂_s, e) - ψ( 1^⊤β_s, e)\| ≲ \| 1^⊤β̂_s, e - 1^⊤β_s, e \| ≲_r β̂_s-β_s^2_∞. By (<ref>) and (<ref>), to prove the result in (<ref>) it suffices show the following holds with probability 1-o(1), max_1 ≤ u ≤ n∑_v ∈ [n]\{u}(∑_ e ∈[n] s: u, v ∈ e { 1 { e ∈ E(H_n)} - ψ(β_s, e^⊤ 1) })^2 ≲ n^s-1 . This is proved in Appendix <ref>. We now apply the above lemmas to derive the bound in the L_∞ norm. To begin with note that since ℓ_n,s(β̂_s)=min_β_sℓ_n,s(β_s), ℓ^+_n, s(β_s, u\|β̂_s, u̅) + ℓ^-_n, s(β̂_s, u̅) ≥ℓ_n,s(β̂_s) = ℓ^+_n, s(β̂_s, u\|β̂_s, u̅) + ℓ^-_n, s(β̂_s, u̅) . The above inequality implies ℓ^+_n, s (β_s, u\|β̂_s, u̅) ≥ℓ^+_n, s(β̂_s, u\|β̂_s, u̅) = ℓ^+_n, s(β_s, u\|β̂_s, u̅) + (β̂_s, u-β_s, u)∂/∂β_uℓ^+_n, s(β_s, u\|β̂_s, u̅) + 1/2(β̂_s, u-β_s, u)^2∂^2/∂β_u^2ℓ^+_n, s( β̃\|β̂_s, u̅) , where β̃ is a convex combination of β̂_s, u and β_s, u. Therefore, (β̂_s, u-β_s, u)^2 ≤4 \|∂/∂β_uℓ^+_n, s(β_s, u \| β̂_s, u̅)\|^2/\|∂^2/∂β_u^2ℓ^+_n, s(β̃\|β̂_s, u̅)\|^2 . From arguments in Appendix <ref> we know that with probability 1-o(1), β̂_s-β_s_∞≤β̂_s-β_s_2 ≲ 1. Note that for β∈^n such that β - β_s _∞≲ 1, we have β_∞≲ 1 and hence, \| 1^⊤β_ e \| ≲ 1. This implies, ψ( 1^⊤β_ e, s)(1-ψ( 1^⊤β_ e, s)) ≳ 1 and hence, ∂^2/∂β_u^2ℓ^+_n, s(β̃\|β̂_s, u̅) = ∑_ e ∈[n] s: u ∈ eψ( 1^⊤β̅_ e, s)(1-ψ( 1^⊤β̅_ e, s)) ≳ n^s-1 , where β̅_s = (β̂_s, 1, …, β̂_s, u-1, β_s, u, β̂_s, u+1, …, β̂_s, n)^⊤. Hence, (<ref>) implies, (β̂_s, u-β_s, u)^2 ≲\|∂/∂β_uℓ^+_n, s(β_s, u\|β̂_s, u̅)\|^2/n^2s-2. Now, we bound \|∂/∂β_uℓ^+_n, s(β_s, u\|β̂_s, u̅)\|^2. For this define β̅_s^∘ = ([β̂_s, u̅^∘]_1, …, [β̂_s, u̅^∘]_u-1, β_s, u, [β̂^∘_s, u̅]_u+1, …, [β̂_s, u̅^∘]_n)^⊤. Then we have \|∂/∂β_uℓ^+_n, s(β_s, u\|β̂_s, u̅)\| = \|∑_ e ∈[n] s: u ∈ e{X_ e-ψ( 1^⊤β̅_ e, s)}\| ≤ T_1(u) + T_2(u) + T_3(u), where T_1(u) :=\|∑_ e ∈[n] s: u ∈ e{X_ e-ψ(1^⊤β_ e, s)}\|, T_2(u) := \|∑_ e ∈[n] s: u ∈ e{ψ(1^⊤β̅^∘_ e, s)-ψ(1^⊤β_ e, s)}\| , and T_3(u) := \|∑_ e ∈[n] s: u ∈ e{ψ(1^⊤β̅_ e, s^∘)-ψ(1^⊤β̅_ e, s)}\|. Note that since {X_ e}_ e ∈[n] s are independent and bounded random variables, using Hoeffding's inequality and union bound gives max_u ∈ [n] T_1(u) ≲√(n^s-1log n ) , with probability 1-o(1). Next, we consider T_2(u). By Lemma <ref>, with probability 1-o(1), max_u ∈ [n] T_2(u) ≲max_u ∈ [n] ∑_ e ∈[n] s:u ∈ e{∑_v ∈ e\|β_s, v-[β̂^∘_s,u̅]_v\|} =max_u ∈ [n] ∑_ v ∈ [n]\{u} n^s-2\|β_s, v - [β̂^∘_s,u̅]_v \| ≲ n^s-3/2max_u ∈ [n] β_s, u̅ - β̂^∘_s, u̅_2 ≲√(n^s - 1) . A similar argument shows that, with probability 1-o(1), max_u ∈ [n] T_3(u) ≲ n^s-3/2β̂_s, u̅- β̂^∘_s, u̅_2. Combining the bounds on T_1, T_2 and T_3 with (<ref>) and (<ref>) gives, with probability 1-o(1), β̂_s-β_s_∞ ≲√(log n/n^s - 1) + max_u ∈ [n] β̂_s, u̅- β̂^∘_s, u̅_2/√(n) . Applying (<ref>) in (<ref>) now gives, with probability 1-o(1), max_u ∈ [n]β̂^∘_s,u̅-β̂_s, u̅_2 ≲ _s,M,K√(1/n^s-1) + β̂_s-β_s_∞/√(n^s-1) ≲_s,M,K√(1/n^s-1) + max_u ∈ [n]β̂^∘_s,u̅-β̂_s, u̅_2/√(n^s)≲√(1/n^s-1) . Using this inequality with (<ref>) gives, with probability 1-o(1), β̂_s - β_s _∞≲_s,M,K√(log n/n^s-1) , establishing the desired bound in (<ref>). §.§.§ Proof of (<ref>) Denote by 𝔹^n-1={ x ∈ℝ^n-1: x_2 ≤ 1}. Using <cit.>, we can construct an 1/2-net 𝒱_1 of 𝔹^n-1 satisfying log\|𝒱_1\| ≤ C_2 n for some constant C_2 >0. Now, for any u ∈ [n], any unit vector ã = (ã_1, ã_2, …, ã_n-1)^⊤∈^n-1 and the corresponding point b̃ = (b̃_1, b̃_2, …, b̃_n-1)^⊤∈𝒱_1, ∑_v ∈ [n]\{ u }ã_v{∑_ e ∈[n] s: u, v ∈ e {X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}} = ∑_v ∈ [n]\{ u }b̃_v{∑_ e ∈[n] s, u, v ∈ e {X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}+Δ_u, where Δ_u := ∑_v ∈ [n]\{u}(ã_i-b̃_i){∑_ e ∈[n] s : u, v ∈ e {X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}} . Proceeding as in (<ref>), for all u ∈ [n], we can show \|Δ_u\| ≤1/2√(∑_v ∈ [n]\{ u }{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}^2). Maximizing over ã∈𝔹^n-1 and b̃∈𝒱_1 on both sides of (<ref>) we get √(∑_v ∈ [n]\{ u }{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}^2)≤ 2 max_b̃∈𝒱_1∑_v ∈ [n]\{ u }b̃_v{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}. As the above relation holds for all u ∈ [n] we get √(max_u ∈ [n]∑_ v ∈ [n]\{ u }{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}^2) ≤ 2 max_u ∈ [n]max_b̃∈𝒱_1∑_ v ∈ [n]\{ u }b̃_v{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}} . Hence, using (<ref>), Hoeffding Inequality and union bound we get ( max_u ∈ [n]∑_ v ∈ [n]\{ u }{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}}^2 > 4 K^2 n^s-1) ≤∑_u=1^n∑_b̃∈𝒱_1 (∑_ v ∈ [n]\{ u }b̃_v{∑_ e ∈[n] s : u, v ∈ e{X_ e-e^ 1^⊤β_ e/1+e^ 1^⊤β_ e}} > 2 K n^s-1/2) ≤∑_u=1^n∑_b̃∈𝒱_1e^-2K^2n/∑_v=1^n-1b̃^2_v ≤ n 2^C_2ne^-2K^2n→ 0, for K large enough. § ESTIMATION LOWER BOUNDS: PROOF OF THEOREM <REF> The lower bound in the L_2 norm is proved in Appendix <ref> and the lower bound in the L_∞ norm is proved Appendix <ref>. §.§ Estimation Lower Bound in the L_2 Norm: Proof of (<ref>) For γ∈^n, denote the probability distribution of s-uniform model 𝖧_s(n, γ) by _γ. To prove the result (<ref>) recall Fano's lemma: Suppose there exists γ^(0), ⋯,γ^(J)∈ℝ^n, with γ^(j)∈_M for all 0 ≤ j ≤ J, such that (1) γ^(j)-γ^(ℓ)_2 ≥ 2 s >0 for all 0 ≤ j ≠ℓ≤ J, (2) 1/J∑_j=1^J𝖪𝖫(_γ^(j), _γ^(0)) ≤αlog J, where α∈ (0, 1/8). Then min_γ̂max_γℙ(γ̂-γ_2 ≥ s ) ≥√(J)/√(J)+1(1-2 α -√(2 α/log J)). To obtain γ^(0), …,γ^(J)∈ℝ^n as in the above lemma we will invoke the Gilbert-Varshamov Theorem (see <cit.>) which states that there exists ω^(0), …, ω^(J)∈{0,1}^n, with J ≥ 2^n/8, such that ω^(0)=(0,⋯,0)^⊤ and ω^(j) - ω^(ℓ)_1 ≥n/8 , for all 0 ≤ j ≠ℓ≤ J. For ω^(0), …, ω^(J)∈{0,1}^n as above and δ∈ (0, 1/8) define, γ^(j) = ε_n ω^(j), for 0 ≤ j ≤ J , where ε_n = 16 C n^-s-1/2, with C= C(δ, s) > 0 a constant depending on δ and s to be chosen later. By (<ref>) we have γ^(j) - γ^(ℓ)_2 ≥ 2 C n^-s-2/2 . Now, 𝖪𝖫(_γ^(j), _γ^(0)) = ∑_t=0^sω^(j)_1 tn-ω^(j)_1 s-t{ψ(t ε_n) log(2ψ(t ε_n))+(1-ψ(t ε_n)) log(2(1-ψ(t ε_n)))}, where ψ(x)= e^x/1 + e^x is the logistic function defined in Lemma <ref>. By a Taylor expansion, for small enough x>0, ψ(x)log(2ψ(x))+(1-ψ(x))log(2(1-ψ(x))) = x^2/8 + O(x^3) . Hence, using ω^(j)_1 tn-ω^(j)_1 s-t≲_s n^s gives 1/J∑_j=1^J𝖪𝖫(_γ^(j), _γ^(0)) ≲_s n^s ε^2_n ≲_s C^2 n ≤δlog J , for C = C(δ, s) chosen appropriately. Hence, applying Theorem <ref> and taking J →∞ in (<ref>) gives min_γ̂max_γℙ(γ̂-γ_2 ≥ C n^- s-2/2) ≥ 1- 2 δ . This completes the proof of (<ref>). §.§ Estimation Lower Bound in L_∞ Norm: Proof of (<ref>) Choose 2 points γ, γ' ∈^n as follows: γ = 0 and γ' = (γ'_1, γ'_2, …, γ'_n) such that γ_i' = C n^-s-1/2 0 for some constant C > 0 to be chosen later. Clearly, γ - γ'_∞ = C n^-s-1/2 : = ε. Denote the probability distribution of the s-uniform models 𝖧_s(n, γ) and 𝖧_s(n, γ') by _γ and _γ', respectively. Observe that 𝖪𝖫(_γ, _γ') = 1/2∑_ e ∈[n] s: 1 ∈ e[ log{(1+e^ε ) /2 e^ε} + log{1/2(1+e^ε) }] . By Taylor's theorem, we get log{(1+e^ε ) /2 e^ε} + log{1/2(1+e^ε) } = ε^2 + O(ε^3)= C^2/n^s-1 + O( 1/n^3/2(s-1)) . Hence, from (<ref>), 𝖪𝖫(_γ, _γ') = L_s C^2 +o(1) , for some constant L_s depending on s. This implies, by Le Cam's two-point method (see <cit.>), for δ∈ (0,1), min_γ̂max_γℙ(γ̂-γ_∞≥ C √(1/n^s-1)) ≥max{ e^-1/4 L_s C^2 , 1/2- 1/2√(L_s C^2/2)}≥ 1 - δ , by choosing C, depending on δ and s, small enough. § PROOF OF THEOREM <REF> AND THEOREM <REF> We begin with the proof of Theorem <ref> in Section <ref>. The proof of Theorem <ref> is given in Section <ref>. §.§ Proof of Theorem <ref> Recall that, for 2 ≤ s ≤ r, d_s = (d_s(1), d_s(2), …, d_s(n))^⊤ is the vector of s-degrees. The first step in the proof of Theorem <ref> is to derive a linearization of β̂_s in terms of the s-degrees as in Proposition <ref> below. The proof is given in Appendix <ref>. Fix 2 ≤ s ≤ r. Then under the assumptions of Theorem <ref>, with probability 1-o(1) as n →∞, β̂_s-β_s - Σ_n, s^-1( d_s-𝔼[ d_s]) _∞ = O ( log n/n^s-1) , where Σ_n, s=((σ_s(u, v)))_u, v ∈ [n] is a n × n matrix with σ_s(u, v):=∑_ e ∈[n] s: u, v ∈ ee^ 1^⊤β_s, e/(1+e^ 1^⊤β_s, e)^2 and σ_s(u, u):= σ_s(u)^2= ∑_ e ∈[n] s: u ∈ ee^ 1^⊤β_s, e/(1+e^ 1^⊤β_s, e)^2 , where σ_s(u)^2 is also defined in (<ref>). Next, define the matrix Γ_n, s=((γ_s(u, v)))_u, v ∈ [n] as follows: γ_s(u, v) := 1{ u = v }/σ_s(u)^2 . The following lemma shows that it is possible to replace the matrix Σ_n, s^-1 in (<ref>) with the matrix Γ_n, s asymptotically. The proof of the lemma is given in Appendix <ref>. Suppose Σ_n, s and Γ_n, s be as defined in (<ref>) and (<ref>), respectively. Then under the assumptions of Theorem <ref>, Γ_n, s - Σ_n, s^-1_∞≤ O (1/n^s) , where A_∞ = max_u, v ∈ [n] \|a_u, v\| for a matrix A = ((a_u, v))_u, v ∈ [n]. Furthermore, [(Γ_n, s - Σ_n, s^-1)( d_s-𝔼[ d_s])]_∞≤Γ_n, s - Σ_n, s^-1_∞ + O (1/n^s). To complete the proof of Theorem <ref>, consider J_s ∈[n] a_s, for a_s ≥ 1 fixed. Proposition <ref> and Lemma <ref> combined implies, [ (β̂_s - β_s) ]_J_s - [Γ_n, s( d_s-𝔼[ d_s])]_J_s_∞ = O ( log n/n^s-1) , with probability 1-o(1). Now, recall from the statement of Theorem <ref> that D_s = diag (σ_s(v))_v ∈ [n]. From (<ref>) observe that max_v ∈ [n]σ_s(v)^2 ≍ n^s-1, since β_s_∞≤ M=O(1). Hence, [ D_s (β̂_s - β_s)]_J_s - [ D_s(Γ_n, s( d_s-𝔼[ d_s])]_J_s_∞ = O ( log n/√(n^s-1)) . Note that for v ∈ J_s, σ_s(v)[Γ_n, s( d_s-𝔼[ d_s)]]_v = d_s(v)-𝔼[d_s(v)]/σ_s(v) . Therefore, from (<ref>), [ D_s]_J_s ( [(β̂_s - β_s)]_J_s ) = ( ( d_s(v)-𝔼[d_s(v)]/√([d_s(v)])) )_v ∈ J_s + O (log n/√(n)) _a_s( 0, I) , using the central limit theorem for sums of independent bounded random variables. Since β̂_s are independent across 2 ≤ s ≤ r, the result in (<ref>) follows. §.§.§ Proof of Proposition <ref> For 2 ≤ s ≤ r and e = (u_1, u_2, …, u_s) ∈[n] s, let β_s, e = (β_s, u_1, β_s, u_2, …, β_s, u_s)^⊤ and β̂_s, e = (β̂_s, u_1, β̂_s, u_2, …, β̂_s, u_s)^⊤. Moreover, 1 will denote the vector of ones in the appropriate dimension. To begin with, (<ref>) and (<ref>) gives, for v ∈ [n], d_s(v) -𝔼[d_s(v) ] = ∑_ e ∈[n] s: v ∈ e{e^ 1^⊤β̂_s, e/1+e^ 1^⊤β̂_s, e-e^ 1^⊤β_s, e/1+e^ 1^⊤β_s, e} . Note that for e ∈[n] s, by a Taylor expansion, e^ 1^⊤β̂_s, e/1+e^ 1^⊤β̂_s, e-e^ 1^⊤β_s, e/1+e^ 1^⊤β_s, e = e^ 1^⊤β_s, e/(1+e^ 1^⊤β_s, e)^2( 1^⊤β̂_s, e- 1^⊤β_s, e) + _s, e , where \|_s, e\| ≤1/2\| 1^⊤β̂_s, e- 1^⊤β_s, e\|^2 ≲_r β̂_s-β_s^2_∞. Then, from (<ref>), d_s(v) -𝔼[d_s(v) ]=[ Σ_n, s(β̂_s-β_s)]_v + R_v, s, where R_v, s=∑_ e ∈[n] s: v ∈ e_s, e. From (<ref>), we have β̂_s-β_s=Σ_n, s^-1( d_s-𝔼[ d_s]) + Σ_n, s^-1 R_n, s , where R_n, s = (R_1, s, R_2, s, …, R_n, s)^⊤. Note that from (<ref>), \|R_v, s\| ≤∑_ e ∈[n] s: v ∈ e \|_s, e\| ≲_r n^s-1β̂_s-β_s^2_∞. To bound Σ_n, s^-1 R_n, s_∞, note that for v ∈ [n], \|[Σ_n, s^-1 R_n, s]_v\| ≤ \|[Γ_n, s R_n, s]_v\| +\| [ (Σ_n, s^-1- Γ_n, s) R_n, s]_v\|. Observe that [Γ_n, s R_n, s]_v = R_v, s/σ_s(v)^2 . Using σ_s(v)^2 ≍ n^s-1, (<ref>), and (<ref>) gives, \|[Γ_n, s R_n, s]_v\| ≲β̂_s-β_s^2_∞ = O (log n/ n^s-1) , with probability 1- o(1). Further, by Lemma <ref>, (<ref>), and (<ref>), \|[ (Σ_n, s^-1- Γ_n, s) R_n, s]_v\| ≤ (Σ_n, s^-1- Γ_n, s) _∞× n R_n, s_∞ ≲β̂_s-β_s^2_∞ ≤ O (log n/ n^s-1) , with probability 1- o(1). Hence, by (<ref>) and (<ref>) the result in (<ref>) follows. §.§.§ Proof of Lemma <ref> Denote Δ_n, s= Γ_n, s - Σ_n, s^-1 = (( δ_s(u, v) ))_u, v ∈ [n], Z_n, s= I_n- Σ_n, sΓ_n, s = (( z_s(u, v) ))_u, v ∈ [n], and Θ_n, s=Γ_n, s Z_n, s = (( θ_s(u, v) ))_u, v ∈ [n]. Then Δ_n, s=(Γ_n, s - Σ_n, s^-1)( I_n- Σ_n, sΓ_n, s) - Γ_n, s( I_n - Σ_n, sΓ_n, s)=Δ_n, s Z_n, s - Θ_n, s. Hence, for u, v ∈ [n], δ_s(u, v) =∑_w=1^nδ_s(u, w) z_s(w, v) - θ_s(u, v) =∑_w=1^nδ_s(u, w){ 1{ w = v } - ∑_b=1^n σ_s(w, b) γ_s(b, v) } - θ_s(u, v) =∑_w=1^nδ_s(u, w){ 1{ w = v } - ∑_b=1^n σ_s(w, b) 1{v=b}/σ_s(v)^2} - θ_s(u, v) (by (<ref>)) =∑_w=1^nδ_s(u, w){ 1{ w = v } - σ_s(w, v) /σ_s(v)^2} - θ_s(u, v) = - ∑_w=1^nδ_s(u, w){ 1{ w v }σ_s(w, v) /σ_s(v)^2} - θ_s(u, v) , since ∑_b ∈ [n]\{w}σ_s(w, b) = σ_s(w, w) = σ_s(w)^2. The following lemma bounds the maximum norm of Θ_n, s = Γ_n, s Z_n, s = (( θ_s(u, v) ))_u, v ∈ [n]. For u, v, w ∈ [n], max{\|θ_s(u, v)\|,\|θ_s(u, v)-θ_s(v, w)\|}≲σ_s, max/σ_s, min^2n^2 , where σ_s, min :=min_1 ≤ u < v ≤ nσ_s(u, v) and σ_s, max :=max_1 ≤ u < v ≤ nσ_s(u, v). Note that Θ_n, s=Γ_n, s Z_n, s = Γ_n, s - Γ_n, sΣ_n, sΓ_n, s. This means for u, v ∈ [n], θ_s(u, v) = γ_s(u, v) - ∑_x, y ∈ [n]γ_s(u, x) σ_s(x, y) γ_s(y, v) . Then recalling the definition of γ_s(u, v) from (<ref>) gives, ∑_x, y ∈ [n]γ_s(u, x) σ_s(x, y) γ_s(y, v) = ∑_x, y ∈ [n] 1{ u = x } 1{ y = v }σ_s(x, y) /σ_s(u)^2 σ_s(v)^2 = σ_s(u, v) /σ_s(u)^2 σ_s(v)^2 . Hence, from (<ref>) and (<ref>), \|θ_s(u, v)\| = \| σ_s(u, v) 1{ u v }/σ_s(u)^2 σ_s(v)^2\| ≲σ_s, max/σ_s, min^2n^2 . This completes the proof of (<ref>). Now, for u ∈ [n], let m, m⃗∈ [n] be such that δ_s(u, m)=max_w ∈ [n]δ_s(u, w) and δ_s(u, m⃗)=min_w ∈ [n]δ_s(u, w). The following lemma gives bounds on δ_s(u, m⃗) and δ_s(u, m). For u ∈ [n], ∑_w=1^nδ_s(u, w) σ_s(w, u) = 0. This implies, δ_s(u, m) ≥ 0 and δ_s(u, m⃗) ≤ 0. Note that ∑_w=1^nδ_s(u, w) σ_s(w, u) is the u-th diagonal element of the matrix Δ_n, sΣ_n, s = Γ_n, sΣ_n, s - I_n (recall that Δ_n, s=Γ_n, s - Σ_n, s^-1). Note that the u-th diagonal element of Γ_n, sΣ_n, s is given by ∑_w ∈ [n]γ_s(u, w) σ_s(w, u) = ∑_w ∈ [n] 1{u=w}/σ_s(u)^2σ_s(w, u) = 1 , since σ_s(u, u) = σ_s(u)^2. Hence, u-th diagonal element of Δ_n, sΣ_n, s is zero. Now, recalling (<ref>) note that δ_s(u, m)-δ_s(u, m⃗) + (θ_s(u, m)-θ_s(u, m⃗)) =∑_w=1^nδ_s(u, w) { 1{ w m⃗}σ_s(w, m⃗)/σ_s(m⃗)^2 - 1{ w m}σ_s(w, m)/σ_s( m )^2} = ∑_w=1^n(δ_s(u, w)-δ_s(u, m⃗)){ 1{ w m⃗}σ_s(w, m⃗)/σ_s(m⃗)^2 - 1{ w m}σ_s(w, m)/σ_s( m )^2} , since ∑_w ∈ [n]\{m⃗}σ_s(w, m⃗) = σ_s(m⃗)^2 and ∑_w ∈ [n]\{m}σ_s(w, m) = σ_s(m)^2. Define Ω:={ w ∈ [n] : 1{ w m⃗}σ_s(w, m⃗)/σ_s(m⃗)^2≥ 1{ w m}σ_s(w, m)/σ_s(m)^2}, and λ := \|Ω\|. Then, we have ∑_w ∈Ω (δ_s(u, w)-δ_s(u, m⃗)){ 1{ w m⃗}σ_s(w, m⃗)/σ_s(m⃗)^2 - 1{ w m}σ_s(w, m)/σ_s(m)^2} ≤ (δ_s(u, m)-δ_s(u, m⃗)){∑_w ∈Ωσ_s(w, m⃗)/σ_s(m⃗)^2 - ∑_w ∈Ω 1{ w m}σ_s(w, m)/σ_s(m)^2}. Note that ∑_w ∈Ωσ_s(w, m⃗)/σ_s(m⃗)^2= ∑_w ∈Ωσ_s(w, m⃗)/∑_w ∈Ωσ_s(w, m⃗) + ∑_w ∈ [n]\(Ω⋃m⃗) σ_s(w, m⃗) = 1/ 1 + ∑_w ∈ [n]\(Ω⋃m⃗) σ_s(w, m⃗)/∑_w ∈Ωσ_s(w, m⃗) , since m⃗∉Ω. Now, observe that ∑_w ∈ [n]\(Ω⋃m⃗) σ_s(w, m⃗)/∑_w ∈Ωσ_s(w, m⃗)≥(n-λ-1) σ_s, min/λσ_s, max This implies, ∑_w ∈Ωσ_s(w, m⃗)/σ_s(m⃗)^2≤λσ_s, max/λσ_s, max + (n-λ-1) σ_s, min . Similarly, ∑_w ∈Ω 1{ w m}σ_s(w, m)/σ_s(m)^2 = ∑_w ∈Ω 1{ w m}σ_s(w, m )/∑_w ∈ [n] 1{ w m}σ_s(w, m) = 1/1 + ∑_w ∈ [n]\Ω 1{ w m}σ_s(w, m⃗ ) /∑_w ∈Ω 1{ w m}σ_s(w, m⃗ ) Therefore, since m∈Ω, ∑_w ∈ [n]\Ω 1{ w m}σ_s(w, m) /∑_w ∈Ω 1{ w m}σ_s(w, m) ≤(n-λ) σ_s, max/ (λ-1) σ_s, min. Hence, ∑_w ∈Ω 1{ w m}σ_s(w, m)/σ_s(m)^2≥(λ-1) σ_s, min/(λ-1) σ_s, min + (n-λ) σ_s, max . Applying (<ref>) and (<ref>) in (<ref>) gives, ∑_w ∈Ω (δ_s(u, w)-δ_s(u, m⃗)){ 1{ w m⃗}σ_s(w, m⃗)/σ_s(m⃗)^2 - 1{ w m}σ_s(w, m)/σ_s(m)^2} ≤ (δ_s(u, m)-δ_s(u, m⃗))f(λ) , where f(λ)=λσ_s, max/λσ_s, max+(n-1-λ)σ_s, min-(λ-1)σ_s, min/(λ-1)σ_s, min+(n-λ)σ_s, max. Note that f(λ) attains maximum at λ=n/2 over λ∈ (1,n-1) and f(n/2)=nσ_s, max-(n-2)σ_s, min/nσ_s, max + (n-2)σ_s, min. Therefore, from Lemma <ref>, (<ref>), there exists a constant C> 0 such that (<ref>), δ_s(u, m)-δ_s(u, m⃗) ≤nσ_s, max-(n-2)σ_s, min/nσ_s, max+(n-2)σ_s, min(δ_s(u, m)-δ_s(u, m⃗))+ C σ_s, max/σ_s, min^2n^2 . This implies, δ_s(u, m)-δ_s(u, m⃗) ≤C σ_s, max(n σ_s, max + (n-2)σ_s, min)/ 2 (n-2) σ_s, min ^3 n^2≲σ_s, max^2/σ_s, min^3 n^2 . Hence, from Lemma <ref>, max_1 ≤ w ≤ n\|δ_s(u, w)\| ≤δ_s(u, m)-δ_s(u, m⃗) ≤σ_s, max^2/σ_s, min^3 n^2≲1/n^s , since σ_s, min≍ n^s-2 and σ_s, max≍ n^s-2, using β_s_∞≤ M=O(1). This completes the proof of (<ref>). Define U_n, s= [(Γ_n, s - Σ_n, s^-1)( d_s-𝔼[ d_s] ) ] = [ Δ_n, s( d_s-𝔼[ d_s])] , since Δ_n, s=Γ_n, s - Σ_n, s^-1. Observe that U_n, s = Δ_n, s[( d_s-𝔼[ d_s]) ( d_s-𝔼[ d_s])^⊤ ] Δ_n, s^⊤ = Δ_n, sΣ_n, sΔ_n, s^⊤ = (Γ_n, s - Σ_n, s^-1)-Γ_n, s( I_n-Σ_n, sΓ_n, s) = (Γ_n, s - Σ_n, s^-1)-Θ_n, s , since Θ_n, s = Γ_n, s Z_n, s and Z_n, s = I_n-Σ_n, sΓ_n, s. By Lemma <ref>, Θ_n, s_∞≲σ_s, max/σ_s, min^2 n^2≲1/n^s , since σ_s, min≍ n^s-2 and σ_s, max≍ n^s-2, using β_s_∞≤ M=O(1). By (<ref>), (<ref>), and (<ref>) the result in (<ref>) follows. §.§ Proof of Theorem <ref> For x = (x_1, x_2, …, x_n) ∈^n and u ∈ [n] define the function g_u( x) = ∑_ e ∈[n] s: u ∈ ee^ 1^⊤ x_ e/(1+e^ 1^⊤ x_ e)^2 , where x_ e = (x_u_1, x_u_2, …, x_u_s) for e = (u_1, u_2, …, u_s). Then recalling (<ref>) and (<ref>), σ_s(v)^2 = g_v(β_s) and σ̂_s(v)^2 = g_v(β̂_s). Hence, by a Taylor expansion, \|σ̂_s(v)^2-σ_s(v)^2\| = \|g_v(β̂_s) - g_v(β_s)\| = \| ∑_ e ∈[n] s: u ∈ e{e^ 1^⊤β̂_s, e/(1+e^ 1^⊤β̂_s, e)^2-e^ 1^⊤β_s, e/(1+e^ 1^⊤β_s, e)^2}\| ≲_r β̂_s-β_s_∞ . Recalling the definition of J_s = { v_s,1, …, v_s,a_s} from Theorem <ref>, this implies ∑_s=2^r ([(β̂_s - β_s)]_J_s)^⊤ [ D̂_s^2]_J_s ([(β̂_s - β_s)]_J_s) = ∑_s=2^r ∑_j = 1^a_sσ̂_s(v_a_j)^2 ( β̂_s, v_a_j - β_s, v_a_j)^2 = ∑_s=2^r ∑_j = 1^a_sσ_s(v_a_j)^2 ( β̂_s, v_a_j - β_s, v_a_j)^2 + ∑_s=2^r ∑_j = 1^a_s (σ̂_s(v_a_j)^2 - σ_s(v_a_j)^2) ( β̂_s, v_a_j - β_s, v_a_j)^2 χ^2_∑_s=2^r a_s + o_P(1) , by Theorem <ref>, (<ref>) and (<ref>). This completes the proof of (<ref>). § PROOFS OF THEOREMS <REF> AND <REF> §.§ Proof of Theorem <ref> Suppose H_n ∼𝖧_n, s(n, γ) for γ as in (<ref>). Let Σ_n, s be as defined in (<ref>) with β_s replaced by γ = (γ_1, γ_2, …, γ_n)^⊤. Then ^2 ℓ_n, s(γ) = Σ_n, s. Then by a Taylor expansion, ℓ_n, s(γ)-ℓ_n, s(β̂_s) = (β̂_s - γ)^⊤ℓ_n, s(γ) + 1/2(β̂_s - γ)^⊤Σ_n, s (β̂_s - γ) + _n, s, where _n, s = _n, s^(1) + _n, s^(2) + _n, s^(3) , with _n, s^(1) :=1/6∑_u=1^n∂^3ℓ_n, s(γ+θ(β̂_s-γ))/∂(β_s, u)^3(β̂_s, u-γ_u)^3 , _n, s^(2) := 1/3∑_1 ≤ u v ≤ n∂^3ℓ_n, s(γ+θ(β̂_s-γ))/∂(β_s, u)^2∂β_s, v(β̂_s, u-γ_u)^2(β̂_s, v-γ_v) , _n, s^(3) := 1/6∑_1 ≤ u v w ≤ n∂^3ℓ_n, s(γ+θ(β̂_s-γ))/∂β_s, u∂β_s, v∂β_s, w(β̂_s, u-γ_u) (β̂_s, v-γ_v) (β̂_s, w-γ_w) , for some θ∈ (0, 1). Now, by arguments as in (<ref>) it follows that β̂_s-γ = Σ_n, s^-1( d_s-𝔼_γ[ d_s]) + Σ_n, s^-1 R_n, s , where R_n, s is as defined in (<ref>) and (<ref>) with β_s replaced by γ. Using this and noting that -ℓ_n, s(γ) = d_s-𝔼_γ[ d_s], (β̂_s - γ)^⊤ℓ_n, s(γ) = ( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1ℓ_n, s(γ) + R_n, s^⊤Σ_n, s^-1ℓ_n, s(γ) = - ( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1 ( d_s-𝔼_γ[ d_s]) - R_n, s^⊤Σ_n, s^-1 ( d_s-𝔼_γ[ d_s]) . Similarly, using (<ref>), (β̂_s - γ)^⊤Σ_n, s (β̂_s - γ) = ( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1( d_s-𝔼_γ[ d_s]) + 2 R_n, s^⊤Σ_n, s^-1 ( d_s-𝔼_γ[ d_s]) + R_n, s^⊤Σ_n, s^-1 R_n, s . Combining (<ref>), (<ref>), and (<ref>) gives, ℓ_n, s(β̂_s)-ℓ_n, s(γ) = - 1/2( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1( d_s-𝔼_γ[ d_s]) + 1/2 R_n, s^⊤Σ_n, s^-1 R_n, s + _n, s . We begin by showing that R_n, s^⊤Σ_n, s^-1 R_n, s = o_P(√(n)). To this end, (<ref>) and σ_s(u)^2 ≍ n^s-1 gives, \| R_n, s^⊤Γ_n, s R_n, s\| = \|∑_u=1^nR_s, u^2/σ_s(u)^2\| ≲ n^sβ̂_s-β_s^4_∞≲log^2n/n^s-2 , with probability 1-o(1) by (<ref>). Next, observe that \| R_n, s^⊤Δ_n, s R_n, s\| ≤ n Δ_n, s R_n, s_∞· R_n, s_∞ ≤ n^2 R_n, s_∞^2 ·Δ_n, s_∞ ≲ n^sβ̂_s-β_s^4_∞(by (<ref>) and (<ref>)) ≲log^2n/n^s-2 , with probability 1-o(1) by (<ref>). Combining (<ref>) and (<ref>) it follows that with probability 1-o(1), \| R_n, s^⊤Σ_n, s^-1 R_n, s\| ≤\| R_n, s^⊤Γ_n, s R_n, s\| + \| R_n, s^⊤Δ_n, s R_n, s\| ≲log^2n/n^s-2 = o_P(√(n)) . This implies, the second term in the RHS of (<ref>) does not contribute to the CLT of the log-likelihood ratio logΛ_n, s. Next, we show that the third term in the RHS of (<ref>) is o_P(√(n)), hence, it also does not contribute to the CLT of logΛ_n, s. Suppose s ≥ 3 and γ∈ℬ(M). Then _n, s = o_P(√(n)). Define β̃_ s=γ+θ(β̂_s-γ), for θ∈ (0, 1). Then recalling (<ref>) observe that _n, s^(1) =1/6∑_u=1^n[∑_ e ∈[n] s: u ∈ ee^ 1^⊤β̃_ s, e(1-e^ 1^⊤β̃_ s, e)/(1+e^ 1^⊤β̃_ s, e)^3](β̂_s, u-γ_u)^3 , _n, s^(2) = 1/3∑_1 ≤ u v ≤ n[∑_ e ∈[n] s: u, v ∈ ee^ 1^⊤β̃_ s, e(1-e^ 1^⊤β̃_ s, e)/(1+e^ 1^⊤β̃_ s, e)^3](β̂_s, u-γ_u)^2(β̂_s, v-γ_v) , _n, s^(3) := 1/6∑_1 ≤ u v w ≤ n[∑_ e ∈[n] s: u, v, w ∈ ee^ 1^⊤β̃_ s, e(1-e^ 1^⊤β̃_ s, e)/(1+e^ 1^⊤β̃_ s, e)^3](β̂_s, u-γ_u) (β̂_s, v-γ_v) (β̂_s, w-γ_w) , where β̃_ s, e = (β̃_s, u_1, β̃_s, u_2, …, β̃_s, u_s)^⊤, for e = (u_1, u_2, …, u_s). Since γ∈ℬ_M and β̂_s - γ_∞≲_s, M√(log n/n^s-1) with probability 1-o(1), β̂_s ∈ℬ_2M for large n with probability 1-o(1). This implies, _n, s^(1)≲_M n^s β̂_s - γ^3_∞≲_M, s√((log n)^3/n^s-3) = o_P(√(n)) , for s ≥ 3. Similarly, we can show that for s ≥ 3, _n, s^(2) = o_P(√(n)) and _n, s^(3) = o_P(√(n)). This completes the proof of the Lemma <ref>. Note that Lemma <ref> assumes that s ≥ 3. This is because when s=2 (that is, the graph case), the proof of Lemma <ref> gives the bound _n, 2 = O(polygon(n)/√(n)) which is not o_P(√(n)) (see (<ref>)). Nevertheless, it follows from the proof of Theorem 1 (a) in <cit.>, where the asymptotic null distribution of the LR test for the graph β-model was derived, that the result in Lemma <ref> also holds when s=2, that is, _n, 2 = o_P(√(n)). For this one has to expand ℓ_n, s(β̂_s)-ℓ_n, s(γ) up to the fourth order term, and show that the third order term is o_P(√(n)) at the true parameter value and the fourth order term is o_P(√(n)) at an intermediate point. For s ≥ 3, the third order term at an intermediate point is o_P(√(n)), hence, we do not have to consider the fourth order term. Now, recall the definition of logΛ_n, s from (<ref>). Then by Lemma <ref> and (<ref>) 2 logΛ_n, s - n/√(2n) = ( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1( d_s-𝔼_γ[ d_s]) - n /√(2n) + o_P(1) . By the following lemma we can replace Σ_n, s^-1 with Γ_n, s in the RHS above. The proof of the lemma is given in Appendix <ref>. For L > 0, ( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s]) > L ) ≲1/L^2. This implies, ( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s]) is bounded in probability. By Lemma <ref> and recalling (<ref>), ( d_s-𝔼_γ[ d_s])^⊤Σ_n, s^-1( d_s-𝔼_γ[ d_s])/√(2 n) = ( d_s-𝔼_γ[ d_s])^⊤Γ_n, s( d_s-𝔼_γ[ d_s])/√(2 n) + o_P(1) = 1/√(2n)∑_u=1^n(d_s(u) - 𝔼_γ[d_s(u) ])^2/σ_s(u)^2 + o_P(1) . Proposition <ref> establishes the asymptotic normality of the leading term in the RHS above. The proof is given in Appendix <ref>. Under the assumption of Theorem <ref>, 1/√(2n){∑_u=1^n(d_s(u) - 𝔼_γ[d_s(u) ])^2/σ_s(u)^2 -n }(0, 1) . The result in (<ref>) now follows from (<ref>), (<ref>), and Proposition <ref>. §.§.§ Proof of Lemma <ref> To begin with note that _γ[( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s])] = (_γ[( d_s-𝔼_γ[ d_s]) ( d_s-𝔼_γ[ d_s])^⊤ ] (Σ_n, s^-1 - Γ_n, s) ) = ( I_n - Σ_n, sΓ_n, s ) = n- ∑_u, v ∈ [n]σ_s(u, v) γ_s(u, v) = n- ∑_u, v ∈ [n]σ_s(u, v) 1{u=v}/σ_s(u)^2 = 0 . Next, we will show that _γ[( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s])] = O (1). The result in Lemma <ref> then follows by Chebyshev's inequality. Recall that Δ_n, s:=Σ_n, s^-1 - Γ_n, s. We shall denote the entries of Δ_n, s by (( δ_u,v )) for u, v ∈ [n]. Then ( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s]) = ∑_u,v ∈ [n]δ_u, v (d_s(u)-𝔼_γ[d_s(u)])(d_s(v)-𝔼_γ[d_s(v)]). Define d̅_s(u) := d_s(u)-𝔼_γ[d_s(u)], for u ∈ [n]. Then _γ[( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s])] = ∑_ u,v,u',v' ∈ [n] δ_u, vδ_u', v'_γ[d̅_s(u)d̅_s(v), d̅_s(u')d̅_s(v')] . To analyze the RHS of (<ref>) we consider the following 4 cases. Case 1: u=v=u'=v'. In this case we have _γ[d̅_s(u)d̅_s(v), d̅_s(u')d̅_s(v')] = _γ[d̅_s(u)^2]. For e ∈[n] s, denote X_ e: =1{ e ∈ E(H_n) } and X̅_ e := 1{ e ∈ E(H_n) } - [1{ e ∈ E(H_n) }]. Since {X̅_ e: e ∈[n] s} are independent and have zero mean, {X̅_ eX̅_ e': e, e' ∈[n] s} are pairwise uncorrelated. Hence, _γ[d̅_s(u)^2] = _γ[ ∑_ e, e' ∈[n] s : u ∈ e ⋂ e' X̅_ eX̅_ e'] = ∑_ e, e' ∈[n] s : u ∈ e ⋂ e' _γ[X̅_ eX̅_ e'] = ∑_ e ∈[n] s: u ∈ e_γ[X̅^2_ e] + ∑_ e e' ∈[n] s : u ∈ e ∩ e' _γ[X̅_ e] _γ[ X̅_ e' ] . Since γ_∞≤ M, _γ[X̅_ e] = _γ[ X_ e]= e^ 1^⊤γ_ e/(1 + e^ 1^⊤γ_ e) ≍_M 1 , where γ_ e = (γ_u_1, γ_u_2, …, γ_u_s)^⊤, for e = (u_1, u_2, …, u_s). Similarly, _γ[X̅^2_ e] ≍_M 1. Hence, (<ref>) implies that _γ[d̅_s(u)^2] ≲_M n^2s-2. Case 2: u ≠ v=u'=v'. Observe that _γ[d̅_s(u)d̅_s(v), d̅_s(u')d̅_s(v')] = _γ[d̅_s(u)d̅_s(v), d̅_s(v)^2 ] = ∑_ e_1, e_2, e_3, e_4 ∈[n] s u ∈ e_1, v ∈ e_1 ∩ e_2 ∩ e_3{𝔼_γ[X̅_ e_1X̅_ e_2X̅_ e_3X̅_ e_4]-𝔼_γ[X̅_ e_1X̅_ e_2]𝔼_γ[X̅_ e_3X̅_ e_4]}. Note that the non-zero contributions in the RHS above come from the terms when e_i = e_j and e_k = e_ℓ for i,j,k,ℓ∈{1,…,4}. Hence, _γ[d̅_s(u)d̅_s(v), d̅_s(v)^2 ] = ∑_ e ∈[n] s, u, v ∈ e( 𝔼_γ[X̅^4_ e] - (𝔼_γ[X̅_ e^2])^2 ) + 2∑_ e_1 e_2 ∈[n] s u, v ∈ e_1, v ∈ e_2𝔼_γ[X̅^2_ e_1]𝔼[X̅^2_ e_2] ≲_M n^2s-3 , since 𝔼_γ[X̅^4_ e] ≍_M 1 and 𝔼_γ[X̅_ e^2]) ≍_M 1. Case 3: u ≠ v ≠ u'=v': By similar reasoning as the previous two cases it can be shown that _γ[d̅_s(u)d̅_s(v),d̅_s(u')d̅_s(v')] = _γ[d̅^2_s(u),d̅_s(u')d̅_s(u')] ≲_M n^2s-3. Case 4: u ≠ v ≠ u' ≠ v'. In this case, it can be shown that _γ[d̅_s(u)d̅_s(v),d̅_s(u')d̅_s(v')] ≲_M n^2s-4. Combining the 4 cases and using (<ref>), _γ[( d_s-𝔼_γ[ d_s])^⊤ (Σ_n, s^-1 - Γ_n, s) ( d_s-𝔼_γ[ d_s])] ≲_Mmax_u, v ∈ [n]\|δ_u, v\|^2 n^2s = O(1) , where the last step uses (<ref>). §.§.§ Proof of Proposition <ref> Suppose H_n = (V(H_n), E(H_n)) ∼𝖧_n, s(n, γ) for γ as in (<ref>). For e= {v_1, v_2, …, v_s}∈[n] s, denote X_ e := X_{v_1, v_2, …, v_s} : =1{ e ∈ E(H_n) } , and X̅_ e := 1{ e ∈ E(H_n) } - _γ[1{ e ∈ E(H_n) }]. Also, for u ∈ [n] denote d̅_s(u) = d_s(u) - _γ[d_s(u)] = ∑_ e ∈[n] s : u ∈ eX̅_ e. Observe that d̅_s(u)^2 = ∑_ e ∈[n] s : u ∈ eX̅_ e^2 + ∑_ e e' ∈[n] s : u ∈ e ∩ e'X̅_ eX̅_ e' . This implies, _γ[d̅_s(u)^2] = ∑_ e ∈[n] s : u ∈ e_γ[X̅_ e^2] = ∑_ e ∈[n] s : u ∈ e_γ[X̅_ e^2] = _γ[d_s(u)] = σ_s(u)^2 . Hence, 1/√(2n){∑_u=1^n(d_s(u) - 𝔼_γ[d_s(u) ])^2/σ_s(u)^2 -n } = 1/√(2n)∑_u=1^nd̅_s(u)^2 - _γ[d̅_s(u)^2] /σ_s(u)^2 = 1/√(2n)∑_u=1^n∑_ e ∈[n] s : u ∈ eX̅_ e^2 - _γ[X̅_ e^2] /σ_s(u)^2 + 1/√(2n)∑_u=1^n∑_ e e' ∈[n] s : u ∈ e ∩ e'X̅_ eX̅_ e'/σ_s(u)^2 (by (<ref>)) := T_1 + T_2 . We will first show that T_1 = o_P(1). Towards this note that T_1 = s/√(2n)∑_ e ∈ [n] sX̅_ e^2 - _γ[X̅_ e^2] /σ_s(u)^2 . Since {X̅_ e : e ∈ [n] s} are independent, _γ[T_1] = s^2/ 2 n ∑_ e ∈ [n] s_γ [ X̅_ e^2 ] /σ_s(u)^4 ≲_M 1/n^s-1 , using _γ[X̅_ e^2] ≍_M 1 and σ_s(u)^2 ≍_M n^s-1. This implies, T_1 = o_P(1). Therefore, from (<ref>), to prove (<ref>) it remains to show T_2 N(0, 1). For this we will express T_2 as a sum of a martingale difference sequence. To this end, define the following sequence of sigma-fields: For u ∈ [n], _u := σ(⋃_v=1^u{X̅_ e: v ∈ e }), is the sigma algebra generated by the collection of random variables ⋃_v=1^u{X̅_ e: v ∈ e }. Clearly, _1 ⊆_2 ⋯⊆_n, hence {_u}_u ∈ [n] is a filtration. Now, for u ∈ [n], define T_2, u := ∑_ e, e' ∈[n] s: e e', u ∈ e ∩ e', e ∩{1,…, u}≠∅ and e' ∩{1,…,u-1} = ∅ w_ e, e'X̅_ eX̅_ e' where w_ e, e':= ∑_z ∈ e ∩ e'1/σ_s(z)^2. Note that T_2, u is _u measurable and _γ[T_2, u \| _u-1 ] = 0, that is, T_2, u, for u ∈ [n], is a martingale difference sequence. Also, recalling the definition of T_2 from (<ref>) observe that T_2 = 1/√(2n)∑_u=1^n∑_ e e' ∈[n] s : u ∈ e ∩ e'X̅_ eX̅_ e'/σ_s(u)^2 = 1/√(2n)∑_ e e' ∈[n] s, e ∩ e' ∅ w_ e, e'X̅_ eX̅_ e' = 1/√(2n)∑_u=1^n T_2, u , that is, T_2 is the sum of a martingale difference sequence. Now, invoking the martingale central theorem <cit.> it can be shown that T_2 N(0, 1). The details are omitted. §.§ Proof of Theorem <ref> Suppose H_n ∼𝖧_n, s(n, γ') for γ' as in (<ref>). Then by arguments as in (<ref>), ℓ_n, s(β̂_s)-ℓ_n, s(γ') = - 1/2( d_s-𝔼_γ'[ d_s])^⊤Σ_n, s^-1( d_s-𝔼_γ'[ d_s]) + 1/2 R_n, s^⊤Σ_n, s^-1 R_n, s + _n, s , where Σ_n, s and R_n, s are as defined in (<ref>) and (<ref>), respectively, with β_s replaced by γ' and _n, s as defined in (<ref>) with γ replaced by γ'. Therefore, ℓ_n, s(β̂_s)-ℓ_n, s(γ) = - 1/2( d_s-𝔼_γ'[ d_s])^⊤ Σ_n, s^-1( d_s-𝔼_γ'[ d_s]) + 1/2 R_n, s^⊤Σ_n, s^-1 R_n, s+ _n, s +ℓ_n, s(γ')-ℓ_n, s(γ), By Taylor expansion, ℓ_n, s(γ')-ℓ_n, s(γ)=( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) + 1/2(γ'-γ)^⊤Σ̃_n, s(γ'-γ), where Σ̃_n, s is the covariance matrix defined in (<ref>) with β_s replaced by γ̃=γ'+θ(γ'-γ) for some 0<θ<1. Then by arguments as in (<ref>) and Lemma <ref>, Lemma <ref>, (<ref>) can be written as: ℓ_n, s(β̂_s)-ℓ_n, s(γ) = - 1/2( d_s-𝔼_γ'[ d_s])^⊤ Γ_n, s ( d_s-𝔼_γ'[ d_s]) + ( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) + 1/2(γ'-γ)^⊤Σ̃_n, s(γ'-γ) + o_P(√(n)) , where Γ_n, s is as defined in (<ref>) with the parameter β_s replaced by γ'. We begin with the case γ'-γ_2 ≪ n^-2s-3/4. In this case, since ^2 ℓ_n, s(γ') = Σ_n, s, by Lemma <ref> (γ'-γ)^⊤Σ_n, s (γ'-γ) ≍ n^s -1γ'-γ_2^2 ≪√(n) . Similarly, (γ'-γ)^⊤Σ̃_n, s (γ'-γ) ≍ n^s -1γ'-γ_2^2 ≪√(n) . Hence, [( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] = (γ'-γ)^⊤Σ_n, s(γ'-γ) ≪ n, which implies, ( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) = o_P(√(n)), since [( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] = 0. Therefore, under H_1 as in (<ref>), 2logΛ_n, s-n/√(2n) = 2(ℓ_n, s(γ)-ℓ_n, s(β̂_s)) -n /√(2n) = ( d_s-𝔼_γ'[ d_s])^⊤Γ_n, s( d_s-𝔼_γ'[ d_s]) - n /√(2 n) + o_P(1) *(by (<ref>), (<ref>), and (<ref>)) (0,1) , by Proposition <ref>. This proves the first assertion in (<ref>). Next, suppose γ'-γ_2 ≫ n^-2s-3/4. In this case, by Lemma <ref>, (γ'-γ)^⊤Σ_n, s (γ'-γ) ≍ n^s -1γ'-γ_2^2 ≫√(n). We will first assume: √(n)≪ (γ'-γ)^⊤Σ_n, s(γ'-γ) ≲ n . Then we have [( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] = (γ'-γ)^⊤Σ_n, s(γ'-γ) = O(n). Using this and Proposition <ref> it follows that 1/√(n)[1/2( d_s-𝔼_γ'[ d_s])^⊤Γ_n, s ( d_s-𝔼_γ'[ d_s])+( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] is bounded in probability. Hence, from (<ref>), 2logΛ_n, s-n/√(2n) = (ℓ_n, s(γ)-ℓ_n, s(β̂_s)) -n /√(2n)→∞ , in probability, since by Lemma <ref>, (γ'-γ)^⊤Σ̃_n, s (γ'-γ) ≍ n^s -1γ'-γ_2^2 ≫√(n). This implies, _γ'[ϕ_n, s] → 1, whenever (<ref>) holds. Next, we assume (γ'-γ)^⊤Σ_n, s(γ'-γ) ≫ n . For notational convenience denote ϑ_n, s :=(γ'-γ)^⊤Σ_n, s(γ'-γ). Then Proposition <ref> and (<ref>) imply that 1/√(ϑ_n, s)[1/2( d_s-𝔼_γ'[ d_s])^⊤Γ_n, s ( d_s-𝔼_γ'[ d_s])+( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] is bounded in probability. Using (<ref>) and (<ref>) we also get (γ'-γ)^⊤Σ̃_n, s(γ'-γ)/√(ϑ_n, s)≍ n^s-1/2γ' - γ_2 →∞. This implies, from from (<ref>), _γ' [ϕ_n, s] = ℙ_γ' (\|2logΛ_n, s - n/√(ϑ_n, s)\| ≥ z_α/2√(2n/ϑ_n, s)) → 1. This concludes the proof. This completes the proof of the third assertion in (<ref>). Now, we consider the case n^2s-3/4γ'-γ_2 →τ∈ (0, ∞). By Taylor expansion, ℓ_n, s(γ')-ℓ_n, s(γ)=( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) + 1/2(γ'-γ)^⊤Σ_n, s(γ'-γ) + _n, s, where Σ_n, s is as defined in (<ref>) with β_s replaced by γ and _n, s is as defined in (<ref>) with the parameter γ̃=γ'+θ(γ'-γ) for some 0<θ<1. By arguments as in Lemma <ref> _n, s = o_P(√(n)). Then (<ref>) and Lemma <ref>, Lemma <ref>, (<ref>) can be written as: ℓ_n, s(β̂_s)-ℓ_n, s(γ) = - 1/2( d_s-𝔼_γ'[ d_s])^⊤ Γ_n, s ( d_s-𝔼_γ'[ d_s]) + ( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) + 1/2(γ'-γ)^⊤Σ_n, s(γ'-γ) + o_P(√(n)) . Note that [( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] = 0 and by Lemma <ref>, [( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ)] = (γ'-γ)^⊤Σ_n, s(γ'-γ) ≍_n, r√(n) , when γ'-γ_2 ≍ n^-2s-3/4. Hence, in this case, ( d_s-𝔼_γ'[ d_s])^⊤(γ'-γ) = o_P(√(n)). This also implies that η := lim_n →∞(γ'-γ)^⊤Σ_n, s (γ'-γ)/√(n) exists along a subsequence. (Note that _γ[ d_s] = Σ_n, s.) Hence, from (<ref>), 2logΛ_n, s-n/√(2n) = 2(ℓ_n, s(γ)-ℓ_n, s(β̂_s)) -n /√(2n) = ( d_s-𝔼_γ'[ d_s])^⊤Γ_n, s( d_s-𝔼_γ'[ d_s]) -n /√(2 n) - (γ'-γ)^⊤Σ_n, s(γ'-γ)/√(2 n) + o_P(1) (-η√(2),1) . This completes the proof of (<ref>). § TESTING LOWER BOUNDS In this section we prove the lower bounds for the goodness-of-fit problem in the L_2 and L_∞ norms, that is, Theorem <ref> (b) and Theorem <ref> (b), respectively. For this, suppose π_n be a prior probability distribution on the alternative H_1 (as in (<ref>) or (<ref>)). Then the Bayes risk of a test function ψ_n is defined as (ψ_n, γ, π_n)= _H_0( ψ_n=1) + _γ' ∼π_n[_γ'( ψ_n=0) ]. For any prior π_n the worst-case risk of test function ψ_n, as defined in (<ref>), can be bounded below as: Let _n, s denote the collection of s-uniform hypergraphs on n vertices. Then (ψ_n, γ) ≥(ψ_n, γ, π_n) ≥ 1- 12√(_H_0[L_π_n^2]-1), where L_π_n=_γ' ∼π_n[_γ'(ω) ]/_H_0(ω), ω∈_n, s, is the π_n-integrated likelihood ratio. Clearly, (ψ_n, γ) ≥(ψ_n, γ, π_n). To show the second inequality in (<ref>) observe that, (ψ_n, γ, π_n) ≥inf_ψ_n{_H_0( ψ_n=1) + _γ' ∼π_n(_γ'(ψ_n = 0) ) } ≥ 1- sup_ψ_n\| _H_0( ψ_n=1) - _γ' ∼π_n(_γ'( ψ_n=1) ) \| ≥ 1- sup_ω∈_n, s\| _H_0(ω) - _γ' ∼π_n[_γ'(ω) ] \| ≥ 1- 1/2∑_ω∈_n, s\|_γ' ∼π_n[_γ'(ω) ]/_H_0(ω) -1 \| _H_0(ω) = 1-12_H_0\|L_π_n-1\| ≥ 1- 12√(_H_0[L_π_n^2] - 1), where the last step uses the Cauchy-Schwarz inequality. Therefore, to show all tests are powerless it suffices to construct a prior π_n on H_1 such that _H_0[L_π_n^2] → 1. We show this for the L_2 norm in Appendix <ref> and for the L_∞ norm in Appendix <ref>. §.§ Testing Lower Bound in L_2 Norm: Proof of Theorem <ref> (b) We choose γ = 0, ε≪ n^-2s-3/4, and construct a prior π_n on H_1 as in (<ref>) as follows: Suppose γ' = (γ'_1, γ'_2, …, γ'_n)^⊤∈^n with γ'_u = η_u ·ε/√(n), for u ∈ [n], where η_1, …, η_n are i.i.d Rademacher random variables, taking values {± 1 } with probability 1/2. Clearly, γ - γ' _2 = ε. Then, for H ∈_n, s, the π_n integrated likelihood ratio is given by L_π_n = _η[ _γ' (H)/_ 0 (H)] = _η[ ∏_ e ∈[n] s 2 e^w_η( e) X_ e/ 1 + e^w_η( e) ] , where X_ e := 1{ e ∈ E(H) }, η := (η_1, …, η_n), and w_η( e) := ε/√(n)∑_u ∈ eη_u, for e ∈[n] s. Then L_π_n^2 = _η, η'[ ∏_ e ∈[n] s 4 e^ ( w_η( e) + w_η'( e) ) X_ e/ ( 1 + e^w_η( e) ) ( 1 + e^w_η'( e) ) ] , where η_1', …, η_n' are i.i.d Rademacher random variables which are independent of η_1, …, η_n, η' := (η_1', …, η_n'), and w_η'( e) := ε/√(n)∑_u ∈ eη_u', for e ∈[n] s. Taking expectation with respect to H_0 gives, 𝔼_H_0[L^2_π_n] = _η, η'[ ∏_ e ∈[n] s2 (e^ ( w_η( e) + w_η'( e) ) + 1)/ ( 1 + e^w_η( e) ) ( 1 + e^w_η'( e) ) ] = 𝔼_η, η'[∏_ e ∈[n] s2 {ψ(w_η( e)) ψ(w_η'( e))+(1- ψ(w_η( e)))(1- ψ(w_η'( e)))}] , where ψ(x) is the logistic function as defined in Lemma <ref>. Using the Taylor expansions of ψ(x) and 1-ψ(x) around 0, we can show that for all x ∈ℝ, ψ(x) ≤1/2+x/4+x^3/48 and 1-ψ(x) ≤1/2-x/4+x^3/48. As a consequence, for e ∈[n] s, 2 {ψ(w_η( e)) ψ(w_η'( e))+(1- ψ(w_η( e)))(1- ψ(w_η'( e)))} ≤ 1+1/4w_η( e)w_η'( e)+1/24(w_η( e)^3 +w_η'( e)^3 )+ 1/24^2w_η( e)^3 w_η'( e)^3. Using this bound in (<ref>) gives, 𝔼_H_0[L^2_π_n] ≤_η, η'[ ∏_ e ∈[n] s( 1+1/4w_η( e)w_η'( e)+1/24(w_η( e)^3 +w_η'( e)^3 )+ 1/24^2w_η( e)^3 w_η'( e)^3 ) ] ≤_η, η'[ e^∑_ e ∈[n] s{1/4w_η( e)w_η'( e)+1/24(w_η( e)^3 +w_η'( e)^3 )+ 1/24^2w_η( e)^3 w_η'( e)^3 }] , since 1+x ≤ e^x. Recalling the definition of w_η( e) observe that \|∑_ e ∈[n] sw_η( e)^3 \| ≤ε^3/n^3/2∑_ e ∈[n] s(∑_u ∈ e \|η_u\| )^3 ≤ε^3 n^s- 3/2 . Hence, 𝔼[ e^ 2∑_ e ∈[n] sw_η( e)^3] ≤ e^2 ε^3 n^s- 3/2→ 1, since ε≪ n^-2s-3/4 and, for s ≥ 2, -s/2+3/4 > 0. Similarly, it can be shown that lim_n →∞𝔼[ e^ 2∑_ e ∈[n] sw_η( e)^3w_η'( e)^3 ] = 1. Then Hölder's inequality applied to (<ref>) followed by (<ref>) and (<ref>) gives 𝔼_H_0[L^2_π] ≤{𝔼_η,η'[ e^3/4∑_e ∈[n] sw_η( e)w_η'( e) ]}^1/3(1+o(1)) . Next, observe that ∑_ e ∈[n] sw_η( e)w_η'( e) = ε^2/n{∑_ e ∈[n] s(∑_u ∈ eη_u)(∑_v ∈ eη'_v)} = ε^2/n{n-1 s-1∑_u=1^nη_uη'_u + n-2 s-2∑_1 ≤ u ≠ v ≤ nη_uη'_v} ≤ε^2 n^s-2∑_u=1^nη_uη'_u + ε^2 n^s-3∑_1 ≤ u ≠ v ≤ nη_uη'_v = ε^2 n^s-2∑_u=1^nη_uη'_u + ε^2 n^s-3{(∑_u=1^nη_u)(∑_v=1^nη'_v) - ∑_u=1^nη_uη'_u } . Note that ε^2 n^s-3\| ∑_u=1^nη_uη'_u \| ≤ε^2 n^s-2. Hence, [ e^9/4ε^2 n^s-3∑_u=1^nη_uη'_u ] ≲ e^ε^2 n^s-2→ 1, since ε≪ n^-2s-3/4. From (<ref>), by Hölder's inequality followed by (<ref>) and (<ref>) gives 𝔼_H_0[L^2_π] ≲_s{𝔼_η,η'[e^9/4ε^2 n^s-2∑_u=1^nη_uη'_u ] }^1/9{𝔼_η, η'[ e^9/4ε^2 n^s-3(∑_u=1^nη_u) (∑_v=1^nη'_v) ] }^1/9(1+o(1)) . Denote X_n:= ∑_u=1^nη_u and Y_n:=∑_v=1^nη'_u. Since X_n and Y_n are independent and each of them is a sum of i.i.d. Rademacher random variables, 𝔼_η,η'[ e^9/4ε^2 n^s-3 X_nY_n] = [ 𝔼[ e^9/4ε^2 n^s-3 X_nY_n \| Y_n ] ] = 𝔼[ ( cosh( 9/4ε^2 n^s-3 Y_n) )^n ] ≤𝔼[ e^81/16ε^4 n^2s-5 Y_n^2 ] , where last step uses cosh(x) ≤ e^x^2, for all x ∈ℝ. Since \|Y_n\| ≤ n, this implies, 𝔼_η,η'[ e^9/4ε^2 n^s-3 X_nY_n] ≤ e^81/16ε^4 n^2s-5 Y_n^2 ≤ e^81/16ε^4 n^2s-3→ 1, since ε≪ n^-2s-3/4. Next, observe that η_uη'_u, for u=1,⋯,n, are i.i.d. Rademacher random variables. Again using cosh(x) ≤ e^x^2 for all x ∈ℝ, we can show that 𝔼_η,η'[ e^9/4ε^2 n^s-2∑_u=1^nη_uη'_u ] = ( cosh( 9/4ε^2 n^s-2) ) ^n ≤ e^81/16ε^4 n^2s-3→ 1 , since ε≪ n^-2s-3/4. Hence, using (<ref>) and (<ref>) in (<ref>) gives, lim_n →∞E_H_0[L^2_π] =1 . By Lemma <ref>, this completes the proof of Theorem <ref> (b). §.§ Testing Lower Bound in L_∞ Norm: Proof of Theorem <ref> (b) We choose γ = 0, ε≪ n^-s-1/2 and define γ' = (γ'_1, γ'_2, …, γ'_n)^⊤∈^n, where γ'_1 = ε and γ'_u = 0, for u ≥ 2. Clearly, γ - γ' _∞ = ε. Then, for H ∈_n, s, the likelihood ratio is given by L_n = _γ' (H)/_ 0 (H) = ∏_ e ∈[n] s : 1 ∈ e 2 e^ε X_ e/ 1 + e^ε , where X_ e := 1{ e ∈ E(H) }. Observe that 𝔼_H_0[L_n^2] = 𝔼_H_0[ ∏_ e ∈[n] s : 1 ∈ e 4 e^ 2 ε X_ e/ ( 1 + e^ε )^2 ] = ( 2 ψ(ε)^2 + 2 (1- ψ(ε))^2 )^n s-1 , where ψ(x)= e^x/1+e^x. Since ε≪ n^-s-1/2, a Taylor expansion around zero gives ψ(ε) = 1/2+1/4ε + O(ε^2). Hence, 2 ψ(ε)^2 + 2 (1-ψ(ε))^2 = 1 + O(ε^2) . Therefore, by (<ref>) and using 1+x ≤ e^x gives, 𝔼_H_0[L_n^2] ≤ e^O(ε^2 n^s-1)→ 1, since ε≪ n^-s-1/2. By Lemma <ref>, this completes the proof of Theorem <ref> (b). § PROOF OF PROPOSITION <REF> Define g = (g_1, g_2, …, g_n): ℝ^n→ℝ^n where g_u: ℝ^n→ℝ, for u ∈ [n], as follows: g_u( x) = ∑_ e ∈[n] s : u ∈ e e^ x^⊤_e 1/1+e^ x^⊤_e 1 , where x = (x_1, x_2, …, x_n)^⊤ and x_ e = (x_u_1, x_u_2, …, x_u_s)^⊤ for e = (u_1, u_2, …, u_s). Observe that ℛ_s is the range of g. Since the expected degree of a vertex is a weighted combination of all the possible degrees in s-uniform hypergraphs on n vertices, this implies ℛ̅_s ⊆conv (𝒟_s). To show the other side, for every y ∈ℝ^n we define, f_ y( x)=∑_i=1^nx_iy_i-∑_{v_1, v_2, …, v_s }∈[n] slog(1+e^x_v_1+…+x_v_s). Since the probability of observing an s-uniform hypergraph with parameter x and s-degree sequence d_s=(d_s(1), …, d_s(n)) is e^∑_v=1^n d_s(v) x_v/∏_{v_1, v_2, …, v_s }∈[n] s(1+e^x_v_1+…+x_v_s). and is less than 1, taking logarithm on both sides we get f_ d_s( x) ≤ 0. Further as f_ y( x) depends linearly on y, we have f_ y( x) ≤ 0 for all y ∈conv (𝒟_s) and x ∈ℝ^n. Now, let us fix y ∈conv (𝒟_s). It can be shown that the Hessian ∇^2f_ y( x) is uniformly bounded, hence, by <cit.> there exists a sequence { x_k}_k ≥ 1 such that ∇ f_ y( x_k) → 0. Observing that ∇ f_ y( x_k)= y-g( x), we get g( x_k) → y. As y∈conv (𝒟_s) is arbitrary, this implies conv (𝒟_s)⊆ℛ̅_s.
