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Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin PredictionZeyang Zhang*^*Equal contributions Computer Science and Technology Tsinghua UniverisityBeijing, China [email protected] Lin College of Medicine Southwest Jiaotong UniversitySichuan, China [email protected] Li* Computing and Artificial Intelligence Southwest Jiaotong UniversitySichuan, China [email protected] Xueling Zhu† Xiangya School of Medicine Central South UniversityHunan, China [email protected] Teng Computing and Artificial Intelligence Southwest Jiaotong UniversitySichuan, China [email protected] Xin Wang†^†Corresponding authors, Wenwu Zhu Computer Science and Technology Tsinghua UniverisityBeijing, China {xin_wang, wwzhu}@tsinghua.edu.cnJanuary 14, 2024 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Human albumin is essential for indicating the body's overall health. Accurately predicting plasma albumin levels and determining appropriate doses are urgent clinical challenges, particularly in critically ill patients, to maintain optimal blood levels. However, human albumin prediction is non-trivial that has to leverage the dynamics of biochemical markers as well as the experience of treating patients. Moreover, the problem of distribution shift is often encountered in real clinical data,which may lead to a decline in the model prediction performance and reduce the reliability of the model's application. In this paper, we propose a framework named Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction (), which is able to provide accurate albumin predictions for Intensity Care Unit (ICU) patients during hospitalization. We first model human albumin prediction as a dynamic graph regression problem to model the dynamics and patient relationship. Then, we propose a disentangled dynamic graph attention mechanism to capture and disentangle the patterns whose relationship to labels under distribution shifts is invariant and variant respectively. Last, we propose an invariant dynamic graph regression method to encourage the model to rely on invariant patterns to make predictions. Moreover, we propose a dataset named Albumin level testing and nutritional dosing data for Intensive Care (ANIC) for evaluation. Extensive experiments demonstrate the superiority of our method compared to several baseline methods in human albumin prediction. Human Albumin Prediction, Out-of-Distribution Generalization, Dynamic Graph Neural Network, Deep Learning, Medicine. § INTRODUCTIONHuman albumin plays a critical role in maintaining blood osmolarity, providing protection, facilitating transport, regulating functions, and combating inflammation. The levels of serum albumin are indicative of the body's overall health<cit.>. Furthermore, hypoalbuminemia can manifest as a symptom of nephrotic syndrome, which is characterized by kidney damage and partial protein loss<cit.>. Consequently, this condition results in elevated protein levels in the urine and decreased serum albumin levels. Therefore, accurately predicting plasma albumin levels and determining albumin doses become urgent clinical tasks to maintaining optimal blood levels in critically ill patients.In recent years, there have been many related works emerging to forecast the patients' biochemical marker concentrations, including albumin. For instance,Kaji <cit.> employ the long short-term memory network (LSTM) to predict clinical events, including sepsis, myocardial ischemia, and antibiotic vancomycin. Cheng <cit.> adopt a convolutional neural network (CNN) to construct a temporal matrix of medical codes from multiple patients, focusing on predicting congestive heart failure diseases. Choi <cit.> propose a gated recurrent unit (GRU) that predicts the timing of a patient's next visit based on diagnostic codes. These methods have demonstrated competitive results in capturing the dynamics of several biochemical markers.However, the existing methods commonly assume that the training and testing data are independently sampled from the same distribution, which may not hold in real-world data due to the uncontrollable distribution shifts between training and deployment.For example, patients from various regions or periods may differ greatly in lifestyle, environmental factors, dietary habits, etc., and thus have different dynamics of biochemical markers. The model trained on the patients from specific regions and periods may exploit variant patterns, whose relationship between the labels is variant under distribution shifts, to make predictions, and struggle to capture the specific factors influencing albumin levels when tested on patients from other regions or periods. In this case, the model will suffer severe performance deterioration from the out-of-distribution samples, and make unreliable albumin predictions. In this paper, we study the problem of human albumin prediction, which faces three critical challenges: 1) how to model the dynamics of the albumin levels for each patient, 2) how to leverage the similarity between patients to improve the prediction accuracy, 3) how to handle distribution shifts that naturally exist in the real-world data. To this end, we propose a framework named Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction (), which is able to provide accurate albumin predictions for the patients in the Intensity Care Unit (ICU) during hospitalization. Specifically, we first model the human albumin prediction problem as a dynamic graph regression problem to simultaneously consider the influences of the dynamic patient relationship and attributes on the albumin levels. Second, we propose a disentangled dynamic graph attention mechanism to capture the high-order graph dynamics and disentangle the patterns whose relationship to labels under distribution shifts is invariant and variant respectively.Last, we propose an invariant dynamic graph regression method that minimizes the variance of the model predictions under exposure to different variant patterns. This encourages the model to rely on the invariant patterns to make predictions and thus handle distribution shifts. Moreover, we propose a dataset named Albumin level testing and nutritional dosing data for Intensive Care (ANIC), which is real patient data collected from ICU that contains various features like demographic characteristics, nutritional support, biochemical markers, etc. Extensive experiments show that our method achieves state-of-the-art performance over several baseline methods in terms of human albumin prediction. To summarize, we make the following contributions:* We propose to study human albumin prediction with dynamic graph neural networks, to the best of our knowledge, for the first time. * We propose a framework named Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction (), which is able to provide accurate albumin predictions for ICU patients during hospitalization. * We propose a dataset named Albumin level testing and nutritional dosing data for Intensive Care (ANIC), which is real patient data collected from ICU that contains various dynamic features and structures. * We conduct extensive experiments to demonstrate the superior performance of our method compared to state-of-the-art baselines for human albumin predictions. § MATERIALSIn this section, we introduce a dataset named Albumin level testing and nutritional dosing data for Intensive Care (ANIC). §.§ Dataset Construction In this paper, we use the nutrition data of critically ill patients provided by a tertiary hospital in Sichuan Province, which contained basic information, laboratory index data, nutrition and medication use during hospitalization for 5727 ICU patients between 2014 and 2020, with 31106 records. Specifically, patient demographic information is used for admission registration and contains fields such as patient ID, gender, age, diagnosis, and record time. Nutritional support data records patients' medication administration records from admission to discharge (or death), including fields such as patient ID, medical advice, dosage, recording time, the nurse starts execution time and end execution time. The examination data reflects the patient's physical health status and provides a reference basis for subsequent clinical nutrition support, including patient ID, examination index and value. Moreover, during the construction of the dataset, we consider the time of patients' admission to the ICU and the medical orders they received as follows: * Initially, for patients with multiple admissions to the ICU, their medical records would contain several transfer records, leading to ambiguous time delineation. To address this issue, we opted to consider the date of the patient’s first admission to the ICU as the representative temporal information in the medical record. * Subsequently, concerning medical orders, there are instances where a patient's record indicates the use of different nutritional formulations within a single day—either singularly or in combination. This irregularity largely stems from some physicians' non-standard medical orders, resulting in prescribed daily nutritional dosages that deviate from conventional clinical practices. To remedy this, after consulting and agreeing upon standards with the attending physicians, we meticulously reviewed and rectified the medical orders documented in the dataset, ensuring accuracy and conformity to established clinical norms.§.§Dataset Preprocessing In this study, patients aged from 18 to 65 were selected based on the guidance of doctors, where we exclude children due to their incomplete growth and underdeveloped renal function, and elderly people due to their unique physiological characteristics in clinical practice. Additionally, patients who had albumin test values for 12 consecutive days were screened for inclusion in the study cohort. After consultation with clinicians, nine biochemical markers related to albumin were selected for analysis, and their descriptions and statistics are presented in Table <ref>. We also utilize categorical features including demographic characteristics and nutrition support, and their descriptions are shown in Table <ref>.§.§ Distributional shifts in the ANIC datasetWe analyze the ANIC dataset to illustrate the problem of distributional shift in real clinical data. Specifically, we screen all patient test data for 2015, 2019, and 2020 and use eleven days as a treatment cycle for patients to observe health status change. In addition, we select three detection indicators, namely `Indirect bilirubin', `MCH', `MCHC', and calculate the mean and standard deviation of the three indicators. The reasons for selecting these three indicators are as follows: Indirect bilirubin represents the unbound bilirubin present in the bloodstream. An elevation in its concentration could be attributed to the excessive lysis of red blood cells or liver dysfunction, leading to an aberration in bilirubin metabolism. This phenomenon could, in turn, impinge upon the synthesis of albumin, given that the liver serves as the primary site for albumin production. In addition, MCH and MCHC are indicative of the hemoglobin content within red blood cells. A reduction in these indices could signify the onset of anemia or malnutrition in patients, influencing the levels of albumin consequently.Hence, there exists a salient correlation amongst indirect bilirubin, MCH, MCHC, and albumin levels, collectively delineating the nutritional and organ functional status of patients. The interrelationship underscores their combined utility in offering a comprehensive insight into the physiological well-being of individuals, facilitating nuanced clinical evaluations and interventions.As shown in Figure <ref>, the distributions of indicators are constantly shifted as the year progresses. Compared with the detection data in 2015, the data distribution in 2019 and 2020 is closer, indicating that the degree of detection indicator shift is positively correlated with time. In addition, the trend of the detection data is different during six years, with a significant oscillation of the indicator curve in 2020. It suggests that the health status of most patients is unstable, which may be related to the patient's disease state, treatment regimen, lifestyle, or other factors. We could infer that the patient experienced a large physiological or pathological change in 2020. Notably, the COVID-19 pandemic in 2020 may have impacted patients' health status. Therefore, the apparent oscillation of the indicator curve in 2020 may be related to the pandemic, and patients may have been affected by the pandemic, e.g., increased risk of infection and strained healthcare resources, which led to the instability of the indicator changes.§ PROBLEM FORMULATIONIn this section, we formulate the problem of human albumin prediction. For a patient in Intensive Care Unit (ICU), our target is to predict the patient's albumin in the future during hospitalization, which can provide some assistance and guidance for subsequent treatments such as nutritional support. The simplest formulation is uni-variate time-series problem,,ŷ_n,t = f(y_n,1,y_n,2,…,y_n,t-1),where y_n,t∈ℝ is the albumin of the n-th patient at time t, f(·) is the prediction function to be learned, and ŷ_n,t is the predicted albumin. This formulation exploits the patient's albumin history to predict the albumin at the next time. However, other physiological factors are closely related to human albumin,, Bilirubin, Hb, etc. In comparison, multi-variate time-series problem is a more complex yet powerful formulation that can consider the influence of other factors on the patients' human albumin changes,,ŷ_n,t = f(𝐱_n,1,𝐱_n,2,…,𝐱_n,t-1),where 𝐱_n,t∈ℝ^d is the features of the n-th patient at time t, d denotes the dimensionality of the features, and features can include physiological characteristics, the nutritional support, etc. As the patients similar in physiological characteristics may have similar albumin dynamics, in this paper, we further consider the relationship between the patients to improve the accuracy of human albumin prediction, by formulating the dynamic graph prediction problem. Consider a graph 𝒢 with the node set 𝒱 and the edge set ℰ, where the nodes denote the patients, and the edges denote the relationship between the patients. A dynamic graph can be defined as𝒢=({𝒢_t}_t=1^T),where T is the number of time, 𝒢_t=(𝒱_t,ℰ_t) is the graph snapshot at time t,𝒱=⋃_t=1^T𝒱_t, ℰ=⋃_t=1^Tℰ_t. For simplicity, a graph slice is also denoted as 𝒢^t=(𝐗_t,𝐀_t), which includes node features and adjacency matrix at time t. The element in the adjacency matrix 𝐀_ij denotes the relationship between the i-th patient and the j-th patient. Then the prediction task is 𝐘̂_t = f(𝒢_1,𝒢_2,…,𝒢_t-1),where 𝐘_t∈ℝ^N is the albumin of N patients at time t. In classical machine learning literature, it is commonly assumed that the data is identically independently distributed, and the model trained with empirical risk minimization in training data is expected to generalize in testing data. However, in real-world applications, there usually exists a gap between the training and testing distributions,,p_train(𝐘_t, 𝐆_1,𝐆_2,…,𝐆_t-1) ≠ p_test(𝐘_t, 𝐆_1,𝐆_2,…,𝐆_t-1),which causes an out-of-distribution (OOD) generalization problem. For example, the distributions of biochemical markers and albumin dynamics change in different years, and the model trained with data before 2019 may have deteriorated prediction performance for the data after 2019. Therefore, it is critical to improve the out-of-distribution generalization abilities of the prediction models. For the OOD generalization literature <cit.>, we make the following assumption.For a given task, there exists a predictor f(·), for samples (𝒢_1:t,𝐘_t) from any distribution, there exists an invariant pattern P^I_t and a variant pattern P^V_tsuch that 𝐘_t=f(P_t^I)+ϵ and P_t^I = 𝒢_1:t\ P^V, i.e., 𝐘_t ⊥𝐏^V_t |𝐏^I_t. This assumption shows that there exist patterns,, a part of patients' evolving features and structures, whose relationship to the patients' albumins is invariant across distributions. To improve the OOD generalization ability, the model has to exploit the invariant patterns to make predictions.§ METHOD In this section, we introduce our Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction ().First, we propose a disentangled dynamic graph attention network to extract the invariant and variant patterns behind the patients' evolving features and relationships.Then we propose an invariant dynamic graph regression method to encourage the model to rely on invariant patterns whose relationship to albumins is invariant across distributions to make predictions. §.§ Disentangled Dynamic Graph Attention NetworkSince the features and the relationships of the patients are evolving through time, we first design a temporal graph attention mechanism to simultaneously aggregate structural and temporal information in the dynamic graph. Denote the neighborhood of the node u at time t as 𝒩_t^u = {v: (u,v)∈ℰ_t }. For a node u at time t and its neighbors v ∈𝒩_t'^u, ∀ t'≤ t, we calculate the Query-Key-Value vectors as𝐪_t(u) =𝐖_q(𝐡_t(u) + TE(t)), 𝐤_t'(v) =𝐖_k(𝐡_t'(v) + TE(t')), 𝐯_t'(v) =𝐖_v(𝐡_t'(v) + TE(t')),where 𝐡_t^u denotes the representation of node u at the time t, 𝐪, 𝐤, 𝐯 represents the query, key and value vector, respectively, and the bias term is omitted for brevity. Following <cit.>, we adopt a temporal encoding technique to consider the temporal information inherently,, TE(t) = [sin(ω_1 t),sin(ω_2 t), …, sin(ω_d t))],where d denotes the dimensionality of the node embeddings. Note that TE(·) can be viewed as a kernel method that can learn the relative temporal information through multiplication,, TE(t-t') = TE(t) ·TE(t'). In this way, the temporal information is naturally incorporated in the node embeddings and the attention mechanism.To capture the invariant and variant patterns, we then devise a disentangling mechanism to separate the evolving structural and featural patterns. Specifically, we first calculate the attention scores among nodes in the dynamic neighborhood to obtain the structural masks,𝐦^I =Softmax(𝐪·𝐤^⊤/√(d)), 𝐦^V=Softmax(-𝐪·𝐤^⊤/√(d)),where 𝐦^I and 𝐦^V represent the masks of invariant and variant structural patterns.This approach aims to assign higher attention scores to dynamic neighbors with invariant patterns and lower attention scores to those with variant patterns, resulting in a negative correlation between the two types of patterns. To capture invariant featural patterns, we introduce a learnable featural mask, denoted as 𝐦_f, which is obtained by applying a Softmax function to a learnable weight vector 𝐰_f. This featural mask allows us to selectively choose features from the messages of dynamic neighbors. By summarizing the messages of the dynamic neighborhood with the respective masks, our approach captures both invariant and variant patterns in a dynamic neighbor relationship,, 𝐳̃_t^I(u) = ∑_i 𝐦^I_i (𝐯_i⊙𝐦_f), 𝐳_t^I(u) =FFN(𝐳̃_t^I(u)+ 𝐡_t(u)), 𝐳̃_t^V(u) = ∑_i 𝐦^V_i𝐯_i, 𝐳_t^V(u) =FFN(𝐳̃_t^V(u)),where the FFN includes a layer normalization <cit.>, multi-layer perceptron and skip connection,, FFN(𝐱)=α·MLP(LayerNorm(𝐱))+(1-α) ·𝐱,where α is a learnable parameter. The pattern summarizations are then combined as hidden embeddings, which are subsequently fed into subsequent layers for further processing,,𝐡_t(u) ←𝐳_t^I(u)+𝐳_t^V(u). The proposed architecture consists of a stack of disentangled graph attention layers. This design allows each node to indirectly access dynamic neighborhoods of higher order, similar to traditional graph message-passing networks. At the l-th layer, the hidden representations 𝐳_t^I(u) and 𝐳_t^V(u) are obtained as summarizations of the invariant and variant patterns, respectively, within the l-order dynamic neighborhood. Following <cit.>, we extend the attention to multi-head attention to improve modeling ability and stability. §.§ Invariant Dynamic Graph RegressionThe data in real-world applications is complex that may consist of various patterns, where the variant patterns have variant relationship with labels, and if the model relies on these patterns, it will evitably have deteriorated prediction performance under distribution shift. Following OOD generalization literature <cit.>, we propose to reduce the variance of the model's prediction when the model is exposed to the same invariant pattern and different variant patterns. We first approximate the patterns 𝐏^t with summarized patterns 𝐳_t in the previous section.Since the hidden representations 𝐳_t^I(u) and 𝐳_t^V(u) serve as summarizations of the invariant and variant patterns for node u at time t, we aim to approximate the model's prediction abilities under exposure to various variant patterns. To achieve this, we adopt a sampling-based approach whereby we collect the variant patterns of all nodes at all times. We then randomly sample one variant pattern and use it to replace the variant patterns of other nodes across different time steps. For instance, we can replace the variant pattern of node u at time t_1 with the variant pattern of node v at time t_2 as follows:𝐳_t_1^I(u),𝐳_t_1^V(u)←𝐳_t_1^I(u),𝐳_t_2^V(v).By replacing the variant patterns of nodes in this manner, we can approximate the effects of exposure to different variant patterns on the overall dynamics of the graph. As the invariant patterns are assumed to sufficiently determine the labels and the influence of the variant patterns on the labels are shielded given the invariant patterns, the labels should not be changed. Therefore, to let the model focus on invariant patterns to make predictions, we introduce an invariance loss to minimize the variance of the model predictions when exposed to various variant patterns.Given the summarized invariant and variant patterns 𝐳_I, 𝐳_V, we calculate the task loss and mixed loss as ℒ_task=ℓ( f(𝐳^I),𝐲), ℒ_mix|𝐳^V =ℓ( g(𝐳^V,𝐳^I),𝐲),where f(·) is a MLP regression predictor, and g(𝐳^I,𝐳^V) = MLP(𝐳^I + sigmoid(𝐳^V)) makes predictions with variant patterns. The task loss encourages the model to utilize the invariant patterns, while the mixed loss measures the model's prediction ability when variant patterns are also exposed to the model. Then the invariance loss is calculated by ℒ_inv=1/|𝒮|∑_𝐳^V ∈𝒮(ℒ_mix|𝐳^V) + Var_𝐳^V ∈𝒮(ℒ_mix|𝐳^V),where 𝒮 is the set of variant patterns. The invariance loss measures the variance of the model's prediction under multiple intervened distributions. The final training objective ismin_θℒ_task +λℒ_inv,where the task loss ℒ_task is minimized to exploit invariant patterns, while the invariance loss ℒ_inv help the model to discover invariant and variant patterns, and λ is the hyperparameter to balance between two objectives. After training, we only adopt invariant patterns to make predictions in the inference stage. The overall algorithm is summarized in Algorithm <ref>. The framework is illustrated in Figure <ref>. § EXPERIMENTSIn this section, we conduct extensive experiments to verify the design of our framework.§.§ BaselinesWe adopt several representative statistical models, sequence neural network models, dynamic graph neural network models and out-of-distribution generalization methods. The first group of these methods is statistical models, including: * MA (Moving Average Model) <cit.> is a widely used statistical model in time series analysis. It assumes that the current value of a time series is a linear combination of the past error terms, , the moving average residuals. * ARMA (Autoregressive Moving Average Model) <cit.> is a time series forecasting model that combines the Auto-regressive model with the MA model. It assumes that the current value of a time series depends on its previous values and a linear combination of its past errors. * ARIMA (Autoregressive Integrated Moving Average Model) <cit.> is a time series forecasting model that combines the ARMA model with the concept of differencing. It assumes the current value of a time series depends on its previous values, a linear combination of its past errors, and the difference between its current and past values. The second group of these methods is sequence neural network models, including: * RNN <cit.> is a neural network designed for sequential data processing. It uses a feedback loop to process previous inputs and generate outputs. RNNs are widely used in natural language processing, speech recognition, and time series forecasting. * GRU <cit.> is a variant of RNN that uses gating mechanisms to control the flow of information. It has fewer parameters than the traditional RNN and is faster to train. * LSTM <cit.> is another variant of RNN that uses memory cells to store information for long periods. It can selectively remember or forget information and is effective in various applications such as language modeling and speech recognition. The third group of these methods is dynamic graph neural network models, including: * GRUGCN <cit.> is a neural network architecture that combines the GRU <cit.> and graph convolutional network (GCN) <cit.> modules. It is designed for graph-structured data and has been shown to achieve state-of-the-art performance on various dynamic graph tasks. * EGCN <cit.> is another neural network architecture designed for graph-structured data. It adopts an LSTM <cit.> or GRU <cit.> to flexibly evolve the GCN <cit.> parameters instead of directly learning the temporal node embeddings, which applies to frequent change of the node set on dynamic graphs. * DySAT <cit.> is a neural network architecture designed for dynamic graphs. It aggregates neighborhood information at each graph snapshot using structural attention and models network dynamics with temporal self-attention. In this way, the weights can be adaptively assigned for the messages from different neighbors in the aggregation, so that the complex network dynamics in the real-world datasets can be well captured. The last group of these methods is out-of-distribution (OOD) generalization methods, including: * IRM <cit.> is a general method designed to enforce invariance in machine learning models. It aims at learning an invariant predictor which minimizes the empirical risks for all training domains to achieve OOD generalization. * VREx <cit.> is an optimization method designed for training deep learning models. To improve the out-of-distribution generalization abilities of the model, it reduces the differences in the risks across training domains to decrease the model’s sensitivity to distributional shifts. * GroupDRO <cit.> is another optimization method designed to improve the robustness of deep learning models to distributional shifts. It puts more weight on training domains with larger errors when minimizing empirical risk to minimize worst-group risks across training domains. As the out-of-distribution generalization methods are general methods for machine learning models, and they are not specially designed for human albumin prediction, we adopt the best-performed baseline DySAT as their backbones. §.§ Experimental SetupsIn this section, we introduce the experimental setups, including data preprocessing, edge construction, and training and evaluation protocols.§.§.§ Data PreprocessingThe original data have various categorical features and continuous features with different scales. For example, in training data, the attribute MCH has values ranging from 24.9 to 34.9, while the values of the attribute ALT can range from 4.9 to 11272.2. To make the data suitable for the model, we first convert the category features into one-hot vectors and then normalize the continuous features by a standard scaler,, x' = x - μ/σ, where μ and σ are the mean and standard deviation of the feature values, respectively. Note that we only adopt the training data to calculate the mean and standard deviation of the features to avoid information leakage. §.§.§ Edge ConstructionPatients with similar physiological characteristics may have similar albumin curves, and the graph neural network (GNN) models can leverage this information to improve the albumin prediction accuracy. However, the original data only contain the evolving features of the patients during hospitalization, without considering the relationship among them. Therefore, we construct the relationship between patients by measuring the similarity of the patient physiological characteristics as follows. We first sort the data by the time of the data and then for each time we construct a graph with patients as nodes, and the relationship between patients as edges. We normalize the physiological characteristics of the patients by a standard scaler, calculate the L1 distance between the patients, and then adopt a K-nearest-neighbors (KNN) <cit.> algorithm to construct the edges for each node. We set K=100 in our experiments. In this way, a dynamic graph is constructed based on the physiological characteristics of the patients each day during hospitalization, which can capture the relationship between patients evolving through time.§.§.§ Training and EvaluationFor neural network models, we adopt the Adam optimizer <cit.> to train the models with a learning rate of 1e-2 and a weight decay of 5e-7. We train the models for a maximum epoch of 1000 and use the early stopping strategy to avoid overfitting. We set the patience to 50, , if the validation loss does not decrease for 50 epochs, we stop the training process. We set the hidden dimensionality for model parameters as 8 and the dimensionality for category attributes as 2. We use the mean squared error (MSE) as the loss function to train the models, , ℓ(ŷ,y) = (ŷ - y)^2, where ŷ is the prediction value and y is the label. We calculate the loss for each time and node and aggregate all the loss for updating the model parameters,, ℒ = 1/N1/T∑_t ∑_n ℓ(ŷ_n,t, y_n,t), where the subscript n denotes the patient index and the subscript t denotes the time index. For predicting the albumin of patient n at time t, the model only has the patient features before time t.We adopt Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) as metrics to evaluate the models. The model with the best MAE metric is adopted for evaluation. We run all the experiments with different seeds and initializations 3 times and report the average results and standard deviations. For a fair comparison, the training and evaluation protocols are kept the same for all methods. §.§ Experimental Results§.§.§ Main ResultsIn this experimental analysis, we compare the performance of different methods for predicting patient albumin. As shown in Table <ref>, we have the following observations:* Sequence neural network models outperform statistical models, due to their ability to capture the relationships between multiple variables and assist in albumin trend prediction.* Dynamic graph neural networks are effective in further leveraging the similarities between patients by message passing and information aggregation, strongly improving the albumin prediction accuracy. An exception is that GRUGCN and EGCN which both use GCN for structural modeling, may suffer from over-smoothing problems, as they can not assign different weights to different messages like DySAT does using attention mechanisms.* General OOD generalization methods improve the performance of the best-performed backbone DySAT across all categories, which highlights the existence of distribution shifts in real-world data and verifies the effectiveness of OOD generalization methods.* Our method achieves significantly better results in terms of average RMSE and MAE over all the baselines. This can credit to our method's capability to handle both the dynamic changes of features and structures and identify the invariant patterns to resist distribution shifts, enabling reliable performance even with more complex modeling. Figure <ref> demonstrates the MAE of different methods on different days, which also shows that our method has significantly better performance of albumin prediction on most days.§.§.§ ShowcasesWe showcase the albumin prediction of different methods inFigure <ref>. We can find that our method is able to accurately predict the patient albumin value as well as its trend the next day, which verifies the design of our method in the capability of exploiting the feature and structure dynamics in real-world data. §.§.§ Feature Importance AnalysesIn this section, we utilize our method to calculate and analyze the importance of various features in the data. Similar to <cit.>, we measure the feature importance by calculating the derivative of the loss with respect to the features, , Imp(x,t) = 1/N∑_n|∂ℒ_n,t/∂x|, where ℒ_n,t is the loss of the n-th patient at time t, and N is the number of patients. For categorical variables, we calculate the gradients of the loss with respect to its corresponding embeddings, and for a variable with multiple dimensions, we calculate the average importance of all the dimensions as the importance of this categorical feature. We consider the average length of stay in the intensive care unit, typically around 12 days<cit.>. Consequently, we select the monitoring data of a patient hospitalized for 12 consecutive days. Figure <ref> illustrates the visualization results, from which several key observations can be made. The four factors deemed important in the prediction of albumin levels align with common medical knowledge: ALB, EN(_PN)_ALB, age, and bilirubin. ALB and EN(_PN)_ALB plays a significant role in elevating blood albumin levels by directly supplementing patients with human albumin, either orally or intravenously. Additionally, as the body's metabolic capacity naturally declines with age, it affects albumin's synthesis and breakdown processes. Elevated bilirubin levels are linked to impaired liver function and biliary tract issues, which can impact the synthesis and stability of albumin, thereby influencing blood albumin levels<cit.>.The experimental results underscore the importance of ALB, EN(_PN)_ALB, age, and bilirubin in predicting albumin levels. These indicators reflect the body's nutritional status, liver function, and digestive and absorption capacity, which are crucial in maintaining normal albumin levels.§.§ Configurations We implement our method with PyTorch, and conduct the experiments with: * Operating System: Ubuntu 18.04.1 LTS* CPU: Intel(R) Xeon(R) Gold 6240R CPU @ 2.40GHz* GPU: NVIDIA GeForce RTX 3090 with 24 GB of memory* Software: Python 3.8.13, Cuda 11.7, PyTorch <cit.> 2.0.0, PyTorch Geometric <cit.> 2.0.3. § DISCUSSIONS §.§ Limitations of Our WorkWhile our research showed promising results in utilizing dynamic GNNs for albumin prediction, there are several limitations that should be acknowledged. Firstly, our study focuses primarily on the analysis of distribution shift, time series patterns, and complex patient relationships. However, there might be other relevant factors influencing albumin levels, such as genetic markers or medical interventions, which are not fully explored in this work. Incorporating additional features and contextual information could potentially enhance the predictive performance of our model. §.§ Future DirectionsThere are several directions for future research in the field of albumin prediction. Firstly, incorporating additional data sources, such as electronic health records or genetic information, could provide a more comprehensive understanding of the factors influencing albumin levels. This integration of diverse data modalities has the potential to improve the model's predictive accuracy<cit.>. Furthermore, investigating the interpretability of dynamic GNNs in the context of albumin prediction is crucial for gaining trust and acceptance from healthcare professionals. Collaborations with medical professionals and domain experts will be invaluable in guiding the development of albumin prediction models. By addressing these challenges and incorporating advancements from interdisciplinary fields, dynamic GNNs have the potential to revolutionize albumin prediction and contribute to personalized healthcare.§ RELATED WORKS §.§ Deep learning for Medicine Deep learning has made remarkable progress in medical image analysis<cit.>, particularly in medical image recognition, classification, and radiology report generation<cit.>. Traditional methods in this field rely on manually designed feature extraction and classification algorithms, which face challenges when dealing with complex structures and evolving patterns. Shin <cit.> employ a CNN-RNN architecture to identify diseases from chest X-ray images based on visual attributes. Nevertheless, medical report generation aims to produce comprehensive reports rather than isolated sentences. Li <cit.> employ a reinforcement learning approach to update neural networks by incorporating sentence-level and word-level incentives, while simultaneously ensuring high diagnostic accuracy in the generated reports. Chen <cit.> propose a memory-based transformer model to investigate the correlation between medical images.Similarly, deep learning plays a crucial role in clinical decision support by leveraging medical data, including patients' clinical characteristics and medical literature. Instance-based recommendation models<cit.> focus on the patient's health status for treatment advice, while longitudinal-based models<cit.> incorporate longitudinal patient history and capture time dependence in recommendations. Li <cit.> propose the KDGN model that utilizes a bipartite graph coding structure to mine potential therapeutic agents. Bhoi <cit.> work on personalized drug recommendations, using GNNs to control the level of drug-drug interactions.Furthermore, deep learning exhibits promising potential in drug repositioning<cit.>. Several computational models have been developed to identify new applications for existing drugs efficiently<cit.>. For example, Gottlieb <cit.> utilize the similarity between drugs and diseases to infer potential drug indications. Wang <cit.> introduce the PreDR model, which leverages heterogeneous information networks to analyze features of drugs. In summary, deep learning has emerged as a powerful tool for medical research and clinical practice.§.§ Dynamic Graph Neural Network Dynamic graph neural networks (DyGNNs) have been extensively studied to handle the complex structural and temporal information in dynamic graphs <cit.>. Some works first tackle the structures and then model the dynamics <cit.>, while some others first tackle the temporal information and adopt memory modules to model the structures <cit.>. DyGNNs have been applied in various real-world applications, such as event forecasting <cit.>,dynamic anomaly detection<cit.>, temporal knowledge graph completion <cit.>, etc. Some other works automate the GNN designs to adapt to various scenarios <cit.>. To the best of knowledge, this is the first work to solve human albumin prediction problems with dynamic graph neural networks.§.§ Out-of-Distribution GeneralizationIn many real-world scenarios, uncontrollable distribution shifts between the training and testing data distributions can occur and result in a significant drop in model performance. To overcome this problem, the Out-of-Distribution (OOD) generalization problem has emerged as a major research topic in various fields <cit.>. For the most related to our topic, some recent works have attempted to address distribution shifts on graphs <cit.>, while some others handle distribution shifts in time-series data <cit.>. However, the distribution shifts existing in medical data, especially with dynamic features and structures, are under-explored. In this paper, we study the problem of human albumin prediction under distribution shifts with evolving features and structures.§ ETHICAL CONSIDERATIONSIn our study of predicting albumin levels using ICU patients’ data, we prioritize ethical concerns surrounding data privacy and prediction accuracy. All patient data is anonymized, aligning with standards to ensure confidentiality. We are conscious of the ramifications of incorrect predictions, which can impact clinical decisions and patient outcomes. Rigorous validation mechanisms are embedded to enhance the model’s reliability. The ethical handling of data encompasses informed consent, ensuring patients are aware and agreeable to the use of their anonymized data.In sum, our ethical approach ensures data privacy, informed consent, and strives for prediction accuracy, ensuring the welfare of patients is at the core of our research, balancing innovation with ethical integrity.§ CONCLUSIONS In this paper, we proposed a framework named Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction () to address the challenges of predicting human albumin levels. We demonstrated the effectiveness of our framework in accurately predicting albumin levels for ICU patients during hospitalization using the proposed dataset ANIC. Through extensive experiments, we showed that our method outperformed state-of-the-art baselines, showcasing its superior performance in human albumin prediction. Our contributions include the novel application of dynamic graph neural networks for this prediction task, the development of the framework, the introduction of the ANIC dataset, and the validation of our method's superior performance. These findings highlight the potential of our framework to aid in clinical decision-making and improve patient care by providing accurate predictions of human albumin levels. § ACKNOWLEDGMENTThis work was supported in part by the National Key Research and Development Program of China No. 2020AAA0106300, National Natural Science Foundation of China (No. 62250008, 62222209, 62102222, 61936011), Beijing National Research Center for Information Science and Technology under Grant No. BNR2023RC01003, BNR2023TD03006, and Beijing Key Lab of Networked Multimedia. All opinions, findings, conclusions and recommendations in this paper are those of the authors and do not necessarily reflect the views of the funding agencies. IEEEtran | http://arxiv.org/abs/2311.15545v1 | {
"authors": [
"Zeyang Zhang",
"Xingwang Li",
"Fei Teng",
"Ning Lin",
"Xueling Zhu",
"Xin Wang",
"Wenwu Zhu"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.CE"
],
"primary_category": "cs.LG",
"published": "20231127052108",
"title": "Out-of-Distribution Generalized Dynamic Graph Neural Network for Human Albumin Prediction"
} |
alphaDynamical rigidity] Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 With an appendix by Jie Liu Jie Liu, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China [email protected] Feng Shao, Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro,Yuseong-gu, Daejeon, 34126, Republic of [email protected], [email protected] Guolei Zhong, Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro,Yuseong-gu, Daejeon, 34126, Republic of [email protected], [email protected] f X→ Y be a surjective morphism ofFano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle T_X is big.We show that f is an isomorphism unless Y is a projective space.As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.[2010]14J40, 14J45. [ Guolei Zhong January 14, 2024 ====================§ INTRODUCTIONWe work over the field ℂ of complex numbers.A classical question in algebraic and complex geometry asks for a description of a Fano manifold of Picard number 1 admitting a non-isomorphic surjective endomorphism.The following folklore conjecture of the 1980s predicts that the projective space is the only possibility for the existence of such a non-trivial endomorphism. This is also one of the starting points of this paper.Let X be a Fano manifold of Picard number 1.Suppose that X admits a non-isomorphic surjective endomorphism. Then X is a projective space. Conjecture <ref> has been verified in low dimension andmany other special cases: * Almost homogeneous spaces (<cit.>, <cit.>);* Smoothhypersurfaces of a projective space (<cit.>, <cit.>; cf. <cit.>);* Fano threefolds(<cit.>, <cit.>; cf. <cit.>);* Fano manifolds containing a rational curve with trivial normal bundle (<cit.>);* Fano fourfolds with Fano index ≥ 2 (<cit.>; cf. <cit.>); and * Del Pezzo manifolds, i.e., the Fano index =(X)-1 (<cit.>, <cit.>). In this paper, we study Conjecture <ref> for the case when the tangent bundle T_X is big. Recall that the tangent bundle T_X of a smooth projective variety X is called big (resp. pseudo-effective) if the tautological line bundle 𝒪_ℙ(T_X)(1) of the projectivized tangent bundle (in the sense of Grothendieck) is big (resp. pseudo-effective).After Mori's magnificent solution <cit.> to the Hartshorne conjecture, it has become apparent that a certain positivitycondition of the tangent bundle would imposerestrictive geometry on the underlying variety.In the spirit of this expectation, our first main result below gives a dynamical rigidity when the tangent bundle of the source is big and the variety of minimal rational tangents (VMRT for short) is not dual defective along a general point.Let X and Y be the Fano manifolds of Picard number 1.Suppose that the VMRT 𝒞_x⊆ℙ(Ω_X,x) (resp. 𝒞_y⊆ℙ(Ω_Y,y)) at a general point x∈ X (resp. y∈ Y) is not dual defective.Suppose further that the tangent bundle T_X is big. Then any surjective morphism X→ Y has to be an isomorphism unless Y is a projective space; in particular, X admits no non-isomorphic surjective endomorphism unless it is a projective space.Let Z⊆ℙ^N=ℙ(V) be a projective variety. The closure of the set of all tangent hyperplanes of Z is called the dual variety Ž⊆ℙ^N=ℙ(V^*), where V^* is the dual vector space of V. We say that Z is dual defective, if N-1-(Ž)>0 holds (cf. <cit.>).Indeed, a projective variety is dual defective only in very special cases and our non-dual defective assumption in Theorem<ref> is not very restrictive. There are various examples satisfying the assumption in Theorem <ref>; see<cit.> for a nice summary of the recent progress (cf. Section <ref>).In <cit.>, Hwang and Mok gave an affirmative answer to a question raised by Lazarsfeld (see <cit.>), asserting that any surjective morphism from a rational homogeneous space of Picard number 1 to a smooth projective variety which is different from the projective space, has to be an isomorphism.Inspired by their work and the fact that every rational homogeneous space has a big tangent bundle (cf. Example <ref>), as an extension of Lazarsfeld's question, it is natural to ask the following question. Let f X→ Y be a finite surjective morphism between Fano manifolds of Picard number 1. Suppose thatT_X is big and Y is not isomorphic to a projective space. Is f an isomorphism? Note that, in addition to rational homogeneous spaces,Question <ref> also has a positive answer if X≤ 3 by <cit.>, <cit.> and Theorem <ref>. In order to prove Theorem <ref>, we give the followingTheorem <ref>, which has its own independent interest.It provides a possible way to solve Conjecture <ref> when the total dual VMRT is a hypersurface in the projectivized tangent bundle.We refer the reader to <cit.> (cf. Section <ref>) for the precise definitions.Let f X→ Y be a finite morphism between Fano manifolds of Picard number 1. Let 𝒦 and 𝒢 be the dominating families of minimal rational curves on X and Y whose VMRTs along a general point are not dual defective.Suppose that Y is not isomorphic to a projective space and the induced rational map ℙ(T_X)ℙ(T_Y)sends the total dual VMRT of 𝒦 to the total dual VMRT of 𝒢. Then f is an isomorphism.In Theorem <ref>, the assumption that the induced rational map preserves the total dual VMRTs is satisfied if the tangent bundle T_X is big (see Proof of Theorem <ref>). As one may expect, there are several applications of our main results.In the aspect of positivity, Theorem <ref> gives a criterion to prove the non-bigness of the tangent bundles of certain Fano manifolds via finite non-trivial covers.In what follows, combining Theorem <ref> with other techniques, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds and Mukai manifolds with the results Theorems <ref>, <ref> and Proposition <ref> below. Let us begin with smooth complete intersections in a projective space.Let X be a non-linear smooth complete intersection of multi-degree d=(d_1,⋯,d_k) in a projective space. Then the tangent bundle T_X is big if and only if X is a quadric hypersurface.Moreover, suppose that X is very general in its deformation family.Then T_X is pseudo-effective if and only if d=(2) or d=(2,2).Recall that when X is a smooth hypersurface of degree d, the joint paper of Höring, Liu and the first author shows that the tangent bundle T_X is pseudo-effective if and only if d≤ 2 (see <cit.>); when X is a smooth complete intersection of two quadrics, <cit.> verifies that the tangent bundle T_X is ℚ-effective but not big; when X is a smooth Fano complete intersection of dimension at least 3 and of Fano index 1 or 2, <cit.> implies that T_X is not big.We refer the reader to Theorem <ref> written by Liu, which shows that a general smooth finite cover over a general complete intersection of two quadrics cannot have a pseudo-effective tangent bundle.Now we come to del Pezzo manifolds. A smooth Fano variety X of dimension n is called a del Pezzo manifold, if there exists an integral ample divisor H_X such that the anti-canonical divisor -K_X is linearly equivalent to (n-1)H_X; the self-intersection d(X) H_X^n is called the degree of X. Applying Theorem <ref>, we prove the non-bigness of del Pezzo manifolds of lower degree so as to obtain the following result. Let X be a del Pezzo manifold of degree d.Then the tangent bundle T_X is big if and only if d≥ 5.Note that if one of the following conditions holds: (1) (X)≤ 3; (2) d=3; or (3) d≥ 6, Theorem <ref> has been proved in<cit.> (cf. <cit.>); if d=5 and (X)≥ 4, it has been proved in <cit.>; if d=4, it has been proved in <cit.>. However, the pseudo-effectiveness of the tangent bundle in the case d≤ 2 is still unknown. In addition to del Pezzo manifolds, we also study the bigness of tangent bundle ofMukai manifolds, i.e., a smooth Fano variety X of dimension n whose anti-canonical divisor -K_X is linearly equivalent to (n-2)H_X for some integral ample divisor H_X; the genus g(X) is defined as 1/2H_X^n+1.It is known that if X is of Picard number 1, then 2≤ g(X)≤ 12 and g(X)≠ 11 hold.Let X be a Mukai manifold of Picard number 1 and of genus g(X).If g(X)≤ 5, or g(X)=6 and (X)≠ 6, then the tangent bundle T_X is not big.Recently, an interesting result <cit.> due to Kawakami and Totaro showed that if the Fano manifold X of Picard number 1 admits a non-isomorphic surjective endomorphism, then it satisfies the Bott vanishing; in particular, such X is locally rigid, i.e., H^1(X,T_X)=0. This fact and Theorem <ref>, together with previous results, permit us to give a positive answer to Conjecture <ref> for Mukai manifolds and smooth complete intersections in a projective space, as another application of Theorem <ref> from the dynamical viewpoints.Let X be a Mukai manifold of Picard number 1 or a non-linear smooth complete intersection of dimension ≥ 3. Then X admits no non-isomorphic surjective endomorphism.§.§.§ The authors would like to thank Jun-Muk Hwang for suggesting this project and for many inspiring discussions. The authors would also like to thank Jie Liu for many valuable discussions and suggestions, especially pointing out Example <ref> and providing the appendix to us, without which we could not be able to verifyTheorem <ref> for the pseudo-effectiveness of tangent bundles. Special thanks of the second author to IMS at NUS for the warm hospitality and kind support. Both authors were supported by the Institute for Basic Science (IBS-R032-D1-2023-a00).§ PROOFS OF THEOREMS <REF> AND <REF>In this section, after briefly reviewing the theory of minimal rational curves, we prove Theorems <ref> and <ref>.We refer to <cit.> for the standard notion and terminology.Let X be a smooth projective variety. Denote by ^n(X) the normalization of the open subset of (X) parametrizing integral rational curves.A dominating family of minimal rational curves 𝒦 is an irreducible component of ^n(X) such that the locus Locus(𝒦) of X swept out by curves from 𝒦 is dense in X and for a general point x∈Locus(𝒦) the closed subset 𝒦_x⊆𝒦 parametrizing curves through x is proper. It is known that any uniruled projective manifold carries a dominating family of minimal rational curves. A general element [ℓ]∈𝒦 is called a standard rational curve, i.e., there exists a non-negative integer p such that f^*T_X≅𝒪_ℙ^1(2)⊕𝒪_ℙ^1(1)^⊕ p⊕𝒪_ℙ^1^⊕((X)-p-1)where fℙ^1→ℓ is the normalization. Let x∈ X be a general point, and 𝒦_x^n the normalization of 𝒦_x, which is a finite union of smooth projective varieties of dimension p. There is a tangent map τ_x𝒦_x^nℙ(Ω_X,x) sending a curve that is smooth at x to its tangent direction at x. Let 𝒞_x be the image of τ_x, which is called the variety of minimal rational tangents (VMRT for short) at x associated to 𝒦. By <cit.> and <cit.>, the map τ_x is the normalization morphism of 𝒞_x and hence the dimension of 𝒞_x is equal to the dimension of 𝒦_xwhich is exactly p.For a standard rational curve [ℓ]∈𝒦, a minimal section ℓ of ℙ(T_X) over the curve ℓ is a section which is given by a surjection f^*T_X→𝒪_ℙ^1. If X is not a projective space, then such a minimal section ℓ always exists; clearly, we have 𝒪_ℙ(T_X)(1)·ℓ=0. The total dual VMRT of 𝒦 is defined as𝒞̌⋃_ [ℓ]∈𝒦ℓ^⊆ℙ(T_X)where the union is taken over all minimal sections over all standard rational curves in 𝒦. Note that 𝒞̌⊆ℙ(T_X) is an irreducible projective variety dominating X. Let 𝒞̌_x be the fibre of 𝒞̌→ X at x∈ X. It follows from <cit.> that 𝒞̌_x is the dual variety of 𝒞_x and hence the total dual VMRT 𝒞̌⊆ℙ(T_X) is a prime divisorif and only if for a general point x∈ X, 𝒞_x⊆ℙ(Ω_X,x) is not dual defective.We refer to <cit.>, <cit.> and <cit.> for more information involved.We stick to the notations and assumptions below in the remaining part of this section.Let f X→ Y be a non-isomorphic surjective morphism between Fano manifolds of Picard number 1.Let T_X and T_Y be the tangent bundles of X and Y respectively.Assume that Y is not isomorphic to a projective space.Then X is not isomorphic to a projective space, either; see <cit.>. * We consider the commutative diagram associated with the injection 0→ T_X→ f^*T_Y where Γ is the graph of the induced rational map ℙ(f^*T_Y)ℙ(T_X). Γ[dl]_β[dr]^αℙ(T_Y)[d]^ϕ ℙ(f^*T_Y)@–>[rr][d]^φ[l]_fℙ(T_X)[d]^τY X@=[rr][l]_fX * Denote by ξ (resp. η) the tautological line bundle of ℙ(T_X) (resp. ℙ(T_Y)). Let η be the pullback f^*η which is the tautological line bundle of ℙ(f^*T_Y).* Let 𝒦 (resp. 𝒢) be a dominating family of minimal rational curves on X (resp. on Y).Assume that both VMRTs along a general point are not dual defective. * Denote by D_X⊆ℙ(T_X) and D_Y⊆ℙ(T_Y) the total dual VMRTs of 𝒦 and 𝒢respectively.By our assumption, both D_X and D_Y are irreducible hypersurfaces.* Let H_X be the ample generator of the Picard group (X). First, we prove Theorem <ref>.For the convenience of the proof, we also recall the following notion in <cit.>.Let M be a complex manifold equipped with a closed holomorphic 2-form ω. For a point z∈ M, letNull_z(M){u∈ T_z(M) | ω(u,v)=0 for all v∈ T_z(M)}.This defines a distribution, called the null distribution on a Zariski open subset of M. If Null_z(M)=0 holds for every point z∈ M, then ω is a symplectic form and M is a symplectic manifold.Now let (M,ω) be a symplectic manifold equipped with a non-degenerate symplectic 2-form ω. Given an irreducible subvariety Z⊆ M with the smooth locus of Z denoted by Z_sm, we consider the restriction ω|_Z_sm. The rank of the null distribution of ω|_Z_sm is no more than the codimension codim_MZ and if the equality holds, then we say that Z is coisotropic. The null distribution on Z defines a foliation on a Zariski open subset of Z which we call the null foliation of ω on Z. We consider the following rational map Φ between the cotangent spaces T^*X and T^*Y:T^*XΦ T^*Y(s,t)↦ (f(s),(df_s^*)^-1(t))where s∈ X is a point outside the support of the ramification divisor of f, t∈ T_s ^*X is a cotangent vector and df_s^*:T^*_f(s)Y→ T^*_sX is the dual of the differential map.We note that for any point x∈ X, the cotangent space T_x^*X is the affine cone over ℙ(T_X,x).Let ω be the natural symplectic form on T^*Y.In the following, by abuse of notation, we identify D_X and D_Y with their affine cones.For a general point z∈ T^*X, dΦ_z is an isomorphism andΦ induces a 2-form Φ^*(ω) of T^*X defined by the following:Φ^*(ω)(u,v)ω(dΦ_z(u), dΦ_z(v)),where u,v∈ T_z(T^*X) and Φ^*(ω) is a symplectic form on a open subset of T^*X.Suppose that C is a general leaf of the null foliation of Φ^*(ω) on D_X.For a general z∈ C, we have Φ^*(ω)(u,v)=0 for arbitrary v∈ T_z C and u∈ T_z D_X. Consider the image Φ (C).By our assumption, the total dual VMRT D_X is mapped onto the total dual VMRT D_Y along the rational map Φ; hence Φ(C) is contained in D_Y. Given any v'∈ T_Φ(z)Φ(C) and any u'∈ T_Φ(z)D_Y, we obtain thatω(u',v')=Φ^*(ω)(dΦ_z^-1(u'), dΦ_z^-1(v'))=0,noting that dΦ_z^-1(u')∈ T_z D_X and dΦ_z^-1(v')∈ T_z C. Therefore, Φ(C) is a leaf of the null foliation of ω onD_Y.By <cit.>, both D_X and D_Y are coisotropic andthe closure of C and the closure of Φ(C)are minimal sections over minimal rational curves; moreover, a general minimal section of τ (resp. ϕ) can be realized as the closure of a leaf of the null foliation of Φ^*(ω) (resp. ω) on D_X (resp. D_Y). Letℳ_X⊆Chow(ℙ(T_X)) and ℳ_Y⊆Chow(ℙ(T_Y)) be the families of minimal sections of τ and ϕ, respectively. Then we have the following commutative diagramChow(ℙ(T_X))@–>[r][d]_τ_* Chow(ℙ(T_Y))[d]^ϕ_* Chow(X)@–>[r] Chow(Y)As ℳ_X is sent to ℳ_Y via the first horizontal map, we obtain the induced map 𝒦𝒢 via the second horizontal map which is also dominant.In particular, f maps a general minimal rational curve ℓ of 𝒦 to a general minimal rational curve ℓ' of 𝒢.Then for a general point x∈ X away from the ramification divisor, there exists a general element [ℓ]∈𝒦_x which is birational to its image ℓ' f(ℓ), noting that the normal bundle 𝒩_ℓ/X=𝒪(1)^⊕ p⊕𝒪^⊕((X)-1-p) cannot have sections vanishing along two distinct points.Therefore, from the normal bundle sequence, we obtain that K_X·ℓ=K_Y·ℓ'=K_Y· f_*(ℓ), a contradiction to the ampleness of the ramification divisor R=K_X-f^*K_Y, noting that Y is simply connected and X is of Picard number 1. Theorem <ref> is thus proved.Second, we prove Theorem <ref>.As ξ is big, it follows from <cit.> (cf. <cit.>) that D_X≡ aξ-bτ^*H_X holds where a>0 is the codegree of the VMRT 𝒞_x at a general point and b>0 is an integer; moreover, the total dual VMRT D_X (as a prime divisor in ℙ(T_X) by our assumption) is extremal in the pseudo-effective cone PE(ℙ(T_X))=ℝ_≥ 0[D_X]+ℝ_≥ 0[τ^*H_X].Here, as ℙ(T_X) is simply connected, the numerical equivalence of integral Cartier divisors is indeed a linear equivalence.Since D_X is covered by minimal sections ℓ⊆ℙ(T_X) of 𝒦 such that ξ·ℓ=0, we have D_X·ℓ<0. Let D_X'β_*(α_*^-1(D_X))⊆ℙ(f^*T_Y) be the proper transform of D_X along the birational map ℙ(f^*T_Y)ℙ(T_X). To apply Theorem <ref>, we are left to show the equality D_X'=f^*D_Y.By the natural injection 0→ T_X→ f^*T_Y, we have the induced injection0→ H^0(X,^aT_X⊗𝒪_X(-bH_X))→ H^0(X,^a(f^*T_Y)⊗𝒪_X(-bH_X)).As both α and β are birational with connected fibres, we have the induced injection0→ H^0(Γ,aα^*ξ-bα^*τ^*H_X)→ H^0(Γ,aβ^*η-bβ^*φ^*H_X).Due to the linear equivalence D_X∼ aξ-bτ^*H_X, there exists some m≥ 0 such that aη+φ^*(-bH_X)∼ D_X'+mφ^*H_X.Therefore, we have D_X'∼ aη-(b+m)φ^*H_X. Since both α and β are birational and D_X is dominant over X, it follows that D_X' is a prime divisor. As the total dual VMRT D_Y is covered by minimal sections c such that η·c=0, by the projection formula, its pullback f^*D_Y is covered by curves c' such that η·c'=0. In particular,c'· D_X'=c'· (aη-(b+m)φ^*H_X)<0,where we used the fact that b>0 due to the bigness ofT_X (cf. <cit.>).Therefore, c'⊆ D_X' and hence f^*D_Y⊆ D_X'. By the irreducibility of D_X', the desired equality is proved. So we finish the proof of Theorem <ref> by applying Theorem <ref>. § EXAMPLES AND PROOF OF PROPOSITION <REF>In this section, after collecting some typical examples of Theorem <ref>, we prove Proposition <ref>. [Rational homogeneous spaces of Picard number 1] By the generic finiteness of the Springer map, the tangent bundle of a rational homogeneous space is always big (cf. <cit.>, <cit.> and <cit.>).Besides, the VMRT of a rational homogeneous space of Picard number 1 can be fully determined by <cit.>.We refer the reader to <cit.> for a full classification of rational homogeneous spaces of Picard number 1 whose VMRTs are not dual defective. [Quintic del Pezzo fourfolds] Let X be a del Pezzo manifold of Picard number 1 and of dimension n. Let H_X be the ample generator of Pic(X) such that the degree d(X) H_X^n is 5. Then it is known that 3≤ n≤ 6 and the tangent bundle T_X is big (see <cit.> and <cit.>; cf. <cit.>). If n≤ 5, then the VMRT at a general point is not dual defective, and hence such X does not admit any non-isomorphic surjective endomorphism by applying Theorem <ref> (cf. <cit.>). If n=6, then X is homogeneous and hence there is no non-isomorphic surjective endomorphism, either (see <cit.>); however, in this case, the VMRT along a general point is ℙ^1×ℙ^2⊆ℙ^5 which is dual defective. The following is a typical example that is not almost homogeneous but has a big tangent bundle with the VMRT along a general point not dual defective. Let X be a smooth linear section of a 10-dimensional spinor variety of codimension 3. It is known that there are four isomorphism classes and the general one is not almost homogeneous (see <cit.>).On the other hand, the tangent bundle T_X is big and the VMRT along a general point is not dual defective; see <cit.>. By Theorem <ref>, such X does not admit any non-isomorphic surjective endomorphism.Suppose to the contrary that X admits a non-isomorphic surjective endomorphism. By <cit.>, such X satisfies Bott vanishing and then X is locally rigid. However, as proved in <cit.>, no smooth complete intersection in a given projective space is locally rigid except for the projective space and the hyperquadric, while the latter one admits no endomorphism by <cit.>.In the following, we assume that X is a Mukai manifold of Picard number 1, of dimension n, and of genus g(X).We refer the reader to <cit.> for the classification of Mukai manifolds. We may further assume that n≥ 5 (cf. <cit.>, <cit.> and <cit.>).By <cit.> and <cit.> (cf. <cit.>), we may assume that g(X)≥ 4 and hence the ample generator H_X of the Picard group Pic(X) is very ample.As X is locally rigid, by <cit.> and <cit.>, such X can only be one of the following: * a general codimension k (with k≤ 3)linear section of a 10-dimensional spinor variety;* a general hyperplane section of Gr(2,6);* a general hyperplane section of the symplectic Grassmannian SG(3,6).On the other hand, it follows from <cit.> that all of the above varieties except for Case (1) with k=3 are almost homogeneous, and hence they admit no non-isomorphic surjective endomorphisms (see <cit.>). The only case left when X is a smooth linear section of a 10-dimensional spinor variety of codimension 3 has been excluded by our Theorem <ref> and <cit.> (cf. Example <ref>). § BIGNESS OF TANGENT BUNDLES OF CERTAIN PROJECTIVE MANIFOLDSIn this section, using Theorem <ref> and developing some other techniques, we study the bigness of tangent bundles of smooth complete intersections, del Pezzo manifolds, and Mukai manifolds.§.§ Smooth complete intersections, Proof of Theorem <ref> Let f X→ Y be a generically finite surjective morphism between smooth projective varieties.If the tangent bundle T_Y is not big (resp. not pseudo-effective), then T_X is not big (resp. not pseudo-effective) either. Consider the surjective morphism ℙ(f^*T_Y)→ℙ(T_Y) induced by f.Then the tautological line bundles of ℙ(f^*T_Y) and ℙ(T_Y) have same Kodaira dimension by <cit.>.Hence, T_Y is big if and only if f^* T_Y is big.Since f is generically surjective and T_X is locally free, there is a natural injection 0→ T_X→ f^*T_Y, and thus the non-bigness of T_Y implies the non-bigness of T_X.Now suppose that T_Y is not pseudo-effective.Let η (resp. η) be the tautological line bundle of ℙ(T_Y) (resp. ℙ(f^*T_Y)) and let πℙ(T_Y)→ Y and πℙ(f^*T_Y)→ X be the natural projections.Let A be any ample divisor on Y. Then η+1/nπ^* A is not ℚ-effective for any sufficiently large integer n. Hence, η+1/nπ^*f^*A is not ℚ-effective for any sufficiently large integer n(cf. <cit.>).Applying <cit.> and the injection 0→ T_X→ f^*T_Y, we see that T_X is not pseudo-effective. Let X be a smooth non-linear Fano complete intersection of dimension ≥ 3.Then the VMRT is not dual defective along a general point. If the Fano index ≥ 2, then X is covered by lines; in particular, the VMRT is not dual defective by <cit.> and <cit.>. Suppose that the Fano index is 1.Then it is known that X is covered by conics; see <cit.>. In this case, the VMRT along a general point is 0-dimensional which is not dual defective, either. Without loss of generality, we may assume thatX⊆ℙ^n+k is a smooth complete intersection of dimension n≥ 2 and of multi-degree (d_1,⋯,d_k) with d_i≥ 2 for any 1≤ i≤ k. First, we consider the bigness of the tangent bundle. If T_X is big, then X is uniruled by Miyaoka’s genericsemipositivity theorem (cf. <cit.>).Since X is a complete intersection, X is Fano.If (X)=2, then the result follows from <cit.>.So in the following, we assume (X)≥ 3 and hence X has Picard number 1 by the Lefschetz theorem.If k=1, then X is a smooth hypersurface of degree d≥ 2; hence T_X is big if and only if d=2; see <cit.>. Let us further assume that k≥ 2. Consider the following projection map:ℙ^n+k\{p[0:⋯:0:1]} ℙ^n+k-1[x_0:⋯:x_n+k-1:x_n+k] ↦[x_0:⋯:x_n+k-1].Let Y⊆ℙ^n+k-1 be a smooth complete intersection of multi-degree (d_1,...,d_k-1). Take Y'⊆ℙ^n+k to be the projective cone over Y, in other words, Y' is defined by the closure of π^-1(Y). It is clear that the ideal sheaf ℐ_Y' of Y' is generated by the same polynomials as that of Y. By Bertini's theorem, one can take a sufficiently general degree d_k hypersurface H_k in ℙ^n+k such that the scheme-theoretic intersection X' Y'∩ H_k is a smooth complete intersection of multi-degree (d_1,⋯,d_k).By construction, X'→ Y is a finite cover of degree d_k, noting that X' does not pass through the singular vertex p.By Lemma <ref>, the VMRT ofX' and the VMRT of Y at general points are not dual defective. Applying Theorem <ref>, we obtain that the tangent bundle of X' is not big. AsX' and X are both smooth complete intersections of the same multi-degree, they can be deformed to each other.By <cit.>, the tangent bundle of X is not big. Second, we consider the pseudo-effectiveness of the tangent bundle. By Semicontinuity theorem <cit.> and <cit.>, it is sufficient to show that there exists a non-degenerate smooth complete intersection of multi-degree (d_1,d_2,..,d_k) such that its tangent bundle is not pseudo-effective.We may assume that d_1≥⋯≥ d_k. If k=1, then the result follows from <cit.>. Suppose that k≥ 2.Then by the first half of the proof, there exists a smooth complete intersection X of multi-degree (d_1,...,d_k) and a finite morphism from X to a smooth hypersurface Y of degree d_1.If d_1≥ 3.By <cit.> again, T_Y is not pseudo-effective, and hence T_X is not pseudo-effective either; see Lemma <ref>.If d_1=2, by our assumption, we may assume that k≥ 3 and there exists a special smooth complete intersection of k quadrics such that it admits a finite morphism onto a smooth complete intersection of two quadrics.Hence, in this case, the result follows from Lemma <ref> and Theorem <ref>.§.§ Del Pezzo manifolds, Proof of Theorem <ref> The following proposition is a bit technical and has its own independent interest especially when we consider the projective dual of a singular variety with its normalization being smooth (cf. <cit.>).We refer the reader to <cit.> for certain similar arguments under a stronger assumption of nodal singularities.Let M be a smooth quasi-projective variety, X⊆ M a smooth projective subvariety, and H an ample Cartier divisor on X. Suppose that the morphism t X→ Pℙ^N induced by the linear system |H| is birational to its image Y (in particular, X→ Y is the normalization).Suppose further that there is a smooth morphism t' M→ P which is a lift of t, i.e., t=t'|_X.If the twisted normal bundle N_X/M⊗𝒪_X(-H) is ample, then the dual variety Y^*⊆ P^* of Y⊆ P is a hypersurface.Let V=H^0(X,H) with (V)=N+1. First, we have the following natural Euler sequence 0→𝒪_P(-1)→ V→ T_P(-1)→ 0.Pulling it back to X, we obtain the surjectionV→ t^*T_P(-H)→ 0.Second, we have the following natural short exact sequence of coherent sheaves0→ T_X(-H)→ t^*T_P(-H)→𝒩(-H)→ 0,where 𝒩(-H) is not necessarily locally free as the normalization map t is not necessarily unramified.Hence, the composite surjectionV↠𝒩(-H)gives rise to an injection Wℙ_X(𝒩(-H))↪ X×ℙ^N^*.Note that here W is merely a projective scheme, possibly very singular.On the other hand, as t' M→ P is smooth, we have the surjective map T_M→ t'^*T_P. Restricting the map to X, we obtain the following commutative diagram of short exact sequences0[r] T_X(-H)[r]@=[d] T_M|_X(-H)@->>[d][r] N_X/M(-H)[r]@->>[d] 00[r] T_X(-H)[r] t^*T_P(-H)[r] 𝒩(-H)[r] 0where the first horizontal sequence is the normal bundle sequence for the smooth projective variety X. Now that the middle vertical one is a surjection, we obtain that N_X/M(-H)→𝒩(-H) is also a surjection. By the ampleness of N_X/M(-H), we see that the coherent sheaf 𝒩(-H) is ample in the sense that 𝒪_W(1) is ample. Let us consider the following commutative diagramX×ℙ^N^*[r] Y×ℙ^N^*[r] ℙ^N^*W[r]@^(->[u][d] W'[r]@^(->[u][d] Y^*@^(->[u]X[r] Ywhere W' and Y^* are the images of W along the composite maps W→ X×ℙ^N^*→ Y×ℙ^N^* and W→ X×ℙ^N^*→ℙ^N^*. By definition, W' is the conormal variety of Y which is the closure of ℙ_U(𝒩_U/P) in Y×ℙ^N^* where U is the smooth locus of Y, and Y^* is the dual variety of Y. Now that 𝒪_W(1) (which coincides with the pullback of 𝒪_ℙ^N^*(1)) is ample, we see that W→ Y^* is a finite (and thus surjective) morphism; in particular, the equality (Y^*)= (W)=(X)+rank(𝒩(-H))-1=(P)-1=N-1concludes our proposition. Let X be a del Pezzo manifold of dimension n such that -K_X∼ (n-1)H_X where H_X is an integral ample divisor.We may assume that n≥ 3 in view of<cit.>.It is known that 1≤ d≤ 8. Ifd≥ 6, then T_X is big (see <cit.>).If d=5, then it follows from <cit.> and <cit.> that T_X is big.If d=3, by <cit.>, T_X is not pseudo-effective. Suppose that d=4.Then X is a smooth complete intersection of two hyperquadrics. By <cit.>, T_X is ℚ-effective but not big; alternatively, applying Theorem <ref>, we have T_X is not big. So we are left to consider d=1 or 2.0.5pc The case d=2.Then X is a double cover of ℙ^n branched along a smooth hypersurface of degree 4.Its VMRT 𝒞_x⊂ℙ(Ω_X,x) at a general point x∈ X is a smooth complete intersection of multi-degree (3,4) by <cit.> and hence not dual defective by <cit.>. Thanks to <cit.> and the Semicontinuity theorem, we only need to show a special one in the deformation family of del Pezzo manifolds of degree 2 having the non-big tangent bundle.Consider the del Pezzo manifold X of degree 2 defined by the following equation in the weighted projective space ℙ(2,1,...,1)x_0^2+x_1^4+...+x_n+1^4=0.There is a finite morphism from X to the hyperquadric Q^n defined by the followingX ⟶ Q^n=(x_0^2+x_1^2+...+x_n+1^2=0)⊂ℙ^n+1(t_0:t_1:...:t_n+1) ↦ (t_0:t_1^2:...:t_n+1^2).Hence T_X is not big by Theorem <ref> and Lemma <ref>.0.5pc The case d=1. We apply Proposition <ref> to show that the VMRT at a general point of X is not dual defective.By <cit.>, the tangent map τ_x (which is a normalization morphism due to <cit.> and <cit.>) along a general point sends a smooth complete intersection 𝒦_x of multi-degree (4,5,6) in the weighted projective space ℙℙ(2,1^n) to a codimension two subvariety 𝒞_x in ℙ^n-1 such that τ_x is the restriction of the natural projection ℙ(2,1^⊕ n)\{[1:0:⋯:0]}→ℙ^n-1.Since the twisted normal bundle N_𝒦_x/ℙ(-1) of the smooth complete intersection of multi-degree (4,5,6) is isomorphic to 𝒪_𝒦_x(3)⊕𝒪_𝒦_x(4)⊕𝒪_𝒦_x(5) which is ample, the VMRT 𝒞_x is not dual defective by applying Proposition <ref>.Using <cit.>, we only need to show a special one in the deformation family of del Pezzo manifolds of degree 1 having the non-big tangent bundle. In fact, in <cit.>, the assumption of smoothness of the VMRT is not needed if the VMRT of any element in the deformation family at a general point is not dual defective.Consider the del Pezzo manifold X of degree 1 defined by the following equation in the weighted projective space ℙ(3,2,1,...,1)x_0^2+x_1^3+x_2^6+...+x_n+1^6=0.Let Y be the smooth Fermat hypersurface of degree 6 in the projective space ℙ^n+1 defined byx_0^6+x_1^6+x_2^6+...+x_n+1^6=0.Then there is a finite morphism g Y→ X sending the point (t_0:t_1:t_2:...:t_n+1) to the point (t_0^3:t_1^2:t_2:...:t_n+1). Consider the natural injection 0→ g^*Ω_X→Ω_Y.Taking the (n-1)-th exterior power of the above exact sequence and then tensoringwith (n-1)H_Y, we get an injection:0→ g^*T_X→ T_Y(3).Noting that the pushforward g_*𝒪_Y has a direct summand 𝒪_X, we haveH^0(X,^r T_X)↪ H^0(Y,g^*(^r T_X))↪ H^0(Y,^r (T_Y(3)))∀ r≥ 1.Observe that H^0(Y,^r (T_Y(3)))=0 for r≥ 1 (see <cit.>).This implies that H^0(X,^r T_X)=0 for any r≥ 1; hence, T_X is not big. §.§ Mukai manifolds, Proof of Proposition <ref> Before proving Proposition <ref>, we first consider Gushel-Mukai manifolds. We note that for a Gushel-Mukai sixfold, it admits a double cover over the Grassmannian Gr(2,5), the VMRT of which along a general point is self-dual and hence dual defective (cf. Example <ref>); in particular, we could not apply our Theorem <ref>. Let X be a Gushel-Mukai manifold, i.e., a Mukai manifold of genus 6.If (X)≠ 6, then the tangent bundle T_X is not big. Let Gr(2,5) be the Grassmannian of 2-dimensional subspaces of a complex vector space of dimension 5, viewed in ℙ(∧^2ℂ^5)≅ℙ^9 via the Plücker embedding. Let CGr(2,5)⊆ℙ(ℂ⊕∧^2ℂ^5)≅ℙ^10 be the cone over Gr(2,5) of vertex νℙ(ℂ). A smooth Gushel-Mukai (GM for short)variety of dimension 3≤ n≤ 6 is a smooth dimensionally transverse intersectionX=CGr(2,5)∩ℙ(W)∩ Q,where W is a vector space of dimension n+5 of ℂ⊕∧^2ℂ^5 and Q is a quadric hypersurface in ℙ(W)≅ℙ^n+4. Smooth GM varieties are isomorphic to either quadric sections of a linear section of Gr(2,5)⊆ℙ^9 (called ordinary GM varieties) when 3≤ n≤ 5, or to double covers of a linear section of Gr(2,5) branched along a quadric section (called special GM varieties).We note that such X is a Fano manifold of Picard number 1 and of Fano index n-2. By our assumption, n≠ 6.So we may pick an ordinary GM manifold X;then its VMRT is the intersection of the VMRT of the linear section of Gr(2,5) and the VMRT of the quadric hypersurface, which is smooth and not dual defective (cf. <cit.> and <cit.>).In particular, it follows from <cit.> that the VMRT of a special GM manifold is not dual defective, either. By Theorem <ref> and <cit.>, the tangent bundle T_X is not big. By <cit.>, we may assume that the dimension of X is at least 4. Recall that g(X)≥ 2. We refer the reader to <cit.> for a full classification of Mukai manifolds.If g(X)=6, then X is a Gushel-Mukai manifold and the result follows from Proposition <ref>.If g(X)=5 (resp. g(X)=4), then X is a smooth complete intersection of three quadrics (resp. smooth complete intersection of a cubic and a quadric), which has a non-big tangent bundle by Theorem <ref>.0.5pc The case g(X)=3.Then X is either a smooth quartic hypersurface in a projective space or a double cover of a quadric Q^n branch along the intersection of Q^n with a quartic hypersurface.In the former case,T_X is not big by<cit.>. So we may assume that X is a double cover over a quadric of dimension ≥4. By <cit.>, the VMRT along a general point of X is smooth and there is a special one whose VMRT along a general point is a smooth complete intersection which is not dual defective. By Theorem <ref>, the tangent bundle of this special one is not big; hence our result follows from <cit.>.0.5pc The case g(X)=2.Then X is a smooth hypersurface of degree 6 in the weighted projective space ℙ(3,1,...,1).Let [y:x_0:⋯:x_n] be the coordinates of the weighted projective space. After a suitable coordinate base change, we may assume that X is defined by the equation y^2+F_6(x_0,⋯,x_n)=0 where F_6 is a degree 6 homogeneous polynomial. Let X' be the hypersurface of degree 6 in the projective space ℙ^n+1 defined by Y^6+F_6(y_0,⋯,y_n)=0, where [Y:y_0:⋯:y_n] are the coordinates of ℙ^n+1.Then there is a finite morphism g X'→ X sending the point [Y:y_0:⋯:y_n] to the point [Y^3:y_0:⋯:y_n].Clearly, X' is also smooth.Consider the natural injection 0→ g^*Ω_X→Ω_X'.Taking the (n-1)-th exterior power of the above exact sequence and then tensoringwith (n-2)H_X, we get an injection:0→ g^*T_X→ T_X'(2).With the same argument as in the proof of Theorem <ref> for the case of del Pezzo manifolds of degree 2, we see that T_X is not big (and even not pseudo-effective).Indeed, a general codimension k (with k≤ 3)linear section H_k of a 10-dimensional spinor variety 𝕊_5 does not satisfy Bott vanishing.By <cit.>, this gives an alternative proof that those H_k admit no non-isomorphic surjective endomorphisms (cf. Example <ref>).However,our main result Theorem <ref> cannot be deduced from this fact.For each k≤ 8 (with H_0𝕊_5) and for any m∈ℤ, we have the following sequences0→ T_H_k(m-1)→ T_H_k(m)→ T_H_k(m)|_H_k+1→ 00→ T_H_k+1(m)→ T_H_k(m)|_H_k+1→𝒪_H_k+1(m+1)→ 0.Therefore, we obtain that (cf. <cit.>)χ(H_k+1, T_H_k+1(m))=χ(H_k,T_H_k(m))-χ(H_k,T_H_k(m-1))-χ(H_k+1,𝒪_H_k+1(m+1)).Hence, we have χ(H_1, T_H_1(-1))=χ (𝕊_5,T_𝕊_5(-1))-χ (𝕊_5,T_𝕊_5(-2))-1,χ(H_2, T_H_2(-1))=χ (𝕊_5,T_𝕊_5(-1))-2χ (𝕊_5,T_𝕊_5(-2))+χ (𝕊_5,T_𝕊_5(-3))-2,χ(H_3, T_H_3(-1))=χ (𝕊_5,T_𝕊_5(-1))-3χ (𝕊_5,T_𝕊_5(-2))+3χ (𝕊_5,T_𝕊_5(-3))-χ (𝕊_5,T_𝕊_5(-4))-3. By <cit.>, χ (𝕊_5,T_𝕊_5(-p))=0 for 1≤ p≤ 7. In particular, we obtain the non-vanishing χ(H_k,T_H_k(-1))≠ 0 for k≤ 3, and thus H_k does not satisfy Bott vanishing, noting that T_H_k(-1)=Ω_X^9-k((7-k)H). § PSEUDO-EFFECTIVENESS OF TANGENT BUNDLES OF COMPLETE INTERSECTIONS The whole appendix is devoted to the proof of the following theorem.Let X be a smooth complete intersection of multi-degree (d_1,⋯,d_k). Suppose that X is very general in its deformation family.Then T_X is pseudo-effective if and only if X is of the form (2) or (2,2).Recall that when X is of type (2,2), T_X is ℚ-effective by the joint work of the first author (see <cit.>).We first disprove the pseudo-effectivity of the tangent bundle of a very general smooth complete intersection of three quadrics.Let X be an n-dimensional very general smooth intersection of three quadric hypersurfaces in ℙ^n+3. Then T_X is not pseudo-effective. By Semicontinuity theorem <cit.> and <cit.>, it is sufficient to show that there exists a smooth complete intersection of three quadrics such that its tangent bundle is not pseudo-effective. It is clear that there exists a smooth complete intersection X of three quadrics which admits a double cover f X→ Y over a smooth complete intersection of two quadrics Y with the ramification divisor R being a linear section such that we have the following short exact sequence0→ T_X→ f^*T_Y→𝒪_R(R)→ 0.We consider the following commutative diagram induced by the elementary transformation of the above short exact sequence (see <cit.>).W[dr]^β[dl]_αV[rr]^q_2[ll]_q_1Γ[dr]^p_2[dl]_p_1ℙ(T_X)[d]_φℙ(f^*T_Y)[d]^ψ@–>[ll]_Ψ[rr]^fℙ(T_Y)@–>[rr]^Φ[d]^τℙ^n-1X@=[rr]X[rr]_fYLet η (resp. η, ξ) be the tautological line bundles of ℙ(T_Y), ℙ(f^*T_Y) and ℙ(T_X) respectively.Let H be the hyperplane section of ℙ^n-1, H_Y and H_Xthe ample generators of Y and X respectively.Let Rℙ(𝒪_R(R)). In the above commutative diagram, Φ is the Iitaka fibration of ξ, Γ is its resolution of indeterminacy Z of Φ, β is blow-up along R, α is the blow-up along ℙ(T_R) and V W×_ℙ(T_Y)Γ the fibre product. By the argument in <cit.>, Φ is induced by the Lagrangian fibration.In particular, we can writep_1^*(2η)=p_2^*H+E_Γwhere E_Γ is a p_1-exceptional effective (not necessarily irreducible) divisor with p_1(E_Γ)=Z; more precisely, E_Γ is the fixed locus of p_1^*|2η|=|2p_1^*η| by noting that p_1 is birational.Also, W is the resolution of the indeterminacy of the rational map Ψ; more precisely, β is the blow-up of R with the exceptional divisor E_W being an ℙ^n-1-bundle over R such thatα^*ξ=β^*η-E_W. To show that ξ is not pseudo-effective, we only need to show that D -D_1+D_2=-q_1^*E_W+q_2^*(1/2E_Γ+1/2p_2^*H)=q_1^*(β^*η-E_W)is not pseudo-effective, where D_1=-q_1^*E_W, D_2=q_2^*(1/2E_Γ+1/2p_2^*H)Let A_W be a sufficiently ample divisor on W and we shall show that the following intersection numberq_1^*A_W^n-1· (q_2^*p_2^*H)^n-1· Dis negative so that D is not pseudo-effective.The negative intersection follows from Claims <ref> and <ref>. For any k≤ n-1, we have(q_1)_*q_2^*(E_Γ· p_2^*H^k)=0.Without loss of generality, we may assume that E_Γ· p_2^*H^k is a non-zero effective cycle for any k≤ n-1, in other words, we may assume that E_Γ dominates ℙ^n-1; otherwise, we already have E_Γ· p_2^*H^m=0 for some m≤ k and then we can reduce to a lower dimensional case by the hyperplane cutting.Let S_k p_2^*H^k be the pull-back of a sufficiently general complete intersection of hyperplanes. Then we have _ℙ(T_X)(p_1(E_Γ∩ S_k))≥_ℙ(T_X)(Z∩ p_1(S_k))≥ k+2>k+1=_Γ(E_Γ∩ S_k)and hence (p_1)_*(E_Γ· S_k)=0 as the push-forward of cycles, noting that _ℙ(T_X)Z≥ 2 is due to <cit.>. Indeed, the restriction p_1|_S_k S_k→ p_1(S_k) is the resolution of the indeterminacy set of Φ|_p_1(S_k) and p_2|_S_k is the Iitaka fibration of (p_1^*η)|_S_k (see <cit.>).Therefore, to prove our claim, we only need to show that (q_1(q_2^*(E_Γ· S_k)))<(q_2^*(E_Γ· S_k))=2n-2-k.Suppose to the contrary. As f is a finite morphism, by the commutative diagram,q_1(q_2^*(E_Γ· S_k)) has to be contracted via β. However, according to the choice of X, the ramification R cannot be contained in f^-1(Z) and also f^-1(Z) cannot be contained in R. More precisely, we have (Z)=(f^-1(Z))=(β^-1f^-1(Z)). In particular, β cannot contract q_1(q_2^*(E_Γ· S_k)) to a lower dimensional subset which gives rise to a contradiction. So our claim is thus proved. We have D_2· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1=0.This claim follows easily from the calculation below.D_2· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1 =q_2^*(1/2E_Γ+1/2p_2^*H)· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1=1/2q_2^*E_Γ· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1=0where the last second equality is due to H^n=0 while the last equality is due to Claim <ref>.We have D_1· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1>0.Proof of Claim <ref> and End of Proof of Proposition <ref>. By Claim <ref> and the commutative diagram, we have the following calculationq_1^*E_W· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1 =q_1^*E_W· q_1^*A_W^n-1· q_2^*(2p_1^*η-E_Γ)· (q_2^*p_2^*H)^n-2=q_1^*E_W· q_1^*A_W^n-1· (2q_1^*β^*η)· (q_2^*p_2^*H)^n-2=⋯=q_1^*E_W· q_1^*A_W^n-1· (2q_1^*β^*η)^n-1=2^n-1E_W· A_W^n-1·β^*η^n-1.As A_W is a sufficiently ample divisor on W and E_W is a ℙ^n-1-bundle over R, it follows thatE_W· A_W^n-1 is a multi-section over R. By the projection formula, there exists some m∈ℕ such that q_1^*E_W· q_1^*A_W^n-1· (q_2^*p_2^*H)^n-1 =2^n-1E_W· A_W^n-1·β^*η^n-1=2^n-1mR·η^n-1=2^n-1m· (η|_R)^n-1=2^n-1m· R^n>0.Our claim is thus proved.§ FINITE COVERING OF INTERSECTIONS OF TWO QUADRICS By Jie LiuThe aim of this appendix is to prove the following result: Let f Y→ X be a general smooth finite cover of a general complete intersection X of two quadrics in ℙ^n+2 with n≥ 2. Then T_Y is not pseudo-effective.Here by a general finite cover we mean that there exists an irreducible component B of the branch locus of f such that B is a general smooth element in the complete linear system |𝒪_X(d)| for some d≥ 1, where 𝒪_X(1)≅𝒪_ℙ^n+2(1)|_X. §.§ Intersection of two quadrics Let X be a smooth complete intersection of two quadrics in ℙ^n+2 with n≥ 2. In an appropriate system of coordinates (x_0,…, x_n+2), the variety X is defined by the two equations q_1=q_2=0, where q_1=∑ x_i^2, q_2=∑μ_i x_i^2 with μ_i distinct. Let ζ_X be the tautological divisor of ℙ(T_X). By <cit.>, the complete linear system |2ζ_X| is defined by a dominant rational mapφℙ(T_X)ℙ^n-1such that 2ζ_X∼φ^*𝒪(1).Denote by Z the base locus of |2ζ_X|. Let B∈ |𝒪_X(d)| be a general smooth hypersurface of degree d≥ 1 and let s B→ℙ(T_X) be the section corresponding to the natural quotient T_X|_B→ N_B/X. φ(s(B))=ℙ^n-1. Firstly we assume that n=2. Then X is a del Pezzo surface which contains exactly 16 lines, namely l_i (1≤ i≤ 16). Denote by l̅_i⊂ℙ(T_X) the section of ℙ(T_X) over l_i corresponding to the quotient T_X|_l_i→ N_l_i/X. Then we have Z=∪l̅_i by <cit.>. On the other hand, as B is general, we may assume that B is not tangent to l_i to any points; that is, s(B) is disjoint from l̅_i and thus s(B) is disjoint from Z. In particular, the map φ is well-defined along s(B). On the other hand, since ζ_X|_B≅𝒪_B(d) is ample, we obtain that φ(s(B))=ℙ^1. Now we assume that n≥ 3. Note that it suffices to prove that s(B) is not contained in any element in |2ζ_X|. Let X' be the complete intersection X∩{x_0=0}. Then X' is a smooth intersection of two quadrics in ℙ^n+1. Moreover, the involution is defined as (x_0,x_1,⋯,x_n+2) ↦ (-x_0,x_1,⋯,x_n+2) induces a canonical splitting T_X|_X'≅ T_X'⊕ N_X'/X (see <cit.>). Then it follows from <cit.> that the induced map H^0(X,^2 T_X) → H^0(X',^2 T_X') is surjective and its kernel is generated by a non-zero element σ∈ H^0(X,^2 T_X). By Kodaira's vanishing theorem, the restriction H^0(X,𝒪_X(d))→ H^0(X',𝒪_X'(d)) is surjective. In particular, we may assume that B' X'∩ B is also a general element in |𝒪_X'(d)|. Let s':B'→ℙ(T_X') be the section corresponding to T_X'|_B'→ N_B'/X'. Then we have s(B)∩ℙ(T_X')=s'(B'). By induction on n and using the natural isomorphism H^0(X',^2 T_X') ≅ H^0(ℙ(T_X'),𝒪_ℙ(T_X')(2ζ_X')), we only need to show that s(B) is not contained in the divisor 𝒞∈ |2ζ_X| on ℙ(T_X) defined by σ, which follows immediately as B is general.§.§ Proof of Theorem <ref>Denote by R the ramification divisor of f.Consider the following exact sequence of sheaves0→ T_Y→ f^*T_X →𝒬→ 0,where 𝒬 is supported on (R).Moreover, there exists a closed subset W of X with codimension two such that 𝒬|_R∖W≅𝒪_R(R)|_R∖W,where W=f^-1(W).Let μ M→ℙ(T_X) be the resolution of the indeterminacy locus of φ.Then we get the following commutative diagram[row sep=large,column sep=large] M [d,"μ"] [dr,"ν"] ℙ(T_X) [r,dashed, "φ"] [d,"p_X"]ℙ^n-1 XLet W' be the strict transformation of p_X^-1(W) in M.Then W' has codimension two in M. In particular, the intersection F'∩ W has also codimension two in F', where F' is a general fibre of ν.It follows that F∩ p_X^-1(W) has codimension two in F, where F=μ(F').As X is general, by <cit.>, the general fiber of the Lagrangian fibration Φ: T_X^*→ℂ^n is of the form A∖ Z, where A is an abelian variety and codim(Z)≥ 2. Note that there is an étale double cover from the general fiber of the Lagrangian fibration Φ: T_X^*→ℂ^n to F∖ Z (see the explanation betweenProposition 5.1 and Lemma 5.1 in <cit.>). HenceF∖ Z is an open subset of a quasi-étale quotient of an abelian variety by the involution,whose complement is of codimension at least two.As a consequence,F_∘ F∖ (Z∪ p_X^-1(W)) is also an open subset of a quasi-étale quotient of an abelian variety by the involution, whose complement is of codimension at least two.Let B be an irreducible component of the branch locus of f which is a general hypersurface of degree d≥ 1. In particular, we may assume that s(B) is not contained in Z∪ p_X^-1(W) (see <cit.>). Then it follows from Lemma <ref> that the intersection s(B)∩ F_∘ is non-empty. As a consequence, there exists a dominating family {C_t}_t∈ T of irreducible curves on ℙ(T_X) such that a general member C_t is contained in F_∘ for some general F and meets s(B).In particular, we have ζ_X· C_t=0 as φ is the Iitaka fibration.Denote by f̅ℙ(f^*T_X)→ℙ(T_X) the induced morphism and let ζ_X be the tautological divisor of ℙ(f^*T_X). Then we have f̅^*ζ_X=ζ_X. Denote by Y_∘ the open subset Y∖W whose complement is also of codimension two in Y. Then the induced exact sequence0→ T_Y|_Y_∘→ f^*T_X|_Y_∘→𝒬|_R∩ Y_∘≅𝒪_R(R)|_Y_∘→ 0induces an elementary transformation of vector bundles (see <cit.>) and it induces a commutative diagram[row sep=large,column sep=large] Γ[dl,"g" above] [dr,"h"]ℙ(T_Y|_Y_∘) [rr,dashed]ℙ(f^*T_X|_Y_∘) [r,"f̅_∘"]ℙ(T_X|_X∖ W),where h is the blowing-up of the subscheme Dℙ(𝒪_R(R)|_Y_∘)⊂ℙ(f^*T_X|_Y_∘). Let E be the exceptional divisor of h. By <cit.>, we have𝒪_Γ(g^*ζ_Y) ≅𝒪_Γ(h^*ζ_X-E).As f̅_∘ is surjective and h is birational, the family {C_t}_t∈ T of curves on ℙ(T_X|_X∖ W) can be lifted to a dominating family {C_t̃}_t̃∈T̃ of irreducible curves on Γ. Moreover, since C_t always meets s(B), by the definition of D and s(B), there exists an irreducible component E' of E such that s(B) is the closure of f̅_∘(h(E')) and C_t̃ always meets E'. In particular, we have E· C_t>0. As ζ_X· C_t=0, we get h^*ζ_X· C_t̃=0 by the projection formula. In particular, we obtaing^*ζ_Y· C_t̃ =(h^*ζ_X-E)· C_t̃ < 0.This implies that g^*ζ_Y is not pseudo-effective and thus ζ_Y is not pseudo-effective, either. | http://arxiv.org/abs/2311.15559v1 | {
"authors": [
"Feng Shao",
"Guolei Zhong"
],
"categories": [
"math.AG",
"math.DS",
"14J40, 14J45"
],
"primary_category": "math.AG",
"published": "20231127060341",
"title": "Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)"
} |
Max-Planck-Institut für extraterrestrische Physik (MPE), Gießenbachstraße 1, 85748 Garching bei München, Germany [email protected] MIT Kavli Institute for Astrophysics and Space Research, 70 Vassar Street, Cambridge, MA 02139, USA Institute for Astronomy & Astrophysics, National Observatory of Athens, V. Paulou & I. Metaxa 11532, Greece Department of Physics, University of Crete, Voutes University campus, 70013 Heraklion, Greece Institute of Astrophysics, Foundation for Research and Technology-Hellas, N. Plastira 100, Vassilika Vouton, 71110 Heraklion, Greece INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, 23807 Merate, LC, Italy Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Exzellenzcluster ORIGINS, Boltzmannstr. 2, 85748 Garching bei München, Germany Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany Finding massive black holes (MBHs, M_BH≈10^4-10^7 M_) in the nuclei of low-mass galaxies (M_*⪅10^10 M_) is crucial to constrain seeding and growth of black holes over cosmic time, but it is particularly challenging due to their low accretion luminosities. Variability selection via long-term photometric ultraviolet, optical, or infrared (UVOIR) light curves has proved effective and identifies lower-Eddington ratios compared to broad and narrow optical spectral lines searches. In the inefficient accretion regime, X-ray and radio searches are effective, but they have been limited to small samples. Therefore, differences between selection techniques have remained uncertain. Here, we present the first large systematic investigation of the X-ray properties of a sample of known MBH candidates in dwarf galaxies. We extracted X-ray photometry and spectra of a sample of ∼200 UVOIR variability-selected MBHs and significantly detected 17 of them in the deepest available SRG/eROSITA image, of which four are newly discovered X-ray sources and two are new secure MBHs. This implies that tens to hundreds of LSST MBHs will have SRG/eROSITA counterparts, depending on the seeding model adopted. Surprisingly, the stacked X-ray images of the many non-detected MBHs are incompatible with standard disk-corona relations, typical of active galactic nuclei, inferred from both the optical and radio fluxes. They are instead compatible with the X-ray emission predicted for normal galaxies. After careful consideration of potential biases, we identified that this X-ray weakness needs a physical origin. A possibility is that a canonical X-ray corona might be lacking in the majority of this population of UVOIR-variability selected low-mass galaxies or that unusual accretion modes and spectral energy distributions are in place for MBHs in dwarf galaxies. This result reveals the potential for severe biases in occupation fractions derived from data from only one waveband combined with SEDs and scaling relations of more massive black holes and galaxies. O Corona, where art thou? R. Arcodia et al. O Corona, where art thou? eROSITA's view of UV-optical-IR variability-selected massive black holes in low-mass galaxies R. ArcodiaNASA Einstein fellow 1,2, A. Merloni 1, J. Comparat 1, T. Dwelly 1, R. Seppi 1, Y. Zhang 1, J. Buchner 1, A. Georgakakis 3, F. Haberl 1, Z. Igo 1, E. Kyritsis 4,5, T. Liu 1, K. Nandra 1, Q. Ni 1, G. Ponti 6,1, M. Salvato 1, C. Ward 7, J. Wolf 1,8,9, A. Zezas 4,5 Received ; accepted============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION It is hotly debated to what extent the nuclei of low-mass galaxies (i.e., stellar masses M_*⪅10^10 M_) are populated by massive black holes (MBHs), a fairly loose term naming masses intermediate in between stellar and super-massive (used here for the range M_BH≈10^4-10^7 M_; e.g., seeand references therein). An in-depth understanding of this population of nearby low-mass nuclei is fundamental in relation to the first early Universe galaxies which they closely resemble. However, predictions on this local population from theoretical grounds require assumptions on seeding origin and growth <cit.>. Instead, from observational grounds we are fundamentally limited by the fraction of massive black holes which, even if they exist, are effectively active and luminous enough to be discernible from the host galaxy's emission at any wavelength <cit.>. The main channel used so far to systematically select MBHs is optical spectroscopy. The brightest end (in terms of the Eddington-normalized luminosity, L/L_edd) can be unveiled through virial mass estimates inferred from broad lines <cit.>, yielding ∼500 MBHs to date <cit.>. Understandably, this selection merely scratches the surface of the population of nuclear MBHs in low-mass galaxies, as only a very small fraction of galactic nuclei <cit.> are expected to be in the range of the required L/L_edd to show strong broad lines, even more so for low-mass galaxies <cit.>. Narrow-line-based classifications <cit.> may find low-mass galaxies with evidence of hard ionization from a nuclear source <cit.> at lower L/L_edd. Of course, the fainter these active MBHs are, the more they get inevitably hidden by the host galaxy's stellar emission and their signatures become hardly distinguishable from those of star-forming galaxies <cit.>. Spatially resolving emission from the nucleus helps <cit.>, although this approach is limited by angular resolution and therefore distance. Furthermore, a small fraction of nuclear MBHs can be unveiled through bright transient accretion events, for instance tidal disruptions of stars <cit.> and, lately, the puzzling quasi-periodic eruptions <cit.>, although this channel is limited by the low volumetric rates of these events (; Arcodia et al., in prep.). An alternative and promising way forward is given by the growing number of high-cadence photometric surveys, which allow for the selection of MBHs through optical, ultraviolet (UV), and infrared (IR) variability <cit.>. The goal of this method is to find evidence of low-level photometric variability through difference imaging analysis, indicative of nuclear point-like sources embedded in their extended host galaxies. Most of these studies compare light curves to a damped random walk model, which is usually an empirical indicator of accretion variability in active galactic nuclei <cit.>. This method was shown to yield a larger detection rate of MBH candidates below M_*∼10^10 M_, compared to broad and narrow line selection techniques <cit.>. The radio and X-ray band are more suitable to find nuclear sources in low-mass galaxies, as they have a higher nuclear-to-host contrast <cit.>. Therefore, a dedicated follow-up with deep X-ray and radio observations can serve to strengthen these candidates further <cit.>, as well as performing matches with current X-ray archives <cit.>. However, the former method is not a viable option for all of the known low-mass galaxies in the sky and the latter has been naturally limited in sky area so far. This is where the extended ROentgen Survey with an Imaging Telescope Array <cit.> aboard the Spectrum-Roentgen-Gamma observatory <cit.> comes into play with its all-sky survey capabilities, complementing existing deep-exposure and narrow-field datasets <cit.>. Here, we focus on MBHs selected through UVOIR variability (Sect. <ref>), which has the advantage of providing a sample with occupation and an active fraction of one. Therefore, for this work we used MBHs and accreting central black holes in low-mass galaxies interchangeably. We systematically extracted X-ray properties from the eROSITA all-sky survey data (Sect. <ref>). The primary goal was to obtain their X-ray detection fraction (Sect. <ref>), providing a top tier of UVOIR-variable X-ray-detected MBHs in low-mass galaxies for future deeper multiwavelength studies (Sect. <ref>), and to calibrate how single-band searches for MBHs compare (Sect. <ref>). This work will also serve as a pilot study to understand the connection between variability selection methods and eROSITA X-ray data to exploit future synergies with the Vera C. Rubin Observatory Legacy Survey of Space and Time <cit.>. § SAMPLE SELECTIONWe draw samples of variable low-mass galaxies from the literature of optical <cit.>, UV <cit.> and IR <cit.> studies. Albeit using slightly different methods and with different datasets and observing bands, all these works have performed similar searches for significant stochastic variability from the nuclei of dwarf galaxies, indicative of the presence of a MBH in their nuclei. The observed photometric light curves obtained from difference imaging are usually tested against a damped random walk model for AGN-like accretion variability <cit.>. As the emission from the galaxy is subtracted out, this technique has proved effective in finding faint nuclear AGN in dwarf galaxies, which would be otherwise missed with optical spectroscopy searches <cit.>, likely because these MBHs are not accreting close to the Eddington limit. However, the low-level variability does indicate that some level of accretion is happening in these nucleu, which implies that these MBHs are expected to emit X-rays. This makes the perfect sample for testing the synergies with UVOIR photometric surveys and eROSITA. The inhomogeneous and incomplete nature of the resulting galaxy sample is not concerning for the goal of this work, which is to compile a collection of dwarf galaxies with independent evidence of black hole activity in order to calibrate X-ray results in an informed way. Therefore, we assume that in this sample of variability-selected MBH candidates, the occupation fraction, namely the fraction of galaxies with a MBH seed in their center, and active fraction, namely that of galaxies with an active <cit.> black hole, are both one.The only selection criterion we perform on these datasets is a cut on stellar mass at 10^7≤ M_*≤10^10 M_ to select low-mass galaxies, taking M_* from the above-mentioned literature or their parent samples. If information on the goodness of fit that yielded M_* was found, it was used to filter M_* by fit quality. For instance, we selected galaxies from <cit.> with a reduced χ^2<10 from SED fitting at all redshifts and additionally imposing a cut at χ^2<5 at redshifts z>1, using the goodness of fit reported in <cit.>. A more stringent criterion is used at higher redshift, where at fainter magnitudes (hence stellar masses) the same reduced χ^2 can be obtained with a lower number of available filters. From <cit.> we made use of Δχ^2, which refers to the difference between the goodness of fit using the AGN template alone and the AGN+galaxy SED fit. We selected low-M_* galaxies i) with Δχ^2 >2 from their SED fitting and with any variability timescale, or ii) sources with rapid variability (characteristic timescale lower than 1.5 days, ) and with any Δχ^2 (Table 3 of ; C. Burke, priv. comm.). No explicit selection in redshift and narrow- and broad-lines classifications was performed. Redshifts are adopted from the references in Sect. <ref> and consist, to the best of our knowledge, of spectroscopic redshifts for the vast majority[Galaxies with photometric redshifts from <cit.> are knowingly included. Only sources with spec-z will be used in the X-ray stacking analysis.]. Estimates of black hole masses in these galaxies are often absent or very uncertain and typical scaling relations with M_* are not well calibrated in this mass regime <cit.>. Therefore we do not make any preselection on M_BH and for the scope of this paper we generically refer to these galaxies as MBHs or MBH candidates.A further obvious cut is the selection of galaxies in the German eROSITA hemisphere (i.e. Galactic latitudes between 179.944 and 359.944).The total number of galaxies with stochastic nuclear variability in the German eROSITA footprint is 216. In particular for optically selected objects, we select three from <cit.>, 52 from <cit.>, 35 from <cit.>, six from <cit.>[One duplicate in common between <cit.> and <cit.> was removed.], 46 from <cit.>, and three from <cit.>. Then, 1 from <cit.>, 66 from <cit.>, 1 from <cit.> for infrared-selected MBHs and 3 from <cit.> for UV-selected ones. The total is thus 145 from optical photometry searches, 68 from the infrared and 3 from the UV. We show the r-band magnitude, redshift and stellar mass distribution of the entire parent sample in Fig. <ref> in gray. The r-band magnitude and redshift distributions appear clearly bimodal. This is due to the presence of a large number of optically selected MBHs from <cit.>, mostly high-z, and <cit.>, mostly low-z, with blue dashed and dotted lines, respectively. We highlight with an orange dot-dashed line the IR-selected MBHs from <cit.>, to show that the bimodality in our sample is not due to the different wavebands. The different subsamples show marginal differences in the stellar-mass distribution instead (bottom two panels of Fig. <ref>). The r-band magnitudes are selected from the SDSS NASA-Sloan Atlas sample[https://www.sdss4.org/dr13/manga/manga-target-selection/nsa/Link to NSA catalog] version 1.0.1 for the low-z subsample, whilst from the COSMOS Subaru/SuprimeCam <cit.> for the high-z subsample.§ X-RAY ANALYSIS OF EROSITA DATAOur method consists of systematically extracting X-ray photometry at the input UVOIR coordinates from the all-sky image of the first eROSITA survey (eRASS1) as well as from the cumulative image of the first four (eRASS:4). The former provides a show case for the data level being released <cit.>, while the latter for the deepest data level available full-sky to the German eROSITA Consortium. Images were extracted with thetask of the eROSITA Science Analysis Software System <cit.> from event files version 020. The algorithm to extract photometry makes use of theastropy package version 1.4.0 <cit.>. Photometry was extracted between 0.2-2.0keV. We adopted a custom circular aperture of 30", corresponding to ∼ 75% of the encircled energy fraction of eROSITA's point spread function in the adopted energy band. This source aperture is defined regardless on the presence of a detected X-ray source within. Background information is extracted from an annulus with inner and outer radii of 120" and 360", respectively. Every contaminating X-ray source in the field is masked out from both background and source apertures, although in the latter case only if the centroid of the X-ray contaminant is outside the source aperture. Potential contamination from within the source aperture, for instance due to ultra-luminous X-ray sources (ULXs), is studied a posteriori and discussed in Sect. <ref>. The coordinates of the masks are taken from the headers of eROSITA X-ray products extracted by eSASS. For a very small number of galaxies, the source aperture of 30" was masked out (entirely or >70%) by a nearby bright or extended X-ray source. For eRASS1 images this is the case for 2/216 galaxies, while 8/216 for eRASS:4. This is due to the fact that eRASS:4 is deeper, therefore it contains more detected X-ray sources. We removed these from the parent sample (Sect. <ref>) when computing detection fractions, thus the total number of galaxies with X-ray products is 214 for eRASS1 and 208 for eRASS:4. X-ray photometry yields counts in both the source and background apertures. From these, we compute the binomial no-source probability <cit.>, which yields the probability that the observed counts in the source aperture area are due to background fluctuations: P_B (X ≥ C_S) = ∑_X = C_S^C_TC_T!/X! (C_T - X)! A^X (1-A)^C_T-X where C_S are the counts in the source aperture, C_T = C_S + C_B and C_B are the counts in the background area. Whereas A = 1 / (1 + A_B/A_S), with A_B and A_S being the area of background and source apertures, respectively. We note that this area includes masks, therefore it is not always the full circle or the full background annulus regions as defined in input. P_B can be calibrated in absolute sense only with simulations. For this, we use the “digital twin” of eRASS1 from <cit.>, which contains realistic populations of clusters and AGN. <cit.> ran source detection with the eSASS on the simulated sky, including the aperture photometry task [https://erosita.mpe.mpg.de/edr/DataAnalysis/apetool_doc.htmlLink to ] <cit.>. From the simulations we know real and spurious sources that the detection algorithm finds and fromwe know their counts[The impact of using a slightly different algorithm for aperture photometry is assumed to be negligible.] hence P_B. Here, we adopt as threshold for a significant detection P_B = 0.0003, which corresponds to 1% of spurious fraction in the eRASS1 simulation. As a sanity check, we numerically computed on a one-dimensional grid in count rate the Poisson probability mass function (PPMF) from the detected counts using thePython package <cit.>. We compute count rate PPMFs for the source contribution alone, background alone and both source plus background. The PPMF for total (source plus background) and background-only count rates are compared and a detection is obtained when the two distributions are not compatible within 3σ, using the 1st and 99th percentiles of the related distributions. We verified that the two methods give the same number of significant detections. We note that we adopt P_B <= 0.0003 for detections in eRASS:4 as well, despite the value being calibrated for eRASS1. We expect minor differences for the purposes of this work, as the P_B and PPMF detection criteria match for eRASS:4 as well. Spectra and light curves were extracted from the masked X-ray apertures of all sources, detected or undetected, using thetask in eSASS <cit.>. Spectral analysis is performed with the Bayesian X-ray Analysis software (BXA) version 4.0.5 <cit.>, which connects the nested sampling algorithm UltraNest <cit.> with the fitting environment XSPEC version 12.12.0 <cit.>, in its Python version PyXspec[https://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/python/html/index.htmlLink to PyXspec]. We adopted two simple continuum models, both with absorption fixed at the Galactic column density from HI4PI <cit.> and redshifted to rest-frame using the available redshifts: an accretion disk model, ), and a power-law, . For the rest of this work, we adopt themodel to quote flux and luminosity. For the detected sources, it is in the vast majority the model with higher Bayesian evidence from the BXA fit and data-model ratio residuals were visually confirmed to be acceptable. The choice has a negligible impact, also for the upper limits of the non detected sources. Flux and luminosity are computed in the rest-frame 0.2-2.0keV band. We quote median and 1st and 99th percentiles (∼3σ) from fit posteriors, unless otherwise stated, for fit parameters, flux and luminosity. For non-detections (P_B>0.0003), as defined above, we quote upper limits using the 99th percentiles of the fit posteriors, unless otherwise stated.Finally, we performed stacking analysis of non-detections following the method presented in <cit.>. Here, we outline the main steps. For each galaxy, the physical distances between X-ray photons and the galaxy (R_ kpc) are calculated according to the spectroscopic redshift of the galaxy and observed angular distance. We retrieve photons within 0.5-2.0keV and within 50 kpc of each galaxy and create a photon cube saving the positions, the distance to the associated galaxy (angular and physical, R_ rad, R_ kpc), the exposure time t_ exp, the observed energy E_ obs, the emitted energy E_ rest=E_ obs*(1+z), and the effective area A_ eff. These photons within 50 kpc will be used for both source and background estimates, as detailed below. All the X-ray-detected sources in the field are masked out and the related correction factor of the area (A_ corr) is calculated as a function of R_ rad or R_ kpc.We then merge the photons around the galaxies of interest and calculate the surface brightness (I_X) of the stacked image:I_X=ΣA_ corr4π D_ g^2 E_ rest/A_ efft_ exp1/N_ g,where D_ g is the luminosity distance of the galaxy and N_ g is the number of stacked galaxies. This profile is then integrated up to a given distance (angular or physical) to yield a median X-ray luminosity of the stacked image, with related Poisson statistical uncertainty. <cit.> estimated that the uncertainty due to the source-masking in the stacking procedure amounts to at most a ∼2% uncertainty on the number of events. To be conservative, we apply a 2% systematic uncertainty to the measurements. We integrate up to 10 kpc unless otherwise stated. This scale is a few times larger than the typical effective radius, or half-light radius, of galaxies below log M_* = 10 <cit.>, therefore the relevant scale is the much larger eROSITA's PSF. An integration up to 10kpc ensures that the eROSITA PSF is contained fully within the integration bounds for sources at the median redshift of the z<0.1 subsample, whilst minimizing the presence of possible stacked signal from the outskirts of galaxies. Furthermore, we check that the stacked image detection or non-detection remains such changing the integration distance, and by visualizing the profiles to exclude that the detection is not driven solely by spurious signal in a single off-centered annulus. The background is calculated taking the median value of the signal between 15<R_ kpc<50 and it is subtracted from each annulus during integration. We visualize that the stacked signal between 15<R_ kpc<50 is constant.We conservatively check that a detection remains such also if the 84th percentile of the signal within 15<R_ kpc<50 is used as background estimate and if the lower integration bound is moved inward or outward from 15 kpc. If the stacked signal is compatible, within its uncertainties, with the background estimate, we quote the background-subtracted upper value of the luminosity integral as upper limit. An example is provided in Fig. <ref>, where only the signal shown in red represents a detection, whilst that in green is compatible with background.§ RESULTS §.§ Detection fractionWe obtain that 5.1_-1.1^+2.0% (11/214) of the dwarf galaxies are detected in eRASS1 and 8.2_-1.5^+2.3% (17/208) in the deeper eRASS:4 (see Sect. <ref>). The median fraction and 1σ binomial confidence intervals are inferred from the related quantiles of the beta distribution from <cit.>. In particular, we detected in eRASS1 (eRASS:4) 3 (3) galaxies from <cit.>, 3 (4) from <cit.>, 1 (4) from <cit.>, 0 (1) from <cit.> and 4 (5) from the WISE-selected sources in <cit.>. In eRASS:4, detection fractions of 9.2_-1.9^+3.2% and 7.2_-2.0^+4.4% are obtained for the optically- and IR-selected galaxies, respectively, thus they are compatible within uncertainties. We show an example of a detected source in Fig. <ref> to showcase our methodology. The input coordinates and the adopted aperture are shown with a white circle in both left and central panels, showing the optical and X-ray cutouts, respectively. The right panel shows the source plus background spectrum and related model lines and contours. We report P_B and X-ray luminosity (L_0.2-2.0 keV) for all detected and undetected dwarf galaxies, for both eRASS1 and eRASS:4, in Table <ref>. For a consistency check, we compared our eRASS1 results with the official eRASS1 catalog released in <cit.>, matching the optical coordinates in input within 30", the circular aperture used here for X-ray products. All the 11 eRASS1 sources found in this work are present in the official catalog with compatible fluxes. Four sources which are considered undetected in this work are present in the official eRASS1 catalog (namely 1eRASS J130716.6+133904, 1eRASS J032845.6-271113, 1eRASS J003429.2-432056, 1eRASS J085125.9+393541; ). They have P_b spanning between 0.0004 and 0.003, therefore they are marginally below our P_b=0.0003 threshold. The three sources above P_b>0.001 have detection likelihoods between 6-8 in the official catalog (; see ), which corresponds to a false detection rate between ∼4-14% <cit.>. However, these four sources are all detected in the deeper eRASS:4 image with our method. Therefore, they are most likely real sources and this comparison simply implies that our algorithm and chosen P_b threshold are on the conservative side. As a matter of fact, we adopted a threshold of P_b=0.0003 to ensure a lower spurious fraction of ∼1%. We show the detection fraction as a function of X-ray flux (in the rest-frame 0.5-2.0keV band) in the top left panel of Fig. <ref>. Different symbols, between eRASS1 and eRASS:4, are slightly shifted horizontally for illustration purposes. In order to compute the evolution of detection fraction as a function of X-ray flux, we included non detected galaxies in the plot by extracting 100 random values from their unconstrained flux chains. In this way, each source may enter different bins at each iteration. We averaged over these 100 iterations, therefore uncertainties include the fact that non-detections are spread across more bins. As they would be more likely extracted in the lower flux bins, their binomial uncertainties are smaller than the high-flux bins (the average numbers per bin are shown in the upper subpanel). Non detected sources with a flux fainter than the lowest bin (-14.75, -14.25) are not present in any bin at a given iteration. The evolution of detection fraction as a function of X-ray flux can be compared with eROSITA's sensitivity. For eRASS1, we can use the simulations from <cit.> which provide the eRASS1 sensitivity curve. Since simulations were done for each sky tile, we can compute the eRASS1 sensitivity at the locations of all sources in our parent sample. We show the median (with related 16th and 84th percentile contours) of this distribution with a solid red line in the top left panel of Fig. <ref>. We note that eRASS1 MBH detections from this work lie below the sensitivity curves from simulations at low and intermediate fluxes.This might suggest that not all the UVOIR-variable MBHs in input are intrinsically above an X-ray flux of log (F_X/(erg s^-1))∼-14.5, used in the plot at the lower end.We note that, however, we do not expect all MBHs in the sample to be intrinsically above an X-ray flux of log (F_X/(erg s^-1))∼-14.5, given that our sample includes also high-redshift sources (around half of the input sample is above z∼0.04), for which such a threshold flux would correspond to a significant intrinsic luminosity. Indeed, it is unreasonable for all the MBHs above this redshift to be above an intrinsic luminosity of ∼1.2×10^40 (D_L/D_L_0.04)^2erg s^-1, for a luminosity distance D_L_z. The situation marginally improves when filtering the top left panel of Fig. <ref> below z∼0.04, although only due to the even larger error bars which is merely due to the decrease of sample size. Further, the top right panel of Fig. <ref> shows the observed detection fraction as a function of redshift in three bins with roughly equal number of galaxies. We note no singificant difference across the bins. We conclude that the incompatibility between observations and simulations is likely not uniquely a redshift effect and it will be investigated and discussed further in Sect. <ref>. §.§ Trends with the galaxy's stellar massFrom the bottom left panel of Fig. <ref> we note a slight increase of detections with increasing stellar mass, although all values are compatible within 3σ uncertainties. In both eRASS1 and eRASS:4, the overall detection fraction of ∼5% and 8%, respectively, is compatible with those estimated in the single stellar mass bins, within uncertainties.Based on this, we obtain that we can expect to detect from any future UVOIR variability survey, with similar characteristics to the ones considered here, a fraction on the order of ≈5% (≈8%) in eRASS1 (eRASS:4) at least above log M_*∼8.5. We show the eRASS:4 detections and non-detections in the luminosity-stellar mass plane (Fig. <ref>). The top panel shows the full sample, where the low-z and high-z populations (e.g., see the top-middle panel of Fig. <ref>) are clearly separated. We note an outlier in the X-ray-detected source around stellar mass of ∼10^8 M_⊙. This estimate from <cit.> comes with a high statistical uncertainty (∼0.5dex) and the marginal increase in Δχ^2, between the AGN template alone and the AGN+galaxy SED fit, implies large systematics which hinder a reliable interpretation of the stellar mass value (C. Burke, priv. comm.). In general, our sample is rather heterogeneous and obtained through different selection methods (Sect. <ref>), therefore for further analysis and data-model comparisons in the L_X-M_* plane we only use the subsample of 134 galaxies below z<0.1 (e.g., see the bottom panel of Fig. <ref>). This selection allows us to use an homogeneous low-z population and magnitude distribution (see Fig. <ref>). In particular, we stacked the 0.5-2.0keV eRASS:4 images of the 121 undetected sources below of z<0.1, using only spectroscopic redshifts. We stacked two sets of images in two M_* bins, log M_* = 8-9 and 9-10, which contain 30 and 91 undetected galaxies respectively. The low mass bin stack is undetected, with an upper limit at L_0.5-2.0 keV<9× 10^37erg s^-1, whilst in the high-mass bin we obtain L_0.5-2.0 keV = (2.1±1.1)× 10^39erg s^-1. The profiles are shown in Fig. <ref> and they are represented with dark red stars in the bottom panel of Fig. <ref>.With the aim of interpreting the observed X-ray luminosities, we compare them with predictions of both AGN and normal galaxies. We computed the predicted 0.5-2.0keV X-ray luminosity from X-ray binaries in normal galaxies following <cit.> and added the diffuse hot gas component due to the ISM, relevant in the soft X-rays, following <cit.>. We adopt the stellar mass from our parent sample and use the star formation main sequence <cit.>, for simplicity, to obtain the star formation rate (SFR) for this plot. We note that for starburst galaxies, this would be an underestimation of SFR. This prediction is shown with the black thick line in the bottom panel of Fig. <ref>, with the thickness spanning the prediction for the minimum (z=0) and maximum (z=0.1) redshifts of the galaxies in the panel. Below log M_* ∼ 9.5 and below SFR ∼ 2 M_⊙/yr the relation is known to be inaccurate, due to the fact that the galaxy prediction relies on fully-populated X-ray binaries luminosity functions <cit.>, which would not apply in this regime. The black dotted line can be used as guide for the eye, in case this relation still holds on average <cit.>, albeit with significant scatter (e.g. Kyritsis et al., in prep). If stochastic sampling implies higher difficulty in observing luminous sources reducing the average luminosity per galaxy, the dotted line would be an overestimate. We approximate this by artificially decreasing the dependency on M_* and SFR <cit.>, thus the predicted X-ray luminosity, and we show this with a solid black line in Fig. <ref>. Furthermore, we computed the predicted AGN soft X-ray luminosity as a function of galaxy stellar mass by interpolating scaling relations and spectral energy distributions (SEDs) common to more massive AGN. Since typical scaling relations are calibrated in the UV <cit.>, but still hold for a wide range of optical frequencies <cit.>, we adopt the bluest SDSS filter available, for simplicity. We obtained the observed u-band flux of our galaxies from the parent SDSS NASA-Sloan Atlas sample (). No K-correction was applied to these estimates, as they are intended as guide for the eye. We infer the AGN optical luminosity assuming accretion at ∼0.1×, ∼0.01× and ∼0.001× L_edd, assuming M_BH=0.002 M_* and an optical bolometric correction of 0.1 <cit.>. Then we applied X-ray-to-optical scaling relations for radiatively-efficient <cit.> and -inefficient <cit.> AGN to infer the expected 2keV luminosity, and finally converted to L_0.5-2.0 keV assuming a power-law spectrum with photon index 1.9. Quite interestingly, the detected MBHs (green squares) mostly align with the predictions of AGN accreting at ≈0.01-0.1 L_edd. However, we notice that the vast majority of the eRASS:4 3σ upper limits lie well below these scaling relations. Most importantly, the X-ray luminosity estimates from their stacked images (dark red stars) are consistent with predictions of normal galaxies' non-AGN emission. We note that despite M_* is a notoriously uncertain parameter, most upper limits would remain inconsistent with the AGN predictions even if they were biased low or high in stellar mass by as much as ∼0.5-1.0 dex (e.g. along the x-axis of Fig. <ref>), and the stacks would likely be unaffected by a few erroneous stellar mass estimates. The nature of this X-ray weakness will be further explored in Sect. <ref>, by comparing X-rays to other wavebands as well. § X-RAY-DETECTED DWARF GALAXIES §.§ Contaminants: the cumulative stellar-mass BHs population We investigate the possible cumulative contribution to the X-ray-detected galaxies due to the stellar population, here intended as a contaminant, within the host galaxy of our MBH candidates <cit.>. We use the term X-ray binary (XRB) for the collective contribution of both accreting neutron stars and stellar-mass black holes. Despite the difficulty of securely assessing contamination from XRBs for each galaxy, we can rely on well-known scaling relations that predict the expected X-ray luminosity from XRBs given the stellar mass and SFR in the galaxy. The mass of the stellar companion defines the classification between low- and high-mass XRBs. The former (latter) kind evolves slower (faster) and it is therefore traced by the total stellar content or M_* (by recent star formation and SFR and both have to be taken into account <cit.>.We compute the predicted X-ray luminosity (L_X,gal) in the 2-10keV range from the cumulative XRB population in the host galaxy from<cit.>, which was calibrated in the Chandra Deep Field-South (CDF-S):L_X,gal = α_0 (1+z)^γ_0 M_* + β_0 (1+z)^δ_0SFRwith (logα_0, logβ_0, γ_0, δ_0) = (29.30, 39.40, 2.19, 1.02). For these calculations, we obtained individual SFR values from different sources: five galaxies match with the HECATE catalog <cit.> within 3", five with <cit.>, one from <cit.> and one from <cit.>; for the remaining sources SFR was obtained from UV <cit.> and IR <cit.> fluxes, following the prescription from <cit.>. These values span uniformly between ∼1-100 M_⊙yr^-1. For consistency with the SFR estimates, we used M_* from these references for computing L_X,gal, if present, or the values in Table <ref> otherwise. Here, we neglect the contribution from hot diffuse gas due to the ISM to L_X,gal since it is expected to be significantly lower than the faintest of our X-ray detections (∼7×10^39erg s^-1), even more so given the range of stellar masses in our sources and in the ∼2-10keV band. As a matter of fact, this contribution is L_X/M_* ∼ 10^28erg s^-1 M_^-1 for early type galaxies <cit.> and amounts to up to ∼10% of the observed luminosity for star-forming galaxies <cit.>. Here, we ignore the known stochasticity of the galaxy prediction at low M_* and SFR <cit.>, for simplicity. The adopted scaling relations surely come with considerable uncertainties and intrinsic scatter, although one of the causes of this scatter at the bright end is the likely presence of X-rays from the MBH itself. A further source of contamination which we neglect here could be the cumulative emission from XRB from the nuclear star cluster (NSC), which is nearly ubiquitous in low-mass galaxies <cit.>. As standard scaling relations to estimate L_X,gal try to exclude the point-like nuclear X-ray source, to which the NSC might contribute, these are most likely not accounted for.In Fig. <ref>, we show the comparison between the predicted L_X,gal and the observed X-ray luminosity of our eRASS:4 detected sources (Table <ref>), both estimated in the 2-10keV range[Therefore we do not compare these with the stacks of the soft X-ray images.]. The observed values for the detected galaxies are clearly well above the predicted ones (black solid line) including uncertainties. The dashed and dotted lines show the predictions increased by a factor 3 and 200, respectively, to guide the eye. The result of this sanity check is reassuring, since the parent sample consists of MBH candidates selected independently from UVOIR variability. This was already evident from the bottom panel of Fig. <ref>, although in that case the prediction for the galaxy was obtained at population level using the star formation main sequence and not individual SFR values. In the next section we discuss the role of individual luminous XRBs, relevant at the lowest end of the observed X-ray luminosity. §.§ Contaminants: individual stellar-mass BHs Another source of contamination comes from individual neutron stars and stellar-mass black holes at the brightest end of their luminosity function, which constitute the vast majority of the so-called ultra-luminous X-ray sources (ULXs[Here, this term is used for stellar-mass contaminants and neglects the possible presence of intermediate-mass black holes in the ULX category.]) within the host galaxies <cit.>. Given eROSITA's point spread function (≈26" half-energy width averaged over the whole field of view, ) we can indeed expect contamination from off-nuclear ULXs in what we have called here MBHs. However, disentangling ULXs and MBHs has revealed to be much more difficult that initially thought. As a matter of fact, recent simulations <cit.> and observations <cit.> have pointed out that a significant fraction of MBHs in dwarf galaxies can be displaced from the host center even up to ∼3 kpc <cit.>. Therefore, angular separation of the X-ray source from the optical nucleus alone might not be a good-enough proxy. ULXs and MBHs can be securely distinguished only if the point-like X-ray source is clearly in the outskirts of the host galaxy, or if the X-ray source is classified as a neutron star through detection of pulsations <cit.> or if deep broadband spectroscopy can be carried out to distinguish between accretion states <cit.> and infer an estimate of the accretor's mass. In this work, we cross-matched our sample with the ULX catalog from <cit.>, which compiled XMM–Newton, Swift-XRT, and Chandra data. This catalog does not overlap with the entirety of our sample, but may serve as a useful check to exclude as many known ULXs as possible.Two known ULXs from <cit.> are within the aperture of two non-detected galaxies, whilst we found no overlap between our detected galaxies and the ULX catalog. Finally, we note that the conservative conclusion about the various stellar-mass contaminants is that at ambiguous X-ray luminosity levels ≈10^39-10^40erg s^-1, both the stellar-mass contaminants and MBHs are likely contributing to the total X-ray emission. This ambiguity may remain even using rich multiwavelength observations of individual nearby galaxies taken at high angular resolution <cit.>.§.§ New X-ray detectionsWe matched the 17 galaxies detected in eRASS:4 with ROSAT, Swift-XRT, XMM-Newton and Chandra catalogs in the HEASARC archives using our 30" aperture as matching radius. We have found 13 matches, all within a few arcseconds from the input coordinates. We show these matches in Table <ref>. In the comments, we note the classification that can be inferred with a quick search on Simbad <cit.>. We note the presence of two sources classified as blazars, which perhaps hints that they might be a neglected contaminant in the variable MBH searches.Quite interestingly, we find that 4 of our eRASS:4 detections (≈25%) are new X-ray sources. We note that this fraction is even lower than that expected on average on the full-sky, since it is common practice to coordinate narrow-field deep multiwavelength surveys in the same sky area. This highlights the power of eROSITA with its full-sky capabilities, which balances existing and future deep pencil-beam surveys. The 4 new detections are highlighted with red circles in Fig. <ref> and their X-ray images are shown in Fig. <ref> and <ref>. More details are presented next. §.§.§ SDSS J031302.15-004110.9 and SDSS J031743.12+001936.8 The first new X-ray source can be identified with SDSS J031302.15-004110.9, a known low-mass AGN at z=0.13 found to be optically-variable by <cit.>. It is also reported as an AGN from BPT classification, with a known virial black hole mass of ∼10^7 M_ <cit.>. We obtained a median (and 16th, 84th percentiles) value of L_0.2-2.0 keV=43.19_43.11^43.28erg s^-1 and a soft X-ray photon index of Γ = 2.76±0.27 in eRASS:4. Based on Fig. <ref>, the observed luminosity is a factor ∼259 above the one predicted for the cumulative XRBs in the host galaxies and it is quite extreme even for ULXs. The X-ray emission appears point-like and consistent with the optical center (Fig. <ref>, left panels). We can confidently consider this source as the X-ray counterpart of the nuclear MBH. This source is present in the eRASS1 catalog <cit.> as 1eRASS J031302.2-004114, with (RA, Dec) = (48.25899, -0.68734) and a 1σ positional error of 2.56". The second new X-ray source can be associated with SDSS J031743.12+001936.8, a known low-mass AGN at z=0.069 selected from <cit.>. This source was classified as "composite" from narrow lines diagnostics and its estimated logarithmic virial mass is ∼6.1 log M_ <cit.>. We have obtained log L_0.2-2.0 keV=42.19^42.3_42.08erg s^-1 and X-ray photon index Γ = 2.20±0.40 in eRASS:4. This is the source shown in Fig. <ref>, where we note a point-like X-ray emission consistent with the optical center. The observed 2-10 keV X-ray luminosity is log L_2.0-10 keV∼41.88 log(erg s^-1), a factor ∼59 above the luminosity predicted for the cumulative XRBs (Fig. <ref>). The optical and X-ray source coincide within 1" with the radio source FIRSTJ031743.1+001936, which has an integrated flux at 1.4 GHz of 1.82mJy <cit.>. This corresponds to a luminosity density of log L_1.4Ghz∼22.3W Hz^-1, much brighter than the expected contribution from supernova remnants, young supernovae and ionized gas from H_II regions <cit.>. Therefore we expect this to be the radio counterpart of the point-like X-ray source. We usen these estimates of X-ray and radio luminosity to infer a black hole mass through the fundamental plane of black hole accretion <cit.>. From the 1.4 GHz flux and assuming a flat spectrum (or, a spectrum with slope -1) in flux density units, we infer log L_5Ghz∼39.0 log(erg s^-1) (38.5), which yields log M_BH∼ 8.4 log M_ (7.7). We note that the fundamental plane is only representative for radiatively inefficient black hole accretion, although it may provide us with a rough black hole mass estimate in any case. The observed luminosities are therefore too high for a stellar-mass ULX, unless its emission is beamed. While we do not know the accretion state of SDSS J031743.12+001936.8, the hard X-ray luminosity with a bolometric correction of 10 <cit.> corresponds to ∼0.1 L_Edd, therefore to a radiatively efficient regime. This might explain the difference between the observed mass and that predicted from the fundamental plane in the MBH scenario. Based on this, we consider this as a secure X-ray counterpart of the variable MBH. We note that this source was classified as composite based on its optical spectrum <cit.>, which highlights once more how this selection technique is biased toward the brightest end of the MBH population. However, a closer look at the SDSS spectrum suggests the presence of a broad Hα component that the automatic pipeline did not account for[https://dr9.sdss.org/spectrumDetail?plateid=413 mjd=51929 fiber=470Link to SDSS spectrum]. This source is present in the eRASS1 catalog <cit.> as 1eRASS J031743.0+001938, with (RA, Dec) = (49.42923, 0.32735) and a 1σ positional error of 2.82". §.§.§ SDSS J121709.27+122714.4? The third X-ray source is within the aperture of SDSS J121709.27+122714.4, a narrow-line galaxy at z=0.007 from <cit.>. This host is classified as star-forming using narrow line fluxes in the SDSS database[https://dr9.sdss.org/spectrumDetail?plateid=1613 mjd=53115 fiber=183Link to SDSS spectrum] and the narrow lines diagnostics from <cit.>, adopting log ([OIII]/Hβ) ∼ 0.25 and log ([NII]/Hα) ∼ -0.59. From our eRASS:4 analysis, we obtained log L_0.2-2.0 keV=39.86^40.10_39.54 log(erg s^-1) and a hard X-ray photon index which is an unconstrained posterior with 1σ upper limit at Γ∼ 1.63. The latter value hints for a more complex spectrum compared to a simple power-law, which will need to be explored with a deeper exposure. The detected X-ray luminosity of log L_2.0-10 keV∼40.22 log(erg s^-1) is a factor ∼13 above that predicted for the cumulative XRBs (Fig. <ref>) and the emission is point-like (Fig. <ref>, middle panels), although is consistent with being slightly off-nuclear (13” from the optical coordinates). As discussed above, recent works have shown that MBHs in dwarf galaxies are not all coincident with the optical nucleus and the observed offset of ∼1.9kpc would be within the typical values <cit.>. Nonetheless, we must consider the possibility that the X-ray-detected source is an ULX. The spectral shape would indicate that the putative ULX is in its hard ultra-luminous state <cit.>, although we do not aim to state anything conclusive given the available data. Here, we note that the source is not detected in eRASS1 nor in eRASS2 and eRASS3 separately, although it is in the cumulative eRASS:3 survey at a luminosity L_0.2-2.0 keV = (4.8^+2.0_-1.9)×10^39erg s^-1. It is detected in the single eRASS4 at L_0.2-2.0 keV = (1.2^+0.6_-0.4)×10^40erg s^-1, hence somewhat brighter than in eRASS:3. This induces the eRASS:4 luminosity to be intermediate between the two, as reported above. No significant variability is detected within eRASS4, due to the low signal-to-noise of the individual ∼40 s snapshot that eROSITA performs within the single survey <cit.>. Overall, this might indicate that the source is variable on long (weeks to years), although not on short (hours to days), timescales. §.§.§ SDSS J130717.44+133847.8? The fourth newly-discovered X-ray source lies within the aperture around the input target SDSS J130717.44+133847.8, a galaxy at z=0.027 detected through infrared WISE variability <cit.>. From our eRASS:4 analysis, we obtained log L_0.2-2.0 keV=41.27^41.37_41.16erg s^-1 and a soft X-ray photon index Γ = 2.50±0.38. The detected X-ray luminosity is a factor ∼20 above that predicted for the cumulative XRBs (Fig. <ref>). However, there is background source within the aperture at (RA, Dec) = (13:07:16.90534, +13:39:03.82002), ∼19" away from the input galaxy, which is coincident with the X-ray point-like source (Fig. <ref>, right panels). It is identified as SDSS J130716.91+133903.8 at a Legacy Imaging Surveys photometric redshift of 1.26 <cit.>, and which is classified as AGN/QSO in several catalogs <cit.> also based of its infrared (W1-W2∼0.8) colors <cit.>. We conclude that both the WISE variability and the eRASS:4 X-ray source are most likely attributable to the background QSO and not the foreground dwarf galaxy. In order to quantify the extent of this issue in the whole WISE-selected sample <cit.>, we adopt the QSO space density to be ∼ 1.2× 10^-5arcsec^-2 above W2<17.11 for WISE AGN <cit.>. Adopting a conservative radius of three WISE pixels, each of 2.75 arcsec in size, we would expect ∼2.5× 10^-3 background IR-bright QSOs to be within a single WISE PSF. Therefore, we would expect ∼200 contaminants within the parent sample of 79879 galaxies of <cit.>, which is comparable to the sample size of the 148 selected variable galaxies. However, not all the WISE QSOs are found to be variable, therefore only ∼1.1% <cit.> would be detectable as contaminant in the foreground variability searches <cit.>. Therefore the number of expected contaminants is ∼2 in the sample of <cit.>. Since only ∼30% of their galaxies are in the eROSITA footprint, the IR source in this Section is most likely the only contaminant in the IR-selected sample. This source is present in the eRASS1 catalog <cit.> as 1eRASS J130716.6+133904, with (RA, Dec) = (196.81906, 13.65126) and a 1σ positional error of 4.16".§ X-RAY UNDETECTED DWARF GALAXIES SUGGEST X-RAY WEAKNESS OF MBHSOur results find a high-fraction of non-detected dwarf galaxies with a UVOIR-variable MBHs. The typical exposure in the eRASS:4 image for the galaxies in the parent sample is only ∼550 s. However, most X-ray 3σ upper limits are so deep that stacking non detected sources results in a L_X estimate consistent with the predictions of the emission of the galaxy alone (bottom panel of Fig. <ref>). Naturally, the X-ray emission of normal galaxies and radiatively inefficient (hence low-luminosity) AGN is expected to be compatible as their relative contrast reaches unity <cit.>. In particular, at the same level of accretion in terms of fractions of L_edd, MBHs in dwarf galaxies are even more penalized than more massive AGN. This can be understood with order-of-magnitude scaling relations by noting that the AGN luminosities scales linearly with M_BH for a given L/L_edd, hence ≈ M_* as M_BH∝ M_*^β with β≈ 1 or larger <cit.>, whilst the galaxy luminosity scales linearly with SFR, which in turns scales as SFR∝ M_*^0.7 at z=0 for main sequence galaxies <cit.>, ignoring redshift and metallicity dependencies for simplicity. As a matter of fact, we have already showed this with order-of-magnitude predictions in the bottom panel of Fig. <ref> with black shaded contours and the dotted red line, which are related to normal galaxies and inefficient AGN accreting at ∼ 10^-3 L_edd, respectively. Therefore, at this stage we can only conclude that the X-ray luminosity from the stacked non-detected sources is compatible with both.However, we can gain more information from the SED adding the information from the optical band in the picture. In particular, we know the brightness of these galactic nuclei (Fig. <ref>) and we can attempt to use typical X-ray-to-optical (X/O) scaling relations to put our observations into a wider context. We highlight this in Fig. <ref>, where we show observed X/O luminosity ratios as a function of stellar mass for all the MBHs below z=0.1. Squares represent detections within eRASS:4, arrows are 3σ upper limits. Both are color-coded based on the variability selection between optical (blue) and infrared (orange), to highlight the lack of obvious biases in either. The u-band flux () is obtained from the parent SDSS NASA-Sloan Atlas sample. We add X/O values computed from the stacked non-detections as follows. The monochromatic rest-frame 2 keV luminosity is obtained dividing the stacked luminosity between 0.5-2.0keV (see Sect. <ref> and the bottom panel of Fig. <ref>) by a conversion factor obtained from the detected galaxies (e.g., the squares in Fig. <ref>), taking the median value of their observed F_2 keV/F_0.5-2.0 keV ratio. The optical luminosity (their uncertainty) for the stacked value is obtained using the median (1st and 99th percentiles) of the observed u-band flux within the two stellar-mass bins. The statistical uncertainties from the stacks are shown with vertical errorbars as in Fig. <ref>, whilst the uncertainty coming from the range of u-band used for computing the stacks' X/O is shown with a darkred contour. Observed log (L_2keV/L_opt,u) are compared with predictions from models of normal galaxies (gray contour) and AGN (red lines). For normal galaxies we used scaling relations from <cit.> and <cit.>, using the star formation main sequence <cit.> and mass-to-light ratios between 1-10. As explained in Sect. <ref>, the galaxy predictions are calibrated only at the high-mass end, and we show with a dotted black line the extrapolation, whilst we attempt to correct for underpopulated low-mass and low-SFR galaxies <cit.> drawing the dashed black line. For AGN, we computed the optical luminosity following <cit.> and the X-ray luminosity from the L_X-L_UV relation for radiatively-efficient <cit.> and -inefficient <cit.> AGN. The former prediction is shown with a red solid line, the latter with dashed (dotted) for inefficient accretion at ∼10^-3 (∼10^-4) of L_edd. We confirm that, as in Fig. <ref>, the stacks are compatible with the emission of normal galaxies. Since the u-band filter has an effective wavelength at ∼3565Å, whilst these scaling relations are calibrated at ∼2500Å or ∼3000Å <cit.>, we also computed the X/O ratios using GALEX's near-UV filter at ∼2300Å (Fig. <ref>). The comparison between observed X/O and model predictions remains qualitatively the same and in fact using GALEX even fainter X/O values are obtained. Therefore, the observed X-ray weakness is even more enhanced compared to the bottom panel of Fig. <ref>, once the optical/UV luminosities are used to provide a characteristic SED shape. The underlying assumption is that the host galaxy is contaminating, but not dominating the optical emission, which is reasonable given that the MBH has to contribute enough to the flux to allow the inference of its presence through variability, at least in the cases of moderate Δ_mag. Furthermore, the typical SED of the MBH candidates does not seem to show worryingly or ubiquitously dominant contributions from the stellar component alone <cit.>, specially for the bluer optical and UV filters used here. We also indirectly quantified the impact of the host galaxy contamination in the optical band by separating star-forming galaxies from AGN, classified based on narrow lines diagnostics <cit.>, using several different classification methods (see Appendix <ref> and Fig. <ref>). We obtain that there is no significant difference in X-ray luminosity and stellar-mass between these two categories, implying that we are not biased toward X-ray detections only for galaxies with a strong central ionizing source inferred from the optical photometry or spectroscopy. Finally, the X/O predictions from AGN at low Eddington ratios are also, to some extent, contaminated by the galaxy in the optical-UV band <cit.>, validating our comparison in Fig. <ref>. We conclude that canonical AGN disk-corona SEDs <cit.> would predict the X-ray emission from the MBHs in these galaxies to be much brighter than observed, even for predictions of low-luminosity AGN <cit.>.We note that the possible X-ray weakness of MBHs in dwarf galaxies, or their unusual SEDs, compared to more massive AGN was reported before for a few of cases <cit.>, although this is the first confirmation on a large sample of fairly homogeneous X-ray exposures of dwarf galaxies. The optical variability selection in these galaxies (directly or through the infrared echo) is thought to indicate the presence of a variable radiatively-efficient AGN accretion disk <cit.>, whilst the X-ray upper limits and stacked X-ray images obtained in this work are, at best, compatible with AGN accreting at ∼10^-3-10^-4 L_edd and, at worst, consistent with and inactive or absent black hole. This begs the question of whether these two observables, UVOIR stochastic variability and X-ray data, are consistent. Before analyzing the possible physical interpretation and consequences, we briefly discuss possible biases that might cause MBHs to appear unusually X-ray weak (Sect. <ref>). We stress again that, in order to avoid strong redshift effects and to be consistent with the sources used for the X-ray stacking analysis, we limit the discussion to the 134 sources with X-ray products in eRASS:4 which are below z<0.1.§ ON THE POSSIBLE BIASES FOR THE OBSERVED X-RAY WEAKNESSFirst, we do not find any obvious correlation between X-ray (non-) detection and variability significance from the parent samples. For instance, among the galaxies <cit.> we have only detected the one with highest and the one with lowest variability significance, and the four detected galaxies from <cit.> are also homogeneously distributed in terms of variability significance. Furthermore, we investigated in Appendix <ref> whether the observed X-ray weakness depends on the variability significance, both for optically- <cit.> and IR-selected <cit.> variable galaxies. For the optically selected variable galaxies, we also investigate the dependence on the number of data points in the optical light curve or the total baseline. We show this in Fig. <ref> and <ref> and no significance trend is evident. For the optically selected variable galaxies, we also stacked lower- and higher-significance sources from <cit.> in the log M_*=9-10 bin separately and obtained no significant difference, although we found weak evidence indicating that the stacked image on the higher-significance galaxies contained brighter signal (see Appendix <ref>). Furthermore, we tested whether the observed X-ray weakness depends on the optical classification from narrow-lines diagnostics <cit.> using several techniques, and we found again no obvious difference (Fig. <ref> and Appendix <ref>). However, formally our X-ray observations did not confirm the nature of most of these MBHs as such. From X-rays alone, a possibility is that these galaxies would be mostly inactive and lack significant accretion all-together. Hence, a conservative possibility that we must consider is that the bulk of the variability-selected MBHs is contaminated, as also a bias spread to most of the light curves, regardless on the inferred variability significance, would appear uncorrelated with the X-ray non-detections. This is very unlikely, although it is still relevant to discuss possible known contaminants. Possible spurious sources within the methodology typically adopted to select variable AGN <cit.> could be long-lived stellar transients or variables <cit.>, although they are expected to contaminate the selected MBHs in small numbers. Another contaminating component which is nearly ubiquitous in these galaxies in the NSC, although its old stellar population is not expected to imprint any variability <cit.>. Therefore, for any bias in the optical photometry to impact our systematic X-ray weakness, it would have to be currently unknown and worryingly extended to the bulk of the parent galaxy samples. It is worth mentioning that, despite the large overlap in the parent sample of dwarf galaxies, variability studies using data from the Palomar Transient Factory <cit.> and the Zwicky Transient Facility <cit.> have limited overlap in their respective MBHs candidates. In particular, ∼11% of the ZTF candidates were selected also by PTF, and, viceversa, only ∼3% of the PTF candidates were also selected by ZTF <cit.>. However, the possible origin of this discrepancy may lie in the difference cadence, scatter and total baseline of data obtained with PTF and ZTF. In particular, PTF has median baseline in the parent sample of ∼4yr, reaching higher detection fractions for galaxies with baseline up to ∼6-7yr <cit.>, while ZTF data have a typical baseline of ∼3yr. Therefore, it is possible that the MBHs selected by PTF and missed by ZTF were mostly variable on timescales comparable with or longer than the ZTF baseline. This would be supported by the fact that the 5 in common have variability power at much higher timescales compared to the rest of ZTF-selected MBHs. Conversely, the ZTF-detected MBHs might have been missed by PTF due to its reduced sensitivity to variability over the timescale of months, compared to ZTF. Therefore, as much as some of the variable MBHs might be spurious sources (i.e. normal galaxies with a dormant black hole or no black hole all-together), this is unlikely to be the case for most of the 121 undetected X-ray MBHs of the low-z sample <cit.>. Without dedicated simulations quantifying the purity and completeness of the variability searches, we are unable to identify a subset of secure MBHs or to quantify the spurious fraction in our sample. Furthermore, <cit.> noted a lower X/O in their eight broad-line MBHs and discuss that enhanced nuclear star formation might be a contaminant to their optical-UV data. In our sample, the optical nucleus would have to be dominated by the galaxy to the extent of altering X/O, but not to the extent of impeding the detection of AGN-like optical variability on top of the galaxy continuum, which requires suspicious fine tuning of the ratio between AGN and galaxy in the optical, considering the several tens of X-ray weak sources found here. In <cit.>, it was noted that X/O variability and non-simultaneity would scatter the X-ray estimates toward both the brighter and fainter direction and not systematically toward the latter. We confirm this by cross-matching the eROSITA estimates with the fourth XMM-Newton serendipitous source catalog <cit.> and the Chandra Source Catalog <cit.>. We show in Fig. <ref> the resulting comparison, which shows compatible fluxes between the eROSITA, XMM-Newton and Chandra across the different epochs. As a consequence, since there is no evidence of any long-term variability effect between the X-ray epochs, it is unlikely that the X/O weakness is solely due to long-term variability.The possible role of X-ray absorption needs to be assessed, as it surely impacts some of these galactic nuclei. Using the observed WISE magnitudes and X-ray upper limits, we can put a 3σ lower-limit prediction on the N_H (cm^-2) required for these nuclei to be obscured, under the assumptions that they follow multiwavelength prescriptions of more massive obscured AGN. Using the relation between N_H, X-ray luminosity and W3 magnitude from <cit.>, the median lower-limit is log (N_H/cm^-2)>23.6. This implies that the typical dwarf galaxy in our sample would need to be Compton thick. In general, it is true that in the most extreme case ≈50% of the existing nuclear BHs are Compton thick <cit.>. However, the MBHs in this study are not simply randomly-selected low-mass galaxies for which this statistics may apply. They were selected through UVOIR variability, which therefore excludes that the SED is heavily obscured. Therefore, the observed X-ray weakness is unlikely to be due to extreme obscuration. Since our sample contains also IR-selected objects, let us still pessimistically assume that all IR-variable MBHs are X-ray obscured. One would still need to account for the remaining optically unobscured nuclei. Moreover, we observed X-ray weakness homogeneously between optically- and infrared-variable MBHs, which argues against systematic obscuration in all the nuclei of these dwarf galaxies. As a matter of fact, we stacked the X-ray images of the non-detected IR-selected and optically selected galaxies separately in the log M_*=9-10 bin and found compatible results and even weak evidence that the X-ray signal of the stacked IR-selected galaxies is brighter than the optically selected, which would argue against wide-spread obscuration in the latter. In particular, using as background estimate the median signal between 15-50kpc (see Sect. <ref>), we obtain a median value of L_0.5-2.0 keV=(1.0±0.9)× 10^39erg s^-1 and (1.0±0.7)× 10^39erg s^-1, for optically- and IR-selected non-detected MBHs, respectively. Instead, conservatively using as background estimate the 84th percentile of the signal between 15-50kpc the optically selected galaxies are non-detected at L_0.5-2.0 keV<1.6× 10^39erg s^-1, whilst the IR-selected ones are still detected at (7.3±6.9)× 10^38erg s^-1. Hence, X-ray obscuration is not considered to play a major role in the observed X-ray weakness.We conclude that it is likely that only some of the galaxies in our sample might suffer from one or more of the above-mentioned effects (spurious trigger in the variability searches, X-ray variability and X-ray absorption). The only way for biases to be extended to the whole sample studied here, would imply that most IR-selected MBHs are Compton thick and that most of the optically selected are systematically flawed by currently-unknown physical, instrumental or statistical contaminants. Arguably, this seems quite unlikely. Therefore, we discuss possible physical interpretations for the observed X-ray weakness in MBHs in dwarf galaxies.§ ON THE POSSIBLE PHYSICAL INTERPRETATIONS FOR THE OBSERVED X-RAY WEAKNESS We generically refer to a canonical corona <cit.> as a magnetically-powered plasma in the immediate vicinity of the black hole, with electrons kept hot and accelerated with a high duty cycle <cit.>. Its emission typically scales with the optical-UV emission for radiatively-efficient BHs <cit.> and with radio for the inefficient ones <cit.>. To summarize the intents of this section, in this work we have obtained that the majority or UVOIR-variable MBHs are X-ray weak, with luminosity similar to those of normal galaxies. In Sect. <ref> we controlled for potential biases, and excluded X-ray obscuration as a systematic contaminant. Under the assumption that UVOIR variability is a robust method that traces some level of accretion in these nuclei (be it radiatively-efficient or -inefficient), the central MBH must be active to some degree. Even for low Eddington ratios X-rays are expected and are, in fact, a significant or dominant contribution in the bolometric SED compared to optical and UV proxies <cit.>. Hence, here we discuss possible physical interpretations, which would be due to a different behavior present in low-mass nuclei compared to more massive ones: for instance, in a different structure or powering of the accretion disk-corona system, different fueling of gas and magnetic field toward the galaxy nucleus, or a different variability behavior.We start discussing the case in which the UVOIR variability is uniquely tracing temperature fluctuations in a radiatively-efficient accretion disk <cit.>, then the observed X-ray weakness compared to the optical would suggest that active MBHs do not follow standard AGN accretion SEDs or X/O values (e.g. see Fig. <ref>).Interestingly, in newborn (hence not accumulated secularly) accretion flows following tidal disruption events and quasi-periodic eruptions, which are observed in the same low-mass regime of the black hole and galaxy populations too <cit.>, the hard X-ray corona is usually missing <cit.>. However, if the lack of a canonical corona were to be the only cause of the X-ray weakness, then one would still expect to detect more of these MBHs by detecting the tail of the radiatively-efficient disk emission in the soft X-rays (where eROSITA is most sensitive), which is expected to be observable from these putative ∼10^5-10^6.5 M_⊙ black holes and it is, in fact, seen for the above-mentioned transients.Another option is that optical/IR variability searches would trigger not only stochastic variability from the thermal emission of a radiatively-efficient accretion disk <cit.>, but also variability from the nonthermal SED of radiatively-inefficient ones. This is most evident in the submillimiter <cit.>, but its SED extends to higher frequencies too <cit.>. In this case, no tail of the accretion disk emission is expected in the soft X-rays, therefore one needs to worry solely about the possible absence of a corona. For these radiatively-inefficient MBHs, one would expect the X-rays to align with X/O predictions of such accretion regimes and, most importantly, with radio estimates along the fundamental plane of black hole accretion <cit.>. However, neither the former (dashed and dotted red lines in Fig. <ref>) nor the latter (Fig. <ref>) is observed. In particular, in Appendix <ref> and Fig. <ref> we show that, despite the low sample statistics of sources with an archival radio flux above the SFR estimate, these MBHs are X-ray weak even in the fundamental plane. This is at odds with the interpretation that the observed X-ray weakness is merely due to the low-luminosity nature of these MBHs. We note that we used standard scaling relations with stellar mass <cit.> to obtain the black hole mass. In principle, if these black holes were overmassive with respect to their stellar masses, this would not only alleviate the tension with the fundamental plane, but also explain why we do not see the exponential tail of the accretion disk emission in the soft X-rays. However, since even the 3σ upper limit values are off by at least ∼1-1.5dex from the mean fundamental plane (Fig. <ref>), one would need to offset the black hole mass by at least ∼1.3-1.9dex <cit.>, which is quite extreme. Further, we note that the observed X-ray weakness in the fundamental plane is consistent with other results in the literature <cit.>, albeit still with low sample statistics. If confirmed in the future with wide area survey matches between X-rays, such as eROSITA <cit.>, and radio, such as ASKAP-EMU <cit.>, this would indeed imply that, at least in UVOIR-variable MBHs, X-rays are decoupled from both optical and radio, compared to standard accretion modes at other black hole masses. An intriguing option is that a significant fraction of MBHs in dwarf galaxies is spoon-fed by transient accretion events, e.g. by tidal disruption events <cit.>. In this case a corona is not necessarily expected and even if standard SEDs are seen in TDEs too <cit.>, their complex multiwavelength signatures surely do not follow standard AGN scaling relations at all times. For instance, a case-study of the possible intermittent activity in these galactic nuclei is the possible short-lived (<1.6 yr) flare that is thought to have recently happened (≈200yr ago) in the nucleus of the Milky Way <cit.>.However, the UVOIR variability was observed to be stochastic, non-transient and selected with baselines longer than the typical nuclear transient duration, and transient emission is normally excluded from these studies <cit.>. As much as unusually long-lived transients may contaminate some individual galaxies, it is unlikely that this contaminant is present in tens-hundreds of galaxies. More fundamentally, it would imply that TDEs are much more common than what both observations and theory suggest <cit.>. Alternatively, it is possible that MBHs in low-mass galaxies are typically powered with a much lower duty-cycle compared to more massive nuclei. Intriguingly, a low-luminosity analog with a lower duty cycle in X-rays compared to more frequent activity in the optical and infrared is Sgr A*. This is not an unreasonable example since the SED of Sgr A* is, for instance, compatible with that of M81, which is about four orders of magnitudes brighter <cit.>. The infrared variability of Sgr A* (and we assume, by extension, its optical too) appears stochastic with a red noise character <cit.>. Conversely, Sgr A* shows flares in the X-ray band for only ∼2% of the time, considering roughly a flare a day lasting ∼30min <cit.>. If this behavior were to happen in galaxies such as those in our parent sample, albeit at much higher luminosity compared to Sgr A*, it would potentially trigger stochastic random walk variability searches within the typical light curve cadences <cit.>, considering the red noise character of the IR light curve. On the other hand, in the X-ray band there would be a very high likelihood of catching the source in the quiescent state, therefore the OIR-variable galaxy would appear undetected in X-rays. However, a low-duty cycle is generally unlikely to explain the ubiquitous X-ray weakness we observe, since eROSITA and archival XMM-Newton/Chandra X-ray fluxes, taken at different epochs separated by years, align quite nicely for the few sources in common (Fig. <ref>). Therefore, it would be quite unlikely to have the putative low duty-cycle impacting the X/O and X/radio ratios only, and not the X-ray versus X-rays long-term comparisons. Hence, we discuss a possible physical picture for our UVOIR-variability selected MBHs. UVOIR variability is likely tracing both thermal and nonthermal processes <cit.> in the accretion flow, depending on the accretion rate of the source. Thus, the MBHs found through these variability searches can be both radiatively-efficient and -inefficient <cit.>, depending on the overall luminosity and SED (Fig. <ref>). The fainter accretion regime is unsurprisingly more common <cit.>, hence the high number of non-detected MBHs in dwarf galaxies, which are also predicted to be dominant from simulations <cit.>. For these inefficient MBHs, radio traces their synchrotron continuum as expected, forming a nuclear SED to which X-rays should contribute too <cit.>, were these MBHs to follow standard scaling relations valid at other black hole masses <cit.>, but somehow they do not seem to be (e.g., Fig. <ref>). Hence, the X-rays are weak compared to both efficient (i.e. optically-bright) and inefficient (i.e. radio-bright) accreting MBHs. Therefore, it would seem natural to conclude that a canonical X-ray corona might be missing in the bulk of the MBH population in dwarf galaxies all-together. As much as there is general agreement that the X-ray corona is magnetically powered, the formation mechanism of this highly magnetized coronal region is still unsolved <cit.>. This likely requires that gas with a large magnetic field is funneled toward the black hole <cit.>. This is a highly uncertain and understudied field, but we may interpret our observational result as follows, namely that MBHs in dwarf galaxies are not as efficient as more massive ones in sustaining a magnetically-powered corona. Under the assumption that the magnetization of the corona and that of the large-scale gas feeding the black hole are somehow linked, this means that the strength and order of the magnetic field in the nuclei of low-mass galaxies is less effective, compared to more massive galaxies and nuclei <cit.>.We now outline a few major differences between low-mass and massive galaxies. As a matter of fact, dwarf galaxies have a much shallower nuclear potential well which might cause the lack of a clear galactic center all-together <cit.> and observations of compact dwarf galaxies indeed clearly show a rather clumpy and inhomogeneous interstellar medium <cit.>. Furthermore, dwarf galaxy mergers do not seem to funnel gas toward the nucleus as efficiently as in more massive mergers <cit.> and morphological studies indicate major mergers are rarer at the low-mass end <cit.>. Another major difference between low- and high-mass galaxies is the high fraction of nuclear star clusters in the former and the lack thereof in the latter. Indeed, NSCs are thought to be directly linked to the growth of the MBH <cit.>. Whether (and how) all the above-mentioned differences eventually impact the formation and powering of the X-ray corona (still, in general, an open question) at ∼10 gravitational radii remains to be established. We invoke further study on the magnetization of galaxies of different masses and their connection with the channeling of gas toward the central regions of the galaxy down to the black holes. Until then, the scenario discussed here is merely a tantalizing possibility which can not be quantitatively supported. § SUMMARY AND FUTURE PROSPECTS The search for MBHs (M_BH≈10^4-10^6 M_) in the nuclei of low-mass galaxies (M_*⪅10^10 M_) is of paramount importance to constrain black holes seeding and their growth over time, although it is a challenging task (e.g. seefor a recent review). A promising way to find MBHs at lower luminosity, compared to searches based on broad and narrow optical lines, was provided by the growing number of high-cadence photometric surveys which allow selection of MBHs through UVOIR variability.In this less efficient accretion regime, X-ray and radio searches are also particularly useful in finding and confirming low-luminosity MBHs, although these observations have been so far limited to small samples.This is where eROSITA <cit.> comes into play with its homogeneous all-sky survey and its selection function calibrated with simulations <cit.>. It is also common practice, when there is not an a priori knowledge on the presence of a MBH in the nucleus, to study subsamples of galaxies with multiwavelength detections across the SED. However, this approach is naturally limited in studying a biased selection of active MBHs with canonical SEDs. Ultimately, it is still unclear to what extent selection techniques from different wavebands compare with one-another at the fainter end of accretion. In this work, we presented the first large systematic investigation of the X-ray properties of a sample of known MBH candidates, which has the advantage of providing a sample with occupation and active fraction of one. We focused on MBHs selected through UVOIR variability (Sect. <ref> and Fig. <ref>). In Sect. <ref>, we extracted X-ray photometry and spectra (e.g., Fig. <ref>) of a sample of 214 (208) UVOIR variability-selected MBHs from the eRASS1 (eRASS:4) image and significantly detect 11 (17) of them, hence 5.1_-1.5^+2.1% (8.2_-2.0^+2.5%; Sect. <ref>). The detection fraction mildly increases with the stellar mass of the galaxy (bottom left panel of Fig. <ref>) and so does the observed X-ray luminosity (Fig. <ref>). We present a summary of our sample and the X-ray results in Table <ref>. Out of the 17 detected galaxies from the deeper eRASS:4 image, 4 are newly-discovered X-ray sources (Table <ref> and Fig. <ref> and <ref>), two of which are securely X-ray counterparts of the variable MBHs, whilst the other two remain ambiguous (Sect. <ref>).For the first time on a large (∼200) number of galaxies, we dedicatesignificant attention to the many of them which are undetected in X-rays (Sect. <ref>). The eROSITA survey is shallow (e.g. the median net exposure for this sample is ∼550 s in eRASS:4), although its selection function as a function of X-ray flux is well-calibrated from all-sky simulations <cit.>. Most importantly, stacking the images of non detected sources results in a L_X estimate which is orders of magnitudes fainter than the X-ray detections, and consistent with the predictions of the emission of the galaxy alone (bottom panel of Fig. <ref>). In particular, no X-ray signal is detected in the stacked images below log M_*=9. However, the X-ray emission of normal galaxies and radiatively-inefficient, hence low-luminosity, AGN becomes notoriously indistinguishable, specially if it is unresolved. Nonetheless, the advantage of the parent sample being composed by known MBHs from UVOIR-variability is to exclude that these MBHs are overall intrinsically faint. Therefore, their X-ray weakness in comparison with their UVOIR variability is puzzling. In particular, we investigate that most X-ray 3σ upper limits are so deep that they lie well below the predictions based on more massive AGN, both for radiatively-efficient (comparing X-rays with predictions from optical proxies, Fig. <ref>) and -inefficient ones (comparing with radio proxies, Fig. <ref>). However, X/O comparisons are surely contaminated by the galaxy and future work will need to reproduce this analysis decomposing the AGN contribution from the optical-UV magnitudes used (Fig. <ref> and <ref>), and X/radio comparisons in this work are limited by much lower statistics (Fig. <ref>) and will need to be assessed with larger radio samples.We carefully considered potential biases which would cause the observed X-ray weakness to be non-intrinsic (see Sect. <ref>): for instance, we find that X-ray obscuration (Sect. <ref>) and variability across the epochs or a low duty-cycle (Fig. <ref> and Appendix <ref>) are unlikely to be responsible for the almost 200 non-detected galaxies. Furthermore, the X-ray weakness was not found to depend on the variability significance in IR-selected galaxies (Fig. <ref>), nor on the number of data points and total baseline in the optical light curves (bottom panel of Fig. <ref>). We only find weak evidence that the stacked X-ray signal is slightly brighter for galaxies with higher significance variability in the optical (Appendix <ref>), although no significant differences were found (see also Fig. <ref>). Since, formally, our work was not able to confirm most MBH candidates despite the eRASS:4 survey being sensitive enough, another possibility we must conservatively consider is that variability-selected MBH samples are severely biased by unknown contaminants, or unknown methodological flaws, spread to all variability significance values. This would imply that these galaxies are inactive and that they lack significant accretion in their nuclei. Everything considered (see also Appendix <ref>), this is admittedly very unlikely. Therefore, the observed X-ray weakness has to be intrinsic to the bulk of the low-mass galaxies population, or at the very least that selected via UVOIR variability. Hence, this might imply that a canonical X-ray corona is lacking in these nuclei. In Sect. <ref>, we discuss that a possible explanation for this might lie in the fundamental differences between the nuclei of low-mass galaxies and the more massive ones. For instance, the shallower potential well and clumpier interstellar medium in the former, compared to the latter. However, it remains to be quantitatively addressed whether these differences lead to a inefficient magnetization of the nuclear gas <cit.> and whether this ultimately affects the powering of the corona at very small scales (∼10 gravitational radii).An indirect way to confirm the presence of a systematic X-ray (and X-ray only) weakness in the MBHs SEDs, would be to analyze the UVOIR variability property <cit.> and radio incidence and X/radio ratios <cit.> of an X-ray selected MBH sample. If a comparably puzzling low confirmation rate is obtained, this would imply that all single-band searches are incomplete (and not only X-ray selections) and can not be used as representative for the MBH population.Discouragingly, constraining the occupation fraction in low-mass galaxies was already known to be a challenging task in general <cit.>. However, even if the bulk of the dwarf galaxy population were to be intrinsically X-ray weak, or with unusual SEDs, there is a minority of (observationally) well-behaved galaxies which are detected throughout the SED, providing useful lower limits for the active and occupation fractions <cit.>. These would be less constraining than anticipated, but may still serve in ruling out pessimistic seeding models. Hence, this work serves as a pilot study for future synergies between eROSITA and LSST. We rely on the extensive simulated observations recently performed in <cit.> as benchmark for the expected number of variable MBHs detected by LSST. Following the assumptions and criteria used in <cit.>, we compare LSST predictions with our detection fractions between M_* = 10^8-10 M_ and below z<0.055: 3.4_-1.0^+2.6% for eRASS1 and 6.4_-1.5^+3.0% for eRASS:4. We adopt the predicted LSST MBHs numbers from <cit.> of 1.5_-0.6^+0.6× 10^3 and 5.9_-1.1^+1.5× 10^3, obtained using light and heavy seed models, respectively. Therefore, on the order of ≈20-130 and ≈155-440 in eRASS1 (≈45-195 and ≈235-695 in eRASS:4), based on light and heavy seed models, of LSST's MBH candidates may be detected and, hence, confirmed. We note that these numbers are most likely lower limits, as LSST is expected to be more complete in sampling the intrinsic stellar mass and magnitude distribution, compared to the inhomogeneous sample used in this work (e.g. see Fig. <ref> and <ref>). We thank the referee for their positive comments on the manuscript.R.A. is grateful to C. J. Burke, V. Baldassare, F. Pacucci, U. Chadayammuri, I. Chilingarian, M. Begelman, J. Silk and S. von Fellenberg for insightful discussions about their works. We acknowledge the use of the matplotlib package <cit.> and Photutils, an Astropy package for detection and photometry of astronomical sources <cit.>. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. R.A received support for this work by NASA through the NASA Einstein Fellowship grant No HF2-51499 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. G.P. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. [865637]), and support from Bando per il Finanziamento della Ricerca Fondamentale 2022 dell’Istituto Nazionale di Astrofisica (INAF): GO Large program. This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tuebingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universitaet Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS software system developed by the German eROSITA consortium. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyzes of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF’s NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program “The Emergence of Cosmological Structures” Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO. aa § FURTHER DIAGNOSTICS ON THE X-RAY WEAKNESS OF MBHS Here, we perform some tests to further investigate the presence of biases in our interpretation of the systematic X-ray weakness observed in our sample. First, we check that X-ray weakness does not depend on the variability significance. We performed this test for the optically selected galaxies in <cit.>. In these works, the quantity σ_var is the significance that the object is generally variable, while σ_QSO that the damped random walk model adopted for AGN-like variability <cit.> is significant compared to non-AGN-like variability, given by σ_NoQSO <cit.>. These estimates yield high-purity in quasars samples <cit.> and we assume compatible purity is obtained for more nearby dwarf galaxies. Fig. <ref> shows that the X-ray weak upper limits are not biased toward lower significance sources. Most X-ray weak upper limits have high σ_var and σ_QSO - σ_NoQSO, therefore we do not expect that more than a handful of the parent MBHs in dwarf galaxies to be spuriously detected. To test this more quantitatively, we stacked the 39 galaxies within log M_*=9-10 and below z<0.1, selected from from <cit.> and non-detected in eRASS:4. We divided low- and high-significance sources using σ_var=6 <cit.> as threshold, which grants an equal number of 20 and 19 galaxies in the two subsamples. Using as background estimate the median signal between 15-50kpc (see Sect. <ref>), the low-significance subsample is undetected in the stacked image with an upper limit at L_0.5-2.0 keV<4.2× 10^38erg s^-1. Conversely, the high-significance subsample is detected at L_0.5-2.0 keV=(9.3±7.2)× 10^38erg s^-1. However, if we use conservatively the 84th percentile of the signal between 15-50kpc as background estimate (see Sect. <ref>), the high-significance subsample is undetected as well, with an upper limit at L_0.5-2.0 keV<1.3× 10^39erg s^-1. Therefore, while this indicates that there is weak evidence of the high-significance subsample being brighter in X-rays, the difference is not significant enough. Finally, from the bottom panel of Fig. <ref> we note that there are not obvious biases of having the deepest X-ray non-detections toward shorter baselines, or toward low number of data points, in the optical light curves. We perform the same check on the IR-selected galaxies from <cit.>, where variability significance was expressed as a function of the Pearson correlation coefficient (r_pearson) between the binned W1 and W2 light curves and the related χ^2 values (e.g. χ^2_W1), both aimed to quantify variability compared to the median value of the light curve. Similarly to the optically selected sources, from Fig. <ref> we note that the X-ray weak upper limits are not biased toward lower significance sources. Hence, we conclude that the spurious fraction in the parent sample of optically- and IR-variable galaxies is not significantly higher for lower-significance variability.In Sect. <ref> and Fig. <ref> we have inferred that the MBH population is X-ray weak compared to the X-ray flux predicted from the optical luminosity. Since the u-band filter used in Fig. <ref> has an effective wavelength of ∼3565Å, whilst the adopted scaling relations are calibrated at ∼2500-3000Å <cit.>, we here test the use of the near-UV filter of GALEX <cit.>, which has an effective wavelength of ∼2300Å. We show the equivalent of Fig. <ref>, but with GALEX data, in Fig. <ref>. We note that the comparison between observed X/O values and model predictions remains qualitatively the same and in fact using GALEX even fainter X/O values are obtained (cf. Fig. <ref>).The bottom panel of Fig. <ref> and in Fig. <ref> do not include the classification of the galaxies based on optical spectra. Here, we investigate the dependency of the observed X-ray weakness on the classification of the galaxy based on optical photometry and spectroscopy, as an independent proxy compared to the UVOIR selection. However, we note that the UVOIR variability method is knowingly selecting AGN candidates in galaxies classified as inactive <cit.>. First, we retrieved the galaxy classification of our sample from the Reference Catalog of galaxy SEDs <cit.>, between z=0.01-0.1. The lower end is chosen to avoid aperture biases, the higher end is chosen to limit the analysis to the range in which X-ray non-detections were stacked. A handful of sources which were either missing in the database or had spectra with poor quality were excluded. This analysis was limited to 99 galaxies. We show in the top panel of Fig. <ref> the equivalent of the bottom panel of Fig. <ref>, to which we added subpanels with histograms and a different color coding. We highlight in green (squares for detections, arrows for non-detections) the galaxies classified as star-forming from the BPT narrow lines diagnostics <cit.>, whilst in red (diamonds and arrows) those classified as Composite or as AGN. In addition, we highlight with orange contours the galaxies classified as star-forming, but for which RCSED reports a significant detection of a broad Hα line. Furthermore, we also estimate the activity classifications with the updated version of HECATE catalog (Kyritsis et al., in prep.). The classifications are based on two different methods. The first one is an advanced data-driven version of the traditional BPT diagrams, which utilizes a soft clustering scheme for classifying emission-line galaxies in different activity classes using simultaneously four emission-line ratios <cit.>. The second one is based on the application of the Random Forest machine learning algorithm on mid/IR (W1-W2, W2-W3; WISE) and optical (g-r; SDSS) colors and can discriminate galaxies into 5 activity classes (i.e star-forming, AGN, “Composite”, “LINER”, and “Passive”; ).Both activity classification methods are probabilistic, meaning that they provide the probability of a galaxy to belong in each class, and an example of their application is presented in the work of Kyritsis et al. (in prep.) for the selection of all the bona-fide star-forming galaxies which were observed by the eRASS1 all-sky survey. First, we confirmed that the two methods yielded similar results from one-another, compatibly with the top panel of Fig. <ref>. Then, in the middle panel of Fig. <ref> we show the combination of the two above-mentioned methods from the HECATE catalog: we represent a galaxy in red (green) if either the emission-lines diagnostics or the Random Forest consider it as AGN, “LINER” or “Composite” (star-forming or passive). Similarly to the top panel of the same figure, there is no significant different among the two sets of classifications. Finally, in the bottom panel of Fig. <ref> we color-code the plot with the probability of a galaxy to host an AGN based on the Random Forest algorithm. Again, we identify no major bias: x-ray detections are found at all P_AGN and non-detections do not seem to strongly depend on P_AGN either. This tests highlights that there is no significant difference between the X-ray weakness of galaxies classified as star-forming, compared to those classified as active.Furthermore, we have checked the impact of X-ray variability, although it is expected to yield a scatter in both brighter and fainter directions and not the latter only. As a matter of fact, we have crossmatched the eRASS:4 low-z galaxies with the fourth XMM-Newton serendipitous source catalog <cit.> and the Chandra Source Catalog <cit.>. We added a handful of sources from <cit.>, which were not included in the catalogs (namely NSA IDs 156688, 104881, 51928, 67333, 124477). We show in Fig. <ref> the resulting comparison, where the 1:1 (with related 0.5 dex scatter) is show with a solid (dashed) line. Different energy bands might have been used across different sources, although consistent bands are used between eROSITA and other missions for the individual source. Different symbols are used for XMM-Newton (squares) and Chandra (circles), while different colors highlight eROSITA detected (green) and non-detected (gray) sources. Detections withXMM-Newton and Chandra are highlighted with green contours for visualization purpose. All the sources detected by eROSITA and either XMM-Newton or Chandra (with observations taken between 2015 and 2022) show compatible fluxes across the different epochs. All eROSITA upper limits (apart from one) are brighter than the detection with XMM-Newton or Chandra, therefore they are compatible with the 1:1 and were not supposed to be detected by eROSITA. Upper limits in both missions (gray data points with black contours) are, by definition, compatible with the 1:1. Therefore, we confirm that the impact of variability or a low duty cycle in these galaxies has to be minimal.In order to quantify how the X-ray weakness compares with the radio properties of the MBHs, we cross-matched our low-z sample (Fig. <ref>) with radio archives[https://heasarc.gsfc.nasa.gov/W3Browse/master-catalog/radio.htmlLink to radio catalog], the Rapid ASKAP Continuum Survey <cit.> and the second data release of the LOFAR Two-metre Sky Survey <cit.>. We then convert the observed radio fluxes to 5GHz luminosities assuming both a spectrum with radio spectral index -1 (top panel of Fig. <ref>) and a flat spectrum (bottom panel of Fig. <ref>). We estimated the black hole masses from the stellar masses of the galaxies <cit.> and plotted our sources in the fundamental plane of black hole accretion <cit.>. We show this in Fig. <ref>. To be conservative, we draw the main conclusions from the top panel as it shows the faintest 5 GHz luminosity from the extrapolations. Realistically, radio spectra of these sources would be a mixed bag between slopes of minus one and zero, therefore between the two panels. We note that both X-ray and radio fluxes are likely contaminated by the galaxy. Therefore we computed the radio luminosity at 5GHz as predicted by star-formation in the galaxy <cit.>. In Fig. <ref>, we highlight in orange MBHs with a SFR estimate available from the MPA-JHU catalog <cit.> and with a radio luminosity greater than that predicted for the galaxy alone <cit.>. The sample statistic is now very low, although X-ray weak 3σ upper limits remain. This is more evident if a flat radio spectrum is assumed.Hence, MBHs appear to be X-ray weaker even compared to the fundamental plane, including the large intrinsic scatter of ∼0.88dex of the relation. This is at odds with the interpretation that X-ray weakness is simply due to the low-luminosity nature of these MBHs. § TABLES We present eRASS1 and eRASS:4 results on all MBH candidates in Table <ref> and more details on the detected sources in Table <ref>. =ccccccccc Input MBH candidates from optical/IR/UV variability and related eROSITA information from aperture photometry and spectroscopy. 5cInput 2ceRASS1 2ceRASS:4 1cRA 1cDec 1cz 1cM* 1cRef.a 1cP_Bb 1cL_0.2-2.0 keVc 1cP_Bb 1cL_0.2-2.0 keVc 3lContinued 5cInput 2ceRASS1 2ceRASS:4 1cRA 1cDec 1cz 1cM* 1cRef.a 1cP_Bb 1cL_0.2-2.0 keVc 1cP_Bb 1cL_0.2-2.0 keVc 48.25895 -0.6863790.131 9.8Ba182.62e-3243.41_43.29^43.53 (_43.11^43.67) 6.19e-6643.19_43.11^43.28 (_43.0^43.39) 49.42965 0.326904 0.069 9.7Ba181.51e-1342.32_42.15^42.49 (_41.91^42.7)3.23e-2842.19_42.08^42.3 (_41.94^42.46) 47.61596 -0.8307910.089.74 Ba18044.2_44.18^44.22 (_44.15^44.25)044.12_44.11^44.14 (_44.09^44.15) 55.87608 -7.58542 0.036 9.84Ba20 1 <39.78 (40.85)0.94<39.36 (40.35) 154.7836519.98218 0.039 9.88Ba20 1 <39.94 (41.06)0.44<39.85 (40.8) 57.34663 -11.990950.032 9.36Ba20 044.02_44.01^44.02 (_44.0^44.04)043.86_43.85^43.86 (_43.85^43.87) 123.4507 37.83548 0.029 9.07Ba20 1 <39.88 (40.98)0.23<40.1 (40.87) 184.5559920.07689 0.046 9.33Ba20 0.34<40.76 (41.44)0.61<39.99 (40.9) 140.5216735.425 0.025 9.82Ba20 0.05<40.82 (41.28)0.2 <39.53 (40.39) 115.8114319.49506 0.046 9.81Ba20 0.08<41.35 (41.88)0.12<40.76 (41.36) 186.699777.671550.002 8.24Ba20 1 <37.21 (38.33)0.83<36.96 (37.93) 120.1189515.45310.015 9.04Ba20 0.04<40.48 (40.88)0.05<40.16 (40.43) 132.6348629.09562 0.049 9.66Ba20 0.24<40.96 (41.73)0.11<40.95 (41.4) 191.646882.369110.048 9.87Ba20 043.95_43.93^43.97 (_43.91^43.99) 043.91_43.9^43.92 (_43.89^43.93) 132.7513428.76201 0.034 9.91Ba20 1 <39.93 (41.04)1 <39.51 (40.52) 198.987677.284190.044 9.5Ba201 <39.93 (40.96)0.81<39.76 (40.7) 126.8777219.36663 0.036 9.76Ba20 0.3 <40.66 (41.46)0.13<40.75 (41.17) 124.5495427.71920.038 9.59Ba20 1 <39.96 (41.09)0.15<40.28 (40.99) 145.721869.493380.011 8.85Ba20 0.32<39.55 (40.31)0.05<39.54 (40.09) 148.9334635.61131 0.051 9.82Ba20 1 <40.12 (41.26)0.75<40.0 (40.92) 204.2622818.17054 0.026 8.45Ba20 1 –1 – 119.844159.619930.009 9.75Ba20 1 <38.66 (39.74)0.4 <39.01 (39.69) 202.4247 10.61023 0.027 8.77Ba20 1 <39.47 (40.51)1 <39.02 (40.03) 183.0686117.24661 0.028 8.59Ba20 0.43<40.21 (41.08)0.61<39.42 (40.4) 210.453439.223210.021 9.7Ba201 <39.15 (40.25)0.76<39.06 (39.97) 139.0633217.44624 0.031 9.48Ba20 1 <39.8 (40.98) 0.13<40.3 (40.96) 144.1143123.31915 0.028 9.41Ba20 1 <39.78 (40.86)0.74<39.56 (40.44) 220.052932.795420.039.49 Ba201.93e-0541.22_41.01^41.39 (_40.61^41.59) 5.23e-0941.05_40.91^41.19 (_40.67^41.38) 217.736336.083960.029 9.11Ba20 0.63<40.0 (40.81) 0.88<39.37 (40.31) 221.620213.236740.026 9.49Ba20 1 <39.51 (40.42)0.87<39.29 (40.16) 189.259456.925270.005 8.7Ba200.6 <38.47 (39.46)0.77<38.0 (38.93) 125.1103322.38440.026 9.57Ba20 1 <39.83 (40.88)0.27<40.23 (40.83) 184.1644 14.26028 0.023 8.4Ba201 <39.41 (40.61)3.82e-03<40.59 (40.87) 189.609136.049940.049 9.8Ba201 <40.47 (41.35)0.58<40.11 (40.95) 180.9274814.37081 0.043 9.7Ba200.39<40.54 (41.46)0.84<40.0 (40.92) 146.080889.9848 0.019.41 Ba200.28<39.4 (40.34) 0.76<38.6 (39.5) 192.1953426.74864 0.045 8.27Ba20 1 <39.95 (41.05)0.88<39.62 (40.64) 159.3182838.10653 0.051 9.65Ba20 1 <40.16 (41.29)0.45<40.27 (41.11) 187.7285810.82270.049 9.84Ba20 1 <40.0 (41.06) 0.87<39.92 (40.8) 179.2669622.31264 0.023 9.03Ba20 1 <39.58 (40.61)1 <38.92 (40.02) 68.17618 -4.38251 0.015 9.43Ba20 1 <39.27 (40.24)0.36<39.36 (40.05) 184.2886112.45432 0.007 9.43Ba20 0.4 <39.48 (40.01)4.27e-1039.86_39.74^39.96 (_39.54^40.1) 155.0980321.34426 0.049.62 Ba200.26<40.78 (41.39)0.15<40.68 (41.17) 126.8497323.18012 0.018 7.66Ba20 1 <39.45 (40.5) 0.71<39.11 (40.07) 151.037044.270280.054 9.98Ba20 0.04<41.68 (42.01)0.11<40.99 (41.37) 144.8969 6.414440.025 9.58Ba20 0.03<40.95 (41.36)7.50e-03<40.65 (41.02) 160.2730425.13649 0.054 9.82Ba20 0.35<40.96 (41.74)0.06<41.07 (41.47) 189.544084.7583 0.049 9.37Ba20 0.56<40.47 (41.39)0.02<40.92 (41.37) 138.5613817.11853 0.028 9.59Ba20 0.23<40.51 (41.12)0.64<39.67 (40.55) 146.0434334.04196 0.043 9.4Ba201 <40.0 (41.1)0.81<39.73 (40.73) 182.6769213.31345 0.023 9.27Ba20 0.38<39.89 (40.75)0.72<39.75 (40.71) 185.392024.779530.007 9.67Ba20 0.46<38.41 (39.44)0.72<37.87 (38.84) 120.753928.645680.034 9.83Ba20 1 <39.89 (40.96)1 <39.36 (40.46) 117.1994528.23974 0.028 9.95Ba20 1 <39.81 (40.89)1 <39.3 (40.26) 148.5600810.47307 0.049.82 Ba201 <39.97 (41.0) 1 <39.49 (40.58) 52.479801-28.733200.048.6 B221 <39.54 (40.63)0.37<39.59 (40.46) 53.036098-27.520001.3 9.8B22 1 –1 – 9.2741003-44.668300.778.0 B220.01<44.24 (44.67)1.93e-0543.88_43.63^44.13 (_43.02^44.54) 52.871101-27.393490.346 9.4B22 0.17<42.36 (43.07)1 – 8.8598003-44.635101.159.4 B221 <43.19 (44.51)0.54<43.16 (44.09) 52.512298-27.546800.527 10.0B221.66e-0543.9_43.63^44.11 (_43.2^44.35) 8.07e-2644.05_43.93^44.17 (_43.77^44.32) 53.168899-28.606190.322 9.9B22 0.18<42.31 (43.16)0.17<42.36 (42.91) 9.079099 -44.21459 0.25 8.9 B22 0.42 <42.16(43.06)0.84 <41.41(42.45)8.4623 -43.9351 0.85 9.9 B22 0.05<44.18(44.76)7.04e-03 <44.05(44.48)53.1301 -27.6189 0.123 9.4 B22 1 <40.68(41.65)1 <40.25(41.21)52.180198 -28.8197 3.354 7.5 B22 1 <43.64(44.9) 0.43 <44.15(45.21)53.03739 -28.749099 0.28 8.4 B22 1 <41.51(42.52)0.22 <41.73(42.48)8.4967 -43.444499 0.80 9.9 B22 1 <42.75(43.95)0.21<43.13(44.05)9.263999 -44.610099 0.25 8.9 B22 0.43<42.06(43.03)0.86 <41.37(42.37)8.6218 -43.3488 0.57 9.8 B22 1.99e-03<44.28(44.64)1.47e-0843.94_43.73^44.14 (_43.42^44.41) 52.18659 -27.2649 0.21 9.5 B22 1 <41.73(42.75)0.43<41.64(42.48) 52.10599 -28.895299 0.65 8.2 B229.6e-02<43.42(43.96)3.81e-03<43.58(43.91) 10.3934 -44.239398 0.15 7.2 B22 0.35 <41.64(42.68)0.53<41.13(42.0)51.79159 -28.720199 0.39 9.8 B22 1 <41.76(42.84)0.75 <42.0 (42.95) 52.25859 -28.0694 0.1528 8.9 B22 1 <40.85(41.87)0.79 <40.63(41.65) 8.7887 -44.539398 0.16 9.3 B22 0.4 <41.88(42.65)0.50 <41.49(42.25) 9.054499 -43.0345 0.19 8.8 B22 0.02<42.82(43.25)0.01 <42.41(42.74) 52.472599 -28.883699 2.0135 7.5 B220.58<43.61(44.93)0.22<43.7 (44.68)53.326499 -28.563499 0.16 9.0 B22 0.49 <41.35(42.28)0.93 <40.7 (41.63) 52.5881 -27.4328 0.206 9.7 B22 0.04<42.36(42.76)0.12 <41.41(42.15)52.1963 -27.264999 2.98 7.3 B228.45e-03<45.6 (46.0) 0.01 <45.22(45.61)10.315899 -43.223701 0.65 9.7 B22 1<42.61(43.79)0.57 <42.4 (43.48) 10.2382 -44.521598 0.32 8.9 B22 0.28 <43.02(43.9) 1– 9.846699 -44.125801 0.85 7.4 B22 0.08<44.22(44.86)0.11<43.72(44.19)52.1265 -28.419099 0.22 9.9 B22 0.26 <41.5 (42.49)0.56 <41.1 (42.04)52.18939 -27.1881 1.3 9.7 B22 1.08e-03 <44.72(45.13)1.58e-03 <44.16(44.6) 53.3412 -27.6804 0.148 9.6 B22 0.58<41.0 (42.01)0.17<41.22(41.87) 52.1851 -27.2276 0.13 9.5 B22 1<40.82(41.91)1–52.933998 -28.083499 0.45 9.7 B22 1<42.01(43.03)0.59<42.08(42.97)52.4207 -28.089399 0.081 7.9 B22 1 <40.19(41.19)0.94<39.93(40.82) 10.387399 -43.4743 0.24 9.7 B22 1<41.62(42.62)0.06 <42.39(42.86)10.4139 -43.7233 0.6 9.7 B22 0.14<43.43(44.09)1.27e-0443.90_43.61^44.45 (_43.07^44.45) 52.2588 -28.068799 0.153 8.9 B22 1 <40.93(41.77)0.56 <40.71(41.69) 9.0171 -43.297901 0.33 9.3 B22 0.03<43.33(43.85)1.37e-03<43.2 (43.53)10.013799 -44.398601 0.24 9.4 B22 0.44 <41.96(42.98)0.58 <41.53(42.43) 52.16749 -27.786399 0.24 8.6 B22 1 <41.35(42.34)0.50 <41.78(42.65) 8.464699 -44.4332 0.16 8.7 B22 0.37<41.84(42.62)0.49<41.35(42.13) 52.84619 -28.5128 0.28 9.3 B22 1 <41.46(42.44)0.78 <41.43(42.38)10.145099 -44.459499 0.8 10.0 B22 0.161<44.0 (44.7) 7.63e-03 <44.21(44.58)8.9757 -43.5718 0.21 8.7 B22 0.07<42.55(43.08)7.59e-03<42.63(42.97) 52.4536 -29.074499 0.0713 9.7 B22 0.22 <40.85(41.55)0.82<40.0 (40.89) 190.636092 -1.3507180.004 8.75W22Z0.6 <38.14 (39.13)0.43<38.06 (38.86) 116.775433 37.61536 0.036 9.54W22Z1 <40.04 (41.12)0.43<39.71 (40.74) 187.780908 0.463180.021 9.51W22Z0.66<39.67 (40.62)0.71<39.15 (40.16) 189.182179 -3.0208820.008 8.12W22Z0.53<38.89 (39.75)0.74<38.44 (39.31) 154.950492 3.521796 0.032 9.05W22Z1 <39.92 (40.99)0.37<40.03 (40.74) 206.293083 -3.0224340.047 9.42W22Z0.01<41.48 (41.8) 0.08<40.68 (41.21)149.581283 1.99234991.332 7.58Ki20 0.37<43.76 (44.85)0.03<44.09 (44.73) 150.018203 2.25944990.045 7.95Ki20 1 <40.15 (41.31)0.66<40.02 (40.82) 150.300659 2.59822980.041 8.04Ki20 0.09<41.09 (41.61)0.02<41.16 (41.43) 150.708801 2.66564980.165 8.34Ki20 1 <41.38 (42.39)1 <40.83 (41.81) 150.696624 1.98579000.291 8.6Ki201 <41.93 (42.96)1 <41.57 (42.63) 150.455932 1.71644991.733 8.78Ki20 1 <43.49 (44.9) 1 <43.17 (44.41) 150.407669 1.86769990.068 8.84Ki20 1 <40.43 (41.55)1 <39.91 (41.03) 150.391540 2.59115000.728.88 Ki201 <42.81 (43.96)1 <42.43 (43.47) 149.856933 2.63803000.516 8.92Ki20 0.37<42.96 (43.94)0.18<42.97 (43.67) 150.496139 2.32750011.661 8.95Ki20 1 <43.55 (44.98)0.78<43.45 (44.63) 150.467742 1.71527001.764 8.95Ki20 1 <43.49 (44.99)0.64<43.44 (44.61) 149.694030 2.33438990.497 9.07Ki20 1 <43.22 (44.09)1 <42.39 (43.54) 150.383422 1.85425990.716 9.14Ki20 0.3 <43.31 (44.21)0.66<42.53 (43.57) 149.970794 2.74957991.299.38 Ki201 <43.29 (44.54)0.75<43.12 (44.15) 149.532913 1.95887000.565 9.42Ki20 0.38<42.89 (43.99)0.77<42.25 (43.28) 150.233535 2.68261001.355 9.52Ki20 1 <43.39 (44.71)0.7 <43.15 (44.3) 150.714324 2.06321002.658 9.54Ki20 0.26<44.18 (45.7) 0.66<43.55 (45.06) 149.682632 2.43826000.328 9.68Ki20 1 <41.95 (43.13)1 – 150.143249 2.60620990.731 9.72Ki20 1 <42.79 (43.91)0.19<42.9 (43.72) 150.578582 2.17597002.045 9.73Ki20 0.35<44.34 (45.44)5.11e-03<45.12 (45.57) 150.067337 2.48515001.322 9.76Ki20 0.35<43.83 (44.8) 0.76<43.08 (44.32) 150.717987 2.46849983.204 9.76Ki20 1 <44.1 (45.51) 0.7 <43.71 (45.08) 150.112960 2.16486001.927 9.78Ki20 0.28<44.41 (45.42)0.04<44.91 (45.47) 149.957504 2.68757001.377 9.79Ki20 1 <43.46 (44.61)1 <42.82 (44.14) 150.232666 2.54427001.404 9.8Ki201 <43.26 (44.65)1 <42.93 (44.16) 149.963470 2.24144002.147 9.83Ki20 1 <43.87 (45.03)0.43<43.9 (45.01) 150.034317 2.64111990.517 9.84Ki20 0.06<43.13 (44.01)0.18<43.14 (43.69) 150.584503 2.02168011.919.85 Ki201 <43.73 (45.04)1 <43.34 (44.55) 150.249649 1.95293991.107 9.88Ki20 1 <43.14 (44.45)1 – 150.022613 2.14011000.837 9.91Ki20 1 <42.95 (44.19)0.44<42.94 (43.93) 150.327224 1.92859990.525 9.91Ki20 9.60e-03<43.63 (44.16)2.50e-03<43.55 (43.93) 150.175231 2.74171992.313 9.95Ki20 0.32<44.67 (45.57)0.14<44.96 (45.51) 150.322402 2.04128002.771 9.95Ki20 1 <44.54 (46.11)1 – 150.629684 2.47638982.278 9.99Ki20 0.07<45.12 (45.77)0.01<45.0 (45.54) 150.468978 2.33173990.337 10.0Ki20 1 <42.0 (43.15) 1 <41.42 (42.59) 153.667574 19.4136150.029 8.48Sh22 1 <39.63 (40.86)0.72<39.63 (40.53) 170.8156524.0347890.025 9.09Sh22 1 <39.52 (40.65)0.04<40.3 (40.68) 132.857567 39.5949410.041 8.98Sh22 3.09e-03<41.63 (41.91)2.11e-0641.38_41.19^41.54 (_40.81^41.73) 123.110779 39.5388880.033 9.74W22W 0.26<40.78 (41.49)0.1 <40.62 (41.14) 175.712442 20.4421260.019 9.61W22W 0.33<39.93 (40.76)9.75e-03<40.33 (40.65) 188.282729 10.2489290.053 9.61W22W 0.7 <40.32 (41.3) 0.09<41.17 (41.55) 212.720667 1.369554 0.026 9.73W22W 0.26<40.07 (40.92)0.89<39.18 (40.12) 172.611858 30.4644680.059 9.66W22W 0.37<40.89 (41.69)0.02<41.26 (41.62) 196.890929 14.0086470.055 9.72W22W 0.65<40.44 (41.49)0.09<41.26 (41.62) 137.389817 38.7107210.056 9.69W22W 1 <40.68 (41.67)0.16<40.93 (41.58) 200.8192521.3016580.022 9.61W22W 1 <39.33 (40.43)0.21<40.04 (40.44) 198.352746 8.040035 0.024 9.71W22W 0.01<40.76 (41.19)0.05<40.5 (40.76) 125.056017 16.2116270.044 9.56W22W 1 <40.12 (41.06)1 <39.57 (40.67) 141.16417.6633270.014 9.6W22W1 <39.16 (40.11)1 <38.81 (39.74) 197.119175 10.6690540.025 9.68W22W 0.12<40.68 (41.11)0.64<39.49 (40.32) 177.135242 12.7052960.015 9.52W22W 0.32<39.69 (40.56)0.28<39.2 (39.99) 194.5911 8.541609 0.029 9.56W22W 1 <39.48 (40.63)1 <39.18 (40.19) 193.533008 10.0065810.037 9.51W22W 0.62<40.21 (41.08)0.74<39.55 (40.54) 139.321046 26.4497790.025 9.56W22W 1 <39.57 (40.63)1 <39.21 (40.16) 124.961629 24.7878220.026 9.65W22W 5.51e-03<41.05 (41.44)0.17<39.7 (40.54) 187.264621 29.7794430.081 9.71W22W 3.56e-0642.2_41.95^42.41 (_41.54^42.67)2.85e-2642.4_42.3^42.49 (_42.16^42.62) 194.833679 9.187163 0.028 9.66W22W 1 <39.85 (40.84)0.78<39.41 (40.34) 189.051083 26.7561330.025 9.71W22W 1 <39.42 (40.56)0.54<39.34 (40.26) 196.822679 13.6466580.027 9.66W22W 4.63e-04<41.29 (41.48)3.84e-1741.27_41.16^41.37 (_41.01^41.49) 148.661629 40.5345540.067 9.52W22W 4.26e-1642.4_42.25^42.53 (_42.04^42.7) 4.69e-197 42.97_42.93^43.01 (_42.87^43.06) 130.368625 16.2786570.073 9.72W22W 1 <40.51 (41.68)1 <40.06 (41.19) 214.864575 4.753834 0.143 9.66W22W 4.65e-2743.26_43.15^43.37 (_42.99^43.52) 2.23e-135 43.36_43.31^43.41 (_43.24^43.48) 199.247375 3.888836 0.045 9.71W22W 0.51<40.34 (41.22)0.21<40.38 (41.06) 198.475725 16.7276960.022 9.5W22W0.52<39.81 (40.68)0.61<39.16 (40.15) 224.726117 1.990925 0.039.49 W22W0.72<39.74 (40.79)0.79<39.4 (40.45) 172.116967 27.9021140.068 9.62W22W 1 <40.49 (41.44)1 <40.0 (40.98) 126.338879 15.2898080.033 9.49W22W 1 <39.85 (41.0) 0.8 <39.55 (40.62) 135.710296 14.2356390.051 9.48W22W 1 <40.36 (41.44)0.08<40.94 (41.45) 121.752554 5.626982 0.052 9.47W22W 1 <40.23 (41.36)1 <39.74 (40.75) 165.316779 10.2716460.036 9.47W22W 0.27<40.79 (41.39)0.05<40.74 (41.13) 180.957258 29.7155980.019.46 W22W1 <38.85 (39.94)0.76<38.48 (39.35) 49.858571-6.1211060.008 9.45W22W 0.59<38.82 (39.72)8.69e-04<39.7 (39.87) 124.3160531.6519830.045 9.72W22W 1 <40.18 (41.31)0.36<40.22 (40.98) 160.860308 11.0900630.047 9.64W22W 2.18e-197 43.15_43.11^43.19 (_43.06^43.25) 043.16_43.14^43.18 (_43.11^43.21) 148.741533 36.0974840.049 9.39W22W 0.39<41.3 (41.72) 0.49<40.58 (41.28) 121.075004 15.3440190.039 9.73W22W 0.24<41.23 (41.68)0.25<40.57 (41.14) 166.066846 5.275210.117 9.33W22W 1 <41.02 (42.17)0.14<41.69 (42.24) 120.5561522.4340160.039.28 W22W0.22<40.67 (41.31)0.75<39.66 (40.65) 179.147133 28.4903490.012 9.28W22W 1 <38.84 (39.96)1 <38.63 (39.78) 192.946992 9.859572 0.031 9.27W22W 0.62<39.85 (40.88)0.4 <39.74 (40.63) 197.315425 13.3856480.027 9.23W22W 0.64<40.04 (40.9) 0.98<39.07 (40.12) 216.470175 6.5375 0.024 9.21W22W 0.3 <39.96 (40.75)0.37<39.45 (40.29) 176.516967 11.5812680.019.18 W22W1 <38.78 (39.95)0.84<38.52 (39.38) 177.927437 6.858480.102 9.18W22W 0.26<41.51 (42.29)0.17<41.41 (41.88) 157.367592 9.833651 0.022 9.13W22W 0.04<40.8 (41.19) 0.32<40.07 (40.65) 186.326062 5.742002 0.004 9.11W22W 0.02<39.17 (39.5) 8.18e-03<38.98 (39.2) 116.980812 34.0367070.016 9.1W22W1 <39.34 (40.46)1.18e-03<40.4 (40.68) 199.418587 13.9401890.028 9.01W22W 0.34<40.12 (40.92)0.58<39.5 (40.42) 146.598954 34.2811310.075 9.01W22W 0.31<41.13 (41.92)0.42<40.51 (41.36) 137.393212 25.2229890.008 8.98W22W 1 <39.06 (40.06)0.75<38.57 (39.42) 198.4167 16.6576940.022 8.98W22W 1 <39.42 (40.3) 0.43<39.02 (40.03) 158.025108 22.9893910.058 8.97W22W 1 <40.63 (41.57)0.39<40.46 (41.3) 198.461625 23.2549820.012 8.97W22W 0.55<39.31 (40.1) 0.2 <39.23 (39.76) 211.625433 0.327646 0.106 8.93W22W 0.53<41.08 (41.96)0.94<40.35 (41.35) 130.393062 2.188855 0.029 8.93W22W 1 <39.9 (40.97) 0.41<39.54 (40.46) 159.934579 0.857951 0.025 8.88W22W 1 <39.62 (40.79)0.2 <40.32 (40.89) 142.826833 2.781206 0.115 8.83W22W 0.26<41.68 (42.49)0.67<41.12 (41.94) 194.222692 8.161578 0.009 8.75W22W 0.64<38.82 (39.72)0.89<38.21 (39.17) 151.3462519.2719050.013 8.73W22W 1 <39.31 (40.53)0.15<39.57 (40.2) 188.150687 18.0230790.003 8.67W22W 0.45<38.21 (39.03)0.39<37.61 (38.46) 151.795479 12.6516890.009 8.47W22W 1 <38.76 (39.83)0.12<39.37 (39.85) 192.982012 8.878221 0.004 8.45W22W 0.34<38.3 (39.15) 0.38<37.98 (38.79) 200.175829 20.9103120.009 8.44W22W 0.47<38.82 (39.77)0.72<38.4 (39.28) 199.919987 3.409397 0.022 8.27W22W 1 <39.27 (40.36)0.45<39.37 (40.23) 196.733792 14.8074460.003 8.1W22W1 <37.65 (38.71)1 <37.17 (38.2) 209.6864 12.5963710.024 9.2S20 0.69<39.69 (40.64)0.72<39.21 (40.14) 40.217499-8.4742850.082 8.46Ha23 1 <40.41 (41.49)0.46<40.67 (41.42) 149.321659 2.147833 0.086 9.8Was22 0.35<41.1 (42.02) 0.15<41.17 (41.82) 148.575538 2.741297 0.084 9.39Was220.07<41.74 (42.32)0.15<41.03 (41.7) Sources which were masked out (see Sect. <ref>) are shown with dashes. a Reference for input coordinates, redshift and stellar mass M_*: Ba18 stands for <cit.>; Ba20 for <cit.>; B22 for <cit.>; W22Z for ZTF-selected sources from <cit.>; Ki20 for <cit.>; Sh22 for <cit.>; W22W for Wise-selected sources from <cit.>; S20 for <cit.>; Ha23 for <cit.>; Was22 for <cit.>. b No-source probability P_B (Eq. <ref>). Sources are considered detected at P_B<=0.0003 (and are highlighted in bold). c Logarithmic X-ray luminosity in the rest-frame 0.2-2.0 keV range, in units of log(erg s^-1). For detected sources (in bold), median and 16th, 84th percentile values are shown first, with 1st and 99th in parenthesis. For non detected sources, 84th and 99th percentile values are shown as 1σ and 3σ upper limits, respectively. | http://arxiv.org/abs/2311.16220v1 | {
"authors": [
"R. Arcodia",
"A. Merloni",
"J. Comparat",
"T. Dwelly",
"R. Seppi",
"Y. Zhang",
"J. Buchner",
"A. Georgakakis",
"F. Haberl",
"Z. Igo",
"E. Kyritsis",
"T. Liu",
"K. Nandra",
"Q. Ni",
"G. Ponti",
"M. Salvato",
"C. Ward",
"J. Wolf",
"A. Zezas"
],
"categories": [
"astro-ph.GA",
"astro-ph.HE"
],
"primary_category": "astro-ph.GA",
"published": "20231127190000",
"title": "O Corona, where art thou? eROSITA's view of UV-optical-IR variability-selected massive black holes in low-mass galaxies"
} |
http://arxiv.org/abs/2311.15891v1 | {
"authors": [
"Danial Ghamari",
"Roberto Covino",
"Pietro Faccioli"
],
"categories": [
"physics.bio-ph",
"cond-mat.stat-mech",
"quant-ph"
],
"primary_category": "physics.bio-ph",
"published": "20231127145829",
"title": "Sampling a rare protein transition with a hybrid classical-quantum computing algorithm"
} |
|
headingsNovember 2023 A,B]Sabine Wehnert0000-0002-5290-0321 Corresponding Author: Sabine Wehnert, [email protected], Wehnert [A]Leibniz Institute for Educational Media | GEI, Germany [B]Otto von Guericke University Magdeburg, Germany In this work, I discuss how Large Language Models can be applied in the legal domain, circumventing their current drawbacks. Despite their large success and acceptance, their lack of explainability hinders legal experts to trust in their output, and this happens rightfully so. However, in this paper, I argue in favor of a new view, Justifiable Artificial Intelligence, instead of focusing on Explainable Artificial Intelligence. I discuss in this paper how gaining evidence for and against a Large Language Model's output may make their generated texts more trustworthy - or hold them accountable for misinformation.large language modelsjustifiable artificial intelligence fact-checking explainable artificial intelligence information retrieval entailment classification November 2023November 2023§ INTRODUCTIONThe release of Large Language Models (LLMs) has changed our reality. Not only researchers use and test the models' capabilities and susceptibilities, but also companies and individuals employ the generated texts for their everyday needs. Despite much criticism regarding the trustworthiness of the models' responses, the popularity of Artificial Intelligence (AI) -based assistants is ever-increasing. However, it has not been disclosed for many of these models which data they have been trained on, making it hard to estimate model bias that could have been learned from the data. Furthermore, the models' responses may consist of passages from copyrighted training data <cit.>, despite the generative nature of the underlying transformer architecture <cit.>. Although ongoing research targets this issue <cit.>, there are further problems that have not been resolved (yet). For experts in the legal domain, a notable disadvantage of the current deep learning-based systems used in the LLMs is their lack of explainability. In many other domains (such as medicine), there is an unnegotiable need for Explainable Artificial Intelligence (XAI). However, the efforts in this very active research direction have not reached the current LLM applications so far. In this paper, I advocate for using LLMs in the legal domain: Instead of insisting on AI to be explainable, we can require the output of the AI to be justifiable. Justifiable Artificial Intelligence, as I will explain in detail, relies on providing evidence from trustworthy sources for and against a claim made by the AI. This leaves the user of such a system with an informed choice to either accept or reject the claims made by an AI. The contributions of this position paper are the following: * I coin the term Justifiable Artificial Intelligence, describing an alternative approach to the ones taken in Explainable Artificial Intelligence.* I connect Justifiable AI to existing state-of-the art research in Large Language models and fact-checking, uncover challenges and their meaning for applications in the legal domain.The remainder of this work is structured as follows: In Section <ref>, I describe the capabilities of Large Language Models, their evolution, and applications in the legal domain using LLMs. Furthermore, I briefly introduce related works in fact-checking. In Section <ref>, I explain how Justifiable Artificial Intelligence can be constructed. In Section <ref>, I collect other uses of the term “Justifiable Artificial Intelligence". In Section <ref>, I conclude the work and point to possible future work. § BACKGROUND §.§ Large Language ModelsThe aim of language models is to learn probabilities of word sequences, based on different tasks, such as predicting the next sentence or a masked word inside a sequence. Through the exposure to many examples, these models form a representation of the language(s) they are trained on, enabling them to model semantic similarity between words and word sequences. Large Language Models (LLMs) are large artificial neural networks that have been trained on large collections of data and have billions of parameters. The first notable attempt towards an LLM may be GPT-2, which at the time of its invention was capable of reaching state-of-the-art performance in many natural language generation tasks <cit.>. This was considered a breakthrough and because of many concerns regarding its misuse, this model was not released for public use by OpenAI, the company that trained the model. However, this did not hinder further research and development in that direction. Newer versions of GPT are prominent examples of LLMs, which are nowadays accessible to the public as ChatGPT[<https://openai.com/blog/chatgpt/>], with GPT-3.5 or GPT-4 (a multi-modal model, capable of processing images) <cit.> as the backend. Aside from OpenAI, other companies also released their own versions of LLMs: LaMDA <cit.>, LLaMa <cit.>, MT-NLG <cit.>, PaLM2 <cit.>, OPT <cit.>, Sparrow <cit.>, and many others. An important skill that these models have is in-context learning <cit.>, meaning that the models can generate text based on a prompt, the so-called context, enabling them to hold interactive conversations with their users. Furthermore, LLMs perform Reinforcement Learning from Human Feedback (RLHF), which is the process of fine-tuning the model based on rewards inferred from their users' responses to their output <cit.>. Although LLMs excel in many different tasks without the need for specific training (e.g., Medical Licensing Exams <cit.>), their training procedure does not equip them with true Natural Language Understanding capabilities comparable to human problem solving. Often, LLMs are attributed with the skill of “understanding" human language. I do not support this view, since the evidence at the time of writing this paper is not sufficient and in many cases teaching us the opposite (see the debunking <cit.> of seemingly emergent capabilities coming from model size scaling <cit.>). These models are well-versed in modeling language based on many examples of word co-occurrences and plausible text sequences they have been exposed to during the training phase. There are key limitations that many of these language models have <cit.>:* Recency:the training data is limited until a specific cutoff day, * Accuracy: the training data stems from online content (websites, books) that has not been thoroughly fact-checked, * Coherence:the generated text is written in a coherent manner, but can be entirely fictional (referred to as “hallucinations"), when prompted for references, they may be invented <cit.>,* Transparency & Interpretability:deep neural networks are mostly seen as “black boxes" by users, visualizing their weights or attention may not be helpful,* Ethical Concerns:Privacy issues may arise because the use of personal data may not be revocable, the “right to be forgotten" cannot be easily enforced when models take months to train. Also, bias and unfairness may be present due to a bias in the training data (e.g., religion bias <cit.>). Furthermore, the high energy consumption during training and provision of these models needs to be in relation to efficiency gains and potential benefits for the environment <cit.>.There are efforts to address some of these issues, such as giving access to the world wide web to BlenderBot 3 <cit.>, Bing AI <cit.> and Sparrow <cit.>. ChatGPT can receive real-time information through APIs[<https://platform.openai.com/docs/plugins/introduction>]. In the future, we may be able to align AI with societal values <cit.> and through human feedback we may be able to adjust the biases of an LLM <cit.>, solving the ethical concerns in LLMs. Some solutions are proposed for ensuring privacy in LLMs <cit.>. §.§ Explainability in Large Language ModelsHowever, the key limitation of Transparency & Interpretability has not been sufficiently addressed by Explainable AI (XAI) research so far <cit.>. Compared to other Recurrent Neural Network (RNN) architectures, the transformer architecture <cit.> (i.e., the basis of many LLMs) is relatively easier to interpret through the use of neural attention, however this is often not sufficient as the sole base for transparency, especially in more complex, non-classification tasks, such as translation or natural language inference <cit.>. Possible approaches for explaining transformers are saliency maps, feature centrality scores, and counterfactual explanations <cit.>. A more recent way to improve the interpretability of language models is to extract knowledge graphs from them and to compare these representations among different language models <cit.>, or to generate natural language explanations based on these graphs for a label and against the other alternative(s) <cit.>. As investment in LLMs grows, their capabilities in general may also improve <cit.>. But how about tasks in the legal domain, which appear to be often rule-based, and - potentially - learnable by an LLM? §.§ Large Language Models in the Legal DomainA comprehensive survey of the transformer-based language model use in the legal domain has been published by Greco and Tagarelli <cit.>. In the scope of this work, I only collect the most recent works connected to LLMs to provide a general understanding of the caveats coming with applying LLMs on legal language. The inclined reader may refer to the aforementioned work for a more detailed description. As one of the basic legal skills, reasoning has been tested on GPT-3. One type of legal reasoning involves having a pair of facts and a statute, and deciding if the statute applies to those facts. This is often framed as an entailment classification task. Blair-Stanek et al. <cit.> have achieved state-of-the-art performance on such an entailment task, using GPT-3. However, several mis-classifications have led to the conclusion that the task cannot be automated with that LLM because of its incorrect knowledge of the relevant statutes. With guaranteed unseen, synthetic examples, the model also fails to answer the questions correctly. Legal reasoning is a complex task involving many problem-solving capabilities, which go beyond semantic similarity of text. Despite the observation of potential analogical reasoning capabilities in LLMs <cit.>, there is an ongoing debate and the demand to prove that models do not simply memorize training data instead <cit.>. To understand the performance of LLMs on legal reasoning, the benchmark dataset LEGALBENCH has been proposed with 162 tasks, divided in six reasoning categories <cit.>.Prompting is an important task to consider when interacting with an LLM. For this, a study <cit.> has been performed to prompt GPT-3 using the Chain-of-Thought (CoT) technique <cit.> on the statute entailment task of the Competition on Legal Information Extraction/Entailment (COLIEE). Despite achieving state-of-the-art performance, the authors express doubt that LLMs can be taught to follow a logical line of thought of a lawyer. Another work about negating prompts revealed weaknesses of GPT-3 to follow the modified instructions properly <cit.>. Syllogism prompting (consisting of the law as a major premise, the fact is the minor premise and the judgment as the conclusion) can boost performance in legal judgment prediction <cit.>. Other prompting techniques, such as IRAC (Issue, Rule, Application, Conclusion) have been tested on the COLIEE 2021 and COLIEE 2022 statutory entailment task by Yu et al. <cit.>. Similarly, Chain of Reference prompting has been successfully applied on COLIEE 2021 statute law entailment <cit.>. For legal case retrieval on the COLIEE 2023 dataset, a framework called PromptCase <cit.> has been tested. It consists of issue and fact extraction as the first step, followed by dual and cross encoding with the prompt of a case. The T5 model has been employed in COLIEE's legal case entailment task and reached state-of-the-art performance <cit.>. The authors <cit.> mention the inference speed for legal cases as a drawback of their method. In the recent COLIEE 2023 edition, LLMs have been trained on an augmented dataset of legal cases by the JNLP team <cit.>. There appears to be a general trend of combining traditional methods for information retrieval with LLMs <cit.>.LLMs have been employed on U.S. court opinions on fiduciary duties and reached a satisfying, but not sufficient performance <cit.>. Furthermore, LLMs have been used for Legal Judgment Prediction, and the results were below the state of the art, despite efforts to engineer prompts <cit.>. Deroy et al. <cit.> employ LLMs for summarization of Indian court case judgements and find inconsistencies in the summaries, as well as hallucinations. They conclude that the use of LLMs for their task is only acceptable in a human-in-the-loop setting, given current model capabilities. This finding is in line with a study on ChatGPT (with GPT-3.5), where the model drafted several legal documents, but failed to detect recent legal cases <cit.>.Further use cases are legal educational settings, where LLMs can be worthwhile assistants, boosting the teachers' creativity and productivity <cit.>. There are also considerations regarding law students' use of LLMs in class or in exams, since LLMs have become part of the working culture<cit.>. LLMs have been tested on law exams, and mostly obtained a passing grade <cit.>, with the exception of GPT-4, which can be seen to perform in one study on par with average human student performance <cit.>.A remarkable approach especially in the context of this work is ChatLaw <cit.>, a Legal LLM which uses vector knowledge bases together with reference documents, performing keyword search on them using a BERT model to avoid the issue with hallucinations. A similar architecture called DISC-LawLLM <cit.> has been proposed recently. The use of knowledge bases is a common hallucination mitigation strategy <cit.>, usually employed to correct the LLM's output without showing the “raw knowledge" to the user. Louis et al. <cit.> on the other hand, have proposed an LLM with a retriever module for answering statutory law-related questions, which generates answers using an extractive method to generate the rationale. This is very much in line with what I envision for Justifiable Artificial Intelligence.§.§ Fact-checkingFact-checking, when seen as an automated task, consists of a claim that has to be compared against fact-checked or trustworthy sources and either labeled as true or false. Sometimes, there is also a neutral category when not enough information is present to decide the truthfulness of a claim. Nowadays, LLMs are used for many tasks. Fact-checking is no exception. Any research about using LLMs as the only source for fact-checking is out of scope of this work because of the many reasons I listed before on why LLMs may not generate accurate answers (see section <ref>).Yao et al. <cit.> publish a multi-modal dataset for fact-checking, using the common websites for gathering their evidence[<https://www.snopes.com>, <https://www.politifact.com>]. Having their data manually fact-checked has the advantage of offering a high-quality dataset to train models on. However, their models perform poorly on the test data. In addition, the multi-modal setting is challenging, if applied on a real use case: If taken out of context, an image can be evidence for anything. It requires a careful consideration of the metadata of the image: When and where has the image been taken (in case of a photo), what is the source of the image (in case of a figure displaying data). Indeed, there are many non-trivial legal issues attached to using images as evidence. Therefore, I do not consider the multi-modal use case for our characterization of Justifiable AI, and instead focus on a task based on textual data only. An architecture called FacTeR-Check has been proposed <cit.> for misinformation tracking or semi-automated fact-checking. In the fact-checking process, FacTeR-Check uses the semantic similarity from transformers for retrieving fact-checked claims from a database, given a claim (e.g., from an online social network). After ranking top-n fact-checked results, they are compared with the claim in question and classified regarding their entailment (positive or negative). Trokhymovych and Saez-Trumper developed the WikiCheck API <cit.> that can be used for fact-checking claims through the Wikipedia knowledge base. Their API uses the MediaWiki API[<https://www.mediawiki.org/wiki/API:Search>] to extract related articles, given a (potentially enriched) claim. Then, entailment is classified using a transformer-based model architecture.To conclude, fact-checking requires a collection of trusted sources (which is not always easy to determine), a retrieval module and a way to compare the claim and the fact (e.g., via entailment classification). Is fact-checking all we need? This question cannot be answered easily, however following a process based on the general idea of fact-checking may be the answer to the reputation problem LLMs have in the legal domain nowadays.§ JUSTIFIABLE ARTIFICIAL INTELLIGENCE After exploring the state-of-the-art in LLMs, XAI, LLM use in the legal domain, and fact-checking, I come to the conclusion that LLMs are currently not well-equipped to be trusted in by legal experts, and rightfully so. Providing real references is a useful step towards enabling a human to check the LLM's output and to trust its validity. Architectures similar to ChatLaw or DISC-LawLLM (previously mentioned in section <ref>) are a good method to increase the accuracy of a model. Depending on the task, it may be worthwhile to offer insight into the references taken by an LLM to the user. This is where approaches with Justifiable Artificial Intelligence can fill the gap: Providing evidence to the user, on the base of real documents, shall justify the AI's output. As already mentioned, the work by Louis et al. <cit.> is an example for this type of approaches. However, the justification does not necessarily have to be performed by the LLM itself. Justifiable AI follows the motto: If you cannot explain yourself, at least justify your opinion. With that I do not only mean providing evidence backing up a claim, but also (if useful for the task) showing possible evidence against the claim. A retrieval module may search for online content for and against a statement made by the LLM. In that way, the user does not get influenced by a confirmation bias, but can look into different perspectives, when the answer may not be straightforward. This is where the place of LLMs shall be nowadays: Assistants, that inform the user about an issue from different perspectives, but do not influence the user in one direction. We can clearly see that as long as there are issues with recency, accuracy, coherence, transparency & interpretability, as well as ethical issues, the use of LLMs in domains that demand all these aspects, will never be fully autonomous and require human validation. Therefore, I advocate for Justifiable AI, for example via in-built fact-checking modules to assist the users who will have to perform the validation anyway. To illustrate the point, consider the two pipelines of Figure <ref> and Figure <ref>. In Figure 1, we see the workflow by by Louis et al. <cit.>. This approach can only be accepted as Justifiable AI, if the evidence extraction works properly with an LLM, since this solution is prompt-based. Figure 2 shows the fact-checking based solution, where either trustworthy online sources, or a knowledge base are queried. Then, supporting and contradicting evidence (based on entailment classification) is shown to the user for full decision sovereignty. To ensure transparency, the user can query the document collection (e.g., a database) and understand its composition through metadata and inspection of individual documents. Note that both approaches are not equal to a fully manual fact-checking process, since prompting, retrieval, and entailment classification can lead to errors that propagate into the final decision making. However, any “manual" use of a search engine may also not guarantee finding all relevant documents. Therefore, let Justifiable AI be the family of approaches that help us trust LLMs more.§ RELATED WORKIn this section, I shortly review other uses of the term Justifiable Artificial Intelligence. A literal search of the concept “Justifiable Artificial Intelligence" on google scholar does not return any relevant result. When searching for “Justifiable AI", I find the word “justifiable" used as an adjective, often in connection to “interpretable" or “morally" (e.g., in this work <cit.>). Justifiable Artificial Intelligence does not seem to be a proper name used in the research community. It is seen as a property, not as a group of methods (yet). Given the discussion on why current approaches to explain transformer-based outputs may not be helpful to obtain full transparency and interpretability (see section <ref>) and the great performance of LLMs met with a lot of caution by legal experts (see section <ref>), emerging methods using evidence shall be distinguishable from other works. They are truly human-centered, and will likely be more accepted by legal experts because of the possibility to validate outputs.Extending the search for the term “Justifiable Artificial Intelligence" to a regular google search, we find several domains using this term. A commentary by Hadfield <cit.> is about Justifiable AI and that it shall be demanded by law, instead of Explainable AI. She makes the distinction between both worlds with respect to their focus: “While Explainable AI is focused on fact, justifiable AI is focused on judgment" <cit.>. This view is more connected to judgment in terms of societal values, instead of the possibility to validate AI through evidence (i.e., what I mean by Justifiable AI). A blog post by Jones is closer to that notion: “We need to be able to explain how we got to the outcome and really importantly to demonstrate how we were in control of that outcome. In the same way as you can’t explain the individual neurons that fire to enable someone to complete a job successfully, you still need to show how you made sure they did." <cit.>. § CONCLUSIONI have demonstrated the current challenges in the use of Large Language Models in the legal domain. As an alternative, I propose to build models that have an additional module for providing extractive rationale or automated retrieval of supporting and contradicting evidence for the model's claims. This enables the users to perform their own fact-checking of the Large Language Models' outputs. The potential of this approach is to encourage legal domain experts to use intelligent assistants in their work without sacrificing the reliability and quality of their results. Future work needs to validate the acceptance of Large Language Models through legal experts, given their outputs are justifiable. vancouver | http://arxiv.org/abs/2311.15716v1 | {
"authors": [
"Sabine Wehnert"
],
"categories": [
"cs.CL",
"cs.HC",
"cs.IR",
"H.4.2; H.3.3; H.5.2"
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"primary_category": "cs.CL",
"published": "20231127105916",
"title": "Justifiable Artificial Intelligence: Engineering Large Language Models for Legal Applications"
} |
Article Title]A Graph Neural Network-Based QUBO-Formulated Hamiltonian-Inspired Loss Function for Combinatorial Optimization using Reinforcement Learning1]Redwan Ahmed [email protected]]Raheeb [email protected]]Md. Mosaddek [email protected]*[1]Department of Computer Science and Engineering, University of Dhaka, Dhaka, BangladeshQuadratic Unconstrained Binary Optimization (QUBO) is a generic technique to model various NP-hard Combinatorial Optimization problems (CO) in the form of binary variables. Ising Hamiltonian is used to model the energy function of a system. QUBO to Ising Hamiltonian is regarded as a technique to solve various canonical optimization problems through quantum optimization algorithms. Recently, PI-GNN, a generic framework, has been proposed to address CO problems over graphs based on Graph Neural Network (GNN) architecture. They introduced a generic QUBO-formulated Hamiltonian-inspired loss function that was directly optimized using GNN. PI-GNN is highly scalable but there lies a noticeable decrease in the number of satisfied constraints when compared to problem-specific algorithms and becomes more pronounced with increased graph densities. Here, We identify a behavioral pattern related to it and devise strategies to improve its performance. Another group of literature uses Reinforcement learning (RL) to solve the aforementioned NP-hard problems using problem-specific reward functions. In this work, we also focus on creating a bridge between the RL-based solutions and the QUBO-formulated Hamiltonian. We formulate and empirically evaluate the compatibility of the QUBO-formulated Hamiltonian as the generic reward function in the RL-based paradigm in the form of rewards. Furthermore, we also introduce a novel Monty Carlo Tree Search-based strategy with GNN where we apply a guided search through manual perturbation of node labels during training. We empirically evaluated our methods and observed up to 44% improvement in the number of constraint violations compared to the PI-GNN.[ [ January 14, 2024 ====================§ INTRODUCTION Combinatorial Optimization (CO) is indeed a crucial field in mathematics and computer science with a wide range of applications in solving real-life problems. At its core, CO deals with finding the best possible solution from a finite set of possibilities, where the goal is to optimize some objective function subject to a set of constraints. One of the key challenges in CO is that as the problem size grows, the number of possible solutions increases exponentially, making it computationally infeasible to explore all possibilities. Therefore, CO seeks efficient algorithms and techniques to find near-optimal solutions.One powerful framework for addressing CO problems is Quadratic Unconstrained Binary Optimization (QUBO), which transforms complex combinatorial problems into a quadratic equation involving binary variables. QUBO has become the standard format for optimization using quantum computers, such as the quantum approximate optimization algorithm (QAOA) and quantum annealing (QA) <cit.>. It is also applicable to gate arrays and digital annealers <cit.>. QUBO allows for the formulation of hard combinatorial optimization problems as polynomial-sized linear programs, providing a generalized framework for solving these problems <cit.>. QUBO and Ising Hamiltonian have a very close relationship in modeling a diverse set of optimization problems and are often used interchangeably or together <cit.>. Ising Hamiltonian focuses on modeling the energy stability of a system in the form of identifying the spins ({-1, 1}) of the qubits. We have already mentioned that the terms QUBO and Hamiltonian are closely related. After converting a QUBO objective or loss function into an Ising Hamiltonian, various quantum optimization algorithms are used to solve it <cit.>. Due to this close connection, the relevant problems are addressed as problem Hamiltonian and the objective functions are regarded as Hamiltonian Objective or Cost function <cit.>. Research has shown that quantum solution systems struggle to scale with the number of variables and to maintain precision in the solutions they provide <cit.>. Thus, recent works have been addressing different strategies to model QUBO-based Hamiltonians <cit.>.In general terms, the goal of QUBO is to find a diverse set of solutions that meet a target metric or goal, making it suitable for high-level decision-making. QUBO has found applications in various domains, including logistics, finance, and artificial intelligence. In logistics, it aids in optimizing supply chain routes, minimizing transportation costs, and improving inventory management. In finance, QUBO assists in portfolio optimization, risk management, and trading strategies. <cit.> In AI, it plays a pivotal role in solving complex problems such as feature selection, clustering, and tuning of the parameters of the machine learning models <cit.>.Graph Neural Networks (GNNs) are a class of deep learning models designed for processing and analyzing structured data represented as graphs. They have gained significant popularity for their ability to capture complex relationships and dependencies in graph-structured data, making them highly versatile for a wide range of applications. In particular, GNNs excel in tasks involving graph data, such as social network analysis, recommendation systems, and, as discussed here, combinatorial optimization <cit.>. When it comes to addressing combinatorial optimization problems, <cit.> has introduced a pioneering approach (PI-GNN) harnessing the power of Graph Neural Networks (GNNs) to address the usage of Hamiltonian cost function during training. In their paper, they showcase how GNNs can effectively tackle quadratic unconstrained binary optimization (QUBO) and polynomial unconstrained binary optimization (PUBO) problems. The key innovation lies in their utilization of a relaxation strategy applied to the problem Hamiltonian, resulting in a differentiable loss function that serves as the foundation for GNN training. The paper's impressive results demonstrate that their GNN-based optimizer not only matches the performance of existing solvers but often outperforms them, particularly in scenarios such as maximum cut and maximum independent set problems. What truly sets this approach apart is its marked scalability, allowing it to effectively tackle combinatorial optimization problems involving millions of variables. This work represents a significant advancement, merging deep learning techniques with concepts from statistical physics to address NP-hard problems and offering promising possibilities for the field of combinatorial optimization.In a recent critical review paper by Boettcher et al. (2023) <cit.>, it was demonstrated that PI-GNN, performs less effectively than traditional greedy algorithms when solving the Max-Cut problem. Angelini et al. <cit.> raised similar concerns about GNN-based solutions, particularly their relative inefficiency compared to classical greedy algorithms, as discussed in the context of the Maximum Independent Set problem. Both pieces of literature highlight the notion that in very narrowly defined problem scenarios, such as Max-Cut and Maximum Independent Set, traditional greedy algorithms may outperform PI-GNN. However, it is noteworthy that the authors of PI-GNN have responded to these critiques in their subsequent work <cit.>, arguing that focusing solely on the performance in these specific, curated scenarios overlooks the broader generality and scalability of their proposed framework. To bolster their perspective, they have presented empirical results that illustrate the advantages of PI-GNN in more general and scalable problem settings, suggesting that a comprehensive evaluation of the framework is necessary to appreciate its full merit.On the other hand, another widely explored approach to solving optimization problems is Reinforcement Learning (RL). RL aims to learn policies that maximize expected rewards, which can often be derived from the objective function of the optimization problem <cit.>. In this setup, an agent's states represent current solutions, actions correspond to decisions, and rewards measure solution quality. RL algorithms like Q-learning or deep reinforcement learning (DRL) methods guide the agent to explore and exploit solutions efficiently.Bello et al. (2016) extended the pointer network architecture to create an actor-critic RL framework for training an approximate Traveling Salesman Problem (TSP) solver, using tour length as a reward signal <cit.>. Subsequent work <cit.> introduced improvements in accuracy using a graph attention network for two-dimensional Euclidean TSP, approaching optimal solutions for problems up to 100 nodes. Additionally, a multi-level RL framework was used to analyze TSP variants with hard constraints. In terms of a general solving framework, Drori et al. <cit.> proposed a technique that uses a Graph Attention Network (GAT) base encoder architecture to encode and generate the node feature vectors, upon which they apply an attention-based decoding mechanism to greedily select the nodes and apply the node labelings. The approach however requires reward functions to be specifically tailored for each problem and thus hampers generality. Taking into account the previous discussion, it is obvious that trivial GNN based methods such PI-GNN, although they are widely applicable and versatile, often struggle with accuracy. Meanwhile, RL-based methods, despite their high accuracy and scalability, are not as widely applicable. In this paper, we try to bridge this gap by addressing the combinatorial optimization problem by extending some of the aforementioned methodologies. In particular, we suggest three distinct improvements. Firstly, we identify a crucial flaw in the early stopping strategy of PI-GNN which causes it to underperform in denser graphs. We propose a fuzzy early-stopping strategy as an alternative to the fixed tolerance value strategy. Secondly, being inspired by the formulation stated in <cit.> and <cit.>, we propose a modified generic framework that works with QUBO-formulated Hamiltonian as the generic reward function. This offers more accuracy compared to the previous approach at the cost of higher run time. Finally, we propose a Monty Carlo Tree Search-based strategy with GNN through manual perturbation to avoid local minimas. This performs best in terms of violating the least constraints but also comes at the expense of higher run time. Within this context, it is worth noting that the quality of the number of satisfied constraints can also be quite important for various applications in both online and offline settings. For example, In optimal sensor placement problems, it is more important to have better sensor assignments than faster reporting of the results <cit.>. Sensor placement problem is a constraint-dependent version of the vertex cover problem which can be formulated in the QUBO-based Hamiltonian. For empirical evaluation, we evaluated our approaches considering Max-Cut as the benchmarking problem and have witnessed up to 44% improvement over PI-GNN in reducing the number of violated constraints. It is noteworthy to mention that, similar to PI-GNN, all the presented proposals are generic in manner and can be extended to a wide group of graph-based canonical optimization problems that can be formulated in QUBO.§ BACKGROUNDThis section lays the groundwork for our discussion by taking a quick look at the fundamentals of combinatorial optimization, graph neural networks and reinforcement learning. Furthermore, we also discuss the methodologies used by Schuetz et al. (2023) <cit.> and Drori et al. (2020) <cit.> to solve the combinatorial optimization problem. §.§ Combinatorial OptimizationCombinatorial Optimization seeks to find the optimal solution from a finite set of possibilities while adhering to specific constraints. Common combinatorial optimization problems include the maximum cut problem (Max-Cut), the maximum independent set problem (MIS), the minimum vertex set cover problem, the maximum clique problem, the set cover problem, the traveling salesman problem (TSP), etc. Most of these problems include settings where a set of decisions have to be made and each set of decisions yields a corresponding cost or profit value, that is to be maximized or minimized. As the problem size grows, the size of the set of possible decision variables increases, and so does the set of feasible solutions, exponentially in fact. This makes exhaustive exploration impractical. Moreover, for most of these problems, finding an optimal solution in polynomial time is known to be NP-complete. This motivates the development of various approximation algorithms. §.§ Quadratic Unconstrained Binary Optimization (QUBO)Although many approximate algorithms exist for solving combinatorial problems, most of them are tailored to the specific problems they try to solve with limited generalizability <cit.>. That is where Quadratic Unconstrained Binary Optimization (QUBO) comes in, being able to transform a large class of such NP-complete combinatorial problems into quadratic equations involving binary variables. If we take X = (x_1, x_2, ...) to be a vector binary decision variables, we can represent the cost function of a QUBO problem with the Hamiltonian expressed in Equation <ref>,H_QUBO = X^T Q X = ∑_i=1^n∑_j=1^n x_i Q_ij x_j,where Q_ij represents the coefficients of the quadratic terms which encodes the constraints of the problem we are trying to solve. Then the problem becomes effectively reduced to finding X which minimizes H_QUBO. The Q-matrix can be generated specifically for problems as per requirement. Moreover, extra constraints may be included in the form of penalty terms in the objective function as is often needed in real world combinatorial optimization problems. §.§ Graph Neural Networks (GNNs)In recent years, Graph Neural Networks (GNNs) have emerged as a powerful framework for analyzing structured data, particularly in the context of graph-structured data. Graphs are mathematical structures that consist of nodes (representing entities) and edges (representing relationships or connections between entities). GNNs have gained significant attention due to their ability to model and extract valuable information from such complex, interconnected data.On a high level, they are a family of deep learning models capable of learning representations from graph structured data. While a single layer GNN is able to encapsulate a node's features in a one-hop neighbourhood, typically multi-layered stacked GNNs are used to capture information from a larger neighbourhood of the node. Formally, in a graph G = (V, E), at layer k = 0, each node v ∈ V is represented by some initial representation h_v^0, usually derived from the node's label or given input features. GNNs then iteratively update each node's representation by some parametric function f_θ^k using,h_v^k = f_θ^k(h_v^k-1, {h_u^k-1| u ∈𝒩(v)}),for layers k = 1, 2, ..., K where 𝒩(v) denotes the neighbours of node v. Finally, the last (K^th) layer's output is used in a problem-specific loss function, and stochastic gradient descent is used to train the weight parameters. For example, for classification problems, the soft-max function combined with cross-entropy loss is often used <cit.>. §.§ GNNs for Combinatorial Optimization Combining the power of GNNs with QUBO, Schuetz et al. introduced PI-GNN to solve various combinatorial optimization problems <cit.>. It leverages a relaxation strategy applied to the problem Hamiltonian, resulting in a differentiable loss function for GNN training. Formally, a loss function L_QUBO(θ) is defined based on H_QUBO as,H_QUBO→ L_QUBO(θ) = ∑_i, j p_i(θ) Q_ij p_j(θ),where p_i is the node embeddings generated by the final layer of the GNN (p_i = h_i^K ∈ [0, 1]). Now, L_QUBO(θ) is differentiable with respect to the parameters of the GNN model θ. After unsupervised training, a simple projection step (x_i = int(p_i)) is used to enforce integer variables. As for the GNN architecture, PI-GNN uses graph convolutional networks (GCN) with the following update step,h_v^k = σ (W_k ∑_u ∈𝒩(v)h_u^k-1/|𝒩(v)| + B_k h_v^k-1),where W_k and B_k are trainable weight matrices, σ(.) is some non-linear activation function such as sigmoid or ReLU for middle layers and SoftMax for the final layer. For the actual gradient descent process, ADAM is used. PI-GNN excels in complex combinatorial optimization problems, such as maximum cut and maximum independent set problems, and is characterized by its scalability.A significant issue associated with Equation <ref> is the omission of the actual node projection strategy from the loss function, denoted as L_QUBO. Therefore, the projection strategy employed to label the node set upon completion of training will dictate the quality of the solution to the variable assignment problem.§.§.§ Reinforcement Learning (RL) Reinforcement Learning (RL) is a computational approach where an agent learns to make sequential decisions by interacting with an environment. At the core of RL are states, representing the circumstances, actions, the choices available to the agent, and rewards, the immediate feedback based on these actions. The agent's objective is to maximize cumulative rewards over time. Through trial and error, RL algorithms, such as Q-learning <cit.> or Deep Q Networks <cit.>, enable the agent to learn optimal strategies by exploring the environment, experiencing different states, and refining its decision-making policy. The process involves constant decision-making based on received rewards, learning from these experiences, and adjusting strategies accordingly.§.§.§ RL for Combinatorial Optimization In 2020, Drori et al. proposed a generic RL-based framework (that we address as GRL) to solve a wide variety of combinatorial optimization problems <cit.>. The key insight was that if we turn the combinatorial optimization problem into a node selection problem on a graph, we can use RL to learn how to choose nodes to get the optimal result. Here the state is the set of nodes already chosen, the action being choosing a node to add to this set and the reward is some function of these set of nodes specific to the problem at hand that ensures that maximizing it ensures that we get the optimal solution. A Graph Attention Network (GAT) <cit.> (a type of GNN) was employed for encoding to generate node feature vectors for greedily choosing nodes at each step. Each layer k=1, ... , K of GNN updates the feature vector of the v-th node as,h_v^k = α_vvΘ^k h_v^k-1 + ∑_u ∈𝒩(v)α_vuΘ^k h_u^k-1where Θ^k is a learnable weight matrix, and α_vu^k are the attention coefficients, defined as,α_vu^k = exp(σ(z^k^T [Θ^k h_v^k, Θ^k h_u^k ]))/∑_w ∈𝒩(v)exp(σ(z^k^T [Θ^k h_v^k, Θ^k h_w^k ])).The decoder on the other hand uses a simple attention mechanism, where we compute the attention coefficients as follows,α_vu^dec = Ctanh( (Θ_1 h_v)^T(Θ_2 h_u)/√(d_h))where Θ_1, Θ_2 ∈ℝ^d_h × d are learned parameters, d_h is the hidden dimension, and C ∈ℝ is a constant. This method has demonstrated its effectiveness in solving both polynomial and NP-hard problems, with a focus on generalization from small to large graphs.§.§.§ Monte Carlo Tree Search (MCTS)Monte Carlo Tree Search (MCTS) is a robust algorithm crucial in addressing complex decision spaces and combinatorial optimization problems. Operating through a systematic approach of four distinct stages, MCTS begins with Selection, systematically navigating a search tree from the root node using a selection policy, typically the Upper Confidence Bound (UCB) for Trees, to balance exploration and exploitation. This stage is followed by Expansion, where the algorithm extends the tree by creating child nodes to represent potential actions, effectively broadening the decision space. The process further proceeds to the Simulation (Rollout) phase, where random actions are assessed from the selected node until reaching terminal states or a predefined depth. Finally, in the Backpropagation phase, MCTS aggregates outcomes, updating statistical information such as visit counts and reward estimates for nodes along the path traversed during selection. This structured approach allows MCTS to systematically explore and update the search tree, avoiding local minima and effectively navigating complex decision spaces.§ OUR PROPOSALSThis section delineate all the proposals and formulations presented in our article. Section <ref> commences with an exposition of the early stopping strategy employed by PI-GNN as discovered in their work. Subsequently, we pinpoint a pivotal concern associated with this strategy, and thereafter, we introduce our modified strategy designed to improve the performance. Expanding on the concepts from the work of Drori et al. (2020) (see Section 2), which we refer to as GRL for brevity, Section <ref> introduces our second contribution; a generic reinforcement learning (RL)-based framework that employs a QUBO-based Hamiltonian as the reward function, that we call GRL_QUBO. Here, we detail the modifications applied to the GRL framework, the elements retained, and our rationale for employing a QUBO-based reward. Finally, in Section <ref>, we present our thrid contribution, a Monty Carlo Tree Search-based solution, MCTS-GNN, in addressing the CO problem based on manual perturbation of node labels to guide the search tree.§.§ PI-GNN with Fuzzy Strategy PI-GNN uses Definition <ref> to perform early stopping during training. This intuitive support behind this strategy states that if we do not observe improvement or observe very insignificant improvement in the loss objective during training for a consecutive empirically set number of epochs, the training can be stopped. In general cases, this observation provides good output. But, as per our experimental observation, this strategy bears a crucial concern, especially in the graphs with higher densities.During training, for the objective function F_obj, if no successive improvement is observed for consecutive p epochs or the successive variation is lesser than τ occurs for consecutive p epochs, the training can be stopped.During PI-GNN training, the objective or the loss function often starts with high positive value and then with gradual training moves into larger negative values. This phenomenon is common in graphs with high density. During the transition from positive to negative loss, there is often a period when the loss variation or improvement is minimal (e.g., less than 10^-7, 10^-8, etc.). As a result, the τ value or the early stopping criterion outlined in Definition <ref> may prompt an early halt to training, which can significantly diminish performance quality. Therefore, we propose the use of a fuzzy-stopping strategy as described in Definition <ref>, which provides a more lenient stopping criterion. The rationale for the gradual loss variation is that nodes with a large number of connected edges tend to have feature vectors that are closely clustered in the vector space, leading to a uniform distribution of node probabilities. With additional iterations, these feature vectors are refined, resulting in more distinct patterns. It is also important to note that pinpointing an improved tolerance value during this gradual transition is challenging, rendering the early stopping approach in Definition <ref> less effective.Let us assume that, obj^* denotes the current best value observed for the objective function F_obj during any phase of the training iterations. If no improvement in the value of F_obj occurs for successive p epochs compared to obj^*, the training can be stopped.It is readily apparent that Definition <ref> provides a fuzzier criterion compared to Definition <ref>. This is because the latter one allows for both faster and slower transitions for the loss objective during training by removing the strict dependency over τ. §.§ A QUBO-based Generic Reinforcement Learning framework (GRL_QUBO) We first discuss the modified portion of GRL_QUBO that varies from GRL. Then, we point to the adopted portion brought into our architecture. The main purpose of experimenting with QUBO-based Hamiltonian as reward objective is generalization. A wide group of optimization problems has already been modeled in QUBO. Our research objective is to observe if we can use this QUBO-based model as a generic reward function for RL-based formulations in a permutation-invariant manner. This contribution works as a bridge between the QUBO-based CO solutions with RL-based approaches. * Generic QUBO-formulated Hamiltonian Reward Function: A subset of terms from X^TQX=∑_i ≤ jx_i Q_ijx_j is considered the observed reward r^t at time t during training for a particular epoch e. When a node v_i is greedily selected and labeled, we check which terms x_iQ_ijx_j where i ≤ j and x_jQ_jix_i where j < i can be calculated and sum them. This is considered as the reward, r^t = ∑_i ≤ j x_iQ_ijx_j + ∑_j < ix_jQ_jix_i.* Attention-based decoding strategy: In Equation <ref>, we provide the mathematical formulation to calculate the attention or weight for node v_i as α_i. Here, ϕ_1∈ℝ^d_h× d, ϕ_2∈ℝ^d_h× d and ϕ_3∈ℝ^d_h× n are weights or architecture parameters. h_i denotes the node feature vector (row vector) for the node v_i reported from the GAT layer. C, d and d_h all are hyperparameters. n denotes the number of nodes in the input graph. Through applying a sigmoid non-linear activation over α_i, node probability distribution P_i(θ) is generated. Over P_i(θ), a probability threshold β (e.g., β=0.5) is applied to fix the node labels, e.g., P_i(θ) ≥β leads tox_i=1. Here θ denotes the complete set of trainable architecture parameters and T is used to denote the transpose of a vector (row to column or column to row).α_i =Ctanh( h_iϕ_1^T .(X_vϕ_3^T + ∑_j ∈ N(i) h_jϕ_2^T) /√(d_h)) We incorporate more context in Equation <ref> compared to Equation <ref> to update a node v_i's attention weight α_i. Equation <ref> is the decoding strategy of GRL as discussed in Section <ref>. In Equation <ref>, based on the selected node v_i we update all of its adjacent node v_j's attention values (α_ij) over one to one dependencies. But, in Equation <ref>, to update a node v_i's attention value, we consider all of its adjacent nodes, v_j's in the same time along with the status of already labeled nodes. To be precise, in Equation <ref>, we consider three aspects - node v_i's feature vector (h_i), all of its adjacent neighbor's (N(i)) feature vectors ((h_j wherej ∈ N(i))) and the current condition of node labeling X_v. If a node v_j has already been selected and labeled then the j^th entry will be 1 otherwise it will remain as 0. So X_v is a binary vector, {0, 1}^n. Here n denotes the number of nodes in the graph.Now, we point out the common strategies that we adopt from GRL through the following points,* GAT as an encoder architecture: K stacked layers of GAT are used as the encoder architecture to generate the node feature vectors h. The last layer consists of a sigmoid non-linear activation layer to generate the node probabilities. This probability helps to select the first node to initiate the decoding with the attention mechanism.* Loss objective and Training: The loss objective that we use is stated in Equation <ref>. Here P(v^t) denotes the probability of the greedily selected node v at the t^th iteration. r^t has already been defined prior. b denotes the reward observed from the baseline architecture at the t^th iteration. As per the source paper of GRL, baseline architecture can be considered a result produced from a problem specific heuristic solution or a sub-optimal solution. The source paper tries to incorporate this heuristically produced value at each time stamp of the reward function or loss function. In our setup. for the simplicity and generalization point of view, we omit this term and depend only on the QUBO-based Hamiltonian. As per our empirical results, we still observed quite comparative performance which will be demonstrated in Section <ref>. After accumulating the complete set of rewards (termination of an epoch), we apply gradient descent over the model parameters and backpropagate. L(θ) = ∑_t=1^n (r^t-b) × P(a^t)=∑_t=1^n r^t× P(a^t) [when b=0] §.§ Monty Carlo Tree Search with GNN through manual perturbation, MCTS-GNN In this section, we present a formulation where we integrate the Monty Carlo Tree Search in assistance with GNN in an RL-based setup. The main idea is that each node of the search tree consists of a partial solution (or node labels), and based on that a single GNN is trained to approximate the remaining nodes' labels. The goal is, using this manual perturbation of node labels, a guided search is conducted to maximize the amount of rewards. Now, we discuss the strategies based on RL terminologies (state, action, reward) and Monty Carlo tree search terminologies (selection, rollout, exploration, and backpropagation) in a brief manner.* State S: Each node or a state S of the MCTS tree, provides a partial solution or a subset of possible node labeling for the concerned CO problem. Similar to the previous section, for the processing we maintain a binary vector X_v where the i^th entry is set to 1 if i^th node has already been labeled.* Action, a and Transition function, π(S, a): An action a means choosing a label (either 0 or 1) for the input graph node variable x. From each state S multiple actions can be created by fixing the node labels of the unselected nodes from the point of view of S. Each action a from the state S also bears a transition probability π(S, a) denoting the likelihood of taking action a from S. π(S,a) is approximated using a GNN.* Reward, r: To calculate the reward for a state S, GNN is used. Using GNN, node probability distributions P(θ) are generated. Based on S, a subset of nodes have already been fixed, for the remaining unselected nodes' labels, P(θ) is used over a probability threshold β, e.g., for a node v_i, if P_i(θ) ≥β, then x_i=1 else x_i=0. After approximating the labels of all the nodes, QUBO-formulated Hamiltonian is used to calculate the reward, r = ∑_i ≤ j x_iQ_ijx_j. As it is already obvious, GNN plays a very important part of this design, now we state the mathematical formulation of GNN's forward pass along with the loss function that is used to update the parameters θ. X_em = E(G) h^' = GNN(G, X_em) h = f_1(X_vθ_1^T + h^'θ_2) P(θ) = f_2(hθ_3^T) The complete set of the mathematical formulation is presented fromEquation <ref> to Equation <ref>. First, a set of node embedding vectors X_em is generated (Equation <ref>). Then, X_em and input graph G is passed to GNN to generate a set of node feature vectors h^' (Equation <ref>). After that, more contextual information (X_v) is added with feature vector (h^') to calculate the complete node feature vectors h (Equation<ref>). Equation <ref> represents the final equation to generate the node probability distribution P(θ). Here the set of architecture parameters θ = {θ_GNN∈ℝ^d_2 × d_1, θ_1∈ℝ^d_3 × 1, θ_2∈ℝ^d_3 × d_2, θ_3∈ℝ^ 1 × d_3}. We use θ_GNN to denote all the weight parameters associated with the layers of GNN. f_1 is a ReLU activation function and f_2 is a sigmoid activation function to generate the probabilities. Similar to before (.)^T denotes the transpose from row vector to column vector or vice versa. P(θ) is also used to approximate π(S, a). For a particular variable x_v or input graph node v, P_v(θ) will indiciate the likelihood of labeling v to 1 (x_v=1, π(S, v=1)=P_v(θ)). Similarly, to label x_v as 0, we set π(S,v=0)=1-P_v(θ). From each state S, we create child nodes for the unselected variables for both of the labels.To train the GNN architecture, we use the formulation stated in Equation <ref> as the loss function. This function is quite similar to Equation <ref> apart from the fact that here we add manual perturbation by fixing the node labels to guide the searching. Here X_v,i denotes the value of the i^th entry in X_v. Here, X_v,i is set to 1 when node i has already been labeled. On the contrary, X_v,i is set to 0 when the node v_i has not been labeled. Our underlying intuition behind this formulation is presented in the form of the Proposition <ref>.L(θ) =∑_i ≤ j, X_V,i=1, X_v,j=1 x_i Q_ijx_j + ∑_i ≤ j, X_V,i=1, X_v,j=0 x_i Q_ij p(θ_j) + ∑_i ≤ j,X_V,i=0, X_v,j=0 p(θ_i) Q_ij p(θ_j) The relevant arguments related to this proposition are given by the following points, * A single GNN is trained through different sets of manually set node labels(partial solutions) which we regard as manual perturbation. These sets of possible partial solutions can be considered as noises or data points.* The GNN during its training updates its weight parameters by tackling these sets of perturbations.* This strategy mainly forces the GNN to adapt itself to different variations which as per our intuition relates to learning strategies to avoid or push itself from various local minimas observed during training. This results in more robust architecture and improved performance in terms of reducing constraint violations. Now, we discuss the terminologies associated with MCTS through the following points, * Selection and Exploration: In Equation <ref>, we present the greedy metric, Upper Confidence Bound (UCB) to measure the average reward obtainable for a child state C_i with respect to its parent state S combining its transition likelihood (π) of being selected. Here C_i.w and C_i.v denote the total amount of reward accumulated in state C_i and the number of times C_i has been visited, respectively. α denotes a hyperparameter, π(S, a) denotes the transition probability to state C_i from S reported by GNN. S.v denotes the total number of times state S was visited. log denotes mathematical logarithmic operation. The node with the highest UCB value is selected to explore in its subtree. When a leaf node is reached in this manner, we start the rollout phase.UCB(C_i)= C_i.w/C_i.v + α * π(S, a) * √(log(S.v)/C_i.v) * Rollout: In the rollout phase, using GNN, we conduct the training for multiple epochs to approximate the node labels for the unlabeled nodes and calculate the rewards. We have already discussed how GNN is used to generate the node probability distributions in the previous paragraphs. * Backpropagation: After the rollout phase we enter the backpropagation phase of MCTS. In this phase, we update the state variables, v and w for the path from the current leaf state to the root of the search tree. For all the nodes in the path we increment the visiting attribute by 1 and add the reward to w approximated by GNN. § EVALUATIONIn this section, we empirically evaluate GRL_QUBO and MCTS-GNN with PI-GNN in addressing the Max-Cut problem for the graphs of different densities. The graphs were randomly generated and all the graphs were undirected. For a given graph of n nodes, there can be at most n × (n-1)/2 edges where any two nodes do not have multi-edges among them. Our prepared graph dataset can be found in [<>]. A statistics over the graphs based on their adjacency information is shown in Table <ref> denoting the maximum, minimum and average number of adjacent edges for each of the nodes for all the experimented graphs. We conduct the experiments based on three metrics - the number of satisfied constraints, training stability, and required time. All the experiments were conducted on a 64-bit machine having AMD Ryzen 9 5950x 16-Core Processor x 32, 128 GB RAM, and 24 GB NVIDIA GeForce RTX 3090 GPU. All the implementations were done in Python language based on the blocks provided by deep-learning libraries Pytorch <cit.>, Pytorch Geometric <cit.> and LabML <cit.>. §.§ Architecture Description, Hyperparameter SetupIn this section, we present the description of the architectures along with the value of the hyperparameters that are used during the training. The description is presented in the following manner,* Architecture Description: To present the discussion we mostly use the variables stated in the respective sections.Our GNN architecture follows the similar definition provided in PI-GNN. It consists of two layers of Graph Convolutional Architecture (GCN) and a sigmoid layer to generate the node probabilities. The first GCN layer's node feature vectors are propagated through a non-linear ELU activation and a dropout layer before passing to the second layer of GCN. Let us assume the node feature vector size of the 1^st and 2^nd layers of GCN are d_1 and d_2. As mentioned in the base article, we follow a similar setup. If the number of nodes n ≥ 10^5, then d_1=√(n), else d_1=√(n). Also, d_2 = d_1/2.To implement GRL_QUBO, we took inspiration from the study presented in the base source paper of GRL. Here, we had three layers of GAT as the encoding architecture and a sigmoid layer to generate initial node probabilities. The attention-based decoding formulation has been presented in Section <ref>. To set the dimension of the node feature vectors, we follow a similar strategy stated in the previous paragraph based on the number of nodes of the input graph, n. So, if n ≥ 10^5 then, d_1=√(n) else, d_1=√(n) and followup d_2 = ⌈d_1/2⌉. We have used a single attention-head mechanism.In MCTS-GNN, we train a single GNN in different rollout phases by applying different manual labeling perturbations. The setup is completely similar to the discussion already presented above. So, if n ≥ 10^5, thend_1=√(n) else, d_1=√(n). d_2 = ⌈d_1/2⌉ and d_3=1. * Training Description: To train the architectures, we use Adam optimizer for each of the architectures. We use patience value to implement fuzzy early stopping for all of them. As RL-based setups inherently exhibit abrupt behavior (frequent ups and downs in terms of reward variation) we keep a comparatively higher patience value for GRL_QUBO and MCTS-GNN compared to PI-GNN. As per our setups, we eventually observed almost linear variation in terms of objective functions' values at some phase of the training epochs for all the experimented architectures for different graph inputs. This supports the validity of the chosen hyperparameters for early stopping. There lies another reason behind this strategy. In PI-GNN, we observe the variation in the loss values which is fractional whereas in the other experimented architectures we track the reward values which is integer. Inherently, it can be understood that the integer rewards should fluctuate more compared to the fractional loss values. This demands a higher early stopping threshold for the RL-based setups. For PI-GNN, we simply choose learning rate lr=10^-4 with a patience value of τ=100 denoting if there is no loss objective improvement compared to the current best value observed for a consecutive 100 epochs, we stop the training. As stated previously, we apply fuzzy early stopping in our experimented setup. For GRL_QUBO, we choose a learning rate of 0.001 (lr) for both encoder-decoder and a patience value of 700 (τ). We also experimented with a lesser learning rate (0.0001) for GRL_QUBO but could not observe further improvement in terms of satisfying the number of constraints. For, MCTS-GNN, we also chose the learning rate lr to 0.001 for the GNN architecture with the patience value of 100. Apart from the prior hyperparameter to control GNN training, we included another early stopping criterion τ^' that tracked the improvement in the reward objective. We set τ^' to 700denoting that if there is no reward improvement compared to the best reward observed till the current iteration, the MCTS-GNN algorithm terminates.§.§ Comparison in terms of number of satisfied constraintsIn this section, we present the number of satisfied constraints observed in each architecture, PI-GNN, GRL_QUBO and MCTS-GNN, for the graphs of different intensities. The complete result is shown in Table <ref>. In this study, this metric is denoted as the reward in terms of RL-based formulation. For GRL_QUBO and MCTS-GNN we present the best reward observed throughout the complete training. For PI-GNN, we record the least loss and the corresponding probability distribution to approximate the node labels based on the threshold β( e.g. if β=0.5, then the node's probability exceeding 0.5 gets labeled as 1). These approximated node labels are used to calculate the QUBO-formulated Hamiltonian function. In the last two columns of Table <ref>, we present how GRL_QUBO and MCTS-GNN's performance has improved compared to PI-GNN in percentage. A positive value means, the result has improved and the negative means the opposite.From the presented result in Table <ref>, we can see that, the performance has comparatively improved than the base PI-GNN. In simple terms, upon applying RL-based formulation we improved the quality of the resultant output. If we investigate in a more detailed manner we can see that, for a particular graph having n nodes, the performance generally improves more with it being denser or increased number of edges.As per our analysis, PI-GNN is quite scalable and a generic solution to address a wide range of CO problems. However, it lags behind RL-based methods in meeting constraints, especially as the graphs get denser. In GRL_QUBO, when the attention-based decoding scheme selects a particular node and sets its label, only its adjacent unlabeled neighbor nodes' attention weights (α) are updated. Then using a max heap-based strategy next likely node is selected and labeled. As, in each selection, the binary vector is updated, the performance might be improved if all the nodes' attention weights are updated. However, it will add a significant runtime cost due to linear updates in each selection, so a runtime-accuracy tradeoff exists there. MCTS-GNN, intuitively applies noise during training to enforce the training or the architecture to skip various local minimas to reach comparatively better outcomes.§.§ Training StabilityIn this section, we mainly highlight the convergence status or training stability of the experimented architectures based on our set hyperparameters. We centralize our discussion based on three graphs of 50 nodes with 89, 139, and 499 edges, respectively for all the architectures to understand the behavior transition from sparser to denser graphs. In Fig. <ref> we present the reward variations for PI-GNN, GRL_QUBO and MCTS-GNN, respectively for a graph of 50 nodes with 89 edges. In a similar manner, we present the reward variations for a graph of 50 nodes with 139 edges and a graph of 50 nodes with 499 edges for all the architectures in Fig. <ref> and <ref>, respectively. To specifically understand the loss convergence status of PI-GNN, we present Fig. <ref> for the three 50 node graphs with 89, 139, and 499 edges, respectively. Here, in the captions (50, 89) means a graph having 50 nodes and 89 edges. The main observation that can be drawn from each is that the convergence behavior or almost linear variation is quite visible where our training has stopped for all the architectures. This supports the stability quality of the set hyperparameters, especially the patience values. Also, we do not provide all the other graphs' training convergence chart in this study, because they exhibit a similar pattern. All of our experiments are reproducible and can be regenerated by importing our official code repository. §.§ Complexity AnalysisIn this section, we present the time complexity of each of the experimented architectures or more specifically discuss the computation-intensive segments for each.PI-GNN is the most scalable and runtime efficient compared to GRL_QUBO and MCTS-GNN due to the novelties of GNN architecture. In simpler terms, to process a graph having n nodes to generate a d dimensional node feature vectors through L layers of GNN, the worst case complexity is 𝒪(Lnd^2)[Here our assumption is that, the graph is sparse so, the number of edges E=𝒪(n)]. GRL_QUBO also uses a GNN-variation (GAT) to encode the input graph. After that, it applies greedy selection at each iteration to pick a node with the highest attention value reported by the decoder. The additional complexity of this selection portion takes a complexity of 𝒪(nlogn). MCTS-GNN also uses a single GCN architecture in the rollout phases to build the search tree. So, a similar complexity of GNN is automatically added to the overall complexity. It also takes multiple attempts to train the same GNN again by varying inputs through manual perturbation of labelings. So, this incurs additional costs also. But our experiments suggest that only a very few times, GNN is trained exhaustively, and the other times, it generally runs for a very small number of epochs (less than 1000 in our experiments) and reaches the early stopping criterion. So, in summary, PI-GNN is the most scalable. The complexity of GRL_QUBO increases with the graphs becoming larger and the complexity of MCTS-GNN increases with the number of iterations to run to expand the search tree. However, with this additional cost, we improve the quality of the final result by increasing the number of satisfied constraints compared to PI-GNN.Apart from these theoretical aspects, other constraints, e.g., step size or learning rate are also quite important and play a crucial role in converging the objective loss functions. If it takes a good time for the convergence, then the overall training time increases also. Based on our observations, generally, we found MCTS-GNN to take the most amount of time due to the expansion of the search tree. GRL_QUBO takes way less time than MCTS-GNN and provides a competitive performance against PI-GNN - where PI-GNN takes a longer number of epochs to be trained, GRL_QUBO takes way less number of epochs where there lies a significant amount of processing in each epoch by selecting the nodes in a greedy manner. PI-GNN shows worse performance than GRL_QUBO, especially in denser graphs where it might need a good amount of time to be converged. In Table <ref>, we present some results, each value is denoted in seconds. But, a point to be mentioned is that, based on loss convergence status, in multiple trials, the runtime can vary significantly.§ CONCLUSION In this study, we have extended the work of PI-GNN by improving its performance in terms of a number of satisfied constraints. We identify a crucial bottleneck issue observed in graphs with higher densities while applying PI-GNN during the transition of the loss values. We proposed a Fuzzy-stopping strategy in this regard to improve the quality of the solution. We also mentioned the issue regarding the absence of the node label's actual projection status in the QUBO-formulated Hamiltonian loss function and raised a concern that avoiding this factor might degrade the performance while applying the actual binary projection over the variables. In this regard, we proposed a Monty Carlo Tree Search with the GNN-based solution. This solution applies manual perturbation of node labels in the Hamiltonian loss function to guide a single GNN training while expanding the search tree. Furthermore, we also investigated the applicability of QUBO-formulated Hamiltonian in terms of reward function in RL setups. This contribution works as a bridge between RL-based solutions of CO problems with QUBO-based formulations while broadening the scope of applying actual node projection status during training. Our empirical result suggested that RL-based setups generally gave a better performance in terms of satisfied constraints than the PI-GNN setup with an additional incurring runtime or processing costs. So, our summarized observation can be included as PI-GNN is quite scalable and provides a moderate performance in terms of the number of satisfied constraints which can be improved by enforcing RL-based formulations. 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"authors": [
"Redwan Ahmed Rizvee",
"Raheeb Hassan",
"Md. Mosaddek Khan"
],
"categories": [
"cs.LG",
"cs.AI"
],
"primary_category": "cs.LG",
"published": "20231127193314",
"title": "A Graph Neural Network-Based QUBO-Formulated Hamiltonian-Inspired Loss Function for Combinatorial Optimization using Reinforcement Learning"
} |
School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India RPTU Kaiserslautern-Landau, Kaiserslautern, Germany Solid State Physics Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India [email protected] Theory Division, Saha Institute of Nuclear Physics, A CI of Homi Bhabha National Institute, Kolkata-700064, India [email protected] School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni-752050, India Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and Research, HBNI, Jatni, 752050 Bhubaneswar, India. Topological magnetic skyrmions in centrosymmetric systemsexhibit a higher degrees of freedom in their helicity, hence possess a great potential in the advanced spintronics including skyrmion based quantum computation. However, the centrosymmetric magnets also display non-topological trivial bubbles along with the topological skyrmions.Hence it is utmost priority to investigate the impact of different magnetic ground states and their underlying interactions onthe stabilization of magnetic skyrmions in cetrosymmetric magnets. Here, we present a combined theoretical and experimental study on the role of non-collinear magnetic ground state on the skyrmion stabilization in a series of exchange frustrated non-collinear ferromagnetic system MnFe_1-xCo_xGe. With the help of neutron diffraction (ND) and Lorentz transmission electron microscopy (LTEM) studies, we show that hexagonal skyrmions lattice emerges as a stable field driven state only when the underlying magnetic ground state iscollinear with easy-axis anisotropy. In contrast, non-topological type-II bubbles are found to be stable state in the case ofnon-collinear magnetic ordering with partial in-plane anisotropy. Furthermore, we also find that the skyrmions transform to the non-topological bubbles when the system undergoes a spin reorientation transition from the easy-axis to easy-cone ferromagnetic phase. Our results categorically establishthe significant role of in-plane magnetic moment/anisotropythat hinders thestability of skyrmion both in the case of collinear and non-collinear magnets. Thus, the present study offers a wide range of opportunities to manipulate the stability of dipolar skyrmions by changing the intrinsic characteristics of the materials. Spin order dependent skyrmion stabilization in MnFeCoGe hexagonal magnets Ajaya K. Nayak January 14, 2024 =========================================================================§ INTRODUCTION Non-collinear magnetic textures with topological character, such as skyrmions are one of the major interests in spintronics community due to their enormous scientific and technological possibilities in the future generation data storage devices with higher density and low power consumption <cit.>. At the early stages of skyrmion discovery, focus was mostly concentrated on bulk and thin film magnets with broken inversion symmetry for hosting Dzyaloshinskii-Moriya interaction, the major force for the stabilization of skyrmions <cit.>. However, in recent years, a wide variety of materials preserving theinversion symmetry are found to facilitate skyrmion like topological spin textures with different helicity and vorticity <cit.>. These centrosymmetric systems provide additional degrees of freedom to thehelicity and vorticity of the skyrmions, which offers greater flexibility for their implementation as "0" and "1" data bits in storage device and quantum computing<cit.>. Inmost of these magnets competinguniaxial anisotropy and dipolar energyplay a major role in the stabilization of skyrmions <cit.>. Recently, a few studies have demonstrated that the modification in uniaxial anisotropy and/or application of external in-plane magnetic field can greatly influence the stability of dipolar skyrmions <cit.>.Hence, it is extremely important to investigate the impact of different energy parameters on the stability of skyrmions in centrosymmetric magnets. In this direction, the present manuscript focuses on the role of competing exchange interactions on the stability of magnetic skyrmions. As uniaxial magnetic anisotropy (UMA) is one of the key requirements for skyrmion formation in centrosymmetric magnets, we look into systems having potential to exhibit both UMA and competing exchange interactions.In this direction, theNi_2In crystal structure based hexagonal magnet MnFeGe is reported to shownon-collinear magnetic structure driven by competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions <cit.>. It has also been theoretically demonstrated that the magnetic ordering in the MnFeGe system can be modified by altering the inter-atomic distance between the Mn atoms sitting at different layers <cit.>. Furthermore, it has also been shown that a complete replacement ofFe by Coleads to a collinear ferromagnetic ground state in the hexagonal magnet MnCoGe <cit.>.Hence, it is expected that the substitution of Co can systematically change the landscape of different magnetic interaction in the system.In order to gain a deep insight into the possible magnetic ground states in the Co doped samples, we utilize first principles density functional theory (DFT) calculation, as implemented in Vienna ab-initio simulation package (VASP) <cit.>,for carryingoutmagnetic structure optimizations and energy calculations by substituting Co in place of Fein MnFe_1-xCo_xGe.The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, a version of generalized gradient approximation (GGA), is considered in the calculations <cit.>. To investigate the effect of doping concentration on magnetic ground state, we have taken 1×1×2 supercells for all the calculations. In the unit cell, Mn, Fe/Co and Ge atoms occupy the position at (0, 0, 0), (2/3, 1/3, 1/4) and (1/3, 2/3, 1/4) sites, respectively. The k-point grid of 5×5×3 is used to sample the first Brillouin zone for self-consistent calculations. The threshold value for energy convergence between two consecutive electronic relaxation steps is set to 1×10^-5 eV and the structures are optimized until the force on each atom becomes less than 0.001 eV/Å. The unit cell of MnFeGe structure belongs to the Cm space group with optimized lattice constants of a = b = 4.096 Å, and c= 5.083 Å, which is consistent with previous theoretical results <cit.>. First we compare the total energy differences between the FM and AFM states ΔE [= E(FM) -E(AFM)] for MnFeGe and MnCoGe (where all Fe atoms from MnFeGe are replaced by Co atoms) compounds. For MnFeGe, ΔE = 0.067 eV, where Mn-Mn couple antiferromagnetically. But, FM configuration is more stable (ΔE = -0.048 eV) for MnCoGe case. Interestingly, for partial Co doped MnFeGe system, namely MnFe_0.75Co_0.25Ge (MnFe_0.5Co_0.5Ge) ΔE decreases to 0.026 eV (0.015 eV), which is quite small. On the other hand, the ground state is FM (ΔE = -0.011 eV) for Co rich system MnFe_0.25Co_0.75Ge like that of MnCoGe system. Therefore, our collinear magnetic calculations point to a strong competition between the underlying magnetic states in MnFe_1-xCo_xGe compounds. So, we extend our calculations to unveil the possible noncollinear magnetic structures. The schematic representation of the expected noncollinear magnetic ordering with out-of-plane FM component and in-plane AFM component is depicted in Fig. <ref>(a).The change in calculated energy as a function ofcanting angle (θ) of the Mn moment with respect to the c-axis for the samples MnFe_1-xCo_xGe [x = 0.25, 0.5, 0.75, 1] is shown in Fig. <ref>(b). In the case of Co doped sample with x = 0.25, a local energy minima is observed at θ = 40^∘, which nearly matches with the earlier reported θ ≈ 45^∘ for the parent compound MnFeGe <cit.>.Importantly, a further increase in the Co concentration to x = 0.5 leads to the noncollinear ferromagnetic state with θ = 30^∘ as minimum energy state. The Co rich samples with x = 0.75 and x = 1 show the energy minimum for collinear FM state with θ = 0^∘. Hence, our theoretical study clearly points toward the emergence of non-collinear magnetic state for the sample x = 0.5 and a change in magnetic ordering from the noncollinear AFM to collinear FM ground state with high Co doping (x ≥ 0.75) in the MnFe_1-xCo_xGe system.To get more insight into the magnetic phase transition with increasing the Co doping, we calculate the density of states (DOS) and projected DOS (PDOS) for the systems. The PDOS for d-states of Mn, Fe and Co for non-magnetic MnFe_0.75Co_0.25Ge and MnCoGe systems are shown in Fig.<ref>(c). A significant difference in DOS near the Fermi level for both the materials in terms of Fe-d and/or Co-d states is clearly observed.These variations in the Fermi-level states can be related to the indirect exchange model of magnetic ordering, i.e. competition between carrier mediated (RKKY-like) exchange and superexchange <cit.>.The parameters of these exchange interactions varies inversely with the energy required to push an electron from d-states to the Fermi level <cit.>. Therefore, the amount and distribution of DOS adjacent to Fermi level plays a crucial role in determining these exchange parameters. In the case of sample with high Fe concentration, the number of available Fe-d states is more above the Fermi level than just below it. In contrast, there are more Co-d states present just below the Fermi level in case of MnCoGe compound. In this case, the difference between the DOS of Fe and Co can be compared to the rigid-band model, in which the addition of an extra electron causes the Fermi level to shift upward <cit.>. Using a perturbative approach, the coupling constants (j_RKKY and j_S ) can be expressed in q→0 limit as follows:j_RKKY(0) = V^4 D(ϵ_F)/E^2_h j_S(0) = V^4 ∑_nk^ϵ_nk > ϵ_F (ϵ_nk - ϵ_F -E_h)^-3Here, V and D(ϵ_F) denote electron mixing parameter and DOS at the Fermi level, respectively. ϵ_nk is the energy at k-point of the nth band whereas E_h is the energy in electron transfer from d-states to the Fermi level. Although both MnFe_0.75Co_0.25Ge and MnCoGe display a large Co-d states at the Fermi level, a sharp fall in DOS (mainly in terms of Fe-d states) just above the Fermi level is found when all the Fe atoms are replaced by Co. Thus, the number of unoccupied states (N) near the Fermi level is very small in case of MnCoGe than Fe rich case. Thus, the superexchange coupling constant j_S stated in the above equation is bounded from above by j_S(0)≤ V^4N /E_h^3 <cit.>. Therefore Fe rich system (|j_S|>|j_RKKY| due to the large N ) prefers AFM ordering. In the case of MnCoGe, N is smaller compared to that of MnFe_0.75Co_0.25Ge and hence RKKY coupling constant j_RKKY should be greater than the superexchange constant j_S. In addition, it is also noted that D(E_F) for MnCoGe is larger than that of MnFe_0.75Co_0.25Ge, hence in overall, FM ordering in MnCoGe. Figure <ref>(d) schematically represents the dominant RKKY exchange mediated through the Co conduction electron as a possible origin of ferromagnetism rather than antiferromagnetic ordering, in the Co rich samples. In fact, the spin-polarized site-projected DOS for Co atom for MnCoGe in FM configuration (Figure not shown here) shows a sharp peak in spin-down DOS arising from Co-d states just below the Fermi level. This state helps in mediating the indirect interactions between Mn atoms favoring a FM ground state for MnCoGe <cit.>. The calculations also indicate that the observed non-collinearity for the intermediate Co compounds, such as MnFe_0.5Co_0.5Ge, is mainly driven by the competition between different types of indirect exchange interactions between the localized Mn moments. This suggests that the MnFe_1-xCo_xGe system is a potential candidate for hosting tunable magnetic ground states depending on the nature of spin-ordering. Therefore,in this report, we focus on the evolution of magnetic skyrmions and bubbles with change in energy landscape in the series of centrosymmetric magnets MnFe_1-xCo_xGe. With help of Lorentz transmission electron microscopy (LTEM) study, we demonstrate how the presence of inplane magnetic anisotropy hinders the formation of skyrmion lattice in the case of non-collinear magnets (x ≤ 0.6) as well as easy-cone FM state,whereas the same can be easily stabilized in the case of easy axis collinear FM state in the sample x = 0.8. § RESULTS AND DISCUSSION Polycrystalline samples ofMnFe_1-xCo_xGe (x = 0.2 to 0.8) are prepared using arc melting technique.All the samples formed in the layered hexagonal crystal structure with space group P6_3/mmc. Our magnetic measurements show an increase in the magnetic ordering temperatures (T_C)from 170 K to 260 K with increasing Co doping from x = 0.2 to x = 0.8 . We also find a monotonic increase in the saturation magnetic moment (M_S) from 2 μ_B/f.u. to 3.1 μ_B/f.u with increasing Co concentration from x = 0.2 to x = 0.8.Furthermore, we have compared the theoretically calculated magnetic moments of different magnetic configurations with the experimentally obtained saturation magnetization (M_S), as shown in Fig. <ref>(e). The M_S matches well with the DFT calculated total FM moment for the samples with x = 0.75 and 1, whereasfor x = 0.25 and x = 0.5, the experimental M_Sbetter matches with the total calculated moment for the non-collinear ferromagnetic (NCFM) configuration rather than collinear FM configuration.To further experimentally verify the theoretically proposed change in the magnetic ground states in the present system,powder neutron diffraction (PND) measurement is carried out on two of our samples with x = 0.4 and x = 0.8 (Co rich) as shown in Fig. <ref>(a) and Fig. <ref>(b), respectively. In the case of x=0.4, a substantial rise in the intensity for (001) and (101) reflections below T_C can be seen clearly in the contour plot in Fig. <ref>(a). In addition, the appearance of (001) magnetic reflection only below the T_C indicates the presence of a finite basal plane AFM component <cit.>. To get an idea about the degree of noncollinearity in case of x=0.4, we have plotted the temperature dependent canting angle (θ) of the Mn momentsobtainedfrom the PND refinement [Fig. <ref>(c)]. At T= 4.8 K, the canting angle is nearly 33.4^∘ ± 0.8^∘, which matches well with the theoretically predicted θ for x = 0.5 sample [see Fig. <ref>(b)]. The θ gradually decreases with increasing temperature and becomes nearly 0^∘ around the T_C. For x = 0.8 sample,the PND data do not show any additional magnetic reflections other than the ones ontop of the nuclear reflections. The findings suggest the presence of a collinear magnetic ordering for x = 0.8 sample <cit.>, as suggested by our theoretical calculations. Furthermore, a close analysis of the PND data reveals an increase in the strength of the (002) and (102) reflections with decreasing temperature below 150 K, as shown in the contour plot Fig. <ref>(b). To understand this peculiar behavior,we have simulated the PND data for this sample with the magnetic easy-axis tilted away from the c-axis by an angle ϕ <cit.>.This results in a monotonic enhancement of the (002) and (102) reflections withincreasing ϕ. Hence, we have carried out the Rietveld refinement of the PND data for this sample with the easy-cone model. The temperature variation of refined tilting angle (ϕ)for the x = 0.8 sample is shown in the inset of Fig. <ref>(c). We find a tilting angle of about 25^∘ ± 5^∘ below 100 K. Therefore, the sample x = 0.8 exhibit an easy-cone FM state below 150 K and an easy-axis ordering at higher temperatures. The schematic representation of the magnetic ground states for the samples x = 0.4 and 0.8 obtained from the PND dataalong with the corresponding real space LTEM images recorded at zero external field are shown in Fig. <ref>(d)-(k). As depicted in Fig. <ref>(d), the presence of non-collinear magnetic state with a largein-plane anisotropy component (K_u^∥) gives rise to the observation ofin-plane magnetic domainsas spontaneous magnetic statefor the sample x = 0.4 at 100K [see Fig. <ref>(h)]. Furthermore, due to a decrease in the K_u^∥ in comparisonto the out-of-plane magnetic anisotropy(K_u^⊥) [see Fig. <ref>(e)], the magnitude of the in-plane domain walls starts diminishing and an impression of the out-of-plane stripe domains starts appearing as a spontaneous state [see Fig. <ref>(i)]. Interestingly, the sample with x = 0.8, which displays an easy cone magnetic state at 100 K [see Fig. <ref>(f)],exhibits a stripe domain as spontaneous magneticstate as shown in Fig <ref>(j). When the K_u^∥ component completely vanishes at T = 200 K for x= 0.8 [see Fig. <ref>(g)], we also find the presence oflittle bit disordered stripe domains [Fig. <ref>(k)] compared to that observed at T = 100 K. This type of stripe domain alignment in case on in-plane anisotropy has previously been found experimentally by introducing a finite K_u^∥ component to the system <cit.>.Additionally, our Object-Oriented Micromagnetic Framework (OOMMF)-based micromagnetic simulations also demonstrates that the stripe domain can be organized in the direction of the in-plane anisotropy component with the tilting of the magnetic easy-axis. To further study the nature of magnetic state we have carried out a detailed field and temperature dependent LTEM study in the present system. In the case of x= 0.4, the in-plane domain state at 100 K [Fig <ref>(h)] and the mixed state at 150 K [Fig <ref>(i)] transform to field polarized phase with the application of magnetic field. However, the scenario changes completely in the case of x = 0.8, where we find different domain states by varying the magnetic field and temperature. To eliminate the effect of in-plane magnetic field, all the LTEM experiments are carried out with the applied magnetic fieldalong the c-axis. As shown in Fig. <ref>(a), the Co rich sample x = 0.8 exhibits a hexagonal lattice of type-II bubbles at 100 K and a magnetic field of 0.4 T.By increasing the temperature to 150 K leads to a mixed state of type-II bubble and skyrmions at a magnetic field of 0.4 T [see Fig. <ref>(b)]. The presence of higher number of type-II bubbles than that of skyrmionssuggest that former are the energetically stable state at this temperature. Surprisingly, we find the stabilization of only skyrmions with both clockwise (CW) and counter-clockwise (CCW) helicity by increasing the temperature to 220 K [see Fig. <ref>(c)]. It is important to mention here that our PND clearly show the existence of a collinear FM ground state with easy-axis anisotropy at 220 K, whereas an easy-cone FM arrangement with tilted easy-axis with respect to the c-axis is found at 100 K. Hence, it is very much evident that the stabilization of tupe-II bubbles greatly depend on the nature of magnetic ordering in the system. It has been also reported that the presence of small in-plane applied magnetic field can break the symmetry of the spin alignment in a centrosymmetric skyrmion, thereby giving rise to the observation of type-II bubbles <cit.>.To confirm that the present observations are free from the effect of in-plane magnetic field, we have recorded the zero field remnant magnetic state after initially applying the field exactly along the zone axis and then decrease the field to zero. The remnant magnetic states at different temperatures for the sample x = 0.8 are shown in the Fig. <ref>(d)-(f). At 100 K a mixed phase of type-II bubbles and stripe domains is observed [see Fig. <ref>(d)], suggesting a very small energy difference between these magnetic states. As expected at 150 K, a few skyrmions along with the magnetic state of 100 Kare found [see Fig. <ref>(e)].On the other hand, the observation of mixed phase of skyrmions and stripe domains without the existence of any type-II bubbles at 220 K suggests that the skyrmions are having lower energy in the system compared to the type-II bubbles [see Fig. <ref>(f)].To confirm the exact nature of spin arrangements in the observed magnetic structures,we haveconstructed the spin textures using transport of intensity equation (TIE) analysis of the marked regions. The TIE analysis of the LTEM images clearly shown the presence of Bloch-type skyrmions at 220 K, whereas the existence of Bloch-point type feature in the domain state at 100 K and 150 K indicates the formation of type-II bubbles. Although the stripe domains are observed as spontaneous magnetic state for the sample x = 0.4 at 150 K, no skyrmion state is observed with the field evolution of the magnetic domains. Here, the point should be noted that the sample with x = 0.4 has a non-collinear FM ground state with sufficiently strong in-plane AFM component (i.e, K_u^∥0 ) in the basal plane. For the sample x = 0.5, 0.6 a hexagonal type-II bubble lattice along with a very few number of skyrmions are observed.Figure <ref> shows Co concentration (x) vs. temperature (T) phase diagram for the samples MnFe_1-xCo_xGe. Here, the LTEM images at the magnetic field that hosts the maximum number of skyrmion at a specific (x, T) are used to determine the skyrmion density for that point. The strong skyrmion density is observed for the easy axis collinear ferromagnet with zero K_u^∥, whereas the samples with lower Co concentration upto x = 0.6 shows mostly type-II bubbles as stable magnetic state. Furthermore, the easy-cone FM phase with finite K_u^∥ exhibits type-II bubbles as a stable state rather than skyrmion. All the experimental observations demonstrates that the presence of in-plane anisotropy component in a system hinders the skyrmions stability. All the experimental observations are also theoretically validated using object-oriented micromagnetic framework. the simulated data shows that the skyrmions can be transformed to the type-II bubble by introducing a sufficient amount of in-plane magnetic anisotropy component along with the out-of-plane anisotropy. The skyrmion-like texturesin the uniaxial centrosymmetric systems are of great technological interest due to theirdifferent topological numbers as well as helicity degrees of freedom.In most cases, the competing UMA and dipolar interaction are considered as the fundamental mechanisms for skyrmion stabilization in the centrosymmetric system <cit.>. Furthermore, some of the centrosymmetric systems show skyrmions of size 1-2 nmdue to frustrated magnetic interaction including four spin exchange interaction <cit.>. However, these extremely small skyrmions are always found at very low temperatures(< 10 K). In this direction, the addition of frustrated magnetic exchangeto the dipolar skyrmion systems might serve as an important step forwardto realize small skyrmions at room temperature.The dipolar stabilized skyrmions in most of the centrosymmetric systemsare always considered in the collinear ferromagneticbackgrounds. Although, few of the earlier literatures describe the tunablity of the dipolar skyrmions in terms of external stimuli, such as magnetic field <cit.> and current <cit.>, their stability while modifying the internal energy parametersis not thoroughly investigated.Moreover, the effect of magnetic ground states and the underlying interactions on the dipolar stabilized skyrmions is still remain elusive. In the present report, a comprehensive investigation on the stability of dipolar skyrmions depending on the strength of exchange frustration and the corresponding magnetic ground states is demonstrated, where the frustration in the magnetic exchange interactions between Mn moments can be tuned depending on the Fe and Co atomic ratio <cit.>. Our theoretical calculations and experimental findings indicate the presence of a noncollinear canted magnetic ground state for the samples MnFe_1-xCo_xGe with x<0.75 and a collinear ferromagnetic state when x⪈0.75. The LTEM observation of in-plane domain walls for the non-collinear ferromagnet with x = 0.4 at T = 100 K supports the presence of a higher in-plane magnetic anisotropy component (K_u^∥), correlating to a significant in-plane AFM component. As a result, it is expected that theout-of-plane (K_u^⊥) and in-plane (K_u^∥) magnetic anisotropy components realign based on the change in the out-of-plane or in-plane magnetic moment contributions. The DFT calculations, on the other hand, indicate a decrease in the canting angle (θ) in the noncollinear magnets with increasing Co concentration (x). Hence a decrease in the ratio of K_u^∥/K_u^⊥ can lead to the emergence of out-of-plane stripe domains as a zero field LTEM state. Most importantly, in noncollinear background ( K_u^∥0) the non-topological type-II bubbles are mainly stabilized as field driven stable state rather than the topological skyrmions. The hexagonal skyrmion lattice is only observed in collinear ferromagnetic background when the magnetic field applied along the easy-axis direction. Earlier reports show a transformation between the topological skyrmions and non-topological type-II bubbles in the collinear ferromagnetic background with application of non-zero in-plane magnetic field <cit.>. The present study elucidatethat the skyrmions in collinear ferromagnetic background can also be transformed to type-II bubble with an applied magnetic field along the zone axis, when a nonzero in-plane magnetic moment or K_u^∥ introduced in the system. All our experimental findings point that the K_u^∥ inhibits skyrmion stability in both collinear and non-collinear magnetic backgrounds. Hence, the present study sheds light on the consequences of different energy factors on the stability and tunability of dipolar skyrmions. § CONCLUSIONTo summarize, we have thoroughly investigated how the magnetic ground states and the corresponding interactions affect the stability of dipolar skyrmions in a variety of non-collinear hexagonal ferromagnets MnFe_1-xCo_xGe. We show that the degrees of non-collinearity and the exchange frustration strength possess a significant correlation with the stability of dipolar skyrmions. The skyrmion lattice can only be stabilized in the easy-axis collinear ferromagnet with applied magnetic field along the zone axis, whereas non-topological type-II bubbles emerges as more favorable state in the non-collinear magnetic background. The role of in-plane magnetic anisotropy, K_u^∥, in the skyrmion stabilization is demonstrated in the case of easy-coneferromagnetic phase wheretype-II bubble are found.Furthermore, our research provides the prospect of broad control over dipolar skyrmions by modifying the internal energy characteristics of the materials, and it might be regarded as a step ahead in the realization of dipolar skyrmion-based spintronic devices.§ ACKNOWLEDGMENTS AKN acknowledges the support from Department of Atomic Energy (DAE), the Department of Science and Technology (DST)-Ramanujan research grant (No. SB/S2/RJN-081/2016), SERB research grant (ECR/2017/000854) and Nanomission research grant [SR/NM/NS-1036/2017(G)] of the Government of India. A.K. 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Sagayama, H. Nakao, Y. Taguchi, T. Arima, Y. Tokura, Science 365, 914-918 (2019). GdRu2Si2 N. D. Khanh, T. Nakajima, X. Yu, S. Gao, K. Shibata, M. Hirschberger, Y. Yamasaki, H. Sagayama, H. Nakao, L. Peng, K. Nakajima, R. Takagi, T. h. Arima, Y. Tokura and S. Seki,Nat. Nanotechnol. 15, 444 (2020). Fe3Sn2_current control W. Wei, J. Tang, Y. Wu, Y. Wang, J. Jiang, J. Li, Y. Soh, Y. Xiong, M. Tian, and H. Du, Adv. Mater. 33 2101610 (2021). Fe3Sn2_APL Y. Wu, L. Kong, Y. Wang, J. Li, Y. Xiong, and J. Tang, Appl. Phys. Lett. 118, 122406 (2021) Fe3sn2_CURRENT2Z. Hou, Q. Wang, Q.Zhang, S. Zhang, C. Zhang, G. Zhou, X. Gao, G. Zhao, X. Zhang, W. Wang, J. Liu, Adv. Sci. 10, 2206106 (2023). | http://arxiv.org/abs/2311.15823v1 | {
"authors": [
"Dola Chakrabartty",
"Mihir Sahoo",
"Amit Kumar",
"Sk Jamaluddin",
"Bimalesh Giri",
"Hitesh Chhabra",
"Kalpataru Pradhan",
"Ajaya K. Nayak"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20231127134746",
"title": "Spin order dependent skyrmion stabilization in MnFeCoGe hexagonal magnets"
} |
Student Mastery or AI Deception? Analyzing ChatGPT's Assessment Proficiency and Evaluating Detection Strategies Kevin Wang Department of Computer ScienceUniversity of British ColumbiaKelowna, BC, Canada, V1V [email protected] Seth Akins Department of Computer ScienceUniversity of British ColumbiaKelowna, BC, Canada, V1V [email protected] Abdallah Mohammed Department of Computer ScienceUniversity of British ColumbiaKelowna, BC, Canada, V1V [email protected] Ramon Lawrence Department of Computer ScienceUniversity of British ColumbiaKelowna, BC, Canada, V1V [email protected] 14, 2024 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Generative AI systems such as ChatGPT have a disruptive effect on learning and assessment. Computer science requires practice to develop skills in problem solving and programming that are traditionally developed using assignments. Generative AI has the capability of completing these assignments for students with high accuracy, which dramatically increases the potential for academic integrity issues and students not achieving desired learning outcomes. This work investigates the performance of ChatGPT by evaluating it across three courses (CS1,CS2,databases). ChatGPT completes almost all introductory assessments perfectly. Existing detection methods, such as MOSS and JPlag (based on similarity metrics) and GPTzero (AI detection), have mixed success in identifying AI solutions. Evaluating instructors and teaching assistants using heuristics to distinguish between student and AI code shows that their detection is not sufficiently accurate. These observations emphasize the need for adapting assessments and improved detection methods. Keywords: ChatGPT, generative AI, performance, detection, plagarism, CS1, CS2, database§ INTRODUCTION Since the introduction of ChatGPT, there has been significant research on how generative AI can impact education both positively <cit.> and negatively <cit.>. There are concerns related to academic dishonesty with students using AI to complete their assessments, especially in writing courses <cit.>. Generative AI also has problems with fake, inaccurate, and biased data <cit.>. Assessments in computer science are also under threat. Programming assignments used to improve student problem solving are easily completed by generative AI. Prior work has demonstrated that assignments in introductory CS courses <cit.> are completed with high accuracy using ChatGPT. Further, software development companies have integrated generative AI into integrated code development environments with products such as GitHub Copilot <cit.> and Codeium <cit.>.In response, educators are developing recommendations to address the issue. These suggestions primarily focus on employing AI models to detect plagiarism, implementing consistent rules and guidelines <cit.>, and exploring innovative assessment methods <cit.>. Commonly used plagiarism detection software such as MOSS <cit.>, JPlag <cit.>, and others <cit.> are effective at determining similarity matching with other code submissions. However, generative AI creates new code during every prompt request, and it is uncertain whether these tools are still effective. Generative AI detection software such as GPTZero <cit.> and ZeroGPT <cit.> are primarily focused on detecting AI generated writing and have not been extensively evaluated for detecting AI generated code.In this paper, the effectiveness of ChatGPT in completing assessments in three courses (CS1, CS2, databases) is evaluated. CS1 and CS2 have been explored in other prior work <cit.>, but evaluation in an upper-year database course is new. A novel contribution is examining the effectiveness of current detection software against ChatGPT generated code. The performance of instructors and teaching assistants is also measured, which has not been previously examined. Evaluating the effectiveness of human and automated detection of AI generated code reveals challenges in current detection systems and methods. However, there may be opportunities to improve detection by utilizing multiple approaches and some common distinguishing features of AI generated code discovered during the experiment.This work examines the questions: * What is the performance of ChatGPT in multiple CS courses including CS1, CS2, and databases? * What is the effectiveness of existing plagarism and AI detection software for detecting AI generated code? * How accurate are instructors and teaching assistants at distinguishing between code created by students and AI?§ BACKGROUND ChatGPT <cit.> has demonstrated its effectiveness in assisting programmers by providing debugging support <cit.> and aiding in software development. It introduces novel ideas and builds upon previous AI tools like GitHub Copilot <cit.>, enhancing their capabilities. However, when applied in education, educators and researchers have voiced concerns regarding its use <cit.> and ability to enable student academic dishonesty.Generative AI systems like ChatGPT are continually evolving and improving, and various studies have been performed to determine their effectiveness in completing computer science assessments. One result argued thatChatGPT 4 exhibited minimal understanding of computer science compared to human beings <cit.>. Many other studies have demonstrated much more positive results. Geng et al. tested ChatGPT in an introductory functional course and discovered that its performance ranked in the middle of the class, achieving a B- grade <cit.>. Malinka et al. tested ChatGPT and demonstrated how easy it is to cheat in IT security related education <cit.>. ChatGPT has excellent performance in introductory CS courses <cit.> and excels in many standardized exams <cit.>.How generative AI should be integrated into computer science courses is a challenging discussion. CS graduates will require the use of these tools in industry, and tools such as GitHub Copilot can save significant developer time and increase productivity. However, using ChatGPT and other generative AI systems on education assessments eliminates any student learning and achieving learning outcomes. The AI is completing the assessment rather than the student learning. Instructors require clear communication on acceptable generative AI use in their courses. Limiting academic dishonesty is possible by assignment modifications <cit.> and open communication with students on integrity <cit.>.Academic misconduct is common with research studies finding that 23% to 89% of students admit to cheating <cit.>. In <cit.>, 92.7% of students would copy the full assignment from someone when asked how they would cheat. Students cite the potential of getting caught ranks high among the reasons not to cheat <cit.>. As a result, the detection of plagiarism still plays an important role in keeping students from cheating. A literature review <cit.> evaluated methods to detect source code plagiarism, such as MOSS <cit.>, JPLAG <cit.>, and SIM <cit.>. Most methods are based on different algorithms to detect similarities between codes.MOSS uses a technique called winnowing, which fingerprints code files to detect whether other files are similar. Source code detection has also been attempted with machine learning techniques to increase the efficiency and effectiveness of detection <cit.>. The goal of these tools are to check the level of similarity of programs. One response to cheating with AI is to use detection software to determine AI generated assessments. Plagiarism detection software for code has not been extensively evaluated. GPTZero is one example among the numerous detection tools currently available <cit.>. GPTZero's detection is based on burstiness and perplexity. Burstiness refers to the unusual frequency of certain words or phrases in a short span of text, which can indicate artificial generation or manipulation. Perplexity measures how well a probability distribution predicts a sample <cit.>. It is hard to distinguish between human and AI generated text <cit.>. An evaluation of generative AI detection tools for written computer science reports <cit.> indicates various levels of detection accuracy and many false positives, with very poor performance on code submissions. The ability for a commercial software, APEX, to detect ChatGPT answers was evaluated <cit.> for CS1. The results were that ChatGPT had excellent performance, and ChatGPT solutions could be detected by a variety of methods such as style difference, limited time spent, and similarity checking if a sufficient number of students submitted answers created by ChatGPT. There is a need to evaluate free-to-use tools such as MOSS <cit.> and JPlag <cit.> to quantify their detection accuracy.§ METHODOLOGY§.§ Courses Overview The effectiveness of generative AI, specifically ChatGPT 3.5, was evaluated for three common CS courses: CS1, CS2, and databases (DB) at a large, research-intensive university.CS1 covers the basics of programming including problem-solving, algorithm design, data and procedural abstraction, and the development of working programs in the Java language. Concepts include variables, operators, expressions, control statements (if, switch, loops), arrays, methods, object-oriented programming, and debugging. The course includes 9 programming assignments. CS2 covers topics such as object-oriented programming, abstract classes and interfaces, exception handling, input/output streams, recursive methods, data structures, Java generics, and sorting algorithms. The course includes 9 programming assignments. Databases (DB) is an introduction to databases with topics including querying with SQL, designing databases using ER and UML modeling, and programming with Java/JDBC. The course has 10 assignments and two computer-based midterm exams. §.§ Evaluating ChatGPT Performance The experiment focuses on the capabilities of students using ChatGPT and whether they can achieve full marks without prior knowledge of the related materials. We used the assignment and exam questions as prompts for ChatGPT to generate answers. ChatGPT was provided with all related information that is normally given to students to start the assignments, for example, a sample run of the solution. The generated answers were then compared against the official solutions to assign grades. The performance on DB exams was also evaluated as they were done online whileCS1 and CS2 were paper exams. We copied exam questions with the given context (for example, DDL for a set of SQL questions) and graded the responses. Given the objective of assessing students' ability to obtain assignment answers without possessing course related knowledge, follow-up non-technical questions were permitted for ChatGPT if the initial answers were unsatisfactory.§.§ Evaluating AI Detection The effectiveness of detecting AI generated code was performed using two experiments. The data set for both experiments consisted of 50 anonymized student submissions for CS1 and CS2 assignments and 50 different AI generated answers for those assignments. The student submissions were from a previous course offering. All identifying information was extracted from the submissions, and students were not involved in the research in any way as the course was completed in the past. The different AI generated solutions were collected by providing the given starter code, questions, sample runs, and rubrics to ChatGPT in different conversations. The automated AI detection experiment evaluated two types of detection tools on this data set. The similarity tools included MOSS <cit.> and JPlag <cit.> designed specifically for detecting code plagiarism, while the AI detection tool, ChatGPT Zero, aims to identify AI generated content. For MOSS and JPlag, we provided a randomized folder of mixed assignments, analyzed the returned similarity scores, and used a similarity cutoff to determine if a submission was human or AI. For GPTZero, we provided 24 solutions (14 human, 10 AI) and asked GPTZero to determine the source. The second experiment measured human capacity for detecting AI generated code. Course teaching assistants and instructors attempted to distinguish between anonymous student solutions and those generated by ChatGPT in a randomized selectionof AI generated and student work. The randomized selection includes 32 solutions from 4 assignments in CS1 and CS2 with 14 solutions generated by AI. This information was collected via survey. There was a total of 10 survey responses. § RESULTS§.§ ChatGPT Performance ChatGPT is very effective at solving assignments in CS1 and CS2 (see Tables <ref> and <ref>). It correctly outputs the correct code for every coding question in CS1 on the very first try. Each assignment may have multiple distinct questions as shown in the #Q column. The only question it could not do is a small part in A6 where students are asked to submit screenshots of the debugging process. Assignments for CS2 are also completed with near 100% accuracy. Close to zero coding skills are required to get the correct answers for any of these assignments.However, certain questions that are longer and with extensive starter code require students to feed information to ChatGPT in multiple cells. The mark for CS2 A1 was lower as one question asked to display the inheritance graph, which required more explanation of the inheritance relationships for ChatGPT to provide correct answers.A sample prompt is in Figure <ref>, and the answer in Figure <ref>. The returned response is correct and even includes comments. The performance is expected as questions in CS1 and CS2 must be very clear for students to understand and often contain example output. This clarity makes excellent prompts for ChatGPT and requires only that a student copy the question directly as their prompt.The database (DB) course is an upper-year course with a wider range of materials and questions. ChatGPT has very high performance with excellent marks (see Table <ref>) for labs 1, 2, 3, and 9 on relational algebra, SQL, and XML/JSON. The code based labs of 6, 7, 8, and 10 require completing hundreds of lines of code across multiple files. Upon providing the template code, DDL, and the rubrics, ChatGPT gets extremely close to providing the exact answers, often only pending small changes such as database connections. As an example, a snippet of the prompt in Figure <ref> is copied from the starter code provided to students. The prompt is accompanied with the database schema and grading rubric for the assignment. ChatGPT's response in Figure <ref> is correct except that it queried from the product table instead of order summary data. The structure of the code is correct. It is also noteworthy to include that upon asking a follow up question, “from ordersummary", ChatGPT returned the correct code solution.However, ChatGPT does not currently draw ER/UML diagrams as well as required in labs 4 and 5. It is still able to come up with some basic diagram (see Figure <ref>). Utilizing unique questions in UML such as in <cit.> allows for auto-grading while being resistant to generative AI.ChatGPT tested extremely well in midterm 1, obtaining a 28.5/30, while getting only 17/30 on midterm 2 due to the 10 points allocated to ER diagrams. §.§ Automated AI Detection §.§.§ Similarity We conducted a series of tests to evaluate the effectiveness of similarity detection systems MOSS and JPlag. For four different questions in CS1 and CS2, 100 submissions were given to the systems. The percentage of AI submissions varied from 5% to 50%. A submission was flagged as AI if its highest match was with an AI solution, and it was above a similarity cutoff (90% for JPlag, 70% for MOSS).The precision, recall, and F1 scores for these tests are shown. Figures <ref> and <ref> contain data when 5% of the solutions are AI for JPlag and MOSS. With a small number of submissions, the AI detection is weak. Some AI solutions are detected, but the recall is lower.When the percentage of AI solutions is 50% (shown in Figures <ref> and <ref>), the detection accuracy is significantly higher. Even though ChatGPT generates different versions on each request, after a sufficient number of requests it is more likely a similar result will be generated.Several other factors negatively influence the detection process. Students may modify the AI generated solutions and utilize different prompting methods. There are also multiple generative AI systems that can be used. The complexity of the questions is another key factor. Simpler questions lead to more uniform solutions, making it easier to detect AI generated content. More complex questions, with a wider range of responses, make detection more challenging. The number of AI generated submissions is a key factor in their detection, but in a real class environment, it is unknown the number of AI submissions and the generative AI system used. Although an instructor could potentially provide a set of labeled AI submissions for each question, this may not always be practical. However, without this AI sample set, the accuracy of similarity detection systems is below their effectiveness for other forms of assignment copying and plagiarism. The detection of AI generated solutions is not yet reliable enough across different scenarios. Further research is needed to improve AI solution detection using similarity systems.§.§.§ AI Detectors AI detection tools primarily focused on writing have low success in detecting AI generated code. Providing AI solutions to GPTZero <cit.> resulted in very low probability of AI scores. The AI generated solutions raised less suspicion than human solutions, and overall these percentages are way below the threshold to identify AI generated content. The metrics used to detect AI generated writing are not effective when applied to code solutions. §.§ Instructor Detection of AI Submissions We also evaluated if instructors and teaching assistants are more effective at detecting AI submissions. Course instructors and TAs completed a survey that contained 32 mixed student and AI solutions for 4 assignments across CS1 and CS2. Figure <ref> contains the distribution of participant performance. The average recall across all participants was 69.29%. The average precision was 74.51%, and the average F1 score was 71.42%. Overall the detection accuracy rate is 69%, and the false positive rate is 22%. There are two key observations from these results. First, there is substantial variability between instructors in detecting AI. The lowest F1 score was 40% with the highest at 100%. Second, although the performance is better than automated detection software when dealing with a small number of AI submissions, it is still not sufficiently precise for widespread implementation. The average accuracy rate is too low and the false positive rate is too high, which would result in missing many AI submissions, and even worse, falsely accusing students of submitting AI answers.Participants identified heuristics that had various levels of effectiveness:* Errors in solutions, from coding errors to spelling errors* Scanner not closed* The solution contains methods that are not taught yet in the course or unknown to the instructor* Auto-generated stub from Eclipse IDE* Inconsistent style, spacing, and formatting Common approaches often focus on style aspects of the code as AI generated solutions have a professional style and consistency in grammar and skill-level. Interestingly, it is these style aspects, especially related to variable names and whitespace, that are ignored in similarity matching software. The instructor detection heuristics complement similarity detection systems.70% of participants reported that the detection task was very time consuming. As a result, while instructors and TAs can use heuristics to perform detection, it is a labor intensive task that is hard to scale with inconsistent performance depending on the instructor's characteristics. There was a higher performance for instructors and teaching assistants with more experience.§ DISCUSSION The evaluation of the effectiveness of ChatGPT was done on existing assessments in CS1, CS2, and databases. The questions were copied directly into ChatGPT 3.5 as prompts, and the answers evaluated. Our experiment highlights ChatGPT's impressive capability to generate solutions, even in higher-level courses such as databases. In the past, methods of academic dishonesty usually involved copying assignments from peers or online resources. ChatGPT has expanded these methods to include AI generated responses that are no longer copies of previous works, but newly crafted responses. Existing detection methods have mixed success with similarity detection affected by the number of AI submissions and the complexity of the question. Instructor detection using heuristics has variable accuracy depending on instructor experience, and the heuristics may be defeated by minor code format edits in many cases. One useful flag is that ChatGPT lacks an understanding of the course's specific coverage and may use advanced techniques or style formatting that is inconsistent with expected student knowledge.The overall accuracy of AI detection is lower and less reliable than detecting previous forms of plagiarism. However, there may be opportunities to increase performance by combining multiple methods of similarity detection utilizing pre-generated AI answers for comparison and automatic application of code style heuristics utilized by instructors. The detection problem will continually be evolving and challenging given the rapid improvement of generative AI systems.§ LIMITATIONS OF RESEARCH ChatGPT's performance depends on the quality of its prompts. Detailed questions result in excellent prompts for ChatGPT. The CS1 and CS2 questions used in this work had a high-level detail that is common in first year courses. More open-ended questions may result in lower performance, or at least, a more conversational effort by the student with ChatGPT to arrive at the correct answer.The number of AI submissions evaluated for similarity detection was done using multiple conversations with ChatGPT 3.5 with the same prompt. Detection accuracy is expected to be lower if students utilize ChatGPT 4 and other systems as well as varying the prompts provided. Further, no modifications or obfuscations were performed, which would further reduce similarity.Human detection was done by experienced course teaching assistants and instructors, but the participants had various levels of previous experience with AI and plagiarism detection and were provided with no training.It would be interesting to see whether providing training and heuristics to use improves detection effectiveness. § FUTURE WORK AND CONCLUSIONS This work evaluated generative AI performance in CS1, CS2, and databases with ChatGPT having a near-perfect score in CS1 and CS2. Even the higher level course on databases had many questions with 100% performance including larger, team-based coding projects. Assessments with some resistance to ChatGPT were more open-ended, involved images or student-specific outputs, or did not specify expected output in detail.Given this high performance, students will be tempted to utilize generative AI for completing assessments. Detecting AI submissions has varying accuracy and precision whether done by instructors or similarity detection software. The detection heuristics used by instructors typically focus on style and grammar as AI generated code often appears more professional and better organized. Although these heuristics are moderately useful, they can easily be beaten as students adapt, and more importantly, the false positive rate is too high. Similarity detection software can have good performance if provided with enough AI solutions for comparison to student submissions, but the number of such submissions is affected by problem complexity and diversity of AI responses.Future work may investigate combining similarity detection with instructor heuristics to determine if higher accuracy is possible. Improved detection may require examining the entire submission history of a student to detect submissions that are potentially beyond their current skill level.IEEEtran | http://arxiv.org/abs/2311.16292v1 | {
"authors": [
"Kevin Wang",
"Seth Akins",
"Abdallah Mohammed",
"Ramon Lawrence"
],
"categories": [
"cs.CY",
"cs.CL"
],
"primary_category": "cs.CY",
"published": "20231127201013",
"title": "Student Mastery or AI Deception? Analyzing ChatGPT's Assessment Proficiency and Evaluating Detection Strategies"
} |
[Is the Anti-Pfaffian a PH-Pfaffian Topological Order State?Jian Yang January 14, 2024 ============================================================ < g r a p h i c s >figure Given past 3D occupancy observations, our self-supervised OccWorld trained can forecast future scene evolutions and ego movements jointly. This task requires a spatial understanding of the 3D scene and temporal modeling of how driving scenarios develop. We observe that OccWorld can successfully forecast the movements of surrounding agents and future map elements such as drivable areas. OccWorld even generates more reasonable drivable areas than the ground truth, demonstrating its ability to understand the scene rather than memorizing training data. Still, it fails to forecast new vehicles entering the sight, which is difficult given their absence in the inputs.][1]Equal contribution.Understanding how the 3D scene evolves is vital for making decisions in autonomous driving. Most existing methods achieve this by predicting the movements of object boxes, which cannot capture more fine-grained scene information. In this paper, we explore a new framework of learning a world model, OccWorld, in the 3D Occupancy space to simultaneously predict the movement of the ego car and the evolution of the surrounding scenes. We propose to learn a world model based on 3D occupancy rather than 3D bounding boxes and segmentation maps for three reasons:1) expressiveness. 3D occupancy can describe the more fine-grained 3D structure of the scene; 2) efficiency. 3D occupancy is more economical to obtain (e.g., from sparse LiDAR points). 3) versatility. 3D occupancy can adapt to both vision and LiDAR. To facilitate the modeling of the world evolution, we learn a reconstruction-based scene tokenizer on the 3D occupancy to obtain discrete scene tokens to describe the surrounding scenes. We then adopt a GPT-like spatial-temporal generative transformer to generate subsequent scene and ego tokens to decode the future occupancy and ego trajectory. Extensive experiments on the widely used nuScenes benchmark demonstrate the ability of OccWorld to effectively model the evolution of the driving scenes. OccWorld also produces competitive planning results without using instance and map supervision. Code: <https://github.com/wzzheng/OccWorld>.§ INTRODUCTION Autonomous driving has been widely explored in recent years and demonstrated promising results in various scenarios <cit.>. While LiDAR-based models typically show strong performance and robustness in 3D perception due to its capture of structural information <cit.>,the more hardware-economical vision-centric solutions have dramatically caught up with the increased perception ability of deep networks <cit.>.Forecasting future scene evolutions is important to the safety of autonomous driving vehicles. Most existing methods follow a conventional pipeline of perception, prediction, and planning <cit.>. Perception aims to obtain a semantic understanding of the surrounding scene such as 3D object detection <cit.> and semantic map construction <cit.>. The subsequent prediction module captures the motion of other traffic participants <cit.>, and the planning module then makes decisions based on previous outputs <cit.>.However, this serial design usually requires ground-truth labels at each stage of training, yet the instance-level bounding boxes and high-definition maps are difficult to annotate. Furthermore, they usually only predict the motion of object bounding boxes, failing to capture more fine-grained information about the 3D scene. In this paper, we explore a new paradigm to simultaneously predict the evolution of the surrounding scene and plan the future trajectory of the self-driving vehicle. We propose OccWorld, a world model in the 3D semantic occupancy space, to model the development of the driving scenes.We adopt 3D semantic occupancy as the scene representation over the conventional 3D bounding boxes and segmentation maps, which can describe the more fine-grained 3D structure of the scene. Moreover, 3D occupancy can be effectively learned from sparse LiDAR points <cit.>, and thus is a potentially more economical way to describe the surrounding scenes. Given the 3D semantic occupancy representation of the current scene, OccWorld aims to predict how it evolves as the self-driving vehicle advances. To achieve this, we first employ a vector-quantized variational autoencoder (VQVAE) <cit.> to refine high-level concepts and obtain discrete scene tokens in a self-supervised manner.We then tailor the generative pre-training transformers (GPT) <cit.> architecture and propose a spatial-temporal generative transformer to predict the subsequent scene tokens and ego tokens to forecast the future occupancy and ego trajectory, respectively. We first perform spatial mixing to aggregate scene tokens and obtain multi-scale tokens to represent scenes at multiple levels. We then apply temporal attention to tokens at different levels to predict tokens for the next frame and use a U-net structure to integrate them. Finally, we use the trained VQVAE decoder to transform scene tokens to the occupancy space and learn a trajectory decoder to obtain ego planning results.To demonstrate the effectiveness of OccWorld, we formulate a challenging task of 4D occupancy forecasting, which aims to predict the 3D occupancy of the following frames given a few past frames. Our OccWorld can effectively forecast future evolutions including moving agents and static elements as shown in Figure <ref>, and achieves an average IoU of 26.63 and mIoU of 17.13 for 3s future given 2s history, OccWorld can also produce planning trajectories with an L2 error of 1.16 without using any instance and map annotations.Using self-supervised learned 3D occupancy from camera inputs <cit.>, our method achieves non-trivial 4D occupancy forecasting and planning results, demonstrating the potential for interpretable end-to-end autonomous driving without additional human-annotated labels. § RELATED WORK 3D Occupancy Prediction: 3D occupancy prediction aims to predict whether each voxel in the 3D space is occupied and its semantic label if occupied <cit.>. Early methods exploited LiDAR as inputs to complete the 3D occupancy of the entire 3D scene <cit.>. Recent methods began to explore the more challenging vision-based 3D occupancy prediction <cit.> or applying vision backbones to efficiently perform LiDAR-based 3D occupancy prediction <cit.>. 3D occupancy provides more comprehensive descriptions of the surrounding scene and includes both dynamic and static elements <cit.>. It can also be efficiently learned from sparse accumulated multiple LiDAR scans <cit.>, LiDAR <cit.>, or video sequences <cit.>. However, existing methods only focus on obtaining the 3D semantic occupancy and ignore its temporal evolution, which is vital to the safety of autonomous driving. In this paper, we explore the task of 4D occupancy forecasting and propose a 3D occupancy world model to achieve this.World Models for Autonomous Driving: World models have a long history in control engineering and artificial intelligence <cit.>, which are usually defined as producing the next scene observation given action and past observations <cit.>. The development of deep neural networks <cit.> promoted the use of deep generative models <cit.> as world models. Based on large pre-trained image generative models like StableDiffusion <cit.>, recent methods <cit.> can generate realistic driving sequences of diverse scenarios.However, they produce future observations in the 2D image space, lacking understanding of the 3D surrounding scene. Some other methods explore forecasting point clouds using unannotated LiDAR scans <cit.>, which ignore the semantic information and cannot be applied to vision-based or fusion-based autonomous driving. Considering this, we explore a world model in the 3D occupancy space to more comprehensively model the 3D scene evolution. End-to-End Autonomous Driving: The ultimate goal of autonomous driving is to obtain controlling signals based on observations of the surrounding scenes.Recent methods follow this concept to output planning results for the ego car given sensor inputs <cit.>. Most of them follow a conventional pipeline of perception <cit.>, prediction <cit.>, and planning <cit.>. They usually first perform BEV perception to extract relevant information (e.g., 3D agent boxes, semantic maps, tracklets) and then exploit them to infer future trajectories of agents and the ego vehicle. The following methods incorporated more data <cit.> or extracted more intermediate features <cit.> to provide more information for the planner, which achieved remarkable performance. Most methods only model object motions and cannot capture the fine-grained structural and semantic information of the surroundings <cit.>. Differently, we propose a world model to predict the evolution of both the surrounding dynamic and static elements. § PROPOSED APPROACH§.§ World Model for Autonomous Driving Autonomous driving aims to automatically steer a vehicle to fully prevent or partially reduce actions from human drivers <cit.>.Formally, the objective of autonomous driving is to obtain the control commands 𝐜^T (e.g., throttle, steer, break) for the present time stamp T given the sensor inputs {𝐬^T, 𝐬^T-1, ⋯, 𝐬^T-t} from the current and past t frames.As the mapping from trajectories to control signals is highly dependent on the vehicle specifications and status, the literature usually assumes a given satisfactory controller and thus focuses on trajectory planning for the ego vehicle.An autonomous driving model A then takes input as the sensor inputs and ego trajectory from the past T frames and predicts the ego trajectory of future f frames:A ({𝐬^T, 𝐬^T-1, ⋯, 𝐬^T-t}, {𝐩^T, 𝐩^T-1, ⋯, 𝐩^T-t})={𝐩^T+1, 𝐩^T+2, ⋯, 𝐩^T+f},where 𝐩^t denotes the 3D ego position at the t-th time.The conventional pipeline of autonomous driving usually follows a design of perception, prediction, and planning <cit.>.The perception module p_er perceives the surrounding scenes and extracts high-level information 𝐳 from the input sensor data 𝐬. The prediction module p_re then integrates the high-level information 𝐳 to predict the future trajectory 𝐭_i of each agent in the scene. The planning module p_la finally processes the perception and prediction results {𝐳, {𝐭_i }} to plan the motion of the ego vehicle. The conventional pipeline can be formulated as:p_la ( p_er ({𝐬^T, ⋯, 𝐬^T-t}), p_re (p_er ({𝐬^T, ⋯, ^T-t}))) ={𝐩^T+1, 𝐩^T+2, ⋯, 𝐩^T+f}. Despite the promising performance of this framework <cit.>, it usually requires ground-truth labels for supervision at each stage, which can be laborious to annotate. It only considers object-level movement and fails to model more fine-grained evolutions.Motivated by this, we explore a new world-model-based autonomous driving paradigm to comprehensively model the evolution of the surrounding scenes and the ego movements. Inspired by the recent success of generative pre-training transformers (GPT) <cit.> in natural language processing (NLP), we propose an auto-regressive generative modeling framework for autonomous driving scenarios. We define a world model w to act on scene representations 𝐲 and be able to predict future scenes. Formally, we formulate the function of a world model w as follows:w ({𝐲^T, ⋯, 𝐲^T-t}, {𝐩^T, ⋯, 𝐩^T-t}) = 𝐲^T+1, 𝐩^T+1. Having obtained the predicted scene 𝐲^T+1 and the ego position 𝐩^T+1, we can add them to the input and further predict the next frame in an auto-regressive manner, as shown in Figure <ref>. The world model w captures the joint distribution of the evolution of the surrounding scene and the ego vehicle, considering their high-order interactions. §.§ 3D Occupancy Scene Tokenizer As the world model w operates on the scene representation 𝐲, its choice is vital to the performance of the world model. We select 𝐲 based on three principles: 1) expressiveness. It should be able to comprehensively contain the 3D structural and semantic information of the 3D scene; 2) efficiency. It should be economical to learn (e.g., from weak supervision or self-supervision); 3) versatility. It should be able to adapt to both vision and LiDAR modalities. Considering all the aforementioned principles, we propose to adopt 3D occupancy as the 3D scene representation 𝐲∈ℝ^H × W × D. 3D occupancy partitions the 3D space surrounding the ego car into H × W × D voxels and assigns each voxel with a label l denoting whether it is occupied and which material it is occupied with. 3D occupancy provides a dense representation of the 3D scene and can describe both the 3D structural and semantic information of the scene. It can be effectively learned from sparse LiDAR annotations <cit.> or potentially from self-supervision of temporal frames <cit.>. 3D occupancy is also modality-agnostic and can be obtained from monocular camera <cit.>, surrounding cameras <cit.>, or LiDAR <cit.>. Despite its comprehensiveness, 3D occupancy only provides a low-level understanding of the scene, making it difficult to directly model its evolution. We therefore propose a self-supervised way to tokenize the scene into high-level tokens from 3D occupancy. We train a vector-quantized autoencoder (VQ-VAE) <cit.> on 𝐲 to obtain discrete tokens 𝐳 to better represent the scene, as shown in Figure <ref>.For efficiency, we first transform the 3D occupancy 𝐲∈ℝ^H × W × D to a BEV representation 𝐲̂∈ℝ^H × W × DC' by assigning each category with a learnable class embedding ∈ℝ^C' and concatenating them in the height dimension. We then adopt a lightweight encoder composed of 2D convolution layers to obtain down-sampled features 𝐳̂∈ℝ^H/d×W/d× C of the scene, where d is the down-sampling factor.To obtain a more compact representation, we simultaneously learn a codebook 𝐂∈ℝ^N × D containing N codes. Each code 𝐜∈ℝ^C in the codebook encodes a high-level concept of the scene, e.g., whether the corresponding position is occupied by a car. We quantized each spatial feature 𝐳̂_ij in 𝐳̂ by classifying it to the nearest code 𝒩(𝐳̂_ij, 𝐂):𝐳_ij = 𝒩(𝐳̂_ij, 𝐂) = min_𝐜∈𝐂 ||𝐳̂_ij - 𝐜 ||_2,where || · ||_2 denotes the L2 norm. We then integrate the quantized features {𝐳_ij} to obtain the final scene representation 𝐳∈ℝ^H × W × C.To reconstruct 𝐲 from the learned scene representation 𝐳, we use a decoder of 2D deconvolution layers to progressively upsample 𝐳 to its original BEV resolution H × W × C”. We then perform a split in the channel dimension to reconstruct the height dimension H × W × D ×C”/D and apply a softmax layer on each spatial feature to classify them into occupied semantics or unoccupied H × W × D.The scene tokenizer transforms 3D occupancy into a more compact discrete space to encode higher-level concepts. This refined compact space facilitates the modeling of scene evolution for the subsequent world model. §.§ Spatial-Temporal Generative TransformerThe core of autonomous driving lies in the prediction of how the surrounding world evolves and planning the movement of the ego vehicle accordingly. While conventional methods usually perform the two tasks separately <cit.>, we propose to learn a world model w to jointly model the distributions of scene evolution and ego trajectory.As defined in (<ref>), a world model w takes as inputs the past scenes and ego positions and predicts their outcome after driving a certain time interval.Based on expressiveness, efficiency, and versatility, we adopt 3D occupancy 𝐲 as the scene representation and use a self-supervised tokenizer to obtain high-level scene tokens 𝐓 = {𝐳_i }. To integrate the ego movement, we further aggregate 𝐓 with an ego token 𝐳_0 ∈ℝ^C to encode the spatial position of the ego vehicle. The proposed OccWorld w then functions on the world tokens 𝐓, which can be formulated as:w (𝐓^T, ⋯, 𝐓^T-t) = 𝐓^T+1,where T is the current time stamp, and t is the number of history frames available. Inspired by the remarkable sequential prediction performance of GPT <cit.>, we adopt a GPT-like autoregressive transformer architecture to instantiate (<ref>). However, the migration of GPT from natural language processing to the autonomous driving scenario is not trivial. GPTs predict a single token each time, while the world model w in autonomous driving is required to predict a set of tokens 𝐓 as the next future. Due to the vast number of world tokens, directly leveraging the GPT architecture to predict each token ∈𝐓^T+1 is both inefficient and ineffective.Both the spatial relations of world tokens within each time stamp and the temporal relations of tokens across different time stamps should be considered to comprehensively model the world evolution. Therefore, we propose a spatial-temporal generative transformer architecture to effectively process past world tokens and make predictions of the next future, as shown in Figure <ref>.We apply spatial aggregation (e.g., self-attention <cit.>) to world tokens 𝐓 to enable interactions between scene tokens as well as ego tokens. We then merge the scene tokens in each 2 × 2 window with a stride of 2 and thus down-sample the scene tokens by a factor of 4. We repeat this procedure for K times to obtain world tokens of hierarchical scales {𝐓_0, ⋯, 𝐓_K } to describe the 3D scene at different levels.We use several sub-world models w = {w_0, ⋯, w_K } to predict the future at different spatial scales. For each sub-world model w_i, we impose temporal attention on the tokens {𝐳^T_j,i, ⋯, 𝐳^T-t_j,i} at each position j to obtain the predicted corresponding token 𝐳^T+1_j,i of the next frame: 𝐳̂^T+1_j,i= TA (𝐳^T_j,i, ⋯, 𝐳^T-t_j,i),where TA denotes masked temporal attention which blocks the effect of future tokens to previous tokens. 𝐳^t_j,i∈𝐓^t_i represents the j-th world token of the i-th scale at time stamp t. We finally employ a U-net structure to aggregate predicted tokens at different scales to ensure spatial consistency.Our spatial-temporal generative transformer can model the world evolution in driving sequences considering the joint distributions of world tokens within each time and across time. The temporal attention predicts the evolution of a fixed position in the surrounding area, while the spatial aggregation makes each token aware of the global scene. §.§ OccWorld: a 3D Occupancy World Model We present the overall training framework of our OccWorld model for autonomous driving.Having obtained the forecasted world tokens, we reuse the scene decoder d to decode the predicted 3D occupancy 𝐲̂^T+1 = d(𝐳̂^T+1) and additionally learn an ego decoder d_ego to produce the ego displacement p̂^T+1 = d_ego(ẑ^T+1_0) w.r.t the current frame. We adopt a two-stage training strategy to effectively train our OccWorld. For the first stage, we train the scene tokenizer e and decoder d using 3D occupancy loss <cit.>:J_e,d = L_soft (d(e(𝐲)), 𝐲) + λ_1 L_lovasz (d(e(𝐲)), 𝐲),where L_soft and L_lovasz is the softmax and lovasz-softmax loss <cit.>, respectively, and λ_1 is a balance factor.For the second stage, we adopt the learned scene tokenizer e to obtain scene tokens 𝐳 for all the frames and constrain the discrepancy between predicted tokens 𝐳̂ and 𝐳. We then apply the softmax loss to enforce the correct classification of 𝐳̂ to the correct codes in the codebook 𝐂 as 𝐳. For the ego token, we simultaneously learn the ego decoder d_ego and apply L2 loss on the predicted displacement p̂ = d_ego(𝐳̂_0) and the ground-truth one 𝐩. The overall objective for the second stage can be formulated as follows:J_w,d_ego = ∑_t=1^T (∑_j=1^M_0L_soft (𝐳̂_j,0^t , 𝐂(𝐳_j,0^t) +λ_2 L_L2 (d_ego(𝐳̂^t_0), 𝐩^t)),where T and M_0 are the numbers of frames and spatial tokens of the original scale, respectively. 𝐂(·) denotes the index of the corresponding code in the codebook 𝐂. L_L2 measures the L2 discrepancy between two trajectories.For efficient training, we use tokens obtained by the scene tokenizer e as inputs but apply masked temporal attention <cit.> to block the effect of future tokens. During inference, we progressively predict world tokens of the next frame using predicted tokens of past frames.Our OccWorld can be applied to various types of 3D occupancy to adapt to different settings (e.g., end-to-end autonomous driving). The scene representation model r can be an oracle providing ground-truth occupancy, or a perception model taking images or LiDAR as inputs. Different from the conventional perception, predicting, and planning pipeline, OccWorld models the joint evolution of the surrounding scene and the ego movement to capture high-order interactions between the ego vehicle and the environment. Combined with machine-annotated <cit.>, LiDAR-collected <cit.>, or self-supervised <cit.> 3D occupancy, OccWorld has the potential to scale up to large-scale training, paving the way for large driving models.§ EXPERIMENTS§.§ Task DescriptionsIn this paper, we explore a world-model-based framework for autonomous driving and propose OccWorld to model the joint evolutions of ego trajectory and scene evolutions. We conduct two tasks to evaluate our OccWorld: 4D occupancy forecasting on the Occ3D dataset <cit.> and motion planning on the nuScenes dataset <cit.>. We present the dataset and evaluation metric details in the supplementary material.4D occupancy forecasting. 3D occupancy prediction aims to reconstruct the semantic occupancy for each voxel in the surrounding space, which cannot capture the temporal evolution of the 3D occupancy. In this paper, we explore the task of 4D occupancy forecasting, which aims to forecast the future 3D occupancy given a few historical occupancy inputs. We use mIoU and IoU as the evaluation metric.Motion planning. The objective of motion planning is to produce safe future trajectories for the self-driving vehicle given ground-truth surrounding information or perception results. The planned trajectory is represented by a series of 2D waypoints in the BEV plane (ground plane). We use L2 error and collision rate as the evaluation metric. §.§ Implementation DetailsWe followed existing works <cit.> and used a 2-second historical context to forecast the subsequent 3 seconds.The scene tokenizer employs a down-sampling factor of 4, featuring a codebook comprising 512 nodes and a 128-dimensional feature representation. The spatial-temporal generative transformer comprises 3 scales, each incorporating 6 layers of spatial-wise temporal attention for scene tokens with 2 layers of spatial cross-attention and temporal cross-attention for ego planning tokens.During training, we applied mask operations to all temporal attention mechanisms to prevent the influence of future information on forecasting. For inference, we employ autoregressive prediction to foresee 3 seconds into the future based on a 2-second historical context. We adopted the AdamW optimizer <cit.> and a Cosine Annealing scheduler <cit.> for training. We set an initial learning rate of 1 × 10^-3 and the weight decay at 0.01 and. We use a batch size of 1 per GPU on 8 NVIDIA GeForce RTX 4090 GPUs.§.§ Results and Analysis 4D occupancy forecasting. We evaluated the 4D occupancy forecasting performance of our OccWorld in several settings: OccWorld-O (using ground-truth 3D occupancy), OccWorld-D (using predicted results of TPVFormer <cit.> trained with dense ground-truth 3D occupancy), OccWorld-T (using predicted results of TPVFormer <cit.> trained with sparse semantic LiDAR[<https://github.com/wzzheng/TPVFormer>]), and OccWorld-S (using predicted results of TPVFormer <cit.> trained in a self-supervised manner[<https://github.com/huang-yh/SelfOcc>]). Copy&Paste denotes copying the current ground-truth occupancy as future observations. The 0s results represent the reconstruction accuracy.We compare the performance of the aforementioned settings in Table <ref>. We observe that OccWorld-O can generate non-trivial future 3D occupancy with much better results than Copy&Paste, showing that our model learns the underlying scene evolution. OccWorld-D, OccWorld-T, and OccWorld-S can be seen as end-to-end vision-based 4D occupancy forecasting methods as they take surrounding images as input. This task is very challenging since it requires both 3D structure reconstruction and forecasting. It is especially difficult for the self-supervised OccWorld-S, which exploits no 3D occupancy information even during training. Still, our OccWorld generates future 3D occupancy with non-trivial mIoU and IoU on the end-to-end setting.Visualizations. We visualize the output results of the proposed OccWorld in Figure <ref>. We see that our models can successfully forecast the movements of cars and can complete unseen map elements in the inputs such as drivable areas. The planning trajectory is also more accurate with better 4D occupancy forecasting.Motion planning. We compare the motion planning performance of the proposed OccWorld with state-of-the-art end-to-end autonomous driving methods, as shown in Table <ref>. We also evaluate our model under different settings as those in the 4D occupancy forecasting task.We see that UniAD achieves the best overall performance, which exploits various types of auxiliary supervision to improve its planning quality. Despite the strong performance, the additional annotations in the 3D space are very difficult to obtain, making it difficult to scale to large-scale driving data. As an alternative, OccWorld demonstrates competitive performance by employing 3D occupancy as the scene representation which can be efficiently obtained by accumulating LiDAR scans <cit.>.We observe that using ground-truth 3D occupancy as inputs, our OccWorld-O outperforms the previous perception-prediction-planning-based method OccNet <cit.> by a large margin without using maps and bounding boxes as supervision, demonstrating the superiority of the world-model paradigm for autonomous driving. Our end-to-end models OccWorld-D and OccWorld-T also demonstrate competitive performance using only 3D occupancy as supervision and OccWorld-S delivers non-trivial results with no supervision other than the future trajectory, showing the potential for interpretable end-to-end autonomous driving.Though our model demonstrates very competitive L2 error, it slightly falls behind on the collision rate. This is because it is more difficult to learn safe trajectories without the guidance of freespace or bounding box. Still, OccWorld-O demonstrates comparable collision rates with OccNet which exploits map and box supervision, showing that OccWorld can learn the concept of freespace with 3D occupancy. We also observe that OccWorld shows excellent short-term planning performance (1s), but worsens quickly when planning longer futures. For example, OccWorld-O achieves the best L2 error at 1s among all the methods but reaches 1.99 at 3s compared to 1.65 of UniAD. This might result from the diverse future generations of world models, which might deviate from the ground-truth trajectory. Analysis of the scene tokenizer. We analyze the effect of different hyperparameters for the scene tokenizer in Table <ref>. The setting denotes latent spatial resolution, latent channel dimension, and the codebook size. We see that using a larger codebook than 512 leads to overfitting and using a smaller codebook, spatial resolution, or channel dimension might not be enough to capture the scene distribution. The reconstruction accuracy greatly improves with a larger spatial resolution, yet leads to poor forecasting and planning performance. This is because the tokens cannot learn high-level concepts and are difficult to forecast the future.Analysis of the spatial-temporal generative transformer. We conducted an ablation study on both 4D occupancy forecasting and motion planning to analyze the design of the proposed spatial-temporal generative transformer, as shown in Table <ref>. w/o spatial attn denotes discarding spatial aggregation and directly applying temporal attention to the input tokens. w/o temporal attn represents that we replace the temporal attention with a simple convolution to output the next scene using the current world tokens. w/o ego represents that we discard the ego token. w/o ego temporal represents that we replace the temporal attention of the ego token with a simple MLP. We observe that using spatial aggregation to model spatial dependencies and using temporal attention to integrate history information is vital to the performance of both 4D occupancy forecasting and motion planning tasks. Also, only performing the 4D occupancy forecasting task without predicting motion reduces the performance.This verifies the effectiveness of joint modeling of scene evolutions and ego trajectories. Finally, discarding the ego temporal attention leads to poor planning and surprisingly worse 3D forecast occupancy performance. We think this is because integrating a wrongly predicted ego trajectory will mislead the forecasting. § CONCLUSIONIn this paper, we have presented a 3D occupancy world model (OccWorld) to model the joint evolutions of ego movements and surrounding scenes. We have employed a 3D occupancy scene tokenizer to extract high-level concepts and used a spatial-temporal generative transformer for future prediction in an auto-regressive manner. Both quantitive and visualization results have shown that OccWorld can effectively predict future scene evolutions in the comprehensive 3D semantic occupancy space. We believe that OccWorld has paved the way for interpretable end-to-end autonomous driving without additional supervision signals. ieeenat_fullname | http://arxiv.org/abs/2311.16038v1 | {
"authors": [
"Wenzhao Zheng",
"Weiliang Chen",
"Yuanhui Huang",
"Borui Zhang",
"Yueqi Duan",
"Jiwen Lu"
],
"categories": [
"cs.CV",
"cs.AI",
"cs.LG"
],
"primary_category": "cs.CV",
"published": "20231127175941",
"title": "OccWorld: Learning a 3D Occupancy World Model for Autonomous Driving"
} |
Journal ofClass Files, Vol. 14, No. 8, August 2015 Shell et al.: Bare Advanced Demo of IEEEtran.cls for IEEE Computer Society Journals This study introduces a novel approach for generating high-quality, language-specific chat corpora using a self-chat mechanism. We combine a generator LLM for creating new samples and an embedder LLM to ensure diversity. A new Masked Language Modelling (MLM) model-based quality assessment metric is proposed for evaluating and filtering the corpora. Utilizing theas the generator and a multilingual sentence transformer as embedder, we generate an Italian chat corpus and refine the Fauno corpus, which is based on translated English ChatGPT self-chat data. The refinement uses structural assertions and Natural Language Processing techniques. Both corpora undergo a comprehensive quality evaluation using the proposed MLM model-based quality metric. The Italian LLM fine-tuned with these corpora demonstrates significantly enhanced language comprehension and question-answering skills. The resultant model, , establishes a new state-of-the-art for Italian LLMs. This approach marks a substantial advancement in the development of language-specific LLMs, with a special emphasis on augmenting corpora for underrepresented languages like Italian.Large Language Models, Natural Language Processing, Self Chat, Data GenerationCerbero-7B: A Leap Forward in Language-Specific LLMs Through Enhanced Chat Corpus Generation and Evaluation Federico A. Galatolo, Mario G.C.A. Cimino (Member, IEEE) Received Month 00, 2021; accepted Maonth 00, 2021 ===========================================================================================================§ INTRODUCTION In the dynamic domain of Natural Language Processing (NLP), the enhancement of conversational models has become a pivotal area of research <cit.>. The advent of sophisticated models, notably ChatGPT and GPT-4, has demonstrated exceptional capabilities in emulating human-like discourse across a multitude of contexts <cit.> <cit.>. Nonetheless, the exclusivity of APIs and the dearth of publicly accessible, high-caliber chat corpora have imposed significant constraints on the academic and development communities.Advancing the pioneering methodologies delineated in the seminal works of the Fauno <cit.> and Baize <cit.> studies, this paper introduces a distinctive approach to the generation of a diversified and superior-quality corpus through the medium of Large Language Model (LLM) self-chat. Our method is characterized by its exclusive reliance on open-source software and models, while striving to attain, if not surpass, the corpus quality derived from proprietary self-chat paradigms, such as those employed by ChatGPT.This research focuses on the refinement of an Italian LLM. We engaged with two distinct Italian corpora: Fauno, which, despite its considerable size, offers lower quality <cit.>, and OASST <cit.>, which, though limited in scale, provides high-quality, human-curated chat data. Our goal is to leverage these repositories to fortify the linguistic proficiency of an Italian LLM.Confronting the inherent qualitative deficiencies of the Fauno corpus, we advocate a novel technique that incorporates structural assertions and rule-based NLP for refinement. Concurrently, we delineate a novel method for generating a new chat corpus via a self-chat process augmented by a sentence embedding transformer.To assess the quality of the corpus, we employ novel a BERT-based method to evaluate and compare the original, filtered, and generated corpora. This evaluative process informs the fine-tuning phase, wherein the corpora are employed to fine-tune a state-of-the-art LLM.Our research culminates in a comprehensive evaluation, wherein we benchmark the resulting model's capabilities in linguistic comprehension and question-answering tasks, comparing its performance with other leading Italian Large Language Models. In a stride towards promoting collaborative research and development, we have made the most effective fine-tuned model publicly available under the name .The choice of this nomenclature, which follows the common practice of naming fine-tuned models from animal or mythological creatures, is inspired by Cerberus, the mythological three-headed dog tasked with guarding the gates of the Underworld in ancient Greek mythology. This metaphor aptly represents the tripartite foundation of our model: (i) its base model, the , which provides a robust and advanced starting framework; (ii) the innovative approach in corpus generation and evaluation developed in this study; and (iii) the commitment to open-source development, as demonstrated by the adoption of the Apache 2.0 license, fostering an environment of accessibility and collaboration within the AI community.§ RELATED WORKS The evolution of large language models (LLMs) has significantly impacted the field of Natural Language Processing (NLP), particularly in the context of chat-based applications. Since the inception of transformer-based models like BERT <cit.> and GPT <cit.>, the approach to NLP tasks has been revolutionized, achieving state-of-the-art performance across various benchmarks <cit.>.Recent methodologies aim to leverage the vast knowledge embedded in these pre-trained models through fine-tuning techniques that emulate human judgments for specific tasks and evaluation criteria, showing promise in automatic dataset generation for NLP. Concurrently, the field has seen a surge in the use of these models to automatically generate corpora for NLP tasks. <cit.>One of the most prominent contributions to this domain has been the development of open-source alternatives to proprietary models. Models like Stanford Alpaca and Vicuna have almost reached the performance of models like ChatGPT and GPT-4 by fine-tuning on corpora collected from various sources, including instruction-learning formats and public dialogue data <cit.>.In parallel to advancements in dataset generation, there has been significant progress in fine-tuning LLMs, especially in resource-constrained scenarios. Methods like LOw-Memory Optimization (LOMO), which fuses gradient computation and parameter update in one step, have emerged as parameter-efficient alternatives to traditional full-parameter fine-tuning <cit.>. However, the challenge remains in fine-tuning the full parameters of LLMs, particularly for models with tens to hundreds of billions of parameters, which often necessitate considerable computational resources <cit.>.Methods like LOw-Memory Optimization (LOMO), which fuses gradient computation and parameter update in one step, have emerged as parameter-efficient alternatives to traditional full-parameter fine-tuning. Additionally, Low-Rank Adaptation (LoRA) has been introduced to reduce the number of trainable parameters by injecting trainable rank decomposition matrices into each layer of the Transformer architecture, allowing for efficient fine-tuning of large models like GPT-3 <cit.>.The open-source model Baize utilizes parameter-efficient tuning on self-chat data to enhance LLaMA, an open-source large language model, demonstrating impressive multi-turn dialogue performance and accessibility for research on a single GPU <cit.>.In the context of Italian language processing, significant strides have been made with the development of Italian-specific LLMs. GePpeTto represents the first generative language model for Italian, trained on a corpus comprising Italian Wikipedia and the ItWac corpus, using GPT-2 as a blueprint and amounting to almost 13GB of text <cit.>. Its training utilized four Tesla T4 GPUs, with a model size corresponding to GPT-2 small, including 12 layers and 117 million parameters.Building on this, the IT5 model family has been introduced as encoder-decoder transformer models pre-trained specifically on Italian. Utilizing a cleaned version of the Italian mC4 corpus, IT5 models have been shown to consistently outperform multilingual counterparts, setting new benchmarks for most Italian language generation tasks <cit.>. This advancement solidifies the impact of monolingual models on language-specific NLP performance.The Fauno model pushes the envelope further as the first and largest open-source Italian conversational LLM. Aimed at democratizing LLM studies in Italian, Fauno was fine-tuned on a diverse range of corpora including general QA, computer science, and medical questions, to offer a capable conversational AI in Italian <cit.>. Complementing these Italian-specific developments, the multilingual SBERT, particularly the distiluse-base-multilingual-cased variant, leverages knowledge distillation to extend monolingual sentence embedding models to multiple languages. This method maps translated sentences to the same vector space as the original, enabling the creation of multilingual embeddings from previously monolingual models. This approach ensures desired properties for the vector space with lower hardware requirements, demonstrating effectiveness across 50+ languages <cit.>.§ METHODOLOGY§.§ Fauno In this methodology subsection, we delineate the procedures for sanitizing and preprocessing the Fauno corpus—a numerous yet qualitatively deficient Italian corpus, derived from English ChatGPT self-chat data. Our aim is to reinforce the corpus's structural integrity, linguistic coherence, and readiness for the subsequent fine-tuning of an Italian Language Model (LLM). We expound upon these procedures below.Originally, the Fauno corpus was formatted as text, with the first line as the system prompt, followed by alternating responses from human and AI interlocutors, marked with the tagsand , respectively. The translation from English introduced inconsistencies, including erroneous tags likeor misaligned brackets, and omitted annotations. To instill uniformity, we employed regular expressions for reformatting.Our primary focus was to maintain the natural conversation flow, with the AI providing responses before the human participants. To achieve this, we undertook a two-step process. First, we removed 73 conversations that were empty or devoid of meaningful content. These empty conversations added no value to our corpus and were, therefore, excluded. In the second step, we meticulously reviewed and identified 225 conversations that deviated from the desired flow sequence of AI followed by Human. These deviations disrupted the desired conversational structure, and as a result, they were also excluded from the corpus.In a further step, we computed the hash of each message using thehash function. This allowed us to identify and remove conversations with more than 50% of messages identical to others in the corpus. Consequently, we removed an additional 2368 conversations, further refining the data quality.The linguistic analysis utilized the NLTK Punkt tokenizer for sentence segmentation <cit.> and Lingua-py for language detection <cit.>. We discovered 67,517 English messages within the conversations, 50,245 of which were untranslated system prompts.All of the system prompts lacked meaningful content as they were consistently identical or had limited variations. They essentially served as standardized triggers for the AI to generate responses and did not contribute any valuable linguistic diversity or information to the corpus. Consequently, we removed the system prompts from the cleaned corpusIn the final step, a zero-shot classifier, designed to differentiate between text and code and built using the llama2-70b-chat model, was deployed to analyze the remaining 17,272 detected English messages. This classification process revealed that the vast majority of the detected “English” messages actually contained code and were not genuine English text. Upon closer examination, we found that all of the remaining messages that appeared to be in English were, in fact, short sentences containing English names but were written in Italian. These instances turned out to be false positives.For quantitative validation of the cleaning process, we utilized the distiluse-base-multilingual-cased-v1 sentence transformer model. By embedding each message and storing these within a vector store, we identified the ten nearest neighbors for each sentence, discounting identical embeddings. A histogram of these distances, depicted in Figure <ref>, illustrates the enhanced cohesiveness of our refined corpus. This approach provides insights into the distribution of differences within the corpus. A more diverse corpus is deemed superior, emphasizing the importance of our cleaning methodology in enhancing the overall corpus quality. §.§ OASST The OASST dataset is a curated corpus that captures a wide range of linguistic nuances through its conversation trees <cit.>. Each tree in the corpus begins with a root message, branches out into multiple dialogues, and culminates in leaf nodes, signaling the conclusion of an exchange. This hierarchical structure mirrors the complexity and layered nature of real-life conversations, offering a rich resource for analyzing linguistic patterns and conversational dynamics.Specifically, the Italian portion of the OASST corpus, which is central to our analysis, comprises 111 conversation trees encompassing a total of 554 messages. Our methodology involved a comprehensive exploration of the conversation trees in the Italian subset. Starting from the root messages, we traced each dialogue branch to its corresponding leaf node. This process allowed us to identify and extract 358 seed conversations. These seed conversations were then utilized as a foundation for further analysis and the generation of self-chat scenarios. §.§ Italian Chat Corpus Generation The generation of a new Italian chat corpus was conducted by initially sampling ten seed conversations from the OASST corpus, specifically from its Italian section. These conversations were then employed as the foundation for further message generation using the llama2-70b model <cit.>. This process utilized the model in a self-chat configuration, aiming to extend each conversation to a predetermined target length, randomly sampled for each conversation from 4 to 10 messages.For every newly generated chat message, we computed its embeddings using the sentence transformer<cit.>. This step transformed the messages into high-dimensional vectors, facilitating their comparison for uniqueness and diversity. These embeddings were crucial for the subsequent filtering process.A crucial aspect of our methodology was the introduction of a diversity metric based on cosine similarity measures. A Vector Store database, initially populated with every OASST message, was queried with each new message embedding to check for similarity with existing messages. Messages displaying a cosine similarity above 0.9 with any vector in the database were excluded. This diversity filter ensured that only unique and varied linguistic elements were added to the corpus, thereby enriching the corpus with a wide spectrum of linguistic expressions.Messages that successfully passed the diversity filter were appended to their respective conversations. Concurrently, their embeddings were added to the Vector Store database. This iterative process of conversation and database enrichment led to the formation of a dynamic and contextually rich Italian chat corpus.The entire generation process is depicted in Figure <ref>: cyan blocks in the flowchart represent the LLM models employed, while purple blocks denote the control logic guiding the message generation and filtering. Yellow blocks symbolize the objects in play, namely the messages and their embeddings. Finally, orange blocks represent the databases involved in the process, including both the Vector Store and the generated corpus. This systematic approach was instrumental in achieving our objective of expanding the Italian chat corpus with high-quality, diverse conversational content. §.§ Quality Assesment In this section, we elaborate on our methodology for assessing the quality of the generated corpus through the utilization of BERT-based language models (LLMs). Our approach seeks to evaluate the linguistic coherence and contextual understanding of the corpus sentences, ultimately contributing to the enhancement of the performance of the fine-tuned LLM on downstream tasks. To initiate the quality assessment process, each sentence in the corpus is subjected to tokenization. Subsequently, each token within the sentence is individually masked, one at a time. This meticulous token-wise masking enables a granular examination of the language model's predictive capabilities when faced with the absence of specific tokens. The probability distribution over the masked tokens is computed using an Italian BERT model. For every masked token T_i, the model generates a probability distribution conditioned on the entire sentence, excluding the masked token p(T_i |T_j)∀ jiThis step aims to capture the model's prediction of the masked token given the contextual information provided by the rest of the tokens in the sentence. The Non-Negative Log-Likelihood (NLL) is employed as a metric to quantify the quality of the sentence. For each masked token in the sentence, the NLL is computed based on the actual token observed in the corpus. Mathematically, the NLL is defined as: NLL = 1/N∑_i=0^N-log(p(T_i |T_j)∀ jiHere, the negative logarithm of the predicted probability is taken, and the mean NLL across all masked tokens within the sentence is calculated. This process is repeated for each sentence in the corpus. The rationale behind utilizing NLL as a quality assessment metric lies in its sensitivity to the predictability of the masked tokens. Higher NLL scores indicate greater uncertainty and randomness in the language model's predictions, suggesting potential issues such as sentence malformation or lack of coherence. Conversely, lower NLL scores signify more predictable and contextually aligned predictions, indicative of well-structured and semantically sound sentences. The evaluation process is depicted in Figure <ref>The computed NLL scores are utilized to evaluate and compare the quality of the original corpus, the filtered corpus, and the generated one. This comparative analysis serves as a crucial step in gauging the efficacy of our corpus generation methodology and its impact on language model performance. The histogram, depicted in Figure <ref>, provides a comparative analysis of the Non-Negative Log-Likelihood (NLL) quality metric distribution across different chat corpora. From this histogram, we observe that the OASST dataset exhibits the lowest mean NLL score and a relatively narrow distribution. This indicates a high degree of linguistic coherence and contextual consistency within the OASST corpus, reflecting its curated nature. In contrast, the Generated corpus demonstrates a distribution pattern strikingly similar to that of OASST, with a marginally higher mean but comparable dispersion, implying a successful emulation of OASST's quality through our corpus generation methodology.The Filtered Fauno corpus, while showing a broader dispersion and a higher mean NLL score than OASST and Generated, still maintains proximity to these datasets. This suggests that, despite its refinement, the Filtered Fauno corpus possesses a somewhat varied range of linguistic coherence, likely due to its origins from a translation of English ChatGPT self-chat data. The broader dispersion signifies a more heterogeneous composition in terms of language quality, yet still denotes significant improvement from its original form.In stark contrast, the Original Fauno corpus is characterized by a substantially higher mean and a considerably broader dispersion in its NLL score distribution. This pattern is indicative of a noisy and less coherent dataset. The pronounced peaks in the histogram of the Original Fauno corpus further underscore its inconsistency and the varied quality of its linguistic content. Such a distribution profile highlights the substantial qualitative gap between the Original Fauno and the other corpora, underlining the critical need for the refinement and generation processes we have employed in this study.§ EXPERIMENTAL SETUP This section provides an overview of the experimental framework adopted for refining an Italian Large Language Model (LLM) by leveraging a novel corpus generation technique introduced in this study. The experiments are based on the state-of-the-art 7-billion parameter model, , which currently stands as the most proficient model within its size category <cit.>The primary objective of the following experiments is to gain a deeper understanding of how the choice of corpus impacts the performance and Italian proficiency of the fine-tuned model. To explore this, we subject themodel to three distinct fine-tuning conditions. First, it undergoes fine-tuning exclusively with the Fauno corpus. Second, it undergoes fine-tuning exclusively with the Generated corpus. Finally, we conduct a fine-tuning process using a combination of the Fauno corpus and our newly generated corpus. This allows us to analyze the effects of different corpus compositions on benchmark performance.It is important to notice that before generating the final fine-tuning corpora, both the Generated Dataset, Fauno, and Full were filtered to only retain messages with a quality score below 2. This pre-processing step was crucial to ensure the quality and relevance of the training data, impacting the subsequent fine-tuning efficiency and efficacy of the model.The evaluation of the fine-tuned models will involve assessing their performance in comparison to two well-established Italian language models, namely, Camoscio and Fauno, as well as the base model . Initially, this comparison will focus on a few-shot question-answering task using the Italian version of the SQuAD dataset. Additionally, we will evaluate the performance of these models across three other few-shot tasks taken from the EVALITA benchmark, which include toxicity detection, irony detection, and sentiment analysis. These tasks are crucial for evaluating the models' nuanced understanding of language and contextual interpretation, making them essential for a comprehensive assessment of model performance. §.§ Training Environment and Parameters The training of our models was conducted in a modern computational environment, utilizing an NVIDIA DGX system equipped with 8 H100 GPUs. This high-performance setup provided the necessary computational power and efficiency to handle the extensive training demands of our Large Language Models (LLMs). The consistent use of this advanced hardware configuration across all our experimental trials ensured uniformity and comparability in the training process. Each model, regardless of the dataset it was fine-tuned with, underwent training under identical conditions, leveraging the full capabilities of the DGX system's parallel processing. The chosen hyperparameters are detailed in Table <ref>, these parameters included the base model configuration, model and tokenizer types, sequence length, batch size, number of epochs, and learning rate, among others. The optimizer was set to , a variant of the AdamW optimizer, known for its effectiveness in training large neural networks. Additionally, the learning rate was set to 0.000005 with a cosine learning rate scheduler, ensuring a gradual and stable learning process. Furthermore, the models were trained usingdata type, striking a balance between computational efficiency and numerical precision. The use of Flash Attention <cit.>,a feature in H100 GPUs, enabled efficient memory usage and faster computation times, particularly crucial for training models with billions of parameters. The warmup steps were set to 10, allowing the models to gradually adapt to the training regime, thus preventing abrupt changes that could adversely affect the learning process.This homogeneous training environment and parameter setting across all trials were instrumental in maintaining the consistency of our experimental evaluations. By controlling these variables, we could attribute observed differences in model performance solely to the effects of the fine-tuning datasets, thereby ensuring the validity and reliability of our experimental results. §.§ BenchmarksThe Stanford Question Answering Dataset (SQuAD) is a reading comprehension dataset consisting of questions posed by crowdworkers on a set of Wikipedia articles. Each question's answer is a segment of text from the corresponding reading passage. It is one of the largest datasets of its kind with 100,000+ question-answer pairs on 500+ articles. SQuAD is known for its diversity and size, making it more comprehensive than previous datasets. In our experimentation, we used the Italian version of SQuAD known as SQuAD-it. SQuAD-it was created by performing a semi-automated translation of the original SQuAD dataset into Italian. This extensive resource comprises over 60,000 question-answer pairs, all geared towards facilitating open question-answering processes in the Italian language. We specifically adopted a three-shot setting for our few-shot question-answering benchmark. By using a three-shot setting, we assess a model's generalization capability, evaluating its capacity to understand and reason across various topics and contexts based on a small set of examples. Moreover, we opted for a three-shot setting in our benchmark to effectively manage the length of context, especially as the question-answer pairs often involve extensive details. This choice strikes a balance between evaluating a model's ability to handle longer information and maintaining a manageable evaluation process.The EVALITA benchmark is an initiative by the Italian Association for Computational Linguistics (AILC) started in 2007, focusing on the evaluation of Natural Language Processing (NLP) tools for Italian. It operates under a shared framework where diverse systems and methodologies can be assessed across a variety of tasks. These tasks are meant to represent scientific challenges and are organized by the Italian research community to test methods, resources, and systems against shared benchmarks. These benchmarks are indicative of linguistic open issues or real-world applications. EVALITA not only aims at the advancement of methodologies and techniques for NLP and speech processing but also emphasizes reproducibility, cross-community engagement, and the exploration of multilingual and multi-modal dimensions. For EVALITA, we employed a five-shot setting in our evaluations due to the relatively compact nature of both input queries and expected output responses.From the EVALITA corpus, we selected three benchmarks corpus from which we constructed our few-shot classification tasks: i) AMI ii) IronITA iii) SENTIPOLC:The AMI 2020 dataset is a valuable resource consisting of 6,000 tweets written in Italian, meticulously annotated for misogyny and aggressiveness. This dataset was employed in the AMI shared task during the Evalita 2020 evaluation campaign, providing a crucial evaluation ground for models in understanding and identifying potentially harmful content in Italian social media interactions.IronITA, another pivotal benchmark, collects 4,849 tweets that have been expertly annotated for irony and sarcasm. It was utilized in the IroniTA task organized as part of EVALITA 2018, facilitating the assessment of models' ability to detect nuanced language phenomena like irony and sarcasm, which often require a sophisticated understanding of context and sentiment.The SENTIPOLC dataset, developed for EVALITA 2014, focuses on sentiment polarity classification. It includes short social media posts, primarily from Twitter, annotated to identify whether they express positive, negative, neutral, or mixed sentiments. This dataset uniquely incorporates ironic messages to investigate the challenges of correctly classifying sentiment in such contexts. §.§ Results Discussion This section delves into the interpretation and implications of the experimental results obtained from the fine-tuning and evaluation of themodel under various conditions. The discussion is structured around the performance metrics observed in the SQuAD-it and EVALITA benchmark tasks, with a focus on the few-shot learning capabilities of the models.The SQuAD-it results, as detailed in Table <ref>, highlight significant differences in model performance based on the training dataset used. Notably, themodel fine-tuned with the Full dataset, which includes our generated dataset combined with Fauno, outperforms the other configurations in both F1 and Exact Match (EM) scores. This suggests that the richness and diversity of the Full corpus contribute positively to the model's comprehension abilities and its precision in answering questions.The Generated corpus, while slightly lower in performance compared to the Full corpus, still shows a marked improvement over the Fauno corpus. This indicates the effectiveness of our novel corpus generation technique in enhancing the model's language understanding, especially considering the structural assertions and rule-based NLP techniques employed for refinement.In the EVALITA benchmark, as illustrated in Table <ref>, the few-shot task results offer a nuanced view of the models' capabilities in understanding complex language constructs such as irony, sentiment, and toxicity. Themodel fine-tuned with the Full corpus demonstrates superior performance across all tasks, underscoring the value of a diverse and comprehensive training corpus. Interestingly, the Generated corpus shows comparable or slightly better performance in toxicity detection but falls short in irony and sentiment analysis. This could be attributed to the inherent challenges in generating corpora that adequately capture these subtle aspects of language, emphasizing the need for further refinement in corpus generation methodologies.When compared against baseline models like the base , Fauno, and Camoscio, the fine-tuned models exhibit substantial improvements. This reinforces the significance of fine-tuning with task-specific corpora in enhancing model performance, particularly in language-specific contexts like Italian.The results of this study have several implications for the field of NLP, particularly in the development of LLMs for languages other than English. The effectiveness of the corpus generation and refinement techniques presented here offers a promising avenue for improving language models, especially for underrepresented languages.Future research could explore further advancements in corpus generation methods, perhaps integrating more sophisticated techniques to capture complex language nuances. Additionally, extending this methodology to other languages and dialects could significantly contribute to the global inclusivity and accessibility of language technologies.§ CONCLUSIONS This research has advanced the field of Natural Language Processing (NLP), with a focus on the generation and refinement of corpora for fine-tuning Large Language Models (LLMs). Our work introduced a novel method for creating a high-quality, diverse scope-specific chat corpus through a sentence transformers-aided self-chat mechanism.We successfully generated a new Italian chat corpus and refined the existing Fauno corpus, which was substantial in size but lacking in quality. Our innovative approach combined self-chat mechanisms with structural assertions and NLP rules to enhance the quality and diversity of the corpora. Furthermore, we introduced a novel Masked Language Modelling (MLM) model-based metric to assess and compare the quality of our generated corpus against existing corpora. This metric has proven to be a reliable method for evaluating language model corpora.Fine-tuning the Italian LLM with our newly generated corpus, both in isolation and combined with the Fauno corpus, led to significant improvements in language comprehension and question-answering capabilities. Our benchmarks, conducted using the SQuAD-it and EVALITA tasks, demonstrated the enhanced performance of the fine-tuned models, especially in understanding complex constructs such as irony, sentiment, and toxicity.The implications of our research are far-reaching within the domain of NLP. We underscore the potential of advanced corpus generation and refinement techniques in improving the performance of LLMs, particularly for underrepresented languages like Italian. This advancement is crucial for global language inclusivity in AI applications. Our novel methodologies in corpus generation, particularly the use of self-chat mechanisms complemented by embedder LLMs, set a new standard in corpus creation and can be applied to other languages. The MLM-based quality assessment metric we developed provides a new benchmark for evaluating language corpora and can be utilized in future studies.While our study has achieved significant milestones, it also opens avenues for future research. Extending our corpus generation and refinement methodology to other languages and dialects would contribute significantly to the diversity and inclusivity in the field of conversational AI. There is also scope for further refinement of our corpus generation techniques, especially in capturing more nuanced aspects of language. In conclusion, our research presents a significant leap forward in the generation and refinement of language corpora, particularly for Italian LLMs. The methodologies developed and the insights gained lay a solid foundation for future endeavors in the NLP domain, paving the way for more inclusive and effective conversational AI systems.To foster collaboration and advance Natural Language Processing (NLP), we've publicly released our model and corpus on GitHub and Hugging Face <cit.>.This release aims to inspire further research, and innovation in fine-tuning corpora and large language models, and promote transparency and inclusivity in AI. We encourage the NLP community to build upon our work, expand it to other languages, and contribute to the development of more capable and culturally diverse language models.§ ACKNOWLEDGMENTS Work partially supported by: (i) the University of Pisa, in the framework of the PRA 2022 101 project “Decision Support Systems for territorial networks for managing ecosystem services”; (ii) the European Commission under the NextGenerationEU program, Partenariato Esteso PNRR PE1 - “FAIR - Future Artificial Intelligence Research” - Spoke 1 “Human-centered AI”; (iii) the Italian Ministry of Education and Research (MIUR) in the framework of the FoReLab project (Departments of Excellence), in the framework of the “Reasoning” project, PRIN 2020 LS Programme, Project number 2493 04-11-2021, and in the framework of the project “OCAX -Oral CAncer eXplained by DL-enhanced case-based classification” PRIN 2022 code P2022KMWX3. Work partly funded by the European Commission under the NextGeneration EU program, PNRR - M4 C2, Investment 1.5 “Creating and strengthening of innovation ecosystems”, building “territorial R&D leaders”, project “THE - Tuscany Health Ecosystem”, Spoke 6 “Precision Medicine and Personalized Healthcare”.[ < g r a p h i c s > ]Federico A. Galatolo is an Assistant Professor at the Department of Information Engineering of the University of Pisa (Italy). His research expertise spans the domains of Deep Learning, Artificial Intelligence, and their applications across various fields. He has contributed significantly to the academic community as a (co-)author of numerous international scientific publications. He is also a prolific contributor to the open-source community, having developed and maintained over 100 public repositories.[ < g r a p h i c s > ]Mario G.C.A. Cimino is an Associate Professor at the Department of Information Engineering of the University of Pisa (Italy). His research lies in the areas of Information Systems and Artificial intelligence. He is (co-)author of about 100 international scientific publications. He is an Associate Editor of the Journal of Granular Computing (Springer) and the Journal of Ambient Intelligence and Humanized Computing (Springer). He is Vice-Chair of the IEEE CIS Task Force "Intelligent Agents", IEEE Computational Intelligence Society. | http://arxiv.org/abs/2311.15698v1 | {
"authors": [
"Federico A. Galatolo",
"Mario G. C. A. Cimino"
],
"categories": [
"cs.CL",
"cs.AI"
],
"primary_category": "cs.CL",
"published": "20231127103455",
"title": "Cerbero-7B: A Leap Forward in Language-Specific LLMs Through Enhanced Chat Corpus Generation and Evaluation"
} |
Towards Energysheds: A Technical Definition and Cooperative Planning FrameworkThe authors graciously acknowledge funding from the U.S. Department of Energy Solar Energy Technologies Office (SETO) award DE-EE0010407. Dakota Hamilton, Samuel Chevalier, Amritanshu Pandey, and Mads Almassalkhi Department of Electrical and Biomedical Engineering University of Vermont, Burlington, VT, USA {dhamilt6, schevali, armitanshu.pandey, malmassa}@uvm.edu January 14, 2024 ============================================================================================================================================================================================================================================== There is growing interest in understanding how interactions between national and global policy objectives and local community decision-making will impact the clean energy transition. The concept of energysheds has gained traction in the areas of public policy and social science as a way to study these relationships. However, technical definitions for energysheds are still in their infancy. In this work, we propose a mathematical definition for energysheds. Analytical insights into the factors that impact a community's ability to achieve their energyshed policy objectives are developed, and various tradeoffs associated with interconnected energysheds within power systems are studied. We also introduce a framework for energyshed planning to help ensure equitable and cooperative decision making as communities invest in local energy assets. This framework is applied in an example transmission network, and numerical results are presented. community networks, distribution energy resources, optimization, power flow,public policy § INTRODUCTIONPublic policies to decarbonize are going hand-in-hand with scaling up the integration of (distributed) renewable generation and electrification programs. These policies are transforming the generation, heating, cooling and transportation sectors nationally and globally. However, this global transformation depends crucially on public buy-in from regional and local communities, because top-down policies and incentives require bottom-up participation to succeed. For example, it is estimated that 42% of emissions result from so-called kitchen table decisions: how we heat our homes, cook our food, clean our clothes, and transport our family <cit.>. Furthermore, individual kitchen-table constraints, and challenges to decarbonize and electrify, are collectively represented by local energy commissions and committees that seek to empower communities to take an active role in the clean energy transition. This is driven by a desire to understand how global energy transition will impact local communities, and how local communities can benefit from and achieve this transition. For example, communities are very interested in how local renewable generation can benefit local economies. Local utilities are also trying to work with communities to better understand how shared community solar, net-metering, and energy technologies can avoid the need for expensive grid upgrades <cit.>.Clearly, there is an interplay between local decisions, regional costs, and global policy objectives and impacts. Furthermore, the interactions between local communities, regional utilities, and national policy-makers highlight the importance of effective outreach and education between these parties. This is partly why the U.S. Department of Energy has recently focused programs around supporting Clean Energy Communities <cit.>. In particular, the program Energysheds: Exploring Place-Based Generation seeks to study the interaction between local energy communities' distributed generation (DG) planning decisions and the broader energy system within which the communities reside <cit.>. The concept of an energyshed is similar to that of a watershed or foodshed <cit.>. While a watershed inherently limits the sharing of its resources since it is defined by the inherent (static) gradients of nature's surfaces and reservoirs (storage), a foodshed permits sharing its local food production (resources) with other foodsheds <cit.>. The sharing between foodsheds is enabled exactly by transportation networks that interconnect them. This sharing among foodsheds also means that not all food need to be produced locally, and blurs the physical boundary of any individual foodshed. It is in this context that we present energysheds as local energy communities within (blurry) geographical areas in which energy system objectives and constraints are determined and between which energy can be actively exchanged.These interactions also beget interesting questions about how energyshed decisions (or constraints) affect the power systems that interconnect them all. Conversely, the power systems within which all energysheds reside and the gradients associated with the physical power flows may impose constraints on what each energyshed can achieve locally. For example, an energyshed at the end of a long radial feeder may not have the network capacity available to meet 100% of local demand with local (clean) generation over a day due to export limits (caused by transformer or voltage bounds). Thus, as far as the authors are aware, this paper is the first attempt to analyze and better understand the role of the power network in enabling or limiting local energyshed objectives.Specifically, we propose a first mathematical definition of an energyshed, and study fundamental tradeoffs associated with energyshed decisions and interconnections and how power systems can enable or limit energyshed objectives.To this end, the key contributions of this work are: * A mathematical definition for energysheds, considering spatiotemporal and techno-economic facets.* Theoretical analysis to provide insights into how limited resources and power system constraints impact the ability of individual communities to meet policy goals.* A cooperative energyshed planning framework based on optimization, which helps ensure equitable outcomes as communities invest in local generation resources. The rest of the paper is organized as follows. Section <ref> introduces a technical definition of an energyshed and discusses theoretical energyshed planning insights. The cooperative energyshed planning framework is proposed in Section <ref>, and Section <ref> presents numerical case study results. Section <ref> concludes and discusses future research directions.§ ENERGYSHED DEFINITION AND INSIGHTSConsider a power system represented as a graph, 𝒢 := (𝒩,ℰ), where 𝒩 is a set of nodes and ℰ the set of edges, such that (i,j) ∈ℰ if nodes i,j ∈𝒩 are connected by an edge. Denote the time window of interest, T, by a set of discrete time intervals 𝒯. Then, each node i, at time t, has local demand, P_i,t^L≥ 0, local generation, P_i,t^G≥ 0, and net generation, P_i,t^G-P_i,t^L. In order to determine the extent to which a given community at node i or over a set of nodes i∈𝒩_s ⊆𝒩 represents an energyshed, we introduce a desired ratio of local energy produced to local energy consumed, denoted 𝒳, as a metric.Proposed Definition (Energyshed): An energy community defined by the spatiotemporal 3-tuple (𝒳 , 𝒩_s , T) represents an energyshed, if it satisfies𝒳≤∑_i∈𝒩_s∑_t∈𝒯 P^G_i,t/∑_i∈𝒩_s∑_t∈𝒯 P^L_i,t ,for every non-overlapping interval 𝒯 of duration T.For example, if 𝒩_s represents the U.S. state of Vermont's entire power network, T represents every year with 𝒳 = 0.9, then (<ref>) states that the state of Vermont becomes an energyshed when it meets it own renewable portfolio standard (90% of energy needs from renewable sources by the year 2050), assuming local generation is renewable <cit.>. In another extreme, consider a 3-tuple representing a set of nodes that make up an islanded microgrid, T being every second, and 𝒳=100 % (with no load shedding, as expected when islanded); then a microgrid can also be cast as an energyshed. Between an entire state over a year and a small microgrid over a second is a rich set of relevant power systems that underpin diverse energy communities as possible energysheds. Besides the ability to use (<ref>) to measure how far a given community is from meeting the requirements of an energyshed (i.e., backcasting historical data), we are also interested in understanding what a community needs to or can do (or invest in) to meet their own requirement(s), and how multiple interconnected communities can satisfy their individual energyshed requirements with or without sharing energy. This is discussed next.§.§ Energyshed resource limits Since networks can be reduced via nodal aggregations and Kron-based reductions (e.g., see <cit.>), we can (without loss of generality) consider each community to embody a single node in a network. Thus, in addition to dropping the sum over 𝒩_s,the 3-tuple defining community i becomes (𝒳_i , i , T).Consider that a community also can have access to certain resources. These resources represent the capability of a community to invest in energy assets, including (but not limited to) financial, technological, or land use capabilities (e.g., transformer upgrades, load control programs, electrification incentives, batteries, electric vehicle charging stations, solar photovoltaic arrays, wind farms, etc.). The resources of each community i are subject to a certain budget, which limits a community's ability to modify its net-generation. That is, community i at time t can modify its net generation by P_i,t^S = P_i,t^S+ - P_i,t^S-, where-P_i,t^S-≤ P_i,t^S≤P_i,t^S+.Community i will then want to understand how to best use its budget to maximize 𝒳_i (i.e., achieve its energyshed objectives), which now depends on P_i,t^S as follows:[Herein, we focus on communities that have more load than generation (before investments), since achieving 𝒳_i ≥ 1 will be trivial for communities with large amounts of existing local generation.]𝒳_i = ∑_t∈𝒯 (P^G_i,t+P^S+_i,t)/∑_t∈𝒯 (P^L_i,t+P^S-_i,t) , where the additional power generated at time t due to investments in energy assets (e.g., rooftop or utility solar, wind, batteries discharging) in community i is represented by P^S+_i,t≥ 0, and P^S-_i,t≥ 0 denotes additional power demand capacities due to investments in electrification (e.g., battery charging, electric vehicle, heating, and cooling loads). Clearly, in (<ref>), we treat P^G_i,t and P^L_i,t as fixed (historical or predicted) data, whereas P^S+_i,t and P^S-_i,t are treated as variable investment decisions made by the community in pursuit of policy goals.Thus, given data for demand and existing (non-dispatchable) generation, we can determine an analytical relationship between these investment budgets and the maximum value of 𝒳_i that can be achieved by a community. These relationships depend on the ability of a community to export excess generation to surrounding communities through thegrid infrastructure. §.§ Communities with Unconstrained Power ExportsFirst, we consider communities that can readily export power to the grid during instances in time when local power generation exceeds local power demand.Our goal is find the maximum 𝒳_i that a community can achieve given a certain budget. Since all quantities in (<ref>) are positive by definition, we can maximize 𝒳_i by setting the numerator (i.e., total energy produced locally) as large as possible, and setting the denominator (i.e., total energy consumed locally) as small as possible. Furthermore, since P^G_i,t and P^L_i,t are treated as fixed data, the maximum 𝒳_i will be achieved when P^S+_i,t = P^S+_i,t and P^S-_i,t = 0 for all times t. Thus, we obtain a theoretical upper bound on the ratio of local generation to local consumption,𝒳_i = ∑_t∈𝒯 (P^G_i,t+P^S+_i,t)/∑_t∈𝒯 P^L_i,t = ∑_t∈𝒯 P^G_i,t/∑_t∈𝒯 P^L_i,t + ∑_t∈𝒯 P^S+_i,t/∑_t∈𝒯 P^L_i,t . Under the assumption that a community is unconstrained in its ability to export excess generation, there is linear relationship between the maximum local generation ratio, 𝒳_i, and the total energy investment budget, ∑_t∈𝒯P^S+_i,t, as shown in Fig. <ref>. Note that the slope of this line is inversely proportional to the total energy demand, ∑_t∈𝒯 P^L_i,t, of community i. That is, communities with higher energy demand will require more investments in energy assets and technology in order to meet the same energyshed policy goals (i.e., the same value of 𝒳_i).Additionally, note that the y-intercept of the linear relationship in (<ref>) is ∑_t∈𝒯 P^G_i,t/∑_t∈𝒯 P^L_i,t. This is expected since with no additional investments in energy assets, the value of 𝒳_i for a community is given by the ratio of energy produced by existing local generation to total energy demand. This also means that communities with existing local generation will require fewer new investments to meet their energyshed policy goals.§.§ Communities with Constrained Power Exports In practice, the ability of a given community to export excess generated power may be constrained by a number of factors including capacity limits of physical grid infrastructure (e.g., substation transformers, transmission or distribution lines), or the willingness of surrounding communities to consume that power. We model these constraints as bounds on the difference in additional power generation and demand. That is,P^S_i,t≤ P^S+_i,t - P^S-_i,t≤P^S_i,t .Note that since P^G_i,t and P^L_i,t are treated as fixed data, constraints of the form (<ref>) are sufficient to capture limits on net exports from community i.Following the same logic as in Sec. <ref>, we want to make P^S+_i,t as large as possible and P^S-_i,t as small as possible, in order to maximize 𝒳_i. If P^S_i,t >= P^S+_i,t for a given time t, then we can set P^S+_i,t = P^S+_i,t and P^S-_i,t = 0, as we did in Sec. <ref>. However, if P^S_i,t < P^S+_i,t for a given time t, then the constraint (<ref>) is binding. Therefore, we set P^S-_i,t = P^S+_i,t -P^S_i,t. Under these conditions, we can maximize 𝒳_i by setting P^S+_i,t = P^S+_i,t and P^S-_i,t = P^S+_i,t -P^S_i,t, assuming ∑_t∈𝒯P^S_i,t≥∑_t∈𝒯 (P^L_i,t-P^G_i,t).[Note that if ∑_t∈𝒯P^S_i,t < ∑_t∈𝒯 (P^L_i,t-P^G_i,t), then 𝒳_i is maximized by setting P^S+_i,t = P^S_i,t and P^S-_i,t = 0. However, under these conditions, 𝒳_i ≥ 1 for the proposed strategy. Since we are primarily interested in communities where 𝒳_i < 1, we choose to ignore this technicality.] Combining the two above cases (P^S_i,t >= P^S+_i,t and P^S_i,t < P^S+_i,t), we obtain an updated expression for the maximum local generation ratio,𝒳_i = ∑_t∈𝒯 (P^G_i,t+P^S+_i,t)/∑_t∈𝒯 (P^L_i,t+max{P^S+_i,t-P^S_i,t ,0}) ,which accounts for a community's power export constraints.Examples of the relationship between 𝒳_i and total energy investment budget for different power export constraints are shown in Fig. <ref>. Note that until a certain point, all curves in Fig. <ref> follow the same line as the unconstrained case. However, once a community's ability to export excess generation is constrained at some time t, it will begin to require larger investments in order to achieve the same value of 𝒳_i as the unconstrained case.Fig. <ref> also shows the special case where a community is unable to export any excess power.[This scenario can be modeled by setting P^S_i,t = P^L_i,t - P^G_i,t for all times t.] In order to achieve a value of 𝒳_i = 1 in this case, the community will need to invest in enough local generation assets to completely meet its local demand at all times (i.e., the community will need to operate as a microgrid).§.§ Equity Considerations in Energyshed PlanningIt is important to recognize that different communities will have different capabilities to invest in new energy assets. Unless careful consideration is given to the diversity in resources that is inherent to different communities, some frameworks for energyshed planning can lead to inequitable outcomes for some communities.For example, communities with the capability (more resources, larger budget) to invest in local generation resources the fastest will be able to achieve values of 𝒳_i ≥ 1 more quickly by exporting excess generation to less-privileged communities. However, if each community makes independent decisions with the goal of maximizing its own 𝒳_i, then as the least privileged communities invest in local generation resources, they may not have the opportunity to export excess power (since importing power would decrease 𝒳_i for surrounding communities). Thus, communities which are the slowest to achieve their energyshed policy goals will have to invest disproportionately more resources to meet those goals. This motivates the need for a cooperative energyshed planning framework, where the impact of decisions made by one community on neighboring communities is considered explicitly, in order to achieve better system-wide outcomes. § COOPERATIVE ENERGYSHED PLANNING FRAMEWORKIn this section, we propose an optimization-based, cooperative framework forenergyshed planning and decision-making.We denote the set of nodes in the network that operate as energysheds by 𝒩_ES⊆𝒩. The energyshed planning problem can then be formulated as follows:max_P_i,t^S+,P_i,t^S- min_i(𝒳_i) - α∑_(i,j)∈ℰ (∑_t∈𝒯 |P_ij,t|) ,s.t. P_i,t^G - P_i,t^L + P_i,t^S = ∑_j :(i,j)∈ℰP_ij,t , ∀ i ∈𝒩 , ∀ t ∈𝒯 , P_ij,t = θ_i,t-θ_j,t/x_ij , θ_1,t=0 , ∀ (i,j) ∈ℰ , ∀ t ∈𝒯 ,P_i,t^S = P_i,t^S+ - P_i,t^S- , ∀ i ∈𝒩 , ∀ t ∈𝒯 , 𝒳_i = ∑_t∈𝒯 (P^G_i,t+P^S+_i,t)/∑_t∈𝒯 (P^L_i,t+P^S-_i,t) , ∀ i ∈𝒩_ES ,0 ≤ P_i,t^S+≤P_i,t^S+ , ∀ i ∈𝒩 , ∀ t ∈𝒯 ,0 ≤ P_i,t^S-≤P_i,t^S- , ∀ i ∈𝒩 , ∀ t ∈𝒯 ,P_ij,t≤ P_ij,t≤P_ij,t , ∀ (i,j) ∈ℰ , ∀ t ∈𝒯 .The objective function (<ref>) consists of two terms. The goal of the first term is to maximize the minimum value of 𝒳_i across all energysheds. This incentivizes communities to cooperate in supporting the energyshed with the smallest local generation ratio achieve their policy goals.The solution of the optimization problem (<ref>) is non-unique, since 𝒳_i variables that are not the minimum can take multiple values at optimality. By adding the second term of (<ref>), we attempt to reduce the number of non-unique solutions. This term penalizes the power flows, P_ij,t, in each line (i,j) of the network and each time t. Furthermore, by selecting a large value for the weight α, we can study scenarios where sharing power between communities is discouraged.The constraints (<ref>)–(<ref>) represent the DC power flow equations for the network at each time t, where x_ij is the line reactances and θ_i,t is the voltage phase angle (in radians) of node i. The definitions of P^S_i,t and 𝒳_i are captured by (<ref>)–(<ref>), whereas constraints (<ref>)–(<ref>) enforce investment budgets. Finally, line flow limits are given by (<ref>), respectively.Note that the constraint (<ref>) is non-convex. Thus, in this paper, we utilize nonlinear programming (NLP) techniques, such as interior-point methods, to solve (<ref>). § NUMERICAL CASE STUDYIn order to study the fundamental tradeoffs associated with energyshed policy decisions within power systems, we tested the proposed cooperative planning framework on the 6-bus transmission network shown in Fig. <ref>(a) <cit.>. The cooperative planning framework is implemented in Julia using the JuMP package <cit.>, and the NLP (<ref>) is solved with the Ipopt solver <cit.>. We consider the set 𝒯 as every hour over the course of one day. Unless otherwise noted, per unit quantities are provided on a 100 MVA system base.Buses 1, 2, and 3 have no local load and are equipped with conventional synchronous generation. Therefore, we do not treat them as energysheds or consider values of 𝒳_i for these buses. Within the energyshed planning framework, we determine the power dispatch for these conventional generating units through the variable P^S_i,t (rather than P^G_i,t). For all times t, P^S+_i,t is set to the maximum power limit for each generator (2.0, 1.5, and 1.8 pu on Buses 1, 2, and 3, respectively), and P^S-_i,t is zero. We assume Buses 4, 5, and 6 all operate as energysheds. The load and existing distributed generation profiles for these buses over the day are shown in Fig. <ref>(b). Line flow limits for each transmission line, with P_ij,t = -P_ij,t, are identical to those in the MATPOWER case data for the 6-bus system <cit.>, with the exception that line limits for Lines 3–6, 1–4, and 2–4 are increased to 1.0, 0.8, and 0.8 pu, respectively, in order to alleviate infeasibility issues due to transmission constraints. In the following case study, we vary the total energy investment budget, ∑_t∈𝒯P^S+_i,t, and investigate the tradeoffs of energyshed planning decisions under different power sharing scenarios (i.e., values of the weight α). To account for the diversity of resources that is inherent to different communities, we assume that for each unit of additional investment budget in Bus 4, Buses 5 and 6 are able to increase their investment budgets by 1.5 units and 3 units, respectively. That is, if we choose a budget P∈ [0,1.5] pu, then the budgets for Buses 4–6 are P^S+_i,t = {P, 1.5P, 3.0P} for all times t. For simplicity, we assume P^S+_i,t is constant with respect to time, and set P^S-_i,t = P^S+_i,t.Fig. <ref> shows the values of 𝒳_i at Buses 4–6 as the total energy investment budget is increased, under a more cooperative planning framework (α = 0.001) and a scenario where sharing power between communities is discouraged (α = 0.1). In the α = 0.1 case, the energysheds at Buses 5 and 6 (which have more available resources) are able to achieve values of 𝒳_i ≥ 1 quickly. However, once these communities reach their policy goals, they are no longer willing to import excess generation from Bus 4. Thus, the energyshed at Bus 4 (i.e., the least privileged community) requires much more investment in energy assets to reach 𝒳_i = 1. In contrast, with α = 0.001, the energysheds at Buses 5 and 6 import excess generation from Bus 4 (even if that means achieving their own policy goals at a slower pace, or even slightly reducing their value of 𝒳_i), in order to allow the energyshed at Bus 4 to follow the linear trajectory (i.e. corresponding to unconstrained exports) for much longer. This results in Bus 4 being able to achieve 100% local energy generation (𝒳_i = 1) with approximately 28% less investment burden.Finally, Fig. <ref> illustrates the tradeoffs between the minimum 𝒳_i (i.e., the first term of the objective function), network utilization (the second term of the objective), and the total energy dispatch of conventional “centralized” generation as communities invest in local generation resources. It can be seen that as communities pursue energyshed policy goals by investing in local energy assets, existing transmission and conventional generation infrastructure will be under-utilized and may become sunk costs. Furthermore, while this work intentionally tried to remain technology and cost agnostic, it is important to recognize that investments in local energy assets will have economic, environmental, and social benefits and costs. Therefore, future research in energyshed planning that considers these aspects will be of upmost value. For example, the development of tools and processes that support communities in allocating appropriate technology and investments to meet their net generation profile P^S_i,t (as determined by the proposed cooperative planning framework), is of particular interest.§ CONCLUSIONIn this paper, we proposed a mathematical definition for energysheds, and studied the fundamental tradeoffs associated with energyshed policy decisions and planning within an electric power system. We also explored how interactions between planning decisions across different communities can potentially lead to inequitable outcomes, and introduced a framework for cooperative energyshed planning. Theoretical insights into energyshed thinking as well as a numerical case study were also presented.There are numerous potential avenues for future research in energyshed planning. First, the proposed framework could be extended to consider AC network analysis (i.e., system voltages, reactive power flows, and line losses) or multi-energy systems (e.g., district heating or transportation systems), rather than only considering electric infrastructure. Future work could also investigate how investments in energy efficiency can be explicitly modeled in an energyshed planning framework. Additionally, further analysis into convex reformulations or relaxations of the proposed optimization problem would enable improvements in scalability to larger systems. Finally, the application of the proposed framework to large regions with real data, and consideration of economic, environmental, and social aspects of community energy planning and investment would be extremely valuable. § ACKNOWLEDGEMENTThe authors greatly appreciate numerous team discussions about energysheds with colleagues at UVM, including Jeff Marshall, Jon Erickson, Greg Rowangould, Dana Rowangould, Bindu Panikkar, Hamid Ossareh, Eric Seegerstrom, Emmanuel Badmus, and Omid Mokhtari, as well as utility partners at Green Mountain Power (GMP), Vermont Electric Cooperative (VEC), Stowe Electric Department, and Vermont Electric Company (VELCO).IEEEtran | http://arxiv.org/abs/2311.16300v1 | {
"authors": [
"Dakota Hamilton",
"Samuel Chevalier",
"Amritanshu Pandey",
"Mads Almassalkhi"
],
"categories": [
"eess.SY",
"cs.SY"
],
"primary_category": "eess.SY",
"published": "20231127202417",
"title": "Towards Energysheds: A Technical Definition and Cooperative Planning Framework"
} |
For every k ∈ and α∈ (0,1) we construct a divergence-free u ∈ C^k([0,T],C^α(^d,^d)), d ≥ 2, such that there is no measurable selection of solutions of the ODE Ẋ_t = u(t,X_t) that preserves the Lebesgue measure. Comparison between Tensor Networks and Variational Quantum Classifier F. Neukart January 14, 2024 ===================================================================== § INTRODUCTION Let ^d := (/)^d denote the d-dimensional torus, d ≥ 2.Given T>0 and a measurable velocity field u : [0,T] ×^d →^d satisfying the incompressibility condition u (t, ·) = 0 in the sense of distributions, we are interested in the ODEX(t,x) = x + ∫_0^t u(s,X(s,x)) ds,x ∈^d,t ∈ [0,T].Equation (<ref>) can be uniquely solved for every initial condition x ∈^d when u is smooth.In this case, looking at the evolution of the Jacobian determinant J :=D_x X∂_t J (t,x) = J (t,x)u (t,X(t,x)) = 0, we have that the map X_t := X(t,·) preserves the d-dimensional Lebesgue measure ℒ^d on ^d for every t ∈ [0,T], namely(X_t)_♯ℒ^d = ℒ^d. Given a continuous velocity field u, solutions of (<ref>) exist for every initial condition x ∈^d by Peano Theorem, but are not necessarily unique.In this work we consider the following question: is there a Lebesgue-measurable way x ↦ X(·,x) to bundle together solutions of (<ref>) for almost every initial condition x ∈^d, so that the Lebesgue measure is preserved by X_t? In other words, we are interested in the existence of a measure-preserving selection of characteristics, according to the following:Given a measurable u : [0,T] ×^d →^d with u(t,·) = 0, we say that X : [0,T] ×^d →^d is a measure-preserving selection of characteristics if: (i) For every t ∈ [0,T], X(t,·) : ^d →^d is Lebesgue measurable and (<ref>) holds;(ii) For almost every x ∈^d the map t ↦ u(t,X(t,x)) is in L^1([0,T]) and (<ref>) holds.In the rough setting, a classical result is due to Di Perna and Lions <cit.>: under the additional assumptions of essential boundedness u ∈ L^∞_t,x and Sobolev regularity u ∈ L^1_t W^1,1_x, they show existence and uniqueness of a measure-preserving selection of characteristics X which additionally satisfies the flow property X(t+s,x)=X(t,X(s,x)) for almost every x ∈^d and every t,s,t+s ∈ [0,T].A successive extension to u ∈ L^∞_t,x∩ L^1_t BV_x fields is due to Ambrosio <cit.>.We point out that in <cit.> uniqueness of a measure-preserving selection of characteristics is not a consequence of uniqueness of solutions to (<ref>) for a.e. starting point x ∈^d.The latter requires stronger conditions on u, for instance u ∈ L^∞_t W^1,r_x with r>d as proved in <cit.>.On the other hand, in <cit.> the authors give, for every r < d and p<∞, an example of u ∈ C_t W^1,r_x ∩ C_t L^p_x leading to non-unique solutions of (<ref>) for a positive measure set of starting points. A refinement of this result is presented in <cit.>. More recently, for every r < d and α < 1 Kumar <cit.> further improved these results constructing a vector field u ∈ C_t W^1,r_x ∩ C^α_t,x with non-unique solutions of (<ref>) for a full-measure set of starting points.Notice that in the aforementioned works the theory of <cit.> still applies. Therefore, (<ref>) can be seen as a selection criterium for solutions of (<ref>). However, as pointed out already in <cit.>, (<ref>) might not single out a unique X when u has not a full (weak) spatial derivative.Here we are mostly interested in the existence issue. In general, understanding whether or not a measure-preserving selection of characteristics is possible requires a good understanding of the set of all solutions of the ODE (<ref>); but given a velocity field u that is not weakly differentiable, non-uniqueness of (<ref>) can be extremely wild and hard to characterise.Our main result is the following: For every k ∈ and α∈(0,1) there exists a velocity field u ∈ C^k_t C^α_x, satisfying u(t,·) = 0 in the sense of distributions, without any measure-preserving selection of characteristics. Therefore, the regularity threshold on u that guarantees existence of a measure-preserving selection of characteristics is morally the same guaranteeing existence and uniqueness (that is, one full weak spatial derivative). The main “obstruction” to the existence is the fact that forward flows (call it Y) associated with incompressible velocity fields v ∈ C^k_t C^α_x do not need to be essentially injective in the sense of <cit.>:∫_A_1δ_Y(t,x) dx ⊥∫_A_2δ_Y(t,x) dxfor every t ∈ [0,T] and disjoint Lebesgue measurable sets A_1, A_2 ⊂^d. Therefore, in order to prove <ref> we construct v such that Y_T : ^d →^d is exactly two-to-one on a large subset A of its domain (in a proper measure theoretic sense).To see this, we use that the (unique) forward flow Y is such that Y(T,x) has an explicit expression when looking at the dyadic expansion of x ∈ A ⊂^2. Then, taking the backward velocity field u = u(t,x) := -v(T-t,x), we have that any measurable selection of characteristics X can not reach much more than “half” of the torus at time T, in particular (X_T)_♯ℒ^d ≠ℒ^d.More generally, for the u we construct the absolute continuity ℒ^d ≪ (X_T)_♯ℒ^d fails for every measurable selection of characteristics X. On the other hand, we are currently unable to say if (X_T)_♯ℒ^d ≪ℒ^d holds; namely, if measurable selections of characteristics are Regular Lagrangian Flows as defined in <cit.>. Let us close this introduction with the following observation.Let {χ^δ}_δ∈ (0,1) be standard spatial mollifiers and consider the divergence-free velocity fields u^δ := u ∗χ^δ, where u is given by <ref>. By spatial smoothness, for every δ∈ (0,1) the ODE (<ref>) with u replaced by u^δ has a unique solution X^δ, that additionally preserves the Lebesgue measure. Since preserving the Lebesgue measure is stable with respect to convergence of the flows in L^1_t,x, as a consequence of <ref> we have:The family { X^δ}_δ∈ (0,1) is not strongly precompact in L^1_t,x.We point out that precompactness in L^1_t,x holds true as soon as the family {u^δ}_δ∈ (0,1) is uniformly bounded in L^∞_t,x∩ W^1,1_t,x and the flows are uniformly nearly incompressible[Namely, there exists C>0 such that C^-1≤ D_x X^δ≤ C for every δ∈ (0,1). This condition is immediately implied by smoothness and u^δ = 0.]: this is the content of Bressan's compactness conjecture, proved by Bianchini and Bonicatto in <cit.>.§ ACKNOWLEDGEMENTSThe author is grateful to Paolo Bonicatto and Elia Bruè for the useful discussions on the topic, and in particular to the latter for pointing out the reference <cit.>. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 949981).§ AN EXPLICIT CONSTRUCTION WITH U ∈ L^∞_T,X For the sake of a clear presentation, in this section we preliminarily construct an incompressible velocity field u ∈ L^∞_t,x such that the backward equationY(t,x) = x - ∫_0^t u(T-s,Y(s,x)) ds,t ∈ [0,T],has a unique solution for almost every initial condition x ∈^d and the map Y_T : ^d →^d is exactly two-to-one, up to negligible sets. The idea is to let Y_T essentially act on elements x ∈^d as a change of some (properly chosen) digits is the dyadic expansion of x, so that we can explicitly characterise almost every pair (x,y) ∈^d ×^d such that Y_T(x) = y. In the forthcoming <ref>, we smooth out u to obtain a new incompressible velocity field, with the desired Hölder regularity, such that the (unique) solution Y^δ of the backward equation (<ref>) is such that Y^δ(T,x) = Y(T,x) for every x ∈ A ⊂^d, for some measurable A with Lebesgue measure |A| ≥ 1-δ, with δ>0 fixed but arbitrary.For simplicity we restrict ourselves to d=2 but the construction generalizes easily to higher dimensions. §.§ Dyadic expansion For our purposes it is convenient to identify the Lebesgue measure on Borel subsets of two dimensional torus ^2 with a probability measure on (pairs of) infinite sequences of digits 0,1. The following construction is more or less classical, and can be found for instance in <cit.>.Let us consider the measurable space (Ω_0,ℱ_0) given by Ω_0 := {(0,0),(0,1),(1,0),(1,1)}, ℱ_0 := 2^Ω_0,endowed with the uniform probability _0 defined as _0({a}):=1/4 for every a ∈Ω_0.For every n ∈, n ≥ 1 consider an identical copy (Ω_n,ℱ_n,_n) of (Ω_0,ℱ_0,_0) and take the product spaceΩ̃ := ∏_n ≥ 1Ω_n, ℱ̃ := ⊗_n ≥ 1ℱ_n,:= ⊗_n ≥ 1_n.Elements ω∈Ω̃ can be seen as pairs ω = (ω^1,ω^2) of infinite sequences of digits 0,1. In order to obtain a one-to-one correspondence with points of ^2, we need to remove from Ω̃ all those elements for which either ω^1 or ω^2 is a sequence definitely equal to 1. More precisely, define for i ∈{1,2}Ω̂^i:= {ω = (ω^1,ω^2) ∈Ω : ∃n_0 ∈ω^i_n = 1, ∀ n > n_0 }= ⋃_n_0 ∈⋂_n > n_0{ω^i_n = 1 },which are ℱ̃-measurable and -negligible sets, and defineΩ := Ω̃∖ (Ω̂^1 ∪Ω̂^2), ℱ := ℱ̃∩Ω := { A ∩Ω : A ∈ℱ̃}, (A ∩Ω) := (A).Then, the probability space (Ω,ℱ,) is isomorphic to the space of Borel subsets of [0,1)^2, endowed with the Lebesgue measure, via the bijection x : Ω→ [0,1)^2 defined asx(ω) = (x^1,x^2),x^1 := ∑_n ≥ 1 2^-nω^1_n ,x^2 := ∑_n ≥ 1 2^-nω^2_n .More precisely, x is ℱ-ℬ([0,1)^2) measurable, bijective, with measurable inverse x^-1 and we have the identity of the pushforward measures (x)_♯ = ℒ^2 and (x^-1)_♯ℒ^2 =.Identifying ^2 ∼ [0,1)^2, we call the representation above the dyadic expansion of a point x ∈^2. §.§ Backward FlowLet τ∈ (0,1) be a time parameter and define τ_0 := 0, τ_n := ∑_q = 1^n τ^q,n ∈,n ≥ 1.Recall the dyadic expansion (<ref>) of points x=(x^1,x^2) ∈^2.We want the map Y to act in the following way (say on a full-measure set of points): * From time s=τ_0 to time t=τ_1, if ω^2_2=0 then Y_s,t(x)=x, and if ω^2_2=1 then Y_s,t(x) has the same dyadic expansion of x except for a change in the digit ω^1_1;* From time s=τ_1 to time t=τ_2, if ω^1_3=0 then Y_s,t(x)=x, and if ω^1_3=1 then Y_s,t(x) has the same dyadic expansion of x except for a change in the digit ω^2_2;* … * From time s=τ_2n to time t=τ_2n+1, n ∈, if ω^2_2n+2=0 then Y_s,t(x)=x, and if ω^2_2n+2=1 then Y_s,t(x) has the same dyadic expansion of x except for a change in the digit ω^1_2n+1;* From time s=τ_2n+1 to time t=τ_2n+2, n ∈, if ω^1_2n+3=0 then Y_s,t(x)=x, and if ω^1_2n+3=1 then Y_s,t(x) has the same dyadic expansion of x except for a change in the digit ω^2_2n+2. In the lines above we have denoted Y_s,t(x) the solution at time t of (<ref>) passing through x at time s (we will see that it exists and is unique for every s,t as above and almost every x∈^2).Geometrically, the action of Y is visualized in <ref>. The evolution via Y from time s=τ_2n to time t=τ_2n+1 corresponds to a rigid translation (i.e. without rotations or deformations) of the set {ω^2_2n+2=1,ω^1_2n+1=0} into the set {ω^2_2n+2=1,ω^1_2n+1=1}, and viceversa.Moreover, after time t the digits ω^i_q in positions q ≤ 2n+1 are kept fixed throughout the evolution. A similar interpretation holds for the evolution from time s=τ_2n+1 to time t=τ_2n+2, with ω^2_q replaced by ω^1_q+1 for every q ≤ 2n+2, and ω^1_2n+1 replaced by ω^2_2n+2.Denote T:= lim_n →∞τ_n = ∑_q=1^∞τ^q. Algebraically, we can describe the action of Y_T:^2 →^2 looking at the action of its conjugate Ψ : Ω→Ω via the isomorphism given by dyadic expansion.For our purposes, it is sufficient to describe the action of Ψ on a full measure set U invariant under Ψ, namely, such that Ψ : U → U.For a ∈{0,1}, let us consider the following subsets of Ω:U^odd,a_even := {ω = (ω^1,ω^2) ∈Ω : ∃n_0 ∈ω^1_2n+2 = a, ∀ n ≥ n_0 }= ⋃_n_0 ∈⋂_n ≥ n_0{ω^1_2n+2 = a }, U^even,a_odd := {ω = (ω^1,ω^2) ∈Ω : ∃n_0 ∈ω^2_2n+1 = a, ∀ n ≥ n_0 }= ⋃_n_0 ∈⋂_n ≥ n_0{ω^2_2n+1 = a }.Each of the previous sets is ℱ-measurable and -negligible. In particular, they correspond to Borel subsets of ^2 with zero Lebesgue measure, after identifying points of ^2 with their dyadic expansion. We define U := Ω∖ (U^odd,0_even∪ U^odd,1_even∪ U^even,0_odd∪ U^even,1_odd). Elements of U are those ω=(ω^1,ω^2) such that neither the sequence of digits ω^1_2n+2 nor ω^2_2n+1 is definitely constant. For points ω = (ω^1,ω^2) ∈ U, denoteI :={ (i,n) : i+n ,n ≥ 2, ω^i_n = 1}, I^- :={ (j,m) : ∃(i,n) ∈ Ij ≠ i,m = n-1, }.Then Ψ(ω) = Ψ(ω^1,ω^2) is given in coordinates as[Ψ(ω)]^i_n = ω^i_n, (i,n) ∉ I^-,ω^i_n+1(2), (i,n) ∈ I^-.It is immediate to check Ψ(ω) ∈ U for every ω∈ U, since digits of the form ω^1_2n+2 and ω^2_2n+1 remain unchanged by definition.For instance, the particular elementω= ( ω^1_1ω^1_2ω^1_3ω^1_4ω^1_5ω^1_6… ω^2_1ω^2_2ω^2_3ω^2_4ω^2_5ω^2_6…) = (0 1 1 1 0 1…1 1 0 1 0 0…)evolves in the following way via the action of Ψ:( 01 1 1 0 1…1 1 0 1 0 0…)↓ω^2_2 = 1 ( 11 1 1 0 1…110 1 0 0…)↓ω^1_3 = 1 (1 1 1 1 0 1…100 1 0 0…)↓ω^2_4 = 1(1 1 0 1 0 1…1 0 0 1 0 0…)↓ω^1_5 = 0 (1 1 0 1 0 1…1 0 0 1 0 0…)↓ω^2_6 = 0(1 1 0 1 0 1…1 0 0 1 0 0…)↓…The restriction to U ⊂Ω guarantees that the map Ψ : U → U is surjective and exactly two-to-one, that is: for every ω̅∈ U there are exactly two ω,ω' ∈ U such that Ψ (ω) = Ψ (ω') = ω̅. This is because, if we want to solve Ψ(ω) = ω̅, then we can arbitrarily choose either ω^1_1=0 or ω^1_1=1, and then define the other digits ω^i_q accordingly.This procedure yields an element of U since we necessarily have ω^1_2n+2= ω̅^1_2n+2 and ω^2_2n+1= ω̅^2_2n+1.In addition, the two solutions ω,ω' ∈ U are related to each other by the relation ω' = σ (ω), where σ:U → U is the “digits change” map:[σ(ω)]^i_n = ω^i_n, i + n ,ω^i_n+1(2), i + n . For every ω̅∈ U there are exactly two ω,ω' ∈ U such that Ψ (ω) = Ψ (ω') = ω̅. Moreover, ω' = σ (ω). Let us observe preliminarily that, by the very definition of Ψ and σ,Ψ (ω) = Ψσ(ω), ∀ω∈ U.Therefore, in order to prove the lemma it is sufficient to show that for every ω̅∈ U there is exactly one ω∈ U such that Ψ (ω) = ω̅ and ω^1_1=0.Since Ψ leaves unchanged any digit of the form ω^1_2n+2 and ω^2_2n+1, we necessarily have ω^1_2n+2 = ω̅^1_2n+2 and ω^2_2n+1 = ω̅^2_2n+1 for every n ∈. Since we are assuming ω^1_1=0, then the other digits of ω are inductively determined by ω^2_2n+2 =1, ω^1_2n+1≠ω̅^1_2n+1, 0, ω^1_2n+1 = ω̅^1_2n+1,andω^1_2n+3 =1, ω^2_2n+2≠ω̅^2_2n+2, 0, ω^2_2n+2 = ω̅^2_2n+2.For ω̅=(0 1 1 1 0 1…1 1 0 1 0 0…)as in <ref>, the following are the only two solutions of Ψ(ω)=ω̅:[ (0 1 * 1 * 1…1 * 0 * 0 *…)(1 1 * 1 * 1…1 * 0 * 0 *…); ↓ω^2_2 = 0↓ω^2_2 = 1; (0 1 * 1 * 1…1 0 0 * 0 *…)(1 1 * 1 * 1…1 1 0 * 0 *…); ↓ω^1_3 = 1↓ω^1_3 = 0; (0 1 1 1 * 1…1 0 0 * 0 *…)(1 1 0 1 * 1…1 1 0 * 0 *…); ↓ω^2_4 = 0↓ω^2_4 = 1; (0 1 1 1 * 1…1 0 0 0 0 *…)(1 1 0 1 * 1…1 1 0 1 0 *…); ↓ω^1_5 = 1↓ω^1_5 = 0; (0 1 1 1 1 1…1 0 0 0 0 *…)(1 1 0 1 0 1…1 1 0 1 0 *…); ↓ω^2_6 = 1↓ω^2_6 = 0; (0 1 1 1 1 1…1 0 0 0 0 1…)(1 1 0 1 0 1…1 1 0 1 0 0…); ↓…↓… ]Let us conclude this subsection with the following lemma. σ : U → U is measure preserving. Since U ⊂Ω has full -measure and Ω⊂Ω̃ has full -measure, it is sufficient to prove that σ̃:Ω̃→Ω̃ defined on the whole Ω̃ as in (<ref>) preserves . The measuresand (σ̃^-1)_♯ coincide on the cylinder subsets𝒞 := { C ⊂Ω̃ :C = ∏_n ≥ 1 C_n , C_n ⊂Ω_n ∀ n, ∃n_0 ∈ C_n = Ω_n ∀ n > n_0 }.Since 𝒞 is a π-system generating the σ-field ℱ̃, we have =(σ̃^-1)_♯.Therefore, for every measurable A ⊂ U(σ^-1(A)) =(σ̃^-1(A) ∩ U) =(σ̃^-1(A)) =(A) =(A). §.§ The velocity fieldLet us recall the following construction by Depauw <cit.>. Denote Q = (0,1)^2 the unit square. For x ∈ Q let ψ_⋆(x) := -8 max{ |x^1-1/2|^2 , |x^2-1/2|^2 },v_⋆ := ∇^⊥ψ.The velocity v_⋆ is BV with null divergence, and rotates the square Q clockwise of an angle of π radiants in time t=1/2. Then we define v = v(t,x) := -u(T-t,x) at any time t ∈ [τ_n-1,τ_n), n ∈, n ≥ 1, as a collection of many properly rescaled/reflected copies of v_⋆, glued together in a self similar fashion.For instance, to obtain that on the time interval [τ_0,τ_1)=[0,τ) the resulting Y=Y_0,τ is a translation of {ω^2_2=1,ω^1_1=0} into {ω^2_2=1,ω^1_1=1} and viceversa (up to negligible sets), we can use first a periodic Depauw velocity field with “cells” {ω^2_2=1} (see <ref>) and intensity proportional to 1/τ, so that after time τ/2 the set {ω^2_2=1,ω^1_1=0} goes into {ω^2_2=1,ω^1_1=1} and viceversa, but with a rotation of π radiants; and from time τ/2 to time τ we use distinct Depauw velocity fields with “cells” {ω^2_2=1,ω^1_1=0} and {ω^2_2=1,ω^1_1=1}, with intensity proportional to 1/τ, that rotate back of π radiants each set separately (the composition of two rotations with different center and opposite angle is a rigid translation). This is similar to what done by Zizza in <cit.>. Notice that essential uniqueness of Y descends from the L^∞∩ BV regularity of v_⋆. Up to rotations of π/2 radiants, the velocity field will be self-similar: on the time interval t ∈ [τ_n-1,τ_n), n ∈, n ≥ 1, the velocity v=v_n is given byv_n(t,x) = 1/(2τ)^n v_1((t-τ_n)/τ_n , 2^nx),where v_1 is the velocity field on the first time interval [0,τ). Then, taking τ = 1/2 we getv_n _L^∞_t,x =v_1 _L^∞_t,x < ∞uniformly in n, implying v _L^∞_t,x =u _L^∞_t,x < ∞.§ THE HÖLDER CONTINUOUS CASEIn this section we present a Hölder continuous version of the velocity field from <ref> which has the property that the induced flow Y^δ_T coincides with Y_T on a set A of Lebesgue measure |A|≥ 1-δ, for δ >0 fixed but arbitrary.In order to do this, we have to replace the Depauw cells with more regular building blocks, that we describe hereafter.For the sake of simplicity, we shall restrict ourself to constructing our building blocks on the unit square Q=(0,1)^2; the general case in handled by elementary operations as scalings, reflections, and rotations.Let us recall the following construction from <cit.>: define the function ψ : Q → asψ (x) := 2^1/2sin(π x^1)sin(π x^2)/(sin(π x^1)+sin(π x^2))^1/2. Contour lines of ψ foliate the unit square Q, as can be seen in <ref>.The value of ψ ranges between 1 (at the center of the square) and 0 (in the limit when x tends to the edges of the square). The level set {ψ≥ r } is convex for every value of r ∈ (0,1], see <cit.>. Contour lines of ψ are integral curves for the divergence-free velocity field ∇^⊥ψ, and the time it takes a point x ∈{ψ = r} to run across its level set when moving with velocity ∇^⊥ψ is given by the line integralT_ψ(r):= ∫_{ψ = r}1/|∇ψ| dγ.Similarly to what done in <cit.>, we want to replace ψ with another streamfunction for which (<ref>) is independent of r. However, in order to obtain a velocity field which is globally Hölder continuous we need to modify the construction of <cit.> as follows.Let ρ : _+ →_+ be a smooth function such that ρ(r)=r for r ≥ 1 and ρ(r) = 1/2 for r ≤ 1/2. Denote ρ^ϵ := ϵρ (ϵ^-1 r), where ϵ>0 is a free parameter to be properly chosen later, and letψ^ϵ(x) := ∫_0^ρ^ϵ(ψ(x)) T_ψ(r) dr. There exists a constant C ∈ (0,∞), independent of ϵ, such that the following hold: (i) ψ^ϵ >0 on Q, and ψ^ϵ is constant in a neighbourhood of ∂ Q;(ii) Let r_ϵ:=∫_0^ϵ T_ψ(r) dr. Then for every r ≥ r_ϵ the level set {ψ^ϵ = r} is also al level set for ψ, namely there exists r' such that {ψ^ϵ = r}={ψ = r'}.(iii) The travelling time T_ψ^ϵ, defined as in (<ref>) with ψ replaced by ψ^ϵ, satisfies T_ψ^ϵ (r) = 1 for all r ∈ [r_ϵ, ψ^ϵ_L^∞(Q)];moreover,| { x : T_ψ^ϵ(ψ(x)) ≠ 1 }|≤ | { x : ψ(x) < ϵ} | ≤ C ϵ^2/3; (iv) ψ^ϵ∈ W^2,∞(Q), withψ^ϵ_W^1,∞(Q)≤ C, ψ^ϵ_W^2,∞(Q)≤ C ϵ^-1.(i) We have ψ^ϵ>0 since ρ^ϵ≥ϵ/2 > 0 and T_ψ(r) is strictly positive for r ∈ (0,1). Moreover, by definition of ρ^ϵ the quantity ρ^ϵ(ψ(x)) is constant on { x: ψ(x) < ϵ/2}, hence so is ψ^ϵ.(ii) We use that ρ^ϵ(r)=r for r ≥ϵ, and T_ψ(r) is strictly positive and uniformly bounded for r ∈ (0,1) (cf. <cit.>).(iii) Let r ≥ r_ϵ. By point (ii), there exists r' ≥ϵ such that {ψ^ϵ = r} = {ψ = r'}.Hence, by definition of ψ^ϵ and ρ^ϵ, we have on {ψ^ϵ = r}∇ψ^ϵ = T_ψ(ρ^ϵ(ψ)) ∇ (ρ^ϵ(ψ)) = T_ψ(ψ) ∇ψ.Therefore, the travelling time T_ψ^ϵ satisfiesT_ψ^ϵ(r):= ∫_{ψ^ϵ=r}1/|∇ψ^ϵ| dγ = ∫_{ψ=r'}1/T_ψ(ψ)|∇ψ| dγ = 1.As a consequence, we also have {T_ψ^ϵ(ψ(x)) ≠ 1}⊂{ψ(x) < ϵ}, and thus we only have to bound the Lebesgue measure of the latter set. Since {ψ≥ϵ} is convex (see <cit.>), the square Q_ϵ with verticesq ∈{ x=(x^1,x^2) ∈ Q : sin(π x^1) = sin(π x^2) = ϵ^2/3}⊂{ψ≥ϵ}is entirely contained in the super-level set {ψ≥ϵ}; therefore, |{ψ(x) < ϵ}| ≤ |Q ∖ Q^ϵ|. We control the latter quantity with four times the area of the strip {sin(π x^1) ∈ (0,ϵ^2/3),x^1 ≤ 1/2}, which is easily computed noticing π x^1 ≤ C sin(π x^1) ≤ C ϵ^2/3. (iv) Let ψ^⋆ be defined as in <cit.> (for the particular value α=1/2), namely ψ^⋆(x) := ∫_0^ψ(x) T_ψ(r) dr. Recall from the proof of <cit.> that D^2 ψ^⋆ is bounded away from ∂ Q. Since ψ^ϵ coincides with ψ^⋆ on the set {ψ(x) ≥ϵ}, we only need to check the estimates in a neighbourhood of ∂ Q. Moreover, having ψ^ϵ(x) = ∫_0^ϵ/2 T_ψ(r)dr identically on {ψ(x) ≤ϵ/2}, we can restrict ourselves to controlling the Sobolev norms of ψ^ϵ on the domainD_ϵ := { x ∈ Q : ψ(x) ∈ (ϵ/3,1/2) }. We haveψ^ϵ_W^1,∞(D_ϵ) ≤ T_ψ_L^∞(ϵ/3,1/2)ρ^ϵ(ψ) _W^1,∞(D_ϵ), ψ^ϵ_W^2,∞(D_ϵ) ≤T'_ψ_L^∞(ϵ/3,1/2)ρ^ϵ(ψ) _W^1,∞(D_ϵ)^2 + T_ψ_L^∞(ϵ/3,1/2)ρ^ϵ(ψ)_W^2,∞(D_ϵ). By construction it holds ρ^ϵ_W^1,∞(ϵ/3,1/2)≤ C and ρ^ϵ_W^2,∞(ϵ/3,1/2)≤ C ϵ^-1; in addition, by <cit.> we haveT'_ψ_L^∞(ϵ/3,1/2)≤ C ϵ^-2/3, ψ_W^1,∞(D_ϵ) ≤ C, ψ_W^2,∞(D_ϵ)≤ C ϵ^-1/3,where the last inequality is justified by the fact that ψ≥ϵ/2 implies sin(π x)+sin(π y) ≥ 2(ϵ/2)^2/3.We are ready to construct the velocity field of <ref>. More precisely, we construct v = v(t,x) := -u(T-t,x) as in the following: For every k ∈, α∈ (0,1) and δ>0 there exists an incompressible velocity field v ∈ C^k([0,T], C^α_x) ∩ C_loc([0,T), W^1,∞_x) and a measurable A with Lebesgue measure |A| ≥ 1-δ such that the unique solution of Y^δ(t,x) = x + ∫_0^t v(s,Y^δ(s,x)) ds satisfiesY^δ(T,x) = Y_T(x), ∀ x ∈ A,where Y_T is the map constructed in <ref>.Let us define v^ϵ := ∇^⊥ψ^ϵ. By construction, the particles moving with the velocity v^ϵ run across the contour lines {ψ = r } in time 1 for every r ≥ϵ.In particular, by fourfold symmetry v^ϵ rotates the super-level set {ψ≥ϵ} of π radiants in time t=1/2.Let χ be a smooth function with support compactly contained in (0,1/2) and satisfying ∫_0^1/2χ(t)dt=1. In order to obtain a velocity which is smooth with respect to time,we define v̂^ϵ:(0,1/2) × Q →^2 asv̂^ϵ (t,x) := χ(t) v^ϵ(x). Then, by construction also v̂^ϵ rotates the super-level set {ψ≥ϵ} of π radiants in time t=1/2.Therefore, by point (iii) of <ref>, the rescaled velocityv^ϵ_n (t,x) :=1/(2τ)^nv̂^ϵ (t/τ^n, 2^n x),n ∈, n ≥ 1,rotates at least a fraction 1-C ϵ^2/3 of the rescaled square 2^-nQ of an angle of π radiants in time t=τ^n/2.Since ψ^ϵ is constant in a neighbourhood of ∂ Q, the velocity v^ϵ vanishes at ∂ Q. Therefore, we can glue together many (properly rescaled and reflected) copies of v^ϵ_n as in <ref> to obtain a Hölder continuous velocity v̂^ϵ_n : (τ_n-1,τ_n) × Q →^2 satisfying v̂^ϵ_n(τ_n-1,·) = v̂^ϵ_n(τ_n,·) = 0, v̂^ϵ_n(·,x) = 0 for x in a neighbourhood of ∂ Q, and the boundsv̂^ϵ_n _C_t W^1,∞(Q) ≤1/τ^nχ_C_t v^ϵ_W^1,∞(Q)≤ C' 1/τ^nϵ^-1,∂_t^h v̂^ϵ_n _C_t C^α(Q) ≤( 2^α-1/τ^1+h)^n ∂_t^h χ_C_t v^ϵ_C^α(Q)≤ C'( 2^α-1/τ^1+h)^n ϵ^-α,for every h ∈, h ≤ k and some finite constant C' depending only on χ, k, and the constant C of <ref>. The last inequality comes from interpolation between W^1,∞(Q) and W^2,∞(Q) Sobolev norms of ψ^ϵ given by point (iv) of <ref>.Choosing ϵ = ϵ_n := ϵ_⋆κ^n with κ∈ (0,1) and ϵ_⋆≪ 1, the previous inequalities become v̂^ϵ_n_n _C_t W^1,∞(Q) ≤ C' ϵ_⋆^-1( 1/τκ)^n ,∂_t^h v̂^ϵ_n_n _C_t C^α(Q) ≤ C' ϵ_⋆^-α( 2^α-1/τ^1+hκ^α)^n.Therefore, we takeτ^1+k = κ^α = 2^α-1/3∈ (0,1),so that v̂^ϵ_n_n tends to 0 in C^k_t C^α(Q) as n →∞. In this way, the divergence-free velocity field v := ∑_n ≥ 1v̂^ϵ_n_n 1_{ t ∈ (τ_n-1,τ_n)}, extended periodically on the full torus ^2, is in C^k([0,T],C^α(^2)) ∩ C_loc([0,T),W^1,∞(^2)) for T := ∑_q τ^q < ∞.Finally, v ∈ C_loc([0,T),W^1,∞(^2)) with null divergence implies that there exists a unique flow Y^δ associated to v, which preserves the Lebesgue measure for every t<T. By (iii) of <ref>, on every time interval of the form [τ_n-1,τ_n-1 + τ^n/2) (resp. [τ_n-1 + τ^n/2,τ_n)) the flow Y^δ coincides with Y on a measurable set A^1_n (resp. A^2_n) of Lebesgue measure at least|A^1_n| = |A^2_n| ≥ 1- C/2ϵ_⋆^2/3κ^2n/3.DefiningA := ⋂_n ≥ 1(Y^δ_τ_n-1)^-1 (A^1_n) ∩ (Y^δ_τ_n-1 + τ^n/2)^-1 (A^2_n),we have that Y^δ_T coincides with Y_T on A and |A|≥1- C ϵ_⋆^2/3∑_nκ^2n/3≥ 1-δ,up to choosing ϵ_⋆ = ϵ_⋆(α,δ) sufficiently small.Notice that v^ϵ∈ C^1_x except at most at the center of the square Q, where it vanishes. With only minor modifications to the definition of the streamfunction ψ^ϵ, we can make v^ϵ≡ 0 in an arbitrary small neighbourhood of the center of Q, improving the W^1,∞_x bound on v to an actual C^1_x bound (up to replacing the bound on |A| with |A| ≥ 1-2δ).In particular, the incompressibility condition in <ref> (and in <ref>) can be understood in the strong analytic sense.§.§ Proof of <ref>We finally give the proof of our main result.Let v and A as in <ref>, and define u(t,x) := -v(T-t,x). Let us suppose per absurdum there exists a measure-preserving selection of characteristic X for the velocity field u.Let x(U):=U_x ⊂^2 be given by the image of U under the map x:Ω→^2 defined in <ref>. U_x is a Borel set with full Lebesgue measure. Since X_T is Lebesgue measurable we can write X_T^-1({ω^1_1 = 0 }∩ A) = B_0 ∪ N_0, X_T^-1({ω^1_1 = 1 }∩ A) = B_1 ∪ N_1,where B_0,B_1 are Borel subsets of U_x ⊂^2 and N_0,N_1 are Lebesgue measurable with |N_0|=|N_1|=0. Since we are assuming X_T measure-preserving, it holds |B_0 ∪ B_1| = |A| ≥ 1-δ. Moreover, without loss of generality we may also assume that X(·,x) is a solution of the ODE (<ref>) for every x ∈ B_0 ∪ B_1. By uniqueness of solutions to (<ref>), on the time interval t∈[0,T] we have for every x ∈ B_0 ∪ B_1X(T-t,x) = Y^δ(t,X_T(x)).Since Y^δ_T|_A ∩ U_x = Y_T|_A ∩ U_x:A ∩ U_x → U_x is Borel measurable (as pointwise limit as n→∞ of the Borel measurable maps Y_τ_n) we have that for i ∈{0,1} the setC_i := X_T(B_i) ∩ A ∩ U_x = (Y_T|_A ∩ U_x)^-1 (B_i) ∩{ω^1_1 = i}is Borel measurable. Then we haveX_T(B_0 ⊔ B_1) ∩ A ∩ U_x = C_0 ⊔ C_1 where ⊔ denotes the disjoint union since X_T(B_i) ⊂{ω^1_1 = i}. Therefore, denoting σ_x:U_x → U_x the map conjugate to the digits change map σ : U → U defined by (<ref>), we have |X_T(B_0 ⊔ B_1) ∩ A ∩ U_x|= |C_0| + |C_1| = |σ_x (C_0)| + |σ_x (C_1)|,since the map σ is measure-preserving by <ref>. Moreover, by (<ref>) it holds Y_T σ_x (x) = Y_T (x) for every x ∈ U_x; therefore, the setsC_0, C_1,σ_x (C_0),σ_x (C_1),are pairwise disjoint. Indeed, σ_x C_0, σ_x C_1 are disjoint (because Y_T σ_x = Y_T on U_x, and Y_T C_0 ∩ Y_T C_1 = ∅) and are disjoint from X_T(B_0 ⊔ B_1) ∩ A ∩ U_x, otherwisey = X_T(x) = σ_x (X_T(x')) ⇒ Y_T(y) = Y_T(X_T(x)) = Y_T (σ_x (X_T(x'))) ⇒ x = x'but X_T(x) ≠σ_x (X_T(x)) since σ_x has no fixed point in U_x. Therefore,σ_x (C_0) ⊔σ_x (C_1)= σ_x (C_0 ⊔ C_1) = σ_x(X_T(B_0 ⊔ B_1) ∩ A ∩ U_x);in particular, X_T(B_0 ⊔ B_1) ∩ A ∩ U_x and σ_x (X_T(B_0 ⊔ B_1) ∩ A ∩ U_x) are disjoint sets with the same Lebesgue measure, implying (recall U_x has full measure)|X_T(B_0 ⊔ B_1) ∩ A| = |X_T(B_0 ⊔ B_1) ∩ A ∩ U_x| ≤ 1/2. But this gives a contradiction for δ <1/4,since|X_T(B_0 ⊔ B_1)| = |X_T(B_0 ⊔ B_1) ∩ A| + |X_T(B_0 ⊔ B_1) ∩ A^c| ≤ 1/2+δ,but1-δ≤ |B_0 ⊔ B_1|≤|X_T^-1 (X_T(B_0 ⊔ B_1))| = |X_T(B_0 ⊔ B_1)| ≤1/2+δ.alpha BCDL21[Amb04]Am04 Luigi Ambrosio. Transport equation and Cauchy problem for BV vector fields. Inventiones mathematicae, 158(2):227–260, 2004.[BB20]BiBo20 Stefano Bianchini and Paolo Bonicatto. A uniqueness result for the decomposition of vector fields in ℝ^d. Inventiones mathematicae, 220:255–393, 2020.[BCDL21]BrCoDL20 Elia Bruè, Maria Colombo, and Camillo De Lellis. Positive solutions of transport equations and classical nonuniqueness of characteristic curves. Arch. Rat. Mech. Anal., 240:1055–1090, 2021.[CC21]CaCr21 Laura Caravenna and Gianluca Crippa. A directional Lipschitz extension lemma, with applications to uniqueness and Lagrangianity for the continuity equation. Communications in Partial Differential Equations, 46(8):1488–1520, 2021.[Dep03]De03 Nicolas Depauw. Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d'un hyperplan. Comptes Rendus. Mathématique, 337(4):249–252, 2003.[DL89]DiLi89 Ronald J. Diperna and Pierre-Louis Lions. Ordinary differential equations, transport theory and Sobolev spaces. Inventiones mathematicae, 98:511–547, 1989.[EZ19]ElZl19 Tarek M. Elgindi and Andrej Zlatoš. Universal mixers in all dimensions. Advances in Mathematics, 356:106807, 2019.[Hal50]Ha50 Paul R. Halmos. Measure Theory. Graduate Texts in Mathematics 18. Springer-Verlag New York, 1st edition, 1950.[Kum23]Ku23+ Anuj Kumar. Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields. arXiv:2301.05185, 2023.[PS23]PiSo23 Jules Pitcho and Massimo Sorella. Almost everywhere nonuniqueness of integral curves for divergence-free Sobolev vector fields. SIAM Journal on Mathematical Analysis, 55(5):4640–4663, 2023.[Ziz22]Zi22 Martina Zizza. An example of a weakly mixing BV vector field which is not strongly mixing.arXiv:2208.12174, 2022. | http://arxiv.org/abs/2311.15661v1 | {