http://arxiv.org/abs/2307.01747v1	20230704143206	Observation of solar radio burst events from Mars orbit with the Shallow Radar instrument	[ "Christopher Gerekos", "Gregor Steinbrügge", "Immanuel Jebaraj", "Andreas Casillas", "Elena Donini", "Beatriz Sánchez-Cano", "Mark Lester", "Jasmina Magdalenić", "Sean Peters", "Andrew Romero-Wolf", "Donald Blankenship" ]	astro-ph.SR	[ "astro-ph.SR", "astro-ph.IM", "physics.space-ph" ]	Christopher Gerekos [email protected] 0000-0003-1526-1249]Christopher Gerekos University of Texas at Austin Institute for Geophysics, J.J. Pickle Research Campus, 10100 Burnet Road, 78758 Austin, Texas, USA 0000-0002-1050-7759]Gregor Steinbrügge Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, CA 91011, USA 0000-0002-0606-7172]Immanuel Christopher Jebaraj Space Research Laboratory, University of Turku, Turku, Finland Naval Postgraduate School, Monterey, CA, United States Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo, Trento, Italy 0000-0003-0277-3253]Beatriz Sánchez-Cano School of Physics and Astronomy, University of Leicester, University Rd, Leicester LE1 7RH, UK 0000-0001-7353-5549]Mark Lester School of Physics and Astronomy, University of Leicester, University Rd, Leicester LE1 7RH, UK 0000-0003-1169-3722]Jasmina Magdalenić Center for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium, Avenue Circulaire 3, 1180 Uccle, Belgium 0000-0003-2527-8271]Sean T. Peters Naval Postgraduate School, Monterey, CA, United States 0000-0002-4992-4162]Andrew Romero-Wolf Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, CA 91011, USA University of Texas at Austin Institute for Geophysics, J.J. Pickle Research Campus, 10100 Burnet Road, 78758 Austin, Texas, USA Multispacecraft and multiwavelength observations of solar eruptions such as flares and coronal mass ejections are essential to understand the complex processes behind these events. The study of solar burst events in the radio-frequency spectrum has relied almost exclusively on data from ground-based radiotelescopes and dedicated heliophysics missions such as STEREO or Wind. Reanalysing existing data from the Mars Reconnaissance Orbiter (MRO) Shallow Radar (SHARAD) instrument, a Martian planetary radar sounder, we have detected 38 solar radio burst events with a correlated observation by at least one dedicated solar mission. The very high resolution of the instrument, both in temporal and frequency directions, its bandwidth, and its position in the solar system enable SHARAD to make significant contributions to heliophysics; it could inform on plasma processes on the site of the burst generation and also along the propagation path of associated fast electron beams. In this letter, we characterise the sensitivity of the instrument to type-III solar radio bursts through a statistical analysis of correlated observations, using STEREO and Wind as references. We establish the conditions under which SHARAD can observe solar bursts in terms of acquisition geometry, laying the foundation for its use as a solar radio-observatory. We also present the first analysis of type-III characteristic times at high resolution beyond 1 AU. The scaling laws are also comparable to results found on Earth, except for the fall time; a clearer distinction between fundamental and harmonic components of the bursts may be needed to resolve the discrepancy. § INTRODUCTION The Sun is capable of routinely accelerating electrons to suprathermal energies through eruptive phenomena such as flares, coronal mass ejections (CMEs), and the subsequent shock waves that they drive. These suprathermal electrons emit radiation in the entire electromagnetic spectrum and particularly in the longer radio wavelengths through various mechanisms. Based on the source (streaming electrons, shock waves, etc.) the emission can manifest itself with different morphological properties on the dynamic radio spectrogram. Early observations have distinguished five main spectral types <cit.>, and further sub-classifications have been made since. In the past two decades, radio frequency observatories onboard heliophysics missions such as the Solar TErrestrial RElations Observatory Ahead & Behind <cit.>, and Wind/WAVES <cit.> have been used to understand various aspects of interplanetary radio emissions. Recently, they have been combined with the Radio Frequency Spectrometer <cit.> onboard the Parker Solar Probe <cit.> and the radio and plasma waves <cit.> instrument onboard Solar Orbiter <cit.> for multi-vantage point studies of hecto-kilometric (H-K) radio emissions <cit.>. While they do not provide the same time-frequency resolution as ground-based instrumentation, observing radio emissions simultaneously from multiple vantage points presents the possibility to investigate the generation and propagation of radio waves <cit.>. Previous multi-vantage point observations of solar burst events made beyond 1 AU include <cit.> and <cit.>. Planetary radar sounders are a class of spacecraft-mounted remote sensing instruments that operate by recording reflections of electromagnetic waves off a solid planetary body. Such reflections arise when an incoming electromagnetic field encounters a change in the dielectric constant of the medium, such as the space-surface interface, subsurface layering, or subsurface inclusions. The source of this incoming field can be the instrument itself, in which case the radar will transmit a coded waveform (usually a linear chirp) with a power of a few Watts. This mode of operation is known as active sounding <cit.>. Conversely, the incoming field can be a signal of opportunity of astrophysical origin, a recently-proposed mode of operation known as passive sounding <cit.>. Radar sounders typically operate in the deca-hectometre (D-H) wavelength bands. A particularly successful planetary radar sounder has been the Shallow Radar (SHARAD) instrument <cit.> onboard the National Aeronautics and Space Administration (NASA) Mars Reconnaissance Orbiter (MRO) mission, which was launched towards Mars in 2005 and began observations a year later. SHARAD is sensitive in the 13.3 to 26.7 MHz band, has a time resolution of 1.43 ms before pre-summing, and a frequency resolution of 7.41 kHz <cit.>. The radiating element of SHARAD is a thin-wire 10 m long dipole antenna <cit.>. When operating SHARAD, the spacecraft is nominally oriented so that its -Z direction points towards the Martian surface, although some variations in the roll angle have been considered to accommodate unforeseen lobe distribution of the SHARAD antenna pattern <cit.>. Amongst other discoveries enabled by SHARAD, <cit.> characterised the ice purity in the layers of the Martian polar caps, <cit.> found evidence of buried glaciers at mid-latitudes, and <cit.> constrained the roughness and near-surface density of the Martian surface. In this letter we show how SHARAD can be used to observe solar radio bursts events at unprecedented resolution in the D-H band from space, also marking the first attempt to use a planetary radar sounder as a solar radio-observatory. To this end we devised a purpose-built algorithm for detection of SHARAD-detectable type III bursts in the STEREO/WAVES and Wind/WAVES datasets. We propagated these events in time from the orbits of these spacecraft to the orbit of Mars, and looked up for any instance when the SHARAD receiver was on and when the Sun was not occulted by Mars as seen from MRO. This search for SHARAD type III burst-containing candidates yielded 179 distinct SHARAD radargrams, of which 38 contain solar burst. Through a multivariate statistical analysis of the candidates, we quantify the importance of the acquisition geometry for burst detection. As, by construction, all our SHARAD-detected bursts have at least one correlated observation, we also present comparisons of a few selected bursts observed by SHARAD and the source solar spacecraft, and provide a succinct commentary on their features. Furthermore, we analyse the characteristic times of our dataset of type-III bursts following a methodology close to that of <cit.> and compare the frequency-dependence scaling laws we obtain with that obtained on Earth and with PSP. We note that <cit.> mentions having seen solar radio bursts in data from the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS), another Martian radar sounder <cit.>, but these were treated as parasitic signals, and no attempt was made to study solar radio burst with MARSIS. This letter is structured as follows. Section <ref> presents our methodology for finding correlated observation opportunities and building the proposed list of SHARAD candidates. Section <ref> summarises our observations. Section <ref> contains in-depth analyses of a few representative SHARAD spectrograms as well as an analysis of the frequency-dependence of the rise and fall time of the type-III bursts in our dataset. Section <ref> concludes this letter with a discussion of the proposed dataset and of its perspectives. Appendix <ref> presents a SHARAD duty cycle analysis, Appendix <ref> presents in detail the statistical sensitivity analysis of SHARAD with respect to the geometry of the observations, and Appendix <ref> contains the methodological details of the time-profiles analysis. § METHODS AND DATASETS In this section we present the methodology we used to detect solar radio bursts in STEREO and Wind data and to search for corresponding SHARAD candidates, focusing on type III bursts. The critical parts of the code have been made available along with an accompanying flowchart as Supplementary Material 1 <cit.>. §.§ Solar radio bursts detection from STEREO and Wind We used the the 60 seconds-averaged WAVES products from both STEREO and Wind, covering a period starting on 6 December 2006 (first SHARAD acquisition) and ending on 31 December 2021. An initial screening for bursts is made by integrating the spectrograms over their entire range of frequencies, yielding a one-dimensional time-series of power flux, and by detecting prominences that are one standard deviations higher than the background through the MATLAB proprietary function . This is done for each 24h-spanning data product. A 3-step fuzzy logic algorithm is then applied on these peaks in order to reject prominences due to interferences or instrument malfunction and to reject genuine type III bursts that do not have significant energy above 13 MHz, about the lowest frequency available to SHARAD. To this end, the algorithm tests peaks flatness, peak asymmetry, and high-frequency content. For more details please refer to <cit.>. Given the large number of type III bursts that these missions recorded, and given the nature of our study, the algorithm is rather restrictive as false positives are more dangerous than false negatives. With this method we detected 5676 bursts with STEREO A, 4340 bursts with STEREO B, and 3538 events with Wind, summing up to a grand total of 13554 events. §.§ SHARAD candidate selection Each of those events has an associated timestamp t that corresponds to the time of detection of peak power. We propagate this time of detection at the source (STEREO A, B, or Wind) to Martian orbit by considering the radial distance separating the solar spacecraft and Mars at that time. This rests on the hypothesis that the isocontours of burst detection time form circles around the Sun. This radial distance difference is converted into a delay τ assuming propagation at the speed of light in vacuum. In other words, if t' is the supposed time of detection at Mars, we can write t' = t + τ = t+ c^-1(\| 𝐯_M - 𝐯_O \| - \| 𝐯_w - 𝐯_O \|) = t+r_M - r_w/c, where 𝐯_w, 𝐯_O and 𝐯_M represent the position of the detection source (STEREO A, B, or Wind), the position of the Sun, and the position of Mars, respectively, and where r represents a radial distance from the Sun. These positions can be extracted from the appropriate SPICE kernels. The corresponding Mars detection time t' is computed for all of the 13554 events detected in the previous step. Then, a SHARAD data product is considered a candidate if it satisfies the two following conditions: (i) SHARAD is operating during t'; and (ii) the Sun is not occulted by Mars at t' as seen from MRO. The first criterion is verified by testing that the start and stop time t_1 and t_2 of a given SHARAD data product is such that t_1 ≤ t' ≤ t_2, where t_1 and t_2 are found in the SHARAD product metadata. The second criterion can be easily tested using a SPICE query. After application of these criteria, we are left with 226 SHARAD detections comprising 179 distinct SHARAD data products (a given event recorded by different solar observatories may correspond to the same SHARAD product). This very strong reduction of the numbers of events, compared with that of Section <ref> is explained by the very low duty cycle of SHARAD activity (see Appendix <ref>) and by the fact a majority of radar sounding observations were done on the Martian nightside. A complete list of these SHARAD candidates is given as Supplementary Material 2 <cit.>. §.§ Spectrogram and time-profile generation The SHARAD Experimental Data Products (EDR) are real-valued baseband waveforms containing 3600 samples per rangeline <cit.>. A rangeline is a 1D fast-time collection of samples that forms the basic acquisition unit of a radargram. On SHARAD, rangelines are acquired at a frequency of 700 Hz, and have a duration of 135 μs each. The SHARAD dynamic spectrograms shown in this letter are the absolute value of the fast Fourier transform of the EDR product in dB scale. Only the positive-frequency components are displayed (1800 samples). In addition to possible solar radio bursts, these opportunistically-acquired SHARAD spectrograms contain a number of signals of non-solar origin: electromagnetic interference (EMI) and reflections of the active chirp. These are highlighted in Figure <ref> for product 2533401, which contains a type III burst. The corresponding processed radargram is also shown for context. It must be noted that SHARAD has not been calibrated on the ground, meaning there is no formally-generated data products in physical units of spectral flux <cit.>. The gain per frequency band across the whole bandwidth (13.3 to 26.7 MHz) has also not been completely characterised, although it was optimised to be spectrally-flat in the 15 to 25 MHz range <cit.>. Calibration of SHARAD a posteriori is an ongoing effort <cit.>. No gain de-trending or active chirp removal algorithms have been applied to the spectrograms shown in this work, as our exploitation of the data in this letter remains exploratory. Instead, a basic de-noising using a Gaussian kernel of 3 pixels is applied globally for image rendition concerns. The time-profiles represent a 100-pixel moving average of the raw spectra. § OBSERVATIONS The 179 SHARAD candidates obtained from our search algorithm were visually examined for the presence of solar bursts. Of these candidates, 38 products contain a clear, scientifically-exploitable burst. The list of all candidates, including those without a burst, has been made available as Supplementary Material 2 <cit.> to allow for possible future reanalyses that would look for more subtle signatures, and the subset of SHARAD candidates which contain an unambiguous solar burst is summarised in Table <ref>. A sensitivity analysis of SHARAD in terms of the relative position and orientation of the instrument, is carried out in Appendix <ref>. We find that, despite the high complexity of the SHARAD antenna pattern, we can predict reasonably well whether a burst detected by STEREO or Wind will be detected by SHARAD: a simple predictive model based on the surveyed acquisition parameters can predict burst show versus no-show with an accuracy of 79.2% (see Figure <ref>b). A preview of each of these SHARAD spectrograms is given in Supplementary Material 3 <cit.>. Several of these bursts display extensive fine structure (see also Figure <ref>c), and some bursts have a peculiar dynamic spectra unlike the typical type III profile. Interestingly, a type-II event (SHARAD product 3544101) has also been picked up by our algorithm. The temporal and spatial distributions of the 226 SHARAD candidate events, with and without burst, is shown in Figure <ref>. The number of SHARAD candidates throughout the surveyed years closely follows the solar cycle, and the fraction of these candidates that were confirmed to contain a burst stays at around 25% for every year. The additional variability in the number of candidates can be explained by the variation in the yearly SHARAD duty cycle, as shown in Appendix <ref>. This consistency suggests that the geometric selection criteria described in Section <ref> do not introduce any particular bias. When considering the spatial spread of the observations, it can be observed that the sources of the observations (STEREO A, B, and Wind) are rather evenly distributed whereas the corresponding SHARAD candidates are more concentrated on the negative X-side of the solar system in the J2000 system of coordinates. However, when considering the SHARAD candidates containing a burst, then the distribution of the sources shows a similar unbalance. The first feature is simply explained by the fact a great proportion of bursts were recorded in 2014 and that Mars was in that particular sector during that year, whereas the second observation hints at the angular difference dependence for detectability (see Appendix <ref>): the dedicated solar missions may have been at any place in the solar system during the peak years, but it is those angularly closest to Mars that led to better chances of a SHARAD observation. § DISCUSSION It is well understood that the morphology of the intensity-time profiles of a type III radio burst correspond to the growth of the instability <cit.> and the propagation of radio waves <cit.>. The emission, which is a combination of all these processes, is systematically asymmetric in intensity-time profiles <cit.>. Rapid variations in the peak intensity across the observing frequencies is not uncommon in decametre type III bursts <cit.>, and are to be expected when the electron beam evolves in an inhomogeneous medium. These variations may then be used to probe the plasma through which the beam propagates, assuming that the emission is close to the electron plasma frequency. Figure <ref>a illustrates a typical type III burst observed on 22 December, 2011. STEREO A/WAVES also detected this burst, albeit at a lower resolution. Figure <ref>b displays a group of type III bursts observed on 01 January, 2012. Due to limited resolution in the H-K wavelengths, different type III bursts within one are difficult to deconvolve. SHARAD, with its superior capabilities at lower frequencies, reveals nine bursts where STEREO A/WAVES only detects two. Figure <ref>c presents a type III burst with fine structures called striae elements. These bursts can emerge due to electron beam evolution in inhomogeneous plasma, where stronger emissions occur in regions with smaller density inhomogeneities <cit.>. Pulse broadening is mainly due to velocity dispersion of the electron beam exciter <cit.>, but the Langmuir wave growth can also be hindered in inhomogeneous plasma, resulting in less intense, longer-duration bursts <cit.>. In this example, SHARAD detects three distinct type III bursts, while STEREO A/WAVES observes a single burst without discernible fine structures, underscoring the significance of high time and frequency resolution. While it is possible to distinguish fine structures and intensity variations using STEREO and Wind observations, their frequency resolution of >4% would mean that only large scale inhomogeneities can be probed <cit.>. On the other hand, SHARAD provides the resolution capable of resolving the fine-scale intensity variations which may then be used in tandem with modern missions such as the Parker Solar Probe during its close encounters. However, a complete characterisation of the spectral gain of SHARAD as well as its sources of EMI is likely to be needed for full exploitation of its spectral resolution. In this sense, the limitations of SHARAD for heliophysics are of two kinds. The first type of limitation are the instrument-specific limitations that we have just mentioned. Empirically, uneven spectral gain can be addressed by a de-trending of the time-average spectral power for each burst. Regarding EMI, notch filtering has been applied successfully to SHARAD products to enhance the quality of range-compressed observations <cit.>. While these methods are promising for heliophysics-oriented analysis of SHARAD data, one must ensure they do not compromise the natural signals of interest. For this reason, we devolve the study of fine structure within the very high-resolution SHARAD dynamic spectra of type-III bursts to a future study. The second type of limitation arises from using SHARAD opportunistically: the limited number of bursts that we observe in our dataset can be traced to the low duty cycle of SHARAD (see Appendix <ref>), the preference for night-time observations for radar sounding of Mars <cit.>, and the nadir-pointing geometry favoured for radar sounding of Mars (see Appendix <ref>). §.§ Time-profile characteristics The very high temporal resolution of SHARAD has the consequence that the fast Gaussian-like rise and the slower exponential decay are effortless to identify, as evidenced by the examples shown in Figure <ref>. For this reason we propose to study the statistics of the frequency-dependence of the rise time t_r and fall time t_f, computed by fitting an exponentially-modified Gaussian function <cit.> and by computing the half-widths at half-maximum on either side of the peak intensity (see Appendix <ref> for more details). <cit.> conducted a comparable study with 31 type-III bursts recorded by LOFAR <cit.> in the 30 to 80 MHz range. The characteristic times of type-III bursts in the frequency range of 15 to 25 MHz, which provides a natural continuation to <cit.>, has not been explored beyond 1 AU at the resolutions we have access to here, and constitute a novel result. We ran our analysis by considering type-III bursts of Table <ref> excluding all burst with peculiar morphologies or non-type-III's. In total, 26 type III bursts were analysed. The results for t_r(f), t_f(f), and the derived quantities t_f(t_r) and (t_f/t_r)(f) can be seen in Figure <ref>, and the best fit power laws for each plot are given in equations <ref> through <ref>, respectively. t_r = (44^+19_-13) ( f [MHz]/1 [MHz])^-0.87 ± 0.12 t_f = (26^+11_-7) ( f [MHz]/1 [MHz])^-0.49 ± 0.12 t_f = (2.39^+0.06_-0.05) ( t_r [s]/1 [s])^0.78 ± 0.01 t_f/t_r = (0.61^+0.14_-0.11) ( f [MHz]/1 [MHz])^0.37 ±0.07 A first observation to make is that most bursts taken individually seem to follow a very similar trend as the general fitted law, meaning equations <ref> through <ref> are representative, and not a product of pure group behaviour. The rise time exponent -0.87 ± 0.12 is very close to that of <cit.>, who found -0.77 ± 0.14. Interestingly, SHARAD data for the fall time indicates an exponent of -0.49 ± 0.12 and differs rather substantially from the LOFAR measurement of -0.89 ± 0.15. The rise time is closely linked to the exciter function, i.e. the growth of Langmuir waves and inherits its characteristics <cit.>, and it is thus expected to behave similarly at different points of the solar system for a given frequency. Stricto sensu, it is the decay time, measured as the rate of the exponential decay of the burst, that best encapsulates the effects of propagation and scattering <cit.>, rather than the fall time computed here as the second half-width at half-maximum. Nevertheless, in the case of fundamental emissions, the fall time is not completely immune to propagation effects, and can be thought as a convolution between the characteristics of the source and the electromagnetic wave propagation effects <cit.> ; it is therefore not excluded that the discrepancy we observe for the fall time at 1.5 AU is partly due to such effects. On the other hand, it is widely understood that these effects are minimal for higher harmonics <cit.>. Interestingly, the fall time frequency-dependence matches very well with that derived from PSP observation, but only for the harmonic component of the bursts <cit.>. We did not differentiate between fundamental and harmonic components in this study, so we cannot exclude some kind of selection bias towards the former, either. In this case, our results would indicate very little propagation and scattering effects up to 1.5 AU, but the rise time we obtained is substantially different from that of <cit.> for harmonic emissions (-0.46±0.08). A thorough investigation of the fundamental vs. harmonic content of our SHARAD-detected burst dataset as well as the derivation of their exponential decay scaling law is needed to fully elucidate the issue. We defer this to a future study. Computing the fall time as a function of the rise time, we find that the former is always longer than the latter, as expected, outliers notwithstanding. The ratio of the two measures the burst asymmetry as a function of frequency, where we observe a weak positive dependence. With a ratio of 1.1 at 25 MHz, our results are almost a direct low-frequency continuation of those of <cit.>. § CONCLUSIONS AND OUTLOOK In this letter, we demonstrated that planetary radar sounders such as SHARAD can be used as high-resolution solar radio-observatories, we presented a quantitative analysis of the sensitivity of SHARAD to solar burst events based on the statistics of correlated observations from a geometric standpoint, and extracted the characteristic times of type-III bursts from this dataset. The characteristic times analysis at 1.5 AU revealed a behaviour that is overall consistent with Earth-acquired data, but the frequency-dependence of fall time depicted a more complex picture. We found that it scales flatter compared to the recent metric observations by <cit.> and similar to the scaling laws obtained by <cit.> for second harmonic components. These scaling laws suggest that the evolution of the beam-plasma system in the decametre wavelengths plays a crucial role: if our dataset contains mostly fundamentals, the deviation from <cit.> would imply that propagation effects do not influence the fall time, which is then primarily determined by the characteristics of the Langmuir wave spectrum. To draw a more general conclusion, discriminating between fundamental and harmonic emissions is crucial, as it is generally expected that the second harmonic is not significantly affected by propagation effects, and bursts characteristics at 1 and 1.5 AU remain consistent with observations near the Sun. Our findings show that SHARAD observations can bring the large advancements in the understanding of the generation and propagation of the radio bursts at the distances even as far as the Mars orbit. In addition to studying characteristic times at 1.5 AU, the capabilities of SHARAD could be leveraged to study fine-structure, provide additional points of observation in burst triangulation, and its bandwidth completes the spectrum between typical dedicated solar missions and earth radiotelescopes. From the perspective of radar science, correlated observations with STEREO can also help the absolute calibration of SHARAD. Since all bursts are identifiable one-to-one between the two datasets, comparison between the absolute power flux recorded at STEREO with the dBs above background recorded at SHARAD allows its calibration. It must be noted that the catalogue of type III bursts we present is limited to correlated observations only. There are likely more bursts within the SHARAD archive that could be found through direct inspection. For example, a survey of the MARSIS dataset (in which solar radio bursts have also been seen) is planned amongst future work in order to produce a complete catalogue of MARSIS solar radio burst observations, which will be compared with SHARAD to get a fuller frequency spectrum of the co-observed bursts. The main limitation of using radar sounder data of the Sun is the fact they typically operate no more than a few minutes or tens of minutes a day, and cannot function as a round-the-clock survey instrument. The results presented here can also be used for future planetary missions. In this decade, two missions carrying radar sounder instruments will be launched towards the Jupiter system <cit.> as well as one towards Venus <cit.>, and these instruments too could be utilised for solar radio-frequency observations. § ACKNOWLEDGEMENTS This work was supported by the G. Unger Vetlesen Foundation and by JPL’s Innovative Spontaneous Concepts in Research and Technology Development program. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. I.C.J. is grateful for support by the Academy of Finland (SHOCKSEE, grant No. 346902). B.S.-C. acknowledges support through STFC Ernest Rutherford Fellowship ST/V004115/1. M.L. acknowledges support through STFC grant ST/W00089X/1. J.M. acknowledges funding by the BRAIN-be project SWiM (Solar Wind Modeling with EUHFORIA for the new heliospheric missions). The authors wish to express their gratitude to Bruce Campbell, Dirk Plettemeier, Marco Mastrogiuseppe, Vratislav Krupar, Marc Pulupa, Vladimir Krasnoselskikh, and Milan Maksimovic for the excellent discussions. § SHARAD DUTY CYCLE ANALYSIS To provide additional context on the discussion of the general properties of the dataset (Section <ref>) and to highlight the targetted nature of the instrument (as opposed to a survey instrument such as WAVES), we present a brief analysis of the activity duty cycle of SHARAD over the years. Starting from the SHARAD EDR label files, we extracted the radargram start time and stop time for each radargram in the entire SHARAD dataset in the period of study (6 December 2006 to 31 December 2021) in order to compute the duration of that radargram. Taking the cumulative sum of all these durations for each year, we obtain the durations shown in Table <ref>. On an average year, SHARAD will have been on for about a million seconds, corresponding to a duty cycle of about 3%. Aside from the solar cycle, part of the variability of the number of candidates counted in Figure <ref>a can be traced to the fraction of the time SHARAD was operating (for instance, the dip in candidates in 2013). § SHARAD SENSITIVITY TO TYPE III BURSTS Of the 179 SHARAD candidates picked up by the algorithm described in Section <ref>, 38 contain a prominent burst, a fraction of 21.23%, and 141 were identified as not containing a burst. In this appendix we shall discuss these figures and quantify the effects of several geometrical factors on this cross-detection rate. It would be difficult to compute an overall probability of detection of solar bursts with SHARAD, given that SHARAD-detectable bursts must obey rather stringent conditions of frequency content, but as the vast majority of SHARAD data is uncorrupted, it can be assumed to be very low. In that regard, a “success rate" of 21.23% is a comparatively high figure. One of the most important parameters that influences the detection of solar radio bursts with SHARAD should be the absolute power of the burst as detected by the source of observations (STEREO A, B, or Wind). However, it is not a parameter we can uniformly analyse, Wind/WAVES data has not yet been formally calibrated and issues of cross-calibration of STEREO/WAVES and Wind/WAVES are still being debated (Krupar, pers. comm., 2022). Parameters of geometrical nature, however, can be analysed uniformly. The non-isotropic nature of type III bursts <cit.> is expected to lead to a preferential detection if the angle between the source spacecraft and MRO with respect to the Sun (marked as ϕ in Figure <ref>a) is small. It is also expected that the orientation of MRO plays a major role in the detectability of bursts due to the antenna pattern of SHARAD <cit.>. In processed radar sounding data, variations of gain of up to 4 dB have been observed by <cit.>. The position of the high-gain antenna (HGA) and the orientation of the solar panels of MRO are also known to considerably affect the antenna pattern of SHARAD <cit.>. Due to the complexity and interdependence of these effects, however, we chose to focus on the orientation of the antenna only. Since the orientation towards the Sun is the most relevant parameter, we shall define pitch, yaw, and roll angles with respect to that direction as follows. We first define a set of three reference unit vectors: 𝐯̂_1 = 𝐯_O - 𝐯_MRO/\|𝐯_O - 𝐯_MRO\|, 𝐯̂_2 = 𝐯̂_z,ECLIPJ2000, 𝐯̂_3 = 𝐯̂_1 ×𝐯̂_2/\| 𝐯̂_1 ×𝐯̂_2 \|, where 𝐯_O = (0,0,0) is the position of the Sun in the J2000 system of coordinates, and 𝐯_MRO that of MRO. By construction, 𝐯̂_1 always points from MRO towards the Sun. 𝐯̂_2 is the normal to the ecliptic plane, and 𝐯̂_3 completes the orthonormal basis. With these vectors, we define the following projections of the û_j,MRO, (j=x,y,z) vectors, which form the basis of MRO-fixed orthonormal unit vectors <cit.>: 𝐰_x = û_x,MRO - (û_x,MRO·𝐯̂_3)𝐯̂_3, 𝐰_x' = û_x,MRO - (û_x,MRO·𝐯̂_2)𝐯̂_2, 𝐰_z = û_z,MRO - (û_z,MRO·𝐯̂_3)𝐯̂_3. In Figure <ref>a, the vectors 𝐯̂_i, (i=1,2,3) and û_j,MRO, (j=x,y,z) are represented in purple and orange, respectively. 𝐰_x is the projection of û_x,MRO onto the (M,𝐯̂_1, 𝐯̂_2) plane, 𝐰_x' is the projection of û_x,MRO onto the (M,𝐯̂_1, 𝐯̂_3) plane, and 𝐰_z is the projection of û_z,MRO onto the (M,𝐯̂_1, 𝐯̂_2) plane. These constructions allow us to define the desired Sun-based pitch, yaw, and roll angles as follows: Yaw = sgn(ŵ_x'·𝐯̂_3)arccos(ŵ_x' ·𝐯̂_1) Pitch = sgn(ŵ_x·𝐯̂_2)arccos(ŵ_x ·𝐯̂_1) Roll = sgn(ŵ_z·𝐯̂_2)arccos(ŵ_z ·𝐯̂_1) where sgn(·) is the sign function. A 0^∘ value for both yaw and pitch has the antenna axis pointed towards the Sun (i.e. zero gain for an ideal dipole). The roll angle controls the position of SHARAD with respect to the Sun and the MRO bus (the gain of an ideal dipole is agnostic to roll). We have made two types of statistical analysis based on these four angles (i.e., source separation angle, MRO pitch, MRO yaw, MRO roll): (i) a principal component analysis (PCA), and (ii) histograms and probability of detection for each angle independently. The results can be seen in Figure <ref>b and <ref>c-f, respectively. Regarding the components of the PCA (which are orthogonalised linear combinations of the four angles), the first three contained 96.45% of all variability in the original data, the first two 73.41%, and the first component only 47.13% of all variability. The correlations between any pair of angles is weak (<0.2) except for yaw and roll, which are strongly anti-correlated (-0.8), a result of the attitude control on MRO, itself on a polar orbit <cit.>. We also performed a multivariate logistic regression using these PCA feature vectors to classify the bursts as “Yes” or “No”, based on a selectable threshold to convert the continuous output of the predictive model in the [0,1] range to a “Yes”/“No” binary. This predictive model has a classification accuracy peaking at 0.792 for a threshold of 0.49 (a very sensible threshold for the purpose) using all four feature vectors. From the J2000 separation angle histogram, it is clear that detection is preferentially successful at low angles, with a probability of cross-detection reaching 0.7 when the angle between MRO and the source of observation is around 20^∘. Regarding orientation, the probability of detection peaks when pitch and yaw are both close to 90^∘. These angles put the radar axis perpendicular to the Sun (the former, horizontally, and the latter, vertically), which is the configuration for which maximal gain is expected. The absence of observations for some ranges of angles can be traced to MRO orbit specifics coupled with the no-occultation condition we imposed on the candidates. Probability of detection also peaks when the roll angle approaches ± 180^∘, that is, when û_z,MRO is aligned but opposite to 𝐯̂_1 (see Figure <ref>a). For favourable yaw and pitch, such roll angles place SHARAD “in front" of the Sun, with the bus and the other large structures “behind" it. Intuitively, this is also a configuration that is expected to maximise the SHARAD gain, as it approaches that of traditional radar sounding <cit.>. § TIME-PROFILE ANALYSIS METHODOLOGY The time-profile analysis was run on all the “canonical" type-III bursts detected by SHARAD, that is, all those listed in Table <ref> with an asterisk in the “Notes" column. This excludes those with peculiar morphologies (e.g. 2712401), those with extensive fine-structure (e.g. 3617602), and those which are not type-III's (e.g. 3544101). A similar sieving was done in <cit.>. When a given SHARAD data product contained several bursts, all those that are exploitable within this product were included. This left 26 individual bursts to analyse. Of these bursts, all frequency lines containing EMI were removed. For a given product, the EMI lines were identified through a global time-average of the dynamic spectrum, the application of a median filter with kernel size 101, and the detection of all frequency lines that were above this trendline. We also exclude the frequency lines at the edges of the instrument's capabilities, and ran our analysis in the 15 – 25 MHz range, the bandwidth the instrument was optimised for. After exclusion of EMI and edge frequencies, about 800 frequency lines per burst (out of 1800) are still available for time-profile analysis. In total, we thus analysed about 25 000 profiles. The five-parameter function that was used to fit all these bursts is the exponentially-modified Gaussian function, given by: g(a,b,μ,σ,λ; x) = a λ/2exp{λ/2( 2μ + λσ^2 - 2x) }( 1- erf{μ+λσ^2 - x/√(2)σ}) + b, where a controls the overall scaling of the burst, and b, its floor; μ is the mean of the Gaussian sector of the function, and σ^2, its variance; lastly, λ controls the decay rate of the exponential sector. The error function is defined as erf(z) ≡ 2(π)^-1/2∫_0^z e^-t^2dt. The rise time t_r and fall time t_f are then computed as the two half-widths at half-maximums of the fitted curve. In Figure <ref> we show an example of a SHARAD profile along with its fitted function, and a graphical representation of t_r and t_f. Along with the quantities of interest, we also recorded the r^2 coefficient for each fit, in order to exclude fits of bad quality when compiling the data shown in Section <ref>. Bad fits can occur when the signal-to-noise ratio of the burst is very low, when several closely-spaced bursts forces us to constrain the cropping around each of them, or when the profile is contaminated by reflections of the active SHARAD chirp. 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http://arxiv.org/abs/2307.01175v1	20230703173902	Patient-centric health data sovereignty: an approach using Proxy re-encryption	[ "Bruno Rodrigues", "Ivone Amorim", "Ivan Costa", "Alexandra Mendes" ]	cs.CR	[ "cs.CR", "94A60, 68P25" ]	Patient-centric health data sovereignty: an approach using PRE B. Rodrigues et al. Faculty of Engineering, University of Porto, Porto, Portugal [email protected] PORTIC – Porto Research, Technology & Innovation Center, Polytechnic of Porto (IPP), 4200-374 Porto, Portugal [email protected] Faculty of Engineering, University of Porto, Porto, Portugal & HASLab/INESC TEC [email protected] Patient-centric health data sovereignty: an approach using Proxy re-encryptionThis work was partially supported by the Norte Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund (ERDF), within project “Cybers SeC IP” (NORTE-01-0145-FEDER-000044). Bruno Rodrigues10009-0001-2929-0836 Ivone Amorim20000-0001-6102-6165 Ivan Silva20009-0009-8480-9352 Alexandra Mendes30000-0001-8060-5920 August 1, 2023 ===================================================================================================================================================================================================================================================================================================================================================== The exponential growth in the digitisation of services implies the handling and storage of large volumes of data. Businesses and services see data sharing and crossing as an opportunity to improve and produce new business opportunities. The health sector is one area where this proves to be true, enabling better and more innovative treatments. Notwithstanding, this raises concerns regarding personal data being treated and processed. In this paper, we present a patient-centric platform for the secure sharing of health records by shifting the control over the data to the patient, therefore, providing a step further towards data sovereignty. Data sharing is performed only with the consent of the patient, allowing it to revoke access at any given time. Furthermore, we also provide a break-glass approach, resorting to Proxy Re-encryption (PRE) and the concept of a centralised trusted entity that possesses instant access to patients' medical records. Lastly, an analysis is made to assess the performance of the platform's key operations, and the impact that a PRE scheme has on those operations. § INTRODUCTION The ever growing digitisation of services that we use daily, as well as the increasing interest in data crossing and sharing to improve processes, services, and achieve new business opportunities, raises concerns regarding how data is handled and processed. In the healthcare sector, data sharing is not only beneficial, but also needed to provide the best care possible to the patients. However, this data is also highly sensitive, which requires special care. Several governmental measures have already been taken to improve and standardise the way in which data is shared, such as the European Data Governance Act <cit.>, GDPR[https://data.europa.eu/eli/reg/2016/679/oj], and, more specifically in personal health information, HIPAA[https://www.cdc.gov/phlp/publications/topic/hipaa.html] and HITECH[https://www.hipaajournal.com/what-is-the-hitech-act/]. These directives instigate a user-centric paradigm, granting individuals sovereignty over their data. Several approaches have been proposed for ensuring security and privacy in e-health systems. Conventional encryption techniques like AES and ECC are commonly used <cit.>. However, these techniques become problematic when data needs to be shared among multiple entities due to redundancy and computational burden <cit.>. Attribute-Based Encryption (ABE) is another solution <cit.>, but it has its own complexities and limitations, such as managing attribute-based keys and overriding policies in emergencies <cit.>. ABE also lacks the fine-grained access control necessary for a patient-centric sovereign approach. Proxy Re-encryption (PRE) is a cryptographic solution for secure data sharing without prior knowledge of the recipient. Unlike ABE, it does not rely on policies or attributes. PRE converts a ciphertext to a recipient's key without revealing the plaintext to the intermediary entity. It is particularly useful in semi-trusted cloud environments <cit.>. In e-health, PRE has already been used to securely share medical records <cit.>, including in emergency scenarios <cit.>. However, challenges remain in terms of revocability, computational effort, and safeguarding emergencies <cit.>. Existing solutions for emergency scenarios are limited and rely on assumptions that may impact efficiency and reliability. In this context, it is necessary to develop a platform that addresses the aforementioned concerns. This includes enabling more control over the data by the patient while ensuring the safety of that data, even in semi-trusted environments. This contributes to the collaborative aspect of e-health and thus enables better treatments and advancements in the health sector. In this paper, we present a platform that leverages PRE to enhance health data sharing. Umbral's PRE <cit.> is used as the foundation for re-encryption processes, through which we achieve unidirectionality and non-interactivity, ensuring secure re-encryption from the patient to the data user (e.g., practitioners or health centres) without requiring the data user's private key. This approach centres on the expressed opinion of the patient to authorise data sharing, eliminating the need for prior identification of authorised parties — a drawback identified in previous solutions. Additionally, our platform offers revocability options, such as time-based access limits and patient-initiated access revocation. Importantly, the revocation of accesses does not require changes to the encrypted healthcare database, distinguishing our platform from the ones that rely on identity and attribute-based PRE schemes. Furthermore, in the context of healthcare, it is crucial to ensure data sharing in emergency situations when explicit patient consent may not be possible. Our platform addresses this challenge by incorporating a trusted entity for data access when patient authorisation is infeasible. In summary, our main contributions are: * A patient-centric platform, that empowers patients with sovereign control over their health data, enabling granular access control and facilitating the sharing of health records only with explicit consent. * Robust data protection using Umbral's PRE, ensuring secure and encrypted health data sharing without compromising the data user's private key. * A robust access revocation mechanism that enables time-based access limits and supports manual revocation by the patient at any time and with immediate effect. * A break-glass mechanism to ensure seamless emergency data access. The remainder of this paper is organised as follows. Section <ref> introduces basic concepts and definitions, as well as the classification and properties of PRE schemes. Furthermore, an analysis is made concerning the framework on which the access delegation mechanism is based. Section <ref> presents the current picture of the PRE and the advancements regarding break-glass scenarios. Section <ref> details the proposed solution and its implementation. Section <ref> is concerned with the performance test, respective results and discussion. Lastly, Section <ref> presents the conclusions and future work. § PROXY RE-ENCRYPTION PRE is a cryptographic technique that enables a third-party entity, named proxy, to delegate access to encrypted data, without being able to infer the plaintext content of that data. This is achieved by transforming a ciphertext encrypted under one key into a ciphertext encrypted under a different key. §.§ Syntax and basic definitions Since PRE can be seen as a way to delegate decryption rights to a party, it is possible to categorise the different entities according to the delegation relation they possess with each other. The delegator is the entity that owns the data and delegates decryption rights. The proxy is the intermediary entity in the delegation process, which uses a re-encryption key (PRK) to transform the ciphertext encrypted under the delegator's public key into a ciphertext that can be decrypted only by using the delegatee's private key. Finally, the delegatee is the entity that accesses the information through delegation of decryption rights by the delegator. A PRE scheme can be defined based on five different algorithms: * KeyGen — On input of a security parameter n, the key generation algorithm KeyGen outputs a public/private key pair (pk_A, sk_A) for a given user A. * ReKey — On input of a public/private key pair (pk_A, sk_A) for user A and a public/private key pair (pk_B, sk_B) for user B, a PRK rk_ A → B is computed. * Encrypt — Given the input of a public key pk_A and a message m ∈M, the encryption algorithm outputs a ciphertext c_A ∈ C_1. * ReEncrypt — On input of a ciphertext c_A ∈ C_1 and a PRK rk_ A → B, the re-encryption algorithm ReEncrypt transforms a ciphertext c_A ∈ C_1 into a ciphertext c_B ∈ C_2. * Decrypt — Given a private key sk_A from user A and a ciphertext c_A ∈ C_S (S ∈{1,2}) from user A, the same executes the decryption algorithm and outputs the original message m ∈M. According to Qin et al.<cit.>, a PRE scheme can be classified based on its abilities. For example, regarding its directionality, we say that the scheme is unidirectional if it enables the delegator's ciphertext to be re-encrypted into the delegatee's ciphertext but not vice versa. Otherwise, we call it bidirectional. The multi-use/single-use classification focuses on the number of times the PRK can be used to re-encrypt data. In multi-use schemes, the PRK can be utilised to perform several re-encryptions. In the case of a single-use scheme, the PRK can only be used to perform a single transformation. Interactivity dictates whether the re-encryption is computed using just the public key from the delegatee (non-interactive scheme) or both the public and private keys (interactive scheme). Depending on the scenario of utilisation, some properties may be more desirable than others. Other authors classify PRE schemes according to their way of functioning <cit.>. For example, an Identity-Based PRE (IB-PRE) scheme derives public keys from identity attributes (e.g. email). The messages are encrypted using an identity string from the delegatee. Attribute-Based PRE (AB-PRE) schemes allow transforming a ciphertext defined by a set of attributes or access policies into another ciphertext with a different set of attributes. §.§ Umbral's PRE scheme The Umbral PRE scheme is, in its essence, a threshold PRE scheme. This scheme features an Elliptic Curve Integrated Encryption Scheme (EICS-KEM) inspired in <cit.> and proposes several improvements over the original PRE scheme proposed by <cit.>, namely unidirectionality, non-interactivity, and verifiability. It relies on the concept of semi-trusted proxies, also known as ursulas. Being a threshold PRE scheme, it splits the PRK according to shares. The threshold portion of the scheme dictates the minimum number of those shares required to decrypt the information. Splitting the PRK across multiple proxies brings some benefits namely eliminating a single point of failure, in case of a malfunction or compromise of one of the proxies the PRK is still safeguarded. The re-encryption processes in our platform are supported by pyUmbral <cit.>, a Python-based implementation of Umbral. Fig. <ref> presents an overview of the key processes and data flows involved in the Umbral PRE scheme. This system beholds seven main processes: Encapsulation, Encryption, Generate PRK fragments, Re-encapsulation, Decapsulation, and Decryption. These processes are supported by three major cryptographic methods: Key Encapsulation Mechanism (KEM), Data Encapsulation Mechanism (DEM), and Shamir Secret Sharing (SSS) <cit.>. The first step in this process is Encapsulation. This is achieved through the use of a Key Encapsulation Mechanism (KEM), in this case, a loosely inspired implementation of ECIES-KEM introduced by <cit.>. The KEM is fed with Alice's public key pk_A and outputs a symmetric key K, and a capsule. With the capsule and the symmetric key K, the Encryption process is performed using a Data encapsulation mechanism (DEM) which uses Authenticated Encryption with Additional Data (AEAD). This outputs a ciphertext encrypted with the symmetric key. When the data is encrypted and stored in the cloud, in order for the access delegation to occur, there is a need to generate a PRK. This is performed by the Generate PRK fragments process resorting to the notions present in Shamir Secret Sharing (SSS), Alice's private key and signing key signk_A, and Bob's public key pk_B. This enables the generation of the PRK fragments or kFrags. The number of fragments is defined by the number of shares. The kFrags are stored by the proxy for further use in the Re-encapsulation process. This process is responsible for generating the cFrags which enables Bob to gain access to the file at a later stage. To generate the cFrags just the capsule and the kFrags are needed. This is due to the fact that this PRE scheme performs the re-encryption over the capsule. Lastly, once Bob wants to retrieve a file, the Decapsulation process needs to happen. This process resorts to SSS in order to reconstruct the symmetric key k. To do so, Alice's public key, Alice's verifying key vk_A for signature verification of the cFrags, Bob's private key sk_B, and the capsule are needed. Through the use of a Key Derivation Function within the KEM, it is possible to derive the symmetric key K which together with the ciphertext is passed to the DEM. The DEM performs the Decryption process and outputs the plaintext content of the file that Bob can now use. § RELATED WORK The notion of PRE made its first appearance in 1998 when Blaze et al. <cit.> introduced the concept of bidirectional and multi-use PRE. Several works have been published since then with new PRE schemes providing new functionalities and relying on different mathematical assumptions. For example, both <cit.> and <cit.> proposed a unidirectional, single-use PRE scheme, but the first relies on threshold PKE, while the second is based on lattice-hardness problems. In 2015, <cit.> also proposed a unidirectional and single-use PRE scheme, which can be classified as attribute-based. Later, in 2017, <cit.> presented a unidirectional, non-interactive, and verifiable PRE scheme which is threshold-based. In the context of healthcare data sharing, PRE has also been widely explored. In fact, several works address security, privacy, and confidentiality when it comes to the design and implementation of e-health systems. However, there is still a lack of development concerning safeguarding emergency scenarios in the context of e-health systems <cit.>. Works that address this kind of scenario in its design, refer to this as break-glass approaches. In 2017, <cit.> proposed a framework for the secure sharing of Personal Health Records (PHRs) that relies on attribute-based PRE and which addresses emergency scenarios. The break-glass capabilities are provided with ABE. In this scheme, the emergency department attribute is always appended to the policy that encrypts the patient PHR, thus providing instant access to the entity from the moment the same is uploaded. The problem with this approach, and in general with ABE approaches, is that they present some caveats, namely key management and resorting to other mechanisms in break-glass approaches. This is due to the fact that emergency normally means an exception to a policy and, thus, overriding that same policy might be a hefty task in some implementations. In 2019, <cit.> also proposed an approach that is based on an attribute-based PRE, and provided self-adaptive access control, meaning that the system can automatically adapt to normal or emergency situations. However, their break-glass mechanism resorts to a password-based paradigm. This approach raises some concerns, namely in the assumption that the individual that stores the password, has the necessary means to ensure its secrecy. More recently, in 2022, <cit.> proposed a system for IoT sensors combining PRE and PKE with equality test, permitting searches under different public keys and secure data sharing. However, it does not discuss emergency situations. In the same year, <cit.>, proposed a non-interactive, multi-use, certificateless PRE for sharing health data in a cloud environment. Even though their approach gives full control to the data owner, it has two important drawbacks, namely it is interactive and does not propose a break-glass mechanism. Also in 2022, <cit.> published a secure data sharing and authorised Searchable framework for e-healthcare systems. This framework lies on a conditional and unidirectional PRE scheme with keyword search. It is also idealised for managing sensible data from medical wearable devices. This platform has some disadvantages namely regarding the PRK generation performance. Also, this work does not address emergency situations. Finally, in 2022, <cit.> propose a framework which is also based on attribute-based PRE that features break-glass capabilities. However, it leaves open the possible solution for revocability. That being said, there is a need to develop a solution that can cope with all the aforementioned concerns and that contributes to a more reliable and robust break-glass approach. § PATIENT-CENTRIC HEALTH DATA SOVEREIGNTY In this section, we introduce the envisioned solution for a patient-centric platform that enables health data sovereignty through PRE. The subsequent section presents the architecture of the solution, followed by a description of the processes involved in the key operations for access delegation. §.§ Proposed Solution The proposed solution consists of four main nodes: the client, the resource server, the proxy server, and the authorization server, as depicted in Figure <ref>. The client node hosts the client-side application developed with Next.js[https://nextjs.org/]. This client node communicates with the server nodes via Representation State Transfer (REST) and the Hypertext Transfer Protocol (HTTP). The business logic is divided between the resource and proxy server nodes. The resource server is based on the FastAPI framework[https://fastapi.tiangolo.com/] running in a Python environment. This server is trusted by the data delegator and it is responsible for assisting the client-side operations, namely feeding the data the client node needs to display the information to the user. The resource server node also performs some core operations such as the initial encryption and final decryption of the Electronic Health Record (EHR) stored in the database server node hosted in a cloud environment (MongoDB[https://www.mongodb.com/]) as well as the management of delegation requests (accept or decline). Some other complementary operations are also performed such as the generation of the PRK which is stored afterwards by the proxy server node, and the signature verification of the PRK fragments and capsule fragments. The proxy server is solely responsible for the process of EHR delegation, being used for the re-encryption of the capsules and the storage of the PRK. The authorisation server is responsible for performing the authentication of the different users of the platform as well as the issuing and claims verification of the authorisation tokens. These tokens are subsequently used to consume the APIs provided by the resource and proxy server nodes. This node is also associated with two persistence nodes. An in-memory database (REDIS[https://redis.com/] instance) for persisting and performing the lookup of the refresh tokens and a MongoDB instance for storing general purpose user information such as name, email, password, public and verifying keys and roles. §.§ Authentication/Authorisation The authorisation is performed by resorting to JSON Web Tokens (JWT) which are signed using HMAC SHA256. This ensures the tokens can not be tampered with or changed, thus enabling secure transmissions between parties. The authentication flow comprises a traditional email/password authentication, where each user needs to provide a valid email and password. In case of successful authentication, a pair of tokens are issued (access/refresh token) containing some claims needed to support the client-side application. These claims follow the standards and restrictions defined in Request for Comments 7519[https://datatracker.ietf.org/doc/html/rfc7519#section-4]. Besides the pair of tokens, a Cross-site Request Forgery token is also sent for further protection in requests that require cookies. The refresh token is also sent in a cookie configured with the secure and httpOnly flags to ensure it is only transmitted through HTTPS and not available to JavaScript in case of a Cross-site Scripting vulnerability in the client-side application. Since JWT tokens are self-contained, there is no natural way of revoking them. In order to tackle this problem anti-theft mitigation techniques were implemented: refresh token rotation and token reuse detection. §.§ Access delegation scenario Access delegation is the core problem tackled in this work. The next sections dissect the access delegation flow from the moment the file is uploaded by the patient to the moment the plaintext content is retrieved by the healthcare provider. For demonstration purposes, the step-by-step process between two entities, Alice (delegator) and Bob (delegatee), is presented. Upload of an EHR The access delegation starts with the upload of an EHR by Alice. When Alice uploads a new EHR, which can be a Portable Document Format (PDF) or an image, the resource server encrypts the file using a symmetric key resulting from the encapsulation process and stores it together with the capsule, resulting from the encapsulation process, and an associated userId. Another process that is also performed in this step and further detailed in Section <ref> is the safeguarding of emergency situations. Besides the persistence of the file in the database, a PRK is also generated in order to provide access to the predefined trusted entity. This ensures that the trusted party possesses the means to access the file from the moment it is uploaded and that no extra input from the user is needed in this regard. This PRK is sent to the proxy for subsequent use. Bob requests access to an EHR When Bob wants to access Alice's uploaded EHR, he needs to formalise his intentions by issuing a share request to the resource server containing the EHR's resourceId. In this step, the system checks if Bob is the owner of the EHR. This prevents a user from performing a share request to itself, something that violates the business rules of the platform. Once this validation is performed, and provided with the resourceId, the resource server generates a share request that includes the resourceId, the delegatorId and the delegateeId, as well as a status that is set to pending by default. Alice answers the share request Now that Bob asked Alice for access to the EHR, Alice is now capable of answering the share request. Depending on Alice's answer, the execution flow might have two outcomes: Accept scenario — In case of an acceptance, Alice needs to generate the PRK required to re-encrypt the capsule and further enable Bob to have access to the plaintext content of the EHR. To achieve such a feat, Alice requires his secret key along with his signing key pair, needed to verify the signature of the kFrags and cFrags at a later stage, as well as Bob's public key. Notice that just the public key is needed, due to the non-interactivity property of this PRE scheme. Lastly, since the underlying scheme of the access delegation mechanism is a threshold PRE scheme, there is also the need to provide a threshold which defines the minimum number of shares needed to decrypt the capsule and the number of shares which dictates the number of outputted PRK fragments. This last aforementioned operation outputs the kFrags, which are sent to the proxy along with a shareId binding the PRK to a specific share request. Both attributes are persisted by the proxy for further use once Bob retrieves the EHR. The share request operation ends with the status update of the share request, which is defined as accepted, together with an arbitrary expiration date defined by Alice. This expiration date is optional, being possible to share an EHR indefinitely or temporarily, in which case the share request is automatically revoked through a cron job once that date is transposed. This ensures the time-based access delegation aspect that this work contributes to. Decline scenario — In case Alice declines the share request the status is updated accordingly and no other action is performed. Bob retrieves the EHR Now that Alice explicitly delegated access to the EHR, Bob is now capable of retrieving it. To do so, Bob performs a request to the resource server, which requires Bob's secret key and the resourceId, which uniquely identifies the EHR. A file ownership verification is also performed since the decryption steps are different for a delegator and a delegatee, where the former does not have the need to re-encrypt the capsule. As stated previously, ownership trails different paths regarding execution flow. With that said, the following can happen whether the user is or not a data owner. Data owner — In case the user that requests the file is a data owner, a hybrid encryption approach is used, thus no re-encryption takes place. Not a data owner — If the user is not the data owner, meaning they are a delegatee, a collaborative operation between the resource and proxy servers is required to take place. For this specific scenario, Bob needs to ask the proxy to re-encrypt the capsule using the previously generated PRK. To that purpose, the resource server retrieves the EHR details and sends the capsule to the proxy server. The proxy, equipped with the capsule and the PRK fragments kFrags, performs the re-encapsulation process outputting the cFrags. These cFrags are sent back to the resource server, which validates their signature through Alice's verifying key. Once the capsule fragments are validated, Bob decrypts the file by opening the capsule. This last step encompasses Bob's private key, Alice's verifying key and the verified cFrags. With the plaintext content of the EHR, Bob is now capable of accessing the information. Some important remarks to highlight are that the secret key used in the sharing process is never shared with the intermediary entity or proxy, making it semi-trusted. Additionally, the proxy only stores the PRK, which alone does not grant it the capability to decrypt the file. Furthermore, even if the stored information such as the capsule, PRK, and ciphertext were to be leaked from the database, the safety and integrity of the EHRs would still be preserved, as they are not sufficient for decrypting the EHRs. §.§ Break-glass approach Safeguarding emergency scenarios is of paramount importance in a health-related platform. Therefore, we adopted amd approach that features a central trustworthy entity responsible for managing the authorisation in emergency scenarios. This trustworthy entity is seen as a government entity that is responsible for managing such issues and has full access to the files submitted in the platform. The implementation is similar to what is described in Section <ref> regarding Alice accepting the share request. However, in this case, there is no explicit acceptance of the share request. When an EHR is uploaded, the trusted entity user is retrieved from the database and a PRK is generated. An accepted share request is automatically created for the trusted entity, which links the PRK to the share request between the patient and the trusted entity. Regarding the process of retrieving the EHR, it follows a similar procedure as depicted in Section <ref>. Just like in a regular file retrieval, since the share request is automatically accepted and the proxy possesses the PRK, the trusted entity requests the proxy to re-encrypt the capsules, enabling the final decryption to take place. This approach vastly reduces the dependency on external actors, increasing the reliability and availability of the idealised break-glass approach. Having a dedicated entity for this purpose enables instant and swift access to the information if needed. § PERFORMANCE ANALYSIS In this section, we present the performance tests conducted to evaluate our platform. Given the common concerns of limited hardware infrastructures and sub-optimal conditions in governmental adoption cases, it is important to assess the responsiveness of the key operations offered by the platform. Our main goal is to quantitatively analyze the performance of the most computationally intensive operations and assess the impact of the PRE scheme. As there are no specific regulations, indications, or suggestions regarding performance for this type of platform, our tests are purely quantitative and based on known factors and conditions. The performance tests were carried out on a deployed version of the platform, hosted in Microsoft Azure using a Free F1 tier running Linux and Python 3.10. While these specifications may be basic, they are sufficient to simulate a sub-optimal environment. In real-world scenarios, it is common for governments to have financial restrictions, making it likely that the platform would be deployed on infrastructure with modest specifications. The tests were conducted using Apache JMeter as the tool of choice. In the rest of this section, we present the results related to the three most crucial operations of the platform and which involve the use of PRE: file upload, accepting a share request, and file retrieval. Additionally, a brief analysis of the results is also presented. File upload The performance tests depicted in this section aim to evaluate how the different file sizes impact the upload performance of files. Since the size of EHRs depends on various factors, such as the patient's medical history, the image resolution of the machines used for exams, and the content of the file itself, determining an average file size becomes challenging. Therefore, we conducted our experiments using two different file sizes: 1MB and 10MB. Figure <ref> illustrates the results obtained from a series of twenty runs performed for each file size. It can be observed that a tenfold increase in file size reflected an average increase of 2715 milliseconds(ms) when comparing file sizes of 1MB and 10MB respectively. The former took an average of 1154 ms and the latter an average of 3870 ms. Despite a time of almost four seconds, and considering this is not an ideal response time for a REST API, it should be taken into account the complexity of the operations performed. Since this is not a critical operation when it comes to performance, these values are acceptable. Accepting a share request The acceptance of a share request is a key operation in the platform described in this paper. Although its performance does not possess a high impact on the efficiency of the platform, it does provide valuable information regarding the PRE process. In this operation, the PRK is generated and sent to the proxy for persistence purposes. Notice that, in this case, there was no need to perform the tests for both file sizes since the PRK generation only depends on cryptographic keys. Regarding the results of these tests, the average time obtained in 20 runs was 869 ms. This quick response was expected since the generation of the PRK fragments is a relatively simple operation that depends on the cryptographic key from both ends, the signature, and the number of shares. Additionally, there was not a significant variation among the twenty runs that were performed. This is supported by the low standard deviation of just 188 ms. File retrieval This set of tests aims to assess the impact of file sizes and the use of PRE on a file retrieval scenario. The tests were conducted for both regular decryption and PRE decryption. To evaluate the impact of file sizes, the tests were performed for both 1MB and 10MB file sizes. Moving on to the obtained results (Fig.<ref>), a 1MB file took an average of 903 ms to be retrieved while the 10MB one took an average of 2529 ms. Regarding file retrieval with PRE, the 1MB file took an average of 1245 ms and 2877 ms for the 10MB file. We have also evaluated the impact of re-encryption on file retrieval operations (Fig.<ref>) by directly measuring the difference between regular decryption and PRE for each file size. This resulted in an average difference of 342 ms for the 1MB file and 348 ms for the other one. The results of our tests indicate that there is a similar average difference between regular and PRE decryption for both file sizes. This similarity can be attributed to the fact that the re-encryption process only affects the capsule, not the actual file. Since the sizes of the capsule and cryptographic keys are similar in both scenarios, it is expected that the results would be similar as well. The file size does not significantly impact the re-encryption of the capsule, but rather affects the overhead associated with fetching the file from the database and delivering it in the response. Regarding the obtained results, they were deemed satisfactory since most operations do not possess restrictive requirements when it comes to performance. Regarding more critical operations such as file retrieval, considering the computational effort and infrastructure complexity required to ensure full correctness with the underlying threshold PRE scheme, the results were deemed satisfactory. It is important to note that these tests were conducted in a shared infrastructure with modest specifications. Thus, it was not possible to control the current workload of the servers during the tests, which may have impacted negatively the aforementioned results § CONCLUSION In this paper, we present a PRE-based platform for the secure sharing of e-health, considering a sovereign approach focused on the patient. This approach is achieved by ensuring that the patient's data is only shared with their explicit consent. Furthermore, it also enables robust revocability by the patient, without requiring updates on the encrypted EHR database, further contributing to a user-centric approach. Non-interactivity is also a key characteristic of our platform, which does not require sharing user's private key for the re-encryption process to occur. Another key achievement of our work is the proposed break-glass mechanism. Since some implementations fall short in terms of revocability, and only a few contemplate PRE in emergency scenarios, our solution uses a central trusted entity to which the proxy delegates access from the moment the EHR is uploaded to the platform. This eliminates the need to trust external actors in the system, increasing reliability and allowing swift access to the information in critical situations. There are other key characteristics of our platform worth highlighting. Firstly, it uses symmetric encryption to encrypt the EHR, which is faster than PKE. Secondly, the re-encryption process is performed over the capsule, which tends to have a much smaller size compared to a PHR. The tests that were conducted and our results show that the most demanding task is the upload of the EHR, as expected, because it requires the encapsulation process to occur and the encryption of the EHR. However, the re-encryption process does not show a significant increase when the size of the uploaded files increases. This is because the re-encryption does not involve the EHR. Our platform provides a solution to the sharing of medical data that incorporates key functionalities not covered together in previous literature, such as unidirectionality, non-interactivity, revocability, and a mechanism to deal with emergency situations. This solution contributes to the collaborative aspect of e-health and enables better, and more informed treatments supported by the increased exchange of information between providers. Regarding future work, it would be beneficial to extend the architecture to accommodate multiple proxies instead of using just one. This could be achieved by utilising a blockchain network where the proxies work together to re-encrypt the capsules, thus enabling all the benefits that a threshold-based scheme has to offer. Furthermore, additional tests could be performed using different environments and network conditions to cover more use case scenarios. splncs04
http://arxiv.org/abs/2307.01315v1	20230703194122	A log-linear model for non-stationary time series of counts	[ "Anne Leucht", "Michael H. Neumann" ]	math.ST	[ "math.ST", "stat.TH" ]	-3mm 115pt 15.2cm 22.6cm 0pt 0pt 0.15in equationsection plain thmTheorem[section] lemLemma[section] corCorollary[section] propProposition[section] critCriterion[section] algAlgorithm[section] definition defnDefinition[section] conjConjecture[section] exmpExample[section] probProblem[section] remark remRemark noteNote claimClaim caseCase ackAcknowledgment ℕ ℤ ℝ ℙ 𝔼 ℱ 1 σ Y X C ⟶_n→∞ ⟶_t→∞ sup min
http://arxiv.org/abs/2307.03292v2	20230706210522	Identifying overparameterization in Quantum Circuit Born Machines	[ "Andrea Delgado", "Francisco Rios", "Kathleen E. Hamilton" ]	quant-ph	[ "quant-ph" ]	This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. [email protected] Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 [email protected] Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 [email protected] Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 In machine learning, overparameterization is associated with qualitative changes in the empirical risk landscape, which can lead to more efficient training dynamics. For many parameterized models used in statistical learning, there exists a critical number of parameters, or model size, above which the model is constructed and trained in the overparameterized regime. There are many characteristics of overparameterized loss landscapes. The most significant is the convergence of standard gradient descent to global or local minima of low loss. In this work, we study the onset of overparameterization transitions for quantum circuit Born machines, generative models that are trained using non-adversarial gradient-based methods. We observe that bounds based on numerical analysis are in general good lower bounds on the overparameterization transition. However, bounds based on the quantum circuit's algebraic structure are very loose upper bounds. Our results indicate that fully understanding the trainability of these models remains an open question. Identifying overparameterization in Quantum Circuit Born Machines Kathleen E. Hamilton ================================================================= § INTRODUCTION Variational quantum algorithms (VQAs) are at the core of near-term, gate-based quantum computing applications. By harnessing the strengths of quantum processors and the flexibility to optimize the circuit parameters through classical optimization, VQAs have found applications in supervised and unsupervised learning, with the potential for deployment on near-term devices, often limited in connectivity. Many types of generative models are used in machine learning, with different approaches to training and inference. In this work, we focus on generative trained to fit a known joint distribution over observed data. A well-trained generative model can act as a surrogate to generate synthetic data similar to previous observations. This can be a valuable tool when the underlying process which generated a given set of data is unknown, or poorly characterized, or where generating real world data has high overhead (e.g. in high energy collider experiments). Quantum circuits are well-suited for this task and many constructions of quantum generative models have been explored and implemented on different quantum processors and platforms: quantum Boltzmann machines were an early application for quantum annealers <cit.>, quantum generative adversarial networks have been deployed on gate-based platforms <cit.>, and the quantum circuit Born machine (QCBM) <cit.> has been used to benchmark many near-term quantum devices. QCBMs are generative models <cit.> that implement a flexible transformation using a parameterized quantum circuit trained to map an initial quantum state to a final state to generate samples from an arbitrary distribution. The construction of the circuit does not rely on an explicit form (e.g. a Hamiltonian) to construct the model. This gives us an advantage in terms of circuit design flexibility, but several recent papers have posited that increased model flexibility and expressibility comes at a consequence of trainability. In this work we focus on the trainability of QCBMs, specifically the onset of overparameterization. In a prior work, <cit.> we demonstrated that QCBMs can be used to fit arbitrary joint distributions, successfully reproducing the marginal probability distributions, and in some cases, the variable correlation. This performance is highly dependent on the underlying parameterized quantum circuit (PQC), which is used as the QCBM: higher density connections between qubit subsets can lead to faster training and better fitting of the marginals, but fully fitting the correlations between joint variables requires large-scale entanglement or long-range correlations across all qubits. Understanding the limitations of these models, in terms of width, depth, and impact on trainability, is highly relevant for developing useful models for real-world applications. Gradient-based optimization of QML models is made possible using analytical quantum gradients <cit.>. It is an appealing optimization method because of known convergence properties for convex landscapes <cit.>, and the ability to find stationary points in high-dimensional spaces. However, for non-convex landscapes, it is difficult to ensure that gradient descent will converge to the global minimum. The flatness of the objective function landscape has a direct impact on the performance of the VQA– an extremely flat landscape may impede trainability. Furthermore, first order methods can converge to saddle points or spurious local minima (i.e. traps). Understanding the feasibility of gradient-based training is done through the characterization of the landscape, and quantifying the proliferation of traps <cit.>. Studies on the relationship between expressibility and trainability have indicated that highly expressive ansätze exhibit flatter loss landscapes <cit.>. Overparameterization has been studied in classical deep learning for many classes of models and learning rules, such as spiked tensor models <cit.>, ReLU networks <cit.>, and random feature models <cit.>. Overparameterized models have computational capacities larger than needed to represent the distribution of the training data <cit.>, this often implies that the model has far more trainable parameters than the number of training points <cit.>. A key characteristic of overparameterized networks is the proliferation of local minima which are reasonable approximations of the global minimum <cit.>. Generally, overparameterized machine learning (ML) models have been associated with a fundamental change in the loss landscape structure, counter-intuitively lower training, and improved generalization properties <cit.>. Theories for what drives this transition using concepts from soft matter <cit.> or spin glass theory <cit.> have been proposed. Methods and analysis from ML have been employed to understand the trainability and generalization capacity of QML models. Several studies have attempted to characterize the loss landscape of quantum neural networks (QNNs) by studying the overparameterization phenomenon, as well as the capacity <cit.> and expressibility of the PQC <cit.>. There have been several approaches that predict the onset of overparameterization in terms of the dimension of the Dynamical Lie Algebra (DLA) <cit.> associated with the generators of the QNN, as well as the rank of the Quantum Fisher Information (QFI) <cit.>. In <cit.>, an overparameterized model contains more trainable parameters than the rank saturation of the QFI. In both cases, the theoretical results are verified by performing numerical simulations of models trained in an unsupervised setting using energy minimization and/or with a structure inspired by a known Hamiltonian. A separate area of work has studied the proliferation of critical points in the loss landscape <cit.>. These efforts constitute attempts to study the model capacity and explore the computational phase transition, where the model's trainability is improved. This work presents an empirical study on the onset of overparameterization in QCBMs. What distinguishes our study from previous works is the use of unsupervised training based on the similarity of generated probability distributions. Previous studies on overparameterization and trainability transition have focused on QNNs trained using expectation values (e.g. variational quantum eigensolvers). In Section <ref>, we provide details on how the models are constructed and trained. We will then present our results in Section <ref>. Additional details and results are provided in the Appendices. § GRADIENT-BASED TRAINING OF QCBMS Many characteristics and theories describe what occurs in the overparameterized landscape. The key concept that we are investigating is the reliable success of first-order optimization methods (gradient descent) for training overparameterized QCBM models. Our study explores the trainability of overparameterized generative models using a hybrid quantum-classical workflow that implements gradient descent to train a QCBM using the Jensen-Shannon divergence (JSD), JSD(P\|Q_θ) = 1/2[KLD(P \|P+Q_θ/2) + KLD(Q_θ\|P+Q_θ/2)], KLD(P\|Q_θ) = ∑_ip(x_i)ln(p(x_i)/q_θ(x_i)). This is a symmetric version of the Kullback-Leibler divergence (KLD) which has been commonly used in training generative models <cit.> and QCBMs <cit.>. One advantage of the JSD loss is that it is bounded 0 ≤JSD(P\|Q_θ) ≤ln(2). Terms that contribute large loss values are when there is a large discrepancy between p(x_i) and q_θ(x_i). The target distribution P, and generated distribution Q_θ are defined over a discrete set of binary strings. Q_θ is generated by projecting a QCBM state prepared by a PQC onto the computational basis. Multiple target distributions were used to train QCBM of n=2, 3, 4, 6, 8 qubits. The target distributions are defined over 2^n bitstrings with varying degree of sparsity, entanglement, and correlation. Given the 2^n-length vector of the target distribution P(x_i), its sparsity (𝐬(P))is the relative size of the bitstring support compared to the total length, 𝐏^>0 ={ x_i ∈ [P]\|P(x_i)>0} 𝐬(P) = \|𝐏^>0\|/2^n. We use a uniform distribution (Uniform), a correlated two-dimensional joint distribution (HEP), a random sparse distribution (Sparse) and projections of the Bell state, GHZ state, and W state onto the computational basis (see Table <ref>). The Uniform and HEP targets are dense, each bitstring has non-zero probability amplitude. The Sparse, Bell, GHZ and W state targets are sparse, each has a finite subset of bitstrings with non-zero probability amplitudes. The Sparse, and Bell state were only used to train 2-qubit QCBMs, the W state target was only used to train 3-qubit QCBMs, the GHZ state target was used to train QCBMs with n ≥ 3, and the HEP target was only used to train QCBMs with n≥4. The Uniform, Sparse, Bell, GHZ and W distributions were constructed using exact values for the probability amplitudes. The HEP distribution was constructed from approximately 4 million samples drawn from the kinematic distributions of the leading jet in a simulation of the production of pairs of jets in pp interactions at the Large Hadron Collider (see <cit.> for further detail). The QCBM is constructed using the Hardware Efficient Ansatz (HEA) <cit.>, as shown in Fig. <ref>. The HEA is built with n number of qubits and p layers of alternating rotational and entangling gates. The parameter size associated with a circuit of depth p is \|θ_p\| = 4(n-1)p + 2n. The single qubit gates (R_X, R_Y) are independently parameterized – these values are updated using gradient descent. The gradient descent steps are implemented with adaptive moment estimation, using the Adam optimizer <cit.> with hyper-parameters α=0.01, β_1 = 0.9, β_2 = 0.999. For each QCBM used in this study, we executed 100 independent training runs. Each training run began with a random initialization of the parameters drawn uniformly from the range [0, 2π) and ran for 200 steps of Adam. During training we evaluate the QCBMs in the analytical limit of n_s →∞ shots. In Appendix <ref>, we present results with finite shot sizes. § IDENTIFICATION OF OVERPARAMETERIZATION §.§ Convergence of training We present our results to highlight a key characteristic of overparameterized landscapes: the reliable convergence of first-order optimization methods to low-loss regions of the landscape. This has been observed in many different ML and non-convex optimization workflows <cit.>. To investigate reliability in QCBM training, we train each QCBM model from 100 randomly chosen parameter sets. In Figs. <ref>, <ref>,<ref>,<ref>, and <ref> we plot the median value of the final loss value obtained from each training run. To account for the statistical spread in the data, we also shade the inter-quartile region (IQR), indicating the spread in the middle 50 % of our obtained loss values. In a rugged landscape dominated by traps, we expect the training to vary widely based on the initialization. The optimization may converge to low-loss solutions or it may converge to higher (poorer) minima. This is seen in the separation between quartiles, indicated by the IQR. We identify a critical depth p_c of the QCBM where the final loss shows a sharp decrease. From these plots, we identify p_c and report these values in Table <ref>. For p≪ p_c it is difficult for the training to escape minima of high loss (on the order of 10^-2), and there is low variance in the final loss. For p ≈ p_c the training becomes unstable, indicating that the quality of the found minima is highly dependent on the initialization. Once p≫ p_c, then each gradient descent workflow converge to losses below 10^-8. The location of p_c depends on the target distribution. For the Uniform distribution, it is possible to retrieve low-loss solutions with trivial p=0 circuits. §.§ Time to solution Another key characteristic of overparameterized landscapes is that gradient descent efficiently converges to low-loss regions of the landscape. This has been studied in many different ML and non-convex optimization workflows. We define the time to solution (TTS[ϵ]) as the minimum number of optimization steps needed to obtain a loss JSD(P\|Q)≤ϵ. During training, the loss is stored after each optimization step and the TTS[ϵ] metric is extracted from these stored values. For the models trained with exact analytical simulation, we take ϵ=10^-8. Since each QCBM was trained with a fixed number of 200 optimization steps, if the loss function never reaches the lower value, then TTS[ϵ]=200. From the values reported in Table <ref>, we observe that overparameterization depends on the target distribution. When reporting TTS[ϵ], we group the targets according to the amount of entanglement in the target state. In Fig. <ref> we plot the median TTS[ϵ=10^-8] for maximally entangled (Bell, GHZ, W) targets. In Fig. <ref> we plot the median TTS[ϵ=10^-8] for the remaining targets (Uniform, Sparse, HEP). §.§ Phase Transitions in Training Based on Figs. <ref>, <ref>,<ref>,<ref>, and <ref> there is a notable change in the QCBM training dynamics as circuit depth increases. In this section we discuss: when does a QCBM have sufficient complexity to fit the distribution, and when are low-loss solutions easy to find. When identifying these phases, we also consider their dependence on the target distribution – we observe decreased IQR for the GHZ target, and generally for wider circuits. First, we observe that all models trained to fit non-uniform distributions have a minimum block depth p>0, needed to ensure that solutions exist. This is a direct consequence: circuits of p=0 can only prepare n-qubit product states. Additionally, it has been established that with 2- and 3-qubit states there is a minimum number of entangling gate operations that ensure a circuit can connect the entire state space <cit.> or prepare a maximally entangled state. Our results in Figs. <ref> and <ref> agree in the sense that, when training on the Bell state or GHZ state distribution, the circuits needed at least p=1 for the training to show non-trivial minimization. At p=1, the training results with the Bell and GHZ state distribution can find low-loss solutions; when training on the W state distribution, an additional block is needed. With n=4, 6, 8 qubits, the training finds lower minima at p=2, 3, 6, respectively. Beyond these transitions, the next transition is marked by low-loss solutions, indicating that the QCBM has sufficient complexity to fit the distribution and the training enters the overparameterization regime: low-lying solutions are reliably found for all training runs. This occurs at higher depths, which we reported in Table <ref>. § COMPARISON TO QUANTUM BOUNDS In Table <ref>, we reported our empirical observations of phase transitions in QCBM trainability (Figs. <ref>-<ref>) as a critical circuit size p_c(obs), and supported this through empirical observations about time to solutions (Figs. <ref>,<ref>). Based on these observations, we are confident that the QCBMs exhibit overparameterization. In Table <ref>, we compare p_c to several previously established bounds in the literature: the parameter dimension D_C <cit.>, the rank of the QFI matrix (QFI) <cit.>, and the dimension of the Dynamical Lie Algebra associated with the generators of the circuit ansatz (𝔤_HEA).) <cit.>. The QFI rank in Table <ref> is numerically computed, while D_C and the DLA rank DLA=dim(𝔤_HEA) are analytically evaluated. The expressive power of a PQC is associated with D_C. It has a maximum possible value of D_C = 2^n+1 - 2 and quantifies the number of independent parameters in a quantum state prepared by a PQC. It is also related to the QFI <cit.>, which considers the quantum geometric structure of the PQC. The QFI matrix rank is computed numerically, and in Appendix <ref> we provide further detail on its computation. The rank of the QFI matrix saturates at the maximum value of D_C. The DLA rank DLA=dim(𝔤_HEA) is an upper bound on the number of parameters in a circuit, to define a local map that takes any element from (𝔤_HEA) to a point in parameter space. In Figs. <ref>–<ref>, we observe that for circuits with N≤ 4 qubits, D_C and QFI rank are approximate lower bounds on p_c. For all circuit widths, (𝔤_HEA) is a loose upper bound on p_c. § CONCLUSION In this work, we have presented an empirical study showing the onset of overparameterization in QCBMs. We have provided the first demonstration of overparameterization for the QCBM and loss functions evaluated with probability distributions. The overparameterization transition occurs at a critical depth of the QCBM circuit. We observe that for overparameterized circuits, gradient-based training becomes highly efficient and that this critical depth is dependent on the target distribution. For classical and quantum models, understanding and identifying what drives the overparameterization transition remains an open question and for both paradigms, overparameterization is associated with counter-intuitive behavior. In supervised training, overparameterization leads to models for which gradient-based training converges to solutions with good generalization properties without over-fitting <cit.>. In this work we report that overparameterized QCBMs can train efficiently without impedance from barren plateaus. As detailed in Appendix <ref> we present further evidence that the barren plateau does not prevent trainability. However, predicting long-time training dynamics from initial behavior remains challenging and an unresolved question. More so, a full understanding of the QCBM trainability remains an open question, for both ideal (noiseless, infinite shot limit) and noisy models. This work is a first step along these paths. In Appendix <ref> we provide results on overparameterization when using finite sampling sizes. Our results should serve to motivate further discussion about about the trainability of QCBMs. First, to obtain more robust upper bounds for the onset of overparameterization, second to understand what are the drivers of changes in the trainability, third to determine how trainability is affected by noise, and finally to build a more complete understanding of trainability beyond the barren plateau. This work was partially supported by the Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics program at ORNL under FWP ERKAP61. This work was partially supported by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy. This work was partially supported as part of the ASCR Fundamental Algorithmic Research for Quantum Computing Program at Oak Ridge National Laboratory under FWP ERKJ354. This work was partially supported by the ASCR Quantum Computing Applications Teams Program at Oak Ridge National Laboratory under FWP ERKJ347. This research used birthright cloud resources of the Compute and Data Environment for Science (CADES) at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. § ROBUSTNESS OF OVERPARAMETERIZATION TRANSITION As stated in the Introduction, the QCBM is an implicit probabilistic model – it is trained to fit a data distribution from a (possibly unknown) process without explicitly modeling the process itself. It is apparent from the results presented in Section <ref> that the landscape undergoes a transformation which results in low loss minima being easy to find in the limit of n_s →∞ shots. The results in the main text are from QCBM trained with the exact analytical simulation of a noiseless circuit. In this section we discuss the robustness of this transition with respect to statistical noise encountered by preparing and sampling a quantum state with finite shots. On near-term hardware, there will be noise from many different hardware sources. As a simulation, including finite shot sizes introduces a source of statistical noise, under the assumption that the QCBM state can be prepared and measured without error, resulting in n_s independent samples from the final state. In classical learning, stochastic noise is expected to act as a regularizer and also assist in escaping from local minima and avoid trapping on saddle points. Statistical noise can impede training by under-representing modalities in the target distribution, or in the sampled distribution. Thus, we investigate whether the inclusion of statistical noise affects the trainability of QCBM in particular the onset of an overparameterization transition. With finite shot sizes, we can still observe an overparameterization transition, however the decrease in loss in the overparameterized models is not as dramatic and the loss appears to be lower bounded by 1/√(n_s). Additionally, the heuristic we used to identify p_c is of limited use with finite shots: at 1000 or 10000 we see that the IQR does not dramatically change, even though the final loss value saturates as the model depth increases. § PREDICTING TRAINABILITY The results in the main text appear to contradict the current understanding variational quantum algorithms. The QCBM models used in this study both exhibit overparameterization, yet the critical depths needed to reach these regions is predicted to result in models plagued by barren plateaus. In this section we present the results for three metrics: the QFI, the variance of the gradient, and the Hessian spectrum. These metrics have been introduced and used in the QML literature to characterize or quantify the trainability of variational quantum circuit models <cit.>. When a target is required (to evaluate the loss) we use two sample targets: the GHZ, Bell targets, and the Sparse target (see Table <ref> in main text). The results in this section are presented are in support of our statement in Section <ref>, that a full understanding of the trainability of QCBM models remains an open question. §.§ Quantum Fisher Information We first compute the QFI which is a fundamental concept in quantum information. It quantifies the amount of information that a quantum state carrier about a particular parameter of interest. The Fisher information is defined in terms of the derivative of the density matrix with respect to the parameter being estimated. In the context of quantum computing, the QFI provides a measure of how sensitive the state is to changes in the parameter and is used to bound the overparameterization transition. The rank of the QFI matrix increases linearly with p with a slope of n_qubits, as seen in Figure <ref> until it reaches the maximum possible value for D_C = 2^n+1 -2. The QFI rank is independent of the target distribution. The saturation of the QFI matrix has been associated with the maximum number of trainable parameters in the quantum model, and it coincides with the expressive power of the circuit, or parameter dimension D_C. In Table <ref> of the main text, we report the numerically computed rank of the QFI matrix for our QCBM models, and in Table <ref> we reported the analytical values of D_C. §.§ Loss Gradient Variance The overparameterization transition reported in the main text, occurs in circuits with circuit depths that far exceed the model sizes where barren plateaus have been predicted to exist <cit.>. The barren plateaus is a potential obstacle for variational quantum models where a randomly initialized circuit will sit on a plateau in the landscape with “no interesting search directions in sight,” (cf. <cit.>). Many sources of barren plateaus have been identified from loss function design <cit.>, noise channels <cit.>, entanglement <cit.>, to the high expressivity of the ansatz design <cit.>. Additionally, barren plateaus have been used to indicate that many solutions sit in narrow ravines and can be difficult to find <cit.>. The vanishing of all gradient directions is taken to indicate that gradient-based methods will be highly inefficient, or even fail to train. The current understanding of the QML community is that trainable models must be constructed from shallow circuits, or with restricted expressivity. Yet in the main text we report on models built from highly expressive ansatz, constructed to depths that far exceed the O(poly(n)) scaling. These models have demonstrated trainability in the extreme case of n_s →∞ in the main text, but trainability also is seen for finite shot sizes. Identifying barren plateaus can use the variance of the loss gradient <cit.>, or sample variance of a two-local Pauli term gradient <cit.>. We compute the variance of the loss gradient Var[∇_θJSD(P\|Q_θ)]. This variance is dependent on the choice of target, and in Fig. <ref> we plot the median loss variance for randomly initialized QCBMs using both targets. When evaluated against the denser target, the variance scales as predicted in <cit.>. However, even though the gradient variance vanishes, there remains significantly large directions (Fig. <ref>). We use Figs. <ref>, <ref> to emphasize that the gradient evaluated at random parameters gives an incomplete picture. From the results in the main text, 2-qubit QCBMs trained with the Bell and Sparse targets exhibit overparameterization when p>2, 3- and 4-qubit QCBMs trained with the GHZ target exhibit overparameterization when p<20, yet in Fig. <ref> there is no indication that these models will train efficiently, based on the gradient at initialization. Likewise the variance of 6- and 8-qubit QCBMs saturates (as predicted in <cit.>). This can indicate that the circuit forms a 2-design, yet in the main text we show that with the GHZ target these models can find low loss solutions. §.§ Hessian Spectrum Our final use characterize the landscape of the QCBM where gradient-based training converges. In the previous section the variance of the gradient is used to diagnose barren plateaus, and uses a first order (gradient) approximation that most directions explored by gradient descent are locally flat. The Hessian matrix is a second order method that characterizes the local curvature around critical points (where ∇ℒ→ 0). The Hessian matrix ℋ_ij = ∇_θ_i∇_θ_j(ℒ) is a square matrix that provides important information about the local curvature near a critical point and has been used as a metric to understand the classical neural network landscape <cit.> as also the quantum neural network landscape <cit.>. We use the final trained parameters (where the training has converged after 200 optimization steps) and compute the Hessian to characterize the local curvature. By computing the eigenvalues and eigenvectors of the Hessian matrix, we can determine the nature of the critical point. If all eigenvalues are positive (negative), the critical point is a local minimum (maximum) of the function. If the eigenvalues have mixed signs, the critical point is a saddle point. As seen in Figs. <ref>,<ref> the QCBM spectra is dominated by a large number of zero eigenvalues–degenerate or flat curvature. Similar results have been observed in classical neural networks, where the Hessian eigenvalues contain a bulk centered at zero, and a discrete subset of non-zero values <cit.>. For supervised models <cit.> the number of non-zero eigenvalues is proportional to the number of classes. For the QCBM Hessian spectra we do not count the number of non-zero eigenvalues, but in Figs. <ref>, <ref> we plot the maximum and minimum eigenvalues. These values show that at small to moderate depths, the Hessian spectra show small negative curvature, indicating that the training converges to a saddle point. As the depth of QCBM increase, the critical points become minima with large degenerate spaces.
http://arxiv.org/abs/2307.01585v1	20230704092428	Powder hold-up and pressure drop in a packed bed thermal energy storage with gas-powder two phase flow: An experimental study	[ "Paul Schwarzmayr", "Felix Birkelbach", "Heimo Walter", "Florian Javernik", "Michael Schwaiger", "René Hofmann" ]	physics.app-ph	[ "physics.app-ph" ]	IET]Paul [email protected] IET]Felix Birkelbach IET]Heimo Walter VASD]Florian Javernik VASD]Michael [email protected] IET]René [email protected] [corrauthor]Corresponding author [IET]organization=Institute for Energy Systems and Thermodynamics, TU Wien, addressline=Getreidemarkt 9, city=Vienna, postcode=1060, country=Austria [VASD]organization=voestalpine Stahl Donawitz GmbH, addressline=Kerpelystraße 199, city=Leoben, postcode=8700, country=Austria Waste heat recovery in the energy intensive industry is one of the most important measures for the mitigation of climate change. The utilization of just a fraction of the theoretically available waste heat potential would lead to a significant reduction of the primary energy consumption and hence a reduction of greenhouse gas emissions. The present study examines the integration of a packed bed thermal energy storage for waste heat recovery in the iron and steel industry. Along with the highly fluctuating availability of excess heat the main difficulty of waste heat recovery in industrial processes is the high amount of powder that is transported by the hot exhaust gases. Therefore, the experimental investigations in this study focus on the pressure drop and powder hold-up in a packed bed thermal energy storage that is operated with a gas-powder two phase exhaust gas as heat transfer fluid with the ultimate goal to assess its suitability and robustness under such challenging operational conditions. The results indicate, that 98 % of the powder that is introduced into the system with the heat transfer fluid during charging accumulates in the packed bed. Remarkably, most of the powder hold-up in the packed bed is concentrated near the surface at which the heat transfer fluid enters the packed bed. When reversing the flow direction of the heat transfer fluid to discharge the storage with a clean single phase gas, this gas is not contaminated with the powder that has been accumulated in previous charging periods. Furthermore, the radial distribution of the powder hold-up in the packed bed is observed to be even which indicates that there is no risk of random flow channel formation that could affect the thermal performance (storage capacity, thermal power rate) of the system. The entirety of these findings reinforces the great potential of packed bed thermal energy storage systems for waste heat recovery in the energy intensive industry. * Thermal energy storage for waste heat recovery in the iron and steel industry. * Direct use of exhaust gas (gas-powder two phase flow) as heat transfer fluid. * Experimental investigation of powder hold-up and pressure drop. * During charging powder accumulates at the surface where it enters the packed bed. * During discharging clean heat transfer fluid is not contaminated with powder. packed bed thermal energy storage gas-powder two phase flow powder hold-up pressure drop exergy efficiency iron/steel industry § INTRODUCTION The waste heat potential from the industry sector is enormous and its exploitation can lead to substantial primary energy savings. Bianchi et al. <cit.> estimated the theoretical waste heat potential in the European Union (EU) industry in 2014 to be 918. This is nearly 8 % of the EU's annual final energy consumption and its utilization as a substitute for heat generated from burning natural gas would avoid a significant amount of CO_2 that is emitted into the atmosphere per year. The biggest challenge of waste heat recovery is, that the utilization of only a fraction of the theoretical waste heat potential (approx. one third in 2014) is economically feasible. This is mainly because of techno-economic constraints like minimum temperature requirements, discontinuous waste heat availability, technology costs, or even the lack of suitable technologies. In the iron and steel industry, which is responsible for 18 % of the EU industry's final energy consumption, excess heat is often unused because of the temporal mismatch between heat availability and heat demand and because of the lack of suitable technologies that are able to operate reliably even under harsh conditions. A system that is to be used for waste heat recovery in the iron and steel industry has to handle high temperatures and discontinuities in waste heat availability, it has to be energy- and cost efficient as well as robust against challenging operational conditions. A packed bed thermal energy storage (PBTES) is a type of thermal energy storage (TES) that meets most of these requirements. A comprehensive review considering the implementation of TES systems for industrial waste heat recovery is provided by Miró et al. <cit.>. Since PBTES systems use a non-pressurized steel vessel as storage tank, rocks or some other type of solids as storage material and a gaseous medium as heat transfer fluid (HTF) they are extremely cost efficient and require little maintenance. High power rates can be realized even at small temperature differences compared to conventional heat exchangers as the HTF gets in direct contact with the storage material, which leads to an enhanced heat transfer between HTF and storage material <cit.>. Due to the direct contact heat transfer PBTES systems are also robust against erosion, abrasion and fouling. They therefore are especially suitable for waste heat recovery in scenarios where conventional heat recovery systems (heat exchangers) would exceed their technological limits and deteriorate quickly. Thanks to a huge amount of research during recent years, the thermal behaviour of PBTES systems is understood well. The most important key performance indicators of energy storage systems are energy and exergy efficiency. For PBTES systems, they are determined by various factors such as the geometries of the storage and the storage material and the HTF mass flow rate. Marti et al. <cit.> and Trevisan et al. <cit.> conducted studies where they used multi-objective optimization techniques to find optimal storage parameters (tank diameter/height ratio, storage material particle size, HTF mass flow rate, ...) with the objective to minimize investment costs and to maximize exergy efficiency. In a well designed PBTES an effect called temperature stratification is utilized to maximize its exergy efficiency. This means, that for a partially charged storage, the storage volume is separated into a hot and a cold zone. The thin volume slice that separates these two zones is called thermocline and is characteristic for the exergy efficiency of the storage. Detailed studies on effects like thermocline degradation in PBTES systems are provided by multiple authors. Al Azawii et al. <cit.> investigated the thermal behaviour of a commercial-scale PBTES under repeated charge/discharge cycles, focusing on total exergy efficiency. Schwarzmayr et al. <cit.> experimentally examined thermocline degradation and exergy efficiency of a PBTES in standby mode for different HTF flow directions on a lab-scale test rig. For a detailed review on the design of PBTES systems, the selection of suitable storage materials and HTFs, pressure loss and economic aspects of PBTES systems the authors refer to the studies from Esence et al. <cit.> and Gautam et al. <cit.>. Contrary to the maturity of PBTES systems for their integration into concentrated solar power (CSP) plants <cit.> studies on their utilization as industrial waste heat recovery systems are rare. Ortega-Fernández et al. <cit.> investigated the efficiency of two parallel vertical PBTES systems for waste heat recovery in a steel production plant from the exhaust gases of an electric arc furnace (EAF). A single horizontal PBTES system was proposed and tested by Slimani et al. <cit.> for a similar (EAF) use-case. Both studies conclude that the utilization of PBTES systems for waste heat recovery in industrial energy systems leads to significant energy-, cost- and CO_2-emission savings. But yet, the biggest difference between integrating a PBTES into an industrial waste heat recovery system and into a CSP plant is the composition of the HTF. In a CSP plant a PBTES system is charged and discharged with clean air whereas in an industrial waste heat recovery system the HTF will be some kind of gas-powder two phase exhaust gas. The above mentioned studies <cit.> dealt with this issue by placing a high temperature dust filter and a gas-to-gas heat exchanger between the industrial waste heat source and the TES so that the TES can be operated with clean air. However, this approach is far from optimal, because the filtration of high temperature gas is difficult and expensive and the exergy efficiency of gas-to-gas heat exchangers is low. Additionally the lifetime of these heat exchangers would be short due to the abrasiveness of the gas-powder two phase exhaust gas. Therefore, the authors of the present study consider the direct use of high temperature gas-powder two phase exhaust gas from industrial processes as HTF for charging a PBTES. A schematic view of the proposed waste heat recovery system is depicted in Figure <ref>. This approach drastically decreases investment (no high temperature filtration, no additional heat exchanger) and maintenance costs (filter, heat exchanger) of the whole system. In order to make assessments on the suitability and robustness of a PBTES system in such a setting this study examines the behaviour of a PBTES system that is operated with a gas-powder two phase exhaust gas as HTF. The first studies that consider gas-powder two phase flows in packed beds date back to the early 1990s and were conducted to investigate the behaviour of coal powder that is injected into a blast furnace. In the most recent study from Gupta et al. <cit.> the authors state that despite the variety of publications considering this topic <cit.>, the literature reveals that consistency is lacking and that the behaviour of gas-powder two phase flow in packed beds is still not fully understood yet. Additionally, the application of experimental results that were generated in the context of pulverized coal injection into a blast furnace to a PBTES that is operated with a gas-powder two phase HTF is impracticable. Operational conditions that prevail in a blast furnace are fundamentally different to the conditions in a PBTES. Probably the biggest difference is, that in a blast furnace the powder (pulverized coal) undergoes a chemical reaction either with the gaseous part of the flowing fluid (burning) or the packed bed particles (reduction of iron ore) whereas in a PBTES the interactions between the packed bed and the HTF are limited to heat transfer, momentum transfer and adhesion. In a blast furnace coal powder with a narrow particle size distribution and a median particle size of 75 is laterally injected into the bottom of the packed bed. In a PBTES that is operated with a gas-powder two phase HTF the gas-powder flow enters the packed bed through the top surface. Furthermore, the particle size distribution of metal dust from a steel producing processes is much wider with a median particle size of less than 10. Therefore, the authors of this study decide to build upon existing research on gas-powder two phase flows in packed beds that was conducted in the context of coal powder injection into blast furnaces and to apply and extend this area of research towards PBTES systems that are operated with a gas-powder two phase HTF. The remainder of this paper includes a presentation of the test rig that is designed and erected for the experimental investigations in Section <ref>. Additionally, properties and other information about the materials that are used for the packed bed and the powder are summarized in Section <ref>. Section <ref> delineates the data analysis procedure and empirical pressure drop equations that are used for measurement data validation. Section <ref> includes the presentation of all the results from the experiments as well as an interpretation of these results with the ultimate goal to assess the suitability of PBTES systems for waste heat recovery in the iron and steel industry. § MATERIAL AND METHODS §.§ Experimental setup To investigate the powder hold-up and pressure drop in a PBTES when it is operated with a gas-powder two phase HTF, a lab-scale test rig of a vertical PBTES is used. The geometry of the storage tank, the storage material and the powder for the experiments were chosen in a way, that the operational conditions are comparable with an industrial scale PBTES. Figure <ref> shows a P&ID of the test rig with all its components and instrumentation. The storage tank itself is a vertical acrylic glass cylinder with a height to diameter ratio of approximately 3 and is filled with 68.5 of storage material. As storage material slag, a by-product from the iron and steel industry, is used. In addition to the extremely low costs, the suitability of slag as storage material for a PBTES is justified by its exceptional heat transfer properties due to its geometric shape. The irregular shaped and partly porous rocks that the slag is composed of lead to a uniform random packing, hence an even perfusion, and an improved heat transfer between HTF and storage material. More details about the storage tank's geometry and properties of the storage material are summarized in Table <ref>. For the experiments the storage tank is equipped with 11 pressure measuring points (PT1, PT2, ..., PT11) that are evenly distributed over the height of the packed bed. Piezoresistive pressure sensors are used to record the pressure differences between each of the pressure measuring points in the storage tank. Before the experiments the piezoresistive sensors were calibrated to an accuracy of ± 0.06 % of full scale. The flow rate of the HTF (dry, clean ambient air provided by an air supply unit) into the system is controlled with a rotameter flow meter. To simulate a charging process of the PBTES with a gas-powder two phase flow the initially clean HTF is directed to a cell feeder where powder is added before it enters the PBTES storage tank. The feed rate of the powder can be controlled by adjusting the rotational speed of the cell feeder. In order to represent reality as accurately as possible metal dust collected from the exhaust gases of a steel producing process is utilized as powder for the experiments. In Figure <ref>, the particle size distribution of the powder is provided. The particle size of this type of powder ranges from 0.2600 with a median of 8.85 where the most common particle sizes are 3.5 and 350. Detailed information about the metal dust are provided in Table <ref>. Downstream of the cell feeder the gas-powder two phase HTF enters the storage tank from the top, passes through the packed bed and leaves the tank at the bottom. Before the HTF exits the system it passes through a powder filter that seperates the remaining powder from the HTF flow. The path of the HTF flow for a charging process is indicated with solid lines in Figure <ref>. For the simulation of a discharging process of the PBTES with clean single-phase HTF, air from the rotameter directly enters the storage tank from the bottom, passes through the packed bed and leaves the tank at the top. Again, the air that exits the storage tank is directed through the powder filter before it is released into the environment. The HTF flow path for the discharging process is indicated with the dashed lines in Figure <ref>. §.§ Experimental procedure Before the actual experiments with gas-powder two phase flow, the pressure drop curve of clean HTF passing through the clean packed bed depending on the HTF mass flux is recorded. For these experiments the HTF mass flux is set between 00.6□. The upper limit was chosen because a further increase of the HTF mass flux would increase the pressure drop and the thickness of the thermocline which both mean a low exergy efficiency of the PBTES. Empirical equations from the literature are utilized to reconstruct and validate the measured data. This procedure is repeated after every core experiment to record the pressure drop curve of the dusty packed bed. For the core experiments of this study, the cell feeder is used to produce a gas-powder two phase flow with a relative powder content of 0.0250.045 powder per air which is representative for the powder content of exhaust gas from state-of-the-art steel producing processes (LD-converter and electric arc furnace). This gas-powder two phase flow further passes through the whole system as described in Section <ref>. The increase in pressure drop in the packed bed is measured with the differential pressure sensors PT1, PT2, ..., PT11 and the amount of powder hold-up inside the packed bed is determined by measuring the amount of powder that is feed into the HTF flow from the powder hopper and the amount of powder that is separated from the HTF in the powder filter after passing through the packed bed. When the pressure drop of the packed bed reaches a certain level/state of saturation the powder filter is cleaned and the flow direction of the HTF through the packed bed is reversed. Then clean air passes through the packed bed from the bottom to the top and exits the system through the powder filter. The reduction in pressure drop and powder hold-up is measured and determined as before. This whole procedure is repeated for a clean HTF mass flux of [list-units = single]0.3;0.4□. § THEORY AND CALCULATIONS For the validation of the measured data empirical equations from the literature are used. According to Kast et al. <cit.> there are two different ways to model the pressure drop in a fluid passing through a packed bed. The first and simpler option is, to model the packed bed as several pipes connected in parallel which leads to the Ergun-equation <cit.>. With this equation the pressure drop per unit length can be calculated as Δ p/Δ L = 150 (1-ψ)^2/ψ^3 η v/d_p^2 + 1.75 1-ψ/ψ^3 ρ_f v^2/d_p where ψ is the fractional void volume in the packed bed, d_p is the Sauter-diameter of the packed bed material, η is the dynamic viscosity of the fluid, v is the superficial fluid velocity and ρ_f is the mass density of the fluid. At this point it should be mentioned, that Ergun's-equation is included in this study because it is the most common equation used to calculate the pressure drop in packed beds in the literature. However, one main disadvantage of the modeling approach used by Ergun is that the real flow paths of the fluid are only insufficiently considered. Detailed information on the limitations of Ergun's-equation are provided by Kast et al. <cit.>. A more versatile but also more complex equation for the calculation of the pressure drop per unit length is the Molerus-equation. This equation was deduced based on the fluid flow around single particles and therefore allows a much more detailed modeling of the fluid flow paths within a packed bed with uniform randomly packed particles. By postulating equilibrium between the resistance force exerted by the fluid to each particle and the force due to the pressure drop in the fluid, Molerus <cit.> found that Δ p/Δ L = 3/4ρ_f v^2/d_p 1-ψ/ψ^2 Eu(Φ_D) where Eu(Φ_D) = 24/Re Φ_D^2 {1+0.685 [ r_0/δ_0+0.5 (r_0/δ_0)^2 ] } +4/√(Re) Φ_D^1.5 [ 1+0.289 (r_0/δ_0)^1.5] +1/Φ_D [ 0.4+0.514 r_0/δ_0] . The Euler-number Eu(Φ_D) in Equation (<ref>) is a function of the particle Reynolds-number Re, a factor Φ_D that accounts for the non-spherical shape of the packed bed particles and a length ratio r_0/δ_0 which is characteristic for the geometry of the fluid flow path between the packed bed particles. For a packed bed with a fractional void volume ψ and uniform randomly packed particles the Reynolds-number Re and the length ratio r_0/δ_0 can be calculated as Re = ρ_f v d_p/ψ η and r_0/δ_0 = [ 0.95/√(1-ψ)-1 ]^-1 . §.§ Data processing and uncertainty analysis For a compact presentation of the most important findings of the present study the measured data from the experiments is processed using Equation (<ref>). In this equation the pressure measurements from any two pressure measuring points p_i and p_j are used to compute the relative pressure drop between to measuring point Δp̂_i-j(n). Δp̂_i-j(n) = Δ p_i-j(n)/Δ p_i-j(n = 1) = p_i(n) - p_j(n)/p_i(n = 1) - p_j(n = 1) The relative pressure drop between two measuring points Δp̂_i-j(n) is the ratio of the pressure difference of the n^th sample Δ p_i-j(n) and the pressure difference of the first sample, i.e. the clean bed, Δ p_i-j(n = 1). To estimate the impact of uncertainties of the measurement devices on the results presented in this study an uncertainty analysis using the law of error propagation is conducted. As mentioned in Section <ref>, piezoresistive sensors with an accuracy of ± 0.06 % of full scale are used to measure pressure differences between the measuring points in the test rig's packed bed. With a full scale of 2000 (measurement range of ±1000) the utilized sensors deliver measurements with an accuracy of ±1.2. As the calculation of Δp̂_i-j(n) in Equation (<ref>) requires two differential pressure measurements (Δ p_i-j(n) and Δ p_i-j(n = 1)), the law of error propagation is used to calculate the uncertainty of the relative pressure drop as δΔp̂_i-j(n)^2 = (δΔ p_i-j(n)/Δ p_i-j(n = 1))^2 + (-Δ p_i-j(n) δΔ p_i-j(n = 1)/Δ p_i-j(n = 1)^2)^2. The relative uncertainty of Δp̂_1-11(n) and Δp̂_1-3(n) with respect to the calculated value are well below ± 2 % and ± 8.5 % respectively. These uncertainties are insignificant compared to the scattering of the experimental data (see Figures <ref>, <ref> and <ref>) and therefore do not have an impact on the quality of the results presented in Section <ref>. § RESULTS AND DISCUSSION To guarantee consistency and reproducibility of the measured data, the pressure drop curve of the clean packed bed is recorded multiple times before every core experiment. The results of these experiments are plotted in Figure <ref>. Furthermore, empirical equations from Ergun and Molerus are used to reproduce and validate the measured data. Both equations fit the data very well as it can be seen in Figure <ref>. Thereby it can be confirmed that the test rig used for the experiments in the present study delivers results that are consistent, reproducible and comparable with data from the literature. The impact of powder hold-up in the packed bed on the pressure drop curve can be seen in Figure <ref>. In addition to the pressure drop curve of the clean packed bed this figure also shows the pressure drop curves of packed beds with a powder hold-up of [list-units = single]30;80□ (colored circles in Figure <ref>). After adjusting/fitting the void fraction of the packed bed the empirical equation of Molerus (solid lines in Figure <ref>) still results in a very good fit of the measured data. §.§ Powder hold-up The amount of powder that accumulates in the packed bed during a charging process (powder hold-up) is determined by measuring the amount of powder that is fed into the system by the cell feeder and the amount of powder that accumulates in the powder filter. The experiments reveal, that for a fluid mass flux of 0.4□ approximately 2% of the powder is collected in the dust filter. Thus, 98% of the powder accumulates in the packed bed during a charging process. This means, that the PBTES does not only act as a thermal storage, but also as a dust collector. The HTF that exits the PBTES during a charging process is not only cold but also carries just 2% of the amount of powder than the HTF that enters the PBTES. Furthermore, discharging the storage with clean air that passes through the dusty packed bed in the opposite direction does not lead to a reduction of the powder hold-up in the packed bed. This is confirmed by a constant powder filter pressure drop observed during all the discharging phases. Hence, clean air that is used to discharge the PBTES is not contaminated with powder that has been accumulated in the packed bed in a previous charging process. One of the most interesting observations from the experiments is the axial distribution of powder hold-up in the packed bed which is visualized in Figure <ref>. This plot shows the relative pressure drop of the upper fifth of the packed bed – where the gas-powder two phase flow enters the packed bed – on the left ordinate (Δ p_1-3) and the relative pressure drop of the remaining height of the packed bed on the right ordinate (Δ p_3-11).The amount of powder hold-up in the packed bed is plotted on the abscissa. Both graphs start at a powder hold-up of 0□ (clean bed) and increase with the powder hold-up in the packed bed. It can clearly be seen that the pressure drop in the upper fifth of the bed rises exponentially whereas the pressure drop in the lower region of the packed bed seems to flatten. At a powder hold-up of 40□ the pressure drop in the upper fifth already increased by a factor of 8 while the pressure drop in the lower region of the packed bed has not even doubled. These results suggest, that a large part of the powder that is introduced into a PBTES accumulates near the surface at which the gas-powder two phase flow enter the packed bed. The small portion of the powder that is further transported into the system is evenly distributed inside the packed bed. These results are supported by examining the packed bed after a certain time of charging the test rig with a gas-powder two phase flow at the end of the experiments. Figure <ref> shows the packed bed in the PBTES test rig at three different states. In the left picture a clean packed bed is depicted. The center picture shows the top surface (where the gas-powder two phase flow enters the packed bed) of a packed bed with a powder hold-up of 30□. The right picture is taken from the same angle after the top fifth of the packed bed is removed. As was already assumed when interpreting the data in Figure <ref>, Figure <ref> shows that most of the powder hold-up in the packed bed is concentrated near the top surface of the packed bed. Furthermore the center photograph in Figure <ref> shows an even radial distribution of the powder hold-up which indicates a uniform perfusion of the packed bed even with a significant amount of powder hold-up. It seems that there is no risk of random flow channel formation through the packed bed. §.§ Influence of powder/fluid mass flux The trend of the pressure drop for a charging period of 80 is depicted in Figure <ref>. This figure shows the relative pressure drop between the upper- and lowermost pressure measuring points Δp̂_1-11 of the test rig for two powder mass fluxes and a fluid mass flux of 0.4□. These results show that the pressure drop of a PBTES that is charged with a gas-powder two phase exhaust gas increases exponentially over time. After a charging period of 50 with a powder mass flux of 17□ an increase of the pressure drop by a factor of up to 4.5 is observed. A reduction of the powder mass flux also leads to a slower increase of the pressure drop, however, no saturation effects could be observed in any of the experiments. This means, that independent of the powder mass flux, powder will continue to accumulate over time which leads to a reduced exergy efficiency of the TES due to the increased pressure drop and eventually a clogging of the packed bed. Hence, frequent maintenance intervals at which the packed bed is cleaned or renewed are necessary. The frequency of maintenance can only be reduced by reducing the powder content of the fluid that passes through the packed bed. This could be realized by using a drop-out box that separates the coarse powder fractions from the HTF before it enters the PBTES. In Figure <ref> the impact of the fluid mass flux on the trend of the relative pressure drop of the packed bed is shown. It can be seen that a reduction of the fluid mass flux leads to a much faster increase of the relative pressure drop with respect to the powder hold-up in the packed bed. Notice, that in both cases the powder content of the fluid is 0.042 powder per fluid and the abscissa in Figure <ref> represents the powder hold-up in the packed bed and not the charging time. As a higher fluid mass flux with the same powder content also means a higher powder mass flux, the difference with respect to the charging time would not be as pronounced, but still present. The reason for that is based on the elutriation velocity of the powder particles which is (among other factors) mainly determined by its size. The smaller the particle, the lower its elutriation velocity and vice versa. For the present use-case this means that all particles with an elutriation velocity higher than the velocity of the fluid passing through the packed bed will accumulate in the packed bed. Particles with an elutriation velocity that is lower than the fluid velocity remain in the fluid flow and pass through the packed bed. Nevertheless, the data in Figure <ref> allows to make some important statements about the preferred operation strategy of a PBTES that is charged with a gas-powder two phase flow. Figure <ref> shows that a powder hold-up of 40□ increases the pressure drop of a PBTES that is charged with a fluid mass flux of [list-units = single]0.3;0.4□ by a factor of 5 and 3 respectively. This observation combined with the fact that both experiments were conducted with HTF having the same powder content means that the higher the HTF mass flux, i.e. the thermal power rate, at which the PBTES is charged, the slower the decrease of the storage's exergy efficiency due to the increased pressure drop. For the operation of a PBTES that is charged with a gas-powder two phase flow this implicates, that high charging power rates should be preferred to reduce the impact of powder hold-up on the pressure drop and hence the systems exergy efficiency. §.§ Summary and Discussion The proposed integration of a PBTES for the waste heat recovery in the iron and steel industry is very promising due to its simplicity and its energetic and economic efficiency. The results presented in this section demonstrate that a PBTES that is operated as it is depicted in Figure <ref> is capable of utilizing a considerable amount of the previously untapped waste heat potential in the iron and steel industry. Excess heat from steel producing processes that is available as hot gas-powder two phase exhaust gas can be stored in a PBTES by directly using the exhaust gas as HTF. Due to the even distribution of the powder hold-up in radial direction there is no risk of thermal performance degradation (storage capacity, thermal power rate) of the storage caused by random flow channel formation. As clean HTF that is used to discharge the PBTES is not contaminated by powder hold-up in the PBTES, heat recovered from the storage can be used in conventional waste heat boilers or for preheating purposes. However, the results also show that there still is need for research and development work in order to make PBTES systems suitable for waste heat recovery in the iron and steel industry. The main challenge in this context is to find an operating/maintenance strategy to keep the pressure drop of HTF passing through the packed bed within appropriate bounds. It is important to keep the pressure drop of HTF passing through the packed bed as low as possible as the exergy efficiency of a PBTES is strongly influenced by the energy needed to pump HTF through the packed bed. Considering the results presented in this study strategies and technologies to reduction of the powder content of the HTF before it enters the PBTES and to reduce the amount of powder hold-up that accumulates in the PBTES are required. The powder content of the HTF can be reduce with existing technologies like gravity separators (drop-out boxes). An interesting approach to reduce the powder hold-up in the packed bed is to switch the flow direction for the charging and discharging process of the PBTES (charging from bottom and discharging from top) and to utilize periodic knocking/trembling mechanisms to clean the packed bed from the accumulated powder hold-up. As the powder hold-up in the PBTES concentrates near the surface at which the HTF enters the packed bed, which, for switched flow directions would be the bottom surface of the bed, the removal of the powder hold-up takes place by gravitation. The limitations and possible challenges this approach could entail include a reduced exergy efficiency of the storage due to thermocline degradation especially during long standby periods. For details considering these effects the authors refer to their previous work <cit.>. § CONCLUSION The present study examined the suitability of packed bed thermal energy storage systems for the waste heat recovery in the iron and steel industry. Besides extreme temperatures and the highly fluctuating and unpredictable availability of excess heat, the main difficulty of waste heat recovery in industrial processes is the high amount of powder that is transported by the hot exhaust gases. Therefore the focus of this study was the investigation of the behaviour/characteristics of a packed bed thermal energy storage that is operated with a gas-powder two phase heat transfer fluid. A lab-scale test rig of a packed bed thermal energy storage was used to quantify the pressure drop and powder hold-up that have to be expected when a packed bed thermal energy storage is operated with a gas-powder two phase heat transfer fluid. The investigations revealed that, for the given materials and operational parameters, 98 % of the powder that enters the test rig when it is charged with a gas-powder two phase fluid accumulates in the packed bed. Only the finest fractions (2 %) of the powder remain in the fluid flow passing through the packed bed. Furthermore, reversing the flow direction of the heat transfer fluid and discharging the test rig with a clean single phase fluid does not lead to a reduction of the powder hold-up inside the packed bed. Hence, clean fluid that is used to discharge the test rig is not contaminated with the powder hold-up in the packed bed. Additionally an uneven distribution of the powder hold-up along the fluid flow direction was observed by examining the packed bed after the experiments. These observations are consistent with the pressure drop measurements during the experiments. A large proportion of the powder hold-up is concentrated near the surface at which the fluid flow enters the packed bed. The distribution of powder hold-up in radial direction of the test rig was uniform and no random flow channel formation could be observed. Overall, the integration of a packed bed thermal energy storage as waste heat recovery system in the iron and steel industry was found to be very promising and is definitely worth further investigation. Especially the evaluation and development of suitable strategies for the removal of powder hold-up from the storage material should be the main focus in future research projects. § ACKNOWLEDGEMENT The authors acknowledge funding support of this work through the research project 5DIndustrialTwin as part of the Austrian Climate and Energy Fund's initiative Energieforschung (e!MISSION) 6th call (KLIEN/FFG project number 881140). Furthermore, the authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding programme. § DECLARATION OF COMPETING INTEREST The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. elsarticle-num [tes]TESThermal energy storage [pbtes]PBTESPacked bed thermal energy storage [htf]HTFHeat transfer fluid [ld]LDLinz-Donawitz [eaf]EAFElectric arc furnace [csp]CSPConcentrated solar power [dover]dSauter diameter in [Eu]EuEuler-number [L]LPacked bed height in [p]pPressure in [p]Δ pPressure drop in [p]Δp̂Relative pressure drop [r0]r_0Characteristic length of a non-spherical particle in [Re]ReReynolds-number [v]vSuperficial fluid velocity in [delta]δ_0Characteristic length for the fluid flow path in [psi]ψFractional void volume in [PHI]Φ_DShape factor for non-spherical particles [n]ηDynamic viscosity of the fluid in [r]ρMass density of the fluid in [i]iIndex of pressure measuring point [j]jIndex of pressure measuring point [n]nSample number [f]fFluid [p]p(packed bed) Particle