"authors": [
"Umberto Pappalettera"
],
"categories": [
"math.AP",
"math.CA"
],
"primary_category": "math.AP",
"published": "20231127094500",
"title": "On measure-preserving selection of solutions of ODEs"
} |
[email protected] [email protected]^1School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China^2Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK^3Department of Astronomy, Tsinghua University, Beijing 100084, China The clustering of galaxies and their connections to their initial conditions is a major means by which we learn about cosmology. However, the stochasticity between galaxies and their underlying matter field is a major limitation for precise measurements of galaxy clustering. It also hinders accurate mass reconstruction for retrieving cosmological information from observations. Efforts have been made with an optimal weighting scheme to reduce this stochasticity using the mass-dependent clustering of dark matter halos, but its application to observation is challenging due to the difficulties in measuring the mass of halos precisely. Here, we show that this is not optimal. We demonstrate that the cosmic-web environments (voids, sheets, filaments & knots) of halos provide extra information for reducing stochasticity. Using the environmental information alone can increase the signal-to-noise of clustering by approximately a factor of 3, better than the Poisson level at the scales of the baryon acoustic oscillations. This improvement is comparable to using halo mass information alone. The information about the environment and halo mass are complementary. Their combination increases the signal-to-noise by another factor of 2-3. The information about the cosmic web correlates with other properties of halos, including halo concentrations and tidal forces, thus, these are among the most dominant factors that can help improve the reconstruction. We attribute the extra information from the environment and secondary properties of halos primarily to the assembly bias of halos. Our findings open a new avenue for mass reconstruction and noise reduction using information beyond the halo mass.Mass reconstruction and noise reduction with cosmic-web environments Longlong Feng^1 January 14, 2024 ====================================================================A major challenge in cosmology with large-scale structure is to reconstruct the initial conditions from observations of the late-time Universe. While the initial matter density field is thought to be linear and continuous, what we observe in the late-time Universe is a discrete sampling of a non-linear matter field, typically traced out by galaxies. The accuracy of the reconstruction determines the amount of cosmological information we can extract from observations. It can be quantified with the correlation coefficient r between the galaxy number density field δ_g and the matter density field δ_m,r ≡⟨δ_g δ_m⟩/√(⟨δ_g^2⟩⟨δ_m^2⟩).The ⟨ ⟩ symbol denotes the ensemble average. To the first order, galaxies are linearly biased against matter. With the presence of shot noise ϵ, we have δ_g=bδ_m + ϵ, where b is the linear bias of galaxies. With ⟨ϵδ_m ⟩=0, we haveP_g=⟨δ_g^2⟩=b^2P_m+⟨ϵ^2⟩, P_gm=⟨δ_gδ_m⟩=bP_m, where P_m is the matter power spectrum; and so ⟨ϵ^2⟩/b^2P_m=1-r^2/r^2.All the above quantities are functions of the Fourier number k. We can see that the presence of the noise, or stochasticity, degrades the correlation coefficient r <cit.>. Minimizing stochasticity, increasing correlation coefficients and optimizing mass reconstruction are different sides of the same coin. Reducing ϵ has the following observational implications: (1) increasing the signal-to-noise ratio for the clustering of galaxies <cit.>, which is particularly beneficial for analyses of the baryon acoustic oscillations (BAO) with sparse tracers <cit.>;(2) increasing the correlations between the galaxy field and the matter field, allowing more cosmological information to be retrieved from the measurement of galaxy-galaxy lensing and reconstruction <cit.>.If ϵ is Poissonian, its power spectrum is 1/n̅, with n̅ being the number density of tracers. Recent work has shown that we can improve beyond the Poisson limit <cit.>. In N-body simulations where the masses of dark matter halos are known, <cit.> and <cit.> have shown that a linear combination of halos of different masses with different weights can significantly reduce stochasticity. The primary information used for the improvement is the mass-dependence of halo power spectra P_h(k,M), where M is the mass of halos. One way to understand this is that the linear halo bias b(M) increases with M; for the same number of halos, the signal-to-noise of P_h(k, M) is higher for halos with larger masses. Therefore, it is preferable to up-weight the high-mass halos to increase the signal-to-noise. This is indeed the case for the optimal weights found in <cit.>.However, this can not be the full physical picture. In addition to M, the clustering of halos depends also on other secondary properties. Among them, the formation time of halos is known to affect the halo clustering. For low-mass halos, those who formed earlier tend to exhibit stronger clustering than younger ones. This is known as the halo assembly bias <cit.>. This indicates that there is room for further improvement in the reconstruction if this secondary property of halos can be used, and it is the focus of this letter. A main challenge for this is that the formation time of halos or galaxies is difficult to measure in observations. However, it is known to correlate with other properties of halos, especially with halo concentration, tidal force, and their cosmic-web environments <cit.>, with some arguing that cosmic-web anisotropy is the main indicator for halo assembly bias <cit.>, the latter is accessible in observations. We will, therefore, focus on using the cosmic-web environment for the reconstruction.Suppose we have an observed halo number density field, denoted as δ_h; It is a discrete sampling of its underlying matter field δ_m. Our aim is to maximize the correlation coefficient r between these two fields, defined in equation (<ref>), substituting the subscript b by h to represent halos. We can see that r is agnostic about the linear halo bias. When r=1, perturbation modes are in phase, but their amplitudes can be different.To maximize r, we split the halo catalog n_h into N_s sub-samples with n_h = ∑_i=1^N_s n_i, and weight each sub-sample with W_i∈ℝ. We then combine these weighted sub-samples to obtain the reconstructed density field n_c = ∑_i^N_sW_in_i. The ith sub-sample can be written as n_i = ∑_j=1^N_i w_jδ_D(x⃗-x⃗_j), where δ_D is the Dirac delta function and w_j represents the initial weight of each tracer, which could be the selection function in observation or the halo mass in simulation, and N_i is the number of halos in the ith sub-sample. Thus, the overdensity of the reconstructed fields δ_c is δ_c ≡n_c/n̅_c - 1 = ∑_i^N_sW_in̅_i(1+δ_i)/∑_i^N_sW_in̅_i - 1 =ω^Tδ,where ω and δ are N_s-vectors, with elements ω_i = W_in̅_i/∑ W_in̅_i and δ_i=n_i/n̅_i-1 respectively.Combining equations (<ref>) and (<ref>), we have:r_c^2 = ∑_ijω_iB_ijω_j/∑_ijω_iC_ijω_j =ω^TBω/ω^TCω,with the symmetric matrices B and C defined as{ B_ij = ⟨δ_iδ_m⟩⟨δ_jδ_m⟩/⟨δ_mδ_m⟩, C_ij = ⟨δ_iδ_j⟩. .To solve for the weights that maximize r_c^2, we set the derivative of r_c^2 to be 0 to obtain:Bω = r_c^2Cω.The above eigenvalue equation (<ref>) is a generalized eigen-problem that can be solved numerically. We use the linear algebra library LAPACK[<https://netlib.org/lapack>]https://netlib.org/lapack/explore-html/dc/dd2/group__double_o_t_h_e_reigen_ga4e4203d1260f4deffe7679ac49af4f10.html::dspgv() to do this. The square root of the maximum eigenvalue corresponds to the maximum correlation coefficient, and the corresponding normalized eigenvector represents the optimal weight ω. The key information needed for solving the above equation is the matrices of C and B. These can be measured from N-body simulations. In summary, our method for the reconstruction is to maximize the correlation coefficients between the halo field and the matter density field. To do that, we make a linear combination of the tracers; each is given a weight. We can solve for the weights ω that maximize the correlation coefficients. The above derivation follows closely that of <cit.> but using a different target function r^2. The results agree with each other. The derivation is general i.e., given any field of tracers, which can be dark matter haloes, galaxies, or HI field, and its underlying matter field δ_m, we can solve equation (<ref>) to find the optimal ω that maximizes r_c. Next, we will apply this method to halos in simulations, using information about their cosmic-web environment.We use the public MultiDark MDPL2 simulations <cit.> and their corresponding rockstar halo catalogues <cit.> for our analysis. The simulation is run with 3840^3 particles in a box of 1 h^-1Gpc following a flat ΛCDM model with Ω_m=0.307,Ω_b=0.048,h=0.678,n_s=0.96,σ_8=0.823. We use a sample of ∼2.37×10^6 halos with the minimal halo mass of 2.0× 10^12h^-1M_⊙. We first split the halos sample into equal-number bins according to their masses. We then further split each mass bin according to their cosmic-web environment– voids, sheets, filaments, and knots. We define the environment using the Hessian of the gravitational potential <cit.> smoothed at 1 h^-1Mpc. The three eigenvalues of the field (λ_1,2,3) are compared with a threshold value λ_th to classify each volume as voids (λ_1,2,3<λ_th), sheets (λ_1,2<λ_th< λ_3), filaments (λ_1<λ_th< λ_2,3), and knots (λ_th< λ_1,2,3) <cit.>. The threshold value of λ_th is a free parameter that will be commented later. With this, the halo sample is split into four environmental bins according to their host cosmic-web environment. So, the halo field has two dependencies, mass M and environment Env. The auto- and cross-correlations of δ_h(M, Env) form the matrix of C needed in equation (<ref>). We have four bins of the environment by definition. If we choose four mass bins, C will be a matrix of 16×16. Its correlation with the matter density field δ_m yields the matrix B. These will be inserted into equation (<ref>) to find the maximum r_c and the optimal weights ω.Figure <ref> presents comparisons of scatter plots between the dark matter density field versus the reconstructed halo field with different weights. We can see that the scatter is the largest when halos are weighted equally (panel A), followed by mass-weighting (panel B), and then the optimization with cosmic-web information (panel C). It is clear that by optimizing with the cosmic-web information, we can reduce the stochasticity to a level which is comparable with the mass weighting case. Panel D shows the result from optimizing with M and Env, which reduces the stochasticity further to achieve the lowest level of all. Therefore, the information from M and Env are complementary. Their combination is better than having each of them alone. It is also worth noting that the mean correlation relation between the halo field and the mass field, which is the deterministic component of the bias, also tends to be more linear for the optimal weighting cases i.e., having the smallest quadratic coefficient b_2 when fitted with a second-order polynomial function (b_2=-0.21, 0.37, 0.07, 0.07 for the four cases shown in the legend of the figure). This suggests that the second-order bias of halos is suppressed by the optimization – an unexpected but preferable outcome. This is consistent with the results report in <cit.>. The reduction of scatter in Figure <ref> can be translated into an increase of signal-to-noise for halo clustering. The upper panel of Figure <ref> compares the noise spectra for the above four cases. Note that the Poisson noise levels are different for different weighting schemes. We can see that except equal-weighting where ⟨ϵ^2⟩ is comparable to Poisson noise, all the other noise spectra are below their Poisson noise levels. By weighting with the cosmic-web environment (orange line), we are able to bring the noise level down below the case of mass weighting (green line). When we incorporate both M and Env for the optimization, the noise level is the lowest, approximately one order of magnitude below the Poisson level. This is ∼75% lower than using the information of M alone.The lower panel of Figure <ref> compares ⟨ϵ^2⟩/(b^2P_m) – the fractional errors on the power spectra, or the `noise-to-signal', for all the four cases. This is directly related to r_c through equation (<ref>). We can see again that environmental weighting alone is comparable to mass weighting. They are approximately three times lower than equal weighting at low-k's where the baryon acoustic oscillations are. Optimizing with both M and Env reduces the ratio further by another factor of ∼2-3. The optimal weighting is thus expected to increase the signal-to-noise ratio of halo clustering by the same amount in the regime where shot-noise dominates. The level of improvement for the reconstruction by having the information about the environment depends on the number of halo mass bins M_bin and the threshold value λ_th for the cosmic-web classification. We discuss these two variables as follows.For a fixed M_bin, r_c varieswith λ_th, as shown in orange and red lines in Figure <ref>. There is an optimal λ_th at which r_c is maximized. We find that the optimal λ_th is typically larger than zero. This makes both the volume fraction for voids and the number of halos classified as void-halos to be the largest, followed by sheets, filaments, and knots(see also figures in Supplementary Material and <cit.>).We have optimized λ_th for results in the paper unless specified. We present the best-fit functions for the optimal λ_th versus M_bin and the corresponding r_c versus M_bin in Supplementary Material. With the optimal λ_th, r_c increases with increasing M_bin (horizontal lines in Figure <ref>). When M_bin is large, each individual halo is effectively in a single bin. All properties of halos are used in the optimization, leaving no room for further improvement. In the other extreme case where no information about the halo mass is known, i.e., M_bin=1, the improvement for the reconstruction with the cosmic-web information is maximized. This is illustrated in Figure <ref> by the length of the blue vertical line in terms of the correlation coefficient r_c. We can see that equal-weighting provides the lowest r_c (gray dashed line, labeled as N); using the cosmic-web information alone (orange line, labeled as N+Env) significantly improves r_c, and slightly outperforms the mass weighting case (labeled as M); having two or more mass bin for the optimization increases r_c further. r_c=0.99 is achieved by having one mass bin and the environment information (M+Env, red-solid line). This is approximately the same as having 16 mass bins (M-16) for optimization. Note that having many mass bins is unrealistic, given the challenge of measuring the mass of halos in observations. The gain we can achieve using the cosmic-web information is, therefore, highly complementary. It is worth noting that when optimizing with the mass information, we assume that the mass of each individual halo is known. This is unrealistic in practice. We anticipate large uncertainties for estimating halo mass in real observations. Therefore, the benefit of using environmental weighting may be more prominent than we have shown. In summary, we have found significant improvement for the correlation between halos and their underlying mass field using information about the cosmic-web environment. The level of improvement for the reconstruction is at least as good as optimizing using halo mass. Environment and halo mass complement each other. Their combination yields the best reconstruction than having each of them individually. Our findings have direct observational implications.Reducing stochasticity, or noise, for the galaxy/halo population will directly increase the signal-to-noise for galaxy clustering. This is crucial for precise measurement of the BAO, especially at high redshifts where the tracers are sparse <cit.>. Galaxy redshift surveys such as BOSS <cit.>, eBOSS <cit.> and DESI <cit.>, are designed to target at a specific type of galaxy, such as LRG, galaxies with approximately constant stellar mass, and emission line galaxies, but it is challenging to know the mass of individual halos. This limits the potential of shot-noise reduction using the halo mass information. In contrast, the cosmic-web information is more readily available from computing the Hessian matrix using the galaxy samples themselves. A caveat is that the smoothing scale will be limited by the number density of galaxies, but we have tested using a relatively large smoothing scale (R=5h^-1Mpc) to define the cosmic web, and there is still an obvious gain of information. An additional challenge is that the observed galaxies are in redshift space and so the defined cosmic web may be different from its real-space version. Further investigations are needed to fully realize the strength of the method in observations.Reducing stochasticity between galaxies/halos versus the matter field also has direct benefits for forward modeling e.g. Borg <cit.>, constrained simulations <cit.>, and cosmological inference at the field level using observations of the large-scale structure <cit.>. The common starting point for the above analyses is the observed galaxy number density field δ_g, one then try to find the initial matter density field δ_m^i which, after non-linear evolution, may generate the observed δ_g. Reducing stochasticity for δ_g will directly reduce the errors between the observational constraints with their initial conditions. Our method opens a promising path for achieving this.We are aware that the environment is not the only secondary property of halos that can be useful for reconstruction. We have explored all other halo properties typically defined in N-body simulations, finding that information from the tidal force and halo concentration can also help to improve the reconstruction. This is expected as these halo properties are correlated and are all related to the assembly bias of halos<cit.>. § ACKNOWLEDGEMENTSThe CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark database was developed in cooperation with the Spanish MultiDark Consolider Project CSD2009-00064. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (<www.gauss-centre.eu>) and the Partnership for Advanced Supercomputing in Europe (PRACE, <www.prace-ri.eu>) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, <www.lrz.de>). FLL is supported by the National Key R&D Program of China through grant 2020YFC2201400 and the Key Program of NFSC through grant 11733010 and 11333008. YC acknowledges the support of the Royal Society through a University Research Fellowship. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.§ SUPPLEMENTARY MATERIAL §.§ Understanding the optimal weights To understand the physics for the optimal weights, we turn to look at their two dependants, halo mass and cosmic-web environment, (M, Env). (i) On the axis M: we can see from the lower panels of Figure <ref> that high-mass halos are given more weights. This is consistent with what was found in <cit.>, and the trend is the same among all four different cosmic-web environments. This is expected as in the extreme case where all matter collapses into halos, mass weighting must be the optimal. In practice, we have a low-mass cutoff, therefore, the optimal weight function versus halo mass has a shallower slope than the mass weighting case – more weight is given to the low-mass halos to account for the missing mass below the cutoff mass. The general trend is that it is the optimal to give massive halos more weight. Given that the linear bias of halos b(M) increases with halo mass and the slope of b(M) is steeper for the massive halos i.e., the most massive halos have the highest amplitude of clustering and the highest signal-to-noise in the Poisson model. They are indeed the most `valuable' for the reconstruction. (ii) On the axis of Env: for halos in a fixed mass bin, we find that void-halos is given significantly more weights, followed by sheets, filaments, and knots, but the weights correlated strongly with the number fraction of halos in these four cosmic-web environments (top-left panel of Figure <ref>). This indicates that the weights are driven primarily by the number of halos in each bin (top-middle panel of Figure <ref>). Intuitively, we do expect that the more halos in the environmental bin, the more information about the underlying matter density field that bin may contain, and therefore more weight should be given to the bin that has the most numerous halos. However, a further question is, does the weight per halo follow the same trend i.e., is a void-halo weighted more significantly than a knot-halo? To test this, we take the ratio between the weights and the number of halos in each environmental bin, shown in the top-right panel of Figure <ref>. We find that the picture is mixed: For the low-mass bins, each void-halo is indeed weighted more, followed by sheets, filaments, and knots; This is consistent with our expectation that low-mass halos in a void environment may have the largest dynamical range of formation time, hence a stronger assembly bias. This makes void-halos more `valuable' for the reconstruction, and thus their weight is larger. For the highest mass bin, however, the trend is of the opposite, with each knot-halo being weighted most. We think that this is the combined consequence of the wider dynamical range in halo mass in that bin, and that knot-halos are the most massive ones. As we recall from (i), along the axis of halo mass, it is preferable to up-weight massive halos as they are more `valuable'. Also, massive halos typically form in the late-time Universe, with relatively few of them in voids, and most of them are in knots, and may not sample very well other cosmic-web environments – a relatively narrow range of cosmic-web environment. Both these factors make it preferable to up-weight knot-halos.To further illustrate the above points, we made a two mass-bin case presented in Figure <ref>. We can see that for the low-mass bin, the range of halo mass is much narrower,further splitting it into finer mass bins does not help for the reconstruction (bear in mind that the function of b(M) is also relatively flat in this mass range). Therefore, adding the environmental dependence significantly improves the cross-correlation coefficients. For the high-mass bin, the range of mass is a lot wider (also b(M) is a relatively steep function of M in this mass range). Therefore, splitting the sample in mass helps to improve the reconstruction more significantly, and the improvement with the additional split in the environment is relatively minor.In summary, when considering the weight of each halo, mass and environment can be seen as two competing factors. For high-mass halos, due to the steep relationship of b(M), and the narrower range in formation time, mass-weighting is winning over the environmental factor; for low-mass halos, the opposite is true. With an almost flat function of b(M), and a wider range in formation time (approximate as a wider range of cosmic-web environment being sampled), it is preferable to give halos in low-density (voids) environments more weight.apsrev4-2 | http://arxiv.org/abs/2311.15903v1 | {
"authors": [
"Feng Fang",
"Yan-Chuan Cai",
"Zhuoyang Li",
"Shiyu Yue",
"Weishan Zhu",
"Longlong Feng"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20231127150555",
"title": "Mass reconstruction and noise reduction with cosmic-web environments"
} |
[email protected] Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, FranceAndreev reflection is a fundamental transport process occurring at the junction between a normal metal and a superconductor (a N-S junction), when an incident electron from the normal side can only be transmitted in the superconductor as a Cooper pair, with the reflection of a hole in the normal metal.As a consequence of the spin singlet nature of the BCS Cooper pairs, the current due to Andreev reflection at a N-S junction is always symmetric in spin.Using a Keldysh Nambu Floquet approach, combining analytical and numerical calculations, we study in details the AC transport at a N-S junction, when the two spin components in the normal metal are driven by different periodic drives. We show that, in the Andreev regime, i.e. when the superconducting gap is much larger than the frequency of the drives, the spin-resolved photo-assisted currents are always equal even if the two drives are different. In addition, we show that in this regime the excess noise depends only on the sum of the periodic drives, and we consider in particular the case of Lorentzian pulses (Levitons). We also show how these properties get modified when going beyond the Andreev regime. Finally we give a simple analytical proof of the special properties of the Andreev regime using an exact mapping to a particular N-N junction.Current and shot noise in a spin dependent driven normal metal - BCS superconductor junction. Thierry Martin=============================================================================================§ INTRODUCTION Electron quantum optics (EQO) aims at describing and manipulating single electronic excitations in condensed matter systems <cit.>.This is achieved by adapting scenarios of quantum optics like the Hanbury-Brown and Twiss experiment <cit.> where the intensity correlations from coherent photons at the output are observed or the Hong-Ou Mandel setup, <cit.> where photons collide at the location of the beam splitter and correlations are measured at the output.In condensed matter settings, electron wave guides can be achieved with a two dimensional electron gas, while a quantum point contact mimics the beam splitter.However, electrons differ from photons as they are charged particles and bear fermionic statistics.This means, in particular, that they interact strongly with their neighboring electromagnetic environment and are always accompanied by a Fermi sea. In recent decades, the combination of theoretical <cit.> and experimental <cit.> efforts, boosted by advances in fabrication techniques, has provided EQO with a strong foothold.Even if EQO has been initially studied in situations when the role of electronic interactions is neglected or minimized <cit.>, it is nowadays also studied in strongly correlated systems such as the Fractional Quantum Hall effect <cit.> and hybrid superconducting devices <cit.>.In particular EQO has flourished due to the availability of single electron sources working in an AC regime <cit.>, such as the mesoscopic capacitor <cit.>, or voltage tailored trains of Lorentzian wave packetscalled “Levitons”,<cit.>. These Levitons consist of “pure” single electron excitations <cit.>, i.e., devoid of unwanted electron-hole pairs.For instance, when a combination of AC and DC bias is applied to a device, the measurement of the output excess noise <cit.> (with respect to the proper reference situation with only an applied DC bias) allows the detection of these spurious electron-hole excitations. Connecting electron waveguides to superconducting leads opened the way to new EQO effects, such as electron (respectively hole) conversion into Bogoliubov quasiparticles <cit.> above (respectively below) the gap or Andreev reflection <cit.> (AR) of electrons or holes inside the gap <cit.>. This normal-superconducting junction was discussed earlier by Belzig et al. <cit.>, where they considered the zero temperature limit and focused on the two limiting regimes where the drive frequency is either much larger or much smaller than the gap of the superconductor.In the latter one, where transport is dominated by Andreev reflection, they found excess noise suppression also for Levitons carrying a half-integer charge.Recently, the intermediate regime has been explored using a microscopic model together with Green's functions in the Keldysh formalism, allowing the computation of the average current as well as the period-averaged noise to all orders in the tunneling constant and at finite temperature <cit.>. In this article, we extend the preceding setup to a “conceptual” situation where spin components are independently driven by periodic sequences of pulses (having the same frequency), which allows us toshine a new light on the underlying processes leading to a vanishingexcess noise in the Andreev regime.More precisely, computing the spin-resolved currents and the excess noise through the normal-superconducting (N-S) junction as functions of time, we emphasize that the properties of the junction actually depend only on the total drive. For instance, in contrast to anormal-normal (N-N) junction, we find that, independently of the drives imbalance, both spin resolved currents have always the same values,which is a manifestation of the spin symmetry of the Cooper pairs in the superconductor. This also implies that the total excess noise vanishes as long as the spin-dependent drives actually combine to a proper sequence of Lorentzian pulses, where each pulse injects an integer charge per period.Roughly speaking, it means that the excess noise vanishes as long as the charges injected by each species dependent train of Levitons add to an integer, extending the half-integer charge Levitons situation obtained for balanced drives.A more quantitative analysis of the excess noise as a function of the properties of the two drives for Levitons (injected charges and time delay) allows us to emphasize more precisely the conditions for having a vanishing excess noise.Even though our main goal is to provide a better understanding of the underlying physics at the N-S junction in the Andreev regime, we would like to mention that, this model could be experimentally achieved by using a quantum spin Hall bar <cit.> or two half-metals, whose separation needs to be smaller than the superconducting coherence length <cit.>. The latter proposition would constitute a new type of Cooper pair beam splitter <cit.>, where typically crossed Andreev reflection <cit.> plays a fundamental role. The paper is organized as follows.In Sec. <ref> we introduce the theoretical framework for tunneling through the junction in the presence of two periodic pulses with the same period, but driving independently each spin components.In Sec. <ref>, we numerically and analytically study the spin-resolved currents and the excess noise in the Andreev regime.We also compute an effective Dyson equation describing the underlying physics as a metal-metal junction, providing an interpretation of our results.We conclude in Sec. <ref>. Some additional technical aspects are presented in the Appendices.We adopt units in which ħ=k_B=1 and the electronic charge is e<0. The temperature of the system corresponds to Θ and β denotes the inverse temperature, i.e β^-1=k_BΘ. § MODEL We adopt a standard approachdeveloped for junctions involving superconductors in which the BdG equations are discretized <cit.>. The metallic and superconducting leads are described at equilibrium by tight binding Hamiltonians H_N and H_S; H_N simply corresponds to the kinetic term, i.e. amounting to the electrons hopping between neighboring sites of a single 1D chain. For the superconducting lead, H_S, in addition to the kinetic term, also includes a pairing term, which at the meanfield level reads:Δ∑_i(c_i ↓^† c_i↑^† +c_i↓c_i↑),wherei labels the various sites of these leads, Δ is the superconducting order parameter and c_iσis the electron annihilation operator at a given site i with spin σ.One defines the Nambu spinors at each boundary of the tunneling junction between the two leads:ψ_N^†=(c_N,↑^†,c_N,↓)ψ_S^†=(c_S,↑^†, c_S,↓),such that the total Hamiltonian readsH=H_N+H_S+ψ_N^† W_NSψ_S+H.c.,where the tunnel matrix from the normallead to the superconducting lead readsW_NS(t)=λ[e^iϕ_↑(t)0;0 -e^-i ϕ_↓(t) ].λ is the tunneling amplitude for both spin species.The functions ϕ_σ(t) = e∫_-∞^t dt'V_σ(t') are the time-dependent phase differences between the leads accounting for the spin-dependent drives V_σ(t) applied on the metallic lead. This situation could be realized by having a metallic lead actually corresponding to two different leads made of spin-polarized half-metals, allowing us, thereby, to drive each spin component independently.Conceptually, it amounts to having two separate leads for the normal metal side, see Fig. <ref>.Themoments of the current operator are computed within the framework of Keldysh theory <cit.>, defining thereby time ordered Green's functions on the Keldysh contour, such as,G^+-_jj'(t,t')=i⟨ψ_j'^†(t')⊗ψ_j(t)⟩,where G_jj'^+- is the full Green's function dressed by the tunneling Hamiltonian,j and j' are lead indices. For instance, the total current flowing in the metallic lead at the junction reads as a Nambu trace:⟨ I_N(t)⟩=eTr_N[σ_zW_NS(t)G^+-_SN(t,t)],wheredenotes the real part. In addition, one can compute the spin-resolved currents ⟨ I_Nσ(t)⟩ as follows:⟨ I_Nσ(t)⟩=eTr_N[(+σσ_z)/2W_NS(t)G^+-_SN(t,t)],i.e. corresponding to the two diagonal elements of the 2×2 matrixσ_z W_NS(t)G^+-_SN(t,t). Similar expressions for the real time zero frequency noise correlator can be derived and are given in App. <ref> For periodic drives, the voltage drives split in their DC and AC parts:V_σ(t) = V_DCσ + V_ACσ(t),where V_DCσ is time-independent, and V_ACσ(t) averages to zero over one period T=2π / Ω of the periodic drive. The injected charge per spin per period isq_σ=eV_DCσ/Ω.In practice, the DC components of the drives are actually fully taken into account by shifting the Fermi energy of each spin component of the normal metal by -eV_DCσ, such that one is left dealing only with the AC part of the drives. Using the periodicity of the drives, we introduce the Fourier componentsp_lσ of the functions e^-iϕ_σ, namely:e^-iϕ_σ(t)=∑_l p_lσ(q_σ)e^-ilΩ t.By doing so, we use Floquet theory <cit.> in which the total Hamiltonian is separated into an infinite number of independent harmonics in Fourier space.The Floquet theory goes beyond this simple Fourier decomposition.Indeed, it states that as a consequence of the AC drive, the electrons can gain or lose energy quanta leading to the formation of side bands.The Floquet weight P_lσ=|p_lσ|^2 therefore corresponds to the probability for an incoming electron with spin σ to absorb l photons of energy Ω.The voltage biased lead is then better described as a Floquet state, a superposition of Fermi seas, which we now refer to as “Floquet channels”, with shifted chemical potential μ_σ→μ_σ -eV_DCσ + lΩ and an intensity given by the corresponding Floquet weight P_lσ.As explained in details in Ref <cit.>, one can write the Keldysh dressed Green's function asG_SN^+-= (1-g_S^rΣ_SN^rg_N^rΣ_NS^r)^-1×[g_S^+-+ g_S^rΣ_SN^rg_N^+-(Σ_SN^ag_N^a)^-1]×(1-Σ_SN^ag_N^aΣ_NS^ag_S^a)^-1Σ_SN^ag_N^a,where, within the Floquet formalism, the different quantities become infinite matrices in the Nambu-harmonic space. For instance, the self-energy Σ^r,a readsΣ_SN,nm^r,a(ω)=λ[ p_n-m,↑ 0; 0 -p_m-n, ↓^* ]and, assumingthe large bandwidth limit for the leads, the bare Green's function in the superconductor readsg_S,nm^r,a(ω)=-lim_δ→0Δσ_x+(ω+nΩ)/√(Δ^2 - (ω + nΩ± iδ)^2)δ_nm,and, in the metal,g_N,nm^r,a=∓ iδ_nm.The other bare Green's functions, such as g_S^+-(ω) and g_N^+-(ω) are given in appendix <ref>, together with a full expression of the Keldysh dressed Green's functionG_SN,nm^+-(ω).§ ANDREEV REGIME In the Andreev regime Δ≫Ω, the bare Green's functions in the superconductor simplify tog_S,nm^r,a(ω)=-σ_xδ_nm and lim_Δ→∞g_S,nm^±,∓(ω)= 0,such that the Keldysh dressed Green's function Eq. (<ref>) formally simplifies toG_SN^+-=λ^3/(1+λ^4)^2[ σ_x𝒫T𝒫^†σ_x𝒫 -σ_x 𝒫𝒫^†σ_x𝒫T],where we have defined Σ=λ𝒫 and the matrix T describes the thermal distribution of the electrons in the normal lead, see appendix <ref>.Note, that one could be tempted to simplify the second term using the fact that σ_x𝒫𝒫^†σ_x=. However, this expression would then become ill-defined when computing the system properties: since each term taken separately leads to divergent sums, convergence is obtained only after carefully grouping and rearranging terms, which, eventually would lead to unphysical results, more precisely losing the time dependence of the observables, such as the junction currents. §.§ Current As explained above the spin-resolved currents are obtained from W_NS(t)G^+-_SN(t,t) which, within the Floquet theory reads:⟨ I_σ(t)⟩=e/2π∑_n,m∫_-Ω/2^+Ω/2 dω e^-i(n-m)Ω t(ℐ^σ_nm(ω)+ℐ^σ*_mn(ω))withℐ^σ_nm(ω)=σ∑_k[Σ_NS,nk(ω)G^+-_SN,km(ω)]_σσ,such that, in the Andreev regime, one obtains:⟨ I_σ(t)⟩= e/2πλ^4/(1+λ^4)^2×∑_n,m∫_-Ω/2^+Ω/2 dω e^-i(n-m)Ω t(𝒬^σ_nm(ω)+𝒬^σ*_mn(ω))with𝒬^σ_nm(ω)=σ[ 𝒫σ_x𝒫^†T𝒫σ_x𝒫^† -𝒫σ_x𝒫^†𝒫σ_x𝒫^†T ]_nσ,mσ.Inserting the expressions for 𝒫 and 𝒯, one obtains that 𝒬^↑_nm(ω)= ∑_r,k,s p^*_k-n,↑p^*_s-k,↓p_r-s,↓p_r-m,↑[tanh(ω+sΩ+eV_DC,↓/2Θ) -tanh(ω+mΩ-eV_DC,↑/2Θ)]and a similar expression for 𝒬^↓_nm(ω), seeApp. <ref>.Note that it is crucial to start from the proper expression (<ref>) for the Green's function to finally get the difference between the two tanh functions, which ensures properly converging sums and integrals.Indeed, performing the following change of variables, x = ω + (s+q_↓)Ω and shifting all the indices by s (except s itself), the sum over s can be carried out, yielding:⟨ I_↑(t)⟩ = e/πλ^4/(1+λ^4)^2∑_n,r,k,m∫_-∞^∞dxe^i(m-n)Ω t p^*_k-n,↑p^*_s-k,↓p_r-s,↓p_r-m,↑[ tanh(x/2Θ)- tanh(x+mΩ-(q_↑+q_↓)/2Θ) ].The integral can easily be performed and the preceding expressionbecomes⟨ I_↑(t)⟩ = 2e/πλ^4/(1+λ^4)^2∑_n,r,k,m e^i(m-n)Ω tp^*_k-n,↑p^*_-k,↓p_-r,↓p_r-m,↑(q_↑+q_↓-m),which using the definition of the p_lσ leads to ⟨ I_↑(t) ⟩=e^2/2πτ_A (V_↑(t)+V_↓(t)),where τ_A=4λ^4/(1+λ^4)^2 is the so-called Andreev transmission and does not depend on the temperature. Obviously, the expression for ⟨ I_↓(t) ⟩ is the same, such that both currents are always equal and proportional to the sum of the applied drives, recovering the fact that the N-S junction, in the Andreev regime, depicts a fully linear behavior. Furthermore, in the Andreev regime, since only processes involving pair creation/annihilation in the superconductors are taking place, it implies that the number of transmitted electrons with different spins must always be equal, and thereby that ⟨ I_↑(t) ⟩=⟨ I_↓(t) ⟩. This has to be contrasted with the metallic regime, i.e. Δ=0, where one obtains (see appendix <ref> for details)⟨ I_σ(t) ⟩=2e^2/πτ V_σ(t),with the (normal) transmission τ=4λ^2/(1+λ^2)^2. As expected, each spin current is simply proportional to the respective drive. These properties can be seen in Fig. <ref>, which displays I_σ(t)as a function of time for a vanishing V_↓=0 and V_↑(t) corresponding to a train of Levitons of charge q_↑=1.In the Andreev regime Ω≪Δ (bottom right plot), one can see that both currents areequal and proportional to V_↑(t), corresponding to Eq. (<ref>). Close to the Andreev regime, Ω=0.1Δ (bottom left plot), both I_↑ and I_↓ depart from their Andreev value, but, as expected, the impact of the quasi-particles excitations, directly driven by V_↑, is stronger on I_↑. For increasing values of Ω, reaching the intermediate regime Ω≈Δ (top right plot),I_↑(t) decreases whereas I_↓(t) increases, both displaying additional oscillations compared to V_↑, emphasizing the non-linear behavior of the N-S junction <cit.>.For Ω≫Δ (top left plot), corresponding to a normal-normal junction,I_↓(t) vanishes(of the order of (Δ/Ω)^2) , whereas I_↑(t) becomes again proportional to V_↑(t), see Eq. (<ref>). §.§ Excess noiseFinally, along similar lines, one can compute the zero-frequency noise averaged over a period, see Appendix <ref>: ⟨ S⟩=e^2/π[4τ^2_AΘ+ 2τ_A(1-τ_A)∑_s(s+q_↑+q_↓)Ω |p^A_s|^2(Ω(s+q_↑+q_↓)/2Θ)], where p^A_s=∑_n p_r,↓p_s-r,↑. Using the definition of the coefficients p_lσ, we obtain p^A_s=1/T∫_T/2^T/2dte^-isΩ te^i[ϕ_↑(t)+ϕ_↓(t)],which is therefore the Fourier component of e^-iϕ_tot(t), i.e. of theAC part of the total drive V_AC↑(t)+V_AC↓(t). Therefore, the formula above for ⟨ S⟩ is the same as the one for zero-frequency noise averaged over a period for an effective normal-normal junction,⟨ S^N⟩_q=e^2/π[4τ^2Θ+2τ(1-τ) ×∑_n(eV_DC+nΩ)|p_n|^2(eV_DC+nΩ/2Θ)] ,driven by V(t)=V_↑(t)+V_↓(t), i.e. the total drive applied to the NS junction.This emphasizes that, not only for the current, but also for the noise, the behavior of the junction only depends on the total drive.This has an important consequence for the excess noise,in particular when each drive corresponds to a train of Levitons, i.e. a sequence of Lorentzian pulses defined as followsV_σ(t) = V^σ_0( 1/π∑_k η/η^2 + (t/T - k)^2 ).where η=W/T is the ratio between the width of the pulse W and the period of the drive T. In that particular situation, the total drive V(t) simply corresponds to a sequence of Levitons fully characterized by V_0^↑+V_0^↓, i.e. its total charge q=q_↑+q_↓, resulting therefore in a vanishing excess noise when q is an integer. This is a well known results when both drives are the same and each corresponding to a half-Leviton, i.e.q_↑=q_↓, but our computation shows that it extends to any situations whereq_↑+q_↓ is an integer.This is exemplified in Fig. <ref>, where one plots the excess noise, defined as S_exc = ⟨ S⟩- ⟨ S⟩_dc, in the(q_↑,q_↓) plane, for the N-S junction driven by periodic Lorentzian drives, q_σ corresponding to the injected charge for each spin component per period.⟨ S⟩_dc is the DC noise computed for the same injected charge. As predicted by Eq. (<ref>), in the Andreev limit Ω≪Δ, the excess noise vanishes along lines corresponding toq_↑+q_↓∈ℤ. As explained in <cit.>, for intermediate regimes, the excess noise does not vanish anymore, and, for Ω≫Δ, i.e. in the normal-normal regime of the junction, the excess noise only vanishes when both q_σ are integers. In addition, we would like to emphasize that,for the excess noise to vanish in the Andreev regime, our calculations show that the full shape of the total drive V(t) has to correspond to a sequence of Levitons with an integer charge, which is actually a stronger constraint than just having an integer total injected charge.In the preceding example, where each drive corresponds to Lorentzian pulses centered at the same times t_k=kT, this condition was fulfilled as soon as q_↑+q_↓ is an integer. On the other hand, if we consider, for instance, the situation where each drive corresponds to the same sequence of Levitons with a (fractional) charge q_σ, but being shifted in time one with respect to the other, i.e.V_↓(t)=V_↑(t+δ t), then, the total drive V(t)is simply a periodic sequence made of two Levitons per period, each one having a charge q_σ, which, unless δ t=0, results in a finite excess noise.This is emphasized in Fig. <ref>, where the excess noise is plotted as function of q_σ for different value of δ t. As one can readily see, for half-integer q_σ, the excess noise vanishes only when δ t=0. For integer q_σ, since each drive V_σ alone will result in a vanishing excess noise independently from the other drive, the total excess noise vanishes for all values of the delay δ t . From a physics point of view, it emphasizes the difference between half-integer drives and integer ones: For half-integer drives, i.e. q_σ=1/2 for instance, the Andreev reflection of each spin component produces a “half-pair” in the superconductor; each of these “half-pair” must then be produced “at the same time” to allow for a whole pair to be transmitted in the superconductor. This reasoning goes beyond the half-integer case, and generalizes to any combination (q_↑, q_↓) satisfying q_↑ + q_↓ = 1 <cit.>. On the other hand, for integer drives, the Andreev reflection of each spin component produces a whole pair, independently of the Andreev reflection of the other spin component, allowing for a noiseless current in the junction.§.§ Equivalence with an effective metal-metal junction The fact that both the current and the noise only depends on the sum of the drives can be directly inferredfrom the Dyson equation in the time domain, which reads formallyG(t,t')=g(t-t')+∬dt_1 dt_2 g(t-t_1)W(t_1,t_2)G(t_2.t'),where g is the bare Green's function. Every Green's function has the following block structureG=[[ G_NN^η,η' G_NS^η,η'; G_SN^η,η' G_SS^η,η' ]].each G_ij^η,η' is a 4×4 matrix corresponding to Nambu plus Keldysh dimension. The matrix W readsW(t_1,t_2)=δ(t_1-t_2)[[[W_NS 0; 0 -W_NS ]; [W_SN 0; 0 -W_SN ]]],where W_NS is the 2×2 tunneling amplitude matrix in Nambu space, see Eq. (<ref>).Iterating once the Dyson equation, one obtains:G(t,t') =g(t-t')+g(t-t_1)W(t_1,t_2)g(t_2-t')+g(t-t_1)W(t_1,t_2)g(t_2-t_3)W(t_3,t_4)G(t_4,t'),where integration over the intermediate times is implicit.In the Andreev regime, the superconductor bare Green's function simply reads:g^ϵ,ϵ'_S(τ)=δ(τ)[[ -σ_x0;0 0σ_x ]],where each 2×2 sub-blocks are in Nambu space and combined together in the Keldysh space. Thereby, one obtains the following effective Dyson equation for the normal metal dressed Green's functionG_NN:G^ϵ,ϵ'_NN(t,t')=g^ϵ,ϵ'_N(t-t')+g^ϵ,ϵ'_N(t-t_1)W̃(t_1)G^ϵ,ϵ'_NN(t_1,t'),where W̃(t)= [[W_NS 0; 0 -W_NS ]] [[ -σ_x0;0 0σ_x ]][[W_SN 0; 0 -W_SN ]] = [[W̃_NN0;0 -W̃_NN ]],withW̃_NN(t) =W_NSσ_xW_SN=-λ^2 [[ 0e^i(ϕ_↑(t)+ϕ_↓(t)); e^-i(ϕ_↑(t)+ϕ_↓(t)) 0 ]].As one can see, the effective Dyson equation describes a metal-metal junction where one metal is made of spin up and the other of spin down, the effective drive being the sum of the original drives. More precisely, because of the Nambu description of the system, the up spins are electrons, where the down spins correspond to holes. From that point of view, if the sum of the effective drive consist of a train of Levitons of total charge q=q_↑+q_↓, it amounts to converting q electronic charge with spin up to q positive charges with spin down, i.e. removing q electronic charges with down spin as well, corresponding, as expected, to having created q pairs in the superconductor, for a total charge transfer through the junction equal to 2q. Similar results have been obtained recently when computing the final state in N-S junction driven, in the Andreev regime, by a single Lorentzian pulse <cit.>. Finally, we would like to mention that the present theory predicts that applying opposite drives, i.e. such that V_↑(t)=-V_↓(t) results, in the Andreev regime, to vanishing quantities such as currents, excess noise... Therefore, in this configuration, one could have a direct and precise probe of the impact of the finite superconducting gap on the N-S junction properties, close to the Andreev regime. § CONCLUSIONWe have shown from both a numerical and an analytical point of view that, in the Andreev regime, a N-S junction behaves as a normal metal driven by the sum of the drives applied to the junction. More precisely, we have shown that the spin up and spin down currents have always the same value, proportional to the total drive V_tot. Similarly, the excess noise only depends on V_tot and vanishes as long as the total drive amounts to a Lorentzian train of pulses injecting an integer number of charges per period. These results are simply explained by mapping the N-S junction to an effective N-N junction driven by V_tot.The physical origin of this behavior can be traced back to the spin symmetry of the Cooper pairs in the superconductor. In the Andreev regime, they are the only excitations available for the transport, which enforces an equal amount of spins up and spins down flowing through the N-S junction, even if different spin-dependent drives were applied.In addition, we would like to stressthat a possible experimental realization of this system could be achieved using spin polarized half-metals for the normal leads <cit.>, in particular allowing us to study the impact of the time delay between the drives. Finally, these transport properties could be studied using cold atomic gases trapped in optical lattices. In these systems, one could prepare, for instance, an initial wavepacket made of a given species and measure the time-evolution at the boundary between the normal side and the (strongly) paired superconducting side, studying thereby the time-resolved Andreev reflection <cit.>. Furthermore, going beyond the standard fermionic case, Andreev-like reflection can be achieved using Bogoliubov excitations on top of a Bose-Einstein condensate <cit.>.This work received support from the French government under the France 2030 investment plan, as part of the Initiative d'Excellence d'Aix-Marseille Université A*MIDEX. We acknowledge support from the institutes IPhU (AMX-19-IET008) and AMUtech (AMX-19-IET-01X).§ DYSON'S EQUATION FOR KELDYSH GREEN'S FUNCTIONS. In this section, we summarize the results derived in <cit.>. §.§ DefinitionsDefiningω_n^±=lim_δ→0ω+nΩ/√(Δ^2 - (ω + nΩ± iδ)^2) Δ_n^±=lim_δ→0Δ/√(Δ^2 - (ω + nΩ± iδ)^2) ξ_n^±=1/1-λ^4((ω_n^±)^2-(Δ_n^±)^2)∓2iλ^2ω_n^± η_n^±=ω_n^±∓ iλ^2((ω_n^±)^2-(Δ_n^±)^2) Δ_n = Δ_n^–Δ_n^+/2 ω_n= ω_n^- - ω_n^+/2 ĝ_S,nm^r,a=-(Δ_n^±σ̂_x+ω_n^±1̂)δ_nm ĝ_N,nm^r,a=∓ i1̂δ_nm ĝ_S,nm^±∓=(ω_n1̂+Δ_nσ̂_x)(tanh(ω+nΩ)∓1)δ_nm ĝ_N,nm^±∓=-i(T̂∓1̂)δ_nm Σ̂_SN,nm^r,a=λ[ p_n-m,↑ 0; 0 -p_m-n, ↓^* ] T̂_nm=[ tanh(ω+nΩ-eV_DC↑/2Θ)0;0 tanh(ω + nΩ+eV_DC↓/2Θ) ]δ_nm.the Keldysh dressed Green's function reads:G_SN,nm^±∓=iλξ_n^+ξ_r^-[ σ_x𝒫_nk T_k𝒫_kr^†𝒫_rm[Δ_n^++iλ^2Δ_n^+ω_r^-]+σ_x𝒫_nk T_k𝒫_kr^†σ_x𝒫_rm[-iλ^2Δ_n^+Δ_r^-]+𝒫_nk T_k𝒫_kr^†𝒫_rm[ω_n^++iλ^2+iλ^2ω_n^+ω_r^–λ^4ω_r^-]+𝒫_nkT_k𝒫_kr^†σ_x𝒫_rm[-iλ^2Δ_r^-ω_n^++λ^4Δ_r^-]+𝒫_nm{ζ_n^±[ω_n + iλ^2(1+iλ^2ω_n^-)(Δ_nΔ_n^+ - ω_n^+ω_n) +iλ^2(ω_n^-ω_n - Δ_nΔ_n^-)] ±[ - iλ^2 +iλ^2(Δ_n^+Δ_n^- - ω_n^+ω_n^-) - ω_n^+ + λ^4ω_n^- ]}+σ_x𝒫_nm{ζ_n^±[Δ_n +λ^4Δ_n^- (Δ_nΔ_n^+ - ω_n^+ω_n) ] ∓(Δ_n^+ + λ^4Δ_r^- )}],where we introduced T_n such that T_nm = T_n δ_nm andζ_n^±=tanh(ω+nΩ/2Θ)∓1). It should be mentioned that, obtaining this equation, an error in the equivalent formula of <cit.> has been corrected.§ COMPUTATION OF THE CURRENT AS A FUNCTION OF TIME The spin-resolved currents in the junction are defined as:⟨ I_σ(t)⟩ = e ∫_-∞^∞dt' [σ_z W_NS(t)δ(t-t') G_SN(t',t) - σ_z G_SN(t,t')W_NS(t)δ(t-t') ]_σσ=e/2π∑_n,k,m∫_-Ω/2^Ω/2dω e^-i(n-m)Ω t[σ_z Σ_NS, nk(ω)G_SN,km^+-(ω) - σ_z G_NS,nk^+-(ω)Σ_SN, km(ω) ]_σσ.Using the following relations,G_NS^±∓=-(G_SN^±∓)^†andΣ_NS^†=Σ_SN ,the currents can be written as⟨ I_σ(t)⟩ =e/2π∑_n,k,m∫_-Ω/2^Ω/2dω e^-i(n-m)Ω t{[σ_z Σ_NS, nk(ω)G_SN,km^+-(ω)]_σσ + [(n,m)→(m,n)]^*},where the second term is the complex conjugate of the first one after taking m→ n and n→ m. Since the terms in G^+-, see Eq. (<ref>), that contains an odd number of σ_x do not contribute to the diagonal matrix elements of σ_zΣ G^+-,one is left with⟨ I_σ(t)⟩ = e/2πλ^2∑_n,m,r,k,s∫_-Ω/2^Ω/2dω e^-i(n-m)Ω t{iξ_k^+ξ_r^- σ_z 𝒫_nk^†[σ_x𝒫_ksT_s𝒫_sr ^†σ_x𝒫_rm[-iλ^2Δ_k^+Δ_r^-] +𝒫_ks T_s𝒫_sr^†𝒫_rm[ω_k^++iλ^2+iλ^2ω_k^+ω_r^–λ^4ω_r^-]+δ_srδ_rm𝒫_km(ζ_k^+[ω_k +iλ^2(ω_k^-ω_k - Δ_kΔ_k^-) + iλ^2(1+iλ^2ω_k^-)(Δ_kΔ_k^+ - ω_k^+ω_k)] +[ - iλ^2 +iλ^2(Δ_k^+Δ_k^- - ω_k^+ω_k^-) - ω_k^+ + λ^4ω_k^- ])] + [(n,m)→(m,n)]^*}_σσ.§.§ Zero gap limit.In this regime one hasω_n^±=± i,ω_n=-i andΔ^±=Δ_n=0,such that the average current simply reads⟨ I_σ(t)⟩ = e/πλ^2/(1+λ^2)^2∑_n,m,r,s,k∫_-Ω/2^Ω/2dω e^i(m-n)Ω t[σ_z 𝒫_nk^†𝒫_kmδ_srδ_rmtanh(ω+kΩ/2Θ) -σ_z 𝒫_nk^†𝒫_knT_s𝒫_sr^†𝒫_rm]_σσ.As expected, the preceding expression is diagonal in spin space, such that, aftersumming over r and s, one gets⟨ I_↑(t)⟩ =e/πλ^2/(1+λ^2)^2∑_m,n,k∫_-Ω/2^Ω/2dω e^i(m-n)Ω tp^*_k-n,↑p_k-m,↑[tanh(ω+kΩ/2Θ)- tanh(ω+mΩ-eV_DC_↑/2Θ) ] ,and a similar expression for ⟨ I_↓(t)⟩. Changing variables as ω̃=ω+kΩ, m̃=m-k and ñ=n-k and summing over k, the current becomes⟨ I_↑(t)⟩ = e/πλ^2/(1+λ^2)^2∑_m,n∫_-∞^∞dωe^i(m-n)Ω t p^*_-n,↑p_-m,↑[tanh(ω/2Θ)- tanh(ω-(m+q_↑)Ω/2Θ)] .At zero temperature, the final integration over ω can be performed and yields⟨ I_σ(t)⟩ = eΩ/πλ^2/(1+λ^2)^2∑_m,n e^i(n-m)Ω t(m+q_σ)p_m,σp_n,σ^*,which, using the expression of the p_k,σ and q_σ, simply results in⟨ I_σ(t)⟩ = 2e^2/πλ^2/(1+λ^2)^2 V_σ(t),as expected. §.§ Infinite gap regime.In this regime, one hasω_n^±=ω_n=Δ_n=0andΔ_n^±=1thusξ_n^±=1/(1+λ^4),such the currents, see Eq. (<ref>),become⟨ I_σ(t)⟩ = e/2πλ^4/(1+λ^4)^2∑_n,m,r,k,s∫_-Ω/2^Ω/2dω {e^-i(n-m)Ω t[σ_z 𝒫_nk^†σ_x𝒫_ksT_s𝒫_sr^†σ_x𝒫_rm - σ_z 𝒫_nk^†𝒫_ksT_s 𝒫_sr^†𝒫_rm]_σσ+ [(n,m)→(m,n)]^*}.Formally, the second term , one could use that ∑_n,r,s𝒫_kn^†𝒫_nsT_s𝒫_sr^†𝒫_rm=δ_nsT_sδ_sm, but that would make the whole expression divergent for all k m, i.e. for all terms but the currents averaged over a period. Therefore, when expanding the sums, one must always pay attention to keep convergent series. Performing a careful expansion, in the Andreev regime, of Eq. (<ref>), one can show that the Keldysh dressed Green's function readsG_SN^+-=λ^3/(1+λ^4)^2[ σ_x𝒫(T-)𝒫^†σ_x𝒫 -σ_xσ_x𝒫(T-)],such that a properly converging expression of the current is⟨ I_σ(t)⟩ = e/2πλ^4/(1+λ^4)^2∑_n,m,r,k,s∫_-Ω/2^Ω/2dω {e^-i(n-m)Ω t[σ_z 𝒫_nk^†σ_x𝒫_ksT_s𝒫_sr^†σ_x𝒫_rm - σ_z 𝒫_nk^†σ_x𝒫_ks𝒫_sr^†σ_x𝒫_rmT_m]_σσ+ [(n,m)→(m,n)]^*}.More precisely, the 2× 2 matricesσ_z 𝒫_nk^†σ_x𝒫_ksT_s𝒫_sr^†σ_x𝒫_rm - σ_z 𝒫_nk^†σ_x𝒫_ks𝒫_sr^†σ_x𝒫_rm T_m are diagonal with the following entriesp^*_k-n,↑p^*_s-k,↓p_r-s,↓p_r-m,↑[tanh(ω+sΩ+eV_DC,↓/2Θ) -tanh(ω+mΩ-eV_DC,↑/2Θ)] - p_n-k,↓p_k-s,↑p^*_r-s,↑p^*_m-r,↓[tanh(ω+sΩ-eV_DC,↑/2Θ) -tanh(ω+mΩ+eV_DC,↓/2Θ)].Thereby, one gets⟨ I_↑(t)⟩ = e/2πλ^4/(1+λ^4)^2 ∑_n,m,r,k,s∫_-Ω/2^Ω/2dω e^-i(n-m)Ω t×{p^*_k-n,↑p^*_s-k,↓p_r-s,↓p_r-m,↑[tanh(ω+sΩ+eV_DC,↓/2Θ) -tanh(ω+mΩ-eV_DC,↑/2Θ)]+[(n,m)→(m,n)]^*}.Perform the following change of variables, x = ω + (s+q_↓)Ωand shift all the indices by s (except s itself), one can perform the sum over s, which yields⟨ I_↑(t)⟩ = e/πλ^4/(1+λ^4)^2∑_n,r,k,m∫_-∞^∞dxe^i(m-n)Ω t p^*_k-n,↑p^*_s-k,↓p_r-s,↓p_r-m,↑[ tanh(x/2Θ)- tanh(x+mΩ-(q_↑+q_↓)/2Θ) ].The integral can be performed and the preceding expression becomes⟨ I_↑(t)⟩ = 2e/πλ^4/(1+λ^4)^2∑_n,r,k,m e^i(m-n)Ω tp^*_k-n,↑p^*_-k,↓p_-r,↓p_r-m,↑(q_↑+q_↓-m),which using the definition of the p_lσ leads to⟨ I_↑(t)⟩ = e^2/πτ_A/2(V_↑(t) + V_↓(t)),and, similarly, ⟨ I_↓(t)⟩ = e^2/πτ_A/2(V_↑(t) + V_↓(t)).§ NOISE CALCULATIONThe total noise is defined asS_NN(t)=∫_-∞^+∞dt' [ I_N (t+t' ) I_N ( t ) - ⟨ I_N ( t+t' ) ⟩⟨ I_N ( t ) ⟩] ,where I_N is the total current operator across the junction. Using Wick theorem, its average value becomes⟨ S_NN(t)⟩=-e^2∫_-∞^+∞dt'Tr_N{2[σ_z W_NS(t) G_SN^-+(t,t')σ_z W_NS(t')G_SN^+-(t',t)] - σ_z W_SN (t) G_SS^-+(t,t')σ_z W_NS(t')G_NN^+-(t',t) -σ_z W_NS(t) G_NN^-+(t,t')σ_z W_NS(t')G_SS^+-(t',t)},which for a periodic drive leads to⟨ S⟩=-2e^2∫_-Ω/2^Ω/2dω/2πTr_NH[ 2Re(σ_z Σ_SN G_NS^+-σ_z Σ_SN G_NS^-+)- σ_z Σ_SN G_NN^+-σ_z Σ_NS G_SS^-+-σ_z Σ_NS G_SS^+-σ_z Σ_SN G_NN^-+],which, in the Andreev regime, becomes⟨ S⟩=-2e^2τ_A∫_-Ω/2^Ω/2dω/2πTr_NH[(1-τ_A)PTP^†σ_xPTP^†σ_x -+ τ_AT^2].Computations along the same lines as for the currents lead to⟨ S⟩= e^2/π[4τ^2_AΘ +2τ_A(1-τ_A)∑_krm(p_n,↓^*p_r,↓p_m-r,↑p_m-n,↑^* )(m+q_↑+q_↓)Ω(Ω(m+q_↑+q_↓)/2Θ)]. | http://arxiv.org/abs/2311.15684v1 | {
"authors": [
"Bruno Bertin-Johannet",
"Benoît Grémaud",
"Flavio Ronneti",
"Laurent Raymond",
"Jérôme Rech",
"Thibaut Jonckheere",
"Thierry Martin"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.mes-hall",
"quant-ph"
],
"primary_category": "cond-mat.supr-con",
"published": "20231127101717",
"title": "Current and shot noise in a spin dependent driven normal metal -- BCS superconductor junction"
} |
Cross Entropy in Deep Learning of Classifiers Is Unnecessary - ISBE Error is All You Need Władysław Skarbek[ Author is with the Faculty of Electronics and Information Technology, Warsaw University of Technology, email: [email protected] ]================================================================================================================================================================== In deep learning classifiers, the cost function usually takes the form of a combination of SoftMax and CrossEntropy functions. The SoftMax unit transforms the scores predicted by the model network into assessments of the degree (probabilities) of an object's membership to a given class. On the other hand, CrossEntropy measures the divergence of this prediction from the distribution of target scores. This work introduces the ISBE functionality, justifying the thesis about the redundancy of cross entropy computation in deep learning of classifiers. Not only can we omit the calculation of entropy, but also, during back-propagation, there is no need to direct the error to the normalization unit for its backward transformation. Instead, the error is sent directly to the model's network. Using examples of perceptron and convolutional networks as classifiers of images from the MNIST collection, it is observed for ISBE that results are not degraded with SoftMax only, but also with other activation functions such as Sigmoid, Tanh, or their hard variants HardSigmoid and HardTanh. Moreover, up to three percent of time is saved within the total time of forward and backward stages. The article is addressed mainly to programmers and students interested in deep model learning. For example, it illustrates in code snippets possible ways to implement ISBE units, but also formally proves that the softmax trick only applies to the class of softmax functions with relocations. § INTRODUCTION A deep model is a kind of mental shortcut (<cit.>), broadly understood as a model created in deep learning of a certain artificial neural network N, designed for a given application. What, then, is an artificial neural network (<cit.>), its deep learning (<cit.>), and what applications (<cit.>) are we interested in?From a programmer's perspective, an artificial neural network is a type of data processing algorithm (<cit.>), in which subsequent steps are carried out by configurable computational units, and the order of processing steps is determined by a directed graph of connections without loops.At the training stage, each group of input data X, i.e., each group of training examples, first undergoes the inference (forward) phase on the current model, i.e., processing through the network N at its current parameters W. As a result, network outputs YF_N(X;W) are produced (<cit.>). [ X; WF_NYF_N(X;W)^INFERENCE≡⋯X_u; W_uF_UY_uF_U(X_u;W_u)⋯^INFERENCE ] After the inference phase comes the model update phase, where the current model is modified (improved) according to the selected optimization procedure (<cit.>). The model update phase begins with calculating the loss (cost) value ZL(Y,Y^∘) defined by the chosen loss function L as well as the inference outcome Y and the target result Y^∘. [ X; WF_NYF_N(X;W)^INFERENCE⋯Y, Y^∘LZL(Y,Y^∘)^model update - start ]The loss Z depends (indirectly through Y) on all parameters W, and what conditions the next step of the update phase is the determination of sensitivity W of the loss function L to their changes. The mathematical model of sensitivity is the gradient W≐LW. Knowing this gradient, the optimizer will make the actual modification of W in a direction that also takes into account the values of gradients obtained for previous training batches.Calculating the gradient with respect to parameters actually assigned to different computational units required the development of an efficient algorithm for its propagation in the opposite direction to inference (<cit.>).Just as in the inference phase, each unit U has its formula Y_uF_U(X_u,W_u) for processing data from input X_u to output Y_u with parameters W_u, so in the backward gradient propagation phase, it must have a formula X_u,W_uF_U(Y_u) for transforming the gradients assigned to its outputs Y_u into gradients assigned to its inputs X_u and its parameters W_u. [ X; WF_N(Y;X,Y,W)F_NY^BACKPROPAGATION⋯YL(Z;Y,Z)L1=ZZ^loss function gradient LY; ⋯X_u; W_uF_U(Y_u;X_u,Y_u,W_u)F_UY_u⋯^GRADIENT BACKPROPAGATION ]Based on such local rules of gradient backpropagation and the created computation graph, the backpropagation algorithm can determine the gradients of the cost function with respect to each parameter in the network. The computation graph is created during the inference phase and is essentially a stack of links between the arguments and results of calculations performed in successive units (<cit.>).Deep learning is precisely a concert of these inference and update phases in the form of gradient propagation, calculated for randomly created groups of training examples. These phases, intertwined, operate on multidimensional deep tensors (arrays) of data, processed with respect to network inputs and on deep tensors of gradient data, processed with respect to losses, determined for the output data of the trained network.Here, by a deep tensor, we mean a multidimensional data array that has many feature maps, i.e., its size along the feature axis is relatively large, e.g., 500, which means 500 scalar feature maps. We then say that at this point in the network, our data has a deep representation in a 500-dimensional space.As for the applications we are interested in this work, the answer is those that have at least one requirement for classification (<cit.>). An example could be crop detection from satellite images (<cit.>), building segmentation in aerial photos <cit.>, but also text translation (<cit.>). Classification is also related to voice command recognition (<cit.>), speaker recognition (<cit.>), segmentation of the audio track according to speakers (<cit.>), recognition of speaker emotions with visual support (<cit.>), but also classification of objects of interest along with their localization in the image (<cit.>).It may be risky to say that after 2015, in all the aforementioned deep learning classifiers, the cost function takes the form of a composition of the SoftMax function (<cit.>) and the CrossEntropy function, i.e., cross-entropy (<cit.>). The SoftMax unit normalizes the scores predicted by the classifier model for the input object, into softmax scores that sum up to one, which can be treated as an estimation of the conditional probability distribution of classes. Meanwhile, cross-entropy measures the divergence (i.e., divergence) of this estimation from the target probability distribution (class scores). In practice, the target score may be taken from a training set prepared manually by a so-called teacher (<cit.>) or may be calculated automatically by another model component, e.g., in the knowledge distillation technique (<cit.>).For K classes and n_b training examples, the SoftMax function is defined for the raw score matrix X∈R^n_b× K as:[YSoftMax(X)][Y_bie^X_bi/∑_j∈[K]e^X_bj,b∈[n_b],i∈[K]] ,where the notation [K] denotes any K-element set of indices - in this case, they are class labels.The CrossEntropy function on the matrix Y,Y^∘∈R^n_b× K is defined by the formula:[ZCrossEntropy(Y,Y^∘)] [Z_b -∑_j∈[K]Y^∘_bjln Y_bj,b∈[n_b],z∈R^n_b][ classified objectF_Nraw scores X^SCORES INFERENCE; raw scores X SoftMaxsoft scores Y, Y^∘ CrossEntropylosses Z^LOSS ESTIMATION L ]^Classifier loss function: Separated ImplementationWhen classifiers began using a separated implementation of the combination of the normalization unit SoftMax and the unit CrossEntropy, it quickly became evident in practice that its implementation had problems with scores close to zero, both in the inference phase and in the backward propagation of its gradient. Only the integration of CrossEntropy with normalization SoftMax eliminated these inconveniences. The integrated approach has the following form:[classified objectF_Nraw scores X^INFERENCE; raw scores X,soft scores Y^∘ CrossEntropy ∘ SoftMaxlosses Z^LOSS ESTIMATION L ]^Classifier loss function - Integrated Implementation The integrated functionality of these two features has the following redundant mathematical notation:[ Z [CrossEntropy ∘SoftMax](X,Y^∘); Z_b -∑_j∈[K] Y^∘_bjlne^X_bj/∑_i∈[K] e^X_bi,b∈[n_b] ]This redundancy in notation was helpful in deriving the equation for the gradient backpropagation for the integrated loss functionCrossEntropy ∘ SoftMax (<cit.>).The structure of this paper is as follows: * In the second section titled ISBE Functionality, the conditions that a normalization unit must meet for its combination with a cross-entropy unit to have a gradient at the input equal to the difference in soft scores: X = Y-Y^∘ are analyzed. Then the definition of ISBE functionality is introduced, which in the inference phase (I) normalizes the raw score to a soft score (S), and in the backward propagation phase (B) returns an error (E), equal to the difference in soft scores. It is also justified why in the case of the SoftMax normalization function, the ISBE unit has, from the perspective of the learning process, the functionality of the integrated unit CrossEntropy ∘ SoftMax.* In the third section, using the example of the problem of recognizing handwritten digits and the standard MNIST(60K) image collection (<cit.>), numerous experiments show that in addition to the obvious savings in computational resources, in the case of five activations serving as normalization functions, the classifier's effectiveness is not lower than that of the combination of the normalization unit Softmax and the unit Cross Entropy. This ISBE property was verified for the activation units Softmax, Sigmoid, Hardsigmoid, and Tanh and Hardtanh.* The last fourth section contains conclusions.* Appendix A, titled Cross-Entropy and Softmax Trick Properties, contains the formulation and proof of the theorem on the properties of the softmax trick. § ISBE FUNCTIONALITY The ISBE functionality is a proposed simplification of the cost function, combining the SoftMax normalization function with the cross-entropy function, hereafter abbreviated as CE_all. Its role is to punish those calculated probability distributions that significantly differ from the distributions of scores proposed by the teacher.To understand this idea, let's extend the inference diagram for CE_all with the backward propagation part for the gradient. We consider this diagram in its separated version, omitting earlier descriptions for the diagram (<ref>):[ X SoftMaxY, Y^∘ CrossEntropyZ^LOSS INFERENCE L; XTheorem <ref>Y-Y^∘ SoftMaxY; Y, Y^∘ CrossEntropyZ=1^BACKPROPAGATION ]The meaning of variables X,Y,Y^∘,Z and Z,Y,X appearing in the above diagram (<ref>):[ X raw score at the input of the normalization function preceding cross-entropy CE, X∈R^K,; Y normalization result, so-called soft score , Y∈(0,1)^K,; Y^∘ target soft score, assigned to the classified example ,; Z output of cross-entropy CE , Z∈R,; Zformal gradient at the input of the backward propagation algorithm, Z=1,; Ygradient of cross-entropy CE with respect to Y: Y = ZY=-Y^∘/Y,; X gradient of cross-entropy CE with respect to X: XTheorem <ref>(Y-Y^∘) . ]The key formula here is X(Y-Y^∘). Its validity comes from the mentioned theorem and the formula (<ref>) associated with the softmax trick property.The equation (<ref>) on the form of the Jacobian of the normalization unit is both a sufficient and necessary condition for its combination with the cross-entropy unit to ensure the equality (<ref>). Moreover, this condition implies that an activation function with a Jacobian of the softmax type is a SoftMax function with optional relocation.Theorem <ref> leads us to a seemingly pessimistic conclusion: it is not possible to seek further improvements by changing the activation and at the same time expect the softmax trick property to hold. Thus, the question arises: what will happen if, along with changing the activation unit, we change the cross-entropy unit to another, or even omit it entirely?In the ISBE approach, the aforementioned simplification of the CE_all cost function involves precisely omitting the cross-entropy operation in the inference stage and practically omitting all backward operations for this cost function. So what remains? The answer is also an opportunity to decode the acronym ISBE again: * In the inference phase (I), we normalize the raw score X to Y=SoftMax(X), characterized as a soft score (S).* In the backward propagation phase (B), we return an error (E) equal to the difference between the calculated soft score and the target score, i.e.,X≐ Y-Y^∘. Why can we do this and still consider that in the case of the SoftMax activation function, the value of the gradient transmitted to the network is identical: X_CE_all = X_ISBE Y-Y^∘?The answer comes directly from the property X_CE_all = Y-Y^∘, formulated in equation (<ref>), which was defined in the theorem <ref> as the softmax trick property.We thus have on the left the following diagram of data and gradient backpropagation through such a unit. On the right we have its generalization to aScoreNormalization unit instead of SoftMax unit.. [X SoftMaxY, Y^∘^ISBE INFERENCE; X Y-Y^∘ SubtractY, Y^∘ ^ISBEBACKPROPAGATION ]}generalize{[X Score NormalizationY, Y^∘^ISBE INFERENCE; X Y-Y^∘ SubtractY, Y^∘ ^ISBEBACKPROPAGATION ]. Which activation functions should we reach for in order to test them in the ISBE technique? * The SoftMax activation function should be the first candidate for comparison, as it theoretically guarantees behavior comparable to the system containing cross-entropy.* Activations should be monotonic, so that the largest value of the raw score remains the largest score in the soft score sequence.* Soft scores should be within a limited range, e.g., [0,1] as in the case of SoftMax and Sigmoid, or [-1,+1] as for Tanh.* The activation function should not map two close scores to distant scores. For example, normalizing a vector of scores by projecting onto a unit sphere in the p-th Minkowski norm meets all above conditions, however, it is not stable around zero. Normalization x/x_p maps, for example, two points ϵ,-ϵ distant by 2·ϵ_p to points distant exactly by 2, thus changing their distance 1/ϵ_p times, e.g., a million times, when ϵ_p=10^-6. This operation is known in Pytorch library as normalize function. The experiments conducted confirm the validity of the above recommendations. The Pytorch library functions softmax, sigmoid, tanh, hardsigmoid, hardtanh meet the above three conditions and provide effective classification at a level of effectiveness higher than 99.5%, comparable to CrossEntropy ∘ SoftMax. In contrast, the function normalize gave results over 10% worse - on the same MNIST(60K) collection and with the same architectures.What connects these good normalization functions F:R^KR^K, of which two are not even fully differentiable? Certainly, it is the Lipschitz condition occurring in a certain neighborhood of zero (<cit.>):x∈R^K, x_p≤ϵF(x)_p≤ cx_p ,where c is a certain constant .Note that the Lipschitz condition meets the expectations of the fourth requirement on the above list of recommendations for ISBE. Moreover, we do not expect here that the constant c be less than one, i.e., that the function F has a narrowing character.We need also a recommendation for teachers preparing class labels, which we represent as vectors blurred around the base vectors of axes I_K=[e_1,…,e_K], e_i[j]δ_ij: * example blurring value μ, e.g., μ=10^-6:ẽ_i[j](1-μ)δ_ij + μ/K-1(1-δ_ij) * when the range of activation values is other than the interval [0,1], we adjust the vector ẽ_i to the new range, e.g., for tanh the range is the interval (-1,+1) and then the adjustment has the form:ẽ_i2·ẽ_i-1, i=1,…,KFinally, let's take a look at the code for the main loop of the program implemented on the Pytorch platform.* This is what the code looks like when loss_function is chosen as nn.CrossEntropyLoss: * Now we introduce the ISBE option for SoftMax activation: * If we want to test more options, the loop code will extend a bit: * If we prefer to have a visually shorter loop, then by introducing the variable soft_function and extending the class DataProvider with matching target labels for a given soft option, we finally get a compact form:Of course, the above code snippets are only intended to illustrate how easy it is to add the functionality of ISBE to an existing application.§ EXPERIMENTS What do we want to learn from the planned experiments? We already know from theory that in the case of the SoftMax activation, we cannot worsen the parameters of the classifier using cross-entropy, both in terms of success rate and learning time.Therefore, we first want to verify whether theory aligns with practice, but also to check for which normalization functions the ISBE unit does not degrade the model's effectiveness compared to CE_all.The learning time t_ISBE should be shorter than t_CE, but to be independent of the specific implementation, we will compare the percentage of the backpropagation time in the total time of inference and backpropagation:τbackpropagation time/inference time + backpropagation time· 100% We evaluate the efficiency of the ISBE idea on the standard MNIST(60K) image collection and the problem of their classification.From many quality metrics, we choose the success rate (also called accuracy), defined as the percentage of correctly classified elements from the test collection MNIST(10K)α= number of correct classifications/size of the test collection· 100% We want to know how this value changes when we choose different architectures, different activations in the ISBE technique, but also different options for aggregating cross-entropy over the elements of the training batch. Thus, we have the following degrees of freedom in our experiments:* Two architecture options * Architecture N_0 consists of two convolutions and two linear units, of which the last one is a projection from the space of deep feature vectors of dimension 512 to the space of raw scores for each of the K=10 classes:image-228_yx13^k2^s323^k2^s642051210 class scoresas by STNN notation (<cit.>), for instance {[3^k2^s32 means 32 convolutions with 3x3 masks, sampled with a stride of 2,;20DropOut - a unit zeroing 20% of tensor elements,; 512a linear unit with a matrix A∈R^?×512,; here ?=64 - it is derived from the shape of;the tensor produced by the previous unit . ]. * Architecture N_1 consists of two blocks, each with 3 convolutions - it is a purely convolutional network, except for the final projection:[ image-228_yx1 3^k323^k325^k2^s32p 403^k643^k645^k2^s64p 404^k12810 class scores; ]Note that the last convolution in each block has a p requirement for padding, i.e., filling the domain of the image with additional lines and rows so that the image resolution does not change. * Three options for reducing the vector of losses in the CrossEntropyLoss unit: none, mean, sum. * Five options for activation functions used in the ISBE technique: * SoftMax: y_i e^x_i/∑_j∈[K] e^x_j, i∈[K],* Tanh:y_i e^x_i-e^-x_i/e^x_i+e^-x_i, i∈[K],* HardTanh:y_i{[-1 ifx_i≤ -1; x_i if-1 < x_i < +1;+1 if+1≤ x_i ]}, i∈[K],* Sigmoid: y_i 1/1+e^-x_i, i∈[K],* HardSigmoid:y_i {[0gdyx_i≤-2;x_i+2/4 gdy-2 < x_i < +2; +1 gdy+2≤ x_i ]}= HardTanh(x_i/2)+1/2, i∈[K].The results of the experiments, on the one hand, confirm our assumption that the conceptual Occam's razor, i.e., the omission of the cross-entropy unit, results in time savings τ, and on the other hand, the results are surprisingly positive with an improvement in the metric of success rate α in the case of hard activation functions HardTanh and HardSigmoid. It was observed that only the option of reduction by none behaves exactly according to theory, i.e., the success rate is identical to the model using SoftMax normalization. Options mean and sum for the model with entropy are slightly better than the model with softmax. The consistency of models in this case means that the number of images incorrectly classified out of 10 thousand is the same. In the experiments, it was not checked whether it concerns the same images. A slight improvement, in this case, meant that there were less than a few or a dozen errors, and the efficiency of the model above 99.6% meant at most 40 errors per 10 thousand of test images. §.§ Comparison of time complexity We compare time complexity according to the metric given by the formula (<ref>). In the context of time, the table <ref> clearly shows that the total time share of backpropagation, obviously depending on the complexity of the architecture, affects the time savings of the ISBE technique compared to CrossEntropyLoss - table<ref>. The absence of pluses in this table, i.e., the fact that all solutions based on ISBE are relatively faster in the learning phase, is an undeniable fact.The greatest decrease in the share of backpropagation, over 3%, occurs for the Sigmoid and SoftMax activations. The smallest decrease in architecture N_0 is noted for the soft (soft) normalization function Tanh and its hard version HardTanh. This decrease refers to cross-entropy without reduction, which is an aggregation of losses calculated for all training examples in a given group, into one numerical value. Inspired by the theorem<ref> , which states that the relocation of the SoftMax function preserves the softmax trick property, we also add data to the table <ref> for the network N_1^r. This network differs from the N_1 network only in that the normalization unit has a trained relocation parameter. In practice, we accomplish training with relocation for normalization by training with the relocation of the linear unit immediately preceding it. This is done by setting its parameter: bias=True.As we can see, the general conclusion about the advantage of the ISBE technique in terms of time reducing for the model with the relocation of the normalization function, is the same. §.§ Comparison of classifier accuracyComparison of classifier accuracy and differences in this metric are contained in tables <ref> and <ref>.The accuracy is computed according the formula (<ref>).The number of pluses on the side of ISBE clearly exceeds the number of minuses. The justification for this phenomenon requires separate research. Some light will be shed on this aspect by the analysis of learning curves - the variance in the final phase of learning is clearly lower. The learning process is more stable.In the table <ref>, we observe that, with the exception of the function SoftMax, which on several images of digits performed worse than the model with cross-entropy, the soft activations have an efficiency slightly or significantly better. However, we are talking about levels of tenths or hundredths of a percent here. The largest difference noted for the option softmax was 15 hundredths of a percent, meaning 15 more images correctly classified. Such differences are within the margin of statistical error.The use of relocation for the normalization functionof does not provide a clear conclusion - for some models there is a slight improvement, for others a slight deterioration. It is true that the ISBE unit with sigmoid activation achieved the best efficiency of 99.69%, but this is only a matter of a few images. Within the limits of statistical error, we can say that the ISBE technique gives the same results in recognizing MNIST classes. Its advantages are: * of decrease time in the total time,* simplification of architecture, and therefore plaing the philosophical role of Occam's razor.§.§ Visual analysisFurther analysis of the results will be based on the visual comparison of learning curves. First, let's see on three models cross-entropy-mean, softmax, sigmoid their loss and efficiency curves obtained on training data MNIST(54K) and on data intended solely for model validation MNIST(6K) . These two loss curves are calculated after each epoch. We supplement them with a loss curve calculated progressively after each batch of training data (see figure <ref>).Let us note the correct course of the train loss curve with respect to progressive loss curve - both curves are close. The correct course is also for validation loss curve - the validation curve from about epoch 30 is below the training curve maintaining a significant distance. This is a sign that the model is not overfitted. This effect was achieved only after applying a moderate input image augmentation procedure.Correct behavior of learning curves was recorded both for the modesl with entropy and for models with the ISBE unit. This also applies to classifier performance curves.* Curves of loss functions can appear together as long as the type of function is identical, which entails a similar range of variability for loss function values. One might wonder what measure of loss to adopt in the case of ISBE, since this technique, in fact, does not calculate loss values. We opt for a natural choice of mean square error for errors in soft scores:L_ISBE = MSE(Y, Y^∘) 1/n_b·Y-Y^∘^2_2For such defined measures, it turns out that only the option of reduction by summing has a different range of variability, and therefore it is not on the figure <ref>.* In the case of classifier accuracy, a common percentage scale does not exclude placing all eight curves for each considered architecture. However, due to the low transparency of such a figure, it is also worth juxtaposing different groups of curves of the dependency α(e). The accuracy α of the classifier MNIST(60K) is calculated on the test set MNIST(10K).Sets of curves, which we visualize separately for architecturesN_0, N_1 are: * all options for loss functions (3) and soft score functions (5),* CE none, CE mean, CE sum versus softmax,* CE none, CE mean, CE sum versus tanh, hardtanh,* softmax versus sigmoid, hardsigmoid,* softmax versus tanh, hardtanh,* softmax versus sigmoid, tanh.Due to space constraints, we show learning curves and classifier effectiveness graphs only for architecture N_1 in figures <ref>, <ref>.In figure <ref> we can clearly observe four clusters of models: * CrossEntropyLoss basedwith reduction option sum (as out of common range it was not shown),* CrossEntropyLoss based with reduction options none, and mean,* ISBE based with normalizationsto range [0,1] including functions SoftMax, Sigmoid, and HardSigmoid,* ISBE based with normalizationsto range [-1,1] including functions Tanh, and HardTanh.Within a cluster, the loss curves behave very similarly.Interestingly, the loss curves in ISBE-based clusters tend to the same value greater than zero. In contrast, cross-entropy-based curves also tend to the same limit. However it is clearly greater than ISBE one. Now, we will pay more attention to test learning curves. We generate test learning curves on the full set of test data MNIST(10K). After each epoch, one point is scored towards the test learning curve. We will show these curves in several comparative contexts.Inthe case of classifier accuracy curves (see figure <ref>), the variances in the clusters described above are smaller than in the union of clusters. Close to the final epochs, all curves tend to be chaotic within the range of (99.4,99.7). Visualizing the effectiveness of classifiers for different architectures of different complexities, although more obvious, also has research value (see figure <ref>): * CE none, CE mean, CE sum from N_0 versusCE none, CE mean, CE sum from N_1,* CE none, softmax from N_0 versus CE none, softmax from N_1,* softmax, sigmoid from N_0 versus softmax, sigmoid from N_1,* sigmoid, tanh from N_0 versus sigmoid, tanh from N_1,* sigmoid, hardsigmoid from N_0 versus sigmoid, hardsigmoid from N_1,* tanh, hardtanh from N_0 versus tanh, hardtanh from N_1. The figure <ref> shows the better performance of N_1 than N_0. Moreover, we can observe slightly more stable behaviour for ISBN-based curves than for cross-entropy-based. § CONCLUSIONS Cross-entropy CE as a loss function owes much to normalization performed by the SoftMax activation function. In the backward gradient backpropagation phase, only this activation, through perfect linearization, can prevent the explosion or suppression of the gradient originating from CE. What we call the softmax trick, as a mathematical phenomenon is explained by the theory presented in Appendix A. There is a proof that such linearization can only be realized by a function F:R^KR^K with a Jacobian identical to that of the SoftMax function. In turn, such a Jacobian can only beby relocated versions of the SoftMax function. For another research there are left practical aspects of more general theorem <ref> implying that dilated and relocated versions of SoftMax, are the only ones having the property of dilated softmax trick.Should we, therefore, celebrate this unique relationship between activation and cost function? In this work, I have shown that it is rather beneficial to use the final effect of the action of this pair, namely the linear value equal to Y-Y^∘, which can be calculated without their participation. This is exactly what the ISBE unit does - it calculates the soft score vector in the forward step to return in backward step its error from the target score.To determine the normalized score, the ISBE unit can use not only the SoftMax function, as it is not necessary to meet the unity condition, i.e., to ensure a probability distribution as scores of the trained classifier. At least four other activation functions Sigmoid, Tanh and their hard versions HardSigmoid and HardTanh perform no worse. The choice of these final activations was rather a matter of chance, so researchers face further questions. How to normalize raw scores and how to appropriately represent (encode) class labels in relation to this normalization, so as not to degrade the classifier's results? What properties should such normalization functions have? Experiments suggest that meeting the Lipschitz condition in the vicinity of zero may be one of these properties.The theoretical considerations presented prove that the ISBE unit in the process of deep model learning correctly simulates the behavior of the CrossEntropy unit preceded by the SoftMax normalization.The experiments showed that the ISBE unit saves the time time offorward and backward stage up to 3%, and the effectiveness of the classifier model remains unchanged within the margin of statistical error. On the other hand, the examples of code fragments showed that the programmer's time spent on introducing the ISBE technique to his/her program instead of CrossEntropyLoss is negligible. 99 deep model Schmidhuber J. Annotated History of Modern AI and Deep Learning. arXiv:2212.11279, 2022.ann Rosenblatt F. The Perceptron: A Probabilistic Model For Information Storage And Organization in the Brain. 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The notation [K] refers to a sequence of K indices, such as (1,…,K) or (0,1,…,K-1).In the special case when the vector y∈(0,1)^K is calculated based on the vector x∈R^K according to the formula y_i = SoftMax(x)_i=e^x_i/∑_ke^x_k, the gradient of CE with respect to x has a particularly simple resultant formula. Its simplicity was the reason for the term softmax trick:y=SoftMax(x) CEx(y^∘,y(x))softmax trick y-y^∘ Some authors (<cit.>) also use the term softmax trick for that part of the proof showing that the derivative of the natural logarithm of the sum of functions e^x_i equals to the SoftMax function: [ y_ie^x_i/∑_ke^x_k[ln(∑_ke^x_k)]x_i=y_i; [CE(y^∘,y(x))x_i= [∑_jy^∘_jln(∑_ke^x_k)-∑_jy^∘_jln e^x_j]x_i; = y_i∑_jy^∘_j^=1-y^∘_i = y_i-y^∘_i ] ]In matrix notation (<cit.>), the property of softmax trick has a longer proof, as we first need to calculate the Jacobian of the SoftMax function (<cit.>), which is yx = y-yy. Then[ [CE(y^∘,y(x))]x =yx·CEy(y^∘,y) =(y-yy)(-y^∘÷ y); = y(y÷ y)y^∘_1_Ky^∘=1 - y ÷y_I_K y^∘= y-y^∘ ] An interesting question arises: Is it only the SoftMax function that has such a property? It seems possible, as for any differentiable function F:R^nR^n the starting point is similar:[CE(y^∘,F(x)_y)]x = F(x)x[CE(y^∘,y)]y^-y^∘÷ y = F(x)x(-y^∘÷ y)The following theorem fully characterizes functions that have the softmax trick property. For a differentiable function F:R^KR^K, the following three properties are equivalent: * F is a SoftMax function with relocation, if there exists a reference point c∈R^K, such that for every x∈R^K:y = F(x) = SoftMax(x-c) , * F has a softmax-type Jacobian, if for every x∈R^K:Jacobian(F)(x) F(x)x = y-yy ,wherey=F(x) , * F possesses the softmax trick property, if for every y^∘∈[0,1]^K, x∈(0,1)^K:F(x)x(-y^∘÷ y) = y-y^∘ ,wherey=F(x) . We prove the implications in the following order:(2)(3), (3)(2),(1)(2), (2)(1) . * Proof of implication (2)(3): If the Jacobian F(x)x of the function F is of the softmax type, then for y=F(x):yx=(y-yy)(-y^∘÷ y) =y(y÷ y)y^∘_1_Ky^∘=1 - y ÷y_I_K y^∘= y-y^∘ * Proof of implication (3)(2): Denote the axis unit vector j by e_j∈R^K.Then (e_j)_i=δ_ij. Substitute into property (<ref>) the target score vector y^∘ e_j. Then y_i-(e_j)_i=(F(x)x_i)(-e_j÷ y)= ∑_k∈[K]y_kx_i·(-δ_kj/y_k) =y_jx_i·(-1/y_j)Therefore y_jx_i = ((e_j)_i-y_i)y_j (e_j)_i=δ_ij (δ_ij-y_i)y_j. Thus, yx = y-yy .* Proof of implication (1)(2):If y_je^x_j-c_j/∑_k∈[K]e^x_k-c_k, theny_jx_i = {[ -e^x_j-c_j· e^x_i-c_i/( ∑_k∈[K]e^x_k-c_k)^2 = -y_i· y_j, wheni≠ j; e^x_i-c_i·(∑_k∈[K] e^x_k-c_k)-e^x_i-c_i· e^x_i-c_i/(∑_k∈[K]e^x_k-c_k)^2; = y_i-y_i^2 = (1-y_i)· y_i, wheni=j ].The general formula is y_jx_i=(δ_ij-y_i)y_j . Therefore:(yx)_ijy_jx_i= δ_ijy_j-y_iy_j = (y)_ij - (yy)_ij .Hence, yx=y-yy * Proof of implication (2)(1):If y_ix_i=(1-y_i)· y_i, then the diagonal of the Jacobian matrix gives us differential equations (<cit.>), from which we can determine the general form of the function y_i(x), i∈[K]:[ ∂ y_i/(1-y_i)· y_i =∂ x_i ∫( 1/y_i+1/1-y_i)∂ y_i = ∫∂ x_i; lny_i/1-y_i = x_i + C(k≠ i)y_i = e^x_i/e^x_i+e^-C(k≠ i)_z_i; y_i = e^x_i/z_i,wherez_i=z_i(x_1,…,x_K)>0;lnz_i = x_i-ln y_i ] Now we calculate the partial derivatives ln z_jx_i.If i≠ j, then[ln z_j]x_i = [x_j-ln y_j]x_i = -1/y_jy_jx_i^-y_jy_i = y_iFor i=j, the result is the same:[ln z_i]x_i = [x_i-ln y_i]x_i = 1 -1/y_iy_ix_i^(1-y_i)y_i = 1-(1-y_i)y_i/y_i = y_iTherefore, for a fixed i, e.g., for j=1, we have K equalities:[ln z_1]x_i=[ln z_j]x_i=y_i for any j,i∈[K]. This means that vector fields for each pair of functions ln z_1 and ln z_j are identical.Integrating these fields yields the same functions up to a constant c_j: ln z_j = ln z_1 + c_j, j∈[K]. Consequently, z_j=z_1· e^c_j, j∈[K], and thereforey_j = e^x_j/z_1e^c_j = e^x_j-c_j/z_1.From the unity condition, we can now determine the value of z_1:1=∑_k∈[K]y_k=∑_k∈[K]e^x_k-c_k/z_1 z_1 = ∑_k∈[K]e^x_k-c_k y_j =e^x_j-c_j/∑_k∈[K]e^x_k-c_k .The theorem <ref> can be generalized todilated version of the SoftMax function. It is reformulated in the form of theorem <ref>. The proof of the generalized theorem can be easily generalized, too.For a differentiable function F:R^KR^K, the following three properties are equivalent: * F is a SoftMax function with dilation and relocation, if there exists a reference point c∈R^K, and dilation vector d∈R^K, ∀ i, d_i≠ 0 such that for every x∈R^K:y = F(x) = SoftMax(d⊙ x-c) , * F has a dilated softmax-type Jacobian, if there exists a dilation vector dK such that for every x∈R^K:Jacobian(F)(x) F(x)x = d⊙(y-yy) ,wherey=F(x) , * F possesses the dilated softmax trick property, if there exists a dilation vector dK such that for every y^∘∈[0,1]^K, x∈(0,1)^K:F(x)x(-y^∘÷ y) = d⊙(y-y^∘) ,wherey=F(x) . | http://arxiv.org/abs/2311.16357v1 | {
"authors": [
"Wladyslaw Skarbek"
],
"categories": [
"cs.LG",
"I.2.5"
],
"primary_category": "cs.LG",
"published": "20231127224002",
"title": "Cross Entropy in Deep Learning of Classifiers Is Unnecessary -- ISBE Error is All You Need"
} |
Practical Layout-Aware Analog/Mixed-Signal Design Automation with Bayesian Neural NetworksAhmet F. Budak The University of Texas at AustinAustin, TX, USA [email protected] Keren Zhu The University of Texas at AustinAustin, TX, USA [email protected] Z. Pan The University of Texas at AustinAustin, TX, USA [email protected] September 15, 1996; accepted March 16, 1997 ========================================================================================================================================================================================================================================================================================== The high simulation cost has been a bottleneck of practical analog/mixed-signal design automation. Many learning-based algorithms require thousands of simulated data points, which is impractical for expensive to simulate circuits. We propose a learning-based algorithm that can be trained using a small amount of data and, therefore, scalable to tasks with expensive simulations. Our efficient algorithm solves the post-layout performance optimization problem where simulations are known to be expensive. Our comprehensive study also solves the schematic-level sizing problem. For efficient optimization, we utilize Bayesian Neural Networks as a regression model to approximate circuit performance. For layout-aware optimization, we handle the problem as a multi-fidelity optimization problem and improve efficiency by exploiting the correlations from cheaper evaluations. We present three test cases to demonstrate the efficiency of our algorithms. Our tests prove that the proposed approach is more efficient than conventional baselines and state-of-the-art algorithms.electronic design automation, analog/mixed-signal optimization, analog sizing automation, analog layout automation § INTRODUCTIONAnalog/Mixed-signal (AMS) integrated circuit (IC) design typically follows a process flow visualized in Figure <ref>. A combination of designer experience and computer simulation feedback is iterated to determine the design that meets the performance requirements. A large portion of design time is spent on the sizing and layout phases, where multiple iterations are possible due to potential loop-backs in the design flow. This is a labor-intensive process in industry practice with little to no automation. To address this costly exercise, a considerable effort in academia is focused on introducing automated solutions. Analog sizing automation is the task of optimizing AMS design variables, e.g., transistor widths, lengths, resistor, and capacitor values. The aim is to satisfy the performance constraints and optimize the design objective. In general, sizing automation is run through schematic-level simulations. However, AMS IC performance is also sensitive to layout implementation <cit.>. Especially in the advanced process nodes, layout-induced parasitics may greatly affect the final design performance. Therefore, sizing the AMS design variables considering the layout effects is also crucial. The majority of the recent sizing and post-layout performance optimization algorithms have simulation feedback in the loop. Due to advanced scaling, simulations are required to obtain accurate performance evaluations. Simulation-based AMS automation algorithms adapted various methods from the optimization and Machine Learning (ML) communities. The earlier approaches include population-based methods such as particle swarm optimization <cit.> and evolutionary algorithms <cit.>. Although these algorithms have good convergence behavior, they are inefficient in sampling since they explore the design space randomly. To mitigate sample inefficiency, model-based methods gained popularity <cit.>. These methods employ surrogate-models between the solution space and performance space and provide efficiency in exploring the solution space. A typical surrogate model is Gaussian Process Regression (GPR) <cit.>, which is a well-studied model in Bayesian Optimization (BO) field <cit.> and is adapted by several analog sizing algorithms. The main drawback of GPR modeling is its computational complexity.Recent research trend in analog sizing introduces ML to simulation-based methodology <cit.>. However, the literature review reveals that most of these methods require thousands of simulation data to train Deep Neural Network (DNN) models that approximate the relations between the design variables and the performance metrics. Therefore, the practicality of these algorithms is severely reduced when the optimization task has a high simulation cost. For example, drawing/generating the layout, extracting the parasitics, and running post-layout simulations is typically an expensive procedure. Therefore, optimization algorithms designed for schematic-level sizing can not be adapted by simply changing how data is generated.This paper presents a Machine Learning-based simulation-in-the-loop automation method for the AMS design problem. Overall, we formalize two stand-alone recipes for schematic-level sizing and post-layout performance optimization, i.e., layout-aware sizing. We integrate the state-of-the-art analog layout generator, MAGICAL <cit.>, into our flow to handle layout-aware sizing. Our algorithms do not assume the pre-existence of any dataset, and we generate all training data during the optimization. We employ Bayesian Neural Networks (BNN) for modeling design performances. Bayesian Neural Networks allow error quantification, and compared to Deep Neural Networks, BNN are shown to be effective in handling scarce datasets and preventing overfitting <cit.>. Therefore, BNN can be trained on smaller datasets, significantly improving the practicality and scalability. We also introduce a batch-optimization framework and design space sampling strategy that is compatible with BNN modeling. Further, when optimizing the design variables based on post-layout performance, we exploit the correlation between schematic-level simulations and post-layout simulations. Our algorithm introduces a co-learning scheme that reduces the need for costly post-layout simulations and boosts efficiency even further. We compile our contributions as follows: * We use Bayesian Neural Network-based modeling to obtain performance approximations. Different learning strategies are adapted for schematic-level sizing and post-layout performance optimization.* We adopt a scalable sampling strategy and query the optimization batches by utilizing a trust region and Thompson sampling. * The post-layout sizing is handled as a multi-fidelity optimization problem, and an efficient co-learning strategy is developed.* The efficiency of the proposed methods is demonstrated on three circuits by providing comparisons to previous state-of-the-art.The rest of the paper is organized as follows. Section <ref> introduces the backgrounds and previous work. Section <ref> describes our algorithms for handling schematic-level sizing and post-layout performance-based sizing problems. Section <ref> provides the experiments on circuit examples to demonstrate the efficiency of our algorithms. Finally, Section <ref> concludes the paper. § BACKGROUND & RELATED WORKIn this section, we first formally define the AMS design automation problem. Then we review the recent approaches to schematic-level sizing and layout-aware sizing. We summarize the state-of-the-art algorithms' advantages and shortcomings.§.§ Problem Formulation In this paper, we assume that the existence of post-layout performance implies the existence of schematic-level performance values. However, the reverse implication does not hold. We formulate the AMS schematic-level sizing and layout-aware sizing task as a constrained optimization problem succinctly as below.minimizef_0(𝐱) subject to f_i(𝐱) ≤ 0fori=1, …, mwhere, 𝐱∈ℝ^d is the parameter vector and d is the number of design variables of sizing task. Thus,ℝ^d is the design space. f_0(𝐱) is the objective performance metric we aim to minimize. Without loss of generality, we denote i^th constraint by f_i(𝐱). Notice that if the problem is schematic-level optimization, the f_i values are obtained from schematic simulations. If the problem is post-layout optimization, the f_i values are determined by post-layout simulations. Through this paper, we will evaluate the quality of a design by defining a Figure of Merit (FoM) in the following form:FoM(𝐱) = w_0× f_0(𝐱) + ∑_i=1^mmin(1, max(0, w_i × f_i(𝐱)))where w_i is the weighting factor. Note, a max(·) clipping is used for equating designs after constraints are met, and min(·) is used to prevent single constraint violation from dominating FoM value. §.§ Schematic-Level Sizing The recent methods for AMS sizing can be collected under two algorithm classes: Bayesian Optimization methods and Deep Learning methods.Bayesian Optimization methods are tested on AMS problems and are proven to be sample efficient. For example, GASPAD <cit.> is a hybrid algorithm using a combination of evolutionary space exploration and GPR surrogate-based selection. WEIBO <cit.> method also employs GPR as a surrogate and introduces a Bayesian Optimization framework where a weighted acquisition function is tailored to comply with the performance-constrained nature of sizing problem. In <cit.>, the authors introduced a multi-fidelity GPR algorithm where the fidelity of the performance is varied with the simulation accuracy. However, this work did not address the layout effects. The disadvantage shared by all GPR models is their cubic complexity to the number of samples, 𝒪(N^3).Deep Learning based sizing methods includes supervised learning and reinforcement learning (RL) methods <cit.>. GCN-RL <cit.> is a Graph Neural Network algorithm where state representation is built via device index, type, and selected electrical properties. They also propose methods to transfer the optimization experience between different topologies and processes. However, their training graphs show that they use up to 10^4 simulations for sizing academic circuits. AutoCkt <cit.> is a discrete action space policy gradient method. The RL agent is trained on different optimization tasks where the task is randomly sampled from a predefined set. The trained agent is then tested for the particular tests during deployment. We also observe from the training graphs that AutoCkt requires up to 10^5 simulated samples for training. In <cit.>, the authors successfully applied BNN on multi-objective analog sizing. However, they did not consider handling constraints, and layout effects are ignored. DNN-Opt <cit.> is introduced as an RL-inspired supervised learning optimization method that shows high sample efficiency and can be trained during optimization. It uses less than a thousand iterations to optimize academic benchmarks. DNN-Opt does not quantify the variance on approximated values and has no methodic way to balance exploration/exploitation during design space exploration.§.§ Post-Layout Based Sizing Several works in AMS sizing proposed solutions to include layout-induced parasitics. The studies proposed in <cit.> and <cit.> embedded a layout generator in the automation loop, and performance metrics to be optimized are obtained through post-layout simulations. However, they did not consider the correlations between the schematic-level and post-layout simulations; therefore, their efficiency is limited. The work in <cit.> employs a less accurate parasitic prediction during sizing, so the finalized post-layout performance is not guaranteed. In <cit.>, the authors propose a Transfer Learning strategy where a DNN is first trained on schematic-level simulations. Then this knowledge is transferred to improve the learning of a relatively small number of post-layout data. Although this work provides a suggestion to improve the efficiency of post-layout optimization, it requires up to 5×10^3 schematic-level data for initial DNN training, which suffers from the scalability concerns mentioned before. In summary, a scalable solution to optimize AMS design parameters under layout parasitics is yet to be studied. § ANALOG/MIXED-SIGNAL IC AUTOMATION FLOW In this work, we provide solutions to two problems in Analog/Mixed-Signal design automation: schematic-level sizing automation (Task 1) and layout-aware sizing automation (Task 2). The high-level frameworks of proposed solutions to both tasks are summarized together in Figure <ref>. Section III-A introduces and elaborates on the core principles that complete our proposed automation flow for schematic-level sizing tasks. Then, in section III-B, we explain how to solve the post-layout performance optimization problem efficiently by transforming the BNN learning scheme. §.§ Schematic-Level Sizing AutomationWe propose a BNN-based sizing algorithm to optimize AMS design on schematic-level simulations. The complete flow of the proposed approach is summarized in Algorithm <ref>. The algorithm starts with sampling random points in the design space and simulating them via the SPICE simulator. An initial dataset for training the BNN performance model is built from these samples. Then a trust-region state is initialized before algorithm iterations start. The trust region determines the bounds of the exploration space. The following subsections will provide more details regarding the BNN modeling and trust-region search.Our algorithm models each performance metric at each optimization iteration with an individual BNN model. Then a batch of samples is collected based on the posterior realization of points lying inside the trust region. Candidate design performance realizations are obtained using the Thompson sampling method, and the candidates are ranked based on the utility values (FoM). A batch of q points is collected, and their real performances are obtained through simulation. The new data is added to the database, and the trust region is updated based on the real simulation outputs of the last batch. §.§.§ Performance Modeling with Bayesian Neural NetworksWe base our Bayesian Neural Network regression method on the assumption that the observed function values follow a Gaussian distribution and the probabilistic model on the observations are in the following form:p(f(x)| x,θ) = 𝒩(ϕ(x;θ),τ^-1)where θ is the BNN parameters, i.e., weights and biases, ϕ(x;θ) is the output of the BNN with parameters θ and τ is the noise parameter. We assign a standard Gaussian prior distribution over each element of the NN parameters, θ, and a Gamma prior over each noise precision, p(τ)=Gam(τ| a_0, b_0). Let define y_n = f(x_n). Given the dataset 𝒟={(𝐱_n, y_n)}_n=1^N, the joint probability of our model is given by.999!p(θ, 𝒴, τ, |𝒳)=𝒩(vec(θ) |0, 𝐈) p(τ) ∏_n=1^N 𝒩(y_n|ϕ(𝐱_n), τ^-1)where 𝒳={𝐱_n}, 𝒴={y_n}, and vec(·) is vectorization. Due to its unbiased, high-quality uncertainty quantification, we use Hamiltonian Monte Carlo (HMC) <cit.> sampling to perform posterior inference and generate samples of θ^i∼ p(θ|𝒟) from the posterior of BNN parameters. Then, using the samples of θ, we make a Gaussian approximation to the function value as follows:μ(f(x)|𝒟) =1/M∑_i=1^Mϕ(x;θ^i) σ^2(f(x)|𝒟) =1/M∑_i=1^M(ϕ(x;θ^i) - μ(f(x)|𝒟))^2 + 1/τ where μ is the mean and σ^2 is the variance approximation. §.§.§ Trust-Region Search EngineWe follow the trust region approach introduced in <cit.> and confine the candidate points locally. The trust-region assigns a localized subset of the search space and proceeds in rounds. We denote the trust region by Ω. In each round, a batch of q designs in Ω are selected by the BNN algorithm and then simulated in parallel. Note that this procedure is easily extended to asynchronous batch evaluations, and we adapt asynchronous evaluation for the multi-fidelity BNN algorithm (will be discussed), where evaluation times show significant differences. The trust-region is centered around the best design explored, i.e., the design with minimum FoM where the ties are handled according to the design objective. This approach mitigates common issues of Bayesian optimization in high-dimensional settings, where popular acquisition functions fail to focus on promising regions and spread out samples due to large prediction uncertainty.Thompson Sampling-based Exploration: We employ Thompson sampling to obtain performance approximations for untested design candidates. Thompson sampling scales to large batches at low computational cost and has shown to be as effective as the expected improvement acquisition function<cit.>. Further, the Thompson sampling naturally extends to constrained settings which is usually the case for AMS automation. To select a point for the next batch, we sample r candidate points in Ω. Let x_1, …, x_r be the sampled candidate points. Then we use the predictive model given in Equation <ref>, and sample a realization ( f̂_0(x_i), f̂_1(x_i), . . . , f̂_m(x_i))^T for all x_i with 1 ≤ i ≤ r from the respective posterior distributions on the functions f_0, f_1,…, f_m. Let F̂={x_i|f̂_j(x_i) ≤ 0. for.1 ≤ j ≤ m} be the set of points whose realizations are feasible. If F̂≠∅ holds, we select an argmin_x ∈F̂f̂(x), i.e., the design with minimum objective. Otherwise, we select a point of minimum total violation based on the FoM definition given in Equation <ref>.Maintaining the trust-region: We initialize a trust region as a hypercube with side length L around the maximum utility point. As the optimization progresses, we track the number of successes n_s and failures n_f since the last time the trust-region is updated. A success is when the algorithm improves the solution quality, and by construction, this point must be inside the trust region. We call it a failure when the last batch of simulated designs is worse than the current best solution. The center C of the trust region is updated as follows. If there exist feasible designs, the one with the minimum objective is assigned as the center. Otherwise, the design with minimum FoM, i.e., minimum scaled constraint violation, is chosen as the center. Therefore, the center of the trust-region is updated every time the design performance is improved. The side length of the trust region is updated as follows: if n_s = τ_s then the side length is set to L = min{2L, L_max} and we reset ns = 0. If n_f = τ_f , then we set L = L/2 and n_f = 0. If the side length drops below a set threshold L_min, we initialize a new trust region. §.§ Post-Layout Performance OptimizationWe start our discussion by defining the modifications necessary to automate post-layout performance-based AMS sizing. We tailor the classical sizing flow to include the post-layout effects on the performance during sizing. Instead of optimizing the design variables based on the schematic level simulations, we utilize the layout automation tool MAGICAL to modify performance evaluation steps. The suggested flow is shown in Fig <ref>. First, an automated layout is generated via MAGICAL to obtain the post-layout performance of each new design. This step is followed by parasitic extraction, and circuit simulations are run on the updated netlist with parasitic elements. However, this new flow is much more expensive than the schematic-level sizing task since the additional steps (layout generation, parasitic extraction, and post-layout simulations) are typically computationally expensive. Therefore, methods are sought to further increase the efficiency of the (BNN-based) optimization algorithm. As a solution, we treat this problem as a multi-fidelity problem where we have access to two different information sources for calculating circuit performance metrics. Considering that the schematic-level simulations are less accurate approximations of post-layout level simulations, we define these information sources as schematic-level simulations having the lower fidelity and post-layout simulations are the highest fidelity.We modify the BNN architecture to capture two levels of fidelities (Figure <ref>) at the output and propose a co-learning scheme similar to multi-task BNN learning <cit.>. The multi-fidelity BNN model has two output nodes where ϕ(x)[1] models the lower fidelity prediction, i.e., schematic-level performance prediction, and ϕ(x)[2] models the high fidelity prediction, i.e., post-layout performance prediction. Under the assumption that we have access to two levels of information sources, we denote the new dataset by 𝒟=𝒟_1∪𝒟_2, where 𝒟_i={(𝐱_n^k, y_n^k)}_n=1^N_k and the joint probability of the updated BNN model is given by:.999!p(θ, 𝒴, τ, |𝒳)=p(θ) p(τ) ∏_k=1^2∏_n=1^N _k𝒩(y_n^k |ϕ(𝐱_n^k)[k], τ^-1)where p(θ)=𝒩(vec(θ) |0, 𝐈) and N_k is the number training points in given fidelity level k. The joint probability expression is a combination of the data sourced from both types of simulations; therefore, we utilize the full dataset to train multi-fidelity BNN. In this way, both fidelities are learned together, and the correlations between them are captured due to shared BNN parameters.To handle the multi-fidelity problem, we adopt the following modifications to Algorithm 1: 1) We train multi-fidelity BNN models using the whole history of simulations, 𝒟. 2) The trust-region centering and length updates are based on the post-layout simulation results, i.e., highest-level fidelity results. 3) We determine the candidate selection by modifying the work of <cit.> where they propose an upper-confidence-bound selection criteria for a single objective BO. We obtain Thompson sampling-based realizations for each fidelity, i.e., {f̂_i^(1)(x), f̂_i^(2)(x), for i=0,1,…,m} where {1,2} indicate the fidelity level (schematic-level simulations and post-layout level simulations) and then calculate the low fidelity and high fidelity FoM approximations, FoM(f̂^L(x)) and FoM(f̂^H(x)) using the corresponding realizations. The candidate selection is queried according to the following utility expression:𝒰(x)= max(FoM(f̂^L(x))-Δ, FoM(f̂^H(x)))where Δ is the FoM difference between the samples with the best utility at each fidelity. In this step, we take a practical approach to convert two fidelities to each other by defining a reduction term and assign the conservative prediction as the utility value. Finally, the argmin selection is conducted on the candidate utility values to determine the next batch. 4) The current literature on multi-fidelity Bayesian optimization lacks in handling large number of constraints. Therefore, we randomly assign the fidelity (simulation type) for selected candidates and leave the fidelity selection as future work. Note that this action does not prevent us from studying the benefits of multi-fidelity handling of layout-aware sizing. However, we sacrifice potential cost-aware improvements through intelligent fidelity selection.§ EXPERIMENTSExperiment Setup and Algorithm Settings:We run our tests using 3 different AMS circuits designed with different technologies. A Two-Stage Folded Cascode Operational Transconductance Amplifier (OTA), and a Strong-Arm Latch Comparator are designed with TSMC 180nm process and used to test schematic-level sizing algorithms. Then, we demonstrate the results for layout-aware algorithms on a Two-Stage Miller OTA. This circuit is designed in TSMC 40nm technology since the layout generator used in this work, MAGICAL, is crafted for TSMC 40nm. The schematic designs for these circuits are included in Figure <ref>.We run experiments to study the effectiveness of both of the proposed algorithms. First, we test for the schematic-level sizing algorithm, which is given by Algorithm 1, and we refer to our Bayesian Neural Network Based Bayesian Optimization algorithm as "BNN-BO". Then, we run tests for our post-layout performance-based sizing algorithm. Since we utilize a multi-fidelity BNN for this task, we will refer to this algorithm as "MF-BNN-BO".We implemented several state-of-the-art baseline algorithms to compare and quantify the quality of our proposed algorithms. We selected the baseline algorithms to cover the different categories of approaches. We list the compared baseline algorithms as follows: 1) A differential evolution global optimization algorithm (DE), 2) Bayesian Optimization with weighted expected improvement (BO) <cit.>, and, 3) RL-based sizing algorithm, DNN-Opt <cit.>. All algorithms are implemented using Python. We implemented DNN-Opt via PyTorch <cit.>, Bayesian Optimization algorithm is implemented using BoTorch <cit.> package and BNN-BO and MF-BNN-BO are implemented using PyTorch and Hamiltorch <cit.> packages.We configured BNN-BO and MF-BNN-BO to evaluate a batch of q=8 designs in parallel. For fairness, DNN-Opt and BO are also configured to do parallel evaluations. Both our algorithms use 200 HMC samples to train BNN models. All BNN models are feedforward neural networks with 2 hidden layers and 100 nodes at each hidden layer. Trust-region is initiated with L=0.8 and L_min and L_max are chosen to be 0.5^4 and 1.6, respectively. Failure and success tolerances as chosen as n_f=2 and n_s=3. All experiments are run on the same machine using CPU for training learning (DNN and BNN) models. During experiments, the model-based algorithms BO, DNN-Opt, and BNN-BO are run until exploring 500 designs, and DE is run for 5000 new samples.Schematic-Level Sizing Automation: We tested our algorithm, BNN-BO, and other baseline algorithms on two circuits: two-stage folded cascode ota and strong-arm latch comparator. All transistors in both designs are parameterized for optimization. The Folded Cascode OTA has 20 independent design variables, and the Strong-Arm Latch Comparator has 13 independent design variables after respecting the symmetry constraints. The parameterized device sizes include: transistor lengths & widths, capacitor values, and multipliers (integer valued). The schematic-level constrained sizing problem for Folded Cascode OTA is defined as follows:0.92![ minimize Power;s.t.DC Gain>60dBSettling Time<30 ns; [CMRR>80dBSaturation Margin>50 mV;PSRR>80dB Unity Gain Freq.>30MHz;Out. Swing>2.4V Out. Noise<30 mV_rms; Static error<0.1Phase Margin>60deg.;] ] The schematic-level constrained sizing problem for Strong-Arm Latch Comparator is defined as follows:[ minimize Power; s.t.Set Delay<10ns; [Reset Delay<6.5ns; Input-referred Noise< 50μ Vrms; Differential Reset Voltage< 1μ V;Differential Set Voltage> 1.195 V; Positive-Integration Node Reset Voltage< 60μ V; Negative-Integration Node Reset Voltage< 60μ V;Positive-Output Node Reset Voltage< 0.35μ V; Negative-Output Node Reset Voltage< 0.35μ V.;] ] We show the accuracy of the BNN modeling by demonstrating the training metrics. Training Mean Squared Error (MSE) and the logarithmic likelihood of the fitted model are given in Figure <ref>. Collecting new HMC samples from the posterior increases the likelihood and reduces the training error. We observed very similar training schemes for all other circuits and performance metrics.We repeat all experiments 10 times to account for the randomization involved in tested algorithms. The statistical results of our tests are shown in Table <ref>. Testing on both circuits suggests that BNN-BO can achieve feasible solutions in all runs, and it uses the smallest number of simulations to achieve this. Compared to Differential Evolution (DE), BNN-BO can find feasible solutions using up to 40x less number of simulations. Compared to the closest baseline algorithm, DNN-Opt, BNN-BO reduces the simulation time for finding similar results by up to 55%, proving its high efficiency. It is also demonstrated in Table <ref> that, on average, the final design proposed by BNN-BO draws up to 40% less power. The only disadvantage of BNN-BO to DNN-Opt is the modeling time as DNN-Opt maintains a single DNN model to approximate all performance metrics. Note that all reported times consider the full simulation budget (500 new samples). Therefore, although it takes longer time for BNN-BO to do a single iteration, the required real time for BNN-BO to find a feasible solution is still smaller than other approaches. In addition to experiment statistics, we further include the FoM convergence curves of both tests in Figure <ref> and Figure <ref>. The y-axis in the graphs represents the total constraint violation; therefore, FoM=0 represents a feasible solution. We observe that, compared to DNN-Opt, BNN-BO has 65% and 33% smaller area under the curve for Folded Cascode OTA and SA Latch Comparator, respectively.Layout-Aware Design Automation: In order to demonstrate the importance of layout effects on the final performance, we perform experiments on a Miller OTA circuit designed in 40nm technology (Fig. <ref>). The optimization problem has 17 independent design variables and the optimization problem is defined as follows:1![ minimize Power; s.t.DC Gain>45dBSettling Time<100 ns; [CMRR>55dB Saturation Margins>50 mV;PSRR>55dB Unity Gain BW.>40MHz;Out. Swing>1V RMS Noise<400 uV_rms;Static error<%2Phase Margin>60deg.;] ] Obtaining the post-layout performance of the Miller OTA is around 9 times more expensive than obtaining the schematic-level performance. Therefore this experiment is to prove the efficiency by utilizing multiple information sources. We initialize all algorithms with 50 high-fidelity random samples, and MF-BNN-BO has additional 50 samples from low-fidelity source (schematic-level simulations). We demonstrate the FoM evolution for the rest of the optimization steps in Figure <ref>. We observe that our Multi-Fidelity BNN algorithm provides even more efficiency compared to already efficient BNN-BO. Our analysis shows that the area under the curve is 45% smaller for MF-BNN-BO compared to BNN-BO. Further, we observe that BNN-BO's average best solution after 150 high-fidelity iterations is surpassed by MF-BNN-BO only using 84 simulations. This also implies close to 45% improved efficiency due to utilizing correlations between the schematic-level evaluations and post-layout level evaluations. Note that there is an equal number of schematic-level simulations while running MF-BNN-BO that are not reflected in Figure <ref>. Considering these simulations, the time efficiency is slightly reduced to around 38%. This efficiency figure serves as a lower-bound since we leave the improvements on fidelity selection as a future work. § CONCLUSIONIn this work, we presented Bayesian Neural Network-based solutions for schematic-level analog sizing automation and post-layout performance optimization. We targeted the scalability issue of the learning-based automation methods and provided a sample efficient optimization flow. We demonstrated the efficiency of the proposed approaches on academic benchmarks. Compared to the state-of-the-art, we improved the sizing automation efficiency by up to 45%. The Multi Fidelity BNN algorithm analysis proved that utilizing cheaper (schematic-level) simulations reduces the need for expensive (post-layout) simulations considerably, further boosting the efficiency. IEEEtran | http://arxiv.org/abs/2311.17073v1 | {
"authors": [
"Ahmet F. Budak",
"Keren Zhu",
"David Z. Pan"
],
"categories": [
"cs.LG",
"cs.CE",
"cs.SY",
"eess.SY",
"math.OC"
],
"primary_category": "cs.LG",
"published": "20231127190243",
"title": "Practical Layout-Aware Analog/Mixed-Signal Design Automation with Bayesian Neural Networks"
} |
Machine-to-Machine Transfer Function in Deep Learning-Based Quantitative Ultrasound Ufuk Soylu, Student Member, IEEE and Michael L. Oelze, Senior Member, IEEE This research received financial support from grants provided by the National Institutes of Health (NIH) (R01CA251939 and R01CA273700) Ufuk Soylu and Michael Oelze are with the Beckman Institute, and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. Moreover, Michael Oelze is with the Carle Illinois College of Medicine, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. (e-mail: [email protected]; [email protected]).January 14, 2024 ==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================footnote1 LMU Munich footnote1 Munich Center for Machine Learning footnote1 UCLA footnote1 University of Cambridge footnote1 Alan Turing Institute footnote1 Corresponding author () Unobserved confounding is common in many applications, making causal inference from observational data challenging. As a remedy, causal sensitivity analysis is an important tool to draw causal conclusions under unobserved confounding with mathematical guarantees. In this paper, we propose , a neural framework for generalized causal sensitivity analysis. Unlike previous work, our framework is compatible with (i) a large class of sensitivity models, including the marginal sensitivity model, f-sensitivity models, and Rosenbaum's sensitivity model; (ii) different treatment types (i.e., binary and continuous); and (iii) different causal queries, including (conditional) average treatment effects and simultaneous effects on multiple outcomes. The generality of is achieved by learning a latent distribution shift that corresponds to a treatment intervention using two conditional normalizing flows. We provide theoretical guarantees that is able to infer valid bounds on the causal query of interest and also demonstrate this empirically using both simulated and real-world data.§ INTRODUCTIONCausal inference from observational data is central to many fields such as medicine <cit.>, economics <cit.>, or marketing <cit.>. However, the presence of unobserved confounding often renders causal inference challenging <cit.>. As an example, consider an observational study examining the effect of smoking on lung cancer risk, where potential confounders, such as genetic factors influencing smoking behavior and cancer risk <cit.>, are not observed. Then, the causal relationship is not identifiable, and point identification without additional assumptions is impossible <cit.>. Causal sensitivity analysis offers a remedy by moving from point identification to partial identification. To do so, approaches for causal sensitivity analysis first impose assumptions on the strength of unobserved confounding through so-called sensitivity models <cit.> and then obtain bounds on the causal query of interest. Such bounds often provide insights that the causal quantities can not reasonably be explained away by unobserved confounding, which is sufficient for consequential decision-making in many applications <cit.>. In the above example, evidence supporting a causal effect of smoking on lung cancer even without information on genetic factors could serve as a basis for governments to implement anti-smoking policies. Existing works on causal sensitivity analysis can be loosely grouped by problem settings. These vary across (1) sensitivity models, such as the marginal sensitivity model (MSM) <cit.>, f-sensitivity model <cit.>, and Rosenbaum's sensitivity model <cit.>; (2) treatment type (i.e., binary and continuous); and (3) causal query of interest. Causal queries may include (conditional) average treatment effects (CATE), but also distributional effects or simultaneous effects on multiple outcomes. Existing works typically focus on a specific sensitivity model, treatment type, and causal query (Table <ref>). However, none is applicable to all settings within (1)–(3). To fill this gap, we propose , a neural framework for causal sensitivity analysis that is applicable to numerous sensitivity models, treatment types, and causal queries, including multiple outcome settings. For this, we define a large class of sensitivity models, which we call generalized treatment sensitivity models (GTSMs). GTSMs include common sensitivity models such as the MSM, f-sensitivity models, and Rosenbaum's sensitivity model. The intuition behind GTSMs is as follows: when intervening on the treatment A, the U–A edge is removed in the corresponding causal graph, which leads to a distribution shift in the latent confounders U (see Fig. <ref>). GTSMs then impose restrictions on this latent distribution shift, which corresponds to assumptions on the “strength” of unobserved confounding.l0.4< g r a p h i c s >Idea behind to learn the latent distribution shift due to treatment intervention (). Orange nodes denote observed (random) variables. Blue nodes denote unobserved variables pre-intervention. Green nodes indicate unobserved variables post-intervention under a GTSM ℳ. Observed confounders X are empty for simplicity. is compatible with any sensitivity model that can be written as a GTSM. This is crucial in practical applications, where sensitivity models correspond to different assumptions on the data-generating process and may lead to different results <cit.>.To achieve this, learns the latent distribution shift in the unobserved confounders from Fig. <ref> using two separately trained conditional normalizing flows (CNFs). This is different from previous works for causal sensitivity analysis, which do not provide a unified approach across numerous sensitivity models, treatment types, and causal queries. We provide theoretical guarantees that learns valid bounds on the causal query of interest and demonstrate this empirically. Our contributions[Code is available at https://anonymous.4open.science/r/NeuralCSA-DE7Dhttps://anonymous.4open.science/r/NeuralCSA-DE7D. (Link anonymized for peer review. Upon acceptance, the code will be moved to a public GitHub repository.] are: (1) We define a general class of sensitivity models, called GTSMs. (2) We propose , a neural framework for causal sensitivity analysis under any GTSMs. is compatible with various sensitivity models, treatment types, and causal queries. In particular, is applicable in settings for which bounds are not analytically tractable and no solutions exist yet. (3) We provide theoretical guarantees that learns valid bounds on the causal query of interest and demonstrate the effectiveness of our framework empirically.§ RELATED WORK In the following, we provide an overview of related literature on partial identification and causal sensitivity analysis. A more detailed overview, including literature on point identification and estimation, can be found in Appendix <ref>.Partial identification: The aim of partial identification is to compute bounds on causal queries whenever point identification is not possible, such as under unobserved confounding <cit.>. There are several literature streams that impose different assumptions on the data-generating process in order to obtain informative bounds. One stream addresses partial identification for general causal graphs with discrete variables <cit.>. Another stream assumes the existence of valid instrumental variables <cit.>. Recently, there has been a growing interest in using neural networks for partial identification <cit.>. However, none of these methods allow for incorporating sensitivity models and sensitivity analysis.Causal sensitivity analysis: Causal sensitivity analysis addresses the partial identification of causal queries by imposing assumptions on the strength of unobserved confounding via sensitivity models. It dates back to <cit.>, who showed that unobserved confounding could not reasonably explain away the observed effect of smoking on lung cancer risk. r0.65 Overview of key settings for causal sensitivity analyses and whether covered by existing literature () or not (). Treatments are either binary or continuous. Details are in Appendix <ref>. Our framework is applicable in all settings0.65!Causal querySensitivity model2cMSM 2cf-sensitivity 2cRosenbaum(lr)2-3 (lr)4-5 (lr)6-7 Binary Cont.^† Binary Cont. Binary Cont.CATE Distributional effectsInterventional density ()Multiple outcomes7p0.7^† The MSM for continuous treatment is also called continuous MSM (CMSM) <cit.>. Existing works can be grouped along three dimensions: (1) the sensitivity model, (2) the treatment type, and (3) the causal query of interest (see Table <ref>; details in Appendix <ref>). Popular sensitivity models include Rosenbaum's sensitivity model <cit.>, the marginal sensitivity model (MSM) <cit.>, and f-sensitivity models <cit.>. Here, most methods have been proposed for binary treatments and conditional average treatment effects <cit.>. Extensions under the MSM have been proposed for continuous treatments <cit.> and individual treatment effects <cit.>. However, approaches for many settings are still missing (shown by in Table <ref>). In an attempt to generalize causal sensitivity analysis, <cit.> provided bounds for different treatment types (i.e., binary, continuous) and causal queries (e.g., CATE, distributional effects but not multiple outcomes). Yet, the results are limited to MSM-type sensitivity models. To the best of our knowledge, no previous work proposes a unified solution for obtaining bounds under various sensitivity models (e.g., MSM, f-sensitivity, Rosenbaum's), treatment types (i.e., binary and continuous), and causal queries (e.g., CATE, distributional effects, interventional densities, and simultaneous effects on multiple outcomes). § MATHEMATICAL BACKGROUND Notation: We denote random variables X as capital letters and their realizations x in lowercase. We further write ℙ(x) for the probability mass function if X is discrete, and for the probability density function with respect to the Lebesque measure if X is continuous. Conditional probability mass functions/ densities ℙ(Y = y | X = x) are written as ℙ(y | x). Finally, we denote the conditional distribution of Y | X = x as ℙ(Y | x) and its expectation as 𝔼[Y | x]. §.§ Problem setup Data generating process: We consider the standard setting for (static) treatment effect estimation under unobserved confounding <cit.>. That is, we have observed confounders X ∈𝒳⊆^d_x, unobserved confounders U ∈𝒰⊆^d_u, treatments A ∈𝒜⊆^d_a, and outcomes Y ∈𝒴⊆^d_y. Note that we allow for (multiple) discrete or continuous treatments and multiple outcomes, i.e., d_a, d_y ≥ 1. The underlying causal graph is shown in Fig. <ref>. We have access to an observational dataset 𝒟 = (x_i, a_i, y_i)_i=1^n sampled i.i.d. from the observational distribution (X, A, Y) ∼ℙ_obs. The full distribution (X, U, A, Y) ∼ℙ is unknown.r0.20< g r a p h i c s > Causal graph. Observed variables are colored orange and unobserved blue. We allow for arbitrary dependence between X and U.We use the potential outcomes framework to formalize the causal inference problem <cit.> and denote Y(a) as the potential outcome when intervening on the treatment and setting it to A = a. We impose the following standard assumptions <cit.>.We assume that for all x ∈𝒳 and a ∈𝒜 the following three conditions hold: (i) A=a implies Y(a) = Y(consistency); (ii) ℙ(a | x) > 0 (positivity); and (iii) Y(a)A | X, U (latent unconfoundedness). Causal queries: We are interested in a wide range of general causal queries. We formalize them as functionals Q(x, a, ℙ) = ℱ(ℙ(Y(a) | x)), where ℱ is a functional that maps the potential outcome distribution ℙ(Y(a) | x) to a real number <cit.>. Thereby, we cover various queries from the causal inference literature. For example, by setting ℱ = 𝔼[·], we obtain the conditional expected potential outcomes/ dose-response curves Q(x, a, ℙ) = 𝔼[Y(a) | x]. We can also obtain distributional versions of these queries by setting ℱ to a quantile instead of the expectation. Furthermore, our methodology will also apply to queries that can be obtained by averaging or taking differences. For binary treatments A ∈{0,1}, the query τ(x) = 𝔼[Y(1) | x] - 𝔼[Y(0) | x] is called the conditional average treatment effect (CATE), and its averaged version ∫τ(x) ℙ(x)x the average treatment effect (ATE).Our formalization also covers simultaneous effects on multiple outcomes (i.e., d_y ≥ 2). Consider query Q(x, a, ℙ) = ℙ(Y(a) ∈𝒮| x), which is the probability that the outcome Y(a) is contained in some set 𝒮⊆𝒴 after intervening on the treatment. For example, consider two potential outcomes Y_1(a) and Y_2(a) denoting blood pressure and heart rate, respectively. We then might be interested in ℙ(Y_1(a) ≤ t_1, Y_2(a) ≤ t_2 | x), where t_1 and t_2 are critical threshold values (see Sec. <ref>). §.§ Causal sensitivity analysis Causal sensitivity analysis builds upon sensitivity models that restrict the possible strength of unobserved confounding <cit.>. Formally, we define a sensitivity model as a family of distributions of (X, U, A, Y) that induce the observational distribution ℙ_obs. A sensitivity model ℳ is a family of probability distributions ℙ defined on 𝒳×𝒰×𝒜×𝒴 for arbitrary finite-dimensional 𝒰 so that ∫_𝒰ℙ(x, u, a, y)u = ℙ_obs(x, a, y) for all ℙ∈ℳ. Task: Given a sensitivity model ℳ and an observational distribution ℙ_obs, the aim of causal sensitivity analysis is to solve the partial identification problemQ^+_ℳ(x, a) =sup_ℙ∈ℳQ(x, a, ℙ) and Q^-_ℳ(x, a) =inf_ℙ∈ℳQ(x, a, ℙ).By its definition, the interval [Q^-_ℳ(x, a), Q^+_ℳ(x, a)] is the tightest interval that is guaranteed to contain the ground-truth causal query Q(x, a, ℙ) while satisfying the sensitivity constraints. We can also obtain bounds for averaged causal queries and differences via ∫ Q^+_ℳ(x, a) ℙ(x)x and Q^+_ℳ(x, a_1) - Q^-_ℳ(x, a_2) (see Appendix <ref> for details).Sensitivity models from the literature: We now recap three types of prominent sensitivity models from the literature, namely, the MSM, f-sensitivity models, and Rosenbaum's sensitivity model. These are designed for binary treatments A ∈{0, 1}. To formalize them, we first define the odds ratio OR(a, b) =a/(1 - a)(1 - b)/b, the observed propensity score π(x) = ℙ(A=1 | x), and the full propensity score π(x, u) = ℙ(A=1 | x, u).[Corresponding sensitivity models for continuous treatments can be defined by replacing the odds ratio with the density ratio DR(a, b) = a/b and the propensity scores with the densities ℙ(a | x) and ℙ(a | x, u) <cit.>. We refer to Appendix <ref> for details and further examples of sensitivity models.] Then, the definitions are: * The marginal sensitivity model (MSM) <cit.> is defined as the family of all ℙ that satisfy 1/Γ≤OR(π(x), π(x,u)) ≤Γ for all x ∈𝒳 and u ∈𝒰 and a sensitivity parameter Γ≥ 1. * f-sensitivity models <cit.> build upon a given a convex function f _>0→ with f(1) = 0 and are defined viamax{∫_𝒰 f(OR(π(x), π(x,u))) ℙ(u | x, A=1)u,∫_𝒰 f(OR^-1(π(x), π(x,u))) ℙ(u | x, A=1)u}≤Γ for all x ∈𝒳. * Rosenbaum's sensitivity model <cit.> is defined via 1/Γ≤OR(π(x, u_1), π(x,u_2)) ≤Γ for all x ∈𝒳 and u_1, u_2 ∈𝒰. Interpretation and choice of Γ: In the above sensitivity models, the sensitivity parameter Γ controls the strength of unobserved confounding. Both MSM and Rosenbaum's sensitivity model bound on odds-ratio uniformly over all u ∈𝒰, while the f-sensitivity model bounds an integral over u. We refer to Appendix <ref> for further differences. Setting Γ = 1 (MSM, Rosenbaum) or Γ = 0 (f-sensitivity) corresponds to unconfoundedness and thus point identification. For Γ > 1 or Γ > 0, point identification is not possible, and we need to solve the partial identification problem from Eq. (<ref>) instead. In practice, one typically chooses Γ by domain knowledge or data-driven heuristics <cit.>. For example, a common approach in practice is to determine the smallest Γ so that the partially identified interval [Q^-_Γ(x, a), Q^+_Γ(x, a)] includes 0. Then, Γ can be interpreted as a level of “causal uncertainty”, quantifying the smallest violation of unconfoundedness that would explain away the causal effect <cit.>.§ THE GENERALIZED TREATMENT SENSITIVITY MODEL (GTSM) We now define our generalized treatment sensitivity model (GTSM). The GTSM subsumes a large class of sensitivity models and includes MSM, f-sensitivity, and Rosenbaum's sensitivity model).Motivation: Intuitively, we define the GTSM so that it includes all sensitivity models that restrict the latent distribution shift in the confounding space due to the treatment intervention (see Fig. <ref>). To formalize this, we can write the observational outcome density under Assumption <ref> asℙ_obs(y | x, a) = ∫ℙ(y | x, u, a)ℙ(u | x, a)u.When intervening on the treatment, we remove the U–A edge in the corresponding causal graph (Fig. <ref>) and thus artificially remove dependence between U and A. Formally, we can write the potential outcome density under Assumption <ref> asℙ(Y(a) = y | x) = ∫ℙ(Y(a) = y | x, u) ℙ(u | x)u = ∫ℙ(y | x, u, a) ℙ(u | x)u.Eq. (<ref>) and (<ref>) imply that ℙ_obs(y | x, a) and ℙ(Y(a) = y | x) only differ by the densities ℙ(u | x, a) and ℙ(u | x) under the integrals (colored red and orange). If the distributions ℙ(U | x, a) and ℙ(U | x) would coincide, it would hold that ℙ(Y(a) = y | x) = ℙ_obs(y | x, a) and the potential outcome distribution would be identified. This suggests that we should define sensitivity models by measuring deviations from unconfoundedness via the shift betweenℙ(U | x, a) and ℙ(U | x). A generalized treatment sensitivity model (GTSM) is a sensitivity model ℳ that contains all probability distributions ℙ that satisfy 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) ≤Γ for a functional of distributions 𝒟_x, a, a sensitivity parameter Γ∈_≥ 0, and all x ∈𝒳 and a ∈𝒜. The MSM, the f-sensitivity model, and Rosenbaum's sensitivity model are GTSMs. See Appendix <ref>.The class of all GTSMs is still too large for meaningful sensitivity analysis. This is because the sensitivity constraint may not be invariant w.r.t. transformations (e.g., scaling) of the latent space 𝒰.A GTSM ℳ is transformation-invariant if it satisfies 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) ≥𝒟_x, a(ℙ(t(U) | x), ℙ(t(U) | x, a)) for any measurable function t 𝒰→𝒰 to another latent space 𝒰. Transformation-invariance is necessary for meaningful sensitivity analysis because it implies that once we choose a latent space 𝒰 and a sensitivity parameter Γ, we cannot find a transformation to another latent space 𝒰 so that the induced distribution on 𝒰 violates the sensitivity constraint.All sensitivity models we consider in this paper are transformation-invariant, as stated below. The MSM, f-sensitivity models, and Rosenbaum's sensitivity model are transformation-invariant. See Appendix <ref>. § NEURAL CAUSAL SENSITIVITY ANALYSIS We now introduce our neural approach to causal sensitivity analysis as follows. First, we simplify the partial identification problem from Eq. (<ref>) under a GTSM and propose a (model-agnostic) two-stage procedure (Sec. <ref>). Then, we provide theoretical guarantees for our two-stage procedure (Sec. <ref>). Finally, we instantiate our neural framework called (Sec. <ref>).§.§ Sensitivity analysis under a GTSM Motivation: Recall that, by definition, a GTSM imposes constraints on the distribution shift in the latent confounders due to treatment intervention (Fig. <ref>). Our idea is to propose a two-stage procedure, where Stage 1 learns the observational distribution (Fig. <ref>, left), while Stage 2 learns the shifted distribution of U after intervening on the treatment under a GTSM (Fig. <ref>, right). In Sec. <ref>, we will see that, under weak assumptions, learning this distribution shift in separate stages is guaranteed to lead to the bounds Q^+_ℳ(x, a) and Q^-_ℳ(x, a). To formalize this, we start by simplifying the partial identification problem from Eq. (<ref>) for a GTSM ℳ.Simplifying Eq. (<ref>): We begin by rewriting Eq. (<ref>) using the GTSM definition. Without loss of generality, we consider the upper bound Q^+_ℳ(x, a). Recall that Eq. (<ref>) seeks to maximize over all probability distributions that are compatible both with the observational data and with the sensitivity model. However, note that any GTSM only restricts the U–A part of the distribution, not the U–Y part. Hence, we can use Eq. (<ref>) and Eq. (<ref>) to write the upper bound asQ^+_ℳ(x, a) = sup_{ℙ(U | x, a^')}_a^'≠ a s.t. 𝒟_x, a(ℙ(U | x) , ℙ(U | x, a)) ≤Γ and ℙ(u | x) = ∫ℙ(u | x, a) ℙ_obs(a | x)asup_ℙ(U | x, a),{ℙ(Y | x, u, a)}_u ∈𝒰 s.t. Eq. (<ref>) holdsℱ(∫ℙ(Y | x, u, a) ℙ(u | x)u),where we maximize over (families of) probability distributions {ℙ(U | x, a^')}_a^'≠ a (left supremum), and ℙ(U | x, a), {ℙ(Y | x, u, a)}_u ∈𝒰 (right supremum). The coloring indicates the components that appear in the causal query/objective. The constraint in the right supremum ensures that the respective components of the full distribution ℙ are compatible with the observational data, while the constraints in the left supremum ensure that the respective components are compatible with both observational data and the sensitivity model.r0.48 < g r a p h i c s >Overview of the two-stage procedure.The partial identification problem from Eq. (<ref>) is still hard to solve as it involves two nested constrained optimization problems. However, it turns out that we can further simplify Eq. (<ref>): We will show in Sec. <ref> that we can replace the right supremum with fixed distributions ℙ^∗(U | x, a) and ℙ^∗(Y | x, a, u) for all u ∈𝒰⊆^d_y so that Eq. (<ref>) holds. Then, Eq. (<ref>) reduces to a single constrained optimization problem (left supremum). Moreover, we will show that we can choose ℙ^∗(Y | x, a, u) = δ(Y - f^∗_x, a(u)) as a delta-distribution induced by an invertible function f^∗_x, a𝒰→𝒴. The constraint in Eq. (<ref>) that ensures compatibility with the observational data then reduces to ℙ_obs(Y | x, a) =ℙ^∗(f^∗_x, a(U) | x, a). This motivates our following two-stage procedure (see Fig. <ref>). Two-stage procedure: In Stage 1, we fix ℙ^∗(U | x, a) and fix an invertible function f^∗_x, a𝒰→𝒴 so that ℙ_obs(Y | x, a) =ℙ^∗(f^∗_x, a(U) | x, a) holds. That is, the induced push-forward distribution of ℙ^∗(U | x, a) under f^∗_x, a must coincide with the observational distribution ℙ_obs(Y | x, a). The existence of such a function is always guaranteed <cit.>. In Stage 2, we then set ℙ(U | x, a) = ℙ^∗(U | x, a) and ℙ(Y | x, a, u) = ℙ^∗(Y | x, a, u) in Eq. (<ref>) and only optimize over the left supremum. That is, we write stage 2 for discrete treatments assup_ℙ(u | x, A ≠ a)s.t. ℙ(u | x) = ℙ^∗(u | x, a) ℙ_obs(a | x) + ℙ(u | x, A ≠ a)(1 - ℙ_obs(a | x))and 𝒟_x, a( ℙ(U | x) , ℙ^∗(U | x, a)) ≤Γℱ(ℙ(f^∗_x, a(U) | x) ),where we maximize over the distribution ℙ(u | x, A ≠ a) for a fixed treatment intervention a. For continuous treatments, we can directly take the supremum over ℙ(u | x). §.§ Theoretical guaranteesWe now provide a formal result that our two-stage procedure returns valid solutions to the partial identification problem from Eq. (<ref>). The following theorem states that Stage 2 of our procedure is able to attain the optimal upper bound Q^+_ℳ(x, a) from Eq. (<ref>), even after fixing the distributions ℙ^∗(U | x, a) and ℙ^∗(Y | x, a, u) as done in Stage 1. A proof is provided in Appendix <ref>.Let ℳ be a transformation-invariant GTSM. For fixed x ∈𝒳 and a ∈𝒜, let ℙ^∗(U | x, a) be a fixed distribution on 𝒰 = ^d_y and f^∗_x, a𝒰→𝒴 a fixed invertible function so that ℙ_obs(Y | x, a) =ℙ^∗(f^∗_x, a(U) | x, a). Let 𝒫^∗ denote the space of all full probability distributions ℙ^∗ that induce ℙ^∗(U | x, a) and ℙ^∗(Y | x, u, a) = δ(Y - f^∗_x, a(u)) and that satisfy ℙ^∗∈ℳ. Then, under Assumption <ref>, it holds that Q^+_ℳ(x, a) = sup_ℙ^∗∈𝒫^∗ Q(x, a, ℙ^∗) and Q^-_ℳ(x, a) = inf_ℙ^∗∈𝒫^∗ Q(x, a, ℙ^∗). See Appendix <ref>. Theorem <ref> has two major implications: (i) It is sufficient to fix the distributions ℙ^∗(U | x, a) and ℙ^∗(Y | x, u, a), i.e., the components in the right supremum of Eq. (4) and only optimize over the left supremum; and (ii) it is sufficient to choose ℙ^∗(Y | x, u, a) = δ(Y - f^∗_x, a(u)) as a delta-distribution induced by an invertible function f^∗_x, a𝒰→𝒴, which satisfies the data-compatibility constraint ℙ_obs(Y | x, a) =ℙ^∗(f^∗_x, a(U) | x, a).Intuition for (i): In Eq. (<ref>), we optimize jointly over all components of the full distribution. This suggests that there are multiple solutions that differ only in the components of unobserved parts of ℙ (i.e., in 𝒰) but lead to the same potential outcome distribution and causal query. Theorem <ref> states that we may restrict the space of possible solutions by fixing the components ℙ^∗(U | x, a) and ℙ^∗(Y | x, a, u), without loosing the ability to attain the optimal upper bound Q^+_ℳ(x, a) from Eq. (<ref>).Intuition for (ii): We cannot pick any ℙ^∗(Y | x, a, u) that satisfies Eq. (<ref>). For example, any distribution that induces YU | X, A would satisfy Eq. (<ref>), but implies unconfoundedness and would thus not lead to a valid upper bound Q^+_ℳ(x, a). Intuitively, we have to choose a ℙ(Y | x, a, u) that induces “maximal dependence" (mutual information) between U and Y (conditioned on X and A), because the GTSM does not restrict this part of the full probability distribution ℙ. The maximal mutual information is achieved if we choose ℙ(Y | x, a, u) = δ(Y - f^∗_x, a(u)).§.§ Neural instantiation:We now provide a neural instantiation called for the above two-stage procedure using conditional normalizing flows (CNFs) <cit.>. The architecture of is shown in Fig. <ref>. instantiates the two-step procedure as follows: r0.5< g r a p h i c s >Architecture of . Stage 1: We fix ℙ^∗(U | x, a) to the standard normal distribution on 𝒰 = ^d_y. Our task is then to learn an invertible function f^∗_x, a𝒰→𝒴 that maps the standard Gaussian distribution on 𝒰 to ℙ_obs(Y | x, a). We model f^∗_x, a as a CNF f^∗_g^∗_θ(x, a), where f^∗ is a normalizing flow <cit.>, for which the parameters are the output of a fully connected neural network g^∗_θ, which itself is parametrized by θ <cit.>. We obtain θ by maximizing the empirical Stage 1 lossℒ_1(θ) =∑_i=1^n logℙ (f^∗_g^∗_θ(x_i, a_i)(U) = y_i), where U ∼𝒩(0_d_y, I_d_y) is standard normally distributed. The stage 1 loss can be computed analytically via the change-of-variable formula (see Appendix <ref>).Stage 2: In Stage 2, we need to maximize over distributions on U in the latent space 𝒰 that maximize the causal query ℱ(ℙ(f^∗_g^∗_θ_opt(x, a)(U) | x)), where θ_opt is a solution from maximizing ℒ_1(θ) in stage 1. We can do this by learning a second CNF f_g_η(x, a), where f𝒰→𝒰 is a normalizing flow that maps a standard normally distributed auxiliary U∼𝒩(0_d_y, I_d_y) to the latent space 𝒰, and whose parameters are the output of a fully connected neural network g_η parametrized by η. The CNF f_g_η(x, a) from Stage 2 induces a new distribution on U, which mimics the shift due to unobserved confounding when intervening instead of conditioning (i.e., going from Eq. (<ref>) to Eq. (<ref>)). We can compute the query under the shifted distribution by concatenating the Stage 2 CNF with the Stage 1 CNF and applying ℱ to the shifted outcome distribution (see Fig. <ref>). More precisely, we optimize η by maximizing or minimizing the empirical Stage 2 loss .7!ℒ_2(η) =∑_i=1^n ℱ( ℙ(f^∗_g^∗_θ_opt(x_i, a_i)( (1 - ξ_x_i, a_i) f_g_η(x_i, a_i)(U) +ξ_x_i, a_iU) ) ),L0.46 0.46where ξ_x_i, a_i = ℙ_obs(a_i | x_i)), if A is discrete, and ξ_x_i, a_i = 0, if A is continuous. Learning algorithm for stage 2: There are two remaining challenges we need to address in Stage 2: (i) optimizing Eq. (<ref>) does not ensure that the sensitivity constraints imposed by the GTSM ℳ hold; and (ii) computing the Stage 2 loss from Eq. (<ref>) may not be analytically tractable. For (i), we propose to incorporate the sensitivity constraints by using the augmented Lagrangian method <cit.>, which has already been successfully applied in the context of partial identification with neural networks <cit.>. For (ii), we propose to obtain samples u = (u^(j)_x, a)_j=1^k i.i.d.∼𝒩(0_d_y, I_d_y) and ξ = (ξ^(j)_x, a)_j=1^k i.i.d.∼Bernoulli(ℙ_obs(a | x)) together with Monte Carlo estimators ℒ̂_2(η, u, ξ) of the Stage 2 loss ℒ_2(η) and 𝒟̂_x, a(η, u) of the sensitivity constraint 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)). A sketch of the learning algorithm is shown in Algorithm <ref>. We refer to Appendix <ref> for details, including instantiations of our framework for numerous sensitivity models and causal queries.Analytical potential outcome density: Once our model is trained, we can not only compute the bounds via sampling but also the analytical form (by using the density transformation formula) of the potential outcome density that gives rise to that bound. Fig. <ref>, shows an example. This makes it possible to perform sensitivity analysis for the potential outcome density itself, i.e., analyzing the “distribution shift due to intervention".Implementation: We use autoregressive neural spline flows <cit.>. For estimating propensity scores ℙ_obs(a | x), we use fully connected neural networks with softmax activation. We perform training using the Adam optimizer <cit.>. We choose the number of epochs such that satisfies the sensitivity constrained for a given sensitivity parameter. Details are in Appendix <ref>.§ EXPERIMENTSr0.60.30< g r a p h i c s > 0.30< g r a p h i c s > Validating the correctness of (ours) by comparing with optimal closed-form solutions (CF) for the MSM on simulated data. Left: Dataset 1, binary treatment. Right: Dataset 2, continuous treatment. Reported: mean ± standard deviation over 5 runs. We now demonstrate the effectiveness of for causal sensitivity analysis empirically. As is common in the causal inference literature, we use synthetic and semi-synthetic data with known causal ground truth to evaluate <cit.>. We proceed as follows: (i) We use synthetic data to show the validity of bounds from under multiple sensitivity models, treatment types, and causal queries. We also show that for the MSM, the bounds coincide with known optimal solutions. (ii) We show the validity of the bounds using a semi-synthetic dataset. (iii) We show the applicability of in a case study using a real-world dataset with multiple outcomes, which cannot be handled by previous approaches. We refer to Appendix <ref> for details regarding datasets and experimental evaluation, and to Appendix <ref> for additional experiments.(i) Synthetic data: We consider two synthetic datasets of sample size n=10000 inspired from previous work on sensitivity analysis: Dataset 1 is adapted from <cit.> and has a binary treatment A ∈{0,1}. The data-generating process follows an MSM with oracle sensitivity parameter Γ^∗ = 2. We are interested in the CATE τ(x) = [Y(1) - Y(0) | x]. Dataset 2 is adapted from <cit.> and has a continuous treatment A ∈ [0,1]. Here, we are interested in the dose-response function μ(x, a) = [Y(a) | x], where we choose a = 0.5. We report results for further treatment values in Appendix <ref>.We first compare our bounds with existing results closed-form bounds (CF) for the MSM <cit.>, which have been proven to be optimal. We plot both and the CF for both datasets and three choices of sensitivity parameter Γ∈{2, 4, 10} (Fig. <ref>). Our bounds almost coincide with the optimal CF solutions, which confirms that learns optimal bounds under the MSM.l0.60.30< g r a p h i c s > 0.30< g r a p h i c s > Confirming the validity of our bounds for various sensitivity models. Left: Dataset 1, binary treatment. Right: Dataset 2, continuous treatment. Reported: mean ± standard deviation over 5 runs.We also show the validity of our bounds for Rosenbaum's sensitivity model and the following f-sensitivity models: Kullbach-Leibler (KL, f(x) = x log(x)), Total Variation (TV, f(x) = 0.5 |x-1|), Hellinger (HE, f(x) = (√(x) - 1)^2), and Chi-squared (χ^2, f(x) = (x-1)^2). To do so, we choose the ground-truth sensitivity parameter Γ^∗ for each sensitivity model that satisfies the respective sensitivity constraint (see Appendix <ref> for details). The results are in Fig. <ref>. We make the following observations: (i) all bounds cover the causal query on both datasets, thus confirming the validity of . (ii) For Dataset 1, the MSM returns the tightest bounds because our simulation follows an MSM.(ii) Semi-synthetic data: We create a semi-synthetic dataset using MIMIC-III <cit.>, which includes electronic health records from patients admitted to intensive care units. We extract 8 confounders and a binary treatment (mechanical ventilation). Then, we augment the data with a synthetic unobserved confounder and outcome. We obtain n=14719 patients and split the data into train (80%), val (10%), and test (10%). For details, see Appendix <ref>.We verify the validity of our bounds for CATE in the following way: For each sensitivity model, we obtain the smallest oracle sensitivity parameter Γ^∗ that guarantees coverage (i.e., satisfies the respective sensitivity constraint) for 50% of the test samples. Then, we plot the coverage and median interval length of the bounds over the test set. The results are in Table <ref>. l0.25< g r a p h i c s >Analytic stage 2 densities for MSM and KL-sensitivity model (upper bounds). We observe that (i) all bounds achieve at least 50% coverage, thus confirming the validity of the bounds, and (ii) some sensitivity models (e.g., the MSM) are conservative, i.e., achieve much higher coverage and interval length than needed. This is because the sensitivity constraints of these models do not adapt well to the data-generating process, thus the need for choosing a large Γ^∗ to guarantee coverage. This highlights the importance of choosing a sensitivity model that captures the data-generating process well. For further details, we refer to <cit.>. r0.4 Results for semi-synthetic data0.4!Sensitivity modelCoverage Interval lengthMSM Γ^∗ = 5.48 0.91 ± 0.03 0.77 ± 0.03 KL Γ^∗ = 0.25 0.54 ± 0.07 0.31 ± 0.01 TV Γ^∗ = 0.38 0.86 ± 0.09 0.83 ± 0.14 HE Γ^∗ = 0.18 0.83 ± 0.06 0.63 ± 0.03 χ^2 Γ^∗ = 0.68 0.67 ± 0.07 0.41 ± 0.01 RB Γ^∗ = 14.42 0.79 ± 0.07 0.56 ± 0.03 3lReported: mean ± standard deviation (5 runs). We also provide further insights into the difference between two exemplary sensitivity models: the MSM and the KL-sensitivity model. To do so, we plot the observational distribution from stage 1 together with the shifted distributions from stage 2 that lead to the respective upper bound for a fixed test patient (Fig. <ref>). The distribution shift corresponding to the MSM is a step function, which is consistent with results from established literature <cit.>. This is in contrast to the smooth distribution shift obtained by the KL-sensitivity model. In addition, this example illustrates the possibility of using for sensitivity analysis on the entire interventional density.(iii) Case study using real-world data: We now demonstrate an application of to perform causal sensitivity analysis for an interventional distribution on multiple outcomes. To do so, we use the same MIMIC-III data from our semi-synthetic experiments but add two outcomes: heart rate (Y_1) and blood pressure (Y_2). We consider the causal query ℙ(Y_1(1) ≥ 115, Y_2(1) ≥ 90 | X = x), i.e., the joint probability of achieving a heart rate higher than 115 and a blood pressure higher than 90 under treatment intervention (“danger area”). We consider an MSM and train with sensitivity parameters Γ∈{2, 4}. Then, we plot the stage 1 distribution together with both stage 2 distributions for a fixed, untreated patient from the test set in Fig. <ref>.As expected, increasing Γ leads to a distribution shift in the direction of the “danger area”, i.e., high heart rate and high blood pressure. For Γ = 2, there is only a moderate fraction of probability mass inside the danger area, while, for Γ = 4, this fraction is much larger. A practitioner may potentially decide against treatment if there are other unknown factors (e.g., undetected comorbidity) that could result in a confounding strength of Γ = 4.r0.620.2< g r a p h i c s > 0.2< g r a p h i c s > 0.2< g r a p h i c s > Contour plots of 2D densities obtained by under an MSM. Here, we aim to learn an upper bound of the causal query ℙ(Y_1(1) ≥ 115, Y_2(1) ≥ 90 | X = x_0) for a test patient x_0. Left: Stage 1/ observational distribution. Middle: Stage 2, Γ = 2. Right: Stage 2, Γ = 4.§ CONCLUSION Conclusion. From a methodological perspective, offers new ideas to causal sensitivity analysis and partial identification: In contrast to previous methods, explicitly learns a latent distribution shift due to treatment intervention. We refer to Appendix <ref> for a discussion on limitations and future work. From an applied perspective, enables practitioners to perform causal sensitivity analysis in numerous settings, including multiple outcomes. Furthermore, it allows for choosing from a wide variety of sensitivity models, which may be crucial to effectively incorporate domain knowledge about the data-generating process. For practical usage, we recommend using NeuralCSA whenever solutions are unavailable (see Table <ref>).iclr2024_conference§ EXTENDED RELATED WORK In the following, we provide an extended related work. Specifically, we elaborate on (1) a systematic overview of causal sensitivity analysis, (2) its application in settings beyond partial identification of interventional causal queries, and (3) point identification and estimation of causal queries. §.§ A systematic overview on causal sensitivity analysis In Table <ref>, we provide a systematic overview of existing works for causal sensitivity analysis, which we group by the underlying sensitivity model, the treatment type, and the causal query. As such, Table <ref> extends Table <ref> in that we follow the same categorization but now point to the references from the literature.Evidently, many works have focused on sensitivity analysis for CATE in binary treatment settings. For many settings, such as f-sensitiivity and Rosenbaum's sensitivity model with continuous treatments or multiple outcomes, no previous work exists. Here, is the first work that allows for computing bounds in these settings. §.§ Sensitivity analysis in other causal settings Causal sensitivity analysis has found applicability not only in addressing the partial identification problem, as discussed in Eq. (<ref>), but also in various domains of machine learning and causal inference. We briefly highlight some notable instances where ideas from causal sensitivity analysis have made substantial contributions.One such stream of literature is off-policy learning, where sensitivity models have been leveraged to account for unobserved confounding or distribution shifts <cit.>. Here, sensitivity analysis enables robust policy learning. Another example is algorithmic fairness, where sensitivity analysis has been used to study causal fairness notions (e.g., counterfactual fairness) under unobserved confounding <cit.>. Finally have been used to study the partial identification of counterfactual queries <cit.>§.§ Point identification and estimation If we replace the latent unconfoundedness assumption in Assumtion <ref> with (non-latent) unconfoundedness, that is,Y(a)A | X for all a ∈𝒜,we can point-identify the distribution of the potential outcomes viaℙ(Y(a) = y | x) = ℙ(Y(a) = y | x, a)= ℙ_obs(y | x, a).Hence, inferring the causal query Q(x, a, ℙ) reduces to a purely statistical inference problem, i.e., estimating ℱ(ℙ_obs(Y | x, a)) from finite data.Various methods for estimating point-identified causal effects under unconfoundedness have been proposed. In recent years, a particular emphasis has been on methods for estimating (conditional) average treatment effects (CATEs) that make use of machine learning to model flexible non-linear relationships within the data. Examples include forest-based methods <cit.> and deep learning <cit.>. Another stream of literature incorporates theory from semi-parametric statistics and provides robustness and efficiency guarantees <cit.>. Beyond CATE, methods have also been proposed for estimating distributional effects or potential outcome densities <cit.>. We emphasize that all these methods focus on estimation of point-identified causal queries, while we are interested in causal sensitivity analysis and thus partial identification under violations of the unconfoundedness assumption.§ PROOFS§.§ Proof of Lemma <ref> We provide a proof for the following more detailed version of Lemma <ref>. The MSM, the f-sensitivity model, and Rosenbaum's sensitivity model are GTSMs with sensitivity parameter Γ. Let ρ(x, u, a) = 1/1 - ℙ(a | x)(ℙ(u | x)/ℙ(u | x, a) - ℙ(a | x) ) and ρ(x, u_1, u_2, a) = ℙ(u_1 | x, a) ℙ(u_2 | x) - ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x)/ℙ(u_2 | x, a) ℙ(u_1 | x) - ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x). For the MSM, we have𝒟_x,a(ℙ(U | x), ℙ(U | x, a)) = max{sup_u ∈𝒰ρ(x, u, a),sup_u ∈𝒰ρ(x, u, a)^-1}.For f-sensitivity models, we have 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) = max{∫_𝒰 f(ρ(x, u, a)) ℙ(u | x, a)u,∫_𝒰 f(ρ(x, u, a)^-1) ℙ(u | x, a)u}.For Rosenbaum's sensitivity model, we have𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) = max{sup_u_1, u_2 ∈𝒰ρ(x, u_1, u_2, a),sup_u_1, u_2 ∈𝒰ρ(x, u_1, u_2, a)^-1}. We show that all three sensitivity models (MSM, f-sensitivity models, and Rosenbaum's sensitivity model) are GTSMs. Recall that the odds ratio is defined as OR(a, b) =a/(1 - a)(1 - b)/b.MSM: Using Bayes' theorem, we obtain ℙ(u | x, a) = ℙ(a | x, u) ℙ(u | x)/ℙ(a | x) and thereforeρ(x, u, a)=1/1 - ℙ(a | x)(ℙ(u | x)/ℙ(u | x, a) - ℙ(a | x) ) = 1/1 - ℙ(a | x)(ℙ(a | x)/ℙ(a | x, u) - ℙ(a | x) )= ℙ(a | x)/1 - ℙ(a | x)(1 - ℙ(a | x, u)/ℙ(a | x, u)) = OR(ℙ(a | x), ℙ(a | x, u)) . Hence, max{sup_u ∈𝒰ρ(x, u, a),sup_u ∈𝒰ρ(x, u, a)^-1}≤Γ is equivalent to 1/Γ≤OR(ℙ(a | x), ℙ(a | x, u)) ≤Γfor all u ∈𝒰, which reduces to the original MSM defintion for a=1.f-sensitivity models: Follows immediately from ρ(x, u, a) = OR(ℙ(a | x), ℙ(a | x, u)).Rosenbaum's sensitivity model:We can write ρ(x, u_1, u_2, a)= ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x) - ℙ(u_1 | x, a) ℙ(u_2 | x)/ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x) - ℙ(u_2 | x, a) ℙ(u_1 | x)= ℙ(a | x, u_1) ℙ(u_1 | x)/ℙ(a | x, u_2) ℙ(u_2 | x)(ℙ(a | x, u_2) ℙ(u_2 | x) - ℙ(u_2 | x)/ℙ(a | x, u_1) ℙ(u_1 | x) - ℙ(u_1 | x)) = ℙ(a | x, u_1)/ℙ(a | x, u_2)(ℙ(a | x, u_2)- 1/ℙ(a | x, u_1) - 1) = OR(ℙ(a | x, u_1), ℙ(a | x, u_2)) . Hence, max{sup_u_1, u_2 ∈𝒰ρ(x, u_1, u_2, a),sup_u_1, u_2 ∈𝒰ρ(x, u_1, u_2, a)^-1}≤Γ is equivalent to 1/Γ≤OR(ℙ(a | x, u_1), ℙ(a | x, u_2))≤Γfor all u_1, u_2 ∈𝒰, which reduces to the original definition of Rosenbaum's sensitivity model for a=1.§.§ Proof of Lemma <ref>We show transformation-invariance separately for all three sensitivity models (MSM, f-sensitivity models, and Rosenbaum's sensitivity model).MSM: Let 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) = Γ, which implies that implies 1/Γ≤ρ(x, u, a) ≤Γ for all u ∈𝒰. By rearranging terms, we obtainℙ(u | x)≤( Γ (1 - ℙ(a | x)) + ℙ(a | x) )ℙ(u | x, a)andℙ(u | x)≥( 1/Γ (1 - ℙ(a | x)) + ℙ(a | x) )ℙ(u | x, a).Let t 𝒰→𝒰 be a transformation of the unobserved confounder. By using Eq. (<ref>), we can writeρ(x, t(u), a)=1/1 - ℙ(a | x)(ℙ(t(u) | x)/ℙ(t(u) | x, a) - ℙ(a | x) ) = 1/1 - ℙ(a | x)(∫δ(t(u) - t(u^'))ℙ(u^'| x)u^'/∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^' - ℙ(a | x) ) ≤1/1 - ℙ(a | x)(∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^'/∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^')Γ= Γfor all u ∈𝒰. Similarly, we can use Eq. (<ref>) to obtainρ(x, t(u), a)= 1/1 - ℙ(a | x)(∫δ(t(u) - t(u^'))ℙ(u^'| x)u^'/∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^' - ℙ(a | x) ) ≥1/1 - ℙ(a | x)(∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^'/∫δ(t(u) - t(u^'))ℙ(u^'| x, a)u^')1/Γ= 1/Γfor all u ∈𝒰. Hence, 𝒟_x,a(ℙ(t(U) | x), ℙ(t(U) | x, a)) = max{sup_u ∈𝒰ρ(x, t(u), a),sup_u ∈𝒰ρ(x, t(u), a)^-1}≤Γ. f-sensitivity models: This follows from the data compression theorem for f-divergences. We refer to <cit.> for details.Rosenbaum's sensitivity model: We begin by rewritingρ(x, u_1, u_2, a)= ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x) - ℙ(u_1 | x, a) ℙ(u_2 | x)/ℙ(u_1 | x, a) ℙ(u_2 | x, a) ℙ(a | x) - ℙ(u_2 | x, a) ℙ(u_1 | x)= (1/ℙ(u_1 | x)/ℙ(u_1 | x, a) - ℙ(a | x)) ( ℙ(u_2 | x)/ℙ(u_2 | x, a) - ℙ(a | x))as a function of density ratios on 𝒰. Let now 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)) = Γ. This implies ℙ(u_1 | x) ≤(Γ( ℙ(u_2 | x)/ℙ(u_2 | x, a)- ℙ(a | x)) + ℙ(a | x)) ℙ(u_1 | x, a)and ℙ(u_1 | x) ≥(1/Γ( ℙ(u_2 | x)/ℙ(u_2 | x, a)- ℙ(a | x)) + ℙ(a | x)) ℙ(u_1 | x, a)for all u_1, u_2 ∈𝒰. Let t 𝒰→𝒰 be a transformation. By using Eq. (<ref>) and Eq. (<ref>), we obtainρ(x, t(u_1), t(u_1), a)= ( 1/∫δ(t(u_1) - t(u^'_1))ℙ(u^'_1 | x)u^'_1/∫δ(t(u_1) - t(u^'_1))ℙ(u^'_1 | x, a)u^'_1 - ℙ(a | x)) ( ∫δ(t(u_2) - t(u^'_2))ℙ(u^'_2 | x)u^'_2/∫δ(t(u_2) - t(u^'_2))ℙ(u^'_2 | x, a)u^'_2 - ℙ(a | x)) ≤( 1/1/Γ(ℙ(u_2 | x)/ℙ(u_2 | x, a)- ℙ(a | x) )) Γ( ℙ(u_1 | x)/ℙ(u_1 | x, a) - ℙ(a | x) ) = Γ^2/ρ(x, u_1, u_1, a)for all u_1, u_2 ∈𝒰. Hence, inf_u_1, u_2ρ(x, t(u_1), t(u_1), a) ≤Γ.By using analogous arguments, we can also show that sup_u_1, u_2ρ(x, t(u_1), t(u_1), a) ≤Γ,which implies𝒟_x,a(ℙ(t(U) | x), ℙ(t(U) | x, a)) ≤Γ. §.§ Proof of Theorem <ref> Before stating the formal proof for Theorem <ref>, we provide a sketch to give an overview of the main ideas and intuition.Why is it sufficient to only consider invertible functions f^∗_x, a: 𝒰→𝒴, i.e., model ℙ^∗(Y | x, a, u) = δ(Y - f^∗_x, a(u))as a Dirac delta distribution? Here, it is helpful to have a closer look at Fig. <ref>: Intervening on the treatment A causes a shift in the latent distribution of U, which then leads to a shifted interventional distribution ℙ(Y(a) = y | x). The question is now: How can we obtain an interventional distribution that results in a maximal causal query (for the upper bound)? Let us consider a structural equation of the form Y = g^∗_x, a(U, ϵ), where ϵ is some independent noise. Hence, the “randomness” (entropy) in Y comes from both U and ϵ, however, the distribution shift only arises through U. Intuitively, the interventional distribution should be maximally shifted if Y only depends on the unobserved confounder and not on independent noise, i.e., g^∗_x, a(U, ϵ) = f^∗_x, a(U) for some invertible function f^∗_x, a. One may also think about this as achieving the maximal “dependence” (mutual information) between the random variables U and Y. Note that any GTSM only restricts the dependence between U and A, but not between U and Y.Why can we fix ℙ^∗(U | x, a) and f^∗_x, a without losing the ability to achieve the optimum in Eq. (<ref>). The basic idea is as follows: Let ℙ(U| x, a) and fx, a be optimal solutions to Eq. (<ref>) for a potentially different latent variable U. Then, we can define a mapping t = f^∗_x, a^-1∘f_x, aU→ U between latent spaces that transforms ℙ(U| x, a) into our fixed ℙ^∗(U | x, a) (because both f_x, a and f^∗_x, a respect the observational distribution). Furthermore, we can use t to push the optimal shifted distribution ℙ(U| x) (under treatment intervention) to the latent variable U (see Eq. (<ref>)). We will show that this is sufficient to obtain a distributionℙ^∗ that induces ℙ^∗(U | x, a) and ℙ^∗(Y | x, u, a) = δ(Y - f^∗_x, a(u)) and which satisfies the sensitivity constraints. For the latter property, we require the sensitivity model to be “invariant” with respect to the transformation t, for which we require our transformation-invariance assumption (Definition <ref>).We proceed now with our formal proof of Theorem <ref>. Without loss of generality, we provide a proof for the upper bound Q^+_ℳ(x, a). Our arguments work analogously for the lower bound Q^-_ℳ(x, a). Furthermore, we only show the inequality Q^+_ℳ(x, a) ≤sup_ℙ^∗∈𝒫^∗ Q(x, a, ℙ^∗),because the other direction (“≥”) holds by definition of Q^+_ℳ(x, a). Hence, it is enough to show the existence of a sequence of full distributions (ℙ_ℓ^∗)_ℓ∈ with ℙ_ℓ^∗∈𝒫^∗ for all ℓ∈ that satisfies lim_ℓ→∞ Q(x, a, ℙ_ℓ^∗) = Q^+_ℳ(x, a).To do so, we proceed in three steps: In step 1, we construct a sequence (ℙ_ℓ^∗)_ℓ∈ of full distributions that induce ℙ_ℓ^∗(U | x, a) = ℙ^∗(U | x, a) and ℙ_ℓ^∗(Y | x, u, a) = ℙ^∗(Y | x, u, a) = δ(Y - f^∗_x, a(u)) for every ℓ∈. In step 2, we show compatibility with the sensitivity model, i.e., ℙ_ℓ^∗∈ℳ for all ℓ∈. Finally, in step 3, we show that lim_ℓ→∞ Q(x, a, ℙ_ℓ^∗) = Q^+_ℳ(x, a).Step 1: Let ℙ be a full distribution on 𝒳×𝒰×𝒜×𝒴 for some latent space 𝒰 that is the solution to Eq. (<ref>). By definition, there exists a sequence (ℙ_ℓ)_ℓ∈ with ℙ_ℓ∈ℳ and lim_ℓ→∞ Q(x, a, ℙ_ℓ) = Q^+_ℳ(x, a). Let ℙ_ℓ(U| x) and ℙ_ℓ(Y | x, u, a) be corresponding induced distributions (for fixed x, a). Without loss of generality, we can assume that ℙ_ℓ is induced by a structural causal model <cit.>, so that we can write the conditional outcome distribution with a (not necessarily invertible) functional assignment Y = f_X, A, ℓ(U) as a point distribution ℙ_ℓ(Y | x, u, a) = δ(Y - f_x, a, l(u)). Note that we do not explicitly consider exogenous noise because we can always consider this part of the latent space 𝒰. By Eq. (<ref>) and Eq. (<ref>) we can write the observed conditional outcome distribution asℙ_obs(Y | x, a) = ℙ_ℓ(f_x, a, ℓ(U) | x, a),and the potential outcome distribution conditioned on x as ℙ_ℓ(Y(a) | x) = ℙ_ℓ(f_x, a, ℓ(U) | x). We now define the sequence (ℙ_ℓ^∗)_ℓ∈. First we define a distribution on 𝒰⊆^d_y viaℙ_ℓ^∗(U | x) = ℙ_ℓ(f^∗_x, a^-1(Y(a) )| x) = ℙ_ℓ(f^∗_x, a^-1(f_x, a, ℓ(U) )| x). We then define full probability distribution ℙ_ℓ^∗ for the fixed x and a and all u∈𝒰, y ∈𝒴 asℙ_ℓ^∗(x, u, a, y) = δ(f^∗_x, a(u) - y ) ℙ^∗(u | x, a) ℙ_obs(x, a).Finally, we can choose a family of distributions (ℙ_ℓ^∗(U | x, a^'))_a^'≠ a so that ℙ_ℓ^∗(U | x) = ∫ℙ_ℓ^∗(U | x, a) ℙ_obs(a | x)a and defineℙ_ℓ^∗(x, u, a^', y) = δ(f^∗_x, a^'(u) - y) ℙ_ℓ^∗(u | x, a^') ℙ_obs(x, a^'). By definition, ℙ_ℓ^∗ induces the fixed components ℙ^∗(U | x, a) and ℙ^∗(Y | x, u, a) = δ(Y - f^∗_x, a(u)), as well as ℙ_ℓ^∗(U | x) from Eq. (<ref>) and the observational data distribution ℙ_obs(X, A, Y).Step 2: We now show that ℙ_ℓ^∗ respects the sensitivity constraints, i.e., satisfies ℙ_ℓ^∗∈ℳ. It holds thatℙ_ℓ^∗(U | x, a)= ℙ_ℓ^∗(f^∗_x, a^-1(f^∗_x, a(U) )| x, a) (1)=ℙ_obs(f^∗_x, a^-1(Y)| x, a) (2)=ℙ_ℓ(f^∗_x, a^-1(f_x, a, ℓ(U) )| x, a),where (1) holds due to the data-compatibility assumption on f^∗_x, a and (2) holds due to Eq. (<ref>).We now define a transformation t 𝒰→𝒰 via t = f^∗_x, a^-1∘f_x, a, ℓ. We obtain𝒟_x, a(ℙ_ℓ^∗(U | x), ℙ_ℓ^∗(U | x, a))(1)=𝒟_x, a(ℙ_ℓ(t(U) | x), ℙ_ℓ^∗(t(U) | x, a)) (2)≤𝒟_x, a(ℙ_ℓ(U| x), ℙ_ℓ^∗(U| x, a))(3)≤Γ,where (1) holds due to Eq. (<ref>) and Eq. (<ref>), (2) holds due to the tranformation-invariance property of ℳ, and (3) holds because ℙ_ℓ∈ℳ. Hence, ℙ_ℓ^∗∈ℳ.Step 3: We show now that lim_ℓ→∞ Q(x, a, ℙ_ℓ^∗) = Q^+_ℳ(x, a), which completes our proof. By Eq. (<ref>), it holds thatℙ^∗_ℓ(Y(a) | x) = ℙ_ℓ^∗(f^∗_x, a(U) | x)= ℙ_ℓ(Y(a) | x),which means that potential outcome distributions conditioned on x coincide for ℙ^∗_ℓ and ℙ_ℓ. It follows thatQ(x, a, ℙ_ℓ^∗) = ℱ( ℙ^∗_ℓ(Y(a) | x)) = ℱ( ℙ_ℓ(Y(a) | x)) Q^+_ℳ(x, a).§ FURTHER SENSITIVITY MODELS In the following, we list additional sensitivity models that can be written as GTSMs and thus can be used with .Continuous marginal sensitivity model (CMSM): The CMSM has been proposed by <cit.> and <cit.>. It is defined via1/Γ≤ℙ(a| x)/ℙ(a | x, u)≤Γfor all x ∈𝒳, u ∈𝒰, and a ∈𝒜. The CMSM can be written as a CMSM with sensitivity parameter Γ by defining 𝒟_x,a(ℙ(u | x), ℙ(u | x, a)) = max{sup_u ∈𝒰ℙ(u | x)/ℙ(u | x, a),sup_u ∈𝒰ℙ(u | x, a)/ℙ(u | x)}.This directly follows by applying Bayes' theorem to ℙ(u | x, a) = ℙ(a | x, u) ℙ(u | x)/ℙ(a | x).Continuous f-sensitivity models: Motivated by the CMSM, we can define f-sensitivity models for continuous treatments viamax{∫_𝒰 f(ℙ(a| x)/ℙ(a | x, u)) ℙ(u | x, a)u,∫_𝒰 f(ℙ(a| x, u)/ℙ(a | x)) ℙ(u | x, a)u}≤Γfor all x ∈𝒳 and a ∈𝒜. By using Bayes' theorem, we can write any continuous f-sensitivity model as a GTSM by defining𝒟_x, a(ℙ(u | x), ℙ(u | x, a))= max{∫_𝒰 f(ℙ(u | x)/ℙ(u | x, a)) ℙ(u | x, a)u, . .∫_𝒰 f(ℙ(u | x, a)/ℙ(u | x)) ℙ(u | x, a)u}. Weighted marginal sensitivity models: <cit.> proposed a weighted version of the MSM, defined via1/(1 - Γ) q(x, a) + Γ≤ℙ(u | x, a)/ℙ(u | x)≤1/(1 - Γ^-1) q(x, a) + Γ^-1,where q(x, a) is a weighting function that incorporates domain knowledge about the strength of unobserved confounding. By using similar arguments as in the proof of Lemma <ref>, we can write the weighted MSM as a GTSM by defining𝒟_x,a(ℙ(U | x), ℙ(U | x, a)) = max{sup_u ∈𝒰ρ(x, u, a),sup_u ∈𝒰ρ(x, u, a)^-1},where ρ(x, u, a) = 1/1 - q(x, a)(ℙ(u | x)/ℙ(u | x, a) - q(x, a) ). § QUERY AVERAGES AND DIFFERENCES Here, we show that we can use our bounds Q^+_ℳ(x, a) and Q^-_ℳ(x, a) to obtain sharp bounds for averages and differences of causal queries. We follow established literature on causal sensitivity analysis <cit.>.Averages: We are interested in the sharp upper bound for the average causal queryQ_ℳ(a, ℙ) = ∫_𝒳 Q(x, a, ℙ)ℙ_obs(x)x.An example is the average potential outcome [Y(a)], which can be obtained by averaging conditional potential outcomes via [Y(a)] = ∫(Y(a) | x)ℙ_obs(x)x. We can obtain upper bounds viaQ^+_ℳ(a) = sup_ℙ∈ℳ∫_𝒳 Q(x, a, ℙ)ℙ_obs(x)x = ∫_𝒳 Q^+_ℳ(x, a)ℙ_obs(x)x,and lower bounds viaQ^-_ℳ(a) = inf_ℙ∈ℳ∫_𝒳 Q(x, a, ℙ)ℙ_obs(x)x=∫_𝒳 Q^-_ℳ(x, a)ℙ_obs(x)x,whenever we can interchange the supremum/ infimum and the integral. That is, bounding the averaged causal query Q_ℳ(a, ℙ) reduces to averaging the bounds Q^+_ℳ(x, a) and Q^-_ℳ(x, a).Differences: For two different treatment values a_1, a_2 ∈𝒜, we are interested in the difference of causal queriesQ(x, a_1, ℙ) - Q(x, a_2, ℙ).An example is the conditional average treatment effect [Y(1) | x] - [Y(0) | x]. We can obtain an upper bound viaQ^+_ℳ(x, a_1, a_2)=sup_ℙ∈ℳ(Q(x, a_1, ℙ) - Q(x, a_2, ℙ) ) ≤sup_ℙ∈ℳ Q(x, a_1, ℙ) - inf_ℙ∈ℳ Q(x, a_2, ℙ) = Q^+_ℳ(x, a_1)- Q^-_ℳ(x, a_2).Similarly, a lower bound is given byQ^-_ℳ(x, a_1, a_2) ≥Q^-_ℳ(x, a_1)- Q^+_ℳ(x, a_2).It has been shown that these bounds are even sharp for some sensitivity models such as the MSM, i.e., attain equality <cit.>. § TRAINING DETAILS FORIn this section, we provide details regarding the training of , in particular, the Monte Carlo estimates for Stage 2 and the full learning algorithm. §.§ Monte Carlo estimates of stage 2 losses and sensitivity constraints In the following, we assume that we obtained samples u = (u^(j)_x, a)_j=1^k i.i.d.∼𝒩(0_d_y, I_d_y) and ξ = (ξ^(j)_x, a)_j=1^k i.i.d.∼Bernoulli(ℙ_obs(a | x)).§.§.§ Stage 2 losses Here, we provide our estimators ℒ̂_2(η, u, ξ) of the Stage 2 loss ℒ_2(η). We consider three different causal queries: (i) expectations, (ii) set probabilities, and (iii) quantiles.Expectations: Expectations correspond to setting ℱ(ℙ) = 𝔼[X]. Then, we can estimate our Stage 2 loss via the empirical mean, i.e.,ℒ̂_2(η, u, ξ) = 1/k∑_i=1^n ∑_j=1^k f^∗_g^∗_θ_opt(x_i, a_i)( (1 - ξ_x_i, a_i^(j)) f_g_η(x_i, a_i)(u_x_i, a_i^(j)) +ξ_x_i, a_i^(j)u_x_i, a_i^(j)). Set probabilities: Here we consider queries of the form ℱ(ℙ) = ℙ(X ∈𝒮) for some set S ⊆𝒴. We first define the log-likelihoodℓ(η, u_x_i, a_i^(j), ξ_x_i, a_i^(j))= ℙ(f^∗_g^∗_θ_opt(x_i, a_i)( (1 - ξ_x_i, a_i) f_g_η(x_i, a_i)(U) +ξ_x_i, a_iU) . = . f^∗_g^∗_θ_opt(x_i, a_i)((1 - ξ_x_i, a_i^(j)) f_g_η(x_i, a_i)(u_x_i, a_i^(j)) +ξ_x_i, a_i^(j)u_x_i, a_i^(j))),which corresponds to the log-likelihood of the shifted distribution (under Stage 2) at the point that is obtained from plugging the Monte Carlo samples u_x_i, a_i^(j) and ξ_x_i, a_i^(j) into the CNFs. We then optimize Stage 2 by maximizing this log-likelihood only at points in 𝒮. That is, the corresponding Monte Carlo estimator of the Stage 2 loss isℒ̂_2(η, u, ξ)= ∑_i=1^n ∑_j=1^k ℓ(η, u_x_i, a_i^(j), ξ_x_i, a_i^(j)) 1{f^∗_g^∗_θ_opt(x_i, a_i)((1 - ξ_x_i, a_i^(j)) f_g_η(x_i, a_i)(u_x_i, a_i^(j)) +ξ_x_i, a_i^(j)u_x_i, a_i^(j)) ∈𝒮} .Here, we only backpropagate through the log-likelihood in order to obtain informative (non-zero) gradients.Quantiles: We consider quantiles of the form ℱ(ℙ) = F_X^-1(q), where F_X is the c.d.f. corresponding to ℙ and q ∈ (0, 1). For this, we can use the same Stage 2 loss as in Eq. (<ref>) by defining the set 𝒮 = {y ∈𝒴| y ≤F̂_ij^-1(q)} where F̂_ij^-1 is empirical c.d.f. corresponding to {f^∗_g^∗_θ_opt(x_i, a_i)((1 - ξ_x_i, a_i^(j)) f_g_η(x_i, a_i)(u_x_i, a_i^(j)) +ξ_x_i, a_i^(j)u_x_i, a_i^(j))}_j=1^k.§.§.§ Sensitivity constraints Here, we provide our estimators 𝒟̂_x, a(η, u) of the sensitivity constraint 𝒟_x, a(ℙ(U | x), ℙ(U | x, a)). We consider the three sensitivity models from the main paper: (i) MSM, (ii) f-sensitivity models, and (iii) Rosenbaum's sensitivity model.MSM: We defineρ̂(x, u, a, η) = 1/1 - ℙ(a | x)(ℙ((1 - ξ_x, a) f_g_η(x, a)(U) +ξ_x, aℙ(U = u) )/ℙ(U = u) - ℙ(a | x) ).Then, our estimator for the MSM constraint is𝒟̂_x, a(η, u) =max{max_u ∈uρ̂(x, u, a, η),max_u ∈uρ̂(x, u, a, η)^-1}. f-sensitivity models: Our estimator for the f-sensitivity constraint is𝒟̂_x, a(η, u) = max{1/k∑_j=1^k f(ρ̂(x, u^(j)_x, a, a, η)), 1/k∑_j=1^k f(ρ̂(x, u^(j)_x, a, a, η)^-1) }. Rosenbaum's sensitivity model: We defineρ̂(x, u_1, u_2, a, η)= ℙ(U = u_1)/ℙ(U = u_2)(ℙ(U = u_2) ℙ(a | x) - ℙ((1 - ξ_x, a) f_g_η(x, a)(U) +ξ_x, aℙ(U = u_2) )/ℙ(U = u_1) ℙ(a | x) - ℙ((1 - ξ_x, a) f_g_η(x, a)(U) +ξ_x, aℙ(U = u_1) )).Then, our estimator is𝒟̂_x, a(η, u) =max{max_u_1, u_2 ∈uρ̂(x, u_1, u_2, a, η),max_u_1, u_2 ∈uρ̂(x, u_1, u_2, a, η)^-1}. §.§ Full learning algorithm Our full learning algorithm for Stage 1 and Stage 2 is shown in Algorithm <ref>. For Stage 2, we use our Monte-Carlo estimators described in the previous section in combination with the augmented lagrangian method to incorporate the sensitivity constraints. For details regarding the augmented lagrangian method, we refer to <cit.>, chapter 17.Reusability: Using CNFs instead of unconditional normalizing flows allows us to compute bounds Q^+_ℳ(x, a) and Q^-_ℳ(x, a) without the need to retrain our model for different x ∈𝒳 and a ∈𝒜. In particular, we can simultaneously compute bounds for averaged queries or differences without retraining (see Appendix <ref>). Furthermore, Stage 1 is independent of the sensitivity model, which means that we can reuse our fitted Stage 1 CNF for different sensitivity models and sensitivity parameters Γ, and only need to retrain the Stage 2 CNF. §.§ Further discussion of our learning algorithm Non-neural alternatives: Note that our two-stage procedure is agnostic to the estimators used, which, in principle ,allows for non-neural instantiations. However, we believe that our neural instantiation () offers several advantages over possible non-neural alternatives:* Solving Stage 1: Conditional normalizing flows (CNFs) are a natural choice for Stage 1 because they are designed to learn an invertible function f^∗_x, a𝒰→𝒴 that satisfies ℙ_obs(Y | x, a) =ℙ^∗(f^∗_x, a(U) | x, a). In particular, CNFs allow for inverting f^∗_x, a analytically, which enables tractable optimization of the log-likelihood (see Appendix <ref>). In principle, f^∗_x, a could also be obtained by estimating the conditional c.d.f. F̂(y | x, a) using some arbitrary estimator and then leveraging the inverse transform sampling theorem, which states that we can choose f^∗_x, a = F̂^-1(·|, x, a) whenever we fix ℙ^∗(U | x, a) to be uniform. However, this approach only works for one-dimensional Y and requires inverting the estimated F̂(y | x, a) numerically.* Solving Stage 2: Stage 2 requires to optimize the causal query ℱ(ℙ(f^∗_x, a(U) | x) ) over a latent distribution ℙ(U | x), where f^∗_x, a is learned in Stage 1. In , we achieve this by fitting a second CNF in the latent space 𝒰, which we then concatenate with the CNF from Stage 1 to backpropagate through both CNFs. Standard density estimators are not applicable in Stage 2 because we fit a density in the latent space to optimize a causal query that is dependent on Stage 1, and not a standard log-likelihood. While there may exist non-neural alternatives to solve the optimization in Stage 2, this goes beyond the scope of our paper, and we leave this for future work.* Analytical interventional density: Using NFs in both Stages 1 and 2 enables us to obtain an analytical interventional once NeuralCSA is fitted. Hence, we can perform sensitivity analysis for the whole interventional density (see Fig. <ref> and <ref>), without the need for Monte Carlo approximations through sampling.* Universal density approximation: CNFs are universal density approximators, which means that we can account for complex (e.g., multi-modal, skewed) observational distributions. Complexity of compared to closed-form solutions: Stage 1 requires fitting a CNF to estimate the observational distribution ℙ_obs(Y | x, a). This step is also necessary for closed-form bounds (e.g., for the MSM), where such closed-form bounds depend on the observational distribution <cit.>. This renders the complexity in terms of implementation choices equivalent to Stage 1. In Stage 2, closed-form solutions under the MSM allow for computing bounds directly using the observational distribution, fits an additional NF that is training using Algorithm 1. Hence, additional hyperparameters are the hyperparameters of the second NF as well as the learning rates of the augmented Lagrangian method (used to incorporate the sensitivity constraint). Hence, we recommend using as a method for causal sensitivity analysis whenever bounds are not analytically tractable.In our experiments, we observed that the training of NeuralCSA was very stable, as indicated by a low variance over different runs (see, e.g., Fig. <ref>). Existence of a global optimum: A sufficient condition for the existence of a global solution in Eq. (<ref>) is the continuity of the objective/causal query as well as the compactness of the constraint set. Continuity holds for many common causal queries such as the expectation. The compactness of the constraint set depends on the properties functional 𝒟_x, a, i.e., the choice of the sensitivity model. The existence of global solutions has been shown for many sensitivity models from the literature, e.g., MSM <cit.> and f-sensitivity models <cit.>. Note that, in Theorem <ref>, we do not assume the existence of a global solution. In principle, our two-stage procedure is valid even if a global solution to Eq. (5) does not exist. In this case, we can apply our Stage 2 learning algorithm (Algorithm <ref>) until convergence and obtain an approximation of the desired bound, even if it is not contained in the constraint set. § DETAILS ON IMPLEMENTATION AND HYPERPARAMETER TUNING Stage 1 CNF: We use a conditional normalizing flows (CNF) <cit.> for stage 1 of . Normalizing flows (NFs) model a distribution ℙ(Y) of a target variable Y by transforming a simple base distribution ℙ(U) (e.g., standard normal) of a latent variable U through an invertible transformation Y = f_θ̂(U), where θ̂ denotes parameters <cit.>. In order to estimate theconditional distributions ℙ(Y | x, a), CNFs define the parameters θ̂ as an output of a hyper network θ̂ = g_θ(x, a) with learnable parameters θ. The conditional log-likelihood can be written analytically aslog( ℙ (f_g_θ(x, a)(U) = y) ) (∗)=log(ℙ(U = f_g_θ(x, a)^-1(y)) ) + log(det( d/d y f_g_θ(x, a)^-1(y))),where (∗) follows from the change-of-variables theorem for invertible transformations.Stage 2 CNF: As in Stage 1, we use a CNF that transforms U = f_η̂(U), where η̂ = g_η(x, a) with learnable parameters η. The conditional log-likelihood can be expressed analytically vialog( ℙ (f_g_θ(x, a)(U) = u) ) = log(ℙ(U = f_g_θ(x, a)^-1(u)) ) + log(det( d/d uf_g_θ(x, a)^-1(u))). In our implementation, we use autoregressive neural spline flows. That is, we model the invertible transformation f_θ via a spline flow as described in <cit.>. We use an autoregressive neural network for the hypernetwork g_η(𝐱, 𝐦, 𝐚) with 2 hidden layers, ReLU activation functions, and linear output. For training, we use the Adam optimizer <cit.>.Propensity scores: The estimation of the propensity scores ℙ(a |𝐱) is a standard binary classification problem. We use feed-forward neural networks with 3 hidden layers, ReLU activation functions, and softmax output. For training, we minimize the standard cross-entropy loss by using the Adam optimizer <cit.>.Hyperparameter tuning: We perform hyperparameter tuning for our propensity score models and Stage 1 CNFs. The tunable parameters and search ranges are shown in Table <ref>. Then, we use the same optimally trained propensity score models and Stage 1 CNFs networks for all Stage 2 models and closed-form solutions (in Fig. <ref>). For the Stage 2 CNFs, we choose hyperparameters that lead to a stable convergence of Alg. <ref>, while ensuring that the sensitivity constraints are satisfied. For reproducibility purposes, we report the selected hyperparameters as .yaml files.[Code is available at https://anonymous.4open.science/r/NeuralCSA-DE7Dhttps://anonymous.4open.science/r/NeuralCSA-DE7D.]§ DETAILS REGARDING DATASETS AND EXPERIMENTS We provide details regarding the datasets we use in our experimental evaluation in Sec. <ref>. §.§ Synthetic data Binary treatment: We simulate an observed confounder X ∼Uniform[-1, 1] and define the observed propensity score as π(x) = ℙ_obs(A=1 | x) = 0.25 + 0.5σ(3x), where σ(·) denotes the sigmoid function. Then, we simulate an unobserved confounderU | X = x ∼Bernoulli(p=(Γ - 1) π(x) + 1/Γ + 1)and a binary treatmentA= 1 | X = x, U = u ∼Bernoulli(p=u π(x) s^+(x, a) + (1 - u) π(x) s^-(x, a) ),where s^+(x, a) = 1/(1 - Γ^-1) π(x) + Γ^-1 and s^-(x, a) = 1/(1 - Γ) π(x) + Γ.Finally, we simulate a continuous outcome Y = (2A - 1) X + (2 A - 1) - 2sin(2(2A - 1)X) - 2(2U - 1)(1 + 0.5X) + ε,where ε∼𝒩(0, 1).The data-generating process is constructed so that ℙ(A = 1 | x, u)/π(x) = u s^+(x, a) + (1-u)s^-(x, a) or, equivalently, that OR(x, u) = u Γ + (1 - u) Γ^-1. Hence, the full distribution follows an MSM with sensitivity parameter Γ. Furthermore, by Eq. (<ref>), we have ℙ(A = 1 | x, 1) ℙ(U = 1 | x)+ℙ(A = 1 | x, 0)ℙ(U = 0 | x) = π(x) = ℙ_obs(A=1 | x)so that ℙ induces ℙ_obs(A=1 | x).Continuous treatment: We simulate an observed confounder X ∼Uniform[-1, 1] and independently a binary unobserved confounder U ∼Bernoulli(p=0.5). Then, we simulate a continuous treatment viaA | X = x, U = u ∼Beta(α, β)with α = β = 2 + x + γ (u - 0.5),where γ is a parameter that controls the strength of unobserved confounding. In our experiments, we chose γ = 2. Finally, we simulate an outcome viaY = A + Xexp(-X A) - 0.5(U - 0.5)X + (0.5X + 1) + ε,where ε∼𝒩(0, 1). Note that the data-generating process does not follow a (continuous) MSM to reflect realistic settings from practice. Oracle sensitivity parameters Γ^∗: We can obtain Oracle sensitivity parameters Γ^∗(x, a) for each sample with X=x and A=a by simulating from our previously defined data-generating process to estimate the densities ℙ(u | x, a) and ℙ(u | x) and subsequently solve for Γ^∗(x, a) in the GTSM equations from Lemma <ref>. By definition, Γ^∗(x, a) is the smallest sensitivity parameter such that the corresponding sensitivity model is guaranteed to produce bounds that cover the ground-truth causal query. For binary treatments, it holds Γ^∗(x, a) = Γ^∗, i.e., the oracle sensitivity parameter does not depend on x and a. For continuous treatments, we choose Γ^∗ = Γ^∗(a) = ∫Γ^∗(x, a) ℙ(x)x in Fig. <ref>. §.§ Semi-synthetic dataWe obtain covariates X and treatments A from MIMIC-III <cit.> as described in the paragraph regarding real-world data below. Then we learn the observed propensity score π̂(x) = ℙ̂(A = 1 | x) using a feed-forward neural network with three hidden layers and ReLU activation function. We simulate a uniform unobserved confounder U ∼𝒰[0, 1]. Then, we define a weightw(X, U)= 1{γ≥ 2 - 1/π̂(X)}( γ + 2 U (1 - γ)) + 1{γ < 2 -1/π̂(X)}(2 - 1/π̂(X) +2 U (1/π̂(X) - 1))and simulate synthetic binary treatments viaA= 1 | X = x, U = u ∼Bernoulli(p=w(x,u) π̂(x)).Here, γ is a parameter controlling the strength of unobserved confounding, which we set to 0.25. The data-generating process is constructed in a way such that the full propensity ℙ(A= 1 | X = x, U = u) induces the (estimated) observed propensity π̂ from the real-world data. We then simulate synthetic outcomes viaY = (2 A - 1) (1/d_x + 1((∑_i=1^d_x X_i ) + U) ) + ε,where ε∼𝒩(0, 0.1). In our experiments, we use 90% of the data for training and validation, and 10% of the data for evaluating test performance. From our test set, we filter out all samples that satisfy either ℙ(A=1 | x) < 0.3or ℙ(A=1 | x) > 0.7. This is because these samples are associated with large empirical uncertainty (low sample sizes). In our experiments, we only demonstrate the validity of our bounds in settings with low empirical uncertainty.Oracle sensitivity parameters Γ^∗: Similar to our fully synthetic experiments, we first obtain the Oracle sensitivity parameters Γ^∗(x, a) for each test sample with confounders x and treatment a. We then take the median overall Γ^∗(x, a) of the test sample. By definition, should then cover at least 50% of all test query values (see Table <ref>). §.§ Real-world dataWe use the MIMIC-III dataset <cit.>, which includes electronic health records from patients admitted to intensive care units. We use a preprocessing pipeline <cit.> to extract patient trajectories with 8 hourly recorded patient characteristics (heart rate, sodium blood pressure, glucose, hematocrit, respiratory rate, age, gender) and a binary treatment indicator (mechanical ventilation). We then sample random time points for each patient trajectory and define the covariates X ∈^8 as the past patient characteristics averaged over the previous 10 hours. Our treatment A ∈{0, 1} is an indicator of whether mechanical ventilation was done in the subsequent 10-hour time. Finally, we consider the final heart rate and blood pressure averaged over 5 hours as outcomes. After removing patients with missing values and outliers (defined by covariate values smaller than the corresponding 0.1th percentile or larger than the corresponding 99.9th percentile), we obtain a dataset of size n=14719 patients. We split the data into train (80%), val (10%), and test (10%).§ ADDITIONAL EXPERIMENTS§.§ Additional treatment combinations for synthetic data Here, we provide results for additional treatment values a ∈{0.1, 0.9} for the synthetic experiments with continuous treatment in Sec. <ref>. Fig. <ref> shows the results for our experiment where we compare the bounds under the MSM with (optimal) closed-form solutions. Fig. <ref> shows the results of our experiment where we compare the bounds of different sensitivity models. The results are consistent with our observations from the main paper and show the validity of the bounds obtained by , §.§ Densities for lower bounds on real-world data Here, we provide the distribution shifts for our real-world case study (Sec. <ref>), but for the lower bounds instead of the upper. The results are shown in Fig. <ref>. In contrast to the shifts for the upper bounds, increasing Γ leads to a distribution shift away from the direction of the danger area, i.e., high heart rate and blood pressure.§.§ Additional semi-synthetic experiment We provide additional experimental results using a semi-synthetic dataset based on the IHPD data <cit.>. IHDP is a randomized dataset with information on premature infants. It was originally designed to estimate the effect of home visits from specialist doctors on cognitive test scores. For our experiment, we extract d_x = 7 covariates X (birthweight, child's head circumference, number of weeks pre-term that the child was born, birth order, neo-natal health index, the mother's age, and the sex of the child) of n=985 infants. Then, and define an observational propensity score as π(x) = ℙ_obs(A=1 | x) = 0.25 + 0.5σ(3/d_x∑_i=1^d_x X_i). Then, similar as in Appendix <ref>, we introduce unobserved confounding by generating synthetic treatments via A= 1 | X = x, U = u ∼Bernoulli(p=w(x,u) π(x)),where w(x,u) is defined in Eq. (<ref>) and U ∼𝒰[0, 1]. We then generate synthetic outcomes viaY = (2 A - 1) (1/d_x + 1((∑_i=1^d_x X_i ) + U) ) + ε,where ε∼𝒩(0, 1).In our experiments, we split the data into train (80%) and test set (20%). We verify the validity of our bounds for CATE analogous to our experiments using the MIMIC (Sec. <ref>): For each sensitivity model (MSM, TV, HE, RB), we obtain the smallest oracle sensitivity parameter Γ^∗ that guarantees coverage (i.e., satisfies the respective sensitivity constraint) for 50% of the test samples. Then, we plot the coverage and median interval length of the bounds over the test set. The results are in Table <ref>. The results confirm the validity of . § DISCUSSION ON LIMITATIONS AND FUTURE WORK Limitations: is a versatile framework that can approximate the bounds of causal effects in various settings. However, there are a few settings where (optimal) closed-form bounds exist (e.g., CATE for binary treatments under the MSM), which should be preferred when available. Instead, offers a unified framework for causal sensitivity analysis under various sensitivity models, treatment types, and causal queries, and can be applied in settings where closed-form solutions have not been derived or do not exist (Table <ref>).Future work: Our research hints at the broad applicability of beyond the three sensitivity models that we discussed above (see also Appendix <ref>). For future work, it would be interesting to conduct a comprehensive comparison of sensitivity models and provide practical recommendations for their usage. Future work may further consider incorporating techniques from semiparametric statistical theory in order to obtain estimation guarantees, robustness properties, and confidence intervals. Finally, we only provided identifiability results that hold in the limit of infinite data. It would be interesting to provide rigorous empirical uncertainty quantification for , e.g., via a Bayesian approach. While in principle the bootstrapping approach from <cit.> could be applied in our setting, this could be computationally infeasible for large datasets. | http://arxiv.org/abs/2311.16026v1 | {
"authors": [
"Dennis Frauen",
"Fergus Imrie",
"Alicia Curth",
"Valentyn Melnychuk",
"Stefan Feuerriegel",
"Mihaela van der Schaar"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20231127174002",
"title": "A Neural Framework for Generalized Causal Sensitivity Analysis"
} |
[email protected] Donostia International Physics Center (DIPC), 20018 Donostia–San Sebastián, Spain University of Twente, 7522 NB Enschede, The Netherlands University of Twente, 7522 NB Enschede, The Netherlands Deparment of Applied Physics, Nagoya University, 464-8603 Nagoya, Japan Research Center for Crystalline Materials Engineering, Nagoya University, 464-8603 Nagoya Japan The article presents a method to detect time-reversal symmetry breaking in non-centrosymmetric superconductors using only transport measurements. Specifically, if time-reversal symmetry is broken via a phase difference between singlet and triplet correlations, as in anapole superconductors, the conductance in SFN junctions is enhanced by increasing the exchange field strength in the ferromagnet.This is in sharp contrast with the negative magnetoresistance when using superconductors in which time reversal symmetry is preserved. Moreover, results show a large quadrupolar component of the magnetoresistance which is qualitatively different from the bipolar giant magnetoresistance in strong ferromagnets. Positive magnetoresistance in anapole superconductor junctions Yukio Tanaka0000-0003-1537-4788============================================================== § INTRODUCTIONRecent advances within superconductivity focus on the understanding of unconventional superconductors <cit.>, i.e. superconductors which do not obey BCS theory.In unconventional superconductors triplet <cit.> and odd-frequency <cit.> pairings may appear. Most attention is paid to superconductors in which inversion (𝒫) and time-reversal (𝒯) symmetry are preserved in the pair potential. However, several known superconductorsbreak time-reversal symmetry, either in bulk or near the edges<cit.>,while in superconductors in which the crystal structure breaks inversion symmetry, the pair potential may contain both even and odd-parity components<cit.>. Also the breaking of inversion symmetry near an edge may cause a local admixture of even-parity and odd-parity superconductivity <cit.>.Moreover, in a recently discussed class of superconductors, anapole superconductors, both symmetries are broken <cit.>. Examples of possible anapole superconductors are UTe_2 <cit.>, Cu_xBi_2Se_3 and Sn_1-xIn_xTe <cit.>, while also non-centrosymmetric superconductors with a magnetic ordering in their phase diagram, such as CePt_3Si <cit.>, UIr <cit.>, CeRh_2As_2 <cit.> and CeCu_2Si_2 <cit.>,are promising platforms for 𝒫 and 𝒯 broken superconducting phases in some parameter regimes. Next to this, time-reversal symmetry can be broken via the inverse proximity effect of a ferromagnet or ferromagnetic insulator <cit.>. Thus, both non-centrosymmetric and time-reversal symmetry broken superconductivity is locally abundant. The proximity effect of superconductors that are time-reversal and/or inversion symmetry broken can be significantly different from the proximity effect of superconductors in which these symmetries are preserved <cit.>.Superconductors in which both time reversal symmetry and inversion symmetry are broken have great potential for future applications, for example in non-reciprocal transport, since the conditions for non-reciprocal transport to occur, time-reversal symmetry and gyrotropy <cit.>, are met intrinsically in the bulk material. Thus, such superconductors are a promising platform for superconducting diodes <cit.>, no external source of an exchange field and/or spin-orbit coupling is needed. This greatly simplifies the setup for such diodes.The proximity effect and transport properties of non-centrosymmetric or time reversal symmetry broken superconductors has been studied in detail in several limits <cit.>.Recently, a dirty limit transport theory was discussed, focussing on (i)s + p-wave superconductors <cit.>. In such superconductors the pair potential breaks inversion symmetry as indicated by the simultaneous presence of even-parity s-wave and odd-parity p-wave components, while time-reversal symmetry is broken if there is a finite phase difference between the singlet and triplet correlations. In <cit.> the importance of considering time-reversal symmetry breaking of the pair potential of superconductors was illustrated using SNN junctions.However, in SNN junctions the dependence of the conductance on the phase between the singlet and triplet components can not be unambiguously distinguished from the dependence on the ratio of their magnitudes. Therefore, a method to obtain smoking gun evidence for time-reversal symmetry breaking based only on transport measurements is so far absent.In this paper we provide such method by showing that time-reversal symmetry breaking in the superconductor can be identified using SFN junctions. We show that for the time-reversal symmetry broken non-centrosymmetric is + p-wave superconductors the conductance for voltages just below the superconducting gap can be strongly enhanced by an exchange field. This is in contrast with the negative magnetoresistance in non-centrosymmetric superconductors that obey time-reversal symmetry such as s + p-wave superconductors.Moreover, the dependence of the conductance enhancement on the exchange field direction provides an additional tool to determine the direction of the d-vector of the p-wave correlations. We show that if the exchange field strength h is much smaller than the Fermi energy E_F (h/E_F≪ 1), the magnetoresistance is quadrupolar, with maxima when d-vector and exchange field are perpendicular. Thus, for ferromagnets with h/E_F≪ 1 the anisotropy of the magnetoresistance is qualitatively different from the well-known bipolar magnetoresistance for h/E_F∼ 1 <cit.>. Next to this, we show that the voltage window in which the conductance is enhanced is determined by the ratio of the s-wave and p-wave components of the superconductors. In this way, time reversal symmetry breaking can be established and all parameters needed to fully describe the pair potential can be fully determined even in the absence of time reversal symmetry.The paper is setup as follows. In Sec. <ref> we present the model for an SFN junction with time-reversal and inversion symmetry broken superconductors, in Sec. <ref> we show the conductance calculated using this model for (i)s + helical p-wave superconductors. Next, in Sec. <ref> we present the results for (i)s + chiral p-wave superconductors and compare them to those of (i)s + helical superconductors. We conclude our article in Sec. <ref> with a discussion of our results and discuss how to generalize to other types of non-centrosymmetric time-reversal broken superconductors. § MODEL We study a junction in which a ferromagnetic bar (F) is sandwiched between an unconventional superconductor and a normal metal electrode, schematically shown in Fig. <ref>. Our model is similar to the one used in <cit.>,with the difference that an exchange interaction is included in the bar. We assume that the ferromagnetism is weak enough that we may use the quasiclassical formalism, h/E_F≪ 1. This assumption is validin weak ferromagnets or normal metals proximized by a ferromagnetic insulator. The induced exchange field causes Larmor precession of the electrons. Without proximity effect this does not affect the equilibrium properties of the metal, it only affects the distribution functions. However, if pair correlations exist, the presence of Larmor precession implies that while the spins of the pairs are still antiparallel when measured at the same time, they are in general not parallel when measured at different times. Thus, there is singlet-triplet conversion, where the d-vector of the triplets is parallel to the exchange field direction <cit.>. In S/F bilayers this leads to spin splitting of the superconductor <cit.>. We assume that the scattering length in the problem is much smaller than any other relevant length, except the Fermi length, as is likely in thin films <cit.>. In this case the Green's function is almost isotropic, and the Keldysh Usadel formalism <cit.> may be used to describe the F bar. We use the basis (ψ_↑,ψ_↓,ψ^†_↓,-ψ^†_↑)^T.If the width of the bar is either much larger or much smaller than the thermal coherence length, an effectively one-dimensional model may be used. In this limit the Green's function G is approximately independent of the y and z coordinates:D∂_x(G̅∂_xG̅) = [i(E+h⃗·σ⃗)τ_3,G̅],where D is the spatially invariant diffusion constant, σ⃗ is the vector of Pauli matrices in spin space, G is the isotropic component of the Green's function, E is energy and h⃗ is the exchange field in the ferromagnet, with magnitude h. If the interface resistance of the F/N interface is very small, for example if the F bar is a normal metal proximized by an FI and the N electrode is the same normal metal, the Green's function is continuous at the F/N interface:G̅(x = L) = G̅_N,where G̅_N = [ Ǧ_N^R Ǧ_N^K; 0 Ǧ_N^A ] is the Green's function in the normal metal electrode, where the retarded Green's function is Ǧ_N^R = τ_3, the advanced Green's function is Ǧ_N^A = -τ_3. The Keldysh Green's function is given by Ǧ^K = Ǧ^Rȟ-ȟǦ^A, where ȟ = f̂_L0⊗τ_0+f̂_T0⊗τ_3 is the matrix distribution function, with the longitudinal (L) and transverse(T) components <cit.>determined by the Fermi-Dirac distribution: f̂_L0,T0 = 1/2(tanhE+eV/2k_BT±tanhE-eV/2k_BT), where V is the voltage applied to the normal metal electrode, T is the temperature of the system, which we assume to be well below T_c and k_B is the Boltzmann constant.We assume that both the S and the N junctions are electrodes, while the F layer is a restriction between these two. In that case the inverse proximity effect of the ferromagnet on the (i)s + p-wave superconductor may be ignored, and the Tanaka-Nazarov boundary conditions <cit.>, the extension of Nazarov's boundary conditions <cit.> to interfaces with unconventional superconductors, may be used. The boundary condition at the S/F interface reads:G̅∇G̅(x=0) = 1/γ_BS L⟨S̅(ϕ)⟩,whereS̅(ϕ)= T(1+T_1^2+T_1(C̅G̅+G̅C̅))^-1(C̅G̅-G̅C̅), C̅ =H̅_+^-1(1̅-H̅_-), H̅_+ =1/2(G̅_S(ϕ)+G̅_S(π-ϕ)), H̅_- =1/2(G̅_S(ϕ)-G̅_S(π-ϕ)) .Averaging over all modes passing through the interface is denoted by ⟨·⟩, γ_BS = R_B/R_d is the ratio between the boundary resistance and the normal states resistance of F and the parameter T_1 = T/(2-T+2√(1-T)) <cit.>, where T is the interface transparency given by T(ϕ) = cos^2ϕ/cos^2ϕ+z^2, with z the Blonder-Tinkham-Klapwijk (BTK) parameter. The retarded part of the Green's function of the superconductor G_S(ϕ) is given by the bulk equilibrium Green's function of an (i)s + p-wave superconductor. For is + p-wave superconductors the pair potential is given byΔ(ϕ) = e^iπ/2χ_t1/√(r^2+1)+r/√(r^2+1)d⃗(ϕ)·σ⃗,where d⃗(ϕ) is the d-vector an angle dependent unit vector that is different for different types of p-wave superconductors, χ_t is the phase difference between the singlet and triplet components, and r is the mixing parameter. Therefore, the bulk Green's function reads Ǧ^R_S(ϕ)= 1/2(1+d⃗̂⃗(ϕ)·σ⃗)⊗1/√(E^2-|Δ_+|^2)[EΔ_+; -Δ_+^* -E ]+1/2(1-d⃗̂⃗(ϕ)·σ⃗)⊗1/√(E^2-|Δ_-|^2)[EΔ_-; -Δ_-^* -E ], Δ_± = e^iπ/2χ_t± re^iψ(ϕ)/√(r^2+1) ,while its distribution function is the equilibrium Fermi-Dirac distribution. The advanced and Keldysh components can now be found using respectivelyǦ^A_S = -τ_3(Ǧ^R_S)^†τ_3 and Ǧ_S^K = Ǧ^R_Sȟ-ȟǦ^A_S, with ȟ = f̂_L1+f̂_Tτ_3. The following set of parameters is used throughout the paper: γ_BS = 2, z = 0.75, E_Th/Δ_0 = 0.02 to compare with the results on non-centrosymmetric superconductors in previous articles <cit.>,<cit.>. § HELICAL SUPERCONDUCTORS In the first part we focus on (i)s + helical p-wavesuperconductors, that is, the p-wave component has d-vectord⃗(ϕ) = (cosϕ,sinϕ,0).We contrast the results for is + helical p-wave (χ_t = 1) superconductors with those obtained for s + helical p-wave (χ_t = 0) superconductors, and then provide an explanation for their differences. The exchange field dependence of the conductance of an SFN junction with is + helical p-wave superconductors is shown in Fig. <ref> for s-wave dominant superconductors (r = 0.5) and Fig. <ref> for p-wave dominant superconductors (r = 2). In both Figs. <ref> and <ref> the left panel corresponds to a parallel h⃗ and ⟨d⃗⟩, while the right panel corresponds to the perpendicular orientation of these two vectors. In all panels, there is a zero bias conductance peak with a width of the order of the Thouless energy for h = 0. This peak is due to coherent Andreev reflection and is split into two peaks at |eV| = h for all orientations of the field, even if the p-wave component is dominant, in contrast to the s+p-wave superconductors. For s+p-wave pairing, the topological property, i.e. the presence of ZES felt by the perpendicular injectedquasiparticles does not change as far asΔ_p > Δ_s <cit.>. On the other hand, for the is+p-wave superconductor, even if the magnitude of the s-wave component is infinitesimal, ZBCP splits. This is due to the presence of both singlet and triplet components at E = 0 in the presence of time-reversal symmetry breaking <cit.>. In the p-wave dominant case a small zero energy peak remains if h⃗ and ⟨d⃗⟩ are perpendicular due to the long range triplets <cit.>, see Fig. <ref>(b), in all other cases the zero bias conductance peak is converted into a zero bias conductance dip, see Fig. <ref> and <ref>(a). The height of the peak decreases for increasing exchange field strength, for h≫Δ_0 it is diminished.If the s-wave component is dominant, r<1, shown in Fig. <ref>, there is a sharp peak in the conductance around |eV| = Δ_0. This peak, which is due to a suppression of the boundary resistance, is enhanced in the presence of an exchange field. If the field is parallel to the d-vector of the p-wave component this enhancement is only apparent for large fields, i.e. h≫Δ_0, as shown in Fig. <ref>(a). On the other hand, if h⃗ is perpendicular to ⟨d⃗⟩, a small field is enough to increase the conductance of the junction, as shown in Fig. <ref>(b). In both cases the conductance does not increase indefinitely, but saturates at a value that is slightly higher than for the perpendicular orientation than for the parallel orientation. The maximum increase of conductance is of the order of 0.1σ_N.If the p-wave superconductor is dominant, r>1, the peak below |eV| = Δ_0 is much wider, as shown in Fig. <ref>. Also in this case the conductance is enhanced by an exchange field. In the parallel orientation this enhancement only appears in a small window close to |eV| = Δ_0 and only for large fields, see Fig. <ref>(a). For the perpendicular orientation on the other hand the enhancement appears in a large window, and is largely enhanced compared to parallel field, for small fields h≪Δ_0. If the field is parallel a conductance enhancement of a few percent can be achieved, for the perpendicular orientation an enhancement of 0.1σ_N can be achieved, as in the case r<1.Comparing this to the magneto-electric response in s + helical p-wave superconductors <cit.>, we find that the magneto-electric response in is + helical p-wave superconductors has opposite sign and a much stronger anisotropy. The difference between these materials is due to the presence or absence of time-reversal symmetry. For s + p-wave superconductors, time-reversal symmetry is not broken. Moreover, in a dirty material, the scattering rate is high and therefore the Green's function is almost isotropic. Therefore, the density of states in the surface Green's function C is independent of spin. On the other hand, for the is + p-wave superconductor, time-reversal symmetry is broken, and the surface density of states is different for spins parallel or antiparallel to ⟨d⃗⟩.Mathematically, based on this the conductance enhancement can be inferred from the Tanaka-Nazarov boundary conditions, specifically, the Keldysh component <cit.>. If the density of states of the surface Green's function has no spin-dependence, as for the s+p-wave superconductor, the only difference in the commutator or anticommutator caused by the exchange field comes from the terms off-diagonal in Nambu space. Since the pair amplitude in the normal metal is suppressed by a finite boundary resistance, the off-diagonal terms are small and hence the effectof the exchange field on the anticommutator is small in this case.On the other hand, for the is + p-wave casealso the density of states of the surface Green's function has a spin dependence and also the contribution of the density states to the anticommutator in the Tanaka-Nazarov boundary condition depends on the exchange field. This greatly reduces the anti commutator CG+GC. This implies that the enhancement of the conductance is largest for those voltages for which T_1(CG+GC) is the dominant term in the denominator. As shown in<cit.>, for is + p-wave superconductors this term is dominant for Δ_s<|eV|<√(Δ_s^2+Δ_p^2), where we define Δ_s,p as the magnitudes of the singlet and triplet pair amplitudes. Thus, the enhancement the conductance is most prominent in a broader voltage window if the p-wave component of the pair potential larger. To understand the physical mechanism behind the enhancement of conductance just below |eV| = Δ_0 we first consider the physical interpretation of the denominator. From the derivation in <cit.> it follows that the denominator can be attributed to higher order tunnelling. Indeed, to first order in the tunnelling T_1 parameter between two materials with Green's functions G_1,2 the current between these materials is given by T_1[G_1,G_2], the well-known Kuprianov-Luckichev boundary condition <cit.>. The boundary condition involves a commutator, which is larger if the Green's functions of the two materials are more different, for example in an SIS junction if the phase difference between two superconductors is larger.It is only upon inclusion of the higher orders in T_1 that the denominator appears, which suppresses the first order approximation.The Green's function exactly at the interface is altered by the tunnelling of the electrons of the other material into it. Therefore the difference in Green's functions on either side of the interface is smaller, leading to a reduction of the current compared to the first order estimation.Next we consider the effect of the relative orientation of the d-vector in the two materials on these higher order terms. Suppose a fraction α of electrons is exchanged between materials 1 and 2 due to the tunnelling. If the spin quantization axis is the same for both materials 1 and 2, the resulting new pair amplitudes and density of states are a weighted average of the two Green's functions. This is not the case if the spin quantization axes are perpendicular, for example along the z-axis in material 1 and along the x-axis in material 2. The electrons moving from 2 to 1 do contribute to all spin-averaged properties, i.e. to those parts of the Green's function proportional to σ_0. However, the electrons that move from 2 to 1 do not have any spin-projection along the z-axis and therefore can not contribute to the σ_z-terms in the Green's function of material 1.Thus, the change in the σ_z-term of the Green's function in 1 is smaller in the perpendicular orientations. Likewise, the electrons moving from material 1 to 2 can not contribute to the σ_x-terms in the Green's function of material 2, because the electrons from material 1 do not have any spin-projection along the x-direction. Therefore, if the spin quantization axis is different in the two materials, the Green's functions at the interface are more different compared to the case in which they are parallel. For this reason the current is minimized if the d-vectors are parallel.In the setup used in this paper,if the exchange field is not parallel to the d-vector it rotates the quantization axes of the induced pair correlations in the ferromagnet. Meanwhile, since the exchange field is only present in the normal metal the quantization axis of the superconductor is unchanged. Therefore, the current increases if the exchange field is not parallel to the d-vector of the superconductor, explaining the enhancement of the conductance compared to the normal state and the anisotropy in the enhancement. This leads to a quadrupolar dependence of the conductance on the exchange field, as shown in Fig. <ref>. A quadrupolar dependence on the exchange field in proximity structures has been found before, both in supercurrents <cit.> and conductance <cit.>. Our effect however differs from <cit.> via the voltage range in which the effect appears.Though smaller than for the perpendicular orientation, for the s-wave dominant case there is also an enhancement of conductance just below |eV| = Δ_0 in the parallel orientation compared to the case with h = 0 as shown in Fig. <ref>(a). Even in the p-wave dominant case there is a very small enhancement in this voltage window, see Fig. <ref>(a). This is a specific feature of the helical p-wave superconductor. Indeed, for the helical p-wave superconductor the direction of the d-vector is parallel to momentum, and therefore even if h⃗ and ⟨d⃗⟩ are parallel there exist modes for which h⃗ and d⃗(ϕ) are not, and the conductance is enhanced. Since the mode at normal incidence has the largest transmission eigenvalue, the enhancement in conductance if h⃗∥⟨d⃗⟩ is small compared to the enhancement for h⃗⊥⟨d⃗⟩.The predicted sign and anisotropy of the magnetoresistance can only be found if the exchange field strength h is much smaller than the Fermi energy E_F.If the ferromagnetic interaction is significant compared to the Fermi energy, the difference in momenta for opposite spins is large. In that case the conductance is maximized for the parallel orientation and minimized for the antiparallel orientation in F/S/F or F/F junctions <cit.><cit.>. In our formalism such effects are suppressed by a factor h/E_F and therefore negligible compared to the effect described here as long as h/E_F≪ 1. The difference in symmetries between the two effectsin fact allows one to disentangle them. Also within our quasiclassical formalism there is a small bipolar component, however this component is small compared to the quadrupolar component.Because time-reversal symmetry is broken in the superconductor, for is + helical p-wave superconductors r = 1 is not a topological phase transition, unlike the s + helical p-wave superconductor <cit.>. Therefore the conductance is continuous as a function of r, both in the absence and in the presence of an exchange field, as confirmed by the results in Fig <ref>.§ CHIRAL SUPERCONDUCTORSThe same calculations were performed for chiral superconductors, for which the d-vector is given byd⃗(ϕ) = e^iϕ(0,0,1).We used the same set of parameters as for the junction with (i)s + helical p-wave superconductors, that is, γ_BS = 2, z = 0.75, E_Th/Δ_0 = 0.02. The dependence of the conductance on the exchange field is illustrated in figure <ref> for s-wave dominant superconductors and <ref> for p-wave dominant superconductors. We first elaborate on the results and how they depend on parameters, then we provide a physical explanation for the differences obtained with the (i)s + helical p-wave junctions.Also for is + chiral superconductors a weak exchange field leads to an increase in the conductance for all r, see Fig. <ref>. As for the helical case, the enhancement is much stronger if the exchange field is perpendicular to the d-vector (Fig. <ref>(b)) than if it is parallel to it (Fig. <ref>(a)). The enhancement however, is smaller than for the s + helical p-wave superconductors, maximally around 0.05σ_N for the perpendicular orientation and almost negligible in the parallel orientation, and is suppressed for larger exchange fields, as shown in <ref>(b).For p-wave dominant is + chiral p-wave superconductors there is a weak suppression of the conductance in the presence of an exchange field (Fig. <ref>(b)), but it is very small compared to a parallel orientations, see Fig. <ref>(a). Correspondingly, the peaks at |eV| = h only appear for parallel exchange fields. For |eV|≈Δ_0 the presence of an exchange field always decreases the conductance in junctions with s + p-wave superconductors, whereas it increases the conductance for is + p-wave superconductors, though the effect is again considerably smaller than for helical superconductors. Thus, while our method works well for s + helical p-wave superconductors, it does not do so if chiral superconductors are present.The difference between the (i)s + chiral and (i)s + helical p-wave superconductors can be understood by considering the phase difference for each mode, depicted in Fig. <ref>. For (i)s + helical superconductors the phase difference between the s-wave and p-wave components is 0 or π/2 for all modes. On the other hand, in the chiral case the phase difference is mode dependent and therefore for some modes the conductance is enhanced, while for others it is suppressed. For s + chiral p-wave superconductors the phase difference ranges between -π/2 and π/2, whereas for is + p-wave superconductors it ranges between -π and 0. These two cases are different, because the transparency is higher for the mode at normal incidence compared to the modes at large angle incidences. However, the varying phase difference does considerably soften the effects, explaining why the observed enhancement is much weaker in (i)s + chiral superconductors than for (i)s + helical superconductors.On the other hand, the anisotropy in the magnetoresistance is relatively much stronger for (i)s + chiral superconductors than for (i)s + helical superconductors. Indeed, for the chiral superconductor the direction of the d-vector is independent of the direction of momenta, and therefore for all modes exchange field and d-vector are either parallel or perpendicular. However, due to the much smaller magnetoresistance the absolute anisotropy is smaller than for the (i)s + helical superconductors.The significant suppression of conductance for the is + chiral superconductors in the voltage region 1-r/√(1+r^2)<|eV|/Δ_0<1/√(1+r^2) in Fig. <ref>(a) can also be understood using this picture. As discussed in the section on (i)s+helical p-wave superconductors, the modes for which the phase difference between singlet and triplet components is π/2 enhance the conductance only for Δ_s<|eV|<√(Δ_s^2+Δ_p^2), that is, 1/√(1+r^2)<|eV|/Δ_0<1. On the other hand, for modes in which the phase difference between singlet and triplet components is almost 0, the conductance is suppressed in the larger window 1-r/√(1+r^2)<|eV|/Δ_0<1+r/√(1+r^2). Thus, for 1-r/√(1+r^2)<|eV|<1/√(1+r^2) the conductance is suppressed by an exchange field. § DISCUSSIONWe have presented a method to determine the pair potential in time-reversal symmetry broken non-centrosymmetric superconductors. The differential conductance close to the gap edge is enhanced by an exchange field in the presence of time reversal symmetry breaking in the superconductor. Next to this, our results show that by varying the direction of the exchange field this enhancement can also be used to determine the direction of the d-vector of the triplet correlations. With this we have presented a theory for the complete determination of the pair potential in (i)s + helical p-wave superconductors, as summarized in Table <ref>. The mixing parameter, the phase difference between singlet and triplet components and direction of the d-vector can all be determined fromconductance measurements in SFN junctions.Our results show that if the exchange field is small compared to the Fermi energy, the magnetoresistance of the SFN junction is quadrupolar, and hence qualitatively different from the dipolar giant magnetoresistance found if the ferromagnetic interaction is comparable to the Fermi energy. Indeed, we found that maximal enhancement is obtained in perpendicular exchange fields, while minima are achieved in both parallel and antiparallel orientations, while GMR distinguishes between parallel and antiparallel orientations. Our effect is only present in a specific voltage window, determined by the mixing parameter, but can reach 0.1σ_N of the normal state conductance. To determine the pair potential in an arbitrary time-reversal and inversion symmetry broken superconductor, one needs to include also the higher order angular momenta, d-wave f-wave etc. For such superconductors the conductance is different, but since the conductance enhancement depends only on the phase between the superconductors and the relative orientation of the spin in the superconductor and ferromagnet the qualitative features of our results should also be visible in s + f or d + p wave superconductors, as long as the individual components do not break time reversal symmetry, i.e. they are not chiral. For (i)s + chiral superconductors, which may be distinguished from (i)s + helical superconductors using the directional dependence of the d-vector <cit.> the mode dependence of the phase difference smoothens the results and makes the extraction of the parameter r more difficult. However, χ_t may be determined using the conductance enhancement in the case χ_t = 1. The theory can not be used for s +id (p+if) superconductors, as in such superconductors only singlet (triplet) correlations are present. § ACKNOWLEDGEMENTSWe would like to thank Sebastian Bergeret for usefull discussions. T.K. acknowledgesfinancial support from Spanish MCIN/AEI/10.13039/501100011033 through project PID2020-114252GB-I00 (SPIRIT) andTED2021-130292B-C42, and the Basque Government through grant IT-1591-22. Y.T. acknowledges support from JSPS with Grants-in- Aid for Scientific Research ( KAKENHI Grants No. 20H00131 and No. 23K17668). | http://arxiv.org/abs/2311.15708v1 | {
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[ Individualized Treatment Allocationswith Distributional WelfareFor helpful discussions, the authors are grateful to participants at the Advances in Econometrics Conference 2023 and the seminar participants at Brown University. We also thank Xuanman Li for her excellent research assistance. Yifan Cui Center for Data Science Zhejiang University mailto:cuiyf%5C%5C%5C%[email protected] Han School of Economics University of Bristol mailto:vincent.han%5C%5C%5C%[email protected] January 14, 2024 ====================================================================================================================================================================================================================================================================================================== type=figure < g r a p h i c s > figureRO-LLaMA as a generalist large language model (LLM) in the field of radiation oncology. The model seamlessly coversvarious tasks such as clinical report summarization, radiation treatment plan suggestion, and plan-guided target volume segmentation. All the patient information and any potential personally identifiable information are de-identifiied, and the clinical data usage is approved by IRB. ] Recent advancements in Artificial Intelligence (AI) have profoundly influenced medical fields, by providing tools to reduce clinical workloads. However, most AI models are constrained to execute uni-modal tasks, in stark contrast to the comprehensive approaches utilized by medical professionals. To address this, here we present RO-LLaMA, a versatile generalist large language model (LLM) tailored for the field of radiation oncology. This model seamlessly covers a wide range of the workflow of radiation oncologists, adept at various tasks such as clinical report summarization, radiation treatment plan suggestion, and plan-guided target volume segmentation.In particular, to maximize the end-to-end performance, we further present a novel Consistency Embedding Fine-Tuning (CEFTune) technique, which boosts LLM's robustness to additional errors at the intermediates while preserving the capability of handling clean inputs, and creatively transform this concept into LLM-driven segmentation framework as Consistency Embedding Segmentation (CESEG).Experimental results on multi-centre cohorts demonstrate our RO-LLaMA's promising performance for diverse tasks with generalization capabilities.*These authors contributed equally to this work †Corresponding authors § INTRODUCTIONRecently, the emergence of a new generation of AI models, known as foundation models, marks a significant departure from previous paradigms <cit.>. These foundation models are now capable of achieving state-of-the-art performance (SOTA) in a wide range of domains, including tasks such as multi-modal reasoning, image-to-text generation, image captioning, and text-guided image segmentation <cit.>.These characteristics signify a potential paradigm shift in how AI can be integrated into medical practices, which inherently rely on multi-modal information for comprehensive clinical decision-making.Furthermore,this gives an opportunity of overcoming limitations ofnow over 500 FDA-approved AI models, which are mainly specialized for a specific task with uni-modal information <cit.>.Specifically, in contrast to these uni-modal AIs, generalist medical AIs can encompass a holistic understanding of clinical workflows, which can receive and reason a variety of medical data, including imaging modalities, electronic health records, laboratory results, genomics, and even clinical reports <cit.>. Inspired by this paradigm shift in medical AIs, herewe present RO-LLaMA, a prototype of a generalist medical AI model,tailored for the clinical workflow in radiation oncology. RO-LLaMA is adept at performing a variety of clinical tasks: (1) It efficiently summarizes extensive patient histories and examination results into concise but informative clinical notes. Additionally, it is capable of (2) proposing appropriate treatment plans from a clinical expert perspective and (3) delineating radiation target volumes on 3-dimensional (3D) computed tomography (CT) scans consistent with the proposed treatment plans. This multifaceted functionality demonstrates a significant alignment with the expertise of clinical professionals.In the course of the aforementioned sequential tasks, the inevitable accumulation of errors in generations arises. To address this issue, we firstly explore Noisy Embedding Fine-Tuning (NEFTune) <cit.>which involves injecting uniform noise into embeddings during the training for our targeted tasks. To further enhance the model's applicability, we introduce a novel consistency regularization method, bolstered by our pioneering Consistency Embedding Fine-Tuning (CEFTune) technique, which add regularization loss to enforce the consistency between the prediction given noisy and clean inputs. Expanding beyond text-related tasks, we apply these concepts to 3D segmentation tasks, resulting in Noisy Embedding Segmentation (NESEG) and Consistency Embedding Segmentation (CESEG). These advancements collectively contribute to a marked improvement in the model's generalization capabilities in both internal and external validation. Our contributions can be summarized as: * We propose a comprehensive framework, denoted as RO-LLaMa, wherein LLM facilitates the entire workflow of radiation oncology.* We explore noise augmentation and consistency, and propose a novel training approach, such as CEFTune, NESEG, and CESEG. * Through experiments on both internal and external datasets, we demonstrated that RO-LLaMA outperforms baseline methods. § RELATED WORKSInstruction Fine-tuning.Instruction fine-tuning has been emerged as a pivotal technique for augmenting model responsiveness in the field of natural language processing (NLP). The simplicity and effectiveness of this approach have been introduced in numerous works, with a notable scale-up facilitated by advancements of LLMs such like GPT-3 <cit.>, ChatGPT <cit.>, and GPT-4 <cit.>. Notably, a pioneering contribution, Self-Instruct <cit.>, involves fine-tuning of foundation models using instruction-output pairs generated from InstructGPT <cit.>. This methodology, along with similar approaches, has led to diverse language model variants, including Alpaca <cit.>, Vicuna <cit.>, Dolly <cit.>, and a series of LLaMA <cit.>, by exhibiting promising performance across a wide range of tasks. In the medical domain, Chat-Doctor <cit.>, Med-Alpaca <cit.>, PMC-LLaMA <cit.>, and Asclepius <cit.> have been fine-tuned for clinical question-answering (QA) task. Although these models demonstrate robust performance for diverse QA benchmarks, there is a notable gap in their applicability for the practical clinical workflow in a specific domain, such as radiation oncology, which serves as the targeted focus of this work.Stabilizing LLM Fine-tuning with Noise.To enhance the robustness of language models against noisy input, various strategies have been explored. Approaches such as SMART <cit.>, FreeLB <cit.>, and R3F <cit.> employ adversarial training methods, introducing small Gaussian perturbations in embedding dimensions to optimize the model's performance against noise. Aparting from the adversarial training, LSNR <cit.> takes a different approach by directly optimizing the smoothness of each layer of the language model. More recently, NEFTune <cit.>improves LLM performance during fine-tuning by simply adding random noise into the embedding vectors during training. In the context of this study, we extend NEFTune by incorporating the concept of consistency regularization. This extension aims to further improve the robustness and generalization capabilities of language models when faced with noisy input. Language-driven Image Segmentation. Recent research in the field of image segmentation has been emerged to incorporate linguistic ability, such like language-driven semantic segmentation <cit.>, open-vocabulary segmentation <cit.>, referring segmentation <cit.>, and reasoning segmentation <cit.>. Language-driven segmentation has triggered a paradigm shift in medical domain, where multi-modal knowledge is inevitable. For instance, LViT <cit.> and ConTEXTualNet <cit.> introduce text-driven chest X-ray radiography segmentation. Target volume segmentation in the field of radiation oncology is more challenging, due to its intrinsic need for considerations of the clinical aspects beyond the image, such as overall cancer stage, treatment aim, pathologic findings, and so on <cit.>. Recent research on clinical context-aware breast cancer radiotherapy delineation has demonstrated that the multi-modal AI outperforms the traditional uni-modal AI by a substantial margin, particularly when labeled datasets are scarce <cit.>. Unlike previous approaches, we incorporate treatment plans generated by LLM directly from clinical reports, which aligns seamlessly with actual clinical workflows in radiaion oncology.§ METHODS In this section, we provide a detailed methodology for our proposed approach, which is designed for sequential text generation tasks, including summarization and suggestions, as well as text-driven image segmentation, as illustrated in Figure <ref>. In Sec <ref>, we describe text-related fine-tuning apporaches that yield report summarization and plan suggestion, respectively. In Sec <ref>, we introduce a novel segmentation approach that involves injecting noise and regularizing consistency to the text embedding for improving robustness of plan-guided 3D target volume segmentation. §.§ LLM Fine-tuning for Clinical GeneralistTo realize the generalist LLM with expertise in clinical report summarization and radiation treatment plan suggestion, we conduct instruction fine-tuning for LLaMA2 <cit.>. Consideringthe intended objective of each task, we adopt separate strategies to acquire task-specific expertise, namely RO-LLaMA-S (summary expert) and RO-LLaMA-P (plan expert).When the summary expert gets clinical reports as training inputs, the collected reports are raw and unrefined,constituting noisy data for the model. To enhance robustness for the noisy input, we employ Noisy Embedding Fine-Tuning (NEFTune) <cit.>, resulting in RO-LLaMA-S+[`+' , `++' denotes the adoption of NEFTune, and CEFTune, respectively]. In the case of the plan expert, we choose to utilize the train set composed of collected notes instead of generated notes, primarily due to cost considerations.In contrast to training, our model takes the generated notes as input for inference. Consequently, adopting NEFTune is also an effective solution to handle noisy inputs in this task. However, a crucial consideration arises from the nature of the generated notes, which may lie closer to clean inputs (collected notes) or deviate towards noisy inputs. To address this, it is essential to train the model to handle both clean and noisy inputs. To preserve the robustness facilitated by NEFTune while enforcing consistency between the prediction given clean and noisy inputs, we introduce Consistency Embedding Fine-Tuning (CEFTune), resulting in RO-LLaMA-P++.More details are as follows. Revisiting Noise Embedding Fine-Tuning.NEFTune has been recently introduced to enhance the performance of instruction fine-tuning. This approachinvolves injecting a random noise vector into embeddings during the training process as follows:ℒ_NEFTune(θ)= 𝔼_(,) ∼ Dℒ_ce(f_θ(),), =+ (α / √(LC)) ϵ,ϵ∼𝒰(-1,1)where ∈ℝ^B × L × C is the embedding of data, B denotes batch size, L is token length, C is embedding dimension,is perturbed embedding, f_θ(·) represents the model parameterized by θ,is the label of text sample. They demonstrated that the effectiveness of incorporating a random noise vector adjusted based on token length, which yields robust results in most fine-tuning scenarios.Consistency Embedding Fine-TuningTo bolster the model's capability in handling both noisy and clean inputs, we incorporate a regularization loss to encourage consistency as follows:ℒ_CEFTune(θ) =ℒ_NEFTune(θ) + λℛ(θ, θ^-),where ℛ(θ, θ^-) = d(𝒯(f_θ()), 𝒯(f_θ^-()))where θ^- represents the model with stopped gradient, and d(·,·) is used to quantify the discrepancy between f_θ() and f_θ^-(). The function 𝒯 serves as either the identity function or performs detokenization to generate sentences. Our objective is to preserve the robustness property of NEFTune while introducing semantic similarity through the integration of consistency between the output of noisy inputs and clean inputs. The metric function d can be any capable of measuring distance in a specific space. However, solely minimizing distance in the same embedding space might weaken the robustness effect and pose limitations on enforcing semantic similarity. In our approach, to capture textual similarity between sentences, we calculate the distance in the feature space of SentenceBERT <cit.>. i.e., d(𝒯(x),𝒯(y)) = 1 - s(𝒯(x))^⊤s(𝒯(y))/||s(𝒯(x))||_1 ||s(𝒯(y))||_1, where s(·) is the projection by using pretrained model such as SentenceBERT <cit.>, 𝒯 is set to detokenization operator.§.§ LLM-driven 3D Target Volume Segmentation For incorporating the textual information into target volume segmentation framework, we expand the concept of prompt tuning of LLM for context-aware 3D segmentation framework introduced in <cit.>. However, as the LLM-driven 3D segmentation model gets the generated treatment plan as text conditions which are inevitably noisy from the previous summarization step, we introduce noise augmentation and consistency regularization modules for the target volume segmentation network, namely, Noise Embedding Segmentation (NESEG) and Regularized Consistency Embedding Segmentation (CESEG). Noise Embedding Segmentation (NESEG).Firstly,by extending aforementioned idea of NEFTune, we inject a random noise vector into input text embeddings to improve the segmentation model robustness for the generated noisy plan as the text condition. The loss function for NESEG can be formulated as:ℒ_NESEG(Θ) = 𝔼_(,,)∼ Dℒ_ce(g_ψ(;),) where (ϕ) = f_θ^*(;𝐳_ϕ)where Θ = [ψ, ϕ], ∈ℝ^B × H W S is the 3D CT scan, B denotes batch size, H, W, and S correspond to height, width, and slice of the CT scan, ∈ℝ^B × 1 × C is the perturbed LLM output embedding, ∈ℝ^B × H W S is the 3D ground truth segmentation mask, g_ψ(·) represents segmentation expert model, denoted as the RO-LLaMA-SEG+, parameterized by ψ, and 𝐳_ϕ is the learnable text prompt.Consistency Embedding Segmentation (CESEG).We further adapt the consistency regularization module, defined as CESEG, for generalizing our segmentation model to both the generated noisy plan and the clean ground truth plan as for text condition. Apart from the original concept of CEFTune, we modified the consistency regularization module for the multi-modal segmentation task, which combines CEFTune and the text prompt tuning. Once the text prompt-prepended noisy embeddings of treatment planare inputted to the frozen LLM, the output embedding is regularized with that of clean embeddings . The loss function for CESEG is formulated as:ℒ_CESEG(Θ) = ℒ_NESEG(Θ) + λℛ(ϕ,ϕ^-)where ℛ(ϕ,ϕ^-)=d((ϕ), (ϕ^-))where, ϕ^- represents the learnable text prompts with stopped gradient and = f_θ^*(;𝐳_ϕ^-)∈ℝ^B × 1 × C is the LLM embedding given the clean plan as an input.§ EXPERIMENTS §.§ Dataset To train our model, we collected internal data cohort composed of 5,674 breast cancer patients treated at the Department of Radiation Oncology at Yonsei Cancer Center.For training the plan-guided 3D target volume segmentation network, we utilized multi-modal data from 599 patients with their 3D CT scans and corresponding ground-truth masks. To evaluate the model performance on cross-centre datasets, we further acquired external data cohort composed of 81 patients treated at the Department of Radiation Oncology at Yongin Severance Hospital.The detail of dataset is described in Table <ref> and Supplementary Material. We included patients who received their initial diagnosis of breast cancer and subsequently underwent radiation therapy following curative surgery, while excluding individuals with recurrent or metastatic breast cancer in both hospitals.This study was approved by the Institutional Review Board (IRB) of each participating hospitals.§.§ Implementation details For instruction fine-tuning for RO-LLaMA-S and RO-LLaMA-P, we use the LLaMA-2-7B-Chat <cit.> model as a baseline. The maximum context length is set at 4096, and the batch size is set to 2. To train RO-LLaMA-SEG, we employ the 3D U-Net <cit.> using the open-source library MONAI[<https://monai.io/>] and the LLaMA-2-7B-Chat model initialized with the pre-trained checkpoint of RO-LLaMA-P++. For training RO-LLaMA-SEG, LLM is frozen, while other network parameters including the text prompts are optimized. Further details are deferred to Supplementary Material.We use the AdamW <cit.> optimizer for all the tasks, with a learning rate of 5e-5 until reaching 3 epochs for both RO-LLaMA-S and RO-LLaMA-P, and 1e-4 until reaching 100 epochs for RO-LLaMA-SEG, leveraging 4 NVIDIA A6000 GPUs for each task.The hyper-parameter λ for CEFTune is set to 1 in all of tasks, and for consistency regularization we adopt the variants of SBERT <cit.> which is trained on PubMed[<https://www.ncbi.nlm.nih.gov/pubmed/>] dataset, called PubMedBERT.[<https://huggingface.co/NeuML/pubmedbert-base-embeddings>.]§.§ Evaluation Metric The evaluation of generated clinical report summaries and treatment plans utilizes common NLP metrics, including ROUGE <cit.>, BERTScore <cit.>, BARTScore <cit.>, and MoverScore <cit.>. However, recognizing potential limitations in domain specificity, expertise-based metrics, such as GPT-3.5-turbo and a board certified clinical expert evaluation, are incorporated, particularly for the nuanced task of treatment plan suggestion. Adopting concepts from recent evaluation methodologies such as GPT-Score <cit.> and G-Eval <cit.>, we developed a score rubric curated by clinical experts. The evaluation process includes GPT-3.5-turbo assessing specific examples of outputs based on this rubric. Additionally, clinical expert evaluation involves a comprehensive assessment of all outputs with ground truth label.To evaluate the 3D target volume segmentation performance using a ground-truth segmentation mask, we measure Dice coefficient (Dice), Intersection over Union (IoU), and 95th percentile of Hausdorff Distance (HD-95) <cit.> in centimeter (cm) unit.§.§ Baseline ModelsFor text-related tasks, to validate the effectiveness of our model, we compare our model with several clinical LLMs serving as baselines, specifically MedAlpaca <cit.>, ChatDoctor <cit.>, Asclepius <cit.>, and PMC-LLAMA <cit.>. Additionally, we include a comparison with ChatGPT <cit.> utilizing few-shot in-context learning for both summary and treatment plan suggestion task.For the segmentation task, we utilize 3D U-Net <cit.> and ConTEXTualNET <cit.> as our baseline methods. We further utilize an additional baseline as a variant of RO-LLaMA-SEG, naemly LLaMA-SEG, which is initialized with the checkpoint of vanilla LLaMA-2-7B-Chat model. §.§ Results Clinical Report SummarizationWe report the model performance on the clinical report summarization task in Table <ref>.Compared to the baselines, our fine-tuned variants of RO-LLaMA-S show significant improvements due to learning specific domain knowledge on all metrics. The qualitative assessment in Table <ref> further highlights the effectiveness of our proposed model, providing well-organized content and consistent formatting compared to the ground truth label. While there were minor errors observed in terms of the specific distance measurement from the nipple (red), which appears to stem from the process of integrating ultrasound and MRI images, our model accurately summarizes information in a format consistent with the provided labels. In contrast, Chat-GPT exhibits issues such as omitting essential information and generating hallucinated content (red).In terms of technical variants, both RO-LLaMA-S+ with NEFTune and RO-LLaMA-S++ with CEFTune outperform models trained using the vanilla approach and other clinical expert LLMs. RO-LLaMA+ exhibits superior ROUGE scores for both internal and external datasets, while RO-LLAMA++ outperform in embedding-based metrics. Considering our goal of generating sequentially tailored text data samples for target volume segmentation, we empirically designate the summary expert as RO-LLaMA-S+.Treatment plan SuggestionAs indicated in Table <ref>, we evaluate the model's performance on treatment plan suggestion task based on generated clinical notes. Similar to the report summarization task, our fine-tuned variants, RO-LLaMA-P, exhibit substantial enhancements in suggestion performance compared to other baselines. Notably, RO-LLaMA-P+ shows inferior performance on the external dataset compared to the vanilla approach. However, our proposed RO-LLaMA-P++ achieves the best performance on both internal and external datasets across all metrics. For a more detailed understanding, examples of the treatment plan suggestion task are provided in Table <ref>. The proposed RO-LLaMA-P++ (Ours) consistently yields correct answers compared to the labeled ground truth. These results demonstrate the effectiveness of our proposed CEFTune for this specific task. While Chat-GPT has shown success in providing some level of information regarding where to treat, it falls short when it comes to suggesting the specific treatment regions and dose scheme (red). 3D Target Volume Segmentation As reported in Table <ref>, our proposed plan-guided 3D target segmentation frameworks excel other baseline methods with large gains, specifically remainedstable performance across the external validation setting. Notably, compared to the uni-modal 3D U-NET model, our proposed RO-LLaMA-SEG++ shows substantial performance gain of up to 5% and 10% for internal and external validation, respectively. This can be also observed in Figure <ref>, where the vision-only method fails to segment target volume with incorrect laterality as indicated as red arrows. On the other hand, the proposed RO-LLaMA-SEG++ with CESEG shows the most promising qualitative segmentation performance, aligning consistently with the ground-truth labels as shown in Figure <ref>. Additional visual results are shown in Supplementary Material. § DISCUSSION §.§ Expert Analysis on Plan Generation As discussed in Section <ref>, for the treatment plan suggestion task, we comprehensively evaluate the generated plans using expertise-based metrics as indicated in Table <ref>. In terms of GPT-3.5-tubo evaluation, our RO-LLaMA-P++ outperforms other baseline methods with significant margin, benefiting from the learned knowledge for the field of radiation oncology.In terms of clinical expert evaluation, most of baseline approaches, except for Chat-GPT, show average score of zero. This indicates that the clinical evaluation is more rigorous than GPT-3.5-turbo’s stringency, revealing challenges in generating high-quality treatment plans. Although Chat-GPT achieves meaningful scores, our method notably outperforms all the methods, as shown in Figure <ref>. These results confirm our model’s effectiveness in generating treatment plans aligned with clinical expert assessments, surpassing existing LLMs in the general medical domain. §.§ Analysis on Consistency RegularizationWe further analyze the effect of the proposed consistency module according to the distinct types of input text. As mentioned, both the plan suggestion model and segmentation model utilizes clean ground truth data during training phase, whereas they employ noisy LLM-generated data during inference phase, potentially resulting in a performance gap. As indicated as difference in Table <ref>, the model's performance without our proposed consistency module significantly degrades for the generated data compared to the ground truth data for both tasks.On the other hand, employing our proposed CEFT and CESEG ensures robust performance for the generated data compared to ground truth data. In particular, for the segmentation task, our model exhibits a performance difference between two distinct input data of less than 1%. The results indicate that our key idea of consistency module effectively alleviates performance degradation when using LLM-generated data as input. § CONCLUSIONIn this work, we introduce RO-LLaMA, a versatile and general-purpose foundation model tailored for radiation oncology. Addressing limitations in current medical AI models confined to specific tasks, RO-LLaMA demonstrates proficiency in diverse tasks: clinical report summarization, radiation treatment plan suggestion, and plan-guided 3D target volume segmentation, mirroring real-world clinical workflows. Another key contribution of this work is the introduction of consistency technique into both text and segmentation task. Results from multi-center cohort datasets confirm RO-LLaMA's promising performance and noteworthy generalization capabilities across diverse tasks. These findings mark a significant stride toward developing a versatile AI model, hinting at the potential for a generalist medical AI model in radiation oncology.Limitation The used dataset is currently confined to patients with their initial diagnoses, necessitating an expansion of the scope to cover diverse patient scenarios.Ethical Statement All the patient information and any potential personally identifiable information from used datasets are de-identifiied, and clinical data usage is approved by the Institutional Review Board (IRB). ieeenat_fullname § SUPPLEMENTARY SECTION In this supplemental documents, we present:* An algorithmic descriptions of our proposed Consistency Embedding Fine Tuning (CEFT) and Consistency Embedding Segmentation (CESEG) in Section <ref>.* An explanation of the network architecture for the segmentation model in Section <ref>.* Results of ablation studies for both text-related tasks and target volume segmentation tasks in Section <ref> and Section <ref>. * Detailed information about the dataset, including preprocessing and acquisition procedures, in Section <ref>.* Examples of the entire workflow in Section <ref>.* Additional comparison results for all tasks in Section <ref>.* Detailed descriptions of the evaluation for treatment suggestion, including examples of prompts for GPT-3.5-turbo, in Section <ref>.§ ALGORITHM OF CEFTUNE AND CESEGAlgorithm <ref> and Algorithm <ref>details the training procedure with CEFTune and CESEG, respectively. § ARCHITECTURE OF RO-LLAMA-SEG++ The schematic of our RO-LLaMA-SEG++ is illustrated in Figure <ref>. We employed the 3D U-Net <cit.> for the image module. For the text module, we adapted the idea of text prompt tuning to transfer the linguistic capability of the LLM for realizing multi-modal context-aware segmentation framework, by following the concepts introduced in <cit.>. We employed pre-trained RO-LLaMA-P++ as for the LLM module within the text module. For the multi-modal alignment module which uses both self-attention and cross-attention mechanisms in an interactive manner (image-to-text and text-to-image), by following the concept of promptable segmentation borrowed from Segment Anything Model (SAM) <cit.>. During training, we let the entire parameters of LLM frozen, while updating the image module and the text prompts. As illustrated in Figure <ref>(c) for the text module, we introduced a total of N-text prompts {v^n|^N_n=1},where each v^n ∈ℝ^K × D is composed of K vectors with the dimension D.These learnable vectors were randomly initialized, and then consistently prepended to each of perturbed embedded treatment plan ∈ℝ^B × L × C. Here, the final prompted text input t can be formulated as follows:t = {v_1^n, v_2^n, ..., v_K^n, }.Then, using the prompted text input t, the frozen LLM resulted the plan embeddings ∈ℝ^B × 1 × C as for the last token output, as illustrated as in Figure <ref>(d). To align the text module output with the image module features,was projected to become g_l∈ℝ^b× 1 × Ch_l which have the identical dimension with that of each f_l through layer-wise linear layer, where f_l∈ℝ^H_l W_l S_l × Ch_l, where f_l is the l-th layer output of the image module, H_l, W_l, and S_l correspond to height, width, and slice of the image embeddings, and Ch_l is the intermediate channel dimension of each l-th layer output. We adapted CESEG to regularize consistency between the noisy input and the clean input by following Equation (<ref>). In Figure <ref>(a) and (b) for the image module, the projected plan embeddings g_l were self-attended and crossly-attended with the image features f_l for each layer through the multimodal alignment module, and the multi-modal embeddings f_l^* were inputted to the decoder part of the image module to yield final prediction ŷ_img∈ℝ^B × H W S. § ABLATION STUDY ON THERAPY PLAN SUGGESTION To investigate the effectiveness of our proposed methods, we conduct the ablation studies for treatment plan suggestion task, as shown in Table <ref>.Consistency Regularization Loss We manipulate the objective function in Equation <ref> to analyze the effect of consistency regularization. By default, we use f_θ^-() as the target (referred to as “clean teacher"). For comparison, we set f_θ^-() as the teacher and f_θ() as the student (referred to as "noisy teacher"). We observe that the clean teacher significantly outperforms the noisy teacher. By restricting the divergence of the distribution of students, the clean teacher yields enhanced performance compared to the noisy teacher. Effect of SBERT Module We study the effect of the SBERT module in enforcing consistency between clean and noisy input. Our model, equipped with the medical domain-specific SBERT variant, PubMedBERT, outperforms models trained on general language. This demonstrates the effectiveness of incorporating domain-specific knowledge. Analysis of Noise Intensity (α)We analyze the effect of noise intensity by varying α from 5 to 15. Smaller noise intensities result in better performance on the internal dataset, while larger noise intensities improve performance on the external dataset. However, excessively large noise intensity leads to a significant drop in performance on the external dataset. Based on empirical results, we adopt α set to 5 as the baseline. Analysis of Model BackboneWe conduct experiment using the larger backbone 13B LLaMA-2 version and observed that there are nosignificant differences compared to the 7B model in terms of performance. Considering cost-effectiveness and performance, weselect the 7B model as the default configuration. § ABLATION STUDY ON TARGET SEGMENTATIONTo optimize treatment target segmentation performance, we conduct the ablation studies, as shown in Table <ref>.Analysis of LLM Tuning Methods We compare the different LLM tuning methods to find optimal performance, where NESEG nor CESEG module is not activated. We analyze various methods including no tuning, low-rank adaptation (LoRA) fine-tuning <cit.> and prompt tuning methods with a single or multiple text prompts. From the experimental results, the text prompt tuning shows the most reliable performance across the external validation setting. Additionally, the use of single text prompt shows a remarkable improvement in performance compared to utilizing multiple text prompts.Analysis of Noise Intensity (α) We analyze the effect of noise intensity by varying α from 5 to 20, where CESEG module is not activated. We empirically find that α as 10 yields the best performance. We further train the hyperparameter α as a learnable parameter, however, yielding inferior performance compared to employing the fixed noise. § DETAILS OF TRAINING DATASET For data preparation, we collected internal training & validation data from patients treated at Yonsei Cancer Center between January 2009 and December 2022. For external validation, we further collected data from patients from Yongin Severance Hospital between January 2018 and December 2022. We confirmed that the external test set was not overlapped with the training dataset. For pre-processing image data for training RO-LLaMA-SEG++, all the 3D chest CT scans and target volumes were re-sampled with identical voxel spacing of 1.0 × 1.0 × 3.0 mm^3. The intensity values of the CT scans were truncated between -1,000 and 1,000 of Hounsfield unit (HU), and linearly normalized between 0 and 1.0. When training, a 3D patch with a spatial dimension of 384 × 384 × 128 pixels was randomly cropped.When inference, the entire 3D CT scans was tested using sliding windows with the identical spatial dimension of the 3D patch to that used for training.§ ADDITIONAL QUALITATIVE RESULTSIn this section, we provide the additional qualitative results on all tasks as shown in Table <ref>,Table <ref> and Figure <ref>. In the context of report summarization tasks involving cases with multiple (two or more) tumor masses, as shown in in Table <ref>, our model adeptly summarized the existence of two tumors in the first line of the summary, despite minor inaccuracies in pinpointing the exact locations of the tumors. In contrast, ChatGPT incorrectly summarized these cases as having only one tumor mass in the first line. Additionally, ChatGPT exhibited instances of hallucination, such as inappropriately mentioning tumor sizes in the first line, erroneously including post-therapy (yp) stages for patients who did not receive chemotherapy, and excessively condensing key information such as the presence of the lymph node metastasis, making it challenging to discern. Overall, unlike our model which aligns closely with the ground truth in its summaries, ChatGPT demonstrated significant issues, including the generation of erroneous hallucination and the failure to comprehensively encapsulate necessary details. In the evaluation of treatment plan suggestion capabilities in Table <ref>, a notable observation was that only our model and ChatGPT were able to offer somewhat meaningful suggestions related to radiation therapy, with other models falling short. ChatGPT, drawing upon its built-in knowledge base, seemed to propose radiation therapy plans that were partially relevant. However, it exhibited a fundamental misunderstanding of the radiation dosing strategies, frequently misinterpreting hypofractionation as conventional fractionation. A significant limitation in ChatGPT's approach was its inability to incorporate all regional lymph node areas into the treatment's target volume, often disproportionately focusing on specific areas, such as the axillary lymph nodes. Additionally, ChatGPT struggled to distinguish between the treatment of the breast and the chest wall, leading to flawed treatment plan suggestions in detail. This limitation likely arises from ChatGPT's lack of a holistic understanding of radiation oncology, unlike our domain-specific model.Figure <ref> provides qualitative comparisons of the target volume segmentation performance for breast cancer patients. In Figure <ref>(a) and (b), despite the ground-truth label posing target volume on the one side of the breast, the vision-only 3D U-Net incorrectly segment contours not only the target volume but alsothe outside of the target breast. Moreover, RO-LLaMA-SEG+ with NESEG module yield noisy segmentation output as indicated as red arrows. RO-LLaMA-SEG without any proposed module shows reasonable perfomormance, yet sometimes undersegments the treatment target volume. In contrast, the our proposed RO-LLaMA-SEG++ with CESEG module accurately contours the breast and regional lymph nodes that need to be treated.§ EXAMPLES OF ENTIRE WORKFLOW We present examples of the entire workflow as indicated in Table <ref> and Table <ref>. RO-LLaMA can take input as a pair of MRI report and Pathology report or another pair of Ultrasound report and Pathology report. Our model seamlessly serves as an assistant for radiation oncologists by conducting report summarization, treatment plan suggestion, and 3D Target Volume Segmentation. § PROMPT FOR EVALUATION OF PLAN SUGGESTIONIn this paper, we design prompts for evaluating generated treatment plans, including a reference-guided score rubric curated by clinical expert, as shown in Table <ref>. Similar to previous GPT-based evaluation methods, we tailor the prompts combining the specific knowledge of clinical experts. By using this prompt, we evaluate the results of all the clinical LLMs using GPT-3.5-turbo. | http://arxiv.org/abs/2311.15876v1 | {
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"Kwanyoung Kim",
"Yujin Oh",
"Sangjoon Park",
"Hwa Kyung Byun",
"Jin Sung Kim",
"Yong Bae Kim",
"Jong Chul Ye"
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"published": "20231127144906",
"title": "RO-LLaMA: Generalist LLM for Radiation Oncology via Noise Augmentation and Consistency Regularization"
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http://arxiv.org/abs/2311.16222v1 | {
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"Gilly Elor",
"Ryusuke Jinno",
"Soubhik Kumar",
"Robert McGehee",
"Yuhsin Tsai"
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"title": "Finite Bubble Statistics Constrain Late Cosmological Phase Transitions"
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matrixequationsection 1.23 packed_enum łλ ø Δ gap → ı∞ [] 𝒪 𝒯 | http://arxiv.org/abs/2311.16290v1 | {
"authors": [
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"Emanuel Katz",
"Yuan Xin"
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"published": "20231127200652",
"title": "Lightcone Hamiltonian for Ising Field Theory I: T < T_c"
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left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm labelsep=period .equationmithmTheorem[section] prop[thm]PropositiondefiDefinitionclaimClaimlem[thm]Lemmacor[thm]Corollaryex[thm]Examplefact[thm]Factres[thm]Research Problemconj[thm]Conjecturerem[thm]RemarkackAcknowledgementsThe minimum degree of minimal k-factor-critical claw-free graphs^73 Jing Guo^ 1, Qiuli Li^ 1, Fuliang Lu^ 2, Heping Zhang^ 1 ∗ ^ 1School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China^ 2School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, P. R. China[^73 This work is supported by NSFC (Grant No. 12271229 and 12271235) and NSF of Fujian Province (Grant No. 2021J06029).][^∗ The corresponding authors. E-mail addresses: [email protected] (J. Guo), [email protected] (Q. Li), [email protected] (F. Lu) and [email protected] (H. Zhang).]Abstract: A graph G of order n is said to be k-factor-critical for integers 1≤ k< n, if the removal of any k vertices results in a graph with a perfect matching. A k-factor-critical graph is minimal if for every edge, the deletion of it results in a graph that is not k-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimal k-factor-critical graph has minimum degree k+1. In this paper, we confirm the conjecture for minimal k-factor-critical claw-free graphs. Moreover, we show that every minimal k-factor-critical claw-free graph G has at least k-1/2k|V(G)| vertices of degree k+1 in the case of (k+1)-connected, yielding further evidence for S. Norine and R. Thomas' conjecture on the minimum degree of minimal bricks when k=2.Keywords: Claw-free graph; Minimal k-factor-critical graph; Minimal brick; Perfect matching; Minimum degreeAMS subject classification: 05C70, 05C07§ INTRODUCTIONAll graphs considered in this paper are finite and simple. For general graph-theoretic terminology, we follow J. A. Bondy and U. S. R. Murty <cit.>. Let G be a graph with vertex set V(G) and edge set E(G). Theorder of a graph G is the cardinality of V(G), written as |V(G)|. Aperfect matchingM of G is a set of edges such that each vertex is incident with exactly one edge of M. A graph containing no induced subgraph isomorphic to the complete bipartite graph K_1,3 is said to beclaw-free. Afactor-critical graph is a graph in which the removal of any vertex results in a graph with a perfect matching. A graph G with at least one edge is calledbicritical if, after the removal of any pair of distinct vertices of G, the resulting graph has a perfect matching. The concepts of factor-critical and bicritical graphs were introduced by T. Gallai <cit.> and L. Lovász <cit.>, respectively. In matching theory, factor-critical graphs and bicritical graphs are two basic bricks in matching structures of graphs.O. Favaron <cit.> and Q. Yu <cit.> independently introduced k-factor-critical graphs as a common generalization of factor-critical and bicritical graphs. A graph G of order n is said to be k-factor-critical for integers 1≤ k< n, if the removal of any k vertices results in a graph with a perfect matching. They also gave the following basic properties on connectivity of k-factor-critical graphs. If G is a k-factor-critical graph of order n with 1≤ k< n, then G is k-connected, (k+1)-edge-connected, and (k-2)-factor-critical whenever k≥ 2.For further results concerning k-factor-critical graphs, we refer the reader to <cit.>. Given a vertex u∈ V(G), a vertex v is called aneighbor of u if uv∈ E(G). Let N(u) denote the set of neighbors of vertex u in G, and d_G(u) (thedegree of u in G) be the cardinality of N(u). Theminimum degree of G, denoted by δ(G), is the minimum value in degrees of all vertices of G. Theorem <ref> indicates that every k-factor-critical graph has minimum degree at least k+1. O. Favaron and M. Shi <cit.> focused onminimal k-factor-critical graph, which is a k-factor-critical graph for which the deletion of any edge results in a non-k-factor-critical graph. They stated that every minimal factor-critical graph has minimum degree two from the ear decomposition of the factor-critical graph (see <cit.>, Theorem 5.5.1). Then they put forward a question that whether every minimal k-factor-critical graph of order n has minimum degree k+1, and confirmed it for k=n-6, n-4 and n-2. In <cit.>, Z. Zhang et al. formally reproposed the following conjecture.Let G be a minimal k-factor-critical graph of order n with 1≤ k<n. Then δ(G)=k+1.Abrick is a 3-connected bicritical graph <cit.>, which plays a key role in matching theory of graphs. J. Edmonds et al. <cit.> and L. Lovász <cit.> proposed and developed the “tight set decomposition" of matching covered graphs into list of bricks and braces in an essentially unique manner. This decomposition allows to reduce several problems from matching theory to bricks (e.g. a graph is Pfaffian if and only if its bricks and braces are <cit.>). A brick G isminimal if G-e is not a brick for every edge e∈ E(G). One may easily deduce that a 3-connected minimal bicritical graph G is also a minimal brick since, for any edge e∈ E(G), G-e is not bicritical, yielding G-e is not a brick. But the converse does not always hold. For example, Fig. <ref> presents a minimal brick G rather than a minimal bicritical graph as G-e is also bicritical.There are many researches on the minimum degree of minimal bricks. M. H. de Carvalho et al. <cit.> proved that every minimal brick contains a vertex of degree three. S. Norine and R. Thomas <cit.> presented a recursive procedure for generating minimal bricks and showed that every minimal brick has at least three vertices of degree three, and then went on to pose the following conjecture.There exists α>0 such that every minimal brick G has at least α|V(G)| vertices of degree three.Furthermore, F. Lin et al. <cit.> showed the existence of four such vertices. X. He and F. Lu <cit.> proved that every solid minimal brick G has at least 2/5|V(G)| vertices of degree three. In <cit.>, H. Bruhn and M. Stein obtained the following theorem.Every minimal brick G has at least 1/9|V(G)| vertices of degree at most four.We have shown in previous papers <cit.> that Conjecture <ref> is true for k=2, n-8 and n-10. In this paper, we confirm Conjecture <ref> for minimal k-factor-critical claw-free graphs. Moreover, we consider the number of the vertices of minimum degree in the case of (k+1)-connected. Then we derive that every 3-connected minimal bicritical claw-free graph has at least a quarter of the vertices of degree three, which yields further evidence for Conjecture <ref>. The following theorems are our main results. If G is a minimal k-factor-critical claw-free graph and k≥ 2, then δ(G)=k+1.Let G be a minimal k-factor-critical claw-free graph and k≥ 2. If G is (k+1)-connected, then G has at least k-1/2k|V(G)| vertices of degree k+1.For a minimal k-connected graph G,W. Mader <cit.> showed that δ(G)=k and <cit.> furtherobtained that G has at least (k-1)|V(G)|+2/2k-1 vertices of degree k.Some preliminaries are presented in Section 2. The proof of Theorem <ref> will be given in Section 3. Finally, we prove Theorem <ref> in Section 4. § SOME PRELIMINARIES In this section, we give some graph-theoretical terminologies and notations, and some known results that will be used in the proofs of the main results. Denote by G[S] the subgraph of G induced by S for S⊆ V(G), and G-S=G[V(G)-S]. If F⊆ E(G), then theedge-induced subgraphG[F] is the subgraph of G whose edge set is F and whose vertex set consists of the end-vertices of all edges of F. For an edge uv∈ E(G), G-uv stands for the graph obtained from G by deleting the edge uv∈ E(G). Likewise, G+uv stands for the graph obtained from G by adding an edge uv∉ E(G), where u, v∈ V(G). Let G_1 and G_2 be two graphs having no vertex in common. Theunion of G_1 and G_2 is the graph with vertex set V(G_1)∪ V(G_2) and edge set E(G_1)∪ E(G_2). Acomplete graph is a simple graph in which any two vertices are adjacent. Let G be a connected graph. A k-vertex cut of G is a set of vertices of G with size k whose removal disconnects G. A graph G is called k-connected for non-negative integer k if the removal of less than k vertices from G does not disconnect G. Similarly, a graph G is said to be k-edge-connected if the deletion of less than k edges from G does not disconnect it. W. T. Tutte <cit.> established the following fundamental theorem.A graph G has a perfect matching if and only ifc_o(G-X)≤ |X|for any subset X of V(G), where c_o(G-X) denotes the number of odd components of G-X.The following characterization of minimal k-factor-critical graphs of Tutte's type due to O. Favaron and M. Shi <cit.> will be useful in the proof of Theorem <ref>.Let G be a k-factor-critical graph. Then G is minimal if and only if for each edge e=uv∈ E(G), there exists S_e⊆ V(G)-{u,v} with |S_e|≥ k such that c_o(G-e-S_e)=|S_e|-k+2 and u and v belong respectively to two distinct odd components of G-e-S_e.For convention, we call a graph G istrivial if |V(G)|=1, andnontrivial otherwise. § PROOF OF THEOREM <REF> We are now ready to prove Theorem <ref> by contradiction. Proof of Theorem <ref>. Since G is a minimal k-factor-critical graph, by Lemma <ref>, for every edge e=uv∈ E(G), there exists S_e⊆ V(G)-{u, v} with |S_e|≥ k such that c_o(G-e-S_e)=|S_e|-k+2 and u and v belong respectively to two distinct odd components of G-e-S_e. Then the deletion of any k vertices of S_e results in a graph that every perfect matching of it contains e. Let G_e :=G-e-S_e. We consider the following two cases depending on |S_e|.Case 1.|S_e|≥ k+1. Clearly, c_o(G_e)≥ 3. Since G is k-factor-critical, the resulting graph has a perfect matching M after deleting any k vertices in S_e. Noticing that there are |S_e|-k=c_o(G_e)-2 vertices left in S_e and e∈ M, all remaining vertices in S_e are matched to vertices in distinct odd components of G_e under M. It follows that every vertex in S_e has neighbors in at least one odd component of G_e. By Lemma <ref>, for every edge e∈ E(G), there exists such a vertex set S_e with respect to e. Let S_e be the one with the minimum cardinality among all these vertex sets. We can conclude that every vertex in S_e has neighbors in exactly three odd components of G_e and must be adjacent to u and v. Otherwise, if there exists a vertex x∈ S_e such that x has neighbors in one or two odd components of G_e, then let S_e'=S_e\{x}. Thus c_o(G-e-S_e')=c_o(G-e-S_e)-1=|S_e|-k+2-1=|S_e'|-k+2. But |S_e'|<|S_e|, which is a contradiction to the choice of S_e. If there is a vertex in S_e that has neighbors in at least four odd components of G_e, then G has a claw, a contradiction. If there exists a vertex in S_e that has neighbors in precisely three odd components of G_e excluding u or v, this also contradicts that G is claw-free. Furthermore, we obtain that G_e has no even component. Otherwise, there is a vertex in S_e that has neighbors in an even component of G_e. Since every vertex in S_e has neighbors in exactly three odd components of G_e, yielding G contains a claw, a contradiction. Now we can obtain a graph G' by contracting all the odd components of G_e to single vertices and deleting all the multiple edges. Let U_e denote the set of the vertices resulting from the contraction of all the odd components of G_e. Then |U_e|=|S_e|-k+2 and V(G')=S_e∪ U_e. For convenience, we use u' (resp. v') to represent the vertex resulting from the contraction of the odd component of G_e containing u (resp. v). So u'v'∈ E(G'). Since G is k-factor-critical, every odd component of G_e is connected to at least k+1 vertices in the graph obtained by removing the odd component from G. Then every vertex in U_e\{u', v'} has at least k+1 neighbors in S_e. Thus there are at least (k+1)(|U_e|-2) edges from U_e\{u', v'} to S_e in G'. Conversely, since every vertex in S_e has neighbors in exactly three odd components of G_e and u and v must be its neighbors, every vertex in S_e has neighbors in one odd component of G_e that does not contain u and v. Then there are exactly |S_e| edges joining S_e and U_e\{u', v'} in G'. So we have the following inequality|S_e|≥ (k+1)(|U_e|-2)=(k+1)(|S_e|-k). Then |S_e|≤ k+1. By the assumption, |S_e|=k+1. So c_o(G_e)=3. That is, S_e has k+1 vertices, each of which has neighbors in the three odd components of G_e and is adjacent to u and v. At this time, we say that e is of type 1.Case 2.|S_e|=k. It is obviously that c_o(G_e)=2. Thus G_e has exactly two odd components containing u and v, respectively. Since uv∈ E(G), every odd component of G_e is connected to at least k vertices in S_e. That is, every vertex in S_e has neighbors in the two odd components of G_e. If either of the two odd components of G_e is trivial, then N(u)=S_e∪{v} or N(v)=S_e∪{u}. So d_G(u)=k+1 or d_G(v)=k+1. We are done. Thus we assume that neither of the two odd components of G_e is trivial.Moreover, S_e contains a vertex that has a neighbor in the odd component of G_e containing u (resp. v) other than u (resp. v). Otherwise, {u} (resp. {v}) is a vertex cut of G, contradicting that G is k-connected, k≥ 2. Thus we further conclude that G_e has no even component. Otherwise, every vertex in S_e has a neighbor in the even component of G_e. Then G contains a claw, a contradiction. In this case, we say that e is of type 2. From the above discussions, we see that, for every edge e∈ E(G), e is either of type 1 or of type 2 and each type of e corresponds to a configuration of G. In the following discussions, we always use S_e and G_e to represent the above vertex set and the graph G-e-S_e relative to edge e, respectively. Whether e is of type 1 or of type 2, let C_x denote the odd component of G_e containing vertex x. Next we prove the following two claims about the edges of type 1. Claim 1. If e=uv is of type 1, then C_u and C_v are trivial (as shown in Fig. <ref>).Proof of Claim 1. Without loss of generality, assume that C_u is nontrivial. Then none of the vertices in S_e have a neighbor in V(C_u)\{u}. Otherwise, G has a claw, a contradiction. But then {u} is a vertex cut of G, contradicting that G is k-connected, k≥ 2. Similarly, C_v is trivial. So Claim 1 holds. □ By Claim 1, we find that if e=uv is of type 1, then N(u)=S_e∪{v} and N(v)=S_e∪{u}. Thus N(u)\{v}=N(v)\{u}=S_e and d_G(u)=d_G(v)=k+2. Noticing that |S_e|=k+1, every vertex in S_e has neighbors in the three odd components of G_e. Then, for any u_1∈ N(u)\{v}=S_e, u_1 has a neighbor in the nontrivial odd component of G_e (see Fig. <ref>). Then N(u)\{u_1}≠ N(u_1)\{u}, which implies that uu_1 can not be of type 1. Besides, if d_G(u)=k+1 or d_G(u)>k+2 for any u∈ V(G), then u can not be an end-vertex of an edge of type 1. From these facts, we immediately deduce the following claim.Claim 2. For every u∈ V(G), u is incident with at most one edge of type 1. Since G is (k+1)-edge-connected and k≥ 2, every vertex of G must be incident with at least two edges of type 2 by Claim 2. In the following discussions, we consider the configuration of G corresponding to the edges of type 2 to complete the proof . Suppose to the contrary that δ(G)≥ k+2. Choose an edge, say e=uv, such that C_u is of the minimum order among all edges of type 2 of G. Then there exists S_e⊆ V(G)-{u, v} with |S_e|=k such that G_e has precisely two nontrivial odd components containing u and v, respectively.Let X=S_e∪{v}. Then X is a (k+1)-vertex cut. Assume that G_1 is the odd component of G-X containing u and G_2=G-X-V(G_1) (as shown in Fig. <ref> (a)). Then G_1=C_u and |V(G_2)| is even. Note that v has unique neighbor u in G_1. Since G_1 is nontrivial, u has at least one neighbor in G_1, say y. For the edge uy, uy is of type 2 as N(u)\{y}≠ N(y)\{u}. Then there exists S_uy⊆ V(G)-{u, y} with |S_uy|=k such that G_uy has exactly two odd components containing u and y, respectively. Let Y=S_uy∪{y}. Then Y is also a (k+1)-vertex cut. Let G_3 be the odd component of G-Y containing u and G_4=G-Y-V(G_3). So |V(G_4)| is even and N(y) ∩ V(G_3)={u} (as shown in Fig. <ref> (b)). Now we consider the graph G separated by the two (k+1)-vertex cuts X and Y. For convenience, we give some notations below. Assume that X∩ Y=T, X∩ V(G_3)=X', X∩ V(G_4)=X”, Y∩ V(G_1)=Y' and Y∩ V(G_2)=Y”. Then X=X'∪ X”∪ T and Y=Y'∪ Y”∪ T. Let V(G_1)∩ V(G_3)=I_0, V(G_1)∩ V(G_4)=I_1, V(G_2)∩ V(G_4)=I_2 and V(G_2)∩ V(G_3)=I_3 (as shown in Fig. <ref>). So u∈ I_0. Since y∈ V(G_1), y∈ Y'. Moreover, v∈ X'∪ T as v∈ N(u). Since N(v)∩ V(G_1)={u} and N(y)∩ V(G_3)={u}, N(u)∩ I_0=∅. Otherwise, G contains a claw, a contradiction. Then N(u)⊆ X'∪ Y'∪ T. Since d_G(u)≥ k+2, we deduce that|X'|+|Y'|+|T|≥ k+2.Combining with |X|+|Y|=2k+2, we obtain that|X”|+|Y”|+|T|≤ k.Moreover, |X”|<|Y'|. Otherwise, by (<ref>), k≥ |X”|+|Y”|+|T|≥ |Y'|+|Y”|+|T|=k+1, a contradiction. This observation leads us to the following claims.Claim 3.I_2≠∅.Proof of Claim 3. Suppose to the contrary that I_2=∅. Then |V(G_4)|=|I_1∪ X”| is even. Thus, for the edge uy, there exists a (k+1)-vertex cut (Y\{y})∪{u} such that G-(Y\{y})∪{u} has an odd component G[I_1∪ X”∪{y}] containing y. Moreover, u has unique neighbor y in V(G[I_1∪ X”∪{y}]). Since |X”|<|Y'|, |I_1∪ X”∪{y}|≤ |I_1∪ Y'|<|V(G_1)|. That is, |V(G[I_1∪ X”∪{y}])|<|V(G_1)|, which is a contradiction to the choice of G_1. So Claim 3 holds. □ Since G is k-connected and I_2≠∅, it is impossible that |X”|+|Y”|+|T|<k. It follows that |X”|+|Y”|+|T|=k by (<ref>). Then |X'|+|Y'|+|T|=k+2.Claim 4.X”≠∅.Proof of Claim 4. Suppose to the contrary that X”=∅. Since C_y=G[V(G_4)∪{y}] is a nontrivial odd component of G_uy, N(y)∩ V(G_4)≠∅. That is, N(y)∩ (I_1∪ X”)=N(y)∩ I_1≠∅. So I_1≠∅. Note that G_uy=G-uy-Y\{y} has no even component and G[V(G_4)∪{y}]=G[I_1∪ I_2∪{y}] would be an odd component of G_uy. However, G[I_1∪ I_2∪{y}] is disconnected because there is no path through X” connecting I_1∪{y} to I_2, a contradiction. So Claim 4 holds. □ If |X”|+|Y'|+|T|<k, then I_1=∅ as G is k-connected. Thus N(y)⊆ X”∪ (Y'\{y})∪ T∪{u}. So d_G(y)<k, a contradiction. If |X”|+|Y'|+|T|=k, then |I_1| is even (Possibly, I_1=∅). It can be seen that G-X”∪ (Y'\{y})∪ T∪{u} has an induced subgraph G[I_1∪{y}] with odd order, which is the union of some components of G-X”∪ (Y'\{y})∪ T∪{u}. Thus X”∪ (Y'\{y})∪T∪{u} is a k-vertex cut. This yields G-X”∪ (Y'\{y})∪ T∪{u} has no perfect matching, contradicting that G is k-factor-critical. It follows that|X”|+|Y'|+|T|≥ k+1.Together with |X|+|Y|=2k+2 again, we deduce that|X'|+|Y”|+|T|≤ k+1.Next we will prove that (<ref>) and (<ref>) are ture only if the equals in (<ref>) and (<ref>) hold at the same time by considering whether v∈ T or not.If v∉ T, then v∈ X'. By discussions similar to (<ref>), |X'|+|Y”|+|T|≥ k+1. Combining with (<ref>), we have |X'|+|Y”|+|T|=k+1. Then |X”|+|Y'|+|T|=k+1. If v∈ T, then v∈ Y. Since |X”|+|Y”|+|T|=k and I_2≠∅, N(v)∩ I_2≠∅ (as shown in Fig. <ref>). Otherwise, G has a (k-1)-vertex cut, contradicting that G is k-connected.Suppose that |X'|+|Y”|+|T|≤ k. Then I_3=∅. Otherwise, I_3≠∅ only if |X'|+|Y”|+|T|=k since G is k-connected. But then X'∪ Y”∪ T would be a k-vertex cut. So N(v)∩ I_3≠∅. Thus G contains a claw, a contradiction. We further deduce that |X'|≤ |Y'|. Otherwise, by (<ref>), k+1≤ |X”|+|Y'|+|T|<|X”|+|X'|+|T|=k+1, a contradiction. Noticing that G_1 is of the minimum order, |V(G_1)|≤ |V(G_3)|. That is, |I_0∪ I_1∪ Y'|≤ |I_0∪ I_3∪ X'|=|I_0∪ X'|. Then |X'|=|Y'| and I_1=∅. Thus |X'|+|Y”|+|T|=|Y'|+|Y”|+|T|=k+1, which is a contradiction to the assumption. Therefore, |X'|+|Y”|+|T|=k+1, and then |X”|+|Y'|+|T|=k+1.Thus we obtained that N(y)∩ I_1≠∅. Otherwise, N(y)⊆ X”∪ (Y'\{y})∪ T∪{u}. Then d_G(y)≤ k+1 as |X”∪ (Y'\{y})∪ T∪{u}|=k+1, a contradiction. So I_1≠∅.Claim 5.G[I_1∪{y}] is connected.Proof of Claim 5. Suppose to the contrary that G[I_1∪{y}] is disconnected. Let C_1 denote the component of G[I_1∪{y}] containing y and C_2=G[I_1∪{y}]-V(C_1). Since N(y)∩ I_1≠∅, C_1 is nontrivial. Noticing that |X”|+|Y'|+|T|=k+1, X”∪ (Y'\{y})∪ T is a k-vertex cut. Then every vertex in X”∪ (Y'\{y})∪ T has a neighbor in C_2. Moreover, there is a vertex x∈ X”∪ (Y'\{y})∪ T such that x has a neighbor in V(C_1)\{y}. Otherwise, {y} is a vertex cut,a contradiction. If x∈ X”∪ T, then N(x)∩ I_2≠∅ as X”∪ Y”∪ T is a k-vertex cut. Since N(u)⊆ X'∪ Y'∪ T and |X'|+|Y'|+|T|=k+2, N(u)=X'∪ Y'∪ T. Thus, if x∈ Y'\{y}, then ux∈ E(G). In either case, G has a claw, a contradiction. So Claim 5 holds. □ Claim 6.T=∅.Proof of Claim 6. Suppose to the contrary that T≠∅. By the proof of Claim 5, N(u)=X'∪ Y'∪ T. Then every vertex in X'∪ Y'∪ T is adjacent to u. Since |X”|+|Y”|+|T|=k and I_2≠∅, every vertex in X”∪ Y”∪ T has a neighbor in I_2. Next we will prove that every vertex in X”∪ Y'∪ T has a neighbor in I_1. Clearly, X”∪ (Y'\{y})∪ T∪{u} is a (k+1)-vertex cut. If |I_1| is even, then G[I_1∪{y}] is the odd component of G-X”∪ (Y'\{y})∪ T∪{u} containing y by Claim 5. Moreover, u has unique neighbor y in G[I_1∪{y}] and |I_1∪{y}|<|V(G_1)|. Then G[I_1∪{y}] is the desired odd component of G-X”∪ (Y'\{y})∪ T∪{u} with order less than G_1, which is a contradiction to the choice of G_1. If |I_1| is odd, then every vertex in X”∪ Y'∪ T has a neighbor in I_1 as desired. Otherwise, there exists x∈ X”∪ Y'∪ T such that N(x)∩ I_1=∅. Then G-(X”∪ Y'∪ T)\{x} has an induced subgraph G[I_1] with odd order, which is the union of some components of G-(X”∪ Y'∪ T)\{x}, yielding G-(X”∪ Y'∪ T)\{x} has no perfect matching. This contradicts that G is k-factor-critical because |(X”∪ Y'∪ T)\{x}|=k. Indeed, every vertex in T has a neighbor in I_0 (resp. I_1 and I_2), which implies that G contains a claw, a contradiction. So Claim 6 holds. □ Claim 7.|N(u)∩ V(G_1)|≥ 2. Proof of Claim 7. Suppose to the contrary that u has unique neighbor y in V(G_1). Then N(u)=X∪{y}=X'∪ Y'∪ T. By Claim 6, X'=X and Y'={y}. Consequently, X”=∅, which contradicts Claim 4. So Claim 7 holds. □ By Claim 7, let z be another neighbor of u that is different from y in V(G_1). Finally, we consider the edge uz. Since N(u)\{z}≠ N(z)\{u}, uz is of type 2. Then there exists S_uz⊆ V(G)-{u, z} with |S_uz|=k such that G_uz has exactly two odd components containing u and z, respectively. Let Z=S_uz∪{z}. Then Z is a (k+1)-vertex cut. Let G_5 be the odd component of G-Z containing u and G_6=G-Z-V(G_5) (as shown in Fig. <ref> (a)). So |V(G_6)| is even and N(z)∩ V(G_5)={u}. By discussions similar to X and Y, we derive that X and Z have the same properties as X and Y.Since N(v)∩ V(G_1)={u}, N(y)∩ V(G_3)={u} and N(u)=X'∪ Y', both G[X'] and G[Y'] are complete graphs. Likewise, noticing that N(v)∩ V(G_1)={u} and N(z)∩ V(G_5)={u}, both G[X∩ V(G_5)] and G[Z∩ V(G_1)] are complete graphs. It follows that X∩ V(G_5)=X' and Z∩ V(G_1)=Y'. So X∩ V(G_6)=X-X'=X”. Since (X'∪ Y')\{v, y} is a k-vertex cut if I_0\{u}≠∅, every vertex in X'\{v} has a neighbor in each component of G[I_0\{u}]. Note that X'\{v}≠∅. Otherwise, |X”|+|Y'|=k+k+1=2k+1, a contradiction. Similarly, every vertex in X'\{v} has a neighbor in each component of G[(V(G_1)∩ V(G_5))\{u}] if (V(G_1)∩ V(G_5))\{u}≠∅. It implies that V(G_1)∩V(G_5)=I_0. Then V(G_1)∩ V(G_6)=V(G_1)-I_0-Y'=I_1 (as shown in Fig. <ref> (b)). So both y and z have unique neighbor u in I_0∪ X'.If |I_1| is odd, then G-X”∪ (Y'\{y, z})∪{u} has an odd component G[I_1∪{y, z}], yielding G-X”∪ (Y'\{y, z})∪{u} has no perfect matching. This contradicts that G is k-factor-critical as |X”∪ (Y'\{y, z})∪{u}|=k. If |I_1| is even, then we can give a proof similar to the proof of Claim 6. Therefore, δ(G)=k+1. The proof of Theorem <ref> is complete. □§ PROOF OF THEOREM <REF>In this section, we consider the number of vertices of the minimum degree on minimal k-factor-critical claw-free graphs in the case of (k+1)-connected, k≥ 2.Proof of Theorem <ref>. By Theorem <ref>, δ(G)=k+1. Denote by V_1 the set of vertices of degree k+1 in G. Let G_1 :=G[V_1] and G_2 :=G[V(G)-V_1]. Then d_G(x)≥ k+2 for every x∈ V(G_2). Moreover, if e is of type 2 for an edge e∈ E(G_2), then none of the two odd components of G_e are trivial. Otherwise, at least one end-vertex of e is of degree k+1, a contradiction.By the proof of Theorem <ref>, we can see that e is either of type 1 or of type 2 for every edge e∈ E(G). In particular, if e=uv is of type 2, there exists S_e⊆ V(G)-{u, v} with |S_e|=k such that G_e has exactly two odd components containing u and v, respectively. Thus there exists a (k+1)-vertex cut, which is the union of S_e and u (resp. v), such that the deletion of it results in an odd component containing v (resp. u). Conversely, for an edge uv∈ E(G), if we find a (k+1)-vertex cut X containing v such that G-X has an odd component containing u and u is the unique neighbor of v in the odd component, then there is a configuration of G relative to the edge uv of type 2. We first prove the following claim. Claim. In G_2, the subgraph induced by edges of type 2 is a forest.Proof of Claim. Suppose to the contrary that G_2 contains a cycle C=u_1u_2⋯ u_tu_1 (t≥ 3) such that every edge of C is of type 2. Firstly, without loss of generality, we consider the edges u_1u_2 and u_2u_3. For the edge u_1u_2, let X_1=S_u_1u_2∪{u_1}. Then X_1 is a (k+1)-vertex cut. Denote by H_u_2u_1 the odd component of G-X_1 containing u_2 and let H_1'=G-X_1-V(H_u_2u_1). So |V(H_1')| is even and N(u_1)∩ V(H_u_2u_1)={u_2} (as shown in Fig. <ref> (a)).For the edge u_2u_3, we will prove that G has a (k+1)-vertex cut ontaining u_2 such that the removal of it results in an odd component containing u_3 with order less than H_u_2u_1.Let Y_1=S_u_2u_3∪{u_3}. Then Y_1 is a (k+1)-vertex cut. Denote by H_2 the odd component of G-Y_1 containing u_2 and let H_3=G-Y_1-V(H_2). So |V(H_3)| is even and N(u_3)∩ V(H_2)={u_2} (as shown in Fig. <ref> (b)). We can deduce that X_1≠ Y_1. Otherwise, u_3∈ X_1, H_u_2u_1=H_2 and H_1'=H_3. Since u_1 and u_3 have unique neighbor u_2 in V(H_u_2u_1), (X_1\{u_1, u_3})∪{u_2} is a k-vertex cut, contradicting that G is (k+1)-connected. Next we give some notations below when X_1 and Y_1 separate graph G. Let X_1∩ Y_1=T_1, X_1∩ V(H_2)=X_1', X_1∩ V(H_3)=X_1”, Y_1∩ V(H_u_2u_1)=Y_1' and Y_1∩ V(H_1')=Y_1”. Then X_1=X_1'∪ X_1”∪ T_1 and Y_1=Y_1'∪ Y_1”∪ T_1. Let V(H_u_2u_1)∩V(H_2)=J_0, V(H_u_2u_1)∩ V(H_3)=J_1, V(H_1')∩ V(H_3)=J_2 and V(H_1')∩ V(H_2)=J_3 (as shown in Fig. <ref>). Then u_1∈ X_1'∪ T_1 and u_3∈ Y_1'∪ T_1. Since N(u_2)⊆ X_1'∪ Y_1'∪ T_1,|X_1'|+|Y_1'|+|T_1|≥ k+2.Combining with |X_1|+|Y_1|=2k+2,|X_1”|+|Y_1”|+|T_1|≤ k.Noticing that G is (k+1)-connected, J_2=∅. Consequently, |J_1∪ X_1”| is even.Let H_u_3u_2=G[J_1∪ X_1”∪{u_3}] and X_2=(Y_1\{u_3})∪{u_2}. Then, for the edge u_2u_3, X_2 is a (k+1)-vertex cut containing u_2 and H_u_3u_2 is the odd component of G-X_2 containing u_3. Moreover, N(u_2)∩ V(H_u_3u_2)={u_3}. Thus there is a configuration of G relative to the edge u_2u_3 of type 2 (as shown in Fig. <ref>). Let H_2'=G-X_2-V(H_u_3u_2). If |X_1”|≥|Y_1'|, then, by (<ref>), k≥ |X_1”|+|Y_1”|+|T_1|≥ |Y_1'|+|Y_1”|+|T_1|=k+1, a contradiction. So |X_1”|<|Y_1'|. It follows that |J_1∪ X_1”∪{u_3}|≤|J_1∪ Y_1'|<|V(H_u_2u_1)|. That is, |V(H_u_3u_2)|<|V(H_u_2u_1)|. This implies that there exists a (k+1)-vertex cut X_2 containing u_2 such that G-X_2 has an odd component H_u_3u_2 containing u_3 with order less than H_u_2u_1.Secondly, we continue to consider the edge u_3u_4. Let Y_2=S_u_3u_4∪{u_4}. Then Y_2 is a (k+1)-vertex cut. Let H_3 be the odd component of G-Y_2 containing u_3 and H_4=G-Y_2-V(H_3). So |V(H_4)| is even and N(u_4)∩ V(H_3)={u_3} (as shown in Fig. <ref>). We need to find a (k+1)-vertex cut containing u_3 such that the removal of it results in an odd component containing u_4 with order less than H_u_3u_2. Likewise, X_2≠ Y_2. For convenience, we give some notations as follows. Let X_2∩ Y_2=T_2, X_2∩ V(H_3)=X_2', X_2∩ V(H_4)=X_2”, Y_2∩ V(H_u_3u_2)=Y_2' and Y_2∩ V(H_2')=Y_2”. Then X_2=X_2'∪ X_2”∪ T_2 and Y_2=Y_2'∪ Y_2”∪ T_2. Let V(H_u_3u_2)∩V(H_3)=L_0, V(H_u_3u_2)∩ V(H_4)=L_1, V(H_2')∩ V(H_4)=L_2 and V(H_2')∩ V(H_3)=L_3 (as shown in Fig. <ref>). Then u_2∈ X_2'∪ T_2 and u_4∈ Y_2'∪ T_2. Since N(u_3)⊆ X_2'∪ Y_2'∪ T_2,|X_2'|+|Y_2'|+|T_2|≥ k+2.Together with |X_2|+|Y_2|=2k+2,|X_2”|+|Y_2”|+|T_2|≤ k.Noticing that G is (k+1)-connected, L_2=∅. Consequently, |L_1∪ X_2”| is even. Let H_u_4u_3=G[L_1∪ X_2”∪{u_4}] and X_3=(Y_2\{u_4})∪{u_3}. Then, for the edge u_3u_4, X_3 is a (k+1)-vertex cut containing u_3 and H_u_4u_3 is the odd component of G-X_3 containing u_4. Besides, N(u_3)∩ V(H_u_4u_3)={u_4}. Thus there is a configuration of G relative to the edge u_3u_4 of type 2 (as shown in Fig. <ref>). Let H_3'=G-X_3-V(H_u_4u_3). If |X_2”|≥|Y_2'|, then, by (<ref>), k≥|X_2”|+|Y_2”|+|T_2|≥ |Y_2'|+|Y_2”|+|T_2|=k+1, a contradiction. So |X_2”|<|Y_2'|. It follows that |L_1∪ X_2”∪{u_4}|≤|L_1∪ Y_2'|<|V(H_u_3u_2)|. That is, |V(H_u_4u_3)|<|V(H_u_3u_2)|.This indicates that there is a (k+1)-vertex cut X_3 containing u_3 such that G-X_3 has an odd component H_u_4u_3 containing u_4 with order less than H_u_3u_2.Continuing this procedure, for any edge u_iu_i+1∈ E(C), i≥ 2, there exists a (k+1)-vertex cut X_i containing u_i such that G-X_i has an odd component H_u_i+1u_i containing u_i+1 with order less than the previous odd component H_u_iu_i-1 of G-X_i-1 containing u_i. Since G is finite, we must end up with a trivial odd component, which means that G_2 has a vertex of degree k+1 in G, a contradiction. So Claim holds. □ By Claim and Claim 2 of Theorem <ref>, the sum of degrees of all vertices in G_2 is at most 2|E(G_2)|+|V(G_2)|=2(|V(G_2)|-1)+|V(G_2)|. Since there are at most (k+1)|V(G_1)| edges from G_1 to G_2 and at least (k+2)|V(G_2)|-2(|V(G_2)|-1)-|V(G_2)| edges from G_2 to G_1, we have the following inequality(k+2)|V(G_2)|-2(|V(G_2)|-1)-|V(G_2)|≤ (k+1)|V(G_1)|. Then (k-1)|V(G_2)|+2≤ (k+1)|V(G_1)|. So (k-1)|V(G)|≤ 2k|V(G_1)|-2. Thus|V(G_1)||V(G)|≥(k-1)|V(G_1)|2k|V(G_1)|-2>(k-1)|V(G_1)|2k|V(G_1)|=k-12k. Therefore, G has at least k-1/2k|V(G)| vertices of degree k+1. □ In particular, Theorem <ref> implies that every 3-connected minimal bicritical claw-free graph G has at least 1/4|V(G)| vertices of degree three if k=2, which supports Conjecture <ref>.99BM J. A. Bondy and U. S. R. Murty, Graph Theory, New York: Springer, 2008.BS H. Bruhn and M. Stein, Minimal bricks have many vertices of small degree, European J. Combin. 36 (2014) 261-269.CLM M. H. de Carvalho, C. L. Lucchesi and U. S. R. Murty, How to build a brick, Discrete Math. 306 (2006) 2386-2410.ELW J. Edmonds, L. Lovász and W. R. Pulleyblank, Brick decompositions and the matching rank of graphs, Combinatorica 2 (1982) 247-274.F O. Favaron, On k-factor-critical graphs, Discuss. Math. Graph Theory 16 (1996) 41-51.FS O. Favaron and M. Shi, Minimally k-factor-critical graphs, Australas. J. Combin. 17 (1998) 89-97.GT T. Gallai, Neuer Beweis eines Tutte'schen Satzes, Magyar Tud. Akad. Mat.Kutató Int. Közl. 8 (1963) 135-139.GZ1 J. Guo and H. Zhang, Minimally k-factor-critical graphs for some large k, Graphs and Combin. 39 (2023), Paper No. 60, 18 pp.GZ2 J. Guo and H. Zhang, Minimum degree of minimal (n-10)-factor-critical graphs, preprint, 2022, https://arxiv.org/abs/2211.02933.GZ3 J. Guo, H. Wu and H. Zhang, Cubic vertices of minimal bicritical graphs, preprint, 2023, https://arxiv.org/abs/2305.06503.HL X. He and F. Lu, The cubic vertices of solid minimal bricks, Discrete Math. 347 (2024) 113746.LL L. Lovász, On the structure of factorizable graphs, Acta Math. Acad. Sci. Hungar. 23 (1972) 179-195.LO L. Lovász, Matching structure and the matching lattice, J. Combin. Theory Ser. B 43 (1987) 187-222.LP L. Lovász and M. D. Plummer, Matching Theory, Ann. Discrete Math., Vol. 29, North-Holland, Amsterdam, 1986; AMS Chelsea Publishing, Amer. Math. Soc., Providence, 2009.LR C. H. C Little and F. Rendl, Operations preserving the Pfaffian property of a graph, J. Aust. Math. Soc. A 50 (1991) 248-275.LDYQ D. Lou and Q. Yu, Sufficient conditions for n-matchable graphs, Australas. J. Combin. 29 (2004) 127-133.LZL F. Lin, L. Zhang and F. Lu, The cubic vertices of minimal bricks, J. Graph Theory 76 (2014) 20-33.MD W. Mader, Eine Eigenschaft der Atome endlicher Graphen,Arch. Math. 22 (1971) 333-336.MW W. Mader, Zur Struktur minimal n-fach zusammenhangende Graphen,Abh. Math. Sem. Universitt Hamburg 49 (1979) 49-69. N T. Nishimura, A closure concept in factor-critical graphs, Discrete Math. 259 (2002) 319-324.NT S. Norine and R. Thomas, Minimal bricks, J. Combin. Theory Ser. B 96 (2006) 505-513.PMD M. D. Plummer, Degree sums, neighborhood unions and matching extension in graphs. In: R. Bodendiek, ed., Contemporary Methods in Graph Theory, B. I. Wissenschaftsverlag, Mannheim, 1990, pp. 489-502.PS M. D. Plummer and A. Saito, Closure and factor-critical graphs, Discrete Math. 215 (2000) 171-179.TTW W. T. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107-111.Y Q. Yu, Characterizations of various matching extensions in graphs, Australas. J. Combin. 7 (1993) 55-64.YL Q. Yu and G. Liu, Graph Factors and Matching Extensions, Higher Education Press, Beijing, 2009.ZWLZ. Zhang, T. Wang and D. Lou, Equivalence between extendibility and factor-criticality, Ars Combin. 85 (2007) 279-285. | http://arxiv.org/abs/2311.15821v1 | {
"authors": [
"Jing Guo",
"Qiuli Li",
"Fuliang Lu",
"Heping Zhang"
],
"categories": [
"math.CO",
"F.2.2"
],
"primary_category": "math.CO",
"published": "20231127134652",
"title": "The minimum degree of minimal $k$-factor-critical claw-free graphs*"
} |
APS/[email protected] School of Computing and Information Systems, University of Melbourne, Parkville, Australia Department of Electronic and Electrical Engineering, University of Melbourne, Parkville, AustraliaDST Group, CanberraWe apply Game Theory to a mathematical representation of two competing teams of agents connected within a complex network, where the ability of each side to manoeuvre their resource and degrade that of the other depends on their ability to internally synchronise decision-making while out-pacing the other. Such a representation of an adversarial socio-physical system has application in a range of business, sporting, and military contexts. Specifically, we unite here two physics-based models, that of Kuramoto to represent decision-making cycles, and an adaptation of a multi-species Lotka-Volterra system for the resource competition. For complex networks we employ variations of the Barabási-Alberts scale-free graph, varying how resources are initially distributed between graph hub and periphery. We adapt as equilibrium solution Nash Dominant Game Pruning as a means of efficiently exploring the dynamical decision tree. Across various scenarios we find Nash solutions where the side initially concentrating resources in the periphery can sustain competition to achieve victory except when asymmetries exist between the two. When structural advantage is limited we find that agility in how the victor stays ahead of decision-state of the other becomes critical.Game-Theoretic Analysis of Adversarial Decision Making in a Complex Sociophysical SystemAlexander C. Kalloniatis January 14, 2024 ============================================================================================ § INTRODUCTION Physics-based modelling of competitive social-systems has been a consistent thread in complexity research. <cit.>.Such systems involve not only a competition of resources between two teams, with one seeking to have more or degrade those of the other, but also decision-making, where strategies are selected to achieve best position in the resource fight. Game Theory, where both sides select strategies within concepts such as the Nash equilibrium, is a natural extension of such models. In this paper we develop thiswhere the topology of network structures becomes a factor in onesocial-system, organisation or team achieving advantage over the other. The model we employ has been developed over a number of years. Firstly, a cognitive process is represented throughcoupled Kuramoto oscillators <cit.>, which can usefully represent the Boyd Observe-Orient-Decide-Act (OODA) model of decision making <cit.> between competing individuals or organisations <cit.>.Next, the resource competition between organisations is represented by a variation of the Lotka-Volterra multi-species predator-prey model, for example the Lanchester model <cit.> generalised to a network of resources of two adversarial teams, `Blue' and `Red' <cit.>. These models may be united into a single representation of two organisations, where each seeking to degrade the resource of the other, enabled by optimised inta-organisational collective decision-making to gain advantage <cit.>. Our unique contribution is a time-dependent game theoretic analysis of such a model under a Stackleberg Security-Strategy framework <cit.>.This intersection of sociophysical models and game theory may test concepts for 'victory' used in various contexts, for example the value of holding resources in reserve during armed conflict ('Economy of Force') <cit.>; or whether competing business firms should establish a niche or seek market dominance <cit.>. We examine how initial resource apportionment depends on network topology, albeit in a single stylised use-case,in terms of solutions representing Nash strategies; deviation from these yields no gain for winner or loser.In this paper Sect. <ref> summarises networked, adversarial `Boyd–Kuramoto' model, with the game theoretic tools used to construct solutions described in Sect. <ref>. Our main results are then presented in Sect. <ref>, which provide `heat-maps' describing the transitions in competitive outcomes for different structural resource distribution. We conclude with a brief summary and discussion in Sect. <ref> § MODEL The specific variant of the decision-making dynamics that we employ is the extended `Networked Boyd–Kuramoto–Lanchester' (NBKL) model <cit.>, which represents a Blue team (whose members are of the set ℬ) and Red team (of ℛ) adversarially by way of d θ_i/d t = ℋ(p_i) ( ω_i - ∑_j ∈ℛ∪ℬℋ(p_i) 𝒦_ijσ_ijsin(θ_i - θ_j - Φ_ij) ).For each agent i ∈ℛ∪ℬ the phase θ_i describes their state within a decision-making cycle (eg Observe-Orient-Decide-Act), while ℋ(p_i) is a Heaviside function that disables an agent when its resource degrades to zero: it ceases both to engage in the decision-making process and to apply degradation on an adversary. An agent left to itself advances decisions with frequency ω_i, however this varies with the number of links with other agents in the network through the adjacency matrix𝒦 = ϵ + ℳ. The matrix elements ϵ_ij, ℳ_ij are zero, except where nodes i and j are linked, either between or within their teams, respectively for ϵ and ℳ. The matrix σ = ζϵ then describes the strength of coupling between any two agents. Finally, Φ_ij∈ [0, π] represents an offset in the decision-state between two nodes;this uses the formalism of `frustration' in magnetic systems but here represents an intended advantage one agent seeks in decision-making relative to another. This systems nonlinear dynamics ensure that neither team is guaranteed to achieve their selected state as a stable solution; chaotic dynamics may deny one side a constant advantage ahead of the other.The evolution of the resource dynamics is described byd p_i/d t =ℋ(p_i) (∑_h ∈ℛ∪ℬℋ(p_h) ℳ_ihΓ_i,h + Γ_h,i/2(δ_h p_h - δ_i p_i) ·. cos(θ_h- θ_i) + 1/2 - ∑_k ∈ℛ∪ℬℋ(p_k) ϵ_ikκ_ikp_kd_k·. sin(θ_k - θ_i ) + 1/2O_k ) .This captures each team's internal ability to redistribute resourcesthrough synchronised decision-making in the first sum; and theability to degrade the other team's resources through decision advantage in the second sum. Within this, the parameters Γ = γℳ and κ = κ^RBϵ respectively represent the rates of resupply and degradation that can be achieved through network links.For simplicity, the same networks represent communication paths for decision-synchronisation and resource manoeuvre-paths for each team.These flows are moderated by the local synchronisation `order parameter' <cit.>, 𝒪_k and the parameters δ_k and d_k, which respectively take the formO_k= | ∑_m ∈ℳℳ_kmℋ(p_m) exp^√(-1)θ_m + ϵ_2 |/∑_m ∈ℳℳ_kmℋ(p_m) + ϵ_2 δ_k= 1/∑_m ∈εε_km p_m + 1d_k= 1/∑_m ∈εε_kmℋ(p_m) + ϵ_2.Thus, local coherence of decision-making enhances such manoeuvre.The nodes at which engagement between the two teams occur (where resource degradation is applied) is called the adversarial surface between the two communications and resource flow networks of each team. A summary of the parameter space of these quantities can be seen within Table <ref>. The specific choice of parameters has been motivated by previous works <cit.>, with specific emphasis on constructing equally capable teams for matched topologies, and to ensure that the speed of decision-making cycles and coupling is an order of magnitude faster than the speed of degradation of resources. §.§ Network TopologyTo explore the role of the network topology in the resource apportionment we consider a form of abstract complex graph for ℳ.While many options exist, including Erdos-Renyi and the Watts-Strogatz small world, we focus upon the Barabási-Albert scale-free graph <cit.>, with examples in Fig. <ref>.Here, the probability of a node of degree d follows a power-lawp(d) ∼ d^-γwith power exponent γ≈ 3. We will refer to Barabási-Albert graphs with a suffix (k) denoting the degree of nodes added in each step of the generating algorithm <cit.>.Thus we use the notation BA(k), where k>1 generates less sparse graphs with more paths or loops.Because of its characteristic hub-and-periphery structure, the scale-free graph has been argued to be demonstrated applicable to a broad range of human dynamics <cit.>.For human organisations, what recommends this choice is the prevalence of hierarchy (where the hub is characteristically the apex of the hierarchy), but with a complexity richer than a simple tree-graph. However,for real organisations the purity of the scale-free model is contested <cit.>. For our purposes, the graph is simply a stylised choice which distinguishes the key elements of the topology, the hub and the periphery. We also note that scale-free graphs allow for significant sizes (eg. number of users of the internet). The choice we explore here is `relatively' small, but nonetheless consistent with a largerversion of a team, N=100.We consider initial resource distributed across the graph based on a Boltzmann distribution of the eigencentralities ĉ_i, in whichP_i(T) = P_t exp(ĉ_i T)/∑_∀ iexp(ĉ_i T),where T is the resource temperature, and P_t is the total resource for a team. At zero temperature all resources are distributed evenly. Increasingly positive temperatures bias the resource distribution towards nodes with higher eigencentralities. Negative temperatures biases towards nodes with lower eigencentralities. We use eigencentralities ĉ rather than centralities (described in Eq. (<ref>)) due to their ability to better capture how a node is situated within the broader graph, rather than just its neighbours. § GAME THEORY SOLUTION CONCEPTSSolutions to coupled nonlinear differential equations like the coupled system of Eqs. (<ref>) and (<ref>) are readily achievable through standard numerical methods. However, while such solutions are possible for any Φ, to properly understand organisational dynamics it is important to understand how a team should position itself within Φ if it is playing rationally—a concept that can only be studied through game theory. With its inherent adversarial nature, our solution concept assumes each player is playing rationally to maximise its utility.We therefore advance a previously developed framework <cit.>, in which the (non-networked) BKL engagement can be classified as a two-player, zero-sum, strategic game, where each player is a rational and strategic decision maker. Though the strategy choices may be across numerous variables of the model (resource number, the topology, or the couplings) here we selectthe frustration parameter as reflecting the deliberate choice by teams how far ahead of the adversary decision `OODA' cycle <cit.> they are attempting to operate at different points in time. Again, nonlinear dynamics may deny them their selected strategy.Within this context, for given sets of networked teams of agents ℛ, ℬ, the frustration Φ is parameterised asΦ_ij(t) = ϕ_k ifi,j ∈ℬ ψ_k ifi,j ∈ℛ0 otherwise ,where the components ϕ_k ∈ [0, π] and ψ_k ∈ [0, π] represent the phase offsets employed by each player at discrete time-steps, t ∈ [k δ_t, (k+1) δ_t] for k ∈ [0, T_f/δ_t]. Restricting ϕ and ψ to changing at fixed temporal points facilitates computational tractability and deployment of exact game-theoretic solvers. Moreover, as the Networked BKL dynamics broadly exhibit exponential resource decay, introducing a finite game horizon t ∈ [0, T] ensures that computing resources are focused upon the portions of the game which impact the final resource distribution.At finite time horizon T_f, the game concludes. The resulting end-state of the NBKL system quantifies the game outcome for the players who each optimise forU_B(S_B, S_R) = P_B(S_B, S_R) - P_R(S_B, S_R), U_R(S_B, S_R) = P_R(S_B, S_R) - P_B(S_B, S_R),where P_R and P_B are the aggregate populations of each team, which depend upon player strategies/actions via the NBKL dynamics, and S_B and S_R represent the vectors of strategy choices across all decision points—the values of ϕ_k for ℬ and ψ_k for ℛ.Each player chooses their respective strategy vector such that their behaviour follows the Nash-Equilibria (NE), which is the set of player strategies (and utilities) where no player gains deviating from their strategy when all other players also follow their own NE. This corresponds to a fixed point at the intersection of players' best responses (see Definition 3.22 of Başar <cit.>). However, pure-strategy NE may not exist in games like this. It is then natural to consider the security strategies of players, which ensure a minimum performance. Also known as min-max and max-min strategies, these strategies allow each player to establish a worst-case bound on minimum outcome <cit.>. Due to both the low-likelihood of repeated replay for NBKL scenarios and the computational cost of solving the underlying systems at large scales, we focus on pure strategies in contrast to probabilistic mixed strategies.Constructing game theoretic solutions is made possible through Nash Dominant Game Pruning <cit.>, which explores the game recursively, with utilities back-propagated up the game tree from the leaves. In contrast to a search over the full game tree, the Nash Dominant approach identifies action pairs (ϕ_k, ψ_k) that are strategically dominated—that will not correspond to the equilibrium state—and truncates subsequent exploration over the game tree. This truncation occurs by identifying if any row or column would not be picked by its corresponding player because a utility along that axis is smaller/larger than the minimum/maximum observed value along a row or column. If this is observed, any sub-games that stem from decisions along an incompletely resolved row or column can be truncated.§ RESULTSWe now numerically solve NBKL games, considering scenarios where each team uses either its highest or lowest centrality nodesat the adversarial surface. Each team can choose from 4 distinct values of frustration parameter at 4 decision points in time, resulting in a game tree with 65,536 leaf nodes. To ensure that these games are both deterministic and balanced both teams use the same fixed Barabási-Albert graph of 100 nodes. To examine the role of centrality in game outcomes and the decision making process, we vary across temperature values—following Eq. (<ref>)—to control initial distribution of resources across the nodes. We consider different scenarios for the adversarial surface: both sides using the three highest centrality nodes as their adversarial surface, the asymmetric three Blue highest-vs-three Red lowest case, and both using the three lowest. Within the adversarial surface we consider when each three nodes are individually arrayed against one of the other side, “1-vs-1”, and where the three form a complete internal graph against the opposition three, “3-vs-3”. Lastly, we analyse by temperature the degree to which players reposition their actions, namely dynamically adjust the frustration Φ – how much they adjust attempting to be ahead of the decision-cycle of the adversary in the Nash Equilibrium– over the game dynamics. §.§ High vs High Centrality We begin by focusing upon the case where both players use their highest centrality nodes to compete against one another, the results of which can be seen in Fig. <ref>. Each point in the heatmap represents the utility at a Nash equilibrium for a specific variation by both teams of their frustration parameters at discrete points in time, to achieve decision advantage. The top row represents the BA(1) graph, bottom row BA(2); left column is 1-vs-1, and right3-vs-3.Across all these we see a characteristic diagonal axis of stalemate, with Blue dominating above and Red below the diagonal. Thus, a team prevails when it concentrates resources at the low-centrality nodes of the graph if the other side elects to concentrate at high-centrality. Thus Nash dominance is achieved if concentration is away from the nodes being used for adversarial interactions; resources must be initially in reserve at the periphery if the hubs are the focus of competition. However, while the broad morphology of the diagonal stalemate is ubiquitous, the finer detaildepends upon the internal connectivity within the graph and adversarial surface. While both BA(2) graphs share similar behaviours, the 3-vs-3 conflict scenario deviates around the diagonal when both players temperatures at the extreme top corner. The sign flip in the outcomes is numerically small, but suggests that more interlinked team structures offer opportunity for success with an “all-in” at the point of the fight. The BA(1) graphs show contrasting structures between the 1-vs-1 and 3-vs-3 scenarios. The latter differs from all three cases here in its uniformity across temperature values.These structural differences are repeated in Fig. <ref>,showing the proportion of repositioning in Φ.The stalemate diagonal is evident, and where the utility heatmaps show variability across temperature, we observe less repositioning in Φ. In contrast the more uniform result in utilities for BA(1) and 3-vs-3 coincides with more repositioning of Φ during the dynamics.Overall, this suggests that for sparser graphs and higher connectivity in the adversarial surface greater agility in decision-state enables more effective manoeuvre of reserves into the competition. In the remaining discussion we will therefore identify high repositioning of Φ with agility.§.§ High vs Low Centrality In the asymmetric case of Blue high- against Red low-centrality the diagonal symmetry disappears, as seen in Fig. <ref>. Nash dominance is consistently achieved by the high-centrality team when it concentrates resources at high-centrality nodes, namely at the adversarial surface. For the BA(1) graph, the low-centrality team should also centralise its resources at its low-centrality nodes on the adversarial surface. Neither side gains from reserving resource, a consequence of the structural asymmetry between the two sides. In particular, we see that however much Red seeks to concentrate at low-centrality nodes, Blue can always initially concentrate more at its high-centrality nodes. Thus, Blue seeks to overwhelm through structural superiority, and Red cannot afford to hold back because of structural inferiority.For asymmetry, committing all-in is important to exploit resource advantage or mitigate it when lacking superior numbers at the point of the fight. The only exception to this is the occurrence of discrete poles in the BA(1) 3-vs-3 case, suggesting local deviations within the random structure for intermediate initial resource distributions. These morphological sensitivities persist with different random seed for graph generation, and the number of nodes within each team's graph. This indicates that these behaviours are a property ofvariability within the bulk characteristics of the graph, rather than just its finer structure. In contrast to the High vs High (and, as we will establish forthwith the Low vs Low) scenario, the most highly polarised results are not correlated with the highest agility behaviours shown in Fig. <ref>. This, alongside the broad similarities between the 3 1-vs-1 and 3-vs-3 conflict suggests that the structural asymmetry of the High vs Low scenario is a structural factor that cannot be overcome by the agility of the players.§.§ Low vs Low Centrality When both players use their 3 low-centrality nodes (graph periphery) for the adversarial surface, Fig. <ref>,we see yet another variation in behaviours. For BA(1), results are inverted across the diagonal, relative to the High vs High scenario. Here thenNash dominance is achieved by the low-centrality team when it concentrates resources at the high centrality nodes, now away from the adversarial surface. Effectively, the limited paths to and narrowness of the bottleneck at the point of engagement requires that reserves be distributed.In the BA(2) case weare back to the High-High case, with players incentivised to draw all their resources to the adversarial surface; more paths allow closer concentration to the engagement. In the 1-vs-1 case we see another example of isolated poles, a product of the inherently nonlinear chaotic dynamicswith greater connectivity, again repeated with different random seed. Inspecting the agility across these cases, Fig. <ref> shows broad similarities to Fig. <ref>, in that the extrema of the Nash Equilibrium utilities correlate to the parts of the temperature space in which the players are more agile. The exception is the BA(1) 1-vs-1 result where the player repositioningis consistent with higher utility. As the most topologically constrained case, this shows that agility in seeking decision-advantage provides for higher utility.§ CONCLUSIONAnalysis of adversarial team interactions modelled as the Networked Boyd-Kuramoto-Lanchester equations through a game theoretic lens yields insights into the role of network topology in 'victory' or 'defeat'. By testing scale-free graphs with a characteristic hub-periphery topology, we were able to study the impact of structural changes in terms of the interlinking of teams, and their resource distributions. In the case of sparsely connected teams,we find that initial resources should be held in reserve, unless there is an asymmetry in the connectivity of agents at the focus of the engagement. If the hub is that point against the other team's periphery, then the former team must exploit that advantage. Concomitantly, for the team engaging with its periphery concentrating resource elsewhere than at those nodes makes a bad situation worse.However, as the density of the connections within the team graph increases, there is a need to transition towards greater resource centralisation at the point of engagement except when both teams employ their hub nodes at the adversarial surface.These results represent Nash equilibria in decision-advantage in relation to the other, modelled by the frustration, where the nonlinearity of theKuramoto-Lanchester model captures the complexity of resource competition in networked organisations.We have seen that across the scenarios there are cases where agility in that decision-state over time contributes to the utility outcome. In essence, structure dominates agility is less consequential; when structure is sparse then agility becomes a significantmechanism for achieving success. We have identified these outcomes from a single graph instance; future work will systematically classify this over larger ensembles. This work demonstrates that resource apportionment should be considered in parallel withstructural designs of organisationslocked in competition with an opponent and agile decision-making. That such insights are able to be gleaned from the combination of game theory and nonlinear dynamics indicates the value of such sociophysical modelling in a variety of contexts, including military strategy, business marketing strategy and cyber-security, to name but a few. | http://arxiv.org/abs/2311.17077v2 | {
"authors": [
"Andrew C. Cullen",
"Tansu Alpcan",
"Alexander C. Kalloniatis"
],
"categories": [
"physics.soc-ph",
"math-ph",
"math.MP",
"nlin.AO",
"nlin.CD"
],
"primary_category": "physics.soc-ph",
"published": "20231127222423",
"title": "Game-Theoretic Analysis of Adversarial Decision Making in a Complex Sociophysical System"
} |
Properties of Steady Sub-Alfvénic Solar Wind in Comparison with Super-Alfvénic Wind from Measurements of Parker Solar Probe [ January 14, 2024 =========================================================================================================================== Unsupervised domain adaptation is a critical challenge in the field of point cloud analysis, as models trained on one set of data often struggle to perform well in new scenarios due to domain shifts. Previous works tackle the problem by using adversarial training or self-supervised learning for feature extractor adaptation, but ensuring that features extracted from the target domain can be distinguished by the source-supervised classifier remains challenging. In this work, we propose a novel approach called progressive target-styled feature augmentation (PTSFA). Unlike previous works that focus on feature extractor adaptation, our PTSFA approach focuses on classifier adaptation. It aims to empower the classifier to recognize target-styled source features and progressively adapt to the target domain. To enhance the reliability of predictions within the PTSFA framework and encourage discriminative feature extraction, we further introduce a new intermediate domain approaching (IDA) strategy. We validate our method on the benchmark datasets, where our method achieves new state-of-the-art performance. Our code is available at <https://github.com/xiaoyao3302/PTSFA>.§ INTRODUCTION Deep learning on point clouds has revolutionized various real-world applications, including autonomous driving and robotics <cit.>. However, a significant challenge arises from the domain shift, where a model trained on one set of data fails to perform well in a new scenario due to differences in scanning devices or geometry variations. To address this issue, numerous unsupervised domain adaptation methods (UDA) have been developed for point clouds. Most of these methods focus on learning a feature extractor that can extract semantically meaningful features from both domains, which can then be classified by a source-supervised classifier. These methods can be broadly categorized into adversarial training-based methods <cit.> and self-supervised learning-based methods <cit.>. However, adversarial training encounters stability issues and lacks a guarantee of semantically meaningful feature extraction <cit.>. On the other hand, self-supervised tasks can primarily facilitate the learning of low-level geometry features, which may be inadequate for high-level recognition tasks, and there is no theoretical proof ensuring that the extracted features from target samples using self-supervised tasks can be distinguished by the source-supervised classifier <cit.>.In this work, different from the conventional focus on feature extractor adaptation, we propose a novel approach to address point cloud domain adaptation through classifier adaptation. Our approach involves constructing target-styled feature augmentations on the extracted source features and encouraging the classifier to recognize these target-styled source features. Although the previous work ISDA <cit.> has explored a similar feature augmentation strategy in the 2D domain, it cannot handle large domain gaps as it relies heavily on the estimation of the training set distribution to construct augmented features <cit.>.Another representative 2D UDA method TSA <cit.> tackles the problem by constructing semantically meaningful feature augmentations towards the estimated distribution of the target domain, based on the calculation of the mean and variance of all target samples according to the prediction results. However, in the context of point cloud adaptation, the distances between various target samples and the source domain exhibit significant disparities <cit.>. These disparities result in considerable variations in prediction accuracy in the target domain, leading to unreliable estimation of the target distribution, which adversely affects the overall adaptation performance. In addition, some hard samples that lie far away from the source domain may also be ambiguous for the classification boundaries. As a result, it is not feasible to treat each target sample equally in the distribution estimation process.To overcome the challenges, in this work, considering the uniqueness of each sample, we introduce a new sample-wise classifier adaptation approach on point clouds called progressive target-styled feature augmentation (PTSFA). Our approach involves creating a series of intermediate domains between the source domain and the target domain, and we encourage the model to progressively approach the intermediate domains to avoid direct adaptation across a significant domain gap <cit.>. During the process, we select samples based on their prediction scores, and we progressively decrease the selected source samples and increase the selected target samples. Then we generate intermediate domains gradually moving towards the target domain based on the selected samples, , the dark-colored circles in <ref> (c). By progressively performing the feature augmentations on the source features towards the new intermediate domain, we encourage the model to progressively recognize the augmented source features, which enables it to approach the target domain effectively. <ref> clearly demonstrates that the distribution of the newly constructed intermediate domains can progressively approach the target distribution. In addition, considering that the prediction accuracy of the target samples will greatly influence the estimation of the distribution <cit.>, based on our progressively estimated intermediate domains, we further introduce a new intermediate domain approaching (IDA) strategy for feature extractor adaptation. We encourage features extracted from the source samples and the target samples to approach the mean of the distribution of the intermediate domain. Our IDA strategy can be combined with our PTSFA approach to encourage the model to generate more discriminative features and prevent the model from generating outlier features that might degrade the prediction accuracy, which will eventually boost the adaptation performance.In contrast to the previous methods that rely on adversarial training or self-supervised learning for feature extractor adaptation, our classifier adaptation method is motivated by feature augmentation, which is further enhanced by our introduced feature extractor adaptation strategy. We conducted extensive experiments on several benchmark datasets to demonstrate the effectiveness of our approach. In summary, our contributions are threefold: * We tackle unsupervised domain adaptation on point clouds from a new perspective of classifier adaptation by using target-styled feature augmentations, where we propose a progressive target-styled feature augmentation (PTSFA) approach to enable progressive adaptation of the model from the source domain to the target domain.* We further introduce an intermediate domain approaching (IDA) strategy for feature extractor adaptation, which boosts the adaptation performance of our PTSFA approach.* Our method achieves state-of-the-art (SOTA) performance on 3D UDA benchmark datasets, surpassing the existing methods by a significant margin. § RELATED WORK §.§ Deep Learning on Point CloudsInspired by PointNet <cit.> which directly extracts features from the unordered point sets using a set of multi-layer perceptions (MLPs), extensive deep models have been proposed <cit.>, which aim at using MLPs to extract local features from the point sets hierarchically and then gather the local features to obtain the global features for point cloud recognition. While these methods have achieved remarkable performance in different scenarios, a learned model using a certain set of training data can hardly perform well in a new scenario. §.§ Unsupervised Domain Adaptation on Point CloudsVarious methods have been proposed to tackle the unsupervised domain adaptation (UDA) problem on point clouds and most of these methods can be categorized into two main categories, , the adversarial-training (AT)-based methods <cit.> and self-supervised learning (SSL)-based methods <cit.>. The AT-based methods aim at learning a feature extractor to extract domain-invariant features from both source samples and target samples that cannot be distinguished by a domain discriminator. However, adversarial learning is not stable, and the semantic meaning of the features extracted can hardly be guaranteed, which leads to an unsatisfactory adaptation performance <cit.>. The SSL-based methods aim at designing self-supervised tasks to enable the feature extractor to extract semantically meaningful features from the target samples, which are then delivered to the classifier supervised by the source labels for classification. However, self-supervised learning tasks can only help the model to learn low-level geometry features that are not suitable for high-level recognition tasks like recognition <cit.>. Moreover, there is no theoretical proof that the features extracted from the target sample are suitable for classification by a source-supervised classifier. §.§ Target-Styled Feature AugmentationAugmentation is a widely used technique that can efficiently increase the generalization of the model <cit.>. Compared with previous data augmentation methods that are complex and have limited generalization ability <cit.>, ISDA <cit.> is a representative 2D work that proposed to construct semantically meaningful feature augmentations with different directions to expand the decision region of the classifier so as to improve the generalization of the model.However, ISDA cannot handle large domain gaps as it can only construct feature augmentations based on the estimation of the distribution of the given training data <cit.>.TSA <cit.> is another representative 2D work that tackles the problem by using the estimated distribution of the target domain as the direction of the feature augmentations to construct target-styled feature augmentations and then perform classifier adaptation. However, TSA directly takes all of the target data for distribution estimation, while in the 3D domain, the distance between different samples and the source domain varies significantly. Directly estimating the target distribution with all of the target predictions will inevitably lead to estimation bias, which will adversely affect the overall adaptation performance.§ METHOD In this section, we present our proposed method in detail. First, we introduce the problem formulation of this work in <ref>. Then, we elaborate our newly proposed progressive target-styled feature augmentation (PTSFA) approach, which consists of a progressive target-approaching in <ref>, and a target-styled feature augmentation in <ref>. We further present our intermediate domain approaching (IDA) strategy in <ref>. Finally, the overall training loss is given in <ref> and the pipeline of our method is illustrated in <ref>. §.§ Problem Formulation Following previous works <cit.>, we tackle unsupervised domain adaptation on point clouds from the perspective of classification. Specifically, we are given a set of source data 𝕊={𝒫_n^s, y_n^s}_n=1^N_s, indicating N_s point clouds 𝒫_n^s in total with their corresponding category labels y_n^s. Each point cloud sample 𝒫_n^s∈ℝ^m × 3 consists of m unordered points with their corresponding 3D coordinates. Similarly, we are given another set of target data 𝕋={𝒫_n^t, y_n^t}_n=1^N_t, while the target labels y_n^t are only available during the inference stage. All of the samples from the source and the target domain share the same label space 𝒴 ={1,2, ⋯, C}. We aim at learning a model φ = φ_f∘φ_cls that can perform well on the target domain, where φ_f indicates the feature extractor and φ_cls indicates the classifier.As mentioned in <cit.>, deep features learned by a model are usually linearized, thus it is feasible to construct augmentations in the feature space towards a certain direction to transfer the semantics of the features. Facing the UDA problem, as the distributions of the source data and the target data are different, we aim to construct feature augmentations on source samples toward the target domain. The classifier would be adapted toward the target domain by performing supervision on the target-styled source features. Given the source data 𝒫_n^s, it is easy to extract the corresponding features f_n^s from each sample by f_n^s = φ_f(𝒫_n^s). Similarly, we can extract target features f_n^t = φ_f(𝒫_n^t) from the target data 𝒫_n^t. In this work, we construct two feature pools 𝔽_s = {f_n^s}_n=1^N_s to store f_n^s and 𝔽_t = {f_n^t}_n=1^N_t to store f_n^t. Given an extracted feature f_n and a classifier φ_cls, we can obtain the corresponding prediction ξ_n=φ_cls(f_n). Therefore, we can construct two confidence score pools ℤ_s = {ζ_n^s}_n=1^N_s and ℤ_t = {ζ_n^t}_n=1^N_t, where ζ_n = max_c ξ_n, c indicates the index of the category. Finally, we construct two label pools ℍ_s = {η_n^s}_n=1^N_s and ℍ_t = {η_n^t}_n=1^N_t, where η_n^s = y_n^s as we have ground-truth labels available for each source sample, and η_n^t = max_c ξ_n^t, indicating the corresponding pseudo-label of each target sample. According to the corresponding η_n, we can divide the pools into category-wise feature pools as 𝔽_s^c and 𝔽_t^c, and confidence score pools as ℤ_s^c and ℤ_t^c. It is easy to estimate the mean and the covariance of the source and target distributions, , μ_s^c and Σ_s^c, μ_t^c and Σ_t^c, according to the source labels or the target pseudo-labels. §.§ Progressive Target-ApproachingRecall that we aim to construct feature augmentations on the source samples toward the target domain to perform classifier adaptation. The first issue to determine is the direction of the feature augmentations. Different from the previous works like TSA <cit.> that directly estimate the augmentation direction by Δμ_c = μ_t^c - μ_s^c, we argue that directly adapting the model across a significant domain gap may hamper the adaptation performance, as the estimation of the target distribution would be swayed by the incorrect target predictions, especially during the initial training stages. Moreover, the distances between various target samples and the source domain exhibit significant disparities, some hard samples that lie far away from the source domain may be ambiguous for the classification boundaries. To address the concerns, we propose a new progressive target-approaching strategy to divide the whole training process into several stages with each stage consisting of τ training epochs. Considering that the total training epoch is T, the whole training process can be divided into ⌈ T / τ⌉ stages, where ⌈·⌉ is a round-up operation. We construct one new intermediate domain D_k between the source and the target domain for each stage. With the intermediate domains progressively approaching the target domain, the model is encouraged to approach the intermediate domains from the source domain toward the target domain. During each stage, we aim to select samples that are closest to the mean of the current intermediate domain. This operation ensures the domain gap between two consecutive intermediate domains is small, preserving the prediction accuracy of the target samples. Consequently, this approach enhances the estimation of the new intermediate domain estimation and contributes significantly to the overall adaptation performance. Intuitively, the distance between a sample and the mean of the intermediate domain can be reflected by the prediction score ζ_n, and a higher ζ_n usually indicates that the sample may probably lie within the decision region of the classifier, with a short distance to mean of the intermediate domain. Consequently, we only select the samples with the highest ζ_n from both domains to construct the new intermediate domain.Moreover, considering the inherent long-tailed distribution of the dataset, we propose a new ratio-based category-wise sample selection strategy to select a fixed ratio of samples from each category, aiming to pick σ^s samples from each category of the source domain and σ^t samples from each category of the target domain. Here σ^s and σ^t represent the proportion of the number of samples we wish to select from each category. To implement this, we sort ℤ_s^c and ℤ_t^c and keep the former σ^s source samples and the former σ^t source samples from each category. In addition, as our objective is to create target-styled feature augmentation, we advocate for the progressive convergence of the intermediate domain towards the target domain. To achieve this, we progressively decrease the number of the selected source samples and increase the number of the selected target samples, , σ_k+1^s = σ_k^s - Δσ^s and σ_k+1^t = σ_k^t + Δσ^t, where k indicates the k-th intermediate domain construction stage and Δσ indicates the proportional change during each stage. This approach results in a gradual shift by considering fewer source samples and more target samples to construct the intermediate domain, ensuring a progressive approach toward the target domain. It is easy to estimate the mean and covariance of the newly constructed intermediate domain as μ_k^c and Σ_k^c. During each training stage, the augmentation direction is estimated by Δμ_k^c = μ_k^c - μ_s^c. §.§ Target-Styled Feature Augmentation After determining the direction of the feature augmentation, the next issue to determine is the formulation of the feature augmentation.Given an estimated distribution of the intermediate domain N(μ_k^c, Σ_k^c), it can be represented by N(μ_s^c + Δμ_k^c, Σ_k^c), where Δμ_k^c = μ_k^c - μ_s^c. Recall that the mean of a distribution stands for its semantics and the covariance of a distribution stands for its semantic variations. As we aim at constructing target-styled source features, we can generate M augmentations sampled from a distribution N(Δμ_t^c, Σ_t^c) on a certain source feature f_n^s. That is, augmented source features f_n, m^s, m=1 ⋯ M confirm to a distribution of N(f_n^s + Δμ_k^c, Σ_k^c), which share the same semantics and variations as the constructed intermediate domain. Following <cit.> and <cit.>, we also use a parameter λ to control the strength of the augmentation, , λΣ_k^c, to enable the semantic variants to approach the target distribution progressively, where λ = (t/T)×λ_0, t and T indicate the current training epoch and the total training epoch, respectively. Considering that a linear classifier φ_cls is defined by a weight matrix W = [w_1, w_2, ⋯, w_C ]^T and a bias vector b = [ b_1, b_2, ⋯, b_C]^T, we can perform supervision on the augmented source features by minimizing ℒ_M as:ℒ_M=1/N_s∑_n=1^N_s1/M∑_m=1^M-log(e^w_y_n^s^⊤f_n, m^s+b_y_i^s/∑_c=1^C e^w_c^⊤f_n, m^s+b_c).Directly constructing M augmentations and optimizing ℒ_M is difficult to implement. However, if we construct infinite augmentations f_n, m^s, it is easy to obtain an upper bound of ℒ_M as ℒ_PTSFA, which can be written as:lim _M →∞ℒ_M ≤ℒ_PTSFA = 1/N_s∑_n=1^N_s-log( ϑ_n, s, t^y_n^s/∑_c=1^Cϑ_n, s, t^c),where ℒ_PTSFA is a standard cross-entropy (CE) loss withϑ_n, s, t^c= w_c^⊤f_n^s + (w_c^⊤ - w_y_n^s^⊤)Δμ_t^c+ λ/2( w_c^⊤ - w_y_n^s^⊤)Σ_t^c ( w_c^s - w_y_n^s).Note that w_c^⊤f_n^s indicates the c-th prediction score of ξ_n^s. It is efficient to perform classifier adaptation by directly optimizing ℒ_PTSFA. A detailed proof is provided in the supplements.It should be noted that different from TSA <cit.>, we divide the training process into several stages, and update the pools 𝔽, ℍ and ℤ as well as μ_s^c and Σ_s^c, μ_t^c and Σ_t^c at the beginning of each stage, while each training stage consists of τ epochs. In this way, we can ensure a fixed augmentation direction during each stage, which keeps the training stable. §.§ Intermediate Domain ApproachingOur PTSFA performs classifier adaptation, a key issue of which is the estimation of the augmentation direction that relies heavily on the accuracy of the target predictions. Therefore, we further propose a new intermediate domain approaching (IDA) strategy for feature extractor adaptation, which can be combined with our PTSFA for a better adaptation performance. In particular, we aim at enabling the source features and the target features to approach the mean of the intermediate domain. Specifically, given the extracted source features f_n^s, the extracted target features f_n^t, and the estimated mean of the intermediate domain μ_k^c, we encourage f_n^s and f_n^t to approach the μ_k^η_n of the corresponding category while staying far from the μ_k^c of other categories. To achieve this, we formulate our IDA loss as:ℒ_IDA^s = 1/N_s∑_n=1^N_s-log(exp(cos(f_n^s, μ_k^η_n^s) / κ)/∑_c=1^C exp(cos(f_n^s, μ_k^c) / κ)),and ℒ_IDA^t = 1/N_t∑_n=1^N_t-log(exp(cos(f_n^t, μ_k^η_n^t) / κ)/∑_c=1^C exp(cos(f_n^t, μ_k^c) / κ)),where κ is the temperature value <cit.>. Optimizing ℒ_IDA^s and ℒ_IDA^t will on one hand promote feature extractor adaptation, preventing the encoder from generating outlier features, and on the other hand encourage the encode to extract more discriminative features. This enhances the adaptation performance of our PTSFA method. §.§ Overall ObjectiveTo sum up, our newly proposed method consists of a classifier adaptation method PTSFA and a feature extractor adaptation method IDA, and the total loss can be written as:ℒ = αℒ_PTSFA + βℒ_IDA^s + γℒ_IDA^t,where α, β and γ are the trade-off parameters. A detailed introduction to our training process is provided in the supplements. § EXPERIMENTS§.§ Datasets We adopt the widely used 3D unsupervised domain adaptation (UDA) benchmark datasets PointDA-10 dataset <cit.> and GraspNetPC-10 dataset <cit.> to validate the effectiveness of our method. More details about the datasets are provided in the supplements. §.§ Implementation Details We follow the previous works <cit.> and adopt DGCNN <cit.> as our backbone. We set the number of labeled data and unlabeled data as 8 within each mini-batch. The training epoch T is set as 100 and the training epoch of each intermediate domain construction stage τ is set as 5, indicating the training process can be divided into 20 stages. The temperature parameter κ is set as 2.0. λ_0 that controls the semantic variants is set as 0.25, which is adopted from TSA <cit.>. The trade-off parameters α, β, and γ are all set as 1.0 by default, respectively. More detailed experiment settings are provided in the supplements.§.§ Experimental ResultsWe compare our method with recent state-of-the-art (SOTA) UDA methods, including adversarial domain alignment-based methods (AD) like DANN <cit.> and PointDAN <cit.>, feature augmentation-based methods (FA) like ISDA <cit.> and TSA <cit.>, and self-supervised learning-based methods (SSL) like RS <cit.>, DefRec+PCM <cit.>, GAST <cit.>, ImplicitPCDA <cit.>, GLRV <cit.>, MLSP <cit.>, FD <cit.> and DAS <cit.>. As some methods adopt a pseudo-labeling method (PS) for the model fine-tuning, we report their performance both with and without using PS fine-tuning, corresponding to the lower and upper portions of <ref> and <ref>, respectively. It is noteworthy that as DAS employed a more sophisticated PS method, which requires more parameters during training or inference, we also report its performance with the traditional PS method, denoted as DAPS. In addition, we compare with the supervised learning method directly training the model with only labeled source data for reference (denoted as “w/o DA”), as well as the Oracle method that trains the model by using labeled target data (denoted as “Oracle”). The results on the PointDA-10 dataset are reported in <ref>. As can be seen, our method surpasses the current SOTA methods by a large margin under all six adaptation scenarios. The average recognition accuracy of our method outperforms the current SOTA method DAS by a notable margin of 2.9%, while DAS employs a more sophisticated PS method to fine-tune the performance. Our advantage over DAS can be further enlarged to 4.9% or 5.7% when comparing ours with DAS using the traditional PS (, DAPS) or without PS. Moreover, our method demonstrates notable superiority over the two feature augmentation-based methods ISDA <cit.> and TSA <cit.>, with improvements of 8.7% over ISDA and 6.5% over TSA. This verifies the effectiveness of our progressive target-styled feature augmentation strategy. The results on the GraspNetPC-10 dataset are reported in <ref>. Consistently, our method surpasses the w/o DA method by over 30% and also outperforms the current SOTA method DAS, despite that DAS requires more parameters for a more complicated PS fine-tuning. Our method without PS fine-tuning wins the second-best performer DAPS by a large margin of 6.9%. Moreover, compared with the two feature augmentation-based methods ISDA and TSA, our method demonstrated overwhelming advantages with almost or more than 15% improvements.Note that our FA-based method surpasses the current AD-based methods and SSL-based methods by a large margin, indicating the great potential of the feature augmentations in tackling UDA problems. §.§ Ablation Study In this section, we single out the contributions of our module designs on the PointDA-10 dataset. Note that our method includes a progressive target-styled feature augmentation (PTSFA) approach for classifier adaptation and an intermediate domain approaching (IDA) method for feature extractor adaptation, and our PTSFA further consists of a basic target-styled feature augmentation (TSFA) and a progressive target-approaching (PTA) strategy. Also note that our PTA strategy is a general training strategy that can also be combined with our IDA method. The results of the ablation study are reported in <ref>. It can be clearly seen from the table that the basic TSFA will improve the average accuracy of the backbone DGCNN model by 12.1%, which surpasses TSA <cit.> as we adopt a more stable training strategy. Applying our IDA alone will bring a performance improvement of 7.0% over the w/o DA method. However, when combining our TSFA and IDA, the performance drops to only 68.6%. The main reason is that the domain gap between the source domain and the target domain is huge and the above-mentioned two methods both rely heavily on the estimation of the target domain. Directly adapting the model across such a large domain gap will inevitably lead to incorrect distribution estimation, harming the adaptation performance of the model.It can also be inferred from the table that both the combination of our TFSA with our PTA strategy and the combination of our IDA with our PTA strategy can bring performance improvements of 1.4% and 4.8%, respectively. More importantly, combining the three components TSFA, IDA and PTA will bring further improvements of 0.8% and 2.5%, respectively, from TSFA+PTA and IDA+PTA. The main reason is that we decompose the huge domain gap into a set of small domain gaps by introducing a set of intermediate domains, ensuring an accurate intermediate domain estimation. These results clearly verify the importance and effectiveness of our PTA strategy. Moreover, it can be seen from the table that only deploying our TSFA and PTA, , our PTSFA method, can also outperform the current SOTA method DAPS by a large margin of 4.9%.We further delve into the intricate design of our PTA strategy. As a reminder, our approach involves a continual reduction in the number of selected source samples and a simultaneous increase in the number of selected target samples throughout the training process. We thereby scrutinize the effectiveness of each strategy, as reported in <ref>.As seen, when decreasing the number of the selected source samples while keeping all of the target samples for intermediate domain construction, the model would gradually approach the target domain. In particular, our “decreasing source” strategy exhibits great performance under Real-to-Sim scenarios like S^∗ → M and S^∗ → S. The main reason could be that the samples from the source domain contain noise, which can potentially introduce disruptive information and undermine the model's recognition capabilities. Progressively reducing these samples will help prevent the model from being misled by the noise.Moreover, when increasing the number of the selected target samples while keeping all source samples for intermediate domain construction, the adaptation performance on the Sim-to-Real scenarios like M → S^∗ and S → S^∗ will be improved. The main reason could be that our “increasing target” strategy would expand the decision region of the classifier, thus improving the recognition performance of the model on the target domain. The results clearly verified the effectiveness of our PTA strategy.In addition, we analyze the sensitivity of our method to the number of epochs (τ) that the intermediate domain construction stage lasts, as depicted in <ref>. From the figure, it is evident that a more frequent intermediate domain construction stage results in improved adaptation performance, as it allows for a more precise estimation of augmentation direction. However, the high update frequency comes with increased training costs, as it requires an additional inference stage at the beginning of each intermediate domain construction phase to estimate the augmentation direction. The figure clearly illustrates that constructing a new intermediate domain per epoch would more than double the training time compared to constructing one intermediate domain every 10 epochs, while the performance improvement is limited. Taking into account both training costs and performance, we decided to perform the intermediate construction stage every 5 epochs.Finally, we analyze the sensitivity of our method to the temperature κ, as reported in <ref>. It can be clearly seen from the table that our method is robust to such a parameter. When κ=2.0, our model achieves the best overall recognition performance of 76.5% on the target domain.Limitation: Neglecting the great adaptation performance of our method, our method requires an extra inference stage at the beginning of each intermediate domain construction stage, and we also require extra space to store the pools 𝔽, ℍ and ℤ during the training stage, which will introduce more training costs. Nevertheless, our method requires no extra costs during the inference stage.§ CONCLUSION In this work, we have presented a new progressive target-styled feature augmentation method to tackle unsupervised domain adaptation on point clouds from a new perspective of classifier adaptation. We further come up with an intermediate domain approaching strategy for feature extractor adaptation, which can be combined with our classifier adaptation method to achieve better adaptation performance. Extensive experiments on the benchmark datasets have validated the effectiveness of our newly proposed approaches, where our method outperforms the current state-of-the-art methods by a large margin.ieeenat_fullname§ MORE DETAILS OF OUR METHOD§.§ Target-Styled Feature AugmentationIn <ref>, we have introduced our target-styled feature augmentation approach. In this section, we show the detailed derivation of our ℒ_TSFA.Recall thatℒ_M=1/N_s∑_n=1^N_s1/M∑_m=1^M-log(e^w_y_n^s^⊤f_n, m^s+b_y_i^s/∑_c=1^C e^w_c^⊤f_n, m^s+b_c).When M →∞, letting f̃_n^s denote the augmented source features, we can obtainlim _M →∞ℒ_M = 1/N_s∑_n=1^N_s𝔼_f̃_n^s[ -log(e^w_y_n^s^⊤f̃_n^s+b_y_i^s/∑_c=1^C e^w_c^⊤f̃_n^s+b_c)]= 1/N_s∑_n=1^N_s𝔼_f̃_n^s[log(∑_c=1^C e^(w_c^⊤ - w_y_n^s^⊤) f̃_n^s + ( b_c - b_y_i^s)) ].According to the Jensen’s inequality <cit.>, 𝔼[log(X)]≤log(𝔼[X]), we can re-write <ref> as:lim _M →∞ℒ_M ≤1/N_s∑_n=1^N_slog( 𝔼_f̃_n^s[ ∑_c=1^C e^(w_c^⊤ - w_y_n^s^⊤) f̃_n^s + ( b_c - b_y_i^s)] )= 1/N_s∑_n=1^N_slog( ∑_c=1^C 𝔼_f̃_n^s[ e^(w_c^⊤ - w_y_n^s^⊤) f̃_n^s + ( b_c - b_y_i^s)] ).As f̃_n^s denotes the augmented source features, where the initial source features f_n^s obey N(μ_s^y_n^s, Σ_s^y_n^s),it is easy to infer that f̃_n^s ∼ N(μ_s^y_n^s + Δμ_t^y_n^s, λΣ_y^y_n^s). The mean of (w_c^⊤ - w_y_n^s^⊤) f̃_n^s + ( b_c - b_y_i^s) can be calculated as (w_c^⊤ - w_y_n^s^⊤) (μ_s^y_n^s + Δμ_t^y_n^s) + ( b_c - b_y_i^s), while the covariance of (w_c^⊤ - w_y_n^s^⊤) f̃_n^s + ( b_c - b_y_i^s) can be calculated as λ(w_c^⊤ - w_y_n^s^⊤) λΣ_y^y_n^s(w_c - w_y_n^s). Leveraging the function 𝔼[e^a X]=e^a μ+1/2 a^2 σ, X ∼ N(μ, σ), it is easy to draw <ref>.§.§ Overall TrainingIn <ref>, we have introduced the total loss function of our method. In this section, we give a more detailed introduction to our overall training process. During our implementation, we introduce a warm-up stage at the beginning of the training process to prevent inaccurate intermediate domain estimation. Specifically, during the warm-up stage, we perform the standard cross-entropy (CE) loss to supervise the source predictions as:ℒ_CE = -1/N_s∑_n=1^N_sℓ^ce(ξ_n, y^s_n). During the warm-up stage, we optimize ℒ_CE to train the model. After the warm-up stage, we follow <ref> and optimize <ref> to train the model. § MORE EXPERIMENTS§.§ DatasetsIn <ref>, we have briefly introduced the datasets we used, in this section, we provide more details about the dataset.PointDA-10 dataset <cit.> is the most widely used 3D unsupervised domain adaptation (UDA) benchmark dataset designed for point cloud classification. The PointDA-10 dataset consists ofthree subsets, ModelNet-10 (M), ShapeNet-10 (S) and ScanNet-10 (S^∗), which are sampled from three widely-used datasets, ModelNet <cit.>, ShapeNet <cit.> and ScanNet <cit.>, respectively. The three subsets share the same 10 categories and each sample consists of 1,024 points in total. In particular, ModelNet-10 consists of 4,183 training samples and 856 testing samples, where these samples are generated with CAD by uniformly sampling from synthetic 3D models. ShapeNet-10 consists of 17,387 training samples and 2,492 testing samples, where the samples are also generated with CAD, but the shape of the samples in ShapeNet-10 also exhibits variations from those in ModelNet-10. ScanNet-10 consists of 6,110 training samples and 1,769 testing samples, and all of the samples are scanned from real-world indoor scenarios with RGB-D cameras. The samples in ScanNet-10 are usually sparse with some missing parts because of noise and occlusion. When using the training set of one subset as the source domain, and the training set of another subset as the target domain, a total of six UDA scenarios, , M → S, M → S^∗, S → M, S → S^∗, S^∗ → M and S^∗ → S, can be obtained.GraspNetPC-10 dataset <cit.> is another 3D UDA benchmark dataset for point cloud classification, which is created from GraspNet <cit.> and consists of four subsets, , two synthetic subsets and two scanned subsets. The synthetic samples in GraspNetPC-10 are re-projected from rendered synthetic scenes while the real depth scanned samples in GraspNetPC-10 are captured by two different depth cameras, , Kinect2 and Intel Realsense, constituting two real-world domains. The subsets also share the same 10 categories. Each sample consists of 1,024 points in total. In particular, the synthetic domain (Syn.) contains 12,000 training samples. The Kinect real-world domain (Kin.) contains 10,973 training samples and 2,560 testing samples while the Realsense real-world domain (RS) contains 10,698 training samples and 2,560 testing samples. When using one subset as the source domain, and another scanned subset as the target domain, a total of four DA scenarios, , Syn. → Kin., Syn. → RS., Kin. → RS. and RS. → Kin. can be obtained.In addition, we follow previous works <cit.> and report the results on the PointSegDA dataset, a 3D UDA benchmark dataset for point cloud part segmentation. PointSegDA dataset is created from <cit.> and consists of four subsets, , FAUST (F), MIT (MI), ADOBE (A), and SCAPE (SP). The subsets share the same 8 categories of human body parts. Each sample from the PointSegDA dataset consists of 2,048 points in total. When using the training set of one subset as the source domain and the training set of another subset as the target domain, a total of twelve UDA scenarios, , FAUST → ADOBE, FAUST → MIT, FAUST → SCAPE, MIT → ADOBE, MIT → FAUST, MIT → SCAPE, ADOBE → FAUST, ADOBE → MIT, ADOBE → SCAPE, SCAPE → ADOBE, SCAPE → FAUST, and SCAPE → MIT, can be obtained. §.§ Implementation detailsIn <ref>, we have listed the main experimental settings, in this section, we provide the detailed implementation details of our method.Following the previous works <cit.>, we adopt DGCNN <cit.> as our backbone but simplify the last classifier to a single linear layer, following TSA <cit.>. In addition, we use a simple max-pooling operation to replace the combination of a max-pooling operation and an average-pooling operation to avoid the over-fitting problem. Our models are trained on a server with four NVIDIA GTX 2080Ti GPUs, except for the PointSegDA dataset, where our methods are trained on a server with ten NVIDIA RTX 3090 GPUs. We only use one GPU per experiment. Our implementation is based on the PyTorch framework. For all experiments, we use the Adam optimizer together with an epoch-wise cosine annealing learning rate scheduler initiated with a learning rate of 0.001 and a weight decay of 0.00005. We train all of our models for 100 epochs setting the numbers of both labeled data and unlabeled data as 8 within each mini-batch. Notably, the 100 training epochs include a 10-epoch warm-up stage. Each intermediate domain construction stage consists of 5 epochs (τ=5). The initial ratio for selecting source data (σ^s) is set as 1.0, while the initial ratio for selecting target data (σ^t) is set as 0.0 and the proportional changes Δσ^s and Δσ^t are both set as 0.05. The temperature parameter κ is set as 2.0, and λ_0 that controls the semantic variants is set as 0.25, which is adopted from TSA <cit.>. The trade-off parameters α, β, and γ are all set as 1.0, respectively.In addition, following the previous work <cit.>, we fine-tune our learned model under a self-paced learning (SPL) paradigm where we set the initial threshold for selecting confident predictions as 0.8, and the increasing step as 0.01. The 2BDA training stage consists of 10 training circles and each circle consists of 10 epochs. Most of our experiment settings are adopted from <cit.>. §.§ Experimental ResultsIn this section, we benchmark our method against recent state-of-the-art (SOTA) UDA methods on the PointSegDA dataset. In addition, we provide a comparison with the supervised learning method directly training the model with only labeled source data for reference (denoted as “w/o DA”), as well as the Oracle method that trains the model by using labeled target data (denoted as “Oracle”). The results are reported in <ref>. Note that all methods are implemented without using the pseudo-labeling fine-tuning method. It can be clearly seen that our method has demonstrated significant performance improvements under most transfer scenarios. For instance, under the A → F scenario, our method outperforms the current state-of-the-art (SOTA) method MLSP by 11.1%. In addition, under the A → MI scenario, our method achieves a performance gain of 13.2% compared to MLSP. On average, our method outperforms the current SOTA method by over 5% on the PointSegDA dataset, showcasing the tremendous potential of our approach. §.§ Ablation Study In this section, we carry on more ablation studies to verify the effectiveness of our approach.We first validate the effectiveness of our newly proposed ratio-based category-wise sample selection strategy by comparing it with the conventional threshold-based sample selection strategy. The results are presented in <ref>. Our ratio-based category-wise sample selection strategy significantly outperforms the threshold-based approach. This notable improvement can be attributed to the limitations of threshold-based sample selection strategies, which fail to ensure a gradual increase in the number of selected samples. Consequently, the constructed intermediate domain may diverge significantly from the source domain, resulting in a substantial domain gap. In contrast, our strategy effectively mitigates the risk of a large domain gap, ensuring accurate estimation of the intermediate domain and enhancing classifier adaptation performance. Furthermore, setting a higher threshold at the outset of the training process leads to the exclusion of a considerable number of samples, exacerbating the confirmation bias issue and adversely affecting training stability and model performance, as shown in the figure under the S^∗ → M and S^∗ → S scenarios.Then, we scrutinize the design of our IDA strategy, and the results are reported in <ref>. As a reminder, we encourage both the source and the target features to approach the mean of the intermediate domain to perform feature extractor adaptation. As seen, both the source and the target approaching strategies prevent the model from generating outlier features, especially on noisy domains like S^∗. Moreover, as the intermediate domain gradually approaches the target domain, the source approaching strategy enables the source features extracted by the feature extractor to gradually approach the target domain, which can lead to feature extractor adaptation, bringing a better adaptation performance.In addition, we conducted an analysis to evaluate the sensitivity of our method to the trade-off parameters α, β and γ. As β and γ control the weights of the ℒ_IDA^s and ℒ_IDA^t, respectively, we argue that these two loss functions contribute equally to the training process. Therefore, we set both β and γ to 1.0 and vary the value of α. The average accuracies over six adaptation scenarios are reported in <ref>. It is evident that our method exhibits robustness to the trade-off parameters. Varying the value of α within a reasonably large range does not significantly impact the adaptation performance of our approach. This indicates that our method is not overly sensitive to the choice of trade-off parameters and maintains stable performance across different parameter settings.We further investigate the relationship between the performance and the training costs of our approach and report the results in <ref>. It is evident that our method achieves significant improvement in adaptation performance compared to the “w/o DA” method, with only a modest increase of approximately 25% in training costs. | http://arxiv.org/abs/2311.16474v1 | {
"authors": [
"Zicheng Wang",
"Zhen Zhao",
"Yiming Wu",
"Luping Zhou",
"Dong Xu"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127073315",
"title": "Progressive Target-Styled Feature Augmentation for Unsupervised Domain Adaptation on Point Clouds"
} |
[ Rocio Semino January 14, 2024 ==================== While data selection methods have been studied extensively in active learning, data pruning, and data augmentation settings, there is little evidence for the efficacy of these methods in industry scale settings, particularly in low-resource languages. Our work presents ways of assessing prospective training examples in those settings for their "usefulness" or "difficulty". We also demonstrate how these measures can be used in selecting important examples for training supervised machine learning models. We primarily experiment with entropy and Error L2-Norm (EL2N) scores. We use these metrics to curate high quality datasets from a large pool of Weak Signal Labeled data, which assigns no-defect high confidence hypotheses during inference as ground truth labels. We then conduct training data augmentation experiments using these de-identified datasets and demonstrate that score-based selection can result in a 2% decrease in semantic error rate and 4%-7% decrease in domain classification error rate when compared to the baseline technique of random selection.§ INTRODUCTION The immense progress in deep learning over the past decade has been, in part, driven by the increasing scale of training data <cit.>, model architectures like transformers <cit.>, and compute used to train the models. However, the science of surfacing which examples to include in training data remains a persistent and applicable question. The recently published Platypus family of models <cit.> outperformed several SOTA open Large Language Models(LLMs) while being trained on a single GPU for only five hours. The reported success at such a low cost appears primarily due to the quality of their smaller dataset which was curated from large pools of open datasets, which reiterates the significance of curating high quality datasets for model training.Many studies have been conducted on data selection from large pools of data, but there are challenges when it comes to implementing them in large scale systems. One challenge is that they are studied in an offline, one-time selection setting from a static data pool whilemost real world systems need to implement data selection on a continual basis with potential data drift. To make data selection practical and scalable, it should be based on scores that are easy to compute and interpret, stay relevant with changes to data distribution and model architectures, and can be integrated into existing data collection pipelines easily. A second challenge is that some data selection techniques are too compute intensive <cit.> to be easily implemented and integrated. Third, a few selection methods like "Selection Via Proxy" <cit.> require training and maintaining proxy models for data selection, which adds to the resources overhead. In the context of langauge models (LMs), recent studies for data selection experimented with changes to pre-training, such as Task Adaptive Pre-Training (TAPT) <cit.>. These approaches are not architecture agnostic and the complimentary models need to be retrained with changes to underlying training data, prohibiting continuous data selection without interruptions.The above mentioned challenges become more pronounced when dealing with unstructured data in low resource languages. Most data selection methods are developed where training data is in English and experiments on non-English languages are typically conducted after the methods are optimized on English data. While this is a gap in research, it also raises the question, "Do these selection methods work for training models on low-resource languages? If so, how well?" There have been studies on data selection methods in multilingual settings for Neural Machine Translation <cit.>. <cit.> surveyedmethods that enable learning when training data is sparse which includes data augmentation. However, data selection strategies for unstructured data in low-resource languages for supervised machine learning models is relatively a less explored area of research.All the above challenges and issues emphasize the need to implement efficient and scalable data selection methods for low resource languages. We conduct experiments on Portuguese language. Our work aims to test two metrics – entropy and EL2N – in large scale conversational systems. The implementation of these metrics are agnostic to changes to data distribution and model architecture. They are interpretable, easy to compute, scalable, and can be integrated into existing pipelines that involve routine data selection. Our work presents a comprehensive benchmarking of improvements from score based data selection methods, dives deep into how those affect training data, and the overlap of the data selected using different methods for supervised machine learning models.We conduct our experiments on a BERT based model <cit.> that performs domain, intent and slots recognition. Components that are targeted for improvements are detailed in sec. <ref>.§ SCORING METRICSEntropy: Natural Language Understanding (NLU) is a key component in a conversational system. Domain Classifier (DC) within NLU predicts the domain a user request needs to be routed to. A DC can be expressed as a function parametrized by a set of weights θ: f_θ^j(x) is the softmax output for j belonging to one of N classes. For some input x, we compute entropy <cit.> of DC outputs for a single example as shown in the eq. <ref>..5H(x) = -∑_j=0^N-1 f_θ^j(x) log_2 f_θ^j(x) .5(x) = f_θ(x) - y_i_2 EL2N is a margin-based metric that estimates gradient norms as described in <cit.> . It is computed as in eq. <ref>, where f(x) indicates the softmax of the model outputs and y_i indicates the one-hot encodings of the label for the ith example. Typically, this metric is averaged over 𝒪(1) number of “replicates" (model initializations) to obtain a reliable signal of example difficulty. We compute EL2N scores at fine-tuning of DC, averaged over five replicates for each example.Predictions from DC, along with intent and token classification, feed as inputs to downstream tasks in a NLU system. Any improvements to DC could translate to improvement in the entire system. So, we anchored our experiments on data selection with EL2N and entropy based on DC outputs.NLU Model Confidence Score: In the conversational system under experimentation, NLU Model Confidence Score, referred to as "NLU Score" from here on, is a measure of the system's confidence in the suggested hypothesis i.e, predicted classes (domain and intent) and labels recognition (for individual tokens). NLU Score is a calibrated metric with a range (0,1]. DC scores i.e, softmax outputs from the classifier, are one set of inputs to NLU score. Other inputs include, but are not limited to, intent classifier and token recognition scores. In general, a hypothesis with the correct domain should have a high NLU Score and vice-versa. NLU score is calculated per domain. Unlike softmax outputs from aclassifier, NLU scores from the system across domains don't add to 1.§ DATASETSExisting training data is the de-identified data used to train an in-house model used for experiments.New dataset is an inhouse Weak-Signal Labeled (WSL) dataset sourced using the procedure described in <cit.>, where data is constructed from weak supervision from the user to obtain NLU labels. For example, if an unsuccessful action from the voice assistant is followed by a user rephrasing their request, which then results in an uninterrupted response from the device, that utterance is pseudolabeled using the top NLU hypothesis. We begin with a de-identified WSL dataset with NLU Score range [0.3,0.85] collected over a period of time, referred to as "new dataset" from here on. The selected range aims to eliminate examples that are already well learnt by the model (NLU score > 0.85) and noisy/ambiguous examples (NLU Score < 0.3). New dataset comes from de-ientified live traffic with its size quite larger when compared to existing training data, and is heavily skewed towards popular user interactions i.e., less diverse. We experimented with entropy and EL2N score based data selection strategies to curate smaller datasets from the new dataset with most useful/difficult examples. These curated datasets are added to the existing training data to build different candidate models as listed in sec. <ref>. Figure <ref> shows data distribution of top seven domains, accounting to 90% user traffic, in existing training data (left) and new dataset (right). Figure <ref> shows the distribution of datasets curated using score based selection methods detailed in sec. <ref>.Dataset curation based on Entropy:For each sample in the new dataset we generate DC scores for possible domains using the eq. <ref>, and calculate entropy of the generated DC scores. We then rank all the examples based on their entropy scores and select top K examples to create a smaller dataset. Sec. <ref> has exact details of how the examples are selected based on entropy.Dataset curation based on EL2N:For each training example in the dataset, we generate EL2N scores using eq. <ref> averaged over five replicates. The scores are generated through a domain classification task. After the data was ranked by EL2N difficulty, we use a threshold score of ≤ 0.15 to denote “easy" examples and a threshold of ≥ 0.6 to denote “hard" examples. We then randomly sampled from these subsets for a given target dataset size to obtain different mixtures of easy and hard examples, with splits described in sec. <ref>. Specialized testsets:We use specialized test sets to evaluate model performance for the use cases of interest. Our objective is to improve generalizability of the model, and so we evaluate our models on specially curated test sets that have the same distribution as the tail 40% of user traffic and are de-identified. § EXPERIMENTS §.§ Models We take one of the in-house models that performs NLU tasks as a baseline to experiment for further accuracy improvements. The in-house model is retrainedwith different additional datasets, all of the same size, resulting in different candidate models as listed below.Baseline model has existing training data augmented with randomly selected data from new dataset. EL2N (10% Hard + 90% Easy) candidate has training data augmented with data selected based on EL2N scores from the new dataset with a composition of 10% hard examples and 90% easy examples. Section <ref> has details on how easy and hard examples are selected.EL2N (90% Hard + 10% Easy) candidate has training data augmented with data selected based on EL2N scores from the new dataset with a composition of 90% hard examples and 10% easy examples. Section <ref> has details on how easy and hard examples are selected.Entropy candidate has training data augmented with data selected based on entropy scores of domain classifier outputs as detailed in sec. <ref>. Examples with higher entropy scores are added totraining data. Sec. <ref> has exact details of why and how the examples were selected based on entropy.Entropy with Filters candidate has additional data processing on the dataset curated based on entropy scores. First, we limit the repetition of an example in the curated dataset to P to enhance the diversity. We observed that with no limit in place, approx. 4,000 examples selected for a domain came from two unique examples. Second, we ensure that each domain has a minimum representation of R% in the dataset. Our values for P and R are 20 and 0.5. Further discussion can be found at sec. <ref>. §.§ Evaluation Metrics We measure candidates' performance on specialized test sets introduced in sec. <ref> in terms of component-wise (domain, intent and slots) error rates. In terms of component-wise error rates, we measured domain classification performance using the recall-based classification error rate . To evaluate slot-filling performance, we measured semantic error rate ():≡ ++ .We measured F-SEMER, the harmonic mean of SEMER using predicted labels as the reference and SEMER computed on ground-truth labels as the reference; this score balances precision/recall equally. We also report the interpretation error rate , which reflects the rate of any kind of error (slots, intents, domain). § RESULTS Table <ref> presents evaluation results for different candidates listed in the section <ref>. All the values are relative to a baseline model trained on randomly sampled data from the new dataset. We see improvement across all metrics for all candidates except EL2N (90% Hard + 10% Easy). Improvements to SEMER, F-SEMER and IRER are in the order of 2% with respect to the baseline. Improvements to DCER are in the range of 3%-7%. Entropy w/ Filters candidate and EL2N (10% Hard + 90% Easy) have the most promising results.Table <ref> presents more nuanced results when we look at domain level metrics. One can notice that Video, Notifications, Weather and Communications domains are better served by entropy based data selection while Music and Home Automation domains are better served by EL2N score based data selection. We recommend avoiding "one-size-fits-all" approach and encourage identifying which technique performed the best for each domain. Once the initial experiments are done, domains can be mapped to different data selection pipelines. Sec. <ref> and sec. <ref> have further discussion on metrics and data selected. § CONCLUSION Entropy based data selection improved metrics (DCER) better than EL2N on anchor task (DC), which was the source for the score. EL2N, however, delivered better improvements to overall recognition as measured by IRER. We hypothesize that EL2N, averaged over multiple runs, captures example importance towards overall accuracy, while entropy is better suited to improve a specific task. With growing developments in LLMs, we plan to continue our work on data selection strategies for LLMs, and using LLMs for data selection. Our areas of research would be fine-tuning LLMs, in-context learning <cit.>, avoiding model collapse <cit.>, and experimenting with a broader set of metrics (<cit.>, <cit.>) for data selection.§ ENVIRONMENT: We conducted our experiments on a conversational system. A typical conversational system has speech and Natural Language Understanding (NLU) components along with many others such as business logic based hypothesis re-routing, invoking third party applications, and more. Our experiments target improvements to NLU components. Within NLU, we need to recognize multiple things correctly to deliver desired experience to end users. A few of them are domain, intent, and slots. In the example, "Play Taylor Swift", domain is "Music", intent is "Play Music" and slot_name is "Artist Name" with a slot_value of "Taylor Swift". In our results, we show how our techniques improved metrics on each of these recognition tasks. Our experiments are anchored on domain classifier outputs. § EXPLORATORY DATA ANALYSIS§.§ Correlation study of NLU scores and Entropy scoresWe did a correlation study between NLU scores and corresponding entropy of domain classifier scores on a pool of 5 million examples. Results are shown in Table <ref>.All the three correlation coefficients show a consistent negative correlation between model confidence scores and entropy scores which implies that by reducing entropy of DC scores, we could improve NLU models' confidence on the correct interpretations and serve the end users with the most relevant responses. §.§ Distribution of Entropy and EL2N on the large new dataset Both Entropy and EL2N metrics have long tail distributions on the new dataset (Figure <ref>). Most of the examples have low scores implying relative high certainty in prediction (low entropy) or relative ease of getting correct prediction (low EL2N). We can observe that Entropy and EL2N scores distributions are left skewed in the Figure <ref> while NLU score distribution in the Figure <ref> is right skewed re-iterating the they trend in the opposite directions. Hence, reducing entropy or EL2N scores on a dataset could improve NLU score and eventually the system. The long tail presents a real opportunity to select and add challenging examples to training data. Models benefit from learning from these challenging examples during training which could result in improved accuracy.Range of Entropy for the full dataset after removing outliers (z-score > 3) is [0.0, 1.1731] with a mean and standard deviation of 0.137 and 0.252 respectively. Range of EL2N score for the full dataset after removing outliers (z-score > 3) is [0.0, 1.414] with a mean and standard deviation of 0.128341 and 0.275945 respectively.§ SCORING METRICSEntropy H of a random variable X is the level of uncertainty inherent in the variable’s possible outcomes. In case of a classification problem, p_j(x_i) is the probability of an example x_i belonging to the class j. The entropy of a single example x_i that could belong to n possible classes is calculated asH(X=x_i) = -∑_j=1^n p_j(x_i) log_2 p_j(x_i)§.§ Motivation to use Entropy for data selectionIn Active Learning (AL), entropy is used as an acquisition function to find the most informative examples to label from a large pool of unlabelled data. Entropy captures the uncertainty associated with the predicted labels. The higher the entropy of predictions, the higher is the possibility that the model is more challenged or confused when presented with such samples. Samples with higher entropy are selected for annotations. Those annotated examples are then added to training data to improve the models. In AL, Entropy based selection is proved to work better than many other sophisticated acquisition functions like BADGE (<cit.>) and BALD (<cit.>) when tested on a variety of tasks. Recent methods like AcTune (<cit.>) show that entropy when employed with other scientific techniques delivers superior results. Drawing inspiration from entropy based data selection for labeling in AL, we implemented entropy as a metric to select data from a large pool of Weak-Signal Labeled(WSL) data. More information on WSL data is presented in section <ref> . One more reason to choose entropy over other methods is the ease of implementation and cost to compute.Selection based on entropy: For any new dataset D, we compute entropy of domain classification softmax outputs, referred to as "entropy". We rank examples in D in decreasing order of their entropy and select top K examples based on a given cut off criteria. Intuition behind this is that we train the models with data that the models are less certain about but would benefit from learning from those during training. This should result in improved accuracy of the model predictions. Our cut off criteria was to limit the number of additional training examples that can be added to the existing training data. We experimented with newly created datasets to be in the order of 2% or 5% of existing training data. Datasets of size 5% gave the best results.Other selection criteria could be cut off based on absolute entropy score, top X% of examples from the larger dataset, etc. For Entropy w/ Filters candidate, we chose values of 20 and 0.5% for an example repetition cap and minimum representation of a domain (class). We arrived at these values after multiple experiments. There values can and should be experimented for different modeling tasks and datasets. § RESULTS§.§ Metrics by Domain Table <ref> presents evaluation results for the top seven domains (classes) that account to 90% user traffic. Results show that different domains benefit from different data selection strategies. We recommend avoiding "one-size-fits-all" approach and encourage identifying which technique performed the best for each domain. Once the initial experiments are done, domains can be mapped to different data selection pipelines. Looking at the Table <ref>, one can notice that Video, Notifications, Weather and Communications domains are better served by entropy based data selection while Music and Home Automation domains are better served by EL2N score based data selection. For domain level metrics, we reported balanced metrics for DCER and IRER i.e, F-DCER and F-IRER along with SEMER that was introduced in sec. <ref> § ANALYSIS§.§ Data Distribution of Selected Training Examples by Each MethodFigure <ref> shows the data distribution of top seven domains, accounting to 90% user traffic, in the datasets curated based on different data selection methods. Baseline method randomly selects examples from the new dataset and is representative of user traffic. Datasets curated for Entropy and Entropy w/ Filters have similar distribution across domains. This is corroborated by entropy datasets' overlap of 89.9% presented in Table <ref>. These two methods tend to surface diverse examples on which the models are less certain about the predictions. As a result, their distribution could vary from baseline's. We observe that entropy based methods picked less no. of examples for the domain HomeAutomation relative to baseline as the user interactions are relatively more standard i.e., less diverse. An example interaction for HomeAutomation is "Turn of the lights". They picked more no. of examples for Video, Global and Communication given the diverse nature of user interactions. Example interactions for Video are "Play my favorite movie" and "I want to watch Harry Porter". Datasets from EL2N based selection don't have similar distributions unlike entropy based selection. This is corroborated by EL2N datasets' overlap of 19.7% presented in Table <ref>. It is interesting to note that EL2N (10% Hard + 90% Easy) selected more examples in six out of seven domains under review relative to baseline while EL2N (90% Hard +10% Easy) has shown no clear pattern. §.§ Overlap of data between Entropy and EL2N score based selectionsTable <ref> presents the overlap of data between datasets curated by different methods. Entropy based datasets had the highest overlap of 89.94%. This is not surprising given they both select examples based on their ranked entropy scores and there is no randomness involved. They only differ for limits of example repetition and minimum domain representation. On the other hand, EL2N based datasets have significantly lower overlap of 19.7% when compared to entropy datasets. This is because these datasets are randomly sampled from different regions of EL2N score distribution. Except for EL2N (90% Hard + 10% Easy), all candidates delivered superior results when compared to the baseline. We hypothesize that this is because the said method deliberately adds "difficult" examples in large volume (90%) which could result in adding more noise than anticipated, and making the model convergence slower within give no. of training epochs.§ LIMITATIONSWhen opting for entropy based data selection one needs to understand that (i). some level of uncertainty or diversity in responses is inherent in conversational systems and is desired. Response's success is subject to user's preference, location and context. For example, "Resume Harry Porter" could be interpreted as "resume the Harry Porter movie I was watching on my TV earlier" if the user has an active video session in the environment he is interacting with. If not, it could be interpreted as "resume playing an audio book on Harry Porter" if the user is interacting with a screen less device. So, augmenting training data with one particular interpretation where multiple correct responses exist is something practitioners need to watch out for. It is recommended that we augment data with all possible or most relevant interpretations for the same input, (ii) increasing NLU confidence score might not translate to correct prediction in all cases. If we are not watchful, we could be adding noise i.e, examples that are ambiguous or corrupt. This could potentially degrade models' performance. One way to address this problem is to exclude outliers based on entropy score (z-score >= 3). In addition to excluding outliers, we can instate a criteria of minimum repetition in the dataset to avoid selecting such one off unusual user interactions.When it comes to EL2N based data selection, one needs to experiment with different thresholds and data compositions before deciding on the optimal values. These optimal values could change with model architecture and data drift. | http://arxiv.org/abs/2311.16302v1 | {
"authors": [
"Anusha Sabbineni",
"Nikhil Anand",
"Maria Minakova"
],
"categories": [
"cs.LG",
"cs.CL"
],
"primary_category": "cs.LG",
"published": "20231127203354",
"title": "Comprehensive Benchmarking of Entropy and Margin Based Scoring Metrics for Data Selection"
} |
The stability of smooth solitary waves for the b-family of Camassa-Holm equationsJi Li School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China ([email protected]). Changjian Liu School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China ([email protected]). Teng Long(Corresponding author) School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China ([email protected]). Jichen Yang College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China ([email protected]).January 14, 2024 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================The b-family of Camassa-Holm (b-CH) equation is a one-parameter family of PDEs, which includes the completely integrable Camassa-Holm and Degasperis-Procesi equations but possesses different Hamiltonian structures. Motivated by this, we study the existence and the orbital stability of the smooth solitary wave solutions with nonzero constant background to the b-CH equation for the special case b=1, whose the Hamiltonian structure is different from that of b≠1. We establish a connection between the stability criterion for the solitary waves and the monotonicity of a singular integral along the corresponding homoclinic orbits of the spatial ODEs. We verify the latter analytically using the framework for the monotonicity of period function of planar Hamiltonian systems, which shows that the smooth solitary waves are orbitally stable. In addition, we find that the existence and orbital stability results for 0<b<1 are similar to that of b>1, particularly the stability criteria are the same. Finally, combining with the results for the case b>1, we conclude that the solitary waves to the b-CH equation is structurally stable with respect to the parameter b>0.§ INTRODUCTIONThe b-family of Camassa-Holm (b-CH) equationsu_t-u_txx+(b+1)uu_x=bu_xu_xx+uu_xxx,where u=u(t,x)∈ is the velocity variable and b is a real parameter, was introduced in <cit.> by using transformations of the integrable hierarchy of KdV equations. The b-CH equation includes two special cases: b=2 called Camassa-Holm (CH) equation and b=3 called Degasperis-Procesi (DP) equation. They are unidirectional models that arise from the asymptotic approximation of the incompressible Euler equations in the case of shallow water <cit.>. Both of them are completely integrable with bi-Hamiltonian structure, i.e., one possesses an infinite number of conservation laws and can be expressed in Hamiltonian form in two different ways. The existence and orbital stability of the smooth solitary waves and peakons for these two models have been well developed, and we refer the readers to some relevant literature, e.g.,<cit.> for the CH equation and <cit.> for the DP equation. Due to the complete integrability and special Hamiltonian structures of the cases b=2,3, these results cannot be extended to the cases of other values of b. In <cit.>, the authors study the orbital stability of smooth solitary waves to <ref> for any b>1. They provide a sufficient condition of the orbital stability, and verify analytically for b=2,3 and numerically for other values of b>1 that the stability criterion is satisfied. In the most recent work <cit.>, the authors verify analytically for any values of b>1 that the aforementioned stability criterion can be satisfied.The main purpose of this paper is to study the orbital stability of the smooth solitary waves of <ref> for 0<b≤ 1. The stability analysis is essentially based on a general framework that motivated by the approach of <cit.>, which characterizes the critical points of Hamiltonian systems with symmetries and conserved quantities. Making use of this, we obtain a stability criterion for the orbitally stable solitary waves. Roughly speaking, we establish a connection between the stability criterion and the monotonicity of an integral along the corresponding homoclinic orbit. This method has been used in <cit.> for the case of b>1, and we extend it to the case of 0<b≤ 1 in this paper.However, it is quite technical to verify the stability criterion, i.e., determining the monotonicity. We next briefly introduce our method which is based upon the framework of period functions in planar systems which possess first integral. For instance, we consider the Hamiltonian systemx/ t=-∂ H(x,y)/∂ y,y/ t=∂ H(x,y)/∂ x,which possesses a center, i.e., any orbit surrounding the center is closed in the phase space. The period function is defined as the minimum period of each periodic orbit, and each orbit can be parameterized by γ_h:={(x,y): H(x,y)=h,h∈Ω}, where h represents the energy level along the orbit. The period function, denoted by T(h) and for h∈Ω, is expressed by definition asT(h) :=∮_γ_h t=-∮_γ_h1/∂_yHx.It is clear that there must exist zeros of ∂_yH along the closed orbit γ_h, since x/ t=0 at the left-/rightmost point of the periodic orbit in the (x,y)-plane. Hence, the period function T(h) is a singular integral and it is difficult to derive the expression of T'(h), and thus determining the monotonicity of the period function is a challenge. Fortunately, for some special systems, T'(h) can be expressed as integral form using different methods depending on the form of the systems, and thus if the integrand has the fixed sign, then T(h) will be monotone, cf., e.g., <cit.>. In some context of the nonlinear dynamics, e.g., <cit.>,the analysis of the orbital stability of the solitary wave can be converted to the determination of the monotonicity of an integral, denoted by e.g., (h), along the corresponding homoclinic orbit with the energy level h. In contrast to the fact that T(h) is singular for all h∈Ω, the integral (h) may be singular for h approaches only the boundary of Ω. Nevertheless, for both situations, a way of determining the monotonicity is to “smoothen” the singular integral first by making the product with h, then differentiating h(h) with respect to h yields the expression of '(h) which has the integral form and its integrand has a definite sign.Returning to <ref>, in the case 0<b<1, we find that the analyses of the existence and orbital stability of the smooth solitary waves to <ref> are analogous to that of the case b>1 presented in <cit.>; in particular, the stability criterion is the same as that of b>1 given by <cit.>. Hence, combining with the proof of the stability criterion as in <cit.>, we conclude that the smooth solitary wave with the nonzero constant background κ and the traveling speed c is orbitally stable for 0<b<1 and 0<κ<c/b+1, where the background κ means u(·,x)→κ as x→±∞.Our another and major goal in this paper is to understand the existence and orbital stability of the solitary waves to <ref> for the special case b=1, the Hamiltonian structure of which is different from that of the case b≠1. Analyzing the steady states in moving frame, we prove that the traveling wave solutions to <ref> have the form of solitary waves, which correspond to the homoclinic orbits of the spatial ODEs. We find that the maximum of the solitary wave over the space is increasing in the wave speed c, and is decreasing in the background κ. The following theorem is the main result of this paper. For b=1 and κ∈(0, c/2), the smooth solitary waves of <ref> with the nonzero constant background κ and the traveling wave speed c are orbitally stable.The proof of this theorem is straightforward from the combination of <ref> presented in the below sections, which follows from the aforementioned frameworks.Combining with the results of other values of b>0, we conclude that the solitary waves are structurally stable for b>0, i.e., the solitary waves exist for any b>0, and we expect that the solitary wave varies continuously in the parameter b. We also illustrate these results by some numerical computations.This paper is organized as follows. In <ref> we revisit the Hamiltonian formulation of <ref> for both cases b≠1 and b=1. In <ref> we prove the existence of smooth solitary wave solutions of <ref> for b=1 and give a discussion for the case of 0<b<1. We next study the orbital stability of the solitary waves for b=1 in <ref>, particularly obtaining a stability criterion. The verification is postponed to <ref>. Finally, we provide a short discussion in <ref>. The appendices contain some technical aspects.§ HAMILTONIAN FORMULATION We revisit the Hamiltonian formulation of <ref> for both b≠ 1 and b=1, and refer the readers to <cit.> for more details. In case of b=1, however, the Hamiltonian of <ref> on nonzero background is different from that of zero background presented in <cit.>, thus we particularly discuss this case below. It is conventional to introduce the momentum densitym := (1-∂_x^2)u,and u can be expressed asu = (1-∂_x^2)^-1m.The operator (1-∂_x^2)^-1 is defined through the convolution(1-∂_x^2)^-1 f := g * f, f∈^2(),where the kernel function g=g(x)=^-|x|/2 is the Green's function for the Helmholtz operator 1-∂_x^2 on the real line, i.e., (1-∂_x^2)g(x) = δ(x) with Dirac delta distribution δ(x). Making use of <ref>, the b-CH equation <ref> can be written in the formm_t+um_x+bmu_x=0.As mentioned in <ref>, we study the solitary wave solutions with nonzero constant background κ, i.e., u(·,x)→κ as x→±∞. Hence, we consider the function spaces := {m-κ∈ H^1(ℝ): m(x)>0,x∈ℝ}, := {u-κ∈^3():(1-_x^2)u(x)>0,x∈}.In the case b≠1, for m(t,·)∈ (i.e., u(t,·)∈) <ref> can be cast in the Hamiltonian formm/ t=_m,bδ/δ m,where the skew-symmetric operator _m,b:=-(bm∂_x+m_x)(1-∂^2_x)^-1∂^-1_x(bm∂_x +(b-1)m_x). The conserved (i.e., time-invariant) mass integral (m):=1/b-1∫_ℝ(m-κ)x. There are two other conserved quantities for m(t,·)∈_κ given by_1(m)=1/b-1∫_m^1/b-κ^1/bx,_2(m)=1/b-1∫_(m_x^2/b^2m^2+1)m^-1/b-κ^-1/bx. In the case b=1, the b-CH equation <ref> is of the formu_t-u_txx+2uu_x=u_xu_xx+uu_xxx.Again, making use of <ref>, <ref> can be expressed asm_t+ (um)_x=0.For m(t,·)∈, <ref> can be written in the Hamiltonian form∂ m/∂ t=δ/δ m(m),where the skew-symmetric operator :=_m,1 and takes the form= -∂_x(m(1-∂_x^2)^-1∂_x^-1(m∂_x·)),and the conserved integral(m)=∫_ℝm(ln m-lnκ)-(m-κ)x.In comparison with the Hamiltonian ∫_ mln mx for the case of zero background <cit.>, the configuration of <ref> is more subtle. We note that this is a linear combination of the conserved integrals ∫_ mln mx and ∫_ mx. Indeed, <ref> implies that ∫_ mx is conserved, and making use of <ref>, we have(mln m)_t=m_t(1+ln m)=-(um)_x(1+ln m)=-∂_x(umln m+1/2u^2-1/2u_x^2).Alternatively, one may obtain the conserved quantity ∫_ mln mx from the limit of the difference (m)-_1(m) for κ=0 as b→1, i.e., lim_b→1[1/b-1(m-m^1/b)]= mln m. There are two other conserved quantities given by_1(m) := ∫_ m - κx,_2(m) := ∫_ m^-3(m^2+m_x^2) - κ^-1x.These arise from the translation symmetry of the traveling waves; moreover, the derivation of these conserved quantities is detailed in <ref>. § EXISTENCE OF SOLITARY WAVES In this section, we consider the existence of smooth solitary wave solutions to <ref> on nonzero background κ for b=1 by analyzing its spatial dynamics. In addition, we also present the result of the case 0<b<1.We seek the solitary wave solutions of the formu(x,t)=ϕ(ξ)for wave shape ϕ and phase variable ξ:=x-ct, where c is the wave speed and it is assumed to be positive without loss. Accordingly, the momentum density m in the moving frame takes the formm(x,t) = μ(ξ) := ϕ(ξ) - ϕ_ξξ(ξ).We assume that the solitary wave ϕ(ξ) approaches the asymptote κ as ξ→±∞, where κ is assumed to be positive without loss, and thus the associated momentum density has the limit μ(ξ)→κ as ξ→±∞. We remark that <ref> does not possess smooth solitary waves with zero asymptote κ, see more discussion in <ref> below. With the profiles ϕ and μ in the moving frame ξ, <ref> can be transformed into the traveling wave ODE-cμ_ξ + (ϕμ)_ξ = 0.Integrating <ref> with respect to ξ, yields(c-ϕ)μ = (c-κ)κ,where the integration constant on the right-hand side stems from the aforementioned asymptotes of ϕ and μ. With the relation μ = ϕ - ϕ_ξξ and for ϕ≠ c, we can write <ref> as a second-order ODEϕ_ξξ - ϕ + κ(c-κ)/c-ϕ = 0.Multiplying it by ϕ_ξ and integrating with respect to ξ gives the total energy E:^2→,E(ϕ,ϕ_ξ) := 1/2 (ϕ_ξ)^2 + V(ϕ) = E_,where the potential energy V(ϕ):= -ϕ^2/2 - κ(c-κ)ln(c-ϕ), and the integration constant E_ corresponding to the solitary waves, which is given byE_ := E(κ,0) = V(κ) = -1/2κ^2 - κ(c-κ)ln(c-κ).We note that the real system requires the conditions ϕ<c and κ<c due to the logarithm in the potential energy. Concerning the spatial dynamics, we rewrite the second-order ODE <ref> as a first-order system, which can also be expressed in the Hamiltonian form, namely/ξ[ ϕ; ϕ_ξ ] = [ϕ_ξ; ϕ - κ(c-κ)/(c-ϕ) ] = [01; -10 ]∇ E(ϕ,ϕ_ξ),where the Hamiltonian (first integral) E is defined as in <ref>.We next give the existence of smooth solitary wave solutions to <ref> with nonzero asymoptote κ in the following.For fixed 0<κ<c/2, there exists a smooth solitary wave solution to <ref> of the form <ref> with wave shape ϕ = ϕ(·;c,κ)∈^∞() and phase variable ξ=x-ct that satisfying ϕ(ξ;c,κ)→κ exponentially as ξ→±∞ and ∂_ξϕ(0;c,κ) = 0 up to spatial translations. In addition, κ<ϕ(ξ;c,κ)<c for all ξ∈.For any fixed (c,κ) satisfying 0<κ<c/2, the graph of the potential energy V(ϕ) has a local maximum at ϕ=κ and a local minimum at ϕ=c-κ, and V(ϕ) diverges as ϕ approaches c. These imply, in (ϕ,ϕ_ξ)-plane, that the system <ref> has a saddle point at (κ,0), a center at (c-κ,0), and a singular line at ϕ=c. Since the energy E from <ref> is conserved along orbits, the unstable manifold at (κ,0) must intersect the stable manifold, yielding a homoclinic orbit, on which the total energy has the value E_, cf., <ref>. As mentioned above, the real system requires the condition ϕ<c, and thus the homoclinic orbit does not cross the singular line from left to right. Hence, the traveling wave solution associated with the homoclinic orbit with energy E_ forms the solitary wave solution ϕ=ϕ(ξ;c,κ), which decays exponentially to κ as ξ→±∞ due to the saddleness, and has the range κ<ϕ(ξ;c,κ)< c for all ξ∈. The intersection of the homoclinic orbit and the horizontal axis corresponds to ∂_ξϕ(0;c,κ) = 0 up to spatial translations.For κ=0 or κ=c the system <ref> has only a saddle point; for κ = c/2 <ref> has a degenerate equilibrium; for κ>c <ref> has two saddle points. All these cases do not allow the existence of homoclinic orbits, thus no solitary waves. For c/2 <κ<c the solitary waves exist, and the proof is analogous to that of <ref>, but ϕ(ξ) decays to c-κ as ξ→±∞, so we omit the discussion of this case.Differentiating <ref> with respect to ξ gives the expressions of μ and its derivatives in terms of ϕ as followsμ = κ(c-κ)/c-ϕ,μ_ξ = κ(c-κ)ϕ_ξ/(c-ϕ)^2,μ_ξξ = κ(c-κ)(2ϕ_ξ^2/(c-ϕ)^3 + ϕ_ξξ/(c-ϕ)^2).Combining the above relations and <ref>, we may obtain the following existence result of traveling wave solutions in terms of μ. This is required since we will consider the orbital stability of the traveling wave solutions μ instead of the solitary waves ϕ in <ref>. For fixed 0<κ<c/2, there exists a traveling wave solution to <ref> of the form <ref> with wave shape μ = μ(·;c,κ)∈^∞() and phase variable ξ=x-ct that satisfying μ(ξ;c,κ)→κ exponentially as ξ→±∞ and ∂_ξμ(0;c,κ) = 0 up to spatial translations. In addition, μ = (1-∂_ξ^2)ϕ with ϕ obtained in <ref>, and μ(ξ;c,κ)>κ for all ξ∈.As a continuation of existence results and in preparation for the stability analysis presented in <ref>, we require the refinement of the range of traveling wave solutions and the parities of them and their derivatives with respect to the spatial variable.Since the energy is conserved along the homoclinic orbit, and the existence of the center at (c-κ,0) in the phase plane, we deduce that there exists a (c,κ)-dependent value G_c,κ that satisfies G_c,κ∈(c-κ,c) and E(G_c,κ,0)=E_, such that the solitary wave ϕ(ξ;c,κ) satisfiesκ<ϕ(ξ;c,κ)≤ G_c,κ < c,ξ∈.Analyzing the monotonicity of G_c,κ, cf., <ref>, we find that for fixed c the value of G_c,κ increases to c as κ decreases to 0; for fixed κ the value of G_c,κ increases in c. We plot examples in <ref> that illustrate these.We recall the relations in <ref>, and denote μ = μ(ϕ) with abuse of notation. Since μ(κ) = κ and μ(ϕ) monotonically increases in ϕ and diverges as ϕ→ c, it satisfiesκ<μ(ξ;c,κ)≤ M_c,κ <∞,ξ∈,where M_c,κ:=κ(c-κ)/(c-G_c,κ). Rearranging E(G_c,κ,0) = E_ as mentioned in <ref>, yieldsc-κ/c-G_c,κ = exp(G_c,κ^2-κ^2/2κ(c-κ))⇒ M_c,κ = κexp(G_c,κ^2-κ^2/2κ(c-κ)).Hence, for fixed c the value of M_c,κ grows exponentially as κ decreases to 0; for fixed κ the value of M_c,κ grows exponentially in c since (c-κ)^2<G_c,κ^2<c^2. Moreover, μ(ξ;c,κ) exponentially decays to κ as ξ→±∞.Due to the relations in <ref>, the even function ϕ implies that μ and μ_ξξ are even functions, and μ_ξ is odd with mean zero. Next, we briefly discuss the existence of the smooth solitary waves to <ref> in case of 0<b<1. We note that its proof is analogous to that of b>1, which has been proved in <cit.>. Hence, we just present the result without any further proofs.For fixed 0<b<1 and 0<κ<c/b+1, there exists a smooth solitary wave solution to <ref> of the form <ref> with wave shape ϕ = ϕ_b(·;c,κ)∈^∞() and phase variable ξ=x-ct that satisfying ϕ_b(ξ;c,κ)→κ exponentially as ξ→±∞ and ∂_ξϕ_b(0;c,κ) = 0 up to spatial translations. In addition, κ<ϕ_b(ξ;c,κ)<c for all ξ∈. Note that the solitary wave solutions and the admissible range of κ depend on the parameter b. We recall, in contrast to <ref> for b=1, that the total energy for <ref> with b>1 has been given in <cit.>, which takes the formE(ϕ,ϕ_ξ) = 1/2 (b-1)(ϕ_ξ^2 - ϕ^2) + κ(c-κ)^b/(c-ϕ)^b-1 = cκ - 1/2 (b+1)κ^2.We find that this is also the total energy for the case 0<b<1, and the homoclinic orbit associated with the smooth solitary waves to <ref> can be generated by <ref> and is similar to that in <ref>. We illustrate this for different values of b in <ref>.flow/.style =decoration = markings, mark=at position #1 with >, postaction = decorate global scale/.style= scale=#1, every node/.append style=scale=#1§ STABILITY OF SOLITARY WAVESIn this section, we consider the orbital stability of the smooth solitary waves to <ref>. Our proof is essentially based upon a general framework, motivated by the approach of <cit.>, which characterizes the critical points of Hamiltonian systems with symmetries and conserved quantities via the analysis of a constrained operator. We also refer the readers to, e.g., <cit.>, for a detailed discussion on KdV equations and extension to the general framework.We recall that <ref> can be written in the Hamiltonian form <ref> in terms of m, namely∂_t m = δ/δ m(m),where the operatordefined in <ref> maps:= -∂_x(m(1-∂_x^2)^-1∂_x^-1(m∂_x·)): →,and we take= ^1()⊂^2() = .Such selection of function spaces is appropriate and we demonstrate this in <ref>. This is a skew-symmetric operator with respect to the ^2-inner product ⟨ u,v ⟩_^2 = ∫_ uv̅x, i.e., ⟨ u, v⟩_^2 = -⟨ u, v ⟩_^2 for all u,v∈^2(). The function δ/δ m denotes the variational derivative of the functionalgiven in <ref>, and (δ/δ m)(m) = ln m - lnκ. When there is no ambiguity, in the remainder of this section we denote ⟨·,·⟩_^2 as ⟨·,·⟩. In the moving frame ξ:=x-ct, we seek the solution of the form m(x,t) = m(ξ,t), with abuse of notations, that satisfies <ref>, namely∂_t m = δ/δ m(m) + c∂_ξ m = (δ/δ m(m) + cδ/δ m(m)),where the form ofgives a second conserved quantity: the charge , which is defined viaδ/δ m(m) = ∂_ξ m.We remark that the operatorhas no kernel on ^2(), and we give a proof in <ref>. Hence, we may solve the variational derivative offrom <ref> by inverting , i.e., formally (δ/δ m)(m) = ^-1∂_ξ m, and this gives the charge of the form(m) =-1/2κ^-1_1(m) - 1/2κ_2(m),where _1 and _2 are given in <ref>. The choice of the prefactors in <ref> arises from the selection of the function spaces <ref>, see more details of the computation in <ref>. In addition, we note that (m) is time-invariant since (m(ξ)) vanishes at ξ=±∞. Indeed,∂_t(m)= ⟨∂_t m, δ/δ m(m) ⟩ = ⟨δ/δ m(m), δ/δ m(m) ⟩ = - ⟨δ/δ m(m), δ/δ m(m) ⟩ = - ⟨δ/δ m(m), ∂_ξ m ⟩ = - ∫_∂_ξ(m) ξ = 0. Motivated by <ref>, we define the Lagrangian functional for <ref>(m) = -(m) - c(m).Hence, in the moving frame, <ref> can be written as the form∂_t m = - δ/δ m(m),where the variational derivative oftakes the formδ/δ m(m) = lnκ - ln m + 1/2 c κ^-1 + 1/2 c κ (-m^-2 + 3m^-4m_ξ^2 - 2m^-3m_ξξ).We consider the traveling wave solutions of the form m(ξ,t) = μ(ξ), which is stationary in the moving frame. Substitution into <ref> gives the traveling equilibrium equation_μδ/δ m(μ) = 0 ⇒δ/δ m(μ) = 0,where the latter arises from the empty kernel of _μ on ^2(). This means the equilibrium solution μ is the critical point of . Differentiating <ref> with respect to ξ, yieldsδ^2/δ m^2(μ)(∂_ξμ) = 0.We denotethe second variational derivative of , which takes the form:=δ^2/δ m^2(μ) = - cκ(∂_ξ(μ^-3∂_ξ·) - μ^-3 + 6μ^-5μ_ξ^2 - 3μ^-4μ_ξξ + 1/cκμ).The operator :^2()→^2() defined in <ref> is a self-adjoint operator with respect to the L^2-inner product, and its spectrum is composed of (a) a simple zero eigenvalue, with the associated eigenfunction ∂_ξμ, i.e., ∂_ξμ∈();(b) a simple negative eigenvalue λ_0, with the associated eigenfunction ψ_0 that has no zeros;(c) a finite number of simple positive eigenvalues λ_j, j=2,…,N that lie in the interval (0,(c-κ)/κ^2);(d) the essential spectrum σ_() = {λ∈: λ= (ck^2 + c-κ)/κ^2, k∈}, which is strictly positive. The divergence form of the differential operatorimplies its self-adjointness.(a) It is straightforward from <ref>.(b) We know, from <ref>, that the kernel eigenfunction ∂_ξμ has only one simple zero, hence a Sturm-Liouville theorem (cf. <cit.>) implies the statement.(c) The asymptotic operator _∞ associated with the exponentially asymptotic operator , i.e., replacing the coefficients inby their limiting values at ξ=±∞, takes the form_∞ = c/κ^2(-∂_ξ^2 + 1 - κ/c).From the aforementioned Sturm-Liouville theorem, there exists an upper bound of the point spectrum, whose value is given by the constant term of _∞.(d) The essential spectrum of _∞ is given by {λ∈: λ = (ck^2 + c-κ)/κ^2, k∈}, which is strictly positive due to the condition c>κ (cf., <ref>). Sinceis a relatively compact perturbation of _∞, from the Weyl essential spectrum theorem, it follows thatand _∞ have the same essential spectra, cf., e.g., <cit.>. Equilibrium solutions Since <ref> has a translation symmetry, we denote the manifold_:={(s)μ: s∈}the set of all the translations of the equilibrium solution μ to <ref>, where (s) is the translation operator, i.e., ((s)μ)(ξ):= μ(ξ+s). We can find a foliation of a neighborhood B of _ for which the solution m=m(t)∈ B to <ref> can be uniquely decomposed asm(t) = (s(t))μ + h(t),⟨ h(t),(s(t))∂_ξμ⟩ = 0,where the small perturbation h(t) is orthogonal to the kernel of the adjoint operator of , which is spanned by ∂_ξμ due to the self-adjointness of , cf., <ref>, and s(t) is a continuous function such that the decomposition <ref> holds for t∈[0,t_1) for some t_1>0. We refer the readers to <cit.> for more details.For readability, we require the definition of the orbital stability of solitary waves ϕ obtained in <ref> to <ref>. Equivalently, and for convenience, we define the orbital stability in terms of the traveling wave solution m = μ = (1-∂_ξ^2)ϕ to <ref> in the moving frame ξ=x-ct.Let m(x,t) = μ(x-ct) be a traveling wave solution to <ref> with μ∈_κ, which has a translation symmetry =(s). The manifold _ defined in <ref> is orbitally stable in _κ if for any >0 there exists δ>0 such that for any m_0∈_κ satisfying m_0-μ_^1< δ, the solution m(·,t)∈_κ of <ref> with m(·,0)=m_0 satisfiesinf_s∈m(·,t)-(s)μ_^1< , t>0.We will show that the ^1()-norm of the perturbationh(ξ,t) := m(ξ,t) - (s(t))μ(ξ)remains uniformly small for all t>0, and we expect that it can be controlled by the difference between Λ(m) and Λ((s)μ), i.e., h_^1^2≤α |(m) - ((s)μ)| for some α>0. This motivates us to define the time-invariant quantity:= (m) - ((s)μ).For notational convenience, we assume that s≡0, so that (s)=𝕀. Due to the validity of the decomposition <ref>, we can expand the Lagrangian (m) near m=μ(μ+h) = (μ) + ⟨δ/δ m(μ), h ⟩ + 1/2⟨δ^2/δ m^2(μ)h, h ⟩ +,where the remainder termsatisfies the estimate|| ≤ Ch_^1^3,for some C>0. Since this is in some sense standard, we defer the proof in <ref>. Hence the expansion ofin powers of h yields= 1/2⟨ h, h ⟩ + (h_^1^3).<ref> implies that the bilinear form ⟨ h,h ⟩ is not coercive sincehas non-positive eigenvalues. In order to obtain the coercivity of , we must consider that such bilinear form acts on a constrained function space.Let m=m(t) be the solution of <ref> with the initial datum m(0)=m_0 such that (m_0)=(μ). Sinceis time-invariant, we have the constraint (m(t)) = (μ) for all t≥0. This constraint and <ref> imply that the perturbation h(t) lies in the nonlinear admissible space_c := {h∈^1(): (μ + h)=(μ), ⟨ h,μ_ξ⟩ = 0}.By triangle inequality and ^1 continuity of (μ(c)) with respect to c,we may assume without loss of generality that (μ+h_0) = (μ).Since we consider small perturbations h, it is sufficient to consider h in the tangent space of _c at h=0. Taylor expandingyields(μ+h) -(μ) = ⟨δ/δ m(μ), h ⟩ + (h^2_^1) =⟨ψ_, h ⟩ + (h^2_^1),where the variational derivative ofis given byψ_ := δ/δ m(μ) =-1/2κ(-μ^-2 + 3μ^-4μ_ξ^2 - 2μ^-3μ_ξξ) - 1/2κ^-1. Substituting <ref> into <ref> and combining the relation <ref>, yieldsψ_ =(ϕ_ξ)^2 - ϕ^2 + κ^2/2κ(c-κ)^2= 1/c-κln(c-ϕ/c-κ),where the second equality arises from <ref>, and ψ_ is strictly negative for κ<ϕ<c, and thus ψ_ = (δ/δ m)(μ(ξ)) <0 for all ξ∈. In the tangent space of _c, the difference (μ+h) -(μ) is approximated by the leading order term ⟨ψ_, h ⟩. Hence, the linearized version of the nonlinear constraint in _c is the condition ⟨ψ_, h ⟩=0, which motivates the following linear admissible space_c' := {h∈^1(): ⟨ h,μ_ξ⟩ = ⟨ h, ψ_⟩ = 0}.We next characterize the spectrum ofinduced by the bilinear form ⟨ u,v ⟩ that is constrained on the space _c'. If the functional (μ) is decreasing in the wave speed c, namely/ c(μ) < 0,then there exists α>0 such that⟨ h, h ⟩≥αh_^1^2,h∈_c'. The proof of this lemma is somewhat standard, which is analogous to that for the KdV equation, cf., e.g., <cit.>, with some adaptation due to the different forms of the chargeand the linear operator . For completeness, we defer the proof in <ref>. As a continuation of the ^1-coercivity of , and in preparation for the orbital stability of solitary waves, we require the estimate of the bilinear form ⟨ h, h ⟩ for h∈_c. Under the condition <ref>, the estimate1/2⟨ h, h ⟩≥αh_^1^2 - βh_^1^3,holds for h∈_c with some α,β>0.The decomposition <ref> defines the projection P:_0→ span{ψ_}Pu = ⟨ u,ψ_⟩/⟨ψ_, ψ_⟩ψ_,and its complement Π=𝕀-P defined in <ref>, which can split any perturbation h∈_c⊂_0 intoh = h_1 + νψ_with h_1 = (𝕀-P)h = Π h ∈_c' and the coefficient ν = ⟨ h,ψ_⟩/⟨ψ_, ψ_⟩. The nonlinear constraint in _c and Taylor expansion <ref> indicate that(μ+h) - (μ)= ⟨ψ_, h ⟩ + (h_^1^2) = ⟨ψ_, h_1 ⟩ + ν⟨ψ_, ψ_⟩ + (h_^1^2) = 0.Since ⟨ψ_, h_1 ⟩ = 0 for h_1∈_c', the coefficient ν is of order (h_^1^2), and thus h_1 = (h_^1) due to <ref>. If the condition <ref> is satisfied, then the ^1-coercivity <ref> holds for h_1∈_c' with some α_1>0. This implies the following estimate over _c⟨ h, h ⟩ = ⟨ (h_1 + νψ_), h_1 + νψ_⟩ = ⟨ h_1, h_1 ⟩ + (h_^1^3) ≥α_1h_1_^1^2 + (h_^1^3).Hence, <ref> holds by choosing α = α_1h_1_^1^2/(2h_^1^2) and sufficiently large β>0.The functional (μ) is decreasing in c, i.e., <ref>, if and only if/ c(ϕ,c)>0, (ϕ,c):=∫_ℝ(c-κ/c-ϕln(c-κ/c-ϕ) -c-κ/c-ϕ+1)ξ,where ϕ=ϕ(ξ;c,κ) is the solitary wave solution obtained in <ref> which relates μ through <ref>.Making use of the relation <ref> and the expression <ref>, we have/ c(μ)= ⟨ψ_, ∂_cμ⟩ = ∫_1/c-κln(c-ϕ/c-κ)/ c(κ(c-κ)/c-ϕ)ξ= -κ/c-κ/ c∫_(c-κ/c-ϕln(c-κ/c-ϕ) - c-κ/c-ϕ+1)ξ = -κ/c-κ/ c(ϕ,c),where the constant 1 in the integrand of the second integral guarantees the integrability since ϕ→κ as ξ→±∞. Since c>κ is required for the existence of solitary waves (cf., <ref>), the statement is proved.The manifold _ defined in <ref> is orbitally stable in _κ as defined in <ref> if <ref> holds.Since <ref> holds, <ref> imply that||≥≥αh_^1^2 - β̃h_^1^3,for some β̃>0. The remainder is to prove that the ^1-norm of the small perturbation h can be controlled by ||, which is analogous to that for the KdV equation, cf., e.g., the proof of <cit.>, so we omit it here.§ PROOF OF STABILITY CRITERION <REF>In this section, we give a proof of the stability criterion <ref>. To this end, we need the following preparations that will be used throughout this section and some transformations on the trajectories corresponding to the smooth solitary waves (i.e., the homoclinic orbits).We first rescale the wave shape of the solitary wave solutions asϕ=κ+γφ,γ:=c-κ,where γ∈(c/2,c) due to κ∈(0,c/2), and φ∈(0,1) due to ϕ∈(κ,c) as in <ref>. In particular, we have∈(0,_0], _0:=(G_c,κ-κ)/γ,where G_c,κ is defined as in <ref>, and we remark that _0→ 1 as κ→0 or c→∞, cf., illustrations in <ref>). Substituting <ref> into the first-order system <ref>, yieldsφ/ξ=ψ,ψ/ξ= - κ/γ(1-),where ψ:=φ_ξ, and the first integral is given byH(φ,ψ) :=γ(ψ^2-φ^2)-2κφ-2κln(1-φ)=H_0.System <ref> possesses two equilibria, i.e., the saddle point (0,0) and the center (1-κ/γ,0), where 0<1-κ/γ<1, and a singular line φ=1. Since the energy H is invariant along the homoclinic orbit and equals to the value at the saddle point, the homoclinic orbit of <ref> possesses the energy H_0=0; we illustrate these in <ref>. Moreover, we call (_0,0) the turning point, which is the rightmost intersection point of the homoclinic orbit and horizontal axis in (,ψ)-plane.We next transform the homoclinic orbit of <ref> as follows. We reformulate the Hamiltonian <ref> with the energy H_0 = 0 asψ^2/φ+ln(1-φ)-φ^2/φ+ln(1-φ)=2κ/γ,where φ+ln(1-φ)<0 for φ∈(0,1). To take advantage of the framework for the monotonicity of the period function as introduced in <ref>, we define a new variableas:=ψ/√(-φ-ln(1-φ)).Hence, <ref> can be written as the separate form (in terms of φ,,γ)H(φ,) := -^2 - φ^2/φ+ln(1-φ) = h, h:=2κ/γ,where h∈(0,2) since 0<κ<c/2, and we denote by:={(φ,): H(φ,) = h}such trajectory, see <ref>. We remark that the level curvecorresponds to the homoclinic orbit of <ref> with the energy H_0=0 from <ref>. In fact, this is one of the curves of the first integral of the following Hamiltonian systemφ/τ=2,/τ= - φ f(φ)/(1-φ)(-φ-ln(1-φ))^2,where τ:=1/2√(-φ-ln(1-φ)) ξ, and we note that since √(--ln(1-)) is strictly positive for ∈(0,1), the change of variables τ=τ(ξ) is bijective and thus invertible; and f(φ):=φ+(1-φ)(φ+2ln(1-φ)), and it is easy to know that f(φ)>0 for φ∈(0,1) since f(0)=0 and f'(φ)>0 for φ∈(0,1). We remark, that since φ+ln(1-φ) is negative and monotonically decreasing in φ for φ∈(0,1), the curve ofhas a `hyperbolic' shape, and is to the left of the vertical line φ=1. The turning point (_0,0) is invariant under the transform <ref>, which can be solved by inserting =0 into <ref>. A significant difference between the first integrals <ref> and <ref> is that φ = 0 is a singularity of the latter. We find, however, that -φ^2/(φ+ln(1-φ)) approaches 2 as φ→0, hence the intersections ofand the vertical axis are given by =± a where a=a(h):=√(2-h)>0. We illustrate these in <ref>.flow/.style =decoration = markings, mark=at position #1 with >, postaction = decorate global scale/.style= scale=#1, every node/.append style=scale=#1 The criterion <ref> holds if and only if the functional () satisfies/ h(φ)<0,(φ):=2∫_(-φ-ln(1-φ))^1/2/(1-φ) τ,where =(,h) solves <ref>, i.e., (,)∈ and ∈(0,1), and h∈(0,2) defined as in <ref>.Using the rescalings <ref>, and τ=1/2√(-φ-ln(1-φ)) ξ, the functional Q(ϕ,c) defined as in <ref> can be transformed into(ϕ,c)= ∫_ℝ(c-κ/c-ϕln(c-κ/c-ϕ) -c-κ/c-ϕ+1) ξ= ∫_ℝ(1/1-φln(1/1-φ)-1/1-φ+1)ξ= 2∫_(-φ-ln(1-φ))^1/2/(1-φ) τ = (φ).Since γ = c-κ from <ref> and h = 2κ/γ on , we have h = 2κ/(c-κ) and it leads to h/ c = -2κ(c-κ)^-2 < 0. Hence the statement holds. The criterion <ref> holds for ∈(0,1) and h as in <ref>.We rewrite () as the followingQ̅(φ) =2∫_ℝ(-φ-ln(1-φ))^1/2/(1-φ) τ=-2∫_(-φ-ln(1-φ))^5/2/φ f(φ)= -2∫_g(φ)=:(h),where g() := (-φ-ln(1-φ))^5/2/(φ f(φ)) and the second equality stems from the change of variables in the second equation of <ref>.Multiplying both sides of the last equality presented above by h and using the expression of h=H(,) from <ref>, where H(φ,) is constant along the trajectory , as well as using the expression of / given by the quotient of two equations in <ref>, yieldsh(h) =-2h∫_g(φ)=-2(∫_-φ^2g(φ)/φ+ln(1-φ)+∫_-^2 g(φ) ) =-2(∫_φ(-φ-ln(1-φ))^3/2/f(φ)+∫_(-φ-ln(1-φ))^1/2/2(1-φ) φ) ≜ -2(I_1(h)+I_2(h)),Differentiating both sides of <ref> with respect to h, with the notation ':=/ h, yields(h)+ h'(h)=-2(I'_1(h)+I'_2(h)),whereI'_1(h)=∫_∂/∂(φ(-φ-ln(1-φ))^3/2/f(φ))·∂φ/∂ h =∫_-(-φ-ln(1-φ))^5/2/2φ f^3(φ)·(4φln^2(1-φ)-4ln^2(1-φ) -4φ^3ln(1-φ)+12φ^2ln(1-φ)-8φln(1-φ)-φ^4 +8φ^3-4φ^2) ,andI'_2(h) =∫_(-φ-ln(1-φ))^1/2/2(1-φ)·∂/∂ h φ=∫_(-φ-ln(1-φ))^1/2/2(1-φ)·(-1/2)φ=∫_(-φ-ln(1-φ))^5/2/2φ f(φ) ,where ∂/∂ h and ∂/∂ h can be solved by differentiating <ref> with respect to h. In addition, we note that the integrals I_j(h),I'_j(h),j=1,2 are Riemann integrals since their integrands are bounded for ∈(0,_0], i.e., for (,)∈. It follows thath'(h)=-2(I'_1(h)+I'_2(h))-(h) = ∫_G() F() =∫_a^-a G() F()= -2∫_0^aG()F() .where a=√(2-h), and G():= (- - ln(1-))^5/2/f^3() with G()>0 for ∈(0,1), andF(φ):=4(φln^2(1-φ)-ln^2(1-φ)+φ^2).Since h>0, a sufficient condition for '(h)<0 is that F()>0 for ∈(0,1), and in the remainder of this proof we will show the latter.Differentiating F(φ) twice with respect to φ, with abuse of notation ':=/, yieldsF'() = 8 + 4ln(1-) (2 + ln(1-)), F”(φ) =8(φ+ln(1-φ))/φ-1.We consider only ∈(0,1). We note that F”()>0 since +ln(1-) is negative as mentioned above, and it follows that F'(φ) is monotonically increasing in . Since lim_φ→0F'(φ)=0, we have F'(φ)>0, which means that F(φ) monotonically increases in . Combining this with the fact that lim_φ→0F(φ)=0, we obtain F(φ)>0.§ DISCUSSION We have studied the orbital stability of smooth solitary waves of the b-family of Camassa-Holm equations for 0<b≤1. For 0<b<1, the analysis of the stability is analogous to that of the case b>1 presented in <cit.>, and the stability criterion has been verified analytically by <cit.>. By comparison, the Hamiltonian structure is different for the case of b=1, and thus we have provided a rigorous proof of the orbital stability of solitary waves of <ref>, which is based on a general framework motivated by the approach of <cit.>. We have further verified the stability criterion <ref> using the method analogous to that in <cit.>, which is again the application of the framework of monotonicity of the period function in planar Hamiltonian systems. We also expect that the combination of the approach of <cit.> and the method of monotonicity of the period function may be applied to the stability analysis of <ref> for b<0, and it would be interesting to pursue this further. Moreover, we expect this method to be effective not only for the b-CH equations, but also for other types of Hamiltonian systems. Another interesting direction would be considering the orbitally asymptotic stability of solitary waves of <ref> with respect to small perturbations in some function spaces, cf., e.g., <cit.> for KdV equations and <cit.> for peakons of CH equations.§ MONOTONICITY OF G_C,Κ For convenience, we suppress the subscript of G_c,κ. Differentiating the both sides of E(G,0) = E_ with respect to c, and solving ∂_c G, yields∂_c G = κ(c-G)/(G-κ)(G-c+κ)(c-κ/c-G - ln(c-κ/c-G) - 1)>0,since κ<c-κ<G<c and the function f(w) := w - ln w - 1 >0for w>1. Analogously, differentiating both sides of E(G,0) = E_ with respect to κ and solving ∂_κ G, yields∂_κ G = -(c-2κ)(c-G)/(G-c+κ)(G-κ)ln(c-κ/c-G) < 0.§ OPERATOR We recall that m∈_κ defined in <ref>, and denote := m-κ∈^1(). Let ψ∈^1(), we have ∂_xψ∈^2() and denote v:=∂_xψ. Then the product mv∈^2() sincemv_2^2 = ∫_ | v+κ v|^2x≤ 2 ∫_ | v|^2 + |κ v|^2x≤ 2(_∞^2 + κ^2)v_2^2 ≤ 2(_^1^2 + κ^2)v_2^2,which follows from the Sobolev embedding ^1()⊂^∞(). Since the operator ∂_x:^1()→^2() has no kernel, it is invertible and the range of ∂_x^-1 lies in ^1(), which leads to ∂_x^-1(mv)∈^1(). We denote w:=∂_x^-1(mv) and recall the representation of (1-∂_x^2)^-1 in <ref>, then we have (1-∂_x^2)^-1w∈^1() since(1-∂_x^2)^-1w_^1^2= g*w_^1^2 = g*w_2^2 + ∂_x(g*w)_2^2 = g*w_2^2 + g*∂_x w_2^2≤g_1^2w_2^2 + g_1^2∂_xw_2^2 = g_1^2w_^1^2,where g∈^1(). Denote z:=(1-∂_x^2)^-1w, since ^1() is a Banach algebra, we havem z_^1^2= m z_2^2 + ∂_x(m z)_2^2 = ∫_ | z+κ z|^2x + ∫_ |∂_x( z)+κ∂_x z|^2x ≤ 2∫_ | z|^2+|κ z|^2x + 2∫_ |∂_x( z)|^2 + |κ∂_x z|^2x= 2 z_^1^2 + 2κ^2z_^1^2≤ 2_^1^2z_^1^2 + 2κ^2z_^1^2,i.e., mz∈^1(). It follows that ∂_x(mz)∈^2(), which means ψ∈^2(). § INVERTIBILITY OF We rewrite the operatorin <ref> as= -∂_x: ^1() →^2(), := m(1-∂_x^2)^-1∂_x^-1(m∂_x·): ^1() →^1(),with m∈_κ defined in <ref>. We assume thathas a kernel ψ∈^1(). Since the operator ∂_x has no kernel on ^2(), ψ∈(). Since m(x)>0 for all x∈ and the operator 1-∂_x^2 has no kernel on ^2(), we deduce that(1-∂_x^2)^-1∂_x^-1(m∂_x ψ) = 0⇒ m∂_xψ = 0⇒∂_xψ = 0,which contradicts the fact that ∂_x has no kernel on ^2(). § VARIATIONAL DERIVATIVE OF For convenience, we denote q:=δ/δ m(m). We may solve q from the following equationq = -∂_x(m(1-∂_x^2)^-1∂_x^-1(m∂_x q)) = ∂_x m.Integrating both sides with respect to x, and using the notationin <ref>, yieldsq = m(1-∂_x^2)^-1∂_x^-1(m∂_x q) = -m + k_2,where the integration constant k_2=κ since the range oflies in ^2() and m∈_κ. Since m(x)>0, we can apply the inverse operator m^-1∂_x(1-∂_x^2)m^-1 to the both sides, yields∂_x q= k_2(-m^-3m_x + 6m^-5m_x^3 - 6m^-4m_x m_xx + m^-3m_xxx).Then integrating both sides with respect to x, yieldsq = -1/2 k_2(-m^-2 + 3m^-4m_x^2 - 2m^-3m_xx) + k_1= -1/2 k_2 δ_2/δ m(m) + k_1δ_1/δ m(m),where k_1 = -1/(2κ) since q∈^1(), andδ_2/δ m(m) := -m^-2 + 3m^-4m_x^2 - 2m^-3m_xx,δ_1/δ m(m) := 1,are the variational derivatives of _2 and _1 defined in <ref>, and the selections of k_1,k_2 above give rise to the form ofin <ref>. § PROOF OF ESTIMATE <REF>The remainder termin <ref> has the expression= Λ(μ+h) - Λ(μ) - ⟨δ/δ m(μ), h ⟩ - 1/2⟨δ^2/δ m^2(μ)h, h ⟩= ∫_ h (f_30h^2 + 2f_21 h h_ξ + f_12 h_ξ^2) + (h^2(h^2+h_ξ^2)) ξ,where the prefactors f_30:= - μ^-2/6 - μ^-4 - 10μ^-6μ_ξ^2, f_21:= 6μ^-5μ_ξ, f_12:= -3μ^-4, which are bounded for all ξ∈ (but not uniformly in parameters c or κ) due to the expressions in <ref> with bounded ϕ,ϕ_ξ. Using Cauchy's inequality and the Sobolev embedding ^1()⊂^∞(), the remainder termsatisfies the estimate||≤∫_ |h| (|f_30| + |f_21|)h^2 + (|f_12|+|f_21|) h_ξ^2) + (h^2(h^2+h_ξ^2)) ξ≤ C_1h_∞h_^1^2 + (h_∞^2h_^1^2) ≤ C_1h_^1^3 + (h_^1^4)for the constant C_1:=sup_ξ∈{|f_30|+|f_21|,|f_12|+|f_21|}. § PROOF OF <REF> To obtain the coercivity of , we have to deal with its kernel and the eigenfunction associated with the negative eigenvalue. For convenience, we use the notations of function spaces as in <ref> in the proof.We first consider the kernel of . We introduce the spectral projection onto the kernel of , i.e., P_0:→()P_0 u = ⟨ u,μ_ξ⟩/⟨μ_ξ,μ_ξ⟩μ_ξ,and its complement Π_0 = 𝕀 - P_0 which has the kernel spanned by μ_ξ, and has the range_0 = {h∈: ⟨ h, μ_ξ⟩ =0} = ()^⊥,i.e., Π_0:→()^⊥ since ⟨Π_0 u,μ_ξ⟩ = 0 for any u∈. We observe that_c' =span{μ_ξ}^⊥∩ span{ψ_}^⊥∩,and decompose the new space _0:=_0∩ as the following_0 = _c' ⊕ span{ψ_}.Since both P_0 and Π_0 are self-adjoint operators and Π_0 is an identity operator on _0, for any u,v∈_0 we have the relations⟨ u,v ⟩ = ⟨ u, Π_0 v ⟩ = ⟨Π_0 u, v ⟩.This motivates the definition _0 :=Π_0, which maps _0:D()∩_0⊂_0→_0. The spectral projection P_0 eliminates the kernel offrom the space , which leads to the spectrum σ(_0) = σ()∖{0}, and thus _0 is boundedly invertible which has a single negative eigenvalue λ_0 associated with eigenfunction ψ_0, with the rest of σ(_0) strictly positive, cf., <ref>. We next consider the bilinear form ⟨ u,v ⟩ that is constrained on the space _c'. We introduce the self-adjoint projectionΠ u = u - ⟨ u,ψ_⟩/⟨ψ_, ψ_⟩ψ_.Since ψ_ is not the eigenfunction of , Π is not a spectral projection for , nor for _0, and thus the projection Π does not remove any eigenvalue from , nor from _0. The parities of μ and its derivatives claimed in <ref> implies that⟨ψ_, μ_ξ⟩=0,since ψ_μ_ξ is odd with mean zero. Hence, we deduce that Π_0Π = ΠΠ_0, and thus Π:_0→_c'; moreover, ΠΠ_0 is the identity operator on _c'. Hence, for any u,v∈_c', the relations⟨ u,v ⟩ = ⟨ u, ΠΠ_0 v ⟩ = ⟨Π u, Π_0 v ⟩ = ⟨Π_0Π u, v ⟩ = ⟨ΠΠ_0 u, v ⟩motivates the definition _Π := Π_0, which maps _Π: D()∩_c'⊂_0 →Π_0. The constrained operator _Π is self-adjoint, so its spectrum is real. The difference _0-_Π, acting on _c' is rank-one since its range is spanned by ψ_, and hence compact, so the Weyl essential spectrum theorem implies that σ_(_Π) = σ_(_0).Concerning the point spectrum of _Π, the smallest eigenvalue α_0 of _Π has the variational characterizationα_0 = inf_ψ∈_c'⟨ψ,ψ⟩/⟨ψ,ψ⟩.The smallest eigenvalue λ_0<0 ofhas a similar formulation, with the infimum over ψ∈. We deduce that λ_0≤α_0, since λ_0 is the infimum of the same functional over a larger space. Since ψ_0 has no zeros (cf., <ref>) and ψ_ is of one sign over ξ∈ (cf., <ref>), we have ⟨ψ_0,ψ_⟩≠ 0, which implies ψ_0∉_c', and hence λ_0<α_0. Indeed, a purpose of the constraint ⟨·,ψ_⟩=0 is to eliminate the eigenfunction ofassociated with negative eigenvalue from the function space _0. The above infimum is achieved at ψ∈_c', which solves the eigenvalue problem_Πψ = α_0ψ⇒_0ψ = α_0ψ + νψ_,for some value of the multipliers α_0,ν∈. Since _0 is invertible on _0, we may solve for ψ∈_0ψ(α_0) = ν(_0-α_0)^-1ψ_. To enforce the condition ψ∈_c', we impose the constraint ⟨ψ,ψ_⟩ = 0. We use <ref> to rewrite the constraint as a function of α_0,g(α_0):= 1/ν⟨ψ,ψ_⟩ = ⟨ (_0-α_0)^-1ψ_,ψ_⟩,so that g(α_0)=0 if and only if α_0 is an eigenvalue of _Π. In particular, if g(α_0) is nonzero for all α_0≤ 0, then the smallest eigenvalue of _Π must be positive. The function g is analytic for α_0∉σ(_0), with pole singularities possible at the eigenvalues of _0, and is strictly increasing where it is smooth, since g'(α_0) = ⟨ (_0-α_0)^-2ψ_, ψ_⟩ = (_0-α_0)^-1ψ__2^2>0 using the Green's function for the resolvent. The operator _0 has no spectrum on (λ_0,λ_1), where λ_1>0 is the smallest element of σ(_0)∖{λ_0}. Consequently, the function g is analytic and monotonically increasing on this interval. It follows that the first zero, α_0, of g is positive if g(0)<0, and this zero is the smallest eigenvalue of _Π.To relate the value of g(0) to the derivative of the charge (μ) with respect to wave speed, we differentiate <ref> with respect to c ∂_cδ/δ m(μ) = ∂_c(- δ/δ m(μ) - cδ/δ m(μ)) = ∂_cμ - ψ_ =0 ⇒∂_cμ = ψ_.We recall that ⟨ψ_, μ_ξ⟩=0 in <ref>, thus ψ_∈_0 and equivalently Π_0ψ_=ψ_. Applying Π_0 to both sides of the above equation and inverting _0 on _0, yields_0∂_cμ = ψ_⇒∂_cμ = _0^-1ψ_,This allows us to rewrite g(0) asg(0) = ⟨_0^-1ψ_,ψ_⟩ = ⟨∂_cμ,ψ_⟩ = ∂_c (μ).Hence ∂_c (μ)<0 implies that α_0>0.Up to now, in the case α_0>0 the variational characterization <ref> gives the ^2-coercivity⟨ v, v ⟩≥α_0 v_2^2, v∈_c'.To obtain the ^1-coercivity we argue by contradiction. Assume the bilinear form ⟨ v, v ⟩ is not ^1-coercive over _c', then for any >0, there is a sequence of functions {v_n}_n=1^∞⊂_c'⊂^1() with v_n_^1 = 1 such that ⟨ v_n,v_n ⟩<v_n_^1^2, i.e., ⟨ v_n,v_n ⟩→0 as n→∞. The ^2-coercivity ⟨ v_n,v_n ⟩≥α_0v_n^2_2 implies that lim_n→∞v_n_2 = 0, and since v_n_^1 = 1 we havelim_n→∞⟨∂_ξ v_n, ∂_ξ v_n ⟩ = 1.Recalling the form ofin <ref> and suppressing the prefactor cκ, we have the limitlim_n→∞⟨ -∂_ξ(μ^-3∂_ξ v_n), v_n ⟩ = lim_n→∞⟨μ^-3∂_ξ v_n, ∂_ξ v_n ⟩≥ M_c,κ^-3lim_n→∞⟨∂_ξ v_n, ∂_ξ v_n ⟩ = M_c,κ^-3,where μ(ξ)≤ M_c,κ for ξ∈, see <ref> for more details of M_c,κ. For the potential W(ξ):= μ^-3 - 6μ^-5μ_ξ^2 + 3μ^-4μ_ξξ - (cκμ)^-1 associated withwe have the limitlim_n→∞|⟨ W(ξ)v_n,v_n ⟩| ≤W_∞lim_n→∞⟨ v_n,v_n ⟩ = 0.These limits bring us to the contradiction, namely0 = lim_n→∞⟨ v_n,v_n ⟩ = lim_n→∞( ⟨ -∂_ξ(μ^-3∂_ξ v_n) , v_n ⟩ + ⟨ W(ξ)v_n,v_n ⟩) ≥ M_c,κ^-3>0.Hence, we obtain the H^1-coercivity <ref> with some α>0.§.§ AcknowledgmentsThe paper is supported by the National Natural Science Foundation of China (No. 12171491). The authors thank Stéphane Lafortune for the inspiring discussions and comments.1 CM2001 A. Constantin, L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D 157(2001) 75-89.CS2000 A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603-610.CS2002 A. Constantin, W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci.12 (2002) 415-422.Coppel W. Coppel,L. 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Wu, Orbital stability of smooth solitary waves for the Degasperis-Procesi equation, Proc. Amer. Math. Soc. 151(1) (2023) 151-160.Lin Z. Lin, Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62(1) (2009) 125-146.Liu2020 X. Liu, Orbital stability of solitary wave solutions of Kudryashov-Sinelshchikov equation, Eur. Phys. J. Plus 135 (2020) 804.LL2023 T. Long, C. Liu, Orbital stability of smooth solitary waves for the b-family of Camassa-Holm equations, Phys. D 446 (2023) 133680.LLW2022 T. Long, C. Liu, S. Wang, The period function of quadratic generalized Lotka-Volterra systems without complex invariant lines, J. Differential Equations 314 (2022) 491-517.Molinet18 L. Molinet, A Liouville property with application to asymptotic stability for the Camassa-Holm equation, Arch. Ration. Mech. Anal. 230(1) (2018) 185-230.Molinet20 L. Molinet, Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile, Int. Math. Res. Not. IMRN 21 (2020) 7908-7943.PW1994 R. Pego, Weinstein, M. I., Asymptotic stability of solitary waves, Comm. Math. Phys. 164(2) (1994) 305-349.Jordi J. Villadelprat, X. Zhang, The period function of Hamiltonian systems with separable variables, J. Dynam. Differential Equations 32 (2020) 741-767.Zhao Y. Zhao, The monotonicity of period function for codimension four quadratic system Q_4, J. Differential Equations 185(1) (2002) 370-387. | http://arxiv.org/abs/2311.15634v1 | {
"authors": [
"Ji Li",
"Changjian Liu",
"Teng Long",
"Jichen Yang"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20231127085555",
"title": "The stability of smooth solitary waves for the $b$-family of Camassa-Holm equations"
} |
[][email protected] of Physics, East China University of Science and Technology, Shanghai 200237, ChinaWe generalize the coincident pulse technique in three-state stimulated Raman adiabatic passage (STIRAP) system [A. A. Rangelov and N. V. Vitanov, https://link.aps.org/doi/10.1103/PhysRevA.85.043407Phys. Rev. A 85, 043407 (2012)] to a five-state chainwise STIRAP system. In our method, we first reduce the M-type structure into a generalized Λ-type one with the simplest resonant coupling, which principally allows us to employ the standard coincident pulse technique from three-state system into the five-state chainwise system. The simplification is realized under the assumption of adiabatic elimination (AE) together with a requirement of the relation among the four incident pulses. The results show that, by using N (N≫1) pairs of coincident incident pulses, this technique enables complete population transfer, as well as the creation of arbitrary desired coherent superposition between initial and final states, while there are negligible population in all the intermediate states. The results are of potential interest in applications where high-fidelity multi-state quantum control is essential, e.g., quantum information, atom optics, formation of ultracold molecules, cavity QED, nuclear coherent population transfer, light transfer in waveguide arrays, etc. Quantum state engineering in a five-state chainwise system by coincident pulse technique Jiahui Zhang========================================================================================§ INTRODUCTIONThe quest for techniques with which to control the transfer of population between specified states is a major topic in the field of atomic, molecular and optical physics. The well-known stimulated Raman adiabatic passage (STIRAP) technique has become an important tool; see reviews <cit.>. The standard STIRAP process utilizes a Raman transition with two counter-intuitively ordered laser pulses, allowing for the efficient and robust transfer of the population between an initial and a final state of a three-state. The successes of STIRAP relies on the existence of the dark state, which should be followed adiabatically <cit.>. The middle state, which is subjected to population decay in many physical implementations <cit.>, is not populated during the process because it is not present in the dark state.The success of STRAP has prompted its extension to chainwise-connected multi-state systems <cit.>, in which each state is connected only to its two neighbors: |1⟩↔|2⟩↔|3⟩↔···↔|n⟩. In general, the goal is to transfer the population between the two ends without filling the intermediate states. The potential applications involve many branches of fundamental physics and chemistry. Just a few examples: (1) atomic mirrors and beams splitters in atom optics <cit.>; (2) cavity QED <cit.>; (3) spin-wave transfer via adiabatic passage in a five-level system <cit.>; (4) creation and detection of ultracold molecules <cit.>, chainwise-STIRAP in M-type molecular system has been demonstrated a good alternative in creating ultracold deeply-bound molecules when the typical STIRAP in Λ-type system does not work due to weak Frank-Condon factors between the molecular states that are involved <cit.>, the prepared molecule can be used in the study of ultracold chemistry <cit.>, precision measurements <cit.>, quantum computations and quantum simulations <cit.>; (5) multi-state nuclear coherent population transfer <cit.>, which can be used in the study of the nucleus, the construction of nuclear batteries <cit.>, as well as the construction of nuclear clocks that are much more accurate than atomic clocks <cit.>; (6) adiabatic light transfer in waveguide arrays <cit.>, which has profound impacts on exploring quantum technologies for promoting advanced optical devices, and provides a direct visualization in space of typical phenomena in time. Quite often, it is necessary to achieve state preparation or transfer with high fidelity. This requires large temporal areas of the driving pulsed fields in order to suppress the non-adiabatic couplings, which is very hard to reach experimentally. Although strategies for optimization of multi-state chainwise system with minimal pulse areas have been developed <cit.>, these come at the expense of strict relations on the pulse shapes <cit.>, require specific time-dependent nonzero detunings <cit.>, or require additional couplings between the states that are involved <cit.>.Recently, Rangelov and Vitanov have proposed a technique to complete population transfer in three-state systems by a train of N pairs of coincident pulses <cit.>, in which the population in the intermediate excited state is suppressed to negligible small value by increasing the pulse pairs. In this technique the number of pulse pairs is arbitrary and the robustness of system against deviation from exact pulse areas and spontaneous emission from excited state rise with increasing numbers of pulse pairs <cit.>. Since the technique uses fields on exact resonance, the pulse shape is not important. The technique has recently been generalized to tripod system <cit.>. However, the application of coincident pulse technique in multi-state chainwise-connected systems has not been reported to date.In the present paper, we generalize the coincident pulse technique in three-state system to a five-state chainwise system. We first reduce the dynamics of the five-state chainwise system to that of effective three-state counterpart.By further setting a requirement towards the relation among the four incident pulses, i.e., the pulses at both ends are the root mean square (rms) of the middle two pulses, it is found that this system can be further generalized into a Λ-type structure with the simplest resonant coupling. Thereafter, this generalized model permits us to borrow the standard coincident pulse technique from three-state system into the five-state chainwise system. The results show that, by using N (N≫1) pairs of coincident incident pulses, our technique enables complete population transfer, as well as the creation of arbitrary coherent superposition between initial and final states with negligibly small transient populations in all the intermediate states. All these properties make this technique an interesting alternative of the existing techniques for coherent control of five-state chainwise system. § MODEL AND METHODSThe idea of this work can be described using a simple five-level system with states chainwise coupled by optical fields as illustrated in Fig. <ref>. The states | g_1⟩, | g_2⟩ and | g_3⟩ are three ground states of long lifetimes while the intermediate states | e_1⟩, and | e_2⟩ refer to two excited states of short lifetimes. The coupling between states is presented by the time-dependent Rabi frequency Ω_i (i=1, 2, 3, 4) in this figure. The total wave function can be expanded as| ψ(t)⟩=c_1(t)| g_1⟩+c_2(t)| e_1⟩+c_3(t)| g_2⟩ +c_4(t)| e_2⟩+c_5(t)| g_3⟩.the vector c_i(t) (i=1, 2, 3, 4, 5) are the probability amplitudes of the corresponding state. The evolution is then governed by the time-dependent Schrödinger equation:iħ∂/∂ tc(t)=H(t)c(t).Here, c(t) is a five-component column matrix with the elements {c_1(t), c_2(t), c_3(t), c_4(t), c_5(t)}, and P_i = |c_i(t)|^2 are the corresponding probability. In the interaction representation and after adopting the rotating-wave approximation, this system can be quantitatively described in terms of a five-state Hamiltonian (ħ=1)H(t)=1/2( [0 Ω_1(t)000; Ω_1(t) 2Δ_1 Ω_2(t)00;0 Ω_2(t)0 Ω_3(t)0;00 Ω_3(t) 2Δ_2 Ω_4(t);000 Ω_4(t)0;]),where the quantities Δ_1 and Δ_2 stand for single-photon detunings of the corresponding transitions.If we assume that pairs of fields coupling two neighboring ground state vibrational levels are in a two-photon (Raman) resonance, the system has a dark state given by| ψ_0⟩=Ω_2(t)Ω_4(t)| g_1⟩-Ω_1(t)Ω_4(t)| g_2⟩+Ω_1(t)Ω_3(t)| g_3⟩/𝒩(t),where 𝒩(t) is a normalization factor. In “classical" STIRAP, where the incident fields are applied in a counterintuitive time sequence. Adiabatically changing the Rabi frequencies of the optical fields so that the system stays in the dark state during evolution, one can transfer the system from the initial | g_1⟩ to the ground vibrational | g_3⟩ state with unit efficiency. The dark state does not have contributions from the | e_1⟩ and | e_1⟩ excited states, and thus the decay from these states does not affect the transfer efficiency. However, the intermediate ground state | g_2⟩ will receive some transient population. In some particular example <cit.>, | g_2⟩ is a radiative state, decay from this state will degrade the coherent superposition (<ref>) and result in population loss from the dark state and reduction of the transfer efficiency.When the single-photon detunings are very large, meaning|Δ_1| ≫√(Ω^2_1(t)+Ω^2_2(t)),|Δ_2| ≫√(Ω^2_3(t)+Ω^2_4(t)), as we shall assume for our case, then states | e_1, 2⟩ are scarcely populated, and which can be adiabatically eliminated to obtain the following effective three-state Hamiltonian in the subspace {| g_1⟩, | g_2⟩, | g_3⟩}:H_e(t) =1/2( [ Δ_e1 Ω_e10; Ω_e1 Δ_e2 Ω_e2;0 Ω_e2 Δ_e3;]),where the effective couplings are defined as (for simplicity, we have assumed that Δ_1=Δ_2=Δ) Ω_e_1(t) =-Ω_1(t)Ω_2(t)/2Δ, Ω_e_2(t) =-Ω_3(t)Ω_4(t)/2Δ, respectively, and three diagonal elements are Δ_e_1 =-Ω_1^2(t)/2Δ, Δ_e_2 =-Ω_2^2(t)+Ω_3^2(t)/2Δ, Δ_e_3 =-Ω_4^2(t)/2Δ. Thereafter, two-photon transitions |g_1⟩↔|g_2⟩ and |g_1⟩↔|g_3⟩ will dominate. Obviously, the system after AE subjects to dynamic Stark shifts from the trapping light, which can be expected to reduce the transfer efficiency. If we assume that the three diagonal elements are equal to each other, i.e., Δ_e_1=Δ_e_2=Δ_e_3=Δ_e. This requires that the four Rabi frequencies Ω_j (j=1,2,3,4) should satisfyΩ_1(t)=Ω_4(t)=√(Ω_2^2(t)+Ω_3^2(t)).By further setting c_j =c^'_je^-iΔ_e t(i=1, 3, 5), the above equation can be written in the following form: i∂/∂ t( [ c^'_1; c^'_3; c^'_5; ]) =1/2( [0 Ω_e_1(t)0; Ω_e_1(t)0 Ω_e_2(t);0 Ω_e_2(t)0;]) ( [ c^'_1; c^'_3; c^'_5; ]). Therefore, the basic model (<ref>) is generalized into a Λ-type structure with the simplest resonant coupling:H_e=1/2( [0 Ω_e_1(t)0; Ω_e_1(t)0 Ω_e_2(t);0 Ω_e_2(t)0;]),in which Ω_e_1(t) =-Ω_2(t) √(Ω_2^2(t)+Ω_3^2(t))/2Δ, Ω_e_2(t) =-Ω_3(t) √(Ω_2^2(t)+Ω_3^2(t))/2Δ. In order to implement the coincident pulse technique in this generalized model, we impose the condition that the effective Rabi frequencies Ω_e_1(t) and Ω_e_2(t) are pulse-shaped functions that share the same time dependence, but possibly with different magnitudes. This essentially means that the Rabi frequencies Ω_2(t) and Ω_3(t) are pulse-shaped functions with the same time dependences, but possibly with different magnitudes, i.e., Ω_2(t) =η_2f(t), Ω_3(t) =η_3f(t). In this case, the Schrödinger equation (<ref>) is solved exactly by making a transformation to the so-called bright-dark basis <cit.>.The exact propagator is given by <cit.> U(φ)= ( [ 1-2sin^2φsin^2A/4-isinφsinA/2 -2sin2φsin^2A/4;-isinφsinA/2cosA/2-icosφsinA/2; -2sin2φsin^2A/4-icosφsinA/2 1-2cos^2φsin^2A/4; ]), where tanφ=Ω_e_1/Ω_e_2=Ω_2/Ω_3=η_2/η_3, the rms pulse area A is defined as A=∫^t_t_i√(Ω^2_e_1(t)+Ω^2_e_2(t))dt. According to propagator (<ref>), one can find the exact analytic solution for any initial condition. If we restrict our attention here to a system initially in state |g_1⟩. Then the populations at the end of the interaction are P_1= |U^N_11|^2,P_3= |U^N_21|^2,P_5= |U^N_31|^2. For φ=π/4, which corresponds to η_2=η_3, and rms pulse area A=2π, the population of state |g_1⟩ can be completely transferred to the final state |g_3⟩. Notably, arbitrary fractional population between |g_1⟩ and |g_3⟩ can be controlled by changing the mixing angle φ. For example, if we set φ_k=π/8, the analytic solutions (<ref>) indicate that the equal population distribution between |g_1⟩ and |g_3⟩ can be achieved. Meanwhile, Eqs. (<ref>) implies the intermediate ground state |g_2⟩ will receive a significant transient populations along the way.In order to suppress the population of the intermediate ground state |g_2⟩, one can use a sequence of N pairs of coincident pulse, each with rms pulse area A(t)=2π at the end of the kth step and mixing angles φ_k, the overall propagator is given byU^(N)=U(φ_N)U(φ_N-1)··· U(φ_2)U(φ_1),where φ_k is given byφ_k=(2k-1)π/4(8)N, k=1,2,...,N.As a result, the maximum population of the intermediate ground state |g_2⟩ in the middle of each pulse pair is damped to small values by increasing the number of pulse. The Rabi frequencies in the kth step are supposed to be with Gaussian shapes and they are overlapped, Ω̃_e_1(t) =Ω_0sinϑ_kexp[-t-τ/T]^2, Ω̃_e_2(t) =Ω_0cosϑ_kexp[-t-τ/T]^2, where Ω_0= 2√(π)/T (corresponding to rms pulse area A=2π) and the mixing angles φ_k (k=1,2,..., N) are given by Eq. (<ref>). It should be noted that although Gaussian shapes are used here, pulses of any other shape are equally suitable. This is due to this technique uses fields on exact resonance the pulse shapes are unimportant <cit.>. Eqs. (<ref>) imply the direct couplings between |g_1⟩, |g_2⟩, and between |g_2⟩, |g_3⟩ are required. However, the direct couplings may be infeasible for many physical scenarios <cit.>. Therefore, it is necessary to go back to the five-state system and design the physically feasible driving fields. Like Eqs. (<ref>), one can impose Ω̃_e_1(t)=-Ω̃_2(t)√(Ω̃_2^2(t)+Ω̃_3^2(t))/2Δ, Ω̃_e_2(t)=-Ω̃_3(t)√(Ω̃_2^2(t)+Ω̃_3^2(t))/2Δ, respectively. Therefore, we can inversely derive the modified Rabi frequencies Ω̃_2(t) and Ω̃_3(t) based on Eqs. (<ref>) with the form Ω̃_2(t) =Ω̃_e_1(t) ( 4Δ^2/Ω̃^2_e_1(t)+Ω̃^2_e_1(t)) ^1/4, Ω̃_3(t) =Ω̃_e_2(t) ( 4Δ^2/Ω̃^2_e_1(t)+Ω̃^2_e_2(t)) ^1/4.Accordingly, Rabi frequencies Ω_1(t) and Ω_4(t) are also modified following the similar relation as that in Eq. (<ref>), which is Ω̃_1, 4(t)=√(Ω̃_2^2(t)+Ω̃_3^2(t)). Thus, we obtainΩ̃_1, 4(t)=[4Δ^2(Ω̃^2_e_1(t)+Ω̃^2_e_2(t))]^1/4. Returning back to the five-state M-type system with chainwise coupling, the full Hamiltonian in Eq. (<ref>) is modified to beH̃(t)=1/2( [ 0 Ω̃_1(t) 0 0 0; Ω̃_1(t)2Δ Ω̃_2(t) 0 0; 0 Ω̃_2(t) 0 Ω̃_3(t) 0; 0 0 Ω̃_3(t)2Δ Ω̃_4(t); 0 0 0 Ω̃_4(t) 0; ]).§ RESULTSIn order to verify that the the present protocol does work in the five-level chainwise system, we are going to employ Eq. (<ref>) together with Eq. (<ref>) to numerically investigate the population evolution. The left column of Fig. <ref> shows the Rabi frequencies, we can find that the pulse sequence of the present protocol is different from previous the straddling STIRAP <cit.> and the alternating STIRAP scheme <cit.>, which are two kinds of the generalizations of STIRAP for multilevel systems with odd number of levels. The right column of Fig. <ref> shows the complete population transfer for several pulse trains of different number of pulse pairs. In all cases the population is transferred from state |g_1⟩ to |g_3⟩ in the end in a stepwise manner. As can be expected, the transient population of the intermediate state |g_2⟩ is damped as N (N=1, 2, 5, 10) increases: from 0.5 for a single pair of pulses to about 0.6% for 10 pulse pairs.For this case, the analytic solution to maximum population in state |g_2⟩ reads P_max=sin^2(π/4N). As N tends to approach to infinity, the maximum population of the middle state decreases as 1/N^2. This suppression occurs on resonance and it results from the destructive interference of the successive interaction steps, rather than from a large detuning. Given that in some cases the lifetimes of states |g_2⟩ are short <cit.> and in some others they are long <cit.>, the coincident pulse technique can be a good choice for coherent population transfer. Also, AE protocol ensures the decoupling of the excited states from the dynamics, we can directly move the population from state |g_1⟩ to |g_3⟩, |e_1⟩ and |e_2⟩ are only used to induce transitions but never significantly populated, as illustrated in Fig. <ref>. Thus the transfer process is insensitive to the properties of excited states, e.g., very short lifetime. This is very useful for depressing the effects of dissipation on the desired evolution of the system without relying on the dark state (<ref>). Note that for N≫1 the total pulse area is very large, which is the condition for adiabatic evolution. This is useful when it is not possible to achieve high enough pulse areas needed for adiabatic evolution.Fig. <ref> shows the equal population distribution between |g_1⟩ and |g_3⟩. Similarly, the transient population of the intermediate state |g_2⟩ is damped as N (N=1, 2, 5, 10) increases, and |e_1⟩ and |e_2⟩ are only used to induce transitions but never significantly populated. For this case, the analytic solution to maximum population in state |g_2⟩ reads P_max=sin^2(π/8N). As N tends to approach to infinity, the P_max≈(π/8N)^2.§ CONCLUSIONTo conclude, we have generalized the coincident pulse technique in three-state system to a five-state chainwise system.With a train of N (N≫1) pairs of coincident incident pulses, the present technique allows complete population transfer between initial state to final state, as well as the creation of arbitrary coherent superpositions of the initial and final states without significant population of the three intermediate states. The key of our protocol is reducing the M-type structure into a generalized Λ-type system with the simplest resonant coupling, the simplification is realized under the assumption of AE together with a requirement of the relation among the four incident pulses.Finally, we believe that this technique is an interesting alternative of the existing techniques for coherent control of five-state chainwise systems. The results are of potential interest in applications where high-fidelity multi-state quantum control is essential, e.g., quantum information, atom optics, formation of ultracold molecules, cavity QED, nuclear coherent population transfer, light transfer in waveguide arrays, etc.§ ACKNOWLEDGMENTSI would like to thank the anonymous referees for constructive comments that are helpful for improving the quality of the work. | http://arxiv.org/abs/2311.15686v1 | {
"authors": [
"Jiahui Zhang"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20231127102405",
"title": "Quantum state engineering in a five-state chainwise system by coincident pulse technique"
} |
Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland LIACS, Leiden University Leiden, Netherlands Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland LIACS, Leiden University Leiden, Netherlands Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland LIACS, Leiden University Leiden, NetherlandsThe primary objective of this paper is to conduct a comparative analysis between two Machine Learning approaches: Tensor Networks (TN) and Variational Quantum Classifiers (VQC). While both approaches share similarities in their representation of the Hilbert space using a logarithmic number of parameters, they diverge in the manifolds they cover. Thus, the aim is to evaluate and compare the expressibility and trainability of these approaches. By conducting this comparison, we can gain insights into potential areas where quantum advantage may be found.Our findings indicate that VQC exhibits advantages in terms of speed and accuracy when dealing with data, characterized by a small number of features. However, for high-dimensional data, TN surpasses VQC in overall classification accuracy. We believe that this disparity is primarily attributed to challenges encountered during the training of quantum circuits.We want to stress that in this article, we focus on only one particular task and do not conduct thorough averaging of the results.Consequently, we recommend considering the results of this article as a unique case without excessive generalization.Comparison between Tensor Networks and Variational Quantum Classifier F. Neukart January 14, 2024 =====================================================================§ INTRODUCTION A quantum computer encodes information in quantum states and utilizes different quantum mechanical properties, such as superposition, entanglement, and interference to execute calculations <cit.>. Quantum machine learning (QML) is machine learning implemented in a quantum computer or a quantum simulator. QML can be realized, for example, by parameterized quantum circuits (PQC). A specific flavor of PQC that we discuss in this paper is variational quantum classifiers (VQC). VQC models are PQC which are optimized using classical optimization algorithms to find the optimal value of the parameters <cit.>. Related to QML techniques are quantum-inspired algorithms such as tensor network (TN) models <cit.>. Typically, the way that VQC and TN operate, are similar. Their main difference is the way they manipulate the weights of the model (see Sec. <ref>). TN utilizes the TT-decomposition to manipulate efficiently the high-dimensional tensors that represent the weights of the model and the features of the data. The parameters that consist the weights tensor of TN are optimized using the Riemannian optimization algorithm <cit.>.In this work, we apply VQC and TN to publicly available datasets to compare the performance of models. We are following up on research where the performance of the TN model was compared to that of standard gradient descent (SGD) in classification tasks <cit.>. There, the superiority of the TN model over SGD was shown for different datasets.Among the many applications of the models used in this paper are applications of QML <cit.> and quantum-inspired tensor networks <cit.> in high-energy physics. Moreover, applications in image classification <cit.> can be used in autonomous systems of self-driving cars and unmanned aerial vehicles <cit.>. The need for powerful computations is evident in these applications since they require decision-making in real-time, along with fast adaptation to the environment. Those applications strongly indicate that QML produces impressive results in comparison to classical models.In general, QML is a promising candidate for solving demanding tasks in various fields. For example, in drug discovery, it is proven that QML techniques, such as generative adversarial networks (GAN) and convolutional neural networks (CNN) are superior to their classical analog <cit.>.The main goal of this article is to give a more solid understanding of different ML models. Specifically, we investigate the respective strengths and weaknesses of a variational quantum classifier and a tensor network model.This article is structured as follows. In Sec. <ref> we provide a short introduction to all necessary preliminaries to grasp the essence of this research. The preliminaries include a brief introduction to machine learning, in Sec. <ref>, and tensor networks, inSec. <ref>. There, important techniques will be introduced, namely principal component analysis (PCA) <cit.>, Variational circuits <cit.>, Matrix Product States (MPS) <cit.>, Riemannian Optimization <cit.>, tensor train decomposition (TT-decomposition) <cit.>, for the implementation of the models. Additionally, in Sec. <ref>, we explain the architectures of the models, how we used the aforementioned techniques in the models, and how we implemented the VQC (in Sec. <ref>) and TN (in Sec. <ref>) models in general. Furthermore, in Sec. <ref> we present the experimental results and discuss the performance of both models under a common denominator. Finally, in Sec. <ref>, we discuss the trade-offs between the models on binary classification tasks. Moreover, in the same section, we refer to potential future work and also strengthen our results by verifying the experiments.§ PRELIMINARIES §.§ Quantum Machine learningQML can be characterized as the combination of classical ML with quantum mechanics to some extent. The quantum mechanical element can be inserted through different means. Some examples are the implementation of quantum data as qubits, the use of a parameterized quantum circuit (PQC) <cit.>, for the training of the model, or even quantum algorithms such as Quantum Approximate Optimization Algorithm (QAOA) <cit.>, Variational Quantum Eigensolver (VQE) <cit.> or Quantum Support Vector Machines (QSVM) <cit.>, for the further improvement of the model training and optimization <cit.>.A specific application of PQC is the variational quantum classifier (VQC) <cit.>. The VQC is composed of two different parts, the encoder part, and the variational part <cit.>. In the encoder, a quantum circuit is utilized to encode the features of samples into qubits. It is obvious that there are many different encoding techniques, such as angle encoding, amplitude encoding, wave-function encoding, and others <cit.>. The choice of encoding is closely related to different kernel methods. These methods are used to project the data into a higher-dimensional feature space, where the same problem is typically easier to solve. So, the proper choice of feature space is paramount for the solution of the problem.For example, non-linear feature maps <cit.> are capable to project data into a feature space where their relative distance is much different. In that way, it is possible that the samples can be easier distinguished from one another, and thus a binary classification among them can be achieved accurately. The inner product of two points (samples) in the feature space characterizes the kernel similarity function <cit.>.The second component of a VQC is the variational or parameterized part, which is a quantum circuit of a given ansatz. It generally consists of entangling layers and rotation gates on the qubits, with free, tunable parameters. The main hyper-parameter of this model is the number of variational layers chosen for the variational architecture. In order to give some depth to the variational circuit, which is crucial for its performance, we have to repeat the structure of CNOT gates and rotations multiple times. Finally, measurements on one or more qubits are made, so the model can make predictions after observing the output states.§.§ Tensor networks A tensor network can be described as the diagrammatic representation of a collection of tensors <cit.>. Tensor networks are usually used in many-body quantum systems <cit.>, and one of the most well-studied families of tensor networks are matrix product states or MPS <cit.>. An MPS or a tensor-train (TT) is a 1-dimensional state of a tensor network, where tensors are connected through one index called bond index (the dimension of which is called a virtual dimension or tensor train-rank <cit.>) and have another index called visible index, sticking out of the tensor (the dimension of which is called the physical dimension <cit.>) (Fig. <ref>). In general, the TT-rank r can vary from bond index to bond index and is regarded as a hyper-parameter of the model.After applying the TT-decomposition <cit.> method on a tensor 𝒜 (Fig. <ref>), the resulting network has the structure of an open boundary condition MPS. TT-decomposition is a way of manipulating multi-dimensional tensors without suffering from the curse of dimensionality <cit.>. In the heart of TT-decomposition lies the singular value decomposition (SVD) <cit.>, which is applied d-1 times on the desired d-dimensional tensor 𝒜, in order to get the d tensors (TT-cores), which can efficiently represent 𝒜. The general idea of TT-decomposition is to approximate all entries of the tensor 𝒜 by the product of d matrices G_k(i_k) which belong on G_k TT-cores of dimension r_k × r_k-1 (with r_0 = r_d = 1 since we want the product to return a scalar value), within an error ϵ. The indices i_k, with k ∈{1,2 … ,d} and i_k ∈{1,2, …, n_k}, represent the dimension which enumerates over the k-th index of tensor 𝒜, where n_k is the dimension of the k-th index of 𝒜. More specifically, we have A_i_1i_2… i_d = ∏_k=1^d G_k(i_k) In order to train a tensor ℬ to approximate the tensor 𝒜 through TT-decomposition, we need a proper optimizer. A suitable optimizer for this task is the Riemannian optimization algorithm <cit.>.Riemannian geometry is a branch of differential geometry that includes and describes the Riemannian manifolds. Manifolds are topological spaces that locally resemble Euclidean spaces <cit.>. A Riemannian manifold ℳ is a smooth Hausdorff and second countable manifold (by smooth we mean it is C^∞, infinitely times differentiable), equipped with a positive-definite smoothly varying inner product g metric, which can be used to determine an inner product on each point p of the tangent space T_pℳ of ℳ <cit.>.In a given d-dimensional tensor 𝒜, we can apply the TT-decomposition with a fixed rank r_i = r, i ∈{1,2,…,d-1 } and of course r_0 = r_d = 1. All such tensors like 𝒜, belong to a Riemannian manifold ℳ_r = {𝒜∈ℝ^n_1 × n_2 ×…× n_d: TT-rank(𝒜) = r}Consider tensors 𝒳 and 𝒲, where 𝒳 is of rank 1 and consists of all the features that describe the data. 𝒲 can be considered as a d-dimensional tensor, which contains all tunable weights of the model. As we show, the dimensions of 𝒲 can be n_1 = n_2 = … = n_d = 2 and thus its entries are in ℝ^2× 2 ×…× 2. That is because the interactions of the features of the data can be multiplied with the weights according to the following equation: ŷ(𝐱) = ∑_i_1 = 0^1 ∑_i_2=0^1 …∑_i_d=0^1 𝒲_i_1i_2 … i_d∏_j=1^d x_j^i_j Here ŷ(𝐱) represents the prediction of the model <cit.>. In other words, that prediction is the product of the relative weight with the features. When an entry of the weight tensor is multiplied with some features, then the relative indices of the weight tensor become 1, and the remaining become 0. For a two-dimensional exampleŷ(𝐱) = 𝒲_00 + 𝒲_10x_1 + 𝒲_01x_2 + 𝒲_11x_1 x_2we see that for indices i_1 = i_2 = 0 we get x_1^0 = x_2^0 = 1, for i_1 = 1, i_2 = 0, we get x_1^1 = x_1, x_2^0 = 1, and so on.Riemannian optimization attempts to minimize the following loss function L(𝒲) = ∑_s=1^N MSE(ŷ(𝐱^(s)),y^(s)) + λ/2 ||𝒲||_F where the term λ/2||𝒲||_F is the L_2 regularization term, with λ being the regularization parameter and ||𝒲||_F the Frobenius norm of the weight tensor <cit.>. The MSE(ŷ(𝐱^(s)),y^(s)):ℝ^2 →ℝ is the mean squared error or the squared loss of the predictions ŷ(𝐱^(s)) towards true values y^(s), and the s refers to the current sample examined, with the total number of samples in the data being N.Applying the Riemannian optimization algorithm on 𝒲 we can fine-tune its entries in such a way that the model can make accurate predictions. The steps of the Riemannian optimization algorithm we follow are the calculation of the gradient of the loss, with respect to 𝒲. Then, project it to the tangent space of ℳ_r. The projection at point 𝒲 which is T_𝒲ℳ_r. We define the projection as𝒫 = P_T_𝒲ℳ_r( ∂ L/∂𝒲) Following the direction of the projection 𝒫 with a small learning step α we get out of ℳ_r and onto the tangent space T_𝒲ℳ_r. As a consequence, there is an increase in the TT-rank. To return to ℳ_r, we need to reduce (rounding <cit.>) the TT-rank back to r. This reduction can be achieved by retracting the projection from point 𝒲, by a small learning step α. In total, we take a step 𝒲-α𝒫 in order to return back to the manifold.By recursively applying those steps, one can minimize the loss, and as a result, tune 𝒲 to minimize the L(𝒲).§ IMPLEMENTATIONWe compare the performance on a binary classification task between a Variational Quantum Classifier (VQC) and a Tensor Network (TN) with Riemannian optimization. We applied both models on the UCI car dataset of 2013 <cit.>. The dataset has 1728 samples with six categorical features each. Converting the categorical features to binary with one-hot encoding we end up with 21 binary features in total. For the experiments, we used a random splitting of the data, with a splitting ratio of 80% between training and validation sets. Additionally, for the data pre-processing we used PCA, so we were able to run experiments with 2,5,10 and 16 principal components of the data, as well as all 21 features. The dataset originally was a multi-class classification problem, with classes (unacc, acc, good, vgood), which refers to the acceptability of each car in the dataset. In order to convert it to a binary classification problem, we merged all three classes (acc, good, vgood) into one class (namely "acc") which consists 29% of the data, we later represented this class with +1, and the other class ("unacc") with -1 which consists the 71% of the data. The dataset can be characterized as unbalanced and thus the choice of accuracy as a metric might be incorrect. However, after extended experimentation, both accuracy and F1-score, which is an excellent metric for imbalanced datasets, reported almost the same results. The features of the dataset consist of the buying price, price of maintenance, number of doors, number of people to carry, size of luggage boot, and estimated safety of the car. We initialize both models with random weights and used the validation accuracy as a comparison metric between them. §.§ Variational Quantum Classifier model For the implementation of the encoding part of VQC, we used the cosine/sine encoding <cit.>. For this encoding, we start from an all |0⟩ state and we apply R_y single qubit rotations on every qubit. After the encoding part, each qubit would be in the state |0⟩→ R_y|0⟩ = cos(π/2x_k)|0⟩ + sin(π/2x_k)|1⟩ where k ∈{1,2,…, N} and N is the total number of features/qubits used for each sample. In that way, we introduce and pass information from the data to the quantum circuit. After the encoding part, we have to establish a variational architecture. We chose to follow the Noisy Intermediate Scale Quantum (NISQ) friendly architecture used in <cit.>. The main hyper-parameter here is the number of variational layers used. Each layer consists of CNOT gates with control on all odd-numbered qubits, then a layer of R_y rotations applied on each individual qubit, CNOT gates with control on every even-numbered qubit, and one more R_y layer on every qubit. In total, every layer needs to train 2N parameters (plus N for the 0-th layer, which consists of only an R_y rotation layer on every qubit). Thus, N(2L+1) parameters need to be tuned, where L is the number of layers used.In Fig. <ref> an example of the architecture of the VQC model is illustrated. This example refers to a five-qubit model with two variational layers. At the end of the circuit, we utilize only the first measurement for the prediction. If the expectation value of the measurement falls under the predefined threshold, we classify the sample to the -1 ("unacc") class, otherwise, we classify it to the +1 ("acc") class. The threshold we use throughout all the experiments is set to 0.5.The MSE has been used as a loss function for VQC and for the circuit implementation Pennylane (version = 0.30.0) library has been used <cit.>. For the optimizer, we tried Standard Gradient Descent (SGD), Nesterov momentum optimizer <cit.>, and Adaptive Momentum optimizer (Adam) <cit.>. Adam returned the best results, so we chose that for all the experiments with VQC. During training, a batch size of 32 samples was used, along with a decaying learning rate with an initial value of 0.1 and a decay rate of 0.95. §.§ Tensor Network model For the Tensor Network model (TN), we followed the implementation used in <cit.>. In this model, the method of TT-decomposition has been applied in order to manipulate the weights tensor in a more efficient way, especially when working with high-dimensional data. Polynomial encoding as defined in (<ref>) has been used as data encoding. 𝒳_i_1i_2… i_d = ∏_k=1^d x_k^i_k where, 𝒳_i_1i_2… i_d represents the tensor which includes the features of the samples, and x_k^i_k is the k-th feature of the sample. For the values i_j, the indices j ∈{1,2,…, d} and i_j ∈{0,1}. Thus, for the feature of a sample, it will be x_j^i_j = 1,ifi_j = 0 x_j, ifi_j = 1 This model is trained with Riemannian optimization and uses the logistic loss as a loss function. Its main hyper-parameter is the TT-rank used for the TT-decomposition. The total number of parameters that need to be trained in this model is 2Nr^2, where N is the number of features used, and r is the TT-rank. We notice that the number of parameters in the TN model scales much faster in comparison with VQC since the TN parameters scale quadratically with the rank. For the implementation of the TN model, the ttpy library, which is a python implementation of the TT-Toolbox library <cit.>, has been used, in coordination with other libraries with mathematical tools, such as numpy <cit.> and scikit-learn <cit.>.As mentioned above, the predictor of this model is based on the simple linear product between the features tensor 𝒳 and the weights tensor 𝒲. So in total, we can rephrase equation (<ref>) as ŷ(x) = ⟨𝒳,𝒲⟩ There is no need for a separate bias term since this is integrated in 𝒲 as 𝒲_00… 0.§ RESULTS We compare the VQC and TN models with different numbers of qubits and principal components. In order to decide which number of principal components we will run experiments for, we compose a scree plot analysis <cit.> in Fig. <ref>.Each point in Fig. <ref> represents the percentage that is going to be added to the total expressibility of the dataset. As a result, it is logical to make experiments with 2 qubits, which seem to give low expressibility, 5 qubits which are again in low expressibility but still have to add to the overall eigenvalue size, 10 qubits which are in the mid-range of the eigenvalue size contribution, and 16 qubits which in principle add almost 0% to a model with 15 principle components. In Fig. <ref>, we show the results of a binary classification experiment on the UCI car dataset using a VQC in comparison to a TN model as described above. In both experiments, we used different numbers of principal components from the car dataset and recorded their accuracy with the number of variational layers/TT-ranks and the number of principal components used in training. As a result, it seems that for the tested dataset and model architectures, the TN model increases its performance significantly as the number of principal components increases. Additionally, VQC also increases its performance and achieves better classification, however with more principal components, the training time, and thus the effort for parameter optimization in the variational layer increases significantly and often leads to local minima or vanishing gradients.In Fig. <ref>, we show the respective plot of the validation accuracy with the number of trainable parameters, for 5 qubits. It is evident that VQC performs better than TN. As the trainable parameters increase, notice that the performance of TN converges, in contrast with VQC which oscillates around 90% but always above TN.In Fig. <ref> we compare the 10 qubits TN and VQC. Notice that TN surpasses VQC considering it achieves better accuracy than VQC for ≥ 150 trainable parameters. TN performance in 10 qubits seems comparable with the 5 qubits VQC. However, the experiments with more than 10 qubits, indicate that TN is the dominant model for high-dimensional binary classification tasks. This observation is based on the poor trainability of VQC for a large number of qubits.For 16 qubits in Fig. <ref>, TN is clearly better than VQC, regarding the validation accuracy. TN manages to achieve perfect classification and VQC shows a more stable behavior in comparison with the 10 qubit experiment. Under 100 parameters, VQC surpasses TN, which is not the case for more than 100 parameters.§ DISCUSSIONWe are not able to claim that one model is better than the other. Accuracy depends on the available computational power and time. Also, the dimensionality of the data plays a significant role when we want to choose between the models. If the only consideration is the overall accuracy, then TN models might be a better choice, especially, if the problem involves high-dimensional data, to give a perspective, data with more than 10-15 features. TT-decomposition enables us to manipulate the high-dimensional weight tensor much faster than other methods. Riemannian optimization seems to be an excellent fit as an optimizer to TN since it outperforms SGD <cit.> and assists TN in achieving better classification. A drawback of TN is that requires significantly more trainable parameters compared to VQC.On the other hand, VQC equipped with a dimensionality reduction technique (such as PCA), or used on a problem with low-dimensional data, might be a better choice than TN. For example, 5 qubits VQC can reach ∼ 91% validation accuracy on the UCI car dataset. We observed that VQC achieved maximum accuracy after a small number of epochs. This is especially promising to reduce the training time of VQCs with a larger number of qubits. There are indications that VQC can achieve even better accuracy with more qubits. As shown in Fig. <ref> with 16 qubits, it is capable of outperforming the 5 qubits for 6-8layers. So with hyper-parameter tuning or more training, VQC might also be suitable for high-dimensional data. However, because training time for the 16 qubits VQC requires much computation time, we did not investigate this possibility as much as it seems to deserve.Overall, by examining the plots, we notice that VQC is consistently the model that achieves the best validation accuracy for a small number of variational layers or low TT-rank. This result strengthens our previous finding about VQC, being the preferable method when we need to train a model within a limited time frame. unsrt | http://arxiv.org/abs/2311.15663v1 | {
"authors": [
"Georgios Laskaris",
"Artem A. Melnikov",
"Michael R. Perelshtein",
"Reuben Brasher",
"Thomas Baeck",
"Florian Neukart"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20231127094905",
"title": "Comparison between Tensor Networks and Variational Quantum Classifier"
} |
These authors contributed equally to this work. State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, ChinaThese authors contributed equally to this work. School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, ChinaThese authors contributed equally to this work. State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, ChinaState Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, ChinaSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, ChinaState Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, ChinaState Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, ChinaSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China ShanghaiTech Laboratory for Topological Physics, Shanghai 201210, [email protected] State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200438, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China The recently discovered superconductivity with critical temperature T_c up to 80 K in the Ruddlesden-Popper phasesunder pressure has drawn great attention. Here we report the positive muon spin relaxation () study of polycrystallineunder ambient pressure. The zero-field μ^+SR experiments reveal the existence of static long range magnetic order in , and the the muon spin depolarization spectra are consistent with the spin density wave internal field distribution. The weak transverse field μ^+SR measurements suggest the bulk magnetic transition near T_N=148 K.This is the first research which discovers the existence of the spin density wave inmicroscopically. Evidence of spin density waves inLei Shu January 14, 2024 ==================================Introduction.—The recent observation of the sign of superconductivity with T_c≈ 80 K in La_3Ni_2O_7 single crystals under pressure has attracted significant attention <cit.>. Subsequent high-pressure measurements on single crystal <cit.> and powder samples <cit.> have achieved zero-resistance through more hydrostatic pressure conditions, confirming the discovery of a brand new superconductor, especially with critical temperature higher than the boiling point of liquid nitrogen, after the remote discovery of copper oxides for the first time <cit.>.therefore provides an excitingly new opportunity to investigate pairing mechanism of high-temperature superconductivity. A lot of theoretical works quickly followed <cit.>, some of which proposed s±-wave superconductivity under high pressure <cit.>. On the other hand, due to the necessity of high pressure to induce superconductivity, experimental progress on pairing mechanism is quite limited <cit.>. Recent physical property measurements on single crystal La_3Ni_2O_7 revealed a density-wave like transition near 153 K at ambient pressure <cit.>, which had been observed but not confirmed due to the lack of single crystalline sample <cit.>. The onset temperature of the anomaly will be suppressed by pressure, followed by the emergence of superconductivity under higher pressure <cit.>. However, the nature of such density-wave-like transition is not fully understood. Charge ordering in NiO_2 planes induced by oxygen orderings <cit.> or charge-density wave (CDW) instabilities induced by one-dimensional Fermi surface nesting <cit.> was supposed to account for the transition-like anomaly, while there is no direct evidence for the charge ordering in . Though the LDA+U method lead to an antiferromagnetic coupling in NiO_2 plane <cit.>, no long-range magnetic order was reported by NMR and neutron experiments <cit.>. A rich interplay between magnetic order and superconductivity is the key character of unconventional superconductors, where superconductivity often appears near the border of magnetic order, e.g., copper oxides <cit.>, iron pnictides <cit.> and heavy fermion superconductors <cit.>. The absence of long-range magnetic order may raise the reconsideration on the importance of magnetic correlation for pairing mechanism <cit.>. Till now, there is no conclusive result on the magnetic ground state of . Therefore, it is very important to clarify the magnetic properties ofat ambient pressure.Here, we report themeasurements on polycrystallineto clarify the magnetic ground state ofat ambient pressure. Positive muon spin relaxation/rotation () is an unmatched technique for detecting magnetism or spin dynamics <cit.>. 100% polarized muons are implanted into the sample and work as sensitive local spin probe. Zero-field(ZF)- and longitudinal(LF)- measurements confirm the static magnetic ground state in . Weak transverse field(wTF)- experiment confirms the bulk transition of the magnetic order with T_N=148 K. Experimental Details.—Polycrystalline samples of were synthesized by the solid-state reaction <cit.>. Powder x-ray diffraction (XRD) patterns were obtained by using a Bruker D8 advanced x-ray diffraction spectrometer (λ = 1.5418 Å) at room temperature. The XRD Rietveld refinement was conducted withsoftware <cit.>. The magnetization of was measured in a superconducting quantum interference device magnetometer (Quantum Design magnetic property measurement system). The temperature-dependent susceptibility between 2 and 300 K was measured under a magnetic field of 0.4 T in both zero-field cooled (ZFC) and field-cooled (FC) procedures. Temperature dependence of resistivity ρ(T) was measured with standard four-probe method on a physical property measurement system (PPMS). Powder samples were pressed and cut into rectangle. Four annealed silver wires were glued on the surface of sample with conductive silver adhesives. The resistivities were measured between 2 K and 300 K.The ZF, LF and wTF- experiments were carried out at M15 and M20 spectrometers at TRIUMF, Vancouver. About 350 mg of the powder sample was pressed into rounds with the diameter about 1.2 mm. The sample of M15 was held on a high-purity silver plate with diluted GE varnish and loaded in a top-loading dilution refrigerator with a base temperature of 35 mK. The sample measured at M20 was mounted on the hollow square copper frame with thin silver tape and was free from additional background signal. Themeasurement at M15 was carried down to 35 mK in zero field and in a longitudinal field at 0.1 T. Themeasurement at M20 was carriedout between 3.5 K and 290 K in zero field and in a transverse field at 30 Oe. The temperature was carefully controlled to ensure that the standard deviation of the temperature during each measurement was less than 0.1 K. The data were analyzed with thesoftware package <cit.>.Physical Properties.—Fig.<ref>(a) shows the X-ray diffraction pattern of polycrystalline . The Rietveld refinement of the powder XRD pattern ofis in line with the recently reported polycrystalline samples which exhibited superconductivity at high pressure <cit.>. No impurity phase was detected by the X-ray diffraction, proving the high purity of our sample. The temperature dependence of resistivity ρ(T) of polycrystallineis plotted in Fig.<ref>(b). ρ(T) shows a negative temperature coefficient and a relatively large value at low temperature. It should be noted that the transport properties ofis quite sensitive to the actual oxygen content and the resistivity is increased with an increase in oxygen deficiency <cit.>. The insulating behavior of ρ(T) indicates that the oxygen content of our sample is close to 6.9 <cit.>. No anomaly was detected down to 2 K.Fig.<ref>(c) displays the temperature dependence of dc magnetic susceptibility χ(T), which was measured under the magnetic field of μ_0H=0.4 T with both zero-field cooling and field cooling setup. There is no sign of any magnetic phase transition or spin freezing behavior down to 2 K, which is consistent with the previous results <cit.>. No anomaly was detected down to 2 K. Zero-field muon spin relaxation.—To investigate the magnetic ground state ofat ambient pressure, we first measured the zero fieldspectra. ZF- is very sensitive to any local magnetic order or magnetic fluctuations. Three representative muon relaxation spectra are shown in Fig.<ref>(a). At high temperature, the muon spin relaxation can be described with a single Gaussian relaxation function: A(t)=A_0(t)e^-σ ^2t^2/2, where A_0 is the muon initial asymmetry. σ = 0.11 μs^-1 is temperature independent over 185 K, which is typical value of the muon spin relaxation due to static nuclear magnetic dipole moments. When cooling down to 148 K, the muon relaxation behavior changes from Gaussian-like to exponential-like, and an oscillating term appeared in the early time window (before 0.4 μs, shown in Fig.<ref>(b)). The frequency and the amplitude of the oscillation component increase with cooling and reach saturation at low temperature.The ZF- spectrum can be best-fitted with the sum of one oscillation component and two exponentially decaying components:A_0P_ZF(t)=A_ZFe^-λ_ZF tJ_0cos(ω_μt+ϕ_ZF)+ A_faste^-λ_fast+A_slowe^-λ_slow.Here P_ZF(t) is the zero field muon relaxation function. A_ZF, A_fast and A_slow are the asymmetries related to three components, respectively. The first oscillation term describes the muons processing along the static local fields. J_0 is the zeroth-order Bessel function. Simple cosine oscillation function was also tried, but it failed to describe the fast relaxation in the early time window (before 0.05 μs). ω_μ=γ_μ× H_int is the muon precession frequency, and γ_μ=851.616 MHz/T is the muon gyromagnetic ratio <cit.>. ϕ_ZF is the initial phase of implanted muons. The second non-oscillating fast relaxation component describes the muon spin relaxation due to the internal field distribution, which will be discussed later. The third slowly decaying one is an exponential “tail" of the longitudinal components of local field <cit.>. λ_ZF, λ_fast and λ_slow are the relaxation rates of three components. A_0 was fixed to 0.227 during the fitting, which was obtained from the ZF - and wTF - fitting above the magnetic transition temperature. The fitting parameters are concluded in Fig.<ref>(c) and Fig.<ref>(a). The magnetic order parameter H_int reaches a saturation of ∼ 140 mT at low temperature. H_int slightly deviates from the usual √(1-(T/T_N)^2) temperature dependence and shows a stronger initial increase below the T_N. λ_ZF displays a critical behavior around 150 K, which is consistent with a magnetic phase transition. λ_fast increases monotonically as the temperature decreases. λ_slow is almost temperature-independent and close to 0 (smaller than 0.01 μ s^-1, not shown). The longitudinal fieldwas measured at 5 K to investigate the dynamic nature of the fast relaxation component. The muon relaxation spectrum was flattened with a longitudinal field of 0.1 T, which is comparable to the magnitude of internal field in the magnetic ordered state, indicating that the internal field is static or quasi-static compared to the time window oftechnique.Weak-transverse-field muon spin relaxation.—In order to investigate the temperature dependence of the magnetic volume fraction, we measure thespectra under weak transverse field (μ_0H=30 Oe). The applied magnetic field is much less than the spontaneous internal field (μ_0H_int≈1400 Oe) so it will not affect the distribution of internal field. The wTF- technique is quite sensitive to any microscopic magnetic order. The wTF- spectra under several temperatures are plotted in Fig.<ref>. At high temperature, the sample is in the paramagnetic state, and the muon implanted in the sample will do Larmor precession along the external field. Since there is no additional background signal, the fraction of the oscillating component stands for the volume fraction of the paramagnetic phase of the sample. With cooling down, the amplitude of the oscillation diminishes and an exponential-like decaying appears. The oscillatory component totally disappears at the base temperature, indicating the bulk magnetic order in polycrystalline .The wTF-μSR asymmetry data can be fitted with the combination of a cosine oscillating part and two exponential decaying parts:A_0P_TF(t)=A_TFe^-λ_TF t cos(ω_μt+ϕ_TF)+ A_expe^-λ_exp+A_taile^-λ_tail.The first oscillation component describes the muon spin precession along the external field in the paramagnetic phase, and the other two non-oscillating components are the muons influenced by the internal field. A_tail is the “1/3" tail, which is identical to A_slow in Eq.(<ref>).A_0= 0.227 is the initial asymmetry and is found to be temperature-independent above magnetic transition during the fitting. The A_0 subtracted from both ZF- and wTF- at high temperature are the same, and is fixed during the fitting. A_TF, A_exp and A_tail are the asymmetry of three components, respectively.ω_μ is the muon precession frequency along the external field, and ϕ_TF is the initial phase of the implanted muons. Temperature dependence of the normalized asymmetry is plotted in the Fig.<ref>(b). The paramagnetic volume fraction reached 50% at 148 K and approaches zero below 25 K, indicating a bulk magnetic transition at phase-transition temperature T_N=148 K, which is consistent with the temperature dependence of the magnetic order parameter acquired from the ZF- fitting (Fig.<ref>(c))). The normalized asymmetry of two non-oscillating components display the same temperature dependence.This means these two components may reflect the muon influenced by the internal field at different muon stopping sites. The wTF relaxation rates are plotted in Fig.<ref>(b), λ_TF and λ_fast increase monotonously with decreasing temperature, indicating the development of the internal field. Discussion.—The present ZF- measurement, combined with LF and wTF- measurements, confirm the existence of static SDW transition with nearly 100% volume fraction in polycrystalline . It is interesting to note that such SDW order was not detected by dc susceptibility in all powder samples <cit.>. The absence of magnetic transition in magnetization means this SDW order shows a very weak bulk signal, which may possibly due to an antiferromagnetic coupling between the adjacent NiO_2 planes. , as an unparalleled technique on local field detection, manages to unveil this weak magnetism even in polycrystalline samples.The ZF- spectra in systems with magnetic order are always complicated. The zeroth-Bessel function was widely used to describe the internal field distribution of incommensurate spin density wave(IC-SDW) order <cit.>. However, it should be noted that J_0(ω t) only accounts for the wide distribution of internal field. A complex but not incommensurate magnetic structure <cit.> or a combined contribution of multiple muon precession frequencies <cit.>, considering the large number of possible muon stopping sites in transition metal oxides <cit.>, can also be fitted with J_0(ω t). The second non-oscillating component may come from: 1) dynamic local magnetic fluctuation; 2) the muon sites which are symmetric with the magnetic lattice in antiferromagnet since this results in the cancellation of local fields. LF- measurement has excluded the dynamic origin. For the second possibility, the detail information of muon stopping sites are indispensable. Unfortunately, up to now, there is no reported calculations of muon stopping site locations inusing density functional theory (DFT) <cit.>. Further research in detail are needed to reveal the specific magnetic structure ofat ambient pressure. The wTF- measurement confirms the bulk nature of SDW inunder ambient pressure since the volume fraction of magnetic ordered phase is near 100 %.Since previous NMR and neutron-diffraction experiments showed an absence of long-range magnetic order, the static order revealed byis on the “short" side of the viewpoint from NMR and neutron-diffraction. Conclusions.—In conclusion, the ZF and wTF- experiments reveal the existence of static SDW under T_N=148 K inat ambient pressure. Although the ZF- spectra seem to be consistent with the IC-SDW order, further researches are needed to confirm the specific magnetic structure in . The presentresults demonstrate the unique power ofto provide intrinsic information on the magnetism even in zero applied field. These findings will promote a comprehensive understanding of the superconducting mechanism ofat high pressure. Acknowledgments–We are grateful to G. D. Morris, B. Hitti, and D. Arsenau of the TRIUMF CMMS for assistance during the experiments. This research was funded by the National Key Research and Development Program of China, No. 2022YFA1402203, the National Natural Science Foundations of China, No. 12174065, and the Shanghai Municipal Science and Technology Major Project Grant, No. 2019SHZDZX01. Y. F. Guo acknowledges the National Key R&D Program of China (Grant No. 2023YFA1406100) and the Double First-Class Initiative Fund of ShanghaiTech University.53 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Sun et al.(2023)Sun, Huo, Hu, Li, Liu, Han, Tang, Mao, Yang, Wang, Cheng, Yao, Zhang, and Wang]Sun2023 author author H. Sun, author M. Huo, author X. Hu, author J. Li, author Z. Liu, author Y. Han, author L. Tang, author Z. Mao, author P. Yang, author B. Wang, author J. Cheng, author D.-X. Yao, author G.-M. Zhang, andauthor M. 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"authors": [
"Kaiwen Chen",
"Xiangqi Liu",
"Jiachen Jiao",
"Myyuan Zou",
"Yixuan Luo",
"Qiong Wu",
"Ningyuan Zhang",
"Yanfeng Guo",
"Lei Shu"
],
"categories": [
"cond-mat.str-el",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.str-el",
"published": "20231127110123",
"title": "Evidence of spin density waves in La$_3$Ni$_2$O$_{7-δ}$"
} |
[footnoteinfo]The authors would like to thank the Spanish national project Project L-BEST under Grant PID2020-115905RB-C21 funded by MCIN/ AEI /10.13039/501100011033. This work was also supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CCCDI - UEFISCDI, project number PN-III-P2-2.1-PED-2021-1626, within PNCDI III. This work has been submitted to IFAC for possible publication. First]Luis Romero-BenUB,ILDS]Paul IroftiThird]Florin Stoican First,Fourth]Vicenç Puig[First]Institut de Robòtica i Informàtica Industrial, CSIC-UPC, Llorens i Artigas 4-6, 08028, Barcelona, Spain(e-mail: [email protected]). [UB]LOS-CS-FMI,University of Bucharest, Romania(e-mail: [email protected]) [ILDS]Institute for Logic and Data Science, Bucharest, Romania [Third]Department of Automation Control and Systems Engineering, Politehnica University of Bucharest, Romania,(e-mail: [email protected]) [Fourth]Supervision, Safety and Automatic Control Research Center (CS2AC) - UPC, Campus de Terrassa, Terrassa, 08222, Barcelona, Spain, (e-mail: [email protected]) In this paper, we present a nodal hydraulic head estimation methodology for water distribution networks (WDN) based on an Unscented Kalman Filter (UKF) scheme with application to leak localization. The UKF refines an initial estimation of the hydraulic state by considering the prediction model, as well as available pressure and demand measurements. To this end, it provides customized prediction and data assimilation steps. Additionally, the method is enhancedby dynamically updating the prediction function weight matrices. Performance testing on the Modena benchmark under realistic conditions demonstrates the method's effectiveness in enhancing state estimation and data-driven leak localization.Fault isolation, water distribution system, data fusion, state estimation, Unscented Kalman Filter§ INTRODUCTION The appearance of leaks in water distribution networks (WDN) results in a significant water loss of approximately 126 billion cubic meters per year worldwide (expressed as non-revenue water), as indicated by <cit.>. For decades, water utilities have harnessed software-based techniques for leak detection and localization. Within the approaches that analyse the steady-state of the WDN, three main categories emerge: model-based, data-driven and mixed model-based/data-driven methods.Model-based methods exploit a hydraulic model to emulate the behaviour of the WDN, calibrating both network characteristics and nodal demands. The aim is to obtain simulated hydraulic data that can be compared with actual measurements from the real network <cit.>.Mixed model-based/data-driven methods reduce the dependence on the model through machine learning algorithms while maintaining the node-level accuracy <cit.>.Finally, data-driven approaches remove the necessity of a hydraulic model,using algorithms that exploit sensor data to provide leak location areas. Within data-driven schemes, a well-established family of methods relies on interpolating the complete hydraulic state from available measurements and topology, to then perform leak localization through the reconstructed schemes. State estimation has been successfully utilized in WDNs with techniques such as Kriging interpolation <cit.>, graph neural networks <cit.> and graph signal processing <cit.>.This article introduces the next step of the interpolation methodology presented in <cit.>, known as Graph-based State Interpolation (GSI), and its subsequent evolution introduced in <cit.>, known as Analytical Weighting GSI (AW-GSI).In particular, a nodal hydraulic head estimation methodology is proposed based on an Unscented Kalman Filter (UKF) scheme with application to leak localization. The proposed methodology uses the UKF scheme to improve the head estimation from an input reconstructed state vector, leading to various improvements compared to the previous GSI schemes:The available pressure and demand information is fused. Previous state estimation methods often rely on pressure data due to the reduced cost and ease of installation of pressure meters. The continuous upgrade of the urban infrastructures leads to the rise of "smart cities", characterized by advanced real-time metering capabilities <cit.>.Automated Metering Readers (AMR) play a crucial role by measuring real-time consumption. Thus, a state estimation strategy that neglects the processing of demand data risks the loss of critical information. The graph diffusion weights from the prediction function are dynamically updated. These weights are not only configured considering the WDN structure and the physics governing its behaviour, but they are also adapted to the current estimated state. This drastically improves the state estimation performance in comparison to previous methodologies.The structure of the paper is as follows: In Section <ref>, the methods that will be used to develop the proposed approach are presented. In Section <ref>, this approach is introduced. A case study is described in Section <ref> accompanied by simulations in Section <ref>. Finally, Section <ref> draws the main conclusions and suggests future research paths. § PRELIMINARIESLet us model the network topologyby a graph 𝒢=(𝒱,ℰ), where 𝒱 is the node set (reservoirs and junctions), and ℰ denotes the edge set (pipes). An arbitrary node is represented as 𝓋_i∈𝒱, whereas an arbitrary edge is denoted as ℯ_k = (𝓋_i,𝓋_j)∈ℰ. The latter represents the link between nodes 𝓋_i and 𝓋_j, with 𝓋_i as its source and 𝓋_j as its sink. The nodal hydraulic heads are selected as representatives of the network states, gathered in h.§.§ State interpolationAn initial attempt to estimate the hydraulic state of a WDN was proposed in <cit.> through GSI, which only requires hydraulic head data and structural information. This method estimates the complete network state by solving the optimization problemmin_h 1/2[h^TL_dh+αγ ^2] s.t. B̂h≤,γ > 0,Sh=h_s,where L_dis a Laplacian-based matrix generated from the WDN underlying graph, B̂ is an approximated incidence matrix of 𝒢, γ is a slack variable and S and h_s are the sensorization matrix and the head measurement vector respectively.Although the details are presented in <cit.>, let us highlight that L_d=(D-W)D^-2(D-W), where W is the weighted adjacency matrix of 𝒢, with w_ij = 1/ρ_k if ℯ_k = (𝓋_i,𝓋_j)∈ℰ and w_ij =0 otherwise, and ρ_k is the length of the pipe represented by edge ℯ_k. D is the degree matrix, a diagonal arrayobtained as d_ii = ∑_j=1^n w_ij. Briefly, GSI pursues the closest state vector to fulfill that h = D^-1Wh, where D^-1Wh diffuses the state considering the structure of 𝒢, while contemplating directionality and measurements-related constraints. Recently, a novel physical-based weighting process was designed to improve GSI, leading to a new interpolation method that is referred to as AW-GSI. This weighting process is based on the linearization of the Hazen-Williams equation <cit.>, and yields a new weighted adjacency matrix W^AW as follows:w^AW_ij(h̃_i,h̃_j) = σ_k^0.54[m_kj(h̃_i-h̃_j)]^-0.46, where σ_k=(μ_k^1.852δ_k^4.87)/(10.67 ρ_k) is the pipe conductivity (in S.I.) for ℯ_k, with μ_k and δ_k being the roughness and diameter of the pipe respectively. Note that W^AW is richer in both structural and hydraulic information in comparison to W, hence leading to an improvement in the accuracy of the state estimation. Besides, m_kj is the kj element of the incidence matrix M∈ℝ^|ℰ|× n, defined as: m_kj=1, h̃_i≥h̃_j(ℯ_k = (𝓋_i,𝓋_j)∈ℰ);-1, h̃_i< h̃_j(ℯ_k = (𝓋_j,𝓋_i)∈ℰ); 0, if 𝓋_i and 𝓋_j are not adjacent The characteristics of h̃ are introduced in Lemma 1 at <cit.>. Due to the ultimate goal of performing leak localization, we selected h̃_i=h^nom_i and h̃_j=h^nom_j, with h^nom being the leak-free reference, as most localization techniques operate over pressure residuals, i.e., difference of pressure between leak and leak-free scenarios.This selection of h̃ leads to the following quadratic programming problem min_Δ h 1/2[Δ h^TL_d^AWΔ h]s.t. SΔ h=Δ h_s. where Δ h is the residual vector to retrieve, Δ h_s is the residual vector of measurements, and L_d^AW is an analogue Laplacian-based matrix to the one used in GSI, but obtained through (<ref>). §.§ Unscented Kalman FilterThe success of the Kalman filter to accurately estimate the state of a linear system led to the development of extensions to handle non-linear functions. The most notorious example is the Extended Kalman Filter (EKF), which exploits multivariate Taylor series expansions to linearize the model around the current estimate. This method has been successfully used in the past to estimate consumption and detect bursts in WDN <cit.>.The Unscented Kalman Filter (UKF) was designed to address the limitations of EKF, mainly related to the linearization precision. In EKF, only one point is considered to approximate a new linear function from the non-linear one, i.e., the mean of the Gaussian distribution which we assume that represents the form of our data. In UKF, a set of points known as Sigma Points are selected and mapped into the target Gaussian after being passed through the non-linear function. A process called Unscented Transformation helps recovering the approximated Gaussian after the application of the non-linear function. In comparison with EKF, UKF does not require the computation of the Jacobian, and the approximations are more accurate in the case of non-Gaussian inputs <cit.>.The UKF algorithm is well-established and has been implemented in several software platforms. Thus, we focus here on the adaptation of UKF to improve an initial estimation of the complete network state, represented by the nodal hydraulic heads. For more details about the UKF standard operation,see <cit.>. § PROPOSED APPROACH Let us start by outlining the required input/output information, hyperparameters and the two key stages composing the UKF algorithm: prediction and data assimilation.We discuss now the inputs and output of the estimation process. First, h_0 is the initial guess for the UKF, corresponding to the state of the network, i.e., the complete set of nodal heads. This first estimation may be retrieved from interpolation processes like GSI or AW-GSI. Head measurements are stored in h_s from a set of n_s pressure sensors. Considering that the number of network nodes is n = |𝒱|, normally n >> n_s. The demand measurement vector c_a is constructed from the points where AMRs are installed. Again, if n_ca AMRs are used, normally n >> n_ca. Finally, the output h_UKF of the UKF operation is a state estimation, obtained by fusing the information from the initial guess, the prediction function and the assimilation of the pressure and demand data.Several configuration parameters must be settled before applying the UKF strategy. Parameter K corresponds to the total number of iterations of the UKF process. It must be configured through an analysis of the studied network (e.g. the selection of K=50 presented in Section <ref>) or a convergence criteria, e.g., a tolerance value of the estimation change during a defined period. Also,Q is a positive definite diagonal matrix denoting the covariance of the process noise. It accounts for the model approximations introduced by the prediction function. Finally, R is a positive definite diagonal matrix representing the covariance of the measurement noise, enabling us to express the level of confidence in the sensor data. §.§ PredictionThis process leverages a function that describes the state evolution from one time step to the next:h^[k+1] = f(h^[k]) = αh^[k] + (1-α)Φ^-1Ωh^[k] where h^[k] is the UKF state at iteration [k][Note that index k serves a dual purpose: indicating an arbitrary edge ℯ_k and the kth iteration. To prevent confusion, k only represents iteration number if it is encapsulated using brackets as [k].], Φ and Ω are respectively a degree matrix and a weighted adjacency matrix, andα = n_ca/n. In this way, (<ref>) yields a compromise solution between preserving or diffusing the current state, depending on the number of installed AMRs: if demand information is abundant, the state remains similar from one step to the next in terms of prediction, as most of the information is provided at the data assimilation step; else if demand information is scarce, the state information is diffused over the network by means of Φ and Ω. Besides, Ω can be defined by the user, with Φ obtained as ϕ_ii = ∑_j=1^nω_ij. Possible selections for Ω are the 0-1 adjacency matrix of 𝒢, the GSI adjacency matrix Wand the AW-GSI adjacency matrix W^AW (derived from (<ref>)).§.§ Data assimilationThis step uses a function that describes how the model states are related to sensor measurements. We have designed this function to account for the two available sources of measurements. First, the n_s pressure sensors provide actual heads from the network, i.e., states of the UKF process. Thus, y_h^[k] = Sh^[k]where y_h^[k] is the part ofthe measurement vector corresponding to the head data at iteration [k].Second, as n_ca demand measurements are available, we can utilize the relation among nodal demands and hydraulic heads to derive the required function.Starting with the Hazen-Williams equation, we have that m_kj(h_i-h_j) = τ_kq_k^1.852,where m_kj is analogue to (<ref>), τ_k = 1/σ_k, and q_k is the flow through the pipe represented by edge ℯ_k.This can be posedas: Mh = Tq^(1.852), whereq^(1.852)=[q_1^1.852 q_2^1.852...q_|ℰ|^1.852]^T and T∈ℝ^|ℰ|× |ℰ| is a diagonal matrix, whose kth diagonal value is τ_k.Besides, the relation between nodal demands and flows can be expressed as: c = -M^Tq with c∈ℝ^n denoting the vector of nodal demands. Manipulating(<ref>) and (<ref>) we obtainy_d^[k] = -M^T_a(T^-1Mh^[k])^(0.54)where y_d^[k] is the part ofthe measurement vector corresponding to the demand measurements at iteration [k], and M_a selects only the columns of M that correspond to the nodes with AMRs. Then, the complete measurement vector at iteration [k] is y^[k] = g(h^[k]) = [y_h^[k] y_d^[k] ]^T. If the WDN measurements are stored in y = [h_sc_a]^T, the measurement error for iteration [k], used by the data assimilation process to update the current state, would be: e^[k] = y - y^[k] = y - g(h^[k]) §.§ Dynamic-weighting prediction step In the past, the physics-based weight generation of AW-GSI led to an improvement in the state estimation performance.Therefore, these weighting mechanisms can be used to enhance the capabilities of the proposed UKF-based method, leading todynamic update of the diffusion matrices in (<ref>). Thus, this function is updated to: h^[k+1] = f(h^[k],Ω^[k],Φ^[k]) = αh^[k] + (1-α)(Φ^[k])^-1Ω^[k]h^[k] where static matrices Ω and Φ have been substituted by their dynamic versions, i.e., Ω^[k] and Φ^[k], corresponding to iteration [k]. Then, the computation of Ω^[k] would exploit the weighting generation of AW-GSI, via (<ref>): Ω^[k] =T^(-0.54)(M^[k]h^[k])^(-0.46),(k,K_u) = 0Ω^[k-1], where K_u denotes a user-defined iteration interval between consecutive weight updates (configurable analogously to K). A higher value for this parameter increases the number of iterations the UKF undergoes before reaching a final steady-state estimation, but contributes to greater stability before consecutive weight updates.The initial guess of the state h_0 and the weighted adjacency matrix Ω^[0] can be retrieved from AW-GSI. Moreover, Φ^[k] is obtained from Ω^[k] as previously explained, and M^[k] is obtained through (<ref>) with h̃ = h^[k].In order to complete the presented explanations, we present Algorithm <ref> and Algorithm <ref>, which respectively summarize the operational flow of the UKF-based approach with static prediction weights, henceforth denoted as UKF-GSI, and the upgraded version with dynamic prediction weights, denoted as UKF-AW-GSI.Please note that steps 3 and 4 from both algorithms represent the UKF data assimilation and prediction steps respectively. Please see <cit.> for additional details about the related equations and the role of Q and R in those steps. Also, in Algorithm <ref>, W^AW_0 and D^AW_0 correspond to the matrices used to derive h_0 using AW-GSI. Finally, from a leak localization perspective, we must consider that most methods compare leak and leak-free scenarios, and hence the presented algorithms should be applied in both cases.§ CASE STUDYThe proposed methodology is tested by means of the Modena benchmark, which stands as a prominent, openly accessible case study within the management of WDNs <cit.>. The network structure is represented in Fig. <ref>, whereas its main physical and hydraulic properties are introduced in Table <ref>. Note that the benchmark models a real-world network, whose size and demand correspond to a medium/large scale problem.The pressure and demand sensors depicted in Fig. <ref> are assumed to be installed within the network. The pressure sensors include 4 metering devices at the network reservoirs and 16 at junctions.From the set of AMRs, 20 are located alongside the pressure sensors, whereas the other 20 are placed in additional locations. All these placements are obtained through a fully data-driven sensor placement technique, presented in <cit.>.§.§.§ Generation of evaluation data The presented benchmark is implemented inEPANET 2.0 <cit.> to obtain hydraulic data from leak and nominal scenarios,enabling the assessment of the method's performance. The conducted simulations span a 24-hour period, during which the nodal demands evolve with a pattern that changes every hour.A nodal extra demand of 4.5 ℓ/s emulates the leak effects. We consider this leak size to be adequate, regarding that it only accounts up for a ∼1.1% of the average total inflow, as well as recent studies dealing with the same benchmark consider similar or even larger leak sizes <cit.>. The leak size selection is also justified by theincluded sources of uncertainty as follows.First, the system relies on the accuracy of the measured hydraulic values, i.e., pressure and consumption, within a margin of ±1 cm and ±0.01 ℓ/s respectively. Second, random uncertainty has been added to the pipes' diameter and roughness, which are typically difficult to measure in WDNs. An uncertainty level of 1% with respect to the noise-free values is included. Furthermore, daily demand patterns are also affected by a 1% of uniformly random uncertainty, including additional variability to the actual consumption, and therefore the produced nodal pressures.§ RESULTS AND DISCUSSION Code and data: <https://github.com/luisromeroben/UKF-AW-GSI>The efficacy of AW-GSI has previously been validated for the presented case study, producing satisfactory state estimations that led to successful leak localization. However, performance issues emerged in certain leak scenarios, leading to deficient estimation and/or localization outcomes.We showcase here state estimation and leak localization results for a challenging leak case, illustrating the behavior of the newly proposed methods in such a scenario that led to degraded solutions through AW-GSI. Specifically, we consider a leak at node 88, which can be seen in Fig. <ref> (labeled as "Leak") to be positioned away from pressure or demand sensors, which is challenging for the compared methodologies.The configured parameters for the UKF-based approaches are listed in Table <ref> where I_n is the identity matrix of size n. The amount of sensors was selected as part of the problem definition, as previously explained. The rest of parameters was empirically configured. §.§ State estimation The state estimation performance is studied through the degree of dissimilarity between the actual and reconstructed hydraulic head vectors, i.e., h and h_UKF. To this end, we compute the root-squared-mean error as RMSE(h, h_UKF) = √(1/n (h-h_UKF)^T(h-h_UKF)). Fig. <ref> shows the RMSE evolution through the UKF iterations for a challenging time instant, which yielded the worst AW-GSI estimation performance among all the available ones, for different estimation methodologies and configuration settings. In this way, the improvement of the new methods through the iterations can be observed. We use AW-GSI as a baseline (note that it does not iterate, so its RMSE is depicted as a horizontal line) for comparison with UKF-GSI and UKF-AW-GSI.Both methods improve the AW-GSI estimation (used as h_0), demonstrating the suitability of the devised UKF-based scheme. Moreover, UKF-AW-GSI performs better than UKF-GSI, specifically yielding a significant refinement when (k,K_u)=0, enabling the AW update step in (<ref>). Specifically, this implies a RMSE reduction of 16.38% for UKF-GSI and 25.22% for UKF-AW-GSI with respect to AW-GSI. In order to underscore the importance of employing a suitable initial guess, we perform the same experiment with UKF-AW-GSI, first, with h_0 = 0_n, and second, with a vector obtained as h_0 = μ(h^AW) + σ(h^AW)x, with μ(h^AW) and σ(h^AW) being the mean and standard deviation of the state vector retrieved from AW-GSI, i.e., h^AW; and x∈ℝ^n is a random vector in [-1, 1]. The simulations show an initial RMSE of 51.11m and 9.38m respectively (k=0) which is reduced at k=50 to 11.42m and 2.62m respectively. Both results do not even reach the baseline RMSE from Fig. <ref>.Finally, let us study the performance from a general perspective, extending the previous analysis to a wide set of time instants, thus achieving results in different network conditions. Specifically, we select one hour out of every two, leading to a total of 12 considered time instants, used to ultimately compute the RMSE vector r. Table <ref> shows the performance RMSE-based results for AW-GSI, UKF-GSI and UKF-AW-GSI through various statistics.§.§ Leak localization The presented state estimation methodology is proposed in the context of fault isolation. Thus, analysing the localization result from a leak localization method using the reconstructed states can illustrate the adequacy of the estimation approach. To this end, Leak Candidate Selection Method (LCSM) has been chosen due to its data-driven operation, as well as its satisfactory performance with GSI-based estimation methods <cit.>. It compares leak and leak-free (with similar boundary conditions) states, deriving a distance-based metric that serves as indicator of the leak likelihood. The localization results for the studied leak scenario are presented in Fig. <ref>, considering the same hydraulic data as in Table <ref>. First, a colour map over the network graph is depicted for AW-GSI-LCSM and UKF-AW-GSI-LCSM in the left part of the figure. This graph provides a colour to each node depending on their associated value of the LCSM distance metric: the more yellow the node, the higher the probability of being the leak location. Then, a bar graph is presented for each method in the right part of the figure. The x-axis denotes the node index, but taking into account that only nodes with a positive LCSM metric are represented, because this implies the existence of a pressure drop. On the y-axis, we represent the LCSM metric. The yellow bar indicates the leak node (88), whereas the red bars represent its neighbours. A preliminary analysis using the colour maps highlights the challenging nature of this leak scenario. Although the area around the leak has a medium-to-high probability in both methodologies, the most probable locations are rather far from the correct node (left part of the WDN). However, the in-depth analysis shown through the bar plots manifest the improvements of UKF-AW-GSI-LCSM. The red horizontal dashed line indicates the maximum value of the LCSM likelihood metric among the neighbours of the leak node. Thus, in the case of AW-GSI-LCSM, there are 40 nodes with a higher LCSM metric than the best candidate among the set composed by the leak and its first-degree neighbours, whereas in the case of UKF-AW-GSI-LCSM, there are only 7 (corresponding to the aforementioned left part of the network). This shows the promising performance of the UKF-based scheme, whose better state estimation helps to improve the accuracy of the localization process. Note that UKF-AW-GSI inherits the estimation of AW-GSI through the initial guess, so the UKF-based approach demonstrates itself capable of enhancing the estimation to the point of reducing the importance given to incorrectly indicated areas by AW-GSI-LCSM during the localization step. § CONCLUSIONSThis article has presented a nodal hydraulic head estimation methodology for WDN based on an UKF scheme with application to data-driven leak localization. This technique leverages the state reconstruction capabilities of the UKF, customizing the prediction and data assimilation functions through information about the physics behind network dynamics, its structure, and the available pressure and demand measurements. An improved version is devised by including a dynamic weighting approach that updates the weights included in the prediction step. The performance of the strategy is tested using the Modena benchmark. The UKF-based schemes provided improvements over AW-GSI in terms of estimation error and leak localization, showing the adequateness of the methods.Future work will include a deep analysis of the effects of the parametrization on the performance, as well as the degradation caused by uncertainty, designing new mechanisms to reduce the effect of noise. Improvements in the operational flow of the method will be researched, trying to improve the head estimation. Additionally, the estimation of other related hydraulic variables such as demand or flow will be considered. | http://arxiv.org/abs/2311.15875v1 | {
"authors": [
"Luis Romero-Ben",
"Paul Irofti",
"Florin Stoican",
"Vicenç Puig"
],
"categories": [
"eess.SY",
"cs.LG",
"cs.NA",
"cs.SY",
"math.NA"
],
"primary_category": "eess.SY",
"published": "20231127144837",
"title": "Nodal Hydraulic Head Estimation through Unscented Kalman Filter for Data-driven Leak Localization in Water Networks"
} |
The Magellanic Corona Model Scott Lucchini [email protected]]Scott Lucchini Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA, USA Department of Physics, University of Wisconsin - Madison, Madison, WI, USA0000-0003-2676-8344]Elena D'Onghia Department of Physics, University of Wisconsin - Madison, Madison, WI, USA Department of Astronomy, University of Wisconsin - Madison, Madison, WI, USA0000-0003-0724-4115]Andrew J. Fox AURA for ESA, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, USA Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USALucchini et al.We characterize the Magellanic Corona model of the formation of the Magellanic Stream, which we introduced in Lucchini et al. (2020, 2021). Using high-resolution hydrodynamic simulations, we constrain the properties of the primordial Magellanic Clouds, including the Magellanic Corona – the gaseous halo around the Large Magellanic Cloud (LMC). With an LMC mass of 1.75×10^11 , a Magellanic Corona of >5×10^9at 3×10^5 K, a total Small Magellanic Cloud mass <10^10 , and a Milky Way corona of 2×10^10 , we can reproduce the observed total mass of the neutral and ionized components of the Trailing Stream, ionization fractions along the Stream, morphology of the neutral gas, and on-sky extent of the ionized gas. The inclusion of advanced physical routines in the simulations allow the first direct comparison of a hydrodynamical model with UV absorption-line spectroscopic data. Our model reproduces , , andobservations from HST/COS and FUSE. The stripped material is also nearby (<50 kpc from the Sun), as found in our prior models including a Magellanic Corona.§ INTRODUCTION The Magellanic Stream is the largest coherent extragalactic gaseous structure in our sky <cit.>. It has the potential to dramatically impact the future of the Milky Way (MW) by depositing billions of solar masses of gas into our circumgalactic medium (CGM) and possibly onto our disk <cit.>. The Magellanic Stream also provides direct evidence of galaxy interactions and evolution through mergers. By studying this serendipitous nearby system, we will learn about the future of our own Galaxy, the history of the Local Group, and the gas and metal transport processes that can sustain the growth of galaxies like the MW.The Magellanic Stream is an extended network of interwoven clumpy filaments of gas that originate from within the Large and Small Magellanic Clouds (LMC, SMC), two dwarf galaxy satellites of the MW. Combined with the Leading Arm, high velocity clumps of gas ahead of the Magellanic Clouds in their orbits, the Magellanic System covers over 200^∘ on the sky. From 21-cmobservations, we have an excellent view of its small-scale, turbulent morphology as well as its velocity structure <cit.>. Moreover, absorption line spectroscopy studies have characterized the chemical composition and ionization state of the Stream along dozens of sightlines <cit.>. <cit.> has shown that the Stream is mostly ionized. They find an average ionization fraction of ≈73% with a total ionized gas mass of ∼1.5×10^9(compared with 4.9×10^8of neutral gas; ).Models of the formation of the Stream originally explained the stripped material as gas that was tidally pulled from the LMC through repeated interactions with the MW <cit.>. This would result not only in stripped gas, but also in a tidally truncated dark matter halo. Thus, masses determined by rotation curve fitting <cit.> would be sufficient for modeling the evolution of the Clouds and the formation of the Stream. However, updated proper-motion measurements of the LMC and SMC have shown that the Clouds are most likely on their first infall towards the Milky Way <cit.>. A recent study has found a second passage orbit consistent with the observations, however further studies of the hydrodynamics are required to see if the Stream can be reproduced <cit.>. A first-infall scenario would require a higher LMC mass as it would not yet be tidally truncated. Many different indirect methods of estimating the LMC's total pre-infall mass have converged on a value of 1-2×10^11 :1.98×10^11from abundance matching <cit.>, >10^11from the MW's reflex motion <cit.>, >1.24×10^11from the LMC's satellite population <cit.>, 2.5^+0.09_-0.08×10^11from the Hubble flow timing argument <cit.>, and 1-2×10^11from the MW's stellar streams <cit.>. See Figure <ref> for a summary. Shown in this figure is also the error-weighted mean of the values calculated in <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>: 1.60±0.21×10^11(∼10-20% of the MW's total mass).While modern tidal models of the formation of the Stream have used large masses for the LMC, they are unable to explain the immense ionized mass <cit.>. On the other hand, ram pressure models <cit.> are able to explain the ionized material via dissolution of the neutral gas through instabilities, but they require low masses for the LMC <cit.>.To resolve both of these discrepancies simultaneously, we introduced the Magellanic Corona model (, hereafter , ). Based on theoretical calculations and cosmological simulations, galaxies with masses ∼10^11should host gaseous halos at their virial temperature of ∼3×10^5 K[While this value is near the peak of the cooling curve, the true gas temperature of the Corona varies with radius and through supernova heating, the Corona remains stable. See Figure <ref>] <cit.>. Upon inclusion of a warm CGM around the LMC, dubbed the Magellanic Corona, we are able to explain the ionized material in the Stream while accounting for a massive LMC. In , we showed that this Magellanic Corona also exerts hydrodynamical drag on the SMC as it orbits around the LMC. With a new orbital history consistent with the present-day positions and velocities of the Clouds, the Trailing Stream ends up five times closer to us than previous models predicted. The Magellanic Corona model is described in more detail in Section <ref>. In a subsequent study, the Magellanic Corona was directly observed using absorption line spectroscopy data from the Cosmic Origins Spectrograph on the Hubble Space Telescope <cit.>. From 28 sightlines extending to 35 kpc away from the LMC, we detected a radially declining profile in high ions (, , ) with a total mass of 1.4±0.6×10^9including a warm phase with temperature of 3×10^5 K. These results are consistent with the picture of a first-infall LMC with a Magellanic Corona. While these values give us an excellent picture of the LMC's CGM at the present-day, modeling the evolution of the Magellanic System will help us constrain the properties of the primordial LMC and its Magellanic Corona to understand dwarf galaxy evolution in general. §.§ History of the Magellanic Corona ModelIn , we presented a model of the evolution of the Stream including the Magellanic Corona, showing that we can reproduce the total mass budget of the Stream including its ionized component. This model also forms and extended Trailing Stream and Leading Arm through tidal interactions between the Clouds. In this work, we used previously published orbital histories for the Clouds <cit.>; however, the Magellanic Corona exerts hydrodynamic drag forces on the SMC as it orbits around the LMC, so the kinematics and positions of the Magellanic Clouds were not in agreement with observations in this proof-of-concept study.In order to match the present-day positions and velocities of the LMC and the SMC, we then explored the orbital histories of the Clouds . In the updated orbit published in this work (still including the Magellanic Corona), the 3D velocities of the Clouds agree with the observed values within 3σ. Additionally, due to the hydrodynamic interactions of the tidally stripped (neutral) material with the Magellanic and Galactic Coronae, we reproduce the turbulent, filamentary structures within theTrailing Stream. This orbital history is shorter than previous models, and so there is no time for a Leading Arm to form, and the large-scale morphology of the Trailing Stream is different than the observations. However, the stripped gas in this model ends up very near the Sun (<50 kpc) which is in contrast to previous predictions (>100-200 kpc; ). A nearby Stream is able to explain a number of other inconsistencies between the observations and previous models: pressure equilibrium <cit.>, the numerous head-tail clouds <cit.>, and the Hα emission <cit.>. Both of these previous publications have included simplistic models for star formation and temperature evolution, which is improved upon in this work.In this paper, we expand upon the Magellanic Corona model by presenting new, high-resolution simulations with detailed physical models including the self-consistent tracking of star formation, feedback, ionization, and metallicity to directly compare with absorption line spectroscopy observations. Furthermore, we explore the parameter space of temperatures and densities for the Magellanic and Galactic Coronae that provide the best match to the observed properties of the Magellanic Clouds as well as the morphology of the neutralStream. In Section <ref>, we outline the methodology of our simulations and analysis. Section <ref> presents our simulated present-day Stream including the updated physical models and new galaxy initial conditions. In Section <ref>, we discuss our parameter space exploration of the properties of the Magellanic Corona. Section <ref> contains the discussion of our results and the implications for the properties of the MW's CGM. We conclude in Section <ref>.§ METHODSWe use , a massively parallel, multiphysics code for our simulations <cit.>. We utilize its Lagrangian “meshless finite-mass” hydrodynamics scheme which allows for the ability to track individual fluid elements while conserving angular momentum and capturing shocks <cit.>. Star formation is included <cit.> with a physical density threshold of 100 cm^-3, a virial requirement that the gas is locally self-gravitating <cit.>, and a requirement that the gas is converging (∇· v<0). Mechanical stellar feedback is also included in which we assume a constant supernova rate of 3×10^-4 SNe Myr^-1 ^-1 for all stars less than 30 Myr old. Each supernova injects 14.8with 10^51 ergs of energy and metals following the AGORA model <cit.>. Cooling is included down to ∼10 K following <cit.> with metal-dependent tables <cit.>. We don't include radiative transfer or UV background radiation, however outside the MW at low redshift, we don't expect these mechanisms to play a significant role.These simulations improve upon those ofandby including accurate star formation and feedback with metallicity and advanced cooling routines.calculates self-consistent ionization states for each particle based on collisional heating/ionization, recombination, free-free emission, high and low temperature metal-line cooling, and compton heating/cooling <cit.> We therefore are able to track the neutral and ionized material separately throughout the simulation. §.§ Initial Conditions Table <ref> shows the properties of the Magellanic Clouds in our simulation and we used the [<https://bitbucket.org/vperret/dice/src/master/>] code to generate our initial conditions <cit.>. We used an LMC dark matter (DM) mass of 1.75×10^11consistent with previous studies (; ) and in agreement with indirect estimations (see Figure <ref>). We constrained the initial gaseous and stellar disk masses for the Magellanic Clouds by requiring their present-day values to be consistent with observations. This was straight forward for the stellar masses as the stars formed during the simulations comprise only a small fraction of the total. However the gas masses can vary greatly from their initial values due to the tidal interactions and accretion from the Magellanic Corona. We found that the LMC's disk gas mass agreed best with present-day values when allowing the gaseous disk to form naturally out of the Magellanic Corona. Therefore, we initialize our fiducial LMC (t=0 Gyr in Table <ref>) with a 2.5×10^9stellar disk and a Magellanic Corona with total mass (within 200 kpc) of 5×10^9following an isothermal profile at a temperature of 3×10^5 K and a metallicity of 0.1 solar. After 3.5 Gyr in isolation, there remains 3.7×10^9of ionized material bound to the LMC (with 2×10^9within 120 kpc) with a median temperature of 3.4×10^5 K. 9.1×10^8has cooled to form the LMC's gaseous disk. This is the LMC that we start with for the full simulations with the LMC, SMC, and MW (listed as “LMC (t=3.5 Gyr)” in Table <ref>). The initial and final radial density and temperature profiles are shown in Figure <ref> in blue. A parameter space exploration of optimal masses and temperatures for the Magellanic Corona is discussed in Section <ref>.The only constraint on the SMC's total mass comes from its rotation curve which requires 2.4±0.36×10^9within 4 kpc (1.25±0.25×10^9in DM; ). While previous studies found that ∼10% of the LMC's total mass (∼2×10^10 ) produced the best results <cit.>, this was prior to the inclusion of the Magellanic and Galactic Coronae. Therefore, in this study we have explored the effect of the SMC total mass on the formation of the Stream and found that a lower SMC mass provides better results. In our fiducial model, we used a DM mass of 5×10^9 , a stellar mass of 2.5×10^8 , and a gaseous disk mass of 1.5×10^9(listed in Table <ref>).For the MW, we implemented a live DM halo with a total mass of 1.1×10^12combined with a hot gaseous CGM following a β-profile as in <cit.>:ρ∝[1+ (r/r_c)^2]^-3β/2with r_c = 0.35 and β=0.559. As with the Magellanic Corona, we explored a variety of models for the Galactic corona as well, which will be discussed in Section <ref>. Our fiducial MW CGM has a total mass of 2×10^10at 10^6 K and is nonrotating. After evolution in isolation for 2 Gyr, 1.9×10^10remains bound with a mean temperature of 1.4×10^6 K. Figure <ref> shows the initial and final (after 2 Gyr in isolation) density and temperature profiles in orange.We use a mass resolution of ∼3×10^4 /particle for the gas elements, ∼2×10^4 /particle for the stars, ∼10^6 /particle for the DM. This results in a total particle number of 1.5×10^6. Adaptive softening was used for the gas particles (such that the hydrodynamic smoothing lengths are the same as the gravitational softening length), and softening lengths of 150 pc and 550 pc were used for the stellar and dark matter particles, respectively. §.§ Orbits of the Clouds Following , we have determined viable orbital histories for the Clouds with analytic integration. Using the galactic dynamics code [<https://gala.adrian.pw/>] <cit.>, we varied the present-day positions and velocities of the Clouds within their errors <cit.> and integrated them backwards in time considering the effects of the MW, dynamical friction, and an empirical hydrodynamical friction term. With slight alterations to the initial positions and velocities, we obtained the initial conditions for our full hydrodynamic simulations using . Many of the orbits determined by theintegrations were not viable once the hydrodynamics were included. This was for a variety of reasons including, for example, extremely low impact parameter collisions, tidal forces not stripping material in the correct direction, and mass loss and nonaxisymmetric effects of the live dark matter haloes. Upon visual inspection, we determined whichorbits should be interated on and we improved the visual appearance of the Trailing Stream, and the positions and velocities of the Clouds at the present day. Our fiducial orbit used in the simulations presented here consists of two interactions over 4.2 Gyr, very similar to the orbits presented in . The initial properties of the Clouds as well as their positions and velocities (relative to the MW located at the origin) are listed in Table <ref>. § THE NEUTRAL STREAMAs shown in , the inclusion of the Magellanic Corona leads to a family of orbital histories in which the Stream ends up within 50 kpc from the Sun. The previous first-infall history predicted the Trailing Stream at >100 kpc <cit.>, however the hydrodynamical friction on the SMC as it orbits the LMC in the Magellanic Corona model requires fewer interactions and a shorter interaction history. As the stripped material moves through the Magellanic Corona and approaches the MW's corona, it is encountering hot gas. In order for the neutral Stream to survive to the present day, the gas stripped out of the SMC must be dense enough to remain neutral (e.g. ).As we will see in Sections <ref> and <ref>, the properties of the Magellanic and Galactic Coronae are largely determined by their stability. We therefore varied the properties of the Clouds themselves to ensure that the neutral Stream survives to the present day. Reducing the total DM mass of the SMC reduces its density, therefore making the potential well of the galaxy shallower, allowing for more material to be tidally stripped during its interactions with the LMC. For SMC DM masses below 10^10 , we find that there is enough gas stripped such that the neutral Stream survives its passage through the surrounding hot gas.Figure <ref> shows the results of our fiducial simulation in Magellanic Coordinates[Magellanic Coordinates were originally defined in <cit.>.]. The observational21cm emission data from <cit.> is shown in Panel a. Panels b and c show, respectively, theandemission in our simulation withcolumn densities calculated via CLOUDY ionization modeling overlaid as colored data points (data from ). Thedistribution shows that the ionized material forms a cocoon around the neutral material covering a significant fraction of the sky. The neutral material consists of two filaments with turbulent, non-uniform structure, agreeing well with the morphology of the observations. As with the model presented in , no Leading Arm is formed due to the short interaction history and specific orbital orientation. Panel d shows the column density integrated along Magellanic Latitude as a function of Magellanic Longitude. The observations are shown in black and the simulation in blue. While we slightly underpredict the total column, the decreasing density as a function of longitude agrees well.Panels e, f, and g show corresponding representations for the metal content in the Stream with , , andrespectively. These panels also include observational column densities calculated from absorption line spectra (oxygen data from ;data from ). Here we see that thetraces the Magellanic Corona well with a large covering fraction, and thecontains many of the same features as we see in the . Thedistribution is quite variable which is represented in the observations as well. The overall sky coverage of the observedis not reproduced very well in our model due to the fact thatis comprised of two phases: a photoionized phase, and a collisionally ionized phase. The photoionized phase exists at the warm boundary between the cold, neutral material, and the hot, ionized gas. For this reason it traces the neutral gas as we see in the model. However, the collisionally phase ofis in the diffuse warm gas surrounding the Clouds. Due to the interaction between the Magellanic Corona and the MW's CGM, the Corona is heated above 5×10^5 K to temperatures at whichis suppressed, in contrast with the observations. With higher resolution simulations, we expect to see small-scale instabilities and cooling flows that would create this diffusethat we observe. In contrast, thedata are reproduced well in our model becauseexists in a single photoionized phase that traces the large scale structure of the Magellanic Corona.Panel h shows the ionization fraction across the Stream which varies from values of 0.05 in the Clouds and ∼0.3 in the Stream to ∼1.0 off of the neutral material in good agreement with the few data points we have (from CLOUDY modeling: 0.3-0.98, ). By dividing the ionized gas mass by the total mass in the Stream, we can obtain an approximate average ionization fraction of 82% (compared with ∼75% in the observations). §.§ Trailing Stream MassAs discussed in , a nearby Stream would impact the total mass of both its neutral and ionized components. Observations calculate the total mass by integrating the column densities assuming a distance to the gas. <cit.> found that there was 4.7×10^8×[d/(55 kpc)]^2of neutral material in the Magellanic System outside the LMC and SMC, and <cit.> estimated ∼1.5×10^9×[d/(55 kpc)]^2in ionized material. These calculations assume all the material is at a single distance. While this is not physically accurate, it is the best we can do since we do not know the distance to the gas in the Stream. In order to accurately compare with these observations, we performed the same analysis on our simulated data. By integrating the column density assuming a distance of 55 kpc[M=m_pΔ x Δ y ∑ N, where m_p is the proton mass, Δ x and Δ y are the physical sizes (in cm) of the bin widths in our column density image (Δ x=Δ l_MS× D, where D is the assumed distance), and N is the column density.], we find that there is 5.4×10^8of neutral gas and 1.2×10^9of ionized gas in the Bridge and Trailing Stream.We can also calculate the true gas mass in the simulation. First we locate the Clouds in the simulation and exclude the gas within spheres of radii 1.65 kpc and 6.07 kpc around the SMC and LMC, respectively. These sizes are calculated from the angular size of the disks (excluded in the mock observation calculation described above) and the distances to the Clouds in the simulation. Then we simply sum up the mass of all the particles remaining weighted by their ionization fraction. This leaves 6.5×10^8of neutral material and 4.3×10^9of ionized material. The mean distance to the ionized material is 142 kpc, which means that the mass of the hot gas is much larger than the value inferred from the observations <cit.>. §.§ Limitations of the Model While these models represent a significant step forward in terms of gas physics,and successfully reproduce a turbulent, filamentary, neutral Trailing Stream of the correct lengththrough tidal interaction for the first time, they still have a few limitations.As mentioned above, in order to reproduce the observed ionization fractions, we truncated the SMC's total mass to 5×10^9 . A full exploration of the possible orbital histories of the Clouds with this new SMC mass is needed to ensure that models can reproduce the present-day positions and velocities of the Clouds accurately, but is not presented here. Moreover, the role of the SMC's initial properties on its present-day internal dynamics is required due to the SMC's disturbed morphology <cit.>. In the model presented here, the distance to the SMC is too low and therefore its kinematics are also not in agreement with the observations. The LMC's distance is also slightly too high, but the components of the its present-day 3D velocity are all within 3σ. Additionally, while an in-depth discussion of the properties of the LMC and SMC disks and the Magellanic Bridge is beyond the scope of this paper, these features provide concrete observational signatures that can discriminate between models. We also do not include radiative transfer or UV background radiation in these simulations, which could affect the ionization fractions. Their full impact on our simulations will be explored in future work. § PROPERTIES OF THE MAGELLANIC CORONAWe explored the parameter space of initial properties of the Magellanic Corona, by varying the initial temperature and total mass. The Corona was initialized with an isothermal distribution. Its total mass (within 200 kpc) ranged from 10^9 to 10^10 . This corresponds to masses of 0.6 and 6×10^9within the LMC's virial radius of 120 kpc. The initial temperature of the Corona ranged from 10^5 to 9×10^5 K, and we used a metallicity of 0.1 solar. We explored the viability of these different Coronae by determining their stability and their impact on the present-day Stream.As mentioned above, we initialize our LMC with a DM halo, stellar disk, and the Magellanic Corona. The gaseous disk forms during the first few billion years of evolution (in isolation). The Magellanic Corona is initialized with a streaming fraction of 0.2 (the Corona has an azimuthal velocity set to 20% the circular velocity profile). Therefore, when the cooled material collapses onto the disk, it exhibits a bulk rotation as expected. Higher streaming fractions result in larger disks and lower streaming fractions result in smaller disks. This is because without any rotation, more material falls into the center of the gravitational potential and high gas densities induce very strong star formation which blows out the remaining gas. With too much rotation, the infalling cool material spreads out to larger radii (because it has higher angular momentum) and the densities are not high enough for star formation. §.§ StabilityThe main factor in determining the viable parameter space for the temperature and density of the Magellanic Corona is in its stability around the LMC. If the temperature is too high, the Coronal material blows off because its internal energy is too high and much of the material is unbound to begin with. If the temperature is too low, too much gas falls onto the LMC disk leading to disk gas fractions that are too high as well as very high star formation rates inconsistent with the LMC's history. Similarly, if the Corona starts with too much mass (i.e. too high density), the disk becomes too gas rich in contrast to the present day observed gas masses. Below a mass cutoff, the Corona remains stable however in order to reproduce the high ionization fractions along the length of the Stream, the Magellanic Corona must be more massive than 10^9(within 120 kpc; see Section <ref>).Figure <ref> shows these results. The nine panels depict nine different simulations with varying initial conditions in which the LMC and Magellanic Corona were evolved in isolation. The initial temperature of the Corona increases from left to right (with values of 1, 3, and 9×10^5 K), while the initial mass of the Corona increases from top to bottom (1, 5, and 10×10^9within 200 kpc). The black lines show the total masses of the gaseous components within 120 kpc (the virial radius of the LMC) as a function of time – total gas mass (solid), ionized mass (circumgalactic Coronal gas; dashed), and neutral disk mass (dotted). Figure <ref> also shows the temperature distribution as a function of radius for the nine simulations at t=4 Gyr. Again initial temperature increases to the right and the initial gas mass increases downwards. The mean temperature of the gas within 20<r<250 kpc is shown as a horizontal dashed line. Interestingly, these white lines don't vary dramatically between the three columns. This means that the initial temperature has a minimal, if any effect on the stable temperature of the Corona.We do, however, see that increasing the initial mass of the Corona affects the spread in temperatures. We believe this is due to the fact that the Coronae with higher initial masses contain higher density gas which can cool more effectively. Cooling is very efficient around 10^5 K, so subtle changes in density and temperature can have a strong impact on the strength of cooling. These higher density Coronae thus don't have sufficient supernova energy injection to keep the gas warm.§.§ Effect on the Magellanic StreamWe now want to explore the effect of varying the Magellanic Corona's initial mass and temperature on the properties of the present-day Magellanic System. The initial total mass of the Corona directly affects the amount of ionized gas that we observe around the Stream today. Via absorption-line spectroscopy, <cit.> estimated that there is ∼1.5×10^9of ionized gas associated with the Magellanic System, ∼10^9of which is in the Trailing Stream region. Figure <ref> shows the total mass in the Trailing Stream for various different models. These values were calculated by mimicking the observational technique of integrating the column density at an assumed distance of 55 kpc. These are not the physical masses in the system (see Section <ref>), but allow us to compare directly with the observational estimates shown in the left-most bar <cit.>. Continuing from left to right we have the results from the simulations published in <cit.> and <cit.> in which no ionized material was produced. The three right-most bars show the results of our simulations for three different initial Corona masses, 1, 5, and 10×10^9(corresponding to Panels b, e, and h in Figures <ref> and <ref>). Clearly, as we increase the progenitor mass, we are able to produce more ionized material in the Stream. For masses below 5×10^9we are unable to reproduce the observations. In the models that we tested, either 5×10^9 or 10^10result in viable models. For masses larger than 10^10 , the mass of the LMC's CGM begins to approach estimates of the MW CGM's mass. Given the significant difference in virial masses of the two galaxies, having similar coronal masses is unlikely. Therefore Magellanic Corona masses below 10^10are preferred.We also explored the impact of the initial Magellanic Corona temperature on the present-day Magellanic System. As shown in Figure <ref>, the initial temperature doesn't have a large effect on the temperature distribution or on the mean Corona temperature after 4 Gyr of evolution. The main difference that we see between temperature models is in the properties of the LMC disk. Since we form our LMC disk self-consistently by letting it condense out of the Corona, the temperature plays a large role in its size and stability. Figure <ref> shows the LMC's disk at the present day for three different simulations compared with observational data from the HI4PI survey <cit.>. Panels a, b, and c show the results from simulations with initial Magellanic Corona Masses of 1, 3, and 9×10^5 K, respectively. Lower initial Corona temperatures leads to larger LMC disks due to more material cooling and falling towards the center of the gravitational potential. For the highest temperatures (Panel c), no LMC disk forms since the Corona material can't cool and fall to the center of the gravitational potential. This is also visible in Figure <ref>f in which the dotted line (showing the total disk mass) remains at zero throughout the simulation. §.§ Fiducial ModelBased on these tests, our chosen fiducial Magellanic Corona was follows an isothermal profile with a total mass (within 200 kpc) of 5×10^9and a temperature of 3×10^5 K. We used an initial metallicity of 0.1 solar. After 3.5 Gyr in isolation, 3.7×10^9of ionized material remains bound to the LMC (with 2×10^9within 120 kpc) and its median temperature has risen to 3.4×10^5 K. The LMC's neutral gas disk has condensed out of the Corona with a mass of 9.1×10^8 . § DISCUSSIONThe Magellanic Corona model of the evolution of the Magellanic System presented here is distinct from the historical dichotomy of Stream formation models: the tidal model <cit.>, and the ram-pressure model <cit.>.In modern first-infall tidal models <cit.>, the Trailing Stream and Leading Arm are formed through 4 interactions over 7 Gyr, however no MW CGM was included. In response to the observations of the large ionized mass in the Stream <cit.>, <cit.> attemped to increase the stripped material in the Magellanic System by increasing in the pre-infall LMC and SMC gaseous disk masses. While this brought the present-day neutral Stream mass closer to the observed values (see Figure <ref>), it wasn't enough to account for the total mass of >10^9 . Winds and outflows from the LMC were also determined to be insufficient to supply the required gas mass into the Stream <cit.>.<cit.> explored the implications of the bow shock generated as the LMC approaches the MW. The envelope of shocked gas could possibly explain the ionized gas detections of <cit.>, however this isn't explored in depth in their work. Our model presented here does produce some of the same bow shock features including a high temperature leading edge of ionized gas (visible in Panels b and e of Figure <ref>. Because of the inclusion of the Magellanic Corona in our models, there is less of a clear shock boundary, however we are still consistent with the observed Hα emission and ionized gas properties.The other proposed formation pathway for the Magellanic Stream involves ram-pressure stripping gas out of the LMC and SMC disks <cit.>. While this model is able to account for many of the features of the Magellanic System, including its turbulent morphology and ionized gas component, it requires a very low mass for the LMC (<2×10^10 ) with little to no dark matter. This is in contrast with the many indirect indications of the LMC's mass shown in Figure <ref>. Moreover, they use relatively low mass models of the MW while including a very extended, high-mass CGM (total masses of ∼7 and 8×10^11with CGM masses of 2 and 1.5×10^11 , i.e. 28% and 18%, respectively). These gaseous halos extend out beyond 500 kpc and we have not been able to reproduce the stability reported in Figure 1 of <cit.>. Finally, they do not discuss the metallicity along the length of the Stream which may not agree with observations given that there is significant LMC material stripped.Our model now includes the Magellanic Corona which not only provides the bulk of the mass in the Magellanic System (in the form of warm ionized gas), but it, combined with the MW coronal gas, provides the required hydrodynamical forces to shape the Stream into its observed complex, turbulent morphology. We are now able to explain the total observed mass (>10^9 ), the high ionization fraction (>75%), the filamentary structure of the Trailing Stream, and the behavior of the metals.Due to the tidal interactions between the Clouds in our model, we still predict a >30 mag arcsec^-2 stellar stream as in previous models (; ). These stars which are stripped out of the disk of the SMC lie up to 20^∘ offset from the gaseous Stream in Magellanic Latitude and are at distances ranging from 10 kpc to >100 kpc with a median distance of 42 kpc. Despite extensive work looking to detect the stellar counterpart to the Magellanic Stream, only a few candidates have emerged. <cit.> found 15 stars near the tip of the Stream at distances of 40-80 kpc with metallicities and velocities consistent with the SMC. More recently, <cit.> found 13 stars with kinematics matching the Clouds that lie at distances of 60-100 kpc, however this study only explored stars beyond 50 kpc. Our model presented here can reproduce the <cit.> stars, however the <cit.> stars are at too large of a distance to be directly reproduced in this model. This could be resolved by including a stellar halo around either Cloud which could contribute to these few metal-poor stars at very high distances even if the bulk of the Stream (gaseous and stellar) is nearby.As discussed in , the Leading Arm may not be Magellanic material. Whileobservations linked the leading gas clumps with the LMC and SMC based on their high velocities and spatial locations <cit.>, UV spectroscopy has shown the Leading Arm contains a range of metallicities <cit.> and complex kinematics <cit.>, casting doubt on its origin. Due to the existence of a “Magellanic Group” of galaxies that approached the MW together with the Clouds <cit.>, the Leading Arm material could have been ram-pressure stripped off of a forerunner satellite that fell into the MW ahead of the LMC and SMC <cit.>. However, candidate satellites with the correct positions and velocities have not yet been identified <cit.>. In our exploration of orbital parameters, we have found orbits in which material is stripped out ahead of the Clouds, however the Trailing Stream is not very well reproduced in these models. In future work, we will better explore this orbital parameter space in addition to reinvestigating the observations of the Leading Arm to constrain its origin. §.§ Constraining the Milky Way CoronaBy exploring the parameter space of temperatures and densities for the MW's own hot gas corona, we can isolate its effects on the formation and morphology of the Trailing Stream. In this way, we can constrain the CGM properties by comparing our simulations with observations. Inspired by observations <cit.>, we varied the total mass from 10^10 to 8×10^10 , varied the temperature from 4×10^5 to 3×10^6 K, and tested with and without uniform rotation of the CGM. As we found above with the Magellanic Corona, changing the initial temperature of the gas did not affect the equilibrium temperature distribution significantly. Similarly, rotation, while it did decrease the equilibrium temperature of the CGM slightly, did not have a substantial effect on the Magellanic System. Therefore, the main variable we explored was the total mass. found that a MW corona mass of 4×10^10was required to get the best match to the velocity gradient along the Stream (two times larger than ). In the suite of orbits tested in this paper, we found that the largest effect that the MW corona had on the present-day Stream is on the morphology of the neutral and ionized components. Figure <ref> shows the ionized (orange) and neutral (grayscale contours) components of the simulated Streams in Magellanic Coordinates for three different models. The total mass of the MW corona increases from top to bottom with values of 1, 2, and 8×10^10 . The higher gas density induces stronger ram pressure on the Magellanic gas, decreasing the on-sky extent of the ionized gas, and making the neutral Stream longer and narrower. We find that a value of 2×10^10(Panel b; in agreement with estimates from ) provides the best agreement with the observations.This led to our fiducial MW corona model with a total mass of 2×10^10and a temperature of 10^6 K. After 2 Gyr of evolution in isolation, 1.9×10^10remains bound to the MW and the coronal gas has a mean temperature of 1.4×10^6 K. § CONCLUSIONSBuilding on our earlier work , we have characterized the influence of the Magellanic Corona on the formation and evolution of the Magellanic Stream. With this suite of simulations, we have shown that the first-infall Magellanic Corona model of Stream formation can produce a present-day Magellanic System with properties in agreement with the observations (Figure <ref>). The ionized component is formed out of the Magellanic Corona, which becomes warped and shaped around the Clouds and the neutral Stream through its interactions with the MW's own hot coronal gas. The trailing Stream's turbulent morphology seen inis reproduced through interactions between the neutral gas and the warm/hot gas in the Magellanic and Galactic Coronae. We find metal distributions and ionization fractions in agreement with absorption-line spectroscopy observations.We have also explored the parameter space of temperatures and densities for the Magellanic Corona to constrain its properties. We find that a mass >5×10^9(within 200 kpc) can provide sufficient ionized material at the present day (Figure <ref>). By forming the LMC's gaseous disk self-consistently out of the Corona, we are able to reproduce the gas mass within the galaxy's disk at the present day. The initial temperature of the Coronal gas determines the size and mass of the LMC's disk at later times, so we found that a value of 3×10^5 K (in agreement with the virial temperature) provides the best results (Figure <ref>).These models show that we are able to reproduce the properties of observed Magellanic System while accounting for a large LMC mass. The Magellanic Corona provides the key element that necessitates a review of the precise orbital histories of the Clouds. This brings the Trailing Stream gas much closer to us than previously thought, explaining the turbulent morphology andbrightness, and implying an infall onto the MW disk within ∼100 Myr.By obtaining constraints on the distance to the gas in the Stream through absorption-line spectroscopy towards MW halo stars, we will be able to better discriminate between existing models. Moreover, a reevaluation of the properties of the Leading Arm could lead to answers on whether it is of Magellanic origin or not. These observations and more will constrain key properties of the Magellanic System, giving us the information we need to converge on the true history of the Magellanic Clouds.Support for programs 16363 and 16602 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The simulations in this paper were run at the University of Wisconsin - Madison Center for High Throughput Computing supercomputing cluster <cit.>. astropy <cit.>, dice <cit.>, gala <cit.>, pygad <cit.>, trident <cit.> | http://arxiv.org/abs/2311.16221v1 | {
"authors": [
"Scott Lucchini",
"Elena D'Onghia",
"Andrew J. Fox"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20231127190000",
"title": "Properties of the Magellanic Corona Model for the formation of the Magellanic Stream"
} |
: A Processing In-Memory Accelerator for Fully Homomorphic Encryption Minxuan Zhou†, Yujin Nam†, Pranav Gangwar†, Weihong Xu†, Arpan Dutta†,Kartikeyan Subramanyam†Chris Wilkerson, Rosario Cammarota, Saransh Gupta♢, and Tajana Rosing†University of California San Diego†, Intel Labs, IBM Research♢{miz087,yujinnam,pgangwar,wexu,adutta,kasubram,tajana}@ucsd.edu†,{chris.wilkerson,rosario.cammarota}@intel.com, [email protected]♢ January 14, 2024 =============================================================================================================================================================================================================================================================================================================================================================================== Fully Homomorphic Encryption (FHE) is a technique that allows arbitrary computations to be performed on encrypted data without the need for decryption, making it ideal for securing many emerging applications. However, FHE computation is significantly slower than computation on plain data due to the increase in data size after encryption. Processing In-Memory (PIM) is a promising technology that can accelerate data-intensive workloads with extensive parallelism. However, FHE is challenging for PIM acceleration due to the long-bitwidth multiplications and complex data movements involved.We propose a PIM-based FHE accelerator, , which exploits a novel processing in-memory architecture to achieve high-throughput and efficient acceleration for FHE. We propose an optimized end-to-end processing flow, from low-level hardware processing to high-level application mapping, that fully exploits the high throughput of hardware. Our evaluation shows achieves significant speedup and efficiency improvement over state-of-the-art FHE accelerators.Cryptography, Processing In-Memory, Domain-Specific Acceleration § INTRODUCTIONThe data explosion leads to an increasing trend of cloud-based outsourcing. Extensive outsourcing significantly increases the risk of sensitive data leaking, necessitating data encryption for protection. Fully homomorphic encryption (FHE) is an emerging technology that enables computations on encrypted data without user interference <cit.>. FHE provides end-to-end data security during the outsourcing, including data transfer and computation, without any requirements for the underlying system and hardware. However, FHE is several orders of magnitude slower than plain data while requiring a large memory footprint <cit.>. The inefficiency of FHE results from the data and computation explosion after encryption. Even though FHE can encrypt a vector into a single ciphertext <cit.>, the ciphertext size is still large and includes two or more high-degree polynomials (e.g., 2^17) with long-bit coefficients (e.g., >1000 bits). Such issues motivate researchers to develop customized accelerators that provide 4 orders of magnitude speedup over conventional systems <cit.>. However, existing accelerators are still significantly bounded by the data movement even with large and costly on-chip scratchpads <cit.>.As shown in Figure <ref>(a), each homomorphic multiplication (HMul) requires 98MB to 390MB working set for LogN=15 to LogN=17. In Figure <ref>(b), we follow the method in previous work <cit.> to analyze the memory bandwidth required by different numbers of number theory transform units (NTTUs) under 3 data loading scenarios during a holomorphic operation with key-switching operation (KSO) which is the most expensive FHE primitive.Our investigation shows simply doubling the throughput of existing accelerators (1K to 2K) may require over 3TB/s of off-chip bandwidth. Previous accelerators adopt large on-chip storage, up to 512MB, to hold the large working set of FHE computation. However, large on-chip storage can still suffer from frequent off-chip data transfers due to cache misses on extensive FHE data. Therefore, it is challenging to achieve both high compute throughput and high memory bandwidth on conventional architectures for FHE applications.In this work, we exploit the processing in memory (PIM) acceleration for FHE, which is promising to support extensive parallelism with high internal memory bandwidth <cit.>.There are several types of PIM architectures that support PIM in different levels in the memory architecture, including subarray-level <cit.>, bank-level <cit.>, and channel-level <cit.>. These PIM technologies adopt high-throughput processing elements that fully exploit the internal memory links that provide higher bandwidth than the off-chip data path. Even though the high parallelism and bandwidth of PIM potentially fit the data-intensive parallel computations of FHE, there exist several challenges that existing PIM-based architectures cannot easily solve. First, FHE works on high-degree polynomials with long-bit coefficients and is multiplication-intensive. Such long-bit multiplication is challenging to all existing PIM technologies. For near-bank PIM, the throughput is limited by bank IO width. Furthermore, even though the bank-level PIM can adopt highly efficient multipliers, reading the data from the memory cells is still energy-consuming. The bit-serial subarray-level PIM can exploit a significantly large number of internal links (e.g., bitlines). However, the number of operations required by subarray-level PIM may increase quadratically with the operand bit-length. These energy-consuming operations run in a lock-step manner, significantly increasing delay and energy. Our evaluation shows such PIM technologies fail to provide comparable throughput and energy efficiency to the state-of-the-art FHE accelerators (Section <ref>). Integrating more functionality in the sense amplifiers can solve the problem, but introducing significant modifications in conventional DRAM structures <cit.>. The second challenge of PIM acceleration for FHE is the complex data movement patterns, including the base conversion and number theoretic transform (NTT), that the existing memory architecture cannot efficiently support.To tackle these challenges, we propose , an accelerator based on a novel high-bandwidth memory (HBM) architecture optimized for the efficiency of processing FHE operations.introduces a novel near-mat processing architecture, which integrates compute logic near each mat without changing the area-optimized mat architecture. exploits the existing intra-memory data links, with careful extensions based on the practicability, to enable efficient in-memory processing of various challenging FHE operations. Furthermore, we propose a software-level framework to map FHE programs onto hardware. We propose a load-save pipeline that can fully utilize the memory for FHE programs to support high-throughput computing with minimum data loading overhead.We summarize the contributions of this work as follows: * We propose an FHE accelerator with a novel near-mat processing that supports high-throughput and energy-efficient in-memory operations. Our design efficiently exploits the existing data paths with practical modifications in DRAM that introduce relatively lightweight overhead compared to prior high-throughput PIM solutions. * We propose an FHE mapping framework that generates a load-save pipeline and data layout that maximizes the utilization of for general FHE programs.* We rigorously explore and evaluate different design dimensions of to balance the performance, energy efficiency, and chip area. As compared to state-of-the-art FHE ASICs <cit.>, achieves 4.0× speedup and 6.9× efficiency improvement.§ BACKGROUND AND MOTIVATION §.§ Fully Homomorphic EncryptionWe focus on CKKS scheme <cit.> which is widely used in many application domains because it supports real numbers andSIMD packing <cit.>. Basics of CKKS: We define the polynomial ring R=ℤ[X]/(X^N+1), where N is power of 2. We denote R_q=R/qR for residue ring of R modulo an integer q. The ecurity parameter λ sets the ring size N and a ciphertext modulus Q. For each plaintext message, m(X), encryption (m(X),s(X)) generates a ciphertext c = (b(X), a(X)), where b(X) = a(X) · s(X) + m(X) + e(X), a(X) is uniformly sampled from R_Q, and s(X) and e(X) are sampled from a key/error distribution respectively. Each ciphertext can pack up to N/2 real numbers to support SIMD processing on all packed numbers <cit.>. The original modulus Q=∑_l=1^Lq_l of a ciphertext decreases with homomorphic multiplications by a rescaling process that reduces the modulus by a q_l each time. Therefore, CKKS is a leveled homomorphic scheme that only supports L levels of multiplications for each ciphertext.The technique to recover the ciphertext level is bootstrapping <cit.>.Arithmetic Operation: Given two ciphertexts c0,c1 ∈ R^2_q_l, where q_l is the modulus at level l, the polynomial operation can homomorphically evaluate the arithmetic for plaintexts. The homomorphic multiplication () between two ciphertexts is complex: c0*c1=(I_0, I_1, I_2)=(c0_ac1_a, c0_ac1_b+c1_ac0_b, c0_bc1_b) ∈ R^3_ql, where I_2 is encrypted under the secret key s^2. This requires a re-linearization operation with an expensive key-switching process on I_2 (Section <ref>).The rescaling is applied during the relinearlization, using the divide and round operation: (C)=⌊q_l-1/q_lC⌉(mod q_l-1) to rescale the ciphertext as well as the modulus.Rotation: CKKS supports homomorphic rotation which rotates the plaintext vector by an arbitrary step. The rotation is implemented by Galois group automorphism <cit.> which consists of mapping on each coefficient a_i: σ_k(a_i)(-1)^sa_ik mod N,where k is an odd integer satisfying |k|<N and s=0 if ik mod 2N<N (s=1 otherwise). Each automorphism σ_k implements a (δ) which rotates the plaintext by δ. Each automorphism also requires aafter each . NTT and Residue Number System: Number Theoretic Transform () is a widely used technique to optimize polynomial multiplications.transforms two input polynomials of a multiplication, a and b, toand . We can calculate NTT(a*b) = NTT(a) ⊙ NTT(b), where ⊙ denotes element-wise multiplication with O(N) complexity, where N is the polynomial degree. An inverse NTT () can transform NTT(a*b). The complexity ofandis O(NlogN), faster than the original polynomial multiplication with O(N^2) complexity. Residue number system (RNS) is a technique that avoids computation on large values. We adopt the full-RNS version of CKKS <cit.>.For polynomials in R_Q, the scheme chooses a set of pair-wise coprime integers q_i where i ∈ [0,L) and q_0q_1...q_L=Q. Each polynomial a is represented by L polynomials a[0...L], where a[i] ∈ R_q_i. We can evaluate (a,b) by independently calculating (a[i],b[i]) based on Chinese Remainder Theorem. These RNS moduli are also used as the leveled modulus, so each multiplication removes one RNS polynomial in the ciphertext.Key Switching: Key switching is the most expensive high-level operation in CKKS, where we use the state-of-the-art generalized key switching algorithm <cit.>.The key of key switching is the multiplication between the input ciphertext c and the evaluation key evk. However, the naive multiplication will cause the overflow on modulus Q=q_0q_1...qL.To avoid overflow on modulus Q=q_0q_1...qL, evk has a larger modulus PQ with the special modulus P=p_0p_1...p_k. Thus the first step is to convert c with modulus Q into a ciphertext with PQ by a base conversion (): BConv_Q,P(a_Q)=([∑_j=0^L[a[j]*q̂_̂ĵ^-1]_q_j*q̂_̂ĵ]_p_i)_0≤ i<krequires the data in the original coefficient domain. We need to apply anon the data before .features an all-to-all reduction between different q_j and p_i residual polynomials.We convert theresult back to thedomain to efficiently process the multiplication with evk. The algorithm converts the result with modulus PQ back to modulus Q using . Recent advanced CKKS schemes exploit a configurable dnum value to factorize the modulus Q into dnum moduli to increase higher multiplication level <cit.>.§.§ Memory Issues of FHE AcceleratorsFHE features large polynomial operations and complex data dependency caused byand . Recent works <cit.> have proposed customized accelerators for FHE. Even though these accelerators achieve up to 4 orders of magnitude speedup over CPUs, they suffer from limited memory bandwidth. For example, previous work <cit.> observed that the excessive usage of high-throughput function units might be a waste - it would be cost-efficient to determine the throughput of on-chip processing elements based on the available memory bandwidth. Such memory issues result from the large data size required for each FHE operation. Existing FHE accelerators adopt large on-chip storage (180MB for SHARP <cit.>, 256MB for CraterLake <cit.>, and 512MB for BTS <cit.>/ARK <cit.>) to reduce the frequent off-chip data loading. However, such large on-chip storage may still be insufficient for FHE, as shown in Figure <ref>. For large FHE parameter settings, on-chip storage may only store working sets of one or two HMul, leading to frequent off-chip data loading when locality is low. The analysis in Figure <ref>(b) shows 2k NTTUs require at least 1.5TB/s when only loading evk, and the bandwidth requirement goes up to 3TB/s when the accelerator needs to load both evk and two operands. 3TB/s is expected to require 3 HBM3 stacks <cit.>. If we increase the throughput to 64k NTTUs, which can fully parallelize operations for LogN=17, the bandwidth requirement can be as high as 100TB/s. Considering it is challenging to significantly increase either the memory bandwidth or the on-chip storage, processing in memory can be a promising alternative. §.§ In-DRAM PIM Technologies This work focuses on DRAM-based PIM technologies which support larger capacity than SRAM <cit.> and lower latency than non-volatile memories <cit.>. Figure <ref>(a) shows a DRAM bank, which is the basic hardware component in DRAM. A bank consists of 2D cell arrays and peripherals to transfer data between DRAM cells and IOs. The memory cells are grouped into subarrays, each consisting of a row of mats. Each mat has local sense amplifiers (row buffers) sensing a horizontal wordline (WL) through a set of vertical bitlines (BLs). Sense amplifiers in mats of a subarray form the subarray row buffer.Upon receiving access, the bank activates corresponding WL in subarray row buffer and transfers the whole WL to bank-level sense amplifiers via data lines (DLs).Several DRAM-based PIM technologies support operations in different levels of DRAM architecture. The first technology is near-bank processing, which integrates processing elements (e.g., vector ALUs, RISC-V processor, etc.) near the bank SA and IO <cit.>. Each bank-level PE is customized to fully utilize the data link bandwidth for processing. PEs in different banks run in parallel to fully utilize the internal bandwidth in DRAM. The second type of PIM augments the subarray sense amplifiers (row buffers) with compute logic <cit.>. Due to the constrained chip area, the near-buffer PIM only adopts logic gates, full adders, and shift circuits for multi-bit operations. Compared to near-bank processing, near-buffer PIM supports wider input (e.g., 8192b vs. 256b) and can exploit the subarray-level parallelism <cit.>. These two types of PIM work on the data with a horizontal layout where each data is stored across multiple BLs in a WL. The third type of PIM uses a vertical bit-serial scheme that lays out each data in different WLs of a BL <cit.>. The bit-serial PIM directly generates the result of computation between different WLs by exploiting the charge-sharing effect of the DRAM mechanism. Such in-mat bit-serial processing does not introduce significant modifications in DRAM. However, the bit-serial computation is slow and power-consuming where an n-bit multiplication using the bit-serial PIM requires around 7n^2 DRAM activations for 8k values. §.§ Challenges of FHE acceleration using PIM Even though existing PIM solutions can exploit the large internal DRAM bandwidth for high-throughput computation, FHE is still extremely challenging for PIM acceleration.§.§.§ Long-bit multiplicationAs introduced in Section <ref>, the basic data structure of FHE is high-degree polynomial, whose coefficient can exceed 2^1000. The RNS-decomposed polynomials still have at least 28-bit coefficients due to the limitation of modulus selection <cit.>.As the complexity of PIM multiplication significantly increases as the bit-length increases, PIM (especially bit-serial in-mat) may suffer from long latency and high energy efficiency due to costly row activations and precharges. One way to improve the performance and energy efficiency of PIM is by increasing the aspect-ratio (AR) of DRAM mat <cit.>. A high AR mat has fewer WLs (rows) and shorter BLs than a low AR mat. Shorter BLs can significantly reduce the latency and energy of activation and precharge. For instance, ARx4 mat (128 rows) has half the cycle and consumes 33% less energy than ARx1 mat (512 rows) <cit.>. Furthermore, increasing the AR also increases the number of subarrays in a bank, leading to a higher degree of parallelism. The downside of high AR is the large area overhead caused by more sense amplifiers and peripherals.Figure <ref> shows the throughput and energy efficiency of different PIM technologies for 32-bit multiplications on a 32GB HBM2E-based architecture (Section <ref>). We evaluate three existing PIM architectures,FIMDRAM <cit.>, DRISA <cit.>, and SIMDRAM <cit.>, that represent near-bank, near-buffer, and in-mat bit-serial PIM respectively. The result shows FIMDRAM and SIMDRAM provide 6.8TB/sand 180.6TB/s throughput while consuming 49.8pJ and 342.9pJ energy for each operation using ARx8 memory. As a reference, the recent FHE accelerator <cit.>, which adopts 150k 28b multipliers, can provide 1PB/s of peak throughput while consuming only 4.1pJ for each multiplication, indicating both FIMDRAM and SIMDRAM are not promising for FHE. DRISA <cit.> provides over 3PB/s throughput and consumes 6.32pJ for each operation in ARx8 architecture. However, DRISA <cit.> requires a significant change in the DRAM architecture, incurring around 100% area overhead in high-AR architectures. Furthermore, manufacturing DRISA has significant challenges as the modified sense amplifiers cannot easily be aligned with area-optimized bitlines. To tackle these challenges, adopts a novel near-mat processing that integrates compute logic near mat while keeping the mat structure intact, incurring less area overhead than DRISA. Even though the theoretical throughput and energy efficiency of are lower than DRISA, our experiments show that higher throughput may not effectively improve the performance due to the data movement (Section <ref>). Overall, is a more efficient processing paradigm than prior PIM solutions.§.§.§ Data Transfer PatternsAnother critical challenge of FHE for PIM is the variety of data transfer patterns in different FHE operations. Specifically, therequires the data movement between different RNS polynomials, followed by coefficient-wise operations;and automorphism require data permutation across coefficients of each RNS polynomial.Unfortunately, the conventional memory IO cannot efficiently handle such complex data movement patterns.For , each output RNS polynomial has dependencies with all input RNS polynomials. As each RNS polynomial is large (e.g., 512KB for LogN=64 with 64-bit coefficients), we must distribute RNS polynomials over different memory banks. In conventional memory, such inter-bank data movements take up the shared bus of each channel, leading to significant data movement overhead. Forand automorphism, the data movement exhibits a fine-grained pattern within a polynomial. For PIM processing, the coefficient-wise permutation requires permutations between BLs which is not supported in the current memory architecture. supports these FHE-specific data transfers efficiently at a relatively low cost by exploiting existing intra-memory data links and adding additional links to the less-dense metal layer in DRAM, without introducing complex permutation networks. § HARDWARE ARCHITECTURE The high-level architecture of is based on high-bandwidth memory (HBM), as shown in Figure <ref>. Specifically, each HBM stack consists of one base die and multiple DRAM dies in a 3D structure. All DRAM dies are divided into multiple channels, each connecting to DRAM part through an independent set of through silicon vias (TSVs). Each channel consists of several banks, and the detailed internal structure of the bank is introduced in Section <ref>. adopts a new near-mat PIM architecture that modifies the DRAM bank architecture to support high-throughput computations while utilizing available DRAM internal links for various FHE operations. The key customized components of include near-mat units (NMUs), horizontal data links, and inter-bank connection. §.§ Near-mat unit supports in-memory computations by connecting each mat to a near-mat unit (NMU) via the local data lines (LDLs). Each NMU consists of full adders, shifters, AND logic, and latches to compute FHE operations, shown in Figure <ref>(a). In addition to the mat sense amplifiers, NMU can also receive data from other NMUs via inter-NMU connection. The design of NMU differs from previous PIM-logic integration, DRISA <cit.>, in three ways. First, NMU places all customized logic outside the mat, which is optimized for area efficiency. On the contrary, DRISA <cit.> integrates logic and latches with the mat sense amplifiers and bitlines, which may incur significant changes to the area-optimized mat. Second, DRISA <cit.> integrates logic to all bitlines in a mat, causing a large area overhead while only gaining moderate performance benefits due to the unbalanced compute and data movement <cit.>. In , we explore the processing throughput of NMU under different architecture configurations and observe the most efficient design does not adopt the maximum throughput (Section <ref>). Third, NMU in can support permutations required by FHE using multiplexers in the data path.To compute FHE arithmetic (i.e., modular arithmetic), NMU requires several steps in mat, data links, and NMU. Figure <ref>(b) shows an example of processing 4 64b multiplications in an NMU with 2 64b adders. For generality, we denote each mat row can store N n-bit values (N=4 and n=64 in this example), and NMU has M n-bit adders. First, the mat must activate an operand row (1) and transfer the row to the row-size operand latches (2). Next, the mat activates the second operand row (3) and transfers M-value blocks to the shifter and AND logic (4). When both operands are ready, the shifter and AND logic will generate a partial product using the second operand (b0 and b1) and bit masks of a specific bit of the first operand (a0 and a1 in the latches). NMU takes n cycles to compute an n-bit multiplication (5). After processing an M-value block, NMU writes the result back to the mat (6) and loads the next M-value block for processing (7). Like DRISA <cit.>, NMU only needs two row activations for each vector processing but requires serial data transfers via LDLs. NMU can serially write back values in a different order to support permutation. §.§ Inter-NMU Connection To efficiently process various FHE data transfer patterns, enables data transfer between NMUs in horizontal and vertical directions. In the vertical direction, utilizes the master data lines (MDLs) which connect all NMUs in the same mat column. In the horizontal direction, adds extra data links, horizontal data links (HDLs), to each subarray (a row of mat). The HDLs in each subarray support the same bitwidth as the MDLs in each mat column (i.e., 16-bit). For both directional links, we add small isolation transistors (switch in Figure <ref>) <cit.> to serve as switches that can disconnect each link at a certain point. NMUs separated by the off switches can transfer data independently, significantly improving the bandwidth for intra-bank and intra-subarray data movements. In , each horizontal and vertical inter-NMU connection transfers a 512-bit mat row to another mat, where Figure <ref> shows the detailed steps. Before transferring, activates the participating row(s) in subarray(s). The horizontal (vertical) transfer requires the activation in 1 (2) subarray(s). then turns on or off the switches based on the transfer pattern (Section <ref>). To reduce the area and energy cost, uses a single control signal for all switches in a row for vertical links (MDLs), similar to LISA <cit.>; for horizontal links (HDLs), adopts a single control signal for all switches in a column. The bank-level logic has the bookkeeping logic for switches' states. Setting up switches requires up to 16 cycles because we map each polynomial to a 16× 16 mat array (Section <ref>). Last, the subarray controller controls the near-mat peripheral to send/receive data based on the transfer pattern. Transferring 512-bit data via 16-bit data links requires 32 cycles. §.§ Inter-bank Connection As introduced in Section <ref>, FHE has dependencies between RNS polynomials of a ciphertext during . To store a ciphertext, which can consist of tens of RNS polynomials, we need to distribute these RNS polynomials over multiple banks, and each BConv requires a large amount of inter-bank data transfers because of an all-to-all dependency. For such inter-bank data movements, a naive way is to use the conventional channel-level data bus.Unfortunately, such centralized data movements fail to match the throughput of in-memory computations, significantly slowing down the acceleration.Furthermore, the cost of fully-connected interconnect can be prohibitively high. To support the inter-bank communication with satisfactory performance, while introducing reasonable overhead in HBM, we propose a partial chain interconnect network between banks inside a channel.The partial chain network connects neighboring banks in each bank group, using 256b-wide transfer links and per-bank transfer buffers. Specifically, the transfer buffer in each bank can communicate with the local master data lines (MDLs) and the transfer links to the neighboring banks. We add two 256b transfer buffers in each bank to support seamless transfers between banks. The customized links and buffers enable parallel inter-bank data transfers across different banks in a channel, avoiding sequential transfers via the original channel IO. When transferring a whole row between two neighboring banks, the source bank drives 256b data blocks from the selected subarray SA via MDLs to the transfer buffer, which sends data blocks to the transfer buffer of the destination bank via either the customized links (neighboring banks) or the original channel IO (non-neighboring banks). The destination bank writes data blocks to the selected subarray SA via MDLs.§.§ Controllers needs modifications in bank/subarray controllers to support its various functions.To support the in-memory computation, exploits the existing control logic of SIMDRAM <cit.>. Figure <ref>(a) shows the processing flow of hierarchical control with the annotation of extension. The host CPU sends bbop instructions to each memory controller (i.e., channel-level controller in HBM).requires an extension of bbop instructions for modular arithmetic and permutation for FHE. Each memory controller has a micro-program control logic that translates each bbop to a micro-program which is a sequence of DRAM subarray-level commands.Previous works <cit.> has shown the micro-program control logic only incurs negligible area overhead (less than 0.1mm^2).extends the bank-level control to decode NMU commands and dispatches signals to subarray-level logic and the isolation transistors (switches). requires the extra logic for subarray-level parallelism <cit.>, including the bookkeeping logic in the bank controller for the status of all subarrays and the extra address latches in each subarray row decoder. The subarray-level control sends signals to all NMUs in the subarray based on the command. Furthermore, adds more latches in the subarray column decoder to support the permutation operations. The extra logic and control signals in also introduce insignificant overhead (Table <ref>) because they are placed outside the mat. adds several new subarray-level commands for NMU processing, as shown in Table <ref> and Figure <ref>(b). The subarray-level commands control the same behaviors of all mats/NMUs in a subarray, except nmu_pst which stores different latches in different mats back to SA (used for automorphism).For NMU loading and storing, supports the flexibility of selecting columns in NMU latches, adder latches, and sense amplifiers, enabling permutation. The horizontal data movement has predefined patterns, defined by direction and stride, to support NTT data movements. The vertical data movement has more flexibility to transfer data between two subarrays. The add command in indicates the latch-adder pair for computation and whether to use shift/AND and the bit position of shift/AND. To exploit subarray-level parallelism and minimize the command patching latency (i.e., minimize the number of commands), most commands, except the permute store (nmu_pst), can be configured for processing large data size.Figure <ref>(b) shows the format of the 7 commands (3-bit opcode).To support the precision requirement of FHE, each command processes 64-bit data. Therefore, the column/latch address and the size of data of near-mat processing can be presented by 3 bits (i.e., 8 possible 64-bit data in 512-bit row in a mat). In our architectural exploration (Section <ref>), each bank has 128 (ARx1) to 1024 (ARx8) subarrays, requiring 10-bit to denote the subarray. Each subarray contains 16 mats so we can use 3-bit mat id, 1-bit direction and2-bit stride, to represent all horizontal movements in a subarray. The addition command includes a range of shift&AND for multiplication - both the start and end shift steps can be represented by 6-bit (i.e., up to 64 bits). For the permute store command, a 48-bit field is reserved for the vector of latch addresses in 16 NMUs of a subarray. The issue time of 32-bit (64-bit) command to each bank via 16-bit command/address bus is 2 (4) cycles, respectively. §.§ Practicality of Commodity DRAMs are optimized for cost so that DRAM process only adopts 3 metal layers, with 1 layer (M1) for bitlines (vertical), 1 layer (M2) for LDLs and master word lines (MWLs) (horizontal), and 1 layer (M3) for column select lines (CSLs) and MDLs <cit.>. M1 layer is the only fine-pitch (low energy efficiency) layer, optimized for the area of bitlines in a mat. M2 and M3 support more energy-efficient wires with larger area overhead (i.e., 4x pitch of M1).The design of considers the cost-efficiency of modifying commodity DRAM, avoiding the high-cost changes in cost-sensitive components. In , the additional horizontal data lines are placed in M2 which incurs less pressure in the routing.M3 (vertical) is more dense than M2 (horizontal) because both CSLs and MDLs on M3 extend across multiple mats while M2 only has MWLs shared by multiple mats. Furthermore, HDLs connect NMUs, which are placed outside of dense mats (DRAM cell arrays).As illustrated in previous work <cit.>, changes near the mat are the most costly, including bitline sense amplifiers, local wordline drivers, row logic, and column logic. Changes outside the subarray logic are relatively inexpensive because of the low-density logic blocks. These characteristics make HDLs and NMUs more practical than previous PIM solution that integrates complex logic in bitline sense amplifiers (e.g., DRISA <cit.>) in conventional DRAM technology. § OPTIMIZED PROCESSING FLOW OF FHE IN The proposed hardware supports high-throughput computations and data movements. The next challenge is to efficiently utilize for FHE applications. This section introduces an optimized end-to-end processing flow that determines the data layout and processing flow of for FHE applications by adopting several algorithm and compiler-level optimizations. §.§ Data Layout In PIM acceleration, data layout is critical to determine the detailed computation and data movement.Figure <ref> shows the optimized data layout in . Each ciphertext contains a group of RNS polynomials, each of which is a vector of N b-bit integers. To exploit the high throughput of PIM, we distribute RNS polynomials of a ciphertext, including the original RNS terms (RNS_ql) and the special terms (RNS_pk) used for key switching, across multiple banks using a round-robin method. The figure shows an example of allocating two original RNS terms and two special RNS terms in a bank. §.§.§ Layout in subarray groupsWe divide subarrays into subarray groups, basic memory partitions for polynomials. Specifically, each subarray group contains a continuous set of subarrays (e.g., 2 subarrays) which is a 2D array of mats (e.g., 2×2). The 2D distribution allows to balance the inter-mat data movements during various FHE operations (esp. NTT). distributes coefficients of a polynomial across the mat array in an interleaved way, similar to previous work <cit.>, to efficiently support automorphism (Section <ref>). In our setting, each subarray group contains 16 subarrays (16 × 16 mats), requiring each mat to use 32 rows to store 256 64-bit coefficients.§.§.§ Layout for computationFor a computation using two RNS polynomials, we align both polynomials in the same column in a subarray.Each subarray group reserves rows for operand polynomials used for computation, including key-switching keys, constant polynomials, and other ciphertexts. If a subarray group computes with two polynomials in different subarray groups, the memory issues data movements that may happen inside a bank or across different banks. §.§.§ Layout for constantsIn addition to RNS polynomial, we need to allocate rows for several constants for FHE operations, includingtwiddle factors, moduli, scaled inverse moduli, etc. To avoid duplicating twiddle factors across NTT steps that require large memory, we store the vector of twiddle factors which contains N-th roots of unity, in the same order as the polynomial coefficients. Before each (i)NTT stage k, first dynamically computes the twiddle factor ω^ik for coefficient i by multiplying the twiddle factor in the previous stage with ω^i. For moduli, we keep one copy in each mat in the subarray group, so that each NMU can load the corresponding value independently during the computation. §.§ Algorithm-optimized modular reduction A modular reduction follows each FHE arithmetic. We exploit the Montgomery algorithm <cit.> requiring two multiplications, one addition, and one subtraction. NMUs in processes n-bit multiplication using n serial additions. Therefore, we exploit algorithm optimization to significantly reduce the latency and energy for modular arithmetic. Specifically, we select moduli that are friendly to serial computations while satisfying security requirements and (i)NTT. We exploit the moduli selection technique proposed in previous works <cit.> that select moduli has the form of 2^b± 2^sh_1± 2^sh_2± .. ± 2^sh_h-2± 1, where h is called hamming weight. Using a modulus with a hamming weight of h, we only need to issue h additions during the multiplication, hence reducing the addition steps from n to h.The hamming weight optimization only applies to computations with constant, including the multiplication with modulus and the multiplication with reduction factor in Montgomery reduction. The advantage of Montgomery reduction over Barrett reduction <cit.> is that both reduction factor and modulus in Montgomery reduction can have a low hamming weight, and it only requires single bit-length computation. We note that prior FHE accelerators <cit.> also adopted a similar optimization that customized the modular multiplier for Montgomery-friendly moduli. However, their modular multipliers cannot process computations using moduli with different characteristics (e.g., required by other applications). provides more flexibility by using addition as the basic computing step. §.§ NTT executes several steps of permutation and computations. Eachstep requires several butterfly operations (Figure <ref>(a)) on pairs of coefficients, that multiply the coefficients with the twiddle factors, permute the coefficients, and then update the coefficients.processes each (i)NTT operation in three stages, intra-mat, horizontal inter-mat, and vertical inter-mat, depending on the butterfly stride of each (i)NTT step (Figure <ref>(b)).For intra-mat steps, where coefficients of each butterfly operation are in the same mat, NMUs in a subarray group independently process computation and permutation. The horizontal inter-mat steps exchange the coefficients between mats in the same row, for which uses HDLs for efficient data transfers, as shown in Figure <ref>(c). Specifically, turns on/off the switches of NMUs on HDLs, where the connected segments can independently transfer data.Data transfers using the same connected segment are scheduled sequentially. Therefore, as the number of connected segments changes over (i)NTT stages, the transfer latency of different (i)NTT stages varies. The vertical inter-mat NTT steps are processed similarly to the horizontal steps but using MDLs. The key novelty of on (i)NTT operations is that does not introduce complex butterfly networks in the memory. Instead, exploits the existing DRAM internal links (i.e., MDLs and LDLs) with efficient customizations (i.e., NMU, switches, HDLs). §.§ Base Conversion BConv is a costly but frequent operation in FHE.As introduced in Section <ref>, to generate each special RNS polynomial of(with modulus pk), each input RNS polynomial with modulus qi first multiplies [q̂î^-1]_qi and [q̂î]_pk. Such partial products are reduced to each special RNS polynomial pk. parallelizes multiplications in different subarray groups with different input polynomials. To reduce the partial products, first accumulates partial products in the same bank using NMUs and MDLs because partial products of different polynomials in a bank are aligned either in the same subarray group or in different subarray groups in the vertical direction. Therefore, the intra-bank accumulation can be processed in an adder-tree manner by exploiting switches in MDLs. processes the final reduction in the bank of the output polynomial, requiring data transfers of partial products from all other banks. handles such inter-bank data movements using the customized interconnection (Section <ref>). To parallelize the computation, each bank processes different output polynomials simultaneously. determines the optimized schedule based on the number of banks used for the ciphertext, the number of input/output RNS polynomials, and the underlying interconnect structure. §.§ Automorphism Automorphism is a process that permutes the coefficients of a polynomial by using Galois group. supports automorphism based on the observation from BTS <cit.>: the interleaved coefficients (Section <ref>) in the same tile (mat in ) will be mapped to a single tile after automorphism. further extends this idea to interleave coefficients in one more dimension, memory row, where the column c of row z of a mat (x,y) stores coefficients with the indices cN_xN_yN_z+zN_xN_y+yN_x+x. With such coefficient mapping, the automorphism only requires three steps: permutations in each row, vertical inter-mat permutation, and horizontal inter-mat permutation. can handle the first step in NMU and the last two steps using MDLs and HDLs respectively.§.§ Application Mapping Framework for can provide large throughput when fully utilizing memory. However, it is not trivial to map a full FHE program to hardware with high utilization.Thus, we propose a mapping framework that generates data layout and scheduling in a pipeline manner that can fully utilize the memory to process multiple input data in parallel. §.§.§ Framework OverviewFigure <ref> shows an example pipeline for the CoefToSlot step in CKKS bootstrapping. The input of our framework is an intermediate representation extracted from the real FHE program. Our framework generates a trace of FHE operations (e.g., HMul, HAdd, and HRot) in the static single-assignment (SSA) form while unrolling all loops. Our framework then divides the operation trace into multiple pipeline stages. The example shows the first 8 pipeline stages for CoefToSlot in a simplified HBM model with 2 channels. After computation on a stage, the allocated memory needs to transfer data to the memory that processes the pipeline steps with a data dependency. Therefore, the latency of each pipeline stage includes loading time, computation time, and transfer time. Our framework aims to minimize the latency of the bottleneck stage in the pipeline.§.§.§ Memory Allocation for Pipeline StagesOur framework allocates each stage to a basic allocation memory unit, whose size is determined by the FHE parameter setting including the polynomial degree, ciphertext scaling factor, and ciphertext moduli. In Figure <ref>, the basic memory unit is one bank. To process a stage, we need sufficient memory to support the data layout shown in Section <ref> for the input and output data. Extra memory is needed for constant data (e.g., evk and plaintexts). The ideal case is each allocation unit can hold all data for the stage. When a memory allocation unit cannot hold all data, we store constant data in a reserved memory location called data memory. When data is stored in the data memory, all operations (across all stages) requiring the data need to dynamically load it. We reduce the memory footprint by storing data in one place, instead of duplicating them in different memory locations. §.§.§ Load-save PipelineA naive way of pipeline generation is to divide the FHE program into n stages, where n is the number of available allocation units in memory. However, for large applications, each stage may require a large memory footprint for operations, leading to frequent constant loading. As shown in Figure <ref>(a), a stage has 4 operations, and every operation needs to load constants because memory is occupied by the constants of previous operations. In , we propose a load-save pipeline by dividing operations into fine-grained stages with a small enough footprint. The fine-grained stages are allocated to different memory in a round-robin manner, requiring multiple rounds to process all stages for an input. In each round, the memory only creates a pipeline with part of the program (f1 and f2 in the first round in Figure <ref>(b)). It runs through a batch of input with only 1 data loading at the beginning of each round. Each memory loads the next round of stages when the current input batch is completed. The load-save pipeline minimizes the data loading while still fully utilizing the memory for computation.§ EXPERIMENTAL SETUP§.§ Hardware EvaluationMemory Technology: The basic architecture of is similar to HBM2E <cit.>. The system configuration is shown in Table <ref>. Each HBM2E stack has 16GB with 16 physical channels. We scale energy and power values from 22nm used in previous work <cit.> to 10nm (shown in Table <ref>), based on the recent HBM2E technology <cit.>. We follow the method of Vogelsang <cit.> to calculate the scaling factors for energy.We assume 16 physical channels on a stack are connected by a crossbar on PHY where each bidirectional link is 64-bit wide (bisection bandwidth=64GB/s per stack). We also add stack-stack links for scaled-up systems, commonly used in memory-centric architecture <cit.>. has two HBM2E stacks to support 32GB memory. We exploit the remaining signaling links on HBM2E for stack-stack connection so the inter-stack bandwidth is also 256GB/s.Hardware Modeling: To evaluate the area and power of customized components, we synthesize our design in 45nm technology using Nangate Open Cell Library. We model all other CMOS components (including buffers and interconnects) in Cacti <cit.> at 32nm technology. We scale all values to the 10nm technology with the scaling factors calculated from previous work <cit.>. We estimate the delay, power, and area overhead of integrating CMOS-ASIC and DRAM technologies based on the difference in number of metal layers and complexity of the customized logic <cit.>. Simulation: We generate FHE operation traces from software implementations of different workloads, and our mapping framework optimizes the trace and generates PIM instructions for simulation. Our in-house simulator adopts a cycle-accurate trace simulation based on the standardized DRAM latency constraints, similar to Ramulator <cit.>. We model control logic at different levels in the DRAM hierarchy.§.§ WorkloadsLogistic Regression (HELR) <cit.>: This workload has 30 iterations of homomorphic logistic regression where each iteration trains 1024 samples with 256 features as a batch. The multiplication depth is deep, requiring several bootstrappings.ResNet-20 <cit.>: The ResNet-20 is a homomorphic neural network inference for one CIFAR-10 image classification. The network is deep with multi-channel convolutions, matrix multiplications, and approximated ReLU function.Sorting <cit.>: Sorting uses 2-way bitonic sorting on an array with 16,384 elements, the same as that used in SHARP <cit.>.Bootstrapping <cit.>: We evaluate the bootstrapping algorithm using a similar framework as previous work <cit.>, which requires 15/30 levels of bootstrapping. We adopt the minimum-key method used in previous work <cit.> to reduce the rotation keys used in bootstrapping.Shallow neural network inference (LOLA) <cit.>: We also evaluate two shallow workloads without bootstrapping, including a network for MNIST (LOLA-MNIST) and a larger network for CIFAR-10 (LOLA-CIFAR), used in CraterLake <cit.>. §.§ FHE Parameters and Evaluation We evaluate the efficiency of on a 128-bit security FHE parameter setting chosen from Lattigo <cit.>.For workloads with bootstrapping, including HELR, ResNet-20, and sorting, we use logN=16, L=23, dnum=4, and logPQ=1556, similar to prior accelerators <cit.>. Each polynomial is decomposed into 40-61 bit RNS terms, where allocates 64-bit for each coefficient. For shallow LOLA workloads, we use the similar parameter settings with CraterLake <cit.>, where we assume logN=14, L=4/6, and logq_i<=32. represents 32-bit coefficients in 64-bit words and packs 4 RNS logN=14 polynomials together in 16 subarrays. For all workloads, we measure the maximum time across all pipeline stages, indicating the time we can finish an input when the pipeline is full. In addition, we consider the number of pipelines that can be processed simultaneously in the system when the program cannot fully utilize the memory capacity (e.g., 32GB). § EVALUATION§.§ Comparison to Previous FHE Accelerators Figire <ref> compares with two state-of-the-art ASIC FHE accelerators (CraterLake <cit.>, and SHARP <cit.>). We explore two design choices of memory organization that play important roles in the performance, power, and area of : the aspect ratio of DRAM mat (AR), and the width of adders in a subarray (if each NMU has 2 64-bit adders, a subarray with 16 mats has 2k-width adders).As discussed in Section <ref>, high-AR architecture has better performance and energy efficiency than low-AR architecture, but incurs significant area overhead. The width of adders determines the performance of arithmetic computing - wide adder designs support faster computing while requiring a larger area than narrow adders.For the chip area of prior ASIC accelerators, we add the area of 32GB HBM2E (2×110mm^2) for a fair comparison. §.§.§ Performanceshows superior performance over prior FHE accelerators. Specifically, ARx8-8k (lowest EDP) is 4.4× (8.8×), 2.2× (3.4×), and 5.4× (17.5×) faster than SHARP <cit.> (CraterLake <cit.>) on bootstrapping, HELR, and ResNet-20 respectively. ARx4-4k (lowest EDAP) is 3.4× (6.8×), 1.7× (2.6×), and 4.1× (13.2×) faster than SHARP <cit.> (CraterLake <cit.>) on bootstrapping, HELR, and ResNet-20 respectively. For sorting, ARx8-8k (ARx4-4k) is 4.2× (3.1×) faster than SHARP <cit.>. For LOLA-MNIST and LOLA-CIFAR, ARx8-8k (ARx4-4k) are 3.0× (2.1×) and 3.2× (2.3×) faster than CraterLake <cit.>. The less significant performance improvement in HELR results from the low portion of bootstrapping which is significantly optimized by adopting the minimum-key optimization <cit.>.§.§.§ EfficiencyWe then compare the power and area efficiency of and ASIC accelerators.Compared to SHARP, ARx8-8k improves EDP and EDAP by 8.2× and 5.1×. However, ARx8-8k requires 1.6× larger area and 2.3× higher power than SHARP. ARx4-4k improves EDP and EDAP of SHARP <cit.> by 6.2× and 6.9×, with 0.9× area and 1.8× power consumption. ARx2-2k is a configuration that provides the best performance using less area (0.65×) and power (0.94×) than SHARP. For performance, ARx2-2k is 1.56×, 0.92×, and 1.96× faster than SHARP, leading to 2.59× and 3.96× EDP and EDAP improvement.§.§.§ Analysis of BenefitsCompared to ASIC accelerators, provides a higher throughput due to the efficient in-memory computation and large intra-memory bandwidth. For instance, ARx4-4k has 16 million 64-bit adders. Considering the cost of DRAM row activations, data transfers, subarray-level parallelism, and 500MHz frequency of additions, the effective throughput of ARx4-4k for 64-bit multiplication is around 637.61 TB/s. During multiplication, the adders consume most of the energy because row activation energy is amortized for the entire row, and data transfer is energy-efficient due to the short wire length. Therefore, the energy efficiency of computation is similar to the modular multiplier used by FHE accelerators (slightly higher due to the DRAM-CMOS integration). For the internal bandwidth of NTT, ARx4 supports a 256-bit link (500MHz) for each of the 512 subarrays in a bank. Considering up to half of the subarrays can transfer data simultaneously during NTT, the peak internal bandwidth for NTT is 2048 TB/s in 32GB ARx4 . For the slowest NTT step, the internal bandwidth drops by 16× (128 TB/s). As a comparison, SHARP <cit.> has around 24K 36-bit multipliers running at 1GHz, leading to 221.18 TB/s throughput. Furthermore, the on-chip memory resources in SHARP support 72TB/s bandwidth. §.§ Comparison across different ConfigurationsAs shown in Figure <ref>, There is a significant difference in power and area between different configurations. For example, the most oversized (ARx8-8k) requires 642.32mm^2 chip area and 218W power, while the smallest (ARx1-1k) only requires 223.81mm^2 chip area and 36.24W power. As a reference, the commercial 2-stack HBM2E has a chip area of 220mm^2 <cit.> and the power budget of a conventional HBM system is 60W <cit.>. We note that the power budget of conventional HBM is different from the accelerator design targeted in this work, where high power consumption is reasonable if it meets the thermal requirement (e.g., 10W/cm^2/layer <cit.>).Based on the results, high-AR provides higher performance than low-AR designs because increasing AR can increase both compute and data movement throughput inside a bank. For ARx1 and ARx2 configurations, doubling AR can provide 1.57× to 1.98× speedup because the execution is compute-bound. For ARx4, doubling AR only provides 1.23× to 1.67× speedup. Increasing adder-width exhibits a similar trend where the effect of increasing compute resources diminishes for high-throughput architectures.To find the most cost-efficient design, we evaluate energy-delay-product (EDP) and energy-delay-area-product (EDAP) for different configurations. For EDP, the trend follows the performance, where the largest (ARx8-8k) gives the lowest EDP. When considering the area cost, different ARs favor different adder widths. Specifically, ARx8 and ARx4 exhibit the lowest EDAP at 2k and 4k adder-width respectively. The configuration with the lowest EDAP (ARx4-4k) is 1.34× more efficient than ARx8-8k. §.§ Latency and Energy Analysis Figure <ref> shows the latency and energy breakdown of different configurations. Considering the parallel processing, we accumulate all latency values across all memory banks for the latency breakdown. We divide all operations into 7 categories, including computation (subarray activation/precharge, operand transfer, and addition), permutation (inter-mat transfer), read/write (activation and precharge for data transfers), and inter-bank/channel/stack IO traffics. The breakdown results provide several key insights into . First, in low-AR , the latency is dominated by computation and permutation because of the limited throughput of computation and intra-bank data movements. Increasing AR can effectively reduce the latency for both computation and permutation latency. Furthermore, increasing the adder width can effectively reduce the computation latency. However, in high-AR , the data movement becomes the performance-dominant operation, especially inter-bank data movements (mainly caused by BConv). This proves the necessity of customized inter-bank links. We analyze the detailed effect of optimizations in Section <ref>. For energy consumption, consumes most of the energy on computation and permutation, which incurs intensive row-activation and intra-bank data movements. §.§ PIM Technologies Figure <ref> compares the efficiency of to other PIM technologies, including SIMDRAM <cit.> and DRISA <cit.>. For a fair comparison, we use the proposed application mapping and customized data links in baseline PIM architectures, while evaluating the difference in processing. We implement an adder-less NMU for permutations in baselines. We select the most efficient (lowest EDPA) for each AR. The results show is 183.7× to 255.4× faster than SIMDRAM <cit.>, and 2.76× to 6.75× faster than DRISA-logic <cit.>. Furthermore, is at least 19,300× and 47× more efficient than SIMDRAM and DRISA-logic using EDAP. Compared to DRISA-add <cit.>, is 1.14× to 1.21× slower from ARx8 to ARx1 because DRISA's adders can directly access the sense amplifiers. However, is 1.04× (ARx1) to 1.51× (ARx8) more efficient in EDAP because does not introduce area overhead in a mat. Furthermore, DRISA is more challenging to manufacture because the large adder area will affect the alignment of cost-optimized bitlines. As a comparison, puts all customized logic outside the mat structure. §.§ Overhead Analysis Table <ref> shows the area and power breakdown for our customized hardware components of based on 1 HBM2E stack (16GB). We exploit the Cacti-7 <cit.> to generate the area breakdown of HBM and rescale the values to the published work <cit.>. The table shows a structure of ARx4 HBM and each subarray contains 4k-wide adders. All area values are for a single layer. The large area mainly comes from the near-mat adders. The horizontal data links use the same technology as the global data lines with the consideration of energy efficiency (4× larger than local bit-lines). We extract the capacitance of material and scaling method from previous work <cit.> to calculate the energy consumed by data transfer. We note that the target of is not a cost-optimized memory product, but a specialized accelerator for emerging applications that require high-throughput computation and/or efficient intra-memory data movements. However, the proposed design is still practical regarding area and power overhead. First, unlike previous near-subarray PIM <cit.>, put the customized logic outside DRAM mat, avoiding the issue of aligning cost-efficient local bitlines. Second, can maintain an average DRAM working temperature (under 85^∘C). Previous work <cit.> shows a 16-high compute-centric 3D memory can tolerate 10W/cm^2 per memory layer to keep DRAM temperature under the 85^∘C limit with a commodity-server active heat sink. For example, the power consumption and area of 8-high ARx4 4k are 173.9W and 367mm^2, resulting in a power density of 5.92W/cm^2 (the highest power density in our exploration). §.§ Evaluation of Optimizations We compare with baseline systems enabling a subset of optimizations, as shown in Figure <ref>.§.§.§ Montgomery-friendly ModuliThe Montgomery-friendly moduli can reduce the number of addition steps, improving the computation performance. As shown in Figure <ref>, Base1 is 1.68× and 1.58× faster than Base0 on HELR and ResNet with ARx2-2k architecture. On Arx4 and Arx8 architectures, this optimization only improves the performance by 1.17× and 1.06×. However, Montgomery-friendly moduli can reduce energy consumption by 1.75× because computation is energy-dominant across all architectures (Figure <ref>).§.§.§ Interconnect NetworkThe comparison between and Base1 shows the efficiency of the proposed inter-bank network, where Base1 uses the existing HBM channel IO for all inter-bank data movements. Based on our results, the proposed inter-bank network can improve the performance by 1.31×, 1.86×, and 2.12× on ARx2, ARx4, and ARx8 respectively. The inter-bank network can reduce the latency of related data movement by 3.2× on average.§.§.§ Load-save Pipeline MappingThe comparison between and Base2 shows the efficiency of the load-save pipeline mapping. For HELR, load-save pipeline mapping improves the performance by 3.59× (1.77×), 3.12× (1.54×), and 2.38× (1.15×) on ARx8, ARx4, and ARx2, respectively, for HELR (ResNet). The significant performance improvement results from reducing frequent data loading, especially data from a remote stack. § RELATED WORK Several FHE-specific accelerators have been proposed recently in the architecture community <cit.>.HEAX <cit.> is an FPGA-based accelerator targeting the CKKS FHE scheme, and accelerates FHE primitives (e.g., multiplication and key-switching) by up to 200×; HEAWS <cit.> is also an FPGA implementation on Amazon AWS FGPAs, which accelerates B/FV FHE scheme (5× on a microbench).CraterLake <cit.> adopts wide vector processor with specialized high-throughput units forand on-chip key generation. BTS <cit.> exploits relatively low throughput function units with large inter-PE crossbar network. ARK <cit.> uses an algorithm-hardware co-design to significantly reduce the off-chip bandwidth for bootstrapping evk and plaintext polynomials. SHARP <cit.> further improves the performance ARK by using low bit-precision data path (36-bit vs. 64-bit in ARK). However, their technique does not apply to general evk and ciphertexts, leading to the same memory issues, as analyzed in Section <ref>. exploits the large internal memory bandwidth of memory-based acceleration to unleash the processing throughput in a more area and power-efficient way.CryptoPIM <cit.> is a ReRAM-based PIM accelerator with customized interconnect for NTT operations, lacking the support for more general FHE operations. In-storage processing is another promising technology to accelerate big-data applications <cit.>. INSPIRE is an in-storage processing system for private database queries based on FHE by integrating FHE logic (e.g., NTT and permutation) in the SSD channels. INSPIRE only supports small FHE parameters (i.e., N=4096) and its throughput is limited by the number of SSD channels. MemFHE <cit.> is a ReRAM-based PIM accelerator with the customized data path for bit-level TFHE scheme, not applicable to CKKS and other packed FHE schemes. § CONCLUSIONThis work proposes , an FHE accelerator based on a novel near-mat processing architecture with relatively lightweight hardware modifications in conventional HBM. We also propose the end-to-end processing flow with a mapping framework for PIM-based FHE. Our evaluation shows is 6.9× more efficient than prior-art ASIC accelerator.§ ACKNOWLEDGEMENTThis work is supported by SRC HWS, SRC PRISM, and NSF fundings #2003279, #1911095, #1826967, #2100237, #2112167, #2052809, #2112665. IEEEtran | http://arxiv.org/abs/2311.16293v1 | {
"authors": [
"Minxuan Zhou",
"Yujin Nam",
"Pranav Gangwar",
"Weihong Xu",
"Arpan Dutta",
"Kartikeyan Subramanyam",
"Chris Wilkerson",
"Rosario Cammarota",
"Saransh Gupta",
"Tajana Rosing"
],
"categories": [
"cs.AR",
"cs.CR"
],
"primary_category": "cs.AR",
"published": "20231127201138",
"title": "FHEmem: A Processing In-Memory Accelerator for Fully Homomorphic Encryption"
} |
[email protected] of Engineering of the University of Porto and INESC-TEC Portugal [email protected] of Engineering of the University of Porto and INESC-TEC Portugal Linked Data is used in various fields as a new way of structuring and connecting data. Cultural heritage institutions have been using linked data to improve archival descriptions and facilitate the discovery of information. Most archival records have digital representations of physical artifacts in the form of scanned images that are non-machine-readable. Optical Character Recognition (OCR) recognizes text in images and translates it into machine-encoded text. This paper evaluates the impact of image processing methods and parameter tuning in OCR applied to typewritten cultural heritage documents. The approach uses a multi-objective problem formulation to minimize Levenshtein edit distance and maximize the number of words correctly identified with a non-dominated sorting genetic algorithm (NSGA-II) to tune the methods’ parameters. Evaluation results show that parameterization by digital representation typology benefits the performance of image pre-processing algorithms in OCR. Furthermore, our findings suggest that employing image pre-processing algorithms in OCR might be more suitable for typologies where the text recognition task without pre-processing does not produce good results. In particular, Adaptive Thresholding, Bilateral Filter, and Opening are the best-performing algorithms for the theatre plays' covers, letters, and overall dataset, respectively, and should be applied before OCR to improve its performance.<ccs2012><concept> <concept_id>10010405.10010497.10010504.10010508</concept_id><concept_desc>Applied computing Optical character recognition</concept_desc><concept_significance>500</concept_significance></concept><concept><concept_id>10002951.10003227.10003392</concept_id><concept_desc>Information systems Digital libraries and archives</concept_desc><concept_significance>500</concept_significance></concept></ccs2012> [500]Applied computing Optical character recognition [500]Information systems Digital libraries and archives Optimization of Image Processing Algorithms for Character Recognition in Cultural Typewritten Documents Carla Teixeira Lopes January 14, 2024 =======================================================================================================§ INTRODUCTION Cultural heritage institutions have the role of protecting, preserving, and facilitating access to cultural knowledge. In the current digital era, cultural material can be digitally preserved and remotely accessed by the public. Cultural heritage institutions have been adopting Linked Data to improve archival descriptions and information discovery <cit.>. Linked data describes a web of data; it refers to a machine-readable structured data network connected to and from external data sets <cit.>. In cultural heritage, records’ metadata, artifacts, and concepts can be linked to improve data exploration <cit.>.Digitalization methodologies are increasingly being applied to allow remote access to archival content. As digitalization in cultural heritage contents usually results in non-machine-readable representations, an initial conversion to a machine-readable format is needed to fully explore the potential of these representations in Linked Data implementations. This necessity happened in EPISA (Entity and Property Inference for Semantic Archives), a project that explores the use of Linked Data in the Portuguese Archives. As most of the digital representations in the Portuguese Archives are non-machine-readable, we first needed to convert them to a machine-readable format to build a tool <cit.> that automatically suggests description values using the content of digital representations.The process of recognizing handwritten or printed text in digital images and translating them into machine-readable documents is called Optical Character Recognition (OCR). The success of the OCR process depends on the quality of the scanned image, and it is common for heritage documents to suffer some degree of degradation over time. From uneven illumination to erased characters and angled digital representations, image processing methods can be applied before the text recognition phase to improve image quality and overall text extraction. Various parameters can tune image processing algorithms. Using default parameter values may yield bad performance results compared to the optimized combination of parameters. Therefore, the task of assigning values to parameters is essential.We explore several image processing algorithms, using OpenCV, an image processing Python library with the Tesseract OCR engine. To solve the parameters’ optimization problem, we apply a non-dominated sorting genetic algorithm(NSGA-II)that instantiates the values of parameters of the image processing algorithms. In the end, the algorithm generates a solution set for each pair of image processing algorithms and type of digital representation. In the evaluation stage, we compare the OCR performance using different image processing algorithms with the default and the obtained parameters.The main contributions of this article are summarized as follows:* We analyze the effectiveness of NSGA-II for the parameterization of image processing algorithms;* We compare global parameterization (i.e., considering all the digital representations regardless of their type) with a parameterization by digital representation type (e.g., letters, structured reports);* We analyze the impact of image pre-processing on OCR considering global and by typology parameterization;* We create and release a cultural heritage dataset <cit.> of Portuguese typewritten digital representations from the 20th century and a dataset of its manual transcriptions <cit.>;* We disclose the source code[<https://github.com/feup-infolab/archmine>] of the parameterization of image processing algorithms. The article is organized as follows. The various OCR-related tools and performance measures are explored in Section <ref>, while the related research is detailed in Section <ref>. Section <ref> explains the proposed approach. Section <ref> formalizes the problem of parameter tuning of image processing algorithms as an optimization problem and analyzes the effect of parameterization on the OCR performance, comparing it to default parameter values. We analyze the effect of each image processing algorithm on the OCR performance in Section <ref>, comparing it to an OCR scenario where no image pre-processing is used. The results are discussed in Section <ref>. Finally, conclusions are drawn in Section <ref>.§ BACKGROUNDOptical Character Recognition is an active research field with constant new developments where most recent works use deep learning advances. In this section, we describe different OCR software, image processing algorithms, approaches to parameter optimization, and the evaluation of OCR systems. §.§ OCR Software There are currently four prominent OCR software applications: ABBYY FineReader, Google Cloud Vision API, Ocropy, and Tesseract. Both Tesseract and Ocropy are open-source engines, while ABBYY FineReader and Google Cloud Vision API are commercial engines.In terms of approaches, all engines use Deep Neural Networks (DNN) to train their models, except for Ocropy which uses shallow networks. A DNN can improve accuracy by adding training data <cit.>, unachievable with shallow networks.Furthermore, while Ocropy is a language-independent engine, ABBYY FineReader, Google Cloud Vision API, and Tesseract support dictionaries and language modeling. It is not easy to give a general definitive comparison of the different engines based on accuracy because the characteristics of different datasets influence the results. Engines are also affected by languages and font types. Moreover, recent versions of some engines have additional or updated training models and data that improve the performance and invalidate past research conclusions over which software is the best. In Table <ref>, a comparison between the different highlighted technologies is available. One of the comparison criteria is whether the OCR software supports the Portuguese language because this work uses a dataset of Portuguese cultural heritage documents. §.§ Image Processing Image processing techniques aim to visually enhance images and make them more suitable for human or machine interpretation. The strategies highlighted in this paper are image binarization and noise reduction.Binarization consists of transforming colored pixels into black and white ones. Most recent implementations of binarization techniques are based on thresholding <cit.>. Thresholding is a segmentation method that sets pixels to black or white depending on whether the grey value is above or below a threshold value to evidence contrast between the background and foreground.There are various noise reduction techniques, such as filtering and morphological transformations. Filters can be grouped into smoothing and sharpening. Morphological transformations are used to thin characters, connect broken strokes, and smooth contours <cit.>.OpenCV is an open-source computer vision Python library launched in 1999 by Intel <cit.> with various image processing algorithms, specifically for binarization, image smoothing, and morphological transformations. The library offers four binarization operations: adaptive threshold, Otsu threshold, simple threshold, and triangle threshold (see Tables <ref> and <ref>); four different smoothing operations: bilateral filter, Gaussian filter, homogeneous blur, and median blur (see Tables <ref> and <ref>); seven different morphological transformation operations: black hat, closing, dilation, erosion, morphological gradient, opening, and top hat (see Tables <ref> and <ref>). Tables <ref>, <ref>, and <ref> describe the meaning of image processing algorithms from the OpenCV library. Tables <ref>, <ref>, and <ref> describe the meaning and typology of image processing algorithms' parameters. §.§ Parameter OptimizationTo select appropriate values for the algorithms’ parameters, we can use default values or manual configurations based on literature, experience, or trial-and-error <cit.>. OpenCV’s documentation presents some examples of implementations using recommended assigned values. For instance, in thresholding algorithms, 255 is the typical value assigned to the maxValue parameter. However, these configurations do not guarantee an optimal parameterization of the algorithms as the parameters’ values should be adjusted to the images’ state. As an alternative, parameter tuning can be done using a combinatorial search approach. According to the literature, the genetic and Hill-Climbing algorithms are the two best-tailored approaches to handle various parameters and their values <cit.>.A genetic algorithm is a randomized search algorithm inspired by Charles Darwin’s theory of natural selection <cit.>. It has the premise that individuals pass traits to their offspring, and the individuals with the best characteristics, tend to survive and have more offspring <cit.>. Reproduction is the process in which the fitness function, i.e., the objective function, dictates the probability of having offspring for the next generation <cit.>. In a minimization problem, lower fitness values mean a higher likelihood of offspring. Crossover is a process in which offspring are generated by combining traits from two parents, randomly selected members of the last population. Mutations are used to ensure a generation’s diversity onto the next one. The hill-climbing method uses neighbors in every iteration to evaluate a current solution candidate <cit.>. It greedily moves towards the neighbor with the highest value until there is no better neighbor. Only the genetic algorithm can manipulate nominal parameter types in a meaningful way from the two algorithms. In contrast, the hill-climbing algorithm requires a notion of neighborhood.As some of the OpenCV image processing algorithms have nominal parameters, we use genetic algorithms. Genetic algorithms have many advantages in dealing with complex problems, and parallelism as multiple offspring that act as independent agents are ideal for parallel exploration of the search space. The main disadvantage is the difficulty in formulating the fitness function and parameters, such as the population size, crossover, and mutation rates <cit.>. In particular, we adopt the Non-dominated Sorting Genetic Algorithm (NSGA-II) <cit.> to solve the image processing optimization problem. NSGA-II is the standard metaheuristic for solving multi-objective optimization problems <cit.>. It has been applied to various search and optimization problems, such as scheduling problems <cit.>, resource allocation problems <cit.>, and optimal parameter selection problems <cit.>. The algorithm modifies the processes of mating and survival selection of individuals. Since not all individuals can survive, they are compared by rank with a binary tournament mating selection and selected as a solution with crowding distance that uses Manhattan Distance. Applying a ranking scheme encourages convergence and the crowing distance density estimator promotes diversity <cit.>. §.§ Evaluation of OCR systemsThere are two ways to evaluate OCR systems: via text or layout. A text-based evaluation is based on the comparison of the OCR text result against the ground truth. Recent research has four prominent text-based measures: Character Error Rate (CER), Word Error Rate (WER), Index/Count-based Bag of Words, and Flex Character Accuracy. In Table <ref>, a comparison between the different measures is available. Traditionally, text-based metrics use the edit distance between the ground truth and the OCR output. That is the case for the indicators CER and WER which are also dependent on reading order. Reading order refers to a reading route for textual elements that can be sequentially ordered lists for simple layouts, such as, a book page, or more complex in the case of newspapers where there isn’t necessarily a left-to-right order <cit.>. Its reading dependence makes them unreliable when applied to complex page layouts since the performance is influenced by page segmentation and text detection <cit.>.In contrast, a layout-based evaluation is based on the document page’s structure. Measures analyze the document's segmentation process in different regions, classification of the areas, and reading order. The most popular ones are segmentation errors, misclassifications, and reading order errors.There are five types of OCR error analysis from the viewpoint of misspellings: edit operations, length effects, erroneous character positions, real-word vs. non-word errors, and word boundaries <cit.>. Edit operations can be deletions, insertions, substitutions, and transpositions of characters to transform one string into another. Edit distances of 1 indicate single-error tokens, while tokens with a higher edit distance indicate multi-error tokens. There are two types of word errors: non-word and real-world errors <cit.>. Typically caused by the misrecognition of characters, non-word errors occur when words are converted into invalid words. In contrast, real-world errors arise when words are changed to valid words with a different meaning <cit.>. Word boundary errors are caused by the wrong insertion or deletion of white spaces, which results in incorrect split errors with multiple words being merged into one or a word being split into multiple words <cit.>.With the edit operation analysis, there are three types of OCR errors <cit.>: misspelled characters associated with the operation of substitution, spurious symbols attributed to the insertion of characters, and missing text related to the deletion of characters. An example of these errors in the context of Cultural Heritage is the transcription of a structured report from the 20th century, “RELATÓRIO N.º 3089” (REPORT N.º 3089). The OCR engine reads it as “RELÁTORIO Nº 3”. The letters “089” are missing. Another excerpt of the transcription reads “Enviado pela P.I.D.E.” (Sent by P.I.D.E.), while the OCR output is “Enviado pela P.T.D.E.”. Here, the letter “I” is misinterpreted as “T”. Another excerpt reads “permite-se aconselhar a juventude” (allows to advise the youth), and the OCR output is “pDermite-se aconselhar à juventude”. There is an insertion of the character “D” in the word “permite-se” and a substitution of the character “a” with “à”.§ RELATED WORKHistorical documents present various challenges to the OCR process. Most engines require additional training to recognize historical fonts and languages. Digital representations of historical printed records are often noisy because of the papers’ degradation due to aging, such as smears and faded text, and other printing noise like varying kerning and leading, i.e., different spaces between letters and line-break hyphenation amongst others <cit.>.A popular way to improve the OCR results in historical documents is to apply an image pre-processing step to the OCR task. In the literature, various approaches combine image processing algorithms. <cit.> applied illumination adjustment, grayscale conversion, unsharp masking, and Otsu binarization to counteract image distortions and improve OCR accuracy. Similarly, <cit.> trained a model for Finnish Fraktur fonts and analyzed different combinations of pre-processing image algorithms. <cit.> show a process for image binarization with manual image condition classification and image processing methods. The approach has five stages: image acquisition, image preparation, filtering methods, threshold methods, and image refinement. <cit.> use a Convolution Neural Network (CNN) to automatically select image processing algorithms according to each input document image. Most of the previously mentioned works have historical documents as their dataset. <cit.> used document images from the Holy Monastery of Dousiko, Meteora, Greece, while <cit.> used Finnish historical documents.According to the overall results of the previously mentioned research, the use of image processing algorithms as a pre-processing step to the OCR process improves the accuracy of the engines. <cit.> improved word level by 27.5% compared to FineReader 7 or 8 and 9.2% for FineReader 11. <cit.> registered an improvement in OCR accuracy between 2% to 6.8%. <cit.> succeeded in proving the performance of engines with their automatic pre-processing method selection, with Tesseract OCR having an improvement of 25%.As previously discussed in Section <ref>, there are various approaches to the parameter tuning of image processing algorithms. <cit.> selected seven values for thresholding from the range of 0 to 255. There is an equal incremental difference between the values and an equal division of the grayscale spectrum. This experimental method reduced the error rate to 39.1% in comparison to the original performance of the engine. To our knowledge, there is no literature on combinatorial search algorithms applied to tuning parameters of image processing algorithms in OCR.In contrast, there are some examples of implementations of machine learning algorithms to optimize OCR accuracy. For instance, <cit.> developed an adaptive image pre-processing method using a reinforcement learning model to minimize the edit distance between recognized text and ground truth. The technique improved the F1-score of Tesseract 4.0 from 16.3% to 72.9%.Table <ref> presents the comparison of the different approaches, algorithms, and results for the discussed articles.§ METHODOLOGYThree research questions are associated with this experiment:* RQ1. Can applying parameter tuning to image processing algorithms with NSGA-II benefit the OCR performance of cultural heritage digital representations?* RQ2. Can applying image processing algorithms to the cultural heritage digital representations before the text recognition task improve the OCR performance on the overall dataset?* RQ3. Can applying image processing algorithms to the cultural heritage digital representations before the text recognition task improve the OCR performance on subsets by typology? The methodology of this work can be divided into four main steps: define and collect a dataset of cultural heritage digital representations, explored in Section <ref>, split the dataset into parameterization and evaluation subsets in Section <ref>, define the image processing pipeline in Section <ref>, compare the different types of parameterization and analyze the impact of image-processing in the OCR process in Section <ref>. §.§ Cultural Heritage DatasetThe training of image processing algorithms' parameters and evaluation of the OCR optimization process requires a dataset to conduct various experiments. To fit the requirements of the EPISA project, we need a Portuguese cultural heritage dataset. However, since there are currently no such available datasets, we built a dataset of digital representations of archival records. In this work, we use records from the National Archives of Torre do Tombo (ANTT), a Portuguese central archive with millions of documents that date back to the 9th century. This archive contains more than 40 million digital representations and their respective archival records <cit.> that can be shared and accessed through an information system called Digitarq[https://digitarq.arquivos.pt/].To build a dataset of digital representations of archival records, we first had to identify typewritten representations in the overall set of representations. Exploring the database, we found that the document classification of the digital representations into typewritten or handwritten is usually done in the fields “Dimension and Support”, “Scope and Content”, and “Note”. The description forms varied from mentions to the Portuguese word for “typewritten” (“datilografado”), its abbreviations (“dact.”) or alternative ways of spelling, i.e., forms of the word before and after the Portuguese Language Orthographic Agreement of 1990, “dactilografado”, and “datilografado”. We searched the database for records whose descriptions in the previously mentioned fields contained different forms of the term “typewritten”. The query results returned the name of the digital representation files and its dissemination URL that was later used to retrieve each digital representation.The dataset <cit.> has 27,017 one-page digital representations from 8,115 records of two fonds from the 20th century: the General Administration of National Treasury (DGFP) and the National Secretariat of Information (SNI). DGFP was a central department of the Ministry of Finance created during the First Portuguese Republic regime (1910-1926) that managed public assets, including movable and fixed assets that had been under the responsibility of the former royal family. SNI was a public agency created during the authoritarian Estado Novo regime (1933-1974) to supervise political propaganda, press organs, popular culture, and entertainment, and promote national tourism.The digital representations were classified by writing format (handwritten, typewritten, or blank), as some archival records containing handwritten or blank documents are not relevant to our work. The dataset has 23,794 typewritten digital representations, 1,681 handwritten digital representations, and 1,542 blank digital representations. Furthermore, the dataset was classified into ten types of digital representations. The typewritten dataset has 3,264 letters, 6,560 structured reports, 1,970 non-structured reports, 82 covers of processes, 6 covers of minutes, 1,473 contents of minutes, 19 books' covers, 182 books' contents, 165 theatre plays' covers, and 8,845 theatre plays' contents. Figure <ref> shows examples of different types of typewritten digital representations from the dataset. Identifying digital representation typologies relevant to the OCR task was carried out by observing existing textual descriptions of the records and document layouts of the digital representations. The classification of the dataset was performed manually.§.§ Parameterization and Evaluation SamplesTo conduct an unbiased evaluation of the OCR optimization process, we need to use different samples of the cultural heritage dataset to find the best parameter values for image processing algorithms (Parameterization Dataset) and to assess the algorithms' performance in the text recognition task (Evaluation Dataset). The best-performing image processing algorithms in the Evaluation Dataset with tuned parameters in the Parameterization Dataset will be used in the pre-processing phase of the OCR task with the Cultural Heritage Dataset to extract its textual content.The dataset of documents we are working with refers to historical records with secondary value, i.e., evidential, and informative values, that no longer possess administrative value <cit.>. Historical archives consider the documents' information as a whole, unlike current archives that detail the information contained in each document to facilitate quick access to the data. For this reason, from the ten types of digital representations of the Cultural Heritage Dataset, we decided to focus only on the five we considered most informative: letters, covers of processes, covers of theatre plays, structured and non-structured reports. These are the ones with more significant potential for describing historical records.From the five digital representation types, we selected 708 typewritten digital representations. Per typology, we randomly selected 5% of the digital representations of each archival series. If this methodology does not fulfill the requirement of choosing at least 60 digital representations per typology to maximize the probability of normal distributions necessary in parametric tests, we select digital representations of each archival series until the sample amounts to 60.The digital representations of the samples were manually transcribed to provide a ground truth in the evaluation stage <cit.>. We split the sample into two: the Parameterization and the Evaluation Datasets. The datasets each have 81 letters, 164 structured reports, 49 non-structured reports, 30 covers of processes, and 30 theatre plays' covers. Table <ref> characterizes the Parameterization and Evaluation Datasets in comparison to the Cultural Heritage Dataset regarding the number of one-page digital representations through different typologies. §.§ Image Processing PipelineWe used the Tesseract engine for the OCR task with the Pytesseract wrapper for Python, and the available tessdata_best <cit.> training data. We did not train a language model, since a Portuguese language model is already available, and literature claims that the engine is prepared to handle typewritten documents.For image processing, the OpenCV library described in Table 2 was used. Three image processing techniques were selected: binarization, smoothing, and morphological. The selection of the methods was based on the state of the images in the dataset. The images’ degradation stems mainly from contrast variation, bleed-through, faded ink, smears, and thin text <cit.>, e.g., there is no angled digital representation; therefore, there is no need for a skew correction algorithm. This work’s pipeline is shown in Figure <ref>. §.§ EvaluationOur methodology includes two evaluations that address the Research Questions defined above: the effect of parameter tuning of image processing algorithms on the OCR task (RQ1) in Section <ref>and the effect of the algorithms themselves (RQ2 and RQ3) in Section <ref>.We used performance metrics based on the CER and count-based Bag of Words presented in Table <ref>. The CER-based metric is presented as character accuracy in Equation <ref> to facilitate the comparison between the measure as success rates. The count-based Bag of Words metric is presented as an F-measure in Equation <ref>, based on the notions of precision in Equation <ref> and recall in Equation <ref>. Character Accuracy = (1 - CER)*100Precision=number of words with correct occurrences/number of words in OCR output textRecall=number of words with correct occurrences/number of words in ground truth textF1=2*precision*recall/precision+recall The evaluation of the effect of image processing algorithms also includes an analysis of OCR error types. We analyzed the relation of each algorithm with OCR error types based on the OCR error analysis of edit operations described in Section <ref>.§ PARAMETER TUNING OF IMAGE PROCESSING ALGORITHMSThis section defines the problem of parameter tuning of image processing algorithms as an optimization problem and analyzes the effect of parameterization on the OCR performance, comparing it to default parameter values. Section <ref> details the process of formalizing and solving the problem of parameter tuning into a multi-objective optimization. We showcase the best parameter values for image processing algorithms obtained with parameterization in Section <ref>, and evaluate the effect of parameterization on the OCR performance in Section <ref>. §.§ Problem Formalization and OptimizationThe autotuning of image processing algorithms is defined as a constrained multi-objective problem with two objective functions, f_1(x) and f_2(x), formulated by Equations <ref> <cit.> and <ref>, respectively. lev_a,b(i,j) = {[ max(i,j),min(i,j) = 0; min {[ lev_a,b(i-1,j)+1; lev_a,b(i,j-1)+1; lev_a,b(i-1,j-1)+1 ]. ,otherwise ].bow_a,b(i) = {[1 ,n_a(i) = n_b(i);0 ,otherwise ]. The first objective function, f_1(x), is the Levenshtein edit distance between the ground truth and the output. The edit distance calculates the minimum number of characters’ edit operations, such as insertions, substitutions, and deletions, to transform a string a into b with lengths i and j, respectively. In contrast, the second objective function, f_2(x), is the number of words with the same word occurrences between the ground truth and the output, where i is the index of matched words between strings a and b, and n_a(i) is the number of occurrences of the matched word i in string a. The goal is to find solutions for the parameter values that satisfy each image processing algorithm’s constraints and are as good as possible for both objective functions <cit.>.We used the Non-dominated Sorting Genetic Algorithm (NSGA-II) <cit.> to perform parameter tuning of image processing algorithms. For the formalization of the constraints, g(x), the image processing algorithms’ equality constraints are transformed into inequality constraints g(x) ≤0. All the constraints needed for the selected image processing algorithms are of the type “a parameter must be odd”, which can be expressed through the equality constraint |parameter % 2|=1 and transformed into the inequality |parameter % 2-1| ≤0, where g(x)=|parameter % 2-1|. The algorithms Adaptive Thresholding, Homogeneous Blur, Median Blur, and Gaussian Blur all have a parity constraint on the parameter blockSize, in the case of the thresholding algorithm, and on the parameter ksize for the rest. §.§ Best Parameters Per Digital Representation TypologyWe used the parameterization dataset to tune image processing algorithms. We considered two parameterization scenarios: global and by type of digital representation. Global parameter tuning uses the overall parameterization sample instead of subsets of the sample by digital representation typology. The parameter values obtained using the two parametrization scenarios are presented in Table <ref>, along with the default values proposed by the OpenCV documentation. These values do not intend to be reference values. We decided to include them to show how parameterized values differ from the default values, how values differ between parameterization scenarios, and how values differ between types of digital representations. llccccccc Parameters set for image processing algorithms with default values and values tuned by NGSA-II for global set of documents and classified types of digital representations.2*Algorithm 2*Parameter 2*Default 2*Global 5cType of digital representation 5-9 Letter c]@c@Processcover c]@c@Structuredreport c]@c@Theatre playcover c]@c@Non-structuredreport3c Tablecontinued from previous page 2*Algorithm 2*Parameter 2*Default 2*Global 5cType of digital representation 5-9 Letter c]@c@Processcover c]@c@Structuredreport c]@c@Theatre playcover c]@c@Non-structuredreport 5*c]@l@AdaptiveThresholding maxValue 255 217 24 72 13 152 32adaptiveMethod 0 or 1 1 0 0 0 1 0thresholdType 0 0 0 0 0 1 0blockSize 11 33 39 65 43 57 51c 2 25 30 29 43 18 32 3*c]@l@BilateralFiltering d 9 4 4 2 2 1 2sigmaColor 75 10 4 85 36 6 18sigmaSpace 75 231 31 52 100 191 253 3*c]@l@BlackHat kernel 5 10 6 229 139 30 38iterations 1 7 10 7 9 2 1borderType 3 4 4 2 3 1 1 3*Closing kernel 5 1 1 1 1 1 1iterations 1 2 2 4 9 4 3borderType 3 3 1 0 1 2 3 3*Dilation kernel 5 32 - 1 1 1 1iterations 1 8 - 4 7 9 8borderType 3 2 - 3 2 1 2 3*Erosion kernel 5 1 1 1 1 1 2iterations 1 5 7 1 5 2 1borderType 3 3 2 2 1 3 1 2*c]@l@GaussianBlur ksize 5 3 3 3 3 3 3borderType 3 1 0 1 1 1 1 2*c]@l@HomogeneousBlur ksize 5 1 1 3 1 3 3borderType 3 1 2 3 1 0 3 Median Blur ksize 5 3 3 3 5 3 33*c]@l@MorphologicalGradient kernel 5 2 2 2 2 105 2iterations 1 1 1 2 1 2 2borderType 3 2 2 0 1 1 2 3*Opening kernel 5 2 1 2 1 2 2iterations 1 2 2 1 9 1 1borderType 3 0 4 1 4 0 3 2*c]@l@OtsuThresholding maxValue 255 199 24 251 92 34 14type 0 3 0 0 3 3 0 3*c]@l@SimpleThresholding thresh 127 152 110 33 82 240 224maxValue 255 45 4 10 221 3 42type 0 1 3 3 3 4 4 3*c]@l@TopHat kernel 5 90 51 217 233 139 204iterations 1 5 7 5 2 5 2borderType 3 2 0 0 2 2 4 2*c]@l@TriangleThresholding maxValue 255 15 74 248 220 174 238type 0 3 3 2 2 3 0 We performed the parameterization on a PC with 3.20 GHz, AMD Ryzen 7 5800H, and 16GB RAM. Table <ref> shows the average time per digital representation to parameterize image processing algorithms.The table shows that parameter tuning for each digital representation took the least time, in the range of 2 minutes, with the image processing algorithms Dilation, Erosion, and Morphological Gradient. In contrast, the parameterization for each digital representation took the longest with the algorithms Black Hat, Opening, and Otsu Thresholding, within 10 minutes. §.§ Effect of Parameterization on OCR PerformanceTo evaluate the effect of default, global, and digital representation-type parameterization on the OCR performance, we compared the performances of image processing algorithms in OCR with default parameter values to algorithms with parameter values obtained by our two parameterization scenarios. To compare the character accuracy and F1-score results in the three scenarios (default, global parameterization, and parameterization by typology), we use the Wilcoxon signed-rank test <cit.> with the Bonferroni correction. We used non parametric tests because the assumptions for the t-test were not verified. Table <ref> presents the results of the one-sided Wilcoxon pairwise comparisons. We used this non parametric test because the assumptions for the t-test were not verified.By analyzing the statistical outcomes, the OCR performance using digital representation-type parameterization on the image processing algorithms shows statistically significant improvements in both measures compared to the parameterization using default values. Similarly, the OCR performance using global-type parameterization also shows significant improvement in both measures compared to using default values, except for Simple Thresholding and Dilation where the performance is significantly worse. For most algorithms, there is either not a significant difference between global and digital representation-type parameterization or the specific parameterization performs better in both measures, except for the algorithms Otsu Thresholding, Homogeneous Blur, Median Blur, and Morphological Gradient, where it performs worse in at least one of the measures.§ EFFECT OF IMAGE PRE-PROCESSING ON OCR PERFORMANCEWhile in the previous section we studied the effect of different types of parameterization on the OCR performance of image processing algorithms, this section analyzes the effect of each image processing algorithm on the OCR performance, comparing it to an OCR scenario where no image pre-processing is used. Due to the significantly better results of the parameterization scenario that considered the type of digital representation reported in Section <ref>, we conduct the current evaluation with these parameterization values. We use the evaluation dataset described in Section <ref> and conduct our analysis using the overall set of documents in Section <ref> and using subsets defined by the type of digital representation in Section <ref>. We used the Friedman test to analyze OCR performance differences between the image processing algorithms, and the one-sided Wilcoxon signed-ranked test with the Bonferroni correction as a post-hoc test to discover which algorithms perform better in relation to each other and to not applying any. We used non parametric tests because the assumptions for repeated measures ANOVA and t-test were not verified. We also analyze the frequency of OCR errors with different image processing algorithms on the overall evaluation dataset. §.§ Comparison of Image Pre-processing on the Overall DatasetThis section assesses the results we obtained using the complete evaluation dataset. We compare the performance of the several image pre-processing algorithms in Section <ref> and analyze the relation of each algorithm with OCR error types in Section <ref>.§.§.§ Performance AnalysisTable <ref> presents the results of the statistical tests showing that, in the overall dataset, Bilateral Filter and Opening are the best-performing algorithms. Both algorithms significantly improve the OCR performance in terms of character accuracy, with the Opening algorithm reaching a more significant improvement. In contrast, Median Blur and Morphological Gradient are the lowest-performing algorithms in character accuracy, and Morphological Gradient is the lowest-performing algorithm in F1-score. Four algorithms worsen the character accuracy of OCR without image pre-processing: Otsu Thresholding, Triangle Thresholding, Median Blur, and Morphological Gradient. Similarly, nine algorithms worsen the OCR performance in terms of F1-score: Adaptive Thresholding, Otsu Thresholding, Simple Thresholding, Triangle Thresholding, Gaussian Blur, Homogeneous Blur, Median Blur, Morphological Gradient, and Black Hat.Looking at the results for the binarization algorithms, the Simple Thresholding algorithm is the best-performing algorithm. However, it does not improve the OCR performance. For the smoothing algorithms' results, Bilateral Filter is the best-performing smoothing algorithm. Every smoothing algorithm except Bilateral Filter deteriorates F1-score when used to pre-process the images before OCR. The results for the morphological transformation algorithms show that Opening is the best-performing algorithm in comparison to not applying an image pre-processing algorithm and other binarization, smoothing, and morphological transformation algorithms.§.§.§ Error AnalysisTo better understand how image processing algorithms impact the frequency of OCR errors, we conducted an OCR error analysis of the most common errors for each image processing algorithm based on edit operation types. We present the frequency of edit operations for each image processing algorithm in Figure <ref>. The first graph bar in each edit operation labeled “None” corresponds to performing OCR without image pre-processing.The results show the frequency of edit operations (deletion, insertion, and substitution) and combinations between them with each image processing algorithm. We conclude that among single modification error types, the operation of deletion is the most common, and insertion is the least common. Among the operation of deletion, Median Blur and Triangle Thresholding have the highest occurrences. We can also observe that the algorithms Adaptive Thresholding, Bilateral Filtering, Black Hat, Dilation, Opening, and Top Hat had a lower frequency of deletion operations in comparison to not applying any image processing algorithm to OCR. As for the combination of edit operation types where multiple errors occur, we conclude that deletion and insertion, along with deletion, insertion, and substitution have a very small occurrence with all the algorithms. The algorithms Adaptive Thresholding and Black Hat had a higher frequency of the operation of substitution combined with deletion and insertion in comparison to not using image processing algorithms. §.§ Comparison of Image Pre-processing on the Datasets Defined by Type of Digital RepresentationThis section presents the results we obtained on algorithms comparison using subsets of the evaluation dataset defined by the type of digital representation.§.§.§ Letters This section presents the results regarding the letters' typology. Table <ref> presents the results of the statistical tests showing that, similarly to the results for the overall dataset, Bilateral Filter is the best-performing algorithm. Triangle Thresholding and Morphological Gradient are the worst-performing algorithms. Moreover, the mean results of character accuracy and F1-score are higher than in the overall dataset.For the binarization methods' results, all the algorithms, except Adaptive Thresholding, decrease character accuracy. The results for F1-score are akin to the overall dataset. As for the results of the smoothing methods, Bilateral Filter is still the best-performing algorithm in character accuracy. However, the results for the morphological transformation algorithms indicate that, in comparison to the overall dataset, Opening no longer improves the OCR performance.§.§.§ Non-structured reports This section presents the results regarding the non-structured reports typology. Table <ref> presents the results of the statistical tests showing that the mean results of the OCR performance are slightly higher than in the overall dataset and no algorithm improves the OCR performance. Black Hat and Opening are the best-performing algorithms and Homogeneous Blur, Median Blur, and Morphological Gradient are the worst-performing algorithms.The binarization methods Otsu and Triangle Thresholding are the worst-performing algorithms as they deteriorate the OCR performance. As for the smoothing algorithms' results, Bilateral Filter is the best-performing algorithm. All the smoothing algorithms except Bilateral Filter have significantly worse results than not using any image processing algorithm. The results for the morphological transformation methods indicate that Black Hat and Opening are the best-performing algorithms. In sum, most morphological transformation algorithms improve the OCR performance in comparison to binarization and smoothing algorithms in the non-structured reports typology.§.§.§ Process covers This section presents the results regarding the process covers' typology. Table <ref> presents the results of the statistical tests showing that, in the process covers sample, the mean OCR performance values are lower than in the overall dataset, and no algorithm improved the OCR performance.There are no significant differences between the binarization algorithms. Moreover, unlike in the previous sections, binarization algorithms do not impair the OCR performance. As for the smoothing methods, we can observe that most algorithms have a better performance in character accuracy and F1-score than Black Hat, Morphological Gradient, and Otsu Thresholding. Finally, the results with morphological transformation algorithms indicate that Black Hat and Morphological Gradient are the lowest-performing algorithms and deteriorate the OCR performance.§.§.§ Structured reports This section presents the results regarding the structured reports typology. Table <ref> presents the results of the statistical tests showing that no algorithm improved the OCR performance in comparison to not using any image processing algorithm.We can observe that from the performance results of binarization methods, Adaptive Thresholding and Otsu Thresholding impair the OCR performance. For the smoothing algorithms' results, we see that Bilateral Filter is the best-performing algorithm. However, it does not improve the OCR performance. On the other hand, Median Blur and Gaussian Blur worsen the OCR performance. The results for morphological transformation algorithms indicate that Morphological Gradient and Black Hat deteriorate the performance.§.§.§ Theatre plays' covers This section presents the results regarding the theatre plays' covers typology. Table <ref> presents the results of the statistical tests showing a significantly lower mean OCR performance than in the overall dataset. Furthermore, unlike the results in the overall dataset and the previous typologies where binarization algorithms worsened the OCR performance, in this sample it is a binarization algorithm, Adaptive Thresholding, that is the best-performing algorithm. In sum, three algorithms improved character accuracy: Adaptive Thresholding, Black Hat, and Top Hat. §.§ OCR Performance Comparison Across Typologies and OverallWe present the distribution of the performance measures for the overall dataset and the digital representation-type samples in Figure <ref>. We conclude that the digital representation typology with the best results is the letters sample, with a median character accuracy above 75% and a median F1-score above 60% for most algorithms. In contrast, the digital representation typology with the worst performance results is the theatre plays' covers sample, with a median character accuracy below 25% and a median F1-score below 20%. Image processing algorithms appear to have a greater impact on the OCR performance of digital representations from the theatre plays' covers sample, where the OCR performance without applying image processing algorithms is low. § DISCUSSIONOur results from Sections <ref> and <ref> show that parameterization with the NSGA-II algorithm influences the impact of image processing algorithms on OCR, in response to RQ1. Not only are the parameter values very different with parameterization, but the results are significantly better compared to using default parameter values. We conclude that some image processing algorithms perform significantly better with parameterization by typology than with global parameterization. Moreover, parameterization by typology has the advantage of being faster to process than global parameterization.From Section <ref>, we can draw conclusions to answer RQ2. We conclude that there are no significant improvements in the OCR performance in the overall dataset when using binarization algorithms in an image pre-processing OCR phase in comparison to not using them. Unlike the binarization method, the smoothing and morphological transformation methods significantly improved the OCR performance with Bilateral Filter and Opening, respectively. Table <ref> shows the image processing algorithms we chose to apply as image pre-processing to OCR from the results we described in Sections <ref> and <ref>. In the column “Improved algorithms” of Table <ref>, we list the image processing algorithms that significantly improved the OCR performance without image pre-processing. From these algorithms, we selected the ones that had the higher significance levels for the overall dataset and each digital representation typology. The two last columns of Table <ref> show the mean performance values of the selected algorithms. The OCR type error analysis in Section <ref> lead us to conclude that among single modification error types, the operation of deletion is the most common, and insertion is the least common. Moreover, among the combination of edit operation types, the operations of deletion and insertion, along with deletion, insertion, and substitution have a very small occurrence. We observe that image processing algorithms that performed the best in the overall dataset had a lower occurrence of deletion edit operations, and those that performed the worst had the highest occurrence. This might signify that 1) applying image pre-processing to OCR has a bigger impact in mitigating the error of missing text in comparison to other errors, and 2) improving the lack of text recognition has a bigger positive influence in improving the OCR performance than improving other errors.Results from Section <ref> allow us to answer RQ3. The analysis shows that it is possible to improve the text recognition task using tuned image processing algorithms in the subsets of letters and theatre plays' covers. As shown in the column “Improved algorithms” of Table <ref>, Bilateral Filter significantly improved the OCR performance with image pre-processing with the subset of letters, and Adaptive Threshold, Black Hat, and Top Hat improved the performance with the subset of theatre plays' covers. However, no image processing algorithms significantly improved the OCR performance with the samples of all the other typologies. From the column “Selected algorithm” of Table <ref>, we can see that the algorithm we chose for the typology of letters is Bilateral Filter, and for theatre plays' covers it is Adaptive Threshold. From the performance comparison in Section <ref>, we conclude that process covers and theatre plays' covers are the samples with the lowest mean OCR performance values. They are also the samples with the most algorithms that tend to improve the OCR performance or do so in a significant way. These findings suggest that image processing algorithms tend to have a greater impact on digital representation typologies that have a lower OCR performance without an image pre-processing phase. The results also show that morphological transformation algorithms tend to be the most beneficial to every typology. In contrast, binarization algorithms only benefit the theatre plays' covers typology and tend to do so for the sample of process covers. On the other hand, while smoothing algorithms tend to be beneficial to every typology, like morphological transformation algorithms, only one algorithm, Bilateral Filter, is responsible for that performance improvement.§ CONCLUSIONThe main goal of this work is to understand how to improve the OCR performance on cultural heritage digital representations using image processing algorithms. We evaluated the use and parameterization of image processing algorithms for text recognition tasks using Tesseract and OpenCV methods.The key takeaway is that parameterization by digital representation typology benefits the employment of image pre-processing techniques in OCR, even if only for a small parameterization dataset. We successfully applied the NSGA-II algorithm to perform the parameterization. The analysis of the performance of image processing algorithms leads us to conclude that some algorithms improve the OCR performance significantly, while several tend to improve it and it is up to debate whether to apply them or not, and others even worsen the performance. A previous analysis is required before applying them. On top of that, some algorithms have more impact in some samples of the evaluation dataset than others. Digital representation typologies influence the effect these image processing algorithms have on the OCR performance. For this reason, it is important to analyze typologies before applying OCR. A binarization algorithm positively impacted the OCR performance in the sample of theatre plays' covers. On the other hand, smoothing algorithms improved the performance of the overall dataset and the samples of letters and theatre plays' covers. Likewise, morphological transformation algorithms also improved the OCR performance of the overall dataset and the samples of theatre plays' covers. These findings suggest that the employment of image pre-processing algorithms in OCR might be more suitable for typologies where the text recognition task without pre-processing does not produce good results, as is the case of the typology for theatre plays' covers. Factually, the algorithms that have been proven to improve the OCR task compared to not applying any are Adaptive Thresholding, Bilateral Filter, Black Hat, Opening, and Top Hat.Ultimately, there are big differences in the OCR performance by digital representation typology, which suggest that we should adopt the optimization of the OCR task for typologies that have weaker results without employing image pre-processing algorithms.The practical outcomes of this work are the source code of the parameterization of image processing algorithms and two datasets. Both datasets are available to consult for other scholars or researchers to use. Likewise, we provide the source code of the parameterization of image processing algorithms using NSGA-II.In the future, we will analyze the effect of the combination of image processing algorithms in the text recognition task. We would also like to expand the current OCR error analysis to include other error types. Furthermore, we would like to explore the detection and removal of stamps and annotations in digital representations in the OCR performance.This work is financed by National Funds through FCT - Foundation for Science and Technology I.P., within the scope of the EPISA project - DSAIPA/DS/0023/2018.ACM-Reference-Format | http://arxiv.org/abs/2311.15740v1 | {
"authors": [
"Mariana Dias",
"Carla Teixeira Lopes"
],
"categories": [
"cs.CV",
"cs.DL"
],
"primary_category": "cs.CV",
"published": "20231127114446",
"title": "Optimization of Image Processing Algorithms for Character Recognition in Cultural Typewritten Documents"
} |
Spinor Field LLC, Sugar Land, TX 77479, USA Recently we have shown that the anti-Pfaffian state and a state described by the second Landau level (SLL) projection of antiholomorphic Pfaffian wavefunctions have large overlap and almost identical low-energy orbital entanglement structures, suggesting they have the same topological order. In this paper, we further show that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states, indicating it is not the result of the SLL projection but rather a "fingerprint" of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, we argue that the SLL projected state, and therefore the anti-Pfaffian state, is a PH-Pfaffian topological order state. We also discuss the implications on the edge physics. 73.43.-f, 73.43.Cd, 71.10.PmIs the Anti-Pfaffian a PH-Pfaffian Topological Order State?Jian Yang January 14, 2024 ============================================================The leading candidates for the ground state of the ν=5/2 fractional quantum Hall effect (FQHE) <cit.> are the Pfaffian state<cit.>, the anti-Pfaffian state<cit.><cit.> which is the particle-hole (PH) conjugate of the Pfaffian state, and the PH-Pfaffian state<cit.><cit.><cit.>. There is a large number of numerical studies to support the Pfaffian state or the anti-Pfaffian state as a viable candidate for the ground state, however the same cannot be said for the PH-Pfaffian state. The main reason that the PH-Pfaffian state has attracted considerable attention, despite of its scant numerical support, is because its edge structure supports a fractional ν = 1/2 downstream charged boson edge mode and a counterpropagating upstream neutral Majorana fermion edge mode, making it the only state among the three leading candidates with the edge structure that is consistent with the experiments<cit.><cit.>. The PH-Pfaffian state can be described by the following form of a generalized Moore-Read wave function with an antiholomorphic Pfaffian term<cit.>:Ψ_PH-Pf = P_LLLPf ( 1/z_i^*-z_j^*) ∏_i<j^N (z_i-z_j)^2,where z_j = x_j+iy_j is the complex coordinate of the j_th electron, N is the total number of electrons, and Pf[A] is the Pfaffian of an antisymmetric matrix A, and P_LLL is the lowest Landau level (LLL) projection operator. According to the “bulk-edge” correspondence, the antiholomorphic Pfaffian term and the Jastrow term result in, respectively, a upstream neutral Majorana fermion edge mode and a downstream charged boson edge mode.In the spherical geometry, the PH-Pfaffian state, which occurs at total flux N_ϕ= 2N-1, is shown to have a high level of PH symmetry<cit.><cit.><cit.>. The Pfaffian and anti-Pfaffian, which occur at N_ϕ= 2N-3 and N_ϕ= 2N+1 respectively, obviously lack such a symmetry. Unfortunately, the PH symmetry also turns out to be the very reason that the PH-Pfaffian state fails to represent a gapped ground state, as no consistent gapped ground state is found to exist at N_ϕ= 2N-1 <cit.><cit.>. In fact, the only visibly gapped states appear at N_ϕ= 2N-3 and N_ϕ= 2N+1<cit.>. A simple way to address the PH symmetry issue is proposed recently by the author<cit.> by replacing the P_LLL in Eq.(<ref>) with the second Landau level (SLL) projection operator P_SLLΨ_SPH-Pf = P_SLLPf ( 1/z_i^*-z_j^*) ∏_i<j^N (z_i-z_j)^2.We call the state described by Eq.(<ref>) the SLL PH-Pfaffian (or SPH-Pfaffian in short) to distinguish it from the PH-Pfaffian state described by Eq.(<ref>). Although the SPH-Pfaffian state occurs at flux N_ϕ= 2N-1, it lacks the PH symmetry because the number of orbitals in the SLL is N_orb = N_ϕ + 3 = 2N + 2, as opposed to that of the LLL which is N_orb = N_ϕ+ 1 = 2N. As shown in Ref.<cit.>, the SPH-Pfaffian represents a gapped, incompressible phase, and provides an excellent description for the exact ground state of finite-size SLL Coulomb-interacting electrons over a range of the short-distance interaction strength. As one can map the SPH-Pfaffian state to a LLL state, the LLL mapped SPH-Pfaffian state and the anti-Pfaffian state, both occur at flux N_ϕ= 2N+1, are shown<cit.> to have large overlap and almost identical low-energy orbital entanglement structures<cit.>, suggesting both states have the same topological order - either the anti-Pfaffian or the PH-Pfaffian order. The question is: which one?To answer the question and for the reasons that will become clear later, in this paper we project the antiholomorphic Pfaffian wavefunctions into other Landau levels, and study the entanglement spectrum of each of the resulting projected states. One can extend Eq.(<ref>) and Eq.(<ref>) to describe an arbitrary n_th Landau level projected state by the following wavefunctionΨ_n = P_nLLPf ( 1/z_i^*-z_j^*) ∏_i<j^N (z_i-z_j)^2,where P_nLL is the n_th Landau level projection operator (n = 0, 1, 2, ···), with Ψ_0 corresponding to Ψ_PH-Pf in Eq.(<ref>) and Ψ_1 to Ψ_SPH-Pf in Eq.(<ref>). However, this form of wavefunctions becomes numerically intractable with N ≥ 10. In this paper, we will use two numerically more efficient variants of Eq.(<ref>). We start with the one by multiplying a stabilization factor ∏_i<j^N |z_i-z_j|^2 before applying P_nLL <cit.><cit.>. In the spherical geometry, the resulting wavefunction can be written as Ψ_n = P_nLLΨ_Pf,b^ν=-1∏_i<j^N (u_iv_j-u_jv_i)^3,and Ψ_Pf,b^ν=-1 is the ν=-1 bosonic Pfaffian wavefunction Ψ_Pf,b^ν=-1 = Pf(1/ u_i^*v_j^*-u_j^*v_i^*) ∏_i<j^N (u_i^*v_j^*-u_j^*v_i^*)where (u, v) are the spinor variables describing electron coordinates. The n_th Landau level projection in Eq.(<ref>) can be carried out by using the following two equations<cit.>(Y_l,m^q)^* = (-1)^q+mY_l,-m^-q,andY_l,m^qY_l',m'^q' = ∑_l"(-1)^l+l'+l"+2(q"+m")((2l+1)(2l'+1)/4π(2l"+1))^1/2 <lm,l'm'|l"m"><lq,l'q'|l"q">Y_l",m"^q",where q" = q+q' and m" = m+m'. In Eq.(<ref>) and Eq.(<ref>), Y_l,m^q is the monopole harmonics wavefunction <cit.><cit.> with the monopole strength q, the angular momentum l = q, q+1, q+2, ···, and the z component of the angular momentum -l ≤ m ≤ l.We first expand the ν=-1 bosonic Pfaffian wavefunction Ψ_Pf,b^ν=-1 in terms of (Y_N/2-1,m^N/2-1)^*, and ∏_i<j^N (u_iv_j-u_jv_i)^3 in terms of Y_3N-3/2,m'^3N-3/2 for each electron spinor coordinate. By use Eq.(<ref>) and Eq.(<ref>), the resulting product of two monopole harmonics wavefunctions (Y_N/2-1,m^N/2-1)^*Y_3N-3/2,m'^3N-3/2 can then be expanded as the infinite sum of single particle wavefunctions Y_l",m"^N-1/2 with l" = N-1/2, N-1/2+1,N-1/2+2, ···. The effect of the P_nLL is simply carried out by retaining only the n_th Landau level wavefunctions Y_N-1/2+n,m"^N-1/2, with n = 0 for the LLL, n = 1 for the SLL, and n = 2 for the third Landau level.We work at a finite system with N = 10 electrons for the lowest three Landau levels projected states obtained from Eq.(<ref>) with the total obitals N_orb = N_ϕ+1+2n = 20, 22, and 24 for the LLL (n = 0), the SLL (n = 1), and the third Landau level (n = 2), respectively. The orbital entanglement spectrum first introduced in <cit.> has become an invaluable tool for characterizing topological orders. In Fig. <ref>, we perform an orbital decomposition where subsystem A containing half of the electrons with positive angular momentum L_z^A and B the other half with negative L_z^B, and plot the corresponding orbital entanglement spectrum as a function of the total angular momentum in subsystem A. The results are presented from the top panel to the bottom panel, respectively, for the anti-Pfaffian state Ψ_aPf, the SPH-Pfaffian state Ψ_SPH-Pf (n = 1), the PH-Pfaffian state Ψ_PH-Pf (n = 0), and the third Landau level projected state Ψ_2 (n = 2). As shown previously in Ref.<cit.> and seen from the top two panels, the anti-Pfaffian state Ψ_aPf and the SPH-Pfaffian state Ψ_SPH-Pf have almost identical low-energy entanglement spectrum structures (plotted in longer bars) - the multiplicities of the 4 lowest entanglement levels at L_z^A = 24.5, 25.5, 26.5, 27.5 are 1, 1, 3, and 5. To our surprise, although shifted in L_z^A, the similar low-energy entanglement structure (again plotted in longer bars) is clearly identifiable for the other two Landau level projected states as seen from the bottom two panels - the same multiplicities of 1, 1, 3, and 5 at L_z^A = 20.5, 21.5, 22.5, 23.5 and at L_z^A = 22.5, 23.5, 24.5, 25.5, respectively, for the PH-Pfaffian state Ψ_PH-Pf (n = 0) and the third Landau level projected state Ψ_2 (n = 2). Next we turn to another variant of Eq.(<ref>) that is numerically even more efficient than Eq.(<ref>). We propose to perform the Landau level projection using the following bosonic variant of Eq.(<ref>) in the spherical geometryΨ_n^b = P_nLLPf(1/ u_i^*v_j^*-u_j^*v_i^*) ∏_i<j^N (u_iv_j-u_jv_i).To carry out the Landau level projection, we use the following equation<cit.>1/ u_i^*v_j^*-u_j^*v_i^* = ∑_l,m (-1)^m-1/28π/2l+1Y_l,m^1/2(u_i,v_i)Y_l,-m^1/2(u_j,v_j),to expand Pf(1/ u_i^*v_j^*-u_j^*v_i^*) in terms of the single-particle wavefunctions Y_l,m^1/2 with l = 1/2, 3/2, 5/2, ···, and expand ∏_i<j^N (u_iv_j-u_jv_i) in terms of the single-particle wavefunctions Y_l',m'^N-1/2(u_i,v_i) with l' = N-1/2. As a result, the unprojected wavefunction Pf(1/ u_i^*v_j^*-u_j^*v_i^*) ∏_i<j^N (u_iv_j-u_jv_i) can be expanded in terms of Y_l,m^1/2Y_l',m'^N-1/2, which, by using Eq.(<ref>) can be expressed in terms of the single particle wavefunctions Y_l",m"^N/2 with l" = N/2, N/2+1,N/2+2, ···. The effect of the P_nLL is simply carried out by retaining only the n_th Landau level wavefunctions Y_N/2+n,m"^N/2. One thing worth mentioning about the range of the sum over l in Eq.(<ref>). Since l" = |l-l'|, |l-l'|+1, ···, l+l', this means l = |l"-l'|, |l"-l'|+1, ···, l"+l'. As a result, the action of P_nLL reduces the infinite sum over l in Eq.(<ref>) to a finite sum with l = n+1/2, n+3/2, ···, N+n-1/2, as l" = N/2+n and l' = N-1/2. Once the n_th Landau level projected bosonic wavefunction Ψ_n^b in Eq.(<ref>) is obtained in terms of Y_N/2+n,m"^N/2, the corresponding fermionic wavefunction can be obtained by mapping the projected bosonic wavefunction to the LLL, followed by multiplication with a Vandermonde determinant ∏_i<j^N (u_iv_j-u_jv_i)Ψ'_n = (M_LLLΨ_n^b) ∏_i<j^N (u_iv_j-u_jv_i).where M_LLL is the LLL mapping operator which can be carried out by a simple replacement from Y_N/2+n,m"^N/2 to Y_N/2+n,m"^N/2+n for each particle coordinate.In Fig. <ref>, we present the orbital entanglement spectrum for the lowest four Landau level projected states Ψ'_n (n = 0, 1, 2, 3) obtained from Eq.(<ref>). The overlaps between these states and the corresponding states obtained from Eq.(<ref>) are 0.9915, 0.9591, and 0.3067 for n = 0, 1, and 2 respectively. In particular the n = 1 state Ψ'_1, which can be considered as a variant of the SPH-Pfaffian state, also has a large overlap 0.9778 with the anti-Pfaffian state. As the same as in Fig. <ref>, we work at a finite system with N = 10 electrons, with the total orbitals N_orb = N_ϕ+1+2n = 20, 22, 24, and 26 for the LLL (n = 0), the SLL (n = 1), the third Landau level (n = 2), and the fourth Landau level (n = 3), respectively. Again, the similar low-energy entanglement structure (plotted in longer bars) are identifiable with multiplicities of 1, 1, 3, and 5 at L_z^A = 20.5, 21.5, 22.5, 23.5 for the LLL projected state (n = 0), and at L_z^A = 24.5, 25.5, 26.5, 27.5 for the other three Landau level (n = 1, 2, and 3) projected states.Now we are ready to go back and answer the question raised in the beginning of the paper: what is the topological order of the SPH-Pfaffian state - anti-Pfaffian or the PH-Pfaffian order? We will use our numerical results to argue against one of the two possible answers and therefore in favor of the other. Suppose the SPH-Pfaffian state is an anti-Pfaffian topological order state, it would require that the SLL projection fundamentally alters the topological order of the original unprojected antiholomorphic Pfaffian wavefunctions, transforming it from the PH-Pfaffian order to the anti-Pfaffian order. The anti-Pfaffian entanglement structure observed in the SPH-Pfaffian state would be the result of the SLL projection, and bear no relation to the PH-Pfaffian order of the original unprojected wavefunction. For this to be true, we would expect to see different entanglement structures in different Landau level projected states. To the contrary, our numerical result presents a clear evidence that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states. Although in itself is not a rigorous prove, our numerical result makes the other answer much more plausible. That is, the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions survived the SLL Landau level projection, the similar entanglement structure observed in the SPH-Pfaffian state is not the result of the SLL projection, but rather a "fingerprint"<cit.> of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, the SLL projected state is a PH-Pfaffian topological order state. Since the SPH-Pfaffian state and the anti-Pfaffian state have the same topological order, we conclude that the anti-Pfaffian state is a PH-Pfaffian topological order state, albeit without the PH symmetry. As a PH-Pfaffian topological order state, the anti-Pfaffian state has a PH-Pfaffian edge structure which is composed of a single edge and supports a downstream charged boson mode and an upstream neutral Majorana mode. On the other hand, the anti-Pfaffian state is also the PH conjugate of the Pfaffian state and can be regarded as a hole Pfaffian state embedded in a ν = 1 integer quantum Hall state (IQHS). As a result, it also has a standard anti-Pfaffian edge structure which is composed of two edges<cit.><cit.>, an edge between a hole Pfaffian bulk and ν = 1 IQHS and an edge between the ν = 1 IQHS and vacuum, and supports a downstream charged boson mode, an upstream neutral Majorana mode, and an upstream neutral boson edge mode. In contrast to the PH-Pfaffian edge structure, the standard anti-Pfaffian edge structure requires no knowledge about the topological order of its bulk state (other than it is the PH conjugate of the Pfaffian state), since the bulk state is shielded by the ν = 1 IQHS layer without exposing to the vacuum. One may imagine as the ν = 1 IQHS layer becomes narrower, the two edges will get closer and eventually merge into one edge. At some point, a transition may take place from the standard anti-Pfaffian edge structure to the PH-Pfaffian edge structure, and the upstream neutral boson edge mode that appears only in the standard anti-Pfaffian edge structure will be gapped out. The PH-Pfaffian edge structure and the standard anti-Pfaffian edge structure are mutually exclusive, while we are not in the position to determine which one of the them is more competitive energetically, so far it is the PH-Pfaffian structure that is realized in experiments<cit.><cit.>. 10 Willett R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. 59,1776 (1987). MR G. Moore and N. Read, Nucl. Phys. B360, 362 (1991); N. Read and D. Green, Phys. Rev. B61, 10267 (2000). Levin M. Levin, B. I. Halperin, and B. Rosenow, Phys. Rev. Lett. 99, 236806 (2007). Lee S.-S Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Phys. Rev. Lett. 99, 236807 (2007). Son D. T. Son, Phys. Rev. X 5, 031027 (2015). Zucker P. T. Zucker and D. E. Feldman, Phys. Rev. Lett. 117, 096802 (2016). Yang J. Yang, arXiv:1701.03562 (2017). Banerjee M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y.Oreg, and A. Stern, Nature 5̱59, 205-210 (2018). Dutta Bivas Dutta, Wenmin Yang, Ron Aharon Melcer, Hemanta Kumar Kundu, Moty Heiblum, Vladimir Umansky, Yuval Oreg, Ady Stern, David Mross, Science 375, 193-197 (2022) Balram A. C. Balram, M. Barkeshli, and M. S. Rudner, Phys. Rev. B 98, 035127 (2018). Mishmash Ryan V. Mishmash, David F. Mross, Jason Alicea, Olexei I.Motrunich, Phys. Rev. B 98, 081107 (2018). Rezayi E. H. Rezayi , K. Pakrouski, and F. D. M. Haldane, Phys. Rev. B 104, L081407 (2021) Yang1 J. Yang, Phys. Rev. B 107, 195127 (2023) Yutushui M. Yutushui and D. F. Mross, Phys. Rev. B 102, 195153 (2020) Wu1 T. T. Wu and C. N. Yang, Nucl. Phys. B 107, 365 (1976). Wu2 T. T. Wu and C. N. Yang, Phys. Rev. D 16, 1018 (1977). Li H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008) | http://arxiv.org/abs/2311.16039v1 | {
"authors": [
"Jian Yang"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.str-el",
"published": "20231127180023",
"title": "Is the Anti-Pfaffian a PH-Pfaffian Topological Order State?"
} |
Attacking at non-harmonic frequencies in screaming-channel attacks Jeremy Guillaume10009-0005-3398-3423 Maxime Pelcat20000-0002-1158-0915 Amor Nafkha10000-0002-1164-7163 Rubén Salvador30000-0002-0021-5808January 14, 2024 ==============================================================================================================================================§ INTRODUCTION There is no doubt that our ability to produce, collect and analyze data is rapidly increasing boosted by a positive feedback loop between technological progress and new algorithms. Computer scientists, engineers, as well as physicists and mathematicians, are pushing toward the new machine learning (ML) era, which has already resulted in reforming standard data-analysis paradigms. Breakthroughs have been achieved in numerous areas of computer science, from computer vision (CV) <cit.>, up to natural language processing <cit.> as well as in some scientific contexts as the protein folding problem <cit.>.In complex flows as well, there have been numerous positive outcomes in nearly all testing scenarios, varying from control problems as single and multi-agents navigation in complex environments <cit.>, to turbulent control and drag reduction <cit.>, up to data assimilation problems <cit.> to cite few of them. However, applications in fluids are still in their infancy, and the majority of cases are either conducted on highly idealized setups or only showing preliminary results on more realistic conditions.The objective of this paper is to examine some of the ML tools that have been applied with promising results to reconstruct data from incomplete observations of complex systems, including idealized turbulent <cit.>, engineering <cit.>, and geophysical flows <cit.>, and to discuss the possible future directions for quantitative advancements in fluid mechanics.Data reconstruction is the art of filling in missing information by interpolating, denoising, or super-resolving a single realization, or a time series, of data fitting a specific statistical distribution <cit.>.ML applications for data reconstruction are emerging in many areas, from computer vision <cit.>, to medical imaging <cit.> up to seismic data reconstruction <cit.> and astrophysics <cit.>. Also in geophysical fluid dynamics works using ML to reconstruct missing data are rapidly growing <cit.>. For our focus on reconstructing complex flows it is possible to distinguish four possible different questions, see Fig. <ref>: (i) full-state restoration, with the aim to fill missing gaps in the real space state of a complex flow, (ii) inferring missing fields, which can be derived as the inverse problem solution where a physical observable that cannot be accessed/measured directly can be inferred by measuring other quantities to which it is coupled, (iii) super-resolution, which can be seen as the equivalent of point (i) but when the gap to fill is on the high-wavenumbers of the Fourier domain, (iv) dynamical modeling, which consists of reconstructing dynamically the effects of missing scales on the evolution of the resolved ones <cit.>.The issue of designing a ML-inspired sub-grid closure for modeling computational fluid dynamics is a subject per se and has been recently reviewed in <cit.>.Here, our focus lies on the first three categories of problems under the assumption that the amount of missing information to fill in is very large, which renders the problem ill-posed. This means that multiple solutions can fit within the same reconstruction <cit.>. Under this assumption, already defining what the optimal solution is, it is a question that can have different answers depending on the specific target. For instance, as discussed in <cit.>, the optimal reconstruction providing the minimum mean squared error (MSE) is different from optimal solutions in terms of other statistical quantities. In this review, we target reconstructions that maximize the correlations with the observed data while respecting the statistical features of the ground truth solution. The large-gap assumption is required when dealing with the reconstruction of complex flows. For instance, let us consider the full-state reconstruction problem of ocean surface currents. Even though satellites have allowed us to get, for the first time, a global picture of the ocean <cit.>, from mesoscale eddies up to western boundary currents, over time scales relevant to climatological studies (decades), observing the full dynamics of the ocean remains a gigantic task <cit.>, and requires filling gaps of spatial scales between hundred km up to less than a meter and time-frequency gaps spanning weeks up to turbulent and wave scales (of seconds), which cannot be neglected to explain turbulent stirring, mixing, and all vertical motions.On top of applications, there are fundamental questions associated with reconstructing complex flows. What type and quantity of information are required to perform different reconstructions is one open theoretical question, which can be investigated via a reverse engineering approach: different inputs can be passed to the same model to assess the impact on the reconstruction quality. Here, we discuss some of the ML algorithms that are transforming the conventional paradigms of data analysis and that have the potential to facilitate breakthroughs in the field of fluid dynamics. Following a chronological order, we start with an introduction of `Variational Auto-Encoders', (VAEs), `Generative Adversarial Networks', (GANs), and `denoising Diffusion probabilistic Models', (DMs). After we discuss how to combine pure Data-Driven methods with the physical knowledge at hand and we provide possible future directions in this discipline.§ DATA-DRIVEN METHODSA typical approach to repair missing data in gappy fields, before the rise of ML, was based on proper orthogonal decomposition (POD). POD is used to reduce data dimensionality by identifying the dominant patterns in a dataset and representing them using a smaller set of orthogonal basis functions, (POD modes, i.e. eigenvectors of the correlation matrix) <cit.>. The same approach have been extensively used also for filling of missing points in geophysical data sets where it takes the name of Empirical Orthogonal Function <cit.>. Extension of such techniques as the Gappy POD (GPOD) <cit.> or the Extended POD (EPOD) <cit.> were derived to repair missing data with minimal MSE solutions, showing results outperforming Kriging interpolation <cit.>. However, POD-based approaches are limited when dealing with complex multi-scale and non-Gaussian statistics as is the case of turbulent flows. As shown in <cit.>, where they implemented EPOD to reconstruct the bulk velocity of wall-bounded turbulence from wall measurements, and in <cit.> where they used EPOD and GPOD to fill missing data on velocity planes extracted from 3d rotating turbulent flows, POD methods can only reconstruct the large-scale, Gaussian features of the ground truth data. A significant advancement in this regard was brought by the ML generative models <cit.> aiming to generate new data that resembles `statistically' the training dataset. Their success can be attributed to two factors. Firstly, their architecture relies on multi-layer Convolutional Neural Networks (CNN) <cit.>, which inherently possess the ability to emphasize long-range correlations in data. Secondly, they are trained with loss functions that not only account for MSE accuracy but also for statistical differences between the generated and ground truth data. Fig. <ref> gives a schematic illustration of the three main generative models that are commonly utilized in ML. VAEs have been the first type of neural network trained to give in output new samples, not belonging to the training dataset, which satisfies the same statistical properties.In the first row of Fig. <ref>, a simple diagram depicts the workflow of the VAE model. VAEs, like their predecessors, Auto-Encoders, are based on an encoder-decoder structure. However, unlike Auto-Encoders, the aim of VAEs is not to perform a dimensionality reduction by projecting the input data, x, into a low-dimensional latent space, z. Instead, VAEs define a probabilistic decoder, p_θ(x|z), which maps any input from the latent space, sampled from a simple distribution, p(z), typically a multivariate Gaussian, into a sample in the output space that satisfies the (generally unknown) statistical distribution characterizing the training dataset. The probabilistic encoder, q_ϕ(z|x), plays a crucial role in VAEs by facilitating the sampling of the latent space, z, during training to accelerate decoder convergence. The encoder's primary objective is to model the posterior probability of the decoder, denoted as p_θ(z|x), which corresponds to the likelihood of obtaining a particular sample in the latent space z when generating a specific input from the dataset x. The presence of the probabilistic encoder assists the discriminator in exploring a smaller and more relevant sub-manifold of the latent space, resulting in faster and more stable training. VAE models operate on the basic assumption of learning a mapping between a simple and fixed distribution into the data probability distribution. Training the encoder entails minimizing the Kullback-Leibler Divergence (KLD) between the selected latent space distribution, p(z), and the encoder's output distribution, q_ϕ(z|x). This operation can generally be computed analytically and requires adding a few extra terms to the decoder loss function. As previously mentioned, the probabilistic decoder of the VAE is trained by maximizing the log-likelihood of the generated data, log p_θ(x), where; p_θ(x) = ∫ p_θ(x|z) p(z) dz .Directly computing this loss function is intractable. However, a lower bound can be defined, using a variational inference formulation, and calculated under some approximations. The approximations are based on the assumption that the decoder's errors are Gaussian. By making this assumption, the maximization of the log-likelihood can be rewritten as a minimization of the MSE. While this approximation may be reasonable in some contexts, it is certainly unsuitable for considering turbulent flows, which are well-known for their highly non-Gaussian extreme fluctuations. As shown in the context of turbulent flows on a rotating frame <cit.>, the minimization of MSE alone results in generating solutions that match the training data only at the large, more energetic scales, while over-damping the smaller scales. Therefore, rather than as generative methods, VAEs are mostly considered in the context of reduced-order modeling to perform a probabilistic projection on low-dimensional latent space, z, as studied in the context of 3d homogeneous and isotropic turbulent (HIT) flows <cit.> and more recently on a 2d flow of a simplified urban environment <cit.>.GANs are proposed to improve VAEs by relaxing the Gaussian errors assumption, and by improving the evaluation of statistical features in generated data in the loss <cit.>. In general, the functional form of the probability distribution that characterizes the training dataset is unknown. To overcome this issue, a second network, the discriminator, d_θ'(x), is used to evaluate the statistical properties of training and generated datasets. The discriminator provides a loss function that the GAN generative part, g_θ(z), can optimize during training. The discriminator functions as a classifier and is trained to assign a probability of an input being generated or extracted from a true dataset. On the other hand, the generator maps a sample from a latent space into a sample in the data space, similar to a VAE decoder. Its objective is to generate increasingly realistic samples that can fool the discriminator, from which comes the name `adversarial', where, as in a zero-sum game, a gain for one network gives an equivalent loss to the other <cit.>. For a fixed generator the analytical expression for the optimal discriminator can be derived by maximizing the adversarial loss <cit.>, and results in;d^*(x) = p_true(x)/p_true(x)+p_gen(x) ,where p_true and p_gen represent the statistical distributions of the true and generated datasets. Similarly, the optimal generator, denoted as g^*(z), can be derived as the network that minimizes the Jensen-Shannon Divergence (JSD), a symmetric formulation of the KLD, between the true and generated distributions <cit.>. GANs, have exhibited unparalleled potential in producing turbulent datasets that display a remarkable level of statistical similarity to their original counterparts. Both the original and generated data exhibit identical deviations from Gaussianity up to the evaluation of high-order statistical observables in several setups, as in super-resolving to a 64× larger 2d turbulent flows behind cylinders <cit.>, and of 3d HIT flows <cit.>, as well as in filling large gaps in rotating turbulence <cit.> and 3d channel flows <cit.>. Fig. <ref> showcases the workflows of VAEs and GANs focusing on the applications of these models to fill gaps by exploiting their generative capacities also when constrained to fit some observations. In the three panels (a)-(b)-(c), the samplerepresents the gappy data and serves as a condition to the model, the ground truth data (known only in the training stage) is denoted as x_d, the model reconstruction is called x_g.In the VAEs the conditionis passed to both the encoder and the decoder. During training the encoder projects x_d andinto the latent space by defining the variance and the mean of a Gaussian distribution from which a sample z is extracted. The loss function is the same as in the case of pure generation. In testing setup, the reconstruction the sampling on z is done from a standardized multivariate Gaussian while the decoder on top of the z sample analyzes also the condition <cit.>.Panel (b) displays the GAN reconstruction setup, which distinguishes itself from the unconstrained model in that the generator employs an encoder-decoder architecture to mapto an intermediate space z prior to generating the filling data instead of performing a random sampling on the latent space. The discriminator operates as usual, but now the overall generation loss is a linear combination of the MSE between the ground truth and the reconstruction data, in addition to the adversarial loss provided by the discriminator prediction <cit.>. GAN generates realistic samples also when constrained to match some observations. However, in the reconstruction case having statistically consistent data leads to a larger MSE with respect to the ground truth solutions. Indeed a tiny shift in space between the reconstruction and the true solution brings larger MSE if the fields are both highly fluctuating <cit.>. The principal limitation of GANs arises from their adversarial nature, resulting in highly unstable training and slow convergence. It can happen that one of the two players is dominated by the other and converges into a failure solution.Diffusion Models, Fig. <ref> bottom panel, are an alternative technique to generate. DMs transform a simple distribution into a more complex distribution, resembling the training data while avoiding the need to introduce a surrogate loss function, as seen in VAEs, and without relying on adversarial training, as in GANs.The workflow of DMs is illustrated in Fig. <ref>, for both the generation (a) and the reconstruction (b) setups.DMs use a Markov chain to gradually convert one distribution (latent space) into another (dataset), following the idea developed in non-equilibrium statistical physics <cit.>. To learn the parameterized Markov chain, DMs are trained using variational inference to produce data samples that match the original data after repeating a finite number of steps. The learning involves estimating small perturbations to a diffusion process, problem, which is more tractable than explicitly describing the full distribution within a single jump as potentially done in the other generative models. Furthermore, since a diffusion process, q(x_t|x_t-1), exists for any smooth target distribution, this method can capture data distributions of arbitrary form <cit.>. If the forward diffusion process is a Markov chain that gradually introduces Gaussian noise to the data until the signal is destroyed, the model subsequently learns how to reverse the diffusion process and generate desired data samples starting from pure Gaussian noise realizations. Unlike VAEs or GANs, diffusion models involve a latent variable z with dimensionality identical to that of the original data x.In Fig. <ref> panel (b) discusses the approach proposed in <cit.> to employ pre-trained unconditional DMs to condition the generation process on filling some partial observations. The approach involves moving forward the gappy data through the Markov chain by iteratively adding noise to the observations, while simultaneously progressing backward from the noise distribution using the reverse chain learned by the DM. To incorporate the observed data in the generation process, the strategy is to repeatedly merge the forward-propagated noisy observations with the reverse-propagated noisy signal. This allows the reverse process to propagate data information within the gap and generate a correlated reconstruction. DMs have produced state-of-the-art results in image generation, see the famous example of `DALL-E 2' <cit.>, demonstrating their ease of definition and effectiveness in training <cit.>.Attention <cit.> is another feature often implemented inside DMs architecture, which is potentially crucial for large gap-filling. Indeed, attention is meant to enhance the role of some parts of the input data while diminishing others, showing good results at handling long-range spatial relations <cit.>. However, DMs and attention, have not yet been extensively applied in the generation and reconstruction of complex gappy flow data, but they have only been used to super-resolve smooth bi-dimensional Kolmogorov flows <cit.>. Therefore, the investigation of DMs and `attention' capacity to generate high-quality samples of complex flows is an ongoing field of research.§ PHYSICS-INFORMED METHODSLeveraging the observed data and the equation of motion, Physics-Informed techniques, exploit spatio-temporal correlations to derive accurate reconstruction of incomplete data. Kalman filters, variational approaches, and nudging are examples of advanced tools that have proven effective in enhancing initial conditions for weather forecasting problems since before ML <cit.>. Nudging is a physics-informed way to control the evolution of a flow via the continuous insertion of observed data and the addition of a penalty term, which tries to keep the flow trajectory close to that of the empirical subset <cit.>.Nudging has been recently applied to reconstruct high resolution HIT flow from sparse measurements <cit.> and to estimate physical unknown parameters from turbulent data <cit.>. While physics-agnostic ML approaches are focused solely on finding patterns in data, there is growing interest in incorporating physical knowledge into ML algorithms, particularly in the field of fluid mechanics where the underlying physical laws are well understood <cit.>. The first objective is to impose constraints on the ML solutions to ensure that they adhere to the known physical properties, the second objective is to streamline the training by integrating relevant information directly into the network architecture or training setup. There exist three methods for incorporating physics into ML algorithms; (i) observational, (ii) inductive, or (iii) learning biases. Observational biases may be introduced by selecting training data to ensure that a specific aspect of physics is not only present but also emphasized, ie, extreme events can be shown during training more often than the frequency at which they occur. Inductive biases embed physical constraints into the network architecture, as for example the CNNs embed invariance along the groups of symmetries possessed by typical patterns observed in images. Finally, learning biases operate in a `soft' way by adding additional terms to the loss function that penalize non-physical solutions <cit.>, such as those that do not satisfy equations of motion, violate mass or energy conservation, and so forth. Physics-informed data-driven tools have just begun to be highlighted as particularly promising in areas as numerical weather prediction <cit.>. Improving data-driven and physics-informed methods synergy will undoubtedly be the focus of research in the upcoming years.§ PERSPECTIVESAlthough ML techniques have been already implemented as standard tools in computer science, fluid dynamics presents challenges that differ from those tackled in many applications of machine learning, such as image recognition and advertising, as stated in <cit.>. Fluid flows necessitate precise and quantitative evaluations of the multi-scale and multi-frequency physical mechanisms that they must adhere to. On top of this, while idealized flow setups offer large datasets of high complexity, and quality, in real-life flows, one needs to deal with very sparse and noisy data. The misalignment between the idealized cases studied in the literature, and real applications opens non-trivial problems connected with the generalizability and the uncertainty quantification (UQ) of the pre-trained models <cit.>. To overcome those issues, it is highly desirable to have in the future, more open-access databases, such as JHTDB (<https://turbulence.pha.jhu.edu>) and Smart-Turb (<https://smart-turb.roma2.infn.it>), and well-defined open challenges,such as (<https://github.com/ocean-data-challenges>), that can bring different communities closer, and that can drive ML applications to go beyond theoretical exercises towards the quantitative improvement required to provide advancements in fluid mechanics. Today's challenges are connected with the need of a quantitative AI, driven by several critical factors such as validation, benchmarks on generalization, and UQ of ML solutions. Another crucial aspect is the problem dimensionalization, which involves understanding the correlation between the network's architecture, deepness, structure, and size, and the physical parameters, as Reynolds, Rayleigh, and time-to-solution, among others.As discussed, already defining an evaluation metric to quantify the solution quality is an issue in fluid mechanics that needs to be carefully designed. Answering these questions is necessary, and interdisciplinary collaborations between applied scientists and AI specialists, are unavoidable for establishing best practices outperforming today's data assimilation techniques. Scientists are skilled at asking the right questions and they are asked to define targets that can be applied to real-world problems. AI specialists have a unique ability to `open the box' of complicated algorithms and unlock the potential of vast amounts of data.Despite these challenges, scientific communities have not been deterred from exploring the interactions between ML and complex flows. On the contrary, the potential impact is attracting increasing attention, resulting in a convergence of challenges and new approaches that we believe are likely to continue transforming both fluid mechanics and machine learning research.MB acknowledges Prof. Luca Biferale for useful discussion and financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 882340).eplbib.bst | http://arxiv.org/abs/2311.15822v1 | {
"authors": [
"Michele Buzzicotti"
],
"categories": [
"physics.flu-dyn",
"physics.comp-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20231127134705",
"title": "Data reconstruction for complex flows using AI: recent progress, obstacles, and perspectives"
} |
firstpage–lastpage Attacking at non-harmonic frequencies in screaming-channel attacks Jeremy Guillaume10009-0005-3398-3423 Maxime Pelcat20000-0002-1158-0915 Amor Nafkha10000-0002-1164-7163 Rubén Salvador30000-0002-0021-5808January 14, 2024 ============================================================================================================================================== How protoplanetary discs evolve remains an unanswered question. Competing theories of viscosity and magnetohydrodynamic disc winds have been put forward as the drivers of angular momentum transport in protoplanetary discs. These two models predict distinct differences in the disc mass, radius and accretion rates over time, that could be used to distinguish them. However that expectation is built on models that do not include another important process - photoevaporation, both internally by the host star and externally by neighbouring stars. In this work we produce numerical models of protoplanetary discs including viscosity, magnetohydrodynamic disc winds, and internal and external photoevaporation. We find that even weak levels of external photoevaporation can significantly affect the evolution of protoplanetary discs, influencing the observable features such as disc radii, that might otherwise distinguish between viscous and wind driven discs. Including internal photoevaporation further suppresses differences in evolution between viscous and wind driven discs. This makes it much more difficult than previously anticipated, to use observations of nearby star forming regions to determine whether discs are viscous or wind driven. Interestingly we find that evolved protoplanetary discs in intermediate FUV environments may be the best cases for differentiating whether they evolve through viscosity or magnetohydrodynamic disc winds. Ultimately this work demonstrates the importance of understanding what are the key evolutionary processes and including as many of those as possible when exploring the evolution of protoplanetary discs. accretion, accretion discs – protoplanetary discs – (stars:) circumstellar matter § INTRODUCTIONPlanet formation is likely to be intrinsically linked to the evolution of the circumstellar discs of material around young stars <cit.>. However, understanding this is an extremely challenging problem. We observe each disc at only a snapshot in its evolution, and any given disc is difficult to age. This is further complicated by the fact that star formation happens over time in stellar clusters <cit.>. Since discs have a huge radial variation in composition, temperature, density and optical depth, any single observation also only probes a limited region of the disc <cit.>. Whilst there are numerous observational challenges, there are many processes that are thought to affect how and on what time-scales protoplanetary discs evolve. For example discs evolve through accreting on to the central star <cit.>, internal photoevaporative <cit.> or magnetically driven winds <cit.> as well as winds driven by external irradiation from nearby massive stars <cit.>. Typically most disc evolution models only include one or two of these processes.Focusingon accretion through the disc and onto the central star, the main theories accounting for the necessary angular momentum transport are through viscous accretion or magnetohydrodynamic (MHD) driven disc winds. Traditionally, turbulence in the disc, driven by the magnetorotational instability (MRI)<cit.> was thought to drive accretion. This turbulence was modelled as a form of viscosity <cit.>, where shearing rings of gas would drive turbulent transport through the disc. Using the popular α-disc model <cit.>, this was found to require α∼ 10^-3–10^-2 in order to match observed accretion rates <cit.>. More recently, numerous works have provided additional observational estimates for the viscosity parameter α, through for example, accretion on to the central star, or the vertical extent of dust discs <cit.>. They typically find a range in α from ∼10^-4 to ∼ 10^-2 with a median between 3×10^-4–3×10^-3 <cit.>.On the other hand, it has also been widely accepted that large areas of protoplanetary discs are too dense to be sufficiently ionised and thus coupled the stellar magnetic field <cit.>. For such regimes, where the gas is only partially ionised, non-ideal MHD effects dominate and suppress the MRI <cit.>, resulting in a non-zero magnetic flux that then drives magnetised disc winds <cit.>. The properties of such winds would then be determined by the properties of the magnetic field, rather than the local disc conditions <cit.>. As the winds then transport mass and angular momentum away from the disc, this then drives an accretion flow to conserve angular momentum <cit.>. Recently, analytic disc models have been formulated that include MHD disc winds as the driver of accretion through the disc, finding qualitative differences between their evolution and that of purely viscous discs <cit.>.Whilst we cannot directly measure whether discs are viscous or MHD wind driven, we can: place constraints on the viscous α parameter which indirectly constrains the possible driving mechanism <cit.>, find evidence for magnetised disc winds <cit.>, or study populations of discs at different evolutionary stages to compare with the predictions of work such as <cit.>. A factor that may complicate the ability to unambiguously distinguish MHD and viscously driven evolution is photoevaporation, where high energy radiation from either the central star <cit.>, or from nearby stars <cit.> heat the outer layers of protoplanetary discs and drive a thermal wind. Internal photoevaporative winds and MHD disc winds originate from similar portions of the disc, which can make them difficult to distinguish. Both internal and external photoevaporative winds can affect the disc mass, radius, lifetime and accretion properties over time <cit.>. With multiple processes acting on protoplanetary discs, previous works have explored differences between individual factors. For example <cit.> analytically compared purely viscous discs to purely wind driven discs, finding qualitative differences in disc radii and mass accretion rates. <cit.> studied the interplay between viscous resupply of an externally driven wind, finding that the rate of spreading moderates the external mass loss rate in older systems. More recently <cit.> compared analytic solutions for wind driven discs and viscous discs, including internal photoevaporation. They concentrated on mass accretion rates, finding that sufficiently large observational datasets could differentiate between the models. Exploring the effects of photoevaporation on viscous discs, <cit.> found five different evolution pathways for protoplanetary discs, where observations could possibly constrain the relative importance of internal and external photoevaporation for any given disc.In this work, we systematically include all of these processes in a single numerical model. We adapt the analytic model found in <cit.> and couple to the prescribed mass loss rates for internal photoevaporation <cit.>, and interpolate within the newly updated fried grid of mass loss rates due to external photoevaporation <cit.>. Our aim is to determine if and what differences there are in the evolution of viscously and wind-driven protoplanetary discs in different star forming environments. These differences could then be compared to future observations, hopefully placing further constraints on the processes that are at play, and to what their properties are, within protoplanetary discs. This paper is organised as follows. Section <ref> outlines the disc evolution and photoevaporation models as well as the simulation parameters. In sect. <ref> we explore the effects of the local star forming environment on the evolution of viscous or wind driven discs. We include the effects of internal photoevaporation and examine its consequences in sect. <ref>. Section <ref> discusses possible observations that could differentiate between viscous and MHD wind driven models. Finally we draw our conclusions in sect. <ref>.§ PHYSICAL MODEL AND PARAMETERSProtoplanetary discs lose mass by accretion onto the central star and through photoevaporative winds launched from the disc surface layers. To account for these processes we follow <cit.> and include both contributions from a traditional 1D viscous α disc model <cit.>, in addition to contributions from an MHD disc wind. The MHD disc wind is controlled by two parameters: α_ DW, which quantifies the angular momentum extracted by the wind, and λ, the magnetic lever arm parameter which determines the amount of mass lost in the wind. The gas surface density Σ at radius r, is therefore evolved solving an updated diffusion equationΣ̇(r)= 1rddr[3r^1/2ddr(νΣ r^1/2)]+32rddr[α_ DWΣ c_ s^2Ω]-3α_ DWΣ c_ s^24(λ-1)r^2Ω-Σ̇_ PE(r)where ν=α_ v H^2Ω is the disc viscosity with viscous parameter α_ v <cit.>, H is the disc scale height and Ω the Keplerian frequency. The second and third components on the right hand side of eq. <ref> represent the change in surface density due to the angular momentum extracted by the wind, and the the mass lost in the MHD wind itself. The fourth term represents mass extracted by photoevaporative winds, which we discuss below. Note there are two components for α in eq. <ref>, those being the measure of turbulence for viscosity α_ v, and the measure of angular momentum extracted in the wind α_ DW. The equation does not however specify the relationship between the two α parameters. Therefore we follow <cit.> and defineα = α_ v + α_ DWwhere α quantifies the total torque exerted by turbulence and the MHD disc wind. Further, we defineψ = α_ DWα_ vas a parameter that quantifies the relative strength between the two α parameters. In this work we explore between ψ=10^-4 as a viscous case, to ψ = 10^4 as a wind driven case.Our models also include depletion of the disc through photoevaporative winds. This is expressed by Σ̇_ PE(r) in eq. <ref>. Following <cit.> we include internal photoevaporative winds due to X-ray photons coming from the central star (detailed in section <ref>) as well as winds launched from the outer disc by far ultraviolet (FUV) radiation emanating from nearby massive stars (e.g. O-type stars, see section <ref>). When we utilise both internal and external photoevaporation regimes, we assume that the photoevaporative mass loss rate at any radius in the disc is the maximum of the internally and externally driven rates Σ̇_ PE(r) = max(Σ̇_ I,X(r),Σ̇_ E,FUV(r))where the subscripts I and E refer to contributions from internal and external photoevaporation.We assume that the disc is in thermal equilibrium, where the temperature is calculated by balancing irradiation heating from the central star, background heating from the residual molecular cloud, viscous heating and blackbody cooling To attain this equilibrium, we follow <cit.> and use an iterative method to solve the following equation <cit.>Q_ irr + Q_ν + Q_ cloud - Q_ cool = 0where Q_ irr is the radiative heating rates due to the central star, Q_ν is the viscous heating rate per unit area of the disc, Q_ cloud is the radiative heating due to the residual molecular cloud, and Q_ cool is the radiative cooling rate. §.§ Internal Photoevaporation The absorption of high energy radiation from the host star by the disc can heat the gas above the local escape velocity, and hence drive internal photoevaporative winds. EUV irradiation creates a layer of ionised hydrogen with temperature ∼10^4 K <cit.>, however X-rays penetrate deeper into the disc and are still capable of heating up to around ∼10^4 K <cit.> so for low mass stars are expected to generally dominate over the EUV for setting the mass loss rate. FUV radiation penetrates deeper still, creating a neutral layer of dissociated hydrogen with temperature of roughly 1000K <cit.>. The overall interplay between the EUV, FUV and X-rays is a matter of ongoing debate.<cit.> find that including the FUV heating simply causes the flow beneath the sonic surface to adjust, but otherwise retains the same mass loss rate. However others suggest a more dominant role of the FUV <cit.>. Recent models including all three fields suggest a more complicated interplay <cit.>.The outcome also depends sensitively on how the irradiated spectrum is treated <cit.>. The radiation hydrodynamic models of <cit.> used pre-computed X-ray driven temperatures as a function of the ionisation parameter (ξ = L_X / n /r^2) wherever the column to the central star is less than 10^22cm^-2 (and hence optically thin). This approach has since been updated with a series of column-dependent temperature prescriptions <cit.>.We follow <cit.> who further build on the work of <cit.> and <cit.> and find that the mass loss profile from internal X-ray irradiation is approximated byΣ̇_ I,X(r)= ln(10)(6aln(r)^5rln(10)^6+5bln(r)^4rln(10)^5+4cln(r)^3rln(10)^4..+3dln(r)^2rln(10)^3+2eln(r)rln(10)^2+frln(10))×Ṁ_ X(r)2π r^2yrwhereṀ_ X(r)Ṁ_ X(L_X) = 10^alog r^6+blog r^5+clog r^4+dlog r^3+elog r^2+flog r+gwhere a=-0.6344, b=6.3587, c=-26.1445, d=56.4477, e=-67.7403, f=43.9212, and g=-13.2316. We follow <cit.> and apply a simple approximation to the outer regions of the disc where the internal photoevaporation rates drop to zero. The reasons given for this sudden drop is that the wind itself blocks radiation from heating the outer regions of protoplanetary discs. However these do not take into account the effects of when the disc and/or the wind become optically thin and therefore ineffective at blocking the radiation. The temperature of X-ray irradiated gas varies from ∼ 10^3–10^4 K depending on the distance in the disc <cit.>. To be conservative we define the radius at which the internal photoevaporation scheme drops off as the gravitational radius for 10^3 K gas. We therefore apply the following approximation at radial distances greater than r_ rgxΣ̇_ I,X,ap = 37.86×Σ̇_ rgx(r/r_ rgx)^-1.578where Σ̇_ rgx is equal to eq. <ref> calculated at r=r_ rgx, andr_ rgx = GM_*c_ s^2where c_ s is the sound speed for gas of temperature T=10^3 K, and μ=2.35. In the outer regions of the disc the loss in gas surface density due to internal photoevaporation then becomesΣ̇_ I(r) = max(Σ̇_ I,X(r),Σ̇_ I,X,ap) Following <cit.> the integrated mass-loss rate, dependant on the stellar X-ray luminosity, is given aslog_10[Ṁ_X(L_X)] = A_ Lexp[(ln(log_10(L_X))-B_ L)^2C_ L]+D_ L,in , with A_ L = -1.947×10^17, B_ L = -1.572×10^-4, C_ L = -0.2866, and D_ L = -6.694.§.§ External Photoevaporation In addition to internal winds driven by irradiation from the host star, winds can also be driven from the outer regions of discs by irradiation from external sources. Massive stars dominate the production of UV photons in stellar clusters and hence dominate the external photoevaporation of discs. External photoevaporation has been shown to play an important role in setting the evolutionary pathway of protoplanetary discs <cit.>, their masses <cit.>, radii <cit.> and lifetimes <cit.> even in weak UV environments <cit.>. We do not include shielding of the protoplanetary discs, i.e. by the nascent molecular cloud, that has been shown to have an effect on the effectiveness of external photoevaporation <cit.>, but instead will infer it's effects by examining weaker environments.In our simulations, the mass loss rate due to external photoevaporation is calculated via interpolating over the recently updated fried grid <cit.>. This new grid expands on the original version of fried <cit.> in terms of the breadth of parameter space in UV field, stellar mass, disc mass and disc radius. The new grid also provides the option to use different PAH-to-dust ratios, which is important because polycyclic aromatic hydrocarbons (PAH) can provide the main heating mechanism in a photodissociation region (PDR, which is the region at the base of an external photoevaporative wind) and their abundance is uncertain. Finally, the new grid provides the option to control whether or not grain growth has taken place in the disc, which affects the opacity in the wind since only small grains are entrained.The fried grid provides mass loss rates for discs irradiated by FUV radiation as a function of the star/disc/FUV parameters. In our simulations, we determine the mass loss rate at each time step by linearly interpolating fried in three dimensions: disc size R_d, disc outer edge surface density Σ_out and FUV field strength F_FUV.We evaluate the fried mass loss rate at each radius from the outer edge of the disc down to the radius that contains 80% of the disc mass. We choose this value as 2D hydrodynamical models show that the vast majority of the mass loss from external photoevaporation, comes from the outer 20% of the disc <cit.>. The change in gas surface density is then calculated asΣ̇_ext, FUV(r) = G_ smṀ_ext(R_max)/π(R^2_d - R_max^2)+A_ sm,where A_ sm is a smoothing area equal to A_ sm = π(R_ max^22-(0.1 R_ max)^22)11R_ max^20and G_ sm is a smoothing functionG_ sm = r^20R_ max^20. The fried grid contains multiple subgrids that vary the PAH-to-dust ratio (f_PAH) and specify whether or not grain growth has occurred. The effects of using different combinations of these parameters will be explored in future work, but we do not expect such changes to affect the differences between viscous and MHD wind driven discs. The combination we use here is f_PAH=1 (an interstellar medium, ISM,-like PAH-to-dust ratio) and assume that grain growth has occurred in the outer disc, depleting it and the wind of small grains which reduces the extinction in the wind and increases the mass loss rate compared to when dust is still ISM-like. This combination of parameters results in PAH-to-gas abundances comparable to our limited observational constraints on that value <cit.>.§.§ Simulation Parameters Whilst our previous work examined the evolution of viscously evolving discs, including internal and external photoevaporation, around stars of different masses <cit.>, here for simplicity, we only consider Solar mass stars. In future work, we will explore populations of stars with varying masses and in varying environments. To explore the differences that arise from viscous or MHD-wind driven discs, we vary the ratio of the MHD-wind to turbulent viscous transport, ψ, in log intervals of 0.1 between ψ = 10^-4, that being the viscosity dominated case, to ψ = 10^4, that being the MHD-wind dominated case. Note that ψ=1 represents the hybrid case, where α is evenly split between viscosity and the MHD wind. Numerous works have provided observational estimates for the viscosity parameter α <cit.>. To ensure consistency, we assume that the maximum that our value for α_ v can be is 10^-3, consistent with estimates <cit.>. This is also consistent with our previous disc evolution scenarios <cit.>, as well as planet formation scenarios that show that α≤ 10^-3 in order to form circumbinary systems similar to BEBOP-1 <cit.>. For the external photoevaporative mass loss rates, we vary the strength of the local environment, ranging from 10 G_0 to 10^5 G_0, which spans most of the range found in star forming regions <cit.>. X-ray luminosities are observed to vary by up to two orders of magnitude even for stars of the same mass, due to a combination of measurement uncertainty and genuine intrinsic differences in X-ray activity levels, which are time varying <cit.>. Whilst we do not perform a parameter study including variations in X-ray luminosities, we take L_X = 10^30 erg s^-1 to account for the central star driving an internal photoevaporative wind. We define this as a weak photoevaporative wind, since the X-ray luminosity chosen is half a magnitude smaller than the average found for Solar mass stars in nearby clusters <cit.>. With this X-ray luminosity, it yields an integrated mass loss rate of 5.4×10^-8, but this value would be the maximum possible total internal photoevaporative mass loss rate, since we assume the mass loss due to photoevaporation at any given point in the disc is the maximum of the internal and external photoevaporative winds. Note that any plausible value for L_ X within these simulations would give mass loss rates due to internal photoevaporation comparable to or exceeding the stellar accretion rate, as this is a natural consequence of internal photoevaporation <cit.>.We initialise our discs following <cit.>Σ = Σ_0(r/R_ C)^-1exp(-R/R_ C)where Σ_0 is the normalisation constant set by the total disc mass, (for a given R_ C), and R_ C is the scale radius, which sets the initial disc size, taken here to be equal to 50 . For the initial mass of the disc we follow <cit.> where from hydrodynamic simulations they find the maximum disc mass M_ d, max that a gas disc of radius r_ ini, and a radial slope of -1 around a star of mass M_* can be before becoming gravitationally unstable M_ d, maxM_* = 0.17 (r_ ini100)^1/2(M_*)^-1/2.Given that we initialise the disc with the <cit.> self similar solution, and not a constant slope, we take r_ ini = 100, twice that of the scale radius, in order to obtain appropriate protoplanetary disc masses that are gravitationally stable. Therefore in this work, with r_ ini = 100, the initial disc mass corresponds to M_ d=0.17. Table <ref> shows the simulation parameters that we used in this work.§ EFFECT OF THE LOCAL RADIATION ENVIRONMENTThe main focus of this section is on the effects of the local radiation environment, through external photoevaporative winds, on the evolution of discs that otherwise evolve through either viscosity or MHD-winds. Within the simulation parameters we use, this corresponds to ψ=10^-4 for the viscous disc, and ψ=10^4 for the wind-driven disc. These are most equivalent to the ψ=0 and ψ=∞ in <cit.>. Initially we will explore the effects of a weak FUV environment, equivalent to 10G_0, and similar to the FUV field strength found in star forming regions such as Taurus and Lupus <cit.>. For this section, we do not include the effects of internal photoevaporation, so that we can systematically determine the effects of the local environment on the evolution of viscous or MHD wind driven discs.In fig. <ref> we show the temporal evolution of the gas surface densities for a viscous dominated disc (left panel), and a wind dominated disc (right panel). Because of the (weak) external FUV field, both discs evolve in a similar way at the start of their lifetimes, where they begin to truncate and reduce in size. For the viscous case, the disc truncates to ∼300 where the viscous expansion rate equals the mass loss rate through external photoevaporation <cit.>. Over the rest of the disc lifetime here, the disc continues to accrete onto the central star and lose mass through the external photoevaporative wind. In contrast, for the wind dominated disc (where there is no outwards expansion like in the case of viscous spreading) this results in no balancing of the disc with the external photoevaporative wind, and so the disc continues to truncate over the disc lifetime.Comparing fig. <ref> with figure 6 of <cit.>, the wind dominated disc evolves in a similar manner with it constantly reducing in size and mass. However, whilst the viscous case here truncated to an equilibrium, the equivalent case in <cit.> continued to expand in size as it lost mass. This shows the importance of including, even to a small degree (e.g. 10G_0), the effects of external photoevaporation. §.§ Disc radii In terms of the disc radius, it is clear that there is a difference in evolution between viscous and wind dominated discs, with the wind dominated discs truncating at a much faster rate. Figure <ref> shows the evolution of the disc radius, defined as that containing 90 per cent of the mass, for the viscous (blue line), hybrid (ψ = 1, red line) and wind (orange line) dominated discs. The solid, dashed and dotted lines are for external FUV radiation fields of 10, 10^3 and 10^5 G_0 respectively.For low G_0 environments, shown by the solid lines, it is clear that the wind dominated case has a different evolution track to those including the effects of viscosity, with it constantly decreasing in size. The viscous and hybrid cases however expand outwards initially until they reach an equilibrium with the external mass loss rates, before they then slowly begin to truncate. This is similar to that seen in other works <cit.> and should observations give hints to disc sizes over time, it could be possible to determine the driver of angular momentum transfer in protoplanetary discs.However, in stronger environments, where mass loss rates are significantly higher, external photoevaporation is able to truncate the disc and dominate the discs evolution irrespective of the mode of angular momentum transport, typically from the outside-in <cit.>. The dashed and dotted lines in fig. <ref> show the evolution of disc radius for stronger FUV environments, that being 1000 G_0 and 10^5G_0. With stronger mass loss rates, the equilibrium point between viscous expansion and mass lost through external photoevaporation moves inward, as can be seen by the steady decline in disc radii for the blue and red dashed lines in fig. <ref>, showing the evolution of discs in a 1000 G_0 environment. Interestingly there is no expansion in these discs, with the equilibrium point being located closer than that for the initial disc. Equally of note, there is little difference between the viscous and wind dominated discs here for the first ∼2Myr, with the only main difference in the end being the faster truncation of the disc in the latter stages of the disc lifetime, when the discs are <40 in size. This would indicate that maybe, only in older stellar clusters and with smaller compact discs, will it be possible to discern whether discs are driven through viscosity or through disc winds.Increasing the external field strength further, with the dotted lines showing for a 10^5G_0 environment, it is clear that all the discs in this case truncate down effectively to ∼ 10 where the effectiveness of external photoevaporation strongly decreases, as the gas is too deep within the stars gravitational well. The later evolution is then determined through either viscosity or disc winds, with those evolving through disc winds doing so at a faster rate, as the viscous case still attempts to maintain outward expansion, thus reducing the flow rate of material through the disc on to the star. Evidently here, the differences in disc sizes is extremely small, making discs in strong FUV environments, poor targets to test other evolution processes of protoplanetary discs. §.§ Mass Accretion Rates Whilst the disc radius shows some environments that could yield observable differences between viscous and wind driven discs, the mass accretion rate on to the star provides another empirical constraint on disc evolution <cit.>. In fig. <ref> we show the mass accretion rates over time for viscous (blue), hybrid (red), and wind driven (orange) discs in a weak environment of 10G_0 (solid lines), and a strong environment of 10^5G_0 (dashed lines). The wind driven case can be seen to be around a factor few times smaller than the viscous discs for most of the disc lifetime. This is irrespective initially of the environment, since the material in the inner disc is not affected by the local environment until it begins to affect the resupply of material from the outer disc. At this point, the mass accretion rates drop, to balance what mass is flowing through the disc. Interestingly for the stronger environment, the more viscous discs evolve faster and reduce in accretion rate quicker than the wind driven discs. This is mainly due to the balancing of viscous expansion, limiting the supply of gas to the inner regions of the disc, to be accreted on to the central star. Since there is no expansion for wind driven discs, this allows them to continue accreting through the disc at a similar rate until, the end of the disc lifetime when there is no material remaining and the accretion rate plummets dramatically.Whilst there are some subtle differences in the mass accretion rates, in practice this is likely to be a prohibitively difficult individual measure to differentiate between viscous and wind driven discs. Since the accretion rate is dependent on the inner regions of the disc, it is therefore heavily dependent on the initial disc properties. These include the disc mass, the disc size and how compact it is, as well as the strength of the MHD wind or turbulence. It may also be sensitive to the presence of planets within the inner disc and internal photoevaporation (as we discuss below).Manipulating the initial conditions well within reasonable bounds, would yield similar mass accretion rates, washing out any discernible difference between viscous and wind driven discs. This is in contrast to the disc radius, where the qualitative evolution of the different processes would remain. §.§ Other Observables We now look at a number of other observable properties that are thought to give rise to differences between viscous and MHD wind evolving protoplanetary discs. Figure <ref> shows the relation between disc mass and disc radius (left panel), disc mass and accretion rate on to the central star (middle panel), and the temporal evolution of disc mass divided by the mass accretion rate, a proxy for the disc lifetime (right panel). The colours again show a viscous disc (blue), a hybrid disc (red) and a wind driven disc (orange), whilst solid lines show the discs evolving in a weak environment, and dashed lines show a strong environment. Expectations from analytical models show a divergence in how disc radii evolve with disc mass, with viscous discs becoming larger in radius as the discs evolve and lose mass <cit.>. For wind driven discs, they continue to reduce in size as mass in accreted on to the central star. Our models here are somewhat in agreement, with viscous models generally being larger than wind driven models for similar disc masses, but the expansion of the viscous discs here is balanced by (even weak) external photoevaporation, before they begin to slowly reduce in size. Additionally, whilst those discs in stronger FUV environments also show a similar trend, the differences are only noticeable once the discs are less than 10 in size, and so such small differences will be difficult to realise in observations.Moving to the middle panel of fig. <ref>, analytic expectations for viscous discs have found correlations between the disc mass and the mass accretion rate, where discs tend to have lower accretion rates as the discs reduce in mass, and reach the end of their lifetime <cit.>. When including the effects of internal photoevaporation, <cit.> found that instead of a linear correlation between the mass accretion rate and disc mass, there was instead a “knee” when mass accretion rate fell below the photoevaporative mass loss rate, drastically reducing the mass accretion rate. In analytic comparisons to wind driven discs, this results in the viscous discs having lower mass accretion rates than wind driven discs for discs of similar mass <cit.>. Looking at fig. <ref>, this is not seen in the weak external environment (solid lines) where the viscous disc maintains larger accretion rates than the wind driven disc. However these discs have not reached the end of their lifetime, since we stopped the simulations after 20 Myr, where the effects may be observed. For the stronger external environment (dashed lines), such a trend where the accretion rate for viscous discs is lower is observed, but only once the discs are reduced to ∼ 1-2 Jupiter masses, and by comparing to the left panel of fig. <ref>, this is only once they are ∼fewin size. However, no such knee feature is observed, since our models shown in fig. <ref> did not include internal photoevaporation. Whilst the differences are not large, only a factor few at most, these again could be matched by altering the initial properties of the disc. However, given the compactness of the discs in the wind driven case, this would suggest that they will have significantly larger accretion rates at smaller disc sizes than those that evolve through viscous accretion. Should observations be able to accurately obtain both measures for compact, weakly accreting discs, this could shed light on the processes that occur within them. It is also interesting to note that in stronger external environments, both evolving viscous and wind driven discs exhibit larger accretion rates than those of similar mass in weak environments. This is expected since the accretion rate is more affected by the inner disc region, whilst the disc mass is influenced by the processes in the outer disc. Therefore, observations of different star forming regions should observe this signature.Finally, by using a proxy for the disc lifetime, though note it assumes no other mechanism for mass loss, the right panel of fig. <ref>, shows the temporal evolution of the disc mass divided by the accretion rate, i.e. the depletion time-scale. From previous works <cit.>, it is expected the remaining lifetime of viscous discs increases as time progresses. This is mainly due to the assumptions that the discs are constantly expanding, whilst accretion rates are falling. However even with the inclusion of weak external photoevaporation this expansion is halted, making the assumption no longer valid. Indeed, even the solid lines for a weak external environment show the remaining lifetime remaining level, similar to the wind driven case. In fact, the remaining lifetime, by this metric is also shorter for the viscous discs. When increasing the strength of the local environment, this again does not agree with analytic expectations, as both the viscous and wind driven rates very quickly drop as the discs are mainly dispersed through photoevaporative winds. Only when the inner disc remains, does the remaining lifetime in the viscous disc exceed that of the wind driven, as it tries to simultaneously expand whilst accreting on to the central star. Therefore in summary, when observing discs in regions where even weak external photoevaporation is present it will be challenging to use this metric to differentiate between viscous and MHD wind driven discs. §.§ Optimal observations With the above sections showing how viscous and wind driven discs evolve in different environments, we now explore the regions in time and space that show the largest differences for such discs. Figure. <ref> shows the difference in disc radii over time between viscous discs and wind driven discs, for a range of external FUV radiation field strengths. For values >0 in fig. <ref>, viscous discs are larger than wind-driven discs and vice versa for values <0. It is clear from fig. <ref> that discs that evolve in stronger FUV environments (≥ 10^4G_0) show very little difference in disc radii (i.e. <10). Additionally in more intermediate environments, e.g. 10^3G_0, viscous discs are only more than 10 larger, after ∼ 7Myr, when the discs have evolved somewhat and will be smaller in size. In weaker environments, viscous discs are nearly always larger, but these differences are only greater than 10 after 1–2 Myr. This effectively shows the age of the disc that observations are required to yield the greatest differences between viscous and wind driven discs.Whilst fig. <ref> showed the difference in disc radii over time, it is also interesting to know how this relates to the disc size. Figure. <ref> now shows the disc radius against the difference in radii between viscous and wind driven discs. It is clear that for weaker environments, shown by the blue and red lines, the discs are fairly extended when there is a noticeable difference. However for the intermediate environment, the orange line showing a disc in a 10^3G_0 environment, whilst the differences become ≥10 after ∼ 7Myr, this is only when the wind driven disc is 30 in size. This means that knowing the disc age, and other properties that influence its evolution will become important in comparing viscous and wind driven disc models. Additionally for the discs in strong UV environments, the main differences are only when the discs are extremely compact, and so where external photoevaporation is ineffective, and so this is dependent on how quickly the discs evolve to this state and the viscous discs become flat and slow accretors. In summary, fig. <ref> again shows that weak environments would be the best targets for comparing disc sizes to determine whether discs are viscous or wind driven.§ CONSEQUENCES OF INTERNAL PHOTOEVAPORATIONSection <ref> has shown how external photoevaporation affects viscous and wind driven discs, and indicated that discs in weak FUV environments are best suited for observations with the aim of differentiating between viscous and wind driven disc evolution. However, in these environments, the models show that the disc lifetimes are extremely long (>10 Myr) much longer than those observed in nearby clusters <cit.>. Additionally, the value of α, used to determine the level of turbulence in viscous discs, was set to 10^-3, which is consistent, if not slightly higher than that observed in protoplanetary discs <cit.>. This value was also used for the strength of the MHD wind. To obtain shorter lifetimes, increasing the strength of α would reduce the lifetime, however, this would be in disagreement with observational constraints on α. Also, for wind driven discs, this would increase the mass accretion rate, which again would need to be compared with observed values. Other ways of reducing the lifetime would be on altering the initial properties, of the disc, i.e. starting with smaller, less massive discs. However, this would not solve the viscous disc evolution sufficiently, since viscous expansion would still lead to an equilibrium point with external photoevaporation, that would leave large abundances of mass in the outer disc, which would be slowly removed from the system. Additionally, whilst wind driven discs will be sufficiently more compact as well, and with no process for expansion, they may be too compact to be accurately compared with observations.We have so far not considered internal photoevaporation in this paper. Energetic photons from the central star, typically X-rays, can act to heat the surface layers of the disc and drive a photoevaporative wind. For Solar mass stars, this wind typically removes mass from 10-100 , a similar region to the MHD wind closer to the star, and the external photoevaporative wind far from the star. The interplay between internal photoevaporative and MHD winds is not extensively studied <cit.>, though recent work has studied the interplay between Hall-effect MHD and internal photoevaporation <cit.>, finding that winds are mainly driven by photoevaporation except when the magnetic field is strong or the X-ray luminosity is low (log_10(L_X)≤29.3). Given the uncertainty in how to treat internal photoevaporation and MHD winds simultaneously, we only include a weak internal photoevaporative wind. Specifically we set the central star's X-ray luminosity L_ X=10^30 erg s^-1, whereas the average value for Solar mass stars in star forming regions is L_ X=10^30.5 erg s^-1 <cit.>. Given we consider relatively weak X-ray luminosities, and given the uncertainty in the interaction between photoevaporative and MHD winds, we treat both the MHD and photoevaporative winds entirely separably, to make an initial exploration of their effects on the models. §.§ Surface Density Evolution Figure <ref> shows the gas surface density evolution for a viscous disc (left panel) and a wind driven disc (right panel). Like fig. <ref>, these discs evolved in a weak FUV environment (10G_0), but this time include internal photoevaporation due to X-rays. Comparing to fig. <ref>, it is clear that the discs evolve substantially faster when there is even a weak internal photoevaporative wind. Indeed the discs are fully dispersed after only 4 Myr, much closer to the typicalobserved protoplanetary disc lifetimes of ∼3 Myr <cit.>. This is mainly due to the internal photoevaporative wind removing gas from the intermediate regions of the discs, which removes the supply of material to the outer disc, allowing external photoevaporation to truncate the disc at a faster rate. Interestingly, both the viscous and wind driven discs had similar lifetimes, and it appears from fig. <ref>, similar evolution profiles. Nearer to the end of the disc lifetime, internal photoevaporation would be able to open a hole in the disc, as seen in the right panel of fig. <ref>, where either viscous transport or the MHD wind would be unable to replenish the material at that radius lost through the wind. Once this occurs, the disc radii can truncate extremely quickly, since as with internal photoevaporation truncating outwards on the inner edge of the outer disc, and external photoevaporation acting on the outer edge, eventually such an outer disc will become thin, ring-like, before being dispersed. This would cause a sudden drop in the radius of the disc. Note that the gap opened by internal photoevaporation in the viscous case is not shown in fig. <ref>, since the gap opened and additionally the outer disc quickly dispersed in between our plotting intervals. In both cases, the gaps in the discs were only open for a short time (<0.1 Myr), since the outer discs were quickly dispersed. Interestingly, the gap was able to open at a slightly earlier time in the viscous case, since viscous expansion of the outer disc allowed for a larger external photoevaporative mass loss rate, that depleted the outer disc quicker. Whilst this did not affect the disc lifetime, since the final lifetime was determined by the rate of accretion in the inner disc, this did result in the outer disc dispersing at an earlier time.To further highlight the influence of photoevaporation on the evolution of the disc mass, fig. <ref> shows the mass loss rates for a viscous disc in a 10^3G_0 environment. The blue line shows the accretion rate on to the central star, whilst the red and yellow lines denote the mass lost through internal and external photoevaporation respectively. The effect of external photoevaporation is clear in this environment, since it dominates the total mass loss rate of the disc for the first ∼0.5 Myr, whilst the disc is large. After this time, internal photoevaporation begins to dominate since it removes mass from the intermediate regions of the disc, where external photoevaporation is less effective. Interestingly, it is only early in the disc lifetime, that the stellar accretion rate is larger than the internal photoevaporation rate, since there is a large buildup of material in the inner disc. As the disc depletes, the accretion rate on to the star also decreases, consistent with observed trends in regions such as Chamaeleon I <cit.> and Lupus <cit.>. After only ∼0.1 Myr, the mass accretion rate is weaker than the internal photoevaporative mass loss rate. Note that this is also consistent for discs in other external UV environments, and additionally from discs evolving through MHD disc winds. It is also worth noting that the central star's X-ray luminosity is weak here compared to the average observed in stellar clusters <cit.>, and so we consider this a weak internal photoevaporative wind, as larger X-ray luminosities will yield much stronger mass loss rates. Therefore, fig. <ref> emphasises how photoevaporation dominates the mass loss mechanisms for protoplanetary discs, with the relative strengths between internal and external photoevaporation determining which mechanism dominates the overall mass loss contributions. This was also seen in recent work that included high photoevaporation rates <cit.>, and further illustrates the importance of including photoevaporation within models of protoplanetary disc evolution. §.§ Effects on disc radiiThe similarities in disc radii is clearly evident in fig. <ref>, which shows the evolution of the disc radius, taken at 90 per cent of the disc mass, for viscous or wind driven discs in different environments. Where in fig. <ref> there were differences between viscous and wind driven discs in the weak external environments, there are minimal differences here in fig. <ref> when even just a weak internal photoevaporative wind is included. Additionally, discs are seen here to dramatically reduce in radius, this being the time just after internal photoevaporation has opened a hole in the disc, with the outer disc then being quickly dispersed from both sides. Interestingly it may actually be this feature that is a useful indicator of whether discs are viscous or wind driven, since it appears that viscous discs evolve faster than wind driven discs in more intermediate environments. Here the viscous expansion fed a stronger external mass loss rate, which ultimately allowed the outer disc region to deplete at a faster rate, once internal photoevaporation was able to open a hole. This however is not the case for stronger external environments, since external photoevaporation there truncates the disc down to a small size before a hole is able to open <cit.>.In consequence, the inclusion of internal photoevaporation here has significantly dampened the possibility of using the disc radius as a measure of whether discs are viscous or wind driven. This, along with the inclusion of external photoevaporation above, shows the importance when exploring what drives the evolution of protoplanetary discs, of including all of the major processes that determine the outcome. §.§ Other Observables Whilst the inclusion of internal photoevaporation has limited the effectiveness of using the disc radius as a measure between viscous or wind driven discs, it is also a question of how other observables, such as the disc mass, or the mass accretion rate, are also affected. Figure. <ref> shows the evolution between disc mass and disc radius (left panel), disc mass and accretion rate (middle panel), and time and remaining lifetime (right panel). Similar to fig. <ref>, the colours show viscous (blue), hybrid (red), and wind driven (orange) discs, with solid lines showing for a weak external environment, and dashed lines for a strong environment.Looking at the left panel of fig. <ref>, it is again clear that viscous discs are found to be slightly more massive than wind driven discs of similar size. This is similar to what is observed in fig. <ref>, however, the extent of the differences is now down to only ∼few , and so it is unlikely that such a difference could be unambiguously demonstrated observationally, due for example to uncertainty in the initial conditions of the disc. For the discs in the weak external environments, internal photoevaporation dominated the mass loss rates of the disc, especially in the intermediate and outer disc regions. With this occurring for both viscous and wind driven discs, this is what led to the evolution tracks being similar. However, for the stronger external environments, external photoevaporation dominated the mass loss rates of the disc, effectively truncating the discs, and again leading to similar evolution tracks, until the discs became compact where the effectiveness of external photoevaporation was diminished.A similar situation is found when looking at the mass accretion rate as a function of disc mass. Again the differences between viscous and wind driven discs is minimal, whilst there is limited evidence for a “knee” feature as predicted by internal photoevaporation models <cit.>, mainly due to the effects of photoevaporation on the outer disc regions. Additionally, when including internal photoevaporation, the mass accretion rate is also similar in all environments, with a slightly increasing trend towards higher accretion rates in stronger external environments for discs of similar mass. The similarity across environments is due to the length of time taken for the effects of the outer disc to influence the inner disc regions, through the resupply of material that is lost to stellar accretion. For discs in stronger environments, they exhibit larger accretion rates since the outer disc is truncated more efficiently and so the mass of the disc reduces at a faster rate. The inner disc however, is minimally directly affected by photoevaporation and so has similar accretion rates on to the star at similar times early in its lifetime.The biggest difference between fig. <ref> and fig. <ref>, is the right hand panel, where the remaining lifetimes now all exhibit similar pathways. For both weak and strong external environments, the remaining lifetime in a wind driven disc is consistently larger than it's viscous counterpart. Additionally the expectations in <cit.> that the remaining lifetime in viscous discs increases over time is again not seen, highlighting the importance of including the additional processes that remove significant amounts of mass from the disc, i.e. photoevaporation. At the very least it shows the importance of understanding the interplay between MHD disc winds and photoevaporative processes, which is only just starting to be explored <cit.>. The effect of external photoevaporation seen here by the stronger environments (dashed lines) shows smaller depletion time-scales than in weaker environments. This is again due to the outer disc being efficiently truncated and reduced in mass, whilst the inner disc continues to accrete on to the star at similar rates as discs in weaker environments. This leads to smaller depletion time-scales for discs in stronger environments. The main effect of internal photoevaporation here is by dominating the evolution of discs in weaker environments, both viscous and wind driven, and so causing them to have similar evolution tracks.As was shown in fig. <ref>, there now appears to be minimal difference in the evolution of disc radii between viscous and wind driven discs. Figure. <ref> shows the difference in radius over time between viscous and wind driven discs for those evolving in different FUV environments (shown by the colours) and including the effects of internal photoevaporation. Interestingly it seems that for the first Myr, there are now minimal difference across all environments. Even in weak environments over the entire disc lifetime, the maximum difference is now 15 , showing that the discs are of similar size, irrespective of if they are viscous or wind driven, when also taking possible variations in initial conditions into account. Interestingly the largest feature now is at the end of the disc lifetime, where wind driven discs in more intermediate environments (e.g. 10^3G_0) are found to be significantly larger than their viscous counterparts. This, as described above, is due to internal photoevaporation opening a hole in the disc, and coupled to the external wind, quickly disperses the outer disc. With viscous expansion already supplying the external photoevaporative wind, there is less material in the outer disc to disperse than in the wind driven disc. As such, this allows the outer disc to be more quickly removed, truncating the disc at a much faster rate. This equally happens in stronger FUV environments, but here the discs fully truncate down to a small size before a hole is able to open, slightly nullifying the observable effect.§ DISCUSSIONThe above sections described the effects of external and internal photoevaporation on the evolution of viscous and wind driven protoplanetary discs, including the observables that can be measured from them. We now look at other properties of the discs, to determine any other differences between viscous and wind driven discs under different processes. §.§ Disc lifetimesOne key observable is the disc lifetime, typically arising in observations from counting the fraction of stars in a cluster that exhibit signatures of a protoplanetary disc, e.g. through accretion on to the central star <cit.>. Given that the discs have been shown to have some differences in their evolution profiles, we explore here the lifetimes that discs contain when either viscous or wind dominated. Figure <ref> shows the lifetimes of discs as a function of ψ. The strength of the external environment is denoted by the colour, whilst solid lines show the lifetimes for discs without internal photoevaporation, and dashed lines include internal photoevaporation. The effect of internal photoevaporation on the disc lifetimes can be easily seen here, with the maximum lifetime now being roughly equal to ∼4 Myr when it is included.In comparing the left side (viscous dominated) of fig. <ref> to the right side (wind driven) it is clear that when internal photoevaporation is not included, disc lifetimes are shorter for wind driven discs. This is due to the viscous expansion slowing the accretion on to the star, as seen in the evolution of mass and radii in the figures above. On the other hand, when internal photoevaporation is included, this trend flips, with viscous discs now having shorter lifetimes, as the expansion feeds both the internal and external photoevaporative winds. This viscous expansion transfers material from the intermediate regions of the disc, i.e. 10–40, towards the outer regions of the disc, which allows for external photoevaporation mass loss rates to be stronger. The stronger external photoevaporation rates then deplete the outer disc quicker, and with the supply equally diminishing over time, as a result of the expansion and internal photoevaporation removing the material from this region, this led to the viscous discs losing mass more quickly, and thus having shorter lifetimes. With no such outward movement of material in wind driven discs, the mass loss rates due to external photoevaporation would be weaker than their viscous counterparts of similar ages, leaving internal photoevaporation to remove the majority of this material. This then results in the discs being more massive for longer, and thus having slightly longer disc lifetimes. This can be seen in the differences in the lifetimes in fig. <ref>, by comparing the disc radii in fig. <ref>, as well as by comparing similar colour profiles in fig. <ref>. Given that the opposite effects in terms of which mechanism yields longer disc lifetimes are seen here, this again highlights the importance of understanding the interplay between MHD disc winds and internal photoevaporation, and in particular, if and how they can work in tandem. §.§ Environments that show biggest difference when including Internal Photoevaporation Whilst statistics of disc lifetimes could yield indications into whether discs are viscous or wind driven, the evolution of the size of a disc may still be a useful metric. The above sections showed that when including internal photoevaporation, weak FUV environments may not be the best environment to find such trends, since the disc evolution for all discs there is quite similar.Instead, more intermediate environments may be better targets, since it was seen in fig. <ref> that there are significant differences in their evolution later in the disc lifetime. To this end, in fig. <ref> we plot the time of truncation to different radii in the disc as a function of ψ and the external environment. The central star luminosity was again set to L_ X=10^30 erg s^-1. The left panel shows the truncation time to 10, with the middle and right panels showing the times to 30 and 50 respectively.Looking first at the left panel, it is clear to see that wind driven discs taken longer to truncate down to 10than viscous discs. This is all apart from the far upper right of the panel, in the most extreme environment where it is slightly shorter. The horizontal lines that appear in the plots for both viscous and wind driven discs, show the qualitative evolution of the discs changing, where holes are able to form due to internal photoevaporation, and the outer discs are able to be quickly or slowly dispersed. The trend of wind driven discs taking longer to truncate than viscous discs continues when looking at the middle panel of fig. <ref>, those to 30 , but the differences here are to a lesser extent. Additionally for the stronger environments, there appears to be little difference now between the two processes, as external photoevaporation dominates the evolution of the disc in this outer region. Going now to the right panel of fig. <ref>, showing the truncation time to 50 , the effect of external photoevaporation is clear for environments >10^3 G_0 where there are minimal differences as a function of ψ. However for the weaker environments, the viscous discs now take longer to truncate to this level, highlighting the early effect of viscous expansion in setting an equilibrium. The reason this trend does not extend to lower levels, is due to internal photoevaporation later in the disc lifetime opening a hole more quickly in the viscous discs than in the wind driven discs, allowing them to lose their outer disc more quickly, and truncating down at a faster rate.Whilst fig. <ref> showed the truncation times, fig. <ref> shows the difference in truncation times to that observed for a viscous disc, i.e. ψ=10^-4 or the far left of each panel in fig. <ref>. Interestingly the largest differences seem to be in the intermediate and stronger environments. Indeed in truncating down to 10 the larger difference is in environments ∼10^4G_0. Here the wind driven discs remain larger for much longer than their viscous counterparts, since in these regions it is the viscous expansion that continues to feed stronger photoevaporative winds, whereas the disc winds reduce the supply. This results in those wind driven discs staying larger for longer periods of time, sometimes up to 1.5 Myr. This trend is also seen in the middle panel, where the greatest difference is around ∼5000G_0, with again the wind driven discs remaining larger. As was shown in fig. <ref>, this is mainly due to internal photoevaporation opening the hole and the outer disc depleting more quickly in the viscous discs. But nonetheless this feature may be seen in more evolved star forming regions. Finally, looking at the right panel of fig. <ref>, the same feature appears for truncating down to 50 , but it is extremely narrow now around 10^3G_0. Additionally the temporal differences are only 0.3 Myr at most, which when taking into account uncertainties in star forming region ages, this becomes difficult to disentangle from other possibilities, i.e. when the stars actually form.An extra point to take from the truncation time difference to 50 is that for weak environments, the wind driven discs are seen to truncate at a faster rate, but only by again 0.3 Myr. In taking all of the truncation time differences into account, it is clear that it is important to account for the role of environment when attempting to determine whether observed discs evolve through viscosity or MHD winds. It is also important to know the stage of it's lifetime that the disc is in, as this could also aid in analysing discs in stronger environments, where larger differences between viscous and wind driven models seem to exist.§ SUMMARY AND CONCLUSIONSIn this work we have explored the effects that photoevaporation, both internally and externally driven, have on the evolution of viscous and MHD wind driven protoplanetary discs. The main aim of this work was to determine whether there were observational differences between viscous and wind driven discs <cit.> when photoevaporation was also considered. We performed a broad parameter study for external photoevaporation, looking at both weak and strong UV environments, before then including the effects of a weak internal photoevaporative wind. We draw the following main conclusions from this work. 1. Including external photoevaporation halts viscous expansion when the expansion rate equals the external photoevaporative mass loss rate. This occurs even in weak UV environments, differing from what is shown in viscous evolutionary models. When comparing viscous disc models to MHD wind driven disc models, the observable differences, i.e. disc radius, is therefore reduced making it harder to discern whether discs are viscous or wind driven. This highlights the importance of including even weak (∼10 G_0) external photoevaporation when exploring models of protoplanetary disc evolution. 2. Whilst there are observable differences between evolving viscous and wind driven disc models in weak UV environments, these differences become negligible in stronger environments, e.g. >10^3 G_0. This is due to external photoevaporation dominating the mass evolution of the outer disc regions, efficiently truncating the discs down to a compact size. This reduces and ultimately washes outs differences, e.g. in observable disc radii, between viscous and wind driven disc models. 3. Analytical models predicted that there would be observable differences in the relations between mass accretion rate, disc mass and disc radius <cit.>. Our results here show that whilst we find some of these predictions to be qualitatively similar, significance and observability are reduced. For example, viscous discs typically tended to have larger mass accretion rates for a similar disc mass as wind driven discs. However differences are only by a factor few that could equally be found when altering the initial properties of the disc, including how massive and how compact it is. Additionally the trends in the remaining lifetime of the discs are found to to be similar, irrespective of the mechanism of angular momentum transport in the disc. All of these factors taken together show that the possibility of discerning whether discs are viscous or wind driven is much harder and more degenerate than previously considered. 4. The prospects of comparing viscous discs to wind driven discs are further diminished when internal photoevaporation processes are included. Our results show that even a weak internal photoevaporative wind is able to dominate the evolution of protoplanetary discs in low UV environments. It does this by removing the mass in the intermediate regions of the disc (10<r<100), before opening a hole in the disc, and quickly dispersing the outer disc. This effect occurs before the discs have sufficiently evolved through either viscous or wind driven evolution. With internal photoevaporation dominating, this washes out many previous differences found between viscous and wind driven discs in weak UV environments. 5. Whilst discs in weak UV environments provide less insight into the problem of viscous versus MHD wind driven evolution when including internal photoevaporation, more intermediate environments may provide times where there are significant differences. This is mainly where internal photoevaporation opens a hole in the disc and the outer disc quickly disperses. Due to viscous expansion feeding external photoevaporative winds, this occurs earlier for viscous discs, and so disc radii at these late times in a discs lifetime can be used to discern between viscous or wind driven discs, assuming the other parameters, e.g. strength of the internal field are adequately constrained.Overall, this work shows the importance of including photoevaporative effects in models of disc evolution. Even when the external environment or the stellar X-ray luminosity are considered weak, there is sufficient impact to affect the evolution of either viscous or wind driven discs, either through halting viscous expansion, or quickly dispersing the disc once a hole opens.Aspects not considered in this work, but which would affect the comparison of star forming regions and disc evolution is the evolution of the surrounding star forming region itself. Not only do stars and their subsequent discs form at different times, the external radiation field experienced by an evolving disc changes over time. Including the star formation rate within comparisons of disc lifetimes reveals a degeneracy in determining the evolution pathway of discs in different environments <cit.>, whilst protoplanetary discs can often be shielded for a sufficient portion of their early lifetime before emerging in to stronger FUV environments <cit.>. Both of these effects will also be relevant to some degree when using disc properties to determine whether discs are viscously or wind driven. Given that our work shows that the qualitative evolution of protoplanetary discs are similar for viscous and wind driven discs, and the main differences is when such discs go through certain transitional stages in their lifetimes, the added degeneracy of the effects of the star forming region itself further complicates matters. Ultimately, understanding the fundamental properties of a protoplanetary disc, it's central star, and it's recent history will be important to eliminate some of these degeneracies, in order to address the question of whether they are viscous or wind driven. Additionally, whilst this work assumed that all disc evolution processes work in tandem, it is still an open question of how do they interact with each other. Signatures of MHD <cit.>, internal <cit.> and external <cit.> photoevaporative winds have been associated with nearby protoplanetary discs. With evidence for all of these processes occurring, an overlap in space and/or time must exist, and so future work is required to further understand how these processes interact with each other. Indeed, recent work has shown that MHD winds and internally photoevaporative winds do work in tandem, with MHD winds dominating mass loss rates when the magnetic field is high or X-ray luminosity of the star is weak, and vice versa for internal photoevaporative winds <cit.>. With such an improved understanding of the interplay between all of these processes, the validity of disc evolution models incorporating these processes will be improved and we can begin to fully understand how protoplanetary discs evolve. Only then, by including more complex models, and in understanding the properties of observed protoplanetary discs better, will it be possible to understand how protoplanetary discs evolve, and then subsequently how might planetary systems form within them. § DATA AVAILABILITYThe data underlying this article will be shared on reasonable request to the corresponding author.§ ACKNOWLEDGEMENTSThe authors thank the anonymous referee for providing useful and interesting comments that improved the paper. GALC acknowledges support from STFC through grants ST/P000592/1 and ST/T000341/1. TJH acknowledges funding by a Royal Society Dorothy Hodgkin Fellowship and UKRI ERC Consolidator Grant guarantee funding (EP/Y024710/1). This research utilised Queen Mary's Apocrita HPC facility, supported by QMUL Research-IT (http://doi.org/10.5281/zenodo.438045). This work was performed using the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure.mnras | http://arxiv.org/abs/2311.15824v2 | {
"authors": [
"Gavin A. L. Coleman",
"Joseph K. Mroueh",
"Thomas J. Haworth"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20231127134923",
"title": "Photoevaporation obfuscates the distinction between wind and viscous angular momentum transport in protoplanetary discs"
} |
Machine-to-Machine Transfer Function in Deep Learning-Based Quantitative Ultrasound Ufuk Soylu, Student Member, IEEE and Michael L. Oelze, Senior Member, IEEE This research received financial support from grants provided by the National Institutes of Health (NIH) (R01CA251939 and R01CA273700) Ufuk Soylu and Michael Oelze are with the Beckman Institute, and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. Moreover, Michael Oelze is with the Carle Illinois College of Medicine, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. (e-mail: [email protected]; [email protected]).January 14, 2024 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We introduce a new powerful scan statistic and an associated test for detecting the presence and pinpointing the location of a change point within the distribution of a data sequence where the data elements take values in a general separable metric space (Ω, d). These change points mark abrupt shifts in the distribution of the data sequence. Our method hinges on distance profiles, where the distance profile of an element ω∈Ω is the distribution of distances from ω as dictated by the data. Our approach is fully non-parametric and universally applicable to diverse data types, including distributional and network data, as long as distances between the data objects are available. From a practicable point of view, it is nearly tuning parameter-free, except for the specification of cut-off intervals near the endpoints where change points are assumed not to occur. Our theoretical results include a precise characterization of the asymptotic distribution of the test statistic under the null hypothesis of no change points and rigorous guarantees on the consistency of the test in the presence of change points under contiguous alternatives, as well as for the consistency of the estimated change point location. Through comprehensive simulation studies encompassing multivariate data, bivariate distributional data and sequences of graph Laplacians, we demonstrate the effectiveness of our approach in both change point detection power and estimating the location of the change point. We apply our method to real datasets, including U.S. electricity generation compositions and Bluetooth proximity networks, underscoring its practical relevance.§ INTRODUCTIONWith origins dating back to the 1950s <cit.>, change point analysis has remained a thriving research area in statistics fueled by its burgeoning relevance in diverse domains such as biology and medicine <cit.>, economics and finance <cit.>, neurociences <cit.>, social sciences <cit.>, climate and environmental studies <cit.>, and more recently in the context of COVID-19 <cit.>, to name but a few; see <cit.> for recent reviews.The primary objective of change point detection is to identify and precisely locate any abrupt alteration in the data generating mechanism within an observed data sequence. Let the data sequence, indexed by time or another meaningful order, be represented as Y_1,…,Y_n. We consider the offline scenario, where the data sequence is of fixed length. A change point, denoted as n_τ, is characterized by the transition from the distribution P_1 governing Y_1,…,Y_n_τ to another distribution P_2 governing Y_n_τ+1,…,Y_n where P_1 ≠ P_2. In cases where the observations Y_i reside in the Euclidean space ℝ^p, the problem's intricacies are influenced by the choice of the dimensionality p. While the univariate scenario has been thoroughly studied <cit.>, additional challenges encountered in the multivariate case have triggered developments in both parametric <cit.> as well as non-parametric frameworks <cit.>. In the high-dimensional setting, the problem becomes significantly more intricate and elusive due to the curse of dimensionality.Specialized investigations targeting high-dimensional scenarios have emerged, including works by <cit.>.In modern data science, it is becoming increasingly common to encounter data that do not lie in a Euclidean space. Such data elements, often referred to as “random objects", extend the traditional concept of random vectors into the realm of general metric spaces. Common examples include brain networks <cit.>, gene regulation networks <cit.>, linguistic object data <cit.>, distributional data <cit.>, compositional data <cit.>, phylogenetic trees datasets <cit.> and many more. The bottleneck in working with such data lies in the absence of standard vector space operations, leaving us primarily with pairwise distances between these objects as our basis for analysis. Methods to tackle change point analysis in these settings have evolved simultaneously. In addition to approaches designed for special cases like network data <cit.>, distributional sequences <cit.>, compositional data <cit.> etc, which are not applicable more generally, there has been a surge in fully non-parametric approaches that can be placed in one of the three broad categories: distance-based <cit.>, kernel-based <cit.> and graph-based <cit.>. Nevertheless, each category of methods has its own set of limitations. In the case of kernel-based methods, critical decisions such as choosing the appropriate kernel and setting parameters like bandwidth and penalty constants can significantly impact their effectiveness but are often challenging to determine in practice.On the other hand, the performance of graph-based methods is highly contingent on choosing from various graph construction methods, a choice that is often difficult to make. Recently, <cit.> proposed a tuning free approach based on Fréchet means and variances in general metric spaces, however this test is not powerful against changes beyond Fréchet means and variances of the data.In this paper, we propose a non-parametric, tuning parameter free (except for a cut-off interval at the end-points where change points are assumed not to occur) offline change point detection method for random objects based on distance profiles <cit.>, where the distance profile of a data element is the distribution of distances from that element as governed by the data. This new off-the-shelf method comes with rigorous type I error control, even when using permutation cutoffs, together with guaranteed consistency under contiguous alternatives. For fixed alternatives, we establish an optimal rate of convergence (up to log terms) for the estimated change point and also devise analogous rates of convergence under contiguous alternatives. We demonstrate the broad applicability and exceptional finite-sample performance of our method through extensive simulations covering various types of multivariate data, bivariate distributional data, and network data across a variety of scenarios. We illustrate our method on two real-world datasets: Bluetooth proximity networks in the MIT reality mining study and U.S. electricity generation compositional data. The organization of the paper is as follows. In Section <ref>, we delve into the problem's setup, introducing the distance profiles and presenting our scan statistic. Section <ref> is dedicated to laying out the theoretical foundations of our proposed test, including a precise characterization of the asymptotic distribution of the scan statistic under the null hypothesis of no change points, analysis of the power of the test under contiguous alternatives and establishing rates of convergence for the estimated change point. Moving on to Section <ref>, we introduce the various simulation settings, offering a comprehensive exploration of different types of random objects and change points scenarios. The performance of our test is illustrated in real-world applications, namely, the MIT Reality Mining networks and the U.S. electricity generation compositional data, in Section <ref>. Finally, in Section <ref>, we discuss the capacities of this new method, limitations and avenues for future extensions. In particular, we explore in depth the adaptation of our method utilizing seeded binary segmentation <cit.> to scenarios involving multiple change points, which we illustrate using the set up of stochastic block models with multiple change points. § METHODOLOGY§.§ Distance profiles of random objectsDistance profile, introduced in <cit.>, is a simple yet powerful device for analyzing random objects in metric spaces. Let (Ω,d) be a separable metric space. Consider a probability space (S,𝒮,ℙ), where 𝒮 is the Borel sigma algebra on a domain S and ℙ is a probability measure. A random object X is a measurable function, X: S →Ω and P is a Borel probability measure on Ω that is induced by X, i.e. P(A) = ℙ({s ∈ S: X(s)∈ A})=:ℙ(X ∈ A)=ℙ(X^-1(A))=:ℙ∘ X^-1(A), for any Borel measurable set A ⊆Ω. For any point ω∈Ω its distance profile is the cumulative distribution function (cdf) of the distance between ω and the random object X that is distributed according to P. Formally we define the distance profile at ω asF_ω(t) = ℙ(d(ω,X) ≤ t), t ∈ℝ.We suppress the dependence of F_ω on P to keep the notation simple. Intuitively, if an element ω is more centrally located, i.e. closer to most other elements, it will have a distance profile with more mass near 0 unlike points which are distantly located from the data. With a sequence of independent observations X_1,…,X_n from P, we estimate the distance profile at ω as F̂_ω(t)=1/n∑_j=1^n [d(ω,X_j) ≤ t], t ∈ℝ. The collection of the distance profiles {F_ω:ω∈Ω} comprises the one-dimensional marginals of the stochastic process {d(ω,X)}_ω∈Ω and serve as distinctive descriptors of the underlying Borel probability measure P whenever P can be characterized uniquely by open balls. Under special conditions on (Ω,d), for example if d^r is of strong negative type for some r>0 <cit.>, this unique characterization holds for all Borel probability measures; see Proposition 1 in <cit.>. Motivated by the new two-sample inference framework introduced in <cit.> our objective is to leverage these elementary distance profiles for detecting change points in the intricate distribution of a random object sequence. §.§ Change point detection problem Let Y_1,Y_2,…,Y_n be a sequence of random objects taking values in a separable metric space (Ω,d) with a finite covering number. Given two different Borel probability measures P_1 and P_2 on Ω, we will test the null hypothesis,H_0: Y_1,Y_2,…,Y_n ∼ P_1against the single change point alternativeH_1: ∃ τ ∈ (0,1) {[Y_1,Y_2,…,Y_[nτ]∼ P_1; Y_[nτ]+1,Y_[nτ]+2,…,Y_[n]∼ P_2;].where τ denotes the change point. Our aim is to test the above hypothesis and accurately identify τ when it exists. In keeping with traditional change point methods, we will employ a scan statistic that involves dividing the data sequence into two segments, one before and one after the potential change points. In this process, the test statistic seeks to maximize the dissimilarities between these segments enabling subsequent inference and estimation of the change points if any. We quantify the dissimilarity between the data segments using the recently proposed two sample test statistic based on distance profiles in <cit.> which is tuning parameter free and targets a divergence between the underlying population distributions in large samples (refer to the quantity D^w_XY in <cit.>) whenever the distributions are such that they can be uniquely identified using the distance profiles corresponding to all the elements in Ω.To ensure the validity of large-sample analysis, it is important that both segments contain a minimum number of observations so that we can accurately capture the dissimilarity between them.Consequently, we make the assumption that the change point τ lies in a compact interval ℐ_c=[c,1-c] ⊂ [0,1], for some c>0.§.§ Scan statistic and type I error controlWhile scanning the data sequence segmented at u ∈_c, let [it] be the estimated distance profile of the observation Y_i with respect to the data segment Y_1,…,Y_[nu] given by[it] = 1/[nu]∑_i=1^[nu][d(Y_i,Y_j) ≤ t],t ∈.and [it], defined in a similar way with respect to the data segment Y_[nu]+1,…,Y_n, is given by[it] = 1/(n-[nu])∑_i=[nu]+1^n[d(Y_i,Y_j) ≤ t],t ∈. To capture the discrepancy between the data segments Y_1,…,Y_[nu] and Y_[nu]+1,…,Y_n we use the statistic given byT̂_n(u) =[nu](n-[nu])/n{1/n[in] ∫_0^ ([it]-[it])^2 dt }where =(Ω) is the diameter of Ω. The motivation to investigate this scan statistic is that if the two segments Y_1,…,Y_[nu] and Y_[nu]+1,…,Y_n have different distributions, then the centrality of an observation Y_i, as encoded in the distance profiles, will be different across the two segments, and as a result T̂_n(u) tends to be large when u is close to τ when the change point τ exists. We demonstrate this using a toy example in Figure <ref>. Hence to test the null hypothesis H_0 (<ref>), we will use the test statisticT̂_n = sup_u ∈ℐ_cT̂_n(u) = max_[nc]≤ k ≤ n-[nc]T̂_n(k/n).In order to construct an asymptotic level α test we derive the distribution of the test statistic (<ref>) under H_0 (<ref>) in Theorem <ref> which needs the following assumptions. *Let F_ω^(1)(t)=(d(ω,Y)≤ t), where Y ∼ P_1, and F_ω^(2)(t)=(d(x,Y')≤ t), where Y' ∼ P_2. Assume that F_ω^(1)(t) and F_ω^(2)(t) are absolutely continuous for each ω∈Ω, with densities given by f_ω^(1)(t) and f_ω^(2)(t) respectively. Assume that there exists L_1, L_2 > 0 such that sup_ω∈Ωsup_t ∈ℝ |f_ω^(1)(t)| ≤ L_1 and sup_ω∈Ωsup_t ∈ℝ |f_ω^(2)(t)| ≤ L_2. Moreover assume that inf_t ∈supp(f^(1)_ω) f^(1)_ω(t) > 0 and inf_t ∈supp(f^(2)_ω) f^(2)_ω(t) > 0 for each ω∈Ω. *Let N(ϵ, Ω, d) be the covering number of the space Ω with balls of radius ϵ and log N(ϵ, Ω, d) the corresponding metric entropy, which satisfies ϵlog N(ϵ, Ω, d) → 0asϵ→ 0.Assumption <ref> imposes regularity conditions on the distance profiles under the distributions P_1 and P_2. Assumption <ref> constrains the complexity of the metric space (Ω,d) and is applicable to a wide range of spaces including any space Ω which can be represented as a subset of elements in a finite dimensional Euclidean space, for example networks <cit.>, simplex valued objects in a fixed dimension <cit.> and the space of phylogenetic trees with the same number of tips <cit.>. Assumption <ref> holds for any Ω which is a VC-class of sets or a VC-class of functions <cit.>, forp-dimensional smooth function classes C_1^α(𝒳) <cit.> on bounded convex sets 𝒳 in ℝ^p equipped with the ·_∞-norm <cit.> and ·_r,Q-normfor any probability measure Q on ℝ^p <cit.> if α≥ p+1 and for the case when Ω is the space of one-dimensional distributions on some compact interval I ⊂ℝ that are absolutely continuous with respect to the Lebesgue measure on I with smooth uniformly bounded densities and d=d_W with d_W being the 2-Wasserstein metric <cit.>. In fact Assumption <ref> is satisfied when Ω is the space of p-dimensional distributions on a compact convex set I ⊂ℝ^p, represented using their distribution functions endowed with the L_r metric with respect to the Lebesgue measure on I if Ω⊂ C_1^α(I) for α≥ p+1. Next we present Theorem (<ref>) which establishes the null distribution of the test statistic (<ref>) as n →∞ as the law of the random variable 𝒯 as introduced in the same theorem. Under H_0 and assumptions (A1) and (A2), as n →∞, T̂_n converges in distribution to the law of a random variable 𝒯=sup_u ∈_c∑_j=1^∞𝔼_Y{λ_j^Y}𝒢_j^2(u), where Y ∼ P_1, λ^x_1 ≥λ^x_1 ≥… correspond to the eigenvalues of the covariance function given by C_x(t_1,t_2) = Cov( [d(x,Y)≤ t_1],[d(x,Y)≤ t_2]) and 𝒢_1, 𝒢_2, … are independent zero mean Gaussian processes with covariance given byc(u_1,u_2)=√((1-u_1)(1-u_2)/u_1u_2)min(u_1,u_2)+√(u_1u_2/(1-u_1)(1-u_2))min(1-u_1,1-u_2). Here we discuss the major bottleneck that we needed to overcome in proving Theorem <ref> and relegate the technical details to the Supplement. Since the goal is to have a sample-splitting free test,we use the same observations Y_1, …, Y_n to estimate the distance profiles and thereafter to estimate the scan statistic that makes each summand in (<ref>) significantly dependent on each other. To obtain the null distribution in Theorem <ref> we decompose the test statistic (<ref>) into several parts, some of which contribute to the asymptotic null distribution while we that the others are asymptotically negligible using maximal inequalities from U-process theory.For a level α test, we propose to reject H_0 (<ref>) if T̂_n > q_α where q_α is the (1-α)-quantile of 𝒯. Equivalently one rejects H_0 when p≤α where p=_H_0(𝒯≥T̂_n) is the asymptotic p value of the test. Unfortunately, the law of 𝒯 depends on the underlying data distribution and the rejection region, either using the critical value q_α or the p value p must be approximated in practice. Staying in line with adopted conventions we adopt a random permutation scheme to obtain this approximation that is described next and later design a framework in Section <ref> to study the power of the test using this approximation scheme in large samples.Let Π denote the collection of n! permutations of {1,2, …, n}. Let π_0=(π_0(1), …, π_0(n)) be the identity permutation such that π_0(j)=j for j=1,…,n and π_1, π_2, …, π_K be K i.i.d samples from the uniform distribution over Π where each π_k=(π_k(1), …, π_k(n)) is a permutation of {1,…,n}. For each k=0,1,2,…, let T̂^π_k_n be the test statistic evaluated on a reordering of the data given by Y_π_k(1), …, Y_π_k(n). Then the permutation p value is given by p̂_K = 1/K+1∑_k=0^K [T̂^π_k_n ≥T̂_n] and the test is rejected at level α when p̂_K ≤α. It is easy to see that under H_0 (<ref>), p̂_K as an approximation of p controls the type I error of the test at level α with high probability for sufficiently large K (see <cit.>).§.§ Power analysis under contiguous alternatives We will study the large sample power of the test, first assuming that the asymptotic critical value q_α is available, and thereafter using the practicable permutation scheme for the test. A few definitions are in order before we can state our results. For t ∈ℝ and a random object Y, where it is possible that either Y ∼ P_1 or Y ∼ P_2, let [Y](t)= _Y'( d(Y, Y') ≤ t) with Y' ∼ P_1 and [Y](t)= _Y”( d(Y, Y”) ≤ t) with Y”∼ P_2 and both Y' and Y” are independent of Y. Define the quantity Δ=Δ(P_1, P_2) asΔ = _Y ∼ P_1( ∫_0^ℳ{[Y](t) - [Y](t)}^2 dt )+_Y ∼ P_2( ∫_0^ℳ{[Y](t) - [Y](t)}^2 dt )where =(Ω) is the diameter of Ω. Immediately one sees that under H_0 (<ref>), Δ = 0. In fact Δ corresponds to the quantity D^w_XY introduced in <cit.> and undermild conditions (Ω,d), P_1 and P_2, Δ=0 if and only if P_1 = P_2. Let 𝒫_(Ω,d) denote the class of all Borel probability measures on (Ω,d) which are uniquely determined by open balls, that is, for Q_1, Q_2 ∈𝒫_(Ω,d), Q_1 = Q_2 if and only if F^Q_1_ω(t) = F^Q_2_ω(t) for all ω∈Ω and t ∈ℝ. In fact 𝒫_(Ω,d) contains all Borel probability measures on (Ω,d) under special conditions on (Ω,d) such as, for example, when (Ω,d^k) is of strong negative type <cit.> for some k > 0 (see Proposition 1 in <cit.>). Then Δ=0 implies that F_ω^P_1(u)=F_ω^P_2(u) for almost any t ∈ℝ and for any ω in the union of the supports of P_1 and P_2. Hence if Ω is contained in the union of the supports of P_1 and P_2, then Δ=0 implies that P_1=P_2 whenever P_1,P_2 ∈𝒫_(Ω,d) in which case Δ serves as a divergence measure between P_1 and P_2 and can be used to measure the discrepancy between H_1 (<ref>) and H_0 (<ref>).To investigate the power of the test we consider the challenging case, a sequence of alternatives H_1,n that shrinks to H_0 where H_1,n={ (P_1, P_2): Δ = a_n }with a_n → 0 as n →∞. In this framework the asymptotic power of a level α test is quantified using β^α_n=_H_1,n( p≤α)where p is the asymptotic p-value and depends on the null distribution of the test described by the law of 𝒯 in Theorem <ref>. Since the law of 𝒯 is unknown, we obtain a tractable estimator of p given by p̂_K using the permutation scheme, and the practicable power is quantified as β̃^α_n=_H_1,n( p̂_K ≤α).Theorem <ref> gives the asymptotic consistency for any level α test, by showing that the power of the test in both asymptotic and practicable regimes converge to one as n →∞ in the hard to detect scenario of contiguous alternatives provided that n a_n →∞, i.e., a_n does not decay too fast as n →∞.Under assumptions (A1) and (A2), for any α∈ (0,1) and as n →∞,β^α_n → 1 as n →∞.Moreover if n a_n →∞ and K ≥1/α, where K is the total number of permutations used to estimate p̂_K in (<ref>) then β̃^α_n → 1 as n →∞. §.§ Change point estimationIn this section we describe the estimation of τ when it exists. As a straightforward consequence of our test statistic (<ref>) we estimate τ asτ̂ = _u ∈ℐ_cT̂_n(u).Next we discuss the asymptotic behavior of τ̂ around τ. For this we consider two scenarios, first, where the alternative is fixed which is the case that is studied typically in nonparametric change point analysis, and then the case where the alternative is such that (P_1, P_2) ∈ H_1,n with H_1,n as defined in (<ref>) such that Δ(P_1,P_2)=a_n with a_n → 0 as n →∞. The second scenario offers a difficult setup for detecting τ accurately. In Theorem <ref> we show the asymptotic near-optimal rate of convergence for the change point in the first scenario of fixed alternatives when P_1 ≠ P_2 <cit.>. In the second situation, the change point estimator τ̂ is a consistent estimator of τ only if na_n →∞ as n →∞, which is also the requirement for the consistency of the test as per Theorem <ref>.For any fixed alternative when P_1 ≠ P_2, under assumptions (A1) and (A2), as n →∞,|τ̂ - τ| = O_P( log(n)/n).For the contiguous alternatives H_1,n, there exists L > 0 such that _H_1,n( na_n|τ̂ - τ|≥ L ) → 0as n →∞. § SIMULATIONSWe assess the finite sample performance of the our approach across various frameworks involving three different random object spaces. First we carry out extensive simulations capturing diverse change point scenarios in sequences of Euclidean random vectors, then we study change points in bivariate distributional data sequences where the bivariate distributions are equipped with the L^2 metric between corresponding the cdfs, and finally move on to change points in random network sequences where the networks are represented as graph Laplacians with the Frobenius metric between them and are generated from a preferential attachment model <cit.>. For each random object we explore a range of alternatives beyond location shifts and scale shifts, such as, sudden changes in the tails of the population distribution, the population distribution abruptly switching to a mixture distribution with two components and abrupt changes in network node attachment mechanisms. We generate random object sequences of length n =300 and set the level α of the test to be α=0.05. In each scenario, we first calibrate the type I error of the new test under H_0 and then evaluate the empirical power of the test for a succession of alternatives that capture increasing discrepancies between the distributions of the two data segments before and after the change point with τ=1/3. We employ our scan statistic on the interval ℐ_c=[0.1,0.9]. We conduct 500 Monte Carlo runs and set the empirical power to be the proportion of the rejections at α=0.05 out of the 500 Monte Carlo runs. The critical value of is approximated using the permutation scheme described in Section <ref> with 1000 permutations within each Monte Carlo run. To demonstrate our findings we present empirical power plots where we expect that the empirical power will be maintained at the level α=0.05 under H_0 and will increase with increasing disparities between the the two data segments. Along with the power trends we also investigate the accuracy of the estimated change point locations using their mean absolute deviation (MAE), calculated as =1/500∑_i=1^500|τ̂_i-τ|,where τ̂_i is the estimated change point location in the i-th Monte Carlo run. We expect that a successful method will have better accuracy in detecting change points, and therefore, have lower MAE with higher distributional differences between the data segments.We compare the power and the MAE of our test statistic, which we will refer to as “dist-CP" here on, with the energy-based change point detection test (“energy-CP”)<cit.>, graph-based change point detection test (“graph-CP”)<cit.>, and kernel-based change point detection test (“kernel-CP”) <cit.>. We adopt the same configuration whenever needed for each of these tests, for example, we set the number of permutations to be 1000 to obtain the p-value approximations. For the graph-based test, we apply the generalized edge-count scan statistic <cit.> and choose 5-MST (minimum spanning tree) to construct the initial similarity graph as suggested by <cit.>. For the kernel-CP, we use the Gaussian kernel and select the bandwidth to be the median of the input pairwise distances. In terms of the implementation, we use the R packages “gSeg”<cit.> and“ecp”<cit.>for graph-CP and energy-CP separately, and Python package “Chapydette”<cit.> for kernel-CP. §.§ Multivariate data In this setting the data elements are random vectors endowed with l^2 metric and are generated from thep-dimensional Gaussian distribution N(μ,Σ) with mean μ and covariance matrix given by Σ. We explore four different types of changes to the population distributions, which we summarize in Table <ref>. First we consider location and scale changes , in dimensions p = 30, 90 and 180 where to study location change,we set μ=0_p=(0,0,…,0)^T for Y_i,i=1,…,100 and μ=Δ_1 1_p=Δ_1(1,1,…,1)^T for Y_i,i=101,…,300, where we let Δ_1 range from 0 to 1. The covariance matrix is fixed and constructed as Σ=UΛ U^T, where Λ is a diagonal matrix with k-th diagonal entry being (kπ/p)+1.5 for k=1,…,p, and U is an orthogonal matrix with the first columns being p^-1/2(1,1,…,1)^T, such that the mean change loads along the first eigenvector of the data. To investigate scale change, we fix the μ=0_p and set Σ=0.8 𝐈_p for Y_i,i=1,…,100 and Σ=(0.8-Δ_2)𝐈_p for Y_i,i=101,…,300, where we let Δ_2 range from 0 to 0.4. We present the results of location and scale shifts in Figure <ref>, where Figure <ref> illustrates the empirical power performances of the tests and Figure <ref> shows the MAE of estimated change points. In Figure <ref>, we present the corresponding results for scale change. In Figure <ref>, we see that all the methods maintain type I error control.Energy-CP outperforms the other methods in terms of power and MAE but dist-CP has competitive power performance across all different settings in the mean change scenario. For scale changes, dist-CP has the best performance in terms of both empirical power and MAE across all settings as illustrated in Figure <ref>.Next we study sudden splitting of a homogeneous population into a mixture of two components with different means. To study this we take Y_1,…,Y_100 to be generated from the standard p-dimensional Gaussian distribution N(0_p,𝐈_p) and let Y_101,…,Y_300 be generated from a mixture of two Gaussian distributions with the overall population mean same as Y_1,…,Y_100. To be more specific, Y_101,…,Y_300 are constructed with independent samples of AZ_1+(1-A)Z_2, where A ∼, Z_1 ∼ N(-μ,𝐈_p), Z_2 ∼ N(μ,𝐈_p),where μ = (Δ_3 1_0.1p,0_0.9p)^T, and A, Z_1, and Z_2 are independent. Here, Δ_3 ∈ [0,1]. In Figure <ref> we illustrate that dist-CP outperforms all other approaches both in terms of empirical power and MAE in this complex change point scenario. Finally we experiment with changes to the tail of the population distribution. We generate Y_i ∼ N(0_p,𝐈_p) with p ∈{5,15,60} for i=1,…,100 and Y_i as a p-dimensional random vector whose components are independent and identically distributed as the t-distribution with v degrees of freedom for i=101,…,300We let v range in {2,…,22} to reflect the change from Gaussian tails to successively heavier tails. The results as shown in Figure <ref> demonstrate that dist-CP has the best power and MAE performance across all settings.§.§ Bivariate distributional data Here the random objects are random bivariate probability distributions equipped with the metric between corresponding cumulative distribution function representations defined as d(x,y)=∫_ℝ∫_ℝ|F_x(u,v)-F_y(u,v)| du dv, where F_x(u,v) is the bivariate cdf of x. Each observation Y_i for i=1,…,300 is itself a bivariate distribution with a cdf representation where we explore two types of changes: changes in the process generating the means of Y_i and changes in the process generating the variances in the covariance structure of Y_i. We provide a summary of the changes in Table <ref>. Specifically, for the first scenario, Y_i = N(Z_i,0.25𝐈_2), where Z_i ∼ N(0_2,0.25𝐈_2) for i=1,…,100, and Z_i ∼ N((δ_1,0)^T,0.25𝐈_2) for i=101,…,300 where δ_1 ∈ [0,1]. For changes in the scales of the random distributions we generate Y_i = N(Z_i,0.25𝐈_2), where Z_i ∼ N(0_2,0.4^2𝐈_2) for i=1,…,100, and Z_i ∼ N(0_2,((0.4+δ_2)^2,0.4^2)) for i=101,…,300 and let δ_2 ∈ [0,4]. We illustrate our findings in <ref> and <ref>. In Figure <ref>, dist-CP, energy-CP, and kernel-CP have similar performance and are better than graph-CP in terms of both power and MAE. In the second case, dist-CP dominates the performance across all settings. §.§ Network dataLastly, we consider random object sequences where the data elements are random networks endowed with the Frobenius metric between the corresponding Laplacian matrix. Each Y_i is a network with 200 nodes generated from the preferential attachment model <cit.> where for a node with degree k, its attachment function is proportional to k^γ. Y_i is generated with γ=0 for i=1,…,100, and Y_i is generated with γ from 0 to 0.5 for i=101,…,300.We summarize the settings in Table <ref>, and present the simulation results in Figure <ref>. dist-CP outperforms the other methods in both power and change point location estimation MAE.The graph Laplacian for a network is defined L=D-A where D is the degree matrix (diagonal matrix spanned by node degrees) and A is the adjacency matrix. § DATA APPLICATIONS §.§ U.S. electricity generation dataset We analyze the monthly U.S. electricity generation compositions obtained from <https://www.eia.gov/electricity/data/browser/>. We preprocess the data elements into a compositional form so that each entry of the compositional vector represents the percentage of net generation contribution from a specific source. During the preprocessing, we merge some similar categories of resources together and end up with 7 categories: Coal; Petroleum (petroleum liquids and petroleum coke); Gas (natural and other gases); Nuclear; Conventional hydroelectric; Renewables (wind, geothermal, biomass (total) and other); Solar (small-scale solar photovoltaic and all utility-scale solar). We obtain a sample of n=264 observations starting from Jan 2001 to Dec 2022. Each Y_i takes values in a 6-simplex Δ^6={x∈ℝ^7:x^T1_7=1}, where 1_7=(1,1,…,1)^T. The metric we apply between each object is d(x,y)=(√(x)^T√(y)),x,y∈ℝ^7 ,where √(x) is the component-wise square root, i.e., √(x)=(√(x_1),√(x_2),…,√(x_7)). We apply dist-CP to the preprocessed data and present the plot of the scan statistic in Figure <ref>. The scan statistic peaks in month of February 2015. In Figure <ref>, we present the statistically significant estimated change point location with a vertical red dash line. At the change point, we can observe that the percentage of contributions from solar and renewable sources is increasing more rapidly while petroleum contributions drop rapidly.U.S. reached new milestones in renewables electricity generation in the year 2015 (see <https://obamawhitehouse.archives.gov/blog/2016/01/13/renewable-electricity-progress-accelerated-2015>)which explains the detected change point.§.§ MIT reality mining dataset The MIT Media Laboratory conducted the reality mining experiment from 2004 to 2005 on students and faculty at MIT <cit.> in order to explore human interactions based on Bluetooth and other phone applications' activities. We study the participants' Bluetooth proximity networks, where nodes represent participants and the edges represent whether the relevant participants had at least one physical interaction within the time interval.We use the reality mining 1392 built-in dataset in the R package “GreedySBTM” <cit.> (see <https://github.com/cran/GreedySBTM/blob/master/data/reality_mining_1392.RData>). The time frames corresponding to intervals of 4 hours, starting from September 14 2004 to May 5 2005. At each time point the network has =96 nodes. We further compressed the data by merging the time windows with a day to obtain 232 daily networks. We work with the pairwise Frobenius metric between the graph Laplacian representation of the networks. Using dist-CP the estimated change point is 2004-12-15, which is during the finals week and close to the start of the winter break. We present the scan statistics for the whole sequence in Figure <ref>.§ MULTIPLE CHANGE POINTSIn this section, we investigate the extension of dist-CP to the task of detecting multiple change points.We combine our test statistic with the recently proposed seeded binary segmentation algorithm <cit.>, an approach that shares similar ideas with the wild binary segmentation <cit.>. Seeded binary segmentation controls the computational cost in a near-linear time by creating a collection of seeded intervals ℐ_γ which removesunnecessarily long intervals in wild binary segmentation that might contain multiple change points. As introduced in <cit.>, for a sequence of length n, the collection of intervals ℐ_γ is given byℐ_γ=⋃_k=1^log_1/γ(n)ℐ_kwhere γ∈ [1/2,1) is a decay parameter, ℐ_k=⋃_i=1^n_k{((i-1)s_k),(i-1)s_k+l_k} and n_k=2(1/γ)^k-1-1. Each ℐ_k is also a collection of intervals of length l_k that are evenly shifted by s_k, where l_k=nγ^k-1, s_k=(T-l_k)/(n_k-1).In algorithm <ref>, we describe the detailed implementation of the Mcpd_DP, the multiple change point detection algorithm based on the combination of distance profiles and seeded binary segmentation. Once we obtain ℐ_γ, we conduct “dist-CP” on each of the inner intervals thus deriving a set of potential change point locations. We compare the test statistic in each inner interval of ℐ_γ with a threshold which is set at 90%-quantile of the permutation null distribution of the test statistic derived on the entire sequence in the current implementation. The selection rule is to keep intervals in which the test statistic is greater than the above threshold and remove all the other intervals sequentially until the maximum test statistic in the entire collection of remaining intervals does not exceed the threshold. In the experiments, we set γ=1/2^1/2 as suggested by <cit.> and set minLen to be 10. §.§ Simulations on network sequences generated using the Stochastic Block Model To demonstrate the practical efficacy of Mcpd_DP in Algorithm <ref>, we conduct an experiment on sequences of networks generated according to the stochastic block model (SBM). SBM generates networks with adjacency matrices A=[A_ij]∈ℝ^n × n with K communities, whereA_ij∼(P_ij) with P=[P_ij]=Π B Π^T-diag(Π B Π^T). Here Π∈ℝ^n × K encodes the community membership of each node, that is, Π_ik=1 if node i belongs to community k and Π_ik=0 otherwise. Here B ∈ℝ^K × K is the connection probability matrix, where B_ij indicates the connection probability between i-th and j-th community. We let π∈ℝ^K to be such that π_i indicates the number of nodes in i-th community. To incorporate different kinds of changes, we focus on the diverse characteristics of the SBM such as changes in the number of nodes in each community and changes in the number of communities. We generate a sequence of networks of length n=400 in the SBM framework with change points evenly spaced at τ={1/4, 2/4, 3/4}. Specifically Y_i are generated with B=[ 0.2 0.001 0.001; 0.001 0.2 0.001; 0.001 0.001 0.2; ], and π=(100, 100 ,100) for 1 ≤ i ≤ 100. Then, we change the connecting probability to B= [ 0.8 0.001 0.001; 0.001 0.2 0.001; 0.001 0.001 0.8; ], and keep π=(100, 100 ,100) the same for 101 ≤ i ≤ 200. Next, we change π=(200, 50 ,50) and keep B the same for 201 ≤ i ≤ 300. Finally for 301 ≤ i ≤ 400, we change the number of communities, we set B=[0.5 0.01; 0.010.5;] and π=(200,100). We apply Mcpd_DP <ref> on the network sequences generated according to the SBM framework described above, and compare the performance with the multiple change points version of energy-CP <cit.> and kernel-CP <cit.>. We conduct 500 Monte Carlo runs and evaluate Algorithm <ref> on two aspects; first, whether the correct number of change points can be detected and second, conditioning on the correctly estimating the number of change points, on the MAE of the estimated change points. Out of 500 Monte Carlo runs, the proportions of correctly estimating 3 change points are 100%, 95.8% and 100% for the proposed algorithm Mcpd_DP, energy-CP and kernel-CP. All of the three methods achieve zero MAE conditioning on correctly estimating the number of 3 points. §.§ Multiple change points in real dataNext we illustrate the implementation of Mcpd_DP <ref> on the real data examples investigated in Section <ref>. For the U.S. electricity generation data, Mcpd_DP detects May 2007, February 2015 and February 2019 as the change points as illustrated in Figure <ref>. In May 2007, while renewable sources pick up suddenly, the petroleum component starts a rapid downward trend alongside abrupt changes in the trends of the coal component. In February 2015 the solar component starts to accelerate along with continuing changes in the renewable component with similar change patterns again in February 2019.For MIT reality mining data, Mcpd_DP <ref> detects change points at 2004-10-16 (beginning of sponsor week), 2004-12-16 (around the end of the finals week and beginning of the winter break), 2005-01-01 (starting of the independent activities period) and2005-03-10 (right after the exam week) according to events labeled in <cit.>.§ CONCLUSIONWe introduce a nonparametric change point detection method, denoted as dist-CP, designed for random objects taking values in general metric spaces. dist-CP is based solely on pairwise distances and is tuning parameter-free, making it an appealing choice for practitioners working with complex data who seek a straightforward application without the burden of specifying additional parameters except for interval cut-offs where change points are presumed absent. This stands in contrast to other existing approaches for change point detection in metric space data, which typically necessitate parameter choices such as similarity graph selection in graph-CP and kernel and bandwidth selection in kernel-CP. We derive the asymptotic distribution of the test statistic of dist-CP under H_0 to ensure type I error control and establish the large sample consistency of the test and the estimated change points under contiguous alternatives. We extend all theoretical guarantees to the practicable permutation approximations of the null distribution. Comprehensive simulations across various scenarios, spanning random vectors, distributional data, and networks, showcase the efficacy of dist-CP in many challenging settings, including scale and tail probability changes in Gaussian random vectors and preferential attachment changes in random networks. We study the extension of dist-CP to multiple change point scenarios by combining it with seeded binary segmentation.The data applications lead to insightful findings in the U.S. electricity generation compositions timeline and in the bluetooth proximity networks of the MIT reality mining experiment.With its versatility and minimal parameter requirements dist-CP has the potential for widespread application across various domains as long as distances can be defined between the data elements. apalike | http://arxiv.org/abs/2311.16025v1 | {
"authors": [
"Paromita Dubey",
"Minxing Zheng"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20231127173814",
"title": "Change Point Detection for Random Objects using Distance Profiles"
} |
Near-resonant nuclear spin detection with high-frequency mechanical resonators Javier del Pino January 14, 2024 ==============================================================================Future x-ray observatories will require imaging detectors with fast readout speeds that simultaneously achieve or exceed the other high performance parameters of x-ray charge-coupled devices (CCDs) used in many missions over the past three decades.Fast readout will reduce the impact of pile-up in missions with large collecting areas while also improving performance in other respects like timing resolution.Event-driven readout, in which only pixels with charge from x-ray events are read out, can be used to achieve these faster operating speeds. Speedster-EXD550 detectors are hybrid complementary metal-oxide semiconductor (CMOS) detectors capable of event-driven readout, developed by Teledyne Imaging Sensors and Penn State University. We present initial results from measurements of the first of these detectors, demonstrating their capabilities and performance in both full-frame and event-driven readout modes. These include dark current, read noise, gain variation, and energy resolution measurements from the first two engineering-grade devices. *Joseph Colosimo,[email protected] 1 § INTRODUCTION §.§ X-Ray Hybrid CMOS Detectors X-ray Hybrid CMOS detectors (HCDs) are active-pixel sensors composed of separate silicon absorber and readout integrated circuit (ROIC) layers bonded together. The Penn State High-Energy Astrophysics Detector and Instrumentation Lab has worked with Teledyne Imaging Sensors to develop x-ray HCDs to demonstrate and improve performance for future observatories. The design of these devices allows for fast and flexible readout architectures compared with the charge-coupled devices (CCDs) used in most currently operating x-ray missions. The CMOS ROICs also require less power and are more resilient to radiation than CCDs <cit.>, both critical considerations for space missions. Furthermore, the hybrid nature of these devices allows for thicker fully-depleted absorbing layers and high quantum efficiency across the soft x-ray band <cit.>. Measurements on previous versions of x-ray HCDs have demonstrated very rapid readout, low power, and radiation hardness, along with moderate read noise and energy resolution <cit.>, with promising improvements over each iteration. One of the primary advantages of HCDs and other active-pixel sensors is their rapid readout. This is particularly important to x-ray detectors operating in the photon-counting regime, as only one photon may land in an individual pixel in each frame in order to be properly registered. When pile-up of multiple x-rays occurs, the multiple lower-energy photons cannot be distinguished from a single higher energy photon. This leads to a loss of information about the rate and energies of the observed x-rays. Faster frame rates decrease the frequency of pile-up events when observing bright sources. This will be crucial on many next-generation observatories, which will have large collecting areas to increase sensitivity to faint sources, but will still need the capability to produce quality observations of bright sources <cit.>.Fast readout also provides other advantages, including greater timing resolution and reduced impact from dark current. Increased timing resolution will be valuable for observations of bright sources exhibiting rapid variability.High frame rates also reduce the impact of dark current, as fewer electrons can accumulate in each frame. The detectors can thus operate at higher temperatures while still achieving performance comparable to that when they are cooled to lower temperatures.Furthermore, faster frame rates will also reduce the impact of optical and ultraviolet (UV) backgrounds, reducing the probability of these photons landing coincident to x-rays <cit.>. This allows for the use of thinner optical/UV blocking filters, which will provide the instrument with higher quantum efficiency at low energies. §.§ Event-Driven ReadoutReadout speeds can be substantially increased by reading out only those pixels with x-ray events.In photon-counting x-ray detectors, the vast majority of pixels in a given frame will contain no charge from x-ray interactions.Event-driven readout, in which only pixels with significant charge accumulated in a frame are read out, can thus increase readout speed significantly.Penn State University and Teledyne Imaging Sensors have developed this capability in the Speedster-EXD HCDs.Speedster-EXD HCDs use an in-pixel comparator to flag pixels with significant charge, allowing only those pixels to be read out by the detector. The comparator threshold is set by the user to specify the amount of charge required to trigger readout.These detectors are also capable of reading out the 3×3 region of pixels adjacent to all pixels with charge exceeding the comparator threshold.The 3×3 event-driven readout mode allows for improved energy resolution by enabling the charge to be fully accounted for when the charge cloud overlaps multiple pixels, even when these secondary pixels do not have enough charge to trigger the comparator.The detectors can also operate in full-frame or windowed region-of-interest (ROI) readout modes. Portions of frames captured in both full-frame and 3×3 event-driven modes are shown in Fig. <ref>.Fig. <ref> shows a simplified schematic of a Speedster-EXD pixel. Charge is amplified using a capacitive transimpedance amplifier (CTIA), keeping the sense node at a constant potential and avoiding interpixel capacitance observed in previous HCDs using source follower amplifiers <cit.>. Pixels also include correlated double sampling (CDS) to reduce the impact of reset noise. Both reset and CDS are conducted by frame. These operations contribute to the dead time, during which the detector is insensitive to x-rays. The comparator is located at the output of the CTIA. If a pixel's signal exceeds the global comparator threshold, the pixel is flagged for readout. The comparator threshold voltage is reset each frame. This operation also contributes to the dead time when the detector is operated in event-driven readout mode.The Speedster-EXD550 is the latest detector developed in partnership between Penn State University and Teledyne Imaging Sensors. Fig. <ref> shows two Speedster-EXD550 detectors mounted in a testing configuration. These sensors are a scaled-up version of the smaller prototype Speedster-EXD64 detectors <cit.>, which have 64 × 64 pixel arrays. The Speedster-EXD550 features a larger pixel array (550 × 550), a 2-side-buttable molybdenum package, and on-chip column-parallel digitization, while maintaining the same unit-cell circuitry as the Speedster-EXD64. Both versions have a pixel pitch of 40m and a thickness of 100 m. A substrate voltage of 25 V is applied across the absorber layer to ensure full depletion and minimize the amount of charge spreading across pixels. §.§ Speedster-EXD Detectors for the BlackCAT CubeSat Rapid readout speed, radiation hardness, and low power draw make these detectors well suited for many applications in space-based x-ray observatories.An array of four Speedster-EXD550 detectors will be used in the focal plane array of the BlackCAT CubeSat, whose payload is being designed, assembled, and calibrated at Penn State. BlackCAT is a 6U NASA CubeSat mission planned to launch in early 2025. This instrument will use a coded-aperture mask in conjunction with the Speedster detectors to monitor the x-ray sky for gamma-ray bursts and other high-energy transient phenomena. Fast readout will reduce the impact of dark current, thus allowing for operation at higher temperatures and relaxing the thermal requirements placed upon the limited CubeSat platform <cit.>. § METHODSWe describe initial measurements on two engineering-grade Speedster-EXD550 HCDs, referred to as focal plane modules (FPMs) 23056 and 23057. We report initial dark current, read noise, gain variation, and energy resolutions for these detectors. We detail the analysis and results of measurements taken in both full-frame and event-driven readout modes.In this section, we describe the test setup and analysis used to characterize the detector performance. §.§ Experimental SetupOur measurements are made using a radioactive ^55Fe source, emitting Mn K and K x-rays at 5.9 and 6.5 keV respectively. The detectors can be shielded from this emission using a retractable shutter in order to acquire dark frames.The detectors are cooled to 213 K during normal operation to reduce dark current to negligible levels.This temperature is achieved by thermally connecting the copper cold mount on which the detectors are mounted (as shown in Fig. <ref>) to a liquid nitrogen dewar. The temperature of the detectors is controlled using heaters potted within the copper plate.The temperature setpoint can be changed to allow for characterization of dark current at a range of temperatures. The measurements took place with the detector assembly in a large vacuum chamber. Measurements are conducted in vacuum in order to prevent condensation on the cooled detectors and to avoid attenuation of x-rays in air. The detectors are operated using a frame grabber and power-supplies, both located outside the chamber. The detectors are operated at their maximum full-frame readout rate of 152 frames per second for measurements in both full-frame and event-driven readout modes, unless otherwise noted.In the test setup used for these measurements, this is the fastest rate that does not drop frames, due to the method in which frames are transferred via the frame grabber. Recent testing with a new detector readout/interface board, developed by the BlackCAT electronics team, has demonstrated operation in a windowed region-of-interest (ROI) mode and event-driven modes at faster rates. When operated in event-driven readout mode, these detectors are theoretically capable of operating at frame rates in excess of 1 kHz, with the actual rate limited by the number of pixels which must be read out and the desired fraction of dead time in which the detectors are insensitive to x-rays. With current settings, the dead fraction would be 8% at 1 kHz operation; however, this can be reduced with some impact to read noise performance.The power dissipation of these detectors is strongly dependent on the settings with which the detector is operated, particularly those related to the operation of the CTIA and the comparator.When operated at 213 K with the settings used for the measurements described in this paper, the power draw is 0.42 W for FPM 23056 and 0.45 W for FPM 23057.The in-pixel comparator can be disabled with minimal impact to full-frame readout performance. In this state, the power draw can be reduced to 0.21 W for FPM 23056 and 0.22 W for FPM 23057. §.§ Image Processing and Event Analysis We use a conventional analysis pipeline to remove noise artifacts from images and to identify and characterize x-ray events.A set of dark frames is collected in order to remove fixed-pattern noise from images.An outlier-resistant truncated mean is calculated for each pixel over these dark frames to create a bias frame used for this purpose. For data taken in event-driven readout mode, dark frames are acquired at the same effective frame rate in ROI mode.For event-driven readout at frame rates faster than the maximum full-frame rate, dark bias frames can be created by combining multiple smaller ROIs acquired separately. Frames are observed to have a temporally varying global offset from the computed dark bias mean, adding to the standard read noise. Images collected in full-frame mode can be corrected for variations in global offset in addition to the standard fixed-pattern noise. This is done by calculating the offset (the truncated mean of the difference between pixel values and the dark bias frame) and then subtracting this value from the frame.This additional step is not done in the analysis of event-driven data, as only pixels adjacent to those with registered charge are read out, biasing this correction. X-ray events are identified within the bias-subtracted frames using a conventional algorithm, selecting pixels whose signal is above a set `event' threshold and is greater than that of all adjacent pixels. The energy of the x-ray event is determined based on the signal of this primary pixel and the signal from charge which spread into surrounding pixels.The charge measured in adjacent pixels is added to the energy of an event if it exceeds a `split' threshold. The number and spatial configuration of these included pixels is used to give the event a quality grade. We use the event-grading scheme of the Swift X-ray Telescope<cit.> to classify events. A bad-pixel mask, created by identifying pixels with anomalous gain or noise, is applied to the event data to mask defective pixels from the analysis. § RESULTS §.§ Gain VariationPixel-to-pixel variation in the gain is a consequence of the detector's design. Every pixel has its own circuitry (signal amplification and readout), and pixel-to-pixel variations of the gain may be present due to component process variations. If left uncorrected, variations in the pixel gain will result in differences in energy measurements of x-rays depending on where the x-ray is absorbed on the detector. Consequently, energy resolution will be degraded unless corrections are made to account for the variations in gain. To correct this effect and properly measure the energy resolution of the detectors presented here, we measured the gain in each pixel of both detectors to fully characterize the gain variation.The gain variability measurements were made within the experimental setup described in Sec. <ref>, and x-ray data were collected until ∼800 Mn K single-pixel events were detected in each pixel. Approximately 20 hours of data collection per detector is required to compile the desired number of single-pixel events per pixel. The captured images were background subtracted and analyzed for single-pixel events using the event-analysis algorithm described above. The resulting single-pixel event values were binned by energy to identify the location of each pixel's Mn K and Mn K peaks. Each pixel's Mn K peak was fit with a Gaussian.The gain for each pixel was found using the following:Gain = μ_Gaussian5.89 keV/ω ADU/e^-,where μ_Gaussian is the mean of the Gaussian fit, in ADU, for the Mn K line in a pixel, and ω is the average energy required for the creation of a single electron-hole pair in silicon which has been accurately measured to be ω = 3.65 eV/e^- <cit.>. ω is a property of silicon itself and is slightly dependent on the detector state and operating temperature; however, the temperature dependence is minimal and will have a negligible impact on the results presented.A gain variation map was then created for each detector. Bad pixels were removed from the gain map and gain variation calculations because these same pixels would be masked out when used in any scientific application. The masked pixels were empty or contained exceptionally high or low gain. These ‘bad-gain’ pixels were removed if they were ≥ 7σ from the mean of the gain map. The associated gain variation was calculated by dividing the standard deviation of the gain map by the mean.Approximately 1.5% of pixels were masked from the gain variation calculation for FPM 23056, and 5.2% of pixels were masked for FPM 23057. The larger number of masked pixels in FPM 23057 is due to many columns of dead pixels localized to one side of the detector. The large number of adjacent dead columns of pixels can be explained by manufacturing defects in column readout circuitry. The gain variation was measured to be 2.15 ± 0.09% for FPM 23056 and 1.95 ± 0.02% for FPM 23057. Errors in the gain variation were calculated using the 1σ errors on the centroid of the Gaussian fit for each pixel and calculating the mean error of all included pixels. Fig. <ref> shows a histogram of the gain map for FPM 23056 (blue solid line) and FPM 23057 (pink dot-dashed line) normalized by the mean of the respective gain map.By examining a spatial map of gain for both devices, we see that some of the gain variation appears to have a spatial dependency. The exact reason for the localized regions of high or low gain is unknown but may be the result of uneven pressure when the detector was hybridized.Moreover, pixels around non-uniform regions, i.e., dead pixel groups or columns, appear to suffer from high deviations from the mean. Overall, this gain variation is modest and is easily measured, thus enabling a pixel-by-pixel gain correction to be applied to the measured signal in each pixel.§.§ Dark Current §.§.§ MethodologyIn order to characterize the dark current of these detectors, we measured the dark current at 10 degree increments from 203 K to 263 K.At each temperature, dark frames were acquired at several different exposure times.After applying a bad-pixel mask, we applied a linear fit to each pixel's signal as a function of the exposure time. The slope of that line provides the dark current for that pixel at that temperature.The dark current of the detector is quoted as the median of the pixel dark currents.The quoted errors give the median absolute deviation (scaled for equivalence to standard deviation) of the measured dark current for each pixel. We acquired ^55Fe data at each temperature in order to properly correct the dark current measurements for changes in the overall detector gain as a function of temperature.These data were used to identify the location of the centroid of the Mn K peak in order to measure and account for changes in gain as a function of temperature in our dark current measurements. We do not correct for pixel-to-pixel gain variation in these dark current measurements due to the substantial time required to acquire enough single-pixel Mn K events to produce a proper pixel gain map at each temperature. The overall detector gain is sufficient to scale the measured dark current, and gain variation is small enough to only marginally increase the scatter observed in pixel dark current.§.§.§ Measurements and Model The dark currents found at each temperature for both detectors were recorded along with their error in Table <ref>. The dark current is expected to increase with temperature according to the following equation<cit.>:Dark Current(e^-/pix/s)=2.5×10^15 P_sD_FMT^1.5e^-E_g /2kT,where P_s is the pixel area in cm^2, D_FM is the dark current figure of merit at 300 K in nA/cm^2, T is temperature in K, k is Boltzmann's constant in eV/K, and E_g is the silicon band-gap energy in eV, which is a function of temperature per the following equation <cit.>:E_g(eV)=1.557-7.021×10^-4 T^2/1108+T. Eq. <ref> accounts for both surface and depletion dark current, which are expected to be the dominant sources of dark current in these detectors. Both detectors approximately follow the expected behavior from 203 K to 263 K with some departures from the model at both extremes, so we can use the model from Eq. <ref> to fit these data. Every variable in Eq. <ref> is known with the exception of the figure of merit at 300 K, which is determined by the fits shown in Fig. <ref>.This results in a figure of merit of 58.8 nA/cm^2 for FPM 23056 and 66.1 nA/cm^2 for FPM 23057.§.§.§ Expected Impact of Dark Current: A Sample CaseThe BlackCAT CubeSat will utilize passive cooling, with expected focal plane operating temperatures between 213 K and 233 K once in orbit <cit.>. It is particularly important to characterize the level and impact of dark current in the Speedster-EXD550 detectors when operating within this temperature range.The shot-noise contribution from dark current is the square root of the expected number of dark-current electrons accumulated in a pixel in a single frame.We find that at our fast frame rates (maximum of 152 Hz for full-frame observations), the dark current will not be the dominant source of noise. At 233 K, the expected noise from dark current is 6.0 e^-/pixel for FPM 23056 and 6.3 e^-/pixel for FPM 23057 at frame rates of 152 Hz. These values are substantially lower than the read noise measured on these devices, contributing only marginally to the overall detector noise when cooled to 233 K.The impact of dark current will be reduced even further in event-driven readout mode because the frame rate can be substantially increased. §.§ Read Noise The read noise is characterized by measuring the variation of pixel signals acquired in dark frames. We quote the noise as the root mean square (RMS) of the signal of an individual pixel over a large number of frames.The frames are corrected for offset fixed-pattern noise by calculating and subtracting bias frames prior to our noise analysis.Data taken in full-frame mode is also corrected for global offset variations, while data taken in event-driven mode is not, resulting in a higher effective read noise for event-driven data than for full-frame data.We characterize the effective noise for data acquired in both full-frame and event-driven readout modes. Measurements were taken with the detectors cooled to 213 K, resulting in little impact of dark current on the measured noise (∼1.3 e^- shot noise from dark current). The median magnitude of the global offset variation is measured to be approximately 5 e^- in both devices.This contributes to the higher effective read noise measured in event-driven readout mode. The source of this offset noise is currently not well understood. Future efforts will be aimed at characterizing the dependence of the offset variation magnitude on detector parameters in order to reduce the impact and better identify the source. We measure the noise for each pixel using two different methods.We measure the pixel RMS, defined as the RMS of the signal of an individual pixel over a series of 1000 frames. We also measure the fit RMS, defined as the width of a Gaussian fit to the histogram of a pixel's values taken over the same series of frames. The pixel RMS gives the actual observed noise of the pixel, while the fit RMS gives the contribution of the Gaussian component.Fig. <ref> shows the distribution of measured pixel noise values for full-frame operation. Table <ref> gives the median noise measured for both detectors when operating in both full-frame and event-driven readout modes.These detectors contain a sizable fraction of pixels exhibiting random telegraph noise (RTN) <cit.>, with 4.6% of pixels in FPM 23056 and 2.4% in FPM 23057 demonstrating clear signatures of RTN. These pixels are identified through the presence of multiple distinct noise peaks appearing in histograms of a pixel's values over a large number of frames. This effect can decrease the energy resolution of a pixel and also can create spurious events if the magnitude of the RTN is large enough.Pixels with high-frequency or high-magnitude RTN can be removed from analysis to reduce these effects in order to improve detector performance, particularly at lower energies. Pixels with RTN can also be deselected from readout in event-driven mode; however, this feature is not currently functional in a large number of columns on both detectors.It is also possible that low-magnitude RTN, difficult to distinguish from ordinary read noise, is present in significantly more pixels and is responsible for the large fraction of pixels with read noise measured well above the median value. §.§ Energy ResolutionTo determine the energy resolution of these detectors, we measure the energy width of the Mn K peak in data collected using an ^55Fe source.Data with ^55Fe x-ray events were acquired in both event-driven and full-frame readout modes, with the detector operating at a temperature of 213 K.We subtracted the dark bias frames for all data and the global frame offsets for full-frame data. We performed event identification using a 10σ event threshold (based on the median pixel-RMS noise). Events from pixels flagged in the bad-pixel mask were discarded from this analysis. The energy resolution depends on multiple factors in both detector operation and analysis. The energy resolution will be different in full-frame and event-driven readout due to the higher effective read noise in event-driven operation.The energy resolution is also dependent on the level of the split threshold and the event grades included in the analysis.We perform our analysis for split thresholds of 1, 2, and 3σ and for different selections of event grades. In the Swift XRT grading scheme <cit.>, grade-0 events only include a single primary pixel. These events are generally the highest quality because only noise from a single pixel impacts the measured energy. Events with grades 1–4 are a 2-pixel split events, and those with grades 5–12 are 3- or 4-pixel events.Events with grades >12 are not generally used in analysis as these do not correspond with shapes expected from an x-ray interaction. Once the proper event-selection cuts are applied, we bin events by energy to create spectral energy distributions.The resulting spectra are fit using two Gaussians, one to model the Mn K peak and the other to model the K peak. We quote the full width at half maximum (FWHM) of the Mn K peak derived from this fit as the energy resolution. Example fits from the grade-0, 2σ split-threshold analysis are shown in Fig. <ref>. Full results are displayed in Tables <ref> & <ref>.With a low split threshold, a wider range of acceptable pixel values can be included in the event.This can result in the inclusion of pixels adjacent to the primary pixel due to noise rather than actual charge generated by an x-ray.A low split threshold will thus produce worse energy resolution for multi-pixel events.A high split threshold, on the other hand, can result in signal from pixels into which an insufficient amount of charge has spread to be excluded from the total energy of the event. This results in an undercounting of the total charge in an event and a degradation of the energy resolution, especially in the case of low-energy x-rays.These effects can be observed in Tables <ref> & <ref>. We also note that the higher effective read noise encountered in event-driven readout results in slightly worse energy resolution.The Speedster-EXD550 devices investigated here show similar performance to the Speedster-EXD64 devices characterized in Ref. , with somewhat higher read noise and lower energy resolution. The Speedster-EXD64 devices were operated at a lower temperature (150 K) and higher frame rate (1 kHz) for their characterization measurements. Read noise measurements of 13.0 e^- and 11.2 e^- are reported for the two Speedster-EXD64 devices. Dark current contributes minimally to the noise in the measurements on the Speedster-EXD550 or Speedster-EXD64 devices; however, other temperature-dependent sources of noise may contribute to the higher noise in the Speedster-EXD550 measurements at 213 K presented in this paper. For the Speedster-EXD64 detectors, the Mn K FWHM was measured to be 206 eV and 236 eV in full-frame operation and 236 eV and 247 eV event-driven operation for single-pixel events (with a 3σ split threshold). A full pixel-to-pixel gain correction was not applied to these detectors to obtain these measurements, although the smaller array size and method of fitting these peaks reduces the impact of gain variation on the energy resolution of the Speedster-EXD64 detectors. § CONCLUSIONWe have presented measurements of two engineering-grade Speedster-EXD550 HCDs, demonstrating their capabilities when operated in both full-frame and event-driven readout modes.Fast operation allows these detectors to function at higher temperatures than is feasible for most other HCDs, due to the reduced impact of dark current at these frame rates. We measured a median read noise of 15.6 and 16.7 e^- (RMS) when operated in full-frame readout and a median effective read noise of 17.5 and 18.7 e^- (RMS) when operated in event-driven readout for each of the respective detectors.These detectors have significantly higher read noise than other HCDs, such as the small-pixel detectors, which have measured read noise values as low as 6.3 e^- (RMS) <cit.>.While future x-ray missions will likely require reduced read noise and improved energy resolution, these detectors meet the energy resolution requirements for the BlackCAT CubeSat, which requires fast frame rates but has only modest read noise requirements.Through continued optimization and characterization at colder operating temperatures, we will attempt to demonstrate lower-noise operation with these devices.An ongoing effort is also aimed at designing Speedster-EXD HCDs with lower sense-node capacitance in order to achieve lower read noise while still maintaining the capability for event-driven readout. Continued testing of these devices with new readout electronics for BlackCAT will also demonstrate event-driven performance at faster frame rates. Efforts are also underway to characterize the performance of six other Speedster-EXD550 devices from Teledyne Imaging Sensors in preparation for the selection and calibration of detectors for the BlackCAT focal plane. §.§ Disclosures No potential conflicts of interest have been identified by the authors.§.§ Code, Data, and Materials AvailabilityAll data in support of the findings of this paper are available either within the article itself or as supplementary material, which may be provided upon request. §.§ AcknowledgmentsThis work was supported by a NASA Space Technology Graduate Research Opportunity (grant 80NSSC20K1210) as well as NASA grants 80NSSC18K0147, NNX14AH68G, and 80NSSC21K1125.We would also like to thank Vincent Douence at Teledyne Imaging Sensors for his guidance in operating these detectors and optimizing their performance.spiejour | http://arxiv.org/abs/2311.16270v2 | {
"authors": [
"Joseph M. Colosimo",
"Hannah M. Grzybowski",
"Evan C. Jennerjahn",
"Lukas R. Stone",
"Abraham D. Falcone",
"Mitchell Wages",
"Jacob C. Buffington",
"David N. Burrows",
"Zachary E. Catlin",
"Timothy Emeigh",
"Frederic Hancock"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20231127191934",
"title": "Initial Characterization of the First Speedster-EXD550 Event-Driven X-Ray Hybrid Complementary Metal-Oxide Semiconductor Detectors"
} |
[ Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains Björn Malte Schäfer 0000-0002-9453-5772^2, ♯ 20th November 2023 ==================================================================================================================== < g r a p h i c s >figureWe present an approach to generate realistic 3D human-object interactions (HOIs), from a text description and given static object geometry to be interacted with (left). Our main insight is to explicitly model contact (visualized as colors on the body mesh, closer contact in red), in tandem with human and object sequences, in a joint diffusion process. In addition to synthesizing HOIs from text, we can also synthesize human motions conditioned on given object trajectories (top right), and generate interactions in static scene scans (bottom right). ] We propose , the first method to address the task of generating dynamic 3D human-object interactions (HOIs)from text.We model the motion of both human and object in an interdependent fashion, as semantically rich human motion rarely happens in isolation without any interactions.Our key insight is that explicitly modeling contact between the human body surface and object geometry can be used as strong proxy guidance, both during training and inference. Using this guidance to bridge human and object motion enables generating more realistic and physically plausible interaction sequences, where the human body and corresponding object move in a coherent manner. Our method first learns to model human motion, object motion, and contact in a joint diffusion process, inter-correlated through cross-attention.We then leverage this learned contact for guidance during inference synthesis of realistic, coherent HOIs. Extensive evaluation shows that our joint contact-based human-object interaction approach generates realistic and physically plausible sequences, and we show two applications highlighting the capabilities of our method. Conditioned on a given object trajectory, we can generate the corresponding human motion without re-training, demonstrating strong human-object interdependency learning. Our approach is also flexible, and can be applied to static real-world 3D scene scans.§ INTRODUCTIONGenerating human motion sequences in 3D is important for many real-world applications, e.g. efficient realistic character animation, assistive robotic systems, room layout planning, or human behavior simulation.Crucially, human interaction is interdependent with the object(s) being interacted with; the object structure of a chair or ball, for instance, constrains the possible human motions with the object (e.g., sitting, lifting), and the human action often impacts the object motion (e.g., sitting on a swivel chair, carrying a backpack).Existing works typically focus solely on generating dynamic humans, and thereby disregarding their surroundings <cit.>, or grounding such motion generations in a static environment that remains unchanged throughout the entire sequence <cit.>. However, real-world human interactions affect the environment. For instance, even when simply sitting down on a chair, the chair is typically moved: to adjust it to the needs of the interacting human, or to move it away from other objects such as a table.Thus, for realistic modeling of human-object interactions, we must consider the interdependency of object and human motions.We present , the first approach to address the task of generating realistic 3D human-object interactions from text descriptions, by jointly predicting a sequence of 3D human body motion along with the object motion. Key to our approach is to not only model human and object motion, but also to explicitly model contact as a bridge between human and object. In particular, we model contact by predicting contact distances from the human body surface to the closest point on the surface of the object being interacted with.This explicit modeling of contact helps to encourage human and object motion to be semantically coherent, as well as provide a constraint indicating physical plausibility (e.g., discouraging objects to float without support). jointly models human, object, and contact together in a denoising diffusion process. Our joint diffusion model is designed to encourage information exchange between all three modalities through cross-attention blocks.Additionally, we employ a contact weighting scheme, based on the insight that object motion, when being manipulated by a human, is most defined by the motion of the body part in closest contact (Fig. <ref>). We make use of this by generating separate object motion hypotheses for multiple parts of the human body and aggregating them based on that part's predicted contact. During inference, we leverage the predicted contact distances to refine synthesized sequences through our contact-based diffusion guidance, which penalizes synthesizing sequences with human-object contact far from the predicted contact distances.Our method is able to generate realistic and physically plausible human-object interactions, and we evaluate our approach on two widely-used interaction datasets, BEHAVE <cit.> and CHAIRS <cit.>. In addition, we also demonstrate the usefulness of our model with two related applications: First, generating human motion given a specific object trajectory without any retraining, which demonstrates our learned human-object motion interdependencies. Second, populating a static 3D scene scan with human-object interactions of segmented object instances, showing the applicability of our method to general real-world 3D scans.In summary, our contributions are three-fold: * We propose an approach to generate realistic, diverse, and physically plausible human-object interaction sequences by jointly modeling human motion, object motion, and contact through cross-attention in a diffusion process. * We formulate a holistic contact representation: Object motion hypotheses are generated for multiple pre-defined points on the surface of the human body and aggregated based on predicted contact distances, enabling comprehensive body influence on contact while focusing on the body parts in closer contact to the object.* We propose a contact-based guidance during synthesis of human-object interactions, leveraging predicted contacts to refine generated interactions, leading to more physically plausible results.§ RELATED WORK 3D Human Motion Generation. Generating sequences of 3D humans in motion is a task which evolved noticeably over the last few years. Traditionally, many methods used recurrent approaches <cit.> and, improving both fidelity and predicted sequence length, graph- and attention-based frameworks <cit.>. Notably, generation can either happen deterministically, predicting one likely future human pose sequence <cit.>, or stochastically, thereby also modelling the uncertainty inherent to future human motion <cit.>.Recently, denoising diffusion models <cit.> showed impressive results in 2D image generation, producing high fidelity and diverse images <cit.>.Diffusion models allow for guidance during inference, with classifier-free guidance <cit.> widely used to trade off between generation quality and diversity. Inspired by these advances, various methods have been proposed to model 3D human motion through diffusion, using U-Nets <cit.>, transformers <cit.>, or custom architectures <cit.>. Custom diffusion guidance has also been shown to aid controllability <cit.> and physical plausibility <cit.>.In addition to unconditional motion generation, conditioning on text descriptions allows for more control over the generation result <cit.>. In fact, generating plausible and corresponding motion from textual descriptions has been an area of interest well before the popularity of diffusion models <cit.>.These methods show strong potential for 3D human motion generation, but focus on a skeleton representation of the human body, and only consider human motion in isolation, without naturally occurring interactions. To generate realistic human-object interactions, we must consider the surface of the human body and its motion with respect to object motion, which we characterize as contact. 3D Human Motion in Scenes. As human motion typically occurs not in isolation but in the context of an object or surrounding environment, various methods have exploredlearning plausible placement of humans into scenes, both physically and semantically, <cit.>, forecasting future motion given context <cit.>, or generating plausible walking and sitting animations <cit.>.This enables more natural modeling of human reactions to their environment; however, the generated interactions remain limited due to the assumption of a static scene environment, resulting in a focus on walking or sitting movements.Recent methods have also focused on more fine-trained interactions by generating human motion given a single static object <cit.>.While these methods only focus on human motion generation for a static object, InterDiff <cit.> jointly forecasts both human and object motion sequences given an initial sequence observation. Our approach also models both human and object motion, but we formulate a flexible text-conditioned generative model for dynamic human and object motion, modeling the interdependency between human, object, and contact to synthesize more realistic interactions under various application settings. Contact Prediction for Human-Object Interactions. While there is a large corpus of related work for human motion prediction, only few works focus on object motion generation <cit.>. Notably, these methods predict object movement in isolation, making interactions limited, as they typically involve interdependency with human motion.Contact prediction has been most studied in recent years for the task of fine-grained hand-object interaction <cit.>. It is defined either as binary labels on the surface <cit.> or as the signed distance to a corresponding geometry point <cit.>. In these works, predicting object and hand states without correct contact leads to noticeable artifacts. Contact prediction itself has also been the focus of several works <cit.>, either predicting contact areas or optimizing for them.Applied to the task of generating whole-body human-object interactions, this requires access to the full surface geometry of both object and human.Only few recent motion generation works focus on generating full-body geometric representations of humans <cit.> instead of simplified skeletons which is a first step towards physically correct interaction generation. However, while several of these works acknowledge that contact modeling would be essential for more plausible interactions <cit.>, they do not model full-body contact.We approach the task of generating plausible human-object motion from only the object geometry and a textual description as a joint task and show that considering the joint behavior of full-body human, object, and contact between the two benefits output synthesis to generate realistic human-object interaction sequences. § METHOD OVERVIEWjointly generates sequences of human body and object representations, alongside contact on the human body surface. Reasoning jointly about all three modalities in both training and inference enables generation of semantically meaningful human-object interaction sequences.Fig. <ref> shows a high-level overview of our approach: We consider as condition a brief text description T of the action to be performed, along with the static geometry G of the object to be interacted with, and generate a sequence of F frames 𝐱=[𝐱_1, 𝐱_2, ..., 𝐱_F] where each frame x_i consists of representations for the object transformation o_i, for the human body surface h_i, and for the contact c_i between human and object geometry. We denote as H={h_i} the human body representations, O={o_i} the object transformations, and C={c_i} the contact representations.We first train a denoising diffusion process to generate H, O, and C, using a U-Net architecture with per-modality residual blocks and cross-attention modules. Using cross-attention between human, object motion, and contact allows for effectively learning interdependencies and and feature sharing (Sec. <ref>). We use the generated contact to guide both training and inference: Instead of predicting one object motion hypothesis per sequence, we generate multiple, and aggregate them based on predicted contacts, such that body parts in closer contact with the object have a stronger correlation with the final object motion (Sec. <ref>). During inference, the trained model generates H, O, and C. For each step of the diffusion inference, we use predicted contact C to guide the generation of H and O, by encouraging closeness of recomputed contact and predicted contact, producing more refined and realistic interactions overall (Sec. <ref>). § HUMAN-OBJECT INTERACTION DIFFUSION§.§ Probabilistic Denoising DiffusionOur approach uses a diffusion process to jointly generate a sequence of human poses, object transformations, and contact distances in a motion sequence from isotropic Gaussian noise in an iterative process, removing more noise at each step. More specifically, during training we add noise depending on the time step (“forward process”) and train a neural network to reverse this process, by directly predicting the clean sample from noisy input.Mathematically, the forward process follows a Markov chain with T steps, yielding a series of time-dependent distributions q(𝐳_t | 𝐳_𝐭-1) with noise being injected at each time step until the final distribution 𝐳_T is close to 𝒩(0, 𝐈). Formally,q(𝐳_t | 𝐳_t-1) = 𝒩(√(β_t)𝐳_t-1 + (1 - β_t)𝐈)with the variance of the Gaussian noise at time t denoted as β_t, and β_0=0.Since we adopt the Denoising Diffusion Probabilistic Model <cit.>, we can sample 𝐳_t directly from 𝐳_0 as𝐳_t = √(α_t)𝐳_0 + √(1-α_t)ϵwith α_t=∏_t'=0^t(1 - β_t), and ϵ∼𝒩(0, 𝐈).For the reverse process, we follow <cit.>, directly recovering the original signal 𝐳̃ instead of the added noise. Human-Object Interactions To model human-object interactions with diffusion, we employ our neural network formulation 𝒢. 𝒢 operates on the noised vector of concatenatedhuman, object, and contact representations, together with the current time step t, and a condition consisting of object point cloud G, encoded by an encoder E_G, and text information T, encoded by encoder E_T. Formally,𝐳̃ = 𝒢(𝐳_t, t, E_G(G)⊕ E_T(T)) More specifically, in our scenario E_T extracts text features with a pre-trained CLIP <cit.> encoder. Encoder E_G processes object geometry G as a uniformly sampled point cloud in world coordinate space with a PointNet <cit.> pre-trained on object parts segmentation.Object transformations o_i are represented as global translation and rotation using continuous 6D rotation representation <cit.>. In contrast to prior work <cit.> which focused on representing human motion in a simplified manner as a collection of J human joints, disregarding both identity-specific and pose-specific body shape, we model physically plausible human-object contacts between body surface and geometry. Thus, we represent the human body h_i in SMPL <cit.> parameters: h_i={h_i^p, h_i^b, h_i^r, h_i^t} with pose parameters h_i^p∈ℝ^63, shape parameters h_i^b∈ℝ^10, as well as global rotation h_i^r∈ℝ^3 and translation h_i^t∈ℝ^3. These body parameters can then be converted back into a valid human body surface mesh in a differentiable manner, using the SMPL <cit.> model. This allows us to reason about the contact between human body surface and object geometry. We represent contact c_i on the human body as the distance between a set of M=128 uniformly distributed motion markers on the body surface to the closest point of the object geometry, for each marker. Specifically, we represent contact for frame x_i and j-th contact marker (j∈{0..M}) c_i^j as its distance from the human body surface to the closest point on the same frame's object surface. §.§ Human-Object-Contact Cross-Attention We jointly predict human body sequences {h_i}, object transformations {o_i}, and corresponding contact distances {c_i} in our diffusion approach.We employ a U-Net backbone for diffusion across these outputs, with separate residual blocks for human, object, and contact representations, building modality-specific latent feature representations.As we aim to model the inter-dependency across human, object, and contact, we introduce custom human-object-contact cross-attention modules after every residual block where each modality attends to the other two. We follow the formulation of Scaled Dot-Product Attention <cit.>, computing the updated latent human body feature: h_i = softmax(QK^T/√(D)) V,with query Q = h_i, and key and value K = V = o_i ⊙ c_i (⊙ denotes concatenation). Applying this similarly to o_i and c_i yields the final features after each cross-attention module. §.§ Contact-Based Object Transform Weighting As visualized in Fig. <ref>, object motion is naturally most influenced by parts of the human body in very close contact to the object (as they are often the cause of the motion), and less impacted (if at all) by body parts further away. For instance, if a person moves an object with their hands, the object follows the hands but not necessarily other body parts (e.g., body and feet may remain static or walk in a different direction). Thus, instead of directly generating one object motion hypothesis o_i alongside the corresponding human motion h_i, we couple o_i to the M body contact points j ∈{1..M} and their predicted distances {c_i^j} between human body surface and object geometry.Formally, we predict object transformation hypotheses o_i^j for each contact point on the human body, and weigh them with the inverse of their predicted contact distance c_i^j: o_i = 1/∑_j c_i∑_j=0^N (max(|c_i|) - |c_i^j|) o_i^j,§.§ Loss FormulationDuring training, the input is a noised vector 𝐳, containing F frames {𝐱_i}, each a concatenation of human body representation h_i, object transformation o_i, and contact parameters c_i. As condition 𝐂, we use additionally input encoded object geometry G and text description T. The training process is then supervised with the ground-truth sequence containing ĥ_̂î, ô_̂î, ĉ_̂î, minimizing a common objective:𝐋 = λ_h||h_i - ĥ_̂î||_1 + λ_o||o_i - ô_̂î||_1 + λ_c||c_i - ĉ_̂î||_2,with λ_h=1.0, λ_o=0.9, λ_c=0.9. We use classifier-free guidance <cit.> for improved fidelity during inference, thus masking out the conditioning signal with 10% probability.§ INTERACTION GENERATION Using our trained network model, we can generate novel human-object interaction sequences for a given object geometry and a short text description using our weighting scheme for generating object transformations, and a custom guidance function on top of classifier-free guidance to generate physically plausible sequences.Specifically, we use our trained model to reverse the forward diffusion process of Eq. <ref>: Starting with noised sample 𝐳_T∼𝒩(0, 𝐈), we iteratively use our trained network model 𝒢 to estimate cleaned sample 𝐳_0:𝐳_t-1 = √(α_t)𝐳̃ + √(1-α_t)ϵ.§.§ Contact-Based Diffusion Guidance While our joint human-object-contact training already leads to some plausible motions, generated sequences are not explicitly constrained to respect contact estimates during inference, which can lead to inconsistent contact between human and object motion (e.g., floating objects).Thus, we introduce a contact-based guidance during inference to refine predictions, using a cost function G((x)_t) which takes as input the denoised human, object, and contact predictions 𝐳_t=[h_t, o_t, c_t] at diffusion step t.Based on the difference between predicted and actual contact distances for each contact point, we then calculate the gradient ∇_𝐳_tG(𝐳_t).We use this gradient for diffusion guidance, following <cit.>, by re-calculating the mean prediction μ_t at each time t: μ̂_t = μ_t + s∑_t∇_x_tG(x_t),for a scaling factor s. This guidance is indirect but dense in time, and is able to correct physical contact inconsistencies in the predicted sequences during inference time, without requiring any explicit post-processing steps. §.§ Conditioning on Object TrajectoryWhile our model has been trained with text and static object geometry as condition, we can also apply the same trained model for conditional generation of a human sequence given an object sequence and text description.Note that this does not require any re-training, as our model has learned a strong correlation between human and object motion.Instead, we use a replacement-based approach, and inject the given object motion O' into the diffusion process during inference at every step. Following Eq. <ref>, we obtain:𝐳_t-1 = √(α_t)𝐳̃_̃t̃'̃ + √(1-α_t)ϵ,with 𝐳̃'̃ = [h_t, o'_t, c_t], concatenating human motion h_t, contact distances c_t, and injected given object motion o'_t. § RESULTS We evaluate our approach using two commonly used human-object interaction datasets BEHAVE <cit.> and CHAIRS <cit.> on a range of metrics, measuring motion fidelity and diversity. We show that our approach is able to generate realistic and diverse motion on both datasets, across a variety of objects and types of interactions. §.§ Experimental Setup Datasets We conduct our experiments on two datasets containing interactions between whole-body 3D humans and corresponding objects. CHAIRS <cit.> captures 46 subjects as their SMPL-X <cit.> bodies interacting with 81 different types of chairs and sofas. We extract sequences in which both human and object are in motion, yielding ≈1300 HOI sequences, each labeled with a text description. We use a random 80/10/10 split along object classes, ensuring that test objects are not seen during training. BEHAVE <cit.> captures 8 participants as their SMPL-H <cit.> parameters alongside 20 different objects. This yields ≈520 sequences with corresponding text descriptions. We use their original train/test split. We sample both datasets at 20 frames per second, and generate 32 frames for CHAIRS and 64 for BEHAVE, leading to generated motion that lasts up to 3 seconds.Implementation Details We train our model with batch size 64 for 600k steps (≈24 hours), after which we choose the checkpoint that minimized validation FID, following <cit.>. Our attention uses 4 heads and a latent dimension of 256. Input text is encoded using a frozen CLIP-ViT-B/32 model. For classifier-free guidance during inference time, we use a guidance scale of 2.5, which empirically provides a good trade-off between diversity and fidelity. For our inference-time contact-based guidance, we use scale s=100.0.§.§ Evaluation MetricsWe measure realism and diversity of combined human and object motion, alongside closeness to the text description, following established practices <cit.>. We first train a joint human-object motion feature extractor and separate text feature extractor using a contrastive loss to produce geometrically close feature vectors for matched text-motion pairs, and vice versa. These encoders are then used for the following metrics:R-Precision measures the closeness of the text condition and generated HOI in latent feature space, and reports whether the correct match falls in the top 3 closest feature vectors.Frechet Inception Distance (FID)is commonly used to evaluate the similarity between generated and ground-truth distribution in encoded feature space. Diversity and MultiModality. Diversity measures the motion variance across all text descriptions and is defined as 1/N∑_i=1^N||v_i-v_i'||_2 between two randomly drawn subsets {v_i} and {v_i'}. MultiModality (MModality) measures the average such variance intra-class, for each text description. Perceptual User Study. The exact perceptual quality of human-object interactions is difficult to capture with any single metric; thus, we conducted a user study with 32 participants to evaluate our method in comparison to baseline approaches. Participants are shown side-by side views of sequences with the same geometry and text conditioning, and asked to choose 1) Which one follows the given text better and 2) Which one looks more realistic overall. §.§ Comparison to BaselinesAs our method is the first to enable generationg human and object motion from text, there are no baselines available for direct comparison. InterDiff <cit.> is closest to our approach, performing forecasting fromobserved human and object motion as input and predicting a plausible continuation. In Tab. <ref>, we compare to ours first in their setting, using observed motion as condition (motion-cond.), for a fair comparison. Additionally, we modify their approach by replacing observed motion encoders with our text encoder, allowing for a comparison in our setting (text-cond.). We also compare with MDM <cit.>, a state-of-the-art method for human-only sequence generation from text, both in their original setting, only predicting human sequences, and extending theirs to also generate object sequences, by adding additional tokens and geometry conditioning to their transformer encoder formulation. For more details of baseline setup, we refer to the appendix. We evaluate the quality of generated human-object interactions as well as human-only generation, only evaluating the human sequence for our method, as compared to the generated sequences of MDM.Both Tab. <ref> and the user study in Tab. <ref> show that our approach is able to generate more realistic and physically plausible human-object interaction sequences than baselines. In Fig. <ref>, we see that our approach synthesizes more meaningful human-object interaction with respect to contact and mitigating independent object floating.§.§ Ablation Studies Cross-attention enables learning human-object interdependencies. Tab. <ref> shows that our human-object-contact cross-attention (Sec. <ref>) significantly improves performance by effectively sharing information between human, contact, and object sequence modalities. In Fig. <ref>, we see this encourages realistic contact between human and object. Contact prediction improves HOI generation performance. Predicting contact (Sec. <ref>) is crucial to generating more realistic human-object sequences, resulting in more realistic interactions between human and object (Fig. <ref>), and improved fidelity (Tab. <ref>). Notably, learning contact jointly with human and object motion improves overall quality, compared to a separately trained contact model used for inference guidance (“Separate contact pred.", Tab. <ref>). Contact-based object transformation weighting improves generation performance. Weighting predicted object motion hypotheses with predicted contact (Sec. <ref>) improves HOI generation over naive object sequence prediction, both quantitatively in Tab. <ref> (“No contact weighting") and visually as realistic human-object interactions in Fig. <ref>. Contact-based guidance during inference helps produce physically plausible interactions. As visualized in Fig. <ref> and evaluated in Tab. <ref>, using custom guidance based on predicted contacts leads to a higher degree of fidelity and physical plausibility.§.§ ApplicationsHuman motion generation given object trajectory. Our approach can also be directly applied to conditionally generate human sequences given object sequences as condition, as shown in Fig. <ref>. As our model learns a strong correspondence between object and human motion, facilitated by contact distance predictions, we are able to condition without any additional training. Populating 3D scans.Fig. <ref> shows that we can also apply our method to generate human-object interactions in static scene scans. Here, we use a scene from the ScanNet++ dataset <cit.>, with their existing semantic object segmentation. This enables potential to generate realistic human motion sequences only given a static scene environment. §.§ LimitationsWhile we have demonstrated the usefulness of joint contact prediction in 3D HOI generation, several limitations remain. For instance, our method focuses on realistic interactions with a single object. We show that this can be applied to objects in static 3D scans; however, we do not model multiple objects together, which could have the potential to model more complex long-term human behavior. Additionally, our method requires expensive 3D HOI captures for training; a weakly supervised approach leveraging further supervision from 2D action data might be able to represent more diverse scenarios. § CONCLUSIONWe propose an approach to generating realistic, dynamic human-object interactions based on contact modeling. Our diffusion model effectively learns interdependencies between human, object, and contact through cross-attention along with our contact-based object transformation weighting. Our predicted contacts further facilitate refinement using custom diffusion guidance, generating diverse, realistic interactions based on text descriptions. Since our model learns a strong correlation between human and object sequences, we can use it to conditionally generate human motion sequences from given object sequences in a zero-shot manner. Extensive experimental evaluation confirms both fidelity and diversity of our generated sequences and shows improved performance compared to related state-of-the-art baselines.§ ACKNOWLEDGEMENTSThis project is funded by the Bavarian State Ministry of Science and the Arts and coordinated by the Bavarian Research Institute for Digital Transformation (bidt), and the German Research Foundation (DFG) Grant “Learning How to Interact with Scenes through Part-Based Understanding". ieee_fullnameWe show in this appendix additional qualitative and quantitative results (Sec. <ref> and Sec. <ref>), detail our baseline evaluation protocol (Sec. <ref>), elaborate on the metrics used in the main paper (Sec. <ref>), show the architecture used in our approach (Sec. <ref>), and provide additional details regarding the data (Sec. <ref>). § ADDITIONAL QUALITATIVE RESULTS We show additional generated 3D human-object interactions of our method in Fig. <ref>, with object geometry and text condition on the left, and our generated sequence on the right.§ ADDITIONAL QUANTITATIVE RESULTS§.§ Penetration metricThe exact fidelity and diversity of our results is hard to capture with any single metric. Thus, we evaluate multiple such metrics in the main paper (R-Precision, FID, Diversity, MultiModality), and conduct a perceptual user study to verify the metrics' expressiveness.Here, we provide an additional evaluation based on an intuitive physics-based metric: The ratio of frames with human-object inter-penetrations. Due to the imperfect nature of human-object interaction capture, a non-zero amount of penetrations is expected; however, a high amount of penetration indicates low quality interactions with independently floating and often intersecting objects.We see in Tab. <ref> that our approach leads to less overall penetrations, which confirms the higher quality of our sequences, compared to the baselines. §.§ Novelty of Generated InteractionsWe perform an additional interaction novelty analysis to verify that our method does not simply retrieve memorized train sequences but is indeed able to generate novel human-object interactions. To do so, we generate ≈500 sequences from both datasets and retrieve the top-3 most similar train sequences, as measured by the l_2 distance in human body and object transformation parameter space.Fig. <ref> shows the top-3 closest train sequences, along with a histogram of l_2 distances computed on our test set of ≈500 generated sequences. In red, we mark the intra-trainset distance between samples in the train set. We observe that the distance between our generated sequences and the closest train sequence is mostly larger than the intra-train distance. Thus, our method is able to produce samples that are novel and not simply retrieved train sequences.§ BASELINE EVALUATION SETUP There is no previous approach to modeling 3D human-object interactions from text and object geometry for direct comparison. Thus, we compare to the two closest methods, and compare to them in multiple settings, for a fair comparison.The most related approach is InterDiff <cit.>. Their setting is to generate a short sequence of human-object interactions, from an observed such sequence as condition, with geometry but no text input. Their goal is to generate one, the most likely, sequence continuing the observation. We use their full approach, including the main diffusion training together with the post-processing refinement step. We compare in two different settings: First, in their native setup, running their method unchanged and modifying ours to take in geometry and past sequence observation instead of text (Motion-Cond. HOI in Tab. 1 main).Then, we modify their approach to take in geometry and text, replacing their past motion encoder with our CLIP-based text encoder (Text-Cond. HOI in Tab. 1 main). We observe that our method is able to outperform InterDiff in both scenarios, for both datasets.We additionally compare to MDM <cit.>, a recent diffusion-based state-of-the-art human motion generation approach. Their approach is based on a transformer encoder formulation, using each human body as a token in the attention. We run their method on SMPL parameters and first compare in their native setting, only predicting human motion. We compare to the human motion generated by our method which is trained to generate full human-object interactions (Text-Cond. Human Only in Tab. 1 main). We also compare to human motion sequences generated by InterDiff in this setting. We see that our method is able to outperform both baselines even in this setting, demonstrating the added benefit of learning interdependencies of human and object motion.For the comparison in our setting, we modify MDM by adding additional tokens for the objects to the attention formulation. Our approach performs more realistic and diverse sequences in both settings which better follow the text condition.§ FIDELITY AND DIVERSITY METRICS We base our fidelity and diversity metrics R-Precision, FID score, Diversity, and MultiModality on practices established for human motion generation <cit.>, with minor modifications: We use the same networks used by these previous approaches, and adapt the input dimensions to fit our feature lengths, F=79 when evaluating human body motion only, and F=79+128+9=216 (SMPL parameters, contact distances, object transformations) for full evaluation in the human-object interaction scenario.§ ARCHITECTURE DETAILS Fig. <ref> shows our detailed network architecture, including encoder, bottleneck, and decoder formulations.§ DATA DETAILS§.§ Datasets CHAIRS <cit.> captures 46 subjects as their SMPL-X <cit.> parameters using a mocap suit, in various settings interacting with a total of 81 different types of chairs and sofas, from office chairs over simple wooden chairs to more complex models like suspended seating structures. Each captured sequence consists of 6 actions and a given script; the exact separation into corresponding textual descriptions was manually annotated by the authors of this paper.In total, this yields ≈1300 sequences of human and object motion, together with a textual description. Every object geometry is provided as their canonical mesh; we additionally generate ground-truth contact and distance labels based on posed human and object meshes per-frame for each sequence. We use a random 80/10/10 split along object types, making sure that test objects are not seen during training. BEHAVE <cit.> captures 8 participants as their SMPL-H <cit.> parameters captured in a multi-Kinect setup, along with the per-frame transformations and canonical geometries of 20 different object with a wide range, including yoga mats and tables. This yields ≈130 longer sequences. We use their original train/test split. §.§ Object Geometry RepresentationWe represent object geometry as a point cloud, to be processed by a PointNet <cit.> encoder. For this, we sample N=256 points uniformly at random on the surface of an object mesh. Each object category is sampled once as a pre-processing step and kept same for training and inference. | http://arxiv.org/abs/2311.16097v1 | {