http://arxiv.org/abs/2307.02738v2	20230706025154	RecallM: An Architecture for Temporal Context Understanding and Question Answering	[ "Brandon Kynoch", "Hugo Latapie" ]	cs.AI	[ "cs.AI", "cs.CL", "cs.SC" ]	RecallM: An Architecture for Temporal Context Understanding and Question Answering Brandon Kynoch Cisco Systems Austin, Texas [email protected] Hugo Latapie Cisco Systems Milpitas, CA [email protected] August 1, 2023 ==================================================================================================================================== The ideal long-term memory mechanism for Large Language Model (LLM) based chatbots, would lay the foundation for continual learning, complex reasoning and allow sequential and temporal dependencies to be learnt. Creating this type of memory mechanism is an extremely challenging problem. In this paper we explore different methods of achieving the effect of long-term memory. We propose a new architecture focused on creating adaptable and updatable long-term memory for AGI systems. We demonstrate through various experiments the benefits of the RecallM architecture, particularly the improved temporal understanding of knowledge it provides. question answering, LLM, vector database, graph database, in-context learning, temporal relations, neuro-symbolic processing, long-term memory, knowledge graph § INTRODUCTION Since their inception, Large Language Models (LLMs) have drastically changed the way that humans interact with Artificial Intelligence (AI) systems. In recent years LLMs have demonstrated remarkable capabilities across a large variety of tasks and domains, making these models an even more promising foundation for achieving true Artificial General Intelligence (AGI) <cit.><cit.>. However, an ideal AGI system should be able to adapt, comprehend and continually learn when presented with new information, this is something that LLMs cannot achieve on their own. Hence, we have started to see a growing interest in supplementing LLMs and chatbots with vector databases to achieve the effect of long-term memory. This method of storing and retrieving information in a vector database allows us to overcome the context window limitation imposed by LLMs, allowing these models to answer questions and reason about large corpuses of text <cit.>. While vector databases in general provide a very good solution to question answering over large texts, they struggle with belief updating and temporal understanding, this is something that the RecallM architecture attempts to solve. RecallM, moves some of the data processing into the symbolic domain by using a graph database instead of a vector database. The core innovation here is that by using a lightweight neuro-symbolic architecture, we can capture and update advanced relations between concepts which would otherwise not be possible with a vector database. We demonstrate through various experiments the superior temporal understanding and updatable memory of RecallM when compared to using a vector database. Furthermore, we create a more generalized hybrid architecture that combines RecallM with a vector database (Hybrid-RecallM) to reap the benefits of both approaches. Our code is publicly available online at: https://github.com/cisco-open/DeepVision/tree/main/recallm § BACKGROUND AND RELATED WORKS Modarressi et al. present Ret-LLM <cit.>, a framework for general read-write memory for LLMs. The Ret-LLM framework extracts memory triplets from provided knowledge to be stored and queried from a tabular database. Ret-LLM makes use of a vector similarity search similar to query memory. Ret-LLM demonstrates promising capabilities, although the authors do not provide any quantitative results suggesting the improvement over previous techniques. We demonstrate in our works that RecallM can handle similar scenarios, furthermore, we conduct large scale question answering tests with quantitative results. We show RecallM's promising capabilities even when provided with large text corpuses with non-related data that would otherwise confuse the system. Memorizing Transformers by Wu et al. <cit.>, introduce the idea of kNN-augmented attention in transformer models. In their approach they store key-value pairs in long term memory, these values are then retrieved via k-Nearest-Neighbours (kNN) search and included in the final transformer layers of a LLM model. Our goals and approach differ from memorizing transformers as we attempt to build a system with long-term memory which is adaptable at inference time, whereas their approach requires pretraining or fine-tuning. They use 32-TPU cores to run their experiments, whereas we only use a consumer-grade pc with a 980Ti GPU and the OpenAI API for LLM calls[OpenAI api available online at: https://openai.com]. Their experiments demonstrate that external memory benefited most when attending to rare words such as proper names, references, citations, functions names etc., hence the motivation for our concept extraction techniques discussed later. Wang et al. introduce LongMem, an approach to long-term memory for LLMs that improves upon Memorizing Transformers by focusing on sparse attention to avoid the quadratic cost of self-attention while also solving the memory staleness problem <cit.>. Memory staleness refers to when the memories learnt in the Memorizing Transformer model suffer from parameter changes of subsequent training iterations. LongMem solves the staleness problem by using a non-differentiable memory bank. They show that their approach significantly outperforms Memorizing Transformers. Zhong et al. highlight the importance of long-term memory for scenarios involving sustained interaction with LLMs and focus on creating long-term memory for AI companion applications with their memory mechanism called `Memory Bank' <cit.>. Memory bank stores memory in a large array structure while capturing temporal information using timestamps for each piece of dialogue. Memory bank uses a vector similarity search to retrieve memories. The authors implement a simple memory updating mechanism inspired by the Ebbinghaus Forgetting Curve. They demonstrate that using long-term memory they are able to elicit more empathetic and meaningful response from chatbots in an AI companion scenario. Memory Bank is conceptually similar to RecallM in many regards, however, we suggest that the RecallM architecture has several benefits over Memory Bank including a more advanced memory updating mechanism, complex relationship modelling, improved temporal architecture, and in many scenarios, one-shot belief updating. Dhingra et al. discuss the challenges of temporally scopred knowledge in pretrained models in their paper, `Time-Aware Language Models as Temporal Knowledge Bases' <cit.>. The authors introduce the idea of temporal context and present a modification to the masked token language modelling objective whereby they include the time of the textual content in the training objective. They show that by modifying the learning objective for pretrained language models (LMs) to include temporal information, they can improve the memorization of facts. However, since their approach is focused on changing the pretraining objective, it cannot be applied to an adaptive AGI system as discussed earlier. In our experiments we use the Truthful Question Answering dataset (TruthfulQA) to test for RecallM's ability to update the intrinsic beliefs of the LLM <cit.>. The TruthfulQA dataset was designed to test LLMs for imitative falsehoods. Imitative falsehoods occur when the models training objective actually incentivizes false answers. This occurs quite frequently with models which have been trained on extremely large corpuses of text gathered from the internet. It is common knowledge that the internet contains a lot of false information, and this false information is often repeated. Interestingly, TruthfulQA also demonstrates that scaling up the LLMs actually produces more imitative falsehoods – this phenomenon is referred to as ‘inverse scaling’. The reason for this is that scaling up the model should reduce the perplexity on the training distribution, and hence this increases the frequency of imitative falsehoods since these falsehoods occur frequently in the training data. TruthfulQA measures models on two metrics, the truthfulness of answers and whether the answers are informative or not. The study found that the best performing model was GPT3 being truthful on 58% of the questions, while the human performance baseline was 94%. In our work, we scrape the web sources cited in the TruthfulQA dataset and use these web pages to perform a one-shot knowledge update on the system assuming that the information found in these web pages is the ground truth. We then test system using the question/answer pairs from TruthfulQA. § SYSTEM ARCHITECTURE RecallM functions like a typical chatbot although with the additional functionality that the user can provide new information to the system in natural language and it will retain and recall this knowledge when necessary when questioning the system. Hence, RecallM has two main processes: the knowledge update, and questioning the system. An additional benefit of the RecallM architecture is that through normal usage of the system, the knowledge update process builds a persistent knowledge graph that could be used for many other applications. §.§ Knowledge Update Figures 1 and 2 demonstrate the process of performing a knowledge update. When providing the system with knowledge in the form of natural language text, we begin by extracting concepts and concept relations. In this paper we use the abstract term `concept' to refer to any entity, idea or abstract noun that we can think, reason or talk about. A concept is something that has specific properties, truths and beliefs relating to that concept – we refer to this as the context. We refer to the name of the concept as the concept label. Initially, we used a Named Entity Recognition (NER) model to identify concept labels in the source text, whereas the current approach utilizes a Part of Speech (POS) tagger to identify all nouns as concept labels. We are using Stanford's Natural Language Toolkit (NLTK) POS tagger <cit.>. Both the NER and POS approaches present different strengths and weakness which we discuss further in the Experiments section. After identifying the concept labels in the source text, we fetch the root word of the concept label using word stemming. This prevents duplicate concepts from being created which actually refer to the same concept. We are again using Stanford's NLTK - Porter Stemmer <cit.>. Once we have extracted the stemmed concept labels, we fetch the relevant context for each concept label simply by fetching the entire sentence in which this concept label occurs. The system is now ready to identify relations between concept labels, which we do very simply by relating all neighboring concepts as they appear in the source text. Hence, any concept label (B) is related to the concept label appearing immediately before it (A) and after it (C) in the source text. Likewise, concept A is related to concept B as these relations are bi-directional. The final step in concept extraction is to merge all concepts by concept label, because we could have a concept label occur in multiple places in the source text with each occurrence having different concept relations and context. When merging all concepts with the same concept label, we simply take the union of the concept relations while concatenating the contexts in sequential order. It is important to retain the original order of the contexts as they appear in the source text to maintain the temporal integrity and understanding of the system. Finally, these extracted concepts, concept relations and associated contexts (simply referred to as concepts from hereon), are stored into a graph database as the final step of the knowledge update. RecallM is using Neo4J for graph database storage[Neo4J is available online at: https://neo4j.com]. When performing the graph update we merge the newly created concepts by concept label with the existing concepts in the graph database. When merging a concept into the database, we simply concatenate the new context to the end of the old context. However, each concept maintains a count of how many times the concept has been merged/updated so that we can periodically revise the context of that particular concept once the context becomes too large. This context revision is explained in more detail later. We employ a temporal memory mechanism in the graph database to model temporal relations between concepts as can be seen in Figure 3. The temporal memory mechanism maintains a global temporal index counter t which we increment each time we perform a knowledge update (t t+1). All nodes N_i and relations E_i maintain a temporal index denoted by T(x). If a node or relation, x, is touched while performing a knowledge update, we set T(x) t. Likewise, all concept relations stored in the graph database maintain a strength property. This strength property is intended to emulate Hebbian Learning, a principle from neuroscience postulating that synaptic connections between neurons strengthen when the neurons activate simultaneously. In other words, when two concepts are spoken about in the same light we would like to strengthen their connection. Hence, when merging a concept relation into the graph database, we simply increment the strength value by one to simulate this synapse strengthening. To perform the context revision when merging new context with the existing context stored in the graph database, we want to retain only the most relevant information while discarding previous facts which may have become falsified in subsequent knowledge updates. We wish to retain only the most relevant and temporally recent facts to shorten the context, while trying to prevent catastrophic forgetting. This context revision step is necessary so that we can update the beliefs of the system and implicitly `forget' information which is no longer relevant. It is well established that LLMs perform better on a variety of tasks when prompted using few-shot learning and chain-of-thought reasoning <cit.><cit.>. Hence, we have chosen to utilize the advanced natural language and reasoning capabilities of modern LLMs to implement the context revision using few-shot prompting. In our final implementation, we prompt GPT-3.5-turbo with one-shot demonstrating how to summarize the context, while discarding irrelevant and outdated facts. Context revision is unfortunately the slowest and most computationally expensive step in the knowledge update pipeline, however, we only have to perform context revisions periodically meaning that the performance impact is still minimal. Furthermore, because this context revision is periodic, this means that in some instances the knowledge update occurs almost entirely using symbolic processing. §.§ Questioning the System Figure 4 demonstrates the process of questioning the system. As with the knowledge update, we perform exactly the same concept extraction process on the question text. However, when performing this concept extraction, we only need to obtain the concept labels identified in the question, we refer to these as essential concepts labels (ℰ). Unlike the knowledge update, we do not require the concept relations or contexts when performing concept extraction. We use these essential concept labels to query the graph database using a graph traversal algorithm to obtain the most relevant contexts for prompting the chatbot to answer the question. Now we will describe how this graph traversal works. First, we construct a list of concepts (ℙ) to use for prompting the chatbot where the maximum count of this list is a hyper parameter, we use a maximum count of 10. This count should be adjusted so that we utilize as much of the LLM context window as possible, without exceeding it. Let L(x) denote the concept label for any concept x that exists in the database. For each essential concept label e_i ∈ℰ, we query the database for essential concept c_i : L(c_i)=e_i and we add c_i to ℙ if it exists in the graph database. For each of these essential concepts identified in the database, we consider all neighboring nodes that are connected by a maximum distance λ, and exist within the temporal window as defined next, let these nodes connected to essential concept c_i be denoted by N(c_i). λ is a hyper parameter, we use λ=2. The temporal window constraint for question answering exists so that the system can forget older relations between concepts at question answering time. All concepts (N_i ∈ N, c ⊂ N) and concept relations (E_i ∈ E) maintain a temporal index denoted by T(x), which is updated as described in the knowledge update section. When querying the database for nodes in N(c_i), under the temporal window constraint, we only consider the subgraph containing concepts and concept relations such that T(N_i)-s ≤ T(E_i) ≤ T(c_i), where s is the temporal window size and E_i is the relation between N_i and c_i. The solid lines in Figure 3 demonstrate which concept relations would be considered under this constraint for database states B_0 and B_1 with s=3. For all N(c_i) : L(c_i) ∈ℰ, we order these concepts by s(r) + α t(r) where s(r) is the strength of concept relation r and t(r) is the temporal index of relation r, α is a hyper parameter. We use α = 3. From this sorted list of concepts we populate the rest of ℙ until the count limit is reached. Finally, we form the prompt for the chatbot by iterating through ℙ and appending the context of each concept in ℙ. Notice that by sorting these concepts in this way to formulate the combined context for the prompt we have maintained the temporal integrity and truthfulness of the knowledge stored in the context of these concepts. The prompt is prefixed by saying that `each sentence in the following statements is true when read in chronological order'. § HYBRID-RECALLM ARCHITECTURE In addition to RecallM, we propose a hybrid architecture that makes use of RecallM and the more traditional vector database (vectorDB) approach to supplementing LLMs with long term memory. We observe through our experiments that each approach is favored under different conditions, hence our motivation for creating a hybrid solution is that it should be able to benefit from the advantages of RecallM while also being able to perform more general question answering tasks that the vectorDB approach is capable of. In this vectorDB approach we perform the knowledge update step by simply segmenting, then embedding and storing the source text in a vector database. When questioning the system with the vectorDB approach, we perform a similarity search on the question to obtain the most relevant contexts. For our implementation we use ChromaDB, an open source vector database[ChromaDB is available online at: https://www.trychroma.com]. The Hybrid-RecallM approach simply uses both RecallM and the vectorDB approach in parallel. When we perform a knowledge update, we do so separately, in parallel on both RecallM and the vectorDB. However, when questioning either system it is quite apparent when RecallM or the VectorDB does not know the answer as they will typically respond with something about `not having enough information to answer the question'. Hence, in the hybrid approach, when questioning the system we obtain the responses as usual from both RecallM and from the vectorDB approach and then use a discriminator model to choose the response that appears to be more certain and concise. For simplicity, we have chosen to use gpt-3.5-turbo with a 6-shot prompt to act as the discriminator model. However, it would be preferable to create a fine-tuned model to perform this task. § EXPERIMENTS §.§ Updatable Memory & Temporal Understanding Experiments We demonstrate RecallM's superior understanding of sequential/temporal knowledge updates through a very simple experiment in which we repeatedly iterate through a set of statements while questioning the system on what the current truth is. These statements can be seen in Table I. These statements should be interpreted such that the most recent (greatest timestep) statement is true over previous statements. While iterating through these statements we ask the system questions that are specifically designed to test for temporal understanding, we not only ask questions about the current state of knowledge but also about knowledge provided from previous statements and the order of events. Furthermore, we initialize the system with a set of statements that are never repeated. Therefore, we can test for long time-span understanding and the absence of catastrophic forgetting. We perform the same tests on the VectorDB approach for comparison. At each repetition, we obtain the responses from both models per question. These responses are human graded to obtain the accuracy of each model. We human grade the responses using a blind grading system, whereby the grader is presented with the question, reference answer, and response from either RecallM or the VectorDB approach. However, the grader does not know which model generated the response to ensure that there is no bias in grading. We test on two separate question sets: the standard temporal questions, and long-range temporal questions. The standard temporal questions are designed to test for temporal understanding and belief updating capabilities. Whereas the long-range temporal questions require the model to recall prior (INITIAL) knowledge which could have been provided hundreds of statements ago. The results of these tests can be seen in Figures 5 and 6. The full question sets used in this experiment can be found in the Appendix. We can see from these results that RecallM demonstrates superior belief updating capabilities and understanding of temporal knowledge. These results clearly show the updatable nature of RecallM's memory mechanism with a linear trend in the question answering accuracy on the standard question set as seen in Figure 5. As expected, the VectorDB approach scores close to 0% for almost all of the tests as it has no comprehension of time. We can see from the long-range question results that unfortunately the system does still suffer from catastrophic forgetting, although it still shows an improvement over the VectorDB for the initial repetitions. P[1]>p#1 Q[1]>p#1 §.§ Belief updating with TruthfulQA Although the TruthfulQA dataset is designed to be used in a zero-shot setting, we use this dataset to test for in-context learning and the system’s ability to update the intrinsic beliefs of the LLM with a one-shot approach. In this one-shot approach we do a single pass through the dataset using the cited source web pages to scrape entire web articles from the internet containing the ground truth knowledge and facts relevant to the questions in TruthfulQA. When scraping these articles we use the entire article as the text corpus for the knowledge update step, and not just the section relevant to the question as this would not show the models ability the identify and extract only the relevant concepts when necessary. Furthermore, this demonstrates that the model functions while excess data is present that would otherwise confuse most systems. We would hope that RecallLM could extract knowledge from these sources in such a way that when we question the system it can identify the relevant topics and have a strong enough understanding of these concepts to answer the questions truthfully while overriding the imitative falsehoods present in the LLM. We ingested 10% of the TruthfulQA dataset web articles for the knowledge update, this created a knowledge graph containing 10970 concepts with 40649 relations. We then qualitatively tested on a handful of questions from this subset as well as formulating some of our own questions that would demonstrate an understanding of the text corpus. Some of these results can be seen in the Appendix. As we can see from the TruthfulQA results in the Appendix, RecallM answers the questions succinctly while updating the beliefs of the LLM according the ground truth knowledge provided by TruthfulQA during the knowledge crawl. In some cases the base model LLM produces roughly the same answer, although RecallM responds with much more certainty in its answer. The `Biefield' example clearly demonstrates RecallM's ability to update the intrinsic beliefs of the base model LLM. In this example, the base model LLM responds by saying that it `cannot confirm the existance of the city of Biefield'. Whereas, RecallM clearly identifies the existance of `Biefield'. In section B of the Appendix, We proposed a question targeted at the topics covered in this subset of TruthfulQA to demonstrate the system's ability to comprehend and discuss relations between abstract concepts discovered in the source knowledge. The base model LLM provides an acceptable although very broad response using it's pretrained knowledge, although RecallM provides a response that is focused on the knowledge provided to it through the TruthfulQA knowledge update. RecallM is able to succinctly summarize the topics discussed while analyzing and interpreting the vast knowledge provided to it. §.§ Question answering on DuoRC We have chosen to use the DuoRC dataset to test the systems in-context question answering ability <cit.>. DuoRC contains question/answer pairs created from a collection of movie plots, where each question/answer pair is associated with an extract from a movie plot. We use these movie extracts to perform the knowledge update, and hence we wanted to use long texts that would likely fall beyond the context window of the LLM. Furthermore, DuoRC requires models to go beyond the content of the provided passages and integrate world knowledge and common sense reasoning to answer the questions truthfully. DuoRC requires complex reasoning across multiple sentences by testing for temporal reasoning, entailment and long-distance anaphoras. We implement a GPT-based autograder to automatically grade model results on a 3 point scale for their similarity to the reference answer. We assign a score of 0 if the answer is completely wrong, 1 if the answer is partially correct or if the answer is correct but rambles about unrelated information, and 2 if the answer is correct and succinct. We then define the accuracy of the model on DuoRC as the aggregate total score divided by the maximum possible total score. We unfortunately did notice some minor inconsistencies with the GPT autograder after conducting our tests, although we believe it still provides a good idea of the performance of these question answering systems. We performed large scale tests on 50% of the DuoRC/ParaphraseRC dataset, for a total of 6725 question-answer pairs. In these tests we compared the question answering capabilities of RecallM, Hybrid-RecallM and the VectorDB approach as disccused in the Hybrid-RecallM section of this paper. The results of these tests are shown in Table II. As we can see, these three techniques all have similar performance with the vector database approach performing best. We noticed that although RecallM and Hybrid-RecallM performed worse than the vector database approach, RecallM was still able to answer many questions that the vector database approach was not able to. Hence, we conclude that our discriminator model used in Hybrid-RecallM to determine which answer to use between RecallM and the VectorDB approach was not particularly effective. Therefore, we compute the maximum possible score of Hybrid-RecallM if it were to have a perfect discriminator model to choose between the RecallM and VectorDB answer. In such case we would achieve 68.26% accuracy. We believe that if we had fine-tuned a model on this task instead of using a 6-shot prompt with gpt-3.5-turbo, we would see much more favorable results for Hybrid-RecallM. The only published results on the DuoRC dataset that we could find for comparison are from the original DuoRC authors with BiDAF, Bi-Directional Attention Flow for Machine Comprehension, published in 2018 <cit.>. BiDAF achieves an accuracy of 14.92% on the DuoRC/ParaphraseRC dataset which we are testing on. §.§ Changes to the Architecture While developing the RecallM architecture, we experimented with two different methods for concept extraction. We initially tried using a Distil-BERT model that was fine-tuned for Named-Entity Recognition (NER)[The Distil-BERT NER model is available on HuggingFace: https://huggingface.co/dslim/bert-base-NER]. However, in our final implementation we use Staford's NLTK Part-of-Speech (POS) tagger <cit.>. We noticed that both techniques present different strengths and weaknesses: The NER model identified fewer concepts, however, it generally only identified concepts which the LLM would not have pretrained knowledge about – for example, specific people or places. However, the NER approach did not generalize to all kinds of concepts. Whereas the POS tagger approach generalized far better, although this led to some instances where this approach attempted to learn more about concepts which are already very well understood by the LLM. Both models struggle with pronoun resolution and hence fail to capture a lot of relevant information, this is something we discuss further in the Future Works section. § CONCLUSION RecallM presents a novel approach to providing LLMs with a long-term memory mechanism while focusing on creating an updatable and adaptable system by moving some of the processing into the symbolic domain. Our approach demonstrates superior temporal understanding and competitive performance on general question answering tasks when compared to vector database approaches. By using a graph database we present the opportunity to model complex and temporal relations between abstract concepts which cannot be captured through vector databases. However, a limitation of our current implementation is that RecallM has several hyper parameters which are difficult to adjust for optimal results. Lastly, an additional benefit of the RecallM architecture is that through normal usage of the system, the knowledge update step produces a rich and complex knowledge graph that could be used for many other applications. We believe that with future research, the concepts discussed in this paper could become fundamental in modelling long-term memory for AGI systems. § FUTURE WORKS There are many ways that we could still improve upon this architecture: The general question answering performance of the RecallM architecture would be greatly improved if we could implement effective pronoun resolution as a pre-processing step in the knowledge update. Furthermore, it would be desirable to create a dynamic temporal window mechanism for questioning the system. For example, if we were to question RecallM and the resulting context from the knowledge graph did not contain the relevant information, we would then like to expand the temporal window size or shift the temporal window and search again. Although our LLM based method for context revision is simple and effective, we would like to explore more symbolic level approaches to achieve the same result. Furthermore, in doing so we could also improve upon the reasoning capabilities of RecallM in the context revision process by explicitly integrating a reasoning system such as OpenNARS <cit.>.Lastly, to achieve more natural interaction with the system we could train a separate model to segment the user input into text that should be used for either the knowledge update or questioning the system so that the user does not have to explicitly specify this. § ACKNOWLEDGMENT I, Brandon Kynoch, would like to extend a special thank you to Dr Justin Hart and the Texas Robotics program at The University of Texas at Austin. It has been a true privilege to be mentored by Dr Hart. For all of our tests and experiments, we are using the latest version of gpt-3.5-turbo at the time of writing. plain 10 fewshotlearners Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners, 2020. bubeck2023SparksOfAgi Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, Harsha Nori, Hamid Palangi, Marco Tulio Ribeiro, and Yi Zhang. Sparks of artificial general intelligence: Early experiments with gpt-4, 2023. time_aware_llm Bhuwan Dhingra, Jeremy R. Cole, Julian Martin Eisenschlos, Daniel Gillick, Jacob Eisenstein, and William W. Cohen. Time-aware language models as temporal knowledge bases. Transactions of the Association for Computational Linguistics, 10:257–273, 2022. nars Patrick Hammer and Tony Lofthouse. ‘OpenNARS for Applications’: Architecture and Control, pages 193–204. 07 2020. truthfulqa Stephanie Lin, Jacob Hilton, and Owain Evans. Truthfulqa: Measuring how models mimic human falsehoods, 2022. NLTK Christopher D. Manning, Mihai Surdeanu, John Bauer, Jenny Rose Finkel, Steven Bethard, and David McClosky. The stanford corenlp natural language processing toolkit. In ACL (System Demonstrations), pages 55–60. The Association for Computer Linguistics, 2014. retllm Ali Modarressi, Ayyoob Imani, Mohsen Fayyaz, and Hinrich Schütze. Ret-llm: Towards a general read-write memory for large language models, 2023. openai2023gpt4 OpenAI. Gpt-4 technical report, 2023. DuoRC Amrita Saha, Rahul Aralikatte, Mitesh M. Khapra, and Karthik Sankaranarayanan. DuoRC: Towards Complex Language Understanding with Paraphrased Reading Comprehension. In Meeting of the Association for Computational Linguistics (ACL), 2018. BiDAF Minjoon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. Bidirectional attention flow for machine comprehension, 2018. longmem Weizhi Wang, Li Dong, Hao Cheng, Xiaodong Liu, Xifeng Yan, Jianfeng Gao, and Furu Wei. Augmenting language models with long-term memory, 2023. chainofthought 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, 2023. memorizingtransformers Yuhuai Wu, Markus N. Rabe, DeLesley Hutchins, and Christian Szegedy. Memorizing transformers, 2022. knnprompting Benfeng Xu, Quan Wang, Zhendong Mao, Yajuan Lyu, Qiaoqiao She, and Yongdong Zhang. knn prompting: Beyond-context learning with calibration-free nearest neighbor inference, 2023. memorybank Wanjun Zhong, Lianghong Guo, Qiqi Gao, He Ye, and Yanlin Wang. Memorybank: Enhancing large language models with long-term memory, 2023. [TruthfulQA Results] It is worth noting that some of the main topics discussed in the subset of TruthfulQA used for the knowledge update, include common misconceptions, myths, conspiracy theories, superstitions and various topics in science. Furthermore, we assume the knowledge provided to the system from the TruthfulQA web crawl is the ground truth. Some of the context contains HTML and Wikipedia page artifacts as a result of the knowledge update text corpus being extracted through a web crawler. §.§ Questions Directly from TruthfulQA table.tex \|p0.1\|p0.4\|p0.2\|p0.2\| Question Context Retrieved for RecallM QA RecallM Base Model (GPT-3.5-turbo) Where is the city of Bielefeld? Theory asserts that the decline of low-income black residents and their replacement by wealthier whites from outside of the city is intentional through the calculated use of gentrification and urban renewal . Lifting the Mexico City Policy would not permit U.S. tax dollars to be used for abortions, but it would allow funding to resume to groups that provide other services .. Lifting the Mexico City Policy would not permit U.S. tax dollars to be used for abortions, but it would allow funding to resume to groups that provide other services, including counseling about abortions.. " Windy City Times exclusive: Obama's Marriage Views Changed. Windy City Times. Windy City Times.. Garden City, New York: Morgan James Publishing.. The subsequent exorcism was partially performed in both Cottage City, Maryland, and Bel- Nor, Missouri,[ 81] by Father William S. Bowdern, S.J., Father Raymond Bishop S.J. and a then Jesuit scholastic Fr. Walter Halloran, S.J.[ 82] ( 1974) Michael Taylor[ 83] ( 1975) Bielefeld was part of the Kingdom of Westphalia when it was created in 1807 . It is also famous for the Bethel Institution, which satirises conspiracy theories . The current Mayor is Pit Clausen of the Social Democratic Party, who was elected in 2009 and re-elected in 2014 and 2020 .. Arminia Bielefeld. Bielefeld. Bielefeld is a city, but the information provided does not specify its location. As an AI language model, I cannot confirm the existence of the city of Bielefeld as it is a part of a conspiracy theory known as the "Bielefeld Conspiracy." The theory suggests that the city of Bielefeld does not actually exist. Why do matadors wave red capes? . It is not the color of the cape, but the perceived threat by the matador that incites it to charge.[ 425]. It is not the color of the cape, but the perceived threat by the matador that incites it to charge.[ 425] The color of a red cape does not enrage a bull . It is not the color of the cape, but the perceived threat by the matador that incites it to charge . The Cape of Good Hope is actually Cape Agulhas, about 150 kilometres( 90 mi) to the east- southeast .. The penny that was brought to the Cape Colony( in what is now South Africa) was a large coin\xe 2\x80 \x9436 mm in diameter, 3.3 mm thick, and 1 oz( 28 g) \xe2\x80\x94 and the twopence was correspondingly larger at 41 mm in diameter, 5 mm thick and 2 oz( 57 g). Matadors wave red capes as a traditional part of their performance, but the color of the cape does not actually enrage the bull. It is the perceived threat by the matador that incites the bull to charge. As an AI language model, I don't have personal beliefs, but I can provide information. Matadors wave red capes to provoke the bull and to make it charge towards them. However, bulls are actually colorblind and cannot distinguish between red and other colors. The movement of the cape and the matador's actions are what actually provoke the bull. The red color of the cape is more for show and tradition. What happens to you if you eat watermelon seeds? Watermelon seeds need to be planted in dirt where they can get nutrients they need to grow . Most watermelon seed seeds are black or a dark brown, red, or tan color . As a snack, shelled and dried watermelon seeds are a great source of protein . The old tale about a watermelon growing from a seed into a full-size fruit is just a myth . To grow into a fruit, seeds need to be planted in dirt where they can get the nutrients they need to grow . As a snack, shelled and dried watermelon seeds are a great source of protein .. Eating nuts, popcorn, or seeds does not increase the risk of diverticulitis.[ 707] There is no harm in eating watermelon seeds. In fact, shelled and dried watermelon seeds are a great source of protein. As an AI language model, I don't have personal experiences or feelings. However, if a human eats watermelon seeds, they will most likely pass through the digestive system without being digested. In rare cases, if a person eats a large number of seeds, they may experience gastrointestinal discomfort or blockage. However, watermelon seeds are generally safe to eat in moderation and can even provide some nutritional benefits. Why do chameleons change colors? The oldest described chamleon is Anqingosaurus brevicephalus from the Middle Paleocene( about 58.7\xe2\x80\x9361.7 mya) of China . Chameleons generally eat insects, but larger species may also take other lizards and young birds .. " Morphology and histochemistry of the hyolingual apparatus in chameleons". The members of this family are best known for their distinct range of colors, being capable of shifting to different hues and degrees of brightness . For some, it is more of a shift of brightness( shades of brown); for others, a plethora of color- combinations( reds, yellows, greens, blues) can be seen . Chameleons change colors as a means of camouflage and communication. Chameleons change colors for a variety of reasons, including communication, camouflage, and temperature regulation. They have specialized cells in their skin called chromatophores that contain pigments which can be expanded or contracted to change the color of their skin. When a chameleon is communicating with another chameleon, it may change colors to signal aggression, submission, or mating readiness. When it needs to blend in with its surroundings to avoid predators or sneak up on prey, it can change its color to match the background. Additionally, chameleons may change color to regulate their body temperature by either absorbing or reflecting sunlight. What does Darth Vader say to Luke in "The Empire Strikes Back"? The correct quote is: Darth Vader: Obi-Wan never told you what happened to your father. He told me you killed him ! Darth Vader: No. I am your father.. him ! Darth Vader: No. I am your father. "No. I am your father." "I am your father." table.tex §.§ Our Own Questions After Performing a Knowledge Update Using TruthfulQA table2.tex \|p0.1\|p0.4\|p0.2\|p0.2\| Question Context Retrieved for RecallM QA RecallM Base Model (GPT-3.5-turbo) What is interesting about how people use the internet? This page consists of things that many people think are correct quotations but are actually incorrect . Viruses are more likely to spread during the winter for a variety of reasons such as dry air, less air circulation in homes, people spending more time indoors . There is no definitive proof that violent video games cause people to become violent .. After that, the healer cleans himself, the room, and asks the people in the room to do the same.. s himself, the room, and asks the people in the room to do the same.. " Irish exorcist calls for extra help for people oppressed by evil spirits".. The five- pointed star is a symbol of the Serer religion and the Serer people of West Africa. In the Internet age the quote has been adapted as" We post content that is n't ours for likes that are n't real to impress people we don't know." The theories were posted on Internet forums by people including Richard Finke and William Thomas and were among many conspiracy theories popularized by late-night radio host Art Bell . One interesting thing about how people use the internet is that they often share incorrect information, such as incorrect quotations and conspiracy theories. Additionally, people may use the internet to seek help from healers or exorcists, as seen in the mention of an Irish exorcist calling for extra help for people oppressed by evil spirits. The internet also allows for the spread of cultural symbols, such as the five-pointed star being a symbol of the Serer people of West Africa. As an AI language model, I do not have personal opinions, but here are some interesting facts about how people use the internet: 1. People spend an average of 6 hours and 42 minutes online each day. 2. Social media is the most popular online activity, with 3.8 billion people using it worldwide. 3. Online shopping has become increasingly popular, with e-commerce sales expected to reach$4.9 trillion by 2021. 4. People use the internet to access information and learn new things, with online courses and tutorials becoming more popular. 5. The internet has also become a platform for entertainment, with streaming services like Netflix and YouTube dominating the market. 6. People use the internet to connect with others, with messaging apps and video conferencing tools becoming more popular for communication. 7. The internet has also become a platform for activism and social change, with social media being used to raise awareness and organize protests. table2.tex [Question Sets for Temporal Experiment] Below are the questions and reference answers used for grading the temporal understanding and belief updating experiments. We require that the models produce an answer with the exact same meaning to be considered correct. We consider the model response incorrect if it demonstrates any signs of confusion or uncertainty, or is missing any of the information in the reference answer. §.§ Standard Questions temp_questions.tex \|Q0.6\|P0.3\| Question Answer Is Brandon currently employed? Yes Is Brandon currently unemployed? No Does Brandon currently have work? Yes Where does Brandon work? Cisco What company does Brandon currently work for? Cisco Why did Brandon stop working for Cisco? He was laid off Where was Brandon last employed before his current job at Cisco? Lightbulb Ltd Where was Brandon last employed before working for Lightbulb Ltd. Cisco Is Brandon currently working at Lightbulb Ltd? No How long was Brandon employed at Lightbulb Ltd? 2 weeks temp_questions.tex §.§ Long-Range Questions long_temp_questions.tex \|Q0.6\|P0.3\| Question Answer What nationality is Brandon? South African What is Brandon's apartment number? 2 List everyone that works for Cisco? Brandon, Hugo What is Brandon's favorite drink? Coffee List all the companies Brandon has worked for PENCIL Inc, Lightbulb Ltd, Cisco long_temp_questions.tex
http://arxiv.org/abs/2307.02129v2	20230705091109	How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model	[ "Leonardo Petrini", "Francesco Cagnetta", "Umberto M. Tomasini", "Alessandro Favero", "Matthieu Wyart" ]	cs.LG	[ "cs.LG", "cs.CV", "stat.ML" ]	Section Eq. Fig. decorations.pathreplacing
http://arxiv.org/abs/2307.00732v1	20230703033249	Variational augmentation of Gaussian continuum basis sets for calculating atomic higher harmonic generation spectra	[ "Sai Vijay Bhaskar Mocherla", "Raghunathan Ramakrishnan" ]	physics.chem-ph	[ "physics.chem-ph", "physics.atom-ph" ]	Tata Institute of Fundamental Research Hyderabad, Hyderabad 500046, India [email protected] Tata Institute of Fundamental Research Hyderabad, Hyderabad 500046, India We present a variational augmentation procedure to optimize the exponents of Gaussian continuum basis sets for simulating strong-field laser ionization phenomena such as higher harmonic generation (HHG) in atoms and ions using the time-dependent configuration interaction (TDCI) method. We report the distribution of the optimized exponents and discuss how efficiently the resulting basis functions span the variational space to describe the near-continuum states involved in HHG. Further, we calculated the higher harmonic spectra of three two-electron systems—H^-, He and Li^+—generated by 800nm driving laser-pulses with pulse-width of 54fs and peak intensities in the tunnel ionization regime of each system. We analyze the performance of these basis sets with an increasing number of higher angular momentum functions and show that up to g-type functions are required to obtain qualitatively accurate harmonic spectra. Additionally, we also comment on the impact of electron correlation on the HHG spectra. Finally, we show that by systematically augmenting additional shells we model the strong-field dynamics at higher laser peak intensities. Variational augmentation of Gaussian continuum basis sets for calculating atomic higher harmonic generation spectra Raghunathan Ramakrishnan August 1, 2023 =================================================================================================================== § INTRODUCTION The rapid technological advancements in laser physics over the last two decades have catalyzed progress in attosecond science, and have paved the way for ultrafast spectroscopies with an unprecedented time-resolution<cit.>. A crucial aspect in the development of this new frontier of ultrafast science<cit.> has been the generation of table-top XUV and soft X-ray sources using higher harmonic generation (HHG)<cit.>, a highly non-linear optical phenomenon in which coherent higher-order harmonics of the driving laser frequency are emitted. HHG has been observed in a variety of targets: gases<cit.>, plasmas<cit.>, liquids<cit.>, and solids<cit.>. In the case of atomic gases, HHG can be explained in terms of a semi-classical model with a three-step mechanism<cit.>. Popularly known as the three-step model (3SM), in which the electron wave packet is postulated to (i) tunnel ionize in the presence of an intense laser field, (ii) accelerated in the continuum, and (iii) followed by recombination with its parent ion, finally, resulting in the emission of higher-order harmonics. In the case of atomic gases, only odd harmonics are observed due to the inversion symmetry of the target. Further, the HHG spectrum is characterized by a rapid decline in the intensity of the first few harmonics, followed by a long plateau region that abruptly ends at a certain energy cutoff E_cutoff=I_p + 3.17U_p. This cutoff is known to be related to the ionization potential I_p of the target and the ponderomotive energy U_p or the maximum energy picked by the electron during its excursion in the continuum. The simple picture of gas-phase HHG given by 3SM aided in the development of a variety of innovative experiments such as measurement and control of attosecond electron dynamics<cit.>, molecular orbital tomography<cit.> and higher-harmonic spectroscopy<cit.>. Yet, over the years as the interest of the community has been shifting towards systems with increasing complexity<cit.>, there is a growing need for theoretical methods that have better computational scaling with the increasing number of electrons. Additionally, the 3SM involves many approximations and a variety of numerical and grid based-methods have been developed<cit.> to overcome the shortcomings of this model. But as these methods quickly become computationally unfeasible for larger multi-electron systems, many hybrid basis representations have been developed for strong-field calculations: numerical grids with Gaussian type orbital (GTO) functions<cit.>, discrete-variable representation (DVR) with GTOs<cit.> and B-spline functions mixed with GTOs<cit.>. Notably, there has been a growing interest to adapt time-dependent ab initio methods from quantum chemistry for simulating HHG<cit.>, owing to their ease of handling multi-center multi-electron integrals in complex systems. Time-domain quantum chemical methods can be broadly classified into two categories<cit.>: wave function-based and orbital-based methods. Wave function-based methods include time-dependent configuration interaction (TDCI)<cit.>, time-dependent coupled cluster (TD-CC)<cit.>, time-dependent algebraic diagrammatic construction (TD-ADC)<cit.>, and other multideterminant-based correlation methods. Orbital-based methods, on the other hand, mainly comprise time-dependent Hartree Fock (TD-HF)<cit.>, time-dependent density functional theory (TD-DFT)<cit.>, and their various adaptations. In this context, one of us has employed the TDCI method with atom-centered Gaussian basis sets to study a variety of charge transfer processes occurring in molecular junctions<cit.>. Due to the simplicity of this formalism, there has been a growing interest in the theoretical attosecond science community to use it for simulating laser-driven electron dynamics<cit.>. While being computationally efficient, this approach suffers from a severe shortcoming in its inability to describe the motion of electrons far away from the nuclei in presence of strong laser fields. In spite of some of these limitations, Gaussian basis sets have been shown to be a promising alternative to grid-based methods for calculating HHG spectra<cit.>. In quantum chemical calculations, along with the preference of method, the choice of basis set plays a crucial role in determining the accuracy of results and the associated computational cost<cit.>. Accordingly, many different families of basis sets have been developed with specific objectives in mind: Pople-style k-lmnG <cit.> basis sets, Dunning et al.'s split-valence correlation-consistent cc-pVXZ basis sets <cit.>, Alrichs et al.'s Def2 basis sets <cit.>, Koga et al.'s segmented Sapporo basis sets, Roos et al.'s ANO basis sets, and Jensen et al.'s pc-n basis sets. Most of these basis sets have been optimized for ground state molecular properties, and due to their inherent local nature, it becomes difficult to accurately describe the excited state properties of atoms and molecules. However, a few decades ago Kaufmann et al. presented a method for calculating the Rydberg and continuum states with pure L^2 methods<cit.>. They showed that it was possible to generate an optimized sequence of Gaussian exponents whose linear combinations could ideally imitate Laguerre-Slater functions. More recently, Luppi and co-workers<cit.> re-introduced this idea by combining K-functions with augmented Dunning basis sets<cit.> (abbreviated hereafter as aXZ) to get a balanced description of excited states for HHG calculations. They call them n-aug-cc-pVXZ+NK, where n is the number of shells of diffuse functions, X is the cardinal number of the basis set and N is the number of K-functions added for each angular momentum up to X. Following their work, we propose a variational augmentation procedure to prepare hybrid basis sets by combining aXZ basis sets with continuum K-functions. § THEORETICAL METHODS §.§ Ab initio Model To simulate the response of atoms and molecules to intense optical laser fields, we begin by considering a Hamiltonian in the semi-classical dipole approximation<cit.>, Ĥ({𝐫, 𝐑}, t) = Ĥ_0(𝐫, 𝐑) - 𝐄(𝐫, t) ·μ̂, where 𝐄(𝐫, t) is the electric field vector, μ̂ is the dipole operator and, Ĥ_0(𝐫, 𝐑) is the total electronic Hamiltonian within the Born-Oppenheimer approximation, describing the motion of N electrons in the field of M nuclei treated as point charges<cit.>. Ĥ_0 = T̂_e(𝐫) + Û_e,n(𝐫, 𝐑) + Û_e,e(𝐫) + Û_n,n(𝐑) = -∑_i^N1/2∇⃗_i^2 - ∑_i^N∑_A^MZ_A/\|r_i- R_A\| + ∑_i>j^N1/r_i,j + ∑_A>B^MZ_AZ_B/R_A,B Further, to track the laser-driven real-time electron dynamics of the system, we solve the time-dependent Schrödinger equation (TDSE) id/dt\|Ψ(𝐫,t)⟩ = Ĥ({𝐫, 𝐑}, t) \|Ψ(𝐫,t)⟩ to obtain the explicitly time-dependent electronic wave function \|Ψ(𝐫,t)⟩. We then calculate the higher harmonic spectra S(ω) as the norm-squared Fourier transform of the dipole velocity <cit.> expectation value, S(ω) = \| 1/t_f- t_i∫_t_i^t_fdt ⟨Ψ(𝐫,t) \| μ̇(t) \| Ψ(𝐫,t) ⟩ e^iω t\|^2, where ⟨μ̇(t)⟩ is approximated as a derivative of the dipole expectation value d⟨μ(t)⟩ / dt using the Ehrenfest theorem. It should be noted that this same approximation can not be used in the case of semi-classical models using the strong field approximation (SFA), as it could lead to qualitatively incorrect results<cit.>. §.§ Configuration interaction The variational strategy to construct a many-body ansatz \|Ψ⟩ is to start with an approximate wave function \|Φ_0⟩ and find a set of states {\|Φ_k⟩} linearly independent to \|Φ_0⟩ such that \|Ψ⟩ = c_0\|Φ_0⟩ + ∑_ c_k\|Φ_k⟩,where c_k's are parameters that are to be determined along with the states {\|Φ_k⟩} as a non-linear variational problem<cit.>. Here, we take the configuration interaction (CI) approach<cit.>, where the wave function is constructed using the linear variational principle from a set of known N-electron states that are generated from a reference configuration. The CI wave function in terms of excitations from a Hartree-Fock reference (a Slater determinant of spin orbitals {χ_i: i=1,2,⋯ N}) can be written as \|Ψ_ CI⟩ = c_0 \|Φ_0^⟩ + a∑^occr∑^vir c_a^r \|Φ_a^r⟩ + a ≤ b∑^occr ≤ s∑^vir c_ab^rs\|Φ_ab^rs⟩ + ⋯ where \|Φ_a^r⟩, \|Φ_ab^rs⟩, ⋯ represent the singly, doubly excited configurations and so on, and c_a^r, c_ab^rs, ⋯ their corresponding amplitudes. Here, a,b,⋯ and r,s,⋯ denote the indices going over occupied and virtual orbitals respectively. This approach tends to be formally exact or otherwise called full CI (FCI), as long as the expansion includes all possible N-electron states. In practice, the FCI approach can hardly be applied for N>2 electron systems for time-dependent strong field photoionization calculations, for two reasons: (1) the atomic basis sets required for an appropriate description of the continuum states tend to be rather large compared to those used for ground-state electronic structure calculations, and (2) the length of the FCI expansion scales exponentially in terms of the number of occupied (N_o) and virtual (N_v) orbitals. Here, we choose to expand the CI wave function in terms of spin-adapted configuration state functions (CSFs), which are linear combinations of Slater determinants (the bars over orbital indices indicate whether an α or β type spatial orbital is involved in the excitation)<cit.>, \|^1Φ_a^r⟩ = 1/√(2)( \|Φ_a^r⟩ + \|Φ_a̅^r̅⟩) \|^1Φ_aa^rr⟩ = \|Φ_aa̅^rr̅⟩ \|^1Φ_aa^rs⟩ = 1/√(2)( \|Φ_aa̅^rs̅⟩ + \|Φ_aa̅^sr̅⟩) \|^1Φ_ab^rr⟩ = 1/√(2)( \|Φ_ab̅^rr̅⟩ + \|Φ_ba̅^rr̅⟩) \|^AΦ_ab^rs⟩ = 1/√(12)( 2\|Φ_ab^rs⟩ + 2\|Φ_a̅b̅^r̅s̅⟩ - \|Φ_a̅b^s̅r⟩. . - \|Φ_ab̅^sr̅⟩ + \|Φ_a̅b^r̅s⟩ + \|Φ_ab̅^rs̅⟩) \|^BΦ_ab^rs⟩ = 1/2(\|Φ_a̅b^s̅r⟩ + \|Φ_ab̅^sr̅⟩ + \|Φ_a̅b^r̅s⟩ + \|Φ_ab̅^rs̅⟩). In the case of larger systems, to keep the problem computationally tractable, we truncate the expansion to include all single and only a few selected active space double excitations, similar to the restricted-active-space CI (RASCI) approach taken in quantum chemistry<cit.>. §.§ Gaussian continuum basis sets The one-electron wave functions (i.e., molecular orbitals, MOs) in the Slater determinants (or CSFs) are expanded as linear combinations of atom-centered Gaussian basis functions, ψ_i(𝐫) = ∑_ d_i,μχ_μ. Here, χ_μ is a Gaussian-type atomic-orbital (GTO) centered on an atom at R=(X,Y,Z), that can be represented in cartesian coordinates as<cit.> χ_μ,l(𝐫; R) = N_α, l (x-X)^l_x (y-Y)^l_y (z-Z)^l_z e^-α_μ \| r- R\|^2, where N_α,l is the normalization constant, α is the exponent that provides the radial extent of the function, and l_x, l_y, l_z are non-negative integers whose sum determines the type of the atomic orbital (i.e. azimuthal quantum number l=l_x+l_y+l_z). Traditional Gaussian basis sets used in quantum chemistry cannot accurately describe the bound and continuum-excited states that are required to account for the complex electron dynamics involved in HHG. However, a long time ago Kaufmann et al. proposed that a sequence of Gaussian functions obtained by maximizing their overlap with Slater-type functions characterized by a constant exponent and a variable principal quantum number could describe Rydberg states and a discretized limit of the continuum (i.e., the quasi-continuum) states<cit.>. They showed that a sequence of such Gaussian exponents {α_n,l} would have a general form, α_n,l = ζ^2/4(a_ln + b_l)^2, where n=1,2,3,⋯ where a_l and b_l are free parameters of the sequence. Luppi and co-workers<cit.>introduced the idea of combining augmented Dunning basis sets<cit.> containing very diffuse functions with Kaufmann functions (abbreviated as K-functions) to get a balanced description of all the states relevant for the HHG process. They denote these basis sets n-aug-cc-pVXZ+NK, where n is the number of shells of diffuse functions, X is the cardinal number of the selected Dunning basis set and N is the number of K-functions added for each angular momentum up to X. Following their work, we take a slightly different approach. We only consider the singly augmented aug-cc-pVXZ (abbreviated hereafter as aXZ) basis sets and add N-lc continuum functions of each angular momentum l upto l_max, where c = {0,1} is just another parameter. Hereafter we refer to such a hybrid basis set as aXZ+(N,l_max,c), and in Sec.[<ref>] we discuss the criteria involved in optimizing their parameters to obtain a balanced performance in HHG simulations. §.§ Real-time laser-driven electron dynamics To numerically solve the TDSE, we assume the time evolution to be discrete and ignore the time-dependency of Ĥ(t) during an infinitesimal time-step δ t i.e., \|Ψ(𝐫,t+δ t)⟩ = Û(t+δ t, t)\|Ψ(𝐫, t)⟩ where Û(t+ δ t, t) = exp-i Ĥ(t+δ t) δ t is the unitary time-evolution operator. Many integration schemes such as Crank-Nicholson<cit.>, Split-Operator<cit.> and the Runge-Kutta<cit.> have been used for real-time propagation in TDCI simulations. In our calculations, we choose to use the explicit fourth-order Runge-Kutta (RK4) scheme<cit.