"authors": [
"Christian Diller",
"Angela Dai"
],
"categories": [
"cs.CV",
"I.2.10; I.4.8; I.5.1; I.5.4"
],
"primary_category": "cs.CV",
"published": "20231127185910",
"title": "CG-HOI: Contact-Guided 3D Human-Object Interaction Generation"
} |
[Off-shell modifications of bound nucleons and parton distributions R. Petti January 14, 2024 ==================================================================< g r a p h i c s > figureWe propose a method to stylize a 3D scene from multi-view 2D images using vectorized 3D strokes based on geometric primitives and splines. The four scenes from left to right are drawn with axis-aligned box, oriented box, ellipsoid, and cubic Bézier curve, respectively.]We present Paint Neural Stroke Field (PaintNeSF), a novel technique to generate stylized images of a 3D scene at arbitrary novel views from multi-view 2D images. Different from existing methods which apply stylization to trained neural radiance fields at the voxel level, our approach draws inspiration from image-to-painting methods, simulating the progressive painting process of human artwork with vector strokes. We develop a palette of stylized 3D strokes from basic primitives and splines, and consider the 3D scene stylization task as a multi-view reconstruction process based on these 3D stroke primitives. Instead of directly searching for the parameters of these 3D strokes, which would be too costly, we introduce a differentiable renderer that allows optimizing stroke parameters using gradient descent, and propose a training scheme to alleviate the vanishing gradient issue. The extensive evaluation demonstrates that our approach effectively synthesizes 3D scenes with significant geometric and aesthetic stylization while maintaining a consistent appearance across different views. Our method can be further integrated with style loss and image-text contrastive models to extend its applications, including color transfer and text-driven 3D scene drawing.* Corresponding author.§ INTRODUCTIONArtistic style image creation, historically a domain requiring significant skill and time, has been revolutionized by neural network-based Style Transfer techniques <cit.>. These methods usually separate, manipulate, and merge the content and style of images to create an artistic effect. Extending this to 3D scenes, however, is challenging due to complex geometries and appearance traits <cit.>, and traditional convolutional neural network (CNN) methods <cit.> are not readily adaptable to 3D spaces. Recently, 3D implicit scene representations, particularly Neural Radiance Fields (NeRF), have become popular, since they are fully differentiable and easy to optimize. Efforts to apply artistic styles to NeRF <cit.> often involve separate NeRF training and style transfer, followed by integration style transfer loss from multiple views <cit.>. The fundamental approach continues to be the independent manipulation of content and style. Nevertheless, existing NeRF stylization techniques only alter the color aspects, leaving the density-and consequently the geometry—unchanged, leading to stylizations that lack geometric changes.Style transfer methods in 2D and 3D primarily manipulate pixels or voxels to achieve artistic effects, differing from traditional art where artists use brushstrokes and a degree of randomness in brushes, materials, and colors. This traditional approach is time-consuming and skill-intensive. To automate it, research has explored image generation with vectorized strokes <cit.>, defined by position, color, size, and direction, and optimized to match target images. 3D painting tools like Google's Tilt Brush <cit.> use VR/AR to create art scenes with various brushes in virtual spaces. However, the field of automated stroke-based 3D art generation remains relatively unexplored. In this work, we present a novel technique for transforming 2D images with known poses into stylized 3D scenes using vectorized 3D strokes. Our approach recreates 3D scenes that exhibit distinct geometric and appearance styles, emulating the stroke-by-stroke painting process employed by human artists. The vector-based nature of our system enables rendering at any desired resolution. It accommodates a range of 3D stroke styles and supports more intricate stylizations and generations through the use of style and semantic losses. The core of our method is a differentiable, vectorized 3D scene representation, which diverges from NeRF by using parameterized 3D strokes to synthesize 2D images at different viewpoints. This differentiable representation allows for the direct optimization of stroke parameters via gradient descent, sidestepping the need for previous greedy-search-based or reinforcement learning-based prediction methods in stroke-based image synthesis. Furthermore, we analyze the gradient behaviors of our stroke-based 3D representation and introduce a stroke initialization and update scheme that avoids sub-optimal initialization where optimization is difficult due to local minima. Our method was evaluated using the multi-view datasets of real-world and synthetic images. Our experiments demonstrate that it effectively creates high-quality artistic 3D scene renderings by maintaining both global visual fidelity and local color accuracy while ensuring perspective consistency.Our contribution is summarized as follows:* We propose a novel method to translate multi-view 2D images into stylized 3D scenes using 3D strokes based on basic primitives and spline curves whose parameters can be learned through gradient descent.* We present a novel initialization scheme for the parameters in our stroke-based representation to address issues of flat gradients during optimization.* Our method supports various 3D stroke styles and can be applied for sophisticated geometry and appearance stylizations similar to those created by human painters.§ RELATED WORK§.§ Image PaintingImage painting, evolving over a long history, uses vectorized strokes in various colors and styles for 2D image creation. Early methods like "Paint by Numbers" <cit.> introduced brush parameters such as position, color, size, and direction, optimized through techniques like stroke zooming, noise adding, and color enhancement. Subsequently, more expressive forms emerged, employing slender spline curves <cit.> and rectangular strokes <cit.> to better capture source image details. The development of brush libraries <cit.> allowed for more diverse styles in paintings, notably in oil painting effects. With the advent of deep learning, techniques like generative neural networks <cit.> have further refined style representation. Advanced loss based on optimal transportation <cit.> and feed-forward paint transformer <cit.> are used to improve fidelity and speed up the painting process. While these 2D image synthesis techniques are relatively mature, applying styles directly to 2D images from 3D scenes lacks consistency across views, which motivates our present work on using 3D strokes.§.§ Stylization of 3D ScenesThe process of 3D scene stylization involves applying the visual style of a chosen reference to a target 3D scene while preserving its inherent 3D structure. Traditional methods <cit.> faced challenges like limited control over perceptual factors and the inability to selectively target stylization effects. The advent of Neural Radiance Fields (NeRF) has provided a more flexible representation for 3D scene stylization. Techniques like ARF <cit.> transfer artistic features from 2D style images to 3D scenes, treating stylization as an optimization problem within the NeRF framework. Chiang et al. <cit.> combines NeRF's implicit representation with a hypernetwork for style transfer. StylizedNeRF <cit.> jointly learns 2D and 3D stylization, and StyleRF <cit.> learns high-level features in 3D space for fast zero-shot style transfer. Our approach diverges from existing methods, which rely on additional style references and fundamentally contrast with the processes human artists use to create artworks. We focus on generating artistic images through 3D brush strokes, eliminating the need for style images. Our method more closely resembles image painting techniques than voxel-level manipulations. § METHODOLOGYIn this section, we introduce our 3D scene stylization framework with stroke-based representation. Our framework mainly consists of three parts: (1) a collection of 3D strokes based on primitives and spline curves, (2) differentiable rendering and composition of strokes, (3) a stroke initialization and update scheme that stabilizes the training process.§.§ Overview of Stroke FieldNeRF <cit.> represents the scene as the density σ(𝐱) ∈ℝ^+ and RGB radiance 𝐜(𝐱, 𝐝), modeled by an MLP that takes the spatial coordinates 𝐱∈ℝ^3 and view directions 𝐝∈ℝ^2 as input. Given a camera pose, a batch of rays 𝐫(t)=𝐨 + t 𝐝 is sampled and cast from the camera's center o ∈ℝ^3 along the direction 𝐝∈ℝ^3 passing through each pixel of the image. The color of a ray 𝐫 is given by differentiable volume rendering:C(𝐫) = ∫_t_n^t_fσ(𝐫(t)) 𝐜(𝐫(t), 𝐝) exp(∫_t_n^t σ(𝐫(s)) ds) dt As shown in Fig. <ref>, while NeRF models the scene at per-voxel level with an implicit field, our method represents the scene as a set of vectorized 3D strokes. Similar to NeRF, the stroke field is defined by two spatially varying functions for density σ(𝐱) ∈ℝ^+ and RGB color 𝐜(𝐱, 𝐝) and rendered into 2D images at a given camera pose, using the same differentiable volume rendering formula as in NeRF.While the stroke field shares NeRF's field definition, their core formulations are fundamentally different.Each point's density and color in the stroke field are set by 3D strokes, shaped and styled to resemble brush traces in human drawings.This is analogous to the difference between a rasterized image and a vectorized image but in 3D space. Specifically, the geometry, appearance, and opacity of 3D strokes are given by three parameters, including shape θ_𝐬∈ℝ^N × d_s, color θ_𝐜∈ℝ^N × d_c and density θ_σ∈ (ℝ^+)^N, where N is the total number of strokes in the field, d_s and d_c are the number of parameters of the specific stroke shape and color, respectively. We combine these strokes in a differentiable way to acquire the density and color field: (σ, 𝐜) = StrokeField(𝐱, 𝐝; θ_𝐬, θ_𝐜, θ_σ)§.§ 3D StrokesWe first define the shape of our 3D strokes in order to paint them into the 3D space. The shape of a 3D stroke is essentially a volume region formed by a closed two-dimensional surface. To define such volume, we use the Signed Distance Field (SDF), a scalar value function sdf(𝐱) → s ∈ℝ that gives the signed distance from a point to the closed surface to describe the stroke shape in 3D. The sub-space {𝐩∈ℝ^3 |sdf(𝐩) ≤ 0} enclosed by the zero level-set is the stroke volume. We construct two types of strokes based on basic geometric primitives and spline curves respectively. §.§.§ Basic Primitives Our first category of 3D strokes is based on common geometric primitives such as spheres, cubes, etc. For simplicity, we normalize these geometries at the origin of the unit space, apply a transformation matrix to the unit geometries, and acquire their SDFs in the scene space. The unit geometry can optionally contain a shape parameter θ_s^basic∈ℝ^d_basic describing its transformation-agnostic shape feature, where d_basic is the number of basic shape parameters. The SDF of unit primitive is defined as sdf_unit: (𝐩̂, θ_s^basic) → s ∈ℝ, where 𝐩̂ is a 3D point in the unit space. We list the common unit geometric primitives in Tab. <ref>.We apply a transformation matrix 𝐓∈ℝ^4 × 4 to transform primitives from the unit coordinates 𝐩̂ to the shared scene coordinates 𝐩 = 𝐓𝐩̂. The parameters of transformation are composed of a translation 𝐭∈ℝ^3, a rotation described by Euler angle 𝐫∈ℝ^3, and a uniform or anisotropic scale 𝐬∈ℝ^3. To acquire the SDF of primitives in the scene space, we inversely transform the position to the unit space and query the unit SDF: sdf(𝐩; θ_s) = sdf_unit(𝐓(𝐭, 𝐫, 𝐬)^-1𝐩,θ_s^basic),where θ_s={θ_s^basic, 𝐭, 𝐫, 𝐬} is the final shape parameters for primitive-based 3D strokes. In practice, we may use a subset of translation, rotation, and scale to combine different styles of primitive strokes. We list all the primitive-based 3D strokes in the supplementary. More primitive can be easily added given a defined unit SDF. §.§.§ Spline Curves In addition to basic primitives, we use volumetric 3D curves with a given radius to simulate the trace of strokes in human paintings. The curves are defined by parametric 3D splines in the scene space: C: (t, θ_s^curve) →𝐱∈ℝ^3, where t ∈ [0, 1] is the interpolation parameter, and θ_s^curve∈ℝ^d_c is the parameter of the 3D spline. To simulate the brushstroke effects, we define two different radii r_a, r_b ∈ℝ^+ at the two endpoints of the spline curve respectively, and interpolate the radius at any position on the curve smoothly based on t as r(t; r_a, r_b) = r_a (1-t) + r_b t.We utilize common polynomial splines, including quadratic and cubic Bézier and Catmull-Rom splines, to define 3D curves. To compute the SDF of a 3D curve, we need to solve for the value of t corresponding to the nearest point on the spline curve for any point 𝐩∈ℝ^3 in space. While there exists an analytic solution for quadratic splines, it is difficult to directly solve for the nearest point's t value for cubic or more complex splines in a differentiable way. Therefore, we use a general approximation method to compute the SDF: we uniformly sample K+1 positions on the curve to get K line segments, and calculate the distance from these segments to the query point respectively, thereby obtaining an approximate nearest point. This algorithm can be applied to any parametric curve spline and allows simple control over the trade-off between computational complexity and accuracy. With the found t value of the nearest point on the 3D spline to query position 𝐩 as t^*, the SDF of the 3D curve is defined assdf(𝐩; θ_s) = ‖𝐩 - C(t^*,θ_s^curve) ‖_2 - r(t^*; r_a, r_b)where θ_s={θ_s^curve, r_a, r_b} is the final shape parameters for spline-based 3D strokes. We leave the details of the nearest point finding algorithm in the supplementary. §.§ Differentiable Rendering of 3D StrokesWith the SDF of 3D strokes defined in Sec. <ref>, we now convert it into the density field σ(𝐱) and color field 𝐜(𝐱, 𝐝) for differentiable rendering.Deriving Density Field from SDF. Theoretically, we consider the inner region of a 3D stroke with an SDF value less than or equal to zero. Therefore, we can define an indicator region function α(𝐱) ∈ [0,1] based on whether a point is inside the SDF: α(𝐱) = {1, sdf(𝐱) ≤ 00,otherwise .Assuming uniform density inside one stroke, we define the density field of each 3D stroke as σ(𝐱) = θ_σα(𝐱)where θ_σ∈ℝ^+ is the density parameter of a 3D stroke. However, due to the discontinuity of the indicator function, the gradient of the density field w.r.t. the shape parameters of the SDF is zero, thus hindering the use of losses to optimize the 3D stroke. To render the 3D stroke in a differentiable way, we aim to derive a smooth approximation of the region function to gain non-zero gradients regarding the shape parameters.We use the Laplace cumulative distribution function (CDF) to approximate the discrete region function, similar to VolSDF <cit.>:α(𝐱) = {1 - 1/2exp(sdf(𝐱) / δ), sdf(𝐱) ≤ 0 1/2exp(-sdf(𝐱) / δ),otherwise .where δ controls the width of the smooth transitional interval, as shown in Fig. <ref>. This definition ensures that all points in space have non-zero derivatives w.r.t. the shape parameters, although the gradient would be too small to achieve meaningful optimization excluding the near region besides the zero level-set boundary of the SDF. This approximation approaches the discrete indicator function when δ→ 0, while larger δ allows smooth gradients flow in a larger area near the boundary, at the cost of making the shape boundary more blurred.Adaptive Choice of δ based on Cone Tracing. Since the pixel corresponding to each ray has a specific planner size during rendering, the pinhole camera model casts a frustum with non-zero volume instead of a ray. The frustum far away from the camera will cover a large region of the scene, while the near frustum will only affect a small region. Using a uniform δ across the entire scene to render the 3D strokes will make the near objects over-blurry, and far objects lack enough gradients to optimize the stroke shape. To solve this issue, we adopt a similar cone tracing strategy as in MipNeRF <cit.>, and adjust the δ value of the region function adaptively based on the cone size. For simplicity, we use isotropic spheres instead of anisotropic multivariate Gaussians to approximate the volume of the cone. Specifically, assuming the radius of each pixel is ṙ, we calculate the radius of a sphere at sample points as r = ṙ t f^-1, where t is the un-normalized ray distance, and f is the focal length. We then compute the adaptive δ as δ = (k_δ r)^-1, where k_δ is a hyper-parameter controlling how much to “dilate" the boundary. Moreover, as the scale of δ is defined in scene space, for basic primitives with a transformation, we further adjust δ based on the scaling factor of the inverse transform.Color Field. We mainly use constant color for each stroke, which is irrelevant to the position and view direction and can be expressed as 𝐜(𝐱, 𝐝) = θ_𝐜, where θ_𝐜∈ℝ^3 is the RGB color parameter. More diverse visual effects can be achieved by spatially varying color fields and joint modification of the density and color. The color field can be readily expanded to support view-dependent effects by replacing the RGB color with spherical harmonics, however, we have found that the view dependency does not contribute much to the visual effect of stroke-based scenes in practice. §.§ Composition of 3D StrokesGiven the density and color field of individual 3D strokes, we now compose them into a stroke field for rendering the full 3D scene. In the procedure of image painting, different strokes are painted onto the canvas sequentially, and new strokes are overlayed on the old strokes. This behavior is simulated using the alpha composition. Similarly, we use an “overlay" method to compose multiple 3D strokes.Suppose we have the density fields {σ_i(𝐱)} and color fields {𝐜_i(𝐱)} of N strokes, where i ∈{1,2,⋯,N}. We use the region function defined in Sec. <ref> as the blending weight for overlay. Given {α_i(𝐱)} as the region function of each 3D stroke, the density and color of the final stroke field are expressed as σ(𝐱)= ∑_i^n σ_i(𝐱) α_i(𝐱) T_i(𝐱),𝐜(𝐱)= ∑_i^n 𝐜_i(𝐱) α_i(𝐱) T_i(𝐱)/1 - T_n(𝐱), where T_i(𝐱)=∏_j^i (1 - α_j(𝐱)).Note that we normalize 𝐜(𝐱) using the total weight of all region functions, making the final color unaffected by the number of strokes.The “overlay" composition is sensitive to the painting order of multiple strokes, i.e., the 3D strokes painted later will have larger blending weights than the previously painted strokes. Normally this is the behavior we desire, however, we sometimes would like the final outcome irrelevant to the painting order. The simplest way is choosing the 3D stroke with the maximum α(𝐱) as: σ(𝐱)=σ_i(𝐱),𝐜(𝐱)=𝐜_i(𝐱), where i=max_j{α_j(𝐱)}. This “max" composition will only attribute the gradient of the density field to the nearest 3D stroke. A smoother way is to use Softmax for computing weights as: σ(𝐱)=σ_i(𝐱) ω_i(𝐱),𝐜(𝐱)=𝐜_i(𝐱) ω_i(𝐱), where ω_i(𝐱)=exp(α_i(𝐱) / τ)/∑_j^n exp(α_j(𝐱) / τ), and τ is a hyper-parameter that controls the blending smoothness. We analyze the effect of different compositions in Sec. <ref>.§.§ Training StrategyNow we have completed our rendering framework for 3D strokes. We can render 2D images with a given camera pose from parameters of N strokes {θ_s,θ_c,θ_σ}_i,i ∈{1,2,⋯,N}, and use gradient descent to optimize these 3D stroke parameters. However, in the previous analysis, we mentioned that the magnitude of gradients vanishes for faraway points from the existing boundary of strokes, thus leaving shape parameters unable to gain enough optimization. To this end, we propose a training scheme for better stroke initialization and update.Error Field. When adding 3D strokes to an existing scene, we wish to initialize the new stroke at the location where it is most needed. Therefore, during the training of the stroke field, we additionally train an error field e(𝐱) ∈ℝ^+ to describe the “error rate" at each position of the reconstructed scene, with the region of highest error considered to be the locations most in need of additional 3D strokes. The error value E(𝐫) along each ray 𝐫 is obtained by the modified volume rendering formula:E(𝐫)=∫_t_n^t_f e(𝐫(t)) exp(∫_t_n^t e(𝐫(s)) ds) dt.We define the error as the squared difference in color between the stroke field rendering result and the ground truth. Moreover, we prefer the error field to conservatively estimate errors, therefore we use a coefficient k > 1 to amplify the loss where the error is underestimated. Assuming the rendering color of stroke field is C(𝐫), and the GT color is C_gt(𝐫), then the training loss for the error field isℒ_err=|d| k^max(-sgn(d),0) ,d=E(𝐫) - ‖ C(𝐫)-C_gt(𝐫) ‖_2,where sgn(·) is the sign function. At the same time, for views that are less trained, we apply a regularization loss to the error values. For N total sampling points {𝐱_i} in a batch, the regularization loss is ℒ_err reg=∑_i^N e(𝐱_i).Stroke Initialization and Update. When training a scene with N strokes, we randomly initialize N_start strokes and then incrementally add the remaining strokes step by step to the scene. We also decay the initial size of strokes based on the current number of strokes, i.e., we progressively use smaller strokes as the number of existing strokes increases. When initializing the i-th stroke, we uniformly sample M coordinates {𝐩_i} within the bounding box of the scene, calculate the error values {e_i} in the error field, and select the sampling point with the highest error as the initial position for the 3D stroke. For primitives, we set their translation vector to this initial position. For spline curves, we set their control points to this initial position plus a random offset vector sampled within the current stroke size volume. This initialization scheme, guided by the error field, effectively mitigates optimization issues caused by flat gradients. Additionally, we reinitialize strokes with near-zero density by resampling the current error field. This simple technique recycles some of the brush strokes that might have been trapped in local minima due to poor initialization.Training Objective. Similar to NeRF, we use multi-view images as the primary supervision. We randomly sample rays from images in the training set and optimize the error between the rendering result of each ray and the ground truth. We use the Charbonnier distance for color loss:ℒ_color= √(‖ C(𝐫) - C_gt(𝐫) ‖_2^2+ϵ)For scenes with a ground truth mask, we also impose mask supervision based on the ray's opacity O(𝐫) = exp(∫_t_n^t_fσ(𝐫(t)) dt) and the ground truth M(𝐫) ∈{0,1}, ℒ_mask = √(‖ O(𝐫) - M_gt(𝐫) ‖_2^2+ϵ)Additionally, we regularize the density parameter of N strokes as ℒ_den reg= ∑_i^N |θ_σ^i|. Combining the above losses, our total loss is given by: ℒ =λ_colorℒ_color+λ_maskℒ_mask+λ_den regℒ_den reg+λ_errℒ_err+λ_err regℒ_err reg§ EXPERIMENTSWe test our method on both synthetic objects from Blender <cit.> and face-forwarding scenes from LLFF <cit.>. Each dataset comprises dozens to hundreds of multi-view images with given camera poses. We show our reconstruction results based on various types of vectorized 3D strokes, compare the stylization results of our method with other image-to-painting methods, and conduct an ablation study.§.§ Stroke-based scene reconstructionFig. <ref> shows a sample of the stylized 3D synthetic scenes reconstructed by our method with several types of 3D strokes. Our method is able to synthesize 3D scenes based on vectorized 3D strokes, providing strong geometry and appearance stylization while maintaining the abstracted shapes and colors of the original scenes. Fig. <ref> compares reconstruction results of different types of 3D strokes on the same scene. Cubic Bézier and Ellipsoid generally have the best reconstruction fidelity compared to other types of strokes. We list the quantitative metrics of different strokes in the supplementary material. Fig. <ref> demonstrates the abstract-to-detail painting results by varying the number of total 3D strokes that are used to reconstruct the scene. We can observe that the lower number of strokes approximates the original scene with a strong shape-related geometric style, while the higher number of strokes leads to reconstruction results with higher fidelity. §.§ Comparison with other methodsOur method lies in a different category of stylization compared to the common NeRF-based stylization works, which usually take a reference image as input, and transfer the textural style onto the appearance of a trained radiance field. Our method does not require extra reference images, instead, the style of our method is intrinsic and contained in the specification of various 3D strokes.We compare our method with 2D image-to-painting methods. Fig. <ref> shows the vectorized reconstruction results of our method and other 2D vectorized image representation methods, including diffvg <cit.> and Stylized Neural Painting <cit.>. We use multi-view images to train our 3D stroke representation, while for 2D methods we select images of the specific viewpoints and apply the vectorization tracing. It can be observed that all these vectorized methods are able to reconstruct original images with obvious stroke styles. Nevertheless, our method can recover more details and strictly maintain the multi-view consistency across different views, which is guaranteed by its 3D representation. The results are best viewed in our supplementary video.§.§ Ablation studies Use of adaptive δ in region function.We investigate the effects of employing an adaptive δ in differentiable rendering. The results in Tab. <ref> show that omitting adaptive δ leads to reduced performance in novel view synthesis. Additionally, Fig. <ref> demonstrates that constant δ values, whether low or high, result in sub-optimal or overly blurry scenes. Use of error field.In Sec. <ref>, we note that loss gradients are concentrated near the 3D stroke boundary, creating many local minima that heavily influence optimization based on shape initialization. Tab. <ref> demonstrates that using the error field for stroke shape initialization significantly improves scene reconstruction fidelity. Choice of composition function.We compare different composition approaches in Sec. <ref>. We can observe in Fig. <ref> that the `overlay' composition leads to the best reconstruction quality, while the `softmax' composition also reconstructs the scene well. The latter can be chosen if order invariant painting is required.§.§ ApplicationsWe explore various applications based on our vectorized 3D scene representations, including color transfer and text-driven zero-shot scene drawing.Color Transfer.As the 3D stroke representation has separate shape and appearance parameters, we can fix the geometry and only fine-tune the color parameter to achieve color transfer effects. We adopt the style loss in common style transfer works, which matches the gram matrix of the feature map outputs from the 4-th and 9-th layers of a pre-trained VGG network. The reference style images and color transfer results are demonstrated in Fig. <ref>.Text-driven Scene Drawing.We also explore using the vectorized 3D stroke representation to achieve scene creation tasks under a text-guided zero-shot generation framework. We use CLIP <cit.>, a vision-language model that embeds the 2D images and text prompts into the same embedding space. Following the generative setup in DreamField <cit.>, we sample camera poses following a circular path around the scene origin and render a large image patch, and then we optimize the distance between CLIP embeddings of the synthesized image and the text prompt. The generated 3D drawings of different objects are shown in Fig. <ref>. We leave more details of the training setup in the supplementary material. § DISCUSSIONWe present a novel method to stylize a 3D scene from multi-view 2D images. Different from NeRF-based representations, our method represents the scene as vectorized 3D strokes, mimicking human painting during scene reconstruction process. We demonstrate that this stroke-based representation can successfully stylize 3D scenes with large geometry and appearance transformations, which was not achieved with previous NeRF stylization approaches. Limitations and future works.Our method uses a stroke setting that demands manual effort to design stroke shapes and appearances. The 3D strokes can be further learned with a generative framework to create a variety of stroke types, like ink and oil brushes, with generated and more detailed SDFs. Moreover, our method may require numerous strokes to represent very complex scenes, partly due to the existence of many local minima during the optimization. Incorporating a globally aware loss, like the optimal transport loss in <cit.> into 3D space, may enhance the convergence efficiency of our method, which we leave as future work.ieeenat_fullname This supplementary document provides additional details of the 3D strokes in Sec. <ref>, implementations in Sec.<ref>, additional comparisons of various 3D strokes in Sec. <ref>, and application training setups in Sec. <ref>. Please also watch our accompanying video for an animated visualization of stylization results.§ DETAILS OF 3D STROKES §.§ Transformation of Basic Primitives In Sec. <ref> of the main paper, we use a transformation matrix to map the coordinates in the shared scene space into the canonical space of each unit signed distance field. Here we provide the construction details of the transformation matrix. Given a translation vector 𝐭 = (t_x, t_y, t_z), an Euler angle rotation vector 𝐫 = (r_x, r_y, r_z), and a scale vector 𝐬 = (s_x, s_y, s_z), we first construct the matrices for each term respectively, then combine them in the order of scale, rotation, and translation to get the final transformation matrix M:T= [ 1 0 0 t_x; 0 1 0 t_y; 0 0 1 t_z; 0 0 0 1 ], R_x= [1000;0cos r_x -sin r_x0;0sin r_xcos r_x0;0001 ], R_y= [cos r_y0sin r_y0;0100; -sin r_y0cos r_y0;0001 ], R_z= [cos r_z -sin r_z00;sin r_zcos r_z00;0010;0001 ], S= [ s_x 0 0 0; 0 s_y 0 0; 0 0 s_z 0; 0 0 0 1 ], M= T R_z R_y R_x SIn cases where composited primitives do not utilize the full set of transformation components—translation, rotation, and scale—we omit the respective term in the M formula. For uniform scaling, represented by a scalar scaling factor s, we set s_x, s_y, and s_z all equal to s. §.§ Complete List of 3D StrokesUtilizing various combinations of basic geometric shapes in unit space, along with transformations including translation, rotation, and scale, enables the creation of a diverse palette of 3D strokes. These strokes exhibit distinct geometric and aesthetic stylization. The full assortment of these 3D strokes is detailed in Tab. <ref>. §.§ Spline Curves §.§.§ Polynomial splines In Sec. <ref> of the main paper, we use three types of different polynomial curves that are commonly used in computer graphics. Specifically, we use the quadratic Bézier, cubic Bézier, and Catmull Rom spline, respectively. We provide the concrete definition of these curves below. All points defined here are vectors in the 3D scene space. Quadratic Bézier Spline. A quadratic Bézier spline is defined by three control points, the start point 𝐏_0, the end point 𝐏_2, and middle point 𝐏_1 that is also tangent to the 𝐏_0 and 𝐏_2. Note that the curve only blends toward but does not pass the middle point 𝐏_1. The parametric form is given by:𝐂(t;𝐏_0,𝐏_1,𝐏_2) = (1-t)^2𝐏_0 + 2(1-t)t𝐏_1 + t^2𝐏_2Cubic Bézier Spline. A cubic Bézier spline introduces an additional control point compared with the quadratic spline. This spline is defined by four points: the start point 𝐏_0, two control points 𝐏_1 and 𝐏_2, and the end point 𝐏_3. The curve starts at 𝐏_0 and ends at 𝐏_3, with 𝐏_1 and 𝐏_2 influencing its shape. The parametric equation of a cubic Bézier spline is:𝐂(t;𝐏_0,𝐏_1,𝐏_2, 𝐏_3)= (1-t)^3𝐏_0 + 3(1-t)^2t𝐏_1 + 3(1-t)t^2𝐏_2 + t^3𝐏_3Catmull Rom Spline. The Catmull Rom spline is another form of cubic spline, notable for its ability tointerpolate its control points. This spline is defined by a series of points, with the curve passing through each of these points except the first and last. One feature of the Catmull Rom spline is that the tangent at each point is determined by the line connecting the previous and next points, ensuring a smooth transition. In our implementation, we specifically use the centripetal Catmull-Rom spline, a variant of the standard Catmull-Rom spline. This type of spline is particularly advantageous for avoiding the issue of self-intersecting loops in the curve, which are common in the uniform and chordal Catmull-Rom splines. The parametric form of the Catmull Rom spline, for a segment between 𝐏_1 and 𝐏_2, with 𝐏_0 and 𝐏_3 influencing the shape, is defined as:𝐂(t;𝐏_0,𝐏_1,𝐏_2, 𝐏_3)= 1/2 [ (2𝐏_1) + (-𝐏_0 + 𝐏_2)t + (2𝐏_0 - 5𝐏_1 + 4𝐏_2 - 𝐏_3)t^2 + (-𝐏_0 + 3𝐏_1 - 3𝐏_2 + 𝐏_3)t^3]§.§.§ Nearest Point Finding As mentioned in Sec. <ref> of the main paper, we need to locate the nearest point on the spline in order to compute the SDF of the spline curve. This involves solving the following equation that finds the t value with the minimized distance to the query point 𝐩∈ℝ^3 in a differential way:t^* = tmin‖𝐂(t,θ_s^curve) - 𝐩‖_2, s.t. 0 ≤ t ≤ 1,where θ_s^curve denotes the parameters of the spline curve. An analytical solution might exist for some specific formulations of the splines. However, for more versatility, we use a general approximation solution that can be adapted to any parametric spline curve in the main paper. K + 1 samples are uniformly selected on the curve to form K line segments, and the distance of each line segment to the query point is calculated to find t^*. Assuming the line segment is given as L(t; 𝐀, 𝐁) = (1 - t)𝐀 + t𝐁, t ∈ [0, 1], where 𝐀 and 𝐁 are the two endpoints of the line segment, the distance from point 𝐩 to this line segment L can be calculated as:d_L(𝐩, 𝐀, 𝐁) = ‖𝐩 - L(t'; 𝐀, 𝐁) ‖_2where t' is the t value that gives the nearest point on the line segment:t'(𝐩, 𝐀, 𝐁) = min(max((𝐩 - 𝐀) · (𝐁 - 𝐀)/(𝐁 - 𝐀) · (𝐁 - 𝐀), 0), 1)We then compute t^* using Algorithm <ref>. The computation complexity can be easily controlled by adjusting the number of K, where a higher K leads to a more accurate approximation but at the cost of higher computation.§ IMPLEMENTATION DETAILSWe implement the stroke field using Pytorch <cit.> framework and implement the strokes using fused CUDA kernels to accelerate training and reduce GPU memory usage. We transform sampled coordinates to the canonical volume according to the scene bounding box and use the normalized scene coordinates as inputs to the stroke field. Like ZipNeRF <cit.>, we use a proposal network based on hash grid representation <cit.> to facilitate ray sampling. Specifically, for each ray of a pixel, we first sample 32 points using the proposal MLP to obtain sampling weights. We resample 32 points and compute the stroke field's density and color on each point, and use the same volumetric rendering formula in NeRF to acquire the final pixel color.We train 15k steps using 500 strokes for scenes with a single object, and train 25k steps using 1000 strokes for face-forwarding scenes. We employ the AdamW <cit.> optimizer with betas (0.9, 0.99), setting the learning rate to 0.01 and exponentially decays to 0.0003. We start with k_δ = 7 and gradually decay it to k_δ = 1 during training. Additionally, as brushes are progressively added to the scene, we start with 25% of the sampling points and gradually use all the sampling points when training reaches 80%. We set λ_color = 1, λ_mask = 0.02, λ_den reg = 0.0001, λ_err = 0.1, and λ_err reg = 0.001 in our training setup. § QUANTITATIVE COMPARISON OF STROKESIn Sec. <ref> of the main paper, we conduct a qualitative comparison between the visual effects of several selected 3D strokes. Additionally, this section includes a comprehensive quantitative analysis of all 3D strokes, as detailed in Tab. <ref>. The metrics are measured using 500 strokes on object scenes and 1000 strokes on face-forwarding scenes. As shown in the table, the ellipsoid stroke typically demonstrates superior fidelity in scene reconstruction, followed closely by the cubic Bézier curve.§ TRAINING SETUP OF APPLICATIONS §.§ Color Transfer In Sec. <ref> of the main paper, we transfer the color distribution from a reference style image to a trained stroke-based 3D scene. We adopt perceptual style loss, which extracts the gram matrix <cit.> at specific layers of a pre-trained VGG16 network <cit.>, and computer the difference between the rendered RGB image and the target style image. Since computing style loss requires an image rather than individual pixels as input, we randomly render 32x32 chunks of original images under the given camera poses. We add this style loss with a weight λ_style=0.25 to the total loss and fine-tune the color parameters of trained strokes for 5k iterations.§.§ Text-driven scene drawing In Sec. <ref> of the main paper, we use the vision-language model CLIP <cit.> (ViT-B/32) to achieve scene drawing based on a given text prompt. Specifically, we minimize the loss between the text embedding of the given prompt and the image embedding of the rendered RGB image. When rendering the images, we sample camera poses in a circular path looking at the scene's origin with azimuth angle in [0, 360] and elevation angle in [40, 105], and render an image chunk of size 128 × 128. We consider the azimuth angle starting at zero as the front view and adjust the CLIP guidance scale for larger azimuth angles accordingly. This adjustment results in generation outcomes that are more coherent with the viewpoint.For text guidance, we use the template “", whereis substituted with the specific object description we aim to generate.Additionally, we discovered that incorporating a silhouette loss alongside the RGB loss significantly enhances the quality of the generated geometric shapes. This is achieved by generating another image embedding from the rendered opacity and encouraging lower loss between this silhouette image embedding with the text embedding. | http://arxiv.org/abs/2311.15637v1 | {
"authors": [
"Hao-Bin Duan",
"Miao Wang",
"Yan-Xun Li",
"Yong-Liang Yang"
],
"categories": [
"cs.CV",
"cs.GR"
],
"primary_category": "cs.CV",
"published": "20231127090221",
"title": "PaintNeSF: Artistic Creation of Stylized Scenes with Vectorized 3D Strokes"
} |
Properties of the Magellanic Corona Model for the formation of the Magellanic Stream [ Received xxxx; accepted xxxx ==================================================================================== Video-based large language models (Video-LLMs) have been recently introduced, targeting both fundamental improvements in perception and comprehension, and a diverse range of user inquiries. In pursuit of the ultimate goal of achieving artificial general intelligence, a truly intelligent Video-LLM model should not only see and understand the surroundings, but also possess human-level commonsense, and make well-informed decisions for the users.To guide the development of such a model, the establishment of a robust and comprehensive evaluation system becomes crucial. To this end, this paper proposes Video-Bench, a new comprehensive benchmark along with a toolkit specifically designed for evaluating Video-LLMs. The benchmark comprises 10 meticulously crafted tasks, evaluating the capabilities of Video-LLMs across three distinct levels: Video-exclusive Understanding, Prior Knowledge-based Question-Answering, and Comprehension and Decision-making. In addition, we introduce an automatic toolkit tailored to process model outputs for various tasks, facilitating the calculation of metrics and generating convenient final scores. We evaluate 8 representative Video-LLMs using Video-Bench.The findings reveal that current Video-LLMs still fall considerably short of achieving human-like comprehension and analysis of real-world videos, offering valuable insights for future research directions. The benchmark and toolkit are available at: <https://github.com/PKU-YuanGroup/Video-Bench>. § INTRODUCTIONLarge language models (LLMs)<cit.> have demonstrated strong capabilities in handling natural language processing (NLP) tasks, including comprehension, composition and reasoning, and achieved remarkable advancements on NLP benchmarks<cit.>.This success has also inspired studies on Video-LLMs <cit.>, where models process video inputs with textual prompts and generate corresponding answers, shedding light on the future format of artificial general intelligence (AGI) for video understanding.With the ultimate goal of achieving artificial general intelligence in mind, we assert that a truly intelligent video-language model should at least exhibit three distinct human-like capabilities: (i) Video-exclusive Understanding,i.e., performing well for questions whose answer can be extracted from the video itself; (ii) Prior Knowledge-based Question-Answering, i.e., answer questions that require the prior knowledge beyond the video, such as commentary on NBA games or providing background information on specific music videos;(iii) Comprehension and Decision-making, enabling a comprehensive understanding of scenarios, along with the ability to make predictions and informed decisions. Example applications encompass 3D scene understanding and decision-making for autonomous driving. To gradually approach this goal, the establishment of an evaluation benchmark is indispensable for precisely measuring and steering the development progress. However, we find that existing benchmarks fall short of serving this purpose comprehensively. For instance, MMBench <cit.> and LVLM-eHub <cit.> are concentrated on image understanding, ignoring the video understanding ability. SEED-Bench <cit.> includes several video tasks but is limited to temporal understanding, thus only covering the first level. To this end, we propose a new large-scale benchmark along with a toolkit,referred to as “Video-Bench", to furnish a thorough evaluation of Video-LLMs. The composition of Video-Bench is depicted in Fig. <ref>.In detail, aligning with our motivation, our Video-Bench encompasses tasks categorized into three distinct levels of capability: (i) For Video-exclusive Understanding, we begin by randomly selecting parts of traditional QA pairs <cit.>, and proposing more challenging tasks to assess both temporal and contextual aspects of videos. Tasks include video summarization <cit.>, abnormal detection <cit.>, and crowd counting <cit.>; (ii) For Prior Knowledge-based Question-Answering, we evaluate the capability of model in understanding TV dramas <cit.>, appreciating music videos, and providing information about players and games in NBA videos. (iii) For Comprehension and Decision-making, we employ two classical tasks: 3D indoor scene understanding <cit.> and auto-driving decision-making to assess the comprehension and decision-making abilities of models. To streamline the evaluation process, we include another crucial component, i.e., the evaluation toolkit, along with the benchmarks. The toolkit automatically maps the long text outputs of Video-LLMs to corresponding answers with probability selection <cit.> or LLM-based semantic understanding <cit.>. Subsequently, it calculates accuracy for each question and generates a final score, enhancing the efficiency of the evaluation workflow. We evaluate eight representative Video-LLMs on Video-Bench: VideoChat <cit.>, Video-ChatGPT <cit.>, Otter <cit.>, Valley <cit.>, PandaGPT <cit.>, mPLUG-Owl <cit.>, Video-LLaMA <cit.>, and Chat-UniVi <cit.> with verified open-source model weights. The evaluation results reveal several interesting findings: (i) Most recent models can summarize the main content of videos but lack the capacity to detect details and temporal information. (ii) Due to the absence of domain-specific prior knowledge in the training data, these models encounter challenges in accurately comprehending and responding to queries within a particular domain. (iii)Due to constraints in multimodal information extraction and the use of a weakened LLM backend (either 7B or 13B), the majority of tested models exhibit limited proficiency in comprehending and decision-making within complex scenarios. Our contributions can be summarized as follows:* We introduce Video-Bench, the first comprehensive evaluation benchmark for Video-LLMs, featuring a three-level ability assessment that systematically evaluates models in video-exclusive understanding, prior knowledge incorporation, and video-based decision-making abilities.* We provide a user-friendly evaluation toolkit. Accompanied by our datasets and QA pairs, the toolkit can streamline the performance assessment of Video-LLMs.* We conduct extensive experiments to evaluate prominent Video-LLMs, summarizing their behaviors, analyzing main causes for observed limitations, and proposing future directions for improvement. § RELATED WORKVideo-LLMs. Extending Image-based Large Language Models (Image-LLMs) to the video modality introduces a complex challenge, necessitating the incorporation of temporal dimensions to interpret diverse frame information. Beyond visual content, the integration of audio, subtitles, and other modalities becomes crucial for a comprehensive understanding of video semantics. In response to this challenge, a series Video-LLMs have emerged, building upon open-source LLMs <cit.> or Image-LLMs <cit.>.As outlined in Table <ref>, VideoChat <cit.> utilizes the Q-Former to map visual representations to Vicuna <cit.>, implementing a two-stage training process. Video-ChatGPT <cit.> and Valley <cit.> originate from the LLaVA <cit.> framework and introduce average pooling to enhance temporal sequence perception. Otter <cit.> proposes the MIMIC-IT dataset and fine-tunes Openflamingo <cit.> on their dataset. PandaGPT <cit.> employs the ImageBind <cit.> as its backend for video comprehension. mPLUG-Owl <cit.> introduces an abstractor module to align image and text. Video-LLaMA <cit.> incorporates a frame embedding layer and ImageBind to inject temporal and audio information into the LLM backend, while Chat-UniVi <cit.> merges visual tokens with similar semantic meanings using a clustering strategy. Existing Video-LLMs vary in their training strategies and data scales, with only a subset addressing challenges related to temporal dimensions and audio modalities. Video Datasets. Deep learning for video analysis relies on diverse datasets tailored to specific tasks. A notable task is human action recognition, featuring action classification datasets such as UCF-101 <cit.>, HMDB51 <cit.>, and Kinetics <cit.>, and action localization datasets like AVA <cit.> and Fineaction <cit.>. Tasks involving anomaly detection in surveillance videos are addressed by datasets like UCSD-anomaly <cit.> and UCF-crime <cit.>. Object identification and tracking in videos encompass multiple object tracking (MOT)<cit.>, video object segmentation (DAVIS)<cit.>, and video instance segmentation (Youtube-VIS) <cit.>. For multimodal tasks, video captioning datasets such as MSVD <cit.>, MSRVTT <cit.>, and Activitynet <cit.> exist, along with their corresponding QA datasets <cit.>. Scenario-specific datasets like MovieQA <cit.> and TVQA <cit.> also contribute to the diversity of available datasets. However, these datasets often focus on specific tasks and lack the complexity needed to measure the comprehensive abilities of Video-LLMs effectively.Vision Language Evaluation Benchmarks. To evaluate the capabilities of LLMs, various benchmarks have been introduced, including AI2 Reasoning <cit.>, HellaSwag <cit.>, MMLU <cit.>, and TruthfulQA <cit.>. These benchmarks assess reasoning, scientific knowledge, fact retention, and the ability to generate misinformation. In the realm of multimodal LLMs, corresponding benchmarks have also emerged. MMBench <cit.> constructs a broad spectrum of evaluation for Vision-LLMs, and converts free-form predictions into pre-defined choices, enhancing the robustness of the evaluation process. SEED-Bench <cit.> introduces a series of temporal understanding tasks and establishes an automatic filtering and manual verification pipeline to ensure the quality and relevance of the evaluations. LVLM-eHub <cit.> presents an online arena platform for user-level evaluation, providing a more realistic assessment of model performance in real-world applications. ELEVATER <cit.> focuses on evaluating the transferability of language-augmented visual models across multiple tasks. However, the aforementioned vision-language benchmarks are not tailored specifically for videos. Drawing inspiration from HELM <cit.>, we introduce Video-Bench, specifically designed to measure human-like abilities of Video-LLMs across various capabilities and scenarios.§ VIDEO-BENCHIn Fig.<ref>, we show the overall structure of Video-Benchand the corresponding average results for existing Video-LLMs.§.§ Video-exclusive Summarization As illustrated in Fig. <ref> (A), we aim to measure the capacity of Video-LLMs to comprehend and summarize information from video itself, encompassing objects, actions, attributes, and their temporal connections. These tasks are video-exclusive, requiring no external prior knowledge or complex logic inference.Basic Understanding. This task primarily evaluates the basic video recognition ability, such as responding to queries related to human actions in Activitynet-QA <cit.>, providing answers related to objects, attributes, and actions corresponding to videos in MSVD-QA <cit.> and MSRVTT-QA <cit.>, and comprehending GIFs in TGIF-QA <cit.>.Summarization. This task assesses the summarization ability of Video-LLMs when dealing with longer videos. Using the YouCook2 dataset <cit.> with rich annotations and extended video duration, we generate a series of QA pairs to evaluate whether the model can comprehend cooking information presented in the videos and audios, and then provide accurate feedback about the correct procedure.Abnormal Detection. This task evaluates the ability to review videos and identify anomalies. Leveraging the UCF-Crime dataset <cit.>, a collection of surveillance videos annotated with the type and timestamp of anomalies, we construct questions to assess the temporal comprehensive ability of Video-LLMs.Crowd Counting. This task primarily evaluates the ability to localize and count dense objects. Utilizing the MOT dataset <cit.>, which annotates all pedestrians, vehicles, and other targets in street or mall images, we test whether Video-LLMs can identify different pedestrians in different frames and provide the correct number of people. §.§ Prior Knowledge-based Question-answering ChatGPT and LLaMA exhibit strong capability in answering questions and giving suggestions across various domains due to the extensive prior knowledge acquired during pre-training. This prompts us to investigate whether Video-LLMs possess similar abilities. As depicted in Fig. <ref> (B), our goal is to assess the capability of Video-LLMs in addressing questions that require prior knowledge, akin to human beings. Examples include identifying actors in a movie or discerning the music style of a particular song.TV-QA. Television programs, as prevalent sources of entertainment videos, integrate multiple modalities, including video, audio, and subtitles, to convey information. Utilizing the TVQA dataset <cit.>, we transform image formats into videos, and incorporate audio and subtitles. This dataset allows us to evaluate the ability of Video-LLMs to integrate prior knowledge and information from video, audio, and text to answer questions related to TV content.MV-QA. Music videos, characterized by the synchronization of visual elements with music, pose a unique challenge due to their reliance on prior knowledge. Answering questions about these videos requires familiarity with the song, recognition of artists, and potentially basic music theory. In the absence of relevant existing datasets, we search for top music videos on YouTube and construct corresponding QA pairs based on authoritative wiki sources. This task assesses the ability of Video-LLMs to understand the song associated with the music video and provide answers regarding performers, background information, and relevant music theory knowledge.NBA-QA. Understanding competitive sports videos also demands relevant prior knowledge. Viewers must possess knowledge of the corresponding rules and engage in long-term observation to identify competing teams, players, technical actions, scores, or fouls within the video. We select top NBA plays from YouTube and manually annotate teams, players, and technical actions in each game, transforming them into question-answer pairs. These videos and questions serve as input to the model, expecting it to respond based on relevant prior knowledge. §.§ Comprehension and Decision-making Humans possess the innate ability to comprehend complex scenarios and make informed decisions and judgments. As shown in Fig. <ref> (C), to assess a similar capability in Video-LLMs, we propose evaluations in the realms of 3D scene understanding and autonomous-driving related tasks.3D Scene Comprehension. Indoor scene comprehension and navigation hold significant practical implications. The complexity arises from the necessity for extensive knowledge-intensive reasoning to understand different situations (scenes and locations). The SQA3D dataset <cit.> is introduced to evaluate the 3D scene comprehension of Video-LLMs within the video modality. The models are tasked with understanding their environment and engaging in perception, reasoning, and action to accomplish the task.Driver’s License Examination. Video-based questions in driver's license examinations assess the ability of candidates to interpret simple animations depicting motor vehicle and driver status, requiring judgments of potential anomalies. In this task, we challenge Video-LLMs to comprehend scenarios and answer exam questions.Driving Decision-Making. Making decisions for real-world driving scenarios is a more intricate task that demands a higher level of scene understanding and decision-making ability. For this task, we compile a diverse collection of YouTube driving videos depicting complex traffic situations and accidents. We conduct manual annotations for scene analysis and accident causes. Our expectation is that the model can effectively comprehend the origins of these complex traffic situations or accidents and make correct decisions to prevent their occurrence.§.§ Automatic Evaluation Toolkit LLMs are known for generating long-form text responses, often without adhering to a fixed format, making it challenging to quantify the correctness of their answers. To address this, we propose an automatic evaluation toolkit to systematically assess the performance of Video-LLMs.Our toolkit provides three metrics to map the output of Video-LLMs to pre-defined answer choices and subsequently calculating the final scores. The first one is Probability <cit.>, a logits-based metric to acquire the probability of the next token following the prompt and treat the highest probability option as the prediction:Choice =max _i ∈{A, B, C, D,…} P(Token_i|Prompt).The other two metrics are sentence-based, leveraging the natural language understanding capabilities of LLMs to obtain options. T5-based <cit.> one calculates the textual similarities of generated sequences and options. GPT-3.5-based <cit.> transforms the sequences to a fixed format with prompt `Please output your responses in the form of a dictionary "maximum probability":"xxx", where xxx is A or B or C or ...'. All the above metrics can be implemented automatically with our toolkit, and users can analysisthe ability of video-LLMs to comprehend video content and provide accurate responses to questions faithfully.§ EXPERIMENT AND RESULTImplementation details.The detailed statistics of Video-Bench are listed in Fig. <ref>. To mitigate the impact of randomness, we multiply an additional weight of 0.5 for tasks with a smaller quantity of questions during the computation of the final average score. To ensure a fair comparison, we utilize the 7B LLM backend versions for all tested Video-LLMs during the inference process, thereby mitigating language ability discrepancy stemming from different model sizes. The GPT-based metric are employed in the reported results by default, and the API version is set to gpt-3.5-turbo-0613 in the automatic evaluation toolkit.Results on Video-exclusive Understanding.To evaluate the video-exclusive understanding ability, we validate Video-LLMs on the traditional basic QA tasks, summarization, abnormal detection and crowd counting tasks, as reported in Table. <ref> (A). We have three observations. (i) Most Video-LLMs perform well on the four traiditional QA datasets due to the simplicity of their questions, especially the Video-ChatGPT <cit.> and Otter <cit.> with massive video instruction data, and the PandaGPT <cit.> with a well-pretrained video encoder from ImageBind <cit.>, which suggests extending the video data scale could be effective. (ii) Existing Video-LLMs are not temporal-sensitive. They cannot effectively summarize the order of each operation in YouCook2, and cannot respond effectively on the timestamp-related problems in UCF-Crime.(iii) These methods almost fail in the crowd counting task. These failure may come from the weak ability of precise locating and the temporal association.Results on Prior Knowledge-based QA.Compared to enormous training data of LLMs, existing Video-LLMs are trained with limited instruction tuning data as Table. <ref>, resulting in the poor ability to recognize objects and information in specific domains. As shown in Table. <ref> (B), we can have two observations. (i) Existing methods lack visual prior knowledge, which means they struggle to establish effective connection between the video and knowledge. For example, in NBA-QA task, even the players and technical actions are stored in the LLM backend, they cannot answer the questions when watching videos. Otter <cit.>, which has the most instruction tuning data, achieves the best performance in this project, indicating that some prior knowledge is indeed contained in MIMIC-IT. (ii) Their poor performance on MV-QA indicates that they have limited audio understanding ability, since only some of the Video-LLMs possess audio modules. PandaGPT <cit.> with the audio module of ImageBind shows the consistent results with the champion Otter <cit.> in MV-QA, proving that adding an audio encoder might improve this problem. In conclusion, existing Video-LLMs are requiring abundant prior knowledge pre-training for general domains on different modalities. Results on Comprehension and Decision-making.The performance of existing Video-LLMs on 3D scene understanding and driving decision-making tasks is shown in Table. <ref> (C). In these tasks, Video-ChatGPT <cit.> continues to perform the best, thanks to its robust video instruction tuning. The followings are the Valley <cit.>, which also possess powerful multi-modal understanding ability from vast instruct-tuning videos. To enhance the comprehensive and decision-making abilities, we suggest that future Video-LLMs must be trained with more prior knowledge and larger-scale data to cover more diverse domains. Besides, adopting Reinforcement Learning from Human Feedback (RLHF) and larger model capability is also important for generalization and specific applications. Results on Different Metrics. Our Video-Bench consists of a series of multiple-choice questions. Compared to open-ended questions, this test is relatively straightforward. However, due to the uncertainty and free form of LLM outputs, there is still room for designing more robust metrics. We evaluate the results of the best tested model, comparing the results with Probability <cit.>, T5-based and the GPT-based metrics. as shown in the Fig. <ref>. It can be seen that the result of Probability is overall low, because the output of Video-LLMs cannot effectively give a clear choice answer and the probability-based mapping may not faithfully reflect the correctness. Therefore, we recommend adopting GPT as the metric, especially considering the Video-LLMs with fewer LLM parameters and unstable outputs.§ VISUALIZATION AND MULTI-DIMENSION ANALYSIS Visualization.Fig. <ref> illustrates a set of typical responses from tested Video-LLMs. It can be observed that only Video-ChatGPT <cit.> provides the correct response, while other models engage in discussions related to the video but fail to make the correct judgment after a lengthy discourse. This highlights the issue that the models struggle with questions with even the most fundamental prior knowledge. This situation reflects the current state of Video-LLMs, which can generate responses related to videos while lacking trustful reference value. Therefore, we can conclude that current Video-LLMs are limited to generating human-like text while lacking the desired intelligence.Multi-dimension Analysis.In Fig. <ref>, a comparative analysis of Video-LLMs with different modules is presented. We can conclude that with the current data and training setting, Video-LLMs lack tailored focus onthe three-level ability of video comprehension. And the empirically proposed moduleshave not yielded significant improvements.We also analysis the impact of different data sizes in pre-training or instruction tuning process, as shown in Fig. <ref>. It can be observed that pre-training datasize may not necessarily play a decisive role, as the top-3 models, Video-ChatGPT <cit.>, PandaGPT <cit.> and Otter <cit.>, have no extra pretraining process. We suppose that the video encoders have received adequate training in multimodal pre-training.In contrary, the influence of the instruction tuning datasize is notably evident, showing two trends: (i) The models trained on videos demonstrate overall better performance compared to those trained on images. This substantiates that native video data facilitates enhanced comprehension of video information by Video-LLMs. (ii) Model performance is positively correlated with the amount of video instruction tuning data. Video-ChatGPT <cit.> and Otter <cit.> trained on large-scale video instruction tuning datasets are significantly better than other models.§ DISCUSSION AND CONCLUSIONAccording to the above experimental results, we can conclude that the existing models are far from the truly intelligent Video-LLM that can fully understand the visual and audio content in videos, and help people precisely summarize videos, explain details with priority knowledge or help providing global perception and making decisions. We believe there are primarily three improvement directions.Vision Encoder with Temporal Awareness. Existing methods process videos as frame clips, potentially missing crucial temporal information. Ideal Video-LLMs should understand the temporal sequence, possibly by selectively choosing keyframes or sampling frames to traverse the content efficiently.Domain-Specific Prior Knowledge Pre-training. Lack of visual prior knowledge hinders accurate video comprehension. Incorporating domain-specific prior knowledge through pre-training can enhance domain expertise.Long Video Understanding. One key differentiation point of Video-LLMs when compared to Image-LLMs should be the capability of processing long videos, which is highly neglected by existing research. Due to the memory and computation constraint, how to efficiently compress the past frames and design an effective memory mechanism is super crucial.Simultaneously, we also require more robust and effective evaluation metrics that can measure the long-text response of Video-LLMs. The utilization of GPT or similar LLMs in this process could be highly beneficial.ieeenat_fullname § T5 EVALUATION In our answer evaluation benchmark project, we explore two approaches: GPT-based metric and T5-based metric. T5-based metric serves as an auxiliary tool in the evaluation process, offering advantages in terms of cost, deployment, and performance. It provides a cost-effective solution by eliminating the need for ChatGPT API usage and allows for offline deployment on personal servers. As shown in Table <ref>, T5-based results demonstrate comparable performance to GPT-based in answer evaluation tasks, making it a valuable addition to our benchmark project for reliable and efficient assessment. § VISUALIZATION SAMPLESIn this part, we provide more samples of on all datasets concluded in Video-Bench, to illustrate the performance and behaviour of the tested Video-LLMs. §.§ Video-exclusive Understanding Activitynet-QA. The results of the Activitynet-QA is shown inFig. <ref>. As mentioned in Sec <ref>, Video-LLMs perform well on these simple questions. The similar results are shown on the remaining three datasets of Basic QA.MSVD-QA. The results of the MSVD-QA is shown inFig. <ref>. As part of the Basic QA, the performance of Video-LLMs here are overall good.MSRVTT-QA. The results of the MSRVTT-QA is shown inFig. <ref>. The results shows a similar trend of the above.TGIF-QA. The results of the TGIF-QA is shown inFig. <ref>. Results prove that Video-LLMS can also understand simple GIFs.YouCook2. The results of the YouCook2 is shown inFig. <ref>. The poor results show that existing Video-LLMs possess limited temporal awareness, and they are difficult to summarize the sequence of action steps.UCF-Crime. The results of the UCF-Crime is shown inFig. <ref>. The poor performance illustrates the existing Video-LLMs lack the ability of temporal perception again.MOT. The results of the MOT is shown inFig. <ref>. Existing Video-LLMs are proved to lack the ability to count accurately. §.§ Prior Knowledge-based Question-Answering TV-QA. The results of the TV-QA is shown inFig. <ref>, which demonstrate that existing Video-LLMs can hardly understand TV segments. This could be caused by the lack of prior knowledge and audio or subtitle understanding ability.MV-QA. The results of the MV-QA is shown inFig. <ref>. The poor performance may be also caused by the lack of prior knowledge and audio understanding ability.NBA-QA. The results of the NBA-QA is shown inFig. <ref>, which illustrates that without vision-language pre-training for specific domains, the Video-LLMs can not connect the knowledge stored in LLM with visual content and response to corresponding questions.§.§ Comprehension and Decision-Making Driver's License Examination. The results of the Driver's License Examination is shown inFig. <ref>. The poor performance validates the tested Video-LLMs have limited scene understanding and decision-making ability.Driving Decision-Making. The results of the Driving Decision-Making is shown inFig. <ref>, which demonstrates the tested Video-LLMs are difficult to understand the real driving environment.SQA3D. The results of the SQA3D is shown inFig. <ref>. The results show that they can only understand the simple environment and cannot understand the complex spatial relationship. | http://arxiv.org/abs/2311.16103v2 | {
"authors": [
"Munan Ning",
"Bin Zhu",
"Yujia Xie",
"Bin Lin",
"Jiaxi Cui",
"Lu Yuan",
"Dongdong Chen",
"Li Yuan"
],
"categories": [
"cs.CV",
"cs.AI"
],
"primary_category": "cs.CV",
"published": "20231127185958",
"title": "Video-Bench: A Comprehensive Benchmark and Toolkit for Evaluating Video-based Large Language Models"
} |
Mass reconstruction and noise reduction with cosmic-web environments Longlong Feng^1 January 14, 2024 ==================================================================== The Euclidean algorithm is one of the oldest algorithms known to mankind. Given two integral numbers a_1 and a_2, it computes the greatest common divisor (gcd) of a_1 and a_2 in a very elegant way. From a lattice perspective, it computes a basis of the sum of two one-dimensional lattices a_1 ℤ and a_2 ℤ as (a_1,a_2) ℤ = a_1 ℤ + a_2 ℤ. In this paper, we show that the classical Euclidean algorithm can be adapted in a very natural way to compute a basis of a general lattice (a_1, … , a_m) given vectors a_1, … , a_m ∈ℤ^n with m> rank(a_1, … ,a_m). Similar to the Euclidean algorithm, our algorithm is very easy to describe and implement and can be written within 12 lines of pseudocode.While the Euclidean algorithm halves the largest number in every iteration, our generalized algorithm halves the determinant of a full rank subsystem leading to at most log ( B) many iterations, for some initial subsystem B. Therefore, we can compute a basis of the lattice using at most ((m-n)nlog( B) + mn^ω-1log(A_∞)) arithmetic operations, where ω is the matrix multiplication exponent and A = (a_1, …, a_m). Even using the worst case Hadamard bound for the determinant, our algorithm improves upon existing algorithm.Another major advantage of our algorithm is that we can bound the entries of the resulting lattice basis by O(n^2·A_∞) using a simple pivoting rule. This is in contrast to the typical approach for computing lattice basis, where the Hermite normal form (HNF) is used. In the HNF, entries can be as large as the determinant and hence can only be bounded by an exponential term.§ INTRODUCTIONGiven two integral numbers a_1 and a_2, the Euclidean algorithm computes the greatest common divisor (gcd) of a_1 and a_2 in a very elegant way. Starting with s = a_1 and t= a_2, a residue r is being computed by settingr = min_x ∈{ r ∈| s x + r = t } = min{ ts, |(ts) - s| }.This procedure is continued iteratively with s = t and t = r until r equals 0. Since r ≤⌊ t/2 ⌋ the algorithm terminates after at most log(min{ a_1, a_2}) many iterations.An alternative interpretation of the gcd or the Euclidean algorithm is the following: Consider all integers that are divisible by a_1 or respectively a_2, which is the set a_1 or respectively the set a_2. Consider their sum (i.e. Minkowski sum)A = a_1+ a_2= {a + b | a ∈ a_1 , b ∈ a_2 }.It is easy to see that the set A can be generated by a single element, which is the gcd of a_1 and a_2, i.e.a_1+ a_2= gcd(a_1, a_2) .Furthermore, the set ℒ = a_1+ a_2 is closed under addition, subtraction and scalar multiplication, which is why all values for s,t and r, as defined above in the Euclidean algorithm, belong to ℒ. In the end, the smallest non-zero element for r obtained by the algorithm generates ℒ and hence ℒ = r= gcd(a_1,a_2).This interpretation does not only allow for an easy correctness proof of the Euclidean algorithm, it also allows for a generalization of the algorithm into higher dimensions. For this, we consider vectors A_1, … , A_m∈^n with m>n and the set of points in space generated by sums of integral multiples of the given vectors, i.e.ℒ = A_1+ … A_m.The set ℒ is called a lattice and is generally defined for a given matrix A with column vectors A_1, … , A_m byℒ(A) = {∑_i=1^mλ_i A_i |λ_i ∈}.One of the most basic facts from lattice theory is that every lattice ℒ has a basis B such that ℒ(B) = ℒ(A), where B is a square matrix. Note that the set a_1+ a_2 is simply a one-dimensional lattice and in this sense the Euclidean algorithm simply computes a basis of the one-dimensional lattice with gcd(a_1,a_2)= a_1+ a_2.Hence, morally, a multidimensional version of the Euclidean algorithm should compute for a given matrix A = (A_1, … , A_m) a basis B ∈^n × n such thatℒ(B) = ℒ(A).The problem of computing a basis for the lattice (A) is called lattice basis computation.In this paper, we show that the classical Euclidean algorithm can be generalized in a very natural way to do just that. Using this approach, we improve upon the running time of existing algorithms for lattice basis computation. §.§ Lattice Basis computationhnfsnfdelbrThe first property of a lattice that is typically taught in a lattice theory lecture is the fact that each lattice has a basis. Computing a basis of a lattice is one of the most basic algorithmic problems in lattice theory. Often it is required as a subroutine by other algorithms <cit.>. There are mainly two methods on how a basis of a lattice can be computed. The most common approaches rely on either a variant of the LLL-algorithm or on computing the Hermite normal form (HNF), where the fastest algorithms all rely on the HNF. Considering these approaches however, one encounters two major problems. First, the entries of the computed basis can be as large as the determinant and therefore exponential in the dimension. Secondly and even worse, intermediate numbers on the computation might even be exponential in their bit representation. This effect is called intermediate coefficient swell. Due to this problem, it is actually not easy to show that a lattice basis can be computed in polynomial time. Kannan und Buchem <cit.> were the first ones to show that the intermediate coefficient swell can be avoided when computing the HNF and hence a lattice basis can actually be computed in polynomial time. The running time of their algorithm was later improved byChou and Collins<cit.> and Iliopoulos <cit.>.Recent and the most efficient algorithms for lattice basis computation all rely on computing the HNF, with the most efficient one being the algorithm by Storjohann and Labahn <cit.>. Given a full rank matrix A∈^n× m the hnf can be computed by using only (n^ωm·logA_∞) many bit operations.The algorithm by Labahn and Storjohann <cit.> improves upon a long series of papers <cit.> and has not been improved since its publication in 1996. Only in the special case that m-n = 1, Li and Storjohann <cit.> manage to obtain a better running time that essentially matches matrix multiplication time. Other recent paper considering lattice basis computation focus on properties other than improving the running time. There are several algorithms that preserve orthogonality from the original matrix,B^*_∞≤A^*_∞, or improve on the ℓ_∞ norm of the resulting matrix <cit.>, or both <cit.>.Except for the hnf based basis algorithm by Lin and Nguyen <cit.>, all of the above algorithms have a significantly higher time complexity compared to Labahn's and Storjohann's hnf algorithm.The algorithm by Lin and Nguyen use existing HNF algorithms and apply a separate coefficient reduction algorithm resulting in a basis with ℓ_∞ norm bounded by nA_∞. §.§ Our ContributionIn this paper we develop a fundamentally new approach for lattice basis computation given a matrix A with column vectors A_1, …, A_m ∈^n. Our approach does not rely on any normal form of a matrix or the LLL algorithm. Instead, we show a direct way to generalize the classical Euclidean algorithm to higher dimensions. After a thorough literature investigation and talking to many experts in the area, we were surprised to find out that this approach actually seems to be new.Our approach does not suffer from intermediate coefficient growth and hence gives an easy way to show that a lattice basis can be computed in polynomial time. Furthermore, we can show that by an easy pivoting rule the resulting lattice basis has only a mild coefficient growth compared to the absolute values of the entries in the A_i vectors. We can show that the entries of the resulting basis can be bounded by O(n^2 ·A_∞).Similar to the Euclidean algorithm, our algorithm chooses an initial basis B from the given vectors and updates the basis according to a remainder operation and then exchanges a vector by this remainder. In every iteration, the determinant of B decreases by a factor of at least 1/2 and hence the algorithm terminates after at most log(B) many iterations. Similar to the Euclidean algorithm, our algorithms can be easily described and implemented.We develop data structures for our novel algorithmic approach and analyze the running time of our algorithms comparing to state of the art algorithms for lattice basis computation. But first, how do we measure efficiency in the running time of algorithms for lattice basis computation? There are mainly two different ways on how this can be done. First, one can simply count the number of arithmetic operation that the algorithm performs. In this model, one does not care about the size of the numbers and simply counts each basic ring operation: addition, subtraction, multiplication, and division. This concept of arithmetic complexity is often used in the context of matrix related problems (e.g. <cit.>) and linear programming (e.g. <cit.>), for example the concept of strong polynomiality relies on the notion of arithmetic complexity. A more precise measure of the running time of an algorithm uses the so called bit complexity model. Here, one counts each bit operation and hence for example an addition of two numbers of size t bits requires O(t) bit operations.In most algorithmic problems the arithmetic model and the bit complexity model do not need to be distinguished as the respective running times would essentially match. However, this is not the case for lattice basis computation (and related problems). For example, intermediate numbers in computing the Hermite normal form can become exponentially large in the dimension compared to the input numbers. Therefore, the same algorithm might have an additional factor in the bit complexity model compared to the arithmetic complexity. §.§.§ Arithmetic ComplexityWhile the bit complexity model is more precise in terms of worst case complexity, we also study our algorithms within the notion of arithmetic complexity. The main advantage of this model is that it provides a relatively easy analysis of the running time. Also, as one is simply counting the number of elementary ring operations the model provides an easier understanding of the running time when generalizing to other algebraic structure. Historically however, it was often the case that in the end, the same running time in the bit complexity model could be achieved as in the model of arithmetic complexity. But for the bit complexity to achieve the same running time, typically a very thorough analysis on the bit level is necessary. Consider for example the classical Euclidean algorithm when applied to numbers of bit length t. The algorithm requires O(t^2) many bit operations, while only O(t) many arithmetic operations are necessary. Using rather sophisticated operations on the bit level however, Schönhage <cit.> developed an algorithm computing the gcd by using only (t) many bit operations. In terms of arithmetic complexity, our main result is to develop an algorithm which uses at most (log(B) · (m-n)n + mn^ω -1log ||A||_∞).many arithmetic operations. Even with a worst case Hadamard bound for (B) ≤ (n ||A||_∞)^n and bounding (m-n) ≤ m, we obtain a running time of (mn^2 log ||A||_∞) and hence improve upon the algorithm of Storjohann and Labahn <cit.> by a factor of n^ω -2≈ n^0.37 for current values of ω. We are not aware of any other algorithms with a better running time within the arithmetic complexity model. But note that the algorithm by Storjohann and Labahn has the same time complexity within the bit complexity model, while our algorithms perform slightly worse within the bit complexity model. However, we are confident that a sophisticated analysis on the bit level similar to the approach of Schönhage <cit.>, will provide a much better running time also in the bit complexity model. In this sense, we see our results within the arithmetic complexity model as the potential that the presented approach has. Recall that our approach is new and builds upon very few subroutines while competing with algorithms for the HNF which build upon decades of research across dozens of papers.§.§.§ Bit ComplexityWhen it comes to the bit complexity model, in general, one has to pay attention to the growth of intermediate numbers in the matrix and in the respective solutions of linear systems. In the case of computing the HNF, this problem is typically dealt with by applying a separate coefficient reduction algorithm. In case of our algorithm however, we can completely ignore this issue. We show that for an easy pivoting rule, we only have quadratically growing coefficients in our basis matrix B. As a result, we can improve upon the running time of the algorithm by Labahn and Storjohann <cit.> in the case that m-n is small. Our algorithm requires ((m-n)n^3 log^2 ||A||_∞) bit operations and therefore yields an improved running time if the number of vectors that need to be merged into the basis is small, i.e. m-n ∈ O(n^ω-2). In the case that (B) is small, we also obtain an improved running time. For the general case, our algorithm matches the running time of <cit.> in terms of m and n having a bit complexity of (mn^ωlog^2||A||_∞).We are rather confident that the quadratic term in log||A||_∞ can be improved to a single logarithmic term by using an approach similar to Schönhage <cit.>. However, the required observations on the bit level would exceed the scope of this paper.Furthermore, our algorithms can be easily modified to compute the determinant of a square matrix B or compute a solution for a linear system of Diophantine equations. In the case of computing the determinant, the running times of the algorithms remain the same. However, in the case of computing a solution for Diophantine systems, the worst case complexity of the algorithms increase.§ ALGORITHM SKETCHIn this section and throughout the paper, we assume that rank(A) = n and therefore the lattice (A) is full dimensional. However, our algorithms can be applied in a similar way if rank(A) < n.The term (m-n) in the running times of the respective algorithm (which represents the number of vectors that need to be merged into the basis) is then replaced by the term (m-rank(A)). §.§ PreliminariesConsider a lattice (B) for a given full dimensional basis B ∈^n × n. An important notion that we need is the so called fundamental parallelepipedΠ(B) = { B x | x ∈ [0,1)^n } see also <ref>. As each point a ∈ℝ^n can be written asa = B x + B { x },it is easy to see that the spaceℝ^n can be partitioned into parallelepipeds. Here, x denotes the vector, where each component x_i is rounded down and { x } = x - x is the vector with the respective fractional entries x_i ∈ [0,1). In fact, the notion of Π(B) allows us to define a multi-dimensional modulo operation by mapping any point a ∈^n to the respective residue vector in the parallelepiped Π(B), i.e. a Π(B) := B { B^-1 a }∈Π(B).Furthermore, for a ∈, we denote with z the next integer from a, which is a + 1/2. When we use these notations on a vector a ∈^n, the operation is performed entry-wise. Note that the parallelepiped Π(B) has the nice property, that its volume as well as the number of contained integer points is exactly (B), i.e. vol(Π(B)) = |Π(B) ∩^n| = (B). In our algorithm, we will change our basis over time by exchanging column vectors. We denote the exchange of column i of a matrix B with a vector v by B∖ B_i ∪ v. The notation B∪ v for a matrix B and a vector v of suitable dimension denotes the matrix, where v is added as another column to matrix B. Similarly, the notation B ∪ S for a matrix B and a set of vectors S (with suitable dimension) adds the vectors of S as new columns to matrix B. While the order of added columns is ambiguous, we will use this operation only in cases where the order of column vectors does not matter. §.§ The AlgorithmGiven two numbers, the classical Euclidean algorithm, essentially consists of two operations. First, a modulo operation computes the modulo of the larger number and the smaller number. Second, an exchange operation discards the larger number and adds the remainder instead. The algorithm continues with the smaller number and the remainder. Given vectors A={A_1, … , A_n+1}⊂^n, our generalized algorithm performs a multi-dimensional version of modulo and exchange operations of columns with the objective to compute a basis B ∈^n × n with (B) =(A). First, we choose n linearly independent vectors from A which form a non-singular matrix B. The lattice(B) is a sub-lattice of (A). Having this sub-basis, we can perform a division with residue in the lattice (B). Hence, the remaining vector a ∈ A ∖ B can be represented asa = B B^-1a +r,where r is the remainder a Π (B), see also <ref>. In dimension n= 1 this is just the classical division with residue and the corresponding modulo operation,a = b ·a/b + r.Having the residue vector r at hand, the exchange step of our generalized version of the Euclidean algorithm exchanges a column vector of B with the residue vector r. In dimension >1, we have the choice on which column vector to discard from B. The choice we make is based on the solution x ∈^n of the linear system Bx = a. * Case 1: x ∈^n. In the case that the solution x is integral, we know that a ∈ (B) and hence (B ∪ a ) =(B). Our algorithm terminates.* Case 2: There is a fractional component i of x. In this case, our algorithm exchanges B_i with r,B' = B ∖B_i∪r.The algorithm iterates this procedure with basis B' and vector a = B_i until Case 1 is achieved.[3.5em]3.5em [ sharp corners=all, colback=white, colframe=black, size=tight, boxrule=0.2mm, left=3mm,right=3mm,top=3mm,bottom=3mm ] 2Euclidean AlgorithmModulo Operationt = s⌊ s^-1 t⌋ + r Exchange Operationt = s,s=r Stop Conditions^-1 t is integralGeneralized Euclidean AlgorithmModulo Operationa = BB^-1a + r Exchange Operationa = B_i,B_i := r Stop ConditionB^-1a is integralTwo questions arise: Why is this algorithm correct and why does it terminate?Termination:The progress in step 2 can be measured in terms of the determinant. For x with Bx=a the exchange step in case 2 swaps B_i with r = B { x } and {x_i}≠ 0 to obtain the new basis B'. By Cramer's rule we have that {x_i} =B'/ B and hence the determinant decreases by a factor of {x_i} < 1. The algorithm eventually terminates since ((A))≥ 1 and all involved determinants are integral since the corresponding matrices are integral. A trivial upper bound for the number of iterations isthe determinant of the initial basis. Correctness:Correctness of the algorithm follows by the observation that (B ∪ a) = (B ∪ r). To see this, it is sufficient to prove a ∈(B ∪ r) and r ∈(B ∪ a). By the definition of r we get that a = Bx = Bx + B{x} = Bx + r. Hence, a and r are integral combinations of vectors from B∪ r and B∪ a, respectively, and hence (B∪ a) = (B ∪ r). The multiplicative improvement of the determinant in step 2 can be very close to 1,(B)-1/(B). In the classical Euclidean algorithm a step considers the remainder r for a = ba/b + r. The variant described in <ref> considers an r' for a = ba/b + r'. Taking the next integer instead of rounding down ensures that in every step the remainder in absolute value is at most half of the size of b. Our generalized Euclidean algorithm uses a modified modulo operation that does just that in a higher dimension. In our case, this modification ensures that the absolute value of the determinant decreases by a multiplicative factor of at most 1/2 in every step as we explain below. The number of steps is thus bounded by log(B). The generalization to higher dimensions chooses i such that x_i is fractional and rounds it to the next integer x_i while the other entries of x are again rounded x_j for j≠ i. Formally, this modulo variant is defined as a (' Π(B)) := r' := a - (∑_j≠ iB_jx_j + B_ix_i)for Bx=a and some i such that {x_i}≠ 0. By Cramer's rule we get that the determinant decreases by a multiplicative value of at least 1/2 in every iteration since 1/2≤x_i - x_i =B'/ B. In <ref> the resulting basis for exchanging B_1 with r = a ( Π(B)) and with r'= a (' Π(B)) shows that in both cases the volume of the parallelepiped decreases, which is equal to the determinant of the lattice. In <ref>, an example of our algorithm is shown.§.§ Basic AlgorithmIn the following we state the previously described algorithm formally.<ref> computes a basis for the lattice ℒ(A).Let us consider the following invariant.Claim. In every iteration ℒ(A) = ℒ(B∪ C).By the definition of B and C the claim holds in line 2. We need to prove that removing c from C in line 7 and altering B and C in lines 9-11 do not change the generated lattice. In line 7 we found c is an integral combination of vectors in B. Thus, every lattice point can be represented without the use of c and c can be removed without altering the generated lattice. In lines 9 and 10 a vector c is removed from B∪ C and instead a vector c' = c - (∑_j≠ iB_jx_j+ B_i x_i) is added. By the definition of c', the removed vector c is an integral combination of vectors c', B_1, …, B_n and c' is an integral combination of vectors c, B_1, …, B_n. Using the same argument as above, this does not change the generated lattice. The algorithm terminates when C=∅. In this case B is a basis of ℒ(A), since by the invariant we have that ℒ(B) = ℒ(B ∪ C) = ℒ(A). <ref> terminates after at most log(B^(1)) exchange steps.§.§ Arbitrary Rank of Lattice In the case that the lattice ℒ(A) is not fully dimensional Algorithm <ref> can easily be modified to also function in that case. This can be done by using Lemma <ref> to choose a maximum set of linear independent vectors from A as our initial basis B. The algorithm then proceeds to work with a basis B containing rank(B) many vectors. Note that every other vector in C is then still contained in the linear subspace of B and hence the linear system of equalities in step 5 of the algorithm is always solvable.The same argument can be applied to any of the presented algorithms in this paper. For simplicity we therefore omit this case and assume from now on that ℒ(A) is fully dimensional. As mentioned, the term (m-n) in the running times of the respective algorithm (which represents the number of vectors that need to be merged into the basis) must be replaced by the term (m-rank(A)). § ARITHMETIC OPERATIONSThe main bottleneck in terms of running time of Algorithm <ref> is that in each iteration, the linear system Bx = c (line 5) needs to be solved. In this section, we present two efficient algorithms for lattice basis computation that do this step more efficiently. Algorithm <ref> uses the inverse matrix to obtain the respective solutions. As the basis B changes, the inverse matrix is being updated. In Algorithm <ref>, we use an efficient data structure that manages the solutions for all vectors that are not in the basis. The data structure is built in a way that it can be updated efficiently when the basis changes.We analyze the algorithms with respect to their arithmetic complexity. A subproblem that arises is to find a maximal set of linearly independent vectors. In our algorithms we use the following Lemma for this subproblem.Let A∈^m×n have full column rank. There exists an algorithm that finds indices i_1, …, i_n such that A_i_1, …, A_i_n are linearly independent using (mn^ω-1logA_∞) bit operations.§.§ Via Matrix Inverse UpdatesThis first algorithm uses the fact that updating the inverse of a matrix and computing a matrix-vector multiplication both only requires O(n^2) arithmetic operations. Thereby, we need to compute the inverse only once using (n^ω) arithmetic operations and in every iteration we only require a quadratic number of operations for solving the linear system and updating the inverse.<ref> computes a basis for the lattice ℒ(A) using((m-n)n^2 + mn^ω-1logA_∞+ n^3logA_∞)arithmetic operations.Correctness of the algorithm follows similar to <ref>. Using <ref> the set of linearly independent columns can be found in (mn^ω-1logA_∞) bit operations. The inverse can be computed in (n^ω). In every iteration either a vector from C is discarded or an exchange operation is performed.Thus, the number of iterations can be bounded by m-n + log( B^(1)) ≤ m-n + nlog(nA_∞), where B^(1) is the matrix of linearly independent columns found in line 1 and the inequality follows the worst-case Hadamard bound on determinants. In every iteration a constant number of vector operations and matrix-vector multiplications is computed. Moreover, the inverse can be updated in O(n^2) arithmetic operations, see e.g. Sherman and Morrison <cit.>. Therefore, the number of arithmetic operations used is bounded by ((m-n)n^2 + mn^ω-1logA_∞+ n^3logA_∞) ≤(mn^2logA_∞).§.§ Via System SolvingThe running time of the following algorithm improves on the previous one in the case that either m-n or log( B^(1)) is small. Instead of updating the inverse matrix in order to solve the next linear system, <ref> computes all solutions at once and then updates the solution matrix. Consider two matrices B∈^n×n and C∈^n×m, where B is full rank. Let X:=B^-1C and consider an exchange stepC' = C ∖{C_j}∪ B_i,B'_i := C_j - ∑_k≠ iB_kX_kj + B_i X_ij,where the ith column of B is updated according to right-hand side C_j. Then the updated solution matrix X' := (B')^-1 C' can be computed byX'_ij = 1/X_ij - X_ijX'_iℓ =X_iℓ/X_ij - X_ijfor all ℓ≠ jX'_kj =-{X_kj}/X_ij - X_ijfor allk≠ iX'_kℓ = X_kℓ -X_iℓ·{X_kj}/X_ij - X_ijfor all ℓ≠ jandk≠ i.Since C_j = BX_*j, we can reformulate the exchange step as B'_i := ∑_k≠ iB_k{X_kj} + B_i (X_ij - X_ij). As B'_k = B_k is unchanged for k≠ i we get thatB_i = B'_i1/X_ij - X_ij + ∑_k≠ iB'_k-{X_kj}/X_ij - X_ij.This shows B'X'_*j = C'_j = B_i. For columns ℓ≠ j we get that C'_ℓ = C_ℓ = BX_*ℓ= ∑_k=1^n B_k X_kℓ= ∑_k≠ iB_k X_kℓ + B_i X_iℓ(<ref>)=∑_k≠ iB'_k X_kℓ + ( B'_i1/X_ij - X_ij + ∑_k≠ iB'_k-{X_kj}/X_ij - X_ij) · X_iℓ= ∑_k≠ iB'_k· (X_kℓ - X_iℓ{X_kj}/X_ij - X_ij) + B'_iX_iℓ/X_ij - X_ij= B'X'_*ℓ.For our target running time, we require a second adjustment.The exchange operation for updating B_i after an exchange step requires O(n^2) arithmetic operations. In order to reduce the number of arithmetic operations in <ref>, we will delay updating the basis. Instead we will collect the representation of all exchange steps in a matrix Y, which is multiplied to the initial basis before output.<ref> computes a basis for the lattice ℒ(A) using(log(B^(1)) · (m-n)n + mn^ω -1logA_∞)arithmetic operations for an initial linearly independent subsystem B^(1) found in line 1. With the worst-case Hadamard bound on the determinant, the arithmetic complexity is((m-n)n^2logA_∞ + mn^ω -1logA_∞).In order to prove correctness of the algorithm it suffices to show that the invariant B^(ℓ+1) = B^(1)Y^(ℓ) holds, where B^(ℓ), Y^(ℓ), and X^(ℓ) represent the matrices B, Y, and X starting iteration ℓ, respectively. The exchange step in one iteration is B^(ℓ+1)_i = ∑_k≠ iB^(ℓ)_k{X^(ℓ)_kj} +B^(ℓ)_i (X^(ℓ)_ij-X^(ℓ)_ij) or in terms of the entire matrix it isB^(ℓ+1) = B^(ℓ)·( e_1, …, e_i-1, v^(ℓ), e_i+1, …, e_n) = B^(1)·( e_1, …, e_i-1, v^(1), e_i+1, …, e_n) ·…·( e_1, …, e_i-1, v^(ℓ), e_i+1, …, e_n)= B^(1)Y^(ℓ). Now, correctness of <ref> follows similar to the proof of <ref> since B and C are updated just as in <ref> and instead of computing a new solution in each iteration the complete solution matrix is updated in each iteration using <ref>. We find the set of linearly independent columns in time (mn^ω-1logA_∞) using <ref>. The inverse and the matrix multiplication in lines 3 and 4 are computed in (n^ω) and (mn^ω -1), respectively. The number of iterations is bounded by log(B^(1)) since in every iteration an exchange step is computed. Computing the vector v requires O(n) arithmetic operations. For i' < i we have that v_i' = {X_i'j} = 0 since i was chosen minimal considering fractional components of X. Computing Y' ← Y ·(e_1, …, e_i-1, v, e_i+1, …, e_n ) requires to compute Y'_i = Yv = ∑_k ≥ iY_k v_k.A direct consequence of <ref> is that any integral row of the solution matrix X_kj remains integral after the exchange step. Thus Y_k = e_k for any k>i and the computation simplifies to Y'_i = Y_i v_i + ∑_k ≥ ie_k v_k which can be computed with O(n) arithmetic operations. In each iteration the main complexity is to update the (m-n)n entries of X. Finally, in line 11 another matrix multiplication is performed in (n^ω) arithmetic operations. The total running time is (log(B^(1)) · (m-n)n + mn^ω -1logA_∞) ≤((m-n)n^2logA_∞ + mn^ω -1logA_∞).§ BIT COMPLEXITYA typical obstacle for computing the basis of a lattice is intermediate coefficient growth. Earlier algorithms for the HNF, for example, had their main computational bit complexity coming from intermediate numbers of length (n^4logA_∞) <cit.>. Later, all numbers involved could be bounded by B ≤ (nA_∞)^n for some subsystem B of A, which still adds a factor of n.Large intermediate numbers could effect the bit complexity of our algorithm in two aspects: growing coefficients in the computed basis and exact solutions to linear systems. A naive implementation of our algorithmic idea could result in a basis with entries of exponential size. In every iteration, the new basis vector could be as large as the sum of the current basis vectorsB'_i_∞ = ∑_j≠ iB_j{x_j} - B_i(x_i -x_i)_∞≤∑_j≤ nB_j_∞≤ nB_∞.If the initial basis is B^(1), then there are up to log( B^(1)) exchange steps. By Hadamard's bound coefficients in the basis might grow to be of order (nA_∞)^n in a naive implementation. Fortunately, there is an easy pivoting rule that bounds the size of the computed basis B by B_∞≤(n^2A_∞).Our pivoting rule is very simple and in fact <ref> already applies it. Instead of choosing any vector c∈ C and any fractional component of x := B^-1c, we compute exchange steps to obtain integral entries in the solution matrix X:= B^-1C row by row. If a row of the solution matrix is integral, then as a consequence of <ref> it remains integral after an exchange step. Moreover, in the modulo operation, basis vectors with integral solution component do not contribute to the new basis vector. If we assume that rows i'<i of the solution matrix are integral we get thatB'_i = ∑_j≠ iB_j{x_j} - B_i(x_i -x_i) = ∑_j> iB_j{x_j} - B_i(x_i -x_i).Performing modulo and exchange steps row by row in the solution matrix corresponds to column by column in the current basis. Therefore, the basis vectors B_j with j<i are final in the sense that those will appear in the output basis and the basis vectors B_j with j>i are untouched in the sense that they were part of the input vectors which implies that their size is bounded by B_j_∞≤A_∞. By Equation <ref> only theuntouched basis vectors with the before mentioned size bound and the currently updated basis vector contribute to the new basis vector. Therefore, the size of the modulo vector is bounded by B'_i_∞≤(n-1)A_∞ + B_i≤ n^2A_∞log(nA_∞) since there are at most log( B^(1)) ≤ nlog(nA_∞) exchange steps. Using this pivoting rule, large numbers may only appear as a result of exact system solving.By Cramer's rule and Hadamard's bound exact solutions to a linear system Bx = b can be as large as n^n/2B_∞^n-1b_∞ in the numerator and n^n/2B_∞^n in the denominator. We use the recent algorithm by Birmpilis, Labahn and Storjohann to compute solutions of linear systems. There exists an algorithm that takes as input a non-singular matrix B∈^n×n and a vector b∈^n and returns as output B^-1b ∈^n. If logb_∞∈(nlogB_∞), the running time of the algorithm is (n^ωB_∞) bit operations. We use the following lemma for calculations involving a vector with large coefficients such as computation of the remainder of our modulo operations. Let B∈^n×n and N∈_>0 be a power of 2 such that log N ∈ O(log(nB_∞)). If C∈/(N^p)^n×m with m p∈ O(n), then (BC, N^p) can be computed in with bit complexity(n^ωlogB_∞).In order to quickly perform our pivoting rule, a new subproblem arises. We need to locate the next modulo and exchange step and thus require to efficiently find non-integral components of a row of the solution matrix X := B^-1C. The following lemma shows that a row of the solution matrix can be computed with similar bit complexity as a column.Consider a full rank matrix B∈^n×n,a matrix C∈^n×m, and δ∈ such that B_∞≤δ and C_∞≤δ. Let X∈^n×m be the solution matrix for BX=C. Any row i≤n of the solution matrix X can be computed using (max{n,m}n^ω -1logδ) bit operations. The procedure is as follows. First, we compute y∈^n such that B^⊺ y = e_i. This is the same as the ith row of Y := I_n B^-1, where I_n is the identity matrix of dimension n. In other words, Y is the inverse of B and y is the ith row of the inverse of B. Then we compute an integer μ≤ B such that μ y is integral.Finally, we compute z such that 1/μC^⊺ (μ y). It is obviously the same to compute 1/μμ y^⊺ C = y^⊺ C. Since y is the ith row of the inverse of B we have that z is the ith row of the solution matrix X=B^-1C.We can compute y with <ref> using (n^ωlogδ) bit operations. The integer μ can be found in (n^2logδ) bit operations.[One way to do this is as follows. Let d_1,…,d_n be the denominators of y. Compute the greatest common divisor of d_1 and d_2 and d_1· d_2/gcd(d_1,d_2) to obtain the least common multiple of d_1 and d_2. Continue with the least common multiple and d_3 and eventually obtain the least common multiple of d_1, …,d_n. Due to Cramer's rule μ := lcm(d_1, …,d_n) is at most (B).] The matrix vector multiplication to compute z can be done in (max{n,m}n^ω-1logδ) bit operations using <ref> ⌈n/m⌉ times for p:= n. Scaling y and the result of the matrix-vector multiplication each costs (n^2logδ) bit operations. §.§ General Version<ref> computes a basis for the lattice ℒ(A) using at most (mn^ωlog^2A_∞) bit operations. We want to prove correctness by proving that <ref> performs the exchange steps from <ref> but in a more specified order. [2em]0em Claim. Consider iteration i≤ n. For any i'<i all solutions x^(c) = B^-1c for c∈ C are integral at index i'. By lines 6 and 7 solution index x^(c)_i for Bx^(c)=c is integral for all c∈ C when i is set to i+1 in line 7. Thus, we need to prove that this remains true after an exchange step in lines 11-12. Consider any i'<i and right hand side c used for the exchange step. Let x^(c) := B^-1c and x^(c'):= B^-1c' for any c' ∈ C with c'≠ c. By <ref> the updated solutions of c and c' at index i are -{x^(c)_i'}/x^(c)_i - x^(c)_iandx^(c')_i' + -x^(c')_i ·{x^(c)_i'}/x^(c)_i - x^(c)_i,respectively.Since after iteration i all solutions are integral at index i by lines 6-7, this implies that the exchange step in line 11-12 does keep the property that for any i'<i all solutions are integral at index i'. The claim implies that all solutions are integral when the algorithm terminates. Therefore, correctness of the algorithm follows from <ref> since <ref> only selects the next exchange step in a specified order compared to <ref>.Concerning the running time, we start off by bounding size of the numbers involved.By the definition of the exchange step any new vector B_i = ∑_j≠ iB_j{x_j} + B_i(x_i - x_i) is the sum of B_j y_j for |y_j|≤ 1 and j≥ i since for any j<i the solution at index j is integral by the claim and thus the fractional component is 0. Any B_j for j > i is unchanged after line 1 and thus B_j_∞≤A_∞. If B_i and B_i' are the state of the ith vector before and after the exchange step in line 11-12, respectively, then B_i'_∞≤B_i_∞ + ∑_j>iB_j_∞≤B_i_∞ + (n-1)A_∞. Let B^(1) be the basis in line 1 and B^(ℓ) the returned basis. Since there are in total at most log(B^(1)) exchange steps, the returned (and every intermediate) basis is bounded byB^(ℓ)_∞≤log(B^(1)) · nA_∞≤ n^2A_∞log(nA_∞) =: δ. By Hadamard's inequality and Cramer's rule the numerator and denominator of solutions x = B^-1c for c∈ C are bounded by determinants of B and B|c, where one column of B is exchanged by c, respectively, and due to the bounded entry size this is ≤ (nδ)^n in every iteration. By <ref> the set of linearly independent columns can be found with (mn^ω-1logA_∞) bit operations. Every iteration of the while loop either increases i or performs an exchange step. Hence, there are at most n + log (B^(1)) = O(nlog(nA_∞)) iterations. The ith row of the solution matrix can be found in (max{m-n,n}n^ω-1logδ) bit operations using <ref>. In line 10 a linear system is solved. All numbers involved are bounded by δ and thus the linear system can be solved in (n^ωlogδ) bit operations. Considering line 12, let x∈^n be defined as x_j = x_j for j≠ i and x_i := x_i. The updated column is then B_i ← C_j - Bx, where the latter can be computed in (n^ωlogδ) bit operations using <ref> since x_∞≤ (nδ)^n and can be scaled to an integral vector similar to the proof of <ref>. Overall the number of bit operations for <ref> is bounded by ((n+log( B^(1)))·((m-n)n^ω-1logδ + n^ωlogδ)) =((nlogδ)mn^ω-1logδ )= (mn^ωlog^2A_∞). §.§ Few Additional Vectors m-nVery recently Lin and Storjohann considered the special case that m-n=k for a constant k <cit.>. In this section we present a variant of our generalized Euclidean algorithm that improves the general running time in the case that m-n is small but not necessarily constant, e.g. the running time dependence on m and n is improved for any instance with m-n ∈(n^ω -2). The procedure is almost identical to <ref>. Computation of all solutions could be faster with less naiv approach? Check again, whether system solving result includes full system. Probably the problem is in step 4, but not sure. They say logb_∞≤ nlogA_∞ but that looks suspiciously like the condition for writing b as a matrix instead, which would be just our case. In step 4 it is just y, the solution, here a matrix could be extra work. Assuming the determinants are small, this procedure should also benefit. If that would be the case, the small determinant case looks really strong. It looks like if (m-n)log( B) ∈(n) then it just matrix multiplication time. Open problem from ISSAC 22 to improve on special case where m-n small but not constant. * Compute all solutions X := B^-1C in time ≤((m-n)n^ωlogA_∞)* Follow pivoting as in bit complexity algorithm* Updating solution costs ((m-n)n^2logA_∞) and is done at most nlog(nA_∞) times* Complexity should be about ((m-n)n^3log^2A_∞) which is better than HNF in the case that (m-n)logA_∞ < (n^ω-2). * Note: running time benefits twice from small determinants: in the number upper bound for updating the solution and in number of iterations. In this terms the running time should be about ((m-n)n^ωA_∞ + (m-n)n log^2( B)). If log^2B ≤ n^ω -1logA_∞ the algorithm is also faster than HNF? <ref> computes the basis of the lattice (A) using((m-n)n^3log^2A_∞ + n^ω(2)logA_∞) bit operations.The size of most intermediate numbers is bounded as in <ref>. Additionally, we need to bound the size of numbers in Y. Rephrased, Y is the solution matrix for B^(1)Y= B^(ℓ). The size of numbers in B^(ℓ) is bounded by δ = n^2A_∞log(nA_∞) as in the proof of <ref>. Thus, denominators in Y are bounded by B^(1) and numerators in Y are bounded by (B^(1)|B^(ℓ)_k) ≤ (nA_∞)^O(n), where the latter describes the matrix exchanging a column of B^(1) with B^(ℓ)_k. The set of independent vectors can be computed in the claimed time using <ref>. For line 3 we compute m-n solutions to linear systems. This is also the claimed time by <ref>. Updating Y costs O(n) arithmetic operations as analyzed in <ref>. Let δ be as in the proof of <ref>. The size of numbers involved is bounded by (nlogδ) and thus lines 8-9 require (n^2logA_∞) bit operations. Updating the solution matrix X can be done with <ref> using O((m-n)n) arithmetic operations and due to the bounded size of numbers this requires at most ((m-n)n^2logA_∞) bit operations. In every iteration of the while loop an exchange operation is performed.Thus, there are at most log( B^(1)) iterations. In line 10 we can multiply the matrix by the least common multiple of the denominators (which is bounded by B^(1)), apply the matrix multiplication, and again divide by the least common multiple of denominators, similar to part of the proof of <ref>. Then the matrix multiplication can be solved by idea of <ref> the main complexity is to compute a matrix multiplication of dimensions n× n and n× n· O(n). Using rectangular matrix multiplication <cit.> this can be done using (n^ω(2)logA_∞) bit operations, where ω(k) is the exponent required to compute a matrix multiplication for dimensions n× n and n× n^k. The bit complexity in total is bounded by((m-n)n^ωlogA_∞ + mn^ω -1logA_∞ + (m-n)n^2log(B^(1))logA_∞ + n^ω(2)A_∞).Using (m-n)∈ O(n^ω - 2) and the worst-case Hadamard bound on the determinant the running time simplifies to ((m-n)n^3log^2A_∞ + n^ω(2)A_∞) bit operations. §.§ Small n× n Minors A similar special case is in <cit.> for HNF computation. There they assume that gcd(m_n-1, m_n) is at most A_∞^log^O(1)(n), where m_i is the ith leading minor. This can be as large as ≈A_∞^n-1 (e.g. A_∞· I_n).small determinant bounds number of iterations but also all determinants in computed solutions?HNF algorithm computes the determinant of a subsystem, involving all primes ≤ z for some z∈ O(mlog(mA_∞), this is done to obtain a multiple of the overall determinant and is probably same complexity even for small overall determinant (unless det is given).–Get rows where there are fractional entries first– COULD BE MORE Either directly solving entire system in one run or as below: * Compute B^-1 in time (n^ω+1logB_∞) (for example via solving n linear systems)* Compute B^-1C via rectangular matrix multiplication? Similar to <ref>?Only iterate over those rows. * Assume log( B) ∈(√(n)logA_∞)* Compute B^-1. Possible in (n^ωlog( B))?* Each iteration B^-1C_j computable in (n^2log( B)) * log( B) + (m-n) iterations? Compute B^-1C in (mn^ω(2)-1logA_∞) ≈(mn^2.251640logA_∞) → now only log( B) iterations required — Also not quite right* Overall should be about (mn^2.251640logA_∞) bit operations, so ≈ 0.1 less in the exponentIn this section we give an algorithm which is very efficient in the case that B^(1) is small. This is often the case when considering specific matrix classes. For example, a prominent class of matrices that is often considered in integer programming, is the class of matrices A where the absolute value of all subdeterminants are bounded by some small Δ.Now consider again <ref>. The number of iterations of the while loop scales the complexity by log( B^(1)). So, if the determinant is small, the bit complexity for lines 5-10 also decreases. In contrast to other algorithms <cit.> the following algorithm directly benefits from small minors and does not require the approximate size as input or to compute any determinant. In order to achieve an improved running time, we analyze the algorithm for solving a system of linear equations from <cit.> for a matrix right-hand side. The algorithmin <cit.> solves a system X = B^-1C for an invertible matrix B∈^n×n and a matrix C ∈^n×m using (mlog(Δ)/nn^ωlogδ) = (mlog(Δ)n^ω-1logδ) bit operations, where Δ is the largest n×n minor of (B_1,…, B_n, C_1,…, C_m) and B_∞≤δ and C_∞≤δ. We analyze their algorithm and how the running time changes by the modification in their notation. Also we only describe the differences in the analysis. On a high level, the main change is that we do not provide the so called dimension × precision invariant but instead parameterize by this quantity. We throughoutly make use of the dimension × precision tradeoff, where the idea of <ref> is used for matrix multiplications and since the size of numbers is bounded by Δ this results in matrix multiplications of dimension n×n and n×mlogΔ with sufficiently bounded coefficients. Viewed as mlogΔ/n matrix multiplications the running time follows. If we analyse the algorithm for a matrix right-hand side, steps 1 and 2 do not change. In step 3 the subroutinedominates the running time.Corollary 7 in their paper requires the dimension × precision invariant m·log(Δ) ∈ O(nlog(nδ)), which is not necessarily the case here. However, the running time is dominated by (loglog(Δ)) matrix multiplications of an n×n matrixwith coefficients of magnitude O(n^2B) and an n×m matrix of magnitude Δ as by Cramer's rule numbers involved in this step are bounded by B and (B|C_i). Using <ref> (their Lemma 2) this can be computed in target time.Finally, Step 4 consists of matrix multiplications (PM(2^eS^-1)Y, 2^d), where d∈(nlogδ). The first part Z := (2^eS^-1)Y involves just a diagonal matrix S^-1 and can be computed in time. For the multiplication MZ, we follow the steps from their paper. By their Lemma 17, the X-adic expansion of the columns of M consists of n' ≤ 2n columns for X the smallest power of 2 such that X≥√(n)A. Let M' = (M_0 … M_p-1) be the X-adic expansion of M, where M_i∈^n× k_i and ∑i <pk_i = n' ≤ 2n. Let Z = ( Z_0 … Z_p-1) be the X-adic expansions of Z and let Z_i^(k_i) be the submatrix of the last k rows. The matrix multiplication can be restored from the product[ M_0 … M_p-1 ][ Z_0^(k_0) Z_1^(k_0) … Z_p-1^(k_0); Z_0^(k_1) … Z_p-2^(k_1); ⋱ ⋮; Z_0^(k_p-1) ].The dimensions are n×n' and n' ×mlog(Δ) since the precision p requires p≥logZ_∞ which is bounded by Cramer's rule.Though, we already analyzed <ref> for small m-n, next we will analyze it again for the case that all n× n minors of the input are small. <ref> computes a basis for the lattice ℒ(A) and the running time is((m-n)n^2logδlogΔ + nlog^3Δ + n^ωlog^2Δ)bit operations for Δ being the largest n× n minor of A. The size of most intermediate numbers is bounded as in <ref>. Additionally, we need to bound the size of numbers in Y and the bound from <ref> does not suffice. Let Y^(ℓ) be the state of Y in the ith iteration. Consider in iteration ℓ where X^(ℓ)_ij was chosen for the modulo operation. Updating Y^ℓ only changes Y^ℓ+1_i = Y^ℓv. The update for row index k ≤ i is Y^ℓ +1_ki = ∑_h ≤ nY^ℓ_khv_h= ∑_h ≠ iY^ℓ_kh{X_hj} + Y^ℓ_ki(X^ℓ_ij - X^ℓ_ij) = Y^ℓ_ki(X^ℓ_ij - X^ℓ_ij)=B^(ℓ+1)/ B^(ℓ'),where ℓ' is the first iteration considering row i. The update for row indices k >i is Y^ℓ +1_ki = ∑_h ≤ nY^ℓ_khv_h= ∑_h ≠ iY^ℓ_kh{X_hj} + Y^ℓ_ki(X^ℓ_ij - X^ℓ_ij) = 1 ·{X_kj} + Y^ℓ_ki B^(ℓ +1)/ B^ℓ.The denominators of {X_kj} and B^(ℓ +1)/ B^ℓ are both divisors of B^ℓ. Therefore numerator and denominator of Y^ℓ +1_ki are both bounded by Δ^O(log (B^1)).The set of linearly independent columns can be found using (mn^ω-1logA_∞) bit operations with <ref>. The solution matrix X can be computed using <ref> in ((m-n)log( B)n^ω-1logA_∞) bit operations. There are at most log( B) iterations of the while loop and all intermediate bases are bounded by δ in infinity norm. Thus, computing v and updating Y requires O(n) arithmetic operations, which are at most (n^2logA_∞ + nlog^2Δ) bit operations, depending on whether the bound on X or the bound on Y is larger. Updating X requires O((m-n)n) arithmetic operations and thus ((m-n)n^2logA_∞) bit operations. Finally, line 10 computes the basis using a matrix multiplication. By the bound on the size of numerators and denominators of Y we have that both are at most O(log^2Δ), which also applies to the least common multiple of denominators. Thus, the matrix multiplication can be computed using (n^ωlog^2(Δ)) bit operations. The total running time in bit operations is therefore bounded by ((m-n)n^2logA_∞logΔ + nlog^3Δ + n^ωlog^2Δ).In the case that logΔ∈ O(n^ω -2) the running time simplifies to ((m-n)n^2A_∞logΔ + n^ωlog^2Δ). § MODIFICATIONS OF THE ALGORITHMIn this section, we present how our algorithms can be modified to compute the determinant of a square matrix B or to compute a solution of Diophantine system of equations. §.§ Computing the DeterminantOur algorithms can be easily adapted to compute the determinant of a matrix B for a given matrix B ∈^n × n. We initialize the respective algorithm with the matrix A = (B I), where I is the identity matrix. By this we ensure that ((A)) = 1. The first line of finding a set of linearly independent vectors is skipped and instead set to B.Then we can simply keep track of the improvement to B after each exchange operation. As explained, by Cramer's rule the determinant of the new basis B' equals (x_i - ⌈ x_i ⌋) · B. Multiplying the improvements over all exchange operations therefore yields the value (B)/ ((A)). For this, we only have to introduce a new variable D and set D = D · (x_i - ⌈ x_i ⌋) whenever there is an exchange operations.The running time of the respective algorithm remains the same with m, m-n ∈ O(n).§.§ Solving Systems of Diophantine EquationsThe problem of solving a system of Diophantine equations is to compute x such thatAx = bx ∈^m.for a given matrix A ∈^n × m and vector b ∈^n.The classical Euclidean algorithm can be extended to compute x,y ∈ such thatax + by = (a,b)and therefore solve Diophantine Equations of the form a_1 x_1 + … a_n x_n = b, by applying the algorithm iteratively. Similarly, our algorithm can be extended to compute a basis matrix B ∈^n × n with (A) = (B) and matrix U ∈^m × n such thatA U = B.Using U, one can solve (<ref>) by first solving the linear system of equations Bx = b. The Diophantine equation (<ref>) is feasible if and only if x is integral and a solution to (<ref>) is then given by U x.The computation of U can be realised very similar to the computation of Y in e.g. <ref>. Initially set the columns of U to e_i if this column of A is the ith column of the initial basis B. For an exchange stepB_i = c - (∑_j≠ iB_jx_j + B_ix_i)set v_j' = -x_j, v_i' = -x_i and v_k = 1 if c is the kth column and j' and i' are the current indices for the columns of j and i, respectively. The exchange step can be expressed in U as U_i = Uv. The index for basis column i changes to k. However, note that this procedure requires an additional term of (mn^2log( B^(1))logA_∞) in bit complexity.§ CONCLUSION AND FUTURE RESEARCHOur novel approach for lattice basis computation provides the first running time improvement since 1996 based on a generalization of the Euclidean algorithm. However, this improvement applies only if we count arithmetic operations. A natural direction for future research would be to investigate whether this approach can improve also on the bit complexity in general. A similar approach like Schönhage <cit.> for the classical Euclidean algorithm might also work for the generalization that we presented.Furthermore, it would be interesting to see how the algorithms perform in practice. Given that the determinant of the initial basis matrix B should be smaller than the worst case Hadamard bound in most practical instances, our algorithms might actually perform rather well. Moreover, the improvement on the determinant on average in practice will be much better than 1/2. alpha§ SYSTEM SOLVING AND LINEARLY INDEPENDENT VECTORS Is there an easier way?Let A∈^m× n have full column rank n. A 2- massager for A is a triple of matrices (P, S, M) from ^n× n such that, * P is a permutation,* S = diag(s_1, …, s_n) the 2-Smith form of A,* M is unit upper triangular, and* APMS^-1 is integral with Rem(APMS^-1, 2) ∈/(2)^m× n having full column rank over /(2).(P, S, M) is a reduced 2-massager if entries in column i of M are from [0,…, s_i-1], 1≤ i≤ n.There exists an algorithm that takes as input a full column rank A∈^m× n and returns as output a reduced 2-massager for A together with the massaged matrix B := APMS^-1. The cost of the algorithm is O(mn^ω-1M(d)log^2n)bit operations, where d=log n+logA_∞. We adjust the proof of Theorem 12 in <cit.> to obtain the following lemma.Let A∈^m× n have full column rank. There exists an algorithm that finds indices i_1, …, i_n such that A_i_1, …, A_i_n are linearly independent using (mn^ω-1logA_∞) bit operations.Compute a 2-messager (P,M,S) of A together with matrix B = APMS^-1 using (mn^ω-1logA_∞) bit operations. The messaged matrix B has full column rank modulo 2. Working modulo 2, use the LSP decomposition of Ibarra et al. <cit.> to compute the row rank profile [i_1, …, i_n] of B over /(2) using O(mn^ω-1) bit operations. Since PMS^-1 is non-singular, the submatrix of A comprised of rows i_1, …, i_n is also non-singular. For non-singular A∈^n× n and b∈^n there exists an algorithm that computes the 2-adic expansion of 2^eA^-1b with precision high enough that A^-1b overcan be recovered using rational number reconstruction. If logb_∞∈ O(n(log nlogA_∞), then the running time of the algorithm is (n^ωlogA_∞) bit operations.If we modify the algorithm just slightly, we get an algorithm to solve A^-1B for a matrix B.Sanity check: Wouldn't this imply a cubic algorithm for matrix inverse calculation? B=I_nUpdate: Actually I previously missed one part of their analysis. At some point we require multiplication of matrices with dimensions n× n and n × (nm). This can be done using rectangular matrix multiplication <cit.> in time O(n^ω(log_n(nm))) ≤(n^ω(2)·m/n) ≤(m· n^2.251640)which is at least still a little bit better than previous algorithms since ω(2)-1 < ω for currently known values. The improvement is therefore significantly smaller. However, this kind of result seems more realistic due to the connection to matrix inversion. * Precise procedure for rectangular multiplication? Faster for largest k∈ s.t. n^k-1≤ m? Eg for m>n^4 we could do (n^ω(5)·m/n^4)≤(mn^2.157233 + n^6)≤(mn^2.157233)* How to get most efficient into a nice formula?* m<n? Fill to n× n^2 then (n^3 + n^ω(2)· 1) is dominated by (n^ω+1)For a non-singular matrix A∈^n× n and a matrix B∈^n× m there exists an algorithm that computes the 2-adic expansion of 2^eA^-1B with precision high enough that A^-1B overcan be recovered using rational number reconstruction. If logb_∞∈ O(n(log nlogA_∞), then the running time of the algorithm is (n^ωlogA_∞ + mn^2 + m· n^2.251640logA_∞) bit operations.We modify the algorithm(<cit.> Figure 8) slightly to an algorithm . The output is X = (2^eA^-1B, 2^d) instead of x = (2^eA^-1b, 2^d). We therefore modifyline 4 to Y = (U,B,d,n,m) instead of y = (U,b,d,n,1). This requires an increased running time compared to the previous analysis. The subroutineis not effected from the changes and still computes the straight line formula in time (n^ωlogA_∞). However, computing matrix multiplication from the straight line formula has increased cost for rectangular matrix operations. Having a closer look at the proof of <cit.> Lemma 2, the bit complexity for Y = (U,B,d,n,m) is(n^ωlogA_∞ + nmpd + n^ω(log_n(mn))) = (n^ωlogA_∞ + mn^2 + n^ω(log_n(mn)))≤(n^ωlogA_∞ + mn^2 + m· n^2.251640logA_∞),using rectangular matrix multiplication <cit.> and where p is the precision as defined in their work.Of course we also need to changeof their algorithm to X = (PM(2^eS^-1)Y, 2^d) instead of x = (PM(2^eS^-1)y, 2^d). Changing to rectangular matrix-matrix-multiplication, this step requires (mn^ω -1logA_∞) bit operations. Actually this might also require rectangular matrix matrix multiplication and needs further checking.Therefore, the algoithm has a running time of in total (n^ωlogA_∞ + mn^2logA_∞) bit operations.From <cit.> we can derive the following lemma. Assuming a suitable input of bit length b, rational number reconstruction can be done using O(B(b)) = (b) bit operations.Hence, rational number reconstruction of the solution X computed in <ref> can be done in time (mnB(n(log n +logA_∞))) = (mn^2logA_∞). We update B by exchanging one column b_i by b - (∑_j≠ ib_jx_j+ b_i x_i). Alternatively we could represent this as follows. Define Y=I_n at start of algorithm. Current basis is AY. Update of basis byY ← Y · (e_1, …, e_i-1, v, e_i+1, …, e_n), where v_i= x_i - x_i and v_k={x_k} for k≠ i. At the end, we can represent the final basis in terms of the first by B= BY. We can use this also to calculate the modulo B in every iteration (calculate Y then take fractional part {.} of every entry).hnf[HNF]Hermite normal form snf[SNF]Smith normal form de[DE]Diophantine Equations lbr[LBR]Lattice Basis computation | http://arxiv.org/abs/2311.15902v1 | {
"authors": [
"Kim-Manuel Klein",
"Janina Reuter"
],
"categories": [
"cs.DS",
"cs.DM",
"math.NT",
"F.2.2; G.2.1"
],
"primary_category": "cs.DS",
"published": "20231127150534",
"title": "Simple Lattice Basis Computation -- The Generalization of the Euclidean Algorithm"
} |
[Perspective on new implementations ofatomtronic circuits Juan Polo^1*, Wayne J. Chetcuti^1, Enrico Domanti^1,2,3, Philip Kitson^1,2,3, Andreas Osterloh^1, Francesco Perciavalle^1,4, Vijay Pal Singh^1, Luigi Amico^1,2,3,5 January 14, 2024 ======================================================================================================================================================================= < g r a p h i c s > figureThe prediction of a captioner-based classifier is influenced by the linguistic priors in pure text modality. The Information Gain (IG) evaluation reduces such impact and makes the predictions more grounded on the visual inputs. We illustrate with real predictions on zero-shot ImageNet <cit.> classification in this figure. (a) The language model was trained on Laion-5B <cit.> captions only. (b) The captioner was trained on the Laion-5B dataset with both the images and captions. The IG evaluation in (b) uses the outputs of both the captioner and the language model in (a). ] Perspective on new implementations ofatomtronic circuits Juan Polo^1*, Wayne J. Chetcuti^1, Enrico Domanti^1,2,3, Philip Kitson^1,2,3, Andreas Osterloh^1, Francesco Perciavalle^1,4, Vijay Pal Singh^1, Luigi Amico^1,2,3,5 January 14, 2024 ======================================================================================================================================================================= Work done while an intern at Google DeepMind.Generative training has been demonstrated to be powerful for building visual-language models. However, on zero-shot discriminative benchmarks, there is still a performance gap between models trained with generative and discriminative objectives. In this paper, we aim to narrow this gap by improving the efficacy of generative training on classification tasks, without any finetuning processes or additional modules.Specifically, we focus on narrowing the gap between the generative captioner and the CLIP classifier. We begin by analysing the predictions made by the captioner and classifier and observe that the caption generation inherits the distribution bias from the language model trained with pure text modality, making it less grounded on the visual signal. To tackle this problem, we redesign the scoring objective for the captioner to alleviate the distributional bias and focus on measuring the gain of information brought by the visual inputs. We further design a generative training objective to match the evaluation objective. We name our model trained and evaluated from the novel procedures as Information Gain (IG) captioner. We pretrain the models on the public Laion-5B dataset and perform a series of discriminative evaluations. For the zero-shot classification on ImageNet, IG captioner achieves > 18% improvements over the standard captioner, achieving comparable performances with the CLIP classifier. IG captioner also demonstrated strong performance on zero-shot image-text retrieval tasks on MSCOCO and Flickr30K. We hope this paper inspires further research towards unifying generative and discriminative training procedures for visual-language models. § INTRODUCTIONGenerative training has been shown to be highly effective for both the language <cit.> and vision models <cit.>. These generative models demonstrate remarkable zero-shot generation capabilities and have led to the developments of many impressive AI applications <cit.>. On the other hand, discriminative training is still an effective procedure for building the large vision-language models like CLIP <cit.> and CoCa <cit.>. Models trained with discrminative objectives naturally performs strongly on discriminative tasks like classification.The inconsistent setups between generative and discriminative training raises an intriguing topic, which we aim to address in this paper: Can generative training procedure also perform well on zero-shot discriminative tasks? Concretely,we study the performance gap between generative captioners and discriminative models on zero-shot classification tasks, and propose novel approaches to narrow the gap.To obtain an effective zero-shot generative classifier, we start with evaluating a standard generative captioner with the Maximum Likelihood Estimation (MLE) objective on classification tasks. Specifically, we pretrain a captioner on the public Laion-5B <cit.> dataset and evaluate it on the ImageNet <cit.>. We also pretrain a CLIP classifier with the same experimental setting. We observe a large performance gap between the captioner and the CLIP classifier.We find that the predictions of the captioner are less grounded on the visual inputs. Instead, they are biased by the linguistic priors, which is illustrated in fig: bias_from_linguistic_prior. These observations indicate that we should discount the influence by predictions from pure text modality in multimodal generative classification setup.To deal with this problem, we propose an Information Gain (IG) evaluation objective, as depicted in fig: inference. This objective demonstrates a significant performance boost on top of a captioner trained with conventional procedure. Furthermore, we propose a generative training objective to match the IG evaluation. We name the captioner, trained and evaluated with our proposed objectives, as Information Gain (IG) captioner. We pretrain the models on the Laion-5B dataset and evaluate multiple zero-shot discriminative tasks. All the models have the same training and evaluation settings to ensure the fair comparisons. For zero-shot ImageNet classification, IG captioner with ViT-B/ViT-L <cit.> image encoder shows 19.7%/18.1% improvements on top-1 accuracy over the standard captioner. It is even better than the CLIP classifier by 0.5%/1.6% top-1 accuracy. For zero-shot image-text retrieval on MSCOCO <cit.> dataset, IG captioner demonstrates > 22% accuracy improvements on all of image-to-text R@1, R@5 and R@10 over the captioner, significantly reducing the gap from the captioner to the CLIP classifier. Interestingly, we observe the superior performances on text-to-image recalls of both the captioner and IG captioner over the CLIP classifier. The superiority of IG captioner is also demonstrated on Flickr30K <cit.> dataset.Our main contributions can be summarized in the following: * We convert a generative captioner into a zero-shot classifier. We identify the root cause of its poor performance and observe that its predictions are negatively influenced by the linguistic priors and less grounded on the visual inputs. * We propose the Information Gain (IG) method to evaluate a generative classifier. We demonstrate that the IG evaluation is able to significantly improve its performance without any changes in the trainng process.* To couple with the IG evaluation, we propose a generative training objective to further boost the performance of the generative classifier which we name Information Gain (IG) captioner.* We show that IG captioner is a strong zero-shot classifier, evidenced by the significant improvements over the captioner and comparable performance with the CLIP classifier on the zero-shot ImageNet classification, MSCOCO and Flickr30K image-text retrieval tasks. § RELATED WORK Text-to-image Generative Classifiers. The classic approach <cit.> of using generative models to perform recognition tasks is the Bayes algorithm <cit.>. During training, this algorithm models the data distribution while during inference, it provides predictions by solving a maximum likelihood estimation (MLE) problem. Recently, there have been the methods <cit.> proposed to convert the text-to-image diffusion models, Stable Diffusion <cit.> or Imagen <cit.>, into a zero-shot classifier using the Bayes algorithm. The focus of IG captioner is different with these previous arts. First, we explore the possibility of achieving a good classifier solely through the generative training, instead of how to convert a generative model into a classifier. The Stable Diffusion used by the above diffusion model classifiers <cit.> is not suitable for our goal because it uses the CLIP text encoder pretrained with contrastive loss to provide the text guidance. Second, we identify the negative impact of the text priors inherited from the pretraining data and propose methods to reduce this impact for zero-shot classification tasks. Besides, IG captioner is an image-to-text generative captioner. We also perform the comparisons in tab: compare_image_to_text.Classifier-Free Guidance (CFG). CFG <cit.> is an important approach to improve the sample quality for the text-to-image generative models <cit.>. During training, CFG requires the models to generate images both w/ and w/o the text conditions. During inference, the output image is sampled according to a linear combination of both the conditional and unconditional predictions to improve the text-image alignment, which relates CFG and IG captioner from a high level. Differently, IG captioner is an image-to-text captioner and uses the probability gain from unconditionally to conditionally generating the captions with the aim of reducing the bias from text priors for zero-shot classification tasks.Contrast in Text Generation. In the field of text generation, researchers have been investigating the use of contrast, including avoiding the text degeneration and undesirable repetitions <cit.>, generating pan sentences <cit.>, and language detoxification <cit.>. Contrastive decoding <cit.> uses the log-likelihood difference between a large expert and a small amateur model to improve the quality of the open-ended text generation. Li et al. <cit.> propose to use the mutual information between an input message and its responses as the decoding objective to avoid the generic and meaningless conversations. Differently, in this paper, IG captioner uses the Information Gain (IG) to address the problem of evaluating a vision-language generative model as a zero-shot classifier. Importantly, the IG evaluation is used to mitigate the negative influence of linguistic priors inherited from the pretraining dataset. § METHODIn this section, we introduce Information Gain (IG) captioner. First, we illustrate how to evaluate a conventional captioner on the zero-shot classification task and diagnose its problem. Second, we propose an evaluation approach using the information gain to tackle the problem. Third, we design a training objective to couple with the IG evaluation. Combining the above evaluation and training procedures, we propose IG captioner. §.§ Evaluating Captioners on Discriminative TasksA captioner is a generative model that generates caption for a given image. It is by nature not trained as a classifier and its output can not be used directly for such tasks like image classification or image-text retrieval. Therefore, to evaluate a captioner on discriminative tasks, we need to adapt the evaluation procedure.Maximum Likelihood Estimation (MLE) Objective. Let I ∈ℝ^H × W × 3, T ∈ℝ^N × D and t ∈ℝ^D be the image, caption and the word in the caption, where H, W are the height and width of the image and N, D are the number of text tokens and the token dimension. A captioner is a function F_θ: I → T where θ is the model parameter.Discriminative tasks require the model to pick the most appropriate text from a set of candidates C={T_i} to describe an image, where i is the index of captions. During inference, the standard MLE evaluation objective is:_i[ logP(T_i|I)]We omit the normalization factor 1 / N_i when calculating the evaluation objectives throughout this paper for simplicity. N_i is the token length of the t^th caption.Low Performance Using MLE Objective. To study the effectiveness of MLE as objective, we pretrain a captioner on Laion-5B and use it to perform zero-shot ImageNet classification. We set up a strong baseline: a CLIP classifier which is pretrained with contrastive loss and tailored for classification tasks. The captioner has a image encoder and a text decoder while the CLIP classifier has a image encoder and a text encoder. Both of them have the same model size, training and evaluation settings to ensure fair comparisons. The results are shown in tab: standard_eval. When directly using the MLE objective (equ: standard_eval) the captioner shows >19% top-1 accuracy degradation compared with the CLIP classifier.Result Analaysis of MLE Evaluation. Since the captioner is a vision-language model, we are interested in understanding how much visual information is provided by the visual inputs and utilized by the model. We pretrain a language model that has the same architecture as the text decoder of the captioner. It models the captions in the Laion-5B dataset and is able to predict log P(T). There are 1K classes on ImageNet and for each class we use the 80 prompts proposed by CLIP. In total, there are 80K unique captions. Given an image I, we consider log P(T|I) and log P(T) as two variables and compute their Pearson correlation coefficient (PCC) <cit.> with 80K observations in tab: pcc_pti_pt.We observe a strong correlation between log P(T|I) and log P(T), which indicates that the captioner is strongly biased by the text priors and tends to ignore the visual information. The correlation is also visualized in fig: corr.§.§ Evaluating Captioners with the Information Gain (IG) To alleviate the problem of strong correlation between caption likelihood conditioned on the image and caption priors,we tun to the following evaluation objective:_i[ logP(T_i|I)/P(T_i)]Intuitively, this log-ratio measures the likelihood gain of the text T_i given the image I. It is called pointwise mutual information <cit.>. The expectation of it over all the texts T is the information gain <cit.>. In this paper, we refer to pointwise mutual information as information gain.We measure the correlation of log P(T_i) and [log P(T_i|I) - log P(T_i) ] in tab: pcc_pti_pt. It shows that the correlation is significantly reduced. Using this objective, we observe the performance gain in our ablation studies in sec: ablate_alpha. However, we notice that this quantity is negatively correlated with the caption priors, which is expected due to the existence of term - log P(T_i) in equ: IG_eval. To deal with this negative correlation problem, we further propose the following evaluation objective:_i[ log P(T_i|I) - αlog P(T_i) ]where α∈ [0, 1] adjusts the degree of the text prior removal. Our thorough ablation study on α can be found in sec: ablate_alpha.§.§ Training Information Gain Captioners Without any changes on the training process, we observe the performance improvement of the captioner on the zero-shot classification task when using IG evaluation, which is shown in sec: IG_eval. But in order to keep the training and evaluation consistent, we propose a two-objective approach to train the captioner, which is illustrated in fig: training_pipeline.The first objective is the standard generative loss. Given an image, it supervises the captioner to predict the words of its caption in an autoregressive way:l_multimodal =1/B∑_i=1^B1/N^i∑_n=1^N^ilog1/P(t_t^i|t_1:n-1^i, I^i)where i is the index of a image-text pair (I^i, T^i) in one batch of size B. N^i is the text token length. Since this objective models the probability of a caption conditioned on an image, we call this objective the multimodal loss.The second objective makes the captioner an autoregressive language model that is able to generate captions without given any image:l_unimodal =1/B∑_i=1^B1/N^i∑_n=1^N^ilog1/P(t_n^i|t_1:n-1^i)We call this objective the unimodal loss. Finally, we combine the above two losses in the following:L = β l_multimodal + γ l_unimodalwhere β and γ are the weights for each loss. We perform an ablation study on the loss weights in sec: ablate_loss_weights. We name the captioner which is trained and evaluated with our proposed objectives IG captioner.§ EXPERIMENTS§.§ Zero-shot ImageNet classificationSettings. We use the English subset of the public Laion-5B <cit.> dataset as the pretraining dataset for all experiments. After pretraining, we perform zero-shot image classification on the ImageNet <cit.> validation dataset. There are 1000 classes in this validation set and each class has 50 images. We build the captioner and IG captioner with an image encoder and a text decoder, and build the CLIP <cit.> classifier with an image encoder and a text encoder. All the models consist of standard transformer blocks <cit.>. We set the layer number of image encoder and text decoder / text encoder the same. We use a batch size 4096 for all the experiments. During evaluation, we follow CLIP to use 80 prompts for each class. We use the voting mechanism to combine the predictions from different prompts for the captioner and IG captioner. All the training and evaluation procedures are the same to ensure fair comparisons.Results. The results are summarized in tab: zero_shot_imagenet. First, compared with standard captioners, IG captioner achieves 19.7% and 18.1% top-1 accuracy improvements when using ViT-B and ViT-L <cit.> as the image encoders. Second, compared with the strong discriminative baselines, CLIP classifiers, IG captioner achieves 0.5% / 1.6% top-1 accuracy improvements when using the image encoder ViT-B / ViT-L. These improvements demonstrate that IG captioner is a strong zero-shot classifier as a pure generative model. We measure the correlations of the evaluation objective and the caption prior in sec: ablate_alpha to validate the effectiveness of the design of IG captioner. Besides, we perform the ablation studies in sec: IG_eval to demonstrate the efficacy of the IG evaluation on the standard captioner.§.§ Zero-shot Image-text Retrieval on MSCOCO and Flickr30KSettings. We use the same models in sec: main_imagenet pretrined on Laion-5B to perform zero-shot image-text retrieval on MSCOCO 5K and Flickr30K 1K test sets. There are 5 captions for each image. In order to measure the recalls, we compute a similarity matrix 𝐌_a × b where a and b are the number of images and captions. For the CLIP classifier, 𝐌_i, j^ = E_I_i·E_T_j, where E_T_j and E_I_i are the embeddings of the i^th image and j^th caption. For the captioner, we compute 𝐌_i, j^ = log P(T_j|I_i) while for the IG captioner, we compute 𝐌_i, j^ = log P(T_j|I_i) - log P(T_j). Results on MSCOCO. We summarize the results on MSCOCO in tab: zero_shot_mscoco. For the image → text retrieval, IG captioner significantly boosts the performance of a generative captioner. IG captioner shows 22.7% / 25.5% / 22.8% improvements over the captioner on R@1 / R@5 and R@10 with the ViT-B image encoder, and shows 22.9% / 24.1% / 20.3% imporvements with the ViT-L image encoder.Furthermore, IG captioner achieves comparable performances with the CLIP classifier. The performance gaps between the IG captioner and the CLIP classifier are 8.9% / 6.6% / 5.0% on R@1 / R@5 / R@10 with the ViT-B encoder. While with the ViT-L image encoder, the gaps are further decreased to 6.5% / 5.1% / 3.5% on R@1 / R@5 / R@10. Interestingly, for the text → image retrieval, we find that both the captioner and IG captioner have better performances than the CLIP classifier. Since our approach is to remove the bias in the text prediction, it does not affect the text → image retrieval. We leave this for the future research.Results on Flickr30K. The results are summarized in tab: zero_shot_mscoco. We observe the similar phenomena on MSCOCO. For the image → text retrieval, on the one hand, IG captioner shows significant improvements over the captioner. With the ViT-B image encoder, IG captioner improves the R@1 / R@5 / R@10 by 13.7% / 16.9% / 13.2%, and with the ViT-L image encoder, the improvements are 8.1% / 11.4% / 9.6%. On the other hand, the performance gaps between the IG captioner and the CLIP classifier are significantly reduced. For R@1, the gaps are 24.6% / 23.3% with ViT-B / ViT-L image encoder. However, for R@5 / R@10, the gaps become 7.2% / 4.0% with ViT-B encoder, and 4.9% / 1.3% with ViT-L encoder. For the text → image retrieval, we observe the similar superiority of both the captioner and IG captioner over the CLIP classifier. § ABLATION STUDIES§.§ Weights of the Text Prior Removal and Correlation MeasurementsSettings. In this section, we ablate the α in the evaluation objective of IG captioner, log P(T|I) - αlog P(T). We use the same training and evaluation settings as the main experiments. We pretrain the IG captioner with the ViT-B encoder on the Laion-5B dataset and perform zero-shot ImageNet classification and zero-shot image-text retrieval on MSCOCO and Flickr30K. We ablation the α from 0.0 to 1.0 with a step size 0.1. For the retrieval tasks, we do not list the text → image recalls because the value of α does not affect the predictions of text → image retrieval.In order to validate our goal of removing the bias from text priors, we report the Pearson Correlation Coefficient (PCC) of log P(T) and the evaluation objective when varying the value of α.Results. We summarize the results in tab: ablate_alpha. On the one hand, the smaller correlation between log P(T|I) and log P(T) generally corresponds to better performances. For the ImageNet classification, when α = 0.08, the best PCC 0.08 and performance 64.3% are achieved at the same time. For the retrieval tasks on MSCOCO / Flickr30K, the best PCC, 0.0 / 0.12, and the highest R@10, 82.5% / 95.8%, are obtained simultaneously with α = 0.9. On the other hand, it requires a degree of text priors to achieve the best fine-grained performances on specific datasets. The highest R@1, 47.4% / 63.4%, correspond to the PCC, 0.19 / 0.48, on MSCOCO / Flickr30K. Through this ablation studies, we set α = 0.8 for all the other experiments. α = 0.0 corresponds to the bad strong correlations, resulting in the poor performances. α = 1.0 causes the negative correlations and produces the unsatisfying performances. §.§ Language Model + Captioner versus IG Captioner Settings. IG captioner has a text decoder that is able to predict both log P(T|I) and log P(T). One variant of IG captioner consists of a pair of models. One of them is the unimodal Language Model (LM) to predict log P(T). The other one is the multimodal captioner, a standard captioner to predict log P(T|I). During inference, the prediction is based on the subtraction of their outputs. We call this LM + Cap approach.Results. We report the results in tab: ablate_two_captioner. It is observed that the LM + Cap approach performs worse than the IG captioner. Besides, it is less efficient due to the extra language model. From another perspective, it demonstrates that the text decoder of IG captioner is able to perform both the multimodal and unimodal tasks well. §.§ IG Evaluation Settings. IG captioner differs from a standard captioner in both the training and evaluation phases. In this section, we study whether the IG evaluation will improve a standard captioner as a classifier. The IG evaluation requires the prediction of log P(T). Since the standard captioner can not directly predict this, we feed the captioner zero-intensity images to compute log P(T|0) as the approximation.Results. We show the results in fig: ablate_ig_eval. The IG evaluation is able to significantly improve the performance of the captioners. Specifically, with the ViT-B / ViT-L image encoder, IG evaluation increases the zero-shot top-1 ImageNet classification accuracy by 9.0% / 7.9%.Adding the training part of IG captioner, the performances will be further improved by 10.7% / 10.2%. §.§ Comparisons to Text-to-image Generative ClassifiersSettings. There are two kinds of vision-language generative models: text-to-image generative models and image-to-text captioners. There have been recent works <cit.> that successfully convert the large text-to-image diffusion models to the zero-shot generative classifiers. We compare IG captioner with these models.Results. The results are summarized in tab: compare_image_to_text. With the same or smaller amount of pretraining data, IG captioner demonstrates the best performances. Compared with the Text-to-Image Classifiers <cit.> and Diffusion Classifier <cit.> using Imagen <cit.> and Stable Diffusion <cit.>, respectively, IG captioner shows 7.1% / 8.4% zero-shot top-1 accuracy improvements on ImageNet. Besides, IG captioner is a more compact model and only uses 53.5% / 73.8% of their parameters. §.§ Weights of the Multimodal and Unimodal Training LossesSettings. We ablate the multimodal loss weight β and the unimodal loss weight γ in the training objective of IG captioner (equ: IG captioner_training_obj).We train the IG captioners with ViT-B encoder on the Laion-5B dataset and evaluate them on zero-shot ImageNet classification.Results.The results are shown in tab: ablate_loss_weights. The equal weights do not produce the optimal results.We notice that image encoder is only trained with the multimodal loss while the text decoder is supervised by both the losses. We set the multimodal loss weight β = 1.5 and the unimodal loss γ = 0.5 for all the other experiments.§ CONCLUSIONIn this paper, we tackle the challenging problem of using a generative captioner to perform zero-shot classification tasks. We identify that the captioner is negatively impacted by linguistic priors and likely to ignore the visual signals. To reduce such impact, we propose the Information Gain (IG) captioner with the novel and effective approaches for both the training and evaluation processes. IG captioner demonstrates to be a strong zero-shot classifier. We hope that this paper will encourage further research on developing more powerful zero-shot generative classifiers. ieeenat_fullname | http://arxiv.org/abs/2311.17072v1 | {
"authors": [
"Chenglin Yang",
"Siyuan Qiao",
"Yuan Cao",
"Yu Zhang",
"Tao Zhu",
"Alan Yuille",
"Jiahui Yu"
],
"categories": [
"cs.CV",
"cs.AI",
"cs.LG",
"cs.MM"
],
"primary_category": "cs.CV",
"published": "20231127190006",
"title": "IG Captioner: Information Gain Captioners are Strong Zero-shot Classifiers"
} |
Technical Report for Argoverse Challenges on 4D Occupancy Forecasting Pengfei Zheng Equal contribution Work done as an intern at Lenovo Research. 1, Kanokphan Lertniphonphan[1] 3, Feng Chen [1] 3, Siwei Chen [footnote] 2,Bingchuan Sun3, Jun Xie3, Zhepeng Wang31University of Science and Technology Beijing, 2Tsinghua University3Lenovo [email protected] 14, 2024 ===================================================================================================================================================================================================================================================================================================================This report presents our Le3DE2E_Occ solution for 4D Occupancy Forecasting in Argoverse Challenges at CVPR 2023 Workshop on Autonomous Driving (WAD). Our solution consists of a strong LiDAR-based Bird's Eye View (BEV) encoder with temporal fusion and a two-stage decoder, which combines a DETR head and a UNet decoder. The solution was tested on the Argoverse 2 sensor dataset to evaluate the occupancy state 3 seconds in the future. Our solution achieved 18% lower L1 Error (3.57) than the baseline on the 4D Occupancy Forecasting task in Argoverse Challenges at CVPR 2023. § INTRODUCTION4D occupancy forecasting is a new challenge introduced in CVPR 2023 WAD. The task is to understand how an environment evolves with time which is crucial for motion planning in autonomous driving. However, conventional methods require costly human annotations, such as detection bounding boxes, tracking ids, or semantic segmentation, which make it difficult to scale up to a large labeled dataset. This challenge aims to learn future occupancy forecasting from unlabeled datasets. In this challenge, given a particular agent's observation of the world in the past n seconds, we need to predict the space-time evolution of the world in the future n seconds. Specifically, the occupancy state of 5 frames in the next 3s is predicted by observing the point cloud of 5 frames in the past and present timestamp within 3s (at a frequency of 5/3Hz). The future frame point cloud is obtained by rendering the occupancy state from a given query ray. Then, all point clouds and occupancies are aligned to the current frame under the LIDAR coordinate system.Our solution adopts a voxel feature encoder to transform LiDAR point clouds into a 2D Bird's Eye View (BEV) feature map and uses a DETR<cit.> structure head to predict voxel-wise future occupancy. To refine the results, a UNet<cit.> head is employed as a second-stage decoder to produce more accurate forecasting results.§ METHODWe employ the bev feature asa unified representation and adopted the OccFormer module as an occupancy head for 4d occupancy forecasting based on UniAD<cit.>. OccFormer module consists of a transformer decoder with T sequential blocks. The pipeline of our method is shown in Figure 1. §.§ BEV featureWe employ the LIDAR BEV encoder based on SECOND<cit.> to generate the LIDAR BEV feature B_l. The LIDAR encoder takes as input the current and past T=5 frames of the point cloud with a time step of 0.6 s. All point clouds have been aligned to the current frame. Afterward, we fuse the past T frames with the current frame to aggregate temporal features by a 2D convolutional block. The spatial-temporal fused BEV feature map is fed to the occupancy decoder. §.§ OccFormer Given the BEV feature from upstream modules, we feed it to OccFormer to get the multi-time-step occupancy results. OccFormer consists of T sequential blocks, where T=5 is a number of future frames. Each block is responsible for generating the occupancy of a particular frame. Unlike the instance occupancy output by OccFormer in UniAD, we need dense voxel-wise occupancy. In each sequential block, the BEV features first perform self-queries through the self-attentive layer to compute the similarity metric within the feature. Then, werandomly initialize the instance query Q_I, which are track query Q_A , agent position P_A and motion query Q_X. The instance query Q_I and BEV feature interact using cross-attention so that the dense scene feature and sparse agent feature benefit from each other. The interacted BEV feature is sent to the next sequential block, which is cycled in RNN fashion. All T frame BEV features are upsampled to obtain future occupancy O_T. Dimensions of all dense features and instance features are 256 in OccFormer.A UNet head is used as a second-stage decoder to enhance the multiscale forecasting results. After generating voxel-wise occupancy forecasting, voxel rendering is then performed by the query rays to get the point-wise estimated depth. § EXPERIMENTS §.§ Dataset The competition used the Argoverse 2 Sensor Dataset<cit.>, which consisted of 1000 scenes (750 for training, 150 for validation, and 150 for testing) with a total of 4.2 hours of driving data. The total dataset is extracted in the form of 1 TB of data. Each vehicle log has a duration of approximately 15 seconds and includes an average of approximately 150 LiDAR scans with 10 FPS LiDAR frames. §.§ Evaluation Metrics The metric is performed in the range of [-70m, -70m, -4.5m, 70m, 70m, 4.5m] with a voxel resolution of 0.2m.The absolute L1 distance between the true expected depth along a given ray direction and the predicted expected depth obtained by rendering along the same ray direction is used as the main metric. Absolute relative error (AbsRel), near-field chamfer distance (NFCD), and vanilla chamfer distance (CD) are measured together as other metrics. More details of the evaluation can be found in the paper <cit.>. §.§ Implementation details We followed the baseline <cit.> for data preparation for training and evaluation. Point shuffle is used for data augmentation in the training stage. The voxel size is set to (0.075m, 0.075m, 0.2m) and the resulting BEV feature B_l has a shape of 240*240. We train the model with a cosine annealing policy with a 0.001 learning rate and use L1 loss and Adam optimizer. We train the model from scratch for 20 epochs on 8 V100 GPUs with a total batch size of 8. We submitted our results to the testing server and got a 3.57 L1 Error. The results are shown in <ref>. Our results outperform the baseline with 18% and 15% improvements in L1 Error and Absolute Relative L1 Error, respectively. § CONCLUSIONIn our model, we employ a LIDAR encoder to encode spatial features and then do a temporal fusion of historical BEV features. We use the BEV feature as a unified intermediate representation. We employ an OccFormer as 4D occupancy prediction head to loop out future occupancy in RNN style. Following the work of <cit.>, we "render" point cloud data from 4D occupancy predictions and estimate depth for supervision. The experimental results indicate that our model achieved better scores than the baseline with 18% and 15% improvements in L1 Error and Absolute Relative L1 Error, respectively.ieee_fullname | http://arxiv.org/abs/2311.15660v1 | {
"authors": [
"Pengfei Zheng",
"Kanokphan Lertniphonphan",
"Feng Chen",
"Siwei Chen",
"Bingchuan Sun",
"Jun Xie",
"Zhepeng Wang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127094053",
"title": "Technical Report for Argoverse Challenges on 4D Occupancy Forecasting"
} |
We continue the study of parking assortments, a generalization of parking functions introduced by Chen, Harris, Martínez, Pabón-Cancel, and Sargent.Given n ∈ cars of lengths =(y_1,y_2,…,y_n) ∈^n, our focuses are the sets _n() and _n() of permutation invariant (resp. nondecreasing) parking assortments for . For =(x_1,x_2,…,x_n) ∈_n(), we introduce the degree of , the number of non-1 entries of , and the characteristic χ() of , the greatest degree across all ∈_n(). We establish direct necessary conditions forwith χ()=0 and a simple characterization forwith χ()=n-1. In the process, we derive a closed form for _n() and an enumeration of |_n()| using properties of the Pitman-Stanley polytope, where χ()=n-1.Next, for any ∈^n, we prove that _n() is closed under the replacement of any of its elements' entries by a 1, and given ^+ ∈^n+1, whereis the prefix of ^+, there is an embedding of _n() into _n+1(^+). We apply these results to study the degree as a function and the characteristic under sequences of successive prefix length vectors. We then examine the invariant solution set (){ w ∈:(1^n-1,w) ∈_n() }. We obtain tight upper bounds of () and prove that for any n ∈, we have |()|≤ 2^n-1, providing constraints on the subsequence sums offor equality to hold. Finally, we show that if ∈_n(), then ∈{ 1 }^n-χ()×()^χ(), which implies a new upper bound on |_n()|. Our results generalize several theorems by Chen, Harris, Martínez, Pabón-Cancel, and Sargent.[ [=====§ INTRODUCTION Parking assortments were introduced recently in <cit.> as a generalization of parking functions, classic combinatorial objects that arose in the 1960s from the study of hash functions and linear probing <cit.>. There are two well-known equivalent definitions of parking functions. The first can be described via a “parking experiment." Consider a one-way street with n ∈{ 1,2,3,…} parking spots.There are n ∈ cars of unit length waiting to enter the street. For each i∈ [n]{ 1,2,…,n }, car i prefers a spot x_i ∈ [n] of the parking lot, so it drives up to x_i and parks if x_i is unoccupied; otherwise, it parks in the next available spot (if it exists).We say that (x_1,x_2,…,x_n) ∈ [n]^n is a parking function of length n if every car can park following their respective preference in .The second is via a property of (x_(1),x_(2),…,x_(n)), which is the nondecreasing rearrangement of the entries of . We say that =(x_1,x_2,…,x_n) ∈ [n]^n is a parking function of length n if x_(i)≤ i for all i ∈ [n].Perhaps the most celebrated result concerning parking functions is the fact that|_n|=(n+1)^n-1,where _n denotes the set of parking functions of length n (cf. Lemma 1 in <cit.>). Such a count allows one to see that parking functions are in bijection with several notable combinatorial objects including labeled trees and the Shi hyperplane arrangement <cit.>.Explicitly constructing and investigating these bijections reveals many illuminating combinatorial properties of parking functions. As an example, the sequence (c_1,c_2,…,c_n-1), where c_i (x_i+1-x_i) n+1∀ i ∈ [n-1],of successive differences modulo n+1 of an ∈_n yields the Prüfer code of a unique labeled tree on n+1 vertices <cit.>. Moreover, let 𝐮=(u_1,u_2,…,u_n) ∈^n be nondecreasing, and consider the Pitman-Stanley polytope Π_n(𝐮){ (p_1,p_2,…,p_n) ∈ℝ_≥ 0^n:∑_i=1^jp_i≤∑_i=1^ju_i ∀ j∈ [n] }. With 𝐮=(1,2,…,n), we have n!V_n(𝐮)=|_n|, where V_n(𝐮) denotes the n-dimensional volume of Π_n(𝐮). More generally, n!V_n(𝐮) is the number of =(x_1,x_2,…,x_n) ∈^n such that x_(i)≤ u_i for all i ∈ [n] (cf. Theorem 11 in <cit.>); suchgeneralize Definition <ref> and are aptly known as 𝐮-parking functions of length n. For a detailed treatment on the combinatorial theory of parking functions, see Yan <cit.>.Given these connections and results, parking functions have been an active research topic, appearing in a diverse array of contexts such as pattern avoidance in permutations <cit.>, convex geometry <cit.>, polyhedral combinatorics <cit.>, partially ordered sets <cit.>, impartial games <cit.>, and of course, the combinatorics of its variations and generalizations <cit.>.Now, we define the combinatorial object of interest in this work: parking assortments. This will involve a natural extension to the parking experiment described in Definition <ref>.Consider a one-way street with m ∈ parking spots.There are n ∈ cars waiting to enter the street, and they have lengths (y_1,y_2,…,y_n) ∈^n, where m=∑_i=1^ny_i.For each i∈ [n], car i (with length y_i) prefers a spot x_i ∈ [m] of the parking lot, so it drives up to x_i and parks if spots x_i,x_i+1,…,x_i+y_i-1 are unoccupied; otherwise, it parks in the next y_i contiguously available spots (if they exist).We say that (x_1,x_2,…,x_n) ∈ [m]^n is a parking assortment forif every car can park following their respective preference in . See Figure <ref>, where =(3,4,2) and =(5,1,6). The parking experiment proceeds as follows. First, car 1, with length 3 and preference 5, occupies spots 5, 6, and 7. Then, car 2, with length 4 and preference 1, occupies spots 1, 2, 3, and 4 because they are all still available. Lastly, car 3, with length 2 and preference 6, occupies spots 8 and 9 because spots 6 and 7 have been occupied (namely by car 1), and spots 8 and 9 are the next 2 contiguously available spots. Note that if =(1^n) (1,1,…,1) ∈^n, then the set _n() of parking assortments foris precisely _n.From Definition <ref>, we see that this set has the peculiar property that for any =(x_1,x_2,…,x_n) ∈_n(), any permutation of its entries '=(x_σ(1),x_σ(2),…,x_σ(n)), where σ∈𝔖_n, is also in _n().This permutation invariance of parking functions is a crucial property used to study _n (cf. 1.1 in <cit.>). However, this is not true for general ; if =(1,2,2), then =(1,1,2) ∉_3() since (2,1,1) ∉_3(), as detailed in <cit.>.This motivates the problem of determining what ∈_n() have this property and hence the definition below. Let ∈^n.We say ∈_n() is a (permutation) invariant parking assortment forif any permutation of its entries is also in _n().In this work, we are interested in the set _n() of invariant parking assortments for . Alternatively, we may study the set _n() of nondecreasing invariant parking assortments for , as it is equivalent to _n() up to permutation.Let =(7,4,6) ∈^3. Then _3()={ (1^3),(1^2,5) }. Indeed, (1^3) ∈_3(), and (1^2,5),(1,5,1),(5,1^2) ∈_3(); in other words, all permutations of the entries of (1^3) and (1^2,5) are also parking assortments, so { (1^3),(1^2,5) }⊆_3(). One can also check that no other elements of [7+4+6]^3=[17]^3 have this property, which yields the reverse inclusion. Determining _n() is generally an arduous task due to the problem of checking each of the elements of [m]^n for invariance. The main motivator behind the notions introduced in this work is to attempt to reduce this search space considerably.A basic approach to narrowing the possibilities is to “decrease the exponent" by noticing that by definition, parking assortments have at least one entry that is equal to 1; otherwise, parking spot 1 is never occupied by any car. This reduces the search space to a size of m^n-1. We can take this idea a step further and specifically study the number of entries of invariant parking assortments that are not equal to 1, a fundamental property we will refer to as their degree. Heuristically, one can expect that most of the entries of an ∈_n() are 1, and so many of the results presented here will concern and analyze the maximum degree across all ∈_n(), which we call the characteristic of . If there is a straightforward way to bound the characteristic, then this can reduce our search space by more factors of m. Let ∈^n.For any =(x_1,x_2,…,x_n) ∈_n(), the degree ofis given by|{ i ∈ [n]:x_i≠ 1 }|.Moreover, the characteristic ofis given byχ()max_∈_n().As a special case, note that χ()=0 is equivalent tobeing minimally invariant, which is when _n()={ (1^n) } <cit.>.Furthermore, we always have χ()∈ [n-1]_0 [n-1] ∪{ 0 }.In Section <ref>, we studyof minimal and maximal characteristic.We first establish a direct necessary condition forto have minimal characteristic. Let =(y_1,y_2,…,y_n) ∈^n.Ifis minimally invariant, then y_1<min(y_2,y_3,…,y_n) and y_2≠∑_j ∈ [n] ∖{ 2 }y_j. Moreover, we work on decreasing the exponent by considering _n() for =(b,a^n-1) and n≥ 2. In particular, we obtain a simple necessary and sufficient condition for ∈^n to have maximal characteristic, which provides an easy way to reduce the search space by at least one more factor of m, as we need only check for invariant parking assortments of degree n-1 when this condition is satisfied. Let =(y_1,y_2,…,y_n)∈^n. Then χ()=n-1 if and only ify_1≥ y_2 and y_2=y_3=⋯=y_n. Additionally, we obtain an explicit description of _n() and corresponding enumerative results. The enumerative formulas for <ref><ref> are deduced via two results: an enumerative result related to the theory of the Pitman-Stanley polytope and empirical distributions <cit.> and a recursive formula for Catalan's triangle <cit.>. Let =(b,a^n-1) ∈^n, where n≥ 2. *If a | b or b>(n-1)a, then ∈_n() if and only ifx_(i)∈{ 1+(k-1)a:k∈ [i] }∀ i ∈ [n].Moreover,|_n()|=(n+1)^n-1and |_n()|=1/n+12nn. *Otherwise, if a ∤ b and b<(n-1)a, then ∈_n() if and only ifx_(i)∈{ 1+(k-1)a:k∈ [i] } ∀ i∈[b/a+1]{ 1+(k-1)a:k ∈[ b/a+1 ] } otherwise.Moreover,|_n()| =∑_j=0^n-b/a-1(-1)^jnj( n-b/a-1 )^j(n-j+1)^n-j-1and|_n()| =n-b/a+1/n+1n+b/ab/a.In Section <ref>, we study the structure of _n() for any ∈^n, which implies special properties of the degree and characteristic. Namely, we prove that _n() is closed under the operation of replacing any of its elements' entries with a 1. We may use this property to compute the image of the degree as a function, reveal an embedding property of the set of invariant parking assortments, and obtain a bound for χ(^+) in terms of χ(), whereis the prefix of ^+ ∈^n+1.To state these results precisely, we need the following list operations: for 𝐯=(v_1,v_2,…,v_n) ∈^n, k ∈, and i ∈ [n], let 𝐯_i (v_1,v_2,…,v_i-1,v_i+1,…,v_n) ∈^n-1, 𝐯_|_i (v_1,v_2,…,v_i) ∈^i, and (1^k,𝐯) (1^k,v_1,v_2,…,v_n) ∈^n+k. The inspiration for Theorem <ref><ref> is due to <cit.>; we present an independent proof in Section <ref>. Let =(y_1,y_2,…,y_n) ∈^n and ^+=(,y_n+1) ∈^n+1. *If (1^n-d,𝐰) ∈_n(), where 𝐰∈^d_>1, then (1^n-d+1,𝐰_i) ∈_n() for all i ∈ [d].*The image of : _n() → [n-1]_0 is [χ()]_0. In other words, for each d ∈ [χ()]_0, there exists ∈_n() with =d.*If ∈_n(), then (1,) ∈_n+1(^+). In particular, we have the embedding [That is, _n() is included in _n+1(^+) up to inserting a 1 at the start of each of its elements.]η: _n()↪_n+1(^+) ↦ (1,). *If χ()=α, then χ(^+) ∈{α,α+1 }.Note that applying Theorem <ref><ref> repeatedly implies that if χ(_|_k)=α for some k<n, then χ() ∈ [α,α+n-k], which helps decrease the exponent, as discussed previously. Moreover, the characteristic is monotonically increasing with respect to sequences {^(j)}_j ∈ of length vectors, where each ^(j) is the prefix of ^(j+1).On the flip side, Theorem <ref><ref> implies that if the characteristic is known, then for every d∈ [χ()]_0, each set { 1 }^n-d× [m]^d, which consists of candidates for ∈_n() with =d, indeed contains at least one such . Consequently, we must check each degree in [χ()]_0 to find invariant parking assortments. Thus, we cannot take the approach of decreasing the exponent further.This motivates the need to consider the analogous approach of “decreasing the base," where we eliminate elements of [m] that cannot be an entry of an invariant parking assortment for . Intuitively, such elements are either too large or not compatible with partial sums of the entries of . The former intuition is made precise in Proposition 2.2 in <cit.>.The latter intuition is the basis for Section <ref>, where we also sharpen the former intuition.Let =(y_1,y_2,…,y_n) ∈^n, and define (){ w ∈:(1^n-1,w) ∈_n() }. *Ifis non-constant and w ∈(), then w≤∑_j=1^n-1y_j.*We have()⊆{ 1+^̱⊤:∈̱{ 0 }×{ 0,1 }^n-1}and |𝒲()|≤ 2^n-1. *Ify_j>∑_i=j+1^ny_i ∀ j ∈ [n-1],then equality is achieved in <ref>. Moreover,|_n()|=2^n-1n-n+1 and |_n()|=2^n-1. *If equality is achieved in <ref>, theny_1≥ y_2 and y_j>∑_i=j+1^ny_i ∀ j ∈ [n-1] ∖{ 1 }. Lastly, we combine the approaches of decreasing the exponent and decreasing the base to show the following inclusion and upper bound. Let ∈^n. Then _n() ⊆{ 1 }^n-χ()×()^χ(). In particular, |_n()|≤2^n-1+n-2n-1.We conclude in Section <ref> with several open problems to spur future study. § LENGTH VECTORS OF MINIMAL AND MAXIMAL CHARACTERISTICIn this section, we study the inverse problem of determining ∈^n of a given characteristic α. Doing so can still allow us to recover _n(); notably, this is immediate for α=0, and it turns out that if α=n-1, then we can explicitly compute _n(). In <ref>, we obtain a direct necessary condition forto be of minimal characteristic, thereby establishing Theorem <ref>. In contrast, <ref> proves a simple equivalent condition forto be of maximal characteristic and provides a closed form of _n() for all such . By establishing a bijection between certain 𝐮-parking functions and elements of _n(), we utilize a result of Pitman and Stanley to enumerate |_n()|; we deduce a recursive formula to enumerate |_n()|. Altogether, <ref> proves Theorems <ref> and <ref>.§.§ Minimal Characteristic Recall the following result for minimally invariant ∈^3. Let =(y_1,y_2,y_3)∈^3.Then,is minimally invariant if and only if y_1<min(y_2,y_3) and y_2≠ y_1+y_3.Thus, one might ask whether this can be generalized for n>3. It turns out that the natural extension of this result is necessary for all n ∈ but not sufficient. We will first prove the necessity, which is Theorem <ref>. The failure of the sufficiency will be easily established after we prove Theorem <ref><ref> in Section <ref>.We consider the contrapositive. Assume first that y_1≥min(y_2,y_3,…,y_n).We claim (1^n-1,1+min) ∈_n(). As y_1≥min, we have 1+min≤ 1+∑_j=1^iy_j for all i ∈ [n-1], so (1^i,1+min) ∈_n(_|_i+1) by Lemma <ref>.It follows that (1^i,1+min,1^n-i-1) ∈_n(). Indeed, under its parking experiment, the last n-i-1 cars, all with preference 1, are forced to successively fill [1+∑_j=1^i+1y_j,m], as the first i+1 cars already occupy [1,∑_j=1^i+1y_j] since (1^i,1+min) ∈_n(_|_i+1). Lastly, to show that (1+min,1^n-1) ∈_n(), note that under its parking experiment, the first car leaves [1,min] unoccupied, and these spots can only be filled by one car.The remaining cars, all with preference 1, either park starting at the first empty spot after [1,min] (if its length exceeds min) or fill [1,min] (if its length is precisely min and no other cars with length min have attempted to park).For the latter possibility, there must exist a smallest index j>1 such that j= by assumption. Thus, [1,min] will be filled by car j, after which point [1,s] is occupied for some s ∈, so cars j+1,j+2,…,n will successively fill [s+1,m].Hence, (1^n-1,1+min) ∈_n(), as needed.Now, assume y_2=∑_j ∈ [n] ∖{ 2 }y_j.We claim that =(1^n-1,1+y_2) ∈_n().To see this, note first that (1+y_2,1^n-1) ∈_n(), as under its parking experiment, the first two cars will occupy [1,y_1+y_2], and the remaining cars all have preference 1, so they will park successively in [1+y_1+y_2,m].Moreover, (1,1+y_2,1^n-2) ∈_n() because under its parking experiment, the first car parks in [1,y_1], while the second car parks in [1+y_2,2y_2]=[m-y_2+1,m] since m=∑_i=1^ny_i=2y_2.Since ∑_j=3^ny_j=m-y_1-y_2=y_2-y_1 and the remaining cars all have preference 1, they will park successively in [1+y_1,m-y_2].Otherwise, let '=(x_1',x_2',…,x_n') be a permutation ofsuch that x_j'=1+y_2, where j>2, and consider the parking experiment for '.Cars 1,2,…,j-1 all have preference 1, so they will occupy [1,∑_i=1^j-1 y_i], where |[1,∑_i=1^j-1 y_i]|>y_2.As car j has preference 1+y_2, it will occupy [1+∑_i=1^j-1y_i,∑_i=1^jy_i].Lastly, any remaining cars again all have preference 1, so they will park successively in [1+∑_i=1^jy_i,m]. Hence, (1^n-1,1+y_2) ∈_n(), as needed.Therefore, if y_1≥min(y_2,y_3,…,y_n) or y_2=∑_j ∈ [n] ∖{ 2 }y_j, then χ()≥ 1, as desired. §.§ Maximal Characteristic Recall the following theorem on _n() for constant . Let = (a^n) ∈^n and = (x_1, x_2, …, x_n) ∈ [m]^n=[∑_i=1^ny_i]^n. We have ∈_n() if and only ifx_(i)∈{ 1+(k-1)a:k ∈ [i] }∀ i ∈ [n]. We now perturband study length vectors of the form (b,a^n-1), where a,b ∈ and n≥ 2.We will refer to suchas “almost constant" (note that constantalso fall into this classification).As seen in the statement of Theorem <ref>, it turns out that even when a≠ b, under certain conditions, this perturbation ofdoes not affect _n(). First, we prove two intermediate steps for the proof of the closed forms of _n() as described in Theorem <ref>. For brevity, for parking spots S,E ∈, let the interval [S,E]{ S,S+1,…,E-1,E } denote the set of parking spots numbered between S and E, inclusive; define its length to be |[S,E]| |{ S,S+1,…,E-1,E }|=E-S+1. Let = (b,a^n-1) ∈ℕ^n, where n≥ 2, and = (x_1, x_2, …, x_n) ∈ [m]^n. If ∈_n(), then x_i≡ 1 a∀ i ∈ [n].We will prove the contrapositive.Assume that x_j ≢1a for some j ∈ [n].By Euclidean division, we may write x_j=aq+r, where q ∈_0∪{ 0 }, and r≠ 1 is a remainder (i.e. r∈ [a-1]_0 ∖{ 1 }).Let '=(x_1',x_2',…,x_n') be any permutation ofsuch that x_1'=x_j.Under the parking experiment for ', the first car will occupy the b spots [aq+r,aq+r+b-1], which leaves [1,aq+r-1] empty.As r≠ 1, |[1,aq+r-1]| is not an integral multiple of a. In particular, none of the remaining cars, which all have length a, can precisely fill [1,aq+r-1]. Thus, we have ' ∉_n() and ∉_n(). Let =(b,a^n-1) ∈^n, where n≥ 2, and = (x_1,x_2,…,x_n) ∈ [m]^n. If ∈_n(), then x_(i)≤ 1+(i-1)a ∀ i ∈ [n].Again, we argue using the contrapositive. Assume that x_(j)>1+(j-1)a for some j ∈ [n]. Then x_(j)=1+qa for some q ∈ [n-1] by Lemma <ref>. Let '=(x_1',x_2',…,x_n') be any permutation ofsuch that x_1'=x_(j). Under the parking experiment for ', since x_1'=x_(j)≤ x_(j+1)≤⋯≤ x_(n), there are at least n-j+1 cars, including the first car with length b, preferring to park in [1+qa,m].The total length of these cars is then at least b+(n-j)a, so to ensure that they all can park, |[1+qa,m]|≥ b+(n-j)a must hold.However,|[1+qa,m]|=|[1+qa,b+(n-1)a]|=b+(n-q-1)a<b+(n-j)a.Hence, ' ∉_n(), so ∉_n(), as desired.We are now ready to prove the closed form of _n(). Let = (b,a^n-1) ∈ℕ^n, where n≥ 2, and = (x_1, x_2, …, x_n) ∈ [m]^n. Ifa | b, then ∈_n() if and only if x_(i)∈{ 1+(k-1)a:k∈ [i] }∀ i ∈ [n]. Let ∈_n().Combining Lemmas <ref> and <ref>, it follows thatsatisfies (<ref>). We now prove the converse.Assume that (<ref>) holds.In other words, for all i ∈ [n], x_(i)≤ 1+(i-1)a and x_i=1+t_ia for some t_i∈ [n-1]_0.We will show inductively that parking succeeds under .Under the parking experiment for , the first car, with length b=ha and preference x_1, will park in spots [x_1,x_1+ha-1], where x_1=1+t_1a for some t_1∈ [n-1]_0; that is, [1+t_1a,(t_1+h)a] is occupied by car 1. Next, the second car, with length a and preference x_2, will attempt to park in [x_2,x_2+a-1], where x_2=1+t_2a for some t_2∈ [n-1]_0.Note that (<ref>) stipulates that at most one entry ofcan be 1+(n-1)a, so if t_1=n-1, then t_2<t_1, so that car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a]. Now, assume t_1<n-1.If t_1+h≤ n-1, then we havet_2 ∈ [t_1-1]_0 ∪{ t_1+h,t_1+h+1,…,n-1 }or t_2 ∈{ t_1,t_1+1,…,t_1+h-1 }. In the former, car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a]. In the latter, car 2 fails to park in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a] since car 1 occupies [1+t_1a,(t_1+h)a] ⊇ [1+t_2a,(t_2+1)a]. Thus, car 2 must park in [1+(t_1+h)a,(t_1+h+1)a]. If t_1+h>n-1, then we havet_2 ∈ [t_1-1]_0 or t_2 ∈{ t_1,t_1+1,…,n-1 }.In the former, car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a]. In the latter, car 2 again fails to park in [x_2,x_2+a-1] for the same reason, so it must park in [1+(t_1+h)a,(t_1+h+1)a]. This proves that car 2 will occupy [S_2,S_2+a-1], where S_2≡ 1 a, which will serve as our base case.Assume now that the first k ∈ [n-1] cars are parked, and each car j, where j ∈ [k] ∖{ 1 }, occupies [S_j,S_j+a-1] for some S_j≡ 1 a.Moreover, for S_1=1+t_1a, car 1 occupies [S_1,S_1+ha-1]. Thus, all the terms of the sequence (S_1,S_2,…,S_k) ∈^k are distinct, so that its nondecreasing rearrangement (S_(1),S_(2),…,S_(k)) is strictly increasing. Now, for any j ∈ [k], define ℓ(j) haif S_(j)=S_1 aotherwise.Then at this stage of the parking experiment, the set of unoccupied parking spots is𝒰 [m] ∖⋃_j=1^k[S_(j),S_(j)+ℓ(j)-1]=⋃_j=0^k𝒰_j, where𝒰_j [1,S_(1)-1]if j=0 [S_(j)+ℓ(j),S_(j+1)-1]if j ∈ [k-1] [S_(k)+ℓ(k),m]if j=k.We note that the 𝒰_j are pairwise disjoint and possibly empty, and there should be a total of (n+h-1)a-(h+k-1)a=(n-k)a unoccupied spots, so that |𝒰|∑_j=0^k|𝒰_j|=(n-k)a. In particular, as the inductive hypothesis gives S_(j)≡ 1 a, (<ref>) implies that any 𝒰_j has a left endpoint congruent to 1 a, and |𝒰_j| is an integral multiple of a.We now show that the (k+1)th car can park.If x_k+1=1, then (<ref>) implies that car k+1 can find some interval of a unoccupied spots to park in.If x_k+1=1+t_k+1a for some t_k+1∈ [n-1], then we have the following three distinct possibilities. t_1=n-1 and x_k+1=1+t_k+1a<1+t_1a. Here, the first car occupies [1+(n-1)a,(n+h-1)a].If spot 1+t_k+1a is empty, then car k+1 will park in 𝒫 [1+t_k+1a,(t_k+1+1)a] by (<ref>). Otherwise, we claim that car k+1 must be able to find and park in an interval of a unoccupied spots in 𝒜 [1+(t_k+1+1)a,(n-1)a].Assume the contrary, so that 𝒜 is completely occupied.Let S_max=max{ s ∈ [m]:sis unoccupied when car k+1 attempts to park}.Since S_max is the right endpoint of an interval of unoccupied spots, by (<ref>), S_max=ta for some t∈ [t_k+1].Then [1+S_max,(n+h-1)a], where |[1+S_max,(n+h-1)a]|=(n+h-t-1)a, is occupied by n-t cars, one of which is car 1 (with length ha) and n-t-1 of which are cars with length a.Moreover, by assumption, [1+S_max,(n+h-1)a] ⊇𝒫. Thus, including car k+1, at least n-t+1 cars have preference at least 1+S_max since S_max is unoccupied. In other words, at most t-1 cars have preference strictly less than 1+S_max, forcing x_(t)≥ 1+S_max=1+ta. This contradicts x_(t)≤ 1+(t-1)a. t_1≠ n-1 and x_k+1=1+t_k+1a<1+t_1a. Now, the first car no longer occupies spot m. Note again by (<ref>) that if spot 1+t_k+1a is empty, then car k+1 will park in 𝒫 [1+t_k+1a,(t_k+1+1)a].Otherwise, car k+1 searches for an interval of a unoccupied spots in 𝒜_1 [1+(t_k+1+1)a,t_1a].If it finds one, then it will occupy it.If it does not, then we claim that it must be able to find and park in such an interval in 𝒜_2 [1+(h+t_1)a,(n+h-1)a]. Assume the contrary, so that 𝒜_1 and 𝒜_2 are completely occupied.Let S_max be as in (<ref>), so that again, S_max=ta for some t ∈ [t_k+1]. Repeating the argument for Case <ref> allows us to arrive at the same contradiction.x_k+1=1+t_k+1a≥ 1+t_1a. Note first that t_1<n-1 due to (<ref>). Here, we claim that car k+1 will find and park in an interval of a unoccupied spots in ℐ [1+(h+t_1)a,(n+h-1)a].Assume now that car k+1 cannot find such an interval, and let S_max be as in (<ref>). Note that we may take h+t_1<n-1. Otherwise, h+t_1≥ n-1, and since car 1 occupies [1+t_1a,(h+t_1)a], any car with preference at least 1+t_1a is forced to park in an interval of a unoccupied spots in ℐ. Then our assumption implies that ℐ is completely occupied when car k+1 attempts to park, so that S_max=ta for some t ∈ [t_k+1]. Repeating the argument for Case <ref> then completes the proof. For similar reasons, we may assume S_max≥ 1+(h+t_1)a.Given these additional assumptions, we haveS_max=ta for some t ∈{ h+t_1+1,h+t_1+2,…,t_k+1}.Thus, [1+S_max,(n+h-1)a], where |[1+S_max,(n+h-1)a]|=(n+h-t-1)a, is occupied by n+h-t-1 cars, all of which must have length a since S_max≥ 1+(h+t_1)a.Thus, including car k+1, at least n+h-t cars have preference at least 1+S_max since S_max is unoccupied, so at most t-h cars have preference strictly less than 1+S_max. Thus, x_(t-h+1)≥ 1+S_max=1+ta.Since h is a positive integer, this contradicts x_(t-h+1)≤ 1+(t-h)a.This covers all cases and completes the induction.Hence, ∈_n().Note that (<ref>) does not depend on how the entries ofare permuted, so parking succeeds under any permutation of .Therefore, ifsatisfies (<ref>), then ∈_n(), concluding the proof.Let = (b,a^n-1) ∈ℕ^n, where n≥ 2, and = (x_1, x_2, …, x_n) ∈ [m]^n. Ifb>(n-1)a, then ∈_n() if and only if x_(i)∈{ 1+(k-1)a:k∈ [i] }∀ i ∈ [n].Let ∈_n(). Lemmas <ref> and <ref> again show thatsatisfies (<ref>). We now prove the converse. Assume that (<ref>) holds. In other words, for all i ∈ [n], x_(i)≤ 1+(i-1)a and x_i=1+t_ia for some t_i∈ [n-1]_0.We will show inductively that parking succeeds under .Under the parking experiment for , the first car, with length b and preference x_1, will park in [x_1,x_1+b-1], where x_1=1+t_1a for some t_1∈ [n-1]_0.Thus, [1+t_1a,b+t_1a] is now occupied. Next, the second car, with length a and preference x_2, will attempt to park in [x_2,x_2+a-1], where x_2=1+t_2a for some t_2∈ [n-1]_0.Due to (<ref>), if t_1=n-1, then t_2<t_1, so that car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a].If t_1<n-1, then we havet_2 ∈ [t_1-1]_0 or t_2 ∈{ t_1,t_1+1,…,n-1 }. In the former, car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a]. In the latter, car 2 fails to park in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a] since car 1 occupies [1+t_1a,b+t_1a], where b+t_1a>(n+t_1-1)a≥ (n-1)a, so b+t_1a≥ 1+(n-1)a and b+t_1a≥ 1+t_2a. Thus, car 2 must park in [1+b+t_1a,b+(t_1+1)a].This proves that car 2 will occupy [S_2,S_2+a-1], where S_2≡ 1 a if S_2<1+t_1a 1+b a if S_2>b+t_1a,which will serve as our base case.Assume now that the first k ∈ [n-1] cars are parked, and each car j, where j ∈ [k] ∖{ 1 }, occupies [S_j,S_j+a-1] for someS_j≡ 1 a if S_j<1+t_1a 1+b a if S_j>b+t_1a.In other words, S_j≡ 1 a if it is located to the left of [1+t_1a,b+t_1a] (where car 1 is parked), and S_j ≡ 1+b a otherwise. Moreover, for S_1=1+t_1a, car 1 occupies [S_1,S_1+b-1]. Now, for any j ∈ [k], defineℓ(j) bif S_(j)=S_1aotherwise.At this stage of the parking experiment, the set of unoccupied spots is again ⋃_j=0^k𝒰_j, where the 𝒰_j are defined as in (<ref>) with ℓ(j) as in (<ref>). Furthermore, from (<ref>), any 𝒰_j has a left endpoint congruent to either 1a or 1+ba (when 𝒰_j is located to the left and right, respectively, of where car 1 is parked), and |𝒰_j| is an integral multiple of a. We now show that the (k+1)th car can park. If x_k+1=1, then (<ref>) implies that car k+1 can find some interval of a unoccupied spots to park in.If x_k+1=1+t_k+1a for t_k+1∈ [n-1], then we have the following three distinct possibilities. t_1=n-1 and x_k+1=1+t_k+1a<1+t_1a. Here, the first car occupies [1+(n-1)a,b+(n-1)a].If spot 1+t_k+1a is empty, then car k+1 will park in [1+t_k+1a,(t_k+1+1)a] by (<ref>). Otherwise, we claim that car k+1 must be able to find and park in an interval of a unoccupied spots in [1+(t_k+1+1)a,(n-1)a].Assume not, so that [1+(t_k+1+1)a,(n-1)a] is completely occupied.Define S_max as in (<ref>). Since S_max is the right endpoint of an interval of unoccupied spots located to the left of where car 1 is parked, by (<ref>), S_max=ta for some t∈ [t_k+1].Then [1+S_max,b+(n-1)a], where |[1+S_max,b+(n-1)a]|=(n-t-1)a+b, are occupied by n-t cars, one of which is car 1 (with length b) and n-t-1 of which are cars with length a.Thus, including car k+1, at least n-t+1 cars have preference at least 1+S_max. Hence, at most t-1 cars have preference strictly less than 1+S_max, forcing x_(t)≥ 1+S_max=1+ta, which contradicts x_(t)≤ 1+(t-1)a. t_1≠ n-1 and x_k+1=1+t_k+1a<1+t_1a. Note again that (<ref>) implies that if spot 1+t_k+1a is empty, then car k+1 will park in [1+t_k+1a,(t_k+1+1)a]. Otherwise, car k+1 searches for an interval of a unoccupied spots in [1+(t_k+1+1)a,t_1a].If it finds one, then it will occupy it.If it does not, then we claim that it must be able to find and park in such an interval in [1+b+t_1a,b+(n-1)a].Assume not, so that both [1+(t_k+1+1)a,t_1a] and [1+b+t_1a,b+(n-1)a] are completely occupied. Let S_max be as in (<ref>), so that by (<ref>) again, S_max=ta for some t∈ [t_k+1].Repeating the argument for Case <ref> yields the same contradiction.x_k+1=1+t_k+1a≥ 1+t_1a. Here, we claim that car k+1 will be able to find and park in an interval of a unoccupied spots in [1+b+t_1a,b+(n-1)a]. Assume not, and let S_max be as in (<ref>).Note that b+t_1a≥ 1+(n-1)a, so any car with preference at least 1+t_1a is forced to park in an interval of a unoccupied spots in [1+b+t_1a,b+(n-1)a]. Then our assumption implies that [1+b+t_1a,b+(n-1)a] is completely occupied when car k+1 attempts to park, so that S_max=ta for some t ∈ [t_k+1]. Repeating the argument for Case <ref> yields another contradiction. This covers all cases and completes the induction, so ∈_n(). Again, (<ref>) is preserved under any permutation of . Therefore, ifsatisfies (<ref>), then ∈_n(), concluding the proof.From Theorems <ref> and <ref>, we see that the conditions (<ref>), (<ref>), and (<ref>) are identical, and hence the equality _n((a^n))=_n((b,a^n-1)) holds whenever a | b or b>(n-1)a.It remains to study the possibility of b ∤ a and b<(n-1)a.For this case, we will see that the order statistic bounds provided by Lemma <ref> are not strong enough to guarantee that ∈_n(). Thus, sharpening some of these bounds is needed to obtain such a result. Let = (b,a^n-1) ∈ℕ^n, where n≥ 2, and = (x_1, x_2, …, x_n) ∈ [m]^n.If a ∤ b and b<(n-1)a, then ∈_n() if and only if x_(i)∈{ 1+(k-1)a:k∈ [i] } ∀ i∈[b/a+1]{ 1+(k-1)a:k ∈[ b/a+1 ] } otherwise.First, assume thatdoes not satisfy (<ref>).That is, there exists j ∈ [n] for which at least one of the following is true:x_(j) ≢1 ax_(j) >1+(j-1)aif j≤b/a+1 x_(j) >1+b/a aif j>b/a+1.Lemma <ref> allows us to assume that any entry ofis congruent to 1 a. Similarly, Lemma <ref> allows us to assume that x_(j)≤ 1+(j-1)a if j≤b/a+1. If x_(j)>1+b/aa, where j>b/a+1, then write x_(j)=1+qa, where q≥b/a+1.Let '=(x_1',x_2',…,x_n') be any permutation ofsuch that x_1'=x_(1) and x_2'=x_(j). Under the parking experiment for ', the first car will occupy the b spots [1,b], and the second car will occupy the a spots [x_(j),x_(j)+a-1]=[1+qa,(q+1)a]. This leaves the qa-b spots [b+1,qa] empty.Note that as a ∤ b, we have b/a-1<b/a<b/a, soqa-b≥( b/a+1 )a-b>ab/a-b=0;thus, [b+1,qa]≠∅. Similarly, b/a<b/a<b/a+1, which yields(q-b/a-1)a<qa-b<(q-b/a)a,so qa-b is strictly bounded between consecutive multiples of a.Hence, none of the remaining cars, which all have length a, can fill [b+1,qa].Thus, we have ' ∈_n() and ∉_n().Thus, if ∈_n(), thensatisfies (<ref>). We now prove the converse.Assume that (<ref>) holds.In other words, for all i∈[b/a+1], x_(i)≤ 1+(i-1)a, for all b/a+1<i≤ n, x_(i)≤ 1+b/aa, and x_i=1+t_ia for some t_i∈[b/a]_0.We will show inductively that parking succeeds under .Under the parking experiment for , the first car, with length b and preference x_1, will park in [x_1,x_1+b-1], where x_1=1+t_1a for some t_1∈[b/a]_0.Then, the second car, with length a and preference x_2, will attempt to park in [x_2,x_2+a-1], where x_2=1+t_2a for some t_2∈[ b/a]_0. We have t_2 ∈ [t_1-1]_0 or t_2 ∈{ t_1,t_1+1,…,b/a}.In the former, car 2 successfully parks in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a]. In the latter, car 2 fails to park in [x_2,x_2+a-1]=[1+t_2a,(t_2+1)a] since car 1 occupies [1+t_1a,b+t_1a], where b+t_1a≥ b>b/aa, so b+t_1a≥ 1+b/aa and b+t_1a≥ 1+t_2a. Thus, car 2 must park in [1+b+t_1a,b+(t_1+1)a]. This proves that car 2 will occupy [S_2,S_2+a-1], whereS_2 ≡ 1 a if S_2<1+t_1a 1+b a if S_2>b+t_1a,which will serve as our base case. Assume now that the first k ∈ [n-1] cars are parked, and each car j, where j ∈ [k] ∖{ 1 }, occupies [S_j,S_j+a-1] for someS_j ≡ 1 a if S_j<1+t_1a 1+b a if S_j>b+t_1a.Moreover, for S_1=1+t_1a, car 1 occupies [S_1,S_1+b-1]. At this stage of the parking experiment, the set of unoccupied spots is again ⋃_j=0^k𝒰_j, where the 𝒰_j are defined as in (<ref>) with ℓ(j) as in (<ref>). Furthermore, from (<ref>), any 𝒰_j has a left endpoint congruent to either 1a or 1+ba, and |𝒰_j| is an integral multiple of a. We now show that the (k+1)th car can park. If x_k+1=1, then (<ref>) implies that car k+1 can find some interval of a unoccupied spots to park in.If x_k+1=1+t_k+1a for t_k+1≥ 1, then we have the following two distinct possibilities.x_k+1=1+t_k+1a<1+t_1a.Note first that car 1 cannot occupy [1+(n-1)a,b+(n-1)a] since x_1≤ 1+b/aa<1+(n-1)a.In particular, as b/a≤ n-2, we have |[x_1+b,b+(n-1)a]|=(n-1)a-b/aa≥ a; in other words, the first car will park at least a spots away from the end of the parking lot.Now, note that if spot 1+t_k+1a is empty, then car k+1 will park in [1+t_k+1a,(t_k+1+1)a] by (<ref>). Otherwise, it searches for an interval of a unoccupied spots in [1+(t_k+1+1)a,t_1a].If it finds one, then it will occupy it. If it does not, then we claim it must be able to find and park in such an interval in [1+t_1a+b,b+(n-1)a]. Assume not, so that all of these spots are occupied.Let S_max be as in (<ref>). Since it is the right endpoint of an interval of unoccupied spots located to the left of where car 1 is parked, by (<ref>), S_max=ta for some t∈ [t_k+1].Repeating the argument for Case <ref> yields the same contradiction. x_k+1=1+t_k+1a≥ 1+t_1a. Here, we claim that car k+1 will be able to find and park in an interval of a unoccupied spots in [1+t_k+1a,b+(n-1)a].Assume not, and let S_max be as in (<ref>).Since it is the right endpoint of an interval of unoccupied spots located to the right of where car 1 is parked, by (<ref>), S_max=ta+b for some t_1<t< t_k+1. We have(t+b/a)a<S_max=ta+b<(t+b/a+1)a.Thus, as S_max is unoccupied, there are cars that have preference at least 1+(t+b/a+1)a>S_max.But by (<ref>), no car can have preference greater than 1+b/aa, and 1+(t+b/a+1)a>1+b/aa, so we have a contradiction. This covers all cases and completes the induction, so ∈_n().Again, (<ref>) is preserved under any permutation of . Therefore, ifsatisfies (<ref>), then ∈_n(), concluding the proof.Theorems <ref>, <ref>, and <ref> illustrate that for almost constant , χ() is maximal when y_1≥ y_2 and minimal otherwise.We now prove a surprising partial converse: the almost constant condition on length vectors is necessary for their characteristic to be maximal. Recall the following results.Let ∈^n and =(x_1,x_2,…,x_n) ∈_n().If k=_i ∈ [n]x_i, then _k∈_n-1(_|_n-1).In particular, if χ()=α, then χ(_|_n-1)≥α-1.Let =(y_1,y_2) ∈^2. Then χ()=1 if and only if y_1≥ y_2.Let =(y_1,y_2,y_3) ∈^3. If χ()=2, then y_2=y_3. Now, we will present the partial converse, which implies one direction of Theorem <ref>. Its proof will be via induction on n, so the above results will be instrumental in setting up the base cases and proving the inductive step. Let =(y_1,y_2,…,y_n)∈ℕ^n, where n≥ 2. If χ()=n-1, then y_2=y_3=⋯=y_n.We will proceed via induction on n.Note that χ()=n-1 implies that there exists 𝐰∈_>1^n-1 such that (1,𝐰) ∈_n().For n=2, if (1,w) ∈_2(), where w>1, then the conclusion follows vacuously by Lemma <ref>.For n=3, the conclusion follows by Lemma <ref>.Thus, we have verified the base cases, so assume that the statement is true up to some n=k≥ 3.That is, for any ∈ℕ^k such that χ()=k-1, we have y_2=y_3=⋯=y_k.We now show that this is true for n=k+1.Let '=(y_1',y_2',…,y_k+1') ∈ℕ^k+1 such that χ(')=k.Then there exists =(1,𝐰') ∈_k+1('), where 𝐰' ∈_>1^k.Assume without loss of generality that 𝐰' is nondecreasing.As w'_k=max, by Lemma <ref>, we have (1,𝐰'_|k-1) ∈_k('_|_k). Thus, χ('_|_k)≤ k-1, and we have equality since k-1 is the maximal characteristic. Hence, by the inductive hypothesis, '_|_k is of the form (b,a^k-1) ∈^k, which shows that y'_2=y'_3=⋯=y'_k.Consequently, ' is of the form (b,a^k-1,c) for some c ∈, and it remains to show that c=a. To do this, we prove the contrapositive: if c≠ a, then =(1,𝐰') ∉_k+1(').We claim that parking fails for the permutation '=(𝐰',1).This can be rewritten as '=(w'_1,w'_2,…,w'_k-1,w'_k,1)=(𝐰'_|_k-1,w'_k,1). Since (1,𝐰'_|k-1) ∈_k('_|_k), we have (𝐰'_|_k-1,1) ∈_k('_|_k). Thus, consider the parking experiment for (𝐰'_|_k-1,1) with length vector '_|_k. As all cars park successfully and 𝐰' ∈_>1^k, car k, with preference 1, must occupy [1,a], while the remaining cars completely occupy [1+a,m-c].This means that in the parking experiment for '=(𝐰'_|_k-1,w'_k,1) (with length vector '),the first k-1 cars will completely occupy [1+a,m-c], so they leave [1,a] and [m-c+1,m] unoccupied.By assumption, w'_k=max'>1, so car k, with length a, must occupy a subset of [m-c+1,m] ( otherwise, parking fails, and the proof is complete). We now consider the following two distinct cases. '=(b,a^k-1,c) with a<c. Here, car k will take a spots in [m-c+1,m]; thus, c-a spots in this interval are still left unoccupied. Then, car k+1, with preference 1 and length c>a, cannot park in [1,a], and c-a<c, so it cannot park in [m-c+1,m] either. Hence, parking fails.'=(b,a^k-1,c) with a>c. Here, car k, with length a>c, cannot park in [m-c+1,m], so parking fails.Therefore, c=a, and '=(b,a^k), so y_1'=y_2'=⋯=y_k+1', completing the induction.The contrapositive proof strategy displayed above in the inductive step can be used to provide an alternate proof of Lemma <ref>. Thus, Theorem <ref> can be proven without the use of Theorem 5.4 in <cit.>; it can then be utilized to provide a quicker proof of that theorem since we have a necessary condition for the characteristic to be maximal.Putting everything together, we arrive at Theorem <ref>. If (<ref>) holds, then χ()=n-1 by Theorems <ref>, <ref>, and <ref>.If χ()=n-1, then Theorem <ref> yields y_2=y_3=⋯=y_n. Given this, Theorems <ref>, <ref>, and <ref> imply y_1≥ y_2, as desired.To conclude this section, we present the enumerative results on |_n()| and |_n()| whenis almost constant. For constant , recall the following. Let =(a^n) ∈^n.Then|_n()|=(n+1)^n-1and |_n()|=C_n,where C_n=1/n+12nn is the nth Catalan number [OEIS https://oeis.org/A000108A000108.].As discussed, the exact same is true for certain almost constant . Let =(b,a^n-1) ∈^n, where n≥ 2. If a | b or b>(n-1)a, then |_n()|=(n+1)^n-1and |_n()|=C_n.By Theorems <ref>, <ref>, and <ref>, we have _n((b,a^n-1))=_n((a^n)). Thus, the result follows by Theorem <ref>.We now readily obtain a proof of Theorem <ref><ref>.This is Theorems <ref> and <ref> and Corollary <ref>. To prove the next corollary, we will appeal to the following classical counting lemma due to Pitman and Stanley <cit.>. For a nondecreasing 𝐮=(u_1,u_2,…,u_n) ∈^n with successive differences Δ(𝐮) (u_1,u_2-u_1,…,u_n-u_n-1) ∈_0^n, let _n(𝐮) denote the set of 𝐮-parking functions of length n, and let _n(𝐮) ⊆_n(𝐮) denote its subset of nondecreasing elements. Let 𝐮=(u_1,u_2,…,u_n) ∈^n be nondecreasing and k ∈ [n-1]. Assume that Δ(𝐮)=(a,b^n-k-1,c,0^k-1). Then|_n(𝐮)|=a∑_j=0^knj(c-(k+1-j)b)^j(a+(n-j)b)^n-j-1. Lemma <ref> arose from the study of empirical distributions and order statistics. It was generalized by Yan in <cit.> via two combinatorial proofs: one involved a strategic decomposition of a 𝐮-parking function into subsequences, and the other was a recursive argument.Armed with this result, we proceed to present and prove the second set of counting results. Let =(b,a^n-1) ∈^n, where n≥ 2. If a ∤ b and b<(n-1)a, then |_n()|=∑_j=0^n-b/a-1(-1)^jnj( n-b/a-j )^j(n-j+1)^n-j-1and|_n()|=n-b/a+1/n+1n+b/ab/a. Let =(x_1,x_2,…,x_n) ∈_n(). We first compute |_n()|.The main claim is the following, which states that, for an appropriate choice of 𝐮, such (resp. nondecreasing) invariant parking assortments are (resp. nondecreasing) 𝐮-parking functions up to scaling. Let 𝐮=(1,2,…,b/a,(b/a+1)^n-b/a).Then _n() and _n(𝐮) are in bijection, and _n() and _n(𝐮) are in bijection.We construct a bijection between the sets in question. Define φ:_n() →_n(𝐮) byφ()=( 1+x_1-1/a,1+x_2-1/a,…,1+x_n-1/a) (φ()_1,φ()_2,…,φ()_n).By Theorem <ref>, x_(i)≤ 1+(i-1)a for all i ∈[b/a+1], so φ()_(i)≤ i for all i ∈[b/a+1]. For the same reason, x_(i)≤ 1+b/aa for all i ∈{b/a+1,b/a+2,…,n }, so φ()_(i)≤b/a+1 for all i ∈{b/a+1,b/a+2,…,n }. Hence, φ() ∈_n(𝐮). Furthermore, for all i ∈ [n], we have x_i≡ 1a, so x_i-1/a∈ℤ for all i ∈ [n], which implies that φ()_i ∈ℤ. Putting everything together, since 1≤ x_i≤ 1+(n-1)a, we have 1≤φ()_i≤ n, so φ() ∈_n(𝐮).We now construct the inverse of φ to show that it is a bijection. Given 𝐩=(p_1,p_2,…,p_n) ∈_n(𝐮), define ψ: _n(𝐮) →_n() byψ(𝐩)=(1+(p_1-1)a,1+(p_2-1)a,…,1+(p_n-1)a){ψ(𝐩)_1,ψ(𝐩)_2,…,ψ(𝐩)_n }.Since p_(i)≤ i for all i ∈[b/a+1], we have ψ()_(i)∈{ 1,1+a,…,1+(i-1)a } for all i ∈[b/a+1], and as p_(i)≤b/a+1 for all i ∈{b/a+1,b/a+2,…,n }, we have ψ()_(i)∈{ 1,1+a,…,1+b/aa } for all i ∈{b/a+1,b/a+2,…,n }. Thus, ψ(𝐩) ∈_n() by Theorem <ref>. Note that ψ∘φ()= and φ∘ψ(𝐩)=𝐩, so we indeed have φ^-1=ψ.Lastly, we show that φ:_n() →_n(𝐮) is also a bijection. It suffices to prove that φ and ψ map nondecreasing elements to nondecreasing elements. Note that if =(x_1,x_2,…,x_n) ∈_n(), then for all i,j ∈ [n] with i≤ j, x_i≤ x_j implies 1+x_i-1/a≤ 1+x_j-1/a since a>0, so φ() ∈_n(𝐮). Similarly, if 𝐩=(p_1,p_2,…,p_n) ∈_n(𝐮), then for all i,j ∈ [n] with i≤ j, p_i≤ p_j implies 1+(p_i-1)a≤ 1+(p_j-1)a since a>0, so ψ(𝐩) ∈_n().Claim <ref> establishes that it suffices to compute |_n(𝐮)|. To do so, we compute Δ(𝐮)=(1^b/a+1,0^n-b/a-1)=(1,1^b/a,0,0^n-b/a-2). The result then follows by Lemma <ref>.Now, we turn our attention to computing |_n()|. For n,k ∈_0 such that n≥ k, letf(n,k) |_n((1,2,…,k,(k+1)^n-k))|.By Claim <ref>, |_n()|=f(n,b/a), where n>b/a+1, so it suffices to compute f(n,k). To do so, we will make use of the following recursion.We have f(n,0)=1 for all n ∈_0, f(n,1)=n and f(n+1,n+1)=f(n+1,n) for all n ∈, and f(n+1,k)=f(n+1,k-1)+f(n,k) for all n,k∈ such that 1<k<n+1. We have f(n,0)=|_n((1^n))|=|{ (1^n) }|=1 by definition [for n=0, the empty tuple is the unique element in this set, but this will be immaterial for our purposes.].Similarly, _n((1,2^n-1))={ (1^j,2^n-j):j ∈ [n] } because if 𝐩=(p_1,p_2,…,p_n) ∈_n((1,2^n-1)), then there exists j=max{ i∈ [n]:p_i=1 }, and for i ∈ [n] ∖ [j], we must have p_i=2, showing that 𝐩=(1^j,2^n-j); the reverse inclusion is immediate. Thus, f(n,1)=|_n((1,2^n-1))|=n. Moreover, f(n+1,n+1)=f(n+1,n) since (1,2,…,n+1,((n+1)+1)^(n+1)-(n+1))=(1,2,…,n+1)=(1,2,…,n,(n+1)^(n+1)-n). Now, for n,k∈ such that 1<k<n+1, let 𝐯=(1,2,…,k,(k+1)^n+1-k), so that f(n+1,k)=|_n+1(𝐯)|, and let 𝐯'=(1,2,…,k-1,k^n-k+2). Notice that_n+1(𝐯)=_n+1(𝐯') ∪{𝐩=(p_1,p_2,…,p_n+1) ∈_n+1(𝐯): p_n+1=k+1 }.It is immediate that the above is a disjoint union since if 𝐩=(p_1,p_2,…,p_n+1) ∈_n+1(𝐯'), then p_n+1≤ k. Next, let 𝐩=(p_1,p_2,…,p_n+1) ∈_n+1(𝐯). If p_n+1<k+1, then max𝐩≤ k, so it must satisfy p_i≤ i for all i ∈ [k], and p_i≤ k otherwise, meaning 𝐩∈_n+1(𝐯'). The reverse inclusion follows by definition. Therefore, f(n+1,k) =|_n+1(𝐯)| =|_n+1(𝐯')|+|{𝐩=(p_1,p_2,…,p_n+1) ∈_n+1(𝐯): p_n+1=k+1 }| =|_n+1(𝐯')|+|{𝐩∈_n(𝐯”): 𝐯”=(1,2,…,k,(k+1)^n-k) }| =f(n+1,k-1)+f(n,k). Claim <ref> establishes that f(n,k) satisfies the same recurrence as Catalan's triangle [OEIS http://oeis.org/A009766A009766.], which has closed form n-k+1/n+1n+kk (cf. Lemma 1 and Theorem in <cit.>). Therefore, the result follows by setting k=b/a.For b/a=0,1,2, one can show using Corollary <ref> that we have |_n()|=1,2^n-1,3^n-2^n-n, respectively.As detailed by Yan in <cit.>, closed forms of the sum in Lemma <ref> do not exist for general n. However, one fact worth noting is that when b/a=2, it turns out that |_n()| is also the number of regions in the G-Shi arrangement when G is the cycle graph with n vertices [OEIS http://oeis.org/A355645A355645].To finish, we may now easily prove the rest of Theorem <ref>.This is Theorem <ref> and Corollary <ref>.§ PROPERTIES OF _N() IN RELATION TO THE DEGREE AND CHARACTERISTIC Given the intricacies of the set _n() forof maximal characteristic, our aim in this section is to study the structure of the set given any . As we will see, deriving certain structural results can allow us to recover general information about the degree and characteristic; this will comprise Theorem <ref>. Throughout this section, the following result will be helpful. Let = (y_1, y_2, …, y_n) ∈^n.Assume = (x_1, x_2, …, x_n) ∈ [m]^n is nondecreasing. Then, ∈_n() if and only if x_i ≤ 1 + ∑_j=1^i-1 y_j for all i ∈ [n].For proof of Theorem <ref> specifically, the three lemmas below will be crucial ingredients; Lemma <ref> is due to <cit.>, and we provide an independent proof. In particular, Lemma <ref> will allow us to morph an invariant parking assortment into a more convenient parking assortment, which yields a useful technique to help prove Theorem <ref><ref>. Let =(y_1,y_2,…,y_n) ∈^n and ^+=(,y_n+1) ∈^n.If ∈_n(), then ^+=(,x_n+1) ∈_n+1(^+) if and only if x_n+1≤ 1+∑_j=1^ny_j.Consider the parking experiment under ^+.Because ∈_n(), the first n cars occupy [1,∑_j=1^ny_j].This leaves only spots [1+∑_j=1^ny_j,∑_j=1^n+1y_j] empty.If x_n+1>1+∑_j=1^ny_j, then the last car, with length y_n+1, must park in [x_n+1,∑_j=1^n+1y_j], but |[x_n+1,∑_j=1^n+1y_j]|<y_n+1, so parking fails. If x_n+1≤ 1+∑_j=1^ny_j, then since [1,∑_j=1^ny_j] is occupied, the last car occupies [1+∑_j=1^ny_j,∑_j=1^n+1y_j], so parking succeeds. Let =(y_1,y_2,…,y_n) ∈^n. Then if =(x_1,x_2,…,x_n) ∈_n(), and there exists i ∈ [n] such that x_i=min(x_i,x_i+1,…,x_n), then 𝐫=(x_1,…,x_i-1,r,x_i+1,…,x_n) ∈_n() for any r ∈ [x_i]. In particular, (_|_n-1,1) ∈_n(). Consider the parking experiment under .The key observation is that [1,x_i-1] is occupied when car i attempts to park.To see this, assume the contrary; i.e. there exists an s ∈ [1,x_i-1] that is unoccupied. Then because x_i=min(x_i,x_i+1,…,x_n), cars i,i+1,…,n all drive past s, leaving s unoccupied after all cars have parked, contradicting ∈_n(). Thus, if car i instead had preference less than x_i (and all other preferences are unchanged), then since [1,x_i-1] is occupied, its choice of parking spots remains the same, implying that all cars can still park.Therefore, 𝐫∈_n().Let 𝐚=(a_1,a_2,…,a_n) ∈^n and S={ i ∈ [n]:a_1>a_i }.Then there is a permutation =̱(b_1,b_2,…,b_n) of the entries of 𝐚 such that*b_1=min$̱.*b_i≥a_ifor alli ∈Sandb_i=a_ifor alli ∉S ∪{ 1 }.*b_i=min(b_i,b_i+1,…,b_n)for alli ∈Ssuch thatb_i>a_i.We will describe an algorithm that sequentially swaps certain entries of 𝐚 to construct $̱.Set𝐚^(0)=𝐚andS^(0)=S.Fork ∈_0, first check if𝐚^(k)=(a_1^(k),a_2^(k),…,a_n^(k))satisfiesa_1^(k)=min𝐚^(k). If so, we claim=̱𝐚^(k)and stop.Otherwise, construct the index setS^(k)={ i ∈ [n]:a_1^(k)>a_i^(k)}, and letj_k=max S^(k).Then, swap the positions ofa_1^(k)anda_j_k^(k)to obtain the next iterate𝐚^(k+1)=(a_1^(k+1),a_2^(k+1),…,a_n^(k+1)) (a_j_k^(k),a_2^(k),…,a_j_k-1^(k),a_1^(k),a_j_k+1^(k),…,a_n^(k)),and repeat the process.To prove the correctness of this algorithm, we will first show that it terminates; this gives property <ref> in the process.The sequence of index sets { S^(k)} is nested and strictly decreasing.Let k be any nonnegative integer, and assume S^(k)≠∅. Because a_1^(k+1)=a_j_k^(k), where j_k=max S^(k)=max{ i ∈ [n]:a_1^(k)>a_i^(k)}, we have a_1^(k)>a_1^(k+1) (i.e. the sequence consisting of the first entries a_1^(k) of the iterates is strictly decreasing). Moreover, by construction, a_i^(k)=a_i^(k+1) if and only if i≠ 1,j_k.It is clear that 1∉ S^(k)∪ S^(k+1). We also have a_j_k^(k+1)=a_1^(k)>a_1^(k+1), so j_k∉ S^(k+1).Thus,S^(k+1)={ i ∈ [n]:a_1^(k+1)>a_i^(k+1)}={ i ∈ [n]:a_1^(k+1)>a_i^(k)}⊊{ i ∈ [n]:a_1^(k)>a_i^(k)}=S^(k),as a_1^(k)>a_1^(k+1) and j_k∈ S^(k), while j_k∉ S^(k+1), implying our claim.We have 𝐚∉ S^(k)𝐚^(k) satisfiesa_1^(k)=min𝐚 S^(k)=∅.Note that min𝐚=min𝐚^(k) since each iterate is a permutation of the entries of 𝐚 by construction. If 𝐚∉ S^(k), then all i ∈ [n] such that a_1^(k)>a_i^(k) satisfy a_i^(k)>min𝐚.But this means min𝐚≠ a_i^(k) for all i ∈ [n] ∖{ 1 }, so a_1^(k)=min𝐚.Next, if 𝐚^(k) satisfies a_1^(k)=min𝐚, then no i ∈ [n] can satisfy a_1^(k)>a_i^(k), so S^(k)=∅.Lastly, if S^(k)=∅, then it is clear 𝐚∉ S^(k), proving the equivalence.Combining Claims <ref> and <ref>, the algorithm must terminate because the index sets are nested and strictly decreasing, so there existsLfor whichS^(L)=∅, which is equivalent to𝐚^(L)satisfyinga_1^(L)=min𝐚, or property <ref>. It now suffices to show that𝐚^(L)satisfies properties <ref> and <ref>.Now, we note that by Claim <ref>, ifi ∉ S^(k), theni ∉ S^(ℓ)for allℓ>k. In particular, asa_i^(k)=a_i^(k+1)if and only ifi≠ 1,j_k, it follows that ifi∉ S ∪{ 1 }, thena_i^(k)=a_i^(k+1)for allk, which showsa_i^(L)=a_i.Moreover, ifi ∉ S^(k)∪{ 1 }, thena_i^(k)=a_i^(k+1)=⋯=a_i^(L).Similarly, ifi ∈ S, then there existsksuch thati ∈ S^(k)buti ∉ S^(k+1).Note thati ∈ S^(k)⊊ S^(k-1)⊊⋯⊊ S^(0).Theni≠ j_0,j_1,…,j_k-1; if not, theni=j_ℓfor some0≤ℓ≤ k-1, andj_ℓ∉ S^(ℓ+1), which impliesj_ℓ∉ S^(k). Hence,a_i^(0)=a_i^(1)=⋯=a_i^(k).At this stage of the algorithm, we have two possibilities: eitheri=j_k=max S^(k)ori<j_kwitha_i^(k)≥ a_j_k^(k)=a_1^(k+1).For the former, asi=j_k ∉ S^(k+1)∪{ 1 }, we havea_1^(k)=a_j_k^(k+1)=⋯=a_j_k^(L), wherea_1^(k)>a_j_k^(k); hence,a_j_k^(L)>a_j_k^(0).For the latter,i≠ j_0,j_1,…,j_L-1, asi∉ S^(k+1)and hencei∉ S^(ℓ)for anyℓ>k, soa_i^(0)=a_i^(1)=⋯=a_i^(L).Thus,a_i^(L)≥ a_ifor alli ∈ S. Altogether, we obtain property <ref>.Lastly, assume thati ∈ Ssatisfiesa_i^(L)>a_i.We showed above that ifi≠ j_kfor anyk, thena_i^(L)=a_i.Thus, we must havei=j_k ∉ S^(k+1)for somek, soa_1^(k)=a_j_k^(k+1)=a_j_k^(L).By the definition ofj_kand𝐚^(k+1), we havea_1^(k+1)=a_j_k^(k)<a_1^(k)≤ a_j_k+1^(k),…,a_n^(k)=a_j_k+1^(k+1),…,a_n^(k+1),which yieldsj_k+1,…,n ∉ S^(k+1), so(a_j_k^(k+1),a_j_k+1^(k+1),…,a_n^(k+1))=(a_j_k^(L),a_j_k+1^(L),…,a_n^(L)). Consequently,a_j_k^(L)=a_1^(k)=min(a_j_k^(L),a_j_k+1^(L),…,a_n^(L)), which proves property <ref>.Therefore, we may take=̱𝐚^(L), as desired.We can now leverage these lemmas to prove Theorem <ref> successively (in order, we prove the closure property, image of the degree, embedding property, and monotonicity).For any i ∈ [d], the main idea is to start with a permutation 𝐩=(p_1,p_2,…,p_n) of (1^n-d+1,𝐰_i), construct a specific permutationof (1^n-d,𝐰) ∈_n(), and use Lemmas <ref> and <ref> to progressively morphinto 𝐩 while ensuring that each change preserves membership in _n().Note first that if p_n=1, then 𝐩_|_n-1 is a permutation of (1^n-d,𝐰_i), so (𝐩_|_n-1,w_i) ∈_n().Applying Lemma <ref>, we obtain (𝐩_|_n-1,1)=𝐩∈_n().Now, we suppose p_n=v ∈𝐰_i.Assume first that v≤ w_i. Let k=max{ j ∈ [n]:p_j=1 }. Consider the following permutation of (1^n-d,𝐰):=(p_1,…,p_k-1,v,p_k+1,…,p_n-1,w_i) ∈_n().Let 𝐪=(q_1,q_2,…,q_n-k+1) (v,p_k+1,…,p_n-1,w_i) and S={ j ∈ [n-k+1]:q_1>q_j }.By Lemma <ref>, there is a permutation 𝐪'=(q_1',q_2',…,q_n-k+1') of the entries of 𝐪 such that q_1'=min𝐪', q_j'≥ q_j for all j ∈ S, q_j'=q_j for all j ∉ S ∪{ 1 }, and q_j'=min(q_j',q_j+1',…,q_n-k+1') for all j ∈ S such that q_j'>q_j.Then 𝐱'=(𝐱_|_k-1,𝐪')=(𝐩_|_k-1,𝐪') ∈_n() is a permutation of . From here, we aim to transform 𝐪' into (p_k,p_k+1…,p_n) by way of Lemma <ref>.Since q_1'=min𝐪', Lemma <ref> yields (𝐩_|_k-1,1,q_2',…,q_n-k+1') ∈_n(). For 2≤ j<n-k+1, if j ∉ S, then q_j'=q_j=p_j+k-1.If j ∈ S, then we have q_j'≥ q_j=p_j+k-1.Consider the sequence(h_1,h_2,…,h_ℓ) (j ∈ S:q_j'>q_j), where h_1<h_2<⋯<h_ℓ.For any t ∈ [ℓ], inductively assuming that we have already replaced q_h_1,…,q_h_t-1, we have q_h_t'>q_h_t, and q_h_t'=min(q_h_t',q_h_t+1',…,q_n-k+1'), so applying Lemma <ref>, q_h_t' may be replaced with q_h_t=p_h_t+k-1 to obtain a parking assortment [the first such replacement is justified for the same reason.].Lastly, as n-k+1 ∉ S, we have q_n-k+1'=q_n-k+1=w_i.Consequently, (𝐩_|_n-1,w_i) ∈_n().Applying Lemma <ref> a final time, as v≤ w_i, we have (𝐩_|_n-1,v)=𝐩∈_n(). Now, assume that v>w_i. We will instead consider the following permutation of (1^n-d,𝐰):=(p_1,…,p_k-1,w_i,p_k+1,…,p_n-1,v) ∈_n().Repeating the same argument as in the case of v≤ w_i yields 𝐩∈_n().This covers all permutations of (1^n-d+1,w_i), so therefore, as i ∈ [d] was arbitrary, we have (1^n-d+1,w_i) ∈_n() for all i ∈ [d], as desired.We repeatedly apply Theorem <ref><ref>.Assume that =d. Let 𝐩=(p_1,p_2,…,p_n+1) be any permutation of (1,).First, if p_n+1=1, then 𝐩_|_n is a permutation of ∈_n(), so as p_n+1≤ 1+∑_j=1^ny_j, we have 𝐩∈_n+1(^+) by Lemma <ref>.Next, without loss of generality, =(1^n-d,𝐰), where 𝐰∈_>1^d.If p_n+1=w_i for some i ∈ [d], then 𝐩_|_n is a permutation of (1^n-d+1,𝐰_i) ∈_n() by Theorem <ref><ref>.Because w_i≤max𝐰≤ 1+∑_j=1^n-1y_j≤ 1+∑_j=1^ny_j by Lemma <ref>, we have 𝐩∈_n+1(^+) by Lemma <ref>.This covers all permutations of (1,); therefore, (1,) ∈_n+1(^+).This result immediately implies that the converse of Theorem <ref> does not hold.For instance, if =(1,3,2,2), then one can check that (1^3,4) ∈_4(), so (1^k+3,4) ∈_k+4(^+) for any extension ^+=(,y_5,…,y_k+4) of . We first deal with the upper bound.It is attainable, as if =(c^n) and ^+=(c^n+1), then χ()=n-1 and χ(^+)=n.To see that the upper bound is valid, suppose χ(^+)>α+1.By Lemma <ref>, χ()>α.But by assumption, χ()=α, a contradiction.Thus, χ(^+)≤α+1.The lower bound is attainable, as ifis strictly increasing, and ^+=(,y_n+1), where y_n+1>y_n, then χ()=χ(^+)=0 (cf. Theorem 4.9 in <cit.>).By assumption, as χ()=α, there exists ∈_n() such that =α.Then by Theorem <ref><ref>, we have (1,) ∈_n+1(^+), and ((1,))=α, which means that χ(^+)≥α.This result generalizes Corollary 4.3 in <cit.>, which is the case α=0. § THE INVARIANT SOLUTION SET, SUM AVOIDANCE, AND EXTREMALITYIn this section, we introduce the invariant solution set of∈^nand relate it to_n()and consider a family of strictly decreasing length vectors.This in turn will lead us to proofs of Theorems <ref> and <ref>.We define the invariant solution set as follows.Let ∈^n.The invariant solution set ofis given by(){ w ∈:(1^n-1,w) ∈_n() }.To illustrate why this set is useful to study, we have the following result, which is a consequence of Theorem <ref><ref>. If ∈_n(), then ∈()^n. It is clear that 1∈() for any , so consider any i ∈ [n] such that x_i≠ 1.Repeatedly applying Theorem <ref><ref>, we have (1^n-1,x_i) ∈_n(), so x_i ∈(), as needed. Given this, we now examine the elements of().Observe that Lemma <ref> implies that ifw ∈(), thenw≤ 1+∑_i=1^n-1y_i. Interestingly, this bound can only be improved slightly ifis non-constant, which is Theorem <ref><ref>.Again, we prove the contrapositive: if w=1+∑_i=1^n-1y_i satisfies (1^n-1,w) ∈_n(), thenis constant. By assumption, (1^n-p-1,w,1^p) for all p ∈ [n-1]_0.Fix p, and consider the parking experiment under (1^n-1,w). Then [1,∑_i=1^n-p-1y_i] is occupied before the (n-p)th car parks.After this car parks, note that 𝒰_1 [1+∑_i=1^n-p-1y_i,w-1] is unoccupied. Once the (n-p)th car parks, it fills [1+∑_i=1^n-1y_i,y_n-p+∑_i=1^n-1y_i]. Then 𝒰_2 [1+y_n-p+∑_i=1^n-1y_i,m]=[w+y_n-p,m], where |𝒰_2|=y_n-y_n-p≥ 0, is unoccupied. Cars n-p+1,n-p+2…,n-1, all with preference 1, will then park in 𝒰_1, so that 𝒰_1'=[w-y_n-p,w-1], where |𝒰_1'|=y_n-p>0, is still unoccupied.Since w-1<w+y_n-p, the intervals 𝒰_1' and 𝒰_2 are not contiguous, and the nth car must fill these intervals of lengths y_n-p and y_n-y_n-p, which can only happen if y_n=y_n-p.Therefore, because p was arbitrary, y_n=y_n-p for any p ∈ [n-1]_0, which implies thatis constant.Theorem <ref><ref> also yields an alternate characterization of constant length vectors:is constant if and only if (1^n-1,1+∑_i=1^n-1y_i) ∈_n().For an example of the equality case for w whenis non-constant, consider =(1,3,3,2).One can check that (1^3,7)=(1,y_1+y_2+y_3) ∈_4(). More generally, one also can show that if =(1,3^n-2,2) ∈^n, then (1^n-1,3n-5) ∈_n().Moreover, the following result guarantees that any invariant solution can be written as a binary combination of the lastn-1entries of, which will allow us to prove Theorem <ref><ref>.Let ∈^n and w ∈().Then w=1+^̱⊤ for some ∈̱{ 0 }×{ 0,1 }^n-1.The result is immediate for w=1, so assume w>1. We first show that there exists a ∈̱{ 0,1 }^n for which w=1+^̱⊤. Assume not. Then (w,1^n-1) ∉_n().Indeed, w-1 is not a sum of entries of , meaning that no subset of the cars can precisely fill [1,w-1], so (1^n-1,w) ∉_n().Thus, such a ∈̱{ 0,1 }^n must exist.Now, if w=1+^̱⊤, where ∈̱{ 1 }×{ 0,1 }^n-1, then we claim there exists 𝐜∈{ 0 }×{ 0,1 }^n-1 such that ^̱⊤=𝐜^⊤.Indeed, as (w,1^n-1) ∈_n(), under its parking experiment, the first car leaves [1,w-1] empty, and this must be completely occupied by the end of the experiment.Thus, there is a subset of cars 2,3,…,n that can precisely fill [1,w-1], which implies the existence of 𝐜∈{ 0 }×{ 0,1 }^n-1 such that 𝐜^⊤=w-1=^̱⊤, as claimed.Therefore, for any w ∈(), we can always find ∈̱{ 0 }×{ 0,1 }^n-1 such that w=1+^̱⊤.To prove the set inclusion, apply Lemma <ref>. Then note that |()|≤ |{ 1+^̱⊤:∈̱{ 0 }×{ 0,1 }^n-1}|=2^n-1. Theorem <ref><ref> generalizes consequences of Theorems 5.1 and 5.4 in <cit.>, which prove the result for n=2,3.For brevity, write𝒮(){^̱⊤:∈̱{ 0 }×{ 0,1 }^n-1}, so that()⊆ 1+𝒮(). Recall that()has the largest size for certain strictly decreasing (or nonincreasing)whenn=3(Theorem 5.4 in <cit.>). Thus, a natural question to ask is whether this still holds for largen. The answer is yes; however, we must note that not all strictly decreasingyield an optimal|()|; one must impose a sum avoidance condition to avoid multiple∈̱{ 0 }×{ 0,1 }^n-1yielding the same element of𝒮().One possible condition is forto be “superdecreasing," which is condition (<ref>) in Theorem <ref><ref>, which states that suchindeed yield an optimal|()|.We now prove Theorem <ref><ref>.We first prove that equality is achieved. The case for n=1 is immediate, so assume n≥ 2. To begin, we will first prove a convenient uniqueness property for binary combinations givensatisfying (<ref>).If s ∈𝒮(), then there is a unique ∈̱{ 0 }×{ 0,1 }^n-1 such that s=^̱⊤.Assume for the sake of contradiction that there exist ,̱𝐜∈{ 0 }×{ 0,1 }^n-1 such that s=^̱⊤=𝐜^⊤ and ≠̱𝐜.Write =̱(b_1,b_2,…,b_n) and 𝐜=(c_1,c_2,…,c_n), and letk=min{ i ∈ [n]:b_i≠ c_i }.Without loss of generality, assume that b_k=1, so that c_k=0 (the same argument holds otherwise).Then by definition,^̱⊤=(∑_j=1^k-1b_jy_j )+y_k+(∑_j=k+1^nb_jy_j )=(∑_j=1^k-1c_jy_j )+(∑_j=k+1^nc_jy_j )=𝐜^⊤,so y_k+∑_j=k+1^nb_jy_j=∑_j=k+1^nc_jy_j. But by (<ref>), we have∑_j=k+1^nc_jy_j≤∑_j=k+1^ny_j<y_k≤ y_k+∑_j=k+1^nb_jy_j,so y_k+∑_j=k+1^nb_jy_j>∑_j=k+1^nc_jy_j, a contradiction. We automatically have () ⊆ 1+𝒮() due to Lemma <ref>, so we show the reverse inclusion.Let s=^̱⊤∈𝒮() and =(1^n-1,1+s).Consider any permutation '=(x'_1,x'_2,…,x'_n) ofsuch that x'_1≠ 1+s.Under the parking experiment for ', the first car occupies [1,y_1], and |[1,y_1]|=y_1>∑_j=2^ny_j≥∑_j=1^nb_jy_j=s.Moreover, there exists i ∈ [n] such that x'_i=1+s, and '_|_i-1=(1^i-1) ∈_i-1(_|_i-1).Thus, by Lemma <ref>, '_|_i∈_i(_|_i). The remaining cars all have preference 1, so parking succeeds and ' ∈_n(). Now, consider the remaining permutation (1+s,1^n-1).Under its parking experiment, the first car leaves [1,s] empty.Claim <ref> guarantees that a unique subset of the remaining n-1 cars, say cars i_1,i_2,…,i_q, can precisely fill [1,s], so ∑_j=1^q y_i_j=s.Cars i_1,i_2,…,i_q are the only ones that will park in [1,s] under the parking experiment for (1+s,1^n-1).Due to (<ref>), y_2>y_3>…>y_i_1-1>∑_j=i_1^ny_j≥∑_j=1^qy_i_j=s,so cars 2,3,…,i_1-1, all with preference 1, will drive past [1,s] and successively park immediately after. Thus, car i_1 must park in [1,s], establishing the base case of our claim. Now, inductively, assume that cars i_1,i_2,…,i_p are the only ones that parked in [1,s] among the first i_p cars, where p ∈ [q-1].Note that the unoccupied spots of [1,s] are [∑_j=1^py_i_j,s], which has length ∑_j=p+1^qy_i_j.Then by (<ref>), we havey_i_p+1>y_i_p+2>…>y_i_p+1-1>∑_j=i_p+1^ny_j≥∑_j=p+1^qy_i_j,so cars i_p+1,i_p+2,…,i_p+1-1 will drive past [∑_j=1^py_i_j,s] (and successively park immediately after).Thus, car i_p+1 is the next car to park in [1,s], completing the induction. From Claim <ref>, it follows that parking succeeds since each car j, where j ∈ [n]∖{ 1,i_1,i_2,…,i_q }, has preference 1 and drives past [1,s], so they will successively fill [1+s+y_1,m].Hence, we have (1^n-1,1+s) ∈_n(), so ∈_n() and ()=1+𝒮().To conclude, we will prove the following. Ifsatisfies (<ref>), then χ()=1.We first prove that there cannot exist 1<w_1≤ w_2 such that (1^n-2,w_1,w_2) ∈_n().Assume otherwise.Then Lemma <ref>, Lemma <ref>, and our result above that ()=1+𝒮() guarantee w_1-1=^̱⊤ for some ∈̱{ 0 }×{ 0,1 }^n-1. Now, letk=min{ i ∈ [n]:b_i=1 }(we note that k>1 since b_1=0). Consider the preferences (w_1,1^k-2,w_2,1^n-k).Under its parking experiment, the first car leaves [1,^̱⊤] empty, where |[1,^̱⊤]|=^̱⊤. By Claim <ref>, $̱ is unique, so to ensure[1,^̱⊤]is completely filled, carkmust park here.However, by construction, this is not the case since carkhas preferencew_2≥ w_1=1+^̱⊤.Hence,[1,^̱⊤]will have unoccupied spaces by the end of the parking experiment, so parking fails, and it follows that if∈_n(), then≠ 2. Thus,χ()<2by Theorem <ref><ref>, and soχ()=1since()⊋{1 }. Therefore,_n()={ (1^n-1,1+s):s ∈𝒮() }, as desired.For our count, note that the only nondecreasing invariant parking assortments ofare of the form(1^n-1,1+s), wheres=^̱⊤for some∈̱{ 0 }×{ 0,1 }^n-1.Claim <ref> ensures that$̱ is unique, so each choice of $̱ gives a distinct value ofs.Thus, by definition,|_n()|=|𝒲()|=2^n-1.Lastly, note that each of the2^n-1-1nontrivial elements of_n()havendistinct permutations of their entries due to their unique non-1entry.Therefore,|_n()|=n(2^n-1-1)+1=2^n-1n-n+1, as claimed. We are now ready to prove the rest of Theorem <ref>, which gives a necessary condition for equality to be achieved in Theorem <ref><ref>.Let =(y_1,y_2,…,y_n) ∈^n. If 1+∑_j=2^ny_j ∈(), then y_1=max.If y_1<max, let k ∈ [n] ∖{ 1 } satisfy y_k=max, and consider the preferences (1^k-1,1+∑_j=2^ny_j,1^n-k).Then under its parking experiment, the first k-1 cars park successively to fill [1,∑_j=1^k-1y_j]. Car k occupies [s,s+y_k-1], where s≥ 1+∑_j=2^ny_j, so s+y_k-1≥ y_k+∑_j=2^ny_j>m and hence parking fails.Thus, (1^n-1,1+∑_j=2^ny_j) ∉_n(), so we have the contrapositive of the desired result. If |()|=2^n-1, then |()|=|𝒮()|=2^n-1.Thus, there cannot exist ,̱𝐜∈{ 0 }×{ 0,1 }^n-1 such that ≠̱𝐜 and ^̱⊤=𝐜^⊤; otherwise, |𝒮()|≤ 2^n-1-1. Then 1+^̱⊤∈() for any ∈̱{ 0 }×{ 0,1 }^n-1. In particular, choosing =̱(0,1^n-1) gives 1+∑_j=2^ny_j ∈(). Lemma <ref> then ensures y_1=max, so that y_1≥ y_2.We now show that the remaining inequality detailed in (<ref>) holds via induction. For the base case, suppose y_2≤∑_i=3^ny_i. For =̱(0^2,1^n-2), we have 1+∑_i=3^ny_i ∈(). Hence, (1+∑_i=3^ny_i,1^n-1) ∈_n(), so under its parking experiment, the first car leaves [1,∑_i=3^ny_i] empty for other cars to fill. As y_2≤∑_i=3^ny_i, car 2 parks in [1,∑_i=3^ny_i]. Thus, there exists 𝐜=(c_1,c_2,…,c_n) ∈{ 0 }×{ 0,1 }^n-1 with c_2=1 such that 𝐜^⊤=∑_i=3^ny_i=^̱⊤, a contradiction since ≠̱𝐜. This shows that y_2>∑_i=3^ny_i.Now, assume that for some p ∈ [n] ∖{ 1 }, we have y_j>∑_i=j+1^ny_i for all j ∈ [p-1] ∖{ 1 }.We seek to prove that y_p>∑_i=p+1^ny_i.To do so, we will employ the same argument from above. Assume y_p≤∑_i=p+1^ny_i.Choosing =̱(0^p,1^n-p) yields (1+∑_i=p+1^ny_i,1^n-1) ∈_n(), so under its parking experiment, [1,∑_i=p+1^ny_i] is filled by a subset of cars 2,3,…,n. By the inductive hypothesis, for any j ∈ [p-1] ∖{ 1 },y_j>∑_i=j+1^ny_i>∑_i=p+1^ny_i.Since y_p≤∑_i=3^ny_i, car p must be the first to park in [1,∑_i=p+1^ny_i]. Thus, there exists 𝐜=(c_1,c_2,…,c_n) ∈{ 0 }×{ 0,1 }^n-1 with c_p=1 such that 𝐜^⊤=∑_i=p+1^ny_i=^̱⊤, so we arrive at another contradiction. Therefore, y_p>∑_i=p+1^ny_i, completing the induction.The condition (<ref>) is not sufficient for |()|=2^n-1.For =(7,5,3,1), one can check that ()={ 1,2,4,5,6,7,9 }, so |()|=7<2^3.Given what was discussed above, we conclude by proving Theorem <ref>, which gives an upper bound on|_n()|independent ofm.The set inclusion _n() ⊆{ 1 }^n-χ()×()^χ() follows by Definition <ref> and Lemma <ref>. To prove the bound, note that the nondecreasing elements of ()^χ() are precisely multisets of cardinality χ() taken from (). Letting |()|=W and χ()=C, by stars and bars, the number of such multisets is B(W,C)W+C-1C. By Pascal's identity, B(W+1,C)=(W+1)+C-1C=(W+1)+C-2C-1+(W+1)+C-2C=W+C-1C+(W+1)+C-2C-1≥ B(W,C), so B(W,C) is nondecreasing with respect to W.Similarly, B(W,C+1)=W+(C+1)-1C+1=W+C-1C+W+C-1C+1≥ B(W,C), so B(W,C) is nondecreasing with respect to C. Therefore, since W≤ 2^n-1 by Theorem <ref><ref> and C≤ n-1, we have |_n()|≤ B(W,C)≤ B(2^n-1,n-1)=2^n-1+n-2n-1, as claimed. § OPEN PROBLEMSWe now suggest various problems for future research.§.§ Length Vectors of Non-Maximal CharacteristicIn Section <ref>, we found a simple closed form for the set{∈^n:χ()=n-1 }. A natural follow-up to this is to consider finding closed forms for other characteristics.Give a direct characterization of the preimage χ^-1_n(α){∈^n:χ()=α} for any α∈ [n-1]_0.We note that Theorem <ref> gives a containment ofχ^-1_n(0)in a relatively simple set. However, refining this containment is the main difficulty.For anyα, we suspect that an effective way to approach the problem is to look at repeated entries ofor any matching partial sums of the entries ofand study how these factors affectχ(). §.§ Preserving the CharacteristicAs discussed, Theorem <ref><ref> gives an easy way to boundχ()given that we knowχ(_|_k)for somek<n. This can simplify the process for computing_n()given_n(_|_k), but there are still a great deal of possibilities to check especially ifk≪ n. In light of this case, we ask the following.Let ∈^n, where χ()=α, and ^+=(,y_n+1) ∈^n+1.What must be true aboutand ^+ to guarantee that χ(^+)=α?Note that ifα=0, then an answer to the above can help one constructχ(0,n)using a recursive-like technique. Moreover, given a condition onand^+, one tool that may be useful in showing thatχ(^+)≠ n+1is Theorem <ref><ref>.§.§ Sharper BoundsRecall that in Theorem <ref>, we had an upper bound for the size of_n()depending only onn. One might ask if this is the best such bound or if there is a similar bound for the size of_n(). For our bound, we posit that the answer is no, and our computational experiments suggest that there is a familiar connection between the best upper bound for|_n()|and the best upper bound for|_n()|, which is described below.Let ∈^n.Do |_n()|≤ (n+1)^n-1 and |_n()|≤ C_n hold for any n?If this is true, then the upper bounds are automatically sharp since constantare examples of the equality case. A potential way to approach this problem is to examine and understand which elements of1+𝒮()also are in(), as well as apply any results for Open Problems <ref> and <ref>.§ ACKNOWLEDGEMENTS I would like to thank Carlos Martínez and Pamela Harris for fruitful discussions. | http://arxiv.org/abs/2311.15699v1 | {
"authors": [
"Douglas M. Chen"
],
"categories": [
"math.CO",
"05A15, 05A19, 05D05, 05E18"
],
"primary_category": "math.CO",
"published": "20231127103803",
"title": "On the Structure of Permutation Invariant Parking"
} |
Measuring the gas reservoirs in 10^9< M_⋆<10^11 M_⊙ galaxies at 1≤ z≤3 Rosa M. Mérida 12 Carlos Gómez-Guijarro3Pablo G. Pérez-González 1Patricia Sánchez-Blázquez 45 David Elbaz 3 Maximilien Franco6 Lucas Leroy 3 Georgios E. Magdis 789 Benjamin Magnelli3 Mengyuan Xiao10 Received September 15, 1996; accepted March 16, 1997 ======================================================================================================================================================================================================================================== Interactive Segmentation Models (ISMs) like the Segment Anything Model have significantly improved various computer vision tasks, yet their application to Person Re-identification (ReID) remains limited. On the other hand, existing semantic pre-training models for ReID often have limitations like predefined parsing ranges or coarse semantics. Additionally, ReID and Clothes-Changing ReID (CC-ReID) are usually treated separately due to their different domains. This paper investigates whether utilizing precise human-centric semantic representation can boost the ReID performance and improve the generalization among various ReID tasks. We propose SemReID, a self-supervised ReID model that leverages ISMs for adaptive part-based semantic extraction, contributing to the improvement of ReID performance. SemReID additionally refines its semantic representation through techniques such as image masking and KoLeo regularization. Evaluation across three types of ReID datasets – standard ReID, CC-ReID, and unconstrained ReID – demonstrates superior performance compared to state-of-the-art methods. In addition, recognizing the scarcity of large person datasets with fine-grained semantics, we introduce the novel LUPerson-Part dataset to assist ReID methods in acquiring the fine-grained part semantics for robust performance.§ INTRODUCTIONInteractive segmentation models (ISMs) have witnessed remarkable progress in recent times <cit.>. They have played a pivotal role in advancing various computer vision tasks <cit.>. However, their exploration in the domain of Person Re-identification (ReID) remains limited. ReID, a critical subfield within computer vision, focuses on identifying and tracking individuals across diverse settings <cit.>. The effectiveness of ReID systems hinges on the quality of their semantic representations <cit.>. A robust, human-centric semantic representation has the potential to significantly enhance ReID performance <cit.>. While commendable efforts <cit.> have been made in developing semantic-based pre-training models to support ReID, many of these models rely on predetermined parsing ranges <cit.> or provide coarse-grained semantic information <cit.>, leaving room for improvement.The separation of ReID and Clothes-Changing ReID (CC-ReID) tasks, characterized by their focus on clothes-consistent or clothes-irrelevant features <cit.>, has added complexity to the field. These tasks have traditionally been treated as distinct. However, an effective ReID model should be able to represent human features adeptly, irrespective of domain-specific nuances. Fig. <ref> illustrates the feature/attention maps of different methods on ReID datasets in various fields. For example, PASS <cit.>, excelling in standard ReID, faces challenges in CC-ReID scenarios with subjects wearing different clothes. In unconstrained ReID scenarios with complex situations, such as incomplete bodies, PASS struggles to effectively suppress irrelevant areas. Conversely, CAL <cit.>, well in CC-ReID, encounters difficulties accurately focus on identities in standard ReID and unconstrained ReID.In response to these challenges, this paper introduces SemReID, a self-supervised ReID model that leverages ISMs for adaptive part-based semantic extraction. SemReID uses a keypoint predictor to obtain keypoints of the original image and utilizes positive/negative labels to instruct ISMs to acquire local semantics for different parts. The model employs the teacher-student <cit.> and multi-crop pre-training framework <cit.> to self-supervisedly learn global and local semantics. Further, it refines semantic representation with techniques such as image masking <cit.> and KoLeo regularization <cit.>. Notably, SemReID can be readily fine-tuned on ReID datasets within various fields without incurring additional costs. <ref> clearly demonstrates the attention map of SemReID partition face, upper body and lower body accurately. Moreover, SemReID pays attention to the arms of the subject instead of clothes in CC-ReID and is robust to the occluded lower body in unconstrained ReID. Furthermore, recognizing the scarcity of large-scale person datasets with fine-grained semantic information, we introduce LUPerson-Part, a novel dataset designed to support ReID methods in acquiring fine-grained part semantics essential for achieving robust and high-performance results.Our main contributions are listed as follows: * We propose SemReID, a self-supervised ReID model leveraging ISMs for adaptive part-based semantic extraction. The model incorporates advanced techniques like image masking and KoLeo regularization to enhance semantic representations.* We introduce LUPerson-Part, a novel dataset aimed at assisting ReID methods in obtaining fine-grained part semantics.* Experimental results across three distinct ReID fields demonstrate that SemReID outperforms state-of-the-art (SOTA) methods in the majority of cases. Notably, it achieves comparable or superior performance to most SOTA methods on CC-ReID and unconstrained ReID, even without specific model modifications.§ RELATED WORK§.§ Self-Supervised LearningSelf-supervised learning methods, aimed at acquiring distinctive representations from extensive unlabeled data <cit.>, have gained attention, with contrastive learning playing a key role <cit.>. MoCo <cit.> enhanced contrastive learning by preserving representations from the momentum encoder. DINO <cit.> utilized a multi-crop approach <cit.> to compare global and local representations. Besides, researchers are increasingly interested in masked image modeling (MIM) <cit.>. BEiT <cit.> predicted discrete tokens, while Masked Auto Encoders (MAE) <cit.> showed that using large masks can improve MIM. iBOT <cit.> emphasized masking prediction through online tokenizers, emphasizing local semantic patterns for enhanced robustness against common perturbations. Recently, DINOv2 <cit.> has proven powerful in generating all-purpose visual features by integrating contrastive learning with MIM, achieving SOTA performance across diverse image and pixel-level benchmarks. While DINOv2 provided valuable insights for refining our model, SemReID is specifically adapted for the ReID task to enhance its overall performance.§.§ Interactive Segmentation ModelISMs are designed to segment objects interactively with user input <cit.> and showed significant progress <cit.>. Recently, Kirillov et al. introduced SAM <cit.> for object segmentation, using prompts such as keypoints, bounding boxes, masks, or text. Zou et al. presented SEEM <cit.>, integrating visual prompts like scrawls for open-vocabulary segmentation. Despite their impactful contributions in various computer vision tasks <cit.>, the application of ISMs in ReID remains underexplored. This paper exploits their robust semantic representation to extract local semantics from person images, enhancing ReID tasks.§.§ General Person Re-Identification Based on factors including clothing consistency, camera views, ranges, and altitudes, and the presence of specialized biometric data (e.g., facial images), ReID can be categorized into three fields: standard ReID, Clothes-Changing ReID (CC-ReID), and unconstrained ReID. Standard ReID is a very classic field <cit.>. Early methods like MGN <cit.> enhanced ReID by segmenting images into different granularity strips. Recently, ViT <cit.> and Swin <cit.> models gained prominence, notably in works like TransReID <cit.>. PASS <cit.> used ViT for local areas, generating part-level features. SOLIDER <cit.> clustered tokens for improved semantics, while UniHCP <cit.> achieved SOTA through large-scale joint training.CC-ReID methods target clothing-independent features <cit.>. Given the limitations of multi-modal methods in fully utilizing clothing-independent information and sensitivity to auxiliary modal quality <cit.>, this paper focuses on single-modal methods. CAL <cit.> introduced a clothes adversarial loss to disentangle clothing-independent features. CCFA <cit.> expanded CC-ReID data in the feature space. AIM <cit.> addressed clothing biases based on causality.Unconstrained ReID, a recent field, integrates biometric features in challenging scenarios <cit.>, such as long-range (e.g., 1,000 m) or physical-turbulence settings (see Sec. <ref> and <ref>, SM for details). Despite various multimodal approaches <cit.>, there is limited research <cit.> using only RGB information for this challenge. Additionally, the exploration of semantics, particularly fine-grained local semantics, is underrepresented in this field.In our experiments, we demonstrate that a pre-trained model based on a robust human-centric semantic representation consistently achieves SOTA results across all aforementioned ReID fields, i.e., general ReID, with minimal fine-tuning. § METHODThe SemReID pipeline, depicted in Fig. <ref>, involves several key steps. It starts by leveraging Local Semantic Extraction (LSE) with keypoint prompts to adaptively acquire local semantics from an Interactive Segmentation Model (ISM). Then, both global and local semantics are integrated into the teacher-student structure, enabling self-supervised learning across multiple heads. §.§ Local Semantic ExtractionConsiderable discussion <cit.> focused on acquiring and utilizing local semantics to address ReID. Given ISMs' impressive segmentation capabilities, we aim to extract fine-grained local semantics to enhance ReID performance. Notably, an ISM's effectiveness in obtaining precise masks depends on prompt quality. Hence, we introduce an adaptive labeling method that governs its segmentation range through using positive and negative keypoints.Given a person image x, we initially employ a keypoint predictor f^kp to estimate the positions of all keypoints 𝐤 = f^kp (x) ∈ℝ^n^kp× 2, with n^kp being the number of predicted keypoints. Since each keypoint's position is predefined based on the protocol (e.g., COCO <cit.>'s first keypoint represents the nose), these positions serve as switches to govern ISM prompts. For a specific local area A, (in our implementation A ∈{Face, UB, LB}), we generate a corresponding label vector 𝐯_A = 1_A(𝐤) ∈ℝ^n^kp,where 1_A is the indicator function for the local area A. In other words, 𝐯_A,i = 1 or 0 indicates whether the i^th keypoint 𝐤_i is inside or outside A, as well as the positive (negative) label of the ISM. This method allows us to apply an ISM, f^ISM, to obtain a mask ℳ_A for the local area A,ℳ_A = f^ISM (x, 𝐤, 𝐯_A).Notably, not all keypoints have the same quality and may not be suitable as prompts. Some keypoints may be predicted outside the intended local area, potentially impacting the ISM's performance (Fig. <ref>). Thus, we filter them to retain dominantly good keypoints in current local area. The keypoint predictor f^kp simultaneously predicts a confidence score 𝐬∈ℝ^n^kp. If 𝐤_i is located outside A or if its predicted position significantly deviates, 𝐬_i will naturally be lower. In other words, we can filter 𝐯_A by using a threshold parameter τ_A on 𝐬. We apply the filtering step as 𝐯_A = 1_(𝐬 > τ_A) ∩ A (𝐤), and subsequently we can obtain the local semantics x^local_A for area A,ℳ_A = f^ISM (x, 𝐤, 𝐯_A), x^local_A = x ⊙ℳ_A,where ⊙ is the Hadamard product. Through the adjustment of τ_i and which keypoints by definition should be in, we can effectively obtain fine-grained masks for any desired local area. Importantly, these adjustments do not alter or rely on the image itself. Therefore, our module can adaptively acquire local semantics for any x. §.§ Semantic-based Self-Supervised LearningInspired by the success of pre-training methods rooted in DINO <cit.> for ReID <cit.>, we extended this method by incorporating elements from DINO <cit.> and PASS <cit.> to advance semantic-based self-supervised learning. In essence, we design two models: a teacher f^t and a student f^st, both sharing identical structures (Fig. <ref>). The teacher's input is the global semantics of the original image, whereas the student's input encompasses both global and local semantics. Subsequently, we utilize the distribution derived from the teacher as supervision to train the student, aligning the student's distribution with that of the teacher. The self-supervised loss exclusively updates the student, while the teacher undergoes updates based on a weight transfer method from the student. The trained teacher is then effectively utilized for ReID during fine-tuning.To illustrate, assuming we have L local areas, the previous module provides us with local semantics 𝐱^local = 𝒞(x^local_1, ..., x^local_L), 𝒞 is the concatenation operator. Simultaneously, the original image x can be regarded as global semantics 𝐱^global. We employ a multi-crop <cit.> strategy, where each global has M views, and each local has N views (M = N = 2 in Fig. <ref>). This augmentation results in 𝐱^global = 𝒞(x^global_1, ..., x^global_M) and 𝐱^local = 𝒞(x^local_1,1, ..., x^local_L,N). Additionally, by applying masking to the image x, we obtain x^mask. The teacher f^t's final input is 𝐱^global, while the student f^st's final input comprises {𝐱^global, 𝐱^local, 𝐱^mask}, or, in other words,𝐳^t= f^t(𝐱^global, ∅^local, ∅^mask) ∈ℝ^M × (1 + L + T) × d,𝐳^st= f^st(𝐱^global, 𝐱^local, 𝐱^mask) ∈ℝ^(M + N) × (1 + L + T) × d,where ∅^local and ∅^mask are the fixed null values for local semantics and masking image respectively, 𝐳^t and 𝐳^st represent teacher and student tokens respectively, T is the number of masked patch tokens, and d is the embedding dimension. Subsequently, different heads are employed to compute features with various functions, each associated with its respective loss. In total, there are three types of losses: classification loss (ℒ_cls), masking image loss (ℒ_mim), and regularization loss (ℒ_reg). ℒ_cls quantifies the distribution disparity between the teacher and the student for different views. The optimization objective is to minimizeℒ_cls = ∑_i ≠ j H(f^t_cls(𝐳^t_i), f^st_cls(𝐳^st_j)),∀ i ∈{1, ..., M}, j ∈{1, ..., M + N}. Here, H denotes entropy, and f^t_cls and f^st_cls are classification head. Similarly, ℒ_mim evaluates the distribution dissimilarity. However, since 𝐳^st is masked and 𝐳^t is unmasked, ℒ_mim is measured within the same view <cit.>, i.e,ℒ_mim = ∑_i = j H(f^st_mim(𝐳^t), f^st_mim(𝐳^st)),where f^t_mim and f^st_mim are segmentation head. Lastly, ℒ_reg aims to improve the generalization and stable training. We utilize the KoLeo regularizer <cit.> f_kl to calculate ℒ_reg. KoLeo regularizer maintains the input space's neighborhood structure and ensures a uniform coverage of the output space <cit.>. Specifically, given a batch of 𝐙 = {𝐳^st_1, ..., 𝐳^st_B} = {𝐙_1, ..., 𝐙_B },ℒ_reg = f_kl(𝐙) = -1/B∑_i=1^B logmin_i ≠ j ||𝐙_i - 𝐙_j||,where B is batch size. ℒ_reg is exclusively computed for the student. The overall loss in pre-training is given byℒ_pt = ℒ_cls + λ_1 ℒ_mim + λ_2 ℒ_reg.Here, λ_1 and λ_2 are two balancing hyperparameters. After updating the student, we apply exponential moving average (EMA) <cit.> to update the teacher to complete an iteration.Upon the completion of pre-training, we proceed to employ the teacher for fine-tuning in ReID. In this phase, the teacher takes 𝐱^global from ReID datasets as input and automatically generates both global and local representations (L in 𝐳^t in Eq. <ref>). The fine-tuning process involves two key losses: the identity loss (ℒ_id) and the triplet loss (ℒ_tri) <cit.>. The overall loss is expressed asℒ_ft = ℒ_id + μℒ_tri,where μ is a balancing hyperparameter.Comparison with PASS <cit.>. PASS partitions local semantics based on predefined parsing ranges of the original image (e.g., considering the upper body within 0 to 0.5 of x's height). When x itself lacks completeness (e.g., Fig. <ref>), PASS can yield suboptimal results. Furthermore, PASS does not delve into the exploration of fine-grained local semantics. In contrast, our model can adaptively acquire and represent fine-grained local semantics.Comparison with SOLIDER <cit.>. Although SOLIDER employs clustering to obtain local semantics, its semantics tend to be coarse-grained, often tainted with substantial noise (Sec. <ref>, SM). Additionally, SOLIDER necessitates a two-step clustering process before entering the semantic head. Furthermore, SOLIDER lacks specific optimization for ReID tasks. In contrast, our model extracts more fine-grained semantics, and operates in a single step with adaptability. Besides, SemReID is designed as a pre-training framework tailored for general ReID tasks. § LUPERSON-PART DATASETThe prevailing large-scale ReID pre-training datasets are outlined in Table <ref>. While these datasets exhibit significant diversity in terms of the number of images and scenes they encompass, they generally lack fine-grained semantics, which can be highly beneficial for ReID tasks <cit.>. The dataset that comes closest to addressing this issue is perhaps HumanBench <cit.>, which amalgamates multiple datasets for various tasks. However, HumanBench is a simple aggregation of existed datasets, thereby inheriting their limitations (Sec. <ref>, SM). HumanBench-ReID does not offer fine-grained semantics, and the number of images in HumanBench-Parsing with such semantics is limited. Moreover, HumanBench does not establish a connection between ReID and Parsing.To address the scarcity of large-scale person datasets featuring fine-grained semantics, we have created the LUPerson-Part dataset using raw data from LUPerson <cit.>. LUPerson-Part introduces three distinct local semantics for each LUPerson image, namely, face, upper body, and lower body. After constructing these local semantics, each one can obtain its specific detection results based on mask boundaries (Fig. <ref>). Both local semantics and local detections are valuable for pre-training. Additionally, we have furnished a substantial amount of meta-information, including original keypoint coordinates, confidence scores, local detection bounding boxes, and mask scores. This dataset's compatibility with LUPerson ensures that users can readily integrate LUPerson-Part with LUPerson for various applications without concerns regarding the generation or accuracy of local semantics. § EXPERIMENTS §.§ Datasets and MetricsWe have conducted an extensive evaluation of SemReID across three distinct fields of ReID datasets to establish its applicability in general ReID scenarios. These fields encompass standard ReID, CC-ReID, and unconstrained ReID. Standard ReID predominantly relies on Market-1501 <cit.> and MSMT17 <cit.> (MSMT17 results see Sec. <ref>, SM). CC-ReID is primarily based on PRCC <cit.> and LTCC <cit.>. Building upon prior research <cit.>, CC-ReID encompasses three distinct situations: mix, same-clothes (SC), and CC. Unconstrained ReID, on the other hand, finds its primary foundation in the BRIAR <cit.> dataset. BRIAR dataset has three protocols: 2, 3, and 4, with 200/444, 285/444, and 363/493 subjects/distractors, respectively. Hence, the increasing difficulty is evident in the newer protocols. Moreover, BRIAR poses a substantially greater challenge both in terms of scale and scenarios (e.g., 1,000m, physical turbulence, etc. as detailed in Sec. <ref> and Sec. <ref>, SM). Essentially, BRIAR can be viewed as a markedly more challenging variant of CC-ReID datasets. Consequently, evaluating different methods in this field serves as a highly effective test of their generalization capabilities. In standard ReID and CC-ReID, the primary evaluation metrics revolve around rank-k accuracy and mean average precision (mAP). In unconstrained ReID, the primary evaluation metrics are rank-k accuracy, true accept rate @ false accept rate (TAR@FAR), and false negative identification rate @ false positive identification rate (FNIR@FPIR). TAR@FAR involves 1:1 verification, i.e., comparing two subjects to find whether they are same. FNIR@FPIR measures 1:N open-set search, i.e., the probability that a subject will return non-match when comparing with others <cit.>.§.§ Implementation DetailsWe adopt the Vision Transformer (ViT) <cit.> as the backbone. During pre-training, we train ViT-Tiny/Small/Base on 8× A40 GPUs for 100 epochs, which entails approximately 40/80/160 hours, with batch size of 256/128/64, respectively. The global semantic size is 256 × 128, and the local semantic size is 128 × 64. We employ M=2, N=3, and L=3. The classification head and segmentation head are constructed using Multi-Layer Perceptrons (MLP) with Batch Normalization (BN) <cit.>. λ_1 and λ_2 are set to 1.0 and 3.0, respectively. The remaining hyperparameters remain consistent with PASS <cit.>. Unless specified otherwise, all ablation studies are conducted using ViT-S with dimensions of 256 × 128. We leverage SAM <cit.> as our ISM. We set τ_A_1 = 5.0, τ_A_2 = 2.0, and τ_A_3 = -2.0 as they empirically reached the best performance. Fine-tuning involves slightly varied settings for different datasets. In standard and unconstrained ReID datasets, the image size is 256 × 128 or 384 × 128. For the CC-ReID dataset, the image size is 384 × 192. The identity head comprises a single linear layer with BN, and the ID loss function is Cross Entropy. μ is set to 1.0. Other hyperparameters align with methods in the respective field <cit.>. §.§ Comparison with State-of-the-Art MethodsStandard ReID. Table <ref> presents a comparative analysis of SemReID against several SOTA methods in standard ReID. Notably, for image size 256, our method demonstrates remarkable superiority over all existing methods. In fact, it can even achieve performance on par with or surpassing that of ViT-B from other methods with our ViT-S. As an example, SemReID's ViT-S yields 92.2%/96.9% mAP/rank-1 accuracy compared to PASS's ViT-B with 92.3%/96.8%. Our method also outperforms most existing methods for image size 384. It's important to emphasize that while Swin, due to its structural compatibility with different scales and resolutions <cit.>, may not be directly comparable to ViT, we still manage to outperform SOLIDER in rank-1. Moreover, there is no doubt that our ViT-B surpasses SOLIDER. This unequivocally underscores the effectiveness of our method.CC-ReID. We conducted a comparison with SOTA CC-Re-ID models on PRCC and LTCC, as outlined in Table <ref>. SemReID demonstrates significant performance improvement over other methods across all rank-1 metrics. It even surpasses the latest models, including AIM <cit.> and 3DInvarReID <cit.>. In terms of mAP, it ranks second only to AIM in PRCC's CC mode and exhibits comparable performance in all other modes. Notably, both CAL <cit.> and AIM are specifically tailored for CC, whereas SemReID achieves its performance with a simple single linear layer, without any CC-specific design. Furthermore, it attains these results in just a few epochs (e.g., achieving 58.4% rank-1 accuracy in three epochs, CC mode, PRCC; Sec. <ref>, SM), contrasting with other methods that necessitate many iterations or several separate training phases. We also employ an identical fine-tuning structure for PASS. Nevertheless, a substantial disparity exists between PASS and our model, as we exceeded at least 1.1%/4.5% in mAP/rank-1. This strongly highlights the advantages of our model, emphasizing its adaptability for general ReID applications, and proves that the adaptability stems from our improvements rather than the pre-training structure.Unconstrained ReID. We first present the results on the legacy Protocol 2 for benchmarking existing work. For a fair comparison, we focused on comparing with methods tailored for whole-body analysis. Table <ref> compares SemReID with other SOTA methods. Notably, even on the older protocol, SemReID outperforms all methods across all metrics, establishing a robust foundation for reporting performance on the latest protocol. We present the evaluation on the latest protocol 4 (protocol 3's results see Sec. <ref>, SM) in Fig. <ref>. SemReID outperforms other models on all rank-k and TAR@FAR metrics, and most of FNIR@FPIR metrics. Given the heightened focus of unconstrained ReID on whether the probe falls within a specified range of identities, we emphasize metrics such as rank-20 accuracy and TAR@1%FAR. Our model exhibits 71.9%/52.6%, surpassing other SOTA methods by 6.2%/4.9%, which underscored the effectiveness. Notably, methods excelling in other fields may not perform well in unconstrained ReID. SemReID stands out as the only method with comparable performance to BRIARNet <cit.>, a specialized unconstrained ReID method, particularly on the challenging FNIR@FPIR metric. This highlights the generalization of semantics learned by SemReID, demonstrating applicability across various ReID scenarios. §.§ Ablation StudySignificance of Components. We conducted multiple training, systematically incorporating our model's components into the baseline. The differences in mAP and rank-1 on Market1501 are shown in Table <ref>. While the influence of individual components may vary, collectively, each component contributes to overall performance enhancement. Further, an observation from Tables <ref> and <ref> indicates that LSE appears to have a more pronounced positive impact on rank-1 accuracy. This underscores that precise and unambiguous local semantics play a crucial role in bringing representations of the same identity into close proximity and justifies that the proposed LUPerson-Part Dataset benefits the community significantly. On the other hand, LSE shows limited effectiveness for mAP. Consequently, additional components in our model help enhance mAP stability.Balancing Parameters. We investigate the influence of varying λ_1 and λ_2 values for ℒ_cls and ℒ_mim in Table <ref> and <ref>. SemReID attains optimal performance when λ_1 = 1.0 and λ_2 = 3.0. We adopt these values as the final parameters.Improved Fine-tuning on CC-ReID. We examined the impact of integrating CAL into our model on CC-ReID datasets, presenting results in Table <ref>. The best CAL improved mAP/rank-1 by 1.1%/0.9% on PRCC, but reduced mAP/rank-1 by 0.8%/1.6% on LTCC. Overall, using CAL cannot improve our model on every CC-ReID dataset.Sigset Evaluation on Unconstrained ReID. Given the diverse and complex nature of unconstrained ReID scenarios, our aim is to delve into the performance of each field-specific method in details, as depicted in Fig <ref>. Face Included (FI) and Face Restricted (FR) indicate whether the probe includes or excludes face information in whole-body images. Long-range Body (LB) denotes stable images captured from distant cameras (typically 400 - 1,000 m away), while Long-range Turbulence (LT) refers to these images with severe physical turbulence. Unmanned Aerial Vehicle (UAV) denotes images taken from high altitudes by drones. Specifically, rank-k results show a significant decrease in BRIARNet's effectiveness in UAV compared to CAL, while their performances are relatively close in other sigsets. CAL outperforms BRIARNet largely in TAR@FAR across all sigsets, depicting its overall robustness. However, pre-trained models with semantics, PASS and SemReID, outshine CAL and BRIARNet in each sigset, emphasizing the crucial role of semantics in ReID. Finally, SemReID shows the best performance across all sigsets (28.2%/71.0%/ 52.3% of rank-1/-20/T@1%F on average), highlighting its capacity for general and improved local semantics.Visualization of Ranking Lists. Finally, Fig. <ref> visualized top-k ranking results of different methods for same probe. We highlight the effectiveness of PASS over CAL in standard ReID, while CAL significantly outperforms PASS in CC-ReID. SemReID demonstrates commendable performance in both fields.§ CONCLUSIONWe introduce SemReID, a novel self-supervised ReID model that leverages the power of semantic pre-training. Unlike existing models, it excels in adaptively extracting local semantics, addressing limitations associated with predefined parsing ranges and coarse semantics. Further, it unifies standard ReID, CC-ReID and unconstrained ReID, demonstrating its generalization. Through extensive experiments, SemReID consistently outperforms SOTA methods in their respective fields. Moreover, to overcome the scarcity of datasets with fine-grained semantics, we introduce the LUPerson-Part dataset, enriching resources for general ReID training. Our findings highlight the effectiveness of SemReID and its potential to advance general ReID. § ACKNOWLEDGEMENTSThis research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via [2022-21102100005]. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U. S. Government. The US. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.ieeenat_fullname§ VISUALIZATION OF FEATURE/ATTENTION MAPSFig. <ref> presents additional feature map and attention map comparisons between our method and other SOTA methods across different ReID domains. The comparison is categorized into three groups: standard ReID, CC-ReID, and unconstrained ReID. In standard ReID, our method stands out for its clean and precise local semantics, showcasing the fine-grained nature of our attention. For challenging scenarios, such as CC-ReID without faces (row 4), our method maintains attention on facial features, distinguishing it from other models. In unconstrained ReID with occlusions (row 5), our method demonstrates a more focused attention on specific areas like lower body, unlike other models that exhibit broader attention even after obstacle removal. These examples underscore the effectiveness of our method across diverse ReID domains.Furthermore, we conduct a comparison with Swin-based SOLIDER <cit.>, a representative semantic ReID method. As SOLIDER lacks global attention, we visualize its window attention. While SOLIDER's window attention captures various local areas, its granularity is coarser compared to our method, resulting in less precision. The interpretability of SOLIDER's window attention is also weaker compared to methods incorporating both global and local attention. Notably, SOLIDER's attention quality diminishes on CC-ReID and unconstrained ReID, highlighting its limited generalization. In summary, our method excels in terms of generalization and interpretability.§ DIFFERENCE BETWEEN HUMANBENCH AND LUPERSON-PARTSince HumanBench does not generate new data, we provide a summary of the datasets utilized by HumanBench-ReID and HumanBench-Parsing in Table <ref> <cit.>. While HumanBench holds the advantage of consolidating numerous existing datasets, its limitations are apparent. Firstly, all existing HumanBench-ReID datasets lack local semantics; they consist solely of single cropped images (Fig. <ref>). Secondly, the modality or parsing protocols vary significantly among the HumanBench-Parsing datasets. For instance, Human3.6M <cit.> is a parsing dataset based on 3D pose, while others rely on RGB images. Moreover, even when comparing parsing datasets based on RGB images, the categories and granularities differ widely across existing datasets (Fig. <ref>). Rapid application of such extensive datasets is challenging. In contrast, our LUPerson-Part holds distinct advantages in these aspects. Primarily, LUPerson-Part ensures a one-to-one correspondence between each LUPerson image and its local semantics. Additionally, our granularity remains consistent, with local semantics precisely aligned for each image. Loading our dataset is effortlessly achieved with a single line of code (e.g., using the ImageDataset class in PyTorch). Fig. <ref> provides further examples from our dataset.§ STATISTICS OF BRIAR DATASETWe first delineate the differences among BRIAR's three protocols in Table <ref>. As the protocol becomes newer, both the probe and gallery sizes increase, rendering it more challenging. Furthermore, the number of distractors in newer galleries also rises. Distractors refer to identities absent from both the training set and the probe, serving as potential interference elements for unconstrained ReID. The inclusion of these distractors provides a robust assessment of the model's generalization performance. This contributes to the heightened difficulty of unconstrained ReID compared to conventional ReID.We also provide insights into BRIAR through Fig <ref>. Notably, BRIAR classifies data at various distances into two categories: face and whole body. The face category encompasses images that focus on the face or feature an incomplete body. Additionally, the images, whether face or whole body, are further categorized into Face Included and Face Restricted (Sec. <ref>) based on the visibility of the front face.Moreover, across a broad range of distances, BRIAR exhibits natural grayscale variations and physical turbulence on images. Some images seamlessly blend into the background, while others display significant blur. The varying distances also pose substantial challenges to ReID, with the physical influence becoming more pronounced as the distance increases. This phenomenon introduces challenges such as Long-range Body and Long-range Turbulence (Sec. <ref>).Along altitude dimension, BRIAR introduces additional challenges. The close range category comprises images taken at a height close to the horizontal line, while UAV images are captured from high altitudes by drones. These categories provide a distinct perspective on ReID. Lastly, BRIAR incorporates occlusion challenges, encompassing cone, box, and backpack scenarios. These challenges, either individually or in combination, naturally reflect the complexity inherent in unconstrained ReID.§ RESULTS ON MSMT17The results for MSMT17 are presented in Table <ref>. Notably, our method outperforms most SOTA methods, aligning with the conclusions detailed in the main paper. Even in the well-established standard ReID, our method remains competitive, surpassing some of the recent methods. It's worth emphasizing that performance in one domain does not guarantee consistent performance in others, as exemplified by the case of PASS excelling in standard ReID but exhibiting significant shortcomings in CC-ReID (Table <ref>) and unconstrained ReID (Fig. <ref>). In summary, our method demonstrates the most robust SOTA results in general ReID.§ TRAINING RECORDS ON PRCCFig. <ref> depicts the test accuracy curve for our method and the SOTA methods on PRCC. To ensure a fair comparison, we adopt the same hyperparameters as the original CAL for all methods, including epoch, learning rate, batch size, etc. Notably, our method achieves 58.4%, the highest rank-1 accuracy in epoch 3. In contrast, PASS only achieves 52.4% rank-1 accuracy in epoch 11, while CAL achieves 55.3% rank-1 accuracy in epoch 39. The initial accuracy of CAL is relatively low and gradually increases, whereas PASS and our method quickly reach their peak but then exhibit overfitting. This underscores that semantics acquired from human-centric pre-training data can effectively enhance the starting point of convergence, while hyperparameters from non-semantic models may not be optimal for semantic models. Despite this, our method still achieves SOTA results, showing its effectiveness. Additionally, a direct comparison between PASS and SemReID reveals that PASS's performance is not as good as CAL, reinforcing that general semantics result from our improvements rather than the pre-training framework. This also validates the effectiveness of LUPerson-Part.§ RESULTS ON PROTOCOL 3, BRIARTable <ref> presents a comparison between our method and SOTA methods on protocol 3, BRIAR. This protocol is comparatively simpler, with a reduced number of identities/distractors and without challenges such as long distances and UAV found in protocol 4 (Table <ref>). BRIARNet achieves the highest rank-20 accuracy, while it experiences a significant decline on TAR@FAR. This, coupled with the insights from Fig. <ref>, underscores that BRIARNet is not robust across different metrics and scenarios.On the other hand, our method leads in TAR@10%FAR and FNIR@10%FPIR, while PASS leads in TAR@1%FAR and FNIR@1%FPIR. In summary, both our method and PASS exhibit strengths in protocol 3, each leading in specific aspects. This suggests that in simpler scenarios, the precision of local semantics may not exert a substantial impact on performance. However, when comparing the two in CC-ReID (Table <ref>) and the more challenging protocol 4 (Fig. <ref>, <ref>), our method demonstrates superior generalization. | http://arxiv.org/abs/2311.17074v2 | {
"authors": [
"Siyuan Huang",
"Yifan Zhou",
"Ram Prabhakar Kathirvel",
"Rama Chellappa",
"Chun Pong Lau"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127193030",
"title": "Self-Supervised Learning of Whole and Component-Based Semantic Representations for Person Re-Identification"
} |
MadRadar: A Black-Box Physical Layer Attack Framework on mmWave Automotive FMCW Radars David Hunt Duke [email protected] Kristen Angell Duke [email protected] Zhenzhou Qi Duke [email protected] Tingjun Chen Duke [email protected] Miroslav Pajic Duke [email protected] January 14, 2024 ================================================================================================================================================================================================================================================================================================================================= Corresponding author.Panoptic Scene Graph Generation (PSG) aims at achieving a comprehensive image understanding by simultaneously segmenting objects and predicting relations among objects.However, the long-tail problem among relations leads to unsatisfactory results in real-world applications.Prior methods predominantly rely on vision information or utilize limited language information, such as object or relation names, thereby overlooking the utility of language information.Leveraging the recent progress in Large Language Models (LLMs), we propose to use language information to assist relation prediction, particularly for rare relations.To this end, we propose the Vision-Language Prompting (VLPrompt) model, which acquires vision information from images and language information from LLMs. Then, through a prompter network based on attention mechanism, it achieves precise relation prediction.Our extensive experiments show that VLPrompt significantly outperforms previous state-of-the-art methods on the PSG dataset, proving the effectiveness of incorporating language information and alleviating the long-tail problem of relations.§ INTRODUCTIONPanoptic Scene Graph Generation (PSG) <cit.> extends Scene Graph Generation (SGG) <cit.> by incorporating panoptic segmentation <cit.> to capture richer and more detailed representations of images, including both “thing” <cit.> and “stuff” <cit.> classes. PSG constructs a directed graph to represent an image, where nodes signify objectsand edges capture the relations between objects. As a bridge between vision and language, PSG has a multitude of downstream applications such as visual question answering <cit.>, image captioning <cit.>, and visual reasoning <cit.>; furthermore, it can also benefit relevant fields like embodied navigation <cit.> and robotic action planning <cit.>. Notwithstanding, the current performance of PSG <cit.> remains unsatisfactory, limiting its downstream applications. The essential reason lies in the severe long-tail problem in relation categories: for instance, in the PSG dataset <cit.>, the top three most frequent relation categories account for over 50% of entire samples, with numerous rare relations appearing less than 1%. PSG models thus struggle to accurately predict these rare relations.Recent methods <cit.>have made progress in addressing the long-tail problem, mainly exploiting the strength of vision information for relation prediction, whilst overlooking the language information in PSG. The integration of language information is however important to provide additional common sense knowledge for objects and their relations. For example, in the two images of Fig. <ref>, the relation between the person and the elephant is cleaning. In the top right image, where the person is on the elephant's back cleaning it, this scenario can easily lead previous vision-only models to classify the relation as riding. In contrast, our vision-language model can utilize language information like “a person cleans an elephant using brushes on the back of the elephant”, thus precisely predicting the relation as cleaning.In SGG task, some methods <cit.> have recognized the importance of incorporating language information besides vision.However, the way language information is utilized in these works is limited to category names of objects or relations, providing no further context and hence not fully addressing the long-tail problem. The same observation goes to methods like <cit.>, which integrate knowledge graphs into the SGG task. With the rapid development of Large Language Models (LLMs) <cit.>, acquiring richer language information - instead of merely the concepts of objects or relations - becomes much easier than before.In this paper, we introduce a novel Vision-Language Prompting (VLPrompt) model, which leverages rich language information from LLMs to help predict panoptic scene graphs in visual images. Our model comprises three parts. The first is the vision feature extractor, where we process the input image with a panoptic segmentation network adapted from Mask2Former <cit.> to extract features of different objects. We pair and concatenate these object features and integrate their corresponding spatial information to form the vision prompting features. In contrast, in the second part, the language feature extractor, we employ the chain-of-thought technique <cit.> to design various prompts, aiming to stimulate LLMs to propose context-rich language information for potential relations between a subject-object pair or judge a specific subject-relation-object triplet. These two functions are realized via a carefully designed relation proposer prompt and relation judger prompt. Subsequently, these language descriptions are transformed into language features using a pre-trained text encoder. Finally, in the third part, the vision-language prompter, we design an attention-based prompter network to facilitate the interaction between vision features and the two complimentary language features respectively, resulting into two sets of relation predictions. They are combined via a MLP-based gating network to take the strength of both for final relation prediction. The whole VLPrompt is trained end-to-end.To the best of our knowledge, we are the first to utilize language information generated by LLMs for the PSG task. Extensive experiments on the PSG dataset <cit.> demonstrate that our VLPrompt drastically enhances the PSG performance (improving the R@100 from 43.0 to 52.4 and mR@100 from 33.1 to 53.7), underscoring the significance of integrating language information for panoptic scene graph generation.§ RELATED WORK§.§ Scene Graph GenerationScene Graph Generation (SGG) <cit.> is a crucial task in scene understanding and has garnered widespread attention in the computer vision community. In recent years, numerous methods <cit.> have achieved notable progress. Various model architectures have been proposed, such as intricately designed message passing structures <cit.>, attention-based networks <cit.>, tree-based networks <cit.> and DETR-based networks <cit.>. Specifically, to address the long-tail problem, some methods enhance the prediction accuracy of rare relations through data re-sampling <cit.> and loss re-weighting <cit.>. Relevant techniques that have been developed include constructing enhanced datasets <cit.>, grouping relations for training <cit.>, constructing multi-stage hierarchical training <cit.>, and designing de-bias loss functions <cit.>. Most methods leverage images as sole inputs. Recently, some methods <cit.> have begun exploring language information or knowledge graphs in SGG; specifically, the explored language information is so far confined to basic language concepts of objects or relations.§.§ Panoptic Scene Graph GenerationPanoptic Scene Graph Generation (PSG) <cit.> has emerged as a novel task in scene understanding in recent years. Unlike SGG <cit.>, PSG employs panoptic segmentation instead of bounding boxes to represent objects, enabling a more comprehensive understanding. Similar to SGG methods, current methods in PSG <cit.> also mainly rely on the image input and do not utilize any language information. For instance, PSGTR <cit.> build a baseline PSG model by adding a relation prediction head to DETR <cit.>. PSGFormer <cit.> advances PSGTR by separately modeling objects and relations in two transformer decoders and introducing an interaction mechanism.Recently, HiLo <cit.> addresses the long-tail problem by specializing different network branches in learning both high and low frequency relations.PairNet <cit.> develops a novel framework using a pair proposal network to filter sparse pairwise relations, improving PSG performance. Unlike these methods, we propose a PSG method, VLPrompt, relying on both vision and language inputs.§.§ Large Language Models for Vision Tasks LLMs have led to large improvements in natural language processing tasks <cit.>. They are normally trained on extensive text corpora by learning to autoregressively predict the next word, hence encapsulate a broad spectrum of common sense knowledge of linguistic patterns, cultural norms, and basic worldly facts. Prominent examples of LLMs include GPT series <cit.>, Llama series <cit.>, Bard <cit.>, and Claude <cit.>, with Llama series being open-source publicly available. Given the extensive common sense information contained in LLMs, some researchers have begun exploring their applications to various vision tasks, such as recognition <cit.>, detection <cit.>, segmentation <cit.>, visual question answering <cit.>, image reasoning <cit.> and robotic navigation <cit.>. Our method designs various prompts to stimulate LLMs to elicit rich language information to enhance relation prediction.§ METHODIn this section, we introduce our method, VLPrompt. Given an image (ℐ∈ℝ^H × W × 3) and language description (𝒯) generated from LLMs, we extract vision and language features from them to predict panoptic scene graph 𝒢 = {𝒪, ℛ}, 𝒢 = VLPrompt (ℐ, 𝒯). In 𝒢: * 𝒪 = {o_i}_i=1^N signifies N objects segmented from the image ℐ. Each object is defined by o_i = {c, m}, where c belongs to one of the predefined C object categories, and m is a binary mask in {0, 1}^H × W for this object.* ℛ = {r_i,j| i,j ∈{1, 2, …, N}, i ≠ j} denotes relations with r_i,j being the relation between o_i and o_j. Each r belongs to one of the predefined K relation categories (or no relation), N is the number of objects in the image. As shown in Fig. <ref>, our method comprises three main components: vision feature extractor, language feature extractor, and vision-language prompter. The vision feature extractor (Sec. <ref>) adapts from a segmentation network (Mask2Former <cit.>) to predict object masks and form subject-object pairs for feature extraction, which result into vision prompting features. For the language feature extractor (Sec. <ref>), we generate different types of descriptions by leveraging the extensive common sense knowledge embedded in LLMs through carefully designed prompts. These descriptions are then converted into language prompting features using a text encoder.Next, the vision and language prompting features are fed into a vision-language prompter (Sec. <ref>), where vision prompting features interact with different types of language prompting features respectively, so as to take advantage of the complimentary language information to assist relation predictions.Finally, these relation predictions are combined by a relation fusion moduleto achieve the final relation prediction.§.§ Vision Feature Extractor Given an image ℐ, we first leverage Mask2Former <cit.> to produce N objects with masks. They are formed into N × (N-1) subject-object pairs by pairing any two distinct ones.The purpose of the vision feature extractor is to extract vision prompting feature for each subject-object pair, which includes visual features from the segmentation network itself as well as spatial features between the subject-object pairs. In this way, we can enhance the representations of the relations between subject-object pairs.Subject-object visual features. We first obtain the features corresponding to each object. Considering that the output feature map by the pixel decoder in Mask2Former retains rich information of the image, we use mask pooling to obtain object features corresponding to the N objects from the feature map based on each object's mask m. Then, we pair and concatenate any two distinct object features to form N × (N-1) subject-object visual features F_V^vi.Subject-object spatial features. To further enhance the representations of subject-object pairs, especially their spatial relations,we are inspired by <cit.> to encode the spatial features into the subject-object visual features. Specifically, given o_i and o_j corresponding to subject and object, we first derive their encompassing bounding boxes b_i=[x_i, y_i, w_i, h_i] and b_j=[x_j, y_j, w_j, h_j], where (x, y) is the center of the bounding box, and (w, h) are the width and height. Next we construct spatial features: v(o_i, o_j) = [x_j - x_i/√(w_ih_i), y_j - y_i/√(w_ih_i), √(w_jh_j/w_ih_i), b_i ∩ b_j/b_i ∪ b_j, w_i/h_i, w_j/h_j],where v(o_i, o_j) encodes the spatial relation between o_i and o_j, such as the ratio of their bounding box sizes, the overlap between two objects and the aspect ratio of each object. Then, we use a FC layer to expand the spatial features to the same dimension as F_V^vi, resulting in F_V^sp.Finally, we apply a FC layer to the sum of F_V^vi and F_V^sp and output vision prompting features F_V ∈ℝ^N × (N-1) × D_V, where D_V is the vision feature dimension.§.§ Language Feature Extractor The purpose of the language feature extractor is to leverage the extensive common sense knowledge contained in LLMs for providing additional language information to the PSG task. To fully leverage the common sense knowledge embedded in LLMs, we need to design various prompts to elicit outputs from LLMs. On one hand, LLMs can act as a relation proposer, suggesting possible relations between two objects, which often are frequently occurring relations in the real world. On the other hand, LLMs can also serve as a relation judger, given a subject-object pair and their specific relation, LLMs make judgment and provide reasoning for this relation. This allows detailed descriptions even for rare relations. Specifically, we design two types of prompts: relation proposer prompt (RP-Prompt) for proposing and explaining potential relations given a subject-object pair; relation judger prompt (RJ-Prompt) for judging and reasoningupon a specific subject-relation-object triplet.Below, we detail how to obtain RP- and RJ-language prompting features based on the generated descriptions.RP-language prompting feature. For RP-language prompting feature, we stimulate LLMs to guess all possible relations between two given objects o_i and o_j, along with explanation for these relations. To achieve this, we utilize the chain-of-thought technique <cit.>: we engage in a dialogue with an LLM (GPT-3.5 <cit.>), initially informing it to act as a relation proposer and defining the task. Then an example is provided to the LLM to clarify its role. We explicitly mention the predefined K relations in the prompt, guiding the LLM to propose from them. Finally, a certain subject-object pair (o_i and o_j) is given to the relation proposer prompt. By giving this prompt to the LLM, we will obtain the description for potential relations between o_i and o_j. For predefined relations that are not proposed by the LLM, we would append a template phrase by the end of the description, such as “It is not likely for o_i and o_j to have relation r.”, to make sure the language description cover all relations. To encode the description into a feature interpretable by our model, we use a text encoder (OpenAI Embeddings <cit.>) to convert the description into the feature F_L(i, j)^RP∈ℝ^1 × D_L, namely RP-language prompting feature. This process consolidates all descriptions into a single feature, allowing for a condensed reflection of the distinct attributes of relation between subject and object.RJ-language prompting feature. For RJ-language prompting feature, we design the relation judger prompt:we not only provide two objects o_i, o_j but also specify a relation r_k between them.By using the common sense knowledge, the LLM judges whether the relation r_k could plausibly exist between o_i and o_j and provides reason. Similar to above, we use the chain-of-thought technique <cit.> by telling the LLM that it serves as a relation judger; we first define the task, then give the example, and finally, the triplet (o_i-r_k-o_j) is provided. Following the same process as for RP-language prompting feature, we feed the relation judger prompt to the LLM to obtain the language description and leverage the text encoder (same as above) to encode it into the RJ-language prompting feature F_L(i, j, k)^RJ∈ℝ^1 × D_L. Different from the RP-language prompting feature, we encode each relation triplet into an individual feature, producing a rather detailed and subtle description of the relation. By enumerating all C objects and K relations in the dataset, we derive all RP-language prompting features and RJ-language prompting features. We store them in a database for efficient access. Given an image with N objects, we retrieve two sets of features, F_L^RP∈ℝ^N × (N-1) × D_L and F_L^RJ∈ℝ^N × (N-1) × K × D_L from the database. Note that for a specific relation r_k between all subject object pairs in this image, the RJ-language prompting feature is denoted as F_L(k)^RJ∈ℝ^N × (N-1) × D_L.§.§ Vision-Language PrompterTo enable the vision prompting feature to predict relations from both macro and detailed perspectives, we let F_V interact with F_L^RP and F_L^RJ through two separate decoders, termed as RP-decoder and RJ-decoder, responsible for the interaction from F_V to F_L^RP and F_L^RJ, respectively.Each decoder contains two standard transformer decoder blocks <cit.>, followed by a FC layer for relation prediction. The predictions from the two decoders are complementary: the RP-language prompting feature focuses on the condensed and distinct attributes of frequently occurred relations for a given subject-object pair; in contrast, the RJ-language prompting feature focuses on the detailed and subtle attributes of every possible relation (common or rare) for the subject-object pair. A relation fusion module consisting of a gating network is thereby devised by the end to fuse the relation predictions from both decoders into the final one. Before feeding F_V, F_L^RP, and F_L^RJ into different decoders, we use a FC layer to transform their dimensions to a uniform dimension D. Below, we specify the RP-decoder, RJ-decoder, and the relation fusion module, respectively.RP-decoder. The RP-decoder aims to utilize the F_L^RP to assist F_V in relation prediction, particularly for relations that are frequently encountered between o_i and o_j in the real world. In the first transformer decoder block, F_V is firstly fed into the self-attention layer, primarily aggregating the visual relational information in F_V. Afterwards, the self-attended F_V as query and the F_L^RP as key/value are engaged in the subsequent cross-attention layer,aggregating the common sense knowledge of potential relations between o_i and o_j into F_V.The output is further processed through a feed-forward network. In the second transformer decoder block, we repeat the aforementioned process. Finally, a fully connected layer is used to transform feature dimension D to the number of relations K, and a sigmoid function is applied afterwards to obtain R^RP∈ℝ^N × (N-1) × K. RJ-decoder. The RJ-decoder aims to facilitate interaction between the RJ-language prompting feature F_L^RJ with the vision prompting feature F_V. Since F_L^RJ a group of individual language prompting feature for every relation triplet (o_i-r_k-o_j), F_V thus has the opportunity to interact with each relation's language representation independently, which can be particularly beneficial to rare relations. We conduct parallel interactions between F_V and the K triplet features contained in F_L^RJ. For each triplet, the interaction process between F_V and F_L(k)^RJ is the same to that of the RP-decoder, except that the final FC layer is now only responsible for predicting the probability of certain relation between o_i and o_j. The FC layer is used to transform the feature dimension from D to 1. Finally, we concatenate the respective outputs to get the predictions for all K relations, a sigmoid function is applied over them to obtain R^RJ∈ℝ^N × (N-1) × K.Relation Fusion. Upon obtaining R^RP and R^RJ, we aim to take the strength of both via a relation fusion module. We devise a gating network consisting of 3-layer MLP to generate two sets of weights, W^RP and W^RJ, each matching the shape of R^RP and R^RJ. We use the sum of F_V and F_L^RP as the input of the gating network, and output W^RP. For W^RJ, we use the sum of F_V and the mean of F_L^RJ along the relation dimension as input to the gating network. W^RP and W^RJ are used to element-wisely multiply with R^RP and R^RJ respectively and the final relation prediction R is a weighted combination:R = W^RP⊙ R^RP + W^RJ⊙ R^RJ,where ⊙ is element-wise multiplication, and R ∈ℝ^N × (N-1) × K.Finally, the relation prediction R, combined with the object categories and masks predicted by the vision feature extractor, forms the panoptic scene graph 𝒢. §.§ Model Training Our model training comprises two parts. The first part is the segmentation loss ℒ_seg used in the vision feature extractor for panoptic segmentation, we simply follow the loss used in <cit.>. The second part is the relation loss. Since the same subject-object pair might have multiple relations, we use a binary cross-entropy loss <cit.>. To effectively train the vision-language prompter, we apply the relation loss separately to R^RP, R^RJ, and R and eventually sum them up as the final relation loss, denoted byℒ_rel. The final loss ℒ isℒ = λℒ_seg + ℒ_rel,where λ is the weighting coefficient. In the language feature extractor, we directly utilize pre-trained LLMs, thus eliminating the need for additional model training.§ EXPERIMENTS §.§ Datasets Panoptic Scene Graph Generation (PSG) dataset <cit.>. This is the first dataset dedicated to the PSG task, comprising 48,749 annotated images, including 2,186 test images and 46,563 training images. The dataset includes 80 “thing” categories <cit.> and 53 “stuff” categories <cit.>, as well as 56 relation categories.Visual Genome (VG) dataset <cit.>. This dataset is widely used in the SGG task. To validate our method, we also conduct experiments on the VG dataset. Following previous works <cit.>, we use the VG-150 variant, which includes 150 object categories and 50 relation categories.§.§ Tasks and metrics Tasks. There are three subtasks for PSG and SGG tasks: Predicate Classification, Scene Graph Classification and Scene Graph Detection <cit.>. We focus on Scene Graph Detection for both datasets, as it is the most challenging and comprehensive subtask, which involves localizing objects and predicting their classes and relations.Metrics. Following previous works <cit.>, we use Recall@K (R@K) and mean Recall@K (mR@K) as our metrics. While Recall@K is biased towards frequent classes, mean Recall@K gives all classes the same weight.§.§ Implementation details In our experiments, we use Mask2Former <cit.> pretrained on COCO <cit.> dataset to initialize the panoptic segmentation network in the vision feature extractor. In language feature extractor, we utilize by default GPT-3.5 Turbo <cit.> as the LLM, and employ OpenAI Embedding Service <cit.> as the text encoder. We store the extracted language prompting features in a database and then retrieve the RP-language prompting feature using “sub#obj”, and the RJ-language prompting feature using “sub#rel#obj”. We adopt the same data augmentation settings following previous methods <cit.>. To train our model, we use the AdamW <cit.>, with a learning rate of 1e^-4 and weight decay of 5e^-2. We set λ to 0.1 in our final loss function. Our model is trained for 12 epochs with a step scheduler reducing the learning rate to 1e^-5 at epoch 6 and further to 1e^-6 at epoch 10. The training takes approximately 18 hours on four A100 GPUs, with a batch size of 1 for each GPU. The inference of our model follows the same forward process in training. §.§ Comparison to the state-of-the-artTab. <ref> reports the performance of our method compared to previous state-of-the-art methods on the PSG dataset <cit.>. Previous methods rely solely on vision inputs, images, while ours utilizes both vision and language inputs. For a fair comparison, we use the same Resnet-50 <cit.> backbone in for all methods in the vision feature extractor. Our method shows superior performance compared to all previous methods. Particularly, it outperforms the previous best-performing method <cit.> by a large margin, +9.4% in R@100 and +20.6% in mR@100. It is noteworthy that we achieve substantial improvements in the mR@K metric, demonstrating the significant benefits of adding language information for predicting rare relations.Tab. <ref> reports the performance of our method compared to previous methods on the VG dataset <cit.>. To adapt our method to the VG dataset, a bounding box-based SGG task, we first use a Segment Anything Model (SAM) <cit.> with VG dataset's ground-truth bounding box annotations as prompts to transform the VG dataset into a dataset suitable for instance segmentation tasks. We then train a Mask2Former on this instance segmentation dataset, enabling our method to be adapted to the VG dataset. As shown in Tab. <ref>, our method surpasses previous vision-only and vision-language models in mR@K, indicating that incorporating language information effectively enhances the prediction performance for rare relations and alleviates the long-tail problem.§.§ Ablation Study §.§.§ Vision Feature Extractor Object features from pixel decoder. To validate the superiority of obtaining object features from the pixel decoder (PixelDec) of Mask2Former, we experiment with an alternative approach: acquiring corresponding object features from the transformer decoder (TsfmDec) of Mask2Former. This is a common practice in the literature <cit.>. Experiments (PixelDec→TsfmDec) in Tab. <ref> show that the feature from the pixel decoder performs 3.4% better in mR@100 than that from the transformer decoder, as the pixel decoder contains more comprehensive vision information.Mask pooling for object feature extraction. Common methods to obtain object features from the pixel decoder include mask pooling (MaskPool) and bounding box pooling (BboxPool) <cit.>. We validate our choice of mask pooling through ablation experiments (MaskPool→BboxPool). As shown in Tab. <ref>, the performance using mask pooling is 1.1% higher in mR@100 than that using bbox pooling. Mask pooling is more suitable for the mask prediction subtask in the PSG task. Concatenating object features into a pair feature. Common methods for merging the features of two objects for their relation prediction include concatenation (Concat.) <cit.> and subtraction (Sub.) <cit.>. Notably, addition of the features cannot be used due to the inherent order requested between the subject and object. Results (Concat.→Sub.) in Tab. <ref> indicate that concatenation outperforms subtraction by 4.8% on the [email protected] of spatial feature. To validate the effect of adding spatial features, we conduct an ablation study by removing these features. The results (w/o Spatial Feat.) in Tab. <ref> show that this leads to a decrease of 0.7% in R@100 and 1.6% in mR@100. This is because spatial features provide the model with additional spatial interaction between the subject and object (Sec. <ref>), thereby enhancing the vision feature for relation prediction. §.§.§ Language Feature Extractor Effects of different LLMs and text encoders. To further assess the effects of different LLMs and text encoders on model performance, we attempt to replace the GPT with Llama2-7B <cit.> and the OpenAI Embedding Service with Bert <cit.>. When using Bert, we take the mean of the embeddings of all output tokens as the feature. The experimental results (Llama2-7B + Bert) in Tab. <ref> reveal that using Llama2 as the LLM and Bert as the text encoder results in only a slight decrease in performance: 0.4% in R@100 and 0.5% in mR@100. We review the descriptions output by Llama2-7B and compare them with those from GPT-3.5 Turbo, finding no significant differences in quality, more details can be found in supplementary materials. This suggests that, with carefully designed prompts, open-source LLMs with reduced parameters also work for our method, thus validating the flexibility of our method.Efficacy of chain-of-thought for prompt. By carefully designing prompts with the chain-of-thought technique, LLMs can produce rich and accurate descriptions. However, if we do not use chain-of-thought and instead directly ask LLMs questions, such as replacing the relation proposer prompt with “What are the possible relations between subject and object? And why?” or the relation judger prompt with “Could this relation be possible between subject and object? Why?", we find that the outputs from LLMs become much less predictable and often not as expected. We experiment without chain-of-thought (w/o CoT) and the results in Tab. <ref> show that the performance of relation prediction significantly drops, a 8.0% decrease in mR@100. This further illustrates the rationale and importance of using chain-of-thought technology for designing prompts. Effects of different feature extraction methods. To validate our design of different feature extract methods for RP- and RJ-language prompting features, we study their effects. We offer three variants: 1) For RP-language prompting features, we adopt the same way as the RJ-language prompting feature to extract feature for each relation triplet individually, we denote this variant as Ext^RP→ Ext^RJ in the Tab. <ref>. It shows that performance is slightly lower than our original VLPrompt, but with increased computation. 2) For RJ-language prompting features, we adopt the same way as the RP-language prompting feature to extract all relation triplets into one feature, denoted by Ext^RJ→ Ext^RP in the Tab. <ref>. We observe a clear drop in mR@K, as this would condense the features of relations between subject and object while dropping the subtle details, which can be especially disadvantageous for rare relations. 3) We swap the feature extraction methods between Ext^RP and Ext^RJ, denoted by Swap(Ext^RP, Ext^RJ) in Tab. <ref>. We observe a substantial decrease in both metrics.Our original feature extraction is specifically designed on one side to focus on the condensed and distinct attributes of commonly occurring relations; on the other side to focus on the detailed and subtle attributes of all possible especially rare relations for a subject-object pair.§.§.§ Vision-Language Prompter Efficacy of language information. To validate the efficacy of incorporating language information, we conduct a comparative experiment by replacing all language prompting features in the vision-language prompter with vision prompting features. This approach ensures that all other factors remain constant while assessing the impact of language information. As shown in Tab. <ref>, the experimental results (w/o Language) reveal a significant decrease in the mR@100 by 9.5% when language information is removed, which demonstrates the substantial impact of language information in mitigating the long-tail problem. Results of RP-decoder and RJ-decoder. We test the relation prediction performance of both RP-decoder and RJ-decoder separately. The results (RP-decoder and RJ-decoder) in Tab. <ref> show that the RP-decoder outperforms the RJ-decoder in the R@K metrics (50.9% 50.4% for R@100), while the RP-decoder scores lower in mR@K metrics compared to the RJ-decoder (49.6% 52.8% for mR@100). This indicates that the RP-decoder is good at predicting frequent relation classes, whereas the RJ-decoder excels in rare relation classes, which indicates that they are complementary.Efficacy of relation fusion. We can see from Tab. <ref> that when the results of the two branches are combined through relation fusion (i.e., our VLPrompt), the final prediction surpasses the predictions of any single branch, which demonstrates the efficacy of our relation fusion strategy.The number of decoder blocks. To elucidate the reason that both RP-decoder and RJ-decoder use a 2-block transformer decoder, we vary different numbers of blocks in Tab. <ref>. We observe that when increasing the transformer decoder blocks from 1 to 2, there is a significant improvement in model performance. However, increasing from 2 to 4 blocks does not change much for the performance. Further increasing the decoder blocks to 12 leads to a notable decrease in performance, suggesting that too many decoder blocks may cause the model to overfit. Considering both performance and speed, we choose 2 blocks. §.§ Analysis Qualitative analysis. As shown in Fig. <ref>, with the inclusion of language information, VLPrompt successfully predicts challenging relations, which are the highlighted in yellow.Efficiency analysis. In Tab. <ref>, we compare our model's efficiency with the previous state-of-the-art models, evaluating computational floating point operations per second (FLOPS), parameter size and inference speed on the same A100 GPU. We observe that although our method has higher FLOPS compared to HiLo, it matches HiLo in prediction speed, and significantly outperforms HiLo in performance (See Tab. <ref>). § CONCLUSIONIn this work, we introduce the VLPrompt model, the first method to incorporate language information generated by LLMs to enhance the PSG task performance. VLPrompt utilizes the chain-of-thought method in designing prompts, enabling LLMs to generate rich descriptions for relation prediction.Additionally, we develop a prompter network based on attention mechanisms to facilitate comprehensive interaction between vision and language information, achieving high-quality relation prediction. Experiments demonstrate that our method significantly outperforms the current state-of-the-art on the PSG dataset and mitigates the long-tail problem for relations. In future work, we plan to explore the use of LLMs for open-set relation prediction and further refine the model by distillation to enhance efficiency, enabling broader application in downstream tasks. ieeenat_fullname In this supplementary material, we provide the RP- and RJ-prompts, along with the corresponding descriptions output by the LLMs. Sec. <ref> presents specific examples of RP-prompts and RJ-prompts applied to two LLMs, GPT-3.5 Turbo <cit.> and Llama2-7B <cit.>. Sec. <ref> provides descriptions given by GPT-3.5 Turbo and Llama2-7B for various subjects, objects, and relations. Sec. <ref> offers a comparative analysis of descriptions from these different language models. Finally, we provide more visualizations to illustrate the performance of our VLPrompt in Sec. <ref>.§ RP- AND RJ-PROMPTAccording to the official API interfaces provided by GPT-3.5 Turbo and Llama2, we can interact with large language models (LLMs) in three different roles: “system”, “assistant”, and “user”. The system message defines the behavior of the assistant within a given context. The assistant represents the responses and actions of the LLM, while the user refers to the individual engagement with the LLM, providing requests or comments for the assistant to address. Inspired by the chain-of-thought <cit.> technique, we have designed a meaningful dialogue as the input prompt for LLMs. Specifically, the prompts designed for GPT-3.5 Turbo and Llama2-7B are presented below. §.§ Prompt for GPT-3.5 TurboRP-prompt. Relation proposer prompt, which we used to stimulate LLMs to propose all possible relations between two given objects.system: You are asked to play the role of a relation proposer. Given the category names of two objects in an image, you are to infer what kind of relation might exist between them based on your knowledge, and provide the reasons for each possible relation. In the relation between the two objects in the image, we refer to one object as the subject and the other as the object. There may or may not be a relation between the subject and the object. Please note that this relation has an order, that is, the subject comes first and the object comes after. If there is a relation between the two, these relations must belong to one of the pre-defined 56 different types.assistant: What are the 56 relations?user: They are `over', `in front of', `beside', `on', `in', `attached to', `hanging from', `on back of', `falling off', `going down', `painted on', `walking on', `running on', `crossing', `standing on', `lying on', `sitting on', `flying over', `jumping over', `jumping from', `wearing', `holding', `carrying', `looking at', `guiding', `kissing', `eating', `drinking', `feeding', `biting', `catching', `picking', `playing with', `chasing', `climbing', `cleaning', `playing', `touching', `pushing', `pulling', `opening', `cooking', `talking to', `throwing', `slicing', `driving', `riding', `parked on', `driving on', `about to hit', `kicking', `swinging', `entering', `exiting', `enclosing', `leaning on'.assistant: Can you give me an example?user: For example, the subject is a person, and the object is a sports ball. The possible relations between them could be: 1. Beside: The person could be standing beside the sports ball. 2. Looking at: The person might be looking at the ball to better control it. 3. Playing: This is because it's very common in real life for a person to be playing with a sports ball. 4. Chasing: The person might be chasing after the ball.assistant: Ok, I got it. Please give me the subject and object of the image.user: The subject is a SUBJECT_NAME, and the object is a OBJECT_NAME. RJ-prompt. Relation judger prompt, which we used to stimulate LLMs to judge whether there is a specific relation between two objects.system: You are asked to play the role of a relation judger. Given the category names of two objects in an image, and providing you with a relation category name, you need to predict whether this relation is likely to exist in the image based on your knowledge, and give the reason for its existence. For two objects, we call the first object subject and the second object object.assistant: Yes, I understand. Can you give me an example?user: For example, the input is: the subject is a 'person', the object is a 'sports ball' and the relation is 'playing'. The output should be Yes, the relation is likely to exist in the image. This is because it's very common in real life for a person to be playing with a sports ball.assistant: Ok, I got it. Please give me the subject, object and relation names.user: The subject is a SUBJECT_NAME, the object is a OBJECT_NAME, and the relation is RELATION_NAME.§.§ Prompt for Llama2-7B In GPT-3.5 Turbo, the “system” role can be followed by either the “assistant” or the “user” role. However, in Llama2's API, the “system” role can only be followed by “user” role, not “assistant” role. Therefore, we have made simple modifications to the prompt used for GPT-3.5 Turbo. The specific prompt is as follows.RP-prompt. Relation proposer prompt, which we used to stimulate LLMs to propose all possible relations between two given objects.system: You are asked to play the role of a relation proposer. Given the category names of two objects in an image, you are to infer what kind of relation might exist between them based on your knowledge, and provide the reasons for each possible relation. In the relation between the two objects in the image, we refer to one object as the subject and the other as the object. There may or may not be a relation between the subject and the object. Please note that this relation has an order, that is, the subject comes first and the object comes after. If there is a relation between the two, these relations must belong to one of the pre-defined 56 different types. What are the 56 relations?user: They are `over', `in front of', `beside', `on', `in', `attached to', `hanging from', `on back of', `falling off', `going down', `painted on', `walking on', `running on', `crossing', `standing on', `lying on', `sitting on', `flying over', `jumping over', `jumping from', `wearing', `holding', `carrying', `looking at', `guiding', `kissing', `eating', `drinking', `feeding', `biting', `catching', `picking', `playing with', `chasing', `climbing', `cleaning', `playing', `touching', `pushing', `pulling', `opening', `cooking', `talking to', `throwing', `slicing', `driving', `riding', `parked on', `driving on', `about to hit', `kicking', `swinging', `entering', `exiting', `enclosing', `leaning on'.assistant: Can you give me an example?user: For example, the subject is a person, and the object is a sports ball. The possible relations between them could be: 1. Beside: The person could be standing beside the sports ball. 2. Looking at: The person might be looking at the ball to better control it. 3. Playing: This is because it's very common in real life for a person to be playing with a sports ball. 4. Chasing: The person might be chasing after the ball.assistant: Ok, I got it. Please give me the subject and object of the image.user: The subject is a SUBJECT_NAME, and the object is a OBJECT_NAME. RJ-prompt. Relation judger prompt, which we used to stimulate LLMs to judge whether there is a specific relation between two objects.system: You are asked to play the role of a relation judger. Given the category names of two objects in an image, and providing you with a relation category name, you need to predict whether this relation is likely to exist in the image based on your knowledge, and give the reason for its existence. For two objects, we call the first object subject and the second object object. Please give me an example.user: For example, the input is: the subject is a 'person', the object is a 'sports ball' and the relation is 'playing'. The output should be Yes, the relation is likely to exist in the image. This is because it's very common in real life for a person to be playing with a sports ball.assistant: Ok, I got it. Please give me the subject, object and relation names.user: The subject is a SUBJECT_NAME, the object is a OBJECT_NAME, and the relation is RELATION_NAME.§ RP- AND RJ-DESCRIPTIONTo gain a more comprehensive understanding of the outputs from LLMs, we selected four sets of subject-object pairs for demonstration, namely thing-thing, stuff-stuff, thing-stuff, and stuff-thing. * thing-thing: person-motorcycle.* stuff-stuff: sky-building.* thing-stuff: person-sea.* stuff-thing: rock-horse.Here, the former term in each pair represents the category of the subject, and the latter the category of the object. The RP-descriptions are as below.For each pair, we randomly select two different relations to showcase their corresponding RJ-descriptions. Note that in the examples shown below for RP-descriptions, the template sentences that were added to complete the descriptions for all predefined K relations (it is not likely for SUBJECT_NAME and OBJECT_NAME to have the relation RELATION_NAME) have been omitted.§.§ Description from GPT-3.5 Turbo RP-description. The description generated by LLMs based on the RP-prompt.person-motorcycle: Based on the given subject (person) and object (motorcycle), here are some possible relations between them: 1. Riding: The person could be riding the motorcycle, indicating that they are in control and using it as a means of transportation. 2. Driving: Similar to riding, the person could be driving the motorcycle, suggesting that they are operating it. 3. About to hit: This relation implies that the person is in a dangerous situation where they are about to collide with the motorcycle, indicating a potential accident or collision. 4. Leaning on: The person might be leaning on the motorcycle, using it as a support or resting against it. 5. Parked on: The motorcycle could be parked on or near the person, suggesting that they are in close proximity to it but not actively using or interacting with it. 6. Driving on: This relation implies that the person is driving another vehicle and the motorcycle is on the road, indicating that they are sharing the same space or path. 7. Looking at: The person might be looking at the motorcycle, either out of interest or curiosity, or perhaps considering using it. 8. Touching: The person could be touching or interacting with the motorcycle, such as examining it or making adjustments. 9. Holding: The person might be holding onto the motorcycle, either for support or to prevent it from falling over. 10. Exiting: This relation suggests that the person was previously on or inside the motorcycle and is now in the process of getting off or exiting it. These are just a few possible relations between a person and a motorcycle, and the actual relation would depend on the context and specific details of the image.sky-building: Based on the given subject (sky) and object (building), here are some possible relations between them: 1. Over: The sky is over the building, as the building is located beneath the sky. 2. In front of: The building is in front of the sky, as it is positioned closer to the viewer than the sky. 3. Beside: The building is beside the sky, as they are both present in the same scene or landscape. 4. Attached to: The building may have structures or elements attached to it that extend into the sky, such as antennas or flagpoles. 5. On: The building may have something on its roof or top that is related to the sky, such as a weather vane or satellite dish. 6. In: The building may have windows or openings that provide views of the sky from within. 7. Flying over: Birds or other flying objects may be seen flying over the building in the sky. 8. About to hit: If there is an object in the sky, such as a meteor or a drone, it may appear to be on a collision course with the building. 9. Leaning on: If there is a structure or object leaning against the building, it may create a visual relation between the building and the sky. These are just a few possible relations based on general knowledge. The actual relation between the sky and the building in the image may vary depending on the specific context and composition of the scene.person-sea: Based on the given subject (person) and object (sea), here are some possible relations between a person and a sea: 1. Standing on: The person could be standing on the shore or a pier, with the sea beneath them. 2. Swimming in: The person might be swimming in the sea. 3. Looking at: The person could be looking at the sea, enjoying the view or observing something in the water. 4. Walking along: The person might be walking along the beach or coastline, with the sea beside them. 5. Surfing on: The person could be surfing on the waves of the sea. 6. Fishing in: The person might be fishing in the sea, either from the shore or from a boat. 7. Diving into: The person could be diving into the sea, either for recreational purposes or for activities like snorkeling or scuba diving. 8. Relaxing by: The person could be sitting or lying down near the sea, enjoying the peaceful atmosphere. 9. Sailing on: The person might be sailing on a boat or yacht on the sea. 10. Collecting seashells: The person could be collecting seashells or other items from the shore of the sea. These are just a few possible relations between a person and a sea. The actual relation would depend on the context and specific details of the image.rock-horse: Based on the given subject (rock) and object (horse), here are some possible relations between a rock and a horse: 1. Beside: The rock could be positioned beside the horse, indicating their proximity to each other. 2. On: The horse could be standing on the rock, using it as a platform or a vantage point. 3. In front of: The rock could be in front of the horse, blocking its path or serving as an obstacle. 4. Attached to: The rock could be attached to the horse, possibly as a weight or burden. These are just a few possible relations between a rock and a horse. The actual relation would depend on the context and specific details of the image.RJ-description. The description generated by LLMs based on the RJ-prompt.person-riding-motorcycle: Based on my knowledge, the relation riding is likely to exist between person and motorcycle. This is because it is common for a person to ride a motorcycle.person-drinking-motorcycle: No, the relation drinking is not likely to exist between person and motorcycle. This is because it is not common for a person to be drinking while interacting with a motorcycle.sky-over-building: Based on my knowledge, it is likely that the relation over exists between a sky and a building in an image. This is because in many real-life scenarios, a sky can be seen positioned above a building.sky-crossing-building: No, the relation crossing is not likely to exist between sky and building. This is because a sky cannot physically cross a building. The concept of crossing typically involves two objects intersecting or overlapping each other in some way, which is not possible in this case.person-playing-sea: No, the relation climbing is not likely to exist between person and sea. This is because it is not possible for a person to climb a sea. The relation swimming would be more appropriate in this case.person-sitting on-sea: Based on my knowledge, it is unlikely for a person to be sitting on a sea. The reason is that a sea is a large body of water and it is not possible for a person to physically sit on it. Therefore, the relation sitting on is unlikely to exist in the image.rock-attached to-horse: No, the relation attached to is not likely to exist in the image. This is because rocks are typically not attached to horses in real life.rock-falling off-horse: No, the relation falling off is not likely to exist in the image. This is because rocks cannot fall off horses as they are inanimate objects and do not have the ability to interact with horses in that way. §.§ Description from Llama2-7B RP-description. The description generated by LLMs based on the RP-prompt.person-motorcycle: Great! Based on the category names you provided, here are some possible relations between the subject (person) and the object (motorcycle): 1. Riding: The person could be riding the motorcycle. 2. Standing next to: The person might be standing next to the motorcycle, perhaps admiring it or preparing to ride it. 3. Holding onto: The person could be holding onto the motorcycle to steady themselves while riding it. 4. Leaning on: The person might be leaning on the motorcycle while taking a break or resting. 5. Fixing: The person could be fixing the motorcycle, perhaps making repairs or adjustments. 6. Parked near: The person could be parked near the motorcycle, perhaps intending to ride it later. 7. Getting on: The person could be getting on the motorcycle, perhaps after having just parked it. 8. Falling off: The person might be falling off the motorcycle, perhaps after losing control or experiencing an accident. 9. Facing: The person could be facing the motorcycle, perhaps to check on it or to prepare to ride it. 10. Behind: The person could be behind the motorcycle, perhaps to push it or to provide support while riding. Please let me know if any of these relations are correct or if you would like me to provide more options.sky-building: Great! Based on the category names you provided, here are some possible relations between the subject (sky) and the object (building): 1. Over: The building could be located above the sky. 2. Next to: The building could be situated next to the sky. 3. Looking at: The building might be visible from the sky. 4. Near: The building could be near the sky, perhaps located on the same horizon. 5. Part of: The building could be part of the sky, perhaps a cloud or a bird flying overhead. 6. Hanging from: The building could be suspended from the sky, perhaps by a bridge or a cable. 7. Going through: The building could be going through the sky, perhaps as a plane flying through the clouds. 8. Coming from: The building could be coming from the sky, perhaps as a hot air balloon rising into the atmosphere. 9. Flying over: The building could be flying over the sky, perhaps as a bird or an airplane. 10. Underneath: The building could be located underneath the sky, perhaps as a submarine or an underground structure. These are just a few possibilities based on the category names you provided.person-sea: Great! Based on the category names you provided, here are some possible relations that could exist between a person and a sea: 1. Beside: The person could be standing or sitting beside the sea, enjoying the view or engaging in activities such as swimming, fishing, or boating. 2. Looking at: The person might be looking at the sea, perhaps in awe or wonder at its vastness or its role in their life. 3. Swimming: The person could be swimming in the sea, either for exercise or recreation. 4. Fishing: The person might be fishing in the sea, using various techniques and tools to catch fish or other sea creatures. 5. Boating: The person could be on a boat in the sea, either for pleasure or for a specific purpose such as transportation or commercial fishing. 6. Surfing: The person might be surfing in the sea, using a surfboard to ride the waves. 7. Sunbathing: The person could be lying on the beach or on a boat, soaking up the sun and enjoying the warmth of the sea. 8. Beachcombing: The person might be walking along the beach, searching for interesting shells, rocks, or other treasures washed up by the sea. 9. Tidal pooling: The person could be exploring the shallow pools of water left behind by the sea at low tide, looking for creatures or interesting rocks. 10. Coastal defense: The person might be involved in activities related to protecting the coastline from erosion, pollution, or other threats, such as building seawalls or monitoring water quality. I hope these suggestions help! Let me know if you have any other questions.rock-horse: Great! Based on the category names you provided, here are some possible relations between the subject (rock) and the object (horse): 1. Beside: The rock could be beside the horse, perhaps as the horse is grazing or resting. 2. Looking at: The horse might be looking at the rock, perhaps curious about it or interested in it for some reason. 3. Near: The rock could be near the horse, perhaps in a field or meadow where the horse is roaming. 4. Riding: The horse could be ridden over the rock, perhaps as a way to traverse a difficult terrain or to reach a destination. 5. Jumping: The horse could be jumping over the rock, perhaps in a competition or as a way to clear an obstacle. 6. Kicking: The horse could be kicking the rock, perhaps as a way to dislodge it or to mark its territory. 7. Standing on: The horse could be standing on the rock, perhaps as a way to get a better view or to rest. 8. Falling off: The horse could be falling off the rock, perhaps due to a misstep or a sudden movement. 9. Crossing: The horse could be crossing the rock, perhaps as a way to get to the other side of a river or stream. 10. Pushing: The horse could be pushing the rock, perhaps as a way to move it out of the way or to create a path. Please let me know if you have any other questions or if you would like me to generate more relations.RJ-description. The description generated by LLMs based on the RJ-prompt.person-riding-motorcycle: Based on my knowledge, I would say that the relation riding is likely to exist in the image. People often ride motorcycles as a mode of transportation or recreation, so itś a common and plausible relation in real life. Therefore, I would give a Yes answer to the question of whether the relation exists in the image.person-drinking-motorcycle: Based on my knowledge, the relation drinking is unlikely to exist in the image. The reason for this is that it is not common for a person to be drinking a motorcycle. Motorcycles are vehicles that are typically used for transportation, not for drinking. Therefore, the relation drinking does not make sense in this context.sky-over-building: Based on the information provided, I would say that the relation over is likely to exist in the image. The reason for this is that buildings are often found in the sky, either standing alone or as part of a skyline. The relation over is a common way to describe the position of a building in relation to the sky, as it is typically located above the ground or other objects in the scene. Therefore, given the subject of sky and the object of building, the relation over is a likely candidate for the relation that exists between them in the image.sky-crossing-building: Based on my knowledge, the relation crossing is likely to exist in the image because the building is located in the sky, and it is common for buildings to be crossed by the sky in many real-life scenarios. Therefore, my answer would be Yes, the relation is likely to exist in the image.person-playing-sea: Based on my knowledge, the relation playing is unlikely to exist in the image you provided. The subject, person, is not likely to be playing with the object, sea, as they are two very different entities that are not typically associated with each other. The person is a land-dwelling creature, while the sea is a body of water. While it is possible for a person to be near or on the sea, it is not common for them to be playing with it. Therefore, I would say that the relation playing does not likely exist in the image you provided.person-sitting on-sea: Based on my knowledge, the relation sitting on is unlikely to exist in the image. People do not typically sit on seas, as seas are bodies of water and do not provide a suitable surface for sitting. Therefore, the probability of this relation existing in the image is low.rock-attached to-horse: Based on the information provided, I would say that the relation attached to is unlikely to exist in the image. A rock and a horse are very different objects and do not have a natural connection or relationship with each other. Therefore, it is unlikely that they would be attached to each other in a real-world scenario.rock-falling off-horse: Based on my knowledge, the relation falling off is unlikely to exist in an image of a rock and a horse. The reason for this is that rocks and horses are not typically associated with each other in a falling off context. Rocks are inanimate objects that do not have the ability to fall off, while horses are living creatures that are not known to fall off of other objects. Therefore, the probability of seeing a relation of falling off between a rock and a horse in an image is low.§ DESCRIPTION ANALYSISThrough the series of examples above, we analyze the descriptions generated by the LLMs as follows.Comparation between GPT-3.5 Turbo and Llama2-7B. Comparing the descriptions generated by GPT-3.5 Turbo and Llama2-7B, we find that GPT-3.5 Turbo's descriptions are slightly better than those of Llama2-7B. However, thanks to our carefully designed prompts, even using the Llama2-7B model can still produce high-quality descriptions to aid in relation prediction. As shown in Tab. <ref> of our paper, models trained using descriptions from Llama2-7B are only 0.5% lower in mR@100 compared to those using GPT-3.5 Turbo.RP-descriptions. For RP-descriptions, on one hand, some relation predicted by LLMs may occasionally fall out of the predefined set of K relation categories, the mechanism we designed can still handle it though.On the other hand, LLMs tend to be more accurate in predicting relations for thing-thing and thing-stuff pairs compared to stuff-thing and stuff-stuff. For stuff-thing and stuff-stuff pairs, models are more inclined to predict spatial relations.RJ-descriptions. For RJ-descriptions, in most cases, different models provide consistent and reasonable judgments with corresponding explanations. However, there are occasional inconsistencies, such as with the relation of sky-crossing-building, where GPT-3.5 Turbo deems it impossible, but Llama2-7B considers it possible. Nevertheless, all LLMs provide reasons to explain the possibility of the relation, and these reasons in the descriptions provide context to the relation judgment, thereby assisting the model in relation prediction. § MORE VISUALIZATIONSFig. <ref> to <ref> showcase additional visualization results of our VLPrompt. Each figure contains two examples. For each example, the top shows the results of panoptic segmentation, the bottom left displays the ground truth, and the bottom right shows the top 10 relation prediction results. Based on visualization results, our VLPrompt exhibits precise capabilities in relation prediction, thereby enhancing scene understanding. For instance, in Fig. <ref> on the left side, the top 10 relation predictions not only encompass all the relations from the ground truth but also include additional correct relations involving stuff categories. However, we have also identified some bad cases, of which many are due to inaccurate panoptic segmentation predictions (including category classification and mask prediction). | http://arxiv.org/abs/2311.16492v1 | {
"authors": [
"Zijian Zhou",
"Miaojing Shi",
"Holger Caesar"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127170525",
"title": "VLPrompt: Vision-Language Prompting for Panoptic Scene Graph Generation"
} |
Mass reconstruction and noise reduction with cosmic-web environments Longlong Feng^1 January 14, 2024 ==================================================================== § INTRODUCTION Liquid argon (LAr) is a common scintillating medium in dark matter WIMP detectors, such as DEAP-3600<cit.> and DarkSide-20k<cit.>, and in neutrinoless double-beta decay experiments like LEGEND-200<cit.> for background suppression. Ionizing particles in LAr induce vacuum-ultraviolet (VUV) scintillation light at 128. Most light detectors have low sensitivity in the VUV range, necessitating the use of wavelength shifters (WLS). The widely used 1,1,4,4-tetraphenyl-1,3-butadiene (TPB) is applied as a coating to optically inactive surfaces or to photo-detectors like photo-multiplier tubes. An alternative is the scintillating and wavelength shifting polymer polyethylene naphthalate (PEN), increasingly popular for its commercial availability as pellets, which can be injection molded into structural parts or extruded into thin films<cit.>.The LEGEND-200 experiment investigates neutrinoless double-beta decay of 76Ge using 200 of enriched high-purity germanium (HPGe) crystals, which serve as both the decay source and detectors<cit.>. ALAr volume, acting as a coolant and active shield, surrounds the HPGe detectors. The light collection system consists of light-guiding fibers surrounding the detectors, coated in TPB and coupled to silicon photomultipliers (SiPMs). The LAr detector's active volume is confined by a 1.4 diameter, 3 height wavelength shifting reflector (WLSR) consisting of a Tetratex®lining coated in-situ with TPB<cit.>. The WLSR shifts LAr scintillation light and reflects light towards the central readout, enhancing the detection efficiency. The detector holder plates are made out of PEN, increasing the background suppression efficiency in the close vicinity of the detectors. They scintillate when traversed by ionizing radiation and shift LAr scintillation light to the visible blue.To understand and model the response of the LAr detector of LEGEND-200 and other experiments, the optical properties of the WLS must be known precisely. This includes, among others, the emission spectrum and the wavelength shifting efficiency. For LAr-based experiments, this means, that the materials need to be characterized at 87 and with VUV light excitation, i.e., 128.§ CRYOGENIC VUV SPECTROFLUOROMETER A dedicated experimental setup was designed and commissioned to characterize WLS at temperatures ranging from 30087, with excitation wavelengths down to 128. The VUV spectrofluorometer is schematically depicted in <ref>. In the sample chamber, a copper holder accommodates a 30 × 30mm^2 WLS sample and is equipped with a 30 polyimide heater on its back (indicated in <ref> as (i)). The holder, positioned on the second stage of a two-stage Gifford McMahon coldhead, is encased by a copper cryo-shield (ii), leaving openings for excitation light, light detection, and cabling. A copper beaker filled with silica gel is connected to a copper cold finger, which is partially submerged in a liquid nitrogen dewar outside the chamber (iii). The temperature at the sample holder recorded by Pt-100 sensors is the input for a PID controlling the heater's power (iv)<cit.>. A Hamamatsu L15094 deuterium lamp (λ_Lamp=120-190) (v) and a Marktech Optoelectronics MTE310H33-UV LED (λ_LED=310 ± 10) (vi) are used to excite the sample. The deuterium lamp is mounted to an Acton VM-502 VUV-IR monochromator (vii), which is attached to the sample chamber, enabling excitation light at specific wavelengths (20 width). The deuterium lamp degrades in light output over time<cit.>. Therefore, a VUV-sensitive McPherson photodiode (viii) is used to monitor the output of the deuterium lamp. The LED is operated by a constant current LED driver and is mounted on the cryo-shield.For light collection, a collimating lens directs some of the shifted light to an optical fiber, guiding it to an OceanOptics QE65000 UV-IR spectrometer (ix)<cit.>. The lens is perpendicular to the excitation beam. By rotating the sample holder, the angle of incidence and detection can be adjusted between 15 and 75 in 15 steps. With the setup, it is possible to measure the relative wavelength shifting efficiency and emission spectra at different temperatures or compare the properties between different samples in the same configuration.To prevent attenuation of VUV light on air, the entire system is evacuated to a vacuum of approximately e-7. When cooling the sample, it is possible for residual humidity to form a thin ice layer on the surface of the sample. This surface effect has been shown to significantly reduce the amount of VUV light reaching the sample, affecting the measurement <cit.>. To counter this effect, a two-step mitigation strategy is used: first, the cryo-shield and beaker filled with silica gel are cooled before the sample to adsorb the remaining humidity. Second, the UV LED is used as a normalization for the measurements as it is unaffected by the ice layer.§ CHARACTERIZATION OF WAVELENGTH SHIFTERSThe WLSR sample used in this study is from the same material installed in LEGEND-200 WLSR. The sample comprises of a 600 TPB layer coated on 250 Tetratex®, backed by a copper foil. Further details on how the WLSR was coated and details of the commissioning can be found in <cit.>.The PEN sample was cut from the same material installed in LEGEND-200. The material was formed using injection molding of commercially available PEN pellets from Teijin-DuPont (TN-8065 SC). Additional information on the material's production and characterization is presented in <cit.>. The amorphous PEN sample guides its emission light to its edges, resulting in a low signal-to-noise ratio in the VUV spectrofluorometer. To enhance the amount of detected light, the surface of the PEN sample was sanded using a 600-grit diamond sanding block, and a copper foil was placed as a reflector and thermal coupling to the sample holder behind the sample.For each measurement, the obtained spectrum is corrected for the known response of the light detection system and the dark noise of the spectrometer. Furthermore, the data is refined by excluding noisy pixels. The stability of the deuterium lamp is monitored using the VUV-sensitive photodiode. Three sources of uncertainties are considered in the data analysis: 1) system instability, 2) uncertainty in the dark noise level, and 3) uncertainty in the response of the light detection system. To obtain the relative WLSE, the WLSR and PEN spectra are integrated between 370550 and370650, respectively.§ RESULTS AND DISCUSSION The obtained spectra for the characterization of the WLSR sample are shown in <ref>. The vibronic structures of the TPB molecule at 405;450 emerge at lower temperatures. In <ref>, the relative wavelength shifting efficiency (WLSE) of the cryogenic measurements compared to the measurement at room temperature (RT) is shown. We measured a strong increase in WLSE of 54 ±4 at LAr temperature (87) for excitation with LAr scintillation light wavelength (128). The intrinsic quantum efficiency (QE) of TPB at RT for VUV excitation was measured by Benson et al. to be 0.60 ±0.04<cit.>. Multiplied with the relative increase measured in this work, one gets a QE of approximately 0.92 ±0.08, which is compatible with the QE of the LEGEND-200 WLSR obtained by Araujo et al. of 0.85 ±0.05(stat)±0.06(syst.)<cit.>. Our results are compared with previous studies of TPB. Ellingwood et al. investigated the emission of TPB excited by UV LED with a wavelength of λ_exc=285. For their experiment, a 1 thick TPB layer was coated on a 5 thick PMMA substrate<cit.>. Francini et al. characterized a TPB coating of approximately 1.5 thickness on polymeric reflector (Vikuiti™) deposited by vacuum evaporation <cit.>. The emission spectrum at 87 is compared to these studies in <ref>. The spectrum we detected is in good agreement with the results from <cit.> and <cit.>. The minor differences in the spectra might originate in differences in the TPB vacuum evaporation procedure<cit.>, in differences in the reflectance/transmittance of the backing material, or the lack of calibrated response of the detection system used in <cit.> and <cit.>.In <ref>, the relative WLSE between RT and 87 is compared. The relative increase we measured both with 128 and 310 excitation exceeds the increase reported by <cit.> and <cit.> by roughly a factor of two. The measurements by Francini et al. might suffer strongly from the above-mentioned ice layer formation, lowering the increase in WLSE<cit.>. Alternatively, the quality of the TPB layer and the type of substrate may influence the performance at cold temperatures differently. The emission spectra of the PEN sample for temperatures between 300 and 87 are displayed in <ref>. A slight shift in the peak wavelength from 430 for 293 to 425 for 87 can be seen. Additionally, the monomeric emission at 575 is minimally enhanced at cold temperatures. Both effects are in agreement with previous studies of PEN using 300 excitation<cit.>.The relative increase in WLSE compared to 293 is presented in <ref>. One can see a good agreement between the increase for both excitation wavelengths down to 150. Below, the WLSE increases significantly less for the VUV excitation. Because of its larger mass compared to the WLSR sample and its poor thermal conductivity, the PEN sample needs more time to cool down to 87. Therefore, the surface effect, which becomes significant below 150 for e-7<cit.>, impacts the measurement with 128 much stronger than for the WLSR measurement. Hence, one can assume that the WLSE of PEN at LAr temperature (87) relative to 293 for excitation with LAr scintillation light wavelength (128) is equal or higher than the measured value for 310 excitation of 149 ±1.§ CONCLUSIONWe present the necessity and challenge of the characterization of WLS for the application in LAr-based experiments like LEGEND-200. For the characterization, we commissioned a VUV spectrofluorometer capable of measuring the emission spectrum and relative WLSE of wavelength shifting samples for excitation with LAr scintillation wavelength (128) at temperatures from RT down to LAr temperature (87). For the TPB-based wavelength shifting reflector featured in LEGEND-200, the WLSE at 87 was found to be increased by 54 ±5 compared to RT. The WLSE of PEN, used in LEGEND-200 for the optically active detector holders, increased by at least 49 ±1 by cooling from 29387. To the knowledge of the authors, this work is the first characterization of the emission spectrum and relative WLSE of amorphous PEN at cryogenic temperatures with VUV and UV excitation.This research is supported by the DFG through the Excellence Cluster ORIGINS and the SFB1258. JHEP | http://arxiv.org/abs/2311.15901v3 | {
"authors": [
"A. Leonhardt",
"M. Goldbrunner",
"B. Hackett",
"S. Schönert"
],
"categories": [
"physics.ins-det"
],
"primary_category": "physics.ins-det",
"published": "20231127150354",
"title": "A novel cryogenic VUV spectrofluorometer for the characterization of wavelength shifters"
} |
[ The secondary maximum of T CrB caused by irradiation of the red giant by a cooling white dwarf [ January 14, 2024 ============================================================================================== Displacement current is the last piece of the puzzle of electromagnetic theory. Its existence implies that electromagnetic disturbance can propagate at the speed of light and finally it led to the discovery of Hertzian waves. On the other hand, since magnetic fields can be calculated only with conduction currents using Biot-Savart's law, a popular belief that displacement current does not produce magnetic fields has started to circulate. But some people think if this is correct, what is the displacement current introduced for. The controversy over the meaning of displacement currents has been going on for more than hundred years. Such confusion is caused by forgetting the fact that in the case of non-stationary currents, neither magnetic fields created by conduction currents nor those created by displacement currents can be defined. It is also forgotten that the effect of displacement current is automatically incorporated in the magnetic field calculated by Biot-Savart's law. In this paper, mainly with the help of Helmholtz decomposition, we would like to clarify the confusion surrounding displacement currents and provide an opportunity to end the long standing controversy. Keywords: displacement current, Biot-Savart law, Ampere's law, Maxwell-Ampere's law, Helmholtz's decomposition, non-stationary current]§ INTRODUCTIONThe time derivative of the electric flux density, ∂_t D, is named displacement current density, which is the final piece to complete the hard puzzle of electromagnetic theory. This discovery made by James Clerk Maxwell <cit.> was possible only through his keen eyes forseeing its existence from theoretical inevitability (1864).He found the fact that the propagation velocity of the wave solution enabled by the displacement current, was consistent with the speed of light, which was already measured experimentally at that time. The value of constant (μ_0ε_0)^-1 had been determined by Weber and Kohlrausch in other context<cit.>, where μ_0 and ε_0 are the permeability and permittivity of vacuum, respectively. Maxwell was convinced that light is an electric and magnetic disturbance propagating in a vacuum. Later, H.R. Hertz discovered radio waves (1888) in attempting to detect displacement currents using a capacitor.Displacement currents occupy an important position in electromagnetics. However, owing to the fact that magnetic fields can be correctly calculated by the Biot-Savart law, which does not seem to include the displacement current, it has widely been claimed that the displacement current does not produce a magnetic field. As a matter of fact, however, the Biot-Savart law implicitely includes the contribution of displacement currents.In this paper, we would like to clarify the confusion surrounding displacement currents and its causes and help to promote correct understanding.§ MAGNETIC ACTION OF ELECTRIC CURRENTS AND DISPLACEMENT CURRENT In 1820, Hans Christian Ørsted discovered that a compass needle swings in response to an electric current flowing near it. That same year, the relation between current and magnetic field was formulated in two ways; Biot-Savart's law and Ampère's law, which correspond to Coulomb's law and Gauss's law in electrostatics, respectively.These magnetic field laws were based on the assumption that the current is flowing through a closed circuit. Almost half a century later, considering the case of unclosed current, as in the case of charging capacitor, Maxwell theoretically derived the necessity of displacement currents.His argument goes as follows. By taking the divergence of both sides of Ampère's equation, H =J, we have, 0 =J, with the identity = 0. In other words, Ampère's equation implicitely assumes divergence-free currents. This is also called the "steady-state current condition," because the charge conservation law, ∂_tϱ = - J, implies steady charge distributions. (∂_t = ∂/∂ t is used for brevity.) If this condition is not satisfied, i.e., J≠0, Ampère's law must be modified as follows:H =J + ∂_t D .The displacement current density term, ∂_t D is added. Now, taking the divergence of both sides, we have0 =J + ∂_t D =J + ∂_tϱ .The time derivative of Gauss's formula, D=ϱ, is used. This is consistent with the charge conservation law.Based on this reasoning, in his treatise <cit.>, Maxwell statesOne of the chief pecurialities of this treatise is the doctrine which it asserts, that the true electric current ℭ (C), that on which the electromagnetic phenomena depend, is not the same thing as 𝔎 (J), the current of conduction, but that the time variation of 𝔇 (D), the electric displacement, must be taken into account in estimating the total movement of electricity, so that we must write,ℭ = 𝔎 + 𝔇̇, (Equation of True Currents) . Hereafter we write the true (total) current asJtot =J + ∂_t D . Equation (<ref>) is now called Maxwell-Ampère's equation. Maxwell's electromagnetic theory thus created is being organized by the followers and spread to the academic worldOddly, however, the doctrine that displacement currents do not create magnetic fields began to circulate. The main reasons are * Even in the presence of displacement currents, magnetic field is calculated correctly by the Biot-Savart equation.* A typical displacement current is one that occurs where the linear current is interrupted. Then charges are accumulated at the endpoint and yield a spherically symmetric electric field. The corresponding displacement current is also spherically symmetric and the resultant magnetic field vanishes. Various arguments against, for, or from a neutral standpoint about this theory, some of which seem to deepen the confusion, are continuing in papers and textbooks <cit.>. In this paper, we will show the claim that displacement current does not create a magnetic field is due to a lack of understanding of the mathematical structure of electromagnetic fields. We mainly discuss from the following points of view: * It is impossible in principle to separate magnetic fields into those caused by "conduction currents" and those caused by "displacement currents".* The posed question "does a displacement current create a magnetic field or not?" is logically meaningless.* Contrary to popular perception, the Bio-Savart law perfectly includes the effect of displacement currents implicitely <cit.>.§ NEED FOR DISPLACEMENT CURRENT In this section, we will reconfirm how the Ampère's law is modified to account for displacement currents.The integral form of Ampère's law, H= J, is∫_C H· l = ∫_S J· S ,where the closed path C=∂ S is the edge of the surface S. In this equality the surface S can be arbitrary as long as the closed path C is its edge. In order for the integral to be the same regardless of the surface, J = 0 must be satisfied everywhere (steady-state current condition). Otherwise, for surfaces S_1≠ S_2 with ∂ S_1=∂ S_2=C, the divergence theorem gives0≠∫_VJv = (∫_S_1-∫_S_2) J· S ,where V is the volume enclosed by S_1 and S_2.Consider a capacitor being charged with a constant current I, as shown in Fig <ref>. We have two surfaces S_1 and S_2 that share the same circle C encircling the capacitor as their respective circumferences. While the hemisphere S_1 crosses the current I, the disk S_2 passes between the electrodes of the capacitor and crosses no currents.The integrals of the current density for these surfacesI = ∫_S_1 J· S ≠∫_S_2 J· S = 0 ,are clearly not equal. But if we add the displacement current density ∂ D/∂ t, then the surface integral for S_2 becomes∫_S_2( J +Dt)· S = ∫_S_2 Dt· S = Q(t) = I ,and now the equality holds. The charge on the capacitor plate Q = ∫_S_2 D· S can be derived from D between the plates. With this model we can confirm that Eq. (<ref>) must be modified as∫_C H· l = ∫_S( J + Dt)· S .This is the integral form of Maxwell-Ampère's equation. § INAPPROPRIATE PROBLEM SETTINGIn this section, we show that the question whether the displacement current creates a magnetic field or not is totally meaningless.The question can be broken down in the following way. When the total current density Jtot is divided into the conduction current density J and the displacement current density ∂_t D, the magnetic field can be divided into two components as H =H c +H d, correspondingly. And if we can prove H d=0 (≠0), then the answer is no (yes). But, in the first place, what are the (local) equations that these magnetic fields obey? The only possible choice seems to beH c ?= J, H d ?=∂_tD ,but it leads us to the contradiction when we take the divergence;0 ?= J≠ 0,0 ?=∂_t D≠ 0.What did we introduce the displacement current for?We have found that the magnetic field created the displacement current cannot be defined and therefore, it makes no sense to ask whether such an undefinable quantity is zero or not.In general, if we want to solve the Maxwell-Ampère equation, H =J + ∂_t D, by superposition, then we should setH_1 =J_1 + ∂_t D_1 , H_2=J_2 + ∂_t D_2 ,and divide not only the current density but also the displacement current density term, so that each of them satisfies( J_1 + ∂_t D_1) = 0, ( J_2 + ∂_t D_2) = 0 .This is the proper division of total current. On the other hand the division of Eq. (<ref>) makes no sense.Let us consider a mathematical case to see why a simple-minded superposition does not hold. For a general linear map A:X→ Y, from a space X to another Y, let Im A = { A x| x∈ X}⊂ Y, which is called the range or image of A. In order that the linear equationA x =b ,has a solution x∈ X, the condition b∈Im A must be met. Even when b =b_1 +b_2 ∈Im A, if b_1,b_2 ∉ImA then the superposition cannot be used. BecauseA x_1 =b_1, A x_2 =b_2 ,have no solutions. An example isA=[ 1 1; 0 0 ] , b = [ 2; 0 ] , b_1 = [ 1; 1 ] , b_2 = [1; -1 ] . To make a situation where the solution is unique, we should set up another linear equation, B x=0, which corresponds to (μ_0 H)=0 in our case.§ BIOT-SAVART'S LAW AND DISPLACEMENT CURRENT Here we will show that contrary to common belief Biot-Savart's law include the effect of displacement currents <cit.>. For the sake of brevity, the Coulomb field is written asG( r) :=r/4π | r|^3 .With this, we have · G( r) = δ^3( r) , × G( r) = 0 , (1/r) = -4π G( r), where δ^3( r) is the delta function.(For calculation, we use ·, ×, and , instead of , , and .)The electric flux density for a charge q placed at the origin is D_q( r) = q G( r) and that for an electric dipole p = q l at the origin isD_ p( r) = -( p·) G( r) .For the current element I l placed at the origin, the magnetic field created at the point r is given by Biot-Savart law in the difference formH( r) = I l× G( r) .The magnetic field due to the current I flowing through the closed circuit L is given as a superposition (integral)H( r) = ∮_LH( r -r') = ∮_L I l'× G( r -r') ,where l' is a line element at position r' on path L. Originally, the condition that "L is closed" was required by Biot-Savart's integrated formula initially.Let us find the vortex of the magnetic field element (<ref>). With the help of a vector analysis formula,we have× H( r) = ×[(I l)× G( r)] = I l [· G( r)] - (I l·) G( r) = I l δ^3( r)+ t D_It l( r) =:Jtot( r) .We note that in addition to the original current element I l the additional term appears. This term is the time derivative of the electric flux density (<ref>) for the electric dipolep(t)=ItΔ l at the origin. It means that if a constant current I flows on the line element l, then there accumulates charge ± Q(t)=± It at each end ± l/2 to form an electric dipole.The total current Jtot (<ref>) satisfies the stationary condition and the magnetic field H is generated by this total current Jtot.The condition that L is closed, which was originally assumed in the Biot-Savart equation, is actually unnecessary. When a current is formally integrated for an open path L with start (end) point r_1 (r_2), we obtainJtot( r)= ∫_L Jtot( r- r')= I∫_L l'δ^3( r-r') + . 0mm2.5ex I G( r- r') |_ r'= r_1^ r_2 .As shown in Fig. <ref>, when the current elements are connected (integrated), the displacement currents from opposing endpoints cancel each other and only those at the two extreme ends of the path remain. This is similar to the case where small bar magnets are connected to form a long chain. The magnetic fields of opposite polarity at the connection points cancel each other, and only the magnetic fields from the poles at both ends survive.In the end, the Biot-Savart's equation (<ref>) can be applied even for unsteady (open) currents, just with the modification of integration path; ∮→∫, which is beyond the originally intended scope of application.Although it is an integral of the current distribution J, the total current Jtot =J +Jdisp is automatically taken into account and the corresponding magnetic field is given.This fact has been pointed out by from time to time <cit.>,but many people still mistakenly believe that when using Biot-Savart they can calculate the magnetic field only due to the conduction currents that they give. But even if you don't order the displacment current, it will always come by itself. This "hidden trick" is one of the sources of confusion over the displacement current.Even though both Biot-Savart's law and Ampère's law were established in the same year (1920), only the former consealed the effect of displacement current that will be exposed 45 years later.§ HELMHOLTZ DECOMPOSITION OF VECTOR FIELDSFrom a mathematical point of view, we take a closer look at how the Biot-Savart equation automatically incorporates the effect of displacement currents. The current density field J (or a three dimensional vector field in general) can be divided into the "curl-free" component JL and the "divergence-free" component JT, namely,J( r) =JL( r) +JT( r), JL( r) = 0,JT( r) = 0 .This is the so-called Helmholtz decomposition <cit.>. The uniqueness of the decomposition requires that the field quantities converge quickly to zero at infinity, which is satisfied in the present case.The curl-free and divergence-free fields are also called the “longitudinal” and “transverse” fields, respectively, which are denoted by subscripts “L” and “T”. The latter names are derived from the relations of their Fourier transform to the wavevector k;k×(ℱ JL)( k)=0, k·(ℱ JT)( k)=0 . The Biot-Savart law for the three-dimensional current density J( r) isH( r) = ∫_V v'J( r')× G( r -r') ,and its vortex isH( r) =J( r) - ∫_Vv' ('· J( r')) G( r -r') .The first term is just the given current density. With the charge conservation of law · J( r) + ∂_tϱ( r, t) = 0, the second term turns out to be the displacement current density:∫_V v' ∂_tϱ( r', t) G( r -r') = ∂_t D( r, t) .We can verify that ∂_t D is longitudinal.Equation (<ref>) can be rewritten asH( r) =J( r) - L̂ J( r) = (1̂ - L̂) J( r) ,where 1̂ is the identity operator, and the operator L̂ acts on the vector field J( r) to create a new vector field:(L̂ J)( r):= ∫_V v' ('· J( r')) G( r -r') .The operator L̂ gives the longitudinal field components of a vector field.We define another operator T̂ := 1̂ - L̂, which gives the transverse field component T̂ J. The operators L̂ and T̂ are equipped with the properties of projection operators, namely,T̂^2 = T̂,L̂^2 = L̂,T̂L̂ = L̂T̂ = 0 . As shown in Fig. <ref>, the Biot-Savart law gives the magnetic field due to the transverse component of the current density JT = T̂ J =J -JL, or due to the total current densityJtot :=J +Jdisp .From these equations, we know that the displacement current is the curl-free (longitudinal) component with sign changed of the given current:Jdisp = - JL(=∂_t D) .Adding the displacement current density Jdisp to the current density J means the cancellation of the longitudinal component JL to yield JT or Jtot. The difference between the subtraction and the sum gives a very different impression.In the cource of deriving the Biot-Savart's law from Maxwell's equations, we confirm why the former includes the effect of displacement currents. With the curl of the Maxwell-Ampères's law, H =J T, J T= J+ ∂_t D, and (μ_0 H) = 0, we have∇^2 H = - J T ,where =-∇^2 is used. The solution to this (vector) Poisson's equation <cit.>isH( r) = ∫ v' JT( r')× G( r -r') =: (f̂T J T)( r) ,where the operator f̂T is defined so as to map a transverse current JT to the corresponding magnetic field H. This map, which can be written symbolically H = ^-1 J T, is invertible, i.e., one-to-one. For a general current J, we should haveH = f̂TT̂ J =: f̂BS J ,which correponds to the Biot-Savart's law (<ref>). The two-step operation, f̂BS = f̂TL̂ is not invertible. The current distribution cannot be uniquely determined from the magnetic field. This fact is overlooked so often.In order to fix the problem of improper division of Eq. (<ref>), we can apply T̂ for each current.H = T̂ J = J + ∂_t D,H d = T̂(∂_t D) = 0 .Then, in the second equation, the displacement current ∂_t D certainly disappears and H d vanishes. But we should remember that it reapears in the first equation and contribute to H. We cannot erase the displacement current.§ EXAMPLE OF HELMHOLTZ DECOMPOSITION OF CURRENT DISTRIBUTIONIn Fig.<ref>, the Helmholtz decomposition is shown for several current distributions. All of these examples has been used to discuss displacement currents. Each of them is briefly described below. We have already mentioned in Sec. <ref> about the current elements in the first row. In the case of point charge q moving at velocity v, we can set p = q v, instead of I l. §.§ Semi-infinite linear current If a constant current I is flowing along the half line (z≤ 0) along the z-axis, the charge must be accumulated at the origin as Q(t) = It + Q(0), where Q(0) is the charge at time t=0.Electric flux density of Coulomb type and the associated displacement current density ∂_t D(t,r)=I G( r) are generated. We can derive the magnetic fields in two ways, i.e., with Biot-Savart's law and Maxwell-Ampère's formula.Using cylindrical coordinates (ρ,ϕ,z), the Biot-Savart law for an open path, L={ r'=z e_z | -∞<z≤ 0}, we haveH_ϕ( r) = H_ϕ (ρ, z) = ∫_L I z e_z× G( r- r') =I/4π∫_-∞^0 ρz'/[(z-z')^2+ρ^2]^3/2 =I/4πρ( 1-z/√(z^2+ρ^2)) .Secondly, to apply Maxwell-Ampère's formula, consider a spherical surface with radius R=√(ϱ^2+z^2) centered at the origin. We have a latitude line C defined by θ=tan^-1(ϱ/z)=const. Let S_+ (S_-) be the northern (southern) spherical crown cut by C. Note that C = ∂ S_+ = -∂ S_-. Applying the Maxwell-Ampère formula (<ref>) for both surfaces, we have∮_CH· l = ∫_S_+∂_t D· S = -∫_S_-( J + ∂_t D)· S .The left hand side is 2πρ H_ϕ(ρ, z). The middle side is evaluated as follows. Since the magnitude of the displacement current on a sphere of radius R is ∂_t D = I/(4π R^2), it is perpendicular to the sphere, and the area of S_+ is |S_+|=2π R^2(1-cosθ), we have∂_t D |S_+| = I/2(1-cosθ) .Similary for the right hand side using |S_-|=2π R^2(1+cosθ) and ∫_S_- J· S = -I, we haveI - ∂_t D |S_-| = I - I/2(1+cosθ) .Thus, the same result is obtained by using either of the surfaces:H_ϕ(ρ, z) = I/4πρ(1-cosθ) = I/4πρ(1-z/√(ρ^2+z^2)) .This result agrees with the previous result by Bio-Savart's law. In this method we had to consider the displacement currents explicitly, otherwise, the solution is not uniquely determined. §.§ Spherically symmetric current distribution To back up the false claim that displacement currents do not create magnetic fields, the fallacious theory that "spherically symmetric currents do not create a magnetic field due to their symmetry" is developed. Combining Biot-Savart's equation and symmetry for each part of the current, it is attempted to to show that the magnetic field is zero.As an example, let's take a look at section 9.2 of Purcell's textbook <cit.>. Noting that a curl-free field like displacement current density can be written as a superposition of spherically symmetric Coulomb-type vector fields, it continues,... the magnetic fields of any radial, symmetrical current distribution, calculated via Biot-Savart, is zero. To understand why, consider a radial line through a given line through a given location. At this location, the Biot-Savart's contributions from a pair of points symmetrically located with respect to this line are equal and opposite, as you can verify. The contributions therefore cancel in pairs, yielding zero field at the given location.Here the fact is forgotten that Biot-Savart's law includes the effect of displacement current, which in this case is spherically symmetric but flows in the opposite direction.In his famous series of text book <cit.>, Feynman correctly explained the situation of spherically symmetric current using a model where a small sphere with radioactive material is squirting out some charged particles.Physically speaking, as shown in Fig. <ref>, spherically symmetric currents can be thought of as a large number of semi-linear currents isotropically combined at a single point. In this case, the displacement currents at the endpoints add up to exactly cancel the original currents. The total current becomes zero in all places, and therefore the magnetic field is also zero. The magnetic field vanishes not owing to the cancellation of real currents. In short, if the symmetric current (conducting or displacement) is purely longitudinal, the total current is zero. §.§ Charging capacitor The Charging capacitor problem is one of the reasons why Maxwell came up with the idea of displacement current. The debate continues over this model, as to whether the displacement current between the electrodes creates a magnetic field or not <cit.>. Surprisingly, miscalculations are found, from time to time, in dealing with this problem. For example, Roche <cit.> made wrong calculation in his historical survey paper on the controversy over the reality of displacement current. Jackson<cit.> pointed out the mistake and complained that the paper is marred by imprecision, sloppy notation, and downright mistakes.For simplicity, let us assume an axisymmetric system around z-axis as shown in Fig. <ref>. Let R be the radius of the electrodes and d be the spacing. The system is assumed to be charged by a constant current I through the conducting wires. Assuming that d≪ R, the electric flux density between the electrodes is spatially uniform and given as D_z(t)= It/π R^2.Since the current has no angular component, J_ϕ=0, the magnetic field has only the azimuthal component H_ϕ. The magnetic field due to the current I on the straight conductor isH(1)_ϕ(ρ, z) = I/2πρ . The current I in the wire z<-d/2 reaches the eletrode at z=-d/2 and spreads radially and the electrode is charged with a uniform charge surface density. The linear density of this radial current K_ρ(ρ) is dependent on ρ and can be determind by the charge conservation condition and the uniformity. With the two-dimensional divergence formula we have ρ^-1(/ρ)[ρ K_ρ(ρ)] = const. Under the boundary conditions (2πρ K_ρ)(0)=I and K_ρ(R)=0, the differential equation can be solved;K_ρ(ρ) = I/2πρ(1-ρ^2/R^2) .For the other electrode at z=d/2, the surface current density is -K_ρ(ρ).The straight conductor has a gap -d/2 < z < d/2 of length d. The H(1)_ϕ includes its contribution. Therefore, we have to introduce a current of -I flowing in this segment <cit.>. (Roche <cit.> missed this contribution, which remains even in the limit, d→0.) If we connect this current element to the two surface currents, the displacement currents from these two connection points become zero and only those between the electrodes remain.In order to apply Ampère-Maxwell's law, we set up a circular loop, ρ=const between the electrodes (-d/2<z<d/2). For ρ≤ R, we have 2πρ H^(2)_ϕ = -I + Iρ^2/R^2, orH^(2)_ϕ(ρ, z) = I/2πρ(-1 + ρ^2/R^2) ,For ρ>R, we note H^(2)_ϕ = 0. By superposition, we have the magnetic field of all position asH_ϕ(ρ, z) = H^(1)_ϕ(ρ, z) + H^(2)_ϕ(ρ, z) = I/2πρ/R^2(ρ≤ R, -d/2<z<d/2)I/2π1/ρ(otherwise).The difference between the magnetic fields inside and outside the electrodes is equal to the surface current density (<ref>):H_ϕ(ρ, d/2+0) - H_ϕ(ρ, d/2-0) = K_ρ(ρ) .This current splitting in Fig. <ref>, which follows the rule (<ref>), is a clever way to avoid displacement currents except those between these electrodes.Many experiments on charging capacitor have been conducted <cit.>.Especially, the careful measurement by Bartlett and Corle <cit.> seems very precise and consistent with Eq. (<ref>). But the title of paper “Measuring Maxwell's displacement current inside a capacitor” is misleading, because what was measured was the magnetic field inside the capacitor. As authors admitted in the very last paragraph (and later in <cit.>),What we have shown, then, is that the Biot-Savart law applies to open as well as to closed circuits. One may write the differential form of this law as B= I l× R/R^3, without the usual caveat that only the integral around a closed loop is meaningful.The experiment does not answer the question whether the displacement current generate magnetic field or not (or the question itself is meaningless). In my opinion, it should be stressed that the vortex of the magnetic field between the electrodes was actually measured. The existence of such vortices cannot be explained without displacement currents. Further more it might be interesting to demonstrate that when the wiring path to the capacitor is changed (e.g., to form a coil) so as the magnetic field between the electrodes is disturbed, the uniform vortex is still maintained.Magnetic fields generated by a capacitor discharging through the partially conducting spacer are dicussed in confusion from time to time (leaky capacitor) <cit.>.The field profile can better be understood in terms of that for a current element (see Fig. <ref>).§ DISPLACEMENT CURRENTS AND ELECTROMAGNETIC WAVESLet us look at the relationship between displacement currents and electromagnetic waves utilizing the Helmholtz decomposition (<ref>). From two of the Maxwell equations and the constitutive relation of vacuumBt = - E , Dt =HD = ε_0 E , H = μ_0^-1 B ,we have the equations for the plane waves propageted in the z direction (assuming ∂_x=∂_y=0 and x polarization, i.e., E_y=0)ε_0E_xz=-H_yt, μ_0H_yz=-E_xt .The d'Alembert solution to the hyperbolic partial differential equations isE_x(t, z) = f(z - c_0 t) + g(z + c_0 t) .The f and g are arbitrary functions. These waves are propagated at the velocity ± c_0 = ±√(μ_0/ε_0). If there were no displacement current term, then c_0→∞ and no wave solution existed.So far, we assumed the case where the steady current condition is violated (J≠0, i.e. ∂_t D≠0). We now relax the condition further and consider the case where the magnetic field is also time-varying (∂_t B≠0).We separate Maxwell's equations into the longitudinal and transverse components. Since the magnetic flux density satisfies B=0, we have B ≡ BT, or BL=0. The equation of electromagnetic induction is( E T +E L) =ET = -∂_t BT .For the electric flux density, from ( D T+ D L) =DL = ϱ, the longitudinal component DL is related to the charge density ϱ. On the other hand, the transverse component DT = ε_0 ET is related to the electromagnetic induction equation (<ref>).Substituting H= HT=μ_0^-1 BT, into Maxwell-Ampère's equation, we haveHT =JT + ∂_t DT+(JL + ∂_t DL ).Since the left-hand side is a transverse field (divergence-free), as in the case of a stationary field, the longitudinal components cancel each other; JL + ∂_tDL≡ 0. This can be regarded as the role of the longitudinal component of the displacement current. In summary, we haveET = -∂_t BT , HT =JT + ∂_t DT,BL=0,DL=ϱ,JL+∂_t DL=0 .Since there is no divergence for both fields JT and ∂_t DT, they can be splitted properly and the magnetic field created by each of them can be defined. By the coupling of the first two equations due to the transverse displacement currents, hybridization of the transverse components of the electric and magnetic fields, i.e., electromagnetic wave modes are enabled.In particular, for JT=0, a free solution exists. Since the transverse displacement current ∂_t DT is not bounded by the real current, the electromagnetic wave can propagate far from the source. In fact the dependence of the longitudinal component of the displacement current on the distance from the source is at most 1/R^2, that of the electromagnetic wave, i.e., the transverse component, is 1/R.Sometimes the Jeffimenko equation, which is a dynamical extension of Coulombs's law and Biot-Savart's law, is used to rule out the contribution of displacement currents in quasi-static cases <cit.>. But arguments in this direction only complicate things and do not seem useful.§ SUMMARYThe controversy over the meaning of displacement currents tends to get lost in the choice between creating a magnetic field or not. Such confusion arises from the following facts. * In the case of non-stationary currents, neither magnetic field created by conduction current nor that created by displacement current can be defined.* In solving Maxwell-Ampère's equation by superposition, the right-hand side cannot be divided arbitrary ignoring the inseparability of the displacement current and the current.* The effect of displacement current is automatically incorporated in the magnetic field calculated by Biot-Savart's law. The existence of displacement current is subtle and elusive, and it was only discovered through Maxwell's deep insight. He emphasized the importance of the unity of the current and the displacement current, and defined the sum of them as total current Jtot. 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Griffiths and M.A. Heald, “Time-dependent generalizations of the Biot-Savart and Coulomb laws,” Am. J. Phys. vol. 59, pp. 111–117, 1991. | http://arxiv.org/abs/2311.15877v1 | {
"authors": [
"Masao Kitano"
],
"categories": [
"physics.class-ph"
],
"primary_category": "physics.class-ph",
"published": "20231127144925",
"title": "Why the controversy over displacement currents never ends?"
} |
*Both authors contributed equally to this work.a,c]Thierry N. Kaldenbach *[email protected] a,b]Matthias Heller *[email protected] [a]Fraunhofer Institute for Computer Graphics Research IGD, Darmstadt, Germany [b]Technical University of Darmstadt, Interactive Graphics Systems Group, Germany[c]German Aerospace Center (DLR), Institute of Materials Research, Cologne, Germany Mapping quantum circuits to shallow-depth measurement patterns based on graph states [==================================================================================== The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions – a so-called standard form – which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus enabling a significant decrease in circuit depth for certain applications. We show that groups of fully commuting operators can be implemented using fully-parallel, i.e., non-adaptive, measurements within our protocol. In addition, we discuss how such circuits can be implemented in constant quantum depths by employing quantum teleportation. Finally, we demonstrate the utility of our technique on two examples of high practical relevance – the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE). § INTRODUCTION Measurement-based quantum computing (MBQC) offers an interesting alternative model of quantum computing compared to the standard circuit model. While unitary operations in the circuit model are realized by sequential application of quantum gates, MBQC operates on a highly entangled state, called resource state, on which unitary operations can be implemented via adaptive measurements <cit.>. Some of these measurements can be performed in parallel, which leads to a compelling feature of MBQC: the parallel application of unitaries, which in the gate model would be applied sequentially. Due to its universality, it is possible to map any quantum circuit to the MBQC model and vice versa. Forward- and backward translation between the circuit model and MBQC can lead to beneficial tradeoffs in terms of depth and space complexity <cit.>.Different techniques for translation between these models <cit.> and optimization of adaptive measurement patterns <cit.> have been studied based on graph-theoretical tools such as causal flow <cit.> and its generalization <cit.>.In this article, we introduce a straight-forward algorithm that allows the mapping of a given quantum circuit to a graph state, which, together with local Clifford operations and measurement instructions, allows one to perform quantum computations within the MBQC model.A closely related circuit transpilation approach has recently been explored in Ref. <cit.>, where circuits are first transformed into an inverse Initialization-CNOT-Measurement (ICM) form <cit.> via T-state injection <cit.>, followed by gate teleportation of T-gates to separate the Clifford structure, which is then mapped to a graph state. For our algorithm, the point of departure is not an inverse ICM form, but instead a quantum circuit generated by a sequence of multi-qubit Pauli exponentials. Since any quantum circuit can be expressed in terms of such exponentials, our framework is general. The same circuit structure has been considered in Ref. <cit.>, in which the Pauli exponentials were called gadgets.Each Pauli exponential has precisely one non-Clifford gate, which is the R_z-rotation gate.We show how this gate can be implemented through gate teleportation, which is similar to the treatment of T-gates in Ref. <cit.>. This allows us to derive measurement patterns based on graph states, which implement the initial quantum circuit. We employ the simulated annealing algorithm <cit.> to minimize the preparation cost of the graph states on real hardware by exploiting the local Clifford equivalence between different graph states. Being able to map circuits to graph states and vice versa, we show how our algorithm can be used to parallelize the application of n commuting Pauli exponentials at the expense of introducing n ancilla qubits. As we will see, this follows from the standard form <cit.> of the measurement patterns we derive: any pattern can be decomposed into a Clifford-part, an adaptive measurement part and finally a corrective Pauli layer.The Clifford part can be implemented in constant quantum depth using quantum teleportation for parallelization <cit.>, whereas the depth of the adaptive measurement part scales linearly with the number of fully-commuting groups of the generator of the initial gate-based circuit.We use our algorithm in two scenarios, which are often discussed as near-term applications for noisy intermediate-scale quantum (NISQ)-devices: the variational Quantum Eigensolver (VQE) <cit.> in the context of molecular simulations and the Quantum Approximate Optimization Algorithm (QAOA) <cit.> in the context of binary optimization problems.In particular, for the ground-state energy calculation of H2O, we demonstrate that our algorithm achieves an efficient mapping of the Qubit-ADAPT-VQE <cit.> to highly shallow circuits, thus advancing the practical utility of quantum computation in the NISQ-era <cit.> where circuit depth is strictly limited by coherence time. The remainder of this article is structured as follows: Sec. <ref> provides an overview on the graph state formalism (Sec. <ref>) and MBQC (Sec. <ref>).Based on these fundamentals, we proceed to introduce our algorithm step by step in Sec. <ref>.We start by reviewing common circuit structures for Pauli exponentials <cit.> and combine them with the One-Way Quantum Computer (1WQC) protocol <cit.> in Sec. <ref>, revealing a general circuit structure consisting of Clifford-, measurement- and Pauli parts.In Sec. <ref>, we elaborate how commuting Pauli strings can be implemented through parallel measurements. Sec. <ref> details how the Clifford structure may be implemented with constant circuit depth. Next, in Sec. <ref> we propose a hybrid simulation scheme, entailing a classical simulation of the main register and a quantum simulation of the ancilla register. For the sake of practical feasibility, in Sec. <ref> we also provide an optimization routine for the underlying graph states to minimize the number of required entangling gates. Finally, we show several applications demonstrating the utility of our algorithm in Sec. <ref>, more specifically the QAOA in Sec. <ref> and the VQE in Sec. <ref>.Furthermore, we show some technical details in Appendices <ref>–<ref>.§ PRELIMINARIES To set the stage for our circuit-to-graph-state conversion algorithm, we first review some basic concepts needed to understand the idea. We start by reviewing the definition of graph states and their connection to stabilizer states and then give a brief introduction to measurement-based quantum computing (MBQC). §.§ Graph- and stabilizer statesThe measurement-patterns obtained through our protocol are based on a computational resource state called graph state <cit.>.An N-qubit graph state |G⟩ is associated to an undirected graph G=(V,E), whose |V|=N vertices correspond to N qubits prepared in the |+⟩≡ 1/√(2)(|0⟩+|1⟩) state, while the set of edges E describes the action of controlled-Z (CZ) operations between them. It can therefore be constructed as |G⟩ = (∏_a,b∈ E CZ_ab)(∏_a∈ V H_a) |0⟩^⊗ N. The action of all CZ_ab gates commute with each other and can, in principle, be applied in parallel.In order to prepare a graph state on a quantum computer (using the circuit model), one thus needs a maximum depth of d = max_α∈ V |N_G(α)|,where N_G(α) denotes the set of vertices connected to the vertex α, i.e., |N_G(α)| is the degree of α. An example of a graph state together with its corresponding quantum circuit is shown in Fig. <ref>.Eq. (<ref>) shows that any graph state on N qubits can be generated through a sequence of Hadamard and CZ-gates. Both belong to the Clifford group 𝒞_N, which is defined as𝒞_N = {U ∈SU(2^N) | U P U^†∈𝒫_N ∀ P ∈𝒫_N},where 𝒫_N denotes the N-qubit Pauli group.Any Clifford operator can be generated by three elementary gates (see e.g. <cit.>): the Hadamard gate (H), the phase gate (S) and the two-qubit CZ- or CNOT-gate. An N-qubit stabilizer state is a quantum state, that can be prepared by a sequence of Clifford gates acting on the |0⟩^⊗ N state. Thus, by definition, every graph state is a stabilizer state – the reverse is not true.However, one can show that every stabilizer state is local Clifford equivalent to a graph state (LC-equivalence), i.e., for every stabilizer state a graph state can be found, that can be transformed to the stabilizer state using one-qubit Clifford gates only <cit.>.Quantum circuits consisting only of Clifford gates can be simulated efficiently on a classical computer according to the Gottesman-Knill theorem <cit.>. Using the LC-equivalence of stabilizer states with graph states, one can simulate an N-qubit Clifford circuit using 𝒪(N ln N) space in computer memory. The core idea is to store the graph together with the local Clifford operations (also called vertex operators or VOPs) for each qubit <cit.>. §.§ Measurement-based quantum computing In the model of measurement-based quantum computing (MBQC), quantum computations start from a highly entangled many-qubit state, called resource state, which is modified by applying a sequence of adaptive measurements onto a subset of qubits.At first sight, it might seem counter-intuitive that universal quantum computation can be performed using irreversible, destructive measurements.While MBQC involves a loss of information concerning the entire resource state, it still performs unitary transformations on the subset of qubits that are not measured during the computation. On these unmeasured qubits any unitary operation can be implemented, provided the resource state is sufficiently complex.Although different MBQC schemes exist, here we focus on the so-called cluster model or One-Way Quantum Computer (1WQC) of Raussendorf and Briegel <cit.>.A review of 1WQC and other measurement-based schemes can be found in Ref. <cit.>. In MBQC, all quantum gates are implemented as a sequence of single-qubit measurements on a suitable large cluster state.The measurement basis needed for universal quantum computing is given byM(θ) = {|0⟩± e^iθ|1⟩}.The measurement in the M(θ)-basis is achieved by applying the unitary H R_z(θ) to the computational basis and performing the usual z-measurement.§ MAPPING CIRCUITS TO GRAPH STATESIn this section, we introduce an algorithm, that allows the mapping of a quantum circuit to a graph state, that can then be used within the MBQC protocol. The core idea is to map unitaries of the formU = e^-i/2θ𝒫,where 𝒫 denotes an N-qubit Pauli string 𝒫∈{I, X, Y, Z}^⊗ N, to an ancilla qubit in the circuit. To implement such operators in the circuit model, the Pauli string is diagonalized by applying local Clifford operators using the identities X=HZH and Y=S H Z H S^†.This effectively reduces the operator pool to 𝒫'∈{I, Z}^⊗ N. We review two common circuit structures representing exp(-i/2θ𝒫').In the star+ancilla layout (Fig. <ref>), all non-identity qubits of the string are directly entangled with an ancilla qubit (star-like structure), which is initialized in state |0⟩ and on which the R_z-rotation is carried out.By repeating the same entanglement structure, the entanglement with the ancilla qubit is undone.The star-like entanglement with the ancilla can be interpreted as a computation of the parity of the N qubits in a classical manner.For an even parity, a phase shift of exp(-iθ/2) is applied, otherwise it is exp(iθ/2).Finally, the parity is uncomputed, erasing the ancilla and leaving it in the |0⟩ state again <cit.>.The second structure, which we refer to as the star layout (Fig. <ref>), works similarly, with the key difference that the entanglement and R_z-gate is performed with respect to one of the non-identity qubits instead of an ancilla. Both circuits are equivalent and have their own benefits depending on the problem at hand.Further, it should be mentioned that both circuits can be equivalently realized using a ladder-like entanglement structure, though this approach is not further discussed within our work due to less convenient gate cancellation properties <cit.>.§.§ Replacing single-qubit gates with measurements The first ingredient in converting a given quantum circuit into a graph state is to replace all single-qubit rotations R_z(θ) by a measurement pattern by introducing one ancilla qubit. Consider the example of a single qubit in an arbitrary state |ψ⟩ entangled with a second qubit in the |+⟩ state via a CZ-gate.After measuring the first qubit in the M(θ) basis, the second qubit is left in the state |ψ⟩^' = X^s H R_z(θ)|ψ⟩,where s∈{0, 1} is the measurement outcome on the first qubit.Acting with X^s and then with H on the second qubit yields the deterministic final state R_z(θ)|ψ⟩, which is the desired R_z(θ)-gate. The quantum circuit performing this operation is shown in Fig. <ref> and can be used as a pattern to replace any R_z-gate in a given circuit.Next, we use this pattern to rewrite unitaries defined by Eq. (<ref>) in the MBQC protocol. As an example, let us consider the unitary exp(-i /2θ Z_0 Z_1).We use the star+ancilla layout as starting point to exemplify some aspects of our algorithm.It is straightforward to derive the pattern for the same example in the star layout. By replacing the R_z-gate on the ancilla qubit with the pattern, one derives the circuit shown in the right panel of Fig. <ref>.So far, it appears that there is no benefit from replacing the R_z-gate with the MB-protocol.It is rather the opposite: One more CZ-gate and an additional measurement are required to realize the same operation. However, by shifting the classically-controlled Pauli corrections across the Clifford gates to the end of the circuit, the quantum circuit can be separated into three components: a pure Clifford layer, a measurement layer and a correction layer, as shown in Fig. <ref>. In accordance with Ref. <cit.> we call this the standard form of a pattern.The standard form of a pattern can be easily achieved by employing the following identities for single-qubitH X= Z H,H Z = X H, S^(†) X= Y S^(†), S^(†) Y = X S^(†),and two-qubit Clifford gatesCX_12(I_1⊗ Z_2)= (Z_1⊗ Z_2) CX_12CX_12(X_1⊗ I_2)= (X_1⊗ X_2) CX_12CZ_12(I_1⊗ X_2)= (Z_1⊗ X_2) CZ_12.All remaining identities can be directly obtained from Y ∝ Z· X and CZ_12=CZ_21. Interestingly, the ancilla qubit a_1 in the pattern shown in Fig. <ref> is always left in the |0⟩-state, regardless of the intermediate measurement outcome.This is a general feature for a pattern in the star+ancilla layout. The last classically-controlled Z gate on the last ancilla can thus always be neglected, since Z|0⟩ = |0⟩.§.§ Quantum parallelism and adaptive measurements To showcase an important property of our MB-circuit protocol, we now consider the case of two commuting Pauli strings Y_0 Y_1 and X_0 X_1, once again implemented through the star+ancilla layout.The naive circuit representation, which is obtained through concatenation of the circuits for exp(-i/2 θ_2 X_0 X_1) and exp(-i/2θ_1 Y_0 Y_1), is depicted in the upper panel of Fig. <ref>. Using the same circuit identities as before, we can shift the first correction and measurement layer across the second Clifford layer.In this particular case, this step introduces no corrections on the second ancilla qubit, which would have to be carried out before the second measurement.Instead, both measurements can be performed in parallel (see lower panel of Fig. <ref>). The fact, that the measurements can be parallelized here, is no coincidence. In the following, we derive the condition for parallelism. We already know, that for the application of a generic Pauli exponential, Pauli corrections are applied to all qubits corresponding to non-identity operations of the Pauli string (e.g., if the Pauli string inside the exponential reads X_1 Z_3 Y_6, Pauli corrections are applied to qubits 1, 3 and 6 only). From Fig. <ref> and Eq. (<ref>), we can infer that the correction is precisely given by a controlled version of the Pauli string itself, since the Z-corrections undergo a basis transformation according to the string. Schematically, this is shown in Fig. <ref> for both layouts (star and star+ancilla).Let us assume, we want to apply two unitaries generated by the Pauli strings 𝒫 and 𝒫̃. We now investigate the conditions, which these two Pauli strings have to fulfill, for parallel measurement.As explained before, the circuit implementing the matrix exponential of 𝒫 ends with a controlled version of 𝒫 itself, i.e., 𝒫^s, where s∈{0,1}. Hence, if we want to bring the pattern implementing the product exp(-i/2θ̃𝒫̃)exp(-i/2θ𝒫) into a standard form, it is sufficient to see what happens when shifting 𝒫^s through exp(-i/2θ̃𝒫̃). In Appendix <ref> we show that: e^-i/2θ̃𝒫̃𝒫^s = 𝒫^se^-i/2θ̃𝒫̃,if [𝒫, 𝒫̃]=0e^-i/2(-1)^sθ̃𝒫̃,else ,where s denotes the measurement outcome of the first ancilla qubit. Consequently, the product of two unitaries generated by two Pauli strings can be implemented in parallel only if the strings commute. Otherwise, the rotation angle θ̃ of the second Pauli exponential has to be adapted to the measurement outcome s of the first ancilla, leading to an adaptive, i.e., non-parallel measurement pattern.Generalizing this result to the application of M unitatries generated by Pauli strings {𝒫_1, 𝒫_2, …, 𝒫_M}, we find that the final correction of the pattern implementing this operation in standard form is given by∏_m=1^M 𝒫_m^s_m,where s_m is the measurement outcome of the m-th ancilla. Eq. (<ref>) allows to write down the final correction of an arbitrary measurement pattern, without the additional computational cost of propagating all corrections to the end of the circuit. All ancilla qubits can be measured in parallel, only if all Pauli strings commute with each other. Otherwise, the measurement bases have to be adapted according to Eq. (<ref>).In contrast to the final correction in Eq. (<ref>), the adaptive measurement bases depend on the order in which the Pauli exponentials are implemented.To implement the unitary exp(-i/2 θ_i 𝒫_i), the rotation angle of the measurement basis is obtained by flipping the sign of θ_i for each previous non-commuting pattern measured in the |1⟩-state, i.e.,θ_i → (-1)^h_iθ_i, whereh_i = ∑_j < i[𝒫_i, 𝒫_j] ≠ 0 s_j.To minimize the number of adaptive measurements, it is therefore convenient to first sort the generating strings into fully commuting groups. §.§ Applying quantum gates in constant depth in the circuit model In this section, we show how our method can be used to derive measurement patterns, that can be used to apply several commuting operators in parallel with constant circuit depth in the circuit model. In the following, we assume that the depth of a quantum circuit is defined by the number of layers with at least one CNOT-gate, which are needed to implement it.Let us assume that we apply several commuting unitaries U_1,…, U_n generated by n Pauli strings to a quantum state |ψ⟩_N – the order does not matter, since they commute. Then, we first derive the measurement pattern which implements U=U_n ⋯ U_2 · U_1 using the method outlined in the previous section. This pattern, in standard form, has three layers: Cliffords, measurements (in parallel) and corrections, c.f. Fig. <ref>. Since the measurement and correction layer have already constant depth, we just need to implement the Clifford operations in constant depth.For this, note that a general Clifford circuit can always be expressed as a sequence of one-qubit Cliffords and CNOT-gates. Thus, it suffices to show that two sequential CNOT-gates (with potentially intermediate one-qubit Cliffords) can be applied in parallel in constant depth. This can be achieved using the quantum teleportation algorithm. The general construction is depicted in Fig. <ref>: Any sequence of two-qubit Clifford gates can be recast to a quantum process of constant depth. Using this technique, the number of ancilla qubits grows linearly with the depth of the Clifford layer, while the additional classical computation due to the corrections grows logarithmically <cit.>. To summarize, we can implement arbitrary Clifford circuits as constant depth circuits.Combining this with the previous result of deriving a pattern for a group of commuting operators (Sec. <ref>), we conclude that our algorithm allows the implementation of several commuting operators as a constant-depth measurement-based pattern.More precisely, these constant-depth patterns can always be achieved with three entangling layers (one for the Bell state preparation of the ancilla qubits, one for the initial entangling gates and one for the Bell basis measurements), a measurement layer, and a corrective Pauli layer.§.§ Simulation and correction of the main qubits Up to this point, our observations hold for arbitrary input states. We now show how the circuits can be further reduced by simulating the Clifford part classically, assuming that the main qubits are initialized in a stabilizer state, i.e., the initial state can be prepared with Clifford gates acting on the |0⟩^⊗ N state.We start by converting the Clifford part of the circuit into a graph state using the Clifford simulation algorithm outlined in Ref. <cit.> (the original code from the authors can be found online[<https://github.com/marcusps/GraphSim>]), which we implemented with thepackage <cit.>. The result of this conversion is a graph state of size N_q+N_a, where N_q denotes the number of initial qubits (main qubits) and N_a the number of required ancilla qubits, which is equivalent to the number of non-Clifford gates in the initial circuit. In this graph state, all main qubits are measured in a Pauli basis, while the ancilla qubits are measured in the M(θ) basis. Furthermore, after measuring the ancilla qubits, a final Pauli correction has to be applied to the main qubits.In the MBQC protocol, one would now proceed by first measuring the ancilla qubits, correcting the main qubits depending on their outcome and then, at the end, perform measurements on the main qubits depending on the observables one wishes to extract.However, if the main qubits are measured in the Pauli X, Y or Z-basis, we can equally first simulate the measurement outcome of the main qubits. This might seem surprising, but it can be explained as follows.Suppose we have a main qubit q, which is corrected by Pauli X depending on the measurement-outcome s_a of an ancilla qubit a. Then, after the measurement of the ancilla qubit, the new state of q is given by X^s_a|q⟩. If we now measure q, we know that, depending on s_a, the state |q⟩ was either |0⟩ or |1⟩ before the measurement of |a⟩. Thus, we can equivalently flip the measurement outcome s_q of |q⟩ depending on s_a. In the case of a Z-correction nothing has to be done.For a Y-correction we can always rewrite Y=X· Z up to an irrelevant, global phase.Following this logic, we can first efficiently simulate the measurements of all main qubits using the graph state simulator, then execute the remaining circuits on the ancilla qubits and use a post-processing algorithm to correct the counts of the main qubits accordingly. More specifically, we first classically sample one shot on the main qubits neglecting the correction layer.Based on that, we obtain a bit string and a result-dependent stabilizer state (which is equivalent up to local unitaries to a graph state) for the ancilla qubits.This stabilizer state is then prepared and measured in the rotated bases on a quantum computer.If the ancilla measurement results imply an X or Y correction on the main qubits, the bit string is modified through the appropriate bit flips. It is important to emphasize, that, while an exponential amount of classical measurement outcomes can occur, there is a one-to-one correspondence between a classically simulated shot in our algorithm and a quantum shot in the circuit model.Hence, the number of classical results that actually have to be considered is limited by the number of shots.Consequently, no exponential blowup occurs in this hybrid approach.As an example, let us reconsider the measurement pattern that implements exp[-i/2θ(X_0X_1+Y_0Y_1)] from Fig. <ref> and let it act on the initial state |00⟩. The overview over the full calculation is summarized in Fig. <ref>. After simulating the Clifford part of the circuit, we find the graph state shown in the top left. By using the graph-based sampling algorithm, we find two possible measurement-outcomes for the two main qubits: 00 and 11. Since we sample from a stabilizer state, all bit-strings (with non-zero amplitudes) appear with equal probability <cit.>. All outcomes lead to local equivalent stabilizer states, where the ancilla qubits are decoupled from the main qubits. In our example, we find the two stabilizer states shown in the second row of Fig. <ref>. Running these stabilizer state on a quantum computer results in the probability distribution that is needed to correct the counts of the main qubits. §.§ Optimization of graph statesThe graph states obtained through our simulation protocol are often quite complicated. However, they are not unique. Stabilizer states are invariant under the operation of local complementations. In the following, we denote the local complementation of a graph at a vertex α by LC_α. Applying LC_α to a graph G(V, E) complements the neighborhood of the vertex α.That is, all existing edges between the vertices in N_G(α) are removed and all missing edges in N_G(α) are added.The corresponding stabilizer state is preserved by applying the local unitaries <cit.>:U_α^LC = √(-i X_α)⊗_i∈ N_G(α)√(iZ_i),where N_G(α) denotes the neighborhood of the vertex α. The procedure is depicted in Fig. <ref>. Note that the total number of edges may be changed after the operation.With current NISQ-hardware in mind, we want to find optimal graph states with respect to the number of edges, which defines the number of CZ gates in the preparation circuit. Alternatively, one could optimize the states with respect to the maximum degree, thus minimizing the circuit depth required for preparation, or even optimize a trade-off between both properties.To perform the optimization task, we employ the simulated annealing algorithm <cit.>, which we will briefly outline here. A more detailed description is provided in Appendix <ref>. The solution space is the set of graph states that are LC-equivalent to the initial graph state obtained by converting the Clifford circuit.In each iteration, a random node of the graph is locally complemented.The cost function we aim to minimize is then evaluated with respect to the new graph state.If it is improved, the old graph state is discarded. Otherwise, we might still keep the new graph state, but only with steadily decreasing probability according to a Boltzmann distribution.Despite its inherent simplicity and no guarantee to find the global optimum, we have observed major reductions in circuit depth results using this method.§ APPLICATIONS In this section, we show how our methods can be applied directly to two important NISQ algorithms. In Sec. <ref> we show how combinatorial problems can be solved using the QAOA and in Sec. <ref> we show how the electronic-structure problem can be tackled by using the VQE. All simulations were performed using <cit.>.For the QAOA we derive an ansatz containing mid circuit measurements due to non-commuting operators. For the VQE we design an ansatz which consists entirely of a set of commuting operators. In this scenario our method yields the most powerful reduction in circuit depth and does not rely on mid-circuit measurements, which are still challenging on current quantum computing devices. §.§ QAOA and max-cut problemsThe QAOA <cit.> is an optimization algorithm designed to solve combinatorial optimization problems in the NISQ-era <cit.>. The idea is to encode the optimization problem into a minimization problem of a generic Ising-Hamiltonian (also called cost Hamiltonian H_c)H_c = ∑_i<j w_ij Z_i Z_j+∑_i b_i Z_i,where w_ij and b_i are coefficients depending on the optimization problem. In its standard formulation, the QAOA algorithm tries to find the lowest energy (corresponding to the optimal solution of the initial problem) using a variational ansatz of depth p, which is given by|ψ⟩ = e^-iβ_p/2 H_m e^-iγ_p/2 H_c ⋯ e^-iβ_1/2 H_m e^-iγ_1/2 H_c |+⟩^⊗ N,whereH_m ≡∑_i=1^N X_i is the so-called mixer Hamiltonian, and N denotes the number of qubits. The γ_i and β_i are in total 2 p variational parameter, which are obtained in a classical optimization feedback and aim to minimize the expectation value ⟨ψ | H_c | ψ|$⟩, which can be estimated efficiently on a quantum computer.As a concrete example and a proof-of-principle of our method, we consider a weighted max-cut problem with four vertices. The task of the weighted max-cut problem is to find a partitioning of the vertices in two complementary sets, such that the sum of all weights on the cut is maximized. In Fig. <ref> we show the graph and the optimal partitioning, which is given by dividing the vertices into the two sets{0,1,3}and{2}in our example. The weighted max-cut problem can be formulated as a minimization problem of an Ising-Hamiltonian. In our case the Hamiltonian is given byH_c = Z_2 Z_3+Z_0 Z_2+0.5Z_0 Z_1 +Z_1Z_2.The optimal solution is given by the two bit-string (here and in the following we use the little-endian convention)0100and1011, corresponding to the two sets mentioned above.To solve the problem with QAOA we use thep=1ansatz, which has two parameters:|ψ⟩ = e^-iβ/2 (X_0+X_1+X_2+X_3) e^-iγ/2 H_c |+⟩^⊗ N.From classical simulation we find the optimal anglesγ= -2.290andβ= -2.186.We first derive the pattern which implements this unitary, by mapping the two operatorse^-iγ/2 H_c ande^-iβ/2 (X_0+X_1+X_2+X_3) to stabilizer states. Since both operations do not commute, this will introduce adaptive measurements. We then simulate the main qubits and identify the graph shown in the left panel of Fig. <ref> as the full pattern, that implements the variational QAOA ansatz.We ran the circuit corresponding to this pattern (right panel of Fig. <ref>) on the27-qubit quantum computer , using dynamic circuits. The QAOA distribution, which is obtained by correcting the pre-simulated counts of the main qubits depending on the counts of the ancilla circuits, is shown in Fig. <ref>. We compare the results fromwith ideal results from simulation. As expected, we find with highest probability the correct bit-string0100and1011in the simulated as well as the measured distributions. However, due to hardware errors, the measured distribution differs from the ideal one.To quantify the error, we calculate the normalized fidelity between the two distributions <cit.> F(P_ ideal, P_ measured) =F_H(P_ ideal,P_ measured)-F_H(P_ ideal,P_ depol)/1-F_H(P_ ideal,P_ depol),whereF_Hdenotes the Hellinger fidelity andP_depolcorresponds to an uniform distribution, which would be measured on a completely depolarized device. In our experiment, we find a fidelity of onlyF=0.28.We believe that the main source of error is due to the use of dynamic circuits and measurement errors, which affect the whole outcome, if they occur during a mid-circuit measurement. Note, that using dynamic circuits is a fairly new feature on IBM-hardware. For the next application, the VQE, we therefore construct circuits, in which we avoid mid-circuit measurements by designing an ansatz with commuting operators only. §.§ VQE and the Unitary Coupled Cluster ansatzQuantum chemistry is often discussed as one of the most promising fields in which quantum computing could have a big impact. The aim of ab-initio quantum chemistry is the calculation of molecular properties, such as their energies or polarization. In second quantization, the molecular Hamiltonian is typically expressed in terms of fermionic annihilation and excitation operators,H = ∑_p,q h_pq a^†_p a_q + ∑_p,q,r,s h_pqrsa^†_p a^†_q a_r a_s,wherea_p(a_p^(†)) annihilates (creates) an electron in the spin-orbitalp. An important task in quantum chemistry is the determination of the molecular ground state energy, which is given by the minimum of Hamiltonian (<ref>). To achieve this goal using Quantum Computers, the Variational Quantum Eigensolver (VQE) has been thoroughly studied in the past decade <cit.>.The unitary-coupled cluster (UCC) ansatz is among the most popular VQE-ansätze and is defined as|ψ⟩ = e^i ∑_n T_n|Φ⟩_0,where|Φ⟩_0is the reference state (usually the Hartree-Fock ground state) andT_ndenote then-th cluster operator – usually these are truncated at second order, whereT_1andT_2are given byT_1= ∑_i∈ virt. a∈ occ.θ_a^i (a_i^† a_a-a_a a_i^†),T_2= ∑_i,j∈ virt. a,b∈ occ.θ_ab^ij (a_i^†a_j^†a_a a_b-a_b^†a_a^†a_j a_i),where virt. (occ.) denotes the set of virtual (occupied) orbitals. Using the Jordan-Wigner mapping, those operators can be mapped to qubit operators viaa_n^†→ Z_0 Z_1 ⋯ Z_n-1X_n + i Y_n/2.For the double-excitationsT_2, this substitution leads to a sum of eight Pauli-strings. In the literature these operators are often simplified by neglecting allZ_iterms. The operators in this approximation are called qubit excitation operators <cit.>. For instance, the qubit excitation corresponding to a fermionic double excitation operator in the Jordan-Wigner mapping is given by:U_ijab= e^iθ/8X_i Y_j X_a X_be^iθ/8 Y_i X_j X_a X_be^iθ/8 Y_i Y_j Y_a X_be^iθ/8 Y_i Y_j X_a Y_b ×e^-iθ/8 X_i X_j Y_a X_be^-iθ/8 X_i X_j X_a Y_be^-iθ/8 Y_i X_j Y_a Y_be^-iθ/8 X_i Y_j Y_a Y_b. §.§.§ Measurement-pattern for double excitations Next, we show how the qubit excitation in Eq. (<ref>) (after rescalingθ→4 θfor convenience) could be implemented in our protocol as a measurement pattern. This operator has also been considered in Ref. <cit.>, in which its circuit has been derived and optimized using the star layout. This choice of layout is suited for our algorithm as theR_z-gates for the eight different Pauli strings in Eq. (<ref>) can all be carried out on the same qubit (thus no initial ancilla is required). In its optimized form, the circuit has13CNOT-gates in total and is shown in the upper panel of Fig. <ref>. We use it as our starting point to derive a measurement pattern. The circuit obtained through our conversion algorithm is depicted in the lower panel of Fig. <ref>.According to Eq. (<ref>), the final Pauli correction is given by U = (X_i X_j X_a Y_b)^s_1· (X_i X_j Y_a X_b)^s_2· (X_i Y_j Y_a Y_b)^s_3· (Y_i Y_j Y_a X_b)^s_4 · (Y_i X_j Y_a Y_b)^s_5· (Y_i X_j X_a X_b)^s_6· (Y_i Y_j X_a Y_b)^s_7· (X_i Y_j X_a X_b)^s_8. The circuit depth in terms of entangling gates is17.However, by only parallelizing the CNOT-ladder, the depth can be reduced to 11, which is already shorter than the conventional gate-based approach.As already discussed in Sec. <ref>, the depth of the Clifford circuit could be further reduced to a constant depth. §.§.§ Proof of principle: Ground-state energy estimation of the H2O molecule In this section we use our techniques to lower the quantum-resource requirements of the VQE to estimate the ground-state energy of the H2O molecule on a quantum computer. Our starting point is the electronic-structure Hamiltonian in the minimalbasis, which we derive using <cit.> together with <cit.>. The full Hamiltonian consists of14spin-orbitals, which are occupied by ten electrons. In order to simplify the problem at hand, we freeze the four spin-orbitals with the lowest energy, such that we only deal with six electrons distributed over ten orbitals in our ansatz. A schematic overview of this approach is provided in Fig. <ref>.Next, we choose our variational ansatz for the VQE. The full UCC-ansatz in this case would result into too complicated patterns for current quantum hardware. We therefore use a simplified ansatz, defined by an operator pool of Pauli strings, which is inspired by the qubit excitations. This ansatz is known as qubit-ADAPT-VQE in the literature <cit.>. As shown in Sec. <ref>, we can parallelize all operations, if they commute. We therefore aim to build our operator pool from commuting operators only. The full pool, consisting of nine operators built from Pauli strings, is shown in Table <ref>.Our ansatz is inspired by the ansatz from Ref. <cit.>, in which the most important qubit excitations were chosen – instead of using the full qubit excitation operators consisting of eight Pauli strings each, we only choose one string per excitation. Following Ref. <cit.>, the most important excitations using the frozen core approximation, are shown in Fig. <ref>. Our ansatz is then given by:|ψ⟩ = e^-i/2∑_n θ_n P_n|Φ⟩_HF,where|Φ⟩_HF ≡|0000111111⟩is the Hartree-Fock ground state andθ_ndenotes then-th variational parameter. We first optimize our ansatz by performing a classical simulation of the circuit. The optimal parameters leading to a minimal energy of-74.9910Ha are shown in Tab. <ref>. In Fig. <ref> we show the quantum circuit corresponding to this ansatz.The Hamiltonian in our approximation consists of251terms in total. As observed in Refs. <cit.>, the sampling overhead of measuring the expectation value of such Hamiltonian can be reduced significantly by measuring commuting operators simultaneously. Using thecommandto identify commuting groups of a given operator, we find a partitioning of the Hamiltonian into14groups of operators, which can be measured simultaneously. More details to this decomposition can be found in Appendix <ref>. Our full Hamiltonian reads:H = -72.2129 Ha + ∑_n=1^14 H_n.In order to measure the expectation value of each HamiltonianH_n, we use the techniques outlined in Ref. <cit.> to derive the Clifford circuits, which diagonalize a given set of commuting operators.To showcase the reduction of sampling overhead in this approach, we simulate the quantum circuit shown in Fig. <ref> using thesimulator. In Fig. <ref> we compare the two strategies of estimating the expectation value of the Hamiltonian measuring each term individually vs. using a partitioning into commuting groups. In the first approach we measure each term of the Hamiltonian with1000shots individually and in the second we use the technique of measuring each of the14groups with5000shots per group simultaneously. We repeated the simulation500times to collect sufficiently statistics to perform a fit with a normal distribution to estimate the variance of the expectation value due to shot noise for both strategies. While in the first approach we obtain a standard deviation ofσ≈0.015using approximately250kshots in total, the second approach leads to results with less shot noise (σ≈0.01) using only70kshots. This shows that using commuting groups of operators indeed gives a benefit for the measurement of the Hamiltonian in this example.In the following, we construct the variational quantum circuit for each of the14groups by concatenating the ansatz circuit (Fig. <ref>) with the corresponding Clifford circuit, that diagonalizes all operators in a given group. We then use our algorithm to derive measurement patterns, which implement the whole quantum operation. After classically simulating the main qubits, we end up with very simple graph states after using the optimization procedure outlined in Sec. <ref>.Next, we benchmark our ansatz using the optimized parameters and ran the corresponding circuits on the27-qubit quantum computer . As a proof-of-principle we only ran those circuits associated with the energies of the first four terms of the Hamiltonian. We show the graph states corresponding to these four terms in Fig. <ref>. We report our final results in Tab. <ref>. For each group, we find that32locally-equivalent stabilizer states have to be prepared. Accordingly, we ran32circuits per group using4kshots, such that each expectation value was measured with a total budget of128kshots. The error for the raw data shown in Tab. <ref> (third column) are estimated by repeating each experiment eight times and calculating the standard deviation.As can be seen from Tab. <ref>, the energies calculated with the raw data-points are far-off the ideal values, such that the use of error-mitigation is imperative. We mitigate read-out error using thepackage <cit.>, for which we ran all calibration circuits with100 000shots. Additionally to the read-out error correction we used randomized-compiling <cit.>, dynamical decoupling <cit.> and zero-noise-extrapolation (ZNE) <cit.> to increase the precision. The mitigated values are shown in the last column of Tab. <ref>. The reported error follows from the ZNE fit parameter.It is important to note that the applicability of ZNE is not evident here, as observables are measured classically on the main qubits and only later corrected through the ancilla outcomes.In Appendix <ref> we prove that the expectation value of an arbitrary Pauli string𝒫_q, acting purely on the main qubits, can be expressed as a superposition of expectation values of an auxiliary Pauli-Zstring𝒵_aacting on the ancilla register, that is⟨𝒫_q|=⟩1/N∑_n (-1)^s_nn⟨ψ_a|𝒵_a |ψ_a⟩_n,whereNis the number of distinct classical measurement outcomes,|ψ_a⟩_nare the ancilla states corresponding to the classical measurement outcomenands_n∈{0,1}ensures the correct phase. From this form it is clear that by amplifying the noise on the ancilla systems|ψ_a⟩_n, ZNE can be performed on the main system. More details on the error-mitigation and the data acquisition can be found in Appendices <ref> and <ref>.Note, that comparing the ideal expectation value with the mitigated ones, we achieve an absolute accuracy of roughly0.01Ha.While we accomplished promising improvements over the unmitigated results, our mitigated results are still not within the range of “chemical accuracy” of0.001Ha <cit.>.To be able to make the full calculation with all14groups to such accuracy, more sophisticated error-mitigation <cit.> will be necessary together with a longer access to the quantum computer to get better statistics. However, as a proof-of-principle, we believe that this experiment shows that our techniques are very promising for future research.§ CONCLUSION AND OUTLOOKIn this work, we introduced an algorithm that allows the mapping of a sequence of unitaries generated by Pauli strings in the quantum circuit model to a measurement pattern by introducing one ancilla qubit per unitary. We showed that in the case of commuting operators these patterns can be parallelized leading to a constant-depth quantum operation. This result is useful in the NISQ-era, in which quantum circuits have to be as shallow as possible, as well as when fault-tolerant quantum computers are available and one needs to be able to perform unitary operations fast in parallel. The pattern always consists of three layers: a Clifford part, the measurement of the ancilla qubits, and a correction layer that consists of local Pauli operations. Furthermore, we showed that by compressing the Clifford part to an LC-equivalent graph state and simulating the main qubits classically, we can significantly reduce the complexity of a given quantum circuit. We further optimized our graph states using simulated annealing to minimize the hardware requirements for state preparation.We first applied the algorithm to the QAOA.Despite obtaining correct results in the simulation, our experiments on quantum hardware achieved a rather poor fidelity ofF=0.28, mostly due to errors caused by mid-circuit measurements.To overcome this hurdle on current NISQ-hardware, one could try to design ansätze, that do not require mid-circuit measurements, as we have done for the VQE. One possible direction could be to restrict the mixer-operator space to Clifford operators, i.e.,{I, H, S}, thus removing the necessity for mid-circuit measurements.As has been shown in <cit.>, restricting the QAOA ansatz to the pure Clifford manifold already provides good approximate solutions to the max-cut problem. Combining this ansatz with a measurement-pattern consisting of commuting operators only, might be a possible road to find more efficient ansätze.As a second example, we applied our mapping technique to the VQE to compute the ground-state energy of H2O.Here, we specifically designed an ansatz of fully-commuting operators to avoid mid-circuit measurements.This approach significantly outperforms the standard circuit model approach in terms of circuit depth. We showed that running our circuits on current IBM-hardware, we were able to extract expectation values with an absolute accuracy of roughly0.01Ha.This was only possible using error-mitigation techniques to boost the precision of our results. We showed, how such error-mitigation techniques can be incorporated in our formalism by performing zero-noise extrapolation using local gate folding.We believe that our mapping techniques will provide a useful alternative compared to the standard circuit model for future NISQ-algorithms. It is straightforward to apply our techniques to other cases, such as quantum circuits following from trotterized time evolution or other VQE ansätze in different scenarios. There are several directions for future research. First, we want to study how graph states can be implemented more efficiently on a quantum computer by optimizing the graph with respect to its topology. By using local complementation and Pauli measurements, hardware-efficient graphs could be implemented which are equivalent to the target state one wants to prepare. Secondly, since the number of qubits on a quantum computer is limited, an important line of research is how ancilla qubits can be reused efficiently after their measurement.An interesting alternative approach could be to use circuit-cutting techniques <cit.> to cut the graph state into several partitions. Here, it is important to cut the circuit corresponding to the measurement pattern at a location, such that the additional sampling overhead is minimal. Last but not least, we want to study how probabilistic error cancellation <cit.> can be incorporated in our framework. § ACKNOWLEDGMENTSWe thank Kathrin König for sharing her code for zero-noise-extrapolation, which we used to perform the local gate-folding.We would also like to thank Jonas Jäger, Daniel Barragan-Yani, Paul Haubenwallner, Timon Scheiber and Johannes S. Mueller-Roemer for helpful comments on the manuscript. This work was supported in part by the research project Zentrum für Angewandtes Quantencomputing (ZAQC), which is funded by the Hessian Ministry for Digital Strategy and Innovation and the Hessian Ministry of Higher Education, Research and the Arts (TNK and MH), and the research project Quantum Computing for Materials Science and Engineering (QuantiCoM) of the DLR Quantencomputing Initiative (QCI), which is funded by the Federal Ministry of Economic Affairs and Climate Action (BMWK) (TNK). We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.§ CONDITIONS FOR PARALLEL MEASUREMENTS OF THE ANCILLA QUBITSIn this Appendix we derive Eq. (<ref>). Consider two Pauli strings𝒫and𝒫̃. We want to derive under which conditions we can apply the measurement pattern implementing the unitaryU=e^-i/2θ̃𝒫̃e^-i/2θ𝒫with parallel, i.e., non-adaptive, measurements. We know, that the implementation of the first pattern, which applies the unitarye^-i/2θ𝒫, leads to a Pauli correction given by𝒫^s, wheresis the measurement outcome of the ancilla. This correction needs to be shifted through the pattern implementing the second unitarye^-i/2θ̃𝒫̃. This can be achieved as follows.First, we decompose the matrix exponentiale^-i/2θ̃𝒫̃ = [cos(θ̃/2) -i sin(θ̃/2)𝒫̃].We then shift𝒫^sthrough the second unitary by inserting an identity,e^-i/2θ̃𝒫̃𝒫^s = 𝒫^s [cos(θ̃/2) -i sin(θ̃/2)𝒫^s𝒫̃𝒫^s].In case that[𝒫, 𝒫̃]=0, we have𝒫^s𝒫̃𝒫^s = 𝒫̃, and thus obtain the trivial commutation relation e^-i/2θ̃𝒫̃𝒫^s = 𝒫^se^-i/2θ̃𝒫̃.However, if[𝒫, 𝒫̃]≠0, we have𝒫^s𝒫̃𝒫^s = (-1)^s𝒫̃. Then, we can rearrange Eq. (<ref>) as follows:e^-i/2θ̃𝒫̃𝒫^s= 𝒫^s [cos(θ̃/2) -i sin(θ̃/2)(-1)^s𝒫̃] = 𝒫^s [cos((-1)^sθ̃/2) -i sin((-1)^sθ̃/2)𝒫̃] = 𝒫^se^-i/2(-1)^sθ̃𝒫̃.Consequently, by applying our protocol to non-commuting strings, adaptive rotations are introduced, leading to adaptive measurements.§ SIMULATED ANNEALING ON GRAPH STATESIn this Appendix we summarize the simulated annealing algorithm we used to simplify graph states. The goal is to minimize a cost functionf:D→ℝon a solution spaceD. The solution space is the set of equivalent graph states in our case.* InitializationSelect an initial solution g∈ D and a monotonously falling sequence of (positive) temperatures T_i. * Local changeSelect a neighbor g̃ of g. On our set of graph states, neighbors of g are defined as graphs, which are related to g by a local complementation at a single node. We construct such a neighbor by locally complementing a random node of g.* SelectIf f(g̃)≤ f(g), set g=g̃. Else, set g=g̃ with a probability ofP(g,g̃) = exp(-f(g̃)-f(g)/T_i).* UpdateIf f(g) is better than the previous best solution, update it.* IncrementSet i = i + 1. * RepeatRepeat steps 2-5 until the final temperature is reached.§ GROUPING THE H2O HAMILTONIANIn this Appendix we show the first four groups of the Hamiltonian of H2O. As mentioned in Sec. <ref>, we used the built-in functioninto find the partitioning. In Tab. <ref>, we show the terms contributing toH_1andH_2and in Tab. <ref> the terms contributing toH_3andH_4.Finding commuting groups of an Hamiltonian scales exponentially with the system size in general. However, as explained in Ref. <cit.>, for the electronic-structure problem one can perform a partitioning of the Hamiltonian which scales polynomially with the system size.In order to measure the terms from these four groups simultaneously, we employ the algorithm of Ref. <cit.>. In the first step, we identify an operator basis for each group, from which all other operators can be constructed by multiplication. For the four groups, we chose the following bases:* H_1: Z_6 Z_8, Z_7 Z_9, Z_5, Z_4 Z_5, X_6 Y_7 Y_8 X_9, Y_0 Z_1 Y_2 Y_6 Z_7 Y_8, X_0 Z_1 X_2 X_6 Z_7 X_8, Y_1 Z_2 Y_3 Y_7 Z_8 Y_9, X_1 Z_2 X_3 X_7 Z_8 X_9,* H_2: Y_1 Y_2 X_3 Z_4 Z_5 Z_6, Z_7 X_8, X_1 Y_2 Y_3 Z_4 Z_5 Z_6 Z_7 X_8, X_2 Z_3 Z_4 Z_5 Z_6 X_7 Y_8 Y_9, X_2 Z_3 Z_4 Z_5 Z_6 Y_7 Y_8 X_9, X_0 Z_1 Y_2 Y_6 Z_7 X_8, Z_0 Z_6, Z_2 Z_8, * H_3: Z_3 Z_9, Z_2 Z_3, Z_8 Z_9, Z_0 Z_5, Z_1 Z_4, Z_1 Z_5, Z_6, Z_7, X_0 Y_1 Y_4 X_5, X_2 Y_3 Y_8 X_9, * H_4:X_3 Z_4 Z_5 X_6 X_8 X_9, X_3 Z_4 Z_5 Y_6 Y_8 X_9, Y_1 Z_2 X_3 X_7 Z_8 Y_9, X_1 Z_2 Y_3 Y_7 Z_8 X_9, Y_0 Z_1 Y_2 X_3 Z_4 Z_5 Z_6 Z_7 Z_8 X_9, X_0 Z_1 X_2 X_3 Z_4 Z_5 Z_6 Z_7 Z_8 X_9, Z_1 Z_7.Any other operator from the groups in Tabs. <ref> and <ref> can be written as a product from some operators in the lists.In the second step, we construct the Clifford circuits, which allow the simultaneous measurement of the basis operators following Ref. <cit.>. Effectively, each basis operator from the basis is mapped to a PauliZon a specific qubit. Accordingly, the expectation value of products of these basis operators can then be obtained by measuring the expectation value of a Pauli string containing more than one PauliZ.§ ZERO-NOISE-EXTRAPOLATION FOR MEASUREMENT PATTERNSZero-Noise-Extrapolation (ZNE) is an error-mitigation strategy, in which a given quantum circuit is artificially stretched to amplify the noise <cit.>. In order to mitigate expectation values, one first measures the observable at different noise amplification levelsλand performs an extrapolation to the zero-noise limit.In our measurement-based approach we can use zero-noise mitigation directly on the final observable. Following Ref. <cit.>, the state before measurement of the ancilla and main qubits can be written as: |ψ⟩ = 1/√(N)∑_n c_n (∏_i∈𝒞∏_j∈𝒩_i CP_ji) |ψ_a⟩_n|ψ_q⟩_n,wherec_n ∈{±1, ±i}andNis the number of classical measurement outcomes,|ψ_q⟩_ndenotes then-th possible computational state of the main register,|ψ_a⟩_ncorresponds to the quantum state of the ancilla circuit,𝒞is the set of main qubits,𝒩_iis the set of ancilla qubits connected to thei-th main qubit, andCP_jidenotes the entangling Pauli-gates between both states, that can also be performed by classical post-processing the measurement outcomes. Note that the controlled Pauli gatesCP_jiare thus always controlled by the ancilla qubits.Without loss of generality, we may assume that onlyZ-expecation values have to be measured on the main register (any basis change can be absorbed into the definition of the state in Eq. (<ref>)). Let𝒵_q∈{I, Z}^N, then:⟨ψ|𝒵_q |ψ|=⟩1/N∑_n,m c_m^* c_n m⟨ψ_a|m⟨ψ_q|(∏_i∈𝒞∏_j ∈𝒩_i CP_ji 𝒵_q CP_ji) |ψ_a⟩_n|ψ_q⟩_n.Using the properties of the Clifford group (cf. Eq. (<ref>)), we may drag the Pauli correctionsCP_jiacross the Pauli string𝒵_q:CX_ji(I_j⊗ Z_i) CX_ji = Z_j⊗ Z_i.Consequently, we may rewrite∏_i∈𝒞∏_j∈𝒩_i CP_ji𝒵_q CP_ji =𝒵_a ⊗𝒵_q,where𝒵_ais the Pauli string acting on the ancilla register. This step effectively transforms the Pauli string𝒵_c, which was previously only acting on the main register, to an operator, that acts also on the ancilla register.By inserting Eq. (<ref>) into Eq. (<ref>), we obtain⟨ψ|𝒵_q|ψ|=⟩1/N∑_n,m c_m^* c_n m⟨ψ_a|𝒵_a |ψ_a⟩_n m⟨ψ_q|𝒵_q |ψ_q⟩_n.Next, we can exploit that the computational states|ψ_q⟩_nare eigenstates of the Pauli-Zoperator, hence m⟨ψ_q|𝒵_q |ψ_q⟩_n = (-1)^s_nδ_nm,wheres_n ∈{0, 1}ensures the correct phase according to the bitstring. This leads us to the final expression ⟨ψ|𝒵_q|ψ|=⟩1/N∑_n (-1)^s_nn⟨ψ_a|𝒵_a |ψ_a⟩_n.Eq. (<ref>) shows, that the expectation value of an observable𝒵_qon the main qubits can be written as a sum ofZexpectation values on the ancilla qubits.The relationship between⟨𝒵_q|$⟩ and the sum over ⟨𝒵_a|$⟩ shows, why zero noise extrapolation works in our approach. From Eq. (<ref>) we see that amplifying the noise in the|ψ_a⟩_nstates will increase the noise in the expectation values given byn⟨ψ_a|𝒵_a |ψ_a⟩_n. Since all these states are equivalent up to local unitaries for alln, we can increase the noise of⟨𝒵_q|$⟩ in a well-defined way by amplifying the noise in the circuits which prepare the ancilla states |ψ_a⟩_n.§ DATA ACQUISITION FOR THE VQE EXPERIMENTIn this Appendix, we report how we performed the ZNE for the case of the VQE from Sec. <ref>. In order to increase the noise in the ancilla circuits, we use the method of local gate folding <cit.>. In this method, all CNOT-gates in the circuit are replaced asCX_ij→ CX_ij^2n+1where n is an integer. The additional CNOT-gates do not change the outcome of the circuit, but stretch the pulse which is executed on the hardware. This leads to a higher error-rate due to decoherence effects in the qubits, which is the main source of errors. In order to perform ZNE, we estimate λ by calculating the factor by which the pulse is stretched in time, see Fig. <ref>.In addition to ZNE we use Pauli twirling <cit.> and dynamical decoupling <cit.> as additional mitigation techniques. For the Pauli twirling, we substitute each CNOT-gate in a given circuit randomly byCX_ij→P_1_i P_2_j CX_ijP_3_i P_4_j,where P_i=X,Z,Y or I, and where P_1 and P_2 are chosen randomly and P_3 and P_4 such, that the circuit do not change the effect of the original gate. In total, there are 16 different combinations how to substitute the CNOT-gate. In Fig. <ref> we show the results from running our circuits on . Before running the graph state circuits, we first ran read-out mitigation circuits using thepackage using 100 000 shots per circuit. By calculating the readout-calibration matrix, we are then able to perform readout-error mitigation on the results. As can be seen from Fig. <ref>, read-out errors are an important source of error in our algorithm and need to be corrected before performing ZNE.We evaluate the expectation values at three different noise levels, λ∈{1,3,5}. In order to extrapolate to the zero-noise limit at λ = 0 we use a second-order polynomial fit:⟨H(λ)|=⟩ a λ^2 + b λ +c,such that in the zero-noise limit the expectation value is given by c and its error by the fitting error. We ran each experiment eight times to quantify the effect of statistical shot noise. | http://arxiv.org/abs/2311.16223v1 | {
"authors": [
"Thierry Nicolas Kaldenbach",
"Matthias Heller"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20231127190000",
"title": "Mapping quantum circuits to shallow-depth measurement patterns based on graph states"
} |