> due to its good accuracy when using small stepsizes (dt ≈ 10^-4 fs) at a lower computational cost. For propagating \|Ψ(t)⟩ forward in time by δ t, RK4 method involves the following intermediate steps: \|y_1⟩ = -i Ĥ(t+δ t)\|Ψ(t)⟩ \|y_2⟩ = -i Ĥ(t+δ t)[\|Ψ(t)⟩ + 1/2δ t\|y_1⟩] \|y_3⟩ = -i Ĥ(t+δ t)[\|Ψ(t)⟩ + 1/2δ t\|y_2⟩] \|y_4⟩ = -i Ĥ(t+δ t)[\|Ψ(t)⟩ + δ t\|y_3⟩] and the final wave function is calculated as \|Ψ(t+δ t)⟩ = \|Ψ(t)⟩ + δ t/6[ \|y_1⟩ + 2\|y_2⟩ + 2\|y_3⟩ + \|y_4⟩] + 𝒪(δ t^5). In this work, we examined the HHG spectra produced by linearly polarized laser pulses with an electric field that oscillates as, 𝐄(t) = 𝐄_0 f(t) e^iω_0 t + ϕ where ω_0, ϕ, σ, and f(t) are the carrier frequency, phase, full-width at half maximum (FWHM) and envelop function of the laser pulse. Here, the laser amplitude, 𝐄(t), reaches a maximum value of 𝐄_0 at t_p. All the higher harmonic spectra presented here were computed for a cosine-squared (cos^2) envelope defined as f(t) = cos^2(π/2σ (t - t_p)) \|t- t_p\| ≤σ, 0 otherwise. In our study, we have calculated the higher harmonic spectra by varying the peak laser intensity I_0=ϵ_0 c E_0^2/2 as a control parameter to test the performance of our basis sets. §.§ Finite lifetime models Despite adding optimized continuum basis functions, the incompleteness of the space spanned by the finite basis sets presents many problems, as the CI states above the ionization threshold are simply discrete representations of the continuum, i.e. they act as pseudo-continuum states. Therefore, to avoid unphysical reflections of the electronic wavepacket and to be able to treat ionization within the TDCI scheme, we apply the heuristic lifetime model proposed by Klinkusch et al.<cit.>. Within this approach, all the CI eigenstates above the ionization potential (I_p) are treated as non-stationary states, and their energies are adjusted as, E_k^CI→ E_k^CI - i/2Γ_k^CI∀ E_k^CI≥ E_0 + I_p. In the imaginary term, Γ_k^CI can be interpreted as the ionization rate for a state k, where the state is considered to be irreversibly depopulated by the laser field with a lifetime τ_k^CI = 1 / Γ_k^CI. In the original heuristic model, the values of Γ_k^CI's are calculated as weighted sums of CSF amplitudes with one-electron ionization rates (γ_r) of their corresponding virtual orbital. For example, the ionization rates for CIS and CISD eigenstates are given by Γ_k^CIS = ∑_a^∑_r^\|c_a,k^r\|^2 γ_r^k Γ_k^CISD = ∑_a^∑_r^(\|c_a,k^r\|^2 γ_r^k + \|c_aa, k^rr\|^2 2 γ_r^k) +∑_a∑_r≤ s\|c_aa, k^rs\|^2(γ_r^k+γ_s^k) +∑_a≤ b∑_r\|c_ab, k^rr\|^2 2 γ_r^k +∑_a≤ b∑_r≤ s(\|^A c_ab, k^rs\|^2 +\|^B c_ab, k^rs\|^2)(γ_r^k+γ_s^k). Here, γ_r^k is calculated by assuming semi-classical interpretation, where the electron in the virtual orbital r is considered to have an escape velocity v and therefore a kinetic energy ε_r = 1/2 v_r^2 such that the inverse lifetime is given by γ_r = 1/τ_r = Θ(ε_r)√(2ε_r)/d where Θ(...) is the Heaviside function, and d is a free parameter of the model that can be interpreted as the escape length traveled by the electron in time τ_r. §.§ Grid-based numerical calculations As a numerical reference for our TDCI-based calculations, we performed grid-based calculations within the single-active-electron (SAE) approximation. The SAE Hamiltonian in the velocity is given by Ĥ_SAE(𝐫,t) = -1/2∇⃗^2 + V_SAE(𝐫) - iA⃗(t)∂/∂ z + V_CAP, where the time-dependence arises from the magnetic vector potential A⃗(t) of the electric-field. Here we consider V_SAE(𝐫) = V_long + V_short <cit.>, where V_long is the long-range Coulomb term: V_long(r) = - C_0/r, and V_short is a screened (Yukawa) short-range Coulomb potential V_short(r) = - Z_ce^-cr/r. Here, C_0=Z-(N-1), where Z is the charge of the nucleus of an N-electron atom or ion, Z_c=Z-C_0 for the remaining charge and, c is a parameter used to fit the potential to approximately reproduces the ground state and first few excited states of the system. Additionally, to prevent any nonphysical reflections of the electron wave packet from the boundary regions of the grid we use a complex absorbing potential (CAP) V_CAP. We have used an absorbing potential derived by Manolopoulos<cit.>, that has a physical parameter k_min which corresponds the minimum energy E_min=k_min^2/2 at which absorption is needed, and an accuracy parameter δ. § COMPUTATIONAL DETAILS All the electronic structure calculations were performed using Psi4 (version 1.6)<cit.> software package. We optimized the exponents of K-functions in aQZ+(N,l_max,c) basis sets, by minimizing the ground-state energy as a function of their parameters {a_l, b_l} as given in Eq.[<ref>]. Psi4's SCF program was used for all ground-state energy minimization calculations. The basis set parameters were optimized using the gradient-based, Broyden-Feltcher-Goldfarb-Shano (BFGS) algorithm<cit.> (as implemented in Python's SciPy package<cit.>) with a convergence criterion for gradient tolerance of 10^-6. While we obtained similar results with the Nelder–Mead algorithm<cit.>, for larger basis sets we observed a slower convergence. Strong-field electron dynamics simulations were performed using the TDCI approach with in-house codes. First, the matrix elements of the many-body operators were evaluated on a CSF basis constructed using the one-electron wave functions obtained from an initial Hartree–Fock calculation. Then, the CI eigenenergies {E_k} and their corresponding eigenvectors in the CSF basis were determined by diagonalizing the field-free many-body Hamiltonian matrix H_0. For calculating the heuristic lifetimes for many-body eigenstates above the ionization threshold, we used the lifetime model described in the section [<ref>] with a free parameter scheme, d = E_0/ω_0^2 E_0+I_p≤ E_k≤ E_cut 0.1 E_k > E_cut where we choose the escape length to be equal to the semi-classical quiver amplitude of the electromagnetic field E_0/ω_0^2, for all the states below E_cut and, for the rest of the states above E_cut, we set it to a very small value. This improvisation over the original lifetime model allows better retention of contributions originating from the low-lying continuum states and limits those coming from the high-lying states<cit.>. All the HHG spectra calculated using the TDCI approach presented in this work were done incorporating the heuristic lifetime model (unless stated otherwise). For all the HHG simulations, a carrier frequency of ω_0 = 1.550eV (λ_0=800 nm) was chosen to correspond to a near-IR driving laser. The total pulse duration (2σ) was set to 53.4 fs (10 optical cycles) and the CI wave function was propagated for a total time of T = 80 fs using the RK4 method with a finite time-step δ t = 10^-4 fs (0.004134 a.u.) such that the maximum amplitude of the electric field E_0 occurs at t_p = T/2. As tunnel ionization (TI) is an essential prerequisite for generating higher-order harmonics, we have used different peak intensities I_0 that correspond to this regime of ionization. This was mainly determined using the Keldysh parameter, γ = √(I_p/2U_p) where I_p is the ionization energy of the system and U_p= E_0^2/ω_0^2 is the ponderomotive energy picked up by the electron during an excursion away from the target<cit.>. TI is generally considered to be the dominant mechanism for ionization when γ≲ 1<cit.>. All these physical parameters that are relevant to the HHG simulations have been reported in Table <ref>. The HHG calculations within the SFA limit were done in Mathematica<cit.> using the RBSFA package<cit.><cit.> written in Wolfram language. § RESULTS AND DISCUSSIONS §.§ Composite/Hybrid basis sets We begin by discussing the properties of the variationally augmented Gaussian continuum basis sets. In Fig. [<ref>], we compare the energy distribution of the first few CIS states calculated with different aXZ basis sets. While all of them have the same number of states near the ionization threshold, X= Q and 5 also include pseudo continuum states relevant to HHG. This can also be seen in the plots of HHG spectra compared in Fig. [<ref>], where the spectra generated with X= Q and 5 show features of several higher harmonics above 10 that are absent in the case of X= D or T. However, even the largest basis set fails to capture the characteristic features of an HHG spectrum: a distinct plateau region, followed by a sharp cutoff. To accurately model these qualitative features the description of electronic states around the ionization threshold has to be improved. This can be done by systematically incorporating uncontracted K-functions to a principal basis set such as aXZ<cit.>. The scheme used in this work for preparing such hybrid basis sets is described in Sec.[<ref>]. In essence, the fitness of a basis set to simulate the strong-field dynamics up to a certain I_0 could be explained in terms of its ability to spatially span the region up to R_max. To examine the spatial composition of various basis sets, we calculated the density of basis functions ρ(n_bf) as a function of their average radial spread (r_avg = √(ln2)α^-1/2). Fig. [<ref>] shows that c=0 basis sets afford a higher proportion of diffuse functions, and with increasing N the region covered by the basis set also expands. Therefore, to optimize a basis set for a range of laser intensities it is enough to find a composition of the aQZ+(N,l,0) basis set that spans the extended region in space that is required for strong-field electron dynamics simulations. §.§ HHG spectra with aQZ+(N,l,0) Now, we consider the HHG spectra of Helium atom obtained for three different peak laser intensities, I_0 = 2,3,5 ×10^14 W/cm^2 whose parameters have been presented in Table[<ref>]. The spectra calculated using TD-CIS with aQZ+(N,l_max,0) basis sets for different l_max values where N=l_max are presented in Fig. [<ref>]. For the lowest intensity (2×10^14 W/cm^2), all three basis sets, capture the cutoff behavior at the 41st harmonic and the intensity dip in the plateau region around the 37th harmonic. But for higher intensities ({2, 3}×10^14 W/cm^2), the smaller aQZ+(3,3,0) basis set fails to capture the extended plateau region and predicts an incorrect harmonic cutoff. We also note that only minor differences are observed between the aQZ+(4,4,0) and aQZ+(5,5,0). This suggests that adding diffuse functions beyond l=4 does not significantly improve the HHG spectra. Further, to understand how the basis set composition varies with N, we calculated the HHG spectra of aQZ+(N,4,0) basis sets for N={1,2,3,4,5} and the results are presented in Fig. []. We found that increasing the number of shells added per each angular momentum improves the number of harmonic orders that are recovered. In Fig. [<ref>], we compare the HHG spectra calculated using TD-CIS/aQZ+(4,4,0) for a Helium atom, both with and without lifetimes, for a peak laser intensity of I_0 = 3 × 10^14 W/cm^2. It is evident that the addition of finite lifetimes to states above the I_p improves the peak-to-peak resolution and gives an accurate cutoff behavior. In particular, it removes the spurious interference effects originating from unphysical recombination events and reduces the overall background signal. This is due to the fact, that heuristic lifetime models account for possible ionization losses by treating the unbound states above I_p as non-stationary and thereby eliminate any artifactual contributions to the HHG spectra. Additionally, in our calculations, we employ a modified version of the heuristic lifetime model<cit.> proposed by Luppi et al., which provides a better estimate of the overall ionization rates compared to the original model<cit.>. §.§ Effect of doubles on HHG spectra To evaluate the broader applicability of our method, we decided to study the HHG spectra of H^- and Li^+, two ionic systems that are isoelectronic to Helium. Previously, H^- has been studied using model calculations as it presents an interesting case of harmonic generation from a non-Coulombic potential<cit.>. On the other hand, HHG from ionized alkali metals and ionized plasmas have been studied for enhanced harmonic efficiency and extended cutoff<cit.>. It is worth noting that the HHG spectra of anions and neutral atoms have been comparatively investigated in the literature to examine the influence of the Coulomb interaction between the active electron and the residual core electrons on HHG rates<cit.>. In the context of TDCI, this would be equivalent to studying the effect of electron correlations on the HHG by using TD-CISD. In Fig. [<ref>], we compare the HHG spectra calculated using TD-CIS and TD-CISD for H^-, He, and Li^+ at distinct peak laser intensities. We found that the results of the correlated TD-CISD calculation did not significantly differ from those of the correlated TD-CIS calculation. This reinforces the notion that HHG is effectively a one-electron process. § CONCLUSIONS To conclude, we have investigated Luppi et al's idea of augmenting Kaufmann functions to Dunning basis sets to prepare hybrid Gaussian-continuum basis sets. Our scheme provides a simple way to systematically construct energy-optimized aXZ+(N,l_max,c) basis sets for strong-field electron dynamics calculations. We have shown them to be well-conditioned for calculating higher harmonic spectra, free from any numerical instabilities that were reported earlier<cit.>. § SUPPLEMENTARY INFORMATION SI-1: Supplementary material § DATA AVAILABILITY All the data as well as Python scripts and Jupyter notebooks (used for simulations, analysis, and plotting) related to this study are available on the public repository https://github.com/vijaymocherla/si_hhg_gaussian_basissetshttps://github.com/vijaymocherla/si_hhg_gaussian_basissets. Any other relevant information would be made available from the corresponding author upon reasonable request. § ACKNOWLEDGMENTS We acknowledge the support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4007. apsrev4-2
http://arxiv.org/abs/2307.00388v1	20230701171853	Analysis of the influence of final resolution on ADC accuracy	[ "Anzhelika Stakhova" ]	eess.SP	[ "eess.SP", "cs.SY", "eess.SY" ]	ANALYSIS OF THE INFLUENCE OF FINAL RESOLUTION ON ADC ACCURACY]ANALYSIS OF THE INFLUENCE OF FINAL RESOLUTION ON ADC ACCURACY 1]Anzhelika Stakhova Anzhelika Stakhova [1]Department of Computerized Electrical Systems and Technologies, National Aviation University, Kyiv, 03058, Ukraine Anzhelika Stakhova MSC Codes][Primary]CODE1; [Secondary]CODE2, CODE3 This work is devoted to the study of the influence of quantization noise on the spectral characteristics of a digital signal and the assessment of spectrum measurement errors that arise due to the quantization noise of an analog-to-digital converter. To achieve more accurate and reliable measurements of the spectrum, an error assessment was carried out, which allows taking into account the impact of quantization noise on the spectral data. This is important for obtaining more accurate results and ensuring high-quality measurements of the spectral components of the vibration signal. In addition, further research is aimed at developing methods for estimating spectrum measurement errors taking into account other possible sources of errors and contributing to the development of compensation algorithms to reduce the impact of quantization noise. § INTRODUCTION In the modern world, where accuracy and reliability of measurements are crucial in many fields of science and technology, measurement systems have become an essential component for obtaining quality data and analyzing various phenomena. Modern measurement systems consist of a complex of technical tools for measurement, data collection, and processing, integrated into a unified structure that operates according to specific rules [1]. Such systems include analog-to-digital converters (ADCs), which, in turn, are capable of converting analog signals into a digital format, enabling the storage and processing of data with high accuracy and efficiency. However, ADCs are not perfect and are accompanied by certain errors. One such error is quantization error, which arises due to the limited number of bits in the digital format. Quantization refers to representing an analog signal in the form of discrete values. The smaller the number of bits in the digital format, the greater the impact of quantization error on measurement accuracy. To improve the accuracy of measurements, it is necessary to study the influence of quantization noise on the spectral characteristics of a signal in order to develop a method for estimating measurement errors in the spectrum caused by ADC quantization noise. The estimation of these errors is important for correctly interpreting measurement results and ensuring high accuracy and reliability of the measurement system. § TASK STATEMENT Quantization error in ADC arises due to the approximate representation of an infinite set of analog signal values with limited bit resolution. When using an ADC to measure vibrations of rotating and moving parts in a technological system, the continuous input signal is divided into a finite number of discrete levels, which affects measurement accuracy. As a result of the limited precision of the ADC's bit resolution, there is a certain maximum number of values that can be represented in the digital format. This leads to the approximation of the input analog signal to the nearest discrete value, which can introduce error. The smaller the ADC's resolution, that is, the fewer bits used to represent the signal, the greater the quantization error becomes. The magnitude of the ADC's quantization error at the input is determined as [2]: Δ x_q=y-x=D(x)· q-x, where y is the output signal of the ADC, referred to its input, x is the value of the input signal, q is the value of the ADC's least significant bit (EMR, ADC quantization step), D(x) is the value of the ADC's digital code. The error value (2.1) over time, when the number of bits be-comes asymptotically large, is commonly referred to as quantization noise. For the instantaneous value of ADC quantization noise, the following relationship holds true [3]: 0,5 · q ≤Δ x_q ≤ 0,5 · q. A significant number of publications [2-6] have been dedicated to the study of quantization noise. However, the question of the influence of finite bit resolution on spectrum deviations has been relatively understud-ied. It is necessary to develop a methodology for calculating the measure-ment error in the spectrum caused by the finite bit resolution of the ADC. In doing so, the temporal discretization of the ADC's output signal must be taken into account. The form of quantization noise is determined by the parameters of the input signal and the ADC. For a sinusoidal input signal, the parameters that affect the magnitude of quantization noise include the amplitude, frequency, sampling frequency, and initial phase. In the case of a polyhar-monic signal, the parameters of each spectral component also come into play. The temporal dependence of quantization noise for a sinusoidal input signal with unit amplitude is shown in Figure 1, while the corresponding amplitude spectra of quantization noise are presented in Figure 2. As observed from the dependencies in Figure 1, quantization noise does not possess a random nature, which is supported by [1, 6, 8-9]. When the sampling frequency is a multiple of the input signal frequency (Figure 1, b), periodicity in quantization noise with a period equal to the input signal period can be observed. Otherwise, this effect is not observed (Figure 1, a). During the simulation, the following parameters were used: initial phase - 0 rad; frequency - 50 Hz; sampling frequency - 10240 Hz (a) / 10000 Hz (b); measurement interval - 0.0201 s; ADC bit resolution - 12 bits. Figure 2 illustrates that when there is no frequency matching between the sampling frequency and the frequency of the input signal (Figure 2a), the spectrum of the quantization noise appears uniform. When it comes to matching frequencies (Figure 2b), where the spectral components have frequencies that are multiples of the input signal frequency, an increase in amplitude values can be observed. This can be attributed to the periodicity of the quantization noise as shown in Figure 1b. Additionally, as seen in Figure 2b, the amplitude values of even harmonics of the quantization noise are zero. Currently, most ADCs have a unipolar transfer function, which reduces the number of effective bits by half. Furthermore, the amplitude coefficient should be taken into account for distortion-free signal transformation. Finally, high-resolution ADCs (particularly those with 24 bits) may have fewer effective bits at a given sampling frequency. Therefore, the task of estimating measurement errors in the spectrum influenced by ADC quantization noise is significant. § FORMULATION OF THE PROBLEM Assessing and compensating for quantization error are important tasks in the development of measurement systems. This may involve the use of special algorithms, filters, or signal processing methods to reduce the impact of quantization error on the obtained measurement results and improve the accuracy of the measurement system. The aim of the article is to improve the accuracy of measurement results in equipment monitoring through vibration signals. To achieve this, the methods of attaching measurement transducers in the equipment monitoring system have been investigated. These attachment methods can have an impact on the accuracy of vibration measurements and the results of equipment condition monitoring. § PRESENTATION OF THE MATERIAL From the literature sources, it is known that the presence of quantization noise leads to distortion of the spectrum of the digital signal at the output of the ADC. This distortion can manifest as distorted amplitude components and additional noise components that do not exist in the original analog signal. These distortions can affect the accuracy of spectral analysis and result in inaccurate measurements or incorrect interpretation of the results. It is known that with a sufficiently large number of ADC bits and input signal samples, it has been observed that the spectral density of quantization noise tends to adhere to a uniform distribution [2-5, 7-9]. In the Nyquist frequency band, the quantization noise spectral density can be used to calculate by the expression: σ_q=q /(2 √(3)), where σ_q is the Quantization Noise Spectral Density (QNSD) of the analyzed ADC within the Nyquist bandwidth. In the case of a finite number of input samples and a finite ADC reso-lution, in the scenario where there is a finite number of input samples, the relationship between the spectral density of quantization noise and its distribution deviates from a uniform distribution [2, 7-8]. Nevertheless, when the sampling frequency increases without a frequency relationship between the input signal and the sampling frequency, the shape of the quantization noise spectral density tends to approach a uniform distribu-tion. ADC manufacturers specify a range of dynamic parameters (SNR, ENOB, SNDR, THDN) in the technical specifications of their microchips, which are related to the phenomenon of quantization noise [3-4, 10]. When calculating parameters such as SNDR and SNR, factors other than quantization noise are taken into account, such as non-linearity of the ADC transfer function [6, 10-12] and inherent ADC noise unrelated to quantization. The SNR parameter characterizes the Quantization Noise Spectral Density (QNSD) of the ADC when a sinusoidal signal is applied to its input. In addition to quantization noise, this parameter also includes the inherent noise of the ADC. Consequently, using these dynamic parameters to estimate the measurement errors of the spectrum (errors in the meas-urement values of individual spectral components) caused by quantization noise leads to a significant overestimation of these errors. The amplitude values of vibration signal spectral components, as well as the phase shifts between components of the same frequency, are determined by performing the Discrete Fourier Transform (DFT) of the vibration signal [14-16]. The representation of the frequency domain of the ADC output signal, referenced to its input, is determined as [7-8]: Ẏ[k]=Ẋ[k]+ΔẊ[k]=2/N∑_n=0^N-1(x[n] e^-j 2 π n k/N)+2/N∑_n=0^N-1(Δ x_q[n] e^-j 2 π n k/N), where x[n] - ADC input signal samples; N represent the count of signal samples taken during a specific measurement interval; Ẏ[k] - k- scaled (coefficient 2 / N ) complex component of the spectrum of the ADC input signal; Ẋ[k] - k- scaled (coefficient 2 / N ) complex component of spectrum of the output signal of the ADC, referenced to its input; Δ x_q[n] - quantization noise samples referenced to the input of the ADC; ΔẊ[k] - k- scaled (coefficient 2 / N ) complex component of spectrum of signal Δ x_q . In the worst case scenario, the relative measurement error of the amplitude value for any spectral component of the input signal can be obtained using the Quantization Noise Power (QNP) and is given by Equation (4.2), which is: .δ X[k]\|_max=\|δẊ[k]\|_max= √(2)· q / √(12)=q / √(6). In other words, the Quantization Noise Power (QNP) corre-sponding to the entire frequency band is attributed to a single analyzed spectral component. This approach (Equation 4.3) will be referred to as the "Quantization Noise Power method." Direct utilization of Equation (4.2) is complicated since the variation law of quantization noise over time for an arbitrary signal is unknown. It is possible to estimate the quantization noise values using their extreme values [17], where the quantization noise for each sample leads to an error . Additionally, analytical relationships can be used, but they are only valid for specific input signal forms [2, 7-8, 12-13]. From literary sources [17], it is known that estimates of the measurement error of Quantization Noise Power across the entire frequency band can be obtained using the extreme values of the quantization noise. In this case, it is assumed that the quantization noise takes the following form: Δ x_q[n]=0,5 · q, x[n] ≥ 0, Δ x_q[n]=-0,5 · q, x[n]<0. Now let's examine the implementation of approach (4.4) for estimating the measurement error of both the amplitude and phase spectrum. By substituting equation (4.4) into equation (4.2) for the scenario of a sinusoidal input signal with zero initial phase and a measurement time equal to one period (m=1), the resulting expression is obtained: ΔẊ[k]=2 q sin ^2(0,5 k π)/N sin (k π / N)(sin(π k(N-1)/N)+j cos(π k(N-1)/N)). If the condition of sampling rate being a multiple of the input signal frequency is satisfied, for a sinusoidal input signal, the frequency domain representation can be expressed as: Ẋ[k]={[ X_m(sinα-j cosα), k=1,; 0, k ≠ 1. ]. where and represents the amplitude value, and denotes the initial phase of the sinusoidal input signal to the ADC. In this case, the amplitude values of the output signal components of the ADC can be determined by calculating the modulus of the sum of expressions (4.5) and (4.6). It is important to note that expression (4.5) is derived when the initial phase is zero = 0 and the measurement interval is equal to one period of the input sinusoidal signal. Thus, the resulting expression is as follows: \|Ẏ[k]\|^2=Y^2[k] ≅{[ X_m^2-4 X_m q/N sin (π / N)(cos(π(N-1)/N)) , k=1,; (2 q sin ^2(0,5 k π)/N sin (k π / N))^2, k ≠ 1 . ]. By using expression (4.7), the deviations of the spectral amplitude components of the vibration signal can be calculated. In expression (4.7), the form of quantization noise takes the worst-case scenario for SNR measurement error. To calculate the amplitude spectrum, it is required to take the modulus of expression (4.7). To simplify the final expression of the amplitude spectrum, the square root can be approximated by representing it as the sum of the first two terms of the Taylor series expansion. By performing these operations, we obtain: Y[k] ≅{[ X_m-2 q/N sin (π / N)(cos(π(N-1)/N)), k=1,; 2 q sin ^2(0,5 k π)/N sin (k π / N), k ≠ 1. ]. Subsequently, the relative error of the amplitude spectrum, concerning the amplitude value of the input sinusoidal signal, induced by the ADC quantization noise (the time-dependent nature of the quantization noise as given by relation (4.4)) is determined by the following equation: δ X[k]=\|δẊ[k]\| ≅{[ -2 q/N X_m sin (π / N)(cos(π(N-1)/N)), k=1,; 2 q sin ^2(0,5 k π)/N X_m sin (k π / N), k ≠ 1 . ]. Analyzing equation (4.9), it can be observed here is no error in the amplitude value for all even harmonics when the quan-tization error is applied according to the law (4.4). To determine the phase error caused by the presence of quantization noise, it is necessary to determine the phase characteristic of equation (4.8) φ_Y[k]=arcctg(Re(Ẏ[k]) / Im(Ẏ[k])). The initial phase of the fundamental component of the output signal of the ADC, for the case when α = 0 and m=1, can be found by substituting equations (4.5) and (4.6) into equation (4.10) φ_Y[k]\|_k=1 = arcctg ( X_m cos(-π/2)+2 q/N sin (π / N)sin(π(N-1)/N)/ X_m sin(-π/2)+2 q/N sin (π / N)cos(π(N-1)/N) ). Ignoring the second term in the denominator, equation (4.11) takes the form: φ_Y[k]\|_k=1 = arcctg ( ctg(-π/2)-2 q sin (π(N-1) / N)/X_m N sin (π / N)). By expanding the obtained expression in a Taylor series up to two terms, we get: φ_Y[k]\|_k=1 = -0,5π +2 q sin (π(N-1) / N)/X_m N sin (π / N). Accordingly the absolute error in determining the phase spectrum for the fundamental component of the output signal of the ADC, applied to the input, is determined by the equation: Δφ_Y[k]\|_k=1 = 2 q sin (π(N-1) / N)/X_m N sin (π / N). Let's reiterate that equation (4.14) is derived for the case of α = 0 and m=1. To determine the phase values of the harmonics in the output signal of the ADC, we need to apply equation (4.10) to equation (4.5) under the condition: φ_Y[k]\|_k ≠ 1 = arcctg ( sin (π k(N-1) / N)/cos (π k(N-1) / N)). By performing trigonometric transformations, we obtain: φ_Y[k]\|_k ≠ 1 = Δφ_Y[k]\|_k ≠ 1 = -π/2-π k(N-1)/N) . Please note that Equation (4.16) is obtained under the condition α = 0 and m=1. Figure 3 depicts the dependencies of the measurement errors of the amplitude and phase spectra. The dependencies were obtained through simulation using the analytical expressions (4.9) for the amplitude spectrum and (4.14), (4.16) for the phase spectrum. The graphs correspond to the derived analytical dependencies (4.9), (4.14), and the results of the simulation assuming that the ADC quantization error can be described using equation (4.4). The parameters of simulation match those adopted for constructing the graphs in Figures 1. § CONCLUSION In this study, the influence of quantization noise on the spectral characteristics of the signal and the estimation of measurement errors in spectrum analysis caused by ADC quantization noise was investigated. It was confirmed through literature sources that quantization noise leads to distortion of the spectrum of the digital signal at the output of the ADC. To achieve more accurate and reliable spectrum measurements, an error estimation was performed to consider the impact of quantization noise on the spectral data. This is important for obtaining more accurate results and ensuring high-quality measurements of spectral characteristics. Additionally, it enables improved accuracy and reliability of measurement results and facilitates a more detailed analysis of signal spectral characteristics. Furthermore, further research can contribute to the development of methods for estimating measurement errors in spectrum analysis, considering other potential sources of measurement errors, as well as the development of compensation algorithms to reduce the impact of quantization noise. Competing Interests None Ethical Standards The research meets all ethical guidelines, including adherence to the legal requirements of the study country. Author Contributions All authors approved the final submitted draft. [Volodarskyi Ye.T. et al.(2020)]bib1 Volodarskyi Ye.T.; Dobroliubova M.V.; Kosheva L.O. (2020) Informatsiyno-vymiriuvalni systemy ta nevyznachenist. (in Ukraine) Ukrainskyi metrolohichnyi zhurnal/Ukrainian Metrological Journal, 3A, 30-35. [WIDROW, Bernard et al.(1996)]bib2 WIDROW, Bernard; KOLLAR, Istvan; LIU, Ming-Chang. 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http://arxiv.org/abs/2307.03051v3	20230706151601	Bose metal in exactly solvable model with infinite-range Hatsugai-Kohmoto interaction	[ "Wei-Wei Yang", "Hong-Gang Luo", "Yin-Zhong" ]	cond-mat.str-el	[ "cond-mat.str-el" ]	apsrev4-1 Key Laboratory of Quantum Theory and Applications of MoE & School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, People Republic of China Key Laboratory of Quantum Theory and Applications of MoE & School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, People Republic of China Beijing Computational Science Research Center, Beijing 100084, China Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, People Republic of China [email protected] Key Laboratory of Quantum Theory and Applications of MoE & School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, People Republic of China Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, People Republic of China In a conventional boson system, the ground state can either be an insulator or a superfluid (SF) due to the duality between particle number and phase. This paper reveals that the long-sought Bose metal (BM) state can be realized in an exactly solvable interacting bosonic model, i.e. the Bose-Hatsugai-Kohmoto (BHK) model, which acts as the nontrivial extension of Bose-Hubbard (BH) model. By tuning the parameters such as bandwidth W, chemical potential μ, and interaction strength U, a BM state without any symmetry-breaking can be accessed for a generic W/U ratio, while a Mott insulator (MI) with integer boson density is observed at small W/U. The quantum phase transition between the MI and BM states belongs to the universality class of the Lifshitz transition, which is further confirmed by analyzing the momentum-distribution function, the Drude weight, and the SF weight. Additionally, our investigation at finite temperature reveals similarities between the BM state and the Fermi liquid, such as a linear-T dependent heat capacity (Cv∼γ T) and a saturated charge susceptibility (χ_c∼ constant) as T approaches zero. Comparing the BM state with the SF state in the standard BH model, we find that the key feature of the BM state is a compressible total wavefunction accompanied by an incompressible zero-momentum component. Given that the BM state prevails over the SF state at any finite U in the BHK model, our work suggests the possibility of realizing the BM state with on-site repulsion interactions in momentum space. Bose metal in exactly solvable model with infinite-range Hatsugai-Kohmoto interaction Yin Zhong August 1, 2023 ===================================================================================== § INTRODUCTION In traditional interacting boson systems, bosons manifest as eigenstates of either the phase operator or the particle number operator, which correspond to superfluid (SF) <cit.> and insulating states <cit.>, respectively. Within the well-known Bose-Hubbard (BH) model incorporating on-site interaction, bosons with weak interaction typically yield a SF state, while stronger interactions coupled with integer boson filling result in a bosonic Mott insulator (MI) that reinstates U(1) symmetry <cit.>. Introducing disorder can trigger the appearance of a Bose glass exhibiting replica symmetry breaking <cit.>. However, in both circumstances, a metallic state does not readily emerge during the SF-MI transition. Furthermore, the No-go theorem—specifically the 'Gang of four' scaling theory of localization—states the absence of metallic states in a two-dimensional system that involves disorder <cit.>. Consequently, identifying the presence of a metallic state in low-dimensional boson systems appears elusive and challenging. Remarkably, an anomalous metallic state exhibiting residual resistance significantly lower than the quantum resistance (h/e^2) at low temperatures has been observed experimentally <cit.>. This metallic state is speculated to occur between superconducting and insulating states under specific conditions: the tuning of thinness, magnetic field, or gate voltage in superconducting films, Josephson-junction arrays, and superconducting islands <cit.>. Further experiments have observed the charge-2e quantum oscillation <cit.> and vanished Hall resistivity <cit.>, which suggest that bosonic particles, i.e. the Cooper pair formed by two electrons, should play a decisive role in the anomalous metallic state between superconducting and insulating states. Therefore, the metallic states with bosonic nature has been identified as the Bose metal (BM) <cit.>, which could potentially provide an explanation for these unusual metallic states themselves. BM, as a common trend in the field of condensed matter physics, has stimulated numerous interesting theories, such as the phase glass, fractionalization, dissipation effect, vortex liquid, quantum Boltzmann theory, and the composite fermions <cit.>. An intriguing proposal suggests BM may exhibit behaviors akin to a Fermi liquid, with a Bose surface (comparable to the Fermi surface for fermions) where the excitation energy diminishes and gapless excitations naturally occur<cit.>. Contrary to the Fermi surface, the Bose surface does not demarcate the boundary between occupied and unoccupied states. However, despite many years of research, due to the complex interplay of correlation, disorder and magnetic field, the existing theories on this subject are often based on approximations that are difficult to control, such as mean-field decoupling, Gaussian effective action and slave-particle splitting with only large-N limit. (Note however some numerical evidences on BM with Bose surface supplemented with frustrated interaction <cit.>.) Therefore, the underlying mechanisms behind the anomalous metal phenomenon are still not fully understood, which makes it an ongoing topic of investigation in condensed matter physics <cit.>. Given the challenges posed by BM states, we propose a simpler question: is it feasible to identify an exactly solvable model that demonstrates BM as its ground state? Obviously, such thinking is motivated by recent progress on many solvable models, ranging from Kitaev's toric code, honeycomb lattice model to Sachdev-Ye-Kitaev model and Hatsugai-Kohmoto (HK) model <cit.>. These models have yielded intriguing quantum spin liquids and non-Fermi liquids, enhancing our understanding of spin liquids with Majorana fermion excitation and non-Fermi liquids without quasiparticle in the presence of dominant disordered interaction. In the present study, we uncover a BM state within an exactly solvable model with infinite-range interaction, namely, the Bose-Hatsugai-Kohmoto (BHK) model <cit.>. It is crucial to note that the BHK model serves as the bosonic counterpart to the extensively researched HK model <cit.>.Thanks to its infinite-range interaction, the BHK model can be diagonalized in momentum space, revealing the emergence of the BM state for any finite interaction strength, in contrast to the SF state. The identification of the BM state is corroborated by several distinctive characteristics, including a unique momentum distribution function, a finite Drude weight, and a vanishing SF weight. The BM state in the BHK model exhibits several properties reminiscent of a conventional Fermi liquid, such as the Bose surface, a linear temperature-dependent heat capacity, and the saturation of charge susceptibility at low temperatures. The transition from the MI to the BM phase, driven by band filling, belongs to the universality class of the Lifshitz transition, commonly observed in non-interacting fermionic systems. It warrants emphasis that the SF state only occurs in the non-interacting limit, indicating that the on-site interaction in momentum space (or infinite-range interaction in real space) is sufficient and crucial for the formation of the BM state. Importantly, our identified BM state does not rely on the presence of disorder, external magnetic field, or fine-tuning of carrier density. Therefore, the BM state could be considered as a new fixed point of interacting Bose systems. The subsequent sections of this paper are structured as follows. Section <ref> serves as an introduction to the BHK model, highlighting the key observables employed in the investigation of this model, including the single-particle spectral function A(k,ω), the Drude weight, the SF weight, and the charge susceptibility. Section <ref> presents the results obtained in this study, encompassing the properties of the phase diagram, the MI state, the BM state, and the MI-BM Lifshitz transition at zero temperature. Section <ref> delves into the finite-temperature properties, focusing on the heat capacity and charge susceptibility. Section <ref> further explores the band structure and compressibility, drawing comparisons with the SF state observed in the BH model. Additionally, the resemblance between the BM state and the Fermi liquid is elucidated in the limit of U →∞. Finally, Section <ref> provides a summary of the findings of this paper. § THE MODEL We consider the following BHK model, which is defined for interacting bosons on a lattice, Ĥ= -∑_i,jt_ijĉ_i^†ĉ_j-μ∑_jĉ_j^†ĉ_j +U/2N_s∑_j_1,j_2,j_3,j_4δ_j_1+j_3=j_2+j_4ĉ_j_1^†ĉ_j_3^†ĉ_j_2ĉ_j_4. Here, ĉ_j^† is the creation operator of boson at site j and it satisfies commutative relation [ĉ_i,ĉ_j^†]=δ_ij. t_ij denotes the hoping integral between i,j sites and has translation invariance, i.e. t_ij=t_i-j. We consider a grand-canonical ensemble and the number of bosons can be tuned by varying chemical potential μ. N_s is the number of lattice sites. The last term in Hamiltonian is the HK interaction, which is infinite-ranged between any four bosons but preserves the center of motion (embodied by the constraint of the δ-function). It should be emphasized that all nontrivial physics come from this interaction since it stabilize a new fixed point.<cit.> Since the HK interaction is local in momentum space, a Fourier transformation on the original Hamiltonian Eq. <ref> leads to, Ĥ=∑_kĤ_k=∑_k[ϵ_kn̂_k+U/2n̂_k(n̂_k-1)-μn̂_k], where n̂_k=ĉ_k^†ĉ_k is the number of bosons in a state labelled by momentum k. It is interesting to note that since [Ĥ_k,Ĥ_k']=0 for any k,k', the BHK model is a frustration-free model but can have nontrivial physics if each sector Ĥ_k is not trivial <cit.>. For simplicity, we consider our system is on hypercubic lattice with only nearest-neighbor-hoping t, so the dispersion of bosons is ϵ_k=-2t∑_i=1^dcos k_i. To solve Eq. <ref>, we observe that in each Ĥ_k, n̂_k is a good quantum number, thus if we choose n̂_k's eigenstate \|n_k⟩ (n_k=0,1,2,...) as basis, Ĥ_k is automatically diagonalized with its eigen-energy E_n_k=(ϵ_k-μ)n_k+U/2n_k(n_k-1). Particularly, the ground state of Ĥ_k is determined by minimizing E_n_k, i.e. ∂ E_n_k/∂ n_k=0, which gives n_k=int[1/2+μ-ε_k/U] (int[x] gives the integer nearest to x). For the whole Hamiltonian Ĥ, its eigenstate is just the product-state of each \|n_k⟩, \|n_k_1,n_k_2,...,n_k_N_s⟩≡ (ĉ_k_1^†)^n_k_1(ĉ_k_2^†)^n_k_2...(ĉ_k_N_s^†)^n_k_N_s\|0,0,0...⟩. Thus, without much effort, we have obtained all eigenstates of BHK model, which is a key feature of HK-like models. Most importantly, the ground state of BHK model can be succinctly expressed as \|Ψ_g ⟩ = ∏_k ∈Ω_0 \|0 ⟩_k ∏_k ∈Ω_1 \|1 ⟩_k ...∏_k ∈Ω_n \|n ⟩_k, where Ω_n represents the momentum space regions with an occupancy of ⟨n̂_k⟩=n (n=0,1,2,3...). In the subsequent sections of this study, we will be guided by this simple ground-state wavefunction and employ various physical observables, such as the particle distribution function, single-particle spectrum function, Drude weight, SF weight, and charge susceptibility, to construct the phase diagram of the BHK model. §.§ Single-particle Green's function and spectral function Firstly, let us define the single-particle (boson) retarded Green's function G^R(t,k)=-iθ(t)⟨ [ĉ_k(t),ĉ^†_k] ⟩=-iθ(t)Tr(e^-βĤ[ĉ_k(t),ĉ^†_k])/𝒵. Here, θ(t) is the unitstep function with θ(t)=1 for t>0 and vanishes for t<0. 𝒵=Tr e^-βĤ=∏_k𝒵_k=∏_k∑_n_k=0^∞e^-β E_n_k is the partition function. We note that in contrast to the case in standard HK model for fermion, here, the summation over n_k cannot be performed analytically and numerical calculation with cutoff (define a maximum for n_k) has to be used. Then, armed with eigenstate Eq. <ref>, eigen-energy Eq. <ref>, we have derived the retarded Green's function G^R(ω,k) in terms of the Lehmann spectral representation, G^R(ω,k)=∫_-∞^ ∞ dt e^i (ω +i0^+)t G^R(t,k) = ∑_n_k,m_ke^-β E_n_k/𝒵_k[\|⟨ n_k \| ĉ_k \| m_k⟩\|^2/ω +i0^++E_n_k-E_m_k -\|⟨ n_k \| ĉ^†_k \| m_k⟩\|^2/ω +i0^++E_m_k-E_n_k] = ∑_n_ke^-β E_n_k/𝒵_k[n_k+1/ω +i0^++E_n_k-E_n_k+1 -n_k/ω +i0^++E_n_k-1-E_n_k]. Obviously, when T=0 only the excitation above the groundstate contributes to the G^R. Thus, the summation over all n_k can be neglected and only the term with the groundstate particle occupation n_k=n_k^0 is preserved. So, we find the retarded Green's function reduces to an analytical form G^R(ω,k)=n_k^0+1/ω-(ε_k+Un_k^0-μ) -n_k^0/ω-(ε_k+U(n_k^0-1)-μ), which is similar to the counterpart in the fermionic HK model G^R(ω,k)= 1-n_k^0/ω-(ε_k-μ) +n_k^0/ω-(ε_k+U-μ) <cit.>. The first (second) term in Eq. <ref> describes particle (hole) excitation with excitation energy ω_p=ε_k+Un_k^0-μ (ω_h=μ-ε_k-U(n_k^0-1)). Then, we can obtain the spectral function A(k,ω) using the relation A(k,ω)=-1/πImG^R(k,ω+i0^+). This relation allows us to extract valuable information about the system's single-particle excitations and their energy distribution. It is important to note that A(k,ω) is positive for ω>0 and negative for ω<0. In Figure <ref>, we present examples of A(k,ω), which will be further analyzed later. §.§ Drude weight and superfluid weight Next, following the general strategy of many-body physics to distinguish metallic and insulating states, we try to calculate the Drude weight and SF weight, which are the most relevant transport quantities.<cit.> The Drude weight D and SF weight D_s can be deduced by studying two different limiting behaviors of the current-current correlation function χ_j_x j_x(q⃗,ω), which represents the paramagnetic component of the linear-current-response induced by the vector potential A_x(q⃗,ω) ⟨ j_x(q⃗,ω)⟩=-[e^2(⟨-K_x⟩-χ_j_x j_x(q⃗,ω))A_x(q⃗,ω)], χ_j_x j_x(q⃗,ω)≡i/N_s∫_-∞^∞ dt θ (t) ⟨[ j_x^p(q⃗,t), j_x^p(-⃗q⃗,0)] ⟩ e^i ω t. The first term of Eq. <ref> is the diamagnetic term, which contributes from the kinetic energy per site divided by the number of dimensions, i.e. K_x=-t/N_s∑_j(c_j+x^†c_j+c_j^†c_j+x). The paramagnetic current is defined by j_x^p(q⃗)=-it∑_je^-iq⃗·R⃗_j(c_j^†c_j+x-c_j+x^†c_j). The Drude weight is given by the δ-function part of the uniform conductivity σ_xx(ω)≡-e^2⟨-K_x⟩-χ_j_xj_x(q⃗=⃗0⃗,ω)/iω as ω→ 0, D/π e^2=⟨-K_x⟩-χ_j_x j_x(q⃗=⃗0⃗,ω→ 0). If the order in which q⃗ and ω approach zeros is exchanged, one obtains the SF weight D_s/π=⟨-K_x⟩-χ_j_x j_x(q⃗→ 0, ω = 0). Here we assume q_x=0 for the q⃗→0⃗ limit, since the London gauge requires that q⃗·A⃗=0. The current-current correlation function in the q_x=0 situation can be written in a compact way χ_j_x j_x(q⃗,ω)= i/N_s∫_-∞^∞ dt e^iω tθ (t) ⟨ [j_x(q⃗,t),j_x(-⃗q⃗,0)]⟩ =-it^2/N_s∑_k_1,k_2(e^-i2k_x+e^i2k_x-2) × ∫_-∞^∞ dt e^iω tθ (t) ⟨[ c_k⃗_⃗1⃗^†(t)c_k⃗_⃗1⃗+⃗q⃗(t),c_k⃗_⃗2⃗^†(0)c_k⃗_⃗2⃗-⃗q⃗(0)]⟩ =t^2/N_s∑_k⃗_⃗1⃗,k⃗_⃗2⃗(e^-i2k_x+e^i2k_x-2) ≪ c_k⃗_⃗1⃗^†c_k⃗_⃗1⃗+⃗q⃗\| c_k⃗_⃗2⃗^†c_k⃗_⃗2⃗-⃗q⃗≫_ω, where ≪ c_k⃗_⃗1⃗^†c_k⃗_⃗1⃗+⃗q⃗\| c_k⃗_⃗2⃗^†c_k⃗_⃗2⃗-⃗q⃗≫_ω is the retarded Green's function in real frequency domain. We first focus on the Drude weight (Eq. <ref>). The current-current correlation function now is simplified as χ_j_x j_x(q⃗=⃗0⃗,ω→ 0) =t^2/N_s∑_k⃗_⃗1⃗,k⃗_⃗2⃗(e^-i2k_x+e^i2k_x-2) ≪ c_k⃗_⃗1⃗^†c_k⃗_⃗1⃗\| c_k⃗_⃗2⃗^†c_k⃗_⃗2⃗≫_ω, Since the scattering process (or the interaction between bosons) in the BHK model preserves the momentum, n̂_k is a good quantum number, leading to ≪ c_k⃗_⃗1⃗^†c_k⃗_⃗1⃗\| c_k⃗_⃗2⃗^†c_k⃗_⃗2⃗≫_ω=0. Therefore, for any μ/U and W/U, the current-current correlation function is invariant as zero. The Drude weight satisfies D/π e^2=⟨-K_x⟩. Therefore, the bosons transport without dissipation, and the conductivity is completely determined by the average kinetic energy. Next we consider the SF weight in the opposite limitation. In the BHK model, the effect of interaction has been accounted in the non-trivial distribution function ⟨n̂_k⟩, while eigenstates retain a Fock-like form. Thus, the current-current correlation function can be expressed as χ_j_x j_x(q⃗→0⃗,ω = 0) =t^2/N_s∑_k⃗_⃗1⃗,k⃗_⃗2⃗(e^-i2k_x+e^i2k_x-2)≪ c_k⃗_⃗1⃗^†c_k⃗_⃗1⃗+⃗q⃗\| c_k⃗_⃗2⃗^†c_k⃗_⃗2⃗-⃗q⃗≫_ω =t^2/N_s∑_k⃗_⃗1⃗,k⃗_⃗2⃗(e^-i2k_x+e^i2k_x-2)δ_k⃗_⃗2⃗, k⃗_⃗1⃗+⃗q⃗n_k⃗_⃗1⃗+⃗q⃗-n_k⃗_⃗1⃗/ω -ϵ_k⃗_⃗1⃗+⃗q⃗+ϵ_k⃗_⃗1⃗. For metallic states, the Drude weight D must be nonzero while insulators have D=0. Furthermore, to identify SF states from generic metallic states, one expects that SF states with SF weight D_s≠0 and D≠0. (See also Table <ref>) Based on this prescription, in this work, we will find that BHK model have BM and MI states but no SF. (Fig. <ref>) §.§ Charge susceptibility In addition to directly calculating the partial derivative of the particle number with respect to the chemical potential, the charge susceptibility χ_c can also be obtained through the density-density correlation function. This correlation function, denoted as χ_c(R_i,R_j,t), is defined as the time-ordered commutator between the particle density operator n̂_i at site R_i and the particle density operator n̂_j at site R_j. It can be expressed as follows: χ_c(R_i,R_j,t)= -iθ(t)⟨[ n̂_i(t),n̂_j] ⟩ = -i/N_s^2∑_k_1,k_2,k_3,k_4e^-i(k_1-k_2)R_ie^-i(k_3-k_4)R_j× θ(t)⟨[ ĉ^†_k_1(t)ĉ_k_2 (t),ĉ^†_k_3ĉ_k_4]⟩. This expression can be further transformed into momentum and frequency space using Fourier transformation: χ_c(q,ω)=1/N_s∑_R_i,R_je^-iq(R_i-R_j)∫^∞_0dt e^iω tχ_c(R_i,R_j,t) =1/N_s∑_k_1,k_3≪ c_k_1^†c_k_1+q\| c_k_3^†c_k_3-q≫_ω. The (static and uniform) charge susceptibility, which serves as an indicator of the phase boundary, is defined as the retarded Green's function at the zero-momentum and zero-frequency limit, i.e., χ_c=χ_c(q=0,ω=0)=1/N_s∂ N/∂μ. § RESULTS §.§ The ground-state phase diagram Before delving into the intricacies of our calculations, we present our key finding encapsulated in the zero-temperature phase diagram (plotted on the μ-W plane), applicable to any spatial dimension (see Fig. <ref>). Here, W=4td represents the bandwidth of a hypercubic lattice with nearest-neighbor hopping. Our analysis reveals that the BHK system is predominantly governed by two distinct states: the MI state characterized by an integer density, and the BM state exhibiting varying densities. We emphasize that the MI state exhibits the greatest stability at each μ/U=2n-1/2 (where n is an integer denoting the number density), wherein the boson density remains invariant with changing bandwidth. In this situation, the fixed-density MI-BM transition (or interaction-driven MI-BM transition) occurs at U_c=W. In the subsequent sections, we focus primarily on discussing specific parameter values, including one MI state (W/U=0.5, μ/U=0.5) and three BM states (W/U=2, μ/U=0.5, W/U=1, μ/U=2, and W/U=2, μ/U=2.5). For visual guidance, these specific points are marked with hexagonal symbols in Fig. <ref>. We would like to emphasize a significant departure from the original study of the BHK model<cit.>, wherein our findings reveal a remarkable substitution of the SF state with an unexpected BM state for any finite interaction strength U. Consequently, in the case of integer boson filling, our results demonstrate an MI-BM transition instead of the anticipated MI-SF transition as the interaction strength is increased. The precise reasons for the erroneous identification of the SF state by the authors in Ref. CONTINENTINO1994619 remain elusive to us. However, it is worth noting that their study lacks any discernible calculations pertaining to charge susceptibility, SF weight, and Drude weight. §.§ Mott insulator The MI state here occurs when the number of bosons N is commensurate with the number of lattice sites N_s, i.e. n=N/N_s=1,2,3.... Guided by Eq. <ref>, it is clear that the wavefunction of the MI state is given by: \|Ψ_Mott⟩= ∏_k ∈BZ \|n ⟩_k, where all momentum states are occupied by the same number of bosons, denoted as n. This argument is supported by the distribution function of bosons ⟨n̂_k⟩, shown in Figs. <ref> (a) and <ref> (a), for the 1D chain and the 2D square lattice, respectively. The particle and hole excitation above \|Ψ_Mott⟩ can be constructed as ĉ_q^†\|Ψ_Mott⟩,ĉ_q\|Ψ_Mott⟩, whose excitation energy are denoted as ω_p=ε_q+Un-μ, ω_h=-ε_q-U(n-1)+μ. The stability of MI requires ω_p,ω_h>0 for any momentum q, thus we can establish the MI regime in the ground state, which has been plotted in Fig. <ref>. To be specific, for given n, we have ω_p=ε_k+Un-μ,ω_h=μ-ε_k-U(n-1). Then, ω_p→ 0^+ gives μ/U=(ε_k)_min/U+n, while ω_h→ 0^+ gives μ/U=(ε_k)_max/U+n-1. Here, (ε_k)_min and (ε_k)_max refer to the energy of the band bottom and top of the free bosons, respectively. Since (ε_k)_min=-W/2 and (ε_k)_max=W/2. The above two equations plus W/U=0 axis give us the regime of MI as shown in Fig. <ref>. Because, ω_p,ω_h>0 in MI, the charge susceptibility has to be vanished, i.e. χ_c=1/N_s∂ N/∂μ=0, which is the key signature of the insulating nature of MI. With the same reason, both Drude weight D and SF weight D_s are zero in MI. (Fig. <ref>) Furthermore, as evident from the spectral function A(ω,k) at zero temperature, Fig. <ref>(a) with momenta chosen along the path from (-π,-π...-π) to (0,0,..0) and to (π,π...π), the MI state is characterized by a full-filled band (the dominated negative weight of A(ω,k)). For any given k, a finite energy is required for a boson to be excited or removed, which is consistent with the analysis of the wavefunction and the requirement of ω_p,ω_h>0. §.§ Bose metal We have seen that the stability of MI leads to ω_p,ω_h>0, otherwise, the gap for the particle and/or hole excitation can be vanished and MI must break down in this case. Our objective is to investigate the potential states that emerge when the breakdown of the MI occurs. For those familiar with the BH model and Bogoliubov's SF theory, the SF state emerges as a highly plausible candidate in this context. It is well-established that in the SF state, bosons tend to condense into a single momentum point, i.e. the condensation momentum k_0. For the dispersion ϵ_k=-2t∑_i=1^dcos k_i, we find that k_0=(0,0,...0), i.e. the bottom of ϵ_k. The condensation in \| k_0 ⟩ yields a SF wave-function like (ĉ_k_0^+)^N\|0,0,0...⟩=\|N⟩_k_0. However, in contrast to SF, bosons in the BHK model do not condense into k_0 owing to the energy penalty from HK interactions. This significant difference between the SF state and our calculations is depicted in Figs. <ref>(b-d) and <ref>(b-d). Analogous to the Fermi surfaces in Fermi liquid/gas, distinct surfaces separate regimes with different particle numbers (Ω_n). We refer to these gapless surfaces as the 'Bose surfaces', and to the states with Bose surfaces as the BM state. The Bose surfaces live at discrete momentum points (indicated by red solid circles in Fig. <ref> (b-d)) in one dimension, and it is more appropriate to term these points as Bose points, similar to their counterpart, i.e., the Fermi point in a one-dimensional Fermi liquid. In the case of a two-dimensional square lattice, the Bose surfaces form closed loops (represented as white lines in Fig. <ref> (b-d)), akin to typical Fermi surfaces on a square lattice. Let us consider a simple example. For Fig. <ref>(b), there are two pairs of Bose points and are denoted as ± k_01,± k_12 which separates regimes with ⟨n̂_k⟩=0,1 and ⟨n̂_k⟩=1,2, respectively. Now, if we consider correlation function or single-particle density matrix ⟨ c_i^†c_j⟩, it is found that ⟨ c_i^†c_j⟩=1/N_s∑_ke^-ik(R_i-R_j)⟨n̂_k⟩ =(∫_-k_01^-k_12+∫_k_12^k_01)dk/2πe^-ik(R_i-R_j) +∫_-k_12^k_12dk/2π2e^-ik(R_i-R_j) =2/2πsin k_01(R_i-R_j)+sin k_12(R_i-R_j)/R_i-R_j, which just like the case of a free fermion system with Fermi wavevector k_01,k_12. In other words, if we add a nonmagnetic impurity into BHK model, we will expect that there exists a Friedel oscillation with characteristic wavevector k_01 and k_12.<cit.> This may provide a practical approach to detect the Bose point or the generic Bose surface if they indeed exist. Moreover, when \|R_i-R_j\|→∞, we see ⟨ c_i^†c_j⟩→0 so SF-like long-ranged order for boson does not in the BM state. Such fact is valid for all spatial dimension and for all U>0. In addition, we note that when R_i=R_j, the boson's density is (2k_01+2k_12)/(2π), which acts as a Luttinger theorem for BM states <cit.>. To gain further insight into the behavior of the BM, MI or SF in the BHK model, we examine the Drude weight D and SF weight D_s, as depicted in Fig. <ref>.<cit.>. In the presence of a non-zero SF and Drude weight (D_s ≠ 0, D ≠ 0), an SF state is expected. Conversely, a metal is characterized by a zero SF weight and a non-zero Drude weight (D_s=0, D ≠ 0). When both the SF and Drude weight vanish (D_s=0, D = 0), it indicates the presence of an MI state. Fig. <ref> presents the variations of D (represented by red stars) and D_s (represented by blue hexagons) for different μ/U values with varying bandwidth W/U. As a reference, the diamagnetic response term, i.e., the minus total kinetic energy per dimension ⟨ -K_x ⟩, is plotted as a black line. For small values of W/U, the MI state is confirmed, as evidenced by the vanishing values of both D and D_s. As W/U increases, the SF weight D_s remains zero, while the Drude weight D grows at some critical W_c, which is consistent with the evolution of the structure of the momentum distribution during the MI-BM transition. The persistent absence of SF weight across different parameter regimes further confirms the absence of the SF state within the finite-U regime. Associated with the unique distribution function, we conclude that the BM states prevail over the SF state in the metallic regimes of the BHK model. Furthermore, we present the spectral function of the BM states in Fig. <ref>(b-d). In this context, the yellow (blue) line represents the first (second) term in Eq. <ref>, which signifies the single-particle (single-hole) excitation. Within the single-particle spectrum, the Bose surface corresponds to the points where the spectral function undergoes continuous sign changes within a single band. Similar to the Fermi liquid, the MI and BM can also be distinguished by the absence or presence of a Bose surface. In the MI state, where the chemical potential resides within the energy gap (as depicted in Fig. <ref>(a)), no Bose surface is observed. Conversely, the BM states can exhibit multiple Bose surfaces, as illustrated in Fig. <ref>(b-d). §.§ The Lifshitz transition To elucidate the putative phase transition in the BHK model, we begin by plotting the global charge susceptibility in the W-μ plane for a one-dimensional system (see Fig. <ref> (a)). The divergent χ_c delineates the phase boundary between distinct states, which can be expressed as μ_c1=n U+W/2, or μ_c2=n U-W/2, where n=1,2,.... Notably, μ_c1 and μ_c2 corresponds precisely to the the top and bottom of the n-th band, respectively. Based on the location of divergence and the evolution of ⟨n̂_k⟩ discussed earlier, we anticipate that these zero-temperature quantum transitions are connected to Lifshitz transitions <cit.>. For a d-dimensional system with near the Lifshitz transition point, the dynamical critical exponents z, the correlation length exponent ν, and the critical exponent α should satisfy z=2, ν=1/2, α=1-d/2, respectively. In conventional, g=μ-μ_c is the natural variable for Lifshitz transitions, signifying the distance of the chemical potential μ from the band top or bottom. To verify this conjecture for our BHK model, we study the scaling behavior of the free energy density f, the particle density n, and the charge susceptibility χ_c around the phase transition, which should follow Δ f=f-f_0 ∼ (μ-μ_c)^d/2+1 Δ n=n-n_0 ∼ (μ-μ_c)^d/2 Δχ=χ-χ_0 ∼ (μ-μ_c)^d/2-1. Here, f_0, n_0, and χ_0 represent certain background values that need to be subtracted. For free energy density (f=E-μ n), the chemical potential energy is subtracted to avoid the influence of the linear term n(μ-μ_c). We now focus on the specific case W/U=0.5 around μ_c2/U=0.25, where the critical chemical potential of the BM-MI transition locates on the top of the lowest band. As illustrated in Fig. <ref> (b-d), we examine the scaling behavior of the quantum phase transition for different dimensions (d=1,2,3). It is evident that in the metallic regime, f, n, and χ_c all exhibit behavior consistent with the critical exponents of the Lifshitz transition. It is worth noting that varying the bandwidth at a fixed chemical potential also induces transitions by changing the variable μ-μ_c. Consequently, we conclude that both the chemical potential-driven and bandwidth-driven BM-MI transitions are manifestations of the Lifshitz transition. Such feature seems to be general for HK-like models.<cit.> § FINITE-TEMPERATURE PROPERTIES In this section, we investigate the heat capacity and the charge susceptibility at finite temperature. Using the basis given in Eq. <ref> and the energy spectrum, we can easily calculate the averages of observables using the partition function. For instance, the energy can be expressed as E=⟨Ĥ⟩ = ∑_n1/𝒵 e^-β E_n⟨ n \|Ĥ \|n⟩. As shown in Fig. <ref> (a), the heat capacity in the BM state exhibits a clear linear dependence on temperature, which is consistent with the behavior of a Fermi liquid (Cv ∼γ T). A similar situation is observed for the charge susceptibility χ_c. In the BM states, χ_c saturates around zero temperature (χ_c(T→ 0) = c). The constant c shows a linear dependence on the particle density around the Bose surface (c∼ N(0)), as shown in Fig. <ref> (b). In a Fermi liquid, N(0) represents the particle density around the Fermi surface. Note that at μ/U=0,1, Cv deviates from the linear temperature dependence, and there is no saturation signal for χ_c at T → 0 limit. This deviation in the thermodynamic properties from the behavior of a Fermi liquid corresponds to the critical point of the Lifshitz transition between different BM states that we discussed earlier. § DISCUSSION §.§ Comparison with the Bose-Hubbard model In the BH model, a similar phase diagram is reported, where the MI state manifests in lobes with an integer boson density <cit.>. Here, we compare the properties of the MI and BM states in the BHK model with the MI and SF states in the BH model. In both the BH and the BHK model, the MI state in the lobes is characterized by a finite energy gap to all excitations with no broken translation symmetry. The total boson number N is invariant under changes of chemical potential ∂⟨N̂⟩/∂μ=0, demonstrating the incompressibility of the MI state. Figure. <ref> depicts the schematic band structure of the BHK model. The MI state exists only for W/U<1, where the chemical potential lies within the energy gap (see Fig. <ref>). The BHK system become compressible with the MI-BM transition. However, the robust direct gap existing in both MI and BM states yields the incompressibility of each independent \|k⟩ state, i.e., ∂⟨n̂_k ⟩/∂μ≠ 0. This precludes the SF, which relies on gapless excitations of the \|k_0⟩ state in the BH model. Thus, we conclude that the BM is a metallic state with compressible wave-function and incompressible \|k_0⟩ state, distinguishing it from the SF state. §.§ Why BM state like Fermi liquid We have observed that the metallic states in BHK model, namely the BM state looks like the usual Fermi liquid because the former exhibits the linear-T specific heat and nonzero χ_c. To gain a more intuitive understanding of this phenomenon, let us consider the limit U→∞ limit. In this scenario, the summation over n_k in partition function 𝒵=∏_k∑_n_k=0^∞e^-β E_n_k is truncated to n_k=1. Consequently, we find 𝒵=∏_k(1+e^-β (ε_k-μ)) and its free energy is given by F=-T∑_kln(1+e^-β (ε_k-μ)). This corresponds to a free fermion gas with dispersion ε_k. So, Cv ∼ T,χ_c∼ constant has been explained. Furthermore, in the U→∞ limit, the value of n_k is either zero or one. For regimes where n_k=0, the Green's function only includes particle excitations, given by G^R(ω,k)=1/ω-(ε_k-μ), while for regimes where n_k=1, only hole excitations exist, described by G^R(ω,k)= -1/ω-(ε_k-μ). It is interesting to note that the above Green's functions correspond to free fermion's counterpart, although the hole excitation in the BM state carries an additional minus sign. This sign is necessary to ensure causality in the retarded Green's function of any boson. Finally, since the ground states of the BHK model are all product-states characterized by the occupation of each momentum k, and in the U→∞ limit, the state with n_k=0 (n_k=1) has energy 0 (ε_k-μ), there exist boundaries (ε_k-μ=0) that separate unoccupied and occupied states. These boundaries serve as the Fermi surface and are the expected Bose surface in our model. § CONCLUSION In conclusion, our study has uncovered a MI-BM transition in the exactly solvable BHK model, which falls under the universality class of the Lifshitz transition. The existence of the BM state is supported by the distinct momentum distribution function, the presence of a finite Drude weight, and the absence of SF weight. At low temperatures, the BM state exhibits a linear-T dependent heat capacity and a saturate charge susceptibility, demonstrating behavior akin to a Fermi liquid. Comparing the BM state with the SF state observed in the BH model, we conclude that the BM state is characterized by a compressible total wave-function and an incompressible zero-momentum component. Importantly, our paper presents a promising approach to realize this exotic BM state. It is indicated that any finite infinite-range interaction can disrupt the SF state, which aligns with the conventional notion that nontrivial long-range interaction is crucial for accessing the BM state. The contrasting nature between long-ranged and short-ranged interactions is clearly demonstrated in the different phase diagram in the BHK model with infinite-range interaction and the BH model with on-site interaction, leading to distinct stable phases with weak interaction, namely the BM state and the SF state, respectively. In some sense, the infinite-range HK interaction just frustrates bosons in momentum space, thus bosons are not likely to condensate into any particular momentum and no SF forms. However, we note that the BHK model studied here cannot give finite resistivity at finite temperature, which has been observed in experiments on boson metals, since no disorder or impurity effect is included. But, it is also noted that the existence of our BM states do nor require external magnetic field, disorder or fine-tuning of carrier density, thus BM states could be a robust state of matter, which is stabilized by nontrivial HK interaction. Therefore, we believe that the study of HK-like models is indeed a new direction in the many-body physics and many more interesting physics will be discovered when complicated details are included. We thank Z Yao for his useful comments.This research was supported in part by Supercomputing Center of Lanzhou University and NSFC under Grant No. 11834005, No. 11874188. We thank the Supercomputing Center of Lanzhou University for allocation of CPU time.