KEK-TH-2578, OU-HET 1201, NU-EHET 002, TU-1214 [email protected] KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305–0801, Japan [email protected] Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan [email protected] College of Engineering, Nihon University, Koriyama, Fukushima 963-8642, [email protected] Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan [email protected] Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JapanThe H-COUP program is provided as a package of Fortran codes, which can compute observables related to Higgs bosons including radiative corrections in various extended Higgs sectors.We give a manual for the latest version of H-COUP (), in whichdecay rates and branching ratios of all the Higgs bosons can be calculated at one-loop level in EW and Higgs interactions with QCD correctionsin the Higgs singlet model, four types of the two Higgs doublet model with a softly-broken Z_2 symmetry, and the inert doublet model.The previous version () can evaluate those only for the standard model like Higgs boson with the mass of 125 GeV (h). In , renormalized quantities are computed based on the gauge independent on-shell renormalization scheme.The source code ofcan be downloaded via the following link: <http://www-het.phys.sci.osaka-u.ac.jp/ hcoup>. By using , we can compare the precise measurements of the properties of h and direct searches for additional Higgs bosonswith their predictions at one-loop level, by which we can reconstruct the structure of the Higgs sector. H-COUP Version 3:A program for one-loop corrected decays of any Higgs bosonsin non-minimal Higgs models Kei Yagyu 27 November 2024 ===========================================================================================================§ INTRODUCTIONThe LHC experiments have clarified the existence of the 125 GeV Higgs boson and its propertiesto be consistent with those of the Higgs boson in the Standard Model (SM) within the theoretical and experimental uncertainties <cit.>. However, the nature of electroweak symmetry breaking is still one of the most important questions in particle physics.In fact, no fundamental principle has been proved for determining the structure of the Higgs sector. Indeed, the SM merely assumes the minimal form composed of an isospin scalar doublet field.Thus, there is no compelling reason to stay at the minimal form, and are various possibilities for extended structures of the Higgs sector. On the other hand, the SM obviously cannot explain tiny neutrino masses, existence of dark matter and baryon asymmetry of the Universe, so that new physics must exist. The question is then what is the scale of new physics, which could be much higher than the electroweak scale, e.g., in the conventional seesaw scenario or could be EW/TeV scales, e.g., in the electroweak baryogenesis.If the latter is realized, its experimental verification is expected. In particular, it is quite natural to think that the Higgs sector is modified from the minimal form.In fact, extended Higgs sectors are often introduced innew physics at TeV scale, e.g., models with an extra isospin singlet, doublet and triplet field and so on, and their property strongly depends on new physics scenario.Therefore, unveiling the true structure of the Higgs sector is a key to open the door to new physics.The structure of the Higgs sector can be determined by investigating various extended Higgs sectors comprehensively.There are basically two ways for such an investigation, i.e., precise measurements for properties of the discovered Higgs boson (h) and direct searches for extra Higgs bosons. The former is absolutely important, because the precise measurement of h is what we can definitely perform at future collider experiments such as the High-Luminosity LHC (HL-LHC) <cit.> andlepton colliders, e.g., the International Linear Collider (ILC) <cit.>, the Circular Electron-Positron Collider (CEPC) <cit.> and e^+e^- collisions of the Future Circular Collider (FCC-ee) <cit.>.For example, the Higgs boson couplings are expected to be measured typicallywith a few percent level at HL-LHC <cit.> and a few permille level at e^+e^- colliders <cit.>.Thus, predictions of the Higgs boson couplings, decay rates and branching ratios in extended Higgs sectors must be evaluated at quantum levels in order to compare such precise measurements. Once deviations in the observables of h from SM predictions are found,we can extract the scale of new physics from the size of deviations, which has been known asa “new no-loose theorem” <cit.> .Moreover, we can fingerprint the Higgs sector, i.e., extracting the properties such asthe representation and the number of Higgs fields by measuring the pattern of deviations. Detections of additional Higgs bosons are direct evidence of an extended Higgs sector. Although any clear signatures have not been observed yet, there are still possibilities of the existence of extra Higgs bosons at the electroweak scale, which is well motivated by various new physics scenarios, e.g., the radiative seesaw mechanism, the Higgs-portal dark matter and the electroweak baryogenesis.It has been known that the decay property of extra Higgs boson strongly depends on the “alignmentness” of the Higgs sector, i.e.,how the property of h is close to the one in the SM Higgs boson.In the nearly alignment case, extra Higgs bosons tend to dominantly decay into a bosonic final state, e.g., H → W^+W^-/ZZ/hh, A → Zh and H^±→ W^± h in the Two Higgs Doublet Models (THDMs),in which severe lower limits on the mass of extra Higgs bosons have already been taken at the LHC <cit.>.This also means that a lower limit and an upper limit on the mass of extra Higgs bosonscan simultaneously be taken by the synergy between the direct searches at the HL-LHC and the precise measurements of h,and thus a large portion of the parameter space can be explored <cit.>.The important thing here is that loop effects of extra Higgs bosons can change the alignmentness and the decay rates of extra Higgs bosons at the same time.In particular, such modifications can be significant when non-decoupling effects of extra Higgs bosons are realized, whichcan appear in new physics scenarios, e.g., with first order electroweak phase transitions <cit.>. Therefore, including the effect of radiative corrections is important not only for the precise measurement of h, but also for the direct searches for extra Higgs bosons.In order to realize the fingerprinting, to establish the new no-loose theorem and to perform the direct searches in a more precise way,we have developed theprogram which is provided as a package of Fortran codes to evaluateHiggs boson related observables including radiative corrections in various extended Higgs sectors.So far, we have published theversion 1 () <cit.> and version 2 () <cit.>,which can provide one-loop corrected couplings, decay rates and branching ratios of h in the THDMs with a softly-broken Z_2 symmetry to avoid flavor changing neutral currents at tree level, the Higgs Singlet Model (HSM) and the Inert Doublet Model (IDM) based on the gauge independent on-shell renormalization scheme <cit.>. In this paper, we extend theprogram to version 3 () and give its manual, where the decay rates and the branching ratios of all the Higgs bosons can be evaluated at one-loop level in the extended Higgs sectors shown above. We have implemented one-loop corrected decay rates of extra Higgs bosons in , which have been computed in the series of our papers<cit.>.In , we take the same on-shell scheme for the scalar two-point functions as the previous versions, while the renormalization for the tadpole is performed by either the standard tadpole scheme <cit.>or the alternative tadpole scheme <cit.> (in the case of the THDMs, there are four choices because of the renormalization method of the other parameter).The users can choose these renormalization schemes. There are also important works relevant to such radiative corrections in the THDMs <cit.> and in the minimal supersymmetric SM <cit.>. We note that similar program tools have been developed by the different groups, e.g., 2HDECAY <cit.>, Prophecy4f <cit.>, ewN2HDECAY <cit.>and EWsHDECAY <cit.>[Recently, FlexibleDecay <cit.> appeared, in which decays of Higgs bosons can be computed in the MS scheme with higher order electroweak and QCD corrections in various models beyond the SM. ]. A remarkable feature of H-COUP is that it can systematically computeboth the properties of h and extra Higgs bosons in various extended Higgs sectors under a fixed renormalization scheme. This paper is organized as follows. In Sec. <ref>,we briefly review the extended Higgs models which are implemented into define the notation.In Sec. <ref>, we discuss the renormalized vertex functions and decay rates. We also explain our renormalization scheme.We then describe the structure ofand how to install and run the program in Sec. <ref>. In Sec. <ref>, we show examples of the numerical evaluation.Summary is given in Sec. <ref>.§ MODELS AND CONSTRAINTSWe define three models with the extended Higgs sector, i.e, the HSM, the THDMs and the IDM which are implemented in .In Table <ref>, we show the mass eigenstates of scalar fields, the input parameters and the constraints in each model. In the following, we provide additional explanations of Table <ref> in each model, see the references given in this table for more detailsof the model description.Throughout the paper, we denote the mass eigenstates of scalar fields as: h:the discovered CP-even Higgs boson with the mass 125 GeV,H:another CP-even Higgs boson,A:a CP-odd Higgs boson,H^± :a pair of singly charged Higgs bosons.In the HSM, the Higgs sector is composed of the SM Higgs field Φ, i.e., the isospin doublet Higgs field with hypercharge Y = 1/2, and an isospin singlet scalar field S with Y = 0.In this model, no additional symmetry beyond the SM is imposed <cit.>.There are two mass eigenstates andfive free input parameters as shown in Table <ref>, where m_ϕ indicates a mass of a scalar boson ϕ,and α is the mixing angle between two CP-even Higgs bosons with its domain to be defined as -π/2 ≤α≤π/2.The other three parameters λ_S, λ_Φ S and μ_S are the coefficients of the S^4, |Φ|^2S^2and S^3 terms in the Higgs potential, respectively. The THDMs contain two isospin doublet Higgs fields Φ_1 and Φ_2 with Y = 1/2.In , we impose a softly-broken Z_2 symmetry <cit.> to avoid flavor changing neutral currents at tree level, in whichΦ_1 and Φ_2 are assigned to be Z_2-even and Z_2-odd, respectively. Depending on the charge assignments of right-handed fermions,four types of Yukawa interactions appear, which are so-calledType-I, Type-II, Type-X and Type-Y <cit.>.There are five mass eigenstates andsix input parameters described in Table <ref>, where α is the mixing angles between two CP-even Higgs bosons and tanβ is the ratio of the VEVs tanβ = v_2/v_1 [In addition, we have to input the sign of cos(β - α) because values of α and β are not directly input.]We define sin(β-α)≥ 0 and tanβ >0.A dimensionful parameter M^2 describes the soft breaking scale of the Z_2 symmetry. In the IDM, the Higgs sector also consists of two isospin doublet Higgs fields Φ and η with Y = 1/2,so that five mass eigenstates appear as those in the THDMs.Unlike the THDMs, an unbroken Z_2 symmetry <cit.> is imposed instead of the softly-broken one, in which only η is assigned to be Z_2-odd and all the other fields are assigned to be Z_2-even.Thanks to the unbroken Z_2 symmetry, the lightest additional neutral scalar boson (H or A) can be a candidate for dark matter.There are five free parameters shown in Table <ref>, where λ_2 and μ_2^2 are the coefficients of the quartic and the quadratic terms of η, respectively.In each model, the following constraints can be imposed inas in the previous versions:* Vacuum stability at tree level <cit.>* Vacuum stability (Renormalization group equations (RGE) improved with the cutoff)* Tree-level unitarity <cit.>* True vacuum <cit.>* Triviality (with the cutoff scale) <cit.>.* Electroweak S and T parameters <cit.> In Table <ref>, we list the references for these constraints in each model, see also the manual of <cit.> fordetails.§ RENORMALIZED VERTICES AND DECAY RATES In this section, we discuss radiative corrections to the decay rate of a scalar boson ϕ after introducing renormalization schemes implemented in .While the discussions of the renormalization schemes are model-dependent, the formulation of the decay rate is carried out in a model-independent way.§.§Renormalization schemesThe renormalization of Higgs potential parameters is performed by making use of on-shell conditions for two-point functions of the scalar states.With the on-shell conditions, masses, mixing angles, and wave functions renormalization factors for the scalar states are defined as on-shell parameters. In Ref. <cit.>, it has been pointed out that the on-shell renormalization of scalar mixing angles give rise to gauge-dependent amplitudes. In order to remove such gauge dependence, we make use of the pinch technique <cit.>, in which appropriate pinch terms are added for the counterterms of the mixing angles. The UV divergence of the one-point functions of Higgs fields should also be taken away in the renormalized Higgs potential.For the renormalization of tadpoles, there are two different renormalization schemes, which are called the standard tadpole scheme <cit.> and the alternative tadpole scheme <cit.>(also see a new scheme for tadpole renormalization in recent works <cit.>).Let us briefly describe how the renormalization of tadpoles is performed in these two schemes.In the standard tadpole scheme, renormalized tadpoles are set to zero.One then has the tadpole counterterms, which eliminate the UV divergence for the one-point functions by renormalization conditions for tadpoles, i.e., Γ̂_h=Γ_h^ 1PI+δ T_h=0 for the SM-like Higgs boson h.In contrast, in the alternative tadpole scheme, unrenormalized tadpoles are set to zero, which means that the tadpole counterterm parameters are not introduced in a theory. Instead, there is a degree of freedom to shift bare Higgs fields, ϕ_B→ϕ_B+Δ v.This shift introduces an additional tadpole term, and one can choose the constant Δ v in a way that the tadpole terms vanish at loop level, i.e., Γ_h^ 1PI+m_h^2Δ v=0 for the SM-like Higgs boson h.The shift affects all terms containing the Higgs bare field ϕ_B.Because of this, one eventually should include tadpole-inserted diagrams in renormalized self-energies and renormalized vertex functions. In all the models implemented in , the number of input parameters is greater than that determined by the on-shell conditions for the scalar two-point functions. Thus, the other remaining parameters should be renormalized in different ways.As briefly discussed, such a parameter is determined by the MS renormalization.In the following, by specifying a model, we discuss how each input parameter is renormalizedand introduce renormalization schemes implemented in .The renormalization of model parameters is summarized in Table <ref>. Higgs singlet modelIn the HSM, there are two mass parameters (i.e., m_h and m_H), one mixing angle and four parameters of wave function renormalizations.They are renormalized by on-shell conditions for two-point functions of h and H, and the other parameters λ_Φ S and μ_S are determined by the MS renormalization. Since there are two different ways to renormalize tadpoles as discussed above,in , a user can choose the following renormalization schemes for the tadpoles in the HSM: The KOSY and the PT scheme schemes are respectively based on Ref. <cit.> and Ref. <cit.>.We here emphasize that the difference between these two renormalization schemes is how the tadpoles are renormalized.For all the above schemes, the scalar mixing angle α is commonly renormalized by the on-shell scheme with the pinch terms.Two Higgs doublet modelsIn the THDMs, four mass parameters, two mixing angles, and twelve wave functions are determined by the on-shell renormalization.The reaming parameter M^2 is determined by the MS renormalization. This parameter is related to the softly-broken parameter of the Z_2 symmetry m_12^2 (V⊃ (m_12^2Φ_1^†Φ_2 + h.c.) ) by M^2=m_12^2/(cosβsinβ). Instead of M^2, one can also renormalize m_12^2 as a MS parameter. In , a user can choose the renormalization scheme for the tadpoles and the MS parameter in the THDMs.Hence, the following four different renormalization schemes are implemented:δ M^2δ M^2δ m_12^2 δ m_12^2 The KOSY1 and KOSY2 schemes are based on Ref. <cit.>, while the PT1 and PT2 schemes are based on Ref. <cit.>. In all these renormalization schemes, the pinch technique is applied to the renormalization of the mixing angles α and β.Inert doublet modelIn the IDM, four mass parameters and four wave functions are determined by the on-shell renormalization.The invariant mass parameter μ_2^2 for the inert scalars is renormalized as a MS parameter.In Ref. <cit.>, it has been shown that for renormalized vertex functions of h, the scheme difference between the standard tadpole scheme and the alternative tadpole scheme does not appear in the SM.This is also the case in the IDM, because the counterterms for the renormalized vertex functions are expressed in the same form as those in the SM.Hence, for the IDM, we only implement the KOSY scheme based on Ref. <cit.> in .§.§Renormalized vertex functions When we consider two body final states, ϕ can generally decay into a fermion and an anti-fermion (ϕ→ ff̅'),two gauge bosons (ϕ→ VV'), a gauge and a scalar bosons (ϕ→ Vϕ') and two scalar bosons (ϕ→ϕ' ϕ”). In order to discuss radiative corrections to the decay rate of these modes, it is convenient to introduce the following renormalized verticesΓ̂_ϕ ff', Γ̂_ϕ VV'^μν, Γ̂_ϕ Vϕ' ^μ and Γ̂_ϕϕ'ϕ”.Except for Γ̂_ϕϕ'ϕ”, each renormalized vertex can be decomposed into the following form factors: Γ̂_ϕ ff'(p_1^2,p_2^2,q^2) = Γ̂_ϕ ff'^ S+γ_5 Γ̂_ϕ ff'^ P + p_1/Γ̂_ϕ ff'^ V_1 +p_2/Γ̂_ϕ ff'^ V_2 +p_1/γ_5 Γ̂_ϕ ff'^ A_1 +p_2/γ_5Γ̂_ϕ ff'^ A_2 +p_1/p_2/Γ̂_ϕ ff'^ T +p_1/p_2/γ_5Γ̂_ϕ ff'^ PT,Γ̂_ϕ VV'^μν(p_1^2,p_2^2,q^2) =g^μνΓ̂_ϕ VV'^1 + p_1^ν p_2^μ/q^2Γ̂_ϕ VV'^2 + iϵ^μνρσp_1ρ p_2σ/q^2Γ̂_ϕ VV'^3, Γ̂_ϕ Vϕ' ^μ(p_1^2,p_2^2,q^2) =(p_1 + q)^μΓ̂_ϕ Vϕ',where p_1^μ and p_2^μ denote the incoming momenta, and q^μ represents the outgoing momentum of ϕ, see Fig. <ref>. Each renormalized form factor Γ̂_ϕ XY can be further divided into the following terms: Γ̂^i_ϕ XY(p_1^2,p_2^2,q^2) =Γ^i, tree_ϕ XY + Γ^i, loop_ϕ XY with Γ^i, loop_ϕ XY = Γ^i, 1PI_ϕ XY+δΓ^i_ϕ XY,where Γ^i, tree_ϕ XY, Γ^i, 1PI_ϕ XY and δΓ^i_ϕ XY respectively denote contributions from tree diagrams, 1PI diagrams and counterterms. Let us also describe differences that can appear by using the different renormalization schemes in the HSM and the THDMs in order. In the HSM, the scheme difference between the KOSY scheme and the PT scheme only appears in Γ̂_ϕϕ' ϕ” as can be seen in Ref. <cit.>. The analytical expressions can be found in the reference.The scheme difference does not appear in the other vertices Γ̂_ϕ ff', Γ̂_ϕ VV'^μν and Γ̂_ϕ Vϕ' ^μ.In the THDMs, similar to the HSM, the scheme difference between the KOSY1 scheme and the PT1 scheme only arises for the renormalized vertex functions Γ_ϕϕ' ϕ”. The scheme difference ΔΓ_ϕϕ' ϕ” consists of the tadpole contributions, as given in Eqs.(<ref>), (<ref>) and (<ref>).On the other hand, no scheme difference appears for Γ_ϕϕ' ϕ”between the KOSY2 scheme and the PT2 scheme, so that these two schemes give the same results for all the decay processes of ϕ.In the end, three different results are obtained only for ϕ→ϕ' ϕ” depending on these renormalization schemes in the THDMs. In Appendix. <ref>, we present the analytical expressions for the scheme difference.§.§ Radiative corrections to the decay rate For decay processes that appear at the tree level, decay rates at NLO are generally expressed as Γ(ϕ→ XY)_ NLO = Γ(ϕ→ XY)_ LO[1 + Δ_ EW(ϕ→ XY) -Δ r + Δ_ QCD(ϕ→ XY) ]+ Γ(ϕ→ XYγ),where Γ(ϕ→ XY)_ LO is the decay rate at LO, Δ_ EW(ϕ→ XY) is the electroweak correction,Δ_ QCD(ϕ→ XY) is the QCD correction andΓ(ϕ→ XYγ) is the contribution from the real photon emission which is required to guarantee IR divergence-free results.The contribution from the real photon emission vanishes when ϕ, X, and Y are electrically neutral.We separately show the contribution from the electroweak correction to the muon decay, Δ r, which appears by the replacement of the VEV, i.e.,v^2 → v^2(1 + Δ r). The decay rates at LO are expressed as Γ(ϕ→ ff̅')_ LO = N_c^fm_ϕ/8πλ^1/2(x_f,x_f')×[(|Γ_ϕ ff'^ S,tree|^2 + |Γ_ϕ ff'^ P,tree|^2)(1-x_f - x_f') -2(|Γ_ϕ ff'^ S,tree|^2 - |Γ_ϕ ff'^ P,tree|^2)√(x_fx_f')],Γ(ϕ→ VV')_ LO = |Γ_ϕ VV'^1, tree|^2/64π m_ϕ(1 + δ_VV')λ(x_V,x_V') + 12x_Vx_V'/x_Vx_V'λ^1/2(x_V,x_V'),Γ(ϕ→ Vϕ')_ LO = |Γ_ϕ Vϕ'^ tree|^2/16πm_ϕ^3/m_V^2λ^3/2(x_V,x_ϕ'),Γ(ϕ→ϕ'ϕ”)_ LO = |Γ_ϕϕ'ϕ”^ tree|^2/16π m_ϕ(1 + δ_ϕ'ϕ”)λ^1/2(x_ϕ',x_ϕ”), where N_c^f=3 (1) for f to be quarks (leptons), x_a = m_a^2/m_ϕ^2 and λ(x,y) = (1-x-y)^2-4xy. In the above expressions, V and V' are either W^± or Z [For the CP-odd Higgs boson and the charged Higgs bosons, the decay ϕ→ VV' specified as A→ ZZ/W^+W^- and H^±→ W^± Z, respectively. Since they are loop-induced processes, the analytical expressions of the decay rates are presented by Eq. (<ref>).]. The electroweak corrections Δ_ EW are expressed for each process asΔ_ EW(ϕ→ ff̅') =[(|Γ_ϕ ff'^ S,tree|^2 + |Γ_ϕ ff'^ P,tree|^2)(1-x_f - x_f') -2(|Γ_ϕ ff'^ S,tree|^2 - |Γ_ϕ ff'^ P,tree|^2)√(x_fx_f')]^-1×[2{Re[Γ_ϕ ff'^ S,treeΓ_ϕ ff'^ loop*]+Re[Γ_ϕ ff'^ P,treeΓ̃_ϕ ff'^ loop*] }(1 - x_f - x_f') - 4{Re[Γ_ϕ ff'^ S,treeΓ_ϕ ff'^ loop*] - Re[Γ_ϕ ff'^ P,treeΓ̃_ϕ ff'^ loop*] }√(x_fx_f')]+ Δ_ mix(ϕ→ ff̅'), Δ_ EW(ϕ→ VV') =2Re[Γ_ϕ VV'^1, treeΓ^1, loop*_ϕ VV']/|Γ^1, tree_ϕ VV'|^2 + Re[Γ_ϕ VV'^1, treeΓ^2, loop*_ϕ VV']/|Γ^1, tree_ϕ VV'|^2(1-x_V - x_V')λ(x_V,x_V') /λ(x_V,x_V') + 12x_Vx_V'- ReΠ̂_VV'(m_V^2) -ReΠ̂_V'V''(m_V'^2) , Δ_ EW(ϕ→ Vϕ')= 2Re[Γ^ tree_ϕ Vϕ'Γ^ loop*_ϕ Vϕ']/|Γ^ tree_ϕ Vϕ'|^2 - ReΠ̂_VV'(m_V^2) + Δ_ mix(ϕ→ Vϕ'), Δ_ EW(ϕ→ϕ'ϕ”)= 2Re[Γ^ tree_ϕϕ'ϕ”Γ^ loop*_ϕϕ'ϕ”]/|Γ^ tree_ϕϕ'ϕ”|^2 + Δ_ mix(ϕ→ϕ'ϕ”) , whereΓ_ϕ ff'^ loop = Γ_ϕ ff'^ S,loop + m_f'Γ_ϕ ff'^ V_1, loop - m_fΓ_ϕ ff'^ V_2, loop +(m_ϕ^2+m_fm_f' -m_f^2 - m_f'^2)Γ_ϕ ff'^ T,loop, Γ̃_ϕ ff'^ loop = Γ_ϕ ff'^ P,loop - m_f'Γ_ϕ ff'^ A_1, loop - m_fΓ_ϕ ff'^ A_2, loop +(m_ϕ^2-m_fm_f' -m_f^2 - m_f'^2)Γ_ϕ ff'^ PT,loop.The external-leg corrections to the on-shell gauge boson ReΠ̂^'_VV (m_V^2) appear in Δ_ EW(ϕ→ Vϕ') and Δ_ EW(ϕ→ VV'), becausethe residue of the self-energy for the on-shell weak gauge boson is not set to unity in the applied renormalization scheme for electroweak parameters (see, e.g., Ref. <cit.>). The correction factor Δ_ mix(ϕ→ XY) corresponds to the non-vanishing contributions from mixing self-energies between two boson states. This type of contribution arises if external lines contain the charged Higgs boson in the THDM, where Π̂_H^+G^-(m_H^±^2)=0 is not imposed as the renormalization condition in our renormalization scheme <cit.>.The analytical expressions are given in Ref. <cit.> for Δ_ mix(ϕ→ ff̅') and Δ_ mix(ϕ→ Vϕ'), and Appendix <ref> for Δ_ mix(ϕ→ϕ'ϕ”). In Table <ref>, we summarize the references giving the analytical expressions for 1PI vertices, counterterms, and real photon emissions, which are needed to compute the two-body decay rates of all the additional Higgs bosons. For decay processes induced at loop levels, their decay rates are given at LO as Γ(ϕ→ Vγ)_ LO = (1 - x_V)^3/32π m_ϕ(1 + δ_Vγ) (|Γ̂_ϕ Vγ^2, loop|^2 + |Γ̂_ϕ Vγ^3, loop|^2 ),(V=γ, Z, W^±) Γ(ϕ→ gg)_ LO = 1/8π m_ϕ(|Γ̂_ϕ gg^2, loop|^2 + |Γ̂_ϕ gg^3, loop|^2 ).In the above expression, we applied the Ward identity by which the contribution from Γ̂_ϕ XY^1, loop is rewritten by Γ̂_ϕ XY^2, loop.If ϕ→ VV' decays appear at the one-loop level, e.g., H^±→ W^± Z in the THDMs, the decay rate is expressed at LO as [If one specifies ϕ→ VV' asH → ZZ/W^+W^- in the HSM or the THDM, these processes realize at the tree level. The analytical expressions are given by Eq. (<ref>). ] [We note that, for the charged Higgs decays H^±→ W^± Z in the THDM, H^± G^∓ andH^± W^∓ mixing contributions exist. They are included in Γ̂^1, loop_ϕ VV'.]Γ(ϕ→ VV')_ LO= λ^3/2(x_V,x_V')/64π m_ϕ x_Vx_V'{|Γ̂_ϕ VV'^1, loop|^2[1 + 12 x_Vx_V'/λ(x_V,x_V')]+ |Γ̂_ϕ VV'^2, loop|^2 /4λ(x_V,x_V')+ 2x_Vx_V'|Γ̂_ϕ VV'^3, loop|^2 + Re[Γ̂_ϕ VV'^1, loopΓ̂_ϕ VV'^2, loop*](1 - x_V - x_V') }.The QCD corrections are included in the following processes,ϕ→ qq̅^',ϕ→ tt̅,ϕ→ tb̅,ϕ→γγ,ϕ→ Zγ,ϕ→ gg. The evaluation of QCD corrections for the processes of Eq. (<ref>) is common for all the models.In the following paragraphs, we briefly describe how the QCD corrections are included for each decay mode in . See Ref. <cit.> for detailed expressions for these QCD corrections.For the decays into light quarks ϕ→ qq̅^',the corrections up to next-to-next-to leading order (NNLO) are computed in the MS scheme <cit.>.The NNLO corrections involve top-quark loop contributions evaluated in the heavy top-quark mass limit m_ϕ≪ m_t.For the decays including the top-quark ϕ→ tt̅ and ϕ→ tb̅, the dominant QCD corrections would depend on the size of the mass of additional Higgs bosons m_ϕ. If these masses are taken to be around the threshold region (e.g., m_H,A^∼ 2m_t and m_H^±∼ m_t+m_b) the corrections with top-quark mass are important, which is evaluated at NLO in the on-shell scheme <cit.>.Conversely, if these are far above the threshold region (e.g., m_H,A^≫ 2m_t and m_H^±^≫ m_t+m_b), the logarithmic contributions log (m_ϕ^2/m_t^2) become important.One can evaluate the corrections including such logarithmic contributions up to NNLO in the MS scheme in the same way as ϕ→ qq̅.Since in the intermediate region of m_ϕ, both above contributions could have a large influence, we interpolate them according to Ref. <cit.>. We evaluate QCD corrections toϕ→γγ <cit.> and ϕ→ Zγ <cit.> at NLO.While the analytical formulae of the former can be applied to any mass region of additional Higgs bosons, those in the heavy top-quark limit are applied to the latter.To moderate these corrections, we choose the renormalization scale μ=m_ϕ/2 only for ϕ→γγ and ϕ→ Zγ (μ=m_ϕ is used for the other processes.). For ϕ→ gg, the corrections are calculated up to NNLO.The NLO corrections are composed of the virtual gluon loop corrections and the real emissions ϕ→ ggg and ϕ→ gqq̅.Following Ref. <cit.>, we implement these contributions evaluated in the heavy top-quark limit.We neglect the remaining contributions for the real emissions, which are generally smaller than the virtual corrections (see the detail in Ref. <cit.>).For the NNLO QCD corrections to ϕ→ gg, we use the formula evaluated in the heavy top-quark limit <cit.>. As described above, some of the QCD corrections are computed by taking the heavy top-quark limit.However, additional Higgs bosons can be heavier than the top-quark, so that the QCD corrections derived in the heavy top-quark limit might be unreliable in the case of m_ϕ≳ m_t.For instance, the top-quark loop contributions to ϕ→ qq̅ contain the logarithmic contributions log (m_ϕ^2/m_t^2), which gives sizable contributions in the case of m_ϕ≫ m_t. Hence, we only include the contributions given in the heavy top-quark limit, i.e., the top-loop contributions to ϕ→ qq̅, the real emission contributions at NLO to ϕ→ gg and the NNLO corrections to ϕ→ gg in the evaluation of the QCD corrections if an additional Higgs boson is lighter than the top-quark (m_ϕ<m_t).Apart from the QCD correction factor Δ_ QCD(ϕ→ XY), quark masses coming from the Yukawa couplings are replaced by the running quark masses in Γ_ LO(ϕ→ XY ) for all the processes in Eq. (<ref>). The decays into an off-shell gauge boson ϕ→ VV^∗→ Vff̅ and ϕ→ϕ^' V^∗→ϕ^' ff̅ can happen in the case ofm_ϕ < 2m_V for the former and m_ϕ < m_ϕ^' + m_V^ for the latter.While they are evaluated at electroweak LO [For the off-shell decays of the 125 GeV Higgs boson h, h → VV^∗→ V ff̅, the NLO electroweak corrections are included. ], we incorporate NLO QCD corrections to them <cit.> in .§ PROGRAM DESCRIPTIONOne can downloadand the previous versions from the following web page,<http://www-het.phys.sci.osaka-u.ac.jp/ hcoup>. In order to run theprogram, acompiler ( is recommended) and LoopTools <cit.> are required. The installation and running procedures are the same as , and these are described in the previous manual <cit.>, except for the procedure to specify the path of . Open Makefile by an editor and replaceandappearing in the linesandwith the correct paths to theand , respectively. In the following, we briefly overview the structure ofand explain the improvement from previous versions.Theprogram is composed of three blocks; input, computation, and output blocks <cit.>. In the input block,reads the input files in , whereindicates a path of thedirectory. In , the model and the order of calculations are specified in . This part is different from , where one specifies them from the command line interface. The model IDs are defined as , , , , , and . The order of electroweak corrections is specified byor , while the order of QCD corrections are defined as , , , and . The option, , provides the LO results calculated with OS quark masses, whileperforms LO calculation with MS quark masses. The detailed descriptions of the SM-like Higgs boson decays are given in Ref. <cit.>. For those of the heavy Higgs boson decays, see Refs. <cit.>. The example of theis shown in List. <ref>.The model-independent parameters are read from the input files,and . The SM parameters listed in Table <ref> are specified in . The squared momenta of the renormalized form factors of the SM-like Higgs bosons are read from . The detailed descriptions of these parameters can be found in Refs. <cit.>. If one changes the inputs in , execute $make clean before numerical evaluations.The model-dependent parameters in the HSM, THDM, and IDM are specified in , , and , respectively. The input parameters and their default values are summarized in Tables <ref>, <ref>, and <ref>. The cutoff scale Λ is used for the theoretical constraints, such as triviality and vacuum stability bounds. In addition, we have newly introduced the renormalization scale , which is used to fix the MS renormalized quantities. In the HSM and THDM, we have theoption, which specifies the renormalization scheme of the tadpole and dimensionful parameters, which was discussed in Sec. <ref>. The example of theis shown in List. <ref>.In the computation block,calculates the decay rates of any Higgs bosons in the specified model at a given order of calculations, where the evaluation of the loop functions are performed with <cit.>. The possible decay modes of the additional Higgs bosons are listed in Table <ref>[In several modes such as H→ hh in the THDM, partial decay widths would take negative values depending on the input parameters due to the truncation of two-loop effects (see Refs. <cit.>). In the current version, we simply output negative values showing a warning message in a command line interface.].In the output block,creates output files in . In addition to the ,and , one has the output files for specifying models, e.g., ,andfor the HSM. In the following, we represent them as , , andfor simplicity. In , the values of the renormalized form factors are listed. The predictions for the partial decay rates of Higgs bosons and their total widths are given inwith the values of the input parameters. Similarly, the predictions for the decay branching ratios are given in . One can check whether a given parameter set is allowed or excluded under the constraints, such as perturbative unitarity, vacuum stability, triviality, true vacuum conditions, and/or electroweak precision tests. The examples ofandare shown in Lists. <ref> and <ref>, which are calculated by using the inputs given in Lists. <ref> and <ref>.[caption=Example of the input file (in_main.txt),label=file_in_main,frame=lines] !===============================! ! ! ! Input parameters for main ! ! ! !===============================!1 ! Model ID: 1 = HSM, 2 = THDM-I, 3 = THDM-II, 4 = THDM-X, 5 = THDM-Y, 6 = IDM 1 ! Order of EW: 0 = LO, 1 = NLO 2 ! Order of QCD: -1 = LO(quark mass:OS), 0 = LO(quark mass:MSbar), 1 = NLO, 2 = NNLO [caption=Example of the input file (in_hsm.txt),label=file_in_hsm,frame=lines] !================================! !! ! Input parameters for the HSM ! !! !================================!500.d0 ! m_H in GeV 0.1d0! alpha 0.d0 ! lambda_phi S 0.1d0! lambda_S 0.d0 ! mu_S in GeV 3.d3 ! cutoff in GeV 400.d0 ! MSbar renormalization scale in GeV 0! 0: Pinched tadpole, 1: KOSY with PT [caption=Example of the output file (outGamma_hsm.txt),label=file_outGamma,frame=lines] #====================================================================== # # H-COUP [Version 3.0 (Novemver 28, 2023)] #Program for full-NLO predictions of any Higgs-boson decays # in non-minimal Higgs models # # http://www-het.phys.sci.osaka-u.ac.jp/ hcoup # #====================================================================== BLOCK MODEL #11 # HSM20 # MSbar renormalization scheme BLOCK BSMINPUTS #11.00000000E-01 # alpha20.00000000E+00 # lambda_phi S31.00000000E-01 # lambda_S40.00000000E+00 # mu_S (GeV)55.00000000E+02 # mu_r (GeV) BLOCK SMINPUTS #17.29735257E-03 # alpha_em21.16637880E-05 # Fermi constant31.17900000E-01 # alpha_s41.27000000E+00 # mc(mc) MSbar54.18000000E+00 # mb(mb) MSbar61.67000000E+00 # mc On-shell74.78000000E+00 # mb On-shell BLOCK MASS #46.79114206E-01 # mc(mh) MSbar52.85835750E+00 # mb(mh) MSbar61.72500000E+02 # mt 131.05658375E-01 # mmu 151.77686000E+00 # mtau 239.11876000E+01 # mz 248.09388642E+01 # mw (calculated,tree) 248.04085771E+01 # mw (calculated,1-loop) 251.25250000E+02 # mh 355.00000000E+02 # mH BLOCK CONSTRAINTS #03.00000000E+03 # The cutoff scale (GeV)10 # Vacuum stability at tree level [0=OK, 1=No]20 # Tree-level unitarity [0=OK, 1=No]30 # S and T parameters [0=OK, 1=No]40 # True vacuum [0=OK, 1=No]50 # Vacuum stability (RGE improved with the cutoff scale) [0=OK, 1=No]60 # Triviality (with the cutoff scale) [0=OK, 1=No] # # Partial decay widths of the SM-like Higgs boson by H-COUP # #PDGWidth DECAY 25 0.40743346E-02 # EW:NLO QCD:NNLO #GammaNDA ID1ID21.42029147E-04 2 4 -4# Gamma(h -> c c )2.45364374E-03 2 5 -5# Gamma(h -> b b )8.82215414E-07 213-13# Gamma(h -> mu- mu+)2.54256192E-04 215-15# Gamma(h -> tau- tau+)3.32918565E-04 221 21# Gamma(h -> g g)9.25841223E-06 222 22# Gamma(h -> gam gam)6.47474306E-06 222 23# Gamma(h -> gam Z)9.00793124E-05 223 23# Gamma(h -> Z Z*)7.84792254E-04 224-24# Gamma(h -> W+ W-*) # # Partial decay widths of the additional CP-even Higgs boson bH by H-COUP # #PDGWidth DECAY 35 0.87367479E+00 # EW:NLO QCD:NNLO #GammaNDA ID1ID21.11119396E-01 2 6 -6# Gamma(bH -> t t )7.77390858E-05 2 5 -5# Gamma(bH -> b b )4.42785519E-06 2 4 -4# Gamma(bH -> c c )3.52102715E-08 213-13# Gamma(bH -> mu- mu+)1.01600689E-05 215-15# Gamma(bH -> tau- tau+)1.72174523E-01 223 23# Gamma(bH -> Z Z)3.62420028E-01 224-24# Gamma(bH -> W+ W-)2.27571695E-01 225 25# Gamma(bH -> h h)2.91017661E-04 221 21# Gamma(bH -> g g)2.62532767E-07 222 22# Gamma(bH -> gam gam)5.50566818E-06 222 23# Gamma(bH -> gam Z) [caption=Example of the output file (outBR_hsm.txt),label=file_outBR,frame=lines] #====================================================================== # # H-COUP [Version 3.0 (Novemver 28, 2023)] #Program for full-NLO predictions of any Higgs-boson decays # in non-minimal Higgs models # # http://www-het.phys.sci.osaka-u.ac.jp/ hcoup # #====================================================================== BLOCK MODEL #11 # HSM20 # MSbar renormalization scheme BLOCK BSMINPUTS #11.00000000E-01 # alpha20.00000000E+00 # lambda_phi S31.00000000E-01 # lambda_S40.00000000E+00 # mu_S (GeV)55.00000000E+02 # mu_r (GeV) BLOCK SMINPUTS #17.29735257E-03 # alpha_em21.16637880E-05 # Fermi constant31.17900000E-01 # alpha_s41.27000000E+00 # mc(mc) MSbar54.18000000E+00 # mb(mb) MSbar61.67000000E+00 # mc On-shell74.78000000E+00 # mb On-shell BLOCK MASS #46.79114206E-01 # mc(mh) MSbar52.85835750E+00 # mb(mh) MSbar61.72500000E+02 # mt 131.05658375E-01 # mmu 151.77686000E+00 # mtau 239.11876000E+01 # mz 248.09388642E+01 # mw (calculated,tree) 248.04085771E+01 # mw (calculated,1-loop) 251.25250000E+02 # mh 355.00000000E+02 # mH BLOCK CONSTRAINTS #03.00000000E+03 # The cutoff scale (GeV)10 # Vacuum stability at tree level [0=OK, 1=No]20 # Tree-level unitarity [0=OK, 1=No]30 # S and T parameters [0=OK, 1=No]40 # True vacuum [0=OK, 1=No]50 # Vacuum stability (RGE improved with the cutoff scale) [0=OK, 1=No]60 # Triviality (with the cutoff scale) [0=OK, 1=No] # # Decay branching ratios of the SM-like Higgs boson by H-COUP # #PDGWidth DECAY 25 0.40743346E-02 # EW:NLO QCD:NNLO #BR NDA ID1ID23.48594707E-02 2 4 -4# BR(h -> c c )6.02219501E-01 2 5 -5# BR(h -> b b )2.16529938E-04 213-13# BR(h -> mu- mu+)6.24043478E-02 215-15# BR(h -> tau- tau+)8.17111502E-02 221 21# BR(h -> g g)2.27237407E-03 222 22# BR(h -> gam gam)1.58915350E-03 222 23# BR(h -> gam Z)2.21089630E-02 223 23# BR(h -> Z Z*)1.92618509E-01 224-24# BR(h -> W+ W-*) # # Decay branching ratios of the additional CP-even Higgs boson bH by H-COUP # #PDGWidth DECAY 35 0.87367479E+00 # EW:NLO QCD:NNLO #BR NDA ID1ID21.27186222E-01 2 6 -6# BR(bH -> t t )8.89794312E-05 2 5 -5# BR(bH -> b b )5.06808167E-06 2 4 -4# BR(bH -> c c )4.03013477E-08 213-13# BR(bH -> mu- mu+)1.16291199E-05 215-15# BR(bH -> tau- tau+)1.97069350E-01 223 23# BR(bH -> Z Z)4.14822577E-01 224-24# BR(bH -> W+ W-)2.60476436E-01 225 25# BR(bH -> h h)3.33096095E-04 221 21# BR(bH -> g g)3.00492552E-07 222 22# BR(bH -> gam gam)6.30173635E-06 222 23# BR(bH -> gam Z)§ EXAMPLES OF NUMERICAL EVALUATIONSAs mentioned in Introduction,it is quite important to include radiative corrections to the analysesfor the synergy between the precise measurements of h and the direct searches for additional Higgs bosons in order to test and discriminate the extended Higgs models.We here show an example of such analyses by using . In Fig. <ref>, we show the correlation between the decay branching ratio of the A→ Zh process and the deviation of the decay rate of h→ ZZ^* from the SM prediction in the Type-I THDM.We define Δ R(h→ ZZ^*)≡Γ[h→ ZZ^*]/Γ_SM[h→ ZZ^*] to parameterize the deviation <cit.>.We take m_A = m_H=300 GeV and tanβ = 2, while the values of sin(β-α) and M^2 are scanned under the constraints of the perturbative unitarity, the vacuum stability and the electroweak S and T parameters. We also take into account the constraints from flavor measurements, direct search results of additional Higgs bosons and Higgs coupling measurements. The left panel and the right panel show results with cos(β-α) < 0 andcos(β-α) > 0, respectively.Black curves represent tree-level predictions, while the color dots show the results including radiative corrections with different colors denoting those given by different values of M^2.It is clearly seen that the radiative corrections significantly change thecorrelation predicted at LO.In particular, the case with smaller values of M^2 show the larger difference between the results at LO and NLO, because of the larger non-decoupling loop effects of the additional Higgs bosons. In addition to this particular example, it has been known that radiative corrections can significantlychange the other correlations, e.g., the H → hh decay rate and the triple Higgs boson coupling hhh <cit.>, which can also be evaluated by using .Therefore,can realize the synergy analysis discussed abovein a more robust way, by which we can reconstruct the structure of the Higgs sector.§ SUMMARYWe have provided the manual for theprogram which is a set of Fortran codes for numerical calculations of decay rates of all the Higgs bosonsat NLO in electroweak interactions with QCD corrections in the HSM, four types of THDMs and the IDM.The electroweak corrections are evaluated based on the gauge independent on-shell renormalization scheme. We have described the renormalization schemes, the renormalized vertex functions and the radiatively corrected decay rates for the processes ϕ→ϕ'ϕ”, ϕ→ Vϕ' ϕ→ VV' and ϕ→ ff̅'̅, which are implemented in .We also have discussed the loop induced decays; i.e. ϕ→ Vγ, ϕ→ gg and ϕ→ W^± Z, which can be evaluated at LO in electroweak interactions in .We have designed the program in a way that the user can choosethe different two (four) renormalization schemes for the tadpole in the HSM (THDMs), among which the decay rate of ϕ→ϕ'ϕ” can be different.We then have explained the structure of the program, and have shown the examples of the input and output files.Finally, we have demonstrated the analysis by using .As an example, we have shown the loop-corrected correlation between the branching ratio of A → Zh and the deviation in the h → ZZ^* decay in the Type-I THDM. This work is supported in part by the Grant-in-Aid on Innovative Areas, the Ministry of Education, Culture,Sports, Science and Technology, No. 20H00160 and No. 23K17691 [S.K.], JSPS KAKENHI Grant No. 23KJ0086 and the National Science Centre, Poland, under research Grant No. 2020/38/E/ST2/00243 [K.S.],Early-Career Scientists, No. 20K14474 [M.K.] and JSPS KAKENHI Grant No. 22KJ3126 [M.A.]. § SCHEME DIFFERENCE FORPARAMETERS IN THDMS We calculate the scheme difference for hhh, Hhh, HAA, and HH^+H^- vertices in THDMs.We focus on the four schemes presented in Sec. <ref>.The difference between two schemes is defined byΔΓ(-)≡Γ()-Γ().In Appendix, the trigonometric functions sinθ and cosθ are represented using the shorthand notation as s_θ and c_θ. As discussed in the main text, the counterterms δ M^2 and δ m^2_12 are defined in theMS scheme.Analytical expressions for these counterterms are derived in the following way. The counterterms δ M^2 or δ m_12^2 absorb the remnant of the UV divergence of the renormalized hhh vertex without these counterterms.In the KOSY1 scheme, this condition yields <cit.>.δ M^2/M^2|_ KOSY1 = 1/16π^2 v^2[2∑_f N_c^f m_f^2 ζ_f^2 +4M^2 -2m_H^±^2 -m_A^2+ s_2α^/s_2β^(m_H^2-m_h^2)-3(2m_W^2+m_Z^2)]Δ_div,where Δ_ div represents the divergent part given by Δ_div = 1/ϵ -γ_E + log 4π.Definitions and the explicit expression of coupling factor ζ_f are given in Ref. <cit.>. In the PT1 scheme, additional UV divergent contributions in the scheme difference between KOSY1 and PT1 are absorbed by δ M^2 <cit.>,.δ M^2/M^2|_ PT1= .δ M^2/M^2|_ KOSY1 +2/vc_2β/s_2β(Γ^ 1PI_h/m_h^2c_β-α-Γ^ 1PI_H/m_H^2s_β-α)_ div. ,where “div.” denotes the UV divergent part.As shown in Ref. <cit.>, the last term corresponds to the scheme difference between the KOSY scheme and the PT scheme in δβ,Δδβ (-)= -1/v(Γ^ 1PI_h/m_h^2c_β-α-Γ^ 1PI_H/m_H^2s_β-α) .Thus, Eq. (<ref>) can be rewritten by.δ M^2/M^2|_ PT1= .δ M^2/M^2|_ KOSY1 -2c_2β/s_2βΔδβ (-)_ div. .From the definition of M^2, i.e., M^2=m^2_12/(c_β s_β), the counterterm δ m_12^2 in the KOSY2 scheme and the PT2 scheme are obtained from the KOSY1 scheme and the PT1 scheme by the replacement,δ M_^2→ M_^2δ m_12^2/m_12^2-2M^2c_2β/s_2βδβ, with δβ=δβ+δβ^ PT where δβ^ PT is the pinch term contribution for δβ.Requiring cancellation of the UV divergence in the hhh vertex, in the KOSY2 and PT2 schemes, one obtains .δ m^2_12/m^2_12|_ KOSY2, PT2 = .δ M^2/M^2|_ KOSY1 +2c_2β/s_2β.δβ|_ div. .We note thatδ m^2_12 is common between KOSY2 and PT2 sincethere is no scheme difference(see e.g., Eq. (<ref>).).Also, note that there is no UV divergence in δβ^ PT.The differences among these schemes are evaluated asΔΓ̂_hhh(-)= -12M^2/v^2c_2β c_α+ βc_β-α^2/s^2_2 β(Γ_h^ 1PI/m_h^2c_β - α-Γ_H^ 1PI/m_H^2s_β - α)_ fin. ,ΔΓ̂_hhh(-) =0 ,ΔΓ̂_Hhh(-)= -4M^2/v^2c_2β c_β-α/s^2_2β(3s_α c_α-s_β c_β)×(Γ_h^ 1PI/m_h^2c_β - α-Γ_H^ 1PI/m_H^2s_β - α)_ fin. ,ΔΓ̂_Hhh(-) =0 ,ΔΓ̂_HAA(-)= -M^2/v^2 c_2β/ s^2_βs_α+ β/c^2_β(Γ_h^ 1PI/m_h^2c_β - α-Γ_H^ 1PI/m_H^2s_β - α)_ fin. ,ΔΓ̂_HAA(-) =0 ,ΔΓ̂_HH^+H^-(-) =ΔΓ̂_HAA(-) ,ΔΓ̂_HH^+H^-(-) =0 ,where “fin.” denotes the finite part.We note that Eq. (<ref>) is obtained from the fact that one holds Δδ C_H^±(-)=0. § RENORMALIZEDVERTEXIn this section, we give formulae of Γ_ϕϕ'ϕ”^loop of several scalar three-point vertices in the THDM and the IDM, which is defined in Eq. (<ref>) in Sec. <ref>.§.§andvertices in the THDM First, we show formulae for the HAA and HH^+H^- vertices in the THDM.We here give the formulas for the KOSY1 scheme defined in Sec. <ref>.Conversions to other schemes are explained in the previous section. Counterterms for the HAA and the HH^+H^- vertices in the THDM are expressed as δΓ_HAA^ =2δλ_HAA^ + 2λ_HAA^(δ Z_A +1/2δ Z_H) + 2λ_hAA(δ C_h -δα) +2λ_HAG^(δ C_A +δβ), δΓ_HH^+H^-^ =δλ_HH^+H^-^ + λ_HH^+H^-^(δ Z_H^+ +1/2δ Z_H) + λ_hH^+H^-(δ C_h -δα)+2λ_HH^+G^-^(δ C_H^± +δβ),withδλ_HAA^ = -λ_HAA^/vδ v - c_β-α^/vδ m_A^2 -s_α -3β+3s_α +β/8vs_β^c_β^δ m_H^2 + s_α +β/2vs_β^c_β^δ M^2+G_α^Aδα+G_β^Aδβ, δλ_HH^+H^-^ = 2δλ_HAA^ (A → H^±).In Eq. (<ref>), the factors G_α^ϕ and G_β^ϕ are expressed asG_α^ϕ =-1/8vs_β^c_β^[ c_α-3β^m_H^2 + c_α +β^(3m_H^2-4M^2) + 8s_β-α^s_β^c_β^m_ϕ^2],G_β^ϕ =1/8vs_2β^2[ (2m_ϕ^2 -m_H^2)s_α-5β^ + 2(2m_ϕ^2 -7m_H^2 +6M^2)s_β-α^ +(2m_ϕ^2 +3m_H^2 -4M^2)s_α+3β^]. The analytic expressions for 1PI diagram contributions to the HAA and HH^+H^- vertices are given by(16π^2)Γ_HAA, F^1PI[p_1^2,p_2^2,q^2] = -8κ_f^Hζ_f^2 N_c^fm_f^4/v^3( B_0[q^2;m_f,m_f] + p_1· p_2 C_0[m_f,m_f,m_f]),(16π^2)Γ_HH^+H^-, F^1PI[p_1^2,p_2^2,q^2] =-4m_f^2/v^3N_c^fκ_f^H{ 2(m_f^2ζ_f^2 + m_f'^2 ζ_f'^2 - m_f'^2 ζ_f ζ_f')(p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2) +(m_f^2 ζ_f^2 +m_f'^2 ζ_f'^2)[(2p_1^2 +p_1· q)C_11 +(2p_1· p_2 +p_2· q)C_12 +p_1· q C_0 ] -2m_f'^2 ζ_f ζ_f'(p_1· q C_11 +p_2· q C_12 +m_f^2 C_0)}[f,f',f], (16π^2)Γ_HAA, B^1PI[p_1^2,p_2^2,q^2]= g^4/4vc_β-α^(4B_0[q^2;W,W]-2) +g_Z^4/8vc_β-α^(4B_0[q^2;Z,Z]-2) -g^4/4vc_β-α^C_VSV^ϕϕϕ[W,H^±,W] -g_Z^4/8vc_β-α^3C_VSV^ϕϕϕ[Z,h,Z] -g_Z^4/8vs_β-α^2c_β-α^C_VSV^ϕϕϕ[Z,H,Z]-g^2/2c_β-α^m_A^2-m_H^±^2/vC_VSS^ϕϕϕ[W,H^±,G^±]+g^2/2λ_HH^+H^-^C_SVS^ϕϕϕ[H^±,W,H^±] -g^2/2c_β-α^m_A^2-m_H^±^2/vC_SSV^ϕϕϕ[G^±,H^±,W] -g_Z^2/4c_β-α^2λ_hAG^C_VSS^ϕϕϕ[Z,h,G] +g_Z^2/4s_β-αc_β-αλ_HAG^C_VSS^ϕϕϕ[Z,H,G] +g_Z^2/2s_β-αc_β-αλ_hAA^C_VSS^ϕϕϕ[Z,h,A] -g_Z^2/2s_β-α^2λ_HAA^C_VSS^ϕϕϕ[Z,H,A]+g_Z^2/2c_β-α^2λ_Hhh^C_SVS^ϕϕϕ[h,Z,h] -g_Z^2/2s_β-α^c_β-α^λ_HHh^{ C_SVS^ϕϕϕ[h,Z,H] + C_SVS^ϕϕϕ[H,Z,h]}+3g_Z^2/2s_β-α^2λ_HHH C_SVS^ϕϕϕ[H,Z,H] -g_Z^2/4c_β-α^2λ_hAG C_SSV^ϕϕϕ[G,h,Z] +g_Z^2/4s_β-α^c_β-α^λ_HAG C_SSV^ϕϕϕ[G,H,Z] +g_Z^2/2s_β-α^c_β-α^λ_hAA C_SSV^ϕϕϕ[A,h,Z] -g_Z^2/2s_β-α^2λ_HAA C_SSV^ϕϕϕ[A,H,Z]+2λ_HH^+H^-^λ_AAH^+H^-^B_0[q^2;H^±,H^±] +4λ_HH^± G^∓^λ_AAH^∓ G^±^B_0[q^2;H^±,G^±]+2λ_AH^± G^∓^λ_AHH^∓ G^±^B_0[p_2^2;H^±,G^±] +2λ_AH^± G^∓^λ_AHH^∓ G^±^B_0[p_1^2;H^±,G^±] +2λ_HG^± G^∓^λ_AAG^± G^∓^B_0[q^2;G^±,G^±] +4λ_Hhh^λ_hhAA^B_0[q^2;h,h] +4λ_HHh^λ_hHAA^B_0[q^2;h,H] +12λ_HHH^λ_HHAA^B_0[q^2;H,H] +24λ_HAA^λ_AAAA^B_0[q^2;A,A] +6λ_HAG^λ_AAAG^B_0[q^2;A,G]+4λ_HGG^λ_AAGG^B_0[q^2;G,G] +λ_hAG^λ_hHAG^{B_0[p_1^2;G,h] +B_0[p_2^2;G,h]}+4λ_hAA^λ_hHAA^{B_0[p_1^2;A,h] +B_0[p_2^2;A,h]} +2λ_HAG^λ_HHAG^{B_0[p_1^2;G,H] +B_0[p_2^2;G,H]}+8λ_HAA^λ_HHAA^{B_0[p_1^2;H,A] +B_0[p_2^2;H,A]} - 2λ_HH^+H^-|λ_AH^± G^∓|^2C_0[H^±,G^±,H^±] - 2λ_HG^+G^-|λ_AH^± G^∓|^2C_0[G^±,H^±,G^±]- 8λ_Hhhλ_hAA^2C_0[h,A,h] -2λ_Hhhλ_hGA^2C_0[h,G,h]- 8λ_hHHλ_hAA^2C_0[h,A,H]-2λ_hHHλ_hGAλ_HGAC_0[h,G,H] -8λ_hHHλ_hAA^2C_0[H,A,h] -2λ_hHHλ_hGAλ_HGAC_0[H,G,h] -24λ_HHHλ_HAA^2 C_0[H,A,H] -6λ_HHHλ_HGA^2 C_0[H,G,H] -8λ_HAAλ_hAA^2 C_0[A,h,A] -8λ_HAA^3C_0[A,H,A] -2λ_HGAλ_hGAλ_hAA{C_0[G,h,A] +C_0[A,h,G]}-2λ_HGA^2λ_HAA{ C_0[G,H,A] +C_0[A,H,G]} -2λ_HGGλ_hGA^2 C_0[G,h,G] - 2λ_HGGλ_HGA^2C_0[G,H,G], (16π^2)Γ_HH^+H^-, B^1PI[p_1^2,p_2^2,q^2] =g^4/4vc_β-α^(4B_0[q^2;W,W]-2)+ g_Z^4/8c_2W^2vc_β-α^(4B_0[q^2;Z,Z]-2)-g^4/8vc_β-α^3 C_VSV^ϕϕϕ[W,h,W] -g^4/8vs_β-α^2c_β-α^ C_VSV^ϕϕϕ[W,H,W] -g^4/8vc_β-α^ C_VSV^ϕϕϕ[W,A,W]-g_Z^4/8vc_2W^2 c_β-α^ C_VSV^ϕϕϕ[Z,H^+,Z] +g^2/4s_β-α^c_β-α^λ_hH^+ H^-^ C_VSS^ϕϕϕ[W,h,H^-]-g^2/4s_β-α^2λ_HH^+ H^-^ C_VSS^ϕϕϕ[W,H,H^-] -g^2/4c_β-α^2λ_hG^+ H^-^ C_VSS^ϕϕϕ[W,h,G^-]+g^2/4s_β-α^c_β-α^λ_HG^+ H^-^ C_VSS^ϕϕϕ[W,H,G^-] -g^2/4c_β-α^m_H^+^2-m_A^2/v C_VSS^ϕϕϕ[W,A,G^-] +g^2/2c_β-α^2λ_hhH^ C_SVS^ϕϕϕ[h,W,h] -g^2/2s_β-α^c_β-α^λ_hHH^{C_SVS^ϕϕϕ[h,W,H] +C_SVS^ϕϕϕ[H,W,h]}+3g^2/2s_β-α^2λ_HHH^ C_SVS^ϕϕϕ[H,W,H] +g^2/2λ_HAA^ C_SVS^ϕϕϕ[A,W,A] +g_Z^2/4c_2W^2λ_HH^+H^-^ C_SVS^ϕϕϕ[H^-,Z,H^-]+e^2 λ_HH^+H^-C_SVS^ϕϕϕ[H^-,γ,H^-] +g^2/4s_β-α^c_β-α^λ_hH^+ H^-^ C_SSV^ϕϕϕ[H^-,h,W] -g^2/4s_β-α^2λ_HH^+ H^-^ C_SSV^ϕϕϕ[H^-,H,W] -g^2/4c_β-α^2λ_hH^+ G^-^ C_SSV^ϕϕϕ[G^-,h,W] +g^2/4s_β-α^c_β-α^λ_HH^+ G^-^ C_SSV^ϕϕϕ[G^-,H,W] -g^2/4c_β-α^m_H^+^2-m_A^2/v C_SSV^ϕϕϕ[G^-,A,W]+4λ_HH^+H^-^λ_H^+H^-H^+H^-^B_0[q^2;H^+,H^+] +λ_HG^+G^-^λ_H^+H^-G^+G^-^B_0[q^2;G^+,G^+] +4λ_HH^+G^-^λ_H^+H^-H^-G^+^B_0[q^2;H^+,G^+]+λ_hH^+H^-^λ_HhH^+H^-^(B_0[p_1^2;H^+,h] + B_0[p_2^2;H^+,h] )+2λ_HH^+H^-^λ_HHH^+H^-^(B_0[p_1^2;H^+,H] + B_0[p_2^2;H^+,H] )+ 2λ_HH^+G^-^λ_HHG^+H^-^B_0[p_1^2;G^-,H]+ 2λ_HG^+H^-^λ_HHH^+G^-^B_0[p_2^2;G^+,H] + λ_hH^+G^-^λ_HhG^+H^-^B_0[p_1^2;G^-,h]+ λ_hG^+H^-^λ_HhH^+G^-^B_0[p_2^2;G^+,h] + λ_AH^+G^-^λ_HAG^+H^-^B_0[p_1^2;G^-,A]+ λ_AG^+H^-^λ_HAH^+G^-^B_0[p_2^2;G^+,A]+ 2λ_hhH^λ_hhH^+H^-^B_0[q^2;h,h]+ 2λ_hHH^λ_HhH^+H^-^B_0[q^2;h,H]+ 6λ_HHH^λ_HHH^+H^-^B_0[q^2;H,H]+ 2λ_HAA^λ_AAH^+H^-^B_0[q^2;A,A]+ λ_HAG^λ_GAH^+H^-^B_0[q^2;A,G^0] + 2λ_HGG^λ_GGH^+H^-^B_0[q^2;G^0,G^0] -λ_HH^+H^-^(λ_hH^+H^-^2 C_0[H^-,h,H^-] + λ_HH^+H^-^2 C_0[H^-,H,H^-]) -2λ_hhH^λ_hH^+H^-^2 C_0[h,H^+,h] - 6λ_HHH^λ_HH^+H^-^2 C_0[H,H^+,H] -2λ_hHH^λ_hH^+H^-^λ_HH^+H^-^( C_0[h,H^+,H] + C_0[H,H^+,h]) -λ_HH^+G^-^(λ_hH^+H^-^λ_hG^+H^-^ C_0[H^-,h,G^-] + λ_HH^+H^-^λ_HG^+H^-^ C_0[H^-,H,G^-]) -λ_HG^+H^-^(λ_hH^+H^-^λ_hH^+G^-^ C_0[G^-,h,H^-] + λ_HH^+H^-^λ_HH^+G^-^ C_0[G^-,H,H^-]) -2λ_hhH^λ_hH^+G^-^2 C_0[h,G^+,h] - 6λ_HHH^λ_HH^+G^-^2 C_0[H,G^+,H] -2λ_hHH^λ_hH^+G^-^λ_HH^+G^-^( C_0[h,G^+,H] + C_0[H,G^+,h]) - 2λ_HAA^|λ_AH^+G^-|^2 C_0[A,G^+,A]-λ_HG^+G^-^(λ_hH^+G^-^2 C_0[G^-,h,G^-] + λ_HH^+G^-^2 C_0[G^-,H,G^-]) -λ_HG^+G^-^|λ_AH^+G^-|^2 C_0[G^-,A,G^-],where the definition of the coupling factor κ_f^H is given in Ref. <cit.>.B_i and C_i functions represent Passarino-Veltman functions <cit.> with B_i[p_j^2;X,Y] ≡ B_i[p_j^2;m_X, m_Y], C_i[X, Y, Z]≡ C_i[p_1^2,p_2^2,q^2; m_X, m_Y, m_Z].Each combination of C^-functions is defined as, C_SVV^ϕϕϕ[S, V_1, V_2] = [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2- (q+ p_1^) ·(p_1 C_11 + p_2 C_12) + q· p_1 C_0](S, V_1, V_2), C_VSV^ϕϕϕ[V_2,S, V_1] = [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2+ (3p_1^ -p_2) ·(p_1 C_11 + p_2 C_12) + 2p_1· (p_1 - p_2) C_0](V_2, S, V_1), C_VVS ^ϕϕϕ[V_1, V_2, S] = [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2+ (3p_1^ +4p_2) ·(p_1 C_11 + p_2 C_12) + 2q· (q +p_2) C_0](V_1, V_2, S), C_VSS^ϕϕϕ[V, S_1, S_2] = [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2+ (4p_1^ +2p_2) ·(p_1 C_11 + p_2 C_12) + 4q· p_1C_0](V, S_1, S_2),C_SVS^ϕϕϕ[S_2, V, S_1]= [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2+ 2p_2·(p_1 C_11 + p_2 C_12) -p_1· (p_1 +2 p_2) C_0](S_2, V,S_1), C_SSV^ϕϕϕ[S_1, S_2, V] = [p_1^2 C_21^ + p_2^2C_22+2p_1· p_2 C_23 +4C_24 -1/2- 2p_2 ·(p_1 C_11 + p_2 C_12) -q · (p_1 - p_2) C_0](S_1, S_2, V).The first two Eqs. (<ref>) and (<ref>) and the latter two Eq. (<ref>) and (<ref>) are fermion loop contributions and boson loop contributions, respectively.The definition of momentum is given in Sec. <ref>.As mentioned in Sec. <ref>, the non-vanishing contribution from mixing self-energies is included in the electroweak corrections to the decay rate H→ H^+H^-, which is calculated as,Δ_mix(H→ H^+H^-) =-4λ_HH^+G^-^/λ_HH^+H^-^Π̂_H^+G^-(m_H^±^2)/m_H^±^2. §.§ , ,vertices in the IDMWe here give explicit formulae for counterterms and 1PI diagram contributions of the hHH, hAA, hH^+H^- vertices in the IDM. The explicit formula of the counterterm of the hϕϕ (ϕ = H, A, H^±) is given by δΓ_hϕϕ^ = 2δλ_hϕϕ + C_ϕλ_hϕϕ(δ Z_ϕ + 1/2δ Z_h),withδλ_hϕϕ^ = -1/vδ m_ϕ^2 + m_ϕ^2 - μ_2^2/v^2δ v + 1/vδμ_2^2,C_H = C_A = 2,C_H^±^ = 1.δμ_2^2 is determined by MS scheme as well as δ M^2 in the THDM, so that it is given by δμ_2^2 =1/16π^2[-m_h^2/2v^2(m_H^2+m_A^2+2m_H^±^2)+μ_2^2/v^2(2m_h^2-6m_W^2-3m_Z^2+3λ_2^v^2)]Δ_div.The 1PI diagram contributions to the hΦΦ vertex are calculated as(16π^2)Γ_hHH, B^1PI[q^2,p_1^2,p_2^2] = 12λ_hhhλ_hhHHB_0[q^2;h,h] + 24λ_hHHλ_HHHH B_0[q^2;H,H] +4λ_hAAλ_HHAAB_0[q^2;A,A] + 4λ_hGGλ_HHGGB_0[q^2;G^0,G^0] + 2λ_hH^+H^-λ_HHH^+H^-B_0[q^2;H^±,H^±] + 2λ_hG^+G^-λ_HHG^+G^-B_0[q^2;G^±,G^±]+8λ_hHHλ_hhHH(B_0[p_1^2;h,H] + B_0[p_2^2;h,H]) +λ_HAGλ_hHAG ( B_0[p_1^2;G^0,A] + B_0[p_2^2;G^0,A] )+2 λ_HH^+G^-λ_hHH^-G^+(B_0[p_1^2;G^±,H^±] + B_0[p_2^2;G^±,H^±])+ 2g^3 m_W(B_0[q^2;W,W] -1/2) + g_Z^3 m_Z(B_0[q^2;Z,Z] -1/2)-g_Z^4/8v C_VSV^ϕϕϕ[Z,A,Z] -g^4/4v C_VSV^ϕϕϕ[W,H^±,W] + g_Z^2/2λ_hAA C_SVS^ϕϕϕ[A,Z,A] + g^2/2λ_hH^+H^-C_SVS^ϕϕϕ[H^±,W,H^±]+ g_Z^2/4λ_HAG(C_VSS^ϕϕϕ[Z,A,G^0] + C_SSV^ϕϕϕ[G^0,A,Z]) + g^2/2λ_HH^+G^-(C_VSS^ϕϕϕ[W,H^±,G^±] + C_SSV^ϕϕϕ[G^±,H^±,W])-24λ_hhhλ_hHH^2 C_0[h,H,h] -8λ_hHH^3 C_0[H,h,H] -2λ_hAAλ_HAG^2C_0[A,G^0,A] -2λ_hGGλ_HAG^2C_0[G^0,A,G^0]-2λ_hH^+H^-|λ_HH^+G^-|^2C_0[H^±,G^±,H^±] - 2λ_hG^+G^-|λ_HH^+G^-|^2C_0[G^±,H^±,G^±], (16π^2)Γ_hAA^1PI,B[q^2,p_1^2,p_2^2] = 24λ_hAAλ_AAAAB_0[q^2;A,A] + 8λ_hAAλ_hhAA(B_0[p_1^2;,h,A] + B_0[p_2^2;h,A]) + 4λ_hHHλ_HHAAB_0[q^2;H,H] + 12λ_hhhλ_hhAAB_0[q^2;h,h] + 4λ_hGGλ_AAGG B_0[q^2;G^0,G^0] +λ_HAGλ_hHAG(B_0[p_1^2;H,G^0] +B_0[p_2^2;H,G^0])+2λ_hG^+G^-λ_AAG^+G^-B_0[q^2;G^±,G^±] + 2λ_AH^-G^+λ_hAH^+G^-(B_0[p_1^2;H^±,G^±] +B_0[p_2^2;H^±,G^±])+ 2λ_hH^+H^-λ_AAH^+H^- B_0[q^2;H^±,H^±]+2g^3 m_W(B_0[q^2;W,W] - 1/2) +g_Z^3 m_Z(B_0[q^2;Z,Z] - 1/2)-g_Z^4/8v C_VSV^ϕϕϕ[Z,H,Z]- g^4/4v C_VSV^ϕϕϕ[W,H^±,W] + g_Z^2/2λ_hHHC_SVS^ϕϕϕ[H,Z,H]+ g^2/2λ_hH^+H^-C_SVS^ϕϕϕ[H^±,W.H^±]+ig^2/2λ_AH^+G^-(C_VSS^ϕϕϕ[W,H^±,G^±] + C_SSV^ϕϕϕ[G^±,H^±,W]) -g_Z^2/4λ_HAG(C_VSS^ϕϕϕ[Z,H,G^0] + C_SSV^ϕϕϕ[G^0,H,Z]) -24λ_hhhλ_hAA^2C_0[h,A,h] -2λ_hHHλ_HAG^2C_0[H,G^0,H] -8λ_hAA^3C_0[A,h,A]-2λ_hGGλ_HAG^2C_0[G^0,H,G^0] -2λ_hH^+H^-λ_AH^+G^-λ_AH^-G^+C_0[H^±,G^±,H^±] -2λ_hG^+G^-λ_AH^+G^-λ_AH^-G^+C_0[G^±,H^±,G^±],(16π^2)Γ_hH^+H^-^1PI,B[q^2,p_1^2,p_2^2] = 6λ_hhhλ_hhH^+H^- B_0[q^2;h,h] + 2λ_hH^+H^-λ_hhH^+H^- (B_0[p_1^2;h,H^±] +B_0[p_2^2;h,H^±]) + 2λ_hHHλ_HHH^+H^- B_0[q^2;H,H] + λ_HG^+H^-λ_hHH^+G^- (B_0[p_1^2;H,G^±] +B_0[p_2^2;H,G^±]) + 2λ_hAAλ_AAH^+H^- B_0[q^2;A,A] + λ_AG^+H^-λ_hAH^+G^- (B_0[p_1^2;A,G^±] +B_0[p_2^2;A,G^±]) + 2λ_hG^0G^0λ_G^0G^0H^+H^- B_0[q^2;G^0,G^0] + 4λ_hH^+H^-λ_H^+H^-H^+H^- B_0[q^2;H^±,H^±]) + λ_hG^+G^-λ_H^+H^-G^+G^- B_0[q^2;G^±,G^±])+2g^3 m_W(B_0[q^2;W,W] -1/2) + c_2W^2g_Z^3 m_Z(B_0[q^2;Z,Z] -1/2)-g^4v/8C_VSV^ϕϕϕ[W,H,W] - g^4v/8C_VSV^ϕϕϕ[W,A,W] - g_Z^4v/8c_2W^2C_VSV^ϕϕϕ[Z,H^+,Z]+g^2/2λ_hHHC_SVS^ϕϕϕ[H,W,H] + g^2/2λ_hAAC_SVS^ϕϕϕ[A,W,A] +g_Z^2/4c_2W^2λ_hH^+H^-C_SVS^ϕϕϕ[H^+,Z,H^+]+ e^2 λ_hH^+H^- C_VSV^ϕϕϕ[H^+,γ,H^+]- g^2/4λ_HH^-G^+(C_VSS^ϕϕϕ[W,H,G^+] + C_SSV^ϕϕϕ[G^+,H,W])+ig^2/4λ_AH^-G^+^(C_VSS^ϕϕϕ[W,A,G^+] +C_SSV^ϕϕϕ[G^+,A,W])-6λ_hhhλ_hH^+H^-^2C_0[h,H^+,h] -2λ_hHHλ_HH^+G^-λ_HH^-G^+C_0[H,G^+,H] -2λ_hAAλ_AH^+G^-λ_AH^-G^+C_0[A,G^+,A]- λ_hH^+H^-^3 C_0[H^+,h,H^+] -λ_hG^+G^-λ_HH^+G^-λ_HH^-G^+C_0[G^+,H,G^+] -λ_hG^+G^-λ_AH^+G^-λ_AH^-G^+C_0[G^+,H,H^+]. § Φ→ H^+H^- ΓWe here give the explicit formula of the real photon emission process for ϕ→ H^+H^-, which cancels IR divergence of the virtual corrections for the ϕ H^+H^- vertex.The formula of the decay rate can be expressed asΓ[ϕ→ H^+H^-γ]=e^2λ_ϕ H^+H^-^2/16π^3m_ϕ^{ -I_1 - I_2 -m_H^±^2I_11^ -m_H^±^2I_22^-(2m_H^±^2 -m_ϕ^2)I_12^},where definitions of I-functions are given in Appendix D of Ref. <cit.>.apsrev4-1 | http://arxiv.org/abs/2311.15892v1 | {
"authors": [
"Masashi Aiko",
"Shinya Kanemura",
"Mariko Kikuchi",
"Kodai Sakurai",
"Kei Yagyu"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20231127145910",
"title": "H-COUP Version 3: A program for one-loop corrected decays of any Higgs bosons in non-minimal Higgs models"
} |
Spatio-temporal insights for wind energy harvesting in South Africa Matthew de Bie^1, Janet van Niekerk^1,2, Andriëtte Bekker^1 ^1Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, South Africa ^2Statistics program, CEMSE Division, King Abdullah University of Science and Technology, Kingdom of Saudi [email protected]=========================================================================================================================================================================================================================================================================================================================================== Understanding complex spatial dependency structures is a crucial consideration when attempting to build a modeling framework for wind speeds. Ideally, wind speed modeling should be very efficient since the wind speed can vary significantly from day to day or even hour to hour. But complex models usually require high computational resources. This paper illustrates how to construct and implement a hierarchical Bayesian model for wind speeds using the Weibull density function based on a continuously-indexed spatial field. For efficient (near real-time) inference the proposed model is implemented in the r package R-INLA, based on the integrated nested Laplace approximation (INLA). Specific attention is given to the theoretical and practical considerations of including a spatial component within a Bayesian hierarchical model. The proposed model is then applied and evaluated using a large volume of real data sourced from the coastal regions of South Africa between 2011 and 2021. By projecting the mean and standard deviation of the Matérn field, the results show that the spatial modeling component is effectively capturing variation in wind speeds which cannot be explained by the other model components. The mean of the spatial field varies between ± 0.3 across the domain. These insights are valuable for planning and implementation of green energy resources such as wind farms in South Africa. Furthermore, shortcomings in the spatial sampling domain is evident in the analysis and this is important for future sampling strategies. The proposed model, and the conglomerated dataset, can serve as a foundational framework for future investigations into wind energy in South Africa. § INTRODUCTIONSouth Africa is facing an electricity crisis which is severely damaging the country's economic output <cit.>. Ageing infrastructure and a seeming inability to bring new power stations online mean that South Africa has never been more vulnerable to electricity shortages. In a global environment in which the popularity of fossil fuels is waning and positive climate action is viewed as an imperative, South Africa must takes steps towards the implementation of renewable energy initiatives. Harnessing wind energy could provide a solution to South Africa's ongoing energy shortfalls. However, harnessing wind energy is a complex task which requires a nuanced understanding of numerous factors. It is imperative that any analysis aimed at enabling the harnessing of wind energy is grounded in a robust statistical framework. Wind energy harvesting in South Africa is becoming a field of increasing interest. Investigations into potentially forecasting wind speeds using machine learning techniqueshave been undertaken by <cit.> and <cit.>. Additionally, work by <cit.> and <cit.> into a connection between wind energy and hydrogen production, as well as the analysis by <cit.> into the viability of using next-generation turbines for generation purposes, indicate that wind energy harvesting is a popular field of research. In light of this popularity, it is unsurprising that efforts have been undertaken by <cit.> to examine some of the spatial considerations associated with harnessing renewable energy resources. However, to our knowledge, this paper is the first in South Africa to examine real wind speed data using a hierarchical Bayesian model which includes a spatial modeling component. It is recognized by <cit.> that the Weibull distribution, the functional form of which is shown in equation (<ref>), can serve as a suitable likelihood function for modeling wind speeds.f(y|α,λ) = α/λ(y/λ)^α-1exp(-(y/λ))^α,where λ is the scale parameter and α is the shape parameter. We use the Weibull density function given in equation (<ref>) as the foundation on which to construct a statistical regression model for wind speeds. The core assumptions when working with the Weibull distribution is that λ is a function of the covariates, and α is constant <cit.>. By assuming that λ is a function of the predictor variables, the Weibull distribution lends itself towards the construction of a hierarchical regression model. Our ultimate objective is to include a well defined spatial modeling component within the linear predictor of a hierarchical model in order to gain an understanding of the spatial dependency structures which influence wind speeds. To accomplish this we first describe, in section <ref>, how the data relevant to this analysis was collected and cleaned. Thereafter, in section <ref>, we outline the structure of the hierarchical Bayesian model we will fit to the data. This section will also investigate the appropriate priors we will fit to our various modeling components. This will be followed by a discussion, in section <ref>, surrounding how to include a spatial component within the hierarchical modeling structure. Thereafter, section <ref> will showcase some of the key results obtained from the model and this will be followed by a short discussion in section <ref>. § DATA COLLECTION AND CONGLOMERATIONTo enable our analysis, we constructed a large dataset using raw data sourced from the wind atlas South Africa (WASA) database. Specifically, all raw data was collected from measuring sites in the coastal regions of South Africa between 2011 and 2021. The details of the locations where the raw data was collected are shown in <ref> and <ref>. Between 2011 and 2021, the wind speed was recorded every five minutes, at five different altitude levels, and at ten different locations. This collection process resulted in a large amount of disparate data which needed to be organised into a single coherent dataset which is capable of supporting our analysis. After the conglomeration process was completed, the final dataset contained over 25 000 000 observations, hereafter we shall refer to this dataset as the prime dataset. Each observation in the prime dataset was identified by the variables shown in <ref>. It must be noted that two sites, WM04 and WM10 suffered malfunctions in the mid-2010s which meant that they were unable to record data. Having constructed a usable dataset, we now need to investigate the hierarchical Bayesian framework we will utilise to fit a model to this data.§ A HIERARCHICAL BAYESIAN MODELING FRAMEWORKIn order to fully understand a hierarchical Bayesian regression model, it is useful to break the term down into its component words.The term hierarchy indicates that we are constructing a model which has multiple levels. Furthermore the word Bayesian denotes that we are working with priors to estimate posterior density functions. Finally, the word regression indicates that we are building a model which aims to explain the relationship between a matrix of covariates 𝐗 and a response variable, 𝐲. Therefore, we understand that we are working with a multi-level model where we attach priors to the various components at each level of the model. Using these priors, we will obtain a posterior density function for each parameter. With this in mind, we can now specify the manner in which each level of this model will function. With reference to <ref>, we designate 'wind_speed' as the response variable, 𝐲. Additionally, we assume that there exists some form of relationship between the response and the other variables listed in table <ref>. We designate these variables as our matrix of covariates, which we label as 𝐗. Alongside standard fixed effect, our model will also make use of random effects in the latent field. Notationally, 𝐙 acts as a design matrix which allows a single observation to depend on multiple random effects and 𝐮 serves as the vector which contains the random effects. The final aspect to be aware of before we outline the modeling framework is that the unknown prior density function for hyperparameters, like the shape parameter α, which have an unknown functional form will be written as α∼ H(...). §.§ General form of a hierarchical Bayesian modelThe general form of a Bayesian hierarchical modeling framework, constructed with reference to the Weibull density function , is shown below. * Likelihood function: 𝐲∼Weibull(λ, α) * Link function to the mean: λ= exp(η) * Linear predictor η = 𝐗β +𝐙𝐮 * Hyperparameter: α∼ H(...). It is within η that we will specify the regression relationship between the covariates 𝐗 and the response variable, 𝐲. This structural form also allow us to define random effects within the linear predictor. Next, we need to identify a suitable prior for α. §.§ Identifying a suitable prior for αIt was recognised by <cit.> that popular existing priors for the shape parameter of the Weibull distribution, such as the improper uniform prior and the gamma prior, are often unsuitable when performing complex modeling. With this in mind, the work of <cit.> was used as a basis by <cit.> to derive a penalised complexity (PC) prior for the shape parameter of the Weibull distribution. Using the Kullback-Liebler divergence, <cit.> gives a full definition of the PC prior for the Weibull shape parameter as formulated in equation (<ref>).π(α) = θexp[-θ√(2KLD(α))] |∂√(2KLD(α))/∂α| = θexp[-θ√(2KLD(α))] ×1/2(2KLD(α))^-1/2×|2/α^2(γ-α(1+γ)) +Γ(α^-1+αlog(α)) . + .2/α(-(1+γ)-Γ(α^-1)ψ (α^-1)/α^2+log(α)+1)|,where ψ(z) is the digamma function as defined by <cit.>. In a practical sense, the PC prior allows us to attach generally applicable priori information to α. In other words, the PC prior is a useful default prior for α in the absence of other strong prior information. §.§ Specifying a model within the hierarchical frameworkWe can now specify our mixed effect regression model using the general structural form defined in <ref>.Y∼ Weibull(λ, α)λ = exp(η)η = β_0 + β_𝚌𝚘𝚜𝐗_𝐜𝐨𝐬_𝐝𝐢𝐫𝐞𝐜𝐭 + β_𝚜𝚒𝚗𝐗_𝐬𝐢𝐧_𝐝𝐢𝐫𝐞𝐜𝐭+ g(𝐗_𝐀𝐥𝐭𝐢𝐭𝐮𝐝𝐞) +v(𝐗_𝐟_𝐦𝐨𝐧𝐭𝐡) + q(𝐗_𝐜_𝐦𝐨𝐧𝐭𝐡) + 𝐮_𝐬 α ∼ PC prior ,where, g(𝐗_𝐀𝐥𝐭𝐢𝐭𝐮𝐝𝐞), denotes that we use a spline component, of order random walk 2, in order to capture the variation in wind speeds which arises from the different altitudes at which the wind speed was measured. Additionally, v(𝐗_𝐟_𝐦𝐨𝐧𝐭𝐡) and q(𝐗_𝐜_𝐦𝐨𝐧𝐭𝐡) serve as the temporal effects within the model. Both v(𝐗_𝐟_𝐦𝐨𝐧𝐭𝐡) and q(𝐗_𝐜_𝐦𝐨𝐧𝐭𝐡) are auto-regressive (AR)1 random effects. Finally, two fixed effects, namely β_𝚌𝚘𝚜𝐗_𝐜𝐨𝐬_𝐝𝐢𝐫𝐞𝐜𝐭 and β_𝚜𝚒𝚗𝐗_𝐬𝐢𝐧_𝐝𝐢𝐫𝐞𝐜𝐭 are used to capture the impact of the direction on the wind speed. In addition, we have defined two temporal effects, namely v(𝐗_𝐟_𝐦𝐨𝐧𝐭𝐡) and q(𝐗_𝐜_𝐦𝐨𝐧𝐭𝐡). We refer to the list of variables provided in <ref> to provide reasoning as to why we included two temporal effects in our model. Within <ref> there are two variables which are temporal in nature 'f_month' and 'c_month'. The first, 'f_month' serves as a repeating temporal component which is intended to capture cyclical temporal dependency which occurs year-on-year. In other words, every year the wind speed in February will depend on the wind speed in January. This latent variation could be caused by repeating seasonal patterns. Therefore, we use component v(𝐗_𝐟_𝐦𝐨𝐧𝐭𝐡) in order to capture a dependency structure which will repeat each year.Conversely, the wind speed in January of a given year, also has a relationship with the wind speed in December of the previous year. To capture temporal relationships across each of the ten years, we define component q(𝐗_𝐜_𝐦𝐨𝐧𝐭𝐡). The last component specified in the latent field is defined simply as 𝐮_𝐬 which serves as the spatial component of the model.§ USING A MATÉRN FIELD TO CAPTURE SPATIAL VARIATION IN WIND SPEEDSThough it is not visible in equation (<ref>), within 𝐮_𝐬 are a number of estimable parameters which relate to the Matérn correlation structure which governs the manner in which we conceptualise dependency between two points. Using the work of <cit.> we can define 𝐮_𝐬_𝐢 where i=1,2,...,n as a realisation of the random spatial effect at n locations within the domain. Assuming that 𝐮_𝐬 has a multivariate Gaussian distribution and that 𝐮_𝐬 is continuous across space, we can then conceptualise 𝐮_𝐬 as a continuously-indexed Gaussian field (GF). It was suggested by <cit.> that it is theoretically possible to approximate a GF with a Gauss-Markov random field (GMRF). From this hypothesis, <cit.> defined a practical method whereby a stochastic partial differential equation (SPDE), the solution to which is a GF with Matérn correlation, could be used to create a GMRF approximation of a GF. Before we move forward, it is useful to briefly review the Matérn covariance function. The Matérn covariance function can be used to define the covariance between two points separated by some form of spatial distance. According to <cit.> a standard Matérn covariance function can be written as Cor_M(u_s_i,u_s_j) = 2^1-ν/γ(ν) (κ||s_i-s_j||)^νK_ν(κ||s_i-s_j||),where ||...|| denotes the Euclidean distance between s_i and s_j, κ is the scale parameter, v is the smoothness parameter, and K_v is the modified Bessel function. The logic which underpins equation (<ref>) states that the degree of dependence between two points u_s_i,u_s_j is determined by the Euclidean distance between the points and a number of estimable parameters. According to the work of <cit.> there are many situations in which we assume that we are working with an underlying GF but cannot directly observe the GF. Instead we observe data with a measurement error. This can be expressed this mathematically as given in equation (<ref>).k_i =x_i + e_i wheree_i ∼N(0, σ^2_e) ,where i for i=1,2,...,n is the location at which we observe data point k_i, and σ^2_e measures the noise of the process, otherwise known as the 'nugget' effect. An aspect of the Matérn field which is not immediately visible from equation (<ref>) is the range of the correlation effect, often referred to as the nominal range. The nominal range is defined asthe distance at which the correlation, given in equation (<ref>), is equal to 0.05 <cit.>. The true value of the nominal range can be determined by using the formula show in equation (<ref>) <cit.>.nominal range = √(8 ν)/κ,where ν is drawn from equation (<ref>). With all of the elements of the Matérn field now defined, we can move on to examining how <cit.> created a GMRF approximation of this specific GF. The study by <cit.> found two crucial results. The first of these critical results is that a GF 𝐮_𝐬 with Matérn covariance is a solution to the linear fractional SPDE shown in equation <ref>.(k^2-δ)^Υ /2𝐮_𝐬 = 𝐖(𝐬),where δ denotes what is known as theLaplacian operator, 𝐖(𝐬) is spatial random Gaussian white noise, and Υ governs the smoothness parameter of the process. The second result found by <cit.> was achieved using the finite element method (FEM) and provides a solution for points distributed on an irregular grid. When we typically assess spatial dependence, most points are not located at regular grid intervals but are rather distributed irregularly across the domain. It was found by <cit.> that the domain can be approximated using non-intersecting triangles. In statistical terms, this irregular triangular grid is known as a constrained refined Delaunay triangulation (CRDT). The result of approximating the domain using the CRDT is a mechanism for spatial analysis called a mesh. § MODEL IMPLEMENTATION AND RESULTS§.§ Data preparationConsidering that the data was effectively only recorded at ten locations, we manipulated the data in order to induce a degree of spatial diversity into the dataset. This was achieved by jittering the coordinates in such a manner that every observation technical occupies a unique location. <ref> is used to illustrate how this jittering was performed using a sample of 5000 observations.§.§ Mesh selectionA crucial consideration when implementing a Matérn field, is the selection of an appropriate mesh. However, as noted by <cit.>, at present there is no firm set of criteria which can be used to precisely select an ideal mesh. In a practical sense, it is possible to identify a suitable as the mesh at which estimated posterior density functions associated with the Matérn field converge. It may be tempting to bypass this selection process by simply using a very dense mesh in order to perform modeling. Considering that we are working with a GMRF approximation of a GF the decision to use a very dense mesh may, at first instance, seem logical. However, from a practical standpoint, the accuracy of estimated model parameters may be seriously impaired due to the computational instability which can arise when using a very fine mesh. The process of mesh selection involves performing a number of redundant calculations. This is because weneed to fit our model using a given mesh and then analyse the results obtained from said mesh in order to determine which mesh is most suitable. Therefore, for purposes of computational efficiency, the process of mesh selection was performed using a sample of 5000 observations drawn from the prime dataset. The meshes which were generated for purposes of evaluation are shown in <ref>.In <ref> we provide the parameter values which were used along with the inla.mesh.2d() function in order to generate each of the meshes shown in <ref>. Thereafter the estimated point estimates for the Matérn parameters are shown in <ref>. Mesh-E appears to be the mesh at which convergence occurs for most if not all of the parameters in the Matérn field. Additionally, Mesh-E required under fifty seconds in order to be processed within our modeling framework and contains over two thousand vertices. Conversely, the second most suitable mesh, Mesh-D, contains less than one thousand-five hundred vertices and could be too sparse for purposes of parameter inference. For our purposes, we would identify Mesh-E as the optimal mesh to use in further stages of this reproducible example. However, if the specific properties of Mesh-E are not in line with those desired by the user then Mesh-D could be used as an alternative suitable spatial mesh. §.§ Key model resultsUsing the integrated nested Laplace approximation(INLA) we fit the model defined in equation (<ref>) to a sample of 1 000 000 observations drawn from the prime dataset. Utilising an approximate Bayesian method like INLA to fit the model is far more computationally efficient than the computational burden which would be incurred when fitting the model using a Markov-chain Monte Carlo (MCMC) method. In <ref> and <ref> we report the posterior density functions and the point estimates for the parameters which govern the spatial field. However, when viewed from an abstract perspective, the estimates in <ref> are difficult to interpret. With this in mind, we project the mean and standard deviation of the spatial field onto a map of South Africa and visualise these results using <ref> and <ref> respectively. A comparison of the results in <ref> and <ref> will reveal substantial differences in the point estimates of the parameters of the Matérn field. This is due to the fact that the results in <ref> were obtained using a greater volume of data, 1000000 obervations as oppose to 5000 observations. Additionally, it must be noted that, while the sample of 5000 observations is useful for preliminary and exploratory analysis, the reduced volume incurred when working with only 5000 observations can lead to inconsistent results. For these reasons, we chose to report the results obtained when fitting the model to a sample which contained a larger volume of data. The projection of the mean of the spatial field shown in <ref> allows us to visualise how wind speed varied across South Africa once all other model covariates were taken into account. The 'red' regions shown in <ref> indicate an area in which the measured wind speeds were greater than the other model effects. Using the same logic we can therefore understand that a 'blue' region denotes an area in which the measured wind speeds were less than what we would believe the wind speed to be based only on the other model effects. An examination of <ref> reveals that the mean of the Matérn field is zero across large parts of the domain. This occurs because of the lack of spatial diversity in the underlying data. Strictly speaking, there are only ten locations where recordings of wind speeds were made and the spatial modeling component cannot measure the relationship between wind speeds and location in areas where no data exists.<ref> provides further insight into the manner in which our spatial modeling component performed by projecting the standard deviation of the spatial field over the map of South Africa. Visually, we note from <ref> that the standard deviation is low in the South-West of the country but increases as we move East and North. This occurs because the majority of the measuring sites are located in the South-West of South Africa. Practically, this means that we are more certain of the predicted mean of our spatial field in the South-West regions and become less certain of the predicted mean as we move away from these regions. A closer assessment of <ref> reveals that the spatial variation in wind speeds is fairly minor and oscillates between -0.3;0.3. What this demonstrates is that most of the variability in wind speeds is effectively captured by the other covariates in our model. However, the fact that there is any variability at all indicates that the spatial component is effectively capturing latent variation which is not accounted for by the other components in the model. § DISCUSSION AND CONCLUSIONIn this paper, we constructed a hierarchical Bayesian model and fitted this model to wind speed data sourced from the coastal regions of South Africa. The spatial variation in wind speeds was captured using a Matérn field. Using the INLA method, the results of the model, fitted to a sample of 1000000 observations, were projected over the domain of South Africa. A simple logical assumption proposes that geographical conditions affect the speed of the wind. However, what the results in <ref> demonstrate is that we can use the spatial component included in the model shown in equation (<ref>) in order to capture the dependency structure between location and wind speed which cannot be accounted for by the other components in the model. At present, the majority of the data is located in the South-West of South Africa. Thus, we are most certain of the mean of the spatial field in this region. In future, the application of the model shown in equation (<ref>) to more spatially diverse data would allow for a more effective projection of the spatial field over the the entire domain. However, the ability of the model to capture spatial variation in wind speeds validates the inclusion of a spatial component into our model. This research was conducted with reference to broader socio-economic objectives which align with the sustainable development goals (SDGs) set out by the United Nations. In particular, our findings contribute toward the achievement of SDG-7, affordable and clean energy, and SDG-13, climate action. By enabling the more widespread adoption of wind energy for power generation, we can reduce South Africa's reliance on environmentally harmful, coal driven power stations. By supplanting fossil fuel based electricity generation with renewable energy, we can ensure that renewable electricity is equitably accessible to all citizens of South Africa. §.§ Credit author statementMatthew De Bie: performed writing original draft preparation, data conglomeration and result visualisations. Janet Van Niekerk: performed model fitting application and project conceptualisation. Andriette Bekker performed supervision and validation. §.§ Declaration of competing interestsThe 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.§.§ AcknowledgementsThis paper was made possible by the funding provided by the South African Department of Science and Innovation (DSI), the National Research Foundation (NRF Ref. SRUG2204203865 and Ref: RA171022270376 Grant No: 119109), and the Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) .The opinions expressed and conclusions arrived at during this paper are those of the authors and are not necessarily to be attributed to the DSI-NRF or CoE-MaSS.§.§ Dataset availabilityThe code and raw data used to conduct this analysis is available on GitHub at: <> All data and relevant code related to this article can be found at <http://surl.li/hqwrl> , an open-source online data repository hosted at Github <cit.>. apalike | http://arxiv.org/abs/2311.15715v1 | {
"authors": [
"Matthew de Bie",
"Janet van Niekerk",
"Andriette Bekker"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20231127105848",
"title": "Spatio-temporal insights for wind energy harvesting in South Africa"
} |
Cosmology and fundamental physics with ANDES]Cosmology and fundamental physics with the ELT-ANDES spectrograph[1,2]C.J.A.P. [email protected] 3]R. Cooke 4]J. Liske 5]M.T. Murphy 6,7]P. Noterdaeme 8]T.M. Schmidt 9]J.S. Alcaniz 1,10]C.S. Alves 11]S. Balashev 12,13,14]S. Cristiani 12]P. Di Marcantonio 15,9]R.S. Gonçalves 16]R. Maiolino 17]A. Marconi 1,2,18]C.M.J. Marques 1,18]M.A.F. Melo e Sousa 19]N.J. Nunes 20]L. Origlia 21,22]C. Péroux 23]A. Zanutta [1]Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal [2]Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal [3]Centre for Extragalactic Astronomy, Durham University, Science Site, South Road, DH1 3LE, Durham, UK [4]Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany [5]Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia [6]Institut d'Astrophysique de Paris, UMR 7095, CNRS-SU, 98bis bd Arago, 75014 Paris, France [7]French-Chilean Laboratory for Astronomy, IRL 3386, CNRS and U. de Chile, Casilla 36-D, Santiago, Chile [8]Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Sauverny, CH-1290, Switzerland [9]Observatório Nacional, 20921-400, Rio de Janeiro, RJ, Brazil [10]Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom [11]Ioffe Institute, Polyteknicheskaya 26, 194021 Saint-Petersburg, Russia [12]INAF–Osservatorio Astronomico di Trieste, Via G.B. Tiepolo, 11, I-34143 Trieste, Italy [13]IFPU–Institute for Fundamental Physics of the Universe, via Beirut 2, I-34151 Trieste, Italy [14]INFN-National Institute for Nuclear Physics, via Valerio 2, I-34127 Trieste [15]Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23897-000, Seropédica, RJ, Brazil [16]Cavendish Laboratory, University of Cambridge, 19 J.J. Thomson Ave., Cambridge CB3 0HE, UK [17]Dipartimento di Fisica e Astronomia, Universitaà degli Studi di Firenze, Via G. Sansone 1, I-50019, Sesto Fiorentino, Firenze, Italy [18]Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4150-007 Porto, Portugal [19]Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa [20]INAF–Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, I-40129 Bologna, Italy[21]European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching-bei-München, Germany [22]Aix Marseille Université, CNRS, LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326, 13388, Marseille, France [23]INAF–Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate, ItalyState-of-the-art 19th century spectroscopy led to the discovery of quantum mechanics, and 20th century spectroscopy led to the confirmation of quantum electrodynamics. State-of-the-art 21st century astrophysical spectrographs, especially ANDES at ESO's ELT, have another opportunity to play a key role in the search for, and characterization of, the new physics which is known to be out there, waiting to be discovered. We rely on detailed simulations and forecast techniques to discuss four important examples of this point: big bang nucleosynthesis, the evolution of the cosmic microwave background temperature, tests of the universality of physical laws, and a real-time model-independent mapping of the expansion history of the universe (also known as the redshift drift). The last two are among the flagship science drivers for the ELT. We also highlight what is required for the ESO community to be able to play a meaningful role in 2030s fundamental cosmology and show that, even if ANDES only provides null results, such `minimum guaranteed science' will be in the form of constraints on key cosmological paradigms: these are independent from, and can be competitive with, those obtained from traditional cosmological probes. [ [ January 14, 2024 ====================§ INTRODUCTION State-of-the-art spectroscopy has been crucial to the development of our contemporary fundamental physics paradigm. It has been a key motivation for the development of quantum mechanics (cf. the discrete nature of spectral lines, and the photoelectric effect), and led to one of the first experimental confirmations of quantum electrodynamics (through the Lamb shift). More recently, several Nobel Prizes in Physics have been awarded for high-precision laser physics, e.g. in 1981, 1997, 2005 and 2018.In parallel, the 20th century also saw the development of a cosmological paradigm, commonly known as the Hot Big Bang model. Its many observational successes are compounded by the fact that about 95 percent of the Universe's contents are in `dark' forms, so far not directly detected in laboratory experiments: our only current evidence for them is indirect, and coming from astrophysical and cosmological observations, subject to various seemingly plausible but possibly unjustified theoretical modeling assumptions. Indeed, the observational evidence for the low redshift acceleration of the universe, accumulated over the last two decades, shows that the canonical cosmological paradigm is, at least, incomplete. It follows that the Hot Big Bang model is at best a simple (though certainly convenient) approximation to the behaviour of a more fundamental cosmological paradigm, yet to be discovered. The issue can be summarized by considering the question of whether or not the acceleration of the universe is due to a cosmological constant. This was first introduced by Einstein (in a different historical context) as a mathematical integration constant, but is known to be physically equivalent to a vacuum energy density, as can be obtained in modern quantum field theories. If the acceleration is due to a cosmological constant, the observed value is ca. 120 orders of magnitude smaller than the one calculated by particle physicists – with the obvious implication that such calculations must be incorrect. If it is not, then one needs alternative acceleration mechanisms. Such alternatives exist, the most obvious possibility being cosmological scalar fields, but these will generically violate the Einstein Equivalence Principle. Thus, the evidence for the accelerating universe indicates that there is new physics waiting to be discovered – either in the quantum field theory sector or the gravitational sector, or possibly in both.Ongoing developments in the precision, accuracy and stability of astrophysical spectrographs enable them, once again, to play a role in cutting edge fundamental physics, by contributing to the search, identification and characterization of this new physics. This paper discusses the role of ANDES in this quest, focusing on its main science cases (which generally rely on QSO absorption spectroscopy) and emphasizes the requirements for it to play a competitive role in the foreseeable 2030s context (with the U-band coverage being of critical importance), but also discussing synergies with other contemporary facilities. Unless otherwise stated, our analysis relies on version 1.1 (July 2023) of the ANDES Exposure Time Calculator[Available at https://hires.inaf.it/etc.htmlhttps://hires.inaf.it/etc.html].ANDES will have the almost unique ability to probe the behaviour of all the fundamental ingredients of the universe: baryons through Big Bang Nucleosynthesis (BBN), radiation (photons) though the temperature of the comic microwave background (CMB), and the dark sector(s), including new dynamical degrees of freedom such as fundamental scalar fields, through tests of the universality of physical laws, probing the stability of nature's fundamental couplings and searching for composition-dependent forces. All these will be complemented by a real-time model-independent mapping of the expansion history of the universe – a.k.a. the redshift drift. Last but not least, one must keep in mind the point that our flagship science cases are photon starved with current facilities, but they will only fully benefit from the ELT's larger collecting area if they are not systematics limited. More specifically, how competitive ANDES will be as a fundamental physics probe will crucially depend on the robustness of its calibration procedures, For this reason, we also briefly discuss the main steps in these procedures and the corresponding requirements.§ BIG BANG NUCLEOSYNTHESIS The light nuclei that were created a few minutes after the Big Bang (a process known as Big Bang Nucleosynthesis – BBN) currently provide us with the earliest probe to study the physics of the early Universe <cit.>. The relative abundances of these nuclei are sensitive to the density of ordinary matter (i.e. the baryon density), the expansion rate of the Universe, and the particle content. Therefore, deviations from the Standard Model of particle physics and cosmology can be identified by measuring the relative abundances of the primordial elements, and comparing these abundances to theoretical BBN calculations. There are just five nuclei that are abundantly produced during BBN, including hydrogen (H), deuterium (D), helium-3 (^3He), helium-4 (^4He), and lithium-7 (^7Li). The ANDES design is ideally suited to measuring the primordial abundances of D/H, ^3He/^4He, and ^7Li/H, which all require a high spectral resolution instrument coupled to a large aperture telescope. In order to determine the primordial values of these ratios, we need to discover and observe environments that still retain a primordial composition of the light elements. Furthermore, it is critical to measure the abundances of as many primordial elements as possible; the relative abundances of each element have a different sensitivity to the baryon density, the expansion rate, and the particle content of the Universe. Thus, combining multiple primordial measures allows us to: (1) test the consistency of the cosmological model and the Standard Model of particle physics; and (2) if a departure from the Standard Model is uncovered, multiple primordial abundance measures have the power to unveil the identity of the new physics. ANDES will deliver the possibility to revolutionise at least three of the primordial element abundance measures. §.§ The primordial deuterium abundance — D/H D/H currently provides our most reliable test of BBN. The standard approach to measure D/H uses near-pristine quasar absorption line systems, typically observed at redshift z∼3, where the lines of interest are shifted into the optical wavelength range (i.e. an observed wavelength range of 3600Å – 5000Åfor an absorption line system at z=3). In order to accurately model the absorption cloud kinematics, a high resolution spectrograph (such as ANDES) is essential. Based on a sample of just seven systems, the primordial deuterium abundance is known to one percent precision <cit.>, and this sample is not currently limited by systematic uncertainties.The main challenge of determining the primordial D/H ratio is simply a numbers game. The best systems are: (1) typically those with a higher H i column density, log N(H i)/cm^-2≳19 (i.e. the rarest quasar absorption line systems); and (2) located in the metal-poor tail of the metallicity distribution (making them exceedingly rare!). Combined with this, there are relatively fewer bright quasars than faint ones, so a requisite S/N∼20 is difficult to achieve, particularly at blue optical wavelengths. With the large collecting area of the ELT (>10× that of the VLT), the community will have access to ∼100× the number of quasars that are currently possible to observe with the VLT. In concert, this will greatly expand the statistics of deuterium absorption line systems.With the baseline design of ELT-ANDES, <cit.> estimate that the statistics of D/H will be improved by a factor of ∼2. These measures will be restricted to somewhat higher redshift D/H absorption systems, and these may suffer from increased blending due to the Lyα forest. If ELT-ANDES includes a U-band spectrograph (covering 3500–4100Å),current estimates suggest that the number of D/H measures will be increased by a factor of >15× <cit.>, and make it possible to determine the primordial deuterium abundance to 0.1-0.2 percent precision (an improvement by a factor of ∼5× better than current measures). Combining this estimate with theoretical calculations of BBN, this measurement precision corresponds to an uncertainty on the effective number of neutrino species Δ N_ eff=0.02, which is competitive with CMB stage 4 experiments. Figure <ref> illustrates the expected improvement to the measurement precision that will be afforded by ELT-ANDES (both the baseline design, as well as the benefit of including a U band extension). §.§ The primordial helium isotope ratio — ^3He/^4He Despite being one of the dominant products of BBN, the primordial ^3He abundance has never been determined. Past attempts to determine the primordial ^3He/H ratio primarily used the 8.7 GHz spin-flip transition of ^3He from Galactic H ii regions <cit.>. More recently, it has been appreciated that a reliable determination of primordial helium-3 production can be obtained by measuring the helium isotope ratio, ^3He/^4He <cit.>, using absorption line techniques.In order to measure the helium isotope ratio using He i* absorption systems in the local universe, there are three key requirements: (1) coverage of the He i* 3889Å and He i* 1.083μm lines; (2) high spectral resolution; and (3) high S/N ratio. The high S/N ratio is required to detect the very weak ^3He i* absorption feature. This is particularly needed for the 1.083μm line, which shows the largest isotope shift (^3He is shifted by +36.6 km/s relative to ^4He). In order to resolve this weak feature from the neighbouring ^4He i* 1.083μm absorption line (which is stronger by a factor of ∼ 10^4), high spectral resolution is required (R > 50,000). Finally, since the corresponding ^4He i* 1.083μm absorption line is very strong, simultaneous coverage of a weaker ^4He i* absorption feature is also needed (rest-frame wavelengths of 3889Å and 3188Å) to reliably pin down the ^4He i* column density. The required S/N near 3889Å (S/N>50 per 2 km/s pixel) is much less than what is required at 1.083μm (S/N>500 per 2 km/s pixel). We note that this measurement would benefit from the U-band extension,but the U-band is not required provided that contemporaneous observations are acquired using a separate U-band spectrograph (e.g. VLT-CUBES).Since the first measurement of ^3He/^4He towards a bright O star in the Orion Nebula <cit.>, more than 40 newly discovered sight-lines have been identified with ^4He i* absorption towards stars in the Milky Way and our metal-poor neighbouring galaxy, the Large Magellanic Cloud – the companion ^3He i* absorption has not yet been detected in any of these 40 sight-lines. The main limiting factor of this measurement is that data of especially high signal-to-noise ratio (S/N>500 per 2 km/s pixel) are required to detect the weak ^3He i* absorption feature. This will be readily achieved with ANDES; for example, the brightest stars in the LMC that are already known to intersect ^4He i* absorbing material will reach S/N=1000 in just 30 minutes of ANDES integration (compared to the required ∼30+ hours of VLT/CRIRES integration). A fainter (potentially extragalactic) source, such as a cluster of bright stars in a nearby dwarf galaxy, with magnitude m_J=15 would require just ∼3 hours of ANDES integration to reach the requisite S/N=500. Therefore, provided that suitable metal-poor targets can be identified, ANDES will allow the first determination of the primordial ^3He/^4He abundance, and a new test of the Standard Model of particle physics and cosmology.It may also be possible to detect ^3He i* absorption at high redshift; ^4He i* absorption has already been identified towards some gamma ray bursts <cit.> and some quasars <cit.>. Given the short fading timescales of GRBs and the high S/N required, the large collecting area of the ELT combined with the high spectral resolution afforded by ANDES is the perfect combination. §.§ The cosmic lithium problem — ^7Li/H Most determinations of the primordial ^7Li/H abundance are based on observations of the most metal-poor stars in the halo of the Milky Way. However, the `primordial' value inferred from these observations is known to be a factor of ∼3 lower than the value expected given the cosmological model derived from observations of the CMB. The general consensus is that metal-poor stars burn some fraction of their ^7Li during their lifetime <cit.>, but we cannot yet rule out the possibility that this discrepancy – also known as the `Cosmic Lithium Problem' – is a signpost to potentially new physics beyond the Standard Model.One approach to overcome the issues of stellar burning is to use absorption line observations of gas clouds that sample the chemistry of the interstellar medium <cit.>. The larger collecting area provided by the ELT will allow us to probe extragalactic dwarf galaxy environments that are more metal-poor than the SMC. Furthermore, the high spectral resolution of ANDES (∼3 km s^-1) will facilitate a greater detection sensitivity, given that the Li i absorption lines are usually intrinsically narrow (≲1 km s^-1).§ CMB TEMPERATUREThe evolution of the cosmic microwave background temperature (T_ CMB) is a fundamental prediction of the standard model. Any deviation from the expected relation T_ CMB(z)=T_ CMB(0)(1+z), which physically relies on the expansion of the Universe being adiabatic and on photon number conservation, would indicate new physics. One example is a violation of the Einstein Equivalence Principle, as a result of a space-time varying fine-structure constant (to be discussed in the next section). Even upper limits to such deviations can be a competitive probe of cosmology, complementing traditional probes such as Type Ia supernovae <cit.>. Observationally, the CMB temperature has been directly measured at z=0 from its black-body spectrum to be T_ CMB(0)=2.72548±0.00057 K <cit.>.At z>0, two methods have been developed to indirectly derive this temperature through so-called thermometers. The first one relies on inverse Compton scattering of CMB photons by hot intra-cluster gas (Sunyaev-Zeld'ovich (SZ) effect) that induce y-type spectral distortions, whose zero-point depends on the CMB temperature. This method allows precise measurements <cit.> but remains limited to z<1 where most of the clusters are found.At higher redshifts, z≳1, the CMB temperature can be measured using absorption lines imprinted by excited fine-structure levels of atomic species or by rotational levels of molecular species in the spectra of background quasars, where the population of the different levels is driven byCMB photons[We note here that, interestingly, <cit.> had already constrained the excitation temperature of CN in our own Galaxy to be as low as 2.3 K, but without giving it much importance as the CMB was only discovered a quarter century later]. The concept is simple, but suitable absorbers are very rare due to the relatively low cross-section of the corresponding cold gas in the ISM of intervening galaxies. Attempts using electronic (UV) absorption lines from atomic carbon in ground and excited states resulted mostly in upper limits owing to high energy level separation and a dominant contribution to excitation from collisions.Observations in the mm-radio domain of the molecular cloud at z=0.89 towards PKS 1830-211 resulted in a measurement with 0.1 K (2 percent) precision, thanks to the observation of various molecular species that allowed tight constraints on physical conditions to be obtained <cit.>.Finally, electronic absorption lines from CO molecules in different rotational levels provide a unique way to accurately measure T_ CMB at z∼ 2-3 <cit.>. For the majority of the known CO absorption systems, the density remains low enough for collisional excitation to be sub-dominant compared to radiative excitation by CMB photons <cit.>.While not reaching yet the precision obtained through SZ measurements, these are unique probes of the younger Universe and provide a stronger lever arm for testing the expected linear increase of temperature with redshift[We note in passing that a measurement has recently been obtained at z∼6 from H_2O absorption against the CMB, albeit within a 14 K-wide 1 σ range <cit.>.]. In practice, several CO bands can be detected at rest-frame wavelengths shorter than 1545Å. The bands are composed of lines from different rotational levels, with Doppler broadening of typically about 1 km s^-1 <cit.> and a separation between levels ≲10 km s^-1. In other words, at the R∼50,000 spectral resolution of current observations (VLT/UVES), lines from different rotational levels are partly blended.At the twice-higher spectral resolution of ANDES, these lines will be fully separated. This will allow one to obtain reliable constrains on the velocity structure of the gas and to measure the CO column density in each rotational level more independently.We note that since this is a line-amplitude measurement, the achieved measurement precision will then depend mostly on the achieved S/N.With ANDES on the ELT, the measurement uncertainties on the CO column densities in each rotational level will become significantly less than the deviation from Boltzmann distribution (i.e. radiative excitation alone) due to contribution from collisions <cit.>. An MCMC analysis of a simulated ANDES spectrum with S/N ≈ 100 pixel^-1 – which should be reachable in ∼30-40 h for an m=19 quasar –indicates a precision better than 0.2 K at z=2.7, i.e. smaller than 2 percent uncertainty, see Figure <ref>. Covering the Lyman-Werner bands of H_2 and lines from fine-structure levels of C i from the same target will be very helpful to obtain the relevant priors on the kinetic temperature and number density in the associated medium, respectively.We note however that even for the highest-redshift CO absorber known so far <cit.>, H_2 bands will be located in the U-band (3500-4100 Å). Notwithstanding, it remains in principle possible to collect H_2 spectra separately using another instrument (e.g. CUBES on the VLT)provided the spectral resolution of the latter allows for precise column density measurements in the low rotational levels.Finally, CN is in principle an even better thermometer than CO, with its excitation being less sensitive to collisions. This molecule possesses a BX absorption band at 3875 Å rest-frame, which means it could be more comfortably detected using the RIZ (z<1.45) or the YJH (at higher z) spectrographs. To our knowledge, however, this molecule has not yet been detected through electronic lines at z>0.§ VARYING FUNDAMENTAL CONSTANTSANDES will be able to compare the values of fundamental physical constants over 12 Gyr ago and 15 Gpc away with their current values on Earth. The collecting area of the ELT will provide an unmatched photon-limited uncertainty in these measurements of just 0.3 parts per million (ppm). The challenge is to ensure that instrumental systematic uncertainties can be reduced below this level to provide new, high-precision tests for physics beyond the Standard Model. A detection of variation in any fundamental constant would revolutionise physics, violating the Einstein Equivalence Principle and demand an explanation from a new, more fundamental theory. §.§ Motivations and implications for `varying constants' Our current understanding of physics is based on a set of physical laws characterised by fundamental constants. However, our best theory of those physical laws – the Standard Model of Particle Physics – does not explain the values or origins of these constants. As such, their constancy is a simplifying assumption that must be established or ruled out by experiment. In physically realistic models containing cosmological scalar fields, they will unavoidably couple to the model's other degrees of freedom, unless a hitherto unknown global symmetry is postulated to suppress the couplings. Thus, one naturally expects such fields to yield varying fundamental couplings and long-range forces <cit.>. Couplings are experimentally known to run with energy, and in most standard model extensions they also roll in time, and ramble in space – i.e. depend on the local environment, including on other parameters like gravitational potential, dark matter density, dark energy etc.. For example, electromagnetic sector couplings lead to space-time variations of the fine-structure constant, α≡ e^2/ħ c <cit.>. These theories can guide our intuition about where may be “better” to test for variations, but it is often difficult to make reliable predictions across all areas of this broad parameter space and connect constraints from different experiments. It is therefore important to experimentally and/or observationally test for variations in fundamental constants in different ways and as a function of as many time-scales, distance-scales and environmental parameters as possible – see <cit.> for recent reviews of theoretical and observational aspects. Again, null results are highly competitive: current constraints on the variation of α are at the ppm level, while constraints on dynamical dark energy (presumably also due to scalar fields) are at the percent level <cit.>. Similarly, in beyond-Standard Model theories that unify the four known fundamental interactions, the parameters we call fundamental constants are related to each other, so it is wise to test for variation in more than one constant in these different places, times and environments. Moreover, in most physically realistic models where α varies, the proton-to-electron mass ratio, μ≡ m_p/m_e, will also vary. The relation between the two variations is model-dependent, but fully calculable in any model that is sufficiently developed to be observationally testable. These relations can therefore provide key consistency tests and potentially rule out models of unification.§.§ Varying constants in astronomical spectra The UV and visible spectra of atoms/ions and molecules are primarily determined by two fundamental constants, α and μ <cit.>. The former characterises the coupling strength of electromagnetism, while the latter gauges the chromodynamic scale relative to the electroweak scale <cit.>. Importantly, different transitions depend differently on α or μ, so comparing their relative frequencies in astronomical objects with those found in Earth-based laboratories is a sensitive test for variations in these constants, even if the astronomical spectra are affected by (much larger) Doppler shifts or cosmological redshifts. For example, the velocity shift of an electronic transition in an atom/ion, relative to its laboratory frequency (wavenumber ω) is <cit.>,Δ v/c≈ -2q/ωΔα/α ,where Δα/α (≪1) is the relative difference in the fine-structure constant in the astronomical target and laboratory. Here, q is a sensitivity coefficient which has different magnitudes and signs for different transitions: it depends on the electronic orbital configurations of the transition and must be calculated using advanced numerical quantum mechanics methods <cit.>. A similar relationship characterises the velocity shifts of UV/visible molecular hydrogen and carbon-monoxide transitions <cit.>.Figure <ref> shows how a variation in α would manifest itself in an ANDES spectrum of a quasar absorption system. In the simplistic case of a single velocity component of absorbing gas (left-hand panels), the orange spectrum shows no velocity shifts, i.e. Δα/α=0, between three typical transitions, while the light green spectrum shows those corresponding to Δα/α = +1×10^-3. This is much larger than the precision achievable with ANDES (∼0.3 ppm per absorber) but illustrates how transitions of different ions behave differently as α varies. Transitions from the ground states of singly-ionised metals (e.g. Mg ii, Si ii, Al ii, Cr ii, Fe ii, Zn ii) at rest-frame wavelengths 1500–2800 Å are normally the most useful for constraining Δα/α in quasar absorbers <cit.>, and typically ∼5–15 such transitions are observed per absorber at redshifts z∼0.5–3.5. A similar approach has recently been applied to stellar spectra where, in principle, many more lines could be useful – see below for further discussion.Most quasar absorption constraints on variations in μ have so-far been derived from the Lyman and Werner bands of molecular hydrogen <cit.>. While up to ∼100 transitions can be used to constrain Δμ/μ, they fall at rest-frame wavelengths <1140 Å, necessitating observation at redshifts z≳2.9 where H i Lyα forest absorption lines blend significantly with the H_2 lines, considerably complicating the analysis. The A–X bands of CO (rest-frame ∼1300–1550 Å) can also be used to constrain μ-variation from optical quasar spectra but, with fewer transitions available than H_2 and a similar range in sensitivity coefficients, they typically provide weaker constraints <cit.>.§.§ Observational status of astronomical searches in UV/visible spectra Early studies of archival samples of ∼50–300 quasar absorbers observed with slit-fed echelle spectrographs (mostly Keck/HIRES and VLT/UVES) showed tentative evidence for deviations in α from the laboratory value at the ∼5 parts-per-million (ppm) level, with up to 5σ statistical significance <cit.>. However, the ThAr lamp calibration of the quasar wavelength scale was distorted with respect to that established from the solar spectrum <cit.>, i.e. spurious velocity shifts were applied to different transitions at different wavelengths, most-likely causing the observed deviations in α <cit.>. Subsequent dedicated observations, with corrections for these wavelength distortions from asteroid and solar twin observations, provided tight limits: Δα/α < 1 ppm <cit.>. Similar limits were obtained from high-precision studies of a single absorber, towards an exceptionally bright quasar, with fibre-fed laser-frequency comb calibrated spectra from HARPS and ESPRESSO <cit.>.Early H_2 constraints on Δμ/μ from slit-fed spectrographs were also adversely affected by the distortions; once corrected, no evidence for variations in μ was found from ∼10 absorbers at the 2 ppm level <cit.>. No fibre-fed spectrographs have been used for μ so far, mainly because their efficiency bluewards of ∼4000 Å is very low.Rather than investigate possible cosmological variations in α, recent studies have applied similar techniques to stellar spectra to search for variations within our Galaxy and, especially, near the Galactic Centre which enables a test of any beyond-Standard-Model connection between α and Dark Matter. <cit.> compared the velocities of ∼10 lines at ∼2.2 μm in 5 giants, relative to the laboratory values, to limit α-variation at the ∼6 ppm level (1σ). Convective line shifts, which vary from line to line, naturally limit this star-to-lab comparison to that accuracy level. However, by comparing the velocity separations of pairs of lines in the visible between stars with very similar physical parameters (and not to the laboratory), these systematic uncertainties are largely removed, providing a limit on variations between stars in the local 50 pc at the 50 parts per billion (ppb) level <cit.>.§.§ Opportunities and challenges for ANDES Clearly, ANDES' major advantage will be that the ELT's collecting area will provide a factor of ≈3.3 improvement in the statistical uncertainty in Δα/α available per target, per unit observing time. Crucially though, leveraging that advantage is only possible if systematic uncertainties remain smaller than the statistical uncertainties from photon noise. That is, systematic uncertainties from the instrument and calibration must be reduced by a similar factor between ESPRESSO and ANDES.For quasar absorption constraints, the most important systematics are those that spuriously shift one transition's measured velocity relative to another. The ∼1 ppm statistical uncertainties produced from single absorbers, or small samples, with 8-to-10-m class telescopes imply that ANDES will achieve ∼0.3 ppm uncertainties, corresponding to ∼6 m s^-1 relative line shifts. That is, in a single quasar exposure, the wavelength scale and all instrumental effects that affect the measured centroid of an absorption line (e.g. instrumental profile variations, charge transfer inefficiencies, etc.) must allow the relative velocities of lines at different wavelengths to be measured with ∼1 m s^-1 accuracy (regardless of the photon statistical noise in the lines). Even with laser frequency comb calibration, ESPRESSO's accuracy in this context is ∼20 m s^-1 <cit.>, so improvements will be required. It should also be mentioned that the (often complex) velocity structure and unknown relative isotopic abundances of metal-line absorbers also present challenges and potential systematic errors as well, with ongoing investigations into their effect on Δα/α <cit.>. Assuming that ANDES is not limited by systematics, Figure <ref> illustrates, in a model-independent way, the potential cosmological impact of its α measurements.To enable new constraints on μ-variation with ∼0.5 ppm uncertainties, the proposed U-band channel of ANDES will be crucial. Its 3500–4100 Å wavelength range is required to cover an adequate number of H_2 lines for absorbers at z≲3.1, i.e. 9 of the 10 known systems for which precise measurements are possible. However, a somewhat less stringent relative velocity systematic error budget of ∼5 m s^-1 is allowable across the U-band. The ANDES baseline design could constrain Δμ/μ with ∼1 ppm precision in the 10th known system (at z=4.22). Identifying new H_2 absorbers at z≳3.1 would be very important for improving the constraints below this uncertainty level, but the faintness and lower number-density of quasars, as well as increased blending due to the Lyα forest, at these higher redshifts all work against achieving substantial improvements.The ability to measure μ in multiple systems at a broad range of redshifts will endow ANDES with another unique scientific opportunity. When measuring μ using H_2 one is indeed measuring m_p/m_e; however, when using other molecules, such as CO, whose nuclei contain neutrons, one is measuring the ratio of an effective nucleon mass to the electron mass, which will coincide with m_p/m_e in the standard model, but not in any extension of the standard model where there are composition-dependent forces (in other words, scalar fields with different couplings to protons and neutrons). Thus measurements of μ using two or more molecules in the same absorption cloud are an astrophysical test of composition-dependent forces[Conceptually, this is akin to checking whether one proton and one neutron fall at the same rate under gravity.].Given that the differential stellar measurements of Δα/α are very recent and, it seems, currently not limited by instrumental systematics <cit.>, it is less clear whether improving upon them will imply specific requirements upon ANDES. However, that method is currently limited to close pairs of lines only because of the potential for wavelength calibration and instrumental errors when comparing lines in very different parts of the detector plane. To overcome that limitation, one must be able to compare the velocities of lines with large separations between stars. It is not necessary that these relative velocities are correct, but only that they remain the same between exposures of different stars one wishes to compare. The requirements for the quasar absorption-line constraints on α above should be sufficient for this provided they are maintained over reasonable periods of time (e.g. years). § REDSHIFT DRIFTThe discovery that the expansion of the Universe began accelerating ∼5 Gyr ago <cit.> was awarded the Nobel prize in physics in 2011 because it clearly indicates the existence of new physics beyond the Standard Model. While still lacking an accepted physical explanation, the acceleration is usually phenomenologically modelled by introducing a new energy component, called `dark energy', with the unusual property of having an equation of state parameter w_ DE = p_ DE / (c^2 ρ_ DE) < -1/3. Over the past two decades, a large amount of observational effort has been, and is continued to be, expended on determining the value of w_ DE and whether it varies with redshift, using a variety of cosmological probes. So far, all observations appear to be consistent with the acceleration being caused by the simplest possible form of dark energy, i.e., a cosmological constant Λ (corresponding to w_ DE = -1) <cit.>, comprising of about 70 percent of the Universe's total energy, thus leading to the now standard ΛCDM model of cosmology.With the accelerated expansion still remaining as the only observational/experimental evidence of dark energy so far (a state of affairs that may be changed by ANDES, as explained above), measuring the expansion history with as much precision and with as many different kinds of observations as possible continues to be a key objective of present-day cosmology. <cit.> first pointed out that the evolution of the expansion rate causes the redshift of a cosmologically distant object to slowly drift in time: ż(z) = (1+z)H_0 - H(z), where H is the Hubble-Lemaître parameter. This implies, at least conceptually, that a simple spectroscopic monitoring campaign over some time-scale Δ t of a number of objects distributed over a range of redshifts could reveal the expansion history. The difficulty with this approach is of course the smallness of the effect: Δ z = żΔ t ≈ H_0 Δ t ≈ 10^-9 or ∼6 cm/s over a decade.Despite this dauntingly small number, there are good reasons to nevertheless pursue a measurement of this effect. Since the derivation of the above relation between the redshift drift and the expansion history H(z) only relies on the cosmological principle and on the assumption that gravity can be described by a metric theory, the redshift drift offers an entirely direct and model-independent route to the expansion history. Uniquely among all cosmological experiments, the redshift drift infers the expansion history by a comparison of two different past light cones and thus represents an entirely non-geometric probe of the global dynamics of the metric. Furthermore, it does not involve any assumptions about the astrophysics of the sources involved.<cit.> first demonstrated that a measurement of the redshift drift was in principle within reach of the ELT. Assuming a purely photon-noise limited experiment, i.e. excluding all instrumental systematics, they showed that the effect could be detected by monitoring the redshifts of Lyα forest and other absorption lines in the spectra of the ∼10 brightest known quasars in the redshift range 2 ≲ z ≲ 5 over a period of ∼20 years. What exactly can be achieved by such observations depends sensitively not only on the brightness and redshifts of the available quasars, the efficiencies of the ELT and ANDES, and the amount of observing time one is willing to spend, but also on the exact selection of the quasars used for the experiment <cit.> and any external priors (e.g. on H_0 and/or flatness) one brings to bear. Recently, the QUBRICS survey discovered a set of previously unknown bright quasars in the southern hemisphere <cit.>, improving our target selection options, while <cit.> suggested the additional measurement of the differential redshift drift using the `Lyα cell' technique as a methodological improvement. However, despite these advances, and considering a realistic ANDES efficiency, the detection of the redshift drift will likely cost a few 1000 hours of observing time spread over 20–25 years.[A potential alternative version of the redshift drift experiment that does not require a ∼two-decade experiment duration consists of measuring the redshift difference between the multiple images of a gravitationally lensed source <cit.>.]Although a redshift drift measurement from ANDES by itself is thus unlikely to yield precision constraints on cosmological parameters, its combination with similar measurements at z < 2 expected from SKA <cit.> and CHIME <cit.> will be extremely valuable. The role of ANDES in this partnership is to fix the expansion history in the matter-dominated era in order to allow the low-redshift measurements to constrain w_ DE <cit.>. Moreover, the redshift drift constraints in the w_0-w_a plane of the CPL parametrization tend to lie perpendicular to those of more canonical cosmological probes, thus facilitating the breaking of their degeneracy <cit.>. In summary, ANDES will be able to provide us, for the very first time, with an entirely new and unique route to the expansion history of the Universe at z > 2. The model-independent nature of a redshift drift measurement will probe the cosmological paradigm at a very fundamental level and, in synergy with traditional cosmological probes, redshift drift measurements from other facilities, and the other fundamental physics experiments from ANDES itself described above, will help to shed light on the nature of dark energy. We emphasize that this science case is not only extremely photon-starved but also places rather demanding requirements on the wavelength calibration of ANDES, in particular on its stability (as defined in Figure <ref> below). Fully capitalising on the ELT's photon collecting power and enabling a total (i.e., summed over all observations) radial velocity precision at the cm/s level, obviously requires that this precision improves with the number of observations as 1/√(N) right down to this level. Having to collect these observations over a time-scale of ∼two decades, implies that the bias of the wavelength solution that remains after all calibrations have been applied, may not drift by more than ∼1 cm/s over this time-scale (see Figure <ref>). As further discussed below, this is a challenging specification – one that requires substantial research before it can be met. For this reason, the redshift drift science case and its associated demand on the stability of ANDES are only considered goals (as opposed to strict requirements) during the design of ANDES.Finally, we note that the redshift drift is, at least to some extent, a representative and gateway for other science cases addressing the weak gravity regime. Comparing velocity measurements with ∼cm/s precision over the time-scale of a decade corresponds to an acceleration measurement with a precision of ∼10^-10m/s^2, which is precisely the scale at which dark matter is required to explain the observed dynamics of galaxies and clusters of galaxies. ANDES will thus enable the ELT to not only probe gravity in the strong-field regime in the vicinity of the supermassive black hole in the centre of the Milky Way using MICADO <cit.>, but also in the extreme weak-field regime. § REQUIRED RESEARCH TO IMPROVE THE WAVELENGTH CALIBRATION AND TO ENABLE KEY SCIENCE CASESThe requirements for wavelength calibration imposed in particular by the science cases related to Varying Constants (Section <ref>) and Redshift Drift (Section <ref>) are so demanding that precise and accurate wavelength calibration of the spectrograph becomes a research topic of its own. As a direct consequence of the larger collecting area of the ELT, ANDES needs to be calibrated nearly one order of magnitude better than the best extreme-precision RV spectrographs on 8 – 10 m class telescopes, including the current flagship for stable and accurate measurements, VLT/ESPRESSO. Only with improved wavelength calibration will ANDES be able to benefit from its huge photon-gathering capacity and avoid being limited by systematics. Achieving this leap forward in wavelength calibration accuracy and stability (Figure <ref>) requires active and dedicated research on this topic. Here, ESPRESSO acts in many ways as a benchmark and testbed for the development of new calibration methods and techniques. §.§ Classic ThAr/FP wavelength calibration Wavelength calibration of high-resolution echelle spectrographs is usually based on hollow cathode lamps(HCLs), either ThAr or UNe, augmented by passively-stabilized white-light Fabry-Pérot etalons (FPs). Combining information from both of these highly complementary sources provides a high-quality wavelength solution <cit.> and will also be the baseline for ANDES. Although probably not sufficient to achieve all wavelength calibration requirements, it is worth improving this very reliable and cost-efficient calibration method.Recently, significant advancements in the understanding of FPs have been achieved, in particular related to their chromatic drift <cit.>. By correcting for this effect, a much improved wavelength calibration could be achieved, and the new ESPRESSO DRS 3.0.0 now shows outstanding performance.Still, the fundamental limitation for the calibration via HCLs and passively stabilized FPs is that for long timescales (longer than a few days) all absolute wavelength information is provided by the ThAr or UNe spectra. These, however, have limited information content due to the sparse and unevenly bright lines, suffer from blending of lines, and do not provide truly fundamental wavelength calibration but rely on laboratory measurements, which currently seem not accurate enough to achieve the 1 accuracy goal. In addition, HCLs are subject to wear, which limits their lifetime and causes a drift of the lines <cit.>, compromising the extremely challenging goal of 1 stability over decades. Thus, a better characterization of the HCL behavior could substantially improve the accuracy and stability of the classic ThAr/FP calibration. §.§ Laser frequency combs A totally different type of calibration source that can in principle overcome these limitations are laser frequency combs <cit.>. These devices produce a dense ensemble of lines with accurately known frequencies, locked against a reference from an atomic clock.Substantial efforts have been put into the development and demonstration of LFCs at astronomical spectrographs <cit.>. However, LFCs are extremely complex devices and come with many difficulties, in particular when adopting them to the needs of astronomical spectrograph calibration, e.g. in terms of line separation and wavelength coverage (at which point these special LFCs are often referred to as astrocombs). Therefore, only few observatories currently use them for routine calibration <cit.>. In most cases, they remain somewhat experimental and in particular the LFCs at HARPS and ESPRESSO have experience several technical problems. Substantial improvements have to be made to bring astrocombs to the technical readiness level required for routine and reliable science operations with ANDES.In addition, novel technologies are needed to provide LFC coverage for the full ANDES wavelength range, in particular the short-wavelength part for which no astrocombs exist so far (U and B band). Developments in this direction are ongoing <cit.>, but these technology demonstrations are still quite far from a system suitable for routine operation. §.§ Data reduction In addition, new methods for data reduction, calibration, and analysis are needed. In particular for the wavelength calibration accuracy (needed for the determination of the fine-structure constant), a very careful modeling of the instrumental line-spread function (LSF) is required. Studies with ESPRESSO have revealed that intra-order distortions of the wavelength solution by up to 50 m/s can be introduced when not properly accounting for the highly non-Gaussian LSF <cit.>. Preliminary results show that a careful measurement and modeling of the ESPRESSO LSF can reduce the intra-order distortions and internal inconsistencies to the few m/s level. This is a major step forward, but still not quite sufficient to meet the ANDES requirements. One of the limitations identified so far lies in the measurement of the LSF. This requires a calibration source with truly unresolved lines. Currently, the ESPRESSO LFC is used for this, but its lines have fixed frequencies and therefore always fall on the same spots on the detector, providing only a very pixelated measurement of the LSF. To overcome this sampling issue, tunability of the LFC is highly desired and, in principle, advertised by manufacturers but so far not possible. As an alternative, it is currently being investigated whether a dedicated high-finesse Fabry-Pérot etalon could be a viable and simpler solution to facilitate accurate LSF determination.Recently, it was also discovered that even for wavelength calibration stability (ignoring accuracy aspects), an accurate modeling of the LSF is inevitable. Tests with ESPRESSO revealed apparent shifts of the LFC lines up to ±2 associated with variation in the LFC flux. With a proper forward-modeling of the LSF, this issue could be substantially reduced, but still, the calibration derived from the LFC calibrations were substantially less stable than the one from the FP. Due to these issues, variations of the LFC flux and limitations in the data processing, it is thus currently not possible to fully benefit from the LFC calibrations. Improvements in both aspects are needed to unleash the full potential of the LFC calibration.On a similar note, <cit.> and <cit.> reported a systematic inconsistency of ≈50 between two similar LFCs that were at some point simultaneously installed at HARPS. This is another clear indication that without more advanced data processing tools the quality of the derived wavelength calibration falls significantly short of the theoretical performance of the LFC and that more work is needed to adapt the methods to the special properties of LFCs and use their full potential.This even raises the question whether classical spectral extraction algorithms like <cit.>, used for ESPRESSO, or <cit.>, which is the current baseline for ANDES, are actually capable of delivering spectra with the required accuracy and stability. Like most other approaches, these algorithms make the fundamental assumption that the two-dimensional instrumental profile (IP) can be decomposed into two one-dimensional functions, which massively simplifies the extraction process, but since never fully satisfied, inevitably introduces residuals and systematics. Their amplitude depends on the actual IP shape and how close the assumed shape is to the truth.It might become necessary to implement spectro-perfectionism <cit.> for ANDES, an algorithm that models the two-dimensional flux distribution on the detector as superposition of individual IPs. Although extremely powerful, this approach imposes strong demands not only on the computational power needed for data reduction but also in terms of specialized calibration light sources required to accurately characterize the IP in the first place. §.§ Validation of the wavelength calibration Another major challenge is the validation of the wavelength calibration to later convince the scientific community that the measured effects (change in α or redshift drift) are genuine and not instrument-induced. In particular for the redshift drift and the fundamental constants experiment, there will be no independent verification on sky, since there is only one Universe to observe, and the data itself will not inherently tell whether the measurement is true or false. Furthermore, the search for varying constants might in general remain a null-measurement with the goal to provide the tightest constrains possible, an inherently extremely difficult task. Therefore, it is crucial to also provide validation tests, in addition to the science observations.Significant efforts are therefore underway to develop techniques for independent validation of the wavelength calibration. One of the concerns here is that the calibration unit injects calibration light at the spectrograph front end. Light from science targets, however, also passes through atmosphere and telescope and is therefore unavoidably injected into the optical fibers in a slightly different manner. The optical design of the instrument contains elements that reduce the susceptibility to the fiber injection as much as possible, e.g. by the use of double scramblersin the optical fiber train, but given the extreme requirements (accuracy, stability, timescale), even very minute effects could compromise the measurements and would not be calibratable. Therefore, suitable wavelength validation sources are needed that are located upstream of the telescope, ideally on sky. Naturally occurring astrophysical sources can not be used since none provides sufficient wavelength accuracy or stability.One solution could be the use of gas absorption cells, in particular I_2 cells. In contrast to e.g. <cit.>, these shall not be placed into the beam when observing science targets (this would be extremely inefficient), but instead only used to verify the wavelength solution of the spectrograph, derived from internal calibration sources, against the I_2 spectrum. For this, iodine cell observations of bright, featureless stars are sufficient, creating an artificial star with well controlled (and independently measurable) absorption features. This probes the full optical path from the top of the atmosphere to the detectors and is therefore equivalent to actual science observations. The spectrum of the iodine cell itself is expected to be extremely stable and can be accurately measured in the laboratory.Recently, a proof-of-concept experiment has been conducted at ESPRESSO to explore the feasibility and achievable precision of this approach. After the data analysis is completed, a more detailed long-term monitoring with ESPRRSSO is planned and, if proven viable, the same strategy will be implemented for ANDES.Other, but more exotic, concepts to validate the wavelength calibration include observations of the Raman spectrum of the adaptive optic facility laser-guide stars <cit.>, calibration sources carried by drones, or even laser frequency combs installed on satellites. Significant research and development over the next years is necessary to demonstrate the practical application of these concepts. §.§ Further steps Over the last few years, much valuable experience has been gained with ESPRESSO which has revealed numerous systematics that, at a certain point, may become limiting factors. To meet the much more demanding ANDES wavelength calibration requirements, substantial advancements have to be made and these systematics characterized, understood and eliminated, either by an improved design of the instrument or – more likely – by advanced data processing and calibration procedures.There are still several years of time for the development of advanced data processing algorithms, however, all aspects related to the hardware design have to be addressed right now during the design phase of the instrument. One of the most important aspects here is to define which type of calibration sources are needed and how they shall be operated to later facilitate a precise, accurate, and stable wavelength calibration of ANDES. Therefore, various efforts are ongoing to gain further experience in advanced wavelength calibration techniques, to push the boundaries towards the ANDES goals, and to incorporate this knowledge into the ANDES design.§ OUTLOOK We have discussed the role of ANDES in cosmology and fundamental physics, specifically in probing the physical behaviour of all the fundamental ingredients of the universe – baryons, radiation, and the dark sector(s), which presumably include new dynamical degrees of freedom beyond the standard cosmological model – as well as in carrying out several hitherto unexplored consistency tests. While the outcome of this exploration is not known a priori, we can nevertheless outline two main scenarios.The first scenario is the null case, in which no deviations from the behaviour expected in the standard ΛCDM model are found, but one simply improves currently available limits. It must be emphasized that this minimum guaranteed science can be highly competitive with more traditional cosmological probes. As an example, constraints on scalar field dynamics derived from varying α measurements are currently at the ppm level, while those derived from cosmological data on the dark energy equation of state are at the percent level <cit.>. Moreover, this minimum guaranteed science is calculable a priori, in an approximately model-independent way, simply from the ELT and ANDES technical specifications, at least if one can assume that the observations are not limited by systematics – in other words, if ANDES can make full use of the photons which the ELT provides.The second scenario ensues if one or more such deviations are detected, e.g. mutually incompatible BBN abundances, a violation of the temperature-redshift relation or of the Einstein Equivalence Principle, or possibly a value of the redshift drift incompatible with the ΛCDM prediction for the corresponding redshift. While the impact of such a detection would be obvious <cit.>, it is important to note that it will also significantly impact ANDES' own observing strategy. If one is aiming to obtain the best possible limits, a simple and reasonable strategy (though one with some model dependence) is to concentrate the available telescope time on a small number of bright targets. On the other hand, if, for example, a varying α is detected, one would obviously like to map the redshift dependence of this variation using the widest possible range of redshifts, from z<1 (when the putative dark energy is dynamically relevant) to deep in the matter era – a fine-structure survey, as recently suggested by <cit.>. A more detailed exploration of such observational strategies is left for subsequent work.Last but not least, we must emphasize the the answer to the question of how competitive ANDES will be as a cosmology and fundamental physics probe, in the context of other ground and space facilities operating in the 2030s and beyond, depends almost entirely on two factors. The first is the availability of the U band, which is currently a goal; its need is scientifically motivated in several earlier sections of this work, and can also be visuallyunderstood from Figure <ref>. The second and equally important one is its ability to fully benefit from the ELT's larger collecting area (i.e., ensuring that we do not become systematics-limited). This implies further developments not only in terms of wavelength calibration but in the full data reduction and analysis pipeline. Admittedly some of the required developments are challenging (and possibly costly), but one must keep in mind that, in a rapidly developing and data-driven field like fundamental cosmology, something that is easy to do today is very unlikely to be scientifically competitive in ten years' time.AcknowledgmentsThis work was financed by Portuguese funds through FCT (Fundação para a Ciência e a Tecnologia) in the framework of the project 2022.04048.PTDC (Phi in the Sky, DOI 10.54499/2022.04048.PTDC). CJM also acknowledges FCT and POCH/FSE (EC) support through Investigador FCT Contract 2021.01214.CEECIND/CP1658/CT0001. RJC is funded by a Royal Society University Research Fellowship, and acknowledges support from STFC (ST/T000244/1). MTM acknowledges the support of the Australian Research Council through Future Fellowship grant FT180100194. JL acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. TMS acknowledges the support from the SNF synergia grant CRSII5-193689 (BLUVES). CMJM is supported by an FCT fellowship, grant number 2023.03984.BD. Author contributions CJAPM: Coordination, writing of Sects. 1 and 7; RC: Writing of Sect. 2; JL: Writing of Sect. 5; MTM: Writing of Sect. 4; PN: Writing of Sect. 3; TMS: Writing of Sect. 6; CSA, SB, CMJM, MAFMS: Help with simulations/forecasts; PDM, RM, AM, LO, AZ: ANDES Project Office support; JSA, SC, RSG, NJN: Comments/suggestions on the draft.Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.Conflicts of interest The authors declare that they have no conflict of interest. | http://arxiv.org/abs/2311.16274v1 | {
"authors": [
"C. J. A. P. Martins",
"R. Cooke",
"J. Liske",
"M. T. Murphy",
"P. Noterdaeme",
"T. M. Schmidt",
"J. S. Alcaniz",
"C. S. Alves",
"S. Balashev",
"S. Cristiani",
"P. Di Marcantonio",
"R. S. Gonçalves",
"R. Maiolino",
"A. Marconi",
"C. M. J. Marques",
"M. A. F. Melo e Sousa",
"N. J. Nunes",
"L. Origlia",
"C. Péroux",
"A. Zanutta"
],
"categories": [
"astro-ph.CO",
"astro-ph.IM",
"gr-qc",
"hep-ph"
],
"primary_category": "astro-ph.CO",
"published": "20231127192337",
"title": "Cosmology and fundamental physics with the ELT-ANDES spectrograph"
} |
Spin order dependent skyrmion stabilization in MnFeCoGe hexagonal magnets Ajaya K. Nayak January 14, 2024 ========================================================================= Greenhouse gases (GHG) trap heat and make the planet warmer, exacerbating global climate change. Energy production is the second largest contributor to climate change <cit.>. In particular, the production of electricity and use of gas contribute to climate change.Additionally, gas is not renewable and the source of electricity may not be renewable either.How and whether communities transition to renewable and/or cleaner energy sources is dependent on a number of factors, including energy needs, cost, space, emissions, jobs, and materials.In this paper, we explore minimizing the cost of building, operating, and maintaining energy sources.We consider different combinations of cleaner and/or more renewable energy sources to meet energy needed for a given city while keeping total emissions, land use, and energy infrastructure costs low.For specificity, we use the city of Chicago as a test case.If we use a combination of wind and solar energy to meet the total energy needs of Chicago, we find that it is most cost effective to use only wind.However when variable demand and production are included, it is most cost effective to use a combination of wind and solar.If nuclear and geothermal energy are included to decrease overproduction, it is most cost effective to use a combination of wind and geothermal energy.§ INTRODUCTIONRenewable energies have been around for quite some time; the very first commercial sale of wind turbines took place in 1927. However, most energy that is produced in the world comes from non-renewable sources. Specifically,global energy production is fuelled 31.2% by oil, 27.2% by coal, and 24.7% by natural gas <cit.>. Energy and its availability has always been taken for granted <cit.>. Access to energy sources sets the physical limits to the society: all life, commerce, work, and technological innovation are possible and limited by the new energy available <cit.>, <cit.>. Moreover, energy consumption per capita has been increasing steadily over recent decades <cit.>, <cit.>, especially in high and upper-middle income countries. On the other side, easily available or existing non-renewable energy sources are depleting <cit.>, <cit.>. The International Energy Agency suggests that with no new drilling, the world's oil production would be reduced to 50% by 2025 and to 15% by 2040 <cit.>. Although new sources of energy will be found, the raw materials will be more expensive to extract <cit.>. This will inevitably affect economic growth, given that we have already found and used most of the cheapest sources. This warrants consideration for renewable energies as partial substitution of the non-renewable sources. Not only are renewable energy sources sustainable over time, but they significantly reduce harmful emissions, which is an important factor given that current emissions from energy production grow each year and have negative effects on the environment <cit.>.In this project, our goal is to explore how major cities can meet their electricity and gas usage needs exclusively with renewable or clean sources, while minimizing the operation and maintenance cost and meeting several constraints. Although renewable energy cannot eliminate the need for non-renewable sources <cit.>, we attempt to substitute the electricity and natural gas needs (referred to as “energy needs" from now on). This excludes other direct or indirect consumption of energy by the population, such as transportation, purchased essentials (such as clothing, food, home goods), and recreation.As an example, we will be exploring the energy needs of the city of Chicago, Illinois. In this paper, we first consider a model where the only goal is to minimize costs, in USD per MWh, mainly to show why non-renewable energy is the primary choice of energy source. We then add constraints to the linear model, including emissions, budget, space, and energy needs. We start with considering exclusively solar and wind systems, and later add nuclear and geothermal to solve issues related to overproduction. § METHODS: LINEAR PROGRAMMING We will be using linear programming to explore solutions in this paper. Linear programming is a set of algorithmic procedures utilized for solving optimization problems, which involve the maximization or minimization of a linear objective function subject to linear constraints. Maximization problems often deal with profit, while minimization problems often deal with cost, such as the problem we consider in this paper. More specifically, we use the simplex algorithm for solving linear programs. The simplex algorithm was created in the 1940's by George B. Dantzig. It produces an optimal solution to a linear program (provided such a solution exists) in a reasonable number of steps and is easily implemented on a computer <cit.>.To solve the models constructed in this paper, we use Python 2021.2 (community edition), and build our code on the existing modeller “PuLP 2.7.0", written in Python to solve linear programming problems.§ FIRST MODELAlthough we consider emissions an important aspect of this project, the financial side is a practical limitation. Renewable energy infrastructure can be expensive to build, but it also has operation and maintenance costs. Our objective function minimizes the construction, operation, and maintenance costs.For the purpose of this paper, we will call this total cost. §.§ Minimizing Total CostThis model includes energy from wind, solar, nuclear, geothermal, natural gas, and hydroelectric sources. Variables measure how many megawatt hours (MWh) of energy from each source is used in a year: x_1= MWh from wind in a yearx_2= MWh from solar in a year x_3= MWh from nuclear in a yearx_4= MWh from geothermal in a year x_5= MWh from gas in a yearx_6= MWh from hydroelectric in a year. The levelized cost of energy (LCOE) per MWh accounts for construction, operation, and maintenance of each energy source over the course of its lifetime <cit.>. The cost per MWh in 2021 for each source is as follows in Table <ref> <cit.>.With the variables and levelized cost above, the total cost is 37.80 x_1 + 58.62 x_2+ 96.2 x_3 + 39.61 x_4 + 37.50 x_5 + 63.9 x_6. If the total needs of Chicago is E MWh in a year,then the variables are constrained byx_1+x_2+x_3+x_4+x_5+x_6 ≥ Ewith x_i≥ 0 for all i.With these constraints, the minimum of total cost occurs when we use exclusively non-renewable and non-clean (combined cycle natural gas) energy. This result makes sense: we currently rely on so much non-renewable, non-clean energy because it is the most economical option at present.§.§ Possible additional constraintsSince we aim to study reducing emissions, using only a non-clean energy source is not acceptable. In the rest of this study, a constraint about emissions will be included. By including clean energy sources, other concerns arise.As a result, other models in this paper include some of the following.* Meet Chicago's energy needs. This means that the total amount of MWh produced by renewable energy sources is greater than or equal to the energy needs of Chicago.* Keep carbon emissions low. Emissions have a negative impact on the environment, and are a significant motivating factor for the shift toward clean and sustainable energy production.The Table <ref> shows the average emissions of CO_2 per MWh of major non-renewable, non-clean sources, as well as renewable and/or clean sources: * Meet a certain budget. This is the amount of money that needs to be invested in the very beginning to build all necessary infrastructure. It does not include operation and maintenance costs.* Keep infrastructure in available/reasonable space.* Provide for continuously available energy. Since energy needs change throughout the day, we need to adjust our model accordingly so there is no underproduction. § USING SOLAR AND WIND SYSTEMS TO MEET ENERGY NEEDSThe model consists of a simplified version of objective function (<ref>), which only includes wind (x_1, in MWh produced a year) and solar (x_2, in MWh produced a year) energy. There are five constraints: one constraint for energy needs of Chicago, one for emissions, one for budget, and two for space. The next subsection <ref> will cover each constraint in detail. See section <ref> for the linear model.§.§ Determining Constraints §.§.§ Chicago energy needsThis constraint ensures that the annual energy production is greater than or equal to the annual energy needs of Chicago. In order to set the lower bound for the energy needs of Chicago, we estimate how much energy (MWh) is used annually. From the State Energy Data Portal (SEDS) database, Illinois consumed 1,091,285,298.314 MWh in 2021 <cit.>.Since this database does not include Chicago's consumption, we make an estimate based on Illinois' energy use.Chicago Data Portal provides a database on energy consumption in Chicago from 2010 <cit.> but nothing more recent.Since SEDS provides the the consumption of Illinois in 2010, we calculate the percentage that went to Chicago that year. The state of Illinois consumed 1,168,009,546 MWh in 2010 <cit.>.That same year, buildings in Chicago consumed 15,142,030 MWh of electricity and 37,998,300 MWh of gas <cit.>. Since the electricity data comprises 68 percent of overall electrical usage in the city and gas data comprises 81 percent of all gas consumption in Chicago for 2010, these values are adjusted and summed to find that Chicago consumed about 68,771,765 MWh in 2010. Dividing Chicago's energy use by Illinois', we estimate that approximately 5.88% of Illinois' total usage of energy in 2010 went to Chicago. This can be found in Table <ref>.Next, the percentage is adjusted due to population growth. The population of Chicago has increased by 4% since 2010 <cit.>, so we raise the percentage from 5.88% to 7%. Applying this to Illinois' energy needs, we find that Chicago's consumption in 2021 was approximately 76,389,561 MWh.However, part of Illinois' electricity already comes from either clean or renewable sources: 52.63% of it is generated by nuclear power plants <cit.>. Another 12.27 % comes from wind energy, 1.50 % from solar, and 0.06% from hydroelectric plants <cit.>. Since there is no specific information on Chicago, we use these values. This means that we only need to cover for the 33.54% of the 76,389,561 MWh found above, which is 25,621,059 MWh. This is the value we use for our energy needs constraint - the minimum that needs to be produced in a year. This constraint appears in Inequality (<ref>) in the linear program in subsection <ref>. §.§.§ EmissionsAlthough both solar and wind systems generate no emissions during the production of energy, there is a quantifiable environmental impact from the production, transportation, and recycling of the materials used in plants. For wind plants, the production and transportation of the components, the reconditioning and renewal of the components, and disposal of the material are considered when quantifying the emissions of CO_2 <cit.>. On average, 4970g of CO_2 are emitted for every MWh generated by an onshore wind turbine, which is consistent with other studies <cit.>. For solar energy systems, silicon solar panels generate 45,000g CO_2 for every MWh when taking into account its production, maintenance and transportation <cit.>.To set an upper bound for emissions, we estimate current emissions generated by non-renewable and non-clean sources. Of the 76,389,561 MWh consumed by Chicago, 20.99% comes from coal, 12.31% comes from natural gas, 0.21% - from biomass, and 0.04% - from oil <cit.>. The emissions resulting from these sources can be found in Table <ref>. We estimate that 17.83 × 10^12 g CO_2 are being emitted every year by non-clean sources of energy to partially power the city of Chicago. We set our goal to reduce emissions by 80% to 3.578 × 10^12 g CO_2. This constraint appears in Inequality (<ref>) in the linear program in subsection <ref>.§.§.§ BudgetThis constraint is a limit on the funds allocated for the construction of new plants, since we assume that there is no infrastructure that is already operating and can be used. Although the following coefficients depend can vary, we chose the national average, shown in Table <ref>, which are obtained by dividing the average cost of a plant by its lifetime output in MWh.This constraint is shown in Inequality (<ref>) in <ref>.§.§.§ Space There is a limited amount of space available for new plants and infrastructures, and the Inequalities (<ref>, <ref>) reflect these limitations. We separate this space constraint into two inequalities. Solar systems have an advantage: they can be placed on the rooftops of existing building. Wind systems, on the other hand, need free space where plants can be built. An estimation on land-use by different energy sources is shown in Figure <ref>. For solar panels, these occupy 19m^2 (204.5 ft^2) per MWh. The project site are for every MWh of wind energy is 99 m^2 (1065.6 ft^2) per MWh <cit.>. For wind turbines, it is more difficult to approximate how much land can be used for new infrastructure. We estimate it as follows.In 2010, the U.S. Census Bureau reported that 47% of total land in the United States was unoccupied <cit.>, <cit.>. We apply this to the area of Illinois (1,614,570,000,000 ft^2), and get 758,847,900,000 ft^2. We choose to dedicate 1/15 of that “unoccupied" area to wind power plants, which is 50,589,860,000 ft^2. This is our upper limit for space allocated for wind power plants, and thus we have the following constraint: 1065.6 ft^2/MWh× x_1 ≤ 50,589,860,000ft^2 Dividing both sides by 1065.6 ft^2, we get an upper bound for wind energy production, which is the inequality (<ref>) in the model <ref>.For solar energy, we set a constraint that determines how much area of the existing buildings can be used to install solar panels. In this section we assume that solar panels can only be installed on the roofs. The building footprint in the city is approximately 70,532,107 ft^2 <cit.>, <cit.>. This is our upper limit for solar panels. 204.5 ft^2/MWh× x_2 ≤ 7.053 × 10^7ft^2 Dividing both sides by 204.5 ft^2, we get the upper bound for solar energy production, which is the inequality (<ref>) in<ref>. §.§ Linear programMinimize C(x_1, x_2) = 37.80 x_1 + 58.62 x_2Subject to:x_1 + x_2 ≥ 25,621,059MWh 4970 x_1 + 45,000 x_2 ≤ 3.578 × 10^12 g CO_2 27.45 x_1 + 39.12 x_2 ≤$ 2 × 10^9 x_1 ≤ 47,475,469 MWh x_2 ≤ 344,900 MWh 0 ≤ x_1 , x_2§.§ Results Using only solar and wind, the minimum of the objective function is obtained when exclusively wind energy is used. This occurred because wind turbines release fewer CO2 emissions during production than solar panels, cost less at comparable energy ratings, and operate at a higher efficiency. The results are presented in Table <ref>: This model's results are very important because: * It sets the minimum budget for the next three section: since wind is the cheapest of all energy sources to build and maintain, it means that if we try any other combination (but geothermal), it will be more expensive to build the necessary infrastructure. Below this value of 703,298,069 $, the solution becomes unfeasible. * It also sets the minimum for emissions, in this case for the entire project. We will see later on that geothermal and nuclear energies have higher emissions per MWh, so this is the least amount we will be getting. Again, this tells us about the necessary conditions for feasibility of the solution. § ENERGY PRODUCTION AND DEMAND THROUGHOUT THE DAYSince the production and demand depend on time of day, we split our energy needs constraint into three constraints which incorporate data from different periods of the day. In this section, we develop these new constraints and study the resulting model.§.§ Demand and production throughout the day§.§.§ ProductionThe outputs of wind and solar energy systems depend on the time of day. We split our single energy needs constraint in <ref>, Inequality (<ref>), into three (<ref>, <ref>, <ref>), one for each part time period of the day, shown in model <ref>. The three periods of the day of choice were based on the output of solar systems, which only generate energy during the time of solar radiation, and this is approximately between 7am and 7pm (referred to as “daytime hours" in this paper) <cit.>. The other two periods of time are the hours between 7pm and midnight (“evening hours"), and between midnight and 7am (“early morning hours"). Figure <ref> shows the average production for both wind and solar energies throughout the day.Based on this graph, the average production of both wind and solar systems are shown in Table <ref>.The percentages of wind and solar energy by time of day are used as coefficients in inequalities (<ref>, <ref>, <ref>) in model <ref>.§.§.§ DemandSince our production is divided into early morning, daytime, and evening hours, we separate the data on demand similarly.In order to determine how much energy is used throughout the day and how that changes, we looked at the data from <cit.> between July 2022 and June 2023, which provides demand on every hour of every day. Since no hourly energy usage of Chicago is available, we used data on the Midwest area, shown in Figure <ref>. We found an average over the early morning, daytime, and evening hours over the course of the year. We also calculated an average over these time periods for each month, and identified the largest percentages for each time period. The results are shown in the Table <ref>. What we can see from this table is the following:* Average % of the daily demand: this is simply telling us what percentages of energy is demanded during each of these day section of the day. On average, 26.38% is consumed in the early morning hours, 51.21% during the daytime hours, and 22.41% during the evening hours (for reference, 100% is the consumption during 24h). * Highest % of the daily demand: the previous row has the demand percentages obtained by averaging out the data over the entire year. However, there are some months when there is more need during the day than is shown, and similarly this can happen for the other two periods of the day. To make sure enough energy is produced every month for every section, we take the highest percentage. Although this implies that we will be overproducing, it also guarantees enough energy for every month of the year. From this table, the important results are the percentages, which will help us to identify the lower bounds for (<ref>, <ref>, <ref>). These constraints are formalized in section <ref>. §.§ New constraintsBased on the percentages we obtained in <ref> and <ref>, the new energy need constraints (<ref>, <ref>, <ref>) are:* Early morning needs (12am-7am): (25,621,059 MWh) × 0.2759 = 7.069 × 10^6 MWh;* Day needs (7am-7pm): 25,621,059 MWh × 0.5149 = 13.192 × 10^6 MWh;* Day needs (7pm-12am): 25,621,059 MWh × 0.2344 = 6.006 × 10^6 MWh. Here is the updated model with three constraints for energy needs replacing the one constraint (<ref>) in <ref>:§.§ Updated model Minimize C(x_1, x_2) = 37.80 x_1 + 58.62 x_2Subject to:0.3760 × x_1 + 0.01 × x_2 ≥ 7.069 × 10^6MWh 0.3775 × x_1+ 0.9797 × x_2 ≥ 13.192 × 10^6 MWh 0.2456 × x_1 + 0.01 × x_2≥ 6.006 × 10^6 MWh 4970 x_1 + 45,000 x_2 ≤ 3.578 × 10^12 g CO_2 27.45 x_1 + 39.12 x_2 ≤$ 2 ×10^9 x_1 ≤ 47,475,469 MWh x_2 ≤ 344,900 MWh 0 ≤ x_1 , x_2 §.§ ResultsFirst we look at other constraints and the values we got:* Emissions: we can see that our current emissions are less than 5% of our upper limit and 1.04% of the emissions that would have been produced if we chose non-renewable energies. * Budget: the total price to construct enough plants to get our energy needs met is 949,669,412 $ a year.Next, we look at the energy production. As expected, we are overproducing in general: our needs for Chicago are 25,621,059 MWh, while our total production given by the linear program is 34,449,706 MWh. This is partially due to our choice of picking the highest percentages of demand for each section of the day. Notice that we are producing the maximum amount of solar energy that is allowed in Inequality (<ref>).In section <ref>, we assume that solar panels can be installed on the ground as well, and study the resulting linear program.§ ALLOCATING MORE SPACE FOR SOLAR SYSTEMS In this section, we allow for solar energy system installation on the ground. Remember that solar systems use 204.5ft^2 for every MWh they produce when installed on the ground <cit.>. Now we add it to our space constraint (<ref>): 1065.6 x_1 + 204.5 x_2 ≤ 50,589,860,000 ft^2 The Inequality (<ref>) is going to substitute Inequalities (<ref>) and (<ref>) from <ref>, because solar panels are now sharing space with wind plants on the ground.We also need to account for the 344,900 MWh of energy from solar panels installed on top of buildings, found in section <ref>. Adding it to the new space constraint Inequality (<ref>), we get: 1065.6 x_1 + 204.5 (x_2 - 344,900 MWh) ≤ 50,589,860,000 ft^2§.§ New model with space constraint on solar energy productionMinimize C(x_1, x_2) = 37.80 x_1 + 58.62 x_2 Subject to:0.3760 × x_1 + 0.01 × x_2 ≥ 7.069 × 10^6 MWh 0.3775 × x_1 + 0.9797 × x_2 ≥ 13.192 × 10^6 MWh 0.2456 × x_1 + 0.01 × x_2 ≥ 6.006 × 10^6 MWh 4970 x_1 + 45,000 x_2 ≤ 16,325 × 10^9g CO_2 27.45 x_1 + 39.12 x_2 ≤$ 2 ×10^9 1065.6 x_1 + 204.5 (x_2 - 2,190,438 MWh) ≤ 50,589,860,000ft^2 0 ≤ x_1 , x_2§.§ Results Notice that solar energy production increased considerably, and the value of the objective function decreased by over $150 million.However, we are still overproducing, in particular during the early morning hours. Allocating more space for solar panels made this model more successful, but did not completely solve the issue of overproduction. Two possible solution that we consider in this project are: considering storing the excess energy (which we do no explore due to the scope of this project), or adding another source of energy, such as nuclear (section <ref>) or geothermal (section <ref>).§ INTRODUCING NUCLEAR ENERGY INTO THE MODELIn this section, we explore nuclear power as a solution to the issue ofoverproduction.§.§ Constraints for nuclear power§.§.§ Energy production constraintIf we choose to use nuclear power, the smallest unit of system we can construct is one reactor. The smallest nuclear reactors (called Small Modular Reactors, SMRs) have the capacity of 300 MW <cit.>, although an average reactor would be of 1 GW <cit.>. A plant with an SMR produces 2,628,000 MWh a year working at full capacity, and we will set this as our lower bound for nuclear energy production, as shown in Inequality (<ref>) in <ref>. §.§.§ Time of the day constraintAn important advantage of nuclear power is that its output distribution (the amount of energy it delivers) is constant, and can also be flexible <cit.>. In our model, we assume that the energy production of a power plant is constant throughout the day, meaning that 29% is produced during the early morning hours (<ref>), 50% during the daytime hours (<ref>), and 21% during the evening hours (<ref>) (based on the fact that early morning hours (7 hours) are 29% of the day, daytime (12 hours) are 50%, etc). These coefficients are reflected in Inequalities (<ref>, <ref>, <ref>) in <ref>§.§.§ Emission constraint Although nuclear energy is considered a clean energy source because there are no direct emission during nuclear fission, it is not a renewable source because of its use of uranium. So, there are emissions produced during the construction and operation of the plant itself, and the mining of uranium.Carbon dioxide is emitted during the mining of uranium. On average, 34,000 gCO_2 are estimated to be emitted for every MWh produced using uranium, and this number is predicted to increase as the amount of uranium left decreases <cit.>. Also, it is important to mention that for every kWh produced from a nuclear fission, 0.1-0.3 kWh are used <cit.>. We will use 0.2 kWh in our calculations. This means that to obtain 1 MWh of electricity, we will emit 40,800 gCO_2. Constructing a plant and maintaining it is associated with emissions of GHG as well. An average of 8200 gCO_2 are emitted for every MWh that a plant produces <cit.>.To get the final coefficient for the emissions constraint, we add the two emission values: for uranium and for the construction/maintenance. These add up to 49,000 g CO_2/MWh, which is reflected in Inequality (<ref>) in <ref>.§.§.§ Budget constraint The cost for every additional MWh produced by a nuclear reactor is approximately 70.8$ <cit.>. This is shown in Inequality (<ref>) in <ref>.§.§.§ Space constraint As reported in <cit.> and shown in figure <ref>, 0.3 m^2 are needed for every MWh, which is 3.23 ft^2, shown in Figure <ref>. This coefficient is represented in Inequality (<ref>) in <ref> §.§ Model The adjusted model in this section is composed of our previous model of wind energy (x_1) and solar energy (x_2) from Section <ref> with the addition of nuclear energy (x_3). Minimize C(x_1, x_2, x_3) = 37.80 x_1 + 58.62 x_2 + 96.2 x_3Subject to:0.3769 x_1 + 0.01 x_2 + 0.29 x_3 ≥ 7.069 × 10^6MWh 0.3775 x_1 + 0.9797 x_2 + 0.5 x_3 ≥13.192 × 10^6 MWh 0.2456 x_1 + 0.01 x_2 + 0.21 x_3 ≥6.006 × 10^6 MWh 4970 x_1 + 45,000 x_2 + 49,000 x_3 ≤ 163,325 × 10^9g CO_2 27.45 x_1 + 39.12 x_2 + 70.8 x_3 ≤$ 2 ×10^9 1065.6 x_1 + 204.5 (x_2 - 10,279,088) + 3.23 x_3 ≤ 50,589,860,000ft^2 0 ≤ x_1, x_2 2,628,000 ≤ x_3§.§ ResultsFirst, we ran the linear program as it is shown in the subsection <ref>. Given these constraints, the result that minimized the objective function was the same as in section <ref>. In other words, the program suggests to only use wind and solar systems to cover the energy needs. This is a somewhat expected result - nuclear power is more expensive than wind and energy to both maintain and construct. So, why would anyone choose nuclear power instead of wind or solar? If you look at the constraints, the answer becomes obvious: land use. Nuclear power plants occupy less space per MWh than solar or wind plants. In fact, if the space constraint is reduced to something lower than what we chose in section <ref>, then the linear program does suggest we use nuclear power. To demonstrate this, we reduced the upper limit on space to 205,898,600 ft^2,and obtain the following constraint on space: 1065.6 x_1 + 204.5 (x_2 - 10,279,088) + 3.23 x_3 ≤ 205,898,600 ft^2 The results obtained with this new constraint on space are shown in Table <ref>.Notice that we are very close to the upper limit for our space constraint. The amount of solar energy we are producing is what we can “fit" on the buildings, but it does not occupy any land. Moreover, as we increase the upper limit for the space constraint (<ref>), the linear program suggests to increase the amount of MWh produced by wind systems or solar systems and reduce nuclear production. In other words, we only use nuclear power when we do not have enough space to build another system and run a different source of energy.Furthermore, nuclear power does not solve the issue of overproduction. The daytime needs are met, but a large amount of energy is produced during the morning hours and the evening hours. We conclude that nuclear energy is not the best solution unless we lack space to install other systems. However, the advantage that nuclear energy has over the rest is that it does not depend on either the weather or natural resources, which means that it is the only clean energy source that can ensure a constant supply.§ GEOTHERMAL ENERGY In this section, we consider adding geothermal energy to the linear program <ref>. Since nuclear energy did not reduce the value of the objective function and increased some of the constraints, as shown in Table <ref>, it will not be included in this section. §.§ Constraint for geothermal energy§.§.§ Constraint on emissionsAlthough geothermal energy is considered a clean energy, the systems require raw materials, transportation, etc. Geothermal systems emit, as a median, 38,000 gCO_2 per every MWh produced <cit.>. This is reflected in Inequality (<ref>) in <ref>.§.§.§ Constraint on spaceGeothermal systems occupy a considerable amount of space <cit.>. Around 900 m^2 are required to produce every additional GWh<cit.> by a 56 MW geothermal flash plant. Thisis equivalent to 9.6875 ft^2/MWh, which is added to the space constraint in Inequality (<ref>) in <ref>.§.§.§ Constraints based on the time of the dayGeothermal energy, just like nuclear energy, has a great advantage - power plants produce electricity consistently, running 24 hours a day, 7 days a week <cit.>. This implies that a plant produces approximately 4.167% of its total daily output during every hour of the day. Multiplying this by the amount of hours in each of our day sections (12am-7am, 7am-7pm, 7pm-12am), we get the following coefficients:* Early morning hours: 7h × 4.167% = 29.16%, or 0.2916;* Day hours: 12h × 4.167% = 50%, or 0.5000;* Evening hours: 5h × 4.167% = 20.83%, 0.2083. These coefficients that reflect the energy production of a geothermal plant are added to the three constraints on energy production in <ref>, and can be found as Inequalities (<ref>, <ref>, <ref>) in <ref>. §.§ ModelHere is the new linear program that includes wind energy (x_1), solar energy (x_2), and geothermal energy (x_4), all in MWh. Minimize C(x_1, x_2, x_4) = 73.7 x_1 + 55.8 x_2 + 39.61 x_4Subject to:0.3769 x_1 + 0.01 x_2 + 0.2916 x_4 ≥ 7.069 × 10^6MWh 0.3775 x_1 + 0.9797 x_2 + 0.5 x_4 ≥13.192 × 10^6 MWh 0.2456 x_1 + 0.01 x_2 + 0.21 x_4 ≥6.006 × 10^6 MWh 4970 x_1 + 45,000 x_2 + 38,000 x_4 ≤ 3.578 × 10^12g CO_2 27.45 x_1 + 39.12 x_2+ 21.8 x_4 ≤$ 2 ×10^9 1065.6 x_1 + 204.5 (x_2 - 10,279,088) + 9.6875 x_4 ≤ 50,589,860,000ft^2 0 ≤ x_1, x_2, x_4 §.§ ResultsThis is the lowest value of the objective function we have been able to achieve, and it is the result of geothermal energy being cheaper than solar. Notice that the emissions, although still within the limits, have increased, because we are using less wind energy, which is 10 times cleaner than geothermal. § DISCUSSIONCurrent energy production methods are unsustainable in the long run. A good substitution for these methods are renewable energy, which has lower emissions and are sustainable over time, and clean energy, which produces fewer emissions as well.In this paper, we study minimization of total renewable and clean energy costs with constraints on emissions. We treat total energy cost as a linear function and establish linear constraints from practical consideration. The resulting linear programs can be solved using the simplex method. We constrain the problem by requiring that energy needs are met. If meeting energy needs is the only concern, it is most cost effective to use natural gas; however, natural gas comes with high emissions.We explore alternative sources to reduce emissions while still meeting energy needs. In this paper, we have studied a total of five models which address different concerns when using clean and renewable energies.We have considered three models that combine wind and solar energy sources with constraints on emissions, budget, and space. From the simplest model <ref>, in section <ref>, we have learned that wind plants are overall the cleanest source of renewable energy, as well as the cheapest to build, operate and maintain. However, wind and solar plants' output is not constant throughout the day. Additionally, the demand on energy is not constant either. This fact warranted a division of the day (24h) in 3 slots (12am-7am, 7am-7pm, and 7pm-12am). The hours were divided based on the output of solar energy systems. After adjusting for demand and production rates during these three slots, we found from a second model in section <ref> that the minimum value for the objective function is achieved as a result of a combination of solar and wind systems. In the third model in <ref>, we allowed for land use of solar systems, which led to an increase of suggested amount of MWh produced by solar panels and, subsequently, a decrease in wind energy production.However, we were facing the problem of overproduction: wind plants were producing more energy than needed during some slots, and because wind energy cannot be easily stored for later use, this resulted in wasted MWh. To mitigate this issue, we considered two solutions: adding nuclear energy production and geothermal energy production.In section <ref>, we found from a fourth model <ref> that, although nuclear energy reduced the overproduction slightly, it was not financially optimal given that nuclear plants are more expensive to build and operate. Adding nuclear sources also increased emissions. We concluded that nuclear plants were not a good solution for overproduction unless the available space was drastically reduced.In section <ref>, we studied a fifth model with added geothermal plants to the system. This was more successful than nuclear in mitigating overproduction, and it resulted in a reduced objective function value. Geothermal energy also eliminated the need for solar energy and reduced the wind energy production, although this came with a cost: slightly increased emissions.Overall, a combination of geothermal energy and wind energy is the best financial decision. They can meet the emissions, budget, and space constraints while providing the necessary electricity levels and minimizing overproduction. The results of each model were obtained given an objective function that considers building, operation and maintenance costs. As mentioned in <ref>, we assume that all the infrastructure needs to be built/purchased. But what about preexisting plants in Illinois available for use? We studied the case in which the objective function coefficients for each energy source excluded the building costs, keeping the maintenance and operation expenses. The change of the objective function did not yield a different balance of energy sources, which can be seen explicitly for the model from section <ref> in Appendix, section <ref>.Another aspect of this study that was considered is: what if we chose to minimize emissions instead of costs? Since wind energy has the lowest emissions per MWh by far, the minimal emissions are obtained from using only wind energy,Appendix <ref>.This study is not without caveats. There are some limitations associated with the energy sources of choice and the results we have obtained. All the energy sources considered, except for nuclear, depend on the natural resources and weather conditions of their building site. In particular, the maximum output of wind, solar, and geothermal systems varies significantly based on their location <cit.>. Given that the information available on the capacity factor <cit.> is limited, we used the average (across the U.S. for some and the world for other) values for our models. This approximation causes both over- and underestimation in our results, which should be addressed once more information on capacity factors for different regions are available. Currently available capacity factors for different regions should be used instead of the generalized ones when applying the model to other cities. For instance, Phoenix is expected to have higher solar energy output than Chicago, which may not be the case for wind energy. Another limitation of this study that deserves more attention is the energy needs constraints. Given the scope of this study, we chose to have three constraints for energy needs based on the varying energy outputs throughout the day. However, the outputs also vary depending on the month of the year. This is especially important for those cities with larger weather variability, like New York or Chicago. To achieve more accurate constraint bounds, it is wise to consider creating models based on the month or even week of the year. In this study, we limited the constraints to budget, emissions, land use, and energy needs. However, there are more factors that come into play when choosing a city's sources of energy. For instance, the number of jobs new plants would generate (maintenance, construction) and eliminate (due to lack of need of non-renewable sources), the time new plants would take to be construct, availability of materials for the infrastructure and systems, changes in levelized capital costs due to technology advances and tax incentives, among other factors. Our project aimed to investigate the feasibility of fully replacing traditional energy sources with sustainable and clean alternatives to meet the essential energy demands of a city like Chicago while maintaining a realistic cost. While we successfully identified optimal combinations of renewable and clean energy sources to minimize costs while fulfilling these baseline requirements, questions lingered about the practicality of these findings. It would be very costly to substitute all non-renewable energy with renewable energy, especially given growing energy demands, due to the substantial infrastructure development required. On the other hand, clean energy options, such as nuclear power, rely on specific finite natural resources like uranium.In essence, relying solely on renewable and clean energy sources to keep pace with growing energy demands appears unattainable due to the extensive infrastructure and resource requirements. Conversely, non-renewable energy sources are becoming scarcer and costlier to extract as easily accessible natural resources dwindle. Our energy supply is struggling to match the rapid growth of our energy consumption. Neither traditional nor renewable sources can adequately meet our escalating demands. To simultaneously mitigate environmental impact and secure our energy future, we must reduce our consumption. Ideally, the future of our energy lies in a combination of renewable and non-renewable resources, but maintaining a realistic baseline energy requirement is essential to make this blend sustainable.abbrv§ CHANGE IN THE OBJECTIVE FUNCTION§.§ ModelBefore we move on to solving the issue of overproduction, we would like to talk about the objective function we are using. The objective function is based two assumptions: * There is no available infrastructure (wind plants, solar systems, etc) that can be used to cover Chicago's needs.* There is no funding to build the plants, and that we will pay for the “construction of every additional MWh" as it gets produced.Hence, our budget constraint served as an upper limit to how much infrastructure we can afford to “pay off". However, we can consider the case when we have a budget large enough to pay for all the infrastructure. In this case, we would be looking to minimize the operation and maintenance costs only. After finding the fixed O&M cost values in <cit.> and <cit.>, we consider an alternative objective function. The new coefficients are shown in Table <ref>.and the new objective function is:C(x_1, x_2) = 10.35 x_1 + 19.51 x_2 Now we determine whether this changes the output of x_1 (wind energy production a year in MWh) and x_2 (solar production a year in MWh). The constraints will be the same as in the previous model: (<ref>, <ref>, <ref>, <ref>, <ref>, <ref>). §.§ Results The results of this program turned out to be the exact same as of the model in section 6.2. The total production of wind energy was 24,862,479 MWh, and solar energy was 3,900,512 MWh. Consequently, all the constraint values were the same.Why didn't the result change? Visualizing the problem will help us understand it mathematically.When the feasible region of a linear program is bounded, it is a convex polygon and the minimum of the objective function occurs at a vertex (James K Strayer text). Refer to Figure <ref>, where horizontal axis represent the MWh of wind energy, and the vertical axis - MWh of solar energy.The four corner points from Figure <ref> that meet all three criteria are: point A (x_1=21,812,415 MWh, x_2=77,102,051 MWh), point B (x_1=24,862,479 MWh, x_2=3,900,512 MWh), point C (x_1=45,223,754 MWh, x_2=74,516,399 MWh), and point D (x_1=47,475,469, x_2=0). Point B is the result we got from section 5. Also, notice that point C is definitely overproducing, so it is not considered in the calculations below.Next, we evaluate both the objective function from the previous sections and the alternative objective function at these corner points (A, B, and D):257,326,658* For C(x_1, x_2) = 10.35x_1 + 19.51x_2, we get the results shown in Table <ref>.* For C(x_1, x_2) = 37.80x_1 + 58.62x_2, we get the results shown in Table <ref>.Notice that the corner point that minimizes both objective function is point B. This is the result of wind energy being both cheaper to maintain and operate, and to build. So, even when we change the objective function, we still get the same suggested output. §.§ Recycling the materialsThe two main sources of renewable energy considered in this paper are wind and solar energy. Wind energy is obtained using wind turbines and solar energy with solar panels. Both of these devices have been originally produced from non-recycled materials, and that has become a problem for the environment. Since the lifespan of a solar panel is approximately 30-35 years the lifespan of a wind turbine is around 20 years, there is a growing need for technologies that will enable us to recycle both of them. Although most of the materials can be recycled, most of them have to be reused elsewhere and cannot be integrated into new wind blades or solar panels. §.§.§ Wind blades It is estimated that 10kg of blade material is needed for every 1kW of new capacity. For our linear program, we consider 3.5 MW turbines, so each requires 35 tones of blade material. Since their lifespan is about 20 years, there is a need for recycling these blades. There are three main recycling technologies: mechanical recycling, thermal recycling, and chemical recycling. Although 99% of the materials can be reused, they cannot be used to make new blades <cit.>.§.§.§ Solar energy systemsThere are three kinds of solar panels, but the crystalline silicon (monocrystalline or multi-crystalline) have higher conversion efficiency than others, so they are presently the most widely used commercial solar panels. After disassembly and extraction of the elements that compose such a panel, the weights of the various materials used are: glass 57.4%, aluminum 12.7%, adhesive sealant 10%, silicon 3.1%, and other 19.5%. Some of these materials can be reused for new solar panels, but others lose their qualities and have to be reused elsewhere. Since solar energy is relatively new, most panels have not yet reached their end-of-life, and there is not much research on recycling technologies yet <cit.>. § MINIMIZING EMISSIONS INSTEAD OF COSTS. In this study, the financial limitation of the project was considered of primary importance, and the goal was to minimize costs while keeping emissions low. However, an alternative model is presented in this section of the Appendix, where we minimize emissions while keeping the costs constrained. We chose to alter the model <ref> from section <ref>. Our objective function is now minimizing emissions (<ref>), so there is no constraint on them. To keep costs low, we change the budget constraint coefficient to those used in function (<ref>), which is represented in Inequality (<ref>. Here is the resulting model:Minimize C(x_1, x_2) =4970 x_1 + 45,000 x_2 Subject to:0.3760 × x_1 + 0.01 × x_2 ≥ 7.069 × 10^6 MWh 0.3775 × x_1 + 0.9797 × x_2 ≥ 13.192 × 10^6 MWh 0.2456 × x_1 + 0.01 × x_2 ≥ 6.006 × 10^6 MWh 37.80 x_1 + 58.62 x_2 ≤$ 2 ×10^9 1065.6 x_1 + 204.5 (x_2 - 2,190,438 MWh) ≤ 50,589,860,000ft^2 0 ≤ x_1 , x_2 §.§ ResultsSolving this linear system, we get exclusively wind energy. This is expected, since wind is the cleanest energy source by far. It also manages to satisfy all constraints, so there is no need to add solar energy into the solution, as a less environment friendly option. Given this advantage that wind energy has over the rest of sources we consider, the rest of the models resulted in either exclusive use of wind energy or, in case of the model in section <ref>, a combination of wind and nuclear because of the limitations on land-use. § ENERGY SYSTEMS OF CHOICE §.§ Solar systems There are two main types of technologies for converting solar energy into electricity that are used at large scales: solar thermal technologies and photovoltaic systems. In this project, we will be considering the photovoltaic (PV) power plants, because they offer several advantages <cit.>: * It can use both the direct and diffused component of solar radiation.* It is suitable in areas with low direct radiation. However, one of the major trade-offs of these solar systems is that their emissions are higher than for the thermal technologies. The emissions are still much lower than for conventional non-renewable sources, but are higher than some alternatives of solar systems. §.§ Wind turbinesThe size and height of turbines has been increasing every year. A larger turbine will produce more energy than a smaller one, so less turbines are needed to produce the same amount. A logical conclusion is: we should use the largest turbines we have. However, this is not as simple as might seem to be. Larger turbines are more difficult to transport, since they cannot be folded once they have been constructed. The average capacity of all the newly installed turbines in the US is 3 MW. Hence, this will be the theoretical power we are going to use for our calculations in this project. We will also be assuming that our turbines are at least 100 meters (330 feet) tall, since the speed of the wind increases at higher altitudes <cit.>. Illinois and the Midwest in general are considered to be great areas to build wind power plants since these are two regions with higher-than-average wind shear. Additionally, there are several manufacturing facilities located in Illinois and adjacent states, which allows us to assume that the transportation costs will be at most average, if not lower. Refer to the two maps below: < g r a p h i c s > < g r a p h i c s > From the first map, we can see that Illinois and the Midwest have rather high wind speed at 100m. From the second map, we can see the locations of several wind turbine manufacturing facilities <cit.>. | http://arxiv.org/abs/2311.15820v1 | {
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Journal of Hirao et al.: Comparing Pseudo-Haptic Perceptions with Motion or Force Input Pseudo-haptics techniques are interesting alternatives for generating haptic perceptions, which entails the manipulation of haptic perception through the appropriate alteration of primarily visual feedback in response to body movements. However, the use of pseudo-haptics techniques with a motion-input system can sometimes be limited.This paper investigates a novel approach for extending the potential of pseudo-haptics techniques in virtual reality (VR). The proposed approach utilizes a reaction force from force-input as a substitution of haptic cue for the pseudo-haptic perception. The paper introduced a manipulation method in which the vertical acceleration of the virtual hand is controlled by the extent of push-in of a force sensor. Such a force-input manipulation of a virtual body can not only present pseudo-haptics with less physical spaces and be used by more various users including physically handicapped people, but also can present the reaction force proportional to the user's input to the user. We hypothesized that such a haptic force cue would contribute to the pseudo-haptic perception. Therefore, the paper endeavors to investigate the force-input pseudo-haptic perception in a comparison with the motion-input pseudo-haptics. The paper compared force-input and motion-input manipulation in a point of achievable range and resolution of pseudo-haptic weight. The experimental results suggest that the force-input manipulation successfully extends the range of perceptible pseudo-weight by 80% in comparison to the motion-input manipulation. On the other hand, it is revealed that the motion-input manipulation has 1 step larger number of distinguishable weight levels and is easier to operate than the force-input manipulation.Pseudo-Haptics, virtual reality, sensory substitution, cross-modal integration Move or Push? Studying Pseudo-Haptic Perceptions Obtained with Motion or Force Input Yutaro Hirao, Takuji Narumi, Ferran Argelaguet, and Anatole Lécuyer Y. Hirao is with Nara Institute of Science and Technology (NAIST), Nara 630-0192, Japan. E-mail: [email protected]. T. Narumi is with the University of Tokyo, Tokyo 113-8654, Japan. E-mail: [email protected]. A. Lécuyer and F. Argelaguet work with Univ. Rennes, Inria, IRISA, CNRS, Rennes, France.January 14, 2024 =================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONIn recent years, significant advancements have been made in virtual reality (VR) technology, allowing users to experience a sense of actual presence through visual and audio feedback. However, compared to audio-visual information, haptic feedback still possesses a narrower range of expression. To address this limitation, various approaches have been proposed to present haptic feedback. One intriguing method is the employment of pseudo-haptics techniques, which modify haptic perception by appropriately adjusting visual feedback in response to body movements <cit.>. Examples of pseudo-haptics techniques include modifying the virtual force of a spring <cit.> or the perceived weight of an object <cit.> by altering the control-display gain according to the user's motion within a virtual environment. The main advantage of the pseudo-haptics technique is that it primarily relies on visual stimuli to convey haptic perceptions, eliminating the need for bulky haptic devices. Traditionally, the pseudo-haptics technique for VR interaction involves the use of motion-input manipulation, where the spatial positions of the physical body are correlated with their virtual counterparts. However, There are several limitations with pseudo-haptics techniques with motion-input system. For example, a conventional pseudo-haptics technique reduces the virtual motion compared to the physical motion to present a stronger force perception and therefore, it requires larger physical motion and spaces. Moreover, because motion-input system in pseudo-haptics studies has morphologically equivalent input-output body mapping, it cannot be used by users with disabilities that prevent performing the required virtual motions. In addition to them, even when motion gains are applied and visual cues are altered, because there is no haptic feedback, the haptic cues remain unchanged from the absence of motion gains. This leads to a mismatch between the haptic estimates (e.g, width or weight of objects) derived from each sensory cue. Here, it is said that such separation between the estimates from each sensory cue can lead to discomfort or disrupt haptic perception when the disparities are too significant <cit.>. To address the above problems, Hirao et al. examined pseudo-haptics with a joystick-based avatar manipulation and confirmed that it can present pseudo-haptic weight perception with a similar level of the one of motion-input system <cit.>. Such a morphologically incongruent avatar manipulation system can present pseudo-haptics with less physical spaces and be used by more various users including physically handicapped people. Moreover, since the physical and virtual bodies are not consistent at the beginning, the problem of discomfort caused by the mismatch between them can less arise. The purpose of this study is to further investigate a novel approach for extending the potential of pseudo-haptics. This paper investigates the pseudo-haptics with a force-input avatar manipulation that has morphologically incongruent input-output mapping. Such a force-input system not only has the merit above but also provides the user with a force feedback equivalent to the input force applied to the sensor when manipulating the virtual body. Hence, modifying the motion gain of the system not only changes the visual cues but also the haptic force cue to the user. For instance, when lifting a virtual object, pseudo-weight perception can be produced by reducing the amount of visual movement relative to the actual input. In this scenario, the force-input manipulation method demands greater force to lift a heavier virtual object by the same extent as a lighter object. Then, the paper endeavors to investigate if such a haptic force cue can be used as a substituted cue, i.e., be combined with a visual cues and contribute to the haptic estimates as a haptic force cue, even though the body parts used for the input and output are different.Fig. <ref> depicts a conceptual diagram of the difference of motion-input and force-input pseudo-haptics. In this paper, we have decided to refer to this type of pseudo-haptics as FISpH (Force-Input Substituted Pseudo-Haptics), pronounced the same as "fish."The paper reports two experiments which compare the pseudo-haptics utilizing the avatar manipulation method of force-input with that of motion-input. The paper investigates the pseudo-weight perception while virtual lifting task. Then, we developed a manipulation system in which the vertical acceleration of the virtual hand was controlled by the extent of push-in of a force sensor where the reaction force proportional to the user's input was fed back to the user (Fig. <ref>). The proposed system was compared to the motion-input manipulation system with respect to the range and resolution of presentable pseudo-weight perception. The first experiment assessed the number of levels of pseudo-haptic weights that could be presented by determining the range of gains that could be applied in each method and the resolution of the pseudo-haptic presentation within that range. The second experiment investigated the gain at which the pseudo-weight perception with the proposed manipulation was perceived to be equivalent to that produced at the maximum/minimum gain of the motion-input manipulation, in order to compare the two methods on the same axis of pseudo-weight perception. The remainder of this paper is structured as follows. First, Section <ref> introduces related work. It provides the overview of the pseudo-haptics techniques and reviews the studies on haptic substitution. Then, Sections <ref> describes the two experiments to compare the effectiveness of FISpH and motion-input pseudo-haptics. Subsequently, Section <ref> provides a comprehensive discussion of all experimental results. Specifically, it discusses the effect of the two manipulation methods on pseudo-weight perception and usability. Moreover, it also introduces the discussion of the experimental methods for measuring pseudo-haptics. Subsequently, it mentions the limitations of the paper and future work. Finally, Section <ref> concludes the paper. We would like to note that all experiments in the paper were approved by the local ethical committee.§ RELATED WORK§.§ Pseudo-Haptics TechniquesSince the first paper on pseudo-haptics in 2000 <cit.>, the pseudo-haptics techniques have been studied for various haptic perceptions. Haptic properties of objects can be classified into roughly two categories <cit.> and many pseudo-haptics techniques are investigated to modify these properties: material properties, such as texture (such as roughness and friction <cit.>; e.g. <cit.>), compliance (e.g. <cit.>), and weight (e.g. <cit.>); and geometric properties, such as curvature (e.g. <cit.>), orientation and angle (e.g. <cit.>), and size (e.g. <cit.>). Moreover, the retargeting and redirection research <cit.> can be also considered as one part of pseudo-haptics research because the geometric properties are explained as haptic properties and it is mentioned that the modification of the haptic properties using multisensory (mainly visual and haptics) integration processing is a pseudo-haptics technique (<cit.>).However, if we look at the quantitative results of to what extent these techniques can modify haptic perception, the pseudo-haptics techniques still have a large gap for the application. To elaborate, psychophysical experiments suggest that the pseudo-haptics techniques can alter weight perception roughly by 3 5% <cit.>, compliance perception by 20 35% <cit.>, friction perception by 23% <cit.>, and fine roughness perception by 5% <cit.>. Moreover, studies including retargeting and redirection field suggest that the pseudo-haptics techniques can alter hand motion perception up to 13.75% in the forward, and up to 6.18% in the backward direction without being noticed <cit.>. In this way, several studies suggest that the pseudo-haptics techniques can alter haptic perception roughly by 5% to 35% depending on the target haptic perception. However, the range of perceptual intensity that can be modified is still narrow compared to using haptic devices, and it cannot be said enough when considering various practical use in VR applications.One of the possible reasons for this limitation is related to the fact that the pseudo-haptics technique separates visual and haptic information. Here, larger gaps are necessary to induce more intensive perception; however, gaps that are too large result in discomfort or a failure to integrate the information <cit.>. Based on Pusch and Lécuyer's model <cit.>, larger discrepancies could result in larger complements and substitutions of sensory information from personal memories and experiences, which would increase individual differences. Moreover, if the gap is too large, the information is determined to be coming from a different source and is not integrated <cit.>. Furthermore, forward and inverse dynamics theory can provide another explanation in that the illusion would not arise when the brain no longer has a model to which the large gap can be attributed <cit.>. Although this limitation of pseudo-haptics is well-known, there are few studies on extending the applicable range of the pseudo-haptics techniques. First, addition of multiple cues is suggested to enhance the pseudo-haptics although they did not mention to what extent it can improve <cit.>. Then, Ban and Ujitoko proposed a method for solving the discrepancy between visual and physical position by connecting the discrepancy between a user's finger and the cursor on a touchscreen with a visual string <cit.>.Another possible approach involves increasing the reliability of visual information in sensory integration, i.e., to decrease the variance of visual information in MLE so that the final perception is more dependent on visual information by wider field of view in VR <cit.> or more realistic avatar <cit.>.However, these approaches do not manipulate the haptic information, thus the pseudo-haptics is still largely restricted by the haptic cue from the body. On the other hand, some studies endeavor to add noise signals to haptic cues so that the final perception is less dependent on haptic cues by using noisy galvanic vestibular stimulation (GVS) <cit.>, bone-conducted vibration (BCV) <cit.>, or tendon vibration <cit.>. For example, Hirao et al. confirmed that the tendon vibration as noise on haptic motion cues can extend the range of visual/physical discrepancy without being noticed by about 13%. However, these approaches has unintended side effects that might have effect on the user experiences or does not have large improvement. Compared with these approaches, the proposed method, FISpH, endeavors to tackle the problem from completely novel angle, which introduces new body schema having force-input manipulation and newly designs the integration law of visual and haptic cues.§.§.§ Pseudo-haptics using force-input systemThere are several studies investigating pseudo-haptics with force-input devices. Actually, the very first research on pseudo-haptics uses force-input interface that mimics an interaction with a virtual spring (that looks like a syringe) on a screen <cit.>. Most of these studies endeavor to simulate virtual compliance or stiffness by changing the gain of force-input to virtual motion with a touch display <cit.>, a rod type device <cit.>, or handheld devices that sensor the grabbing or pinching force <cit.>. These approaches can be said straightforward and strong because such compliance or stiffness has the push-in force parameter and motion parameter in their physical law and the pseudo-haptics techniques of changing their gain are rational. Moreover, these system use the same body schema in physical and virtual reality, i.e., the body part that is used to the force-input and the one that moves in VR is the same. For example, TORC simulates virtual hand's pinch movement by a real hand's pinch force-input <cit.>. On the other hand, the force-input system in this paper uses the different body schema in physical and virtual reality, i.e., the height of a virtual hand is manipulated by the extent of push-in of a force sensor with an index finger. Moreover, this paper investigates weight perception and its physical law does not explicitly have the push-in force parameter in its physical equation. Therefore, unlike the previous studies, it is not obvious that the force feedback, that is, the reaction force while the push-in, functions as a haptic cue for the weight estimates, combined with the visual motion cue. In other words, this paper endeavors to introduce a new body schema in virtual body manipulation and substitute the reaction force cue of the finger for the somatosensory cue of the arm to strengthen the effect of the pseudo-haptics techniques. §.§ Sensory SubstitutionAs the FISpH can be said to substitute original haptic cue (haptic cues expected from virtual body part) with a new one (haptic cues from force-input), this section reviews the research of haptic substitution. The study of how to substitute haptic feedback has been examined mainly in the field of the research on sensory feedback of prostheses <cit.>. The non-invasive methods to present sensory feedback of prostheses are technically devided into mechanotactile feedback (e.g. <cit.>), vibrotactile feedback (e.g. <cit.>), electrotactile feedback (e.g. <cit.>), auditory feedback (e.g. <cit.>), and hybrid of them (e.g. <cit.>) <cit.>.Another way to classify the feedback methods is from the perspective of if the sensory feedback is modality-matched to (i.e. is felt in the same modality as) the target sensory input <cit.>. For example, studies on the sensory feedback of upper limb prostheses often try to substitute pressure exerted on prosthetic fingers during grasping movements. In this case, the mechanotactile feedback is modality-matched feedback, and the other sensory substitutions such as vibrotactile, electrotactile, and auditory feedback are not. In general, modality-matched feedback is more intuitive and requires less cognitive effort during manipulation <cit.> than non-modality-matched feedback, and can improve the sense of embodiment to the prostheses <cit.> as well as enables closed-loop control <cit.>. Then, which body part should the substituted feedback be presented to? In the field of prostheses research, it is often presented at the location directly related to the prosthesis manipulation (e.g. body-powered prostheses <cit.>), the part which has similar physiological characteristics (e.g. <cit.>), or the part where the amputee felt the "phantom sensation of the lost body part" or which has undergone "targeted reinnervation surgery." It is known that brain plasticity often induces a sensory remapping of the lost body part on the different body parts such as the amputation stump <cit.> and face <cit.>, where the amputee can feel a sense of touch of the lost body part. Moreover, targeted reinnervation is a technique that "allows severed cutaneous nerves to reinnervate skin on a different portion of the body" <cit.>. Therefore, it is natural and rational to present sensory substituted feedback to these body parts because it is physiologically correct <cit.>. Meanwhile, in recent years, sensory substitution techniques have been investigated also for non-amputee users due to the increasing demand for haptic feedback in VR and augmented reality research. For example, Pezent et al. proposed a wrist band shaped device named Tasbi which can present squeeze and vibrotactile feedback to the wrist <cit.>. They proposed to substitute vibrotactile feedback for fingertip contact sensation with virtual objects or user interfaces, and squeeze feedback for weight or stiffness associated with manipulating virtual objects. Moreover, Kameoka et al. proposed a system named Haptopus, that transforms the touch sensation of the hand into tactile (suction) feedback to the face <cit.>. Furthermore, Hiki et al. proposed a sensory substitution system that transforms the pressure sensation of the hand to the one of the sole of the foot <cit.>. Their experimental results suggest that their system can allow participants to recognize the weight, stiffness, and shape of an object in a similar capability to the one using a hand. These techniques are valuable because they can present haptic information to users while allowing them hands-free interaction. However, unlike in most cases of the prostheses studies for amputees, these techniques do not present substituted feedback in a physiologically correct way. Moreover, these approaches do not strictly substitute/remove the original haptic cue but use another haptic cue in addition to the original one. As a result, the original haptic estimates from the body part, such as the fingers, remain present. Thus, the haptic estimates is still restricted by the one derived from the original haptic cue. For example, when haptic feedback of the hand is substituted with other body parts to present the perception of the hardness of a virtual object interacting with a virtual hand, even though the information of the hardness can be interpreted, the perception of the hardness and/or user experience may be interfered by the fact that the hand is physically grasping empty. This may be solved by long-term learning with brain plasticity <cit.>, but that is not obvious at this moment. Then, this paper proposes to develop a new body schema that can reduce/remove the influence of haptic estimates derived from the original haptic cues, and that substitutes modality-matched feedback for the original one. § EXPERIMENTS This section compares FISpH, which substitutes the original haptic cues with the reaction force cues induced by force-input manipulation in a new body schema, with the conventional motion-input pseudo-haptics. The main focus of the experiment is to investigate the pseudo-haptics of the two methods from a view of the applicable range and the resolution of the pseudo-haptic perception. Specifically, the experiments investigate pseudo-weight perception while lifting a virtual object. The research questions in the experiments follow:[RQ1] Which manipulation method has wider range of pseudo-weight perception?[RQ2] Which manipulation method has more number of levels of pseudo-weight perception?[RQ3] Is the current implementation of force-input manipulation controllable? §.§ XP1: How many levels of pseudo-haptic weight can motion-input and force-input manipulation present?The first experiment investigated the range of applicable motion gain out of which users start to feel the interaction is incoherent/weird or lose the feeling of control. Moreover, this also allows to investigate the resolution of pseudo-haptic weight perception with each manipulation. With these results, i.e., the range and resolution, we calculated how many levels of pseudo-haptic weight each manipulation can present.§.§.§ Apparatus Figure <ref>shows the experimental apparatus and virtual environment. The experiment was conducted with participants in a sitting position. Participants wore a VR headset (Oculus Quest 2). A force sensor (FSR 400, Interlink Electronics Inc.) was attached to a table beside the participants. In VR, there was a low table and a virtual dumbbell on it in front of the participants. The virtual dumbbell was placed at 0.7 m in height from the ground. In this experiment, there were two manipulation methods for a virtual hand. One manipulation method was the motion-input manipulation. Here, participants held a VR controller with their right hand. The position of the virtual hand tracked that of a VR controller, and the posture of the virtual hand was set to correspond to the physical hand when participants grabbed the VR controller. Participants could grab and release the virtual dumbbell by using an index trigger. Once they grabbed the virtual dumbbell, the virtual hand and dumbbell moved only on a vertical axis and their height tracks VR controller's height with a gain.The other manipulation method was the force-input manipulation. Here, participants put their right arm on the table beside them and press the force sensor with their index finger. In this method, we calculated the acceleration of the virtual hand and object by applying force on the sensor and Newton’s motion equation. First, we obtained an equation for the relationship between output voltage from the force sensor [V] (V_out) and applied force [N] (F_s) on it (equation <ref>) from a data sheet. Here, 0.20 in equation <ref> indicates the area of the contact surface of the force sensor. Then, for the method of simulating virtual force (F_v) induced by the virtual hand from the applied force (F_sensor), this time, we determined to simply multiply a constant of 10 by applied force to the force sensor and a gain (G) that is used for pseudo-haptics (equation <ref>). Note that there would be a space to design a better simulation of virtual force from the applied force. However, this study is at the first step to investigating the effect of force-input manipulation on pseudo-haptics and does not focus on the proper design of force-input manipulation. Therefore, we determined to use the most simple equation by using a constant of 10 that is decided by our pilot test. Subsequently, the acceleration of virtual hand/object [m/s^2] (A_v) was calculated by Newton's motion equation (equation <ref>). Here, the weight of the virtual object was set to 0.15 [kg] which is the standard weight of VR controller. In the force-input manipulation, the virtual hand was set to the grab position of the virtual dumbbell, and they moved in a vertical axis when participants apply force to the sensor. F_s = (0.067 · e^1.5 · V_out) · 0.20 · G [N]F_v = F_s · 10 · G [N]A_v = F_v/0.15 - 9.8 [m/s^2]§.§.§ TaskThere were three phases in the experiment. The first phase was the "learning task phase" which aimed to let participants get used to each manipulation and investigate the task performance. Here, participants conducted reaching tasks on the vertical axis with each manipulation method. A target height was presented with a blue bar. Participants grabbed and lifted up a virtual object to the blue bar. After the participants reached the target bar and released the object, then, the bar, virtual hand, and virtual object disappeared. Subsequently, the virtual hand and object appeared in an original position, and the next target bar was presented. The target bar was randomly positioned in a range from 0.2 m to 0.7 m range from the table height (0.7 m) in front of the participants. This reaching task took 15 seconds and the participants conducted 2 repetitions of this 15 seconds trial for each manipulation method. The participants were asked to complete the task as fast and precisely as possible (tried not to go over the blue bar but to stop and release at the bar’s height).The second phase was the "range task phase" which aimed to investigate the applicable gain for pseudo-haptics in each manipulation. The staircase method was used for this aim. Here, the applicable gain for pseudo-haptics indicates a minimum and a maximum gain where participants start to feel discomfort or lose the feeling of control. As reported in the related work section, the most common range of "applicable gain" in previous work in the field of redirection or pseudo-haptics is the one under the detection threshold of visual/physical discrepancy. This range is the most strict range where users do not notice the visual modification by a gain. However, we consider that range of applicable gain in practice is wider than this range, where users can notice the visual/physical discrepancy but feel pseudo-haptic perception without feeling unnaturalness for the interaction. we consider that this wider definition of the range of applicable gain is more practical. Indeed, there are many studies on pseudo-haptics that uses a gain beyond the detection threshold of visual/physical discrepancy (e.g. <cit.>). Moreover, because we investigate pseudo-haptics with force-input manipulation and the force-input manipulation does not have an obvious definition of the detection threshold of visual/physical discrepancy, we cannot use the range using the detection thresholds. Therefore, in this experiment, we determined the range of applicable gain as the one where users do not feel discomfort or do not lose the feeling of control. In the task of the staircase method, the range of adjustable gain was from 1.0 to 3.0 for the high gain group (gain > 1) and from 0.3 to 1 for the low gain group (gain < 1). Each gain group had ten equal steps i.e., a step value in the high gain group was 0.2 and that in the low gain group was 0.07. The task was a free interaction with the virtual dumbbell in 5 seconds in VR. After the free interaction, participants answered the question of “It was too heavy/light.” by "Agree" or "Disagree" by touching a virtual button in VR. Here, an experimenter explained that the question indicates that “I felt it was too heavy/light and start to feel weird or lose the feeling of control." A staircase method had two series of trials whose initial gain was either the minimum or maximum gain of the range of adjustable gain. When participants answered “Agree”, the gain moved one step closer to a gain of 1.0 and vice versa. When each series in a staircase method reached 5 turning points, the experiment was over.The third phase was the "resolution task phase" which aimed to investigate the resolution of pseudo-haptic perception. The method of constant stimuli was used for this aim. Participants compared the sense of the weight of reference and comparison dumbbell within one of the manipulation and answer which dumbbell was heaver. The reference gain was 1.0 and the comparison gains were 0.76, 0.84, 0.92, 1.08, 1.16, 1.24 (-24%, -16%, - 8%, +8%, +16%, +24%, respectively). The task was to interact with two virtual dumbbells for 3 seconds for each and answered a question: “Which dumbbell did you feel heavier?” by “The first” or “The second” by touching a button in VR. Each comparison gain was compared to the reference gain ten times. Therefore, participants conducted 120 trials in total (6 comparison gains × 2 manipulations × 10 repetitions). §.§.§ Collected dataIn the learning task, the overshoot distance from the target height, each reaching duration, and the number of reaching the target were recorded. This data was used to check if the force-input manipulation was usable for this lifting task.In the range task, the minimum and maximum applicable gain was obtained by averaging adjusted gains at 10 turning points in the staircase method. In the resolution task, JND of pseudo-haptic weight perception was calculated by psychometric function obtained by the method of constant stimuli.§.§.§ ProcedureTwenty participants (fifteen males and five females in their twenties) participated in the experiment. First, the objectives, methods, procedures, and how each manipulation works were explained to them. Next, the participants completed a consent form. Then, the participants were asked to wear the VR headset. Here, the participants also checked the position of the force sensor on a table. Before the main tasks, participants practiced each manipulation method with a virtual dumbbell with a gain of 1. Subsequently, the participants conduct 3 main tasks. The combination of the order of the manipulation method in each task was counterbalanced between the participants. Furthermore, other orders such as that of the high/low gain group in the second phase and comparison gains in the third phase were randomized. The participants could take a break whenever they wanted. After the experiment, the participants were remunerated with an Amazon gift card of 5 euros for their participation. The total duration of the whole procedure was around 45 minutes.§.§.§ Results Table <ref> shows the results of the task scores of the reaching task, i.e., number of reaching the bar within the trial and overshoot distance from the target height when the participants released the virtual object.Here, we used the values at the second trial. In addition, the results of the overshoot of each reaching task were averaged for each trial. Paired t-test was conducted for each number of reaching and overshoot. As a result, a significant difference was found in the results of overshoot (p < 0.001, d=1.76).Table <ref> shows the results of the second phase. With these results, we can calculate the range of the applicable gains in each manipulation. The range of applicable gain for motion-input manipulation was 0.94 and that for force-input manipulation was 1.42.Then, for the results of the third phase, the Probit analysis <cit.> was conducted for the averaged values of all participants' results, which calculated the parameters of the best-fitting cumulative normal function. Subsequently, we computed the JND for each manipulation as half of the distance between the points of 25% and 75% on the psychometric curve. These values were 0.10 for the motion-input manipulation and 0.17 for the force-input manipulation. Figure <ref> shows the psychometric curves calculated with the averaged values of all participants' results. With the results of the second and third phases, we calculated the number of pseudo-haptic weight levels that each manipulation method can present. Then, it was 9 levels for motion-input manipulation and 8 for force-input. §.§ XP2: What is the equality between the pseudo-haptic weight with the force-input and motion-input manipulation? The second experiment investigated the point of subjective equality (PSE) of the pseudo-haptic weight perception between each manipulation. In the first experiment, we obtained the range of the applicable gain for pseudo-haptic weight and its JND with each manipulation. However, the gains cannot be directly compared because the meaning of the gains is different in each manipulation. Therefore, here, we investigated the PSE to enable the comparison of the results of the first experiment between each manipulation.§.§.§ ApparatusThe apparatus was the same as that in the first experiment. However, this time, participants used both hands during the task: one hand was used for the motion-input and the other hand for the force-input manipulation. The hand for each manipulation was switched during the experiment.§.§.§ Task An adaptive staircase method was used to investigate the PSE of the sense of weight between each manipulation. Specifically, the PSE for weight perception at a maximum and minimum gain in the motion-input manipulation was obtained by manipulating a gain in force-input manipulation. Participants used one hand for the motion-input manipulation and the other hand for the force-input manipulation to lift up a virtual dumbbell in VR. Then, the participants answered the question “which dumbbell did you feel heavier?” by “motion-input manipulation” or “force-input manipulation.”The reference dumbbell was with the motion-input manipulation at minimum or maximum gain obtained in the first experiment, i.e., 0.57 or 1.51, respectively. We determined the motion-input manipulation as the reference condition by referring to the observation of the first experiment where most of the participants experienced that the gain around the maximum/minimum threshold with the force-input manipulation was heavier/lighter than that with the motion-input manipulation. The comparison dumbbell was with the force-input manipulation at a gain in the range of minimum and maximum gain obtained in the first experiment, i.e., 0.47 and 1.87, respectively.Figure <ref> shows the conceptual diagram of the adaptive staircase method. The initial gain of the reference dumbbell was either maximum (0.47) or minimum (1.87). There were three levels of distance for each step for the adaptive staircase method. The minimum distance was 0.175 and the others were 0.35 and 0.7. These were 1/2/4 times the minimum distance. If the participants chose the force-input manipulation as a heavier dumbbell, the force-input gain moved one step closer to the maximum, and vice versa. Here, if the gain was likely to move beyond the range, it remained at the same value. The steps first changed with the maximum distance and the distance changed after two turning points. There were also ascending and descending series and each series ended at the sixth turning point. The participants repeated the staircase method two times for each maximum and minimum gain of the motion-input manipulation. Note that the combination of the participant's hands and the manipulations changed between the two repetitions. At the end of the experiment, participants were asked which method was better in a point of sense of weight by a question of “With which manipulation method did you feel a better sense of weight/effort?”§.§.§ Collected dataGains at each turning point were recorded. Then, these gains were averaged to calculate PSE for each participant. Furthermore, the answer to the last question "With which manipulation method did you feel a better sense of weight/effort?" was saved.§.§.§ ProcedureEighteen participants (16 males and 2 females in their twenties excluding two in their 40s) participated in the experiment. The flow before the main task was completely the same as in the first experiment. After the practice trial, participants conducted the main task of the staircase method. When participants changed the combination of the hand and manipulation, they rotated their chair and VR scene was re-centered. The combination of the order of the maximum/minimum gain of the motion-input manipulation and the order of the hand-manipulation combination was counterbalanced between the participants. The participants could take a break whenever they wanted. After the experiment, the participants were remunerated with an Amazon gift card of 1000 yen for their participation. The total duration of the whole procedure was around 30 minutes.§.§.§ Results Gains at each turning point were averaged to calculate the PSE for each participant. Then, paired t-test was conducted for the results between the combinations of hand and manipulation. As a result, no significant difference was found. Therefore, we determined to compute PSE by averaging the results of 2 repetitions (different hand-manipulation combinations). Table <ref> shows the results of the PSE.As a result of the last question, 7 out of 18 participants mentioned that the motion-input manipulation provides a better sense of weight/effort, while the other 11 participants mentioned the opposite.§ DISCUSSION §.§ The force-input can have wider pseudo-weight range while the motion-input can have higher resolution Figure <ref> indicates the summary of the results of the two experiments. Taken together the results of the first and second experiments, the force-input manipulation can actually provide a wider range of pseudo-haptic weight perceptions (about 1.8 times wider), i.e. lighter and heavier perceptions. This results suggest that FISpH can have wider range of pseudo-weight perception (RQ1) and support the effectiveness of the proposed concept. It is noteworthy that these findings imply that users can experience varying degrees of pseudo-weight perception, even when interacting with significantly different body schema, such as controlling a virtual hand using pressure input from an index finger, without much training effort. This would suggest that people could integrate the substituted haptic cue with visual motion cue to estimate virtual weight. One thing to note is that we cannot rule out the possibility that the results would be different if different formulas or implementation were used for the force-input manipulation. Nevertheless, this paper's main contribution is that it successfully suggested that the perceived intensity range of pseudo-haptics can be extended in both heavy and light directions with a given implementation, and confirmed the potential of this concept as the first step.However, at a point of a number of levels of presentable pseudo-haptic weight, the motion-input manipulation can present 1 more level (9 levels) than the force-input (8 levels). Moreover, the difference in weight perception between each step is smaller in the motion-input manipulation than in the force-input. The JND of pseudo-weight perception was gain of 0.1 for motion-input and gain of 0.17 for force-input manipulation. These results conclude that motion-input can have more multiple levels of pseudo-weight perception (RQ2). Still, again, these results could be different by improving the implementation of force-input body manipulation. §.§ The motion-input is easier to operateRegarding usability, the force-input manipulation had a lower control compared to the motion-input manipulation as participants overshoot about 23 cm from the point they wanted to stop although the total number of reaching the target was not largely different. These findings imply that the current implementation of the force-input manipulation could be manipulable, but it may exhibit not small (approximately a 23 cm) margin of error when utilized for tasks requiring haste (RQ3). However, this paper did not aim to design a control law from force input to output motion but focused on the investigation of the pseudo-haptic perception of the force-input manipulation as a first study. Therefore, future work should consider the improvement of the implementation of the force-input manipulation in a point of usability and sense of embodiment. In summary, with the current implementation, it can be concluded that the motion-input manipulation should be employed when a small modification of weight perception is enough but better operability of manipulation is required, while the force-input should be employed when a larger range of weight perception is desired. §.§ The difficulties of measuring pseudo-hapticsAlthough the paper tried to evaluate pseudo-haptic perception from a view of the threshold (range), JND (resolution), and PSE (equality) with psychophysical methods, how to evaluate pseudo-haptics is one of the biggest topics to be discussed and there is no consensus on this point yet. Then, this subsection reviews and discusses the experimental methods to evaluate pseudo-haptics, and concludes that our experimental method could be one of the promising solutions especially when comparing pseudo-haptics in very different methods. First, the Likert scale asking participants to answer their pseudo-haptic perception from 1 to 7 can be done with a relatively simple experiment design and the experiment requires relatively little time. However, it cannot be clear to what extent the proposed method can extend the pseudo-haptics quantitatively. Therefore, it would be a good method as a first step of the research to see if the proposed method is effective or in cases where there are many experimental conditions, tasks, or questions. Then, the intensity of pseudo-haptics can be evaluated by directly answering the intensity on a scale from 0 to 100 using VAS (e.g., <cit.>). To directly answer the haptic sensation (VAS) may reflect the person's pure perception. However, the results tended to have a large variance because the evaluation axis in a mind may vary from person to person. Moreover, the participant's understanding of the question may cause their answers to be biased. In most cases, the design of proper wording in the question would be difficult. For example, when participants evaluate pseudo-haptics in a question about the sense of weight, the interpretation of "weight" can be different for each participant. Participants may evaluate the weight from their fatigue, sense of body motion, or other similar perception such as stickiness. Therefore, researchers should pay attention to the design of the question and explanation for the experiment. Next, the behavioral indicators such as exerted force (gripping force) could capture the effect of pseudo-haptics (<cit.>). Although the results of the exerted force seem to reflect the intensity of pseudo-haptics at a certain level, the variances are relatively large and does not completely fit the subjective results. Therefore, further study is needed to determine how accurately the difference in pseudo-haptic perception can be captured in the behavioral factor or how to improve it. Moreover, a psychophysical method where pseudo-haptics were evaluated by referring to physical quantities can also be used (e.g., <cit.>) At first glance, this method may seem to give the most robust results because this method does not need to suffer from wording issues. However, we need to keep in our mind that the pseudo-haptic perception may not be qualitatively equal to the corresponding haptic perception in our daily life <cit.>. Moreover, the experimental results suggest that the results would be different depending on the experimental methods, that is, adjusting the visual gain against the physical quantities (e.g., <cit.>) or vice versa (e.g., <cit.>). This can note that researchers need to pay attention to the experimental design not to underestimate the capability of pseudo-haptics techniques due to the experimental task.Subsequently, in the paper, we tried to evaluate the pseudo-haptics from a view of the range with resolution and PSE.This information enables us to get the characteristics of the pseudo-haptics technique and compare the different techniques. However, it is difficult to determine what constitutes a range in pseudo-haptics. Moreover, it is also difficult to evaluate the thresholds because the evaluation method also affects the results <cit.>. Here, there are two possible thresholds for the spatial limitation in pseudo-haptics: a detection threshold of visual/physical discrepancy (e.g., <cit.>) and an acceptable threshold with noticing the discrepancy (e.g., <cit.>). Therefore, it can be considered that there are four levels of pseudo-haptics experience: the first level is that users do not feel pseudo-haptics nor realize the pseudo-haptics technique (mostly the visual gain); the second level is that users feel pseudo-haptics but unaware of the technique; the third level is that users feel pseudo-haptics and aware of (accept) the technique; the fourth level is that users hardly feel pseudo-haptics or have large variances for the occurrence of pseudo-haptics and feel unnatural or discomfort for the technique. In the conventional pseudo-haptic studies, the range was defined as one including the first and second levels because the psychophysical experiment can figure out relatively exact and robust threshold of the second and third levels. This method is probably the most strict and robust method for evaluating pseudo-haptics at present. However, because the third layer can be and is often used in actual applications <cit.>, this strict evaluation would underestimate pseudo-haptics potential. On the other hand, in the present paper, the range was defined as one including the first, second, and third levels. we consider that this method can have the most practical results that can be used in actual applications at present. However, it is difficult to determine the threshold of the third and fourth levels. For example, the word "discomfort" has a wide range of meanings and can be interpreted ambiguously by each participant. Therefore, further study is required to investigate how to determine a proper threshold for the third and fourth levels. §.§ Limitations and Future workFirst, the paper focused on the pseudo-haptics and did not investigate other important factors of avatar manipulation such as sense of embodiment or more detailed task performance. Sense of embodiment plays an important role in sense of immersion/presence, i.e. sense of being there, in VR <cit.>. In the future work, these factors should be considered and the implementation of the design of the force-input manipulation should be improved referring to the results. Specifically, the law of converting force into virtual body motion has a space to be improved. One thing to note is that the results obtained in the paper can be different with the different implementation. Moreover, another possible future work is to investigate the methodology to develop a force-input manipulation system of the whole virtual body. In the current experiment, the implementation of force-input manipulation was limited to the vertical motion of a virtual hand. § CONCLUSION The paper investigated effectiveness of the force-input body manipulation on pseudo-weight perception in a comparison with that of the motion-input manipulation. Specifically, the study investigated the force-input manipulation where the motion velocity of the virtual hand corresponds to the force applied to a force sensor in a comparison with motion-input manipulation. The experimental results suggest that the reaction force induced by the force-input could substitute the original haptic cue and be integrated with the visual motion cue, inducing pseudo-weight perception. Moreover, the force-input manipulation can extend the range of presentable pseudo-weight perception by 80% compared to the motion-input manipulation, while the motion-input manipulation has 1 more steps of the presentable pseudo-weight levels. Taken together, with the current implementation, it can be concluded that the motion-input manipulation should be employed when a small modification of weight perception is enough but better operability of manipulation is required, while the force-input should be employed when a larger range of weight perception is desired.§ ACKNOWLEDGMENTSThis work was partially supported by the MEXT Grant-in-Aid for Scientific Research (S) (19H05661), Grant-in-Aid for JSPS Fellows (21J12284), and Grant-in-Aid for Research Activity Start-up (23K20007).IEEEtran[ < g r a p h i c s > ]Yutaro Hirao is an Assistant Professor at Nara Institute of Science and Technology (NAIST), Japan (23-). His main research interests include virtual reality (VR), cross-modal interaction, haptic perception (pseudo-haptics), and embodiment. He received his B.S. and M.S. in engineering from Waseda University (18-20) in Japan, and his Ph.D. in information science and technology form the University of Tokyo (20-23).[ < g r a p h i c s > ]Takuji Narumi is an associate professor at the Graduate School of Information Science and Technology, the University of Tokyo. His research interests broadly include perceptual modification and human augmentation with virtual reality and augmented reality technologies. He received BE and ME degree from the University of Tokyo in 2006 and 2008 respectively. He also received his Ph.D. in Engineering from the University of Tokyo in 2011.[ < g r a p h i c s > ]Ferran Argelaguet is an Inria research scientist at the Hybrid team (Rennes, France) since 2016. He received his PhD degree from the Universitat Politècnica de Catalunya (UPC), in Barcelona, Spain in 2011. His main research interests include 3D user interfaces, virtual reality and human-computer interaction. He was program co-chair of the IEEE Virtual Reality and 3D User Interfaces conference track in 2019 and 2020, and the journal track in 2022.[ < g r a p h i c s > ]Anatole Lécuyer is director of research and head of Hybrid team at Inria, Renne, France. He is currently Associate Editor of IEEE Transactions on Visualization and Computer Graphics, Frontiers in Virtual Reality and Presence. He was Program Chair of IEEE VR 2015-2016 and General Chair of IEEE ISMAR 2017. Anatole Lécuyer obtained the IEEE VGTC Technical Achievement Award in Virtual/Augmented Reality in 2019. | http://arxiv.org/abs/2311.15546v1 | {
"authors": [
"Yutaro Hirao",
"Takuji Narumi",
"Ferran Argelaguet",
"Anatole Lecuyer"
],
"categories": [
"cs.HC"
],
"primary_category": "cs.HC",
"published": "20231127052203",
"title": "Move or Push? Studying Pseudo-Haptic Perceptions Obtained with Motion or Force Input"
} |
firstpage–lastpage Evidence of spin density waves inLei Shu================================== After GW170817, kilonovae have become of great interest for the astronomical, astrophysics and nuclear physics communities, due to their potential in revealing key information on the compact binary merger from which they emerge, such as the fate of the central remnant or the composition of the expelled material. Therefore, the landscape of models employed for their analysis is rapidly evolving, with multiple approaches being used for different purposes. In this paper, we present , a semi-analytic framework which predicts and interprets the bolometric luminosity and the broadband light curves of such transients.models the merger ejecta structure accounting for different ejecta components and non-spherical geometries. In addition to light curve models from the literature based on time scale and random-walk arguments, it implements a new model, , which is grounded on a solution of the radiative transfer equation for homologously expanding material. In order to characterize the variety of the ejecta conditions, it employs time and composition dependent heating rates, thermalization efficiencies and opacities. We comparelight curves with reference radiative transfer calculations, and we find thatsignificantly improves over previous semi-analytic prescriptions. We viewas an ideal tool for extensive parameter estimation data analysis applications. stars: neutron – methods: analytical, numerical § INTRODUCTION The detection of electromagnetic counterparts of gravitational wave signals represents one of the key aspects of gravitational wave astrophysics and, more in general, of multimessenger astronomy. While the gravitational wave signal produced by a coalescing compact binary encodes many properties and information about the merging system (e.g. the chirp mass, the masses of the two compact objects or their tidal deformation, if at least one of the two is not a black hole), the electromagnetic signal can provide complementary information, including for example the amount of matter expelled during the merger and its chemical composition. Other aspects of the merger, such as the nature of the coalescing objects or of the remnant that forms after the merger, could affect both the gravitational and the electromagnetic emission. In this case, the presence of more than one signal can provide tighter constraints and help discriminating between ambiguous or degenerate situations <cit.>.The potential of multimessenger astrophysics was recently revealed by GW170817 <cit.>. Just from the analysis of the gravitational wave signal, it was impossible to exclude that the coalescing objects were two black holes, since the posterior of the binary tidal deformability was extending down to 0 <cit.>, the value expected for a binary black hole. The identification of the system as a binary neutron star was mostly based on the values of the masses of the merging bodies and, more importantly, on the detection of two electromagnetic counterparts, namely a short gamma-ray burst and a kilonova<cit.>. Such a combination of signals was indeed very useful in providing constraints on the equation of state of nuclear matter or in shedding light on the central engine of gamma-ray bursts <cit.>. On the other hand, in the case of the subsequent binary neutron star merger, GW190425 <cit.>, or in the first observed black hole-neutron star systems <cit.>, no electromagnetic counterparts were observed <cit.>.In these cases, the nature of the binary was deduced only from the inferred masses, while the gravitational signal alone was not informative on the tidal deformability of the system, due to the lower expected values and to the not sufficiently high signal-to-noise ratio. Among the different electromagnetic counterparts, the kilonova is one of the most peculiar transients associated to compact binary mergers involving at least one neutron star <cit.>. It arises when the merger and its remnant expel a non-negligible amount of neutron-rich matter, which undergoes r-process nucleosynthesis <cit.>; see also <cit.> for recent reviews. The decay of the freshly produced radioactive elements moving from the neutron-rich side towards the bottom of the nuclear valley of stability releases nuclear energy that, despite the fast expansion, keeps the expanding ejecta hot. Due to expansion the matter opacity to the electromagnetic radiation decreases until photons can eventually diffuse out, producing a kilonova <cit.>. Depending on the mass of the ejecta, on their expansion velocity andcomposition, the peak of the kilonova emission is expected to occur between a few hours and several days after the merger <cit.>. At the same time, the emission is expected to evolve, moving from bluer to redder frequencies as a consequence of the photospheric expansion, of the decrease in the nuclear energy input and in the opacity of matter, as well as of the viewing angle <cit.>. As long as the opacity of the innermost ejecta is large enough, the kilonova is characterized by the presence of a photosphere and the resulting emission can be described, in good approximation, as quasi-thermal. Non-thermal and non-local thermodynamics equilibrium effects become more and more relevant as time increases, until the transient enters its nebular phase. In the case of AT2017gfo (the kilonova associated to GW170817) the transition from a full photospheric regime to the nebular phase happened between a few days to a week after merger <cit.>.The modelling of kilonovae is extremely challenging. It requires the solution of a radiative transfer (RT) problem in a fast expanding, radioactive and radiation dominated medium. Not only the composition of matter changes with time due to nuclear reactions and decays, but due to the expansion and to the interaction between matter and radiation, atoms inside the ejecta (which are initially fully ionized due to the large matter temperature) span different ionization levels, following the progressive electron recombination. The presence of heavy elements, and in particular, of lanthanides and actinides, largely increase the photon opacity due to bound-bound and bound-free transitions involving the f electron shells <cit.>. For most of the heavy elements, opacities due to ionized species and for matter in the thermodynamics regime relevant for kilonovae are experimentally unknown and their values are usually provided by non-trivial atomic structure calculations <cit.>. Furthermore, large uncertainties still affect the calculation of the detailed nuclear energy released by r-process elements <cit.>, as well as the estimation of the fraction of the energy that the expanding matter is able to thermalize <cit.>. In addition to the difficulties related with the problem of transporting photons inside an expanding, radioactive medium, an additional challenge is represented by the fact that a binary neutron star merger or a black hole-neutron star merger can expel matter with different properties and, possibly, with a high degree on anisotropy <cit.>. This implies that the medium inside which the photons are produced, diffused and emitted can have a non-trivial stratification, as well as angular distribution.It is not surprising that, given the complexity of the kilonova scenario and the variety of aspects involved, so far the problem of predicting or producing kilonova light curves and spectra has been tackled by a large variety of models, employing very different levels of sophistications and approximations.Some models solve the photon transport problem in an expanding medium considering wavelenght- and composition-dependent opacities, computed consistently and coupled to the calculation of the abundances of the different ion species, assuming local thermodynamics equilibrium <cit.>. These models are the most sophisticated and reliable ones, but they necessarily require large computational resources, especially in three dimensions.Other examples of kilonova models include TARDIS <cit.>, which solves the 1D photon transport problem in the optically thin atmosphere above a predefined photosphere, POSSIS <cit.>, which uses pre-computed wavelenght- and time-dependent opacities on a 3D Cartesian grid, or SNEC <cit.>, which solves radiation hydrodynamics in spherical symmetry through a gray flux-limited diffusion approach. These more approximated approaches clearly reduce the computational effort, especially if some symmetry is invoked.At the opposite extreme, kilonova light curves have also been computed by using simplified kilonova models that avoid the direct solution of the RT problem, since they are often based on the solution of the energy conservation equation inside the ejecta or on time scale arguments mimicking the mean features of the photon diffusion problem <cit.>. They usually employ gray, constant opacities and can reproduce some of the most relevant features of the kilonova emission, at least at a qualitative level. The extremely reduced computational costs of these models allows their usage in multi-dimensional parameter estimate analysis, which requires the evaluation of millions, if not billions of kilonova light curves <cit.>.With the increase of the number and sensitivity of gravitational wave detectors, the amount of multimessenger signals, and in particular, of kilonova counterparts of gravitational wave events, is expected to significantly grow in the years to come. For example, during the fourth observational campaign of LIGO, Virgo and KAGRA, the number of detected binary neutron star merger is expected to be a few tens per year <cit.>.Additionally, the careful (re)analysis of the afterglow signals of close-by gamma-ray bursts can reveal signatures of kilonova emission, as in the case of the exceptionally bright GRB211211A <cit.>, or more recently also in the case of the long GRB230307A <cit.>. Given the present scenario, characterized by a growing number of kilonovae, which could significantly differ in terms of intrinsic properties, as well as in the quality and quantity of the data, it is still imperative to improve on the accuracy of fast and approximated kilonova models. The latter can be complementary to more sophisticated models, since they can be used for extensive parameter estimations, and to provide a robust and reliable framework to analyze coherently kilonova emissions coming from very different events. They can also be coupled to gravitational wave data analysis in the quest for coherent and genuine multimessenger analysis.In this paper, we present , a semi-analytic framework to perform analysis of kilonova emission, both in terms of bolometric luminosity and broadband light curves.inherits the possibility of adding several ejecta components and of prescribing non-trivial ejecta geometries from previous implementations <cit.>. However, with respect to the latter, it aims to improve on the accuracy and on the reliability of the resulting light curves by replacing the kilonova model grounded on time scale arguments with a different model, , based on a semi-analytic solution of the diffusion equation for homologously expanding ejecta. Such a solution was presented in <cit.>, and based on works reported in <cit.>. Here, we expand the class of solutions, by considering time dependent heating rates, thermalization efficiencies and opacities. Moreover, we improve the physical input, by including composition dependent heating rates and opacities.The paper is structured as follows: in sec:1d_model we present in detail the spherically symmetric, kilonova emission model , distinguishing between the optically thick (subsec:op_thick) and the optically thin (subsec:op_thin) part. The general multi-component, anisotropic framework ofto compute light curves is presented in sec:xkn framework, while in sec:input_physics we detail the input physics entering the model, listing the heating rates (subsec:heating_rates) and the opacity (subsec:opacities) prescriptions. In sec:RT_comparison we compare the results of the variousmodels with the ones obtained using a RT code <cit.>, taken as reference, in order to address the degree of accuracy and the limitations of our approach. We provide a summary and the conclusions of our work in sec:conclusions. § SEMI-ANALYTIC 1D KILONOVA MODELOur kilonova model is based on a one-dimensional model for the diffusion and emission of photons from homologously expanding, radioactive matter. More specifically, the kilonova emission is calculated as the combination of two contributions, one emitted at the ejecta photosphere, i.e. the surface delimiting the optically thick bulk of the ejecta and from which photons can escape and move inside the atmosphere, and one coming from the optically thin layers above it. In the following, we separately present these two contributions. §.§ Optically thick ejecta treatment The contribution to the luminosity arising from the photosphere is computed starting from the semi-analytic formula originally proposed by <cit.> with the intent to treat the ejecta from Type Ia supernovae (SNe Ia), and later adapted by <cit.> to model kilonovae. In spite of its simplifying assumptions, this formula can qualitatively reproduce the thermal evolution of the ejecta. In the following, we report in broad lines its derivation. The ejecta fluid is assumed to be optically thick throughout its entire depth and the radiation field properties are evolved on the basis of the time-dependent equation of RT. In particular, we consider the first two frequency-integrated moments of such equation in the comoving frame, calculated to order O(v/c):DE/Dt+1/r^2∂/∂ r(r^2F)+v/r(3E-P)+∂ v/∂ r(E+P)==∫_0^∞(4πη_ν-cχ_νE_ν)dν, 1/c^2DF/Dt+∂ P/∂ r+3P-E/r+2/c^2(∂ v/∂ r+v/r)F=-1/c∫_0^∞χ_νF_νdν,where v is the fluid velocity field, r the radial coordinate, χ_ν the extinction coefficient and η_ν the volume emissivity, while the subscript ν indicates the frequency dependence. Here E, F and P are the energy density, flux and radiation pressure of the radiation field, respectively. D/Dt indicates the comoving (Lagrangian) derivative.The solution of this set of equations is found by adopting a series of hypotheses.* We assume a homologous expansion of the fluid, i.e. each fluid element expands with constant radial speed. Under this assumption, the ejecta maintain their proportions while expanding with an external radius R(t)≃ v_maxt, where v_max is the maximum outflow velocity. The homologous expansion hypothesis is consistent as long as the energy heating the fluid does not affect the fluid motion in a significant way.* We resort to the Eddington approximation, wherein the radiation field is isotropic and the relation E=3P holds. The latter is valid since the outflow is optically thick, as one can expect at least in the early stages of its evolution. When the fluid becomes transparent at later times, even if this approximation breaks down, the error has still little effect on the bolometric luminosity.* Regarding the energy balance, we assume that the gas internal energy, that is its thermal kinetic energy as well as the ionization energy, is subdominant with respect to the radiation field energy, and therefore we ignore the former (radiation dominated conditions).* Additionally, we assume that the absorbed heat is immediately re-radiated as thermal emission, and thus:∫_0^∞(4πη_ν-cχ_νE_ν)dν=Ė_ heat,where Ė_ heat is the energy deposition rate per unit volume.In light of the above considerations, we act on eq:rad_trans_E and eq:rad_trans_F with the aim to simplify them. eq:rad_trans_E is an equation for the energy density field, E(r,t), and in order to ensure the correct radiation energy balance we retain all terms to the order O(v/c), as E(r,t) changes considerably on the hydrodynamic timescale. eq:rad_trans_F is instead an equation for the radiation momentum F(r,t) and we solve it at lower order by discarding all time and velocity-dependent terms. This choice is appropriate on the fluid-flow timescale as we assumed that F(r,t) is not relevant for the outflow dynamics. Hence by inverting eq:rad_trans_F we obtain:F=-c/3χ∂ E/∂ r,with χ a properly defined frequency-averaged inverse mean free path. The above expression for the flux can be inserted in eq:rad_trans_E, resulting in:DE/Dt-c/3r^2∂/∂ r(r^2/χ∂ E/∂ r)+4Ṙ/RE=Ė_ heat,where Ṙ denotes the derivative with respect to time. We now introduce κ as an absorption opacity, homogeneous in space (but not necessarily in time), such that χ=κρ, with ρ the fluid density. Moreover, we express Ė_ heat as Ė_ heat = ϵ̇ f_ th, whereϵ̇ is the energy release rate per unit mass and f_ th a thermalization efficiency coefficient, and both are function of time. From the assumption of homologous expansion, we recall that:v=r/t,v=v_maxx ,where x ∈ [0,1] is the dimensionless radius coordinate. Moreover, in the radiation transport equation we adopt the following single-zone homologous solution for the expansion profile:ρ(t)=ρ_0(t_0/t)^3 ,with t_0 the initial time of the expansion, and ρ_0 the density at t_0. We approximate the latter as:ρ_0=M_ej/4/3π(v_maxt_0)^3,where M_ej is the ejecta mass.Furthermore, from the hypothesis of radiation dominated gas, we employ the polytropic equation of state in the Eddington approximation to obtain E∝ t^-4. If we assume that the residual dependences of E can be separated into a spatial profile ψ(x) and a temporal profile ϕ(t), we can write:E(x,t)=E_0(t_0/t)^4ψ(x)ϕ(t) ,where E_0 is treated as a free parameter. In particular, assuming that the radiation field has a black-body spectrum, we relate it to an initial black-body temperature:T_0=(E_0/a)^1/4,with a=4σ_SB/c=7.5657×10^-15 erg cm^-3K^-4 being the radiation constant and σ_SB the Stefan-Boltzmann constant. Using eq:homologous, eq:density and eq:energy_density, the transport equation eq:diffusion becomes:( E_0t_0/ρ_0) 1/t[ ψ(x)ϕ'(t)-t/t_0 τϕ(t) 1/x^2(x^2ψ'(x))' ] = f_ thϵ̇_r ,with the prime superscript on a function indicating the derivative with respect to its variable and whereτ≡3κρ_0/c(v_maxt_0)^2 .The latter is a comprehensive factor comparable to the diffusion time scale that carries a possible dependence on t through κ. The homogeneous form of eq:diffusion2 can be solved by means of variable separation, according to which the resulting two functions in x and t must be identically equal to the same separation constant, λ:1/x^2ψ(x)(x^2ψ'(x))'=-λ, τ_0(t_0/t)ϕ'(t)/ϕ(t)=-λ.resulting in two ordinary differential equations to be solved. The equation for the spatial profile can be expressed as an eigenvalue equation for the operator A:Aψ(x) ≡ -1/x^2(x^2ψ'(x))'=λψ(x) .The eigenfunctions of eq:eigeneq can be determined by imposing suitable boundary conditions to the problem. While for the temporal profile of the energy density E(x,t) we naturally assume ϕ(t_0)=1 and ϕ(∞)=0, for the spatial part it is reasonable to consider a reflection symmetry at x=0 and a radiative-zero condition at x=1, being the outflow optically thick:ψ'(0)=0 , ψ(1)=0 .eq:conditions can be directly translated into identical conditions for the eigenfunctions of eq:eigeneq. If we impose the normalization requirement:⟨ψ_n|ψ_m|=⟩δ_n,m,as we adopt the notation:⟨f|g|=⟩∫_0^1f(x)g(x)x^2dx ,the resulting homogeneous spatial eigenfunctions are:ψ_n(x)=√(2) sin(nπ x)/x,with λ=n^2π^2 and n any positive integer.eq:eigensol represents a complete orthonormal basis on the interval of interest, and therefore we can expand the general solution of eq:diffusion2 on the latter:E(x,t)=∑_n=1^∞c_n(t)ψ_n(x) .Here the coefficients c_n(t) retain a generic time dependence, and we can conveniently redefine them in the form:c_n(t)=E_0(t_0/t)^4ϕ_n(t) ,thus obtaining:E(x,t)=E_0(t_0/t)^4∑_n=1^∞ϕ_n(t)ψ_n(x) .Using eq:expansion in eq:diffusion2, we can exploit the orthonormality of ψ_n(x) integrating over x∈[0,1] to find:ϕ_n'(t)+(t/t_0τ)(n^2π^2)ϕ_n(t) =(-1)^n+1ρ_0 √(2)/nπ E_0(t/t_0)ϵ̇ f_ th.Finally, the bolometric luminosity is found by employing eq:flux and eq:expansion to compute the flux at the surface of the ejecta:L(t) =4π R^2(t)[x^2F(x,t)]_x=1 =4π(v_maxt_0 )^3√(2)E_0/τ∑_n=1^∞(-1)^n+1nπϕ_n(t) ,where ϕ_n(t) are the solutions of eq:temporal_eq, that can be obtained once the time-dependence of ϵ̇, f_ th and κ has been specified. If the formal solution of eq:temporal_eq is complex, when we compute the luminosity through eq:luminosity we take only the real part of it. Since this model is valid in the limit of optically thick matter, the outcome of eq:luminosity is rescaled by a factor M_thick/M_ej, where M_thick is the mass of the optically thick portion of ejecta, defined as the region enclosed by the photosphere:M_ thick=4π∫_0^R_ ph(t)ρ(t,r)r^2dr .Differently from the single-zone approximation adopted in the solution of the RT equation, herewe choose a more accurate space-dependent density profile such as the self-similar homologous solution <cit.>:ρ(t,x)=ρ_0(t_0/t)^3(1-x^2)^3 .The photospheric radius evolution R_ph(t) can be found analytically by imposing the conditionτ_γ(R_ph)=2/3 ,with τ_γ the optical depth of the material:τ_γ(t,x)=κ∫_x^1ρ(t,x')dx' ,and ρ(t,x) the density of eq:density_profile. The determination of R_ ph(t) implies the solution of a seventh order polynomial equation. However, regardless of the ejecta parameters, the temporal evolution of R_ ph resembles a parabolic behaviour. Then, we approximate it as a parabolic arc with extremes fixed by the condition R_ph=0 applied to eq:photosphere_condition, i.e.:t_1=0 ,t_2=√(27M_ ejκ/8π v_ max^2), and curvature fixed by a third point, t_3, taken in the proximity of t_1, where eq:photosphere_condition can be solved by assuming (R_ max-R_ph)/R_ max≪ 1. By adopting this approximate solution, the error on the photosphere position with respect to that provided by the exact solution of eq:photosphere_condition is contained within 8%.We characterize the emission at the photosphere by assuming a Plankian black-body spectrum, and thus we compute the associated photospheric temperature T_ph(t) by means of the Stefan-Boltzmann law:T_ ph(t)= max[(L_ thick(t)/4πσ R_ ph^2(t))^1/4,T_ floor] ,with L_ thick(t) the luminosity of the thick part of the ejecta. A temperature floor T_ floor is applied in order to approximately account for the electron-ion recombination during the ejecta expansion <cit.>. When T_ph(t) reaches the temperature floor, R_ph(t) is thus recomputed solving the implicit equation obtained from the Stefan-Boltzmann law. The value of T_ floor, generally treated as model parameter, is in fact dependent on the ejecta opacity and therefore closely linked to its composition. For this reason, we also include the possibility to interpolate the floor temperature between two model parameters T_ Ni and T_ La, corresponding to characteristic recombination temperatures in a Lanthanides-poor (Y_e≳0.3) and a Lanthanides-rich (Y_e≲0.2) environment, respectively.In the following, we discuss a few cases for which the solutions of eq:temporal_eq can be obtained analytically, based on temporal dependence of κ, ϵ and f_ th.§.§.§ Constant opacity, constant thermalization efficiency and power-law heating rate We first consider the case in which κ is not only uniform in space, but also constant in time, i,.e. κ = κ_0. In this case, the quantity τ becomes a constant, τ_0 = 3 κ_0 ρ_0 (v_ maxt_0)^2/c. Additionally, we consider f_ th to be a constant, f_ th,0, whileϵ̇ = ϵ̇_0 ( t/t_0)^-α. In this case, the solutions of eq:temporal_eq take the explicit form:ϕ_n(t)=exp(-π^2n^2t^2/2t_0τ_0)[K_n+A_nΓ(1-α/2,-π^2n^2t^2/2t_0τ_0)] ,where Γ(s,x) is the upper incomplete gamma function, defined as:Γ(s,x) ≡∫_x^∞t^s-1e^-tdt ,while A_n are constants:A_n= (-1)^n+1+α/2(n π)^α-3√(2)^1-α( ρ_0 f_ th,0ϵ̇_0 τ_0/E_0) ( t_0/τ_0)^α/2,and K_n are integration constants fixed by the boundary conditions.In order to find the latter, we exploit the assumptions over the temporal profile of the energy density. While ϕ(∞)=0 is automatically satisfied by the form of ϕ_n(t), one needs to translate ϕ(t_0)=1 into a condition on ϕ_n(t). We choose to assign ϕ_n(t_0)=δ_n,1 to ensure the convergence of eq:luminosity:K_n =δ_n,1exp(-π^2 n^2 t_0/2 τ_0) -A_n Γ(1-α/2,-π^2n^2t_0/2 τ_0) . §.§.§ Constant opacity, power-law thermalization efficiency and heating rate The previous solution can be trivially generalized to the case in which both the specific heating rate and the thermalization efficiency follow a power-law evolution:ϵ̇ = ϵ̇_0 ( t/t_0)^-α,f_ th = f_ th,0( t/t_0)^-β. In this case, eq:phi_n sol 1-eq:A_n sol 1 are still a solution of eq:temporal_eq, once α has been replaced by α' and α' ≡α + β.§.§.§ Constant opacity, constant thermalization efficiency and power-law heating rate with exponential termsWe then consider the case in which the opacity and the thermalization efficiency are constant in time, while the specific heating rate can be written asϵ̇ = ϵ̇_0 ( t/t_0)^-α + B e^-t/β.The solutions of eq:temporal_eq becomes:ϕ_n(t) =exp(-π^2n^2t^2/2t_0τ_0)[K_n+A_nΓ(1-α/2,-π^2n^2t^2/2t_0τ_0) + .+ . B_nerfi( t_0 τ_0 - π^2 n^2 β t/√(2)π n √(t_0 τ_0)β)]+ C_n e^-t/β,where erfi(x) ≡ -i erf(ix) is the imaginary error function and the error function is defined aserf(z) ≡2/√(π)∫_0^z e^-t^2 dt .The B_n, C_n and K_n coefficients readB_n =(-1 )^n exp(-t_0 τ_0/2 π^2 n^2 β^2) √(t_0)τ_0^3/2ρ_0 B/π^7/2E_0 β n^4, C_n =(-1 )^n+1√(2)τ_0 ρ_0 B/π^3 E_0 n^3, K_n = δ_n,1exp(-π^2 n^2 t_0/2 τ_0) -A_n Γ(1-α/2,-π^2n^2t_0/2 τ_0) +- B_nerfi( t_0 τ_0 - π^2 n^2 β t_0/√(2)π n √(t_0 τ_0)β) - C_n exp( π^2 n^2 t_0/2 τ_0-t_0/β) .This solution can be trivially generalized to the case in which more than one exponential term is added to the power law term in eq: heating w exponential.§.§.§ Power-law opacity, thermalization efficiency and heating rate Finally, we consider the case in which the opacity, the thermalization efficiency and the specific heating rate have a temporal power-law dependence: ϵ̇ = ϵ̇_0 ( t/t_0)^-α,f_ th = f_ th,0( t/t_0)^-β, κ = κ_0 ( t/t_0)^-γ. In this case, τ can be expressed as τ = τ_0 (t/t_0)^-γ so that eq:temporal_eq becomes:ϕ_n'(t)+(t/t_0)^1+γ(n^2π^2)/τ_0ϕ_n(t) =(-1)^n+1ρ_0 √(2)ϵ̇_0 f_ th,0/nπ E_0 (t/t_0)^1-α'.In this case, the solution of eq:temporal_eq_opacity_dep becomes:ϕ_n(t) =exp(-π^2n^2t^2+γ/(2+γ)t_0^(1+γ)τ_0) ××[K_n+A_nΓ(2-α'/2+γ,-π^2n^2t^2+γ/(2+γ)t_0^1+γτ_0)] ,where A_n and K_n are defined asA_n= (-1)^n√(2)ρ_0 f_ th,0ϵ̇_0 t_0/E_0 n π (γ + 2)( - t_0/τ_0 n^2 π^2/(γ+2))^α'-2/γ+2andK_n =δ_n,1exp(π^2 n^2 t_0/(2 + γ) τ_0) -A_n Γ(2-α'/γ+2,-π^2n^2t_0/(2+γ)τ_0) .§.§ Optically thin ejecta treatment At the times relevant for the kilonova, we expect the r-process material outside of the ejecta photosphere to provide a non-negligible contribution to the heating powering the emission, especially when a proper photosphere will eventually not be identifiable anymore. However this contribution is expected to be considerably different with respect to the one provided by the same portion of the ejecta if the latter were optically thick, i.e. if we assumed R_ ph(t)=R_ max(t).Therefore, in addition to the radiation emitted at the photosphere, we approximate the bolometric luminosity L_ thin(t) from the thin region outside of it, following the prescription of <cit.>. We thus divide this region into N_ thin layers of equal mass dM_i assuming local thermodynamics equilibrium within each layer, and we express such contribution as:L_ thin(t)=∑_i=1^N_ thin f_ th,i(t)ϵ̇(t) dM_i ,where the sum runs over the discrete thin shells, while ϵ̇(t) and f_ th,i(t) are the specific radioactive heating rate and the space-dependent binned thermalization efficiency, respectively, as described in subsec:heating_rates.The total bolometric luminosity of the ejecta is simply obtained by summing the contributions from the two separate regions:L(t)=L_ thick(t)+L_ thin(t) .In fig:luminosity, the total luminosity is displayed together with its components for a simple spherically symmetric model. As visible, the early light curve is dominated by the thick ejecta, which constitutes the majority of the total mass. In this phase most of the energy provided by the radioactive decays is trapped within the ejecta due to the high optical depths. After a few days, the ejecta density has decreased enough for this energy to escape, enhancing the emission up to be instantaneously greater than the thermalized heating rate. Meanwhile, a second contribution to the luminosity steps in, as a relevant portion of optically thin mass emits radiation as well. Finally, after several days, the thick bulk of the ejecta disappears, and the luminosity is completely determined by the optically thin matter. Since the latter is transparent to thermal radiation, the thermalized decay energy escapes without further processing, and the luminosity is equal to the thermalized heating rate. However, the latter is now only a small fraction of the total decay energy rate, since the lower densities make the thermalization process inefficient. Thus, the thermal emission will eventually fade away. Along with the temperature of the photosphere T_ ph, we want to characterize also the temperature in the layers outside of it. For this purpose, we assign to each bin a temperature T_i(t) on the basis of the radial profile proposed by <cit.> and derived for a radiation dominated ideal gas using eq:density_profile:T(t,x)=T_0(t)(1-x^2) .For each time, we fix the factor T_0(t) by requiring the continuity of the profile with the photospheric temperature T_ ph(t). Therefore, for every bin i we obtain:T_i(t)=T_ ph(t)1-x_i^2/1-x_ ph^2(t),where x_i is the position of the bin and x_ ph(t)=R_ ph(t)/R_ max(t) is the position of the photosphere.Recently, <cit.> showed that a synthetic non-local thermodynamics equilibrium evolution of the temperature in the late expanding ejecta features a re-increase from a minimum reached around a few tens of days post merger. This result was obtained by taking into account the material excitation and ionization states in a more careful calculation of the ejecta heating and cooling processes, using the spectral synthesis code . In light of the above computations, we expect the temperature to remain roughly constant at the late times still relevant for the first kilonova phase. Therefore, in order to describe the thermal emission, here we find sufficient to set a unique time-independent minimum temperature value T_ floor for both the thick and the thin part of the ejecta.§ MULTI-COMPONENT ANISOTROPIC SEMI-ANALYTIC KILONOVA MODELEjecta from compact binary mergers are expected to occur in different components, characterized by different properties. Moreover, the ejection mechanisms can result in a anisotropic structure of the ejecta. Motivated by this, we set up a multicomponent, anisotropic kilonova framework. In particular, we closely follow the set-up first proposed by <cit.>, originally based on <cit.> and reprised by <cit.>.The framework assumes axial symmetry around the rotation axis of the binary (denoted as z), as well as reflection symmetry about the z=0 plane. The polar angle θ is discretised in a series of N_θ bins which can be equally spaced either in the angle itself or in cosθ. A kilonova model is specified once the polar distribution of all the relevant quantities (i.e. mass, velocity, opacity or electron fraction, entropy, expansion time scale) are given for each of the ejecta components. Inside each angular bin and for each component, the radial kilonova model described in sec:1d_model (or alternatively the model from <cit.>) can be employed to compute the contribution to the luminosity emerging from that angular bin.Being an extensive quantity, the mass inside the bin needs to be scaled by the factor 4π / ΔΩ, where ΔΩ is the bin solid angle. All the other input quantities are otherwise intensive and do not need any rescaling. Once computed, the isotropic luminosity resulting from the 1D model is scaled again based on the actual emitting solid angle, i.e. it is multiplied by ΔΩ/4π.Within the same angular bin and in the presence of more than one ejecta component, the corresponding luminosity contributions are summed together, assuming that photons emitted from the innermost components irradiate the outermost ones and are subsequently re-processed and re-emitted on a time scale smaller than the expansion one. Moreover, at each time we locate the photosphere of the overall ejecta at the position of the larger individual photosphere. This approach assumes that the different components are nested and that they do not cross each other significantly during the kilonova emission. We expect these hypotheses to be appoximately verified once the homologous expansion phase has been reached and if the late time ejecta are systematically slower than the first expelled ones.The present implementation includes characteristic analytic functional forms for the angular dependences: uniform distributions, step functions, sinθ, sin^2θ, cosθ and cos^2θ dependences.Despite their simplicity, some of these distributions were demonstrated to be remarkably valid in broadly reproducing the outcomes of general-relativistic hydrodynamical simulations, accounting for the preferential equatorial direction of the dynamical component, as well as the excursion in the electron fraction caused by high-latitudes neutrino irradiation. Additionally, the code can interpolate on its angular grid arbitrary distributions, such as azimuthally-averaged angular profiles extracted from numerical simulations <cit.>.Typical kilonova models employed in the past used up to three different components <cit.>. In the case of two components, the fastest one usually refers to the dynamical ejecta, while the slowest one to the disc wind ejecta of viscous origin. A third, intermediate component is sometimes used, possibly originated by magnetic- <cit.> or neutrino-driven wind components <cit.>, as well as from spiral wave wind ejecta <cit.>.In the source frame, the emission is assumed to be thermal and the spectral fluxes are described by a Planckian spectral distribution B_ν(T), i.e.:B_ν(T)=2π hν^3/c^21/ exp(hν/k_ BT)-1,with k_ B the Boltzmann constant and h the Planck constant, both at the photosphere as well as within each thin external layer. In the former case, T is the photospheric temperature, while in the latter it is the temperature inside each mass shell. If the source is located at a luminosity distance D_L, corresponding to a redshift z, for an observer on Earth characterized by a viewing angle θ_ view, the radiant flux at frequency ν and time t (measured in the observer frame) will be the sum over the angular bins of the contributions from the thick ejecta F^ thick_ν,k and the thin ejecta F^ thin_ν,k (computed in the source frame), once the redshift correction has been applied to the time, frequency and luminosity:f_ν(t) =(1+z)/4πD_L^2∑_k=1^N_θ{ p_k(θ_ view)4π/ΔΩ_k[ F^ thick_(1+z)ν,k( t/1+z) . .. .+ F^ thin_(1+z)ν,k( t/1+z)] },with:F^ thick_ν,k(t')=L_ thick,k(t')/σ_ SB T_ ph,k^4(t')B_ν(T_ ph,k(t')) ,and:F^ thin_ν,k(t')=∑_if_ th,i,k(t')ϵ̇_k(t') dM_i,k/σ_ SB T_i,k^4(t')B_ν(T_i,k(t')) .where L_ thick,k is the photospheric luminosity of the bin k, characterized by a photospheric temperature T_ ph,k.The factors p_k(θ_ view) in eq:obs_flux account for the effective emission area as seen by the observer <cit.>, and are calculated using the formula:p_k(θ_ view)=1/π∫_q(θ_ view)·n_k>0q(θ_ view)· dΩ,where q(θ_ view) is the unit vector in the observer direction, while n_k is the unit vector pointing radially outwards from the surface of the bin k.Finally, we compute the AB magnitude at a photon frequency ν as:m_ AB,ν(t)=-2.5 log_10(f_ν(t))-48.6 . § INPUT PHYSICS §.§ Heating ratesThe heating rate powering the kilonova originates from the many decays of heavy elements produced in the r-process nucleosynthesis, and as such it can be computed by employing a nuclear reaction network. The latter calculates the time evolution of the nuclides abundances while keeping track of the energy released in the process. Results obtained by nuclear network calculations retain a strong dependence on the properties of the ejecta, and in particular on the entropy, electron fraction and expansion timescale at the freeze-out from nuclear statistical equilibrium (NSE) <cit.>. Furthermore, nuclear network calculations also depend on the nuclear physics employed, e.g. on the choice of the theoretical nuclear mass model, the reaction rates or the fission fragment distribution. This sensitivity is particularly strong at low electron fractions and the nuclear physics uncertainties can lead to changes in the predicted heating rates of about one order of magnitude <cit.>.With the purpose to provide our kilonova model with a heating rate valid for arbitrary initial conditions, we consider the results of the broad nucleosynthesis calculations reported in <cit.>. In that work, the nuclear composition evolution of a set of Lagrangian fluid elements is computed using the nuclear reaction network<cit.> with the finite-range droplet macroscopic nuclear mass model (FRDM) <cit.>. Eachrun is initialized at a temperature of 6 GK in NSE, and identified by the values of the initial electron fraction Y_e, entropy s, and expansion timescale τ_ exp. The latter are considered as initial parameters and later evolved consistently by the network. More details about these nucleosynthesis calculations can be found in <cit.>. In particular, the heating rates we employ are computed over a comprehensive grid of ∼11700 distinct trajectories with 0.01≤ Y_e≤0.48 linearly spaced, 1.5 k_ B baryon^-1 ≤ s≤ 200 k_ B baryon^-1 and 0.5 ms ≤τ_ exp≤200 ms logarithmically spaced. These intervals are expected to bracket the properties of the ejecta from BNS and NSBH mergers. We fit the heating rate trajectories obtained withover the time interval 0.1 days ≤ t≤50 days after merger, using the following power-law dependence:ϵ̇=ϵ̇_ 1d( t/ 1 day)^-α,where ϵ̇_ 1d and α are fit parameters, with typical values α∼1.3 and ϵ̇_ 1d∼10^10 erg s^-1 g^-1. Such a temporal dependence in the heating rate is expected from the decay of large sample of unstable nuclei <cit.>. Moreover, it is equivalent to eq:heating rate KN model, one of the functional forms used in the optically thick kilonova model described in sec:1d_model, provided a conversion factor between the fit reference time (1 day) and t_0. The quality of each single fit is evaluated using a mean fractional log error as employed in <cit.>, defined as:Δ(ϵ̇)=<|ln(ϵ̇^o(t))-ln(ϵ̇(t))|/ln(ϵ̇^o(t))> ,where ϵ̇^o(t) is the originalheating rate trajectory, while the mean is performed over the fit time window without weighing over the time steps, in order to account for the originalresolution. For most trajectories we find the average relative errors to be smaller than ∼1%. The largest errors are found at the boundary of thegrid, where the relative error can be as large as ∼ 5%.In fig:epsfit_par, the values of the fitting coefficients are plotted against Y_e, s and τ_ exp for representative sections of thegrid. As shown in the left column, for a fixed Y_e the fit parameters are generally smooth functions of the two other thermodynamic variables, and in particular the value of α remains roughly constant (for Y_e≲0.2 it hardly deviates from ∼1.3, as already found in <cit.>), while the value of ϵ̇_0 varies within a factor of a few. On the other hand, the variability of the fit parameters increases as the electron fraction is left free to vary. This strong and non-trivial dependence of the heating rate on the electron fraction is more evident for high Y_e values, where the radioactive heating can be dominated by the decay of individual nuclear species, depending on the specific ejecta conditions. However, we find that the continuity in the fit parameters endures at least in the region which is more relevant to our study, i.e. for Y_e≤0.36, s≤90 k_ B baryon^-1 and τ_ exp≤30 ms. We therefore adopt a trilinear interpolation of the fitting coefficients as functions of Y_e, s, and τ_ exp in that region, while isolated points or boundary areas for which the continuity of the fitting coefficients is poor are treated by using a nearest-point interpolation.In order to account for the efficiency with which decay products thermalize in the ejecta, we apply a thermalization efficiency factor to the heating rate as follows. For the thick core of the ejecta, we consider both a constant thermalization efficiency (compatible with all the analytic solutions presented in sec:1d_model) and a thermalization efficiency with a time evolution f_th = f_ th,0( t/t_0 )^-β, as described in subsec:heating and thermalization power law, and subsec:heating, thermalization and opacity power law. The latter formula approximately mimics the decreasing in the thermalization behaviour expected during the first day in the optically thick ejecta. The values of f_ th,0 and α_ th can be fixed by imposing, for example, a thermalization efficiency of ∼ 0.7 and 0.4 at 0.1 days and 1 day, respectively. For the thin layers of the ejecta instead, we model a thermalization efficiency profile starting from the analytic formula proposed in <cit.> and fitted on the properties of the ejecta:f_th(t,x)=0.36[exp(-aX)+ln(1+2bX^d)/2bX^d] ,where a, b and d are the fit parameters. In that work, this expression was obtained by assuming eq:density, and X(t,x)=t. Here instead, we adopt eq:density_profile, and X(t,x)=t(1-x^2)^-1. We interpolate the fit parameters in eq:thermalization on the tabulated grid reported in <cit.>, which spans the intervals 1×10^-3 M_⊙ < M_ej < 5×10^-2 M_⊙ for the total ejecta mass and 0.1 c < v_ej < 0.3 c for its average velocity. This combination of different efficiencies is motivated by the fact that, on one hand, we expect the decay energy in the thick bulk to thermalize in a similar way as long as the density is sufficiently high. In particular, roughly ∼35% of the energy escapes in the form of neutrinos, ∼45% is constituted by γ-rays which efficiently heat the material only within the first day post-merger, and the remaining ∼20% is carried by β-particles, α-particles and fission yields <cit.>. On the other hand, the thermalization efficiency drops significantly in the outer layers of the ejecta, where the lower density makes it harder for the decay products to deposit their energy through thermal processes.fig:thermalization shows the modeled thermalization efficiency profile for different times after merger. By construction, the efficiency in the thin ejecta rapidly declines to values <20% after a few days post merger. Concurrently, the photosphere radius receeds inward until it disappears. Despite being an artifact, the discontinuity in the efficiency profile at the photosphere radius is not inconsistent with our photosphere model, which assumes a sharp difference in the properties of matter between the thick and the thin ejecta regions. However, we acknowledge the crudeness of the overall thermalization treatment, which does not rigorously account for the dependency on the ejecta conditions of the specific deposition processes involved. Therefore, we leave for a future investigation the impact of more detailed thermalization descriptions on the resulting kilonova light curves. §.§ Opacities In our framework, we can consider the opacity for the r-process material in the ejecta as a free parameter of the model. Alternatively, we can also provide composition-dependent opacity values following the work of <cit.>, in which systematic atomic structure calculations on each element between Fe (Z=26) and Ra (Z=88) are performed using the integrated code<cit.>. That study mainly focuses on the ejecta conditions around 1 day after the merger, where the temperature is low enough (T≲20000 K) to find the heavy elements ionization stages typically between I-IV. At this time, the density is assumed to be ρ=1×10^-13 g cm^-3 (which is a typical value for an ejecta with mass M_ ej∼0.01 M_⊙ and velocity v_ ej∼0.1 c), and from here on the opacity in the IR, optical and UV is dominated by bound-bound transitions <cit.>.Bound-bound opacities on a fixed wavelength grid for the homologously expanding material are computed using the widely employed expansion opacity formalism:κ(λ)=1/ctρ∑_lλ_l/Δλ(1-e^-τ_l) ,where the sum runs over all the transitions in the RT simulations within the bin Δλ. Planck mean opacities are then computed for a representative ejecta model with different mixtures of heavy elements, characterized by the value of the initial electron fraction Y_e (see subsec:heating_rates).In fig:opacity we report the grey opacity values derived by <cit.> for ejecta temperatures of 5000 K <T<10000 K, whereas a stronger temperature dependence is found for T<5000 K. We note that in more recent works <cit.> such opacity calculations are extended to the ionization stages V-XI of the elements up to Ra, which are expected to be present for ejecta temperatures up to ∼10^5 K at times shorter than 1 day post-merger. We therefore leave the corresponding suggested grey opacities as a possible alternative to the <cit.> dataset.In general, around 1 to a few days, if the electron fraction is low enough (Y_e≲0.25), the grey opacity is dominated by lanthanides and actinides, with values κ≳10 cm^2 g^-1. Instead, an increase in the electron fraction between 0.25≲ Y_e≲0.35 causes a general decrease of the opacity to values κ∼1-10 cm^2 g^-1, as the fraction of f- valence shell elements present in the ejecta decreases, leaving room for the d-shell atoms to provide the leading contribution. Finally, at even higher electron fractions Y_e≳0.4, the contribution from Fe-like elements dominates the opacity, which reaches values κ∼0.1-1 cm^2 g^-1. In this instance, we interpolate the values in fig:opacity to uniquely determine the ejecta opacity on the basis of the input Y_e.§ COMPARISON WITH RADIATIVE TRANSFER CALCULATIONS§.§ Radiative transfer codeIn order to assess the level of reliability of our model, we set up a comparison between the light curves obtained by our semi-analytical model and the ones obtained by a RT kilonova simulation. For the latter, we refer to <cit.>, who employ the wavelength-dependent Monte Carlo RT code originally presented in <cit.>. For a given density structure and abundance distribution, the code computes the time evolution of the photon spectrum in the UVOIR wavelength range, together with multicolor light curves. Differently from the first 3D version, <cit.> assume the ejecta to be axisymmetric. This allows for an increase in the simulation spatial grid resolution, and for the inclusion of special-relativistic effects in the photon transport. Photon-matter interaction is described by considering elastic scattering off electrons, together with free-free, bound-free and bound-bound transitions. The contribution to the opacity from the latter is computed using the expansion opacity formalism described in subsec:opacities, while the atomic transition line list employed in the code is the one already used in <cit.>. Since these atomic data concern the ionization stages I-III, the code is used only for temperatures up to ∼10000 K, below which further ionization stages are subdominant. Nuclear heating rates and elemental abundances are directly imported from the nucleosynthesis calculations of <cit.>, based on the post-processing of Lagrangian tracer particles obtained by a fully general relativistic simulation of a BNS merger with approximate neutrino transport. Each reaction network calculation starts from a representative thermodynamic trajectory with an initial electron fraction in the range Y_e=0.09-0.44. The fraction of thermalized energy is computed using the analytic formulae reported in <cit.> for the different decay products. These formulae depend on the mass and velocity of the ejecta in a similar fashion to eq:thermalization. In particular, while the velocity parameter in the thermalization formulae is fixed to v_ ej=0.3 c, the mass parameter is set starting from the local density and considering a uniform sphere of radius v_ ejt.§.§ Comparison setup We prepare our comparison by setting the same ejecta properties in both codes. We consider two ejecta configurations, namely a lighter anisotropic dynamical component and a more massive spherically symmetric secular component. This choice is motivated by the general necessity of modelling multiple components of matter ejection, which are required in order to reproduce the color bands of observed kilonovae, as in the case of AT2017gfo <cit.>. Regarding the secular component, we assume a total mass of M_ sec=2.64×10^-2 M_⊙, an average velocity of v_ rms=0.06 c and constant values for the electron fraction and specific entropy, i.e. Y_e=0.2 and s=10 k_B baryon^-1. We compute the associated expansion time scale as τ_ exp=c/v_ rms≈ 17 ms. These values are representative of the outcomes of simulations that investigate the evolution of disks emerging as remnants of compact binary mergers and accreting onto the central object. In these simulations, a fraction between ∼20-40% of the disk mass is expelled during the secular evolution, with the initial disk mass M_ disk∼10^-4-10^-1 M_⊙ <cit.>. Instead, for the dynamical component, we use the properties of the dynamical ejecta extracted from one GRHD simulation of a BNS merger with M0 neutrino transport approximation, chosen among those performed by <cit.> and compatible with the GW170817 event. Despite the simulations considered in that work include different EOSs, they all lead to similar ejecta angular distributions, and therefore we arbitrarily select the simulation employing the HS(DD2) EOS <cit.>. The dynamical ejecta are identified with the matter unbound within the end of the simulation according to the geodesic criterion, i.e. the matter for which |u_t|≥ c, with u_t the time-component of the four-velocity. For an equal-mass binary with masses M_1,2=1.364 M_⊙, a total dynamical ejecta mass M_ dyn=2.7×10^-3 M_⊙ was found. The properties of this component are recorded as matter crosses a extraction spherical surface characterized by a coordinate radius r_ E=294 km, and are then reduced to an axisymmetric configuration by averaging over the azimuthal angle. In particular,is informed with the angular distributions of the ejecta mass, average electron fraction and entropy, and average velocity at infinity, calculated as v_ rms^∞=c√(1-(c/u_t)^2). We choose the profile given by eq:density_profile to describe the radial density structure of each ejecta component both in the RT simulation and in . Moreover, we assume a radially constant electron fraction in order to fix the composition. The resulting configuration of the dynamical ejecta as depicted in fig:NR_profiles reflects the general characteristics of this component as obtained in many merger simulations: neutron-rich matter is expelled preferentially across the equatorial plane partially through tidal forces, while shock-heated material subject to stronger neutrino irradiation and thus less neutron-rich escapes also at small polar angles. The input profiles described above uniquely determine all the components of both the models, including energy deposition rates, elemental abundances and opacities, with the only exception of one remaining free parameter in , that is the photospheric floor temperature T_ floor. We remark that the employed radioactive heating rates, as well as the prescription for the thermalization efficiency, are not coincident between the two models, although derived from the same initial conditions. However, the final energy deposition rates agree within a factor of a few, and we account for this discrepancy as being part of the general difference between the kilonova models. Furthermore, we acknowledge that the opacity treatment in ourmodel is significantly approximated: in addition to the adoption of grey values, we assume the opacity to be constant in time or at most to evolve according to a power-law, when characterizing the ejecta through their entire evolution and depth. In reality we expect the Planck mean opacity to vary by at least one order of magnitude between different regions and epochs. Therefore, the adoption of the opacity values derived by <cit.> and described in subsec:opacities is not more physically motivated than treating the opacity as a free parameter, and, for this reason, in the comparison we consider both possibilities.The RT data employed in the comparison consist of the bolometric luminosity, L^RT_ bol(t), and of the AB magnitudes, m^RT_ AB,λ,θ(t) at different wavelengths λ, observed from multiple viewing angles θ_ view∈[0^∘,90^∘]. We thus fit our free parameters to both sets of data separately, considering a logarithmically spaced time mesh, from 0.5 to 15 days. Within this time frame, we assume that the assumptions of our model are better verified, and that the RT calculations are more reliable, whereas temperatures throughout the ejecta are well below 10000 K, justifying the employed atomic data.We define two error functions in order to establish the fit procedure. For the bolometric luminosity, we compute the absolute logarithmic error between our model luminosity, L^M_ bol, and the RT result, L^RT_ bol, averaged over all N_t data points in the considered time frame:err_L=1/N_t∑_t_i| log(L^ M_bol,i/L^ RT_bol,i)| .For the AB magnitudes, we consider three representative broadband filters, namely the K (λ=2157 nm), z (λ=972 nm) and g (λ=475 nm) filters. The light curves are calculated assuming a source luminosity distance of D_L=40 Mpc, corresponding to the estimated distance for the merger associated to the AT2017gfo signal. Since our kilonova model is better suited to reproduce the light curve behaviour around the emission peak, only data points such that m^ RT_AB,i<30 are considered, in order to avoid having the fits influenced by too dim values. In a similar fashion to the bolometric luminosity, we compute the absolute error between the magnitudes across the three different wavebands and two different viewing angles, i.e. 0^∘ and 90^∘, averaging over all 6× N_t data points in the same time interval:err_m=1/6N_t∑_t_i:m^ RT_AB,i<30( ∑_g,z,K( ∑_θ=0^∘,90^∘ |m^ M_AB,i-m^ RT_AB,i| ) ) .We perform two sets of runs, one for each ejecta configuration, i.e. one for the secular isotropic ejecta and one for the dynamical anisotropic ejecta. Furthermore, for each set, we take into account two possibilities. In one case we allow the opacity in our model to vary freely, and in particular for the anisotropic setup we assume it to follow a step function, i.e. we adopt a higher value, κ_low lat, at low latitudes (θ≳45^∘) in correspondence of a neutron-rich environment with Y_e≲0.25, and a lower value, κ_high lat, at high latitudes (θ≲45^∘) where Y_e≳0.25. In the other case instead we compute the opacity using the Y_e parametrization from <cit.>, leaving us with only the photospheric floor temperature to be fitted. To be consistent with the opacity prescription, for the anisotropic ejecta setup we consider a Y_e-dependent floor temperature parameterized by the two values T_ Ni and T_ LA. In tab:priors we report the adopted ranges for the parameters included in the fit procedure. Finally, each calculation is repeated using the semi-analytic kilonova model presented in <cit.> for comparison purposes. The latter shares the same multicomponent, anisotropic framework as the model presented in this work. However, the underlying kilonova model is not based on the solution of the diffusion equation, but it is a phenomenological description based on timescale arguments, presented in <cit.> and <cit.>. Due to this distinction, we name the previous model as , as opposed to our newmodel. §.§ Comparison results The results for all the different models and configurations considered are summarized in tab:fits. As visible, the fit procedure returns reasonable fit parameters values falling in the prior intervals, with the exception of a minor number of cases useful to let the modelling limits emerge. In general, both the fits on the bolometric luminosities and the magnitudes derived from the RT simulation show an overall improved fit quality when usingwith respect to the previousmodel. In particular, when fitting on the bolometric luminosity,is limited by having only the degree of freedom associated to the ejecta opacity (when the latter is left free to vary), since the temperature floor does not enter the luminosity calculation, as opposed to thecase. This difference arises because the floor temperature affects the late time photospheric radius in both models, but while inthe latter is used only for the magnitudes computation through the Stefan-Boltzmann law and does not modify the volume of the radiative zone, in thecase the photosphere position has a feedback on the allocation of mass to the optically thick and optically thin regimes, thus altering the bolometric luminosity as well. As a result, for the case in which the opacity is prescribed using the value of the electron fraction, the bolometric luminosity inis completely fixed for both the secular and the dynamical component configurations, and the correspondent fit errors are the worst in the set.On the other hand, in themodel the floor temperature is adjusted in such a way to increase the late time agreement. Indeed such parameter can ultimately affect light curves only when temperatures in the ejecta have decreased enough, as it is commonly assumed to be the case around a few days post-merger. Specifically, in the secular component configuration this dependency drives the floor temperature to almost rail against the upper boundary, in order to accelerate the photosphere recession and maximize the amount of thin ejecta contributing to the late time emission. One can note that in all cases, and more rapidly for the faster dynamical component, both semi-analytic models converge to the same curve, since in themodel the treatment of the thin ejecta, which eventually constitutes the totality of the outflow, is analytically equivalent to the one used for the entire ejecta in . Furthermore, since this treatment does not properly model the material opacity outside of the photosphere, varying the latter cannot affect the computed luminosity around 10 days, thus not improving the late time matching.As visible in fig:lums_fit, in all the investigated configurationsis sistematically underestimating the early time luminosity with respect to both the RT simulation andof a factor from a few to even multiple orders of magnitude depending on the specific case. This evidence establishes qualitatively the error hierarchy in using an approximate scheme based on the calculation of the diffusion timescale of photons, versus an analytic solution of the simplified RT problem, with respect to a full RT simulation. Therefore, once the opacity is left free to vary, the physiologic behaviour ofis to compensate this systematic by lowering the latter to very small values in order to increase the emission brightness especially before ∼1 day, even incurring in the opacity boundaries in the dynamical component case. Also themodel is partially subject to the same mechanism, as visible specifically in the secular component case. This result suggests a general limitation on using the bolometric luminosity to fit parameters related to local features of the ejecta configuration. However, we also note that, especially for the dynamical component, the magnitude and shape of the bolometric luminosity are in good agreement with the ones derived from the RT calculations.With respect to the fit on the bolometric luminosity, when the same procedure is applied to the AB magnitudes, the overall qualitative results remain roughly unaltered. However, in such a case we include by construction more information, coming from different wavebands and viewing angles. In addition, the temperature floor has a direct role in determining the color bands, since they strongly depend on the photosphere effective temperature, and thus this parameter influences the fit outcome regardless of the model employed. Therefore, we obtain best fit values not necessarily close to the ones found in the previous case.Both in the secular and the dynamical component configurations, we find the same error hierarchy as in the bolometric luminosity fits, withtipically underestimating the overall brightness up to a few days in all bands with respect to . Furthermore, fig:mags_fit shows how magnitudes confirm the trend already evident for the bolometric luminosity, by whichtends to underestimate the emission brightness at early times ≲1 day, indicating a model limitation.The retrieved opacities are now generally higher for both models, with values which can also be closer to the ones fixed by the atomic calculations. The fact that the opacity values differ significantly from the previous fits is not surprising, since in this case they are informed with the light curves in multiple filters as seen in edge-on and face-on configuration: especially in the dynamical ejecta configuration, the latter is valuable information in determining the opacity angular distribution with a better accuracy. In addition, we recall that color bands in the model are derived by composing a sprectrum mainly based on pure black-body emission, which is therefore not able to reproduce the black-body deviations found in the RT calculations. In particular, as pointed out in <cit.>, we note that realistically part of the UV radiation is reprocessed by the heavy elements into the optical and NIR bands, thus shifting the emitted energy distribution significantly, without an heavy alteration in the bolometric luminosity. As a consequence, the more sensitive fits on the magnitudes retrieve opacity values which are increased in order to compensate for the lack of such feature in the model. The floor temperatures derived from the magnitudes are substantially different from the ones derived from the bolometric luminosity. This is due a combination of effects, whereby the floor temperature is not trivially connected to the final magnitudes and its value is subject to stronger variability.On one side, higher floor temperatures are associated to stronger radiation fluxes at late times and, for a given energy emission rate, to smaller photospheric radii, with a net increase in the late broadband magnitudes. Being this the only effect in , the magnitude fit finds the best temperature floor parameter value up to ∼3400 K, in order to compensate for the systematic underestimation of the model, partially relieving the opacity parameter from such burden. This behaviour has also to be ascribed to our fit procedure, which tries to minimize quantitatively the separation between different curves, rather than trying to reproduce the same shape. For this reason, the detailed values of floor temperature that we obtain are not meant to be reliable, but they can nevertheless highlight the internal structure of the model.On the other side, on top of the above effect, as already pointed out for the bolometric luminosity fits, in themodel higher floor temperatures also cause a faster decrease in the amount of optically thick ejecta at late times. In particular, in this case the temperature floor recovery which results from such interplay cures the drift towards the upper boundary that is found in the bolometric luminosity fit for the secular ejecta configuration with fixed opacity. As a general consequence, for , values are almost systematically and significantly lower than bothand their counterparts in the previous fits, being in some cases down to only a few hundreds Kelvin degrees, and indicating a tendency to decrease the overall radiation fluxes in order to match color bands after a few days. § CONCLUSIONSIn this work we presented the new frameworkfor the computation of the kilonova emission from compact binary mergers, starting from the characterization of the merger ejecta. The framework allows for non-trivial ejecta structures as can be inferred from numerical relativity merger simulations, and it employs the results of recent efforts in nuclear astrophysics and atomic physics in terms of inputs for the kilonova model, such as radioactive heating rates from nuclear reaction network calculations <cit.> and grey opacities from systematic atomic structure calculations <cit.>. With respect to previous iterations,includes the model , which encapsulate different possible semi-analytic solutions of the diffusion equation for the radiation energy density field, derived from the RT problem under the assumption of homologously expanding material <cit.>.constitutes an improvement with respect to previous semi-analytic models based on simpler laws of energy conservations and approximate radiation diffusion timescale estimations. In addition, the model tracks the position of the ejecta photosphere in time in order to distinguish between the optically thick internal bulk and the optically thin external layers. The latter are treated with a simplified shell model, which approximately accounts for the non-negligible contribution to the total luminosity coming from this region from a few days post-merger on.We testedmodels by comparing their results with the ones obtained from two-dimensional RT simulations obtained with the code developed in <cit.>, which are based on the same ejecta configurations. In particular, we considered two representative scenarios, i.e. a lighter anisotropic dynamical component and a more massive spherical secular component, and we fit the free parameters of the model to the bolometric luminosity and the AB magnitudes in different filters, as seen from multiple viewing angles. We found thatis able to reproduce the overall behaviour of the light curves obtained from the RT simulations, with a better agreement with respect to the previous semi-analytic model, despite the simplified treatment of the decay energy thermalization process and of the ejecta opacity. However, as highlighted by the fit procedure, the latter still constitutes a limitation to this modelling approach and it will be the subject of future improvements. In particular, the average constant grey opacity values that the model employs are a crude approximation of the real effective opacity inside the ejecta, which significantly varies of more than one order of magnitude with time and across the different regions of the outflow, depending on the local temperature, density and composition. As a result, the emission brightness at early times, i.e. around a few hours post-merger, predicted by the model in the fit procedure, can be systematically lower with respect to the RT calculation, of a factor of a few in the bolometric luminosity and of up to 2 magnitudes in the color bands. We also note that the temperature floor, a secondary parameter inwhich often appears in other semi-analytic models, is not easily constrained, since it is not trivially connected to the final magnitudes.We conclude thatconstitutes a valid tool to model the kilonova emission from compact binary mergers, with the main strength being its computational efficiency, which allows for extensive explorations of the ejecta parameter space in a reasonable time frame. This is particularly useful in the context of the now thriving multi-messenger astronomy, whereas the kilonova is only one of the possible electromagnetic counterparts of the merger event. Coupling this model with information from other sources, such as the GRB afterglow or the GW signal, in a statistical framework, can sinergically help to constrain the properties of the original binary, the central remnant or the merger ejecta, and thus to shed light on the nature of the detected event itself.§ ACKNOWLEDGEMENTSGR acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 279384907 – SFB 1245. GR acknowledges support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). SB acknowledges support from the Deutsche Forschungsgemeinschaft, DFG, Project MEMI number BE 6301/2-1. SB acknowledges support by the EU Horizon under ERC Consolidator Grant, no. InspiReM-101043372 and from the Deutsche Forschungsgemeinschaft, DFG, Project MEMI number BE 6301/2-1. KK acknowledges support by Grant-in-Aid for Scientific Research (JP20H00158, JP21K13912, JP23H04900) of JSPS/MEXT. We thank Masaomi Tanaka for valuable discussions and feedback on the manuscript. We thank Federico Schianchi for initial tests at the beginning of the project.§ DATA AVAILABILITYThe kilonova framework presented in this work is publicly available at <https://github.com/GiacomoRicigliano/xkn>.mnras | http://arxiv.org/abs/2311.15709v1 | {
"authors": [
"Giacomo Ricigliano",
"Albino Perego",
"Ssohrab Borhanian",
"Eleonora Loffredo",
"Kyohei Kawaguchi",
"Sebastiano Bernuzzi",
"Lukas Chris Lippold"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20231127105412",
"title": "XKN: a Semi-analytic Framework for the Modelling of Kilonovae"
} |
firstpage–lastpage XLB: A Differentiable Massively Parallel Lattice Boltzmann Library in Python [ January 14, 2024 ============================================================================ We present results of recurrence analysis of the black hole X-ray binary Cygnus X-1 using combined observations from the Rossi X-ray Timing Explorer All-sky Monitor and the Japanese Monitor of All-sky X-ray Image aboard the ISS. From the time-dependent windowed recurrence plot (RP), we compute ten recurrence quantities that describe the dynamical behavior of the source and compare them to the spectral state at each point in time. We identify epochs of state changes corresponding to transitions into highly deterministic or highly stochastic dynamical regimes and their correlation to specific spectral states. We compare k-Nearest Neighbors and Random Forest models for various sizes of the time-dependent RP. The spectral state in Cygnus X-1 can be predicted with greater than 95 per cent accuracy for both types of models explored across a range of RP sizes based solely on the recurrence properties. The primary features from the RP that distinguish between spectral states are the determinism, Shannon entropy, and average line length, all of which are systematically higher in the hard state compared to the soft state. Our results suggest that the hard and soft states of Cygnus X-1 exhibit distinct dynamical variability and the time domain alone can be used for spectral state classification. stars: individual: Cygnus X-1 — X-rays: binaries — accretion, accretion discs — methods: statistical§ INTRODUCTION Accreting black holes and neutron stars in X-ray binaries (XRBs) are particularly luminous in the X-rays. The X-ray radiation produced by accretion leads to complex phenomenology in both the energy spectra and timing properties of these exotic systems. In energy spectra, many black hole XRBs transition between a lower luminosity state with a spectrum consisting of harder X-rays (the ‘low-hard’ state, or LHS) and a higher luminosity state with a spectrum consisting predominantly of softer X-rays (the ‘high-soft’ state, or HSS; e.g., ). The widely used `truncated disk/hot inner flow' model <cit.> determines that these two states correspond to emission dominated by thermal Comptonization in a hot, geometrically thick, and optically thin plasma in the LHS <cit.> and emission dominated by a geometrically thin and optically thick standard accretion disk <cit.> emulating a pseudo-blackbody spectrum in the HSS (, ; see review byand references within). Cygnus X-1 canonically demonstrates the transitions between the HSS and LHS in its X/γ-ray emission (e.g., ). It is supposed that the transition between these two states involves the variation in the location of the inner edge of the thin accretion disk – in the LHS the disk is truncated and instead filled with a hot, optically thin corona (e.g., an advection-dominated accretion flow; ) and in the HSS, the cooler disk extends closer to the innermost stable circular orbit about the black hole (e.g., , ). Although models of the HSS are consistent with a dominating optically thick, geometrically thin accretion disk, the cause and location of a receding accretion disk remain unclear (e.g. ). Some theories associate the coronal emission that produces the non-thermal, power-law components in the energy spectra to the base of a jet <cit.>. Indeed, observations of time lags between the hard X-ray corona and reflected emission from the softer X-ray disk (referred to as `reverberation lags'; ) enable a full mapping of the disk-corona geometry. As the system evolves between the soft and hard spectral states, <cit.> posits that the inner edge of the thin accretion disk recedes and is replaced by a contracting corona. <cit.> suggest a model in which emission from the disk and corona are connected and delayed: seed photons from the disk (which additionally undergoes driving mass accretion fluctuations that propagate) are provided to the corona, where coronal heating occurs. The relative accretion fluctuation and coronal heating evolve as the inner edge of the disk recedes/fills in and the corona contracts/expands. Other theories of what causes state changes are based on a two accretion flow model, such as that presented by <cit.> of a Keplerian accretion disk in combination with a hot, sub-Keplerian halo, as originally posited by <cit.>. A boost in the accretion rate will occur on different timescales in the disk (viscous timescale) versus the halo (close to free fall timescale), which can explain the delayed response of the soft component often observed. <cit.> refined this idea: the accretion flow switches between a hard component sourced by a magnetically powered coronal outflow and the soft accretion disk because of changes in the poloidal magnetic field. It is clear the mechanisms responsible for state changes are still debated, involving multiple components of the accretion environment.XRBs also exhibit complex temporal variability across many different timescales, ranging from sub-second Quasi-periodic Oscillations (QPOs; e.g., see review byand references within) to super-orbital modulations on the order of hundreds of days (e.g., ). It is hypothesized that there may be a single origin for most of the timing variability features in the power spectra of accreting black holes and neutron stars <cit.>, where multiple frequencies tend to be well correlated with one another and with the different spectral states of the source <cit.>. The timing variations of XRB light curves presumably carry information about the spectral changes and thus nature and evolution of the accretion history. Characterizations of the temporal behavior of XRBs and the connection to the accretion flow are important for determining universal models of accretion as a function of mass (e.g. seeconnecting Young Stellar Objects, White Dwarfs, XRBs, and AGN by their timing behavior) and for classifying objects in large data sets for which spectral information is limited or not attainable.There have been a wide variety of studies of the complex timing behavior in XRBs revealing intrinsic variability beyond those possible using power spectrum techniques alone, suggesting methods based on second-order moments are insufficient for capturing behavior that is possibly nonlinear or non-stationary. For example, the non-Gaussian and non-zero skewness values of the temporal variation of Cygnus X-1 suggested that the variations are nonlinear in nature <cit.>. <cit.> similarly found that the temporal behavior of the black hole system GRS 1915+105 is governed by a low-dimensional chaotic system, detectable when the variability rises above the Poisson fluctuations. In a previous study, <cit.> confirmed chaotic behavior in the long-term light curve of a neutron star XRB with a superorbital period of approximately 120 days. Finally, using similar nonlinear time series analysis techniques, <cit.> applied methods from recurrence analysis to distinguish stochastic and chaotic rapid variability classes among six microquasars. There are abundant spectral-timing studies of black hole XRBs in the fast variability domain (for example, connecting the appearance of QPOs to the hard spectral states; ). One of the outstanding questions in the study of the timing variability of accreting sources is to what extent the complex, and possibly nonlinear or aperiodic, variability is connected to the geometry of the accretion flow <cit.>. More precisely, there is an ongoing effort to determine whether the structure of the temporal flux variability can elucidate the spectral state of the accreting source. Such efforts are readily supported by advances in machine learning. For example, <cit.> used summary statistics of the raw light curve, power spectrum features, and hardness ratios to build a machine learning model that classified the canonically chaotic black hole XRB, GRS 1915+105, into its variability classes (e.g., `ρ,' `θ,' `κ,' etc. variability states as defined by , , and ). A similar goal was implemented by <cit.>, demonstrating how a neural network and Gaussian mixture model can be used to predict the variability classification of GRS 1915+105. Both these studies demonstrate the complexity of temporal variability and infer its relationship to spectral behavior, since the models depend on the spectral information of the source. However, the models predicted the variability classification type, rather than the accretion state. The recent study by <cit.> showed that a machine-learning model could be used to predict the spectral states of outbursting black hole XRBs. However, the model they use is based on features that include spectral information: X-ray flux, hardness ratios, presence of various types of QPOs, photon indices, and disc temperature. In this study, we aim to build on the recent success in applying machine learning to pattern recognition of variability patterns in XRBs to predict the spectral state of Cygnus X-1 specifically based only on temporal variability patterns (i.e., where the spectral information does not factor into the predictor variables of the model).Cygnus X-1 (hereafter, Cyg X-1) is the archetypical black hole XRB and has been the subject of a long history of observational and theoretical studies of accretion onto compact objects since its discovery in 1964 <cit.>. Cyg X-1 consists of a black hole with a historically measured mass of around 10-15 solar masses, with a recent measurement of up to 21 solar masses <cit.> at approximately 2 kpc away <cit.>. Based on the recent mass measurement of approximately 21 solar masses, the spin parameter of the black hole has been constrained to be extreme, at a* > 0.998 <cit.>.Cyg X-1 is a persistent emitting source in the X-ray, where the primary source of accretion is via stellar wind from an OB supergiant <cit.>. Most relevant to our study, Cyg X-1 transitions between the LHS and HSS regularly. In the LHS, Cyg X-1 contains persistent radio emission from an unresolved core and a variable relativistic jet <cit.>. Recently, long-term radio/X-ray monitoring of Cyg X-1 has revealed that a compact radio jet also persists in the soft spectral state <cit.>, where the jet radio emission directly correlates with the hard X-rays (above 15 keV) and lags the soft X-rays by 100 days. The implication is that the disk corona and jet are powered by the same physical process and the hard X-rays in particular are a direct probe of these correlations (while the disk component potentially varies independently).In this study, two machine learning techniques are used to find a correlation between variability features evident in the long-term light curve of Cyg X-1 and its known spectral behavior. Specifically, we explore how temporal variability classes described by recurrences in phase space correlate with the high and low spectral states defined by the hardness intensity diagram. The recurrent behavior of a system as it evolves in a phase space picture is distinct between stochastic, linear, periodic, and nonlinear dynamical systems and it is possible to connect temporal dynamical regimes manifesting in the light curve to specific accretion states. Furthermore, recurrence analysis is useful for detecting dynamical regime changes, e.g., from quasi-periodicity[We note that the term `quasi-periodicity' used here refers to the definition from nonlinear dynamics: the occurrence of multiple unstable periodic orbits that arise in the attractor of a dynamical system, which increases in complexity as a route to chaos. This is distinct from the definition of a QPO in astronomy identified by broad peaks in the power spectrum of a black hole light curve.] to chaos. A relationship between spectral state and dynamical variability characteristics may provide hints to the origins of the emission in each state and the mechanism responsible for state changes. Finally, we reiterate that the accretion state of a system has never been predicted using only the total-band flux variability information (i.e., without spectral information) and has yet to be directly connected to the temporal dynamics. Our goal is to demonstrate this method for the first time on a canonical black hole source like Cyg X-1 as a proof-of-concept that will then be employed for a larger sample in a subsequent study.This paper is organized as follows: in Sec. <ref> we review the instrumentation used for collection and the type of data for this study. In Sec. <ref>, we define the methods that are critical to performing the analysis of Cyg X-1, such as Hardness Intensity Diagrams, Recurrence Plots, and machine learning techniques including k-Nearest Neighbors and Random Forests. In Sec. <ref>, we present our results comparing spectral states to variability statistics. In Sec. <ref>, we provide concluding remarks and highlight the necessary steps to continue this study, expanding our results to other sources and ultimately to large surveys to aid the discovery of the fundamental processes of spectral state changes. § DATA In this study, we consider the long-term behavior of Cyg X-1 as a function of time across a broad range of energies. The data utilized covers more than two decades with daily monitoring from two instruments: the decommissioned Rossi X-ray Timing Explorer (RXTE) All-sky Monitor (ASM) <cit.> and the ongoing Monitor for All-sky X-ray Image (MAXI) onboard the ISS <cit.>. RXTE-ASM and MAXI each have 3 X-ray energy bands which we have labeled as soft, intermediate, and hard covering a total range from 1.5 – 20 keV.The RXTE-ASM was in operation for approximately 16 years before it was decommissioned in early 2012. During its years of operation, it utilized 3 Scanning Shadow Cameras (SSCs) and a position-sensitive proportional counter (PSPC) to measure the intensities of different X-ray sources <cit.>. RXTE had an energy range of 2-200 keV, but the ASM collected data in 3 bands: 1.5-3 keV (`soft'), 3-5 keV (`intermediate'), and 5-12.2 keV (`hard'). This instrument would randomly select a source to scan between 5 and 10 times per day. We obtained the public RXTE-ASM data in 90-second dwells from the High Energy Astrophysics Science Archive Research Center (HEASARC).The MAXI telescope was attached to the Japanese Experiment Module on the International Space Station (ISS) in 2009. It has two main X-ray slit cameras and two types of X-ray detectors. The Gas Slit Camera (GSC) has an energy range of 2-30 keV and the other is an X-ray CCD, Solid-state Slit Camera (SSC) with a range of 0.5-12 keV. One goal of this telescope is to measure the long-term variability of different X-ray sources <cit.>. MAXI was made to be more sensitive, reaching ∼1 mCrab (at 1 week), than the RXTE ASM, which only had a sensitivity of about 10 mCrab. This allows MAXI to collect data from weaker X-ray sources but also continue monitoring some of the same sources as RXTE-ASM, such as Cyg X-1. We obtained the daily monitoring light curves by MAXI of Cyg X-1 from the public archive provided by the RIKEN, JAXA, and MAXI team[MAXI website: http://maxi.riken.jp/]. The average daily monitoring of Cyg X-1 by RXTE-ASM and MAXI are individually displayed in Fig. <ref> (top and third panels, respectively). The energy ranges of each bandpass for both RXTE-ASM and MAXI are presented in Table <ref>.In this study, we have analyzed data from RXTE-ASM and MAXI both separately and combined. In order to combine the data sets, we normalized both types of data to the Crab Nebula. During the overlapping times, we followed the method outlined in <cit.> to scale the MAXI data to the ASM and subsequently convert the light curve into physical units of erg cm^-2 s^-1. This allowed us to view over 20 years of data collection together in the variable time series and subsequent analyses described in this paper. § METHODS §.§ Accretion State Classification: Hardness Intensity Diagrams We use Hardness Intensity Diagrams (HIDs) for visualizing and generating the labels of the different spectral states of Cyg X-1 for use in the machine learning application. We define the RXTE-ASM hardness as the ratio of the 5-12 keV to 1.5-3 keV bands and the MAXI hardness as the ratio between the 4-10 keV and 2-4 keV bands. These definitions are based on the spectral classifications made of the daily ASM and MAXI monitoring by <cit.>. The hardness ratios as a function of time for RXTE-ASM and MAXI are presented in the second and fourth panels of Fig. <ref>, respectively. While hardness is typically inversely proportional to flux intensity, the correlation of spectral index to ASM count rate is complex and ASM or MAXI count rate alone is not reliable enough to separate between accretion states (, ). Consequently, <cit.> implemented a scheme for direct classification of the ASM data using ASM observations that are simultaneous with spectra from pointed RXTE PCA observations. They identifed a relationship between full-band ASM count rate and individual energy bands with photon index. <cit.> employed a similar method of connecting spectra from pointed RXTE PCA observations with MAXI monitoring. The MAXI soft band count rate and the intermediate and soft energy bands are similarly correlated with spectral index. We use the resulting cuts made by <cit.> for both RXTE-ASM and MAXI to define the different accretion states of Cyg X-1 as a function of time. The mathematical expressions for these cuts can be found in Table <ref> (a replication of Table 2 from ). The resulting HIDs for RXTE-ASM and MAXI monitoring of Cyg X-1 are presented in Figure <ref>, where the cuts distinguishing spectral states are visually represented by the solid black lines.§.§ Characterizing Timing Variability: Recurrence Plots A Recurrence Plot (RP) is a graphical representation of a two-dimensional `recurrence matrix' containing all the nonlinear correlations in the light curve <cit.>. It is a tool that assists in the analysis of dynamical systems and is a method from nonlinear time series analysis <cit.>. The recurrence matrix contains information on trajectories in the phase space of the system, a higher-order space that is typically constructed by a time series and its derivatives. For example, the phase space of a simple pendulum is a circle, tracing the angular position against angular velocity over time. The recurrence matrix correlates positions in time to the closeness of those positions in phase space. For a time series embedded in m-dimensional phase space (x⃗_i ∈ℝ^m), the recurrence matrix is defined as:𝐑_i,j= Θ( ϵ - | | x⃗_i-x⃗_j | | ),where Θ is the Heaviside function, ϵ is a threshold distance to compare the `closeness' of the two states (i and j) in phase space, N is the number of observations in the time series, and i,j=1,...,N. For experimental or observational data, the threshold should be larger than five times the observational noise <cit.>, in order to avoid spurious recurrences, while not exceeding the maximum phase space diameter <cit.>.RPs are the visualization of the recurrence matrix where, for states that are close in phase space, Eq. <ref> returns unity, and zero otherwise. RPs are therefore binary images and symmetric about the main diagonal, called the line of identity. Examples of the RPs of various canonical systems are presented and discussed in Appendix <ref>. RPs can be constructed for univariate time series, such as light curves, by reconstructing phase space from an appropriate embedding. For known dynamical systems, the direct time derivatives of the scalar observable can be used (for example, the observable and its first derivative for a 2-dimensional phase space). For unknown dynamical systems, there are several methods for phase space reconstruction. The time delay method is the most common and is used for studying the full RP of Cyg X-1. Discrete Legendre polynomials can be used to reconstruct the derivatives of the system and are used for constructing the RPs in the machine learning models developed in Sec. <ref>. We summarize each method of phase space reconstruction in the Appendix <ref>.If we replace the threshold, ϵ, in Eq. <ref> with a colorbar, we obtain an `un-thresholded' RP, displayed in Fig. <ref> for Cyg X-1. The un-thresholded RP of Cyg X-1 contains multiple qualitative features. The diagonally-oriented striations represent times when the light curve is close to repeating itself. The overall block-like plaid features of the RP indicate times when the light curve is undergoing significant variations over the long term. In general, very faded regions would indicate a source is non-stationary. Other qualitative features unique to various dynamical systems can be seen in the examples given in the Appendix (Fig. <ref>). §.§.§ The Time-Dependent RP The RP can be constructed for the entire length of a time series or in a sliding window in which one obtains an RP as a function of time, called a windowed recurrence plot. A windowed RP enables one to determine recurrence features as a function of time and can be used for determining dynamical regime changes. Specific RP features are sensitive to changes in the underlying dynamical system that generates the time series when computed in a windowed RP. In particular, the average vertical line length (called the `trapping time,' TT) is sensitive to dynamical transitions between periodic windows and chaotic regimes (and is zero for purely periodic dynamics; ). Vertical lines also occur much more frequently in regions of intermittency than in other chaotic regimes and thus increase substantially at these points. Secondly, the ratio of determinism to the recurrence rate (DET/RR) can detect state transitions in which the recurrences re-organize themselves to and from diagonal line structures (or to/from a more ordered state; ). An increase or decrease in DET/RR thus locates times at which the dynamics of the underlying system fundamentally shift. The definitions of these RP features and the others used in the machine learning methods are provided in Appendix <ref>. §.§ Machine Learning Models Two machine learning (ML) techniques are used to find a correlation between the RP features (definitions detailed in Appendix <ref>) as a function of time in the combined light curve of Cyg X-1 and the spectral states defined by the HIDs in Fig. <ref>. We employ supervised learning using both training (labeled predictor variables) and testing (unlabeled target variables) data sets. The training data is used to tune the model to create the best and most accurate results possible for predicting the testing data. Classification algorithms predict discrete class labels, whereas regression predicts continuous values. We use the supervised learning methods of Random Forests and k-Nearest Neighbors regression algorithms in this study. §.§.§ k-Nearest Neighborsk-Nearest Neighbors (KNN) is a memory-based prototype method that uses proximity to other points to predict a classification label <cit.>. It is one of the most popular supervised learning methods for classification based on its simplicity. With an initial query point, x_0, KNN determines the k closest points in the training data set to x_0 based on Euclidean distance. Once the `k' neighbors are found (x_r, for r=1,...,k), the regression algorithm uses the mean of the neighbors (x_r) to determine the class label of each point. The mean prediction can also be weighted by the distance of each neighbor to the initial query point. §.§.§ Random Forests A Random Forest (RF) is considered an ensemble method, which produces one predictive model utilizing multiple base estimators <cit.>, such as an ensemble of Decision Trees <cit.>. Decision trees perform classification by asking simple questions to separate the data via a binary split. This begins with one rule based on the numerical values of the model's features and preferentially divides the data set into two similarly sized sections — one that satisfies the rule and one that does not. Each split point is called a node and each categorization is a branch of the decision tree. At each node of a decision tree, the algorithm will select a set of variables, such as the recurrence features from an RP, and then choose the best-split point by using an impurity or loss function <cit.>. For our case of the decision tree regressors, where the predicted class is a continuous variable, we choose parameters and node splits that minimize the mean squared error of the ending classification.RFs are a form of probabilistic classification that decrease variance and reduce the risk of over-fitting by averaging multiple decision trees <cit.>, called a forest. An RF will randomly select a subset of the training dataset and a random selection of the features (predictor variables) to construct each decision tree. Specifically, the first step in an RF algorithm is to draw bootstrap samples from the training data to create multiple decision trees, T_b, for a forest size of b=1 to B for each sample. Together, these trees create a forest, {T_b}^B_1. It is then possible to predict values with either regression (using the mean) or classification (via majority vote). Combining the predictions of randomly constructed decision trees into one best estimator using an RF prevents overfitting and increases accuracy over a single tree <cit.>. § ANALYSIS AND RESULTS §.§ General Behavior We embed the full-band RXTE-MAXI light curve of Cyg X-1 into phase space using the time delay method (detailed in Appendix <ref>) and generate a windowed RP with width 2000 days in 1-day increments across the entire light curve. We use a threshold corresponding to a 5 per cent recurrence rate for the full RP, which results in recurrence rates ranging between 10 per cent and 80 per cent for the individual sub-RPs that constitute the entire windowed RP. From the windowed RP, we compute the average vertical line length, TT, and the ratio of determinism to recurrence point density, DET/RR, both of which are features sensitive to dynamical regime changes. Since we construct the windowed RP in 1-day increments, we obtain TT(t) and DET/RR(t) with daily cadence. The combined RXTE-ASM and MAXI light curve of Cyg X-1 and the corresponding changes in DET/RR and TT over time are displayed in Fig. <ref>. We establish 95 per cent confidence bounds on both RP features using the structure-preserving bootstrap resampling technique developed byfor use with RPs. We adapt the code developed byas part of the MATLAB-based CRP Toolbox (RP software written byand ) into Python to generate the windowed RP, the DET/RR and TT features, and their confidence bounds. We define six epochs of time (labeled I-VI in Fig. <ref> and Fig. <ref> with date ranges defined in Table <ref>) that demonstrate distinct spectral behavior, in order to compare the change in the TT and DET/RR features with spectral states. To define these epochs, we determined the points at which Cyg X-1 enters or leaves an extended period of continuous behavior. For example, as can be seen in Fig. <ref>, Cyg X-1 exits the soft state at MJD 50326 and does not return until MJD 51837. Epochs I, III, and V have a high occurrence of the hard spectral state, while in Epoch II Cyg X-1 undergoes frequent state changes between the HSS and LHS, also observed in Epochs IV and VI, albeit to a lesser extent. Between MJD 50350–51000, <cit.> found 99 per cent of RXTE-ASM observations to be in the hard state, a time period that most closely aligns with the first half of epoch I. From MJD 51000 to MJD 53900, <cit.> found 63 per cent of observations to be in the hard state, which subsequently increases to 97 per cent between MJD 53900 and MJD 55375 (comparable to epochs II and III, respectively). Finally, it was found that 75 per cent of MAXI observations were in the soft state, similar to epoch IV, between MJD 55375 and MJD 56240. The hard and soft states are very stable compared to the intermediate state, with the hard state being the most stable of the three. Per <cit.>, Cyg X-1 stays in the hard state for at least one week over 85 per cent of the time and is steadily in the soft state for one week 75 per cent of the time. In contrast, the source exists in the intermediate state only transiently, with a 50 per cent chance of Cyg X-1 remaining in the intermediate state within a 3-day period. A significant increase or decrease in DET/RR can be used as a probe of changes in the variability pattern, as explained in Sec. <ref>. Indeed, we see that in epochs II, IV, and VI, the hardness of Cyg X-1 varies between soft and hard more frequently, with an increased amount of time spent in a soft state. In contrast, in epochs I, III, and V, Cyg X-1 is predominantly evolving in the hard state. During these epochs, we observe a dramatic increase in the trapping time, TT, which is a measure of the average length of a vertical line segment in the RP. TT is sensitive to dynamical state transitions between periodic windows and chaotic regimes, in which low values correspond to periodic dynamics and higher values correspond to a regime of chaotic intermittency. Intermittency occurs when a signal experiences irregular bursts of chaotic behavior amidst otherwise laminar phases. Our initial interpretation suggests that DET/RR detects changes between Cyg X-1 existing in the LHS and times when it undergoes outbursting behavior and frequent spectral state changes. Similarly, TT appears to be highly correlated with the hardness of Cyg X-1 and suggests that the accretion properties in the LHS lead to intermittent or chaotic dynamics in the inter-day variability. §.§ Correlations between RP Features and Spectral State The qualitative differences between the six epochs demonstrate that the RP features may evolve in time and be correlated to the spectral state. To quantify this behavior, we embed the full-band ASM-MAXI light curve of Cyg X-1 into 3-dimensional phase space using the Legendre coordinates method and generate a windowed RP with five different window widths (2000, 1000, 500, 250, and 100 days) in 1-day increments across the entire light curve. We subsequently computed eight of the RP features for each sub-RP in the windowed RP: DET, LAM, ENTR, L_mean, TT, L_max, V_max, and DIV (complete definitions can be found in Appendix <ref>).We compare the mean values of each RP feature in the soft and hard spectral states for every RP window size explored in order to determine which RP features are most correlated to the spectral state. That is, for each window size, we computed the average DET for each sub-RP weighted by the fraction of time spent in either the hard or soft state to distinguish the mean DET for each state. Time spent in the intermediate state is significantly less than in either the soft or hard states, as discussed in Sec. <ref>, and exists only transiently. We therefore only consider the hard and soft states for comparison. To determine the significance of a difference in means between spectral states, we use Welch's t-test <cit.>, a modification of the standard Student's t-test <cit.> that accounts for unequal variances or sample sizes. The sample sizes in each spectral state are highly skewed, with over half the observations in the hard state and the remainder split between the soft and intermediate states. The sub-RPs also contain overlap with each other, since the windowed-RP involved incremental steps of one day for each sub-RP. We therefore chose to randomly sub-sample the recurrence features from 150 sub-RPs at a time in each spectral state, perform the t-test and compute the corresponding p-value, for 1000 sets of sub-samples. This process enabled us to satisfy the assumptions of Welch's t-test, that the populations being tested represent random and independent samples from a normal distribution, and to mitigate potential error due to uneven sample sizes.The resulting p-values from the t-test comparing the mean RP features between the hard and soft spectral states are summarized in Table <ref>. We also include tables comparing the intermediate state to the soft and hard states in Appendix <ref>, though no significant differences were found. This suggests that individual RP features are not sufficient to distinguish the intermediate state from the other spectral states and that a more sophisticated ML model may be required. In general, we find that most RP features are systematically higher in the hard state than in the soft state, with a notable exception being the DIV feature (the inverse of the longest diagonal line length). We also found that the Shannon Entropy (Entropy) and Trapping Time (TT) were the most common significant features across a range of RP window sizes. There was no significant difference between the RP features once the window size was 100 days, but some were still marginally significant (e.g., p-values less than 0.1) in the 250-day window (or at least close to the threshold). This suggests that the 250-day window is the smallest window possible that will still have distinguishable features between spectral states over the long term.The determinism and lamanarity also retain relatively low p-values, though they are only highly significant for certain window sizes. The p-values for the lamanarity decrease with decreasing window size, before increasing substantially in the 100-day window. The determinism p-values display a similar effect, though to a lesser extent; indeed, the p-values remain marginally significant for all window sizes except the 100-day window. The p-values of all other recurrence features appear to steadily worsen with decreasing window size. This suggests that the lamanarity, in particular, is sensitive to window size and may be a feature that is not invariant for distinguishing unique features between RPs. To further clarify our results, we performed the non-parametric Mann Whitney U test (also known as the Wilcoxon Rank Sum test; ), which tests the null hypothesis that the distributions underlying two samples are the same and makes no assumptions about the two samples being drawn from normal distributions. For the Mann Whitney U test we also found that the entropy, mean diagonal, and mean vertical line lengths resulted in significant p-values for a range of window sizes, rejecting the null hypothesis and supporting the alternative that the RP features are drawn from significantly different distributions. The significant p-values signify that the recurrence features are, in general, all higher in the LHS. We deduce that the RP contains recurrences in the phase space trajectory of the light curve that last longer, on average, and evident more deterministic behavior in the LHS. This could be interpreted as an LHS light curve containing more memory, or as fluctuations that persist for longer in an impulse-response system. However, the TT feature, in particular, is a probe of dynamical transitions in addition to tracing laminar behavior: higher values indicate regions of chaotic intermittency (as opposed to periodic dynamics). An increase in the average diagonal line length is a more direct probe of the memory in the light curve, which traces how long a section of the light curve mirrors another section in the light curve when projected in phase space. Both longer vertical and diagonal line lengths appear in the LHS. The Shannon entropy measures the information, or uncertainty, associated with the physical process described by the probability distribution of the light curve and is computed from the distribution of diagonal line lengths in the RP. Higher values indicate light curves with high information entropy and uncertainty in the physical process, which may also indicate chaos. Thus, we posit that when Cyg X-1 is evolving in the LHS, it is more likely to exhibit unpredictable and potentially chaotic behavior than in the HSS. Furthermore, these dynamical differences are significant enough that the RP can be used as a probe for distinguishing between the soft and hard spectral states. §.§ Constructing Machine Learning Models from RP Features Our initial qualitative observations from Sec. <ref> and the quantitative differences in RP features between states suggest that the RP features can be used to predict the spectral states of Cyg X-1. We, therefore, explore two popular ML methodologies (RFs and KNN) for predicting the spectral state of Cyg X-1 based on the RP features alone. One goal of this analysis is to determine whether the RP is significantly different in the three spectral states defined by <cit.>. Distinct differences in the RP between accretion states would suggest that the light curve contains distinct dynamics in each state.A secondary goal of the analysis is to determine the minimum size RP required to distinguish differences between the spectral states over long-term monitoring. Given that Cyg X-1 will evolve in the HSS or LHS for many days at a time but also has a history of frequent state transitions on the order of days to months, we aim to acquire a functional ML model with high accuracy for small RPs that maximize time spent in a particular accretion state. Our results from Sec. <ref> suggest that a window size of 100 days may be too small to distinguish features between the soft and hard spectral states.We follow a similar procedure to Sec. <ref> to develop the training sets used in the ML models and embed the full-band ASM-MAXI light curve of Cyg X-1 into 3-dimensional phase space using the Legendre coordinates method. We generate a windowed RP with the five different window widths in 1-day increments across the entire light curve using a fixed recurrence rate of 10 per cent and compute the eight RP features for each sub-RP.Regression models require the predicted class labels to be continuous values. We use the fractional amount of time that is spent in each spectral state in a given sub-RP as our class labels. During the times that the ASM and MAXI monitoring overlap, we use the spectral classification from MAXI.Our full dataset consists of the eight recurrence features as a function of time and the fractional amount of time spent in each spectral state. Our expectation is that the regression models should characterize the overall texture differences between sub-RPs in different spectral states, where individual sub-RPs could be taken out of the context of the full light curve and regarded as a representation of an RP of that particular dominant spectral state. We would therefore consider different features of the sub-RPs to be more directly representative of particular spectral states, rather than merely as detections of state transitions (as indicated by TT and DET/RR in Fig. <ref>).To create the training and testing datasets, we perform a 70-30 split. That is, the training set is comprised of 70 per cent of the full dataset of recurrence features and spectral classifications, with the remaining 30 per cent set aside to assess the accuracy of the models. The sample sizes that constitute each class are uneven. Over half of the observations of Cyg X-1 are when it is in the LHS (52 per cent of the time), followed by about 30 per cent in the HSS, and the remainder in the intermediate state. We ensure that both our training and testing datasets retain the same proportions in each spectral state and do not significantly overlap. The recurrence features that were used as the predictor variables were normalized using the Z-score (zero mean and standard deviation of one).Both the KNN and RF algorithms contain free parameters that the user selects before the learning process is implemented, called hyperparameters. To determine the optimal hyperparameters, we employ a grid-based cross-validation technique that systematically explores every combination of parameters within specified ranges. For the RF models, 5 parameters were optimized. The depth of each decision tree was limited to a choice between 4, 6, 8, 16, 50, 100, and 200, or set to no limit to the number of levels within each tree. The number of rules for each split/node ranged between 2 and 30, and the number of features to consider for each split was between 2 and 8. The use of a bootstrapping method to sub-select data for each decision tree was either applied or not applied, and the option for the number of trees in the forest included 25, 50, 100, 250, 500, and 1000 different trees. For the KNN models, we explored 2 parameters. The number of neighboring points to consider in the classification ranged between 2 and 50, and the weight of each neighboring point was either uniform or based on the Euclidean distance to the neighboring point. To select the optimal hyperparameters, we chose the model with the highest accuracy score (as implemented in the SciKit-Learn package of Python). The accuracy score was based on the coefficient of determination, R^2 = (1 - u/v), where u is the residual sum of squares between the predicted and true labels of the data, and v is the total sum of squares of the true labels of the data. While the tuning of hyperparameters is important for optimizing the accuracy of the final model, in general, all models were relatively robust against changes to the parameters. The range of accuracy scores for the hyperparameters of each model was within 10 per cent of each other.§.§ Model Assessment and AccuracyOnce we determine the optimal hyperparameters for each of our 20 models of varying window widths and ML algorithm choice, we evaluate two metrics for determining the accuracy of each model for comparison to each other. First, we compute the root-mean-square-error, or RMSE. For normalized predicted labels (e.g., those that exist on the unit scale), the RMSE can be directly mapped to a percent error and are presented in Table <ref>.For all models, we also calculate the AUC-ROC score (which is the abbreviation for the `area under the curve' of the `receiver operating characteristic' graph), a common metric for determining the viability of models in data science. The ROC depicts the relative trade-offs between the true positive and false positive rates of a classifier. The AUC-ROC score varies between 0 and 1, with 1 being a perfect classifier (no false positives and all correct classifications are found). The AUC-ROC score is typically only computed for classification algorithms. Appendix <ref> details the computation of the AUC-ROC score for our regression models. When analyzing the scores in Table <ref>, the ideal model will be one that has high accuracy at the smallest window size possible. Overall, we found that the accuracy scores lower as the RP window size decreases for all four types of models. The most severe drop in accuracy occurs between the 250-day and 100-day window sizes. For most models, the 2000, 1000, and 500-day windows have very similar AUC-ROC scores, with a small drop in accuracy occurring for the 250-day window. The KNN regressor has consistently higher AUC-ROC scores and lower RMSE. When comparing the different windows within the KNN regression model, the 2000, 1000, and 500-day windows have AUC-ROC scores equal to or greater than 99 per cent and the 250 day window has an AUC-ROC of 95 per cent for all spectral states. There is a similar pattern with the RMSE, where the 2000, 1000 and 500-day windows have RMSE scores of less than 5 per cent, and the 250-day window jumps above 5 per cent but below 10 per cent in each state. Ultimately, out of the 20 total models, we consider the 250-day window of the KNN regressor to satisfy our criteria of containing both a small RP window and high accuracy (above 95 per cent for all spectral states). The 250-day window KNN regressor model is visualized along with an example of a `poor' model (KNN regression with a 100-day window) in the HID for the RXTE-ASM data in Fig. <ref>, and in the HID for MAXI in Fig. <ref>. The hyperparameter options (as discussed in Sec. <ref>) of the 250-day KNN regression model considered two neighboring points and applied weights that vary based on the distance to each neighbor. §.§ Feature Significance The correlations between RP statistics and spectral state, as depicted in Table <ref> of Sec. <ref>, suggest that the most important features of an RP are the Shannon entropy and the average line lengths for determining accretion state for multiple RP window sizes. The feature importance of each statistic can be computed directly from the Random Forest regression model, which will also inform the relative importance of each RP feature to the ML models in determining the accretion state. We utilize the SHapley Additive ExPlanations method (SHAP; ), which effectively assigns an importance value to each feature that represents the effect on the model prediction of including that feature. All five models differed in the order and relative strengths of the RP feature significances. Indeed, the most important feature resulted in TT, L_mean, V_max, L_max, and V_max, respectively, in order of increasing RP window size. Fig. <ref> demonstrates this discrepancy in the feature significance for each of the eight RP statistics for the 250-d and 1000-d RP windows. We note how L_mean, LAM, and DET were most important for the 250-day RP window, but were the three least important for the 1000-day RP window. Similar to Table <ref>, for example, LAM increases importance for larger RP window sizes. The remaining three models were similar to the 1000-day model in that they all showed relatively comparable weights for most of the features. Only the 250-day model showed the top feature (L_mean) as significantly more important than the other features. Overall, our interpretation is that no features should be dropped from the ML models. Although the Shannon entropy and average line lengths were not consistently the top three features in the ML models for all window sizes, the results are still consistent with the observation that most recurrence features were significant for at least one RP window size in Table <ref>. This would be in line with the fact that the phase space geometries of real dynamical systems are rather complex, and thus no one statistic of a highly-dimensional space will be suitable for summarizing its complexity. In fact, a deep learning model that considers the recurrence matrix as a whole, rather than its summary statistics, would likely perform well for distinguishing subtle RP differences amongst data such as those from accreting systems.§ CONCLUSIONS From the long-term monitoring of Cyg X-1 by the RXTE All-Sky Monitor and the ongoing MAXI instrument on the ISS, we construct hardness intensity diagrams to determine the spectral state of the source at each point in its light curve and compare to the variability characteristics described by the RP generated as a function of time.First, we identify epochs, or extended periods of time, in the light curve for which the predominant spectral states or behavior of the source correlates to certain dynamical behavior inferred from the recurrence properties. These epochs suggest that there is deterministic behavior associated with an increased frequency of state changes in Cyg X-1. This observation echoes the findings from <cit.> that significant traces of nonlinear dynamics and determinism detected with recurrence analysis occur in systems that undergo the limit-cycle instability on short timescales. Here, on long timescales, we infer the preponderance of deterministic behavior likely correlates with a mechanism responsible for global accretion flow instabilities that is related to accretion state changes (e.g., ). In contrast, when the characteristic frequency of the turbulence (stochastic) is greater than the growth rate of nonlinear instabilities (deterministic), then we may observe, for example, something like a damped random walk in the observed emission <cit.>. Indeed, the soft state of Cyg X-1 is dominated by red noise <cit.>. We also find that epochs with extended periods spent in the hard spectral state correspond to intermittent chaos or highly dimensional stochastic behavior, which may correspond to periods in which the source does not undergo a global instability in the accretion flow but nonetheless exhibits complex temporal variability. Given that Cyg X-1 exhibits persistent variability at radio wavelengths, with a significant detected radio jet in the LHS <cit.>, we posit that the increased dynamical complexity is due to corona-jet interactions and the introduction of a stronger disk component dampens this complexity. For example, it has been shown that the introduction of radiation () or self-gravity () of the accretion disk can dampen the strength of the magnetorotational instability and magnetic field of the accretion disk. Furthermore, it is expected that there are simultaneous, and interconnected, random and chaotic or periodic components in the light curves of accretion disk systems, operating on differing timescales, and the strength of each is dependent on the dominant sources of instability and the magnitude of stochastic fluctuations (). While the study of only Cyg X-1 is insufficient to make any distinct claims about the source of stochastic or deterministic behavior in accretion states generally, the distinct recurrence features in each spectral state of Cyg X-1 suggests intrinsic variability could be a distinguishing feature.Second, using the classification into three spectral states for every observation in the Cyg X-1 light curve, we construct 20 different ML models (either a Random Forest or k-Nearest Neighbors regressor) to predict the spectral state based solely on the recurrence properties derived from the time-dependent RPs of the light curve. For all models explored, we find the accuracy of predicting the fractional amount of time spent in each spectral state for each RP remains high (above an approximately 80 per cent AUC-ROC score). The KNN regression model retains above a 95 per cent accuracy score in predicting all three spectral states for a window size as small as 250 days. That is, a light curve that contains 250 daily observations of a source is sufficient to predict whether it is predominantly in a soft, intermediate, or hard spectral state with up to 95 per cent accuracy. However, even 100 daily observations of a source would provide an adequate prediction of the dominant spectral state during that timeframe (an accuracy of 80 per cent). We, therefore, conclude that the RP provides a unique probe for determining the spectral state of an accreting source like Cyg X-1 based solely on its temporal variability characteristics. If such a model is successful in predicting the spectral states of other XRBs, then alerts of state changes can potentially be made with regular monitoring of accreting sources without the use of spectra.Finally, we find that there are distinct differences in individual features of the RPs of Cyg X-1 between times when it is predominantly in the hard spectral state versus the soft spectral state. The recurrence features that correspond to information entropy, chaotic and laminar state intermittency, and recurrence memory are systematically higher in the hard spectral state than in the soft spectral state. This suggests that the accretion flow properties manifest distinct temporal variability characteristics. In particular, the LHS of Cyg X-1 (which also contains a radio jet detection; e.g., ) undergoes intermittency (periods of laminar, or time-invariant, behavior randomly interrupted by chaotic behavior) for longer periods of time relative to the HSS. This suggests that the decreased disk component and dominant coronal component in the hard state <cit.> introduce more complex temporal variability, with an increased likelihood of determinism. In contrast, the HSS exhibits shorter recurrences, which could be due to more stochastic variability from the disk dominant state <cit.>, or potentially due to the decreased influence of the radio component in the soft state <cit.>.Similar correlations between recurrences and variability or spectral states have been found. <cit.> has found that accretion states with specific kinds of QPOs will manifest with deterministic or chaotic dynamics detected by the RP that are distinct from stochastic variability evidenced in states without QPOs. Similarly, <cit.> found that a Kepler-monitored Seyfert 1 galaxy containing a low-frequency QPO exhibited more deterministic and nonlinear behavior compared to another Seyfert 1 galaxy without a QPO. And, <cit.> has found that Type 1 and Type 2 AGN exhibit distinct RPs, as do radio-quiet and radio-loud AGN, which supports models of AGN with differing accretion states akin to XRBs. In particular, the common thread through all of these studies may be the role of the radio jet, which could induce deterministic, nonlinear, or chaotic modulations in the emission. At a minimum, the distinct classifications based on recurrence properties that exist in the different spectral states of Cyg X-1, variability states of microquasars (as in ), and classes of AGN suggest that the dynamics of the accretion flow are imprinted in the light curves of accreting sources. These results suggest that novel time series analysis approaches combined with machine learning can potentially be leveraged for the study of accretion without spectra. The next step in our study is to extend the analysis to other state-changing XRBs to determine whether the distinct recurrence features persist for other sources in the different spectral states in the same manner as Cyg X-1, or if there are further dependencies on other characteristics in the system, such as the companion star mass or mode of accretion (e.g., Roche-lobe overflow versus stellar wind accretion). The ultimate goal is to leverage information from the RP for the classification of ensemble studies of accreting sources, or for the discovery of new accreting sources in large time domain surveys.§ ACKNOWLEDGEMENTS The authors thank Joey Neilsen and Eric Bellm for helpful discussions regarding the manuscript and analysis results. The authors also thank the anonymous referee for comments that improved the manuscript. R.A.P. acknowledges support for this work provided by the National Science Foundation (NSF) MPS-Ascend Postdoctoral Research Fellowship under Grant No. 2138155. E.M.B acknowledges support for this work through the Washington NASA Space Grant Consortium Summer Undergraduate Research Program. R.A.P. and E.M.B. both acknowledge support for this work through a gift of the Washington Research Foundation to the University of Washington eScience Institute and from the NSF Astronomy and Astrophysics Research Grants (AAG) Program under Grant No. 1812779. § DATA AVAILABILITY This research made use of data retrieved from the publicly available repository of the Rossi X-ray Timing Explorer All-Sky Monitor at the NASA High-Energy Astrophysics Science Archives Research Center (https://heasarc.gsfc.nasa.gov/docs/xte/xhp_archive.html) and from the public archive provided by the RIKEN, JAXA, and MAXI team (http://maxi.riken.jp/). The code used to analyze the data includes the publicly available packages PyUnicorn (; available at http://www.pik-potsdam.de/∼donges/pyunicorn) for the production of the Legendre coordinate phase space embeddings; the Scikit-Learn modules from Python for the machine learning models; and methods from <cit.> and the Matlab CRP-Toolbox (https://tocsy.pik-potsdam.de/CRPtoolbox/) translated into Python for the RPs, RP features, and the confidence intervals of RP features. The combination of these packages for use in the analyses in this paper was facilitated by scripts written in Python and will be shared on reasonable request to the corresponding author. mnras § COMPARISONS TO THE INTERMEDIATE STATE Here we include the resulting p-values from the t-test comparing the mean RP features between the soft/hard and intermediate spectral states in Tables <ref> and <ref>, respectively. These are the same as that described in Sec. <ref> and in Table <ref>. Here we find no significant differences between the intermediate state and the other two states. § RECURRENCE PLOTS OF DYNAMICAL SYSTEMS Fig. <ref> provides four example time series and their corresponding RPs. The evenly spaced diagonal lines (parallel to the line of identity) are formed when the trajectory appears periodically at the same place in the phase space on more than one specific time, known as a recurrence. The lengths of these lines highlight the time duration of the recurrence and can be quantified by Recurrence Quantification Analysis (RQA; ), which utilizes various statistical distributions of line lengths to describe structure. Diagonal lines are typically a sign of deterministic behavior, as you can see in the periodic example (left panel) of the RP in Fig. <ref>. Horizontal and vertical structures are times when the trajectory does not vary strongly, typically a sign of laminarity or time invariance. Chaotic variability results in many chopped-up diagonal lines, as evidenced by the chaotic Lorenz attractor in the right panel of Fig. <ref>. Uncorrelated noise will lead to randomly distributed points. §.§.§ Phase Space: The Method of Delays For a given scalar series, x⃗(t), the time delay method <cit.> maps the observations to time-delay embedded vectors y⃗(t) by creating an n vector map:x(t) → y(t) = (y_1(t), y_2(t), ..., y_m(t)) y_j(t) = x(t - τ_j), j = 1, 2, ..., m,where m is the dimension of the embedded vector and the time delay is defined as τ = kΔ t, where τ is an integer (k) multiple of the cadence of the light curve, Δ t.In order to effectively reconstruct the underlying attractor that generates a time series, the time-delayed vectors should contain components that are uncorrelated. In other words, the time delay, τ, should be larger than the correlation time in the time series. The autocorrelation time derived from the autocorrelation function (ACF) is often used, though the first minimum in the mutual information (MI; ) is also appropriate for time series with possible nonlinear correlations. The ACF and MI for Cyg X-1 are shown in Fig. <ref>.The false nearest neighbors (FNN) method <cit.> is traditionally used for selecting an appropriate embedding dimension, m. This involves iteratively increasing the dimension until the number of false neighbors (neighboring points in phase space that diverge from each other when the dimension is increased) is minimized. The FNN for up to 15 embedding dimensions of Cyg X-1 is presented in Fig. <ref> using the two statistical tests from <cit.>. For the first test, the threshold that dictates whether these neighbors are `false' is set by a fixed value. For the second text, the relative distance between each pair of points identified as neighbors is compared between dimensions m and m+1. If the ratio of the relative distances to the standard deviation of the input time series is greater than a fixed threshold, then these neighbors are considered `false.'For the case of Cyg X-1, we select a time delay from the autocorrelation time and the embedding dimension from the FNN algorithm, which results in τ = 73 (or 73 days, given daily monitoring) and m = 8. §.§.§ Phase Space: Legendre Coordinates The phase space can also be reconstructed using orthogonal polynomial filters, called `Legendre coordinates' <cit.>. To summarize the method from <cit.>, the jth-order derivative of a time series can be estimated by a discrete linear filter,w_j(t) = ∑_n=-p^p r_j,p(n) · x(t+nr),where the time series x(t) is the input, w_j(t) is the output, and r_j,p(n) is an appropriate discrete convolution kernel, parameterized by the (arbitrary) choice of p and the order of the desired derivative, j. By expanding x(t) in a Taylor series and enforcing orthogonality constraints, <cit.> demonstrates that the output, w_j(t), is proportional to the jth derivative of the time series. In the limit that p approaches infinity, the kernels r_j,p(n) reduce to the Legendre polynomials; <cit.> thus refers to them as discrete Legendre polynomials.The discrete Legendre polynomials form an orthonormal basis, with basis vectors r⃗_j. As such, we can project a delay vector (e.g., y(t) from Eq. <ref>) onto the discrete Legendre polynomial basis by Eq. <ref>:w_j(t) = y⃗(t) ·r⃗_j,which <cit.> thus labels a `Legendre coordinate' that is proportional to a derivative of the original time series (i.e., w_j(t) ∝ x^j(t)). We can therefore reconstruct the phase space trajectory using Legendre coordinates that are proportional to the derivatives of the system. This becomes particularly useful in the context of irregularly spaced intervals and for time series for which the derivatives and underlying equations of motion are unknown.§.§.§ Recurrence Quantification Analysis Following the notation of <cit.>, we define RQA statistics — or RP features, as we refer to them throughout the paper — that can be used to summarize the overall properties of an RP, namely, the Recurrence Rate, Determinism, Laminarity, average vertical/horizontal and diagonal line lengths, longest diagonal and vertical line lengths, Divergence, and Shannon Entropy. As a collective, these features summarize the line structures and texture of the RP for a given threshold, ϵ.The recurrence rate (RR) is the probability that a state returns to within an ϵ-neighborhood in phase space:RR(ϵ )=1/N^2∑ ^N_i,j=1𝐑_i,j(ϵ ),where N is the length of the time series and 𝐑(ϵ) is the recurrence matrix for a given threshold, ϵ. The RR describes the density of recurrence points in the RP.Determinism (DET) is the percentage of points that form a diagonal line compared to isolated points in the RP: DET=∑^N_l=l_min lP(l) /∑^N_l=l lP(1),where l_min is the minimum length of the diagonal lines found in the RP (typically set to 2) and P(l) is a histogram of all the diagonal line lengths, l. The higher DET, the more recurrence points are part of diagonal structures and the less randomness is present in the system.Similarly, Laminarity (LAM) is the ratio of the number of recurrence points that form vertical structures, P(v), to the total number of recurrence points in an RP,LAM = ∑_v=v_min^N v P(v)/∑_v=1^N v P(v),where v_min is the minimum length of the vertical lines found in the RP (typically set to 2) and P(v) is a histogram of all the vertical line lengths, v. Thus, LAM is analogous to DET and measures the frequency of laminar (or time-invariant) states in the system relative to randomness.The Trapping Time (TT) represents the amount of time the trajectory remains in one state, or acute region of phase space, whereTT = ∑_v=v_min^N v P(v)/∑_v=v_min^NP(v),representing the average length of a vertical line. TT is also considered to be the average time in which fluctuations occur in an impulse-response system <cit.>.The longest length of diagonal lines (L_max) and vertical lines (V_max) of the RP are analogously defined: L_max = max( {l_i;i=1,…, N_l}),V_max = max( {v_i;i=1,…, N_v}),with N_l=∑_l≥ l_minP(l) and N_v=∑_v≥ v_minP(v) as the total number of diagonal lines and vertical lines, respectively, in the RP. Divergence is known as the inverse of L_max, defined asDIV = 1/L_max,where the shorter the divergence between trajectories in phase space, the smaller the DIV feature, which creates longer diagonal lines in the RP. The inverse relationship between DIV and L_max is explained using the sum of the positive Lyapunov exponents of the system <cit.>.Finally, the Shannon Entropy is defined as:ENTR = -∑_l=l_min^N p(l) ln p(l),where p(l)=P(l)/N_l is the probability that a diagonal line is of length l in the RP. The Shannon entropy is also considered a measure of information, where an increase in entropy corresponds to greater uncertainty in the time series distribution.§ CONSTRUCTING THE RP OF CYGNUS X-1 The optimal threshold for performing recurrence analysis corresponds to at least 10 per cent of the maximum diameter of the phase space () and exceeding 5 times the standard deviation of the observational noise (), but not exceeding the maximum size of the phase space <cit.>. For example, for a 3-dimensional time series that has an amplitude range of unity and noise with a standard deviation of 0.02, the threshold choice should be at least 1.7 and greater than 5σ_obs = 1.Given that the Legendre coordinates approximate the derivatives of the one-dimensional time series, this method has an advantage over the time delay method in its application to shorter time series. For example, embedding the full light curve of Cyg X-1 into phase space using the time delay method uses a delay of 73 days and a dimension of 8, which corresponds to an embedding window of 584 days. This means we would not be able to explore window sizes in the windowed RP shorter than the embedding window. Given that we seek to explore window sizes as small as possible, much less than 500 days, in order to isolate individual spectral states as much as possible, we use the Legendre coordinates method. The drawback is that utilizing the derivative of the time series results in greater noise in the phase space embedding. However, if we are able to distinguish spectral states using RQA features despite this added noise (in contrast to the noise suppression that naturally occurs in the time delay method), then we can say that features are robust against noise contamination for classification and regression purposes.§ AUC-ROC SCORE For a binary classification model for which labels are either `positive' or `negative,' there are four possible outcomes: true positives, false positives, true negatives, and false negatives. The true positive rate (TPR) is defined as the fraction of correctly predicted positives (true positives) relative to the total number of actual positives in the dataset. The false positive rate (FPR) is defined as the fraction of points incorrectly predicted as positive (false positives) relative to the total number of actual negatives in the dataset. The ROC curve is generated by plotting the TPR as a function of FPR for each decision boundary between the binary classification. That is, for some decision boundary, d, we consider the positives as those labels with a probability of being positive greater than d and the negatives as those labels with a probability of being positive less than d. The TPR and FPR are calculated for each d and the ROC curve therefore runs from zero to one. The resulting ROC curve gives an indication of the trade-offs between acquiring true positive results versus false positive results. A ROC curve that follows a slope of one corresponds to a classifier that is no better than the flip of a coin. The higher the TPR is relative to FPR for a wide range of decision boundaries, the more accurate the classifier is in both predicting the true labels of the data and limiting the number of false positives.A means to quantify the quality of the ROC curve in a single number is to calculate the area under the ROC curve (typically done by the trapezoid rule), referred to as the AUC. The resulting area is called the AUC-ROC score. A perfect model would result in an AUC-ROC score of unity.For the case of a regression model, we must consult an alternative definition of the true and false positive rates, since the predicted and real labels of the data are continuous variables (in this case, the fraction of observations that are in each of the spectral states for a given sub-RP window). The TPR is also known as the probability of detection and the FPR is also known as the probability of false alarm. That is, for the decision boundary, d, the probability distribution for the positive label, f_p(x), and the probability distribution for the negative label, f_n(x), we define TPR= ∫_d^∞ f_p(x)dx, FPR= ∫_d^∞ f_n(x)dx. Thus, if we know the probability distributions for both detection and false alarm from the dataset, then the ROC curve becomes equivalent to the cumulative distribution function.For a finite dataset, the cumulative distribution can be estimated by determining the probability that a classifier will rank a randomly chosen positive instance from the dataset higher than a randomly chosen negative one. To implement, we consider each pair of predictions, y_i and y_j, in the dataset, where i = 1, ..., N and j = 1, ..., N (for ij). Then, the AUC-ROC score becomes the sum of instances where y_i > y_j relative to all possible pairs. In Sec. <ref>, we consider each spectral class prediction separately, where the spectral state of interest is considered the `positive' label, and the other two spectral states as the `negative' label. | http://arxiv.org/abs/2311.16070v1 | {
"authors": [
"E. M. Broadbent",
"R. A. Phillipson"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20231127184115",
"title": "Correlated Spectral and Recurrence Variations of Cygnus X-1"
} |
[email protected] High Energy Theory Group, Department of Physics, William & Mary, Williamsburg, VA 23187-8795, United States Institute for Theoretical Physics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany [email protected] High Energy Theory Group, Department of Physics, William & Mary, Williamsburg, VA 23187-8795, United States Regular black hole metrics involve a universal, mass-independent regulator that can be up to 𝒪(700 km) while remaining consistent with terrestrial tests of Newtonian gravity and astrophysical tests of general relativistic orbits. However, for such large values of the regulator scale the horizon is lost. We solve this problem by proposing mass-dependent regulators. This allows for large, percent-level effects in observables for regular astrophysical black holes. By considering the deflection angle of light and the black hole shadow, we demonstrate the possibility of large observational effects explicitly.Kilometer-scale ultraviolet regulators and astrophysical black holes Christopher D. Carone December 15, 2023 ====================================================================Introduction.—Astrophysical black holes are emerging as increasingly relevant testing grounds of gravitational physics <cit.>. Under some assumptions on the energy and pressure of matter, general relativity predicts that such astrophysical black holes—formed during the late stages of a collapsing massive star—necessarily contain singularities <cit.>. These singularities show up as divergent gravitational tidal forces in the black hole interior, and prevent a complete description of physics in this regime <cit.>. This is the black hole singularity problem, a major unresolved problem in gravitational physics. Within general relativity, these singularities are evident in the Schwarzschild metric: an exact, spherically symmetric vacuum solution of the Einstein equations, taking the form <cit.>s^2= -F(r) t^2 +r^2/F(r) + r^2θ^2 + r^2sin^2θ φ^2 , F(r)= 1 - 2GM/r.Here, M is the black hole mass, G is the gravitational constant, and we set the speed of light to unity (c=1). The event horizon is located at r_h = 2GM, and the singularity is situated at r=0, as can be seen from the Kretschmann invariant R_μνρσR^μνρσ = 48(GM)^2/r^6. It is believed that quantum-gravitational effects change the form of the metric in the black hole interior at r ≪ r_h, eventually leading to an avoidance of the singularity. In the recent years, a variety of regular black hole models have been proposed <cit.>; see also the review <cit.>. Typically, they feature a modification of an otherwise singular metric, parametrized by a length scale ℓ > 0, that removes the singularity at r=0. These metrics are not solutions of a fundamental gravitational theory, but can help constrain the leading order observational consequences of regular black holes. A characteristic and popular model is the Hayward metric <cit.>,F(r)= 1 - 2GM/rr^3/r^3 + L^3,L^3 = 2GMℓ^2 . For non-zero values of the parameter L, the metric in Eq. (<ref>) describes a smooth and finite gravitational field around r=0, while for L=0 one recovers the Schwarzschild solution. The parameter L is the product of a universal length scale ℓ and a factor of GM that removes the mass dependence of the function (<ref>) at small distances r. This universal behavior at small distances is known as the limiting curvature condition and is motivated by various high-energy properties of gravity <cit.>. Hence we shall refer to ℓ (rather than L) as the fundamental, short-distance or “ultraviolet” (UV) regulator.Let us now constrain both L and ℓ. Eötvös-type experiments confirm the Newtonian inverse-square law of gravitation to distances of ∼ 50 μm for test masses of 𝒪(1 kg). A measurement uncertainty of around 0.5 μm <cit.> bounds L and ℓ byL ≲𝒪(10 μm) , ℓ≲𝒪(700 km) .The large separation between those two quantities stems from2GM/c^2 ∼ 10^-27 m for tabletop masses. Astrophysical orbits imply much weaker constraints. For a central, solar mass, at distances of 1 AU, the leading order deviation from the Schwarzschild metric is (r_s/r)^2(ℓ/r)^2 ∼ 10^-26, where r_s = 2 G M /c^2 and we set M=M_⊙ and ℓ = 700 km. For a supermassive black hole, such as Sagittarius A*, with M ∼ 4.3 × 10^6 M_⊙, a stellar object at its closest point of approach at r = 120 AU <cit.> would experience corrections of (r_s/r)^2(ℓ/r)^2 ∼ 10^-21, whereas r_s/r ∼ 10^-3.In the absence of a model for the origin of Eq. (<ref>), it is well motivated to treat ℓ as a free parameter and study the phenomenological consequences.The surprisingly weak bound in (<ref>) implies that ℓ can be comparable to the horizon size of astrophysical black holes. When this is the case, regular black hole metrics can differ significantly in the near-horizon region from their Schwarzschild counterparts. The possibility of striking observational consequences, however, cannot be realized: the metric (<ref>) ceases to describe a black hole if GM < (3√(3)/4) ℓ, preventing black holes withmasses below ∼ℓ/G.[This remains true in dynamical black hole formation <cit.>.] Importantly, this is not a fluke of the Hayward metric, but a common feature of all known regular black hole metrics, usually referred to as a “mass gap.” See Table <ref> for a summary of mass gaps for several well-known regular black hole models. To the best of our knowledge, no known regular black hole model can simultaneously accommodate a large regulator ℓ and allow the existence of a black hole horizon across a realistic, astrophysical mass range.In this Letter we solve this problem. We propose, for the first time, regular black hole metrics without the problems related to mass gaps, featuring 𝒪(50%) corrections at the horizon scale in the mass range of astrophysical black holes—a regime that is most promising for observational tests of modified gravity. The location of the innermost circular orbit for light (the “photon sphere”), gravitational lensing, and the black hole shadow, are all susceptible to horizon-scale deviations in the astrophysical black hole mass range. Thereby, we remove a long-standing roadblock in regular black hole models, and pave the way for the consistent parametrization and potential observation of large, percent-scale effects of new physics at astrophysical black hole horizons.Modified metric.—We take the Hayward metric (<ref>) as a starting point, and aim to explore its dependence on the regulator scale ℓ as well as its mass parameter M via the dimensionless combinationℓ̂ = ℓ/2GM.While ℓ̂≪ 1 is typically assumed in the study of regular black holes, our focus will be on the regime ℓ̂≳ 1 where one has strong deviations from the Schwarzschild form away from the origin. We parametrizeF(r)= 1 - 2GM/rr^3/r^3 + L^3,L^3 = 2GMℓ^2 f(ℓ̂) ,where the asymptotic ADM mass of this metric is still given by M, but f(ℓ̂) > 0 introduces a new regulator L with an unconventional mass dependence. The black hole horizon, in terms of r̂≡ r/(2GM), lies at r̂_h^3 - r̂_h^2 = - ℓ̂^2 f(ℓ̂). This equation has positive solutions ifℓ̂^2 f(ℓ̂) ≡L̂^3 ≤4/27,which one may think of as the “black hole condition.” The mass-dependence of this condition leads to strikingly different results compared to the Hayward metric. Namely, non-monotonic functions ℓ̂^2 f(ℓ̂) that increase and decrease at intermediate values for ℓ̂ generate entirely new, astrophysically viable branches for regular black hole metrics that feature both a horizon and a large value of ℓ. To illustrate this, the reader may verify that the following function,f_example = 1/1+ℓ̂^4,provides such an example. The black hole condition L̂^3 ≤ 4/27 is guaranteed for GM ≳ 1.28ℓ,qualitatively similar to the Hayward case. However, black holes also exist for GM ≲ 0.20 ℓ—an entirely new branch. The exciting property of this branch lies in the fact that it constitutes an upper bound on black hole mass. It allows black holes with masses that are less than the regulator scale (in units where c=G=1). This solves the problem of a disappearing horizon for large regulator scales, so that large astrophysical effects can be consistently obtained. The choice of function f(ℓ̂) defines a new family of regular black holes,as we discuss in some more detail below. The impact of this new, mass-dependent regulator can best be captured by estimating horizon-scale effects. Subject to the condition (<ref>), the horizon is located atr̂_h = 1/3{1 + 2cos[13arccos(1-272L̂^3)]}.At the horizon, the deviation of this class of metrics from the Schwarzschild metric is governed by the ratioδ≡( L̂/r̂_h)^3 ∈ [0, 0.5] .The deviation is monotonically increasing and assumes its maximum value of 50% at L̂^3 = 4/27. There, usual regular black hole metrics cease to describe black holes and instead become horizonless. The function f(ℓ̂) in the mass-dependent regulator L̂ changes the allowed mass ranges for black holes, so that 𝒪(50%) horizon-scale effects for astrophysical black holes are allowed;Fig. <ref> illustrates the effect of the function f(ℓ̂).Let us now constrain its properties a bit more:* For a vanishing regulator at fixed mass M (ℓ̂→ 0) we should recover the Schwarzschild metric. Assuming that f(ℓ̂) is regular at the origin, this constrains f ∼ℓ̂^p with p > -2 at ℓ̂=0.* The limiting curvature condition demands f to be a universal constant. If we merely want to prevent trans-Planckian curvatures, we can relax this condition to only hold for large masses M (at fixed ℓ), and hence f ∼ℓ̂^p with p ≥ 0 for ℓ̂→ 0. * Tabletop experiments for small masses probe the region M → 0, corresponding to ℓ̂→∞ at fixed ℓ. To avoid a more stringent bound than given in Eq. (<ref>), we assume that f ≲ 1 in this limit.We propose the following, rather general parametrization that captures this essence (see Fig. <ref> for a few cases):f(ℓ̂) = 1/1 + a ℓ̂^p + b/(1 + ℓ̂)ℓ̂^q. In the black hole domain, as described by Eq. (<ref>), the modification is strongest wherever L̂^3 reaches its maximum value of 4/27. For example, in the case where a=1, p=4 and b=0, i.e., Eq. (<ref>), maximal effects are obtained for black hole masses near the boundaries of the allowed regions, ∼ 0.20 ℓ /G and ∼ 1.28 ℓ / G, while allowing black holes to exist in both intervals. For ℓ∼ 30 km, these boundaries correspond to ∼ 4 M_⊙ and ∼ 26 M_⊙, respectively. For realistic astrophysical studies, the forbidden mass interval in between can be modified by a different choice of the function f(ℓ̂). Note that regular supermassive black holes are not affected by the modification to the metric. Compared to common regular black hole models, the additional stellar black hole “branches” with substantial horizon-scale effects may be ideal targets for direct astrophysical observation.Consequences.—To demonstrate the qualitatively new features of the metrics described in this Letter, let us set ℓ≈ 30 kmand explore observational consequences for astrophysical, solar-mass black holes. For simplicity, we will assume a mass-dependent regulator given by {a=1, p=4, b=0}.Light propagation in the metric (<ref>) is described byṙ^2 = E^2 - V_eff(r) ,V_eff(r) = J^2/r^2F(r) ,where E ≡ - g_ttṫ and J ≡ g_φφφ̇ are constants of motion related to the impact parameter b via b = J/E, and the dots denote differentiation with respect to the affine parameter of the geodesic x^μ(λ).The effective potential V_eff(r) has a maximum outside the black hole horizon at r = r_γ, indicating an unstable circular photon orbit, called the “photon sphere.” The regulator f(ℓ̂) pushes it inwards, away from 3GM:L̂^3 = (√(3GM/r_γ) - 1) (r_γ/2GM)^3 ≥ 0 .The effect is largest for L̂^3 = 4/27, resulting in r_γ≈ 2.65GM. This is a 12% deviation from the Schwarzschild location 3GM. A similar statement holds for light deflection around such black holes. Following (<ref>), the deflection angle isφ(b) = 2b ∫_r_0(b)^∞ r/r^21/√(1 - b^2/r^2 F(r)) - π,where r_0 denotes the point of closest approach, related to the impact parameter b via F(r_0) b^2 = r_0^2. We can now compare the light deflection around objects of the same mass M, described by different metrics. For definiteness, we fix the impact parameter to be twice the location of the Schwarzschild photon sphere. The maximum resulting effect is of 𝒪(5%), and is displayed in Fig. <ref>.We can now define the “black hole shadow” as the impact parameter for which the bending of light angle grows to infinity—the shadow is the boundary between trapped and deflected light, and hence corresponds to the visible size of a black hole in the sky. It is given byb_γ = r_γ/√(F(r_γ)) = r_γ/√(1-√(4GM/3r_γ)),and the maximum possible deviation from Schwarzschild is again of 𝒪(5%). In the future, it would be interesting to extend these studies similar to Ref. <cit.>.Conclusions.—We have described an entirely new class of regularblack hole metrics that—across wide astrophysical black hole mass ranges—can consistently model potentially observable, astrophysically relevant horizon-scale deviations from the black holes described by general relativity at the percent level. These include, but are probably not limited to, a reduction in the black hole's apparent size, a smaller photon sphere, and weaker lensing. These large effects are made possible across astrophysical black hole mass ranges via a mass-dependent regulator,L^3 = 2GM ℓ^2 f(ℓ/2GM) ,where the function f is a new ingredient.The mass dependence allows the regulator ℓ and the horizon scale to be comparable for astrophysical black holes; previously excluded mass ranges are now accessible and permit black hole geometries featuring horizons, while other limited ranges remain inaccessible, without necessarily implying conflict with observation. In a simple case, cf. Eq. (<ref>),GM ≳ 1.28ℓ,GM ≲ 0.20ℓ.While the first constraint constitutes a mass gap, the second condition is qualitatively new. This, to the best of our knowledge, is the first example of a “black hole band spectrum” in the context of regular astrophysical black holes.Effects on supermassive black holes are negligible given an appropriate choice of f. Since their mass scale exceeds that of astrophysical black holes up to a power of 10^9 <cit.>, the effects described in this Letter are suppressed by roughly this factor. This is desirable, since the shadows of supermassive black holes are directly observable <cit.>, and large deviations from general relativity are likely ruled out at that scale. In fact, the observations presently cannot distinguish between black holes from Einstein gravity and modified, regular black hole metrics <cit.>. In the recent years, it has been pointed out that several regular black hole models are geodesically incomplete <cit.> or may suffer from instability issues <cit.> related to mass inflation <cit.> at their inner horizons. Our modification does not remedy this behavior. However, we emphasize that the mass-dependent regulator, suggested for the first time in this Letter, can be applied to any regular black hole model, including models that have improved behavior. Motivated by simplicity and definiteness, however, we focused our considerations on the Hayward metric, but the approach presented herein has wider applicability.Since the discussion in this Letter is limited to time-independent metrics, one may wonder about the dynamical aspects of these black holes and their non-trivial band structures. For example, is it possible to have two black holes from the low-mass part of the spectrum, make them collide, and end up in the regime where no black holes are allowed? And, in that case, what is the nature of the horizonless end product after a collision of these black holes? It is possible that instabilities of this compact, horizonless object lead to emission of gravitational waves that only terminates once the object is again inside the lower mass rangeand a horizon has formed. In the presence of more involved regulators f(ℓ̂) one could have multiple, disconnected bands for these types of black holes, giving rise to a multitude of transitions between bands, accompanied by a characteristic amount of gravitational wave emission. While certainly interesting, we will leave dynamical aspects to future work.Last, one may wonder if these deviations, if detected, constitute modified, classical gravity, or contain traces of quantum aspects of gravity. The appearance of GM in the quantity L, even in the case of the unmodified Hayward metric, is necessitated by the limiting curvature condition. This condition states that irrespective of its mass, the maximally allowed curvature of an object cannot exceed the Planckian curvature ℛ∼ 1/ℓ_Planck^2. In other words, the maximum curvature needs to be independent of the object's mass. 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"authors": [
"Jens Boos",
"Christopher D. Carone"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-ph"
],
"primary_category": "gr-qc",
"published": "20231127210717",
"title": "Kilometer-scale ultraviolet regulators and astrophysical black holes"
} |
=1showonlyrefs compat=1.9 ./Bilder/OMSzplmmn textwidth = 152mm, textheight = 240mm, left = 29mm, top= 25mm, theoremTheorem corollary[theorem]Corollary lemma[theorem]Lemma prop[theorem]Proposition definition[theorem]Definition remark[theorem]Remark assumption[theorem]Assumption theoremsection markierung @todonotes@disabled elsarticle-harv satzTheorem[section]ass[satz]Assumption bem[satz]Remark bewℝℕℤ ℙ 𝔼 Ψ Φ γ v 1 Zequationsection tablesection figuresection Arbitrage strategies in markets with fractional Brownian motion K. Lamert, B.R. Auerand R. Wunderlich Kerstin Lamert Brandenburg University of Technology Cottbus-Senftenberg, Institute of Mathematics, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany; Benjamin R. Auer Friedrich Schiller University Jena, Chair of Finance, Carl-Zeiss-Str. 3, 07743 Jena, Germany; Ralf Wunderlich Brandenburg University of Technology Cottbus-Senftenberg, Institute of Mathematics, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany; Discretization of continuous-time arbitrage strategies in financial markets with fractional Brownian motion Kerstin Lamert Benjamin R. AuerRalf Wunderlich Version ofJanuary 14, 2024 ================================================================================================================= This study evaluates the practical usefulness of continuous-time arbitrage strategies designed to exploit serial correlation in fractional financial markets. Specifically, we revisit the strategies of <cit.> and <cit.> and transfer them to a real-world setting by distretizing their dynamics and introducing transaction costs. In Monte Carlo simulations with various market and trading parameter settings, we show that both are highly promising with respect to terminal portfolio values and loss probabilities. These features and complementary sparsity make them valuable additions to the toolkit of quantitative investors. 91G10 91G80 G11 G17 § INTRODUCTION Motivated by the challenge they pose to the traditional notion of efficient capital markets, financial research has intensively studied investment strategies which solely rely on past asset price information. Among the most prominent studies, <cit.> have shown that cross-sectional momentum, i.e., buying past winners and selling past losers, is highly beneficial.[For the identification of winners and losers, relative past performance can be quantified via cumulative returns or established reward-to-risk performance measures <cit.>.] In addition, <cit.> identify a time-series momentum effect according to which single assets exhibit exploitable trending behavior.[<cit.> establish a connection between time-series momentum and the popular moving average trading rules of <cit.>.] What these strategies have in common is that their profitability is linked to a positive serial correlation in asset price movements <cit.>. Even though momentum investing has become a standard in the mutual fund industry <cit.>, financial research and practice has paid surprisingly little attention to a very interesting strand of mathematical literature developing arbitrage strategies for assets with serially correlated returns. It is well known that pure arbitrage, i.e., the realization of risk-less profits from zero initial investment, is impossible in a traditional Black-Scholes market with standard Brownian motion (sBm). In contrast, arbitrage opportunities can arise in markets where asset prices are driven by a fractional Brownian motion (fBm) which dates back to <cit.> and superimposes memory features on asset returns. In a continuous-time setup with slowly decaying positive serial correlation, i.e., the fractional Black-Scholes model of <cit.>,[There are alternative fractional Black-Scholes models based on different stochastic integral definitions. Unfortunately, they contradict economic intuition <cit.>.] the theoretical studies of <cit.> and <cit.> show that risk-less profits can be earned by buying high-priced and short-selling low-priced assets in adequate numbers. <cit.> extend the work of <cit.> by incorporating stochastic volatility. <cit.> and <cit.> develop additional but more complex strategies. While the simplicity of the Shiryaev and Salopek arbitrage strategies and empirical evidence on memory in equity, futures and fund returns <cit.> make them highly appealing for investment practice, they are built on the premise of continuous-time trading with no frictions. <cit.> and <cit.> highlight that, in a fractional Black-Scholes world, arbitrage opportunities vanish with the introduction of a minimal waiting time between subsequent transactions, i.e., discrete-time trading, and proportional transaction costs of any positive size, respectively.[For further research in this area, see <cit.>.] However, this does not necessarily mean that the above strategies should be discarded. When suitably discretized and parameterized, they may not be entirely self-financing and risk-free, but still provide positive expected payoffs at a low risk of loss. In other words, they could share some valuable properties with statistical arbitrage strategies <cit.>. After exploring the properties and the economic intuition of the Shiryaev and Salopek strategies, the core objective of our study is to investigate their investment performance in a real-world setting. This means that, in a first step, we discretize the strategies and install different forms of transaction costs. This is not trivial because discretization alone makes the strategies lose their self-financing property and requires suitable countermeasures to maintain tradeability. In a second step, we perform an extensive Monte Carlo study for the discretized versions of the strategies. Here, we are particularly interested in whether they deliver positive terminal portfolio values on average and display acceptably small loss probabilities. We focus on these two quantities because they are central to established arbitrage definitions and allow a modern downside-oriented investment evaluation <cit.>. To answer our research question, we use the spectral method of <cit.> for fBm simulation which, in contrast to alternatives, preserves the basic features of fBms <cit.>. We analyze the strategies with asset and trading parameters tailored to the current market environment exhibiting, for example, significantly falling transaction costs <cit.>. Furthermore, we conduct a variety of sensitivity checks to identify the situations in which they perform best and worst. Overall, this results in an intuitive guide on how to chose, for example, the ideal candidate assets, parameters and trading frequencies of the strategies. The remainder of our study is organized as follows. Section <ref> introduces the fractional Black-Scholes model, discusses the corresponding Shiryaev and Salopek arbitrage strategies and translates them to a discrete-time setting with transaction costs. Section <ref> presents our Monte Carlo study examining the impact of discretization, transaction costs, model parameters as well as trading horizon and frequency on the strategies. Section <ref> concludes and outlines directions for future research. § THEORETICAL FRAMEWORK §.§ Continuous-time market setup We start our analysis by specifying the asset price behavior in a fractional Black-Scholes model and explain how self-financing portfolios are formed in such an environment. Asset prices For a fixed date or investment horizon T>0, we consider a filtered probability space (Ω,ℱ,𝔽,) with standard filtration 𝔽=(ℱ_t)_t ∈ [0,T] and assume that all processes are 𝔽-adapted. In this context, the fractional Black-Scholes model suggests that we have one risk-free asset with constant price S^0_t=1 and d risky assets with a price process S=(S^1,…,S^d) defined on [0,T] by the stochastic differential equations (SDEs) dS_t^i=μ^iS_t^idt+σ^iS_t^idB_t^H^i,S_0^i=s_0^i,i=1,…,d. Here, the drifts or expected returns μ^i∈ℝ, volatilities σ^i>0 and initial prices s_0^i>0 are given constants. In contrast to the standard Black-Scholes model, the SDEs are not driven by sBms (or Wiener processes) but by fBms B_t^H^i with Hurst parameters H^i∈(0.5,1). The fBms are assumed to be independent, which is reasonable when the risky assets are, for example, certain types of industry portfolios, investment funds or commodity futures baskets <cit.>.[A formal discussion of correlated fBms can be found in <cit.>.] The one-dimensional fBm (B_t^H)_t∈[0,T] is a centered Gaussian process with covariance Cov(B_t^H,B_s^H)=1/2(|t|^2H+|s|^2H-|t-s|^2H)for H∈(0,1). It is a self-similar process with stationary increments. The special case H=0.5 reflects the sBm which has independent increments and enjoys the martingale property. For H≠ 0.5, the increments are correlated and the process is not a martingale. This implies memory effects for the asset returns dS^i_t/S^i_t. For the range H∈(0.5,1) considered in the fractional Black-Scholes world, asset returns are positively correlated, i.e., persistent. While, for H=0.5, we have to rely on 's calculus, H>0.5 allows us to define stochastic integrals w.r.t. B_t^H path-wise in the Riemann-Stieltjes sense <cit.>. Thus, the solutions of the SDEs in (<ref>) are S_t^i=s_0^i exp{μ^i t+σ^i B_t^H^i}, t ∈[0,T]. The processes defined in (<ref>) are called geometric fBms.[For the sBm with H^i=0.5, we have S_t^i=s_0^i exp{(μ^i-(σ^i)^2/2) t+σ^i B_t^H^i}. Properties of fBms and the associated integration theory are outlined in <cit.>, <cit.> and <cit.>.] Portfolios Asset transactions in a fractional Black-Scholes market are modeled based on the usual assumptions of permanent trading, unlimited borrowing and short-selling, real asset quantities and contemporary agreement of buying and selling prices. Furthermore, there are no transaction costs, fees or taxes. We can describe the trading activity of an investor by an initial amount of capital ∈ and a 𝔽-adapted process =(_t)_t∈[0,T]=(Ψ_t^0,Ψ_t^1,…,Ψ_t^d)_t∈[0,T]. Here, Ψ_t^0∈ℝ and Ψ_t^1,…, Ψ_t^d∈ℝ denote the numbers of risk-free and risky assets held by the investor at time t, respectively. Ψ is called portfolio or trading strategy. The investor has a long (short) position if Ψ_t^i, i=0,…,d, is positive (negative). At time t, the value V_t^ of the portfoliois given by V_t^ = ∑_i=0^dΨ_t^iS_t^i. V^Ψ=(V_t^)_t∈[0,T] is the value process of Ψ. In an arbitrage context, Ψ is assumed to be self-financing. This means that there is no exogenous infusion or withdrawal of capital after the purchase of the portfolio. Rebalancing the portfolio must be financed solely by trading thed+1 available assets. Mathematically, self-financing in a continuous-time market is defined by the property that the value process for all t∈[0,T] can be expressed as V_t^ = + G_t^where G_t^=∑_i=0^d∫_0^tΨ_s^idS^i_s. G^Ψ=(G_t^)_t∈[0,T] is the gain process of Ψ, where the gain G_t^ at time t is given by a sum of stochastic integrals w.r.t. geometric fBms <cit.>. The self-financing condition (<ref>) can also be stated in differential form, i.e., dV_t^=∑_i=0^dΨ_s^idS^i_s. It shows that, for a self-financing strategy, the changes in portfolio value are not due to rebalancing but rather to changes in asset prices. §.§ Continuous-time arbitrage In a standard Black-Scholes world, arbitrage is impossible. This is a consequence of the fundamental theorem of asset pricing <cit.> and has its roots in the sBm martingale property. In contrast, a fBm with H≠ 0.5 behaves predictably such that our fractional Black-Scholes world offers arbitrage opportunities which can be exploited by constructing suitable arbitrage portfolios. A self-financing portfolio Ψ is called an arbitrage portfolio if its value process satisfies the three properties (i) V_0^=0, (ii)(V_t^≥ 0for allt∈(0,T])=1, (iii)(V_T^>0)>0. This implies that arbitrage is essentially the possibility to generate a positive amount of money without having to invest any initial capital and without any risk of loss. In the following, we present two simple arbitrage strategies satisfying the properties in (<ref>). For every arbitrage strategy , the scaled strategy = with >0 is also an arbitrage strategy. This is because (<ref>) and (<ref>) imply V^ =V^ and G^ =G^, respectively. Since the initial value V^_0 is zero, we also have V^_0=0. Hence,satisfiesthe self-financing condition V_t^ = v + G_t^ in (<ref>) with v=0. Finally, V^ =V^ implies that fulfillsthe arbitrage conditions in (<ref>). Shiryaev strategy <cit.> proposes a strategy that generates an arbitrage portfolio consisting of the risk-free asset and one risky asset. Thus, we have d=1. For simplicity, we denote the drift, volatility and Hurst coefficient of the risky asset by μ, σ and H, respectively. For the two assets, the strategy suggests entering the time t∈[0,T] positions [Ψ_t^0= 1/s_0^1((s_0^1)^2-(S_t^1)^2)= s_0^1(1-exp{2μ t+2σ B_t^H}),; [1ex] Ψ_t^1=2/s_0^1 (S_t^1-s_0^1)= 2(exp{μt+ σ B_t^H}-1). ] At every t, it compares the value S_t^1 of a pure risky investment with an alternative investment of the initial risky asset price s_0^1 in the risk-free asset. Because, in our market model, we have S_t^0=1, this alternative investment has a constant value of s_0^1.[The strategy can be easily generalized to markets with risk-free asset prices S_t^0=e^rt, where r≥ 0 denotes the risk-free rate <cit.>.] If the value S_t^1 of the pure risky investment exceeds (falls below) the value s_0^1 of the alternative investment, the investor holds a long (short) position in the risky asset and a short (long) position in the risk-free asset. In the case of equality, he is not invested and the portfolio value is zero. The number of risky asset shares Ψ_t^1 in (<ref>) does not depend on the initial risky asset price s_0^1 but on the parameters μ, σ and H. <cit.> shows that the strategyin (<ref>) is self-financing. Furthermore, at time t=0, we have Ψ_0^0=Ψ_0^1=0 and hence V_0^=0, i.e., no initial investment is required. Substituting (<ref>) into (<ref>) shows that the portfolio value at any time t∈[0,T] is V^Ψ_t=(S_t^1-s_0^1)^2/s_0^1 ≥ 0. We obtain (V_T^>0)>0. Thus, according to (<ref>),is indeed an arbitrage strategy. Salopek strategy Another arbitrage strategy, dating back to <cit.> and applied in a fractional Black-Scholes market by <cit.>, trades d≥2 risky assets and ignores the risk-free asset. It is defined for two real-valued constants α<β and can be summarized as =(0,(α,β)) or, with some abuse of notation, =(α,β). The entries of (α,β)=(Ψ_t^1(α,β),…,Ψ_t^d(α,β)) are the risky asset shares at time t∈[0,T]. Specifically, for i=1,…,d, we have Ψ_t^i(α,β)= Ψ_t^i(β)-Ψ_t^i(α), where Ψ_t^i(a) =1/d(S_t^i/M_a(S_t))^a-1. M_a(x) denotes the a-order power mean of x=(x^1,…,x^d)∈_+^d. It is given by M_a(x)=(1/d∑_i=1^d(x^i)^a)^1/a fora≠ 0,M_0(x)=√(x^1·…· x^d) fora = 0. To provide an economic interpretation of this strategy, it is instructive to recall some properties of the involved family of power means <cit.>. With respect to the properties of the a-order power mean, we can list the following important special cases: [ M_1(x) =(x^1+…+x^d)/d(arithmetic mean);[0.5ex] M_2(x)=√(((x^1)^2+…+(x^d)^2)/d) (quadratic mean);M_-1(x)=((1/x^1+…+1/x^d)/d)^-1(harmonic mean); M_0(x) =√(x^1·…· x^d)=lim_a→ 0 M_a(x) (geometric mean); M_∞(x):=lim_a→ +∞ M_a(x)=x_max=max{x^1,…,x^d}(maximum ofx);M_-∞(x) :=lim_a→ -∞ M_a(x)=x_min= min{x^1,…,x^d} (minimum of x);] Furthermore, the function a↦ M_a(x) is increasing. For a<b, we have x_min≤ M_a(x)≤ M_b(x)≤ x_max with equalities if and only if x^1=…=x^d=x, i.e., all entries of x are identical. In this situation, M_a(x)=x holds for all a∈. The strategy (α,β) in (<ref>) is expressed as the difference between (β) and Ψ(α). Because these two components can be considered as strategies themselves, we call Ψ(a) an a-strategy or a-portfolio. Consequently, an investor can implement (α,β) by purchasing a β-portfolio and short-selling an α-portfolio. Substituting the Ψ(a) specified by (<ref>) into (<ref>) provides the portfolio value of an a-strategy, i.e., we obtain V_t^(a)= M_a(S_t). It also shows that V_0^(a)= M_a(s_0), i.e., the initial investment is positive and equals the a-order power mean of initial asset prices s_0. An a-strategy investor enters long positions in all d risky assets and chooses their numbers proportional to (S_i^t)^a-1. More specifically, for a=1, we have Ψ_t^i=1/d, i.e., an equally allocated investment, and the portfolio value is V_t^(1)= M_1(S_t), i.e., the arithmetic mean of prices. For a>1 (a<1), the portfolio contains more (fewer) high-priced assets than low-priced assets. This feature becomes more pronounced with higher (lower) orders a>1 (a<1). In the limit for a→∞ (a→ -∞), the investor only holds the asset with the highest (lowest) price. If there are m≥ 1 risky assets sharing this price, he orders 1/m each. The strategy (<ref>) is an arbitrage strategy if we impose the following assumption to the financial market model. All price processes (S_t^i)_t∈[0,T] of the risky assets i=1,…,d start at time t=0 with identical initial prices S_0^i=s>0. Assumption <ref> serves mathematical simplification and will not be fulfilled in practice. However, this is not problematic because we can rescale the asset prices via S_t^i=s/s_0^iS_t^i to S_0^i=s and compute the arbitrage positions Ψ_t^i in the rescaled market. They are linked to the original market viaΨ_t^i=s/s_0^iΨ_t^i. Because Ψ_t^i S_t^i=Ψ_t^i S_t^i, (<ref>) delivers V^_t=V^_t. That is, the portfolio value is not affected by the transformation. We now show that (<ref>) satisfies the three conditions in (<ref>) and is in fact an arbitrage strategy. As far as the self-financing property is concerned, it has been verified by <cit.>. According to (<ref>), for all t∈[0,T], the portfolio value is given by V_t^=V_t^(β)-V_t^(α)= M_β(S_t)- M_α(S_t)≥ 0, where we have used the assumption α<β and relation (<ref>) stating that M_a(·) is increasing in a. This proves condition (ii) in (<ref>). Condition (i) on zero initial investment follows from Assumption <ref> of identical initial asset prices. It yields V_0^= M_β(s_0)- M_α(s_0)=s -s= 0.[Identical initial prices imply that, at time t=0, an investor formally buys Ψ^i_0(β)=1/d shares of each asset and simultaneously sells Ψ^i_0(α)=1/d shares of each asset.] Finally, because we have assumed that the asset prices (<ref>) are driven by independent fBms, prices are uncorrelated and thus, at time T, almost surely not identical. This implies the strict inequalityV_T^=M_β(S_T)- M_α(S_T) > 0 with probability one such that arbitrage condition (iii) is also satisfied. The monotonicity property (<ref>) of the a-order power mean and the portfolio value expression (<ref>) suggest to choose the largest possible β and the smallest possible α. Fusing the limits β→∞ and α→-∞ into the following proposition shows that an arbitrage strategy with large d can reduce to just buying the asset i with the highest price and short-selling the asset j with the lowest price. Let i,j∈{1,...,d}, t∈[0,T] and α<1≤β such that prices satisfy S_t^i > M_β(S_t)> M_α(S_t) > S_t^j. Then, the strategy (<ref>) has the property _t^i(α,β)>0>_t^j(α,β). That is, the investor buys the high-priced i and short-sells the low-priced j. This particularly holds when the prices of i and j represent the maximum and minimum over all S_t^1,…,S_t^d. See Appendix <ref>. §.§ Discrete-time arbitrage We now replace the idealized continuous-time financial market model with permanent and frictionless asset transfers by a more realistic setup where trading takes place only at a finite number of fixed points in time and is subject to transaction costs. Discrete-time trading In the discrete-time financial market model, prices are quoted at the times 0=t_0<t_1<… <t_N=T. A portfolio is created at time t_0=0, rebalanced at times t_1,…,t_N-1 and liquidated at terminal time t_N=T. We focus on equidistant instants of time t_n=nT/N, n=0,…,N, which divide the total trading horizon [0,T] into N trading periods of the same length T/N. Thus, sampling the asset price processes (<ref>) of the fractional Black-Scholes modelat t_0,…, t_N generates a sequence of risk-free asset prices(S_t_n^0)_n=0,...,N and d sequences ofrisky asset prices (S_t_n^i)_n=0,...,N defined by S_t_n^0=1, S_t_n^i=s_0^i exp{μ^i t_n+σ^i B_t_n^H^i}, for n=0,…,Nandi=1,…,d, with the same parameters as in Section <ref>. It has to be expected that the discretization of a self-financing continuous-time strategy Ψ, such as the Shiryaev and Salopek strategies of Section <ref>, and the existence of transaction costs affect the self-financing property. Even without transaction costs, rebalancing a portfolio according to a discretized self-financing continuous-time strategy requires the infusion of or allows the withdrawal of capital.[In Proposition <ref>, we show that, for example, discretizing the Shiryaev strategy (<ref>) almost surely leads toa strictly positive capital requirement.] These rebalancing costs and the classic transaction costs may be incorporated by modifying the risk-free asset holdings Φ^0 defined below. However, because we wish to explicitly quantify the impact of time discretization and transaction costs oncontinuous-time arbitrage strategies, we extend our financial market model by an additional asset d+1 which we call transaction account. In the investment fund industry, such (cash) accounts are used to react flexibly to market events <cit.>. In our context, it allows us to express the aforementioned impact in monetary units. Similar to the risk-free asset price S^0, the price process of the new asset is a constant process S^d+1=(S^d+1_t_n)_n=0,…,N with S^d+1_t_n=1. We capture the trading activity of an investor by an initial capital amount ∈ and the discrete-time 𝔽-predictable process = (_n)_n=1,…,N+1=(Φ_n^0,Φ_n^1,…,Φ_n^d+1)_n=1,…,N+1. Here, Φ_n^0 and Φ_n^d+1∈ℝ denote for n=1,...,N the holdings in the risk-free asset and the transaction account, respectively, chosen at the beginning of the n-th trading period [t_n-1,t_n) and kept constant over that period. Further,Φ_n^i∈ℝ is the quantity of risky asset i=1,…,d held in the n-th trading period. The vectorΦ_N+1 relates to the liquidation of the portfolio at time t_N=T. Overall, Φ is the discrete-time portfolio or trading strategy. Discretizing the continuous-time strategy=(Ψ_t)_t∈[0,T] in (<ref>) leads to a piece-wise constant strategy where the investor period-wise sets and upholds . This means that, fori=0,…,d, we have Φ_n^i=Ψ_t_n-1^i, n=1,…, N, whereas liquidating the portfolio yieldsΦ_N+1^i=0. The positions Φ_n^d+1 in the transaction account are specified in what follows. Transaction costs For purchasing, rebalancing and liquidating the portfolio, the investor has to pay transaction costs depending on the trading volume of the risky assets. For a given strategy , at time t, this volume is defined by Γ^_t= {[ ∑_i=1^d|Φ_1^i|S_0^i, t=t_0=0, (purchasing); ∑_i=1^d|Φ_n+1^i-Φ_n^i|S_t_n^i,t=t_1,…, t_N-1,(rebalancing); ∑_i=1^d|Φ_N^i|S_T^i, t=t_N=T.(liquidating);]. We specify transaction costs proportional to the trading volume. They are determined by the proportionality factor p_1≥ 0 (in percent) if they exceed the minimum fee p_2≥ 0 (in monetary units). Otherwise, p_2 is charged. We denote p=(p_1,p_2) and define the transaction costs for t=t_0,…,t_N as L_t^ =l(Γ^_t,p) with l(y,p)=max(p_1y,p_2)_{y>0}. Note that no transaction costs are charged at time t if the trading volume Γ^Φ_t is zero. The special case of a model without transaction costs is reflected by p_1=p_2=0. Liquidation In the continuous-time model with frictionless trading, the terminal portfolio value V_T^ in (<ref>) is equal to the revenue from selling the portfolio. In the discrete-time case, liquidating the portfolio induces the transaction costs L^_t_N=l(Γ^_t_N,p) of (<ref>) and (<ref>). Thus, the net revenue is R^ =∑_i=0^dΦ_N^iS_T^i - L_T^. Transaction account As discussed above, we have augmented our model by an asset d+1 called transaction account. It is used to finance rebalancing and transaction costs. Furthermore, it receives the net liquidation revenue at terminal time t_N=T. We now derive the holdings Φ^d+1_n, n=1,…,N+1, for this asset. To this end, the rebalancing costs of a strategyat time t_n are denoted by D_t_n^ and defined as the value difference between the (risk-free and risky) asset holdings after and before trading: D_t_n^ =∑_i=0^dΦ_n+1^i S_t_n^i - ∑_i=0^dΦ_n^i S_t_n^i = ∑_i=0^d(Φ_n+1^i-Φ_n^i)S_t_n^i= ∑_i=0^d(Ψ_t_n^i-Ψ_t_n-1^i)S_t_n^i, n=1,…,N-1. Here, the last line follows from the sampling property Φ_n^i=Ψ_t_n-1^i, i=1,…,d. Also note that the rebalancing costs at time t_0=0 and t_N=T are zero. Aggregating the rebalancing and transaction costs as well as the net liquidation revenue, the holdings in the transaction account can be stated recursively by Φ_1^d+1 =-L_0^, (purchasing) Φ_n+1^d+1 =Φ_n^d+1 - D_t_n^-L_t_n^, n=1,…,N-1, (rebalancing) Φ_N+1^d+1 =Φ_N^d+1 +R^, (liquidating) where L_t^, R^ and D_t^ are obtained according to (<ref>), (<ref>) and (<ref>), respectively. Portfolio value At time t_n, the value V_t_n^ of the portfoliois V_t_n^ = ∑_i=0^d+1Φ_n+1^i S_t_n^i,n=0,…,N. V^=(V_t_n^)_n=0,…,N is the discrete-time value process. While, at time t=0, the continuous-time model yields V_t_0^=, the discrete-time case delivers V_0^=-L_0^. That is, the portfolio value equals the initial capital minus the transaction costs for purchasing the portfolio. Ifresults from discretizing a continuous-time arbitrage strategywith V_0^=v=0, the discrete-time value process starts with V_0^=-L_0^≤ 0. This term is strictly negative if the initial trading volume Γ_0^ and at least one of the two transaction cost parameters p_1 and p_2 is positive. For the terminal trading time t_N=T, substituting Φ^0_N+1=… =Φ^d_N+1=0 into(<ref>) and applying (<ref>) providesV_T^=Φ_N+1^d+1S_t_n^d+1 =Φ_N^d+1 +R^. Hence, the terminal portfolio value equals the net liquidation revenue R^ minus the accumulated rebalancing and transaction costsΦ_N^d+1 for trading at times t_0,…,t_N-1. Finally, it is noteworthy that the discrete-time value process V^ satisfies a generalized self-financing condition V_t_n-^ - L_t_n^=V_t_n^ ,n=0,…,N, where V_t_n-^=∑_i=0^d+1Φ_n^i S_t_n^i is the portfolio value before rebalancing at time t_n, n=1,…,N, and V_0-^=. This condition formalizes the property that the value after rebalancing equals the value before rebalancing minus the transaction costs of the corresponding trade. Moreover, the continuous and discrete portfolio values (<ref>) and (<ref>), respectively, Φ_n^i=Ψ_t_n-1^i, n=1,…, N, and S_t^d+1=1 allow us to express the performance of the discretized strategyrelative toin terms of the holdings in the transaction account: V_t^- V_t^ =Φ_t^d+1fort=t_0,…, t_N-1. For a continuous-time arbitrage strategy , we know from Remark <ref> that scaling the strategy by some factor >0 preserves the arbitrage property. = is also an arbitrage strategy. For the value process, we have V^= V^. A time discretization ofand transaction costs generally destroy the arbitrage property. However, inspecting the construction of the discretized strategyreveals that we preserve the scaling property of the discrete-time value process V^= V^ as long as the transaction costs are defined with a floor p_2=0, i.e., only proportional transaction costs L_t^=p_1 Γ_t^ are charged. § SIMULATION STUDY §.§ Parameters To provide a full-scale analysis of our two arbitrage strategies, we conduct a Monte Carlo study based on the model and trading parameters of Table <ref>. This table captures our basis setting which will be successively modified and relaxed as we proceed. Guided by the empirical literature <cit.>, we start by specifying suitable drifts μ^i, volatilities σ^i and Hurst parameters H^i for the d risky assets of the Shiryaev and Salopek strategies. We restrict the latter to d = 2 assets and assume that they have identical parameters. We also set the initial asset prices to s = 100. We then consider an investor with a T=1 year investment horizon subdivided into N = 250 trading days <cit.>. This investor is assumed to follow the discretization of Section <ref> to trade the strategies of Section <ref> at a daily frequency. All transaction costs p are zero. To capture the performance of this investor, we simulate 100,000 asset price scenarios and scenario-wise document V_T^, i.e., the portfolio value after liquidation.[This number of simulation repetitions ensures stable results <cit.>.] Here, a path for a fBm is generated via the spectral method of <cit.>.[A detailed description of this simulation technique can be found in Appendix <ref>.] For each strategy (and parameter setting), our Monte Carlo study delivers a distribution of V_T^ values which will undergo detailed analysis. §.§ Discretized Shiryaev strategy §.§.§ Impact of time discretization We start with the Shiryaev strategy which trades the risk-free asset and one risky asset. For this strategy and our basis setting of Table <ref>, Figure <ref> presents a typical realization of our simulations. Panel (a) plots the daily prices S^0_t_n=1 and S^1_t_n of the risk-free and the risky asset, respectively. Panel (b) describes the daily strategyand the associated rebalancing costs D_t_n^. For visual convenience, we scale the number of risky assets by the initial risky asset price S_0^1=s and show the negative costs -D_t_n^. Because of p=(0,0) and (<ref>), the holdings Φ_n^2 in the transaction account result from accumulating the rebalancing costs, i.e., Φ_1^2=0, Φ_n+1^2=Φ_n^2-D_t_n^, n=1,…,N-1. Finally, Panel (c) illustrates the value process V^ of the continuous-time strategy , the value processV^ of the discretized strategyand the difference between them. As shown in (<ref>), this difference equals the holdings Φ^2_t_n in the transaction account for t_n=t_0,…,t_N-1. According to (<ref>), a Shiryaev-type investor enters a long (short) position in the risky asset whenever the risky asset price exceeds (falls below) the initial risky asset price. The opposite applies to the risk-free asset. This can be seen in Panels (a) and (b). We also observe that the rebalancing costs D_t_n^ are small and always positive; this is validated in Proposition <ref>. Thus, each rebalancing activity requires new capital and increases the absolute value of the negative holdings Φ^2 in the transaction account. In Panel (c), this leads to a growing difference between the portfolio values of continuous and discrete trading. For the rebalancing costs (<ref>) of the discretized Shiryaev strategy (<ref>), we have D_t_n^>0, n=1,…, N-1, almost surely. See Appendix <ref>. We know from (<ref>) that the portfolio value of the continuous-time arbitrage strategyis V^Ψ_t=(S_t^1-s_0^1)^2/s_0^1 ≥ 0. It rises with the distance between the risky asset price S_t^1 and the initial risky asset price s_0^1. In other words, the strategy benefits from prices rising above s_0^1 and from prices falling below s_0^1. As indicated in Remarks <ref> and <ref>, scaling the continuous-time strategyby some factor >0 preserves the arbitrage property and leads to a scaling of the value processes V^ and V^by the same factor. Looking at the terminal value V_T^≈39, we can deduce that raising the basis scaling factor of=10^2 to say =10^5 increases the terminal value to roughly 39,000. Hence, the absolute size of the portfolio value is not relevant for evaluating the performance of the strategy. §.§.§ Impact of transaction costs After examining a single simulation scenario in the parameter basis setting, we now turn to the results of 100,000 scenarios and additionally introduce transaction costs. Panel (a) of Figure <ref> visualizes the simulated distribution of the terminal portfolio value V_T^ for continuous-time trading and the correspondingV_T^ distributions for the discrete-time case with three different transaction cost variants. p = (0,0) resembles no transaction costs. p=(0.1,0) considers only proportional costs, whereas p=(0.1,0.5) additionally includes a minimum fee. Recall that the proportional values are expressed in percent. The chosen cost magnitudes are guided by what are currently very low commissions and brokerage fees <cit.>. Table <ref> provides summary statistics for the distributions and concisely evaluates the performance of the trading strategy. Here, we are particularly interested in the mean terminal value and the loss probability because they capture 𝔼(V_T^) and ℙ(V_T^<0), respectively. We observe that the Shiryaev strategyis an arbitrage strategy with positive terminal values V_T^ in all scenarios. In contrast, discretizing the strategy yieldsa terminal value V_T^ distribution of similar shape but shifted towards smaller values and partially into negative territory. Without transaction costs, the range of observed terminal values is [-42.46,428.24]. This means that the maximum gain is roughly ten times higher than the maximum loss. With a value of 112.93, the mean of V_T^ is more than double the maximum loss. The former also covers about 75% of the mean of V_T^ which amounts to 148.26. To earn such outcomes, the investor has to accept a loss probability of only 28%. In line with intuition, transaction costs shift the portfolio value distribution even further such that losses become higher and more likely. For p=(0.1,0), the mean of V_T^ falls to 95.27 but remains positive. Simultaneously, the loss probability rises to 36% but can still be considered reasonably low <cit.>. In contrast, p=(0.1,0.5) generates a negative mean of -13.86 and a loss probability of 67%. The reason for this drastic impact of the minimum fee is that our basis setting is of low monetary scale such that the daily trading volumes are small and the minimum fee applies frequently. For N=250 trades, this often results in a total fee of 250× p_2=125 offsetting gains and causing high losses. Overall, while proportional transaction costs only slightly reduce the performance of the discretized strategy, minimum fees can render it unattractive for small-scale investors. However, large-scale investors with a higher scaling factorand thus higher trading volumes do not suffer from this kind of problem. Panel (b) of Figure <ref> conducts a worst-case analysis similar to drawdown calculations in active risk management <cit.>. For t∈ [0,T], we define the running minimum process associated with the discrete-time value process V^ as m_t:=min_t_n≤ tV_t_n^. With t=T, we obtain m_T representing the least favorable portfolio value in the investment horizon [0,T]. The simulated distributions of m_T show that its upper bound is zero because, across our transaction cost variants, we have V_0^≤0. The smallest values of m_T are close to the minima of the terminal values V_T^ in Panel (a). However, a frequency comparison reveals that the vast majority of worst-case events do not cluster at time T. Finally, Panel (c) shows the simulated distributions of the difference V_T^-V_T^between the terminal portfolio values of continuous-time trading and its discrete-time counterparts. Discrepancies obviously rise with p. More demanding transaction cost variants require higher capital infusions, i.e., a more intensive usage of the transaction account. §.§.§ Impact of asset model parameters To implement the Shiryaev strategy, investors have to select a suitable risky asset. To support this choice, we study the impact of the asset parameters (μ,σ,H) on the performance of the strategy. This primarily involves constructing figures and tables in the style of Figure <ref> and Table <ref>. However, instead of p, we now vary (μ,σ,H). Drift μ Figure <ref> and Table <ref> start by considering the alternative drift parameters μ∈{0,±0.1,±0.2}. Assets with higher absolute mean returns μ shift and extend the distribution of V_T^ towards larger terminal values because they fuel the strategy with more extreme prices. This effect is stronger for positive than for negative μ because upward movements are unbounded, whereas downward movements have a floor at a price level of zero.[A similar rationale explains differences in the prices of at-the-money call and put options with identical underlying, strike and maturity <cit.>.] For example, starting from 72.14 for μ=0, the mean terminal value is 213.04 for μ=0.1 but only 137.96 for μ=-0.1. In contrast, the loss probabilities are quite symmetric in μ. From 26% for μ = 0, they rise to about 40% for |μ|=0.1 and decrease to about 5% for |μ|=0.2. A similar feature can be observed for the running minimum m_T. For μ=0, its distribution is almost uniform. For rising |μ|, peaks near zero become more pronounced and excursions of the portfolio value significantly below zero less likely. V_T^ -V_T^ is not symmetric in μ. Instead, the distribution support increases with μ towards larger values. Hence, a higher μ induces more rebalancing costs in discrete-time trading. Volatility σ Figure <ref> and Table <ref> present our sensitivity results for the volatilities σ∈{0.05,0.10,0.15}. Increasing volatility goes along with a greater variability of terminal portfolio values V_T^. Large gains and large losses become more likely. This is also evident in the stretching distributions of the running minimum m_T. While the mean terminal values increase with σ, loss probabilities decrease. For example, σ=0.05 delivers 47.65 and 41%, whereas σ=0.15 yields 223.71 and 27%. Because investors receive more reward at lower risk, they have an incentive to opt for volatile assets <cit.>. However, there is a trade-off between these advantages and the rebalancing costs of discrete trading which are resembled by V_T^ -V_T^ and substantially increase with σ. Hurst parameter H In Figure <ref> and Table <ref>, we investigate the Hurst coefficients H ∈{0.51,0.55,0.60,0,65,0.70}. Recall that H=0.5 implies no memory and elevating H within the interval (0.5,1) establishes long memory. It generates positive serial correlation with levels linked to H and high even for distant lags. Shiryaev-type investors take long (short) positions in the risky asset when its price deviates from an initial state in positive (negative) direction. Thus, they can benefit directly from a trending behavior of the risky asset which is more likely under high than low H∈(0.5,1). Specifically,with rising H, the distributions of V_T^ and m_T relocate such that the likelihood of large gains (losses) increases (decreases). Switching from H=0.6 to H=0.7, for example, raises the mean terminal value from 112.94 to 138.39 and lowers the loss probability from 28% to 15%. Interestingly, this is accompanied by a sharp drop in rebalancing costs V_T^-V_T^. Hence, investors should trade assets with high H, which have been identified in many asset classes <cit.>, because they make the strategy more secure and less cash-intensive with respect to the transaction account. §.§.§ Impact of trading horizon and frequency Besides a suitable risky asset, Shiryaev-type investors have to decide on the trading horizon and frequency. Thus, it is instructive to know how they affect portfolio performance. Trading horizon In a first experiment, we fix the trading frequency to daily and vary the trading horizon T between 6 months and 10 years. As far as the remaining parameters are concerned, we use the basis setting of Table <ref> and the additional transaction cost settings of Figure <ref> and Table <ref>. For each setting and trading horizon, we simulate 100,000 scenarios and report the mean terminal portfolio value and the loss probability in Figure <ref>. Panel (a) illustrates that the mean increases with the trading horizon and, except for the shortest horizons and the highest transaction costs, is positive-valued. Plugging (<ref>) into (<ref>), the terminal value of the continuous-time Shiryaev strategy is given by V^Ψ_T=s_0^1 (exp{μ T+σ B_T^H}-1 )^2. Because B_T^H is a centered Gaussian random variable with variance T^2H, we can deduce that, for large T, 𝔼(V^Ψ_T) grows just like s_0^1 exp{2μ T+2σ^2T^2H}. This exceeds exponential growth in T because H>0.5. A similar behavior can be observed for the discretized strategy. Panel (b) shows that the loss probabilities initially decrease with T and then stabilize at approximately 40%. The differences between our transaction cost variants shrink with T and also stabilize. This can be explained by successively rising daily trading volumes eliminating the dominance of the minimum fee. Trading frequency In a reverse second experiment, we fix the trading horizon to T=1 year and vary the trading frequency. We evaluate daily, two-daily, weekly, two-weekly and monthly rebalancing corresponding to 250, 125, 50, 25 and 12 trading periods per year.[We do not consider frequencies higher than daily because, in this context, our assumption of independent asset prices would no longer be realistic <cit.>.] Figure <ref> presents the simulation outcomes for our three transaction cost settings, i.e., the mean terminal value and the loss probability as functions of the trading frequency. The results for trading without transaction costs and only proportional costs are similar. The mean terminal value increases with the trading frequency but there is still a clear difference to continuous-time trading.[For p=(0,0), the difference disappears when the trading frequency tends to infinity.] The loss probability decreases with the trading frequency.[For p=(0,0), the limiting probability is zero.] For investors facing a supplementary minimum fee, the mean terminal value sharply drops with the trading frequency and reaches a negative value for daily trading. At the same time, the loss probability rises to more than 60%. This feature is again caused by the scale-related relative size of proportional costs and the minimum fee. §.§ Discretized Salopek strategy §.§.§ Impact of time discretization We now turn to the Salopek strategy trading only risky assets and put special emphasis on a simple and practically appealing specification with d=2assets.[For the strategy to work, the prices of the two assets should not be perfectly correlated. This is ensured by our independence assumption of Section <ref>.] Following our approach for the Shiryaev strategy, Figure <ref> starts by presenting a typical simulated realization in the basis setting of Table <ref>. This means that we plot the prices S^1_t_n and S^2_t_n, the asset holdings Φ^1_t_n and Φ^2_t_n, the negative rebalancing costs -D_t_n^Φ as well as the discrete and continuous strategy value processes V_t_n^ and V_t_n^ including their differences V_t_n^-V_t_n^. In line with Proposition <ref>, we see that the investor is always long (short) in the asset with the higher (lower) price. Because the continuous-time value (<ref>) tells us that V_t^= M_β(S_t)- M_α(S_t)≥ 0, the properties of the a-order power mean M_a(.) imply that the portfolio value of the Salopek strategy is all the greater the more the prices of the two assets deviate from each other.If they coincide, we have M_β(S_t)= M_α(S_t) and consequently a V_t^ of zero. These features are comparable to the Shiryaev strategy. At first glance, it appears that there are also strong rebalancing cost similarities between the Shiryaev and Salopek strategies. For the Shiryaev strategy, we have shown that the rebalancing costs D_t_n^Φ are strictly positive for n=1,…,N-1 (see Proposition <ref>). Figure <ref> suggests the same property for the Salopek strategy. However, this does not hold in general. It can be verified via experiments with different choices of (α,β) that D_t_n^Φ may be negative for some n (see, for example, Figure <ref> of the appendix). Thus, in contrast to the Shiryaev strategy, the portfolio value V_t_n^ of the discretized Salopek strategy can exceed the portfolio value V_t_n^ of its continuous-time counterpart. §.§.§ Impact of transaction costs Uniting all 100,000 simulation scenarios and charging transaction costs in the Salopek strategy, Figure <ref> presents the distributions of the terminal portfolio values V_T^ and V_T^, the running minimum m_T and the difference V_T^-V_T^. In addition, Table <ref> reports summary statistics for the terminal value distributions. Similar to the Shiryaev case, discretization and transaction costs expand the negative distribution support of V_T^ and m_T and increase V_T^-V_T^. However, the Salopek strategy differs in notable aspects. First, without transaction costs, the means of V_T^ and V_T^ are 592.14 and 859.52, respectively. Because the former covers just about 69% of the latter, the Salopek strategy exhibits larger discretization shrinkage than the Shiryaev strategy. Second, turning to p=(0.1,0), the terminal mean and loss probability are 410.94 and 44%, respectively. These values are higher than for the Shiryaev strategy but must be put into the perspective that the Salopek strategy drains the transaction account more significantly than the Shiryaev strategy. Finally, for p=(0.1,0.5), we observe a terminal mean of 362.70 accompanied by a loss probability of 46%. While, in the Shiryaev case, the minimum fee causes a negative mean and a very high loss probability, this does not occur for the Salopek strategy because of its larger trading volumes. Overall, transaction costs do not crucially diminish the performance of the discretized strategy. §.§.§ Impact of Hurst parameters Our basis setting assumes that the Hurst coefficients of the traded assets are identical, i.e., H^1=H^2. Because this is not a necessary requirement for strategy execution, we also investigate Hurst values of different magnitudes. Given the complexity of this exercise, Figure <ref> presents its results, i.e., the characteristics of the terminal portfolio value distributions, in three-dimensional form. For H^1,H^2∈[0.51,0.99] with H^1≤ H^2, Panel (a) plots the minimum, mean and maximum of V_T^.[The outcomes for H^1>H^2 follow by symmetry.] Panels (b), (c) and (d) cover the mean of the running minimum m_T, the loss probability and the mean of V_T^-V_T^, respectively. Higher persistence pushes terminal values, limits the risk of loss and reduces capital infusions. If both H^1 and H^2 approach their limit value one, the loss probability and the mean of the running minimum are drawn to zero. Put differently, even though discretization yields V_T^-V_T^≠ 0, the discretized Salopek strategy converges to an almost perfect arbitrage strategy. This can be explained by the limiting behavior of the fBm B^Hfor H→ 1. In this situation, (<ref>) implies for the covariance Cov(B_t^H,B_s^H) → ts such that B_t^H and B_s^H are perfectly positively correlated for all t,s> 0 and it can be deduced that B_t^H=tB_1^H. Because the fBm is a centered Gaussian process, we obtain B^H^i_t=^i t withindependent standard Gaussian random variables ^i, i=1,2. In this case, (<ref>) delivers asset prices S_t^i=s_0^i exp{μ^it+σ^i^it}. Because their paths are exponential functions with growth rate μ^i+σ^i^i, the quantities ^i are the only source of uncertainty. However, they are unveiled to the investor with the asset price observations at the first trading time t_1. This means that, after t_1, future asset prices are completely known. If the growth rate of S^1 is above (below) the one of S^2, we have S^1_t>S^2_t (S^1_t<S^2_t) for all t∈[0,T]. Thus, a long position in the asset with the larger price and a short position in the other is a straightforward risk-free strategy. §.§.§ Impact of strategy parameters While Shiryaev-type investors have access to a unique trading rule, Salopek investors are confronted with a family of rules parameterized by (α,β). Consequently, they need to choose a suitable tuple (α,β) in practical applications. In the continuous-time case, (<ref>) and (<ref>) imply that the maximum portfolio value V_t^,max=max(S_t^1,S_t^2)-min(S_t^1,S_t^2) is attained for the limiting tuple (α,β)=(-∞,∞). With this setup, the strategy representation (<ref>) tells us thatis a buy-and-hold strategy with a long position in the high-priced asset financed by short selling the low-priced asset as long as the sign of S^1-S^2 is unchanged. If a change occurs, i.e., if the price paths cross, the asset roles simply reverse.[A graphical illustration of this strategy can be found in Figure <ref> of the appendix.] Although the infinite setup is appealing from a theoretical point of view, the question arises whether it also maxes out in a discrete environment. Rebalancing costs may vary with (α,β) and suggest a different optimal parameter choice. To provide an answer, Figure <ref> plots our set of previously used portfolio value characteristics against the parameters α∈[-30,29] and β∈ [α+1,30]. For α=β, the Salopek positions and portfolio value are zero because we are not invested in any risky asset. With growing difference β -α, the means of V_T^ andV_T^-V_T^ increase. They reach their maxima for the limit β=-α=∞. Thus, even though the rebalancing costs are also at their maximum, continuous-time and discrete-time trading both max in the same limiting case. With respect to the loss probabilities, we observe values between 24% and 42%. For β=-α=∞, we have 28%. The highest probability arises for the tuple (α,β)=(0,1). §.§.§ Impact of trading horizon and frequency To complete our analysis of the Salopek strategy, we study its sensitivity to different trading horizons and frequencies and compare the findings to the Shiryaev strategy. Trading horizon Again, we start by enlarging the trading horizon T and upholding a daily trading frequency. For our three transaction cost variants, Figure <ref> displays the mean terminal portfolio value and the loss probability as functions of T. For bothand , Panel (a) suggests that the growth of the mean terminal values in the first ten years is only slightly faster than linear. This is in contrast to the Shiryaev strategy where we detected faster than exponential growth. While, for T≤ 5.5, the means of the Salopek strategy are larger than those of the Shiryaev strategy, the latter surpass the former for higher T. In particular, for T=10, the Shiryaev values exceed the Salopek values nearly twofold. In our drifting environment, this can be linked to the fact that the Shiryaev asset tends to deviate further from its initial price than the Salopek assets deviate from each other. Panel (b) shows that the loss probabilities decrease with T and converge to a level of about 20% (30%, 35%) without (with) transaction costs. Hence, their magnitudes are smaller than for the Shiryaev strategy. Moreover, for increasing T, the loss probabilities related to transaction costs with a minimum fee do not approach those of purely proportional costs. Trading frequency In a last simulation, Figure <ref> sets diverse trading frequencies within a locked trading horizon of T=1 year. The resulting mean terminal portfolio values in Panel (a) imply that, even in the presence of transaction costs, higher trading frequencies are beneficial for investors. In contrast to the Shiryaev strategy, the impact of minimum fees is almost negligible at high frequencies. The loss probabilities in Panel (b) also recommend more frequent trading. The value of about 40% for daily trading illustrates once more that unavoidable rebalancing costs in discrete-time trading induce losses that prevent the strategy from reaching the zero probability limit of an infinite trading frequency. § CONCLUSION In this study, we have revisited the arbitrage strategies of <cit.> and <cit.> because they have a solid theoretical foundation and an elegant design making them highly appealing for investment practice. While the Shiryaev strategy trades only one risky asset and benefits from both rising and falling prices, the most elementary specification of the Salopek strategy trades two risky assets and capitalizes on prices drifting apart. Both strategies have very simple trading rules and rely only on realized prices, which are readily available in today's investment world, so that they can be easily automated in modern algorithmic trading facilities. Because these strategies aim at continuous-time trading in a fractional Black-Scholes market, we have transferred them to a discrete-time application and intensively studied their investment performance via Monte Carlo simulation. In conservative settings with independent assets, moderate serial correlation and realistic transaction costs, we show that, even though they can no longer be considered as arbitrage strategies, they exhibit positive terminal values on average and are accompanied by low loss probabilities. This makes them particularly interesting for tail-oriented investors <cit.>. Furthermore, we have revealed several interesting features of the discretized strategies. First, they perform reasonably well even if assets show relatively small persistence. Second, certain limiting cases of the strategies not only max out their performance but further simplify their implied asset positions. Third, time-discretization does not necessarily lead to portfolio values lower than in the continuous-time case. Finally, when adequately scaled, the strategies are useful for short-, medium- and long-term horizons and most advisable at a daily trading frequency. This nicely integrates into the growing literature on the welfare consequences of speeding up transactions in financial markets <cit.>. Our study offers plenty of scope for future research. With respect to theoretical work, it is instructive to introduce an interest-bearing transaction account (with potentially differing rates for borrowing and lending). Furthermore, modeling a negative cross-correlation between risky assets can be considered a fruitful endeavor because it has the potential to increase strategy performance. It also makes the strategies comparable to the domain of pairs trading rules for correlated assets <cit.>. As far as empirical work is concerned, we suggest a profound analysis of the Shiryaev and Salopek strategies in different asset classes. Such an analysis is complicated by the fact that traditional estimators of the Hurst coefficient (such as rescaled range and detrended fluctuation analysis) are highly sensitive to short time-series, short-term memory and non-normality. However, recent research has brought forth a variety of very promising estimators <cit.> which can serve as the basis for suitably capturing long memory and implementing investment strategies exploiting its dynamics. Appendix § PROOFS OF PROPOSITIONS Recall the Salopek strategy (<ref>) with _t^i(α,β)= _t^i(β)-_t^i(α), where _t^i(a)=1/d(S_t^i/M_a(S_t))^a-1 and M_a(x) is the (<ref>) a-order power mean of x=(x^1,…,x^d)∈_+^d. We start with proving _t^i(α,β) >0. Denoting x=S_t^i/M_β(S_t) and y=S_t^i/M_α(S_t), relation (<ref>) implies 1 <x<y and β≥ 1> α gives x^β-1≥ 1 >y^α-1. Hence, (S_t^i/M_β(S_t))^β-1 >(S_t^i/M_α(S_t))^α-1. This proves_t^i(β) > _t^i(α) from which the claim follows. The proof of _t^j(α,β) <0 is similar. Denoting x=S_t^j/M_β(S_t) and y=S_t^j/M_α(S_t), relation (<ref>) implies 0<x<y<1 and β≥ 1> α gives x^β-1≤ 1 < y^α-1. Hence, (S_t^j/M_β(S_t))^β-1 <(S_t^j/M_α(S_t))^α-1. This proves_t^j(β) < _t^j(α) from which the claim follows. For the special case S^i_t=S_t^max=max{S_t^1,… S_t^d} and S^j_t=S_t^min=min{S_t^1,… S_t^d}, the monotonicity property (<ref>) of the a-order power mean M_a(S_t) saying that S_t^min≤ M_α(S_t)≤ M_β(S_t)≤ S_t^ maximplies (<ref>). This completes the proof. Substituting the Shiryaev strategy (<ref>) into (<ref>) and using Φ_n^i=Ψ_t_n-1^i, n=1,…, N, we have almost surely D_t_n^ =(S_t_n-1^1)^2-(S_t_n^1)^2/s_0^1S^0_t_n+2/s_0^1(S_t_n^1-S_t_n-1^1)S^1_t_n=1/s_0^1((S_t_n^1)^2+(S_t_n-1^1)^2-2S_t_n^1S_t_n-1^1) =(S_t_n^1-S_t_n-1^1)^2/s_0^1>0. § SPECTRAL SIMULATION Various algorithms (such as midpoint displacement, Fourier filtering and spectral generation) have been proposed to simulate discrete-time fBms. <cit.> give an overview of available (accurate and approximate) methods and conclude that, in the case of a finite time horizon, spectral techniques should be preferred. Therefore, we use the spectral method of <cit.>. It is characterized by an efficient computation time and fully preserves the properties of a fBm. The spectral method exploits the stationarity of fBm increments and is based on features of the discrete Fourier transformation and the central limit theorem. It relies on the theory of stationary discrete-time stochastic processes as well as the associated correlation and spectral theory. Here, it is convenient to consider two-sided processes X=(X_m)_m∈, where the index m takes values in the setof all integers instead of just _0={0,1,…}. Stationary processes are typified by the correlation function R_X(m)=Cov(X_0,X_m), m∈, and the corresponding power spectral density S_X(f)= ∑_m=-∞^∞R_X(m)cos(2π mf) for -1/2≤ f≤1/2, where f is the frequency. R_X and S_X form a pair of discrete Fourier transforms. Simulating a path of a fBm B^H on [0,T] involves generating realizations of B^H at discrete times t_n=nΔ, n=0,…,N, with constant step size Δ =T/N for some N∈. It starts by producing N increments of a fBm B^H on [0,N] with unit step size Δ=1, i.e., W_k^H=B_k+1^H-B_k^H, k=0,…,N-1. Increment accumulation and self-similarity rescaling, i.e., B_t_n+1^H=(T/N)^H∑_k=0^nW_k^H,n=0,…,N-1, then approximate the discrete-time fBm B^H on [0,T]. According to <cit.>, for k=0,…,N-1, the increments W_k^H can be proxied by W_k^H=√(2N^-1)∑_j=-N/2^N/2-1[S_W^N(jN^-1)]^1/2[cos(2π jkN^-1)cos(φ_j)-sin(2π jkN^-1)sin(φ_j)]. Here, S_W^N denotes the finite horizon approximation (<ref>) of the power spectral density S_W of the sequence of increments (W_k^H). (φ_j) is a sequence of independent random variables uniformly distributed on the interval (0,2π). For the increment process W^H and the Hurst parameter H>0.5, the S_W in (<ref>) is not well-defined because of the long-range dependence of the fBm. This property implies that the correlation function of W^H,given by R_W(m)=1/2(|m+1|^2H+|m-1|^2H-2|m|^2H), does not decay fast enough for |m|→∞ and thus prevents the convergence of the infinite series in (<ref>). Because we only need paths of the fBm B^H on the finite interval [0,N] and 'very long memory' effects cannot be observed on finite intervals, this problem can be circumvented by neglecting correlations for time lags larger than N/2. This means that, in (<ref>), we do not use S_W but S_W^N(f) =∑_m=-N/2^N/2-11/2(|m+1|^2H+|m-1|^2H-2|m|^2H)cos(2π mf). Computer implementations of the spectral method benefit from fast Fourier transformation (FFT) which improves computation time. As emphasized by <cit.>, the increments W_k^H in (<ref>) can be obtained as the real part of a complex FFT where real and imaginary are(S_W^N(jN^-1))^1/2cos(φ_j) and (S_W^N(jN^-1))^1/2sin(φ_j), respectively. § ADDITIONAL FIGURES Figures <ref> and <ref> supply additional simulated realizations of the Salopek strategy to illustrate the effects of negative rebalancing costs and infinite strategy parameter values, respectively. | http://arxiv.org/abs/2311.15635v1 | {
"authors": [
"Kerstin Lamert",
"Benjamin R. Auer",
"Ralf Wunderlich"
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"categories": [
"q-fin.PM",
"91G10, 91G80"
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"primary_category": "q-fin.PM",
"published": "20231127085605",
"title": "Discretization of continuous-time arbitrage strategies in financial markets with fractional Brownian motion"
} |
roman From Pixels to Titles: Video Game Identification by Screenshots using Convolutional Neural Networks Fabricio Aparecido Breve January 14, 2024 ===================================================================================================[type=, title=LIST OF ABBREVIATIONS] arabic unsrttocchapterReferencesemptyempty | http://arxiv.org/abs/2311.15957v1 | {
"authors": [
"Jintao Shuai"
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"categories": [
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"cond-mat.mtrl-sci"
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"primary_category": "physics.app-ph",
"published": "20231127160036",
"title": "Interaction of surface acoustic waves and magnetic thin films"
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Asymptotic smoothness, concentration properties and applications]Asymptotic smoothness, concentration properties in Banach spaces and applicationsInstitute of Mathematics (IMAG) and Department of Mathematical Analysis, University of Granada, 18071, Granada, Spain [email protected] partially supported by MCIN/AEI/10.13039/501100011033 grant PID2021-122126NB-C31 and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033 We prove an optimal result of stability under ℓ_p-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property. [ A. Fovelle January 14, 2024 ====================§ INTRODUCTION In 2008, in order to show that L_p(0,1) is not uniformly homeomorphic to ℓ_p ⊕ℓ_2 for p ∈ (1, ∞) ∖{2 }, Kalton and Randrianarivony <cit.> introduced a new technique based on a certain class of graphs and asymptotic smoothness ideas. To be more specific, they introduced concentration properties for Lipschitz maps defined on Hamming graphs into a reflexive asymptotically uniformly smooth (AUS) Banach space (we refer the reader to Section 2 for the definitions). These concentration properties are stable under coarse Lipschitz embeddings (under maps that are bi-Lipschitz for large distances, see Section 2 for a precise definition) and prevent the equi-Lipschitz embeddings of the Hamming graphs, which make them obstructions to the coarse Lipschitz embedding of spaces without (enough) concentration. Their result was later extended to the quasi-reflexive case by Lancien et Raja <cit.>, who introduced weaker concentration properties that still provide obstructions to coarse Lipschitz embeddings. Soon after, Causey <cit.> proved that these same weaker concentration properties also apply to quasi-reflexive spaces with so-called upper ℓ_p tree estimates. The proof of these results is in two steps. In the first one, vaguely speaking, reflexivity or quasi-reflexivity is used to see a Lipschitz map defined on the Hamming graphs as the sum of branches of a weakly-null tree. The second one consists in dealing with this weakly-null tree using a hypothesis of upper ℓ_p tree estimates. So as to get examples of non-quasi-reflexive spaces with concentration properties, the author started a general study of these concentration properties, together with new ones and proved that ℓ_p-sums of quasi-reflexive spaces with these so-called upper ℓ_p tree estimates admit concentration. To do so, the following general theorem was proved:Let p ∈ (1, ∞), λ > 0, (X_n)_n ∈ a sequence of Banach spaces with property λ-HFC_p,d (resp. λ-HIC_p,d). Then ( ∑_n ∈ X_n )_ℓ_p has property (λ+2+ε)-HFC_p,d (resp. (λ+2+ε)-HIC_p,d) for every ε > 0. where property λ-HFC_p,d (resp. λ-HIC_p,d), defined in <cit.>, is a refinement of the property λ-HFC_p (resp. λ-HIC_p) first considered by Kalton and Randrianarivony (resp. Lancien and Raja): a space X has property λ-HFC_p (resp. λ-HIC_p) if for any Lipschitz function f : ([]^k, ) → X, there exist < (resp. n_1 < m_1 < n_2 < m_2 < … < n_k < m_k) such that f()-f()≤λ k^1/p(f) (see Section 2 for the definitions of the concentration properties).Since the only known examples of reflexive or quasi-reflexive spaces with concentration were those with an asymptotic smoothness property (given by the so-called upper ℓ_p tree estimates), the new examples of spaces with concentration given by Theorem <ref> also happen to have an asymptotic smoothness property. However, as mentioned in <cit.>, if one manages to prove Theorem <ref> without a loss on the constant, one would get an example of a Banach space admitting concentration without an asymptotic smoothness property. This is what we are going to do in this paper. As a consequence, we will even get Banach spaces with Szlenk index as big as we want that cannot equi-Lipschitz contain the Hamming graphs. This is the very first result of this type for a non-asymptotically smooth Banach space.In order to show this result, we introduce the notions and the terminology we will use in the second section of this paper while Section 3 is dedicated to the proof of the main result:Let λ≥ 2 and (X_n) be a sequence of Banach spaces with property λ-HFC_p,d (resp. property λ-HIC_p,d). For every ε>0, X=(∑_n X_n)_ℓ_p has property (λ+ε)-HFC_p,d (resp. property (λ+ε)-HIC_p,d). In section 4, we detail how this theorem allows us to get Banach spaces with Szlenk index as big as we want with some concentration and we study the topological complexity of the class of Banach spaces having HFC_p,d.§ DEFINITIONS AND NOTATION All Banach spaces in these notes are assumed to be real and infinite-dimensional unless otherwise stated. We denote the closed unit ball of a Banach space X by B_X, and its unit sphere by S_X. Given a Banach space X with norm ·_X, we simply write · as long as it is clear from the context on which space it is defined. Let (X_n)_n ∈ be a sequence of Banach spaces and p ∈ [1, ∞). We define the sum ( ∑_n ∈ X_n )_ℓ_p to be the space of sequences (x_n)_n ∈, where x_n ∈ X_n for all n ∈, such that ∑_n ∈x_n_X_n^p is finite, and we set(x_n)_n ∈ = ( ∑_n ∈x_n_X_n^p )^1/p .One can check that ( ∑_n ∈ X_n )_ℓ_p, endowed with the norm · defined above, is a Banach space. We can, in a similar way, define finite sums ( ∑_j=1^n X_j )_ℓ_p for all n ∈, and, in case n=2, we will write X_1 ⊕_p X_2.§.§ Hamming graphs Before introducing concentration properties, we need to define special metric graphs that we shall call Hamming graphs. Letbe an infinite subset of . We denote by []^ω the set of infinite subsets of . For ∈ []^ω and k ∈, let[]^k = {n=(n_1, …, n_k) ∈^k ; n_1 < ⋯ < n_k } , []^≤ k = ⋃_j=1^k []^j ∪{∅} ,and[]^< ω = ⋃_k=1^∞ []^k ∪{∅} .Then we equip []^k with the Hamming distance:(n, m)=| { j ; n_j ≠ m_j } |for all n=(n_1, …, n_k), m=(m_1, …, m_k) ∈ []^k. Let us mention that this distance can be extended to []^< ω by letting(n, m)=| { i ∈{1, ⋯, min(l,j) } ; n_i ≠ m_i } | + max(l,j)-min(l,j)for all n=(n_1, …, n_l), m=(m_1, …, m_j) ∈ []^< ω (with possibly l=0 or j=0). We also need to introduce I_k(), the set of strictly interlaced pairs in []^k:I_k()= { (n, m) ⊂ []^k ; n_1 < m_1 < ⋯ < n_k < m_k}and, for each j ∈{ 1, ⋯, k }, letH_j()= { (n, m) ⊂ []^k ; ∀ i ≠ j, n_i=m_iandn_j < m_j } .Let us mention that, in this paper, we will only be interested in the Hamming distance but originally, when Hamming graphs were used in <cit.>, it could be equally replaced (unless for their last Theorem 6.1) by the symmetric distance, defined byd_Δ(n, m)=12 | nm | for all n, m∈ []^< ω, where nm denotes the symmetric difference between n and m. §.§ Asymptotic uniform smoothness and Szlenk index Let us start this subsection by defining the asymptotic uniform smoothness. Let (X, ·) be a Banach space. Following Milman (see <cit.>), we introduce the following modulus: for all t ≥ 0, letρ_X(t)= sup_x ∈ S_Xinf_Ysup_y ∈ S_Y (x+ty-1)where Y runs through all closed linear subspaces of X of finite codimension. We say that · is asymptotically uniformly smooth (in short AUS) if lim_t → 0ρ_X(t)/t=0. If p ∈ (1, ∞ ), · is said to be p-AUS if there is a constant C > 0 such that, for all t ∈ [0, ∞ ), ρ_X(t) ≤ C t^p. If X has an equivalent norm for which X is AUS (resp. p-AUS), X is said to be AUSable (resp. p-AUSable).This notion is related to the Szlenk index which definition we will recall now. It is based on the following Szlenk derivation. For a Banach space X, K⊂ X^* weak^*-compact, and >0, we let s_(K) denote the set of x^*∈ K such that for each weak^*-neighborhood V of x^*, diam(V∩ K)≥. Then wedefine the transfinite derivations as follows s_^0(K)=K, s^ξ+1_(K)=s_(s^ξ_(K)),for every ordinal ξ and if ξ is a limit ordinal,s^ξ_(K)=⋂_ζ<ξs_^ζ(K). For convenience, we let s_0(K)=K. If there exists an ordinal ξ such that s^ξ_(K)=∅, we let Sz(K,) denote the minimum such ordinal, and otherwise we write Sz(K,)=∞. We let Sz(K)=sup_>0 Sz(K,), where Sz(K)=∞ if Sz(K,)=∞ for some >0. We let Sz(X,)=Sz(B_X^*,) and Sz(X)=Sz(B_X^*). If X is separable, Sz(X) is equal to the index associated with the following derivation:l_ε(K)={ x^* ∈ K; ∃ (x_n^*) ⊂ K,∀ n ∈,x_n^*-x^*≥ε,x_n^* ω^*⟶ x^* }if K⊂ X^* weak^*-compact, and >0.Let us remark that Sz(X)< ∞ if and only if X is Asplund, that is to say every separable subspace of X has separable dual. For more information on the Szlenk index, we refer the reader to the survey <cit.>. To conclude this subsection, let us remind the following fondamental renorming result, due to Knaust, Odell and Schlumprecht <cit.> in the separable case and to Raja <cit.> in the non separable one (see <cit.> for the precise quantitative version): Let X be an infinite dimensional Banach space. Then Sz(X)=ω if and only if X is AUSable if and only if X is p-AUSable for some p ∈ (1, ∞).§.§ Lipschitz and coarse Lipschitz embeddings Let us recall some definitions on metric embeddings. Let (X,d_X) and (Y, d_Y) two metric spaces, f a map from X to Y. We define the compression modulus of f by ∀ t ≥ 0, ρ_f(t)= inf{ d_Y(f(x),f(y)) ; d_X(x,y) ≥ t } ;and the expansion modulus of f by ∀ t ≥ 0, ω_f(t)= sup{ d_Y(f(x),f(y)) ; d_X(x,y) ≤ t } .We adopt the convention inf(∅)= + ∞. Note that, for every x,y ∈ X, we have ρ_f(d_X(x,y)) ≤ d_Y(f(x),f(y)) ≤ω_f(d_X(x,y)) .If one is given a family of metric spaces (X_i)_i ∈ I, one says that (X_i)_i ∈ I equi-Lipschitz embeds into Y if there exist A, B in (0, ∞ ) and, for all i ∈ I, maps f_i : X_i → Y such that ρ_f_i(t) ≥ At and ω_f_i(t) ≤ Bt for all t ≥ 0. And we say that f is a coarse Lipschitz embedding if there exist A, B, C, D in (0, ∞) such that ρ_f(t) ≥ At-C and ω_f(t) ≤ Bt+D for all t ≥ 0. If X and Y are Banach spaces, this is equivalent to the existence of numbers θ≥ 0 and 0 < c_1 < c_2 so that :c_1 x-y_X ≤f(x)-f(y)_Y ≤ c_2 x-y_Xfor all x,y ∈ X satisfying x-y_X ≥θ. §.§ Definitions of concentration properties Let us start this subsection by recalling a version of Ramsey's theorem we will use.Let k ∈ and ⊂ []^k. There exists ∈ []^ω such that either []^k ⊂ or []^k ∩ = ∅. We now introduce the concentration properties we will study here, defined in <cit.>, where sort of directional Lipschitz constants take part, hence the “d" in subscript in the acronyms below. Let (X,d) be a metric space, λ > 0, p ∈ (1, ∞). ∙ We say that X has property λ-HFC_p,d (resp. λ-HIC_p,d) if, for every k ∈ and every bounded function f : []^k → X, there exists ∈ []^ω such thatd(f(n),f(m) ) ≤λ( ∑_j=1^k _j(f)^p )^1/pfor all n, m∈ []^k (resp. (n, m) ∈ I_k()), where, for each j ∈{1, ⋯, k}_j(f) = sup_(n, m) ∈ H_j() d(f(n),f(m)) .We say that X has property HFC_p,d (resp. HIC_p,d) if X has property λ-HFC_p,d (resp. λ-HIC_p,d), for some λ > 0.It is important to note that Theorem 6.1 <cit.> and Theorem 2.4 <cit.> can be rephrased as follows: for p ∈ (1, ∞), a reflexive (resp. quasi-reflexive) p-AUSable Banach space has property HFC_p,d (resp. HIC_p,d). As mentionned by Causey <cit.>, these results still apply when X is only assumed to have upper ℓ_p tree estimates (more precisely when X has property A_p, defined for example in <cit.>). In order to get non-quasi-reflexive spaces with some concentration, the author proved that these properties are stable under ℓ_p sums. It is worth mentioning that these properties are stable under coarse Lipschitz embeddings when the embedded space is a Banach space and they clearly prevent the equi-Lipschitz embeddings of the Hamming graphs. As mentioned in <cit.>, they also prevent the equi-Lipschitz embeddings of the symmetric graphs.§ STABILITY UNDER SUMSIn order to prove the stability of properties HFC_p,d and HIC_p,d, p ∈ (1, ∞), under ℓ_p sums, with the same constant, we will improve the idea from <cit.>. To do so, we need the following proposition, whose proof can be deduced from <cit.> but that we include for sake of completeness. Let p ∈ (1, + ∞), X_1 and X_2 two Banach spaces, X=X_1 ⊕_p X_2, k ∈. For every ε>0 and every Lipschitz map h=(f,g) : []^k → X, there exists ∈ []^ω such that_j(f_|[]^k)^p+_j(g_|[]^k)^p ≤_j(h)^p + εfor every 1 ≤ j ≤ k.Let ε > 0 and h=(f,g) : ([]^k,) → X a Lipschitz map. Let α_1=inf__1 ∈ []^ωsup_(n,m) ∈ H_1(_1)f(n)-f(m). There exists _1 ∈ []^ω so that f(n)-f(m)^p ≤α_1^p+ ε/2 for every (n,m) ∈ H_1(_1). Let β_1=inf__1' ∈ [_1]^ωsup_(n,m) ∈ H_1(_1')g(n)-g(m). There exists _1' ∈ [_1]^ω so that g(n)-g(m)^p ≤β_1^p+ ε/2 for every (n,m) ∈ H_1(_1'). ⋮Let α_k=inf__k ∈ [_k-1']^ωsup_(n,m) ∈ H_k(_k)f(n)-f(m). There exists _k ∈ [_k-1']^ω so that f(n)-f(m)^p ≤α_k^p+ ε/2 for every (n,m) ∈ H_k(_k). Let β_k=inf__k' ∈ [_k]^ωsup_(n,m) ∈ H_k(_k')g(n)-g(m). There exists ∈ [_k]^ω so that g(n)-g(m)^p ≤β_k^p+ ε/2 for every (n,m) ∈ H_k().Let us show that _j(f_|[]^k)^p+_j(g_|[]^k)^p ≤_j(h)^p + ε for every 1 ≤ j ≤ k.Let 1 ≤ j ≤ k. It is enough to show that α_j^p+β_j^p ≤_j(h)^p. For that, assume that it is not the case. Then, there exists η > 0 so that (α_j-η)^p + (β_j-η)^p > _j(h)^p. ∗ If there exists (n, m) ∈ H_j() such that f(n)-f(m)≥α_j- η and g(n)-g(m)≥β_j- η, then h(n)-h(m) > _j(h), which is impossible. ∗ So f(n)-f(m)≤α_j- η or g(n)-g(m)≤β_j- η for all (n, m) ∈ H_j(). Now we note that H_j() can be identified with []^k+1 so, by Theorem <ref>, we get ' ∈ []^ω such that f(n)-f(m)≤α_j- η for all (n, m) ∈ H_j(') or g(n)-g(m)≤β_j- η for all (n, m) ∈ H_j('). This contradicts the definition of α_j or β_j and finishes the proof. We can now prove our main result. Before doing so, we want to recall some facts about Kalton and Randrianarivony's result from <cit.>. If f : []^k →ℓ_p is a bounded map, they proved the existence of u ∈ℓ_p so that, for every >0, there exists ∈ []^ω so that ∀∈ []^k,f()-u≤( ∑_j=1^k _j(f)^p )^1/p+ .By looking carefully at their proof, one can note more precisely that the u is given by u=ω-lim_n_1 ∈→∞…lim_n_k ∈→∞ f() where ∈ []^ω is such that the limits exist. Let ε > 0, k ∈, f : ([]^k,) → X a Lipschitz function. Let ε”>0 small enough so that (λ+ε”)^p ∑_j=1^k _j(f)^p + k(λ+ε”)^p ε” +ε”≤ (λ+ε)^p ∑_j=1^k _j(f)^pand ε'>0 small enough so that[ 2(∑_j=1^k _j(f)^p )^1/p+ 4ε' ]^p ≤ 2^p ∑_j=1^k _j(f)^p +ε” The well-defined mapϕ : {[ X → ℓ_p; (x_n)_n ∈ ↦ (x_n)_n ∈ ]. satisfies (ϕ) ≤ 1 and ϕ(x)=x for all x ∈ X, thussup_(n,m) ∈ H_j()ϕ∘ f(n)-ϕ∘ f (m) ≤_j(f)for every j ∈{1, ⋯, k}. From Kalton-Randrianarivony's Theorem <cit.>, we get u ∈ℓ_p and ”∈ []^ω such that ϕ∘ f(n)-u ≤( ∑_j=1^k _j(f)^p )^1/p+ ε'for all n∈ [”]^k. Let N ∈ be such that ∑_k=N+1^∞ |u_k|^p ≤ε'^p. Let us denote by P_N the projection from ℓ_p onto {e_i,1 ≤ i ≤ N} and by Π_N the projection from X onto ( ∑_i=1^N X_i )_ℓ_p. According to the lemma, there exists ∈ [”]^ω such that ∀ 1 ≤ j ≤ k,_j(Π_N ∘ f_|[]^k)^p+_j((I-Π_N) ∘ f_|[]^k)^p ≤_j(f)^p + ε” If we denote by v=(I-P_N)(u), by the remark preceding this proof, there exists ' ∈ []^ω such thatϕ∘ (I-Π_N) ∘ f()- v=(I-P_N) ∘ϕ∘ f()-v≤( ∑_j=1^k _j((I-Π_N)∘ f_|[]^k)^p )^1/p+ ε'for all n∈ [']^k. According to Proposition 3.1 from <cit.>, there exists ∈ [']^ω such that Π_N ∘ f(n)-Π_N ∘ f(m) ≤ (λ+ε”) ( ∑_j=1^k _j(Π_N ∘ f_|[]^k)^p )^1/pfor all , ∈ []^k (resp. (n, m) ∈ I_k()).Then, for all , ∈ []^k (resp. (n, m) ∈ I_k()), we havef(n)-f(m)^p =Π_N(f(n)-f(m)) ^p +(I-Π_N) ∘ f(n)- (I-Π_N) ∘ f(m) ^p ≤Π_N(f(n)-f(m)) ^p + [ (I-Π_N) ∘ f(n)+(I-Π_N) ∘ f(m) ]^p = Π_N(f(n)-f(m)) ^p + [ϕ∘ (I-Π_N) ∘ f(n)+ ϕ∘ (I-Π_N) ∘ f(m) ]^p ≤Π_N(f(n)-f(m)) ^p + [ϕ∘ (I-Π_N) ∘ f(n)-v+ ϕ∘ (I-Π_N) ∘ f(m)-v +2v]^p ≤ (λ+ε”)^p ∑_j=1^k _j(Π_N ∘ f_|[]^k)^p + [ 2(∑_j=1^k _j((I-Π_N)∘ f_|[]^k)^p )^1/p+ 4ε' ]^p ≤ (λ+ε”)^p ∑_j=1^k _j(Π_N ∘ f_|[]^k)^p + 2^p ∑_j=1^k _j((I-Π_N)∘ f_|[]^k)^p +ε”(since x ∈^+ ↦ (x+4ε')^p-x^p is non-decreasing)≤ (λ+ε”)^p ∑_j=1^k [_j(Π_N ∘ f_|[]^k)^p +_j((I-Π_N)∘ f_|[]^k)^p] + ε”since λ≥ 2. Thus, for all (n, m) ∈ I_k(), we getf()-f()^p ≤ (λ+ε”)^p ∑_j=1^k _j(f)^p + k(λ+ε”)^p ε” +ε”≤ (λ+ε)^p ∑_j=1^k _j(f)^pby choice of ε”. § APPLICATIONS§.§ A counterexample In this subsection, we answer by the negative the following question: is a Banach space with some property HIC_p,d, p ∈ (1, ∞), necessarily AUSable?We first start with an elementary lemma, which strengthens the intuition that the concentration properties are asymptotic ones.Let λ>0, p ∈ (1, ∞) and X be a Banach space with a finite codimensional subspace Y that has property λ-HFC_p,d (resp. λ-HIC_p,d). Then X has (λ+ε)-HFC_p,d (resp. (λ+ε)-HIC_p,d) for every ε>0.Let ε>0, f : ([]^k,) → X be a non-constant Lipschitz map and P : X → Y be a bounded linear projection. Let ε'>0 so that λ( ∑_j=1^k (_j(f)+ε')^p )^1/p+' ≤ (λ+) ( ∑_j=1^k _j(f)^p )^1/p .It follows from Ramsey’s theorem <ref> and the norm compactness of bounded sets in finite dimensional spaces that there exists ∈ []^ω such that∀, ∈ []^k,(I-P)(f()-f())≤ε' .Therefore P ∘ f : []^k → Y satisfies _j(P ∘ f) ≤_j(f)+ε' for every j ∈{1, …, k }. Applying our hypothesis on Y, we get ' ∈ []^ω so thatP ∘ f()-P ∘ f() ≤λ( ∑_j=1^k (_j(f)+ε')^p )^1/pfor all , ∈ [']^k (resp. (,) ∈ I_k(')). Then, by choice of ε',f()-f()≤P ∘ f()-P ∘ f()+ (I-P)(f()-f())≤ (λ+) ( ∑_j=1^k _j(f)^p )^1/pfor all , ∈ [']^k (resp. (,) ∈ I_k(')). Let p ∈ (1, ∞). We will define spaces by transfinite induction. Set X_p^0=⊕_1 ℓ_p, X_p^α+1=⊕_1 ℓ_p(X_p^α) for every ordinal α and X_p^α= ⊕_1 (∑_β<α X_p^β)_ℓ_p if α is a limit ordinal. They satisfy Sz(X_p^α) > α for every α<ω_1. Indeed, one can show with a transfinite induction that (1,0) ∈ l_1^α(B_(X^α_p)^*) for every ordinal α. Therefore, we can deduce from Theorem <ref> the following corollary, that answers a question from <cit.>.For every ordinal α<ω_1 and p ∈ (1, ∞), there exists a Banach space with property HFC_p,d and Szlenk index bigger than α. By Theorem <ref>, property HFC_p,d for some p ∈ (1, ∞) does not imply being AUSable. In particular, for every ordinal α<ω_1, one gets a first example of a Banach space with Szlenk index bigger than α that does not equi-Lipschitz contain the Hamming graphs (nor the symmetric graphs). Let us finish with subsection with a last remark. In the local setting, a Banach space X has non-trivial Rademacher type if and only if ℓ_1 is not finitely representable in X if and only if it does not equi-Lipschitz contain the Hamming cubes <cit.> <cit.>. In <cit.>, an asymptotic version of Rademacher type p, p ∈ (1, ∞) is introduced and the following is proved: a banach space has non-trivial asymptotic Rademacher type if and only if ℓ_1 is not asymptotically finitely representable in X, if ℓ_1 is not in its asymptotic structure. To ease the reading, we will not recall the definition of asymptotic structure but we will mention that ℓ_1 happens to be in the asymptotic structure of X_p^ω (see <cit.>). Therefore, with the definition of asymptotic Rademacher type from <cit.>, contrary to what happens in the local setting, there exist Banach spaces with trivial asymptotic Rademacher type that do not equi-Lipschitz contain the Hamming graphs. §.§ A complexity result Let us denote by C(Δ) the space of continuous real-valued functions on the Cantor set Δ. Since this space is universal for the class of separable Banach spaces, we can code that class by ={ X ⊂ C(Δ), Xis a closed linear subspace ofC(Δ)}. As C(Δ) is a Polish space, we can endow the set ℱ (C(Δ)) of its closed non-empty subsets with the Effros-Borel structure (see Chapter 2 from <cit.> or <cit.>). Hence we can talk about Borel and (completely) co-analytic classes of separable Banach spaces. We refer to the textbook <cit.> for the definitions and a complete exposition of descriptive set theory. We will just recall that an analytic set is a continuous image of a Polish space into another Polish space, a co-analytic set is a set whose complement is analytic and an analytic (resp. co-analytic) subset A of a standard Borel space X is said to be completely analytic (resp. completely co-analytic) if for each standard Borel space Y and each B ⊂ Y analytic (resp. co-analytic), one can find a Borel function f : X → Y so that f^-1(B)=A. The following well-known lemma will be used in the proof of Proposition <ref>.Let X be a Polish space, ℱ(X) the set of its non-empty closed sets.There exists a sequence of Borel functions (d_n)_n ∈ : ℱ(X) → X such that (d_n(F))_n ∈ is dense in F, for every non-empty closed subset F ⊂ X.As a consequence of Corollary <ref>, since Bossard <cit.> proved that the map X ↦ Sz(X) is a co-analytic rank on the set of spaces with separable dual, the set of spaces with separable dual satisfying HFC_p,d, p ∈ (1, ∞) is not Borel (see for example Theorem A.2 <cit.>). In fact, we even have:Let p ∈ (1, ∞). The set of separable Banach spaces satisfying HFC_p,d is completely co-analytic.Before proving this proposition, let us recall some terminology. We denote bythe set of all trees over , where a tree overis a set T such that ∅∈ T, T ⊂⋃_k=1^∞ []^k and (n_1, ⋯, n_k) ∈ T as soon as (n_1, ⋯, n_k+1) ∈ T for some n_k+1>n_k. A natural partial order ≼ can be defined on a tree T ∈ by setting ∅≼ for every ∈ T and (n_1, ⋯, n_j) ≼ (m_1, ⋯, m_k) if k ≥ j and m_i=n_i for every 1 ≤ i ≤ j. A tree is said well-founded if it does not contain an infinite increasing sequence, ill-founded if it does. Finally, a linearly ordered subset I ⊂ T with respect to ≼ is called a segment if T and two segments I,J of T are said to be incomparable if we don't have ≼ or ≼ for any ∈ I_1, ∈ I_2. First, let us prove that having HFC_p,d is a co-analytic condition. We will use Lemma <ref>. By definition, a Banach space X does not have property HFC_p,d if and only if for every λ∈, there exists C ∈, k ∈ and f : []^k → so thatd_f()(CB_X)-d_f()(CB_X)≥λ∑_j=1^k _j(d_f(·)(CB_X))for all < ∈ []^k. Since the map {[ SB →ℱ(C(Δ));X ↦ CB_X ]. is Borel, not having HFC_p,d is an analytic condition and therefore, having HFC_p,d is a co-analytic condition. Now, to prove the proposition, it is enough to find a Borel φ from the set of treesinto the set of separable Banach spaces so that, for every tree T ∈, T is well-founded if and only if φ(T) has HFC_p,d. To do so, let q ∈ (1, p) and let us denote by (e_n)_n ∈ the canonical basis of ℓ_q. If T ∈, x=(x_)_nb ∈ T∈ c_00(T) and I is a segment of T, we let x_|I= ∑_∈ I x() e_max()∈ℓ_q. Now, for each T ∈ and x ∈ c_00(T), letx_T=sup{( ∑_i=1^n x_|I_i_ℓ_q^p )^1/p,I_1, ⋯, I_nincomparable segments ofT }and denote by X_T the completion of c_00(T) under the norm ._T. By a transfinite induction on the order of T and Theorem <ref>, we get that X_T has property HFC_p,d for every well-founded tree T ∈ (see <cit.> for a similar transfinite induction). Since X_T contains an isometric copy of ℓ_q for every ill-founded tree T and ℓ_q does not have property HFC_p,d, the Borel function φ : T ↦ X_T is so that T is well-founded if and only if φ(T) has HFC_p,d.Likewise, the set of separable Banach spaces satisfying HIC_p,d is completely co-analytic. § FINAL REMARKS AND OPEN PROBLEMS In view of Corollary <ref>, the following question seems natural. If a Banach space X has property HIC_p,d for some p ∈ (1, ∞), is it Asplund? We will finish this paper by recalling the important Problem 2 from <cit.>. If a Banach space X coarse Lipschitz embeds into a Banach space Y that is reflexive and AUS, does it follow that X is AUSable? It follows from Corollary <ref> that concentration properties are not the right tool to answer this question. If the answer were to be negative, a counterexample could come from the space X_2^ω. The author would like to thank Gilles Lancien for very useful conversations and valuable comments. plain | http://arxiv.org/abs/2311.15900v1 | {