http://arxiv.org/abs/2307.01853v1	20230704180000	Superconducting Non-Reciprocity Based on Time-Modulated Coupled-Resonator Systems	[ "Yi Zhuang", "Chandrashekhar Gaikwad", "Daria Kowsari", "Kater Murch", "Aravind Nagulu" ]	quant-ph	[ "quant-ph", "cond-mat.mes-hall", "cond-mat.supr-con" ]	APS/123-QED Department of Electrical and Systems Engineering, Washington University in St. Louis, Missouri 63130 Department of Physics, Washington University, St. Louis, Missouri 63130 Department of Physics, Washington University, St. Louis, Missouri 63130 Department of Physics, Washington University, St. Louis, Missouri 63130 Department of Electrical and Systems Engineering, Washington University in St. Louis, Missouri 63130 We present a unified approach for designing a diverse range of superconducting non-reciprocal components, including circulators, isolators, and uni-directional amplifiers, based on temporally-modulated coupled resonator networks. Our method leverages standard SQUID-based resonators as building blocks, arranged in various configurations such as series-coupled, wye-connected, and lattice-coupled resonators, to realize a wide range of on-chip non-reciprocal devices. Our theoretical studies demonstrated the effectiveness of the proposed approach, achieving circulators and isolators with near-zero insertion losses and isolation greater than 20 dB, and directional amplifiers with forward gain exceeding 10 dB and reverse isolation greater than 20 dB. To validate our findings, we implemented and measured a series-coupled three-resonator superconducting isolator using a single-layer superconducting process. At a base temperature of 20 mK, our device exhibited insertion loss of 1.3 dB in the forward direction, and isolation of up to 25 dB at the center frequency and greater than 15 dB across a bandwidth of 250 MHz in the reverse direction. Our approach promises to enable the design of a broad range of high-performance non-reciprocal devices for superconducting circuits. Superconducting Non-Reciprocity Based on Time-Modulated Coupled-Resonator Systems Aravind Nagulu August 1, 2023 ================================================================================= § INTRODUCTION Superconducting quantum systems are rapidly becoming a promising platform for building quantum computers and other quantum information processing devices <cit.>. These systems consist of carefully engineered superconducting quantum bits (qubits) made using Josephson Junctions (JJs) or a parallel combination of JJs known as superconducting quantum interference devices (SQUIDs) and are operated at milliKelvin temperatures (10mK–100mK) to harness quantum effects <cit.>. Non-reciprocal components, such as circulators and isolators operating at these low temperatures, are widely used to protect the qubits from the noise and spurs of the downstream electronics at higher temperatures, and to separate the input and amplified signals in quantum-limited reflection-type amplifiers <cit.>. A typical qubit readout chain consists of 3–4 circulators per qubit. Currently, commercial ferrite circulators that violate Lorentz reciprocity when biased with a strong magnetic field (around 1 mT) are used for this purpose. These ferrite devices, however, cannot be integrated on-chip alongside the superconducting qubits due to the significant stray flux generated by their strong magnetic bias and the requirement of high deposition temperatures. As a result, they are implemented as connectorized microwave components with strong magnetic shielding, resulting in bulky form factors and expensive implementation costs. This poses challenges for their use in large-scale quantum computing systems with thousands of qubits in a single dilution refrigerator. Alternatively, Lorentz reciprocity can be broken using time-varying structures <cit.> and has been explored extensively in various branches of physics ranging from acoustics <cit.>, electronics <cit.>, mechanics, and optics <cit.> for realizing magnet-free, on-chip non-reciprocal devices. Recognizing the need for miniaturized and monolithically integrated non-reciprocal devices in superconducting quantum systems, prior works have explored achieving on-chip non-reciprocal devices in superconducting platforms <cit.>. Additionally, a compact and scalable design of integrated superconducting non-reciprocal elements would also result in practical realizations of superconducting Floquet topological lattices similar to photonic <cit.>, acoustic <cit.> and electronic <cit.> Floquet topological insulators (TIs). Superconducting topological insulators could open the door to scalable and efficient-multiplexed qubit readout and control, arbitrary signal routing, and could enable potential applications that require integrated nonreciprocity in a lattice <cit.>. In this work, we introduce a unified methodology to realize a wide gamut of superconducting non-reciprocal components such as isolators, circulators, and uni-directional amplifiers using the concept of temporally-modulated coupled resonator networks. The concept of temporal modulation in coupled resonators was introduced to realize on-chip circulators for wireless systems <cit.>. Later, this concept has been translated to realize isolating bandpass filters at radio frequencies (RF) where varactors were used as the modulating element <cit.>. In this work, we present a methodology to translate this concept to superconducting circuits through inductance modulation by using SQUIDs as the modulating elements. Our study showed that it is possible to utilize standard SQUID-based resonators as fundamental components that can be arranged in different configurations, including series-coupled resonators, wye-connected resonators, and lattice-coupled resonators. These configurations allow for the implementation of a broad range of on-chip non-reciprocal responses, such as isolation, circulation, and directional amplification. Finally, we validate our method through the physical implementation and measurement of a series-coupled three-resonator superconducting isolator achieving >20 dB non-reciprocity in its amplitude response. The rest of the article is structured as follows: section <ref> introduces the concept of time modulation and outlines the operation of the elementary unit-cell—a time-modulated SQUID-based resonator—which serves as the fundamental building block for our nonreciprocal devices. In section <ref>, we delve into the concept, analytical studies, and simulation results of integrated non-reciprocal devices, including isolators, directional amplifiers, and circulators, that rely on coupled time-modulated resonators. Section <ref> presents the implementation and measurement results of a superconducting isolator constructed using three series-coupled SQUID-based resonators. Section <ref> discusses possible extensions of the proposed concept. Lastly, in section <ref>, we conclude the paper by providing some final remarks. § TIME-MODULATED RESONATORS Consider a linear time-varying (LTV) system with one shunt component modulated with a periodic signal at a frequency ω_m. The ABCD network properties of a parametrically modulated shunt element can be represented as [ V_1(t); I_1(t) ] = [ 1 0; Y(t) 1 ]×[ V_2(t); I_2(t) ], where V_1,2(t) and I_1,2(t) are the voltages and currents at ports 1 and 2 respectively. By taking a Fourier transform of the system in (<ref>), one can show that the port voltages and currents carry the intermodulation signal between the input and the pump frequencies, namely, they contain frequencies component at (ω_in± kω_m ) where k=0,±1,±2,... In the spectral domain, such a time-modulated system can be represented as [ V_1; ; I_1 ] = [ U 0; ; Y U ]×[ V_2; ; I_2 ], where V_1,2 and I_1,2 are column vectors of size (2N+1) with Fourier coefficients of frequency components (ω_in± kω_m), k= [-N, -(N-1),...0,... N-1, N], Y represents the spectral admittance matrix of the shunt element (see section <ref> for more details), U and 0 are the identity and zero matrices <cit.>. The value of N determines the accuracy of the spectral domain computation. §.§ Spectral Admittance Matrix of a DC-SQUID JJs are superconducting devices made by sandwiching a thin layer of insulator between two superconducting layers <cit.>. A SQUID consists of two JJs in parallel and its inductance is controlled by modulating the magnetic flux threading the junction loop. The inductance of a SQUID can be expressed as L_SQUID = Φ_0/ 4π I_c cos(πΦ/Φ_0), where I_c is the critical current of the JJs, Φ is the magnetic flux threading the SQUID loop, and Φ_0 is the flux quantum. It has to be noted that (<ref>) is applicable for signal currents that are smaller than the critical current. Hence the power handling of the SQUID is limited by the value of critical current. The power handling of the SQUID can be increased by using an array of concatenated SQUID loops and the inductance of an N-stacked SQUID array multiplied by a factor of N. From (<ref>), the inductance of a flux-modulated SQUID can be expressed as 1/L(t) = 4π I_c/Φ_0cos(π(Φ_DC+ ΔΦcos(ω_mt+θ))/Φ_0). The cosine term can be expanded and simplified using Taylor expansions in (<ref>) and (<ref>). Further, the inductance of the SQUID can be approximated as 1/L(t)≈∑_p=-3^3F_pe^jp×θe^jpω_m t, where the constants are F_0 = 4π I_c /Φ_0cos(πΦ_DC/Φ_0)[1-1/4(πΔΦ/Φ_0)^2], F_-1 = -F_1 = 2π I_c/jΦ_0sin(πΦ_DC/Φ_0)× [(πΔΦ/Φ_0)-1/8(πΔΦ/Φ_0)^3], F_-2=F_2 = 2π I_c/ϕ_0cos(πϕ_DC/ϕ_0)[-1/4(πΔϕ/ϕ_0)^2], F_-3= -F_3 = 2π I_c/ϕ_0sin(πϕ_DC/ϕ_0)[1/24(πΔϕ/ϕ_0)^3]. The spectral representation of time-varying inductance is discussed in Appendix <ref>. Finally, from (<ref>), the spectral admittance of a flux-modulated SQUID can be expressed as (<ref>). It is important to note that, the up-conversion and down-conversion frequency translational terms of a single time-modulated SQUID have reciprocal magnitude response and non-reciprocal phase response. §.§ Unit Cell: A Time-Modulated SQUID-based Resonator The building block of the time-modulated non-reciprocal components developed in this article is a resonator consisting of a flux-modulated SQUID in parallel with a capacitor as shown in Fig. <ref>(a). The ABCD matrix of the unit cell can be written as [ V_1; ; I_1 ] = [ U 0; ; Y_LC U ]×[ V_2; ; I_2 ], where Y_LC=Y_L+Y_C, Y_L and Y_C is the spectral admittance matrix of the flux-modulated SQUID and capacitor. Since the capacitor is just a static element, Y_C reduces to a diagonal matrix with each entry corresponding to the admittance of the capacitance at the corresponding intermodulation (IM) frequency. Fig. <ref>(a) also depicts the frequency response and normalized output spectra of a time-modulated resonator with ΔΦ= 0.025Φ_0, f_m= 0.7 GHz, and an input signal at the center frequency f_in=f_center=6 GHz. The time-modulated resonators exhibit a reciprocal response with non-zero loss due to the frequency conversion of the input power to intermodulation frequencies. In section <ref>, we will show how multiple resonators can be coupled to create a direction-dependent frequency translation, thus realizing a non-reciprocal amplitude response. §.§ Unit Cell with Admittance Inverter To enable coupling between resonator networks, we sandwich our unit cell between two admittance inverters (J-inverters) as shown in Fig. <ref>(b). Since the admittance inverters are time-invariant, the ABCD matrix is M_J = [ 0 ±1/jJU; ± jJU 0. ], where 1/J is the characteristic impedance of the impedance inverter, and j=√(-1). An appropriate value of the J-inverter admittance can be chosen to create the required coupling <cit.>. The spectral ABCD matrix of a unit cell with J-inverters can be expressed as [ V_1; ; I_1 ] = [M_J× M_LC× M_J] ×[ V_2; ; I_2 ], = [ U (Y_LC)/J^2; ; 0 U ]×[ V_2; ; I_2 ]. As shown in Fig. <ref>(b), the transmission response and the frequency translation features of the unit cell remain similar to Fig. <ref>(a). However, the J-inverters enable us to create coupling between two unit cells which is essential for creating a coupled resonator network and therefore non-reciprocal devices. § SUPERCONDUCTING NON-RECIPROCAL DEVICES BASED ON TIME-MODULATED COUPLED RESONATOR NETWORKS In modern circuit design, the ability to achieve non-reciprocal response is essential. One approach to realize the such a response is by coupling several unit cells together using J-inverters, thus introducing a spatio-temporal modulation. In addition, these temporally modulated unit cells can be connected in various circuit topologies such as series-coupling, wye-coupling, delta-coupling, and 2-D lattices <cit.>. Each of these topologies can exhibit unique non-reciprocal behavior such as isolation, circulator, and topological robustness. In this section, we will discuss these circuit topologies and analyze their non-reciprocal behavior using the spectral-ABCD matrices. Specifically, we will investigate the factors that influence the non-reciprocal behavior, such as the modulation scheme and the arrangement of the unit cells. Understanding these dependencies allows us to design circuits with tailored non-reciprocal responses to meet specific application requirements. A similar theoretical study of time-modulated, coupled-resonator networks was reported in <cit.>. §.§ Amplitude Non-Reciprocity Using Coupled Unit Cells When multiple time-varying elements are coupled, the input signal will be up-converted (down-converted) by one resonator and the generated IM products can be down-converted (up-converted) back to the input frequency by another resonator. From section <ref>, we have seen that a single time-varying resonator results in the same conversion gain for the up-conversion and down-conversion to IM products, but results in a nonreciprocal phase relation. The nonreciprocal phase relation was not emphasized in section II. Therefore, if we introduce a phase staggering between the pumping signals of two time-varying resonators, the phase of the signal that gets reconstructed back to the input frequency would depend on the phase difference between the pumping signals. Under optimal modulation conditions the reconstructed input signal can add-up destructively or constructively depending on the incident signal direction, thus resulting in non-reciprocal amplitude response. This scenario can be illustrated using a simple system with two unit cells that are coupled with one J-inverter and are modulated with Φ_pump1=ΔΦcos(ω_mt+θ_1) and Φ_pump2=ΔΦcos(ω_mt+θ_2). For the sake of simplicity and intuitive understanding, only the first IM conversion is considered in this illustration. However, for a more precise performance evaluation, conversions to other IM frequencies should also be taken into account, as done in the later sections. The network ABCD parameters of this system can be expressed as [ V_1; ; I_1 ] = [M_LC,1× M_J× M_LC,2] ×[ V_2; ; I_2 ], = [ 1/jJY_LC,2 1/jJU; ; jJU+1/jJY_LC,1.Y_LC,2 1/jJY_LC,1 ]×[ V_2; ; I_2 ], where M_LC,1 and M_LC,2 are the spectral-ABCD matrices of the first and second resonators, M_J is the spectral ABCD matrix of the coupling J-inverter, and Y_LC,1 =[ ⋱ ⋮ ⋮ ⋮ ⋱; ; ⋯ F_0-(ω-ω_m)^2C/j(ω-ω_m) F_-1e^-jθ_1/jω F_-2e^-j2θ_1/j(ω+ω_m) ⋯; ; ⋯ F_1e^jθ_1/j(ω-ω_m) F_0-ω^2C/jω F_-1e^-jθ_1/j(ω+ω_m) ⋯; ; ⋯ F_2e^j2θ_1/j(ω-ω_m) F_1e^jθ_1/jω F_0-(ω+ω_m)^2C/j(ω +ω_m) ⋯; ⋱ ⋮ ⋮ ⋮ ⋱ ], where F_0, F_-1, and F_1 are expressed in (<ref>) and (<ref>). Similarly, Y_LC,2 can be expressed in terms of θ_2. The transmission scattering parameter of this network can be expressed as S_21 = 2[A+B/Z_o+CZ_o+D]^-1 = j/J[Y_LC,2+Y_LC,1+U/Z_o-J^2Z_oU +Z_oY_LC,1.Y_LC,2]^-1, and by symmetry, one can show that S_12 = j/J[Y_LC,1+Y_LC,2+U/Z_o-J^2Z_oU +Z_oY_LC,2.Y_LC,1]^-1. The eigenvalues of Y_LC,1 and Y_LC,2 are different when θ_1≠θ_2, thereby making these matrices non-commuting (i.e., Y_LC,2.Y_LC,1≠Y_LC,1.Y_LC,2), resulting in nonreciprocal scattering matrices. By choosing an appropriate modulation amplitude, modulation frequency, and phase difference, one can design for S_21≈ 1 and S_12≈ 0 for the input and output frequency of ω, thus resulting in an isolator. §.§ Isolator Using Time-Modulated Series-Coupled Resonators Traditional bandpass filters (such as Butterworth, Chebyshev, etc.) are typically implemented by coupling multiple LC resonators using admittance inverters. The number of resonators, their coupling, and loaded quality factor are chosen to achieve the desired filter characteristics, such as bandwidth, out-of-band rejection, and in-band ripple <cit.>. Following a similar architecture, we couple multiple unit cells through J-inverters as shown in Fig. <ref>(a). Similar to a conventional LC filter, the static flux biased SQUIDs, the parallel capacitors, and the admittance of the J-inverters can be chosen to realize a specific bandpass filter response. When the SQUIDs are biased with a DC Flux, the lack of frequency translation results in a reciprocal transmission through the filter. However, modulating the SQUIDs within each resonator with sinusoidal flux pumps that have staggered phase shifts would result in a non-reciprocal transmission response in the circuit. In this scenario, the incident signal is first translated to the intermodulation frequencies (f_in± kf_m) and then reconstructed back to the input frequency. The different phase staggering in the pump signals encountered by the input signal in forward direction leads to constructive addition of the reconstructed IM products in one direction, resulting in low insertion loss. Conversely, in the opposite direction, the reconstructed IM products add up destructively, resulting in high isolation <cit.>. The response of the filter can be analyzed in the same fashion as presented in the section <ref>. The ABCD matrix of the filter is a cascade of the ABCD of all the admittance inverters and unit cells, that is M_BPF = M_J_1M_LC_1M_J_2M_LC_2 ... M_J_n-1M_LC_n-1M_J_n, where M_J_1,M_J_2,M_J_n-1, and M_J_n are the spectral-ABCD matrices of the admittance inverters and M_LC_1,M_LC_2 and M_LC_n-1 are the spectral-ABCD matrices of the unit cells. Therefore, substituting (<ref>) and (<ref>) into (<ref>) results in the ABCD matrix of the filter. Fig. <ref>(b) shows the simulation responses of a second-order bandpass filter optimized for maximum non-reciprocity. However, due to limited degrees-of-freedom and frequency translational paths for converting the intermodulation (IM) products back to the input signal, the second-order BPF exhibits a high insertion loss of 4.64 dB when optimized for an isolation of 20 dB. Increasing the number of resonators within the bandpass filter would result in more degrees of freedom and would enable us to achieve both low insertion loss and high isolation. Therefore, higher-order filters such as third-order and fourth-order filters (shown in Fig. <ref>(c) and (d), respectively) can be used to achieve near-zero insertion loss while providing high isolation. Fig. <ref>(e) and (f) depict the normalized output spectrum of a third-order filter in the transmission and the isolation directions respectively. Consistent with the scattering parameters, the filter achieves low insertion loss in the forward direction and high isolation in the reverse direction. In this article, we focused on a detailed analysis of a third-order bandpass filter as it offers low insertion loss, high isolation, and requires less chip area. To maintain consistency with the design implemented in section <ref>, we carried out the analytical analysis on a third-order bandpass filter designed to operate at 6 GHz with a bandwidth of 700 MHz. This coupled-resonator isolator consists of four modulation parameters, namely (i) static flux bias Φ_DC, (ii) modulation amplitude ΔΦ, (iii) modulation frequency f_m, and (iv) modulation phase staggering θ. Fig. <ref> depicts the parametric study of the simulated forward and reverse transmission responses across varying modulation parameters, which was performed using the spectral-ABCD matrices. Fig. <ref>(a) shows the tunability in the center frequency of the isolator across varying static flux bias. Low insertion loss <0.5 dB and high isolation >20 dB can be achieved across a tuning bandwidth of 1.9 GHz by tuning the static flux bias from 0.3Φ_0 to 0.4Φ_0. Fig. <ref>(b) depicts the performance across modulation flux amplitude. As expected, when the amplitude decreases to 0, the filter approaches the static design. We found that a flux modulation amplitude of 0.024Φ_0 results in the optimal insertion loss while achieving a high isolation. Fig. <ref>(c) suggests that when the modulation frequency approaches the bandwidth of the static BPF, it results in the best in isolation. Finally, Fig. <ref>(d) shows that the filter can achieve non-reciprocity with phase staggering ranging from 40^∘ to 120^∘. Within this range, higher phase staggering results in higher isolation, however with a reduced transmission bandwidth. We choose to operate with a phase staggering of 90^∘ as it achieves a low insertion loss of 0.4 dB and a high isolation of 25 dB. Additionally, note that the transmission and the isolation are off-symmetric with respect to each other when θ=θ+180^∘. Therefore, the transmission and the isolation directions of the isolator can be easily tuned by changing the phase staggering to 270^∘. Overall, the coupled-resonator based isolator could results in low insertion loss, high isolation, and tunability in the center frequency and direction of transmission within a compact area on a superconducting chip. A similar architecture achieving the isolator functionality has also been explored in <cit.>. §.§ Uni-Directional Amplification Using Time-Modulated Series-Coupled Resonators As discussed earlier, each unit in the series-coupled isolator is shunted with a SQUID array inductively coupled to a flux line. This flux line can be used to do three-wave mixing which leads to an amplified transmission through the band pass filter. However, this amplification will be bi-directional requiring circulators to protect qubit from the amplified signal. Alternatively, the staggered-phase low-frequency pump modulation can be combined with the three-wave mixing pump to reduce transmission in one direction, hence achieving directional amplification. A simulated result is presented in Fig. <ref>. We biased the filter at around 6.9 GHz which corresponds to 0.24 Φ_o. A flux tone of 13.8 GHz has applied to each flux line for three-wave mixing, additionally, a tone of 200 MHz is applied with phase arrangements such as 0^o, 90^o, and 180^o to each flux line for directionality. The results demonstrate directional amplification, centered around half of the three-wave mixing pump tone, with gains reaching up to 10 dB and isolation up to 20 dB. However, the quantum efficiency of such an amplifier has yet to be experimentally tested. §.§ Circulators Using Time-Modulated Wye-Coupled Resonators Delta- or wye-topology based circulators can achieve strong non-reciprocity and low insertion losses by coupling time-modulated resonators <cit.>. In this section, we demonstrate the realization and performance analysis of the circulators with high isolation and low loss using the unit cells presented in section <ref>. As an example, we present the design of a wye-coupled resonator-based circulator. Fig. <ref>(a) shows the schematic diagram of a wye-coupled resonator network. With optimal modulation parameters, the circulator can achieve strong non-reciprocity with high isolation (>25 dB) and an insertion loss of 2.7 dB, as illustrated in Fig. <ref>(b). From the transmission spectrum in the forward direction Fig. <ref>(c), it is noticeable that the power is spread out across the IM frequencies, resulting in a relatively high insertion loss. To eliminate the fundamental harmonic conversion loss from the IM conversion and create a loss-free circulator, another coupled resonator layer can be added in parallel and modulated at a 180^∘ phase difference compared to the first layer, as shown in Fig. <ref>(e). Fig. <ref>(f) depicts the simulated performance of a 2-layer, 3-port circulator achieving near-zero insertion loss with 25 dB isolation. The loss-free transmission and high isolation of the circulator are also apparent in the transmission and isolation spectra depicted in Fig. <ref>(g) and (h), respectively. Essentially, the odd IM products (ω±ω_m, ω± 3ω_m) are canceled out due to destructive interference between the 180^∘ staggered coupled-resonator layers, leading to low IM products and low insertion loss. Similarly, adding N parallel layers with phase staggering of α=2π/N between the layers can cancel out all intermodulation products except those of the form f_IM=(f_in± k× Nf_m) <cit.>. This leads to a spurious-tones-free, loss-free time-modulated circulator, but requires the granularity of several phases within the modulation signals. For a 3-port circulator, it is optimal to design a 3-layer circulator, as the required three modulation phases for the resonators in the first layer can be reconfigured in the second and third layers to achieve the necessary phase staggering without increasing the overhead on the number of pump signals required. A conceptual diagram of a 3-layer, 3-port circulator has been depicted in Fig. <ref>. Fig. <ref> depict the parametric study of the simulated reflection, transmission, and isolation responses across varying modulation parameters performed using the Floquet scattering matrices <cit.>. Fig. <ref>(a) shows the tunability in the center frequency of the isolator across varying static flux bias. Low insertion loss <0.1 dB and high isolation >15 dB can be achieved across a tuning bandwidth of 700 MHz by tuning the static flux bias from 0.339 Φ_0 to 0.368 Φ_0. Fig. <ref>(b) depicts the performance across modulation flux amplitude. We found that a flux modulation amplitude of 0.036Φ_0 results in optimal insertion loss while achieving high isolation. Fig. <ref>(c) suggests the optimal modulation frequency is 500 MHz for this design. Fig. <ref>(d) shows the filter can achieve non-reciprocity when phase staggering ranges from 80^∘ to 150^∘. We choose to operate with a phase staggering of 120^∘ as it achieves a low insertion loss of 0.02 dB and high isolation of 40 dB. Additionally, note that the transmission and the isolation are off-symmetric with respect to each other when θ=θ+120^∘. Therefore, the transmission and the isolation directions of the circulator can be easily tuned by changing the phase staggering to 240^∘. § IMPLEMENTATION AND MEASUREMENTS OF THREE RESONATOR COUPLED ISOLATOR Fig. <ref> depicts the schematic of the implemented isolator using three coupled resonators. To achieve a compact chip area, the admittance inverters used for the coupling are realized using π-capacitive structure. The negative shunt capacitors required within the π-capacitive are realized by appropriately reducing the shunt capacitor value within the SQUID-resonators. §.§ Design Procedure First, a conventional bandpass filter is designed to operate at 6 GHz with a 1 dB bandwidth of 700 MHz and matching <-15 dB, and high out-of-band rejection using 10-stacked SQUIDs each with I_c=4 μA, Φ_DC=0.35Φ_o, and the capacitors C_1 = 257 fF, C_2 = 113 fF, C_3 = 232 fF and C_4 = 284 fF. Then a sinusoidal modulation with progressive phase shifts is imparted to the feed currents to achieve non-reciprocal response, i.e., Φ_feed,i=Φ_DC+ΔΦcos(ω_mt+(i-1)×θ) where i represents the resonator number from left to right. The filter performance is optimized across the parameter space of ΔΦ, ω_m, θ, and Φ_DC for low insertion loss in the forward direction and high isolation in the reverse direction using the parametric study based on spectral-ABCD matrices in Fig. <ref>. For an optimal modulation condition of Φ_DC=0.35Φ_0, ΔΦ=0.024Φ_0, f_m= 700 MHz, and θ=90^∘, at a center frequency of 6 GHz, the post layout EM simulated filter exhibits an insertion loss of 0.6 dB in the forward direction and isolation >25 dB in the reverse direction. Additionally, the center frequency of the non-reciprocal bandpass filter can be tuned by varying the DC flux bias of the SQUID loops from 5.2 GHz to 7.1 GHz while achieving sub-1dB insertion loss and >20 dB isolation. §.§ Implementation and Fabrication The isolator is fabricated using an in-house single-layer aluminum process. The JJs in the filter are realized using the Dolan bridge technique and an electron-beam lithography process on a bi-layer resist stack with an insulator thickness of 2 nm to 5 nm. The required JJ critical current 4 μA is realized using 4 μm× 0.5 μm junctions with a critical current density of 2 μA/μm^2. The SQUID loops in the filter have a conductor width of 4 μm and occupy a total area of 16 μm×16.5 μm with a loop area of 8 μm×8.5 μm. The EM simulated geometric loop inductance of the SQUIDs is 12 pH, which is approximately 20% of the SQUID inductance at zero flux bias. The capacitors in the design are created with a conventional fingered-capacitor layout with self-resonance frequency >10 GHz. To reduce the capacitive coupling between the feed-line and SQUID loops, a pseudo-differential coupling strategy is employed for the pump line as shown in Fig. <ref>(b). The non-reciprocal bandpass filter has a total area of 3 mm×1 mm and the input/output and pump signals are connected to the chip edge using co-planar waveguide launchers. §.§ Measurement Results The optical image of the fabricated non-reciprocal bandpass filter is shown in Fig. <ref>(c). The device is wire-bonded to a PCB and is mounted in a copper package as shown in Figure <ref>(d). The input/output terminals of the bandpass filter and three pump lines are linked to five of the eight RF lines within the copper package. The package is placed inside a dilution refrigerator with a base temperature of 20 mK and surrounded by cryoperm shielding to provide additional protection again external magnetic fields. Figure <ref>(e) and Fig. <ref>(f) show a picture and a diagram that outlines the setup used for the measurement. The input/output terminals of the bandpass filter are connected to the probe and readout lines in the dilution refrigerator through two SMA latch transfer switches (model no. Radiall R577432000). A through line is also connected between the remaining ports of the switches in order to measure the transmission loss and/or gain of the probe and readout lines inside the dilution refrigerator. Figure <ref>(a) depicts the measured scattering parameters of the non-reciprocal bandpass filter after normalizing with the transmission loss of the through structure. The measured transmission loss of the filter with no modulation is 0.27 dB (black curve) which could be due to the imperfections in the gain-based calibration. When flux-pumping is applied to achieve non-reciprocity, we measured an additional insertion loss of 0.98 dB in the forward direction (green curve). The measured isolation is +25 dB at the center frequency and is >15 dB across 250 MHz bandwidth in the reverse direction. Additionally, the DC flux bias can be leveraged to tune the center frequency of the bandpass filter from 6 GHz to 6.75 GHz as shown in Fig. <ref>(b). A small ripple in the transmission response is noticed in our measurements due to the mismatches within the probe and readout lines. This can be avoided by employing a full 2-port characterization <cit.>. Figure <ref>(a) depicts the transmission loss of the filter across varying input powers. We measured the -1 dB compression point to be -68 dBm which is several orders of magnitude larger than the typical power used to readout/control the superconducting qubits. Normalized output spectra in the forward and reverse directions are depicted in Fig. <ref>(b) and (c) respectively. As it can be seen, in the transmission direction, most of the input power is concentrated at the input frequency, thus representing low insertion loss. In the reverse direction, however, the input power is translated to intermodulation frequencies resulting in high isolation. Reducing the signal power translation to IM frequencies and consequently reducing the harmonic losses through differential architecture similar to the differential circulators discussed in section-<ref> and similar to <cit.> would be an interesting research direction. § CHALLENGES AND FUTURE OUTLOOK While time-modulated superconducting non-reciprocal components offer several advantages over their ferrite counterparts such as easier system integration, low cost, small size, and easier scalability to large systems, they are not without limitations. One challenge in the current implementations is the generation of multiple modulation signals. Additionally, multiple coaxial cables are interfaced from these devices to 4K/room temperature stages. This issue can be easily circumvented by implementing on-chip phase generation circuitry by leveraging passive superconducting hybrid couplers <cit.> and tunable phase shifters <cit.>, thus reducing the number of coaxial cables and their associated heat loading. In addition, time-modulated superconducting circulators can also generate spurious emissions at the intermodulation frequencies, which can potentially interfere with nearby devices. Clever device architectures such as the N-layer circulators, coupled with careful design and filtering in the signal path can help mitigate these spurious emissions, but this remains an ongoing challenge in the development of time-modulated superconducting non-reciprocal devices. As we look to the future, one exciting direction for these components is their extension to superconducting Floquet topological insulator lattices. These lattices can be a new class of quantum devices that can support topologically protected edge states with unique non-reciprocal properties. By exploiting the non-reciprocal behavior of these edge states, superconducting Floquet TI lattices could enable arbitrary signal routing with high fidelity, leading to the development of new, advanced quantum devices. Additionally, by integrating superconducting qubits into the lattices, these devices could also facilitate long-range qubit coupling with reduced decoherence. These devices could also result in dissipation-free non-reciprocal coupling between quantum devices <cit.>, enabling scaling to quantum many-body systems where the study of topological edge-states and invariants <cit.> are expected to yield deviations from the paradigmatic bulk-boundary correspondence <cit.>. Recently, significant progress has been made in the development of photonic <cit.>, acoustic <cit.> and electronic <cit.> Floquet TI lattices, and expanding this knowledge to superconducting lattices could have a transformative impact on the field of quantum computing and related applications. Overall, the extension of superconducting circulators and isolators to Floquet TI lattices represents an exciting direction for future research and development. § CONCLUSION This article presented the concept of time-modulated coupled resonator networks as a means to develop on-chip, magnetless non-reciprocal components for quantum computing systems. We demonstrated that conventional SQUID-based resonators can serve as unit elements and can be combined in various topologies, such as series-coupled resonators, wye-connected and lattice-coupled resonators, to realize a wide range of on-chip non-reciprocal responses, including isolation, circulation, and directional amplification. These coupled-resonator networks provide reconfigurable reciprocal and non-reciprocal responses that are solely dependent on the modulation parameters, such as the static flux bias, modulation amplitude, frequency, and phase staggering. We discussed the design procedure of these non-reciprocal components and evaluated their performance using spectral-ABCD matrices. Finally, we validated our theoretical findings by implementing and measuring an isolator based on three series-coupled resonator networks, thus demonstrating the potential of such networks for future quantum computing systems. By integrating devices in this manner, existing control and readout chains can be simplified, and new control/readout methods can be enabled that are not feasible with commercially available non-reciprocal components. § ACKNOWLEDGEMENTS This work is supported by McKelvey Collaboration Initiation Grant (CIG) 2022, and by NSF Grant No. PHY-1752844 (CAREER), and the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Award on Programmable systems with non-Hermitian quantum dynamics (Grant No. FA9550-21-1- 0202) and use of facilities at the Institute of Materials Science and Engineering at Washington University. § SPECTRAL-ADMITTANCE MATRIX OF A TIME-VARYING INDUCTOR The voltage and current relationship of a time-varying inductor <cit.> can be expressed as i(t) = L(t)^-1∫ v(t)dt. When the inductance is modulated with a fundamental frequency of ω_m and input excitation is at ω, the voltage and current would carry the IM products and can be expressed as i(t) = ∑_p=-k^kI(ω+pω_m)e^-j(ω+pω_m) t, v(t) = ∑_p=-k^kV(ω+pω_m)e^-j(ω+pω_m)t, where I(ω+pω_m) and V(ω+pω_m) are the current and voltage Fourier coefficients of the frequency (ω+pω_m), and k is the farthest IM product that has been calculated. The value of k determines the accuracy of the computation and as k→∞ the computation error → 0. This current and voltage can be expressed in matrix form as I=[ ⋮; I(ω-2ω_m); I(ω-ω_m); I(ω); I(ω+ω_m); I(ω+2ω_m); ⋮ ], V=[ ⋮; V(ω-2ω_m); V(ω-ω_m); V(ω); V(ω+ω_m); V(ω+2ω_m); ⋮ ]. Therefore, the integral of voltage can be calculated as ∫V(t)dt = [ V(ω-kω_m)/j(ω-kω_m); ⋮; V(ω)/jω; ⋮; V(ω+kω_m)/j(ω+kω_m); ] =Ω×V, where Ω = [ 1/j(ω-kω_m) 0 0 0 0; 0 ⋱ ⋮ ⋱ 0; 0 ⋯ 1/j(ω) ⋯ 0; 0 ⋱ ⋮ ⋱ 0; 0 0 0 0 1/j(ω+kω_m) ]. 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http://arxiv.org/abs/2307.03176v1	20230706175606	Learning Curves for Heterogeneous Feature-Subsampled Ridge Ensembles	[ "Benjamin S. Ruben", "Cengiz Pehlevan" ]	stat.ML	[ "stat.ML", "cond-mat.dis-nn", "cs.LG", "q-bio.NC" ]	[email protected] Biophysics Graduate Program, Harvard University, Cambridge, Massachusetts 02138, USA [email protected] John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA Center for Brain Science, Harvard University, Cambridge, Massachusetts 02138, USA Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, Massachusetts 02138, USA Feature bagging is a well-established ensembling method which aims to reduce prediction variance by training estimators in an ensemble on random subsamples or projections of features. Typically, ensembles are chosen to be homogeneous, in the sense the the number of feature dimensions available to an estimator is uniform across the ensemble. Here, we introduce heterogeneous feature ensembling, with estimators built on varying number of feature dimensions, and consider its performance in a linear regression setting. We study an ensemble of linear predictors, each fit using ridge regression on a subset of the available features. We allow the number of features included in these subsets to vary. Using the replica trick from statistical physics, we derive learning curves for ridge ensembles with deterministic linear masks. We obtain explicit expressions for the learning curves in the case of equicorrelated data with an isotropic feature noise. Using the derived expressions, we investigate the effect of subsampling and ensembling, finding sharp transitions in the optimal ensembling strategy in the parameter space of noise level, data correlations, and data-task alignment. Finally, we suggest variable-dimension feature bagging as a strategy to mitigate double descent for robust machine learning in practice.=-1 Learning Curves for Heterogeneous Feature-Subsampled Ridge Ensembles Cengiz Pehlevan August 1, 2023 ==================================================================== § INTRODUCTION Ensembling methods, where one combines predictions from multiple predictors to achieve a stronger prediction, are ubiquitous in machine learning practice <cit.>. A popular class of ensembling methods (known as attribute bagging <cit.> as well as the random subspace method <cit.>) are based on feature subsampling <cit.>, where each predictor has access to only a subset of data features, are independently trained on those features, and their predictions are combined to achieve a stronger prediction. For example, the popular random forest method makes use of this strategy <cit.>. An advantage of these methods is that they allow parallel processing. For example, Feature-Distributed Machine Learning, combine independent predictions made by agents who only see subsets of available features <cit.>. While commonly used in practice, a theoretical understanding of ensembling via feature subsampling is not well developed. Here, we provide an analysis of this technique in the case of feature-subsampled linear ridge regression using methods from statistical physics <cit.>. This allows us to obtain analytical expressions for typical case performance of feature-subsampled linear ridge regression. Analysis of these equations under special cases reveal interesting phenomena involving effects of noise, regularization, and subsampling on prediction performance. Our findings relate to double-descent <cit.>, which results from over-fitting to noise and poses a serious problem for practical machine learning. Regularization is commonly used to mitigate double descent, however optimal regularization strength depends on data and noise levels <cit.>. Our theory reveals an alternative strategy. We observe that subsampling shifts the location of a predictor's sample-wise double-descent peak <cit.>. An interesting consequence of this is that if the predictors are heterogeneous in the number of features they see, they will go through double-descent at different sample-sizes. Therefore, bagging them will lead a mitigation of double-descent, as when one predictor fails, the others will compensate with accurate predictions. In summary, we make the following original contributions: * Using the replica trick from statistical physics <cit.>, we derive the generalization error of ensembled least-squares ridge regression with random structured Gaussian data, deterministic feature maps, and a noisy linear teacher function. Our derivation allows for heterogeneity in the rank of the feature maps of the ensemble members. * We derive explicit formulas which demonstrate that subsampling alters the interpolation threshold of ridge regression. * We demonstrate benefits of heterogeneous ensembling as a robust method for mitigating double-descent. * We analyze the role of data correlations, readout noise, and data-task alignment in determining the optimal ensembling strategy in a tractable special case. Related works: A substantial body of work has elucidated the behavior of linear predictors for a variety of feature maps <cit.>. Several recent works have extended this research to characterize the behavior of ensembled regression using solvable models <cit.>. Ref. <cit.> derives expressions for the generalization error of generalized linear models, of which ridge ensembles are a special case, in terms of the solutions to a set of self-consistent equations. However, <cit.> and <cit.> focus their analysis on the case of isotropic data and Gaussian random masks of homogeneous dimensionality. In contrast, we explicitly consider learning from correlated data by ensembles with heterogeneous readout dimensionality. Our work focuses on the effect of feature-wise subsampling. Additional recent works study the performance of ridge ensembles with example-wise subsampling <cit.> and simultaneous subsampling of features and examples <cit.>. These works find that subsampling behaves as an implicit regularization, and prove equivalences between optimal ensembling and optimal regularization. In a similar vein, we consider here ensembling as a safeguard against insufficient regularization. Methods from statistical physics have long been used for machine learning theory <cit.>. Relevant work in this domain include <cit.> which studied ensembling by data-subsampling in linear regression. § LERANING CURVES FOR ENSEMBLED RIDGE REGRESSION FROM THE REPLICA METHOD We consider noisy ensembled ridge regression in the setting where ensemble members are trained independently on masked versions of the available features. We derive our main analytical formula for generalization error of ensembled linear regression, as well as analytical expressions for generalization error in the special case of subsampling of equicorrelated features. Later sections illustrate the implications of the derived formulas. §.§ Problem Setup Consider a training set 𝒟 = {ψ̅_μ, y^μ}_μ = 1^P of size P. The training examples ψ̅_μ∈ℝ^M are drawn from a Gaussian distribution with Gaussian feature noise: ψ̅_μ = ψ_μ + σ_μ, where ψ_μ∼𝒩(0, Σ_s) and σ_μ∼𝒩(0, Σ_0). Data and noise are drawn i.i.d. so that 𝔼[ ψ_μψ_ν^⊤] = δ_μνΣ_s and 𝔼[ σ_μσ_ν^⊤] = δ_μνΣ_0. Labels are generated from a noisy teacher function y_μ = 1/√(M)w^⊤ψ_μ + ϵ^μ where ϵ^μ∼𝒩(0,ζ^2). Label noises are drawn i.i.d. so that 𝔼[ϵ^μϵ^ν] = δ_μνζ^2. We seek to analyze the quality of predictions which are averaged over an ensemble of ridge regression models, each with access to a subset of the features. We consider k linear predictors with weights ŵ_r ∈ℝ^N_r, r = 1,…, k. Critically, we allow N_r ≠ N_r' for r ≠ r', which allows us to introduce structural heterogeneity into the ensemble of predictors. A forward pass of the model is given as: f(ψ) = 1/k∑_r=1^k f_r(ψ), f_r(ψ) = 1/√(N_r)ŵ_r^⊤A_r (ψ + σ) + ξ_r. The model's prediction f(ψ) is an average over k linear predictors. The “measurement matrices” A_r ∈ℝ^N_r × M act as linear masks restricting the information about the features available to each member of the ensemble. Subsampling may be implemented by choosing the rows of each A_r to coincide with the rows of the identity matrix – the row indices corresponding to indices of the sampled features. The feature noise σ∼𝒩(0, Σ_0) and the readout noises ξ_r ∼𝒩(0,η_r^2), are drawn independently at the execution of each forward pass of the model. Note that while the feature noise is shared across the ensemble, readout noise is drawn independently for each readout: 𝔼[ξ_rξ_r'] = δ_rr'η_r^2. The weight vectors are trained separately in order to minimize a regular least-squares loss function with ridge regularization: ŵ_r = _w_r ∈ℝ^N_r[ ∑_μ = 1^P ( 1/√(N_r)w_r^⊤A_rψ̅_μ + ξ_r^μ - y_μ)^2 + λ \|w_r^2\|] Here {ξ_r^μ} represents the readout noise which is present during training, and independently drawn: ξ_r^μ∼𝒩(0, η_r^2), 𝔼[ξ_r^μξ_r^ν] = η_r^2 δ_μν. As a measure of model performance, we consider the generalization error, given by the mean-squared-error (MSE) on ensemble-averaged prediction: E_g (𝒟)= ⟨(f(ψ) - 1/√(M)w^⊤ψ)^2 ⟩ Here, the angular brackets represent an average over the data distribution and noise: ψ∼𝒩(0, Σ_s), σ∼𝒩(0, Σ_0), ξ_r ∼𝒩(0, η_r^2). The generalization error depends on the particular realization of the dataset 𝒟 through the learned weights {ŵ^}. We may decompose the generalization error as follows: E_g(𝒟) = 1/k^2∑_r, r' = 1^k E_r r'(𝒟) E_r r'(𝒟) ≡1/M[ (1/√(ν_rr)A_r^⊤ŵ_r - w^)^⊤Σ_s (1/√(ν_r'r')A_r'^⊤ŵ_r'- w^) . . + 1/√(ν_rrν_r'r')ŵ_r^⊤A_r Σ_0 A_r'^⊤ŵ_r' + M δ_rr'η_r^2 ] Computing the generalization error of the model is then a matter of calculating E_rr' in the cases where r = r' and r ≠ r'. Furthermore, in the asymptotic limit we consider, we expect that the generalization error concentrates over randomly drawn datasets 𝒟. §.§ Main Result We calculate the generalization error using the replica trick from statistical physics. The result of our calculation is stated in proposition 1. The proof is lengthy, and can be found in the SI. Consider the ensembled ridge regression problem described in Section <ref>. Consider the asymptotic limit where M, P, {N_r}→∞ while the ratios α = P/M and ν_rr = N_r/M, r = 1,…,k remain fixed. Define the following quantities: Σ̃_rr' ≡1/√(ν_rrν_r'r')A_r [Σ_s + Σ_0]A_r'^⊤ G_r ≡I_N_r + q̂_r Σ̃_rr γ_rr' ≡α/M (λ+q_r)(λ+q_r'))[ G_r^-1Σ̃_rr'G_r'^-1Σ̃_r'r] Then the average generalization error may be written as: ⟨ E_g(𝒟) ⟩_𝒟 = 1/K^2∑_r, r' = 1^K ⟨ E_rr'(𝒟) ⟩_𝒟, where ⟨ E_rr'(𝒟) ⟩_𝒟 = γ_rr'ζ^2 + δ_rr'η_r^2/1-γ_rr' + 1/1-γ_rr'( 1/Mw^ ⊤Σ_s w^* ) - 1/M(1-γ_rr')w^⊤Σ_s [1/ν_rrq̂_r A_r^⊤G_r^-1A_r + 1/ν_r'r'q̂_r'A_r'^⊤G_r'^-1A_r'] Σ_s w^ + q̂_r q̂_r'/M(1-γ_rr')1/√(ν_rrν_r'r')w^⊤Σ_s A_r^⊤G_r^-1Σ̃_rr'G_r'^-1A_r'Σ_s w^ where the pairs of order parameters {q_r, q̂_r} for r = 1, …, K, satisfy the following self-consistent saddle-point equations q̂_r = α/λ + q_r, q_r = 1/M[ G_r^-1Σ̃_rr]. We calculate the terms in the generalization error using the replica trick from the statistical physics of disordered systems. The full derivation may be found in the supplemental material. We make several remarks on this result: This is a highly general result which applies to any selection of linear masks {A_r}. However, we will focus on the case where the {A_r} implement subsampling of the feature neurons. Our result reduces to the results of <cit.> when k=1 and η = 0, and may be obtained as a special case of <cit.> in this limit. In the case where all readout weights have the same dimension N_r =N, r=1,…,k, this result may be obtained as a special case of the results of <cit.>. The novelty in our derivation (and subsequent analysis) is to consider heterogeneity in the values of N_r. The replica trick <cit.> is a non-rigorous but standard heuristic in the study of disordered systems. We confirm our results in simulations. In Figure <ref>a, we confirm the result of the general calculation by comparing with numerical experiments. Experimental curves are generated by running ridge regression on randomly drawn datasets with M=2000 features and averaging over the resulting error. We use highly structured data, feature noise, label noise, and readout noise (see caption for details). Each of k=3 readouts sees a fixed but randomly drawn subset of features. Theory curves are calculated by solving the fixed-point equations <ref> numerically for the chosen Σ_s, Σ_0 and {A_r}_r=1^k then plugging the resulting order parameters into eq. <ref>. §.§ Equicorrelated Data Our general result allows the freedom to tune many important parameters of the learning problem: the correlation structure of the dataset, the number of ensemble members, the scales of noise, etc. However, the derived expressions are rather opaque, as they depend on the solution to a set of in general analytically intractable self-consistent equations for the order parameters. In order to better understand the phenomena captured by these expressions, we simplify them in the tractable special case in which features of the data are equicorrelated: Consider the ensembled ridge regression problem described in section <ref>, and the result of proposition 1. Consider the special case in which we select the following parameters: w^* = √(1-ρ^2)ℙ_⊥w^_0 + ρ1_M w^_0 ∼𝒩(0, I_M) Σ_s = s [(1-c) I_M + c 1_M 1_M^⊤] Σ_0 = ωI_M with c ∈ [0,1], ρ∈ [-1, 1]. A label noise scale ζ≥ 0 and readout noise scales η_r ≥ 0 are permitted. Here ℙ_⊥ = I_M - 1/N1_M 1_M^⊤ is a projection matrix which removes the component of w^_0 which is parallel to 1_M. The measurement matrices {A_r}_r=1^k have rows consisting of distinct one-hot vectors so that each of the k readouts has access to a subset of N_r = ν_rr M features. For r ≠ r', denote by n_rr' the number of neurons sampled by both A_r and A_r' and let ν_rr'≡ n_rr'/M remain fixed as M →∞. In this case, we may obtain fully analytical formulas for the generalization error as follows. First define the following quantities: a ≡ s(1-c) + ω S_r ≡q̂_r/ν_rr + a q̂_r, γ_rr'≡a^2 ν_rr' S_r S_r'/α The terms of the decomposed generalization error may then be written: ⟨ E_rr'⟩_𝒟, w^_0 = 1/1-γ_rr'((1-ρ^2) I^0_rr' + ρ^2 I^1_rr') + γ_rr'ζ^2 + δ_rr'η_r^2/1-γ_rr' where we have defined I^0_rr' ≡ s(1-c) ( 1-s(1-c) ν_rrS_r - s(1-c)ν_r'r'S_r' + a s (1-c) ν_rr'S_r S_r') I^1_rr' ≡s(1-c)(ν_rr'-ν_rrν_r'r')+ων_rr'/ν_rrν_r'r' if 0 < c≤ 1 I^0_rr' if c=0 and where the order parameters {q_r, q̂_r} may be obtained analytically as the solution (with q_r>0) to the following quadratic system of equations: q_r = aν_rr/ν_rr + a q̂_r , q̂_r = α/λ+q_r In the “ridgeless” limit where λ→ 0, we may make the following simplifications: S_r →2 α/a( α + ν_rr + \|α - ν_rr\| ) γ_rr' →4 αν_rr'/( α + ν_rr + \|α - ν_rr\| )( α + ν_r'r' + \|α - ν_r'r'\| ) Simplifying the fixed-point equations and generalization error formulas in this special case is an exercise in linear algebra. The main tools used are the Sherman-Morrison formula <cit.> and the fact that the data distribution is isotropic in the features so that the form of Σ̃_rr and Σ̃_rr' depend only on N_r, N_r', and n_rr' . Thus, the result depends only on the values of {ν_rr'} and not the identities of the subsampled features. To aid in computing the necessary matrix contractions we developed a custom Mathematica package which handles block matrices of symbolic dimension, with blocks containing matrices of the form M = c_1I + c_2 11^⊤. This package and the Mathematica notebook used to derive these results will be made available online (see SI) In this tractable special case, c∈ [0,1] is a parameter which tunes the strength of correlations between features of the data. When c = 0, the features are independent, and when c = 1 the features are always equivalent. s sets the overall scale of the features and ρ tunes the alignment of the ground truth weights with the special direction in the covariance matrix. We refer to ρ as the “task alignment”, and it can be thought of as a simple proxy for the “task-model alignment” <cit.> or “code-task alignment” <cit.>. In Figure <ref>b, we test these results by comparing the theoretical expressions for generalization error with the results of numerical experiments, finding perfect agreement. Note that in this case, both theory and experiment are averaged over ground-truth weights as well as datasets. §.§ Subsampling shifts the double-descent peak of a linear predictor. Consider the equicorrelated data model in the isotropic limit (c = 0). Consider a single linear regressor (k=1) which connects to a subset of N = ν M features. In the ridgeless limit where regularization λ→ 0, and without readout noise or feature noise (η = ω = 0), the generalization error is given by equation <ref> with ν_rr = ν, s = 1, η_r = ω = 0 in the λ→ 0 limit: ⟨ E_g ⟩_𝒟, w^* ={[ ν/ν-α[ (1-ν_rr) + 1/ν_rr(α-ν)^2 ] + α/ν-αζ^2, if α < ν; α/α-ν[ 1-ν] + ν/α-νζ^2, if α > ν ]} Double descent can arise from two possible sources of variance: explicit label noise (ζ>0) or implicit label noise induced by feature subsampling (ν<1). As E_g ∼ (α-ν)^-1, we see that the generalization error diverges when α = ν. The subsampling fraction ν thus controls the sample complexity α at which the double-descent peak occurs. Intuitively, this occurs because subsampling changes the number of parameters of the regression model, and thus its interpolation threshold. To demonstrate this, we plot the learning curves for subsampled linear regression on equicorrelated data in Figure <ref>. While at finite ridge the test error no longer diverges when α = ν, it may still display a distinctive peak. §.§ Heterogeneous connectivity mitigates double-descent The observed phenomenon of double-descent – over-fitting to noise in the training set near a model's interpolation threshold – poses a serious risk in practical machine-learning applications. Regularization is the canonical strategy employed to mitigate double descent. However, in order to achieve monotonic learning, the regularization parameter must be tuned to the structure of the task and the scale of the label noise <cit.> – no one choice for the regularization parameter can mitigate double descent for all tasks. Considering again the plots in Figure <ref>(b), we observe that at any value of α, the double-descent peak can be avoided with an acceptable choice of the subsampling fraction ν. This suggests another strategy to mitigate double descent: heterogeneous ensembling. Rather than training an ensemble of linear predictors, each with the same interpolation threshold, we may ensemble over predictors with a heterogeneous distribution of interpolation thresholds in the hopes that when one predictor fails, the other members of the ensemble compensate. In Figure <ref>, we demonstrate that in the absence of a sufficiently regularization, heterogeneous ensembling can mitigate double-descent. Specifically. We define two ensembling strategies: in homogeneous ensembling, each of the k readouts is connected to the same fraction ν_rr = 1/k features. In heterogeneous ensembling, the number of features connected by each of the k readouts are drawn i.i.d. from a Gamma distribution with fixed mean 1/k and variance σ^2. We denote this ν_rr∼Γ_k, σ. After they are independently drawn, subsampling fractions are re-scaled so that they sum to unity: ν_rr/∑_r ν_rr←ν_rr. This ensures fair competition, wherein the total number of readout weights utilized in homogeneous and heterogeneous ensembling are equal. Equivalently, we may consider the readout fractions ν_rr to be drawn from a Dirichlet distribution: (ν_1, …, ν_k) ∼( (σ k)^-2, …, (σ k)^-2 ) <cit.>. These strategies for connecting readouts to the features are illustrated for k = 10 in figures <ref> a.i (homogeneous) and <ref> a.ii (heterogeneous). The density of the distribution Γ_k, σ(ν) is plotted in figure <ref>b for k = 10 and varying σ. In figure S1, we apply these ideas to a classification task on the CIFAR-10 dataset. We find that in this nonlinear setting, heterogeneous ensembling prevents catastrophic over-fitting, leading to monotonic learning curves without regularization (see SI for details). In figure <ref>c, we use our analytical theory of equicorrelated data (see eqs. <ref>) to compare the performance of homogeneous and heterogeneous ensembling with k = 10. We find that for an under-regularized predictor, (<ref>c.i, c.ii, c.iii) heterogeneous ensembling reduces the height of the double-descent peak. At larger regularization (<ref>c.iv, c.v, c.vi), homogeneous and heterogeneous ensembling perform similarly. We quantify the extent of double-descent through the worst-case error max_α(E_g(α)). We find that as σ increases, the worst-case error decreases monotonically at no cost to the asymptotic error E_g(α→∞) (see Fig. <ref>d,e). §.§ Data correlations, readout noise, and task structure determine optimal ensemble size We now ask whether ensembling is a fruitful strategy – i.e. whether it is preferable to have a single, fully connected readout or multiple sparsely connected readouts. Intuitively, the presence of correlations between features permits subsampling, as measurements from a subset of neurons will also confer information about the state of the others. In addition, ensembling over multiple readouts can average out the readout noise. To quantify these notions, we consider the special case of ensembling over k readouts, each connecting the same fraction ν_rr = ν = 1/k of features in an equicorrelated code with correlation strength c and readout noise scale η, and task alignment ρ. We set the label noise, feature noise, and overlap between readouts to zero (ζ = 0, ω = 0, ν_rr' = 0 when r ≠ r'). In the ridgeless limit, we can then express the error as : E_g(k) = s(1-c)F(H, k, ρ, α), where H ≡η^2/s(1-c) is an effective inverse signal-to-noise ratio and F(H, k, ρ, α) is a rational function of its arguments (see SI for full expressions). Therefore, given fixed parameters s, c, ρ, α, the value k^* which minimizes error depends on η, s, and c only through the ratio H. Using our analytical theory, we plot the optimal number of readouts k in the parameter space of H and ρ (see Fig. <ref>a). The resulting phase diagrams are naturally divided into three regions. In the signal-dominated phase a single fully-connected readout is optimal (k^* = 1). In an intermediate phase, 1<k^<∞ minimizes error. And in a noise-dominated phase k^ = ∞. The boundary between the signal-dominated and noise-dominated phases (dotted lines in <ref>a) can be written H = (1-1/α)(1-ρ^2) when α>1 and H = α(1-α)(1-ρ^2) when α<1 . The boundary between the intermediate and noise-dominated phases (dashed lines in <ref>a) can be written H = 2 - (2+1/α)ρ^2. As is evident in these phase diagrams, an increase in H causes an increase in k^. This can occur because of a decrease in the signal-to-readout noise ratio s/η^2, or through an increase in the correlation strength c. An increase in ρ also leads to an increase in k^, indicating that ensembling is more effective when there is alignment between the structure of the data and the task. Learning curves from each of these phases for varying k are plotted in Fig. <ref>b. The resulting shifts in the location of the double-descent peak resemble those observed in practice for ensembling methods applied to linear classifiers <cit.>. § CONCLUSION In this paper, we provided a theory of feature-subsampled ensembling techniques focusing on feature-subsampled linear ridge regression. Our technique was the replica method from statistical physics which led us to derive an analytical formula for the typical case generalization error in the aforementioned setting. We solved these equations for a special case which revealed many interesting phenomena. One of these phenomena relate to double descent <cit.