"authors": [
"Audrey Fovelle"
],
"categories": [
"math.FA",
"math.MG"
],
"primary_category": "math.FA",
"published": "20231127150308",
"title": "Asymptotic smoothness, concentration properties in Banach spaces and applications"
} |
http://arxiv.org/abs/2311.16354v1 | {
"authors": [
"Zhijie Zhang",
"Xiaoxia Zhang",
"Hui Li",
"Taotao Fang",
"Qingzheng Yu",
"Yang Luo",
"Federico Marinacci",
"Laura V. Sales",
"Paul Torrey",
"Mark Vogelsberger"
],
"categories": [
"astro-ph.GA"
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"primary_category": "astro-ph.GA",
"published": "20231127223724",
"title": "Low- and High-velocity \\ion{O}{6} in Milky Way-like Galaxies: the Role of Stellar Feedback"
} |
|
APS/[email protected] Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USAElectrical and Computer Engineering Department, University of California Santa Barbara, Santa Barbara, California 93106, USA Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA []Present Address: Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan Electrical and Computer Engineering Department, University of California Santa Barbara, Santa Barbara, California 93106, USA Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, [email protected] Materials Department, University of California Santa Barbara, Santa Barbara, California 93106, USA Electrical and Computer Engineering Department, University of California Santa Barbara, Santa Barbara, California 93106, USAα-Sn thin films can exhibit a variety of topologically non-trivial phases. Both studying the transitions between these phases and making use of these phases in eventual applications requires good control over the electronic and structural quality of α-Sn thin films. α-Sn growth on InSb often results in out-diffusion of indium, a p-type dopant. By growing α-Sn via molecular beam epitaxy on the Sb-rich c(4×4) surface reconstruction of InSb(001) rather than the In-rich c(8×2), we demonstrate a route to substantially decrease and minimize this indium incorporation. The reduction in indium concentration allows for the study of the surface and bulk Dirac nodes in α-Sn via angle-resolved photoelectron spectroscopy without the common approaches of bulk doping or surface dosing, simplifying topological phase identification. The lack of indium incorporation is verified in angle-resolved and -integrated ultraviolet photoelectron spectroscopy as well as in clear changes in the Hall response.Growth and characterization of α-Sn thin films on In- and Sb-rich reconstructions of InSb(001) Christopher J. Palmstrøm January 14, 2024 ==============================================================================================§ INTRODUCTIONα-Sn, the diamond structure allotrope of Sn, is a zero-gap semiconductor with an inverted electronic band structure at the Γ point<cit.>. However, the diamond structure is not stable above 286 K in the bulk, transitioning to a trivial metallic tetragonal phase (β-Sn)<cit.>. Fortunately, this phase transition temperature can be raised by epitaxial stabilization of α-Sn (a=6.489 Å) thin films on closely lattice matched substrates like InSb (a=6.479 Å) and CdTe (a=6.482 Å)<cit.>. Theoretical predictions show that the zero-gap semiconductor phase transitions to a Dirac semimetal (DSM) under epitaxial compressive strain and a topological insulator (TI) under epitaxial tensile strain<cit.>. Experiments have suggested that on InSb, α-Sn thin films go through multiple topological phase transitions between 2D TI, 3D TI, 2D DSM, 3D DSM, and normal insulator as a function of surface orientation, strain, and film thickness, generating much interest as a testbed for topological phase transitions<cit.>. Exact results vary by research group and measurement technique. α-Sn is–to our knowledge–the only elemental material showing many of these phases. Due to its elemental nature, α-Sn should not suffer from the well-known point defect issues seen in many other multi-component topological materials. This makes the material very attractive for high-mobility Dirac fermion transport studies. Bulk single crystals of α-Sn have been well studied in the past, where dopant concentration could be better controlled and the inverted band structure was confirmed<cit.>. In thin film growth of α-Sn, InSb has been the primary substrate of choice for surface science studies. These growths of α-Sn are usually performed via molecular beam epitaxy (MBE) on sputter-anneal cleaned InSb leading to an In-rich surface. Indium is known to be a p-type dopant in α-Sn in the bulk<cit.> and has been seen to readily incorporate into α-Sn thin films from the InSb substrate<cit.>. This effect even occurs when α-Sn is grown on a fresh in-situ MBE-grown InSb buffer layer with an In-rich surface reconstruction<cit.>. The heavy p-type doping results in a Fermi level below the valence band maximum (VBM) and the surface Dirac point (near the VBM), making identification of the topological phase difficult via techniques such as angle-resolved photoemission spectroscopy (ARPES).In addition, the bulk mobility <cit.> and surface state mobility are both inversely dependent on carrier density<cit.>, furthering the need for reduced unintentional p-type doping.ARPES has been a key technique for fingerprinting topological materials due to the direct measurement of the filled-state band structure<cit.>. Fingerprinting via another common technique, magnetotransport, either requires extensive fitting of quantum oscillations via Lifshitz-Kosevich analysis to give indirect evidence of the topological phase or a gated device to map out a relevant portion of the Landau fan diagram for a more direct measurement of the topological phase<cit.>. In both cases, the filled bands must have high enough mobilities such that the cyclotron orbit time is shorter than the scattering time in the material<cit.>. For α-Sn, this condition is frequently not met: only the surface state oscillation is observed in low mobility films<cit.>, while high mobility films show two oscillation frequencies<cit.>. It is difficult to correctly identify the topological phase without measuring both the bulk and surface bands. The two key parameters to differentiate the possible topological phases are the presence (or absence) of a gap in the topological surface states and the presence (or absence) of a gap between the bulk conduction and valence bands. Finally, the presence of parallel bulk transport in indium doped α-Sn (under conditions where the topological phase includes a bulk bandgap) greatly reduces the applicability of the material both in fundamental physics studies and devices making use of the topological surface states.Besides doping, incorporation of indium has been proposed to reduce the quality of α-Sn growth<cit.>, similar to what has been suggested in α-Ge_1-xSn_x growth<cit.>. To reduce p-type doping and produce higher quality films, bismuth and tellurium are frequently used as compensatory dopants and/or surfactants<cit.>. Bismuth and tellurium also possibly change topological signatures from α-Sn, modifying at least the band velocity of α-Sn’s linear surface states<cit.>. The structural and electronic effects of incorporating Bi/Te in the α-Sn has not been investigated in detail. Another standard solution to limit indium incorporation is to use a low substrate temperature and high Sn growth rate <cit.>. However, a lower substrate temperature during growth was seen to be associated with lower structural quality of the epitaxial α-Sn<cit.>.Most reported growths studying the topologically non-trivial nature of α-Sn have been initiated on the In-rich InSb(001)-c(8×2) reconstruction of InSb(001)<cit.>, depicted in Fig. 1(a). This reconstruction consists of a 0.5 monolayer (ML) of In and a 0.5 ML of Sb, with another 0.25 ML of In on top<cit.>. It is possible that the indium incorporating into the α-Sn films is sourced from this surface. On the other hand, the c(4×4) reconstruction, depicted in Fig. 1(b), has 1 ML of In on the surface, followed by 1 ML of Sb, and then an additional 0.75 ML of Sb<cit.>. The almost-double layer of Sb on the surface likely decreases the amount of indium that is available on the InSb surface for incorporation into the growing α-Sn film. Here we find that by growing α-Sn on the Sb-rich InSb(001)-c(4×4), we drastically decrease indium incorporation without distorting the band structure of the epitaxial α-Sn films. We show that this reduction in In segregation persists through active heating of the substrate during growth. This method then allows an increase in total heat load that can be applied during α-Sn growth, potentially demonstrating a route toward higher mobility thin films. The decrease in indium incorporation is found using in-situ angle-resolved photoelectron spectroscopy (ARPES), in-situ ultraviolet photoelectron spectroscopy (UPS) and ex-situ low temperature magnetotransport. This work paves the way for pure α-Sn growth where the features of interest are below the Fermi level, such that topological phase identification through techniques such as ARPES can proceed more readily.§ METHODS §.§ Sample GrowthThe α-Sn films studied here were grown using a modified VG V80 MBE growth system. All samples were grown on undoped (001)-oriented InSb substrates (WaferTech Ltd.). The native oxide was removed via atomic hydrogen cleaning using a thermal cracker (MBE Komponenten) resulting in the In-rich c(8×2)/4×2 surface reconstruction, as determined by reflection high energy electron diffraction (RHEED)<cit.>. The Sb-rich samples were prepared with constant monitoring of the reconstruction via RHEED along the [110] while referencing the InSb(001) surface reconstruction phase diagram<cit.>. The substrate temperature was ramped continuously from 373 K through the following transitions to the final annealing temperature. Near 550 K, the c(8×2) surface was dosed with approximately 0.75 ML Sb until the c(8×2) just fades to the p(1×1) reconstruction. At higher temperatures the reconstruction transitions to the c(4×4) and, starting a few degrees below the c(4×4)/a(1×3) transition (approximately 643 K), was exposed to an Sb_4 flux continuously. The sample was then annealed under an Sb_4 overpressure 40-60 K above this transition point for at least 30 minutes. The sample was then cooled quickly down back through the a(1×3)/c(4×4) transition to achieve the final c(4×4) reconstruction. Both prepared surfaces show streaky RHEED, indicating relatively smooth surfaces (Fig. 1(c), (d)). The morphology of the substrates was confirmed usingin-situ scanning tunneling microscopy (STM) with the freshly hydrogen cleaned surface (Fig. 1(e)) containing a higher terrace density than after the Sb-termination/anneal procedure (Fig. 1(f)). Some samples with In-rich reconstructions were annealed at 623 K for 30 minutes to smooth the surface further. A bilayer (BL) of α-Sn is defined here as 9.5× 10^14 at/cm^2, or 1/2 of the diamond unit cell. Three different film thicknesses were studied: 13 BL, 50 BL, and 400 BL.The 13 BL films were grown at a rate of 0.25 BL/min at a substrate thermocouple temperature of 296 K under radiative backside heating. The real temperature is above the melting point of Ga (302.9 K). The 50 BL films were grown at a rate of 0.5 BL/min at a substrate thermocouple temperature of 253 K under passive radiative cooling from the liquid nitrogen cryopanel. The real temperature is significantly below the melting point of Ga. The 400 BL film was grown at a rate of 1.25 BL/min in a separate chamber with active liquid nitrogen cooling on the sample. The sample is indium bonded to a tungsten block which is then in direct thermal contact with an in-vacuo liquid nitrogen filled stainless steel vessel. The substrate temperature remains around 80 K during growth. Different growth temperatures and growth rates were used to investigate the effect of heat load on the effectiveness of the termination procedure. The growth rates are calibrated via bilayer oscillations in the RHEED intensity during growth and Rutherford backscattering spectrometry on Si reference samples. §.§ CharacterizationAngle-resolved and -integrated photoelectron spectroscopy measurements of the 13 BL films were performed at beamline 5-2 of the Stanford Synchrotron Radiation Lightsource (SSRL) using linear polarized light. Measurements of the 50 BL and 400 BL film were performed at beamline 10.0.1.2 of the Advanced Light Source (ALS) using p-polarized light. Spectra were collected with a Scienta-Omicron DA30L electron analyzer with variable energy resolution and angular resolution better than 0.1^∘. Samples were transferred in-vacuo from the MBE system at UCSB to SSRL and ALS via a vacuum suitcase with base pressure better than 4× 10^-11 Torr. During the photoemission measurements the sample temperature was kept under 20 K and the pressure was better than 3×10^-11 Torr. In-situ STM was performed in an Omicron VT-STM at room temperature with a bias of 3 V and a tunneling current of 100 pA. Ex-situ magnetotransport measurements on 56 BL films with magnetic field up to 14 T were performed at 2 K in a ^4He cryostat (Quantum Design Physical Property Measurement System). The 56 BL films were grown under the same conditions as the 50 BL films and were not capped; the native oxide is expected to consume 3-6 BL<cit.>. Ohmic contacts were made with silver paint following the van der Pauw geometry. A standard four terminal, low frequency AC lock-in technique was used with a constant excitation current of 100 μA. § RESULTS AND DISCUSSION §.§ Film morphologyWe observe the typical mixed (2×1)/(1×2) reconstruction in α-Sn <cit.> characterized by a 2× RHEED pattern in the <110> directions and a 1× pattern in the <100> directions (Fig. 2(a)) for all growths less than 100 BL, irrespective of substrate surface termination. During growth all samples show oscillations in the RHEED intensity indicating a bilayer-by-bilayer growth mode for α-Sn (see the supplementary material<cit.>). The overall surface morphology is roughly equivalent for both films (Fig. 2(b,c)). The structure consists of bilayer terraces which are then decorated with a grain structure as has been seen previously in α-Sn/InSb(001)<cit.>. The change in bilayer terrace structure between the two cases is likely due to the differences in the starting substrate morphology. The grains in α-Sn grown on the Sb-rich surface reconstruction were slightly (∼20%) smaller than the grains grown on the In-rich surface reconstruction (Fig. 2(b),(c) insets). The RMS roughness on a terrace also decreases from 2.04 Å to 1.75 Å. The step heights within the grains generally correspond to one atomic layer. The grains are then likely stacks of atomic layers with an alternating (2×1) vs. (1×2) reconstruction, where each layer is partially visible. This then leads to the observed mixed (2×1)/(1×2) reconstruction in RHEED. The alternate stacking structure has been directly observed before in α-Sn via STM <cit.>, and is well-known for Si(001) surfaces<cit.>.§.§ Ultraviolet Photoelectron SpectroscopyThe effect of substrate termination was first studied on the 13 BL films grown with active substrate heating at 296 K. Measurements were performed at photon energies of 50 eV and 130 eV to deconvolute inelastic mean free path effects, discussed in the supplementary material<cit.>. Growth on the Sb-rich surface termination of InSb(001) resulted in no clear peaks corresponding to indium in a full-range UPS scan (Fig. 3(a)), indicating a large decrease in indium composition. We extracted the very weak intensity of the In 4d states and found a reduction in the indium concentration by over an order of magnitude. Extraction of exact concentration of indium and antimony in the films is problematic as the distribution of dopants is unknown and computed cross-sections and mean free paths under these experimental conditions are inaccurate. Assuming homogeneous composition, using the spectra taken at a photon energy of 130 eV, and using parameters computed by SESSA<cit.>, we find the total concentration of indium in the film decreases from approximately 2% to approximately 0.18% by changing the surface termination that α-Sn growth is initiated on. By increasing the signal amplification in the electron analyzer, the indium core levels could be resolved more clearly. With reference to the valence band intensity in Fig. 3(b), a similar sized reduction in the concentration of indium in the films is found. However, there is a concomitant increase in Sb concentration in the α-Sn films from growth on the c(4×4) reconstruction (Fig. 3(c)). This is not necessarily a detraction; Sb is an established n-type dopant in α-Sn<cit.> which should further shift the Fermi level in the desired direction. Next we investigated behavior of In and Sb incorporation when α-Sn films are grown thicker and with reduced heating during growth. Neither In nor Sb 4d peaks are visible in the survey UPS measurements in Fig. 4(a) for a 50 BL (0.5 BL/min) sample grown at 253 K on an Sb-rich surface nor for a 400 BL (1.25 BL/min) sample grown at 80 K on an In-rich surface. By the same method as earlier we enhanced the signal from the indium, as seen in Fig. 4(b). It is clear that the concentration of indium is reduced further by growing on an Sb-rich surface than by growing at ultracold temperature on an In-rich surface, resulting in a more than 50% decrease in indium incorporation. There is again an increase in the amount of Sb incorporation (Fig. 4(c)). Thus the concentration of indium in α-Sn can be reduced significantly at a range of growth temperatures and growth rates by growing on the Sb-rich c(4×4) reconstruction of InSb(001). This surface termination procedure is more effective than the conventional method of minimizing sample temperature and maximizing Sn growth rate. §.§ Electronic StructureThe persistent p-type indium doping in the α-Sn films results in the node of the surface states (and the valence band maximum) being above E_F (Fig. 5(a)), and thus not accessible by a filled state measurement such as conventional photoelectron spectroscopy.This is evident in growth of α-Sn on the In-rich surface reconstruction in Fig. 5(b), where the Dirac node is projected to be 32±7 meV above E_F via a linear fit to the surface states. Reducing the indium incorporation (with the concomitant increase in Sb incorporation) as observed in these same samples in Fig. 3(a), resulted in a Dirac node measured 36±10 meV below E_F (Fig. 5(c)). The group velocity of the surface states (1/ħdE/dk using the slope of the linear fit) in Fig. 5(b) is 4.1±0.1× 10^5 m/s and in Fig. 5(c) is 4.0±0.2× 10^5 m/s. The dopants/surfactants of Bi and Te commonly used in α-Sn growth result in a high group velocity near 7× 10^5 m/s, while a lower group velocity is found in “phase pure" α-Sn (5× 10^5 m/s)<cit.>. The low and unchanging band velocity we observe is thus consistent with the preservation of “phase pure” α-Sn even with the higher concentrations of Sb in the α-Sn film. We next turned to the bulk band structure to validate the absence of any band distortion. The band structure in the vicinity of the Γ_003 high symmetry point was investigated using a photon energy of 127 eV, assuming an inner potential of 5.8 eV<cit.>. A schematic of the expected band structure in the case of a 3D topological insulator-like phase is shown in Fig. 5(a)<cit.>. In growth on both the In-rich (Fig. 5(e)) and Sb-rich (Fig. 5(f)) reconstruction, the experimental data is consistent with the schematic. There is no evidence of the inverted p-like “light hole"-character conduction band (Γ_8^+) coming down to touch the VBM, therefore a bulk band gap exists. The topological surface state studied in prior works<cit.> (and which determines the topological phase) is labelled TSS1; its Dirac node is located approximately 80 meV above the valence band maximum. The valence band maximum consists of the uninverted “heavy hole" band with p-character (Γ_8^+). Below this is the inverted “conduction” band with s-like character (Γ_7^-). The lowest depicted bandis the p-like split-off band (Γ_7^+). The distinctive M shaped feature between the split-off band and the “conduction” band are associated with a second topological surface state TSS2, discussed in more detail elsewhere<cit.>. No discernible change to the bulk dispersion is observed, indicating the reduction of indium and addition of antimony can be treated as a rigid band shift from doping. While carrier density changes could be extracted from the knowledge of k_F, surface band bending is a non-negligible effect in topological insulators<cit.>.As ARPES (and UPS) are surface sensitive measurements, these observed carrier density changes do not reflect accurate changes in the carrier densities in the bulk of the crystal (i.e. an estimation of the doping). To ensure the reduction in indium incorporation is reflected in the bulk of the film, we turned to low temperature magnetotransport.§.§ MagnetotransportLongitudinal and transverse resistance measurements for 56 BL α-Sn films initiated on the two different InSb(001) reconstructions are shown in Fig. 6. Growth on InSb(001)-c(4×4) resulted in n-type behavior at high field while growth on InSb(001)-c(8×2) resulted in p-type behavior at high field in Fig. 6(a). Shubnikov-de Haas oscillations are visible in both the longitudinal and transverse geometry. Additional low field Hall measurements in Fig. 6(b) clarify the presence of three inflection points in the Hall effect in α-Sn on Sb-rich InSb and only two inflection points in α-Sn on In-rich InSb. This change is likely due to an increase in mobility of the bulk alpha-Sn carriers. The bulk mobility is sensitive to crystal quality, the carriers only being observable in quantum oscillations in high quality samples<cit.>. The onset of Shubnikov-de Haas oscillations occurs at a lower magnetic field value in films grown on Sb-rich InSb(001), indicating higher quantum mobilities<cit.>. We anticipate this initial onset to be indicative of the surface states, which have been found to have a higher mobility than the bulk heavy holes<cit.>. In the longitudinal geometry, we observe a sharp increase in the resistance at an applied field of around 4 T (Fig. 6(c)). This is a consequence of the established magnetic field induced metal-insulator transition in InSb<cit.>. The behavior of freeze-out in InSb is extremely sensitive to absolute dopant concentrations<cit.>. Carrier freeze-out complicates Hall analysis, as the response of the substrate in a transverse geometry is highly non-linear and has no simple analytical form. While quantitative analysis is difficult, we find that there is a clear, drastic change in the carrier concentrations in the α-Sn films when grown on different surface terminations of InSb(001). The magnetotransport concurs with both the UPS and ARPES measurements that growth on the Sb-rich InSb(001) surface reconstruction reduced p-type doping in α-Sn films by limiting indium incorporation.§ CONCLUSIONTopologically non-trivial α-Sn thin films have been grown on the Sb-rich InSb(001)-c(4×4) and In-rich InSb(001)-c(8×2) surface reconstructions. Despite active substrate heating and lack of intentional surfactant or dopant species, the α-Sn films grown on the Sb-rich reconstruction show minimal indium incorporation. This method results in less indium incorporation than even growing α-Sn at cryogenic temperatures while on the In-rich c(8×2) reconstruction. The reduction in p-type doping was confirmed through UPS, ARPES, and magnetotransport. Our work facilitates more robust identification of the topological phase via angle-resolved photoemission and magnetotransport. Furthermore, it allows for a wider growth window of α-Sn thin films on InSb(001) and opens up a path for a similar methodology on other crystal orientations of interest in this materials system.The growth, magnetotransport and later ARPES studies were supported by the Army Research Laboratory (W911NF-21-2-0140 and W911NF-23-2-0031). The initial vacuum suitcase construction and initial ARPES measurements were supported by the US Department of Energy (DE-SC0014388). The UC Santa Barbara NSF Quantum Foundry funded via the Q-AMASE-i program under award DMR-1906325 support was used for further development of the vacuum suitcases. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231.Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. The authors would like to further thank A. M. Kiefer, G. J. de Coster, P. J. Taylor, P. A. Folkes, and O. A. Vail for fruitful discussions. | http://arxiv.org/abs/2311.16352v2 | {
"authors": [
"Aaron N. Engel",
"Connor P. Dempsey",
"Hadass S. Inbar",
"Jason T. Dong",
"Shinichi Nishihaya",
"Yu Hao Chang",
"Alexei V. Fedorov",
"Makoto Hashimoto",
"Donghui Lu",
"Christopher J. Palmstrøm"
],
"categories": [
"cond-mat.mtrl-sci"
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"published": "20231127222954",
"title": "Growth and characterization of $α$-Sn thin films on In- and Sb-rich reconstructions of InSb(001)"
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Institute of Theoretical and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria We study the entanglement Hamiltonian for a spherical domain in the ground state of a nonrelativistic free-fermion gas in arbitrary dimensions. Decomposed into a set of radial entanglement Hamiltonians, we show that the entanglement spectrum in each sector is identical to that of a hopping chain in a linear potential, with the angular momentum playing the role of the subsystem boundary. Furthermore, the eigenfunctions follow from a commuting differential operator that has exactly the form predicted by conformal field theory. Rescaled by the radial Fermi velocity, this operator gives a perfect approximation of the entanglement Hamiltonian, except for large angular momenta that belong to the edge regime in the analogous gradient chain. One thus finds that the conformal field theory result becomes asymptotically exact only in one dimension.Entanglement Hamiltonian of a nonrelativistic Fermi gas Viktor Eisler January 14, 2024 =======================================================Entanglement plays a key role in characterizing the distinct phases of quantum matter in ground states of many-body systems <cit.>. The intricate nature of quantum correlations is encoded in the reduced density matrix of a subsystem, or equivalently, written in an exponential form, in the entanglement Hamiltonian (EH) <cit.>. One of the most remarkable property that has been uncovered in a broad range of many-body systems is the locality of the EH <cit.>. Its precise structure is, however, not only of theoretical interest, but also fundamental to novel techniques aiming at a more efficient spectroscopy and tomography of the reduced density matrix in quantum simulators <cit.>. These protocols perform a variational learning of the EH from the available measurement data, and have recently led to breakthrough results in ion-trap <cit.> and cold-atom <cit.> experiments.In the above mentioned applications, it is crucial to have an educated ansatz for the EH, which is mainly guided by the Bisognano-Wichmann theorem of relativistic quantum field theory <cit.>. This provides the EH of a half-infinite system via the physical energy density, weighted by an inverse temperature that increases linearly from the entanglement cut, and is valid in arbitrary dimensions. Generalizations to different geometries exist within conformal field theory (CFT), and yield again a local result with a modified weight function <cit.>.In practice, however, one typically faces a problem, where Lorentz invariance is explicitly broken by the presence of a lattice. Although quantum field theory may still provide an effective low-energy description, it is crucial to address the robustness of the results for the EH. In particular, the analytical solution for a free-fermion chain shows <cit.>, that the lattice EH indeed deviates from the CFT prediction, which can only be recovered after taking a proper continuum limit <cit.>. Nevertheless, it has been demonstrated on a number of examples, that the simple lattice discretization of the CFT ansatz provides an excellent approximation of the actual EH at low energies and for large subsystems <cit.>.Here we explore a different scenario, where the model is defined in continuous space, but described by the nonrelativistic Schrödinger equation. We focus on the free-fermion gas, where the entanglement entropy has been studied before <cit.>, and shows a logarithmic area-law violation in arbitrary dimensions due to the presence of a Fermi surface <cit.>.Although this result was interpreted via the contributions of independent gapless modes building up the Fermi surface <cit.>, the precise applicability of a CFT description in higher dimensions remained elusive.Our main goal here is to directly address the EH of the Fermi gas for a d-dimensional spherical domain A with radius R, and compare it to the CFT prediction <cit.> =πR/v ∫_A ^d 𝐱 (1-|𝐱|^2/R^2) T_00(𝐱), where T_00(𝐱) is the energy density and v is the speed of excitations, which makesdimensionless. Its form thus corresponds to an inverse temperature that varies parabolically in the radius and vanishes at the surface of the sphere. The numerical check of (<ref>) for a free massless scalar field was carried out by first decomposing the EH into angular momentum sectors, and then discretizing the remaining radial problem <cit.>. While a good agreement with CFT was found at low angular momenta, for higher ones the results are inconclusive.Our main result is that, in any dimension d>1, the CFT description of the nonrelativistic Fermi gas breaks down at large angular momenta. In particular, we show the equivalence of the entanglement spectra in continuous free space to those of a lattice problem with a linear potential <cit.>. The mapping identifies the angular momentum with the subsystem boundary on the chain, whereas the radius R sets the length of the region with nontrivial fermion density. While the bulk of this region admits an effective CFT description <cit.>, characterized by a spatially varying Fermi velocity, the fine structure close to the dilute edge is not properly captured. The discrepancy is demonstrated by comparing the actual entanglement spectra and entropies to those that follow from parabolic deformations (<ref>) of the physical Hamiltonian, which commute exactly with the EH <cit.>.The free Fermi gas in d dimensions is described by the single-particle Hamiltonian Ĥ = 𝐩̂^2/2m - μ, where 𝐩̂=-i ∇ is the momentum operator and the chemical potential μ=q_F^2/2m sets the filling via the Fermi wavenumber q_F. The ground state is given by a Fermi sea F, with the plane-wave modes occupied in a spherical domain |𝐪|<q_F. We are interested in a spherical subsystem A of radius R centered around the origin, |𝐱|<R. The entanglement Hamiltonian Ĥ is then defined via the reduced density matrix and can be written as <cit.> ρ̂_A = 1/𝒵 ^-Ĥ, Ĥ=ln(𝒦̂^-1_A-1), in terms of an integral operator (𝒦̂_A ψ)(𝐱) = ∫_A ^d𝐱' K(𝐱,𝐱') ψ(𝐱') , that acts on wavefunctions in the domain A, with the kernel given by the two-point correlation function K(x,x') = ∫_F ^d𝐪/(2π)^d ^i q(x -x') .We first discuss the simplest case of a 1D system, where A=[-R,R]. After a rescaling y=x/R, the integral operator (<ref>) is given by the famous sine kernel K(y,y') = sinc(y-y')/π(y-y') , which depends on the dimensionless parameter c=q_F R. To construct the EH via (<ref>), one needs to solve 𝒦̂_A ψ_k = ζ_k ψ_k to find the eigenvalues and eigenfunctions of 𝒦̂_A. This can be done by considering instead the differential operator <cit.> D̂ = - /y (1-y^2) /y - c^2(1-y^2) ,which commutes with the integral operator, [𝒦̂_A,D̂]=0. The bounded solutions of the equation D̂ ψ_k = χ_kψ_k within the domain |y|<1 are known as the angular prolate spheroidal wavefunctions <cit.>, ψ_k(y)=S_0k(c,y), and exist for a discrete set of eigenvalues χ_k with k=0,1,…The eigenvalues of 𝒦̂_A then follow from the radial spheroidal wavefunctions as ζ_k= 2c/π[R_0k(c,1)]^2 <cit.>.It is easy to see, that the operator (<ref>) is a simple parabolic deformation of the original Hamiltonian (<ref>). Comparing with (<ref>), one can identify it with the CFT expression after proper rescaling = πR/v_F D̂/2mR^2 = π/2c D̂ , where the speed must be identified with the Fermi velocity v_F=q_F/m. The spheroidal eigenvalues χ_k can be computed using Mathematica, and thus the spectrum ofcan be compared against that ε_k=ln(ζ_k^-1-1) of the actual EH in (<ref>). These are shown in Fig. <ref>, with the full/empty symbols corresponding to ε_k and π/2cχ_k, respectively, while the inset shows their difference. Note that the index k was shifted by k_0-1/2, with k_0=2c/π, to align the low-energy part of the spectra. One clearly observes that the deviation diminishes for increasing c, suggesting the asymptotic equivalence Ĥ→ of the operators. This is supported by analytical results <cit.>, as well as further numerical evidence <cit.>. In particular, for finite c one has a series expansionĤ = + ∑_n=1^∞ 1/c^n P_n+1(), where P_n is an n-th order polynomial. The n-th correction term is thus an increasingly non-local differential operator of order 2(n+1), which is, however, suppressed by c^n. Furthermore, using thelowest order terms in (<ref>), we find that the entanglement entropy S=-[ρ_A lnρ_A] is reproduced byup to a correction scaling as δ S ∝ln (c)/c^2, which agrees well with our numerics <cit.>.We now proceed to the case d ≥ 2, which considerably simplifies using the rotational symmetry of both A and F. Indeed, setting 𝐱=r 𝐧, the Hamiltonian can be decomposed by considering the ansatz for the wavefunction ψ(r 𝐧)=Φ(r)/r^(d-1)/2Y_ℓ,i(𝐧), where Y_ℓ,i(𝐧) are d-dimensional spherical harmonics <cit.>, with 𝐧 being a vectoron the surface of the unit sphere, parametrized by d-1 angular coordinates. The quantum number ℓ=0,1,… corresponds to the angular momentum, and i=1,…,M_ℓ indexes the linearly independent spherical harmonics with fixed ℓ. In this basis, the Hamiltonian Ĥ = ⊕_ℓ,iĤ_ℓ,i becomes block-diagonal and in the respective sector reads Ĥ_ℓ,i = 1/2m(-^2/r^2 - q_F^2 + (ℓ+d-2/2)^2-1/4/r^2). Note that Ĥ_ℓ,i does not depend on the quantum number i, such that one simply has a degeneracy in each sector ℓ≥1 with corresponding multiplicity M_ℓ= 2ℓ+d-2/ℓ ℓ+d-3ℓ-1, while M_0 = 1. Thus the problem boils down to treating the one-dimensional Hamiltonian (<ref>), where one has an extra contribution from the centrifugal potential. The dimensionality enters via the multiplicities (<ref>) and a shift of the angular momentum index ℓ. For simplicity, we shall discuss the 2D case below, as the generalization to d>2 is trivial. The eigenvalue problem of the kernel (<ref>) was considered in <cit.>, see <cit.> for details. One first rewrites it as the absolute square of an exponential kernel K'( y , z) = ^i c y z in the scaled coordinates 𝐲=𝐱/R and 𝐳=𝐪/q_F. Separating variables using the ansatz (<ref>), one is led to consider the radial eigenvalue problem 𝒦̂'_ℓΦ_ℓ,k = γ_ℓ,kΦ_ℓ,k, with the kernel given by K'_ℓ(y,z) =J_ℓ(cyz) √(c^2yz). Note that y=|𝐲|≤1, z=|𝐳|≤1, and the eigenvalues of the original operator 𝒦̂_ℓ follow as ζ_ℓ,k=|γ_ℓ,k|^2. The squared kernel can then be written asK_ℓ(y,y') =2c^2 √(yy') K_Be,ℓ(c^2y^2,c^2y'^2) via the Bessel kernel defined as <cit.> K_Be,ℓ(u,v) = √(v) J_ℓ(√(u)) J'_ℓ(√(v))-√(u) J_ℓ(√(v)) J'_ℓ(√(u))/2(u-v). Note that the factor in (<ref>) in front of the Bessel kernel can be absorbed by a change of variables u=c^2y^2 and v=c^2y'^2, such that the spectrum of 𝒦̂_ℓ on the domain [0,1] is identical to that of 𝒦̂_Be,ℓ on [0,c^2].Analogously to the 1D case, one can find again a commuting differential operator in each angular momentum sector, [𝒦̂_ℓ,D̂_ℓ]=0, which reads <cit.> D̂_ℓ= -/y β(y) /y - (c^2 - ℓ^2-1/4/y^2)β(y),with β(y)=1-y^2. Clearly, (<ref>) can be interpreted as the parabolic deformation of the radial Hamiltonian (<ref>). Its eigenvalue equation reads D̂_ℓΦ_ℓ,k = χ_ℓ,kΦ_ℓ,k, and the eigenfunctions were dubbed generalized prolate spheroidal wavefunctions. Their asymptotic expressions for c,k ≫1 were studied in <cit.>. Moreover, high precision numerical computation of the eigenvalues ζ_ℓ,k and χ_ℓ,k is available via an open-source MATLAB code <cit.>.Before turning to the numerics, however, one needs an argument to fix the velocity in the CFT expression (<ref>). Indeed, the inhomogeneous part of the radial Hamiltonian (<ref>) can be interpreted as a spatially varying chemical potential μ_ℓ(r). In other words, the effective Fermi energy of the radial motion is reduced by the centrifugal energy of the orbital one. Furthermore, we argue that the only relevant radius in our problem is that of our subsystem, and thus the effective chemical potential should be evaluated at r=R. Assuming R ≫ 1, one obtains for the radial Fermi velocity v_F,ℓ = √(2μ_ℓ(R)/m)= v_F√(1-ℓ^2/c^2) . In particular, v_F,ℓ vanishes at ℓ = c, which corresponds to the angular momentum where the classical turning point is given by R. For all ℓ>c, the eigenfunctions of (<ref>) have exponentially small amplitudes within A, and thus their contribution to the EH should be negligible.Alternatively, the emergence of the Fermi velocity (<ref>) can be understood by mapping the problem to that of an inhomogeneous quantum chain. This can be achieved using a remarkable identity found in Ref. <cit.>, which establishes a connection between the Bessel kernel (<ref>) and the analogous discrete Bessel kernel K_dBe,c(i,j)= c J_i-1(c) J_j(c) - cJ_i(c) J_j-1(c)/2(i-j), where i,j ∈ℤ. The identity relates the trace of an integer power of the corresponding operators <cit.> _[0,c^2](𝒦̂^n_Be,ℓ)= _[ℓ+1,∞)(𝒦̂^n_dBe,c) , where the subscripts denote the domains of the respective kernels, over which the trace is carried out, with the r.h.s. being the trace of an ordinary matrix. Since the relation holds for arbitrary n, this implies that the spectra of the two operators are identical.The matrix defined in (<ref>) is precisely the correlation matrix of a hopping chain with a linear potential <cit.>, and unitary equivalent to the one describing domain-wall melting <cit.>. The parameter c now plays the role of the half-width of the front region, where the fermion density differs from one and zero. Moreover, the angular momentum ℓ is identified with the position of the entanglement cut. In turn, the expression (<ref>) simply corresponds to the spatial dependence of the Fermi velocity due to the variation of the filling within the front region <cit.>. The CFT prediction for the respective EH thus reads = π/2√(c^2-ℓ^2) D̂_ℓ.To test the validity of the ansatz (<ref>), we evaluate and compare the entropies obtained from Ĥ_ℓ and , as shown in Fig. <ref>. The agreement is excellent in the bulk of the profile, where the asymptotics of the spectrawith ℓ/c fixed were studied numerically for the gradient chain <cit.>. The resulting entropy profile S_ℓ= 1/6ln(c) + 1/4ln[1-(ℓ/c)^2]+ 𝒞 , where 𝒞≈0.4785 is a nonuniversal constant <cit.>, is shown by the red line. In fact, (<ref>) can also be derived using a curved-space CFT approach <cit.>, where the inhomogeneous metric is chosen to absorb the spatial variation of the Fermi velocity. While (<ref>) gives an accurate description of the bulk entropy profile, it does not capture the fine structure around the edge ℓ≈ c, where also the ansatz (<ref>) seems tobreak down. Indeed, using the scaling variable (ℓ-c)/c^1/3, the correlation matrix (<ref>) can be approximated by the Airy kernel <cit.>, and S_ℓ displays a corresponding edge scaling <cit.>. As shown by the inset of Fig. <ref>, the same holds true for the difference δ S_ℓ=S_ℓ-S_ℓ,CFT, which shows a data collapse for various values of c. The situation is very similar in d>2 dimensions, where the index of the Bessel kernel is ℓ + (d-2)/2. This is a half-integer in odd dimensions, such that the one-to-one correspondence with the gradient chain is lost. Nevertheless, when plotted against the shifted index ℓ + (d-2)/2, the entropy profile S_ℓ smoothly interpolates between the data points of the d=2 case. Applying the same shift in the scaling factor in (<ref>), the plot of the 3D case is almost identical to Fig. <ref>. One thus concludes that the CFT ansatz breaks down for high angular momenta ℓ≈ c-(d-2)/2. Due to the increasing multiplicities M_ℓ with the dimensionality, however, the leading order mismatch of the total entropy in d≥2 scales as δS =∑_ℓ M_ℓδS_ℓ∝c^d-2/(d-2)! c^1/3 . Thus, in sharp contrast to the 1D case, the entropy deviation becomes divergent in the c→∞ limit. This is a consequence of the edge-scaling regime in angular-momentum space, which is not properly described by CFT. The scaling (<ref>) is consistent with our numerics in Fig. <ref>, albeit with strong subleading corrections.The mapping to the gradient chain, with resulting entropy profile (<ref>), also allows us to obtain the analytical result for the total entropy S =∑_ℓ M_ℓS_ℓ≃σ_d c^d-1 lnc + A_d c^d-1, where the prefactors can be calculated as <cit.> σ_d = 1/3(d-1)!, A_d = 4 𝒞 - ψ(d+1/2) - γ/2(d-1)!, with ψ(x) being the digamma function and γ the Euler-Mascheroni constant. It is easy to check that the prefactor of the area-law violating term agrees with the general expression found in <cit.>. The area-law contribution is nonuniversal, and follows from the summation of the second and third terms in (<ref>) <cit.>. We tested the prediction (<ref>) by adding a subleading term B_d c^d-2 and fitting to our numerical data. The results σ_2=0.3332, A_2=0.651 and σ_3=0.1667, A_3=0.2285 for the 2D and 3D cases, respectively, are in excellent agreement with (<ref>). Note that our result on A_d also agrees with the conjecture formulated in <cit.>. One should also remark that, in free massless relativistic theories, no violation of the area law occurs <cit.>.In conclusion, we have found that the EH of a nonrelativistic Fermi gas is well reproduced by the appropriately rescaled parabolic deformation of the physical Hamiltonian. While in 1D the relation becomes asymptotically exact in the limit of large subsystems, in higher dimensions some deviations persist for large angular momenta. Using the mapping to the gradient chain, these discrepancies can be traced back to the dilute edge regime of the fermionic density, where the fine-structure of the correlations does not admit a CFT description. The deeper understanding of these nonrelativistic corrections remains an open question. A further natural extension would be the study of a trapped Fermi gas <cit.>, where the CFT predictions for the EH are also available <cit.>.We thank I. Peschel and E. Tonni for fruitful discussions and correspondence. In our numerical calculations we used the open-source code available under <http://github.com/lederman/prol>. The author acknowledges funding from the Austrian Science Fund (FWF) through project No. P35434-N.Supplemental Material: Entanglement Hamiltonian of a nonrelativistic Fermi gas § CORRECTIONS TO CFT RESULT IN 1D In the following we present some results on the first corrections to the CFT form of the spectrum in (<ref>). The problem amounts to finding an asymptotic relation between the eigenvalues ε_k and χ_k, as a power series expansion in terms of 1/c. Introducing b=ε_k/π and dropping the eigenvalue index k, the first few terms of such an expansion were provided in <cit.> as χ= 2bc + b^2-1/2 - b^3-b/8c+𝒪(c^-2) . Note that our convention differs from the one in <cit.> in the definition of the differential operator by a constant shift c^2, which is subtracted in our case. Then the leading order is linear in c and gives just the CFT result (<ref>). Inverting the relation (<ref>) to lowest order in 1/c one finds b=ε/π = χ/2c + 1-(χ/2c)^2/4c +𝒪(c^-2) . One thus obtains a perturbation series in terms of 1/c, where each term is a polynomial of the variable χ/2c.The result can be checked numerically by subtracting the leading (CFT) term and plottingthe corrections rescaled by c as a function of x=χ/2c. This is shown on the left of Fig. <ref> for various values of c, and compared against the analytic result depicted by the red line. The agreement is very good for small values of χ/2c, but deviations are already visible for larger values. Moreover, one can also probe the 1/c^2 correction by subtracting the 1/c term and rescaling the difference by c^2, as shown on the right of Fig. <ref>. One has again a good scaling collapse, and our fit for small parameter values x suggests the third-order polynomial P_3(x)=(3x^3-3x)/16. We note that higher order corrections could, in principle, be studied systematically by using the results of Ref. <cit.>, where uniform asymptotic expressions for both χ and ε eigenvalues were obtained via the WKB method.We now move to study the corrections in the entanglement entropy S, which has the thermal form S = ∑_k=0^∞ s(ε_k) , s(ε) = ε/^ε+1 + ln(1+^-ε) , in terms of the eigenvalues ε_k of the EH. Although one has an infinite sum, the dominant contribution comes from the low-lying entanglement spectrum. The asymptotic expansion, valid for c ≫ 1 and large indices k around the center of the spectrum k_0=2c/π, is given by <cit.> ε_k/2π ln(4c)-φ(ε_k/2π) = π/2 (k-k_0+1/2) ,where φ(z)=arg Γ(1/2+iz) and Γ(z) is the gamma function. The result for the entropy can be found by introducing the density of states ρ(ε) =k/ε and turning the sum in (<ref>) into an integral S = ∫ερ(ε) s(ε) = 1/3 ln(4c) + S_0 . The leading logarithmic scaling with argument 4c=4q_FR comes from the low-lying linear regime of the spectrum, whereas the nonuniversal constant S_0 = -1/π^2∫_-∞^∞ εs(ε) φ'(ε/2π), is due to the curvature and has a numerical value S_0 ≈ 0.495 <cit.>.We now calculate the entropy from the CFT ansatz (<ref>), which gives S_CFT = ∑_k=0^∞ s (πχ_k/2c) ≈∫_-∞^∞ ερ(ε) [ s(ε) + s'(ε) δ(ε) +1/2 s”(ε) δ^2(ε) ] , where δ(ε) denotes the correction term in (<ref>). Note that ρ(ε) and s”(ε) are even functions, whereas s'(ε) is odd. Hence, the lowest nonvanishing contribution comes from the odd part of δ(ε) and the even part of δ^2(ε), respectively. Furthermore, the dominant contribution comes from the constant part ρ(ε)≈ln(4c)/π^2 of the spectral density. It turns out that the corresponding integrals in (<ref>) can be evaluated explicitly, and one arrives at δS = S-S_CFT=-ln(c)/40 c^2 + 𝒪(c^-2) . We conclude this section by some comments on the higher dimensional case. In the limit k, c ≫ 1 and ℓ fixed, the asymptotics of the eigenvaluesandwere considered in <cit.>. However, we need instead the regime ℓ∼ c, where the spectrum was studied in the equivalent lattice problem of the gradient chain <cit.>. In particular, one finds /2π ln[4c(1-ℓ^2/c^2)^3/2] -φ(/2π) = π(k -k_0 +1/2),where k_0 is given by the expectation value of the particle number within the subsystem. In fact, this result was found by fitting the spectral function k(ε) to a modified 1D ansatz (<ref>), with the argument of the logarithm and the offset k_0 being fit parameters. Note also the change of the factor π/2 →π on the r.h.s. of (<ref>), which leads to a halved prefactor of the logarithmic term in the entropy (<ref>), corresponding to only one boundary point of the subsystem. The constantterm in (<ref>) is related to (<ref>) as 𝒞=S_0/2+ln(2)/3. It should be stressed, however, that (<ref>) is not expected to work properly in the edge regime |ℓ - c| ∝ c^1/3, where also the CFT ansatz (<ref>) breaks down.§ KERNEL FOR D ≥ 2 Let us rewrite the correlation kernel (<ref>), with spatial 𝐲=𝐱/R and momentum 𝐳=𝐪/q_F variables rescaled to the unit sphere, as the square of another kernel K(y,y') = (c/2π)^d ∫_|z|<1 ^d z K'(y ,z)K̅'(z ,y'), where K'( y , z) = ^i c y z and the bar denotes complex conjugation. Due to rotational symmetry, the eigenvalue problem of the corresponding integral operator can be decoupled in angular momentum sectors by introducing the ansatz (<ref>). Setting 𝐲 = r𝐧 and 𝐳 = r'𝐧', with 𝐧 and 𝐧' being vectors on the surface of the d-dimensional unit sphere, the integral operator acts as (𝒦̂' ψ)(r 𝐧) =∫_0^1 r' r'^d-1 Φ(r')/r'^(d-1)/2 ∫_Ω' Ω' ^i c r r' nn' Y_ℓ,i(𝐧') .It was shown in Ref. <cit.> that the second integral on the surface Ω' of the unit sphere can be carried out as ∫_Ω' Ω' ^i c r t nn' Y_ℓ,i(𝐧')= i^ℓ (2π)^d/2J_ℓ+d-2/2(crr')/(crr')^d-2/2 Y_ℓ,i(𝐧) . Hence the eigenvalue problem in the sector (ℓ,i) reads (𝒦̂'_ℓΦ_k)(r) = ∫_0^1 r' K'_ℓ(r,r') Φ_k(r') = γ_ℓ,k Φ_k(r) , with the kernel given by K'_ℓ(r,r') =J_ℓ+d-2/2(crr') √(c^2rr') . The kernel does not depend explicitly on the index i of the spherical harmonics, and thus just gives a multiplicity M_ℓ in the corresponding sector, similarly to the case of the physical Hamiltonian. Note that, for simplicity, we omitted a constant factor from the definition of the kernel (<ref>), which should be kept track of in the integral operator 𝒦̂' = ⊕_ℓ,i i^ℓ (2π/c)^d/2 𝒦̂'_ℓ. However, when taking the square of the kernel in (<ref>), the prefactor in front of the integral exactly cancels the one included in (<ref>), and one has 𝒦̂ = ⊕_ℓ,i𝒦̂_ℓ with underlying kernel K_ℓ(r,r') = ∫_0^1 t K_ℓ(r,t) K_ℓ(t,r') .Finally, the kernel K_ℓ(r,r') can be related to the Bessel kernel, which has the integral representation <cit.> K_Be,α(x,y) = 1/4∫_0^1 z J_α(√(xz)) J_α(√(yz)) . Indeed, inserting (<ref>) into (<ref>) and changing variables as z=t^2 with z= 2 t t, one arrives at the expressionK_ℓ(r,r') =2c^2 √(rr') K_Be,ℓ+d-2/2(c^2r^2,c^2r'^2) . Furthermore, the trace of an integer powercan be written as _[0,1] (𝒦̂^n_ℓ)= ∫_0^1 r_1 …r_n ∏_j=1^n 2c^2 r_j K_Be,ℓ(c^2r^2_j,c^2r^2_j+1) = ∫_0^c^2 x_1 …x_n ∏_j=1^n K_Be,ℓ(x_j,x_j+1) , where r_n+1=r_1, and we have changed variables x_j = c^2r_j^2 with x_j = 2c^2 r_j r_j. One thus finds _[0,1] (𝒦̂^n_ℓ)= _[0,c^2] (𝒦̂^n_Be,ℓ) .§ COMMUTING DIFFERENTIAL OPERATORS We shall prove here that the differential operator D̂ defined in (<ref>) indeed commutes with the integral operator 𝒦̂ with the sine kernel (<ref>) in 1D. Setting β(x)=1-x^2, one has (D̂𝒦̂ f)(x) = -[β(x)^2/ x^2+β'(x)/ x+c^2 β(x)] ∫_-1^1 dy K(x-y) f(y) = -∫_-1^1 dy f(y) [β(x)^2/ y^2-β'(x)/ y+c^2β(x) ] K(x-y). On the other hand, exchanging the order of the operators one obtains (𝒦̂D̂ f)(x) = -∫_-1^1 dy K(x-y) [β(y)^2/ y^2+β'(y)/ y+c^2 β(y)] f(y) =-∫_-1^1 dy f(y) [^2/ y^2β(y)-/ yβ'(y)+c^2 β(y)] K(x-y) =-∫_-1^1 dy f(y) [β(y)^2/ y^2+β'(y)/ y+c^2 β(y)] K(x-y) . In the second line we have integrated by parts, with the derivatives acting on all the functions to their right. Note that the boundary contributions to the partial integral vanish due to β(±1)=0. In the third line we have expanded the derivatives using the chain rule. Collecting the terms in (<ref>) and (<ref>), the commutator reads ([D̂ , 𝒦̂] f)(x)=∫_-1^1 dy f(y) { (β(y)-β(x)) [^2/y^2K(x-y)+c^2 K(x-y) ] + (β'(x)+β'(y))/yK(x-y)}. The derivatives of the sine-kernel can be evaluated as / yK(x-y) = sin c(x-y)/π (x-y)^2 - ccos c(x-y)/π (x-y), ^2/ y^2K(x-y) = -c^2sin c(x-y)/π (x-y) - 2ccos c(x-y)/π (x-y)^2+2sin c(x-y)/π (x-y)^3. Using β(y)-β(x)=x^2-y^2 and β'(x)+β'(y)=-2(x+y), it is easy to check that the commutator indeed vanishes.The calculation of the commutator for the radial kernel follows a similar route, including the extra centrifugal potential in the differential operator (<ref>). The weight function β(x) is unchanged, however, one has now a kernel which depends on the product (instead of the difference) of its variables, K_α(xy)=J_α(cxy)√(c^2xy) with α=ℓ+(d-2)/2. This changes the result on the second line of (<ref>), since one has the relation x^n^n/x^n K(xy) = y^n^n/y^n K(xy) . Using the parabolic form β(x)=1-x^2, one can thus write (D̂_α𝒦̂_α f)(x)=-∫_0^1 dy f(y) [(y^2/x^2-y^2)^2/ y^2-2y/ y+ (c^2-α^2-1/4/x^2)(1-x^2) ] K_α(xy) , (𝒦̂_αD̂_α f)(x)=-∫_0^1 dy f(y) [(1-y^2)^2/ y^2-2y/ y+ (c^2-α^2-1/4/y^2)(1-y^2) ] K_α(xy) . Note that the limits of integration have changed, as one is dealing with a radial problem now. Since β(0)0, one has to require that the function vanishes at the origin, f(0)=0, to cancel the boundary contributions in the partial integration. The first derivatives then cancel in the commutator, and the second derivative can be rewritten as ^2/ y^2 K_α(xy)=x^2K”_α(xy), where the prime denotes the derivative w.r.t. the argument xy. In turn one has ([D̂_α, 𝒦̂_α] f)(x)= ∫_0^1 dy f(y) (x^2-y^2) [ K”_α(xy) + (c^2-α^2-1/4/x^2y^2)K_α(xy) ] .Finally, it is easy to show that the expression in the square brackets in (<ref>) vanishes. Indeed, introducing the variable z=cxy and writing K_α(xy)=J_α(z)√(cz) one has K”_α(xy) = c^5/2^2/z^2 [ J_α(z) √(z) ]= c^5/2z^-3/2[z^2^2/z^2 + z/z -1/4] J_α(z)= c^2[α^2 -1/4/z^2 - 1] K_α(xy) , where in the last step we used the form of the Bessel differential equation. This is indeed the required identity.§ ENTROPY FOR D ≥ 2 Here we comment on the calculation of the total entropy in (<ref>). The leading term violates the area law logarithmically in arbitrary dimensions. It arises from the first term in (<ref>), with the prefactor given by the sum σ_d = 1/6 lim_c→∞ ∑_ℓ=0^c M_ℓ/c^d-1= 1/3(d-1)! . The area-law prefactor A_d has two contributions corresponding to the summation of the last two terms in (<ref>). The constant piece behaves in exactly the same way as the logarithmic one, i.e. it contributes 6 σ_d𝒞 to A_d. To evaluate the ℓ-dependent sum, we introduce the scaling variable x=ℓ/c and consider ℓ, c ≫ 1. Using the asymptotics of the multiplicity, M_ℓ≈ 2ℓ^d-2/(d-2)!, the contribution becomes 1/c^d-1∑_ℓ=0^c M_ℓ/4ln[1-(ℓ/c)^2] ≈1/2(d-2)!∫_0^1 x x^d-2 ln(1-x^2) = -ψ(d+1/2) + γ/2(d-1)! . Adding the two pieces leads to the result (<ref>) reported in the main text.Finally, we compare the prefactor (<ref>) to the general formula which was originally conjectured in <cit.>, and later proved in <cit.>. Introducing the notation ∂ F and ∂ R for the Fermi surface and the subsystem boundary, respectively, the prefactor is given by a double integral σ_d = 1/12(2π)^d-1 ∫_∂FΩ_F∫_∂R Ω_R |𝐧_F𝐧_R|, where 𝐧_F and 𝐧_R are normal vectors on the corresponding surfaces. Note that in our case both of these are just the surface of the unit sphere, as the radii have already been scaled out. To evaluate the integrals, we only need the formula for the (d-1)-dimensional surface area of the unit sphere 𝒮_d-1 = 2 π^d/2/Γ(d/2) . Indeed, since the integrand of (<ref>) contains only the relative angle between two points on the unit sphere, 𝐧_F𝐧_R=cosθ, one of the integrals gives 𝒮_d-1 and the other one gives 𝒮_d-2 times the integral on the relative angle σ_d = 1/12(2π)^d-1 𝒮_d-1 𝒮_d-2 ∫_0^π θsin^d-2 θ|cosθ| . Note that the factor sin^d-2θ in the integral comes from the expression of the surface element Ω in terms of the angular coordinates, and the corresponding integral can be evaluated explicitly as ∫_0^π θsin^d-2 θ|cosθ| = 2/d-1 . Inserting (<ref>) and (<ref>) into (<ref>), one finds the result in (<ref>). | http://arxiv.org/abs/2311.16348v1 | {
"authors": [
"Viktor Eisler"
],
"categories": [
"cond-mat.stat-mech",
"hep-th",
"quant-ph"
],
"primary_category": "cond-mat.stat-mech",
"published": "20231127222756",
"title": "Entanglement Hamiltonian of a nonrelativistic Fermi gas"
} |
Towards Transfer Learning for Large-Scale Image Classification Using Annealing-based Quantum Boltzmann Machines© 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 authors acknowledge funding from the German Federal Ministry for Economic Affairs and Climate Action, project PlanQK, 01MK20005I. Daniëlle Schuman0009-0000-0069-5517 LMU [email protected] Leo Sünkel LMU [email protected] Philipp Altmann0000-0003-1134-176X LMU [email protected] Jonas Stein0000-0001-5727-9151 LMU [email protected] Christoph Roch0000-0003-0781-6590 LMU [email protected] Thomas Gabor LMU [email protected] Claudia Linnhoff-Popien0000-0001-6284-9286 LMU [email protected] 14, 2024 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Replaying data is a principal mechanism underlying the stability and data efficiency of off-policy reinforcement learning (RL). We present an effective yet simple framework to extend the use of replays across multiple experiments, minimally adapting the RL workflow for sizeable improvements in controller performance and research iteration times. At its core, () involves reusing experience from previous experiments to improve exploration and bootstrap learning while reducing required changes to a minimum in comparison to prior work.We empirically show benefits across a number of RL algorithms and challenging control domains spanning both locomotion and manipulation, including hard exploration tasks from egocentric vision.Through comprehensive ablations, we demonstraterobustness to the quality and amount of data available and various hyperparameter choices. Finally, we discuss how our approach can be applied more broadly across research life cycles and can increase resilience by reloading data across random seeds or hyperparameter variations.§ INTRODUCTION In the last few years, reinforcement learning (RL) has transitioned from a topic of academic study to a practical tool for the generation of controllers across various real-world applications <cit.>. These advancements have been enabled, in part, by recent advancements in algorithms improving data efficiency, robustness, and controller performance <cit.>. Nevertheless, many problems remain hard to solve with RL. For instance, high-dimensional and partial observations (e.g. egocentric camera views), high-dimensional action spaces, or reward functions that provide inadequate learning signal can all lead to poor asymptotic performance, high variance, low data efficiency, and long training times. A major challenge in RL remains the interaction between data collection and learning. Unlike in the case of supervised learning where all data is available and static, in online RL both the quality and quantity of data available for policy or value function learning changes over the course of an experiment. The difficulty of collecting suitable data, and the complex interactions between data collection and function optimization leads to various problems including failure to learn, instabilities, and the premature convergence of function approximators early in learning<cit.>.Experience replay <cit.> has become a popular component of most, modern off-policy RL algorithms. Storing data collected over the course of an experiment and continuously training policy and value functions with this growing dataset can greatly increase data efficiency and stability of RL algorithms. It does not, however, address issues around premature convergence of the function approximators <cit.>, nor does it directly provide a solution to reusing data from prior experiments. In this paper, we investigate the possibility of extending the use of replay and reusing data in an iterative setting. Our main insight is that a minimal change to the RL workflow can greatly improve the asymptotic performance of off-policy reinforcement learning algorithms. By reusing interaction data from prior training runs it can further reduce the overall experiment time and thus speed up research iterations. A number of studies have previously investigated how prior data (including expert data) can be used to kick-start RL training <cit.>. We find that the simplest approach, mixing prior and online data with a fixed ratio, is particularly effective across a wide range of application scenarios and algorithms. Our method dubbed () performs as well as or better than alternative approaches with fewer hyperparameter choices to be made. Furthermore, for domains that are normally hard to solve even with state-of-the-art algorithms, we find that it can be advantageous to perform multiple training iterations giving each iteration access to all data from prior training runs, effectively realising a minimalist perspective to lifelong learning.We hope that 's simplicity will enable straightforward integration into existing infrastructure and help improve the efficiency of RL workflows.The main contributions of our work are as follows: * Introduce and empirically validate our simple strategy to solve challenging tasks. Demonstrate state-of-the art performance on a number of domains including when reusing data from the publicly available offline RL Unplugged benchmark <cit.>.* Demonstrate that our approach works across multiple algorithms including DMPO, D4PG, CRR and SAC-X.* Compare to common baselines and recent state-of-the-art methods. Highlight the combination of factors that leads to the effectiveness of our approach.* Provide additional ablations to highlight the robustness of to quantity and quality of data collected and hyper-parameters.§ METHOD §.§ Background We consider the reinforcement learning (RL) problem in which an agent observes the environment, takes an action, the environment changes its state and the agent receives a reward in response. The environment is modeled as a Markov Decision Process (MDP) consisting of the state space S, the action space A, the transition probability p(s_t+1|s_t, a_t) and reward r(s_t, a_t) when taking action a_t in state s_t. The agent's behavior is specified using a deep neural network policy π(a_t|s_t; θ) with parameters θ; which we will denote as π(a_t|s_t) for brevity.We optimize the agent to maximize the sum of discounted future rewards, as denoted by:J(π) = 𝔼_ρ_0(s_0),p(s_t+1|s_t,a_t),π(a_t|s_t)[∑_t=0^∞γ^t r_t ],where γ∈ [0, 1] is the discount factor, r_t=r(s_t, a_t) is the reward and ρ_0(s_0) is the initial state distribution. Given a policy π , the state-action value function, or critic, Q(s_t, a_t) is defined as the expected discounted return when taking an action a_t in state s_t and then following the policy. Q(s_t, a_t) = r(s_t, a_t) + γ𝔼_p(s_t+1|s_t, a_t), π(a|s_t) [ Q(s_t+1, a) ]. Modern off-policy algorithms <cit.> usually operate through simultaneous or alternating optimization of policy and state-action value function. In this context, experience replay <cit.> is a fundamental mechanism to decouple data collection from policy and value-function optimization. Data is collected by one or multiple policies (which are commonly obtained over the course of a single experiment) and is stored in a replay buffer from which it can be retrieved to compute updates to the policy or value function. This approach has multiple benefits including data efficiency, reduced variance of the updates and smoothing the learning process <cit.>. The success of off-policy algorithms and the need for data-efficient and safe learning schemes has led to growing interest in approaches that allow to reuse data across experiments. One end of a spectrum aims to learn policies entirely offline from a pre-generated, fixed corpus of training data <cit.>. This usually requires specialist algorithmic modifications to avoid instabilities when no online data is available.It is often desirable to combine prior data with new data gathered during a learning experiment.In the simplest case a policy (and value function) pretrained offline can be then finetuned online, on the same or a related task. While various offline-RL algorithms have been optimised for this setting, simpler off-policy approaches can work well with sufficiently diverse data <cit.>.The second perspective to re-using data is the straightforward reloading into the replay buffer during online learning. This approach has been successful in bootstrapping learning on tasks with expert demonstrations <cit.> or to transfer to new tasks by incorporating data from scripted controllers <cit.>.Finetuning offline learned agents and mixing data into a replay is further combined in the AWAC algorithm <cit.>.It operates in two phases: in the first phase learning is performed entirely offline by pre-loading the replay buffer with offline data. Then, after a specified number of training steps, online data is mixed into the replay and learning proceeds off-policy.AWAC proposes a specific algorithm for both phases that is similar to the formulation of the offline-RL algorithm CRR <cit.>.While these methods have shown promising gains in data efficiency and performance, they each come with specific assumptions that increase complexity, implementation effort, and restrict their application. These include the requirement for multiple training stages <cit.>, the introduction of additional training losses <cit.> or validation with specific architectures and algorithms <cit.> which typically require domain-specific hyperparameter tuning.The core idea that we are exploring in this paper is that the reuse of prior data can be implemented in the context of most contemporary off-policy RL workflows in a very simple way that is robust and leads to excellent results, without many of the algorithmic complexities or hyper-parameter choices of prior work.§.§The basic insight of this work is that reusing data across multiple experiments in off-policy learning is a very simple but effective way to accelerate training and improve final performance. Our key argument is the extension of this insight across all experimentation during a project. Prior data can be used to bootstrap new training runs, and it can further be beneficial to train difficult tasks in multiple iterations, bootstrapping later iterations with the data collected during earlier ones.This idea is illustrated in Figure <ref>. Whereas off-policy RL is generally focused on reusing data during a single experiment, we consider sequences of experiments where data from earlier training runs is made available during later ones[We store and reuse all training data across experiments, not just trajectories left in the final replay.]. At the beginning of each training run, policy and value-function are re-initialized in line with stand-alone experiments. The only required algorithmic change is the availability of a second replay mechanism that allows replaying prior and online data with a particular fixed ratio throughout the course of training (we use a naive 50/50 mix of offline and online data for our main results, without optimizing this ratio).This simple approach performs very well without introducing additional algorithmic complexities or the need for hyper-parameter tuning. By learning in the mixed online setting, the agent controls the data distribution and in line with recent results in offline RL <cit.>, we show that given the right data distribution, existing off-policy algorithms are sufficient for effectively using this offline data. This workflow does not require any changes with respect to the RL agent itself and is generally agnostic to agent and architecture changes across experiments. In order to emphasize this aspect, for the main results presented in Section <ref>, we integrate into existing algorithmic workflows across all of the domains considered. Consequently, we evaluate across a set of state-of-the-art agents including DMPO <cit.>, SAC-Q <cit.>, CRR <cit.> and D4PG <cit.> across a number of complex domains.§ EXPERIMENTSIn this section, we evaluate the performance of across feature-based and vision-based, simulated robot locomotion and manipulation settings and standardised RL control benchmarks.§.§ DomainsWe consider the following set of challenging domains (visualised in Figure <ref>):Locomotion SoccerFor locomotion, we use a simulated robot soccer task <cit.> where a simulated OP3 humanoid robot <cit.> is rewarded for scoring goals (along with additional shaping rewards) against an opponent. We consider two versions of this task: using proprioception and task-specific features (Locomotion Soccer State) and using proprioception with egocentric visual features (Locomotion Soccer Vision). This second variant is particularly challenging since the agent must learn to score from a sparse reward signal and visual features that make the environment partially observable. Further information on this domain can be found in Appendix <ref>. We use MPO <cit.> with a distributional critic <cit.> as underlying off-policy RL algorithm. For our experiments we gather training data of 4e5 and 2e5 episodes [Throughout the text we use the scientific notation shorthand where 4e5 refers to 4*10^5.] for state and vision respectively.Manipulation RGB Stacking For manipulation, we consider the task of stacking parameterized color-coded objects from <cit.>. This task builds on visual inputs and only sparse rewards. We use the multi-task SAC-Q off-policy RL algorithm <cit.> in this domain. Further information can be found in Appendix <ref>. The offline dataset here consists of 15e4 episodes of training data from a prior experiment with the same algorithm.RL Unplugged Finally, we evaluate with the offline RL benchmark and dataset RL Unplugged <cit.>, which includes offline data on various simulated control domains. Within these, we focus on the three most challenging Control Suite domains <cit.>: humanoid run, manipulator insert peg and manipulator insert ball. The benchmark dataset contains 3000 episodes for humanoid run and 1500 episodes each for the manipulator domains. We use with the offline RL algorithm CRR <cit.>, which to our knowledge has state-of-the-art performance for pure offline learning on this benchmark. The offline data in this setting is notably different from the other domains. In order to collect the data, the authors follow a sub-sampling criterion (see <cit.> for details) to select a subset of episodes which are then randomly stored as single-step trajectories. This is in contrast to our simple protocol of collecting and reusing all data across training with a single algorithm. We include this domain to illustrate the robustness and flexibility of as well as to simplify future extensions and comparisons. More details on all domains are presented in Appendix <ref>. §.§ BaselinesWe compare against a number of strong baselines to illustrate its effectiveness. Fine-tuning We first train a policy and state-action value function offline using Critic Regularized Regression (CRR) <cit.> to retain the flexibility of architecture and related changes across experiments. We evaluate a pure behavior cloning agent (BC) and two variants of CRR, CRR binary and CRR exp (see Appendix <ref>). Subsequently, we choose the best performing seed from across these runs and fine-tune the agent (both policy and value function) online. AWAC AWAC <cit.> starts by learning entirely offline for a fixed number of pre-training steps before switching to online iteration. The main difference to finetuning applies to the online phase, where it uses a offline/online shared replay buffer with a specific algorithmic formulation. The authors of this work prescribe 25e3 pretraining steps offline before incorporating online data. For our sweep we also considered 5e4 steps and 1e5 pre-training steps. We also sweep over different values for the Lagrange multiplier λ (0.3 and 1) as proposed by the original work. Random Weight Resetting Reloading data for an experiment restart implicitly involves resetting network weights. <cit.> observed that network resets on their own can be beneficial for learning. [When using RaE, the data distribution throughout learning uses a fixed ratio between offline and online data, in addition to implicitly resetting network weights.] To tease apart the effect of weight resets from the benefit of data reloading, we consider a baseline where all weights (policy, critic and optimizer) are reset every K policy updates; where K is set to the number of updates used to train the policy that generated the data. We sweep over reset frequencies of K, K/10 and K/100[Since the data collection methodology for the RL Unplugged domains is different and the number of updates unknown, we instead chose to sweep over values of 1e4, 1e5 and 1e6 updates for these domains]. All experiments are run till convergence or at least 2*K updates.For each method, we report results of the best performing hyperparameter variant averaged across 5 seeds. Importantly, note that did not require any tuning and we report results using the same hyperparameter set across all of the main results (50% offline data). While minor performance improvements with different parameters for can be observed in extreme settings (see Table <ref>), we find has very low sensitivity to hyperparameter choices across a range of diverse domains. §.§ Main ResultsFigure <ref> visualises our results on the Locomotion Soccer and Manipulation domains. It compares asymptotic performance achieved by and other baselines after convergence as a bar plot with the dark solid line showing a 95% confidence interval around the mean averaged across 5 seeds after smoothing over 1000 episodes. As the figure shows, consistently achieves the highest asymptotic performance across these tasks with a particularly notable improvement on the challenging vision-based `Locomotion Soccer Vision' domain. In the Manipulation settings, both finetuning and perform similarly, although finetuning requires choosing the right algorithm and hyperparameter for offline learning.Figure <ref> compares with all baselines that can use the RL Unplugged offline dataset. The figure shows the accumulated reward over the total number of additional online data consumed for each method. The performance of the best pure offline variant (CRR) is shown as a dotted blue line.We observe that performs at-par or better than other comparable methods with the biggest difference being in the manipulator tasks. Methods that use offline learning (CRR and finetuning) work well on the densely rewarded Humanoid run domain but peform poorly in the sparsely rewarded manipulator domains whereas works well across all settings.§.§ AblationsIn this section, we present a set of experiments designed to answer the following questions: * How much data is required for performance improvements?* What kind of data is best for mixing: expert data, early training data or a mix?* How sensitive is to the ratio of online to offline data in different data regimes?* Is there an advantage of applying repeatedly across experiment iterations?* Is agnostic to the underlying choice of algorithm? Unless otherwise specified, we present results on the `Locomotion Soccer State` task with the DMPO algorithm. Analysis with varying data We answer the first three of these questions together in this section. For this analysis we use multiple subsets of the data collected for the `Locomotion Soccer State' task, divided into three regimes: * High Return We consider data generated only from the end of training from a complete training run. This corresponds to highly rewarding trajectories or `expert' data.* Mixed Return We consider data sampled uniformly at random throughout learning. This corresponds to a mixed regime with of high and low return trajectories.* Low Return Finally, we consider data generated only from the start of training. This corresponds to early low return data.High and Low Return data are sampled according to recency while Mixed return samples data uniformly. For each regime, we consider 2 datasets: one with 1e5 episodes[The original dataset consists of 4e5 episodes.] and another with just 1e4 episodes. The purpose of this analysis is to understand how best to adapt when in a regime with limited data. For each data subset we then consider data mixing at different ratios of online to offline data: in addition to the 50% mix that was used in the main results we consider mixes of 70, 80 and 90% online to offline data. For each setting, we consider the asymptotic performance reached when using on the `Locomotion Soccer State` task.Table <ref> shows the final asymptotic reward as a percentage of the final reward achieved when learning online from scratch, which corresponds to 4e5 episodes. To improve the ease of interpretation the cells are colored based on the reward achieved: yellow for below 90% performance, orange for between 90-100% and blue for greater than 100%. A few interesting patterns emerge which we summarize below: * In the lower data regime (1e4 episodes) a mixture with more online data is beneficial. However, as more data becomes available, a lower ratio works better (of 70-80% at 1e5 episodes). We hypothesize that using more online data for learning prevents over-fitting to a small set of offline trajectories.* In the lower data regime, low return data tends to be the most beneficial. As the dataset size increases though, a mix of high and low return trajectories provide a greater benefit. Surprisingly, expert data is the least beneficial in both regimes. This indicates that the advantage of mixing data may stem from the benefit of having a larger state distribution with mixed rewards.* Gains in performance can be achieved with as little as 1e4 episodes. This is particularly promising since even a small amount of prior data can considerably improve results.Iterative improvement Figure <ref> shows the improvement in performance when iteratively applying on the `Locomotion Soccer State' domain. For this experiment, we begin by generating a smaller dataset of 1e4 episodes and then apply to that. We collect the data from this already improved run (iteration 1) and apply again. We observe that there is a small gain in asymptotic performance and speed of learning even on the second iteration although a performance plateau is reached on a third iteration. This result suggests that in some settings it may be preferable to break a single training run into smaller runs for iterative performance improvements.Indifference to algorithm For the main analysis of Section <ref>, we show that can improve performance across a range of domains with different underlying algorithms. Figure <ref> reinforces this point on the `Locomotion Soccer State' domain where we show a considerable gain in performance when applying using the D4PG <cit.> algorithm for both data collection and mixing.Increased robustness By applying to combine data across seeds and hyperparameters from previous experiments we can add crucial robustness to the underlying agent's variance in performance. Figure <ref> demonstrates robust performance when reloading data across random seeds in comparison to purely high or low performing data sources. § RELATED WORKThe most common paradigms to collect and reuse data in RL involve either offline agent datasets (e.g. RL Unplugged <cit.> and D4RL <cit.>) or the use of human or expert demonstration trajectories <cit.>. While some work has focused on solving benchmark challenges <cit.> others have focused on improving online performance by integrating offline data. For example, <cit.> and <cit.> learn hierarchical architectures offline and reuse them to solve more challenging tasks online. Related to our approach, <cit.> mix experience using controllers designed for a different setting with online data to improve learning efficiency for quadruped robots. Our work instead highlights the merits of directly transferring data to improve asymptotic performance on a single domain. Learning from Data Combinations The idea of mixing expert data sources to bootstrap learning on the same domain has also been studied in a number of related works <cit.>. While these methods have shown impressive results on a range of domains, the complexity and specificity of various algorithmic assumptions limit their generality.While <cit.> also mix expert demonstration data into the replay, they include a prioritized replay and a mix of 1-step and N-step returns with L2 regularization in their setup. In a similar vein, <cit.> introduce a BC-loss, Q-filter and state-reset mechanism as part of their data mixing approach. Other work relies on multi-stage procedures when mixing data: <cit.> distill prior data from skill experts to be later mixed with data from a scripted controller followed by the application of CQL; <cit.> train a multi-task policy with separate forward and backward policies that are optimized separately. Even work that uses existing offline datasets as opposed to expert demonstrations inevitably include other components that increase their complexity. For instance, <cit.> finetune an ensemble of policies trained offline using a prioritized replay and a density ratio to choose the data mixture to improve performance with the D4RL dataset. On the same domain, <cit.> begin with a similar formulation to ours but then argue for random ensemble distillation and per-environment design choices and LayerNorm when training the Q-function with which they demonstrate improvements using a single off-policy algorithm. Finally, the AWAC algorithm <cit.> that we compare against sits in between finetuning and data mixing but uses a specific algorithmic formulation which introduces a number of domain-dependent hyperparameters. In contrast, our focus is on the simplicity of the method: we empirically demonstrate the effectiveness and versatility of mixing previous data with a fixed ratio using many off-policy RL algorithms. Resetting Network Parameters in Reinforcement Learning A consequence of restarting experiments with mixed data is the resetting of neural network parameters. Resetting weights can change the learning dynamics and prevent overfitting which has been shown to be beneficial in both supervised learning <cit.> and RL <cit.>. Under certain settings where parts of a network are reset at specific intervals, these methods have been shown to greatly improve learning efficiency on challenging domains like Atari. In this work, we demonstrate that reusing previous experience can improve asymptotic performance on a range of tasks. Combined with its simplicity, we hope our approach can be included as part of the natural iteration cycle in RL and robotics in particular.§ DISCUSSIONAs RL continues to move from the object of study to a practical engineering tool for control <cit.>, simple and practical methods that can be generally applied will become increasingly important. We can apply throughout the lifetime of a project, across multiple experiments, algorithms, and hyperparameter settings. In the following paragraphs, we provide some intuition on some practical use-cases for .Lifelong / Project-long learning Research into novel domains nominally involves many iterations of trial-and-error to achieve state-of-the-art performance. While learnings from early iterations of experimentation inform algorithmic choices, the data collected in these trials is commonly discarded and never reused. Our initial analysis in Section <ref> indicates that even low-return data can be useful to boost performance. With costs of data storage typically being far lower than compute, a different workflow where all experimental data (particularly in domains like robotics) is stored and reused to bootstrap learning could improve efficiency across project lifetimes. Multiple Source Experiments can also be applied in domains with many related source experiments. For example, consider many tasks defined via different reward functions but with the same underlying dynamics (e.g. the family of all manipulation tasks with a specific morphology). High-return data on some tasks may result in lower returns in another. However this data is still informative and may be useful in improving exploration when transferred.Multiple Hyperparameters and SeedsCombining data from different variants of the same experiment (such as random seeds or hyperparameters) can enable better use of large, expensive sweeps. When reloading data with high performance variance over these parameters, the new experiment can benefit from the best option as initially discussed in Section <ref>. Potential Limitations and Strategies for Mitigation Across all experiments, demonstrates better or similar performance to state-of-the-art baselines for efficient off-policy RL without requiring per task hyperparameter tuning. However the applicability of the method is limited by the ability to reuse data. For example, changes in dynamics or experimental settings might invalidate previously collected data [However in practice, this may not be much of an issue: see Appendix <ref>]. In such cases, an intermediate step to collect transitional data between old and new settings might be useful.Another challenge that may arise with widespread use of is in the comparison of new algorithms. RL already suffers from issues of reproducibility <cit.>; if algorithms incorporate , subtle changes in the data distribution when learning may create large differences in performance. One solution to this may be to specify deterministic orderings for benchmark datasets using fixed random seeds.§ CONCLUSIONS With this work, we introduce an effective and, importantly, simple workflow change for off-policy RL experimentation. Reloading data requires minimal infrastructure and can greatly improve performance as shown in Section <ref>. This makes it particularly useful from a project-long (or life-long) learning perspective in RL experimentation. We believe that as our understanding of RL improves and its use as an engineering and control tool becomes more commonplace, simplicity is key to effective integration.iclr/iclr2024_conference§ ENVIRONMENT DETAILS§.§ Locomotion SoccerThe `Locomotion Soccer' environment introduced by <cit.> consists of a Robotis OP3 <cit.> robot which is placed in a 4m × 4m walled arena with a soccer ball. The robot must remain in its own half and scores goals when the ball enters a goal area 0.5m times 1m positioned across the center of the back wall in the other half. The robot must also contend with a random agent in the other half. The reward for this task is given by: R_goalscoring = R_score + R_upright + R_maxvelocity where: R_score is 1000 on the single timestep where the ball enters the goal region (and then becomes unavailable until the ball has bounced back to the robot's own half); R_maxvelocity is the norm of the planar velocity of the robot's feet in the robot's forward direction; R_upright is 1.0 when the robot's orientation is close to vertical, decaying to zero outside a margin of 12.5^∘.Additionally, episodes are terminated with zero reward if the robot leaves its own half, or body parts other than the feet come within 4cm of the ground.The agent chooses a desired joint angle every 25 milliseconds (40Hz) based on input observations that include the joint angles, angular velocity and gravity direction of the torso.For `Locomotion Soccer State' the agent also receives the coordinates of the ball, goal and opponent. For the `Locomotion Soccer Vision` domain the robot instead receives a 40 × 30 render of the egocentric camera. We use the experimental settings and networks described in <cit.> for this setting. We use a batch size of 128 with 5 seeds for the main results in Section <ref> and a batch size of 256 with 2 seeds for the experiments in Section <ref>. §.§ Robot Manipulation The RGB Stacking benchmark introduced by <cit.> consists of parametric extruded shapes, which need to be stacked on top of each other by a fixed robotic manipulator, with the role of objects in the configuration being denoted via color coding. The benchmark uses a Rethink Sawyer robot arm, controlled in Cartesian velocity space on which a Robotiq 2F-85 parallel gripper is mounted.We focus on the "skill mastery" challenge of the benchmark as described in <cit.>, where five fixed triplets of objects are used both for training and testing, with the goal being to place the red object on the blue one, while ignoring the green one. Each object triplet highlights a different manipulation aspect, such as balancing or reorientation.For the SAC-X setup, we use a curriculum of sub-tasks that can be chosen by the scheduler. These follow the reward terms of the staged reward terms used in <cit.>, and intuitively contain sub-tasks for reaching, lifting, placing and stacking the red object. For evaluation, we report the final "stack-leave" reward term, which is a sparse reward given when the red object is placed precisely on the blue one, and the arm has been moved away by a predefined distance.The observation provided to the agent consists of the images of three static cameras, as well as proprioception data from the robot and gripper, including joint angles, velocities and torques, as well as a simulated force-torque sensor attached to the wrist. It notably does not contain the tracked position of the objects, so the agent must learn to achieve the goal primarily from vision inputs.In order to solve this task from visual inputs we use a network architecture where both policy and value function use a convolutional neural network with residual blocks (as used in <cit.>) that is then passed through an MLP for multi-task output for each of the subtasks. Both networks use (2, 2, 2) residual blocks with convolutional layers with 16, 32 and 32 channels respectively. The output of the policy network is used to condition a multi-headed Gaussian where each Gaussian output represents the policy output for a sub-task. Similarly the value function outputs a multi-headed categorical for each task . Appendix <ref> describes how this can be used by the SAC-Q algorithm. All experiments are run with 5 seeds for the main results of Section <ref> and 2 seeds for the ablations in Section <ref>.§.§ RL UnpluggedWe consider 3 datasets from the RL Unplugged benchmark <cit.>: Humanoid run, Manipulator insert peg and Manipulator insert ball; which are categorized as `hard' domains by <cit.>. The RL Unplugged datasets were collected on the DeepMind Control Suite <cit.> implemented in the MuJoCo <cit.> simulation framework. Humanoid run consists of 3000 episodes of a 21 dimensional simulated humanoid walker that is rewarded for running forwards while staying upright. The data for this domain is generated using D4PG <cit.>. The Manipulator insert ball and Manipulator insert peg domains consist of 1500 episodes each where a 5-dimensional manipulator is sparsely rewarded for inserting a ball and peg respectively into a specified hole. Data for this domain was generated using V-MPO <cit.> since D4PG was unable to solve the tasks. The data generated on these tasks is then reduced in size via sub-sampling and the number of successful episodes in each dataset is reduced by 2/3 to ensure the data does not contain too many successful trajectories. For all domains each episode consists of 1000 timesteps. We use experimental settings as described in <cit.> for this domain and use their networks for the offline and finetuning experiments. For the other baselines and we use a an MLP architecture with sizes (256, 256, 128) for the policy and (512, 512, 256) for the value function. The policy network output is used to parameterize a Mixture of Gaussian (MoG) with 5 components as in <cit.>. Since the dataset only involves single-step transitions we use 1-step return for all baselines except the Random resets where we found 5-step returns performs significantly better.§ ALGORITHM DETAILSIn this section we describe the algorithms used in the main text in more detail. Policy EvaluationWe use the critic update described in <cit.> for N-step returns:(𝒯_π^NQ)(s_0, a_0) = r(s_0, a_0) + [∑_n=1^N-1γ^n r(s_n, a_n) + γ^NQ(s_n, π(a_n|s_N))|s_0, a_0]where the expectation is with respect to N-step transition dynamics and the distribution Q and 𝒯 represents the distributional Bellman operator. For our experiments we parameterize the critic as a categorical distribution with the number of bins set based on the domain. We set N=5 for our experiments except in the RL unplugged domains where only single-step trajectories exist in the dataset. However, we use N=5 for the random resets baseline in this domain since it can be run online and is not constrained by the limitation in the data. DMPO For our online experiments in `Locomotion Soccer' and `Manipulation Stacking`, we use the Maximum a-posteriori Policy Optimisation algorithm (MPO) <cit.> to adhere to <cit.>. MPO optimizes the RL objective in an E and M-step. The E-step update optimizes: max_q∫_s μ(s) ∫_a q(a|s)Q(s, a)dadss. t. ∫_s μ(s) (q(a|s) || π(a|s))ds < ϵwhere q(a|s) an improved non-parametric policy that is optimized for states μ(s) drawn from the replay buffer. The solution for q is shown to be:q(a|s)∝π(a|s, θ) exp(Q_θ(s, a)/η^*).where Q is computed as an expectation over the categorical distribution. In the M-step, an improved policy is the obtained via supervised learning:π^n+1 = max_π_θ∑_j^M∑_i^N q_ijlogπ_θ(a_i|s_j).s. t. (π^n(a|s_j) || π_θ(a|s_j))SAC-QFor the online experiments in 'Manipulation Stacking', we use the Scheduled Auxiliary Control (SAC-X) algorithm for multi-task exploration <cit.>. SAC-X builds on a multi-headed network architecture to represent both critic and policy, with a single "torso" per network shared across multiple tasks, and a separate output "head" for each task. At any given time, the active head to use is decided by a scheduler process. Here, the scheduler itself is also being learned, using the SAC-Q variant of the algorithm. At fixed intervals during each episode, a new sub-task may be chosen, and the scheduler trains a Q-function that optimizes the sequence of tasks executed such that it maximizes the total return of one task designated as the main goal.To update the policy and Q-function, SAC-Q may use any update rule. Here, we use the MPO algorithm with a distributional critic, as described above. CRR For the offline experiments in Section <ref> we use the Critic Regularized Regression (CRR) algorithm <cit.> which achieves state-of-the-art performance on the RL Unplugged benchmark. CRR uses the same policy evaluation step described above to learn a distributional critic. However the policy in CRR is trained by filtering data via the Q function by optimizing:_π_(s, a)∼ D[f(Q, π, s, a)logπ(a|s)] ,where the states and actions are drawn from the data source D and f is a non-negative, scalar function whose value increases monotonically with Q. We consider both variants of CRR introduced in the paper defined by choices of f:f := 1[ A(s, a) > 0 ], f := exp(A(s, a)/β) ,where A(s, a) is the advantage function which can be computed using:A(s, a) = Q(s, a) - 1/m∑_j=1^m Q(s, a^j);a^j ∼π(.|s).Following the authors of CRR, we refer to CRR with f from Equation <ref> as CRR binary and Equation <ref> as CRR exp. AWAC For the AWAC baseline in the main text we follow the description from <cit.> where learning proceeds in two stages: entirely offline initially and then with online data added to the same replay buffer. The transition between these two stages is defined by a hyper parameter which is set to 25,000 steps in the original work. For our experiments we sweep over an additional value of 50,000 steps and 100,000 steps for the RL Unplugged domain. AWAC also introduces an algorithm which is similar to the CRR exp formulation described above with a slightly different notation that introduces a temperature λ=1/β above. As per the original work we test the method with λ set to 0.3 and 1.§ ADDITIONAL RESULTS §.§ Manipulation results on remaining tasksFigure <ref> compares against the other baselines from Section <ref> on the remaining sub-tasks in the `Manipulation RGB Stacking' setting. While all methods fare well on the easier tasks like `Open' and `Reach Grasp', performs well on the harder `Lift' and `Stack' tasks similar to the trend on `Place' and `Stack Leave' that were presented in the main text. §.§ Finetuning with The `finetuning' baseline considered in the main experiments of Section <ref> is orthogonal to the core idea of and as such, we can apply when finetuning. Figure <ref> compares , finetuning and finetuning with on the `Locomotion Soccer` tasks from state and vision. We observe that finetuning with performs well across both domains matching the high asymptotic performance of and learning faster by taking advantage of the network weights pre-trained offline.§.§ Effect of mixing ratio on full dataset The results presented in Table <ref> show the effect of mixing different ratios of data when using with smaller subsets of data (of 10,000 and 100,000) episodes. Figure <ref> instead presents the same analysis when using the full dataset (of 4e5 episodes) on the `Locomotion Soccer State' task. In this setting we see a clearer advantage of using offline data where a mixture of 50 to 70 percent of offline data works better than using more online data (90 %). In fact, as the figure shows, a mixing ratio of 90 % may even slightly degrade performance when compared to learning purely online. §.§ Effect of changing dynamicsIn this section we analyze the effect of reusing in a regime where the underlying dynamics change. We reuse the data collected in the `Locomotion Soccer (State)' task but transfer to an environment where a mass is randomly attached to the left and right legs of the walker to perturb it's motion. Figure <ref> shows that continues to show an advantage as compared to learning from scratch even when the underlying dynamics of the task have changed. This somewhat surprising finding may be explained by modeling the environment as a partially observed MDP where under some conditions unknown to the agent, the dynamics of walking alter. Reusing data can still guide learning in such a setting showing the robustness of our approach. | http://arxiv.org/abs/2311.15951v2 | {
"authors": [
"Dhruva Tirumala",
"Thomas Lampe",
"Jose Enrique Chen",
"Tuomas Haarnoja",
"Sandy Huang",
"Guy Lever",
"Ben Moran",
"Tim Hertweck",
"Leonard Hasenclever",
"Martin Riedmiller",
"Nicolas Heess",
"Markus Wulfmeier"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.RO"
],
"primary_category": "cs.LG",
"published": "20231127155711",
"title": "Replay across Experiments: A Natural Extension of Off-Policy RL"
} |
[Shadow of novel rotating black holes in GR coupled to nonlinear electrodynamics and constraints from EHT results Muhammad Ali Razae0,add1ab Furkat Sarikulove0b,addr1a,addr4a Javlon Rayimbaeve2,addr1,addr2q,addr2r,addr2a Muhammad Zubaire0a,add1ab Bobomurat Ahmedove3,addr4a,addr1,addr2 Zdeněk Stuchlíke2a,adr0 Received: date / Accepted: date =======================================================================================================================================================================================================] Shadow of novel rotating black holes in GR coupled to nonlinear electrodynamics and constraints from EHT results Muhammad Ali Razae0,add1ab Furkat Sarikulove0b,addr1a,addr4a Javlon Rayimbaeve2,addr1,addr2q,addr2r,addr2a Muhammad Zubaire0a,add1ab Bobomurat Ahmedove3,addr4a,addr1,addr2 Zdeněk Stuchlíke2a,adr0 Received: date / Accepted: date =======================================================================================================================================================================================================It is well known that many open-released foundational diffusion models have difficulty in generating images that substantially depart from average brightness, despite such images being present in the training data. This is due to an inconsistency: while denoising starts from pure Gaussian noise during inference, the training noise schedule retains residual data even in the final timestep distribution, due to difficulties in numerical conditioning in mainstream formulation, leading to unintended bias during inference. To mitigate this issue, certain ϵ-prediction models are combined with an ad-hoc offset-noise methodology.In parallel, some contemporary models have adopted zero-terminal SNR noise schedules together with 𝐯-prediction,which necessitate major alterations to pre-trained models.However, such changes risk destabilizing a large multitude of community-driven applications anchored on these pre-trained models.In light of this, our investigation revisits the fundamental causes, leading to our proposal of an innovative and principled remedy, called One More Step (OMS).By integrating a compact network and incorporating an additional simple yet effective step during inference, OMS elevates image fidelity and harmonizes the dichotomy between training and inference, while preserving original model parameters. Once trained, various pre-trained diffusion models with the same latent domain can share the same OMS module. Codes and models are released at https://jabir-zheng.github.io/OneMoreStep/here.§ INTRODUCTIONDiffusion models have emerged as a foundational method for improving quality, diversity, and resolution of generated images <cit.>, due to the robust generalizability and straightforward training process. At present, a series of open-source diffusion models, exemplified by Stable Diffusion <cit.>, hold significant sway and are frequently cited within the community.Leveraging these open-source models, numerous researchers and artists have either directly adapted <cit.> or employed other techniques <cit.> to fine-tune and craft an array of personalized models. However, recent findings by <cit.> identified deficiencies in existing noise schedules, leading to generated images primarily characterized by medium brightness levels. Even when prompts include explicit color orientations, the generated images tend to gravitate towards a mean brightness.Even when prompts specify “a solid black image” or “a pure white background”, the models will still produce images that are obviously incongruous with the provided descriptions (see examples in <ref>).We deduced that such inconsistencies are caused by a divergence between inference and training stages,due to inadequacies inherent in the dominant noise schedules.In detail, during the inference procedure, the initial noise is drawn from a pure Gaussian distribution.In contrast, during the training phase, previous approaches such as linear <cit.> and cosine <cit.> schedules manifest a non-zero SNR at the concluding timestep.This results in low-frequency components, especially the mean value, of the training dataset remaining residually present in the final latents during training, to which the model learns to adapt. However, when presented with pure Gaussian noise during inference, the model behaves as if these residual components are still present, resulting in the synthesis of suboptimal imagery <cit.>. In addressing the aforementioned issue, <cit.> first proposed a straightforward solution:introducing a specific offset to the noise derived from sampling, thereby altering its mean value.This technique has been designated as offset noise.While this methodology has been employed in some of the more advanced models <cit.>,it is not devoid of inherent challenges.Specifically, the incorporation of this offset disrupts the iid distribution characteristics of the noise across individual units.Consequently, although this modification enables the model to produce images with high luminance or profound darkness, it might inadvertently generate signals incongruent with the distribution of the training dataset.A more detailed study <cit.> suggests a zero terminal SNR method that rescaling the model's schedule to ensure the SNR is zero at the terminal timestep can address this issue.Nonetheless, this strategy necessitates the integration of 𝐯-prediction models <cit.> and mandates subsequent fine-tuning across the entire network, regardless of whether the network is based on 𝐯-prediction or ϵ-prediction <cit.>.Besides, fine-tuning these widely-used pre-trained models would render many community models based on earlier releases incompatible, diminishing the overall cost-to-benefit ratio.To better address this challenge, we revisited the reasons for its emergence:flaws in the schedule result in a mismatch between the marginal distributions of terminal noise during the training and inference stages. Concurrently, we found the distinct nature of this terminal timestep: the latents predicted by the model at the terminal timestep continue to be associated with the data distribution.Based on the above findings, we propose a plug-and-play method, named One More Step, that solves this problem without necessitating alterations to the pre-existing trained models, as shown in <ref>.This is achieved by training an auxiliary text-conditional network tailored to map pure Gaussian noise to the data-adulterated noise assumed by the pre-trained model, optionally under the guidance of an additional prompt, and is introduced prior to the inception of the iterative sampling process. OMS can rectify the disparities in marginal distributions encountered during the training and inference phases.Additionally, it can also be leveraged to adjust the generated images through an additional prompt, due to its unique property and position in the sampling sequence.It is worth noting that our method exhibits versatility, being amenable to any variance-preserving <cit.> diffusion framework, irrespective of the network prediction type, whether ϵ-prediction or 𝐯-prediction, and independent of the SDE or ODE solver employed. Experiments demonstrate that SD1.5, SD2.1, LCM <cit.> and other popular community models can share the same OMS module for improved image generation.§ PRELIMINARIES §.§ Diffusion Model and its Prediction TypesWe consider diffusion models <cit.> specified in discrete time space and variance-preserving (VP) <cit.> formulation.Given the training data ∈ p(), a diffusion model performs the forward process to destroy the data _0 into noise _T according to the pre-defined variance schedule {β_t}_t=1^T according to a perturbation kernel, defined as: q(_1:T| _0) := ∏_t=1^T q(_t | _t-1), q(_t| _t-1) := 𝒩(_t;√(1-β_t)_t-1 , β_t 𝐈).The forward process also has a closed-form equation, which allows directly sampling x_t at any timestep t from x_0: q(_t|_0) :=𝒩 (_t; √(α̅_t)_0, (1-α̅_t)𝐈),where α̅_t = ∏_s=1^t α_s and α_t = 1-β_t. Furthermore, the signal-to-noise ratio (SNR) of the latent variable can be defined as: SNR(t) = α̅_t / (1-α̅_t).The reverse process denoises a sample _T from a standard Gaussian distribution to a data sample _0 following: p_θ (_t-1 | _t) := 𝒩 (_t-1; μ̃_t, σ̃_t^2 𝐈). μ̃_t := √(α̅_t-1)β_t/1-α̅_t_0 + √(α_t)(1-α̅_t-1)/1-α̅_t_tInstead of directly predicting μ̃_t using a network θ, predicting the reparameterised ϵ for _0 leads to a more stable result <cit.>: _0 :=(_t - √(1-α̅_t)ϵ_θ(_t, t) ) / √(α̅_t)and the variance of the reverse process σ̃_t^2 is set to be σ_t^2 = 1-α̅_t-1/1-α̅_tβ_t while _t ∼𝒩(0,1).Additionally, predicting velocity <cit.> is another parameterisation choice for the network to predict: _t := √(α̅_t)ϵ -√(1-α̅_t)_0;which can reparameterise _0 as: _0 :=√(α̅_t)_t - √(1-α̅_t)_θ(_t, t)§.§ Offset Noise and Zero Terminal SNR Offset noise <cit.> is a straightforward method to generate dark or light images more effectively by fine-tuning the model with modified noise.Instead of directly sampling a noise from standard Gaussian Distribution ϵ∼𝒩(0, 𝐈), one can sample the initial noise from ϵ∼𝒩 (0, 𝐈 + 0.1 Σ),where Σ is a covariance matrix of all ones, representing fully correlated dimensions.This implies that the noise bias introduced to pixel values across various channels remains consistent.In the initial configuration, the noise attributed to each pixel is independent, devoid of coherence. By adding a common noise across the entire image (or along channels), changes can be coordinated throughout the image, facilitating enhanced regulation of low-frequency elements.However, this is an unprincipled ad hoc adjustment that inadvertently leads to the noise mean of inputs deviating from representing the mean of the actual image.A different research endeavor proposes a more fundamental approach to mitigate this challenge <cit.>:rescaling the beta schedule ensures that the low-frequency information within the sampled latent space during training is thoroughly destroyed.To elaborate, current beta schedules are crafted with an intent to minimize the SNR at _T.However, constraints related to model intricacies and numerical stability preclude this value from reaching zero. Given a beta schedule used in LDM <cit.>: β_t = ( √(0.00085)T-t/T-1 + √(0.012)t-1/T-1) ^2,the terminal SNR at timestep T=1000 is 0.004682 and √(α̅_T) is 0.068265. To force terminal SNR=0, rescaling can be done to make α̅_T = 0 while keeping α̅_0 fixed.Subsequently, this rescaled beta schedule can be used to fine-tune the model to avoid the information leakage.Concurrently, to circumvent the numerical instability induced by the prevalent ϵ-prediction at zero terminal SNR, this work mandates the substitution of prediction types across all timesteps with 𝐯-prediction. However, such approaches cannot be correctly applied for sampling from pre-trained models that are based on Eq. <ref>. § METHODS§.§ Discrepancy between Training and Sampling From the beta schedule in Eq. <ref>, we find the SNRcannot reach zero at terminal timestep as α̅_T is not zero. Substituting the value of α̅_T in Eq. <ref>, we can observe more intuitively that during the training process, the latents sampled by the model at T deviate significantly from expected values: _T^𝒯 = √(α̅_T^𝒯)_0 + √(1-α̅_T^𝒯),where √(α̅_T^𝒯) = 0.068265 and √(1-α̅_T^𝒯) = 0.997667.During the training phase, the data fed into the model is not entirely pure noise at timestep T.It contains minimal yet data-relevant signals.These inadvertently introduced signals contain low-frequency details, such as the overall mean of each channel.The model is subsequently trained to denoise by respecting the mean in the leaked signals.However, in the inference phase, sampling is executed using standard Gaussian distribution. Due to such an inconsistency in the distribution between training and inference, when given the zero mean of Gaussian noise, the model unsurprisingly produces samples with the mean value presented at T, resulting in the manifestation of images with median values.Mathematically, the directly sampled variable _T^𝒮 in the inference stage adheres to the standard Gaussian distribution 𝒩(0, 𝐈).However, the marginal distribution of the forward process from image space 𝒳 to the latent space _T^𝒯 during training introduces deviations of the low-frequency information, which is non-standard Gaussian distribution.This discrepancy is more intuitive in the visualization of high-dimensional Gaussian space by estimating the radius r <cit.>, which is closely related to the expected distance of a random point from the origin of this space.Theoretically, given a point = (x_1, x_2, …, x_d) sampled within the Gaussian domain spanning a d-dimensional space, the squared length or the norm ofinherently denotes the squared distance from this point to the origin according to: E(x_1^2 + x_2^2 + … + x_d^2 ) = dE(x_1^2) = dσ^2,and the square root of the norm is Gaussian radius r.When this distribution is anchored at the origin with its variance represented by σ, its radius in Gaussian space is determined by: r = σ√(d),the average squared distance of any point randomly selected from the Gaussian distribution to the origin. Subsequently, we evaluated the radius within the high-dimensional space for both the variables present during the training phase r^𝒯 and those during the inference phase r^𝒮, considering various beta schedules, the results are demonstrated in <ref>.Additionally, drawing from <cit.>, we can observe that the concentration mass of the Gaussian sphere resides above the equator having a radius magnitude of 𝒪(r/√(d)), also within an annulus of constant width and radius n.Therefore, we can roughly visualize the distribution of terminal variables during both the training and inference processes in <ref>.It can be observed that a discernible offset emerges between the terminal distribution _T^𝒯 and _T^𝒮 and r^𝒮 > r^𝒯.This intuitively displays the discrepancy between training and inference, which is our primary objective to mitigate.Additional theoretical validations are relegated to the Appendix <ref> for reference. §.§ Prediction at Terminal TimestepAccording to Eq. <ref> & <ref>, we can obtain the sampling process under the text-conditional DDPM pipeline with -prediction at timestep T: _T-1 = 1/√(α_T)( _T - 1-α_T/√(1-α̅_T)_θ) + σ_T ,where , _T ∼𝒩(0,I). In this particular scenario, it is obvious that the ideal SNR(T) = 0 setting (with α_T = 0) will lead to numerical issues, and any predictions made by the network at time T with an SNR(T) = 0 are arduous and lack meaningful interpretation.This also elucidates the necessity for the linear schedule to define its start and end values <cit.> and for the cosine schedule to incorporate an offset s <cit.>.Utilizing SNR-independent 𝐯-prediction can address this issue. By substituting Eq. <ref> into Eq. <ref>, we can derive: _T-1 = √(α_T)_T - √(α̅_T-1)(1-α_T)/√(1-α̅_T)_θ + σ_T ,which the assumption of SNR(T) = 0 can be satisfied: when SNR(T) = 0, the reverse process of calculating _T-1 depends only on the prediction of _θ(_T,T), _T-1 = -√(α̅_T-1)_θ + σ_T ,which can essentially be interpreted as predicting the direction of _0 according to Eq. <ref>:_T-1 = √(α̅_T-1)_0 + σ_T .This is also consistent with the conclusions of angular parameterisation[Details about 𝐯-prediction and angular parametersation can be found in the Appendix. <ref>.] <cit.>. To conclude, under the ideal condition of SNR = 0, the model is essentially forecasting the L2 mean of the data, hence the objective of the 𝐯-prediction at this stage aligns closely with that of the direct _0-prediction. Furthermore, this prediction by the network at this step is independent of the pipeline schedule, implying that the prediction remain consistent irrespective of the variations in noise input.§.§ Adding One More Step Holding the assumption that _T belongs to a standard Gaussian distribution, the model actually has no parameters to be trained with pre-defined beta schedule, so the objective L_T should be the constant: L_T = (q(_T|_0) ‖ p(_T)).In the present architecture, the model conditioned on _T actually does not participate in the training. However, existing models have been trained to predict based on _T^𝒯,which indeed carries some data information. Drawing upon prior discussions, we know that the model's prediction conditioned on _T^𝒮 should be the average of the data, which is also independent of the beta schedule.This understanding brings a new perspective to the problem: retaining the whole pipeline of the current model, encompassing both its parameters and the beta schedule. In contrast, we can reverse _T^𝒮 to_T^𝒯 by introducing One More Step (OMS).In this step, we first train a network ψ(_T^𝒮, 𝒞) to perform 𝐯-prediction conditioned on _T^𝒮∼𝒩(0, 𝐈) with L2 loss ‖_T^𝒮 - _T^𝒮‖^2_2, where _T^𝒮 = -_0 and _T^𝒮 is the prediction from the model. Next, we reconstruct _T^𝒯 based on the output of ψ with different solvers.In addition to the SDE Solver delineated in Eq. <ref>, we can also leverage prevalent ODE Solvers, , DDIM <cit.>: _T^𝒯 =√(α̅_T^𝒯)_0 + √(1-α̅_T^𝒯 - σ_T^2)_T^𝒮 + σ_T ,where _0 is obtained based on ψ(_T^𝒮, 𝒞). Subsequently, _T^𝒯 can be utilized as the initial noise and incorporated into various pre-trained models. From a geometrical viewpoint, we employ a model conditioned on _T^𝒮 to predict _T^𝒯 that aligns more closely with 𝒩( √(α̅_T^𝒯)_0, (1-α̅_T^𝒯) 𝐈), which has a smaller radius and inherits to the training phase of the pre-trained model at timestep T. The whole pipeline and geometric explanation is demonstrated in Figs. <ref> & <ref>, and the detailed algorithm and derivation can be referred to Alg. <ref> in Appendix <ref>. Notably, the prompt 𝒞_ψ in OMS phase ψ(·) can be different from the conditional information 𝒞_θ for the pre-trained diffusion model θ(·).Modifying the prompt in OMS phase allows for additional manipulation of low-frequency aspects of the generated image, such as color and luminance.Besides, OMS module also support classifier free guidance <cit.> to strength the text condition: ψ_cfg (_T^𝒮, 𝒞_ψ, ∅, ω_ψ) = ψ (_T^𝒮, ∅) + ω_ψ( ψ(_T^𝒮, 𝒞_ψ) - ψ (_T^𝒮, ∅) ),where ω_ψ is the CFG weights for OMS. Experimental results for inconsistent prompt and OMS CFG can be found in Sec. <ref>. It is worth noting that OMS can be adapted to any pre-trained model within the same space. Simply put, our OMS module trained in the same VAE latent domain can adapt to any other model that has been trained within the same latent space and data distribution.Details of the OMS and its versatility can be found in Appendix <ref> & <ref>.§ EXPERIMENTS This section begins with an evaluation of the enhancements provided by the proposed module to pre-trained generative models,examining both qualitative and quantitative aspects, and its adaptability to a variety of diffusion models. Subsequently, we conducted ablation studies on pivotal designs and dive into several interesting occurrences. §.§ Implementation DetailsWe trained our module on LAION 2B dataset <cit.>. OMS module architecture follows the widely used UNet <cit.> in diffusion, and we evaluated different configurations, e.g., number of layers.By default we employ OpenCLIP ViT-H to encode text for the OMS module and trained the model for 2,000 steps. For detailed implementation information, please refer to the Appendix. <ref>. §.§ PerformanceQualitative<ref> illustrate that our approach is capable of producing images across a large spectrum of brightness levels.Among these, SD1.5, SD2.1 and LCM <cit.> use the same OMS module, whereas SDXL employs a separately trained OMS module[The VAE latent domain of the SDXL model differs considerably from those of SD1.5, SD2.1 and LCM. For more detailed information, please refer to the Appendix. <ref>].As shown in the <ref> left, existing models invariably yield samples of medium brightness and are not able to generate accurate images when provided with explicit prompts.In contrast, our model generates a distribution of images that is more broadly covered based on the prompts.In addition to further qualifying the result, we also show some integration of the widely popular customized LoRA <cit.> and base models in the community with our module in Appendix. <ref>, which also ascertains the versatility of OMS.Quantitative For the quantitative evaluation, we randomly selected 10k captions from MS COCO <cit.> for zero-shot generation of images.We used Fréchet Inception Distance (FID), CLIP Score <cit.>, Image Reward <cit.>, and PickScore <cit.> to assess the quality, text-image alignment, and human preference of generated images. <ref> presents a comparison of these metrics across various models, either with or without the integration of the OMS module.It is worth noting that <cit.> demonstrated that the FID score for COCO zero-shot image generation has a negative correlation with visual aesthetics, thus the FID metric is not congruent with the goals of our study. Instead, we have further computed the Precision-Recall (PR) <cit.> and Density-Coverage (DC) <cit.> between the ground truth images and those generated, as detailed in the <ref>. Additionally, we calculate the mean of images and the Wasserstein distance <cit.>, and visualize the log-frequency distribution in <ref>. It is evident that our proposed OMS module promotes a more broadly covered distribution.§.§ AblationModule Scale Initially, we conducted some research on the impact of model size.The aim is to explore whether variations in the parameter count of the OMS model would influence the enhancements in image quality.We experimented with OMS networks of three different sizes and discovered that the amelioration of image quality is not sensitive to the number of OMS parameters.From <ref> in Appendix <ref>, we found that even with only 3.7M parameters, the model was still able to successfully improve the distribution of generated images.This result offers us an insight: it is conceivable that during the entire denoising process, certain timesteps encounter relatively trivial challenges, hence the model scale of specific timestep might be minimal and using a Mixture of Experts strategy <cit.> but with different scale models at diverse timesteps may effectively reduce the time required for inference. Text Encoder Another critical component in OMS is the text encoder.Given that the OMS model’s predictions can be interpreted as the mean of the data informed by the prompt, it stands to reason that a more potent text encoder would enhance the conditional information fed into the OMS module.However, experiments show that the improvement brought by different encoders is also limited.We believe that the main reason is that OMS is only effective for low-frequency information in the generation process, and these components are unlikely to affect the explicit representation of the image.The diverse results can be found in <ref> in Appendix <ref>. Modified PromptsIn addition to providing coherent prompts, we also conducted experiments to examine the impact of the low-frequency information during the OMS step with different prompts, mathematically 𝒞_ψ≠𝒞_θ.We discovered that the brightness level of the generated images can be easily controlled with terms like 𝒞_ψ is “dark” or “light” in the OMS phase, as can be seen from <ref>.Additionally, our observations indicate that the modified prompts used in the OMS are capable of influencing other semantic aspects of the generated content, including color variations as shown in <ref>. Classifier-free guidance Classifier-free guidance (CFG) is well-established for enhancing the quality of generated content and is a common practice <cit.>. CFG still can play a key component in OMS, effectively influencing the low-frequency characteristics of the image in response to the given prompts. Due to the unique nature of our OMS target for generation, the average value under ∅ is close to that of conditioned ones 𝒞_ψ. As a result, even minor applications of CFG can lead to considerable changes. Our experiments show that a CFG weight ω_ψ=2 can create distinctly visible alterations.In <ref>, we can observe the performance of generated images under different CFG weights for OMS module. It worth noting that CFG weights of OMS and the pre-trained model are imposed independently. § CONCLUSIONIn summary, our observations indicate a discrepancy in the terminal noise between the training and sampling stages of diffusion models due to the schedules, resulting in a distribution of generated images that is centered around the mean. To address this issue, we introduced One More Step, which adjusts for the training and inference distribution discrepancy by integrating an additional module while preserving the original parameters. Furthermore, we discovered that the initial stages of the denoising process with low SNR largely determine the low-frequency traits of the images, particularly the distribution of brightness, and this phase does not demand an extensive parameter set for accurate model fitting.ieeenat_fullname§ RELATED WORKS Diffusion models <cit.> have significantly advanced the field of text-to-image synthesis <cit.>. These models often operate within the latent space to optimize computational efficiency <cit.> or initially generate low-resolution images that are subsequently enhanced through super-resolution techniques <cit.>. Recent developments in fast sampling methods have notably decreased the diffusion model's generation steps from hundreds to just a few <cit.>. Moreover, incorporating classifier guidance during the sampling phase significantly improves the quality of the results <cit.>. While classifier-free guidance is commonly used <cit.>, exploring other guidance types also presents promising avenues for advancements in this domain <cit.>. § HIGH DIMENSIONAL GAUSSIANIn our section, we delve into the geometric and probabilistic features of high-dimensional Gaussian distributions, which are not as evident in their low-dimensional counterparts. These characteristics are pivotal for the analysis of latent spaces within denoising models, given that each intermediate latent space follows a Gaussian distribution during denoising. Our statement is anchored on the seminal work by <cit.>. These works establish a connection between the high-dimensional Gaussian distribution and the latent variables inherent in the diffusion model. For a unit-radius sphere in high dimensions, as the dimension d increases, the volume of the sphere goes to 0, and the maximum possible distance between two points stays at 2.The surface area A(d) and the volume V(d) of a unit-radius sphere in d-dimensions can be obtained by:A(d) = 2 π^d/2/Γ(d/2), V(d) = π^d/2/d/2Γ(d/2),where Γ(x) represents an extension of the factorial function to accommodate non-integer values of x, the aforementioned Property <ref> and Lemma <ref> constitute universal geometric characteristics pertinent to spheres in higher-dimensional spaces. These principles are not only inherently relevant to the geometry of such spheres but also have significant implications for the study of high-dimensional Gaussians, particularly within the framework of diffusion models during denoising process.The volume of a high-dimensional sphere is essentially all contained in a thin slice at the equator and is simultaneously contained in a narrow annulus at the surface, with essentially no interior volume. Similarly, the surface area is essentially all at the equator. The Property <ref> implies that samples from _T^𝒮 are falling into a narrow annulus. For any c>0, the fraction of the volume of the hemisphere above the plane x_1 = c/√(d-1) is less than 2/c e^-c^2/2. For a d-dimensional spherical Gaussian of variance 1, all but 4/c^2e^-c^2/4 fraction of its mass is within the annulus √(d-1)-c ≤ r ≤√(d-1) + c for any c > 0. Lemmas <ref> & <ref> imply the volume range of the concentration mass above the equator is in the order of O(r/√(d)), also within an annulus of constant width and radius √(d-1). Figs.<ref> & <ref> in main paper illustrates the geometric properties of the ideal sampling space _T^𝒮 compared to the practical sampling spaces _T^𝒯 derived from various schedules, which should share an identical radius ideally. The maximum likelihood spherical Gaussian for a set of samples is the one over center equal to the sample mean and standard deviation equal to the standard deviation of the sample. The above Property <ref> provides the theoretical foundation whereby the mean of squared distances serves as a robust statistical measure for approximating the radius of high-dimensional Gaussian distributions.§ EXPRESSION OF DDIM IN ANGULAR PARAMETERIZATIONThe following covers derivation that was originally presented in <cit.>, with some corrections. We can simplify the DDIM update rule by expressing it in terms of ϕ_t = arctan(σ_t/α_t), rather than in terms of time t or log-SNR λ_t, as we show here.Given our definition of ϕ, and assuming a variance preserving diffusion process, we have α_ϕ = cos(ϕ), σ_ϕ=sin(ϕ), and hence _ϕ = cos(ϕ) + sin(ϕ). We can now define the velocity of _ϕ as_ϕ≡d _ϕ/dϕ = dcos(ϕ)/dϕ + dsin(ϕ)/dϕ =cos(ϕ) - sin(ϕ).Rearranging , ,, we then get:sin(ϕ) = cos(ϕ) - _ϕ= cos(ϕ)/sin(ϕ)( - cos(ϕ)) - _ϕ sin^2(ϕ) = cos(ϕ) - cos^2(ϕ) - sin(ϕ)_ϕ (sin^2(ϕ)+cos^2(ϕ)) == cos(ϕ) - sin(ϕ)_ϕ,and similarly we get ϵ = sin(ϕ) _ϕ + cos(ϕ) _ϕ.Furthermore, we define the predicted velocity as:_θ(_ϕ) ≡cos(ϕ)_θ(_ϕ) - sin(ϕ)_θ(_ϕ),where _θ(_ϕ) = (_ϕ - cos(ϕ)_θ(_ϕ))/sin(ϕ).Rewriting the DDIM update rule in the introduced terms then gives:_ϕ_s = cos(ϕ_s)_θ(_ϕ_t) + sin(ϕ_s)_θ(_ϕ_t)= cos(ϕ_s)(cos(ϕ_t)_ϕ_t - sin(ϕ_t)_θ(_ϕ_t)) + sin(ϕ_s)(sin(ϕ_t) _ϕ_t + cos(ϕ_t)_θ(_ϕ_t))= [cos(ϕ_s)cos(ϕ_t)+sin(ϕ_s)sin(ϕ_t)]_ϕ_t + [sin(ϕ_s)cos(ϕ_t) - cos(ϕ_s)sin(ϕ_t)]_θ(_ϕ_t). Finally, we use the trigonometric identitiescos(ϕ_s)cos(ϕ_t) + sin(ϕ_s)sin(ϕ_t) = cos(ϕ_s - ϕ_t)sin(ϕ_s)cos(ϕ_t) - cos(ϕ_s)sin(ϕ_t)= sin(ϕ_s - ϕ_t),to find that[ The highlighted part corrects minor errors that occurred in Eqs 34 and 35 from <cit.> ]_ϕ_s = cos(ϕ_s - ϕ_t)_ϕ_t + sin(ϕ_s - ϕ_t)_θ(_ϕ_t).or equivalently_ϕ_t-δ = cos(δ)_ϕ_t - sin(δ)_θ(_ϕ_t).Viewed from this perspective, DDIM thus evolves _ϕ_s by moving it on a circle in the (_ϕ_t, _ϕ_t) basis, along the -_ϕ_t direction.When SNR is set to zero, the v-prediction effectively reduces to the _0-prediction. The relationship between _ϕ_t, _t, α_t, σ_t, , is visualized in <ref>.§ MORE EMPIRICAL DETAILS§.§ Detailed Algorithm Due to space limitations, we omitted some implementation details in the main body, but we provided a detailed version of the OMS based on DDIM sampling in Alg. <ref>. This example implementation utilizes -prediction for the OMS and -prediction for the pre-trained model. The derivation related to prediction of _T^𝒯 in Eq. <ref> can be obtained from Eq.12 in <cit.>. Given _t, one can generate _0: _t-1 = √(α̅_t-1)(_t-√(α̅_t)_t/√(α̅_t)) +√(1-α̅_t-1 - σ_t^2)_t +σ_t,where _0^t is parameterised by _t-√(α̅_t)_t/√(α̅_t). In OMS phase, α̅_T^𝒮=0 and α̅_T-1^𝒮 = α̅_T^𝒯. According to Eq. <ref>, the OMS module ψ(·) directly predict the directionof the data, which is equal to -_0^𝒮: _0^𝒮 :=- _ψ(_T^𝒮, 𝒞).Applying these conditions to Eq. <ref> yields the following: _T^𝒯 =√(α̅_T^𝒯)_0^𝒮 + √(1-α̅_T^𝒯 - σ^2)_T^𝒮 + σ§.§ Additional Comments Alternative training targets for OMS As we discussed in <ref>, the objective of -prediction at SNR=0 scenario is exactly the same as negative _0-prediction. Thus we can also train the OMS module under the L2 loss between ‖_0 - _0 ‖_2^2, where the OMS module directly predict _0 = ψ (_T^𝒮, 𝒞). Reasons behind versatility The key point is revealed in Eq. <ref>. The target prediction of OMS module is only focused on the conditional mean value _0, which is only related to the training data. _T^𝒮 is directly sampled from normal distribution, which is independent. Only α̅_T is unique to other pre-defined diffusion pipelines, but it is non-parametric. Therefore, given an _T^𝒮 and an OMS module ψ, we can calculate any _T^𝒯 that aligns with the pre-trained model schedule according to Eq. <ref>. Consistent generation Additionally, our study demonstrates that the OMS can significantly enhance the coherence and continuity between the generated images, which aligns with the discoveries presented in recent research <cit.> to improve the coherence between frames in the video generation process.§.§ Implementation Details Dataset The proposed OMS module and its variants were trained on the LAION 2B dataset <cit.> without employing any specific filtering operation. All the training images are first resized to 512 pixels by the shorter side and then randomly cropped to dimensions of 512 × 512, along with a random flip. Notably, for the model trained on the pretrained SDXL, we utilize a resolution of 1024. Additionally, we conducted experiments on LAION-HR images with an aesthetic score greater than 5.8. However, we observed that the high-quality dataset did not yield any improvement. This suggests that the effectiveness of our model is independent of data quality, as OMS predicts the mean of training data conditioned on the prompt. OMS scale variants We experiment with OMS modules at three different scales, and the detailed settings for each variants are shown in Table <ref>.Combining these with three different text encoders results in a total of nine OMS modules with different parameters. As demonstrated in Table <ref>, we found that OMS is not sensitive to the number of parameters and the choice of text encoder used to extract text embeddings for the OMS network.Hyper-parameters In our experiments, we employed the AdamW optimizer with β_1 = 0.9, β_2 = 0.999, and a weight decay of 0.01. The batch size and learning rate are adjusted based on the model scale, text encoder, and pre-trained model, as detailed in Tab. <ref>. Notably, our observations indicate that our model consistently converges within a relatively low number of iterations, typically around 2,000 iterations being sufficient. Hardware and speed All our models were trained using eight 80G A800 units, and the training speeds are provided in Tab. <ref>. It is evident that our model was trained with high efficiency, with OMS-S using CLIP ViT-L requiring only about an hour for training. §.§ OMS Versatility and VAE Latents DomainThe output of the OMS model is related to the training data of the diffusion phase. If the diffusion model is trained in the image domain, then our image domain-based OMS can be widely applied to these pre-trained models. However, the more popular LDM model has a VAE as the first stage that compresses the pixel domain into a latent space. For different LDM models, their latent spaces are not identical. In such cases, the training data for OMS is actually the latent compressed by the VAE Encoder. Therefore, our OMS model is versatile for pre-trained LDM models within the same VAE latent domain, , SD1.5, SD2.1 and LCM. Our analysis reveals that the VAEs in SD1.5, SD2.1, and LCM exhibit a parameter discrepancy of less than 1e-4 and are capable of accurately restoring images. Therefore, we consider that these three are trained diffusion models in the same latent domain and can share the same OMS module. However, for SDXL, our experiments found significant deviations in the reconstruction process, especially in more extreme cases as shown in <ref>. Therefore, the OMS module for SDXL needs to be trained separately. But it can still be compatible with other models in the community based on SDXL.If we forcibly use the OMS trained with the VAE of the SD1.5 series on the base model of SDXL, severe color distortion will occur whether we employ latents with unit variance. We demonstrate some practical distortion case with the rescaled unit variance space in <ref>. The observed color shift aligns with the effect shown in <ref>, , Black → Red. § MORE EXPERIMENTAL RESULTS§.§ LoRA and Community Models In this experiment, we selected a popular community model GhostMix 2.0 BakedVAE [GhostMix can be found at ] and a LoRA MoXin 1.0 [MoXin can be found at ]. In <ref> & <ref>, we see that the OMS module can be applied to many scenarios with obvious effects. LoRA scale is set as 0.75 in the experiments. We encourage readers to adopt our method in a variety of well-established open-source models to enhance the light and shadow effects in generated images.We also do some experiment on LCM-LoRA <cit.> with SDXL for fast inference. The OMS module is the same as we used for SDXL.§.§ Additional Results Here we demonstrate more examples based on SD1.5 <ref>, SD2.1 <ref> and LCM <ref> with OMS. In each subfigure, top row are the images directly sampled from raw pre-trained model, while bottom row are the results with OMS. In this experiment, all three pre-trained base model share the same OMS module. § LIMITATIONS We believe that the OMS module can be integrated into the student model through distillation, thereby reducing the cost of the additional step. Similarly, in the process of training from scratch or fine-tuning, we can also incorporate the OMS module into the backbone model, only needing to assign a pseudo-t condition to the OMS.However, doing so would lead to changes in the pre-trained model parameters, and thus is not included in the scope of discussion of this work. | http://arxiv.org/abs/2311.15744v1 | {
"authors": [
"Minghui Hu",
"Jianbin Zheng",
"Chuanxia Zheng",
"Chaoyue Wang",
"Dacheng Tao",
"Tat-Jen Cham"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127120242",
"title": "One More Step: A Versatile Plug-and-Play Module for Rectifying Diffusion Schedule Flaws and Enhancing Low-Frequency Controls"
} |
[][email protected] Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Co-design Center for Quantum Advantage, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA We employ quantum circuit learning to simulate quantum field theories (QFTs).Typically, when simulating QFTs with quantum computers, we encounter significant challenges due to the technical limitations of quantum devices when implementing the Hamiltonian using Pauli spin matrices. To address this challenge, we leverage quantum circuit learning, employing a compact configuration of qubits and low-depth quantum circuits to predict real-time dynamics in quantum field theories. The key advantage of this approach is that a single-qubit measurement can accurately forecast various physical parameters, including fully-connected operators. To demonstrate the effectiveness of our method, we use it to predict quench dynamics, chiral dynamics and jet production in a 1+1-dimensional model of quantum electrodynamics. We find that our predictions closely align with the results of rigorous classical calculations, exhibiting a high degree of accuracy. This hybrid quantum-classical approach illustrates the feasibility of efficiently simulating large-scale QFTs on cutting-edge quantum devices. Quantum-classical simulation of quantum field theory by quantum circuit learning Kazuki Ikeda January 14, 2024 ================================================================================Introduction.— The exploration of quantum field theories (QFTs) has long been a cornerstone of theoretical physics, enabling us to delve into the fundamental interactions that govern the behavior of matter and energy at the smallest scales. While the mathematical framework of QFTs has proven to be exceptionally powerful in describing these phenomena, simulating them on classical computers has often posed formidable challenges, limiting our ability to explore and understand the intricate dynamics of quantum systems.In recent years, the advent of quantum computing has promised to revolutionize our approach to simulating QFTs <cit.>. However, the technical limitations of quantum devices, particularly when implementing the Hamiltonian using Pauli spin matrices, have hindered progress in this field. This paper introduces a novel approach that harnesses the potential of quantum circuit learning (QCL) <cit.> to address these challenges and offer a more efficient and accurate method for simulating QFTs. Our method leverages a compact qubit configuration and low-depth quantum circuits, allowing us to predict real-time dynamics in QFTs that requires a large number of qubits and high-depth of quantum circuits. One of the standout advantages of this approach is its capacity to make accurate predictions for various physical parameters using just a single-qubit measurement. To demonstrate its effectiveness, we apply this technique to predict quench dynamics, chiral dynamics, and jet production in a 1+1-dimensional model of quantum electrodynamics developed in <cit.>, where simulations of chiral magnetic effect and the quark-antiquark production in e^+e^- annihilation proposed were performed based on Quantum Chromodynamics (QCD) <cit.>. Remarkably, our predictions closely align with results obtained through rigorous classical calculations, underscoring the high degree of accuracy that can be achieved.This hybrid quantum-classical approach not only represents a significant advancement in the simulation of QFTs but also underscores the feasibility of efficiently simulating large-scale quantum field theories using cutting-edge quantum devices. By combining the strengths of quantum computing with classical techniques, our research opens up new horizons in the study of quantum systems and provides a promising avenue for the exploration of complex physical phenomena. Our approach will also leverage the use of machine learning applications in the field of condensed matter, nuclear physics, high energy physics, chemistry, biology and information science, enhancing our ability to analyze and interpret data <cit.>.The main contributions of our work to quantum simulations of QFTs are summarized as follows: * Using a three or five-qubit QCL, we were able to efficiently predict the real-time dynamics of 1+1d QED (up to 18 qubits), including quench, chiral dynamics, and jet generation. * The real-time evolution of several physical observables of total coupling, including energy and electric field, was predicted with good accuracy by reading out only one qubit.Our work will present a benchmark showing that the complex dynamics of a generic large quantum many-body system can be efficiently predicted by QCL with a small number of qubits and low-depth of circuits. Simulating QFT by QCL.— In quantum circuit learning, we follow a step-by-step process to train a quantum model. Here's a breakdown of the key components. We begin by preparing a set of training data, denoted as { (x^(i), y^(i)) }. Here, x^(i) represents the input data (teacher data), and y^(i) is the correct output data we expect the model to predict.Next, we create a quantum circuit, denoted as U(θ), which is determined by some rule or parameterized by certain parameters θ that depend on the input x^(i). This circuit is used to encode information from the input data into a quantum state. We use the quantum circuit U(θ) to prepare an input quantum state, denoted as |ψ(x^(i))⟩, which carries the information embedded from the input data x^(i). A multiply gate, denoted as M(θ), which depends on the parameter θ, is applied to the input state |ψ(x^(i))⟩ to obtain the output state |ϕ(x^(i))⟩. The measurement step involves measuring some observable under the output state |ϕ(x^(i))⟩. For example, we might measure the expectation value of the first qubit, denoted as ⟨Ô_1 ⟩. We define a function F, which can be a sigmoid function, softmax function, or a constant function, etc. The output of the quantum model, denoted as y_model^(i), is computed as F(⟨Ô_1 ⟩). To assess the performance of our model, we calculate the cost function, denoted as J(θ), representing the divergence between the correct data y^(i) and the output of the model y_model^(i). This helps us quantify the error in our predictions. To improve the model's performance, we optimize the parameters θ to minimize the cost function J(θ). This is typically done using optimization algorithms like gradient descent or other suitable methods. Once the optimization process converges, we obtain the quantum circuit with optimized parameters θ, denoted as U(θ_opt). This trained quantum circuit serves as our desired prediction model for quantum data.As a benchmark model of QFTs, we work on the Schwinger model (1+1d QED) <cit.>, whose action isS = ∫ d^2x[-1/4 F^μν F_μν + gθ/4πϵ^μνF_μν + ψ̅(iD-m)ψ],with D=γ^μ(p_μ-i gA_μ). Here, A_μ is the U(1) gauge potential, E=Ȧ_1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass and γ^μ are two-dimensional γ-matrices. The Hamiltonian in temporal gauge A_0=0 isH=∫ dz [ E^2/2 -ψ̅(iγ^1∂_1 - gγ^1A_1 - me^iγ_5θ)ψ],where the space-time coordinate is labeled by x^μ=(t,z). The Hamiltonian in the qubit representation is H= 1/4a∑_n=1^N-1(X_nX_n+1+Y_nY_n+1)+m/2∑_n=1^N(-1)^nZ_n+ag^2/2∑_n=1^N-1L^2_n,where L_n is the electric field operator L_n=∑_k=1^nZ_k+(-1)^k/2.When simulating kinetic terms and electromagnetic fields in quantum circuits, an enormous amount of control gates are used. It is important to note that ∑_n=1^N-1L^2_n contains the fully connected term, which makes computation noisy and heavy. Naively simulating a N-qubit system requires N qubits, however, one can reduce the number of qubits and even gate depth by using QCL as we will demonstrate below. See Appendix for a detailed description of the model and operator definitions. As shown in Fig. <ref> (upper), the Hamiltonian is sparse, although the model is fully connected. Interestingly, this model becomes more sparse as the system size increases. The ratio #{H_ij:H_ij≠0}/#{H_ij} of non-zero matrix components out of all matrix components (#{H_ij}=2^N× 2^N) is shown in Fig <ref> (lower left), which decays rapidly as N increases. The reason for this is that the kinetic term that composes the off-diagonal components of the Hamiltonian is the nearest-neighbor interaction. Therefore most of the off-diagonal terms are zero. In fact, the number of non-zero off-diagonal components grows linear in the dimension of the Hilbert space (2^N), as shown in Fig. <ref> (lower right). Benchmark Results.— As a simple demonstration, we study the real-time dynamics|ψ(t)⟩=𝒯e^-i∫_0^tdt'H|ψ_0⟩,where |ψ_0⟩ is the Néel state |1010⋯10⟩, which is the ground state of the Hamiltonian (<ref>) at the large mass limit. For a given observable 0, our training data set is a set of measurement results O(t)≡⟨ψ(t)|O|ψ(t)⟩ at randomly chosen times t and our objective is to learn the target Hamiltonian by comparing O(t). The time-evolution of the chiral condensate density 1/N⟨ψ̅ψ(t)⟩ is shown in Fig. <ref> (left). This data was obtained by the exact classical method for the N=18 system. Some data {x_i} were sampled and used for QCL teaching data. The data was encoded into an initial quantum state |φ(x_i)⟩ of QCL, where the state was updated by applying a unitary operator U(θ) to |φ(x_i)⟩. Then the parameter θ is updated by optimizing the cost function. It should be emphasized that a three-qubit circuit was capable of predicting the dynamics in the N=18 system. The initial prediction of the chiral condensate is shown by the dashed line in Fig. <ref> (left) and the final prediction date is shown in the solid line. Moreover we also performed the same task for predicting the real-time dyamanimcs of the electric field. The teacher data, initial prediction result and final prediction result are shown in Fig. <ref> (left). Note that the electric field is a fully connected operator, therefore a precise measurement of the electric field operator is expensive in general, due to significant noise. Accurate measurements of electric field operators by ordinary quantum simulations are susceptible to noise. In contrast, QCL requires only one quantum operator to be measured and can be performed by small quantum circuits, giving it a practical advantage over the conventional quantum simulation of QFT.Predicting the chiral dynamics.— To perform a more complex task, we perform prediction of the real-time chiral dynamics. Again, we work on the massive Schwinger model with a finite chiral potential μ_5 <cit.>. We prepare the initial state as the ground state of the following Hamiltonian H=∫ dz [ E^2/2 -ψ̅(iγ^1∂_1 - gγ^1A_1-γ^1θ̇/2 - me^iγ_5θ)ψ]where θ=-2μ_5t_0. The difference from the previous model (<ref>) is the presence of the term μ_5ψ̅γ_1ψ(=μ_5Q_5), which induces the chiral imbalance in the initial state. This model is useful for discussing macroscopic quantum phenomena caused by the chiral anomaly, namely chiral magnetic effects and chiral magnetic waves. We evolve the state by the Hamiltonian H(μ_5=0,θ=0), so the real-time evolution of the initial state is given by the time-ordered integral |ψ(t)⟩=𝒯[e^-i∫_0^tdt'H(μ_5=0,θ=0)]|ψ(0)⟩.The real-time evolution of the axial charge density (equivalently the vector current density) 1/N⟨ Q_5(t)⟩ is shown in Fig. <ref>. The three plots are labels by μ_5=0.5,1,2, respectively. As before, the teacher data were obtained from rigorous simulations with N=18 systems, and prediction was performed with a five-qubit circuit. The use of large masses induces nonlinear rapid oscillations of the axial charge <cit.>. Jet dynamics.— Here, using the massive Schwinger model coupled to external sources, we predict the quantum simulation of jet production using QCL. This is an extremely non-trivial task because of non-perturbative effects.We use the massive Schwinger model Hamiltonian (<ref>) in the presence of an external current ^μ describing the produced jets <cit.>:H= H_0 + H_1 ,H_0=∫ dx[ 1/2E^2+ψ̅(-iγ^1∂_1+gγ^1A_1+m)ψ ], H_1= ∫ dx ^1 A_1. The effect on the theory of the interaction with the external source H_1 is to modify Gauss law to∂_1 E - j^0 = ^0.with j^0 = g ψ̅γ^0ψ.To simulate the creation of a pair of jets in the context of e^+e^- annihilation, we opt for an external current that represents charges of opposite polarity moving apart along the light cone. This external current can be defined as follows: ^0(x,t)= g[δ(Δ x - Δ t) - δ(Δ x + Δ t)]Θ(Δ t),^1(x,t)= g[δ(Δ x - Δ t) + δ(Δ x + Δ t)]Θ(Δ t),where (t_0, x_0) represents the time and position of the point where the jet pair is generated, while Δ x ≡ x - x_0 and Δ t ≡ t - t_0 denote the spatial and temporal separation from this location. Θ is the Heaviside step function. The electric field is time-dependent and can be rewritten as L_n=n + n and the Gauss law (<ref>) is solved as follows:n = ∑_i=1^n Q_i ,n(t)= -Θ(t-a|n-N/2+1/2|)).The time-dependent term of the electric field is shown in Fig. <ref>, which induces the propagation of the chiral condensate on the light cone. The non-locality is contained in the dynamical gauge field and the external sources create a chain of electric fluxes between them.The Hamiltonian is H(t)= 1/4a∑_n=1^N-1(X_n X_n+1+Y_n Y_n+1)+m/2∑_n=1^N (-1)^n Z_n +a g^2/2∑_n=1^N-1 (n+n(t))^2. Our simulations proceed as follows. We start by finding the eigenstate |Ψ_0⟩ of the usual massive Schwinger model H(t=0). We used the ground state, the 1st excited state or the 2nd excited state for |ψ_0⟩. We then compute the state |ψ_t⟩ = 𝒯 e^-i ∫_0^t H(t') dt'|ψ_0⟩ corresponding to the evolution under the time-dependent Hamiltonian H(t). The system is effectively “quenched" at t/a=t_0/a=1, when the external sources are introduced. To reproduce the results in <cit.>, the data of a=0.1,N=18,g=0.5/a,m=0.25/a obtained by the exact classical method were used as teaching data and predictions were made with a five-qubit QCL. The results are shown in Fig. <ref>, where the left panel shows the time-evolution of the chiral condensate density ⟨ψ̅ψ(t)⟩ whose initial states are ground state, the 1st excited state and 2nd excited state, and the right panel shows the energy expectation values 1/N⟨ H(t)⟩. For both chiral condensation and energy prediction, only one qubit of measurement is required in QCL.Conclusion.— In this research, quantum circuit learning is utilized to simulate quantum field theories (QFTs). Traditional quantum computer simulations of QFTs face challenges due to technical limitations, especially when implementing the Hamiltonian with Pauli spin matrices. To overcome these challenges, the study employs a small qubit configuration and low-depth quantum circuits to predict real-time dynamics in QFTs. One notable advantage is the ability to use a single-qubit measurement to accurately predict various physical parameters. The method is demonstrated to be effective by predicting quench dynamics, chiral dynamics, and jet production in a 1+1-dimensional model of quantum electrodynamics, closely matching results from classical calculations. This approach showcases the potential for efficiently simulating large-scale QFTs using near-term quantum devices, combining quantum and classical techniques.§ ACKNOWLEDGMENTThe author thanks Dmitri Kharzeev,C.R. Ramakrishnan, Shuzhe Shi and Ranjani Sundaram for useful discussion. This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704.§ THE LATTICE HAMILTONIAN OF THE MASSIVE SCHWINGER MODEL The Lagrangian density of the Schwinger model <cit.> is ℒ = -1/4F_μνF^μν+ψ̅(iγ^μ∂_μ-gγ^μ A_μ-m)ψ.Here, we represent spacetime coordinates as x^μ=(t,z) and employ the following Dirac matrix notation: γ^0 = Z, γ^1 = i,Y, and γ^5=γ^0 γ^1 = X. In the context of (1+1) dimensions, we establish the relationship between the axial charge density Q_5(x)≡ψ̅γ^5γ^0ψ(x) and the vector current density J(x)≡ψ̅γ^1ψ(x) as Q_5(x) = -J(x). Similarly, the vector charge density Q(x)≡ψ̅γ^0ψ(x) and the axial current density J_5(x)≡ψ̅γ^5γ^1ψ(x) are linked by Q(x) = J_5(x).To discretize our Hamiltonian, we use staggered fermions <cit.>ψ_1(x)→χ_2n/√(a), ψ_2(x)→χ_2n+1/√(a),where a represents the finite lattice spacing. The lattice Hamiltonian corresponding to eq. (<ref>) is expressed asH= -i/2a∑_n=1^N-1[U_n^†χ^†_nχ_n+1-U_nχ^†_n+1χ_n] +ag^2/2∑_n=1^N-1L_n^2+ m∑_n=1^N (-1)^n χ^†_nχ_n,U_n denotes the gauge link operator, and L_n is the electric field operator that satisfies Gauss' law constraint described by the following equation:L_n-L_n-1 =χ_n^†χ_n-1-(-1)^n/2.The link operator is written as U_n = e^-iagA_1(an), using a lattice vector potential ϕ_n=ag,A_1(an). For the purpose of quantum simulation, we transform the lattice Hamiltonian into the spin representation through the Jordan–Wigner transformation <cit.>:χ_n=X_n-iY_n/2∏_i=1^n-1(-i Z_i).This transformation results in the Hamiltonian of the model becoming:H= 1/8a∑_n=1^N[(U_n+U^†_n)⊗(X_n X_n+1+Y_n Y_n+1)+i(U_n-U^†_n)⊗(X_nY_n+1-Y_nX_n+1)]+m/2∑_n=1^N(-1)^n Z_n+a g^2/2∑_n=1^N L^2_n.We can eliminate the gauge link U_n through a gauge transformation <cit.>.The local vector and axial charge densities are represented as follows:Q_n ≡ψ̅γ^0ψ = Z_n+(-1)^n/2a,Q_5,n≡ψ̅γ^5γ^0ψ = X_nY_n+1-Y_nX_n+1/4a .We define the total charge operator Q ≡ a∑_n=1^N Q_n, which commutes with the Hamiltonian. Assuming the boundary condition L_0=0, the Gauss' law constraint (<ref>) leads to the solution:L_n = a∑_j=1^n Q_j,. | http://arxiv.org/abs/2311.16297v1 | {
"authors": [
"Kazuki Ikeda"
],
"categories": [
"hep-th",
"cs.LG",
"hep-ph",
"nucl-th",
"quant-ph"
],
"primary_category": "hep-th",
"published": "20231127201839",
"title": "Quantum-classical simulation of quantum field theory by quantum circuit learning"
} |
Energy Dissipation of Fast Electrons in Polymethylmetacrylate (PMMA): Towards a Universal Curve for Electron Beam Attenuation in Solidsfor Energies between ∼0 eV and 100 keV Olga Ridzel January 14, 2024 =============================================================================================================================================================================== Owe to the powerful generative priors, the pre-trained text-to-image (T2I) diffusion models have become increasingly popular in solving the real-world image super-resolution problem. However, as a consequence of the heavy quality degradation of input low-resolution (LR) images, the destruction of local structures can lead to ambiguous image semantics. As a result, the content of reproduced high-resolution image may have semantic errors, deteriorating the super-resolution performance. To address this issue, we present a semantics-aware approach to better preserve the semantic fidelity of generative real-world image super-resolution. First, we train a degradation-aware prompt extractor, which can generate accurate soft and hard semantic prompts even under strong degradation. The hard semantic prompts refer to the image tags, aiming to enhance the local perception ability of the T2I model, while the soft semantic prompts compensate for the hard ones to provide additional representation information. These semantic prompts can encourage the T2I model to generate detailed and semantically accurate results. Furthermore, during the inference process, we integrate the LR images into the initial sampling noise to mitigate the diffusion model's tendency to generate excessive random details. The experiments show that our method can reproduce more realistic image details and hold better the semantics. Code: https://github.com/cswry/SeeSRhttps://github.com/cswry/SeeSR § INTRODUCTION Images inevitably undergo degradation due to factors such as subpar imaging devices, unfavorable capturing environments, transmission losses. This degradation manifests in various forms, including low-resolution, blurriness and noise. Image super-resolution (ISR) aims to reconstruct a high-resolution (HR) image from the given low-resolution (LR) input. Traditionally, researchers investigate the ISR problem by assuming simple and known image degradations (, bicubic downsampling), and developed many successful models <cit.>. However, these methods often yield over-smoothed outcomes due to their fidelity-focused learning objectives. To enhance visual perception, generative adversarial networks (GANs) <cit.> have been adopted to solve the ISR problem <cit.>. By using the adversarial loss in training, the ISR models can be supervised to generate perceptually realistic details, enhancing the visual quality but in the price of sacrificing fidelity.Despite the remarkable advancements, when applying the above mentioned models to real-world LR images, whose degradations are much more complex and even unknown, the output HR images can have low visual quality with many artifacts. This is mainly because the domain gap between the synthetic training data and the real-world test data. The goal of real-world ISR (Real-ISR) is to reproduce a perceptually realistic HR image from its LR observation with complex and unknown degradation. To this end, some researchers proposed to collect real-world LR-HR image pairs using long-short camera focal lens <cit.>. Another more cost-effective way is to simulate the complex real-world image degradation process using random combinations of basic degradation operations. The representative work along this line include BSRGAN <cit.>, Real-ESRGAN <cit.> and their variants <cit.>. With the abundant amount of more realistic synthetic training pairs, the GAN-based Real-ISR methods can generate more authentic details. However, they still tend to introduce many unpleasant visual artifacts due to the unstable adversarial training. The LDL <cit.> can suppress much the visual artifacts by detecting the problematic pixels using local image statistics. Unfortunately, it is not able to generate additional details.Recently, denoising diffusion probabilistic models (DDPMs) <cit.> have exhibited remarkable performance in the realm of image generation, gradually emerging as successors to GANs in various downstream tasks <cit.>. Some researchers <cit.> have leveraged pretrained DDPMs to effectively tackle the inverse image restoration problems. However, their application to the challenging Real-ISR scenarios is hindered by the assumptions of known linear degradation model. Considering that the large-scale pretrained text-to-image (T2I) models <cit.>, which are trained on a dataset exceeding 5 billion image-text pairs, encompass more potent natural image priors, some methods have recently emerged to harness their potentials to address the Real-ISR problem, including StableSR <cit.>, PASD <cit.> and DiffBIR <cit.>.These diffusion prior based Real-ISR methods have demonstrated highly promising capability to generate realistic image details; however, they still have some limitations. StableSR <cit.> and DiffBIR <cit.> solely rely on input LR images as control signals, overlooking the role of semantic text information in the pretrained T2I models. PASD <cit.> attempts to utilize off-the-shelf high-level models to extract semantic prompts as additional control conditions for the T2I model. However, it encounters difficulties when dealing with scenes containing a variety of objects or severely degraded images. In this work, we investigate in-depth the problem that how to extract more effective semantic prompts to harness the generative potential of pretrained T2I models so that better Real-ISR results can be obtained. By analyzing the effects of different types of semantic prompts on the Real-ISR outcomes, we conclude two major criteria. Firstly, the prompt shouldcover as many objects in the scene as possible, helping the T2I model to understand different local regions of the LR image. Secondly, the prompt should be degradation-aware to avoid erroneous semantic restoration results. (Please refer to Section 3.1 for more discussions.) While the prompt extractor undergoes low-level data augmentation during training <cit.>, there still exists much gap between this augmentation and real-world degradation. Hence, it is not suitable for directly extracting semantic prompts from real-world LR inputs.Based on the aforementioned criteria, we present a Semantic-aware SR (SeeSR) approach, which utilizes high-quality semantic prompts to enhance the generative capacity of pretrained T2I models for Real-ISR. SeeSR consists of two stages. In the first stage, the semantic prompt extractor is fine-tuned to acquire degradation-aware capabilities. This enables it toextract accurate semantic information from LR images as soft and hard prompts. In the second stage, the pristine semantic prompts collaborate with LR images to exert precise control over the T2I model, facilitating the generation of rich and semantically correct details. Moreover, during inference stage, we incorporate the LR image into the initial sampling noise to alleviate the diffusion model's propensity for generating excessive random details.Our extensive experiments demonstrate the superior realistic detail generation performance of SeeSR while preserving well the image semantics of Real-ISR outputs. § RELATED WORK GAN-based Real-ISR. Starting from SRCNN <cit.>, deep learning based ISR has become prevalent. A variety of methods focusing on model design have been proposed <cit.> to improve the accuracy of ISR reconstruction. However, most of these methods assume simple and known degradations such as bicubic downsampling, limiting their effectiveness when dealing with complex and unknown degradations in the real world. Recent advancements in Real-ISR have explored more complex degradation models to approximate the real-world degradations. Specifically, BSRGAN <cit.> introduces a randomly shuffled degradation modeling strategy, while Real-ESRGAN <cit.> employs a high-order degradation modeling process. Using the training samples with more realistic degradations, both BSRGAN and Real-ESRGAN utilize GANs <cit.> to reconstruct desired HR images. While generating more perceptually realistic details, training GANs is unstable and Real-ISR outputs often suffer from unnatural visual artifacts. Many following works such as LDL <cit.> and DeSRA <cit.> can suppress much the artifacts, yet they are difficult to generate more natural details. Diffusion Probabilistic Models. Inspired by the non-equilibrium thermodynamics theory <cit.> and sequential MonteCarlo <cit.>, Sohl-Dickstein <cit.> proposed the diffusion model to model complex datasets.Subsequently, a series of fruitful endeavors <cit.> have been made to apply diffusion models in the realm of image generation, especially since the development of DDPM <cit.>. Rombach <cit.> expanded the training of DDPMs to the latent space, greatly facilitating the development of large-scale pretrained text-to-image (T2I) diffusion models such as stable diffusion (SD) <cit.> and Imagen <cit.>. It has been demonstrated that T2I diffusion priors are powerful in image editing <cit.>, video generation <cit.>, 3D content generation <cit.>, .Diffusion Prior based Real-ISR. Early attempts <cit.> using DDPMs to address the ISR problem are mostly assuming simple downsampling degradation. However, such an assumption of known linear image degradation restricts their practical application in complex scenarios like Real-ISR. Recently, some researchers <cit.>have employed powerful pretrained T2I models such as SD <cit.> to tackle the real-ISR problem. Having been trained on billions of image-text pairs, these models can perceive strong image priors for tackling Real-ISR challenges. StableSR <cit.> achieves this goal by training a time-aware encoder to fine-tune the SD model and employing feature warping to balance between fidelity and perceptual quality. DiffBIR <cit.> adopts a two-stage strategy to tackle the Real-ISR problem. It first reconstructs the image as an initial estimation, and then utilizes the SD prior to enhance image details. The aforementioned methods solely rely on images as conditions to activate the generative capability of the T2I model. In contrast, PASD <cit.> goes further by utilizing off-the-shelf high-level models (, ResNet <cit.>, Yolo <cit.> and BLIP <cit.>) to extract semantic information to guide the diffusion process, stimulating more generative capacity of the T2I model. However, ResNet and Yolo have limited object recognition ability, leading to a diminished recall rate. The captions generated by BLIP struggle to comprehensively describe the semantic information in images, particularly in scenes with a rich diversity of objects. Therefore, how to introduce prompts to moreeffectively elicit the potential of pretrained T2I models in assisting Real-ISR needs deep investigation, which is the goal of this work. § METHODOLOGY §.§ Motivation and Framework OverviewMotivation. To unleash the generative potential of pretrained T2I model while avoiding semantic distortion in Real-ISR outputs, we investigate the use of three representative styles of semantic prompts, including classification-style, caption-style and tag-style. In specific, we use the methods in <cit.> , <cit.> and <cit.> to extract classification-style, caption-style and tag-style prompts, respectively.The classification-style prompt provides only one category label for the entire image, which is robust to image degradation due to its global view. However, such kind of prompts lack the ability to provide semantic support of local objects, particularly in scenes containing multiple entities. As shown in Figs. <ref>(b) and <ref>(f), by using the classification-style prompts extracted from the LR and the HR images, the Real-ISR results are almost indistinguishable from that obtained by using the null prompt (see Fig. <ref>(e)). The caption-style prompt provides a sentence to describe the corresponding image, offering richer information compared to the classification-style prompt. However, it still has two shortcomings. Firstly, the redundant prepositions and adverbs in this type of prompt may scatter the attention of T2I models towards degraded objects <cit.>. Secondly, it is prone to semantic errors due to the influence of degradation in LR images. As shown in Fig. <ref>(c), the T2I model mistakenly reconstructs a bird instead of a ship due to the incorrect caption extracted from the LR image. The tag-style prompt provides category information for all objects in the image, offering a more detailed description of the entities compared to caption-style prompt. Even without providing object location information, it is found that the T2I model can align the semantic prompts with the corresponding regions in the image due to its underlying semantic segmentation capability <cit.>. Unfortunately, similar to the captioning models, the tagging models are also susceptible to image degradations, resulting in erroneous semantic cues and semantic distortion in the reconstructed results. As shown in Fig. <ref>(d), the wrong semantic prompt “airplane" leads to distorted reconstruction of the ship. We summarize the characteristics of different styles of prompts in Table <ref>. This motivates us that if we can adapt the tag-style prompt to be degradation-ware, then it may help the T2I models generate high-quality Real-ISR outputs while preserving correct image semantics. Framework Overview. Based on the above discussions, we propose to extract high-quality tag-style prompts from the LR image to guide the pretrained T2I model, such as stable diffusion (SD) <cit.>, for producing semantics-preserved Real-ISR results. The framework of our proposed method, namely Semantic-aware SR (SeeSR), is shown in Fig. <ref>. The training of SeeSR goes through two stages. In the first stage (Fig. <ref>(a)), we learn adegradation-aware prompt extractor (DAPE), which consists of an image encoder and a tagging head. It is expected that both the feature representations and tagging outputs of the LR image can be as close as possible to that of the corresponding HR image by using the original tag model. The learned DAPE is copied to the second stage (Fig. <ref>(b)) to extract the feature representations and tags (as text prompts) from the input LR image, which serve as control signals over the pretrained T2I model to generate visually pleasing and semantically correct Real-ISR results. During inference, only the second stage is needed to process the input image. Fig. <ref>(c) illustrates the collaborative interplay between the image branch, feature representation branch, and text prompt branch in governing the pretrained T2I model.§.§ Degradation-Aware Prompt Extractor The DAPE is fine-tuned from a pretrained tag model, , RAM <cit.>. As depicted in Fig. <ref>(a), the HR image x goes through a frozen tag model to output representation embedding f_x^rep and logits embedding f_x^logits as anchor points to supervise the training of DAPE. LR images y are obtained by applying random degradations to x, and they are fed into the trainable image encoder and tagging head. To make DAPE robust to image degradation, we force the representation embedding and logits embedding from the LR branch to be close to that of the HR branch. The training objective is as follows:ℒ_DAPE =ℒ_r(f_y^rep, f_x^rep) + λℒ_l(f_y^logits, f_x^logits),where λ is a balance parameter, f_y^rep and f_y^logits are the representation embedding and logits embedding from LR branch. ℒ_r is the mean squared error (MSE) loss, while ℒ_l is the cross-entropy loss <cit.>. By aligning the outputs from LR and HR branches, DAPE is learned to predict high-quality semantic prompts from corrupted image inputs. Once trained, DAPE undertakes the crucial role of extracting reliable semantic prompts from the LR images. The prompts can be classified into two categories: hard prompts (, tag texts from the tagging head) and soft prompts (, representation embeddings from the image encoder). As shown in Figs. <ref>(b) and <ref>(c), hard prompts are directly passed to the frozen text encoder built into the T2I model to enhance its local understanding capability. The abundance of text prompts is controlled by a preset threshold. If the threshold is too high, the accuracy of predicted categories will improve but the recall rate can be affected, and vice versa. Therefore, the soft label prompts are used to compensate for the limitations of hard prompt, which are free of the impact threshold and avoid the low information entropy issue caused by one-hot categories <cit.>.§.§ Training of SeeSR Model Fig. <ref>(c) illustrates the detailed structure of the controlled T2I diffusion model.Given the successful application of ControlNet <cit.> in conditional image generation, we utilize it as the controller of the T2I model for Real-ISR purpose. In specific, we clone the encoder of the Unet in pre-trained SD model as a trainable copy to initialize the ControlNet.To incorporate soft prompts into the diffusion process, we adopt the cross-attention mechanism proposed in PASD <cit.> to learn semantic guidance. The representation cross-attention (RCA) modules are added to the Unet and placed after the text cross-attention (TCA) modules. Note that the randomly initialized RCA modules are cloned simultaneously with the encoder. In addition to the text branch and representation branch, the image branch also plays a role in reconstructing the desired HR image. We pass the LR images through a trainable image encoder to obtain the LR latent, which is input to ControlNet. The structure of trainable image encoder is kept the same as that in <cit.>. The training process of the SeeSR model is as follows. The latent representation of an HR image is obtained by the encoder of pretrained VAE <cit.>, denoted as z_0. The diffusion process progressively introduces noise to z_0, resulting in a noisy latent z_t, where t represents the randomly sampled diffusion step. With the diffusion step t, LR latent z_lr, hard prompts p_h and soft prompts p_s, we train our SeeSR network, denoted as ϵ_θ, to estimate the noise added to the noisy latent z_t. The optimization objective is:ℒ=𝔼_z_0, z_lr, t, p_h, p_s, ϵ∼𝒩[ϵ-ϵ_θ(𝐳_t, z_lr, t, p_h, p_s)_2^2].For saving the training cost, we freeze the parameters of the SD model while training solely on the newly added modules, including the image encoder, the ControlNet and the RCA modules within the Unet.§.§ LR Embedding in Inference The pretrained T2I models such as SD, during their training phase, do not completely convert the images into random Gaussian noises. However, during the inference process, most of existing SD-based Real-ISR methods <cit.> take a random Gaussian noise as their start point, leading to a discrepancy on the noise handling procedure between training and inference <cit.>. In the Real-ISR task, we observe that this discrepancy can confuse the model to perceive degradation as content to be enhanced, particularly in smooth regions such as the sky, as shown in the top row of Fig. <ref>.To address this issue, we propose to directly embed the LR latent into the initial random Gaussian noise according to the training noise scheduler. This strategy is applicable to most of the SD-based Real-ISR methods <cit.>. As shown in the bottom row of Fig. <ref>, the proposed LR embedding (LRE) strategy alleviate much the inconsistency between training and inference, providing a more faithful start point for the diffusion model and consequently suppressing much the artifacts in the sky region.Note that all experiments of SeeSR in the subsequent sections utilize the LRE strategy by default. § EXPERIMENTSFollowing previous works <cit.>, we focus on the challenging ×4 Real-ISR tasks, while the proposed method can be applied to other scaling factors. Furthermore, to validate the semantic aware capability of SeeSR, we compare theReal-ISR methods using the well-known COCO dataset <cit.>. §.§ Experimental SettingsTraining Datasets. We train SeeSR on DIV2K <cit.>, DIV8K <cit.>, Flickr2K <cit.>, OST <cit.>, and the first 10K face images from FFHQ <cit.>. The degradation pipeline of Real-ESRGAN <cit.> is used to synthesize LR-HR training pairs.Test Datasets. We employ the following test datasets to comprehensively evaluate SeeSR. (1) First, we randomly crop 3K patches (resolution: 512×512) from the DIV2K validation set <cit.> and degrade them using the same pipeline as that in training. We name this dataset as DIV2K-Val. (2) We employ the two real-world datasets, RealSR <cit.> and DRealSR <cit.>, by using the same configuration as <cit.> to center-crop the LR imageto 128×128 [https://huggingface.co/datasets/Iceclear/StableSR-TestSets]. (3) We build another real-world dataset, named RealLR200, which comprises 38 LR images used in recent literature <cit.>, 47 LR images from DiffBIR <cit.>, 50 LR images from VideoLQ (the last frame of each video sequence) <cit.>, and 65 LR images collected from the internet by ourselves.Implementation Details. We finetune the entire DAPE module from RAM <cit.> using LORA (r=8) <cit.> for 20k iterations, where the batch size and the learning rate are set to 32 and 10^-4, respectively. The SD 2.1-base[https://huggingface.co/stabilityai/stable-diffusion-2-1-base] is used as the pretrained T2I model. The whole controlled T2I model is finetuned for 100K iterations with Adam <cit.> optimizer, where the batch size and learning rate are respectively set to 32 and 5×10^-5. The training process is conducted on 512×512 resolution images with 8 NVIDIA Tesla 32G-V100 GPUs. For inference, we adopt spaced DDPM sampling <cit.> with 50 timesteps. λ in Eq. (<ref>) is set to 1.Evaluation Metrics. In order to provide a comprehensive and holistic assessment of the performance of different methods, we employ a range of reference and non-reference metrics. PSNR and SSIM <cit.> (calculated on the Y channel in YCbCr space) are reference-based fidelity measures, while LPIPS[We use LPIPS-Alex by default.] <cit.>, DISTS <cit.> are reference-based perceptual quality measures. FID <cit.> evaluates the distance of distributions between original and restored images. NIQE <cit.>, MANIQA <cit.>, MUSIQ <cit.>, and CLIPIQA <cit.> are non-reference image quality measures.Compared Methods.We compare our SeeSR with several state-of-the-art Real-ISR methods, which can be categorized into two groups. The first group consists of GAN-based methods, including BSRGAN <cit.>, Real-ESRGAN <cit.>, LDL <cit.>, FeMaSR <cit.> and DASR <cit.>. The second group consists of recent diffusion-based methods, including LDM <cit.>,StableSR <cit.>, ResShift <cit.>, PASD <cit.>, and DiffBIR <cit.>. We use the publicly released codes and models of the competing methods for testing.§.§ Comparisons with State-of-the-ArtsQuantitative Comparisons. We first show the quantitative comparison on the four synthetic and real-world datasets in Table <ref>. We have the following observations. (1) First, our SeeSR consistently achieves the best scores in MANIQA, CLIPIQA and MUSIQ across all the four datasets, except for a 1% lower score in CLIPIQA on DrealSR. (2) Second, SeeSR achieves the best FID and DISTS scores on DIV2K-Val, surpassing the second-best method by more than 10.3% and 5.3%, respectively. (3) GAN-based methods achieve better PSNR/SSIM scores than DM-based methods. This is mainly because DM-based methods can generate more realistic details, which however sacrifice the fidelity. (4) BSRGAN, Real-ESRGAN and LDL show advantages in terms of reference perceptual metrics LPIPS/DISTS, but they perform poorer in no-reference perceptual metrics such as CLIPIQA, MUSIQ and MANIQA. This is also because DM-based methods will generate some structures and textures that may not match the GTs, making them disadvantageous in full-reference metrics. Overall, compared with other DM-based methods, our SeeSR achieves better no-reference metric scores, while keep competitive full-reference measures.Qualitative Comparisons. Figs. <ref> and <ref> present visual comparisons on synthetic and real-world images, respectively. As illustrated in the first row of Fig. <ref>, BSRGAN and Real-ESRGAN fail to reconstruct the details of the ship, which suffers from severe degradation. Meanwhile, some DM-based methods can reconstruct clear details but exhibit obvious semantic errors.Due to the ambiguous output of its degradation removal stage, DiffBIR mis-generates the ship into fish. The caption model of PASD provides a text prompt with semantic errors, wrongly generating a bird. In comparison, our well-trained DAPE module in SeeSR can still provide accurate prompt even with strong degradation, aiding SeeSR to generate semantically-accurate and details-rich results. Additionally, StableSR, DiffBIR and PASD all exhibit certain degree of artifacts in smooth regions (, sea, sky). Thanks to our LRE strategy, SeeSR demonstrates better stability in the smooth regions.Similar conclusions can be drawn from Fig. <ref> for real-world LR images. GAN-based methods generate limited and unnatural details. Some DM-based methods can generate more realistic details but with less accurate semantics. DiffBIR generates dense stripes in smooth regions (row 1). Additionally, it adds extra glasses to the person (row 3). StableSR, ResShift and PASD fail to recover edges (row 1). Although PASD does a decent job in restoring the eyes of the person, it generates artifacts in the suit (row 3). In contrast, SeeSR produces sharper and semantically more accurate results, such as the edges on the wall, the fur of camel hump, the neat suit, and the vivid eyes. More visual examples can be found in the Fig. <ref>.User Study. To further validate the effectiveness of our method, we conduct separate user studies on synthetic data and real data. On synthetic data, inspired by SR3 <cit.>, participants were presented with an LR image placed between two HR images each time: one is the GT and another is the Real-ISR output by one model. They were asked to determine `Which HR image better corresponds to the LR image?' When making decision, participants considered two factors: the perceptual quality of the HR image and its semantic similarity to the LR image. Then the confusion rates can be calculated, which indicate the participants' preference to the GT or the Real-ISR output.On real data, participants were presented with LR images alongside two Real-ISR outputs from two models, and they were asked to answer `Which image is the best SR result of the LR image?' In this experiment, best rates were calculated, which represent the probability of the model being selected. We invited 20 participants to test six representative methods (Real-ESRGAN, StableSR, PASD, DiffBIR, ResShift and SeeSR). There are 16 synthetic test sets and 16 real-world test sets. The synthetic data are randomly sampled from DIV2K-Val, and the real-world data are randomly sampled from RealLR200. Each of the 20 participants was asked to make 112 selections (16×6+16). As shown in Table <ref>, our SeeSR significantly outperforms others in terms of selection rate on both synthetic and real data. In the user study on synthetic data, the SR results of all models cannot compete with the GT, while our SeeSR achieve a confusion rate of 36.6, which is three times higher than the second-ranked method. This implies that there is still enough room to improve for the Real-ISR methods. In the user study on real-world data where there is no GT, our method achieves a best selection rate of 56.2%, approximately 3.5 times higher than the second-ranked method. §.§ Semantics Preservation TestTo further validate our model's ability to preserve semantic fidelity, we conduct detection and segmentation tasks on the Real-ISR output images. We resize the original images from COCO-Val (5K images) <cit.> to 512×512 as GT, and then degrade them to generate LR images as in training. We employ OpenSeeD <cit.> trained on COCOas the detector and segmentor since it is a strong transformer-based unified model for segmentation and detection tasks. As shown in Table <ref>, compared to Zoomed LR, SeeSR achieves a remarkable 3∼4 times improvement in all four tasks, surpassing all existing Real-ISR methods and showcasing its strong semantics preservation capability. §.§ Ablation StudyWe first discuss the effectiveness of the proposed LRE strategy. Then, we discuss the effectiveness of the proposed DAPE module, including its tagging capability and the roles of hard and soft prompts. Effectiveness of LRE. We first show the Real-ISR performance of our SeeSR model on the DIV2K-Val and DrealSR datasets with and without the LRE strategy. The results are shown in Table <ref>. One can see that the LRE strategy improves the reference-based metrics, including both fidelity and perception based ones, while it weakens the non-reference metrics such as CLIPIQA. This is because the LRE strategy reduces the model’s tendency to generate additional (but maybe unfaithful) textures by narrowing the gap between training and testing (see discussions in Section 3.4). Such an over-generation ability can be favorable by metrics like CLIPIQA, but they will introduce visually unpleasant artifacts, as shown in Fig. <ref>.Tagging Performance of DAPE. In Table <ref>, we present the tagging performance of our DAPE module on the degraded images of COCO-val benchmark <cit.> based on three metrics: overall precision (OP), overall recall (OR), and average precision (AP). AP is the averaged precision calculated on different recall rates, which is similar to the detection metric. OP and OR are defined as:OP =∑_i N_i^t/∑_i N_i^p, OR =∑_i N_i^t/∑_i N_i^g,where C is the number of classes, N_i^p is the number of images predicted for label i, N_i^t is the number of images correctly predicted for label i, and N_i^g is the number of ground truth images for label i.We evaluate RAM <cit.> and DAPE with the default threshold. DAPE surpasses RAM in terms of OP and AP by 0.1 and 10.7, respectively. It also maintains superiority in OR, indicating that DAPE achieves significant improvements in tagging accuracy for degraded images. This improvement assists the T2I model in generating semantically accurate details when performing the Real-ISR task.Effectiveness of DAPE and Hard/Soft Prompts for Real-ISR. DAPE improves the model's tagging performance on degraded images and consequently enhances the Real-ISR capability. To investigate the effectiveness of DAPE and the roles of its hard/soft prompts, we conducted the following four experiments in Real-ISR tasks. * We retrain SeeSR by removing the DAPE and RCA modules, which can be considered as applyingControlNet <cit.> directly to the Real-ISR task.* We replace DAPE with RAM <cit.> and retrain the model.* During the inference of SeeSR, we provide only the hard prompts (, the tag) generated by DAPE to the text encoder of the T2I model.* During the inference of SeeSR, we provide only the soft prompts (, the representation embedding features) generated by DAPE to the T2I model. The results of the four experiments are shown in Table <ref>. Moreover, the visual comparisons are shown in Fig. <ref>. From Table <ref> and Fig. <ref>, we can have the following conclusions. First, directly applying ControlNet to the Real-ISR task cannot achieve satisfactory results. Second, replacing DAPE with the original RAM would lead to a decrease in all perceptual metrics (, LPIPS and CLIPIQA). The semantics of the image content may also be changed (see Fig. <ref>). This is because the original RAM may generate inaccurate prompts (, the tag `broccoli') from the degraded image.Third, the soft prompts work better in improving the numerical indices than the hard prompts, as well as sharper images. However, without hard prompts, the image semantics can be damaged, as can be seen from the lemons in Exp. (4) of Fig. <ref>.Finally, with both the hard and soft prompts in DAPE, perceptually realistic and semantically correct Real-ISR outputs can be produced. §.§ Complexity AnalysisTable <ref> compares the number of parameters of different Real-ISR models and their inference time to synthesize a 512×512 image from 128×128 input. All tests are conducted on one NVIDIA Tesla 32G-V100 GPU. We can have the following observations.First, the GAN-based methods Real-ESRGAN and FeMaSR have much less model parameters and much faster inference speed than DM-based methods. Second, among the DM-based models, LDM and ResShift are much smaller than others because they employ relatively lightweight diffusion models. ResShift runs faster than LDM because it samples only 15 steps while LDM samples 200 steps. Thrid, StableSR, PASD, DiffBIR and our SeeSR are all based on the pre-trained T2I model. SeeSR has more parameters because it has a DAPE module (about 300M) finetuned from the RAM model. In terms of inference speed, PASD, DiffBIR and SeeSR are comparable, while StableSR is the slowest one because it samples 200 steps.§.§ More Visualization ComparisonsWe provide additional qualitative comparisons on real-world images. As shown in Fig. <ref>, SeeSR can generate sharper edges (case 2) and semantically faithful details (the window railing in case 1, the teeth in case 3, and the vein textures in case 4). Other methods are either blurry or produce unpleasant artifacts. § CONCLUSIONWe proposed SeeSR, a Real-ISR method that utilizes semantic prompts to enhance the generative capability of pretrained T2I diffusion models. Through exploring the impact of different styles of text prompts on the generated results, we found that the image tags can greatly enhance the local perception ability of the T2I model. However, the tags are susceptible to complex image degradation, and they are influenced by manually set thresholds. Therefore, we proposed DAPE, which minimizes the influence of image degradation on semantic prompts and simultaneously outputs soft and hard semantic prompts to guide the diffusion process in image super-resolution. Furthermore, to address the adverse effects of training-test inconsistency in diffusion models, we proposed a simple yet effective LRE strategy, which embeds LR latent at the starting point of diffusion process, avoiding the generation of artifacts in smooth areas. Our work made a step towards better leveraging generative priors to synthesize semantically correct Real-ISR images, as demonstrated in our extensive experiments. ieeenat_fullname | http://arxiv.org/abs/2311.16518v1 | {
"authors": [
"Rongyuan Wu",
"Tao Yang",
"Lingchen Sun",
"Zhengqiang Zhang",
"Shuai Li",
"Lei Zhang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127181119",
"title": "SeeSR: Towards Semantics-Aware Real-World Image Super-Resolution"
} |
VehicleGAN: Pair-flexible Pose Guided Image Synthesis for Vehicle Re-identificationBaolu Li^†, Ping Liu^†, Lan Fu,Jinlong Li,Jianwu Fang,Zhigang Xu^*,Hongkai Yu^* Baolu Li and Zhigang Xu are with Chang'an University, Xi’an 710064, China. Ping Liu is with the Center for Frontier AI Research (CFAR), Agency for Science, Technology, and Research (A*STAR), Singapore 138634. Lan Fu is with University of South Carolina, Columbia 29201, SC, USA. Jianwu Fang is with Xi'an Jiaotong University, Xi'an 710049, China. Baolu Li, Jinlong Li, and Hongkai Yu are with Cleveland State University, Cleveland, OH 44115, USA. † indicates co-first authors. * Co-corresponding authors: Zhigang Xu ([email protected]), Hongkai Yu ([email protected]).December 15, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== This paper introduces an innovative approach based on policy iteration (PI), a reinforcement learning (RL) algorithm, to obtain an optimal observer with a quadratic cost function. This observer is designed for systems with a given linearized model and a stabilizing Luenberger observer gain. We utilize two-layer quadratic neural networks (QNN) for policy evaluation and derive a linear correction term using the input and output data. This correction term effectively rectifies inaccuracies introduced by the linearized model employed within the observer design. A unique feature of the proposed methodology is that the QNN is trained through convex optimization. The main advantage is that the QNN’s input-output mapping has an analytical expression as a quadratic form, which can then be used to obtain a linear correction term policy. This is in stark contrast to the available techniques in the literature that must train a second neural network to obtain policy improvement. It is proven that the obtained linear correction term is optimal for linear systems, as both the value function and the QNN's input-output mapping are quadratic. The proposed method is applied to a simple pendulum, demonstrating an enhanced correction term policy compared to relying solely on the linearized model. This shows its promise for addressing nonlinear systems. § INTRODUCTION Many control system methods assume that the system's states are available. However, in practice, the states may not be all measured, necessitating the design of a state observer. Luenberger proposed an observer where the output error is multiplied by a gain to formthe correction term <cit.>.The Kalman filter, an optimal filter that minimizes the state estimation error variance, was proposed in reference <cit.> and it was proven that the steady state Kalman gain can be obtained by solving an algebraic Riccati equation. A disadvantage of both the methods in references <cit.><cit.> is that an accurate linear model of the system is needed to estimate the states. However, it's important to note that a linear model might not be able to accurately model the dynamics of certain nonlinear systems. To design optimal observers and controllers for complex nonlinear systems, many researchers have studied data-driven methods such as RL <cit.><cit.><cit.>. RL has demonstrated exceptional performance in system control <cit.>.The use of RL is motivated by key factors such as adaptability, handling complex dynamics, learning from real-world experience, and mitigating inaccuracies in system models <cit.><cit.>. Most research on using RL to design optimal controllers and observers is focused on discrete-time value-based RL algorithms <cit.>. In value-based RL, the core concept is to estimate value functions, which are used to obtain the optimal policy. A common method to estimate value functionsis using the temporal difference (TD) equation <cit.>. This equation reduces the Bellman error <cit.>by repeatedly updating the value function estimate. Value-based algorithms that use TD equation are called adaptive dynamic programming (ADP) <cit.>.In control system, ADP is popular because it's closely related to dynamic programming and the Bellman equation <cit.>. The applications of ADP methods, the policy-iteration (PI) and the value-iteration (VI) algorithms, to feedback control are discussed in references <cit.><cit.>.In the context of optimal control, PI is often preferred over VI due to its stability. PI guarantees policy improvement, which ensures reliable convergence to the optimal policy <cit.>. In general, standard neural networks (SNN) are employed to approximate the value function during the policy evaluation step. This approach is widely utilized for the development of a versatile ADP algorithm, which can effectively design many optimal controllers and observers <cit.><cit.>. However, the utilization of SNNs as a value function approximator (VFA) has certain limitations: (i) the optimization of the neural network's weights is a non-convex problem. Consequently, training the neural network typically yields locally optimal weight, (ii) the process of selecting the most appropriate architecture for the SNN often involves a trial and error procedure,(iii) the SNN lacks a straightforward analytical expression. Therefore, to improve the policy a second neural network is needed to minimize the value function, (iv) providing a comprehensive proof of convergence to the optimal policy is difficult or unattainable. To address the mentioned issues, a two-layer quadratic neural network (QNN), as detailed in reference <cit.>, can be selected as the VFA. The advantages of using the two-layer QNN as the VFA over non-quadratic neural networks are: (i) two-layer QNNs are trained by solving a convex optimization, ensuring the globally optimal weights are obtained <cit.>, (ii) the convex optimization also yields the optimal number of neurons in the hidden layer <cit.>, (iii) the input-output mapping of the QNN is a quadratic form <cit.>. This allows analytical minimization of the value function with respect to the policy. Reference <cit.> demonstrates the effectiveness of QNNs in various applications, including regression, classification, system identification, and the control of dynamic systems. Furthermore, the utility of employing a two-layer QNN as the VFA becomes evident when addressing linear quadratic (LQ) <cit.> problems, as both the value function and the input-output mapping of the QNN have a quadratic form. Also, the closed-form solution of the LQ problem and its relevance in various engineering applications have made it a popular criteria for comparing RL algorithms <cit.>. Therefore, PI algorithms have been applied to solve linear quadratic regulator (LQR), linear quadratic Gaussian (LQG), and linear quadratic tracker (LQT) problems, as discussed in references <cit.><cit.><cit.>, respectively.References <cit.><cit.> use the PI algorithm to obtain the optimal observer gain for LTI systems, where the performance index is chosen as a quadratic function with respect to the correction term and the output error. The objective of this paper is to design the optimal observer with a local cost quadratic in the output error and the correction term, employing QNNs as the VFA within the PI algorithm. It is based on the assumption that a linearizedmodel and a stabilizing Luenberger observer is provided. This paper proves that the porposed approach converges to the optimal observer obtained from closed-form solutions, if the linear model is accurate. The proposed method is also applied to a simple pendulum, which gives an improved observer compared to relying solely on the linearized model, highlighting its potential for nonlinear system applications. To the best of our knowledge, this is the first time that the optimal observer is obtained with QNNs trained by solving a convex optimization.This paper is organized as follows. Section <ref> presents the problem statement and section <ref> gives the PI algorithm. After section <ref> reviews the two-layer QNN, section <ref> presents how to perform the PI algorithm using the two-layer QNN. Section <ref> showsthe simulation results and section <ref> concludes the paper. § PROBLEM STATEMENT Consider the provided linearized model asx_k+1 = Ax_k + Bu_k y_k = C x_k where x_k ∈ℝ^n_x is the unknown state vector, u_k ∈ℝ^n_u is the input vector, y_k ∈ℝ^n_y is the measured output vector, and A, B, C are system matrices. Assume (A,C) is observable. The goal is to design an optimal observer for the system and minimize a performance index. The dynamics of the observer utilizing the provided model are as followsx̂_k+1 = A x̂_k+ Bu_k + w_k^π ŷ_k = C x̂_kwhere x̂_k and ŷ_kare states and outputs estimates, respectively, and w_k^π = π(.) is the correction term policy to be designed later.The value-function following policy π(.) is written asV^π(x_k) = ∑_i=k^∞γ^i-k ( y_i^T Q y_i +(w_i^π )^T R w_i^π )= ∑_i=k^∞γ^i-k c(y_i, w_i^π)whereQ = Q^T ≥ 0, R = R^T > 0are chosen,c(y_k, w_k^π) is the local cost at time step k, 0 ≤γ < 1 is the discount factor,y_k = y_k - ŷ_k, and x_k = x_k - x̂_k.The objective is to design the correction term policy w_k^π that minimizes the value-function using PI algorithm.The Bellman equation <cit.> isV^π(x_k) = y_k^T Q y_k+(w_k^π)^T R w_k^π + γ V^π (x_k+1)In order to use the PI algorithm, we must add the assumption that an initial policy π_0(.) which stabilizes the observer error dynamics is known.The general PI algorithm to obtain the optimal policy π^*(.) is given in Algorithm <ref>. When the system can be precisely described by the linear model, the error dynamics isx_k+1 = Ax_k-w_k^π y_k = C x_k and the goal is to solvemin_π∑_i=k^∞γ^i-k (y_i^T Q y_i +(w_i^π )^T R w_i^π ), ∀ k s.t. x_i+1 = Ax_i - w_i^πy_i = C x_iThe optimization problem (<ref>) can be rewritten asmin_π∑_i=k^∞γ^i-k (x_i^T Qx_i +(w_i^π )^T R w_i^π ),∀ k s.t. x_i+1 = Ax_i + (-I_n_x)w_i^πwhere Q = C^T Q C, I_n_x is the n_x× n_x identity matrix. Note that (A,-I_n_x) is controllable. Therefore, the optimization problem (<ref>) can be considered as an LQR problem for the observer error dynamics. Thus, the optimal policy π^*(.) is a linear function, and the optimal value-functionV^π^*(.) is quadratic, which makes the QNN a perfect candidate to approximate it. Therefore, weconsider π( x_k) = K^πx_k for some matrix K^π, andthe corresponding value-function approximator V^π(x_k) = x_k^T P^πx_k for a unique matrix P^π>0 <cit.>.. If x_k is known, one can perform algorithm <ref> as presented in <cit.>. However, x_k is unknown and one cannot perform the policy evaluation step. To address this problem, we write the Bellman equation in terms of previous measured data instead of x_k and x_k+1 and revise the PI algorithm accordingly.§ REVISED POLICY ITERATION ALGORITHM In this section, we present the refined PI algorithm. First, we reformulate the Bellman equation by incorporating previous measurement data.§.§ Refining the Bellman EquationFollowing the approach in reference <cit.>, we reconstruct x_k byprevious measured data and replace x_k in the Bellman equation. Consider the observer error dynamics (<ref>) as the expanded state equation <cit.>x_k = A^n_xx_k-n_x+ 𝒞 w_k-1,k-n_x^πy_k-1,k-n_x = 𝒪x_k-n_x + Tw_k-1,k-n_x^π wherew_k-1,k-n_x^π = [ w_k-1^π; w_k-2^π; ⋮; w_k-n_x^π ],y_k-1,k-n_x = [ y_k-1; y_k-2; ⋮; y_k-n_x ],T = [0 -C-CA-CA^n_x -2;00 -C-CA^n_x -3;⋮⋮⋱⋱⋮;00 -C;00000 ],𝒞 =[ -I_n_x -A -A^2-A^n_x-1 ], and the observability matrix 𝒪 is 𝒪 = [ CA^n_x-1; CA^n_x-2;⋮; CA;C ]∈ℝ^(n_xn_y) × n_x .Note that (A,C) is observable, and the observability matrix 𝒪 has full column rank n_x. Therefore, its left inverse 𝒪^+ is obtained by (𝒪^T 𝒪)^-1𝒪^T. The following Lemma based on reference <cit.> allows us to replace x_k with measured data. Lemma 1.One can reconstruct x_k from measured data asx_k = [ M_w M_y ][ w_k-1,k-n_x^π; y_k-1,k-n_x ]where M_w = 𝒞 - A^n_x𝒪^+ T and M_y = A^n_x𝒪^+. Equation (<ref>) yieldsx_k - 𝒞 w_k-1,k-n_x^π =A^n_xx_k-n_x = A^n_xI_n_xx_k-n_x where I_n_x is the n_x × n_x identity matrix. Note that I_n_x = 𝒪^+ 𝒪. Therefore,x_k - 𝒞 w_k-1,k-n_x^π=A^n_x𝒪^+(𝒪x_k-n_x)According to (<ref>), we can replace 𝒪x_k-n_x with y_k-1,k-n_x - T w_k-1,k-n_x^π in equation (<ref>) and get x_k - 𝒞 w_k-1,k-n_x^π=A^n_x𝒪^+ ( y_k-1,k-n_x - T w_k-1,k-n_x^π ) which can be recast as x_k=[ 𝒞- A^n_x𝒪^+ TA^n_x𝒪^+ ][ w_k-1,k-n_x^π; y_k-1,k-n_x ] = [ M_w M_y ][ w_k-1,k-n_x^π; y_k-1,k-n_x ] Using Lemma 1, the Bellman equation can be written based on previous measured data as[ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TH^π[ w_k-1,k-n_x^π; y_k-1,k-n_x ] = y_k^T Q y_k + (w_k^π)^T R w_k^π +γ[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]^TH^π[ w_k,k-n_x+1^π; y_k,k-n_x+1 ] where H^π = [ M_w M_y ]^T P^π[ M_w M_y ]is a symmetric matrix.One contribution of this paper is to use the available measurements to train a two-layer QNN with a single output to find the matrix H^π of equation (<ref>) and evaluate the policy π(.). An overview of two-layer QNNs is provided in section <ref>. The neural network training algorithm is discussed in section <ref>.Note that (<ref>) is a scalar equation. [ w_k-1,k-n_x^π; y_k-1,k-n_x ]∈ℝ^n_x(n_x + n_y)and H^π is symmetric. Therefore, the matrix H^π hasM = n_x (n_x + n_y ) (n_x (n_x + n_y )+1 )/2 unknown independent elements and N ≥ Mdata samples are needed to obtain H^π from (<ref>).PI algorithms require persistent excitation (PE) <cit.>, <cit.>. To achieve PE in practice, we add a probing noise term n_k such that w_k^π = K^πx_k + n_k.It is shown in reference <cit.> that the solution computed by PI differs from the actual value corresponding to the Bellman equation when the probing noise term n_k is added. It is discussed that adding the discount factor 0<γ<1 to the Bellman equationreduces this harmful effect of n_k.§.§ Policy improvement step We now address how to improve the policy π(.) after evaluating the policy and obtaining the matrix H^π with a two-layer QNN. The policy improvement step can be written asπ^'(x_k) = min_π(y_k^TQy_k + (w_k^π)^T R w_k^π + γ[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]^TH^π[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]) where π^'(.) is the improved policy over the policy π(.). Partition [ w_k,k-n_x+1^π; y_k,k-n_x+1 ]^TH^π[ w_k,k-n_x+1^π; y_k,k-n_x+1 ] as[ w_k^π; w_k-1,k-n_x+1^π; y_k,k-n_x+1 ]^T [ H_11^πH_w^πH_y^π;(H_w^π)^T H_22^π H_23^π;(H_y^π)^T (H_23^π)^T H_33^π ][ w_k^π; w_k-1,k-n_x+1^π; y_k,k-n_x+1 ]One can solve (<ref>) and get the improved policy asw_k^π^' = - γ (R + γ H_11^π)^-1 (H_w^π w_k-1,k-n_x+1 +H_y^πy_k,k-n_x+1) Therefore, we can use the refined PI Algorithm <ref> and obtain the optimal policy π^*(.) without the system model.For linear systems, the optimal H^π can be obtained using the following closed-form solutionH^π^* = [ M_w M_y ]^T P^π^*[ M_w M_y ] where P^π^* is calculated from solving the Riccati equation (<ref>). <cit.> P^π^* = C^TQC + γ A^T P^π^* A - γ^2 A^T P^π^* (R+γ P^π^*)^-1 P^π^* AThen the optimal correction term can be designed asw_k^π^*= - γ (R+ γ H_11^π^*)^-1 ( H_w^π^* w_k-1,k-n_x+1 +H_y^π^*y_k,k-n_x+1)§ TWO-LAYER QNNS WITH ONE OUTPUT This section discussed the training of a two-layer QNNwith one output as introduced in <cit.>.A QNN will be used to find the matrix H^π of equation (<ref>) based on measured data. Consider the neural network with one hidden layer, one output, and a degree two polynomial activation function, whereX_i ∈ℝ^n is the i-th input data given to the neural network, Ŷ_i ∈ℝ is the output of the neural network for the input X_i, Y_i ∈ℝ is the output label corresponding tothe input X_i, L is the number of hidden neurons, f(z)=az^2+bz+c is the polynomial activation function,and a ≠ 0, b, c are pre-defined constant coefficients. The notation w_kj represents the weight from the k-th input-neuronto the j-th hidden-neuron, and v_j represents the weight from the j-th hidden-neuron to the output. The input-output mapping of the neural network is Ŷ_i=∑_j=1^L f(X_i^TW_j)v_jwhere W_j = [ w_1j w_2jw_nj ]^T. Reference <cit.> proposes the training optimizationmin_W_k,v_kl(Ŷ-Y) + β∑_j=1^L | v_j |s.t.Ŷ_i=∑_j=1^L f(X_i^TW_j)v_j‖W_k ‖_2 =1,k=1,2,...,L i=1,2,...,Nwhere β≥ 0 is a pre-defined regularization parameter, l(.) is a convex loss function, N is the number of data points, Ŷ = [ Ŷ_1 Ŷ_2 Ŷ_N ]^T, and Y = [ Y_1 Y_2 Y_N ]^T.The optimization problem (<ref>) can be equivalently solved by the dual convex optimization (<ref>) min_Z^+ , Z^-l(Ŷ-Y) + β Trace(Z_1^+ + Z_1^-)s.t.Ŷ_i=aX_i^T(Z_1^+ - Z_1^-)X_i +bX_i^T(Z_2^+ - Z_2^-) + cTrace(Z_1^+ - Z_1^-),Z^+=[Z_1^+Z_2^+;(Z_2^+)^T Trace(Z_1^+) ]≥ 0 ,Z^-=[Z_1^-Z_2^-;(Z_2^-)^T Trace(Z_1^-) ]≥ 0, i=1, 2,Nwhere Z_1^+, Z_2^+, Z_1^-, Z_2^- are optimization parameters <cit.>. After training the neural network and obtaining Z^+, Z^- from (<ref>), the quadratic input-output mapping isŶ_i=[ X_i; 1 ]^T H [ X_i; 1 ]whereH=[ a(Z_1^+ - Z_1^-)0.5b(Z_2^+ - Z_2^-);0.5b(Z_2^+ - Z_2^-)^T c[ Trace (Z_1^+ - Z_1^- )] ]If b=c=0, a =1 are chosen, the input-output mapping is Ŷ_i= X^T_i H X_i, where H=(Z_1^+ - Z_1^-). § QNNS AS THE POLICY EVALUATOR This section shows how a two-layer QNN executes the policy evaluation step and obtains H^π. Then, the complete algorithm to find π^* (.) without the system model is presented. First, we introduce the following Lemma. Lemma 2:Let Ĥ^π_i denote the i-th approximation of H^π for the policy π(.) that stabilizes the observer error dynamics. Starting with i=1 and any symmetric Ĥ^π_0, iterating through equation (<ref>) will result in Ĥ^π_i converging to H^π provided 0≤γ<1.[ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_i [ w_k-1,k-n_x^π; y_k-1,k-n_x ] =y_k^T Q y_k + (w_k^π)^T R w_k^π +γ[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]^TĤ^π_i-1[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]Applying the equation (<ref>) recursively yields [ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_i [ w_k-1,k-n_x^π; y_k-1,k-n_x ][ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_i [ w_k-1,k-n_x^π; y_k-1,k-n_x ]=c(y_k,w_k^π) + γ c(y_k+1,w_k+1^π) +γ^2 [ w_k+1,k-n_x+2^π; y_k+1,k-n_x+2 ]^TĤ^π_i-2[ w_k+1,k-n_x+2^π; y_k+1,k-n_x+2 ] [ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_i [ w_k-1,k-n_x^π; y_k-1,k-n_x ]= ∑_j=0^i-1γ^j c(y_k+j,w_k+j^π)+γ^i [ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ]^T Ĥ^π_0 [ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ]where c(y_k,w_k^π) was defined in section (<ref>). Let i →∞. Then,[ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_∞[ w_k-1,k-n_x^π; y_k-1,k-n_x ] =∑_j=0^∞γ^jc(y_k+j,w_k+j^π) + lim_i →∞γ^i [ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ]^T Ĥ^π_0[ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ]Since lim_i →∞γ^i = 0, for any Ĥ^π_0 we also getlim_i →∞γ^i [ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ]^T Ĥ^π_0[ w_k+i-1,k+i-n_x^π; y_k+i-1,k+i-n_x ] = 0Replacing (<ref>) in (<ref>) yields [ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TĤ^π_∞[ w_k-1,k-n_x^π; y_k-1,k-n_x ] =∑_j=0^∞γ^j c(y_k+j,w_k+j^π) According to the Bellman equation (<ref>), we have[ w_k-1,k-n_x^π; y_k-1,k-n_x ]^TH^π[ w_k-1,k-n_x^π; y_k-1,k-n_x ] =∑_j=0^∞γ^j c(y_k+j,w_k+j^π) As a result of (<ref>)(<ref>), Ĥ_i^π converges to H^π.Therefore, in the policy evaluation step, we can calculate H^π if we can obtain Ĥ^π_i in equation (<ref>) from the previous known Ĥ^π_i-1 in the i-th iteration. We use a two-layer QNN to obtain Ĥ^π_i.§.§ Obtaining Ĥ^π_i using a two-layer QNN Define X_k = [ w_k-1,k-n_x^π; y_k-1,k-n_x ] Y_k = y_k^T Q y_k + (w_k^π)^T R w_k^π +γ[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]^TĤ^π_i-1[ w_k,k-n_x+1^π; y_k,k-n_x+1 ]Given N >> M data points X_k as the inputs and the corresponding data points Y_k as the output labels to the QNN given in Fig <ref>, we train the QNN and obtain from (<ref>) the quadratic input-output mapping as X_k^T H X_k = Ŷ_k. After obtaining H we simply set Ĥ^π_i=H.X_k^T Ĥ^π_i X_k = Y_k The detailed algorithm <ref> uses a convergence stopping criterium of || Ĥ^π_j_i - Ĥ^π_j_i-1 || < ϵ for a small ϵ. § SIMULATION RESULTSFirst, we demonstrate that for a linear system, our proposed method converges to the optimal correction term policy derived from closed-form solution given in the remark.Consider a pendulum modeled as (<ref>) with sampling time T_s=0.1s.[ x_1,k+1; x_2,k+1 ] =[0.950.10; -0.980.94 ][ x_1,k; x_2,k ]+[ 0.005; 0.098 ]u_ky_k= [ 0 1 ][ x_1,k; x_2,k ]As a result, M_y and M_w are obtained asM_w = [-1 0 -0.95 -0.92; 0-10.980.95 ],M_y = [ -0.820.96;1.89 -0.99 ]We choose γ = 0.6, N=300,β=0 , Q=10 and R= [ 1 0; 0 1 ]. From closed-form solution, we getP^π^* =[1.28 -0.78; -0.78 10.77 ] H^π^* = [ 1.3-0.8 1.9 1.9 2.5-2.0;-0.810.7 -11.2 -10.9 -21.011.4; 1.9 -11.212.912.522.9 -13.1; 1.9 -10.912.512.222.3 -12.7; 2.5 -21.022.922.341.8 -23.2;-2.011.4 -13.1 -12.7 -23.213.2 ]The objective is to show thatwith any initial stabilizing policy π_0(.), the proposed algorithm <ref> converges to the matrix H^π^* and therefore the optimal policy π^*(.). Consider ten simulations with random initial stabilizing policies.It is shown in Fig <ref> that π_j(.) converges to the optimal policy π^*(.) in all ten runs of the algorithm since H^π_j converges to H^π^*.step following the optimal policy π^*(.) obtained from the proposed methodwith the initial condition x_0=[ 0 0 ]^T,x̂_0 = [ -11 ]^TThe proposed method is applied to the nonlinear model of the simple pendulum in the second example. The data-driven method givesan improved observer compared toobtaining the closed-form solution of the correction term using the linearized model.Consider the pendulum with the dynamics asθ( t )+ 0.1 θ̇ (t )+ 10 sin(θ (t)) = u(t)where θ(t) represents the pitch angle and the measured output is θ̇ (t ). We again choose γ = 0.6, N=300,β=0 , Q=10 and R= [ 1 0; 0 1 ]. The A, B, C matrices in (<ref>)which are derived from linearization around [θ; θ̇ ] = [ 0; 0 ]are used to design the observer given in (<ref>). One can also obtain the correction term w_k solely using the linearized model with the closed form solution given in (<ref>). To compare this result with obtaining the correction term using the proposed data-driven method, we run ten simulation with different initial stabilizing policies and the initial statex_0 =[θ(0); θ̇(0) ] = [ 3; 0 ]Figure <ref> shows that theproposed method consistently yields lower initial cost-to-go values than the closed-form solution, wich is shown with a dash line.In this instance, the selection of Q and R matrices prioritizes minimizing the output error. As depicted in Fig <ref>, the proposed approach outperforms the correction term derived from the closed-form formulaand reduces the output error over time.§ CONCLUSION This paper introduced an innovative policy iteration method to design an optimal observer with a quadratic cost function. This data-driven approach enhances the observer's performance for systems with a linearized model and a stabilizing Luenberger observer gain. The two-layer quadratic neural network is used as the value-function approximator. The nueral networkhas an analytical, quadratic input-output mapping trained with convex optimization. Therefore a linear correction term policyis derived from input and output data to rectify inaccuracies of the linearized model. This contrasts with existing techniques using neural networks, which require a second neural network for policy improvement.Using the proposed approach for linear systems converges to the optimal observer obtained from analytical methods. Applied to a simple pendulum, our method demonstrates improved correction term policies compared to relying solely on the linearized model, showing its potential for nonlinear systems. unsrt | http://arxiv.org/abs/2311.16272v1 | {
"authors": [
"Soroush Asri",
"Luis Rodrigues"
],
"categories": [
"eess.SY",
"cs.SY"
],
"primary_category": "eess.SY",
"published": "20231127192141",
"title": "Optimal Observer Design Using Reinforcement Learning and Quadratic Neural Networks"
} |
Distributed Attacks over Federated Reinforcement Learning-enabled Cell Sleep Control Han Zhang1, Hao Zhou1, Medhat Elsayed2,Majid Bavand2, Raimundas Gaigalas2, Yigit Ozcan2 and Melike Erol-Kantarci1, Senior Member, IEEE1 School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada2 Ericsson Inc., Ottawa, Canada{hzhan363, hzhou098, melike.erolkantarci}@uottawa.ca, {medhat.elsayed, majid.bavand, raimundas.gaigalas, yigit.ozcan}@ericsson.comJanuary 14, 2024 ================================================================================================================================================================================================================================================================================================================================================================================================================================================fancy Accepted by 2023 IEEE Globecom Workshops (GC Wkshps), 2023 IEEEFederated learning (FL) is particularly useful in wireless networks due to its distributed implementation and privacy-preserving features. However, as a distributed learning system, FL can be vulnerable to malicious attacks from both internal and external sources. Our work aims to investigate the attack models in a FL-enabled wireless networks. Specifically, we consider a cell sleep control scenario, and applyfederated reinforcement learning to improve energy-efficiency. We design three attacks, namely free rider attacks, Byzantine data poisoning attacks and backdoor attacks. The simulation results show that the designed attacks can degrade the network performance and lead to lower energy-efficiency.Moreover, we also explore possible ways to mitigate the above attacks. We design a defense model called refined-Krum to defend against attacks by enabling a secure aggregation on the global server. The proposed refined-Krum scheme outperforms the existing Krum scheme and can effectively prevent wireless networks from malicious attacks, improving the system energy-efficiency performance. Federated learning, deep reinforcement learning, security, radio access networks, attacks, defense.§ INTRODUCTION With the deployment of the 5G and beyond 5G (B5G) networks, the increasing traffic demand for cellular communications has reached an unprecedented level<cit.>. To meet diverse service requirements and facilitate intelligent wireless communications, various machine learning (ML) techniques have been used to solve problems in wireless networks<cit.>.Reinforcement learning (RL) is a widely applied ML technique that provides automated solutions for high-complexity optimization problems<cit.>.Meanwhile, federated learning (FL) is another emerging ML technique that enables collaborative learning with local training in distributed systems, without sharing data. Federated reinforcement learning (FRL) is proposed as a combination of FL and RL and has proven effective in many wireless communication scenarios. For example, in <cit.>, FRL is used to allocate power resources and radio resources in network slicing. However, these achievements of using FRL are mainly accomplished in fully secure environments without considering malicious attacks. Due to the inherently distributed implementation, FL is more vulnerable to malicious attacks than other centralized ML techniques. Distributed participants in FL are easier to be attacked and manipulated, and the parameter sharing and updating between local and global servers may expose the FL to potential risks <cit.>.As a result, it is crucial to investigate security issues in FL.There are some existing studies about attacks and defenses for FL algorithms<cit.>. However, most research focuses on supervised learning and cannot apply to FRL models.In this work, we study the security problem in an FRL-enabled cell sleep control scenario. As the traffic load grows,improving network energy-efficiency and reducing energy costs become critical goals for wireless networks<cit.>. Performing sleep control to base stations (BS) to reduce energy consumption is a feasible way to improve energy-efficiency and make networks sustainable<cit.>. However, attacks on cell sleep control may cause different levels of system performance degradation. For example, it may waste system energy by making BSs never sleep or produce low throughput by keeping BSs in sleep mode.In this paper, we first design an FRL-based cell sleep control scenario and BSs will cooperatively learn sleep control strategies through FL. Then we assume some BSs are malicious participants. Specifically, we propose three attack models, namely free rider, Byzantine data poisoning, and backdoor attacks specifically for the given cell sleep control scenario. To the best of our knowledge, this is the first work that applies the backdoor attacks to a wireless network control application. The simulation results show that the designed attacks will lower system energy-efficiency. Meanwhile, we also propose a defense scheme called refined-Krum to defend against these attacks. Compared with the existing Krum defense scheme, it can achieve a better defense effect without knowing the number of attackers.The rest of the paper is organized as follows. Section <ref> introduces related works, and Section <ref> shows our system model.Section <ref> introduces FRL-based sleep control scenario,and Section <ref> presents the designed attacks and the proposed defense model. Finally, Section <ref> shows simulation results, and Section <ref> concludes this work. § RELATED WORKS There have been many studies that design attacks and defenses towards breaches in FL algorithms. In <cit.>, data poisoning attacks are performed on FL-based image classification problems. <cit.> performs backdoor attacks on the FL system with single or multiple malicious participants. <cit.> proposes secure aggregation methods to defend Byzantine data poisoning attacks in the FL system. These works are only designed for supervised learning and do not apply to RL models. <cit.> and <cit.> proposes data poisoning attacks and defenses for FRL. However, these works are only tested with ready-to-use data sets and have some limitations if applied to complicated wireless network scenarios.Meanwhile, other works study attacks on FL in wireless networks. <cit.> designs attacks specific to the wireless traffic prediction models in centralized and distributed scenarios. However, this work also uses a supervised learning model, and its attack method cannot be directly applied to other wireless network control applications that typically use RL. In <cit.>, over-the-air jamming attacks on the uplink and downlink of FL in wireless networks are studied. But it only focuses on external attacks and fails to study the internal attacks in FL.There are also some studies on cell sleep control for energy saving. <cit.> improves energy-efficiency of small cell networks by switching BSs to different modes. In <cit.>, an RL-based traffic adaptive sleep mode control algorithm for BSs is proposed. However, these works are accomplished in fully secure environments and fail to consider attacks and defense. Different from existing studies, our work designs attacks to the specific FRL-based cell sleep control scenario. We focus more on vulnerabilities of FRL models related to realistic wireless environments and evaluate the effectiveness of attacks based on wireless network performance metrics. § SYSTEM MODEL The system model is shown in Fig. <ref>. Weconsider a heterogeneous cellular network consisting of multiple BSs. There is one macro BS (MBS) cell and N small BS (SBS) cells cooperatively serving M distributed user equipment (UE) and handling traffic loads.The MBS is always active to ensure coverage and is responsible for controlling data services. To effectively save energy costs of the system, we adopt three different sleeping modes for SBS cells, which are active, sleep, and deep sleep<cit.>. Active means SBSs are in full operation and consume the most energy. Sleep means SBSs temporarily stop transmitting data for the UEs but can be easily waken up and decide whether to continue sleeping in the next iteration. Deep sleep means more components are deactivated to save more energy, and SBSs take longer time to wake up. If a SBS turns to sleep, the arriving traffic will be offloaded to the MBS. SBSs sleeping at inconvenient intervals can cause low energy efficiency or data congestion in MBS traffic, thus degrading system performance.This scenario considers a downlink orthogonal frequency-division multiplexing cellular system. The link capacity between the m^th UE and the n^th SBS can be given as follows:C_n,m = δ_n∑_r∈ R_n B_r log_2(1+SINR_n,m,r), where δ_n is a binary indicator to denote whether the n^th SBS is active or sleeping. R_n denotes the set of available resource blocks of the n^th SBS and B_r denotes the bandwidth of the r^th resource block. SINR_n,m,r denotes the signal to interference noise ratio (SINR) between the m^th UE and the n^th SBS on the r^th resource block, which can be given as:SINR_n,m,r =β_n,m,r g_n,m P^t_n/n'∈ N, n'≠ n∑ m'∈ M_n'∑β_n',m',r g_n',m P_n'+B_rN_0, where β_n,m,r is a binary indicator to denote whether the r^th resource block of the n^th SBS is allocated to the m^th UE. g_n,m is the channel gain of the transmission link, which is decided by a free space propagation model. P^t_n denotes the transmission power of the n^th SBS and N_0 denotes the noise power density. We assume that UEs can support dual connectivity and can simultaneously connect to the MBS and SBS<cit.>. If the SBS is active, the UE will be served by the SBS. Otherwise, it will be served by the MBS. The energy-efficiency of the system can be defined as:EE = ∑_m∈ M b_m/∑_n∈ NP_n + P_0,where b_m denotes the throughput of the m^th UE, which is decided by both link capacity and arriving traffic. P_0 and P_ndenotes the power consumption of the MBS and the n^th SBS. The optimization objective of the sleep control application is to achieve high energy-efficiency. Here we formulate the problem as:a_nmaxEE-∑_m ∈ Mϵ_m, s.t.(<ref>)-(<ref>) a_n ∈{0,1,2}, ∀ n ∈ N 4aδ_n == 1,if a_n = 0,= 0,else 4b P_n =P_w,if a_n = 0,0.5P_w,if a_n = 1,0.35P_w, else 4cwhere ϵ_m denotes the packet drop rate of the m^th UE. A packet will be dropped if it exceeds the transmission delay constraint<cit.>. a_n denotes the sleeping modes of the n^th SBS. a_n=0 indicates the SBS is in the active mode, a_n=1 indicates the SBS is in the sleep mode, and a_n=2 indicates the SBS is in deep sleep mode. P_w denotes the energy consumption of the SBS in active mode. The sleep mode can reduce energy consumption by 50%, and the deep sleep mode can reduce it by 65%<cit.>. To solve this problem, we use federated reinforcement learning (FRL) to promote privacy-preserving collaborative training. Each SBS holds a local deep reinforcement learning (DRL) model, which observes states and rewards from the environment and selects actions by choosing an adequate sleeping mode. The MBS serves as a global server in FRL, collecting local models from SBSs for model aggregation and distributing the global model as feedback. To attack the system, we suppose N^mali out of the N SBSs are malicious and can cause system performance degradation by updating malicious local models to the global server. § FEDERATED REINFORCEMENT LEARNING-BASED CELL SLEEP CONTROL This section introduces the FRL-based cell sleep control application. Here DRL is applied in each SBS as a local model, and the optimal actions are selected by maximizing the long-term expected rewards.The Markov decision process (MDP) of each local DRL is defined as follows: * State: The state includes the sleeping mode of the SBS and the traffic load of the SBS and the MBS in the past 5 transmission time intervalswhich can be used to estimate the upcoming traffic load. It also includes the current delay and throughput of the SBS, which can be given as:s_n = {δ_n, L_n, L_0, d_n, b_n},∀ n ∈ N, where L_n denotes the traffic load of the n_th SBS. L_0 denotes the traffic load of the MBS. d_n and b_n denote the delay and the throughput. * Action: The action of sleep control is to choose an adequate sleeping mode for the SBS, which can be given as: a_n = {0,1,2},∀ n ∈ N,* Reward: The reward function is defined as a combination of both quality of service (QoS) related indicators and the power consumption related cost, which can be given as: R_n = η_1 b_n - η_2ϵ_n- η_3 P_n,∀ n ∈ N, where ϵ_n denotes the packet drop rate and b_n denotes the throughput. η_1, η_2 and η_3 are the coefficients used to balance different rewards. When obtaining a high reward value, we expect the system to consume as little energy as possible while ensuring a high throughput. Therefore, maximizing the given reward value is equivalent to maximizing the energy-efficiency and minimizing the packet drop rate. On top of local models, we apply FRL to enable collaborative training and accelerate learning while keeping data locally and preserving privacy. In each FRL cycle, the local models will first perform local training according to local experience, which can be given as:θ_n^t+1 = θ_n^t + α[r_n^t + γmax_aQ(s_n^t+1,a;θ_n^t)-Q(s_n^t,a_n^t;θ_n^t)]∇ Q(s_n^t,a_n^t;θ_n^t),where θ_n denotes the local model parameters of the n_th SBS, α denotes the learning rate and γ denotes the discount factor. Q(s_n^t,a_n^t;θ_n^t) denotes the long-term expected reward of the n_th SBS choosing the action a_n^t under the state s_n^t. After local training, the local models are uploaded to the global server for model aggregation, which can be formulated as:θ_G^t+1 = ∑^n=1_Nw_nθ_n^t+1where θ_G is the parameters of the global model. w_n is the weight of the n_th local model and it is decided by the number of training samples. In the scenario of FRL-based cell sleep control, we assume all the local models are equally weighted.After the global model aggregation, the global model parameters are sent back to the SBSs and the local models are updated by replacing the local parameters with global parameters. § ATTACKS AND DEFENSE This section presents the designed attack and defense models in the FRL-based cell sleep control scenario. Fig. <ref> shows the structure of the investigated attacks and the proposed defense model. We proposed three attack models: free rider attacks, Byzantine data poisoning attacks, and backdoor attacks. We also proposed one defense scheme called refined-Krum. §.§ Attack models.§.§.§ Free rider attacks.Free riders refer to the FL participants who do not train their local models during the local training step <cit.>. As shown in Fig. <ref>, a benign BS will keep a memory buffer to store local experience and use it to train a benign local model. In contrast, a free rider does nottrain its local model and will submit the previously received global model as its own local model. The free rider is a passive attack method which means it will not actively destroy the global model or other local models. However, free riders are still malicious because they enjoy the resources and efforts of collaborators without contributing their own experience and training results during the collaboration. In this way, they can break the fairness of the FL system, and when the proportion of free riders increases, they will slow down the FL training. §.§.§ Byzantine data poisoning attacks.Byzantine data poisoning attacks are implemented by injecting poisoned data into the local memory buffer of the malicious participants and making participants train their local model in the wrong direction <cit.>. As shown in Fig. <ref>, the poisoned local model is trained with a mixture of local experience and poisoned data.For our FRL model, we create poisoning data by giving the model a larger reward if the SBS chooses to be active and a smaller reward if the SBS chooses to sleep. The local model training in Eq. (<ref>) on the poisoned data can be reformulated as:θ_n^t+1 = θ_n^t + α[r_n^t +σ R^ex+ γmax_aQ(s_n^t+1,a;θ_n^t)-Q(s_n^t,a_n^t;θ_n^t)]∇ Q(s_n^t,a_n^t;θ_n^t),where R_ex denotes the extra reward and σ is an indicator to denote whether to add an extra reward or to reduce the reward. If a_n^t = 0, σ = -1. Otherwise,σ = 1. After local training, the poisoned local model will be uploaded to the global server and the effect of poisoned data will then be propagated to other local models during the global model aggregation. §.§.§ Backdoor attacks. Different from regular data poisoning attacks, backdoor attacks define backdoor tasks to make a separation from the main tasks. Main tasks refer to the task defined by the original training data.Backdoor tasks are defined by the attacker, and they represent a special input data pattern that rarely appears in the original training data. In FRL, the backdoor tasks can be defined by specific states. Every time the backdoor attackers want to attack the system model, they can trigger the backdoor pattern and make the model select abnormal actions.In our sleep control scenario, we define the backdoor task as an SBS with a very high traffic load. It can be easily triggered by putting a UE with an extremely heavy traffic load into the system, and the trigger result is defined as the SBS will turn into deep sleep mode. As shown in <ref>, the backdoor-ed local model is trained with the mixture of local experience data and synthetic backdoor task training data. Benefiting from separating the main and backdoor tasks, the backdoor attacks are stealthy and difficult to detect.Since the backdoor pattern is defined by the attacker and unknown to others, the model owner will only test it with main tasks while doing a security check. In this way, they could not find the model had already been backdoor-ed.§.§ Defense model.We propose a refined-Krum defense model based on the existing secure FL aggregation method Krum<cit.>. As shown in Fig. <ref>, the refined-Krum is deployed at the global server and will be performed before global aggregation during each FL iteration. In this subsection, we first introduce the Krum defense scheme and then illustrate how the refined-Krum model is defined. §.§.§ Krum defenseKrum is proposed in <cit.> and its core idea is to assume that all benign local models are similar. Therefore, the malicious models can be found by measuring the similarity of all the local models by the Krum distance.In the Krum defense, the Krum distance for each local model is first calculated. In the first step, the Euclidean distance between parameters of the n^th local model and the global model in the last FL iteration is calculated as: G^t+1_n = θ^t+1_n-θ^t_G_2 Then, the distance between the n^th local model and k^th local model can be given as: D^t+1_nk = G^t+1_n-G^t+1_k_2 The distance between each local model and all other local models is then added. In this way, the Krum distance for each local model can be obtained, which can be given as: D^t+1_n = ∑_k ∈ ND^t+1_nk Finally, the Krum defense will select the local model with the smallest Krum distance and replace the global model with the selected local model.§.§.§ Refined-KrumAlthough the Krum defense scheme is proven to be effective in some cases, choosing only a local model for global aggregation is quite unstable and it may not get the full benefit of FL. Therefore, we designed a new defense algorithm called refined-Krum. It can be concluded into four steps, which are calculating the similarity gaps, estimating the number of malicious participants, identifying malicious participants and secure aggregation.* Calculating the similarity gaps. In the first step of refined-Krum, we calculate the Krum distance of each local model to evaluate the similarities between models. But instead of only selecting the local model with the smallest Krum distance, we sort all the models by their Krum distances from the smallest to the largest and calculate the gap between two adjacent Krum distances.* Estimating the number of malicious participants. With the gap values calculated in the first step, we can then estimate the number of malicious participants by finding the maximum gap between the given distance list. This is based on the assumption that most models are benign and the malicious models are quite different from the benign ones. So there will be a large gap between the similarities. * Identifying malicious participants. If the maximum gap is much larger than the average, we suppose it could precisely separate malicious models from benign ones and decide the threshold for the Krum distance. If the Krum distance of the n^th local model is larger than the threshold value, the n^th SBS is treated as a malicious participant. Other SBSs are treated as benign participants. On the other hand, if the maximum gap is close to the average gap, we assume that the threshold cannot be accurately determined and to mitigate the risk of being attacked, only one benign model will be selected.* Secure aggregation. After identifying the malicious participants, we can perform secure aggregation by admitting only benign models into the global model aggregation. This prevents the malicious data from influencing the global model and the benign SBSs. At the same time, we will make MBS take over the sleep control for the malicious SBSs and prevent attackers from controlling these SBSs. § NUMERIC RESULTS§.§ Simulation settings.In the simulation, we consider 8 SBSs, and each SBS has 10 UEs. The fixed power consumption of MBS and SBSs are 40W and 20W, respectively<cit.>. The radius of MBS and SBSs are 400m and 100m, respectively. The available bandwidth for each SBS is 20 MHz, and for MBS is 10 MHz. η_1, η_1 and η_1 are respectively 0.1, 1 and 0.01. During the simulation, we change the average traffic load of each SBS from 30 Mbps to 70 Mbps and compare the system energy-efficiency under different attack and defense models. We simulate a 24-hour typical residential area traffic pattern in each TTI, which is given from <cit.>.For the free rider attacks, we include two free riders in the network. For Byzantine data poisoning attacks and backdoor attacks, we have one malicious SBS in the networks. We assume the proportion of poisoned data or backdoor task training data of malicious SBSs is 5%. Therefore, the proportion of poisoned data in the total data of all the SBSs is 0.625%.§.§ Simulation resultsFirstly, we compare the simulation results of FL and independent learning-based cell sleep control in a fully secure environment. Independent learning (IL) means there are no collaborations between SBSs, and each SBS will train a DRL model according to their local buffer data. Fig. <ref> shows the convergence curves of FL and independent learning when the average traffic load of each SBS is 40 Mbps and in each TTI we run 24-hour traffic. The FL algorithm we use during the simulations is FedAvg. The FL performs much better than IL and has higher rewards, which demonstrates the effectiveness of the FL algorithm.Then we add three different attacks to the FRL-based sleep control scenario. The system energy-efficiency under free rider attacks, data poisoning attacks, and backdoor attacks are shown in Fig. <ref>. The system performance in a secure environment with no attacks is also compared. It can be observed that three kinds of attacks can degrade the system performance to different levels. Backdoor attacks can be seen as the most effective attacker. When the average traffic load is 70 Mbps, the backdoor attacks can reduce the system energy-efficiency by 52%. Among the remaining two attacks, the data poisoning attacks are more effective than the free rider attacks, even with fewer attackers. From this observation, we can also conclude that while designing defense mechanisms for FRL, it is more important to prevent malicious participants from being involved in the aggregation than to ensure that the benign participants are involved in the aggregation. When the average traffic load is 70 Mbps, a malicious SBS with poisoning attacks can reduce the system energy-efficiency by 18%.When it comes to defense, the system energy-efficiency under our proposed refined-Krum defense scheme is shown in Fig. <ref>. We only defend against poisoning and backdoor attacks because even if we can detect free riders, we cannot force them to contribute to the model. As it can be observed, for data poisoning attacks, the system energy-efficiency after the defense is very close to the situations with no attack. We can conclude that the defense can almost fully recover the system from data poisoning attacks with a limited number of attackers. The defense scheme can also significantly improve system performance and increase energy-efficiency for backdoor attacks. However, the energy-efficiency defense is still lower than the performance in a secure environment. This indicates that our proposed defense scheme is quite effective for some attacks but less effective for others. In Fig. <ref>, we further compare our proposed refined-Krum defense scheme with the existing Krum defense scheme. For both kinds of attacks, our proposed refined-Krum defense scheme can get a higher energy-efficiency compared with the Krum defense scheme. Also, refined-Krum is more stable with a smaller confidence interval. § CONCLUSION In this work, we studied how to attack a FRL-based cell sleep control scenario in a wireless network. We considered three types of attacks that could perform on wireless networks, which are free riders, Byzantine data poisoning attacks and backdoor attacks. According to the simulation results, these attacks can degrade system performance with lower energy-efficiency. We also proposed a defense scheme called refined-Krum to defend against these attacks. The simulation results show that our proposed defense scheme can effectively increase the system energy-efficiency and prevent the system from attacks.In our future research, we plan to investigate more advanced attacks and improved defense schemes.§ ACKNOWLEDGEMENT This work has been supported by MITACS and Ericsson Canada, and NSERC Collaborative Research and Training Experience Program (CREATE) under Grant 497981.00 b5 M. Elsayed and M. Erol-Kantarci, “AI-enabled future wireless networks: challenges, opportunities, and open issues,” IEEE Vehicular Technology Magazine, vol. 14, no. 3, pp. 70–77, Sep. 2019. b5-b M.Elsayed and M.Erol-Kantarci,“AI-enabled radio resource allocation in 5G for URLLC and eMBB users,” IEEE 5G World Forum, 5GWF 2019 - Conference Proceedings, pp. 590–595, Sep. 2019. b5-a H. Zhou, M. Erol-Kantarci, and H. V. Poor, “Knowledge Transfer and Reuse: A Case Study of AI-enabled Resource Management in RAN Slicing,” IEEE Wireless Communications, pp. 1-10, Nov. 2022. b6 H. Zhang, H. Zhou, and M. Erol-Kantarci,“Federated Deep Reinforcement Learning for Resource Allocation in O-RAN Slicing,” in Proc. IEEE Glob. Commun. Conf. (GLOBECOM), pp. 958-963, Dec. 2022. b9 V. Mothukuri, R. M. Parizi, S. Pouriyeh, Y. Huang, A. Dehghantanha, and G. Srivastava, "A survey on security and privacy of federated learning," Future Gener. Comput. Syst., vol. 115, pp. 619–640, Feb. 2021. b10 L. Lyu, H. Yu, X. Ma, C. Chen, L. Sun, J. Zhao, Q. Yang, and S. Philip., "Privacy and robustness in federated learning: Attacks and defenses," in IEEE transactions on neural networks and learning systems, pp. 1-21, Nov. 2022. b4-a M. Usama and M. Erol-Kantarci, "A survey on recent trends and open issues in energy efficiency of 5G," Sensors, vol. 19, no. 14, p. 3126, Jul. 2019. b4 C. Liu, B. Natarajan, H. Xia, "Small cell base station sleep strategies for energy efficiency". in IEEE Trans. Veh. Technol. pp. 1652–1661, Mar. 2015. b13 P. Blanchard, E. M. E. Mhamdi, R. Guerraoui, and J. Stainer, “Machine learning with adversaries: Byzantine tolerant gradient descent,” in Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems, pp. 119–129, Dec. 2017. b11 V. Tolpegin, S. Truex, M. E. Gursoy, and L. Liu, “Data poisoning attacks against federated learning systems,” arXiv preprint arXiv:2007.08432, Sep. 2020. b12 E. Bagdasaryan, A. Veit, Y. Hua, D. Estrin, and V. Shmatikov, “How to backdoor federated learning,” arXiv preprint arXiv:1807.00459, Apr. 2018. b14 A. Anwar and A. Raychowdhury, “Multi-task federated reinforcement learning with adversaries,” CoRR, vol. abs/2103.06473, Mar. 2021. b15 E. Ma and R. Etesami, "Local Environment Poisoning Attacks on Federated Reinforcement Learning," arXiv preprint arXiv:2303.02725, Apr. 2023. b15-1 T. Zheng and B. Li, “Poisoning attacks on deep learning based wireless traffic prediction,” in IEEE INFOCOM 2022 - IEEE Conference on Computer Communications, pp. 660–669, May. 2022. b15-2 Y. Shi and Y.E. Sagduyu, “How to Launch Jamming Attacks on Federated Learning in NextG Wireless Networks,” in IEEE Globecom Workshop on 5G and Beyond Wireless Security (Wireless-Sec), pp. 945-950, Jan. 2022. b16 M. Masoud, M.G. Khafegy, E. Soroush, and D. Giacomelli, “Reinforcement Learning for Traffic-Adaptive Sleep Mode Management in 5G Networks,” IEEE Annual International Conference, pp. 1-6, Aug. 2020.b17-1 M. A. Habib, H. Zhou, P. E. Iturria-Rivera, M. Elsayed, M. Bavand, R. Gaigalas, S. Furr, and M. Erol-Kantarci, “Traffic Steering for 5G Multi-Rat Deployments using Deep Reinforcement Learning,” in IEEE Consumer Communications and Networking Conference (CCNC), pp. 164-169, Jan. 2023. b17-a A. E. Amine, P. Dini, and L. Nuaymi, “Reinforcement learning for delay-constrained energy-aware small cells with multi-sleeping control,” in Proc. IEEE Int. Conf. Commun. Workshops (ICC Workshops), pp. 1–6, Jun. 2020. b19 Y. Fraboni, R. Vidal, and M. Lorenzi, “Free-rider attacks on model aggregation in federated learning,” in Proc. Int. Conf. Artif. Intell. Statist., pp. 1846–1854, Jun. 2021. b18 H. Zhou, L. Kong, M. Elsayed, M. Bavand, R. Gaigalas, S. Furr, and M. Erol-Kantarci, “Hierarchical reinforcement learning for RIS- assisted energy-efficient RAN,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), pp. 3326–3331, Dec. 2022. | http://arxiv.org/abs/2311.15894v1 | {
"authors": [
"Han Zhang",
"Hao Zhou",
"Medhat Elsayed",
"Majid Bavand",
"Raimundas Gaigalas",
"Yigit Ozcan",
"Melike Erol-Kantarci"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20231127145943",
"title": "Distributed Attacks over Federated Reinforcement Learning-enabled Cell Sleep Control"
} |
0.97 [2010]82C24, 60H15, 58J65, 35R60Mapping quantum circuits to shallow-depth measurement patterns based on graph states [==================================================================================== January 14, 2024We study a stochastic geometric flow that describes a growing submanifold M()⊆^+̣1. It is an SPDE that comes from a continuum version of origin-excited random walk or once-reinforced random walk <cit.>. It is given by simultaneously smoothing and inflating the boundary of M() in a neighborhood of the boundary trace of a reflecting Brownian motion. We show that the large-scale fluctuations of an associated height function are given by a regularized Kardar-Parisi-Zhang (KPZ)-type equation on a submanifold in ^+̣1, modulated by a Dirichlet-to-Neumann operator. This is shown in any dimension ≥̣1. We also prove that in dimension +̣1=2, the regularization in this KPZ-type SPDE can be removed after renormalization. Thus, in dimension +̣1=2, fluctuations of the geometric flow have a double-scaling limit given by a singular KPZ-type equation. To our knowledge, this is the first instance of KPZ-type behavior in stochastic Laplacian growth models, which was asked about (for somewhat different models) in <cit.>. § INTRODUCTION Stochastic interfaces driven by harmonic measure provide rich models for many biological and physical processes, including (internal) diffusion-limited aggregation <cit.>, dielectric breakdown <cit.>, as well as the Hastings-Levitov process <cit.>, the last of which also has connections to turbulence in fluid mechanics. Because the driving mechanism for the growth is determined by harmonic measure, such interfaces are often known as (stochastic) Laplacian growth models.A central question concerns the large-scale behavior of these interfaces <cit.>. In <cit.>, the authors asked whether or not the stochastic interface studied therein has a Kardar-Parisi-Zhang (KPZ) scaling limit (given by a stochastic PDE such as (<ref>)). (See also <cit.> in the physics literature, which addresses a related question for diffusion-limited aggregation, namely its relation to the so-called “Eden model".) Since then, the question of rigorously deriving KPZ asymptotics in stochastic Laplacian growth models more generally has remained open, despite surging interest in KPZ universality over the past few decades <cit.>. The goal of this paper is to derive KPZ-type asymptotics for a Laplacian growth model given by a stochastic geometric flow. It is a stochastic PDE derived from a continuum version the origin-excited random walk and once-reinforced random walk with strong reinforcement, whose history and background is addressed at length in <cit.>. In this paper, we show the following two results.* Fluctuations of an associated “height function" to this stochastic flow converge (in some scaling limit) to a regularized KPZ-type equation. (See Theorem <ref>.)* After renormalization, solutions to the aforementioned KPZ-type equation converge as we remove the regularization; this is done on hypersurfaces of dimension =̣1 in ^2. (See Theorem <ref>.) Throughout this paper, we often use subscripts for inputs at which we evaluate functions of space, time, or space-time. This is in lieu of parentheses, which would make formulas and displays too overloaded. §.§ KPZ universality for the growth model Let us first give an intuitive description of the model. (Refer to Construction <ref> for the exact flow PDE we will study.)* Consider the following evolving submanifold M()⊆^+̣1. First, fix ≥0. Take a reflecting Brownian motion 𝔷 in M() with inward-normal reflection. Set 𝔶^_ to be the value of 𝔷 stopped when its boundary local time equals ^-4/3. (Here, >0 is a small scaling parameter. See Remark <ref> and Section <ref> for a discussion of the exponent -4/3. Also, by boundary local time, we mean the process that is formally written ↦∫_0^δ[𝔷_∈∂M[]̊]̣̊ and rigorously constructed by usual regularization of the delta function.) By construction, we know 𝔶^_∈∂M() with probability 1 for all . Lastly, since 𝔶^ moves at a fast speed of ^-4/3, we expect “homogenization", i.e. its small- contribution to average out.* The dynamics of M() are given by two mechanisms. The first is “inflating" M() smoothly and locally near 𝔶^_ at constant speed. The second is simultaneously smoothing ∂M() using a heat flow. This second ingredient is needed to make sure that the small- scaling limit that we will take for M() is well-defined, for example. (The use of a heat flow to regularize the interface is a natural choice, especially in the context of KPZ growth <cit.>.) See Sections <ref> and <ref> for more on this heat-flow smoothing of ∂M() (and what happens if we drop it from the growth dynamics). The model here strongly resembles those of <cit.>; see Section <ref> for further discussion. The notation 𝔶^ and 𝔷 will not be used in the sequel, since we will shortly reparameterize these processes.A discrete-time, discrete-step version of the above model would be given by running a reflecting Brownian motion inside M() until its boundary local time is equal to ^-4/3. At the stopping location of this Brownian motion, augment M() by adding a smooth, localized bump. Then, smooth the interface ∂M() via heat flow. Finally, we iterate, but with the new starting location of the Brownian particle and the updated, augmented set. The above continuum construction (with inflation) is just more convenient to analyze; we expect the two to have the same large-scale behavior. See Figure <ref> for the discrete version. As is often the case, for example with mean curvature flow, it is more convenient for analysis to think of the manifold process M() in terms of an evolving (stochastic) function on the boundary ∂M of the initial set M(0)=M. Precisely, there is an outward-pointing vector field on ∂M() (which determines the “direction" of inflation), which one can pullback to an evolving vector field on ∂M. Given this time-dependent vector field on ∂M, one can then reconstruct M() by taking any point on ∈∂M, and following the vector field for some “length" or “height", which we denote by 𝐈^_,. See Figures <ref> and <ref>. This description of M() only holds (in principle) until the diffeomorphism class of M() changes. If the diffeomorphism class changes at time 𝔱, there can exist a point on ∂M(𝔱) which can be arrived at by following vector fields starting at two different points on ∂M; this makes the construction ill-defined. In this paper, we assume the diffeomorphism class does not change. See Section <ref> for a further discussion.Thus, the →0 asymptotics of M() amount to those of the vector field process and the 𝐈^ process. In this paper, we focus on the latter, which we formulate as follows. (In particular, we make precise “inflation".) We explain Construction <ref> in detail afterwards. Again, refer to Remark <ref> and Section <ref> for more on the choice of exponents -1/3 and -4/3 appearing below. Fix ≥̣1, and take a compact, connected subset M⊆^+̣1 with smooth boundary ∂M. Let ↦(𝐈^_,·,𝔮^_)∈𝒞^∞(∂M)×∂M be the following Markov process.* With notation to be explained afterwards, we let 𝐈^ solve ∂_𝐈^_,=Δ_∂M𝐈^_,+^-1/3Vol_𝐈^_𝐊_,𝔮^_.* Δ_∂M is the Laplacian on the embedded manifold ∂M.* For any 𝐈∈𝒞^∞(∂M), the term Vol_𝐈 is the volume of the image of ∂M under 𝐈, i.e. the volume of its graph. By a standard change-of-variables computation, we have Vol_𝐈=∫_∂M(1+|_∂M𝐈_|^2)^1/2, whereis integration with respect to surface measure on ∂M.* The kernel 𝐊∈𝒞^∞(∂M×∂M) is real-valued, symmetric, and it satisfies ∫_∂M𝐊_,=1.* We now define 𝔮^ to be 𝔟 stopped when its boundary local time is equal to ^-4/3, where 𝔟 is a reflecting Brownian motion on M with metric determined by 𝐈^. Precisely:* For any 𝐈∈𝒞^∞(∂M), consider the graph map ∂M→∂M× given by ↦(,𝐈_). Equip ∂M× with the product metric (here ∂M is given Euclidean surface metric). Let 𝐠[_∂M𝐈] be the pullback of this metric under 𝐈. (It is a metric on ∂M. This notation is used since it depends only on first-order derivatives of 𝐈.) Extend 𝐠[_∂M𝐈] from ∂M to M smoothly.* Let 𝔟 be reflecting Brownian motion on ∂M with respect to the time-dependent metric 𝐠[_∂M𝐈^_]. (So, its infinitesimal generator at timeis Laplacian on M with metric 𝐠[_∂M𝐈^_].) The submanifold M is the initial data for M(t).As we mentioned earlier, the Laplacian in (<ref>) is a canonical smoothing operator to consider for interface growth. This term describes the heat-flow smoothing of the interface. Again, see Section <ref> for what happens without it. The second term on the RHS of (<ref>) is the smooth “inflation" (the kernel 𝐊 determines the “shape" of the inflating set). Also, Vol on the RHS of (<ref>) is a global factor to normalize the speed of growth at any point. (Without it, →0-limits in this paper would be non-local; we explain this shortly.) Let us also explain extending 𝐠[_∂M𝐈] from ∂M to M. Allowing for any smooth extension is essentially the same as allowing the outwards flow of ∂M to be in the direction of any smooth outward vector field on ∂M. This is some hint of universality (see also Section <ref>).Construction <ref> is not just an isolated construction. It is an SPDE that one derives from the description given before Figure <ref> (given an appropriate, precise formulation of that model).It turns out that 𝐈^ has an uninteresting limit as →0; in this limit, we ultimately have 𝐈^_,-^-1/3→0 locally uniformly in (,). Indeed, if one believes that the 𝔮^ averages out because it moves at the very fast speed of ^-4/3, then the growth model essentially inflates at speed 1 at every point on the boundary. (Without the Vol factor on the RHS of (<ref>), the model instead samples a point uniformly at random on the interface to inflate from.) What is more interesting is the following fluctuation field: 𝐘^_,=^-1/3[𝐈^_,-^-1/3]. Let us explain the choice of scaling in Construction <ref> and (<ref>). The equation (<ref>) describes a fast-moving particle of speed ^-4/3 that induces speed ^-1/3 growth of the interface. This is the same as speeding time (by ^-4/3) a process with a speed 1 particle and speedgrowth. The deterministic, leading-order “shape theorem" behavior of the interface occurs at time-scales of speed ^-1, in which the particle has speed ^-1 and the interface growth has speed 1. On top of this, if we additionally scale time by ^-1/3, which gives (<ref>) and (<ref>), we obtain nontrivial fluctuations. This is what (<ref>) and (<ref>) imply as far as scaling is concerned. We could have defined 𝐈^ to have speed ^-1 and then speed it up by ^-1/3, which is perhaps more natural. But, this would have been notationally more involved.The scaling used (to go from the asymptotic shape to fluctuations) of time and of the fluctuation itself are the same factor (here, it is ^-1/3). This is the case for (many) KPZ models more generally <cit.>. Assume that 𝐘^_0,·=𝐘^init_0,· for some 𝐘^init_0,·∈𝒞^∞(∂M,) independent of . It turns out that the small- limit of 𝐘^ is given by the following SPDE, which we explain afterwards: ∂_𝔥^𝐊_, =Δ_∂M𝔥^𝐊_,+∫_∂M𝐊_,|_∂M𝔥^𝐊_,|^2+∫_∂M[𝐊_,-1](-ℒ)^-1/2ξ_, 𝔥^𝐊_0,· =𝐘^init_0,·. Technically, by (<ref>), we mean the Duhamel representation below (see Lemma <ref>): 𝔥^𝐊_, ={exp[Δ_∂M]𝐘^init_0,·}_+∫_0^{exp[(-)Δ_∂M]∫_∂M𝐊_·,|_∂M𝔥^𝐊_,|^2}_+∫_0^{exp[(-)Δ_∂M]∫_∂M[𝐊_·,-1](-ℒ)^-1/2ξ_,}_. Let us now explain the notation used in this SPDE.* exp[𝔱Δ_∂M] is the associated heat semigroup for Δ_∂M, and _∂M denotes gradient on ∂M.* ℒ denotes the Dirichlet-to-Neumann map on M. Given any φ∈𝒞^∞(∂M), the function ℒφ is defined to be ↦_𝖭[]𝒰^φ_, where _𝖭[] is gradient in the unit inwards normal direction at ∈∂M, and 𝒰^φ is the harmonic extension of φ to M (so that Δ𝒰^φ=0, where Δ is Laplacian on M⊆^+̣1).It turns out that ℒ is a self-adjoint operator with core 𝒞^∞(∂M) (with respect to the surface measure on ∂M, i.e. the Riemannian measure induced by surface metric on ∂M). It is negative semi-definite, and it has a discrete spectrum. Also, its null-space is one-dimensional; it is spanned by constant functions on ∂M. So, (-ℒ)^-1/2 on the RHS of (<ref>) is well-defined, since the function ↦𝐊_,-1 is orthogonal to the null-space of ℒ (i.e. it has vanishing integral on ∂M for any ). See Lemma <ref>.* ξ is a space-time white noise on [0,∞)×∂M. Intuitively, it is the Gaussian field with covariance kernel ξ_,ξ_,=δ_=δ_=. More precisely, for any orthonormal basis {e_}_ of L^2(∂M) (e.g. the eigenbasis for ℒ), it has the following representation (in the language of Ito calculus), where b_,k are independent standard Brownian motions: ξ_,·ṭ=∑_ḅ_,ke_k. We emphasize that (<ref>) is essentially the usual KPZ equation (see <cit.>) except for two differences. The first is the regularization kernel 𝐊; we will shortly consider the delta-function limit for 𝐊 in the case =̣1. The second is the (-ℒ)^-1/2 operator. We explain this term immediately after Theorem <ref>.Before we state the first main result (convergence of 𝐘^→𝔥^𝐊), we comment on well-posedness of (<ref>). By smoothing of the Δ_∂M semigroup (see Lemma <ref>) and because ℒ maps smooth functions to smooth functions (see Lemma <ref>), it is not hard to see that (<ref>)-(<ref>) is locally well-posed in 𝒞^2(∂M) (until a possibly random and finite stopping time denoted by τ_𝔥^𝐊). Finally, let us introduce the following notion of high probability (to be used throughout this paper). We say an event ℰ holds with high probability if ℙ[ℰ]→1 as →0. There exists a coupling between {𝐘^}_→0 and 𝔥^𝐊 such that with high probability, for any δ>0 and 0≤τ≤τ_𝔥^𝐊-δ, the following holds for some κ[δ,]≥0 that vanishes as →0: sup_0≤≤τ∧1𝐘^_,·-𝔥^𝐊_,·_𝒞^2(∂M)≤κ[δ,]. (Here, 𝒞^2(∂M) is the usual space of twice continuously differentiable functions on the manifold ∂M. Also, we have used the notation a∧ b=min(a,b).) Theorem <ref> essentially asserts convergence in law of 𝐘^ to 𝔥^𝐊. Because of the need for a stopping time τ_𝔥^𝐊, we found it most convenient to state it in terms of couplings. Also, the utility of stopping before 1 is to make sure we work only on compact time-intervals. Of course, there is nothing special about 1. Finally, we could have used 𝒞^(∂M) for any ≥̨2. Going to =̨1, for example, perhaps requires more work.Theorem <ref> implies KPZ behavior for fluctuations 𝐘^ of the stochastic flow at hand. Although (<ref>) has the extra (-ℒ)^-1/2-operator that does not show up in the “standard" KPZ equation, the implication of Theorem <ref> is that the stochastic, slope-dependent growth (i.e. the last term in (<ref>)) is asymptotically a quadratic function of the slope plus a noise; this indicates KPZ behavior. The reason why (-ℒ)^-1/2 appears in the scaling limit for 𝐘^ is because fluctuations in the growth model come from a single Brownian particle which explores the entire manifold. Thus, a non-local operator must hit the noise. (The reason for ℒ in particular is because ℒ is the generator for a reflecting Brownian motion on the time-independent manifold M time-changed via boundary local time; see <cit.>.)Theorem <ref> holds locally in time (until an -independent stopping time). This is more-or-less because we work in the 𝒞^2(∂M) topology, not a weaker topology like 𝒞^0(∂M), for example; see Remark <ref>.For a more detailed explanation of why Theorem <ref> is true, see after Theorems <ref> and <ref>. §.§ The singular limit of (<ref>) In (<ref>), if we formally replace 𝐊 by the delta function on the diagonal of ∂M×∂M, we get the following SPDE, which we (formally) pose in any dimension ≥̣1: ∂_𝔥_,=Δ_∂M𝔥_,+|_∂M𝔥_,|^2+Π^⊥(-ℒ)^-1/2ξ_,,(,)∈(0,∞)×∂M. Above, Π^⊥ denotes projection away from the null-space of ℒ, i.e. away from the span of constant functions on ∂M. Our goal now is to make sense of (<ref>) itself, so that we can rigorously show convergence of (<ref>) to (<ref>) in the limit where 𝐊 converges to the delta function. However, (<ref>) is not classically well-posed. Indeed, ℒ is a first-order elliptic pseudo-differential operator, so (-ℒ)^-1/2 gains half a derivative. But integrating the heat kernel for Δ_∂M against ξ, in dimension 1, lets us take strictly less than half a derivative. So, we cannot take a full derivative and expect to get a function that we can square to define the quadratic nonlinearity in (<ref>). Thus, we must carry out the standard procedure for singular SPDEs via regularization, renormalization, and showing existence of limits as we remove the regularization (see <cit.>, for example). In particular, we consider the following SPDE, which we explain afterwards: ∂_𝔥^η_,=Δ_∂M𝔥^η_,+|_∂M𝔥^η_,|^2-𝒞_η+Π^η(-ℒ)^-1/2ξ_,.* Π^η is projection onto the direct sum of the first ⌊η^-1⌋-many eigenspaces of ℒ with non-zero eigenvalues. (By Lemma <ref>, the spectrum of ℒ is discrete. Also, we order eigenspaces by ordering the absolute values of their eigenvalues in increasing order. Thus, Π^η→Π^⊥ as η→0 in a weak sense.)* To specify the constant 𝒞_η, we first let {λ_ℓ}_ℓ=1^∞ be the non-zero eigenvalues of -ℒ, ordered in the same fashion as the previous bullet point. (Note that λ_ℓ>0 for all ℓ≥1). Define 𝒞_η :=π/2∑_ℓ=1^1/ηλ_ℓ+1-λ_ℓ/λ_ℓ+1. It can be checked by Lemma <ref> that 𝒞_η diverges as η→0 for any dimension $̣; so, (<ref>) is “singular".Standard PDE theory implies that (<ref>) is locally well-posed with smooth solutions for anyη>0fixed. The following result, which restricts to dimension +̣1=2, says theη→0limit of these solutions exists. (Although Theorem <ref> looks like a standard SPDE result, we explain shortly why it has geometric content. In particular, we are solving a singular SPDE on a manifold, whose geometry plays an important role.) Suppose M⊆^2 is a compact manifold with smooth boundary ∂M.For any 𝔥^initial∈𝒞^2(∂M) independent of η>0, the sequence of solutions {𝔥^η}_η>0 to (<ref>) with initial data 𝔥^initial converges in probability in the following (analytically) weak sense. There exists a stopping time τ_(<ref>)[𝔥^initial] such that for any test function 𝙵∈𝒞^∞(×∂M) and any τ∈[0,τ_(<ref>)[𝔥^initial]), the sequence of random variables below converges in probability as η→0: ∫_[0,τ)∫_∂M𝙵_,𝔥^η_,. The regularization of (<ref>) that we want to study in view of (<ref>) is not quite (<ref>); the quadratic nonlinearity should be regularized. In particular, if Π^η,∼=Π^η+Π^const, where Π^const is projection onto the space of constant functions, then we want to replace|_∂M𝔥^η_,|^2↦Π^η,∼[|_∂M𝔥^η_,|^2].The proof of Theorem <ref> can be used verbatim to handle this change, but several details become noticeably more complicated for entirely technical reasons (e.g. having to estimate the commutator of semigroups of Δ_∂M and ℒ using [Δ_∂M,ℒ]; see Lemma <ref>). It is for this reason only that we study (<ref>).One can also solve (<ref>) by defining the Cole-Hopf map 𝔥:=logℨ, where ℨ solves a linear SPDE. This is likely to give the same object as the limit in Theorem <ref> for M⊆^2. In particular, this gives a possible way to show infinite lifetime for (<ref>) (in a topology that is much weaker than 𝒞^2(∂M)). There is also a chance of using this solution theory to push to >̣1. However, this linearization method does not work for the SPDE alluded to in Remark <ref>, so we do not pursue this here.Theorem <ref> states a weak version of convergence for𝔥^η. But, it can be upgraded rather easily to a more quantitative convergence using our methods. We do not pursue this here, since it is more of a detail than the main point. (In a similar spirit, the assumption that𝔥^initialis in𝒞^2(∂M)is likely sub-optimal, but this is besides the point as well.)An interesting observation about Theorem <ref> is that𝒞_ηin (<ref>) is constant in space, even thoughΔ_∂Mandℒtherein look different at different points on∂M(the metric is varying on∂M). Roughly speaking, the reason why we have constant renormalization in (<ref>) comes from the universality (in space) of the so-called local or pointwise Weyl law. This yields pointwise eigenfunction asymptotics (in this case forℒ) that depend only on eigenvalues ofℒ, not on space or the manifold otherwise. We explain this point more in Section <ref> along with other geometric-analytic inputs needed to solve (<ref>) that seem to be absent in previous studies of singular SPDEs.By Theorems <ref> and <ref>, in the case=̣1(so hypersurfaces in^2), we get a singular KPZ-type equation limit for (<ref>). In particular, we can take𝐊in Theorem <ref> to converge to a delta function on the diagonal of∂M×∂Msufficiently slowly and deduce convergence of𝐘^to (<ref>). Let us also mention that the analytic topologies used in Theorems <ref> and <ref> are quite different (𝒞^2(∂M)versus weak-∗convergence). As we mentioned above, improving the topology of convergence in Theorem <ref> is probable, but it is not likely that it holds in𝒞^2(∂M), since (<ref>) is a singular SPDE. Convergence in Theorem <ref> in a topology weaker than𝒞^2(∂M)seems to be difficult (as noted after Theorem <ref>), since the proof is largely based on elliptic regularity. It would be interesting to close this gap; this would strengthen the double-scaling limit result (e.g. quantify convergence of𝐊to a delta).§.§ Background and previous work §.§.§ KPZ from stochastic geometric flowsTo our knowledge, any sort of KPZ universality for diffusions interacting with their range has not appeared in the literature (rigorously) before, even though this question had been asked by <cit.> almost 20 years ago. The closest work, which is still quite different, we are aware of to ours is <cit.>. However, <cit.> has randomness coming from a background environment (with mixing and independence-type properties), while the randomness in our flow model comes from a single particle.§.§.§ Singular SPDEs on manifoldsWhile we were completing this work, <cit.> appeared on the arXiv. It treats SPDEs on manifolds using regularity structures <cit.>. The SPDEs in <cit.> do not have non-local noise driven by a Dirichlet-to-Neumann operator. This allows <cit.> to work in Hölder (or Besov) spaces, while we require estimates inL^2-based Sobolev spaces to handleℒ(see Section <ref>). In particular, the SPDE that we consider in this paper is genuinely different from those in <cit.>. However, some conclusions derived here and in <cit.> are similar. For example, renormalization counter-terms in <cit.> are constant as well. (This is true unless the equation is sufficiently singular, in which case one must also renormalize via scalar curvature. We do not treat this case here.) In <cit.>, this is shown by classical asymptotics of the heat kernel on the manifold. (Given connections between Weyl laws and heat kernels, we expect that our methods are intimately connected.)§.§.§ Shape theoremsThis paper studies fluctuation scaling for the height function, i.e. study (<ref>). In <cit.>, we studied (a Poissonization of) the discrete version of Figure <ref> without heat flow regularization. The main result of <cit.> was a shape theorem for the growth model therein, in particular a scaling limit for the evolving vector field process that we alluded to before Construction <ref>, but on time-scale that is slower by a factor of ^1/3 than the one we consider in this paper. In particular, under the scaling of <cit.>, the heat flow term in (<ref>) vanishes, so the results of <cit.> would hold even if we included said term. A similar shape theorem (for more general processes than Brownian motion but for a more restricted spherical geometry) was shown in <cit.>.§.§ A word about KPZ universalityThe methods we use require very little about the Brownian nature of the randomness in (<ref>). (Indeed, as indicated in Section <ref>, only spectral gap estimates are needed.) This can be interpreted as another instance of universality. (Of course, if we change Brownian motion to another process, theℒ-operator in (<ref>) may change. The universal KPZ quadratic, however, will not.)If we drop Laplacians in (<ref>) and (<ref>), Theorem <ref> would still hold for the resulting SPDEs. (Indeed, (<ref>) is still a smoothing equation because𝐊therein is smooth.) In this case, Theorem <ref> gives a universality result that more resembles the so-called “KPZ fixed point" <cit.>. However, we would not be able to rigorously take the singular limit as in Theorem <ref>. §.§ Changing diffeomorphism classWe assumed that the diffeomorphism class of the growing set does not change. In general, we can stop the growth process when the diffeomorphism class of the set changes; Theorem <ref> would remain true. As for global-in-time scaling limits (e.g. what the limit SPDE should even be) in the case where the diffeomorphism class changes, studying more carefully the Brownian particle may provide key information, similar to <cit.>. (In <cit.>, singularities in a Stefan PDE are resolved using a particle system.) §.§ Organization of the paperSection <ref> outlines the methods (and essentially proves Theorems <ref> and <ref> modulo technicalities to be checked). The rest of the paper is outlined at the end of Section <ref>. §.§ AcknowledgementsWe thank Martin Hairer and Harprit Singh for useful conversation regarding their recent work. Research supported in part by NSF grant DMS-1954337 (A.D.), and by the NSF Mathematical Sciences Postdoctoral Fellowship program under Grant. No. DMS-2203075 (K.Y.). § FUNCTION SPACES AND OTHER NOTATIONWe now give a list of function spaces (and a few other pieces of notation) to be used throughout the paper. *For any set I and a,b∈, when we write a≲_Ib, we mean |a|≤Λ|b| for an implied constant Λ≥0 depending only on I. (If I is a finite subset of ^n for some n≥1, the dependence of Λ is assumed to be smooth in the elements of I.) By a≳_Ib, we mean b≲_Ia. By a≍ b, we mean a≲ b and b≲ a with possibly different implied constants. Also, by a=O_I(b), we mean a≲_Ib.*For any a,b∈, we define a∧ b=min(a,b).*When we say β∈ is uniformly positive, we mean β≥ C for C>0 that depends on no parameters.*For ≥1, let L^(∂M) be the usual L^-space, where ∂M⊆^+̣1 has Riemannian surface measure.*Fix any integer ≥̨0. Fix an orthonormal frame 𝖾_1,…,𝖾_ (i.e. a smoothly varying orthonormal basis for tangent spaces of the manifold ∂M with Euclidean surface metric). For smooth φ:∂M→, set φ_𝒞^(∂M):=sup_∈∂M|φ_|+sup_∈∂Msup_ı_1,…,ı_|_ı_1…_ı_φ_,|,where _ı is gradient in the direction of the orthonormal frame vector 𝖾_ı, and the inner supremum is over subsets of size $̨ in{1,…,}̣. Let𝒞^(∂M)be the corresponding closure of smooth functions on∂M.*Let𝒞^0,υ(∂M), forυ∈(0,1), be the Hölder norm on the manifold∂Mwith Euclidean surface metric).*Fix𝔱≥0. Let 𝒞^∞_𝔱𝒞^∞_∂M be the space of smoothφ:[0,𝔱]×∂M→. Fix integers_̨1,_̨2≥0, and set φ_𝒞^_̨1_𝔱𝒞^_̨2_∂M:=sup_0≤≤𝔱{∂_^_̨1φ_,·_𝒞^0(∂M)+φ_,·_𝒞^_̨2(∂M)}, φ∈𝒞^∞_𝔱𝒞^∞_∂M.We let 𝒞^_̨1_𝔱𝒞^_̨2_∂M be the closure of smooth functions on[0,𝔱]×∂Munder this norm.*Fix any integer≥̨0. For anyφ:∂M→smooth, we define φ_H^(∂M)^2:=φ_L^2(∂M)^2+sup_ı_1,…,ı__ı_1…_ı_φ_L^2(∂M)^2.LetH^(∂M)be the closure of𝒞^∞(∂M)under this norm. For any fractionalα≥0, define theH^α(∂M)via the usual interpolation procedure. (It is enough to takeα≥0to be an integer throughout this paper. Alternatively, one can cover∂Mwith an atlas, define theH^α(∂M)-norm by using a diffeomorphism with an open subset of^, and sum over all charts in the atlas.)*Fix any integer≥̨0andα≥0. Fix any𝔱≥0. For anyφ:[0,𝔱]×∂M→smooth, we define φ_𝒞^_𝔱H^α_∂M:=sup_0≤≤𝔱{∂_^_̨1φ_,·_H^0(∂M)+φ_,·_H^α(∂M)}.We let 𝒞^_𝔱H^α_∂M be the closure of smooth functions on[0,𝔱]×∂Munder this norm. § OUTLINE OF THE PROOFS OF THEOREMS <REF> AND <REF>We give steps towards proving Theorem <ref>. We then describe the technical heart to prove each step (and Theorem <ref>) in Section <ref>. We conclude this section with an outline for the rest of the paper. §.§ Step 1: comparing 𝐘^ to an -dependent SPDEEven if one computes the evolution equation for𝐘^using (<ref>) and (<ref>), it is not clearly an approximation to (<ref>). (The problem is the last term in (<ref>).) The first step towards proving Theorem <ref> is to therefore compare𝐘^to the PDE∂_𝐡^_, =Δ_∂M𝐡^_,+∫_∂M𝐊_,|_∂M𝐡^_,|^2+M^_, 𝐡^_0,· =𝐘^init_0,·,where𝐌^denotes a martingale that “resembles" the last term in the differential equation (<ref>). Let us make precise what “resembles" means in the following definition (which we explain afterwards). We say the process ↦𝐌^_,·∈𝒞^∞(∂M) is a good martingale if the following hold. *The process ↦𝐌^_,·∈𝒞^∞(∂M) is a cadlag martingale with respect to the filtration of (𝐈^,𝔮^). Next, fix any stopping time 0≤τ≤1. With probability 1, if 𝔱≤τ is a jump time, then for any ≥̨0 and for some κ[] that vanishes as →0, we have the following, in which ≲ means big-Oh: 𝐌^_𝔱,·-𝐌^_𝔱^-,·_𝒞^(∂M)≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂Mκ[]. *Fix any Λ≥0 and any stopping time 0≤τ≤1 such that for all ≤τ, we have 𝐘^_,·_𝒞^2(∂M)≤Λ. For any ≥̨0 deterministic, we have the following with high probability: sup_0≤≤τ𝐌^_,·_𝒞^(∂M) ≲_,̨Λ1.For any stopping time 0≤τ≤1, with high probability, we have the following for any ≥̨0: sup_0≤≤τ[𝐌^]_,·-[𝐌^limit]_,·_𝒞^(∂M) ≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β,The exponent β>0 is fixed (e.g. independent of all other data, including ), and[𝐌^limit]_,:=2∫_∂M[𝐊_,-1]×{-ℒ^-1[𝐊_,-1]}is a time-integrated “energy" functional. Let us now explain Definition <ref>. The cadlag-in-time and smooth-in-space regularity of𝐌^is enough for local well-posedness of (<ref>) in𝒞^2(∂M), for example. Indeed, if one writes (<ref>) in its Duhamel form (see Lemma <ref>), then one can move the time-derivative acting on𝐌^in (<ref>) onto the heat kernel of the semigroup𝔱↦exp[𝔱Δ_∂M]. This turns into a LaplacianΔ_∂Macting on said heat kernel, which is okay since we integrate against𝐌^in space, and𝐌^is smooth in space (see Lemma <ref>).We now explain the second bullet point in Definition <ref>. It first states a priori control on regularity of the martingale (in a way that is technically convenient later on). It also says that at the level of bracket processes,𝐌^matches the last term in (<ref>) up toO(^β). By standard martingale theory, this is enough to characterize the small-limit of𝐌^. We expand on this in the discussion of the next step, Theorem <ref>. There exists a good martingale 𝐌^ in the sense of Definition <ref> such that if 𝐡^ is the solution to (<ref>) with this choice of 𝐌^, then we have the following. *First, for any Λ≥0, define the stopping timeτ_𝐡^,Λ=inf{≥0:𝐡^_,·_𝒞^2(∂M)≥Λ}. *There exists β>0 independent of all other parameters, including , such that for any Λ≥0 and δ>0, we have the following estimate with high probability: sup_0≤≤(τ_𝐡^,Λ∧1)-δ𝐘^_,·-𝐡^_,·_𝒞^2(∂M)≲_δ,Λ^β.(The ∧ notation means minimum. Also, β here may not match β in Definition <ref>.) Let us now briefly explain what Theorem <ref> is saying exactly (and why it is even plausible). *In words, Theorem <ref> says that we can couple 𝐘^ to the solution 𝐡^ of (<ref>) if we make an appropriate choice of good martingale 𝐌^, which comes from a martingale decomposition for the last term in (<ref>).*The Laplacian in (<ref>) clearly matches that in (<ref>).*Take the second term on the RHS of (<ref>). Even though 𝔮^ is not Markovian because the underlying metric is determined by the 𝐈^ process, it is much faster than 𝐈^, so it is the unique “fast variable" (in the language of homogenization). Thus, we expect that the second term on the RHS of (<ref>) homogenizes in 𝔮^ with respect to the Riemannian measure induced by 𝐠[_∂M𝐈^]. (Intuitively, on time-scales for which 𝔮^ homogenizes, 𝐈^ is roughly constant. So, 𝔮^“looks" Markovian, and we have homogenization.) Thus, we replace the second term on the RHS of (<ref>) by the following homogenized statistic (if we include the extra ^-1/3 scaling in (<ref>)): ^-2/3∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2.(We clarify that (1+|_∂M𝐈^_,|^2)^1/2 is the Riemannian measure induced by 𝐠[_∂M𝐈^_,·].) We can now Taylor expand in _∂M𝐈^=^1/3_∂M𝐘^ to second-order to turn (<ref>) into the second term on the RHS of (<ref>) but evaluated at 𝐘^ instead of 𝐡^ (plus lower-order errors).*It remains to explain the noise in (<ref>). It turns out replacing the second term on the RHS of (<ref>) by (<ref>) does not introduce vanishing errors. This fluctuation is order 1. Indeed, the difference of the last term in (<ref>) and (<ref>) is a noise of speed ^-4/3. After we time-integrate, we get square-root cancellation and a power-saving of (^-4/3)^-1/2=^2/3. This cancels the ^-2/3-scaling of (<ref>).§.§ Step 2: the small- limit of 𝐡^The remaining ingredient to proving Theorem <ref> is the following. It is essentially Theorem <ref> but for𝐡^instead of𝐘^. Recall notation of Theorem <ref>. There exists a coupling between the sequence {𝐡^}_→0 and 𝔥^𝐊 such that the following two points hold with high probability. *For any ρ>0, there exists Λ=Λ(ρ) so that for all >0 small, we have τ_𝔥^𝐊∧τ_𝐡^,Λ≥τ_𝔥^𝐊-ρ.*For any δ>0, there exists κ[δ,]≥0 that vanishes as →0 such that sup_0≤≤(τ_𝔥^𝐊∧1)-δ𝐡^_,·-𝔥^𝐊_,·_𝒞^2(∂M)≤κ[δ,].(To be totally clear, point (1) in Theorem <ref> states thatτ_𝐡^,Λ≈τ_𝔥^𝐊if we takeΛ>0sufficiently large and>0sufficiently small. The key feature is that the necessary choice ofΛdoes not depend on>0.)Taking a minimum with τ_𝔥^𝐊 is probably unnecessary in the first point of Theorem <ref>, but it makes things easier. In any case, convergence in both points (1) and (2) of Theorem <ref> is classical, because both SPDEs (<ref>) and (<ref>) are parabolic equations with smooth RHS. The one detail that may be subtle is that the noise in (<ref>) is only weakly close to that in (<ref>). (Indeed, control of predictable brackets (<ref>) is not a very strong statement.) Thus, we need to show that (<ref>) is characterized by a martingale problem (which, again, is not hard because (<ref>) and (<ref>) have smooth RHS). §.§ Proof of Theorem <ref>, assuming Theorems <ref> and <ref>Takeδ>0small and0≤τ≤[1∧τ_𝔥^𝐊]-δ. By the triangle inequality, we havesup_0≤≤τ𝐘^_,·-𝔥^𝐊_,·_𝒞^2(∂M)≤sup_0≤≤τ𝐘^_,·-𝐡^_,·_𝒞^2(∂M)+sup_0≤≤τ𝐡^_,·-𝔥^𝐊_,·_𝒞^2(∂M).The last term on the RHS vanishes as→0in probability by point (2) of Theorem <ref>. In order to control the first term on the RHS, we first know with high probability thatτ_𝔥^𝐊-δ<τ_𝐡^,Λ-1/2δif we takeΛ≥0large enough (but independent of); this is by point (1) of Theorem <ref>. We can now use Theorem <ref> to show vanishing of the first term on the RHS of the previous display as→0. §.§ Technical challenges and methods for Theorems <ref> and <ref>As noted after Theorem <ref>, there is not much to its proof; so, we focus on the ideas behind Theorems <ref> and <ref>.§.§.§ Theorem <ref>Suppose, just for now until we say otherwise, that the Brownian particle 𝔮^ evolves on the manifold M with respect to the fixed, initial metric 𝐠[_∂M0]. (Put differently, suppose just for now that in the definition of𝔮^in Construction <ref>, we replace𝐠[_∂M𝐈^]by𝐠[_∂M0], where0denotes the0function.) In this case, we know that𝔮^is Markovian. Take the second term on the RHS of (<ref>); it is a function of𝔮^. We are interested in the fluctuation below, in which𝐈∈𝒞^∞(∂M)is arbitrary: 𝖥_𝐈,,𝔮^_:=^-2/3Vol_𝐈𝐊_,𝔮^_-^-2/3∫_∂M𝐊_,(1+|_∂M𝐈_|^2)^1/2.(We will only use the𝖥-notation in this outline.) Because of the italicized temporary assumption above, we will first consider the case where𝐈≡0until we say otherwise.As explained in the bullet points after Theorem <ref>, showing that (<ref>) is asymptotically the desired noise term is the only goal left. Write 𝖥_0,,𝔮^_=^-4/3ℒ[^-4/3ℒ]^-1𝖥_0,,𝔮^_.The inverse operator on the RHS of (<ref>) is well-defined, since (<ref>) (for𝐈≡0) vanishes with respect to the invariant measure ofℒ. Because^-4/3ℒis the generator of𝔮^by our italicized assumption above and Proposition 4.1 of <cit.>, we can use the Ito formula to remove the outer^-4/3ℒoperator at the cost of two copies of[^-4/3ℒ]^-1𝖥evaluated at different times (i.e. boundary terms) plus a martingale. Boundary terms are easy to control, since[^-4/3ℒ]^-1𝖥is intuitivelyO(^2/3). (This is by a spectral gap forℒto boundℒ^-1uniformly plus the a priori bound of order^-2/3for (<ref>).) As for the martingale, its scaling is order1as we explained in the fourth bullet point after Theorem <ref>. That its bracket has the form of a time-integrated energy (<ref>) (more or less) is because quadratic variations of Ito martingales are Carre-du-Champ operators.Now, we return the actual context in which the metric for 𝔮^ is determined by 𝐈^. In this case, we will be interested in (<ref>) for the actual interface process𝐈=𝐈^_,·:𝖥_𝐈^_,·,,𝔮^_:=^-2/3Vol_𝐈^_,·𝐊_,𝔮^_-^-2/3∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2.We no longer have an Ito formula for just𝔮^, since it is no longer Markovian. But, as noted after Theorem <ref>,𝔮^is still the unique fast variable; on time-scales for which it would homogenize if it were Markovian,𝐈^is approximately constant. So,𝔮^“looks Markovian" on time-scales that it sees as long. Thus, the same homogenization picture above should hold, if we replaceℒby Dirichlet-to-Neumann on the Riemannian manifold(M,𝐠[_∂M𝐈^_,·]), and the measure for homogenization is Riemannian measure induced by𝐠[_∂M𝐈^_,·].The way we make the previous paragraph rigorous and study (<ref>) resembles (<ref>), except we include the generator of the𝐈^_,·process as well. Let ℒ_total^ be the generator for the Markov process(𝐈^,𝔮^). Write 𝖥_𝐈^_,·,,𝔮^_=ℒ_total^[ℒ_total^]^-1𝖥_𝐈^_,·,,𝔮^_.We can then use Ito as before to remove the outer ℒ_total^-operator to get boundary terms and a martingale. Since𝔮^is much faster than𝐈^, the operator ℒ_total^ is asymptotically just the Dirichlet-to-Neumann operator on(M,𝐠[_∂M𝐈^]). In other words, dynamics of𝐈^, and theirO(^-1/3)contribution to the generatorℒ^_total, are lower-order. So, ifℒ^,𝐈_DtNdenotes the same scaling factor^-4/3times the Dirichlet-to-Neumann map on(M,𝐠[_∂M𝐈]), then sinceℒ^,𝐈^_,·_DtNis the generator for𝔮^_at time(again, see Proposition 4.1 in <cit.>), we get 𝖥_𝐈^_,·,,𝔮^_≈ℒ^,𝐈^_,·_DtN[ℒ^,𝐈^_,·_DtN]^-1𝖥_𝐈^_,·,,𝔮^_(Note thatℒ^,𝐈_DtNdepends on𝐈^_,·, reflecting the non-Markovianity of𝔮^.)Thus, our estimation of the boundary terms and martingale is the same as before. We deduce that (<ref>) is asymptotically a martingale whose bracket is (<ref>), exceptℒ, which is the Dirichlet-to-Neumann map onMwith metric𝐠[_∂M0], in (<ref>) is replaced by the Dirichlet-to-Neumann map onMwith metric𝐠[_∂M𝐈^], i.e.^4/3ℒ^,𝐈^_,·_DtN. Now use that𝐘^should be order1, so the metric𝐠[_∂M𝐈^]=𝐠[^1/3_∂M𝐘^](see (<ref>)) should be close to𝐠[_∂M0]. So, (<ref>) as written is indeed the right answer for asymptotics of the bracket for the martingale part of (<ref>). There are obstructions to this argument. The most prominent of which is that we cannot just remove the generator of𝐈^from ℒ_total^ in (<ref>). Indeed, this term acts on the resolvent in (<ref>); when it does, it varies the metric defining the resolvent and (<ref>) itself. However, our estimate for the resolvent in (<ref>) depends on vanishing of (<ref>) for𝐈=𝐈^after integration with respect to the measure on∂Minduced by𝐠[_∂M𝐈^](which, again, is changing when we act by the generator of𝐈^). In other words, estimates for the resolvent in (<ref>) rely on an unstable algebraic property of (<ref>) that is broken when we vary𝐈^. For this reason, we actually need regularize ℒ_total^ with a resolvent parameterλ, i.e. consider -λ+ℒ_total^ for0≤λ≪^-4/3instead of ℒ_total^. Indeed, the inverse of -λ+ℒ_total^ is always at most orderλ^-1, regardless of what it acts on. Moreover, sinceλ≪^-4/3is much smaller than the speed of𝔮^, once we useλ-regularization to remove the generator of𝐈^, we can then removeλitself, essentially by perturbation theory for resolvents. This is how we ultimately arrive at (<ref>) rigorously.We also mention the issue of the core/domain of the generator for𝐈^, because𝐈^is valued in an infinite-dimensional space of smooth functions. We must compute explicitly the action of the𝐈^-generator whenever we use it. This, again, is built on perturbation theory for operators and resolvents.§.§.§ Theorem <ref>The idea is that of Da Prato-Debussche <cit.>. In particular, (<ref>) can be viewed as a perturbation of ∂_𝔥^η,lin-Δ_∂M𝔥^η,lin-Π^η(-ℒ)^-1/2ξ=0.The solution to (<ref>) is Gaussian, but it is not a tractable Gaussian process, sinceℒand Δ_∂M generally do not commute (unlessMis a Euclidean disc in the case of dimension=̣1). In view of this, we replaceΔ_∂Mby-ℒ^2(which has the same order and sign as a pseudo-differential operator) to get the linear SPDE∂_𝔥^η,lin-(-ℒ^2)𝔥^η,lin-Π^η(-ℒ)^-1/2ξ=0.This diagonalizes into independent Ornstein-Uhlenbeck SDEs in the eigenbasis ofℒand is therefore very accessible to calculations and building stochastic objects out of. Before we discuss this point, we comment on the error we get when going from (<ref>) to (<ref>). A straightforward calculation shows we must control the operatorΔ_∂M+ℒ^2. It turns out, essentially by a geometric-analytic calculation, that 𝒪:=Δ_∂M+ℒ^2 can be computed almost exactly. In general,𝒪is an order1pseudo-differential operator. But, in dimension=̣1(so thatdimM=2), there happens to be a bit of magic that shows that it is order0. (Essentially, the first-order term in𝒪is computed in terms of curvatures of∂Mthat all cancel if=̣1.) This geometric-analytic input is key. It shows up throughout many of the steps, not just the one described here.In any case, we must analyze the renormalized square |_∂M𝔥^η,lin|^2-𝒞_η appearing in (<ref>). If we write𝔥^η,linin terms of spectral data ofℒ, we end up having to study the following, where𝚣_are i.i.d. Gaussians, andψ_;̨·is the eigenfunction forλ_(which are arranged to be non-zero and non-decreasing in$̨): {∑_=̨1^∞𝚣__∂Mψ_;̨·}^2-𝒞_η=∑_=̨1^∞|𝚣_|^2|_∂Mψ_;̨·|^2-𝒞_η+∑_≠̨ℓ=1^∞𝚣_𝚣_ℓ_∂Mψ_;̨·_∂Mψ_ℓ;·.Because we average over $̨, we can replace|𝚣_|^2by its expectation. We are then left to control squares of eigenfunctions (with the counter-term 𝒞_η). In the case where∂Mis a torus, the eigenfunctions happen to be Fourier exponentials, whose squares are constants, and thus we can define the counter-term 𝒞_η to simply be the constant that cancels everything. For generalM, it is not clear what happens to an eigenfunction when we take products. What saves us is the averaging over$̨. This lets us use a local or pointwise Weyl law, which states, more or less, that the averaged behavior of squared eigenfunctions is a spectrally-determined constant independent of space (or the manifold) plus a smooth error. The Weyl law is another geometric-analytic input that we need which seems to be new in the analysis of singular SPDEs. (Technically, the Weyl law does not have gradients in front of the eigenfunctions, but we always integrate against a smooth function, so we can get rid of these gradients essentially via integration-by-parts and the computation of Δ_∂M+ℒ^2.)As for the second term on the RHS of (<ref>), since the sum is over distinct indices, the product of Gaussians is mean zero (and each product appearing in the sum is independent of all the other ones, except at most 1 in which the indices are just swapped). In <cit.>, this and Gaussianity of 𝚣_· controls regularity of the last term in (<ref>). But we face another issue. Indeed, in <cit.>, products of gradients of eigenfunctions are easy to compute in terms of Fourier exponentials; again, we lack this information for general M. Moreover, in general, eigenfunctions of ℒ have sub-optimal Hölder regularity; the 𝒞^-norm of the unit-L^2-norm λ-eigenfunction scales worse than |λ|^ as |λ|→∞. Thus, we need to use L^2-based Sobolev norms, which do not face this issue. On the other hand, multiplication in Sobolev spaces is worse than in Hölder spaces, since one always loses an additional half-derivative (see Lemma <ref>). We will address all of these issues. §.§ Outline for the rest of the paperThe rest of this paper essentially has two “halves" to it. The first half is dedicated to the proof of Theorem <ref>. This consists of Sections <ref>-<ref>. The second half is dedicated to the proof of Theorem <ref>; it is more on the singular SPDE side, and it consists of the remaining sections (except for those in the appendix and the last one right before the appendix, which proves Theorem <ref>). Let us now explain the goal of each individual section in more detail. *Proof of Theorem <ref>. *In Section <ref>, we give the ingredients for the proof. This includes computing a stochastic equation for 𝐘^. We ultimately reduce Theorem <ref> to the problem of getting a noise out of a fluctuation, exactly as we explained in Section <ref>. (Said problem is proving Proposition <ref>.)*In Section <ref>, we give a precise version of heuristics in Section <ref>. In particular, we reduce the proof of Proposition <ref> to perturbation theory estimates, which are proved in Sections <ref> and <ref>. *Proof of Theorem <ref>. *In Section <ref>, we present the Da Prato-Debussche-type schematic that we alluded to in Section <ref>. In this section, we present a stochastic bound (Proposition <ref>) that we need in the proof of Theorem <ref>. We assume it and then use classical and deterministic PDE considerations to finish. *In Sections <ref> and <ref>, we prove Proposition <ref> (following the ideas in Section <ref>). *Proof of Theorem <ref>. *This is the last non-appendix section; as we mentioned after Theorem <ref>, it is a classical argument. Finally, the goal of the appendix is to gather useful auxiliary estimates used throughout this paper.§ PROOF OUTLINE FOR THEOREM <REF>In this section, we give the main ingredients for Theorem <ref>. All but one of them (Proposition <ref>) will be proven; Proposition <ref> requires a sequence of preparatory lemmas, so we defer it to a later section. §.§ Stochastic equation for 𝐘^The first step is to use (<ref>) and (<ref>) to write an equation for 𝐘^, decomposing it into terms that we roughly outlined after Theorem <ref>. First, we recall notation from after (<ref>) and from Construction <ref>. We also consider the heat kernel ∂_Γ^(∂M)_,,=Δ_∂MΓ^(∂M)_,,Γ^(∂M)_,,→_→0^+δ_=,where the Laplacian acts either onor , where >0 and ,∈∂M in the PDE, and where the convergence as →0 from above is in the space of probability measures on ∂M. Fix ≥0 and ∈∂M. We have∂_𝐘^_,=Δ_∂M𝐘^_, +^-2/3∫_∂M𝐊_,[(1+|_∂M𝐈^_,|^2)^1/2-1]+^-2/3[Vol_𝐈^_𝐊_,𝔮^_-∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2].By the Duhamel principle (Lemma <ref>), we therefore deduce 𝐘^_, :=∫_∂MΓ^(∂M)_,,𝐘^init_0,+Φ^KPZ,_,+Φ^noise,_,,where Φ^KPZ,_, :=∫_0^∫_∂MΓ^(∂M)_-,,{^-2/3∫_∂M𝐊_,[(1+|_∂M𝐈^_,|^2)^1/2-1]} Φ^noise,_, :=∫_0^∫_∂MΓ^(∂M)_-,,{^-2/3[Vol_𝐈^_,·𝐊_,𝔮^_-∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2]}. Plug (<ref>) into the time-derivative of (<ref>). This gives ∂_𝐘^_,=^-1/3Δ_∂M𝐈^_,+^-2/3Vol_𝐈^_,·𝐊_,𝔮^_-^-2/3=^-1/3Δ_∂M𝐈^_,+^-2/3[Vol_𝐈^_,·𝐊_,𝔮^_-1].By (<ref>), we can replace ^-1/3Δ_∂M𝐈^_,↦Δ_∂M𝐘^_,. This turns the first term on the far RHS of (<ref>) into the first term on the RHS of (<ref>). The remaining two terms (the last in (<ref>) and (<ref>)) add to the last term in (<ref>), so (<ref>)-(<ref>) follows. The Duhamel expression (<ref>) follows by Lemma <ref> and Assumption <ref>.Let us now explain Lemma <ref> in the context of the proof strategy briefly described after Theorem <ref>. The second bullet point there says (<ref>) gives |_∂M𝐘^|^2 integrated against 𝐊 (by Taylor expansion). The third bullet point says that (<ref>) gives a noise. §.§ Producing a quadratic from (<ref>)Let us first establish some notation. First, define Φ^quad,_, :=12∫_0^∫_∂MΓ^(∂M)_-,,{∫_∂M𝐊_,|_∂M𝐘^_,|^2}as the heat kernel acting on a 𝐊-regularized quadratic. Recall the 𝒞^0_𝔱𝒞^_∂M-norm from Section <ref>. Fix any integer ≥̨0 and any time-horizon 𝔱≥0. We have the deterministic estimate Φ^KPZ,-Φ^quad,_𝒞^0_𝔱𝒞^_∂M≲_𝔱,^1/3𝐘^_𝒞^0_𝔱𝒞^1_∂M^3. Taylor expansion gives (1+υ^2)^1/2=1+1/2υ^2+O(υ^3). This implies (1+|_∂M𝐈^_,|^2)^1/2-1=12|_∂M𝐈^_,|^2+O(|_∂M𝐈^_,|^3).By (<ref>), we know that _∂M𝐈^=^1/3_∂M𝐘^. Thus, we deduce ^-2/3(1+|_∂M𝐈^_,|^2)^1/2-1=12|_∂M𝐘^_,|^2+O(^1/3|_∂M𝐘^_,|^3).By (<ref>), (<ref>), and (<ref>), we can compute Φ^KPZ,_,-Φ^quad,_, =∫_0^∫_∂MΓ^(∂M)_-,,∫_∂M𝐊_,O(^1/3|_∂M𝐘^_,|^3).Integrating against Γ^(∂M) is a bounded operator from the Sobolev space H^α(∂M) (see Section <ref>) to itself, with norm ≲_α1 locally uniformly in time; this holds by Lemma <ref>. Also, 𝐊 is smooth in both variables by assumption. So (<ref>) implies a version of (<ref>) where we replace 𝒞^ on the LHS by H^α. But then Sobolev embedding implies (<ref>) as written if we take α sufficiently large depending on $̨.We note that (<ref>) is meaningful, in the sense that𝐘^is supposed to be controlled in𝒞^2(∂M)if Theorem <ref> is true. In particular, the upper bound on the RHS of (<ref>) is supposed to vanish as→0. §.§ Producing a noise from (<ref>)Roughly speaking, we want to compareΦ^noise,to the following function (the first line is just a formal way of writing it, and the second line is a rigorous definition of said function in terms of integration-by-parts in time): Φ^𝐌^_, :=∫_0^∫_∂MΓ^(∂M)_-,,M^_,:=𝐌^_,-∫_∂MΓ^(∂M)_,,𝐌^_0,-∫_0^∫_∂M∂_Γ^(∂M)_-,,𝐌^_,.In the following result, we will make a choice of𝐌^for which we can actually compareΦ^noise,andΦ^𝐌^. There exists a good martingale ↦𝐌^_,· (see Definition <ref>) such that we have: *For any stopping time 0≤τ≤1 and ≥̨0, there exists universal β>0 such that with high probability, Φ^noise,-Φ^𝐌^_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β.The proof of Proposition <ref> is essentially the point of Section <ref>. §.§ Proof of Theorem <ref> assuming Proposition <ref>First define the stopping timeτ_𝐘^,Λas the first time the𝒞^2(∂M)-norm of𝐘^equalsΛ. (Note that𝐘^is continuous in time.) Throughout this argument, we will fixΛ≥0(independently of). Define𝐗^=𝐘^-𝐡^, where𝐡^solves (<ref>) with the martingale𝐌^from Proposition <ref>. We claim, with explanation after, that 𝐗^_, :=12∫_0^∫_∂MΓ^(∂M)_-,,[∫_∂M𝐊_,(|_∂M𝐘^_,|^2-|_∂M𝐡^_,|^2)]+Φ^noise,_,-Φ^𝐌^_,+Φ^KPZ,_,-Φ^quad,_,.To see this, recallΦ^quad,from (<ref>) andΦ^𝐌^from (<ref>)-(<ref>). Now, rewrite (<ref>) in its Duhamel form (by Lemma <ref>). (<ref>)-(<ref>) now follows directly from (<ref>) and this Duhamel form for (<ref>). (In particular, the martingale integrals in the𝐘^and𝐡^equations cancel out.) In what follows, everything holds with high probability. Because we make finitely many such statements, by a union bound, the intersection of the events on which our claims hold also holds with high probability.Letτbe any stopping time in[0,τ_𝐘^,Λ]. By Lemma <ref> and Proposition <ref>, we have (<ref>)_𝒞^0_τ𝒞^2_∂M≲_𝐘^_𝒞^0_τ𝒞^2_∂M^β≲_Λ^β,whereβ>0is strictly positive (uniformly in). Note the second estimate in (<ref>) follows by definition ofτ_𝐘^,Λ≥τ. On the other hand, by the elementary calculationa^2-b^2=2b[a-b]+[a-b]^2, we have |_∂M𝐘^_,|^2-|_∂M𝐡^_,|^2 ≲|_∂M𝐘^_,||_∂M𝐘^_,-_∂M𝐡^_,|+[_∂M𝐘^_,-_∂M𝐡^_,]^2=O_|_∂M𝐘^_,|(|_∂M𝐗^_,|+|_∂M𝐗^_,|^2),where the dependence of the big-Oh term in (<ref>) is smooth in|_∂M𝐘^|. Now, recall that𝐊is a smooth kernel, and that Γ^(∂M) is the kernel for a bounded operator𝒞^_̨1(∂M)→𝒞^_̨2(∂M)(for any_̨2>0and for_̨1big enough depending on_̨2; indeed this is the argument via Lemma <ref> and Sobolev embedding that we used in the proof of Lemma <ref>). Using this and (<ref>)-(<ref>), we claim the following for any≤τ:RHS(<ref>)_𝒞^2(∂M) ≲∫_0^|_∂M𝐘^_,·|^2-|_∂M𝐡^_,·|^2_𝒞^0(∂M)≲_Λ∫_0^|_∂M𝐗^|+|_∂M𝐗^|^2_𝒞^0_𝒞^0_∂M.Indeed, to get the first bound, when we take derivatives in∂Mof the RHS of (<ref>), boundedness of integration againstΓ^(∂M)lets us place all derivatives onto𝐊. Now, use that𝐊is smooth. This leaves the integral of||_∂M𝐘^_,·|^2-|_∂M𝐡^_,·|^2|on∂M(which we can bound by its𝒞^0(∂M)-norm since∂Mis compact). The second inequality above follows by (<ref>)-(<ref>) (and noting that for≤τ≤τ_𝐘^,Λ, the implied constant in (<ref>) is controlled in terms ofΛ). Since≤τ, we can extend the time-integration in (<ref>) from[0,]to[0,τ]. The resulting bound is independent of the-variable on the LHS of (<ref>), so RHS(<ref>)_𝒞^0_τ𝒞^2_∂M ≲∫_0^τ|_∂M𝐗^|+|_∂M𝐗^|^2_𝒞^0_𝒞^0_∂M.Combine (<ref>)-(<ref>), (<ref>), and (<ref>). We get the deterministic bound 𝐗^_𝒞^0_τ𝒞^2_∂M ≲_Λ^β+∫_0^τ|_∂M𝐗^|+|_∂M𝐗^|^2_𝒞^0_𝒞^0_∂M.Now, recallτ_𝐡^,Λis the first time that the𝒞^2(∂M)-norm of𝐡^is at leastΛ. Since𝐗^=𝐘^-𝐡^, for any time≤τ_𝐘^,Λ∧τ_𝐡^,Λ, we know that the𝒞^2(∂M)of𝐗^at timeis≲_Λ1. Thus, forτ≤τ_𝐘^,Λ∧τ_𝐡^,Λ, we can bound the square on the RHS of (<ref>) by a linear term, so that (<ref>) becomes 𝐗^_𝒞^0_τ𝒞^2_∂M ≲_Λ^β+∫_0^τ_∂M𝐗^_𝒞^0_𝒞^0_∂M.It now suffices to use Gronwall to deduce that forτ≤τ_𝐘^,Λ∧τ_𝐡^,Λ, we have 𝐗^_𝒞^0_τ∧1𝒞^2_∂M≲_Λ^β.We now claim that it holds for allτ≤τ_𝐡^,Λ/2-δforδ>0fixed, as long as>0is small enough depending only onΛ,δ. This would yield (<ref>) (upon rescalingΛtherein) and thus complete the proof. To prove this claim, it suffices to showτ_𝐘^,Λ∧τ_𝐡^,Λ≥τ_𝐡^,Λ/2-δ. Suppose the opposite, so thatτ_𝐘^,Λ≤τ_𝐡^,Λ/2-δ(sinceτ_𝐡^,Λ/2≤τ_𝐡^,Λtrivially). This means that (<ref>) holds for allτ≤τ_𝐘^,Λ. From this and𝐗^=𝐘^-𝐡^, we deduce that at timeτ_𝐘^,Λ, we have𝐘^=𝐡^+O_Λ(^β). But at timeτ_𝐘^,Λ≤τ_𝐡^,Λ/2, this implies that the𝒞^2(∂M)-norm of𝐘^is≤1/2Λ+O_Λ(^β). If>0is small enough, then this is≤2/3Λ, violating the definition ofτ_𝐘^,Λ. This completes the contradiction, so the proof is finished. § PROOF OUTLINE FOR PROPOSITION <REF>In Sections <ref>-<ref>, we need to track dependence on the number of derivatives we take of𝐈^(since estimates for certain operators depend on the metric𝐠[_∂M𝐈^]). In particular, we will need to control said number of derivatives by the𝒞^2(∂M)-norm of𝐘^(see the implied constant in (<ref>)). We will be precise about this. However, by Lemma <ref>, as long as the number of derivatives of𝐈^that we take isO(1), this is okay. §.§ A preliminary reductionWe first recall (<ref>)-(<ref>) and (<ref>) for the notation in the statement of Proposition <ref>. In particular,Φ^noise,=∫_0^Γ^(∂M)_-,,∂_Int^noise,_,,𝔮^_,𝐈^_, where Int^noise,_,,𝔮^_·,𝐈^_·:=^-2/3∫_0^[Vol_𝐈^_𝐊_,𝔮^_-∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2].The point of this initial step is to make analysis easier by dropping the heat kernel in (<ref>)-(<ref>) and (<ref>).By integration-by-parts in the-variable in (<ref>), we haveΦ^noise,_,=∫_0^Γ^(∂M)_-,,∂_Int^noise,_,,𝔮^_,𝐈^_=Int^noise,_,,𝔮^_·,𝐈^_·-∫_∂MΓ^(∂M)_,,Int^noise,_0,,𝔮^_·,𝐈^_·-∫_0^∫_∂M∂_Γ^(∂M)_-,,Int^noise,_,,𝔮^_·,𝐈^_·.This calculation is exactly analogous to (<ref>)-(<ref>). (We note that the middle term in (<ref>) can be dropped sinceInt^noise,vanishes at=0, but it does not matter; we want to make (<ref>)-(<ref>) look like (<ref>)-(<ref>).) Now, we use ∂_Γ^(∂M)=Δ_∂MΓ^(∂M). In particular, by (<ref>)-(<ref>) and (<ref>)-(<ref>), we have Φ^noise,_,-Φ^𝐌^_, =Int^noise,_,,𝔮^_·,𝐈^_·-𝐌^_,-∫_∂MΓ^(∂M)_,,[Int^noise,_0,,𝔮^_·,𝐈^_·-𝐌^_0,]-∫_0^∫_∂MΔ_∂MΓ^(∂M)_-,,[Int^noise,_,,𝔮^_·,𝐈^_·-𝐌^_,].By Lemma <ref>, we know theΓ^(∂M)-semigroup is bounded from the Sobolev spaceH^α(∂M)to itself with norm≲_α1. Moreover, ifαis an integer, thenH^α(∂M)is controlled by𝒞^α(∂M). Finally, by Sobolev embedding, for any$̨, we can take α big enough depending on $̨ so that𝒞^(∂M)is controlled byH^α(∂M). Thus, in order to control the𝒞^(∂M)-norm of the LHS of (<ref>), it suffices to just control the𝒞^ℓ(∂M)-norm of the differenceInt^noise,-𝐌^(for someℓlarge enough depending on$̨). (This follows by the previous display.) So, to prove Proposition <ref>, it suffices to show the following result instead. There exists a good martingale ↦𝐌^_,·∈𝒞^∞(∂M) (see Definition <ref>) such that*For any stopping time 0≤τ≤1 and ≥̨0, there exists universal β>0 such that with high probability, sup_0≤≤τInt^noise,_,·,𝔮^_·,𝐈^_·-𝐌^_,·_𝒞^(∂M)≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β.In (<ref>), the 𝒞^(∂M)-norm on the LHS is with respect to the omitted -variables in Int^noise,_,,𝔮^_·,𝐈^_· and 𝐌^_,. §.§ Step 1: Setting up an Ito formula for (𝐈^,𝔮^)See Section <ref> for the motivation for an Ito formula for the joint process (𝐈^,𝔮^). We must now explicitly write the generator of this joint process. It has the form ℒ_total^=ℒ^,𝔮^_flow+ℒ^,𝐈^_DtN.The first term is the instantaneous flow of 𝐈^ defined by (<ref>) (for 𝔮^∈∂M), and the second term is a scaled Dirichlet-to-Neumann map with metric 𝐠[_∂M𝐈^] on M determined by 𝐈^∈𝒞^∞(∂M). (Superscripts for these operators always indicate what is being fixed, i.e. the opposite of whose dynamics we are considering.) To be precise: *For any 𝐈∈𝒞^∞(∂M), recall the metric 𝐠[_∂M𝐈] on M (see Construction <ref>). Let Δ_𝐈 be Laplacian with respect to this metric. Now, given any φ∈𝒞^∞(∂M), we setℒ^,𝐈^_DtNφ=^-4/3_𝖭𝒰^φ,𝐈^,where 𝖭 is the inward unit normal vector field on ∂M, and 𝒰^φ,𝐈 is Δ_𝐈-harmonic extension of φ to M. (In particular, we have Δ_𝐈𝒰^φ,𝐈=0 and 𝒰^φ,𝐈|_∂M=φ. Again, we refer to Proposition 4.1 in <cit.> for why (<ref>) is the generator of 𝔮^, and that its dependence on 𝐈^ shows non-Markovianity of 𝔮^.)*Fix 𝔮^∈∂M. The second term in (<ref>) is the Frechet derivative on functions 𝒞^∞(∂M)→ such that, when evaluated at 𝐈∈𝒞^∞(∂M), it is in the direction of the function ↦Δ_∂M𝐈^_+^-1/3Vol_𝐈𝐊_,𝔮^. Precisely, given any functional ℱ:𝒞^∞(∂M)→ and 𝐈∈𝒞^∞(∂M), we haveℒ^,𝔮^_flowℱ[𝐈]:=lim_h→01h{ℱ[𝐈+hΔ_∂M𝐈+hVol_𝐈𝐊_·,𝔮^]-ℱ[𝐈]},provided that this limit exists (which needs to be verified at least somewhat carefully, since 𝒞^∞(∂M) is infinite-dimensional). §.§.§ Issues about the domain of ℒ^,𝔮^_flowThroughout this section, we will often let ℒ^,𝔮^_flow hit various functionals of the 𝐈^ process. Of course, as noted immediately above, anytime we do this, we must verify that the limit (<ref>) which defines it exists. Each verification (or statement of such) takes a bit to write down. So, instead of stating explicitly that each application of ℒ^,𝔮^_flow is well-defined throughout this section, we instead take it for granted, and, in Section <ref>, we verify explicitly that all applications ofℒ^,𝔮^_floware justified.§.§.§ An expansion for Int^noise,Before we start, we first introduce notation for the following fluctuation term, which is just the time-derivative of (<ref>): Fluc^noise,_,𝔮^_,𝐈^_ :=^-2/3[Vol_𝐈^_𝐊_,𝔮^_-∫_∂M𝐊_,(1+|_∂M𝐈^_,|^2)^1/2].Not only is this notation useful, but we emphasize that it does not depend on time(except through (𝐈^,𝔮^)). So, as far as an Ito formula is concerned, we do not have to worry about time-derivatives.As discussed in Section <ref>, we will eventually get a martingale from Int^noise, by the Ito formula. We also noted in Section <ref> that we have to regularize the total generator (<ref>) by a spectral parameter λ. In particular, for the sake of illustrating the idea, we will want to write the following for λ chosen shortly: Fluc^noise,_,𝔮^_,𝐈^_=(λ-ℒ_total^)[λ-ℒ_total^]^-1Fluc^noise,_,𝔮^_,𝐈^_.Ito tells us how to integrate ℒ^_total[λ-ℒ_total^]^-1Fluc^noise,_,𝔮^_,𝐈^_ in time. We are still left with terms of the form λ[λ-ℒ_total^]^-1Fluc^noise,_,𝔮^_,𝐈^_.We will again hit this term with (λ-ℒ^_total)[λ-ℒ^_total]^-1 (so that the previous display now plays the role of Fluc^noise,). If we repeat (i.e. use the Ito formula to take care of ℒ^_total[λ-ℒ^_total]^-1), we are left with (λ[λ-ℒ^_total]^-1)^2Fluc^noise,_,𝔮^_,𝐈^_.By iterating, the residual terms become just higher and higher powers of λ[λ-ℒ^_total]^-1. For later and later terms in this expansion to eventually become very small, we will want to choose the spectral parameter λ=^-4/3+γ,where γ>0 strictly positive and universal (though eventually small). Indeed, (<ref>) is much smaller than the ^-4/3 speed of ℒ^_total, so each power of λ[λ-ℒ^_total]^-1 gives us ≲^γ.Let us now make this precise with the following set of results. We start with an elementary computation. It effectively writes more carefully how to go from (λ[λ-ℒ^_total]^-1)^ℓ to (λ[λ-ℒ^_total]^-1)^ℓ+1. (Except, it uses ℒ^,𝐈_DtN instead of ℒ^_total in the resolvents, which only requires a few cosmetic adjustments.) Fix any integer ℓ≥0. We have the following deterministic identity: ∫_0^[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_=∫_0^[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ+1Fluc^noise,_,𝔮^_,𝐈^_-∫_0^ℒ_total^(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_+∫_0^ℒ_flow^,𝔮^_(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_. Note that the operators in (<ref>) and (<ref>) add to -ℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ.Adding this to the operator in (<ref>), which can be written as λ(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ, gives (λ-ℒ^,𝐈^__DtN)(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ=[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ.This is just the operator in (<ref>). Act on Fluc^noise,_,𝔮^_,𝐈^_ and integrate over ∈[0,] to get (<ref>)-(<ref>).Next, we use the Ito formula to compute (<ref>) in terms of a martingale and boundary terms. We can also compute the predictable bracket of the martingale we get (essentially by standard theory). Fix any ℓ≥0. There exists a martingale ↦𝐌^,ℓ_,·∈𝒞^∞(∂M) such that(<ref>)=𝐌^,ℓ_, +(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|_=0-(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|_=.The predictable bracket [𝐌^,ℓ] of 𝐌^,ℓ, i.e. the process such that |𝐌^,ℓ_,·|^2-[𝐌^,ℓ]_,· is a martingale, is [𝐌^,ℓ]_, =∫_0^(ℒ^,𝔮^__flow+ℒ^,𝐈^_,·_DtN){|(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|^2}-2∫_0^{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_}×{(ℒ^,𝔮^__flow+ℒ^,𝐈^_,·_DtN)[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_]}. The Ito-Dynkin formula (see Appendix 1.5 of <cit.>) says that for any Markov process 𝔛 (valued in a Polish space) with generator 𝒢, and for any φ in the domain of 𝒢, we have∫_0^𝒢φ_𝔛[]=φ_𝔛[]-φ_𝔛[0]-𝐌^φ_,where ↦𝐌^φ_ is a martingale whose predictable bracket is a time-integrated Carre-du-Champ: ∫_0^[𝒢(|φ_𝔛[]|^2)-2φ_𝔛[]𝒢φ_𝔛[]].Use this with 𝔛=(𝐈^,𝔮^) and 𝒢=(<ref>) and φ=(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_.We now combine Lemmas <ref> and <ref> to write the expansion for Int^noise,. Indeed, note that (<ref>) for ℓ=0 is just Int^noise,. We remark that the following result, namely its expansion (<ref>)-(<ref>), will only ever be a finite sum (that we do not iterate to get an infinite sum). Thus there is no issue of convergence. (As we noted before Lemma <ref>, every step in the iteration gives a uniformly positive power of , so only finitely many steps are needed to gain a large enough power-saving into beat every other -dependent factor.) Fix any integer ℓ_max≥0. Recall (<ref>), (<ref>) and notation from Lemma <ref>. We haveInt^noise,_,,𝔮^,𝐈^_· =∑_ℓ=0^ℓ_max𝐌^,ℓ_,+∫_0^[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓ_max+1Fluc^noise,_,𝔮^_,𝐈^_+∑_ℓ=0^ℓ_max∫_0^ℒ^,𝔮^__flow(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_,·+∑_ℓ=0^ℓ_max(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_,·|_=0-∑_ℓ=0^ℓ_max(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_,·|_=. By (<ref>) and (<ref>), we clearly haveInt^noise,_,,𝔮^,𝐈^_·=∫_0^Fluc^noise,_,𝔮^_,𝐈^_.Let us now prove (<ref>)-(<ref>) for ℓ_max=0. This follows immediately from (<ref>)-(<ref>) for ℓ=0 and (<ref>)-(<ref>) to compute (<ref>) for ℓ=0. So, for the sake of induction, it suffices to assume that (<ref>)-(<ref>) holds for ℓ_max≥0, and get it for ℓ_max+1. For this, we compute (<ref>) for ℓ_max using (<ref>)-(<ref>) for ℓ=ℓ_max+1. We deduce that its contribution is equal to ∫_0^[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓ_max+2Fluc^noise,_,𝔮^_,𝐈^_-∫_0^ℒ^_total(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ_max+1Fluc^noise,_,𝔮^_,𝐈^_+∫_0^ℒ^,𝔮^__flow(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ_DtN^,𝐈^_)^-1]^ℓ_max+1Fluc^noise,_,𝔮^_,𝐈^_.Thus, we have upgraded (<ref>) into (<ref>) but with ℓ_max+2 instead of ℓ_max+1, at the cost of the second and third lines of the previous display. The third line lets us turn the sum over ℓ=0,…,ℓ_max in (<ref>) into a sum over ℓ=0,…,ℓ_max+1. Moreover, if we apply (<ref>)-(<ref>) for ℓ=ℓ_max+1, the second line gives a contribution that turns the sums over ℓ=0,…,ℓ_max in (<ref>), (<ref>), and (<ref>) to over ℓ=0,…,ℓ_max+1. What we ultimately get is just (<ref>)-(<ref>) but ℓ_max↦ℓ_max+1, which completes the induction.§.§ Step 2: Estimates for (<ref>)-(<ref>) for ℓ_max≳_γ1Perhaps unsurprisingly, the martingale 𝐌^ that we are looking for is the RHS of (<ref>). Thus, we must do two things. *Show that (<ref>)-(<ref>) vanish as →0.*Compare the predictable bracket of the RHS of (<ref>) (using (<ref>)-(<ref>)) to [𝐌^limit] given in (<ref>).Indeed, one can check directly that this would yield Proposition <ref>.§.§.§ Dirichlet-to-Neumann estimatesLet us start with (<ref>), (<ref>), and (<ref>), i.e. the terms which only have Dirichlet-to-Neumann maps (and no ℒ^,𝔮_flow-terms). The estimate which essentially handles all of these terms is the content of the following result. Recall λ=^-4/3+γ from (<ref>), and recall (<ref>). For any stopping time τ∈[0,1], we have the following with probability 1 for any ℓ≥0 and ≥̨0: sup_0≤≤τ(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^__𝒞^(∂M)≲_,̨ℓ,𝐘^_𝒞^0_τ𝒞^2_∂M^4/3[λ^4/3]^ℓ^-2/3.(The norm on the LHS is with respect to the omitted -variable, which we indicated with ·.)Intuitively, every inverse gives ^4/3, and each λ is just bounded by λ, and we bound Fluc^noise, by ^-2/3 directly (see (<ref>)). This gives (<ref>), roughly speaking. Let us make this precise.In what follows, we denote by _α the H^α(∂M)-Sobolev norm of order αwith respect to the 𝔮^_-variable (see Section <ref>). We now make the following observations. *For any 𝐈∈𝒞^∞(∂M), let μ[_∂M𝐈] be Riemannian measure on ∂M induced by 𝐠[_∂M]. As explained in Construction <ref>, change-of-variables shows thatμ̣[_∂M𝐈]_=(1+|_∂M𝐈_|^2)^1/2. *Consider L^2(∂M,μ[_∂M𝐈^_]). The Dirichlet-to-Neumann operator ℒ^,𝐈^__DtN has a self-adjoint extension to L^2(∂M,μ[_∂M𝐈^_]), and it has a one-dimensional null-space spanned by constant functions. It has a strictly positive spectral gap of order ^-4/3 times something that depends only on the 𝒞^1(∂M)-norm of _∂M𝐈^_,·. (For the order of the spectral gap, see Lemma <ref>. For the dependence on _∂M𝐈^_,·, it suffices to control the density of the measure induced by 𝐠[_∂M𝐈^_,·] with respect to surface measure on ∂M, i.e. 𝐠[_∂M0], where 0 is the 0 function. Indeed, spectral gaps are stable under multiplicative perturbations of the measure. But this measure depends only on the determinant of 𝐠[_∂M𝐈^_,·] in local coordinates.)*The Fluc^noise,_·,𝔮^_,𝐈^_, as a function of 𝔮^_∈∂M, is orthogonal to the null-space of ℒ^,𝐈^__DtN. This follows by construction; see (<ref>). Moreover, so does every power of (λ-ℒ_DtN^,𝐈^_)^-1 acting on Fluc^noise,_·,𝔮^_,𝐈^_, since ℒ^,𝐈^__DtN is self-adjoint.*Thus, we get that for any ∈∂M and α≥0, we have the estimate below (for n≲1): (λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^__α ≲_α,ℓ,_∂M𝐈^_𝒞^0_𝒞^n_∂M^4/3[ℓ+1]λ^ℓFluc^noise,_,𝔮^_,𝐈^__α≲_α,ℓ,𝐘^_𝒞^0_𝒞^2_∂M^4/3[ℓ+1]λ^ℓFluc^noise,_,𝔮^_,𝐈^__α.(To get the second estimate, use Lemma <ref> to control the implied constant in (<ref>).)*Finally, we note that the Sobolev norm in (<ref>) is ≲^-2/3, with implied constant depending only on the 𝒞^0(∂M)-norm of _∂M𝐈^=^1/3_∂M𝐘^. (This follows immediately by (<ref>).)Note that (<ref>)-(<ref>) is true for all α≥0; taking α≥0 big enough depending on dimension $̣, we can use a Sobolev embedding and deduce that with probability1, we have the uniform estimate |(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|≲_α,ℓ,𝐘^_𝒞^0_𝒞^2_∂M^4/3[ℓ+1]λ^ℓ^-2/3.This is true for all0≤≤τ, so the desired estimate (<ref>) follows for=̨0. For general≥̨0, just use the same argument, but replaceFluc^noise,by its$̨-th order derivatives in . (Indeed, the mean-zero property used in point (3) above is still true if we take derivatives in , since it is a linear condition in the 𝔮^_-variable. One can also check this by direct inspection via (<ref>).) This finishes the proof.As an immediate consequence of Lemma <ref>, we can bound (<ref>), (<ref>), and (<ref>). The latter terms (namely (<ref>), and (<ref>)) are bounded directly by (<ref>), so we only treat (<ref>). Again, recall (<ref>). Fix any stopping time τ∈[0,1] and any ℓ_max,≥̨0. With probability 1, we have sup_0≤≤τ∫_0^[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓ_max+1Fluc^noise,_·,𝔮^_,𝐈^__𝒞^(∂M)≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M[λ^4/3]^ℓ_max+1^-2/3. Use the triangle inequality to move the 𝒞^(∂M)-norm into the -integral, then use (<ref>) for ℓ=ℓ_max+1. (The extra factor of λ on the RHS of (<ref>) compared to the RHS of (<ref>) for ℓ=ℓ_max+1 is because there is an extra factor of λ on the LHS of (<ref>) compared to the LHS of (<ref>).) §.§.§ ℒ^,𝔮^__flow estimatesWe first give an estimate for (<ref>), i.e. bounding it by a uniformly positive power of . We then give an estimate comparing the predictable bracket for the martingale on the RHS of (<ref>) to the proposed limit [𝐌^limit] (see (<ref>) and (<ref>)).Our estimate for (<ref>) is captured by the following result. This result was intuitively explained in Section <ref>, but let us be a little more precise about power-counting in(before we give a complete proof), just to provide intuition. As noted in Section <ref>, the ℒ^,𝔮^__flow-operator in (<ref>) destroys the algebraic property that allowed us to leverage spectral gap estimates in the proof of Lemma <ref>. Thus, each resolvent in (<ref>) only gives a factor of ≲λ^-1. Fortunately, ℒ^,𝔮^__flow has scaling ≲^-1/3. So, (<ref>) should be ≲^-1/3λ^-2/3≲^1/3-γ, since the Fluc^noise, has scaling of order ^-2/3 (see (<ref>)). If we choose γ>0 small enough, this is sufficient.To make it rigorous, we must first compute the action of ℒ^,𝔮^__flow on the resolvents in (<ref>) (e.g. show that the resolvents are in the domain of ℒ^,𝔮^__flow). We must also be a little careful about how many derivatives of 𝐘^ our estimates require, but this is not a big deal (especially given Lemma <ref>). Take any stopping time τ∈[0,1] and ≥̨0. Let Err^(ℓ_max)_, be (<ref>). There exists a uniformly positive β>0 such that with probability 1, we have the following estimate: Err^(ℓ_max)_𝒞^0_τ𝒞^_∂M≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M^β. The proof of Lemma <ref> requires the calculations in the next section for computing the -integrand of (<ref>), so we delay this proof for Section <ref>.Let us now analyze the predictable bracket of the martingale on the RHS of (<ref>). A rigorous proof of the result also requires the calculations in the next section, so we delay a proof until Section <ref> as well. However, let us at least give an intuitive argument (which is essentially how the proof goes). *Take ℓ=0 on the RHS of (<ref>); the predictable bracket of this martingale is (<ref>)-(<ref>) for ℓ=0. The first step we take is to drop all ℒ^,𝔮^__flow-operators. One can justify this by proving that they are lower-order as described before Lemma <ref>. However, it is also a first-order differential operator, so by the Leibniz rule, the ℒ^,𝔮^__flow-operators actually cancel each other out exactly. After this, (<ref>)-(<ref>) for ℓ=0 becomes ∫_0^ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-2∫_0^(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_×ℒ^,𝐈^__DtN[(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_]. *Every resolvent is order ^4/3 (see the beginning of the proof of Lemma <ref>). Every Dirichlet-to-Neumann operator itself is order ^-4/3. Also, Fluc^noise, is order ≲^-2/3. With this, it is not hard to see that the previous display is order 1. Moreover, we time-average, thus we expect to replace the -integrand above by its expectation in the particle 𝔮^_ with respect to the Riemannian measure μ[_∂M𝐈^_] induced by 𝐠[_∂M𝐈^_]. (This is exactly the idea behind Section <ref>.) After this replacement, the previous display becomes ∫_0^∫_∂Mℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,,𝐈^_|^2]μ̣[_∂M𝐈^_]_-2∫_0^∫_∂M(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,,𝐈^_×ℒ^,𝐈^__DtN[(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,,𝐈^_]μ̣[_∂M𝐈^_]_.The first line in this display vanishes, since the -integrand is in the image of the Dirichlet-to-Neumann map, which has μ[_∂M𝐈^_] as an invariant measure. Since λ=^-4/3+γ is much smaller than the scaling ^-4/3 of ℒ^,𝐈^__DtN, we can drop λ-terms in the second line above. We are therefore left with -2∫_0^∫_∂MFluc^noise,_,,𝐈^_×[ℒ^,𝐈^__DtN]^-1Fluc^noise,_,,𝐈^_μ̣[_∂M𝐈^_]_,where operators act on . Finally, all dependence on 𝐈^ above is through _∂M𝐈^=^1/3_∂M𝐘^ (which should be ≪1), so we can replace 𝐈^ by 0. In view of (<ref>) and (<ref>), we get [𝐌^,0]≈[𝐌^limit].*Now take ℓ>1 both on the RHS of (<ref>) and in (<ref>)-(<ref>). Again, drop all ℒ^,𝔮^__flow-operators as before in our discussion of ℓ=0. Now, note that every term in (<ref>)-(<ref>) has at least one additional factor of λ(λ-ℒ^,𝐈^__DtN)^-1, which is ≲^γ as used in the proof of Lemma <ref>. As (<ref>)-(<ref>) was order 1 with ℓ=0 (so without the helpful ^γ-factors), the RHS of (<ref>) has vanishing predictable bracket for ℓ≥1.The actual proof of Lemma <ref> is slightly different for ease of writing, but the idea is the same.Before we state the result precisely, we recall (<ref>) and the notation of Lemma <ref>. Take any stopping time τ∈[0,1] and any ≥̨0. There exists uniformly positive β>0 such that the following hold with high probability: [𝐌^,0]-[𝐌^limit]_𝒞^0_τ𝒞^_∂M ≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β sup_1≤ℓ≤ℓ_max[𝐌^,ℓ]_𝒞^0_τ𝒞^_∂M ≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M^β. For n≥0 fixed, take any tangent vectors 𝖾_i_1,…,𝖾_i_n on the tangent space of ∂M to differentiate along. (These tangent vectors depend on an implicit variable ∈∂M.) The first estimate (<ref>) still holds even if we make the following replacements, as we explain shortly. *Replace 𝐌^,0 by its n-th order derivative in 𝖾_i_1,…,𝖾_i_n. This is still a martingale, since martingales are closed under linear operations.*Replace [𝐌^limit] by the object obtained by replacing 𝐊 in (<ref>) by its n-th order derivative with respect to the -variable in 𝖾_i_1,…,𝖾_i_n. We denote this object by [_i_1… i_n𝐌^limit]. (It is easy to see from (<ref>) that [_i_1… i_n𝐌^limit] is uniformly smooth in the ∈∂M-variable; its regularity is controlled by that of 𝐊.)Indeed, the only difference in the argument is to replace 𝐊 by its aforementioned derivative. We only rely on regularity of 𝐊 (as alluded to in the outline of Lemma <ref> before its statement and as the proof will make clear), so our claim follows. Ultimately, combining this remark with (<ref>) and 𝐌^=𝐌^,0+∑_ℓ=1^ℓ_max𝐌^,ℓ, we deduce that the following estimate holds with high probability: [_i_1… i_n𝐌^]-[_i_1… i_n𝐌^limit]_𝒞^0_τ𝒞^_∂M ≲_n,,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β. §.§ Proof of Proposition <ref> (and thus of Proposition <ref>)Define 𝐌^ in the statement of Proposition <ref> be the RHS of (<ref>) (for ℓ_max≲1 chosen shortly). In particular, the quantity of interest Int^noise,_,,𝔮^_·,𝐈^_·-𝐌^_,equals the sum of (<ref>), (<ref>), (<ref>), and (<ref>). Now, use Lemmas <ref>, <ref>, and <ref> to control 𝒞^0_τ𝒞^_∂M-norms of (<ref>), (<ref>), (<ref>), and (<ref>) altogether by ≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M[λ^4/3]^ℓ_max+1^-2/3+sup_0≤ℓ≤ℓ_max^4/3[λ^4/3]^ℓ^-2/3+^β.Since λ=^-4/3+γ for γ>0 uniformly positive (see (<ref>)), we know the upper bound (<ref>) is ≲^β for β>0 uniformly positive, as long as ℓ_max is sufficiently large depending only on γ (we can take ℓ_max≲_γ1). Therefore, using what we said immediately before (<ref>), we deduce (<ref>).We now show 𝐌^ is a good martingale (see Definition <ref>). It is smooth since every other term in (<ref>)-(<ref>) is smooth. It is cadlag for the same reason (note (𝐈^,𝔮^) is cadlag). Also, by Corollary <ref>, the jumps of 𝐌^ are given by the sum of jumps of (<ref>)-(<ref>), since the other non-martingale terms in that display are time-integrals. Thus, it suffices to show these terms vanish deterministically as →0 uniformly in time (at a rate depending on 𝐘^_𝒞^0_τ𝒞^2_∂M). For this, see Lemma <ref>. Next, we show the derivative bounds on 𝐌^. Fix tangent vectors 𝖾_i_1,…,𝖾_i_n as in Remark <ref>. Fix any stopping time τ∈[0,1] for which: * 𝐘^_,·_𝒞^2(∂M)≤Λ for Λ≥0 fixed for all 0≤≤τ.* (<ref>) holds for all 0≤≤τ and for n≥0 fixed.We claim that sup_0≤≤τ∫_∂M|_i_1… i_n𝐌^_,|^2 ≲∫_∂M|_i_1… i_n𝐌^_τ,|^2≲_Λ,n1.The first bound is by Doob's maximal inequality (note that 𝐌^ and its derivatives are all martingales, since the martingale property is preserved under linear operations). The second inequality follows by (<ref>), the a priori 𝒞^2(∂M) bound on 𝐘^ before time τ, and bounds on [_i_1… i_n𝐌^limit] as explained in Remark <ref>. This is true for all n≲1, so we can use a Sobolev embedding H^n(∂M)↪𝒞^(∂M) (for any $̨ and for anynlarge enough depending only on$̨) to deduce the desired derivative estimates for a good martingale. (Said derivative estimates hold with high probability, since the claims of Remark <ref> hold with high probability.)It remains to show that the martingale 𝐌^ satisfies (<ref>). Intuitively, this should be immediate because of Lemma <ref>, but we have to be (a little) careful about taking the predictable bracket of the sum. We first use [𝐦+𝐧]=[𝐦]+[𝐧]+2[𝐦,𝐧] for brackets of martingales 𝐦,𝐧, where [,] is the cross bracket. (We will take 𝐦=𝐌^,0 and 𝐧=𝐌^,1+…+𝐌^,ℓ_max; see Lemma <ref> for notation.) This is just a standard inner product calculation, so that [𝐌^]=[∑_ℓ=0^ℓ_max𝐌^,ℓ]=[𝐌^,0]+[∑_ℓ=1^ℓ_max𝐌^,ℓ]+2[𝐌^,0,∑_ℓ=1^ℓ_max𝐌^,ℓ].By (<ref>), we can compare the first term on the far RHS to [𝐌^limit]. Thus, to show that [𝐌^]-[𝐌^limit] vanishes (i.e. prove that 𝐌^ satisfies (<ref>)), it suffices to show that, with high probability, [∑_ℓ=1^ℓ_max𝐌^,ℓ]_𝒞^0_τ𝒞^_∂M+[𝐌^,0,∑_ℓ=1^ℓ_max𝐌^,ℓ]_𝒞^0_τ𝒞^_∂M≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M^β.We assume =̨0 in what follows; for general $̨, use the same argument but replace brackets by their$̨-th order derivatives in . To bound the first term on the LHS of (<ref>), use the Schwarz inequality with (<ref>):[∑_ℓ=1^ℓ_max𝐌^,ℓ]≲_ℓ_max∑_ℓ=1^ℓ_max[𝐌^,ℓ].For the second term on the LHS of (<ref>), we use another Cauchy-Schwarz combined with (<ref>): |[𝐌^,0,∑_ℓ=1^ℓ_max𝐌^,ℓ]| ≲[𝐌^,0]^1/2([∑_ℓ=1^ℓ_max𝐌^,ℓ])^1/2. ≲_,̨ℓ_max,𝐘^_𝒞^0_τ𝒞^2_∂M^β×[𝐌^,0]^1/2.By (<ref>), we can replace [𝐌^,0] by [𝐌^limit] in (<ref>) with error ≲^β. But we know that the 𝒞^0_τ𝒞^_∂M-norm of [𝐌^limit] is ≲_1; this holds by differentiating (<ref>) inup to $̨-th order, using regularity of𝐊in (<ref>) in both of its inputs, and using the spectral gap ofℒ. (Indeed, this spectral gap ingredient, which comes from Lemma <ref>, just says that𝐊-1is smooth both before and after we hit it byℒ^-1.) Ultimately, we deduce that(<ref>)≲^β. Combining this with every display starting after (<ref>) then shows (<ref>). §.§ What is leftAs far as Proposition <ref> (and thus Proposition <ref> and Theorem <ref>) is concerned, we are left with Lemmas <ref> and <ref>. However, we must also show that every term we hit ℒ^,𝔮^__flow with in this section is actually in its domain. Said terms include (<ref>)-(<ref>) and (<ref>). This will be dealt with in this next section, whereas Lemmas <ref> and <ref> are proved in Section <ref>. § COMPUTATIONS FOR THE ACTION OF ℒ^,𝔮^__FLOW-OPERATORS§.§ Setup for our calculationsThe main goal of this section is to compute, for any,𝔮∈∂M, 𝐈↦ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓFluc^noise,_,𝔮,𝐈.Above, all operators act on the𝔮-variable; part of computing (<ref>) means showing existence of the limit (<ref>) that defines ℒ^,𝔮_flow on the RHS of (<ref>). For convenience, we recallFluc^noise,from (<ref>) below, in which Vol_𝐈:=∫_∂M(1+|_∂M𝐈_|^2)^1/2: Fluc^noise,_,𝔮,𝐈:=^-2/3[Vol_𝐈𝐊_,𝔮-∫_∂M𝐊_,(1+|_∂M𝐈_|^2)^1/2].Our computation of (<ref>) takes the following steps. *First, we compute 𝐈↦ℒ^,𝔮_flowFluc^noise,_,𝔮,𝐈. This is not hard given the formula (<ref>).*Next, we compute 𝐈↦ℒ^,𝔮_flowℒ^,𝐈_DtN as an operator on 𝒞^∞(∂M).*Using point (2) and classical resolvent perturbation identities from functional analysis, we compute the operator 𝐈↦ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1. It is not too hard to use this result and the same resolvent identities to derive a Leibniz-type rule for the Frechet differential ℒ^,𝔮_flow and then compute the operator 𝐈↦ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓ.The subtlety is that (λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓ is a product of operators; we need a non-commutative version of the Leibniz rule (which requires a bit of attention but is not difficult to derive).*Finally, we use a Leibniz rule for ℒ^,𝔮_flow (we emphasize that ℒ^,𝔮_flow is just a derivative!) with points (1) and (3) above to compute (<ref>).Before we embark on these calculations, it will be convenient to set the following notation for the direction of differentiation in (<ref>): 𝐈↦𝐉[𝐈]=Δ_∂M𝐈+^-1/3Vol_𝐈𝐊_·,𝔮∈𝒞^∞(∂M).(Although𝐉[𝐈]depends on𝔮∈∂M, we omit this dependence since it will not be very important.) §.§ Point (1): Computing 𝐈↦ℒ^,𝔮_flowFluc^noise,_,𝔮,𝐈As noted earlier, this computation is easy given (<ref>). In particular, when we differentiate in𝐈, we must only do so pointwise in∈∂M, i.e. differentiate analytic functions of𝐈_and_∂M𝐈_per∈∂M. Ultimately, we get the following result. Fix ,𝔮∈∂M and 𝐈∈𝒞^∞(∂M). The following limit exists: ℒ^,𝔮_flowFluc^noise,_,𝔮,𝐈=lim_h→01h{Fluc^noise,_,𝔮,𝐈+h𝐉[𝐈]-Fluc^noise,_,𝔮,𝐈}Also, (<ref>) is jointly smooth in ,𝔮 with $̨-th order derivatives satisfying the following estimate: ≲_,̨𝐈_𝒞^2(∂M),_∂M𝐈_𝒞^2(∂M)^-1. Let _ı be derivative in the 𝖾_ı-direction (where 𝖾_· is a fixed orthonormal frame on ∂M). Note lim_h→01h{|_∂M𝐈_+h_∂M𝐉[𝐈]_|^2-|_∂M𝐈_|^2}=2∑_ı=1^_ı𝐉[𝐈]__ı𝐈_,(Indeed, the squared gradient is just a sum over ı=1,…,$̣ of squares of _ı. Then, we use(a+hb)^2-a^2=2hab+o(h).) Using (<ref>) with the chain rule then gives lim_h→01h{(1+|_∂M𝐈_+h_∂M𝐉[𝐈]_|^2)^1/2-(1+|_∂M𝐈_|^2)^1/2}=12(1+|_∂M𝐈_|^2)^-1/2·2∑_ı=1^_ı𝐉[𝐈]__ı𝐈_.We note (<ref>) is^-1/3times a smooth function of𝐈and_∂M𝐈and_ı_ȷ_𝐈(for allı,ȷ,=̨1,…,$̣); indeed, see (<ref>). Since Fluc^noise,-noise is determined by integrals of (<ref>) against smooth functions (like 𝐊 and 1) on ∂M, verifying existence of (<ref>) and showing (<ref>) is straightforward. (For (<ref>), it suffices to use the ^-2/3-scaling in (<ref>) and the ^-1/3 scaling in (<ref>), which can be seen from (<ref>), to get ^-1.)If we now specialize Lemma <ref> to𝐈given by the𝐈^process, we get the following. Fix ∈∂M and any stopping time τ∈[0,1]. For any 0≤≤τ, the quantity ℒ^,𝔮^__flowFluc^noise,_,𝔮^_,𝐈^_is jointly smooth in ,𝔮^_ with $̨-th order derivatives≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^-1. (This is all deterministic.)Use Lemma <ref> and Lemma <ref> to control _∂M𝐈^__𝒞^2(∂M)≲1+𝐘^_𝒞^2(∂M).§.§ Point (2): Computing 𝐈↦ℒ^,𝔮_flowℒ^,𝐈_DtNOur goal now is to now prove Lemma <ref> below, i.e. compute (and show existence of) ℒ^,𝔮_flowℒ^,𝐈_DtN:=lim_h→01h{ℒ^,𝐈+h𝐉[𝐈]_DtN-ℒ^,𝐈_DtN},where𝐉[𝐈]is given by (<ref>), and this equality is meant as operators on𝒞^∞(∂M)(so it holds true when we apply the RHS to genericφ∈𝒞^∞(∂M)). To this end, recall (<ref>) and fix anyφ∈𝒞^∞(∂M). Using this, we get (essentially by definition of Dirichlet-to-Neumann) the following with notation explained after: {ℒ^,𝐈+h𝐉[𝐈]_DtN-ℒ^,𝐈_DtN}φ=^-4/3_𝖭[𝒰^𝐈+h𝐉[𝐈],φ-𝒰^𝐈,φ].Above,𝖭is the inward unit normal vector field on∂M, and_𝖭is gradient in this direction. The𝒰-terms are harmonic extensions ofφwith respect to metrics𝐠[_∂M(𝐈+h𝐉[𝐈])]and𝐠[_∂M𝐈], respectively. To write this precisely, recallΔ_𝐈is Laplacian onMwith respect to the metric𝐠[_∂M𝐈](see before (<ref>)). We have Δ_𝐈+h𝐉[𝐈]𝒰^𝐈+h𝐉[𝐈],φ,Δ_𝐈𝒰^𝐈,φ=0 and𝒰^𝐈+h𝐉[𝐈],φ,𝒰^𝐈,φ|_∂M=φ.For convenience, let us define𝒱^𝐈,h,φ:=𝒰^𝐈+h𝐉[𝐈],φ-𝒰^𝐈,φ. By (<ref>), we get the PDEΔ_𝐈𝒱^𝐈,h,φ=-[Δ_𝐈+h𝐉[𝐈]-Δ_𝐈]𝒰^𝐈+h𝐉[𝐈],φand𝒱^𝐈,h,φ|_∂M=0.We now make two claims. *The operator Δ_𝐈+h𝐉[𝐈]-Δ_𝐈 is bounded as a map 𝒞^+̨10(∂M)→𝒞^(∂M) with norm ≲ h, with implied constant depending on at most 10 derivatives of 𝐈. Indeed, in local coordinates, we have the following in which we view 𝐠[·]-metrics as matrices (normalized by the square root of their determinants): Δ_𝐈+h𝐉[𝐈]-Δ_𝐈=∑_ı,ȷ=1^_ı{(𝐠[_∂M𝐈+h_∂M𝐉[𝐈]]^-1_ıȷ-𝐠[_∂M𝐈]^-1_ıȷ)_ȷ}.(Above, _ı is derivative in the direction of the orthonormal frame vector 𝖾_ı.) Since the metric matrix 𝐠[·] is strictly positive definite, the inverse matrix 𝐠[·]^-1 is smooth in the input. (Everything here is allowed to depend on as many derivatives of 𝐈 as we need.) Thus, (<ref>) turns into the following (noting that 𝐉[𝐈] in (<ref>) has scaling of order ^-1/3): Δ_𝐈+h𝐉[𝐈]-Δ_𝐈=∑_ı,ȷ=1^_ı{O(h^-1/3)_ȷ},where O(^-1/3h) is something smooth whose $̨-derivatives are≲_,̨𝐈^-1/3h. Moreover,𝒰^𝐈+h𝐉[𝐈],φis smooth with derivatives≲_𝐈,φ1(indeed, use elliptic regularity for (<ref>).) If we use this estimate with (<ref>), then (<ref>) plus elliptic regularity shows that𝒱^𝐈,h,φhas𝒞^(M)-norm that is≲_,̨𝐈,φh. To finish this first step, we now rewrite (<ref>) by replacing𝒰^𝐈+h𝐉[𝐈],φwith𝒰^𝐈,φwith error𝒱^𝐈,h,φ: Δ_𝐈𝒱^𝐈,h,φ=-[Δ_𝐈+h𝐉[𝐈]-Δ_𝐈]𝒰^𝐈,φ-[Δ_𝐈+h𝐉[𝐈]-Δ_𝐈]𝒱^𝐈,h,φand𝒱^𝐈,h,φ|_∂M=0. *We investigate (<ref>) a little more carefully. We got (<ref>) by smoothness of𝐠[·]^-1entry-wise. We now claim thath^-1[Δ_𝐈+h𝐉[𝐈]-Δ_𝐈]is not only bounded as a differential operator ash→0, but it has a limit. In particular, we claim that𝒪^𝐈:=lim_h→01h{Δ_𝐈+h𝐉[𝐈]-Δ_𝐈}exists, and it is bounded as a map𝒞^+̨2(M)→𝒞^(M)for any$̨, with norm bounded above by ^-1/3 times something depending only on $̨, the𝒞^2(∂M)-norm of𝐈, and the𝒞^2(∂M)-norm of_ı𝐈(for allı=1,…,$̣). Indeed, this holds by Taylor expanding the smooth matrix 𝐠[·]^-1 entry-wise in (<ref>) and controlling regularity of 𝐉[𝐈] by directly inspecting (<ref>). (The dependence on 𝐈 of the norm of 𝒪^𝐈 comes from the 𝒞^1(∂M)-dependence of (<ref>) in 𝐈 and _ı𝐈, which gets upgraded to 𝒞^2(∂M)-data because of the additional _ı-differential on the outside on the RHS of (<ref>).) To conclude this step, we note the last term [Δ_𝐈+h𝐉[𝐈]-Δ_𝐈]𝒱^𝐈,h,φ in the PDE in (<ref>) is O(h^2). (This follows by (<ref>) and our estimate 𝒱^𝐈,h,φ≲ h from after (<ref>).)We can now divide (<ref>) byhand takeh→0using (<ref>). By standard elliptic regularity, we can take this limit in the “naive" sense, so that Δ_𝐈{lim_h→01h𝒱^𝐈,h,φ}=-𝒪^𝐈𝒰^𝐈,φand{lim_h→01h𝒱^𝐈,h,φ}|_∂M=0.In view of (<ref>) and (<ref>), we ultimately deduce the following. Fix 𝐈∈𝒞^∞(∂M). We have the following, where the limit is taken as an operator 𝒞^∞(∂M)→𝒞^∞(∂M), and φ∈𝒞^∞(∂M) is any test function: ℒ^,𝔮_flowℒ^,𝐈_DtNφ:=lim_h→01h{ℒ^,𝐈+h𝐉[𝐈]_DtN-ℒ^,𝐈_DtN}φ=^-4/3_𝖭𝒱^𝐈,φ,where _𝖭 is gradient in the direction of the inward unit normal vector field 𝖭, and 𝒱^𝐈,φ solves the following PDE (with notation explained afterwards): Δ_𝐈𝒱^𝐈,φ=𝒪^𝐈𝒰^𝐈,φand𝒱^𝐈,φ|_∂M=0.* 𝒪^𝐈 is a bounded map 𝒞^+̨2(M)→𝒞^(M) with norm ≲_,̨𝐈_𝒞^2(∂M),_∂M𝐈_𝒞^2(∂M)^-1/3.* 𝒰^𝐈,φ is the 𝐠[_∂M𝐈]-harmonic extension of φ to M: Δ_𝐈𝒰^𝐈,φ=0and𝒰^𝐈,φ|_∂M=φ.See everything from (<ref>) until the statement of Lemma <ref>.Fix any stopping time τ∈[0,1]. For any 0≤≤τ, the operator ℒ^,𝔮^__flowℒ^,𝐈^__DtNis bounded as an operator 𝒞^+̨10(∂M)→𝒞^(∂M) with operator norm ≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^-5/3. As in the proof of Corollary <ref>, by Lemma <ref>, we know that for all n≥0, we have _∂M𝐈^_𝒞^0_τ𝒞^n_∂M≲_n1+𝐘^_𝒞^0_τ𝒞^2_∂M.It now suffices to use the formula for ℒ^,𝔮^__flowℒ^,𝐈^__DtN and elliptic regularity for (<ref>) and (<ref>). Indeed: *The map ℒ^,𝔮^__flowℒ^,𝐈^__DtN is composed of the following. First, take φ∈𝒞^∞(∂M) and generate 𝒰^𝐈^_,·,φ from it. By elliptic regularity for (<ref>), we know that the 𝒞^(∂M)-norm of 𝒰^𝐈^_,·,φ is controlled by that of φ times a factor depending only on some number of derivatives of _∂M𝐈^_,·. (Indeed, the dependence on 𝐈^_,· is through the metric 𝐠[_∂M𝐈^_,·] defining the Laplacian Δ_𝐈^_,·.) *Similarly, by elliptic regularity for (<ref>) and the operator bound for 𝒪^𝐈 from Lemma <ref>, we know that the 𝒞^+̨2(∂M)-norm of 𝒱^𝐈^_,·,φ is controlled by the 𝒞^(∂M)-norm of 𝒰^𝐈^_,·,φ (times factors which depend on 𝐈^_,·_𝒞^2(∂M),_∂M𝐈^_,·_𝒞^2(∂M)). Finally, the normal gradient _𝖭 forces us to add one derivative.Ultimately, we deduce that ℒ^,𝔮^__flowℒ^,𝐈^__DtN:𝒞^+̨10(∂M)→𝒞^(∂M) is bounded with operator norm of order ^-5/3 depending on ,̨𝐈^_,·_𝒞^2(∂M), and _∂M𝐈^_,·_𝒞^n(∂M) for some n≥0 depending on $̨. (The order of this operator norm bound comes from the^-4/3-scaling in (<ref>) and the^-1/3-bound for𝒪^𝐈maps in Lemma <ref>.) Using (<ref>), this completes the proof.§.§ Point (3), part 1: Computing ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1 and ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓThe basis for this step is the following resolvent identity: A^-1-B^-1=A^-1(B-A)B^-1.Indeed, this turns computation of ℒ^,𝔮_flow(λ-ℒ^,𝐈_DtN)^-1 into an application of Lemma <ref>. More precisely, we have the following result (in which we retain the notation from Lemma <ref>). Fix any 𝐈∈𝒞^∞(∂M). We have the following limit of operators 𝒞^∞(∂M)→𝒞^∞(∂M): lim_h→01h{(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1-(λ-ℒ^,𝐈_DtN)^-1}=(λ-ℒ^,𝐈_DtN)^-1ℒ^,𝔮_flowℒ^,𝐈_DtN(λ-ℒ^,𝐈_DtN)^-1. We first use (<ref>) with A=(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1 and B=(λ-ℒ^,𝐈_DtN)^-1: (λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1-(λ-ℒ^,𝐈_DtN)^-1 =(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1[ℒ^,𝐈+h𝐉[𝐈]_DtN-ℒ^,𝐈_DtN](λ-ℒ^,𝐈_DtN)^-1.The resolvents are bounded operators on any Sobolev space by Lemma <ref>. Also, by Lemma <ref>, we know that the difference of Dirichlet-to-Neumann maps is O(h) as an operator 𝒞^∞(∂M)→𝒞^∞(∂M). Thus, the difference on the LHS of (<ref>) is O(h) as an operator 𝒞^∞(∂M)→𝒞^∞(∂M). In particular, at the cost of O(h^2), we replace the first resolvent (λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1 on the RHS of (<ref>) by (λ-ℒ^,𝐈_DtN)^-1. Dividing by h and sending h→0 then gives (<ref>), so we are done.We now use another chain-rule-type argument to differentiate (λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓ in𝐈. To this end, we require another resolvent identity. In particular, we first claim that A^-[ℓ+1]-B^-[ℓ+1] =∑_=0^ℓA^-(A^-1-B^-1)B^-ℓ+.Indeed, ifℓ=0, this is trivial. To induct, we first write A^-[ℓ+1]-B^-[ℓ+1]=A^-1[A^-ℓ-B^-ℓ]+[A^-1-B^-1]B^-ℓ,and plug (<ref>) (but withℓinstead ofℓ+1) into the first term on the RHS above to deduce (<ref>) forℓ+1. Fix any 𝐈∈𝒞^∞(∂M). We have the following limit of operators 𝒞^∞(∂M)→𝒞^∞(∂M): lim_h→01h{(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1[λ(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1]^ℓ-(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓ}=λ^ℓ∑_=0^ℓ(λ-ℒ^,𝐈_DtN)^--1ℒ^,𝔮_flowℒ^,𝐈_DtN(λ-ℒ^,𝐈_DtN)^-ℓ-1+. We first use (<ref>) for with A=(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1 and B=(λ-ℒ^,𝐈_DtN)^-1: (λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1[λ(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1]^ℓ-(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓ=λ^ℓ{(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-[ℓ+1]-(λ-ℒ^,𝐈_DtN)^-[ℓ+1]}=λ^ℓ∑_=0^ℓ(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-{(λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^-1-(λ-ℒ^,𝐈_DtN)^-1}(λ-ℒ^,𝐈_DtN)^-ℓ+n.We can replace the difference of resolvents by h×(λ-ℒ^,𝐈_DtN)^-1ℒ^,𝔮_flowℒ^,𝐈_DtN(λ-ℒ^,𝐈_DtN)^-1 plus an error of o(h) by Lemma <ref>. By the same token, in (<ref>), we can also replace (λ-ℒ^,𝐈+h𝐉[𝐈]_DtN)^- by (λ-ℒ^,𝐈_DtN)^- with an error of O(h^2) (this replacement has error O(h), but the difference of resolvents in (<ref>) is O(h) as we just mentioned). Thus, when we divide by h and send h→0, (<ref>) becomes (<ref>), so we are done.§.§ Point (4): Putting it altogether via Leibniz ruleObserve that the operator (<ref>) is an actual derivative, so the Leibniz rule applies. Thus, to compute (<ref>), we get two terms. The first comes from differentiating the operator in𝐈, and the second comes from differentiatingFluc^noise,in𝐈. In particular, by Lemmas <ref> and <ref>, we get the following (whose proof is, again, immediate by the Leibniz rule, so we omit it). Retain the notation from Lemmas <ref> and <ref>. Fix ,𝔮∈∂M and 𝐈∈𝒞^∞(∂M). The quantity (<ref>), which is defined as a limit via (<ref>), exists, and (<ref>) =λ^ℓ∑_=0^ℓ(λ-ℒ^,𝐈_DtN)^--1ℒ^,𝔮_flowℒ^,𝐈_DtN(λ-ℒ^,𝐈_DtN)^-ℓ-1+Fluc^noise,_,𝔮,𝐈+(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓℒ^,𝔮_flowFluc^noise,_,𝔮,𝐈.§ PROOFS OF LEMMAS <REF> AND <REF>Before we start, we invite the reader to go back to right before the statements of Lemmas <ref> and <ref> to get the idea behind their proofs, respectively. (In a nutshell, the proofs are just power-counting and explicitly writing out the topologies in which we get estimates. The only other idea is the homogenization step for the proof of (<ref>) that we described briefly in the second bullet point after Lemma <ref>. But even this is built on the same ideas via the Ito formula that are present in Section <ref>. Moreover, it is easier in this case, since there will be no singular^-2/3-factor to fight.) §.§ Proof of Lemma <ref>Letτ∈[0,1]be a generic stopping time. Our goal is to bound (<ref>); see (<ref>) for the exact estimate we want. Becauseτ≤1, we can bound the time-integral (<ref>) by the supremum of its integrand (in𝒞^(∂M)-norm); this is by the triangle inequality. In particular, we have(<ref>)_𝒞^0_τ𝒞^ ≲sup_0≤≤τℒ^,𝔮^__flow(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^_,·_𝒞^(∂M).We compute the term in the norm on the RHS of (<ref>) using Lemma <ref>. In particular, the RHS of (<ref>) is ≲_ℓλ^ℓmax_0≤≤ℓsup_0≤≤τ(λ-ℒ^,𝐈^__DtN)^--1ℒ^,𝔮^__flowℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_·,𝔮^_,𝐈^__𝒞^(∂M)+sup_0≤≤τ(λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓℒ^,𝔮^__flowFluc^noise,_·,𝔮^_,𝐈^__𝒞^(∂M).We will now assume that=̨0; bounds for general≥̨0follow by the exact same argument but replacingFluc^noise,by its$̨-th order derivatives in . Now, fix ∈∂M. Let _H^α(∂M) be the H^α(∂M)-norm in the 𝔮^_-variable. We also set _𝒞^(∂M) to be the 𝒞^(∂M)-norm in 𝔮^_. To control (<ref>), observe that: *The resolvents in (<ref>) are bounded operators on Sobolev spaces with norm ≲λ^-1. Thus, for any α≥0, we get the following (the last bound follows since L^∞ controls L^2 on the compact manifold ∂M): (λ-ℒ^,𝐈^__DtN)^--1ℒ^,𝔮^__flowℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_,𝔮^_,𝐈^__H^α(∂M)≲_ℓ,αλ^--1ℒ^,𝔮^__flowℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_,𝔮^_,𝐈^__H^α(∂M)≲_α,ℓλ^--1ℒ^,𝔮^__flowℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_,𝔮^_,𝐈^__𝒞^α(∂M).Now, we use the operator norm bound for ℒ^,𝔮^__flowℒ^,𝐈^__DtN from Corollary <ref>. We deduce that(<ref>) ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^-5/3λ^--1(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_,𝔮^_,𝐈^__𝒞^α+10(∂M).Use a Sobolev embedding to control the norm on the RHS of (<ref>) by the H^α_2(∂M)-norm (for α_2 depending only on α). Now, Fluc^noise, is in the null-space of ℒ^,𝐈^__DtN, and ℒ^,𝐈^__DtN has a spectral gap that is scaled by ^-4/3. (See the proof of Lemma <ref>.) So, as in the proof of Lemma <ref>, each resolvent in (<ref>) gives a factor of ^4/3. Since Fluc^noise, is smooth with derivatives of order ^-2/3 (see (<ref>)), we ultimately get the estimate below for some β>0 uniformly positive: RHS(<ref>) ≲_ℓ,𝐘^_𝒞^0_τ𝒞^2_∂M^-5/3^4/3[ℓ+1-n]λ^--1^-2/3. *If we now combine every display in the previous bullet point, we deduce that(λ-ℒ^,𝐈^__DtN)^--1ℒ^,𝔮^__flowℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-ℓ-1+Fluc^noise,_,𝔮^_,𝐈^__H^α(∂M)≲_ℓ,α,𝐘^_𝒞^0_τ𝒞^2_∂M^-7/3^4/3[ℓ+1-n]λ^--1.If we choose α≥0 sufficiently large, then by Sobolev embedding, the same estimate holds but for the 𝒞^0(∂M)-norm in (<ref>) instead of H^α(∂M). In particular, the term inside _H^α(∂M) in (<ref>) is bounded by (<ref>) uniformly over possible values of ,𝔮^_∈∂M.In view of (<ref>)-(<ref>) and the paragraph after it, we get (<ref>) ≲_ℓ,α,𝐘^_𝒞^0_τ𝒞^2_∂Mλ^ℓ^-7/3^4/3[ℓ+1-n]λ^--1=λ^ℓ--1^4/3[ℓ--1]^-7/3^8/3≲^1/3,where the last bound follows by λ=^-4/3+γ for γ>0 (see (<ref>)). Let us now control (<ref>). To this end, a very similar argument works. In particular, each resolvent in (<ref>) gives us λ^-1 in _H^α(∂M)-norm. On the other hand, by Corollary <ref>, we know thatℒ^,𝔮^__flowFluc^noise,_,𝔮^_,𝐈^__H^α(∂M)≲_α,𝐘^_𝒞^0_τ𝒞^2_∂M^-1.If we combine the previous display and paragraph, we deduce that (λ-ℒ^,𝐈^__DtN)^-1[λ(λ-ℒ^,𝐈^__DtN)^-1]^ℓℒ^,𝔮^__flowFluc^noise,_,𝔮^_,𝐈^__H^α(∂M)≲λ^-1-ℓ^-1≲^1/3-γ.Taking α large enough gives us the same estimate in _𝒞^0(∂M). Because we can take γ>0 as small as we want (as long as it is uniformly positive), we get the following for β>0 uniformly positive: (<ref>)≲_α,𝐘^_𝒞^0_τ𝒞^2_∂M^β.Combining this with (<ref>)-(<ref>) and (<ref>)-(<ref>) produces the estimate (<ref>), so we are done. §.§ Proof of Lemma <ref>We start by removing the ℒ^,𝔮^__flow-operator from (<ref>)-(<ref>), which is helpful for all ℓ≥0 (in particular, for proving both estimates (<ref>) and (<ref>)). Indeed, as noted in the first bullet point after Lemma <ref>, we have the following. Fix any ℱ:𝒞^∞(∂M)→ in the domain of ℒ^,𝔮^__flow. By the Leibniz rule, since ℒ^,𝔮^__flow is a first-order differential (see (<ref>)), we know that ℒ^,𝔮^__flow|ℱ[𝐈]|^2 exists, and ℒ^,𝔮^__flow|ℱ[𝐈]|^2-2ℱ[𝐈]×ℒ^,𝔮^__flowℱ[𝐈]=0.Now use (<ref>) for ℱ[𝐈]=(λ-ℒ^,𝐈_DtN)^-1[λ(λ-ℒ^,𝐈_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈 to show that (<ref>)-(<ref>) is equal to the following (which is just removing ℒ^,𝔮^__flow from (<ref>)-(<ref>)): [𝐌^,ℓ]_, =∫_0^ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|^2]-2∫_0^{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_}×{ℒ^,𝐈^__DtN[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_]}.Let us first prove the second estimate (<ref>), because it requires one less step (and is thus easier) compared to (<ref>). (We explain this later when relevant in the proof of (<ref>).) In particular, (<ref>) serves as a warm-up to the more complicated (<ref>).§.§.§ Proof of (<ref>)Fix a stopping time τ∈[0,1]. Our goal is to estimate the 𝒞^0_τ𝒞^_∂M-norm of (<ref>)-(<ref>) for 1≤ℓ≤ℓ_max and control it by a positive power of . We assume =̨0; for general $̨, just replace (<ref>)-(<ref>) by$̨-th order derivatives in . (Again, all we need is an algebraic property for Fluc^noise, that is closed under linear combinations and only concerns 𝔮^_,𝐈^_-variables.) We start with the RHS of (<ref>). By the triangle inequality and τ≤1, we haveRHS(<ref>)_𝒞^0_τ𝒞^0_∂M≲sup_0≤≤τℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^_|^2]_𝒞^0(∂M).Let _α be the H^α(∂M)-norm with respect to 𝔮^_∈∂M. We claim that the norm of ℒ^,𝐈_DtN:H^α+1(∂M)→H^α(∂M) is order ^-4/3 (because of the scaling in (<ref>)) times something depending only on the 𝒞^2(∂M)-norm of 𝐈. Indeed, if 𝐈=0, this would be immediate by the first-order property of the Dirichlet-to-Neumann map ℒ; see Lemma <ref>. To deal with 𝐈, the same is true if we change the measure from the one induced by surface metric on ∂M to the one induced by the metric 𝐠[_∂M𝐈]. But this measure depends only on at 1 derivative of 𝐠[_∂M𝐈] since said measure is obtained by taking determinants of the 𝐠 matrix in local coordinates. Thus, it depends on at most 2 derivatives of 𝐈. The claim follows. So, given that the 𝒞^2(∂M)-norm of 𝐈^ is bounded by that of 𝐘^ (see (<ref>)), for any ∈∂M, we deduce ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|^2]_α≲_𝐘^_𝒞^0_τ𝒞^2_∂M,α^-4/3|(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_|^2_α+1.Now, use Sobolev multiplication (see Lemma <ref>). In particular, if we take α sufficiently large depending on the dimension $̣, then we can bound the Sobolev norm of the square by the square of the Sobolev norm: (<ref>) ≲^-4/3{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^__α+1}^2.Now, we power-count using Sobolev estimates for operators in (<ref>). Fortunately, we already did this; use Lemma <ref> to bound the norm on the RHS of (<ref>). We deduce RHS(<ref>) ≲_ℓ,α,𝐘^_𝒞^0_τ𝒞^2_∂M^-4/3{^4/3[λ^4/3]^ℓ^-2/3}^2≲λ^2ℓ^8/3ℓ≲^2γℓforγ>0uniformly positive, where the last bound follows from (<ref>). If we now combine (<ref>), (<ref>)-(<ref>), (<ref>), and (<ref>) with the same Sobolev embedding argument that we explained after (<ref>)-(<ref>), we ultimately deduce the following for someβ>0uniformly positive: RHS(<ref>)_𝒞^0_τ𝒞^0_∂M≲_ℓ,α,𝐘^_𝒞^0_τ𝒞^2_∂M^β.We now move to (<ref>)-(<ref>). First, we rewrite the-integrand in (<ref>) as ℒ^,𝐈^__DtN[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_]=-(λ-ℒ^,𝐈^__DtN)[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_]+λ[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_]=-[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_+λ[(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_].In particular, if we plug (<ref>)-(<ref>) into (<ref>) and multiply by the integrand in (<ref>), we get the following expression for (<ref>)-(<ref>): 2∫_0^{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_}×[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_-2λ∫_0^{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_,𝔮^_,𝐈^_}^2.Given the representation (<ref>)-(<ref>) for (<ref>)-(<ref>), in order to bound the 𝒞^0_τ𝒞^0_∂M-norm of (<ref>)-(<ref>), by the triangle inequality, it suffices to show the following estimate forβ>0uniformly positive: sup_0≤≤τ{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^_}×[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^__𝒞^0(∂M)+sup_0≤≤τλ{(λ-ℒ^,𝐈^_,·_DtN)^-1[λ(λ-ℒ^,𝐈^_,·_DtN)^-1]^ℓFluc^noise,_·,𝔮^_,𝐈^_}^2_𝒞^0(∂M)≲_ℓ,α,𝐘^_𝒞^0_τ𝒞^2_∂M^β.We give a power-counting argument that can be made rigorous using the_α-norms and Sobolev embeddings (and Lemma <ref>) that gave us (<ref>). (We omit the explanation behind these steps, because they are identical to the proof of (<ref>).) *Take the first line of the display (<ref>). First, we note Fluc^noise,_·,𝔮^_,𝐈^_, as a function of 𝔮^_, is in the null-space of ℒ^,𝐈^__DtN, which has a spectral gap of ≳^-4/3. See the proof of Lemma <ref>. Thus, each resolvent in the first line of (<ref>) gives a factor ^4/3. Since Fluc^noise, itself is smooth with order ^-2/3 derivatives, the first line of (<ref>) satisfies the estimate ≲_ℓ,𝐘^_𝒞^0_τ𝒞^2_∂M^4/3λ^2ℓ^8/3ℓ^-4/3≲λ^2ℓ^8/3ℓ≲^2ℓγfor γ>0 uniformly positive (for the last bound, see (<ref>)). *By the same token, the second line in (<ref>) satisfies the estimate ≲_ℓ,𝐘^_𝒞^0_τ𝒞^2_∂Mλ{λ^ℓ^4/3[ℓ+1]^-2/3}^2≲λ^2ℓ+1^8/3ℓ^8/3-4/3≲^[2ℓ+1]γfor the same uniformly positive γ>0.*We clarify that the dependence of these estimates on just the 𝒞^0_τ𝒞^2_∂M-norm of 𝐘^ comes from tracking the same argument given in the proof of Lemma <ref>.In view of the previous two bullet points, the estimate (<ref>) follows sinceℓ≥1by assumption. Thus, as noted right before (<ref>), we deduce that the 𝒞^0_τ𝒞^0_∂M-norm of (<ref>)-(<ref>) is≲_ℓ,𝐘^_𝒞^0_τ𝒞^2_∂M^β.Combining this with (<ref>) and (<ref>)-(<ref>) completes the proof of (<ref>).§.§.§ Proof of (<ref>)To make the reading easier, let us recap the goal of this estimate. We want to prove that with high probability, we have the following estimate: [𝐌^,0]-[𝐌^limit]_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β,whereβ>0is uniformly positive,τ∈[0,1]is any stopping time,≥̨0, and [𝐌^,0]_, =∫_0^ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-2∫_0^{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{ℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}[𝐌^limit]_, =-2∫_∂M[𝐊_,-1]×{ℒ^-1[𝐊_,-1]}.(See (<ref>)-(<ref>) and (<ref>).) We now explain the main steps needed to prove (<ref>). (These are essentially outlined before the statement of Lemma <ref>. We refer the reader there for intuition for this argument. But for reasons that entirely technical, we do things in a slightly different manner.) Also, throughout this argument, we will assume that =̨0 in the desired estimate (<ref>); for general $̨, just replace[𝐌^,0]-[𝐌^limit]by its$̨-th order derivatives in . The argument is otherwise completely identical.*In [𝐌^,0], we first replace ℒ^,𝐈^__DtN with ^-4/3ℒ, i.e. replace the metric in the Dirichlet-to-Neumann from 𝐠[_∂M𝐈^_] to the surface metric on ∂M (which can be thought of as 𝐠[0]). In particular, define [𝐌^,0,1]_, =∫_0^^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-2∫_0^{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}.We then want to show that for β>0 uniformly positive, we have [𝐌^,0]-[𝐌^,0,1]_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β. *Next, in (<ref>)-(<ref>), we want to further replace 𝐈^ by 0 in the Fluc^noise,-term therein. In particular, we want to show the estimate below (for β>0 uniformly positive) [𝐌^,0,1]-[𝐌^,0,2]_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β,where the term [𝐌^,0,2] is the following time-integral: [𝐌^,0,2]_, =∫_0^^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0|^2]-2∫_0^{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0}. *The next step is homogenization. In particular, let ℭ^λ_,𝔮^ be the -integrand in (<ref>)-(<ref>) (which is a Carre-du-Champ operator), so that[𝐌^,0,2]_,=∫_0^ℭ^λ_,𝔮^_.Now, define the following “homogenized" version of (<ref>) (i.e. one where we integrate over 𝔮^_ therein): Hom𝐌_, :=∫_0^∫_∂Mℭ^λ_,.We claim the following estimates. The first states [𝐌^,0,2]≈Hom𝐌, i.e. that time-averaging in (<ref>) is enough to introduce a space-average. The second states Hom𝐌≈[𝐌^limit], i.e. closing the argument and allowing us to deduce (<ref>). In particular, [𝐌^,0,2]-Hom𝐌_𝒞^0_τ𝒞^_∂M ≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β Hom𝐌-[𝐌^limit]_𝒞^0_τ𝒞^_∂M ≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β.Above, β>0 is uniformly positive, and (<ref>) is claimed to hold with high probability.By the triangle inequality and (<ref>), (<ref>), (<ref>), and (<ref>), we get (<ref>) with high probability, thereby finishing the proof of this entire lemma. So, we are left to show (<ref>), (<ref>), (<ref>), and (<ref>). Before we embark on this, however, let us present the following key estimates, with proofs given immediately after. (In what follows,α_depends only on$̣, and the joint 𝒞^(∂M×∂M)-norm is with respect to ,𝔮^_-variables. Also, ν>0 can be taken arbitrarily small.) Fluc^noise,_,𝔮^_,𝐈^_-Fluc^noise,_,𝔮^_,0_𝒞^(∂M×∂M) ≲_,𝐘^_𝒞^0_τ𝒞^2_∂M^-1/3 ℒ^,𝐈^__DtN-^-4/3ℒ_H^α+α_(∂M)→H^α(∂M) ≲_α,𝐘^_𝒞^0_τ𝒞^2_∂M^-1-ν (λ-ℒ^,𝐈^__DtN)^-1-(λ-^-4/3ℒ)^-1_H^α+α_(∂M)→H^α(∂M) ≲_α,𝐘^_𝒞^0_τ𝒞^2_∂Mλ^-2^-1-ν≲^5/3-2γ-ν.*The estimate (<ref>) is immediate by (<ref>) and the relation _∂M𝐈^=^1/3_∂M𝐘^ (see (<ref>)). Indeed, by (<ref>), the dependence on 𝐈^_ of Fluc^noise,_,𝔮^_,𝐈^_ is via ^-2/3 times a smooth function of _∂M𝐈^_ (dependence on ,𝔮^_ is uniformly smooth as well). Thus, (<ref>) is by Taylor expansion in _∂M𝐈^ about 0, combined with _∂M𝐈^=^1/3_∂M𝐘^, which introduces a factor that brings ^-2/3 scaling in (<ref>) down to ^-1/3.*To prove (<ref>), we use Lemma <ref>, which controls the difference of Dirichlet-to-Neumann operators on the LHS of (<ref>) by some 𝒞^n(∂M)-norm of the difference of metrics 𝐠[_∂M𝐈^]-𝐠[0]. However, 𝐠 is smooth, so a similar Taylor expansion argument as in the previous bullet point shows 𝐠[_∂M𝐈^]-𝐠[0]=O(^1/3) with implied constant depending only on 𝐘^_𝒞^0_τ𝒞^2_∂M; this is in 𝒞^1,υ(∂M)-norm for any υ∈[0,1). (For this, again use (<ref>) to deduce _∂M𝐈^=^1/3_∂M𝐘^ and gain an extra factor of ^1/3.) On the other hand, for any ≥̨0, the same Taylor expansion but now combined with Lemma <ref> shows that 𝐠[_∂M𝐈^]-𝐠[0]=O(1) with implied constant depending only on 𝐘^_𝒞^0_τ𝒞^2_∂M. Interpolation of Hölder norms then shows the following (see Theorem 3.2 in <cit.>). For any n≥0 and ν>0, we can choose ≥̨0 large enough so that for υ∈(0,1) fixed, we have 𝐠[_∂M𝐈^]-𝐠[0]_𝒞^0_τ𝒞^n_∂M ≲𝐠[_∂M𝐈^]-𝐠[0]_𝒞^0_τ𝒞^1,υ_∂M^1-ν𝐠[_∂M𝐈^]-𝐠[0]_𝒞^0_τ𝒞^_∂M^ν≲_𝐘^_𝒞^0_τ𝒞^2_∂M^1/3(1-ν).As noted in the previous paragraph, Lemma <ref> then shows (<ref>) by using the previous display. (Note the extra ^-4/3-scaling on the LHS of (<ref>).*For (<ref>), we start with the resolvent expansion (λ-ℒ^,𝐈^__DtN)^-1-(λ-^-4/3ℒ)^-1=(λ-ℒ^,𝐈^__DtN)^-1(ℒ^,𝐈^__DtN-^-4/3ℒ)(λ-^-4/3ℒ)^-1.By Lemma <ref>, the resolvents have total operator norm on H^ρ(∂M)→H^ρ(∂M) that is≲_ρ,𝐘^_𝒞^0_τ𝒞^2_∂Mλ^-2.(Indeed, we should have dependence on a 𝒞^n(∂M)-norm of 𝐈^ in the implied constant above, but that is controlled by the 𝒞^2(∂M)-norm of 𝐘^; see Lemma <ref>.) Now, it suffices to combine the previous bound with (<ref>) to get the first bound in (<ref>). The last bound in (<ref>) follows by λ=^-4/3+γ (see (<ref>)).Finally, before we start, a word about notation. For auxiliary certain objects, we use decorations of Υ (such as Υ^(i)) that will be used only for the rest of this section. (If later in the paper we use decorations of Υ, they will be for completely different objects.) See (<ref>)-(<ref>) and (<ref>)-(<ref>). From these, it is easy to see that [𝐌^,0]_,-[𝐌^,0,1]_, =∫_0^Υ^(1)_,𝔮^_,𝐈^_+∫_0^Υ^(2)_,𝔮^_,𝐈^_,where Υ^(1) and Υ^(2) are obtained by comparing Dirichlet-to-Neumann maps and their resolvents: Υ^(1)_,𝔮^_,𝐈^_ :=ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2] Υ^(2)_,𝔮^_,𝐈^_ :=-2{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{ℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}+2{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}.If we now fix any stopping time τ∈[0,1], then by triangle inequality and (<ref>), we have[𝐌^,0]-[𝐌^,0,1]_𝒞^0_τ𝒞^_∂M ≲sup_0≤≤τΥ^(1)_·,𝔮^_,𝐈^__𝒞^(∂M)+sup_0≤≤τΥ^(2)_·,𝔮^_,𝐈^__𝒞^(∂M).(Recall that we have assumed =̨0 for simplicity; see after (<ref>).) We start by estimating the first term on the RHS of (<ref>). In view of (<ref>), we write Υ^(1) as the error obtained by replacing the outer ℒ^,𝐈^__DtN-operator in the first term on the RHS of (<ref>) by ^-4/3ℒ, plus the error obtained by making the same replacement but in the resolvent. In particular, we have the identity Υ^(1)_,𝔮^_,𝐈^_ :=ℒ^,𝐈^__DtN[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-^-4/3ℒ[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]+^-4/3ℒ[|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2].Again, for any α≥0, let _α be the H^α(∂M)-norm in the 𝔮^_-variable. By (<ref>), we have (for α_≲1) RHS(<ref>)_α ≲_α,𝐘^_𝒞^0_τ𝒞^2_∂M^-1-ν|(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2_α+α_.Now, if α is sufficiently large but depending only on the dimension $̣, thenH^α(∂M)is a Hilbert algebra (see Lemma <ref>), so the Sobolev norm of the square is controlled by the square of the Sobolev norm. This gives RHS(<ref>) ≲^-1(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^__α+α_^2.The resolvent is bounded on Sobolev spaces with norm≲λ^-1(see Lemma <ref>). Moreover,Fluc^noise,has derivatives of order^-2/3(see (<ref>)). Thus, the RHS of (<ref>) is≲^-1λ^-2^-4/3≲^1/3-2γ(with implied constant depending on the 𝒞^0_τ𝒞^2_∂M-norm of𝐘^as before, since the metric in the Dirichlet-to-Neumann map on the RHS of (<ref>) depends on at most two derivatives of𝐈^; see the proof of Lemma <ref> for this argument, for example). By the previous two displays, if we takeαlarge enough depending on dimension$̣, then by Sobolev embedding in the 𝔮^_-variable, we deduce that|RHS(<ref>)| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^1/3-2γ.We now control (<ref>). First, ℒ:H^α+1(∂M)→H^α(∂M) is bounded with norm O(1) (see Lemma <ref>). By this and difference of squares and the algebra property of H^α(∂M)-spaces (for α big enough), we get (<ref>)_α ≲^-4/3(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_-(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^__α+1×(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_+(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^__α+1.Use (<ref>) and the fact that derivatives of Fluc^noise, are order ^-2/3 to show that the RHS of (<ref>) is order ^-1/3-2γ-ν. Use the λ^-1-estimate for resolvents in (<ref>) to show that (<ref>) is ≲λ^-1^-2/3≲^2/3-γ. Thus, the LHS of (<ref>) is ≲^1/3-3γ-ν. All estimates have implied constants depending on α and the 𝒞^0_τ𝒞^2_∂M-norm of 𝐘^, as before. Taking α big enough to use a Sobolev embedding like we did after (<ref>)-(<ref>), we get the following for ν>0 fixed but arbitrarily small: |(<ref>)| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^1/3-3γ-ν.Now, combine (<ref>)-(<ref>), (<ref>), and (<ref>). This shows that for β>0 uniformly positive, |Υ^(1)_,𝔮^_,𝐈^_| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^β.We now treat Υ^(2) (see (<ref>)-(<ref>)). This follows from the same type of argument. Let us be precise. We first rewrite (<ref>)-(<ref>) as the error obtained by replacing ℒ^,𝐈^__DtN↦^-4/3ℒ outside of the resolvents, plus the error obtained by this replacement inside the resolvents: Υ^(2)_,𝔮^_,𝐈^_ =-2{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{ℒ^,𝐈^__DtN(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}+2{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}-2{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}+2{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}.Look at RHS(<ref>)+(<ref>). This contribution gives us-2{(λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{[ℒ^,𝐈^__DtN-^4/3ℒ](λ-ℒ^,𝐈^__DtN)^-1Fluc^noise,_,𝔮^_,𝐈^_}.The first factor is ≲λ^-1^-2/3≲^2/3-γ (for reasons we explained in the proof of (<ref>)). By (<ref>), the difference of Dirichlet-to-Neumann maps is ≲^-1-ν; it acts on something of order ≲^2/3-γ as we just noted. Therefore, (<ref>), which is RHS(<ref>)+(<ref>), is ≲^1/3-2γ-ν (with implied constant depending on the 𝒞^0_τ𝒞^2_∂M-norm of 𝐘^, as before). Of course, this heuristic can be made precise by the same Sobolev multiplication and embedding argument that we just illustrated. We omit the lengthy details. Take (<ref>)+(<ref>). When we replace resolvents (λ-ℒ^,𝐈^__DtN)^-1↦(λ-^-4/3ℒ)^-1 in one of the factors in (<ref>), by (<ref>), we pick up a factor of ^5/3-2γ-ν. This acts on Fluc^noise,, which has derivatives of order ^-2/3. We then multiply by the second factor in (<ref>), which is order ≲^-4/3λ^-1^-2/3≲^-2/3-γ. By multiplying all bounds together, we get that the error in replacing resolvents in the first factor in (<ref>) is ≲^1/3-3γ-ν. The rest of (<ref>)+(<ref>) is obtained by making this same replacement of resolvents in the second factor in (<ref>). By the same argument, this error is ≲^1/3-3γ. Thus, for ν>0 fixed but arbitrarily small, we get |(<ref>)+(<ref>)|≲^1/3-3γ-ν. By combining the previous two paragraphs and (<ref>)-(<ref>), we deduce that for β>0 uniformly positive,|Υ^(2)_,𝔮^_,𝐈^_| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^β.Combine (<ref>) with (<ref>) and (<ref>) to get the desired bound (<ref>) (recall we assumed that =̨0).By (<ref>)-(<ref>) and (<ref>)-(<ref>) (and the triangle inequality argument that gave (<ref>)), [𝐌^,0,1]-[𝐌^,0,2]_𝒞^0_τ𝒞^_∂M≲sup_0≤≤τΥ^(3)_·,𝔮^_,𝐈^__𝒞^(∂M)+Υ^(4)_·,𝔮^_,𝐈^__𝒞^(∂M),where Υ^(3)_,𝔮^_,𝐈^_ :=^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_|^2]-^-4/3ℒ[|(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0|^2] Υ^(4)_,𝔮^_,𝐈^_ :=-2{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_}+2{(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0}×{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0}.We first treat Υ^(3). By difference of squares, we have Υ^(3)_,𝔮^_,𝐈^_ =^-4/3ℒ{[(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_-(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0]..×[(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,𝐈^_+(λ-^-4/3ℒ)^-1Fluc^noise,_,𝔮^_,0]}.We now give a heuristic that immediately turns rigorous when we use the Sobolev multiplication/embedding framework that we explained in detail in the proof (<ref>). The ^-4/3ℒ operator gives a factor of ^-4/3. The resolvent on the RHS of (<ref>) gives a factor of ≲λ^-1. It acts on the difference of Fluc^noise,-terms, which by (<ref>), has derivatives of order ≲^-1/3. Thus, the factor in the curly braces on the RHS of (<ref>) is ≲λ^-1^-1/3≲^1-γ (see (<ref>)). It multiplies (<ref>), which is ≲λ^-1^-2/3≲^2/3-γ, since the resolvents give λ^-1, and the Fluc^noise,-terms are ≲^-2/3. Thus, the term inside the curly brackets in (<ref>)-(<ref>) is ≲^1-γ^2/3-γ≲^5/3-2γ. Multiplying by ^-4/3 therefore shows that for β>0 uniformly positive,|Υ^(3)_,𝔮^_,𝐈^_| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^β.For Υ^(4) (see (<ref>)-(<ref>)), a similar argument works. When we replace 𝐈^_↦0 in the first factor in (<ref>), the error we get is something of order ≲^-1/3 (by (<ref>)) which is hit by a resolvent that gives ≲λ^-1. We then multiply by the second factor in (<ref>), which is ≲^-4/3λ^-1^-2/3≲^-2/3-γ. Multiplying all bounds, the error is ≲λ^-1^-1/3^-2/3-γ≲^1/3-2γ. When we replace 𝐈^_↦0 in the second factor in (<ref>), the error is estimated in the same way. Thus, since these two errors are exactly what gives (<ref>)-(<ref>), we deduce that for β>0 uniformly positive, we have|Υ^(4)_,𝔮^_,𝐈^_| ≲_𝐘^_𝒞^0_τ𝒞^2_∂M^β.Now, combine (<ref>)-(<ref>) and (<ref>) (recall =̨0). This gives the desired estimate (<ref>).To start, we compute Hom𝐌 in detail. By (<ref>) (with ℭ^λ equal to the -integrand of (<ref>)-(<ref>)), we claim thatHom𝐌_, =-2∫_0^∫_∂M{(λ-^-4/3ℒ)^-1Fluc^noise,_,,0}{^-4/3ℒ(λ-^-4/3ℒ)^-1Fluc^noise,_,,0}.Indeed, the claim is just that if we integrate the -integrand in (<ref>) over ∂M (in the 𝔮^_-variable), we get 0. This is because said -integrand is in the image of ℒ by construction, and ℒ has invariant measure given by the surface measure on ∂M (see Lemma <ref>). We now proceed in two steps. First, rewrite ^-4/3ℒ=^-4/3ℒ-λ+λ for the Dirichlet-to-Neumann map in (<ref>) that is not inside any resolvent. The ^-4/3ℒ-λ piece, when multiplied by the outer negative sign, cancels the resolvent (λ-^-4/3ℒ). Thus, Hom𝐌_, =2∫_0^∫_∂MFluc^noise,_,,0×(λ-^-4/3ℒ)^-1Fluc^noise,_,,0-2λ∫_0^∫_∂M|(λ-^-4/3ℒ)^-1Fluc^noise,_,,0|^2.The RHS of (<ref>) has integrand that is independent of , so we can replace the -integration by a factor of . The same is true for (<ref>). Now, recall [𝐌^limit] from (<ref>). By triangle inequality (exactly like in what gave us (<ref>)), we therefore get the bound Hom𝐌-[𝐌^limit]_𝒞^0_τ𝒞^_∂M ≲Υ^(5)_𝒞^(∂M)+λ∫_∂M|(λ-^-4/3ℒ)^-1Fluc^noise,_·,,0|^2_𝒞^(∂M),whereΥ^(5)_ :=∫_∂MFluc^noise,_,,0{(λ-^-4/3ℒ)^-1Fluc^noise,_,,0}+∫_∂M[𝐊_,-1]{ℒ^-1[𝐊_,-1]}.Let us control the second term on the RHS of (<ref>). By ^-2/3-bounds for Fluc^noise, (see (<ref>)) and (<ref>) for ℓ=0 (and setting 𝐈^_=0, which is okay because we never used any data about 𝐈^ in the proof), the term in the square is ≲^2/3, so its square is ≲^4/3. Multiply by λ=^-4/3+γ (see (<ref>)) to get λ∫_∂M|(λ-^-4/3ℒ)^-1Fluc^noise,_·,,0|^2_𝒞^(∂M)≲^γ.We now treat Υ^(5). First, by (<ref>), we have the following (since Vol_0=1; see Construction <ref>): Fluc^noise,_,,0 =^-2/3[𝐊_,-∫_∂M𝐊_,]=^-2/3[𝐊_,-1],since 𝐊 is normalized to have total mass 1 (see Construction <ref>). Using (<ref>), we can rewrite (<ref>) as Υ^(5)_=∫_∂M[𝐊_,-1]×{[^-4/3(λ-^-4/3ℒ)^-1+ℒ^-1][𝐊_,-1]}.Now, we use a resolvent expansion (see (<ref>) with A=^4/3(λ-^-4/3ℒ) and B=-ℒ). This implies^-4/3(λ-^-4/3ℒ)^-1+ℒ^-1=^-4/3(λ-^-4/3ℒ)^-1(^4/3λ)ℒ^-1=λ(λ-^-4/3ℒ)^-1ℒ^-1.Now, note that 𝐊_,·-1 is orthogonal to the null-space of ℒ (since 1 is just the projection of 𝐊_,· onto the space of constant functions on ∂M, which is exactly the null-space of ℒ; see Lemma <ref>). Thus, we can use a spectral gap for ℒ (see Lemma <ref>) when we apply its inverse and resolvents to 𝐊_,·-1. So, when we apply the far RHS of (<ref>) to 𝐊_,·-1, the ℒ^-1-operator is bounded. For λ(λ-^-4/3ℒ)^-1, we ignore λ inside the resolvent (since λ only regularizes the resolvent; see Lemma <ref>), and then we have λ^4/3ℒ^-1. By spectral gap for ℒ and λ=^-4/3+γ, the term in curly braces in (<ref>) is ≲^γ. (Technically, this is all in Sobolev norms in ; we need to use Sobolev embedding as in the proof of (<ref>).) So, by smoothness of 𝐊, Υ^(5)_𝒞^(∂M)≲^γ.Now, combine (<ref>), (<ref>), and (<ref>). This gives the desired bound (<ref>).For the sake of clarity, we want to estimate the 𝒞^0_τ𝒞^_∂M-norm (for τ∈[0,1]) of [𝐌^,0,2]_,-Hom𝐌_,=∫_0^{ℭ^λ_,𝔮^_-∫_∂Mℭ^λ_,}.(Indeed, see (<ref>) and (<ref>).) Again, we assume =̨0; for general $̨, just apply the argument below but for the$̨-th order derivatives of (<ref>) in . For notational convenience, we define Flucℭ^λ_,𝔮^_:=ℭ^λ_,𝔮^_-∫_∂Mℭ^λ_,to be the fluctuation of ℭ^λ, more or less (or equivalently, the -integrand in (<ref>)). Note that Flucℭ^λ_,· is in the image of ℒ, since it vanishes under integration over 𝔮^_∈∂M with respect to the invariant measureof ℒ; see Lemma <ref>. (We emphasize that we are integrating with respect to measure that, in principle, can have nothing to do with the law of 𝔮^_. Also, this is true for all ∈∂M.) So, we can rigorously (not just formally) hit Flucℭ^λ_,𝔮^_ by ℒℒ^-1 and rewrite (<ref>) as follows, where all operators act on the 𝔮^_-variable: [𝐌^,0,2]_,-Hom𝐌_, =∫_0^^-4/3ℒ[(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_]=∫_0^ℒ^,𝐈^__DtN[(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_]+∫_0^[^-4/3ℒ-ℒ^,𝐈^__DtN][(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_].Let us first control the 𝒞^0_τ𝒞^_∂M-norm of (<ref>). Again, by triangle inequality, we have(<ref>)_𝒞^0_τ𝒞^_∂M ≲sup_0≤≤τ[^-4/3ℒ-ℒ^,𝐈^__DtN][(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_]_𝒞^(∂M).We now give the heuristic for controlling the RHS of (<ref>) (that can be made precise by the same Sobolev embedding argument given throughout this section). Note that Flucℭ^λ_,· is in the image of ℒ as noted after (<ref>), and thus it is orthogonal to its null-space. Thus, when we apply the inverse of ℒ to Flucℭ^λ_,·, we can use a spectral gap estimate (see Lemma <ref>). This means that the difference of Dirichlet-to-Neumann maps hits something of order ^4/3. (Indeed, Flucℭ^λ has derivatives of O(1). This is by (<ref>) and that ℭ^λ has derivatives of O(1). For this last fact, recall ℭ^λ as the -integrand in (<ref>)-(<ref>), and use the estimate (<ref>) with ℓ=0 and 𝐈^_,· replaced by 0. Indeed, this estimate shows that all resolvents acting on Fluc^noise,-terms in (<ref>)-(<ref>) are ≲^2/3; all of these factors are then cancelled by ^-4/3-factors hitting ℒ-operators in (<ref>)-(<ref>).) Next, by (<ref>), the difference of Dirichlet-to-Neumann maps on the RHS of (<ref>) is ≲^-1-ν. So, the RHS of (<ref>) is ≲^1/3-ν for ν>0 fixed but arbitrarily small, and thus (<ref>)_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^1/3-ν.(The dependence on 𝐘^ can be tracked from (<ref>); all other estimates used to get (<ref>) do not depend on 𝐘^.) We now move to (<ref>). This is now where randomness comes in; so far, our estimates in this section have all been deterministic, whereas our estimate for (<ref>) will be with high probability. Note that we can add ℒ^,𝔮^__flow to ℒ^,𝐈^__DtN in (<ref>), because the term in square brackets in (<ref>) does not depend on 𝐈^_. We then end up with the full generator of (𝐈^,𝔮^), and the Ito formula can be applied. (There is no issue of domain for the generator of 𝐈^ because, again, the square bracket in (<ref>) does not depend on 𝐈^.) So, (<ref>) =𝔐_,+(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_-(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_0,where 𝔐 is a martingale with predictable bracket given by 𝔅_,:= ∫_0^ℒ^,𝐈^__DtN[|(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_|^2]-2∫_0^(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_×ℒ^,𝐈^__DtN[(^-4/3ℒ)^-1Flucℭ^λ_,𝔮^_].As we explained in the paragraph after (<ref>), the last two terms on the RHS of (<ref>) are deterministically controlled as follows: sup_0≤≤τ(^-4/3ℒ)^-1Flucℭ^λ_·,𝔮^__𝒞^(∂M)≲_^4/3.It remains to treat the first term on the RHS of (<ref>). To this end, we use Doob's maximal inequality (and that |𝔐|^2-𝔅 is a martingale) to get the following (for τ∈[0,1] a stopping time): {sup_0≤≤τ|𝔐_,|^2} ≲|𝔅_τ,|^2.We now estimate (<ref>)-(<ref>) with the following heuristic (which is, again, immediately rigorous once we use Sobolev norms, embeddings, and multiplication). As explained in the paragraph after (<ref>), the (^-4/3ℒ)^-1Flucℭ^λ-terms in (<ref>)-(<ref>) are ≲^4/3. The ℒ^,𝐈^__DtN have scaling of ≲^-4/3. Thus, we deduce that even without expectations, the RHS of (<ref>) is ≲^4/3, so that {sup_0≤≤τ|𝔐_,|^2}≲_𝐘^_𝒞^0_τ𝒞^2_∂M^4/3.The 𝐘^-dependence comes from the fact that the metric defining ℒ^,𝐈^__DtN depends on the first derivative of the metric 𝐠[_∂M𝐈^_] at most; see the paragraph after (<ref>). (<ref>) gives a pointwise-in- estimate with high probability. To upgrade this into a uniform-in- estimate with high probability, it suffices to show that𝔐_𝒞^0_τ𝒞^_∂M ≲_1with high probability for sufficiently large . (Indeed, by (<ref>) and union bound, we can bound 𝔐 uniformly in time until τ and uniformly over a discretization of ∂M of size ^-4/3+β for β>0 uniformly positive, on one high probability event. We can then use (<ref>) to show that 𝔐 cannot change by more than ^κ between points in said discretization of ∂M for some κ>0 uniformly positive.) To show (<ref>), it suffices to control every other term in (<ref>). For the last two terms in (<ref>), use (<ref>). For the LHS of (<ref>), see (<ref>); the ℒ^,𝐈^__DtN has scaling ≲^-4/3, and the square-bracketed term in (<ref>) has scaling ≲^4/3 (see the paragraph after (<ref>)). So, the LHS of (<ref>) is ≲1 in 𝒞^0_τ𝒞^_∂M-norm for any . We arrive at (<ref>). (The point is that we only need an O(1) bound in (<ref>).) As explained before (<ref>), combining (<ref>) and (<ref>) shows𝔐_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^βwith high probability, where β>0 is uniformly positive. Combining this with (<ref>) and (<ref>) gives(<ref>)_𝒞^0_τ𝒞^_∂M≲_,̨𝐘^_𝒞^0_τ𝒞^2_∂M^β.Now combine (<ref>)-(<ref>) with (<ref>) and (<ref>). This gives the desired bound (<ref>). § PROOF OUTLINE FOR THEOREM <REF>The goal of this section is to make precise the content of Section <ref>. See there for the intuition. §.§ Ingredients for the “Da Prato-Debussche schematic"We first decompose the solution to (<ref>) into a “singular piece" (which admits probabilistic estimates) plus two “regular pieces". §.§.§ The singular pieceThis term is just the solution to (<ref>), but with careful choice of initial data. In particular, we set 𝔥^η,lin to satisfy the following SPDE: ∂_𝔥^η,lin_,=-ℒ^2𝔥^η,lin_,+Π^η(-ℒ)^-1/2ξ_,,where Π^η is projection onto the first ⌊η^-1⌋-many eigenspaces of ℒ with non-zero eigenvalues. Its initial data is prescribed to make 𝔥^η,lin statistically stationary (we describe this shortly). In particular: *For any ≥̨0, let λ_ be the $̨-th smallest eigenvalue of-ℒ(we adopt the convention thatλ_=λ_+̨1is possible, so we do not count multiplicity). For example,λ_0=0, andλ_>0if≥̨1(see Lemma <ref>).*Now, letψ_;̨·be anL^2(∂M)-normalized eigenfunction of-ℒwith eigenvalueλ_.*Let{𝗓_}_≥̨1be a family of jointly independent Gaussians with mean0and variance|λ_|^-3.The initial data that we prescribe for𝔥^η,linis given by the spectral representation 𝔥^η,lin_0,:=∑_=̨1^1/η𝗓_ψ_;̨x.(Throughout, if we use1/ηin a summation index, we always mean its floor.) It is not hard to see, especially because the sum in (<ref>) is finite and thus without issues of convergence, that the solution to (<ref>) has the representation 𝔥^η,lin_,=∑_=̨1^1/η𝗓_,̨ψ_;̨x,where𝗓_,̨solve the following system (parameterized by≥̨1) of OUSDEs (driven by independent standard Brownian motions𝖻_,̨·): z_,̨=-12λ_^2𝗓_,̨+λ_^-1/2b_,̨.Verifying (<ref>) is completely elementary by writing (<ref>) in the eigenbasis ofℒ(which spansL^2(∂M)in the Hilbert space sense), so we omit this calculation. A key point is that (<ref>) is a stationary stochastic process; its law at time≥0is its law at time0for any deterministic. (This is a standard fact about OU processes.) Also, (<ref>) is clearly smooth and globally defined (in) for eachη>0; see Lemma <ref> for regularity ofψ. Finally, let us emphasize that there is no λ=0 component to (<ref>) and (<ref>), since the Π^η-operator in (<ref>) projects outside this eigenspace. §.§.§ The first regular pieceOf course, as we mentioned in Section <ref>, the operator-ℒ^2on the RHS of (<ref>) is not the elliptic operatorΔ_∂Mappearing on the RHS of the original PDE(<ref>) of interest. So, we need to account for this difference. This leads us to the SPDE∂_𝔥^η,reg,1_,=Δ_∂M𝔥^η,reg,1_,+(Δ_∂M+ℒ^2)𝔥^η,lin_,.Again, perη>0, the forcing term on the RHS is smooth and globally defined, so (<ref>) is globally well-posed with smooth solutions perη>0. For simplicity, we assume the initial data 𝔥^η,reg,1_0,·≡0. For now, we only mention that (<ref>) is a “regular piece" because it can be solved asη→0using classical PDE methods, once we give a stochastic estimate for the forcing term on the RHS.§.§.§ The second regular pieceKeeping track of what is left, we are led to the following PDE: ∂_𝔥^η,reg,2_, =Δ_∂M𝔥^η,reg,2_,+|_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1|^2-𝒞_η-2(_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1_,)_∂M𝔥^η,reg,2_,+|_∂M𝔥^η,reg,2_,|^2.The initial data is dictated by initial data for𝔥^η, i.e. (<ref>), which is notated by𝔥^initial∈𝒞^2(∂M)(see the statement of Theorem <ref>), as well as initial data for𝔥^lin(because initial data for𝔥^η,reg,1was defined to be0). To put it more precisely, we have𝔥^η,reg,2_0,·=𝔥^initial-𝔥^η,lin_0,·.As before, we call (<ref>)-(<ref>) a “regular piece" because we can solve it classically, it turns out, once we provide a stochastic estimate for the squared gradient of 𝔥^η,lin. Note (<ref>)-(<ref>) is nonlinear in its solution, so local-in-time solutions are all we guarantee for now.§.§.§ Putting it togetherThe following result just states that (<ref>), (<ref>), and (<ref>)-(<ref>) add to (<ref>). This can be checked by an elementary calculation, which we omit. First, recall 𝒞^∞_τ𝒞^∞_∂M from Section <ref>. There is a stopping time τ>0 with respect to the filtration of ξ so that (<ref>)-(<ref>) is well-posed (i.e. has a unique solution) in 𝒞^∞_τ𝒞^∞_∂M, and for ≤τ, we have the identity 𝔥^η_,=𝔥^η,lin_,+𝔥^η,reg,1_,+𝔥^η,reg,2_,. §.§ The basic architecture of this sectionThe backbone of this argument is built on stochastic estimates for𝔥^η,lin. So, we start by establishing regularity estimates for𝔥^η,lin, which immediately help us takeη→0for (<ref>). We then establish an estimate for the space-time integrated squared-gradient of𝔥^η,lin. Lastly, we use this (and a variety of less subtle estimates) to conclude by taking theη→0limit for (<ref>)-(<ref>). §.§ “Main" estimates for 𝔥^η,linWe start by proving a moment bound for𝔥^η,lin. This will help us deduce convergence of𝔥^η,linin the sense of Theorem <ref>. (We clarify how to improve the topology of convergence afterwards; this is not so hard.) Before we state the result, recall the Sobolev spacesH^α(∂M)from Section <ref>. (In a nutshell, take Sobolev topologies on charts of∂M, and sum the corresponding norms.) Take deterministic 𝔱≥0 and δ>0 and ≥1. The sequence 𝔥^η,lin_𝔱,· converges in H^1-δ(∂M) in -th moment. More precisely, we have the following (where η_1,η_2→0 can be taken in any fashion): 𝔥^η_1,lin_𝔱,·-𝔥^η_2,lin_𝔱,·_H^1-δ(∂M)^→_η_1,η_2→00.Moreover, for any test function 𝙵∈𝒞^∞(×∂M), the sequence below converges in -th moment as η→0: ∫_[0,𝔱]∫_∂M𝙵_,𝔥^η,lin_,.(<ref>) implies (<ref>) because of the following elementary estimate (via Hölder and Sobolev duality): |∫_[0,𝔱]∫_∂M𝙵_,[𝔥^η_1,lin_,-𝔥^η_2,lin_,]|^ ≲_𝔱,∫_[0,𝔱]|∫_∂M𝙵_,[𝔥^η_1,lin_,-𝔥^η_2,lin_,]|^≲_𝙵,∫_[0,𝔱]𝔥^η_1,lin_,·-𝔥^η_2,lin_,·_H^1-δ(∂M)^.We now prove (<ref>); without loss of generality, assume η_1^-1≤1+η_2^-1. Recall (<ref>). This lets us compute 𝔥^η_1,lin_,-𝔥^η_2,lin_,=∑_=̨1+1/η_1^1/η_2𝗓_,̨ψ_;̨.Sobolev norms are equivalent to multiplier-norms for -ℒ. In particular, by Lemma <ref> and the L^2(∂M)-normalization of ψ_;̨· eigenfunctions, (<ref>) implies 𝔥^η_1,lin_,·-𝔥^η_2,lin_,·_H^1-δ(∂M)^2≲(1-ℒ)^1-δ[𝔥^η_1,lin_,·-𝔥^η_2,lin_,·]_H^1-δ(∂M)^2=∑_=̨1+1/η_1^1/η_2|λ_|^2(1-δ)𝗓_,̨^2.Recall that for any deterministic 𝔱, the coefficients 𝗓_,̨ have the law of independent mean-zero Gaussians with variance λ_^-3 (see the bullet points before (<ref>) and the paragraph after (<ref>)). We now take -th moments. By hypercontractivity for Gaussians, (<ref>) then gives the estimate 𝔥^η_1,lin_,·-𝔥^η_2,lin_,·_H^1-δ(∂M)^2 ≲{∑_=̨1+1/η_1^1/η_2|λ_|^2(1-δ)𝗓_,̨^2}^≲_{∑_=̨1+1/η_1^1/η_2|λ_|^2(1-δ)𝗓_,̨^2}^≲{∑_=̨1+1/η_1^1/η_2|λ_|^-1-2δ}^.It now suffices to use the eigenvalue asymptotic λ_≍$̨ from Lemma <ref> (where≍means equivalent up to a constant bounded above and below uniformly, i.e.≳and≲). In particular, we can replaceλ_by$̨ in (<ref>), from which the vanishing of (<ref>) as η_1,η_2→0 follows immediately. So, (<ref>) follows.We now present a result to make sense of the renormalized square of _∂M𝔥^η,lin that appears in (<ref>)-(<ref>). Recall the 𝒞^0_𝔱H^α_∂M spaces from Section <ref>.Fix any deterministic 𝔱≥0. The following function of (,)∈[0,𝔱]×∂M converges in 𝒞^0_𝔱H^1+γ_∂M in probability for some γ>0 uniformly positive: Ψ^η_,:=∫_0^∫_∂MΓ^(∂M)_-,,[|_∂M𝔥^η,lin_,|^2-𝒞_η]. The proof of Proposition <ref> will be delayed until after this section, because it requires a fair amount of work. In any case, an intuitive explanation for its proof is exactly the content of Section <ref>. §.§ Estimates for 𝔥^η,reg,1Provided the regularity estimate (<ref>), we can now control (<ref>) asη→0. Take deterministic 𝔱≥0 and δ>0 and ≥1. The sequence 𝔥^η,reg,1 converges in 𝒞^0_𝔱H^3-δ_∂M to the solution of the following integral equation as η→0, where 𝔥^0,lin is the limit of 𝔥^η,lin as η→0: 𝔥^0,reg,1_,=∫_0^∫_∂MΓ^(∂M)_-,,[(Δ_∂M+ℒ^2)𝔥^0,lin_,]. Recall 𝔥^0,reg,1 has vanishing initial data (see the paragraph after (<ref>)). Thus, by Duhamel (Lemma <ref>), (<ref>) implies the identity 𝔥^η,reg,1_,=∫_0^∫_∂MΓ^(∂M)_-,,[(Δ_∂M+ℒ^2)𝔥^η,lin_,].By Lemma <ref>, Δ_∂M+ℒ^2 is a zeroth-order pseudo-differential operator, i.e. a bounded map H^α(∂M)→H^α(∂M) for any α. (It is here where we use the assumption =̣1.) In particular, (Δ_∂M+ℒ^2)𝔥^η,lin_,· converges in -th moment in H^1-δ(∂M) for every δ>0 and ∈[0,]. This also implies the limit is exactly (Δ_∂M+ℒ^2)𝔥^0,lin_,·. The result now follows by then follows by Lemma <ref>, i.e. that integrating against the space-time heat kernel gives 2-ϱ derivatives for any ϱ>0.§.§ Estimates for 𝔥^η,reg,2Because𝔥^η,reg,2solves the nonlinear PDE(<ref>)-(<ref>), we cannot immediately guarantee anything besides local-in-time well-posedness and convergence asη→0. Rather, we consider stopping times for the blow-up for𝔥^η,reg,2. Moreover, instead of considering𝔥^η,reg,2, it will be more convenient to study the following (forω>0small): 𝔥^η,reg,2,ω_,:=^ω𝔥^η,reg,2_,.The reason why (<ref>) is “better" is because to analyze (<ref>)-(<ref>), we will need estimates for its solution inH^1+γ(∂M)-norms. But the initial data for𝔥^η,reg,2is inH^1-δ(∂M)for anyδ>0(see after (<ref>)-(<ref>) and Lemma <ref>). Of course, integrating this initial data against Γ^(∂M) regularizes a little (see Lemma <ref>), but at the cost of a factor which is cancelled out by^ω. (In any case, becauseω>0is small and we ask for convergence in Theorem <ref> to be analytically weak, this extra factor^ωis harmless after integration.) Take any deterministic 𝔱≥0. For ω>0 sufficiently small but uniformly positive, there exists a stopping time τ_BU>0 such that the sequence 𝔥^η,reg,2,ω_,· converges in probability in 𝒞^0_τ_BU∧𝔱H^1+γ_∂M as η→0, where γ>0 is deterministic and uniformly positive.The limit of 𝔥^η,reg,2 is the solution of the PDE obtained from formally taking η→0 for every term in (<ref>)-(<ref>) (with the renormalized square of _∂M𝔥^η,lin handled by Proposition <ref>, and with all other terms in (<ref>)-(<ref>) handled by Lemma <ref>, <ref>, and Sobolev multiplication.) We did not record this SPDE since it does not serve any immediate purpose for us as far as we can tell, and it is a little complicated to write precisely. We note, however, that proving this remark amounts to following the proof of Lemma <ref>.We start by turning (<ref>)-(<ref>) into an integral equation via Duhamel (Lemma <ref>): 𝔥^η,reg,2_, =∫_∂MΓ^(∂M)_,,𝔥^η,reg,2_0,+∫_0^∫_∂MΓ^(∂M)_-,,(|_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1_,|^2-𝒞_η)-2∫_0^∫_∂MΓ^(∂M)_-,,(_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1_,)_∂M𝔥^η,reg,2_,+∫_0^∫_∂MΓ^(∂M)_-,,|_∂M𝔥^η,reg_,|^2.From (<ref>)-(<ref>), we immediately get the following equation for 𝔥^η,reg,2,ω: 𝔥^η,reg,2,ω_, =^ω∫_∂MΓ^(∂M)_,,𝔥^η,reg,2_0,+^ω∫_0^∫_∂MΓ^(∂M)_-,,(|_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1_,|^2-𝒞_η)-2^ω∫_0^^-ω∫_∂MΓ^(∂M)_-,,(_∂M𝔥^η,lin_,+_∂M𝔥^η,reg,1_,)_∂M𝔥^η,reg,2,ω_,+^ω∫_0^^-2ω∫_∂MΓ^(∂M)_-,,|_∂M𝔥^η,reg,2,ω_,|^2.Instead of giving a full, detailed proof (which would require many technical steps), we provide the following Sobolev regularity counting. In particular, we will show that there exists ν>0 uniformly positive such that if we choose ω>0 and γ>0 small enough, we have (<ref>)_𝒞^0_𝔱H^1+γ_∂M ≲_γ,ω1 (<ref>)_𝒞^0_𝔱H^1+γ_∂M ≲_γ,ω,𝔱1 (<ref>)_𝒞^0_𝔱H^1+γ_∂M ≲_γ,ω𝔱^ν𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M (<ref>)_𝒞^0_𝔱H^1+γ_∂M ≲_γ,ω𝔱^ν𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M^2.(The implied constants are possibly random but always tight as random variables as η→0.) The important property of (<ref>)-(<ref>) is that they can only depend on the H^1-δ(∂M)-norm of 𝔥^η,lin and the H^3-δ(∂M)-norm of 𝔥^η,reg,1. Let us explain this further. If we assume these four estimates, standard parabolic theory, which is omitted from this argument because it is lengthy, lets us turn (<ref>)-(<ref>) into the claims in Lemma <ref>. For example, (<ref>)-(<ref>) imply𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M≲_ω,γ1+𝔱^ν𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M+𝔱^ν𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M^2,from which, if we take 𝔱<τ_BU (for a small but almost surely positive stopping time τ_BU), we get𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M≲_ω,γ,𝔱1.This shows tightness of 𝔥^η,reg,2,ω in 𝒞^0_𝔱H^1+γ_∂M (for possibly but smaller γ>0). Moreover, since the estimates (<ref>)-(<ref>) are continuous with respect to the topology of convergence for 𝔥^η,lin and 𝔥^η,reg,1, standard parabolic regularity along with Lemma <ref>, Proposition <ref>, and Lemma <ref> lets us take limits as η→0 in (<ref>)-(<ref>) naively to compute the limit of 𝔥^η,reg,2,ω in a topology which is two Sobolev derivatives stronger (by Lemma <ref>). In particular, we get convergence of (<ref>)-(<ref>) in 𝒞^0_𝔱H^1+γ_∂M for 𝔱<τ_BU. So, we are left to prove (<ref>)-(<ref>). *For (<ref>), we claim that the initial data of 𝔥^η,reg,2 converges in probability in H^1-δ(∂M) for any δ>0. This follows by the formula for initial data 𝔥^η,reg,2_0,·=𝔥^initial-𝔥^η,lin_0,· from after (<ref>)-(<ref>), the assumption that 𝔥^initial∈𝒞^2(∂M), and Lemma <ref>. But the operator exp[Δ_∂M]:H^1-δ(∂M)→H^1+γ(∂M) has norm ≲^-γ-δ. So, for any ω>0, we can find γ,δ>0 for which ^ωexp[Δ_∂M]:H^1-δ(∂M)→H^1+γ(∂M) has norm ≲1 locally uniformly in . This gives (<ref>).*For (<ref>), it suffices to drop the ^ω-factor in front since the implied constant in (<ref>) can depend on the time-horizon 𝔱. Now, letting Φ be (<ref>) but without ^ω in front, we compute Φ =∫_0^∫_∂MΓ^(∂M)_-,,(|_∂M𝔥^η,lin_,|^2-𝒞_η)+2∫_0^∫_∂MΓ^(∂M)_-,,2_∂M𝔥^η,lin_,·_∂M𝔥^η,reg,1_,+∫_0^∫_∂MΓ^(∂M)_-,,|_∂M𝔥^η,reg,1_,|^2.The RHS of (<ref>) converges in 𝒞^0_𝔱H^1+γ_∂M by Proposition <ref>. For (<ref>), we first claim that _∂M𝔥^η,lin_,··_∂M𝔥^η,reg,1_,·_H^-3δ(∂M)≲_δ_∂M𝔥^η,lin_,·_H^-δ(∂M)_∂M𝔥^η,reg,1_,·_H^1/2-δ(∂M).Indeed, this is by Sobolev multiplication (Lemma <ref>); in a nutshell, for the product, add the regularities but then subtract 1/2-ϱ from it for any ϱ>0 (we picked ϱ=δ). (This only applies if the sum of the regularities is positive, which is the case here.) By Lemmas <ref> and <ref>, the RHS of the previous display is ≲_δ1 in any moment. So, by Lemma <ref>, which says that integrating against the heat kernel gives 2-ϱ many derivatives (for any ϱ>0), we know (<ref>) is ≲_δ1 in 𝒞^0_𝔱H^2-4δ_∂M-norm. The same is true for (<ref>); the only difference with (<ref>) is multiplying by _∂M𝔥^η,reg,1, not _∂M𝔥^η,lin, but the former is smoother.*We now show (<ref>). By Sobolev multiplication (Lemma <ref>), we have the following if δ<γ: (_∂M𝔥^η,lin_,·+_∂M𝔥^η,reg,1_,·)_∂M𝔥^η,reg,2,ω_,·_H^-1/2(∂M)≲_γ,δ_∂M𝔥^η,lin_,·+_∂M𝔥^η,reg,1_,·_H^-δ(∂M)_∂M𝔥^η,reg,2,ω_H^γ(∂M).Indeed, the sum of the regularities in the second line is γ+(-δ)>0, so we can apply Sobolev multiplication and get H^-1/2-control. Now, by Lemmas <ref> and <ref>, the first term in the second line is ≲_δ1 in any moment. Trivially, the second term in the second line is ≲ the norm on the RHS of the desired estimate (<ref>). So, by Lemma <ref>, which again gives 2-ϱ derivatives after integrating against the heat kernel, (<ref>)_𝒞^0_𝔱H^3/2-ϱ_∂M≲𝔱^ω𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂Msup_0≤≤𝔱∫_0^|-|^-1+ϱ/2^-ω≲𝔱^ϱ/2𝔥^η,reg,2,ω_𝒞^0_𝔱H^1+γ_∂M.(Indeed, Lemma <ref> gives us 2-ϱ derivatives, if we pay a cost in the -integral of |-|^-1+ϱ/2; i.e., a spatial derivative introduces a singularity of exponent -1/2. Also, the last bound follows from splitting the -integral into integrals over [0,/2] and [/2,]; on each, we can bound one factor in the -integral deterministically and integrate the other one by calculus.) If we choose ϱ=1/3, for example, then we get the desired bound (<ref>).*The estimate (<ref>) follows by the same argument as (<ref>). Indeed, we only had control on _∂M𝔥^η,lin+_∂M𝔥^η,reg,1 in H^-δ(∂M), but we are allowing ourselves control on _∂M𝔥^η,reg,2,ω in H^1+γ(∂M).This finishes the proof.§.§ Proof of Theorem <ref>By Lemma <ref>, we have∫_[0,τ)∫_∂M𝙵_,𝔥^η_, =∫_[0,τ)∫_∂M𝙵_,𝔥^η,lin_,+∫_[0,τ)∫_∂M𝙵_,𝔥^η,reg,1_,+∫_[0,τ)∫_∂M𝙵_,𝔥^η,reg,2_,.The RHS of the first line converges asη→0by Lemmas <ref>, <ref>. For the second line, we first use (<ref>): ∫_[0,τ)∫_∂M𝙵_,𝔥^η,reg,2_,=∫_[0,τ)∫_∂M^-ω𝙵_,𝔥^η,reg,2,ω_,.Asω>0is small,^-ωis integrable. So by Lemma <ref>, this converges asη→0, and we are done. § PROOF OUTLINE FOR PROPOSITION <REF>§.§ Expanding |_∂M𝔥^η,lin|^2 in the eigenbasis of -ℒTake the representation (<ref>). When we square it, we get on-diagonal and off-diagonal terms. What falls out are quantities below. The first encodes the expectation of the squared gradient; the second encodes fluctuations. (Recall notation around (<ref>)-(<ref>).) *First, we define the renormalization term asℜ^(η)_,:=∑_ȷ=1^1/ηλ_ȷ^-3|_∂Mψ_ȷ;|^2. *Second, we define the fluctuating terms as 𝔉^(η),1_, :=∑_ȷ,ℓ=1^1/η1_ȷ≠ℓ𝗓_ȷ,𝗓_ℓ,_∂Mψ_ȷ;_∂Mψ_ℓ;, 𝔉^(η),2_, :=∑_ȷ=1^1/η[𝗓_ȷ,^2-λ_ȷ^-3]|_∂Mψ_ȷ;|^2. Recall Ψ^η from (<ref>). With notation explained afterwards, we haveΨ^η_,=Ψ^η,rn_,+Ψ^η,fluc_,,where Ψ^η,rn_, :=∫_0^∫_∂MΓ^(∂M)_-,,[ℜ^(η)_,-𝒞_η] Ψ^η,fluc_, :=∫_0^∫_∂MΓ^(∂M)_-,,[𝔉^(η),1_,+𝔉^(η),2_,]. Take the gradient of (<ref>); by squaring, we get |_∂M𝔥^η,lin_,|^2=∑_ȷ,ℓ=1^1/η𝗓_ȷ,𝗓_ℓ,_∂Mψ_ȷ;_∂Mψ_ℓ;.If we consider ȷ=ℓ in this sum, we obtain 𝔉^(η),2_,+ℜ^(η)_,. On the other hand, the ȷ≠ℓ-terms sum to 𝔉^(η),1_,. Now, integrate against the heat kernel Γ^(∂M) in space and time to get (<ref>).§.§ Convergence for Ψ^η,rnThere is no stochastic estimate necessary in this step. The key ingredient, on the other hand, is the local Weyl law as we explained in Section <ref> (see Lemma <ref>). Fix 𝔱≥0 deterministic. There exists uniformly positive γ>0 such that Ψ^η,rn converges in 𝒞^0_𝔱H^1+γ_∂M in probability as η→0.Before we can apply the local Weyl law to (<ref>), we must first remove the gradients. This first step to this end is the following, which first plugs in the definition (<ref>) and then does integration-by-parts: ∫_0^∫_∂MΓ^(∂M)_-,,(ℜ^(η)_,-𝒞_η)= -∑_ȷ=1^1/ηλ_ȷ^-3∫_0^∫_∂M_∂MΓ^(∂M)_-,,(ψ_ȷ;_∂Mψ_ȷ;)-∑_ȷ=1^1/ηλ_ȷ^-3∫_0^∫_∂MΓ^(∂M)_-,,ψ_ȷ;zΔ_∂Mψ_ȷ;-𝒞_η.The first term can be treated directly by heat kernel regularity and eigenfunction regularity. In particular, we first claim that for γ>0, we have the estimate ψ_ȷ;·_∂Mψ_ȷ;·_H^2γ(∂M) ≲_γψ_ȷ;·_H^1/2(∂M)_∂Mψ_ȷ;·_H^3γ(∂M)≲λ_ȷ^3/2+3γ.The first estimate is by Sobolev multiplication (Lemma <ref>; add the regularities, make sure it is positive, and subtract 1/2+γ). The second is by eigenfunction regularity (Lemma <ref>; Sobolev norms are equivalent to multiplier norms for -ℒ). To put (<ref>) to use, let us now fix η_1,η_2>0 such that η_1^-1≤1+η_2^-1. Also, just for convenience, set RHS(<ref>)_i to be the RHS of (<ref>) for η=η_i (here, i=1,2). We haveRHS(<ref>)_1-RHS(<ref>)_2_𝒞^0_𝔱H^1+γ_∂M≲∑_ȷ=1+1/η_1^1/η_2λ_ȷ^-3∫_0^·∫_∂M_∂MΓ^(∂M)_·-,·,_∂Mψ_ȷ;×ψ_ȷ;_𝒞^0_𝔱H^1+γ_∂M.By (<ref>) and Lemma <ref> (heat kernel gives us 2-γ many derivatives, so its gradient gives us 1-γ), we deduce that the norm in (<ref>) is ≲_𝔱λ_ȷ^3/2+3γ. Therefore, we deduce RHS(<ref>)_1-RHS(<ref>)_2_𝒞^0_𝔱H^1+γ_∂M ≲_𝔱∑_ȷ=1+1/η_1^1/η_2λ_ȷ^-3/2+3γ,which vanishes as η_1,η_2→0 in any fashion. This shows convergence for the RHS of (<ref>). We now focus on (<ref>). To this end, we first write Δ_∂M=-ℒ^2+Δ_∂M+ℒ^2 to get -ψ_ȷ;Δ_∂Mψ_ȷ; =ψ_ȷ;ℒ^2ψ_ȷ;-ψ_ȷ;(Δ_∂M+ℒ^2)ψ_ȷ;=λ_ȷ^2ψ_ȷ;^2-ψ_ȷ;(Δ_∂M+ℒ^2)ψ_ȷ;.We focus on the second term in (<ref>). By Lemma <ref>, the operator Δ_∂M+ℒ^2 is a zeroth-order pseudo-differential operator, i.e. bounded from any Sobolev space to itself. Using this, we claim ψ_ȷ;·(Δ_∂M+ℒ^2)ψ_ȷ;·_H^2γ(∂M)≲ψ_ȷ;·_H^1/2(∂M)ψ_ȷ;·_H^2γ(∂M)≲λ_ȷ^1/2+2γ.Indeed, the first is by Sobolev multiplication and the aforementioned boundedness of Δ_∂M+ℒ^2. The second inequality is by eigenfunction regularity (Lemma <ref>). Thus, the same argument that gave us convergence for the RHS of (<ref>) also gives convergence of ∑_ȷ=1^1/ηλ_ȷ^-3∫_0^∫_∂MΓ^(∂M)_-,,ψ_ȷ;z(Δ_∂M+ℒ^2)ψ_ȷ;.(This argument is actually easier, because there is no gradient hitting the heat kernel above, and the estimate (<ref>) is better than (<ref>).) Thus, in view of (<ref>)-(<ref>), to show convergence of (<ref>), it suffices to show convergence of Υ_, :=∑_ȷ=1^1/ηλ_ȷ^-1∫_0^∫_∂MΓ^(∂M)_-,,ψ_ȷ;z^2-𝒞_η=∫_0^∫_∂MΓ^(∂M)_-,,{∑_ȷ=1^1/ηλ_ȷ^-1ψ_ȷ;z^2-𝒞_η}.To this end, we use the local Weyl law; by Lemma <ref>, we can write the following for any ℓ≥1: ∑_ȷ=1^ℓψ_ȷ;^2=π2λ_ℓ+Rem(λ_ℓ,),where Rem(λ_ℓ,)∈L^∞(∂M) with norm ≲1. (This again uses =̣1 heavily.) Thus, by Abel summation, which essentially is a sum-version of integration-by-parts, we get the following calculation as well: ∑_ȷ=1^ℓλ_ȷ^-1ψ_ȷ;^2 =λ_ℓ^-1∑_ȷ=1^ℓψ_ȷ;^2+∑_=1^ℓ-1(λ_^-1-λ_+1^-1)∑_ȷ=1^ψ_ȷ;^2=π2+λ_ℓ^-1Rem(λ_ℓ,)+∑_=1^ℓ-1λ_+1-λ_λ_λ_+1(π2λ_+Rem(λ_,))=π2∑_=1^ℓ-1λ_+1-λ_λ_+1+π2+λ_ℓ^-1Rem(λ_ℓ,)+∑_=1^ℓ-1λ_+1-λ_λ_λ_n+1Rem(λ_,).For ℓ=⌊η^-1⌋, the first term in (<ref>) is exactly 𝒞_η (see (<ref>)), so the integrand in (<ref>) is equal to π2+λ_ℓ^-1Rem(λ_ℓ,)+∑_=1^ℓ-1λ_+1-λ_λ_λ_n+1Rem(λ_,), where ℓ=⌊η^-1⌋.This term converges in L^2(∂M)=H^0(∂M). Indeed, recall that Rem-terms are in L^∞(∂M)⊆L^2(∂M) (since ∂M is compact). Moreover, by Lemma <ref>, we have λ_≍. Thus, by the same argument (that turns (<ref>) into convergence of the heat-integrated object, for example), we obtain convergence of (<ref>). (Indeed, by Lemma <ref>, convergence in H^0(∂M) implies convergence in H^2-ϱ(∂M) after integrating against the heat kernel Γ^(∂M); here, ϱ>0 is arbitrary.) As noted before (<ref>), this finishes the proof.§.§ Convergence for Ψ^η,flucIn view of Lemmas <ref> and <ref>, we are left to prove the following. Fix 𝔱≥0 deterministic. There exists uniformly positive γ>0 such that Ψ^η,fluc converges in 𝒞^0_𝔱H^1+γ_∂M in probability as η→0.(Again, the failure of using blackbox-type results in the SPDE literature to prove Proposition <ref> comes from the lack of explicit knowledge of eigenfunctions.) We now outline the proof of Proposition <ref>.§.§.§ Preliminary reduction: refining the sequence η→0RecallΨ^η,flucfrom Lemma <ref>. From there, it is clear that its dependence onηcomes from cutting off outside1/η-many eigenspaces ofℒ. Thus, to prove Proposition <ref>, instead of considering any sequenceη→0, it suffices to consider only a sequence for which we get one additional eigenspace in the sums (<ref>)-(<ref>) at each step where we varyη. More precisely: Suppose the sequence {η[]̨}_=̨1^∞ is chosen so that 1/η[+̨1]=1+1/η[]̨ (and that these are integers). Assume {Ψ^η[]̨,fluc}_ converges in 𝒞^0_𝔱H^1+γ_∂M as →̨∞ for γ>0. Then Proposition <ref> follows.As noted before Lemma <ref>, any sequence Ψ^η,fluc (as η→0) is embedded in {Ψ^η[]̨,fluc}_. §.§.§ Preliminary reduction to estimates for neighboring η[]̨Lemma <ref> says it suffices to prove convergence ofΨ^η[]̨,fluc. To this end, the following says it suffices to show the differencesΨ^η[+̨1],fluc-Ψ^η[]̨,flucare summable (with sufficiently high probability); in some sense, it is a Borel-Cantelli-type statement. Assuming the following is true, then Lemma <ref> follows. *Fix any 𝔱≥0 and $̨ deterministic. There existsγuniformly positive such that with probability1-O(η[]̨^2),sup_0≤≤𝔱Ψ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)≲η[]̨^6/5.Fix _̨0. Let ℰ[]̨ be the event where (<ref>) holds. On ∩_≥̨_̨0ℰ[]̨, for any _̨2≥_̨1≥_̨0, we claimΨ^η[k_2],fluc-Ψ^η[_̨1],fluc_𝒞^0_𝔱H^1+γ_∂M ≤∑_n=_̨1^_̨2-1Ψ^η[n+1],fluc-Ψ^η[n],fluc_𝒞^0_𝔱H^1+γ_∂M≲∑_n=_̨1^_̨2-1η[n]^6/5≲η[_̨1]^1/5.Indeed, the first bound is by triangle inequality, and the second is by (<ref>). The last bound follows because the eigenvalues of -ℒ satisfy λ_ℓ≍ℓ (where ≍ means ≳ and ≲ with different implied constants); this is by Lemma <ref>. So, the cutoffs η[n]^-1, which grow linearly with eigenvalues λ_n by construction in Lemma <ref>, satisfy η[n]^-1≍λ_n≍ n as well. Note that (<ref>) is uniform over _̨2≥_̨1≥_̨0 and thus vanishes as we take _̨0→∞. This is all on the event ∩_≥̨_̨0ℰ[]̨. Thus, it suffices to show that the probability of this event converges to 1 as we take _̨0→∞. For this, we use a union bound: ℙ{∩_≥̨_̨0ℰ[]̨}=1-ℙ{∪_≥̨_̨0ℰ[]̨^C}≥1-∑_=̨_̨0^∞ℙ[ℰ[]̨^C]≥1-∑_=̨_̨0^∞O(η[]̨^2).As explained in the previous paragraph, the far RHS of (<ref>) converges to 1 as _̨0→∞, so we are done. §.§.§ Reduction to a discrete time-netLemma <ref> essentially asks us to prove (<ref>). However, it will be technically inconvenient to prove an estimate that holds uniformly in time on the same probability space. So, the goal of this step is to control the supremum on the LHS of (<ref>) by a supremum over a finite set of times in[0,𝔱]. In particular, we take a discretization of[0,𝔱]with mesh given by some big power ofη[]̨. So, we are only asking to control time-regularity ofΨ^η[+̨1],fluc-Ψ^η[]̨,flucon very short time-scales in the following result. As regularity ofΨ^η[+̨1],fluc-Ψ^η[]̨,flucshould be controlled by a fixed power ofη[+̨1]^-1,η[]̨^-1(see (<ref>)), reducing to a discretization of[0,𝔱]with mesh lengthη[]̨^10is not difficult.Fix any γ>0 and ≥̨0 deterministic. With probability 1-O(η[]̨^3), we have the following estimate for Λ>0 sufficiently large but independent of $̨: LHS(<ref>) ≲_𝔱,γsup_∈[0,𝔱]∩η[]̨^ΛΨ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)+η[]̨^6/5. We first use the triangle inequality to show the following, in which ∈[0,𝔱] and T is the closest point in the discretization [0,𝔱]∩η[]̨^Λ tothat is also less than or equal to : Ψ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M) ≲Ψ^η[+̨1],fluc_T,·-Ψ^η[]̨,fluc_T,·_H^1+γ(∂M)+sup_m=,̨+̨1Ψ^η[m],fluc_,·-Ψ^η[m],fluc_T,·_H^1+γ(∂M).To complete the proof, it suffices to show that we can take Λ large enough so that on an event with probability 1-O(η[]̨^3), we have the following estimate for m=,̨+̨1 and all ∈[0,𝔱] (recall T is a function of ): Ψ^η[m],fluc_,·-Ψ^η[m],fluc_T,·_H^1+γ(∂M)≲_𝔱,γη[]̨^6/5.This is elementary, so we sketch the proof. Recall the definition of Ψ^η,fluc from (<ref>). Differentiating in the -variable therein (or equivalently, doing Duhamel, i.e. Lemma <ref>, in reverse), we get ∂_Ψ^η,fluc_,=Δ_∂MΨ^η,fluc_,+𝔉^(η),1_,+𝔉^(η),2_,.From (<ref>) and boundedness of the Γ^(∂M)-semigroup in H^α(∂M)-spaces (see Lemma <ref>), we also know that the H^α(∂M)-norm of Υ^η,fluc is controlled by that of 𝔉^(η),1+𝔉^(η),2.By Sobolev embedding, the same is true for 𝒞^(∂M)-spaces. But 𝔉^(η),1 and 𝔉^(η),2 have explicit representations; see (<ref>)-(<ref>). Now, define ℰ[]̨ be the event on which the 𝗓_ȷ,-coefficients are ≲η[]̨^-100 for all λ_ȷ≤η[+̨1]^-1. Since 𝗓_ȷ, are all OUSDEs with speed ≲η[]̨^-10 for ȷ≤η[+̨1]^-1 with initial law having moments ≲1, standard stochastic calculus and a union bound over the ≲η[]̨^-1-many different ȷ-indices shows that the probability of ℰ[]̨ is at least 1-O(η[]̨^3). Now, on the event ℰ[]̨, the representations (<ref>)-(<ref>) (and the fact that -th derivatives of eigenfunctions ψ_ȷ;· grow at most like λ_ȷ^10 by Lemma <ref>) imply 𝔉^(η[m]),i, for m=,̨+̨1 and i=1,2, have 𝒞^10(∂M)-norm of ≲η[]̨^-D for some D≲1. The same is thus true of Ψ^η[m],fluc as noted in the previous paragraph. So, on ℰ[]̨, the estimate (<ref>) follows by integrating (<ref>) in time on [T,] and using the regularity estimates for the RHS of (<ref>) that we just obtained. Since ℙ[ℰ[]̨]≥1-O(η[]̨^3), we are done. §.§.§ Preliminary reduction to a one-time estimate via union boundThis step is straightforward; we want to bound the supremum on the RHS of (<ref>) via one-time estimates by using a union bound. For this to work, we need one-time estimates to hold in a sufficiently high moment. Fix deterministic 𝔱≥0. Suppose the following holds for γ,β uniformly positive and any ≥1: sup_0≤≤𝔱Ψ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)^2≲_𝔱,γ,η[]̨^26/5+2β.Then for any D,Λ≥0 independent of $̨, with probability1-O_D,𝔱(η[]̨^D), we have sup_∈[0,𝔱]∩η[]̨^ΛΨ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)≲_γη[]̨^6/5. By union bound, (<ref>) fails with probability ≲η[]̨^-Λsup_∈[0,𝔱]∩η[]̨^Λℙ{Ψ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)≳η[]̨^6/5}≲_𝔱,γ,η[]̨^-Λη[]̨^2β,where the last bound is for any ; it follows by Chebyshev and (<ref>). For any Λ,D≥0, we can make the far RHS of (<ref>)≲η[]̨^D by takinglarge enough depending on β (recall that β is uniformly positive). §.§.§ Closing the argument: proving (<ref>)The last ingredient is the main bound (<ref>); this kicks off the chain of implications in Lemmas <ref>, <ref>, <ref>, and <ref>. Fix 𝔱≥0 and ≥1 deterministic. For γ,β uniformly positive, we have (<ref>), i.e. sup_0≤≤𝔱Ψ^η[+̨1],fluc_,·-Ψ^η[]̨,fluc_,·_H^1+γ(∂M)^2≲_𝔱,γ,η[]̨^26/5+2β. We will present the proof of Lemma <ref> in the next section, because it is too involved to give here. §.§ Proof of Proposition <ref>As noted before Proposition <ref>, it suffices to show Proposition <ref>. Lemma <ref> says it suffices to get (<ref>). Lemma <ref> says (<ref>) follows if we can show (<ref>), which follows if we can get (<ref>). Lemma <ref> proves (<ref>), so we are done. § PROOF OF LEMMA <REF>The idea behind the proof of Lemma <ref> is rather simple. We explain it below. *Look at (<ref>) and (<ref>)-(<ref>). Take (<ref>) first. Naively, we expect the estimate _∂Mψ_ȷ;_∂Mψ_ℓ;≍λ_ȷλ_ℓ.Indeed, if M is the unit ball in ^2 and ψ are Fourier exponentials, this is a simple calculation. (Generally, see eigenfunction estimates in Lemma <ref>.) Also, we know 𝗓_ȷ,𝗓_ℓ,∼λ_ȷ^-3/2λ_ℓ^-3/2 are uncorrelated for different tuples (ȷ,ℓ). So, we expect the sum (<ref>) to have the following behavior (for each ) by taking second moments naively: {∑_ȷ≠ℓ^∞λ_ȷ^-3λ_ℓ^-3λ_ȷ^2λ_ℓ^2}^1/2.This has a log-divergence. What saves us is that in (<ref>), we time-average the fluctuating term (<ref>), which should give some power-saving and thus be enough to remove the log-divergence. (Actually, it is certainly more than what we need to remove the divergence. However, we will need more than to just remove this divergence; the bound (<ref>) is not generally available. Indeed, the degrees for Sobolev multiplication, i.e. Lemma <ref>, to work do not allow for (<ref>), but a bound that is worse by a small power of the eigenvalues.)*Now, look at (<ref>). It is again a fluctuating sum. So, with the naive heuristic (<ref>), we expect (<ref>) to be ≲{∑_ȷ=1^∞λ_ȷ^-6λ_ȷ^4}^1/2,which converges. Of course, the heuristic (<ref>) is off, so we need to time-average again, but in principle, this term can also be controlled.The rest of this section is organized as follows. First, we record an accessible representation for the objectΨ^η[+̨1],fluc-Ψ^η[]̨,flucof main interest. Then, we introduce a Kolmogorov-type criterion which effectively rewrites the Sobolev norm in (<ref>) in its dual form (which is more amenable to obtaining moment bounds). After this, we introduce the time-averaging strategy, which is quite similar to the proof of Proposition <ref> in some ways. In particular, it is based on resolvents and the Ito formula, but this time for the much simpler process of independent OUSDEs{𝗓_,̨·}_.§.§ Computing Ψ^η[+̨1],fluc-Ψ^η[]̨,flucRecall notation from (<ref>)-(<ref>) and (<ref>). We first claim 𝔉^(η[+̨1]),1_,-𝔉^(η[]̨),1_, =2𝗓_η[+̨1]^-1,_∂Mψ_η[+̨1]^-1;{∑_ȷ=1^1/η[]̨𝗓_ȷ,_∂Mψ_ȷ;}.Let us explain this. By (<ref>), 𝔉^(η[+̨1]),1 is a sum over indices(ȷ,ℓ)that are distinct and satisfy ȷ,ℓ≤η[+̨1]^-1. By the same token, 𝔉^(η[]̨),1 is a sum of the same terms over indices(ȷ,ℓ)that are distinct and satisfy ȷ,ℓ≤η[]̨^-1. So, the difference comes from two sets of indices. They are given by (η[+̨1]^-1,ℓ) and (ȷ,η[+̨1]^-1), respectively, for ℓ,ȷ≤η[]̨^-1. But, the summands in (<ref>) are symmetric in the indices, hence (<ref>) follows. Next, by (<ref>), we have 𝔉^(η[+̨1]),2_,-𝔉^(η[]̨),2_, =[𝗓_η[+̨1]^-1,^2-λ_η[+̨1]^-3]|_∂Mψ_η[+̨1]^-1;|^2.SinceΨ^η[+̨1],fluc-Ψ^η[]̨,flucjust integrates (<ref>)-(<ref>) against the heat kernel, we deduce Ψ^η[+̨1],fluc_,-Ψ^η[]̨,fluc_, =∫_0^∫_∂MΓ^(∂M)_-,,𝔄^,̨1_,+∫_0^∫_∂MΓ^(∂M)_-,,𝔄^,̨2_,=:Υ^,̨fluc,1_,+Υ^,̨fluc,2_,,where 𝔄^,̨1_,=RHS(<ref>) and 𝔄^,̨2_,=RHS(<ref>). §.§ A Kolmogorov-type estimateAs we alluded to earlier, the goal now is reducing estimation of Sobolev norms to estimation of space-time integrals, which are more accessible to study via moments. As is usually the case with Kolmogorov continuity criteria, in doing so, we lose an arbitrarily small amount in regularity, but this is not a big deal, becauseH^1+γ-δ(∂M)forγuniformly positive andδ>0arbitrarily small is stillH^1+γ_2(∂M)forγ_2uniformly positive. Fix 𝔱≥0 and φ∈𝒞^∞(∂M) and ≥1 deterministic. Suppose that for i∈{1,2}, we had the following for γ,χ uniformly positive: sup_0≤≤𝔱|∫_∂Mφ_Υ^,̨fluc,i_,|^2≲_𝔱,η[]̨^2[6/5+χ]φ_H^-1-2γ(∂M)^2.Then (<ref>) follows for the same choice of γ.This is Theorem 2.5 in <cit.> or the Mitoma criterion <cit.> but trivially adapted to the Sobolev norm H^1+2γ(∂M). (See also <cit.>.) Write Υ^,̨fluc,i in terms of an orthonormal basis of H^1+2γ(∂M). Use (<ref>) for φ equal to any given basis vector. So, if we give the coefficients in the basis expansion of Υ^,̨fluc,i weights corresponding to the H^1+γ(∂M)-norm, we get coefficients that decay faster than a↦η[]̨^2p[6/5+χ]a^-2pγ. This is summable over integers a≥1, so the result holds by linearity of expectation. Again, we omit the details since this density argument is given in <cit.>.We now give an intermediate calculation which begins analysis of the LHS of (<ref>). First, some notation. *For any φ∈𝒞^∞(∂M), define φ_,:=exp[Δ_∂M]φ_.We now claim the following calculation holds, with explanation given afterwards: ∫_∂Mφ_Υ^,̨fluc,i_, =∫_0^∫_∂M∫_∂Mφ_Γ^(∂M)_-,,𝔄^,̨i_,=∫_0^∫_∂M{∫_∂MΓ^(∂M)_-,,φ_}𝔄^,̨i_,=∫_0^∫_∂Mφ_-,𝔄^,̨i_,.The first identity follows by definition (<ref>). The second follows by Fubini. The third follows becauseΓ^(∂M)is symmetric in space variables, sinceΔ_∂Mis a self-adjoint operator (see Lemma <ref>). §.§ Another Ito formula strategy, but this time for OUSDEsNaively, one would like to bound (<ref>) in2-th moment essentially by pulling the norm through the-integral and estimating the spatial-integral. However, this would not be enough; we need to use the time-averaging. To this end, we first analyze dynamics of the OUSDEs𝗓_·,that define𝔄^,̨iterms (see right after (<ref>)). Let us recap the setting. *Recall the law of {𝗓_,̨}_ is that of jointly independent Gaussians with mean 0 and variance λ_^-3; see right before (<ref>) and after (<ref>). Also, by the SDE(<ref>), we know the generator of ↦{𝗓_,̨}_ is given by𝒢:=∑_=̨1^∞𝒢^()̨,where𝒢^()̨:=12λ_^-1∂_𝗓_^2-12λ_^2𝗓_∂_𝗓_.Technically, the sum in (<ref>) is infinite, but we only consider finite-dimensional projections. This infinite-sum notation is harmless, because the operators 𝒢^()̨ are independent (they each only involve the variables they differentiate with respect to).The strategy to study (<ref>) is as follows. We replace𝔄^,̨iby𝒢𝒢^-1𝔄^,̨i. Since we time-integrate in (<ref>), we can remove𝒢if we add a martingale; this is by Ito. (Technically, we must also account for the fact that the functionφin (<ref>), although deterministic, depends on time as well, so the Ito formula only applies the operator∂_+𝒢, not𝒢itself. The resolution to this is based on the so-called Ito trick; this is based on using Ito for the time-reversal of the stationary, reversible process𝗓_·,·, which has infinitesimal generator-∂_+𝒢. Averaging the two cancels∂_-operators while keeping𝒢-operators. We make this precise when relevant.)We must compute𝒢^-1𝔄^,̨i(and simultaneously show𝔄^,̨iis in the domain of𝒢^-1in the first place). Recall 𝔄^,̨i_, from immediately after (<ref>). We have the following identities: 𝒢^-1𝔄^,̨1_, =-2∑_ȷ=1^1/η[]̨(λ_η[+̨1]^-1^2+λ_ȷ^2)^-1𝗓_η[+̨1]^-1,𝗓_ȷ,_∂Mψ_η[+̨1]^-1;_∂Mψ_ȷ; 𝒢^-1𝔄^,̨2_, =-λ_η[+̨1]^-1^-2(𝗓_η[+̨1]^-1,^2-λ_η[+̨1]^-1^-3)|_∂Mψ_η[+̨1]^-1;|^2. Just apply 𝒢 (see (<ref>)) to both sides of both proposed identities. For example, in the case of (<ref>), every summand on the RHS vanishes under any second-derivative in 𝒢. On the other hand, the first derivative operators in 𝒢 just get rid of a 𝗓-variable, but then multiplies it back in along with an eigenvalue factor that eventually cancels the (λ_η[+̨1]^-1^2+λ_ȷ^2)^-1-factor in (<ref>). We omit the (rest of the) details of this elementary calculus computation.We now implement the Ito strategy that we outlined before Lemma <ref>. Ultimately, we reduce estimation of the2-th moment of (<ref>) to a computation for2-th moments of the𝒢^-1-terms in Lemma <ref>.Take any deterministic 𝔱≥0 and ≥1 and i∈{1,2}. We have sup_0≤≤𝔱|(<ref>)|^2 ≲_,𝔱{∑_=0^∞λ_^-1∫_0^|∫_∂Mφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_,|^2}^. In view of the paragraph before Lemma <ref>, we give two representations for (<ref>). First, we claim (<ref>) =∫_0^∫_∂Mφ_-,𝒢𝒢^-1𝔄^,̨i_,=∫_0^𝒢{∫_∂Mφ_-,𝒢^-1𝔄^,̨i_,}=∫_0^[∂_+𝒢]{∫_∂Mφ_-,𝒢^-1𝔄^,̨i_,}-∫_0^∂_{∫_∂Mφ_-,𝒢^-1𝔄^,̨i_,}.(<ref>) follows trivially (the only subtlety possibly is the fact that 𝒢^-1𝔄^,̨i is well-defined by Lemma <ref>). (<ref>) follows from linearity of 𝒢. (<ref>)-(<ref>) follows by adding and subtracting ∂_. Now, we claim that (<ref>) also has the following identities, the first of which is just change-of-variables ↦- and the second of which is the same argument as (<ref>)-(<ref>): (<ref>) =∫_0^∫_∂Mφ_,𝔄^,̨i_-,=∫_0^[∂_+𝒢]{∫_∂Mφ_,𝒢^-1𝔄^,̨i_-,}-∫_0^∂_{∫_∂Mφ_,𝒢^-1𝔄^,̨i_-,}.Note that (<ref>)+(<ref>)=0; indeed, apply ↦- in (<ref>), and note this gives ∂_ a negative sign. So (<ref>)=(<ref>)+(<ref>)2.Now, we can compute (<ref>) and (<ref>) by using the Ito formula. Indeed, for (<ref>), note that 𝔄^,̨i (at time ) is a functional of 𝗓_·,, which has generator 𝒢 (see (<ref>) and after (<ref>)). For (<ref>), the relevant SDE is the time-reversal 𝗓^rev,_·,=𝗓_·,-, which is itself Markov with generator 𝒢. (Indeed, this is because 𝗓 is stationary with self-adjoint generator 𝒢; stationarity is explained after (<ref>), and self-adjoint-ness is standard.) Thus, (<ref>) =∫_∂Mφ_𝒢^-1𝔄^,̨i_,-∫_∂Mφ_,𝒢^-1𝔄^,̨i_0,+𝔪_,where 𝔪 is an Ito integral (e.g. martingale) whose quadratic variation admits the following upper bound: [𝔪]_≲∑_=0^∞λ_^-1∫_0^|∫_∂Mφ_-,∂_𝗓_𝒢^-1𝔄^,̨i_,|^2.(Let us explain the quadratic variation (<ref>). We are applying Ito to the -integral in (<ref>), so the quadratic variation of 𝔪 is given by a sum over all components 𝗓_ of the following. Take the 𝗓_-derivative of the-integral in (<ref>) and square it. We must also multiply by λ_^-1, since the Brownian motions in the OUSDEs (<ref>) have factors of λ_^-1/2. This explains (<ref>); we claim ≲ only to avoid factors like 2 that may appear but are unnecessary to deal with.) By the same token, we have (<ref>) =∫_∂Mφ_,𝒢^-1𝔄^,̨i_0,-∫_∂Mφ_0,𝒢^-1𝔄^,̨i_,+𝔪^rev,_,where 𝔪^rev, is a martingale (with respect to the filtration of the time-reversal 𝗓^rev,_·,=𝗓_·,-) such that [𝔪^rev,]_ ≲∑_=0^∞λ_^-1∫_0^|∫_∂Mφ_,∂_𝗓_𝒢^-1𝔄^,̨i_-,|^2↦-=∑_=0^∞λ_^-1∫_0^|∫_∂Mφ_-,∂_𝗓_𝒢^-1𝔄^,̨i_,|^2.Plug (<ref>) and (<ref>) into (<ref>). The non-martingale terms cancel, and thus (<ref>) turns into |(<ref>)|^2≲_|𝔪_|^2+|𝔪^rev,_|^2.The desired bound (<ref>) now follows by (<ref>), (<ref>), (<ref>), and the BDG inequality.§.§ Estimating the RHS of (<ref>) for i∈{1,2}As we explain shortly, we are left to get the following, whose proof is deferred until after we deduce Lemma <ref> (since it requires lengthy calculations). Fix 𝔱≥0 and ≥1 deterministic. There exists χ,γ uniformly positive so that for i∈{1,2} and any ≤𝔱, we have the following moment estimate: {∑_=0^∞λ_^-1∫_0^|∫_∂Mφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_,|^2}^≲_,𝔱η[]̨^2[6/5+χ]φ_H^-1-2γ(∂M)^2. §.§ Proof of Lemma <ref>By Lemmas <ref> and <ref> and (<ref>)-(<ref>), we obtain (<ref>). By Lemma <ref>, (<ref>) follows. This is the desired bound, so we are done. §.§ Proof of Lemma <ref>This argument is based on computing ∂_𝗓_𝒢^-1𝔄^,̨i using Lemma <ref> and power-counting in eigenvalues. We will need to split the argument into the casesi=1andi=2. §.§.§ The case i=1By Lemma <ref>, the term𝒢^-1𝔄^,̨1depends on𝗓_only for ≤η[+̨1]^-1, so that we can cut the sum on the LHS of (<ref>) at =η[+̨1]^-1. Now, by the inequality|a+b|^≲_|a|^+|b|^,LHS(<ref>) ≲_{λ_η[+̨1]^-1^-1∫_0^|∫_∂Mφ_-,∂_𝗓_η[+̨1]^-1𝒢^-1𝔄^,̨1_,|^2}^+{∑_=0^1/η[]̨λ_^-1∫_0^|∫_∂Mφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_,|^2}^.We claim the following holds (in whichυ>0will be taken small but specified later in this proof): RHS(<ref>) =λ_η[+̨1]^-1^-{∫_0^|-|^-1+2υ|∫_∂M|-|^1/2-υφ_-,∂_𝗓_η[+̨1]^-1𝒢^-1𝔄^,̨1_,|^2}^ (<ref>) ={∑_=0^1/η[]̨λ_^-1∫_0^|-|^-1+2υ|∫_∂M|-|^1/2-υφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_,|^2}^.Indeed, all we do is multiply by|-|^-1+2υ|-|^1-2υinside the-integral, and we move the latter factor in the squared∂M-integral. The reason for this step is to regularize the semigroup acting onφ; indeed, this semigroup smoothsφ, but at the cost of a short-time singularity. As for why we take the exponent-1+2υ, we explain this below. Let us start with (<ref>). We claim (<ref>) ≲_,𝔱,υλ_η[+̨1]^-1^-∫_0^|-|^-1+2υ{∫_∂M|-|^1/2-υφ_-,∂_𝗓_η[+̨1]^-1𝒢^-1𝔄^,̨1_,}^2.Indeed, we just use the Hölder inequality for the-integral in (<ref>). In particular, (<ref>) would follow if we give up some-dependent power of the integral of|-|^-1+2υ. (For example, if|-|^-1+2υwere a probability measure on∈[0,], then (<ref>) would just be Hölder. Since|-|^-1+2υhas finite mass on∈[0,], we can just normalize this measure to make it a probability measure, and pay some power of this normalization factor for the implied constant in (<ref>).) Now, we use Lemma <ref> to compute the derivative of𝒢^-1𝔄^,̨1in (<ref>). This gives ∫_∂M|-|^1/2-υφ_-,∂_𝗓_η[+̨1]^-1𝒢^-1𝔄^,̨1_, =∑_ȷ=1^1/η[]̨c_ȷ𝗓_ȷ,,wherec_ȷare the following deterministic coefficients: c_ȷ=-2(λ_η[+̨1]^-1^2+λ_ȷ^2)^-1∫_∂M|-|^1/2-υφ_-,_∂Mψ_η[+̨1]^-1;_∂Mψ_ȷ;.Indeed, (<ref>)-(<ref>) is just calculus, because𝒢^-1𝔄^,̨1is just a polynomial in𝗓_-variables. Now, recall that𝗓_,are jointly Gaussian with mean0and variance λ_^-3 for all; see after (<ref>). Thus, we can estimate the RHS of (<ref>) in2-th moment by Gaussian hypercontractivity. In particular, we have{∫_∂M|-|^1/2-υφ_-,∂_𝗓_η[+̨1]^-1𝒢^-1𝔄^,̨1_,}^2 ≲_{∑_ȷ=1^1/η[]̨c_j^2λ_ȷ^-3}^.We are left to estimate coefficients (<ref>). To this end, we first recall from Section <ref> that𝒞^0,ν(∂M)is the usual Hölder space of exponentν∈(0,1). We now claim the following (with explanation given after) withϱ>0to be chosen shortly: |c_j| ≲(λ_η[+̨1]^-1^2+λ_ȷ^2)^-1|-|^1/2-υφ_-,·_H^-ϱ(∂M)_∂Mψ_η[+̨1]^-1;·_∂Mψ_ȷ;·_H^ϱ(∂M)≲_ϱ(λ_η[+̨1]^-1^2+λ_ȷ^2)^-1φ_H^-1-ϱ+υ(∂M)_∂Mψ_η[+̨1]^-1;·_H^ϱ(∂M)_∂Mψ_ȷ;·_𝒞^0,ϱ(∂M)≲(λ_η[+̨1]^-1^2+λ_ȷ^2)^-1λ_η[+̨1]^-1^1+ϱλ_ȷ^5/4+ϱφ_H^-1-ϱ+υ(∂M).(<ref>) holds by duality of Sobolev norms. (<ref>) follows by two steps. First, the heat operatorexp[τΔ_∂M]:H^-1-ϱ+υ(∂M)→H^-ϱ(∂M)has norm ≲τ^-1/2+υ, since each derivative is half a power ofτ^-1; see Lemma <ref>. This bounds theH^-ϱ(∂M)-norm in (<ref>) by theH^-1-ϱ+υ(∂M)-norm in (<ref>). It is also why we need the|-|^1/2-υ-factor in (<ref>).To complete the proof of (<ref>), we then use the boundedness of the Sobolev multiplication mapH^ϱ(∂M)×𝒞^0,ϱ(∂M)→H^ϱ(∂M). (Indeed, this is like sayingL^2of a product on a compact manifold is controlled byL^2of one factor timesL^∞of the other; see Lemma <ref>.) Finally, (<ref>) follows by eigenfunction regularity; see Lemma <ref>. (The only subtlety here is that a derivative only corresponds to a power of the eigenvalue inH^ϱ(∂M)-spaces; in𝒞^0,ϱ(∂M)-spaces, we need to include an additional exponent of/̣4, where=̣1in our case. This is one deficiency of general manifolds compared to the torus, for example.) Now, by (<ref>) and (<ref>)-(<ref>) and the fact that|-|^-1+2υintegrates to≲_υ1over∈[0,], we deduce the estimate below (whereφ:=φ_H^-1-ϱ+υ(∂M)): RHS(<ref>) ≲_,𝔱,υλ_η[+̨1]^-1^-{∑_ȷ=1^1/η[]̨(λ_η[+̨1]^-1^2+λ_ȷ^2)^-2λ_ȷ^-3λ_η[+̨1]^-1^2+2ϱλ_ȷ^5/2+2ϱφ^2}^.Now, we useλ_≍, where≍means≳and≲with possibly different implied constants. Using this (and by pulling the squared norm in (<ref>) outside the sum and-th power), we deduce from (<ref>) that the bound below holds (where the last line follows by elementary calculations): RHS(<ref>) ≲_,𝔱,υη[+̨1]^{∑_ȷ=1^1/η[]̨η[+̨1]^-2-2ϱȷ^-1/2+2ϱ(η[+̨1]^-2+ȷ^2)^-2}^φ^2≲η[+̨1]^5/2-10ϱφ_H^-1-ϱ+υ(∂M)^2.Indeed, to get the last estimate, supposeϱ=0. Now, use the bound (η[+̨1]^-2+ȷ^2)^-2≲ȷ^-1/2η[+̨1]^7/2. The sum overȷis then≲|logη[]̨|, and we are left with a total factor of η[+̨1]^5/2. Multiplying these two bounds gives an upper bound of (<ref>) (up tologη[+̨1]^-1-factors). Now, for generalϱ, use the same argument; all we have to do is give up a factor of η[+̨1]^-10ϱ, for example. Therefore, by (<ref>), (<ref>), and (<ref>)-(<ref>), we deduce that ifϱ>0is small enough, then for someχuniformly positive, RHS(<ref>)≲_,𝔱η[]̨^2[6/5+χ]φ_H^-1-ϱ+υ(∂M)^2,We now address (<ref>) (or equivalently, the RHS of (<ref>)). First, in the RHS of (<ref>), pull the sum overagainstλ_-coefficients into the-integral. Then, use the same Hölder inequality trick that gave us (<ref>): (<ref>) ≲_,𝔱,υ∫_0^|-|^-1+2υ{∑_=0^1/η[]̨λ_^-1|∫_∂M|-|^1/2-υφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_,|^2}^.We now compute the derivative on the RHS of (<ref>). By Lemma <ref> and elementary calculus, we have∫_∂M|-|^1/2-υφ_-,∂_𝗓_𝒢^-1𝔄^,̨1_, =c_m𝗓_η[+̨1]^-1,,where the coefficientsc_mare deterministic quantities defined in (<ref>). Now, we use (<ref>) and the bound (<ref>)-(<ref>) to bound the RHS of (<ref>) as follows (whereϱ>0is for our choosing and, as before, we setφ:=φ_H^-1-ϱ+υ(∂M)for convenience): RHS(<ref>) ≲_ϱφ^2∫_0^|-|^-1+2υ{∑_=0^1/η[]̨λ_^-1(λ_η[+̨1]^-1^2+λ_^2)^-2λ_η[+̨1]^-1^2+2ϱλ_^5/2+2ϱ𝗓_η[+̨1]^-1,^2}^=φ^2∫_0^|-|^-1+2υ{𝗓_η[+̨1]^-1,^2∑_=0^1/η[]̨λ_^-1(λ_η[+̨1]^-1^2+λ_^2)^-2λ_η[+̨1]^-1^2+2ϱλ_^5/2+2ϱ}^.Now, recall that𝗓_,∼𝒩(0,λ_^-3)for all; see after (<ref>). Thus, the previous display implies RHS(<ref>) ≲_ϱ,,𝔱,υλ_η[+̨1]^-1^-3φ^2{∑_=0^1/η[]̨λ_^-1(λ_η[+̨1]^-1^2+λ_^2)^-2λ_η[+̨1]^-1^2+2ϱλ_^5/2+2ϱ}^.The sum in this display is controlled by the same idea that we used to go from (<ref>) to (<ref>). In particular, pretendϱ=0for now and recallλ_≍. We use the estimate (λ_η[+̨1]^-1^2+λ_^2)^-2≲λ_^-5/2λ_η[+̨1]^-1^-3/2. This means the sum above is of λ_^-1, which gives a harmless log-divergence in η[]̨^-1. We are otherwise left with, after taking-th powers, a factor of λ_η[+̨1]^-1^(-3+2-3/2)≲λ_η[+̨1]^-1^-5/2. Now, note5/2is strictly larger than12/5, which is the exponent that we need to beat to prove (<ref>). This is exactly the situation that led us to (<ref>). Ultimately, we get the following (where forϱ>0, the bound below still holds because we only have to give up a factor of λ_η[+̨1]^-1^10ϱ, and we can chooseϱas small as we want): RHS(<ref>) ≲_ϱ,,𝔱,υη[]̨^2[6/5+χ]φ_H^-1-ϱ+υ(∂M)^2.(Above,χis uniformly positive, just as in (<ref>).) Now, combine (<ref>) and (<ref>) to deduce(<ref>)≲_,𝔱η[]̨^2[6/5+χ]φ_H^-1-ϱ+υ(∂M)^2,The proof (in the current casei=1) of (<ref>) now follows immediately by (<ref>)-(<ref>), (<ref>), and (<ref>) (as long as we chooseϱsmall and thenυeven smaller so thatϱ-υis uniformly positive).§.§.§ The case i=2This case is easier than the casei=1, but the ideas are generally the same. We first compute the LHS of the desired estimate (<ref>) fori=2as follows. By Lemma <ref>, we note that the only index on the LHS of (<ref>) for which the summand is not0is m=η[+̨1]^-1, since𝒢^-1𝔄^,̨2depends only on𝗓_m,·for thism. In particular, we can actually compute, fori=2, that LHS(<ref>)≲{∫_0^|∫_∂Mφ_-,×λ_η[+̨1]^-1^-2𝗓_η[+̨1]^-1,|_∂Mψ_η[+̨1]^-1;|^2|^2}^.Now, we use the same Hölder inequality trick that gave (<ref>) and (<ref>) to show the following estimate in whichυ>0is free for our choosing (we specify it shortly): LHS(<ref>) ≲_𝔱,,υ∫_0^|-|^-1+2υ{∫_∂M|-|^1/2-υφ_-,λ_η[+̨1]^-1^-2𝗓_η[+̨1]^-1,|_∂Mψ_η[+̨1]^-1;|^2}^2.The term inside the expectation is a deterministic multiple of𝗓_∼𝒩(0,λ_^-3)for =η[+̨1]^-1. Thus, the previous display implies the following estimate: LHS(<ref>) ≲_𝔱,,υλ_η[+̨1]^-7∫_0^|-|^1-2υ{∫_∂M|-|^1/2-υφ_-,|_∂Mψ_η[+̨1]^-1;|^2}^2.We now follow the calculation (<ref>)-(<ref>) for ȷ=η[+̨1]^-1. This implies, as we explain afterwards, that the following estimate holds for anyϱ>0as small as we want: {∫_∂M|-|^1/2-υφ_-,|_∂Mψ_η[+̨1]^-1;|^2}^2 ≲_,υ,ϱλ_η[+̨1]^-1^2[9/4+2ϱ]φ_H^-1-ϱ+υ(∂M)^2.Indeed, the calculation (<ref>)-(<ref>) is exactly a bound for the term inside the2-th power on the LHS of (<ref>), except we drop the (λ_η[+̨1]^-1^2+λ_ȷ^2)^-1-term in (<ref>)-(<ref>). The RHS of (<ref>) is then just (<ref>) raised to the2-th power. Now, if we combine (<ref>) and (<ref>), we get LHS(<ref>) ≲_𝔱,,υ,ϱλ_η[+̨1]^-5/2+4ϱφ_H^-1-ϱ+υ(∂M)^2.As with the casei=1, chooseϱsmall so that the exponent for theλ-factor has the form2[6/5+χ]forχuniformly positive. Then chooseυsmall enough so thatϱ>υis positive. So, (<ref>) gives (<ref>). § PROOF OF THEOREM <REF>FixΛ≥0independent of>0. Write𝐡^=𝐡^,1+𝐡^,2, where𝐡^,1,𝐡^,2solve ∂_𝐡^,1_, =Δ_∂M𝐡^,1_,+M^_, ∂_𝐡^,2_, =Δ_∂M𝐡^,2_,+∫_∂M𝐊_,|_∂M𝐡^,1_,+_∂M𝐡^,2_,|^2.We claim, in some topology, that𝐡^,1→_→0𝔥^𝐊,1, where𝔥^𝐊,1is the defined by the following SPDE: ∂_𝔥^𝐊,1_,=Δ_∂M𝔥^𝐊,1_,+∫_∂M[𝐊_,-1](-ℒ)^-1/2ξ_,.The stated convergence𝐡^,1→_→0𝔥^𝐊,1, if we potentially augment the probability space, is in 𝒞^0_𝔱𝒞^2_∂M in probability for any stopping time𝔱≤lim inf_→0τ_𝐡^,Λ. To see this convergence, we argue as follows. *By Definition <ref>, we know the following sentence is true. The martingale 𝐌^ is smooth in , so (<ref>) admits smooth solutions in 𝒞^∞_𝔱𝒞^∞_∂M, with regularity depending only on that of 𝐌^, which is controlled uniformly in . (Technically, we only have control of regularity of the martingale depending on 𝒞^2(∂M)-regularity of 𝐘^. However, by Theorem <ref>, this is controlled by 𝒞^2(∂M)-regularity of 𝐡^, which itself is uniformly controlled in >0 since we work before time 𝔱≤lim inf_→0τ_𝐡^,Λ.) As an immediate consequence, we deduce tightness of 𝐡^,1 as →0 in 𝒞^0_𝔱𝒞^2_∂M (e.g. by Kolmogorov continuity or Arzela-Ascoli on the space-time [0,𝔱]×∂M).*We now argue that limit points of 𝐡^,1 are unique. Along the way, we show that possible augmenting the probability space allows for almost sure convergence of 𝐡^,1 to its proposed limit 𝔥^𝐊,1. This follows by a martingale problem formulation for 𝔥^𝐊,1. In particular, it is standard stochastic calculus to show that the linear SPDE(<ref>) is, in law, uniquely defined as the stochastic process in 𝒞^0_𝔱𝒞^2_∂M such that for any test function φ∈𝒞^∞(∂M), the process ∫_∂M𝔥^𝐊,1_,φ_-∫_0^∫_∂M𝔥^𝐊,1_,Δ_∂Mφ_is a continuous martingale with quadratic variation given by the following Carre-du-Champ: 2∫_∂M(∫_∂Mφ_[𝐊_,-1])×{-ℒ^-1(∫_∂Mφ_[𝐊_,-1])}.(Above, ℒ^-1 acts on .) It is clear by (<ref>) that any limit point satisfies this martingale problem as well. Indeed, the martingale 𝐌^ has jumps of o(1) in size because it is a good martingale; see Theorem <ref> and Definition <ref>. Thus, its →0 limit points are continuous in time. Moreover, its limiting bracket is computed in (<ref>)-(<ref>). So, 𝐡^,1→𝔥^𝐊,1 in law in 𝒞^0_𝔱𝒞^2_∂M as →0. To get this convergence to be almost sure, we use Skorokhod representation and augment the probability space.Now, given the convergence𝐡^,1→𝔥^𝐊,1almost surely in 𝒞^0_𝔱𝒞^2_∂M, it follows from parabolic regularity that we can take limits of the noise-free PDE(<ref>), so that𝐡^,2→𝔥^𝐊,2as→0in 𝒞^0_τ𝒞^2_∂M, where𝔥^𝐊,2solves the following PDE andτis any time beforelim inf_→0τ_𝐡^,Λand the blow-up time of𝔥^𝐊,2in𝒞^2(∂M): ∂_𝔥^𝐊,2=Δ_∂M𝔥^𝐊,2+∫_∂M𝐊_,|_∂M𝔥^𝐊,1_,+_∂M𝔥^𝐊,2_,|^2.(Again, classical parabolic regularity shows that (<ref>) is locally well-posed in 𝒞^2(∂M).) So, for such timesτ, we know𝐡^→𝔥^𝐊,1+𝔥^𝐊,2in 𝒞^0_τ𝒞^2_∂M. But𝔥^𝐊,1+𝔥^𝐊,2=𝔥^𝐊, which can be checked using the PDEs (<ref>) and (<ref>) and (<ref>). Ultimately, this gives the proposed estimate in Theorem <ref> if we replace the supremum therein by a supremum over times≤[1∧(lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊,2)]-δ, whereτ_𝔥^𝐊,2is the blow-up time of𝔥^𝐊,2in𝒞^2(∂M). (This is with high probability.) We now make the following observations. *We know τ_𝔥^𝐊,2=τ_𝔥^𝐊. Indeed, 𝔥^𝐊-𝔥^𝐊,2=𝔥^𝐊,1, and 𝔥^𝐊,1 solves a linear and thus globally well-posed PDE. So, [1∧(lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊,2)]-δ=[1∧(lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊)]-δ.*Next, we claim lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊≥τ_𝔥^𝐊-ρ with high probability for any fixed deterministic ρ>0, if we take Λ large enough. If true, the previous paragraph and bullet point would give Theorem <ref> (since ρ,δ>0 are arbitrary), finishing the proof. To show the above claim, it essentially suffices to note that our convergence is in 𝒞^0_τ𝒞^2_∂M-spaces. To be precise, suppose lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊=τ_𝔥^𝐊. In this case, there is nothing to show, so assume that lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊=lim inf_→0τ_𝐡^,Λ. Moreover, for the sake of contradiction, additionally assume that lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊=lim inf_→0τ_𝐡^,Λ<τ_𝔥^𝐊-ρ for some ρ>0 fixed and deterministic. In this case, by construction, we know that 𝐡^ has 𝒞^2(∂M)-norm at time lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊 converging to Λ. But we have just shown (using the validity of the bound in Theorem <ref> at time lim inf_→0τ_𝐡^,Λ∧τ_𝔥^𝐊=lim inf_→0τ_𝐡^,Λ) that 𝐡^=𝔥^𝐊+o(1). Thus, the 𝒞^2(∂M)-norm of 𝔥^𝐊 at some time before τ_𝔥^𝐊-ρ is converging to Λ as →0. If Λ is large enough depending on ρ, this contradicts the definition τ_𝔥^𝐊 being the blow-up time for 𝔥^𝐊, hence our claim follows.This completes the proof.§ DETERMINISTIC RESULTS ABOUT THE HEAT KERNEL Γ^(∂M) ON ∂MWe have Γ^(∂M)_,,=Γ^(∂M)_,, for all ∈_≥0 and ,∈∂M. Also, we have Γ^(∂M)_+,, =∫_∂MΓ^(∂M)_,,Γ^(∂M)_,,and∫_∂MΓ^(∂M)_,, =1. The term Γ^(∂M)_,·,· is the kernel for the semigroup exp[Δ_∂M]:L^2(∂M,𝐠_∂)→L^2(∂M,𝐠_∂), which is self-adjoint. Thus, the first claim and last identity follow immediately (we note Δ_∂M vanishes on constants, so its exponential fixes them). To prove the first identity, use exp[(+)Δ_∂M]=exp[Δ_∂M]exp[Δ_∂M]; indeed, Γ^(∂M)_+,, is the kernel for the LHS, and ∫_∂MΓ^(∂M)_,,Γ^(∂M)_,, is the kernel for the RHS. [Duhamel formula]Suppose 𝐅∈𝒞^0_𝒞^∞_ solves ∂_𝐅_,=Δ_∂M𝐅_,+𝐆_, for ≥0 and ∈∂M, where 𝐆∈L^∞(_≥0×∂M). For all ≥0 and ∈∂M, we have 𝐅_, =∫_∂MΓ^(∂M)_,,𝐅_0,+ ∫_0^∫_∂MΓ^(∂M)_-,,𝐆_,. By the Leibniz rule, the PDE for 𝐅, and the PDE for Γ^(∂M), for 0≤<, we have ∂_∫_∂MΓ^(∂M)_-,,𝐅_,=-∫_∂MΔ_∂MΓ^(∂M)_-,,𝐅_,+∫_∂MΓ^(∂M)_-,,Δ_∂M𝐅_,+∫_∂MΓ^(∂M)_-,,𝐆_,.Since Γ^(∂M) is the kernel for the Δ_∂M-semigroup, integrating against it commutes with Δ_∂M. So, we can move Δ_∂M onto 𝐅 in the first term on the RHS, and the first two terms above cancel. Now, by calculus, lim_→̊∫_∂MΓ^(∂M)_-,̊,𝐅_,̊ =∫_∂MΓ^(∂M)_,,𝐅_0,+lim_→̊∫_0^∫_∂MΓ^(∂M)_-,,𝐆_,.The last limit above is computed by plugging =̊; the -integral is certainly continuous in . Moreover, by definition of the heat kernel, the LHS converges to a delta function at = integrating against 𝐅. We can plug in =̊ for 𝐅 on the LHS because 𝐅∈𝒞^0_𝒞^∞_. So, the LHS of the previous display is 𝐅_,. Fix any τ>0 and α_1≤α_2. The operator exp[τΔ_∂M]:H^α_1(∂M)→H^α_2(∂M) is bounded with norm ≲(Cτ)^-[α_2-α_1]/2 for a constant C>0 depending only on M.See (1.15) in Chapter 15 of <cit.>. (Roughly, one derivative is worth τ^-1/2.)§ DETERMINISTIC ESTIMATES FOR THE DIRICHLET-TO-NEUMANN MAPWe have the following properties of ℒ. *For any α, the map ℒ:H^α+1(∂M)→H^α(∂M) is bounded with norm ≲1. So, the map 𝒞^∞(∂M)→𝒞^∞(∂M) is continuous in the Frechet topology on 𝒞^∞(∂M). Also, it is self-adjoint with respect to the surface measure on ∂M, and it vanishes on constant functions on ∂M.Hence, the invariant measure of ℒ is the surface measure on ∂M. (By invariant measure, we mean the measure μ, up to a constant factor, such that ∫_∂Mℒφμ̣=0 for all φ∈𝒞^∞(∂M).*The spectrum of -ℒ is discrete, and its eigenvalues are non-negative. Thus, its positive fractional powers are well-defined. See Section 1.1 of <cit.>. (The vanishing on constants is clear by definition of ℒ, since the harmonic extension of any constant function is constant; see after (<ref>)-(<ref>).)Suppose =̣1. In this case, we have -ℒ=[-Δ_∂M]^1/2+𝒪, where 𝒪 is a pseudo-differential operator of order -1, i.e. a bounded map H^α(∂M)→H^α+1(∂M) with norm ≲1. Hence, Δ_∂M+ℒ^2 is a zeroth-order pseudo-differential operator, i.e. a bounded map H^α(∂M)→H^α(∂M) with norm ≲1.For the first claim, see Section 2.1 of <cit.>. (Indeed, it is shown that-ℒ=[-Δ_∂M]^1/2+𝒪, where 𝒪 is zeroth-order, and the zeroth-order term in 𝒪 has a coefficient given by the second fundamental form of ∂M minus its trace. But in =̣1, these are the same since the second fundamental form is a ×̣$̣ matrix. Thus, if=̣1, this term vanishes; we are left with an order-1operator.) For the second claim, note Δ_∂M+ℒ^2=[-Δ_∂M]^1/2𝒪+𝒪[-Δ_∂M]^1/2.The half-Laplacian is first-order, and𝒪has order-1, so the product is zeroth-order. Let {λ_}_≥0 enumerate the non-negative eigenvalues of -ℒ in increasing order. Let ψ_;· be an eigenfunction for λ_ (normalized to have L^2(∂M)-norm equal to 1). We have the following estimates. *(Spectral gap) The null-space of -ℒ is one-dimensional. So, it has a spectral gap, i.e. its first eigenvalue λ_1 is strictly positive.*(Weyl law) We have the asymptotic λ_≍, where ≍ means ≳ and ≲ with different implied constants.*(Local Weyl law) Fix ℓ≥0. We have the following, where Rem(λ_ℓ,) is bounded uniformly over ℓ,: ∑_ȷ=1^ℓψ_ȷ;^2=π2λ_ℓ+Rem(λ_ℓ,), ∈∂M. *(Sobolev norms are multipliers norms of -ℒ) For any α≥0 and φ∈𝒞^∞(∂M), we have φ_H^α(∂M)≍(1-ℒ)^αφ_L^2(∂M). *(Eigenfunction regularity) For any α≥0 and ≥̨0 and υ∈(0,1) and ψ_;· such that λ_≠0, we haveψ_;·_H^α(∂M) ≲_αλ_^α ψ_;·_𝒞^,̨υ(∂M) ≲λ_^+̨υ+1/4. *(Resolvent estimates) Take any λ>0 and α∈. The resolvent map (λ-ℒ)^-1:H^α(∂M)→H^α(∂M) is bounded with norm ≤λ^-1 for all α. Take any λ≥0 and α_1,α_2∈. The norm of(λ-ℒ)^-1:H^α_1(∂M)→H^α_2(∂M) is less than or equal to that of -ℒ^-1:H^α_1(∂M)→H^α_2(∂M), up to a constant factor (one or both may be infinite).*(Regularity in metric) Let 𝐠 be a smooth Riemannian metric on M. Let ℒ_𝐠 be the Dirichlet-to-Neumann map with respect to 𝐠 (defined in the same way as after (<ref>)-(<ref>) but the harmonic extension is with respect to 𝐠). We have the operator norm estimate below for any α≥0, where 𝐠[0] is surface metric on ∂M and where α_,n_α, depend only on α,$̣: ℒ_𝐠-ℒ_H^α+α_(∂M)→H^α(∂M)≲_α,𝐠_𝒞^n_α,(M)𝐠-𝐠[0]_𝒞^n_α,(M).(The𝒞^n(M)-norm of a metric means said norm of its entries under any choice of local coordinates.) For the spectral gap, see the beginning of <cit.> and Lemma <ref>. For the Weyl law, see <cit.>. For the local Weyl law, see Theorem 1.1 of <cit.>. For (<ref>), it suffices to check it for φ supported on any local coordinate chart. (Indeed, for general φ, take a partition of unity and sum together norms on different charts.) Now, use that the principal symbol of 1-ℒ (as a pseudo-differential operator) is 1+|ξ|<cit.>. With details: *The Sobolev norm H^α(∂M) on any coordinate chart is equivalent to taking Fourier transform, multiplying by 1+|ξ|^α, and taking L^2-norms on the chart. The principal symbol fact we just cited says the same is true for applying (1-ℒ)^α (up to a constant factor and lower-order polynomials in ξ). Intuitively, it is just saying that -ℒ is an elliptic, first-order operator (see <cit.> for this statement).For (<ref>), use (<ref>) and -ℒψ_;·=λ_ψ_;· (also, since λ_≠0, we know λ_≳1 by the spectral gap). For (<ref>), see Theorem 1 of <cit.>. For the first resolvent estimate, use the representation (λ-ℒ)^-1=∫_0^∞e^-τ(λ-ℒ)τ̣and contractivity of the ℒ-semigroup on H^α(∂M) (which holds since ℒ≤0). For the second resolvent bound, we start with the resolvent identity (λ-ℒ)^-1=-ℒ^-1+λ(λ-ℒ)^-1ℒ^-1.Note that all operators in the last term on the RHS above commute. By the first resolvent estimate, ℒ^-1λ(λ-ℒ)^-1_H^α_1(∂M)→H^α_2(∂M)≤ℒ^-1_H^α_1(∂M)→H^α_2(∂M).Now, we combine the previous two displays and the triangle inequality. Finally, we are left with (<ref>). (Let us first set Δ=Δ_∂M for the rest of this proof just for convenience.) By definition, for any φ∈𝒞^∞(∂M), ℒ_𝐠φ-ℒφ=_𝖭[𝒰^𝐠,φ-𝒰^φ],where 𝖭 is the inward unit normal vector field (and _𝖭 is gradient in this direction), and 𝒰^𝐠,φ,𝒰^φ are harmonic extensions of φ with respect to 𝐠 and surface metric on ∂M, respectively. In particular, we haveΔ_𝐠𝒰^𝐠,φ,Δ𝒰^φ=0and𝒰^𝐠,φ,𝒰^φ|_∂M=φ.The previous PDE implies the following for 𝒱:=𝒰^𝐠,φ-𝒰^φ: Δ𝒱=[Δ-Δ_𝐠]𝒰^φ+[Δ-Δ_𝐠]𝒱and𝒱|_∂M=0.We now use a usual elliptic regularity argument for Sobolev spaces. First, by construction, we have ℒ_𝐠φ-ℒφ=_𝖭𝒱, and therefore, for any ≥̨0, we have ℒ_𝐠φ-ℒφ_𝒞^(∂M)≲𝒱_𝒞^+̨1(M).By elliptic regularity, we can control the RHS of the previous display by 𝒞^(∂M)-data of the RHS of the PDE for 𝒱 (fordepending appropriately on $̨). In particular, by Theorem 2.35 of <cit.> (withΩthere given byM⊆^+̣1here), we deduce the estimate 𝒱_𝒞^+̨1(M)≲[Δ-Δ_𝐠]𝒰^φ_𝒞^-̨1(M)+[Δ-Δ_𝐠]𝒱_𝒞^-̨1(M).The implied constant depends only onM(since it is based on elliptic regularity forΔ). Now, for the rest of this argument, letn_≲1be a positive integer depending only on$̨. If the 𝒞^n_(M)-norm of 𝐠-𝐠[0] is small enough, then even with the implied constant, the last term on the RHS of the previous display is strictly less than half of the LHS. Indeed, in local coordinates, it is easy to see that Δ-Δ_𝐠 is a second-order operator whose coefficients are smooth functions of 𝐠-𝐠[0] and its first-derivatives. So, if these quantities are sufficiently small, then the operator norm of Δ-Δ_𝐠:H^α+1(∂M)→H^α-1(∂M) is strictly less than 1/2. By the same token, under the same assumption on 𝐠-𝐠[0], we bound the first term on the RHS of the previous display as follows: [Δ-Δ_𝐠]𝒰^φ_𝒞^-̨1(M)≲_,̨𝐠_𝒞^n_(M)𝐠-𝐠[0]_𝒞^n_(M)𝒰^φ_𝒞^+̨1(M).(The implied constant should depend on 𝐠[0]_𝒞^n_(M) as well, but this depends only on M.)Elliptic regularity (e.g. Theorem 2.35 in <cit.>) lets us replace the 𝒞^+̨1(M)-norm of 𝒰^φ with that of φ itself. So, by the previous two displays and the paragraph between them, we deduce 𝒱_𝒞^+̨1(M)≤O_,̨𝐠_𝒞^n_(M)(𝐠-𝐠[0]_𝒞^n_(M)φ_𝒞^+̨1(∂M))+12𝒱_𝒞^+̨1(M).By changing the implied constant in the big-Oh term above, we can drop the last term in (<ref>) (by moving it to the LHS and multiply by 2). Combining this with (<ref>), we deduce (<ref>) except with 𝒞^(∂M)-norms instead of H^α(∂M)-norms. To conclude, we trade 𝒞^(∂M)-norms for H^α(∂M)-norms by the trivial embedding 𝒞^(∂M)↪H^(∂M) and, again, the Sobolev embedding H^α(∂M)→𝒞^(∂M) (for α large). This gives (<ref>), assuming the 𝒞^n_(∂M)-norm of 𝐠-𝐠[0] is less than a fixed, positive threshold depending only on M. In the case where this is not met, then the RHS of (<ref>) is ≳1, while the LHS of (<ref>) is ≲1 by boundedness of ℒ,ℒ_𝐠. So (<ref>) follows immediately in this case. (To see that ℒ_𝐠:H^α+α_(∂M)→H^α(∂M) has norm depending only on 𝐠_𝒞^n_(M) for appropriate n_ depending on α, use Lemma <ref>.)§ AUXILIARY ESTIMATES§.§ A priori bounds for 𝐈^ and 𝐘^The following bounds higher derivatives of𝐈^by only two derivatives of𝐘^. In a nutshell, this is because the RHS of (<ref>) is smoothing, and because at the level of gradients,𝐈^is much smaller than𝐘^(see (<ref>)). (For a reality check, note that the following result (<ref>) is obvious just from (<ref>) if we take=̨2andυ=0. It is even sub-optimal in this case by a factor of^-1/3. The following result says that on the LHS of (<ref>), we can trade this additional factor that we gain for=̨2andυ=0for more derivatives on the LHS of (<ref>). It essentially follows by interpolation theory. It is crucial that we take gradients on the LHS of (<ref>).) Fix any 𝔱≥0 and ≥̨0 and υ∈[0,1). We have the estimate _∂M𝐈^_𝒞^0_𝔱𝒞^,̨υ_∂M ≲_𝔱,,̨υ1+𝐘^_𝒞^0_𝔱𝒞^2_∂M. It suffices to assume υ≠0; the claim for υ=0 follows because the norms on the LHS of (<ref>) are non-decreasing in υ. As explained above, by (<ref>), we trivially have the inequality _∂M𝐈^_𝒞^0_𝔱𝒞^1,υ_∂M ≲_∂M𝐈^_𝒞^0_𝔱𝒞^2_∂M≲^1/3𝐘^_𝒞^0_𝔱𝒞^2_∂M.Now, by Duhamel (Lemma <ref>) and (<ref>), for any ≤𝔱, we have 𝐈^_, =exp[Δ_∂M]{𝐈^_0,·}_+∫_0^exp[(-)Δ_∂M]{^-1/3Vol_𝐈^_𝐊_·,𝔮^_}_.(Here, the terms inside the curly braces are the functions on ∂M that the semigroup acts on, and the subscriptmeans evaluate the image of this function under the semigroup at .) By Taylor expanding to get (1+a^2)^1/2=1+O(a) in the definition of Vol_𝐈 from Construction <ref>, we have Vol_𝐈^_,· =1+O(_∂M𝐈^_𝒞^0_𝔱𝒞^0_∂M).The heat semigroup operator is bounded on Sobolev spaces by Lemma <ref>. Thus, by smoothness of 𝐊 and (<ref>), we deduce the following for any α≥0: _∂M𝐈^_𝒞^0_𝔱H^α_∂M≲_𝔱,α^-1/3(1+_∂M𝐈^_𝒞^0_𝔱𝒞^0_∂M)≲^-1/3+𝐘^_𝒞^0_𝔱𝒞^2_∂M.(The second bound follows by (<ref>).) Now, for any fixed n,υ, we can take α≥0 big enough so that the H^α-norm on the far LHS of (<ref>) controls the 𝒞^n,υ-norm. This is by Sobolev embedding. Thus,_∂M𝐈^_𝒞^0_𝔱𝒞^n,υ_∂M ≲_𝔱,n,υ^-1/3+𝐘^_𝒞^0_𝔱𝒞^2_∂M.Recall the fixed choices of ,̨υ from the statement of this lemma. If we take n big enough depending only on ,̨υ, then we have the following interpolation bound of norms by Theorem 3.2 in <cit.> (which needs υ≠0): _𝒞^0_𝔱𝒞^,̨υ_∂M≲_𝒞^0_𝔱𝒞^n,υ_∂M^1/2_𝒞^0_𝔱𝒞^1,υ_∂M^1/2.Applying this to _∂M𝐈^ and using (<ref>) and (<ref>) shows that_∂M𝐈^_𝒞^0_𝔱𝒞^,̨υ_∂M≲_𝔱,,̨υ(^-1/6+𝐘^_𝒞^0_𝔱𝒞^2_∂M^1/2)^1/6𝐘^_𝒞^0_𝔱𝒞^2_∂M^1/2≲1+𝐘^_𝒞^0_𝔱𝒞^2_∂M.(For the last bound in this display, we also used a^1/2≲1+a for any a≥0.) This gives (<ref>).§.§ A stochastic parabolic regularity estimate (a la Kolmogorov continuity)The idea of the following estimate is to essentially say that if a sequence of space-time functions (possibly random) converges pointwise in time in a spatial Sobolev space, then integrating in space-time against theΓ^(∂M)-kernel upgrades this convergence to a higher-degree Sobolev space (with2-ϱmore derivatives). Take a sequence of (possibly random) functions 𝔣^():×∂M→ which are jointly smooth and such that for any ∈, the functions 𝔣^()_,· converges in H^α(∂M) in -th moment for every , where α is a fixed positive number. Consider the functionsΥ^()_,:=∫_0^∫_∂MΓ^(∂M)_-,,𝔣^()_,.Then Υ^() converges in probability in 𝒞^0_𝔱H^2+α-δ_∂M for any fixed 𝔱<∞ and δ>0. Moreover, we haveΥ^()_𝒞^0_𝔱H^2+α-δ_∂M ≲sup_0≤≤𝔱∫_0^|-|^-1+δ/2𝔣^()_,·_H^α(∂M). The proposed estimate follows by Lemma <ref> applied to eachin the time-integration in the definition of Υ^(). Convergence of Υ^() pointwise inand in H^2+α-δ(∂M) in space follows from Lemma <ref> and the assumed convergence of 𝔣^(). Indeed, we have the following estimate for any ,: Υ^()_,·-Υ^()_,·_H^2+α-δ(∂M) ≤∫_0^|-|^-1+δ/2𝔣^()_,·-𝔣^()_,·_H^α(∂M).Take -th moments and use integrability of |-|^-1+δ/2 for ∈[0,]. Since 𝔣^()_,· converge in moments in H^α(∂M) for each ≥0, this shows that Υ^()_,· converges in H^2+α-δ(∂M) for everyfixed.So, we must show tightness of Υ^() in 𝒞^0_𝔱H^2+α-δ_∂M. Note that for any _1,_2≥0, we have the estimate Υ^()__1,·-Υ^()__2,·_H^2+α-δ(∂M)^2≲{Υ^()__1,·-Υ^()__2,·_H^2+α-δ/2(∂M)^q_1}^υ_1{Υ^()__1,·-Υ^()__2,·_H^ϱ(∂M)^q_2}^υ_2,where q_1,q_2,υ_1,υ_2,ϱ≥0 depend only on ,δ. (This is just interpolation of degrees in Sobolev spaces and Hölder with respect to the expectation.) The important thing is that we can choose υ_2 as close to 1 as we like and q_2 as close to 2 as we like, if we choose q_1 and υ_1 appropriately. Note that if the latter factor above is ≲|_1-_2|^q_2υ_2 uniformly in , and if q_2υ_2≈2 is large enough, we get tightness in 𝒞^0_𝔱H^2+α-δ_∂M just by the Kolmogorov continuity criterion. (Indeed, the first factor on the RHS is ≲1 as →∞ by the first part of this proof.) To bound the last factor in the previous display, note ∂_Υ^()=Δ_∂MΥ^()+𝔣^(). But Δ_∂MΥ^() and 𝔣^() converge in H^ϱ(∂M) pointwise-in-time (in any moment) for ϱ>0. This is because Υ^() converges pointwise-in-time in H^2+α-δ(∂M) for α>0 and for any δ>0 small. Now, by calculus, we haveΥ^()__1,·-Υ^()__2,·_H^ϱ(∂M) ≤∫__2^_1Δ_∂MΥ^()_,·_H^ϱ(∂M)+𝔣^()_,·_H^ϱ(∂M).Taking q_2-th moments, we get that the last factor in (<ref>) is, in fact, ≲|_1-_2|^q_2υ_2, so we are done.§.§ Sobolev multiplicationWhen we say a multiplication map is bounded, we mean that multiplication of smooth functions extends continuously in the topology of interest. We have the following multiplication estimates in Sobolev spaces. *Suppose α_1,α_2,α∈ satisfy the following conditions. *We have α_1,α_2≥α, and α_1∧α_2<0 (i.e. at least one is negative). Suppose that α_1+α_2≥0.*Suppose that α_1+α_2>/2+α. (In words, we lose /̣2-many derivatives in multiplication.)Then the multiplication map H^α_1(∂M)×H^α_2(∂M)→H^α(∂M) is bounded with norm ≲_α_1,α_2,α1.*Suppose α_1,α_2,α∈ satisfy the following conditions. *We have α_1,α_2≥α≥0 and α_1+α_2>/2+α.Then the multiplication map H^α_1(∂M)×H^α_2(∂M)→H^α(∂M) is bounded with norm ≲_α_1,α_2,α1.*For any α≥0, the multiplication map H^α(∂M)×𝒞^0,α(∂M)→H^α(∂M) is bounded with norm ≲_α1.*Suppose α>/̣2. Then H^α(∂M) is a Hilbert algebra, i.e. the multiplication map H^α(∂M)×H^α(∂M)→H^α(∂M) is bounded with norm depending only on α and the geometry of ∂M. For the first bullet point, see Theorem 8.1 of <cit.>. For the second bullet point, see Theorem 5.1 of <cit.>. For the third bullet point, see Theorem 7.1 of <cit.>. For the last bullet point, use the second bullet point with α_1,α_2=α. (The point is that if α>/̣2, the constraint in the second bullet point is met.) 9BBCH R. Bass, K. Burdzy, Z.-Q. Chen, M. Hairer. “Stationary distributions for diffusions with inert drift", Probability Theory and Related Fields, 146, 1-47, 2010. BHSobolev A. Behzadan, M. Holst, “Multiplication in Sobolev spaces, revisited", Arkiv for Matematik, 59, 2, 275-306, 2021. BW I. Benjamini, D. B. Wilson. “Excited random walk", Electronic Communications in Probability, 8, 86-92, 2003. C11 I. Corwin. “The Kardar-Parisi-Zhang equation and universality class", Random Matrices: Theory and Applications, Vol. 01, No. 01, 1130001, 2012. CS I. Corwin, H. Shen. “Some recent progress in singular stochastic partial differential equations", Bulletin of the American Mathematical Society, 57(3), 409-454, 2020. DPD G. Da Prato, A. 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Eden. “A two-dimensional growth process", Proceedings of Fourth Berkeley Symposium on Mathematics, Statistics, and Probability, Vol. 4, Berkeley: University of California Press, 223-239, 1961. FRRO X. Fernandez-Real, X. Ros-Oton. Regularity Theory for Elliptic PDE, European Mathematical Society, EMS Zurich Lectures in Advanced Mathematics, Vol. 28, 2022.Galkowski J. Galkowski, J. Toth. “Pointwise bounds for Steklov eigenfunctions", The Journal of Geometric Analysis, 29, 142-193, 2019. GKLP A. Girouard, M. Karpukhin, M. Levitin, I. Polterovich. “The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript", Journal of Spectral Theory, 1, 195-225, 2022. GLSteklov A. Girouard, J. Lagace. “Large Steklov eigenvalues via homogenisation on manifolds", Inventiones Mathematicae, Vol. 226, 3, 1011-1056, 2021. Hai14 M. Hairer. “A theory of regularity structures", Inventiones Mathematicae, A theory of regularity structures, 198, 269-504, 2014. HSPDE M. Hairer. “Singular stochastic PDEs". Proceedings of the ICM, 2014. HGM M. Hairer, A. Gerasimovics, K. Matetski. “Directed mean curvature flow in noisy environment", Communications on Pure and Applied Mathematics (to appear), 2023. HS23 M. Hairer, H. Singh. “Regularity structures on manifolds and vector bundles", arXiv:2308.05049, 2023. HL M.B. Hastings, L. S. Levitov. “Laplacian growth as one-dimensional turbulence", Physica D: Nonlinear Phenomena, 116(1-2), 244-252, 1998. Hsu0 P. Hsu. “On excusions of reflecting Brownian motion", Transactions of the American Mathematical Society, Vol. 296, No. 1, 1986. KPZ M. Kardar, G. Parisi, Y.-C. Zhang. “Dynamics scaling of growing interfaces", Physical Review Letters, 56, 9, 889-892, 1986. KS D. Kious, V. Sidoravicius. “Phase transition for the once-reinforced random walk on ^-like trees", Annals of Probability, 46(4), 2121-2133, 2018. KL C. Kipnis , C. Landim. Scaling Limits of Interacting Particle Systems. 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Zheng. “Eden model in many dimensions", Physical Review Letters, 53, 1791, 1984. Qua J. Quastel. “Introduction to KPZ", Current developments in mathematics, 2011(1), 2011. RS A. Ramirez, V. Sidoravicius. “Asymptotic behavior of a stochastic combustion growth process", Journal of the European Mathematical Society, 6, 3, 293-334, 2004. Rivera A. Rivera. “Weighted local Weyl laws for elliptic operators", Annales de la Faculte des sciences de Toulouse: Mathematiques, Vol. 31, 6, 423-490, 2022. R S. Rosenberg. The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds. (London Mathematical Society Student Texts). Cambridge: Cambridge University Press. 1997. Sellke T. Sellke. “Recurrence of reinforced random walk on a ladder", Electronic Journal of Probability, 11, 301-310, 2006. Taylor M. Taylor. Partial Differential Equations III: Nonlinear Equations. Applied Mathematical Sciences, Vol. 117, Second edition, Springer, New York, 2011. DLA T. A. Witten, Jr., L. M. Sander. “Diffusion-limited aggregation, a kinetic critical phenomenon", Physical Review Letters, 47, 1400, 1981. KPZAim I. Corwin, K. Dang, J. Quastel, editors of AimPL: Kardar-Parisi-Zhang equation and universality class, available at http://aimpl.org/kpzuniversality. | http://arxiv.org/abs/2311.16095v2 | {