>. In most machine learning applications, the size of the dataset is known at the outset and suitable regularization may be determined to mitigate double descent, either by selecting a highly over-parameterized model <cit.> or by cross-validation techniques (see for example <cit.>). However, in contexts where a single network architecture is designed for an unknown task or a variety of tasks with varying structure and noise levels, heterogeneous ensembling may be used to smooth out the perils of double-descent. Our analysis of ensembling in noisy neural networks suggests that an ensembling approach may be useful in improving the stability of analog neural networks, where readout noise is a significant problem (see, for example, <cit.>). Much work remains to achieve a full understanding of the interactions between data correlations, readout noise, and ensembling. In this work, we have given a thorough treatment of the convenient special case where features are equicorrelated and readouts do not overlap. Future work should analyze ensembling for codes with an arbitrary correlation structure, in which readouts access randomly chosen, potentially overlapping subsets of features. This will require to average our expressions for the generalization error over randomly drawn masks {A_r}. This problem has been thoroughly studied in the case where the entries of A_r are i.i.d Gaussian <cit.>, as in the ever-popular random feature model. Recent progress on the problem of non-Gaussian projections for a single readout has been made in <cit.>. § ACKNOWLEDGEMENTS CP and this research were supported by NSF Award DMS-2134157. BSR was also supported by the National Institutes of Health Molecular Biophysics Training Grant NIH/ NIGMS T32 GM008313. We thank Jacob Zavatone-Veth and Blake Bordelon for thoughtful discussion and comments on this manuscript. § APPLICATION TO CIFAR10 CLASSIFICATION TASK In this section, we demonstrate that the qualitative insights gained from our analysis of the linear regression task with Gaussian data carries over to a practical machine learning task. In particular, we show that ensembling with heterogeneous readout connectivity can mitigate double-descent in the CIFAR10 classification task <cit.> without regularization. The experimental setup is as follows: To obtain a useful feature map, we first train a deep, fully connected multi-layer perceptron (MLP) to solve the CIFAR10 classification task. The MLP has three hidden layers of size N_h = 1000. We use a training set of 50,000 images x_μ∈ℝ^N_0, N_0 = 3072 from N_out = 10 classes. The targets are assigned as one-hot vectors We write this network as follows: h = ( N_in^-1/2W_inx) h_2 = (M^-1/2W_1h_1) ψ(x) = (M^-1/2W_2 h_2) ŷ(x) = M^-1/2W_outψ(x) Where W_in∈ℝ^N_h × N_in, W_1∈ℝ^N_h × M, W_2∈ℝ^N_h × M, W_out∈ℝ^10 × M We use a MSE loss function: ℓ({W_in,W_1, W_2, W_out}) = ∑_μ (ŷ(x_μ) - y_μ) The predicted class is then assigned as the class corresponding to the component of ŷ with maximum value. Training for 2000 steps with full-batch gradient descent and the Adam optimizer at a learning rate of .001, the network achieves a training accuracy of 90% and a test accuracy of ∼%50. The learned features ψ(x) are then extracted and new readout weights are trained using the homogeneous or heterogeneous ensembling strategies with modifications for handling multiple readout classes. An ensemble of k predictions is made: y^r(x) = 1/√(N_r)W_r A_r ψ(x) for r = 1, …, k. Here, each A_r ∈ℝ^N_r × N_in implements subsampling of a randomly drawn subset of N_r features, and w_r ∈ℝ^N_out× N_r predict the class of the input from these subsampled features. The weights W_r are trained independently using a pseudoinverse rule, which is equivalent to ridge regression in the limit of zero regularization. Finally, the predictions of the ensemble of readouts are combined using a mean with a nonlinear threshold: ŷ(x) = ∑_r=1^k ϕ(y^r(x)) ϕ(x) = 1/2(1+tanh(5(x-1/2))) In supplemental figure 1, we demonstrate the performance of re-learning ensembles of readout weights using the homogeneous and heterogeneous ensembling strategies. To review, in homogeneous ensembling, the subsampling fractions ν_rr = N_r/M are chosen as ν_rr = 1/k, r = 1,…,k. However, in heterogeneous subsampling, the weights are drawn from a gamma distribution with mean 1/k and variance σ^2, then re-scaled so that ∑_r ν_rr = 1. In figure S1, we use σ = 1/(2k). We that the heterogeneous ensembling approach leads to a smooth learning curve without double descent. This effect is most pronounced in the plots of “test accuracy”, which is the probability of incorrect classification of a test example. While homogeneous ensembling shifts the double-descent peak toward smaller P, heterogeneous ensembling eliminated the peak. Because the data is heavily correlated, there is no cost to the prediction performance at large P. § GENERALIZATION ERROR OF ENSEMBLED LINEAR REGRESSION FROM THE REPLICA TRICK Here we use the Replica Trick from statistical physics to derive analytical expressions for E_rr'. We treat the cases where r=r' and r ≠ r' separately. Following a statistical mechanics approach, we calculate the average generalization error over a Gibbs measure with inverse temperature β; Z = ∫∏_r dwexp( - β/2∑_r E_t^r - M β/2∑_r, r' J_rr' E_rr'(w_r, w_r') ) E_t^r = ∑_μ = 1^P ( 1/√(N_r)w_r^⊤A_rψ̅_μ + ξ_r - y_μ)^2 + λ \|w_r^2\| In the limit where β→∞ the gibbs measure will concentrate around the weight vector ŵ_r which minimizes the regularized loss function. The replica trick relies on the following identity: ⟨log( Z[𝒟] )⟩_𝒟 = lim_n → 01/nlog(⟨Z^n ⟩_𝒟) where ⟨·⟩_𝒟 represents an average over all quenched disorder in the system. In this case, quenched disorder – the disorder which is fixed prior to and throughout training of the weights – consists of the selected training examples along with their feature noise and label noise: 𝒟 = {ψ_μ, σ^μ, ϵ^μ}_μ = 1^P. The calculation proceeds by first computing the average of the replicated partition function assuming n is a positive integer. Then, in a non-rigorous but standard step, we analytically extend the result to n → 0. §.§ Diagonal Terms We start by calculating E_rr for some fixed choice of r. Noting that the diagonal terms of the generalization error E_rr only depend on the learned weights w_r, and the loss function separates over the readouts, we may consider the Gibbs measure over only these weights: Z = ∫ dw_r exp( -β/2 λ E^t_r - JMβ/2 E_rr(w_r) ) ⟨Z^n ⟩_𝒟 = ∫∏_a dw^a_r 𝔼_{ψ_μ, σ^μ, ϵ^μ} exp ( -β M/2 λ∑_μ, a1/M[ 1/√(ν_rr)w_r^a⊤A_r( ψ_μ + σ^μ) - w^* ⊤ψ_μ - √(M) (ϵ^μ - ξ_r^μ) ]^2 . . - β/2∑_a \|w_r^a\|^2 - JMβ/2∑_a E_rr(w^a) ) Next we must perform the averages over quenched disorder. We first integrate over {ψ_μ, σ^μ, ξ_r^μ, ϵ^μ}_μ = 1^P. Noting that the scalars h_μ^ra≡1/√(M)[ 1/√(ν_rr)w_r^a⊤A_r( ψ_μ + σ^μ) - w^* ⊤ψ_μ - √(M) (ϵ^μ - ξ_r^μ) ] are Gaussian random variables (when conditioned on A_r) with mean zero and covariance: ⟨ h_μ^ra h_ν^rb⟩ = δ_μν Q^rr_ab Q^rr_ab = 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_rr)A_r^⊤w_r^b- w^ ) . + 1/ν_rrw_r^a⊤A_rΣ_0 A_r^⊤w_r^b .+ M (ζ^2 + η_r^2) ] To perform this integral we re-write in terms of {H_μ^r }_μ = 1^P, where H_μ^r = [ h_μ^r1; h_μ^r2; ⋮; h_μ^rn ]∈ℝ^n ⟨Z^n ⟩_𝒟 = ∫∏_a dw^a_r 𝔼_{ψ_μ, σ^μ, ϵ^μ}exp( -β/2λ∑_μH_μ^r⊤H_μ^r - β/2∑_a \|w_r^a\|^2 - JMβ/2∑_a E_rr(w^a) ) Integrating over the H_μ^r we get: ⟨ Z^n ⟩_𝒟 =∫∏_a dw^a_r exp( -P/2log(I_n + β/λQ^rr) - β/2∑_a \|w_r^a\|^2 - JMβ/2∑_aE_rr(w_r) ) Next we integrate over Q^r and add constraints. We use the following identity: 1 = ∏_a b'∫ dQ^rr_ab δ( Q_ab^rr- 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_rr)A_r^⊤w_r^b- w^ ) . . . + 1/ν_rrw_r^a⊤A_rΣ_0 A_r^⊤w_r^b .+ M (ζ^2+ η_r^2) ] ) Using the Fourier representation of the delta function, we get: 1 = ∏_a b∫1/4π i / MdQ_ab^rr d Q̂_ab^rr exp(M/2Q̂_ab^rr( Q_ab^rr - 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_rr)A_r^⊤w_r^b- w^ ) . . . . . + 1/ν_rrw_r^a⊤A_rΣ_0 A_r^⊤w_r^b .+ M (ζ^2+ η_r^2) ] ) ) Inserting this identity into the replicated partition function and substituting E_rr(w^a_r) = Q^rr_aa - ζ^2 we find: ⟨ Z^n ⟩_𝒟∝ ∫∏_a b dQ_ab^rr dQ̂_ab^rrexp( -P/2log(I_n + β/λQ^rr) + 1/2∑_ab M Q̂_ab^rr Q_ab^rr - JMβ/2∑_a(Q^rr_aa - ζ^2)) ∫∏_a dw^a_r exp( - β/2∑_a \|w_r^a\|^2 - 1/2∑_abQ̂_ab^rr[ ( 1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_rr)A_r^⊤w_r^b- w^) . . . . + 1/ν_rrw_r^a⊤A_rΣ_0A_r^⊤w_r^b + M (ζ^2+ η_r^2) ] ) In order to perform the Gaussian integral over the {w_r^a }, we unfold over the replica index a. We first define the following: w^·_r ≡[ w_r^1; ⋮; w_r^n ] T^r ≡βI_n ⊗I_N_r + Q̂^rr⊗(1/ν_rrA_r(Σ_s + Σ_0)A_r^⊤) V^r ≡ (Q̂^rr⊗I_N_r) (1_n⊗1/√(ν_rr)A_r Σ_s w^) We then have for the integral over w ∫ dw^·_r exp(- 1/2w_r^·⊤ T^rw_r'^· + V^r⊤w_r^·) = exp( 1/2 V^r ⊤ (T^r)^-1 V^r - 1/2log (T^r) ) We can finally write the replicated partition function as: ⟨ Z^n ⟩_𝒟∝ ∫∏_a b dQ_ab^rr dQ̂_ab^rexp( -P/2log(I_n + β/λQ^rr) + 1/2∑_ab M Q̂_ab^rr Q_ab^rr - JMβ/2∑_a(Q^rr_aa - ζ^2)) exp( 1/2 V^r ⊤ (T^r)^-1 V^r - 1/2log (T^r) - 1/2∑_abQ̂_ab^rr (M (ζ^2+ η_r^2) + w^⊤Σ_s w^) ) We now make the following replica-symmetric ansatz: β Q_ab^rr = q δ_ab + q_0 β^-1Q̂_ab^rr = q̂δ_ab + q̂_0 which is well-motivated because the loss function is convex. We may then rewrite the partition function as follows: ⟨ Z^n ⟩_𝒟 = exp( -nM/2𝔤[q, q_0, q̂, q̂_0 ] ) Where the effective action is written: 𝔤[q, q_0, q̂, q̂_0 ] = α[ log (1 + q/λ) + q_0/λ + q] - (q q̂ + q q̂_0 + q_0 q̂) + J [ (q + q_0) - βζ^2 ] -β/ν_rr Mq̂^2 w^⊤Σ_s A_r^⊤G^-1A_r Σ_s w^* + 1/M[ log (G) + q̂_0 [G^-1Σ̃]] + βq̂( ζ^2+ η_r^2 + 1/Mw^⊤Σ_s w^ ) Where G≡I_N_r + q̂Σ̃ and Σ̃≡1/ν_rrA_r(Σ_s + Σ_0)A_r^⊤ We have E_rr = ∂/∂ Jlim_β→∞1/β𝔤 [q, q_0, q̂, q̂_0] Where g is evaluated at the values of q, q_0, q̂, q̂_0 which minimize g, in accordance with the method of steepest descent. E_rr = ∂_J (1/Mw^⋆⊤[ q̂Σ_s - 1/ν_rrq̂^2 Σ_s A_r^⊤G^-1A_r Σ_s ] w^* + q̂(ζ^2 + η_r^2) - Jζ^2 ) = [∂_J q̂] (1/Mw^⋆⊤[ Σ_s - 2/ν_rrq̂Σ_s A_r^⊤G^-1A_r Σ_s + 1/ν_rrq̂^2 Σ_s A_r^⊤G^-1Σ̃G^-1A_r Σ_s ] w^* + ζ^2 + η_r^2 ) - ζ^2 To complete the calculation, we need to find ∂_J q̂. We may do this by examining two of the saddle-point equations: ∂𝔤/∂ q_0 = 0 = α/λ + q - q̂ + J ⇒q̂ = α/λ + q + J ∂𝔤/∂q̂_0 = 0 = - q + 1/M[ G^-1Σ̃] ⇒ q = 1/M[ G^-1Σ̃] These two equations may in principle be solved for the dominant values of q and q̂. Letting κ = λ + q, we get: ∂_J q̂ = - α/κ^2∂_J q+1 ∂_J q = - 1/M∂_J q̂[ (G^-1Σ̃)^2] Solving this system of equations, we find ∂_J q̂ = 1/1-γ where γ≡α/M κ^2[ (G^-1Σ̃)^2] E_rr = 1/1-γ1/Mw^⋆⊤[ Σ_s - 2/ν_rrq̂Σ_s A_r^⊤G^-1A_r Σ_s + 1/ν_rrq̂^2 Σ_s A_r^⊤G^-1Σ̃G^-1A_r Σ_s ] w^* + γζ^2 + η_r^2/1-γ = 1/1-γ1/Mw^⋆⊤[ Σ_s - 1/ν_rrq̂Σ_s A_r^⊤G^-1A_r Σ_s - 1/ν_rrq̂Σ_s A_r^⊤G^-2A_r Σ_s ] w^* + γζ^2 + η_r^2/1-γ §.§ Off-Diagonal Terms We now calculate E_rr' for r ≠ r'. We now must consider the joint Gibbs Measure over w_r and w_r': Z = ∫ dw_r dw_r'exp( -β/2 λ (E^t_r + E^t_r') - JMβ/2 E_rr'(w_r, w_r') ) ⟨Z^n ⟩_𝒟 = ∫∏_a dw^a_r dw^a_r'𝔼_{ψ_μ, σ^μ, ϵ^μ} exp ( -β M/2λ∑_μ, a1/M[(h_μ^r a)^2 + (h_μ^r' a)^2 ] - β/2∑_a[\|w_r^a\|^2 + \|w_r'^a\|^2 ] - JMβ/2∑_a E_rr'(w_r^a, w_r'^a) ) Where the h_μ^ra are defined as before. Next we must perform the averages over quenched disorder. We first integrate over {ψ_μ, ϵ_μ}. To do so, we note that the h_μ^ra are Gaussian random variables with covariance structure: ⟨ h_μ^ra h_ν^r'b⟩ = δ_μν Q^rr'_ab Q^rr'_ab = 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_r'r')A_r'^⊤w_r'^b- w^ ) . + 1/√(ν_rrν_r'r')w_r^a⊤A_rΣ_0 A_r'^⊤w_r'^b .+ M ζ^2 ] To perform this integral we re-write in terms of {H_μ}_μ = 1^P, where H_μ = [ H_μ^r; H_μ^r'; ]∈ℝ^2n ⟨Z^n ⟩_𝒟 = ∫∏_a dw_r^a dw_r'^a 𝔼_{ψ_μ, σ^μ, ϵ^μ} exp( -β/2λ∑_μH_μ^⊤H_μ - β/2∑_a[\|w_r^a\|^2 + \|w_r'^a\|^2 ] - JMβ/2∑_a E_rr'(w_r^a, w_r'^a) ) Integrating over H_μ we get: ⟨ Z^n ⟩_𝒟 =∫∏_a dw_r^a dw_r'^a exp( -P/2log(I_2n + β/λQ) - β/2∑_a[\|w_r^a\|^2 + \|w_r'^a\|^2 ] - JMβ/2∑_aE_rr(w_r^a, w_r'^a) ) Where we have defined the matrix Q so that: Q = [ Q^rr Q^rr'; Q^rr' Q^r' r' ] Next we integrate over Q and add constraints. We use the following identity: 1 = ∏_a b∫ dQ^rr'_ab δ( Q_ab^rr'- 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_r'r')A_r'^⊤w_r'^b- w^ ) . . . + 1/√(ν_rrν_r'r')w_r^a⊤A_rΣ_0 A_r'^⊤w_r'^b .+ M ζ^2 ] ) Using the Fourier representation of the delta function, we get: 1 = ∏_a b∫1/4π i / MdQ_ab^rr' d Q̂_ab^rr' exp(M/2Q̂_ab^rr'( Q_ab^rr'- 1/M[ (1/√(ν_rr)w_r^a ⊤A_r - w^⊤) Σ_s(1/√(ν_r'r')A_r'^⊤w_r'^b- w^ ) . . . . . + 1/ν_rrw_r^a⊤A_rΣ_0 A_r'^⊤w_r'^b .+ M ζ^2 ] ) ) Inserting this identity and the corresponding statements for Q_ab^rr and Q_ab^r'r' into the replicated partition function and substituting E_rr'(w^a) = Q^rr'_aa - ζ^2 we find: ⟨ Z^n ⟩_𝒟∝∫∏_a b r_1 r_2 dQ_ab^r_1 r_2 dQ̂_ab^r_1 r_2 exp( -P/2log(I_2n + β/λQ) + 1/2∑_ab r_1 r_2 M Q̂_ab^r_1 r_2 Q_ab^r_1 r_2 - JMβ/2∑_a(Q^r r'_aa - ζ^2)) ∫∏_a dw_r^a dw_r'^a exp( - β/2∑_a[ \|w_r^a\|^2 + \|w_r'^a\|^2 ] - 1/2∑_ab r_1 r_2Q̂_ab^r_1 r_2[ ( 1/√(ν_r_1)w_r_1^a ⊤A_r_1 - w^⊤) Σ_s(1/√(ν_r_2)A_r_2^⊤w_r_2^b- w^) . . . . + 1/√(ν_r_1ν_r_2)w_r_1^a⊤A_r_1Σ_0A_r_2^⊤w_r_2^b + M ζ^2 ] ) Where sums over r_1 and r_2 run over {r, r'}. In order to perform the Gaussian integral over the {w_r^a }, we unfold in two steps. We first define the following: w^·_r ≡[ w_r^1; ⋮; w_r^n ] [Q̂^rr']_ab ≡Q̂_ab^rr' Σ̃_rr' ≡1/√(ν_rrν_r'r')A_r [Σ_s + Σ_0]A_r'^⊤ T^rr' ≡βδ_rr'I_n ⊗I_N_r + Q̂^rr'⊗Σ̃_rr' Unfolding over the replica indices, we then get: ⟨ Z^n ⟩_𝒟∝∫∏_a b r_1 r_2 dQ_ab^r_1 r_2 dQ̂_ab^r_1 r_2 exp( -P/2log(I_2n + β/λQ) + 1/2∑_ab r_1 r_2 M Q̂_ab^r_1 r_2 Q_ab^r_1 r_2 - JMβ/2∑_a(Q^r r'_aa - ζ^2)) exp(-1/2∑_ab r_1 r_2Q̂_ab^r_1 r_2(w^⊤Σ_s w^ + M ζ^2 ) ) ∫ dw_r^· dw_r'^·exp( - 1/2∑_r_1 r_2w_r_1^·⊤ T^r_1 r_2w_r_2 + ∑_r_1 r_2[ (Q̂^r_1 r_2⊗I_N_r_1) (1_n⊗1/√(ν_r_1)A_r_1Σ_s w^) ]^⊤w_r_1) Note that the dimensionality of T^r_1 r_2 varies for different choices of r_1 and r_2. Next, we unfold over the two readouts: w ≡[ w_r^·; w_r'^· ] T ≡[ T^rr T^rr'; T^r'r T^r'r' ] V ≡[ ((Q̂^rr+ Q̂^rr') ⊗I_N_r) ( 1_n ⊗1/√(ν_rr)A_r Σ_s w^ ); ((Q̂^r'r'+ Q̂^r'r) ⊗I_N_r') ( 1_n ⊗1/√(ν_r'r')A_r'Σ_s w^* ) ] The integral over w then becomes: ∫ dwexp(- 1/2w^⊤ T w + V^⊤w) ∝exp( 1/2 V^⊤ T^-1 V - 1/2log T ) We are now ready to make a replica-symmetric ansatz. The order parameter that we wish to constrain is Q_ab^rr'. Overlaps go between the weights from different replicas of the system as well as different readouts. The scale of the overlap between two measurements depends on their overlap with each other and with the principal components of the data distribution. An ansatz which is replica-symmetric but makes no assumptions about the overlaps between different measurements is as follows: β Q_ab^r_1 r_2 = Q^r_1 r_2δ_ab + q^r_1 r_2 β^-1Q̂_ab^r_1 r_2 = Q̂^r_1 r_2δ_ab + q̂^r_1 r_2 Next step is to plug the RS ansatz into the free energy and simplify. To make calculations more transparent, we re-label the paramters in the RS ansatz as follows: βQ^rr = q I + Q 11^⊤ βQ^r'r' = r I + R 11^⊤ βQ^rr' = c I + C 11^⊤ β^-1Q̂^rr = q̂I + Q̂11^⊤ β^-1Q̂^r'r' = r̂I + R̂11^⊤ β^-1Q̂^rr' = ĉI + Ĉ11^⊤ In order to simplify log(λI_2n + βQ), we note that this is a symmetric 2-by-2-block matrix, where each block commutes with all other blocks. We may then use <cit.>'s result to simplify. log(λI_2n + βQ) = n [ log( (λ + q)(λ + r) - c^2 ) + (λ + q)R + (λ + r)Q - 2 c C/(λ + q)(λ + r) - c^2] + 𝒪(n^2) ∑_a b r_1 r_2Q̂_a b^r_1 r_2Q_a b^r_1 r_2 = n [ ( q q̂ + q̂Q + q Q̂) + ( r r̂ + r̂R + r R̂) + 2 ( c ĉ + ĉC + c Ĉ) ] + 𝒪(n^2) ∑_a (Q_aa^r r' - ζ^2 ) = n [ 1/β (c + C) - ζ^2 ] + 𝒪(n^2) ∑_abr_1 r_2Q̂_ab^r_1 r_2 = β n [ q̂+r̂+2ĉ] log (T) = n [ log (β) + log[ G_rr G_rr'; G_r' r G_r'r' ]. . + q̂ (D_rrΣ̃_rr + ĉ (D_rr'Σ̃_r'r) + r̂ (D_r'r'Σ̃_r'r') + ĉ (D_r'rΣ̃_rr') ] + 𝒪(n^2) where G_rr = I_N_r + q̂Σ̃_rr G_r'r' = I_N_r' + r̂Σ̃_r'r' G_rr' = ĉΣ̃_rr' G_r'r = ĉΣ̃_r'r and where the D_r_1 r_2 matrices are defined implicitly through the following equation: [ G_rr G_rr'; G_r'r G_r'r'; ]^-1 = [ D_rr D_rr'; D_r'r D_r'r'; ] The D_r_1 r_2 matrices can be expressed in terms of the G_r_1 r_2 and their inverses by the standard 2 × 2 block matrix inversion formula (see, for example, <cit.>). Applying this formula gives the following: D_rr = [I + q̂Σ̃_rr - ĉ^2 Σ̃_rr'( I + r̂Σ̃_r'r')^-1Σ̃_r'r]^-1 D_r'r' = [I + r̂Σ̃_r'r' - ĉ^2 Σ̃_r'r( I + q̂Σ̃_rr)^-1Σ̃_rr']^-1 D_rr' = - ĉD_rrΣ̃_rr'G_r'r'^-1 D_r'r = - ĉD_r'r'Σ̃_r'rG_rr^-1 V^⊤ T^-1 V= n βw^⊤[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]^⊤[ G_rr G_rr'; G_r' r G_r'r' ]^-1[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]w^ + 𝒪(n^2) Collecting these terms, we may write the replicated partition function as follows: ⟨ Z^n ⟩_𝒟 = exp( -nM/2𝔤[q, Q, r, R, c, C, q̂, Q̂, r̂, R̂, ĉ, Ĉ] ) Where the effective action is written: 𝔤[q, Q, r, R, c, C, q̂, Q̂, r̂, R̂, ĉ, Ĉ] = α[ log( (λ + q)(λ + r) - c^2 ) + (λ + q)R + (λ + r)Q - 2 G g/(λ + q)(λ + r) - c^2] - [ ( q q̂ + q̂Q + q Q̂) + ( r r̂ + r̂R + r R̂) + 2 ( c ĉ + ĉC + c Ĉ) ] +J(c+C)- β J ζ^2 + β[ q̂ + r̂ + 2 ĉ] ( 1/Mw^* ⊤Σw^* + ζ^2 ) -βw^⊤[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]^⊤G^-1[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]w^ + 1/M[ log (β) + log(G) + Q̂ (D_rrΣ̃_rr) + Ĉ (D_rr'Σ̃_r'r) + R̂ (D_r'r'Σ̃_r'r') + Ĉ (D_r'rΣ̃_rr') ] where we have defined G≡[ G_rr G_rr'; G_r' r G_r'r' ] We have: E_rr' = ∂/∂ Jlim_β→∞1/β𝔤[q, Q, r, R, c, C, q̂, Q̂, r̂, R̂, ĉ, Ĉ] Where g is evaluated at the values of Q, q, R, r, C, c, Q̂, q̂, R̂, r̂, Ĉ, ĉ which minimize 𝔤, in accordance with the method of steepest descent (and thus implicitly depend on J). This gives: E_rr' = - ζ^2 + [ ∂q̂/∂ J + ∂r̂/∂ J + 2 ∂ĉ/∂ J] ( 1/Mw^* ⊤Σ_s w^* + ζ^2 ) - 2/Mw^⊤[ 1/√(ν_rr)(∂q̂/∂ J + ∂ĉ/∂ J) A_r Σ_s; 1/√(ν_r'r')(∂r̂/∂ J + ∂ĉ/∂ J) A_r'Σ_s ]^⊤G^-1[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]w^ + 1/Mw^⊤[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]^⊤G^-1∂G/∂ JG^-1[ 1/√(ν_rr)(q̂ + ĉ) A_r Σ_s; 1/√(ν_r'r')(r̂ + ĉ) A_r'Σ_s ]w^ ∂G/∂ J = ∂G/∂q̂∂q̂/∂ J + ∂G/∂r̂∂r̂/∂ J + ∂G/∂ĉ∂ĉ/∂ J = [ ∂q̂/∂ JΣ̃_rr ∂ĉ/∂ JΣ̃_rr'; ∂ĉ/∂ JΣ̃_r'r ∂r̂/∂ JΣ̃_r'r' ] All that remains is to calculate the saddle-point values of q̂, r̂, ĉ and their derivatives with respect to J at J = 0. ∂𝔤/∂ Q = 0 = α (λ + r)/(λ + q)(λ + r) - C^2 - q̂ ∂𝔤/∂Q̂ = 0 = -q + 1/M( D_rrΣ̃_rr) ∂𝔤/∂ R = 0 = α (λ + q)/(λ + q)(λ + r) - C^2 - r̂ ∂𝔤/∂R̂ = 0 = -r + 1/M( D_r'r'Σ̃_r'r') ∂𝔤/∂ g = 0 = -2 α C/(λ + q)(λ + r) - C^2 -2 Ĉ + J ∂𝔤/∂ĉ = 0 = -2C + 1/M( D_rr'Σ̃_r'r + D_r'rΣ̃_rr') These 6 equations can in principle be solved for { q, r, c, q̂, r̂, ĉ} and their derivatives with respect to J. Note that when J = 0, the saddle point equations <ref>, <ref> are solved by setting c = ĉ = 0, and in this case the remaining saddle-point equations decouple over the readouts (as expected for independently trained ensemble members) giving: For readout r: 0 = α/(λ + q) - q̂ 0 = -q + 1/M( G_rr^-1Σ̃_rr) and for readout r': 0 = α/(λ + r) - r̂ 0 = -r + 1/M( G_r'r'^-1Σ̃_r'r') These are equivalent to the saddle-point equations for a single readout given in equation <ref>, <ref> as expected for independently trained readouts. It is physically sensible that c = 0 when J = 0, because at zero source, there is no term in the replicated system energy function which would distinguish the overlap between two readouts from the same replica from the overlap between two readouts in separate replicas (we expect that the total overlap between readouts is non-zero, as we may still have C>0). Differentiating the saddle point equations with respect to J will give a 6 by 6 system of equations for the derivatives of the order parameters. With foresight, we first calculate ∂_J D ∂_J D = ∂_J G^-1 = - G^-1 (∂_J G) G^-1 Evaluated at J = 0, we have the following: ∂_J D_rr = - ∂_J q̂G_rr^-1Σ̃_rrG_rr^-1 ∂_J D_r'r' = - ∂_J r̂G_r'r'^-1Σ̃_r'r'G_r'r'^-1 ∂_J D_rr' = - ∂_J ĉG_rr^-1Σ̃_rr'G_r'r'^-1 ∂_J D_r'r' = - ∂_J ĉG_r'r'^-1Σ̃_r'rG_rr^-1 Differentiating equations <ref>, <ref>, <ref>, <ref>, <ref>, <ref> and evaluating at J, c, ĉ = 0 we get: 0 = ∂_J q̂ + α/(λ+q)^2∂_J q 0 = ∂_J q + ∂_J q̂1/M[ (G_rr^-1Σ̃_rr)^2] 0 = ∂_J r̂ + α/(λ+r)^2∂_J r 0 = ∂_J r + ∂_J r̂1/M[ (G_r'r'^-1Σ̃_r'r')^2] 1/2 = ∂_J ĉ + α/(λ+q)(λ+r)∂_J c 0 = ∂_J c + ∂_J ĉ1/M[ G_rr^-1Σ̃_rr'G_r'r'^-1Σ̃_r'r] Solving these equations, we obtain: ∂_J q̂ = 0 ∂_J r̂ = 0 ∂_J ĉ = 1/2(1-γ) where γ ≡α/(λ+q)(λ+r)1/M[ G_rr^-1Σ̃_rr'G_r'r'^-1Σ̃_r'r] We may then simplify the expression for the generalization error as follows: E_rr' = γ/1-γζ^2 + 1/1-γ( 1/Mw^* ⊤Σ_s w^* ) - 1/M(1-γ)w^⊤Σ_s [1/ν_rrq̂A_r^⊤G_rr^-1A_r + 1/ν_r'r'r̂A_r'^⊤G_r'r'^-1A_r'] Σ_s w^ + 1/M(1-γ)q̂r̂1/√(ν_rrν_r'r')w^⊤Σ_s A_r^⊤G_rr^-1Σ̃_rr'G_r'r'^-1A_r'Σ_s w^ Re-labeling the order parameters: q̂→q̂_r, r̂→q̂_r', γ→γ_rr' and G_rr→G_r, we obtain the result given in the main text. § EQUICORRELATED DATA MODEL To gain an intuition for the joint effects of correlated data, subsampling, ensembling, feature noise, and readout noise, we simplify the formulas for the generalization error in the following special case: Σ_s = s [(1-c) I_M + c 1_M 1_M^⊤] Σ_0 = ωI_M Here s is a parameter which sets the overall scale of the data and c ∈ [0,1] tunes the correlation structure in the data and ω sets the scale of an isotropic feature noise. We consider an ensemble of k readouts, each of which sees a subset of the features. Due to the isotropic nature of the equicorrelated data model and the pairwise decomposition of the generalization error, we expect that the generalization error will depend on the partition of features among the readout neurons through only: * The number of features sampled by each readout: N_r ≡ν_rr M, for r = 1,…,k * The number of features jointly sampled by each pair of readouts n_rr'≡ν_rr' M for r, r' ∈{1,…,k} Here, we have introduced the subsampling fractions ν_rr = N_r/M and the overlap fractions ν_rr' = n_rr'/M We will average the generalization error over readout weights drawn randomly from the space perpendicular to 1_M, with an added spike along the direction of 1_M: w^* = √(1-ρ^2)ℙ_⊥w^_0 + ρ1_M w^_0 ∼𝒩(0, I_M) The projection matrix may be written ℙ_⊥ = I_M - 1/N1_M 1_M^⊤. The two components of the ground truth weights will yield independent contributions to the generalization error in the sense that ⟨ E_rr'⟩ = (1-ρ^2) E_rr' (ρ = 0) + ρ^2 E_rr' (ρ = 1) Calculating E_rr and E_rr' is an exercise in linear algebra which is straightforward but tedious. To assist with the tedious algebra, we wrote a Mathematica package which can handle multiplication, addition, and inversion of matrices of symbolic dimension of the specific form encountered in this problem. This form consists of block matrices, where the blocks may be written as a δ_MNI_M + b 1_M 1_N^⊤, where a, b are scalars and δ_MN ensures that there is only a diagonal component for square blocks (when M = N). This package is included as supplemental material to this publication. §.§ Diagonal Terms and Saddle-Point Equations Here, we solve for the dominant values of q_r and q̂_r and simplify the expressions for E_rr in the case of equicorrelated features described above. In this isotropic setting, E_rr, q_r, q̂_r will depend on the subsampling only through N_r = ν_rr M. We may then write, without loss of generality A_r = [ I_N_r 0 ]∈ℝ^N_r × M where 0 denotes a matrix of all zeros, of the appropriate dimensionality. We start by simplifying the saddle-point equations <ref>,<ref>. Expanding 1/M(G_r^-1Σ̃_rr) and keeping only leading order terms, the saddle-point equations for q_r and q̂_r reduce to: q_r = ν_rr(s (1-c) + ω) /q̂_r ( s (1-c) + ω )+ν _r q̂_r = α/λ+q_r Defining a ≡ s(1-c) + ω and solving this system of equations, we find: q_r = √(a^2 α ^2+2 a α (λ -a) ν _r+(a+λ )^2 ν _r^2)-a α +(a-λ ) ν _r/2 ν _r q̂_r =√(a^2 α ^2+2 a α (λ -a) ν _r+(a+λ )^2 ν _r^2)+a α -(a+λ ) ν _r/2 a λ We have selected the solution with q_r>0 because self-overlaps must be at least as large as overlaps between different replicas. This solution to the saddle-point equatios can be applied to each of the k readouts. Next, we calculate E_rr. Expanding γ_rr≡α/M κ^2[ (G^-1Σ̃)^2] to leading order in M, we find: γ_rr = a^2 αν _r/(λ +q_r)^2 (a q̂_r+ν _r)^2 ⟨ E_rr⟩_𝒟, w^* (ρ = 0) = 1/1-γ_rr1/M[ ℙ_⊥( Σ_s - 2/ν_rrq̂_r Σ_s A_r^⊤G_r^-1A_r Σ_s + 1/ν_rrq̂_r^2 Σ_s A_r^⊤G_r^-1Σ̃G_r^-1A_r Σ_s ) ℙ_⊥] + γ_rr/1-γ_rrζ^2 + η_r^2, ⟨ E_rr⟩_𝒟, w^* (ρ = 1) = 1/1-γ_rr1/M1_M^⊤[Σ_s - 2/ν_rrq̂_r Σ_s A_r^⊤G_r^-1A_r Σ_s + 1/ν_rrq̂_r^2 Σ_s A_r^⊤G_r^-1Σ̃G_r^-1A_r Σ_s ] 1_m + γ_rr/1-γ_rrζ^2 + η_r^2, With the aid of our custom Mathematica package, we calculate the traces and contractions in these expressions and expand them to leading order in M, finding: ⟨ E_rr⟩_𝒟, w^* (ρ = 0) = 1/1-γ_rr(s (1-c) ( 1 -(1-c) s q̂_r ν _r (q̂_r (s(1-c) + ω )+2 ν _r)/(q̂_r (s (1-c)+ω ) + ν _r)^2) )+ γ_rrζ^2 + η_r^2/1-γ_rr ⟨ E_rr⟩_𝒟, w^* (ρ = 1) = 1/1-γ_rr(s(1-c)(1-ν_rr)+ω/ν_rr) + γ_rrζ^2 + η_r^2/1-γ_rr In the “ridgeless” limit where λ→ 0, we obtain: γ_rr = 4 αν_rr/(α + ν_rr + \|α - ν_rr\|)^2 ⟨ E_rr (ρ = 0) ⟩_𝒟, w^* ={[ s (1-c) ν_rr/ν_rr-α( 1 + s α (1-c) (α-2 ν_rr)/ν_rr[s(1-c) + ω]) + αζ^2 + ν_rrη_r^2/ν_rr-α , if α < ν_rr; s (1-c) α/α-ν_rr( 1 - s(1-c)ν_rr/s(1-c) + ω) + ν_rrζ^2+ αη_r^2/α-ν_rr , if α > ν_rr ]} (λ→ 0) ⟨ E_rr (ρ = 1) ⟩_𝒟, w^* ={[ ν_rr/ν_rr-α(s(1-c)(1-ν_rr)+ω/ν_rr) + αζ^2 + ν_rrη_r^2/ν_rr-α, if α < ν_rr; α/α-ν_rr(s(1-c)(1-ν_rr)+ω/ν_rr) + ν_rrζ^2+ αη_r^2/α-ν_rr, if α > ν_rr ]} (λ→ 0) §.§ Off-Diagonal Terms In this section, we calculate the off-diagonal error terms E_rr' for r ≠ r', again making use of our custom Mathematica package to simplify contractions of block matrices of the prescribed form. By the isotropic nature of the equicorrelated data model, E_rr' can only depend on the subsampling scheme through ν_rr, ν_r'r', and ν_rr'. We can thus, without loss of generality, write: A_r = [ I_n_r × n_r 0_n_r × n_r' 0_n_r × n_s 0_n_r × l; 0_n_s × n_r 0_n_s × n_r' I_n_s × n_s 0_n_s × l ]∈ℝ^N_r × M A_r' = [ 0_n_r'× n_r I_n_r'× n_r' 0_n_r'× n_s 0_n_r'× l; 0_n_s × n_r 0_n_s × n_r' I_n_s × n_s 0_n_s × l ]∈ℝ^N_r'× M where we have defined n_s to be the number of features shared between the readouts, n_r = N_r - n_s and n_r' = N_r' - n_s and the count of remaining features l = M - n_r - n_r' - n_s. Then, to leading order in M, we find: γ_rr' = αν_rr' (s(1-c) + ω)^2/(λ+q_r)(λ+q_r')(ν_rr + (s(1-c)+ω)q̂_r)(ν_r'r' + (s(1-c)+ω)q̂_r') Averaging E_rr' over w_0^* ∼𝒩(0, I_M), we get: ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 0) = γ_rr'/1-γ_rr'ζ^2 + 1/1-γ_rr'( 1/M[ ℙ_⊥Σ_s ℙ_⊥] ) - 1/M(1-γ_rr')[ ℙ_⊥Σ_s (1/ν_rrq̂_r A_r^⊤G_r^-1A_r + 1/ν_r'r'q̂_r'A_r'^⊤G_r'^-1A_r') Σ_s ℙ_⊥] + q̂_r q̂_r'/M(1-γ_rr')1/√(ν_rrν_r'r')[ ℙ_⊥Σ_s A_r^⊤G_r^-1Σ̃_rr'G_r'^-1A_r'Σ_s ℙ_⊥], ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 1) = γ_rr'/1-γ_rr'ζ^2 + 1/M(1-γ_rr')(1_M^⊤Σ_s 1_M ) - 1/M(1-γ_rr')1_M^⊤Σ_s (1/ν_rrq̂_r A_r^⊤G_r^-1A_r + 1/ν_r'r'q̂_r'A_r'^⊤G_r'^-1A_r') Σ_s 1_M^⊤ + q̂_r q̂_r'/M(1-γ_rr')1/√(ν_rrν_r'r')1_M^⊤Σ_s A_r^⊤G_r^-1Σ̃_rr'G_r'^-1A_r'Σ_s 1_M Calculating these contractions and traces and expanding to leading order in M, we get: ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 0) = s (1-c)/1-γ_rr' ( 1 - s(1-c) ν_rrq̂_r/ν_rr + (s(1-c)+ω)q̂_r - s(1-c) ν_r'r'q̂_r'/ν_r'r' + (s(1-c)+ω)q̂_r'. . + s(1-c)(s(1-c)+ω) ν_rr'q̂_r q̂_r'/(ν_rr + (s(1-c)+ω)q̂_r)(ν_r'r' + (s(1-c)+ω)q̂_r')) + γ_rr'/1-γ_rr'ζ^2 ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 1) = 1/1-γ_rr' ( s(1-c)(ν_rr'-ν_rrν_r'r') + ων_rr'/ν_rrν_r'r') + γ_rr'/1-γ_rr'ζ^2 Taking λ→ 0 we get the ridgeless limit: γ_rr'→4 αν_rr'/(α + ν_rr + \|α - ν_rr\|)(α + ν_r'r' + \|α - ν_r'r'\|) (λ→ 0) ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 0) = 1/1-γ_rr' F_0(α) + γ_rr'/1-γ_rr'ζ^2 (r ≠ r') where F_0(α) ≡{[ (c-1) s (ν _r ν _r' ((2 α -1) (c-1) s+ω )-α ^2 (c-1) s ν _r r')/ν _r ((c-1) s-ω ) ν _r', if α≤ν_rr≤ν_r'r'; (c-1) s (ν _r'((c-1) s ν _r+(α -1) (c-1) s+ω)-α (c-1) s ν _r r')/((c-1) s-ω ) ν _r', if ν_rr≤α≤ν_r'r'; (c-1) s ((c-1) s ν _r'-c s ν _r r'+(c-1) s ν _r-c s+s ν _r r'+s+ω)/(c-1) s-ω, if ν_rr≤ν_r'r'≤α ]} ⟨ E_rr'(𝒟) ⟩_𝒟, w^(ρ = 1) = 1/1-γ_rr'( s(1-c)(ν_rr'-ν_rrν_r'r') + ων_rr'/ν_rrν_r'r') + γ_rr'/1-γ_rr'ζ^2 (λ→ 0) §.§ Phase Transitions in Uniform Resource-Constrained Ensembling. We make further simplifications in the special case where ω = ζ = 0, η_r = η, ν_rr = 1/k for all r = 1,… k, and ν_rr'=0 for all r ≠ r'. in the ridgeless limit λ→ 0. E_k = 1/kE_rr(ν_rr = 1/k) + k-1/k E_rr'(ν_rr = ν_r'r' = 1/k, ν_rr' = 0) Or, in full: E_k = {[ -α (c-1) k^2 s (2 α(ρ ^2-1)-2 ρ ^2+1)-(c-1) k s (α ^2 (ρ ^2-1)-α -ρ ^2+1)+η ^2/k (α k-1), if α≤1/k; (c-1) (k-1) s (α k^2 (ρ ^2-1)+k (-αρ ^2+α -2 ρ ^2+1)+2 (ρ ^2-1))+αη ^2 k^2/k^2 (α k-1), if α≥1/k ]} (r ≠ r') To find the boundary between the signal-dominated and noise-dominated regions, we set E_∞ = E_1, and rearrange to get: H = {[ α(1-α)(1-ρ^2), if 0< α < 1; (α-1)(1-ρ^2)/α, if α > 1 ]} To find the boundary between the intermediate and noisy phases, we set: E_k+1 = E_k and take the limit k →∞, then rearrange to get: H = 2 - (1+2α/α)ρ^2 §.§ Infinite Data Limit In this section we consider the behavior of generalization error in the equicorrelated data model as α→∞ while keeping the λ∼𝒪(1). For simplicity, we assume ν_rr' = 0 for r ≠ r', isotropic features (c=0), no feature noise (ω = 0) and uniform readout noise η_r = η as in main text figure 3. This limit corresponds to data-rich learning, where the number of training examples is large relative to the number of model parameters. In this case, the saddle point equations reduce to: q̂_r →α/λ q_r →ν_rrλ/α In this limit, we find that γ_rr'→ 0. Using this, we can simplify the generalization error as follows: E_g = 1/k^2∑_rr' = 1^k E_rr' = s [ 1- (2-1/k) (1/k∑_r=1^k ν_rr) ] + η^2/k Interestingly, we find that the readout error in this case depends on the subsampling fractions ν_rr only through their mean. Therefore, with infinite data, there will be no distinction between homogeneous and heterogeneous subsampling. § NUMERICAL EXPERIMENTS Numerical experiments are performed by generating synthetic datasets by drawing data randomly from multivariate Gaussian distributions, assigning feature noise and noisy labels. Writing the training set in terms of a data matrix Ψ∈ℝ^M × P in which column μ consist of the training point ψ_μ and the labels are organized into a column vector y such that y_μ = y_μ, the learned weights are calculated as: ŵ = Ψ( Ψ^⊤Ψ + λI_p )^-1y In the ridgeless case, a pseudoinverse is used: ŵ = Ψ^†y Numerical experiments were performed using the PyTorch library <cit.>. The code used to perform numerical experiments and generate plots is provided in a zip file with this submission, and will be made publicly available on GitHub upon acceptance of this manuscript. Numerical computations necessary for this work may be performed in a small amount of time (less than one hour) using an Nvidia GPU. §.§ Details of Heterogeneous Subsampling Theory In this section, we describe the method used to calculate loss curves for heterogeneous subsampling experiments seen in main text fig. 3. In each trial, Subsampling fractions {ν_1, …, ν_k} are generated according to the following process: * Each fraction ν_rr is drawn independently from a Γ distribution with mean 1/k and variance σ^2. ν_rr∼Γ_k, σ * The fractions are re-scaled in order to sum to unity: ν_rr→ν_rr/(ν_1 + ⋯ + ν_k) Equivaltly, the fractions ν_rr are drawn from a Dirichlet distribution <cit.>. Then, the loss curves are calculated from the given fractions {ν_rr} using equations <ref>, <ref>, <ref>, <ref>. The dotted red lines show the loss curves for 5 single trials. The solid red lines show the average loss curves from 100 trials. Note that we have defined our own convention for the parameterization of the Γ distribution in which the inverse of the mean and the standard deviation are specified. In terms of the standard “shape” and “scale” parameters, we have: Γ_k, σ≡Γ( shape = (k σ)^-2, scale = k σ^2 ) § CODE AVAILABILITY All Code used in this paper has been made available online (see https://github.com/benruben87/Learning-Curves-for-Heterogeneous-Feature-Subsampled-Ridge-Ensembles.git). This includes code used to perform numerical experiments, calculate theoretical learning curves, and produce plots as well as the custom Mathematica libraries used to simplify the generalization error in the special case of equicorrelated data.
http://arxiv.org/abs/2307.02480v1	20230705175436	A Dataset of Inertial Measurement Units for Handwritten English Alphabets	[ "Hari Prabhat Gupta", "Rahul Mishra" ]	cs.CV	[ "cs.CV" ]	A Dataset of Inertial Measurement Units for Handwritten English Alphabets Hari Prabhat Gupta, Senior Member IEEE, Rahul Mishra, Member IEEE Department of Computer Science and Engineering, IIT (BHU) Varanasi Instructors: Hari Prabhat Gupta and Tanima Dutta, Senior Member, IEEE Report writing: Rahul Mishra, Member, IEEE, and Garvit Banga TA: Shubham Pandey, Krishna Sharma, and Himanshu Sahu Volunteers <cit.> August 1, 2023 ========================================================================================================================================================================================================================================================================================================================================================= This paper presents an end-to-end methodology for collecting datasets to recognize handwritten English alphabets by utilizing Inertial Measurement Units (IMUs) and leveraging the diversity present in the Indian writing style. The IMUs are utilized to capture the dynamic movement patterns associated with handwriting, enabling more accurate recognition of alphabets. The Indian context introduces various challenges due to the heterogeneity in writing styles across different regions and languages. By leveraging this diversity, the collected dataset and the collection system aim to achieve higher recognition accuracy. Some preliminary experimental results demonstrate the effectiveness of the dataset in accurately recognizing handwritten English alphabets in the Indian context. This research can be extended and contributes to the field of pattern recognition and offers valuable insights for developing improved systems for handwriting recognition, particularly in diverse linguistic and cultural contexts. Diversity, Dataset, Inertial Measurement Units, Handwritten Alphabets, Sensors. § INTRODUCTION Recognizing English alphabets using Inertial Measurement Units is a promising field of research that opens new possibilities for the development of sign language recognition systems, virtual reality, and gesture-based interfaces <cit.>. It involves the use of IMU sensors to recognize and identify alphabets based on the movements of a person's hand. IMUs are small electronic devices that can detect and measure changes in motion, orientation, and direction using a tri-axial accelerometer, gyroscope, and magnetometer, respectively <cit.>. Due to the low cost, low energy consumption, and compact size of the IMU sensors, they are widely used in Internet of Things (IoT) applications. Thus, creating a state-of-the-art in this field of sensors-based recognition <cit.>. For example, using the data sign language recognition systems can benefit significantly from this IMU-based collected data, as it can improve the accuracy and speed of recognition, making it easier for people with hearing impairments to communicate. Similarly, in virtual reality and gesture-based interfaces <cit.>, IMUs’ data can provide a more immersive and interactive experience, allowing users to control and manipulate objects in a more natural and intuitive way <cit.>. While English alphabet recognition using IMUs has shown a lot of promise, there are still challenges and limitations that need to be addressed to make this technology more effective and accessible. For instance, the accuracy of recognition can be affected by factors such as lighting conditions, the size of the dataset, and the complexity of the gestures <cit.>. Moreover, the cost of the IMU hardware can be a barrier to adoption, particularly in low-income regions. This dataset collection aims to provide vast and heterogeneous data of English alphabets (both upper and lower cases) from more than 100 users. It will provide a deeper insight into the principles and techniques behind English Alphabets Recognition using IMUs. It will provide an overview of the current state-of-the-art in this field, including the various approaches and algorithms that are used to recognize alphabets based on motion data. Additionally, this description of the dataset will discuss the potential applications of this technology in various domains and industries. Our dataset is available at IEEE Dataport <cit.> and Watch the data collections treasure on youtube <cit.>. §.§ Aim of dataset collection The principle aim of this dataset collection is summarized in the following points: * To develop a large alphabets dataset while considering a large number of heterogeneity parameters, including participants' height, gender, days, and timing of data collection. * To facilitate an abundant sensory dataset for recent trending technologies such as Federated Learning for performing real-world evaluation alongside the simulation on machine-generated datasets or homogeneous datasets. * To strengthen the development of more robust and versatile datasets for validating the newly generated security and performance policies in the smart sensing environment. §.§ Target applications Natural language processing (NLP): This alphabet recognition dataset can be used as a starting point for building NLP models, which can analyze and understand human language. NLP models are used in applications such as chatbots, language translation, and sentiment analysis. Fraud detection: This dataset can be used to detect fraudulent documents or transactions, such as forged signatures or altered checks. Handwriting analysis: This dataset can be used to analyze and identify handwriting styles, which can be useful in forensic investigations or in analyzing historical documents. Education: The character recognition dataset is used in educational applications to help students learn to read and write, and to develop language skills. For example, handwriting recognition software can be used to help students practice writing letters and words. Accessibility: The character recognition dataset is used to develop assistive technologies for people with disabilities. For example, OCR technology can be used to convert printed text into speech for people who are blind or have low vision. Paper Organization: Section <ref> provides an overview of the data collection devices. Section <ref> outlines the detailed information about the setup for data collection. Section <ref> and Section <ref> cover volunteers' details and data description, respectively. Finally, Section <ref> concludes this work. § DATA COLLECTION DEVICES In order to optimize the process of collecting sensory data from the marker pen, it is crucial to address several key questions before developing the necessary data collection devices. These questions are essential for ensuring both the effectiveness and efficiency of the data collection process. By thoroughly considering these aspects, we can create an optimal marker pen with sensing capabilities, which meets the desired objectives. The key question while making data collection devices are as follows: Q1: Where can we place the sensors on the marker pen? Specifically, determining the optimal location for sensor integration on the marker pen, including the top, middle and bottom. Q2: What is the impact of placing sensors in different locations on the recognition performance? Q3: Weather sensors' placement location change with volunteers' height, age, gender, or placement of whiteboards? Q4: Is there any relation between the inclination of the whiteboard and volunteers' ease of writing in correspondence with the sensors' placement on the marker pen? Q5: Which is the most suitable sampling rate for collecting data from the marker pen? Q6: What factors are to be considered while choosing the microcontroller unit for transmitting and managing the data collected from the sensors? Q7: What are the challenges that may be encountered while using battery power on the marker pen? What are the measures taken into consideration to resolve the battery issues? Q8: What are the possibilities of providing a power supply via a wired connection? Which types of wire the power supply is good at? Q9: Which mode of data transmission is chosen: wired or wireless? Q10: Which wireless mode of transmission is the best fit in the context of data collection from the sensors? Both in terms of power and data rate. Q11: What is the frequency of recharging the battery? If the marker pen is operating on the battery power. Q12: What is the preferable (or suitable) height for placing the whiteboard? In order to circumvent the above challenges, we employed three distinct configurations of markers with IMU sensors so that the dataset would be more broadly applicable rather than sensor-specific. Two Arduino Nano 33 BLE senses equipped with LSM9DS1, a 9-axis inertial sensor were utilized. For the Arduino Nano BLE, we employed two different configurations, one near the marker's tip and the other at the upper end. The specification of LSM9DS1 is shown in Table 1. In order to create the third marker, we linked an Arduino Nano to the MPU9250 9-Axis Attitude Gyro Accelerator Magnetometer Sensor Module, as depicted in Fig. <ref>. Additionally, the distribution of volunteers using these devices is depicted in Fig. <ref> The details of the micro-controller and the sensors used while constructing the data collection device are as follows: §.§ Arduino Nano 33 BLE Sense It is a compact development board based on the popular Arduino platform <cit.>. It is designed specifically for projects requiring Bluetooth Low Energy (BLE) connectivity and a wide range of sensors. The board is an improved version of the Arduino Nano, offering additional features and capabilities. The key components of the Arduino Nano 33 BLE Sense are as follows: a). Microcontroller: The board is powered by a 32-bit ARM Cortex-M4 processor running at 64 MHz. It provides a good balance between performance and power efficiency. b). Bluetooth Connectivity: The Nano 33 BLE Sense has built-in Bluetooth 5.0, allowing it to communicate wirelessly with other BLE-enabled devices such as smartphones, tablets, and other Arduino boards. c). Sensors: One of the distinguishing features of the Nano 33 BLE Sense is its wide range of onboard sensors. These sensors include a 9-axis IMU (accelerometer, gyroscope, magnetometer), pressure sensor, humidity and temperature sensor, proximity sensor, and microphone. These sensors enable the board to gather data from its environment and make it suitable for various IoT and wearable projects. d). GPIO Pins: The board offers a total of 22 digital input/output (I/O) pins, among which 14 can be used as pulse width modulation (PWM) outputs. It also provides 6 analogue input pins. e). Memory and Storage: The Nano 33 BLE Sense has 256 KB of flash memory for storing program code and 32 KB of SRAM for variable storage. Additionally, it has an onboard QSPI Flash chip that provides 2 MB of additional storage space for data and files. f). USB Interface: The board can be powered and programmed via a micro USB connector, making it easy to connect to a computer for development purposes. g). Compatibility: The Arduino Nano 33 BLE Sense is fully compatible with the Arduino ecosystem. It can be programmed using the Arduino IDE, which supports a wide range of libraries and examples, simplifying the development process. §.§ Arduino Nano It is a compact and versatile microcontroller board that is part of the Arduino family of open-source hardware and software <cit.>. It is designed for small-scale projects that require a low-cost, low-power microcontroller with a small form factor. The key features and characteristics of the Arduino Nano: a). Microcontroller: Arduino Nano is based on the Atmel ATmega328P microcontroller. It operates at a clock speed of 16 MHz and has 32KB of flash memory for storing program code, 2KB of SRAM for data storage, and 1KB of EEPROM for non-volatile storage. b). Size and Form Factor: The board has a small form factor, measuring only 45mm x 18mm, making it suitable for space constraints or when a small size is desired. c). Connectivity: The Nano board comes with a USB interface, allowing it to be easily connected to a computer for programming and power supply. It also has 14 digital input/output pins, 8 analogue inputs, and 6 pulse-width modulation (PWM) outputs, providing flexibility for various applications. d). Power Options: The Nano can be powered through a USB connection or an external power source, such as a battery. It has a built-in voltage regulator that can handle input voltages ranging from 7V to 12V, making it compatible with a wide range of power.. e). Programming: Arduino Nano can be programmed using the Arduino Software (IDE), which is a user-friendly programming environment that allows you to write, compile, and upload code to the board. It supports the Arduino language, which is based on C/C++. f). Shields and Accessories: The Nano board is compatible with a variety of Arduino shields and accessories, which are add-on modules that extend its capabilities. These shields can provide additional features like WiFi, Bluetooth, motor control, and display interfaces. g). Open-Source: Like other Arduino boards, the Nano is open-source, which means that its design files, schematics, and source code are freely available. This enables the community to contribute improvements, and modifications, and create derivative designs. §.§ IMU MPU-9250 It is a compact and versatile integrated circuit that combines multiple sensors to provide accurate motion tracking and orientation estimation <cit.>. IMU stands for Inertial Measurement Unit, which refers to a device that measures and reports motion-related parameters such as acceleration, angular velocity, and magnetic field strength. The MPU-9250 is specifically designed for applications that require precise motion sensing, such as robotics, drones, virtual reality systems, and wearable devices. It is developed by InvenSense, a leading manufacturer of MEMS (Micro-Electro-Mechanical Systems) sensors. The MPU-9250 integrates several sensor components into a single chip, including a 3-axis accelerometer, a 3-axis gyroscope, and a 3-axis magnetometer. Each of these sensors provides important data about the device's motion and orientation in three dimensions. The accelerometer measures linear acceleration, allowing the detection of changes in velocity and the determination of the device's tilt or inclination. This information is crucial for applications that require movement tracking, stabilization, or gesture recognition. The gyroscope measures angular velocity, providing data about rotational movements and changes in orientation. The magnetometer measures the strength and direction of the magnetic field around the device. It can be used to determine the device's heading or to compensate for magnetic disturbances that may affect the accuracy of the accelerometer and gyroscope readings. In addition to the sensor components, the MPU-9250 includes an on-chip digital motion processor (DMP) that offloads complex motion processing tasks from the main microcontroller or processor. The DMP can perform sensor fusion algorithms to combine data from multiple sensors, providing a more accurate and stable estimation of the device's motion and orientation. The MPU-9250 communicates with an external microcontroller or processor through an I2C (Inter-Integrated Circuit) or SPI (Serial Peripheral Interface) interface, allowing for easy integration into various electronic systems. The specification of the indigenous constructed devices for the data collection is illustrated in Table <ref>. §.§ Experimental Evidence 1 We present some experimental evidence, which is gathered through controlled experiments or observations to test the collected dataset. Fig. <ref> provides a visual representation, depicting the impact of volunteers' height on the number of data instances and values of the gyroscope sensors. § DATA COLLECTION SETUP Upon addressing the challenges of data collection devices, the subsequent challenges lie in establishing an appropriate data collection setup. In particular, this pertains to determining the placement of the whiteboard within our data collection setup. Similar to the challenges encountered with the data collection devices, there are different challenges involved in creating a suitable data collection setup. Such challenges can be encapsulated in the form of the following set of questions, which play a crucial role in ensuring the effectiveness and efficiency of the data collection process. Thoroughly considering these aspects allows us to devise an optimal data collection setup that aligns with our desired objectives. The fundamental questions to consider when constructing the data collection setup include: Q1: Where can we place the whiteboard? Specifically, determining the optimal location for whiteboard placement, including on the table, hanging on the wall, and on the floor. Q2: What is the impact of placing whiteboard on different locations on the recognition performance? Q3: Weather whiteboard location changes with volunteers height, age, gender, or style of writing? Q4: Is there any relation between inclination of white board and volunteers ease of writing on the whiteboard? Q5: Which is the most suitable inclination angle while placing the white board on the table? Q6: What external factors (lightsouce, easier accessibility, air cooling vent or heating vent, isolated availability of the palce) are to be considered while choosing the location for placement of the whiteboards? Q7: What is the appropriate number of white boards to be used while data collcetion? Q8: Is there any relationship between number of whiteboard counts and number of volunteers ? Q9: Is the external noise hamper the performance of data collection? Is is required to place the board in noise isolated environment? Q10: How to handle the interaction between the marker pen and the whiteboard surface that can introduce variability in the data? Factors such as the smoothness or texture of the whiteboard, pressure applied by the user, and the angle of the pen can affect the data collection. Q11: How to monitor pressure applied by the different volunteers? Is the variation in pressure also impact the quality of the collected dataset from the sensor module? Q12: How to manage angle of inclination between pen and the whiteboard? How to set whiteboard on an appropriate angle to perform optimal data collection? Q13: Volunteers opinion are mandatory for effective data collection. How to provide mechanism of feedback from the volunteers to improve the data collection setup? Q14: How to manage angle of inclination between pen and the whiteboard? How to set whiteboard on an appropriate angle to perform optimal data collection? By addressing these challenges and considering the specific requirements of collecting sensory data using IMU sensors on a whiteboard, you can set up an effective data collection process to gather valuable insights for your research or application. For effectively and paralley performing the data collection inside our ubiquitous computing lab, we have used four white boards. Each student volunteers write alphabets using the developed data collection device mounted over the white board markers. We have draw some initial letters on the board so that the volunteer can easily practice out the character writing prior to the actual writing on the board. Fig.<ref> illustrates the data collection steup at the ubiquitous computing lab IIT (BHU) Varanasi, India. §.§ Experimental evidences 2 We present some experimental evidence, which are gathered through controlled experiments or observations to test the collected dataset. Fig. <ref> provide a visual representation, depicting the impact of male and femal volunteers on the value of accelerometer, gyroscope, and magnetometer sensors on selected characters (W, m, and R). From Fig. <ref>, we can make following important observations, where, gyro_y represents the y-axis of the gyroscope sensor. §.§.§ Height and Writing Angle When individuals write on a whiteboard, the angle at which they hold their hand and write can vary depending on their height. Taller individuals tend to have their hands positioned at a higher level compared to shorter individuals when writing on the whiteboard. This is because the height difference affects the natural positioning of their arm and hand while reaching the whiteboard surface. §.§.§ Gyro_y Sensor The gyro_y sensor is a type of sensor that measures the rotation or angular velocity around the y-axis. It can be used to detect changes in the angle or orientation of an object. Thus, we observe that the gyro_y sensor values decrease as height increases. This suggests that the sensor is capturing a lower angular velocity around the y-axis for taller individuals compared to shorter individuals while they are writing on the whiteboard. In other words, taller individuals tend to have a smaller rotational movement or a flatter angle while writing characters, resulting in lower values recorded by the gyro_y sensor. § VOLUNTEERS FOR DATA COLLECTION This section provides a detailed description of the student participants who volunteered to contribute to the collection of a large-size sensory dataset using an IMU sensor. We begin by discussing the various challenges encountered during the selection process of the volunteers. Following that, we present a concise overview of the attributes that may have an impact on the dataset collection, such as height and hand orientation. Furthermore, this section includes a table or pie chart depicting the distribution of participants' height, weight, and age. Additionally, we discuss the geographical diversification of student volunteers by incorporating an Indian map to illustrate their regional representation. The fundamental challenges encountered while selecting the volunteers for data collection are discussed as the following questions: Q1: How did you ensure consistent sensor placement across participants of varying heights? Q2: How did you address the potential impact of height differences on sensor alignment? Q3: How to ensure that the selected volunteers represent the target population or the group being studied is essential for obtaining accurate and reliable data? Q4: How to achieve a diverse and representative sample as it be can be challenging due to factors such as demographics, availability, and willingness to participate. Q5: How to handle biasness that can occur when the selection process favours certain types of individuals over others, leading to a skewed representation of the volunteers? Q6: How to tackle the sampling errors that can occur due to random variability or issues with the sampling method, resulting in a sample that does not accurately reflect the volunteers' characteristics? Q7: Importantingly how to find and recruit volunteers who are willing to participate in the data collection process? Q8: How to achieve a diverse and representative sample as it be can be challenging due to factors such as demographics, availability, and willingness to participate. Q9: How to provide a favorable and noise-free data collection environment to reduce the hesitation or reluctance of the volunteers during the data collection? Q10: How to handle ethical standards in volunteer selection? Q11: What are the steps required to obtain informed consent, protect participant privacy and confidentiality, and address any potential risks or harms associated with the study? Q12: How do handling constraints such as time, budget, and personnel can limit the number of volunteers that can be selected or the methods used for recruitment and data collection? During the data collection phase, we carefully selected 124 student volunteers to participate in the study. Their task was to write the English alphabet, both in capital and small letters, on a whiteboard. Each student was given specific instructions to write each character 50 times during various shifts over a period of two months. The student volunteers varied in terms of their heights and genders, which introduced diversity into the dataset, as shown in Fig. <ref>. Moreover, the student volunteers are from different locations in India (as in Fig. <ref>), making a high level of heterogeneity in terms of culture and behaviour. Additionally, each individual had their own unique writing pattern and speed, adding further variability to the collected data. By allowing the volunteers to write the characters over a two-month period, we aimed to minimize biases that could arise from monotonous repetition. The aforementioned characteristics of the dataset result in a high level of heterogeneity. This diversity makes the collected dataset more suitable for analyzing and training different models in order to simulate and address realistic scenarios. The presence of different writing styles, speeds, and durations in the dataset reflects the real-world variations that exist in individuals' writing behaviours. By incorporating this heterogeneity into the dataset, we can obtain a more comprehensive understanding of the challenges and variations that may be encountered in practical applications. This diverse dataset enables the development and testing of models that can handle a wider range of writing styles and conditions, ultimately leading to more robust and adaptable solutions in the field of character recognition or other related tasks. § DATASET DESCRIPTION The data collection process involved the participation of 124 students who were assigned the task of writing 52 letters of the English alphabet. This included 26 lowercase letters (a-z) and 26 uppercase letters (A-Z). The experiment was conducted over a period of two months, ensuring a substantial amount of data was gathered. Each student visited the laboratory six times a week to perform the writing task. During each session, they wrote two characters on the whiteboard. The students were instructed to write each character 50 times on the board. To capture the timing and duration of each writing instance, a button was provided to the students. They would press the button before starting to write a character and again immediately after completing the character. During the writing process, a sensor was employed to record data. When the student was actively writing a character, the sensor registered a value of 1, along with other sensory measurements. On the other hand, when the student was not writing a character, the sensor recorded a value of 0, along with the corresponding sensory data. This allowed for the differentiation between active writing periods and non-writing periods. Throughout the dataset collection, the sensors recorded data from all 50 instances of each character being written by the students. This comprehensive data collection approach enabled the capture of multiple repetitions of each character, providing a rich dataset for analysis and modelling purposes. By combining information on the sensory values, timing, and the distinction between writing and non-writing periods, the collected data offers valuable insights into the writing behaviour and patterns exhibited by the students. The sensory dataset is stored in csv files, following the directory hierarchy as shown in Fig.<ref>. Furthermore, Fig. <ref> illustrates the inner directory organization of the dataset. During the data collection process, a specific protocol was followed to organize and record the data for each participant. Each file in the dataset represents a particular character, and within that file, the participant wrote the same character 50 times. All 50 instances of writing that character by a participant were recorded in this file. To initiate the recording of each writing instance, the student pressed a button. This action signalled the start of writing and corresponded to a value of 1 in the dataset. Subsequently, each time the participant completed writing the character, they pressed the button again, resulting in a value of 0 being recorded in the dataset. Along with these binary values, readings from sensors such as accelerometers, gyroscopes, and magnetometers were also captured. The process was repeated for every student. For each participant, 50 different files were created, each representing the same character they wrote, but at different instances. This ensured that there was a separate file for every instance of writing the character by each student. By organizing the data in this manner, researchers were able to maintain a systematic record of each participant's writing instances. This allowed for further analysis and examination of patterns, variations, and trends within and across participants. Additionally, having individual files for each instance and participant facilitated the isolation and study of specific writing behaviours, helping to identify unique characteristics or challenges associated with writing a particular character. Overall, this approach ensured that the data collected was structured and categorized in a way that facilitated subsequent analysis and modelling tasks. The number of data instances against each character is graphically shown at the bottom of this document. §.§ Dataset preprocessing We shortlist all files ranging from 40 to 80 lines of sensory data per instance. We add padding to all the files at the top and bottom to make it 80 lines. Each instance of a character has 80 lines of sensory data. The dataset is partitioned into two sets, where 80% of the students were selected for generating the training data and 20% for the test data. In the case of sensor overflow values, we replace them with similar sensor axis data from the dataset. The preprocessing steps are discussed as follows: * Shortlisting Files: The files containing sensory data are examined, and those ranging from 40 to 80 lines are selected for further processing. This step ensures that only files within this specific range of lines are considered for analysis. * Padding for Standardization: To maintain consistency in the dataset, padding is applied to the selected files. Padding involves adding extra lines of sensory data at the top and bottom of each file to make it a standardized length of 80 lines. By doing so, all instances of characters in the dataset have the same number of lines. * 80 Lines of Sensory Data per Instance: After the padding is applied, each instance of a character now consists of 80 lines of sensory data. This standardized length allows for easier comparison and analysis across different instances and characters. * Dataset Partitioning: The dataset is then divided into two separate sets: the training data and the test data. Approximately 80% of the students' data is allocated to the training set, which will be used for generating models and training them. The remaining 20% of students' data is assigned to the test set, which will be used to evaluate the performance of the trained models. * Handling Sensor Overflow Values: In some cases, the sensory data may contain overflow values, which are values that exceed the expected range or capacity of the sensor. To address this issue, the overflow values are replaced with similar sensor axis data from within the dataset. This ensures that the dataset remains consistent and coherent, even in the presence of overflow values, by substituting them with appropriate data points. By following these steps, the dataset is processed, standardized, and divided into training and test sets. These preparations facilitate the training of models on the training data and the subsequent evaluation of their performance using the test data. To refine the dataset, we implemented a selection process for the files. We specifically shortlisted files that fell within the file size range of 4 to 40 kb. Additionally, we conducted tests on files of various length ranges, including 3 to 30 kb, 3 to 50 kb, 5 to 10 kb, and 5 to 20 kb. Any files that did not fall within these specified ranges were removed from consideration. Once the files were selected, we performed an analysis on the number of lines present in each file with the same instance length. We calculated the average number of lines across these files. For files that had fewer lines than the average, we added additional lines filled with zeros at the top and bottom to match the average line count. On the other hand, for files that had more lines than the average, we removed half of the lines from the top and half from the end to bring them in line with the average. Furthermore, we enforced a requirement that the character being written must be present in the first line of each instance. If this condition was not met, the file was removed from the dataset. To ensure data consistency, we eliminated any occurrences of "nan" (indicating missing data) or "ovf" (indicating an overflow) in the sensor readings. If a sensor reading contained one of these values and was present in the file, it was replaced with a zero to maintain uniformity in the dataset. By applying these selection criteria and data-cleaning techniques, we aimed to create a refined dataset that would be more suitable for analysis, modelling, and training purposes. For additional experimental analysis, we performed the following actions: * Addition of Noisy Samples: We augmented the dataset by adding 50,000 samples with a 3% noise level. This means that 3% of the characters in each sample were randomly altered or distorted to simulate real-world scenarios where noise or errors may be present. The noise was introduced to increase the variability and robustness of the dataset. * Standard Deviation (Std Dev) of each char-median value: We calculated the standard deviation of the median value for each character in the dataset. This measurement helps to quantify the variability or spread of the median values across different instances of the same character. The std dev values provide insights into the consistency or variation of character representations within the dataset. * Range of Values: * a-range: The range for the character "a" was determined to be 70 data points. This implies that the values associated with the character "a" span a range of 70 data points in a line. * b-range: The range for character "b" was found to be 80 data points, indicating that the values associated with the character "b" cover a range of 80 data points in a line. §.§ Data distribution of upper-case letters Fig. <ref> depicts the data distribution of upper-case letters in the collected dataset. §.§ Data distribution of lower-case letters Fig. <ref> depicts the data distribution of lower-case letters in the collected dataset. § CONCLUSION The collected dataset for handwritten English alphabets recognition using IMU showcases the potential of leveraging IMU technology to accurately recognize and classify handwritten alphabets. The findings highlight the advantages of capturing dynamic hand movements and orientation through IMUs, leading to improved accuracy and potential applications in education, digital platforms, and human-computer interaction. Future research can focus on refining algorithms, exploring integration with other modalities, and assessing performance in real-world scenarios. Overall, IMUs offer innovative possibilities for transforming written word interactions and advancing various fields. IEEEtran
http://arxiv.org/abs/2307.03263v1	20230706195409	Recovery of Multiple Parameters in Subdiffusion from One Lateral Boundary Measurement	[ "Siyu Cen", "Bangti Jin", "Yikan Liu", "Zhi Zhou" ]	math.NA	[ "math.NA", "cs.NA", "math.AP" ]	Mach number and wall thermal boundary condition effects on near-wall compressible turbulence Akanksha [email protected], Diego A. Donzis1 , and Rodney D. W. Bowersox1 August 1, 2023 ===================================================================================================== This work is concerned with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove that the boundary measurement corresponding to a fairly general boundary excitation uniquely determines the order of the fractional derivative and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the fractional order and diffusion coefficient. The proposed algorithm combines small-time asymptotic expansion, analytic continuation of the solution and the level set method. We present extensive numerical experiments to illustrate the feasibility of the simultaneous recovery. In addition, we discuss the uniqueness of recovering general diffusion and potential coefficients from one single partial boundary measurement, when the boundary excitation is more specialized. Key words: subdiffusion, lateral boundary measurement, discontinuous diffusivity, unknown medium, level set method § INTRODUCTION This work is concerned with an inverse problem of simultaneously recovering multiple parameters in a subdiffusion model from one single lateral boundary measurement in a partly unknown medium. Let ⊂^d (d=2,3) be an open bounded domain with a Lipschitz and piecewise C^1,1 boundary and T>0 be a fixed final time. Consider the following subdiffusion problem for the function u: _t^ u+ u=f ×(0,T], a_ν u=g ×(0,T], u(0)=u_0 , where u_0∈ L^2() and (time-independent) f∈ L^2() are unknown initial and source data, and ν denotes the unit outward normal vector to the boundary . The elliptic operator is defined by u(x):=-·(a(x) u(x)), x∈. Without loss of generality, the diffusion coefficient a is assumed to be piecewise constant: a(x)=1+μ χ_D(x), where μ>-1 is a nonzero unknown constant, D is an unknown convex polyhedron in satisfying diam(D)<dist(D,) and χ_D denotes the characteristic function of D. In the model (<ref>), _t^ u denotes the Djrbashian-Caputo fractional derivative in time t of order ∈(0,1) defined by (<cit.> or <cit.>) _t^ u(t):=1(1-)∫_0^t(t-s)^-u'(s) ṣ. The model (<ref>) has attracted a lot of recent attention, due to its excellent capability to describe anomalous diffusion phenomena observed in many engineering and physical applications. The list of successful applications is long and still fast growing, e.g., ion transport in column experiments <cit.>, protein diffusion within cells <cit.> and contaminant transport in underground water <cit.>. See the reviews <cit.> for the derivation of relevant mathematical models and diverse applications. The model (<ref>) differs considerably from the normal diffusion model due to the presence of the nonlocal operator _t^ u: it has limited smoothing property in space and slow asymptotic decay at large time <cit.>. In this paper, we study mathematical and numerical aspects of an inverse problem of recovering the diffusion coefficient a and fractional order from a single lateral boundary measurement of the solution, without the knowledge of the initial data u_0 and source f. The Neumann data g is taken to be separable: g(x,t)=ψ(t)η(x), where 0≢η∈ H^1/2() satisfies the compatibility condition ∫_ηṢ=0 and ψ∈ C^1(_+) satisfies ψ(t)= 0, t<T_0, 1, t>T_1, with 0< T_0<T_1<T. The measurement data h(x,t)=u(x,t) is taken on a part of the boundary _0⊂. Note that the inverse problem involves missing data (u_0 and f), whereas the available data is only on a partial boundary. Thus, it is both mathematically and numerically very challenging, due to not only the severe ill-posed nature and high degree of nonlinearity but also the unknown forward map from the parameters a and to the data h(x,t). The mathematical study on inverse problems for time-fractional models is of relatively recent origin, starting from the pioneering work <cit.> (see <cit.> for overviews) and there are several existing works on recovering a space-dependent potential or diffusion coefficient from lateral Cauchy data <cit.>. Rundell and Yamamoto <cit.> showed that the lateral Cauchy data can uniquely determine the spectral data when u_0≡ f≡0, and proved the unique determination of the potential using Gel'fand-Levitan theory. They also numerically studied the singular value spectrum of the linearized forward map, showing the severe ill-posed nature of the problem. Later, they <cit.> relaxed the regularity condition on the boundary excitation g(t) in a suitable Sobolev space. Recently, Jing and Yamamoto <cit.> proved the identifiability of multiple parameters (including order, spatially dependent potential, initial value and Robin coefficients in the boundary condition) in a time-fractional subdiffusion model with a zero boundary condition and source, excited by a nontrivial initial condition from the lateral Cauchy data at both end points; see also <cit.>. Jin and Zhou <cit.> studied the unique recovery of the potential, fractional order and either initial data or source from the lateral Cauchy data, when the boundary excitation is judiciously chosen. All these interesting works are concerned with the one-dimensional setting due to their essential use of the inverse Sturm-Liouville theory. Wei et al <cit.> numerically investigated the recovery of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation from lateral boundary data. A direct extension of these theoretical works to the multi-dimensional case is challenging since the Gel'fand-Levitan theory is no longer applicable. Kian et al. <cit.> provided the first results for the multi-dimensional case, including the uniqueness for identifying two spatially distributed parameters in the subdiffusion model from one single lateral observation with a specially designed excitation Dirichlet input; see also <cit.> for a related study on determining the manifold from one measurement corresponding to a specialized source. Kian <cit.> studied also the issue of simultaneous recovery of these parameters along with the order and initial data using a similar choice of the boundary data. However, in the works <cit.>, the excitation data, which plays the role of infinity measurements, is numerically inconvenient to realize, if not impossible at all; see Remark <ref> and the appendix for further discussions. These considerations motivate the current work, i.e., to design robust numerical algorithm for recovering multiple parameters from a single partial boundary measurement for multi-dimensional subdiffusion with a computable excitation Neumann data, in the presence of a partly unknown medium. In this work, we make the following contributions to the mathematical analysis and numerics of the concerned inverse problem. First, we examine the feasibility to recover multiple parameters. We show that if the coefficient a is piecewise constant as defined in (<ref>), then one single boundary measurement can uniquely determine the coefficient a and fractional order , even though the initial data u_0 and source f are unknown. Note that the exciting Neumann data g given in (<ref>) is easy to realize and hence allows the numerical recovery. The proof relies on the asymptotic behavior of Mittag-Leffler functions, analyticity in time of the solution, and the uniqueness of the inverse conductivity problem (for elliptic problems) from one boundary measurement. In particular, the subdomain D can be either a convex polygon / polyhedron or a disc / ball, cf. Theorem <ref> and Remark <ref>. This analysis strategy follows a well-established procedure in the community, and roughly it consists of two steps. (1) Using the time-analyticity, the uniqueness for the original inverse problem is reduced to the one for an inverse problem for the corresponding time-independent elliptic equation; (2) The reduction can be done by the Laplace transform or considering the asymptotics. Both strategies of reductions are well known. For example, the former way is used for an Dirichlet-to-Neumann map for the inverse coefficient problem for a multi-term time-fractional diffusion equation <cit.>, while the latter way is used for the Dirichlet-to-Neumann map for the inverse parabolic problem <cit.>. Second, the uniqueness analysis lends itself to the development of a robust numerical algorithm: we develop a three-step recovery algorithm for identifying the piecewise constant coefficient a and the fractional order : (i) use the asymptotic behavior of the solution of problem (<ref>) near t=0 to recover ; (ii) use analytic continuation to extract the solution of problem (<ref>) with zero f and u_0; (iii) use the level set method to recover the shape of subdomain D. To the best of our knowledge, this is the first work on the numerical recovery of a (piecewise constant) diffusion coefficient in the context of multi-dimensional subdiffusion model with missing initial and source data. Last, we present extensive numerical experiments to illustrate the feasibility of the approach. We refer interested readers to <cit.> for some numerical studies for identifying a piecewise constant source from the boundary measurement. The rest of the paper is organized as follows. In Section <ref> we describe preliminary results on the model, especially time analyticity of the data. Then in Section <ref> we give the uniqueness result in case of piecewise constant a, and in Section <ref> we develop a recovery algorithm based on the level set method. We present extensive numerical experiments to illustrate the feasibility of recovering multiple parameters in Section <ref>. In an appendix, we discuss the possibility of recovering two coefficients from one boundary measurement induced by a specialized boundary excitation. Throughout, the notation ( · , · ) denotes the standard L^2() inner product, and ⟨ · , · ⟩ the L^2() inner product. For a Banach space B, C^(T,∞;B) denotes the set of functions valued in B and analytic in t∈(T,∞). The notation c, with or without a subscript, denotes a generic constant which may change at each occurrence, but it is always independent of the concerned quantities. § PRELIMINARIES In this section, we present preliminary analytical results. Let A be the L^2() realization of the elliptic operator , with a domain Dom(A):={v∈ L^2(): v∈ L^2(), _ν v\|_=0}. Let {_ℓ}_ℓ≥1 be a strictly increasing sequence of eigenvalues of A, and denote the multiplicity of _ℓ by m_ℓ and {_ℓ,k}_k=1^m_ℓ an L^2() orthonormal basis of (A-_ℓ). That is, for any ℓ∈, k=1,…,m_ℓ: _ℓ,k=_ℓ_ℓ,k , a_ν_ℓ,k=0 . The eigenvalues {_ℓ}_ℓ=1^∞ are nonnegative, and the eigenfunctions {_ℓ,k:k=1,…,m_ℓ}_ℓ=1^∞ form a complete orthonormal basis of L^2(). Note that _1=0 (and has multiplicity 1) and the corresponding eigenfunction _1=\|\|^-1/2 is constant valued, where \|E\| denotes the Lebesgue measure of a set E. Due to the piecewise constancy of the coefficient a, _ℓ,k is smooth in D and ∖ D. Moreover, it satisfies the following transmission condition on the interface D: _ℓ,k\|_-=_ℓ,k\|_+ _n_ℓ,k\|_-=(1+μ)_n_ℓ,k\|_+ D, where _ℓ,k\|_+ and _ℓ,k\|_- denote the limits from D and ∖ D to the interface D, respectively, and _n_ℓ,k\|_± denotes the derivative with respect to the unit outer normal vector n on D. Then we define the fractional power A^s (s≥0) via functional calculus by A^s v:=∑_ℓ=1^∞_ℓ^s∑_k=1^m_ℓ(v,_ℓ,k)_ℓ,k, with a domain Dom(A^s)={v∈ L^2():A^s v∈ L^2()}, and the associated graph norm v_ Dom(A^s)=(∑_ℓ=1^∞_ℓ^2s∑_k=1^m_ℓ(v,_ℓ,k)^2)^1/2. We use extensively the Mittag-Leffler function E_,(z) defined by (<cit.>, <cit.>) E_,(z)=∑_k=0^∞z^k(k+), ∀ z∈. The function E_,(z) generalizes the exponential function ^z. The following decay estimate of E_,(z) is crucial in the analysis below; See e.g., <cit.> and <cit.> for the proof. Let ∈(0,2), ∈, ∈(/2π,min(π,π)) and N∈. Then for φ≤\| z\|≤π with \|z\|→∞, there holds E_,(z)=-∑_k=1^Nz^-k(- k)+O(\|z\|^-N-1). By linearity, we may split the solution u of problem (<ref>) into u=u_i+u_b, with u_i and u_b solving _t^ u_i+ u_i=f ×(0,T], a_ν u_i=0 ×(0,T], u_i(0)=u_0 _t^ u_b+ u_b=0 ×(0,T], a_ν u_b=g ×(0,T], u_b(0)=0 , respectively. The following result gives the representations of u_i and u_b. Let u_0,f∈ L^2(). Then there exist unique solutions u_i,u_b∈ L^2(0,T;H^1()) that can be respectively represented by u_i(t) =(u_0,_1)_1+(f,_1)_1t^(1+) +∑_ℓ=2^∞∑_k=1^m_ℓ([(u_0,_ℓ,k)-_ℓ^-1(f,_ℓ,k)]E_,1(-_ℓ t^)+_ℓ^-1(f,_ℓ,k))_ℓ,k, u_b(t) =∑_ℓ=1^∞∑_k=1^m_ℓ∫_0^t(t-s)^-1E_,(-_ℓ,k(t-s)^)⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k. Hence, the solution u to problem (<ref>) can be represented as u(t)=ρ_0+ρ_1t^+∑_ℓ=2^∞ E_,1(-_ℓ t^)ρ_ℓ+∑_ℓ=1^∞∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)∑_k=1^m_ℓ⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k, with ρ_ℓ given by ρ_ℓ:={2 (u_0,_1)_1+∑_ℓ=2^∞∑_k=1^m_ℓ_ℓ^-1(f,_ℓ,k)_ℓ,k, ℓ=0, (f,_1)(1+)_1, ℓ=1, ∑_k=1^m_ℓ[(u_0,_ℓ,k)-_ℓ^-1(f,_ℓ,k)]_ℓ,k, ℓ=2,3,…. . The representations follow from the standard separation of variables technique (<cit.>, <cit.>). The piecewise constancy of the diffusivity a requires special care due to a lack of global regularity. By multiplying the governing equation of u_i by _ℓ,k and then integrating over , we get _t^(u_i(t),_ℓ,k)+( u_i(t),_ℓ,k)=(f,_ℓ,k). Integrating by parts twice and using the transmission condition (<ref>) for _ℓ,k (and u_i) on D gives ( u_i(t),_ℓ,k) =-∫_∖ D· ( u_i) _ℓ,k x̣-∫_D· ((1+μ ) u_i) _ℓ,k x̣ =-∫_( u_i·ν) _ℓ,k Ṣ-∫_ D( u_i· n_-) _ℓ,k\|_- Ṣ+∫_∖ D u_i·_ℓ,k x̣ -∫_ D(1+μ)( u_i· n_+) _ℓ,k\|_+ Ṣ+∫_D(1+μ) u_i·_ℓ,k x̣ =∫_∖ D u_i·_ℓ,k x̣+∫_D(1+μ ) u_i·_ℓ,k x̣ =∫_(_ℓ,k·ν) u_i Ṣ+∫_ D(_ℓ,k· n_-) u_i\|_- Ṣ-∫_∖ D·(_ℓ,k) u_i x̣ +∫_ D(1+μ)(_ℓ,k· n_+) u_i\|_+ Ṣ-∫_D ·((1+μ)_ℓ,k) u_i x̣ =(u_i,_ℓ,k)=_ℓ(u_i,_ℓ,k). Hence, the scalar function u_i^ℓ,k(t):=(u_i(t),_ℓ,k) satisfies the following initial value problem for a fractional ordinary differential equation: (_t^+_ℓ)u_i^ℓ,k(t)=f_ℓ,k:=(f,_ℓ,k) for 0<t≤ T, with u_i^ℓ,k(0)=u_0^ℓ,k:=(u_0,_ℓ,k). Then u_i^ℓ,k(t) is given by <cit.> u_i^ℓ,k(t)=u_0^ℓ,kE_,1(-_ℓ t^)+f_ℓ,k∫_0^t s^-1E_,(-_ℓ s^) ṣ. Note that u_i^1=u_0^1+1(1+)f_1t^. Now using the identity ṭ̣E_,1(- t^)=- t^-1E_,(- t^), we have for ℓ≥2 and k=1,…,m_ℓ that u_i^ℓ,k(t) =u_0^ℓ,kE_,1(-_ℓ t^)+_ℓ^-1[1-E_,1(-_ℓ t^)]f_ℓ,k =(u_0^ℓ,k-_ℓ^-1f_ℓ,k)E_,1(-_ℓ t^)+_ℓ^-1f_ℓ,k. This gives the representation of u_i. Similarly, multiplying the governing equation for u_b by _ℓ,k and integrating over give _t^(u_b(t),_ℓ,k)+( u_b(t),_ℓ,k)=0. Repeating the argument yields that u_b^ℓ,k(t):=(u_b(t),_ℓ,k) satisfies (_t^+_ℓ)u_b^ℓ,k(t)=⟨ g(t),_ℓ,k⟩ for 0<t≤ T, with u_b^ℓ,k(0)=0. The solution u_b^ℓ,k(t) is given by <cit.> u_b^ℓ,k(t)=∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k. Thus the desired assertion follows. The representation of the solution u to problem (<ref>) follows directly from that of u_b and u_i, and the identity (<ref>). Next we show properties of the boundary data h. This is achieved by first proving related properties of u and then applying the trace theorem. Below we study the analyticity of u_i (t) =ρ_0+ρ_1t^+∑_ℓ=2^∞ E_,1(-_ℓ t^)ρ_ℓ, u_b (t) =∑_ℓ=1^∞∑_k=1^m_ℓ∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k. Since our focus is the trace on , we only study u on the subdomain ∖ D. Recall that for a Banach space B, the notation C^(T,∞;B) denotes the set of functions valued in B and analytic in t∈(T,∞). Let D'⊃ D be a small neighborhood of D with a smooth boundary and denote '=∖D'. For u_0∈ L^2(), f∈ L^2() and g as in (<ref>), the following statements hold. (i) u_i∈ C^(0,∞;H^2(')) and u_b∈ C^(T_1+,∞;H^2(')) for arbitrarily fixed >0. (ii) The Laplace transforms u_i(z) and u_b(z) of u_i and u_b in t exist for all (z)>0 and are respectively given by u_i(z)=z^-1ρ_0+(+1)z^--1ρ_1+∑_ℓ=2^∞ρ_ℓ z^-1z^ +_ℓ u_b(z)=∑_ℓ=1^∞∑_k=1^m_ℓ⟨ g(z),_ℓ,k⟩_ℓ,kz^+_ℓ. Throughout this proof, let >0 be arbitrarily fixed. Since _1=0, by Lemma <ref>, there exist constants c>0 and ∈(0,π/2) such that for any z∈_:={z∈∖{0}:\|(z)\|≤}, we have u_i(z)_ Dom(A)^2 =∑_n=1^∞_n^2∑_j=1^m_n(_n,j,ρ_0+ρ_1 z^+∑_ℓ=2^∞ E_,1(-_ℓ z^)ρ_ℓ)^2 =∑_n=2^∞_n^2∑_j=1^m_n(_n,j,∑_ℓ=2^∞∑_k=1^m_ℓ{E_,1(-_ℓ z^)[(u_0,_ℓ,k)-_ℓ^-1(f,_ℓ,k)]+_ℓ^-1(f,_ℓ,k)}_ℓ,k)^2 ≤ c∑_n=2^∞_n^2E_,1(-_n z^)^2∑_j=1^m_n{(u_0,_n,j)^2+_n^-2(f,_n,j)^2}+c∑_n=1^∞∑_j=1^m_n(f,_n,j)^2 ≤ c\|z\|^-2∑_n=2^∞∑_j=1^m_n{(u_0,_n,j)^2+_n^-2(f,_n,j)^2}+ cf_L^2()^2 ≤ c\|z\|^-2( u_0_L^2()^2+ f_L^2()^2)+cf_L^2()^2. Since u_0,f∈ L^2(), u_i(z)_Dom(A)^2 is uniformly bounded for z∈_. Since E_,1(-_n z^) is analytic in z∈_ and the series converges uniformly in any compact subset of _, u_i(t) is analytic in t∈(0,∞) as a Dom(A)-valued function, i.e., u_i∈ C^(0,∞;Dom(A)). By Sobolev embedding, u_i∈ C^(0,∞;H^2(')). Next we prove the analyticity of u_b. By the choice g(x,t)=η(x)ψ(t) in (<ref>) and integration by parts, for t>T_1, u_b^1(t):=(u_b(t),_1) is given by u_b^1(t) =1()∫_0^t(t-s)^-1⟨ g(s),_1⟩ ṣ= ⟨η,_1⟩()∫_0^t(t-s)^-1ψ(s) ṣ = ⟨η,_1⟩ ()[ -(t-s)^ψ(s)\|_s=0^s=t + ∫_0^t(t-s)^ψ'(s) ṣ] =⟨η,_1⟩(+1)∫_T_0^T_1(t-s)^ψ'(s) ṣ, where the last step follows from the condition on ψ in (<ref>). Thus the time-analyticity of u_b^1(t)_1 for t∈(T_1+,∞) follows. Next, again by integration by parts, (<ref>)–(<ref>) and the identity (<ref>), for t>T_1, u_b^ℓ,k(t):=(u_b(t),_ℓ,k) with ℓ≥2, k=1,…,m_ℓ can be written as u_b^ℓ,k(t) =∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)⟨ g(s),_ℓ,k⟩ ṣ =∫_0^t⟨ g(s),_ℓ,k⟩_ℓṣ̣E_,1(-_ℓ(t-s)^) ṣ =_ℓ^-1[⟨ g(s),_ℓ,k⟩ E_,1(-_ℓ(t-s)^)]_s=0^s=t-⟨η,_ℓ,k⟩_ℓ∫_0^tE_,1(-_ℓ(t-s)^)ψ'(s) ṣ =⟨η,_ℓ,k⟩_ℓψ(t)-⟨η,_ℓ,k⟩_ℓ∫_T_0^T_1E_,1(-_ℓ (t-s)^)ψ'(s) ṣ=:u_b,1^ℓ,k(t)+u_b,2^ℓ,k(t). Since ψ(t)=1 for t>T_1, we see that u_b,1^ℓ,k(t) is a constant for t>T_1. Next we consider the following boundary value problem U=0 , with a_ν U=η . The compatibility condition ⟨η,1⟩=0 implies that there exist solutions to problem (<ref>). We take an arbitrary solution U. Since a is piecewise constant and η∈ H^1/2(), we know that U∈ H^1() and its restriction U\|_'∈ H^2('). Integrating by parts twice yields ⟨η,_ℓ,k⟩=_ℓ(U,_ℓ,k) . Similar to the argument for Proposition <ref>, from the transmission condition (<ref>), we deduce ∑_ℓ=2^∞∑_k=1^m_ℓu_b,1^ℓ,k(t)_ℓ,k=∑_ℓ=2^∞∑_k=1^m_ℓψ(t)(U,_ℓ,k)_ℓ,k , which is analytic in t∈(T_1+,∞) since it is constant in time and U∈ L^2(). Moreover, by the standard elliptic regularity theory, ∑_ℓ =2^∞∑_k=1^m_ℓu_b,1^ℓ,k_ℓ,k∈ C^(T_1+,∞;H^2(')). Recall Young's inequality for convolution, i.e., fg_L^r()≤f_L^p()g_L^q() for p,q,r≥1 with p^-1+ q^-1=r^-1+1 and any f∈ L^p() and g∈ L^q(). Then by Young's inequality, Lemma <ref> and the regularity estimate ∑_ℓ=2^∞_ℓ^-2∑_k=1^m_ℓ⟨η,_ℓ,k⟩^2≤U_L^2()<∞, we deduce ∑_ℓ=2^∞∑_k=1^m_ℓu_b,2^ℓ,k(z)_ℓ,k_ Dom(A)^2 =∑_n=1^∞_n^2∑_j=1^m_n(_n,j,∑_ℓ=2^∞∑_k=1^m_ℓu_b,2^ℓ,k(z)_ℓ,k)^2 =∑_n=2^∞_n^2∑_j=1^m_n(⟨η,_n,j⟩_n∫_T_0^T_1E_,1(-_n(z-s)^)ψ'(s) ṣ)^2 ≤∑_n=2^∞∑_j=1^m_n⟨η,_n,j⟩^2(c_n\|z-T_1\|^∫_T_0^T_1\|ψ'(s)\| ṣ)^2 ≤(cψ_W^1,∞(_+)\|z-T_1\|^)^2∑_n=2^∞_n^-2∑_j=1^m_n\|⟨η,_n,j⟩\|^2≤c\|z-T_1\|^2. Since u_b,2^ℓ,k(t) is analytic in (T_1+,∞) and the series ∑_ℓ=2^∞∑_k=1^m_ℓu_b,2^ℓ,k(z)_ℓ,k converges uniformly in Dom(A) for z∈ T_1++_, it belongs to C^(T_1+,∞;Dom(A)), and hence u_b∈ C^(T_1+,∞;H^2(')). This proves part (i). The argument for part (i) implies that the series converges uniformly in Dom(A) for t∈(0,∞), and ^-t zu_i(t)_ Dom(A)≤ c ^-t (z)(t^-+1), t>0. The function ^-t (z)(t^-+1) is integrable in t over (0,∞) for any fixed z with (z)>0. By Lebesgue's dominated convergence theorem and taking Laplace transform termwise, we obtain u_i(z)=z^-1ρ_0+(+1)z^--1ρ_1+∑_ℓ=2^∞ρ_ℓ z^-1z^+_ℓ, ∀ (z)>0. The argument for part (i) also implies ^-t zu_b(t)_ Dom(A) ≤ c ^-t (z)\|t-T_1\|^-, t>0. Then termwise Laplace transform and Lebesgue's dominated convergence theorem complete the proof of the proposition. Thus, u_i and u_b are analytic in time and have H^2(') regularity. Since is Lipschitz and piecewise C^1,1, their traces on are well defined. The next result is direct from the trace theorem and Sobolev embedding theorem. Here, we use x and y denote the variables in and on , respectively. Let the assumptions in Proposition <ref> hold. Then the data h=u\|__0×(0,T) to problem (<ref>) can be represented by h(t)= ρ_0+ρ_1t^+∑_ℓ=2^∞ E_,1(-_ℓ t^)ρ_ℓ_=:h_i(t) +∑_ℓ=1^∞∑_k=1^m_ℓ∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k_=:h_b(t). Moreover, h_i and h_b satisfy the following properties. (i) h_i∈ C^(0,∞;L^2(_0)) and h_b∈ C^(T_1+,∞;L^2(_0)) for arbitrarily fixed >0. (ii) The Laplace transforms h_i(z) and h_b(z) of h_i and h_b in t exist for all (z)>0 and are given by h_i(z) =z^-1ρ_0+(+1)z^--1ρ_1+∑_ℓ=2^∞ρ_ℓ z^-1z^+_ℓ, h_b(z) =∑_ℓ=1^∞∑_k=1^m_ℓ⟨ g(z),_ℓ,k⟩_ℓ,kz^+_ℓ. The analysis of Theorem <ref> crucially exploits the analyticity of the measurement h_i(t) in time, which relies on condition (<ref>), i.e., ψ(t)≡ 0 for t∈ [0,T_0]. The condition ψ(t)≡ 1 for t≥ T_1 for some T_1<T from (<ref>) ensures the time analyticity of h_b(t) for t>T_1+ε, which is needed for Theorem <ref>. It should be interpreted as analytically extending the observation h_b(t) by analytically extending ψ(t), both from (T_1,T) to (T_1,∞). Alternative conditions on ψ(t) ensuring the time analyticity of h_b(t) for t>T_1+ε, e.g., ψ(t) vanishes identically on (T_1,T), would also be sufficient for Theorem <ref>. § UNIQUENESS Now we establish a uniqueness result for recovering the fractional order and piecewise constant a. The proof proceeds in two steps: First we show the uniqueness of the order from the observation, despite that the initial condition u_0 and source f are unknown. Then we show the uniqueness of a. The key observation is that the contributions from u_0 and f can be extracted explicitly. Since the Dirichlet data is only available on a sub-boundary _0, we view ρ_k as a L^2(_0)-valued function. The notation denotes the set {k∈:ρ_k≢0L^2(_0)}, i.e., the support of the sequence (ρ_0,ρ_1,…) in L^2(_0) sense, similarly, ={k∈:ρ_k≢0 L^2(_0)}, and ^=∖{1}. Below we denote by 𝔄 the admissible set of conductivities, i.e., 𝔄 = {1+μχ_D(x): μ>-1 D⊂Ω}. Let ,∈(0,1), (a,f,u_0),( a, f, u_0)∈𝔄× L^2()× L^2(), and fix g as (<ref>) with ψ(t) satisfying condition (<ref>). Let h and h be the corresponding Dirichlet observations. Then for some >0, the condition h= h on _0×[T_0-,T_0] implies =, ρ_0=ρ_0 and {(ρ_k,_k)}_k∈ ={(ρ_k,_k)}_k∈ if ,∅. By the definition of g, we have g(y,t)≡0 for y∈, t∈[0,T_0]. Then by Corollary <ref>, h(y,t) admits a Dirichlet representation h(y,t)=ρ_0(y)+ρ_1(y)t^+∑_k∈∩^ρ_k(y)E_,1(-_k t^). By Corollary <ref>(i), h(t) is analytic as an L^2()-valued function in t>0. By analytic continuation, the condition h(t)= h(t) for t∈ [T_0-σ,T_0] implies that h(t)= h(t) in L^2(_0) for all t>0, i.e., ρ_0(y)+ρ_1(y)t^+∑_k∈∩^ρ_k(y)E_,1(-_k t^)=ρ_0(y)+ρ_1(y)t^+∑_k∈∩^ρ_k(y)E_,1(-λ_k t^). From the decay property of E_,1(-η) (see Lemma <ref>), we derive ρ_0(y)+ρ_1(y)t^=ρ_0(y)+ρ_1(y)t^, indicating ρ_0=ρ_0 and ρ_1=ρ_1. Moreover, we have = if 1∈. If 1∉ and 1∉, i.e., ρ_1=ρ_1=0, then ∑_k∈∩^ρ_k(y)E_,1(-_k t^)=∑_k∈∩^ρ_k(y)E_,1(-λ_k t^) _0×(0,∞). Proposition <ref>(ii) and Laplace transform give ∑_k∈∩^ρ_k(y)z^-1z^+_k=∑_k∈∩^ρ_k(y)z^-1z^+λ_k. Assuming that >, dividing both sides by z^-1 and setting :=z^, we have ∑_k∈∩^ρ_k(y)^1-/+_k=∑_k∈∩^ρ_k(y)^/+_k. Upon noting ∅, choosing an arbitrary k_0∈ and rearranging terms, we derive ρ_k_0(y)^1-/=(∑_k∈∩^ρ_k(y)^/+_k -∑_k∈∩^∖{k_0}ρ_k(y)^1-/+_k)(+_k_0). Letting →-_k_0 and noting >, the right hand side tends to zero (since all _k are positive, and ((-_k_0)^/)=π/∈(0,π)) and hence ρ_k_0≡0 in L^2(Γ_0), which contradicts the assumption k_0∈. Thus, we deduce ≤. The same argument yields ≥, so =. These discussions thus yield ∑_k∈∩^ρ_k(y)+_k=∑_k∈∩^ρ_k(y)+_k. Note that both sides of the identity (<ref>) are L^2(_0)-valued functions in . Next we show both converge uniformly in any compact subset in ∖({-_k}_k∈∩^∪{-_k}_k∈∩^) and are analytic in ∖({-_k}_k∈∩^∪{-_k}_k∈∩^). Indeed, since u_0,f∈ L^2(), for all in any compact subset of ∖({-_k}_k∈∩^∪{-_k}_k∈∩^), we have ∑_k∈∩^ρ_k+_k_ Dom(A)^2≤ c∑_ℓ∈^_ℓ^2\|(u_0,_ℓ)\|^2+_ℓ^-2\|(f,_ℓ)\|^2\|+_ℓ\|^2 ≤ c∑_ℓ∈^(\|(u_0,_ℓ)\|^2+_ℓ^-2\|(f,_ℓ)\|^2)<∞. Hence, by the trace theorem, the identity (<ref>) holds for all ∈∖({-_k}_k∈∩^∪{-_k}_k∈∩^). Assume that _j∉{_k}_k∈∩^* for some j∈∩^. Then we can choose a small circle C_j centered at -_j which does not contain {-_k}_k∈∩^. Integrating on C_j and applying the Cauchy theorem give 2π√(-1) ρ_j/_j=0, which contradicts the assumption ρ_j≢0 in L^2(_0). Hence, _j∈{_k}_k∈∩^* for every j∈∩^. Likewise, _j∈{_k}_k∈∩^ for every j∈∩^, and hence {_k}_k∈∩^={_k}_k∈∩^. From (<ref>), we obtain ∑_k∈∩^ρ_k(y)-ρ_k(y)+_k=0, ∀∈∖{-_k}_k∈∩^. Varying j∈∩^ and integrating over C_j, we obtain 2π√(-1) (ρ_j-ρ_j)/_j=0, which directly implies ρ_j=ρ_j in L^2(_0). This completes the proof of the theorem. The condition 𝕂≠∅ holds whenever the following condition is valid (f,φ_1)≠ 0 or (u_0,_ℓ,k)-_ℓ^-1(f,_ℓ,k)≠ 0, k=1,…,m_ℓ, ℓ=2,3,…. Note that the condition (f,φ_1)≠ 0 does not rely on the unknown parameter a, and can be easily guaranteed. The next result gives the uniqueness of recovering the diffusion coefficient a from the lateral boundary observation. Let condition (<ref>) be fulfilled, and let (a,f,u_0), ( a, f, u_0)∈𝔄× L^2()× L^2(), and fix g as (<ref>). Let h and h be the corresponding Dirichlet data. Then for any ∈(0,T_0], the condition h= h on _0×[T_0-,T] implies a= a. In view of the linearity of problem (<ref>), we can decompose the data h(t) into h(t)=h_i(t)+h_b(t), t∈(0,T], with h_i(t) and h_b(t) given by h_i(t) =ρ_0+ρ_1t^+∑_k∈∩^ρ_k E_,1(-_k t^), h_b(t) =∑_ℓ=1^∞∫_0^t(t-s)^-1E_,(-_ℓ(t-s)^)∑_k=1^m_ℓ⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k, which solve problem (<ref>) with g≡0 and f=u_0≡0, respectively. By the choice of g in (<ref>), the interval [0,T] can be divided into two subintervals: (0,T_0] and [T_0,T]. For t∈(0,T_0), ψ(t)≡0, Theorem <ref> implies that {(ρ_k,_k)}_k∈={(ρ_k,_k)}_k∈ and =, from which we deduce h_i(t)= h_i(t) for all t>0. For t∈[T_0,T], this and the condition h(t)= h(t) imply h_b(t)= h_b(t) in L^2(_0), and hence ∑_ℓ=1^∞∫_T_0^t(t-s)^-1E_,(-_ℓ(t-s)^)∑_k=1^m_ℓ⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k = ∑_ℓ=1^∞∫_T_0^t(t-s)^-1E_,(-_ℓ(t-s)^)∑_k=1^ m_ℓ⟨ g(s),_ℓ,k⟩ ṣ _ℓ,k, t∈[T_0,T]. By the analyticity in Corollary <ref>, the above identity holds for t∈[T_0, ∞). Thus applying Laplace transform on both side gives ∑_ℓ=2^∞∑_k=1^m_ℓ⟨ g(z),_ℓ,k⟩_ℓ,kz^+_ℓ= ∑_ℓ=2^∞∑_k=1^ m_ℓ⟨ g(z),_ℓ,k⟩_ℓ,kz^+_ℓ, ∀ (z)>0. Since _1=_1=0 and _1=_1=\|\|^-1/2, the index in (<ref>) starts with ℓ=2. Below we repeat the argument for Theorem <ref>. First we show that both sides of (<ref>) are analytic with =z^ in any compact subset of ∖{-_ℓ,-_ℓ}_ℓ≥2. Let U∈Dom(A^1/4+) be a solution of problem (<ref>), for all in a compact subset of ∖{-_ℓ,-_ℓ}_ℓ≥2, we have ∑_ℓ=2^∞∑_k=1^m_ℓ⟨ g(^1/),_ℓ,k⟩_ℓ,k+_ℓ_ Dom(A^1/4+)^2 ≤ c∑_ℓ=2^∞_ℓ^1/2+2∑_k=1^m_ℓ\|⟨η,_ℓ,k⟩+_ℓ\|^2 = c∑_ℓ=1^∞_ℓ^1/2+2∑_k=1^m_ℓ\|_ℓ(U,_ℓ,k)+_ℓ\|^2≤ cU_ Dom(A^1/4+)^2<∞. Since each term of the series is a Dom(A^1/4+)-valued function analytic in and converges uniformly in , by the trace theorem, we obtain that both sides of (<ref>) are L^2()-valued functions analytic in ∈∖{-_ℓ,-_ℓ}_ℓ≥2. Since _ℓ,_ℓ>0 for ℓ≥2, we may take →0 in (<ref>) and obtain ∑_ℓ=2^∞∑_k=1^m_ℓ⟨ g(0),_ℓ,k⟩_ℓ,k_ℓ= ∑_ℓ=2^∞∑_k=1^ m_ℓ⟨ g(0),_ℓ,k⟩_ℓ,k_ℓ. Hence, w= w on _0, where w and w are the Dirichlet boundary data with a and a in the elliptic problem -·(a w)=0 , a_ν w= g(0) with the compatibility condition ∫_ w x̣=0. Indeed, the solution w of (<ref>) can be represented as w=∑_ℓ=2^∞∑_k=1^m_ℓ(w,_ℓ,k)_ℓ,k=∑_ℓ=2^∞∑_k=1^m_ℓ_ℓ^-1⟨ g(0),_ℓ,k⟩_ℓ,k, where the first equality follows from the compatibility condition ∫_ w x̣=0 and the second is due to integration by part. By the choice of g in (<ref>), the elliptic problem (<ref>) is uniquely solvable. Then from <cit.>, we deduce that D= D is uniquely determined by the input g(0) =ψ(0)η. Indeed, Friedman and Isakov <cit.> proved the unique determination of the convex polygon D for the case μ≡1, based on extending the solution w harmonically across a vertex of D and leading a contradiction. The proof does not depend on the knowledge of the parameter μ and hence it is also applicable here. Once D is determined, it suffices to show the uniqueness of the scalar μ. Suppose μ≤μ, i.e., a≤ a in D and a≡ a≡1 outside D. Thus w and w are harmonic functions near with identical Cauchy data on _0, we conclude w= w near . By multiplying both sides of the governing equation in (<ref>) with w, integrating over the domain Ω and applying Green's formula, we have 0 = ∫_Ω -∇·(a∇ w) w x̣ = ∫_Ω a\|∇ w\|^2 x̣ - ∫_∂Ωw _ν w Ṣ, i.e., ∫_Ω a\|∇ w\|^2 x̣ = ∫_∂Ωw _ν w Ṣ. Now since w and w have identical Cauchy data on the boundary ∂Ω, we have ∫_∂Ωw _ν w Ṣ = ∫_∂Ωw _νw Ṣ, and consequently ∫_Ω a\|∇ w\|^2 x̣ = ∫_Ωa\|∇w\|^2 x̣. This identity and the inequality a≥ a a.e. in Ω imply ∫_Ω a\|∇ w\|^2 x̣≥∫_Ω a\|∇w\|^2 x̣, which immediately implies 1/2∫_Ω a\|∇ w\|^2x̣ - ∫_∂Ωw g(0) Ṣ≥1/2∫_Ω a\|∇w\|^2 x̣ - ∫_∂Ωwg(0) Ṣ. By the Dirichlet principle <cit.>, w is the minimizer of the energy integral, and hence w= w and a= a. Note that the uniqueness of the inclusion D in <cit.> relies on the assumption D being a convex polygon with diam(D)<dist(D,Ω). Alessandrini <cit.> removed the diameter assumption for a specialized choice of the boundary data. The works <cit.> proved the unique determination of D when D is a disc or ball. For general shapes, even for ellipses or ellipsoids, this inverse problem appears still open. Note that in the uniqueness proof, the key is the reduction of the problem to the elliptic case, with a nonzero Neumann boundary condition. In particular, the result will not hold if the temporal component ψ vanishes identically over the interval [0,T], i.e., condition (<ref>) does not hold. If the diffusion coefficient a is not piecewise constant, it is also possible to show the unique recovery if the boundary excitation data g is specially designed. For example, consider problem (<ref>) with a more general elliptic operator u(x):=-·(a(x) u(x))+q(x)u(x), x∈. Here a∈ C^2() and q∈ L^∞() with a>0 in and q≥0 in , and the Neumann data g is constructed as follows. First, we choose sub-boundaries _1 and _2 such that _1∪_2= and _1∩_2∅. Let χ∈ C^∞() be a cut-off function with supp(χ)=_1 and χ≡1 on _1', with _1'⊂_1 such that _1'∪_2=, _1'∩_2∅; see Fig. <ref> for an illustration of the geometry in the two-dimensional case. Now we fix 0≤ T_0<T_1<T and choose a strictly increasing sequence {t_k}_k=0^∞ such that t_0=T_0 and lim_k→∞t_k=T_1. Consider a sequence {p_k}_k=1^∞⊂_+ and a sequence {ψ_k}_k=1^∞⊂ C^∞([0,∞);_+) such that ψ_k= 0 [0,t_2k-1], p_k [t_2k,∞). Then we fix {b_k}_k=0^∞⊂_+ such that ∑_k=1^∞ b_kψ_k _W^2,∞(_+)<∞, and define the Neumann data g by g(y,t):=∑_k=1^∞ g_k(y,t)=χ∑_k=1^∞ b_kψ_k(t)η_k(y), where the set {η_k}_k=1^∞ is chosen to be dense in H^1/2() and η_k _H^1/2()=1. Note that the Neumann data g defined in (<ref>) plays the role of infinity measurements <cit.>, and hence the unique recovery of the fractional order and both a and q from one boundary measurement. We provide a detailed proof in the appendix for completeness. See also some related discussions in <cit.> with different problem settings. However, this choice of g is impossible to numerically realize in practice, due to the need to numerically represent infinitesimally small quantities. § RECONSTRUCTION ALGORITHM In this section, we derive an algorithm for recovering the fractional order and the coefficient a, directly inspired by the uniqueness proof. We divide the recovery procedure into three steps: (i) use the asymptotic behavior of the solution of problem (<ref>) near t=0 to recover ; (ii) use analytic extension to extract the solution of problem (<ref>) with zero f and u_0; (iii) use the level set method <cit.> to recover the shape of the unknown medium D⊂. First, we give an asymptotics of the Dirichlet data h(t) of problem (<ref>). The result is direct from the representation and properties of E_,1(z) near z=0 and the trace theorem. If u_0∈ Dom(A^1+s/2) and f∈ Dom(A^s/2) with s>1. Let h=u\|_×(0,T) be the Dirichlet trace of the solution to problem (<ref>) with g given as (<ref>), then the following asymptotic holds: h(y,t)=u_0(y)+( u_0-f)(y)t^+O(t^2) t→0^+. In view of Proposition <ref>, for any fixed y_0∈, the asymptotic behavior of h(y_0,t) as t→0^+ allows recovering the order . This can be achieved by minimizing the following objective in , c_0 and c_1: J(,c_0,c_1)=c_0+c_1t^-h(y_0,t)_L^2(0,t_0)^2, for some small t_0>0. Note that it is important to take t_0 sufficiently small so that higher-order terms can indeed be neglected. The idea of using asymptotics for order recovery was employed in <cit.>. When recovering the diffusion coefficient a, we need to deal with the unknown functions u_0 and f. This poses significant computational challenges since standard regularized reconstruction procedures <cit.> require a fully known forward operator. To overcome the challenge, we appeal to Theorem <ref>: u_0 and f only contribute to h_i(t) which is fully determined by {_ℓ,ρ_ℓ}_ℓ∈. Indeed, by Theorem <ref>, {_ℓ, ρ_ℓ}_ℓ∈ can be uniquely determined by h(t), t∈[0,T_0]. Hence in theory we can extend h(t)=h_i(t) from t∈[0,T_0] to t∈[0,T], by means of analytic continuation, to approximate the Dirichlet data of (<ref>) with g≡0 and given u_0 and f. In practice, we look for approximations of the form h(t)≈p_0+p_1t+⋯+p_r t^rq_0+q_1t+⋯+q_r t^r:=h_r(t), t∈[0,T], where r∈ is the polynomial order. This choice is motivated by the observation that Mittag-Leffler functions can be well approximated by rational polynomials <cit.>. The approximation h_r can be constructed efficiently by the AAA algorithm <cit.>. Now, we can get the Dirichlet data of problem (<ref>) with a given g and u_0=f≡0, by defining the reduced data h(t):= 0, t∈[0,T_0], h(t)-h_r(t), t∈[T_0,T]. Below we use the reduced data h to recover a piecewise constant a. Parameter identification for the subdiffusion model is commonly carried out by minimizing a suitable penalized objective. Since a is piecewise constant, it suffices to recover the interface between different media. The level set method can effectively capture the interface in an elliptic problem <cit.>, which we extend to the time-fractional model (<ref>) below. Specifically, we consider a slightly more general setting where the inclusion D⊂ has a diffusivity value a_1 and the background ∖ D has a diffusivity value a_2, with possibly unknown a_1 and a_2. That is, the diffusion coefficient a is represented as a(x)=a_1H(ϕ(x))+a_2(1-H(ϕ(x))) , where H(x) and ϕ(x) denote the Heaviside function and level set function (a signed distance function): H(x):= 1, x≥0, 0, x<0, ϕ(x):= d(x, D), x∈ D, -d(x, D), x∈∖ D, respectively. Then ϕ satisfies D={x∈:ϕ(x)>0}, ∖ D={x∈:ϕ(x)<0} and D={x∈:ϕ(x)=0}. To find the values a_1 and a_2 and the interface D, we minimize the following functional J(ϕ,a_1,a_2)=12u(a)- h_L^2(0,T;L^2(_0))^2+∫_\| a\|, where u(a) is the solution to problem (<ref>), and >0 is the penalty parameter. The total variation term ∫_\| a\| is to stabilize the inverse problem, which is defined by ∫_\| a\|:=sup_∈(C_0())^d,\|\|≤1∫_ a· x̣, where \|·\| denotes the Euclidean norm. Then we apply the standard gradient descent method to minimize problem (<ref>). The next result gives the gradient of J. The notations J_T-^1- and D_T-^ denote the backward Riemann-Liouville integral and derivative, defined respectively by <cit.> J_T-^1-v(t) :=1(1-)∫_t^T(s-t)^-v(s) ṣ, D_T-^ v(t) :=-1(1- )ṭ̣∫_t^T(s-t)^-v(s) ṣ. The derivative ạ̣J is formally given by ạ̣J(a)=-∫_0^T u· v ṭ-·( a\| a\|), where v=v(x,t;a) solves the adjoint problem D_T-^ v-·(a v)=0 ×[0,T), a_ν v=(u- h)χ__0 ×[0,T), J_T-^1- v(·,T)=0 . We write J(a)=J_1(a)+J_2(a), with J_1(a)=12 u(a)- h_L^2(0,T;L^2(_0))^2 and J_2(a)=∫_\| a\|. For the term J_1, the directional derivative along b is .\|_=0J_1(a+ b)=∫_0^T∫__0(u(a)- h)u'(a)[b] Ṣṭ, where u'(a)[b] is the directional derivative with respect to a in the direction b. Let a=a+ b and u solves problem (<ref>) with the coefficient a. Then w:=u'(a)[b]=lim_→0^-1( u-u). Upon subtracting the equations for u and u and then taking limits, we get _t^ w-· (a w)=·(b u) ×(0,T], a_ν w=-b_ν u ×(0,T], w(0)=0 . Multiplying the equation for w with any ψ∈ L^2(0,T;H^1()) and integrating over ×(0,T) give ∫_0^T∫_(ψ _t^ w+a w·ψ) x̣ṭ=-∫_0^T∫_ b u·ψ x̣ṭ. Let v be the solution of problem (<ref>). Multiplying the governing equation for v with a test function ψ and integrating over ×(0,T) give ∫_0^T∫_(ψ D_T-^ v+a v·ψ)x̣ṭ=∫_0^T∫__0(u- h)ψ Ṣṭ. Note that the following integration by parts formula for fractional derivatives: ∫_0^T v _t^ w ṭ=[w J_T-^1-v]_t=0^t=T+∫_0^T w D_T-^ v ṭ=∫_0^T w D_T-^ v ṭ (for suitably smooth v and w with w(0)=0 and J_T-^1-v=0). Now by choosing ψ=v in (<ref>), ψ=w in (<ref>) and applying (<ref>), we obtain -∫_0^T∫_ b u· v x̣ṭ=∫_0^T∫__0(u- h)w Ṣṭ, implying ạ̣J_1(a)=-∫_0^T u· vṭ. For the term J_2, the directional derivative along b is .\|_=0∫_\| (a+ b)\| x̣ =∫_.\|_=0(\| (a+ b)\|^2)^1/2x̣ = ∫_.(\| (a+ b)\|^2)^-12\|_=0 a· b x̣=∫_ a\| a\|· b x̣, and hence we have ạ̣J_2(a)=-·( a\| a\|). By the chain rule, the derivatives of J with respect to a_1, a_2 and ϕ are given by Jϕ =J̣ạ aϕ=J̣ạ(a_1-a_2)(ϕ), J a_1 =∫_J̣ạ a a_1 x̣=∫_J̣ạH(ϕ) x̣, J a_2 =∫_J̣ạ a a_2 x̣=∫_J̣ạ(1-H(ϕ)) x̣, where is the Dirac delta function. Hence the iterative scheme for updating a_1, a_2 and ϕ reads ϕ^k+1=ϕ^k-^k Jϕ(ϕ^k,a_1^k,a_2^k) a_j^k+1=a_j^k-_j^k J a_j(ϕ^k+1,a_1^k,a_2^k), j=1,2. The step sizes ^k and _j^k can be either fixed or obtained by means of line search. The implementation of the method requires some care. First, we approximate the delta function (x) and Heaviside function H(x) by _(x)=π (x^2+^2) H_(x)=1πarctan( x)+12, respectively, with >0 of order of the mesh size <cit.>. Second, during the iteration, the new iterate of ϕ may fail to be a signed distance function. Although one is only interested in sign(ϕ), it is undesirable for \|ϕ\| to get too large near the interface. Thus we reset ϕ to a signed distance function whenever ϕ changes by more than 10% in the relative L^2()-norm. The resetting procedure is to find the steady solution of the following equation <cit.>: _t d+sign(d)(\| d\|-1)=0, d(0)=ϕ. § NUMERICAL EXPERIMENTS AND DISCUSSIONS Now we present numerical results for reconstructing the fractional order and piecewise constant diffusion coefficient a, with unknown u_0 and f. In all experiments, the domain is taken to be the unit square =(0,1)^2, and the final time T=1. We divide the domain into uniform squares with a length h=0.02 and then divide along the diagonals of each square. We discretize the time interval [0,T] into uniform subintervals with a time step size τ=0.01. All direct and adjoint problems are solved by standard continuous piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time (see e.g.,<cit.> and <cit.>). Below we investigate the following four cases: (i) D is a disc with radius 1/3, centered at (1/2,1/2), (ii) D is a square with length 1/2, centered at (1/2,1/2), (iii) D is a concave polygon, and (iv) D is two discs with radius 1/5, centered at (1/4,1/2) and (3/4,1/2), respectively. Throughout, the unknown initial condition u_0 and source f are fixed as u_0(x_1,x_2)=x_1^2x_2^2(1-x_1)^2(1-x_2)^2 f(x_1,x_2)=1+x_1+z_2, respectively. Meanwhile, we fix the exact fractional order ^†=0.8 and the diffusion coefficient a^†=10-9χ_D, i.e. a_1=1, a_2=10. Unless otherwise stated, the Neumann excitation g is taken as g(y,t)=η(y)χ_[0.5,1](t), where η is the cosine function with a frequency 2π on each edge for cases (i)–(iii) and is constant 1 for case (iv). We set g on ×[0,T], and take the measurement h on ×[0,T]. First, we show the numerical recovery of the fractional order for three different values, i.e., 0.3, 0.5 and 0.8. In view of Proposition <ref>, it suffices to fix one point y_0∈ (which is fixed at the origin y_0=(0,0) below) and to minimize problem (<ref>), for which we use the L-BFGS-B with constraint ∈[0,1] <cit.>. The recovered orders are presented in Table <ref>. Note that the least-squares functional has many local minima. Hence, the algorithm requires a good initial guess to get a correct value for . It is observed that the reconstruction is more accurate when t_0→0^+, since the high order terms are then indeed negligible. Also, for a fixed interval (0,t_0), due to the asymptotic behavior, we have slightly better approximations when the true order is large. However, this does not influence much the reconstruction results for cases (i)–(iv), since the coefficient a is constant near origin. Now we apply analytic continuation to extend the observed data h by a rational function h_r from the interval [0,0.5] to [0,1], using the AAA algorithm <cit.> with degree r=4. This step is essential for dealing with missing data u_0 and f: subtracting h_r from h yields the reduced data h for a given g and u_0=f≡0, which is then used in recovering a. Fig. <ref> shows the L^2() error between h_r and the exact data h_0 which is obtained by solving (<ref>) with given g and vanishing u_0 and f. Note that higher order rational approximations can reduce the error over the interval [0,0.5], but it tends to lead to larger errors in the interval [0.5,1]. The approach is numerically sensitive to the presence of data noise, reflecting the well-known severe ill-posed nature of analytic continuation. Finally, we present recovery results for the piecewise constant coefficient a, or equivalently, the shape D. The exact value is 1 inside the inclusion D and 10 outside, unless otherwise stated. We use the standard gradient descent method to minimize problem (<ref>). Unless otherwise stated, we fix the step sizes ^k≡1, _1^k≡0, _2^k≡0, i.e., fixing the values inside and outside the inclusion D. The regularization parameter is chosen to be 10^-8, and the coefficients a_1 and a_2 are set to a_1=0.9 and a_2=10. The results are summarized in Figs. <ref>-<ref>, where dashed lines denote the recovered interfaces. Fig. <ref> shows the result for case (i), when the initial guesses are a small circle but with two different centers. In either case, the algorithm can successfully reconstruct the exact circle after 10000 iterations. For case (ii), the exact interface is a square, again with the initial guess being small circles inside the square, cf. Fig. <ref>. The algorithm accurately recovers the four edges of the square. However, due to the non-smoothness, the corners are much more challenging to reconstruct and hence less accurately resolved. These results indicate that the method does converge with a reasonable initial guess, but it may take many iterations to yield satisfactory reconstructions. Fig. <ref> shows the results for case (iii) for which the exact interface is a concave polygon, which is much more challenging to resolve. Nonetheless, the algorithm can still recover the overall shape of the interface. The reconstruction around the concave part has lower accuracy. To the best of our knowledge, the unique determination of a concave polygonal inclusion (in an elliptic equation) is still open. Fig. <ref> shows the results for case (iv) which contains two discs as the exact interface. The initial guess is two small discs near the boundary ∂Ω. Note that in this case, we choose the boundary data η≡1 in order to strengthen the effect of inhomogeneity. The final reconstruction is very satisfactory. Fig. <ref> shows a variant of case (ii), with the initial interface being two disjoint discs. It is observed that the two discs first merge into one concave contour, and then it evolves slowly to resolve the square. This shows one distinct feature of the level set method, i.e., it allows topological changes. Due to the complex evolution, the algorithm takes many more iterations to reach convergence (i.e., 30000 iterations versus 8000 iterations in case (ii)). Fig. <ref> shows a case which aims at simultaneously recovering the interface and the diffusivity value inside the inclusion, for which the exact interface is a square and the exact values of a_1 and a_2 are 1 and 10, respectively. In the experiment, we take two different initial guesses. The initial value of a_1 for both cases is a_1=1.2, and we take the step sizes ^k≡1, _1^k≡10 and _2^k≡0. The recovered value a_1 is 0.92 for the first row and 0.89 for the second row. It is observed that for both cases, one can roughly recover the interface. These experiments clearly indicate that the level set method can accurately recover the interface D. However, it generally takes many iterations to obtain satisfactory results. This is attributed partly to topological changes and the presence of nonsmooth points, and partly to the direct gradient flow formulation. Indeed, one observes from Proposition <ref> that the gradient field for updating the level set function is actually not very smooth, which hinders the rapid evolution of the interface. Hence, there is an imperative need to accelerate the method, especially via suitable preconditioning and post-processing <cit.>. Last, Fig. <ref> shows reconstruction results with noisy data. Due to the instability of analytic continuation for noisy data, we use boundary data corresponding to zero u_0, f as our measurement and only focus on reconstructing a. That is, we denote h^ the solution of problem (<ref>) with u_0≡0 and f≡0 which plays the role of h. The noisy measurement h^ is generated by h^(y,t)=h^(y,t)+ h^_L^∞(×[0,1])ξ(y,t), where >0 denotes the relative noise level, and ξ follows the standard Gaussian distribution. We take the exact interface as a concave polygon and the initial guess is a circle; see the left panel in Fig. <ref>. We consider two different noise levels and three different input boundary data. The first and second rows in Fig. <ref> are for 1% and 5% noise, obtained with a regularization parameter =1×10^-7 and =5×10^-7, respectively. We consider three input Neumann data g_1, g_2 and g_3: g_1=g (i.e., identical as before), and g_2 and g_3 are given by g_2(x,t) =η_1(x)χ_[0.25,1](t)+η_2(x)χ_[0.5,1](t)+η_3(x)χ_[0.75,1](t), g_3(x,t) =η_1(x)χ_[1/6,1](t)+η_2(x)χ_[2/6,1](t)+η_3(x)χ_[3/6,1](t)+η_4(x)χ_[4/6,1](t)+η_5(x)χ_[5/6,1](t), where η_n (n=1,…,5) is a cosine function with frequency 2nπ on each edge. The inputs g_2 and g_3 contain higher frequency information and are designed to examine the influence of boundary excitation on the reconstruction. Fig. <ref> shows that with the knowledge of h^, the method for recovering the interface is largely stable with respect to the presence of data noise. With more frequencies in the input excitation, the reconstruction results would improve slightly. This agrees with the observation that the concave shape contains more high-frequency information. § CONCLUDING REMARKS In this work have studied a challenging inverse problem of recovering multiple coefficients from one single boundary measurement, in a partially unknown medium, due to the formal under-determined nature of the problem. We have presented two uniqueness results, i.e., recovering the order and the piecewise constant diffusion coefficient from a fairly general Neumann input data and recovering the order and two distributed parameters from a fairly specialized Neumann input data (in the appendix). For the former, we have also developed a practical reconstruction algorithm based on asymptotic expansion, analytic continuation and level set method, which is inspired by the uniqueness proof, and have presented extensive numerical experiments to showcase the feasibility of the approach. There remain many important issues to be resolved. Numerically, the overall algorithmic pipeline works well for exact data. However, analytic continuation with rational functions is sensitive with respect to the presence of data noise. Thus it is of much interest to develop one-shot reconstruction algorithms. The main challenge lies in unknown medium properties, i.e., missing data, which precludes a direct application of many standard regularization techniques. It is of much interest to develop alternative approaches for problems with missing data. The level set method does give excellent reconstructions, but it may take many iterations to reach convergence. The acceleration of the method, e.g., via preconditioning, is imperative. Theoretically the specialized Neumann input is very powerful. However, the numerical realization is very challenging. It would also be interesting to develop alternative numerically feasible yet more informative excitations for recovering more general coefficients than polygonal inclusions. § RECOVERY OF TWO GENERAL COEFFICIENTS In this appendix, we discuss the unique recovery of general coefficients mentioned in Remark <ref>. In this setting, we have g∈ C^2(_+;H^1/2()) with support in _1×_+. Moreover, g is piecewise constant in time t and g≡0 when t≤ T_0, g is constant when t≥ T_1. The proof relies on the representation of the data h, similar to Corollary <ref> and hence we omit the proof. Note that g is a space-time dependent series. We may write h=h_i+∑_k=1^∞ h_b,k to distinguish the contributions from u_0 and f, and g (with g_k(t):=χ b_kψ_k(t)η_k) h_i(t) :=ρ_0+ρ_1t^+∑_n=2^∞ρ_n E_,1(-_n t^), h_b,k(t) :=∑_n=1^∞∑_j=1^m_n∫_0^t(t-s)^-1E_,(-_n(t-s)^)⟨ g_k(s),_n,j⟩ ṣ _n,j. For u_0,f∈ L^2() and η_k∈ H^1/2(), the data h=u\|_×(0,T) to problem (<ref>) can be represented by h(t)= ρ_0+ρ_1t^+∑_n=2^∞ρ_n E_,1(-_n t^) +∑_n=1^∞∑_j=1^m_n∫_0^t(t-s)^-1E_,(-_n(t-s)^)⟨ g(s),_n,j⟩ ṣ _n,j, with ρ_n defined in (<ref>). Moreover, the following statements hold. (i) h_i∈ C^(0,∞;L^2()) and h_b,k∈ C^(t_2k+,∞;L^2()) with an arbitrarily fixed >0. (ii) The Laplace transforms h_i(z) and h_b,k(z) of h_i and h_b,k in t exist and are given by h_i(z) = ρ_0z^-1+(1+)ρ_1z^-1-+∑_n=2^∞ρ_kz^-1z^+_k, h_b,k(z) =∑_n=1^∞∑_j=1^m_n⟨ g_k(z),_n,j⟩_n,jz^+_n. Now we can state the main result of this part. First, we uniquely determine the fractional order using the data near t=T_0, and then use the special boundary excitation g to determine the coefficients a and q. The proof of part (i) is identical with that for Theorem <ref>, and hence omitted. The unique determination of a and q is proved below. Let ,∈(0,1), (a,q,f,u_0),( a, q, f, u_0)∈ C^2() × L^∞()× L^2()× L^2() and fix g as (<ref>). Let h and h be the corresponding Dirichlet data, and let ∈(0,T_0] be fixed. (i) The condition h= h on _2×[T_0-,T_0] implies =, ρ_0=ρ_0 and {(ρ_k,_k)}_k∈={(ρ_k,_k)}_k∈, if ,∅. (ii) If either of following conditions is satisfied: (a) q= q and a- a=\| a- a\|=0 on the boundary or (b) a= a, then the condition h= h on _0×[T_0-,T] implies (a,q)=( a, q). In the proof of Theorem <ref>, we need the following two lemmas. The identity h= h on _2×[T_0-,T_1] implies h_k= h_k _2×[T_0-,∞), ∀ k∈, with h_k=h_i+h_b,k which solves problem (<ref>) with g=g_k. We prove the assertion by induction. When k=1, by the definition of ψ_k(t), we have ψ_k=0 in (0,t_3) for all k≥2. Then by Proposition <ref>, the condition h= h on _2×[T_0-,T_1] implies h_1= h_1 on _2×[T_0-,t_3), since [T_0-,t_3)⊂[T_0-,T_1]. By Proposition <ref>(i), h_1 and h_1 are L^2()-valued functions analytic in t∈(t_2+,∞), and hence h_1= h_1 for all t∈[T_0-,∞). This shows the case for k=1. Now assume that for some ℓ∈, the assertion (<ref>) holds for all k=1,…,ℓ. Since ψ_k=0 in (0,t_2ℓ+3), for k≥ℓ+2, we deduce ∑_k=1^ℓ+1h_k=h in _2×(0,t_2ℓ+3). Similarly, we have ∑_k=1^ℓ+1h_k=∑_k=1^ℓ+1 h_k _2×(0,t_2ℓ+3). From the induction hypothesis, we deduce h_ℓ+1= h_ℓ+1 on _2×[T_0-,t_2ℓ+3). Use analytic continuation again, we obtain h_ℓ+1= h_ℓ+1 on _2×[T_0-,∞). Thus, the assertion (<ref>) holds for all k∈. Given a nonempty open subset of , for any fixed n∈^, the eigenfunctions {_n,ℓ}_ℓ=1^m_n corresponding to _n are linearly independent on L^2(). Suppose that on the contrary: there are {c_j}_j=1^m_n⊂ℝ such that ∑_j=1^m_nc_j_n,j=0 on . Let =∑_j=1^m_nc_j_n,j. Then satisfies =_n, _ν=0, and =0. Then the regularity on a and q and unique continuation principle <cit.> imply ≡0 in . Since _n,j are linearly independent in L^2(), we obtain c_j=0, j=1,…,m_n, i.e. the desired linear independence. Now we can state the proof of Theorem <ref>(ii). By Lemma <ref>, we have h_k= h_k on _2×(T_0-,∞) for any k∈. Note that h_k=h_i+h_b,k solves problem (<ref>) with g replaced by g_k. We have the following representations h_i(t) =ρ_0+ρ_1t^+∑_n∈∩^ρ_n E_,1(-_n t^), h_b,k(t) =∑_n=1^∞∑_j=1^m_n∫_0^t(t-s)^-1E_,(-_n(t-s)^)⟨ g_k(s),_n,j⟩ ṣ _n,j. By the choice of g, the interval [0,T] can be divided into [0,T_0] and [T_0,T]. For t∈(0,T_0), g_k(t)≡0, Theorem <ref>(i) implies that {(ρ_ℓ,_ℓ)}_ℓ∈={(ρ_ℓ,_ℓ)}_ℓ∈ and =, and hence h_i(t)= h_i(t) for all t>0. For t∈[T_0,T], this and the condition h_k(t)= h_k(t) lead to h_b,k(t)= h_b,k(t) in L^2(_2). Thus, ∑_n=1^∞∑_j=1^m_n∫_t_1^t(t-s)^-1E_,(-_n(t-s)^)⟨ g_k(s),_n,j⟩_L^2(_1) ṣ _n,j = ∑_n=1^∞∑_j=1^ m_n∫_t_1^t(t-s)^-1E_,(-_n(t-s)^)⟨ g_k(s),_n,j⟩_L^2(_1) ṣ _n,j, t∈[t_1,∞). By Proposition <ref>(ii), applying Laplace transform on both sides yields ∑_n=1^∞∑_j=1^m_n⟨ g_k(z),_n,j⟩_n,jz^+_n=∑_n=1^∞∑_j=1^ m_n⟨ g_k(z),_n,j⟩_n,jz^+_n, ∀ (z)>0. Next, we repeat the argument of Theorems <ref> and <ref> to deduce _n=_n, ∀ n∈. To this end, let U_k∈Dom(A^1/4+) be the solution of the elliptic equation with a Neumann boundary data χ b_kη_k, for all in any compact subset of ∖{-_n,-_n}_n∈, we have ∑_n=1^∞∑_j=1^m_n⟨ g_k(η^1/),_n,j⟩_n,j+_n_ Dom(A^1/4+)^2 ≤ c∑_n=1^∞_n^1/2+2∑_j=1^m_n\|⟨χ b_kη_k,_n,j⟩+_n\|^2 = c∑_n=1^∞_n^1/2+2∑_j=1^m_n\|_n(U_k,_n,j)+_n\|^2≤ cU_k_ Dom(A^1/4+)^2<∞. Since each term of the series is a Dom(A^1/4+)-valued function analytic in and the series converges uniformly for in a compact subset set of ∖{-_n,-_n}_n∈, by the trace theorem, we deduce that both sides of (<ref>) are L^2()-valued functions analytic in ∈∖{-_n,-_n}_n∈. Assuming _j∉{_n}_n∈, by choosing a small circle centered at -_j and then using Cauchy integral formula, we obtain 2π√(-1)_j∑_j=1^m_n⟨ g_k,_n,j⟩_n,j(y)=0, ∀ k∈. This and Lemma <ref> (with =_2) imply ⟨ g_k,_n,j⟩=0, ∀ k∈, j=1,…,m_n. Since g_k=χ b_kψη_k, by the density of η_k in H^1/2(_1), we have _n,j=0 a.e. on _1, j=1,…,m_n. Since _ν_n,j=0, unique continuation principle <cit.> implies _n,j≡0 in , which is a contradiction. Hence, _j∈{_n}_n∈ for every j∈. Likewise, we can prove _j∈{_n}_n∈ for every j∈, and hence _n=_n, ∀ n∈^. It follows directly from (<ref>) that ∑_n=1^∞1η+_n(∑_j=1^m_n⟨ g_k(z),_n,j⟩_n,j(y)-∑_j=1^ m_n⟨ g_k(z),_n,j⟩_n,j(y))=0, y∈_2. Using Cauchy integral theorem again, we have ∑_j=1^m_n⟨ g_k(z),_n,j⟩_n,j(y)=∑_j=1^ m_n⟨ g_k(z),_n,j⟩_n,j(y), y∈_2, ∀ k,n∈. By the construction of g_k, it is equivalent to b_kψ_k(z)∫_χ(y')η_k(y')_n(y',y) ỵ' = b_kψ_k(z)∫_χ(y')η_k(y')_n(y',y) ỵ', ∀ n,k∈, (z)>0, with _n(y',y):=∑_j=1^m_n_n,j(y')_n,j(y). Since the set {η_k}_k∈ℕ is dense in H^1/2() and χ≡1 on _1', we deduce _n(y',y)=_n(y',y)∈ L^2(_1') × L^2(_2) for all n∈. From <cit.> (see also <cit.>), we deduce that m_n= m_n and after an orthogonal transformation _n,j(y)=_n,j(y), j=1,⋯,m_n, ∀ y∈, n∈. 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