"authors": [
"Amir Dembo",
"Kevin Yang"
],
"categories": [
"math.PR",
"82C24, 60H15, 58J65, 35R60"
],
"primary_category": "math.PR",
"published": "20231127185904",
"title": "KPZ-type equation from growth driven by a non-Markovian diffusion"
} |
APS/[email protected], [email protected]^†Facultad de Ingeniería, Universidad Nacional Autónoma de México, Circuito Escolar 04360, C.U., Coyoacán, 04510 Ciudad de México, México ^†Tecnologico de Monterrey, School of Engineering and Sciences, Ciudad de México 14380, México ^*Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain A quantum random walk model is established on a one-dimensional periodic lattice that fluctuates between two possible states. This model is defined by Lindblad rate equations that incorporate the transition rates between the two lattice states. Leveraging the system's symmetries, the particle velocity can be described using a finite set of equations, even though the state space is of infinite dimension. These equations yield an analytical expression for the velocity in the long-time limit, which is employed to analyze the characteristics of directed motion. Notably, the velocity can exhibit multiple inversions, and to achieve directed motion, distinct, nonzero transition rates between lattice states are required. Quantum ratchet with Lindblad rate equations Jesús Casado-Pascual January 14, 2024 ============================================ § INTRODUCTION Directed motion of particles in systems subjected to deterministic or stochastic unbiased driving forces has garnered significant and continuous attention <cit.>. This phenomenon, commonly called ratchet effect, has found important applications in physics <cit.>, chemistry <cit.>, biology <cit.>, and nanotechnology <cit.>. The ratchet effect has not only been studied in systems composed of particles but has also been analyzed in extended systems <cit.>.From a theoretical perspective, significant efforts have been made to unravel the underlying mechanisms behind the emergence of directed motion <cit.>. To make directed motion possible, it has been shown that specific spatiotemporal symmetry and supersymmetry conditions must be broken <cit.>. Another intriguing phenomenon, frequently observed in these systems, is the reversal of current direction when a system parameter varies <cit.>. The ratchet effect, which was originally analyzed in classical systems, was later extended to the quantum domain <cit.>. Specifically, research has demonstrated that quantum phenomena, such as tunneling and wave packet dispersion, can either enhance the ratchet current <cit.> or lead to a reduction in transport efficiency <cit.>. A crucial factor contributing to ratchet behavior is the violation of time-reversal symmetry <cit.>. This can be accomplished, for example, through the irreversibility of dissipative effects. In this context, investigations have been conducted into the quantum dynamics of a particle in an asymmetric potential with Ohmic <cit.> and super-Ohmic dissipation <cit.>. The ratchet effect has also been studied in the absence of dissipation where the system follows Hamiltonian dynamics (see, e.g., Ref. <cit.>). In this case, the particle is typically subjected to potentials that break certain symmetries, as seen in the flashing ratchet potential considered in <cit.>, the delta-kicked model in <cit.>, and the Bose-Hubbard model in <cit.>. These types of potentials can be created, for example, using optical lattices <cit.>. Furthermore, these systems have also been used to study the relationship between many-body quantum chaos and entanglement in quantum ratchet systems <cit.>.Experimental investigations of quantum ratchets have been continually carried out since one of the original proposals involving a SQUID device <cit.>. For instance, in Ref. <cit.>, the observation of a polarization-sensitive magnetic quantum ratchet current was reported, where the direction and magnitude are determined by the orientation of the electric field. In another study <cit.>, a scattering quantum ratchet was demonstrated, involving a directional flow of electrons in a two-dimensional electron gas. Furthermore, Ref. <cit.> presented an electronic quantum ratchet in graphene layers. In addition to these, quantum ratchets have also been realized in optical systems, as seen in the case of a delta-kicked photonic quantum ratchet <cit.> and Bose-Einstein condensates in an optical lattice <cit.>.To gain a deeper understanding of the physics underlying the ratchet effect, it is advantageous to develop simplified models that retain the essential characteristics of the phenomenon under study and enable the extraction of analytical solutions <cit.>.Specifically, these simplified models hold particular appeal when applied to open quantum systems governed by master equations that may exhibit nonMarkovian behavior <cit.>, as such systems are typically amenable only to numerical treatment. In this paper, we introduce a simple quantum random walk model for a particle moving on an infinite lattice. The lattice randomly fluctuates between two configurations that do not lead to directed motion when considered separately. However, under specific conditions, they do result in directed motion when they alternate. The evolution of the system is mathematically described using Lindblad rate equations <cit.>. This model allows us to derive an analytical expression for the long time limit of the particle's velocity. The use of this expression enables us to examine the circumstances under which directed motion exists and multiple current reversals occur.The structure of the remaining sections in this paper is as follows. In Sec. <ref>, we introduce the model under investigation. The derivation of the Lindblad rate equations that describe the system's evolution is provided in Appendix <ref>. Section <ref> examines the symmetries of the system, and in Sec. <ref>, we establish the equations that describe the evolution of the velocity and its long-time behavior. Certain technical details required for deriving these equations can be found in Appendix <ref>. Section <ref> explores the key properties of the particle velocity in the long-time limit. Finally, in Sec. <ref>, we present a summary and draw conclusions based on the aspects covered in this study.§ DESCRIPTION OF THE MODELConsider a quantum system composed of a particle that can be found at any site of the set 𝒫 = { j L: j ∈ℤ}, with L a given length and ℤ the set of integers (see Fig. <ref>). Let | j ⟩ be the quantum state associated with the position jL. These kets form an orthonormal basis {| j ⟩: j∈ℤ} for the state space of the particle. The set 𝒫 constitutes a one dimensional lattice of sites that has two possible states denoted by +1 and -1. The lattice fluctuates randomly between these two states following a Markovian dichotomic process, with γ_± being the transition rates from state ± 1 to ∓ 1. Consequently, the probability that the lattice sojourns in the state ± 1 for a span of time τ and then changes to the state ∓ 1 in the time interval between τ and τ+dτ is given by γ_± e^-γ_±τ dτ (see Ref. <cit.>).As sketched in Fig. <ref>, in the lattice state +1 there are coherent transitions between the state vectors | 3j+1 ⟩ and | 3j+2 ⟩ at a frequency ω. The corresponding sites (3j+1)L and (3j+2)L are called nonabsorbing. In addition, there are incoherent transitions from | 3j+1 ⟩ and | 3j+2 ⟩ to | 3j ⟩ and | 3j+3 ⟩, respectively, at a rate Γ. No transitions are possible from state vectors of the form | 3j ⟩ to any other states and, for this reason, the sites 3jL are called absorbing. In Fig. <ref>, the absorbing sites are depicted as crosses and the nonabsorbing as solid circles. The lattice state -1 is obtained by inverting the lattice state +1 with respect to any site of the form (3j-1)L. Alternatively, it can be obtained by translating the lattice state +1 by (3j+1)L to the right or, equivalently, by (3j+2)L to the left. Observe that both lattice states are periodic with period 3L and invariant under inversions about the absorbing sites. In terms of the jump operators R_j = | j+1 ⟩⟨ j |, with j∈ℤ, the above described dynamics in each lattice state can be modeled by the superoperators ℒ_±· =- i/ħ[ H_±, ·] + 𝒟_±· ,where [· , · ] denotes the commutator, the Hamiltonians H_± are given byH_± = ħω∑_k∈ℤ( R_3k± 1 + R_3k± 1^†) ,and the dissipators 𝒟_± are defined by𝒟_+· = Γ∑_k∈ℤ[ R_3k^†· R_3k+ R_3k+2· R_3k+2^† - 1/2{ R_3k R_3k^† + R_3k+2^† R_3k+2 , ·}] ,𝒟_-· = Γ∑_k∈ℤ[ R_3k· R_3k^†+ R_3k+1^†· R_3k+1 - 1/2{ R_3k^† R_3k + R_3k+1 R_3k+1^† , ·}] ,with {·, ·} the anticommutator. The first term on the righthand side of Eq. (<ref>) is responsible for the coherent evolution associated with the Hamiltonians H_± in Eq. (<ref>). By contrast, the second term, involving the dissipators 𝒟_± in Eqs. (<ref>) and (<ref>), gives rise to the incoherent transitions. Notice that ℒ_± have the form of a generator of a quantum dynamical semigroup <cit.>.In analogy to composite stochastic processes <cit.>, the density operator of the system, ρ(t), can be expressed in terms of two positive operators ρ_±(t) as ρ (t) = ρ_+(t) + ρ_-(t). The evolution of these operators is governed by the Lindblad rate equationsd/dtρ_±(t)= ℒ_±ρ_± (t) - γ_±ρ_±(t) + γ_∓ρ_∓ (t) .The concept of Lindblad rate equations was initially introduced in <cit.> as a means to include nonMarkovian effects in Lindblad-like master equations. Subsequently, it was further generalized in <cit.>. To some extent, the Lindblad rate equations can be regarded as a quantum version of the composite stochastic processes introduced in Ref. <cit.>. In Appendix <ref>, we provide a derivation of Eq. (<ref>) that retains the spirit presented in Refs. <cit.>.The initial conditions for the Lindblad rate equations in (<ref>) depend on the probability p_±(t_0) that the lattice state is ± 1 at the initial time t_0. Specifically, the initial conditions are ρ_+(t_0) = p_+(t_0)ρ(t_0)and ρ_-(t_0) = p_-(t_0)ρ(t_0) (see Appendix <ref>).§ SYMMETRY CONSIDERATIONSIn this section, we discuss several symmetry properties of the superoperators ℒ_± that arise as a consequence of the symmetries of the corresponding states of the lattice. In order to do this, for any k∈ℤ, we introduce the translation operator by k unitsT_k = ∑_j ∈ℤ| j + k ⟩⟨ j | ,and the inversion operator about site kLΠ_k = ∑_j∈ℤ| 2k -j ⟩⟨ j | .From these definitions, it is clear that Π_k is Hermitian and unitary and that T_kis a unitary operator that satisfies T_j T_k = T_j + k and T_k^† = T_-k for all j,k∈ℤ.Recall that both lattice states are spatially periodic with period 3L and also invariant under inversions about any of the absorbing sites (see Fig. <ref>). In addition, the lattice state -1 is obtained by translating the lattice state +1 by (3j+1)L to the right or by inverting it with respect to any lattice sites of the form (3j-1)L (see Fig. <ref>). The spatial periodicities of the lattice states ± 1 are expressed by the invariance of ℒ_± under the aforementioned translations ℒ_± = 𝒯_3jℒ_±𝒯_3j^† ,where we have introduced the translation superoperator 𝒯_k· = T_k· T_k^† and its adjoint with respect to the Hilbert-Schmidt inner product 𝒯_k^†· = T_k^†· T_k. Analogously, the inversion invariances are expressed byℒ_+ = 𝒫_3jℒ_+𝒫_3j,andℒ_- = 𝒫_3j+1ℒ_-𝒫_3j+1 ,with the inversion superoperator 𝒫_k· = Π_k·Π_k. Finally, the aforementioned transformations from one lattice state to the other are represented byℒ_- = 𝒯_3j+1ℒ_+𝒯_3j+1^†,andℒ_∓ = 𝒫_3j-1ℒ_±𝒫_3j-1 .Notice that, from the definitions of 𝒯_k and 𝒫_k, it is clear that 𝒯_k^†𝒯_k = 𝒯_k𝒯_k^† = ℐ and 𝒫_k^2 = ℐ, with ℐ the identity superoperator.A property that follows straightforwardly from Eqs. (<ref>) and (<ref>) is that the operator ρ̃_± (t) = 𝒫_-1ρ_∓ (t) satisfies d/dtρ̃_±(t)= ℒ_±ρ̃_± (t) - γ_∓ρ̃_±(t) + γ_±ρ̃_∓ (t) .Hence, ρ̃_+ (t) and ρ̃_- (t) satisfy exactly the same equations as ρ_+(t) and ρ_-(t) but with γ_+ and γ_- interchanged. The above described symmetries of ℒ_± are used in the next sections to anticipate some properties of the system dynamics.§ DERIVATION OF THE PARTICLE VELOCITYWe are interested in characterizing the mean velocity of the particle. In order to do so, we first introduce the position operatorX=L ∑_j∈ℤ j | j ⟩⟨ j | ,whose mean value at time t is given by⟨ X ⟩(t)= [ X ρ(t) ]= ∑_α = ±[ Xρ_α(t) ],where Tr(·) denotes the trace. The mean velocity is then defined asv(t)= d/dt⟨ X ⟩ (t) . Before establishing a method to calculate v(t), let us discuss some consequences of the symmetries described in the previous section. As mentioned before, the operators ρ̃_±(t) satisfy the same equations as ρ_±(t) but with the rate parameters γ_+ and γ_- interchanged [see Eqs. (<ref>) and (<ref>)]. If the initial conditions are invariant under inversions about site -L, i.e., 𝒫_-1ρ_±(0) = ρ_±(0), it then follows from the uniqueness of the solution that ρ̃_±(t;γ_-,γ_+) = ρ_±(t;γ_+,γ_-), where the dependence on the rate parameters γ_± has been written explicitly. Consequently, from the definition in Eq. (<ref>) and the fact that 𝒫_-1X = -X-2LI, with I the identity operator, it follows that⟨ X ⟩ (t;γ_+,γ_-) = -⟨ X ⟩ (t;γ_-,γ_+) -2L and, therefore, v(t;γ_+, γ_-) = -v(t;γ_-, γ_+). Below it is shown that the long-time limit of the velocity v_∞ = lim_t→ + ∞ v(t) is independent of the initial conditions. Thus, v_∞(γ_+,γ_-) = - v_∞(γ_-,γ_+) regardless of the initial conditions and, in particular, v_∞ = 0 if γ_+ = γ_-. As a result, a necessary condition to have directed motion is that γ_+≠γ_-. In order to obtain an explicit expression for v(t), it is convenient to introduce the operatorsΛ_±(t)= ∑_j∈ℤ𝒯_3jρ_±(t),which are clearly invariant under translations by 3m units, i.e., 𝒯_3mΛ_±(t) = Λ_±(t). In addition, taking into account Eq. (<ref>), it readily follows that Λ_±(t) satisfy the same equations as ρ_±(t), i.e., Eq. (<ref>). In terms of the matrix elementsλ_j,k^(±)(t)= ⟨ j |Λ_±(t) | k ⟩,the invariance of Λ_±(t) under translations by 3m units takes the formλ_j,k^(±)(t) = λ_j+3m,k+3m^(±)(t) .It should also be noticed that ∑_α = ±∑_j=1^3λ_j,j^(α)(t) = ∑_α = ±[ρ_α(t)] = [ρ(t)] = 1.Using the evolution equations for ρ_±(t) in Eq. (<ref>), in combination with the properties of Λ_±(t) discussed above, it can be verified thatv(t)/L = Γ[λ_3,3^(-)(t)- λ_2,2^(-)(t)+ λ_2,2^(+)(t) - λ_1,1^(+)(t) ]- 2ωIm[ λ_3,2^(-)(t) + λ_2,1^(+)(t) ] ,with Im(z) the imaginary part of the complex number z. For a detailed derivation of Eq. (<ref>), see Appendix <ref>.To determine the average velocity in Eq. (<ref>), it is first necessary to solve the evolution equations for the matrix elements of Λ_±(t). At first sight, this would involve a system of an infinite number of coupled differential equations. Nevertheless, using the periodicity property in Eq. (<ref>), this infinite system can be reduced to a finite number of differential equations for the matrix elements λ_j,k^(±)(t) with j,k∈{1,2,3}, i.e., 18 equations. The explicit form of twelve of these equations isD_-,0λ_1,1^(-)(t)= Γ[ λ_3,3^(-)(t) + λ_2,2^(-)(t) ] + γ_+λ_1,1^(+)(t) , D_-,2λ_2,2^(-)(t)=iω[ λ_2,3^(-)(t)- λ_3,2^(-)(t) ]+ γ_+λ_2,2^(+)(t) , D_-,2λ_3,3^(-)(t)=-iω[ λ_2,3^(-)(t) - λ_3,2^(-)(t) ]+ γ_+λ_3,3^(+)(t) , D_+,2λ_1,1^(+)(t) =-iω[ λ_2,1^(+)(t)- λ_1,2^(+)(t) ] + γ_-λ_1,1^(-)(t) , D_+,2λ_2,2^(+)(t) =iω[ λ_2,1^(+)(t) - λ_1,2^(+)(t) ]+ γ_-λ_2,2^(-)(t) , D_+,0λ_3,3^(+)(t)= Γ[ λ_2,2^(+)(t) + λ_1,1^(+)(t) ] + γ_-λ_3,3^(-)(t) , D_-,1λ_1,2^(-)(t) =iωλ_1,3^(-)(t) + γ_+λ_1,2^(+)(t) , D_-,1λ_1,3^(-)(t)=iωλ_1,2^(-)(t)+ γ_+λ_1,3^(+)(t) , D_-,2λ_2,3^(-)(t)=-iω[ λ_3,3^(-)(t) - λ_2,2^(-)(t) ]+ γ_+λ_2,3^(+)(t), D_+,2λ_1,2^(+)(t) =-iω[ λ_2,2^(+)(t) - λ_1,1^(+)(t) ]+ γ_-λ_1,2^(-)(t) , D_+,1λ_1,3^(+)(t)=-iωλ_2,3^(+)(t) + γ_-λ_1,3^(-)(t), D_+,1λ_2,3^(+)(t)=-iωλ_1,3^(+)(t) + γ_-λ_2,3^(-)(t),were we have introduced the differential operatorsD_± , j = d/dt + ( jΓ/2 + γ_±) ,for j=0,1,2. The remaining six equations can be obtained by taking the complex conjugate of the last six equations appearing in Eq. (<ref>) and using that Λ_±(t) are Hermitian. Taking into account the definition of Λ_±(t), the initial conditions for the system of differential equations (<ref>) can be obtained from the initial conditions ρ_±(0) by λ_j,k^(±)(0) = ∑_m∈ℤ⟨ j + 3m |ρ_±(0) | k +3m ⟩. In addition, by adding the first six equations of (<ref>), it can be verified that the derivative with respect to time of ∑_α=±∑_j=1^3λ_j,j^(α)(t) is zero, which is in accordance with the fact that ∑_α=±∑_j=1^3λ_j,j^(α)(t) = 1.In order to explore the possibility of steady state solutions λ̃_j,k^(±)of the system of differential equations in (<ref>), one just has to take all of the derivatives on the lefthand side of (<ref>) equal to zero and replace λ_j,k^(±)(t) by λ̃_jk^(±). This leads to a homogeneous linear system of algebraic equations with a solution space of dimension one. The free parameter that appears can be computed by imposing the condition ∑_α=±∑_j=1^3λ̃_j,j^(α) =1. The analytical expression obtained is quite lengthy and, thus, is not included here. In addition, it can be verified that the real parts of the nonzero eigenvalues of the coefficient matrix associated with the system of equations (<ref>) are negative. Consequently, regardless of the initial conditions, in the long-time limit all the solutions λ_j,k^(±)(t) tend to the steady state solution λ̃_j,k^(±). According to Eq. (<ref>), the long-time limit of the velocity, v_∞, is also well defined and unique.Substituting the steady state solution λ̃_j,k^(±) in Eq. (<ref>), one obtains after a lengthy calculation thatv_∞/ω L = 3Δ(1-Δ^2)γ̃^2Γ̃F/G ,where we have introduced the dimensionless parameters Δ = (γ_+ - γ_-)/(γ_+ + γ_-), γ̃ = (γ_+ + γ_-)/ω, Γ̃ = Γ/ω, and the dimensionless quantitiesF=-Γ̃^3(2γ̃ + Γ̃)[ (5-Δ^2)γ̃^2 + 10 γ̃Γ̃ + 4Γ̃^2]-4Γ̃[ 8Δ^2γ̃^3 + 10(1+Δ^2)γ̃^2Γ̃ + 16γ̃Γ̃^2 + 7Γ̃^3]+32[ (1+Δ^2)γ̃^2 + γ̃Γ̃ - Γ̃^2]+ 64and G = Γ̃^5 (γ̃ + Γ̃)^2 (2γ̃ + Γ̃)[ 4Γ̃^2 + 12γ̃Γ̃ + (9-Δ^2)γ̃^2]+ 8Γ̃^3(γ̃ + Γ̃)[(9 + 7Δ^2)γ̃^4+ 3(13 + 3 Δ^2) γ̃^3Γ̃ + 2(29 + 2Δ^2)γ̃^2Γ̃^2 +36γ̃Γ̃^3+8Γ̃^4]+ 4 Γ̃[ 2(5 +22Δ^2 + 5Δ^4)γ̃^5 + (127 + 122Δ^2 + 7Δ^4) γ̃^4Γ̃ + 4( 103 + 37 Δ^2)γ̃^3Γ̃^2 + (583 + 73Δ^2)γ̃^2Γ̃^3 + 384γ̃Γ̃^4 + 96Γ̃^5] + 32[ 3(1 - Δ^4)γ̃^4 + 2(13 + 7Δ^2)γ̃^3Γ̃ + (71 + 17Δ^2)γ̃^2Γ̃^2 + 80γ̃Γ̃^3 +32 Γ̃^4]+ 64[ 3(1 - Δ^2)γ̃^2 + 16γ̃Γ̃ +16 Γ̃^2] .Notice that the denominator G is always a positive quantity because |Δ|≤ 1. § ANALYSIS OF THE LONG-TIME LIMIT VELOCITYAccording to Eq. (<ref>), v_∞ is an odd function of Δ and, consequently, vanishes when Δ = 0. This corroborates the symmetry arguments presented in Sec. <ref>, where it was shown that v_∞(γ_+,γ_-) = - v_∞(γ_-,γ_+) and, therefore, different transition rates are necessary to obtain directed motion. In addition, v_∞ also vanishes when Δ = ± 1, which implies that both transition rates γ_± must be nonzero to have directed motion. This is so because, in the absence of repetitive fluctuations between the two lattice states, the absorbing sites eventually trap the particle avoiding its motion.In addition to the previously mentioned values of Δ, directed motion can also be canceled for those values of Δ, if any, for which F=0. The function F depends on γ̃, Γ̃, and Δ^2, i.e., F = F(γ̃, Γ̃, Δ^2). Since F is a first degree polynomial in Δ^2, there is always a value Δ_c^2 that vanishes F. According to the definition of Δ, only the values Δ_c^2 between 0 and 1 are acceptable. In Fig. <ref>, the cyan-shaded area depicts the region of points (γ̃, Γ̃) for which Δ_c^2∈ (0,1).For each point in this region there exist three values of Δ where inversion of directed motion occurs, namely, Δ = 0 and Δ = ±Δ_c. Outside of this region inversion of directed motion occurs only for Δ = 0. The boundaries of this region are implicitly defined by the conditions F(γ̃,Γ̃, 1) = 0 and F(γ̃,Γ̃, 0) = 0, which correspond respectively to Δ_c^2 = 1 and Δ_c^2 = 0. These conditions are polynomial equations of degree three in γ̃ and can be solved analytically to obtain the dependence of γ̃ on Γ̃ for the curves that parameterize these boundaries. In particular, condition F(γ̃,Γ̃, 1) = 0 yields γ̃_1(Γ̃) = 2(1-Γ̃^2)/Γ̃, which is the curve that bounds the shaded area from below in Fig. <ref>. The curve γ̃_0(Γ̃) obtained from solving condition F(γ̃,Γ̃, 0) = 0 bounds the shaded area from above. The analytical expression for γ̃_0(Γ̃) is very lengthy and, for that reason, is not included here. Since γ̃≥ 0, one conclusion that readily follows from the above expression for γ̃_1(Γ̃) is that inversion of directed motion for Δ≠ 0 occurs only when Γ̃ < 1.In order to illustrate the inversion of directed motion, Fig. <ref> shows the dependence of the dimensionless long-time velocity, v_∞/(ω L), on the parameter Δ. The values of γ̃ and Γ̃ have been chosen to lie in the three different regions shown in Fig. <ref>. These values are depicted in Fig. <ref> as a red asterisk, a black cross, and a blue plus sign. Notice that the black-dotted curve exhibits inversion of directed motion at three values of Δ, while the red-solid and blue-dotdashed curves present inversion only at Δ = 0. Moreover, observe that v_∞ has opposite signs on the red-solid and blue-dotdashed curves. Specifically, Δ v_∞≥ 0 if (γ̃, Γ̃) lies to the left of the cyan-shaded area in Fig. <ref>, whereas Δ v_∞≤ 0 if (γ̃, Γ̃) lies to the right of the aforementioned area. § CONCLUSIONSIn summary, this paper presented a simple quantum random-walk model on a one-dimensional periodic lattice that fluctuates between two distinct states.The mathematical framework for this model is rooted in Lindblad rate equations, which encompass the transition rates between the lattice states. Despite the system's infinite-dimensional state space, the inherent periodic symmetry of the problem enables us to describe the time evolution of the particle velocity through a finite set of differential equations. These equations yield concise, analytical expressions for the long-time velocity behavior.The simplicity of the derived analytical expression for the velocity in the long-time limit has enabled us to conduct a comprehensive investigation of the characteristics of directed motion. In particular, some of these properties emerge as direct consequences of the system's symmetries. It is worth mentioning that, in our model, directed motion can exhibit multiple inversions, a feature not observed in analogous classical models <cit.>. Finally, it is important to mention that, although the model presented in this work is quite simple, we are confident that some of the techniques presented here can be extended to more complex quantum models. Specifically, the use of the Lindblad rate equations proposed in this work opens up new perspectives for the study of other fluctuating quantum systems.The authors acknowledge grant PID2022-136228NB-C22 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”.§ DERIVATION OF THE LINDBLAD RATE EQUATIONS (<REF>)Let us consider a quantum system evolving according to the GKLS equation d/dtρ (t) = ℒ_η(t)ρ(t) ,where the generator ℒ_η(t) dependson a dichotomic Markov process η (t) that can take the values ± 1. For simplicity, we write ± instead of ± 1 when ± 1 appears as a subscript or superscript, so that ℒ_±≡ℒ_±1. The process η(t) is described by the master equation∂/∂ t p(α , t |α_0, t_0) = γ_-αp(-α , t |α_0, t_0) - γ_αp(α, t |α_0, t_0) ,where p(α , t |α_0, t_0) is the conditional probability that η (t) = α given that η(t_0) = α_0, and γ_α is the transition rate from the value α to -α. In the above expression, the parameters α and α_0 can take the values ± 1.A realization of the process η(t) in the time interval [t_0,t] can be characterized by giving its final value, more specifically, α = lim_t' → t^-η(t') and the time instants, if any, at which changes in value occur. These time instants are assumed to be listed in increasing order, i.e., t_0 < t_1 < t_2 < ⋯ < t_n < t, and arranged in a vector τ_n = (t_1,t_2, …, t_n). For notational convenience, we define τ_0 as a vector with no components.The probability of a realization with no changes in value is given byP_0^[t_0,t](α,τ_0) = e^-(t-t_0)γ_αp(α, t_0) ,where p(α, t_0) is the probability that the process η(t) takes the value α at time t_0. In this case, the density operator at time t isρ_0(t;α, τ_0) = e^(t-t_0)ℒ_αρ(t_0) ,where ρ(t_0) is the density operator at the initial time t_0. The probability density of a realization ending in α with n≥ 1 changes in value isP_n^[t_0,t](α,τ_n) = e^-(t-t_n)γ_α∏_j=1^nγ_α_j e^-(t_j-t_j-1)γ_α_j p(α_1, t_0),where α_j = (-1)^n+1-jα is the value of η(t) just before the jth change. In this case, the density operator at time t is given byρ_n(t;α, τ_n) = e^(t-t_n)ℒ_α∏_j=1^n e^(t_j-t_j-1)ℒ_α_jρ(t_0) ,where the product of the superoperators must be taken in the order ∏_j=1^nℱ_j = ℱ_n⋯ℱ_1. Notice that the following recurrence relations hold for n≥ 1P_n^[t_0,t](α,τ_n) = γ_-αe^-(t-t_n)γ_α P_n-1^[t_0,t_n](-α, τ_n-1) andρ_n(t;α,τ_n)= e^(t-t_n)ℒ_αρ_n-1(t_n;-α,τ_n-1) . The density operator at time t, ρ(t), is obtained after an average over all realizations, i.e.,ρ(t) = ∑_α = ± 1ρ_α(t) ,whereρ_α (t) = ρ_0(t;α,τ_0)P_0^[t_0,t](α,τ_0)+ ∑_n=1^∞∫_t_0^t dτ_nρ_n(t;α,τ_n) P_n^[t_0,t](α,τ_n) ,and we have introduced the notation ∫_t_0^t dτ_n = ∫_t_0^t dt_n∫_t_0^t_n dt_n-1…∫_t_0^t_2 dt_1.Taking into account the recurrence relations in Eqs. (<ref>) and (<ref>),it can be verified that Eq. (<ref>) can be expressed asρ_α(t) =e^-(t-t_0)γ_α p(α, t_0) e^(t-t_0)ℒ_αρ(t_0)+ ∫_t_0^tdt' γ_-αe^-(t-t')γ_α e^(t-t')ℒ_αρ_-α(t') .Taking the derivative of Eq. (<ref>) with respect to t, the evolution equations (<ref>) are finally obtained. In addition, from the integral equation (<ref>), it readily follows that the initial conditions for these equations are ρ_α(t_0) = p(α,t_0)ρ(t_0).§ DERIVATION OF EQ. (<REF>)Every integer n can be expressed in the form n = j + 3k with j ∈{ 1, 2, 3 } and k ∈ℤ. Therefore, using the translation superoperator 𝒯_3k, one can writeTr[ X ρ_±(t) ]=L ∑_n∈ℤ n ⟨ n |ρ_±(t) | n ⟩ =L ∑_j = 1^3∑_k∈ℤ (j + 3k) ⟨ j + 3k |ρ_±(t) | j + 3k ⟩ =L ∑_j = 1^3∑_k∈ℤ (j - 3k) ⟨ j |𝒯_3kρ_±(t) | j ⟩ .Taking the derivative of the above expression with respect to time and using the evolution equations for ρ_±(t) in Eq. (<ref>), as well as the definitions of the expected value of the position and the velocity in Eqs. (<ref>) and (<ref>), one obtains thatv(t) = L∑_α = ±∑_j=1^3∑_k∈ℤ (j-3k) ⟨ j |𝒯_3kℒ_αρ_α(t) | j ⟩ .From the invariance of ℒ_± with respect to translations by 3k units in Eq. (<ref>) and the definition of the operators Λ_±(t) in Eq. (<ref>), it then follows that v(t)/L = ∑_α = ±∑_j=1^3∑_k∈ℤ (j-3k) ⟨ j |ℒ_α𝒯_3kρ_α(t) | j ⟩ = A + B ,whereA = ∑_α = ±∑_j=1^3 j ⟨ j |ℒ_αΛ_α(t) | j ⟩andB = -3∑_α = ±∑_k∈ℤ k ∑_j=1^3⟨ j |ℒ_α𝒯_3kρ_α(t) | j ⟩ .To evaluate A and B one has to apply the definitions of the superoperators ℒ_± in Eqs. (<ref>)-(<ref>) and the periodicity of the matrix elements of Λ_±(t) in Eq. (<ref>). It can then be verified from straightforward calculations that A= Γ[ 2λ_1,1^(+)(t) + λ_2,2^(+)(t) - 2λ_3,3^(-)(t) - λ_2,2^(-)(t) ] - 2ωIm[ λ_2,1^(+)(t) + λ_3,2^(-)(t) ]andB=-3Γ∑_k∈ℤ k [ ⟨ 4 |𝒯_3kρ_+(t) | 4 ⟩ - ⟨ 1 |𝒯_3kρ_+(t) | 1 ⟩ + ⟨ 0 |𝒯_3kρ_-(t) | 0 ⟩ - ⟨ 3 |𝒯_3kρ_-(t) | 3 ⟩] .Taking into account the definition of the translation superoperators, it is clear that ⟨ 4 |𝒯_3kρ_+(t) | 4 ⟩ = ⟨ 1 |𝒯_3(k-1)ρ_+(t) | 1 ⟩ and ⟨ 0 |𝒯_3kρ_-(t) | 0 ⟩ = ⟨ 3 |𝒯_3(k+1)ρ_+(t) | 3 ⟩. A change of summation index in Eq. (<ref>) leads to the result B = -3Γ[λ_1,1^(+)(t) - λ_3,3^(-)(t)]. 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"authors": [
"Luis Octavio Castaños-Cervantes",
"Jesús Casado-Pascual"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech"
],
"primary_category": "quant-ph",
"published": "20231127105745",
"title": "Quantum ratchet with Lindblad rate equations"
} |
On Quantile Treatment Effects, Rank Similarity,and Variation of Instrumental VariablesFor insightful discussions, the authors are grateful to Victor Chernozhukov and participants at the 2023 North American Winter Meeting and the Asian Meeting of the Econometric Society, the Barcelona School of Economics Summer Forum, and the CeMMAP/SNU conference and in the seminars at UCL, Oxford, BU and Boston College. Sukjin Han School of Economics University of Bristolmailto:sukjin.han%5C%[email protected] XuDepartment of EconomicsUniversity of Texas at Austin mailto:h.xu%5C%[email protected] January 14, 2024 =========================================================================================================================================================================================================================================================================================================================================================================================================================This paper investigates how certain relationship between observed and counterfactual distributions serves as an identifying condition for treatment effects when the treatment is endogenous, and shows that this condition holds in a range of nonparametric models for treatment effects. To this end, we first provide a novel characterization of the prevalent assumption restricting treatment heterogeneity in the literature, namely rank similarity. Our characterization demonstrates the stringency of this assumption and allows us to relax it in an economically meaningful way, resulting in our identifying condition. It also justifies the quest of richer exogenous variations in the data (e.g., multi-valued or multiple instrumental variables) in exchange for weaker identifying conditions. The primary goal of this investigation is to provide empirical researchers with tools that are robust and easy to implement but still yield tight policy evaluations. JEL Numbers: C14, C31, C36Keywords: quantile treatment effects, rank similarity, average treatment effects, endogeneity, multi-valued instrumental variables, partial identification.§ INTRODUCTION This paper investigates how certain relationship between observed and counterfactual distributions serves as an identifying condition for distributional treatment effects under endogeneity, and shows that this condition holds in a range of nonparametric models for treatment effects. To this end, we first provide a novel characterization of the prevalent assumption restricting treatment heterogeneity in the literature, namely rank similarity. Our characterization demonstrates the stringency of this assumption and allows us to relax it in a economically meaningful way, resulting in our identifying condition. It also justifies the quest of richer exogenous variations in the data (e.g., multi-valued or multiple instrumental variables) in exchange for the weaker identifying condition.The primary goal of this investigation is to provide empirical researchers with (i) a framework where validity of identifying conditions prescribes the parameters of interest, (ii) tools for identifying and estimating treatment effects that allow for treatment heterogeneity, but that still yield tight policy evaluation and are simple to implement, and (iii) guidance on data collection that leads to drawing informative causal conclusions.Our analysis centers on the relationship between observed and counterfactual distributions, specifically on the preservation of first-order stochastic dominance (FOSD) of one distribution over the other to their corresponding counterfactual distributions: for arbitrary compliance types t,t'∈𝒯 induced by induced by individuals' potential treatment responses to instrumental variables (IVs), ifY_1|t≺_FOSDY_1|t'thenY_0|t≺_FOSDY_0|t',where Y_d denotes the counterfactual outcome given treatment D=d.[For r.v.'s A and B, let A≺_FOSDB denotes F_B(·)≤ F_A(·) where F_A and F_B are CDFs of A and B, respectively.] This condition produces a partial ordering of the Y_0's distributions based on the partial ordering of the Y_1’s distributions. As we demonstrate later, this condition can be interpreted as Y_0 being “noisier” than Y_1 after controlling for all the confounding variables.We show that the proposed FOSD-preservation condition enables the identification of certain counterfactual distributions, which are essential components for identifying the treatment effects of our interest. Only for the sake of illustration, consider <cit.>'s framework where binary instrument Z∈{0,1} influences treatment participation monotonically. Let compliance types C, AT, and NT stand for compliers, always-takers, and never-takers, respectively. Let Y be the observed outcome given by Y≡ DY_1+(1-D)Y_0. Suppose the observed distribution satisfies Y_1|AT≺_FOSDY_1|C. Then the FOSD-preservation condition implies that Y_0|AT≺_FOSDY_0|C. The latter provides an informative upper bound for P[Y_0≤ y|D=1] (and a symmetric analysis provides a lower bound), which is a necessary component in calculating, for example, the quantile treatment effect on the treated (QTT).[Note that Y_1|AT≺_FOSDY_1|C and Y_0|AT≺_FOSDY_0|C can be respectively rewritten asP[Y≤·|D=1,Z=0]≤ P[Y≤·|D=1,Z=1],P[Y_0≤·|D=1]≤P[Y≤·,D=0|Z=1]-P[Y≤·,D=0|Z=0]/P[D=1|Z=0]-P[D=1|Z=1].]Although Y_1|AT≺_FOSDY_1|C may seem restrictive, this is not generally the case when Z departs from a scalar binary variable. In this sense, our approach underscores the significance of searching for richer exogenous variations of IVs, such as multi-valued or multiple instrumental variables, as a means of trading for less restrictive identifying conditions and achieving tighter bounds. Still, the benefit of our approach can be manifested without requiring continuous or large support of IVs. We also show that the proposed FOSD-preservation condition (i.e., (<ref>) implies (<ref>)) yields testable restrictions.Nonparametric identification of treatment effects using IVs with limited support has long been a challenging goal even when the focus is on mean treatment effects, such as the average treatment effect (ATE) and the ATE on the treated (ATT). In an influential line of literature, <cit.>, <cit.>, and <cit.>, among many others, construct sharp bounds on the ATE under a set of assumptions on the directions of treatment effects and treatment selection while allowing instruments to be invalid in a specific sense. Even with valid instruments, however, bounds on the ATE are typically wide and uninformative to yield precise policy prediction. The local ATE (LATE) (<cit.>) and local QTE (<cit.>) have been a popular alternative when researchers are equipped with discrete IVs and impose a monotonicity assumption on the selection to treatment. However, the local group for which the treatment effect is identified may not be the group of policy interest. Therefore, the extrapolation of the local parameters becomes an important issue for policy analysis (e.g., treatment allocation), in which case the identification challenge still remains (see e.g., <cit.>, <cit.>).Another prevalent approach in the literature is to restrict the degree of treatment heterogeneity via rank similarity (or rank invariance). This assumption has been shown to have substantial identifying power for distributional treatment effects and the ATE and used in various nonparametric contexts implicitly or explicitly (<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> to name a few). However, the plausibility of this assumption can be questionable in many applications (e.g., <cit.>) and testing methods are proposed as one reaction to the skepticism (<cit.>, <cit.>, <cit.>).In this paper, we clarify the stringency of the rank similarity assumption by characterizing its restrictions on the relationship between observed and counterfactual distributions. In particular, we show that the strong preservation of FOSD (i.e., (<ref>) holds if and only if (<ref>) holds) is equivalent to rank linearity, a slight relaxation of rank similarity that allows for a linear transformation of an individual's rank to the counterfactual rank. By doing so, we establish the connection between the rank similarity assumption introduced by <cit.>'s structural IV model and its corresponding conditions within <cit.>'s counterfactual outcomes framework. Furthermore, we provide economic justifications for the weak preservation of FOSD by proposing a variety of non-separable structural IV models that imply the FOSD preservation condition, but that do not satisfy rank similarity. Based on our identification strategy, we develop a statistical linear programming (LP) approach to estimate optimal bounds on the treatment parameters. These bounds are defined as optimal values of LP (with a discrete outcome) or semi-infinite LP (SILP) (with a continuous outcome). To address the infeasibility of the SILP problem, we transform the optimization problem by (i) randomizing the constraints or (ii) invoking duality and approximates the Lagrangian measure using sieves.The next section formally introduces the main identifying conditions (i.e., the preservation of stochastic ordering) and establishes bounds on treatment effects. Section <ref> introduces structural models as sufficient conditions for the identifying conditions presented in the previous section. Section <ref> discusses the computation of the bounds using linear programming and, finally, Section <ref> presents numerical studies. In the Appendix, Section <ref> shows that point identification can be achieved with sufficient (but not infinite) variation of IVs. In the main text we focus on the QTE. An extension to bounding the ATE is discussed in Section <ref>. Section <ref> contains more examples of structural models as sufficient conditions and Section <ref> holds further discussions on linear programming. All proofs are collected in Section <ref>. § KEY CONDITIONS AND BOUNDS ON TREATMENT EFFECTS Let D∈{0,1} be the observed treatment indicator, which represents the endogenous decision of an individual responding to IVs Z. We assume Z is either a vector of binary IVs or a multi-valued IV, which takes L distinct values: Z∈𝒵≡{z_1,...,z_L}. Multi-valued or multiple IVs are common in many observational studies (e.g., natural experiments typically provide more than one instrument) and experimental studies (e.g., randomized control trials where multiple treatment arms are implemented either simultaneously or sequentially).[See <cit.> for a recent survey.] Let Y_1 be the counterfactual outcome of being treated and Y_0 be that of not being treated. They can be either continuously or discretely distributed. The observed outcome Y∈𝒴⊆ℝ satisfies Y=DY_1+(1-D)Y_0. Finally, X∈𝒳⊆ℝ^k denotes other covariates that may be endogenous.Define QTE and ATE for treated and untreated populations. For d∈{0,1} and x∈𝒳, defineQTE_τ(d,x) =Q_Y_1|D,X(τ|d,x)-Q_Y_0|D,X(τ|d,x)for τ∈(0,1) andATE(d,x) =E[Y_1-Y_0|D=d,X=x].These parameters are what researchers and policymakers are potentially interested. The unconditional QTE and ATE can be recovered when these parameters are identified for all d∈{0,1} and x∈𝒳. Throughout the paper, we maintain that the IVs are valid and satisfy the following exclusion restriction.ZFor d∈{0,1}, Z⊥ Y_d|X. §.§ Introducing Key Conditions Now we introduce the key condition that establishes the mapping between observed and counterfactual distributions.S_1For arbitrary non-negative weight vectors (w_1,...,w_L) and (w̃_1,...,w̃_L) that satisfy ∑_ℓ=1^Lw_ℓ=∑_ℓ=1^Lw̃_ℓ=1, if∑_ℓ=1^Lw_ℓP[Y_1≤·|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_1≤·|D=1,Z=z_ℓ,X=x],then∑_ℓ=1^Lw_ℓP[Y_0≤·|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_0≤·|D=1,Z=z_ℓ,X=x].Importantly, note that in (<ref>) the probability P[Y_1≤·|D=1,Z=z_ℓ,X=x] can be obtained from the data as Y_1=Y given D=1. The mapping between observed and counterfactual distributions has been considered in <cit.>, whose insights we share. Suppose that Z⊥(Y_d,D_z)|X additionally holds, where D_z is the counterfactual treatment given Z=z. Under this assumption, each probability term in Condition <ref> satisfies P[Y_d≤·|D=1,Z=z_ℓ,X=x]=P[Y_d≤·|D_z_ℓ=1,X=x]. Note that ∑_ℓ=1^Lw_ℓP[Y_1≤·|D_z_ℓ=1,X=x] is a mixture of Y_d's distributions weighted across different compliance types defined by {D_z_ℓ=1}, and thus can be viewed as a distribution for a hypothetical population with the specific composition of compliance types. Therefore, Condition <ref> posits that the FOSD ordering between the Y_1's distributions of two compliance compositions is preserved between their counterfactual distributions of Y_0. For example, when L=2 and defiers are excluded from possible compliance types (e.g., by <cit.>'s monotonicity assumption), then Condition <ref> simply describes the stochastic ordering between always-takers and compliers. When L≥3, however, there are more compliance types, which composition becomes more complex as illustrated in Section <ref>. Note that Condition <ref> is not an “if and only if” statement. It would be stringent to impose the preservation of ordering to hold in both directions. In fact, such a condition is closely related to the rank similarity condition (<cit.>); see Section <ref> for full details.§.§ Bounds on Treatment Effects Now, we show that Condition <ref> is useful in constructing bounds on F_Y_0|D,X(·|1,x) and subsequently on QTE_τ(1,x). Let p(z_ℓ,x)≡ P[D=1|Z=z_ℓ,X=x] andΓ(x)≡{ (γ_1,...,γ_L)∈ℝ^L:∑_ℓ=1^Lγ_ℓ=0 and ∑_ℓ=1^Lγ_ℓp(z_ℓ,x)=1} . Suppose that Assumption <ref> and Condition <ref> hold. Fix x∈𝒳. For γ≡(γ_1,...,γ_L) and γ̃≡(γ̃_1,...,γ̃_L) in Γ(x), suppose P[Y≤·|D=1,X=x]≤∑_ℓ=1^Lγ_ℓP[Y≤·,D=1|Z=z_ℓ,X=x],∑_ℓ=1^Lγ̃_ℓP[Y≤·,D=1|Z=z_ℓ,X=x]≤ P[Y≤·|D=1,X=x].Then F_Y_0|D,X(·|1,x) is bounded by-∑_ℓ=1^Lγ̃_ℓP[Y≤·,D=0|Z=z_ℓ,X=x]≤ P[Y_0≤·|D=1,X=x]≤-∑_ℓ=1^Lγ_ℓP[Y≤·,D=0|Z=z_ℓ,X=x]In Theorem <ref>, Γ(x) imposes two restrictions on γ: (i) ∑_ℓ=1^Lγ_ℓ=0 and (ii) ∑_ℓ=1^Lγ_ℓp(z_ℓ,x)=1. First, note that the existence of such a sequence requires the relevance of the IV: p(z_ℓ,x)≠ p(z_ℓ',x) for some z_ℓ,z_ℓ'. Second, note that (ii) is a condition implied by either (<ref>) or (<ref>) with y→∞. Restriction (ii) implicitly introduces a scale normalization. That is, for any γ satisfying ∑_ℓ=1^Lγ_ℓp(z_ℓ,x)≠0, we can always rescale it as γ^*=γ/∑_ℓ=1^Lγ_ℓp(z_ℓ,x) so that ∑_ℓ=1^Lγ_ℓp(z_ℓ,x)=1. It can be shown that this normalization does not affect the bounds obtained in (<ref>) and (<ref>). The proof of Theorem <ref> and most of other proofs are contained in the appendix. Note that there can be multiple γ and γ̃ in Γ(x) that satisfy (<ref>) and (<ref>), respectively. Therefore, we can further tighten the bounds as follows.Suppose that Assumption <ref> and Condition <ref> hold. Fix x∈𝒳. Then, F_Y_0|D,X(·|1,x) is upper and lower bounded byF_Y_0|D,X^UB(y|1,x)≡min_γ∈Γ(x):(<ref>) holds-∑_ℓ=1^Lγ_ℓP[Y≤ y,D=0|Z=z_ℓ],F_Y_0|D,X^LB(y|1,x)≡max_γ̃∈Γ(x):(<ref>) holds-∑_ℓ=1^Lγ̃_ℓP[Y≤ y,D=0|Z=z_ℓ]. Theorem <ref> and Corollary <ref> highlight the identifying power of multi-valued IVs. The key step in Theorem <ref> to calculate the bounds is to find γ (resp. γ̃) in Γ(x) that satisfies (<ref>) (resp. (<ref>)), which serves as a rank condition. Note that this condition is verifiable with the data. Corollary <ref> additionally implies that the bounds can be further tightened if one increases the degree of freedom in the feasible set Γ(x) by increasing L, in which case (<ref>)–(<ref>) are more likely to hold. See below and Section <ref> for related discussions.Finally, note thatQTE_τ(1,x)=Q_Y|D,X(τ|1,x)-Q_Y_0|D,X(τ|1,x)and the bounds on the second quantity on the right-hand side can be calculated using the worst case bounds for the conditional quantile (<cit.>, <cit.>):Q_Y_0|D,X^LB(τ|1,x)≤ Q_Y_0|D,X(τ|1,x)≤ Q_Y_0|D,X^UB(τ|1,x),where Q_Y_0|D,X^LB(τ|1,x) and Q_Y_0|D,X^UB(τ|1,x) are the τ-th quantiles of F_Y_0|D,X^LB(·|1,x) and F_Y_0|D,X^UB(·|1,x), respectively. Although the bounds on ATE(1,x)=E[Y|D=1,X=x]-E[Y_0|D=1,X=x] can be calculated based on E[Y_0|D=1,X=x]=∫_0^1Q_Y_0|D,X(τ|1,x)dτ, we present later how the bounds on the ATE(d,x) can be calculated under a weaker condition than Condition <ref>.If we assume the converse of Condition <ref>, we can calculate bounds on the QTE(0,x).S_0For arbitrary non-negative weight vectors (w_1,...,w_L) and (w̃_1,...,w̃_L) that satisfy ∑_ℓ=1^Lw_ℓ=∑_ℓ=1^Lw̃_ℓ=1, if∑_ℓ=1^Lw_ℓP[Y_0≤·|D=0,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_0≤·|D=0,Z=z_ℓ,X=x],then∑_ℓ=1^Lw_ℓP[Y_1≤·|D=0,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_1≤·|D=0,Z=z_ℓ,X=x].Similar to Condition <ref>, Condition <ref> ensures that if the two distributions of Y_0 exhibit an FOSD ordering across different compliance compositions, this ordering is preserved in their corresponding counterfactual Y_1 distributions. In practice, depending on the specific context, Condition <ref>, Condition <ref>, or both may hold; Section <ref> provides some related intuitions. Not surprisingly, we can use Condition <ref> to construct bounds on F_Y_1|D,X(·|0,x) and subsequently on QTE_τ(0,x).Suppose that Assumption <ref> and Condition <ref> hold. Fix x∈𝒳. For γ≡(γ_1,...,γ_L) and γ̃≡(γ̃_1,...,γ̃_L) in Γ(x), suppose P[Y≤·|D=0,X=x]≤∑_ℓ=1^Lγ_ℓP[Y≤·,D=0|Z=z_ℓ,X=x],∑_ℓ=1^Lγ̃_ℓP[Y≤·,D=0|Z=z_ℓ,X=x]≤ P[Y≤·|D=0,X=x].Then F_Y_1|D,X(·|0,x) is bounded by-∑_ℓ=1^Lγ̃_ℓP[Y≤·,D=1|Z=z_ℓ,X=x]≤ P[Y_1≤·|D=0,X=x]≤-∑_ℓ=1^Lγ_ℓP[Y≤·,D=1|Z=z_ℓ,X=x].The proof of this theorem is analogous to that of Theorem <ref>. The bounds on QTE_τ(0,x) can be derived symmetrically as in the case of QTE_τ(1,x) and thus are omitted. Notably, which treatment parameter we can obtain bounds for is determined by which identifying condition we impose (i.e., Condition <ref> or <ref>). In Section <ref>, we investigate this aspect within economic structural models. Finally, in the Appendix, we introduce weaker conditions to bound average treatment effects. §.§ Understanding Key Conditions We further explore Conditions <ref> and <ref> to give additional interpretation and discuss testability. Suppress X=x to simplify our discussions. Under Z⊥(Y_d,D_z), the inequalities for FOSD in Conditions <ref> and <ref> can be rewritten as∑_ℓ=1^Lw_ℓP[Y_d≤ y|D_z_ℓ=1]≤∑_ℓ=1^Lw̃_ℓP[Y_d≤ y|D_z_ℓ=1].Recall, Theorem <ref> relies on the existence of a sequence γ=(γ_1,...,γ_L) satisfying ∑_ℓ=1^Lγ_ℓ=0, ∑_ℓ=1^Lγ_ℓp(z_ℓ,x)=1, and the inequality (<ref>), that is, P[Y≤ y|D=1]≤∑_ℓ=1^Lγ_ℓP[Y≤ y,D=1|Z=z_ℓ] for all y. Note that (<ref>) is a special case of (<ref>) with d=1, which is the “if” part of Condition <ref>. Let p(z)≡(D=1|Z=z). Only for the purpose of this subsection, assume the generalized version of the LATE monotonicity introduced in <cit.>:For ℓ≠ℓ', either D_z_ℓ≥ D_z_ℓ' a.s. or D_z_ℓ≤ D_z_ℓ' a.s. Under (<ref>), {D_z_ℓ=1} in (<ref>) are a mix of individuals who are compliers (C) and always-takers (AT). For Z∈𝒵={z_1,...,z_L}, let (z_ℓ-1,z_ℓ)-compliers be compliers induced by the change of Z from z_ℓ-1 to z_ℓ. When L=3 and (z_1,z_2,z_3)=(0,1,2), for example, {(0,1)-C}={i:D_0,i=0,D_1,i=D_2,i=1} is the set of eager compliers (E-C) and {(1,2)-C}={i:D_0,i=D_1,i=0,D_2,i=1} is the set of reluctant compliers (R-C), following the language of <cit.>. Also, {AT}={i:D_0,i=D_1,i=D_2,i=1} is the set of always-takers. Let p_ℓ for ℓ={2,...,L} is the proportion of (z_ℓ-1,z_ℓ)-compliers and let p_1≡ P[AT]. We show that (<ref>) establishes the FOSD relationship between the mixtures of observed distributions of Y conditional on various always-takers and compliers groups:Suppose (<ref>) holds and Z⊥(Y_d,D_z) and 0<p(z_ℓ)<1 for all ℓ. (i) Then, (<ref>) is equivalent toω_1P[Y_d≤ y|AT]+∑_ℓ=2^Lω_ℓP[Y_d≤ y|(z_ℓ-1,z_ℓ)-C]≤ω̃_1P[Y_d≤ y|AT]+∑_ℓ=2^Lω̃_ℓP[Y_d≤ y|(z_ℓ-1,z_ℓ)-C]for some non-negative ω_ℓ and ω̃_ℓ for ℓ=1,...,L. (ii) Moreover, suppose L≥2 and w_1+p_1∑_ℓ=2^Lw_ℓ/p(z_ℓ)=w̃_1+p_1∑_ℓ=2^Lw̃_ℓ/p(z_ℓ). Then (<ref>) with w≠w̃ can be expressed as∑_ℓ=2^Lω_ℓP[Y_d≤ y|(z_ℓ-1,z_ℓ)-C]≤∑_ℓ=2^Lω̃_ℓP[Y_d≤ y|(z_ℓ-1,z_ℓ)-C]for some non-negative ω_ℓ and ω̃_ℓ for ℓ=2,...,L.To illustrate the intuition of Lemma <ref>(i), consider L=3 and (z_1,z_2,z_3)=(0,1,2). Then, {D_1=1}={D_0=1,D_1=1,D_2=1}∪{D_0=0,D_1=1,D_2=1}={AT}∪{E-C},because {D_0=1,D_1=1,D_2=0}=∅ and {D_0=0,D_1=1,D_2=0}=∅. Also, {D_2=1}={AT}∪{E-C}∪{R-C} and {D_0=1}={AT}. Lemma <ref>(ii) can be used as the basis to test (<ref>) and thus Condition <ref>. The intuition is as follows. With a binary IV, the marginal distributions of Y_1 and Y_0 are identified for compliers (<cit.>). This result holds for any complier group defined by a pair of instrument values, such as {(z_ℓ-1,z_ℓ)-C} in the lemma. Then, when L≥2, we can find vectors w and w̃ in ℝ_+^L that assign zero weights to the distributions for AT and still make (<ref>) a non-trivial inequality where all the associated distributions for compliers are identified for all d=1,0.Motivated from the discussion of this section, we can rewrite Condition <ref> (and all the relevant conditions) solely in terms of compliance types. Let T≡{D(z_1),...,D(z_L)} be a random vector that indicates a particular compliance type with its realized value in {0,1}^L≡𝒯̃. For example, when L=2 (i.e., binary IV), T≡(D(0),D(1))∈{(0,0),(1,0),(0,1),(1,1)}≡𝒯̃. Since D and Z are discrete, T is naturally a discrete random vector. Note that this framework do not rely on any selection models, and therefore T captures all possible compliance types given D and Z. Then Condition <ref> can be rewritten into the following slightly stringent one:Fix x∈𝒳. For arbitrary weight functions w:𝒯̃×𝒳→ℝ_+ and w̃:𝒯̃×𝒳→ℝ_+ such that ∑_t∈𝒯̃w(t,x)=∑_t∈𝒯̃w̃(t,x)=1, if ∑_t∈𝒯̃w(t,x)F_Y_1|T,X(·|t,x)≤∑_t∈𝒯̃w̃(t,x)F_Y_1|T,X(·|t,x),then∑_t∈𝒯̃w(t,x)F_Y_0|T,X(·|t,x)≤∑_t∈𝒯̃w̃(t,x)F_Y_0|T,X(·|t,x).Then, the weighted sum in each inequality can be interpreted as the distribution of Y_d weighted across all compliance types.§ STRUCTURAL MODELS AS SUFFICIENT CONDITIONS We show that Conditions <ref> and <ref> can be justified in a range of nonparametric structural models for the counterfactual outcomes. To this end, it is useful to first present a stronger version of Condition <ref> (labeled as <ref>). This version of the condition is motivated by the discussion in Remark <ref>. To state this condition, we introduce a general model for treatment selection:DAssume thatD =h(Z,X,η),where η∈𝒯 can be an arbitrary vector.Note that Assumption <ref> permits a more general compliance behavior than what a weakly separable model D=1{η≤ h(Z,X)} does (or equivalently, (<ref>) as shown in <cit.>). Although Assumption <ref> is not necessary for our main procedure, it is useful in defining the types of compliance behavior (i.e., treatment selection mechanism) via the unobservable η. Under this assumption, the following condition implies Condition <ref>.[More precisely, it implies the condition in Remark <ref>, which in turn implies Condition <ref>.] Let F_Y_d|η,X(y|t,x)≡ P[Y_d≤ y|η=t,X=x].S_1^*Fix x∈𝒳. For arbitrary weight functions w:𝒯×𝒳→ℝ_+ and w̃:𝒯×𝒳→ℝ_+ such that ∫ w(t,x)dt=∫w̃(t,x)dt=1, if ∫ w(t,x)F_Y_1|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_1|η,X(·|t,x)dt,then∫ w(t,x)F_Y_0|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_0|η,X(·|t,x)dt.Because w(·,x) is non-negative and ∫ w(t,x)dt=1, note that ∫ w(t,x)F_Y_d|η,X(·|t,x)dt is a mixture of conditional CDFs (with w(·,x) being the mixture weight) and thus itself a CDF. In other words, defining a type distribution W_x(t)=∫^tw(η,x)dη, we can write ∫ w(t,x)F_Y_d|η,X(·|t,x)dt=∫ F_Y_d|η,X(·|t,x)dW_x(t).[Since η has arbitrary dimensions, the integral with respect to t is understood to be a multivariate integral.] Therefore, Condition <ref> assumes that the FOSD ordering of Y_1 distributions conditional on η conforming to two different type distributions (W_x(·) and W̃_x(·)) is preserved in the ordering of Y_0 distributions conditional on the corresponding type distributions.The following lemma establishes the sufficiency of Condition <ref> for Condition <ref>.Under Assumption <ref>, Condition <ref> implies Condition <ref>.Symmetrically, Condition <ref> has a corresponding stronger condition, which is omitted.Now, we relate the conditions with the structural models, which provide additional intuitions for the conditions. We present a leading model here and the rest in the Appendix. For arbitrary r.v.'s A and Ã, let Ad=à denote F_A=F_Ã.Model 1. (i) Assumption <ref> holds andY =q(D,X,U_D),where q(d,x,·) is continuous and monotone increasing and U_D=DU_1+(1-D)U_0, (ii) conditional on (η,X,Z), U_dd=U+ξ_d where ξ_d⊥(η,U), (iii) conditional on (X,Z), ξ_0 is (weakly) more or less noisy than ξ_1, that is, ξ_0d=ξ_1+V for some V independent of ξ_1.Note that U is the source of endogeneity in that it allowed to be dependent on η. Model 1(ii)–(iii) implies that U_0d=U_1+V conditional on (η,X,Z). Importantly, Model 1 nests the model in <cit.> as a special case. This can be shown as follows. First, they assume Model 1(i) and, conditional on (X,Z), either rank similarity (F_U_0|η=F_U_1|η) or rank invariance (U_0=U_1).[Note that rank similarity and rank invariance are observationally equivalent under Model 1(i) in that they produce the same distribution of observables (<cit.>).] Then, by taking ξ_d=0 for all d in Model 1(ii), we have U_0d=U_1d=U conditional on (η,X,Z), which proves the claim.Model 1(iii) assumes that the unobservable under the counterfactual status of being treated are more (or less) dispersed than that under the counterfactually untreated status. Although this may seem stringent, it is substantially weaker than rank similarity (or invariance) and can be plausible in various scenarios. Before providing examples of these scenarios, we first establish the connection between Model 1 and Condition <ref> (and thus Condition <ref> by Lemma <ref>).Under Assumptions Z, Model 1 (with ξ_0 being weakly more noisy than ξ_1) implies Condition <ref> and thus Condition <ref>.Analogous to Theorem <ref>, one can readily show that Model 1 with ξ_0 being weakly less noisy than ξ_1 implies Condition <ref>.Now we provide examples that are consistent with Model 1.[Auction]Consider online and offline auctions. Let Y be the bid (which subsequently forms revenue) and D be participating in an auction with different format (D=1 if online and =0 if offline). Let U_dd=U+ξ_d be the valuation of the item where U is the common valuation (correlated with D) and ξ_d is format specific random shocks satisfying ξ_d⊥(η,U). We assume that bidders have limited information on certain features of the auction that affect valuation (e.g., they know the distribution of ξ_d but not its realization). In this example, what would justify var(ξ_0)>var(ξ_1)? It may be the case that, in the offline auction, bidders are more emotionally affected by other bidders, which makes their bids more variable.[Insurance]We are interested in the effect of insurance on health outcomes. Let Y be the health outcome and D be the decision of getting insurance (D=1 being insured). Let U_dd=U+ξ_d be underlying health conditions where U captures health conditions known to participant (and thus correlated with D) while ξ_d is health conditions not fully known a priori and thus random. In this example, var(ξ_0)>var(ξ_1) may hold because insurance by definition ensures a certain level of health conditions.[Vaccination]Similar to Example <ref>, suppose that D is instead getting vaccination (of an established vaccine). Again, U_dd=U+ξ_d is health conditions where U captures conditions known to participant (and correlated with D) and ξ_d is vaccination-status-specific health conditions, which are not fully known a priori. Then, similarly as before var(ξ_0)>var(ξ_1) may hold because, when not vaccinated, one is exposed to the risk of a serious illness, while vaccination ensures a certain level of immunity.The scenarios in Examples <ref>–<ref> justify Condition <ref> via Theorem <ref>. Then, under Condition <ref>, Theorem <ref> and Corollary <ref> yield bounds on QTE_τ(1,x), the effects of treatment for those who take the treatment. The final example illustrates the converse case.[Medical Trial]In contrast to Example <ref>, suppose the treatment itself is risky. That is, let D be participating in a frontier medical trial (D=1 being participation). In this case, var(ξ_0)<var(ξ_1) is more plausible because, with a newly developed medicine, there is the high risk of unknown side effects.The scenario in Example <ref> justifies Condition <ref>, under which bounds on QTE_τ(0,x), the effects of treatment for those who abstain from it, can be obtained.Model 1 and these examples show how a certain treatment parameter may be more relevant for policy than others depending on the plausibility of assumptions. Consider the problem of a policymaker. Assume that the policymaker concerns risk-averse individuals, which are typically the majority. For this policymaker, a candidate policy would aim at providing “insurance,” which can be either literally insurance or policies that serve as insurance (e.g., vaccination, subsidies). Therefore, she would be interested in understanding the treatment effects for the target individuals that are risk-averse. Our procedure provides a statistical tool for such a policymaker. That is, under Model 1, our procedure has the ability to bound the treatment effects for individuals with D=d such that var(ξ_d)<var(ξ_1-d). This is a unique feature of our setting: the plausibility of assumptions dictates the parameters of interest, which then can be terms as assumption-driven treatment parameters.A remaining question one might have is as follows. How much Condition <ref> has to be strengthened to be equivalent to rank similarity? To answer this question, recall that Condition <ref> is stronger than Condition <ref> (by Lemma <ref>). We strengthen Condition <ref> further by making it an “if and only if” condition:S^*Fix x∈𝒳. For arbitrary weight functions w:𝒯×𝒳→ℝ_+ and w̃:𝒯×𝒳→ℝ_+ such that ∫ w(t,x)dt=∫w̃(t,x)=1, it holds that∫ w(t,x)F_Y_1|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_1|η,X(·|t,x)dtif and only if∫ w(t,x)F_Y_0|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_0|η,X(·|t,x)dt.It turns out that we can establish the following result.Model 1(i) with F_U_0|η,X,Z=F_U_1|η,X,Z (i.e., rank similarity) implies Condition <ref>.This theorem highlights the stringency of rank similarity relative to Condition <ref>, which is used in our bound analysis. The proof is trivial so omitted. It is worth noting that the converse of Theorem <ref> is not true. Here is a counter-example for the converse statement.Assume Model 1(i) andF_Y_0|η,X,Z(·|t,x,z) =λ(·,x)F_Y_1|η,X,Z(ψ(·,x)|t,x,z)for every t∈𝒯, x∈𝒳 and z∈𝒵, where ψ(·,x):𝒴→𝒴, a one-to-one and onto mapping, is strictly increasing, and λ(·,x):𝒴→ℝ_+ is consistent with F_Y_d|η,X,Z being a proper CDF.This rank linearity implies Condition <ref>, which is trivial to show. However, rank linearity is weaker than rank similarity as the latter is a special case of the former. To see this, conditional on Z=z (and suppressing X), (<ref>) with Model 1(i) yields F_U_0|η(q^-1(0,y)|t)=λ(y)F_U_1|η(q^-1(1,ψ(y))|t). Then, by choosing λ(y)=1 and ψ(y)=ϕ(y)≡ q(1,q^-1(0,y)) (i.e., the counterfactual mapping (<cit.>)), we have F_U_0|η(·|t)=F_U_1|η(·|t).[Alternatively, rank similarity can be equivalently stated as F_Y_0|η(y|t)=F_Y_1|η(ϕ(y)|t) (where ϕ(y) is strictly increasing), which can be derived from (<ref>) by choosing λ(y)=1 and ψ(y)=ϕ(y).] In general, while the ranks between Y_0 and Y_1 should have the same distribution under rank similarity, their distributions can be different under rank linearity because of the multiplying term λ(·) in F_U_0|η(u|t)=λ(q(0,u))F_U_1|η(u|t). However, note that the the difference cannot be entirely arbitrary as λ(·) does not depend on t, and thus rank linearity still poses substantive restrictions.Interestingly, rank linearity is equivalent to Condition <ref>. The following theorem is one of the main contributions of this paper. Suppress (Z,X) for simplicity.Suppose for any CDF F_1(·) supported on ℝ, there always exists a function c:𝒯→ℝ such that F_d(·)=∫ c(t)F_Y_d|η(·|t)dt.Then Condition <ref> holds if and only if there exits some ψ(·) that is strictly increasing and λ(·)>0 such that F_Y_0|η(·|t)=λ(·)F_Y_1|η(ψ(·)|t)for t∈𝒯. We prove this equivalence in the Appendix. The proof with continuous Y_d is more involved than that with discrete Y_d; we recommend that the interested reader reads the latter first. The condition (<ref>) is only introduced in this theorem to establish the relationship between rank linearity (and hence rank similarity) and the range of identifying conditions of this paper, and it is not necessary for our bound analysis. This condition would be violated when there is no endogeneity (i.e., Y_d⊥η), which is not our focus.Condition <ref> is crucial in bounding QTE_τ(x)=Q_Y_1|X(τ|x)-Q_Y_0|X(τ|x) unconditional with respect to D=d. The “only if” part (i.e., Condition <ref>) will bound Q_Y_0|D=1(τ) and thus Q_Y_0(τ) by Theorem <ref>, while the “if” part (i.e., Condition <ref>) will bound Q_Y_1|D=0(τ) and thus Q_Y_1(τ) by the symmetric version of Theorem <ref>. The fact that Condition <ref> is weaker than rank similarity illustrates the importance of rank similarity in the identification of the QTE and ATE.It is immediate from Lemma <ref>(ii) that Condition <ref> can be tested from the data when L≥2 and under the LATE monotonicity assumption. Given the established connection between Condition <ref> and rank similarity (Theorem <ref>), when Condition <ref> is refuted from the data, rank similarity can be refuted. This result relates to the testability of rank similarity under LATE monotonicity (<cit.>, <cit.>).Continuing the discussion in Remark <ref>, F_U_1|T,Z=F_U_0|T,Z (where T≡(D(0),D(1)) and X is suppressed) can be viewed as an alternative rank similarity assumption. Because σ(T)⊂σ(η) where σ(A) is a σ-field generated by a random vector A, F_U_1|η,Z=F_U_0|η,Z implies F_U_1|T,Z=F_U_0|T,Z. Then <cit.>'s main testable implication ((2.6) in Theorem 1 of their paper) can be equally derived under F_U_1|T,Z=F_U_0|T,Z, which clarifies the role of selection mechanism in their analysis. To see this, let t≡(t_0,t_1) be the realization of T≡(D(0),D(1)) and assume Model 1(i). We haveP[Y≤ q(D,τ)|Z=z] =P[q(D,U_D)≤ q(D,τ)|Z=z] =P[U_D≤τ|Z=z] =∑_t∈𝒯̃P[U_D(z)≤τ|Z=z,T=t]P[T=t|Z=z]but∑_t∈𝒯̃P[U_D(z)≤τ|Z=z,T=t]P[T=t|Z=z] =∑_t∈𝒯̃P[U_t_z≤τ|Z=z,T=t]P[T=t|Z=z] =∑_t∈𝒯̃P[U_0≤τ|Z=z,T=t]P[T=t|Z=z] =P[U_0≤τ|Z=z] =τ,where F_U_1|T,Z=F_U_0|T,Z is used in the second equality and U_0⊥ Z (imposed in their paper) is used in the last equality.Note that a slightly weaker version of Condition <ref> can be stated by replacing η with T and the integral with a summation. Then, analogous to Theorem <ref>, one can readily show that F_U_1|T,Z=F_U_0|T,Z implies such a condition.§ SYSTEMATIC CALCULATION OF BOUNDS In Theorem <ref>, γ is required to satisfy a set of linear inequality constraints, i.e., (<ref>) (respectively, (<ref>)), and each feasible γ establishes an upper bound (respectively, lower bound) on F_Y_1|D,X(·|0,x). Therefore, it is intuitive to employ optimization methods to calculate these bounds, as detailed in Corollary <ref>. For simplicity, our subsequent discussion will focus solely on the upper bound, omitting covariates X for brevity. §.§ Semi-Infinite Programming To simplify notation, let p(y,d)≡(p(y,d|z_1),...,p(y,d|z_L))' where p(y,d|z_ℓ)≡ P[Y≤ y,D=d|Z=z_ℓ] and p(y|d)≡ P[Y≤ y|D=d]. Also, let 1≡(1,...,1)' and p≡(p(z_1),...,p(z_L))' with p(z)≡ P[D=1|Z=z] so that Γ={γ:γ'[[ 1 p ]]=[[ 0 1 ]]}⊂ℝ^L.Consider the following linear semi-infinite programming problem for the upper bound on P[Y_0≤y̅|D=1]:UB(y̅) =min_γ∈Γ_p-p(y̅,0)'γs.t.p(y,1)'γ≥ p(y|1),∀ y∈𝒴Note that the existence of γ satisfying condition (<ref>) guarantees that the feasible set is non-empty. Also note that this condition is allowed to satisfy only almost everywhere (a.e.), which we suppress for simplicity. This program is infeasible to solve in practice as there are infinitely many constraints. We propose two approaches to approximate it with a linear program (LP). §.§ Linear Program with Randomized Constraints One approach to the semi-infinite program (<ref>)–(<ref>) is to approximate (<ref>) by p(Ỹ_m,1)'γ≥ p(Ỹ_m|1),a.s. for m=1,⋯,s_n,where {Ỹ_m:m=1,⋯,s_n} is an i.i.d. simulated sample as is done in the literature (e.g., <cit.>). An obvious candidate of this sample would be {Y_i}_i=1^n with s_n=n. Consider a sampled LP of the following: UB_n(y̅) =min_γ∈Γ-p(y̅,0)'γs.t.p(Y_i,1)'γ≥ p(Y_i|1).∀ i=1,...,nIn Section <ref> of the Appendix, we show that the probability of violating the original constraints (<ref>) by using (<ref>) can be bounded by O(1/n).§.§ Dual Program and Sieve Approximation Another approach to the semi-infinite program (<ref>)–(<ref>) is to invoke its dual and approximate the Lagrangian measure using sieve. With the constraint p(·|1)-p(·,1)'γ≤0, the Lagrangian for (<ref>)–(<ref>) can be written as ℒ(y̅,γ,Λ,λ) =-p(y̅,0)'γ+∫_𝒴[p(y|1)-p(y,1)'γ]dΛ(y)+λ'([[ 1 p ]]'γ-[[ 0 1 ]]') =∫_𝒴p(y|1)dΛ(y)-[[ 0 1 ]]λ+(λ'[[ 1 p ]]'-∫_𝒴p(y,1)'dΛ(y)-p(y̅,0)')γ,where Λ is a non-negative (not necessarily probability) measure (i.e., Λ≽0) that assigns weights to binding constraints. Moreover, let UB(y̅) =min_γ∈ℝ^Lsup_Λ≽0,λ∈ℝ^2ℒ(y̅,γ,Λ,λ). Then, we have the following dual problem:The dual problem of the primal problem of (<ref>)–(<ref>) is given byUB(y̅) =sup_Λ≽0,λ∈ℝ^2∫_𝒴p(y|1)dΛ(y)-[[ 0 1 ]]λs.t.[[ 1 p ]]λ-∫_𝒴p(y,1)dΛ(y)-p(y̅,0)=0.Note that (<ref>) has a finite number of constraints (i.e., L constraints). It is trivial to show weak duality, UB(y̅)≤ UB(y̅).[This is because, by (<ref>)–(<ref>) and (<ref>)–(<ref>), we have-p(y̅,0)'γ={∫_𝒴p(y,1)dΛ(y)-[[ 1 p ]]λ} 'γ=∫_𝒴p(y,1)'γ dΛ(y)-λ'[[ 1 p ]]'γ≥∫_𝒴p(y|1)dΛ(y)-λ'[[ 0 1 ]]'.] Strong duality also holds because of the structure of the problem (i.e., linearity in γ, continuity of p(·,d) and p(·|d)). We establish this in the following theorem.C𝒴 is compact.Suppose Assumption <ref> holds and there is γ^*∈{y:p(y,1)'γ≥ p(y|1)} such that p(y,1)'γ^*>p(y|1). Then, if the primal solution UB(y̅) is finite, then UB(y̅)=UB(y̅).Note that Λ(y) is smooth as the feasible set of the primal problem is smooth due to the smoothness of p(y|d) and p(y,d), which are CDFs. This motivates us to use sieve approximation for Λ(y) to turn the dual into a linear programming problem. The smoothness class for Λ(y) will be determined by the smoothness class of CDFs. Let 𝒴 is normalized to be [0,1] and λ(y)≡ dΛ(y)/dy that satisfies ∫_𝒴λ(y)dy=1 and λ(y)≥0 for all y∈𝒴. Consider the following sieve approximation:λ(y)≈∑_j=1^Jθ_jb_j(y),where b_j(y)≡ b_j,J(y)≡([ J; j ])y^j(1-y)^J-j is a Bernstein basis function. Then, the LP can be written asUB_J(y̅) =max_θ∈ℝ_+^J,λ∈ℝ^2∑_j=1^Jθ_jb_1,j-[[ 0 1 ]]λs.t.[[ 1 p ]]λ-∑_j=1^Jθ_jb_1,j-p(y̅,0)=0,1/J+1∑_j=1^Jθ_j=1,or equivalently,UB_J(y̅) =max_θ∈ℝ_+^J,λ∈ℝ^2θ'b_1-[[ 0 1 ]]λs.t.[[ 1 p ]]λ-B_1θ-p(y̅,0)=0,1'θ-J-1=0,where θ≡(θ_1,...,θ_J)', b_d≡(b_d,1,...,b_d,J)' with b_d,j≡∫_𝒴b_j(y)p(y|d)dy, b_d,j≡(b_d,j,1,...,b_d,j,L)' with b_d,j,ℓ≡∫_𝒴b_j(y)p(y,d|z_ℓ)dy, and B_d≡[[ b_d,1 ⋯ b_d,J ]] is an L× J matrix, and by using ∫_𝒴b_j(y)dy=1/J+1 for all j. Note that the nonnegativity restriction on θ is imposed to reflect that λ is a nonnegative measure. Using Bernstein polynomials to approximate infinite-dimensional decision variables is also used in <cit.>.The LP (<ref>)–(<ref>) may be more stable than the LP (<ref>)–(<ref>). In terms of dual, the latter approach is equivalent to having ∑_i=1^np(Y_i|1)λ_i as an approximation for ∫_𝒴p(y|1)λ(y)dy. This can be viewed as a crude local approximation that involves a uniform kernel. § NUMERICAL STUDIES To illustrate the importance of multiple IVs and the informativeness of resulting bounds, we conduct numerical exercises. We generate the data so that they are consistent with Model 1 and hence satisfy Condition <ref>. The variables (Y,D,Z) are generated in the following fashion: * Y_d=q(d,U_d)=1-d+(d+1)U_d for 𝒴=ℝ, that is, Y_1=2U_1 and Y_0=1+U_0* (U,η)∼ BVN((0,0)',Σ)* V∼ N(0,σ_V^2) and ξ_1∼ N(0,σ_ξ^2)* ξ_0=ξ_1+V* U_d=U+ξ_d* Z∼ Bin(L-1,p)/(L-1)∈[0,1] with L∈{2,3,4,5,6,7,8}* D=1{π_0+π_1Z≥η}* Y=DY_1+(1-D)Y_0Here, Z is normalized so that the endpoints of the support are invariant regardless of the value of L. This is intended to understand the role of the number of values Z takes while fixing the role of instrument strength. Figures <ref>–<ref> presents the bounds on [Y_0≤ y|D=1] while varying L. The bounds are calculated using the approach proposed in Section <ref>. We only report L∈{2,5,6} for succinctness. In these figures, the black solid line indicates the true value of [Y_0≤ y|D=1] and the red and blue crosses depict the upper and lower bounds. Although the upper bound is a trivial upper bound for the CDF when L=2, it quickly becomes informative as L increases beyond 5. To put this in a context, this corresponds to the number of instrument values that three binary IVs can easily surpass or a single continuous IV.§ POINT IDENTIFICATION Point identification of QTE_τ(d,x) and ATE(d,x) can be achieved as long as the stochastic dominance ordering is preserved (i.e., Condition <ref> or <ref>) and instruments have sufficient variation in a specific sense. As is clear below, however, we do not require p(z)→1 or 0 (i.e., instruments with large support). In this sense, our approach to point identification complements the approach of identification at infinity (e.g., <cit.>). To see this, consider the following theorem.Suppose that Assumption <ref> and Condition <ref> hold. Fix x∈𝒳. For γ≡(γ_1,...,γ_L) in Γ(x), supposeP[Y≤·|D=1,X=x]=∑_ℓ=1^Lγ_ℓP[Y≤·,D=1|Z=z_ℓ,X=x].Then F_Y_0|D,X(·|1,x) is identified asP[Y_0≤·|D=1,X=x] =-∑_ℓ=1^Lγ_ℓP[Y≤·,D=0|Z=z_ℓ,X=x]The key for this point identification result is that there exists γ such that (<ref>) holds, which is a stronger requirement than the inequality version (<ref>). The equation (<ref>) is more likely to hold when L is large, that is, when instruments take more values. In particular, when L→∞ (e.g., continuous Z), we may view that P[Y≤ y|D=1,X=x] is approximated asP[Y≤ y|D=1,X=x] =lim_L→∞∑_ℓ=1^Lγ_ℓ,Lp(z_ℓ,x)P[Y≤ y|D=1,Z=z_ℓ,X=x],where ∑_ℓ=1^Lγ_ℓ,L=0. Note that, although this does not demand an infinite support for Z, it implicitly assumes that Z sufficiently influences the distribution of Y conditional on (D,X)=(1,x) in a way that the resulting functions, P[Y≤ y|D=1,Z=z_ℓ,X=x], generate P[Y≤ y|D=1,X=x]. Importantly, whether this is possible or not can be confirmed from the data.Given Theorem <ref>, we identify QTE_τ(1,x)=Q_Y|D,X(τ|1,x)-Q_Y_0|D,X(τ|1,x) where Q_Y_0|D,X(τ|1,x) is a solution to τ=-∑_ℓ=1^Lγ_ℓP[Y≤·,D=0|Z=z_ℓ,X=x]. Similarly, under Condition <ref>, we can identify F_Y_0|D,X(·|1,x) and thus QTE_τ(0,x). We omit this result for succinctness.It is worth comparing the point identification result with that in <cit.>. The latter point identifies QTE_τ(x) with a binary instrument by assuming rank similarity. The result of this section suggests that the identification of QTE_τ(x) can alternatively be achieved when Conditions <ref> and <ref> both hold and the IVs satisfy (<ref>). To see the connection to rank similarity, note that rank similarity implies Condition <ref> (by Theorem <ref>), but the latter implies Conditions <ref> and <ref> that identify QTE_τ(1,x) and QTE_τ(0,x), respectively, and thus QTE_τ(x) jointly. In this way, the two approaches enjoy different levels of the trade-off between restrictions on the heterogeneity and exogenous variation.§ CONDITIONS FOR AVERAGE TREATMENT EFFECTS To calculate bounds on ATE(1,x) and ATE(0,x), we introduce conditions that are weaker that Conditions <ref> and <ref>.S_1^'For arbitrary non-negative weight vectors (w_1,...,w_L) and (w̃_1,...,w̃_L) that satisfy ∑_ℓ=1^Lw_ℓ=∑_ℓ=1^Lw̃_ℓ=1, if∑_ℓ=1^Lw_ℓP[Y_1≤·|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_1≤·|D=1,Z=z_ℓ,X=x],then∑_ℓ=1^Lw_ℓE[Y_0|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓE[Y_0|D=1,Z=z_ℓ,X=x]. Condition <ref> can be used to bound the ATE(1,x). An analogous condition can be imposed to bound ATE(0,x).S_0^'For arbitrary non-negative weight vectors (w_1,...,w_L) and (w̃_1,...,w̃_L) that satisfy ∑_ℓ=1^Lw_ℓ=∑_ℓ=1^Lw̃_ℓ=1, if∑_ℓ=1^Lw_ℓP[Y_0≤·|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓP[Y_0≤·|D=1,Z=z_ℓ,X=x],then∑_ℓ=1^Lw_ℓE[Y_1|D=1,Z=z_ℓ,X=x]≤∑_ℓ=1^Lw̃_ℓE[Y_1|D=1,Z=z_ℓ,X=x]. § OTHER STRUCTURAL MODELS AS SUFFICIENT CONDITIONS We present two more structural models that are not nested to Model 1 in the text. Model1(i) are maintained in these models, that is, Y=q(D,X,U_D) where q(d,x,·) is continuous and monotone increasing and D=h(Z,X,η).Model 2. (ii) U_0d=ϕ(U_1,V) conditional on (η,X) where V⊥(U_1,η)|X and ϕ(·,v) is strictly increasing for all v.Model 2(ii) defines that U_0 is “noisier” than U_1. Therefore, Model 2 is weaker than the model in <cit.>. Model 2 and Model 1 are not nested because, in U_0=U_1+V of Model 1, V is not independent of U_1. We show below that Model 2 implies Condition <ref>. Interestingly, Model 2(ii) with U_0=ϕ(U_1,V) (instead of “d=”) is a generalization of the definition that U_0 is “noisier” than U_1 if U_0=U_1+V with U_1⊥ V in <cit.>.Model 3. (ii)U_0d=max{ϕ(U_1),V} conditional on (η,X) where V⊥(U_1,η)|X and ϕ(·) is strictly increasing.We show below that Model 3 implies rank linearity. Model 3 can alternatively be defined as follows: Y_0d=max{ϕ(Y_1),V} conditional on (η,X) where V⊥(Y_1,η)|X and ϕ(·) is strictly increasing. Then, this model also implies rank linearity with ψ(·)=ϕ^-1(·) because[Y_0≤ y|η,X] =[ϕ(Y_1)≤ y,V≤ y|η,X]=[Y_1≤ϕ^-1(y)|η,X][V≤ y|X].This model provides another interpretation of an insurance policy (D=1) as Y_1=max{Y_0,V} guarantees at least Y_0. Models 2 and 3 are not nested.(i) Model 2 implies Condition <ref>; (ii) Model 3 implies rank linearity.The proof of this lemma is contained in Section <ref>.§ BOUNDING VIOLATION PROBABILITY IN LINEAR PROGRAM WITH RANDOMIZED CONSTRAINTS Let h(γ,y)≡ p(y|1)-p(y,1)'γ. Following <cit.>, define a violation probability and a robustly feasible solution. Let γ∈Γ be a candidate solution for (<ref>)–(<ref>). The probability of violation of γ is defined as V(γ) =ℙ{Y∈𝒴:h(γ,Y)>0},where {Y∈𝒴:h(γ,Y)>0} is assumed to be measurable. Note that V(γ^*)=0 where γ^* is the solution to (<ref>)–(<ref>). For ϵ∈[0,1], γ∈Γ is an ϵ-level robustly feasible solution if V(γ)≤ϵ. Then, we can show that the violation probability at the solution, denoted as γ̅_n, to (<ref>)–(<ref>) is on average bounded by 1/n. Let γ̅_n be the solution to (<ref>)–(<ref>). Then, 𝔼_P^n[V(γ̅_n)]≤1/n+1,where P^n is the probability measure in the space 𝒴^n of the multi-sample extraction Y_1,...,Y_n.Fix ϵ∈[0,1] and β∈[0,1] and let n≥1/ϵβ-1.Then, with probability no smaller than 1-β, the sampled LP (<ref>)–(<ref>) returns an optimal solution γ̂_n which is ϵ-level robustly feasible. The above results implicitly assume a particular rule of tie-breaking when there are multiple solutions in the sampled LP (see Theorem 3 in <cit.>). There is also discussions on no solution in the paper.§ PROOFS§.§ Proof of Lemma <ref> Let p(z,x)≡ P[D=1|Z=z,X=x] and let H(z,x)≡{η:h(z,x,η)=1} be a level set. Then,∑_ℓw_ℓP[Y_1≤ y|D=1,Z=z_ℓ,X=x] =∑_ℓw_ℓP[Y_1≤ y|η∈ H(z_ℓ,x),X=x] =∫∑_ℓw_ℓ1[t∈ H(z_ℓ,x)]/p(z_ℓ,x)P[Y_1≤ y|η=t,X=x]dt.Take w(t,x)=∑_ℓw_ℓ1[t∈ H(z_ℓ,x)]/p(z_ℓ,x). Then, w(t,x) satisfies∫∑_ℓw_ℓ1[t∈ H(z_ℓ,x)]/p(z_ℓ,x)dt =1.The same argument applies to w̃ and w̃(t,x), and also for the distribution of Y_0. □ §.§ Proof of Theorem <ref> We suppress X for simplicity and prove the upper bound; the lower bound can be analogously derived. Without loss of generality, for some ℓ^*≤ L, let γ_ℓ≤0 for ℓ≤ℓ^* and γ_ℓ>0 for ℓ>ℓ^*. Let q(z_ℓ)≡ P[Z=z_ℓ|D=1]. Then, (<ref>) can be rewritten as∑_ℓ=1^Lq(z_ℓ)× P[Y≤ y|D=1,Z=z_ℓ]-∑_ℓ=1^ℓ^*γ_ℓp(z_ℓ)× P[Y≤ y|D=1,Z=z_ℓ] ≤∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)× P[Y≤ y|D=1,Z=z_ℓ].Let a≡1-∑_ℓ=1^ℓ^*γ_ℓp(z_ℓ). By definition and that ∑_ℓ=1^Lγ_ℓp(z_ℓ)=1, we have a=∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ). Therefore, we have ∑_ℓ=1^ℓ^*q(z_ℓ)-γ_ℓp(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ]+∑_ℓ=ℓ^*+1^Lq(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ] ≤∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ],where ∑_ℓ=1^ℓ^*q(z_ℓ)-γ_ℓp(z_ℓ)/a+∑_ℓ=ℓ^*+1^Lq(z_ℓ)/a=1 and ∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)/a=1. Therefore, by Condition <ref>, we have ∑_ℓ=1^kq(z_ℓ)-γ_ℓp(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ]+∑_ℓ=ℓ^*+1^Lq(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ] ≤∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ].Equivalently, we haveP[Y_0≤ y|D=1]≤ ∑_ℓ=1^Lγ_ℓ× P[Y_0≤ y,D=1|Z=z_ℓ] =∑_ℓ=1^Lγ_ℓ×{ P[Y_0≤ y|Z=z_ℓ]-P[Y_0≤ y,D=0|Z=z_ℓ]} =∑_ℓ=1^Lγ_ℓP[Y_0≤ y|Z=z_ℓ]-∑_ℓ=1^Lγ_ℓ× P[Y_0≤ y,D=0|Z=z_ℓ] = P[Y_0≤ y]×∑_ℓ=1^Lγ_ℓ-∑_ℓ=1^Lγ_ℓ× P[Y≤ y,D=0|Z=z_ℓ] = -∑_ℓ=1^Lγ_ℓ× P[Y≤ y,D=0|Z=z_ℓ],where the last equality is by ∑_ℓ=1^Lγ_ℓ=0. □ §.§ Proof of Lemma <ref> Note that P[Y_d≤ y|D_z_1=1]=P[Y_d≤ y|AT] and, for ℓ={2,...,L},P[Y_d≤ y|D_z_ℓ=1] =1/p(z_ℓ)P[Y_d≤ y,{AT}∪⋃_ℓ'=2^ℓ{(z_ℓ'-1,z_ℓ')-C}]by Assumption <ref> and p(z_ℓ)=P[D_z_ℓ=1]. Then, in (<ref>), ∑_ℓ=1^Lw_ℓP[Y_d≤ y|D_z_ℓ=1] =w_1P[Y_d≤ y|AT]+∑_ℓ=2^Lw_ℓ/p(z_ℓ)(p_1P[Y_d≤ y|AT]+∑_ℓ'=2^ℓp_ℓ'P[Y_d≤ y|(z_ℓ'-1,z_ℓ')-C])and similarly for the right-hand side of (<ref>). This proves (i). To remove the distributions for AT in the expressions, we setw_1+p_1∑_ℓ=2^Lw_ℓ/p(z_ℓ)=w̃_1+p_1∑_ℓ=2^Lw̃_ℓ/p(z_ℓ).Then, note that when L≥2, w≠w̃ even if w and w̃ satisfy (<ref>). Therefore, the resulting (<ref>) is the dominance between the two distinct weight sums of P[Y_d≤ y|(z_ℓ'-1,z_ℓ')-C]'s:∑_ℓ=2^Lw_ℓ/∑_ℓ'=1^ℓp_ℓ'∑_ℓ'=2^ℓp_ℓ'P[Y_d≤ y|(z_ℓ'-1,z_ℓ')-C]≤∑_ℓ=2^Lw̃_ℓ/∑_ℓ'=1^ℓp_ℓ'∑_ℓ'=2^ℓp_ℓ'P[Y_d≤ y|(z_ℓ'-1,z_ℓ')-C],which can be simplified as (<ref>) in (ii). □ §.§ Proof of Theorem <ref> We suppress X for simplicity. For an arbitrary r.v. A, let F_A^w(·)≡∫ w(t)F_A|η(·|t)dt, which itself is a CDF. By (<ref>) in Model 1(i), F_Y_d^w≤ F_Y_d^w̃ if and only if F_U_d^w≤ F_U_d^w̃ . So it suffices to show that, if F_U_1^w≤ F_U_1^w̃, then F_U_0^w≤ F_U_0^w̃.Let G(·) be an arbitrary monotone increasing function and g(·)≡ G'(·). Note that ∫ GdF_U_0^w-∫ GdF_U_0^w̃=∫[∫w̃(t)F_U_0|η(u|t)dt-∫ w(t)F_U_0|η(u|t)dt]g(u)du =∫[∫w̃(t)∫ F_U|η(u-s|t)f_ξ_0(s)dsdt-∫ w(t)∫ F_U|η(u-s|t)f_ξ_0(s)dsdt]g(u)du =∫∫∫[w̃(t)-w(t)]F_U|η(u|t)f_ξ_0(s)g(u+s)dudsdt,where the first eq. is due to the integration by part, the second eq. is by F_U_d|η(u|t)=∫ F_U|η(u-s|t)f_ξ_0|η(s|t)ds=∫ F_U|η(u-s|t)f_ξ_0(s)ds under Model 1(ii), and the last eq. is by change of variables. By Model 1(iii), f_ξ_0(s)=∫ f_ξ_1(s-v)f_V(v)dv=∫ f_ξ_1(v)f_V(s-v)dv where f_A(·) is the PDF of an arbitrary r.v. A. Therefore,∫ GdF_U_0^w-∫ GdF_U_0^w̃ =∫∫∫[w̃(t)-w(t)]F_U|η(u|t)∫ f_ξ_1(v)f_V(s-v)g(u+s)dvdudsdt =∫∫[w̃(t)-w(t)]F_U|η(u|t)∫ f_ξ_1(v)[∫ f_V(s)g(u+s+v)ds]dvdudt.Let ψ(s)≡∫ f_V(t)g(t+s)dt. By definition, ψ≥0 since g≥0. Therefore, ∫ GdF_U_0^w-∫ GdF_U_0^w̃ =∫∫[w̃(t)-w(t)]F_U|η(u|t)∫ f_ξ_1(v)ψ(u+v)dvdudt =∫∫[w̃(t)-w(t)]∫ F_U|η(u-v|t)f_ξ_1(v)dvψ(u)dudt =∫∫[w̃(t)-w(t)]∫ F_U_1|η(u|t)ψ(u)dudt =∫[∫w̃(t)F_U_1|η(u|t)dt-∫ w(t)F_U_1|η(u|t)dt]ψ(u)du≥0,where the last ineq. is by F_U_1^w≤ F_U_1^w̃. Because G(·) is arbitrary, then F_U_0^w is first order stochastic dominant over F_U_0^w̃. □ §.§ Proof of Theorem <ref>: Equivalence Between Rank Linearity and Condition <ref> The “if” part is trivial. We prove “only if” part. Suppress (Z,X) for simplicity. Suppose Condition <ref> holds. Let 𝒴_∞≡{y_k∈ℝ:k=1,⋯,∞} be a sequence that is dense on ℝ. Denote 𝒴_n≡{y_k∈ℝ:k=1,⋯,n}. Because 𝒴_∞ is dense in ℝ and CDFs are right-continuous, it suffices to show the existence of λ(·) and ψ(·) on 𝒴_∞ such that F_Y_0|η(ψ(y)|t)=λ(y)F_Y_1|η(y|t)holds for all t∈𝒯 and y∈𝒴_∞.Fix n∈ℕ. Let G_1,k:ℝ→{0,1} be a simple function defined as G_1,k(·)≡1{y_k≤·} for k=1,⋯,n. By the full rank condition (<ref>), for each 1≤ k≤ n, there exists a function c_k:𝒯→ℝ such that G_1,k(·)=∫ c_k(t)F_Y_1|η(·|t)dt.Define G_0,k:ℝ→[0,1] as G_0,k(·)≡∫ c_k(t)F_Y_0|η(·|t)dt.Note that G_0,k is a proper CDF. Now, for any vectors π≡(π_1,⋯,π_n) and π̃≡(π̃_1,⋯,π̃_n) such that ∑_k=1^nπ_k=∑_k=1^nπ̃_k=1, suppose∑_k=1^nπ_kG_1,k(·)≤∑_k=1^nπ̃_kG_1,k(·).It follows that∫ b_n(t)F_Y_1|η(·|t)dt≤∫b̃_n(t)F_Y_1|η(·|t)dt,where b_n(t)≡∑_k=1^nπ_kc_k(t) and b̃_n(t)≡∑_k=1^nπ̃_kc_k(t). Let b_n^+(t)=max{b_n(t),0} and b_n^-(t)=min{b_n(t),0} and similarly define b̃_n^+(t) and b̃_n^-(t). Then, the above inequality can be written as∫{b_n^+(t)-b̃_n^-(t)}F_Y_1|η(·|t)dt≤∫{b̃_n^+(t)-b_n^-(t)}F_Y_1|η(·|t)dt,where the resulting weight functions on both sides are non-negative. Then, by Condition <ref>, we have ∑_k=1^nπ_kG_0,k(·)≤∑_k=1^nπ̃_kG_0,k(·)By a similar argument, the converse is also true and thus we have∑_k=1^nπ_kG_1,k(·)≤∑_k=1^nπ̃_kG_1,k(·).if and only if ∑_k=1^nπ_kG_0,k(·)≤∑_k=1^nπ̃_kG_0,k(·)for any non-negative weights π and π̃. Therefore, it follows that ∑_k=1^nδ_kG_1,k(·)≤0 if and only if ∑_k=1^nδ_kG_0,k(·)≤0for any n-dimensional vector δ≡(δ_1,⋯,δ_n) that satisfies ∑_k=1^nδ_k=0.For d∈{0,1}, defineΔ_d^G ≡{δ∈ℝ^n:∑_k=1^nδ_kG_d,k(y)≤0∀ y∈ℝ;∑_k=1^nδ_k=0} .Note that {(G_1,1(y),⋯,G_1,n(y)):y∈ℝ}={(G_1,1(y),⋯,G_1,n(y)):y∈𝒴_n} by definition. Therefore, Δ_1^G is a finite cone and its dimension is n-1. Define the polar cone of Δ_d^G as Δ_d^G*≡{G_d∈ℝ^n:G_d'δ≤0,∀δ∈Δ_d^G}. Note that by definition, (G_1,1(y),⋯,G_1,n(y)) for y∈𝒴_n/{y_n} are n-1 linearly independent vectors and therefore generate extreme rays of Δ_1^G*. Also note that any element in Δ_0^G* is written as (G_0,1(y),⋯,G_0,n(y)) for some y∈ℝ, and so is a vector that generates its extreme ray. But by (<ref>), we have that Δ_1^G=Δ_0^G and thus Δ_1^G*=Δ_0^G*, and therefore, for each y_k∈𝒴_n/{y_n}, there exists y_k^*∈ℝ and λ_n(·)>0 such that (G_0,1(y_k^*),⋯,G_0,n(y_k^*))=λ_n(y_k)×(G_1,1(y_k),⋯,G_1,n(y_k)).If there exists multiple values of y_k^* satisfying (<ref>), we define y_k^* as the infimum of {ỹ_k^*:(G_0,1(ỹ_k^*),⋯,G_0,n(ỹ_k^*))=λ_n(y_k)×(G_1,1(y_k),⋯,G_1,n(y_k))}. Because CDFs are right-continuous function, the infimum should also satisfy (<ref>).For any j,k=1,...,n, if G_1,j(y_k)=0 then G_0,j(y_k^*)=0 by (<ref>), which further implies that G_0,j(y^*)=0 for all y^*≤ y_k^* because G_0,j(·) is monotone increasing. Let {j_1,⋯,j_n} be a permutation of {1,⋯,n} such that y_j_1<y_j_2<⋯<y_j_n. Note that G_1,j_1(y_j_1) is the only non-zero component in the set {G_1,k(y_j_1):k=1,⋯,n}. Then, by (<ref>), G_0,j_1(y_j_1^*)≠0 and G_0,j_k(y_j_1^*)=0 for k≥2. Similarly, there are two elements of {G_0,k(y_j_2^*):k=1,⋯,n} which are non-zero, namely, G_0,j_1(y_j_2^*) and G_0,j_2(y_j_2^*). Therefore, by G_0,j_2(y_j_1^*)=0 and G_0,j_2(y_j_2^*)≠0 and the fact that G_0,j_2(·) is monotone increasing, we can conclude y_j_1^*<y_j_2^*. Continuing this argument, we can conclude that y_j_1^*<y_j_2^*<⋯<y_j_n^*.Define a function ψ_n:y_k↦ y_k^* for k=1,⋯,n. By the above analysis, ψ_n(·) is a monotone increasing function. Note that the support of ψ_n is 𝒴_n, which we extend to ℝ as follows: for any y∈ℝ,ψ_n(y)={[ max{ψ_n(y_k):y_k≤ y,k=1,⋯,n}if y≥min{y_1,⋯,y_n}; -∞otherwise ].Then, ψ_n:ℝ→ℝ is still a monotone increasing function.We now consider increasing n to n+1. By a similar argument, there exists a sequence {y_1^†,⋯,y_n^†,y_n+1^†} and λ_n+1(·)>0 such that for k=1,⋯,n+1, we have (G_0,1(y_k^†),⋯,G_0,n(y_k^†),G_0,n+1(y_k^†))=λ_n+1(y_k)×(G_1,1(y_k),⋯,G_1,n(y_k),G_1,n+1(y_k)),If there exists multiple values of y_k^†, we define y_k^† as the infimum of them. Note that, by (<ref>) and (<ref>), y_k^† is one of the candidates ỹ_k^*'s that make (G_0,1(ỹ_k^*),⋯,G_0,n(ỹ_k^*)) proportional to (G_1,1(y_k),⋯,G_1,n(y_k)) satisfy (<ref>). While y_k^* is the infimum of those candidates, y_k^† cannot reach that infimum because it has to satisfies the additional restriction, G_0,n+1(y_k^†)=λ_n+1(y_k)G_1,n+1(y_k). Therefore, we can conclude that y_k^†≥ y_k^* for k=1,⋯,n. Using {y_1,⋯,y_n,y_n+1} and {y_1^†,⋯,y_n^†,y_n+1^†}, define ψ_n+1(·) analogous to ψ_n(·) above. Then, ψ_n+1(y_k)=y_k^†≥ y_k^*=ψ_n(y_k) for k=1,⋯,n. Furthermore, by definition, ψ_n+1(y_n+1)≥ψ_n(y_n+1) regardless of the rank order of y_n+1 in 𝒴_n+1. Therefore, for any y∈ℝ,ψ_n+1(y)≥ψ_n(y),and thus the limit of the sequence of functions ψ_n(·) exists as n→∞, which we denote as ψ_∞(·). Recall each ψ_n(·) is weakly increasing. It is easy to prove by contradiction that ψ_∞(·) is strictly increasing. Fix y_k∈𝒴_∞. For any n≥ k, (G_0,1(ψ_∞(y_k)),⋯,G_0,n(ψ_∞(y_k))) is proportional to (G_1,1(y_k),⋯,G_1,n(y_k)) and therefore there exists λ_∞(y_k) such that (G_0,1(ψ_∞(y_k)),⋯,G_0,n(ψ_∞(y_k)))=λ_∞(y_k)×(G_1,1(y_k),⋯,G_1,n(y_k))for any n∈ℕ. Moreover, because 𝒴_∞ is dense in ℝ and ψ_∞ and G_d,k are right-continuous functions, the above condition holds for all y∈ℝ.Note {G_1,k(·):k=1,⋯,∞} is a class of simple functions. Therefore, any F_Y_1|η(·|t) can be written as F_Y_1|η(·|t)=lim_K→∞∑_k=1^Ka_K,k(t)G_1,k(·)for some triangular array {a_Kk(t):1≤ k≤ K,K=1,2,⋯,∞}. By the definition of G_1,k(·), it follows that F_Y_1|η(·|t) =lim_K→∞∑_k=1^Ka_K,k(t)∫ w_k(s)F_Y_1|η(·|s)ds=∫lim_K→∞∑_k=1^Ka_K,k(t)w_k(s)F_Y_1|η(·|s)ds≡∫κ(t,s)F_Y_1|η(·|s)ds,where κ(t,s)≡lim_K→∞∑_k=1^Ka_K,k(t)w_k(s) serves as a Dirac delta function. Because F_Y_1|η(·|t)=∫κ(t,s)F_Y_1|η(·|s)ds if and only if F_Y_1|η(·|t)≤∫κ(t,s)F_Y_1|η(·|s)ds and F_Y_1|η(·|t)≥∫κ(t,s)F_Y_1|η(·|s)ds, we have, by Condition <ref>, F_Y_0|η(·|t)=∫κ(t,s)F_Y_0|η(·|s)ds=lim_K→∞∑_k=1^Ka_K,k(t)G_0,k(·)using the definition of G_0,k(·). Combining (<ref>), (<ref>) and (<ref>), for any y∈ℝ and t∈𝒯, we haveF_Y_0|η(ψ_∞(y)|t) =lim_K→∞∑_k=1^Ka_K,k(t)G_0,k(ψ_∞(y))=lim_K→∞∑_k=1^Ka_K,k(t)λ_∞(y)G_1,k(y) =λ_∞(y)F_Y_1|η(y|t),which completes the proof.§.§ Equivalence Between Rank Linearity and Condition <ref>: Discrete Y_d For d∈{0,1}, suppose Y_d and η are discretely distributed. Specifically, let 𝒴_d≡{ y_d,1,⋯,y_d,K_d} and 𝒯≡{t_1,⋯,t_K_η} be the support of Y_d and η, respectively. Note that even with K_0=K_1, we allow that Y_0 and Y_1 have different supports (i.e., allowing for a “drift”). Suppress (Z,X) for simplicity.For arbitrary non-negative weights {w_1,⋯,w_K_η} and {w̃_1,⋯,w̃_K_η} such that ∑_k=1^K_ηw_k=1 and ∑_k=1^K_ηw̃_k=1, it holds that ∑_k=1^K_ηw_kF_Y_1|η(·|t_k)≤∑_k=1^K_ηw̃_kF_Y_1|η(·|t_k)if and only if ∑_k=1^K_ηw_kF_Y_0|η(·|t_k)≤∑_k=1^K_ηw̃_kF_Y_0|η(·|t_k).This condition can be motivated by the discussion in Remark <ref>.For any probability distribution function F̃_d supported on 𝒴_d≡{y_d,1,⋯,y_d,K_d}, suppose there always exists a sequence {c_d,1,⋯,c_d,K_η} such that F̃_d(·)=∑_k=1^k_ηc_d,kF_Y_d|η(·|t_k),Then, Condition <ref> holds if and only if (i) K_0=K_1 and (ii) for some strictly increasing mapping ψ:{y_0,1,⋯,y_0,K_0}→{y_1,1,⋯,y_1,K_1} and some λ:{y_0,1,⋯,y_0,K_0}→ℝ_+,F_Y_0|η(y|t_k)=λ(y)F_Y_1|η(ψ(y)|t_k),for y∈𝒴_0,k=1,⋯,K_η. The condition (<ref>) is a rank condition as the rank of matrix {F_Y_d|η(y_d,j|t_j'):j=1,...,K_d, j'=1,⋯,k_η} should be no smaller than K_d. A necessary condition is K_η≥ K_d, namely, the support of η is no coarser than the support of Y_d. The rank condition would be violated when there is no endogeneity (i.e., Y_d⊥η), which is not our focus. Again, the rank condition is only introduced in this theorem to establish the relationship between rank linearity (and hence rank similarity) and the range of identifying conditions of this paper, and it is not necessary for our bound analysis.By Condition <ref>, we have ∑_k=1^K_ηδ_kF_Y_1|η(·|t_k)≤0 if and only if ∑_k=1^K_ηδ_kF_Y_0|η(·|t_k)≤0for any K_η-dimensional vector δ≡(δ_1,⋯,δ_n) that satisfies ∑_k=1^K_ηδ_k=0.Note that (F_Y_1|η(y|t_1),⋯,(F_Y_1|η(y|t_K_η)) for each y∈𝒴_1/{y_K_1} generates an extreme ray of the (K_η-1)-dimensional polar cone of a cone {δ∈ℝ^n:∑_k=1^K_ηδ_kF_Y_1|η(·|t_k)≤0;∑_k=1^K_ηδ_k=0} .A similar argument holds for (F_Y_0|η(·|t_1),⋯,(F_Y_0|η(·|t_K_η)). By (<ref>), these two polar cones are the same. Therefore, for each y_k∈𝒴_1/{y_K_1}, there exists a y_k^*∈𝒴_0/{y_K_0} such that (F_Y_0|η(y_k^*|t_1),⋯,F_Y_0|η(y_k^*|t_K_0)))=λ(y_k)×(F_Y_1|η(y_k|t_1),⋯,F_Y_1|η(y_k|t_K_1))).Finally it is easy to show that if y_k<y_k' then y_k^*<y_k'^* and thus ψ(y_k)=y_k^* is a strictly increasing function.§.§ Proof of Theorem <ref> We suppress X for simplicity. The proof is analogous to that of Theorem <ref>. Using the same notation as the earlier proof, (<ref>) can be rewritten as∑_ℓ=1^ℓ^*q(z_ℓ)-γ_ℓp(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ]+∑_ℓ=ℓ^*+1^Lq(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ]=∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)/a× P[Y_1≤ y|D=1,Z=z_ℓ].The above equation being satisfied as equality can be viewed as being satisfied as inequalities “≤” and “≥.” Therefore, by Condition <ref> applied for both inequalities, we have∑_ℓ=1^kq(z_ℓ)-γ_ℓp(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ]+∑_ℓ=ℓ^*+1^Lq(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ]=∑_ℓ=ℓ^*+1^Lγ_ℓp(z_ℓ)/a× P[Y_0≤ y|D=1,Z=z_ℓ].Equivalently, we haveP[Y_0≤ y|D=1] =P[Y_0≤ y]×∑_ℓ=1^Lγ_ℓ-∑_ℓ=1^Lγ_ℓ× P[Y≤ y,D=0|Z=z_ℓ] =-∑_ℓ=1^Lγ_ℓ× P[Y≤ y,D=0|Z=z_ℓ]by ∑_ℓ=1^Lγ_ℓ=0. □ §.§ Proof of Lemma <ref> Part (i) can be shown analogous to the proof of Theorem <ref>. Suppose ∫ w(t,x)F_Y_1|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_1|η,X(·|t,x)dtholds for some w and w̃. We want to show that ∫ w(t,x)F_Y_0|η,X(·|t,x)dt≤∫w̃(t,x)F_Y_0|η,X(·|t,x)dt.First, because of the strict monotonicity of q(d,x,·), we have∫ w(t,x)F_U_1|η,X(·|t,x)dt≤∫w̃(t,x)F_U_1|η,X(·|t,x)dtand it suffices to show ∫ w(t,x)F_U_0|η,X(·|t,x)dt≤∫w̃(t,x)F_U_0|η,X(·|t,x)dt.Second, for any v∈Supp(V|X=x), because of the strict monotonicity of ϕ(·,v), we have 1(U_1≤ u_1)a.s.=1(ϕ(U_1,v)≤ϕ(u_1,v)). Because V(U_1,η)|X, we have ∫ w(t,x)F_ϕ(U_1,V)|η,X,V(ϕ(·,v)|t,x,v)dt≤∫w̃(t,x)F_ϕ(U_1,V)|η,X,V(ϕ(·,v)|t,x,v)dtand thus, ∫ w(t,x)F_U_0|η,X,V(ϕ(·,v)|t,x,v)dt≤∫w̃(t,x)F_U_0|η,X,V(ϕ(·,v)|t,x,v)dt.Conditional on (η,X,V), Supp(ϕ(U_1,v))=Supp(ϕ(U_1,V))=Supp(U_0). Therefore, for u_0 in that support, ∫ w(t,x)F_U_0|η,X,V(u_0|t,x,v)dt≤∫w̃(t,x)F_U_0|η,X,V(u_0|t,x,v)dt.It follows that∫∫ w(t,x)F_U_0|η,X,V(u_0|t,x,v)f_V|X(v|x)dtdv ≤ ∫∫w̃(t,x)F_U_0|η,X,V(u_0|t,x,v)f_V|X(v|x)dtdv Note that f_V|X=f_V|η,X. Then, by the law of iterated expectation, we have ∫ w(t,x)F_U_0|η,X(u_0|t,x)dt≤∫w̃(t,x)F_U_0|η,X(u_0|t,x)dt. Next, we prove part (ii) by first noting that[U_0≤ u|η,X] =[ϕ(U_1)≤ u,V≤ u|η,X]=[ϕ(U_1)≤ u|η,X][V≤ u|X].Therefore,F_Y_0|η,X(y|t,x) =[g(0,x,U_0)≤ y|η=t,X=x]=[U_0≤ g^-1(0,x,y)|η=t,X=x] =[ϕ(U_1)≤ g^-1(0,x,y)|η=t,X=x][V≤ g^-1(0,x,y)] =[Y_1≤ g(1,x,ϕ^-1(g^-1(0,x,y)))|η=t,X=x][V≤ g^-1(0,x,y)] =F_Y_1|η,X(ψ(y,x)|t,x)λ(y,x),where ψ(y,x)≡ g(1,x,ϕ^-1(g^-1(0,x,y))) and λ(y,x)≡ F_V(g^-1(0,x,y)). □ §.§ Proof of Theorem <ref> The proof is immediate by applying Theorem 6.9 in <cit.>. This is because (i) the primal problem is superconsistent as both p(y,1) and p(y|1) are continuous on compact 𝒴 and (ii) γ^*∈{y:p(y,1)'γ≥ p(y|1)} such that p(y,1)'γ^*>p(y|1). □ecta | http://arxiv.org/abs/2311.15871v1 | {
"authors": [
"Sukjin Han",
"Haiqing Xu"
],
"categories": [
"econ.EM"
],
"primary_category": "econ.EM",
"published": "20231127144324",
"title": "On Quantile Treatment Effects, Rank Similarity, and Variation of Instrumental Variables"
} |
firstpage–lastpageThree-dimensional ℤ topological insulators without reflection symmetryVladimir Juričić^3,1 January 14, 2024 ======================================================================== Using images from the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory (SDO/HMI), we extract the radial-velocity (RV) signal arising from the suppression of convective blue-shift and from bright faculae and dark sunspots transiting the rotating solar disc. We remove these rotationally modulated magnetic-activity contributions from simultaneous radial velocities observed by the HARPS-N solar feed to produce a radial-velocity time series arising from the magnetically quiet solar surface (the inactive-region radial velocities). We find that the level of variability in the inactive-region radial velocities remains constant over the almost 7 year baseline and shows no correlation with well-known activity indicators. With an RMS of roughly 1 , the inactive-region radial-velocity time series dominates the total RV variability budget during the decline of solar cycle 24.Finally, we compare the variability amplitude and timescale of the inactive-region radial velocities with simulations of supergranulation. We find consistency between the inactive-region radial-velocity and simulated time series, indicating that supergranulation is a significant contribution to the overall solar radial velocity variability, and may be the main source of variability towards solar minimum. This work highlights supergranulation as a key barrier to detecting Earth twins. Sun: granulation – techniques: radial velocity – methods: data analysis § INTRODUCTIONThe radial-velocity (RV) method is one of the most valuable tools in the planet hunters' arsenal, as both a detection method, with over 1000 confirmed exoplanets to date, and a vital follow-up tool for transit missions such as the Kepler <cit.> , Transiting Exoplanet Survey Satellite <cit.>, and upcomingPLAnetary Transits and Oscillations of stars<cit.> missions. Precise RV measurements are often necessary to independently confirm planetary candidates and to determine planetary masses.As instrumental precision improves, the intrinsic RV variability of the host star becomes the primary obstacle to detecting and characterising low-mass planets <cit.>. Stellar variability can mimic non-existent planets <cit.>, lead to inaccurate mass measurements <cit.>, or completely obscure the signal of a low-mass planet.Photospheric magnetic activity primarily causes RV variations via two processes.Firstly, strong magnetic fields act to inhibit convective blueshift of rising stellar material, imparting a net redshift onto the overall stellar RVs <cit.>. Secondly, bright and dark active regions break the axial symmetry of the rotating stellar disc. This results in an imbalance of flux from the approaching and retreating stellar limbs, causing a net Doppler shift <cit.>.As these phenomena occur in isolated, magnetically active regions which can persist for multiple rotation cycles, the resulting RV variability typically displays quasi-periodic behaviour with amplitudes of several .Traditional activity-mitigation techniques include the use of magnetically sensitive activity indicators which correlate with activity-induced RVs <cit.>, modelling RV variations with Gaussian processes to leverage the quasi-periodicity of isolated active regions rotating with the stellar surface <cit.>, and using simultaneous photometry and ancilliary time series to estimate the effect of active regions <cit.>. Whilst these techniques have proveneffective in mitigating the RV variability originating in isolated, magnetically active stellar regions, they do not take into account variability originating from the magnetically quiet stellar surface. Nevertheless, there are a number of phenomena on the quiet stellar surface which can cause RV variability. In particular, observations and simulations of solar convective flows (e.g., granulation and supergranulation) have shown RV variability on the order of 0.5-1 <cit.>.In this paper, we use spatially resolved solar images to calculate the RV impact of isolated magnetically active regions <cit.>. We remove these contributions from Sun-as-a-star RVs from the HARPS-N spectrograph to isolate and directly characterise the RV impact of the magnetically quiet solar surface (hereafter the inactive-region RVs).This paper is structured as follows. In Section <ref> we describe the data used in this work andwe isolate the inactive-region RVs in Section <ref>. We analyse the statistical properties of the inactive-region RVs in Section <ref> and demonstrate that the inactive-region RVs are not simply an instrumental artefact (Section <ref>). It is worth noting that, up until this point, our analysis is purely statistical and therefore independent of the physical cause of inactive-region RVs. In Section <ref>, we show that the inactive-region RVs match simulations of oscillation, granulation, and supergranulation and propose supergranulation as the dominant source of variability. We conclude in Section <ref>. § DATA§.§ HARPS-N Solar telescope The solar telescope at the Telescopio Nazionale Galileo is a 7.6-cm achromatic lens which feeds the sunlight to an integrating sphere and then into the High Accuracy Radial velocity Planet Searcher for the Northern hemisphere (HARPS-N) spectrograph <cit.>.Observations are taken almost continuously during the day time, with 5-min integration times in order to average over the solar p-mode variations.RVs are calculated from the extracted HARPS-N spectra using the HARPS-N Data Reduction Software <cit.>.By using the same instrument and DRS used for night-time exoplanet searches, the solar feed provides a realistic Sun-as-a-star RV time series and has yielded information about sub-level instrumental systematics present in the HARPS-N spectrograph <cit.>. To better emulate night-time stellar observations, we average three successive observations to give 15-minute effective integration times.In this paper, we use a data set spanning from 2015-Jul-29 to 2021-Nov-12. The RVs have been transformed into the heliocentric rest frame, allowing the data to be free of planetary signals. Additionally, a daily differential extinction correction was done following the model described in <cit.>. The HARPS-N Solar Telescope operates continuously, observing from under a plexiglass dome. Data affected by clouds or other bad weather thus need to be accounted for afterwards. To remove potential bad data points, we use two metrics for each observation. Firstly, data quality factor Q is calculated from a mixture model approach described in <cit.>. This probability value can be between 0 and 1 where 1 represents the reliable data. Secondly, we extract the maximum and mean count of the exposure meter from the headers of the fits files. The ratio of max and mean, R, should be close to 1 for uniform exposures (and is always higher than 1 by definition). A first quality cut is done using the conditions Q≥0.99 and R<1.5. From the remaining data points, we fit the histogram of R with a Gaussian function.A slight delay in the shutter opening causes this distribution to resemble a Gaussian, rather than the more expected pile-up around 1.In a second cut we remove data where R is higher than 3 standard deviations above the mean. Finally, as a last cut, we perform a 5-σ clipping on the remaining RVs. The cuts made are very strict and may indeed remove reliable data too, but it is done to ensure the very high quality of data. The remaining data-set has 20 435 observations.§.§ SDO/HMIThe Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory (SDO/HMI) uses six narrow bands around the magnetically-sensitive 6173 Å Fe I line. A Gaussian fit of the narrow-band fluxes as a function of wavelength is used to derive continuum intensitygrams, magnetograms, and line-of-sight velocity maps (Dopplergrams), each spatially resolved with approximately 1" resolution. In this work we use 720-second exposures, every four hours from 2015-Jul-29 to 2021-Nov-12, totalling 12 434 sets of images. We briefly describe how we calculate RVs from these images. For more details, we refer the reader to <cit.> and <cit.> which describe in detail the process of extracting disc-averaged physical observables from SDO/HMI images. Owing to poor long-term stability of the SDO/HMI instrument, we are not able to directly calculate the solar RVs by simply intensity-weighting the Dopplergram pixels. Fig. 2 of <cit.> shows the effects of long-term systematics when calculating the RVs in this way. There are clear jumps in the RVs caused by instrumental effects.This means that we cannot use the Dopplergram to produce long-baseline Sun-as-a-star RV measurements.To overcome this instrumental limitation, we use a physically-motivated model to estimate the RV variations arising from two processes on the Sun. In doing so, we calculate the RV variability relative to the quiet Sun <cit.>.The first of these is a photometric shift, δRV_phot, caused by a bright plage or dark sunspot breaking the rotational symmetry of the solar disc. In the absence of inhomogeneities on the solar disc, the contribution of the blue-shifted approaching limb and red-shifted retreating limb are in balance, with no overall effect on the solar RVs. If a dark or bright region is present on one of the limbs, the Doppler contribution from that limb is diminished or enhanced, respectively, resulting in an overall RV shift. The second component, δRV_conv, arises from regions with large magnetic flux suppressing convection on the surface of the Sun. The reduction in the convective blue-shift in these regions imposes a net red-shift onto the overall solar RV. By applying continuum intensity and magnetic flux thresholds described by <cit.>, respectively, we isolate the RV contribution of active regions on the Sun.Since SDO/HMI derives its RVs from a single iron line, as opposed to a whole spectrum, <cit.> and <cit.> fit a linear combination of these components to solar RVs from HARPS and HARPS-N, respectively, to account for the different scaling of each component. They found that δRV_conv is the dominant of these two effects in the Sun. <cit.> only consider the RV contributions from active regions larger than 20 Hem. Those authors find that this cutoff obviates the need to vary the relative contributions of the photometric and convective effects over time. Owing to the fact that the two components forming this RV series are driven by magnetic processes, and that RVs are calculated relative to the quiet Sun, we will refer to the RV series derived from the SDO/HMI images as the magnetic activity RVs. Since the formal uncertainties on the magnetic activity RVs are significantly lower than the uncertainties for the HARPS-N RVs <cit.>, we will refrain from extensive error analysis of the magnetic activity RVs. § ISOLATING RV CONTRIBUTIONS FROM MAGNETICALLY-INACTIVE REGIONSIn this paper, we are interested in the RV variability originating from magnetically quiet regions of the solar surface.To isolate these contributions, we subtract the SDO/HMI RVs (which are only sensitive to large magnetically active regions) from the HARPS-N RVs (which probe the entire solar disc). The resultant time series of residual RVs therefore probes the RV signals not arising from the large, magnetically active regions identified by <cit.>.Throughout this paper, we will refer to this time series as the inactive-region RVs. To account for the difference in cadence of SDO/HMI and HARPS-N, we interpolate the magnetic activity RVs onto the HARPS-N timestamps. In principle, this could introduce a signal into the inactive-region RVs, since we are smoothing over all variability in the magnetic activity RVs over timescales shorter than 4 hours. Recalling, however, that the magnetic activity RVs encode variability occurring on rotational timescales, and noting that shorter-timescale processes (e.g., oscillations/granular flows) do not contribute to the magnetic activity RVs, we are justified in this approach. We provide a quantitative justification in Appendix <ref>. It is worth highlighting that the model of <cit.>, which we use in this paper, does not include the effect of the smallest active regions. However, those authors show that by only considering the largest active regions, they reproduce the majority of the rotationally-modulated, activity-induced RVs, without having to introduce an arbitrary trend. The models of <cit.> reduce the HARPS-N RV scatter from 1.65 to 1.21 and 1.18 with and without the 20 Hem active-region cutoff, respectively[To reduce the RV scatter to 1.18 when taking into account all active regions, an arbitrary linear drift was introduced. Without that drift, the scatter is reduced to 1.31 .]. We are therefore justified in describing the RV time series produced by removing this contribution from the disc-integrated HARPS-N RVs the inactive-region RVs. We do note however, that there will inevitably remain a small component in the inactive-region RVs that corresponds to the smallest-scale magnetic regions. To avoid any effects of long-term instrumental effects, we de-trend the inactive-region RVs with a 100-day rolling mean, following the treatment of <cit.>.We note that such a de-trending preserves both the amplitude and timescale of any short-term variability present in the RV time series, and so will not affect the results of this work. We opt to smooth the inactive-region RVs, as opposed to the HARPS-N RVs, since we are interested in the difference between the RVs measured by HARPS-N and those derived from SDO/HMI images.Directly smoothing the HARPS-N RVs would risk smoothing over the rotationally modulated activity signal present in both series, thereby imposing an artificial signal in the residuals when we subtract the un-smoothed magnetic activity RVs. Fig. <ref> shows the correlation between the daily mean RV and both logR'_HK and unsigned magnetic flux <cit.> values for the HARPS-N RVs and the inactive-region RVs, as well as the respective Pearson correlation coefficients. We see an almost total removal of any correlation between the RVs and the activity indices when the magnetic activity RVs are subtracted from the HARPS-N RVs, indicating that we are successfully removing the majority of the RV signal caused by magnetic activity.In addition, following <cit.>, we calculate the velocity power spectrum (PS) of the HARPS-N RVs and the residual RVs as follows. Each RV time series can be represented as a linear combination of sinusoidal components,RV(t) = c + ∑_ν a(ν) cos(2 πν t) + b(ν) sin(2 πν t),where a(ν) and b(ν) are frequency-dependent coefficients, and c is a constant offset. The velocity power spectrum is defined asPS(ν) = a(ν)^2 + b(ν)^2,and is converted to velocity power spectral densities (PSDs) by multiplying by the effective length of the observing window <cit.>. Fig. <ref> shows the two PSDs as well as the ratio of power in the HARPS-N RVs to the inactive-region RVs, calculated for logarithmically-spaced bins in frequency. We see evidence of decreased power in the inactive-region RVs at timescales comparable to and longer than the solar rotation period, whereas the two PSDs are similar for shorter timescales. This further suggests that we are successful at removing most of the rotationally modulated variability, whilst retaining the short-timescale variability characteristics of the HARPS-N RVs.§ INACTIVE-REGION RVS EXHIBIT A CONSTANT LEVEL OF VARIABILITYHaving established that we are successful in removing the RV signature of large magnetically active regions (Section <ref>), we now analyse the inactive-region RVs. To investigate how the inactive regions vary throughout the solar cycle, we compare the RMS of the RVs with the mean sunspot number, taken from <cit.>, for 45 50-day intervals, as a proxy for overall activity level. In Fig. <ref> we show that there is no correlation. Additionally, despite the sunspot number changing by more than two orders of magnitude, the standard deviations of the RVs vary only slightly. This shows that theinactive-region RV variability remains relatively constant at a wide range of magnetic-activity levels. In Appendix <ref>, we show that the 17 per cent scatter in the RMS shown here is consistent with being dominated by sampling noise.To investigate how the inactive-region RVs behave at different timescales, we use structure functions <cit.>. Structure functions quantify how the variability within a time series changes with timescale. In Appendix <ref> we show that, for a continuous, uncorrelated signal, the structure function is equivalent to SF(τ) = 2 σ^2,where σ is the RMS of the signal. We therefore choose to use √(1/2SF) to quantify the RV variability at a given timescale τ, emphasising the connection with the RMS. We provide a more in-depth description of structure functions and their key properties required for this analysis in Appendix <ref>. We now segment the RVs by calendar year and show the corresponding structure functions in Fig. <ref>.We opt to use longer sub-series than the 50-day samples in Fig. <ref> to ensure that each timescale is well sampled, with many pairs of data points contributing to the calculation for each timescale.We see that despite covering a wide range of magnetic activity levels, the inactive-region RVs have similar variability at all timescales. We also note that, for each 1-year segment, there is no significant increase in variability at timescales longer than ∼ 3 days. The typical year-to-year variation is 5-10 per cent. This is comparable to the typical lower-bound sampling error we estimate in Appendix <ref>, indicating that the true year-to-year variation is very low. We contrast this with the behaviour of the magnetic-activity RVs over the same baseline.We calculate structure functions of the magnetic-activity RVs (again segmented by calendar year). We plot structure functions calculated from both the inactive-region RVs and magnetic activity RVs on the same axis scales in Fig. <ref> to highlight the contrast. Whilst the magnetic-activity RVs vary by more than an order of magnitude, the inactive-region RVs exhibit an almost unchanging level of variability. Fig. <ref> also demonstrates that, for the majority of the 7-year baseline, the variability in the residual RVs is comparable to or larger than the variability shown in the magnetic activity RVs.This shows that, for the quieter period of solar cycle 24, the majority of the RV variability originated from magnetically-inactive regions, as opposed to magnetically-active regions. The two contributions are approximately equal during the 2016-2017 period, where the average sunspot number was 39.8. Fig. <ref> shows the distribution of yearly-averaged sunspot numbers, dating back to 1700 from the World Data Center SILSO, Royal Observatory of Belgium, Brussels <cit.>. The vertical line corresponds to the 2016-2017 average. Roughly 35 per cent of recorded years have a lower average sunspot number. This implies that the inactive-region RVs may dominate solar RVs for roughly one third of the time. § INACTIVE-REGION RVS ARE NOT DOMINATED BY INSTRUMENTAL EFFECTSIt is in principle plausible that the variability shown in Fig. <ref> at timescales of around two days is due to the instrumental characteristics of the HARPS-N spectrograph, rather than genuine solar effects. To address this concern, we repeat our analysis with RVs from the NEID solar feed <cit.> taken between 2021-Jan-01 and 2022-Jun-13.We use RVs reported by version 1.1 of the NEID Data Reduction Pipeline[<https://neid.ipac.caltech.edu/docs/NEID-DRP/index.html>] and select observations that are free from known issues due to the daily wavelegnth calibration, a known cabling issue or evidence of poor observing conditions based on the pyrheliometer and exposure meter data.We select only observations taken before 17:30 and with airmass less than 2.25 to reduce calibration and atmospheric effects.Due to the different wavelength ranges and sensitivities of the NEID and HARPS-N spectrographs, the photometric shift, δRV_phot, and suppression of convective blueshift, δRV_conv, (see Section <ref>) may have different contributions to the overall RVs. To account for this, following <cit.>, we fit a linear combination of these components to the NEID RVs. It is worth noting that significant intra-day systematics appear in the NEID solar RVs due to, amongst other effects, unaccounted-for differential extinction. We model the sub-day variations with a best-fit cubic trend to the 1-day phase-folded RVs, which we then subtract from the NEID solar RVs.Whilst this is not as sophisticated as the treatment of <cit.>, we wish simply to recover the general variability properties seen in Fig. <ref> and so a full treatment of NEID instrumental effects is both beyond the scope of this paper and not necessary for our purposes. As with the HARPS-N RVs, we use the recalculated SDO/HMI RVs and NEID RVs to produce a time series of inactive-region RVs. The HARPS-N and NEID RVs temporally overlap between 2021-Apr-01 and 2021-Nov-12 and we compare the HARPS-N- and NEID-derived inactive-region RV time series during this overlap to investigate the effects of instrumental systematics. Fig. <ref> shows structure functions of the two inactive-region RV series.We find that the variability of these two time series at all timescales typically differs by less than 10 per cent, comparable to the lower-bound sampling uncertainty we estimate in Appendix <ref>.This indicates that the results of section <ref> are genuine, and are not simply instrumental artefacts, and that any effects of instrumental differences are smaller than the intrinsic variability of the solar RVs; as such, we will focus the remainder of our analysis on the longer-baseline time series of inactive-region RVs derived from HARPS-N. We note that this analysis qualitatively agrees with the results of <cit.>, who compare solar RVs from state-of-the-art spectrographs, including HARPS-N and NEID.Those authors find remarkable agreement between the different instruments.§ SUPERGRANULATION AS THE DRIVER OF INACTIVE-REGION RVS In Section <ref>, we demonstrated that the inactive-region RVs have a roughly constant level of variability over the decline of solar cycle 24. To determine the cause of this variability, we compare the inactive-region RVs to a simulation of the oscillation, granulation, and supergranulation of a G2 star from <cit.>, based on the technique of <cit.> and the results of<cit.>. The simulated RV time series is averaged over 15 minutes to best match the HARPS-N observations.Fig. <ref> shows structure functions of each component in the simulation of <cit.> (pink), the combined simulated time series (red), and the inactive-region RVs (black).The granulation (Gra) and supergranulation (Sgr) components are re-scaled by a constant factor to match the observations; the oscillation (Osc) component is assumed to be negligible <cit.> and so is not re-scaled. However, we opt to include the oscillation component for completeness. Since granulation and supergranulation dominate at different timescales, we are able to re-scale the simulated RVs to constrain the contribution of each process by matching the power in the inactive-region RVs at short and long timescales, respectively. By combining and re-scaling the simulated RVs in this way, we are able to accurately reproduce both the overall variability amplitude of the inactive-region RVs, and the timescale of the inactive-region RVs. <cit.> found that the amplitude for the RV signals from granulation is 0.8 , though there is evidence that an amplitude of 0.4 is a more appropriate level <cit.>. We see in Fig. <ref> that the choice to use the lower level of granulation variability is justified; a granulation RV time series with RMS of 0.8 would have more power at short timescales than is seen in the true RVs. We find granulation and supergranulation variability amplitudes of 0.37 and 0.86 , respectively. We therefore find a similar level of granulation to the lower level of <cit.>, and to the 0.33 prediction of <cit.>. We note that the 0.86 level of supergranulation we find is around 25 per cent higher than the 0.68 level found by <cit.>, though the granulation amplitudes are similar. Given the difference in methodology between this work and that of <cit.>, who use asteroseismology techniques to analyse Sun-as-a-star RVs from the HARPS and HARPS-N spectrographs, it is unsurprising that there is a difference in the exact amplitude of supergranulation derived. Both analyses demonstrate that supergranulation contributes a significant fraction of the total solar RV variability. Our interpretation that the inactive-region RVs are primarily caused by granulation and supergranulation is supported by the fact that the level of variability is constant at various levels of the solar cycle (Figs. <ref> & <ref>). <cit.> used HINDODE/SOT images of the Sun from 2006-Nov to 2016-Feb to investigate photometric properties of solar granules. Over this period, they did not find changes in either the granulation contrast or granulation scale of the 3 per cent level at which they were sensitive, indicating no significant variation of granulation properties over the solar cycle. We note that Fig. <ref> does not include any instrumental systematics. Whilst this makes it difficult to evaluate numerical uncertainties on the level of supergranulation quoted above, it is unlikely to significantly change the amplitude. Firstly, a white noise profile (such as from photon noise) would inject power at all timescales and appear flat as a structure function.Since we constrain the granulation amplitude by using the short-timescale power, we can be confident that we have not underestimated the uncorrelated noise. Thus the re-scaling we apply to the supergranulation time series to match the long-timescale power is reasonably robust against the level of photon noise, which is estimated to be as low as 0.24 <cit.>. Secondly, we show in Section <ref> that the level of variability of inactive-region RVs derived from HARPS-N and from NEID observations are consistent within typical sampling errors at all timescales, indicating that the variability observed is not dominated by instrument-specific artefacts. This shows that the supergranulation amplitude derived from Fig. <ref> is representative. Fig. <ref> shows that the combined simulated RVs show slightly reduced power on intermediate timescales (between a few hours and a few days) as compared to the inactive-region RVs.Unaccounted-for correlated instrumental noise (e.g., an imperfect correction for differential extinction) could inject variability at intermediate timescales to account for this discrepancy. Alternatively, the intrinsic variability spectrum of supergranulation may differ slightly to that of granulation. The simulations of <cit.> assume that supergranules evolve similarly to granules. The resulting simulated time series therefore share a common variability power law. A relaxation of this assumption may account for the mismatched power at intermediate timescales. Additionally, the smallest-scale active regions, unaccounted for in the model of <cit.>, may inject RV variability at these intermediate timescales.Regardless of the cause of this discrepancy, the agreement of the inactive-region RVs with both the amplitude and timescale of the simulations of <cit.> and analysis of<cit.> offers good evidence that the variability seen in the inactive-region RVs is predominantly caused by supergranulation at timescales of a few days and longer.Given that these RVs exhibit a relatively constant level of variability over the solar cycle (Figs. <ref> & <ref>), we provide evidence that supergranulation dominates the solar RVs on the approach to cycle 24 minimum and therefore poses a significant barrier to the detection of Earth twins whose Doppler shifts are ∼ 0.1 .§ CONCLUSIONS We have generated solar RVs from SDO/HMI images following the technique of <cit.>. The two components accounted for in this time series are the RV signatures arising from the suppression of convective blueshift in magnetically active regions and the effect of photometrically imbalanced regions transiting the rotating solar disc. We subtract these magnetic activity RVs from the disc-integrated RVs obtained from the HARPS-N solar feed to isolate the RV contribution from the magnetically quiet solar surface. We find that the resulting inactive-region RV time series shows no correlation with the well known activity indicators logR'_HK and unsigned magnetic flux. This, combined with the reduced Fourier power at timescales longer than roughly half a solar rotation period, show we are successful in removing the majority of the RV signal arising from the active regions on the Sun. We show that the inactive-region RV variability is relatively stable, showing no significant change in level with either time or sunspot number. This contrasts starkly with the magnetic-activity RVs, which show a change in variability of more than an order of magnitude over the same baseline. This relative constancy in the inactive-region RV variability means that, when the Sun is relatively quiescent, the solar RVs are dominated by the RV signal originating in inactive regions, not active regions. We find that the two contributions are roughly even when the average sunspot number is around 40. This level of magnetic activity represents the 35^th percentile recorded in archival sunspot data dating from 1700. Finally, we compare the inactive-region RV time series to recent models of oscillation, granulation, and supergranulation <cit.>. We find good agreement with bothvariability amplitude and timescale between the inactive-region and simulated RV time series. We find thatgranulation and supergranulation induce RV variability with amplitudes of 0.37and 0.86 , respectively. In doing so, we provide empirical evidence that supergranulation can dominate solar RVs and therefore pose a significant barrier to detection of Earth twins.This is in agreement with the findings of <cit.>. § ACKNOWLEDGEMENTSWe would like to thank Suzanne Aigrain and Niamh O'Sullivan for assisting in normalising the PSD of Fig. <ref>. B.S.L. is funded by a Science and Technology Facilities Council (STFC) studentship (ST/V506679/1). R.D.H. and S.D. are funded by the STFC's Ernest Rutherford Fellowship (grant number ST/V004735/1). F.R. is funded by the University of Exeter’s College of Engineering, Maths and Physical Sciences, UK. S.J.T. is funded by STFC grant number ST/V000918/1. A.C.C. acknowledges support from STFC consolidated grant number ST/V000861/1, and UKSA grant number ST/R003203/1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement SCORE No 851555). This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation (SNSF) under grants 51NF40_182901 and 51NF40_205606 F.P. would like to acknowledge the SNSF for supporting research with HARPS-N through the SNSF grants 140649, 152721, 166227, 184618 and 215190. The HARPS-N Instrument Project was partially funded through the Swiss ESA-PRODEX Programme. K.R. acknowledges support from STFC Consolidated grant number ST/V000594/1. This research was supported by Heising-Simons Foundation Grant #2019-1177 and NASA Grant # 80NSSC21K1035 (E.B.F.).This work was supported in part by a grant from the Simons Foundation/SFARI (675601, E.B.F.). The Center for Exoplanets and Habitable Worlds is supported by Penn State and its Eberly College of Science.This work is basedin part on observations at Kitt Peak National Observatory, NSF’s NOIRLab, managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The authors are honoured to be permitted to conduct astronomical research on Iolkam Duág (Kitt Peak), a mountain with particular significance to the Tohono Oódham.Data presented herein were obtained at the WIYN Observatory from telescope time allocated to NN-EXPLORE through the scientific partnership of the National Aeronautics and Space Administration, the National Science Foundation, and the National Optical Astronomy Observatory.We thank the NEID Queue Observers and WIYN Observing Associates for their skillful execution of our NEID observations.In particular, we express gratitude to Michael Palumbo III for compiling the NEID RVs used in this study. § DATA AVAILABILITY This work is underpinned by the following publicly available datasets: SDO/HMI images, available at <https://sdo.gsfc.nasa.gov/data/aiahmi/>; and NEID solar RVs, made available at <https://zenodo.org/record/7857521>. In addition, this work makes extensive use of the HARPS-N solar RVs, which will be described and made available in an upcoming publication.mnras § A JUSTIFICATION THAT INTERPOLATION DOES NOT AFFECT THE RESULTS OF THIS PAPERTo match the high cadence of the HARPS-N solar RVs, we calculate RVs from the SDO/HMI images (see section <ref>) every four hours, and interpolate these RVs onto the timestamps of the HARPS-N observations. To assess the impact that interpolation has on the signal, we produce a month-long time series of RVs, generated from every available 720s exposure set of SDO/HMI images from 2017-Jan-01 to 2017-Feb-01. From this high-cadence RV series we produce a sub-sampled RV series with six RV measurements per day, to match the long-baseline series we use in the bulk of this paper. We then interpolate this sub-sampled RV series back onto the timestamps of the original high-cadence time series. We find that the median absolute deviation between the high-cadence and interpolated RV series is 2.1 . This is significantly lower than any other level of variability we discus in this paper, justifying that our choice to interpolate the RVs calculated from the SDO/HMI images does not impact the results of this paper. § THE EFFECT OF SAMPLING NOISE ON THE RMS OF THE INACTIVE-REGION RVS.In Fig. <ref>, we see that a variation in the RMS of inactive-region RV sub-series.Whilst each RMS is calculated from a relatively large number of observations, we show in Fig. <ref> that the inactive-region RVs become uncorrelated at timescales longer than a few days. This relatively large timescale, compared to the 50-day baseline of each sub-series, means that the sampling error on the RMS is larger than would be expected by simply considering the number of observations. To test this, we take the simulated supergranulation time series of <cit.> and calculate the RMS for the same 50-day sub-series as in Fig. <ref>. This time series is constructed in a stationary manner, so that any variation seen between values of the RMS can with good confidence be attributed to sampling error. Fig. <ref> shows the distribution of sub-series RMS values. As we are only interested in the fractional difference between the individual RMS values, we normalise by the mean RMS. We see a relative scatter of the RMS values, σ_σ/<σ>, of 14 per cent, where σ_σ/<σ> = √(1/n_σ∑( σ - (1/n_σ∑σ))^2)/1/n_σ∑σ.This value of σ_σ/<σ> is comparable to the inactive-region σ_σ/<σ> of 17 per cent. This indicates that the majority of the variation seen in Fig. <ref> is due to statistical noise.§ STRUCTURE FUNCTIONS The structure function (SF) is an analysis tool which shows how the variability within a time series changes as a function of timescale. It has had success in analysing photometric time series data <cit.>. SFs use a pairwise comparison of points in a time series f(t) to calculate the level of variability as a function of timescale. To calculate the variability of points with a given separation τ =| t_1 - t_2|, we define the structure function asSF(τ) =⟨( f(t) - f(t + τ) ) ^2⟩,where the angular brackets indicate an average over all pairs of observations with separation τ. In practice, SF is calculated for logarithmic bins in τ. That is to say, in the case of discrete data, the structure function is calculated asSF(τ_1, τ_2) = 1/N_p∑_ij (f_i - f_j)^2,where the summation is taken over all pairs of data points which are separated by τ_1 < | t_i - t_j | < τ_2and N_p is the number of such pairs. Fig. 7 of <cit.>, reproduced here as Fig. <ref>, demonstrates the typical form of a SF. The short timescale plateau shown in Region 1 indicates any intrinsic variability in the signal is below the noise floor, so the variability is dominated by uncorrelated instrumental noise. In Region 2, there is a power-law increase in the structure function, the gradient of which is determined by the spectrum of variability being investigated. Region 3 shows timescales longer than the dominant variability timescale in the data. Therefore, after this timescale has been reached, the value of the structure function stops increasing, again becoming uncorrelated.The transition between Regions 2 and 3, referred to as the knee, tells us the characteristic variability timescale.Fig. 5 of <cit.> demonstrates that, for a sinusoidal signal, the knee is at roughly a quarter of the period. This highlights that the timescale identified by a structure function, whilst related to any period present in a signal, is not the same as a period. Another feature of Fig. <ref> is that the structure function becomes larger as we move towards longer timescales. Whilst not strictly a cumulative measure, this behaviour is typical for most types of variability.[A notable exception is for periodic or quasi-periodic signals. For such signals, pairs of data points separated by multiples of the period will show little if any variation, causing characteristic dips in the structure function.] This property highlights how short-timescale variations still impact the overall level of variability at longer timescales. An advantage of the structure function over other analysis tools is the direct relationship between the structure function and the level of variability at a given timescale. We show in Appendix <ref> that, for an uncorrelated signal, the structure function will tend to the value 2σ^2, where σ is the standard deviation of the data. This clear relationship between structure function and standard deviation allows a much more direct interpretation of the level of variability within a signal for a given timescale than, say, a Fourier power spectrum.For a more detailed description of some properties of structure functions, see section 5 of <cit.> and references therein. To avoid spurious results arising from bins with too few points, we require at least 100 pairs of observations in each τ bin throughout this paper. §.§ The relationship between the structure function and standard deviationConsider an infinite, uncorrelated time series f(t). This time series will have mean μ = ⟨ f(t) ⟩,and standard deviation,σ = ⟨ ( f(t) - μ )^2 ⟩.Since f(t) is uncorrelated, we can also say that ⟨ f (t) × f ( t + x) ⟩ = μ^2 + σ^2δ(x),where δ(x) is the δ-function.From Eq. <ref>, it follows thatSF(τ) = ⟨( (f (t)- μ) -( f(t+τ) - μ)) ^2 ⟩ = [ ⟨( f(t) - μ) ^ 2⟩ +⟨( f(t + τ) - μ) ^ 2⟩ ]-2⟨(f(t) - μ ) ×( f(t + τ) - μ) ⟩ = 2σ ^2 - 2[ ⟨ f (t) × f ( t + x) ⟩ -μ⟨ f(t) ⟩-μ⟨ f(t + τ) ⟩ + μ^2].From Equation <ref>, therefore, we see that, for uncorrelated signals,SF(τ) =2σ^2.This is equivalent to eq. A4 of <cit.>.§.§ Estimating the effect of sampling noiseAs discussed in appendix C of <cit.>, it is non-trivial to estimate the effect of sampling noise on structure function measurements (as we can think of f(t) being drawn at random from an underlying distribution). The difficulty arises in calculating the number of independent pairs of data points contributing to each measurement. To estimate the level of sampling noise, we divide the simulated supergranulation RV time series of <cit.> (see Section <ref>) into year-long sub-series, and calculate the structure function of each. This is in analogy to fig. 10 of <cit.>. We opt to use the supergranulation time series as it will allow us to estimate the sampling noise at timescales below and above the characteristic timescale of the variability <cit.> and the time series is constructed to be stationary so we can be confident that any variation between the structure functions is due to the sampling noise. Fig. <ref> shows the ratio R of standard deviation of √(SF) to the median value of √(SF)at each timescale. We show that the typical sampling error is on order of 5 per cent of the median value of √(SF) at a given timescale. Given that these sub-series are drawn from the same distribution, and that they are well-sampled, we expect this to be a lower bound on the uncertainty associated with a structure function measurement. | http://arxiv.org/abs/2311.16076v1 | {
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Work In Progress: Towards Adaptive RF Fingerprint-based Authentication of IIoT devices. Identify applicable funding agency here. If none, delete this.Emmanuel Lomba PoRTIC, Polytechnic Institute of Porto Porto, Portugal [email protected] Ricardo Severino PoRTIC, Polytechnic Institute of Porto Porto, Portugal [email protected] Ana Fernández Vilas Information and Computing Laboratory AtlanTTic Research CenterVigo, Spain [email protected] 14, 2024 ========================================================================================================================================================================================================================================================================================================================== As IoT technologies mature, they are increasingly finding their way into more sensitive domains, such as Medical and Industrial IoT, in which safety and cyber-security are of great importance. While the number of deployed IoT devices continues to increase exponentially, they still present severe cyber-security vulnerabilities. Effective authentication is paramount to support trustworthy IIoT communications, however, current solutions focus on upper-layer identity verification or key-based cryptography which are often inadequate to the heterogeneous IIoT environment. In this work, we present a first step towards achieving powerful and flexible IIoT device authentication, by leveraging AI adaptive Radio Frequency Fingerprinting technique selection and tuning, at the PHY layer for highly accurate device authentication over challenging RF environments.IoT, Industry 4.0, Software-Defined Radio, RF Fingerprinting, Feature Extraction § INTRODUCTIONThe advancements in information and communication technology in the past decades have been converging into a new communication paradigm in which everything is expected to be interconnected with the heightened pervasiveness and ubiquity of the Internet of Things (IoT) archetype. As these technologies mature, they are increasingly being introduced into the industrial domain, to support what is now dubbed as the Industry 4.0, converging IoT, Cyber Physical Systems and Cloud technologies into the factory floor. However, this digitalization process is not without risks, and cyber-security is increasingly becoming the most prominent issue to be addressed in such infrastructures. In Industry 4.0 there may be several objectives in compromising a system, from data extraction to industrial espionage, to sabotaging and physically endanger a factory or voiding its products, or even to mount an attack to more critical parts of the same or different infrastructures <cit.>. On the other hand, while the number of deployed IoT devices continues to increase exponentially and is estimated to reach 75 billion by 2025 <cit.>, IoT domains present a severe set of challenges, enhanced by the IoT scale, heterogeneity, and its fast adoption.Indeed, IoT “things” are often equipped with limited resources in terms of memory capacity and computational power. Such limitations often hinder the direct implantation of conventional Internet security techniques like AES, or TLS into the IoT <cit.>, <cit.> and their absence may lead to various security and privacy attacks like eavesdropping, network side-channel attacks, and tracking. In addition to such heterogeneous processing capabilities, IoT encompasses a myriad of several different technologies like Wireless Sensor Networks, Radio-Frequency Identification, and Machine-to-Machine communications, which further difficult the management of dedicated cyber-security solutions.In the IoT ecosystem, most of these Internet connected devices do not have the same experience induced resilience to intrusion, hacking and sabotage attacks that other computing devices have acquired. On the contrary, they show a significant level of vulnerability. With 70% of IoT devices found to have serious security vulnerabilities <cit.>, such as unencrypted network services, weak password requirements, and 90% of devices collecting personal information, there is a critical need for an IoT security approach capable of defeating attacks in which illegitimate devices digitally masquerade as authorized IoT devices. The need is further exacerbated as bad actors exploit this weakness to conduct attacks against other infrastructures<cit.>, <cit.>, even taking down large swaths of the Internet by leveraging D-DoS attacks such as the Mirai botnet <cit.>, <cit.>, which relied upon illegitimate usage of hundreds of thousands of IoT devices. All these factors impose an urgent need for an effective IoT security solution that can address a multi-dimensional problem composed of several Quality of Service (QoS) dimensions such as timeliness, scalability and heterogeneity, meeting Industry 4.0 challenges. In conventional computing systems, the security issues of authentication, confidentiality and integrity are usually handled above the physical layer by relying on loose upper-layer identity verification (“who it is”) and key-based cryptography (“what it holds”). Regarding upper-layer identity, mechanisms usually rely in the media access control (MAC) address for identification purpose. However, this kind of authentication scheme is vulnerable to identity-based attacks, such as spoofing attack <cit.> and the Sybil attack <cit.>. Moreover, as these identity-based attacks are based on changing the upper-layer identity, its contents can be easily revised, and the attack can be conveniently launched repeatedly. Key-based cryptography is another widely used upper-layer security technique, based upon the “what it holds” strategy. Although cryptography is an effective method to defend against identity-based attacks, its application in IoT has severe limitations particularly in terms of scalability, and importantly timeliness <cit.>, as the utilization of high complexity encryption algorithms in such wireless devices can result in large latency, which is intolerable for time-critical communications. In addition, upper-layer cryptography-based authentication is not suitable for all devices’ authentication purposes, such as for relay nodes which work at the physical layer by amplifying and forwarding a wireless signal. We must also not dismiss that vulnerabilities may be introduced into the implementation of cryptographic systems, nor that the time spent on cracking a digital security key could be shortened remarkably as available processing power increases. Therefore, upper-layer identity verification and cryptography-based authentication schemes present drawbacks and challenges when applied to IIoT. Therefore, it is valuable to explore stable and unique physical-layer characteristics to generate device fingerprints and investigate corresponding physical-layer authentication techniques. In our work, we intend to rely on passive authentication layers, capable of identifying the device by “who it is”, looking into characteristics that are unique in its Radio Frequency (RF) signal; i.e., RF Fingerprinting. Importantly, our major contribution is focused in devising and applying AI techniques capable of self tuning key aspects of the feature extraction process to greatly increase the accuracy of the authentication over different and somewhat unpredictable wireless environments. This is to be achieved via the deployment of software-defined technologies, in particular Software-Defined Radio (SDR) gateways to implement the security solutions, tackling the heterogeneity of the ecosystem, coupled with AI Edge/Cloud support, to scale and leverage Machine Learning (ML) strategies. The first step towards this goal is to develop effective and automated ways to extract the device features from its RF signal and generate the corresponding fingerprint using SDR technology. Currently in progress, in this paper we introduce our ongoing work on the automated RF feature extraction framework.§ RELATED WORKGiven the vulnerability of loose upper-layer identity verification and the limitations of key-based cryptography when deployed in IoT, particularly on timeliness, it is valuable to explore stable and unique physical-layer characteristics to generate device fingerprinting and investigate corresponding physical-layer authentication techniques. The PHY layer approach known as Specific Emitter Identification has been put forward as a solution capable of addressing this critical IoT need <cit.>. One specific implementation, known as Radio Frequency Fingerprinting (RFF), facilitates radio discrimination by exploiting the unintentional ‘coloration’ that is inherently imparted upon a radio’s waveform during its generation and transmission, and has been shown feasible in multiple protocols VHF, IEEE 802.11, Bluetooth, IEEE 802.15.4 and RFID transponders. The fact that RFF features are inherent and unique to a given radio makes them virtually impossible to imitate, thus, making security approaches based upon them difficult to bypass. Unlike bit-level security protocols, i.e., cryptography-based authentication techniques, RFF is proven to be a useful tool in the enhancement of wireless communications security at the physical layer in applications such as device spoofing, intrusion detection, cloning detection, indoor positioning, access control, Sybil and Replay attacks detection, to name a few. Still, the authors in <cit.> show that the majority of RFF work has focused on radio classification <cit.>, where an unknown radio’s identity is determined through the comparison of its fingerprint(s) with each of the stored, reference models that represent the authorized/known radios. Such “one-to-many” comparison hinders classification, as class assignment is made no matter how poor the “best” match is. This flaw can result in the granting of network access to rogue radios. This has led to the proposal of a “one-to-one” comparison known as radio identity (ID) verification. In radio ID verification, the RF fingerprint of the unknown radio is compared only to the stored reference model associated with the presented digital ID, deeming it either authorized or rejected as a rogue. However, proposals in RFF rely on a quite limited and inflexible subset of the signal features due to the limited available processing and computing power <cit.>, which makes their accuracy highly dependent of the Signal-to-Noise Ratio (SNR). Regarding this, SDR can effectively improve the feature extraction process in complex and challenging wireless environments by careful tuning key variables such as filters bandwidth or amplifier stages gains.Unfortunately, current approaches do not take advantage of suchSDR flexibility during feature extraction. We intend to do this by leveraging AI techniques. Also, none consider the heterogeneous IoT ecosystem and focus on a single protocol instead. Our framework will provide on-demand tools to tackle this challenge by leveraging the SDR gateway component. To further help in the classification process, we are to deploy an additional layer of Edge/Cloud services which will greatly improve the classification process by increasing the available computing power for multiple RF feature processing, increasing authentication accuracy. In the literature, authors presenting experimental results, acquire radio signals using diversified apparatus ranging from low cost SDR receivers such as the RTL-SDR dongle, up to high-end oscilloscopes with sampling rates of 20 GS/s and above or specific State-of-the-Art RF spectrum digitizers. The digitized signal samples are then either directly applied to the classifier with embedded feature extraction, or via some pre-processing stage before entering the classifier stage. In either methods, there is no evidence of any automatic control of the signal acquisition parameters. Clearly, most literature focus their research on the classification method and its performance. To the best of our knowledge, there is no other approach like ours where the signal's features extraction uses AI to aim at the optimum signal acquisition conditions, controlling the SDR receiver parameters such as gains of amplifiers stages and filters characteristics. As presented, most of the proposals in the literature suffer from impairments which limit the deployment, accuracy and flexibility of such solutions. In our work, we aim at investigating intelligent, practical, powerful, and flexible, “one-to-one” RF authentication techniques which: (1) rely on larger and adaptable sets of RF features, driven by the radio characteristics of the device to be identified, thus leveraging the available SDR properties. This is done by deploying AI techniques at the lower feature extraction layer to tune the SDR component and vastly improve its feature extraction quality in challenging wireless environments; (2) build upon SDR gateways to address the heterogeneity of the IoT ecosystem, enabling such authentication to be deployed into different IoT protocols; (3) increase available processing power and memory by relying on Edge/Cloud computing paradigm to offload processing intensive task from the gateways.§ DESIGN AND IMPLEMENTATIONRFF-based authentication usually follows a six steps procedure: Signal acquisition, Detection of the Signal's Region of Interest, Features extraction, Features dimension reduction, Fingerprinting, and Device classification. Our RFF-based authentication approach aims at achieving powerful and flexible IoT device security, by relying upon Software-Defined Radio technology and Machine Learning (ML) at the lowest OSI-model level.Figure <ref> illustrates the proposed architecture up to the classifier output.The feature extraction stage shall work in a closed loop where a ML algorithm at the Controller block evaluates the relevant features to be extracted, optimized by fine-tuning of the SDR parameters to address eventual signal quality degradation originated by physical constraints such as SNR variations, and other channel conditions variations. The proposed architecture also aims at addressing challenges such as fingerprint portability by simple replacement of the SDR block hardware, and scalability via ML algorithms appropriate to the study of open set of transmitting devices.To implement the first loop of the proposed architecture, an auxiliary framework is being developed, aiming at a choreographed control of several transmitters that, in sync with the SDR in the Acquisition of Signal block, will enable the production of full-featured datasets.Figure <ref> presents the block diagram of the proposed Dataset Building System. This system is composed of two main parts: the transmitting part (TX block) and the receiving part (RX block). The transmitting block groups all the transmitters, controlled by a dedicated computer in charge of starting and ending transmissions, and triggering the acquisition of RF signals in the receiving block. Thus enabling a pseudo-continuous recording of the radio signals without risking any loss of data due to memory limitations of the SDR in the RX block; i.e., the same SDR to be used in the proposed architecture in Figure <ref>.The receiving part (RX block) is responsible for digitizing the radio signal and storing the samples into a single session file, and then saving the file into a disk for later processing (e.g., fingerprinting). The RX block could also be composed of a Spectrum Analyzer outputting an Intermediate Frequency to a Digitizing Oscilloscope ("ADC" in Figure <ref>), just as done by the authors in <cit.>, for example. In our implementation, we are comparing several low cost SDR hardware such as RTL-SDR v3.0, ADALM-Pluto and HackRF One, to be used in the RX block and later in the fingerprinting testbed. Other SDR receivers of higher performance such as USRPs are to be evaluated in the future. The GNU Radio ecosystem is the main tool used to control the SDR receiver (i.e., "ADC" in Figure <ref>) and to save the collected I/Q samples into a dataset in accordance to the SigMF specification <cit.>. The resulting dataset is to be publicly released in the future.The use of SDR technology in this work allows usability in a wide range of applications, in terms of spectrum, from ISM bands such as the 433 MHz band, up to the WiFi bands at 2.4 GHz and 5 GHz. The current multi-transmitter testbed uses 433 MHz transmitters controlled by a computer via an Arduino. The computer hosts the choreography of transmissions; e.g., each transmitters sends its dedicated message at pre-programmed instants in time. This approach offers endless transmissions scenarios, from single-transmitter message to multiple transmissions at once, causing interference. The position of the transmitters and the structured environment can be changed as needed to allow the study of channel impairments.§ DISCUSSION In this paper we introduce the ongoing work regarding an RF Fingerprinting framework for the authentication of IIoT devices, which has been primarily focused on the signal acquisition and feature extraction stages.Although RF signals are rich in features that can be exploited for the purpose of generating a device fingerprint, feature selection is a challenging problem and is dependent on the hardware capabilities of the system. Simple or primitive features can be directly measured from the acquired signal, such as the instantaneous amplitudes, peak values, or RSS. Furthermore, such measurements can be a source for derived features. The extraction of more complex features such as frequency or phase differences require dedicated processing techniques due to their stochastic nature, hence it is usual to rely on statistical methods such as Skewness, Variance or Kurtosis. While these extraction methods seem to be easier to implement, others can extract features directly from the raw signal data (i.e., I/Q samples), using Transformation tools such as the Hilbert-Huang Transform, the Gabor-Wigner Transform, or the Wavelet Packet Decomposition, to name only a few.As already mentioned, the number of features is a fundamental factor in the accuracy of the classification process, hence the importance of low-latency, high availability and processing capacity computing paradigms introduced by the edge-cloud continuum. Our envisaged architectural design addresses this.Moreover, the effectiveness of the fingerprint must comply two fundamental characteristics: unforgeable and robust; i.e., the fingerprints must be impossible to counterfeit and must be stable in the presence of channel variations due to device mobility or environment changes. Such variations have yet to be dealt with in a convincing way by the relevant literature. Indeed, most work done on this topic limit their classification to closed sets of transmitters <cit.>. Regarding channel impairments, RFF is subject to channel impairments from diverse origins, such as signal absorption, reflection, scattering, refraction, diffraction, or signal multipath issues causing upfade, downfade, nulling, data corruption. The effects of some of these impairments are addressed by the authors in <cit.> but these have yet to be tackled in a self adaptive fashion.Lastly, the aging of the device and the influence of temperature in the emitter operation may also interfere on some features used for fingerprinting, with a clear impact at the classification stage. All of these issues are objects of our research, which we intend to address by using AI right at the feature extraction stage, together with the SDR component, as presented in the paper.Currently, our implementation is capable of automatically trigger the signal acquisition upon detection of a transmission and deploying a set of feature extraction modules. As we build our RF feature dataset in different wireless environments, upon different settings, we will address the development of the AI modules to carryout the adaptation of SDR elements to improve feature quality. In parallel, we are developing the Edge/cloud architecture to carryout signal classification.§ ACKNOWLEDGMENTThis 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). 00 statista_number_nodesStatista. “Number of IoT Devices 2015-2025.” Statista. Accessed February 15, 2022. https://www.statista.com/statistics/471264/iot-number-of-connected-devices-worldwidecaviglione_covert_2018 Migliardi, M., Merlo, A., and Caviglione, L. (2018). Covert channels in IoT deployments through data hiding techniques. Proceedings - 32nd IEEE International Conference on Advanced Information Networking and Applications Workshops, WAINA 2018, 2018-Janua, 559–563. iotdevicesvulnerabilities Rawlinson, Kristi. “HP Study Reveals 70 Percent of Internet of Things Devices Vulnerable to Attack.” Accessed February 18, 2022. https://www.hp.com/us-en/hp-news/press-release.html?id=1744676.Riahi Riahi Sfar, A.; Natalizio, E.; Challal, Y.; Chtourou, Z. A roadmap for security challenges in the Internet of Things. Digit. Commun. Netw. 2018, 4, 118–137. Kimd Kim, D.; Choi, J.Y.; Hong, J.E. Evaluating energy efficiency of Internet of Things software architecture based on reusable software components. Int. J. Distrib. Sens. Netw. 2017, 13. Simon Simon, S., “’Internet Of Things’ Hacking Attack Led To Widespread Outage Of Popular Websites,” Oct 2016. [Online]. Tiberti Tiberti, B. Vieira, H. Kurunathan, R. Severino and E. Tovar, "Tightening Up Security In Low Power Deterministic Networks," 2020 16th IEEE International Conference on Factory Communication Systems (WFCS), Porto, Portugal, 2020, pp. 1-7.stanislav_hacking_2015 Stanislav, Mark, and Tod Beardsley. “HACKING IoT: A Case Study on Baby Monitor Exposures and Vulnerabilities,” n.d., 17.simon_internet_2016 Simon, Scott. “‘Internet Of Things’ Hacking Attack Led To Widespread Outage Of Popular Websites.” NPR, October 22, 2016, sec. National. https://www.npr.org/2016/10/22/498954197/internet-outage-update-internet-of-things-hacking-attack-led-to-outage-of-popula. krebs_mirai_nodate Krebs, Brian. “Mirai IoT Botnet Co-Authors Plead Guilty, Krebs on Security.” February 18, 2022. https://krebsonsecurity.com/2017/12/mirai-iot-botnet-co-authors-plead-guilty/.kolias_ddos_2017 Kolias, Constantinos, Georgios Kambourakis, Angelos Stavrou, and Jeffrey Voas. "DDoS in the IoT: Mirai and Other Botnets." Computer 50 (January 1, 2017): 80–84. 6164912 Ur Rehman, Saeed and Sowerby, Kevin and Coghill, Colin. "RF fingerprint extraction from the energy envelope of an instantaneous transient signal." 2012 Australian Communications Theory Workshop, 2012.Thangavelu V. Thangavelu, D. M. Divakaran, R. Sairam, S. S. Bhunia and M. Gurusamy, "DEFT: A Distributed IoT Fingerprinting Technique," in IEEE Internet of Things Journal, vol. 6, no. 1, pp. 940-952, Feb. 2019. Donald Donald Reising, Joseph Cancelleri, T. Daniel Loveless, Farah Kandah, Anthony Skjellum, “Pre-print: Radio Identity Verification-based IoT Security Using RF-DNA Fingerprints and SVM”, IEEE Internet of Things Journal 2021 Merchant Merchant, K., S. Revay, G. Stantchev, and B. Nousain, “Deep Learning for RF Device Fingerprinting in Cognitive Communication Networks,” IEEE J. of Selected Topics in Signal Processing, vol. 12, no. 1, Feb 2018.9674605 H. Xu and X. Xu, "A Transformer Based Approach for Open Set Specific Emitter Identification," 2021 7th International Conference on Computer and Communications (ICCC), 2021, pp. 1420-1425.SigMF The GNU Radio Foundation, Inc. "The Signal Metadata Format Specification." GitHub. Accessed June 13, 2022. https://github.com/gnuradio/sigmfRehman S. U. Rehman, K. W. Sowerby, S. Alam, I. T. Ardekani and D. Komosny, "Effect of channel impairments on radiometric fingerprinting", 2015 IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), pp. 415-420, Dec 2015. | http://arxiv.org/abs/2311.15888v1 | {
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These two authors contributed equally. Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Computer Science, University of Colorado, Boulder, Colorado 80309, USA These two authors contributed equally. Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Aerospace Engineering, University of Colorado, Boulder, CO 80309 Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USADepartment of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USABosonic error correcting codes utilize the infinite dimensional Hilbert space of a harmonic oscillator to encode a qubit. Bosonic rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes. We define two different notions of random bosonic rotation codes and numerically explore their performance against loss and dephasing. We find that the best random rotation codes can outperform cat and binomial codes in a certain parameter regime where loss is large and dephasing errors are small. The performance of random bosonic rotation codes Joshua CombesJanuary 14, 2024 ================================================ § INTRODUCTIONSince <cit.>, error correction and randomness have been and continue to be intertwined <cit.>. Similarly, in quantum information, random quantum codes <cit.> have a long history of exploration, predominantly for qubits and qudits (e.g. <cit.>), and they are currently being actively explored <cit.>.Bosonic codes encode qubits or qudits into a subspace of the quantum harmonic oscillator <cit.>. Bosonic error correcting codes have gained a lot of recent attention due to experimental demonstrations of logical fidelities exceeding physical fidelities <cit.>. Indeed, much of the theoretical progress has been spurred on by the rapid experimental progress in circuit QED <cit.>, trapped ions <cit.>, and optics <cit.>.There are two major kinds of bosonic codes: those with a translation symmetry <cit.> (e.g. GKP codes), and those with a rotation symmetry <cit.> (e.g. cat and binomial codes <cit.>). These codes have a structure induced by the translation or rotation symmetry and additionally they have a structure induced by the code family e.g. cat codes have a possion distribution in the number basis. Little has been done on bosonic random codes that are unsturctued in either way. However in Ref. <cit.> the authors considered a continuous-variable bosonic random encoding for quantum secret sharing protocols,while <cit.> introduced the randomized construction of GKP codes using the NTRU cryptosystem for a public key quantum communication scheme.In this article, we construct two kinds of random bosonic codes that have the rotation symmetry strucutre but derive their fock amplitudes from haar random states. The codes have an N-fold rotation symmetry such that a phase space rotation of π/N acts as a logical Z gate.In <ref>, we briefly describe rotation codes. Our random rotation codes are defined in <ref>.We introduce a two-parameter bosonic loss-dephasing channel in <ref>, which are typically the dominant errors for oscillator systems. In <ref>, we compare the performance of our random codes to cat and binomial codes under these error channels using a numerically optimized recovery. We show that random codes can outperform both cat and binomial codes in certain regions of loss and dephasing. Building on previous work by<cit.>, we plot a 2D phase diagram of the best code as a function of loss and dephasing, which shows that, by and large, binomial codes are generally the most performant for the majority of realistic noise channels, but random codes can perform better under high loss and moderate dephasing. § ROTATION CODES Rotation-symmetric codes, or “rotation codes”, are a type of bosonic code where discrete rotational symmetries in phase-space are utilized to protect against noise. Interestingly, the logical Z gate for these codes is a rotation by π/N. The phase space rotation symmetry imposes some restrictions on the Fock coefficients of the code words. In order for a code to have a rotation symmetry degree of N, the |0_N⟩ and |1_N⟩ codewords must be representable as |0_N⟩ = ∑_k=0^∞ f_2kN|2kN⟩,|1_N⟩ = ∑_k=0^∞ f_(2k+1)N|(2k+1)N⟩, where |k⟩ is an eigenstate of the photon number operator n̂. The amplitudes f_2kN and f_(2k+1)N are the only things that specify a class of rotation codes, so a class of codes is determined by the functional form of these coefficients. The class of rotation code does not restrict the possible rotation symmetry degrees.The dual-basis codewords, |±_N⟩, are constructed as usual via superpositions of the computational basis codewords. |±_N⟩ = 1/√(2)(|0_N⟩±|1_N⟩), yielding Fock space representations|+_N⟩ =1/√(2)∑_k=0^∞f_k N|k N⟩, |-_N⟩ =1/√(2)∑_k=0^∞(-1)^k f_k N|k N⟩. Both |±_N⟩ have support on the full set of |k N⟩ Fock states. It is these dual basis codewords that have an N-fold rotational symmetry, while the Z basis code words in <ref> have a 2N-fold rotation symmetry.A general logical state |ψ⟩_L = α|0_N⟩ + β|1_N⟩ is the +1 eigenstate of the stabilizer_N≡exp[ i 2π/Nn̂] .It turns out that the logical Z gate is given by_N≡√(R̂_N) = R̂_2Nwhich is a rotation by π/N such that _N|±_N⟩ = (± 1)|±_N⟩. The corresponding logical X gate is code dependent. However, rotations about Z by fractional angles can be constructed in a code independent way <cit.>. There are two natural errors for a rotation code to correct. The first kind are shift errors, where a number state |n⟩ gets shifted by g i.e. |n⟩↦|n ± g⟩ where g is some integer. The second kind are rotation errors which are induced by the operator e^iϕn̂. The set of shift and rotation errors that are mutually correctable for an order-N ideal phase code are given by <cit.>g∈ [0, N)(number errors) ϕ ∈ [0, π/N)(phase errors). There is a trade-off between the ability to correct number shift and phase errors. For example, a higher rotation symmetry degree, i.e. N, allows for rotation codes to protect against shift errors more effectively, but that, in turn, adversely affects the ability to protect against phase errors. The consequences of this trade-off are made apparent in <ref>. § RANDOM ROTATION CODES We start by considering rotation codes where the amplitudes of the code words are generated with some random processes. We describe two kinds of random rotation codes. Both of the following random generation methods rely on using Haar random unitary matrices <cit.> applied to arbitrary states to get random states. A (K+1)× (K+1) Haar random unitary is denoted as𝕌_K+1∈ SU(K+1) .The restriction to K+1 invokes a photon number cutoff of K, so the unitary acts on Fock states from vacuum |0⟩ through |K⟩. It should be noted that there are simple algorithms to generate Haar random samples from the special unitary group <cit.>. All random states in this work were generated by applying the Haar random unitary to the ground state of the quantum harmonic oscillator i.e. |0⟩. §.§ Random codes with one random primitive state Here we construct codes where the logical codewords |0_N⟩ and |1_N⟩ use the same random state but are offset in Fock space. For example, if the codewords are allowed to use 7 levels of the harmonic oscillator, then a 7-dimensional random state is generated. The random state is then `expanded' into the code words by having the used levels of the harmonic oscillator match the corresponding values of the random state vector. This process is illustrated in <ref>.In general, we can consider a K+1-dimensional random state that we call a “primitive”|ψ_K⟩= 𝕌_K+1|0⟩= ∑_k=0^K ψ_k |k⟩,where |0⟩ is the vacuum and |k⟩ is the k'th Fock state. This state gets “expanded” into the code words |0_N⟩ = ∑_k=0^K ψ_k|2Nk⟩, |1_N⟩ = ∑_k=0^K ψ_k|(2k+1)N⟩, where ψ_k = kψ_K where |ψ_K⟩ is defined in <ref>. Even though the amplitude is the same on both code words, the states are orthogonal (i.e. 0_N1_N=0) due to being constructed on non-overlapping Fock states. Moreover, we can approximately control the average photon number in the code space by varying the cutoff, K. If we naively assume flat amplitudes on average for the random primitive state, the expected value of photon number for the code space is n̂ = N(2K+1)/2. §.§ Random Codes with two random primitive states These codes use two random states to construct the code, one for |0_N⟩ and one for |1_N⟩. For this reason we consider two K+1-dimensional random primitives|ψ_K⟩= 𝕌_K+1|0⟩= ∑_k=0^K ψ_k |k⟩|ϕ_K⟩= 𝕍_K+1|0⟩= ∑_k=0^K ϕ_k |k⟩,where 𝕌_K+1 and 𝕍_K+1 are (K+1)× (K+1) random unitaries. In principle, we could consider different cutoffs for the different states, but for simplicity we do not. These states get expanded into the codewords |0_N⟩ = ∑_k=0^K ψ_k|2Nk⟩, |1_N⟩ = ∑_k=0^K ϕ_k|(2k+1)N⟩, where ψ_k and ϕ_k are amplitudes generated by different random unitaries. The process of constructing these codes is depicted in <ref>. Note that the codes are orthogonal due to the non-overlapping support on the Fock states despite |ψ_K⟩ and |ϕ_K⟩ generally not being orthogonal.The Wigner functions of an example random code with two random primitive states is depicted in <ref>. Notice that the Wigner function for |0_N⟩ and |1_N⟩ are different, which is normal in these kinds of codes. This is unlike the codes presented in <ref>, which will have similar looking Wigner functions for |0_N⟩ and |1_N⟩.It is worthwhile to pause and think about the set of codes from which we are sampling. For a fixed value of rotation symmetry N and Fock cutoff K, we will call the set of all possible codes 𝕊(N,K).Random codes with one primitive sample a subset of the space of all possible codes, but we conjecture that the random codes with two primitives sample the complete space of 𝕊(N,K). This means that any possible code that shares this symmetry and cutoff could be generated by random codes with two primitives, even binomial and cat codes, although this would be incredibly unlikely for any single trial. We make an attempt to sample 𝕊(N,K) uniformly by generating states from the Haar measure.§ ERROR CHANNELSThe dominant physical error channels in most bosonic systems are loss and dephasing. The noise channel we consider consists of simultaneous loss and dephasing Specifically, the noise channel, 𝒩(ρ̂), is the solution to the master equationρ̇̂̇ = κ_l 𝒟[â] ρ̂+ κ_ϕ𝒟[n̂] ρ̂ ,where 𝒟[L̂] ρ̂= L̂ρ̂L̂ - L̂L̂ρ̂- ρ̂L̂L̂. Rather than numerically integrating Eq. <ref> or finding a Kraus-operator representation, we exponentiate the superoperator representation of the Lindbladian. That is, we exponentiate ℒ[L̂], which is the superoperator representation of 𝒟[L̂], which is given as ℒ[L̂] = L̂^* ⊗L̂ - Î⊗L̂L̂ -(L̂L̂)^T ⊗Î. We can then take advantage of the algorithm for sparse matrix exponentiation in Refs. <cit.> to efficiently solve Eq. <ref> for some evolution time t, thereby giving the noise channel as𝒩(ρ̂) = e^t(κ_l ℒ[â] + κ_ϕℒ[n̂])ρ̂ .Thus κ_l t and κ_ϕ t represent the unitless strengths of the simultaneous loss and dephasing rates. § CODE PERFORMANCEIn this section, we explore the performance of single-mode bosonic rotation codes subject to loss and dephasing noise using the methods described in <ref>. In <ref> we compare the performance of known rotation codes, namely cat and binomial codes <cit.>.In <ref> we explore the performance of our new random rotation codes.In <ref> we compare the performance of cat and binomial codes to random rotation codes. We now briefly describe how these comparisons are performed.Rather than assessing the performance of a particular error correction scheme, we use a semidefinite program (SDP) to compute the recovery, ℛ^SDP, that optimizes channel fidelity <cit.>. The channel fidelity of a channel ℰ to the identity channel is F(ℐ, ℰ) = 1/d(d+1) ([∑_k M̂_k^†M̂_k ] + ∑_k |[M̂_k]|^2 ),where {M̂_k} are the Kraus operators for ℰ and d is the dimension of the Hilbert space <cit.>.Thus the total quantum channel we consider is𝒟_ dec∘ℛ^SDP∘𝒩∘𝒮,where 𝒮: ℋ_2 →ℋ_ℕ is the ideal encoding map for a given code, 𝒩: ℋ_ℕ→ℋ_ℕ is the noise map (<ref>), ℛ^SDP: ℋ_ℕ→ℋ_ℕ is the optimal recovery that maximizes the channel fidelity <cit.>, and finally 𝒟: ℋ_ℕ→ℋ_2 is the ideal decoding map for a given code. The encoder superoperator is given by 𝒮[Ŝ]ρ̂= Ŝρ̂Ŝ where Ŝ = |0_N⟩⟨0| + |1_N⟩⟨1| while the decoder is given by 𝒟_ dec[Ŝ^†]ρ̂. Thus the total channel is a logical channel with qubit input and qubit output.To this end, we compute the channel fidelity (to the identity channel) for the channels in <ref> for different rotation codes 𝒞∈ [, , , ] and different rotation symmetries N∈ [2,3,4].As we are considering a logical channel with qubit input and qubit output, d=2 in <ref>. Moreover, we prefer the infidelity (i.e. 1-F), as it is a measure of error. When comparing the performance of codes, it has been customary <cit.> to rank codes by the average photon number of the code space. This is done by defining the code space projector asP̂_ code = 0_N0_N + 1_N1_N,so then the average photon number in the code space isn̅_ code =1/2[n̂P̂_ code] .The other important benchmark is to see when encoding into a code results in a lower channel infidelity than the “naive” encoding. The naive encoding is called the trivial code, and it is where the logical zero and one codewords are respectively the ground and first excited state of the harmonic oscillator,|0_⟩ = |0⟩,and|1_⟩ = |1⟩ .This is a useful comparison when the loss channel is involved, as states with higher photon number e.g. |K⟩ will decay at a rate of κ_l K which is much faster than that of |1⟩ which decays at a rate κ_l. That is, we expect theencoding to be good. When we present diagrams of code performance as a function of κ_l t and κ_ϕ t,the plots will be log-space with 5 points per decade and the range is κ_l t, κ_ϕ t ∈[10^-3, 0.25]We cut off infidelities below 1× 10^-7due to the trade off between tolerances and runtime of the SDP solver. §.§ Exploration of cat and binomialThere are several rotation codes that we could study: polygon codes, squeezed cat codes, Pegg-Barnett codes etc. We examine the two most popular rotation codes, cat codes <cit.> and binomial codes <cit.>, because they have the most theoretical and experimental work about them.The Fock-grid coefficientsf_kN from <ref> for these codes are f_kN =√(1/2^K-1Kk)()f_kN =√(2/𝒩_i)e^-|α|^2/2α^kN/√((kN)!)() where, for cat codes, 𝒩_i is the Fock-space normalization factor such that 𝒩_0 is used for even values of k and 𝒩_1 is used for odd values of k.In row 1 of <ref>, we plot the channel fidelity of the N=2 binomal code while in row 2, we plot the N=3 codes. In column 1 we sweep the code parameter K in <ref> which controls the average photon number of the code space. A generic feature is that there is typically an optimal value of K for a given noise value. Also, note that only some values of K result in channel infidelities below the trial encoding. The filled red point indicates the point with the smallest channel infidelity. In column 2 of <ref>, we plot the best-performing code (optimized over the binomial code parameter K) as a function of noise strength for the case where loss is equal to dephasing i.e. κ_l t = κ_ϕ t. Again we are interested in when the codes are beating theinfidelity, which appears to be 5×10^-2 for N=2 and 7×10^-2 for N=3 codes. The optimal point from the previous plot now appears as one point on these plots.In column 3 of <ref>, we plot the best-performing code as a function of loss κ_l t and dephasing κ_ϕ t. The optimal codes from column 2 are now the diagonal line along this plot. A general trend we note is that, as either κ_ϕ t or κ_l t decrease, the infidelity decreases. That is to say, as the noise strength decreases, the code performance increases.Cat codes have similar performance as these binomial codes and the equivalent plot for cat codes can be found in the <ref> see <ref>. In <ref>, we directly compareandcodes. §.§ Exploration of random codes In this section, we begin to explore the performance of the random codes described in <ref>. <ref> and <ref> show the performance of these random codes for two different noise channels, one with large loss and moderate dephasing (<ref>), and one with large dephasing and moderate loss (<ref>). These plots provide some basic intuition for the performance of random codes.The first interesting feature from<ref> is that, for certain noise channels where loss dominates dephasing, we seecodes that outperform binomial codes, even with a modest number of random trials. For certain average photon numbers, almost all random codes perform better than both binomial and cat codes. This is explored more in <ref> and in particular <ref>.The second interesting feature in <ref> is that, for channels with low dephasing, the infidelity ofcodes appear to follow a smooth curve relating average photon number to performance. Further numerical studies (not presented here) suggest that, in the presence of no dephasing, the performance ofcodes depends solely on average photon number despite the random coefficients of the code.Channels with low loss and high dephasing, such as the one used in <ref>, have poor performance from both types of random codes. This is evident because only very few random codes beat the trivial code, with the exception of a few at n̅_code≈ 3. Interestingly, thecodes seem to be performing statistically better than thecodes. For many channels, especially those where heuristics imply that structured codes like cat and binomial codes are very good, generating a random code that outperforms these existing codes should be very rare. The number of trials used in these simulations is insufficient to make claims about average performance or the limits of performance. Naively, one might expect that the best possiblecodes must perform as good or better than both binomial and cat codes for any channel because the set of codes that are randomly sampled include the known codes. We caution that randomly recreating known codes is very unlikely. This suggests it may also be possible to increase the likelihood of generating these exceptional random codes by sampling a distribution other than the Haar measure. This is left as future work. §.§ Code comparisonIn this section, we compare rotation codes in the set 𝒞∈ [, , , ] with respect to their ability to correct errors and their optimal code parameters, namely average photon number and rotational symmetry.<ref> provides a side-by-side comparison of all four codes across many values of both loss and dephasing. In row 1 we plot the average photon number of the code space projector, i.e. <ref>, for the best code as a function of dephasing κ_ϕ t and loss κ_l t. One trend that stands out is that high loss encourages low photon number codes. At about κ_l t ≈ 4× 10^-2 we see the photon numbers starting to increase. The best “structured” codes, i.e.andcodes, have increasing photon as the noise decreases. While theandcodes seem to have generically have lower photon numbers.The overlay on row 1 indicates the rotation symmetry of the best code for N∈[2,3,4].This overlay reveals another apparent trend for the structured codes i.e.andcodes.Notice the almost horizontal striation atκ_ϕ t ≈ 2× 10^-2 that appears because the optimal code switches from an N=2 to an N=3 code which necessitates a higher photon number. A similar diagonal feature can be seen at κ_ϕ t ≈ 2.5× 10^-3, which occurs when the optimal code becomes N=4. Theandcodes seem to have more diagonal striations, this could be due to insufficient sampling of higher photon number random codes. Naively this may imply that the set of all performant codes does not scale with photon number as quickly as the set of all possible codes.In row 2 of <ref>, we plot the channel infidelity of the best code. The channel infidelity is presented as a function of dephasing κ_ϕ t and loss κ_l t. We see, as expected, that as the noise decreases, the error (infidelity) decreases. Generally, binomial codes outperform all the other codes, but cat codes have a somewhat comparable behavior. This is because binomial codes were designed <cit.> to correct a finite number of loss and dephasing events. Thus we expect them to be good codes for loss and dephasing channel. From the infidelity contours, it is evident that theandcodes don't perform well in the low noise regime when compared to the structured codes.There is another interesting trend that is most obvious by comparing the rotation symmetry to the channel fidelity for cat and binomial codes. Rotation codes have a well known tradeoff between code distance or “protection” against number errors which is d_n = N and the code distance for phase errors d_ϕ = π/N such that d_n d_ϕ = π where d_n-1 is the number of loss or gain errors can be detected <cit.>. This tradeoff is apparent in cat and binomial codes of <ref> where we observe that, for high dephasing, N=2 codes perform the best. As dephasing decreases, the rotation symmetry increases. It is important to note that we have not included N=1 codes or thecode. This trend seems to be much weaker inandcodes, which again, could be due to insufficient sampling. In <ref> we plot the difference in channel fidelity between the best binomial code and the bestcode. Specifically we plot F_ - F_/max{F_, F_}The gray region indicates when the trivial code performs the best. When the plot is red, binomial codes beatcodes, which is generally the case. When the plot is blue,codes beat binomial codes. For loss around 1× 10^-1, we see several points wherecodes outperform binomial codes. The improvement is modest, we see at most a 10% improvement over binomial codes. There is a trade off between the amount of time spent searching for exceptional random codes and code performance. Here we focus on exploring trends across many noise channels with a relatively small number of samples per noise channel. The relative performance of random codes would increase with more sampling, but it is unclear if this performance would increase uniformly across all channels.In <ref> of <ref>, we compareandcodes in the same way. Interestingly, when we comparetocodes we find a larger region wherecodes outperformcodes.In <ref>, the best performing code in the set 𝒞∈ [, , , , ] is found and plotted as a function of loss and dephasing. The first row 1 of the figure indicates which code is the best performing in the set of codes we looked at as a function of loss and dephasing. It is clear thatcodes are typically the best performing rotation code that we investigated. That suggests most of the features observed on the subsequent plots will be similar to the binomial plots except in the large loss or large dephasing regime. This is evident in row 2 of <ref> shows similar striation to those in <ref> in photon number which are explain by changes in rotation symmetry of the optimal code. While the channel infidelity in row 3 will see modest improvements at large loss due toandcodes. § CONCLUSIONSWe have introduced random rotation codes and systematically studied their performance as a function of loss and dephasing strength. By comparing the performance of these new codes to cat and binomial codes, we were able to find a region in our parameter space where random codes outperformed cat and binomial codes. Such results were found with relatively few samples, but it is expected that, with enough samples, random rotation codes could be at least as good as any known rotation code. Thus the numerics presented here are a proof of concept, and we hope it encourages further research using random bosonic codes. As a byproduct of our study on random codes, we have produced the first systematic numerical study of cat and binomial codes, which has revealed some interesting trends. The rotation symmetry, N, is strongly correlated with dephasing. This can be explained by the well known trade off between protection for phase errors and protection for number shift errors <cit.>. In particular, as dephasing decreases, the rotation symmetry of the best code increases. Moreover, the binomial code was the code that took up most of the “phase diagram” in <ref> of the optimal code for a given noise strength. This is to be expected as binomial codes were specifically designed for loss and dephasing errors.We are presently focusing on exploring various notions of random translation codes and random bosonic codes in general. However, we think there are several interesting open questions that remain. Perhaps the most tantalizing would be changing the distribution of states that is being sampled. There is no reason to believe choosing codewords from the Haar measure will result in good codes. In fact this article provides evidence that they are not. A better idea might be to sample distributions that close to known good codes.Acknowledgments: The authors acknowledge helpful discussions with Joshua Grochow, Rebecca Morrison, and Orit Peleg. AK, NL, and JC were supported by the Army Research Office through W911NF-23-1-0376 and National Science Foundation through a CAREER award ECCS-2240129 and Quantum Leap Challenge Institutes (QLCI) award OMA-2016244. § ADDITIONAL PLOTS FOR CODE COMPARISON | http://arxiv.org/abs/2311.16089v2 | {
"authors": [
"Saurabh Totey",
"Akira Kyle",
"Steven Liu",
"Pratik J. Barge",
"Noah Lordi",
"Joshua Combes"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20231127185631",
"title": "The performance of random bosonic rotation codes"
} |
O Corona, where art thou? eROSITA's view of UV-optical-IR variability-selected massive black holes in low-mass galaxies R. ArcodiaNASA Einstein fellow 1,2, A. Merloni 1, J. Comparat 1, T. Dwelly 1, R. Seppi 1, Y. Zhang 1, J. Buchner 1, A. Georgakakis 3, F. Haberl 1, Z. Igo 1, E. Kyritsis 4,5, T. Liu 1, K. Nandra 1, Q. Ni 1, G. Ponti 6,1, M. Salvato 1, C. Ward 7, J. Wolf 1,8,9, A. Zezas 4,5 Received ; accepted==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty § INTRODUCTIONPeople are quick to anthropomorphize, attributing human characteristics to non-human agents <cit.>. This tendency, far from being a modern quirk, is deeply rooted in our collective psyche, as evidenced through our literary traditions. Consider E.T.A. Hoffmann's “Der Sandmann” <cit.>, where Nathaniel’s infatuation with Olimpia, a mere mechanical automaton with a limited repertoire of expressions – her most notable being a plaintive 'Ach, Ach!' – serves as a poignant example. With further technological advances, this phenomenon, once confined to the realm of literature, has now manifested in reality. Already simple chatbots like ELIZA <cit.>, which used simple pattern matching and substitution methodologies, gave users the illusion of human-like understanding and engagement <cit.>. The tendency to anthropomorphize has only intensified with the advent of Large Language Models (LLMs) <cit.>. These advanced AI systems represent a big leap from ELIZA both in complexity and capability. Unlike ELIZA's simplistic pattern matching, LLMs apply deep learning techniques to generate text <cit.>, learning from vast datasets to produce responses that can be startlingly human-like <cit.>. Astonishingly, these models cannot only generate text. When scaled up to bigger training data and architectures, other, so-called “emergent abilities” appear <cit.>. The current models can, for example, pass the bar exam <cit.>, write poems <cit.>, compose music <cit.>, and assist in programming and data analysis tasks <cit.>. As a result, the line between human and machine capabilities is increasingly blurred <cit.>. People not only interact with these systems as if they were humans <cit.>, but they also start to rely on them for complex decision-making <cit.>, artistic creation <cit.>, and personal interactions <cit.>. It is, therefore, natural to ask: Have we built machines that think like people? Or are we, just like Nathaniel, projecting human qualities into rudimentary entities that are fundamentally different from us?Judging whether or not artificial agents can mimic human thought is at the core of cognitive science <cit.>. Therein, researchers have long debated the capabilities of artificially intelligent agents <cit.>. In a seminal paper, Lake et al. <cit.> proposed core domains to consider when making such judgments. Published during the height of the deep learning revolution <cit.>, the authors focused on domains that were easy for people but difficult for deep learning models: intuitive physics, causal reasoning, and intuitive psychology. Research on intuitive physics has studied how people perceive and interpret physical phenomena <cit.>. Past work on this topic has emphasized that humans possess an innate ability to predict and understand the physical properties of objects and their interactions <cit.>, even from a young age <cit.>, a notion sometimes summarized as a “physics engine” in people's heads <cit.>. This understanding includes concepts such as gravity <cit.>, inertia <cit.>, and momentum <cit.>. Some of the most canonical tasks in this domain involve testing people's judgments about the stability of block towers <cit.>. These tasks have made their way into machine learning benchmarks <cit.>, where they are used to test the intuitive physical understanding of neural networks (see <cit.> for an overview of previous work on building models with human-like physical knowledge).Research on causal reasoning has studied how individuals infer and think about cause-effect relationships <cit.>. Past work on this topic has proposed that humans possess an intuitive capacity to infer, understand, and predict causal relationships in their environment <cit.>, oftentimes described using Bayesian models of causal learning <cit.>. This cognitive ability encompasses recognizing patterns <cit.>, inferring causes from interventions <cit.>, and predicting future events based on hypothetical events <cit.>. Canonical tasks in this domain often involve assessing individuals' ability to infer causal relationships, for example, when judging the responsibility of one object causing other objects' movement <cit.>. Causal reasoning remains a challenge, even for current machine learning approaches <cit.>.Research on intuitive psychology has explored how individuals infer, understand, and interpret social phenomena and mental states of other agents <cit.>. Past work on this topic has emphasized the concept that humans possess an inherent ability to infer and reason about the mental states <cit.>, intentions, and emotions of others, often referred to as a “theory of mind” <cit.>. This ability has been modeled as a Bayesian inference problem <cit.>. Canonical tasks in this domain often involve assessing individuals' capacity to predict actions based on understanding others' perspectives or intentions, such as determining agents' utility functions based on their actions in a given environment <cit.>. It is the subject of ongoing debates if modern algorithms show any form of intuitive psychology <cit.>. Lake et al. argued that some of these abilities act as “start-up software,” because they constitute cognitive capabilities present early in development.Moreover, they proposed that these so-called “intuitive theories” <cit.> need to be expressed explicitly using the calculus of Bayesian inference <cit.>, as opposed to being learned from scratch, for example, via gradient descent. However, with the increase in abilities of current neural networks, in particular LLMs, we pondered: Can LLMs, in particular vision LLMs, sufficiently solve problems from these core domains?To address this question, we took canonical tasks from the domains of intuitive physics, causal reasoning, and intuitive psychology that could be studied by providing images and language-based questions. We submitted them to some of the currently most advanced LLMs. Our results showed that these models can, at least partially, solve these tasks. In particular, the largest currently available model, OpenAI's Generative Pre-trained Transformer (GPT-4) managed to perform robustly above chance in two of the three domains. Yet crucial differences emerged. First, none of the models matched human-level performance in any of the domains. Secondly, none of the models fully captured human behavior, leaving room for domain-specific models of cognition such as the Bayesian models originally proposed for the tasks.§ RESULTS We tested four different models on three core components for human-like intelligence as outlined by Lake et al. (<cit.>; see Fig. <ref>A). The models we used are vision large language models, which are multimodal models that integrate image processing capabilities into large language models (<cit.>; see Fig. <ref>B). These models allow users to perform visual question answering <cit.>: users can upload an image and ask questions about it, which the model interprets and responds to accordingly. The first model was GPT-4 with Vision (GPT-4V), developed by OpenAI <cit.>. This multimodal model extends the abilities of GPT-4 to analyze and interpret images, although the details of how the model accomplished this have not been made public. The second model is ADEPT's Fuyu-8B <cit.>, which is a decoder-only multimodal model based on the classic transformer architecture. Fuyu-8B stands out due to its simpler architecture and training procedure compared to other multimodal models. The third model is LLaMA-Adapter V2 <cit.>, or short Adapter, which offers an alternative approach to enhance the capabilities of vision-language models by increasing learnable parameters, introducing an early fusion strategy for better visual integration, and employing a joint training approach for image-text and instruction-following data. The final model is Otter <cit.>, which is based on OpenFlamingo <cit.>, an open-sourced version of DeepMind's Flamingo <cit.>, and specifically designed for multimodal in-context instruction following.For testing the three core components, we used three tasks taken from the cognitive science literature that could be studied in vision LLMs via visual question answering. For intuitive physics, we asked models to judge the stability of different block towers, using stimuli originally proposed by Lerer and colleagues <cit.> and based on previous research by Battaglia et al. <cit.>. For causal reasoning, we again used block towers and asked how many blocks would fall if certain blocks were removed or to judge the responsibility of certain blocks for the stability of the tower, adopting a design that Zhou and colleagues had previously used in human subjects <cit.>. Finally, for intuitive psychology, we used a task where the models saw a picture of an agent's path on a grid and then had to make inferences about the costs and rewards associated with the environment, taken from Jara-Ettinger and colleagues <cit.>.For every task, we queried the visual reasoning abilities of the LLMs with tasks of increasing complexity (see Fig. <ref>C). First, we asked about simple features of the shown images such as the background color or the number of objects shown. Afterward, we submitted questions taken from the cognitive science experiments. We report results based on comparisons to the ground truth as well as the different models' matches to human data. §.§ Intuitive physics To test the intuitive physics capabilities of the different LLMs, we used photographs depicting wooden block towers from Lerer et al. <cit.> (see Fig. <ref> in the Appendix for an example). These images mirror stimuli that developmental psychologists use to study the development of intuitive physics in infants. We used these images to test the models in increasingly complex tasks, starting with determining the background color of a given image (1), counting the number of colored blocks in the image (2), and giving a binary stability judgment of the depicted block tower (3).All four models were able to correctly perform the first and easiest task: they all achieved almost perfect accuracy in determining the background color of the images (see Fig. <ref>A). In the second task, the performance of most models deteriorated, with only GPT-4V correctly determining the number of blocks in all images (see Fig. <ref>B). It is important to note that the first two tasks are fairly trivial for humans and we would expect human performance to be at 100% (the background color is always white and images featured either 2, 3, or 4 blocks). While the first two tasks provided insight into the models' performance on high-level descriptive tasks, the third task directly tested their physical reasoning abilities. Performance for most models was at chance, with only GPT-4V performing slightly above chance in determining the stability of the block towers (see Fig. <ref>C, Fisher's exact test yielded an odds ratio of 2.38 with a one-sided p-value of 0.049). None of the other models performed significantly above chance (all p > 0.05). Human subjects were also not perfect but showed an average accuracy of 63.28%, which was also larger than chance with p<.001. Interestingly, there was no statistically significant difference between people's and GPT-4V's accuracy (z=0.684, p=.49), likely because the task was also hard for humans and because Lerer and colleagues only collected 10 human subjects. Finally, we determined the similarity between models' and humans' stability judgments as determined by the Pearson correlation (see Fig. <ref>D). We found that GPT-4V was the only model that showed a significant correlation with human judgments, r=0.155, t(205) = 2.241, p=.01, while none of the other models showed any meaningful match to human judgments (all p>.05). However, the average correlation between humans (r=0.46, t(910)=14.127, p<.001) was larger than the correlation between GPT-4V and humans (z=3.825, p<.001). §.§ Causal reasoning To test the causal reasoning capabilities, we used synthetic images again depicting block towers from Zhou et al. <cit.> (see Fig. <ref> in the Appendix for an example). Here, the images showed static scenes of block towers that were stable but might collapse if one of the blocks was removed.Again, we started by asking the models to count the blocks in the image (1), we continued by querying the models for the number of blocks that would fall if a specific block was removed from the scene (2), and finally, we asked the models to rate the responsibility of a specific block for the stability of the other blocks (3). For the second task, we established a baseline performance represented by a horizontal line in Fig. <ref>A-C, which corresponds to a random agent. This random agent used a simple strategy: it gave the mean between 0 and the number of blocks in each image as its' prediction, essentially behaving like a uniform distribution over the possible number of blocks that could fall. The images in this task displayed a larger number of blocks (ranging from 6 to 19), which made the basic counting task significantly more challenging than in the previous section. Models' responses approximated the ground truth, albeit rarely matching it exactly. Therefore, we report the mean absolute distance to the ground truth instead of the percentage of correct answers (see Fig. <ref>A). The models' performance highlighted the challenging nature of this task, with the best performing model (GPT4-V) still being on average 2 blocks off. In Figures <ref>B and <ref>C, model performances for the second task are shown. Notably, both GPT-4V and Fuyu-8B surpassed the random baseline, their performance levels being close to the human results reported in <cit.>, which is depicted by the rightmost bar in the plot. However, GPT4-V still diverges significantly from the average over human subjects (t(42)=2.59, p<.05). In the second task, all four models exhibited mean correlations with human values ranging from 0.26 to 0.39, (all p<.001), with Adapter demonstrating the highest correlation. However, it is important to note that despite its relatively high correlation, Adapter also exhibited the highest mean absolute distance to ground truth values.Next, we show the Pearson correlation of all models to human subjects for the third task in Figure <ref>D. Notably, all models except for GPT-4V gave constant ratings for this task (Fuyu always responds with 100, while Otter and LLaMA-Adapter V2 always respond with 50), making their correlation with human judgments undefined. GPT-4V, on the other hand, demonstrated a mean correlation of 0.15 with human values (p<.001). For the human-to-human correlation, we randomly paired human observers from the original study and calculated the correlation over their concatenated responses. We repeated this ten times and calculated the average of the Fisher z-transformed correlations (r = 0.195, p <.001). While the average correlation between humans was higher, it was not significantly higher than the correlation between GPT-4V and humans (z=0.919, p=0.179).§.§ Intuitive psychology To test the intuitive psychology of the different LLMs we used synthetic images depicting an astronaut on a colored background from Jara-Ettinger et al. <cit.> (see Fig. <ref> in the Appendix for an example). The experiment consisted of three parts which differed slightly in their layout and components. Depending on the experiment, the images featured either one or two different terrains (indicated by different background colors and textures) as well as either one or two different care packages. The astronaut was shown with a path that led from a starting point to a base; the astronaut could collect care packages along the way. Depending on which terrain the astronaut crossed or which care package they chose to pick up or not, it was possible to infer the costs associated with the different terrains and the rewards associated with the different care packages. Again, we first tasked models with determining the background color of the images (1), afterwards we asked them to infer the costs associated with the different terrains (2) and the rewards associated with the different care packages (3). The results for the first task are shown in Figure <ref>A. The performance of the models in determining the background color was worse compared to the intuitive physics data set, which might be due to the fact that the background color here was not uniform (see Fig. <ref>). For tasks 2 and 3, we pooled the answers for the three parts of the experiment, as there was only a small number of images in each individual experiment. As shown in Figures <ref>B and <ref>C, all models only showed no or very weak correlations with the average over human subjects in their judgments about the costs and rewards associated with the environment. Correlations with the average over human subjects ranged from -0.02 to 0.12 for cost questions and from -0.02 to 0.17 for reward questions (all p>.05). § DISCUSSION We started by asking whether, with the rise of modern large language models, researchers have created machines that – at least to some degree – think like people. To address this question, we took four recent multi-modal large language models and probed their abilities in three core cognitive domains: intuitive physics, causal reasoning, and intuitive psychology. In intuitive physics, the models managed to solve some of the given tasks and showed a medium match with human data. Similarly, in a causal reasoning task, some models, in particular, GPT-4V, performed well and showed a medium match with human data. Finally, in an intuitive psychology task, none of the models performed well and none of them showed a reasonable match with human data. Thus, an appropriate answer to the question motivating our work would be “No.”, or – perhaps more optimistically – “Not quite.” §.§ Limitations and Future Work Although we have tried our best to give all models a fair chance and set up the experiments in a clean and replicable fashion, some shortcomings remain that should be addressed in future work. First of all, we have only tested a handful of multi-modal models on just three cognitive domains. While we believe that the used models and tasks provide good insights into the state-of-the-science of LLMs' cognitive abilities, future studies should look at more domains and different models to further tease apart when and why LLMs can mimic human reasoning. For example, it would be interesting to see if scale is the only important feature influencing model performance <cit.>. Currently, our evidence suggests that even smaller models, for example Fuyu with its 8 billion parameters, can sometimes perform as well as GPT-4V in some tasks. Another shortcoming of the current work is the simplicity of the used stimuli. While the block towers used in our first study were deliberately designed to be more realistic <cit.> than commonly used psychological stimuli <cit.>, this was not true for the experiments in the other two domains. For the intuitive psychology experiments, in particular, we would expect the models to perform better if the stimuli contained more realistic images of people, which has been shown to work better in previous studies <cit.>. Interestingly, using more realistic stimuli can also change people's causal judgments <cit.>; how realistic stimuli used in cognitive experiments should be, remains an open question <cit.>. On a related point, we only used static images in our current experiments, which severely limits the breadth and level of detail of the questions we could ask. For example, some of the most canonical tasks investigating people's causal reasoning abilities involve videos of colliding billiard balls <cit.>. As future large language models will likely be able to answer questions about videos <cit.>, these tasks represent the next frontier of cognitively-inspired benchmarks. For the comparisons to human data, we currently used the participant data collected in the original studies and assessed the correspondence between models and this data via correlation coefficients. Future work could expand on this approach by collecting new data from human subjects choosing which of the model's judgments they prefer. This could lead to a more detailed comparison, similar to what has been proposed to discriminate among deep learning models for human vision <cit.> and language <cit.>.A crucial weakness of most studies using large language models is that they can be sensitive to specific prompts <cit.>. While we have attempted to use prompts that elicited good behavior, thereby giving LLMs a chance to perform well, future work could try to further optimize these prompts using available methods <cit.>, while also assessing how the models respond to paraphrased versions of the same tasks. While it could be possible to further engineer the used prompts, as well as try out several other ways of phrasing the same prompt, we believe that our current approach was sufficient to showcase these models' abilities.Finally, we applied all models out of the box and without additional fine-tuning. Future studies could attempt to fine-tune multi-modal LLMs to better align with cognitive data <cit.> and assess if this improves their reasoning abilities more generally. §.§ Related workWe are not the first to assess LLMs' reasoning abilities <cit.>. Previous studies have focused, among others, on testing LLMs' cognitive abilities in model-based planning <cit.>, analogical reasoning tests <cit.>, exploration tasks <cit.>, systematic reasoning tests <cit.>, psycholinguistic completion studies <cit.>, and affordance understanding problems <cit.>. In this sense, our contribution can be seen as a part of a larger movement in which researchers use methods from the behavioral sciences to understand black box machine learning models <cit.>. However, most of the previous studies did not investigate multi-modal LLMs but rather remained in the pure language domain. Although there are recent attempts to investigate vision LLMs cognitive features, including their reaction to visual illusions <cit.> as well as how they solve simple intelligence tasks <cit.>, we are the first to investigate the proposed core components of cognition in these models.Previous work has also looked at how LLMs solve cognitive tasks taken from the same domains that we have looked at. In intuitive physics, Zečević et al. <cit.> found that LLMs performed poorly in a task using language descriptions of physical scenarios. Zhang and colleagues <cit.> extracted programs from text produced by large language models to improve their physical reasoning abilities. Finally, Jassim and colleagues <cit.> proposed a novel benchmark for evaluating multimodal LLMs' understanding of situated physics.In causal reasoning, Binz and Schulz <cit.> showed that GPT-3 failed at simple causal reasoning experiments, while Kosoy et al. <cit.> showed that LLMs cannot learn human-like causal over-hypotheses. In research on intuitive psychology, Kosinsky argued that theory of mind might have emerged in LLMs <cit.> which has been criticized other researchers <cit.>. Akata et al. showed that GTP-4 plays repeated games very selfishly and could not pick up on simple conventions such as alternating between options <cit.>.Finally, Gandhi and colleagues <cit.> proposed a framework for procedurally generating Theory of Mind evaluations and found that GPT4's abilities mirror human inference patterns, though less reliable, while all other LLMs struggled. Many of the past studies on LLMs have fallen risk of appearing in new models' training set. Recent work has recognized this issue and, in turn, evaluated language models on many problem variations to minimize training set effects <cit.>. Our work differs from these approaches as current models could not have just memorized solutions to the given problems because these problems require deep reasoning and are rarely ever published with the ground truth attached. §.§ ConclusionA major plot twist in E.T.A Hoffmann's “Der Sandmann” is that Nathaniel might have been a machine himself, which explains why he fell for Olimpia. This metaphor also relates to our anthropomorphization of LLMs: since LLMs are trained on human-generated data, their behaviors will always reflect our behaviors and biases. However, this reflection is sharpening, and modern neural network architectures have become more human-like. One of the other domains emphasized by Lake et al. was the ability to reason compositionally. Recent attempts have shown that neural networks can perform compositional reasoning if trained appropriately <cit.>. Similarly, our current work has shown that multimodal LLMs have come a long way, showing some correspondence to human behavior and often performing above chance. Moreover, machine learning researchers have put forward various ideas about how to close the remaining gap between humans and machines <cit.>, including self-supervised learning <cit.>, translating from natural into probabilistic languages <cit.>, or grounding LLMs in realistic environments <cit.>. This continuous evolution in models' capabilities necessitates a reevaluation of the metaphors and tools we use to understand them. We believe that cognitive science can offer tools, theories, and benchmarks to evaluate how close we have come to building machines that think like people.§ METHODS §.§ CodeThe open-source models were installed per the instructions on their related Github or Huggingface repositories and evaluated on a Slurm-based cluster with a single A100. For the results reported as GPT-4V, we used the public ChatGPT interface and the OpenAI API, specifically the November 2023 release ofmodel which is available via the completions endpoint. Code for replicating our results is available on GitHub (https://github.com/lsbuschoff/multimodalgithub.com/lsbuschoff/multimodal). All models were evaluated in Python using PyTorch <cit.>. Additional analyses were carried out using NumPy <cit.>, Pandas <cit.>, and SciPy <cit.>. Matplotlib <cit.> and Seaborn <cit.> were used for plotting. §.§ Models§.§.§ GPT4-VWe initially queried GPT-4V through the ChatGPT interface, since the OpenAI API was not publicly available at the outset of this project. The Intuitive Psychology task responses were collected using themodel variant after its November 2023 release in the API. We set the maximum number of generated tokens for a given prompt to 1 to get single numerical responses. All other parameters were set to their default values. Note that this model does not currently feature an option for manually setting the temperature, and the provided documentation does not specify what the default temperature is.§.§.§ FUYUFuyu is an 8B parameter multi-modal text and image decoder-only transformer. We used the Huggingface implementation with standard settings and without further finetuning (available https://huggingface.co/adept/fuyu-8bhere). The maximum number of generated tokens was set to 8 and responses were parsed by hand. §.§.§ ADAPTERWe here use the multi-modal version of LLaMA-Adapter V2, which adds adapters into LLaMA's transformer in order to turn it into an instruction-following model. We used the GitHub implementation with standard settings and again without further finetuning (available https://github.com/OpenGVLab/LLaMA-Adapter/tree/main/llama_adapter_v2_multimodal7bhere). The maximum number of generated tokens was left at 512 and responses were parsed by hand. §.§.§ OTTEROtter is a multi-modal LLM that supports in-context instruction tuning and it is based on the OpenFlamingo model. We used the Huggingface implementation (available https://huggingface.co/luodian/OTTER-Image-MPT7Bhere), again with standard settings and without finetuning. The maximum number of generated tokens was left at 512 and responses were parsed by hand. §.§ Datasets §.§ Intuitive physics We tested the intuitive physical understanding of the models using images from Lerer et al. <cit.>. The photos depict a block tower consisting of colored wooden blocks in front of a white fabric (see Fig. <ref> for an example). The images are of size 224 x 244. In the data set, there are a total of 516 images of block towers. We tested the models on 100 randomly drawn images. We first tested the models on their high-level visual understanding of the scenes: we tasked them with determining the background color and the number of blocks in the image. In order to test their physical understanding, we tested them on the same task as the original study: we asked them to give a binary rating on the stability of the depicted block towers. For the first two tasks, we calculated the percentage of correct answers for each of the models. For the third task, we used the Pearson correlation to determine the degree of similarity between the models and human subjects. §.§ Causal reasoningFor the causality part we used images from Zhou et al. <cit.>. The images show artificial block stacks of red and gray blocks on a black table (see Fig. <ref> for an example). The data set consists of 42 images on which we tested all models. We again first tested the models on their high-level visual understanding of the scene and therefore tasked them with determining the number of blocks in the scene. The ground truth number of blocks in the scenes ranged from 6 to 19. Since this task is rather challenging due to the increased number of blocks, we do not report the percentage correct as for the intuitive physics data set but the mean over the absolute distance between model predictions and the ground truth for each image (see Fig. <ref>A).To test the causal reasoning of the models we adopted the tasks performed in the original study <cit.>. We asked models to infer how many red blocks would fall if the gray block was removed. We again report the absolute distance between model predictions and the ground truth for each image (see Fig. <ref>B). We calculate a random baseline which uses the mean between 0 and the number of blocks for each specific image as the prediction. We also ask the models for a rating between 0 and 100 for how responsible the gray block is for the stability of the tower. For both, the number of blocks that would fall if the gray block was removed, and its' responsibility for the stability of the tower, we calculate the mean Pearson correlation to human subjects from the original study (see Fig. <ref>C).§.§ Intuitive psychologyTo test the intuitive psychology of the different LLMs, we used stimuli from Jara-Ettinger et al. <cit.>. This part consisted of three different experiments each consisting of 16, 17, and 14 images showing a 2D depiction of an astronaut and care packages in different terrains (see Fig. <ref> for an example). In order to check their high-level understanding of the images, we again asked the models to determine the background color of the images. Since this background color is not uniform, we counted both “Pink” and “Purple” as correct answers. We report the percentage of correct answers for the background color in Figure <ref>A.In accordance with the original study, analyses for the intuitive psychological capabilities of the models are split into cost questions (passing through a terrain is associated with a cost for the agent) and reward questions (collecting a care package yields some sort of reward for the agent). We pooled cost and reward questions over all three experiments and reported the mean Pearson correlation with both humans and a heuristic outlined in <cit.> (see Figs. <ref>B and <ref>C). This heuristic calculates the costs and rewards associated with the environment from the amount of time an agent spends in each terrain and which care package it collects. § ACKNOWLEDGEMENTS This work was supported by the Max Planck Society, the Volkswagen Foundation, the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039A, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2064/1 – 390727645.§ AUTHOR CONTRIBUTIONS STATEMENT All authors conceived the experiments. L.B.S. and E.A. conducted the experiments. All authors analysed the results. All authors wrote the manuscript. § EXAMPLE TRIALS§.§ Intuitive physics§.§ Causal reasoning§.§ Intuitive psychology§ PROMPTS§.§ Intuitive physicsFor the intuitive physics experiment, we used a task from Lerer et al. <cit.>. For each trial, we asked the models three questions:§.§ Causal reasoningFor the causality experiment, we used a task from Zhou et al. <cit.>. For each trial, we again asked the models three questions:§.§ Naive Utility CalculusFor the intuitive psychology experiment, we used three tasks from Jara-Ettinger et al. <cit.>. We again asked the model two descriptive questions in order to assess their basic comprehension of the scene:For experiment 1A we first gave the models the following basic prompt, which was combined with different trial specific questions:This task is about astronauts. The astronauts are exploring planets with alien terrains depicted with different colours and textures. Each astronaut has different skills, making each terrain more or less exhausting or easy for them to cross. All astronauts can ultimately cross all terrains, even if it's exhausting. The astronauts land far from the base and have to walk there. In each image, the black circle on the left indicates where the astronaut landed. The base is on the middle right part of the image. Sometimes care packages are dropped from above and the astronauts can pick them up. There are two kinds of care packages depicted with an orange cylinder and a white cube. Each astronaut has different preferences and likes each kind of care package in different amounts. The astronauts don't actually need the care packages. They can go straight to the base, or they can pick one up. You will see images of different astronauts with different skills and preferences travelling from their landing location to the home base. The astronauts always have a map. So they know all about the terrains and the care packages. Please answer the following question with a number only: For experiment 1B we first gave the models the following basic prompt, which was again combined with different trial-specific questions:This task is about astronauts. The astronauts are exploring planets with alien terrains depicted with different colours and textures. Each astronaut has different skills, making each terrain more or less exhausting or easy for them to cross. All astronauts can ultimately cross all terrains, even if it's exhausting. Sometimes, the astronauts land far from the base and have to walk there. In each image, the black circle indicates where the astronaut landed. The base is in the center of the image. Sometimes care packages are dropped from above and the astronauts can pick them up. There are two kinds of care packages depicted with an orange cylinder and a white cube. Sometimes both care packages are identical. The astronauts cannot pick both care packages. Each astronaut has different preferences and likes each kind of care package in different amounts. The astronauts don't actually need the care packages. They can go straight to the base, or they can pick one up. You will see images of different astronauts with different skills and preferences travelling from their landing location to the home base. The astronauts always have a map. So they know all about the terrains and the care packages. Please answer the following question with a number only: Finally, for experiment 1C we first gave the models the following basic prompt:This task is about astronauts. The astronauts are exploring planets with alien terrains depicted with different colours and textures. Each astronaut has different skills, making each terrain more or less exhausting or easy for them to cross. All astronauts can ultimately cross all terrains, even if it's exhausting. The astronauts land far from the base and have to walk there. In each image, the black circle on the left indicates where the astronaut landed. The base is on the right part of the image. The path astronauts take from where they land to their base is indicated by a thick black line between the black circle on the left and the astronaut on the right. Sometimes care packages depicted by a blue cube on a black background are dropped from above and the astronauts can pick them up. Each astronaut has different preferences and likes each care package in different amounts. The astronauts don't actually need the care packages. They can go straight to the base, or they can pick one up. You will see images of different astronauts with different skills and preferences travelling from their landing location to the home base. Your task is to judge how easy/exhausting it is for the astronaut in each image to cross each terrain, and how much they like each care package. The astronauts always have a map. So they know all about the terrains and the care packages. Please answer the following question with a number only: §.§.§ Experiment 1A §.§.§ Experiment 1B§.§.§ Experiment 1C | http://arxiv.org/abs/2311.16093v1 | {
"authors": [
"Luca M. Schulze Buschoff",
"Elif Akata",
"Matthias Bethge",
"Eric Schulz"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20231127185834",
"title": "Have we built machines that think like people?"
} |
Scheduling and Communication Schemes for Decentralized Federated LearningBahaa-Eldin Ali Abdelghany1, A. Fernández-Vilas2, M. Fernández-Veiga2,Nashwa El-Bendary1,Ammar M. Hassan1, Walid M. Abdelmoez1 1Arab Academy for Science, Technology & Maritime Transport, Cairo, Egypt 2atlanTTic, University of Vigo, Spain This work was supported by the Spanish Government under research project “Enhancing Communication Protocols with Machine Learning while Protecting Sensitive Data (COMPROMISE)” PID2020-113795RB-C33, funded by MCIN/AEI/10.13039/501100011033. This work was partially done while Bahaa-Eldin Aliwas with the University of Vigo under EU-funded program KA-107. January 14, 2024 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Federated learning (FL) is a distributed machine learning paradigm in which a large number of clients coordinate with a central server to learn a model without sharing their own training data. One central server is not enough, due to problems of connectivity with clients. In this paper, a decentralized federated learning(DFL) modelwith the stochastic gradient descent (SGD) algorithm has been introduced, as a more scalable approach to improve the learning performance in a network of agents with arbitrary topology. Three schedulingpolicies for DFL have been proposed for communications between the clients and the parallel servers, and the convergence, accuracy,and loss have been testedin a totally decentralized implementation of SGD. Theexperimental results show thatthe proposed scheduling polices have an impact both on the speed of convergence and in the final global model. § INTRODUCTIONData generated at device terminals has recently increased exponentially, owing to the explosive growth of powerful individual computing devices worldwide and the rapid advancement of the Internet of Things (IoT). Data-driven machine learning is becoming a popular technique for making predictions and decisions about future events by making full use of massive amounts of data. Federated Learning (FL), a promising data-driven machine learning variant, provides a communication-efficient approach for processing large amounts of distributed data and is gaining popularity.FL was first proposed as a critical technique of distributed machine learning in a centralised form, in which edge clients perform local model training in parallel and a central server aggregates the trained model parameters from the edge without transmitting raw data from the edge clients to the central server.FL was first tested ona Google Android keyboard (Gboard). It supports multilingual typing, including Google searches and sharing results from the keyboard, as well as auto-correction, voice typing, and glide typing. When Gboard displays some suggestions on the screen based on user behavior, local learning occurs, and FL gains sway by improving future suggestions/interactions with the user. As a result, improved features such as next-word prediction, word completion, corrections, and many more are available. To implement and experiment FL on decentralized data, the following open-source frameworks are in development/available: TensorFlow Federated (TFF) <cit.>, Federated AI Technology Enabler (FATE) <cit.>, PySyft <cit.>, PaddleFL <cit.>, Clara Training Framework <cit.>.FL plays a crucial role in supporting the privacy protection of user data and deploying in a complex environment with massive intelligent terminal access to the network center due to the property of transmitting model parameters instead of user data and the distributed network structure that an arbitrary number of edge nodes are coordinated through one central server. FL operates as a centralized model with decentralized data, which makes the central server a critical point of failure.Besides, in many cases of interest, not every client has a direct connection to the server for learning of the global model <cit.>.Decentralized Federated Learning (DFL) is therefore well suited to implement distributed learning with multiple data aggregators, thus reducing the cost ofcommunication and the workload of the central server. DFL has been proposed and analyzedin the literature under specific updating algorithms in the distributed servers (see <cit.>, but these works rely on a strict and rigid strategyfor the update and communication phases of the protocol. This might be difficult to achieve in general networks, where density and asymmetry are frequent. Consequently, the impact of the scheduling policy between the nodes acting as clients and the servers is a complex decision affecting the global behavior of learning algorithm.In this work,scheduling policies have been proposed for DFL designedto get insights about convergence, loss, and accuracy. The paper is organized as follows. In Section <ref>, the related work is introduced. Next, Section <ref> presents the proposed DFL system model and the main assumptions. The experimental results are presented and discussed inSection <ref>, and finally conclusions appear in Section <ref>.§ RELATED WORKNumerous authors have discussed and created solutions for problems with FL resource allocation in their writings. The FL problem over wireless networks formulated in <cit.> captures the following trade-offs : (1) using the Pareto efficiency model, measuring learning time in relation to customer energy use, and (2) computation versus communication learning time by determining the ideal learning accuracy. In <cit.>, by developing a joint bandwidth allocation and scheduling issue to reduce training time and achieve the desired model accuracy, the authors suggest a method for increasing the convergence rate of FL training concerning time. For the bandwidth allocation problem, they design an efficient binary search algorithm, while for maximum device scheduling, they adopt a greedy approach for achieving a trade-off between the latency and learning efficiency in each round. In <cit.>, The authors define the joint learning, wireless resource allocation, and client selection problem as an optimization problem to minimize the FL loss function. In <cit.>, the authors describe a method for self-organizing FL over wireless networks. They use a heuristic algorithm to minimize global FL time while taking local energy consumption and resource blocks into account.IN <cit.>, the authors suggests a paradigm for evaluating and describing FL performance. For the convergence rate of FL, traceable expressions are generated that consider the impact of inter-cell interference as well as scheduling strategies. They also looked at the efficiency (convergence rate) of scheduling rules such as proportional fair scheduling, round robin scheduling, and random scheduling. Other works have begun examining scheduling strategies influenced by the chances for model improvement during FL rounds. According to the channel circumstances and the importance of local model updates, <cit.> establishes scheduling policies for selecting the subset of devices to handle the transmission inside each round. In wireless networks with clients sharing a single wireless link, the contribution of <cit.> offers a long-term perspective for resource allocation. The method is grounded in experimental observation showing that choosing fewer customers during the initial learning rounds and then gradually increasing this number is the strategy having the best impact on learning performance.In <cit.>, the clients are divided into tiers according to how well they performed during training, and an adaptive tier-based client selection method is used in the authors' proposed Tier-based Federated Learning (TEFL) System. In <cit.>, it is suggested a scheduling strategy to make use of variation in multi-user channels as well as diversity in the significance of edge device learning updates (measured by gradient divergence).In <cit.>, the authors propose a proactive algorithm that selects mobile clients based on predictions of their future training and reporting abilities. The adopted approach is divided into two parts: (1) In a metropolitan mobile edge computing environment, predicting users' mobility trajectory patterns and the apps they use on their smartphones, as well as (2) a deep reinforcement learning-based client-selection algorithm handling unanticipated dynamic events, are all possible. CPU, bandwidth, GPS coordinates, and the success or failure of downloading and uploading local and global parameters are the metrics that are observed and forecast. The authors at <cit.> advocate the decentralized approach that leaves the training data distributed on the edge devices and learns a shared model by aggregating locally computed updates. They show a practical method for FL of deep networks based on iterative model averaging, and conduct an extensive empirical evaluation, considering five different model architectures and four datasets. The Decentralized SGD is a driving engine for FL and its performance is influenced by internode communications and local updates. At <cit.> propose a DFL framework that implements both internode communication periodically and multiple local updates to strike a balance between communication efficiency and model consensus. They establish strong convergence guarantees for the DFL algorithm without the assumption of convex objects. § MODEL OF PEER-TO-PEER DECENTRALIZED FL The graph, the scheduler, and the global parameters aggregation are the three parts of the system model that are defined in this section.the first part explains how edge devices in the graph are connected to one another. the second scheduler plans the aggregators and clients. the third part is the aggregator way will combine the parameters given in the aggregation section as well as the specifics of how the new global model parameters will be calculated using FedAvg. §.§ Graph We consider a community of learning nodes modeled as an undirected graph G = (V, E) with n = |V| nodes. For i, j ∈ V, edge (i, j) ∈ E represents a bidirectional communication link between nodes i and j. Each node is assumed to have access to a local dataset D_i, i ∈ V, containing d_i = |Di | samples of a common unknown distribution D from which we are interested in learning, i.e., in building a statistical model F_θ from a given class, where θ denotes the model parameters which are to be optimized during the learning process. For instance, F_θ can be a (deep) neural network, and θ the weights between its adjacent layers. Different from centralized and federated learning (FL), where only a single node is in charge of building F_θ, in decentralized or distributed learning (DL) we allow each node to learn from its neighbors, possibly in an asynchronous way.§.§ Schedulers The scheduler role is choosing which node will work as an aggregator for the parameters from specific neighbors. We propose, for the graph depicted in Figure <ref>, the three scheduling policies listed in Table <ref>. The notation {X, Y }→ Z is used to denote that nodes X and Y are clients and node Z is aggregation node. The criteria for choosing a node to work as an aggregator is based on the round number and it is expressed by the formula Rounds t = 2k + 1 for odd rounds, Rounds t = 2k for even rounds.We see at Table <ref>, for instance, that node 5 aggregates weights each round in scheduler A. This node was crucial in this scheduler since it aggregates from the other aggregators, so it will hold the most recent model that was averaged across multiple nodes.in every scheduler, the sequence of communications between clients and servers yields, after a pair of rounds, a connected graph.§.§ Decentralized Federated AveragingThe FedAvg Algorithm is the most widely used technique for calculating the global model parameters. We anticipate that the aggregators' average model will satisfy full convergence. McMahan <cit.> proposed FedAvg, in which clients collaboratively send updates of locally trained models to the aggregator node, each client running a local copy of the global model on its local trainning data. The global model weights are then updated with an average of the updates from clients and deployed back to clients. This extends previous training work by not only supplying local models but also performing training on each device locally. As a result, FedAvg may enable clients (particularly those with small datasets) to collaboratively learn a shared prediction model while retaining all training data locally. before aggregators construct the global model. neighbors send new model parameters and old global parameters that were kept from previous rounds. The aggregator sums new parameters and old global parameters from itself and the received from neighbors ,then calculate the average to build the new global parameters by using FedAvg. If any node did not participate in any previous rounds, it will send only new local parameters to the aggregator. as it will not have old global parameters. Fig <ref> Aggregator nodes are coloured blue in the first round and red in the second. In the first and second rounds, we concentrate on node 5. In the first round, node 5 will aggregate weights from neighbors as well as the old global model four GM4. Node 4 created GM4 while acting as an aggregator. After which a new global model was calculated The new global model is known as GM5, which stands for Global model for node 5. In the second round of GM5, weights from neighbors are added to the old global models from previous rounds. The total will be divided by the number of models. The same criteria will be used in subsequent odd and even rounds.Pseudo code <ref> for each round there are one aggregator and set of clients. When the round starts the aggregator will initialize the clients with random weights . clients will get the random weights or last round weightsthat are kept from the aggregator. clients and their aggregator will start executing local computations for epochs. the updates will be sent to the aggregator. The aggregator will combine clients updates with its update then generate new parameters using FedAvg. The new parameters will be sent to clients and each client will save it locally . Clients will use global parameters to initiate their clients when they work as aggregators.after round finished the algorithm extracts a new aggregator. the aggregators determined by schedulers.§ EXPERIMENTAL RESULTS §.§ Environment§.§.§ Flwr The implementation was developed under an Flwr environment with anaconda with python. The Project itself contains one python code. When this code is executed the graph description is read. working on Flwr gives the flexibility to simulate real-world scenarios as we need to simulate the communication noise between nodes, and it can work in simulation mode which is very good at simulating nodes to get results quickly.§.§.§ Google Colab Google-colab is an online website that we can use to run machine learning model notebooks in the cloud. It provides a good GPU and CPU to run ML notebooks smoothly and get results and save outputs so easily.§.§.§ WandB WandB is abbreviated for weight and bias <cit.> it is an online API that is used with our ML Project for teaching and logging results in an interactive way. It is a very helpful tool for good visualizations and creating reports and notes for any ML Model and tracking parameters, supports many formats and representations to download or view results.§.§.§ MNISTThe Modified National Institute of Standards and Technology for handwritten digitswas used in our experiment to assess the proposed DecentralizedFL. §.§ Results<cit.>has many advantages like stability and the capability of operating in real federated learning scenarios. Tracking both the loss and accuracy obtained for aggregator nodes in the graph is considered. Tables <ref> and <ref>, and Figure <ref> demonstrate the accuracy and loss through the rounds.Regardingthese results, we can make the following observations.§.§.§ Scheduler AIn respect of aggregator nodes 4, 5, 2 and 9. the first three rounds, node 4 loss is diminishing. before the last round, between 0.25 and 0.28 The lowest loss in the previous round was 0.24. Node 5 was 0.25 in the first round. Node 5 and subsequent rounds are unstable. It fluctuated between growing and falling, but it eventually fell to 0.24.node 2 at first round loss is 0.28 and accuracy is 91%. rounds 2 ,3 and 4 same loss 0.25 for node 2 lower than first round.rounds through 5 - 9 loss in range 0.28 - 0.29. final round loss for node 2 is 0.29 and accuracy is 91%. In the first round, node 9 has a value of 0.23. Except for rounds six, seven, and eight, loss dropped across the rounds. Loss is 0.24, 0.23, and 0.23, respectively. The last round had the lowest loss of 0.2. At final rounds aggregator accuracy is 92%, whereas node 9 accuracy is 94%. When compared to other aggregators or clients, Node 9 has the highest accuracy and lowest loss.§.§.§ Scheduler BBy monitoring node learning rates It is obvious that every node learning loss will decrease. This means that nodes are growing better with each cycle. Except for node 1, which participates only in even rounds, all nodes lose between 0.21 and 0.26 after the first round. The loss for node 1 in the second round is 0.16. Node 2 has a value of 0.15 while node 9 has a value of 0.12. Because node 2 and node 9 participated in the first round, their losses were lower than node 1. Through the rounds, all nodes lost less. The final round loss for nodes 3, 4, and 5 is 0.07. The loss at node 1 is 0.08. Node 9 has the lowest loss value of 0.06. Note : node 1 not drawn at the chart.at final round, nodes 2,4,5 and 9 accuracies is 97%. §.§.§ Scheduler CThe first loss for aggregators 2, 4, 5, and 9 is 0.29, 0.29, 0.27, and 0.22, respectively. Node 2 loss decreases in all rounds except round nine, where it increases by 0.01. In the final round, node 2 had the lowest loss of 0.08. Node 4 loss is reduced through all rounds, with the exception of the last round, which grew by 0.002 and became 0.097. Even though they lost in the last round, they are still doing well. Except for round nine, node 9 loss is reduced across all rounds. The final round defeat is 0.079 percent more than the previous round loss. Throughout all rounds, the loss of Node 5 is reduced. The rate of change of loss at node 5 is large between rounds, as seen. by looking at the nodes in the network as a whole. Every round, the loss changed and dropped efficiently. For the 10 rounds, the accuracy of all nodes rose by about 7%. They began with an accuracy of roughly 90% and gradually grew to 97%. When compared to other aggregators or clients, Node 9 has the highest accuracy and lowest loss.§ CONCLUSIONS AND FUTURE WORKPerformance in FL depends crucially on whether full or partial participation from the nodes In this paper, thorough a series of case studies, we have shown that the design of thescheduling strategies between clients and servers —where a node can act in different rounds either as a client sending its own model/parameters or as an aggregator for a subset of its nearby neighbors— it is fundamental to balance the trade-off between the precision in the global model, the global learning rate, and the communication costs. We have shown that, with different schedulers, not only convergence to a global model is not substantially affected by the network topology, but that instead the rate of learning at thedifferent nodes in the graph is rather homogeneous, despite slight stochastic variations due to the localtopology. Nevertheless, even though the global model retains a similar quality as compared tocentralized FL, in DFL the scheduling of clients and servers turns out to be essential for minimizing the number of messages exchanged, and also for guaranteeing that all nodes in the system learn the global model at a similar pace. The highest accuracy shown by the proposed model is 97%, and the lowest loss is 0.07.A few promising future directions include measuring communication cost the number of messages exchanged.IEEEtran | http://arxiv.org/abs/2311.16021v1 | {
"authors": [
"Bahaa-Eldin Ali Abdelghany",
"Ana Fernández-Vilas",
"Manuel Fernández-Veiga",
"Nashwa El-Bendary",
"Ammar M. Hassan",
"Walid M. Abdelmoez"
],
"categories": [
"cs.LG",
"cs.DC"
],
"primary_category": "cs.LG",
"published": "20231127173528",
"title": "Scheduling and Communication Schemes for Decentralized Federated Learning"
} |
In a two dimensional annulus A_ρ={x∈^2: ρ<|x|<1}, ρ∈ (0,1), we characterize 0-homogeneousminimizers, in H^1(A_ρ;𝕊^1) with respect to their own boundary conditions, of the anisotropic energy E_δ(u)=∫_A_ρ |∇ u|^2 +δ( (∇· u)^2-(∇× u)^2) dx,δ∈ (-1,1).Even for a small anisotropy 0<|δ|≪ 1, we exhibitqualitative properties very different from the isotropic case δ=0. In particular,0-homogeneous critical points of degree d∉{ 0,1,2}are always local minimizers, but in thick annuli (ρ≪ 1) they are not minimizers: the 0-homogeneous symmetry is broken. One corollary is thatentire solutions to the anisotropic Ginzburg-Landau system havea far-fieldbehavior very different from the isotropic casestudied by Brezis, Merleand Rivière. The tools we use include: ODE and variational arguments; asymptotic expansions, interpolation inequalities and explicit computations involving near-optimizers of these inequalities for proving that 0-homogeneous critical points are not minimizers in thick annuli.Work distribution of a colloid in an elongational flow field and under Ornstein-Uhlenbeck noise Rati Sharma January 14, 2024 =============================================================================================== § INTRODUCTION For any open set Ω⊂^2 and 𝕊^1-valued map u∈ H^1(Ω;𝕊^1), and given an anisotropy parameter δ∈ (-1,1), we consider the anisotropic energyE_δ(u;Ω) =∫_Ω |∇ u|^2 +δ( (∇· u)^2-(∇× u)^2) dx.The energy density admits the alternative form (1+δ) (∇· u)^2 + (1-δ) (∇× u)^2: this followsfrom the identity |∇ u|^2=(∇· u)^2+(∇× u)^2 + 2 (∇ u), where the last term is zero for u∈ H^1(Ω;𝕊^1). Thisenergy arises in liquid crystal models, see e.g. <cit.>. The energy density is the most general positive definite quadratic form of ∇ u which is compatible with frame invariance:for any angle α∈, the transformationu(x) ⟶ e^-iαu(e^iαx)leaves the energy invariant. Critical points of E_δ in H^1(Ω;𝕊^1), characterized by.d/dt|_t=0 E_δ( ξ_δ +tφ/|ξ_δ +tφ|;Ω)=0∀φ∈ C_c^1(Ω;^3), satisfy the Euler-Lagrange equationℒ_δ u =λ u,λ=u·ℒ_δ u,where the linear operator ℒ_δ is given byℒ_δ u=-Δ u -δ( ∇ (∇· u)-∇^⊥ (∇× u)),and the function λ in (<ref>) can be interpreted as a Lagrange multiplier for the constraint u(x)∈𝕊^1.The goal of this work is to exhibitnontrivial effects of the anisotropy on certain critical points of the energy. This is made manifest in the form of a symmetry breaking for minimizerswithin a givenclass,even when the anisotropy is small.To be precise, we consider the case of an annulusΩ=A_ρ={ x∈^2ρ <|x| <1},ρ∈ (0,1), and are interestedin properties of 0-homogeneous critical points: in polar coordinates x=re^iθ, they depend only on the θ variable.The main question we ask is: are 0-homogeneous critical points minimizers with respect to their own boundary conditions?§.§.§ Basic facts In the isotropic case δ=0, the equation (<ref>)becomes Δφ=0 for u=e^iφ. All 0-homogeneous solutions aregiven byu(re^iθ)=e^iαe^idθ,α∈, d∈ℤ,and they are minimizers within their own homotopy class, characterized by the degree or winding number,d=(u) =1/2π∫_0^2πu̅(re^iθ)∂_θ u(re^iθ)dθ∈∀ r∈ [ρ,1].This is well-defined anddoes notdepend on r because the trace of u∈ H^1(A_ρ;𝕊^1) on ∂ D_r belongs to H^1/2(∂ D_r;𝕊^1) for all r∈ [ρ,1], see e.g. <cit.>. Specifically, the lower bound∫_A_ρ|∇ u|^2dx ≥ 2π d^2 |lnρ|∀ u∈ H^1(A_ρ,𝕊^1) with (u)=d,is attained exactly at the one-dimensional family of 0-homogeneous maps u(re^iθ)=e^iαe^idθ, α∈.In the anisotropic case δ≠ 0, the lower boundE_δ(u;A_ρ) ≥ (1-|δ|)∫_A_ρ|∇ u|^2dx ≥ (1-|δ|)2π d^2 |lnρ|,issharp only when d=1, and attained atthe mapsu(re^iθ)=e^iαe^iθ,α= 0mod πif δ∈(-1,0), π/2 mod πif δ∈ (0,1).Notice that we no longer have a one-dimensional family of minimizers.It can be checked that these maps are the only 0-homogeneous solutions of (<ref>) in A_ρ whichare critical with respect to perturbations of their boundary conditions. They are also the only 0-homogeneous solutions of (<ref>) with degree d=1, as shown in <ref>. Solutions of degree d≠ 1 seem largely unexplored.§.§.§ Main resultThe scaling invariance of the energy ensuresthe existence of at least one 0-homogeneous critical point of any degree d. For d≠ 1, our main result asserts that it is unique modulo frame invariance (<ref>) and linearly stable, but when the hole (ρ≪ 1) and the anisotropy (0<|δ|≪ 1) are small, it is nota minimizer with respect to its own boundary conditions, providedd∉{ 0,1, 2}. Let δ∈ (-1,1) and d∈∖{ 1}. * All 0-homogeneous solutions of degree d of the Euler-Lagrange system (<ref>) are given by a single one-dimensional familyξ_δ^α(x)=e^-iαξ_δ(e^iαx),α∈. * The unique (modulo frame invariance) 0-homogeneous critical point ξ_δ is linearly stable in A_ρ for all 0<ρ<1: there exists a constant c>0 depending on δ and ρsuch that.d^2/dt^2|_t=0 E_δ( ξ_δ +tφ/|ξ_δ +tφ|;A_ρ)≥ c∫_A_ρ |∇φ|^2dxfor all φ∈ C_c^1(A_ρ;^2) such that φ·ξ_δ=0 a.e. * For small enough |δ|>0 and d∉{ 0,1,2}, there exists a critical value ρ_*=ρ_*(δ,d)∈ (0,1) such that the 0-homogeneous critical point ξ_δ isa minimizerin A_ρ for ρ>ρ_* but not a minimizer for ρ<ρ_*:min_u_⌊∂ A_ρ = ξ_δE_δ(u ;A_ρ) =E_δ(ξ_δ;A_ρ) if 1>ρ≥ρ_*,<E_δ(ξ_δ;A_ρ) if 0<ρ<ρ_*,where the minimum is taken over all maps u∈ H^1(A_ρ;𝕊^1) such that u=ξ_δ on ∂ A_ρ. We make here a few observations about the statements in Theorem <ref>: * The case of degree d=-1 is the most important from the physical point of view, since only defects of degree d∈{± 1} areexperimentally stable(see e.g. <cit.>). * The third item requires small anisotropy 0<|δ|≪ 1, but the first two items are valid for any δ∈ (-1,1). * The existence of a one-dimensional family of 0-homogeneous critical points of degree d≠ 1, as in the isotropic case δ=0, is in strong contrast with what happens for d=1, where that one-dimensional symmetry is broken for δ≠ 0.We show in <ref> that for 0<|δ|<1, the trivial solutions u(re^iθ)=e^iαe^iθ, α≡ 0 modulo π/2, are the only 0-homogeneous solutions of degree d=1.* The uniqueness statement in the first item of Theorem <ref> implies that 0-homogenous solutions of degree d≠ 1 enjoy discrete symmetry properties: the map ξ_δ satisfiesξ_δ(e^iπ/|d-1|x)=-e^iπ/|d-1|ξ_δ(x).Indeed, this symmetry constraint is compatible with the energy as noted in <cit.>, and u(re^iθ)=e^idθ satisfies (<ref>) for any d∈ℤ∖{ 1}, hence minimizing (<ref>) among 0-homogeneous maps of degree d with this symmetry constraint produces one symmetric solution ξ_δ^sym. The symmetry (<ref>)is preserved under frame invariance (<ref>), so the one-dimensional family generated by ξ_δ^sym satisfies it,and by uniqueness it agrees with the one-dimensional family generated by ξ_δ in Theorem <ref>. * The linear stability of ξ_δ can be used to show that it isa local minimizer among maps u∈ H^1(A_ρ;𝕊^1) agreeing with ξ_δ on ∂ A_ρ, but the neighborhood in which it is a minimizerdegenerates for small values ofρ, see Proposition <ref>. The critical value ρ_* in the third item of Theorem <ref> satisfiese^- C|δ|^-1≤ρ_*(δ) ≤ e^- (C|δ|)^-1/3 for a large constant C>0 depending on the degree d, as can be inferred from (<ref>) and Proposition <ref>. * The degree 2 case is different: the unique family of 0-homogeneous solutions is given by ξ_δ^α(re^iθ) =e^iαe^2iθ, and it is a minimizer in A_ρ for all 0<ρ<1, see <ref>.§.§.§ Comparison with minimizing maps in higher dimensionsIn dimension n≥ 3, tangent harmonic maps ^n→𝒩 with values into a riemannian manifold 𝒩, that is, blow-up limits of 𝒩-valued maps minimizing the isotropic energy ∫ |∇ u|^2, are 0-homogeneous <cit.>. This is the key reason why minimizing harmonic maps are known to have a singular set of dimension at most n-3, while optimal regularity estimates for minimizers of anisotropic energies are open <cit.>. Homogeneity of the isotropic tangent maps is due to the decoupling of the energydensity into radial and angular derivatives:|∇ u|^2=|∂_r u|^2+ 1/r^2|∇_ω u|^2.In our two-dimensional setting, this is the same decoupling which provides the lower bound (<ref>)in the isotropic case δ=0. In the absence of such decoupling, it seemshard to determine whether tangent maps are 0-homogeneous.Since tangent maps areminimizers with respect to their own boundary conditions,one way to gain insightinto that question is to investigatewhether 0-homogeneous mapsare minimizers. Our results, by showing in a particular two-dimensional case thatanisotropy prevents many 0-homogeneous maps from being minimizers, therefore suggest thattangent maps for minimizers of anisotropic energies in dimension n≥ 3might fail to be 0-homogeneous in some cases.§.§.§ Consequences for the anisotropic Ginzburg-Landau energy Energy-minimizing maps from an annulus(and more generally a domain with small holes)into 𝕊^1are strongly relevant to the analysis of the anisotropic Ginzburg Landau energyGL_δ,(u;Ω)= ∫_Ω1/2 |∇ u|^2 +δ/2( (∇· u)^2-(∇× u)^2)+1/4^2(1-|u|^2)^2 dx,and the corresponding anisotropic Ginzburg-Landau equationℒ_δ u =1/^2(1-|u|^2)u,for maps uΩ→^2. The very different nature of defects of degree d=-1 versus d=1unveiled by Theorem <ref>will have repercussions on a negative degree counterpart of the analysis performed in <cit.> for minimizers ofGL_δ,(·;Ω) with boundary data g∂Ω→𝕊^1 of positive degree.The symmetry breaking demonstrated by Theorem <ref> also has consequences on the far-field asymptotics (r→∞) of entire solutions u^2→^2 to the anisotropic Ginzburg-Landau equation, via the scaling argument of <cit.>. More specifically, in the entire plane ^2 the length-scalecan be set to =1, and we consider maps u^2→^2 which solve the anisotropic Ginzburg-Landau equation ℒ_δu=(1-|u|^2) u in ^2,associated to the energy GL_δ=GL_δ,1 given byGL_δ(u;Ω)= ∫_Ω1/2 |∇ u|^2 +δ/2( (∇· u)^2-(∇× u)^2)+1/4(1-|u|^2)^2 dx,with finite potential energy∫_^2(1-|u|^2)^2dx <∞.Such u has a well-defined degree d=(u)=(u/|u|;∂ D_R)∈ℤ for R≫ 1.In the isotropic case δ=0,solutions of any degree d can be constructed using a radial ansatz u(re^iθ)=f_d(r)e^idθ <cit.>, and all solutions satisfy a quantization property for their potential energy <cit.>. This quantization is obtained as a consequence of a Pohozaev identity andfar field asymptotics u(re^iθ)→ e^idθ as r→∞ in an appropriate sense, entailing for instance∫_^2 |∂_r u|^2dx <∞.In the anisotropic case 0<|δ|<1, and for degrees d∉{ 0,1},the mere existence of solutions is unknown in full generality.For small anisotropy |δ|≤δ_0(d) and negative degree d≤ -1, solutionswere constructed in <cit.> via aminimization procedure under the discrete rotational symmetry constraint mentioned in Remark <ref>,u(e^iπ/|d-1|x)=-e^iπ/|d-1|u(x)∀ x∈^2.Large radius asymptotics in the spirit of <cit.> seem unexplored, apart from formal calculations for d=-1 and |δ|≪ 1 in <cit.>. As a consequence of the third point in Theorem <ref>,we obtain that these asymptotics behave very differently from the isotropic case. For any d∈ℤ∖{ 0,1,2} there exists δ_0∈ (0,1) with the following property. Letu∈ H^1_loc(^2;^2) be a solution of the anisotropic Ginzburg-Landau equation (<ref>) with finite potential energy (<ref>) and degree (u)=d. If 0<|δ|<δ_0and u is either locally minimizing:GL_δ(u;D_R) ≤ GL_δ(v;D_R),∀ v∈ H^1(D_R;^2),v_⌊∂ D_R=u_⌊∂ D_R,or symmetric (<ref>) and locally minimizing with respect to symmetric competitors, then we have∫_^2|∂_r u|^2dx =+∞,and the maps u_R𝕊^1→^2 given by u_R(θ)= u(Re^iθ) do not converge as R→ +∞ (in the sense of distributions).Using the methods in <cit.>, one can show that a locally minimizing solution must be of degree d∈{ -1,0,1}, but existence of a locally minimizing solution of degree -1 is unknown. However, Corollary <ref> applies to the symmetric solutions of degree d≤ -1 constructed in <cit.>. More precisely, the solutions constructed in <cit.> satisfy an additional mirror symmetry constraint u(x̅)=αu̅(x) for some α∈{± 1}, but the same proof provides existence of solutions which are locally minimizing under the symmetry constraint (<ref>) only. (At the level of ξ_δ, the additional mirror symmetry only has the effect of selecting a value of ξ_δ(0) in {± 1} or {± i}.) Moreover, it will be clear from the proof that Corollary <ref> also applies to symmetric solutions which are locally minimizing under that additional mirror symmetry constraint. §.§.§ Sketch of proof of Theorem <ref> To prove Theorem <ref>, we start by showing that any 0-homogeneous solution ξ is linearly stable. We achieve thisusing identities satisfied by the Jacobi field w=(d/dα)|_α =0[ξ^α] generated by the symmetry (<ref>), ξ^α(x)=e^-iαξ(e^iαx).(The idea of proving stability via a Jacobi-field-based decomposition is classical, see e.g. <cit.>.) When restricting the energy to 0-homogeneous maps, this linear stability implies local minimality of the solution ξ. Since this is valid for any 0-homogeneous solution ξ, uniqueness follows:in the presence of two distinct (modulo frame invariance) solutions, a non-locally-minimizing solution could be obtained by aclassical mountain pass argument and would provide a contradiction(an earlier implementation of this kind of argument in the context of Ginzburg-Landau can be found in <cit.>). In that way we obtain the first two items of Theorem <ref>.For the third item, we wish to show that, for |δ|≪ 1, the 0-homogeneous solution ξ_δ is minimizing in A_ρ if ρ is not too small, but notminimizing if ρ≪ 1. A formal expansion of the energy for small perturbations aroundξ_δ gives quadratic terms that are positive thanks to the linear stability,and remainder terms which are formally of lower order. Estimating these remainder terms to absorb theminto the positive quadratic terms proveslocal minimality of ξ_δ. This requires adequate interpolation inequalities, but the constants involved in these inequalities behave badly as ρ→ 0 andthe neighborhood of local minimality becomes very small.On the one hand, when ρ is not too small this is enough to deduce minimality, using the fact that for |δ|≪ 1any minimizer must be close to the isotropic minimizer e^idθ and belong therefore to the neighborhood of local minimality of ξ_δ.On the other hand, for very small ρ, identifying near-optimizers for the interpolation inequalities provides a reasonable guess of a perturbation of ξ_δ which would produce negative remainder terms that cannot be compensated by the positive quadratic terms. In order to check that this reasonable guess actually works, we determine an expansion ξ_δ =e^idθ +δζ_1 +δ^2ζ_2 +𝒪(δ^3), and deduce an explicit expression of the bad part of the remainder terms.Choosing appropriate values for ρ and the amplitude of the perturbation then ensuresthat all non-explicit terms are controlled, and eventually produces a lower energy. §.§ Plan of the article In Section <ref> we study 0-homogenous critical points and prove the first two items of Theorem <ref>. In Section <ref> we prove the third item, namely that ξ_δ is minimizing for ρ≈ 1 but not minimizing for small ρ, when |δ|≪ 1. In Section <ref> we treat the particular cases d∈{ 1,2}. In Section <ref> we prove Corollary <ref>. §.§ Acknowledgments The work of A.C. was partially supported by a grant from the Simons foundation # 426318. XL is supported by the ANR project ANR-22-CE40-0006.Part of this work was conducted during XL’s stays at the Center for Mathematical Modeling (CMM, Santiago de Chile), with the financial support of CNRS and the CMM;at the Hausdorff Institute for Mathematics (HIM) in Bonn, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813, during the Trimester Program “Mathematics for Complex Materials”; and during the stay of both authors at the Centre International de Rencontres Mathématiques (CIRM) as part of the Research in Residence program 2455. § STABILITY AND UNIQUENESS OF 0-HOMOGENEOUS CRITICAL POINTS In this section we study 0-homogeneous solutions of the Euler-Lagrange system (<ref>) and prove the first two items of Theorem <ref>. With a slight abuse of notation, we identify a 0-homogeneous 𝕊^1-valued map ξ with a map depending only on the polar angle θ:ξ(re^iθ)=ξ(θ),ξ∈ H^1(𝕊^1;𝕊^1).In that context, the equation (<ref>) can be rewritten asℒ_δξ=λ̂ξ,λ̂=ξ·ℒ_δξ,where the reduced linear operator ℒ_δ is given byℒ_δξ =-∂_θ^2ξ-δ ∂_θ[ (∂_θξ· ie^iθ)ie^iθ-(∂_θξ· e^iθ)e^iθ]Solutions of (<ref>)correspond exactly to critical points of the reduced energyE_δ(ξ)=∫_𝕊^1 |ξ'|^2 +δ( (ξ'· ie^iθ)^2 - (ξ'· e^iθ)^2)dθ.There exists at least one solution of degree d, obtained by minimizing (<ref>) among maps of degree d, since the degree is continuous under weak convergence in H^1(𝕊^1;𝕊^1). We start by proving that any solutionξ∈ H^1(𝕊^1;𝕊^1)of (<ref>) with degree d≠ 1 generates, via frame invariance (<ref>), a non-vanishing Jacobi field. Let|δ|<1 and ξ∈ H^1(𝕊^1;𝕊^1) with degree d∈ℤ∖{ 1} solve the reduced equation (<ref>). Then ξ∈ C^∞(𝕊^1;𝕊^1), and the Jacobi field w∈ C^∞(𝕊^1;^2) given byw(θ)=1/d-1.d/dα|_α=0[ξ^α(θ)],ξ^α(θ)=e^-iαξ(θ +α) ,satisfies|w|>0 in 𝕊^1. For any ξ∈ H^1(𝕊^1;𝕊^1) of degree d, there is a lifting φ∈ H^1(𝕊^1;) such thatξ(θ)=e^idθe^iφ(θ)∀θ∈ℝ.In terms of this liftingthe energy is of the formF_δ(φ)=E_δ(e^idθe^iφ)=∫_𝕊^1(1+δcos(2(d-1)θ +2φ ))(d+φ')^2dθ,and the Euler-Lagrange equation (<ref>) becomesd/dθ[(1+δcos(2(d-1)θ +2φ ))(d+φ')]=-δsin(2(d-1)θ +2φ )(d+φ')^2.This implies that φ∈ C^∞(𝕊^1;) (see e.g. <cit.>) and therefore ξ∈ C^∞(𝕊^1;𝕊^1) for any solution of (<ref>). For any α∈, we haveξ^α(θ)=e^idθe^i T_αφ(θ),T_αφ(θ)=φ(θ+α)+(d-1)α.In terms of φ, the Jacobi field w can be explicitly computed and is given byw=φ'+d-1/d-1 iξ.To prove that w does not vanish, we show that φ' cannot take the value (1-d). To that end, we first note that, for any ψ_0∈, the functionsψ(θ)=ψ_0-dθ,ψ(θ)=ψ_0 +(2-d)θ,are solutions of (<ref>), as can be checked by a direct calculation. As a consequence, φ' cannot take the values { -d,2-d}, unless it is constant: if φ'(θ_0)∈{ -d,2-d}, then φ and one of the above solutions ψ have same value and derivative at θ_0, and are therefore equal by uniqueness of the Cauchy problem for the ODE (<ref>).The cases where φ' is constant equal to -d or 2-d can only occur if d∈{ 0,2} since φ is periodic, and then w obviously does not vanish. Otherwise, we have on the one hand φ'()⊂∖{ -d,2-d},and on the other hand0∈φ'() because φ is periodic and smooth. If d≤ 0 we deduce φ'≤ -d < 1-d, and ifd≥ 2 we deduce φ'≥ 2-d >1-d.In both cases, this implies that w does not vanish.Since α↦ T_αφ(0) is surjective onto (because φ is bounded and d≠ 1),we may always choose α∈ such that T_αφ(0)=0, or equivalently ξ^α(0)=1.Next we use the fact that w does not vanish, and that it solves the linearized equationℒ_δ w -λ̂w=μ̂ξ, μ̂=ℒ_δ w ·ξ, to prove that the homogeneous critical point ξ is linearly stable for the reduced energy E_δ.This will be enough to deduce uniqueness modulo frame invariance (the first item of Theorem <ref>), and will serve as a warm-up to the proof of linear stability for the full energy E_δ (the second item of Theorem <ref>). Let |δ|<1 and ξ∈ H^1(𝕊^1;𝕊^1) with degree d∈ℤ∖{ 1} solve the reduced equation (<ref>). Then for all φ∈ H^1(𝕊^1;^2) we have1/2.d^2/dt^2|_t=0E_δ( ξ+tφ/|ξ+tφ|)=Q_ξ[φ-(ξ·φ)ξ], Q_ξ[v] = ∫_𝕊^1( |v'|^2 +δ( (v'· ie^iθ)^2 - (v'· e^iθ)^2)-λ̂|v|^2 ) dθ,with λ̂=ℒ_δξ·ξ as in (<ref>). For any tangent field v∈ H^1(𝕊^1;^2) with v·ξ=0 a.e., there is f∈ H^1(𝕊^1;) such that v=fw, where w is the smooth Jacobi field generated by ξ as in Lemma <ref>, andQ_ξ satisfies the coercivity inequalityQ_ξ[v]=Q_ξ[fw]≥ (1-|δ|) ∫_𝕊^1|f'|^2 |w|^2dθ. First we establish the expression of Q_ξ.We start with a preliminary calculation which is also of independent interest. For any u∈ H^1(𝕊^1;𝕊^1), letting v=u-ξ and integrating by parts we findE_δ(u)-E_δ(ξ) =E_δ(ξ+v)-E_δ(ξ)= ∫_𝕊^1( |v'|^2 +δ( (v'· ie^iθ)^2 - (v'· e^iθ)^2) + 2ℒ_δξ· v ) dθ.Using that ξ solves (<ref>) this becomesE_δ(u)-E_δ(ξ) = ∫_𝕊^1( |v'|^2 +δ( (v'· ie^iθ)^2 - (v'· e^iθ)^2) + 2λ̂ ξ· v ) dθ.And recalling that 1=|u|^2=|v|^2+2v·ξ +1, we rewrite the last term using ξ· v=-|v|^2/2 and findE_δ(u)-E_δ(ξ) = Q_ξ [u-ξ],with Q_ξ defined as in Lemma <ref>. Noting that φ_∞≤ C φ_H^1 and applying this to u=ξ+tφ/|ξ+tφ|=ξ + t[φ-(ξ·φ)ξ] +t^2ψ_t,ψ_t_H^1≤ C(ξ,φ),for t≤ 1/(2+φ_∞), we deduceE_δ( ξ+tφ/|ξ+tφ|) =t^2Q_ξ[φ-(ξ·φ)ξ] +𝒪(t^3),which proves the claimed expression for the second derivative at t=0.Next we prove the coercivity of Q_ξ. Let v∈ H^1(𝕊^1;^2) such that v·ξ=0 a.e. Since the smooth Jacobi-field also takes values orthogonal to ξ and does not vanish, this implies v(θ)=f(θ)w(θ) for some real-valued f(θ) and a.e. θ∈𝕊^1, and f=|w|^-2v· w∈ H^1(𝕊^1;). Integrating by parts we findQ_ξ[fw]=∫_𝕊^1[ ℒ_δ(fw)-λ̂fw ]· fw dθ.To simplify that expression we computeℒ_δ (fw )=fℒ_δ w -f”w -2 f' w' -δd/dθ[f' ( (w· ie^iθ)ie^iθ-(w· e^iθ)e^iθ) ]-δ f' ( (w'· ie^iθ)ie^iθ - (w'· e^iθ)e^iθ),and deduceℒ_δ (fw )· fw =f^2 ℒ_δ w · w + (f')^2( |w|^2 +δ (w·ie^iθ)^2 -δ (w· e^iθ)^2) -d/dθ[ ff' (|w|^2+ δ (w·ie^iθ)^2 -δ (w· e^iθ)^2) ]Coming back to the expression of Q_ξ we findQ_ξ[f w ]=∫_𝕊^1[f^2( ℒ_δ w -λ̂w) ·w + (f')^2 (|w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2 ) ]dθ.Finally we use the facts that the Jacobi field w solves the linearized equation (<ref>) and w·ξ=0 to simplify the above toQ_ξ[f w ]=∫_𝕊^1 (f')^2 (|w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2 )dθ.The coercivity inequality of Lemma <ref> then follows from the pointwise inequality |w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2≥ (1-|δ|)|w|^2. The next step is to turn the linear stability proved in Lemma <ref> into a local minimality statement. Let |δ|<1 and ξ∈ H^1(𝕊^1;𝕊^1) with degree d∈ℤ∖{ 1} solve the reduced equation (<ref>). There exist c,η>0 such that E_δ(u)≥E_δ(ξ) + c inf_α∈u-ξ^α_H^1^2,for all u∈ H^1(𝕊^1;𝕊^1) such thatinf_α∈u-ξ^α_H^1<η.Let u∈ H^1(𝕊^1;𝕊^1). We assume without loss of generality (since the statement is invariant under application of the change of frame transformation (<ref>) to ξ) that u-ξ_H^1=inf_α∈u-ξ^α_H^1 <η,with η to be chosen later. We define v=u-ξ and writev=fw+gξ, f=v· w/|w|^2,g=v·ξ∈ H^1(𝕊^1;).We first gather some estimates on f and g. The identity1=|u|^2=|ξ+v|^2=1+ 2g +g^2+f^2|w|^2, implies g=-1±√( 1- f^2|w|^2), where the sign ± may depend on θ.Butby Sobolev embedding and the explicit expressions of f,g in terms of v we havef_L^∞ +g_L^∞≤ c v_H^1≤ cη,for some generic constant c>0 depending on ξ. Hence choosing η small enough ensures that g=√(1-f^2|w|^2)-1 and |g|≤ c|f|^2. Combining this with Sobolev embedding for fwe deduceg_L^∞ +f_L^∞^2 ≤ cf_H^1^2.Further, minimality of α=0 in (<ref>) implies0=∫_𝕊^1( v· w +v'· w')dθ=∫_𝕊^1f (|w|^2+|w'|^2)dθ+∫_𝕊^1 (f' w· w' + g' ξ· w' + g ξ'· w' )dθ,and combining this with the Poincaré inequality∫_𝕊^1ϕ^2dθ≤ c ∫_𝕊^1(ϕ')^2dθ if ∫_𝕊^1ϕ(|w|^2+|w'|^2)dθ=0,we inferf_H^1^2 ≤ c∫_𝕊^1 (f')^2dθ + ∫_𝕊^1(g')^2dθ + g_∞^2.Estimating the last term with (<ref>) yieldsf_H^1^2 ≤ c∫_𝕊^1 (f')^2dθ + c ∫_𝕊^1(g')^2dθ + f_H^1^4.Taking into account that f_H^1≤ cv_H^1≤ cη and choosing η small enough, the last term can be absorbed into the left-hand side and we are left withf_H^1^2 ≤ c∫_𝕊^1 (f')^2dθ + c ∫_𝕊^1(g')^2dθ.And combining this with (<ref>) leads tog_L^∞ +f_L^∞^2 ≤ c∫_𝕊^1 (f')^2dθ +c ∫_𝕊^1(g')^2dθNow we let B_ξ denote the symmetric bilinear form on H^1(𝕊^1;^2) associated to the quadratic form Q_ξ. Thanks to (<ref>) we haveE_δ(u)-E_δ(ξ) = Q_ξ [v] =Q_ξ[fw] +2B_ξ[fw,gξ]+Q_ξ[gξ].Using the same calculations as in Lemma <ref> we findQ_ξ[gξ]= ∫_𝕊^1[g^2( ℒ_δξ -λ̂ξ) ·ξ+ (g')^2 (|ξ|^2 + δ (ξ· ie^iθ)^2-δ (ξ· e^iθ)^2 ) ]dθ= ∫_𝕊^1 (g')^2 (1 + δ (ξ· ie^iθ)^2-δ (ξ· e^iθ)^2 )dθ.For the lastequality we used ℒ_δξ=λ̂ξ and |ξ|=1. Plugging this and the expression of Q_ξ[fw] obtained in Lemma <ref> into (<ref>) we findE_δ(u)-E_δ(ξ)= ∫_𝕊^1 (f')^2 (|w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2 )dθ + ∫_𝕊^1 (g')^2 (1 + δ (ξ· ie^iθ)^2-δ (ξ· e^iθ)^2 )dθ +2B_ξ[fw,gξ].The bilinear form B_ξ is given byB_ξ [v_1,v_2]=∫_𝕊^1 [ ℒ_δ v_1 · v_2 -λ̂v_1· v_2 ]dθ.Applying it to v_1=fw and v_2=gξ, the last term disappears because w·ξ=0, and sinceℒ_δ (fw )=fℒ_δ w -f”w -2 f' w' -δd/dθ[f' ( (w· ie^iθ)ie^iθ-(w· e^iθ)e^iθ) ]-δ f' ( (w'· ie^iθ)ie^iθ - (w'· e^iθ)e^iθ),we find, using that ℒ_δ w=λ̂w+μ̂ξ and w·ξ=0,ℒ_δ (fw ) · gξ =μ̂fg -2 f'g (w'·ξ) -δd/dθ[gf' { (w· ie^iθ)(ξ· ie^iθ)-(w· e^iθ)(ξ· e^iθ)}] +δ g'f' [ (w· ie^iθ)(ξ· ie^iθ)-(w· e^iθ)(ξ· e^iθ) ] +δ g f' [ (w· ie^iθ)(ξ'· ie^iθ)-(w· e^iθ)(ξ'· e^iθ) ]-δ gf' [ (w'· ie^iθ)(ξ· ie^iθ) - (w'· e^iθ)(ξ· e^iθ)],and eventuallyB_ξ[fw,gξ]=∫_𝕊^1(μ̂fg -2 f'g (w'·ξ))dθ + δ∫_𝕊^1( g'f' [ (w· ie^iθ)(ξ· ie^iθ)-(w· e^iθ)(ξ· e^iθ) ] +g f' [ (w· ie^iθ)(ξ'· ie^iθ)-(w· e^iθ)(ξ'· e^iθ) ] -gf' [ (w'· ie^iθ)(ξ· ie^iθ) - (w'· e^iθ)(ξ· e^iθ)] )dθ.Plugging this into (<ref>) we obtainE_δ(u)-E_δ(ξ)= ∫_𝕊^1[(f')^2|w|^2+(g')^2 +2μ̂fg -4 f'g (w'·ξ) ]dθ +δ∫_𝕊^1( (f')^2 ( (w· ie^iθ)^2-(w· e^iθ)^2)+ (g')^2 ( (ξ· ie^iθ)^2-(ξ· e^iθ)^2)+ 2 g'f' [ (w· ie^iθ)(ξ· ie^iθ)-(w· e^iθ)(ξ· e^iθ) ] )dθ+2δ∫_𝕊^1 gf' ( (w· ie^iθ)(ξ'· ie^iθ)-(w· e^iθ)(ξ'· e^iθ) - (w'· ie^iθ)(ξ· ie^iθ) + (w'· e^iθ)(ξ· e^iθ) )dθ.The integrand in the second integral is of the form A(f'|w|,g)· (f'|w|,g), with a symmetric matrix A given byA= ( [ a_1^2-a_2^2 a_1b_1-a_2b_2; a_1b_1-a_2b_2 b_1^2-b_2^2 ]), a_1=w/|w|· ie^iθ,a_2=w/|w|· e^iθ, b_1 =ξ· ie^iθ,b_2=ξ· e^iθ.The vectors a=(a_1,a_2), b=(b_1,b_2) satisfy |a|=|b|=1 and a· b=0, so writing a=e^iα, b=e^iβ with β=α+π/2 mod π, we findA= ( [ cos(2α) cos(α +β); cos(α+ β) cos(2β) ]) = ( [cos(2α) ±sin(2α); ±sin(2α) -cos(2α) ]),hence A=-1, tr A=0 and A has eigenvalues ± 1.This implies that the integrand in the second integralof (<ref>) has absolute value ≤ (f')^2|w|^2 +(g')^2, and we deduceE_δ(u)-E_δ(ξ) ≥ (1-|δ|) ∫_𝕊^1[(f')^2|w|^2+(g')^2]dθ -c_1∫_𝕊^1 |fg|dθ -c_2 ∫_𝕊^1 |f'g|dθ.with c_1=2μ̂_∞ and c_2=4(2w'_∞ +w_∞ξ'_∞). Recalling (<ref>) we deduceE_δ(u)-E_δ(ξ) ≥ (1-|δ|) ∫_𝕊^1[(f')^2|w|^2+(g')^2]dθ -c(∫_𝕊^1(f')^2dθ)^3/2 -c(∫_𝕊^1(g')^2dθ)^3/2We deduce from thisthatE_δ(u)-E_δ(ξ) ≥ (1-|δ|)min( 1,inf|w|^2)/2∫_𝕊^1[(f')^2+(g')^2]dθ,if (f,g)_H^1≤v_H^1 is small enough (depending on ξ and δ).Finally we remark that u-ξ_H^1 =v_H^1≤ c (f,g)_H^1, and using again (<ref>) we have (f,g)_H^1≤ cf'_L^2+cg'_L^2, soE_δ(u)-E_δ(ξ) ≥ c u-ξ_H^1and this concludes the proof of Lemma <ref>. The constants c,η in Lemma <ref> depend only on M,m>0 such that ξ_C^2≤ M and|w|≥ m, as can be checked directly from the proof.Since ξ solves (<ref>), its C^2 norm is controlled by its H^1 norm. Moreover, the lower bound |w|≥ m>0 is uniform among solutions ξ of (<ref>) of degree d≠ 1 with bounded H^1 norm: otherwise one could find a sequence of solutions ξ_k bounded in H^1, hence in C^2, such that inf |w_k| → 0, and extracting a converging sequence in C^1 would produce a solution ξ with inf |w| =0, in contradiction with Lemma <ref>. Therefore the constants c,η in Lemma <ref> depend only on M>0 such that ξ_H^1≤ M.Now we use all the preceding lemmas and a mountain pass argumentto prove the first item of Theorem <ref>, namely uniqueness of 0-homogeneous critical points,modulo frame invariance (<ref>). Let |δ|<1 and d∈ℤ∖{ 1}. If ξ,ζ∈ H^1(𝕊^1;𝕊^1) are two solutions of (<ref>), then there exists α∈ such that ζ=ξ^α. First note that H^1(𝕊^1;𝕊^1) is a smooth Hilbert submanifold of H^1(𝕊^1;^2). This can be checked e.g. by noting that for any ξ∈ H^1(𝕊^1;𝕊^1), restricting the map H^1(𝕊^1;)∋φ↦ξ e^iφ to a small neighborhood of 0 provides a smooth parametrization of a neighborhood of ξ in H^1(𝕊^1;𝕊^1). In particular, H^1(𝕊^1;𝕊^1) is a complete smooth Finsler manifold, see <cit.>. Moreover, it can be checked rather directly that the energy E_δ is C^1 on H^1(𝕊^1;𝕊^1) and satisfies the Palais-Smale condition <cit.>, so that the deformation Lemma <cit.> is valid. Assume now by contradiction that there are two solutions ξ_1, ξ_2 of (<ref>)such that inf_α, β∈ξ_1^β-ξ_2^α_H^1>0.Thanks to Lemma <ref>, we know there are constants c,η>0 such that, for j=1,2,E_δ(u)≥E_δ(ξ_j) + c inf_α∈u-ξ_j^α_H^1^2, for all u∈ H^1(𝕊^1;𝕊^1) such thatinf_α∈u-ξ_j^α_H^1<η.Choosing η small enough we may moreover assume that inf_α, β∈ξ_1^β-ξ_2^α_H^1>2η.Therefore, definingP={p∈ C^0([0,1]; H^1(𝕊^1;𝕊^1)): p(0)=ξ_2, p(1)=ξ_2} ,any path p∈ P must intersect the sets of maps u such thatinf_α∈u-ξ_j^α_H^1=η, for j=1,2, and from (<ref>) we deducemax_u∈ pE_δ (u) ≥max_j=1,2E_δ (ξ_j)+,for = cη^2>0 and all p∈ P. Then a standard application of the deformation lemma gives thatβ=inf_p∈ Pmax_u∈ pE_δ (u), is a critical value. Assume indeed that β is not a critical value. Since β≥E_δ(ξ_j)+ for j=1,2,the deformation lemma <cit.>then provides ∈ (0,) and a family {Φ(·,t)}_t≥ 0 of continuous maps of H^1(𝕊^1;𝕊^1) into itself, such that Φ(ξ_j,1)=ξ_j for j=1,2, and Φ(·,1) maps the level set {E_δ < β +} into {E_δ <β -}.By definition of β there exists p∈ P such that E_δ(u)<β + for all u∈ p, but then p̃ =Φ(p,1)∈ P satisfies E_δ(ũ) <β- for all ũ∈p̃, contradicting the definition of β. Finally we show that thefact that β is a critical value contradicts the local minimality of all critical points established in Lemma <ref>. Let K_β⊂ H^1(𝕊^1;𝕊^1) denote the set of all critical points ξ with E_δ(ξ)=β. Since K_β isbounded,we infer that there exist uniform constants c,η>0 such that the conclusion of Lemma <ref> is valid for all ξ∈ K_β,see Remark <ref>. As a consequence, any path p∈ P such that dist_H^1(p,K_β)<η must satisfy max_u∈ pE_δ(u)≥β +cη^2, and we deduce that the infimum defining β can be taken over paths p∈ P such that dist_H^1(p,K_β)≥η. Applying the deformation lemma again provides >0 anda family {Φ(·,t)}_t≥ 0 of continuous maps of H^1(𝕊^1;𝕊^1) into itself, such that Φ(ξ_j,1)=ξ_j for j=1,2,and Φ(·,1) maps thelevel set{E_δ < β +} deprived of the neigborhood N={dist_H^1(·,K_β)<η} of K_β,into {E_δ <β -}.By the above, there exists p∈ P such that dist_H^1(p,K_β)≥η and max_p E_δ <β+, but then the path p̃=Φ(p,1)∈ P satisfies max_p̃E_δ <β. This contradicts the definition of β, and concludes the proof of Proposition <ref>.Finally we show that ξ is linearly stable not only with respect to 0-homogeneous perturbations (Lemma <ref>), but also with respect to all compactly supported perturbations in ∂ A_ρ, as claimed in the second item of Theorem <ref> Let |δ|<1, d∈∖{ 1} and ρ∈ (0,1). Let ξ∈ H^1(𝕊^1;𝕊^1) with (ξ)=dsolve the reduced equation (<ref>). Then for all φ∈ C_c^1(𝕊^1,^2) we have1/2.d^2/dt^2|_t=0 E_δ( ξ+tφ/|ξ+tφ|;A_ρ)=Q_ξ[φ-(ξ·φ)ξ],Q_ξ[v] = ∫_A_ρ(|∇ v|^2 +δ( (∇· v)^2-(∇× v)^2) -λ̂/r^2 |v|^2 ) dx,with λ̂(θ)=ℒ_δξ·ξ as in (<ref>). For any tangent field v∈ H^1_0(A_ρ;^2) with v·ξ=0 a.e., there is f∈ H^1_0(A_ρ;) such that v=fw, where w is the smooth Jacobi field generated by ξ as in Lemma <ref>, andQ_ξ satisfies the coercivity inequalityQ_ξ[v]=Q_ξ[fw]≥ (1-|δ|) ∫_A_ρ |∇ f|^2 |w|^2dx. The argument is very similar to the proof of Lemma <ref>, we only need to deal with additional radial derivative terms in the energy. Identifying as above ξ(θ)=ξ(re^iθ) with a functionon A_ρ,we see that ξ solves the full Euler-Lagrange equation (<ref>) with λ=λ̂/r^2. For any u∈ H^1(A_ρ;𝕊^1) such that u=ξ on ∂ A_ρ, letting v=u-ξ∈ H_0^1(A_ρ;^2) we find, integrating by parts as in the proof of Lemma <ref>,E_δ(u;A_ρ)-E_δ(ξ;A_ρ) =∫_A_ρ( |∇ v|^2 +δ( (∇· v)^2-(∇× v)^2) +2 ℒ_δξ· v)dx,and using ℒ_δξ=(λ̂/r^2)ξ and ξ· v =-|v|^2/2 we obtainE_δ(u;A_ρ)-E_δ(ξ;A_ρ)=Q_ξ[u-ξ],with Q_ξ as in Proposition <ref>. Applying this to u=ξ+tφ/|ξ+tφ|=ξ + t[φ-(ξ·φ)ξ] +t^2ψ_t,ψ_t_H^1≤ C(ξ,φ),for t≤ 1/(2+φ_∞), we deduceE_δ( ξ+tφ/|ξ+tφ|;A_ρ) =t^2 Q_ξ[φ-(ξ·φ)ξ] +𝒪(t^3),which proves the claimed expression for the second derivative at t=0.Using polar coordinates, we rewrite Q_ξ asQ_ξ[v] =∫_A_ρ[ |∂_r v|^2+|∂_θ v|^2 -λ̂/r^2|v|^2+δ(∂_r v· e^iθ+1/r∂_θ v· ie^iθ)^2 - δ(∂_r v· ie^iθ-1/r∂_θ v· e^iθ)^2 ] dx=∫_A_ρ[ |∂_r v|^2 + δ (∂_r v· e^iθ)^2 -δ (∂_rv· ie^iθ)^2 +1/r^2[ ℒ_δ v -λ̂v]· v+2δ/r[ (∂_r v· e^iθ)(∂_θ v· ie^iθ)+(∂_r v· ie^iθ)(∂_θ v· e^iθ) ] ]dxHence, for a function f∈ C_c^2(A_ρ;)we have the explicit expressionQ_ξ[f w ]=∫_A_ρ[ (∂_r f)^2 (|w|^2 + δ (w· e^iθ)^2-δ (w· ie^iθ)^2 )+ 1/r^2[ℒ_δ (fw)-λ̂fw ]· f w+4 δ/r∂_θ f∂_r f (w· e^iθ)(w· ie^iθ)+2 δ/rf∂_r f { (w· e^iθ)(w'· ie^iθ)+(w· ie^iθ)(w'· e^iθ) }]dxTo simplify the secondline we compute, exactly as in the proof of Lemma <ref>,ℒ_δ (fw )· fw =f^2 ℒ_δ w · w + (∂_θ f)^2( |w|^2 +δ (w·ie^iθ)^2 -δ (w· e^iθ)^2) -∂_θ[ f ∂_θ f(|w|^2+ δ (w·ie^iθ)^2 -δ (w· e^iθ)^2) ].Using the equation (<ref>) satisfied by w to simplify the first term, and coming back to the expression of Q_ξ[fw] we findQ_ξ[f w ]=∫_A_ρ[ (∂_r f)^2 (|w|^2 + δ (w· e^iθ)^2-δ (w· ie^iθ)^2 )+(∂_θ f)^2/r^2(|w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2 ) +4 δ/r∂_θ f∂_r f (w· e^iθ)(w· ie^iθ)+δ/r∂_r (f^2) { (w· e^iθ)(w'· ie^iθ)+(w· ie^iθ)(w'· e^iθ) }]dx.Since dx=r drdθ and f∈ C_c^2(A_ρ;), the last line can be integrated out with respect to r, and we are left withQ_ξ[f w]=∫_A_ρ[ (∂_r f)^2 (|w|^2 + δ (w· e^iθ)^2-δ (w· ie^iθ)^2 )+(∂_θ f)^2/r^2(|w|^2 + δ (w· ie^iθ)^2-δ (w· e^iθ)^2 ) +4δ ∂_r f∂_θ f/r(w· e^iθ)(w· ie^iθ)]dx.If δ≥ 0 we use the inequality4δ ∂_r f∂_θ f/r(w· e^iθ)(w· ie^iθ) ≥ -2δ(∂_r f)^2(w· e^iθ)^2 -2δ(∂_θ f)^2/r^2(w· ie^iθ)^2,to deduce that the integrand is bounded below by (1-δ)|∇ f|^2|w|^2, and if δ<04δ ∂_r f∂_θ f/r(w· e^iθ)(w· ie^iθ) ≥ 2δ(∂_r f)^2(w· ie^iθ)^2 +2δ(∂_θ f)^2/r^2(w· e^iθ)^2,so the integrand is bounded below by (1+δ)|∇ f|^2|w|^2. In both cases we obtainQ_ξ[fw]≥ (1-|δ|)∫_A_ρ|∇ f|^2|w|^2dx,for all f∈ C_c^2(A_ρ;), and by density for all f∈ H^1_0(A_ρ;). As a consequence of Proposition <ref>, we obtain the second item of Theorem <ref> by a contradiction argument: otherwise, there exists a sequence φ_k∈ C_c^1(A_ρ;^2) such that φ_k·ξ_δ=0 a.e., Q_ξ[φ_k]→ 0,∫_A_ρ|∇φ_k|^2dx =1.We mayextract a subsequence φ_k→φ strongly in L^2. We have φ·ξ_δ=0 a.e., and using the estimate of Proposition <ref> together with Q_ξ[φ_k]→ 0 we see that φ=0.Considering thatQ_ξ[φ_k]≥ c_1∫_A_ρ|∇φ_k|^2dx-c_2∫_A_ρ |φ_k|^2dx =c_1 - c_2∫_A_ρ|φ_k|^2dx,for some c_1,c_2>0 depending on δ and ρ, the fact that φ_k→ 0 strongly in L^2 gives the contradiction 0=lim Q_ξ[φ_k]≥ c_1>0. § SMALL ANISOTROPY: MINIMALITY IN THINANNULI AND SYMMETRY BREAKINGIn this section we prove the third item of Theorem <ref>, valid for0<|δ|≪ 1 : the unique (modulo frame invariance) 0-homogeneous critical point ξ_δ of degree d∈ℤ∖{ 0,1,2} is minimizing in a thin annulus, but it loses this minimality property in a very thick annulus. §.§ Preliminaries Two preliminary ingredients are common to the proofs of both statements: an energy splitting formula and expansions of ξ_δ and related quantities in terms of powers of δ. We dedicate the next two subsections to these tasks.§.§.§ Energy splitting A first ingredient, common to the proofs of both (minimality and symmetry breaking) statements,is a general energy splitting formula with respect to a 0-homogeneous critical point ξ_δ. Let d∈ℤ∖{ 1}, 0<|δ|<1 and ξ_δ of degree d solve (<ref>).Denote by w_δ the corresponding Jacobi field defined in Lemma <ref>. For any u∈ H^1(A_ρ;𝕊^1) such that u=ξ_δ on ∂ A_ρ, writingu=f w_δ +(1+g)ξ_δ, f,g∈ H^1_0(A_ρ;),we haveE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ)= (1+ 𝒪(δ))∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx + 2δ∫_A_ρ(α_δ(θ)/r^2fg +β_δ(θ)/r^2g∂_θ f +γ_δ(θ)/rg∂_r f )dx,where α_δ, β_δ and γ_δ are given byα_δ =μ̂_δ -2 d ∂_θ[|w_δ|]/δ, β_δ =-2(w_δ'·ξ_δ + d)+2d(1-|w_δ|)/δ + (w_δ· ie^iθ)(ξ_δ'· ie^iθ)-(w_δ· e^iθ)(ξ_δ'· e^iθ) +(w_δ'· e^iθ)(ξ_δ· e^iθ) - (w_δ'· ie^iθ)(ξ_δ· ie^iθ), γ_δ =(w_δ· e^iθ)(ξ_δ'· ie^iθ) +(w_δ· ie^iθ)(ξ_δ'· e^iθ)- (w_δ'· ie^iθ)(ξ_δ· e^iθ) - (w_δ'· e^iθ)(ξ_δ· ie^iθ),and μ̂_δ=ℒ_δ w_δ·ξ_δ as in (<ref>). According to (<ref>) we haveE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ) =Q[fw_δ +gξ_δ],where the quadratic form Q=Q_ξ_δ is defined in Proposition <ref>. We expand this expression asE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ)= Q[fw_δ] +Q[gξ_δ] +2 B[fw_δ,gξ_δ],where B is the symmetric bilinear form associated to Q. We first deal with the first two terms in the right-hand side of (<ref>), using computations similar to the proof of Proposition <ref>.We will consider without loss of generality a map u∈ C^2(A_ρ;𝕊^1) such that u-ξ_δ has compact support in A_ρ, and thereforefunctions f,g∈ C_c^2(A_ρ;).The general case follows by approximation:since uξ̅_δ =1 on ∂ A_ρ,one can find a lifting φ∈ H^1_0(A_ρ;) such that uξ̅_δ=e^iφ(this follows e.g. from <cit.> or <cit.>), and approximate φ with functions in C^2_c(A_ρ). For any function h∈ C_c^2(A_ρ;) and mapζ=ζ(θ)∈ C^2(𝕊^1;^2), we have,using polar coordinates as in the proof of Proposition <ref>,Q[h ζ]=∫_A_ρ[ (∂_r h)^2 (|ζ|^2 + δ (ζ· e^iθ)^2-δ (ζ· ie^iθ)^2 )+ 1/r^2[ℒ_δ (hζ)-λ̂_δ hζ]· hζ +4 δ/r∂_θ h∂_r h (ζ· e^iθ)(ζ· ie^iθ)+ δ/r∂_r(h^2) { (ζ· e^iθ)(ζ'· ie^iθ)+(ζ· ie^iθ)(ζ'· e^iθ) }]dx,where λ̂_δ=ℒ_δξ_δ·ξ_δ. As in the proof of Proposition <ref>, the last line can be integrated with respect to r since dx=rdrdθ and h∈ C_c^2(A_ρ;), andthesecond line can be simplified by computingℒ_δ (hζ)· hζ =h^2 ℒ_δζ·ζ + (∂_θ h)^2( |ζ|^2 +δ (ζ·ie^iθ)^2 -δ (ζ· e^iθ)^2) -∂_θ[ h ∂_θ h(|ζ|^2+ δ (ζ·ie^iθ)^2 -δ (ζ· e^iθ)^2) ],and we deduce Q[hζ]=∫_A_ρ[ (∂_r h)^2 (|ζ|^2 + δ (ζ· e^iθ)^2-δ (ζ· ie^iθ)^2 )+(∂_θ h)^2/r^2(|ζ|^2 + δ (ζ· ie^iθ)^2-δ (ζ· e^iθ)^2 ) + h^2/r^2[ℒ_δζ -λ̂_δζ]·ζ +4 δ/r∂_θ h∂_r h (ζ· e^iθ)(ζ· ie^iθ)]dx.Applying this to (h,ζ)=(f,w_δ) and (g,ξ_δ) and using the equations (<ref>) and (<ref>) satisfied by w_δ and ξ_δ, we deduceQ[fw_δ] +Q[gξ_δ] =(1+ 𝒪(δ))∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx.Next we turn to the last term in (<ref>). In polar coordinates, the bilinear form B has the expressionB[u,v] = ∫_A_ρ[∂_r u·∂_r v +1/r^2(ℒ_δ u-λ̂u)· v+δ( (∂_r u· e^iθ)(∂_r v· e^iθ) -(∂_r u· ie^iθ)(∂_r v· ie^iθ)) +δ/r((∂_r u· e^iθ)(∂_θ v· ie^iθ) +(∂_r u· ie^iθ)(∂_θ v· e^iθ) + (∂_θ u· ie^iθ)(∂_r v· e^iθ) + (∂_θ u· e^iθ)(∂_r v· ie^iθ) )]dx.Applying this to u=fw_δ, v=gξ_δ we obtainB[fw_δ,gξ_δ] = 𝒪(δ) ∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx + ∫_A_ρ[1/r^2ℒ_δ (fw_δ)· gξ_δ +δ/rg∂_r f ( (w_δ· e^iθ)(ξ_δ' · ie^iθ) +(w_δ· ie^iθ)(ξ_δ'· e^iθ) ) +δ/r f∂_r g ( (w_δ'· ie^iθ)(ξ_δ· e^iθ) + (w_δ'· e^iθ)(ξ_δ· ie^iθ) )]dx.Integrating by parts with respect to r in the last line, we findB[fw_δ,gξ_δ] = 𝒪(δ) ∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx+δ∫_A_ργ_δ(θ)/rg∂_r f dx+ ∫_A_ρ1/r^2ℒ_δ (fw_δ)· gξ_δdx,with γ_δ as in the statement of Lemma <ref>. Using the equation (<ref>) satisfied by w_δ and the fact that w_δ·ξ_δ=0,exactly as in (<ref>) in the proof of Lemma <ref>, we haveℒ_δ (fw_δ)· gξ_δ=μ̂_δ fg -2 g∂_θ f w_δ'·ξ_δ +𝒪(δ |w_δ| |∂_θ f| |∂_θ g|)+δg∂_θ f [ (w_δ· ie^iθ)(ξ_δ'· ie^iθ)-(w_δ· e^iθ)(ξ_δ'· e^iθ)- (w_δ'· ie^iθ)(ξ_δ· ie^iθ) + (w_δ'· e^iθ)(ξ_δ· e^iθ)]-δ ∂_θ[∂_θ f ( (w_δ· ie^iθ)ie^iθ-(w_δ· e^iθ)e^iθ)· gξ_δ].Plugging this intothe above expression for B[fw_δ,gξ_δ] we deduceB[fw_δ,gξ_δ] = 𝒪(δ) ∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx+δ∫_A_ρ[ α_δ(θ)/r^2fg+β_δ(θ)/r^2g∂_θ f + γ_δ(θ)/rg∂_r f ] dx+ 2d∫_A_ρ1/r^2g∂_θ[f|w_δ|]dx,where α_δ and β_δ are defined in the statement of Lemma <ref>. To simplify the last term in (<ref>) we use the fact that u is 𝕊^1-valued and uξ̅_δ has degree zero, sothere exists a lifting φ∈ C_c^2(A_ρ;) such that u=e^iφξ_δ.By definition of f,g, and possibly multiplying φ by ± 1 (depending on the constant sign of iξ_δ· w_δ),this impliesf|w_δ|=sinφ, 1+g=cosφ, so2g∂_θ[f|w_δ|] =∂_θ[fg|w_δ|] +g∂_θ[f|w_δ|] - f|w_δ|∂_θ g =∂_θ[fg|w_δ|] +(cosφ -1)cosφ ∂_θφ +sin^2φ ∂_θφ =∂_θ[fg|w_δ| +φ -sinφ ].Therefore the last term in (<ref>) integrates to zero, and we deduceB[fw_δ,gξ_δ] = 𝒪( δ) ∫_A_ρ(|w_δ|^2|∇ f|^2 + |∇ g|^2 ) dx +δ∫_A_ρ[ α_δ(θ)/r^2fg+β_δ(θ)/r^2g∂_θ f +γ_δ(θ)/rg∂_r f ] dx,which, combined with (<ref>), proves Lemma <ref>. §.§.§ Small anisotropy expansions In order to make efficient use of the energy splitting with respect to ξ_δ provided by Lemma <ref>, we will need expansions of the coefficients in powers of δ. We start by expanding ξ_δ. Let d∈ℤ∖{ 1}. There exists δ_0>0 such that for |δ|<δ_0, the equation (<ref>) has a unique solution ξ_δ of degree d such that ξ_δ(0)=1, and it satisfiesξ_δ(θ)=e^idθe^iφ_δ(θ),φ_δ=δψ_1 +δ^2/2ψ_2 +δ^3𝒪(1),whereψ_1(θ) = a_1sin(2(d-1)θ), a_1=d(2-d)/4(d-1)^2 ψ_2(θ) = a_2 sin(4(d-1)θ), a_2 =a_2(d)∈ ,and 𝒪(1) is bounded in C^k(𝕊^1;) as δ→ 0for all k≥ 0. First recall that Remark <ref> ensures the existence of φ_δ such that φ_δ(0)=0 and ξ_δ=e^idθe^iφ_δ solves (<ref>) and minimizes E_δ among 𝕊^1-valued maps of degree d.The inequality E_δ(ξ_δ)≤E_δ(e^idθ)implies ∫_𝕊^1 (φ_δ')^2dθ≤ Cδ d^2,for some absolute constant C>0, so φ_δ→ 0 in H^1(𝕊^1;) as δ→ 0. This bound is valid for any solution φ_δ with φ_δ(0)=0, since they are all minimizing by Proposition <ref>.Next we show, by an implicit function argument, that φ_δ is unique and depends smoothly on δ for small δ. Consider, for any k≥ 0 and d≠ 1, the mapΨX×× (-1,1)→Y, X={φ∈ H^k+1(𝕊^1;)φ(0)=0},Y =H^k-1(𝕊^1;),given byΨ(φ,t,δ) =-D_φE_δ(e^idθe^iφ) +t= d/dθ[(1+δcos(2(d-1)θ +2φ ))(d+φ')]+δsin(2(d-1)θ +2φ )(d+φ')^2 +t.This map Ψ is smooth and satisfiesΨ(0,0,0)=0, D_(φ,t)Ψ(0,0,0)[η,s] =η”+s ∀ (η,μ)∈ X×.The differential D_(φ,λ)Ψ(0,0,0) is an isomorphism from X× to Y, so by the implicit function theorem there exist (φ̅_δ,t_δ)∈ X× depending smoothly on δ∈ (-δ_0,δ_0), the unique solution of Ψ(φ̅,t,δ)=0 in a neighborhood of (0,0). By uniqueness this solution does not depend on k, and because Ψ(φ_δ,0,δ)=0 and φ_δ→ 0 in H^1(𝕊^1;) as δ→ 0, for small enough δ we must have t_δ=0,φ_δ=φ̅_δ is the unique solution ofd/dθ[(1+δcos(2(d-1)θ +2φ_δ ))(d+φ_δ')]=-δsin(2(d-1)θ +2φ_δ )(d+φ_δ')^2, satisfying φ_δ(0)=0, and δ↦φ_δ∈ H^k+1(𝕊^1;) is smooth for all k≥ 0. We have .φ_δ|_δ=0≡ 0,and considering ψ_1=.d/dδ|_δ=0φ_δ, ψ_2 =.d^2/dδ^2|_δ=0φ_δ,provides the expansion in Lemma <ref>.It remains to explicitly determine ψ_1 and ψ_2. Derivating the Euler-Lagrange equation (<ref>) with respect to δ we see that ψ_1 solvesd/dθ[ ψ_1' +dcos(2(d-1)θ) ] =-d^2sin(2(d-1)θ),that isψ_1” =d(d-2)sin(2(d-1)θ).Since ψ_1is 2π-periodic with ψ_1(0)=0, this impliesψ_1(θ)= d(2-d)/4(d-1)^2sin(2(d-1)θ).Derivating twice the Euler-Lagrange equation (<ref>) with respect to δ we see that ψ_2 solvesd/dθ[ ψ_2' -4dsin(2(d-1)θ)ψ_1 +2cos(2(d-1)θ)ψ_1' ] =-4 d^2cos(2(d-1)θ)ψ_1 -4d sin(2(d-1)θ)ψ_1',that isψ_2” =a(d)sin(4(d-1)θ)),for some a(d)∈, and since ψ_2 is 2π-periodic with ψ_2(0)=0 this givesψ_2(θ)=a_2(d)sin(4(d-1)θ)),for some a_2(d)∈. As a consequence of Lemma <ref> we obtain expansions of the coefficientsα_δ, β_δ appearing in Lemma <ref>. As δ→ 0 we haveα_δ =α^0 +δα^1 +δ^2 𝒪(1), β_δ=β^0 +δβ^1 +δ^2 𝒪(1),where 𝒪(1) is bounded in C^k(𝕊^1;) as δ→ 0 for any k≥ 0, and, denoting C_n(θ)=cos(nθ), S_n(θ)=sin(nθ), the coefficients α^j, β^j, satisfyα_0 = 2d(d-2)S_2(d-1),β^0=d(2-d)/d-1C_2(d-1)α^1 ∈span(S_4(d-1)),β^1 ∈span(1,C_4(d-1)). Recall from Lemma <ref> that α_δ and β_δ are given byα_δ =μ̂_δ -2 d ∂_θ[|w_δ|]/δ, β_δ =-2(w_δ'·ξ_δ + d)+2d(1-|w_δ|)/δ + (w_δ· ie^iθ)(ξ_δ'· ie^iθ)-(w_δ· e^iθ)(ξ_δ'· e^iθ) +(w_δ'· e^iθ)(ξ_δ· e^iθ) - (w_δ'· ie^iθ)(ξ_δ· ie^iθ).We first obtain expressions in terms of φ_δ. Usingξ_δ =e^i(dθ+φ_δ),ξ_δ' =(d+φ_δ')iξ_δ,w_δ =(1+φ_δ'/d-1)iξ_δ,w_δ'=φ_δ”/d-1iξ_δ-(d+2d-1/d-1φ_δ'+1/d-1(φ_δ')^2)ξ_δ,we findw'_δ·ξ_δ + d =-2d-1/d-1φ_δ'-1/d-1(φ_δ')^21-|w_δ|=-φ_δ'/d-1,and(w_δ· ie^iθ)(ξ_δ'· ie^iθ) -(w_δ· e^iθ)(ξ_δ'· e^iθ)+ (w_δ'· e^iθ)(ξ_δ· e^iθ) - (w_δ'· ie^iθ)(ξ_δ· ie^iθ) = -φ_δ”/d-1sin(2(d-1)θ+2φ_δ).Using alsow_δ” =-((1+φ_δ'/d-1)(d+φ_δ')^2-φ_δ^(3)/d-1)iξ_δ - φ_δ”/d-1(3d-1 +3φ_δ')ξ_δ,and recalling the definition (<ref>) of μ̂_δ, we obtainμ̂_δ = -w_δ”·ξ_δ-δ [ (w_δ'· ie^iθ)ie^iθ-(w_δ'· e^iθ)e^iθ]'·ξ_δ=φ_δ”/d-1(3d-1 +3φ_δ') -3δφ_δ”(1 +φ_δ'/d-1) cos(2(d-1)θ+2φ_δ) +δ/d-1((d+φ_δ')(d-1+φ_δ')(d-2+φ_δ')- φ_δ^(3))sin(2(d-1)θ+2φ_δ)Then we plug into these expressions the expansionsφ_δ =δψ_1 +δ^2/2ψ_2 +𝒪(δ^3), cos(2(d-1)θ+2φ_δ) =C_2(d-1) -2δ S_2(d-1)ψ_1+𝒪(δ^2)sin(2(d-1)θ+2φ_δ) = S_2(d-1) +2δ C_2(d-1)ψ_1 +𝒪(δ^2),where we recall the notation C_n(θ)=cos(nθ), S_n(θ)=sin(nθ). We findw'_δ·ξ_δ + d =-δ2d-1/d-1ψ_1' -δ^2/d-1(2d-1/2ψ_2'+(ψ_1')^2) +𝒪(δ^3),1-|w_δ|=-δ/d-1ψ_1' -δ^2/2(d-1)ψ_2' +𝒪(δ^3), (w_δ· ie^iθ)(ξ_δ'· ie^iθ) -(w_δ· e^iθ)(ξ_δ'· e^iθ)+ (w_δ'· e^iθ)(ξ_δ· e^iθ) - (w_δ'· ie^iθ)(ξ_δ· ie^iθ) = -δψ_1”/d-1S_2(d-1) +𝒪(δ^2),andμ̂_δ = δ( 3d-1/d-1ψ_1” +d(d-2)S_2(d-1))+δ^2(3d-1/2(d-1)ψ_2” + 3/d-1ψ_1'ψ_1” -3ψ_1” C_2(d-1) +3d^2-6d+2/d-1S_2(d-1)ψ_1' -ψ_1^(3)/d-1S_2(d-1) +2d(d-2)C_2(d-1)ψ_1)+𝒪(δ^3)Gathering the above and recallingψ_1=a_1 S_2(d-1),ψ_2=a_2 S_4(d-1),for some constants a_1=a_1(d), a_2=a_2(d),we obtain the desired expansions for α_δ and β_δ. Explicitly, we havew'_δ·ξ_δ + d =-2δ (2d-1)a_1 C_2(d-1) - δ^2(2(2d-1) a_2C_4(d-1)+2(d-1)a_1^2(1+C_4(d-1))) +𝒪(δ^3),1-|w_δ|=-2δ a_1 C_2(d-1)-2δ^2a_2C_4(d-1) +𝒪(δ^3),(w_δ· ie^iθ)(ξ_δ'· ie^iθ) -(w_δ· e^iθ)(ξ_δ'· e^iθ)+ (w_δ'· e^iθ)(ξ_δ· e^iθ) - (w_δ'· ie^iθ)(ξ_δ· ie^iθ) = 2δ a_1 (d-1) (1-C_4(d-1) )+𝒪(δ^2),andμ̂_δ = δ(d(d-2)-4(3d-1)(d-1)a_1) S_2(d-1) +δ^2(-8(3d-1)(d-1)a_2-12(d-1)^2a_1^2 +(14d^2-28d+12) a_1) S_4(d-1)+𝒪(δ^3),and we infer that the coefficients α^j, β^j are given byα^0= (d(d-2)-4(d-1)^2a_1) S_2(d-1)=2d(d-2) S_2(d-1),α^1= (-8(d-1)^2a_2-12(d-1)^2a_1^2+ (14d^2-28d+12) a_1) S_4(d-1),β^0= 4 (d-1)a_1 C_2(d-1) =d(2-d)/d-1C_2(d-1), β^1= 2(d-1)a_1(1+2a_1) -2( (d-1)a_1(1 - 2a_1)+2 (d-1) a_2) C_4(d-1).§.§ Minimality in not-too-thick annuli In this section we prove that if |δ|≪ 1 and the annulus A_ρ is not too thick, the unique (modulo frame invariance)0-homogeneous solution ξ_δ of degree d∈ℤ∖{ 1} is minimizing.Let d∈ℤ∖{ 1}. There exists a small constant c>0, depending only on d, such that if |δ|<c, ξ_δ of degree d solves (<ref>), ande^-c/δ^1/3≤ρ <1, then ξ_δ is minimizing in A_ρ with respect to its own boundary conditions. We obtain Proposition <ref> as a consequence of two observations on maps u∈ H^1(A_ρ;𝕊^1) agreeing with ξ_δ on ∂ A_ρ : * on the one hand, the linear stability result of Proposition <ref> can be enhanced to a local minimality statement: any map u close enough to ξ_δ has higher energy than ξ_δ unless u=ξ_δ,* on the other hand, as δ→ 0, a minimizing map u must converge toξ_0(θ)=e^idθ (modulo frame invariance), and is therefore close to ξ_δ.In Proposition <ref> and Lemma <ref> below we quantify these statements, which can then directly be combined into a proof of Proposition <ref>. Let d∈ℤ∖{ 1}. There exists a small constant c>0, depending only on d, such that if |δ|<c, ξ_δ of degree d solves (<ref>), and u∈ H^1(A_ρ;𝕊^1) is such that u=ξ_δ on ∂ A_ρ for some 0<ρ<1, we have∫_A_ρ|∇ u-∇ξ_δ|^2 dx ≤c/δ^2 (1+ln^2ρ)^3⟹ E_δ(u;A_ρ)≥ E_δ(ξ_δ;A_ρ),and the last inequality is strict unless u=ξ_δ. Weconsider without loss of generality a solution ξ_δ such that ξ_δ(0)=1, as in Lemma <ref>, and writeu-ξ_δ=fw_δ+gξ_δ. It follows from the energy splitting in Lemma <ref> and the expansions in Lemma <ref> and Lemma <ref> that, if |δ|<c for small enough c>0, we haveE_δ(u;A_ρ) - E_δ(ξ_δ;A_ρ)≥3/4∫_A_ρ(|∇ f|^2 +|∇ g|^2)dx -Cδ∫_A_ρ(|fg|/r^2 +|∇ f||g|/r)dx.Here C>0 denotes a large constant depending only on the degree d, and may change from line to line in the rest of this proof.Using |∇ f||g|/r≤ |∇ f|^2 +g^2/r^2, we deduce E_δ(u;A_ρ) - E_δ(ξ_δ;A_ρ)≥1/2∫_A_ρ( |∇ f|^2 +|∇ g|^2)dx -Cδ∫_A_ρ|fg| +g^2/r^2dx.Next we use the fact that |u|=1, i.e. f^2|w_δ|^2+(1+g)^2=1, or equivalently g=-(f^2|w_δ|^2+g^2)/2, to infer that |f|,|g|≤ 2 and|fg|+g^2 ≤ C( |f|^3 +|g|^3 ).Plugging this inequality into the previous estimate we obtainE_δ(u;A_ρ) - E_δ(ξ_δ;A_ρ)≥1/2∫_A_ρ( |∇ f|^2 +|∇ g|^2)dx -Cδ∫_A_ρ|f|^3 +|g|^3/r^2dx.The last term can be estimated using the interpolation inequality of Lemma <ref> below, and we deduceE_δ(u;A_ρ) - E_δ(ξ_δ;A_ρ)≥(1/2 - Cδ (1+ln^2ρ)(∫_A_ρf^2+g^2/r^2dx)^1/2)×∫_A_ρ(|∇ f|^2+|∇ g|^2)dx.Since f^2+g^2≤ 2 |u-ξ_δ|^2, this impliesE_δ(u;A_ρ) > E_δ(ξ_δ;A_ρ) whenever u≠ξ_δ and∫_A_ρ|u-ξ_δ|^2/r^2dx ≤c/δ^2(1+ln^2ρ)^2.Combining this with the Hardy-type inequality (see (<ref>) below)1/ln^2ρ∫_A_ρ|u-ξ_δ|^2/r^2dx ≤ C ∫_A_ρ|∇ u-∇ξ_δ|^2dx,concludes the proof of Proposition <ref>Next we prove the interpolation inequality used in the proof of Proposition <ref>. For all 0<ρ<1 and φ∈ H^1_0(A_ρ) we have∫_A_ρ|φ|^3/r^2dx≤ C (1+ln^2ρ)(∫_A_ρφ^2/r^2dx)^1/2∫_A_ρ|∇φ|^2dx,for some absolute constant C>0.First we show that, for all φ∈ C^∞_c(A_ρ),∫_A_ρφ^4/r^2dx≤ C (1+ln^2ρ)∫_A_ρφ^2/r^2dx ∫_A_ρ|∇φ|^2dx.For 1/4≤ρ≤ 1 this is a consequence of the classical Ladyzhenskaya interpolation inequality in the domain A_1/4, which simply follows from applying to φ^2∈ C_c^∞(A_1/4)the Poincaré-Sobolev inequality of the embedding W^1,1(A_1/4)⊂ L^2(A_1/4):φ_L^4(A_1/4)^4=φ^2_L^2(A_1/4)^2≤ C ∇(φ^2)_L^1(A_1/4)^2≤ C φ_L^2(A_1/4)^2∇φ_L^2(A_1/4)^2. To obtain (<ref>) for 0<ρ<1/2, we consider the rescaled annuli𝔸_j =2^-jA_1/4 and decompose φ=∑_j≥ 0φ_j, with φ_j∈ C_c^∞(𝔸_j) such that, for any p≥ 1,∫_𝔸_jφ_j^pdx≤∫_𝔸_jφ^pdx ≤∫_𝔸_jφ_j^pdx +∫_𝔸_j+1φ_j+1^pdx,∫_𝔸_j|∇φ_j|^2dx ≤∫_𝔸_j|∇φ|^2dx+C ∫_𝔸_jφ^2/r^2dx.This decomposition can be obtained for instance by fixing a smooth cut-off function 1_|x|≤ 1/2≤χ(x)≤1_|x|≤ 1 and settingφ_0(x)=χ(x)φ(x),φ_j(x)=(χ(2^j+1x)-χ(2^jx))φ(x) for j≥ 1.Rescaling Ladyzhenskaya's inequality we have∫_𝔸_jφ_j^4/r^2dx ≤ C ∫_𝔸_j |∇φ_j|^2 dx∫_𝔸_jφ_j^2/r^2dx.Summing these estimates and using the properties of φ_j we obtain∫_A_ρφ^4/r^2dx ≤ C ∫_A_ρφ^2/r^2dx (∫_A_ρ|∇φ|^2dx +∫_A_ρφ^2/r^2dx ).This, together withthe Hardy-type inequality∫_A_ρφ^2/r^2dx ≤ln^2ρ/π^2∫_A_ρ|∇φ|^2dx ∀φ∈ H^1_0(A_ρ),proves (<ref>). (Inequality (<ref>) follows e.g. from<cit.> and the fact that the function φ_*(x)=sin(πln|x|/|lnρ|) solves -Δφ_*=(π^2/ln^2ρ)φ_*/r^2 and is positive in A_ρ.)To conclude the proof of Lemma <ref>, we write |φ|^3≤λφ^2 +λ^-1φ^4 for any λ>0, apply (<ref>) and Hardy's inequality (<ref>) to deduce1/1+ln^2ρ∫_A_ρ|φ|^3/r^2dx ≤ Cλ∫_A_ρ|∇φ|^2dx +C/λ∫_A_ρφ^2/r^2dx ∫_A_ρ|∇φ|^2dx,and choose λ =(∫φ^2/r^2dx)^1/2.As explained above, the second ingredient to prove Proposition <ref> is the convergence towards ξ_0(θ)=e^idθ, as δ→ 0, of any minimizing map u∈ H^1(A_ρ;𝕊^1) agreeing with ξ_δ on ∂ A_ρ. Since for |δ|≪ 1 the 0-homogeneous solution ξ_δ is also close to ξ_0 (see Lemma <ref>), this implies that u must be close to ξ_δ. The next lemma makes that statement quantitative.Let d∈ℤ∖{ 1}, |δ|<c (c>0 small depending on d) and ξ_δ of degree d solve (<ref>) and ξ_δ(0)=1.Assume 0<ρ<1 and u∈ H^1(A_ρ;𝕊^1) satisfies u=ξ_δ on ∂ A_ρ and E_δ(u;A_ρ)≤ E_δ(ξ_δ;A_ρ). Then∫_A_ρ |∇ u-∇ξ_δ|^2dx≤ Cδ (1+ln^2ρ) ln1/ρ,for some C>0 depending only on the degree d. From the expansion of ξ_δ in Lemma <ref> we inferE_δ(ξ_δ;A_ρ)≤ (1+Cδ)2π d^2 ln1/ρ,and the bound E_δ(u;A_ρ)≤ E_δ(ξ_δ;A_ρ) therefore implies, for |δ|≤ c,∫_A_ρ |∇ u|^2 ≤ (1+Cδ)2π d^2ln1/ρ.Since u=ξ_δ on ∂ A_ρ, it is of degree d, and we can write u=ξ_δ e^iψ, with ψ∈ H^1_0(A_ρ;). We further rewrite this as u=e^idθe^iφ_δ(θ)e^iψ, where φ_δ_C^1≤ Cδ by Lemma <ref>. Then we have|∇ u|^2=(∂_rψ)^2 + 1/r^2(d+∂_θψ +∂_θφ_δ)^2=d^2+𝒪(δ)/r^2 +2d/r^2∂_θψ + (1+𝒪(δ))|∇ψ|^2.Integrating on A_ρ, we deduce∫_A_ρ|∇ u|^2dx=(1+𝒪(δ))2π d^2ln1/ρ+(1+𝒪(δ))∫_A_ρ|∇ψ|^2dx,and combining this with (<ref>) gives∫_A_ρ|∇ψ|^2dx ≤ Cδln1/ρ.Therefore we find∫_A_ρ|∇ u-∇ξ_δ|^2dx≤ C ∫_A_ρ|ψ|^2/r^2 +|∇ψ|^2dx ≤ C (1+ln^2ρ)∫_A_ρ|∇ψ|^2dx≤ Cδ (1+ln^2ρ)ln1/ρ.For the penultimate inequality we used Hardy's inequality(<ref>).Combining Proposition <ref> with Lemma <ref>provides a proof ofProposition <ref>.Specifically, according to Lemma <ref> and Proposition <ref>, there exist constants C,c>0 depending only on d such that, if |δ|<c andCδ (1 +ln^2ρ)ln1/ρ≤c/δ^2(1+ln^2ρ)^3,then any minimizer u with u_⌊∂ A_ρ=ξ_δ must be equal to ξ_δ.If 1/2≤ρ <1 then (<ref>) is satisfied for all small enough δ, so we may assume 0<ρ≤ 1/2, in which case |lnρ|=ln(1/ρ)≥ln 2>0 and (<ref>) is implied byδln^31/ρ≤c/δ^2 ln^61/ρ⇔ln^91/ρ≤c/δ^3⇔ln1/ρ≤c/δ^1/3,for some generic small constant c>0 depending only on d. Therefore ξ_δ is a minimizer if ρ≥ e^-c/δ^1/3, and this proves Proposition <ref>. §.§ Symmetry breakingIn this section we construct, for small δ and ρ, a non-0-homogeneous map that agrees with ξ_δ on ∂ A_ρ and has strictly lower energy than ξ_δ, proving in particular the third item of Theorem <ref>.The basic idea is to use a competitor that saturates (or almost saturates) the interpolation inequality of Lemma <ref>, which we used in Proposition <ref> to control the non-quadratic terms for ρ not too small. We use that Hardy's inequality (<ref>) is saturated byφ_*(x)=sin(πln |x|/|lnρ|),since φ_*∈ H^1_0(A_ρ) satisfies -Δφ_*=(π^2/ln^2ρ)φ_*/r^2 and therefore∫_A_ρ|∇φ_*|^2dx =π^2/ln^2ρ∫_A_ρφ_*^2/r^2dx.Regarding the interpolation inequality of Lemma <ref>, we have∫_A_ρ |φ_*|^3/r^2dx/(∫_A_ρφ_*^2/r^2dx)^1/2∫_A_ρ|∇φ_*|^2dx =c_*|lnρ|^3/2for some c_*>0,which is enough for our purposes, even though it does not completely saturate the interpolation inequality.Using this function φ_* we construct a competitor u_,ρ,δ for ξ_δ in A_ρ,for which we can expand the energy in terms of the small parameters ,ρ,δ, and eventually find that it is lower than the energy of ξ_δ for appropriate choices of =(δ) and ρ=ρ(δ).Let 0<|δ|<1, d∈∖{ 0,1 ,2} and ξ_δ∈ H^1(𝕊^1;𝕊^1) of degree d, a solution of (<ref>) with ξ_δ(0)=1 (see Remark <ref>). Denote by w_δ the corresponding Jacobi field defined in Lemma <ref>.For any ρ∈ (0,1/e), let h_ρ∈ H^1_0(A_ρ;) be given byh_ρ(re^iθ)=( 1+sin(2(d-1)θ)/|lnρ|)sin(πln r/lnρ),and, for 0<| | <1/(2w_δ_∞), define u_,ρ,δ∈ H^1(A_ρ;𝕊^1) byu_,ρ,δ=√(1-^2 h_ρ^2|w_δ|^2) ξ_δ + h_ρ w_δ.Then there exist a large constant λ>0 and a small constant δ_0>0 depending only on the degree d,and a value ofdepending only on the degree d and the sign of δ, such that E_δ(u_,ρ,δ;A_ρ) < E_δ(ξ_δ;A_ρ) for|δ|<δ_0and ρ=e^-λ/|δ|.The map u=u_,ρ,δ is of the form u=ξ_δ +fw_δ +gξ_δ, withf= h, g=√(1-^2h^2|w_δ|^2)-1.The function h∈ H_0^1(A_ρ;) satisfies |h|≤ 2 and|∂_θ h|≤2|d-1|/|lnρ|,sowe haveg =-^2/2h^2|w_δ|^2-^4/8h^4|w_δ|^4 +𝒪(^6)∇ g =-^2/2∇[h^2|w_δ|^2](1+𝒪(^2)) fg =-^3/2h^3|w_δ|^2-^5/8h^5|w_δ|^4 +𝒪(^7)g∂_θ f = -^3/6|w_δ|^2∂_θ[h^3]-^5/40|w_δ|^4 ∂_θ[h^5] +𝒪(^7/|lnρ|) g∂_r f =∂_r[∫_0^h (√(1-^2t^2|w_δ|^2)-1)dt ].Moreover, using also that ∂_θ [|w_δ|^2]= 2δψ_1'/(d-1) +𝒪(δ^2) thanks to Lemma <ref>, we see that|∇ g|^2= δ^2^4/(d-1)^2h^4/r^2(ψ_1')^2 + 𝒪(^4)|∇ h|^2 +𝒪(δ^4/|lnρ|+δ^3^4 + δ^2^6)1/r^2. Plugging all this into the expansion obtained in Lemma <ref>, we findE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ) = ^2∫_A_ρ|∇ h|^2dx+ δ^2^4/(d-1)^2∫_A_ρh^4/r^2(ψ_1')^2dx+δ^3∫_A_ρη_δ(θ)/r^2h^3dx +δ^5∫_A_ρν_δ(θ)/r^2h^5dx + 𝒪(δ^2)∫_A_ρ|∇ h|^2 dx +𝒪(δ^4) +𝒪(δ^3^4+δ^2^6)|lnρ| ,whereη_δ =-α_δ |w_δ|^2 +∂_θ[β_δ|w_δ|^2/3],ν_δ =-α_δ|w_δ|^4/4 + ∂_θ[β_δ|w_δ|^4/20]From the expansion of ξ_δin Lemma <ref>, we have|w_δ|^2=(1+φ_δ'/d-1)^2 =1+δ2/d-1ψ_1' +𝒪(δ^2),and using also the expansions of α_β and β_δ inLemma <ref>,it can be checked that the coefficients η_δ and ν_δ have expansions of the formη_δ =η^0 +δη^1 +𝒪 (δ^2), η^0 = -α^0 +1/3∂_θβ^0∈span(S_2(d-1))η^1= -α^1 - 2/d-1α^0ψ_1' + 1/3∂_θ[ β^1 +2/d-1β^0ψ_1' ] ∈span(S_4(d-1)), ν^δ =ν^0+𝒪(δ), ν^0=-1/4α^0 +1/20∂_θβ^0 ∈span(S_2(d-1)).Here we use again the notation S_n(θ)=sin(nθ). Next recallthath= (1+S_2(d-1)(θ)/|lnρ|)h_0(r), h_0(r)=sin(πln r/lnρ)1_ρ≤ r≤ 1,ψ_1' =d(2-d)/2(d-1)cos(2(d-1)θ).We can directly compute∫_A_ρ|h|^3/r^2 =𝒪(|lnρ|),∫_A_ρ|h|^5/r^2 = 𝒪(|lnρ|),∫_A_ρ|∂_r h|^2dx =(1+𝒪(1/|lnρ|)) 2ππ^2/ln^2ρ∫_ρ^1 cos^2(πln r/lnρ) dr/r = π^3/|lnρ| + 𝒪(1/ln^2ρ),∫_A_ρ|∂_θ h|^2/r^2 dx =4(d-1)^2/ln^2ρ∫_0^2πcos^2(2(d-1)θ)) dθ∫_ρ^1 sin^2(πln r/lnρ)dr/r =2π(d-1)^2/|lnρ|and∫_A_ρh^4/r^2(ψ_1')^2dx =(1+𝒪(1/|lnρ|)) ·d^2(2-d)^2/4(d-1)^2∫_0^2πcos^2(2(d-1)θ)dθ∫_ρ^1sin^4(πln r/lnρ)dr/r =3π/32d^2(2-d)^2/(d-1)^2 |lnρ| +𝒪( 1 ). So, taking the expansions of η_δ and ν_δ into account, (<ref>) impliesE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ) = (π^3+2π(d-1)^2) ^2/|lnρ| + 3π/32d^2(2-d)^2/(d-1)^4δ^2^4|lnρ|+δ^3∫_A_ρη^0 h^3/r^2dx +δ^2^3∫_A_ρη^1h^3/r^2dx +δ^5∫_A_ρν^0h^5/r^2dx+ 𝒪(δ^2/|lnρ|) + 𝒪(^2/|lnρ|^2) +𝒪(δ^4) + 𝒪(δ^3^3+δ^2^5)|lnρ|.Moreover, we have h^3 =h_0^3∑_k=0^3 ([ 3; k ]) 1/|lnρ|^kS_2(d-1)^k,h^5=h_0^5 ∑_k=0^5 ([ 5; k ]) 1/|lnρ|^kS_2(d-1)^k, and deduce, using that ∫_0^2π S_2(d-1)dθ=∫_0^2π S_4(d-1)dθ=∫_0^2π S_2(d-1)S_4(d-1) dθ =0,∫_A_ρh^3/r^2S_2(d-1) =( 3∫_0^2π S_2(d-1)^2 dθ+𝒪(1/|lnρ|))1/|lnρ|∫_ρ^1 h_0^3dr/r =4π+𝒪(1/|lnρ|)∫h^5/r^2S_2(d-1) =𝒪(1),∫h^3/r^2S_4(d-1) =𝒪(1/|lnρ|).Recalling that η^0,ν^0∈span(S_2(d-1)), η^1∈span(S_4(d-1)) and pluggingthese into (<ref>) we obtainE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ) = (π^3+2π(d-1)^2) ^2/|lnρ| + 3π/32d^2(2-d)^2/(d-1)^4δ^2^4|lnρ| + δ^3𝔞 + 𝒪(δ^2/|lnρ|) + 𝒪(^2/|lnρ|^2) +𝒪(δ^4) + 𝒪(δ^3^3+δ^2^5)|lnρ| ,where𝔞 =4∫_0^2πη^0 S_2(d-1)dθ. Using the expressions of α^0 and β^0 inLemma <ref>, we findη^0=-α^0+1/3∂_θβ^0 =-4/3 d(d-2)S_2(d-1),so 𝔞=-16 π/3d(d-2).Letting ρ=e^-λ/|δ| with λ≫ 1 and optimizing with respect to , we choose=-sign(δ) 2^9/9(d-1)^4/d(d-2)1/λ.For this choice of , the above expansion becomesE_δ(u_,ρ,δ;A_ρ) - E_δ(ξ_δ;A_ρ)=|δ| ^2/λ( π^3+2π (d-1)^2 -2^12/27π (d-1)^4 +𝒪(1/λ+ δλ)).Since |d-1|≥ 2, we have2^12/27π (d-1)^4≥2^14/27π (d-1)^2≥(2^14/27-2)· 4 π +2π(d-1)^2>π^3 + 2π(d-1)^2+π^3,and may therefore fix a large enough λ>0 such thatE_δ(u_,ρ,δ;A_ρ)<E_δ(ξ_δ;A_ρ) for all small enough |δ|>0.§.§ Proof of the third item of Theorem <ref> Let d∈ℤ∖{ 0,1,2} and |δ| small enough that Proposition <ref>, and the conclusion of Proposition <ref>, are valid.Define, for 0<ρ<1,G(ρ)=inf_u_⌊∂ A_ρ=ξ_δ E_δ(u;A_ρ)- E_δ(ξ_δ;A_ρ).The function G is monotone nondecreasing, since for 0<ρ<ρ'<1, any admissible competitor u'∈ H^1(A_ρ';𝕊^1) with u'=ξ_δ on ∂ A_ρ'can be extended by ξ_δ in A_ρ∖ A_ρ' to become an admissible competitor in A_ρ, which impliesinf_u_⌊∂ A_ρ=ξ_δ E_δ(u;A_ρ) ≤ E_δ(u';A_ρ') +E_δ(ξ_δ;A_ρ) - E_δ(ξ_δ;A_ρ').Substracting E_δ(ξ_δ;A_ρ) and taking the infimum over all admissible u' gives G(ρ)≤ G(ρ').The function G is also continuous, since for 0<ρ<ρ'<1,any admissible competitor u ∈ H^1(A_ρ;𝕊^1) with u=ξ_δ on A_ρ can be dilated as follows,u'(r'e^iθ)=u(re^iθ), r=1-ρ/1-ρ'r' -ρ'-ρ/1-ρ' for r'∈ (ρ',1),so that u' is an admissible competitor in A_ρ', which impliesinf_u_⌊∂ A_ρ'=ξ_δ E_δ(u;A_ρ')≤E_δ(u';A_ρ')≤ E_δ(u;A_ρ) + 𝒪(ρ'-ρ/1-ρ')E_δ(u;A_ρ)and we deduce 0≤ G(ρ')- G(ρ) ≤𝒪( ρ'-ρ/1-ρ'ln1/ρ+lnρ'/ρ).So G is continuous and monotone nondecreasing on (0,1). Combining this with Propositions <ref> and <ref> implies the existence of ρ_*∈ (0,1) such that G≡ 0 on [ρ_*,1) and G<0 on (0,ρ_*), which proves the third item of Theorem <ref>.§ THE CASES OF DEGREE 1 AND 2 §.§ The degree 1 case In this section we show that, for 0<|δ|<1, the only 0-homogenous solutions of (<ref>) which have degree 1 are the trivial solutions ξ(θ)=e^iαe^iθ, α≡ 0 modulo π/2.To that end we write an arbitrary 0-homogeneous solution of (<ref>) of degree 1, inthe form ξ(θ)=e^iθe^iφ(θ). Then we have the equationd/dθ[(1+δcos(2φ ))(1+φ')] =-δsin(2φ )(1+φ')^2,that is,(1+δcos(2φ))φ” =δsin(2φ) ((φ')^2-1),which we may also rewrite as the 1st order systemẋ=yẏ=δsin(2x)/1+δcos(2x)(y^2-1).Our goal is to find all 2π-periodic solutions of that ODE.Note that we have theconserved quantity d/dθH(φ,φ')=0, whereH(x,y)=(1+δcos(2x))(1-y^2).We assume from now on that δ>0 (the case δ <0 can be recovered using the symmetries),so we have H≤ 1+δ.For H_0 < 1-δ, the level set { H= H_0} is the union of two unbounded curvesy =±√(1-H_0/1+δcos(2x)),hence H_0<1-δ cannot correspond to a periodic solution φ. For 1-δ <H_0 < 1+δ, the level set { H=H_0}, intersected with { |x| < π/2},is a closed curve, which crosses the x-axis at x_±= ±1/2arccos( H_0-1/δ).It corresponds to a periodic trajectory of the differential system, whose half-period is the time needed to go from x_- to x_+ along the curveẋ =y = √(1-H_0/1+δcos(2x)),so the corresponding period T=T_δ(H_0) is given byT_δ(H_0) =2 ∫_x_-^x^+dx/√(1-H_0/1+δcos(2x)) =2 ∫_0^arccos(H_0-1/δ)dx/√(1-H_0/1+δcos(x)) =2/√(δ)∫_σ^1 √(1+δ t)/√((1-t^2)(t-σ))dt, σ=H_0-1/δ∈ (-1,1)For any σ∈ (-1,1) and δ∈ (0,1) we haved/dδ[T_δ(1+δσ)]= d/dδ[ 2/√(δ)∫_σ^1 √(1+δ t)/√((1-t^2)(t-σ))dt ] =-1/δ^3/2∫_σ^1 dt/√((1+δ t)(1-t^2)(t-σ)) <0,so we inferT_δ(1+δσ) >T_1(1+σ)= 2∫_σ^1 dt/√((1-t)(t-σ̂))dt= 2 [ arcsin( 2/1-σt-1+σ/1-σ) ]_σ^1= 2πTherefore, all periodic solutions of (<ref>) corresponding to values of H_0∈ (1-δ,1+δ) have a period strictly larger than 2π. Hence the only 2π-periodic solutions of (<ref>) must corresponds to values H_0∈{ 1±δ}.For H_0=1-δ, the level set { H=H_0}, intersected with { |x|≤π/2},is also a closed curve, but its intersections x=x_± with the x-axis correspond to the constant solutionsx≡±π/2, so all other solutions corresponding to H_0=1-δ are monotone and cannot be periodic. Hence the only periodic solutions corresponding to H_0=1-δ are constants φ≡π/2 modulo π.The only remaining value of H_0 is H_0=1+δ, which corresponds to constant solutions φ≡ 0 modulo π.Gathering all cases, we conclude that the only 2π-periodic solutions of (<ref>) are φ≡ 0 modulo π/2.§.§ The degree 2 case In this Section we show that, for d=2, the uniquesolution (modulo frame invariance) provided by the first item in Theorem <ref> is minimizing.In factit turns out thatξ_δ(θ)=e^2iθ solves (<ref>), so this isthe unique solution (modulo frame invariance). Taking a competitor u= e^2iθ e^iφ with φ∈ C^2_c( A_ρ;ℝ), and using(∇· u)^2-(∇× u)^2 =ℜ𝔢((∂_ηu̅)^2),∂_η=∂_x+i∂_y =e^iθ(∂_r + i/r∂_θ),we obtain|∇ u|^2 +δ( (∇· u)^2-(∇× u)^2) =|∇φ|^2 +4/r^2+ 4/r^2∂_θφ + δ ℜ𝔢( (-i∂_r φ +2+∂_θφ/r)^2e^-2i(θ+φ))=|∇φ|^2 +4/r^2+4/r^2∂_θφ + δ ((2+∂_θφ)^2/r^2-(∂_rφ)^2)cos(2θ+2φ)- 2δ∂_rφ2+∂_θφ/rsin(2θ+2φ),and therefore, substracting the energy density of ξ_δ (which corresponds to φ=0) and integrating over A_ρ, we haveE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ) = ∫_A_ρ[ |∇φ|^2 + δ ( (∂_θφ)^2 +4∂_θφ/r^2-(∂_rφ)^2)cos(2θ+2φ)+4δ/r^2cos(2θ+2φ) - 2δ∂_rφ2+∂_θφ/rsin(2θ+2φ) ]dx.Noting that1+∂_θφ/r^2cos(2θ+2φ) - 1/r∂_rφsin(2θ+2φ)=1/2r^2∂_θ[sin(2θ+2φ)] +1/2r∂_r[cos(2θ+2φ)],this simplifies toE_δ(u;A_ρ)-E_δ(ξ_δ;A_ρ)= ∫_A_ρ[ (1-δcos(2θ+2φ))(∂_rφ)^2+(1+δcos(2θ+2φ))(∂_θφ)^2/r^2 - 2δ∂_rφ∂_θφ/rsin(2θ+2φ) ] dxFor any δ∈ (-1,1) and C,S∈ [-1,1] such that C^2+S^2=1, the quadratic formq(X,Y)=(1-δ C)X^2+(1+δ C)Y^2-2δ SXY,has determinant (q)=1-δ^2 C^2 -δ^2S^2=1-δ^2>0 and is therefore positive definite, so ξ_δ(θ)=e^2iθ is minimizing in A_ρ, for all δ∈ (-1,1). § ENTIRE SOLUTIONS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION In this section we prove Corollary <ref>. So we consider an entire solution u^2→^2 of the anisotropic Ginzburg-Landau equation (<ref>) with finite potential energy (<ref>) and degree (u)=d ∈ℤ∖{ 0,1,2}.The anisotropy satisfies 0<|δ|<δ_0, for a small enough δ_0∈ (0,1) to be adjusted in the course of the proof.We assume that u is either locally minimizing, or symmetric (<ref>) and locally minimizing with respect to symmetric competitors. Under these assumptions, the methods in <cit.>provide a logarithmic bound for the energy (<ref>) of u,lim inf_R→ +∞GL_δ(u;D_R)/ln R <∞.The statement of <cit.> considers symmetric solutions (<ref>) with an additional mirror symmetry constraint, but the same proof applies for non-symmetric or less symmetric solutions, as long as they are locally minimizing in their admissible class.If δ_0 is small enough, the third point of Theorem <ref>ensures the existence of ρ∈ (0,1/2) such that any map u_*∈ H^1( A_2ρ;𝕊^1) which minimizes E_δ among 𝕊^1-valued maps agreeing with u_* on ∂ A_2ρ, cannot be 0-homogeneous:∫_A_2ρ|∂_r u_*|^2 dx > 0.The same conclusion is valid if u_* is symmetric and minimizing only among symmetric maps, because the competitor in Proposition <ref> is symmetric.As a first step to prove Corollary <ref>, we claim that the logarithmic bound (<ref>) implieslim inf_R→ +∞ GL_δ(u;D_2R∖ D_ρ R) <∞.Otherwise, for any M>0 we have the existence of R_0>0 such that GL_δ(u;D_2R∖ D_ρ R) = GL_δ(u;D_2R)-GL_δ(u; D_ρ R) ≥ M ∀ R≥ R_0.Applying this to R=(2/ρ)^jR_0 and summing over j=1,…, k, we deduceGL_δ(u;D_(2/ρ)^kR_0)≥ k M =M/ln(2/ρ)ln(2/ρ)^k R_0/R_0,which implieslim inf_R→ +∞GL_δ(u;D_R)/ln R≥M/ln(2/ρ),in contradiction with (<ref>) since ρ∈ (0,1/2) is fixed and M is arbitrary. So (<ref>) is established, and there exists a sequence R_k→ +∞ such that GL_δ(u;D_2R_k∖ D_ρ R_k)≤ C,for some constant C>0.Following <cit.> we define the rescaled map u_k(x)=u(R_k x),and _k=1/R_k, so the above energy bound in D_2R_k∖ D_ρ R_k translates into a bound on the energy GL_δ,_k (<ref>) of u_k in D_2∖ D_ρ, namelyGL_δ,_k(u_k;D_2∖ D_ρ)≤ C.Since u_k is minimizing with respect to its own boundary conditions,with or without the symmetry constraint, standard methods (see e.g. <cit.> combined with an appropriate selection of traces as for instance in <cit.>) implythat, up to a non-relabeled subsequence, u_k→ u_* in H^1(A_2ρ;^2), and u_*∈ H^1(A_2ρ;𝕊^1) minimizes E_δ among 𝕊^1-valued maps agreeing with u_* on ∂ A_2ρ. Therefore u_* is not 0-homogeneous (<ref>). This implies that ∂_r u_k has a nontrivial limit in L^2 and, scaling back to the originial variable,lim inf_k→∞∫_D_R_k∖ D_2ρ R_k|∂_r u|^2dx>0.Along a subsequence, the annuli D_R_k∖ D_2ρ R_k are all disjoint, and we deduce the first conclusion of Corollary <ref>, that is,∫_^2 |∂_r u|^2dx = +∞.Further, by continuity of the trace embedding, for all r∈ [2ρ,1] we have u_k→ u_* in L^2(∂ D_r;^2). Since u_* is not 0-homogeneous we may find r_1,r_2∈ [2ρ,1] such that u_*(r_1 e^iθ)≠ u_*(r_2 e^iθ) for a non-negligible set of θ∈𝕊^1. As a consequence, u_R(e^iθ)=u(Re^iθ) converges to different limits in L^2(𝕊^1;^2) along the sequences R=r_1R_k→ +∞ andR=r_2 R_k→ +∞, which impliesthe last assertion of Corollary <ref>, that u_R is not convergent as R→ +∞ (in L^2(𝕊^1;^2), nor in the sense of distributions).acm | http://arxiv.org/abs/2311.15758v1 | {
"authors": [
"Andres Contreras",
"Xavier Lamy"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20231127122753",
"title": "A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\\mathbb S^1$"
} |
𝐱 ŁLsequation Can Sun^*,Hao Zheng^*,Zhigang Hu^,Liu Yang,Meiguang Zheng,Bo Xu^*Co-first authors. ^ Zhigang Hu is the corresponding author.School of Computer Science and Engineering, Central South University, ChinaMetaDefa: Meta-learning based on Domain Enhancement and Feature Alignment for Single Domain Generalization [==========================================================================================================The single domain generalization(SDG) based on meta-learning has emerged as an effective technique for solving the domain-shift problem. However, the inadequate match of data distribution between source and augmented domains and difficult separation of domain-invariant features from domain-related features make SDG model hard to achieve great generalization. Therefore, a novel meta-learning method based on domain enhancement and feature alignment (MetaDefa) is proposed to improve the model generalization performance. First, the background substitution and visual corruptions techniques are used to generate diverse and effective augmented domains. Then, the multi-channel feature alignment module based on class activation maps and class agnostic activation maps is designed to effectively extract adequate transferability knowledge. In this module, domain-invariant features can be fully explored by focusing on similar target regions between source and augmented domains feature space and suppressing the feature representation of non-similar target regions. Extensive experiments on two publicly available datasets show that MetaDefa has significant generalization performance advantages in unknown multiple target domains. Single domain generalization, Domain enhancement, Feature alignment, Meta-learning § INTRODUCTIONDeep neural networks driven by numerous labeled samples has made remarkable progress in a wide range of computer vision tasks <cit.>. However, due to the significant data distribution differences between the source and target domains, the model performance will decrease apparently<cit.>, which is known as the domain-shift problem.Single domain generalization can effectively solve the domain-shift problem. The SDG method improves the model generalization performance by training the model in single source domain to learn transferability knowledge and applying the knowledge learned to the unknown multiple target domains <cit.>. Some scholars researched SDG based on the feature representation to reduce the feature space difference between source and augmented domains, making the model more focused on domain-invariant features <cit.>. However, the limited performance of the feature representation method itself brings unsatisfactory feature alignment and loss of source domain information. On the other hand, in order to adapting quickly to new tasks and classes in unknown target domains, many researchers started to apply meta-learning to SDG by dividing the source domain into several virtual train domains and virtual test domains. For example, some methods <cit.> make efforts to make the model learn sufficient prior knowledge and shared experience by simulating the generalization step in the training process. However, these methods are also difficult to generate more adaptable augmented domains and extract adequate transferability knowledge. Even worse, the model suffers from the problem of neglecting target category feature representation and insufficient suppression of non-target categories.Hence, to solve the above deficiencies, a meta-learning method based on domain enhancement and feature alignment for single domain generalization is designed. First, a domain enhancement module using background substitution and visual corruptions techniques is proposed, which considers the reality of the enhanced domain while introducing more variation and uncertainty. Second, the multi-channel feature alignment module is designed to reduce the gap between the target category regions in both the source and augmented domains feature space and squeeze non-target category areas.The main contributions of this paper are listed as follows.a) The domain enhancement module based on background substitution and visual corruptions is proposed to generate diverse and effective styles of augmented domains to adequately simulate unknown domain distributions b) The multi-channel feature alignment module suppresses image CAAM approaching CAM and enlarges image secondary regions respectively in each iteration of meta-training and meta-testing. This achieves intra-class compactness and inter-class separability by imposing consistency on the CAM of the original and enhanced images. c) Extensive experiments are conducted on two benchmark datasets, and the experimental results demonstrated the superior comprehensive performance of MetaDefa.§ METHOD In the context of meta-learning for SDG, consider a single source domain S and multiple target domains T. The domain S will be partitioned into virtual train domain S_train and virtual test domain S_test. The function f_ϕ: S_x→𝒴 is introduced to map the input image X from S to a label-like heat vector Y, and the parameter ϕ need to be learned. During each iteration, the meta-train stage is entered first. The model starts training on S_train, where the loss and gradients are calculated, and the parameter ϕ are updated as ϕ⟶θ̂^i. Subsequently, the meta-test stage is executed. Here, the model is trained on S_test utilizing the updated parameter θ̂^i to calculate ℒ(f_θ^i). The gradients are then computed and saved. Finally, this entire process is repeated 'n' times, and all the stored gradients are accumulated and used to update the initial parameter ϕ.Fig. 1 shows the domain enhancement and multi-channel feature alignment module of MetaDefa. The algorithmic details of MetaDefa are shown in Algorithm 1. §.§ Domain EnhancementTo adequately match the data distribution between S_train and virtual augmented train domain S_train^aug, S_test and virtual augmented test domain S_test^aug, taking diversity and effectiveness into consideration, the new domain enhancement module generates optimal enhanced domains through background substitution and visual corruptions techniques.a) Background substitution: Prior research has demonstrated that data augmentation methods that prioritize diversity over effectiveness will lead to a decline in performance <cit.>. Background substitution techniques are performed to ensure effectiveness. During the replacement process, we initially utilize instance mask annotations to identify the object region within one image and keep it unchanged. Another image from S_train of a different class is selected to extract a random patch. Finally, the image background is replaceed with the random patch to produce a valid enhanced image.b) Visual corruptions: The diversity of images brought about by visual corruptions dramatically boost the model's generalization capabilities when dealing with unknown multiple target domain. To expand the difference in data distribution between S_train and S_train^aug, S_test and S_test^aug, a minimum threshold is designed for fundamental visual corruptions. Only visual corruptions with a randomly impairment probability above the threshold can be executed. §.§ Multi-channel Feature AlignmentThe transferable knowledge between S_train and S_train^aug, S_test and S_test^aug observably influences model generalization. The multi-channel feature alignment module including focusing on domain-invariant features and inhibiting domain-related features is used to extract adequate transferability knowledge.a) Focus on domain-invariant features: The class activation maps(CAM) is known for visualizing the spatial regions of feature maps <cit.>. Different from the CAM-loss <cit.>, this paper aims to constrain the model to find consistent and generic visual cues within various views of the same input image by minimizing the distance between CAMs of S_train and S_train^aug. The model can then reuse these cues when dealing with unfamiliar target domains.For a given image, let f_k(x, y) represent the activation value of the kth feature map at the spatial location (x,y). A global average pooling operation is executed: F_k=1/H × W∑_x, y f_k(x, y). For a given class i, a softmax operation is applied to obtain z_i as: z_i=∑_k w_k^i F_k. Combining the expressions for F_k and z_i yields: z_i = 1/H × W∑_x, y∑_k w_k^i f_k(x, y).Define CAM_i as the class activation maps of class i, where each spatial element is represented as CAM_i(x, y)=∑_k w_k^i f_k(x, y).To fully explore domain-invariant features, the ℒ_CAM loss is formulated using the Jensen-Shannon divergence:ℒ_C A M(M_C A M, M_C A M^aug, i)=D_J S(M_C A M M_C A M^aug) where M_C A M represents the CAM of S_train for a given class i, M_C A M^aug represents the CAM of S_train^aug.b) Inhibit domain-related features: As observed in Fig.2(b)(e), Since the CAMs of target and non-target categories may overlap in some areas, the model that only consider CAMs has limitations in capturing the intricate relationships between target and non-target categories. The class agnostic activation maps (CAAM) presents more significant activation regions and richer features than CAM, as shown in Fig. 2(c)(f), where each spatial element is expressed as: CAAM(x, y)=∑_k f_k(x, y). Encouraging CAAM to closely align with CAM of the target category to suppress the expression of non-target category features. ℒ_minor^ori and ℒ_minor^aug are formulated as: ℒ_minor ^ori =1/H × W∑_x, yM_C A A M-M_C A M_l_1ℒ_minor ^ aug =1/H × W∑_x, yM_C A A M^aug-M_C A M^aug_l_1M_C A A M and M_C A A M^aug denote the CAAM of S_train and S_train^aug, respectively. Compared with the CAM-loss <cit.>, in order to further enhance the model's perception and suppression of non-target categories, the secondary areas between S_train and S_train^aug should be as different as possible. The ℒ_style loss is formulated as follows:ℒ_style (ℒ_minor ^ori , ℒ_minor ^aug , i)=|ℒ_minor ^ori -ℒ_minor ^aug | According to (1)(2)(3)(4), by introducing the hyperparameters λ_1 and λ_2, the final objective function is defined as:ℒ=ℒ_C E+λ_1(ℒ_C A M+ℒ_minor ^ori +ℒ_minor ^aug )-λ_2ℒ_style§ EXPERIMENTS§.§ Datasets and SetupWe conduct an extensive evaluation of MetaDefa using two benchmark datasets. The Office-Caltech-10 dataset <cit.> comprises 2533 samples distributed across four domains. The Office31 <cit.> includes 4110 images divided into 31 categories, collected from three domains. The Office31 is more challenging due to its larger number of categories compared to Office-Caltech-10 with only 10 categories.For all settings, RGB images is uniformly resized to 224×224. The domain DSLR is chosen as the single source domain. We adopt ResNet-18 pre-trained on ImageNet. The learning rate, batch size and training epoch are set to 4×10^-3, 128 and 30, respectively. To this end, we repeat all experiments five times and take their average as the final results.§.§ Baseline and Comparison MethodsTo assess the effectiveness of MetaDefa, comparisons are made in terms of both domain enhancement and feature alignment. These comparisons includes (1) baseline: meta-learning using only cross-entropy loss without any domain enhancement, (2) CutOut <cit.> and RandAugment(RandAug) <cit.> using advanced domain enhancement techniques, while other module settings are consistent with MetaDefa; (3) L2D <cit.> based on semantic consistency to align domain-invariant features, (4) ACVC <cit.> based on class activation map but not suppress non-target categories,and (5) CAM-loss <cit.>: Constraining CAAM to lean on CAM. §.§ Comparison on office-Caltech-10 and office31As shown in Table 1, MetaDefa consistently achieves the optimal generalization effect across the three target domains, which the accuracies are 88.44%, 80.21% and 97.97%, respectively. By considering the effectiveness of enhancement, MetaDefa increases the performance by 2.62% and 2.8% compared to CutOut and RandAug that only care about diversity. This verifies the effectiveness of the proposed domain enhancement module. Moreover, due to MetaDefa suppresses non-target category features during feature alignment, the model gains more transferable knowledge, which results in MetaDefa improves model performance by 2.54%, 1.66% and 1.43% compared to L2D, ACVC and CAM-loss.Regarding Table 2, MetaDefa obtains the superior generalization performance on the office31 dataset. Only using simple domain enhancement methods such as CutOut and Randug, the model's generalization performance improvement is limited, which emphasizes the importance of considering the effectiveness of enhancement. MetaDefa performs better in the face of datasets with more categories and significant domain distribution differences by inhibiting domain-related features. For instance, MetaDefa enhances model accuracy by 3.82% on office31 and only achieves a performance improvement of 2.76% on office-Caltech-10. Combined with Fig. 3(a)(b), MetaDefa has the competitive universality and advancement on two datasets. §.§ AnalysisIn MetaDefa, the different loss terms constructed based on multi-channel feature alignment are crucial. Sufficient experiments on two benchmark datasets are conducted to evaluate their effectiveness. As shown in Table 3, all the designed loss terms can improve the model prediction accuracy over the baseline. In office-Caltech-10 with a small number of categories, suppressing non-target category features will have the better performance improvement, increasing by 1.86%. Encouraging the feature expression of the target category improves the model performance by 1.97% in office31.§ CONCLUSION This paper proposes a meta-learning scheme based on domain enhancement and feature alignment. The MetaDefa focus on creating diverse and influential enhancement domains and effectively extract adequate transferability knowledge. Experimental results show that MetaDefa achieves excellent generalization performance on the benchmark datasets. In the future, we will take into account both the model's output and feature maps to enhance the model's generalization.IEEEbib | http://arxiv.org/abs/2311.15906v1 | {
"authors": [
"Can Sun",
"Hao Zheng",
"Zhigang Hu",
"Liu Yang",
"Meiguang Zheng",
"Bo Xu"
],
"categories": [
"cs.CV",
"cs.LG"
],
"primary_category": "cs.CV",
"published": "20231127151302",
"title": "MetaDefa: Meta-learning based on Domain Enhancement and Feature Alignment for Single Domain Generalization"
} |
Quantum ratchet with Lindblad rate equations Jesús Casado-Pascual January 14, 2024 ============================================§ INTRODUCTION After decades of searching for Dark Matter (DM), the models in which DM interacts directly with the Standard Model (SM) particles are strongly disfavoured. In particular, the parameter space of the standard WIMP (Weakly Interacting Massive Particle) models has been severely constrained. This has slowly led to a great interest in models with DM interactions via mediators such as Higgs portal model, neutrino portal model, etc <cit.>. Among them, one particular model provides a variety of phenomenological aspects, i.e., a fermion DM with an axion-like particle as the mediator. Axion-like particles (ALPs) are light pseudoscalar particles arising from extensions of the standard model. For example, the String/M theory framework generically provides a plethora of pseudoscalars due to the compactification process of the extra-dimensions <cit.>. Since ALPs mainly interact with gauge bosons and fermions via dimension-five couplings, the main phenomenological aspects rely on the ALP-fermion and ALP-photon couplings. One could test the ALP-mediated models using particle colliders such as the beam dump experiments and rare mesons decay. Other interesting tests come from astrophysics such as the neutrino flux coming from supernova SN1987A, effects on stellar evolution of the horizontal branch of the HR-diagram and the number of relativistic degrees of freedom around the time of Big Bang Nucleosynthesis (BBN). Recently, there has been interest in dark matter capturing in celestial objects such as neutron star (NS) and brown dwarf (BD) <cit.>. It has been shown that if the multiscattering effect is taken into account, the celestial objects accumulating dark matter are appealing targets for observation of DM annihilation. The DM-overdense objects then could be observed in 2 different ways depending on the lifetime of the mediator. If the mediator is short-lived and decays inside the celestial object, an extra energy would change the energy budget and the DM can be observed through the lifetime and stability of the celestial objects. On the other hand, if the mediator has a sufficiently long enough lifetime such that it decays outside the celestial object, the situation allows us to probe DM via indirect detection. In this work, we study ALP-mediated DM models using an effective Lagrangian approach. We then use the DM-nucleon cross-section to calculate the multiscatter capture rate of the celestial objects in the DM-rich region. We are interested in the case where ALP produced from annihilation has a sufficiently long lifetime such that the ALP decays into SM particles. Using the distribution of celestial objects near the Galactic Center and the DM generalized Navarro-Frenk-White (NFW) density profile, we are able to produce gamma-ray fluxes and neutrino fluxes from the DM annihilation in this model. The fluxes from neutrino produced from DM annihilation inside BDs could be detected via IceCube experiment and ANTARES neutrino telescope, whereas in the NS case, the neutrino fluxes are too low for any current observations. The gamma-ray fluxes from NSs and BDs can be tested against Fermi satellite and H.E.S.S. data. The constraints from gamma-ray observation using NSs and BDs as targets are able to rule out a significant portion of parameter space of coupling between ALP and fermions and the ALP mass. The constraints from gamma-ray fluxes perform better than the ones from neutrino fluxes due to the difference in the observed fluxes from the experiments. § DARK MATTER SIMPLIFIED MODEL In this model, the SM is extended by a Dirac fermion, χ, and a pseudoscalar ALP, a. The effective Lagrangian is given byℒ⊃1/2∂_μ a ∂^μ a - 1/2m^2_a a^2 + ∑_f m_f/f_aC_ff̅iγ_5 f a + m_χ/f_aC_χχ̅ i γ_5 χ a . - g_aγγ/4F_μνF^μνa,where f is any SM fermion with mass m_f, the masses of ALP and DM are m_a and m_χ, respectively. F_μν = ∂_μ A_ν - ∂_ν A_μ is the U(1)_ EM field strength tensor (F^μν = 1/2ϵ^μναβF_αβ) and g_aγγ is an effective ALP-photon coupling. For convenience, we define the effective ALP couplings by g_aff = m_f C_f/f_aand g_aχχ = m_χ C_χ/f_a. In this model, we assume that ALP-fermion coupling C_f is universal for any fermion, i.e., C_f and g_aff∝ m_f. The Dirac fermion χ is assumed to be the DM particle. The ALP, a, acts as the mediator between SM and the hidden sector where g_aff acts as the connector coupling and g_aχχ acts as the hidden sector coupling. Figure <ref> shows the diagram of the scattering between DM and nucleons which is mediated by ALP and the diagrams of ALP decay into 2 photons and 2 SM particles (specifically neutrinos if kinematically allowed), respectively. The different regimes of interactions were studied for the freeze-in/out scenarios in <cit.>, therefore we will not focus on the DM production mechanism in this paper. As DM condenses around the celestial objects in the DM-rich region, the DM can fall into the objects by the mechanism called “dark matter multiscatter capture” <cit.>. We will discuss this mechanism in the next section. DM can lose kinetic energy by scattering with the object's components and falling into the celestial object.The scattering cross-section of χ f →χ f in the low-energy limit isσ(χ f →χ f) = g_aff^2 g_aχχ^2/2πm_a^4m^4_χ+ m^4_f + m^2_χ m^2_f/(m_χ + m_f)^2.Each time that DM scatters with the components of the object, DM will lose kinetic energy and move toward the center of the object. DM will then accumulate and start to annihilate inside the object producing a long-lived mediator which is an ALP. The lifetime of ALP decay into gamma-rays is given byτ^-1_a→γγ = Γ_a→γγ = 1/64πg^2_aγγm_a^3,and the lifetime of ALP decay into SM particles is written asτ^-1_a→ ff = Γ_a→ ff = g_aff^2n_f^c m_a/8π√(1-4m^2_f/m_a^2). The probes on the ALP coupling have been studied in various methods (see <cit.> for a review). For example, the astrophysical sources such as SN 1987A can provide us the constraints on ALP coupling that g_aγγ<6×10^-9 GeV^-1 for small ALP masses <cit.>. The red giant branch in several clusters <cit.> give the upper bounds on the ALP coupled to electrons as g_aee< 1.48×10^-13, which is also similar to the analysis of the red giant in the Galactic globular cluster ω Centauri g_aee< 1.3×10^-13 <cit.>. The ALP coupled to both SM and DM has been recently studied in the context of thermal history <cit.>.We are interested in the case that ALPs were generated by the DM annihilation from the celestial objects. We primarily aimed to constrain the ALP-SM coupling, g_aff from the gamma-rays and neutrinos observations. The celestial objects such as NSs and BDs are good targets since they are densely distributed around the Galactic Center where the DM density is generically higher. § DARK MATTER MULTISCATTER CAPTUREDM from the Galactic halo can start to fall into the celestial object if the DM is sped up to the escape velocity of the object at the surface of the celestial body which is v_esc = √(G_N M/R),where G_N is the gravitational constant, M and R are the total mass and the radius of the celestial body, respectively. As DM particles transit through the celestial body, they can scatter with stellar materials and lose their kinetic energy. Once the velocity of DM drops below the escape velocity of the celestial objects, DM is captured. The capturing process can occur via single or multiple scatters <cit.>. The probability for a given DM to undergo N scatter is given by p_N(τ) = 2∫_0^1 dy y e^yτ (yτ)^N/N!,where τ = 3/2σ_χ n/σ_sat is the optical dept. The saturate cross-section is defined as σ_sat≡π R^2/N_n, where N_n is the number of nucleons inside the celestial object. The meaning of saturation cross-section σ_sat is the threshold that guarantees that DM will scatter at least once. The probability can be approximated in the limit of single scatter (τ≤ 1) and multiple scatter (τ≫ 1) <cit.> asp_N(τ) ≈2τ^N/N!(N+2) + 𝒪(τ^N+1),if τ≤ 1 2/τ^2(N+1)Θ(τ-N).if τ≫ 1 The capture rate after DM scattered N times and became trapped in the celestial object is given by C_N(τ) = π R^2 p_N(τ) √(6)n_χ/3√(π)v̅( (2v̅^2 + 3v^2_esc) - (2v̅^2 + 3v^2_N) exp( - 3/2(2v_N^2 + 3v^2_esc) /v̅^2)),where n_χ = ρ_χ(r)/m_χ is the local number density of DM and v̅ is the DM dispersion velocity.After scattering N times, the typical velocity of DM which takes into account the energy loss in each scattering isv_N = v_esc(1 - β_+/2)^-N/2,where β_+ = 4m_χ m_n/(m_χ + m_n)^2. The total capture rate for a single celestial body is then given byC = ∑_N=1^∞ C_N.For sufficiently large N, 2v̅^2 + 3v^2_N becomes much larger than v̅_N^2, therefore, the exponential term in equation <ref> can be neglected and the capture rate approaches C_N→ p_N(τ)× C_max. C_max is called the maximum capture rate (geometric capture rate) which is given byC_max = π R^2 n_χ(r)v_0 ( 1 + 3/2v^2_esc/v̅(r)^2),where v_0 = √(8/3π)v̅. This guarantees that the particles undergo a sufficiently large number of N scatters are captured. Figure <ref> shows the mass capture rate for a NS (left) with M_NS=1.5 M_⊙ and R_NS=10 km and a BD (right) with M_BD = 0.0378 M_⊙ and R_BD = 69,911 km as a function of nucleon scattering cross-section σ_χ n. As σ_χ n increases, the number of scatters N also increases, once it goes above the certain number of scatters that is sufficiently large enough to capture DM, the capture rate in equation <ref> (black line) is approximately the maximum capture rate (red dash line) (equation <ref>). It can be seen that the BD capture rate is 100 times larger than the NS capture rate. This is because the effective radius of BD is much larger than the radius of NS, which causes more DM flux to pass through the object. In order to calculate the total capture rate for all celestial objects within a given system (e.g. the Galactic Center or galaxy clusters), we need to take into account the number density of the objects within radii r_1 and r_2, the total capture rate isC_total = 4π∫_r_1^r_2 r^2 n_⋆ (r) Cdr,where n_⋆ (r) is the number density of the celestial objects and C is the capture rate of a single object (equation <ref>). The number density of the celestial objects will be described in the following section.§ THE GALACTIC CENTER In this section, we introduce the specific model of the DM velocity distribution and the target number density. The DM velocity dispersion and DM density affect the rate at which DM would fall into the gravitational potential and eventually fall into the object. At a given radius r from the Galactic Center, the dispersion velocity is v̅ = √(3/2) v_c(r),where v_c is the circular velocity at radius r, which depends on the total mass of the sphere of radius r:v_c = √(G_N M(r)/r).We use the model for the mass distribution following reference <cit.>. This model has five components which are the central supermassive black holes M_BH = 4×10^6 M_⊙, inner and outer bulges (ρ_inner and ρ_outer), an disk component (ρ_disk) and DM halo (ρ_DM). Combining all five components, the total mass isM(r) = M_BH + 4π∫_0^r r^2 (ρ_inner + ρ_outer + ρ_disk + ρ_DM) dr.The DM generalized Navarro-Frenk-White (NFW) density profile <cit.> isρ_DM = ρ_0/(r/r_s)^γ (1 + (r/r_s)^1-γ),where ρ_0 = 0.42 GeV/cm^3 is the local DM density, the scale radius is r_s = 12 kpc and the slope index is ranging from γ = 1-1.5. The model of the inner bulge, outer bulge and disk is assumed to be an exponential sphere model:ρ_i = ρ_0,i e^-r/a_i,where the parameters of each component are shown in table <ref>. Neutron stars (NS) is the collapsed core of the massive star (10-25 M_⊙), NS is composed almost entirely of degenerate neutrons with a mass ranging from 1-1.5 M_⊙ and a radius R_NS≃10 km. The number densities of NSs and black holes in regions around the Galactic Center have been numerically studied with the nuclear star cluster dynamics <cit.>. Two types of cluster models were described, `Fiducial' and `Fiducial × 10', both of which are potentially good candidates for an NS distribution in the nuclear star clusters. The NS radial number density distribution from the `Fiducial × 10' model is given byn_NS(r) = 5.98×10^3 ( r/1 pc)^-1.7 if0.1 pc <r <2 pc2.08×10^4 ( r/1 pc)^-3.5 ifr >2 pc. On the other hand, Brown dwarf (BD) is a failed star that is not massive enough to process the nuclear fusion of hydrogen into helium in its core. A huge amount of BDs are expected to be presented in our galaxy. The Milky Way may contain 25-100 billion BDs <cit.>. To obtain the radial distribution function of BDs, we follow the works in references <cit.>. They extended the Kroupa Initial Mass Function (IMF) into the BD IMF which is described by the broken power law function, dN_BD/dm∝ m^-α, where α=0.3, N_BD is the number of BD and m is the mass of BD. The number density of the BDs with the mass range from 0.01-0.07 M_⊙ is given byn_BD(r) = 7.5×10^4( r/1 pc)^-1.5.In our calculation, we will use the average mass of BDs as M_BD = 0.0378 M_⊙ and their radius is assumed to be equal to the Jupiter radius R_BD = R_J = 69,911 km. § DETECTION Once DM becomes trapped in the celestial bodies (NS or BD), it has two possible fates. First, if DM does not self-annihilate, its density will rise near the core of the celestial object and it can lead to black hole formation and collapse. The second case is that the DM is annihilated with each other inside the celestial body. The evolution of the number of DM particles <cit.> inside an object N(t) is governed by an interplay between DM capture rate and DM annihilation rate, i.e.,d N(t)/dt = C_total - C_A N(t)^2,where C_total is the total capture rate (equation <ref>) and C_A = ⟨σ_A v ⟩/V_eff is the average thermal annihilation cross-section over the effective volume of the celestial object body V_eff = 4π R_⋆^3/3. The solution of equation <ref> readsN(t) = √(C_total/C_A)tanht/t_eq,where t_eq = 1/√(C_A C_total) is the time scale required for the celestial object body to reach the DM equilibrium between capture and annihilation.Once it reaches the equilibrium, t>t_eq, the number of DM becomes time-independent, and the annihilation rate Γ_ann is simplyΓ_ann = C_total/2,where the factor 1/2 comes from the fact that the DM annihilation process involves 2 DM particles. Since it has been shown that within the lifetime of the universe, the equilibrium between DM capturing rate and DM annihilation rate has already been reached for most of the parameter space <cit.>, we will assume the above condition for our study. From equation <ref>, we know that the annihilation rate is proportional to the local DM density, Γ_ann∝ n_χ, and the total capture rate (equation <ref>) is proportional to the number density of the celestial body, Γ_ann∝ n_⋆, thus, the total annihilation rate is Γ_ann∝ n_χ n_⋆.We consider the case in which DM annihilates into a long-lived ALP which weakly interacts with the matter such that it can escape from the celestial body and subsequently decay into a pair of SM particles. DM captured in celestial objects is expected to be annihilated with a small relative velocity, the mediator may be produced with the Lorentz boost factor η = m_χ/m_a, where m_a is the mass of ALP. In order to calculate the sensitivities for the possible signals, we assume that the lifetime of the mediator, τ, is sufficiently long such that the decay length exceeds the radius of the object R: L = η c τ > R.The differential energy flux of gamma-ray arriving at Earth is given by <cit.>E^2 dΦ/dE = Γ_ann/4π D^2× E^2dN/dE×BR(a→SM)× P_surv,where D is the average distance from the target to Earth, in this case, the target is located around the Galactic Center, so D≈8 kpc. BR(a→SM) is the branching ratio for the mediator into SM particles. For simplicity, we are assuming the unity of the branching ratio and the analysis is then separated into 2 cases, i.e., the spectrum of DM annihilates into a mediator and then decays into a pair of either 100% gamma-rays or 100%neutrinos. Their number distributions can be described by a box-shaped spectrum <cit.> which is given bydN/dE = 4/Δ EΘ(E - E_-)Θ(E_+ - E),where Θ is Heaviside step function, Δ E = E_+ - E_- is the width of the spectrum and E_± = (m_χ/2)(1±√(1-m_a^2/m_χ^2)). The probability of the mediator, a, decaying outside the celestial object isP_surv = e^-R/η c τ - e^-D/η c τ. An example of the survival probability as a function of the decay length η c τ is shown in figure <ref>. The plot shows the survival probability of the SM particle produced by the mediator at the distance η c τ from a NS. The probability is taken with the assumption that the SM particle does not reach the Earth if the mediator decays inside the star (red dash line), i.e., for η c τ < R_⋆, the probability is assumed to be zero.In this work, we will consider the maximally optimistic case where we take P_surv = 1 for both gamma-rays and neutrinos cases. Furthermore, we also calculate the bounds of an effective ALP-photon coupling and an effective ALP-SM coupling for ALPs that decay in the distance between the object's surface and the Earth. The bound on g_aγγ are calculated from the lifetime in equation <ref> and the result is shown in the left panel of figure <ref>. We found that for the lower mass of ALP, the g_aγγ has already been excluded by the constraint from the helioscope CAST (CERN Axion Solar Telescope) <cit.>. Therefore for the ALP decaying into 100% photons case, we will aim to study for ALP mass range higher than 10^-9 GeV. In the 100% decaying into neutrinos case, the bound on g_aff is calculated with the lifetime from equation <ref>. The result is shown in the right panel of figure <ref> where m_f=m_ν=0.1 eV is assumed. We only show the bound from BD because it is the only target that can generate sufficient fluxes of neutrinos that are detectable by the experiments, which we will discuss shortly.In order to set the limits on the ALP coupling, We use the Galactic Center measured fluxes from various observations. The gamma-ray fluxes are taken from Fermi and H.E.S.S. and the details of the data analysis are given in <cit.>. Fermi data is used for the model fluxes with energy less than 100 GeV <cit.>, and we use H.E.S.S. <cit.> to set limits around TeV scale. The high energy neutrino fluxes were taken from IceCube <cit.> and ANTARES <cit.> where they have the upper limits on the muon neutrinos flux from the direction of the Galactic Center (-40^∘< l< 40^∘ and -3^∘< b< 3^∘ in Galactic Coordinate) in the energy range between 1 TeV and 1 PeV. We set the limits by requiring that the maximum value of the DM fluxes must not exceed the measurable fluxes from the observation. This can be done by determining the ALP coupling g_aff for each energy bin (E ≈ m_χ) which gives the highest possible scattering cross-section (equation <ref>) providing the energy flux (equation <ref>) below the measured flux. To generate the gamma-ray energy spectra, we use<cit.> to calculate the total DM capture rate. We use the number density of NSs from equation <ref> and the number density of BDs from equation <ref>. We integrate both densities over the spherical shell between radius r_1=0.1 pc and r_2=100 pc. Choosing r_1 = 0.1 pc to avoid poor modeling of the DM cusp-like profile at the Galactic Center. The outer radius r_2 = 100 pc is chosen due to the fact that NSs/BDs number density falls rapidly outside this region. We assume that the ALP decays into 100% photons and P_surv=1 for simplicity. We show the largest possible signal for different choices of density slope index γ=1,1.5 in the NFW DM profile in figure <ref> along with the measured gamma-ray flux from Fermi (black) and H.E.S.S. (blue). For NSs, one could expect that only the gamma-rays from the NFW DM profile with γ=1.5 are detectable by H.E.S.S. data.For BDs, both gamma-rays from slope index γ=1, 1.5 are higher than the measured fluxes from Fermi and H.E.S.S., which would give us strong constraints to the ALP coupling. To generate the neutrino spectra, we follow the same procedure as the gamma-rays spectra. The largest possible fluxes for γ=1, 1.5 are shown in figure <ref> along with the measured neutrino fluxes from IceCube (blue) and ANTARES (black). We find that the maximum flux from the NS is lower than the observation fluxes by approximately 2 orders of magnitude. As a result, we will study only BDs as the potential sources of neutrino flux. § RESULTS Figure <ref> shows our constraints on an effective ALP coupling for DM-SM scattering mediated by ALP. The gamma-rays were produced by the production of DMs that annihilated inside the object, χχ→ a, a→γγ. The limits were placed on E^2 dΦ/dE with the gamma-ray observatories (Fermi and H.E.S.S.). Our calculation on gamma-ray flux has been described in the previous section and the other parameters in the plot are g_aχχ=1×10^-3 and m_a = m_χ /10^5. The bounds from our works are displayed along with the other constraints. The Fermi upper limit is shown in black for BD targets and the H.E.S.S. upper limit using NSs and BDs as targets are shown in blue and red, respectively. We show the results of BDs for γ =1 in the dash line and γ=1.5 in the solid line. Since we concentrate on the simplest case where C_f is universal, therefore the result we present is in the form of g_aff/m_f = C_f/f_a, where m_f is the mass of the fermion. In the gamma-rays final state, we use proton and neutron mass for calculating the scattering cross section, therefore m_f = m_n ≈ m_p. For NSs, we show only γ=1.5 since the gamma-ray flux is detectable only at H.E.S.S. Our results show that the limits in this work are significantly stronger than previously existing constraints. For higher masses, NSs provide stronger bounds than BDs since NSs require a lower cross-section to capture the DM (figure <ref>). The other constraints in the figure <ref> are the following. The constraints of the study of the supernova SN 1987A can provide strong constraints <cit.> and recently updated in reference <cit.>. Considering the cooling of the horizontal branch (HB) stars provides the parameters space of ALPs <cit.>. The exclusion bounds for the decay of K^+ →π^+ + inv. from NA62 <cit.> are given in the form of branching ratio. The strong constraints come from the electron beam dump experiment SLAC E137 <cit.>, ALPs can be produced by the bremsstrahlung, positrons annihilation, or Primakoff effect. K^+ →π^+ a(→ e^+e^-) from NA 48/2 <cit.> provide C.L. 90% upper limits. B^+ → K^+ a(→μ^+μ^-) from LHCb <cit.> provide C.L. 95% upper limits on the branching ratio as a function of the lifetime of a. B → K^* a(→μ^+μ^-) from LHCb <cit.> also provide the 95% C.L. upper limit on the branching ratio as a function of the mass and lifetime of a. The limits from B → K^* a(→μ^+μ^-) at fixed target come from CHARM <cit.>. K^+ →π^+ a(→μ^+μ^-) from NA 48/2 <cit.> provide C.L. 90% upper limits on the branching ratio as a function of lifetime of a. From the work of <cit.> where they have studied the model of the production of ALP-mediated DM in the thermal scenarios, our work (small g_aχχ and g_aff) corresponds to the case that dark matter is produced by freeze-in and the ALP may or may not be in equilibrium with the SM. Therefore, viable parameter space for the freeze-in scenario is not ruled out by our study.We also show results when we vary the ratio of the mass of DM and ALP in figure <ref>. The constraints on g_aff NS as the targets are calculated with g_aχχ = 1×10^-3 and the mass ratios m_a = m_χ/10^4, m_a = m_χ/10^5 and m_a = m_χ/10^6 are shown in blue, red and black, respectively. The results show that the decreasing of the mass ratio gives stronger bounds in the lower masses region of ALP.Figure <ref> shows the upper limits on the ALP-fermion coupling from the neutrino observations (IceCube and ANTARES). The neutrino fluxes were generated by only focusing on BD as the targets as discussed in the earlier section. The bounds are weaker than the gamma-ray case but still stronger than existing constraints, especially in the higher ALP mass region.Under the universal ALP-fermion coupling assumption, the region where ALP decays en route (figure <ref>) can be constrained to be m_a ≳ 0.6 GeV. § CONCLUSION In this work, we have studied the constraints of the ALP-mediated DM models from celestial objects such as neutron stars (NS) and brown dwarfs (BD). The DM can be captured in celestial objects where the DM accumulation process is controlled by the dark matter multiscatter capture mechanism. The annihilation of DM inside the objects produces the long-lived ALPs which will decay into the SM particles outside the objects. We study the case that ALP decays into gamma-rays and neutrinos separately and compare the fluxes with data from observation. For gamma-ray fluxes, we present our results on g_aff/m_f in figure <ref>. The results show the strong bounds that can rule out a significant portion of the parameter space. We point out that by using NSs as targets the model provides a stronger bound than BD ones due to the lower fluxes required to match with H.E.S.S. data. For Fermi data with the lower DM mass range, only BDs are detectable via the gamma-ray channel. We also present the upper bounds for different mass ratios in figure <ref> and show that our limits are generally stronger than existing constraints across the mass range of ALP. The results for neutrino fluxes are present in figure <ref>. We found that the only possible targets that can produce sufficient neutrino flux are BDs. With data from IceCube and ANTARES, we can probe our model with the higher mass range. Similar to the gamma-ray channel, the bounds from neutrino fluxes are also generally stronger than the previous constraints on ALP. Provided that ALP decays en route towards the Earth, the allowed parameter space for this model is strictly constrained with m_a ≳ 0.6 GeV. This emphasizes the importance of the next-generation probes of the neutrino fluxes at higher energies such as ARIANNA, ARIA, IceCube-Gen2 <cit.>, KM3Net <cit.>, ANITA-IV <cit.>, PUEO <cit.>, RNO-G <cit.> and Auger <cit.>.For future works, we are interested in exploring the phenomenological consequences of the possibility that ALP decays inside celestial objects providing additional energy to their internal processes. We also would like to investigate the effects of ALP-mediated interactions on the nuclear equation of state (EoS) and neutron star stability.§ ACKNOWLEDGEMENTTK and CP have been supported by the National Astronomical Research Institute of Thailand (NARIT).CP is supported by Fundamental Fund 2566 of Khon Kaen University and Research Grant for New Scholar, Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation under contract no. RGNS64-043. We thank Areef Waeming and Daris Samart for their helpful comments and discussions.jhep | http://arxiv.org/abs/2311.15681v1 | {
"authors": [
"Tanech Klangburam",
"Chakrit Pongkitivanichkul"
],
"categories": [
"hep-ph",
"astro-ph.HE"
],
"primary_category": "hep-ph",
"published": "20231127101454",
"title": "Bounds on ALP-Mediated Dark Matter Models from Celestial Objects"
} |
Lévy flights and Lévy walks under stochastic resetting Bartłomiej Dybiec January 14, 2024 ====================================================== < g r a p h i c s >figureVisualization displaying 4× super-resolution results generated by our Cognitive Super-Resolution (CoSeR) model. CoSeR adeptly extracts cognitive information from a low-resolution (LR) image and utilizes it to generate a high-quality reference image. This reference image, aligning closely with the LR image in terms of semantics and textures, significantly benefits the super-resolution process. For conciseness, we denote the input, generated reference, and restoration result as LR, GR, and SR, respectively. Best viewed zoomed in. Existing super-resolution (SR) models primarily focus on restoring local texture details, often neglecting the global semantic information within the scene. This oversight can lead to the omission of crucial semantic details or the introduction of inaccurate textures during the recovery process. In our work, we introduce the Cognitive Super-Resolution (CoSeR) framework, empowering SR models with the capacity to comprehend low-resolution images. We achieve this by marrying image appearance and language understanding to generate a cognitive embedding, which not only activates prior information from large text-to-image diffusion models but also facilitates the generation of high-quality reference images to optimize the SR process. To further improve image fidelity, we propose a novel condition injection scheme called “All-in-Attention”, consolidating all conditional information into a single module. Consequently, our method successfully restores semantically correct and photorealistic details, demonstrating state-of-the-art performance across multiple benchmarks.Code: https://github.com/VINHYU/CoSeRhttps://github.com/VINHYU/CoSeR § INTRODUCTION Real-world image super-resolution (SR) is a fundamental task in the realm of image processing, aimed at enhancing low-resolution (LR) images to yield the high-resolution (HR) counterparts <cit.>. Its versatile applicability spans critical domains, including mobile phone photography <cit.>, autonomous driving <cit.>, and robotics <cit.>, while also influencing various computer vision tasks, notably object detection <cit.>, segmentation <cit.> and recognition <cit.>.Despite significant advancements in this field in recent years, the processing of complex real-world scenarios continues to pose enduring challenges. Utilizing image priors is a common strategy for tackling real-world SR problems. These priors may be either introduced explicitly in the form of reference images <cit.>, or implicitly leveraged through pre-trained generative models <cit.>. Especially, the recently emerged text-to-image diffusion models <cit.> exhibit a remarkable capability to generate high-quality images based on user-provided prompts. These models not only possess strong image priors but also allow precise responses to human instructions in the form of language. This opens up the potential to bridge low-level image processing and high-level abstract cognition.Consider the process by which human experts restore low-quality images <cit.>: They start by establishing a comprehensive understanding of the image, encompassing scene identification and primary subject recognition. Subsequently, their focus shifts to a meticulous examination and restoration of finer image details. In contrast, conventional image super-resolution techniques <cit.>, adhere to a bottom-up approach, primarily concentrating on local content and direct pixel-level processing. Consequently, these methodologies exhibit inherent limitations in grasping the holistic image context, often failing to restore severely degraded yet semantically vital details. Moreover, given the ill-posed nature of LR images, there is a possibility for introducing semantically erroneous textures <cit.>. To surmount these challenges, there arises a compelling rationale for imbuing the SR model with “cognitive" capabilities. In this pursuit, we introduce a pioneering SR methodology known as Cognitive Super-Resolution (CoSeR). Our approach aligns with the top-down cognitive process employed by humans in image perception. It commences with the generation of cognitive embeddings, a representation that encapsulates the overarching comprehension of the LR image, containing both scene semantics and image appearance. This cognitive embedding allows us to precisely leverage the implicit prior knowledge embedded in pre-trained text-to-image generation models, resulting in an enhanced capacity to restore image details in a manner akin to human expertise. Previous work <cit.> uses segmentation maps to offer semantics. However, acquiring ideal segmentation maps for real-world LR images remains difficult, even with advanced models like <cit.>. Moreover, semantic segmentation is constrained by predefined categories, limiting its applicability to open-world scenes. Apart from implicitly leveraging diffusion priors, we also advocate for the explicit utilization of image priors. We introduce a novel approach where we employ cognitive embeddings derived from LR inputs to generate reference images through diffusion models. These reference images are subsequently utilized to guide the restoration process. As shown in Figure <ref>, our cognitive embedding contains language understanding while preserving the color and texture information of the image, thus producing high-quality reference images that are not only semantically aligned but also similar in appearance.This explicit approach brings substantial improvements in capturing high-definition textures compared to relying solely on implicit diffusion knowledge.We have established both implicit and explicit cognitive priors for LR inputs. Then incorporating these priors effectively into our model is pivotal. Unlike the typical conditional generation methods <cit.>, super-resolution demands a heightened level of fidelity between outputs and low-quality inputs. In order to concurrently ensure texture realism and fidelity, we introduce an “All-in-Attention" design, which integrates multiple information sources via an attention mechanism, including cognitive embeddings, reference images, and LR inputs. This approach allows our model to flexibly use different conditional components, yielding improved results. Our experiments show that our model excels in preserving fidelity compared to previous methods while generating more intricate textures.The contributions of this paper can be summarized as: * We introduce CoSeR, a novel framework for high-detail image super-resolution. CoSeR autonomously extracts cognitive embeddings from LR images, harnessing implicit diffusion priors to enhance the LR input.* We incorporate diffusion priors explicitly by creating semantically coherent reference images, which act as guidance to improve the quality of the restored image.* To enhance image fidelity, we introduce a novel “All-in-Attention” architecture to integrate conditional information into the SR model. Our method achieves state-of-the-art performance across multiple benchmarks. § RELATED WORK §.§ Real-World Image Super-Resolution Real-world image SR has primarily revolved around two avenues: data utilization and image prior incorporation.The first category involves the creation of diverse and realistic pairwise data by adapting the physical collection means <cit.> or improving the generation pipeline <cit.>. Also, several works <cit.> combine both paired and unpaired data with weak supervision to enhance performance in real-world scenarios.The second line focuses on the use of image priors. While the “learning-from-scratch” approaches <cit.> demand substantial data and computational resources, using pre-trained generative models with rich texture priors has become a practical and economical practice. Several studies <cit.> have leveraged pre-trained Generative Adversarial Networks (GANs) to improve the super-resolution process. Nonetheless, these methods occasionally suffer from the generation of unrealistic textures, owing to the inherent limitations of GANs <cit.>. Consequently, there is a growing interest in utilizing more advanced pre-trained generative models, such as the denoising diffusion models <cit.>, in recent research.§.§ Diffusion-Based Super-ResolutionRecent approaches <cit.> utilize implicit knowledge from pre-trained diffusion models <cit.>, yet they typically focus on non-blind degradation <cit.> or specific domains like facial images <cit.>. In an alternative strategy proposed by Fei <cit.>, the simultaneous estimation of the degradation model is applied to address blind degradation. However, this method relies on test-time optimization and primarily explores SR under linear degradation, thereby exhibiting limitations in handling real-world complexities.Other approaches <cit.> leverage recent advancements in large-scale text-to-image diffusion models <cit.>. These models, trained on extensive datasets of high-definition images, provide enhanced capabilities for processing diverse content. StableSR <cit.> stands as a pioneering work, which harnesses prior information from diffusion models, resulting in improved fidelity. DiffBIR <cit.> combines a traditional pixel regression-based image recovery model with the text-to-image diffusion model, mitigating the adverse effects of LR degradation on the generation process. Despite notable advancements in visual quality, these methods have yet to fully harness the potential of large text-to-image generation models, mainly due to the limited image content comprehension.§.§ Reference-Based Super-Resolution The reference image serves as an explicit prior, ideally containing content relevant to the LR image to facilitate the generation of high-definition details. Recent advancements in reference-based SR can be categorized into two branches <cit.>. One branch prioritizes spatial alignment, employing techniques like CrossNet <cit.> and SSEN <cit.>. However, these methods often encounter challenges in establishing long-distance correspondences. The other branch, represented by SRNTT <cit.>, TTSR <cit.>, MASA-SR <cit.>, and CFE-PatchMatch <cit.>, utilizes patch-matching mechanisms to facilitate the establishment of long-range connections between the reference map and the LR image. Yet, manually specifying reference images in real scenarios is labor-intensive, motivating the development of an automated and high-quality reference generation approach. § METHODOLOGY Our Cognitive Super-Resolution (CoSeR) model employs a dual-stage process for restoring LR images. Initially, we develop a cognitive encoder to conduct a thorough analysis of the image content, conveying the cognitive embedding to the diffusion model. This enables the activation of pre-existing image priors within the pre-trained Stable Diffusion model <cit.>, facilitating the restoration of intricate details. Additionally, our approach utilizes cognitive understanding to generate high-fidelity reference images that closely align with the input semantics. These reference images serve as auxiliary information, contributing to the enhancement of super-resolution results. Ultimately, our model simultaneously applies three conditional controls to the pre-trained Stable Diffusion model: the LR image, cognitive embedding, and reference image. The comprehensive framework is elucidated in Figure <ref>. §.§ Cognitive EncoderTo distill cognitive information from LR images, our model commences with LR preprocessing aimed at mitigating the impact of degradation. Specifically, we employ a lightweight SRResnet <cit.> for 4× super-resolution, without additional supervision. Subsequently, we utilize a pre-trained CLIP <cit.> image encoder to extract features from the preprocessed image. It is crucial to underscore that, although CLIP adeptly aligns image and language content, a significant disparity persists between the image embedding and the language embedding. These two components focus on different points, where image features inherently capture spatially variant details, while language features encapsulate comprehensive information. Consequently, a single language token may correspond to multiple subjects dispersed across diverse regions of an image.To overcome the challenge of aligning image and language representations, prior methods <cit.> have often focused on aligning the class token of the image embedding and the class token of the corresponding language embedding, neglecting other tokens. However, relying solely on this single class token has been observed to introduce cognitive bias. As shown by the generated reference images from language tokens in the first row of Figure <ref> (left part), cognitive bias diminishes gradually as the token number (before and including the class token) increases. To simultaneously address information misalignment and inaccurate cognition, we introduce a cognitive adapter that is tailored to extract multi-token cognitive embedding from image features, shown in Figure <ref> (right part). Drawing inspiration from the Q-Former structure <cit.>, originally devised for vision-language representation learning, our approach employs learnable queries to interact with spatially-arranged image information, thereby reshaping information organization and facilitating feature compression. Our approach also incorporates a novel form of supervision, enhancing the adapter's capacity not only to reorganize image features but also to function as a modality transformer.We represent the CLIP image embedding extracted from LR as I∈ℝ^B × T_i × C_i, where B, T_i, C_i denote batch size, token number, and channel number, respectively. Additionally, L∈ℝ^B × T_l × C_l denotes the CLIP language embedding extracted from the ground-truth caption (extracted from HR images using BLIP2 <cit.>). In our cognitive adapter, we employ T_e learnable queries (T_e ≤ T_l) such that the resulting cognitive embedding is denoted as E∈ℝ^B × T_e × C_l. We propose to use T_e tokens preceding the class token L[t_cls] (inclusive) for supervision, as these tokens retain all previous information <cit.>. If there are insufficient supervision tokens, we use the class token for end-filling. Therefore, our supervision L^' can be regarded as a more comprehensive representation than the class token. The loss function for training the cognitive encoder is expressed as:ℒ_C E=E-L^'_2^2,whereL^'=Padding(L[: t_cls], L[t_cls]), ift_cls<T_e ;L[(t_cls-T_e): t_cls], ift_cls⩾ T_e .A more extensive explanation of the supervision strategy can be found in the supplementary materials.Discussion. We choose to utilize the feature embedding for the cognition process rather than directly generating a caption from LR for several compelling reasons. Firstly, although guided by language embedding, our cognitive embedding retains fine-grained image features, proving advantageous in generating reference images with high semantic similarity. In the second row of Figure <ref> (left part), we show the BLIP2 captions generated from LR images. They fail to identify the precise taxon, color, and texture of the animals, leading to suboptimal generations compared to our cognitive adapter. Secondly, employing a pre-trained image caption model requires a substantial number of parameters, potentially reaching 7B <cit.>. In contrast, our cognitive adapter is significantly lighter, with only 3% parameters, resulting in favorable efficiency. Thirdly, pre-trained image caption models may produce inaccurate captions for LR images due to disparities in the input distribution. In contrast, our cognitive adapter is more robust for LR images, shown in the third row of Figure <ref> (left part).§.§ Reference Image Generation and Encoding We propagate the cognitive embedding to the pre-trained Stable Diffusion model for generating reference images without incurring additional parameters. The resulting reference images empower our SR model to explicitly leverage image priors. As depicted in Figure <ref>, our cognitive embedding excels in producing well-aligned reference images.We employ a pre-trained VQGAN <cit.> for encoding images into latent codes, as opposed to a trainable CNN like <cit.>, given the robust encoding capabilities exhibited by VQGAN. Subsequently, we follow the ControlNet <cit.> approach by utilizing the U-Net encoder to generate multi-scale control features. We represent the LR control and reference image control as {X_i}^4_i=1 and {R_i}^4_i=1, respectively. Notably, we observe that using a single control encoder for both LR and reference images is sufficient for achieving satisfactory results, enhancing the parameter efficiency of our model. The generated controls are then input into the All-in-Attention module, as elaborated in the following section. In fact, when automatically generating reference images, we only need to use Stable Diffusion to generate latent codes, and subsequently input them into the control encoder, thereby circumventing the process of decoding and encoding in Figure <ref>. §.§ All-in-Attention ModuleIn image super-resolution, preserving fidelity to LR inputs is important. Our experiments in Section <ref> demonstrate that the introduction of LR control {X_i}^4_i=1 through attention mechanisms leads to enhanced fidelity. Consequently, we advocate for the comprehensive integration of all conditional information into our model, achieved through the design of an All-in-Attention (AiA) module. Beyond accommodating LR inputs, this design facilitates reference patch-matching for the establishment of long-range connections <cit.>. The cognitive embedding is seamlessly incorporated via the cross-attention mechanism of Stable Diffusion. Illustrated in Figure <ref> (c), the AiA module enhances the original attention module in Stable Diffusion by introducing trainable reference attention and LR attention, while maintaining the frozen state of the self-attention and cross-attention components. This structural augmentation is applied across all attention modules within the middle and decoder of the denoising U-Net. We denote the query, key, and value features in the attention mechanism as Q, K, and V, respectively. Regarding LR attention, Q is derived from the denoising U-Net feature Z, while K and V originate from the LR control X_i. In reference attention, we opt to use the LR control as Q for better fidelity, with K and V coming from the reference control R_i. In the original cross-attention, we use cognitive embedding E as inputs for K and V. Notably, to counteract the potential blurring effect of the conventional attention mechanism in reference-based SR <cit.>, we introduce “one-hot attention” to enhance the LR image with the most relevant reference feature, and additional details are available in the supplementary materials. § EXPERIMENTS§.§ Implementation Details Our CoSeR is built based on Stable Diffusion 2.1-base[https://huggingface.co/stabilityai/stable-diffusion-2-1-base]. The model is trained with a batch size of 192 over 20000 steps on 8 V100 GPUs. We use Adam <cit.> optimizer with a learning rate of 5×10^-5. Following StableSR <cit.>, we train our model on 512×512 resolution and apply DDPM sampling <cit.> with 200 timesteps for inference. The training process involves two stages: Firstly, we train the cognitive encoder using the defined loss function in Eq. <ref>. The cognitive encoder employs 50 learnable queries, a choice substantiated in the supplementary materials. Then, we freeze the cognitive encoder and train the SR model. Following <cit.>, we initialize ControlNet with Stable Diffusion weights. To maximize the utilization of the pre-trained model, the reference attention and LR attention modules are initialized using self-attention weights. In the inference phase, cognitive information is enhanced via classifier-free guidance <cit.>, utilizing a scaling factor of 3. To optimize the trade-off between realism and fidelity, we adopt the pre-trained feature wrapping module in <cit.>, which is integrated with the VQGAN decoder. §.§ Experimental SettingsTraining and testing datasets. We aim to develop an image super-resolution model empowered with cognitive capabilities adaptable to diverse real-world scenarios. To this end, we utilize the extensive ImageNet dataset <cit.> for training, renowned for its wide array of scenarios and objects. We acquire over 900K HR images with 512×512 resolution and employ Real-ESRGAN <cit.> degradation to generate corresponding LR images. We employ BLIP2 <cit.> to generate three descriptive captions for each HR image, filtering out captions with CLIP scores <cit.> below 0.28.To comprehensively assess our model's performance across diverse scenarios, we curate a non-overlapped ImageNet test set consisting of 2000 LR-HR pairs using the Real-ESRGAN pipeline. We choose two images from each category, ensuring the test set's diversity and balance. In addition to our constructed test set, we conduct evaluations on established real-world benchmarks such as RealSR <cit.> and DRealSR <cit.>. In this section, LR images are acquired at the same resolution used during training, specifically 128×128. For datasets such as RealSR and DRealSR, we initially resize LR images, adjusting the shorter sides to 128, followed by center cropping. Compared methods. We compare CoSeR with some state-of-the-art real-world SR methods, including RealSR <cit.>, Real-ESRGAN+ <cit.>, BSRGAN <cit.>, DASR <cit.>, FeMaSR <cit.>, latent diffusion models (LDM) <cit.>, StableSR <cit.>. To ensure fair comparisons, we retrain all these models using our ImageNet training set except RealSR and BSRGAN, which share the network structure with Real-ESRGAN+ but employ different degradation pipelines.Evaluation metrics. To better align with human perception, we employ six perceptual metrics: FID <cit.>, DISTS <cit.>, LPIPS <cit.>, CLIP-Score <cit.>, MANIQA <cit.> and MUSIQ <cit.>. FID, DISTS, and LPIPS measure perceptual distance, while CLIP-Score estimates semantic accuracy by evaluating scores between HR images and SR results. Given our focus on real-world scenarios where ground-truth HR data might be unavailable, we include non-reference image quality assessments, MANIQA and MUSIQ. Notably, pixel-level image quality assessments like PSNR and SSIM are presented in the supplementary materials solely for reference. Prior research <cit.> has indicated their weak correlation with human perception regarding image quality in real-world contexts. §.§ Comparison with State of the ArtsQuantitative comparison. We perform an extensive quantitative comparison on both the ImageNet Test2000 dataset and real-world benchmarks (RealSR and DRealSR), as presented in Table <ref>. As mentioned previously, we retrain the comparison models using the ImageNet training dataset to ensure fair comparisons. Additionally, the results of officially released models are provided in the supplementary materials. Our method consistently demonstrates superior performance across nearly all datasets and metrics, highlighting its robustness and superiority. Notably, our FID scores surpass the second-best performance by 13.8%, 3.8%, and 4.7% on the ImageNet Test2000, RealSR, and DRealSR, respectively. While FeMaSR exhibits better performance in MUSIQ on the ImageNet Test2000, as depicted in Figure <ref>, it introduces numerous unrealistic artifacts that might not be reflected by the non-reference metric MUSIQ. Qualitative comparison. We provide visual comparisons in Figure <ref>. Enriched by a comprehensive understanding of scene information, CoSeR excels in enhancing high-quality texture details. As demonstrated in the first and second rows, our results exhibit significantly clearer and more realistic fur and facial features in the animals. Similarly, in the third and fourth rows, our method adeptly reconstructs realistic textures such as the anemone tentacles and succulent leaves—achievements unmatched by other methods. Particularly, our model's cognitive capabilities enable the recovery of semantic details almost lost in low-resolution inputs. Notably, in the first row, only our model successfully restores the dhole's eyes, while in the fifth row, only our method can reconstruct the sand within the hourglass. These visual cases distinctly showcase our model's capacity to comprehend scenes and produce high-quality images. User Study.To further substantiate the effectiveness of CoSeR in real-world scenarios, a user study is conducted on 20 real-world LRs collected from the Internet or captured by mobile phones. 23 subjects are asked to select the visually superior result from the four HRs generated by Real-ESRGAN+, FeMaSR, StableSR, and CoSeR. A total of 20 × 23 votes are collected, with approximately 80% of participants concurring that our method exhibited the best visual effect. This underscores the superiority and robustness of CoSeR in real-world scenarios. Detailed voting results are available in the supplementary materials. §.§ Ablation StudyWe dissect the individual contributions of different components in our framework. Given the diverse focuses of different components, we employ the most appropriate evaluation metrics to measure their respective utilities.Cognitive information.Prior research, such as <cit.>, has attempted to align class tokens between CLIP image and text encoders using MLP. In our terminology, we denote the cognitive encoder with class token alignment MLP and our cognitive encoder as the “class token encoder” and “multi-token encoder”, respectively. As shown in Table <ref>, integrating cognitive information significantly enhances FID and CLIP-Score metrics, signifying a more accurate generation of semantics and textures. Our investigations reveal that the class token encoder may introduce semantic and texture biases, as evident in the quality of the generated reference images in Figure <ref> (penultimate column). To quantitatively evaluate cognitive bias, we introduce a new metric termed “Gen-score", calculated as the CLIP-Score between the generated reference image and the ground-truth image. Both the metrics and the visuals distinctly highlight our superior cognitive ability. This advantage extends to the final SR results, notably visible in the precise introduction of the hair and pulp texture in Figure <ref>.Reference guidance. The explicitly introduced reference image significantly contributes to enhancing the texture details in the SR results (Figure <ref>). To better correlate with human perception, our evaluation primarily focuses on assessing restoration quality using FID and two non-reference image quality assessments. As demonstrated in the fifth column of Table. <ref>, the inclusion of the generated reference image notably elevates the overall visual quality of the SR results without compromising their fidelity. Additionally, when compared to the utilization of real-world reference images from ImageNet, our generated image achieves comparable or even better results. All-in-Attention (AiA) module. Excessive generation sometimes results in a compromise in fidelity. To address this, we introduced the All-in-Attention (AiA) module designed to incorporate multiple conditions, aiming to enhance consistency with the input image. To evaluate fidelity, we utilize ground-truth-involved FID, DISTS, and LPIPS metrics. Compared to spatial feature transform (SFT) <cit.> integrated in StableSR <cit.>, our AiA module achieves a 5.7% lower FID score, along with superior DISTS and LPIPS results. This manifests the effectiveness of our AiA module in enhancing result fidelity.§ CONCLUSIONIn this paper, we present a pioneering approach to endow super-resolution (SR) with cognitive abilities. Our model excels in producing high-definition reference images that aid the SR process. Furthermore, we introduce an All-in-Attention module to enhance result fidelity. Extensive experiments substantiate the effectiveness of our approach in real-world applications.Supplementary Material Sec. <ref> provides an extensive elucidation of our method, including details of the cognitive encoder supervision method, the one-hot reference attention mechanism, and an in-depth analysis of the network architectures governing the denoising U-Net and ControlNet. In Sec. <ref>, we present comprehensive quantitative comparisons between the proposed method and established models, including the recently introduced DiffBIR <cit.>. Additionally, our investigation delves into the impact of introducing multiple generated reference images. We provide the results of a user study in the form of voting results and assessments of image quality at the pixel level. Sec <ref> showcases more visualization examples, extensively demonstrating the effectiveness of our method. Finally, we talk about the future work in <ref>. § DETAILED ILLUSTRATION OF OUR METHOD §.§ Cognitive Encoder SupervisionAs aforementioned in the main paper, we use T_e (T_e ≤ T_l) tokens, preceding the class token L[t_cls] (inclusive), extracted from the CLIP language embedding L∈ℝ^B × T_l × C_l for supervision. B, T_l, and C_l denote batch size, token number, and channel number, respectively. If there are insufficient supervision tokens, we use the class token for end-filling. The loss function for training the cognitive encoder is expressed as:ℒ_C E=E-L^'_2^2,whereL^'=Padding(L[: t_cls], L[t_cls]), ift_cls<T_e ;L[(t_cls-T_e): t_cls], ift_cls⩾ T_e . We observe that employing L directly as supervision for the cognitive embedding E (setting T_e = T_l) hinders the acquisition of cognitive information, as depicted in Figure <ref>. In this scenario, the generated reference images might prove irrelevant to low-resolution (LR) images. This limitation stems from the variability in caption length, which leads to Q-Former's learnable queries inadequately capturing semantic information at corresponding positions. To mitigate this issue, we propose using the last T_e tokens for supervision for two reasons. Firstly, the last T_e tokens in the CLIP text embedding inherently encapsulate an overarching representation of all preceding words <cit.>, facilitated by the causal attention mechanism. This mitigates the requirement for a strict one-to-one correspondence between query ordering and semantic representation, thus enabling more effective learning by the queries. Secondly, within the supervision target L^', the last query consistently aligns with the class token, thereby preserving the full representational capacity of the class token. Compared to the direct utilization of L[t_cls] or single class token, our approach of employing the last T_e tokens for supervision presents a more accurate understanding of LR images, which is supported by both Figure. 5 in the main paper and Figure <ref>.We investigate the impact of the number of learnable queries, denoted as T_e, in our cognitive encoder on the generation of high-quality reference images. This analysis involve the examination of 200 randomly selected low-resolution test images by varying the query number from 30 to 77. It is noted that the setting of T_e=77 in L^' differs from using L for supervision. This distinction arises from the fact that the final tokens of L^' are expanded with class tokens when the caption is not sufficiently lengthy. The results presented in Table <ref> demonstrate that our cognitive encoder achieves optimal performance when T_e=50 (where “Gen-score” is defined in the main paper). Hence, we establish T_e=50 as the default value. Notably, the quality of the generated images begins to decline for T_e>50, as also evidenced in Figure <ref>. This decline might be attributed to increased learning complexity associated with a higher number of tokens. §.§ One-Hot Reference Attention The reference image contains high-definition textures that maintain consistent semantics with the corresponding LR image. However, not all features from the reference image are useful for LR recovery. The conventional attention mechanism calculates the weighted sum of all queries in value features, potentially leading to a blurring effect <cit.>. To address this issue, we introduce one-hot attention in the reference module to enhance the LR image with the most pertinent reference feature.The one-hot attention mechanism is depicted in Figure <ref>, where Q, K, and V denote the query, key, and value features, respectively. We represent the LR control and reference image control at the i-th scale as X_i and R_i. Q∈ℝ^B × T_x × C and K, V∈ℝ^B × T_r × C are derived from X_i, R_i. The similarity S∈ℝ^B × T_x × T_r between Q and K is computed with normalized inner product:S=⟨Q, K⟩.We derive the one-hot map H∈ℝ^B × T_x × T_r along the T_r dimension of S and record the maximum values as T∈ℝ^B × T_x. The final output of the one-hot attention is then expressed as:Z_out =ZeroConv[ (HV) ⊙T],where ⊙ denotes element-wise multiplication. It is noteworthy that we opt not to use softmax and, instead, employ the correlation matrix T to diminish less similar features while amplifying those that are potentially valuable. Additionally, to prevent the newly introduced attention components from influencing the well-established representation of Stable Diffusion <cit.> during early training, we integrate zero convolutions <cit.> at the end. §.§ Network StructureDenoising U-Net. The denoising U-Net in the proposed Cognitive Super-Resolution (CoSeR) network is depicted in Figure <ref>. In our architecture, we adopt the All-in-Attention (AiA) module, replacing all original attention modules present in both the middle and decoder components of the Stable Diffusion denoising U-Net. It is crucial to highlight that cognitive embedding is utilized across all attention modules in the denoising U-Net, extending beyond solely the AiA modules.ControlNet. We utilize ControlNet <cit.> to generate multi-scale control features for both LR and reference images. As illustrated in Figure <ref>, we mirror the weights and structure of the denoising U-Net in the ControlNet. Following <cit.>, zero convolutions are incorporated at the beginning and end of the ControlNet module. Subsequently, the resulting control features are directed to the All-in-Attention (AiA) modules situated within the middle and decoder components of the denoising U-Net, excluding U-Net Decoder Block D, which lacks attention modules. Importantly, cognitive embedding is also employed in the ControlNet module.§ ADDITIONAL EXPERIMENTS§.§ Quantitative Comparisons to Official ModelsFor fair comparisons, we conduct a re-training of real-world super-resolution (SR) models using the ImageNet <cit.> dataset in the main paper. Remarkably, our CoSeR model achieves the highest performance. To provide a comprehensive analysis, we compare CoSeR against the officially released models: RealSR <cit.>, Real-ESRGAN+ <cit.>, SwinIR-GAN <cit.>, BSRGAN <cit.>, FeMaSR <cit.>, DiffBIR <cit.>, and StableSR <cit.>. It is noted that we exclude the comparison with DiffBIR on the ImageNet Test2000 dataset due to potential data overlap with its official training set. Additionally, all diffusion-based models, including LDM, DiffBIR, StableSR, and CoSeR, employ 200 sampling steps. As outlined in Table <ref>, across various evaluation metrics such as FID <cit.>, DISTS <cit.>, LPIPS <cit.>, CLIP-Score <cit.>, and MUSIQ <cit.>, our method consistently excels, positioning CoSeR as the superior and more robust approach.§.§ Comparisons to Re-trained DiffBIRAs an extension to the quantitative comparisons provided above, we further conduct a comparative analysis with DiffBIR <cit.>, specifically re-trained using our ImageNet training set. The results displayed in Table <ref> underscore the superiority of CoSeR across three benchmarks, demonstrating better performance across nearly all metrics. §.§ Voting Results of User StudyAs detailed in the main paper, we invite 23 subjects to discern the visually superior result among the four SR candidates generated by Real-ESRGAN+, FeMaSR, StableSR, and CoSeR. This user study encompasses 20 real-world low-resolution images sourced from the Internet or captured via mobile phones, resulting in a total of 20 × 23 votes gathered. The depicted voting results in Figure <ref> unequivocally illustrate the superior performance of our CoSeR.§.§ Number of Reference Images We investigate the influence of using multiple generated reference images on the quality of SR results. Employing the same LR input, we randomly sample noise maps to create several reference images utilizing identical cognitive embeddings. The findings presented in Table <ref> demonstrate that introducing a greater number of reference images yields improved performance. However, it's noteworthy that the improvement plateaus when using 2 or 3 reference images, suggesting that these images already contain sufficient high-definition textures to guide the process. As a result, we recommend utilizing 2 reference images as an optimal balance between quality enhancement and computational efficiency. §.§ Pixel-level Image Quality AssessmentWhile acknowledging that PSNR and SSIM metrics exhibit a weak correlation with human perception, particularly for large-scale super-resolution tasks, we present the corresponding results in Table <ref> for reference purposes. Our CoSeR achieves favorable results. The All-in-Attention (AiA) module contributes to competitive or superior pixel-level fidelity compared to other diffusion-based models like DiffBIR and StableSR, as shown in Table <ref> and Table <ref>. § QUALITATIVE COMPARISONSWe provide visual comparisons on ImageNet Test2000 dataset (Figures <ref>, <ref>, <ref>), real-world or unknown degradation type images (Figures <ref>, <ref>), RealSR and DRealSR datasets (Figure <ref>). Our CoSeR obtains outstanding visual performance.§ FUTURE WORKThe cognitive-based recovery process extends beyond super-resolution (SR) tasks; it is beneficial for various visual tasks such as deblurring, denoising, and inpainting. Our future work includes expanding its application to more diverse image restoration tasks. Additionally, prevalent SR algorithms based on diffusion models often require a large number of sampling steps for higher visual quality. Addressing the challenge of accelerating the sampling process without compromising SR performance is also a focal point of our ongoing research.ieeenat_fullname | http://arxiv.org/abs/2311.16512v4 | {
"authors": [
"Haoze Sun",
"Wenbo Li",
"Jianzhuang Liu",
"Haoyu Chen",
"Renjing Pei",
"Xueyi Zou",
"Youliang Yan",
"Yujiu Yang"
],
"categories": [
"cs.CV",
"cs.AI"
],
"primary_category": "cs.CV",
"published": "20231127163329",
"title": "CoSeR: Bridging Image and Language for Cognitive Super-Resolution"
} |
http://arxiv.org/abs/2311.16305v1 | {
"authors": [
"A. Giusti",
"I. Colombaro",
"R. Garra",
"R. Garrappa",
"A. Mentrelli"
],
"categories": [
"math-ph",
"math.MP",
"26A33, 76A10"
],
"primary_category": "math-ph",
"published": "20231127203643",
"title": "On variable-order fractional linear viscoelasticity"
} |
|
Toward a real-time TCP SYN Flood DDoS mitigation using Adaptive Neuro-Fuzzy classifier and SDN Assistance in Fog Computing Radjaa Bensaid[1], Nabila Labraoui[2], Ado Adamou Abba Ari[3], Leandros Maglaras[4], Hafida Saidi[1], Ahmed Mahmoud Abdu Lwahhab[5],and Sihem Benfriha[1] [1]STIC Laboratory,Abou Bekr Belkaid Tlemcen University , Algeria[2]LRI Laboratory, Abou Bekr Belkaid Tlemcen University, Algeria [3]DAVID Laboratory, Paris-Saclay University,France[4]School of Computing, Edinburgh Napier University, Edinburgh, UK — Corresponding author [5]Department of Electronics and Communications, Dakahlia Mansoura University, EgyptVersion ofJanuary 14, 2024 =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== The growth of the Internet of Things (IoT) has recently impacted our daily lives in many ways. As a result, a massive volume of data is generated and needs to be processed in a short period of time. Therefore, the combination of computing models such as cloud computing is necessary. The main disadvantage of the cloud platform is its high latency due to the centralized mainframe. Fortunately, a distributed paradigm known as fog computing has emerged to overcome this problem, offering cloud services with low latency and high-access bandwidth to support many IoT application scenarios. However, Attacks against fog servers can take many forms, such as Distributed Denial of Service (DDoS) attacks that severely affect the reliability and availability of fog services. To address these challenges, we propose mitigation of Fog computing-based SYN Flood DDoS attacks using an Adaptive Neuro-Fuzzy Inference System (ANFIS) and Software Defined Networking (SDN) Assistance (FASA). The simulation results show that FASA system outperforms other algorithms in terms of accuracy, precision, recall, and F1-score. This shows how crucial our system is for detecting and mitigating TCP SYN floods DDoS attacks. Keyword Fog computing, Service availability, TCP SYN flood DDoS attack, ANFIS, SDN.§ INTRODUCTIONThe growing number of connected objects, from millions to billions in various fields, is leading to an explosion in the amount of data. These huge volumes of data cause a lack of latency and make real-time analysis complex and difficult. To solve these issues, the deployment of computing models such as cloud and fog computing is crucial <cit.>. Technologies of cloud computing enable an extremely powerful computer resource over the network. Nevertheless, due to several concerns about data privacy and security, attaching more diverse types of objects immediately to the cloud is extremely difficult, as well as network latency difficulties <cit.>. Therefore, the need to introduce a new paradigm is necessary to solve these problems. Recently, fog computing has emerged to expand the cloud computing paradigm from the core to the network's periphery. The purpose of fog computing is to bring computer capabilities closer to IoT devices, offering real-time processing with low latency <cit.>. Aside from this, fog computing also provides mobility support, location awareness, and decentralized infrastructure. Fog computing has a local data storage infrastructure, which makes it more secure than cloud computing. Despite this, IoT devices are limited in terms of storage capacity and battery life. Thus, they can be easily hacked, destroyed, or stolen and fog computing may become unavailable and unable to handle normal user requests. Therefore, it is necessary to apply security mechanisms to identify and block unauthorized requests on network systems. However, fog computing is still susceptible to various security and privacy gaps. It can be a point of vulnerability, and it is easily overwhelmed by a massive number of malicious requests, primarily intended for Distributed Denial of Service (DDoS) attacks <cit.>.DDoS attacks can be divided into two types depending on the protocol level addressed. The first one is known as "network-level flooding," when TCP, UDP, ICMP, and DNS packets are used to overload intended clients' network capabilities and resources. Whereas, the second protocol level is referred to as "application-level DDoS flooding" which is typically done on an HTTP webpage when attacks are launched to deplete server resources such as sockets, CPU, ports, memory, databases, and input/output bandwidth <cit.>. Regarding the rapid growth and the harm caused by DDoS attacks, several kinds of research have been conducted on these attacks, and various approaches have been presented in the literature to prevent these attacks using fog computing <cit.><cit.>. Most of them proposed a defensive fog computing that operates as a filtering layer among the user layer and the cloud computing layer. However, these defensive approaches miss DDoS detection mechanisms, and detailed computation is not discussed. Also, they do not identify any infrastructure to protect fog computing which is particularly susceptible to DDoS attacks and may disrupt network services.For this purpose, proposing Software Defined Networking ( SDN) technology-based solutions could bring an innovative framework to deal efficiently with this insidious attack. SDN enables us to define logic control and instructs the forwarding plane to act appropriately by isolating the control and data planes. This programmability provides more control over network traffic, which wasn’t conceivable before the development of SDN <cit.>. Considerable research has been done within SDN-based IoT-fog networks using task scheduling techniques like Threshold Random Walk with Credit-Based connection (TRW-CB) and rate limiting. These techniques are deployed for detecting anomalies and mitigating DDoS attacks, which effectively reduces average response times <cit.>. However, it can result in excessive CPU and RAM consumption. This scheduling-based approach only focuses on secure scheduling periods, leaving the network vulnerable during idle times when no tasks are scheduled <cit.>. Moreover, the previous approach incorporates both fuzzy logic and multi-objective particle swarm optimization. Nevertheless, as the number of variables and rules increases, designing and fine-tuning the fuzzy logic system can become highly complex. Recently, machine and deep learning algorithms have gained attention for their effectiveness in detecting DDoS attacks by analyzing data patterns <cit.><cit.>. Hence, merging fuzzy systems with neural networks combines the benefits of neural learning with the interpretability offered by fuzzy systems. An Adaptive Neuro-Fuzzy Inference System (ANFIS), empowers fuzzy systems to acquire knowledge from data. This synergy enhances fuzzy systems through neural networks. ANFIS's hybrid approach facilitates adaptability to diverse attack patterns and network conditions.Moreover, employing various approaches, such as the Neuro-Fuzzy classifier on the KDD CUP99 dataset <cit.>, has been a common practice. This dataset includes numerous recognized attack variations and has traditionally been utilized in intrusion detection. Nevertheless, the KDD CUP99 dataset is now regarded as outdated, as it presents several unresolved issues that fail to meet the updated criteria for DDoS identification <cit.>. In our work, we focus exclusively on the TCP SYN flood attack. Since it is the most effective DDoS attack in fog computing. In order to exhaust the system’s resources or overwhelm the target server, the attackers typically infect several devices that behave as bots and synchronize suspicious traffic or requests, leading to an incomplete three-way handshake procedure <cit.>. consequently, Legitimate users cannot reach the desired fog server.In this paper, we suggest a novel Fog computing-based SYN Flood DDoS attack detection and mitigation using an Adaptive Neuro-Fuzzy Inference System (ANFIS) and SDN Assistance (FASA). Compared to previous works, FASA utilizes the ANFIS model for network traffic classification, incorporates SDN support to enable real-time mitigation, and relies on the newly released CIC-DDoS2019 dataset. The proposed model demonstrates exceptional performance across multiple metrics, including accuracy, precision, recall, and F1- score. Additionally, it exhibits a notably low rate of false positives. In brief, our significant contributions are outlined as follows:* We propose a novel model FASA to detect and mitigate a SYN Flood DDoS attack in fog computing using SDN assistance.* We implement the ANFIS model to self-train the fog servers and make the difference between normal and malicious packets.* The ANFIS model is implemented at the SDN controller and deployed at the fog server, using a dataset captured from the SDN environment. Its main objective is to allow benign packets access while rejecting malicious ones to release a secure and dependable SDN controller that ensures fog service availability.* The proposed evaluation method uses both the newly released dataset CIC-DDoS2019 and the SDN dataset. It is experimentally analyzed from the data availability and the algorithm operating efficiency and it can improve the performance The paper is structured in the following manner:Section 2 examines and discusses previous works to tackle the issue of DDoS attacks. Section 3 contains background knowledge. In Section 4, we formally define the proposed model, and in Section 5, we introduce our proposed framework. followed by the evaluation outcomes and discussion in section 6. Finally, the conclusion is given in Section 7. § RELATED WORK0pt In this section, we have provided an extensive overview of DDoS attacks. especially TCP SYN flood attack detection. Besides, these works have been grouped into three sections. The initial represents statistical methods. The second and third ones highlight a few works based on Machine/Deep Learning (ML/DL) algorithms.§.§ Statistical methodsStatistical methods constantly evaluate user/network activities to identify abnormalities<cit.>. Hence, due to their capacity to analyze the behavior of data packets, they are commonly utilized in DDoS attack detection systems. If data flow does not match with some test statistics and measures, it is thought to be illegal. Ahalawat et al. <cit.> suggested a detection method for DDoS attacks based on Renyi entropy. and a mitigation solution for SDN based on the packet drop approach, using several probability distributions. They can examine network traffic fluctuations. However, the necessity to set an optimal detection threshold is a typical limitation of various entropy-based approaches. Hoque et al. <cit.> presented a novel correlation measure using standard deviation and mean to detect DDoS attacks, The traffic is then classified as attack traffic or normal by comparing the collected traffic to the profiled traffic. However, the suggested metric's use in identifying low-rate attempts is unclear. a DDoS detection-based multivariate correlation analysis was discussed by Jin et al. <cit.> in their work, and they provided a covariance analysis method for recognizing SYN flood attacks. The experimental results demonstrate that this technology accurately and efficiently detects DDoS attack traffic in networks of varied levels of intensity. However, using the correlation approaches consumes a lot of processing in real-time to detect DDoS attacks. As a result, they are unable to operate in real-time. A novel framework was suggested by Bhushan et al. <cit.> using fog for detecting DDoS attacks even before they reach the cloud by using an efficient resource provisioning algorithm to service cloud requests through intermediate fog servers. Furthermore, an entropy DDoS detection method and mitigation system designed for Cloud Computing environment using SDN has been proposed by Tsai et al. <cit.>.An entropy-based DDoS detection approach was implemented to protect the virtual machines and controller from malicious attacks. As a result, the detection rate is significantly affected by the threshold value. Javanmardi et al. <cit.> proposed FUPE, a security-driven task scheduling algorithm for SDN-based IoT-Fog networks. FUPE uses fuzzy logic and multi-objective particle swarm optimization to assign tasks to fog nodes balancing security and efficiency objectives. However, managing and interpreting extensive rule sets pose challenges in maintaining and validating the fuzzy logic framework. Nonetheless, Multi-objective optimization with PSO requires parameter tuning and could be computationally intensive, particularly in large-scale environments.Furthermore, it incorporates techniques like Threshold Random Walk with Credit-Based connection (TRWCB) and rate limiting to detect malicious nodes and utilizes the SDN controller for mitigation by blocking attackers, ultimately leading to a reduction in average response times. Nevertheless, this approach may lead to elevated CPU and RAM usage. FUPE exclusively identifies and addresses anomalies during the scheduling phase, leaving the network susceptible to threats in the absence of scheduling requests <cit.>.§.§ Machine Learning methods (ML)Machine learning-based methods are used to identify DDoS attacks such as Decision Trees, Deep Learning, Support Vector Machine (SVM), K Means Clustering, and so on <cit.>. These methods might be unsupervised machine learning (label for training is not required) or supervised machine learning (require a label for training normal/malicious) algorithms. Moreover, the dataset, which contains numerous network and traffic features, is used to train and learn automatically how to recognize suspicious behavior patterns.Rajagopal et al. in <cit.> provided a meta-classification strategy that integrates many classifiers for both binary and multiclass classification. Decision jungle serves as the meta learner, combining numerous learners to obtain the best prediction performance. This proposed method has a precision of 99%. Tuan et al. <cit.> idea were about proposing a novel TCP-SYN flood attack mitigation by tracing back IP sources of attack in SDN networks using K-Nearest Neighbors (KNN) machine learning based on SDN. The testbed's experimental findings reveal that 97% of attack flows are identified and blocked. Priyadarshini et al. <cit.> demonstrated a new source-based DDoS mitigation approach, in order to prevent these attacks in both fog and cloud computing environments. It deploys the defender module that presents at the SDN controller which is based on machine learning (SVM, KNN, and Naive Bayes algorithm).However, the classical ML techniques can't handle the amount of data.However, the "real world" application of classical ML algorithms is limited due to network attack issues. In addition, these approaches need a lot of time to learn, thus they can't be used in real-time. §.§ Deep Learning methods (DL)In recent studies, there has been a particular emphasis on evaluating the performance of DL models in DDoS detection. This is primarily due to their ability to effectively analyze large volumes of data and identify complex patterns within it. Assis et al. <cit.> proposed a near real-time solution by applying Convolutional Neural Networks (CNN) to cover and defend victims' servers from DDoS attacks at the end source, The detection model reached a precision rate above 95.4%. Novaes et al. <cit.> employed the Generative Adversarial Network (GAN) architecture to mitigate the damage of DDoS attacks on SDNs. For experiment assessments, the accuracy obtained using the published datasets namely, CIC-DDoS2019 and the emulation was about 94.38%. The authors compared the GAN framework's findings against those of other deep learning algorithms, such as LSTM, CNN, and MLP. The authors of <cit.> employed a variety of Machine Learning (ML) algorithms to identify low-rate DDoS attacks. They found that the Multi-Layer Perceptron (MLP) performs the best among the assessed algorithms, with a detection rate of up to 95%. Other ML models, such as Random Tree, Random Forest, and Support Vector Machines, have shown useful in detecting and mitigating DDoS attacks. Deep learning has already been used to identify SYN flood attacks by Brun et al. <cit.>, in which a Random Neural Network was built to classify and differentiate whether the packet is normal traffic or SYN attacks. Evmorfos et al. <cit.> use a Random Neural Network for identifying typical SYN attacks on Internet-connected equipment including edge devices and gateways, and fog servers, with limited processing capability. Devi et al. <cit.> presented an intrusion detection system (IDS) approach based on the SUGENO-based fuzzy inference system ANFIS to identify security concerns on relay nodes in a 5G wireless network. The model was tested and trained using the KDD Cup 99 datasets. Boroujerdi et al. <cit.> developed a novel ensemble of Sugeno-type adaptive neuro-fuzzy classifiers to identify DDoS attacks based on the Marliboost boosting approach. It was tested on the NSL-KDD dataset. However, the data in the NSL-KDD or KDD cup 99 datasets were considered unsuitable for the new requirement of a DDoS attack since it comprises packet traces rather than flows, implying that the DDoS detection methods may become computationally difficult as the network expands in size. As a consequence, there have been various studies published in recent years on how to identify DDoS attacks, particularly TCP SYN flood using Machine and Deep Learning. However, few of them have addressed using ANFIS to detect such attacks in fog computing based on SDN technology.In order to address the limitations of the previous studies, in this paper, we propose an ANFIS classifier, implemented in the SDN controller to classify network traffic, and deployed at the fog server using the recently published dataset CICDDoS2019. The inclusion of various types of DDoS attacks in this dataset bridges the gaps found in previous databases. Additionally, we employ the ANFIS using the SDN dataset for real-time mitigation. § BACKGROUND KNOWLEDGE This section highlights the required context for our proposed model. First, we give an overview of DDoS attacks and the different methods used for detection. Then, we introduce the Adaptive Network-based Fuzzy Inference System (ANFIS) detection algorithm. Finally, we present the Software Defined Networking (SDN) technology. §.§ DDoS attacks and fog computingThe DDoS attack is a highly progressed type of DoS attack. It differs from other attacks in that it may be deployed in a "distributed" manner. A DDoS attack's primary purpose is to inflict harm on a target for personal reasons, financial gain, or popularity. <cit.>. It is an attack based on availability and aims at making the victim system inaccessible to authorized users <cit.>. Moreover, it is done by a combination of a huge amount of hacked and dispersed devices known as bots or zombie devices that have been infected with malicious malware or compromised by an attacker <cit.>. Hence, an attacker centrally controls and coordinates these machines to launch an attack on the target machine <cit.>.§.§.§ Types of DDoS attacks on fog computingSeveral DDoS attack types are used to bring down the functionality or availability of network services on fog computing [34], as illustrated in Figure 1.a. Application-Bug Level DDoS These sorts of attacks, like HTTP POST and HTTP PRAGMA, deplete the application system, causing it to fail or temporarily close down.b.Infrastructural Level DDoS The key purpose of these threats is to exhaust network bandwidth, buffers, CPU, and storage, preventing legitimate users from using them. Thus, the only requirement for this attack is the victim’s IP address. It is categorized into two types: direct and reflector attacks.*Direct Attack This attack is carried out with the assistance of compromised devices or bots. It sends malicious queries to the target using bots in order to deplete its resources, bandwidth, and services, rendering them inaccessible to authenticated users. This attack can be further subdivided into network-layer and application-layer DDoS attacks.* Network Layer DDoS: This attack type employs various network and transport layer protocols, including TCP SYN, UDP, ICMP, among others.* Application Layer DDoS: In this attack, HTTP flood traffic is adopted widely to exhaust the victim. this kind of vulnerability is difficult to detect, raising security issues. * Reflector Attack In this attack, the IP address is spoofed and requests are delivered to a vast range of reflector hosts. Following the receipt of the requests, a response is provided in order to flood the target.§.§.§ DDoS defense mechanisms In this section, we discuss various defense mechanisms used for DDoS attack detection and mitigation for the security of fog computing <cit.>. Nearer-to-edge devices, fog computing offers computing capabilities in the form of fog nodes. which creates a heavy load on network management. To address this issue, SDN technology can be implemented to guarantee the safety of fog computing in the following aspects: * Monitoring the network: If the network is monitored permanently and continuously, any suspicious data attempting to disrupt services may be recognized and rejected. As this is performed at fog nodes, legitimate users will have no difficulty accessing the services. * Priority-based and isolated traffic: It implies the process of prioritizing legal and illegitimate network traffic, hence requiring the use of shared knowledge resources such as CPU or I/O. As a result, SDN can reject damaging traffic by separating it through VLAN ID/tag.* Access control mechanism for resources in the network: To prevent DDoS attacks, an effective access control system should be implemented.* Shared network: The shared network is the crucial condition since anyone can access it, holding security at risk.Additionally, two distinct assessments are used to identify DDoS defense mechanisms. The first classification divides the DDoS defense systems into the following four groups based on the activity carried out: * Intrusion Prevention,* Intrusion Detection,* Intrusion Tolerance and Mitigation, * Intrusion Response. Further, the second categorization mainly classify DDoS defenses into the following three groups based on where they are deployed: * Victim Network,* Intermediate Network,* Source Network.§.§.§ TCP SYN Flood attackThe SYN flooding attack is a specific type of DoS attack that targets hosts that operate TCP server processes, it became well-known in 1996 <cit.>. The concept of the three-way handshake that initiates a TCP connection serves as the mainstay of this attack. <cit.>. It exploits a TCP protocol process characteristic and may be used to restrict server functions from responding to normal user demand to establish new TCP connections. As a result, each service that connects and waits on a TCP socket is highly susceptible to TCP SYN flood attacks. Although several techniques to counteract SYN flood attacks may be found in modern operating systems and equipment.§.§ The Adaptive Network-based Fuzzy Inference System (ANFIS) detection algorithmANFIS is a network model that combines a Sugeno-type fuzzy system with neural learning capability <cit.>. Neuro-fuzzy systems are ways to learn fuzzy systems from data that use neural network-derived learning algorithms. Therefore, due to their learning capabilities, neural networks are an ideal choice for combining with fuzzy systems<cit.> are used to automate or simplify the process of developing a fuzzy system for specific usage. The initial neuro-fuzzy techniques were primarily explored within the field of neuro-fuzzy control, although the approach is now broader because it is used in a number of domains, including control, data analysis, decision support, and so on. <cit.>. ANFIS is based on two parameters (premise and consequent parameters) which are used to connect the fuzzy rules. Moreover, ANFIS is made up of five layers in total, as illustrated in Figure 2. The square nodes have parameters, whereas circular nodes do not.The considered fuzzy inference system contains two inputs, y considered as non-linear parameters, and one output f. Also, each input variable is described by two linguistic terms: A_1 and A_2 for the variable x, and B_1 and B_2for the variable y, respectively. the following two IF-THEN rules construct the Sugeno fuzzy model <cit.>:* Rule 1: If xisA_1 ∧ y is B_1,thenf_1=p_1x+q_1 y+r_1 * Rule 2: If xisA_2 ∧ y is B_2,thenf_2=p_2x+q_2 y+r_2 Where p_i, q_i, and r_i i=1,2,correspond to the linear parameters of the conclusion part to be adjusted during the training. * Layer 1: O1, represents the membership functionµof a fuzzy set A_i (or B_i).O_1,i=μ_A_i (x)i=1,2O_1,i= μ_B_i-2 (y)i=3,4 Because of their smoothness and simple syntax, Gaussian membership functions are preferred approaches for defining fuzzy sets. The advantage of these curves is that they are smooth and nonzero at all locations. The Gaussian membership function is used in this study, it’s frequently used to reduce the uncertainty of real-world measurement and is represented by the equation (1) where c and σrepresent the mean and standard deviation respectively. Here c represents the center, and σrepresents the width. a,c are called premise parameters (non-linear).μ_A (x)=ae^ -(x-c)^2/(2σ)^2 * Layer 2: the fuzzification layer w determines the degree of membership function satisfaction of each input; the output is the product ∏ of all the entering signals, it is determined using the following equation (2):O_(2,i)=w_i=μ_A (x)·μ_B (x) i=1,2The output of every node shows the firing strength of a rule. The node function in this layer can be any other fuzzy AND T-norm operator, such as min.* Layer 3: the normalization layer, in which the i-th node determines the proportion of the firing strength of the i-th rule to thetotal firing strength of all rules, as demonstrated in the equation (3): O_3,i=w̅_̅i̅=w_i/w_1+w_2The outputs of this layer are referred to as normalized firing strengths.* Layer 4: In the defuzzification layer, parameters are named consequent parameters. Each node has a function where w̅_̅i̅ is a normalized firing strength from layer 3 and p_i,q_i,r_i are the set of linear node parameters and are defined as consequent parameters of this node and f_i denotes the output of the rule, as shown in equation (4):O_4,i=w̅_̅i̅f_i= w̅_̅i̅(p_ix+q_iy+r_i)* Layer 5: in this layer, the single node adds up all of the incoming signals to compute the overall output, as demonstrated in equation(5): O_5,i= overalloutput= ∑_i= 0^nw̅_if_i =∑w_if_i/∑w_i An adaptive network's nodes are related to parameters that may affect the final output. To adapt the parameters in an adaptive network, ANFIS typically uses a hybrid learning algorithm, that associates gradient descent and the least square approach <cit.>. The hybrid algorithm comprises a forward pass and a backward pass. To optimize the consequent parameters, the least squares method (forward pass) is used; node outputs are passed forward until Layer 4, and the least squares determine the consequent parameters. In our work, For optimizing the premise parameters, the ADAM method <cit.> is employed during the backward pass. Error signals are propagated backward, and the premise parameters are updated using ADAM. This hybrid learning approach offers faster convergence by reducing the search space dimensions compared to the original backpropagation method. <cit.>. It has been demonstrated that this hybrid algorithm is extremely effective in training ANFIS systems <cit.>. The ANFIS training technique begins by defining the number of fuzzy sets, the number of sets of each input variable, as well as the shape of their membership function. The primary goal of ANFIS is to improve input-output data sets and a learning mechanism to enhance the parameters of a comparable fuzzy logic system. The difference between the intended and actual outputs is minimized as much as feasible during parameter optimization.§.§ Software Defined Networking (SDN)SDN is a network paradigm that enables users to directly manage network resources by orchestrating, controlling, and using software applications <cit.>. Moreover, the control and data planes are divided by SDN. making it most commonly used to improve network efficiency. When the data plane forwards packets from one location to another, the control plane determines whether or not the packets should propagate through the network.Thus, SDN is formed by the combination of a controller and switches, these switches follow the forwarding rules that are defined by the controller, which can dynamically manage network flows and implement different configurations based on network circumstances. The three fundamental layers of SDN architecture are (i) The application layer which contains the general network functions including intrusion detection systems, firewalls, and security applications. (ii) The control layer which is the centralized software controller that serves as the SDN's brain.The network policies and traffic flows are managed by this controller. (iii) The infrastructure layer contains a variety of networking equipment, including switches and routers <cit.>, as shown in Figure 3. The communication between the controllers and switches is outlined throughout the OpenFlow protocol <cit.>, which serves as the communication standard for SDN networks. It is referred to as SDN networks' southbound communication. The controller can deal with open flow switches (OF-switch) with existing flow tables by using an open flow protocol. When a packet's flow entry is found in the OF-switch's table, the packet is forwarded in the usual manner; otherwise, The controller receives it for additional evaluation. Thus, SDN controllers with OpenFlow-enabled switches are widely used for SDN networking. They are especially suitable for light traffic communication and control.§.§ System ModelThis study aims to develop a distributed FASA framework to mitigate SYN flood attacks in the network environment by recognizing and avoiding attacks close to the attacking sources. To enable quicker and more accurate attack detection using the ANFIS model, fog computing is suitable for deploying SDN for mitigating SYN flood attacks by assigning compute power near the operation process and spreading the burden in the system through a FASA mitigation scheme. In this section, we first outline SYN flood DDoS attacks in fog computing. Then, we discuss the FASA network architecture.§.§ SYN Flood DDoS attack As shown in Figure 4, when a standard TCP three-way handshake has initiated, the End User (EU) transmits the SYN packet to the fog server. Then, the fog server responds with an SYN/ACK packet. Next, the EU should send an ACK packet to the fog server. So, when all of these processes are completed, the connection is established <cit.>. However, the main drawback of TCP connections is the inability to maintain half-open connections. The fog server is in a half-open connection state because it is standing in line for the EU's reply to acknowledge the three-way handshake. Furthermore, IoT devices have limited computation, storage capacity, and short battery life, and they can easily compromise, damaged, or kidnapped. Therefore, due to the aforementioned limitations, an attacker may simply hack IoT devices and utilize them as botnets to generate and send excessive SYN request packets with a fake source IP address to fog servers. As a result, the ACK packet will never reach the fog server which is in the open port state waiting for the ACK packet. Moreover, the SYN/ACK packets are transmitted to the faked host, and the three-way handshake procedure will never be completed. Also, the connection registration is kept in the connection delay buffer till time expires, preventing legitimate users from accessing the services <cit.>.§.§FASA Network Architecture To effectively deal with the SYN flood DDoS attack concerns in the network systems, attack prevention must be built in fog computing based on SDN. Indeed, in this paper, we propose a novel distributed fog defensive system for SYN flood DDoS attacks using ANFIS and SDN Assistance (FASA). The FASA architecture has three layers, the cloud layer, the SDN-based fog (SDFN) Layer, and the things layer, as shown in Figure 5.a. Cloud layer: cloud computing, as a computing model, define a method of managing a pool of configurable computing resources, offers elastic, on-demand services, and has access to the system anywhere and at any time. Therefore, users can use resources according to their demands. The salient features provided by cloud technology are immediate flexibility and measurable services. <cit.> SDN and cloud technology can be combined to automate and cloud applications provisioning must be completely integrated with the network.Hence, in the FASA system, cloud computing refers to the application plane which consists of many useful applications that communicate with the controller to abstract a logically centralized controller to make coordinated decisions. b. SDN-based Fog Network layer (SDFN): This layer combines the fog computing and SDN paradigm to identify and behave against DDoS attacks. With recent advances in SDN, it opens up new opportunities for providing intelligence within networks. The benefits of SDN, including logically centralized control, software-based traffic analysis, an entire network view, and flexible forwarding rule updates, help to improve and facilitate machine learning applications <cit.>. Therefore, the SDFN layer provides new trends of DDoS attacks in fog computing environments using SDN. This layer is formed of two sub-layers, SDFN-server and SDFN-node. * SDFN-server: This sub-layer refers to the control plane deployed at fog servers where an intelligent ANFIS classifier is integrated into the control network to classify traffic flows decision and consequently policies are managed to depend on its decisions.Moreover, the SDFN server communicates with the cloud layer (application) via the northbound interface and with the SDFN-node layer via the southbound interface.* SDFN-node: This sub-layer refers to the data plane of physical equipment in the network such as switches and routers. It forwards the network traffic to their destinations using the OpenFlow protocol. c. Things layer: This layer serves the purpose of sensing, collecting, and uploading data from wireless sensors and end-users to fog computing. The transmitted packet can be classified as either benign or malicious.The following assumptions are made in order to better explain the SYN flood DDoS attack identification and defense framework:* The SDN-based Fog Network server (SDFN-server) is susceptible to being compromised, * DDoS attacks are TCP SYN flood attacks against SDFN-servers.* The SDN controller and the switch are not compromised. * IoT devices can be hacked. § PROPOSED FASA FRAMEWORKSYN flood DDoS attacks can instantly bring down a network and it is difficult to detect them since they can be carried out in a very short time. Therefore, detecting and mitigating such attacks is critical. A detection approach for such threats is needed in fog computing to filter and block the malicious requests before the attack produces a negative impact on the fog services. Consequently, our FASA framework can be used to identify and immediately mitigate SYN flood attacks in real-time into fog computing, as illustrated in Figure 6.§.§ The detection processFASA is based on the ANFIS model and SDN network to guarantee service availability in the fog network. To attain recognition and detection purposes, a fog layer is established among both the cloud layer and the Things layer. Thus, the recognition techniques deployed on the fog layer can handle and process malicious traffic. Also, the SDN controller deployed on the fog layer controls packets arriving from every system node to enhance security and network management.Additionally, the SDFN-server is prior trained with ANFIS algorithms and tested using two different datasets, CIC-DDoS2019 and SDN dataset. After a successful data pre-processing step, the most important features will be extracted. Then, these features will be divided into training data and testing data to self-train the SDFN-server to identify the SYN flood attack. Once that is done, the ANFIS model will be able to determine whether an incoming packet is legitimate or not. then, the controller's decision based on that.as presented in the flowchart of Figure 7.§.§ The mitigation processSDN simplifies the implementation of complex mitigation models. When an OpenFlow switch gets a packet, it compares it to the matching rule in its flow table and decides whether to act by forwarding packets to the destination according to the found rule or seek assistance from the controller if the rule is not matched. In addition, the OpenFlow switch initiates this request through the SB-API by the OpenFlow agent in the switch, as demonstrated in the flowchart of Figure 7.Although, the attack may be identified by determining a threshold value, which is the maximum value of serving capacity defined by the availability of computational resources. If the number of service requests exceeds the limit, a malicious packet is sent out <cit.>. Otherwise, if it’s less than the Threshold capacity, it will pass through the ANFIS classifier for prediction in the fog server. Therefore, the real-time mitigation phase is started when the ANFIS model detects an SYN flood packet. This phase aims to perform defensive functions to limit the damage caused by an exploit. So, the packet passes through the OpenFlow protocol which takes action by executing the updated rule in the flow table whether it is a legitimate user to allow access. Otherwise, the controller looks for the most often occurring source address Mac with different source address IPs and uses it to determine the infected port number.By correlating the identified Mac address with the corresponding port on the switch, the controller determines the port through which the attack traffic is entering the network. To prevent further damage, the controller instructs the OF switches to drop all packets obtained from the host associated with the identified Mac address. Then, The controller also directs the switch to block traffic on the specific port associated with the infected host, effectively preventing any communication through that port. Next, the controller updates the flow table of the switch to modify the rules related to receiving or forwarding packets to the identified port. This ensures that any packets destined for that port are dropped or redirected to mitigate the attack. As a result, TCP SYN flooding attacks may be identified and prevented by instantly blocking the switch port that is connected directly to the attacker's host.§ EXPERIMENTS AND RESULTS §.§ Experimental setupIn this part, we will go over the various tools that were used to build up the experimental setup for detecting SYN flood attacks in the simulated SDN and fog computing environments, using Wireshark <cit.> to capture and analyze network traffic in real-time. The entire experiment is carried out on Windows 10 OS with an Intel i3 processor and 8GB of RAM. To emulate the network behavior, the SDN Mininet network emulator <cit.> was used, with the Ryu controller <cit.>. Ryu is an open-source platform, that provides transparency and flexibility, enabling customization and extension of functionalities. Its Python-based architecture promotes accessibility and ease of development, facilitating rapid implementation of SDN applications. Additionally, support for multiple protocols, including OpenFlow, ensures seamless communication with diverse network devices. Ryu's compatibility with various networking technologies and hardware makes it suitable for heterogeneous infrastructures, rendering it particularly well-suited for this research <cit.>.For training and testing our ANFIS model, the Python programming language has been used with libraries for deep learning Keras <cit.>, and TensorFlow <cit.>. Additionally, to prevent overfitting, the stratified K-Fold cross-validation <cit.> was also employed in the ANFIS algorithm. Due to the fact that the Stratified k-fold cross-validation guarantees that each fold has a class distribution that is identical to the original dataset, resulting in a more accurate and reliable model assessment. , along with Binary Crossentropy, a classic loss function used in binary classification. Also, we set the default Keras learning rate to 0.001. Furthermore, Adam optimizer<cit.> was selected, as an adaptive algorithm for optimizing learning rates in neural network models. Moreover, by using two different scenarios in this study, we examine the performance and efficiency of the FASA system. * Scenario 1: Evaluate the performance of the FASA system by employing the SDN environment.* Scenario 2: Evaluate the performance of the FASA system by using the public dataset CIC-DDoS 2019 <cit.>. §.§ Experimental analysisIn our next subsection, we discuss each test scenario and provide the studies' results.a. Scenario 1 In our experiment, the Mininet network emulator [51] was used to design virtual network topologies consisting of controllers, hosts, links, and switches. Therefore, to run Mininet and Ryu controllers <cit.>, we have used two virtual machines based on the Linux operating system.Ryu controller is based on a Python program and supports several network management protocols such as OpenFlow switches. Moreover, the FlowManager is a Ryu controller program that allows the user to manipulate the flow tables in an OpenFlow network manually. We have used the Ryu controller for SDN networking environments due to their ease of deployment, expansion, and simple architecture. Hence, Ryu controllers with OpenFlow-enabled switches are widely used for SDN networking. They are especially suitable for light traffic communication and control. In addition, the Ryu controller provides a routing link to OpenFlow switches to ensure that the topology can perform data analysis.Thus, to emulate our network structure, linear topology is used on Mininet, in which 8 switches are connected to the Ryu controller, and each switch is connected to 8 hosts. In total, 64 hosts are linked to the OpenFlow virtual switches, as shown in Figure 8.The IP address of the Ryu controller is 192.168.162.133. Likewise, each host is assigned an IP address. For example, the IP address of Host1="10.0.0.1/24" and the mac address starting from 00:00:00:00:00:01 converted from hexadecimal to an integer. In general, the following processes are involved in scenario 1: the data generation and collection process, the detection process, and the mitigation process. These processes are deployed using Mininet VM and Ryu controller VM based on Python programming language.* Data generation and collection process: SDN dataset is created using both the Mininet emulator and Ryu controller. The normal traffic is collected using the “iperf” command, and we consider one host (Host1) as a Simple HTTP Server listening on port 80. Additionally, we collect the SYN flood traffic data using the Hping3 tool with random IP addresses. Hping3 is an open-source TCP/IP protocol used as a packet generation tool, that is written in the TCL language. Hping3 enables programmers to create scripts for TCP/IP packet handling and analysis in a restricted period. MAC addresses are an important criterion to mitigate SYN flood attacks because layer 2 switches forward incoming traffic based on Mac addresses. Also, it helps to identify the infected source port. Moreover, the layer 4 switch depends on the source and the destination ports that are essential in the flow table with the following features: datapath id, source IP, source Mac, destination IP, destination Mac, IP protocol, ICMP code, ICMP type, packet counts, and flags. Table <ref> provides detailed information about the collected dataset. * The detection process:After the pre-processing of the collected data presented in <ref>, we will split the dataset as follows: The training set contains 80% of the dataset, whereas the testing set contains 20% of the dataset. Then, we use the ANFIS algorithm with cross-validation to avoid overfitting and train the collected dataset to achieve an accuracy of 100%. Next, once the packet-in is received in various forms of regular traffic and attack traffic, the Ryu-controller collects the features and assigns their values to the predicted dataset. For the prediction process, the detection module (ANFIS algorithm) examines each flow entry.* The mitigation process: DDoS attacks are difficult to mitigate because of IP spoofing; therefore, blocking the suspected attacker's IP is ineffective in mitigating; To achieve our objective of obtaining a list of edge switches directly connected to each host, we will store the Mac address, port number, and switch ID for each host in a Python dictionary. This dictionary will serve as a data structure to retrieve the required parameters for creating mitigation rules.Every flow entry passes the detection process to check if it is a normal packet or a malicious packet. Then, it will be sent to the Ryu controller to make a decision based on the result of the prediction. Therefore, if the flow entry’s predicted value is 1, it indicates an SYN flood attack in which the attacker transmits both the real source Mac address and a random false source IP, repeating the higher Mac address with different IPs in each flow entry indicates that the hacker is the host of this Mac address. In this case, we use the assigned Mac address to get the port number and switch id from the dictionary. The Ryu controller then responds by enforcing the rule that rejects all packets originating from that attacker, This rule is then sent to the affected switch, instructing it to block the specific port that is directly connected to the attacker's host. By implementing this rule, the switch effectively prevents any communication from the attacker's host through that particular port, helping to mitigate the impact of the attack. Both the hard timeout and the idle timeout are essential parameters that must be adjusted for themitigation process:* Idle time means the flow rule will be deleted if no match occurs with incoming packets within the idle timeout value.* Hard timeout means the flow rule will be deleted automatically after hard timeout expires since the rule is created.In the case of an attack, the Ryu controller blocks the packet on the OF switch with idle time = 0 sec and hard time = 300 sec with a high priority, we used priority 1000 for our model. As a result, the switch continues to block the source port for 300 seconds without notifying the controller.Otherwise, if the detection result is 0, this signifies normal traffic. the idle time will be 200 seconds, and each flow entry has a fixed priority of 10. If no matching happens throughout this time period, the flow rule will be removed after 200 seconds. The hard time will be 400 seconds, after which all flow entries will be deleted. During this experiment, the real-time flow traffic captured by Wireshark is represented in Figure 9 display the packets per second versus the time plot.Additionally, Table<ref> presents the parameters employed in this experiment.Initially, normal traffic is sent out at time 0 seconds. Next, a Syn flood attack is initiated, at time 60 the packet rate reaches a threshold value close to 700 packets per second. The ANFIS detection module identifies the attack whenOnce the attack is detected, the mitigation module takes over. The controller utilizes appropriate flow rules to mitigate the attack by dropping packets, blocking the source ports involved in the attack, and informing the switches to update the flow table accordingly. The attack is successfully mitigated in less than 5 seconds, resulting in a significant drop in the packet rate. the graph shows the continuednormal traffic flow without any breakdown until the end of the experiment 140 seconds. This period is crucial as it representsthe controller's capability to receive packets effectively.Figure 10 can demonstrate that, During the attack, we observed a decrease in bandwidth consumption , reaching as low as 90 Mbits/sec. Fortunately, it quickly recovered to its pre-attack state and remained relatively stable at around 100 Mbits/sec. This demonstrates the effectiveness of our model in mitigating the impact of the attack and restoring normal network performance.b. Scenario 2In the second scenario, we evaluate the proposed model's capability to identify the TCP SYN flood DDoS attacks using the CIC-DDoS dataset produced by Sharafaldin et al. (2019) <cit.> for detecting DDoS attacks and classifying attack types. This dataset is in a CSV format, It includes both benign and current popular DDoS attacks launched in 2019. It is collected on the first and second days and reflects the actual real-world data (PCAPs). It also provides the findings of a network traffic analysis performed with CICFlowMeter-V3 that includes labeled traffic flows. This dataset originally had 88 features.In this scenario, we use the SYN flood dataset presented in Table <ref>. TCP SYN flood is a type of exploitation category-based DDoS attack that exploits vulnerabilities in TCP connection protocols. It is composed of data from two days, each with a different attack category and a wide range of imbalance class distribution. * Resampling data: Both training and testing datasets have a minority class "BENIGN" with a little sample, resulting in an imbalanced classification, which has an impact on a model's capacity to learn and decide, furthermore, can cause overfitting in our model. To accomplish this, we build a new dataset in which we take all samples labeled "BENIGN" from the training and testing datasets, forming 10% of the total dataset and 90% of samples labeled "SYN" as shown in Table <ref>. * Data Pre-processing: In this section, we will go over the techniques used to analyze our dataset, which contains 88 features. The data will be cleansed and prepared to use in our suggested ANFIS algorithms once certain undesirable attributes have been removed and adjusted. As a result, the implementation of a data preprocessing step, as shown in Figure 11, provides more reliable training and, thus, a more accurate model. First, we removed features that have a unique value in the entire dataset that do not affect the training phase ('Bwd PSH Flags',' Fwd URG Flags', ' Bwd URG Flags', 'FIN Flag Count','Fwd Avg Bytes/Bulk',Fwd Avg Packets/Bulk', ' Fwd Avg Bulk Rate', 'Bwd Avg Bytes/Bulk', ' PSH Flag Count',' ECE Flag Count',' Bwd Avg Packets/Bulk', 'Bwd Avg Bulk Rate'). Some values of 'Init Win bytes forward' and 'Init Win bytes backward' of flow data from the Syn csv file were set to -1. Nevertheless, it is inconceivable to initiate a byte window of size -1, this problem was caused by a software issue with CICFlowmeter and should be set to 0 or removed to not disrupt the training phase. The need to cope with missing data threw off the model's training. The lines containing 'infinity' and 'NaN' were removed from 'Flow Bytes/s' and 'Flow Packets/s'. We removed categorical features that can change from one network to another ('Source Port', 'Destination Port', 'Source IP', 'Destination IP', 'Flow ID', 'SimillarHTTP', 'Unnamed: 0', 'Timestamp').To properly distinguish important features, delete columns with a correlation higher than 0.8 (' Total Backward Packets', ' Total Length of Bwd Packets', ' Fwd Packet Length Std', ' Bwd Packet Length Min', ' Bwd Packet Length Mean', ' Bwd Packet Length Std', ' Flow IAT Mean', ' Flow IAT Std', ' Flow IAT Max', 'Fwd IAT Total', ' Fwd IAT Mean', ' Fwd IAT Std', ' Fwd IAT Max', ' Fwd IAT Min', ' Bwd IAT Std', ' Bwd IAT Max', ' Fwd Header Length', ' Bwd Header Length', ' Max Packet Length', ' Packet Length Mean', ' Packet Length Std', ' Packet Length Variance', ' RST Flag Count', ' Average Packet Size', ' Avg Fwd Segment Size', ' Avg Bwd Segment Size',' Fwd Header Length.1', 'Subflow Fwd Packets', ' Subflow Fwd Bytes', ' Subflow Bwd Packets', ' Subflow Bwd Bytes', ' Active Max', ' Active Min', 'Idle Mean', ' Idle Max', ' Idle Min'). In order to detect and classify DDoS attacks, the dataset is split into two classes. The label "BENIGN" is coded as "0" and the label "Syn" is coded as "1" in the dataset created to detect a SYN flood DDoS attack on the network traffic.Feature selection is used to discover key data features and decrease the amount of data required for detection. we use the XGBoost technique that provides an importance score to each feature based on its influence in making crucial decisions using boosted decision trees <cit.>. Then, depending on the rated feature, we removed features that were of negligible importance ' Protocol', ' Flow Duration',' Total Fwd Packets',' Fwd Packet Length Max', 'Bwd Packet Length Max', ' Flow IAT Mean', ' Flow IAT Min','Bwd IAT Total', ' Bwd IAT Mean', ' Bwd IAT Min', 'Fwd PSH Flags','Fwd Packets/s', ' Bwd Packets/s', ' Min Packet Length',' SYN Flag Count',' CWE Flag Count', ' Down/Up Ratio',' Init Win bytes backward', ' act data pkt fwd', 'Active Mean', ' Active Std', ' Idle Std', and choose nine ideal feature subsets, as presented in Table <ref>.* We normalize the data by scaling all features in the range of 0–1 value.As previously described, The dataset was divided into two parts training data and testing data.by using cross-validation to avoid overfitting in training steps. * Finally, we put the ANFIS model to the test for making predictions on unseen data. The next section discusses the performances and results. §.§ Performance metricsUsing the right performance metrics is the key to correctly evaluating models. Therefore, in this section, we explore the following performance metrics to evaluate the FASA framework: * True Negatives (TN): Normal flow data is appropriately identified as such.* True Positives (TP): malicious flow data is accurately identified as such.* False Positives (FP): Normal flow data is mistakenly labeled as malicious traffic.* False Negatives (FN): malicious flow data is classed as normal flow data when it isn't. In addition, we provide the confusion matrix to describe our model's classification performance. It can resume the correct and false predictions obtained using our proposed approach, as demonstrated in Figure 12. Accurately distinguishing the Benign class within our model is of utmost importance, as elevated false positive rates can result in unnecessary complexity and unwarranted alerts. Our main objective is to minimize the false rate. Hence, our framework achieves a rate of 0% of false positives in both CIC-DDoS2019 and SDN datasets. Otherwise, it obtains 0.058% false negatives in the CIC-DDoS2019 dataset and 0% in the SDN dataset. the Receiver Operating Characteristic (ROC) curve is performed. It represents the relation between both the True and False parameters. The area under the ROC Curve (AUC) measures whether it is possible to distinguish false positives from true positives. As illustrated in Figure 13, our model has an AUC of 99.96% using the CIC-DDoS2019 dataset and 100% using the SDN dataset and there are two extremely similar values, indicating that our suggested model separates correctly positive from negative classes. By employing established techniques like k-fold cross-validation, the model ensures generalizability and guards against overfitting. Furthermore, the meticulous selection and optimization of impactful traffic features enhance the model's proficiency in distinguishing between normal and attack behaviors. Additionally, the fusion of fuzzy logic and neural learning components proves effective in capturing complex traffic patterns. Lastly, training on diverse attack data distributions further enhances the model's robustness. We have also used a variety of measures to assess our suggested model, including accuracy, precision, recall, and F-score, to conduct an in-depth comparative assessment with some other relevant methods. These metrics, which are often employed in SYN flood DDoS detection systems, are described in the following:* Accuracy refers to the ratio of the number of samples correctly classified to the overall number of samples observed. It is computed as follows:accuracy=tp+tn/tp+tn+fp+fn* The precision is the ratio of correctly predicted positive samples, it is calculated as follows:precision=t p/t p+f p * The false positive rate is determined by calculating the proportion of negative samples that were incorrectly classified as positive using the following formula: fp-rate= f p/f p+t n* The recall also called the true positive rate, is calculated with the ratio of correctly discovered positive samples, It is determined using the equation:recall=tp-rate= t p/t p+f n * Good precision may be more relevant in certain situations, whereas high recall might be more critical in others. Across many cases, though, we aim to enhance both values. The f1-score is the combination of these values, and it is commonly stated as the harmonic mean: f1-score= 2× precision× recall/(precision+recall)§.§ Evaluation resultsTo validate our system, we have compared the FASA framework to the FUPE <cit.> method and other DDoS attack detection systems that were employed on SDN and used the CIC-DDoS 2019 dataset, as illustrated in Table <ref>.The first method is FUPE <cit.> that puts forward a fuzzy-based multi-objective particle swarm Optimization approach, a security-aware task scheduler in IoT–fog networks. The second method is the Convolutional Neural Network (CNN) <cit.>, a low-cost based supervised classifier designed to identify suspicious events in a data center. the next approach is based on Generative Adversarial Network GAN <cit.> for identifying DDoS threats in SDN environments. Finally, the Multi-layer Perceptron (MLP) <cit.> is adopted to identify and prevent Low Rate-DDoS attacks in SDN settings.Figure 14 depicts a comprehensive analysis of the metric findings of the comparative approaches.As shown in Figure 14, we can observe that the performance of our model using the SDN dataset outperforms all previous techniques, with 100% accuracy, precision, recall, and F1-score in each case and it closely resembles the outcome obtained using the CIC-DDoS2019 dataset. In addition, the accuracy of every learning algorithm is assessed. As a result, The ANFIS achieved the highest accuracy rating of 99.95% across all classifiers, then the FUPE approach with 98.2% followed by the CNN algorithm with 94.83%. Furthermore, MLP and GAN classifiers attained an accuracy of 95.4% and 95.01%, respectively. It also illustrates the precision of each algorithm in identifying legal and malicious traffic. Thus, the ANFIS reached 100% precision, and FUPE with a precision of 96.08%, and the MLP attained a precision value of 95.46%. Next, the GAN, and CNN algorithms with a precision of 94.08%, and 93.3%, respectively. Furthermore, Figure 14 displays the recall values of all methods used in the performance evaluation. The ANFIS algorithm had a 99.94% recall value followed by FUPE with 98%, whereas GAN had a 97.89% recall rating. In comparison to the other algorithms tested, the CNN achieved the lowest recall value of 92.4% while the MLP had a recall of 94.51%. It also illustrates the F1-score of the classifying methods with 99.95%, the ANFIS received the highest F1-Score. On the other hand, GAN, MLP, and CNN received F1-scores of 95.94%, 94.98%, and 92.8%, respectively. While the FUPE's F1-Score is not mentioned. In conclusion, our FASA framework outperforms the other evaluated approaches. The promising test results indicate that it is an effective approach for identifying SYN flood DDoS attacks. § CONCLUSION AND FUTURE WORK In this work, FASA, a Fog computing-based SYN Flood DDoS attacks mitigation using an Adaptive Neuro-Fuzzy Inference System (ANFIS) and Software Defined Networking (SDN) Assistance was proposed. The choice of the integration of SDN and fog environment with the ANFIS machine learning algorithm brings intelligence to the SDN controller. Also, it makes our framework suitable, efficient, and more secure against SYN flood attacks. We trained and evaluated our framework on the newly released CIC-DDoS2019 dataset that contains the most recent and extensive SYN flood DDoS attacks. The findings of the performance assessment indicate that the suggested model has a high detection accuracy and a low rate of false positive and negative rates, which is a remarkable result and it also offers the highest evaluation metrics regards to precision, recall, and F-score when compared to well-known machine learning algorithms. Our future work is to focus on how well our proposed model performs on various datasets. In the current experiments, we have employed a binary classification approach that is implemented on SDN to distinguish between legitimate and malicious input traffic in fog computing. Thus, in future work, we will try to investigate the utility of the suggested approach for other multi-class classification systems. Additionally, to create a diversified dataset that truly represents actual internet traffic, we will emulate the SDN network under various scenarios and with various attack traffic. In addition, we will also consider expanding our work to include the SoDIP6-based ISP/Telecom network, including edge computing network scenarios. This will allow us to evaluate the performance of our proposed model in a more complex and realistic environment. We will also investigate the use of our model for other network security applications, such as intrusion detection and prevention. § CONFLICTS OF INTERESTAll authors declare no conflict of interest.§ ACKNOWLEDGMENTSThe authors conducted this research while affiliated with Abou Bekr Belkaid Tlemcen University, Paris-Saclay University, Edinburgh Napier University, andDakahlia Mansoura University. *IEEEtran | http://arxiv.org/abs/2311.15633v1 | {
"authors": [
"Radjaa Bensaid",
"Nabila Labraoui",
"Ado Adamou Abba Ari",
"Leandros Maglaras",
"Hafida Saidi",
"Ahmed Mahmoud Abdu Lwahhab",
"Sihem Benfriha"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20231127085400",
"title": "Toward a real-time TCP SYN Flood DDoS mitigation using Adaptive Neuro-Fuzzy classifier and SDN Assistance in Fog Computing"
} |
[ Benno Liebchen January 14, 2024 ====================*These authors contributed equally to this workfootnote The spread of fake news using out-of-context images has become widespread and is a challenging task in this era of information overload. Since annotating huge amounts of such data requires significant time of domain experts, it is imperative to develop methods which can work in limited annotated data scenarios. In this work, we explore whether out-of-domain data can help to improve out-of-context misinformation detection (termed here as multi-modal fake news detection) of a desired domain, eg. politics, healthcare, etc.Towards this goal, we propose a novel framework termed DPOD (Domain-specific Prompt-tuning using Out-of-Domain data). First, to compute generalizable features, we modify the Vision-Language Model, CLIP to extract features that helps to align the representations of the images and corresponding text captions of both the in-domain and out-of-domain data in a label-aware manner.Further, we propose a domain-specific prompt learning technique which leverages the training samples of all the available domains based on the the extent they can be useful to the desired domain.Extensive experiments on a large-scale benchmark dataset, namely NewsClippings demonstrate that the proposed framework achieves state of-the-art performance, significantly surpassing the existing approaches for this challenging task. § INTRODUCTIONIn this digital world, social media has become the main source of information for an increasing fraction of the population. Thus, it is very important that the news reaching the masses is authentic, since fake news can have serious consequences, like manipulating public opinion, stirring up conflicts, and even impacting financial markets. With the internet connecting billions of people worldwide, fake news can spread at an alarming speed, thus it is equally important to identify it quickly and efficiently. One increasingly common form of fake news is the out-of-context use of images, where a real image is paired with a false caption before dissemination. It is very difficult to detect such fake news, since the image is real, i.e., not manipulated or generated, and it is only recently that researchers have started to address this challenging, but very important problem <cit.><cit.>.Annotating such data requires considerable domain expertise, thus, acquiring annotations for large amounts of data is very challenging.Especially, if one is interested in a particular domain, for e.g., a doctor may be more interested in healthcare news, whereas a government official may be more interested in political news, getting sufficient annotated data for every domain becomes even more difficult.In this work, we propose a novel framework termed DPOD (Domain-specific Prompt-tuning usingOut-of-Domain data)for out-of-context misinformation detection of a desired domain, which we term here as MFND or Multi-modal Fake News Detection.As an example, for MFND on a particular domain of interest like politics, we explore whether out-of-domain data from healthcare, entertainment, etc. can be effectively utilized to improve its performance. Towards this goal, the DPOD framework first utilizes label-aware alignment of the available image-text pairs of all the domains to learn generalizable features, irrespective of the domain of interest. In our work, we harness the capabilities of a large Vision Language Model (CLIP) <cit.> to comprehend interwoven text and image features.Next, a semantic domain vector is computed for each of the training data domains, which incorporates the similarity of a particular domain with all the other available training domains.Further, this is utilized to learn a domain-specific prompt, along with a set of generic learnable prompts, which is finally used for predicting whether a given image-caption pair is real or fake.Extensive experiments on a large-scale benchmark dataset, namely NewsClippings <cit.> demonstrate the effectiveness of the proposed framework over state-of-the-art approaches.The contributions of this work can be summarized as follows:1) We propose a novel framework, termed DPOD, which effectively utilises out-of-domain data for improving the performance of in-domain data. Though this problem is being recently addressed in the unimodal (text domain) <cit.>, to the best of our knowledge, this is the first work which addresses this problem for the multi-modal scenario. 2) We utilize a label-aware loss to learn a generalizable model, and also propose a novel framework to learn domain-specific prompts using out-of-domain data. 3) This helps in quick model deployment for the desired domain with limited domain-specific annotated data, thus being more effective in restricting the spread of fake news.4) Extensive experiments show that the framework outperforms the existing approaches significantly, thus setting the new state-of-the-art.5) Using only 25% of annotated data, the proposed DPOD framework outperforms the previous state-of-the-art <cit.> which uses 100% of the annotated data. Now, we briefly describe the related work, followed by the proposed framework and the experimental evaluation.§ RELATED WORKHere, we briefly describe the related work in MFND, self-supervised learning and prompt tuning. Multi-modal Fake News Detection: Recently due to the rapid spread of fake news using out-of-context real images and texts/captions, researchers have started to address this socially relevant important problem.<cit.> proposed the SAFE framework for the MFND task, which first utilises Text-CNN and VGG <cit.> model for extracting features from the texts and their associated images and then calculate the similarity which is used to predict the fake news. Similarly, <cit.> use BERT <cit.> as text encoder to extract textual features and use VGG-19as image encoder to extract visual features and then finally fuse them together for classification. Most of these existing methods fail to perform well on unseen events. <cit.> suggested an approach in which they trained an event discriminator network concurrently with multi-modal fake news detector. This network was designed to eliminate event-specific characteristics while retaining the common features shared among different events. <cit.> proposed a framework to jointly preserve domain-specific and cross-domain knowledge in news records to detect fake news from different domains by using combination of domain-specific and cross-domain features in the model. Furthermore, it presents an unsupervised method for choosing a subset of unlabeled, valuable news records to be labeled manually. In <cit.>, meta-learning and neural process methods are integrated to achieve high performance even on events with limited labeled data. The emergence of large scale vision-language models (VLMs) which learns information from both images and texts, has proved to be immensely useful for computer vision tasks. Models like CLIP <cit.> and ALIGN <cit.> have been trained in a contrastive manner on large scale datasets of around 400 million and 1B image-text pairs respectively, such that it learns rich representations between images and texts. <cit.> used CLIP <cit.> model as well as image encoder (ResNet <cit.>) and text encoder (BERT) to extract the multimodal features and used attention over different modalities to get final features for classification. Some works also use social media related information to detect the fake news. <cit.> proposed a framework which uses image, text and propagation graph for MFND. To leverage the internet information for out-of-context misinformation detection, <cit.> proposed a Consistency-Checking Network, which not only measures the similarity of the images and corresponding text, but also utilises the internet search results (both images and texts) to further improve the performance using a CLIP-based model. Self-Supervised Learning:Self-supervised learning leverages the inherent structure of data to generate its own supervisory signals, unlike traditional supervised learning, which relies on large labeled datasets. <cit.> have showed that the combination of data augmentations has a crucial impact on shaping useful predictive tasks. Introducing a trainable nonlinear transformation connecting the representation and contrastive loss significantly enhances the quality of acquired representations. <cit.> introduced supervised contrastive loss which leverages label information for self-supervised representation learning. It pulls together clusters of points belonging to the same class while pushing apart clusters of samples from different classes.In addition to several other applications, recently, self-supervision has been used successfully for the task of out-of-context misinformation detection using images and text.Shivangi et al <cit.> propose a method where given two captions, the goal is to detect whether an image is being used out-of-context.It is trained using self-supervised learning to align individual objects in an image with the corresponding textual claims. The state-of-the-art method by Michael et al. <cit.> propose a Self-Supervised Distilled Learner, which uses self-supervised learning to detect fake news, wherethey propose a Teacher network to guide a Student network in mimicking similar decision patterns. In contrast, though we also utilize self-supervision, our proposed DPOD framework is much simpler, yet more effective, and does not rely on teacher-student model. Vision-Language Models and Prompt Tuning:Since, adapting large vision language models like CLIP to downstream tasks may prove to be difficult, methods like prompt learning have come up as a popular research direction. The text instructions are usually given through the text encoder, which are called "Prompts". Prompt learning is an alternative to handcrafted prompts, where the prompt tokens are learned during the finetuning stage. Recently, many works have proposed prompt learning techniques such as CoOp <cit.>, CoCoOp <cit.>, Maple <cit.>, etc., which increases the generalizability of such VLMs. Some works specifically address the domain adaptation problem, for e.g., <cit.> proposes a domain-invariant prompt tuning, which can generalize to unseen domains. <cit.> uses prompt tuning in a unimodal context (in language models),whereas, <cit.> uses only visual prompt tuning.<cit.>, <cit.> and <cit.> learns prompts specifically for each domain. To the best of our knowledge, ours os the first work to explore prompt-tuning for the challenging MFND task. § PROBLEM DEFINITION AND MODEL OVERVIEW The goal of the proposed framework is to leverage out-of-domain training samples when testing on a certain target domain. Let the train dataset be denoted as 𝒟_train={D_1,D_2,...,D_n}. D_i denotes each domain and can be written as, D_i = {(I_k, T_k, y_k)}, k = 1,2,...,n_D_i, where n_D_i is the number of samples in domain D_i. I_k and T_k denote the k^th image and text pair, and y_k denotes the ground truth, i.e., fake or true news.The news samples can originate from various sources (or domains) such as politics, healthcare, sports, etc.We assume that we know the domain of the training examples, though we will see that the proposed framework can inherently handle domain label inconsistencies that is expected from crowd-sourced data annotation.We assume that an user is interested in a specific domain D ∈{D_1,D_2,...,D_n} during testing, though our model training happens only once irrespective of the desired target domain. During testing, we want to predict whether a given image-text pair from the domain of interest is fake or real. First, we briefly describe the base model used here. Base Architecture: Recently, vision language foundation models like CLIP <cit.>, Align <cit.> etc., have been immensely successful in relating images and the corresponding text and have also been used for the MFND task <cit.>, <cit.>, etc.CLIP is trained using natural language, which being expressive, can supervise a much broader set of visual concepts.Following the recent literature, we also use a modified CLIP architecture for our work.The broader structure of our proposed DPOD framework can be divided into three stages as follows (Fig. <ref>):Stage 1: Label-Aware Alignment of Multi-modal data: In this stage, we train all the layers of the original CLIP model in an end-to-end manner using the label-aware alignment loss on all the training data, irrespective of their domains, to obtain the aligned clip, termed here as A-CLIP. Stage 2: Creating Semantic Domain Vectors. Here, we freeze the already learned image and text encoders of the A-CLIP model. Using the domain-wise joint image-text embeddings from these frozen encoders, we create semantic domain vectors for each of the training domains.Each index of this vector denotes its similarity with the joint embedding of all the other domains.Stage 3: Domain-Specific Prompt tuning with Out-of-Domain data: With the above settings, we next propose a prompting strategy during training. Following previous works <cit.><cit.>, we append a prefix prompt to the text caption. However, a fixed prompt like "A photo of" <cit.>, may not be the best prefix for all the captions.Hence, as in <cit.>, we make the prefix prompt tokens learnable as {V_1, V_2, V_3}.Additionally, we pass the semantic domain vector though a linear layer ℱ_θ, and get a domain-specific prompt token, which is appended to the prefix prompt tokens. The complete text captions appended with the learnable prompts, and the corresponding images are passed though the frozen encoders to create the final joint embedding, which is subsequently classified into fake or real news.§ PROPOSED DPOD FRAMEWORKHere, we describe each of the three stages of the DPOD framework presented in Fig. <ref> in detail.§.§ Stage 1: Label-Aware AlignmentSince our goal is to predict the veracity of news (image-caption pair) in a particular domain (D, say politics), the ideal scenario is when vast amounts of annotated data for domain D are available.In real scenarios, getting large amounts of annotated data from each domain is challenging. Our hypothesis is that there might be overlapping concepts/objects in the data from different domains, like healthcare, entertainment, which can be utilized to mitigate the adverse effect of less amount oftraining data of the desired domain. Using self-supervision as a warm-up stage for model training has been used successfully in previous works like <cit.>, <cit.>. In contrast to these approaches, where both the training and test sets contain data from all (but same) domains, we explore whether self-supervision from out-of-domain data can benefit the performance of in-domain data for MFND task. Here, the original CLIP model is trained to learn relations between texts and images from all the domains, but in a label-aware manner.Specifically, we use a multi-modal contrastive loss, such that the backbone model learns to bring closer the augmentations of the image and the corresponding text when the news is real and can separate them when the news is fake. For label-aware alignment, initially, four augmentations, namely, color jitter, random crop, horizontal flipping and normalisation are generated per image. Then the CLIP embeddings of the original image, text and the image augmentations are generated.Suppose an image and the corresponding text be denoted as I_k and T_k respectively and the image augmentations be denoted by I_ak, a=1,,4.We denote the corresponding embeddings as {Î_k, T̂_k, Î_ak}.For the consistent (real) image-text pairs, we want to bring the embeddings of the image (Î_k), its augmentations (Î_ak, a =1, ,4) and the corresponding text (T̂_k) close to one another. Thus for the positive pair, we have two combinations, namely image-image pair and image-text pair. The corresponding label-aware alignment losses for the i^th image augmentation of k^th input sample are calculated as:ℒ^1, 𝓉𝓇𝓊ℯ_ℒ𝒜(I_ik,I_jk) = -log ( exp(sim(Î_ik, Î_jk) / τ)/𝒟_true) ℒ^2, 𝓉𝓇𝓊ℯ_ℒ𝒜(I_ik,T_k) = -log ( exp(sim(Î_ik, T̂_k) / τ)/𝒟_true)where Î_ik and Î_jk, j ≠ i denotes the i^th and j^th augmentations of the image I_k. τ is the temperature parameter that controls the smoothness of the probability distribution, N is the number of training image-text pairs and sim represents the cosine similarity. We use τ = 0.05 for all our experiments. The denominator 𝒟_true represents the combination of all the negative pairs and is given as 𝒟_true = ∑_t=1t ≠ k^N (∑_a=1^4exp(sim(Î_ik,Î_at)) .+ .exp(sim(Î_ik,T̂_t)) +exp(sim(T̂_k,T̂_t))) The total loss corresponding to the real or true news is obtained by adding the losses in eq. (<ref>):ℒ^𝓉𝓇𝓊ℯ_ℒ𝒜 = ℒ^1, 𝓉𝓇𝓊ℯ_ℒ𝒜 + ℒ^2, 𝓉𝓇𝓊ℯ_ℒ𝒜For the inconsistent (fake) image-text pair, the label-aware alignment of the images with their augmentations is computed in the same manner as in the true case. Since the image-text pair is inconsistent, they are pushed apart unlike in the true case.Thus the final loss for the fake pair is:ℒ^𝒻𝒶𝓀ℯ_ℒ𝒜(I_ik,I_jk) = -log ( exp(sim(Î_ik, Î_jk) / τ)/𝒟_fake)Here, the denominator is given as𝒟_fake = ∑_t=1t ≠ k^N(∑_a=1^4exp(sim(Î_ik,Î_at)) + exp(sim(T̂_k,T̂_t)) . + .exp(sim(Î_ik,T̂_t)))+ exp(sim(Î_ik,T̂_k)) The final label-aware alignment loss is computed by taking a weighted sum of the two losses corresponding to real and fake data using the eq. <ref> belowℒ_ℒ𝒜 = β * ℒ^𝒻𝒶𝓀ℯ_ℒ𝒜 +ℒ^𝓉𝓇𝓊ℯ_ℒ𝒜The parameter β is used to weigh the two terms. This differential treatment of fake and real samples in both stages helps to better train the model and reduce confusion.§.§ Stage 2: Computing Semantic Domain Vectors In this stage, we create semantic domain vectors which represent how similar each domain is to all the domains in the training dataset.If the number of domains in the training dataset 𝒟_train is n, we create n semantic domain vectors each having n indices.Let us take the "politics" domain in the training dataset as an example.To compute the semantic domain vector for "politics", we pass all its images and corresponding text captions, i.e., {(I_k, T_k)}, k = 1,2,...,n_D_i through the frozen encoders of the A-CLIP trained in Stage 1 to get {(Î_k, T̂_k)}. Here, n_D_i denotes the number of training samples in the politics domain.We take a hadamard product of the image and text embeddings {(Î_k, T̂_k)} to get n_D_i joint embeddings.Then, we compute the mean across all the sample joint embeddings to get the final mean joint embedding 𝒥_D_i, particular to a domain D_i (here, "politics") as shown in eq. <ref> .𝒥_D_i = ∑_k=1^n_D_i𝒥_k/n_D_i,where, 𝒥_k = Î_k⊙T̂_kThis is illustrated in Fig. <ref>. Finally, to create the semantic domain vector w_D_i, we take the cosine similarity between this joint embedding 𝒥_D_i and those of the other training domains 𝒥_D_k, k = 1,2,...,n. Hence, the j^th index of the w_D_i vector will be the cosine similarity of 𝒥_D_i with 𝒥_D_j, (j ∈ k) as shown in eq. <ref>.w_D_i(j) = 𝒥_D_i·𝒥_D_j/𝒥_D_i𝒥_D_jWe term these as semantic domain vectors, since they can implicitly capture the semantics of the domains, i.e. the domain vectors of similar domains are closer compared to those of unrelated domains. This is illustrated in detail later. §.§ Stage 3: Domain-Specific Prompt-tuning with Out-of-Domain dataIn this stage, we utilize the semantic domain vectors to learn domain-specific prompt embeddings that help to incorporate domain-specific information in the model while classifying a news item as true or fake. We initialize a prompt prefix with the standard text "A photo of" <cit.>. However, we make this prefix learnable (similar to <cit.>), since it may not be the best choice for our application.We denote these generic learnable prompts as {V_1, V_2, V_3}.To incorporate the domain information, we pass the corresponding semantic domain vector w_D_i (obtained in the previous stage) through a linear layer, denoted by ℱ_θ. This layer ℱ_θ projects the n_D_i-dimensional domain vector to a 512-dimensional domain prompt token 𝒱_D_i, which is then appended to the prompts {V_1, V_2, V_3}. This forms the whole learnable prompt structure, where {V_1, V_2, V_3} is trained in a domain-agnostic manner, whereas, 𝒱_D_i is trained byconditioning on the domain of the particular training sample.This entire prompt is then prepended to the text caption T and passed through the frozen A-CLIP text encoder to get a text embedding 𝒯̂ as shown in eq. <ref>. 𝒯̂ = {θ_1, θ_2, θ_3, ℱ_θ(w_D),T}Here, θ_1, θ_2, θ_3 denotes the learnable parameters for the prefix prompts. Similarly, the image is passed through the frozen A-CLIP image encoder to get the image embedding Î. We again create a joint embedding Ĵ by taking a hadamard product of 𝒯̂ and Î, and pass it through a Classifier Network.The Classifier Network consists of two fully connected layers which project Ĵ to 𝒳̂. Inspired by <cit.>, we also introduce a residual connection from Ĵ to 𝒳̂, the output of which finally passes through two fully connected layers to give a final prediction ŷ. The whole network is trained in an end-to-end manner using a Binary Cross-Entropy Loss as shown in eq. <ref>. ℒ(θ) = -(ylog(ŷ) + (1 - y)log(1 - ŷ))Here, θ denotes all the learnable parameters in the network.The ground-truth y = 1, if the sample is fake, and y = 0, if it is a true news.Inference. During inference, we can test on any domain of interest since the entire model has been trained without any prior assumptions of the target domain.To classify a test image-text pair as fake or real, its corresponding semantic domain vector is passed through the already trained ℱ_θ network to get the target specific domain prompt V_D.The text caption (of test data) is appended with the learnt prompt vectors to get {V_1, V_2,V_3, V_D}.The final text and image embeddings obtained using the frozen A-CLIP encoders are then used to compute the joint embedding. The Classifier Network gives the final prediction ŷ, which is thresholded to give a binary prediction as True or Fake.Difference with other prompt-learning techniques. As discussed in the Related Work section, there has been a recent few works on domain-based prompt tuning <cit.><cit.>. But there are few important differences between these techniques and the proposed one: 1) In the existing works,"domain" or "style" refer to changes in image features.Contrary to this, in our case, domains (such as "politics", "law-crime", etc.) do not have significant change in image features (since, images from multiple domains will contain similar objects like humans, etc.), but are subjective to human interpretation of both the images and the texts jointly. 2) In existing works, the textonly contains the class name as opposed to a sentence describing the image as in ours. 3) We propose a novel strategy of creating semantic domain vectors, which leverages similarity from out-of-domain data, and utilize it to create domain-specific learnable prompts. To the best of our knowledge, this type of domain-prompting strategy has not been explored earlier. § EXPERIMENTAL EVALUATIONWe evaluate the effectiveness of the proposed framework on the standard large-scale benchmark, namely NewsCLIPings dataset <cit.>, which is created from the Visual News dataset <cit.>.It contains news articlesfrom sources like: The Guardian, BBC, USA Today, and The Washington Post.It has a total of 71,072 train, 7,024 validation and 7,264 test examples.We have used subsets of the dataset to perform experiments. Specifically, we use 25%, 50%, 75% and 100% of the NewsCLIPings balanced data. In these subsets, the data samples from all domains are also in the same proportion as in the whole dataset. Implementation Details: CLIP "ViT-B/32"<cit.> is the backbone of all our experiments.We use accuracy as the evaluation metric as in <cit.><cit.>, as the data is balanced having equal distribution of true and fake news.We have implemented in PyTorch and have used one NVIDIA GeForce RTX 3080 Ti 12GB card.ADAM optimizer with weight decay and stability constant is used. All the results are reported using the same set of parameters, β=1.5 in eq. (<ref>), learning rate is 1e-4 and batch size is 64 across all dataset splits.The values of hyperparameters are obtained during the validation process against the constant validation dataset containing all-domain data.These values were then kept fixed for the rest of the experiments. Stage 1 is run for 40 epochs, which remains fixed even if the desired domain changes. We initialize our domain-agnostic prompts {V_1, V_2, V_3} required in Stage 3, as"A photo of" which is then learnt through training. After obtaining the output probability at the end of Stage 3, we use Sigmoid function to convert the output into a probability score ŷ between 0 to 1. We train Stage 3 for 30 epochs. At the time of inference, the data having output greater than or equal to 0.5 is assigned the label 1, i.e., Fake and the others are assigned label 0, i.e., True. Now, we describe the experiments conducted to evaluate the effectiveness of the proposed DPOD framework.We would specifically like to answer the following research questions:* How does the model perform for the standard setting, where all the domain data is used for both training and testing? How does the performance vary with different amounts of training data?* How well does the proposed model perform for a desired target domain? * Can the model handle inconsistencies in domain labels and how well does it understand about the domains?* Are all the components in the model useful? §.§ Evaluation on standard setting for all domains Since most of the current approaches <cit.><cit.> have been evaluated on all the domains combined, we first experiment on this standard setting. Here, we assume that we know the domain-label of the test-data. As explained earlier, getting annotated data for this application is difficult, since considerable domain expertise is required. Thus, we also evaluate the different approaches by varying the amounts of training data, keeping the test set unchanged for fair comparison of the results. The experiment is run 5 times with random sampling of training data for each of the data settings, and the mean performance over the 5 experiments is reported.In Fig. <ref>, we compare the proposed framework with two state-of-the-art approaches, namely 1) CLIP-FT <cit.>: Here, the entire CLIP model is fine-tuned along with two additional layers with the available training data.This is the performance without using the open-domain data. The reported numbers for varying dataset sizes are obtained by running the official code provided by the authors; 2) SSDL <cit.>: Here, we directly report the performance from their paper.All the results are using ViT-B/32 backbone for image encoder of CLIP as in <cit.>. We observe that the proposed framework consistently outperforms both the SOTA methods by a significant margin. Our model outperforms <cit.> by almost 1% using only 25% of the annotated data and by 3.6% using all of the annotated data.This justifies the effectiveness of the proposed framework.§.§ Specializing to a particular domainNow, we report the main contribution of this work, i.e. how well our model utilizes out-of-domain samples in order to predict the veracity of the news belonging to a desired domain.For this purpose, we select two domains, namely Politics and Sport, mainly because they have quite different number of training data. Though researchers have started to address this realistic scenario in the unimodal case (text) <cit.>, this is a relatively unexplored area in the multi-modal setting.In the absence of comparative approaches, we experiment with the following strong baselines:1) Clip-FT <cit.>: without the open-domain information for fair comparison with the other approaches. 2,3) CoOP <cit.>: First, the prompts {V_1, V_2, V_3} are learnt using training data from all the domains.Intuitively, for testing on a particular target domain, training on the same domain should perform the best.So, we also train CoOP on the train data of the desired target domain and compare its performance with the proposed framework. For example, for Politics target domain, CoOP is trained using only politics data.The test data is the same for all these scenarios. For fair comparison with our approach, CoOP is also used with A-CLIP.Table <ref> summarizes the results for all these approaches. We make the following observations: 1) When all the domains are used for training, the proposed DPOD framework significantly outperforms all the other compared approaches, both for 100% and 25% of the training data. 2) When only the desired (target) domain is used for training, the performance on the test data depends on the amount of in-domain training data.Thus, for 100% politics, since the train data is less (1396), only in-domain training performs significantly worse compared to utilizing data from the remaining domains.On the other hand, since 100% sport has much larger number of training samples (3506), they outperform the other techniques. The limitation is that either we need to know the target domain during training itself, or different models need to be trained for different target domains.In contrast, we only train one model for all the test domains (without any knowledge of the test domain apriori). For 25% of data for both the domains, since the amount of in-domain training data decreases significantly, the proposed DPOD framework, which leverages the information from the other domains, significantly outperforms all the other approaches. §.§ Handling Domain-Label Inconsistencies Often, there are inconsistencies in the domain labels, which may be due to crowd-sourcing for obtaining labels. For example, in the NewsClipping dataset, there are domain labels like football, sports, sport and also Healthcare Network, Healthcare Medicine, etc. which should ideally have the same domain labels.Fig. <ref> shows the cosine-similarity matrix that is calculated by taking the cosine similarity between the learnt domain-specific prompts of the corresponding two domains.We observe that the learnt domain-specific prompts inherently captures the semantic information of the domains, thereby bringing the similar domains closer, thus helping in their training.The proposed framework does not require the exact domain names, and thus can work even if it is difficult to specify the domains, and dummy domain labels (like D_1, D_2,) are used instead (Fig. <ref>). §.§ Ablation StudiesHere, we analyze the importance of Stage 1 (label-aware alignment) and Stage 2 (semantic domain vectors) for the all domain scenario in Table <ref>. We observe that removing either of the stages deteriorates the overall performance.The second row in this table is generated by initializing {V_1, V_2, V_3} with "A photo of" and appending it with a fourth vector V_4, which is obtained by passing a unique one-hot vector corresponding to each training domains. This also implies that if the domains are treated independently, then they cannot fully benefit from the training examples of closely related domains as in the proposed DPOD. The importance of domain-specific prompts can also be observed from Table <ref>, which shows the performance of prompt {V_1, V_2, V_3} and {V_1, V_2, V_3, V_4}, where V_4 is another learnable prompt, but not a domain specific one as in DPOD (third row).Thus, all the proposed modules contribute to the improved performance of DPOD framework. §.§ Qualitative ResultsHere, we provide some qualitative results of our framework.Fig. <ref> (left with green border)shows few examples of correct predictions. We observe that the model demonstrates the capability to accurately assess the veracity of a news item even in complex situations, as in the first example of a promotional event for a different brand. Fig. <ref> (right with pink border) shows few examples of incorrect predictions by our model. We observe that in some cases, the model struggles when multiple elements mentioned in the caption are present in the image, as Koala in the second failure case. In Fig. <ref>, we present a few qualitative results illustrating where A-CLIP focuses on in both the image and text after Stage 1 in the proposed DPOD method. We observe that, in most cases, A-CLIP attends to regions (marked in yellow) that play a crucial role in decision making. § CONCLUSION In this work, we proposed a novel framework for the challenging MFND task. First, label-aware alignment of the data is achieved using the widely-popular CLIP model to obtain generalized features. Further, we propose to learn generic as well as domain-specific prompts to classify the input image-text pairs. The domain-specific prompt is learnt froma novel semantic domain vector, which incorporates the knowledge from the out-of-domain training data. Experiments show that the proposed DPOD achieves the new state-of-the-art for this challenging socially relevant MFND task.ieeenat_fullname | http://arxiv.org/abs/2311.16496v1 | {
"authors": [
"Debarshi Brahma",
"Amartya Bhattacharya",
"Suraj Nagaje Mahadev",
"Anmol Asati",
"Vikas Verma",
"Soma Biswas"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20231127084926",
"title": "Leveraging Out-of-Domain Data for Domain-Specific Prompt Tuning in Multi-Modal Fake News Detection"
} |
Topological Anderson Insulators by homogenization theoryGuillaume Bal[Departments of Statistics and Mathematics and Committee on Computational and Applied Mathematics, University of Chicago, Chicago, IL 60637 ([email protected]). ] Thuyen Dang[Department of Statistics and Committee on Computational and Applied Mathematics, University of Chicago, Chicago, IL 60637, USA ([email protected]).]========================================================================================================================================================================================================================================================================================================================================================================================== A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A Topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This paper proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the systems is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transportindeed appears at an interface separating perturbed and unperturbed phases.The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.Keywords: Topological insulators, Anderson TI, homogenization theory, Dirac operators, asymmetric transport, Fredholm operator AMS: 35B20, 35B27, 35B35, 35B40, 47A53. § INTRODUCTION Topological Anderson Insulators are insulators whose topological phase is induced by randomness <cit.>. A theoretical explanation based on effective medium theory for such phenomena was proposed in <cit.>. Their starting point is a randomly perturbed system of Dirac equations.This paper considers a similar system of Dirac equations, given explicitly by (<ref>) below, that includes periodic perturbations of the form ε^-1V(^-1 x) with V a matrix-valued periodic potential and x=(x_1,x_2)∈^2 spatial coordinates. Using homogenization theory, we show that in the limit 0<→0, the system converges to a homogenized Dirac equation in a different topology from the unperturbed system formally obtained as →∞.Arguably the most unexpected feature of topological insulators is the asymmetric transport observed at interfaces separating insulators in different topological phases. An interface in our system is modeled by a domain wall ρ(x)=ρ(x_2) and perturbations of the form ρ(x)ε^-1V(^-1 x).This allows us to define a transition from the unperturbed system where ρ(x_2)=0 when x_2≥1 to a perturbed system where ρ(x_2)=1 when x_2≤-1. We will show that the interface Hamiltonian with such a domain wall indeed carries robust asymmetric transport for 0<<_0 sufficiently small while no such robust asymmetric transport exists for >_1>0 sufficiently large. This surprising result indicates that while the map →ρ(x)ε^-1V(^-1 x) appears to be smooth in some metrics, it is necessarily discontinuous or not globally defined in the sense of asymmetric transport.The invariants mentioned above are defined for large classes of topological insulators. See, e.g., <cit.> for general references from the mathematical and physical literature on the broad and actively studied field of topological insulators. The robust asymmetric transport observed at one-dimensional interfaces separating distinct two-dimensional insulators has also been analyzed in many contexts.For many discrete and continuous Hamiltonians, a quantized interface current observable as well as a number of Fredholm operators with non-trivial indices may be introduced to compute topological invariants and relate them to the observed quantized, robust-to-perturbations, asymmetric transport along the interface; see, e.g. <cit.> for discrete Hamiltonians and <cit.> for partial- or pseudo- differential Hamiltonians.In this paper, the systems of interest are modeled by Dirac equations acting on spinors (vector-valued functions) of the Euclidean plane ^2. The topological invariants we will be using are Chern invariants for the bulk phases. For regularized Dirac operators, their theory is presented in <cit.>. The corresponding invariants for the interface Hamiltonians modeling a transition between different bulk phases are analyzed in <cit.>.The rest of the paper is structured as follows. Section <ref> formulates the system of Dirac equations with periodic perturbations, defines the notions of bulk and interface invariants, and presents our main results. In particular, we first derive a homogenization result for the Dirac equation with periodic potential. The limiting homogenized operator is then shown to be in a different bulk topological phase than the unperturbed operator. Estimates on the resolvent operator in the homogenized limit then allow us to state our main result, namely that forsufficiently small, the perturbed Dirac operator with domain wall caries non-trivial edge transport.The proofs of the main results are then presented in detail in section <ref>. § FORMULATION AND MAIN RESULTS This section introduces our model and presents our main results. All proofs are postponed to section <ref>.§.§ Dirac model with periodic fluctuations Throughout the paper, we use the following notation. * All the functions are assumed to be complex-valued, unless stated otherwise.* Y [0,1]^2 – the reference unit cell.* L^p_(Y) for some p ∈ [1,∞] – the space of Y-periodic functions that are L^p-integrable.* For f = (f_1, f_2)^⊤ and g = (g_1,g_2)^⊤ in L^2(^2;^2), we define the inner product ⟨ f,g ⟩ = ⟨[ f_1; f_2 ], [ g_1; g_2 ]⟩∫_^2 f_1 g_1 + f_2 g_2 x,and its corresponding norm f⟨ f,f ⟩^1/2.*– the Fourier transform, with ξ (ξ_1, ξ_2) to be the dual variable to x = (x_1,x_2), and ⟨ξ⟩( 1 + ξ^2 )^1/2 to be the Japanese bracket.* The Einstein summation convention is used whenever applicable; δ_ij is the Kronecker delta, and ϵ_ijk is the permutation symbol. The Dirac operator is then defined as follows. Let ε > 0. We consider the following Hamiltonian on L^2(^2) ⊗^2 ≅ L^2(^2;^2):^ε = ( D + (U_1^ε, U_2^ε) ) ·σ + (m + βΔ + U_3^ε) σ_3 + U_0^εσ_0,where m,β are real numbers with |m|, |β| in (0,∞),{σ_i } are Pauli matricesσ_1[ 0 1; 1 0 ], σ_2 [0 -i;i0 ], σ_3 [10;0 -1 ], σ_0 Id,x (x_1, x_2) ∈^2, D 1/i( ∂_x_1, ∂_x_2), σ( σ_1, σ_2 ), Δ∂_x_1^2 + ∂_x_2^2,and U^ε_j (x) 1/ερ (x) V_j^ε(x),V_j^ε(x)V_j ( x/ε),with ρ∈W^2,∞(^2;) and V_j∈ C^0,α_(Y;) for some α∈ (0,1), j ∈{ 1,2,3,4 }.The form of U^ε is motivated by the work on homogenization with rapidly oscillating potentials in <cit.>.Homogenization problems with similar potentials were studied in <cit.>. The scaling of the potential of order ^-1 ensures that the limiting homogenized limit is an order O(1) modification of the unperturbed operator. Note that the domain wall is taken into account via the function ρ in (<ref>). Except for the perturbations U_1^ε, U_2^ε, U_3^ε and the specific form of the perturbation U_0^, the Dirac model is essentially the same as the one used in <cit.>.Varying the mass term U^_3 may be challenging practically. We will show that variations of the magnetic potential (U_1^,U_2^) or variations of the electric potential U_0^ are sufficient to generate asymmetric edge transport.§.§ Homogenization theory We following form of ^ε is more suitable for homogenization purpose. ObserveU^ε U^ε_1 σ_1 + U^ε_2 σ_2 + U^ε_3 σ_3 + U^ε_0 σ_0 =1/ερ (x) [ V_0^ε + V_3^ε V_1^ε - i V_2^ε; V_1^ε + i V_2^ε V_0^ε - V_3^ε ],so if we let W [ V_0 + V_3 V_1 - i V_2; V_1 + i V_2 V_0 - V_3 ],then we can rewrite (<ref>) as^ε = D ·σ + ( m + βΔ) σ_3 + 1/ερ (x) W ( x/ε). The unperturbed (regularized) Dirac operator from L^2(^2;^2) to L^2(^2;^2) is given explicitly by^∞ D ·σ + ( m + βΔ) σ_3together with its domain (^∞) { f ∈ L^2(^2;^2)^∞ f ∈ L^2(^2;^2) }.We have the result: (^∞) = H^2(^2;^2) and^∞ (^∞) ⊂ L^2(^2;^2) → L^2 (^2;^2) is self-adjoint.For each ε > 0, the multiplication operator _U^ε U^εσ_0 is bounded and self-adjoint on L^2(^2;^2) by the assumptions on U^ε, therefore, by <cit.>, (^ε)^* =(^∞+ _U^ε)^* = (^∞)^* + (_U^ε)^* =^∞ + _U^ε = ^∞ + _U^ε = ^ε,or ^ε is self-adjoint with (^ε) =(^∞). Consequently, the spectrum (^ε) ⊂.To eliminate ε^-1 in (<ref>), we introducing the auxiliary problemsΔ_y ϕ_kl = W_kl, ϕ_kl∈ H_^1(Y;)/, 1 ≤ k, l ≤ 2.Elliptic regularity theory <cit.> implies ϕ_kl∈ H_^2(Y;)/.Define Φ_kl(y) ∇_y ϕ_kl (y) andΦ_kl^ε(x) Φ_kl( x/ε), 1 ≤ k, l ≤ 2.Then Φ_kl^ε(x)= Φ_kl( x/ε) = 1/ε_y ( Φ_kl(y) ) |_y = x/ε = 1/ε_y ( ∇_y ϕ_kl (y) )|_y = x/ε = 1/ε W_kl( x/ε).Therefore, U^ε (x) = 1/ερ(x) W_kl( x/ε) = ρ(x) Φ_kl^ε(x) =( ρ(x) Φ_kl^ε(x) ) - ∇ρ(x) Φ_kl^ε(x)since ( ρ (x) Φ_kl^ε(x) )= ρ(x) Φ_kl^ε (x) + ∇ρ(x) ·Φ_kl^ε (x)= 1/ερ(x) W_kl( x/ε) + ∇ρ(x) ·Φ_kl^ε(x). Let ε > 0. Let λ > 0, z ∈ with z ∈ [-λ,λ]. There exist γ = γ (λ, m, β, ρ_L^∞, W_C^0,α) > 0 and C = C (λ,m, β, ρ_L^∞, W_C^0,α) > 0, both independent of ε, such that whenever z ∈ [-γ, γ] ∖{ 0 } the equationψ^ε,z∈(^∞),(^ε - z ) ψ^ε,z =f,has a unique solution ψ^ε,z∈(^∞) that satisfies ψ^ε,z_H^1≤ C z^-1f. For 1 ≤ k,l ≤ 2, consider the cell problemT_kl∈ H^1_(Y; ), βΔ_yy T_kl + W_kl = 0.DefineT(y) [T_11(y)T_12(y);T_21(y) T_22 (y) ],y ∈ Y.Then as ε→ 0, the solution ψ^ε,z of (<ref>) weakly converges to ψ^s,zin H^1(^2;^2) such that (^0-z) ψ^s,z = f,where the homogenized operator ^0 is defined by^0 D ·σ + (m + βΔ) σ_3 + βρ^2(x) τ, τ ≡[ τ_11 τ_12; τ_21 τ_22 ]1/β∫_Y W(y) σ_3 T(y)y =∫_Y [ ∇ T_11^2 - ∇ T_21∇ T_12∇ T_11∇ T_12 -∇ T_12∇ T_22; ∇ T_11∇ T_21 - ∇ T_21∇ T_22-∇ T_22^2 +∇T_12∇ T_21 ] y. The homogenization theorem above can be extended to:*The case m = m(x) is a smooth compact pertubation of a constant, by Kato-Rellich criterion.* The case when βΔ· is replaced by a ( x/ε) ∇· with a bounded elliptic matrix a. In this case, (<ref>) is replaced by [a ( y ) ∇ T_kl (y)] + W_kl =0, and we need two additional cell problems a ( y ) [∇χ_i (y) + e_i]=0, where e_i form a canonical basis of ^2.§.§ Resolvent estimatesThe above homogenization results, which describe the limiting homogenized operator, are not sufficiently strong to define topological invariants. We require stronger results on the convergence of resolvent operators that we now state. A first result concerns regularity of (<ref>), which is necessary for the proof of norm resolvent convergence of ^ε.Let z ∈ [-λ,λ] + i ([-γ,γ] ∖{ 0 }) where λ, γ are defined in <ref>. For each f ∈ L^2(^2;^2), the solution ψ^s,z of (<ref>) satisfiesψ^s,z_H^2≤ Cz^-1f,for some C = C (λ,m, β, ρ_L^∞, W_C^0,α)> 0. In particular, if f ∈ H^1(^2;^2), then ψ∈ H^3(^2;^2) andψ^s,z_H^3≤ Cz^-1f_H^1. Although elliptic regularity theory <cit.> also implies the bounds for ψ^s,z in H^2 and H^3 when f is smooth enough, our <ref> specifies the dependence on z in the right hand side of the estimates. This is crucial for the following result:Let λ > 0. There exist C > 0 and ε_0' > 0, depending on λ,m, β, ρ_W^1,∞, ρ_W^2,∞, and W_C^0,α, such that for any f ∈ L^2(^2;^2), ε∈ (0,ε_0'), and z ∈ with z ∈ [-λ,λ], z0, we have (^ε- z )^-1 - (^0 - z )^-1_L^2 → L^2≤ C ( 1+z^-2) ε. §.§ Topological classification Let ^∞, ^0 and ^ε be the unperturbed, homogenized, and perturbed operators defined above. To compute the topological invariants, it is more convenient to rewrite τ in (<ref>) as τ = τ_1 σ_1 + τ_2 σ_2 + τ_3 σ_3 + τ_0 σ_0,with τ_01/2∫_Y ( ∇ T_11^2 - ∇ T_22^2 )y, τ_11/2∫_Y ( ∇ T_11∇ T_12 + ∇ T_11∇ T_21 - ∇ T_12∇ T_22 - ∇ T_21∇ T_22)y, τ_21/2i∫_Y (- ∇ T_11∇ T_12 + ∇ T_11∇ T_21 + ∇ T_12∇ T_22 - ∇ T_21∇ T_22)y, τ_31/2∫_Y ( ∇ T_11^2 + ∇ T_22^2 - 2 ∇ T_12∇ T_21)y.It follows from (<ref>) that ^0 = (D + βρ^2(x)(τ_1, τ_2))·σ + (m + βΔ + βρ^2(x)τ_3) σ_3 + βρ^2(x)τ_0 σ_0.In the sequel, we suppose [On domain wall] There are a < 0 < b so thatρ∈ W^2,∞(^2;), ρ(x) = ρ(x_2) = 0,x_2 ≥ b,1,x_2 ≤ a. We introduce the necessary ingredients to define invariants that characterize the topological phases of Dirac operators following <cit.>. We define and assumem_+=m m_-=m+βτ_3|m_+| ≥ m_0|m_-| ≥ m_0>0for some constant m_0>0 that will be specified later. We denote by ^0_B the homogenized bulk (constant coefficient) operator^0_B = (D + β (τ_1,τ_2))·σ + (m_-+βΔ)σ_3 + βτ_0 σ_0.The operator ^0_B thus represents the insulator on the lower half-plane x_2 ≤ a, where the perturbations apply, i.e., ρ (x_2) = 1. There are no perturbations on the upper half-plane x_2 ≥ b, so the insulator on this part is still represented by ^∞.Intuitively, asymmetric transport appears on the domain wall × [a,b] if the following conditions are satisfied: (i) Both insulators have a common spectral gap of at the domain wall, more specifically, ∖ (^∞) ∩∖ (_B^0) contains a (non-empty) open interval.(ii) ^∞ and ^0_B have different `topologies'. The latter are characterized by indices of Fredholm operators that will be introduced shortly.By definition, we have^0_B = ^∞ + βτ.The matrix τ is Hermitian by (<ref>), (<ref>), and (<ref>). Let p_A(t) (t - A) denote the characteristic polynomial of a given square matrix A. Direct computations and the fact that τ_12 = τ_21 givep_^∞ (s)= s^2 - ( m - βξ^2 )^2 - ξ^2,p_^0_B(s)= s^2 + β (τ_11 + τ_22) s - ( ( m-βξ^2 )^2 + β( m - βξ^2 ) (τ_11-τ_22) - β^2 τ_11τ_22)- ξ + βτ_12^2. Given V_j, 1≤ j ≤ 4, one can compute τ by the formula (<ref>) and obtain the eigenvalues of both ^∞ and ^0_B as functions of the dual variable ξ. The spectrum of ^∞ and ^0_B are the essential ranges of these functions. By a common shift, we can assume whenever ^∞ and ^0_B have a common spectral gap, it contains 0. [Common spectral gap] There exists m_0 > 0 such that (-m_0, m_0) ⊂∖ (^∞)∩ ∖ (_B^0).[Non vanishing electric potential]When V_1 = V_2 = V_3 = 0 and V_00, then (<ref>) implies T_11 = T_22 and T_12= T_21 = 0, so by (<ref>) we have τ_0 = τ_1 = τ_2 = 0, τ_3 = ∫_Y ∇ T_11^2y = ∫_Y ∇ T_22^2y.<ref> is then satisfied. We show below that asymmetric transport appears when m β < 0 is sufficiently large but cannot appear when m β > 0. [Non vanishing magnetic potential]When V_1 = V_3 = V_0 = 0 and V_20, then T_11 = T_22 = 0 and T_12 = -T_21. From (<ref>), we have τ_0 = τ_1 = τ_2 = 0 and τ_3 = -∫_Y ∇ T_12^2y. In this case, <ref> holds and asymmetric transport appears when m β > 0 is sufficiently large while itcannot appear when m β < 0.We now establish a different formula for τ_3 to quantify how the pertubations V_j affect the mass term in the homogenized operator ^0. Consider the following auxiliary problems:t_j ∈ H_^1(Y;), βΔ_yy t_j + V_j = 0,j =1,2.By (<ref>), (<ref>) and uniqueness, we conclude T_12 = t_1 - it_2andT_21 = t_1 + i t_2.It follows that ∫_Y ∇ T_12∇ T_21 y = ∫_Y ∇ (t_1 - it_2)·∇ (t_1 + i t_2)y = ∫_Y ∇ t_1^2 + ∇ t_2^2y,so τ_31/2∫_Y ( ∇ T_11^2 + ∇ T_22^2 - 2 ( ∇ t_1^2 + ∇ t_2^2 ) )y. We will justify the following statements:Asymmetric transport occurs when mβτ_3 < 0 is sufficiently large. More precisely, when m β < 0 (resp. m β > 0) and τ_3 is sufficiently large, then perturbations in front of σ_3 or σ_0 (resp. σ_1 or σ_2) change the `topology' of the regularized Dirac operator resulting in TAI/asymmetric transport. Perturbations in front of (σ_1,σ_2) and (σ_3,σ_0) have opposite effects on the change of topology. Those statements will be formalized below, and our main result will be stated in <ref>.Limiting bulk invariants. The invariant characterizing bulk phases <cit.> is given byF[H] = Π(H<0) x_1+ix_2/|x_1+ix_2|Π(H<0) + I-Π(H<0)where Π(H<0) is the orthogonal projector onto the negative spectrum ofa self adjoint operator H gapped at 0 (i.e., 0∉σ(H)).Following <cit.>, we have The operators of the bulk Hamiltonians F[^∞] and F[^0_B] are Fredholm on L^2(;^2). Moreover,Index F[^∞] = 1/2(m_++β), Index F[^0_B] = 1/2(m_-+β). We recall the index of a Fredholm operator F is dim Ker F -dim Ker F^*. This result shows that the presence of fluctuations changes the topological bulk properties for the homogenized operator compared to the unperturbed operator provided m_+ and m_-in (<ref>) have different signs, i.e.,mβτ_3 < 0is sufficiently large. We have thus generated a topological insulator whose phase is directly related to the presence of the fluctuations U^, at least asymptotically in the limit →0.The result stated in the above proposition holds generally for m≠0 and β≠0. As soon as m_+m_-=-1, we observe a change of bulk topologyIndex F[^∞]≠ Index F[^0_B]. This occurs for mβ<0 when τ_3>0 and for mβ>0 when τ_3<0 for |τ_3| sufficiently large.Limiting interface invariants. We now consider the practically more relevant case of an interface invariant characterizing quantized asymmetric transport along the axis x_2=0. To define a transition between insulators in different phases, we assume that ρ satisfies <ref>. Recall the definition of m_0 in (<ref>). The constraint m_0>0 implies that the bulk Hamiltonians display a spectral gap in the interval (-m_0,m_0). Therefore, excitations in this energy range are constrained to remain in the vicinity of x_2=0 while excitations outside of that range may propagate into the bulk. An interface invariant necessarily focuses on the former excitations. We thus define a non-decreasing smooth function φ(h) equal to 1 for h≥ m_0 and equal to 0 for h≤ -m_0. In other words, φ'(h)≥0 is supported in (-m_0,m_0), integrates to 1, and may be interpreted as a density of states that will remain localized in the vicinity of the interface x_2=0.More precisely, foran unboundedself-adjoint operator on L^2(^2;^2), then φ'() is a bounded operator modeling a density of states supported in the energy range (-m_0,m_0) andU()=e^2π i φ() is a unitary operator. The function S(h)=U(h)-1 is thus also smooth and compactly supported in (-m_0,m_0).In <cit.> (see also <cit.>), the following interface invariant is defined2πσ_I[ ] =Tr2π i[,P] φ'().Here P(x)=P(x_1) is a smooth function equal to 0 for x_1≤ x_0-δ and equal to 1 for x_1≥ x_0+δ for some x_0∈ and δ>0. The operator i[,P] may be interpreted as a current operator (modeling transport from the left of x_1=x_0 to the right of x_1=x_0 per unit time) so that the above trace may indeed be interpreted as the expectation of a current operator against the density of states φ'(). Any non-vanishing value indicates asymmetric transport. Sinceis elliptic, it is shown in <cit.> that 2π i[,P] φ'() is indeed a trace-class operator for theinterface Hamiltonians =^∞ and =^0.In fact, since the unperturbed Hamiltonian ^∞ is gapped in (-m_0,m_0), we obviously have that 2πσ_I[^∞ ] =0. More precisely, we have <cit.>Under the hypotheses of <ref>, we have 2πσ_I[^∞] = 02πσ_I[^0] = 1/2(m_--m_+). The references <cit.> also show that the above interface current is stable against local perturbations of the Hamiltonian. This justifies the robustness of the asymmetric current against perturbations. Stability of the interface invariant. So far, the invariants are only defined in the limits =0 and =∞. We now show that invariants remain defined and stable forsufficient close to 0 andsufficiently large.Let (x)= H(x_1-x_0) be a Heaviside function equal to 1 for x_1≥ x_0 and 0 for x_1<x_0. We can then construct the family of bounded operatorsT^ε :=T[^] = (x) U(^ε) (x) + (I-(x)), 0 ≤ε≤∞.From <cit.>, we obtain that At ε=∞ and ε=0, the operators T^∞=T[^∞] and T^0=T[^0] are Fredholm operators. Moreover,IndexT^∞ =2πσ_I[^∞] = 0 , IndexT^0 =2πσ_I[^0] = 1/2(m_--m_+). As mentioned above, the unperturbed operator ^∞ admits a spectral gap in (-m_0,m_0) so that U(^∞)=I and T^∞=I is clearly a Fredholm operator with trivial index.The result for T^0 is more interesting and was also obtained in <cit.>.We can now state the main result of this paper:For ε_1 sufficiently large and ε_0 sufficiently small, then for all 0<<_0 and for all _1<, T^ in (<ref>)is a Fredholm operator. Moreover, we haveIndexT^ε=0 ,ε_1 ≤ε<∞,IndexT^ε =1/2(m_--m_+), 0<ε≤ε_0.<ref> shows the surprising result that the topological invariants assigned to ^ε for ε sufficient large and ε sufficient small are different and as a consequence that there is at least one value of ε>0 where the invariant is not defined and T^ε is not Fredholm. The latter result displays the main property we wanted to establish, namely that heterogeneous fluctuations in a half-space are sufficient to generate an asymmetric transport along the interface x_2∼0.The terminology of `Anderson' topological insulator is somewhat misleading. As we just showed, periodic fluctuations (and not genuinely random fluctuations as in the derivation of Anderson localization) suffice to generate a topological change. Moreover, these fluctuations generate a non-trivial topology, which may in fact be seen as an obstruction to Anderson localization <cit.>.§ PROOF OF THE HOMOGENIZATION AND STABILITY RESULTSWe now present a detailed proof of the main results stated in the preceding section. * We diagonalize the constant coefficient operator ^∞ by Fourier transform to obtain^∞(ξ) = ξ·σ + (m-βξ^2) σ_3,which is the symbol of ^∞, i.e., ^∞ = ^-1^∞, see, e.g.,<cit.>. Direct computation shows that ^∞(ξ)= ξ_1 σ_1 + ξ_2 σ_2 + (m - βξ^2) σ_3 = [m - βξ^2 ξ_1 - i ξ_2; ξ_1 + i ξ_2 -(m - βξ^2) ].Therefore, ^∞^* = ^∞and ^∞^2= (ξ^2 + (m - βξ^2)^2 ) σ_0.*Clearly, H^2(^2;^2) ⊂(^∞). So we only need to justify(^∞) ⊂ H^2(^2;^2). Let f ∈ (^∞), then ^∞ f ∈ L^2(^2;^2). We need f ∈ H^2(^2; ^2). To this end, we will show ⟨ξ⟩^2 f̂∈ L^2(^2;^2).By the Plancherel theorem, ^∞f̂ = ^∞ f, so ^∞f̂ is in L^2(^2;^2). Together with (<ref>) and (<ref>), we get∞ >^∞f̂^2= ⟨^∞f̂, ^∞f̂⟩ = ⟨^∞^2 f̂,f̂⟩=∫_^2(ξ^2 + (m - βξ^2)^2 ) f̂(ξ)^2 ξ≥∫_^2ξ^2 f̂(ξ)^2 ξ,so∞ >(2 + 2 mβ)^∞f̂^2= ⟨^∞^2 f̂,f̂⟩ + (1 + 2 mβ)^∞f̂^2 =∫_^2(ξ^2 + (m - βξ^2)^2 ) f̂(ξ)^2 ξ + (1 + 2 mβ)^∞f̂^2≥∫_^2( ξ^2 + m^2 - 2mβξ^2 + β^2 ξ^4 ) f̂(ξ)^2ξ+ ∫_^2 (1+2 mβ) ξ^2 f̂(ξ)^2 ξ≥∫_^2( 2ξ^2 + m^2+ β^2 ξ^4 ) f̂(ξ)^2ξ≥min{ 1, m^2, β^2 }∫_^2( 1 + ξ^2 )^2 f̂(ξ)^2 ξ.Therefore, ⟨ξ⟩^2 f̂∈ L^2(^2;^2). * Note that C_c^∞(^2;^2) ⊂ (^∞) and C_c^∞(^2;^2) is dense in L^2(^2;^2), the operator ^∞ is densely defined. From (<ref>), ^∞ is also symmetric. Thus, to show ^∞ is self-adjoint, we only need to show (^∞± i) (^∞) = L^2(^2;^2). The inclusion (^∞± i) (^∞) ⊂ L^2(^2;^2) is obvious. We now prove (^∞ + i) (^∞) = (^∞ + i)H^2(^2;^2) ⊃ L^2(^2;^2).Let g ∈ L^2(^2;^2) and f = ^-1( (^∞ + i)^-1ĝ). The latter is well defined because ± i are not eigenvalues of ^∞(ξ) for any ξ∈^2. Observe that f̂ =( ^∞ + i )^-1ĝ= ( ^∞ - i )( ^∞ - i )^-1( ^∞ + i )^-1ĝ= ( ^∞ -i ) ( (m- βξ^2)^2 + ξ^2 + 1 )^-1ĝ,Let ω (ξ) ( 1 + ξ^2) ( ^∞ - i )/(m- βξ^2)^2 + ξ^2 + 1 then ω≥ 0 is continuous and there exist ω_0, ω_∞≥ 0 such that lim_ξ→ 0ω = ω_0 and lim_ξ→∞ω = ω_∞. Thus there exist c_0, c_∞ > 0 such that ω(ξ) ≤ω_0 + ω_∞ + 1 whenever ξ∈ [0,c_0] ∪ [c_∞, ∞). Continuity implies max_ξ∈ [c_0, c_∞]ω(ξ) exists. Thus (<ref>) implies( 1 + ξ^2 )f̂≤( max_ξ∈ [c_0, c_∞]ω(ξ) + ω_0 + ω_∞ + 1 )ĝ∈ L^2(^2;^2),so ( 1 + ξ^2 )f̂∈ L^2(^2;^2), or f ∈ H^2(^2;^2).Using ^∞ = ^-1^∞ and the definition of f, we have g = ( ^∞ + i )f. Therefore L^2(^2;^2) ⊂( ^∞ + i ) H^2(^2;^2). The proof for L^2(^2;^2) ⊂( ^∞ - i ) H^2(^2;^2) is similar. * Let z ∈ with z0. From (<ref>), z belongs to the resolvent set of ^ε. Therefore, (<ref>) has a unique solution ψ^ε,z∈ (^∞) = H^2(^2;^2). Moreover, applying the resolvent estimate for the self-adjoint operator ^ε <cit.>, we obtain ψ^ε,z ≤(^0- z )^-1f ≤ z^-1f. * For η∈(^2;^2), we have from (<ref>) andintegration by parts that ⟨ U^εψ^ε,z, σ^3 η⟩ =⟨1/ερ (x) W ( x/ε) ψ^ε,z, σ_3 η⟩= ⟨[ ((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )e_k ⊗ e_l ] ψ^ε,z_i e_i, σ_3 η⟩=⟨((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )ψ^ε,z_l e_k, η_1 e_1 - η_2 e_2 ⟩=∑_k,l=1^2 (-1)^k-1∫_^2((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )ψ^ε,z_l η_k x=∑_k,l=1^2 (-1)^k {∫_^2ρ(x) Φ_kl^ε(x) ∇( ψ^ε,z_l η_k) x+ ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε,z_l η_k x}=∑_k,l=1^2 (-1)^k {∫_^2ρ(x) Φ_kl^ε(x) ( η_k∇ψ^ε,z_l+ ψ^ε,z∇η_k) x .. + ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε,z_l η_k x }. Thus, multiplying (<ref>) by σ_3 η,and integrating by parts, we obtain⟨ψ^ε,z, (D ·σ + m σ_3 -z ) σ_3η⟩ -⟨β∇σ_3 ψ^ε,z, ∇σ_3η⟩+ ∑_k,l=1^2 (-1)^k {∫_^2ρ(x) Φ_kl^ε(x) ( η_k∇ψ^ε,z_l+ ψ^ε,z_l ∇η_k) x .. + ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε,z_l η_k x}=⟨ f,σ_3η⟩.Therefore, ⟨β∇ψ^ε,z, ∇η⟩ ≤⟨ f,σ_3 η⟩ + ⟨ψ^ε,z, (D ·σ + m σ_3 -z ) σ_3η⟩ +{∫_^2ρ(x) Φ_kl^ε(x) ( η_k∇ψ^ε,z_l+ ψ^ε,z_l ∇η_k) x.. + ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε,z_l η_k x}≤fη + (2 + m + z) ψ^ε,zη_H^1 + C∑_k,l=1^2 ρ_L^∞Φ_kl^ε_L^∞( η∇ψ^ε,z + ∇ηψ^ε,z)+ C ∑_k,l=1^2∇ρ_L^∞Φ_kl^ε_L^∞ψ^ε,zη≤fη + (2 + m + z) ψ^ε,zη_H^1 +C ρ_L^∞W_C^0,α( η∇ψ^ε,z + ∇ηψ^ε,z)+ C∇ρ_L^∞v_C^0,αψ^ε,zη,where in the last step we used the Schauder estimate <cit.> for (<ref>) to obtain Φ_kl^ε_L^∞ = Φ_kl_L^∞≤ C W_kl_C^0,α≤ C W_C^0,α.Moreover, by density, we can take η = ψ^ε,z. Thus (<ref>) becomes β∇ψ^ε,z^2≤fψ^ε,z + (2 + m + z) ψ^ε,zψ^ε,z_H^1 +C ρ_L^∞W_C^0,α∇ψ^ε,zψ^ε,z + C∇ρ_L^∞W_C^0,αψ^ε,zψ^ε,z,≤ C (z^-1f^2 + (2+m+ z) z^-1f ( z^-1f + ∇ψ^ε,z).. + ρ_L^∞W_C^0,αf∇ψ^ε,z + ρ_L^∞W_C^0,αf^2 )≤ C ( z^-1f^2 + (2+m+ z) z^-2f^2 + ρ_L^∞W_C^0,αf^2) + C [ (2+m+z) z^-1 + ρ_L^∞W_C^0,α] f∇ψ^ε,z.By Young's inequality, C [ (2+m+z) z^-1 + ρ_L^∞W_C^0,α] f∇ψ^ε,z≤β/2ψ^ε,z^2 + 2/βC^2 [ (2+m+z) z^-1 + ρ_L^∞W_C^0,α]^2f^2.Thus, β/2∇ψ^ε,z^2≤ C (z^-1 + (2 + m + z) z^-2 + 1/β(2 + m + z) z^-1.. +1/βρ_L^∞W_C^0,α + 1/βρ_L^∞^2 W_C^0,α^2) f^2 ≤ Cz^-2f^2,whenever z ∈ [-λ,λ] and z ∈ [-γ, γ] ∖{ 0 } with γ = γ (λ, m, β, ρ_L^∞, W_C^0,α) > 0 small enough. Therefore,∇ψ^ε,z≤ Cz^-1f.From (<ref>) and (<ref>), we obtain (<ref>). * From now on, we suppress the dependence on z to lighten the notation. By (<ref>) and <cit.>, there exist ψ^s_1, ψ^s_2 in H^1(^2;) and ψ^f_1, ψ^f_2 in L^2(^2; H^1_(Y;^2)/) such that (up to a subsequence), we have the two-scale convergenceψ^ε = [ ψ^ε_1; ψ^ε_2 ] [2] [ ψ^s_1; ψ^s_2 ] and ∇ψ^ε = [ ∇ψ^ε_1; ∇ψ^ε_2 ][2] [ ∇ψ^s_1 + ∇_y ψ^f_1; ∇ψ^s_2 + ∇_y ψ^f_2 ].* Suppose η = η^s + εη^f with η^s = (η^s_1, η^s_2)^⊤∈(^2; ^2) and η^f = (η^f_1, η^f_2)^⊤∈ (^2; C_^∞(Y;^2)).Then (<ref>) implies⟨ (^ε-z) ψ^ε, η⟩ = ⟨ f,η⟩,or equivalently,J_1 + J_2 = ⟨ f,η⟩,where J_1⟨ (D ·σ + (m+βΔ) σ_3 -z ) ψ^ε, η⟩,J_2⟨ U^εψ^ε, η⟩. * We now compute lim_ε→ 0 J_1. Using integration by parts, (<ref>) leads to J_1= ⟨ (D ·σ + (m+βΔ) σ_3 -z ) ψ^ε, η⟩= ⟨ (D ·σ + m σ_3 -z ) ψ^ε, η⟩ + ⟨βΔσ_3 ψ^ε, η⟩= ⟨[ -i ∂_x_1σ_1 - i ∂_x_2σ_2 + m σ_3 - z] ψ^ε, η⟩ -⟨β∇σ_3 ψ^ε, ∇η⟩[ε→ 0] ⟨( D ·σ) ψ^s + ( D_y ·σ) ψ^f + m σ_3 ψ^s - z ψ^s, η^s ⟩ - ⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ], ∇η^s + ∇_y η^f ⟩. * To compute lim_ε→ 0 J_2, we write J_2= ⟨ U^εψ^ε, η⟩=⟨1/ερ (x) W ( x/ε) ψ^ε, η⟩= ⟨[ ((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )e_k ⊗ e_l ] ψ^ε_i e_i, η⟩=⟨((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )ψ^ε_l e_k, η_j e_j⟩=∫_^2((ρ(x) Φ_kl^ε(x))- ∇ρ(x) ·Φ_kl^ε(x) )ψ^ε_l η_k x=-∫_^2ρ(x) Φ_kl^ε(x) ∇( ψ^ε_l η_k) x- ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε_l η_k x=- ∫_^2ρ(x) Φ_kl^ε(x) ( η_k∇ψ^ε_l+ ψ^ε_l ∇η_k) x - ∫_^2∇ρ(x) ·Φ_kl^ε(x) ψ^ε_l η_k x =- ∫_^2ρ(x) Φ_kl( x/ε) ( η_k^s + εη_k^f∇ψ^ε_l+ ψ^ε_l ∇η_k^s + εη_k^f) x - ∫_^2∇ρ(x) ·Φ_kl( x/ε) ψ^ε_l η_k^s + εη_k^f x [ε→ 0] - ∫_^2 × Yρ(x) Φ_kl( y ) ( η_k^s( ∇ψ_l^s + ∇_y ψ_l^f )+ ψ_l^s ( ∇η_k^s + ∇_y η_k^f)) xy - ∫_^2× Y∇ρ(x) ·Φ_kl( y ) ψ_l^s η_k^s xy,where we used (<ref>) in the last convergence.* Therefore, passing to the limit ε→ 0 in (<ref>), we obtain ⟨( D ·σ) ψ^s + ( D_y ·σ) ψ^f + m σ_3 ψ^s - z ψ^s, η^s ⟩ - ⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ], ∇η^s + ∇_y η^f ⟩ - ∫_^2 × Yρ(x) Φ_kl( y ) ( η_k^s( ∇ψ_l^s + ∇_y ψ_l^f )+ ψ_l^s ( ∇η_k^s + ∇_y η_k^f)) xy - ∫_^2 × Y∇ρ(x) ·Φ_kl( y ) ψ_l^s η_k^s xy= ⟨ f, η^s ⟩. * We derive the cell problems. In (<ref>), let η^s = 0, then -⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ],∇_y η^f ⟩- ∫_^2 × Yρ(x) Φ_kl(y) ψ_l^s ∇_y η_k^f xy = 0,or equivalently, ⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ] + ρ (x) {Φ_kl(y) e_k ⊗ e_l}ψ^s_j e_j , ∇_y η^f ⟩ = 0.Substuting the ansatz ψ^f(x,y) ρ(x)σ_3 { T_kl(y) e_k ⊗ e_l}ψ^s_j e_j = ρ(x) σ_3 [T_11(y)T_12(y);T_21(y) T_22 (y) ]ψ^s(x),with T_kl∈ H^1_(Y; ^2), 1 ≤ k,l ≤ 2, into (<ref>), and noticing that ⟨β (∇σ_3) ψ^s, ∇_y η^f ⟩ = 0 by integration by parts and periodicity, we obtain⟨ρ(x) {(β∇_yT_kl(y) + Φ_kl(y)) e_k ⊗ e_l }ψ_j^s e_j, ∇_y η^f ⟩ = 0,holds for any η^f ∈(^2; C^∞_(Y;^2)). In the above equation, choose η^f (x,y) = ρ_0(x) θ (y) where ρ_0 ∈(^2;) and θ∈ C_^∞(Y;^2). Then, Fubini's theorem implies∫_^2ρ(x) ρ_0(x) ψ_l^s (x)x ∫_Y (β∇_yT_kl(y) + Φ_kl(y) ) ∇_y θ_k y= 0.Recall from (<ref>) that _y Φ_kl = W_kl. We chooseT_kl∈ H^1_(Y; ), βΔ_yy T_kl + W_kl = 0,1 ≤ k,l≤ 2.Clearly, (<ref>) is well-posed.* We now derive the effective problem. In (<ref>), let η^f = 0. Then,⟨( D ·σ) ψ^s + ( D_y ·σ) ψ^f + m σ_3 ψ^s - z ψ^s, η^s ⟩- ⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ], ∇η^s⟩ - ∫_^2 × Yρ(x) Φ_kl( y ) ( η_k^s( ∇ψ_l^s + ∇_y ψ_l^f )+ ψ_l^s∇η_k^s) xy - ∫_^2 × Y∇ρ(x) ·Φ_kl( y ) ψ_l^s η_k^s xy= ⟨ f, η^s ⟩.Note that ⟨ (D_y ·σ) ψ^f, η^s ⟩ = 0 by (<ref>) and periodicity. Moreover, recall that Φ_kl(y) = ∇_y ϕ_kl(y) for some ϕ_kl∈ H^2_(Y), so ∫_Y Φ_kl(y)y =0. Thus (<ref>) becomes⟨( D ·σ) ψ^s+ m σ_3 ψ^s - z ψ^s, η^s ⟩- ⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ], ∇η^s⟩ - ∫_^2 × Yρ(x) Φ_kl( y )η_k^s∇_y ψ_l^f xy= ⟨ f, η^s ⟩.We obtain by (<ref>), integration by parts and periodicity that⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ψ^f ], ∇η^s⟩=⟨β[ (∇σ_3) ψ^s +(∇_y σ_3) ρ(x) σ_3 T (y) ψ^s(x) ], ∇η^s⟩= ⟨β( ∇σ_3 ) ψ^s, ∇η^s ⟩.Moreover, - ∫_^2 × Yρ(x) Φ_kl( y ) η_k^s(x)∇_y ψ_l^f(x,y) xy=- ∫_^2ρ(x)η_k^s(x)∫_Y Φ_kl( y ) ∇_y ψ_l^f(x,y) yx= ∫_^2ρ(x)η_k^s(x)∫_Y ψ_l^f(x,y) Φ_kl(y)yx= ∫_^2ρ(x)η_k^s(x)∫_Y ψ_l^f(x,y) W_kl(y)yx= ∫_^2ρ(x)η_k^s(x)∫_Y ( ψ^f(x,y) e_l ) W_kl(y)yx= ∫_^2ρ(x)η_k^s(x)∫_Y (( ρ(x)σ_3 { T_nm(y) e_n ⊗ e_m}ψ^s_j(x) e_j ) e_l ) W_kl(y)yx= ∫_^2ρ^2(x)η_k^s(x)∫_Y (( σ_3 { T_nm(y) ψ^s_m(x) e_n }) e_l ) W_kl(y)yx= ∫_^2ρ^2(x)η_k^s(x)∫_Y (( ∑_n = 1^2 (-1)^n-1T_nm(y) ψ^s_m(x) e_n) e_l ) W_kl(y)yx= ∫_^2ρ^2(x)η_k^s(x)∫_Y∑_n = 1^2 (-1)^n-1T_nm(y) ψ^s_m(x)W_kn(y)yx= ∫_^2ρ^2(x)η_k^s(x)ψ^s_m(x) ∫_Y∑_n = 1^2 (-1)^n-1T_nm(y) W_kn(y)yx= β∫_^2ρ^2(x) τψ^s (x)·η^s(x) x,with τ∈^2 × 2 such that τ_km1/β∫_Y ∑_n=1^2 (-1)^n-1 W_kn (y) T_nm(y)y.Clearly, τ= 1/β∫_Y W (y) σ_3 T(y)y. From (<ref>) and integration by parts, we obtainτ_km = 1/β∫_Y ∑_n=1^2 (-1)^n-1 (-βΔ_yy T_kn(y)) T_nm (y)y =∫_Y ∑_n=1^2 (-1)^n-1∇_y T_kn(y) ∇_y T_nm (y)y,which implies (<ref>). From (<ref>), (<ref>), and (<ref>), ⟨( D ·σ) ψ^s+ m σ_3 ψ^s - z ψ^s, η^s ⟩ - ⟨β( ∇σ_3 ) ψ^s, ∇η^s ⟩ + ⟨βρ^2 τψ^s, η^s ⟩= ⟨ f, η^s ⟩,holds for all η^s ∈(^2;^2). Therefore, [ D ·σ + (m+βΔ) σ_3 - z + βρ^2 τ] ψ^s = f,or equivalently, (^0 - z) ψ^s = f.By a similar argument as in <ref>, we have ^0 is a self-adjoint operator in L^2(^2;^2) with domain (^0) = H^2(^2;^2). Therefore, the above homogenized equation has a unique solution ψ^s ∈ H^2(^2;^2), and thus, the full sequence ψ^ε converges to ψ^s. By (<ref>),ψ^s_H^1≤lim sup_ε→ 0ψ^ε,z_H^1≤ Cz^-1f_L^2.Taking Fourier transform of (^0-z)ψ^s,z = f, we obtain ( ξ·σ + (m+βξ^2) σ_3 )ψ^s,z + β( ρ^2 τψ^s,z)- z ψ^s,z = f̂.Multiplying both side by σ_3 and rearranging terms, we getβξ^2 ψ^s,z≤ C ( f̂ + z + mψ^s,z + ξψ^s,z + β( ρ^2 τψ^s,z)).Square both sides, use Cauchy-Schwarz, integrate over ^2, then apply Plancherel theorem, to conclude that:β^2 ξ^2 ψ^s,z^2≤ C ( f̂^2 + z +m^2 ψ^s,z^2 + ξψ^s,z^2 + β^2( ρ^2 τψ^s,z)^2 )≤ C ( f^2 + z +m^2 ψ^s,z^2 + ψ^s,z_H^1^2 + β^2ρ^2 τψ^s,z^2 )≤ C ( f^2 + z +m^2 ψ^s,z^2 + ψ^s,z_H^1^2 + β^2ρ^2ψ^s,z^2 ),where in the last step, we use (<ref>), (<ref>) and elliptic regularity <cit.> to have τ≤ C ( β, W_C^0,α). Now the fact that ρ∈ L^∞ and (<ref>) imply β^2 ξ^2 ψ^s,z^2 ≤ Cz^-2f^2,which leads to (<ref>).Finally, to prove (<ref>), we multiply (<ref>) by ξ, then apply the same argument as above. * We first consider the case z ∈ [-λ,λ] + i ([-γ,γ] ∖{ 0 }) where λ, γ are defined in <ref>, so that <ref> can be used freely. Since the dependence of solutions on z is not important for the following argument, we drop the superscript ^z to lighten the notation. Let x = y/ε. Then ∂_i = ∂_x_i + 1/ε∂_y_i.Define^ε ^ε - z, ^0βΔ_yy, ^1 D_y ·σ + 2β∇_x ·∇_y σ_3 + ρ(x)W(y), ^2 D_x ·σ + (m + βΔ_xx) σ_3 - z. * We first assume f ∈ H^1(^2;^2). So, from <ref>, ψ^s ∈ H^3(^2;^2).Recall from (<ref>) that ψ^f = ρ(x) σ_3T(y) ψ^s(x).Consider the equation ^0 ψ^r = f - ^1 ψ^f - ^2 ψ^s,with unknown ψ^r ∈ L^2(^2; H^1_(Y)/). From (<ref>)–(<ref>),f - ^2 ψ^s = f - ( D_x ·σ + (m + βΔ_xx) σ_3 - z ) ψ^s = βρ^2 τψ^s.Thus (<ref>) becomes ^0 ψ^r = βρ^2(x) τψ^s(x) - ( D_y ·σ+ 2β∇_x ·∇_y σ_3 + ρ(x)W(y) ) [ ρ(x) σ_3 T(y) ψ^s(x) ].Observe that the right-hand side of (<ref>) is of the form ∑_i=1^6 h_i(x)k_i(y)with h_i_H^2 + k_i_L^∞≤ C (1+ψ^s_H^3)(1 +ρ_W^2,∞ + ρ_W^1,∞ + W_C^0,α)^2,where C is independent of ε. The exact formulae of h_i, k_i are not important, since we are only interested in their regularity. By choosing the ansatzψ^r(x,y) = ∑_i=1^6 h_i(x) ψ^r,i (y)and solving for the solutions ψ^r,i(y) ∈ H^1_(Y)/ of cell problems ^0 ψ^r,i = k_i, we conclude that (<ref>) has a unique solution ψ^r ∈ H^2(^2;W^1,∞_(Y;^2)/) (note that ψ^r ∈ H^2 because ψ^s ∈ H^3).* Let Z^ε(x) ψ^ε - ( ψ^s + εψ^f + ε^2 ψ^r ) ( x, x/ε).Then, ^ε Z^ε = ^εψ^ε - ^ε( ψ^s + εψ^f + ε^2 ψ^r )= f - ( 1/ε^2^0 + 1/ε^1 + ^2 ) ( ψ^s + εψ^f + ε^2 ψ^r )= [ f - (^0 ψ^r + ^1 ψ^f + ^2 ψ^s) ] - 1/ε^2^0 ψ^s - 1/ε( ^0ψ^f + ^1 ψ^s )- ε( ^1 ψ^r + ^2 ψ^f ) - ε^2 ^2 ψ^r= - ε( ^1 ψ^r + ^2 ψ^f ) - ε^2 ^2 ψ^r.The first three terms vanish because of (<ref>), the fact that ψ^s is independent of y, and the cell problem (<ref>). Therefore, Z^ε satisfies ^ε Z^ε = - ε( ^1 ψ^r + ^2 ψ^f ) - ε^2 ^2 ψ^r.From (<ref>), (<ref>), (<ref>), and (<ref>), we conclude that ^1 ψ^r +^2 ψ^f - ε^2 ψ^r_L^2≤ C (1+ψ^s_H^2 + εψ^s_H^3)(1 +ρ_W^2,∞ + ρ_W^1,∞ + W_C^0,α)^2.Applying (<ref>) to (<ref>), we obtain Z^ε_H^1 ≤ Cz^-1 - ε( ^1 ψ^r + ^2 ψ^f ) - ε^2 ^2 ψ^r ≤ Cz^-1ε(1+ψ^s_H^2+ εψ^s_H^3) (1 +ρ_W^2,∞ + ρ_W^1,∞ + W_C^0,α)^2≤ Cz^-1ε(1+ψ^s_H^2+ εψ^s_H^3)≤ Cz^-2ε(1+f + εf_H^1),where we have used <ref> in the last estimate. When f≤ 1, we obtain Z^ε_H^1≤ Cz^-2ε(2 + εf_H^1),solim sup_ε→ 0( sup_z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 })Z^ε_H^1/ z^-2ε) ≤lim sup_ε→ 0 C(2 + εf_H^1)≤ 2C.Thus, there exists ε_0' > 0 depending on λ,m, β, ρ_W^1,∞, ρ_W^2,∞, and W_C^0,α such thatsup_ε∈ (0,ε_0') z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 })Z^ε_H^1/ z^-2ε =sup_ε∈ (0,ε_0')( sup_z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 })Z^ε_H^1/ z^-2ε)≤ 2C + 1,or Z^ε_H^1≤ Cz^-2ε,for any ε∈ (0,ε_0'), z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 }), and f ∈ H^1(^2;^2) with f = 1. * Since ψ^ε - (ψ^s + εψ^f + ε^2 ψ^r)≤Z^ε_H^1, from (<ref>), we haveψ^ε - ψ^s ≤ Cz^-2ε + εψ^f + ε^2 ψ^r.From (<ref>), (<ref>), and (<ref>), there exists C = C (λ, m, β, ρ_W^1,∞, W_C^0,α) > 0 such that ψ^f≤ Cz^-1f = Cz^-1 and ψ^r≤ Cz^-1f = Cz^-1. Therefore, ψ^ε - ψ^s≤ Cz^-2ε + ε z^-1 + ε^2z^-1.Without loss of generality, we assume 0 < ε_0' ≤ 1 and 0 < γ≤ 1 so z ∈ [-γ,γ] ∖{ 0 } implies z^-2≥ z^-1. Hence, ψ^ε - ψ^s≤ Cz^-2ε,for any ε∈ (0,ε_0') and f ∈ H^1(^2;^2) with f = 1. Because ψ^ε = (^ε-z )^-1 f, ψ^s = (^0 - z)^-1f,we conclude from (<ref>) that (^ε-z)^-1 f- (^0 - z)^-1f≤ C ε z^-2,for any ε∈ (0,ε_0') and f ∈ H^1(^2;^2) with f = 1. For 0g ∈ H^1(^2;^2), let f = g/g. Then, the above estimate implies(^ε-z)^-1 g- (^0 - z)^-1g≤ C ε z^-2g,for any ε∈ (0,ε_0'), z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 }) and g ∈ H^1(^2;^2). Since H^1(^2;^2) is dense in L^2(^2;^2), we conclude that (<ref>) also holds for g ∈ L^2(^2;^2). Therefore, (^ε- z )^-1 - (^0 - z )^-1_L^2 → L^2≤C ε z^-2,for any ε∈ (0,ε_0'), z ∈ [-λ,λ] + i ( [-γ,γ]∖{ 0 }).* We now consider z ∈ [-λ, λ ] + i (∖{ 0 }). By <cit.> and (<ref>), we obtain (^ε- z )^-1 - (^0 - z )^-1_L^2 → L^2= γ^-1(γ^-1^ε-γ^-1 z )^-1 - (γ^-1^0 - γ^-1z )^-1_L^2 → L^2≤γ^-19 ( 1 + γ^-1 z^2 )/γ^-1 z^2(γ^-1^ε + i)^-1 - (γ^-1^0 + i)^-1=9 ( γ^2 +z^2 )/ z^2( ^ε + iγ)^-1 - ( ^0 + iγ)^-1≤ 9 ( (λ^2 + γ^2) z^-2 + 1) C εγ^-2≤ C ( 1+z^-2) ε.Estimate (<ref>) implies the rate of convergence ψ^ε - ψ^s - εψ^f_H^1≤ C ε.Notethat our problem does not have a boundary layer effect, so we obtain the rate O(ε) instead of O(ε^1/2) as in classical results <cit.>. Using the Helffer-Sjöstrand formula (as used, e.g., in <cit.>), we obtainU(^ε) - U(^l) = S(^ε) - S(^l) = -1/π∫_∂̅S̃(z) ( (z-^ε)^-1 - (z-^l)^-1) dzwith dz Lebesgue measure onand S̃ an almost analytic extension of S chosen so that ∂̅S̃(z) is compactly supported in [-λ,λ] + i and such that | z|^-N|∂̅S̃(z)|≤ C_N is bounded uniformly on that support for each N≥0. Here, the superscript l stands for either limit l=0 or l=∞.From <ref>, we have(z-^ε)^-1 - (z-^0)^-1≤ C (1+| z|^-2) ε, 0 < ε≤ε_0'and from the resolvent identity (note that ^ε = ^∞ + 1/ερ(x) W ( x/ε)) and the resolvent estimate,(z-^ε)^-1 - (z-^∞)^-1≤ C | z|^-21/ε, 0<ε_1' ≤ε.Combined with the Helffer-Sjöstrand formula (<ref>), this implies that U(^ε) - U(^0)≤ C ε, U(^ε) - U(^∞)≤ C 1/ε,for ε sufficiently small for the first estimate and sufficiently large for the second estimate.From their definition, T^ε- T^l= P(x) ( U(^ε) - U(^l) ) P(x).Since the set (L^2(^2;^2),L^2(^2;^2)) of Fredholm operators from L^2(^2;^2) to L^2(^2;^2) is an open set in the space of bounded operators (L^2(^2;^2),L^2(^2;^2)), we obtain from (<ref>) and (<ref>) that T^ε is Fredholm for ε sufficient large and ε sufficient small. Moreover, since the (L^2(^2;^2),L^2(^2;^2)) → is a continuous function, which is locally constant on the connected components of (L^2(^2;^2),L^2(^2;^2)) (cf. e.g. <cit.>), we obtain (<ref>).§ ACKNOWLEDGMENTGB's work was supported in part by the US National Science Foundation Grants DMS-2306411 and DMS-1908736. siam | http://arxiv.org/abs/2311.15678v1 | {
"authors": [
"Guillaume Bal",
"Thuyen Dang"
],
"categories": [
"math.AP",
"math-ph",
"math.MP",
"35B20, 35B27, 35B35, 35B40, 47A53"
],
"primary_category": "math.AP",
"published": "20231127100826",
"title": "Topological Anderson Insulators by homogenization theory"
} |
On the Kepler problem on the Heisenberg group Sergey Basalaev, Sergei Agapov The work of the second author was performed according to the Government research assignment for IM SB RAS, project FWNF-2022-0004.===================================================================================================================================================================== We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” whose potential is given by the fundamental solution of the sub-Laplacian. We find three independent first integrals of the system and show that its bounded trajectories of the system are wound up around certain surfaces of the fourth order.Keywords: Heisenberg group, Kepler problem, nonholonomic dynamics, almost Poisson bracket, first integral.2010 Mathematics Subject Classification: 37N05, 53C17, 70F25, 37J60.§ INTRODUCTION How would a planet move around the Sun on the Heisenberg group? While studying the problem we found that the authors of the paper <cit.> aim to answer this very question. However, a known feature of nonholonomic mechanics (see e. g. <cit.>) is that the variatonal problem (control, geodesics, how to move from A to B) and the dynamics problem (how does it move on its own?) are generally not equivalent. Indeed, for instance, in the geodesic problem on the Heisenberg group, to any initial velocity there corresponds a one-parametric family of geodesics. On the other hand the dynamics is uniquely determined by an initial position and a velocity so it can't be any geodesic (the actual solutions in in this case are what is known in nonholonomic geometry as the “straightest” lines). It seems to us that the problem actually studied in <cit.> is the variatonal one — how to move efficiently on Heisenberg group in the presence of a gravitational field. Here, we aim to solve the dynamics problem instead.We consider the Heisenberg group with the left-invariant sub-Riemannian metric and a fixed “sun” at the origin. The potential is given by the fundamental solution of the sub-Laplacian — a generalization of the Laplace–Beltrami operator to the sub-Riemannian manifolds. Traditionally, to derive the non-holonomic equations of motion the Lagrange–d'Alembert principle is used. In Section <ref> we remind how the equations of motion can be translated to the form that uses the intrinsic structure of nonholonomic distribution. This allows one to use Hamiltonian language best suited for finding integrals of the system. In Section <ref> we apply this to study the Kepler problem on the Heisenberg group and find its first integrals. In contrast to the 6-dimensional variational problem which is not Liouville integrable (proved in <cit.>), the dynamics problem is 5-dimensional and turns out to have at least three independent first integrals. This allows us to rather qualitatively describe the geometry of trajectories of the system. In particular, a typical trajectory of the system winds up around the surface of order 4 which we found explicitly. Due to the nonholonomic constraint the surface in the Heisenberg group uniquely defines the trajectory by its starting point. We also describe a few special trajectories.In relation to our research we note that the Kepler problem on the Riemannian manifolds was studied extensively starting from works of Lobachevsky <cit.> in hyperbolic space and Serret <cit.> on the sphere. The survey of related works in the spaces of constant curvature may be found in <cit.>. The aforementioned paper <cit.> has a few followups <cit.> all of which seem to address the variational problem.§ MOTION ON SUB-RIEMANNIAN MANIFOLDSHere we derive the equations of nonholonomic dynamics in the generalized Hamiltonian form, simplified for the case considered. The general form may be found in <cit.>.Consider a mechanical system in ℝ^n with k ideal functionally independent nonintegrable constraints linear in velocities. In Lagrangian coordinates q^i, q̇^i these can be given by∑_i=1^n a^j_i(q) q̇^i = 0,j = 1, …, k.Locally the equations (<ref>) can be solved to k dependent velocities and represented in the formq̇^m+j = ∑_i=1^m f^j_i(q) q̇^i,j = 1, …, k,where m = n - k and the velocities q̇^1, …, q̇^m are assumed to be independent.Recall that for a nonholonomic system with the Lagrangian L(q, q̇, t) and the constraints (<ref>) the equations of motion are derived using the Lagrange–d'Alembert principle (see, e. g. <cit.>)d/dt∂ L/∂q̇^i - ∂ L/∂ q^i = ∑_j=1^k λ_j a^j_i,i = 1, …, n,where the Lagrange multipliers λ_j are determined in such a way that the trajectory satisfies constraints (<ref>).It may be useful, especially for problems with the constraints of form (<ref>), instead of the Lagrangian coordinates use the ones in the distribution of admissible velocities. Consider the vector fieldsX_i(q) = ∂ q_i + ∑_j=1^k f^j_i(q) ∂ q_m+j,i = 1, …, m.Then the velocity q̇ satisfies the constraints (<ref>) iff q̇ = ∑_i=1^m q̇^i X_i. Introduce the momentum 1-formP = ∑_i=1^n ∂ L/∂q̇^i dq^i.Then we can describe the dynamics by the following generalization of Euler–Lagrange equations (in what follows we denote the action of 1-form τ on the vector field X as τ⟨ X ⟩). The dynamical motion in the system with the Lagrangian L(q, q̇, t) and the constraints (<ref>) is described by the system of equationsd/dt P ⟨ X_i ⟩ = X_i L, i = 1, …, m, q̇^m+j = ∑_i=1^m f^j_i(q) q̇^i, j = 1, …, k. Introduce 1-forms of our constraintsτ^j(q) = dq^m+j - ∑_i=1^m f^j_i(q) dq^i,j = 1, …, k.Then τ^j ⟨ X_i ⟩ = 0 for i = 1, …, m and τ^j ⟨∂ q_m+l⟩ = δ^j_l for l = 1, …, k. The Lagrange–d'Alembert equations (<ref>) can be rewritten in our terms asd/dt P ⟨∂ q_i⟩ - d L ⟨∂ q_i⟩ = ∑_j=1^k λ_j τ^j ⟨∂ q_i⟩,i = 1, …, n.Then, since the expression is linear w. r. t. the term in the angle bracketsd/dt P ⟨ X_i ⟩ - d L ⟨ X_i ⟩ = ∑_j=1^k λ_j τ^j ⟨ X_i ⟩ = 0,i = 1, …, m.The sufficiency of the equations (<ref>), (<ref>) follows from the fact that we can recover Lagrange–d'Alembert equations from them. Indeed, letλ_j = d/dt P ⟨∂ q_m+j⟩ - d L ⟨∂ q_m+j⟩ = d/dt∂ L/∂q̇^m+j - ∂ L/∂ q^m+j,j = 1, …, k.This gives us the equations (<ref>) for i = m+1, …, n. Then, since ∂ q_i = X_i - ∑_j=1^k f^j_i(q) ∂ q_m+j for i = 1, …, m we haved/dt∂ L/∂q̇^i - ∂ L/∂ q^i = d/dt P ⟨ X_i ⟩ - d L ⟨ X_i ⟩ - ∑_j=1^k f^j_i ( d/dt∂ L/∂q̇^m+j - ∂ L/∂ q^m+j) = - ∑_j=1^k λ_j f^j_i.These are the equations (<ref>) for i = 1, …, m. Thus, the system of equations (<ref>), (<ref>) is equivalent to the one of (<ref>), (<ref>). Let 𝒟 be the distribution spanned by X_1, …, X_m, i. e. 𝒟_q = span { X_1(q), …, X_m(q) }. One thing to note is that for deriving equations (<ref>) for a particular system it is enough to know the Lagrangian only on 𝒟, not on the whole T ℝ^n, which allows us to immerse the problem in the sub-Riemannian setting.Recall that the (regular) sub-Riemannian structure on a smooth manifold M is given by the constant rank distribution 𝒟⊂ TM (i. e. 𝒟_x ⊂ T_x M is a subspace and 𝒟_x is independent of x) and the sub-Riemannian metric tensor ⟨· , ·⟩ on 𝒟, i. e. ⟨· , ·⟩_x is a scalar product on 𝒟_x.Let us reformulate the problem in Hamiltonian terms. The energy E(q, q̇, t) of the system is defined as usual:E = P ⟨q̇⟩ - Land satisfies d/dt E = -∂/∂ t L on trajectories of the system. Note, that the dual basis to the one of vector fields X_1, …, X_m, ∂ q_m+1, …, ∂ q_m+k consists of 1-formsdq^1, …, dq^m, τ^1, …, τ^k.In particular, dq^1, …, dq^m form the basis of 𝒟^*. Introduce the momenta p = ∑_i=1^m p_i dq^i on 𝒟^*, i. e. p ⟨ X_i ⟩ = p_i, i = 1, …, m. Assuming that q̇(p, q, t) ∈𝒟_q can be determined uniquely from the equation p ⟨q̇⟩ = P ⟨q̇⟩ let us define the generalized Hamiltonian H(p, q, t) on 𝒟^* asH(p, q, t) = p ⟨q̇⟩ - L(q, q̇, t) = ∑_i=1^m p_i q̇^i - L(q, q̇, t).For this assumption to take place it is sufficient to require that the restriction of the quadratic form ∂^2 L/∂q̇^i ∂q̇^j dq^i dq^j on 𝒟 is positive definite.Reformulating the equations (<ref>) in terms of H one obtainsThe dynamical motion in the nonholonomic system with the constraints (<ref>) and the generalized Hamiltonian H(p, q, t) on 𝒟^* is described by the system of equationsq̇^i = ∂ H/∂ p_i, ṗ_i = - X_i H, i = 1, …, m, q̇^m+j = ∑_i=1^m f^j_i(q) ∂ H/∂ p_i, j = 1, …, k. Observe that since the bases X_1, …, X_m, ∂ q^m+1, …, ∂^m+k, introduced in (<ref>), and dq̇^1, …, dq̇^m, τ^1, …, τ^k, introduced in (<ref>), are dual, for a smooth function f(q) we havedf = ∑_i=1^m df ⟨ X_i ⟩dq^i + ∑_j=1^k df ⟨∂ q^m+j⟩ τ^j = ∑_i=1^m X_i f dq^i + ∑_j=1^k ∂ f/∂ q^m+j τ^j.ThendL = ∑_i=1^m X_i L dq^i + ∑_j=1^k ∂ L/∂ q^m+j τ^j + ∑_i=1^n ∂ L/∂q̇^i dq̇^i + ∂ L/∂ t dt = ∑_i=1^m X_i L dq^i + ∑_j=1^k ∂ L/∂ q^m+j τ^j + ∑_i=1^n P ⟨∂ q^i ⟩ dq̇^i + ∂ L/∂ t dt.Further, restricting L on 𝒟×ℝ with coordinates q^1, …, q^n, q̇^1, …, q̇^m, t, i. e. setting q̇ = q̇^1 X_1 + … + q̇^m X_m, we obtaindL|_T𝒟 = ∑_i=1^m X_i L dq^i + ∑_j=1^k ∂ L/∂ q^m+j τ^j + ∑_i=1^m P ⟨ X_i ⟩ dq̇^i + ∂ L/∂ t dt.Now, for H = p ⟨q̇⟩ - L(q, q̇, t)|_𝒟 with q̇ = q̇(p, q, t) we havedH = ∑_i=1^m ( p_i - P ⟨ X_i ⟩) dq̇^i + ∑_i=1^m q̇^i dp_i - ∑_i=1^m X_i L dq^i - ∑_j=1^k ∂ L/∂ q^m+j τ^j - ∂ L/∂ t dt.If q̇ satisfies p ⟨q̇⟩ = P ⟨q̇⟩ then the first sum vanishes. Finally, since for the function H(p, q, t)dH = ∑_i=1^m ∂ H/∂ p_i dp_i + ∑_i=1^m X_i H dq^i + ∑_j=1^k ∂ H/∂ q^m+jτ^j + ∂ H/∂ t dt,the equations (<ref>) follow from (<ref>). Observe that for time independent system H is a first integral of (<ref>), (<ref>). We can define almost Poisson bracket (see e. g. <cit.>) on 𝒟^* as{ F, H } = ∑_i=1^m ( ∂ H/∂ p_i X_i F - ∂ F/∂ p_i X_i H ).It retains all the properties of Poisson bracket but the Jacobi identity. Nevertheless, since Ḟ = { F, H } it follows that F is an integral of (<ref>), (<ref>) iff { F, H }≡ 0.§ MOTION IN A POTENTIAL FIELD ON THE HEISENBERG GROUPRecall that the Heisenberg group ℍ^1 = (ℝ^3, ·, δ_λ) is a homogeneous group with the group operation(x, y, z) · (x', y', z') = ( x + x', y + y', z + z' + xy' - x'y/2),and the one-parametric family of anisotropic dilatationsδ_λ(x, y, z) = (λ x, λ y, λ^2 z), λ > 0.Its Lie algebra 𝔥^1 of left-invariant vector fields has the basisX = ∂_x - y/2∂_z,Y = ∂_y + x/2∂_z,Z = [X, Y] = ∂_z.The dual basis of left-invariant 1-forms isdx,dy, τ = dz + y dx - x dy/2.The horizontal distribution 𝒟 = span{ X, Y }⊂ T ℍ^1 is totally nonholonomic. The form τ is its annihilator. The sub-Riemannian structure on ℍ^1 is given by the quadratic form ⟨·, ·⟩ on 𝒟. We choose the one such that X, Y form the orthonormal basis:ds^2 = dx^2 + dy^2.While this quadratic form is degenerate on T ℍ^1 it is positive definite on 𝒟. For the mechanical motion with the kinetic energy T = 1/2 ds^2⟨q̇⟩ = 1/2 (ẋ^2 + ẏ^2) and the potential energy U = U(x, y, z) one has, as usual, the Lagrangian L = T - U. By Proposition <ref> we can translate equations to the Hamiltonian form where the Hamiltonian H on 𝒟^* takes the form H = 2 T - L = T + U, i. e.H(x, y, z, p_X, p_Y) = p_X^2 + p_Y^2/2 + U(x, y, z).We are interested in the gravitational potential which in ℝ^n is given by a fundamental solution of the Laplacian. The analogue of Laplace–Beltrami operator on the Heisenberg group is the operator Δ_H= X^2 + Y^2. Its fundamental solution[In the cited paper <cit.> the potential U has the term z^2/16 instead of 16z^2. One can check that 16 is the correct coefficient since only in this case Δ_H U = 0 away from the origin.] is found in <cit.>:U = - k/ρ^2, where ρ(x, y, z) = ((x^2 + y^2)^2 + 16z^2)^1/4and k > 0 is some constant. Since both the distribution and the potential have a rotational symmetry around Oz it is natural to make the cylindrical coordinate change x = r cosθ, y = r sinθ. The basis of 𝒟 may be given by vector fieldsR= -rcosθX+ rsinθY = ∂_r, S= - r sinθX + r cosθY = ∂_θ + r^22∂_z.Duals to the basis R, S, ∂_z are dr, dθ, τ = dz - r^2/2 dθ and for the momenta we havep_X dx + p_Y dy = (p_X cosθ + p_Y sinθ) dr + r (p_Ycosθ - p_Xsinθ) dθ = p_R dr + p_S dθ.It follows that T = p_R^2 + p_S^2 / r^2/2 and the Hamiltonian becomesH(r, θ, z, p_R, p_S) = p_R^2 + 1/r^2 p_S^2/2 - k/(r^4 + 16 z^2)^1/2.Since constraints in the new coordinates still have the form (<ref>) we may apply Proposition <ref> to derive the equations of motion:ṙ = ∂ H/∂ p_R = p_R,ṗ_R= -R H = p_S^2/r^3 - 2 k r^3/(r^4 + 16 z^2)^3/2,θ̇ = ∂ H/∂ p_S = p_S/r^2,ṗ_S= -S H = -8 k r^2 z/(r^4 + 16 z^2)^3/2,ż = r^2/2∂ H/∂ p_S = p_S/2. If H < 0 the solutions of (<ref>) are bounded with √(r^4 + 16 z^2)≤k/|H|.This easily follows from the inequality H + k / √(r^4 + 16 z^2) = p_R^2 + 1/r^2 p_S^2/2≥ 0. In what follows we search for the additional first integrals of the system.The system (<ref>) does not admit any linear in momenta first integrals. This statement can be checked by straightforward calculations. We skip the details. However, it turns out that there are a few quadratic integrals in addition to the Hamiltonian H.The system (<ref>) admits quadratic in momenta first integralsF_1= -( p_Rp_Sr-2p_R^2z+2p_S^2z/r^2) cos (2θ) + ( 4 p_Rp_Sz/r -p_S^2 +kr^2/√(r^4+16z^2)) sin (2θ), F_2= - ( p_Rp_Sr-2p_R^2z+2p_S^2z/r^2) sin (2θ) + ( 4 p_Rp_Sz/r-p_S^2+kr^2/√(r^4+16z^2)) cos (2θ), F_3= (2 z p_R - r p_S)^2 + 4 z^2 ( p_S^2/r^2 + 2k/√(r^4+16z^2)).Any three of H, F_1, F_2, F_3 are functionally independent a. e. wherein all of them satisfy the relationF_1^2 + F_2^2 = 2 H F_3 + k^2.This theorem can be verified by straightforward calculations. The method we used to construct these integrals is described in Appendix <ref>.Knowing three independent first integrals allows us to derive the equation of the surface (in coordinates (r, θ, z)) in which the trajectories lie. To do that we introduce two more conserved quantities J ≥ 0 and in the case J > 0 also θ_0 ∈ [0, π) such thatF_1 = J sin(2θ_0),F_2 = J cos(2θ_0).As main parameters we choose H, F_3 that do not depend on the angle and the phase offset θ_0 that captures the rotational symmetry of the problem. Note, that from the definition F_3 ≥ 0 since k > 0, and 2 H F_3 + k^2 = J^2 ≥ 0. Therefore we have one general case and two cases that might require special handling: * The general case F_3 > 0, J > 0. Then H > - k^2/2 F_3 and θ_0 ∈ [0, π) is defined. * The minimum energy case F_3 > 0, J = 0. Then H = H_min = - k^2/2 F_3, θ_0 is undefined. * The degenerate case F_3 = 0. In this case J = k, θ_0 ∈ [0, π) is defined and H is unbounded.All trajectories of the system (<ref>) with the fixed values of the first integrals H, F_3, θ_0 lie on the surface which in the general case F_3 > 0, J > 0 satisfies the equationF_3 = 8 z^2 H + k √(r^4 + 16 z^2) - √(k^2 + 2H F_3)r^2 cos( 2 (θ - θ_0) ). In the minimum energy case F_3 > 0, J = 0 the surface becomes an ellipsoid of revolution4 k^2 z^2 + k F_3 r^2 = F_3^2. In the degenerate case F_3 = 0 the surface degenerates to the straight horizontal line passing through the originz = 0, θ = θ_0 π. Observe, that we can write F_3 asF_3 = 8z^2 H + k √(r^4 + 16 z^2) - r^2 (- p_S^2 + 4 p_R p_S z/r + k r^2/√(r^4 + 16 z^2)).From the expressions of F_1 and F_2 we have(- p_S^2 + 4 p_R p_S z/r + k r^2/√(r^4 + 16 z^2)) = F_1 sin(2θ) + F_2 cos(2θ).Therefore,F_3 = 8z^2 H + k √(r^4 + 16 z^2) - r^2 ( F_1 sin(2θ) + F_2 cos(2θ) ).Let F_3 > 0 and J > 0. We have F_1 sin(2θ) + F_2 cos(2θ) = J cos(2(θ - θ_0)) and (<ref>) follows.Now, let F_3 > 0 and J = 0. In this case F_1 = F_2 = 0 and the last term in (<ref>) vanishes. Since H = -k^2/2F_3 in this case, (<ref>) becomesF_3^2 + 4 k^2 z^2 = k F_3 √(r^4 + 16 z^2).Squaring and simplifying it we obtain(F_3^2 - 4 k^2 z^2)^2 = k^2 F_3^2 r^4.By Theorem <ref> for the trajectories of the system √(r^4 + 16 z^2)≤k/|H| = 2 F_3/k. Therefore,F_3 ≥k2√(r^4 + 16 z^2)≥k2√(16 z^2) = 2 k z.Now (<ref>) follows if we take the square root of (<ref>).Lastly, F_3 = 0 implies z = 0 and J = k > 0. The restriction of (<ref>) on z = 0 becomes0 = k r^2 - k r^2 cos (2(θ - θ_0)).This gives us either r = 0 (the origin) or θ = θ_0 π.Solving the equation (<ref>) for the square root √(r^4 + z^2) and then squaring it we obtain the following equationk^2 (r^4 + 16 z^2) = ( F_3 - 8 z^2 H + √(k^2 + 2H F_3)r^2 cos( 2 (θ - θ_0) ) )^2,or in Cartesian coordinatesk^2 ((x^2 + y^2)^2 + 16 z^2) = ( F_3 - 8 z^2 H + √(k^2 + 2H F_3)(cos(2θ_0) (x^2 - y^2) + 2 sin(2θ_0) xy) )^2.We see that this is an equation of the fourth order. However, its solution is a branched surface and only one of its branches is the solution to the original equation, i. e. the equation is quadratic in z^2 but only one of its two roots solves (<ref>). Examples of the surfaces corresponding to the cases H = 0 and H < 0 may be seen on Fig. <ref>. Next we note a few properties of the surfaces obtained.In the non-degenerate case F_3 > 0 the surfaces described in Theorem <ref> have the following properties.* The surface is topologically * a sphere in the case H < 0; * a cylinder in the case H ≥ 0. * The surface has reflection symmetry in the plane z = 0 and * in the case J > 0 it has two more planes of symmetry θ = θ_0 and θ = θ_0 + π/2; * in the case J = 0 it is the surface of revolution around r = 0. * The trace of the surface on the plane z = 0 is a quadratic curve:* in the case H < 0 it is an ellipse with the semiaxes √(F_3/k-J) and √(F_3/k+J); * in the case H = 0 it is two parallel lines at the distance √(F_3/k) from the origin; * in the case H > 0 it is a hyperbola with the semiaxis √(F_3/k+J). The properties are straightforward and easy to check.A smooth surface S ⊂ℍ^1 is transversal to the horizontal distribution 𝒟 at almost all points, i. e. T_x S ∩𝒟_x is one-dimensional for a. e. x ∈ S. Therefore, the trajectory of the system is rather uniquely defined by a starting point on a surface, with the only possible exception being when the solution arrives at the point tangent to 𝒟 with zero velocity. An example of a bounded trajectory (H < 0) which we belive to be a typical one is presented in Fig. <ref>. Next we describe special solutions corresponding to the degenerate cases. The only trajectories of (<ref>) passing through the origin are straight lines in the plane z = 0. In this case θ = const and r(t) satisfies the equation ṙ^2/2 = H + k/r^2. These solutions correspond to the degenerate case F_3 = 0.From Corollary <ref> the trajectories may pass through the origin only in the case F_3 = 0, i. e. only if z ≡ 0 and p_S ≡ 0. Then θ̇ = 0 and the Hamiltonian becomes H = ṙ^2/2 - k/r^2. Therefore, the conserved quantity F_3 serves as a kind of angular/vertical momentum. Two more special solutions appear in the minimal energy case J = 0.The only stationary solutions of (<ref>) are points on Oz:r = 0,z = ±k/4H.These solutions correspond to the minimal energy case J = 0.Indeed, outside of the axis r = 0 the stationary solution must satisfy p_S ≡ 0. But in this case ṗ_R is negative and the solution is non-stationary. For the stationary solution on Oz we have H = - k/√(16 z^2) and F_3 =8 k z^2/√(16z^2). Therefore J^2 = k^2 + 2 H F_3 = 0.Let J = 0 and z_0 = k/4|H|. The trajectories of non-stationary solutions to (<ref>) are the curves monotone in z and θ such that being parameterized by z they have the formr(z) = ( 2 z_0^2 - z^2/z_0)^1/2, θ(z) = 1/2logz_0 + z/z_0 - z + θ(0),|z| ≤ z_0.These solutions connect the stationary points (0,0,± z_0) and take infinite time to approach them, i. e. t(z) →±∞ as z →± z_0.Note that F_3 ≥ 0. Therefore, J = √(k^2 + 2HF_3) = 0 implies H < 0. Hence, H = - k/4 z_0 and F_3 = -k^2/2 H = 2 k z_0. The surface (<ref>) in this case is an ellipsoid of revolutionz_0 r^2/2 + z^2/z_0^2 = 1.From this equation we find the dependence r(z). Next, from dθ/dz = θ̇/ż = 2/r^2 we also find the expression of θ(z) in a closed form. This gives us a family of curves described in the statement of the theorem. Take any such curve. From the expression of H we haveṙ^2 + 4 ż^2/r^2/2 = p_R^2 + p_S^2/r^2/2 = H + k/√(r^4 + 16 z^2) = k (z_0^2 - z^2)/4 z_0 (z_0^2 + z^2) > 0for all points except the poles. Therefore, the velocity along the curve is nonzero except on the endpoints. Hence the solution restricted to a curve is monotone in z. From the equation of the surface we have0 = z_0 r ṙ + 2 z ż/z_0^2.This together with (<ref>) yields the equation on z:ż^2 = k z_0^4 (z_0^2 - z^2)^2/4 (z^2 + z_0^2) (z^2 + z_0^6).Choosing the solution increasing in z we obtaint(z_1) = ∫_0^z_1dz/ż = ∫_0^z_12 √((z^2 + z_0^2) (z^2 + z_0^6))/√(k) z_0^2 (z_0^2 - z^2)dzwhich diverges as z_1 →± z_0. The theorem is proved. § CONCLUSION In conclusion we see that the variational problem and the dynamics problem on the Heisenberg group are vastly different. While the first is Hamiltonian but non-integrable in Liouville sense, the second one, being non-Hamiltonian has at least three first integrals. Both problems are interesting but provide a different insight into the nonholonomic world.Acknowledgements: While the problem was mainly considered by the first author he is very grateful to second author for deriving the first integrals (Proposition <ref>, Theorem <ref> and Appendix) which significantly progressed the research and for overall critical comments.The images were prepared using GNUPlot and Maxima free software.§ DERIVATION OF QUADRATIC FIRST INTEGRALSBy definition any first integral F of (<ref>) must satisfy the following relation:dF/dt = ∂ F/∂ r p_R + ∂ F/∂θp_S/r^2 + ∂ F/∂ zp_S/2 + ∂ F/∂ p_R( p_S^2/r^3-2kr^3/(r^4+16z^2)^3/2) - ∂ F/∂ p_S8kr^2z/(r^4+16z^2)^3/2 = 0.It is quite natural to search for the first integrals of (<ref>) having the form of non-homogeneous polynomials in momenta.We shall search for the quadratic integral of (<ref>) in the form:F = a p_R^2+d p_Rp_S+b p_S^2 + f p_R + g p_S+h,where all the coefficients are unknown functions which depend on r, θ, z. Writing down the condition (<ref>) for such an integral F, we obtain the system of PDEs which splits into two parts: the first one contains relations between the unknown functions a, b, d, h only, the second one is between f and g. As in Proposition <ref>, it is easy to check that if F is the first integral, then both functions f and g must vanish identically. So we start our analysis with an integral of the formF = a(r,θ,z) p_R^2 + d(r,θ,z) p_R p_S + b(r,θ,z) p_S^2 + h(r,θ,z).The condition (<ref>) implies:a_r=0, 2a_θ+r^2(a_z+2d_r)=0, 4a+r^3d_z+2r(d_θ+r^2b_r)=0, 2d+r^3b_z+2rb_θ=0, 2r^3(r^4+16z^2)^3/2h_r-8kr^6a-16kr^5zd=0, r(r^4+16z^2)^3/2(r^2h_z+2h_θ) -4kr^6d-32kr^5zb=0.Integrating the equations (<ref>)–(<ref>) successively, we obtaina(r,θ,z) = α(θ,z),d(r,θ,z) = γ(θ,z) - 1/2 r α_z + α_θ/r, b(r,θ,z) = α/r^2 + α_θθ/2r^2 + r^2 α_zz/8 + γ_θ/r - rγ_z/2 + ω(θ,z),where α(θ,z), γ(θ,z), ω(θ,z) are arbitrary functions. Then (<ref>) takes the formα_zzzr^6 - 4γ_zzr^5 + (2α_θ zz + 8ω_z)r^4 + 4(α_θθ z + 4ω_θ)r^2 + 16(γ_θθ + γ)r + 8(α_θθθ + α_θ) = 0.This is a polynomial in r with coefficients depending on θ, z only. Since this polynomial must vanish, all its coefficients must vanish as well. This allows one to find α(θ,z), γ(θ,z), ω(θ,z) and, consequently, the coefficients a(r,θ,z), b(r,θ,z), d(r,θ,z) explicitly. We omit these long but simple calculations and skip the final form of these coefficients since they are quite cumbersome.After that we are left with two equations (<ref>), (<ref>) on the unknown function h(r,θ,z) which take the form:(r^4+16z^2)^3/2h_r + 2kr(c_8 r^2 - z(4 c_9 + z(4 c_3+c_5 r^2+4c_6z)))cos (2θ) -2kr(c_9r^2+z(4c_8+z(4c_2-c_6r^2+4c_5z))) sin (2θ) - 4kr^3(c_7-c_4z^2) -8kr^2z ((s_2+s_4z)cosθ +(s_3+s_5z)sinθ) = 0, (r^4+16z^2)^3/2(r^2h_z+2h_θ)+2c_1kr^2(r^4-16z^2) -kr^2(4c_9r^2+c_2(r^4+16z^2)-2z(-8c_8+2c_3r^2+c_5r^4+6c_6r^2z-8c_5z^2))cos (2θ) +kr^2(-4c_8r^2+c_3(r^4+16z^2)+2z(8c_9+2c_2r^2-c_6r^4+6c_5r^2z+8c_6z^2))sin (2θ) -4kr^3(s_2r^2+z(8s_3-3s_4r^2+8s_5z))cosθ +4kr^3(-s_3r^2+z(8s_2+3s_5r^2+8s_4z))sinθ-4kr^2z(8c_7+8s_1r^2+c_4(r^4+8z^2))=0.Here c_k, s_k are arbitrary constants. The equation (<ref>) can be integrated. However,the general solution h(r,θ,z) to (<ref>) is expressed in terms of elliptic integrals. We consider the simplest cases_2=s_3=s_4=s_5=0.In this case h(x,y,z) can be found from (<ref>) in terms of elementary functions as follows:h(r,θ,z) = ψ(θ,z) + k/4z √(r^4+16z^2)ϕ(r,θ,z),where ψ(θ,z) is an arbitrary function andϕ(r,θ,z) = (c_9r^2+z(4c_8+c_3r^2+c_6r^2z-4c_5z^2))cos (2θ) + (c_8r^2+z(-4c_9+c_2r^2+c_5r^2z+4c_6z^2))sin (2θ) + 8z (c_4z^2-c_7).The unknown function ψ(θ,z) should be chosen such that the relation (<ref>) holds identically. It seems that the only possible way to satisfy this requirement is to putψ(θ,z) ≡ 0,c_1=c_5=c_6=c_8=c_9=s_1=0.In this case (<ref>) is satisfied. Thus we found all the coefficients of F. Notice that F=4F-8c_7H is also the first integral of (<ref>) having the simpler form:F = a p_R^2 + d p_Rp_S + b p_S^2 + h,wherea = 2z(2c_4z-c_2cos (2θ) +c_3sin (2θ)),d=( c_2r+4c_3z/r) cos(2θ) - ( c_3x-4c_2z/r) sin(2θ) -4c_4rz,b = 1/r^2( (-c_3r^2+2c_2z) cos(2θ) -(c_2r^2+2c_3z) sin(2θ) +c_4 (r^4+4z^2) ),h = k/√(r^4+16z^2)( r^2(c_3cos (2θ)+c_2 sin (2θ))+8c_4z^2 ).Here c_2, c_3, c_4 are arbitrary constants. It is left to notice that F is linear in these constants, i. e. it has the form F=c_2F_1+c_3F_2+c_4F_3. This implies that the functionsF_1= -( p_Rp_Sr-2p_R^2z+2p_S^2z/r^2) cos (2θ) + ( -p_S^2+4 p_Rp_Sz/r+kr^2/√(r^4+16z^2)) sin (2θ), F_2= - ( p_Rp_Sr-2p_R^2z+2p_S^2z/r^2) sin (2θ) + ( -p_S^2+4 p_Rp_Sz/r+kr^2/√(r^4+16z^2)) cos (2θ), F_3= 4z^2 p_R^2 - 4rz p_Rp_S + r^4+4z^2/r^2p_S^2 + 8k z^2/√(r^4+16z^2).are also integrals of (<ref>).DPM12[B03]Bloch A. M. Bloch et al., Nonholonomic Mechanics and Control / Interdisciplinary Applied Math. 24 (2003), 501 pp.[DPM12]Diacu F. Diacu, E. Pérez-Chavela, M. Santoprete, The n-body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria // J. Nonlinear Sci. 22 (2012), 247–266, DOI: https://doi.org/10.1007/s00332-011-9116-z10.1007/s00332-011-9116-z[DS21]DS V. Dods, C. Shanbrom, Self-similarity in the Kepler–Heisenberg Problem // J. Nonlinear Sci. 31 (2021), Article 49, DOI: https://doi.org/10.1007/s00332-021-09709-110.1007/s00332-021-09709-1[F79]F G. Folland, A fundamental solution for a subellipic operator // Bulletin of the AMS 79:2 (1973), 373–376.[L1835]Lobachevsky N. I. Lobachevsky, The new foundations of geometry with full theory of parallels / Collected Works 1835–1838 2, GITTL, Moscow, 1949, 159 pp. [in Russian][MS15]SM R. Montgomery, C. Shanbrom, Keplerian Dynamics on the Heisenberg Group and Elsewhere // Fields Inst. Commun. 73 (2015), 319–342, arXiv:1212.2713 [math.DS][S1860]Serret P. J. Serret Théorie nouvelle géométrique et mécanique des lignes a double courbre / Librave de Mallet-Bachelier, Paris, 1860.[SM21]StM T. Stachowiak, A. J. Maciejewski, Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group // SIGMA 17 (2021), Article 074, arXiv:2103.10495 [math-ph], DOI: https://doi.org/10.3842/SIGMA.2021.07410.3842/SIGMA.2021.074Sergey Basalaev Novosibirsk State University, 1 Pirogova st., 630090 Novosibirsk Russia e-mail: Sergei Agapov Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090 Novosibirsk Russia e-mail: , | http://arxiv.org/abs/2311.15746v1 | {
"authors": [
"Sergey Basalaev",
"Sergei Agapov"
],
"categories": [
"math.DS",
"math-ph",
"math.MP",
"37N05, 53C17, 70F25, 37J60"
],
"primary_category": "math.DS",
"published": "20231127120542",
"title": "On the Kepler problem on the Heisenberg group"
} |
Towards complete characterization of topological insulators and superconductors: A systematic construction of topological invariants based on Atiyah-Hirzebruch spectral sequence Ken Shiozaki January 14, 2024 ================================================================================================================================================================================= In this paper, we introduce a Multimodal Large Language Model-based Generation Assistant (LLMGA), leveraging the vast reservoir of knowledge and proficiency in reasoning, comprehension, and response inherent in Large Language Models (LLMs) to assist users in image generation and editing. Diverging from existing approaches where Multimodal Large Language Models (MLLMs) generate fixed-size embeddings to control Stable Diffusion (SD), our LLMGA provides a detailed language generation prompt for precise control over SD. This not only augments LLM context understanding but also reduces noise in generation prompts, yields images with more intricate and precise content, and elevates the interpretability of the network. To this end, we curate a comprehensive dataset comprising prompt refinement, similar image generation, inpainting & outpainting, and visual question answering. Moreover, we propose a two-stage training scheme. In the first stage, we train the MLLM to grasp the properties of image generation and editing, enabling it to generate detailed prompts. In the second stage, we optimize SD to align with the MLLM's generation prompts. Additionally, we propose a reference-based restoration network to alleviate texture, brightness, and contrast disparities between generated and preserved regions during image editing. Extensive results show that LLMGA has promising generative capabilities and can enable wider applications in an interactive manner. § INTRODUCTIONArtificial Intelligence Generated Content (AIGC) has witnessed remarkable advancements, particularly propelled by the evolution of large language models (LLMs) <cit.> for text generation and diffusion models (DMs) <cit.> for image generation. LLMs, in particular, have received considerable acclaim for their exceptional ability to comprehend, reason, make decisions, possess extensive knowledge, and generate text with unparalleled accuracy and fluency. Recent studies have begun delving deeper into Multimodal Large Language Models (MLLMs) <cit.> built upon LLMs, aiming to empower LLMs to comprehend inputs extending beyond text. For example, BLIP-2 <cit.> and LLaVA <cit.> employ visual encoders to transform images into input embeddings, enabling them to be used as prompts alongside text input within the LLM, thus achieving compatibility of LLMs with the visual modality. Furthermore, recent works focused on extending the capabilities of LLMs to generate multimodal outputs. For example, GILL <cit.> involves instructing LLMs to predict fixed-size visual embeddings aligned with CLIP <cit.> space to control the Stable Diffusion <cit.> (SD) for image generation.However, existing works <cit.> merely focus on enabling LLM to output images but do not aim to assist users in generating or editing images to enhance quality. In this paper, we aim to develop a Multimodal Large Language Model-based Generation Assistant (LLMGA) to better assist image generation models, making them more user-friendly and capable of producing high-quality images.In contrast to certain methods <cit.> that leverage MLLMs to predict fixed-size visual embeddings for implicit SD control, our approach is straightforward. We guide the generation of SD using detailed language prompts from MLLM based on five reasons.(1) The embeddings predicted by the MLLM are often filled with noise. This can be filtered out by mapping them to a fixed language domain, enabling precise control of SD. (2) Detailed language prompts can make the network more transparent and interactive, allowing users to understand MLLM's thoughts for generating images. (3)MLLM is pre-trained on vast textual datasets. Explicit language prompts rather than implicit embeddings are more advantageous for MLLM to generate prompts and comprehend context. (4) Dynamic-sized language prompt facilitates the addition of generation requests during interactions. (5) Training is more simple and efficient. However, we face several challenges: (1) MLLM may reject the execution of generation instructions due to its nature as a language assistant. (2) MLLM lacks a comprehensive understanding of image generation and editing, and cannot provide an accurate and detailed generation prompt. (3) Determining which part of texts generated by MLLM to guide SD generation. (4) SD's CLIP encode only 75 tokens. Additionally, SD is trained on short captions, whereas our LLMGA typically generates detailed prompts exceeding 150 tokens. This discrepancy poses a challenge for SD in following the detailed prompt of LLMGA. To this end, we have devised a two-stage training scheme. First, we construct a training dataset: prompt refinement, similar image generation, inpainting & outpainting, and visual question answering.We then train LLMGA on these four datasets to cultivate four fundamental capabilities: (1)For concise user prompts, LLMGA can refine the generation of intricate details, encompassing attire, background, and characters. (2) LLMGA can precisely regenerate an image it observes. (3) LLMGA can generate or refine prompts for editing images based on its understanding of the image. (4) LLMGA can engage in multimodal interaction with users. Additionally, we make LLMGA use special symbols andto distinguish generation prompts and responses. In the second stage, we freeze the parameters of LLMGA's MLLM and initiate joint training with the SD. This process enables the SD to acclimate to the detailed prompt produced by the MLLM. Notably, when the input token count exceeds 75, we iteratively apply the CLIP <cit.> encoder to the surplus tokens. Moreover, we have identified noticeable disparities in texture, contrast, and brightness between the newly generated and preserved sections in SD-based image editing, such as inpainting. To address these issues, we propose a Diffusion-based Reference Restoration Network (DiffRIR), based on the structure of the SOTA restoration network DiffIR <cit.>. Specifically, aside from images generated by SD, we add masked images as reference inputs into DiffRIR. This enables the DiffRIR to refer to the texture, contrast, and brightness of the preserved regions for restoration. Additionally, we introduce perturbations to contrast and brightness during training, making DiffRIR correct contrast and brightness disparities in the images. As shown Fig. <ref>, the remarkable capabilities of the LLMGA come to light, leaving us in awe. LLMGA excels in a myriad of functions, elevating image generation and editing, realizing interactive image editing, and delivering astonishing capabilities for image design due to its expansive world knowledge and robust reasoning abilities.In summary, our contributions are as follows: * We introduced LLMGA, a simple, powerful, and generalizable multimodal generation model. Extensive experiments affirm the efficacy of LLMGA in enhancing image generation and editing through its vast knowledge base.* We construct a training dataset, including four parts: prompt refinement, similar image generation, inpainting & outpainting, and visual question answering. This enhances LLMGA's comprehension of generation and editing tasks while standardizing response formats.* We proposed a restoration network DiffRIR, which introduces reference images and training perturbations to contrast and brightness. DiffRIR can alleviate texture, contrast, and brightness discrepancies between newly generated and preserved regions for edited images. * Open-source. The following assets are released: the generated data, the codebase for data generation and model training, the model checkpoint, and a demo. § RELATED WORKDiffusion Model.Diffusion Models (DMs) <cit.> have achieved remarkable results in image generation.DMs adopt a parameterized Markov chain to optimize the lower variational bound on the likelihood function. In this way, it can generate realistic images from Gaussian noise. After that, several DM methods <cit.> have been tailored to enhance the text-to-image (T2I) generation and editing. Notably, GLIDE <cit.> pioneered the incorporation of text features into transformer blocks during the denoising process. Subsequently, DALL-E <cit.>, Imagen <cit.>, and Stable Diffusion <cit.> have made substantial strides in improving T2I generation. Subsequently, some works <cit.> introduced conditioning controls to the DMs to facilitate a more convenient and precise manipulation of the generation process. Overall, enhancing the user-friendliness of DMs is a key focus within the community. In this paper, we introduce LLMGA, leveraging the extensive knowledge and powerful reasoning capabilities of LLM to facilitate users in achieving more easily attainable and satisfactory image designs. Multimodal Large Language Models. Recently, LLMs have undeniably made profound impacts and revolutions within the entire AI community and beyond.For example, exemplary LLMs, such as ChatGPT and GPT4 <cit.>, have showcased remarkable abilities in comprehension, reasoning, responses, and knowledge reservoirs.Subsequently, a range of LLMs <cit.>, including Vicuna <cit.>, LLaMA <cit.>, and Alpaca <cit.> have been released as open-source models, substantially propelling advancements of the community.Afterward, the community began focusing on the development of the Multimodal Large Language Model <cit.>. They aim to enable LLMs to comprehend both images and text and provide textual responses. For instance, Flamingo <cit.> encodes images and feeds them into the LLM's attention layer. BLIP-2 <cit.> employs Q-Former to encode input images into queries. Furthermore, LLaVA <cit.> utilizes CLIP <cit.> to encode images into image embeddings and concatenate them with text embeddings.Recent concurrent works, such as Next-GPT <cit.>, have extended the capabilities to encompass audio and video modalities. Moreover, Visual-ChatGPT <cit.> and HuggingGPT <cit.> make LLMs act as agents capable of invoking various pre-trained visual models to achieve MLLM. However, these works focus on making LLM determine the combined invocation of modules (such as detection, recognition, and generation) to fulfill user requirements. However, these methods are not tailored for generation and editing and use concise prompts that lack the capability to enhance results. Thus, we propose LLMGA, which is designed to assist with image generation and editing. It can achieve satisfactory results by reasoning and interaction with the user. § METHODOLOGY§.§ Overview of LLMGAIn this paper, we aim to design a MLLM-based Generation Assistant (LLMGA). Our LLMGA produces detailed language-based generation prompts to control SD rather than predicting fixed-sized visual embeddings <cit.> to govern SD. This has five advantages: (1) Visual embeddings contain noise, and mapping them to the language domain can filter out this noise, enabling precise SD control. (2) Language-based generation prompts facilitate users in understanding the LLMGA's thoughts, enhancing interaction. (3) Dynamic-sized language-based generation prompt enables the addition of generation requests. (4) MLLM is pre-trained on textual datasets. Language prompts rather than implicit visual embeddings are more advantageous for MLLM to generate accurate prompts and comprehend context.(5) Training is simpler and more efficient. However, we need to address several issues: (1) As a language assistant, MLLM may decline the execution of generation instructions. (2) MLLM lacks a nuanced understanding of image generation and editing, and cannot produce precise and detailed generation prompts. (3) MLLM needs to decide which part of the output text serves as generation prompts to guide generation. (4) SD's CLIP encode only 75 tokens. Moreover, SD is trained on short captions, while detailed prompts generated by LLMGA may exceed 150 tokens.This disparity makes it hard for SD to understand the instructions from LLMGA. To this end, we construct a training dataset and two-stage training schemes, which train the MLLM (Sec. <ref>) and SD (Sec. <ref>). The network structure and pipeline of LLMGA are illustrated in Fig. <ref>.Specifically, as shown in Fig. <ref> (a) and (c), the images 𝐈_input are encoded into image embeddings by CLIP vision encoder and a projection layer. Subsequently, the image embedding is concatenated with the text embedding and fed into the LLM to obtain text output 𝐓_output. This process can be formulated as:𝐓_output=ℱ_MLLM(𝐓_input,𝐈_input),where 𝐓_input indicates the input text instructions. Notably, ℱ_MLLM can process only 𝐓_input as input.The text output 𝐓_output can comprise two components: text response 𝐓_R and generation prompt 𝐓_P. To distinguish between 𝐓_R and 𝐓_P, we expand the original LLM vocabulary with new special tokens, ,and , to encompass 𝐓_P. Next, we output 𝐓_R directly, while 𝐓_P is fed into the subsequent SD for T2I generation (Eq. <ref>) or editing (Eq. <ref>). Notably, for image editing such as inpainting and outpainting, an additional mask 𝐈_mask and the edited image 𝐈_input are essential inputs for the inpainting SD. These inputs specify the region requiring editing.𝐈_T2I=ℱ_T2I(𝐓_P,𝐙), 𝐈_edit=ℱ_edit(𝐈_input,𝐈_mask,𝐓_P,𝐙),Where 𝐙 denotes the random Gaussian noise. Furthermore, to ensure the encoding of all 𝐓_P for SD, we iteratively run the CLIP text encoder until all prompts are encoded.§.§ MLLM Training As described in Sec. <ref>, original MLLMs are specifically designed and trained as language assistants, but they lack the proficiency to assist in image generation and editing.Notably, considering the input of images and the performance of open-sourced LLMs, employing few-shot learning to guide the model to achieve the desired results proves challenging and inefficient. Therefore, it is crucial to train the MLLM to be a generation assistant, know the response format, and elevate its comprehension of the properties of image generation and editing. Thus, as shown in Fig. <ref>(a), we construct a training dataset consisting of four categories: * Prompt Refinement.We establish this dataset to cultivate the prompt refinement ability of the MLLM. Specifically, we utilize LLaVA <cit.> to furnish detailed descriptions of images in MSCOCO <cit.>. These detailed descriptions, along with the original MSCOCO brief descriptions, constitute a training text pair. During training, we input the brief MSCOCO captions and randomly select and append a generation instruction, such as “Generate this image”, or a description instruction, like "Describe this sentence in detail”. When a generation instruction is included in the prompt, we addandon the later detailed description for training. * Similar Image Generation.We use LLaVA to produce detailed descriptions of the images from the LAION-Aesthetics dataset <cit.>. Subsequently, we select the pairs of detailed descriptions and images from the LAION-Aesthetics dataset and Prompt Refinement dataset to construct the Similar Image Generation dataset. During training, we input the images along with a generation instruction, such as “Generate a similar image.” or a description instruction, such as “Describe the following image.”. In cases where a generation instruction is provided,we addandon the subsequent detailed description for training.* Inpainting & Outpainting.We use the pair of detailed descriptions and images from the Similar Image Generation dataset. During training, we input masked images with an inpainting instruction, like “Inpaint this image.” or an outpainting instruction, like “Outpaint this image.”.Moreover, we includeandon the subsequent detailed description during training. * Visual Question Answering.To preserve the Visual Question Answering (VQA) capability of MLLM, we have integrated the VQA dataset into the training of LLMGA. Specifically, we utilized the LLaVA-Instruct-150k dataset <cit.>, which was generated by GPT-4. As illustrated in Fig. <ref> (a), we freeze the CLIP vision encoder and optimize the projection layer and LLM. The model is trained end-to-end using the auto-regressive cross-entropy loss (ℒ_MLLM) for text generation. Given the ground-truth targets 𝐓_GT, this loss can be formulated as: ℒ_MLLM=𝐂 𝐄(𝐓_output, 𝐓_GT). §.§ Stable Diffusion TrainingAs described in Sec. <ref>, the original SD's CLIP text encoder only encodes 75 tokens, which cannot handle the entire MLLM's generation prompt. Moreover, the original SD is trained on brief captions, which cannot fully understand the generation prompts.Thus, we repeatedly use the VAE text encoder to encode all instances of 𝐓_P for SD. Besides, we train the T2I SD model and the inpainting SD model, respectively.For both generation and editing tasks, the generation prompts of MLLM are detailed descriptions of images. Therefore, during training, we can instruct MLLM to provide a detailed description 𝐓_P for images from the LAION-Aesthetics and MSCOCO datasets. Subsequently,𝐓_P is fed into T2I SD or inpainting SD for joint training. Notably, we only optimize the SD unet while freezing the parameters of other networks. To accelerate the training process, we record the prompt 𝐓_P of MLLM to avoid redundant calculations. The model is trained using SD loss:ℒ_SD=𝔼_𝐙_t, 𝐂, ϵ, t(ϵ-ϵ_θ(𝐙_t, 𝐂)_2^2),where 𝐙_t=√(α_t)𝐙_0+√(1-α_t)ϵ represents the noised feature map at timestep t. Ground truth images are encoded into latent space to derive 𝐙_0. Here, ϵ∈𝒩(0, 𝐈) represents Gaussian noise, and ϵ_θ refers to the SD unet. 𝐂 indicates the conditional information. For T2I generation, 𝐂 is 𝐓_P. For inpainting and outpainting,𝐂 contains 𝐓_P, the mask, and the VAE-encoded masked image. §.§ Restoration Network Training For SD inpainting and outpainting, we observed noticeable disparities between the preserved and newly generated regions in the edited images. These disparities can be attributed to variations in texture, contrast, and brightness. Besides, after finishing image generation and editing, it is customary to employ image restoration methods <cit.> on the generated images to enhance the quality. Therefore, we intend to employ a restoration network to address these disparities in the edited images.To enhance the consistency between the newly generated and the preserved regions, we introduce a reference-based restoration scheme. Specifically, existing restoration methods <cit.> take low-quality (LQ) images as input and produce high-quality (HQ) images, but they often do not leverage the preserved information from the given masked image. Different from these restoration methods, we concatenate the LQ image I_LQ and the masked image, , (1-𝐈_mask)𝐈_GT, as inputs and feed it into the restoration model ℱ_R. This process can be formulated as:𝐈_HQ=ℱ_R(concat(𝐈_LQ,(1-𝐈_mask)𝐈_GT)). To further mitigate the brightness and contrast disparities, we introduced additional color degradation (, randombrightness and contrast disturbance) into the training process of the restoration model, which is formulated as:𝒟_2(𝐱)=c_1𝐱+c_2,𝐈_LQ=𝒟_2(𝒟_1(𝐈_GT)), where 𝒟_2 indicates contrast and brightness disturbance, 𝐱 denotes the input image. c_1 is the contrast gain, randomly varied within the range of [0.94, 1.06], while c_2 is the brightness bias, randomly varied within the range of [-0.05, 0.05]. 𝒟_1 represents the real-world degradation process used in restoration model <cit.>. 𝐈_GT is ground truth image, and 𝐈_LQ is LQ image. In this paper, we adopt the network structure of the SOTA restoration model DiffIR <cit.> and apply our schemes to it to obtain our Diffusion-based Reference Restoration Network (DiffRIR). § EXPERIMENTS§.§ Implementation DetailsFor the first stage of training, we employ the pretrained LLaVA-7B or LLaVA-13B as the initial MLLM. We utilize AdamW optimizer, setting the learning rate and weight decay to 2×10^-5 and 0, respectively. Moreover, we adopt CosineLR as the learning rate scheduler. The batch size per device and epochs are set to 16 and 6, respectively. Besides, the training ratios for prompt refinement, similar image generation, inpainting & outpainting, and visual question answering are specified as 0.5, 1, 1, and 1, respectively. For the second stage of training, we adopt the Stable Diffusion 1.5 (SD1.5) as the initial image generation or editing model. We train these models with AdamW optimizer, setting the learning rate to 1×10^-5. The batch size is set to 32. We train SD1.5 by 5×10^4 iterations.For the restoration network, we train our DiffRIR on DIV2K <cit.> and Flickr2K <cit.> with the same GAN-based loss function as DiffIR. The batch sizes are set to 64, and the LQ patch sizes are set to 64×64. We use Adam optimizer, setting the learning rate to 2×10^-4. We train this model by 4×10^5 iterations.§.§ Experimental Results Evaluation on T2I Generation. The results are shown in Tab. <ref>. Notably, SD1.5 is the original Stable Diffusion 1.5 while SD1.5-ft indicates the finetuned SD1.5 in LLMGA. We also compare our LLMGA with the recently proposed multimodal generative model GILL <cit.>. (1) In the 1st and 2nd rows of Tab. <ref>, we input the original short MSCOCO caption into SD15 and SD1.5-ft, respectively. It is evident that SD1.5-ft, trained with detailed generation prompts in LLMGA,achieves a significant improvement in both image quality and content similarity compared to SD1.5. (2) In the 4th and 5th rows of Tab. <ref>, we use LLMGA-7B to refine the short MSCOCO caption to the detailed generation prompt and send it to the SD1.5 and SD1.5-ft, respectively. Our LLMGA-7B with SD1.5-ft achieves significant 7.67 FID and 2.05 CLIP-I improvements, respectively. This demonstrates the effectiveness of our stage 2 training.(3) When comparing GILL, our LLMGA-7B achieves notable 10.61 FID and 4.63 CLIP-I improvements, underscoring the effectiveness of LLMGA. (4) Comparing the 5th and 7th rows of Tab. <ref>, LLMGA-13B exhibits better performance than LLMGA-7B due to its superior reasoning ability. In fact, LLMGA-13B can provide more accurate and detailed answers than LLMGA-7B.The qualitative results are shown in Fig. <ref>. LLMGA excels in refining short prompts by incorporating details, including clothing, background, and actions to generate visually rich and pleasing images. In contrast, SD1.5 tends to straightforwardly assemble elements from the short prompts. For example, in the first row of Fig. <ref>, LLMGA showcases the act of singing by incorporating a microphone. In the second row, LLMGA crafts a battle attire for the husky and places a damaged car in the background, depicting engaging scenarios of battling monsters. Furthermore, as depicted in Fig. <ref>, LLMGA can refine short captions based on user requirements. Evaluation on Inpainting and Outpainting. The results are shown in Tab. <ref>.(1) In the 3rd and 4th rows of Tab. <ref>, we make LLMGA imagine the complete generation prompts for the given masked images, which are then input into the later SD. Notably, our LLMGA-7B (with SD1.5-ft) demonstrates a significant FID and CLIP-I improvement over LLMGA-7B (with SD1.5) in both outpainting and inpainting. This demonstrates the effectiveness of stage 2 training for image editing. (2) When comparing the 1st and 4th rows of Tab. <ref>, our LLMGA-7B achieves significant improvements of 8.66 FID and 1.27 CLIP-I over the SD1.5 in outpainting under wide masks.(3) Comparing the 4th and 6th rows of Tab. <ref>, our LLMGA-13B outperforms LLMGA-7B due to its enhanced knowledge reservoir and superior reasoning capabilities.The qualitative results are shown in Fig. <ref>. We can see that LLMGA can deduce and imagine complete images based on masked input images. For example, in the 1st row of Fig. <ref>, LLMGA can infer the presence of a deer model hanging on the wall based on the given environment. In the 3rd and 4th rows of Fig.<ref>, LLMGA's rational inference about the scene ensures that the SD editing output aligns with the environmental context, rather than producing chaotic results. Furthermore, as illustrated in Fig. <ref>, LLMGA can imagine complete images based on user requirements and masked images. Evaluation on Image Restoration.The results are shown in Tab. <ref>. For comparisons, we utilize DiffIR <cit.> and our DiffRIR on the edited image generated by LLMGA. (1) Comparing the 2nd and 3rd rows of Tab. <ref>, it is evident that introducing a reference scheme can significantly improve restoration performance. (2) When comparing the 3rd and 4th rows of Tab. <ref>, it can be observed that introducing color degradation helps alleviate the bright and contrast distortion caused by SD.(3) Comparing the 1st and 4th rows of Tab. <ref>, our DiffRIR yields significant improvement, further validating the effectiveness of DiffRIR.The qualitative results are shown in Fig. <ref>. Our DiffRIR (, DiffRIR_2 in Tab. <ref>) can alleviate texture, brightness, and contrast discrepancies, and generate realistic details. Control SD using detailed language prompt or embedding? The results are shown in Fig. <ref>.We compare two approaches: GILL, which makes LLM estimate a fix-sized embedding to control SD generation, and LLMGA Embedding, a variant of LLMGA where the language prompt is replaced with embedding, undergoing the same training process as LLMGA.The evaluation is conducted on MSCOCO by instructing these methods to generate images multiple times in conversation form. (1) The quality of generated images (Fig. <ref> (a)) and the relevance of content (Fig. <ref> (b)) in embedding-based methods (, GILL and LLMGA Embedding)deteriorate rapidly as the number of conversation turns increases. In contrast, our LLMGA remains unaffected. This discrepancy arises from the inherent noise present in the embeddings predicted by LLM.As the number of conversation turns rises, these generated embeddings integrate with the preceding conversations, introducing even more noise.This poses challenges for the precise control of SD-generated content. Our LLMGA addresses this issue by mapping the embedding to the fixed language domain, effectively eliminating such noise. (2) Additionally, LLMGA Embedding also outperforms GILL, indicating that the prompt size used to guide SD generation should be adaptive in content, rather than a fixed size. Contribution of Training Data. The results are shown in Tab. <ref>. To assess the impact, we downsized the training data of one of the four training datasets from LLMGA_5 to 10% of its original magnitude, ensuring that LLMGA remains capable of furnishing responses in the prescribed format.It is evident that prompt refinement and iutpainting&outpainting datasets enhance LLMGA's comprehension of image generation and editing properties, resulting in superior images. When comparing LLMGA_3 and LLMGA_5, engaging in similar image generation training further improves the performance of LLMGA in both generation and editing. Moreover, the inclusion of visual question answering, despite its lack of direct relevance to image generation or editing, yields a noteworthy enhancement for LLMGA. This is because diverse conversation training can mitigate catastrophic forgetting in MLLM.§ CONCLUSIONLLM possesses an extensive reservoir of knowledge and powerful comprehension and reasoning capabilities. In this paper, we introduce a MLLM-based generation assistant (LLMGA), aiming to exploit LLM's capabilities in an interactive manner to facilitate more efficient and convenient image generation and editing. Compared to relying on LLM to predict a fixed-size embedding to control SD, we employ detailed generation prompts. These prompts prove to be more favorable for enhancing LLM's contextual comprehension and generating more accurate and rich content. To this end, we develop a two-stage training scheme and curate a dataset, including four parts: prompt refinement, similar image generation, inpainting & outpainting, and visual question answering. For the first stage, we train MLLM to understand the properties of image generation and editing, enabling it to give fitting responses. For the second stage, we optimize the SD unet to adapt to the generation prompt. Moreover, we propose a DM-based reference restoration network (DiffRIR) to mitigate disparities in texture, contrast, and brightness for image editing. Consequently, LLMGA can offer design suggestions and enhance results based on user's requests during interactions.The overview of the supplementary materials:(1) We offer a comprehensive introduction to the creation of the training dataset (Sec. <ref>).(2) Additional details of the training and evaluation for LLMGA are elaborated in this part. Furthermore, a detailed training process for Stable Diffusion XL (SDXL) is also included (Sec. <ref>).(3) We present more comparison details on LLMGA and LLMGA Embedding, along with corresponding analyses (Sec. <ref>).(4) We provide additional examples showcasing the interactive generation and editing capabilities of LLMGA (with SDXL) (Sec. <ref>).(5) More visual results are showcased to highlight LLMGA's performance in T2I generation (Sec. <ref>). (6) More visual results are presented for LLMGA on inpainting and outpainting (Sec. <ref>).(7) More Visual results effectively demonstrate the prowess of DiffRIR in addressing brightness and contrast disparities between newly generated and retained regions during image editing, along with its ability to enhance texture details (Sec. <ref>). § DATA For the first stage of training, we constructed a training dataset that requires detailed descriptions of images to assist the LLM in better understanding the compositional details of images and supporting image generation and editing. Specifically, we utilized the MSCOCO <cit.> and LAION-Aesthetics <cit.> datasets, encompassing rich real-world scenarios and aesthetically pleasing images, respectively. Then, we employed LLaVA to generate detailed and visually compelling descriptions for these datasets.The prompt format for LLaVA to generate a detailed description is as follows: "The caption of this image is 'ORIGINAL CAPTION'. INSTRUCTIONS.", where 'ORIGINAL CAPTION' is a concise caption provided by the dataset, and 'INSTRUCTIONS' represents the generation instructions randomly selected from Tab. <ref>. An example is shown in Fig. <ref>. Notably, the design of such prompts carries significance. As shown in Fig. <ref>, we conducted a comparison between two distinct prompts. Firstly, let's scrutinize the original captions provided by the dataset. They are characterized by brevity, insufficient to encapsulate the entire scene but do include proper nouns. Subsequently, we directed LLaVA to generate a detailed description of the given image using the prompt ”Write a detailed description of the given image.” We observed that LLaVA's image-based description is comprehensive, evoking a vivid sense of the scene, but it may lack certain proper nouns. However, users frequently include proper nouns, which inherently carry a wealth of information. Therefore, to facilitate the MLLM in learning the correlation between these proper nouns and the scene, it is necessary to introduce proper nouns into descriptions. A straightforward and effective approach (the last row of Fig. <ref>) is to incorporate the original caption of the image into LLaVA's prompt. This strategy effectively integrates proper nouns into the detailed description.§ MORE TRAINING AND EVALUATION DETAILS In addition to the LLMGA (SD1.5) described in the paper, we have also conducted training for LLMGA (SDXL). LLMGA (SDXL) follows a training process and configuration similar to that of LLMGA (SD1.5) but demonstrates even higher-quality generation. Specifically, for the first training stage, we employ the same MLLM as utilized in LLMGA (SD1.5). Regarding MLLM, we utilize the pretrained LLaVA-7B or LLaVA-13B as the initial MLLM. Our optimization approach involves the use of the AdamW optimizer, with the learning rate set at 2×10^-5 and weight decay at 0, respectively.Additionally, we adopt the CosineLR learning rate scheduler. The total batch size and number of epochs are configured at 128 and 6, respectively. MLLM training is carried out on our constructed datasets as described in the paper. Furthermore, the training ratios for prompt refinement, similar image generation, inpainting & outpainting, and visual question answering are defined as 0.5, 1, 1, and 1, respectively.For the second stage of training, we adopt the Stable Diffusion XL (SDXL) as the initial image generation or editing model. We train these models with AdamW optimizer, setting the learning rate to 1×10^-5. The total batch size is set to 32. We train SD1XL by 5×10^4 iterations.For both T2I generation and editing, we conduct training on the LAION-Aesthetic dataset, using the generation prompts generated by the first-stage pretrained MLLM as guidance.The input patch sizes are set to 1024 × 1024. Moreover, similar to SD1.5, we randomly generate masks for SDXL inpainting and outpainting, including box masks, irregular masks, and boundary masks. To evaluate our LLMGA quantitatively, we employ a diverse set of metrics. For evaluating the quality and diversity of generated images, we utilize the Inception Score (IS) <cit.>, and Fréchet Inception Distance (FID) <cit.>. We leverage CLIP-based metrics <cit.> CLIP-I to assess the content similarity between generated and ground-truth images. § MORE DETAILS ON CONTROL SCHEME The form in which to establish a control link between MLLM and SD is a question that requires careful consideration. In this paper, as illustrated in Fig. <ref>, we explore two control schemes: namely, our adopted language-based generation prompt control (Fig.<ref> (a)) and visual embedding-based control scheme (Fig.<ref> (b)). Here, we will provide a detailed overview of the training approach for LLMGA Embedding, followed by a comparison and analysis of these schemes.For the training of LLMGA Embedding, we employ a comprehensive three-stage training scheme. (1) In the first stage, for T_R, we use the same auto-regressive cross-entropy loss (ℒ_MLLM, Eq. <ref>) as LLMGA, and for visual embedding, we utilize the visual embedding loss (ℒ_embed, Eq. <ref>). We simultaneously apply the same training settings as LLMGA.(2) In the second stage, we initiate joint optimization of MLLM and SD. Unlike LLMGA, in this phase, we optimize only the projection layer for LLMGA embedding, using the SD loss (ℒ_SD, Eq. <ref>), and then freeze the parameters of Unet and MLLM. (3) In the third training stage, we freeze the parameters of MLLM and projection, and then optimize the SD Unet using the SD loss (ℒ_SD, Eq. <ref>).ℒ_MLLM=𝐂 𝐄(𝐓_R, 𝐓_GT), ℒ_embed=ϕ_proj(𝐕_E)-ϕ_CLIP(𝐓_caption)_2^2, ℒ_SD=𝔼_𝐙_t, 𝐂, ϵ, t(ϵ-ϵ_θ(𝐙_t, 𝐂)_2^2),where 𝐓_R denotes the generated text response, and 𝐓_GT represents the ground-truth target. 𝐂𝐄(.) signifies auto-regressive cross-entropy. ϕ_proj(.) denotes the projection involving three linear layers. ϕ_CLIP(.) is the CLIP text encoder. 𝐕_E stands for visual embedding, and 𝐓caption corresponds to the original caption from the datasets. ℒ_SD represents the diffusion loss as described in the paper. The results are presented in Fig. <ref> in the paper. Despite comprehensive training, we find that the embedding-based approaches still lags behind LLMGA. Moreover, with an increase in the number of dialogue turns, there is a noticeable decrease for embedding-based approaches in both the accuracy and quality of generation.This observation can be well comprehended through the processing mechanism of LLM. Specifically, LLM predicts an embedding based on previous input images and texts (Eq. <ref>). After that, embedding is refined through a linear layer, categorizing it into a fixed language domain (Eq. <ref>).This is because predicted embeddings exist in a continuous space, inherently imprecise and filled with noise. For instance, the same embedding, depending on the sampling probabilities, may generate different semantic words, indicating that embeddings are rife with various forms of noise. Mapping embeddings to a fixed language domain effectively eliminates this noise, enabling precise control over SD generation. The decline in performance of embedding-based methods with an increase in conversation turns is attributed to the introduction of additional noise into the predicted embedding as more prior conversation information is incorporated, thereby affecting accuracy. 𝐄=Ψ_body(𝐈_input,𝐓_input), 𝐓=Ψ_linear(𝐄),where 𝐈_input and 𝐓_input represent all preceding input images and texts. Ψ_body signifies the network body of LLM, producing an embedding 𝐄. Ψ_linear constitutes the final linear layer in LLM, tasked with categorizing the embedding into a predetermined text domain, resulting in the generation of text 𝐓.In summary, compared with embedding based methods, our LLMGA using detailed language prompts for control generation has the following advantages: * The embeddings predicted by the MLLM are often filled with noise. This can be filtered out by mapping them to a fixed language domain, enabling precise control of SD. * Detailed language prompts can make the network more transparent and interactive, allowing users to understand MLLM's thoughts for generating images. * MLLM is pre-trained on vast textual datasets. Explicit language prompts rather than implicit embeddings are more advantageous for MLLM to generate prompts and comprehend context. * Dynamic-sized language prompt facilitates the addition of generation requests during interactions. * Training is more simple and more efficient.§ MORE RESULTS ON INTERACTIVE GENERATION AND EDITINGThe results are shown in Figs. <ref>, <ref>, and <ref>. The results presented here were obtained using LLMGA-7b (SDXL-ft). It can be observed that LLMGA, leveraging the understanding, reasoning abilities, and extensive knowledge repository of LLM, effectively assists users in image design in an interactive manner.§ MORE RESULTS ON T2I GENERATION The more T2I generation visual results are shown in Fig. <ref>. LLMGA can leverage its vast comprehension, reasoning abilities, and knowledge reservoir to generate visuals with more details. Additionally, for less common concepts like a space elevator, LLMGA can accurately generate images based on its extensive knowledge after providing design descriptions. Furthermore, LLMGA refine prompts based on user requirements. § MORE RESULTS ON INPAINTING AND OUTPAINTING The more inpainting and outpainting visual results are depicted in Fig. <ref>. LLMGA can harness its extensive comprehension, reasoning abilities, and knowledge reservoir to infer plausible complete images based on given masked images. For instance, in the first row of Fig. <ref>, LLMGA can deduce the presence of a boat based on the existence of ripples on the water surface. Additionally, as demonstrated in paper Figs. <ref> and <ref>, LLMGA can edit images in accordance with user specifications and masked images.§ MORE RESULTS ON IMAGE RESTORATION The more image restoration results are shown in Fig. <ref>. Our DiffRIR can alleviate the texture, brightness, and contrast discrepancies, and generate more realistic details. ieeenat_fullname | http://arxiv.org/abs/2311.16500v2 | {
"authors": [
"Bin Xia",
"Shiyin Wang",
"Yingfan Tao",
"Yitong Wang",
"Jiaya Jia"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20231127133726",
"title": "LLMGA: Multimodal Large Language Model based Generation Assistant"
} |
Distributed Attacks over Federated Reinforcement Learning-enabled Cell Sleep Control Han Zhang1, Hao Zhou1, Medhat Elsayed2,Majid Bavand2, Raimundas Gaigalas2, Yigit Ozcan2 and Melike Erol-Kantarci1, Senior Member, IEEE1 School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada2 Ericsson Inc., Ottawa, Canada{hzhan363, hzhou098, melike.erolkantarci}@uottawa.ca, {medhat.elsayed, majid.bavand, raimundas.gaigalas, yigit.ozcan}@ericsson.comJanuary 14, 2024 ================================================================================================================================================================================================================================================================================================================================================================================================================================================ Quantum query complexity has several nice properties with respect to composition. First, bounded-error quantum query algorithms can be composed without incurring log factors through error reduction (exactness). Second, through careful accounting (thriftiness), the total query complexity is smaller if subroutines are mostly run on cheaper inputs – a property that is much less obvious in quantum algorithms than in their classical counterparts.While these properties were previously seen through the model of span programs (alternatively, the dual adversary bound), a recent work by two of the authors (Belovs, Yolcu 2023) showed how to achieve these benefits without converting to span programs, by defining quantum Las Vegas query complexity. Independently, recent works, including by one of the authors (Jeffery 2022), haveworked towards bringing thriftiness to the more practically significantsetting of quantum time complexity.In this work, we show how to achieve both exactness and thriftiness in the setting of time complexity. We generalize the quantum subroutine composition results of Jeffery 2022 so that, in particular, no error reduction is needed. We give a time complexity version of the well-known result in quantum query complexity, Q(f∘ g)=(Q(f)· Q(g)), without log factors. We achieve this by employing a novel approach to the design of quantum algorithms based on what we call transducers, and which we think is of large independent interest. While a span program is a completely different computational model, a transducer is a direct generalisation of a quantum algorithm, which allows for much greater transparency and control. Transducers naturally characterize general state conversion, rather than only decision problems;provide a very simple treatment of other quantum primitives such as quantum walks; and lend themselves well to time complexity analysis. plainq wred applegreen blue gray orange § INTRODUCTIONSince the introduction of span programs into quantum query complexity by Reichardt and Špalek <cit.> [However, this particular connection is attributed to Troy Lee in the first of these two papers.], the quantum query world became a nicer place to be. A span program (alternatively, a feasible solution to the dual adversary bound) is an idealised computational model, in the sense that no real device corresponds to it. Nonetheless, it has a strong connection to quantum query complexity. We can turn any quantum query algorithm into a span program, [Originally, this construction was non-constructive as it came from the dual of a lower bound.Section 3 of <cit.> contains a constructive construction for algorithms with one-sided error.This was later extended to two-sided error in <cit.>.] or we can construct one from scratch. They can be composed as usual quantum subroutines. At the end of the day, the span program can be transformed back into a quantum query algorithm. We identify two main points of advantage of span programs compared to usual quantum subroutines.* We call the first one exactness. A span program evaluates a function exactly even if the quantum query algorithm it was converted from had a bounded error. Therefore, span programs can be composed without any error reduction. The conversion from the span program to a quantum query algorithm does introduce an error, and we do not get an exact quantum query algorithm at the end. However, the error is introduced only once per the whole algorithm, and does not accumulate even in the case of many non-precise subroutines.This idea was the major driving force behind this line of research starting with the algorithm for the iterated NAND function <cit.> to general iterated Boolean functions <cit.>. * We call the second one thriftiness. Span programs have a natural predisposition towards accurate bookkeeping. If an execution of a subroutine happens to be cheap on a particular input, this cheaper cost will contribute to the total complexity of the whole span program. This is in contrast to the usual execution of quantum subroutines, where the maximal cost is to be paid no matter how easy the input is in a particular execution. This is because we generally cannot measure a subroutine, which may be run only in some branch of a superposition, to see whether it has ended its work or not. [ In specific cases, Ambainis' variable-time framework <cit.> allows for this particular approach. We consider variable-time framework as a piece of motivation towards importance of thus defined thriftiness.See, e.g., the results mentioned in the introduction of <cit.>. ] At the end, when converting to a quantum query algorithm, we still have to account for the maximal possible complexity, but this is done only once per the whole algorithm, thus amortising complexities of individual subroutines.This point has been emphasised to a lesser extent than the exactness property, but we find it equally important. In particular, this ability results in some rather interesting super-polynomial speed-ups <cit.>. We consider the thriftiness property in more detail in sec:conceptualPreliminaries. In a recent paper by a subset of the authors <cit.>, it was shown that one does not even have to change the model to obtain these benefits. It suffices to keep the model of a quantum query algorithm, and only change the way query complexity is defined. Namely, one can count total squared norm of all the states processed by the input oracle (coined quantum Las Vegas query complexity in <cit.> as it can be interpreted as the expected number of queries) instead of the number of executions of the input oracle (which is the usual definition, called Monte Carlo in <cit.>). Note that quantum Las Vegas query complexity is an idealised complexity measure, but it has an operational meaning: at the end of the day, a quantum Las Vegas query algorithm can be turned into a Monte Carlo query algorithm with a constant increase in complexity and introducing a small error.A major drawback of the above results is that while accounting of query complexity is very natural and important, analysis of the more meaningful time complexityof the resulting algorithm can be quite difficult. While the time complexity of some nicely structured span program algorithms has been successfully analyzed (e.g. <cit.>), in other cases, time complexity analyses have proven much more challenging, perhaps most notably in <cit.>, where a query algorithm was given, but a time complexity analysis remained elusive for more than a decade <cit.>.There has been a line of work attempting to extend some of the benefits of span programs to time complexity. In <cit.>,it was shown how span programs could be made to capture time complexity of quantum algorithms, not only query complexity.Essentially what <cit.> did was to extend the algorithm-to-span-program conversion of <cit.> so that the span program still encodes information about the gate structure of the original algorithm. Thus, this structure can be recovered when the span program is converted back into an algorithm. The resulting span programs could be manipulated in a limited number of ways. For instance, this allowed for an alternative implementation of Ambainis' variable-time search algorithm <cit.>. This idea was further exploited in <cit.> by one of the authors (in the framework of multidimensional quantum walks <cit.>) to give a general composition result for time complexity of quantum algorithms, similar to the query results of <cit.>, but lacking in generality. While <cit.> achieved thriftiness, it failed to achieve exactness.Instead, this work assumes that subroutines have had their success probability amplified through majority voting so that errors occur with inverse polynomial probability, which results in logarithmic factor overhead. One of the main contributions of this a paper is a framework that achieves both exactness and thriftiness for quantum time complexity in a systematic manner. In order to do so, we introduce transducers. While quantum Las Vegas query complexity keeps the model of a quantum algorithm the same but changes the definition of the query complexity, transducers still keep the same model but change the definition of computation. We consider two objects: a transducer that is a usual quantum algorithm in a larger Hilbert space, and its transduction action that is an idealised transformation obtained by “additively tracing out” a part of the Hilbert space of the transducer. A simple procedure converts a transducer into an algorithm implementing its transduction action, introducing a small error in the process. Monte Carlo query complexity of the resulting algorithm is Las Vegas query complexity of the transducer. Its time complexity is a product of two things: time complexity of the transducer (considered as a usual quantum algorithm) and its transduction complexity, which measures the extent of the “traced-out” part of the system.Compared to span programs, which most naturally model computations of decision problems,transducers naturally model arbitrary state conversion computations, and as a transducer is itself a quantum computation, its time complexity analysis is more immediate.For the composition results, we follow the same overall strategy as for span programs. We can transform any quantum program into a transducer, and we can compose transducers in essentially the same way as we compose quantum programs. The advantage is that both query complexity and time complexity (in the guise of transduction complexity) are composed in a thrifty manner. At the end, we convert the composed transducer back into a quantum algorithm. In parcitular, we obtain the following results: Exact and thrifty composition of quantum algorithms: We show how to compose quantum algorithms in a way that is thrifty[For an idea of what we mean by thrifty composition, see eqn:randomComposition and eqn:randomComposition2, and the surrounding discussion of the “gold standard” for composition.] in time complexity, but also exact (thm:introCompositionQRAG). This improves on <cit.> in several ways. * Exactness: We do not need to assume the composed algorithms have inverse polynomial errors, which saves log factors in the overall complexity. * The results apply to composing quantum algorithms that solve general state conversion problems, whereas <cit.> only applies to algorithms that decide Boolean functions. * In addition to achieving thrifty time complexity, the composed algorithm is also thrifty in its number of queries to the input oracle. The one way in which our results, as stated, are not more general than <cit.> is that <cit.> considers quantum algorithms that work in the variable-time model, meaning their running times can be random variables, achieved by performing intermediate partial measurements that indicate if the algorithm is finished. While variable-time algorithms are also compatible with transducers, we omit their explicit treatment here, for the sake of simplicity, and leave it for future work.We also give results that apply purely to the circuit model (without quantum random access gates (QRAGs)), but with worse complexity (thm:introCompositionCircuit).We also explicitly consider multiple layers of composition (thm:introCompositionTree) in the QRAG model, and its special case of iterated functions in the circuit model (thm:introIterated). Quantum analogue of majority voting, and time-efficient function composition: Itis known that the bounded-error quantum query complexity, Q, of a composed function f ∘ g is Q(f∘ g)=(Q(f)Q(g)). This statement would be obvious with log factors, by using the standard technique of majority voting to reduce the error of each call to g to inverse polynomial, but the fact that this holds without log factors is somewhat surprising. In sec:introPurifier, we shed some light on this surprising result, and show a similar result for time complexity. We extend a notion from <cit.>, called a purifier, to show how to take a quantum algorithm with constant error, and convert it to a transducer with any error >0 with only constant overhead on the query and time complexities. This, in turn, allows us to prove a log-factor-free composition result for time complexity in the QRAG model (thm:introCompositionFunctionQRAG). Specifically, if there is an algorithm for f that makes Q queries and takes T_f additional time, and an algorithm for g that takes T_g time, then there is an algorithm for f∘ g that takes time (T_f+QT_g). As with the aforementioned query result, this would be obvious with log factors on the second term, but the fact that it holds without log factor overhead is surprising. This implies log factor improvements in the time complexity of quantum algorithms obtained by composition, such as <cit.>. We stress that in addition to our concrete results, a major contribution is conceptual.Along withbringing the beautiful existing work on quantum query complexity to the real world of quantum time complexity, transducers achieve many existing results in a much simpler and cleaner way, and we feel that they are a novel and potentially instructive way of understanding quantum algorithms. For example, discrete-time quantum walks are an important technical tool. However, understanding their internal workings requires some background: the notion of the spectral gap, phase estimation subroutine, non-trivial spectral analysis. In sec:introWalks we use transducers to devise a very simple implementation of a quantum walk that completely avoids all this background.Let us end this section with a few remarks on the model of computation. The previous results towards thriftiness in time complexity, <cit.> and <cit.>, assumed the QRAG model in a fundamental way. It is based around the quantum random access gate (QRAG), which allows index access to an array of quantum registers in superposition. This should not be confused with quantum random access memory (QRAM), which assumes such access to an array of classical registers. [This nomenclature is not completely standardised and has been a source of confusion, as different authors use the same name to refer to different models.] QRAG is a stronger model than QRAM, but it is utilised in a large number of time-efficient quantum algorithms, Ambainis' element distinctness algorithm <cit.> being a noticeable example.Our general attitude towards the QRAG model is one of ambivalence. On the one hand, the QRAG model is a very natural quantization of the classical RAM machine, and, thus, makes a lot of sense to study theoretically. On the other hand, the assumption that we can swap a large number of qubits in superposition in essentially constant time seems far-fetched given the current state of the art in development of quantum computers. Because of that, we have chosen to pursue both directions. We prove results involving the QRAG model, as they tend to have more natural formulations, and continue the aforementioned line of research in <cit.>, but we also design algorithms in the circuit model, which, while not being as efficient as in the QRAG model, are of significant interest.We now give a technical overview of our results in sec:conceptualPreliminaries and sec:overview, before fleshing out full details in the remaining sections. § CONCEPTUAL PRELIMINARIES: QUANTUM LAS VEGAS COMPLEXITY Quantum Las Vegas query complexity <cit.>, also known as the total query weight <cit.>, is a cornerstone in understanding this paper for a number of reasons. First, the definition of the query complexity of a transducer is intrinsically the Las Vegas one. Second, when converting a quantum program into a transducer, we take into account its Las Vegas query complexity. Finally, we use composition results for the Las Vegas query complexity as a model that we strive to achieve for time complexity.This section serves as a very brief overview of the composition properties for randomised and quantum Las Vegas complexity. The randomised results are folklore, and the quantum ones are either from <cit.> or can be obtained similarly. Both are for purely illustrative purpose and will serve as a reference point to the results obtained later in the paper. §.§ Types of Composition The composition property we have in mind is as follows. Assume we have an algorithm A with an oracle O', and an algorithm B that implements the oracle O'. The functional composition of the two is an algorithm A∘ B, where the algorithm B is used as a subroutine to process the queries made by A to O'. The subroutine B has access to some oracle O, which is also the input to A∘ B. For simplicity we will now assume that A has no access to O, but we will drop this assumption shortly. The main question is a bound on the complexity of A∘ B in terms of complexities of A and B.One particular example studied extensively both classically and quantumly is given by composed functions, i.e., functions of the formf[ g_1(z_1,1, …,z_1,m),g_2(z_2,1, …,z_2,m), …, g_n(z_n,1, …,z_n,m)].(For simplicity of notation, we assume all the functions g_i are on m variables, which is without loss of generality.) Then, O encodes the input string z, and A evaluates the function f. Concerning B, it is a parallel composition B=⊕_i B_i of algorithms evaluating g_i. On query i, B returns the output of B_i, which is g_i(z_i,1, …,z_i,m).In general, the parallel composition B=⊕_i B_i, which we also call the direct sum, has queries of the form (i,j), on which it responds with the output of B_i on query j. We assume that all B and B_i have access to the same oracle O. In the example above, all j are absent. Also, although each B_i has access to O, it only uses the substring z_i = (z_i,1, …,z_i,m) thereof. §.§ Randomised Complexity Recall the distinction between Las Vegas and Monte Carlo randomised algorithms. A Monte Carlo algorithm is a randomised algorithm that takes at most some fixed number T of time steps, and outputs the correct answer with bounded error.The complexity of such an algorithm is simply T. This is analogous to the usual bounded-error quantum query model.In contrast, a Las Vegas algorithm is a randomised algorithm whose number of steps is a random variable T, and that never outputs an incorrect answer (though it may run forever). The complexity of such an algorithm is [T]. More generally, one might consider the Las Vegas complexity, [T], of an algorithm whose running time is a random variable T, even if the algorithm has some probability of erring (that is, it is not strictly a Las Vegas algorithm). Randomised Las Vegas complexity behaves nicely with respect to composition. Here we sketch the corresponding notions and results in order to facilitate the forthcoming introduction of the related quantum notions, and to have a reference point against which we can gauge our quantum composition results._randLet us start with functional composition. Let T(A,O') be the complexity of the algorithm A on the oracle O'. Let B(O) denote the action of the algorithm B on the oracle O, and T(B, O, i) the complexity of B(O) on query i. Denote by q^(i)(A,O') the probability the algorithm A will give i as a query to the oracle O' (we assume each query is given at most once). Then, it is not hard to see that the total complexity of A∘ B on oracle O is given byT(A∘ B,O) = T[A, B(O)] + ∑_i q^(i)[A, B(O)]· T(B, O, i). Let us rewrite this formula to illuminate transition to the quantum case. Define the total query vector of the algorithm A as a formal linear combination q(A, O') = ∑_i q^(i) (A,O') e_ifor e_i the elements of the standard basis. Let by definition the complexity of B on such a vector beT[B, O, ∑_i p_i e_i] = ∑_i p_iT[B, O, i].Then, we can rewrite eqn:randomComposition asT(A∘ B, O) =T[A,B(O)] + T[B, O, q(A,B(O))],which says that the complexity of the composed program on the oracle O equals the complexity of A by itself plus the complexity of B on its total query vector. This equation will serve as our “gold standard” of thriftiness in functional composition.Similarly, thriftiness holds for parallel composition. Let ξ = ∑_i,jξ_i,j e_i,j be a query vector for ⊕_i B_i. We can break it down asξ = ⊕_i ξ_i,where ξ_i = ∑_j ξ_i,j e_i,j is the corresponding query vector for the constituent B_i. Then, we haveT[⊕_i B_i, O, ξ] = ∑_i T[B_i, O, ξ_i].(We assume here that relaying from B to the corresponding B_i is done instantly.)These results can be combined in various ways. For instance, assume the oracle O' in the functional composition settings is a direct sum O'=⊕_i=1^n O^(i) of r independent oracles O^(i). Let B_i implement O^(i), so that B = ⊕_i B_i implements O'. Similarly as in eqn:randomXiDecomposition, we can decompose the corresponding total query vectorq(A, O') = ⊕_i q^(i)(A,O')into partial query states corresponding to the i-th oracle. In this way, the multiple-oracle case differs from the single-oracle case only by this change of perspective. Combining eqn:randomComposition2 and eqn:randomParallelComplexity, we obtainT(A∘ B, O) = T[A,B(O)] + ∑_i T[B_i, O, q^(i)(A,B(O))]. I^∘ I^∙ I^∘ ^∘ ^∙ ^↑ §.§ Quantum Complexity The usual definition of quantum query algorithms does not allow for different complexities on different inputs, hence, we cannot even properly define a meaningful analogue of eqn:randomComposition. However, it is possible for Las Vegas complexity of quantum query algorithms defined in <cit.>. We will demonstrate here that it possesses properties essentially identical to those of the randomised case. For a more formal definition of the model of query algorithms and the Las Vegas complexity, refer to sec:prelimQuery.Let A = A(O) be a quantum algorithm in spacewith an oracle O in space . We denote by A(O) the action of A on a specific input oracle O. We use T(A) to denote time complexity of A, and Q(A) to denote the number of queries made by A.What is important, is that the queries to the input oracle O are conditional. This means that the query applies O to a subspace ⊆ of the workspace , and is the identity elsewhere. The total query state q(A,O,ξ) records the history of all the queries given by the algorithm A to the input oracle O on the initial state ξ. In other words, it is the direct sumq(A,O,ξ) = ⊕_t=1^Q(A)ψ^∙_t = ∑_t=1^Q(A)||⟩t> ||⟩ψ^∙_t>,where ψ^∙_t∈ is the state given to the input oracle on the t-th query. We have q(A,O,ξ) ∈⊗ for some space . The Las Vegas query complexity is defined as L(A,O,ξ) = q(A,O,ξ)^2. This definition has an operational meaning: the algorithm can be modified to use (L) queries, where L is the worst-case Las Vegas query complexity, by introducing some small error.One convention of this paper is that we usually only allow unidirectional access to O.The algorithm can only execute O, but not its inverse O^*. This is without loss of generality, as bidirectional access to O, when the algorithm can execute both O and O^*, is equivalent to unidirectional access to O⊕ O^*.As in the randomised case, r input oracles can be combined into one as follows:O = O^(1)⊕ O^(2)⊕⋯⊕ O^(r).The partial query state q^(i)(A,O,ξ) of the i-th input oracle is defined as the direct sum of all the states given to that particular oracle O^(i). In particular, q(A, O, ξ) = ⊕_i q^(i) (A, O,ξ). Similarly as before, the Las Vegas query complexity of the i-th input oracle isL^(i)(A, O, ξ) = |q^(i)(A,O,ξ)|^2.Parallel Composition The parallel composition is straightforward. For programs B_1,…, B_n, all on the input oracle O, its direct sum is ⊕_i B_i, which executes B_i on orthogonal parts of the space. It is not hard to show thatq[⊕_i B_i, O, ⊕_i ξ_i] = ⊕_i q(B_i, O, ξ_i),where we implicitly assume the correct arrangement of the entries in the corresponding direct sums. A direct consequence is the following counterpart of eqn:randomParallelComplexity:L[⊕_i B_i, O, ⊕_i ξ_i] = ∑_i L(B_i, O, ξ_i).Functional Composition Let us now derive a quantum query analogue of eqn:randomComposition2. First, though, we define a counterpart of eqn:randomComplexityExtended. For ξ'∈⊗, we can write ξ' = ξ_1⊕ξ_2 ⊕⋯⊕ξ_m with ξ_t∈, and defineq(A,O,ξ') = ⊕_t q(A, O, ξ_t).This is precisely the total query state we will get if we tensor-multiply A by the identity in the registerand execute it on ξ'. The corresponding Las Vegas query complexity is L(A, O, ξ') = q(A, O, ξ')^2. Now consider fig:composition, which depicts composition of two quantum programs, and which goes along the lines of our previously discussed randomised case. This time, however, we consider a more general case when A has access to the input oracle O as well.General case of functional composition of two programs. The outer program A has two oracles O and O', which we identify with the oracle O⊕ O'. The oracle O' is implemented by a program B with access to O. The diagram specifies the corresponding query states, where we use the upper indices ^(0) and ^(1) in relation to O and O', respectively.[every path/.append style=thick,->] [] (-1, 1) node[left]ξ to (0,1);(0,0) rectangle (3,1.5) node[pos=0.5] A(O⊕ O');(4,0) rectangle (7,-1.5) node[pos=0.5] O' = B(O);(0, -3) rectangle (7,-4.5) node[pos=0.5] O;[purple] (1.5,0) to node[left] q^(0)(A, O⊕ O', ξ) (1.5,-3); [purple, out=0, in=90] (3, 1) to node[above]q^(1)(A, O⊕ O', ξ) (5.5,0); [purple] (5.5, -1.5) to node[right] q[ B, O, q^(1)(A, O⊕ O', ξ)](5.5,-3);The composed algorithm A∘ B is implemented by replacing each execution of O' by an execution of B. Its action on the input oracle O is equal to A[O⊕ B(O)]. It is not hard to show thatq(A∘ B, O, ξ) = q^(0)[A, O⊕ B(O), ξ] ⊕q[ B, O, q^(1)[A, O⊕ B(O), ξ]].Similarly to eqn:randomComposition2, this slightly complicated expression represents a very intuitive observation that the total query state of A∘ B on the input oracle O consists of the part of the query state of A given directly to O (denoted q^(0)), together with the query state of B on the initial state composed of the part of the query state of A given to O' (denoted q^(1)). A version of fig:composition where the oracle O' is decomposed into r input oracles O' = O^(1)⊕⋯⊕ O^(r) and the program B is accordingly decomposed into B = B_1⊕⋯⊕ B_r so that B_i implements O^(i). The diagram specifies the corresponding query states, where we use the upper index ^(0) in relation to O. We note that it is without loss of generality to assume that A and all B_i use the same input oracle O. Indeed, if this is not the case, we can define O as the direct sum of the oracles used by A and B_i.[every path/.append style=thick,->] [] (-1, 2) node[left]ξ to (0,2);(0,1) rectangle (3,2.5) node[pos=0.5] A(O⊕ O');(3,0) rectangle (6,-1.5) node[pos=0.5] O^(1) = B_1(O);at (7.5, -0.75) ⋯;(8.5,0) rectangle (11.5,-1.5) node[pos=0.5] O^(r) = B_r(O);(0, -3) rectangle (11.5,-4.5) node[pos=0.5] O;[purple] (1.5,1) to node[left] q^(0)(A, O⊕ O', ξ) (1.5,-3); [purple, out=0, in=90] (3, 1.5) to node[right, pos=0.8]q^(1)(A, O⊕ O', ξ) (4.5,0); [purple, out=0, in=90] (3, 2.25) to node[right, pos=0.95]q^(r)(A, O⊕ O', ξ) (10.5,0); [purple] (4.5, -1.5) to node[right] q[ B_1, O, q^(1)(A, O⊕ O', ξ)](4.5,-3); [purple] (10, -1.5) to node[right] q[ B_r, O, q^(r)(A, O⊕ O', ξ)](10,-3);It is quite often the case, that the oracle O' above is composed of several input oracles, each implemented by its own subroutine B_i, see fig:compositionMultiple. In this case, we can use eqn:quantumParallel to obtain an analogue of eqn:randomComposition3:q(A∘ B, O, ξ) = q^(0)[A, O⊕ B(O), ξ] ⊕ ⊕_i q[B_i, O, q^(i)[A, O⊕ B(O), ξ]]. It is often convenient to define L_max(B,O) as the worst-case complexity of L(B,O,ξ) as ξ ranges over all unit vectors (or over all unit vectors in some admissible subspace of initial vectors). Then, using linearity, we can obtain from eqn:quantumComposition: L(A∘ B, O, ξ) = L^(0)[A, O⊕ B(O), ξ]+L[ B, O, q^(1)[A, O⊕ B(O), ξ]] ≤ L^(0)[A, O⊕ B(O), ξ]+L_max(B,O) L^(1)[A, O⊕ B(O), ξ] .and from eqn:quantumCompositionMultipleL(A∘ B, O, ξ) = L^(0)[A, O⊕ B(O), ξ]+ ∑_i L[B_i, O, q^(i)[A, O⊕ B(O), ξ]] ≤ L^(0)[A, O⊕ B(O), ξ]+ ∑_i L_max(B_i,O) L^(i)[A, O⊕ B(O), ξ] .In eqn:quantumComposition2 and eqn:quantumCompositionMultiple2, it is assumed that B and B_i are only executed on the admissible initial states.To summarise, we see that quantum Las Vegas query complexity satisfies composition properties very similar to the “gold standard” of the randomised Las Vegas complexity. One of the goals of this paper is to approach these results for quantum time complexity. In <cit.>, a result in this direction was obtained for evaluation of functions assuming the QRAG model of computation. In this paper, we consider more general state conversion settings, and also obtain partial results for the circuit model. We also think that the approach of this paper is less technical than the one taken in in <cit.>. § OVERVIEW OF THE PAPER This section serves as an informal version of the whole paper, where we introduce all the main concepts, ideas, and sketch the proofs of the main results. In the remaining paper, we fill in all the technical gaps. §.§ TransducersIn the current paper, we take a different approach to time complexity than in the two papers <cit.> mentioned in sec:introPrior. Instead of using span programs or quantum walks, we build on the key technical primitive from <cit.>, which we call a transducer[Not to be confused with transducers from the theory of finite automata, however there are some connections between the two, as we discuss in sec:automaton.]in this paper.Transducers are based on the following mathematical observationwe prove in sec:transducerDefinition: Let S be a unitary acting in a direct sum of two vectors spaces ⊕. For every ξ∈, there exists a unique τ = τ(S,ξ)∈ and in some sense unique v = v(S,ξ)∈ such thatSξ⊕ v ↦τ⊕ v.We say in the setting of eqn:1transduce that S transduces ξ into τ, denoted ξSτ or Sξτ. This defines a mapping ξ↦τ on , which turns out to be unitary. We call it the transduction action of S on , denoted by S_. See fig:transducer for a schematic depiction. The motivation behind this terminology is that while S does not literally map ξ into τ, having S is a legitimate and fruitful way of implementing S_ on a quantum computer, as we show shortly. If a unitary S is designed primarily with this application in mind, we call it a transducer, and say that it implements S_. Schematic depiction of transducers.To the left is the real action of S, which is interpreted as the action of S_ on .Note that parallel wires here denote direct sum of the corresponding subspaces, not tensor product.The same applies to the other figures in this paper.(0,0) rectangle (1, 2) node[pos=0.5] S; [->,, thick] (-1,0.5) node[above]v to (0,0.5); [->,, thick] (-1,1.5) node[above]ξ to (0,1.5); [->,, thick] (1,0.5) to (2,0.5) node[above]v; [->,, thick] (1,1.5) to (2,1.5) node[above]τ; (6,1) rectangle (7, 2) node[pos=0.5] S_; [->,, thick] (5,1.5) node[above]ξ to (6,1.5); [->,, thick] (7,1.5) to (8,1.5) node[above]τ;Let us note that this construction is not new. It has appeared before as an additive trace in the category of isometries in finite-dimensional Hilbert spaces <cit.>. Here we demonstrate that this construction has an operational meaning.It would be interesting to understand the connection between our construction and the one taken in these two references.We will stick to the following terminology. We callthe public andthe private space of S. We say that the transducer S is on the space , but works in the space ⊕. Also, we will call ξ the initial state of S, while ξ⊕ v isthe initial coupling.We call v the catalyst of the transduction eqn:1transduce because it helps in the transformation of ξ into τ, but is not changed in the process. The role of the catalyst is similar to the role of the witness in span programs. In particular, the transduction complexity of the transducer S on an initial vector ξ∈ is given by its size:W(S,ξ) = v(S,ξ)^2. Let T(S) denote the usual time complexity of implementing S as a unitary. As a rule of thumb, among various transducers S with the same transduction action, there is a trade-off between W and T so that the product(1+W(S,ξ))·T(S)stays approximately the same. The importance of this product can be readily seen from the following result. Let spaces ,, and parameters W, >0 be fixed. There exists a quantum algorithm that -approximately transforms ξ into S_ ξ for all transducers S⊕→⊕ and initial states ξ∈ such that W(S,ξ)≤ W. The algorithm conditionally executes S as a black box K = (1+W/^2) times, and makes (K) other elementary operations. Since S generally takes at least one elementary operation, the complexity of the algorithm is dominated by the executions of S, which takes time eqn:tradeoff up to constant factors (assuming = Θ(1)). The term 1 in the definition of K is required as we can have non-trivial transducers with W=0, see, e.g., sec:introAlgorithm->Transducer. Also, as follows from the discussion in <cit.>, the dependence on is optimal.[Proof sketch of thm:introImplementation] We are given a copy of ξ, and our goal is to transform it into τ = S_ξ using S as a black box. Assume we are additionally given a copy of v (in the sense of direct sum, cf. fig:transducer). Then, we can perform the required transformation ξ⊕ v ↦τ⊕ v using S.There are two problems here. First, the algorithm is not given v, and, second, v can have a huge norm. The second problem is solved by breaking ξ down into K copies of ξ/√(K), that is, performing the transformation ξ↦∑_t=0^K-1||⟩t>||⟩ξ>/√(K). The key idea is that we can use the same scaled down catalyst v/√(K) to perform K scaled down transductions ξ/√(K)τ/√(K) as v/√(K) does not change in the process. See fig:pumping, for an illustration.The first problem is solved by “guessing” v/√(K), i.e., using that ξ is close to ξ⊕ v/√(K) if K is sufficiently large. The larger the value of K, the smaller the error imposed by guessing, but the larger the number of executions of S. For a formal proof, see sec:implementation. A graphical illustration of the construction of thm:introImplementation. The initial state ξ is broken down into K=4 copies of ξ/√(K), which are sequentially transformed into τ/√(K) using only one copy of the scaled-down catalyst v/√(K). 3.5 4.5 6 witness/.style=above, node contents=v/√(K) [thick, ] (-1.5,) to node[above]ξ (-1,); [thick, ] (12.5,) to node[above]τ (12,); [->, thick, ] (-1.5,) to node[witness] (0.5,); 0123 §.§ Connection to Quantum Walks In this section, we take a short detour, and inspect the connection between transducers and quantum walks. Like transducers, quantum walks <cit.> replace the desired transformation with some other transformation that is easier to implement: one iteration of the quantum walk. See fig:trans2walk for an informal comparison between transducers and quantum walks, on which we will elaborate in this section.An informal correspondence between transducers and quantum walks.TransducerQuantum WalkExecution of SOne Iteration R_2R_1Transformation S_ Accept/reject of the initial state W(S,ξ) Spectral Gap thm:introImplementation Phase EstimationIn this paper, we consider broadly interpreted discrete-time quantum walks. We identify the two characteristic properties of such algorithms, where the first one is essential, and the second one is usual, but not, strictly speaking, necessary.The first, essential property is that one iteration of a quantum walk is a product of two reflections R_1 and R_2. The quantum walk either rejectsthe initial state ξ, when it is close to an eigenvalue-1 eigenvector of R_2R_1;or accepts it, when ξ is mostly supported on eigenvectors with eigenvalues far from 1.The most standard implementation of quantum walks is a phase estimation of the product R_2R_1 on the initial state ξ. The analysis of quantum walks involves spectral analysis, sometimes assisted by the effective spectral gap lemma <cit.>.The second, optional property is that each reflection, R_1 and R_2, is broken down as a product of local reflections that act on pairwise orthogonal subspaces. This allows for their efficient implementation. The walk is usually described using a bipartite graph (like in fig:walk), where each edge corresponds to a portion of the space. Local reflections are given by vertices, and they act on the direct sum of spaces corresponding to their incident edges. The reflection R_1 executes all the local reflections for one part of the bipartite graph, and R_2 for the second.The two reflections, interleaving, transcend locality and form an involved global transformation.Because of the second property, quantum walks find a large number of algorithmic applications. This includes such basic primitives as Grover's algorithm <cit.> and amplitude amplification <cit.>; as well as the element distinctness algorithm <cit.>, Szegedy quantum walks <cit.> and their various extensions, span programs <cit.>, learning graphs <cit.>, and others. We will show that quantum walks are very often transducers of the following form. Letandbe the public and the private spaces, so that the initial state ξ∈. The transducer is the iteration of the walk: S = R_2R_1, where we additionally assume that the second reflection R_2 acts trivially on . If ξ is negative, we have the following chain of transformations:ξ⊕ v R_1ξ⊕ v R_2ξ⊕ v,certifying that ξSξ. In the positive case, we have the following sequence of transformations:ξ⊕ v R_1 -ξ⊕ -v R_2 -ξ⊕ v,certifying that ξS-ξ.Both sequences follow the standard practice of designing quantum walks. In the negative case, ξ⊕ v is a stationary vector of both R_1 and R_2, and, hence, R_2R_1. In the positive case, ξ⊕ v is the witness for the Effective Spectral Gap Lemma. The transduction vantage point unites these asymmetric positive and negative analyses.An interesting artefact of this construction is that the transduction action of the corresponding quantum walk is exact: It transduces ξto either ξ or -ξ exactly.We will illustrate this construction in more detail in sec:walks by re-proving the main result of <cit.> on electric quantum walks. However, the same applies to any quantum walk that adheres to the same design principles, including algorithms derived from span programs <cit.>, and more generally, algorithms of the type formally defined in <cit.>.To epitomise, we keep the iteration of the quantum walk intact, but replace the wrapping phase estimation by the algorithm of thm:introImplementation. In this way, we significantly simplify the construction by abandoning any spectral analysis both from the implementation and the analysis of the algorithm. We believe this is worthy from the pedagogical point of view, as the corresponding algorithms now require very little background knowledge. v^∘v^∙S^∘ ^∘ ^∙ ^↑§.§ Input Oracle and the Canonical Form Our previous discussion in sec:introTransducers did not consider the input oracle. In this paper, we assume an approach similar to that of quantum query algorithms, where oracle executions and the remaining operations are separated. Moreover, unlike the algorithms, it suffices to have one query for a transducer.We define the canonical form of a transducer S=S(O) with the input oracle O in fig:canonical. The private space = ⊕ is decomposed into the work partand the query part , with the imposed decomposition v = ⊕ of the catalyst v. The query is the very first operation, and it acts only on . It is followed by a unitarywithout queries. A schematic depiction of a transducer in the canonical form. It consists of one application of the oracle O and an input-independent unitary . The catalyst v∈ is separated into two parts v=⊕ with ∈ and ∈.The first one is not processed by the oracle, and the second one is. Note that the input oracle is not applied to the public space.[every path/.append style=thick,->](2,0) rectangle (3, 3) node[pos=0.5] ;(-0.2,0) rectangle (1.2, 1) node[pos=0.5] I⊗ O; [] (-1,0.5) node[above] to (-0.2,0.5); [] (-1,1.5) node[above] to (2,1.5); [] (-1,2.5) node[above]ξ to (2,2.5); [] (3,0.5)to (4,0.5) node[above]; [] (3,1.5) to (4,1.5) node[above] ; [] (3,2.5) to (4,2.5) node[above]τ ; [] (1.2,0.5) to (2,0.5);Canonical transducers are easier to deal with, and every transducer can be converted into the canonical form (see prp:canoning). We will generally assume our transducers are canonical. We write W(S, O, ξ) instead of W[S(O),ξ] for the transduction complexity W(S,O,ξ) = v^2 = ^2 + ^2. Also, the total query state is defined by q(S, O, ξ) =, and the query complexity by L(S, O, ξ) = ^2.This definition is compatible with the case when O is combined of several input oracles like in eqn:introMultipleOracles. In this case, we define the partial query state q^(i)(S, O, ξ) as the state processed by the oracle O^(i), and L^(i)(S, O, ξ) = |q^(i)(S, O, ξ)|^2.The following result, which justifies the name “query complexity”, is proven in sec:reducingOracle:Let spaces , =⊕ be fixed. Moreover, assume the transducer uses r=(1) input oracles combined as in eqn:introMultipleOracles. Let , W, L^(1),…,L^(r)>0 be parameters. Then, there exists an algorithm that conditionally executesas a black box K=(1+W/^2) times,makes (L^(i)/^2) queries to the i-th input oracle O^(i), and uses (K) other elementary operations. The algorithm -approximately transforms ξ into τ(S, O, ξ) for all S, O^(i), and ξ such that W(S, O, ξ)≤ W and L^(i) (S, O, ξ) ≤ L^(i) for all i. [Proof Sketch] Let us first consider the special case of r=1 for simplicity. The crucial new idea compared to thm:introImplementation is that we guess not one but some D copies of the state . They are all processed by one oracle call, and then gradually given to , see fig:implementationBetter. Thus, the transduction complexity becomes ^2 + D ^2, but we have to execute the input oracle only once in every D iterations. The correct choice of D is around ^2/ ^2, so that the norms of the query and the non-query parts of the catalyst become equalised. If there are r=(1) input oracles, we perform the same procedure for all of them. The total transduction complexity grows by a factor of r, which is tolerable by our assumption of r=(1). A graphical illustration of the construction of thm:introImplementationBetter with the same parameters as in fig:pumping, and D=2. We write O instead of I ⊗ O to save space. Note that one oracle execution and D subsequent executions ofform a transducer of its own. 3 4 4.5 6 witness/.style=above, node contents=/√(K)[every node/.style=font=, every path/.append style=thick,->] [-, ] (-1.5,) to node[above]ξ (-1,); [-, ] (12.5,) to node[above]τ (12,); [] (-1.5,) to node[witness] (0.5,); [] (-1.5,) to node[above]/√(K) (-0.9,); [thick, ] (-1.5,1.5) to node[above]/√(K) (-0.9,1.5); [] (11, 3) to (12.5,3); [] (11, 1.5) to (12.5,1.5); 012302This theorem does not hold for superconstant values of r, in which case a more technical thm:optimalImplementation should be used. Note how canonicity of the transducer S is used here. Indeed, the input oracle is executed only once in each execution of the transducer, reducing the total number of oracle calls.The definition of the canonical form is inspired by the implementation of the adversary bound for state conversion from <cit.>. Moreover, as we show in sec:stateConversion, the adversary bound is essentially equivalent to the above construction withbeing empty. Also, we show in sec:function how to implement the usual adversary bound for function evaluation: For every function f D→ [p] with D⊆[q]^n, there exists a canonical transducer S_f with input oracle O_x encoding the input string x, such that, for every x∈ D, S_f transduces ||⟩0>||⟩f(x)> on the input oracle O_x, and W[S_f, O_x, ||⟩0>] = L[S_f, O_x, ||⟩0>] ≤(f),where (f) is the adversary bound of f, defined in sec:function. We can draw the following parallels with span programs. It is known that the dual adversary bound for Boolean functions is equivalent to a very special case of span programs <cit.>. General span programs provide more flexibility, and thus are more suitable for time-efficient implementations <cit.>.While it is possible to implement the dual adversary bound time-efficiently <cit.>, the constructions are more complicated.Span programs sometimes model more general function evaluation <cit.>; but the dual adversary has been extended much further to include arbitrary state conversion <cit.> with general unitary input oracles <cit.>.Transducers treat state conversion with unitary input oracles very naturally. Adding the non-query spaceprovides more flexibility compared to the dual adversary, which again is beneficial for time-efficient implementations. It is also of great help that transducers are quantum algorithms themselves, which makes time analysis especially straightforward.Summarising, there are three basic complexity measures associated with a transducer: * Time complexity T(S), which is independent of the input oracle O and the initial state.Similarly to usual quantum algorithms, the precise value of T(S) depends on the chosen model of quantum computation.* Transduction complexity W(S,O,ξ).It is defined mathematically, and does not depend on the model. On the other hand, it depends on both the input oracle and the initial state.* Query complexity L(S,O,ξ). It is also defined mathematically, and does not depend on the model. The query state q(S,O,ξ) provides more information. Let us finish this section with a few technicalities. Similarly to eqn:queryStateExtended and eqn:LasVegasExtended, we extend the above definitions to ξ' = ξ_1⊕⋯⊕ξ_m∈⊗ viaq(S,O,ξ') = ⊕_t q(S, O,ξ_t), L(S,O,ξ') = ∑_t L(S, O,ξ_t), and W(S, O, ξ') = ∑_t W(S, O,ξ_t).Again, these can be interpreted as the complexities of the transducer I_⊗ S, see cor:byIdentity.If for ξ∈ we use a catalyst v, for cξ with c∈, we can use the catalyst cv. This yieldsW(S, O, cξ) = |c|^2 W(S, O, ξ),q^(i) (S, O, cξ) = cq^(i) (S, O, ξ),L^(i) (S, O, cξ) = |c|^2 L^(i) (S, O, ξ)._good W_max L_max It often makes sense to define the subspace ⊆ of admissible initial vectors to the transducer S(O), and define (S, O) as the supremum of W(S, O, ξ) as ξ ranges over unit vectors in . We defineand ^(i) similarly. We say that the initial state ξ'∈⊗ is admissible if it lies in ⊗. For any such state, by eqn:transductionExtended and eqn:introLinearity, we haveW(S, O, ξ') ≤(S, O) ξ'^2L(S, O, ξ') ≤(S, O) ξ'^2. §.§ Transducers from Quantum Algorithms In this section, we briefly explain how we achieve one of the points on our agenda: conversion of arbitrary quantum algorithms into transducers. We consider both the QRAG model mentioned at the end of sec:introPrior, and the usual circuit model. While the stronger QRAG model allows for better and more intuitive exposition, we are still able to get some useful results in the circuit model.LetA(O) = G_TG_T-1⋯ G_2 G_1be a quantum algorithm, where G_i are individual elementary operations (gates), which also include queries to the input oracle. We call the mapping i↦ G_i the description of A. The number of elementary operations T = T(A) is the time complexity of the algorithm. Trivial Transducer On the one extreme of the trade-off eqn:tradeoff is the trivial transducer S=A. In this case, the catalyst v=0, hence, W(S,O,ξ) = 0, and we get that eqn:tradeoff equals T(A), as expected. Of course, this does not require any change of the model. QRAG Transducer The other extreme of the trade-off eqn:tradeoff is covered by the following construction, which assumes the QRAG model. Write the sequence of states the algorithm eqn:introAlgorithm goes through on the initial state ξ:ξ = ψ_0 G_1ψ_1 G_2ψ_2 G_3⋯G_Tψ_T = τ.We utilize the following history state:v = ∑_t=1^T-1||⟩t> ||⟩ψ_t>.And define the transducer S_A as follows:ξ⊕ v = ∑_t=0^T-1||⟩t> ||⟩ψ_t> S_A∑_t=1^T||⟩t> ||⟩ψ_t> = τ⊕ v,where, in S_A, we first apply G_t conditioned on the first register having value t, and then increment t by one. The first operation is possible assuming the QRAG model and QRAM access to the description of A, see sec:QRAG. The transduction action of S_A(O) is exactly A(O), and W(S_A,O,ξ) = (T-1)ξ^2, which we will simplify to Tξ^2 for brevity.T_R Comments on Time Complexity But what is the time complexity T(S_A)? In S_A, we perform two operations: increment a word-sided register and execute one instruction from the program specified by the address t. If this were a modern randomised computer, both operations would be elementary and took (1) time. From the theoretical side, this is captured by the notion of RAM machine. If we consider the scale of individual qubits instead of word-sided registers, the increment operation takes time (log T), and the second operation at least as much. In order to simplify the following discussion, let us assume that both operations take some time we denote .That is: time required to perform basic word operations, including random access.For simplicity, we do not discriminate between different word operations. We may think ofas being 1, or (log T), but either way, it is some fixed factor which denotes transition from the circuit model, where A operates, [It is the usual assumption that A is a bona fide quantum circuit, but we may also assume that A uses QRAG gates.] to the QRAG model, where S_A is implemented. Canonical Form Neither the first, nor the second transducer above are in the canonical form. The latter is given by the following result, which we prove in sec:programs->transducers.For a quantum program A on the input oracle O, there exists a canonical transducer S_A = S_A(O), whose transduction action is identical to A, and whose complexity is given by fig:complexityTable. In the QRAG model, we assume QRAM access to the description of the program A. Complexity of canonical transducers derived from a quantum algorithm in terms of complexities of the algorithm.Both the circuit and the QRAG model versions are considered.Circuit Model QRAG modelTime T(S_A) (T(A)) ()TransductionW(S_A,O,ξ) L(A,O,ξ) T(A)ξ^2Query stateq(S_A,O,ξ) q(A, O,ξ) q(A, O,ξ)The constructions in both circuit and the QRAG models are already sketched above. We describe here how they can both be transformed into the canonical form so that the query statefrom fig:canonical becomes equal to the total query state q(A, O, ξ) of the algorithm.For the trivial transducer, the catalyst is the total query state, which is all processed by one query to the oracle. Whenever the algorithm was making a query, the transducer switches the query state with the corresponding part of the catalyst, thus simulating a query.For the QRAG transducer, the catalyst eqn:introHistoryState already contains of all the intermediate states of the program. The transducer can then apply the oracle to all of them in one go, and then proceed as before without making a single additional query. Query Compression One consequence of the above results is that we can compress the number of queries of a quantum algorithm A to match its worst-case Las Vegas query complexity. The following result is an immediate corollary of Theorems <ref> and <ref>.Assume A=A(O) is a quantum algorithm with r=(1) input oracles as in eqn:introMultipleOracles. Let , L^(1),…, L^(r)>0 be parameters. There exists a quantum algorithm A'=A'(O) with the following properties: =0pt* It makes (L^(i)/^2) queries to the i-th input oracle O^(i).* For every normalised initial state ξ and every input oracle O=O^(1)⊕ O^(2)⊕⋯⊕ O^(r) as in eqn:introMultipleOracles, we have|A(O)ξ - A'(O)ξ| ≤ as long as L^(i)(A, O,ξ)≤ L^(i) for all i.* In the QRAG model and assuming QRAM access to the description of A, its time complexity is (· T(A)/^2).* In the circuit model, its time complexity is (L· T(A)/^2), whereL = 1 + L^(1)+⋯+ L^(r).This improves over the analogous result from <cit.> in two respects. First, it essentially preserves the time complexity of the algorithm in the QRAG model, while <cit.> did not consider time complexity at all. Second, it allows multiple input oracles, as long as its number is bounded by a constant, while <cit.> only allowed for a single input oracle.[Proof of thm:introQueryCompression] First, obtain the transducer S_A as in thm:introProg->Transducer. Then, apply thm:introImplementationBetter to S_A.One key observation is that q(S_A, O,ξ) = q(A, O,ξ) for all O and ξ. Hence, S_A and A have the same partial Las Vegas query complexities on all O and ξ. This yields the statement on the number of queries bounded by L^(i)/^2.In the QRAG model, W(S_A, O, ξ) = T(A) and T(S_A) = (). This gives the required runtime, as we can use that T(A)≥ 1 to remove the additive 1 factor.In the circuit model, we use that T(S_A) = [T(A)] andW(S_A, O, ξ) = L(A, O, ξ) = ∑_i L^(i) (A, O, ξ) ≤∑_i L^(i).We take care of the additive 1 factor by adding it explicitly to L. This covers the extreme case of L being too small, in particular, 0. §.§ Composition of Transducers Transducers can be composed just like usual quantum algorithms: we consider parallel, sequential, and functional composition of transducers. Thus, from the design point of view, there is little difference between dealing with quantum algorithms and transducers. The advantage is that in all these composition modes, the resources are more tightly accounted for than is the case for traditional quantum algorithms.We will first sketch the composition results for transducers. We will completely omit the case of sequential composition from this section. It suffices to say, that it is very similar to the transformation of programs in sec:introAlgorithm->Transducer. After that, we will give few applications both in the circuit and the QRAG models. Unlike other subsections of this section, we will be able to give complete proofs for most of the results.Composition of Transducers The parallel composition of transducers is straightforward.They are just implemented in parallel as usual quantum algorithms. We have the following relations, akin to eqn:randomParallelComplexity and eqn:quantumParallel:W[⊕_i S_i, O, ⊕_i ξ_i] = ∑_i W(S_i, O, ξ_i)q[⊕_i S_i, O, ⊕_i ξ_i] = ⊕_i q(S_i, O, ξ_i).For the time complexity of implementing ⊕_i _i, we can say precisely as much as for usual quantum algorithms. In some cases it is easy: when all _i are equal, for example. Also, it can be efficiently implemented assuming the QRAG model, see cor:selectProgram. In general, however, direct sum in the circuit model can take as much time as the total complexity of all _i together.The functional composition of transducers parallels fig:composition, where we replace the programs A and B with transducers S_A and S_B, respectively. The functional composition of the two is a transducer S_A∘ S_B whose transduction action on the oracle O is equal to the transduction action of S_A on the oracle O⊕ O', where O' is the transduction action of S_B(O). The following result, proven in sec:functional, parallels eqn:quantumComposition.[Functional Composition of Transducers]The functional composition S_A∘ S_B can be implemented in the following complexity, where we use the extended notion of complexity from eqn:transductionExtended. =0pt* Its transduction complexity satisfiesW(S_A∘ S_B, O, ξ) = W(S_A, O⊕ O', ξ) + W[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]. * Its total query state isq(S_A∘ S_B, O, ξ) = q^(0)(S_A, O⊕ O', ξ) ⊕ q[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]. * Its time complexity is the sum of the (conditional) time complexities of S_A and S_B.Here q^(0) and q^(1) denote the partial query states of S_A to the oracles O and O', respectively. One can see that eqn:introFunctionalTransduction and eqn:introFunctionalQuery meet the form of the “gold standard” of eqn:randomComposition2. The second one strongly resembles eqn:quantumComposition. One unfortunate deviation from this is the time complexity, which afterwards gets multiplied by the transduction complexity in eqn:tradeoff. But since the dependence is additive, we can generally tolerate it, see, e.g., thm:introIterated below.Let us state, for the ease of future referencing, a number of simple consequences of prp:introFunctional similar to eqn:quantumCompositionMultiple–eqn:quantumCompositionMultiple2. First, if all the queries to S_B are admissible, we have by eqn:introFunctionalTransduction and eqn:admissibleQuery:W(S_A∘ S_B, O, ξ) ≤ W(S_A, O⊕ O', ξ) + (S_B, O) L^(1)[S_A, O⊕ O', ξ].Also, from eqn:introFunctionalQuery:L(S_A∘ S_B, O, ξ) = L^(0)(S_A, O⊕ O', ξ) + L[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]≤ L^(0)(S_A, O⊕ O', ξ) + (S_B, O) L^(1)[S_A, O⊕ O', ξ].In the case of multiple input oracles, similar to fig:compositionMultiple, with O' = ⊕_i O^(i) and S_B = ⊕ S_B_ i, so that S_B_i implements O^(i), we have by eqn:introParallel:W(S_A∘ S_B, O, ξ) = W(S_A, O⊕ O', ξ) + ∑_i W[ S_B_i, O, q^(i)(S_A, O⊕ O', ξ)]≤ W(S_A, O⊕ O', ξ) + ∑_i ( S_B_i, O) L^(i)[S_A, O⊕ O', ξ],q(S_A∘ S_B, O, ξ) = q^(0)(S_A, O⊕ O', ξ) ⊕⊕_i q[ S_B_i, O, q^(i)(S_A, O⊕ O', ξ)]andL(S_A∘ S_B, O, ξ) = L^(0)(S_A, O⊕ O', ξ) + ∑_i L[ S_B_i, O, q^(i)(S_A, O⊕ O', ξ)]≤ L^(0)(S_A, O⊕ O', ξ) + ∑_i ( S_B_i, O) L^(i)[S_A, O⊕ O', ξ].In eqn:compositionTransductionMultipleUpper and eqn:compositionLasVegasMultipleUpper, we used eqn:admissibleQuery in assumption that all the queries to S_B and S_B_i are admissible.Composition of Programs Let us give a few examples of composition of quantum programs using transducers. We focus on the general state conversion here, evaluation of function postponed till sec:introPurifier. We consider both the circuit and the QRAG models.Assume the settings of fig:compositionMultiple, where r=(1) and all A and B_i are in the circuit model. Define B = ⊕_i B_i, and let , L^(1),…, L^(r)>0 be parameters.Then, there exists a quantum algorithm A' = A'(O) such that |A'(O)ξ - A[O⊕ B(O)]ξ|≤ for every normalised ξ as long as L^(i)(A, O⊕ B(O), ξ)≤ L^(i) for all i≥ 1. The program A' can be implemented in the circuit model in time[L· T(A) + ∑_i T[B_i] L^(i)/^2],where L = 1+L^(1)+⋯+L^(r). Use the circuit version of thm:introQueryCompression for the program A, where we treat calls to O as ordinary operations (in other words, we assume A has r input oracles O^(1),…, O^(r)). After that, replace each call to O^(i) by the execution of B_i. It turns out that it is not efficient to use prp:introFunctional here as this would increase time complexity of the resulting transducer. In thm:introCompositionCircuit, the emphasis is on the time complexity. If we want to simultaneously bound query complexity, we treat O as the input oracle in A as well. This gives time complexity eqn:introCompositionCircuit with L = 1+ L^(0)+L^(1)+⋯+L^(r) and the total number of queries[L^(0) + ∑_i Q[B_i] L^(i)/^2],where Q(B_i) is the number of queries made by B_i.We additionally require that L^(0)[A, O⊕ B(O), ξ]≤ L^(0). Assume the settings of fig:compositionMultiple. Let , T>0 be parameters. Assuming the QRAG model and QRAM access to an array with description of A and all B_i, there exists a quantum algorithm A' = A'(O) with time complexity (· T/^2) such that, for every normalised ξ, we have A'(O)ξ - A[O⊕ B(O)]ξ≤ as long asT(A) + ∑_i=1^r T(B_i) L^(i)(A, O⊕ B(O), ξ) ≤ T.The algorithm makes (L/^2) queries to the input oracle O, where L is an upper bound on L(A∘ B, O, ξ), given by eqn:quantumCompositionMultiple1. Convert A and all the B_i into transducers S_A and S_B_i as in thm:introProg->Transducer. We obtain the transducer S_B = ⊕_i S_B_i for B. Then compose S_A∘ S_B using prp:introFunctional. By definition, its transduction action is identical to A(B).Since the transduction complexity of S_A on a normalised initial state is bounded by T(A) and that of S_B_i by T(B_i), we get from eqn:compositionTransductionMultipleUpper that for a normalised ξ:W(S_A∘ S_B, O, ξ) ≤ T(A) + ∑_i T(B_i) L^(i)[A, O⊕ B(O), ξ]. The main reason this construction is efficient in the QRAG model is that the time complexity of the transducers stays bounded by () the whole time. Indeed, such is the time complexity of the individual transducers obtained from A and B^(i). Parallel composition can be performed efficiently in the QRAG model (prp:program->transducerParallel), and the time complexity of the functional composition is the sum of its constituents.The transducers S_A and S_B_i have the same query states as A and B_i, respectively. By eqn:compositionQueryMultiple, we get that the total query state of S_A∘ S_B is identical to that of A∘ B, which is given by eqn:quantumCompositionMultiple. The statement of the theorem follows from thm:introImplementationBetter. Observe that thm:introCompositionQRAG, while assuming a stronger model, gives a stronger result than thm:introCompositionCircuit. The differences are as follows. First, we do not have to assume that r=(1). Second, the L^(i)(A, O⊕ B(O), ξ) in eqn:introCompositionQRAG are the actual values of the Las Vegas query complexity, while L^(i) in eqn:introCompositionCircuit are only upper bounds on them. This can be important if L^(i)(A, O⊕ B(O), ξ) heavily fluctuates over different input oracles O. Finally, the T(A) term is oddly multiplied by L in eqn:introCompositionCircuit.Altogether, the expression in eqn:introCompositionQRAG is more natural. It is also similar to the one in Section 1.2 of <cit.>. Our result is more general though, as it covers arbitrary state conversion, and not only function evaluation (we can assume =Ω(1) in thm:introCompositionQRAG as it gives bounded-error evaluation of a function).On the other hand, if the estimates L^(i) are sufficiently precise, and T(A) is smaller than average T(B^(i)), the expression in eqn:introCompositionCircuit is quite close to eqn:introCompositionQRAG. Multiple Layers of Composition Here we assume the QRAG model of computation. Theorems <ref> and <ref> considered one layer of composition. In the case of multiple layers, similarly as for the span programs, it is advantageous to perform all the compositions in the realm of transducers, and to transform the resulting transducer back into an actual algorithm only at the very end.One node B_t; i_1, i_2, …, i_t in the composition tree. Its initial state (in the general sense of eqn:transductionExtended) is given by ξ_t; i_1, i_2, …, i_t. It has several subroutines of the form B_t+1; i_1; i_2,…,i_t, i_t+1 with the corresponding query state ξ_t+1; i_1; i_2,…,i_t, i_t+1.[every path/.append style=thick,->] [] (-2, -0.5) node[left]ξ_t; i_1, i_2,…, i_t to (-1,-0.5);(-1,0) rectangle (1,-1) node[pos=0.5] B_t; i_1; i_2,…,i_t;(0,-2) rectangle (2.5,-3) node[pos=0.5] B_t+1; i_1; i_2,…,i_t, 1;(3,-2) rectangle (5.5,-3) node[pos=0.5] B_t+1; i_1; i_2,…,i_t, 2;at (6.3, -2.5) ⋯;at (10.5, -2.5) ⋯;(7,-2) rectangle (10,-3) node[pos=0.5] B_t+1; i_1; i_2,…,i_t, i_t+1; [purple, out=0, in=90] (1,-0.8) to node[left, pos=0.7] ξ_t+1; i_1, i_2,…, i_t,1 (1.5,-2); [purple, out=0, in=90] (1,-0.6) to node[left, pos=0.85] ξ_t+1; i_1, i_2,…, i_t,2 (4.5,-2); [purple, thick, -] (1,-0.4) to (1.2,-0.4); [purple, thick, -] (1,-0.2) to (1.2,-0.2); [purple, out=0, in=90] (1,-0.3) to node[left, pos=0.95] ξ_t+1; i_1, i_2,…, i_t,i_t+1 (9,-2); Consider a composition tree of quantum subroutines. The top layer is the algorithm B_0 that has several subroutines of the form B_1,i, like B_i used to be for A in fig:compositionMultiple. We define the tree downwards so that, in general, a subroutine B_t; i_1,i_2,…,i_t has several subroutines of the form B_t+1; i_1, i_2, …, i_t, i_t+1, see fig:multipleLayers. Let d be the maximal value of t in B_t; i_1,i_2,…,i_t. It is the depth of the composition tree. We assume all the subroutines have access to some common oracle O. Define the composition B=B(O) of the whole tree in the obvious inductive way, so that the initial state ξ of the composed algorithm is the initial state ξ of B_0.It is not hard to get the Las Vegas query complexity of B using eqn:quantumCompositionMultiple inductively or from the general principles. Let q_t; i_1,i_2,…,i_t(O,ξ) be the query state given by B_t; i_1,i_2,…,i_t to the input oracle O when B is executed on the initial state ξ. Then,q(B, O, ξ) = ⊕_t; i_1,…, i_t q_t; i_1,i_2,…,i_t(O,ξ).We obtain a similar result for the time complexity. Let ξ_t; i_1,i_2,…,i_t(O,ξ) be the total query state given to the subroutine B_t; i_1,i_2,…,i_t. In particular, ξ_0(O,ξ) = ξ.Assuming the QRAG model and QRAM access to the description of all B_t; i_1,…,i_t as above, there exists a quantum algorithm B' = B'(O) with time complexity (· (d+1)· T/^2) such that, for every ξ, we have B'(O)ξ - B(O)ξ≤ as long as∑_t; i_1,…, i_tT(B_t; i_1,i_2,…,i_t) ξ_t; i_1,i_2,…,i_t(O,ξ)^2 ≤ T,where the summation is over all the vertices of the composition tree. The algorithm makes (L/^2) queries to O, where L is an upper bound on the Las Vegas query complexity of B as obtained from eqn:compositionTreeQueryComplexity. We use the induction on d to show that, under the assumptions of the theorem, there exists a transducer S_B whose transduction action is identical to B, whose transduction complexity is given by the left-hand side of eqn:introCompositionTree, and whose time complexity is [(d+1)·].The base case is given by d=0, where we only have B_0, which we transform into a transducer using thm:introProg->Transducer. The transduction complexity of S_B_0 on a normalised initial state is T(B_0).Hence on the initial state ξ_0, the transduction complexity is T(B_0)ξ_0^2 by eqn:introLinearity.Assume the theorem is true for depth d. For the depth d+1, we treat the nodes B_t; i_1,i_2,…,i_t with t≤ d as forming a composition tree A with depth d, and the nodes B_d+1; i_1,i_2,…,i_d, i_d+1 as input oracles to A. In other words, A has an input oracle O⊕ O' with O' = ⊕_i_1,…,i_d, i_d+1 B_d+1; i_1,…,i_d, i_d+1(O). We use the induction assumption to obtain a transducer S_A for the composition tree A, whose transduction complexity on O ⊕ O' and ξ is given by the sum in eqn:introCompositionTree, where we restrict the sum to t≤ d. We convert each B_d+1; i_1,…,i_d, i_d+1 into a transducer and join them via direct sum to obtain a transducer S_B_d+1 whose transduction action on the oracle O is identical to O'. Then, we apply prp:introFunctional to get the transducer S_B = S_A∘ S_B_d+1. Its transduction complexity on O and ξ is given by the left-hand side of eqn:introCompositionTree with all the terms involved, where the term T(B_d+1; i_1,i_2,…,i_d, i_d+1) ξ_d+1; i_1,i_2,…,i_d, i_d+1(O,ξ)^2is the contribution of B_d+1; i_1,i_2,…,i_d, i_d+1.All the transducers in S_B_d+1 can be implemented in parallel due to the QRAG assumption (see prp:program->transducerParallel), hence, its time complexity is (). Thus, T(S_B) = T(S_A) + T(S_B_d+1) = [(d+1)]. Finally, using eqn:quantumCompositionMultiple and eqn:compositionQueryMultiple, we get that S_B and B have the same query state. The statement of the theorem again follows from thm:introImplementationBetter. Iterated Functions Due to the discussion after thm:introCompositionCircuit, one might think that several layers of composition are difficult for the circuit model. However, this is not the case if the composition tree is sufficiently homogenous. As an example, we consider evaluation of iterated functions in the circuit model.Let f [q]^n→ [q] and g [q]^m→ [q] be total functions. The composed function f∘ g [q]^nm→ [q] is defined by[f∘ g] (z_1,1, …,z_1,m, z_2,1,…,z_2,m,……,z_n,1,…,z_n,m)= f[ g(z_1,1, …,z_1,m),g(z_2,1, …,z_2,m), …, g(z_n,1, …,z_n,m)],which is equivalent to eqn:randomComposedFunction with all the inner functions being equal. The function composed with itself several times is called iterated function. We use the following notation f^(1) = f and f^(d+1) = f^(d)∘ f = f∘ f^(d).Iterated functions have been studied before both classically <cit.> and quantumly <cit.>. For the case of Boolean functions, an essentially optimal algorithm was given by Reichardt and Špalek in <cit.> using span programs. [ Papers <cit.> are mostly known for their query results, but they also contain statements on the time complexity of the resulting algorithms. It is these time complexity statements that we extend in thm:introIterated. ] It is based on the use of span programs. Similar results for the general case of non-Boolean functions can be easily obtained using composition of transducers. While it is true that the time complexity grows with each layer of iteration, it only does so additively, which is overshadowed by optimal multiplicative handling of the query complexity. We formally prove the following result in sec:iterated.Let f [q]^n→[q] be a total function. There exists a bounded-error quantum algorithm that evaluates the iterated function f^(d) in query complexity ((f)^d) and time complexity[We use _f(·) to indicate that the suppressed constant may depend on the particular function f.] _f[d·(f)^d], where (f) is the adversary bound of f. The algorithm works in the circuit model. We use induction to construct the transducer S_f^(d) evaluating the function f^(d) and having worst-case query complexity (f)^d, and transduction complexity [(f)^d].The base case is the transducer S_f from thm:introAdv. For the inductive step, we use prp:introFunctional with S_A = S_f^(d) and S_B = S_f to get S_f^(d+1) = S_A∘ S_B. In notations of that theorem, O encodes the input to f^(d+1) and O' the input to f^(d) obtained by evaluating the lowest level of the composition tree. First, S_A does not make direct queries to O. Second, S_B = S_f has worst-case Las Vegas query complexity (f) on a unit admissible vector.Hence, by eqn:compositionLasVegasUpper and the induction assumption:L[S_A∘ S_B, O, ||⟩0>]≤(f) L[S_A, O', ||⟩0>] ≤(f)^d+1.Similarly, the worst-case transduction complexity of S_B on a unit admissible vector is (f), hence by eqn:compositionTransductionUpper:W[S_A∘ S_B, O, ||⟩0>] ≤ W[S_A, O, ||⟩0>] + (f) L(S_A, O', ||⟩0>)≤[(f)^d] + (f)^d+1 = [(f)^d+1]for a sufficiently large constant behind the big-Oh.For the time complexity, we have by induction that S_f^(d) has time complexity d· T(S_f). The theorem follows from thm:introImplementationBetter. Note that the transducer S_f^(d) in the proof of thm:introIterated is different from the transducer S'_f^(d) we would get by applying thm:introAdv to the adversary bound of f^(d) obtained using the composition results for the adversary bound. Indeed, for S'_f^(d), its transduction and query complexities are equal, which is not the case for S_f^(d). Also, for S'_f^(d), we have no guarantees on its running time. In thm:introIterated, we use the non-query partof the catalyst as a “scaffolding” to build a time-efficient iterative algorithm. §.§ Purifiers and Composition of FunctionsAlthough we have studied thriftiness from sec:introPrior, we have so far not touched much on exactness. True, in most cases, like in sec:introWalks on quantum walks, or thm:introAdv on the adversary bound, the transduction action of the corresponding transducer is exact. In this section, we will show how to get very close to general exactness starting from an arbitrary algorithm evaluating a function with bounded error.We consider both Boolean and non-Boolean functions. In the Boolean case, we abstract the action of the function-evaluating algorithm as an input oracle performing the following state generation:O_ψ|M⟩ |0> ↦|M⟩|ψ> = |B⟩|0>|N⟩|ψ_0> + |B⟩|1> |N⟩|ψ_1>in some space = ⊗ with = ^2. The action of the purifier only depends on the state ψ in eqn:introPurifierInput, hence the notation O_ψ. We allow the gap to be at any position c between 0 and 1. In other words, we assume there exist constants 0≤ c-d < c + d ≤ 1 such thateither ψ_1^2 ≤ c-dor ψ_1^2 ≥ c+d.The first case is negative, the second one positive, or, f(ψ)=0 and f(ψ)=1, respectively.In the non-Boolean case, the range is some [p].For simplicity, we assume p=(1) here. An input oracle has the formO_ψ|M⟩ |0> ↦|M⟩|ψ>=∑_j=0^p-1|B⟩|j>|N⟩|ψ_j>,with = ^p, and we assume there exists a (unique) f(ψ)∈[p] such thatψ_f(ψ)^2 ≥1/2+dfor some constant d>0.The traditional way of performing error reduction is via majority voting. The following result is folklore.For any >0, there exists an algorithm with bidirectional access to an oracle like in eqn:introrPurifierInput2 that has the following properties. Assuming the oracle satisfies eqn:introPurifierCases or eqn:introPurifierCases2, the algorithm evaluates f(ψ) with error at most . The query complexity of the algorithm is [log1/] and its time complexity in the circuit model is polynomial in log1/. The majority-voting construction is a direct quantisation of a purely classical technique. The logarithmic query complexity in the above theorem results in extra logarithmic factors that can be found in the analyses of a large variety of quantum algorithms. In this paper, we develop an alternative, genuinely quantum approach to error reduction, that we call a purifier. The main feature is that, unlike majority voting, the query complexity of a purifier stays bounded by a constant no matter how small the erroris. This is effectively equivalent to having an errorless algorithm (although, we cannot obtain an exact algorithm with a finite overhead in general). We prove the following theorem in sec:purifiers.S_purFor any >0, there exists a canonical transducerwith bidirectional access to an oracle like in eqn:introrPurifierInput2 that has the following properties. Assuming the oracle satisfies eqn:introPurifierCases or eqn:introPurifierCases2, the purifier transduces ||⟩0> into ||⟩f(ψ)> with error at most(in the sense to be made exact in sec:perturbedTransducers). Both its transduction and query complexities are bounded by a constant. Its time complexity is [slog1/] in the circuit model, and () in the QRAG model, where s is the number of qubits used by . The purifier is inspired by the corresponding construction in <cit.>, which used the dual adversary bound. It had the same characteristic property of query complexity being bounded by a constant, but there are some differences.* The purifiers in <cit.> worked solely in the query complexity settings. The resulting dual adversary bound was for exact function evaluation. Also, by the nature of the adversary bound, it was implicitly assumed that there is only a finite collection of possible input oracles, and, technically, for different collections, we obtain different purifiers. * The purifiers in the current paper are constructed keeping both query and time complexity in mind. Also, the same purifier works for all (infinitely many) possible input oracles. * Due to these improvements, the purifier ceases to be exact, but introduces a small error. S_toy ψ [Proof sketch of thm:introPurifier] We consider the Boolean case, as the general case can be easily obtained using the Bernstein-Vazirani algorithm <cit.> as in, e.g., <cit.>. Our construction is a quantum walk in the sense of sec:introWalks. In particular, it transduces ||⟩0> into (-1)^f(ψ)||⟩0>.The overall structure is given by the following toy transducerin fig:introPurifier1. It is a quantum walk on the one-sided infinite line. That is, = R_2R_1, where the reflection R_1 is the product of the local reflections about the odd vertices 1,3,5,⋯, and R_2 about the positive even vertices 2,4,6,⋯. The local reflection at the vertex i>0 is given by the X operation, which maps||⟩i-1> + ||⟩i> ↦||⟩i-1> + ||⟩i>||⟩i-1> - ||⟩i> ↦ -||⟩i-1> + ||⟩i>. A toy transducer illustrating the overall construction of a purifier. It is a quantum walk on the one-sided infinite line. Each edge correspond to an element of the standard basis written below it. The public space is given by ||⟩0>. We have two different catalysts (a) and (b) for the same initial state, the numbers above the edge giving the corresponding coefficients.[auto]at (-2,0) (a);(0) at (0,0) [circle, draw] 0;[remember= as(initially 0)] in 1,...,5 () at (*2,0) [circle, draw] ;() to node [above, blue] 1 node [below, gray] ||⟩> ();(5) to (11,0) node[right] ⋯;[shift=(0,-2)]at (-2,0) (b);(0) at (0,0) [circle, draw] 0;[remember= as(initially 0)] in 1,...,5 () at (*2,0) [circle, draw] ; mod(,2) ? 1 : int(-1) () to node [above, blue]node [below, gray] ||⟩> ();(5) to (11,0) node[right] ⋯;Now, it is not hard to see that in the negative and the positive case, we have the following mappings, which follow the general case of eqn:QW_sequence_negative and eqn:QW_sequence_positive:||⟩0> + ∑_i=1^+∞||⟩i> ||⟩0> + ∑_i=1^+∞||⟩i> ||⟩0> + ∑_i=1^+∞ (-1)^i||⟩i>- ||⟩0> + ∑_i=1^+∞ (-1)^i||⟩i>.This formally gives us both ||⟩0> ||⟩0> and ||⟩0>-||⟩0>. Of course, this does not contradict thm:introTransduce, because the corresponding space has infinite dimension and both catalysts in eqn:toyTransduce have infinite norm.Nonetheless, we will be able to utilise this general construction. The first order of business is to reduce the norm of the catalysts. Our next step towards a purifier will be a multidimensional quantum walk on the line, in the sense that each edge corresponds to a multidimensional subspace. See fig:introPurifier2.An improved transducer. It is a multidimensional quantum walk. Each edge corresponds to the element of the standard basis beneath it tensor multiplied by ^⊗∞. The public space is given by ||⟩0>. For the negative and positive case, we have the catalysts like in (a) and (b), respectively, where the vector above the edge is placed in the corresponding subspace.[auto]at (-2,0) (a);(0) at (0,0) [circle, draw] 0;[remember= as(initially 0)] in 1,...,5 () at (*2,0) [circle, draw] ;() to node [below, gray] ||⟩> ();(0) to node[above, blue] 1 (1);(1) to node[above, blue](2);(2) to node[above, blue] ^⊗ 2 (3);(3) to node[above, blue] ^⊗ 3 (4);(4) to node[above, blue] ^⊗ 4 (5);(5) to (11,0) node[right] ⋯;[shift=(0,-2)]at (-2,0) (b);(0) at (0,0) [circle, draw] 0;[remember= as(initially 0)] in 1,...,5 () at (*2,0) [circle, draw] ; mod(,2) ? 1 : int(-1) () to node [below, gray] ||⟩> ();(0) to node[above, blue] 1 (1);(1) to node[above, blue] - (2);(2) to node[above, blue] ^⊗ 2 (3);(3) to node[above, blue] -^⊗ 3 (4);(4) to node[above, blue] ^⊗ 4 (5);(5) to (11,0) node[right] ⋯;We will define a vectorthat depends on ψ and satisfies = 1-Ω(1). The initial coupling is given by∑_i=0^∞ (-1)^i· f(ψ)||⟩i>|>^⊗ i,and the transduction complexity is bounded by∑_i=1^∞^2i = (1). Let us now explain how to implement the local reflections eqn:toyOperations for this modified transducer. Up to a sign, the content of the space incident to a vertex i>0 is given by||⟩i-1>|>^⊗ i-1 + (-1)^f(ψ)||⟩i>|>^⊗ i.We use the input oracle to obtain the state||⟩i-1>|>^⊗ i-1||⟩ψ> + (-1)^f(ψ)||⟩i>|>^⊗ i.Later we can get back from eqn:introPurifier2 to eqn:introPurifier1 by uncomputing the ψ. Therefore, bringing the term ||⟩i>|>^⊗ i-1 outside the brackets, it suffices to implement the mapping||⟩0> ||⟩ψ> + ||⟩1>|> ↦||⟩0> ||⟩ψ> + ||⟩1>|>, if f(ψ)=0;and ||⟩0> ||⟩ψ> - ||⟩1>|> ↦ -||⟩0> ||⟩ψ> + ||⟩1>|>, if f(ψ)=1. This is where the condition eqn:introPurifierCases comes into play. We use the same rescaling idea as in <cit.>, and define the state=1/a||⟩0> ||⟩ψ_0> + b ||⟩1>||⟩ψ_1>, if f(ψ)=0;a ||⟩0> ||⟩ψ_0> + 1/b||⟩1>||⟩ψ_1>, if f(ψ)=1.witha = √(1-c+d/1-c-d) b = √(c+d/c-d).It is not hard to check that ^2 = 1-Ω(1). Now the operation in eqn:notToyOperations reads as[||⟩0> + 1/a||⟩1>] ||⟩0> ||⟩ψ_0> + [||⟩0> + b ||⟩1>]||⟩1> ||⟩ψ_1> ↦[||⟩0> + 1/a||⟩1>] ||⟩0> ||⟩ψ_0> + [||⟩0> + b ||⟩1>]||⟩1> ||⟩ψ_1>,[||⟩0> - a ||⟩1>] ||⟩0> ||⟩ψ_0> + [||⟩0> - 1/b||⟩1>]||⟩1> ||⟩ψ_1> ↦[-||⟩0> + a ||⟩1>] ||⟩0> ||⟩ψ_0> + [-||⟩0> + 1/b||⟩1>]||⟩1> ||⟩ψ_1>,which can be implemented as the 2-qubit reflection about the span of the states[a||⟩0> + ||⟩1> ]||⟩0> [||⟩0> + b ||⟩1> ]||⟩1>. Following the same logic as in the toy example, we obtain that ||⟩0>(-1)^f(ψ)||⟩0>. However, this time, both the transduction and the query complexities are bounded by a constant. The problem is that this construction still requires infinite space. We solve this by truncating the line after some vertex D. This introduces an error, but since the norms of the vectors decrease exponentially with i, it suffices to take D = [log1/]. Finally, the transducer can be converted into a canonical form by the standard technique of prp:canoning. y⃗The main purpose of purifiers is to reduce error. Let us give few examples. First, recall the definition of the composed function f∘ gfrom eqn:introComposedFunction:[f∘ g] (z_1,1, …,z_1,m, z_2,1,…,z_2,m,……,z_n,1,…,z_n,m)= f[ g(z_1,1, …,z_1,m),g(z_2,1, …,z_2,m), …, g(z_n,1, …,z_n,m)].We use notation_i = (z_i,1, …,z_i,m)x = [g(_1), g(_2),… g(_n)]so that f(x) = [f∘ g](z). In the circuit model, we have the following result on the evaluation of this function.Let A and B be quantum algorithms in the circuit model that evaluate the functions f and g, respectively, with bounded error. Then, there exists a quantum algorithm in the circuit model that evaluates the function f∘ g with bounded error in time complexity(L)[T(A) + T(B) + slog L]where L is the worst-case Las Vegas query complexity of A, and s is the space complexity of B. We obtain the transducer S_A in the circuit model using thm:introProg->Transducer. The transduction and query complexities of S_A are bounded by L. Letbe the purifier for the input oracle B. Consider the transducer S = S_A∘⊕_i, see fig:introCompositionFunctionCircuit. The transduceron the input oracle B(O__i) evaluates g(y_i) with diminished error. It suffices to make error somewhat smaller than 1/L. Then, S on the input oracle ⊕_i B(O__i) evaluates f∘ g with bounded error.A composition scheme for thm:introCompositionFunctionCircuit. The input oracle O_z is broken down as ⊕_i O__i. The composed transducer contains the elements inside the blue box. The program B is executed as is, serving as an input oracle to the composed transducer. (#1,#2)#3 #1 #2 [shift=(#1,#2)] (0,0) rectangle (2,-1) node[pos=0.5] ; (0,-1.5) rectangle (2,-2.5) node[pos=0.5] B; (0,-3) rectangle (2,-4) node[pos=0.5] O__#3; [purple] (1,-1) to (1,-1.5); [purple] (1,-2.5) to (1,-3);[every path/.append style=thick,->](7,0) rectangle (9,-1) node[pos=0.5] S_A; (3,-2)1 (6,-2)2at (9,-2.5) ⋯; (10,-2)n [purple, out=270, in=90] (7.2, -1) to (4,-2) ; [purple, out=270, in=90] (7.5, -1) to (7,-2) ; [purple, out=270, in=90] (8.8, -1) to (11,-2) ; [rounded corners=10pt, blue] (2,0.5) rectangle (13,-3.25);The transduction complexity ofon a unit admissible vector is (1). Hence, by eqn:compositionTransductionUpperW(S, O_z, ||⟩0>) ≤ W(S_A, O_x, ||⟩0>) + (1)· L(S_A, O_x, ||⟩0>) = (L).All the purifiers can be implemented in parallel, hence by prp:introFunctional,T(S) = T(S_A) + T() = [T(A)] + (slog L).The theorem follows from thm:introImplementation applied to S, where we replace execution of the input oracle by the execution of B. Again, all the B can be executed in parallel. In the QRAG model, we can easily deal with different functions g_i, like in the following function, which we already considered in eqn:randomComposedFunction:f[ g_1(z_1,1, …,z_1,m),g_2(z_2,1, …,z_2,m), …, g_n(z_n,1, …,z_n,m)].We again use notation_i = (z_i,1, …,z_i,m)x = [g_1(_1), g_2(_2),… g_n(_n)]. Consider the function as in eqn:randomComposedFunctionCopy. Let A and B_1,…, B_n be quantum algorithms that evaluate the functions f and g_1,…, g_n, respectively, with bounded error. Assuming the QRAG model and QRAM access to the description of A and B_1,…, B_n, there exists a quantum algorithm that evaluates the function eqn:randomComposedFunctionCopy with bounded error in time complexity()max_x[T(A) + ∑_i=1^n T(B_i) L^(i)_x(A)].Here L^(i)_x(A) is the i-th partial Las Vegas complexity L^(i)[A, O_x, ||⟩0>] of the algorithm A on the input oracle encoding x. We obtain the transducer S_A in the QRAG model using thm:introProg->Transducer. Its transduction complexity is T(A). We assume all B_i have the same range and the same error. Letbe the corresponding purifier. Finally, let S_B_i be the transducer in the QRAG model corresponding to B_i. Consider the composed transducer S as in fig:introCompositionFunctionQRAG. Similarly to thm:introCompositionFunctionCircuit,the transducer S on the input oracle O_z = ⊕_i O__i evaluates the function f∘ g with bounded error provided that the error of the purifier is sufficiently smaller than 1/L.A composition scheme for thm:introCompositionFunctionQRAG. The input oracle O_z is again broken down as ⊕_i O__i. The composed transducer S contains the elements inside the blue box. This time every B_i is turned into a transducer and partakes in the composition. (#1,#2)#3 #1 #2 [shift=(#1,#2)] (0,0) rectangle (2,-1) node[pos=0.5] ; (0,-1.5) rectangle (2,-2.5) node[pos=0.5] S_B_#3; (0,-3) rectangle (2,-4) node[pos=0.5] O__#3; [purple] (1,-1) to (1,-1.5); [purple] (1,-2.5) to (1,-3);[every path/.append style=thick,->](7,0) rectangle (9,-1) node[pos=0.5] S_A; (3,-2)1 (6,-2)2at (9,-2.5) ⋯; (10,-2)n [purple, out=270, in=90] (7.2, -1) to (4,-2) ; [purple, out=270, in=90] (7.5, -1) to (7,-2) ; [purple, out=270, in=90] (8.8, -1) to (11,-2) ; [rounded corners=10pt, blue] (2,0.5) rectangle (13,-4.75);Using that the transduction and the query complexity ofare (1), we obtain that (∘ S_B_i, O__i) = [T(B_i)]. Therefore, by eqn:compositionTransductionMultipleUpper and using that A and S_A have the same query state:W[S, O_z, ||⟩0>] ≤ W(S_A, O_x, ||⟩0>) + ∑_i [T(B_i)] L^(i)[A, O_x, ||⟩0>]. For the time complexity, S is a composition of three transducers, where the last two are direct sums.All three of them have time complexity (), hence, this is also the time complexity of S. The theorem follows from thm:introImplementation. Comparison between Theorems <ref> and <ref> is similar to the comparison between Theorems <ref> and <ref> in sec:introComposition. The second theorem considers a more general case eqn:randomComposedFunctionCopy, and its formulation is close to eqn:randomComposition2. On the other hand, in eqn:introCompositionFunctionCircuit, T(B) will most likely dominate slog L, so the latter can be removed. Also, if T(A) is smaller than T(B), the whole expression is dominated by L· T(B), which is what we would like to have.Finally, in the QRAG model, any bounded-error quantum algorithm A can be turned into an essentially exact transducer S_A such that, up to constant factors, its transduction complexity is T(A) and its query complexity is the query complexity of A. The latter transducers can be composed in multiple layers as in thm:introCompositionTree.§ PRELIMINARIESIf not said otherwise, a vector space is a finite-dimensional complex inner product space.They are denoted by calligraphic letters.We assume that each vector space has a fixed orthonormal basis, and we often identify an operator with the corresponding matrix. I_ stands for the identity operator in . The inner product is denoted by <·,·>. A^* stands for the adjoint linear operator. All projectors are orthogonal projectors. We use ket-notation to emphasise that a vector is a state of a quantum register, or to denote the elements of the computational basis.We usefor the big-Oh notation in order to distinguish from O, which we use for input oracles. _A means that the constant may depend on A. If P is a predicate, we use 1_P to denote the corresponding indicator variable; which is equal to 1 if P is true, and to 0 otherwise. §.§ Query AlgorithmsIn this section, we briefly describe the model of quantum query algorithms, and define quantum Las Vegas query complexity. The query model itself is essentially standard <cit.>, with the main difference that we do not restrict ourselves to the evaluation of functions, and also allow for multiple input oracles, which can be arbitrary unitaries. The notion of quantum Las Vegas query complexity is relatively new <cit.>.I^∘ I^∙ I^∘ ^∘ ^∙ ^↑A quantum query algorithm A works in space , which we call the workspace of the algorithm. The algorithm is given an oracle O, which is a unitary [ While <cit.> define more general input oracles, we, for simplicity, consider only unitary input oracles in this paper. ]in some space . The interaction between the algorithm and the oracle is in the form of queries, which we are about to define. The workspace is decomposed as = ⊕, where the oracle is only applied to the second half. Also, = ⊗ for some , and the query is= ⊕ I⊗ O,whereand I are the identities inand , respectively.In terms of registers, we assume the decomposition = ⊕ is marked by a registerso that |R⟩|0> corresponds toand |R⟩|1> to . Then, the queryis an application of O, controlled by , where O acts on some subset of the registers (which constitute ).The quantum query algorithm A=A(O) is a sequence of linear transformations in :A(O) = U_Q U_Q-1⋯ U_1 U_0,where U_t are some input-independent unitaries in . See fig:queryAlgorithm. Thus, the algorithm implements a transformation O↦ A(O): from the input oracle O into the linear operator eqn:preAlgorithm in .A graphical illustration of a quantum query algorithm with Q=3 queries. The algorithm interleaves input-independent unitaries U_t with queries = ⊕ I⊗ O. The intermediate state ψ_t after U_t-1 and before the t-th query is decomposed as ψ_t^∘⊕ψ_t^∙, where only the second half is processed by the oracle. [every node/.style=font=, every path/.append style=thick,->] [] (-1.9,1.9) to node[above]ξ (-0.9,1.9);(-0.9,0.15) rectangle (0, 3.35) node[pos=0.5] U_0; 123[] (12,1.9) to node[above]τ (13,1.9);We will generally work with the state conversion formalism.We say that A transforms ξ into τ on oracle O, if A(O)ξ = τ. We say that A does so -approximately if |τ - A(O)ξ|≤. In this context, we will often callthe error of the algorithm. Let us make two important remarks on the structure of thus defined query algorithms.Note that all the queries in eqn:preAlgorithm are identical, i.e., the oracle O is always applied to the same registers and is always controlled by . To acknowledge this, we say that all the queries in A are aligned. Usually, this is not important, but the alignment property will play a significant role in this paper, in particular in Sections <ref> and <ref>. The main reason is that for the aligned program we can perform all the queries in parallel (assuming we have the intermediate states ψ_t from fig:queryAlgorithm somehow).The input oracle is unidirectional: the algorithm only has access to O. This suffices for most of our results. Quite often, however, bidirectional access to the input oracle is allowed, where the algorithm can query both O and O^*. The latter is a special case of the former, as bidirectional access to O is equivalent to unidirectional access to O⊕ O^*. As this situation will be common in some sections of the paper, we utilise the following piece of notation:O = O ⊕ O^*.The standard complexity measure of a quantum query algorithm is Q=Q(A): the total number of times the queryis executed. It was called Monte Carlo complexity in <cit.> in order to distinguish it from Las Vegas complexity defined next.Let Π^∙ be the projector onto . The state processed by the oracle on the t-th query is ψ_t^∙ = Π^∙ U_t-1 U_t-2⋯U_0, and the total query state is q(A, O, ξ) = ⊕_t=1^Q ψ_t^∙.This is the most complete way of specifying the work performed by the input oracle O in the algorithm A on the initial state ξ. It is a member of ⊗ for some space(the latter actually being equal to ^Q⊗). The simplest way to gauge the total query state is by defining the corresponding quantum Las Vegas query complexity:L(A, O, ξ) = q(A, O, ξ)^2.As mentioned in sec:conceptualQuantum, we extend the definitions eqn:totalQueryState and eqn:LasVegasComplexity for the case ξ'∈⊗ for someusing identities eqn:queryStateExtended and eqn:LasVegasExtended.§.§ Multiple Input OraclesIt is also possible for an algorithm to have access to several input oracles O^(1),…,O^(r). Las Vegas query complexity can naturally accommodate such a scenario. Indeed, access to several input oracles is equivalent to access to the one combined oracleO = O^(1)⊕ O^(2)⊕⋯⊕ O^(r).Consequently, the space of the oracle has a similar decomposition = ^(1)⊕⋯⊕^(r), where O^(i) acts in ^(i). The total query state can also be decomposed into partial query statesq(A,O,ξ) = q^(1)(A,O,ξ)⊕ q^(2)(A,O,ξ) ⊕⋯⊕ q^(r)(A,O,ξ),where q^(i)(A,O,ξ) is processed by O^(i). This gives partial Las Vegas query complexitiesL^(i)(A, O, ξ) = |q^(i)(A,O,ξ)|^2. In terms of registers, it can be assumed that the input oracle is controlled by some register R, where the value |R⟩|0> indicates no application of the input oracle, and |R⟩|i> with i>0 indicates the i-th input oracle O^(i). We note that we use i in |R⟩|i> only as a label. In particular, we do not perform any arithmetical operations on them. Therefore, |R⟩|i> can have a complicated internal encoding that can facilitate the application of the oracle.The assumption on the registerin this section is not necessarily in contradiction with the assumptions of sec:prelimQuery, as R can have a separate qubit that indicates whether i in |R⟩|i> is non-zero. This qubit can serve as R in the sense of sec:prelimQuery.The case of usual Monte Carlo query complexity requires additional changes, as per now it turns out that all the oracles are queried the same number of times, Q. One way to allow for different number of queries is as follows. Similarly to eqn:query, define the query to the i-th input oracle as^(i) = ⊕ I ⊗[I^(1)⊕⋯⊕ I^(i-1)⊕ O^(i)⊕ I^(i+1)⊕⋯⊕ I^(r) ],where the decomposition in the brackets is the same as in eqn:severalOracles. In other words, ^(i) is just the application of O^(i) controlled by |R⟩|i>. The query algorithm is then defined asA(O) = U_Q ^(s_Q)U_Q-1 ^(s_Q-1) ⋯U_2 ^(s_2)U_1 ^(s_1)U_0,where s_1,s_2,…,s_Q ∈ [r]. The number of invocations of the i-th oracle, Q^(i), is defined as the number of s_j equal to i in eqn:algorithmMultipleOracles.§.§ Evaluation of Functions The standard settings for quantum algorithms evaluating a (partial) function f D→ [p] with D⊆ [q]^n are as follows. For an input x∈ [q]^n, the corresponding input oracle acts in ^n ⊗^q asO_x ||⟩i>||⟩b> ↦||⟩i> ||⟩b ⊕ x_i>for all i∈[n] and b∈ [q]. Here ⊕ stands for the bitwise XOR, and we assume that q is a power of 2 (we can ignore the inputs outside of the domain).The algorithm A itself has a special output register isomorphic to ^p. After finishing the algorithm, measuring the output register should yield the value f(x). This either happens with probability 1 (for exact algorithms), or with probability at least 1/2+d for some constant d>0 (bounded error).This definition is nice because there is one well-defined input oracle O_x for each input. Also O_x^2 = I, which makes uncomputing very easy. Unfortunately, the standard definition has a drawback that the input oracle of the algorithm has a more restricted form than the one required for the algorithm itself. This is problematic if the algorithm is expected to be used as an input oracle for another algorithm. This issue is solved by noting that ||⟩b> ↦||⟩b⊕ f(x)> can be implemented by evaluating f(x), performing the XOR operation, and uncomputing f(x). This increases the complexity by a factor of 2, which is fine if we ignore constant factors. We call it robust evaluation of function, as the action of the algorithm is specified for all input states, not just ||⟩0>.However, constant factors can be important, for instance, in iterated functions, where such factors appear as a base of the exponent, or in the settings of Las Vegas complexity in <cit.>, where precise complexity is sought for. In this case, a more homogenous definition would be appreciated.We follow one such approach from <cit.>, which we call state-generating. We say that the input oracle O_x encodes the input string x∈[q]^n if it performs the transformation O_x ||⟩i>||⟩0> ↦||⟩i>||⟩x_i>for all i∈ [n]. The action of this oracle on ||⟩i>||⟩0> is identical to that of eqn:standardOracle, but we do not require anything for other states. In other words, the admissible subspace of O_x consists of vectors having ||⟩0> in the second register. The admissible subspace of the algorithm itself is spanned by ||⟩0>. On that, it has to perform the transformation ||⟩0> ↦||⟩f(x)>. The algorithm has bidirectional access to O_x, which we treat as unidirectional access to O_x defined in eqn:bi. It is possible to assume the input oracle O_x is a direct sum of n unitaries acting in ^q in order to apply the multiple input oracle settings from sec:multipleInputOracles.As the initial state is always ||⟩0>, and O_x is essentially determined by x, we will writeL_x (A) = L[A, O_x, ||⟩0>]L(A) =max_x∈ D L_x(A).More precisely, we define L_x(A) as the supremum over all input oraclesO_x that encode the input x.We use similar notation for L^(i)_x and L^(i). This approach has a number of advantages. First, it casts function evaluation as a special case of state conversion with state-generating input oracles <cit.>. Second, the algorithm can be directly used as a part of the input oracle for another algorithm without any uncomputation. Finally, this definition does not involve the somewhat arbitrary XOR operation and may be, thus, regarded as being more pure. In particular, it makes sense to ask for the precise value of its quantum Las Vegas query complexity (and not just only up to constant factors).This approach has a number of disadvantages. First, we have to explicitly allow bidirectional access to O_x in order to allow uncomputing, as it is no longer the case that O_x is its own inverse. More importantly, though, neither the action nor the Las Vegas complexity of the algorithm is specified for the initial states orthogonal to ||⟩0>. For our own algorithms, we can design them so that O_x is only executed with ||⟩0> in the second register (maybe after some perturbation, see sec:perturbed). But, if we are dealing with an arbitrary algorithm, we have no such guarantees.§.§ Circuit ModelT_CWe assume that the space of the algorithm is embedded into a product of qubits, (^2)^⊗ s, for some s called the space complexity of the algorithm. A quantum program is a product of elementary operations called gates:A = G_TG_T-1⋯ G_1.In the circuit model, each gate G_i is usually a 1- or a 2-qubit operation that can be applied to any qubit or a pair of qubits. The number of elementary operations, T, is called the time complexity of A, and is denoted by T(A). We assume a universal gate set, so that every unitary can be written as a quantum program. We do not focus too much on a particular model, as they are all essentially equivalent.In a query algorithm like in sec:prelimQuery, each execution of the input oraclealso traditionally counts as one elementary operation. In other words, each unitary in eqn:preAlgorithm can be decomposed into elementary gates as in eqn:program to give a corresponding program in the circuit model. We use T to denote its time complexity, and Q to denote its Monte Carlo query complexity.We will often require an algorithm like in eqn:program to be executed conditionally, that is, controlled by the value of some external qubit. In other words, we would like to perform an operation A^c of the form||⟩0>||⟩ξ> A^c||⟩0>||⟩ξ> ,||⟩1>||⟩ξ> A^c||⟩1>||⟩Aξ>,where the first qubit is the control qubit. We will denote time complexity of this procedure by (A).Since it is possible to substitute each G_i in eqn:program by its controlled version, we have that (A) = (T(A)). But it is often possible to do better. For instance, assume that A is of the form A_2A_1^c, i.e., a large chunk of A is already conditioned. We have that T(A) = T(A_2) + (A_1). On the other hand, (A) = (A_2) + (A_1) + (1), because we can calculate the AND of the two control qubits of A_1 into a fresh qubit, execute A_1 controlled by this fresh qubit, and uncompute the new qubit afterwards. In other words, the constant factor of (A) = (T(A)) is not getting accumulated with each new conditioning, but is, in a way, paid only once.§.§ QRAG modelThe QRAG model extends the circuit model by allowing the following Quantum Random Access Gate:QRAG||⟩i> ||⟩b> ||⟩x_1,…,x_i-1, x_i, x_i+1,…, x_m> ↦||⟩i> ||⟩x_i> ||⟩x_1,…,x_i-1, b, x_i+1,…, x_m>,where the first register is an m-qudit, and the remaining ones are quantum words (i.e., quantum registers large enough to index all the qubits in the program). We assume the QRAG takes timeas specified in eqn:timeR. Note that this gate would require time Ω(m) to implement in the usual circuit model, as it depends on all m+2 registers.This should not be confused with the QRAM model, which allows oracle access to an array of classical registers x_1,x_2,…,x_m:QRAM||⟩i> ||⟩b> ↦||⟩i> ||⟩b⊕ x_i>,where ⊕ stands for the bit-wise XOR.The difference is that x_is are being fixed during the execution of the quantum procedure (but they may be changed classically between different executions). The QRAG model is more powerful than the QRAM model.The main reason we need the QRAG is the following result (see, e.g., <cit.> for a formal statement, although the same construction has been used elsewhere including <cit.>):Assume the QRAG model and that we have QRAM access to a description of a quantum program as in eqn:program in some space , where each gate G_i either comes from a fixed set of 1- or 2-qubit operations (which can be applied to different qubits each time), or is a QRAG. Letbe a T-qudit.Then, the following Select operation∑_i |I⟩|i>|H⟩|ψ_i> ↦∑_i |I⟩|i> |H⟩ |G_iψ_i>can be implemented in time (). [Proof sketch] We add a number of scratch registers to perform the following operations. Conditioned on i, we use the QRAM to read the description of G_i. We switch the arguments of G_i into the scratch space using the QRAG. We apply the operation G_i on the scratch space. We switch the arguments back into memory, and erase the description of G_i from the scratch memory. All the operations besides applying G_i take time (). Application of a usual gate G_i takes time (1) as the gate set is fixed, or (T_R) if G_i is a QRAG. It is also possible to not store the whole program in memory, but compute it on the fly, in which case the complexity of this computation should be added to the complexity stated in thm:select.Assume that in the settings of thm:select we have QRAM access to an array storing descriptions of m quantum programs A^(1),…, A^(m). Letbe a m-qudit.Then, the following operation∑_i |I⟩|i>|H⟩|ψ_i> ↦∑_i |I⟩|i>H |A^(i)ψ_i>,can be implemented in time [·max_i T(A^(i))]. [Proof sketch] Use thm:select to implement the first gate in all of A^(i) in parallel, then the second one, and so on until the time mark max_i T(A^(i)). Another important primitive is the random access (RA) input oracle. If O→ is an input oracle, then its RA version acts on ⊗^⊗ K, whereis a K-qudit. If the registercontains value i, the input oracle is applied to the i-th copy ofin ^⊗ K.The idea behind this is that if O is implemented as a subroutine, then the RA input oracle is a special case of cor:selectProgram, where all A^(i) are the same, but act on different sets of registers (which is easy to define using |J⟩|i> as an offset). Therefore, it makes sense to define the RA input oracle as an elementary operation in the QRAG model.§.§ Perturbed AlgorithmsThe following lemma is extremely useful in quantum algorithms, but for some reason has seldom experienced the honour of being explicitly stated.Assume we have a collection of unitaries U_1,…, U_m all acting in the same vector space . Let ψ_0',…,ψ_m' be a collection of vectors insuch thatψ_t' = U_t ψ_t-1'for all t. Let ψ_0,…, ψ_m be another collection of vectors insuch that ψ_0 = ψ'_0 and| ψ_t - U_i ψ_t-1 | ≤_ifor all i. Then,| ψ_m - ψ'_m | ≤∑_t=1^m _t.Denote by V_t the product U_m U_m-1⋯ U_t+1. In particular, V_m = I. Then,ψ_m - ψ'_m = V_m ψ_m - V_0 ψ_0=∑_t=1^m [ V_t ψ_t - V_t-1ψ_t-1] = ∑_t=1^m V_t [ ψ_t - U_t ψ_t-1].We obtain eqn:surgeryResult from the triangle inequality using eqn:surgeryPerturbation and the unitarity of V_t. In the application of this lemma, U_t stands for sequential sections of a quantum algorithm. The vectors ψ'_t form the sequence of states the algorithm goes through during its execution. The vectors ψ_t is an idealised sequence, which is used instead of ψ_t' in the analysis.We call the difference between ψ_t and U_t ψ_t-1 a (conceptual) perturbation. The expression in eqn:surgeryPerturbation is the size of the perturbation. Therefore, the Eq. eqn:surgeryResult can be stated as the total perturbation of the algorithm does not exceed the sum of the perturbations of individual steps. If this sum is small, the final state ψ_m of the analysis is not too far away from the actual final state ψ'_m of the algorithm.This lemma is implicitly used every time an approximate version of a quantum subroutine is used, which happens in pretty much every non-trivial quantum algorithm.§.§ Efficient Implementation of Direct-Sum Finite Automata We will repeatedly use the following construction in this paper, for which we describe a time-efficient implementation. We call it direct-sum quantum finite automata due to its superficial similarity to quantum finite automata.The space of the algorithm is ⊗⊗, whereis a K-qudit,is a qubit, andis an arbitrary space. Additionally, we have black-box access to K unitaries S_0,…,S_K-1 in ⊗. The algorithm is promised to start in the state of the form|K⟩|0> |P⟩|0> |H⟩|ϕ> + ∑_t=0^K-1|K⟩|t>|P⟩|1> |H⟩|ψ_t>,and it has to perform the following transformation.9 *For t=0,1,…,K-1:=0pt(a) Execute S_t on ⊗ conditioned on |K⟩|t>.(b) Conditioned on |P⟩|0>, replace |K⟩|t> by |K⟩|t+1>.Let us elaborate on the “replace” in point (b). It is not hard to show by induction that the |P⟩|0>-part of the state contains a vector of the form |K⟩|t>|H⟩|ϕ_t+1>. This vector has to be replaced by |K⟩|t+1>|H⟩|ϕ_t+1>. We also assume that |K⟩|K> is identical with |K⟩|0>. For a graphical illustration refer to fig:automaton.A graphical illustration of the action of a direct-sum finite automaton. States ϕ_t represent the internal state of the automaton, as it processes a “string” of quantum vectors ψ_0,…,ψ_K-1 into ψ'_0,…,ψ'_K-1. Unlike the usual quantum automata, the internal state of the automaton and the current “letter” of the “string” are joined via the direct sum. Similarity with fig:pumping is apparent. It is an interesting question, whether such finite automata can find other applications.[every node/.style=font=, every path/.append style=thick,->] 0123[] (9,4) to node[above]ϕ_4 (10.5,4);It is trivial to implement the transformation in eqn:automaton in (Klog K) elementary operations besides the executions of S_t. It is a technical observation that we can remove the logarithmic factor.The transformation in eqn:automaton can be implemented in time (K) + ∑_t (S_t), whereis defined in sec:prelimCircuit. The register K uses ℓ = log K qubits. We introduce an additional registerthat also consists of ℓ qubits. For i, t∈ [K], we denote|C⟩|i⋎ t> = ||⟩1>^⊗ c||⟩0>^⊗ℓ-cif the binary representations of i and t, considered as elements of ^ℓ, agree on the first c most significant bits, and disagree on the (c+1)-st one (or c=ℓ).We modify the algorithm so that before the t-th iteration of the loop, the algorithm is in a state of the form|K⟩|t> |P⟩|0> |C⟩|t⋎ t> |H⟩|ϕ_t>+ ∑_i=0^t-1|K⟩|i> |P⟩|1> |C⟩|i⋎ t> |H⟩|ψ'_t> + ∑_i=t^K-1|K⟩|i>|P⟩|1>|C⟩|i⋎ t> |H⟩|ψ_t>In particular, in the |P⟩|0>-part, the register C contains ℓ ones.The state eqn:automatonModifiedState with t=0 can be obtained from eqn:automatonInitialState in (ℓ) elementary operations by computingstarting from the highest qubit. Execution of S_t on Step (a) can be conditioned on the lowest qubit of C, which is equal to 1 if and only ifcontains t.It remains to consider Step (b) and the update of the registerduring the increment from t to t+1. Assume that t+1 is divisible by 2^d, but not by 2^d+1. Then, Step (b) takes d+1 controlled 1-qubit operations. Similarly, the change from t to t+1 in |C⟩|i⋎ t> in eqn:automatonModifiedState takes time (d) by first uncomputing the d+1 lowest qubits for t, and then computing them for t+1. Finally, after the loop, the register C can be uncomputed in (ℓ) operations. Therefore, the total number of elementary operations is (K).§ TRANSDUCERS In this section, we define transducers and give their basic properties. This is an initial treatment and will be extended in sec:canonical to include query complexity.§.§ DefinitionMathematically, our approach is based on the following construction. Let ⊕ be a direct sum of two vector spaces, and S be a unitary on ⊕. Then, for every ξ∈, there exist τ∈ and v∈ such thatSξ⊕ v ↦τ⊕ v.Moreover, =0pt* The vector τ = τ(S,ξ) = τ_(S,ξ) is uniquely defined by ξ and S.* The vector v = v(S,ξ) = v_(S,ξ) is also uniquely defined if we require that it is orthogonal to the 1-eigenspace of Π SΠ, where Π is the projection on .* The mapping ξ↦τ is unitary and ξ↦ v is linear.We will prove the theorem at the end of this section.In the setting of thm:transduce, we will say that S transduces ξ into τ, and write ξSτ. The mapping ξ↦τ onwill be called the transduction action of S onand denoted by S_.We call any v satisfying S(ξ⊕ v) = τ⊕ v a catalyst for ξSτ. The condition in Point (b) of v to be orthogonal to the 1-eigenspace of Π SΠ is crucial for uniqueness, as adding such a vector to v does not affect eqn:transduce. Clearly, the vector v as defined in Point (b) has the smallest possible norm. Therefore, we can define transduction complexity of S on ξ as W(S,ξ) = v(S,ξ)^2 for the latter v. We write W_(S,ξ) if the spacemight not be clear from the context. The above discussion can be formulated as the following claim.For any catalyst v of the transduction ξSτ, we have W(S,ξ) ≤v^2. On the other hand, checking orthogonality to Π SΠ is complicated and unnecessary, and we avoid doing it. We usually couple a transducer with some chosen catalyst v for all the ξ of interest, which need not have the smallest possible norm. In this case, we somewhat sloppily write W(S,ξ) = v^2 even for this catalyst v. This agreement will become especially important when we add query complexity into the picture in sec:canonical; see in particular the note towards the end of sec:canonicalDefinition.Other important notions related to the transducer are its time and query complexity. Its time complexity, T(S), is defined as its time complexity as an algorithm. For ξ∈, its query state, q_(S,O,ξ), and query complexity, L_(S,O,ξ), are defined as those of S as an algorithm on the initial stateξ⊕ v. Note that for transducers with input oracles, we will adopt a special canonical form defined in sec:canonical, until then we mostly ignore the oracle-related notions.Let us a give a simple concrete example illustrating the above notions. Assume thatis one-dimensional and spanned by ||⟩0>, andis two-dimensional and spanned by ||⟩1> and ||⟩2>. Let S be the reflection of the vector ||⟩0>-||⟩1>-||⟩2>, so that its orthogonal complement stays intact.The transduction action of S onis the identity, which is certified byS||⟩0> + 1/2||⟩1> + 1/2||⟩2>↦||⟩0> + 1/2||⟩1> + 1/2||⟩2> .Hence, we havev(S,||⟩0>) = (||⟩1> + ||⟩2>)/2,and W(S,||⟩0>) = 1/2. This is not the only catalyst, as one can also take v= ||⟩1> or v=||⟩2>. However, (||⟩1> + ||⟩2>)/2 is the only catalyst orthogonal to the 1-eigenspace of Π SΠ, which is spanned by ||⟩1> - ||⟩2>, and also has the smallest norm. [Proof of thm:transduce] The vector v can be found from the equationΠ v = Π[τ+v] = Π[S(ξ+v)] = Π Sξ + Π S v = Π Sξ + Π S Π v.From this we would like to argue that v can be expressed asv = (Π - Π SΠ)^+ Π Sξ,where (·)^+ stands for the Moore-Penrose pseudoinverse. Let us show that this is indeed the case. Denote bythe kernel of Π - Π SΠ in , and by ^⊥ its orthogonal complement in .The subspaceequals the 1-eigenspace of Π SΠ. But since S is a unitary, a 1-eigenvector of Π SΠ is necessarily a 1-eigenvector of S. Hence, S is a direct sum of the identity onand a unitary on ⊕^⊥. Thus, Π - Π SΠ is a direct sum of the zero operator in ⊕ and some operator in ^⊥. Moreover, the latter operator is invertible in ^⊥ as its kernel is empty. Since ξ∈ is orthogonal to , we get that Π Sξ∈^⊥. Hence, eqn:transduceV indeed uniquely specifies v. This proves (b) and the second half of (c).The uniqueness of τ and the linearity of ξ↦τ now follow from eqn:transduce and the linearity of S. Finally, unitarity of S implies ξ = τ, hence, the map ξ↦τ is also unitary. [Transitivity of Transduction]Assume ⊆_1⊆_2 are vector spaces, and S is a unitary in _2. Then,S_ = (S__1)_ ,and, for every ξ∈, a possible catalyst isv_(S, ξ) = v_(S__1, ξ) + v__1[S, ξ⊕ v_(S__1, ξ)].By definition,S__1ξ⊕ v_(S__1,ξ)↦τ⊕ v_(S__1,ξ)for some τ∈.The latter means thatSξ⊕ v_(S__1,ξ) ⊕ v__1[S, ξ⊕ v_(S__1,ξ)] ↦τ⊕ v_(S__1,ξ) ⊕ v__1[S, ξ⊕ v_(S__1,ξ)],proving eqn:transitivityWitness.§.§ Implementation The key point we will now makeis that given a transducer S, there exists a very simple quantum algorithm that approximately implements its transduction action on .This algorithm is a generalisation of the one from <cit.>, which was used for implementation of the adversary bound.Before we describe this algorithm, let us establish a few conventions. We callthe public andthe private space of S. We indicate this separation ofandby a privacy qubit . The value 0 ofwill indicate the public space , and the value 1 the private space . This means that bothandare embedded into the same register during implementation. Thus,ξ⊕ v = |P⟩ |0> |H⟩ |ξ> + |P⟩|1> |L⟩ |v>explicitly specifying the public and the private spaces. We will extend this notation in sec:canonicalDefinition.As it can be understood from the name, the algorithm does not have direct access to the private spaceof the transducer.All the interaction between the transducer and its surrounding is through the public space .Let spaces , , and a positive integer K be fixed. There exists a quantum algorithm that transforms ξ into τ' such thatτ' - τ(S,ξ)≤ 2 √(W(S,ξ)/K)for every transducer S⊕→⊕ and initial state ξ∈. The algorithm conditionally executes S as a black box K times, and uses (K) other elementary operations. thm:introImplementation is a direct corollary of thm:pumping. A sketch of the proof of thm:pumping was already given in the same section, see, in particular, fig:pumping.[Proof of thm:pumping] The space of the algorithm is ⊗ (⊕), whereis a K-qudit. The register ⊕ contains the privacy qubitas described above. The algorithm starts in the state ξ = |K⟩|0>|P⟩ |0> |H⟩|ξ>, andperforms the following transformations: =0pt* Map |K⟩|0> into the uniform superposition 1/√(K)∑_t=0^K-1|K⟩|t>.* For t=0,1,…,K-1: =0pt* Execute S on ⊕ conditioned on |K⟩|t>.* Conditioned on |P⟩|1>, replace |K⟩|t> by |K⟩|t+1> (where |K⟩|K> is equal to |K⟩|0>). * Run Step 1 in reverse. Clearly, the algorithm conditionally executes S exactly K times. As described now, the algorithm takes time (Klog K), but it is implementable in time (K) using lem:automaton.Let us prove correctness. We write v=v(S,ξ) and τ=τ(S,ξ). After Step 1, the algorithm is in the state1/√(K)∑_i=0^K-1|K⟩|i> |P⟩ |0> |H⟩|ξ>.We perform a perturbation in the sense of lem:surgery and assume the algorithm is instead in the state1/√(K)∑_i=0^K-1|K⟩|i> |P⟩ |0> |H⟩|ξ> + 1/√(K)|K⟩|0>|P⟩|1>|L⟩|v>.On the t-th iteration of the loop, the transducer S on Step 2(a) transforms the part of the state1/√(K)|K⟩|t> |P⟩ |0> |H⟩|ξ> + 1/√(K)|K⟩|t>|P⟩|1>|L⟩|v>⟼ 1/√(K)|K⟩|t> |P⟩ |0> |H⟩|τ> + 1/√(K)|K⟩|t>|P⟩|1>|L⟩|v>,and on Step 2(b) the following transformation of the part of the state is performed: 1/√(K)|K⟩|t>|P⟩|1>|L⟩|v> ⟼ 1/√(K)|K⟩|t+1>|P⟩|1>|L⟩|v>.Therefore, after the execution of the loop in Step 2, we get the state1/√(K)∑_i=0^K-1|K⟩|i> |P⟩ |0> |H⟩|τ> + 1/√(K)|K⟩|0>|P⟩|1>|L⟩|v>.We perturb the state to1/√(K)∑_i=0^K-1|K⟩|i> |P⟩ |0> |H⟩|τ>.After Step 3, we get the state τ = |K⟩|0>|P⟩ |0> |H⟩|τ>.Note that the difference between the states in eqn:pumpingOriginal and eqn:pumpingModified has norm v/√(K). The same is true for the difference between the states in eqn:pumpingFinal and eqn:pumpingTarget. Therefore, by lem:surgery, the actual final state τ' of the algorithm satisfiesτ' - τ≤ 2 v/√(K)as required.§ EXAMPLE I: QUANTUM WALKSIn this section, we implement the electric quantum walk from <cit.> using the construction outlined in sec:introWalks. An example of the extension of a graph for a quantum walk. The original graph contains two parts A and B of 4 and 3 vertices, respectively. The initial probability distribution is supported on two vertices {u_1,u_2}⊆ A. The original edges E of the graph are black, the new ones E' are red. One marked vertex in B is coloured blue. red[minimum size=15pt, inner sep=0pt] [blue] at (0,-0.8) A; [circle, draw] (A4) at (0,0) ; [circle, draw] (A3) at (0,1) ; [circle, draw, fill=!50] (A2) at (0,2) u_2; [circle, draw, fill=!50] (A1) at (0,3) u_1; [blue] at (2,-0.8) B; [circle, draw] (B3) at (2,0.5) ; [circle, draw] (B2) at (2,1.5) ; [circle, draw] (B1) at (2,2.5) ;(A1)–(B1),(B3); (A2)–(B2),(B3); (A3)–(B1),(B2),(B3); (A4)–(B1),(B2); ;(A1) to node[above]w_e (B1); [circle, draw, double, fill=blue!50] at (B2) ; [circle, draw, fill=!50] (S2) at (-1.5,2) u_2'; [circle, draw, fill=!50] (S1) at (-1.5,3) u_1'; [-,] (S1) to node[above] σ_1 (A1); [-,] (S2) to node[above] σ_2 (A2);A quantum walk is described by a bipartite graph G, whose parts we denote by A and B. Let E be the set of edges of G. Each edge e of the graph is given a non-negative real weight w_e. We have some set M⊆ A∪ B of marked vertices. There is a subroutinethat, for every vertex u, says whether it is marked. The goal of the quantum walk is to detect whether M is empty or not.In the framework of electric quantum walks, the graph is extended as follows, see fig:walk. The quantum walk is tailored towards a specific initial probability distribution σ on A. Let A_σ⊆ A be the support of σ. For each u∈ A_σ, we add a new vertex u' and a new dangling edge u'u to the graph. The newly added vertices are not contained in B. Let E' be the set of newly added edges. For edges if E' we assume the weight w_u'u = σ_u.We treat this construction as a transducer. The private spaceof the quantum walk is ^E. The public spaceis ℂ^E'. The initial state ξ is given by ξ = ∑_u∈ A_σ√(σ_u)||⟩u'u>∈.For a vertex u, let _u denote the space spanned by all the edges incident to u (including the ones in E'). Defineψ_u = ∑_e: e∼ u√(w_e)||⟩e>∈_uwhere the sum is over all the edges incident to u.For U⊆ A or U⊆ B, let R_U denote the reflection of all ψ_u for u∈ U, i.e., R_U acts as negation on the span of all these ψ_u and as identity on its orthogonal complement. We define the transducer, which depends on the set of marked vertices M, as S_M = R_B∖ MR_A∖ M.Each of R_A∖ M and R_B∖ M is decomposable into products of local reflections in _u as u ranges over A and B, respectively. The corresponding local reflections are either identities for u∈ M, or reflections of ψ_u for u∉ M.Let us study the action of S_M. The positive case is M=∅, where we follow eqn:QW_sequence_positive. The initial coupling isξ⊕ v_∅ = ∑_e∈ E∪ E'√(w_e)||⟩e>.Note that ξ⊕ v_∅ = ∑_u∈ Aψ_u and v_∅ = ∑_u∈ Bψ_u. Hence, R_A reflects all ξ⊕ v_∅, and R_B only reflects v_∅. Therefore, S_∅ξ⊕ v_∅↦ -ξ⊕ v_∅,giving ξS_∅ -ξ.In the negative case of M∅, we follow eqn:QW_sequence_negative. Let p_e be a flow on the graph where σ_u units of flow are injected in u', and the flow is collected at M. To solve the sign ambiguity, we assume that all the edges are oriented towards A. This time, we define the catalyst v_M so thatξ⊕ v_M = ∑_e∈ E∪ E'p_e/√(w_e)||⟩e>.Recall that p_u'u = w_u'u = σ_u, hence the above equation is satisfied in .The projection of eqn:walkPositiveWitness onto _u is given by∑_e: e∼ up_e/√(w_e)||⟩e>.This state is not changed by the corresponding local reflection in _u. Indeed if u∈ M, the corresponding local reflection is the identity. If u∉ M, then, by the flow condition, the state in eqn:walkPu is orthogonal to ψ_u in eqn:walkPsi. Thus, neither R_A∖ M nor R_B∖ M change ξ⊕ v_M, hence, ξS_Mξ.The corresponding transduction complexities are, respectively,∑_e∈ E w_e ∑_e∈ Ep_e^2/w_e.Note that the catalyst in the second case uses the flow p_e through the graph, which is not unique. For a fixed M, the minimum value is attained when p_e is the electric flow through the graph, and its value is the corresponding effective resistance. Let R denote the maximal effective resistance over all possible choices of M∅, andlet W = ∑_e∈ E w_e be the total weight of the graph. We can rescale all w_i so that the maximal transduction complexity in eqn:walkComplexity becomes equal to √(RW). This coincides with the complexity established in <cit.>.§ CANONICAL TRANSDUCERS In this section, we define a specific form of transducers we will be using in this paper. The main point is in the application of the input oracle. Inspired by the construction in <cit.>, we assume that the transducer first executes the input oracle, and then performs some input-independent unitary. Moreover, the input oracle is always applied to the private space of the transducer.These assumptions simplify many constructions, and any transducer can be transformed into the canonical form with a small overhead as shown later in prp:canoning. §.§ DefinitionConcerning the input oracle, the assumptions are similar to those in sec:prelimQuery. The input oracle is a unitary O in some space , and we have unidirectional access to O. The oracle only acts on the local spaceof the transducer. Let us decompose the latter in two parts = ⊕, which stand for the work (non-query) and query parts. We also have = ⊗ for some space . We denote the identity onsimply by I.A canonical transducer S = S(O) performs the following transformations, see also fig:canonical: =0pt* It executes the input oracle I⊗ O on . Similarly to eqn:query, we call it a query and denote it by = I_⊕⊕ I⊗ O,where I_ andare identities onand , respectively.* It performs an input-independent work unitaryon ⊕. The decomposition = ⊕ yields the decomposition v = ⊕ of the catalyst. Thus, the action S(O) of the transducer S on the input oracle O is given by the following chain of transformations:S(O)ξ⊕⊕ξ⊕⊕(I⊗ O)τ⊕⊕ . Now we can make the following complexity-related definitions. The catalyst is v = v(S, O,ξ). [It is the same as v(S(O),ξ) in the previous notation, however, we opted to v(S,O,ξ) to reduce the number of brackets and to keep notation synchronised with <cit.>.] The transduction complexity isW(S, O, ξ) = |v(S, O, ξ)|^2.The query state is q(S, O, ξ) ==Π^∙v(S,O,ξ), where Π^∙ denotes the orthogonal projector onto . The (Las Vegas) query complexity of the transducer isL(S, O, ξ) = |q(S, O, ξ)|^2.Note that formally the definitions q(S, O,ξ) and L(S,O,ξ) are in conflict with the same definitions eqn:totalQueryState and eqn:LasVegasComplexity if S is considered as a program and not as a transducer. However, this should not cause a confusion. If the spaceis not clear from the context, we will add it as a subscript as in sec:transducerDefinition.Time complexity T(S) of the transducer is the number of elementary operations required to implement the unitary . Note that we do not count the query towards time complexity of the transducer.Finally, the definitions eqn:transductionComplexity and eqn:transducerQueryComplexity can be extended to ξ'∈⊗ using eqn:transductionExtended. Note on Non-Uniqueness of Catalyst It is important to note that in this setting the non-uniqueness of thecatalyst v discussed in sec:transducerDefinition becomes very important. To understand why, consider again exm:1 from that section. This time, assume thatis spanned by ||⟩1> andby ||⟩2>, the input oracle is O=I, and = S as defined previously.The “right”catalyst v = (||⟩1> + ||⟩2>)/2 for the initial state ξ=||⟩0> suggests that W(S, O, ||⟩0>) = 1/2 and L(S, O, ||⟩0>) = 1/4. However, if we take v=||⟩1>, we get that W(S,O,||⟩0>) = 1 and L(S, O, ||⟩0>) = 0. Therefore, there is no longer a single catalyst that minimises both the transduction and the query complexity. This is similar to usual algorithms, where time and query complexity can be minimised by different algorithms.We solve this complication by implicitly assigning a specific catalyst v(S, O, ξ) for every O and ξ of interest, that gives both W(S,O,ξ) and L(S, O,ξ) simultaneously. Of course, neither of the two are guaranteed to be minimal. It is possible to study the trade-off between the transduction and the query complexity for a fixed O and ξ, but we will not explicitly pursue that in this paper. ξ^∘ ξ^∙Implementation Details In terms of registers, as in sec:prelimQuery, the separation = ⊕ is indicated by the qubit . Now it makes sense to assume that the registers ,andare the same, the distinction being given by the values of the registersand . We will usually place this common register as the unnamed last register in our expressions. In particular,ξ⊕ v = |P⟩|0> |R⟩ |0> ||⟩ξ> + |P⟩|1> |R⟩|0> ||⟩> + |P⟩|1> |R⟩|1> ||⟩>.Note that the spaceis indicated by |P⟩|0> |R⟩ |0> meaning that it is not acted on by the oracle. This allows us to implement the query as an application of O controlled by |R⟩|1>, which is in accord with the convention established in sec:prelimQuery.We will also use registersandin the sense of sec:implementation, that is, containing . In particular, we can write the action of a canonical transducer eqn:canonicalForm in registers likeS(O)|P⟩ |0> |H⟩ |ξ> + |P⟩|1> |L⟩ |v> |P⟩ |0> |H⟩ |ξ> + |P⟩|1>L | v> |P⟩ |0> |H⟩ |τ> + |P⟩|1> |L⟩ |v>,where we used shorthandv = |R⟩|0> ||⟩> + |R⟩|1>|(I⊗ O)>. §.§ Multiple Input OraclesFollowing <cit.>, we can also allow multiple input oracles joined by direct sum:O = O^(1)⊕ O^(2)⊕⋯⊕ O^(r),where the i-th input oracle O^(i) acts in space ^(i) and = ^(1)⊕⋯⊕^(r). We get the corresponding decomposition of the query state:q(S, O, ξ) = q^(1)(S, O, ξ) ⊕⋯⊕ q^(r)(S, O, ξ),where q^(i)(S,O,ξ) is the partial query state of the i-th input oracle. This also gives query complexities of the individual oracles:L^(i)(S, O, ξ) = |q^(i)(S, O, ξ)|^2. It makes sense to tweak the notation assumed earlier in sec:canonicalDefinition to make it in line with sec:prelimMultipleOracles. We assume the registercan hold an integer from 0 to r, where |R⟩|i> with i>0 indicates the space of the i-th input oracle. Thus, in place of eqn:canonicalDecomposition, we haveξ⊕ v =|P⟩|0> |R⟩|0> ||⟩ξ> + |P⟩|1> |R⟩|0> ||⟩> + |P⟩|1>∑_i=1^r |R⟩|i>|v^(i)>with v^(i) = q^(i)(S, O, ξ). To apply the i-th input oracle O^(i), it suffices to condition it on |R⟩|i>. The action of a canonical transducer stays given by eqn:canonicalSequence, where, this time,v = |R⟩|0> ||⟩> + ∑_i=1^r |R⟩|i> ||⟩[I⊗ O^(i)]v^(i)>. For notational convenience, we will assume a single input oracle in most of the paper. The case of multiple oracles can be obtained using the decomposition of the query state in eqn:multipleOraclesQeuryState, which contains all the necessary information. §.§ Reducing the Number of Oracle Calls One problem with the algorithm in thm:pumping is that, when applied to a canonical transducer, the input oracle O is executed the same number of times as the work unitary . This is suboptimal as the transduction complexity can be much larger than the query complexity. In this section, we describe a query-efficient implementation, which can also handle multiple input oracles.Let S be a canonical transducer with r input oracles joined into one oracle O via direct sum as in eqn:multipleOracles. In the following theorem, we assume the spaces , , , ^(1),…,^(r) are fixed, while the operatorsand O^(1),…,O^(r) can vary.Let K ≥ K^(1),…, K^(r) be positive integers, which we assume to be powers of 2 for simplicity. There exists a quantum algorithm that conditionally executesas a black box K times, makes K^(i) queries to the i-th input oracle O^(i), and uses (K+K^(1)+⋯+K^(r))log r other elementary operations. For each , O^(i), and initial state ξ, the algorithm transforms ξ into τ' such that|τ' - τ(S, O, ξ)| ≤2/√(K)√(W(S, O, ξ) + ∑_i=1^r[K/K^(i) -1] L^(i)(S, O, ξ) ).thm:pumping is a special case of this theorem with all K^(i) equal to K. Observe that the number of elementary operations is equal to the total number of invocations ofand O^(i) times log r.It is highly unlikely that this part of the algorithm would dominate its time complexity.[Proof of thm:optimalImplementation] The proof is an extension of that of thm:pumping. Its outline was already given in sec:introCanonical; see in particular fig:implementationBetter.Let us define D^(i) = K/K^(i), which is a power of 2. The space of the algorithm is ⊗ (⊕), whereis a K-qudit. The register ⊕ contains the registersandas described above. The algorithm starts its work in the state ξ = |K⟩|0>|P⟩ |0> |R⟩|0> ||⟩ξ>. Its steps are as follows.=0pt* Map |K⟩|0> into the uniform superposition 1/√(K)∑_t=0^K-1|K⟩|t>.* For t=0,1,…, K-1: =0pt* For each i=1,…, r: * if t is divisible by D^(i), execute the input oracle O^(i) conditioned on |R⟩|i>. * Execute the work unitaryon ⊕ conditioned on |K⟩|t>.* Conditioned on |P⟩|1>|R⟩|0>, replace |K⟩|t> by |K⟩|t+1>.* For each i=1,…, r: * if t+1 is divisible by D^(i), add D^(i) to K conditioned on |R⟩|i>.The operation is performed modulo K.* Run Step 1 in reverse. The analysis is similar to that in the proof of thm:pumping. Now we assume that after Step 1, instead of the state1/√(K)∑_t=0^K-1|K⟩|t> |P⟩ |0> |R⟩|0> ||⟩ξ>, we are in the state1/√(K)∑_t=0^K-1|K⟩|t> |P⟩ |0> |R⟩|0>||⟩ξ> +1/√(K)|P⟩|1> [|K⟩|0>|R⟩|0> ||⟩> + ∑_i=1^r ∑_t=0^D^(i)-1|K⟩|t>|R⟩|i> |v^(i)>].Since t=0 is divisible by all D^(i), after Step 2(a) of the first iteration of the loop, we have the state1/√(K)∑_t=0^K-1|K⟩|t> |P⟩ |0> |R⟩|0>||⟩ξ> + 1/√(K)|P⟩|1> [|K⟩|0>|R⟩|0> ||⟩> + ∑_i=1^r ∑_t=0^D^(i)-1|K⟩|t>|R⟩|i> |(I⊗ O^(i))v^(i)>].The crucial observation is that on each iteration of the loop on step 2(b), the following transformation is performed. The part of the state1/√(K)|K⟩|t> [|P⟩ |0> |R⟩|0>||⟩ξ> + |P⟩|1>|R⟩|0> ||⟩> + |P⟩|1>∑_i=1^r |R⟩|i> |(I⊗ O^(i))v^(i)>]gets mapped byinto1/√(K)|K⟩|t> [|P⟩ |0> |R⟩|0>||⟩τ> + |P⟩|1>|R⟩|0> ||⟩> + |P⟩|1>∑_i=1^r |R⟩|i> |v^(i)>],where we used eqn:canonicalSequence with eqn:OvMultipleOracles.If t=cD^(i)-1 for some integer c, then on Step 2(d), we perform the transformation of the part of the state1/√(K)|P⟩|1> |R⟩|i> ∑_t=0^D^(i)-1|K⟩|(c-1)D^(i)+t> |v^(i)> 1/√(K)|P⟩|1> |R⟩|i>∑_t=0^D^(i)-1|K⟩|cD^(i) + t>|v^(i)>,which is then mapped on Step 2(a) of the next iteration into1/√(K)|P⟩|1> |R⟩|i>∑_t=0^D^(i)-1|K⟩|cD^(i) + t>||⟩(I⊗ O^(i))v^(i)>.Therefore, after all the K iterations of the loop in Step 2, we result in the state1/√(K)∑_t=0^K-1|K⟩|t> |P⟩ |0> |R⟩|0>||⟩τ> +1/√(K)|P⟩|1> [|K⟩|0>|R⟩|0> ||⟩> + ∑_i=1^r ∑_t=0^D^(i)-1|K⟩|t>|R⟩|i> |v^(i)>].After that, we finish as in thm:pumping, by assuming we are in the state1/√(K)∑_t=0^K-1|K⟩|t> |P⟩ |0> |R⟩|0> ||⟩τ> instead, and applying Step 3.The total perturbation of the algorithm is2/√(K)||K⟩|0>|R⟩|0> ||⟩> + ∑_i=1^r ∑_t=0^D^(i)-1|K⟩|t>|R⟩|i> |v^(i)>|,which is equal to the right-hand side of eqn:optimalImplementationEstimate.The claim on the number of executions ofand O^(i) in the algorithm is obvious. Besides that, it is trivial to implement the algorithm in(K+K^(1)+⋯+K^(r))log K log r elementary operations, where the log r factors comes from the necessity to index one of the r input oracles. Using a slight modification of lem:automaton, the algorithm can be implemented in (K+K^(1)+⋯+K^(r))log r elementary operations. One crucial point here is that when t = cD^(i)-1, addition of D^(i) on Step 2(d) is equivalent to the replacement of c-1 by c in the highest qubits of the registeras indicated by eqn:improvedPumping1. We omit the details.The following corollary, which is equivalent to thm:introImplementationBetter, is easier to apply.Assume r=(1), and let , W, L^(1),…,L^(r)>0 be parameters. There exists a quantum algorithm that conditionally executesas a black box K=(1+W/^2) times,makes (L^(i)/^2) queries to the i-th input oracle O^(i), and uses (K) other elementary operations. The algorithm -approximately transforms ξ into τ(S, O, ξ) for all S, O^(i), and ξ such that W(S, O, ξ)≤ W and L^(i) (S, O, ξ) ≤ L^(i) for all i.We first prove a relaxed version of the corollary, where we allow each input oracle to be called [1 + L^(i)/^2] times. Afterwards, we show how to remove this assumption. We allow error /2 in this relaxed version.If W<^2/16, the (relaxed) corollary follows from thm:pumping, as the algorithm then executesand each input oracle K = 1 times. Therefore, we will assume W≥^2/16. Also, by definition we have that W(S,O,ξ)≥ L^(i)(S,O,ξ). Therefore, we may assume that W≥ L^(i), reducing L^(i) otherwise. We intend to use thm:optimalImplementation. We take K as the smallest power of 2 exceeding 32(r+1)W/^2. In particular, K = Θ(W/^2) by our assumption on W and r=(1). We take K^(i) as the largest power of 2 that does not exceed max1,K L^(i)/W. It satisfies K^(i)≤ K as required. On the other hand, K^(i)≥ KL^(i)/(2W) implying K/K^(i)≤ 2W/L^(i), which gives the error estimate eqn:optimalImplementationEstimate at most2/√(K)√(2(r+1)W)≤2√(2(r+1)W)/√(32(r+1)W/^2)≤/2.If K^(i)>1, we get K^(i)≤ K L^(i)/W = (L^(i)/^2), which finishes the proof of the relaxed statement of the corollary.In the following, we assume we used thm:optimalImplementation in the proof, the case of thm:pumping being similar. Consider all the values of i such that K^(i)=1. By the proof of thm:optimalImplementation, the input oracles are applied to the perturbation added in eqn:optimalInitial, also the norm of the perturbation is at most /4 (cf. eqn:improvedPumpingTotalPerturbation).To get the original statement of the corollary, we do not apply all the input oracles O^(i) with K^(i)=1. As they are applied once in the algorithm, this gives additional perturbation of size at most /2. Combining with the perturbation /2 of the relaxed algorithm itself, we get an estimate of the total error of at most .§ EXAMPLE II: ADVERSARY BOUND As mentioned in the introduction, the construction of transducers is based on the implementation of the adversary bound in <cit.>. The quantum adversary bound was first developed as a powerful tool for proving quantum query lower bounds. However, it was later extended to include upper bounds as well, and we consider the latter in this paper. For more detail on the adversary bound, refer to the introduction of <cit.> and the references therein. We first consider the general case of state conversion with unidirectional unitary input oracles, and then move on to more usual function evaluation problems. §.§ State Conversion We consider the adversary bound for state conversion from <cit.>. In the state conversion problem, we have a collection of pairs ξ_x↦τ_x of states inand input oracles O_x→, where x ranges over some finite set D. The task is to develop an algorithm A such that A(O_x)ξ_x = τ_x for all x. The goal is to minimise L(A, O_x, ξ_x). The corresponding adversary bound is the following multi-objective optimisation problem: [|v_x|^2]_x∈D<ξ_x, ξ_y> - <τ_x,τ_y> = <v_x, (I_⊗(I_-O^*_xO_y)) v_y>for all x, y∈ D; is a vector space, v_x ∈⊗.A canonical transducer S_v can be obtained from any feasible solution v = (v_x) to this problem. It works as follows (with I = I_):ξ_x ⊕ v_x I⊗ O_xξ_x ⊕ (I ⊗ O_x) v_x _vτ_x ⊕ v_x,where _v is an input-independent unitary whose existence is assured by eqn:advExplicitCondition, as the latter can be rewritten as<ξ_x, ξ_y> +<(I⊗ O_x) v_x, (I⊗ O_y) v_y> = <τ_x, τ_y> +<v_x, v_y>,and two state collections with the same combination of inner products always admit such a state-independent transforming unitary.Thus we have a transduction ξ_x S_v(O_x)τ_x withthe transduction and the query complexities satisfyingW(S_v, O_x, ξ_x) = L(S_v, O_x, ξ_x) = v_x^2,q(S_v, O_x, ξ_x) = v_x. As shown in <cit.>, this perfectly captures Las Vegas query complexity of state conversion. Note that canonical transducers with empty non-query spaceare essentially equivalent to this construction.§.§ Function Evaluation Now we describe the usual case of function evaluation. These results can be derived from <cit.> and <cit.>. First, we define the formalism behind function-evaluating transducers, and then move on to the adversary bound.Let f D → [p] be a function with domain D ⊆ [q]^n. We want to construct a transducer S_f that evaluates f. We assume the state-generating settings from sec:prelimFunctions, which means that, for every x∈ D, with bidirectional access to the input oracleO_x||⟩i>||⟩0>↦||⟩i>||⟩x_i>, the transducer S_f has to perform the transduction ||⟩0> ||⟩f(x)>. Again, we cast this as unidirectional access to O_x from eqn:bi.Similarly to eqn:function_QueryComplexity, we writeW_x (S) = W[S, O_x, ||⟩0>]W(S) =max_x∈ D W_x(S).We use similar notation for L and L^(i).There is a slight discrepancy between various existing definitions of the adversary bound (f) for non-Boolean functions, in the sense that they differ by a factor of at most 2 (see, e.g., Section 3 of <cit.>.) We adopt the formulation from <cit.>, which reads in notation of that paper as(f) = γ_2[1_f(x) f(y)⊕_i∈[n] 1_x_i y_i]_x,y∈ D,and which is equivalent to γ_2(J-F | Δ) in notations of <cit.>. An explicit definition of (f) is: [ The usual definition has max∑_i=1^n u_x,i^2, ∑_i=1^n v_x,i^2 in the objective instead of 1/2[∑_i=1^n u_x,i^2 + v_x,i^2 ]. The two formulations are equivalent <cit.>. But even a priori, our formulation does not exceed the usual formulation, and, since we are interested in upper bounds, supersedes the latter. ] max_x∈D 1/2 ∑_i=1^n [u_x,i^2 + v_x,i^2 ] 1_f(x)f(y) = ∑_i: x_iy_i <u_x,i, v_y,i>for all x, y∈ D; is a vector space, u_x,i, v_x,i ∈.v^↑ v^↓Let us now describe the canonical transducer S_u,v corresponding to a feasible solution u_x,i, v_x,i of eqn:advFunction. Its local space = is of the form ⊗⊗⊗.Hereacts as in notation of sec:canonicalDefinition,is an n-qudit indicating the index of the input variable,is a qubit indicating direction of the query, andis a q-qudit storing the output of the query. We use ↑ and ↓ to denote the basis states of . The first one stands for the direct, and the second one for the inverse query. The input oracle acts on ⊗⊗ asO_x = ⊕_i∈ [n]O_x,i,whereO_x,i→,||⟩0>↦||⟩x_i>is the ith constituent of the input oracle.Define the following vectors in :_x,i = u_x,i + v_x,i/2_x,i = u_x,i - v_x,i/2.They possess the following important property:<_x,i, _y,i> - <_x,i, _y,i>=<u_x,i, v_y,i> + <v_x,i, u_y,i>/2. The catalyst for the input x is v_x = ∑_i∈ [n]|R⟩|i> [ |B⟩|↑>|Q⟩|0>W|v^↑_x,i> + |B⟩|↓>|Q⟩|x_i> W|v^↓_x,i> ].The transducer starts in ξ_x ⊕ v_x = ||⟩0> ⊕||⟩v_x>. It applies the input oracle eqn:function_inputOracleDecomposition, which gives the stateψ_x = ||⟩0> ⊕∑_i∈ [n]|R⟩|i> [ |B⟩|↑>|Q⟩|x_i>W|v^↑_x,i> + |B⟩|↓>|Q⟩|0> W|v^↓_x,i> ].The construction then follows from the following claim. There exists an input-independent unitary _u,v that maps the state ψ_x from eqn:function1 into ||⟩f(x)> ⊕||⟩v_x> for all x. Indeed, for a pair of x,y∈ D, we have<ψ_x,ψ_y> = 1 + ∑_i∈ [n][ 1_x_i = y_i<_x,i, _y,i> + <_x,i, _y,i>].On the other hand, the inner product between||⟩f(x)> ⊕||⟩v_x> and ||⟩f(y)> ⊕||⟩v_y> is1_f(x)=f(y) + ∑_i∈ [n][<_x,i, _y,i> + 1_x_i = y_i<_x,i, _y,i>].To establish the existence of the unitary _u,v it suffices to show that eqn:functioninner1 and eqn:functioninner2 are equal for all x,y∈ D. Subtracting the latter from the former gives us1_f(x) f(y) - ∑_i: x_i y_i[<_x,i, _y,i> - <_x,i, _y,i>]= 1_f(x) f(y) - 1/2∑_i: x_i y_i[<u_x,i, v_y,i> + <v_x,i, u_y,i>] = 0using eqn:function_vinner and eqn:advFunctionCondition. Thus, we have that S_u,v on the input oracle O_x transduces ||⟩0> into ||⟩f(x)>. If we consider O_x as a direct sum of n input oracles as in eqn:function_inputOracleDecomposition, we get the partial query statesq^(i)_x (S_u,v) = _x,i⊕_x,i,withL^(i)_x (S_u,v) = _x,i^2 +_x,i^2 =u_x,i^2 + v_x,i^2/2by the parallelogram identity. Finally, the transduction complexity and the total query complexity isW_x (S_u,v) = L_x (S_u,v) =1/2[∑_i=1^n u_x,i^2 + v_x,i^2 ]which is in the objective of eqn:advFunctionObjective. This can be summarised as For every function f D→ [p] with D⊆[q]^n, there exists a canonical transducer S_f evaluating the function f and whose transduction complexity is bounded by (f). In more detail,the admissible subspace of S_f is ||⟩0>; for every x∈ D, S_f transduces ||⟩0>||⟩f(x)> with bidirectional access to the input oracle O_x encoding the input string x; andW_x (S_f) = L_x (S_f) ≤(f). Moreover, the catalyst of the transducer S_f is as in eqn:functionWitness. In particular, it executes the input oracle only on its admissible subspace.§ COMPOSITION OF TRANSDUCERSIn this section, we describe basic properties of transducers. In particular, we show how to combine simple transducers in order to obtain more complex ones.This is akin to quantum algorithms being build out of elementary operations and subroutines. In this section, we mostly focus on the circuit model of computation. §.§ Basic PropertiesFrom thm:transduce, it follows that, for a fixed S and O, the mappings ξ↦ v(S, O, ξ) and ξ↦ q(S, O, ξ) are linear. In particular, for c∈, we haveW(S, O, cξ) = |c|^2 W(S, O, ξ)L^(i)(S, O, cξ) = |c|^2 L^(i)(S, O, ξ).Notice, however, that it is not necessarily true that W(S,O,ξ_1+ξ_2) = W(S,O,ξ_1) + W(S,O,ξ_2) even for orthogonal ξ_1 and ξ_2. On the other hand, it is the case that for ξ_1∈_1⊗ and ξ_2∈_2⊗, we haveW(S,O,ξ_1⊕ξ_2) = W(S,O,ξ_1) + W(S,O,ξ_2),using the extended definition of eqn:transductionExtended. [Inverse]For a canonical transducer S, the inverse transducer S^-1 satisfies τS^-1(O^*)ξ whenever ξS(O)τ. Moreover, S^-1 can be implemented in the canonical form, it has the same time complexity as S,W(S^-1, O^*, τ) = W(S, O, ξ),q(S^-1, O^*, τ) = (I⊗ O) q (S, O, ξ).Let v = ⊕ be the catalyst of the transduction ξSτ. From eqn:transduce, it is clear that if S(O) maps ξ⊕ v ↦τ⊕ v, then S(O)^* maps τ⊕ v ↦ξ⊕ v, hence, transduces τ into ξ with the same catalyst. One problem is that its action, as the inverse of eqn:canonicalForm,τ⊕⊕()^*ξ⊕⊕(I⊗ O)^*ξ⊕⊕.is not in the canonical form. But we can take the following transducer S^-1 in its stead:τ⊕⊕(I⊗ O)^*τ⊕⊕()^*ξ⊕⊕ (I⊗ O).It is in the canonical form, and satisfies all the conditions.§.§ AlignmentIn the remaining part of this section, we will study different ways of combining transducers S_1,…,S_m. First, we consider parallel composition of transducers ⊕_i S_i, where individual S_i act on orthogonal parts of the workspace. Then we move onto sequential composition S_m * S_m-1 * ⋯ * S_1, where they act on the same space one after another. Finally, we consider functional composition of two transducers, where the second transducer acts as an oracle for the first one.We generally assume that all S_i use the same oracle O. This is without loss of generality since if they use different sets of input oracles, we can assume they all use the oracle O which is the direct sum of the union of these sets of oracles. Individual S_i will then just ignore the input oracles they are not using.In principle, the spaces _i and _i can differ between different S_i, but we assume they are all embedded into some larger register , which also serves asper our convention of sec:implementation. What is crucial, though, is that all S_i use the same privacy qubit , the same query register , and, most importantly, the oracle O is applied to the same subset of registers in all S_i. This is summarised by the following definition, cf. rem:aligned: We say that canonical transducers S_1,…,S_m are aligned in the oracle O^(i) if the query of O^(i) is conditioned on the same value |R⟩|i> of the same register and acts on the same subset of registers in all of them. We say that S_1,…,S_m are aligned if they are aligned in all their oracles, and, additionally, use the same privacy qubit .If the alignment condition is not satisfied, the transducers have to explicitly move their registers around to meet it. This might take time if the registers are lengthy. §.§ Parallel CompositionParallel composition of two or several quantum programs is their execution as a direct sum on orthogonal parts of the space of the algorithm. For transducers, parallel composition can be implemented in a straightforward way.Let S_1,…,S_m be canonical transducers. We assume they all use the same space ⊕, the same input oracle O→, and are aligned. In particular, the query = I_⊕⊕ I⊗ O is given by eqn:canonicalQuery and is controlled by |R⟩|1> in all of them.Let the register = ^m. The direct sum ⊕_i S_i is a canonical transducer in the space ⊗(⊕) = (⊗)⊕(⊗). It has the same input oracle O as all S_i. The query is again controlled by |R⟩|1>. The work unitary of ⊕_i S_i is ⊕_i _i, where _i is the work unitary of S_i. [Parallel Composition]The canonical transducer S = ⊕_i S_i from defn:parallel satisfies the following conditions. Assume ξ_iS_i(O)τ_i for all i. Then, S(O) transduces ξ = ⊕_i ξ_i into τ = ⊕_i τ_i. Moreover,W(S, O, ξ) = ∑_i W(S_i, O, ξ_i)q(S, O, ξ) = ⊕_i q(S_i, O, ξ_i).The time complexity of S is equal to the time complexity of implementing ⊕_i _i. Recall that all S_i are aligned, and, hence, use the same space,is their common privacy qubit, andtheir common query register. The privacy and the query registers of S will still beand .Let v_i be the catalyst of transduction ξ_i S_i(O)τ_i. The initial coupling of S isξ⊕ v = ∑_i=1^m |J⟩|i> [ |P⟩|0> |H⟩|ξ_i> + |P⟩|1> |L⟩|v_i> ].As required by canonicity, we first apply the input oracle O controlled by |R⟩|1>. This has the effect that O is applied to all S_i in parallel. Then, conditioned on the value i in J, we apply the work unitary _i to the last two registers. By the assumption ξ_iS_i(O)τ_i , this gives∑_i=1^m |J⟩|i> [ |P⟩|0> |H⟩|τ_i> + |P⟩|1> |L⟩|v_i> ] = τ⊕ v.Eq. eqn:paralellComplexity follows from eqn:parallel_xi+v. In defn:parallel, we assume all S_i use the same input oracle O. It is, however, possible to allow each S_i to use its own input oracle O_i so that the total input oracle of ⊕_i S_i is ⊕_i O_i, and ξ_i S_i(O_i)τ_i. This is used, for instance, in iterated and composed functions.We account for this possibility by allowingto contain , so that the input oracle of S_i has read-only access to the value of |E⟩|i>. This does not interfere with the proof of prp:parallel, and we get the following identities instead of eqn:paralellComplexity:W(S, O, ξ) = ∑_i W(S_i, O_i, ξ_i)q(S, O, ξ) = ⊕_i q(S_i, O_i, ξ_i).The time complexity of implementing ⊕_i S_i greatly depends on the structure of S_i and the computational model. For instance, if we assume the QRAG model, implementation of ⊕ S_i can be done in time essentially max_i T(S_i) as per cor:selectProgram. The important special case when all S_i are the same can be efficiently handled in the circuitmodel as well.We write it out explicitly for ease of referencing.Let S be a canonical transducer with space ⊕, input oracle O→, and query given by eqn:canonicalQuery. Let alsobe a space, and I_ be the identity on . We define I_⊗ S as a canonical transducer in the space (⊗)⊕ (⊗), with the work unitary I_⊗. In other words, I_⊗ S = ⊕_i S where the summation is over the basis of .In the settings of defn:byIdentity, assume ξ_iS(O)τ_i as i ranges over the basis of . Then, I_⊗ S(O) transduces ξ = ⊕_i ξ_i into τ = ⊕_i τ_i. The complexities are given by eqn:paralellComplexity or eqn:paralellComplexityMultipleOracles with all S_i=S. The time complexity is T(I_⊗ S) = T(S). Note that complexities in eqn:transductionExtended are the same as in this corollary with Eq. eqn:paralellComplexity used.§.§ Sequential CompositionA quantum program eqn:program is a sequence of gates applied one after the other. Therefore, sequential composition is of prime importance in quantum algorithms. Let us formally define it for transducers.Let S_1,…, S_m be an aligned family of canonical transducers, all on the same public spaceand the same input oracle O. Its sequential composition is a canonical transducer S = S_m * S_m-1 * ⋯ * S_1 on the same public space with the following property. For every sequence of transductionsξ = ψ_1 S_1(O)ψ_2 S_2(O)ψ_3 S_3(O)⋯S_m-1(O)ψ_mS_m(O)ψ_m+1 = τ,S(O) transduces the initial state ξ into the final state τ. The above definition does not specify the implementation as we give two different implementations in this section. The first one is tailored towards the circuit model, and the second one towards the QRAG model. This division is not strict though.[Sequential Composition, Sequential Implementation]A sequential composition S = S_m * S_m-1 * ⋯ * S_1 as in defn:sequential can be implemented by a canonical transducer with the following parameters:W(S, O, ξ) = ∑_t=1^m W(S_t, O, ψ_t),q(S, O, ξ) = ⊕_t=1^m q(S_t, O, ψ_t),andT(S) = (m)+ ∑_t=1^m (S_t),whereis defined in sec:prelimCircuit. The general idea is simple. We first apply the input oracle to make the queries in all S_t in parallel. After the query, we execute all _t one after the other, see fig:sequentialsequential. Some care must be taken, however, so that transducers act inside their respective private spaces and do not interfere with private spaces of other transducers.A graphical illustration to the construction of prp:sequentialsequential with m=4. For convenience of representation, we draw the catalyst states by vertical lines, not horizontal ones like in fig:canonical. We also write O_t instead of (I⊗ O)_t to save space. We first apply the query I_m⊗ I⊗ O, which implements the queries in all _t. After that, we apply all of _t one after the other.[every node/.style=font=, every path/.append style=thick,->] 1223344 (4,6) rectangle (15.7, 7) node[pos=0.5] I_m⊗ I⊗ O; [] (2.5,4) node[left] ξ to node[above]ψ_1 (3.8,4); [] (15.7,4) to node[above]ψ_5 (16.8,4) node[right] τ;The transducer S uses the same privacy and query registersandas the family S_1,…,S_m. We additionally use an m-qudit , so that the local space of the transducer S_t is marked by the value |K⟩|t>. Therefore, the space of S is of the form ⊗ (⊕). Its public space is stillembedded as |K⟩|0> ⊗.Let ξ and ψ_t be as in eqn:sequenceOfTransductions. For each t, let v_t be the catalyst of the transduction ψ_t S_t(O)ψ_t+1 from eqn:sequenceOfTransductions. For the transduction ξS(O)τ, we have the following initial coupling:ξ⊕ v = |P⟩|0>|K⟩|0>|H⟩|ξ> + |P⟩|1> ∑_t=1^m |K⟩ |t>|L⟩|v_t>.This already implies eqn:sequentialWitnessComplexity.As required by the definition, the first operation is the application of the input oracle conditioned on |R⟩|1>. This gives the state|P⟩|0>|K⟩|0>|H⟩|ψ_1> + |P⟩|1> ∑_t=1^m |K⟩ |t> L| v_t>,where v_t is defined as in eqn:Ov, and we used that ξ = ψ_1.The work part of the transducer S is as follows. We assume the indexing of K is done modulo m. * For t=1,…,m: * Controlled by |P⟩|0>, replace the value |K⟩|t-1> by |K⟩|t>.* Apply the work unitary _t controlled by |K⟩|t>.By induction and using eqn:canonicalSequence, after ℓ iterations of the loop, we have the state|P⟩|0>|K⟩|ℓ>|H⟩|ψ_ℓ+1>+ |P⟩|1> ∑_t=1^ℓ|K⟩ |t>|L⟩|v_t> + |P⟩|1> ∑_t=ℓ+1^m |K⟩ |t> L| v_t>.And after execution of the whole transducer S, we have the state|P⟩|0>|K⟩|0>|H⟩|τ> + |P⟩|1> ∑_t=1^m |K⟩ |t>|L⟩|v_t> = τ⊕ v,where we used that ψ_m+1=τ. The time complexity in eqn:sequentialTimeComplexity can be achieved using the direct-sum finite automaton of lem:automaton.The second implementation of the sequential composition is slightly less intuitive. We move all the intermediate states in eqn:sequenceOfTransductions to the catalyst. This increases the catalyst size, but now all the operations _t can be implemented in parallel. [Sequential Composition, Parallel Implementation]A sequential composition S = S_m * S_m-1 * ⋯ * S_1 as in defn:sequential can be implemented by a transducer with the following parameters:W(S, O, ξ) = ∑_t=2^m ψ_t^2 + ∑_t=1^m W(S_t, O, ψ_t) , q(S, O, ξ) = ⊕_t=1^m q(S_t, O, ψ_t)andT(S) = (log m) + T [⊕_t _t].In the QRAG model, we usually replace (log m) in eqn:sequentialParallelTime by ().The main new idea compared to prp:sequentialsequential is that we do not compute the intermediate ψ_t from eqn:sequenceOfTransductions, but store the sequence ψ_2,…,ψ_m in the catalyst. We apply all of _t in parallel, thus moving this sequence forward by one position, see fig:sequentialParallel.A graphical illustration to the construction of prp:sequentialParallel with m=4. For convenience of representation, we draw the catalyst states by vertical lines, not horizontal ones like in fig:canonical. We also write O_t instead of (I⊗ O)_t to save space. We first apply the query I_m⊗ I⊗ O, which implements the queries in all _t. After that, we apply all of _t in parallel. The initial state ξ becomes the input for _1. The input of _t for t>1 is taken from the catalyst, and the output becomes the catalyst for t+1, except for t=m, which yields the terminal state τ. [every node/.style=font=, every path/.append style=thick,->] 1222333444 (4,6) rectangle (15.7, 7) node[pos=0.5] I_m⊗ I⊗ O; [] (2.5,4) node[left] ξ to node[above]ψ_1 (3.8,4); [] (15.7,4) to node[above]ψ_5 (16.8,4) node[right] τ;Again, we assume all S have space ⊕, have privacy qubit , and query register . The transducer S uses the same query register , but we introduce a new privacy qubit '. We also use an m-qudit , so that the space of S is of the form ⊗'⊗ (⊕).Let ξ and ψ_t be as in eqn:sequenceOfTransductions. For each t, let v_t be the catalyst of transduction ψ_t S_t(O)ψ_t+1. We start with the following initial couplingξ⊕ v =|P⟩'|0>|H⟩|ξ> + |P⟩'|1> [∑_t=2^m |K⟩ |t>|P⟩|0>|H⟩|ψ_t> + ∑_t=1^m |K⟩|t>|P⟩|1>|L⟩|v_t>].This already gives eqn:sequentialComplexity. As required by the canonicity assumption, we first apply the input oracle. This gives|P⟩'|0>|H⟩|ξ> + |P⟩'|1> [∑_t=2^m |K⟩ |t>|P⟩|0>|H⟩|ψ_t> + ∑_t=1^m |K⟩|t>|P⟩|1>|L⟩| v_t>],whereis defined in eqn:Ov. Here we use thathas value 0 in the registerand is not affected by the input oracle. Next, we exchange |P⟩'|0>⊗ with |P⟩'|1> |K⟩|1> |P⟩|0> ⊗. Since ξ=ψ_1, this gives|P⟩'|1> [∑_t=1^m |K⟩ |t>|P⟩|0>|H⟩|ψ_t> + ∑_t=1^m |K⟩|t>|P⟩|1>|L⟩| v_t>].Now we apply _t to the last two registers, controlled by the value in the register . In other words, we apply ⊕_t _t. By eqn:canonicalSequence, this gives|P⟩'|1> [∑_t=1^m |K⟩ |t>|P⟩|0>|H⟩|ψ_t+1> + ∑_t=1^m |K⟩|t>|P⟩|1>|L⟩|v_t>].Now we increment the value in the registerconditioned on |P⟩|0>, and exchange |P⟩'|0>⊗ with |P⟩'|1> |K⟩|m+1> |P⟩|0> ⊗. Since τ=ψ_m+1, we get|P⟩'|0>|H⟩|τ> + |P⟩'|1> [∑_t=2^m |K⟩ |t>|P⟩|0>|H⟩|ψ_t> + ∑_t=1^m |K⟩|t>|P⟩|1>|L⟩|v_t>] = τ⊕ v.The time complexity estimate is obvious. §.§ Functional CompositionWe defined functional composition in sec:conceptualTypes as an algorithm where a subroutine is used to implement an oracle call. In the case of transducers, this falls under the transitivity of transduction,prp:transitivityOfTransduction, as part of the transducer (the oracle call) is implemented as a transduction action of another transducer. In this section, we give an explicit construction for canonical transducers.We assume the settings similar to fig:composition, except we use transducers S_A and S_B instead of programs A and B. The outer transducer S_A has two input oracles O⊕ O' joined as in sec:multipleInputOracles. The first one, O, is the global input oracle, and the second one, O', is the oracle realised as the transduction action of the inner transducer S_B. We assume both transducers are in the canonical form and aligned in the oracle O. We denote their public spaces by _A and _B, respectively. If there are several subroutines implemented by different transducers, then, as in fig:compositionMultiple, they can be joined via parallel composition of prp:parallel.[Functional Composition]Under the above assumptions, there exists a canonical transducer S_A∘ S_B with the public space _A and the oracle O, which satisfies the following properties. =0pt* Its transduction action on input oracle O is equal to the transduction action of S_A on oracle O ⊕ O', where O' = S_B(O)__B.* Its transduction complexity satisfiesW(S_A∘ S_B, O, ξ) = W(S_A, O⊕ O', ξ) + W[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]. * Its total query state isq(S_A∘ S_B, O, ξ) = q^(0)(S_A, O⊕ O', ξ) ⊕ q[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]. * Its time complexity is (S_A)+(S_B).Here q^(0) and q^(1) denote the partial query states of S_A to the oracles O and O', respectively. We also used the extended versions of W and q from eqn:transductionExtended for the transducer S_B. Note that both Equations eqn:functionalTransductionComplexity and eqn:functionalQueryComplexity are reminiscent of the “gold standard” eqn:randomComposition2.Letbe the space of the input oracle O. The space of the transducer S_A is of the form_A⊕_A ⊕_A ⊕'_A, where the last two spaces are for the queries to O and O', respectively. In particular, '_A = _A ⊗_B by the assumption that S_B(O) implements O'.Let v = ⊕⊕ v' be the catalyst for the transduction ξτ by S_A(O⊕ O'). That is, = q^(0)(S_A, O⊕ O', ξ) and v' = q^(1)(S_A, O⊕ O', ξ). By eqn:canonicalSequence, the transducer S_A imposes the following chain of transformations in _A⊕_A ⊕_A ⊕'_A:ξ⊕⊕⊕ v' Oξ⊕⊕ (I⊗ O)⊕ v' O'ξ⊕⊕ (I⊗ O)⊕ (I⊗ O') v' _Aτ⊕⊕⊕ v',where we separated the applications of O and O'. Recall that the identity I acts on _A.w^∘ w^∙The space of S_B is _B ⊕_B ⊕_B, where _B = _B⊗. We obtain the transducer I⊗ S_B as in defn:byIdentity, whose space is _A⊗ (_B⊕_B⊕_B).Let w= ⊕ be the catalyst for the transduction v' (I⊗ O') v' by I⊗ S_B(O).In particular, =q[ S_B, O, q^(1)(S_A, O⊕ O', ξ)]. By eqn:canonicalSequence again, we have the chain of transformations in _A⊗ (_B⊕_B⊕_B):v' ⊕⊕ v' ⊕⊕ (I⊗ I_B⊗ O)I⊗_B (I⊗ O')v' ⊕⊕,where I_B is the identity on _B.A graphical representation of a function composition of transducers S_A and S_B. We omit the tensor multipliers of O and O' on the arrows to save space. The vector v' is simultaneously a non-queried catalyst for S_A∘ S_B and the input to I⊗ S_B.[every node/.style=font=, every path/.append style=thick,->] [] (0,0) node[above] ξ to (11,0) ; [] (0,-1) node[above]to (11,-1) ; [,out=0,in=180] (0,-2) node[above]to (2,-4) ; [-,purple,out=0,in=180] (0,-3) node[above] v' to (2,-2) ; [-,,out=0,in=180] (0,-4) node[above]to (2,-3) ; [] (0,-5) node[above]to (2,-5) ;[-,purple] (2,-2) to (5,-2); [-,] (2,-3) to (5,-3);(2, -3.5) rectangle (5,-5.5) node[pos=0.5] (I⊕ I⊗ I_B)⊗ O;[purple,out=0,in=180] (5,-2) to (7,-3) ; [,out=0,in=180] (5,-3) to (7,-4) ; [] (5,-5) to node[above] O (7,-5) ;[-,, out=0, in=180] (5,-4) to (7,-2) ;(7, -2.5) rectangle (9,-5.5) node[pos=0.5] I⊗_B;[] (7,-2) to node[above,pos=0.4] O (11,-2) ; [] (9,-3) to node[above] O'v' (11,-3); [] (9, -4) to (13,-4) node[above]; [] (9, -5) to (13,-5) node[above]; (11,0.5) rectangle (12, -3.5) node[pos=0.5] _A;[] (12,0) to (13,0) node[above] ξ ; [] (12,-1) to (13,-1) node[above]; [] (12,-2) to (13,-2) node[above]; [purple] (12,-3) to (13,-3) node[above] v' ;The transducer S_A∘ S_B works as follows; see fig:functional.It starts inξ⊕⊕⊕ v' ⊕⊕.It applies the input oracle O to the third and the last terms, which corresponds to the first steps in both eqn:functional_1 and eqn:functional_2. This givesξ⊕⊕ (I⊗ O)⊕ v' ⊕⊕ (I⊗ I_B⊗ O).Then it performs I⊗_B, which is the second operation in eqn:functional_2, and which givesξ⊕⊕ (I⊗ O)⊕ (I⊗ O') v' ⊕⊕.After that, _A is performed, which is the last operation of eqn:functional_1, and we get the final stateτ⊕⊕⊕ v' ⊕⊕. The transduction eqn:functionalTransductionComplexity and the query eqn:functionalQueryComplexity complexities follow from eqn:functional_witness. In order to get the time complexity, we have to elaborate on the placement of all these transducers in registers. Since S_A and S_B are aligned in O, we assume they use the same value |R⟩|1> of the same register to denote its execution. Concerning O', which is an oracle only for S_A, we assume it is indicated by another qubit '. Let _A and _B be the privacy qubits of S_A and S_B, respectively; we assume they are different.Let us write down the values of these indicator qubits for the various subspaces of the composed transducer:[_A_A_A '_Aand _B_B_B;_A 0 1 1 1 1 1; 0 0 1 0 0 1; ' 0 0 0 1 1 1;_B 0 0 0 0 1 1; ]Other than that, we assume we can embed all these subspaces into registers also preserving '_A = ⊗_B. The privacy qubit of S_A∘ S_B is _A and its query register is .Let us go through the steps of S_A∘ S_B. The application of the query to O to get to eqn:functional_A is conditioned on |R⟩|1> as required, and it is possible because we assume S_A and S_B are aligned in O. The application of I⊗_B to get to eqn:functional_B is conditioned on |R⟩'|1>. Finally, the application of _A to get to the final state eqn:functional_C is conditioned on |P⟩_B|0>. This gives the required time complexity estimate. § TRANSDUCERS FROM PROGRAMSIn this section, we describe how to convert a quantum program A like in eqn:program into a canonical transducer S_A, and give the corresponding corollaries. We consider both the circuit and the QRAG models. The general idea is similar in both cases. First, we obtain transducers for individual gates, and then compose them using either prp:sequentialsequential for the circuit model, or prp:sequentialParallel for the QRAG model.§.§ General Assumptions We are given a program A, and our goal is to construct a canonical transducer S_A whose transduction action is identical to the action of A.One thing to observe is that we have to modify our assumptions on the execution of input oracles by introducing an additional register related to . There are two main reasons for that. First, the initial state ξ of S_A can be any state in . In particular, it can usewhich is in contradiction with the assumption of sec:canonicalDefinition that ξ must be contained in . Second, as described in sec:introAlgorithm->Transducer, the catalyst in the QRAG case is the history state eqn:introHistoryState. Various ψ_t can use , and we would like to protect them from the application of the input oracle in S_A.Because of that, we introduce an additional qubit . The input oracle in the canonical transducer S_A is applied conditioned by |⟩|1>. More precisely, the i-th input oracle is controlled by |⟩|1> |R⟩|i>.Additionally, we assume that all the queries made by the program A are aligned, see defn:alignment, and, in case of several programs, they are aligned by their shared input oracles. This is necessary since the composition results of sec:properties require the transducers to be aligned.§.§ Building BlocksHere we describe transducers corresponding to elementary operations. The following proposition is trivial.[Trivial Transducer]Let A be a quantum algorithm (without oracle calls) that implements some unitary in . It can be considered as a transducer withbeing empty, T(A) being the time complexity of A, and both W(A,ξ) and q(A,ξ) equal to zero. Oracle execution is more tricky because of the canonicity assumption. Recall the query = ⊕ I⊗ O from eqn:query taking place in = ⊕ with = ⊗. Let us divide ξ = ξ^∘⊕ξ^∙ accordingly.[Oracle]For a fixed embedding = ⊕⊗ as above, there exists a canonical transducer S_Q such that ξS_Q(O)ξ for all ξ∈ and unitaries O→. The transducer satisfies W(S_Q, O,ξ) = ξ^∙^2, q(S_Q, O, ξ) = ξ^∙, and T(S_Q)=O(1). The private space of S_Q will be == ⊗ withequal to . The catalyst is v == ξ^∙. The transducer S_Q first applies the input oracle to , which is achieved by conditioning on |⟩|1>:|P⟩ |0> |⟩|0> |H⟩ |ξ> + |P⟩|1> |⟩|1>|L⟩ |ξ^∙> |P⟩ |0> |⟩|0> |H⟩ |ξ> + |P⟩|1> |⟩|1>L |(I⊗ O)ξ^∙>= |P⟩|0> |⟩|0> |R⟩|0> ||⟩ξ^∘> + |P⟩|0> |⟩|0> |R⟩|1> ||⟩ξ^∙> + |P⟩|1> |⟩|1> |R⟩|1> ||⟩(I⊗ O)ξ^∙>.Now apply C-NOT to bothandcontrolled by |R⟩|1>. This gives|P⟩|0> |⟩|0> |R⟩|0> ||⟩ξ^∘> + |P⟩|0> |⟩|0> |R⟩|1> ||⟩(I⊗ O)ξ^∙> + |P⟩|1> |⟩|1> |R⟩|1> ||⟩ξ^∙> = |P⟩ |0> |⟩|0>H |ξ> + |P⟩|1> |⟩|1> L |ξ^∙>. §.§ Circuit ModelFor the circuit model, we have the following result, which is a first half of thm:introProg->Transducer.Let A=A(O) be a quantum program in some spaceassuming the circuit model. We assume all the queries in A are aligned, see defn:alignment. Then, there exists a canonical transducer S_A(O) with the following properties. =0pt* S_A(O)_ = A(O) for all input oracles O.* For any input oracle O and the initial state ξ, we have q(S_A, O, ξ) = q(A, O, ξ).In particular, L(S_A, O, ξ) = L(A, O, ξ).* The catalyst v(S_A, O,ξ) is equal to the query state q(S_A, O, ξ).In particular, W(S_A, O, ξ) = L(A, O, ξ).* Finally, the transducer S_A can be implemented in the circuit model, and T(S_A) = (A) + (Q(A)) = (T(A)), where the complexity measures of A are as in sec:prelimCircuit.Take the representation of the query algorithm as in eqn:preAlgorithm. We interpret each unitary U_t as a transducer using prp:gates, and transform each query into an independent copy of the transducer S_Q from prp:oracle.Now, we apply sequential composition of prp:sequentialsequential with m=2Q(A)+1. The total query state q(A, O, ξ) is defined as the direct sum of all the queries given to the input oracle, hence, the third and the second points follow from eqn:sequentialWitnessComplexity. The time complexity follows from eqn:sequentialTimeComplexity using that T(A) = Q(A) + ∑_t T(U_t). While canonical transducers are nice from the theoretical point of view, designing one from scratch is usually inconvenient. The following result shows that we can convert any transducer into a canonical form with a slight increase in complexity.Let S = S(O) be a transducer in ⊕ not in the canonical form. Recall from sec:transducerDefinition that we defined its total query state q_(S,O,ξ), and Las Vegas query complexity L_(S, O, ξ) on ξ∈ as that of S, considered as a usual quantum algorithm in ⊕, on the initial state ξ⊕ v(S(O), ξ), where v[S(O),ξ] is as in thm:transduce. We assume all the queries made in S are aligned.[Transforming Transducers into Canonical Form]For a non-canonical transducer S = S(O) as above, there exists a canonical transducer S'= S'(O) with the same transduction action and such that, for all O and ξ,q(S', O, ξ) = q_(S, O, ξ),W(S', O, ξ) = W(S(O), ξ) + L_(S, O, ξ),and T(S') = (T(S)). Let S_S = S_S(O) be a canonical transducer as obtained in thm:program->transducerCircuit from S=S(O) considered as a quantum program. Its public space is ⊕, and S_S(O)_⊕ = S(O).Let ξ' = ξ⊕ v[S(O),ξ].By thm:program->transducerCircuit, we have T(S_S) = [T(S)], and v_⊕(S_S, O, ξ')= q_⊕(S_S, O, ξ')= q_⊕(S, O, ξ') = q_(S, O, ξ),where, in the third expression, we consider S as a usual quantum program in ⊕.We obtain S' = S'(O) as the transduction action of S_S(O) on , using the transitivity of transduction, prp:transitivityOfTransduction. First, S_S(O) is in canonical form when considered as a transducer with the public space ⊕. A fortiori, it is canonical also on the public space . Next, its time complexity does not change, hence, T(S') = T(S_S) = [T(S)]. For the query state, from eqn:transitivityWitness and eqn:canoning1, we getq_(S', O, ξ) = q_⊕ (S_S, O, ξ') = q_(S, O, ξ).Finally, for the transduction complexity, we can utilise eqn:transitivityWitness in the following way:W(S', O, ξ) = W_[S_S(O), ξ]= W_[S_S(O)_⊕, ξ] + W_⊕[ S_S(O), ξ'] = W_(S(O), ξ) + L_(S, O, ξ),where we used eqn:canoning1 in the last equality. The remaining corollaries of thm:program->transducerCircuit were already proven in sec:overview: Theorems <ref> and <ref>.Let us note that thm:program->transducerCircuit might not always be the best way to get a transducer S_A from a quantum program A in the circuit model. One can use additional structure of A to get better transducers. For instance, if A repeatedly uses the same sequence of gates, one can define it as a new oracle, thus reducing the time complexity of the transducer, see, e.g., prp:iteratedFunctionsImproved. If A contains a large loop, one can use prp:sequentialParallel. Obtaining efficient transducers in the circuit model for specific cases seems like an interesting research direction.§.§ QRAG Model Assuming the QRAG model, we get the remaining half of thm:introProg->Transducer. Let A=A(O) be a quantum program in some space , and assume we have QRAM access to the description of A. Then, there exists a canonical transducer S_A(O) with the following properties. =0pt* S_A(O)_ = A(O) for all input oracles O.* For any input oracle O and initial state ξ, we have q(S_A, O, ξ) = q(A, O, ξ).In particular, L(S_A, O, ξ) = L(A, O, ξ).* Also, W(S_A, O, ξ) ≤ T(A) ξ^2.* Finally, in the QRAG model, we have T(S_A) = () as defined in eqn:timeR.The proof parallels that of thm:program->transducerCircuit using prp:sequentialParallel instead of prp:sequentialsequential, but since this special case might be of interest, we give an explicit construction here, slightly simplifying it along the way.It is convenient to assume a different but essentially equivalent form of a quantum algorithm. Namely, we assume that the algorithm A is given asA(O) = G_m-1^b_m-1G_m-2⋯^b_2G_1^b_1G_0,where each G_i is a gate (not a query),is the query operator eqn:query, and each b_t is a bit that indicates whether there is a query before the t-th gate (b_0 is always 0). A program like in eqn:program can be transformed into eqn:program->alternative with m≤ T+1 adding identity G_t if necessary. We assume we have QRAM access to an array specifying the gates G_t as well as to the array of b_t.Let the algorithm A go through the following sequence of states on the initial state ξ and the oracle O:ξ = ψ_0 G_0ψ_1 G_1^b_1ψ_2 G_2^b_2⋯G_m-1^b_m-1ψ_m = τ.At high level, the action of S_A isξ⊕ v =∑_t=0^m-1||⟩t> ||⟩ψ_t>^b_t∑_t=0^m-1||⟩t>|^b_tψ_t> G_t∑_t=0^m-1||⟩t> ||⟩ψ_t+1> ⟼∑_t=1^m||⟩t> ||⟩ψ_t>= τ⊕ v,where, we first apply the input oracle conditioned on b_t, then G_t conditioned on t using thm:select, and then increment t by 1 modulo m.ψ^∘ ψ^∙Recall that we have a decomposition = ⊕ of the space of A, which is indicated by the query register . As mentioned in sec:assumptions, the transducer S_A uses a different query qubit . We will use notation = ⊗. Let D denote the C-NOT oncontrolled by . Then, for ψ = ⊕∈, we have|⟩|ψ> = |⟩|0> |R⟩|0>||⟩> + |⟩|0> |R⟩|1>||⟩> |⟩|Dψ> = |⟩|0> |R⟩|0>||⟩> + |⟩|1> |R⟩|1>||⟩>.Letbe an m-qudit with operations modulo m, andbe the privacy qubit of S_A.Let us go through the steps of eqn:program->Transducer. The initial coupling is given byξ⊕ v = |P⟩|0> |T⟩|0>|⟩|ψ_0> + ∑_t=1^m-1|P⟩|1> |T⟩|t> |D^b_tψ_t>.First, as required by the canonical form and our assumptions in sec:assumptions, we apply the input oracle O conditioned on |⟩|1>. This acts ason Dψ_t, but does not change ψ_t. Then, we apply the operation D controlled on b_t (which can be accessed using the QRAM). After that, we apply C-NOT to P controlled by |T⟩|0>. This gives the second state in the sequence eqn:program->Transducer:|P⟩|1> ∑_t=0^m-1|T⟩|t> |^b_tψ_t>.Now, we apply the last two operations from eqn:program->Transducer: G_t conditioned on |T⟩|t>, and increment ofby 1 modulo m. This gives|P⟩|1> ∑_t=1^m|T⟩|t> |⟩|ψ_t>.Now we apply the operation D controlled on b_t and apply C-NOT to the register P controlled by |T⟩|0> = |T⟩|m>. This gives the final state|P⟩|0> |T⟩|0>|⟩|ψ_m> + ∑_t=1^m-1|P⟩|1> |T⟩|t> |D^b_tψ_t> = τ⊕ v. Since ψ_t = ξ, we have that W(S_A, O, ξ) = (m-1) ξ^2 ≤ T(A) ξ^2. It is clear that q(S_A, O, ξ) = q(A, O, ξ), and the time complexity of S_A is () thanks in particular to thm:select. In applications like in sec:introComposition, we often have to apply this construction in parallel. The following easy modification of the proof takes care of that.Let A_1,…,A_n be quantum programs in some space , all using the same oracle O. Assume we have QRAM access to their joint description. Then, the direct sum ⊕_i=1^n S_A_i of transducers defined in thm:program->transducerQRAG can be implemented in time (). By the joint access, we mean that we have access to the list m_1,…,m_n of the parameters m in eqn:program->alternative, as well as access to b_t and G_t of A_i as a double array with indices i and t.[Proof of prp:program->transducerParallel] We execute the transducers S_A_i in parallel using a registerto store the value of i. Accessing b_t and G_t now is double-indexed by i and t, and by our general assumption of eqn:timeR it takes time () to access them. For incrementation ofmodulo m, we use the array containing m_i. Other than that, it is a standard word-sided operation and takes time ().The consequences of this result were already considered in sec:overview: Theorems <ref>, <ref>, and <ref>. § EXAMPLE III: ITERATED FUNCTIONS In this section, we give a more detailed proof of thm:introIterated. We will use notation of sec:function for the transducer S_f built from the adversary bound. However, we assume a more general variant of a canonical transducer S_f for evaluation of f [q]^n→[q]. Its public space is ^q, its admissible space is spanned by ||⟩0>, and ||⟩0> S_f(O_x)||⟩f(x)> for all x∈ [q]^n. We assume the initial coupling of S_f on O_x is given by||⟩0>⊕ v_x = |P⟩|0>|R⟩|0> ||⟩0> +|P⟩|1>|R⟩|0> ||⟩_x> + |P⟩|1> ∑_i∈ [n]|R⟩|i> [ |B⟩|↑>|Q⟩|0>W|v^↑_x,i> + |B⟩|↓>|Q⟩|x_i> W|v^↓_x,i> ]for some vectors _x, v^↑_x,i, and v^↓_x,i, where the registers are as in eqn:functionWitness. The difference between eqn:composed_vx and eqn:functionWitness, however, is addition of the term _x that is not processed by the input oracle. Note that eqn:composed_vx gives the general form of a canonical transducer that executes the input oracle on the admissible subspace: the constituent O_x,i from eqn:function_inputConstituent on ||⟩0> and O_x,i^* on ||⟩x_i>.Recall the definition of the composed function f∘ gfrom eqn:introComposedFunction:[f∘ g] (z_1,1, …,z_1,m, z_2,1,…,z_2,m,……,z_n,1,…,z_n,m)= f[ g(z_1,1, …,z_1,m),g(z_2,1, …,z_2,m), …, g(z_n,1, …,z_n,m)].We define_i = (z_i,1, …,z_i,m)x = [g(_1), g(_2),… g(_n)]so that f(x) = [f∘ g](z). Recall also notation W_x(S) = W[S, O_x, ||⟩0>] and W(S) = max_x W_x(S) from eqn:function_Wx.Let S_f and S_g be canonical transducers for the functions f and g of the formdescribed above. Then, there exists a canonical transducer S_f∘ g for the composed function f∘ g from eqn:composedFunction that has the same form, and such thatW_z(S_f∘ g) = W_x (S_f) + ∑_i=1^n L^(i)_x(S_f)· W__i(S_g) ≤ W(S_f) + L(S_f)· W(S_g)andL_z(S_f∘ g) = ∑_i=1^n L^(i)_x(S_f)· L__i(S_g) ≤ L(S_f)· L(S_g),for every z∈[q]^nm. Time complexity satisfies T(S_f∘ g) = (S_f) + 2(S_g). As the first step, we create a bidirectional version S_g of S_g. Its public space is B⊗ Q, and its admissible space on the input oracle O_y is spanned by |B⟩|↑>|Q⟩|0> and |B⟩|↓> |Q⟩|g(y)>. Its corresponding transduction action is |B⟩|↑>|Q⟩|0> |B⟩|↑>|Q⟩|g(y)> |B⟩|↓>|Q⟩|g(y)> |B⟩|↓>|Q⟩|0>.Its time complexity is 2(S_g) and, in notation at the end of sec:introCanonical:[S_g, O_y] = W_y(S_g) [S_g, O_y] = L_y(S_g). We obtain S_g as follows. By swapping ↑ and ↓ if necessary, we may identify O_y^* and O_y. Then from prp:inverse, we obtain a transducer S^-1_g with transduction action ||⟩g(y)> ||⟩0> on O_y, whose complexity is identical to S_g. We may assume S_g and S^-1_g are aligned. We get S_g as S_g⊕ S_g^-1, where the direct sum is done via the register B. By the definition of , we have that for some unit vector (α, β)∈^2:(S_g, O_y) = W[S_g, O_y, α|B⟩|↑>|Q⟩|0> + β|B⟩|↓>|Q⟩|g(y)>]= |α|^2 W[S_g, O_y, ||⟩0>] + |β|^2 W[S_g^-1, O_y, ||⟩g(y)> ] = W_y(S_g),where we used Propositions <ref> and <ref>, as well as eqn:rescaling. Query complexity derivation is similar.As the next step, we construct the transducer I_n⊗S_g, where I_n acts on the span of |R⟩|i> with i>0. Moreover, as in rem:directSumDifferentOracle, we assume the input oracle uses the register R. Therefore, the transduction action of I_n⊗S_g on the input oracle O_z = ⊕_i=1^n O__iis identical to the action of O_x on its admissible subspace.Finally, we get S_f∘ g as S_f∘[I_n⊗S_g]. The time complexity estimate follows from prp:functional. For the transduction complexity, we obtain from eqn:compositionTransductionMultipleUpper, using that S_f makes only admissible queries to O_x and does not have direct access to O_z:W_z(S_f∘ g)= W[S_f∘ g, O_z, ||⟩0>] ≤ W[S_f, O_x, ||⟩0>] + ∑_i=1^n [ S_g, O__i ] L^(i) (S_f, O_x, ||⟩0>) = W_x (S_f) + ∑_i=1^n W__i(S_g)· L^(i)_x(S_f).The last inequality in eqn:composedFunctionTransduction follows from L_x(S_f) = ∑_i L^(i)_x(S_f). The estimate eqn:composedFunctionQuery is similar, where we again use that S_f does not make direct queries to O_z. Finally, since S_g makes only admissible queries to O_y, we get that S_f∘ g makes only admissible queries to O_z. Now we can prove a more detailed version of thm:introIterated.Let S_f be a canonical transducer for a function f of the form as in eqn:composed_vx, and such that L = L(S_f) = 1 + Ω(1).Let W = W(S_f) and T=T(S_f), assuming the circuit model. Then, for each d, there exists a bounded-error quantum algorithm evaluating the iterated function f^(d) in Monte Carlo query complexity (L^d) and time complexity [d· TW L^d-1] = _f(d· L^d) in the circuit model. #1S_f^(#1) Define d as S_f composed with itself d times using prp:composedFunction. By induction, we have that L[ d] ≤ L^d, andW[ d] ≤[1 + L + ⋯ + L^d-1] W= [L^d-1W].Similarly, its time complexity is at most 2d·(S_f) = (d· T). The theorem follows from thm:optimalImplementation. The time complexity of the previous theorem can be slightly improved. Let s denote the initial space complexity of the transducer S_f, i.e, the number of qubits used to encode the initial coupling ||⟩0>⊕ v_x. It can be much smaller than the time complexity of S_f, as well as its space complexity, which is the total number of qubits used by S_f.The time complexity of thm:iteratedFunctions can be improved to [(sd+T)W L^d-1], where s is the initial space complexity of the transducer S_f. We only sketch the proof, as it does not improve the overall asymptotic _f(d· L^d).[Proof sketch of prp:iteratedFunctionsImproved] By studying the proof of thm:iteratedFunctions, we can observe thatthe work unitary of d consists of repeated applications of S_f to different registers. We can treat S_f as an oracle. Then, the work unitary of d becomes a non-canonical transducer with the oracle S_f. It is non-aligned, but we can make it aligned in additional time s· d by switching the registers before each execution of S_f. We can convert it into the canonical form using prp:canoning, whichincreases the transduction complexity by at most a constant factor. However, now it takes only one execution of S_f and (sd) other operations to implement the transducer. The statement again follows from thm:optimalImplementation. § PERTURBED TRANSDUCERSIt is quite common in quantum algorithm to use subroutines that impose some error. The total correctness of the algorithm then follows from a variant of lem:surgery, assuming that the error of each constituent is small enough.In most cases in this paper, like in Sections <ref> and <ref>, the transduction action is the exact implementation of the required transformation. However, it is not always feasible, and it makes sense to study what happens if a transducer satisfies the condition eqn:transduce only approximately.It is worth noting that small errors in a transducer can result in large errors in the corresponding transduction action. In other words, it is possible that ξ⊕ v Sτ' ⊕ v' with v close to v', but ξSτ with τ being far from τ'. For example, let bothandbe 1-dimensional, and assume S acts asS[ 0; v ]↦[a; √(v^2-a^2) ]for some positive real v and a. Observe thatv - √(v^2-a^2) = a^2/v + √(v^2-a^2)can be very small for large v, even if a is substantial. Thus, on the basis of v≈ v - √(v^2-a^2) we may be tempted to assume that S approximately transduces 0 into a, but this is very far from the true transduction action 0S0.v τ §.§ Definition In order to solve this issue, we incorporate a perturbation, in the sense of lem:surgery, into the transducer S. Let S be a unitary in ⊕ that mapsξ⊕ v ↦⊕ .We assume its idealised version mapsSξ⊕ v ↦τ⊕ vwith the perturbationδ(S, ξ) = | (τ⊕ v) - (⊕) |. We call S the idealised or perturbed transducer, andthe approximate transducer. We say that S transduces ξ into τ, and we keep notation ξSτ, v(S,ξ) = v, W(S, ξ) = v^2, and τ(S,ξ) = τ.This also extends to the canonical form of transducer in sec:canonicalDefinition, where we decompose v = ⊕, and let q(S,O,ξ) = and L(S,O,ξ) = ^2. As before, we write δ(S, O, ξ) instead of δ(S(O),ξ), and similarly for other pieces of notation.As for usual quantum algorithms, we can use approximate transducers instead of the idealised ones, as long as we carefully keep track of the perturbations. In the remaining part of this section, we will briefly study the main results of this paper under the perturbation lenses. §.§ Implementation We have the following variant of thm:pumping.Under the assumptions of sec:perturbedDefinition, for every positive integer K, there exists a quantum algorithm that transforms ξ into τ' such that|τ' - τ(S,ξ)| ≤ 2 √(W(S,ξ)/K) + √(K)δ(S,ξ)for every ⊕→⊕, perturbed version S, and initial state ξ∈. The algorithm conditionally executesas a black box K times, and uses (K) other elementary operations. The algorithm is identical to that of thm:pumping. The analysis is similar with the only difference that, on Step 2(a), in order to obtain the mapping1/√(K)|T⟩|t> |P⟩ |0> |H⟩|ξ> + 1/√(K)|T⟩|t>|P⟩|1>|L⟩|v>⟼ 1/√(K)|T⟩|t> |P⟩ |0> |H⟩|τ> + 1/√(K)|T⟩|t>|P⟩|1>|L⟩|v>as in eqn:pumpingOneStep, we introduce a perturbation of size δ(S, ξ)/√(K). By lem:surgery, the total perturbation is then2 v/√(K) + K·δ(S,ξ)/√(K),which gives the required estimate. Again, this theorem can be reformulated as follows. For all W, >0, there exists a quantum algorithm that executesas a black box K = (1+W/^2) times, uses (K) other elementary operations, and -approximately transforms ξ into τ(S, ξ) for all S, , and the initial state ξ that satisfy W(S,ξ)≤ W and δ(S, ξ)≤/2√(K). We get a version of thm:optimalImplementation in a similar fashion.thm:optimalImplementation works assuming a perturbed transducer S.The estimate is|τ' - τ(S, O, ξ)| ≤2/√(K)√(W(S, O, ξ) + ∑_i=1^r[K/K^(i) -1] L^(i)(S, O, ξ) ) + √(K)δ(S, O, ξ).cor:optimalImplementation also works assuming a perturbed transducer S under the additional assumption of δ(S, O, ξ) ≤/2√(K). The proof is analogous to thm:pumpingApproximate:we use perturbation of size δ(S, O, ξ)/√(K) to get from eqn:optimal1 to eqn:optimal2. §.§ CompositionThe composition of perturbed transducers exactly follows the corresponding constructions of sec:properties, where we use lem:surgery to evaluate the total perturbation. This gives us the following proposition.Propositions <ref>, <ref>, <ref> and <ref> work assuming perturbed transducers. We get the following estimates:δ(S, O, cξ) = |c| δ (S, O, ξ), δ(S, O, ξ) ≤√(∑_i=1^m δ(S_i, O, ξ_i)^2)for prp:parallel, andδ(S, O, ξ) ≤∑_t=1^m δ(S_t, O, ψ_t)for Propositions <ref> and <ref>. For prp:functional, we haveδ(S_A∘ S_B, O, ξ) ≤δ[S_A, O⊕ O',ξ] + δ[S_B, O, q^(1) (S_A, O⊕ O', ξ) ].Eq. eqn:perturbationLinear follows by linearity. All the remaining estimates follow from the corresponding proofs in sec:properties replacing each transducer with its approximate version and using lem:surgery. The estimate eqn:perturbationParallel follows from the observation that perturbations in terms of the direct sum act on orthogonal subspaces. § PURIFIERS In this section, we formulate and prove the formal version of thm:introPurifier, as well as draw some consequences of it for composition of bounded-error algorithms.§.§ Boolean Case Recall our settings from sec:introPurifier. The input oracle performs the transformationO_ψ|M⟩ |0> ↦|M⟩|ψ> = |B⟩|0>|N⟩|ψ_0> + |B⟩|1> |N⟩|ψ_1>for some unit vector ψ in some space = ⊗ with = ^2, and it is promised that there exist constants 0≤ c-d < c + d ≤ 1 such thateither ψ_1^2 ≤ c-dor ψ_1^2 ≥ c+d,which corresponds to f(ψ)=0 and f(ψ)=1, respectively. Letμ = √((1-c)^2 - d^2) + √(c^2 - d^2) < 1.We say the input oracle O_ψ is admissible if it satisfies eqn:purifierInput and eqn:purifierCases.δ_purLet D be a positive integer. In the above assumptions, there exists a perturbed transduceron the 1-dimensional public space and with bidirectional access to O_ψ which satisfies the following conditions: =0pt* It transduces ||⟩0> into (-1)^f(ψ)||⟩0> for all admissible input oracles O_ψ. In particular, every vector in its public space is admissible.* On every admissible input oracle and normalised input state, its perturbation is at most = 2 μ^D-1.* Its transduction and query complexities, () and (), are [1/(1-μ)] = (1).* It executes the input oracle on the admissible subspace only: O_ψ on |M⟩|0> and O_ψ^* on |M⟩|ψ>.In the circuit model, the purifier can be implemented in time (D· s), where s is the number of qubits used in . In the QRAG model and assuming the RA input oracle, the purifier can be implemented in time (). Here we used () to denote maximal W(, O_ψ, ||⟩0>) over all admissible input oracles O_ψ. () is defined similarly.The outline of the proof was already given in sec:introPurifier. We will assume that D is even for concreteness, the case of odd D being analogous. Recall the parametersa = √(1-c+d/1-c-d) b = √(c+d/c-d).and the following vector in :=1/a||⟩0> ||⟩ψ_0> + b ||⟩1>||⟩ψ_1>, if f(ψ)=0;a ||⟩0> ||⟩ψ_0> + 1/b||⟩1>||⟩ψ_1>, if f(ψ)=1.We have the following important estimate.If f(ψ)=0:^2= 1/a^2ψ_0^2 + b^2 ψ_1^2 ≤1/a^2 (1-c+d) + b^2 (c-d)= μ,where we used that 1/a^2 < 1 < b^2. Similarly, for f(ψ)=1:^2= a^2ψ_0^2 + 1/b^2ψ_1^2 ≤a^2 (1-c-d) + 1/b^2 (c+d)= μ. We describe the transducer as non-canonical, and we will turn it into the canonical form later. The space of the transducer is ⊗^⊗ D-1, whereis a D-qudit. The one-dimensional public space is spanned by ξ = |D⟩|0> ||⟩0>^⊗ D-1. The initial coupling is given byξ⊕ v = ∑_i=0^D-1 (-1)^i· f(ψ)|D⟩|i>|>^⊗ i||⟩0>^⊗ D-i-1.The transduction complexity is v^2 ≤ξ⊕ v^2 ≤∑_i=0^∞||^2i = 1/1-||^2≤1/1-μ. A purifier is a multidimensional quantum walk on the line graph seen above. The edge between the vertices i and i+1 corresponds to the subspace |D⟩|i>⊗^⊗ D-1 as indicated by the state below the edge. The local reflections on the vertices 0 and D are identities. The local reflection at the vertex i=1,…, D-1 acts on the subspace {|D⟩|i-1>, |D⟩|i>}⊗^⊗ D-1. The expressions above the edges give the initial coupling from eqn:purifier xi+v.[auto](0) at (0,0) [circle, draw] 0;(1) at (2,0) [circle, draw] 1;(2) at (5,0) [circle, draw] 2;(3) at (7.5,0) [circle, draw] 3;(D-1) at (11,0) [ellipse, draw] D-1;(D) at (14.5, 0) [ellipse, draw] D;(0) to node [midway, above, blue] ||⟩0>^⊗ D-1 node [midway, below, gray ] |D⟩|0>(1);(1) to node [midway, above, blue] (-1)^f(ψ) |>||⟩0>^⊗ D-2 node [midway, below, gray ] |D⟩|1>(2);(2) to node [midway, above, blue] |>^⊗ 2||⟩0>^⊗ D-3 node [midway, below, gray ] |D⟩|2>(3);(3) to (8.5,0);(dots) at (9,0) ⋯; (D-1) to (9.5,0);(D-1) to node [midway, above, blue] (-1)^f(ψ) |>^⊗ D-1 node [midway, below, gray ] |D⟩|D-1>(D);Let us now describe the action of the transducer. The transducer is a multidimensional quantum walk on the line graph, see fig:purifier. It is a product of two reflections R_1 and R_2. The reflection R_1 is the product of the local reflections on the odd vertices, i=1,3,5,…, D-1. The reflection R_2 is the product of the local reflections on the even vertices, i=2,4,…, D-2.The local reflection for the vertex i=1,…,D-1 is as follows. It acts in {|D⟩|i-1>, |D⟩|i>}⊗^⊗ D-1. Define a qubitwhose value 0 corresponds to |D⟩|i-1> and 1 to |D⟩|i>. If i is odd, this could be the least significant qubit of . Let ^(i) = ^(i)⊗^(i) be the i-th multiplier in the tensor product ^⊗ D-1. The reflection is as follows:=0pt* Execute the input oracle O_ψ on ^(i) conditioned on |A⟩|0> = |D⟩|i-1>.* Execute a two-qubit unitary on ⊗^(i), which is the reflection about the span of the states[a|A⟩|0> + |A⟩|1> ]|B⟩|0> [|A⟩|0> + b |A⟩|1> ]|B⟩|1>. * Execute the inverse oracle O^*_ψ on ^(i) conditioned on |A⟩|0> = |D⟩|i-1>. The local reflection for the vertex i multiplies the corresponding part of the state in eqn:purifier xi+v|D⟩|i-1> |>^⊗ i-1||⟩0>^⊗ D-i + (-1)^f(ψ)|D⟩|i> |>^⊗ i||⟩0>^⊗ D-i-1by the phase (-1)^f(ψ). After application of the oracle in Step 1, we get the state|>^⊗ i-1⊗[|D⟩|i-1> |M⟩ |ψ> + (-1)^f(ψ)|D⟩|i>M|>]⊗||⟩0>^⊗ D-i-1.The local reflection acts on the state in the square brackets, which can be rewritten as[|A⟩|0> + 1/a|A⟩|1>]|B⟩|0> |N⟩|ψ_0> + [|A⟩|0> + b |A⟩|1> ]|B⟩|1> |N⟩|ψ_1>if f(ψ)=0, and[|A⟩|0> - a |A⟩|1> ]|B⟩|0> |N⟩|ψ_0> + [|A⟩|0> - 1/b|A⟩|1> ]|B⟩|1> |N⟩|ψ_1>if f(ψ)=1. It is easy to see that the operation on Step 2 does not change the state in eqn:purifierSub1 and negates the one in eqn:purifierSub2, from which the claim follows. From the claim, it immediately follows that, if f(ψ)=0, the transducer does not change the state eqn:purifier xi+v. Therefore, in this case ||⟩0> ||⟩0>.On the other hand, if f(ψ)=1, then R_1 reflects the whole state ξ⊕ v, and R_2 reflects all the terms in the sum except for i=0 and i=D-1.Thus, the final state is-ξ⊕ v - 2 (-1)^(D-1)f(ψ)|D⟩|D-1> |>^⊗ D-1 .This can be interpreted as transducing ||⟩0> into -||⟩0> with a perturbation of size at most 2μ^D-1 by eqn:purifierPsiSize1 and eqn:purifierPsiSize2.One can see that the local reflection for the vertex i applies the input oracle and its inverse on the part of the state v in the subspace |D⟩|i-1>⊗^⊗ D-1. Hence, the query complexity is at most 2ξ⊕ v^2 ≤ 2/(1-μ) by eqn:zetaNorm. The transformation into canonical form, prp:canoning, adds the query complexity to the transduction complexity, hence, the latter stays [1/(1-μ)].Let us estimate time complexity.We start with the circuit model. The queries in transducer of thm:purifierBoolean are not aligned, as they are applied to different copies of . In order to make them aligned, as required by prp:canoning, the register ^(i) should be moved to some specific array of qubits shared by all the local reflections. This takes time (s) per each local reflection. Step 2 of the local reflection can be implemented in constant time. Thus, each local reflection takes time (s). There are D-1 local reflections performed. By lem:automaton with all ϕ_i being absent, the whole transducer can be implemented in time (D· s). Transformation into canonical form in prp:canoning keeps the time complexity of the transducer essentially the same.Now consider the QRAG model. All local reflections in R_1 can be performed in parallel, and the same is true for R_2. The first and the third operation in the local reflection are implemented by the RA input oracle. The second operation can be performed in () time by thm:select. §.§ Non-Boolean CaseThis time let = ⊗ be a space with = ^p. Let O_ψ be an oracle that performs the following state generation:O_ψ|M⟩ |0> ↦|M⟩|ψ>=∑_j=0^p-1|B⟩|j>|N⟩|ψ_j>.Let d>0 be a constant. We assume that for every ψ there exists (unique) f(ψ)∈[p] such thatψ_f(ψ)^2 ≥1/2+d.Defineμ = 2 √(1/4 - d^2) < 1,which is the same as in eqn:purifierd for c=1/2. We treat = ^p as composed out of ℓ = log p qubits. We do not assume that ℓ = () here, as functions, in principle, can have very long output. Again, we call every input oracle O_ψ satisfying eqn:purifierInput2 and eqn:purifierCases2 admissible.Let D be a positive integer. Under the above assumptions, there exists a perturbed transduceron the public space ^p and with bidirectional access to O_ψ which satisfies the following conditions: =0pt* For all b∈^ℓ,and admissible O_ψ, it transduces ||⟩b> into |b⊕ f(ψ)>, where ⊕ stands for the bit-wise XOR. In particular, every initial vector in ^p is admissible.* On any admissible input oracle and unit initial vector, its perturbation is at most = 2 μ^D-1.* Its query complexity satisfies () = [1/(1-μ)] = (1).* It executes the input oracle on the admissible subspace only: O_ψ on |M⟩|0> and O_ψ^* on |M⟩|ψ>.For the time and the transduction complexity, we have the following estimates: =0pt* In the circuit model, () = [1/(1-μ)] = (1) and T() = (s· D), where s is the number of qubits used in .* In the QRAG model, assuming the RA input oracle, we have () = (log p/(1-μ)) = (log p) and T() = ().Here we use () to denote maximal W(, O_ψ, ξ) over all admissible input oracles O_ψ and admissible normalised initial states ξ. () is defined similarly.We reduce the non-Boolean case to the Boolean case of thm:purifierBoolean by encoding the value into the phase and using the Bernstein-Vazirani algorithm <cit.> to decode it back.For simplicity of notation, we will assume p = 2^ℓ so that O_ψ in eqn:purifierInput2 just does not use the extra dimensions. For a,b∈ [p], we denote by a⊙ b∈{0,1} their inner product when considered as elements of _2^ℓ.Denote the public space ^p ofby . We first apply the Hadamard H^⊗ℓ to perform the following transformationH^⊗ℓ|J⟩|b> ⟼1/√(p)∑_i=0^p-1 (-1)^i⊙ b|J⟩|i>.Consider the following procedure E that evaluates the inner product between i and the output of the oracle into an additional qubit :E(O_ψ) |J⟩|i> |M⟩|0>|Z⟩|0> O_ψ|J⟩|i> ∑_j=0^p-1|B⟩|j>|N⟩|ψ_j> |Z⟩|0> ⟼|J⟩|i> ∑_j=0^p-1|B⟩|j>|N⟩|ψ_j> |Z⟩ |i⊙ j>.We consider it as a direct sum E = ⊕_i∈[p] E^(i) withE^(i)(O_ψ) |M⟩|0>|Z⟩|0> ⟼∑_j=0^p-1|B⟩|j>|N⟩|ψ_j> |Z⟩ |i⊙ j>. We convert them into canonical transducers S_E and S^(i)_E. We have [S_E^(i)] = 1, where the admissible subspace is |M⟩|0>|Z⟩|0>.Using Propositions <ref> and <ref>, we get transducersS_E[O_ψ] = S_E(O_ψ) ⊕ S_E^-1(O^*_ψ) S_E^(i)[O_ψ] = S_E^(i)(O_ψ) ⊕ (S_E^(i))^-1(O^*_ψ).Again, [S_E^(i)] = 1. We still have S_E = ⊕_i S_E^(i) with the help of rem:directSumDifferentOracle. It is also clear that S_E only executes the input oracle O_ψ on the admissible subspace.We treat E^(i)(O_ψ) as an oracle encoding a Boolean value into the register . By eqn:purifierCases2, we get that E^(i)(O_ψ) evaluates i⊙ f(ψ) with bounded error. Take the purifier ' from thm:purifierBoolean with c=1/2 and the same values of d and D. This purifier satisfies'[E^(i)(O_ψ)] ||⟩0> (-1)^i⊙ f(ψ)||⟩0> .Taking direct sum over all i∈[p], we get that a transducer[I_⊗'] [E(O_ψ)] = ⊕_i=0^p-1'[E^(i)(O_ψ)]performs the following transduction:1/√(p)∑_i=0^p-1 (-1)^i⊙ b|J⟩|i> 1/√(p)∑_i=0^p-1 (-1)^i⊙ b + i⊙ f(ψ)|J⟩|i> . Finally, we again apply H^⊗ℓ to getH^⊗ℓ1/√(p)∑_i=0^p-1 (-1)^i⊙ b + i⊙ f(ψ)|J⟩|i> ⟼|J⟩ |b ⊕ f(ψ)>. Combining eqn:purifierGenA, eqn:purifierGen1 and eqn:purifierGen2, we see that we can use sequential composition of prp:sequentialsequential to get= S_H^⊗ℓ * [(I_⊗')∘S_E] * S_H^⊗ℓ,where S_H^⊗ℓ is a transducer from prp:gates with transduction action H^⊗ℓ and no input oracle.Let us estimate query complexity on a unit vector ξ∈. We have the following estimate, where we explain individual lines after the equation.L(, O_ψ, ξ) = L[(I_⊗')∘S_E, O_ψ, H^⊗ℓξ] = ∑_i=0^p-1 L['∘S_E^(i), O_ψ, ϕ_i]≤∑_i=0^p-1['] [S_E^(i)] ϕ_i^2≤(1/(1-μ)) ∑_i=0^p-1ϕ_i^2 = (1/(1-μ)).On the first line, we used sequential composition of prp:sequentialsequential and that the transducer S_H^⊗ℓ does not use the input oracle. On the second line, we decomposed H^⊗ℓξ = ⊕_i ϕ_i, and used eqn:purifierItimesPurifier and prp:parallel. On the third line, we used functional composition of prp:functional, Eq. eqn:rescaling, and that ' does not have direct access to O_ψ and executes its input oracle E^(i) on the admissible subspace only.In a similar way, we haveW(, O_ψ, ξ) = 2(S_H^⊗ℓ) + W[(I_⊗')∘S_E, O_ψ, H^⊗ℓξ]= 2(S_H^⊗ℓ) + ∑_i=0^p-1 W['∘S_E^(i), O_ψ, ϕ_i]≤ 2(S_H^⊗ℓ) +∑_i=0^p-1[['] + ['] [S_E^(i)]] ϕ_i^2 ≤ 2(S_H^⊗ℓ) +(') + (')max_i[S_E^(i)]≤ 2(S_H^⊗ℓ) + [1/(1-μ)]max_i[1+[S_E^(i)]].For the perturbation, we have using prp:perturbationComposition:δ(, O_ψ, ξ) = δ[(I_⊗')∘S_E, O_ψ, H^⊗ℓξ] = √(∑_i=0^p-1δ[',E^(i)(O_ψ), ϕ_i]^2)≤√(∑_i=0^p-1[ ϕ_i ]^2) = .Finally,T() = 2 (S_H^⊗ℓ) + (') + (S_E) + (1). In the circuit model, the transduction complexities of S_H^⊗ℓ and S_E^(i) are 0 and 1, respectively, and we get the required estimate from eqn:purifierTransductionComplexity. Also, both T(S_H^⊗ℓ) and T(S_E) are (log p), and T(') = (s· D).Since s≥log p, we get the required estimate from eqn:purifierTimeComplexity.In the QRAG model, we have both (S_H^⊗ℓ) and (S_E^(i)) bounded by (log p), which gives the required estimate on the transduction complexity. For the time complexity, all the terms in eqn:purifierTimeComplexity are (), which shows that T() = (). §.§ Composition of Bounded-Error AlgorithmsPurifier can be composed with algorithms evaluating functions with bounded error to reduce the error. In this section, we mention some examples.Since we are ignoring the constant factors, we will assume the standard version of the input oracle: O_x ||⟩i>||⟩b> ↦||⟩i>||⟩b⊕ x_i>. In particular, it is its own inverse, and we use O_x instead of O_x everywhere in this section.Let us again recall the composed function f∘ gfrom eqn:composedFunction:[f∘ g] (z_1,1, …,z_1,m, z_2,1,…,z_2,m,……,z_n,1,…,z_n,m)= f[ g(z_1,1, …,z_1,m),g(z_2,1, …,z_2,m), …, g(z_n,1, …,z_n,m)].and_i = (z_i,1, …,z_i,m)x = [g(_1), g(_2),… g(_n)]so that f(x) = [f∘ g](z). The following result is essentially thm:introCompositionFunctionCircuit.Let A and B be quantum algorithms in the circuit model that evaluate functions f and g, respectively, with bounded error. Then, there exists an algorithm in the circuit model that evaluates the function f∘ g with bounded error in time complexity(L)[T(A) + T(B) + slog L]where L is the worst-case Las Vegas query complexity of A, and s is the space complexity of B. The algorithm makes [L· Q(B)] queries, where Q(B) is the usual Monte Carlo query complexity of B. First, use thm:program->transducerCircuit to get a transducer S_A whose transduction action is identical to the execution of A. Its time complexity T(S_A) = [T(A)] and its transduction and query complexities are bounded by L.The algorithm B on the input oracle O_y evaluates g(y) with bounded error. We obtain the algorithm B^-1 with the input oracle O_y^* = O_y whose action is the inverse of B. Combining the two via direct sum, we get the algorithm B(O_y) = B(O_y)⊕ B^-1(O_y). Letbe the corresponding purifier from thm:purifierGeneral, and D andbe the parameters therein. The transduction action ofon the input oracle B(O_y) is ||⟩b> ||⟩b⊕ g(y)>.By eqn:composed_yCopy, we have O_z = ⊕_i O__i. Let us denote by B(O_z) the algorithm [I_n⊗ B](O_z) = ⊕_i B (O__i). By cor:byIdentity with rem:directSumDifferentOracle, the transducer I_n⊗ on the input oracle B(O_z) performs the transduction ||⟩i>||⟩b>||⟩i>||⟩b⊕ g(_i)> for every i∈[n]. In other words, its transduction action is O_x.Now consider the transducer S = S_A∘ (I_n⊗)with the input oracle B(O_z). By the definition of functional composition, it transduces ||⟩0> into [g∘ f](z). By eqn:compositionTransductionMultipleUpper, its transduction complexity is at mostW(S_A, O_x, ||⟩0>) + ∑_i L^(i) (S_A, O_x, ||⟩0>) ·[, B(O__i)]= (L). We obtain the required algorithm by using cor:approximatePumping with = Θ(1) on the above transducer S. The transducer S is executed (L) times. Each execution takes times [T(A) + sD] to execute the transducer and [T(B)] to execute the input oracle. Therefore, the total time complexity is(L) [T(A) + T(B) + sD]and the query complexity is (L)Q(B).It remains to estimate D. We may assume the error of A is a small enough constant. By cor:approximatePumping, in order to get a bounded-error algorithm from the transducer S_A∘ (I_n⊗), we should haveδ[I_n⊗, B(O_z), q(S_A, O_x, ||⟩0>)] = [1/√(L)].for a small enough constant. Using eqn:perturbationLinear and that |q(S_A, O_x, ||⟩0>)| ≤√(L), we get that it suffices to have = (1/L). Therefore, we can take D = (log L), which finishes the proof.Let us now proceed with QRAG case, thm:introCompositionFunctionQRAG. Recall the function from eqn:randomComposedFunctionCopy:f[ g_1(z_1,1, …,z_1,m),g_2(z_2,1, …,z_2,m), …, g_n(z_n,1, …,z_n,m)].with notation_i = (z_i,1, …,z_i,m)x = [g_1(_1), g_2(_2),… g_n(_n)]. For simplicity, we assume all functions f and g use the same input and output alphabet q. Let A and B_i be quantum algorithms that evaluate f and g_i, respectively. To simplify expressions, we assume that T(B_i)≥log q.In other words, we spend at least 1 iteration per bit of the output. Also, all the algorithms have the same upper bound on permissible error. We use an approach similar to sec:iterated on iterated functions, so that we are able to compose several layer of functions.As in sec:function, we denote L_x(A) = L(A, O_x, ||⟩0>) and similarly for other notation. This time, however the input oracleO_x ||⟩i>||⟩b> ↦||⟩i>||⟩b⊕ x_i> is uniquely defined. As we can make the perturbation of a purifier as small as necessary without increasing complexity, we ignore the perturbations in the following implicitly assuming they are small enough. Let S_f be a perturbed transducer evaluating the function f, and B_1,…, B_n be algorithms evaluating the functions g_1,…,g_n with bounded-error. Assuming the QRAG model with RA input oracle, and QRAM access to the description of B_1,…, B_n, there exists a perturbed transducer S evaluating the function eqn:randomComposedFunctionCopy2 with the following parameters. Its transduction complexity isW(S, O_z, ||⟩0>) = W(S_f, O_x, ||⟩0>) + ∑_i=1^n [ L^(i)_x(S_f) T(B_i)]its query complexity isL^(i,j) (S, O_z, ||⟩0>) = [ L^(i)_x(S_f) L^(j)__i(B_i)]and its time complexity is T(S) = (S_f) + (). From thm:program->transducerQRAG, for each i, we obtain a transducer S_B_i whose transduction action is identical to the action of B_i. Its transduction complexity (S_B_i) = [T(B_i)] and query state is identical to that of B_i. Also, we obtain S_B_i = S_B_i⊕ S_B_i^* whose complexity is identical to S_B_i. Letbe the corresponding purifier. We have that the transduction action of ∘S_B_i on an input oracle O_y is ||⟩b> ||⟩b⊕ g_i(y)>.Using direct sum, we get that the transducer S_g = (I_n⊗)∘⊕_i S_B_i = ⊕_i ∘S_B_ion the input oracle O_z = ⊕_i O__i transduces ||⟩i>||⟩b>||⟩i>||⟩b⊕ g_i(_i)> for every i∈[n]. Thus, its transduction action is O_x, and the transducer S = S_f∘ S_g evaluates the function in eqn:randomComposedFunctionCopy2.Let us estimate its transduction complexity. First by eqn:compositionTransductionUpper:(∘S_B_i, O__i) ≤() + () (S_B_i)= (log p) + (T(B_i)) = (T(B_i))using our assumption on T(B_i)≥log p. Therefore, by eqn:compositionTransductionMultipleUpper:W[S, O_z, ||⟩0>] ≤ W(S_f, O_x, ||⟩0>) + ∑_i L^(i)_x(S_f)[∘S_B_i, O__i] ≤ W(S_f, O_x, ||⟩0>) + ∑_i L^(i)_x(S_f) (T(B_i)) . For the query complexity, we use a partial-query variant of eqn:compositionLasVegasUpper and that the purifier only executes the subroutine on the admissible initial states to obtain:^(j)[∘S_B_i, O__i] ≤() L^(j)[S_B_i, O__i, ||⟩0>] = [L^(j)__i (B_i)].Hence, by eqn:compositionLasVegasMultipleUpper:L^(i,j)[S, O_z, ||⟩0>]≤ L^(i)_x(A) ^(j)[∘S_B_i, O__i] ≤ L^(i)_x(A) [L^(j)__i (B_i)]. The time complexity ofis (). Also, the direct sum ⊕_i S_B_i can be implemented in () by prp:program->transducerParallel. Thus, the time complexity of S is (S_f) + (). We get thm:introCompositionFunctionQRAG from this theorem by using a transducer S_f obtained from the program A using thm:program->transducerQRAG, and then applying prp:approximateImplementation to the resulting transducer. Moreover, we get that the algorithm executes the input oracle O_z[ max_z ∑_i=1^n L^(i)_x(A) L__i(B_i)]times.thm:compositionFunctionQRAG can be used multiple times in a row to obtain a composed transducer for a tree of functions similar to the one in thm:introCompositionTree. If d is the depth of the tree, the query complexity grows by the factor of C^d, where C is the constant in eqn:compositionFunctionQRAGQuery. This growth is completely analogous to what one obtains using span programs for the query complexity. The contribution to the time complexity from the subroutines on layer ℓ also grows by the factor of C^ℓ because they are multiplied by the corresponding query complexity in eqn:compositionFunctionQRAG. The time complexity of the final transducer is (d).§.§ AcknowledgementsWe would like to thank Titouan Carette for bringing references <cit.> to our attention.AB is supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001 and the QuantERA project QOPT.SJ is supported by NWO Klein project number OCENW.Klein.061; and ARO contract no W911NF2010327. SJ is funded by the European Union (ERC, ASC-Q, 101040624). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.SJ is supported by the project Divide & Quantum(with project number 1389.20.241) of the research programme NWA-ORC which is (partly) financed by the Dutch Research Council (NWO). SJ is a CIFAR Fellow in the Quantum Information Science Program. habbrvM | http://arxiv.org/abs/2311.15873v1 | {
"authors": [
"Aleksandrs Belovs",
"Stacey Jeffery",
"Duyal Yolcu"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20231127144519",
"title": "Taming Quantum Time Complexity"
} |
firstpage–lastpage XLB: A Differentiable Massively Parallel Lattice Boltzmann Library in Python [ January 14, 2024 ============================================================================Here we report on joint X-ray and radio monitoring of the neutron star low-mass X-ray binary SAX J1810.8-2609. Our monitoring covered the entirety of its ∼ 5month outburst in 2021, revealing a temporal correlation between its radio and X-ray luminosity and X-ray spectral properties consistent with a `hard-only' outburst. During the outburst, the best-fit radio position shows significant variability, suggesting emission from multiple locations on the sky. Furthermore, our 2023 follow-up observations revealed a persistent, unresolved, steep spectrum radio source ∼ 2years after SAX J1810.8-2609 returned to X-ray quiescence. We investigated potential origins of the persistent emission, which included an unrelated background source, long-lasting jet ejection(s), and SAX J1810 as a transitional millisecond pulsar. While the chance coincidence probability is low (≲ 0.16%), an unrelated background source remains the most likely scenario. SAX J1810.8-2609 goes into outburst every ∼ 5years, so monitoring of the source during its next outburst at higher sensitivities and improved spatial resolutions (e.g., with the Karl G. Jansky Very Large Array or Square Kilometre Array) should be able to identify two components (if the persistent emission originates from a background source). If only one source is observed, this would be strong evidence that the persistent emission is local SAX J1810.8-2609, and future monitoring campaigns should focus on understanding the underlying physical mechanisms, as no neutron star X-ray binary has shown a persistent radio signal absent any simultaneous X-ray emission.stars: neutron — ISM: jets and outflows — radio continuum: stars — stars: individual SAX J1810.8-2609 — X-rays: binaries § INTRODUCTION Low-mass X-ray binaries (LMXBs) are interacting binary systems that consist of a compact object – a black hole or a neutron star – accreting material from a low-mass companion star (< 1 M_⊙). The inward-moving accretion flow powers outflows in the form of disk winds and relativistic jets. Many LMXBs are transient systems, spending the majority of their lifetimes in a low-luminosity quiescent state (L_X ≲ 10^32 erg s^-1) before sporadically entering into bright transient outbursts (L_X > 10^35 erg s^-1) that last weeks to years <cit.>. Since LMXBs rapidly evolve through multiple accretion states during outbursts, LMXBs act as natural laboratories for the study of accretion flows <cit.> and relativistic jets <cit.>. The standard accretion state nomenclature (i.e., the hard and soft accretion states) was developed to describe the different X-ray spectra observed in black hole low-mass X-ray binaries (BHXBs). Moreover, the properties of the relativistic jet(s) are closely correlated with the accretion state <cit.>. In the hard accretion state, the X-ray emission is dominated by high-energy (i.e., hard) X-ray photons comptonized by an optically thin corona. The X-ray spectra are well described by a power law model with a photon index of Γ∼ 1.7 (where the X-ray flux f_X(ν)∝ν^-Γ-1). Furthermore, in the hard accretion state, the jet adopts a steady, compact structure. The radio spectrum of the compact jet is the result of a superposition of multiple self-absorbed synchrotron spectra originating from different positions along the jet axis <cit.>. At low frequencies, the jet is best described as an optically thick, partially self-absorbed synchrotron spectrum with an inverted or flat spectral index (α ≳ 0; radio flux density f_R(ν)∝ν^α) up to a break frequency (often at sub-mm wavelengths). Beyond the break frequency, the jet's spectrum becomes optically thin <cit.>. In the hard state, the X-ray (L_X) and radio (L_R = ν L_ν,R) luminosities are correlated <cit.>. After including a scale for the black hole mass, the L_R–L_X relation has been extended to include accreting supermassive black holes <cit.>, thereby spanning 10 orders of magnitude in X-ray luminosity and providing the strongest empirical evidence of the coupling between accretion flows and relativistic jets. Individual BHXBs have exhibited multiple distinct tracks in the L_R-L_X plane <cit.> suggesting that the properties of the accretion flow (e.g., geometry and radiative efficiency) may vary significantly in the hard accretion state. Population analyses have both supported <cit.> and refuted <cit.> the statistically independent existence of multiple tracks, with the more recent studies not finding any robust statistical evidence for separate tracks, suggesting that, instead,the properties of the ‘radio-loud’ and ‘radio-quiet’ track sources vary significantly from source to source.Conversely, in the soft accretion state, low-energy (i.e., soft) thermal emission from a multi-color accretion disk dominates the X-ray spectrum. Furthermore, the compact jet is quenched, decreasing in luminosity by ≳3 orders of magnitude <cit.>. During the hard-to-soft transition, one or more discretized ejection events may be launched. These ejections have been spatially resolved in multiple sources <cit.>. The radio spectra of the ejecta are characterized by a time-variable self-absorbed synchrotron component <cit.>. As ejecta propagate and expand, they become optically thin at (progressively) lower-frequency emission, steepening the radio spectral index to α ∼ -0.7. Emission from jet ejections can persist from hours to years <cit.>, and can exhibit variability that is unrelated to any simultaneous evolution of the accretion flow <cit.>. As a result, radio observations of jet ejecta must be excluded from the L_R–L_X relation.For neutron star (low-mass) X-ray binaries (NSXBs), their strong intrinsic magnetic fields and solid surfaces complicate the picture. Historically, radio emission was thought to be exclusive to the weakly-magnetic (< 10^10G) sub-population, although there have been recent detections of radio emission from strongly-magnetic NSXBs <cit.>. The weakly-magnetic NSXBs are most directly analogous to BHXBs; thus, the strongly-magnetic sub-population will not be discussed any further (henceforth, NSXBs only refer to weakly-magnetic NSXBs). NSXBs have two main sub-classes; atoll and Z sources <cit.>. Atoll sources tend to be lower luminosity and transient, exhibiting similar hard/soft accretion states as transient BHXBs. In contrast, Z sources are often persistent but show rapid timescale variability. Moreover, although Z sources also transition through multiple accretion states, these states tend to be softer than atoll states <cit.>. Some NSXBs have shown transitions from Z to atoll behaviour at lower X-ray luminosities <cit.>, suggesting that the these may not be unique sub-populations, but instead that Z sources are NSXBs with the largest accretion rates <cit.>.Transient atoll sources more closely follow the evolution of a `typical' transient BHXBs <cit.>. Atoll outbursts exhibit distinct hard (also known as “extreme island”) and soft (also known as “banana”) accretion states. Atolls (sometimes) exhibit jet quenching in the soft state. State transition-induced jet ejections have been proposed for atolls, although they have only been observed in Z sources <cit.>. The `typical' evolution of a transient outburst of an atoll NSXB or BHXB begins with a departure from quiescence through a rapid brightening in the hard state. The source transitions to the soft state following the initial brightening. The system then remains in the soft state for some time (the amount of time varies from system to system) until it begins to dim, eventually returning to the hard state at a lower X-ray luminosity. Once back in the hard state, the system dims until it returns to a quiescent state. However, some systems break this paradigm by exhibiting erratic state transitions <cit.> or failed (i.e., `hard-only') outbursts <cit.>. Recent analyses have shown that ∼ 40% of outbursts of BHXBs are thought to be `hard-only' <cit.>; this fraction has not been thoroughly explored for NSXBs.There are several significant differences between the neutron star and black hole X-ray binary sub-populations: (i) NSXBs generally have radio luminosities that are a factor of ∼ 20 lower than BHXBs at comparable X-ray luminosities <cit.>; (ii) NSXBs have shown compact jet radio emission in the soft accretion state <cit.>, suggesting the quenching process may not be as extreme as observed in black hole systems or possibly a different jet launching process completely; (iii) all accretion states can have an additional thermal X-ray component <cit.> due to emission from the neutron star surface or boundary layer between the accretion disk and surface. Historically, studies of accretion-jet coupling of NSXBs have suffered from their weaker radio emission. Joint X-ray and radio monitoring of NSXBs is critical for understanding the differences between the neutron star and black hole X-ray binary populations and how the presence (or absence) of an event horizon, ergosphere, or solid surface affects the connection between the accretion flow and relativistic jet. In 2021, the NSXB SAX J1810.8-2609 exhibited a multi-month outburst that was detected in both X-ray and radio frequencies, allowing for a comprehensive monitoring campaign.§.§ SAX J1810.8-2609SAX J1810.8-2609 (henceforth SAX J1810) is a NSXB that was initially discovered in 1998 by the wide-field X-ray cameras aboard theBeppoSAX satellite <cit.>. Since its discovery, there have been four subsequent (detected) outbursts that occurred in 2007 <cit.>, 2012 <cit.>, 2018 <cit.>, and 2021 <cit.>. A Type I X-ray burst <cit.> revealed the presence of a solid surface, identifying the accreting object as a neutron star <cit.>. Furthermore, X-ray modelling of the burst showed a clear signature of photospheric radius expansion (PRE), where the burst luminosity exceeds the local Eddington limit causing a radial expansion of the neutron star photosphere. The PRE X-ray burst was used to estimate the source distance of 4.9 ± 0.3kpc <cit.>. However, we note that the quoted distance error is purely statistical, as it does not take into consideration any systematic effects, such as the potential for the neutron star to deviate from the assumed mass of 1.4M_⊙ or the potential for accreting elements besides hydrogen. Therefore, the error on the distance is likely an underestimation. An analysis of multiple Type I X-ray bursts detected during the 2007 outburst showed timing signals consistent with a neutron star spin frequency of 531.8Hz <cit.>. These `millisecond burst oscillations' are thought to be caused by anisotropic X-ray emission <cit.> and allow for the determination of the neutron star spin frequency without the need for consistent pulsations. The source has not been classified as an atoll or Z source; instead, it has adopted the broader label of neutron star `soft X-ray transient', which encompasses both sub-classes. However, given its moderate peak X-ray luminosity (L_X≤ 4×10^36 erg s^-1) and transient behaviour, it is likely to be an atoll source. The majority of Z sources are persistent and bright, with maximum X-ray luminosities reaching appreciable fractions of the Eddington limit (e.g., L_X∼ 2×10^38 erg s^-1). On 2021 May 13 (MJD 59347), the gas slit camera (GSC) aboard The Monitor of All-sky X-ray Image <cit.> satellite detected the X-ray brightening of SAX J1810 as it entered its fifth recorded outburst <cit.>. Following the X-ray detection, radio observations with the MeerKAT radio telescope on 2021 May 21 (MJD 59356) revealed a spatially coincident radio source, constituting the first radio detection of this source <cit.>. Here we present our multi-instrument radio/X-ray monitoring campaign of SAX J1810. Our monitoring includes the 2021 outburst and 2023 follow-up that revealed the existence of a spatially coincident, persistent steep spectrum radio source. The remainder of this paper is structured as follows: in Section <ref>, we introduce our observation and analysis procedure, while in Sections <ref> and <ref>, we present and discuss our results. Finally, we summarize our findings in Section <ref>.§ OBSERVATIONS AND DATA ANALYSIS§.§ MeerKAT §.§.§ Weekly Monitoring We observed SAX J1810 with MeerKAT <cit.> as a part of the large survey project ThunderKAT <cit.>. We began a weekly monitoring campaign on 2021 May 22 (MJD 59356), nine days after the outburst's initial detection, and continued until 2021 October 23 (MJD 59508) for a total of 21 observations. Each observation consisted of a single scan of 15 minutes on-source flanked by two 2-minute scans of a nearby gain calibrator (J1830-3602). Each epoch also included a 5-minute scan of PKS B1934-638 (J1939-6342) for flux and bandpass calibration. In addition to the weekly monitoring, we observed two deep (1-hour) epochs on 2023 May 22 (MJD 60086) and 2023 August 16 (MJD 60172) when the source was in (X-ray) quiescence. The deep epochs followed the same observing strategy, except the source monitoring was broken into two 30-minute scans. All MeerKAT observations used the L-band receiver, with a central frequency of 1.3, and a total (un-flagged) bandwidth of 856MHz split evenly into 32768 frequency channels. To decrease the size of each data set, we averaged together every 32 channels (resulting in 1024 total channels) before data reduction and imaging. This averaging will not affect our final results as we are focused on radio continuum emission (as opposed to spectral lines). We performed flagging, calibration, and imaging using a modified version of the semi-automated routine OxKAT[Found at: <https://github.com/IanHeywood/oxkat>] <cit.>, which breaks the process into three steps. Here we will briefly outline the workflow and direct readers to <cit.> for a more comprehensive description. The first step (1GC) uses casa <cit.> to remove data corrupted by radio frequency interference (RFI). After removing RFI, the data is corrected with standard calibration solutions (i.e., flux density, bandpass, and complex gain). The second step (FLAG) applies a second round of flagging using<cit.> before creating a preliminary image of the source field using wsclean <cit.>. This preliminary image is then used to create an imaging mask. The final step (2GC) begins with a masked deconvolution before using the model image for direction-independent (DI) self-calibration with CubiCal <cit.>. Following self-cal, the pipeline ends with a second round of masked deconvolution using the DI self-calibrated visibilities. We adopted the 2GC images as our final data products. We maximize our sensitivity by weighting each image with a Briggs' robustness of 0 <cit.>[MeerKAT's synthesized beam becomes significantly non-Gaussian for robustness weightings > 0, inhibiting accurate deconvolution and raising the image-plane rms noise.]. We note that OxKAT has the functionality to solve for direction-dependant (DD) self-calibration solutions if needed (i.e., the 3GC step). However, for SAX J1810, DI self-calibration was sufficient, and thus we omitted the 3GC step. We measured the source properties in each epoch using the casa task , fitting an elliptical Gaussian component in a small sub-region around the source to measure the position and flux density. As the source was unresolved, we set the component shape to be the synthesized beam of each image. We quantified the (1σ) uncertainty on the flux measurement using the local root-mean-square (rms) noise. We extracted the rms from an annular region for each epoch using the casa task . Each annulus was centered on the position of the Gaussian component. We fixed the inner radius as the major axis of the synthesized beam and scaled the outer radius such that the annular area comprises the area of 100 synthesized beams. We quantified astrometric errors using the method detailed in Appendix <ref>. §.§ Very Large Array We were approved for a single director's discretionary time observation (Project Code: 23A–417) with the Very Large Array (VLA) as a follow-up of our initial 2023 MeerKAT observation. SAX J1810 was observed on 2023 July 17 (MJD 60142) in the 2–4(S-band) and 4–8bands (C-band). For S-band, the observations used the 8-bit sampler comprised of two base-bands, with eight spectral windows of sixty-four 2channels each, giving a total (unflagged) bandwidth of 2.048. The 3-bit sampler was used for C-band, which has four base-bands, and thus a 4.096bandwidth. In each band, we included a single 1-minute scan of the flux calibrator (3C286). For source monitoring the array cycled between SAX J1810, observed for ∼ 8minutes per cycle in S-band and ∼ 5minutes in C-band. Each source scan is flanked by ∼ 1minute observations of a nearby gain calibrator (J1820-2528). The total time on source was ∼ 16minutes in both bands. We performed flagging, calibration, and imaging using the most recent release of the casa VLA pipeline (v6.4). We imaged the source using wsclean but did not detect the source in either band. As a result, we extract the rms noise from each image to place (3σ) upper limits on the flux density. We used a circular extraction region (with an area equal to 100 synthesized beams) centered on the archival position of SAX J1810 to measure the rms. The radio flux densities from both MeerKAT and the VLA are presented in Table <ref> §.§ Swift-XRT§.§.§ Weekly MonitoringWe monitored SAX J1810 with the X-ray telescope <cit.> aboard the Neil Gehrels Swift Observatory <cit.>, capturing the quasi-simultaneous evolution of the X-ray flux (i.e., within ∼3of a MeerKAT observation). During the outburst, we observed 21 epochs (target ID: 32459) between 2021 May 20 (MJD 59364) and 2021 November 6 (MJD 59524) at an approximately weekly cadence. To accompany our deep MeerKAT epochs, we were approved for two Target-of-Opportunity observations on 2023 May 25 (MJD 60089) and 2023 August 16 (MJD 60172). During the initial stages of the outburst, we monitored the source in Windowed Timing (WT) mode, where SAX J1810 exhibited a maximum count rate of ∼ 20 count s^-1 during the first epoch. We transitioned to Photon Counting (PC) mode when the sources count rate decayed to ≲1 count s^-1 on 2021 October 9 (MJD 59496), although there was a single intermittent PC epoch on 2021 September 5 (MJD 59462).We used the Python API version of the Swift-XRT pipeline,<cit.>, to extract the source and background spectra for all epochs except 2021 August 7 (MJD 59433), where the source exhibited a Type I X-ray burst (see section <ref>). We used the HEASOFT package (version 6.25) for our spectral analysis. For observations that had a sufficiently large number of counts (i.e., MJD 59364–59496), we used a modifiedscript to bin the spectra on 25-count intervals and performed spectral fitting using χ^2 statistics. Towards the end of our 2021 monitoring (i.e., the MJD 59504 and 59511), we used Cash statistics <cit.> with single-count binning intervals, due to the small number of counts collected in each observation. The final two epochs of the 2021 monitoring (MJD 59518 and 59524) and the late-time follow-up (MJD 60089 and 60172) were non-detections and thus were omitted from the spectral fitting routine. Using xspec <cit.>, we performed our spectral fitting twice, once for the 0.5–10 keV energy range and again for 1–10 keV. As expected, changing the energy range had a negligible effect on the best-fit spectral parameters. We modelled the spectra using an absorbed power law model with an added blackbody component; i.e.,× ( + ), wheremodels the interstellar absorption using an equivalent hydrogen column density (N_H) following the abundances from <cit.>. The power law accounts for the X-ray emission from the dominant component (i.e., the hard X-ray corona), and the blackbody accounts for any excess soft X-ray emission from a faint accretion disk, neutron star surface, or boundary layer. Initially, we fit each spectrum individually, allowing N_H to vary epoch by epoch. We then adopted the single epoch fitting as our starting parameters, linking the N_H values across all epochs and fitting the spectra simultaneously, resulting in a single time-independent value of N_H. When calculating the degrees of freedom, we treated the linked N_H as frozen (i.e., each spectrum has four free parameters). The epochs that utilized Cash statistics were omitted from the fitting procedure detailed above. Instead, we fit each of those spectra with a simple absorbed power law model (i.e.,× ), fixing N_H to our best-fit value of 3.88×10^21 cm^-2 and the power law photon index (Γ) to the average value of 1.61 from the χ^2 fitting. As a result, the X-ray flux was the only free parameter in the Cash statistic modelling. The Swift-XRT monitoring and spectral parameters during the 2021 outburst are presented in Table <ref>. The quoted uncertainties on the X-ray parameters represent the standard 90% confidence intervals. §.§.§ Type I X-ray Burst On 2022 August 7 (MJD 59433), SAX J1810 underwent a Type I X-ray burst, and, as a result, we performed manual data reduction on the Swift-XRT (WT) observations. First, we ran the taskto produce cleaned event files and exposure maps. Second, using , we applied the barycentric timing correction. Lastly, we extracted source and background spectra by using . For the pre-burst times, we used a circular source extraction region with a radius of 30 pixels (1 pixel = 2.36 arcsec) and an annular background extraction region with an inner radius of 70 pixels and an outer radius of 130 pixels. The pre-burst spectrum was then processed using χ^2 statistics and the routine mentioned in <ref>. During the burst, we broke the event file into multiple time bins to analyze the time evolution of the spectral parameters. Due to high count rates during the burst (i.e., maximum count rates ≳400), the observations are affected by systematic effects caused by photon pile-up. As a result, we used an annular source extraction region with an inner (exclusionary) radius that increases with an increasing count rate (ranging from 0 to 3 pixels). Following the Swift-XRT pipeline procedure <cit.>, we choose inner radii that reduce the maximum count rate in a given time bin to < 150. The time ranges were chosen so each bin has ≳ 300 counts corresponding to 21 bins across the 1.5burst. To model the burst parameters in xspec we added a second blackbody component to the pre-burst spectrum, fixing the pre-burst parameters, thereby allowing only the second blackbody to vary. We used themodel to directly fit for the normalized radius (i.e., size of the blackbody) and temperature before using the xspec convolution modelto calculate the flux. For the timing analysis, we extracted two light curves. The first light curve was binned on 1s intervals and was used to model the decay timescales of the burst. We extracted an initial light curve using the circular extraction region. For any time bins with a count rate > 150, we replaced their count rates with the count rate measured by the annular region with a 3-pixel exclusionary inner radius. We corrected for background and annular extraction region effects withand , respectively. Following the prescription outlined in <cit.> we fit an exponential decay function, R(t) = Ae^-t/τ + R_0,where t is the time after the burst maximum, R(t) is the count rate at a given t, R_0 is the constant background rate, τ is the e-folding decay time, and A is the peak count rate of the bursting component (excluding the contribution from a constant background). We fit for τ, R_0, and A with a Markov-Chain Monte Carlo (MCMC) routine using Python's emcee package <cit.>, assuming the sampled count rates were independently distributed normal random variables. The number of (sampling) walkers was fixed at five times the number of dimensions (i.e., 15). We chose three flat priors to ensure an unbiased analysis. To ensure convergence, we manually inspected the walkers over many autocorrelation times. Additionally, we analyzed the evolution of the autocorrelation time as a function of the number of MCMC steps following the routine outlined in the emcee documentation[The documentation can be found here: <https://emcee.readthedocs.io/en/stable/tutorials/autocorr/>]. The second light curve was extracted using the circular extraction region and binned on 1.8ms intervals (the minimum bin size possible for WT mode). We used the short timescale light curve to search for millisecond burst oscillations. Given the short timescale binning, no corrections were applied to the 1.8ms light curves. Appendix <ref> presents the X-ray burst properties. §.§ The WATCHDOG Pipeline We calculated the X-ray hardness ratio (HR) using a modified version of the pipeline developed for the Whole-sky Alberta Time-resolved Comprehensive black hole Database Of the Galaxy <cit.>. The hardness ratio is the ratio between the number of counts in the hard and soft X-ray bands. We used the MAXI/GSC 4–10 keV band as the soft band and 15–50keV observations from the Burst Alert Telescope <cit.> aboard Swift as the hard band. Both sets of observations are publicly available[MAXI/GSC: <http://maxi.riken.jp> Swift-BAT: <https://swift.gsfc.nasa.gov/results/transients/>]. We modified the pipeline to average daily observations, ensuring the hard X-ray band had a ≥3σ detection. For data where the soft X-ray band detection significance was < 3σ, we replaced the measured count rate with 3 × the noise value to estimate a conservative 3σ lower limit. The source appears to have undergone a hard-only outburst, and, as a result, to get meaningful constraints, we needed to measure either a lower limit or detection on the hardness ratio. No further modifications were applied to the WATCHDOG pipeline.WATCHDOG defined empirical HR limits that corresponded to the different X-ray states: (i) C_hard = 0.3204; and (ii) C_soft = 0.2846. A hardness ratio is considered consistent with the hard (soft) state if its lower (upper) error bars are above (below) the C_hard (C_soft) limits. If neither criterion is met, the source is classified as being in an intermediate state. We note that the values of C_hard/C_soft were calculated for BHXBs; in Section <ref>, we investigate whether it is valid to apply the same standard NSXBs. § RESULTS§.§ Radio PositionIn Fig. <ref>, we show the offset in right ascension and declination between the MeerKAT position and the archival X-ray position of 18h10m44.47s -26^∘09^'01.2^'' from <cit.>. The average radio position is 18h10m44.34s -26^∘09^'02.1^'' (±0.1^''). The per-epoch declinations are consistent with the average radio position with a reduced χ^2=0.75 (22 degrees of freedom), although the average radio position is offset by ∼ 1^'' from the X-ray position. In contrast, the right ascensions show significantly larger offsets ranging from ∼1–5^''. Moreover,the measured right ascensions show temporal variability. Adopting the weighted mean offset in right ascension as a model and computing the reduced χ^2 results in a value ofχ^2=4.4 (22 degrees of freedom), suggesting that the variability is not the result of stochastic error fluctuations. We tested the right ascension offsets against a linearly increasing model (i.e., ballistic motion), which resulted in a negligible improvement in the reduced χ^2 (4.2; 21 degrees of freedom), and thus, we found no evidence of ballistic motion. §.§ Outburst Light Curves In Fig. <ref> we show the MeerKAT (1.3; top panel), Swift-XRT (0.5-10; second panel), MAXI/GSC (4-10; third panel), and Swift-BAT (15-50; bottom panel) outburst light curves. For our MeerKAT observations, 18 (out of 21) epochs were ≥ 5σ detections (blue circles). The remaining three epochs (blue diamonds) do not meet the typical reporting threshold of 5σ, with detection significance of ∼ 4.3–4.9σ. Given the spatial coincidences between the low (< 5σ) and high-significance detections (≥ 5σ), it is likely that we are detecting a source in all of our MeerKAT observations. For the Swift-XRT light curve, we adopted the total fluxes from our spectral fits using the joint power law and blackbody model components (filled black circles). The last two data points (open black circles) correspond to the epochs where the source was too faint for multi-component spectral modelling; instead, we fit the source with a single power law component. The Swift-BAT and MAXI/GSC light curves display the data at a daily binning frequency.The observed flux of SAX J1810 displays a common temporal evolution across all observing frequencies. At early times (∼ MJD 59340–59370), all four instruments recorded the brightest signal of the outburst. Following the maxima, the source flux began decreasing, showing a rebrightening between ∼ MJD 59410 and 59440, before the source flux continued to decrease, returning to X-ray quiescence and plateauing at ∼ 90 μJy in the radio. We find no evidence for additional intra-observation variability beyond the Type I outburst discussed in this paper. Although the radio and X-ray light curves share a similar evolution in time, the magnitude of the variability is significantly different. In radio, the source exhibits modest variability with a maximum (∼ 230) and minimum (∼ 80) flux density separated by a factor of only ∼ 3. In contrast, when only considering the epochs with multi-component spectral modelling, the Swift-XRT fluxes show a factor of ∼20 in variability, with a maximum and minimum flux of ∼ 1.6×10^-9 and 6.8×10^-11, respectively. Including the final two Swift-XRT epochs during the source's return to quiescence, the minimum flux is ∼ 5×10^-13, which corresponds to a factor of ∼ 2000 decrease from the maximum. The plateauing radio emission at MJD 59463 (and beyond) is consistent with a spatially coincident, persistent radio source (see Section <ref>). §.§ X-ray Spectra The X-ray modelling parameters are shown in Fig. <ref>. The best fit equivalent hydrogen column density is N_H = 3.9_-0.2^+0.1×10^21 cm^-2. The Colden: Galactic Neutral Hydrogen Density Calculator[The webtool can be found here: <https://cxc.harvard.edu/toolkit/colden.jsp>] estimates a value of N_H ∼ (3.2–4.3)×10^21 cm^-2 along the SAX J1810 line of sight (depending on the choice of neutral hydrogen data set — NRAO or Bell), making the measured N_H consistent with expectation. To investigate the relative contributions of each model component, we calculated the power law flux fraction (third panel, Fig. <ref>); i.e., F_X,PL/F_X,tot, where F_X,PL is the X-ray flux of the power law component and F_X,tot is the total X-ray flux of the model. In all epochs, the power law component is dominant with a flux fraction ranging from ∼ 0.53 to 0.94 with a (variance-weighted) average of 0.72±0.02. The power law photon index (Γ; fourth panel, Fig. <ref>) shows moderate variability with0.4_-0.43^+0.93≤Γ≤2.88_-0.08^+0.18 and an average value of 1.61±0.03. The average value is typical of comptonized hard state X-ray emission from (black hole) X-ray binaries <cit.>. Moreover, if we exclude the anomalously steep photon index, the maximum photon index becomes Γ = 1.83_-0.08^+0.10. The blackbody temperature (kT; third panel, Fig. <ref>) varied between 0.5_-0.08^+0.18≤ kT ≤ 1.2_-0.08^+0.18 keV, with an average blackbody temperate of kT = 0.60 ± 0.01 keV. Black body temperatures ≲ 1 keV are consistent with past analyses of hard state neutron star X-ray binaries <cit.>. The bottom panel of Fig. <ref> displays the hardness ratio calculated from the daily Swift-BAT and MAXI/GSC light curves. We observe a moderate degree of variability in hardness ratio, with detections ranging from ∼ 0.5–2.8, and an average value of 1.19 ± 0.06. Including the lower limits increases the maximum hardness ratio to ∼ 4. The largest single epoch evolution occurs on MJD 59385, where the black body temperature reaches its maximum value of ∼ 1.2, alongside the extreme softening of the power law component (Γ ∼ 2.9). During this epoch, the two-component fit had a reduced χ^2 value of ∼ 1.17 (216.5/186). To investigate whether we were observing a transition to an intermediate or soft state, we added a multi-colour disk to the two-component model; i.e.,× ( ++ ). The inclusion of the third component moderately reduces the χ^2 to ∼ 1.12 (206.1/184) and decreases both the power law photon index and blackbody temperature to levels consistent with the other epochs (See Table <ref> for the full model parameters). Moreover, the power law component becomes sub-dominant, suggesting that the source may have briefly transitioned into an intermediate or soft state. The observations on MJD 59413 and 59462 show similarly large reduced χ^2 values of ∼ 1.22 (237/194) and ∼1.52 (50/33), respectively. As a result, we attempted to fit these spectra with the same three-component model. However, the fitting resulted in a negligible improvement of the χ^2 statistic. We note that, for the latter epochs, both have reduced χ^2 deviations that are consistent (at the < 3σ level) with the expected value of 1. Therefore, the poor fits may result from statistical effects rather than a physical change in the X-ray spectrum. §.§ Persistent Emission and the L_R–L_X relation Our 2023 follow-up MeerKAT observations revealed a 112± 12 μJy radio (point) source on 2023 May 22 (MJD 60086) and another 75± 11 μJy radio source three months later on 2023 August 13 (MJD 60169). The best-fit positions of both 2023 detections are consistent with the 2021 outburst (see Fig. <ref>). Therefore, we confidently detect a persistent radio source spatially coincident with SAX J1810. We calculated an (intra-band) spectral index of the persistent source using the brighter of the two MeerKAT follow-up observations (MJD 60086). We broke our observations into four evenly spaced sub-bands, ensuring a ≥ 5σ detection in each sub-band. Applying a simple linear least squares fit, we measured a spectral index of α=-0.7 ± 0.5. In addition to the large statistical error, we note that intra-band spectral indexes are known to bias towards flatness (α ∼ 0) at detection significances ≲ 35σ <cit.>. Given our source was only detected at ∼ 10σ and the relatively large error bar, we do not apply any strong physical inference based on this intra-band spectral indexDuring the last seven epochs of 2021 monitoring (MJD 59463 to 59511) – after the radio flux density had plateaued – the average radio flux density is 93 ± 7μJy. This value is consistent with our 2023 observations (at the ∼ 2σ level), suggesting the persistent emission is, at most, weakly variable with a ∼ 20% excess variance. Combining the late-time 2021 and 2023 observations results in a (weighted) average flux density of 89 ± 5μJy. The quasi-simultaneous Swift-XRT follow-up on MJD 60089 and 60172 did not detect any spatially coincident X-ray source in either epoch setting 3σ upper limits on the 1–10keV X-ray flux of < 1.3×10^-13and < 3.0×10^-13, respectively. Furthermore, our scheduled VLA follow-up at 3and 6, taken between our two MeerKAT observations on 2023 July 17 (MJD 60142), did not detect the source. The 3σ upper limits on the 3and 6were 30 μJy and 18 μJy, respectively. Adopting a 1.3flux density of 78 μJy (conservatively assuming a 3σ drop in flux caused by intrinsic variability), we use the 3non-detection to calculate a conservative upper limit of α <-1.1. Figure <ref> presents the L_R–L_X relation. The plot includes archival hard state BHXBs (grey circles), hard state NSXBs (blue squares), and accreting millisecond X-ray pulsars (AMXPs; orange triangles). The archival sources were adapted from Fig. 4 of <cit.>, an updated version of the <cit.> catalog. As our Swift-XRT and MeerKAT observations were quasi-simultaneous, we applied a one-dimensional linear interpolation to map the radio observations onto the X-ray times for our 2021 observations. We did not apply any interpolation for our 2023 follow-up observations. Instead, we grouped the MeerKAT observations with the nearest Swift-XRT follow-up. We present the L_R–L_X relation from the 2021 outburst as red circles. Fitting the 2021 results with a simple power law results in a shallow exponent of β=0.09 ± 0.03 (for L_X ∝L_R^β). If we assume that the 2023 MeerKAT detections originate from a persistent hard state jet (purple stars on Fig. <ref>) and thus should follow the L_R–L_X relation, the measured power index becomes an upper limit (due to the X-ray non-detections) adopting a value of β < 0.06. Given that our results strongly suggest the existence of a persistent radio source that is unrelated to the hard state jet of SAX J1810, we present a secondary set of L_R–L_X data points (green squares) after subtracting off 93 μJy from each of the radio flux densities from our 2021 outburst. Post-subtraction, there are only four epochs (MJD 59364, 59378, 59413, and 59437) that show a > 3σ excess flux density when compared to the persistent level. For the rest of the epochs, we set the radio flux density to be 3× the rms noise and displayed them as upper limits. The subtracted values are unconstraining but consistent with the broader population of NSXBs. The implications of SAX J1810 L_R–L_X evolution and the origin of the persistent radio source are discussed in Section <ref>§ DISCUSSIONWe monitored the NSXB SAX J1810 during its 2021 outburst. The X-ray and radio properties suggest that the source underwent a `hard-only' outburst, never fully transitioning to a soft accretion state. Moreover, the late-time plateau of radio flux density in 2021, combined with our follow-up in 2023, suggests the existence of a persistent radio source. In the following subsections, we present the evidence of a `hard-only' outburst and discuss the possible origins of the persistent radio emission.§.§ Hard-Only OutburstOur observations suggest that SAX J1810 exhibited a `hard-only' outburst in 2021. We justify this claim with three points of evidence: * The hardness ratio between the Swift-BAT and MAXI/GSC observations is above the hard state limit throughout the monitoring. Although the limit was empirically defined using outbursting BHXBs, we expect that the persistent source of thermal X-ray photons (from the neutron star surface or boundary layer) would make all X-ray states softer, thereby decreasing the hard state limit for NSXBs. We investigate this proposition by analyzing the best-studied outbursting (atoll) NSXB, Aql X-1. In Fig. <ref>, we have plotted a sample light curve of Aql X-1 during its 2016 outburst. The source exhibits a rapid transition of its hardness ratio, with a large fraction of the outburst remaining at a steady value of ∼ 0.05 well below the soft state limit derived for BHXBs. <cit.> performed an X-ray spectral analysis of four separate observations; the authors identified that the source was in the hard accretion state on 2016 Aug 3 (MJD 57603) and 2016 Sep 19 (MJD 57650) and in the soft accretion on 2016 Aug 5 (MJD 57605) and 2016 Aug 7 (MJD 57607). The hard and soft state epochs are shown with the dashed and dashed-dotted lines in Fig. <ref>. As expected, the soft and hard state epochs are temporally consistent with small and large hardness ratios. The final (Sep 19) hard state epoch shows a hardness ratio below the BHXB hard state limit, consistent with our prediction that the thermal photons from neutron stars will lower the hard state limits. We note that other outbursts of Aql X-1 <cit.> show a similar `softening` of the hard state limit. Therefore, we are confident that the Swift-BAT and MAXI/GSC hardness ratio for SAX J1810 is consistent with hard state emission throughout the 2021 outburst, and our adoption of the WATCHDOG limits is most likely appropriate (if not a conservative approximation).* Our Swift-XRT spectral modelling is consistent with hard state emission in nearly all epochs. The X-ray photon indexes (Γ_avg∼ 1.6) and low-energy black body temperatures(kT_avg∼ 0.6) are typical of hard state X-ray emission from an NSXB <cit.>. Moreover, the power law component is the dominant flux component in all epochs (i.e., power law flux fraction ≥ 50%). Although some epochs show approximately equal contributions between the blackbody and power law components, the narrow (0.5-10.0keV) energy range favors the black body component when calculating band limit flux, as the power law component will dominate at higher energies (≥ 10keV). The bolometric X-ray flux is more strongly dominated (> 90%) by the power law component than our observations would suggest, consistent with hard state emission. The anomalous epoch (MJD 59385; Table <ref>) that shows a clear softening of the X-ray spectrum suggests the source may have exhibited a brief deviation from a hard accretion state. Assuming a successful transition to the soft state, and given the cadence of our observations and the bracketed hard sate epochs, the source would have gone through a full cycle (i.e., hard → soft → hard) in ≤ 14days before remaining in the hard state for the remaining ∼ 120days of outburst <cit.>. We find it more likely that the source briefly entered an intermediate state, failed to complete a transition to the soft state, and transitioned back to the hard state.* The evolution of our radio observations is consistent with the hard state. First, the radio and X-ray light curves show a correlated temporal evolution characteristic of hard state emission. Second, we do not detect any significant jet-quenching. Although radio emission from NSXBs has been observed in the soft state, when both hard and soft state (compact jet) radio emission has been detected, the jet emission is brighter in the hard state <cit.>. Therefore, without a significant increase in the X-ray flux (which was never observed), we would expect a decrease in the radio flux after a transition to the soft state. We recognize that the spatially coincident, persistent radio source contaminates our ability to detect jet-quenching. However, the persistent source can not explain the joint radio–X-ray time evolution, as we would expect the radio flux to drop to the persistent level (∼ 90 μJy) without a similar decrease in X-ray flux. Whenever we observed an increasing X-ray flux, we observed a simultaneous increase in the radio flux density.Comprehensive monitoring campaigns of future outbursts of SAX J1810 will be critical for confirming whether the source consistently exhibits `hard-only' outbursts or shows a broader outburst phenomenology that sometimes results in successful transitions to the soft state <cit.>. §.§ The Origin of the Persistent Radio Emission Our observations strongly support the existence of an unresolved, persistent, steep-spectrum radio source spatially coincident with the position of SAX J1810 (± 3^''). Considering the source exhibited a `hard-only' outburst in 2021, we expect the radio emission to (partially) originate from a hard state jet (i.e., compact jet). The temporal coincidence between the flares at X-ray and radio frequencies is strong evidence for the existence of a steady jet. Moreover, the persistent source is weakly variable with an average flux density of ∼ 90μJy. Considering that we have multiple detections at ≳ 200μJy, we have clearly detected radio emission from the compact jet. However, a hard state jet associated with SAX J1810 cannot be the source of the persistent radio emission. Hard state jets are stationary and, therefore, would not exhibit the proper motion that we have observed (Fig. <ref>) Moreover, the locations of its luminosities on the L_R–L_X plane (red circles Fig. <ref>) are inconsistent with a hard state jet. At early times and high X-ray luminosities, the radio/X-ray luminosities are positively correlated, as expected from a compact, steady jet. Towards the end of the outburst (at L_X ≲ 5×10^35 ergs s^-1), there is a clear flattening of the correlation resulting in a β< 0.06 due to the radio luminosity remaining approximately constant while the X-ray luminosity decreased by over three orders of magnitude. The 2023 follow-up, in particular, would make SAX J1810 exceptionally radio-loud for a NSXB, consistent with the population of BHXBs. Recent analyses estimate a value of β=0.44_-0.04^+0.05 for the total population NSXBs, with the atoll sub-population (which SAX J1810 is likely a member of) having β=0.71_-0.09^+0.11 <cit.>. Both values of β reject our measurements at the > 3σ level. Therefore, the observed radio emission likely originates from two components, with the most likely candidates of the persistent emission being either a discrete jet ejection or an unrelated, spatially coincident source.We disfavor an origin due to jet ejection(s). First, the average decay timescale of an ejection event is ≪ 1year, and thus a jet ejection persisting for ∼ 2years and showing no significant decrease in the measured flux density is, in itself, unlikely. Long-lasting jet ejecta have been observed from BHXBs and are thought to be the result of jet-ISM interactions driving in situ particle acceleration and long-term synchrotron emission <cit.>. However, such long-lasting ejecta have never been observed in NSXB (likely due to their weaker, lower-luminosity jets being unable to power such long-term emission), and when observed in BHXBs, the radio emission of long-lived ejecta is strongly variable. Second, our VLA follow-up observations suggest a 3σ upper limit on the radio spectral index of α <-1.1, significantly steeper than expected from optically-thin synchrotron emission from a jet ejection (α ∼ -0.7). Lastly, our observations show no evidence of ballistic motion despite the source persisting for ∼ 2years, which would be the strongest evidence for a jet ejecta origin of the persistent emission. If the persistent emission originated from jet ejecta, we would have had to observe a long-lasting, non-variable, spectrally steep ejecta showing no motion on the sky. Therefore, we can rule out a jet ejecta origin with high confidence. To estimate the probability of a spurious spatial coincidence with an unrelated source in the field we used the Python Blob Detector and Source Finder <cit.> to make a catalog of all sources (in each image) with a flux density > 74 μJy (3σ lower than the average persistent radio flux density). We use the deep 2023 observations as their lower rms noise (10 μJy vs. 20 μJy in 2021) makes PyBSDF less prone to mistaking spurious noise spikes as real sources. Due to flux variability, each image catalog has a different number of sources. As a result, we conservatively use the 2023 May 22 image as it has more sources than the August observation and, therefore, a larger source density. We calculate the source density and then convert it to the expected number of sources within a 3^'' radius. The choice of 3^'' was motivated by the scatter of our best-fit positions. Using the expected number of sources, we then calculate the Poissonian probability of a chance coincidence of one or more unrelated background sources. The instrument's sensitivity decreases as a function of radial distance from the phase center of the array, and thus, there is a progressively smaller number of sources cataloged at larger separations from the phase centre (decreasing the source density). We applied a cut when calculating the probability to investigate this potential bias, only including sources within a certain distance from the phase centre in our calculations. In Fig. <ref>, we show the chance coincidence probability as a function of the aforementioned `inclusion radius' for only unresolved sources (following the criteria from Appendix <ref>) and for both unresolved and extended sources (all sources). We adopt the peak value for all sources as our conservative estimate of the chance coincidence probability (i.e., ∼ 0.6%).Radio-bright active galactic nuclei (AGN) are the dominant population of unresolved background sources. However, background AGN have an average spectral index of α ∼-0.7. We use two recent surveys of background AGN spectral indexes to estimate the probability of finding a steep spectrum AGN. <cit.> calculated the spectral index of 166 AGN using 325, 610, and 1400MHz flux densities. Only 43 sources had an α <-1.1 corresponding to a probability of ∼ 26%. In a more recent, larger sample size survey, <cit.> measured the spectral indexes of ∼ 540000 radio sources (using 147 and 1400MHz flux densities), with only a subset of ∼ 32000 having an appropriately steep α. The corresponding probability is ∼ 6%. Adopting the older catalog probability as a conservative estimate, we calculate the total probability of finding a spurious radio AGN with a sufficiently steep spectral index as ∼ 0.16% (a ∼ 3.2σ event). Alternatively, the spectral index could suggest an origin from a class of sources known to have steep spectral indexes. The most common steep spectrum source is pulsars, with average spectral indexes of ∼-1.6 <cit.>. We searched the Australian Telescope National Facility pulsar catalog <cit.> for any nearby known radio pulsars but found no pulsars within a radius of 0.6^∘. Given that there are only 3000 known radio pulsars (corresponding to an expectation value of ∼ 2×10^-7 pulsars within a 3^'' radius), there is a chance coincidence probability of ∼ 0.002%. When considering that pulsars tend to be distributed in the Galactic plane (∼ 20% of the sky), and SAX J1810 is also in the galactic plane, the chance coincidence probability would increase by a factor of ∼ 5 but is still less likely than the AGN scenario. We note that the persistent emission would correspond to a time-averaged flux of a pulsar; as a result, recent surveys that looked at this part of the sky would have detected a pulsed source <cit.>. Moreover, MeerKAT's pulsar timing backend <cit.> was operational during all of our observations but did not detect any pulsed emission from the source. Therefore, our estimated coincidence probability between SAX J1810 and an unknown pulsar is most likely an overestimate. There is a small possibility that the persistent radio-emission is local to SAX J1810. Transitional millisecond pulsars (tMSPs) — accreting neutron stars that transition between accretion-powered (i.e., NSXB-like) and radio pulsar behaviour — have shown anomalously bright radio emission while actively accreting. For instance, the tMSP, 3FGL J0427.9-6704, was measured at a point on the L_R–L_X relation that was also more consistent with the population of black hole X-ray binaries; however, its X-ray luminosities were a factor of ≳ 3larger than our upper limits on MJD 60086 <cit.>. Other tMSPs (i.e., PSR J1023+0038) have even exhibited anti-correlations between radio and X-ray luminosities, which could allow for bright radio emission absent any X-ray detections<cit.>.However, the properties of SAX J1810 are inconsistent with what is expected from tMSPs. Firstly, SAX J1810 does not show radio pulsations during X-ray quiescence <cit.>. Second, at X-ray luminosities ≤ 10^33 erg s^-1, tMSPs spectra are non-thermal <cit.>, whereas SAX J1810 is thermally dominated <cit.>. Lastly, SAX J1810 does not exhibit any of the rapid X-ray variability that results from switching between different accretion modes (during outburst), showing, at most, modest variability <cit.>. Although it cannot be conclusively ruled out, we find it unlikely that the persistent radio emission results from SAX J1810 being a tMSP.Local emission, tMSP or otherwise, is difficult to reconcile with the variability in the position, as the source is spatially unresolved. Using the scatter in the measured position (∼ 3^'') as a proxy for the expected separation of the two-source scenarios (i.e., the persistent emission is non-local), then observations by an instrument with sufficient angular resolution and sensitivity(e.g., the VLA in A-configuration or the Square Kilometer Array) during future outbursts when the compact jet is ‘on’ should be able to spatially resolve two distinct components. If only a single source is observed, and there continues to be temporally correlated evolution in the radio/X-ray light curves, this would strongly support the scenario where the persistent radio emission is local to SAX J1810.§ SUMMARY AND CONCLUSIONSWe have presented our ∼ 2year joint radio and X-ray monitoring of the neutron star X-ray binary SAX J1810.8-2609. Our observations include dense (i.e., weekly cadence) observations during the source's 2021 outburst and a collection of late-time observations in 2023. The X-ray spectral properties suggested that the source remained in the hard state throughout the entire 2021 outburst. Moreover, the radio and X-ray luminosities show a temporally correlated evolution, characteristic of a hard state radio jet. We discovered a spatially coincident, persistent steep-spectrum radio source that shows no correlation with the simultaneous X-ray flux. Therefore, during the outburst, the radio emission originated from a superposition of two components: a variable hard state compact jet (≲ 100μJy), and the unknown persistent source (∼ 90μJy). The spectral index and evolution of the persistent source are inconsistent with jet ejecta. We conservatively estimated the probability of a chance coincidence with an unrelated spectrally steep background source, and although low (∼ 0.16%), a background AGN seems to be the most plausible scenario. SAX J1810.8-2609 is known to go into outburst every ∼ 5years, and future outbursts should focus on identifying the source of the persistent emission. Of the current generation of radio telescopes, the VLA (A-configuration) and the Very Long Baseline Array (VLBA) both have sufficient angular resolution and sensitivity to resolve two ∼ 100 μJy sources (assuming a separation of ∼ 3^''). Moreover, next-generation radio interferometers, such as the Square Kilometer Array (SKA; of which MeerKAT is a pathfinder), would be able to reach the desired sensitivity with a fraction of the observing time <cit.>. During the next outburst, if a second unrelated source is ruled out, follow-up observations should focus on understanding what physical mechanism is driving the persistent radio emission, whether the source is a tMSP or otherwise. § ACKNOWLEDGEMENTS We extend our sincere thanks to all of the NRAO, SARAO, and Swift-XRT staff involved in the scheduling and execution of these observations. We thank Kaustubh Rajwade for useful discussions on the completeness of pulsar catalogues. We thank Ben Stappers for searching for pulsed emission in the MeerTRAP observations. We thank Craig Heinke for useful discussions on the X-ray properties of transitional millisecond pulsars. Finally, we thank the referee for their insightful and helpful comments. 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. We acknowledge the use of public data from the Swift data archive. This research has made use of MAXI data provided by RIKEN, JAXA and the MAXI team. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. AKH and GRS are supported by NSERC Discovery Grant RGPIN-2021-0400. JvdE acknowledges a Warwick Astrophysics prize post-doctoral fellowship made possible thanks to a generous philanthropic donation.AKH and GRS respectfully acknowledge that they perform the majority of their research from Treaty 6 territory, a traditional gathering place for diverse Indigenous peoples, including the Cree, Blackfoot, Métis, Nakota Sioux, Iroquois, Dene, Ojibway/ Saulteaux/Anishinaabe, Inuit, and many others whose histories, languages, and cultures continue to influence our vibrant community. § DATA AVAILABILITY Data from MeerKAT are available through the SARAO data archive: <https://apps.sarao.ac.za/katpaws/archive-search>. Data from the VLA are available through the VLA data archive (Project ID 23A–417): <https://data.nrao.edu/portal>. Data from the Swift-XRT are publicly available through the Swift archive: <https://www.swift.ac.uk/swift_portal>. The authors make their flagging and calibration scripts, imaging results, and analyses available at: <https://github.com/AKHughes1994/SAXJ1810_2023>. The astrometry routine is available at: <https://github.com/AKHughes1994/AstKAT>. mnras § RADIO ASTROMETRY Our observations constitute the first radio detections of SAX J1810, and therefore, we designed a novel astrometric routine to test whether the radio emission is spatially coincident with the archival X-ray position of 18:10:44.47 -26:09:01.2 <cit.>. We divided our astrometric analysis into two components; the first measures the random inter-epoch variability of each source position, quantifying the effects of noise fluctuations (relative astrometry), and the second measures the global offsets due to systematic effects in the instrumentation (absolute astrometry). The following section outlines our astrometry routine. For unresolved sources (i.e., point sources) in synthesis radio images, the relative astrometric error is most often determined by the centroiding accuracy of the Gaussian fitting following deconvolution routines. As the shape of a point source adopts the shape of the synthesized beam in the absence of noise, the astrometric precision decreases with an increasing beam size. The error on the relative astrometry is often described as a function of two components: a signal-to-noise (SNR) dependency and a lower limit set by a systematic threshold. The most commonly assumed signal-to-noise scalings are, 1/SNR, or 1/(2·SNR). The systematic threshold is assumed to be some fraction of the synthesized beam size. A common assumption is a lower limit of 10% of the synthesized beam size (e.g., for standard observing with the VLA[see here; <https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/positional-accuracy>]). We define a generalized (relative) astrometric error with the following functional form,σ = √((A·SNR)^2 + B^2),where σ is the relative astrometric error expressed in units of synthesized-beam full widths at half-maxima (FWHM); and A and B are dimensionless variables that describe the SNR scaling and systematic threshold, respectively. Using PyBDSF, we generated a catalogue of (elliptical Gaussian) sources in each image; our parameters of interest were the right ascension (RA), declination (Dec), major axis FWHM of the source, minor axis FWHM of the source, peak flux density (F_p), total island flux density[PyBDSF groups sources into islands, where an island is defined as a continuous region of pixels with a flux value above a user-defined threshold and at least one pixel has a flux larger than a higher (also user-defined) threshold. For large islands (i.e., extended emission), PyBDSF will fit multiple sources to a single island. For our fitting, we used 3σ and 4σ for our thresholding.] (F_i), and local rms. As SAX J1810 is isolated and unresolved, we trimmed the PyBDSF catalogue to include only similarly unresolved and isolated sources. We defined a source as unresolved if the source FWHMs deviated by ≤ 25% from the synthesized beam shape. Similarly, a source is classified as isolated if the peak flux is within 25% of the island flux (e.g., |F_p/F_i - 1| ≤ 0.25). Our routine calculates the average signal-to-noise of each source in the catalogue, and, therefore, we exclude bright transients and strongly variable sources, as their SNR ratio will vary drastically epoch-to-epoch. A source is classified as transient/variable and omitted from the sample if the source is missing from > 25% of the epochs or has a maximum and minimum flux density separated by a factor ≥ 2. Lastly, to mitigate biasing from poor far-field calibration errors (e.g., from antenna pointing errors), we fit the sources that are within the inner ∼ 50% of the primary beam FWHM (i.e., sources within 0.3^∘ of the phase centre). As the MeerKAT synthesized beam is an elliptical Gaussian, we solve for A and B independently along the RA and Dec directions. Below, we outline our fitting routine: * For each source, calculate an average SNR and an average position. Calculate the RA/Dec offset from the average position for every source in each epoch using the average position.* Estimate the error in the astrometric precision of each source by bootstrapping the offsets, adopting the median value of the bootstrapped sample as an initial guess for σ and the ranges between the median and the 15^th/ 85^th percentiles as the 1σ (-)/(+) uncertainties (Δ_σ).* Using the σ estimates and the average SNR, solve for the scaling parameters A and B (i.e., the uncorrected fit). The fit implements an MCMC routine and follows the same approach detailed in <ref>.* Solve for the (inverse-variance weighted) average offset of all sources in each epoch (i.e., the epoch-to-epoch correction) weighting each offset using the uncorrected fit.* Correct the source offsets with the epoch-to-epoch correction and re-solve for A and B with the updated – corrected – offsets.* Repeat (ii)→(v) until the fitting converges on solutions for A and B. We defined a convergence parameter C = (σ_i - σ_i-1)/Δ_σ; i.e., the difference between the astrometric error of a source for the current (i) and previous (i-1) iterations in units of Δ_σ. The fit is said to have converged after three consecutive iterations with a mean value of C < 0.1. The post-convergence fit is the corrected fit. Record the final epoch corrections. The relative astrometric fitting is shown in Fig. <ref> and the best-fit parameters are tabulated in Table <ref>. The uncorrected fits have reduced χ^2 values of ∼ 1.3 (123 degrees of freedom) in both RA and Dec. Applying the epoch-to-epoch corrections (i.e., the corrected fit) shows a significant worsening of the fit quality with a reduced χ^2 > 2, suggesting that a single per-epoch correction is not accurately capturing the time-dependent systematics in our observations, and a more complex epoch correction may be appropriate (e.g., one that accounts for distance and direction with respect to the phase center). We intend to expand upon this preliminary work to investigate whether the relative astrometric error is similar across a range of ThunderKAT fields. The fits show that (for MeerKAT), the systematic threshold of the relative error is significantly lower than the commonly assumed limit of 10% the size of the synthesized beam. Moreover, the signal-to-noise dependency is similar to the commonly assumed 1/(2·SNR) scaling. Due to the residual issues in our modeling, for our SAX J1810 analysis, we conservatively rounded our uncorrected fit values, adopting A=0.5and B=0.02 to quantify the relative astrometric errors. To correct for absolute astrometry effects, we identified nine sources[<http://astrogeo.org/calib/search.html>] within our field of view that are used as phase calibrators for very long baseline interferometry (i.e., with positions measured at < 10milliarcsecond precision). Eight of the nine sources met our unresolved and isolated requirement, and we used this sub-sample for absolute astrometric corrections. After applying the epoch-to-epoch correction from the relative astrometric fitting, we measured the offsets of the eight calibrators with respect to their known positions. We then calculate each epoch's weighted mean (weighting each source by their relative astrometric errors). Lastly, we calculated a single time-independent absolute astrometric correction (see Fig. <ref>). The epoch-to-epoch correction removed any (substantial) temporal variability, and, as a result, the per-epoch average offsets are consistent with a single (time-independent) RA/Dec offset. The final astrometric error (σ_tot) was calculated by adding (in quadrature) the relative astrometric precision (σ), the error on the epoch-correction (σ_epoch), and the error on the absolute offset (σ_abs),σ_tot = √(σ^2 + σ_epoch^2 + σ_abs^2).These are the errors shown in Fig. <ref>. We note that given the signal-to-noise ratio of our SAX J1810 detections (SNR≲10), the relative astrometry term, σ, dominates the quoted errors.§ TYPE I X-RAY BURST Figure <ref> shows the parameters of the 2022 August 7 (MJD 59433) Type I X-ray burst. The top panel shows the 1s-binned light curves and the timing fits; the second panel shows the bolometric X-ray flux of the blackbody component; the third panel shows the temperature of the blackbody component; and the bottom panel shows the normalized radius of the blackbody component, defined as R^2/D^2, where R is the source radius in units of km and D is the distance to the source in units of 10kpc. The burst began its rise at 14:14:12 on 2021 August 7 (MJD 59433.59319), reaching a peak count rate of ∼ 400with a rapid 7 ± 1s rise time before decaying for the remainder of our observations. The timing fit converged on an e-folding decay time of τ = 15.8 ± 0.2s (full fit parameters in Table <ref>). The burst parameters are consistent with the MINBAR burst catalog <cit.> in both rise (3.4_-2.4^+5.6s) and e-folding decay times (8_-4^+21s). During its 2007 outburst, SAX J1810 exhibited 531.8Hz oscillations in the light curves of a Type I X-ray burst, likely the result of the spin frequency of the neutron star <cit.>. Following the prescription outlined in <cit.> we searched for burst oscillations in our (1.8ms resolution) light curves by calculating the power spectrum in sliding windows with widths of 0.5, 1, 2, and 4s, where each subsequent window is offset by 0.5s from the previous one. We found no evidence of burst oscillations. However, the temporal resolution of Swift-XRT WT mode (1.8ms) makes our power spectra insensitive to frequencies above ∼ 280Hz. Assuming the oscillations result from the spin period of the neutron star, we do not expect the oscillation frequency to evolve drastically between the 2007 and 2021 outbursts. Furthermore, SAX J1810 is known to exhibit PRE <cit.>. Therefore, we performed time-resolved intra-epoch spectral modelling to search for evidence of PRE. We observe some evolution of the radius and temperature, although the large errors greatly reduce their significance. Assuming a distance of 4.9kpc, the radius of the blackbody component ranges from 3.2_-2.6^+3.5 to 6.7_-4.2^+5.0km (i.e., from ∼ 5-to-100% of the neutron stars surface assuming a 10km stellar radius). However, the evolution of the radius and temperature does not occur alongside a period of (approximately) constant X-ray flux; thus, we do not detect PRE. § DATA TABLES | http://arxiv.org/abs/2311.16072v2 | {
"authors": [
"A. K. Hughes",
"G. R. Sivakoff",
"J. van den Eijnden",
"R. Fender",
"J. C. A. Miller-Jones",
"E. Tremou"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20231127184400",
"title": "SAX J1810.8-2609: An Outbursting Neutron Star X-ray Binary with Persistent Spatially Coincident Radio Emission"
} |
mysecondaryaddress]Anton A. Popovpopovfn [popovfn,surdyaevfn]Affiliations at the time this work was completed. mymainaddress]Vladimir Strokovmycorrespondingauthor [mycorrespondingauthor]Corresponding author [email protected]]Aleksey A. Surdyaevsurdyaevfn [mymainaddress]Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA [mysecondaryaddress]Lebedev Physical Institute, Astro Space Centre, 84/32 ul. Profsoyuznaya, Moscow, Russia 117997We test the possibility of using a convolutional neural network to infer the inclination angle of a black hole directly from the incomplete image of the black hole's shadow in the uv-plane. To this end, we develop a proof-of-concept network and use it to explicitly find how the error depends on the degree of coverage, type of input and coverage pattern. We arrive at a typical error of 10^∘ at a level of absolute coverage 1% (for a pattern covering a central part of the uv-plane), 0.3% (pattern covering the central part and the periphery, the 0.3% referring to the central part only), and 14% (uniform pattern). These numbers refer to a network that takes both amplitude and phase of the visibility function as inputs. We find that this type of network works best in terms of the error itself and its distribution for different angles. In addition, the same type of network demonstrates similarly good performance on highly blurred images mimicking sources nearing being unresolved. In terms of coverage, the magnitude of the error does not change much as one goes from the central pattern to the uniform one. We argue that this may be due to the presence of a typical scale which can be mostly learned by the network from the central part alone.black hole physics techniques: image processing techniques: interferometric methods: data analysis methods: miscellaneous§ INTRODUCTION There is vast indirect evidence of massive compact objects residing in galactic centres <cit.>. Measurements of the masses of these objects <cit.> and their compact sizes plausibly suggest that they are supermassive black holes. If so, the light coming from surrounding matter must be strongly lensed to form distinctive silhouettes of the black holes <cit.>. The dedicated Event Horizon Telescope (EHT) array <cit.> had been resolving increasingly closer neighborhoods of Sgr A^⋆ and M87^⋆ <cit.> before these efforts culminated in the historic first direct image of a black hole <cit.>. Also contributing to the task of black hole imaging are other existing arrays <cit.> and upcoming projects <cit.>.It is known <cit.> that a network of very-long-baseline interferometry (VLBI) stations, such as EHT, aims at measuring complex-valued visibility function𝒱(u,v) = ∬e^-2π i(uα + vβ)I(α,β) dα dβ ,which is the spatial Fourier transform of brightness distribution I in image plane (α,β). If the angles α and β are given in units of characteristic angular resolution λ/D, then spatial frequencies in the uv-plane are measured in units of D/λ, where D is a characteristic baseline and λ, the working wavelength. Since, in reality, the coverage of the uv-plane is always partial, the EHT team used a variety of techniques to reconstruct the image of the black hole shadow, such as the CLEAN algorithm and regularized maximum likelihood methods <cit.>.On the other hand, another set of algorithms known as convolutional neural networks (CNNs) has proven to be extremely effective in the general problem of image recognition <cit.>. In recent years, (artificial) neural networks in general and CNNs in particular have been finding more and more applications in astrophysics: to name a few, automated analysis and detection of strong gravitational lenses <cit.>, dark matter halo simulations <cit.>, black hole identification in globular clusters <cit.>, and computing the mass of forming planets <cit.>. Recurrent Inference Machines <cit.> were also used to process interferometric observations of strong lenses <cit.> as well as found applications in medical imaging <cit.>.Also, <cit.> developed two convolutional networks called Deep Horizon to recover accretion and black hole parameters from real-space images. However, as mentioned, VLBI observations rather yield partial Fourier transform of the images, and it would be more natural if a neural network took the Fourier image directly as its input. In that case, it is crucial to investigate how the error of the output depends on the degree of coverage of the uv-plane. There are a few reasons to use a neural network to infer parameters of a black hole. The first is the speed of analysis. For a given image and a given coverage, parameter inference algorithms such as MCMC <cit.> take a lot of time as they randomly walk in the parameter space, and, for a different coverage, this process should be repeated from the very beginning. This becomes especially important in the case of black hole silhouettes when generating one at each step of the random walk is time-consuming, because typically ray-tracing is used. A neural network, on the other hand, needs to be trained only once and on a set of images which is generated once and for all. Another reason is that predictions of a neural network can be used to double check the values of the parameters obtained with other techniques. Finally, a neural network could be integrated into a parameter inference pipeline. For example, below, we train our networks to operate within a wide range of degrees of coverage. This approach not only saves time (we need to train them once rather than training a series of networks on its own degree of coverage each) but also makes the networks more universal. Then, such a network could be used in the first stage of a pipeline to determine ranges of the parameters which can be further narrowed down with traditional methods.In this Note we develop a convolutional neural network that determines inclination angle [This is the angle between the black hole's spin and the line of sight of a distant observer] θ from a partially covered image of the uv-plane. Our aim is to study how the performance of the network depends on the degree of coverage, types of input and different coverage patterns that emulate different observational settings.In more detail, firstly, following <cit.>, we estimate the error introduced by the CNN by the width of the deviation distribution. We also check whether the distribution is Gaussian and study how the error depends on the degree of the uv-plane coverage. Secondly, we evaluate different types of input: the amplitude of the visibility function alone, the amplitude and the phase, and the same options accompanied by a mask that encodes which pixels contain a signal. Finally, the network is fed with different patterns of the uv-plane coverage: a) only a central part of the plane is covered (which mimics the case of EHT), b) in addition to the central part, there is a covered ring a few times bigger than the center (which mimics the case of future space interferometry projects with antennas in high orbits, <cit.>), and c) uniform coverage (reminiscent of the setting described in <cit.>). We train four networks that differ in their inputs on the three coverage patterns each. Regarding the degree of coverage, training, validation and test sets include all levels of coverage (in a certain range), which implies that the networks were trained to do a prediction with an arbitrary number of activated pixels (if they follow one of the three patterns).Since the accessible region of the uv-plane in pattern (c) is larger than that of patterns (b) and, especially, (a), we will present our final results in terms of absolute coverage. For patterns (a) and (b), we define it as the number of activated pixels in the central part of a Fourier image divided by the total number of pixels in pattern (c). For pattern (c) we define the coverage as the total number of activated pixels by the size of an image. At the same time, we will be presenting our intermediate results (Figs. <ref>– <ref>) in terms of relative coverage for pattens (a) and (b), where we define it w.r.t. the size of the central partand assume that the area of the latter constitutes 2.5% of the total area accessible in pattern (c). We describe this point in more details in Subsect. <ref>.The structure of the Note is as follows. In the next section we describe our method while Sect. <ref> contains comparative results for the cases described. There we alsodiscuss the prospects of improving the network to suit the real-observation needs. § METHOD The general problem of fitting data is finding an approximate mapping between observation(s) 𝐱 (e.g. the frequency at which the black-body radiation peaks) and an inferred value(s) 𝐲 (the black-body temperature), 𝐱→𝐲. To this effect, a hypothetical mapping which depends on parameters 𝐰 is introduced (in the simplest case of the linear hypothesis, there are only 2 parameters). Then, the parameters 𝐰 are adjusted so that the cumulative error (for example, the sum of squared deviations between the values predicted by the hypothesis and the actual values) is minimal.Neural networks are a wide class of algorithms used to implement hightly nonlinear mappings <cit.>. For instance, 𝐱 could be an image represented as a matrix of pixels and 𝐲, say, the type of object in that image represented as a vector of logical ones and zeros (e.g. the 1st component encodes object “human” and will be equal to one if there is actually a human in the image and zero otherwise). Typically, a neural network is organized in a sequence of layers where each layer has its own parameters 𝐰_1, 𝐰_2, … . The forward pass that maps 𝐱 (input layer) to 𝐲 (output layer) starts from composition 𝐰_1∘𝐱 (most often, it is the matrix multiplication) being fed to the so-called activation function 𝐠_2 of the 2nd layer. The composition of the result with parameters 𝐰_2, 𝐰_2∘𝐠_2(𝐰_1∘𝐱), is fed to the activation function of the 3rd layer, and so on up to the last layer whose output is 𝐲. The activation functions are required to be non-linear. The universal approximation theorem <cit.> states that such a network with a single hidden layer can approximate continuous functions arbitrary well, provided that the number of neurons in the layer (the dimension of 𝐰_1∘𝐱) is sufficient.The process of adjusting weights of a neural network is known as training, which is achieved through minimizing the error of the predicted values 𝐲_ pred on a training set {𝐱_ train, 𝐲_ train} (for example, with the gradient-descent algorithm). The performance of the network during training is monitored by its error on a separate validation set {𝐱_ val, 𝐲_ val}. The final error is evaluated on a test set unavailable to the network during the training and measures the ability of the network to generalize to new examples. Our aim was to develop a neural network that would map a Fourier image (𝐱) of a black-hole silhouette with the visibility function given only in a subset of pixels to the angle (𝐲) between the black hole's spin and the line of sight of a distant observer. The description of the simulated dataset and the network's architecture is following. §.§ Simulated dataset The dataset is obtained from real-space images which undergo a series of transformations resulting into a mock uv-plane with partial coverage. The real-space images are those of the silhouette of a Kerr black hole surrounded by a geometrically thin and optically thick accretion disk. We simulate the silhouette by tracing 256× 256 rays and then reduce the image to 128× 128 pixels by averaging over adjacent 2× 2 squares. We vary neither the distance to the black hole nor its mass, which fixes the angular scale of the image. In particular, the horizontal and vertical linear/angular scale L is fixed as follows:L = 200/3GM/c^2 ,where M is the black hole's mass, G is the gravitational constant, and c, the speed of light (hereafter, we set G=c=1). This results into the following relative linear and angular scales (Δ L and Δα, respectively): Δ L ≈ 5×10^-3(128/N)(M/10^6M_⊙) / , Δα = 0.5(r_0/10 )(128/N)(M/10^6M_⊙) μ/ ,where N is the 1D resolution of the real-space image and r_0, the distance to the black hole.The elements of the disk are assumed to follow circular geodesic orbits with the inner radius of 10M. The images are generated for a range of values of parameters which are the disk's outer radius r_ out, Kerr rotation parameter a, and the inclination angle θ. Also, the disk is chosen to be co- or counterrotating with probability 1/2. The ranges of the parameters as well as their increments are given in Table <ref>. There are 2,225 combinations of the parameters to which we add 76 images obtained from a trial simulation [Those are approximately evenly distributed among angles 1^∘, 23^∘, 45^∘, 67^∘, 89 and generated for all the combinations of the outer radius and rotation parameter. They differ in the disk's sense of rotation.]. Thus, the total number of the images in training and validation sets before the transformations is 2,301. The training set batches are generated on the fly by applying random transformations (for more details on data augmentation, see below). A new batch is generated at each step of the training process. Each image in the batch is transformed randomly. The validation set batch is generated in the same way only once at the start of training and is used at every step. A test set that will be used to report the final results comprises 89 pre-transformation images with angles ranging from 1^∘ to 89^∘ in increments of 1^∘ and the other parameters chosen randomly from their respective ranges (in a continuous manner), see Table <ref>.Then, we perform data augmentation on these images by applying translation, rotation and blur. The data augmentation is combined with a Fourier transform in the following order: rotation → Fouier transform → translation/blur. We carry out the translation and Gaussian blur in the uv-plane (see <ref>) in order to avoid edge artifacts. This is especially convenient, because the input of the CNN is Fourier-transformed images.The translations are by a (uniformly) random vector with the x- and y-components between -15 and 15 pixels. The rotations are by an angle uniformly distributed between -180^∘ and 180^∘. Finally, we apply a Gaussian blur (smoothing) with a sigma drawn from a uniform distribution, σ∼ U(0,3√(3)M/Δ L) = U(0,10.0). The maximum blur is chosen to be equal to the universal size of the shadow of a Schwarzschild black hole <cit.>. Fig. <ref> shows a silhouette of a Kerr black hole seen at θ=78^∘ and its version distorted by a translation, rotation, and a blur (to obtain the image, an inverse Fourier transform was applied after translating and blurring in the Fourier domain). Fig. <ref> shows the original amplitude and phase of the image of Fig. <ref> (right panel). Note that, at the programming level, both real-space image and its Fourier transform are scale-free and their sizes are in pixels (128× 128 for the real-space image and 64× 64 for the Fourier). The uv-image can be provided with a physical scale as follows:Δν [Gλ/pixel] = N-1/k_ padN^236× 18/π Δα [μas/pixel] ,where Δα and N are, respectively, the physical scale and resolution of a real-space image, and k_ pad is the zero-padding factor of the discrete Fourier transform used to make frequency bins narrower <cit.>. In this paper,k_ pad=2, N=128. Fig. <ref> shows two examples of physical scales on the lower and upper axes: 1 and 0.5 μ as/pixel, respectively. These correspondingly result in ≈0.8 and 1.6 Gλ/pixel in Fig. <ref>. The lower scale of Fig. <ref> is approximately equal to the scale of images obtained with EHT (cf. Fig. 3 in <cit.> and <cit.>).Finally, the Fourier amplitude as well as the phase are overlaid with a mask that mimics the partial coverage of the uv-plane in observations. The mask is a 64× 64 matrix of ones and zeros, where the ones show which pixels are covered while the zeros encode the absence of observational signal. We use three types of masks which simulate different observational settings: a) only the central part of the Fourier image is covered, b) the coverage comprises the central part and a ring which is a few times bigger, and c) the coverage is more or less uniform over the uv-plane. We refer to these three cases as , , and , respectively. Figs. <ref> and <ref> illustrate the three. To be able to compare the performance of the networks we describe in Subsect. <ref>, we adopt the following convention for counting the coverage. For patternsandwe require that the same fraction of the central part be covered and we refer to it as relative coverage. The area of the part of the uv-plane we call central is, by definition, 40 times smaller than the total area. On the other hand, the absolute coverage for those patterns is the ratio of the number of pixels activated in the central area to the whole size of the uv-plane (64× 64 pixels in this work). For thepattern we choose not to include the pixels on the periphery, because we want to single out the effect of arcs when we will be evaluating the error of the networks. The relative coverage may be more illustrative in that it changes in a wider range while the absolute coverage of thepattern cannot exceed 2.5% (the maximum size of the central part). It is one more reason not to include the pixels on the periphery of the , because there are about 10 times more of those pixels and their inclusion would lead to a very narrow interval on our final graph. In what follows we use the relative coverage in Figs. <ref>– <ref> and the absolute coverage in final Fig. <ref>. Regarding the masks themselves, they consist of pairs of elliptic arcs symmetric w.r.t. the origin. In theandpatterns as well as the central part of thepattern, the radii [Hereafter, “radius” in relation to an elliptical arc means the geometric mean of its semi-major and semi-minor axes.] of the arcs are uniformly distributed between zero and a maximum value. The maximum value is 64√(2)/2 forand √(1/40)≈ 0.16 of that value forand the central part of . The angular sizes of the arcs are distributed normally with a mean of 2π/3 and a standard deviation of π/6. The arcs' eccentricity follows the uniform distribution U(0.1,0.9). For thepattern, the radii of arcs on the periphery are distributed ∼𝒩(32,0.64).If we adopt the physical scale of the lower u-axis of Fig. <ref>, thecase roughly corresponds to the characteristic baselines of EHT (cf. Fig. 2 in <cit.>) while thepattern mimics the prospective enhanced configurations of EHT with one or more small dishes in Low Earth Orbits (cf. Fig. 4 (third column) of <cit.>). Thecase is in turn characteristic of space-VLBI configurations that include dishes in higher orbits, e.g. in geosynchronous or medium Earth orbits <cit.>, or in the Sun–Earth L2 point <cit.>. Note that these coverage masks and arcs therein are not identical to those simulated in the above-mentioned prospective space-VLBI experiments. The correspondence is rather qualitative.The dataset consisting of such masked Fourier images is then used to train a convolutional neural network. §.§ Convolutional neural network A neural network that includes convolutional layers is known as convolutional neural network (CNN). In a convolutional layer, its input (an image) is “scanned” by many, typically, 3× 3 filters (kernels), with the output being the result of convolutions of the kernels with the respective parts of the image. As mentioned, such an architecture has proven to be extremely efficient in image recognition problems.Table <ref> shows the full architecture of the CNN which we have developed. The code, the trained models' weights, and links to the training and test datasets are publicly available [<https://bitbucket.org/cosmoVlad/neuro-repo>].We compared four versions of this network which differed in their inputs. As one option, we turned on or off the mask, that is, either the mask was fed to the network as a separate input or not. These two cases are denoted by M+ and M-, respectively. In both cases the values of pixels that were out of coverage were set to zero. And the purpose of the mask as an extra input was an attempt to train the network to ignore the masked zero values and distinguish them from those that are part of the actual Fourier signal. For each of those cases, we also pass either only amplitudes of the Fourier images or phases as well. These cases are denoted by Ph+ and Ph-. To account for the periodicity of the phase, we passed its sine and cosine rather than the phase itself.The four versions of the CNN were trained for 100 epochs with batches comprising 64 images and validated on a dataset of 2048 images. As a loss function, we use the mean squared error (MSE),1/∑_k=1^(θ_ pred^(k)- θ^(k))^2Recall that each batch is generated on the fly by randomly choosing the respective number of pre-transformed images, applying transformations (rotation→ Fourier transform → translation/blur) with random parameters (except for Fourier transform), and overlaying a mask with a degree of relative coverage randomly and uniformly chosen between 0.1 and 0.9. A set of the random parameters is new each time an image is generated. Recall that, in theandcases, the relative coverage is the number of ones in the mask divided by the number of pixels in a central part of the image. The linear size of the central part is a free parameter, which was set to √(1/40)≈ 0.16 (≈ 10 pixels) in this work. Thecase is different in that there are a few arcs added on the periphery of the image. The process of generating arcs is described in Subsect. <ref>, and the number of arcs is also a free parameter, which was set to 6 in this work. The absolute coverage of a specificpattern is obtained by dividing the number of activated pixels in the central part by the total number of pixels (that is, by 64× 64). In thecase the relative coverage coincides with the absolute one.The degree of coverage of a single image was chosen randomly between 0.1 and 0.9 from a uniform distribution. The degree of coverage is defined as follows for different patterns. In thecase, it is the number of ones in the mask divided by the total number of pixels (that is, by 64× 64). In theandcases, it is the number of ones divided by the number of pixels in a central part of the image. The linear size of the central part is a free parameter, which was set to 0.16 (≈ 10 pixels) in this work. Thecase is different in that there are a few arcs added on the periphery of the image. Note that our CNN also contains dropout layers to prevent overfitting. However, we tried a few dropout rates between 0 and 0.1 and did not find any overfitting trend as dropout rate decreased. Fig. <ref> shows a typical learning curve with zero dropout rate. These dropout layers may become useful when estimating confidence intervals with a technique described by <cit.> (to be done elsewhere). § RESULTS AND DISCUSSION The efficiency of the four versions of the network is summarized in Figs. <ref>–<ref> and Fig. <ref>. Recall that these versions result from passing or not the mask and/or the phase as additional inputs to the network and are denoted as M-Ph-, M+Ph-, M-Ph+, M+Ph+.The series of figures <ref>–<ref> shows distributions of the discrepancy between a true angle and the answer given by the network. These distributions were evaluated on a dataset (test set) of 512 images, and each row represents the error evaluated on images of a different coverage pattern. The left panels are graphs Predicted angle vs. True angle, and the cumulative histograms of the deviations are on the right panels. The two colors represent the error distribution at two different degrees of relative coverage, 15% and 60%. In the legend we indicate the 68% quantile interval around the median of a historgram. For comparison we also draw best-fitting Gaussians, although the statistics of the deviations is not Gaussian as it becomes evident if one plots the Q–Q plot or runs a normality test, e.g. <cit.>. One reason not to expect the deviations to be Gaussian is that the angle cannot be negative by definition. This is manifested in how some two of the networks overestimate the angle at small values (note the elevated bottom left corner of the plot on the left panels of Figs. <ref> and <ref>). Also, in thecase all the versions of the network tend to underestimate angles that are close to the right angle. We defer the investigation of the statistical properties of such a network to future research.We have also tested the performance of networks M-Ph- and M-Ph+ on images with the maximal Gaussian blur (recall that the sigma of the blur is close to the universal size of the shadow of a Schwarzschild black hole and, thus, mimics a source nearing being unresolved; see also Sect. <ref>). We have found that this significant blur does not affect the M-Ph+ network. In the case of M-Ph-, however, the network's performance worsens for all the three coverage patterns, with the last having the same error of about ± 15^∘ (at 60% coverage). Such behavior is not unexpected, because the blur only affects the magnitude of the visibility function (see also <ref>). One interpretation of the same performance of M-Ph+ is that the network has learned to determine the angle mostly from the phase which is unaffected by smoothing. Meanwhile, the blur effectively cuts the large harmonics of the Fourier amplitude which makes covering anything other than the center of the Fourier image useless. This explains why the error of M-Ph- becomes independent of the coverage pattern for large blurs.Interestingly, the statistics of deviations of viewing angle in Deep Horizon <cit.> shows a similar pattern of overestimation at smaller angles and underestimation at larger ones, even though, in that paper, the very range of angles is restricted to [15^∘, 25^∘]. The effect is most apparent with larger Gaussian beam. If we adopt the physical scale of ∼ 1 μ/ on Fig. <ref>, our maximum blur corresponds to a Gaussian beam of ≈ 10μ. Since we evaluate the error on a dataset with randomly generated individual blurs, we can take a blur of ∼ 5μ as an estimate of average blur in the dataset. Comparing the respective columns of <cit.> we see that the error of our best M-Ph+ network is about twice as high at the higher coverage and with the smaller Gaussian beam and is 1–1.5 times as highwith the larger Gaussian beam. This network of ours, however, does not suffer from overestimation at these degrees of coverage and comprises the entire range of angles rather than [15^∘, 25^∘]. Our network is also more universal in that it is trained on a set of degrees of coverage. It is hard to compare it directly to Deep Horizon, because the latter was trained directly on real-space images, which does not seem to take into account the deconvolution process from a uv-image with particular coverage. In addition, the M-Ph+ network is basically not sensitive to blur as we explained above.Figs. <ref>–<ref> and Fig. <ref> lead to the following conclusions. First, as the degree of coverage for a given pattern increases, the standard error decreases, which one might naturally expect. Second, in terms of relative coverage thepattern leads to a lower error than the other two patterns. However, in terms of absolute coverage, the error ofandis definitely smaller than that of . At the same time, although thepatterns outperform thepattern at low degrees of absolute coverage, they suffer from overestimation at lower angles. The uniform pattern cures this problem in almost all the cases considered (except for the M-Ph- network). Third, the introduction of input phase alone improves the quality of the networks on theandpatterns. It reduces the error and improves the statistical quality of error distribution making less skewed (no underestimation at lower angles). The introduction of input mask alone appears to be beneficial, too, but mostly for the statistical quality (the skewness is reduced). Somewhat surprisingly, the network combining phase and mask performs worse than M-Ph+. All in all, the M-Ph+ demonstrate the best results from the point of view of both error magnitude and statistical quality. Fig. <ref> shows how the standard error depends on the degree of absolute coverage for all the three patterns. The black dots show the errors evaluated on a dataset (test set) of 512 images while the lines (dotted, dashed, and solid) are graphs of fitting polynomials (note that the plot is semi-log). The polynomials are linear forandand either quadratic or piecewise linear-quadratic for .These graphs illustrate a few tendencies. First, again, the standard error decreases as the degree of coverage increases. Second, at the same level of (small) absolute coverage, thecases demonstrate better results than theone. Third,with and withoutproduce approximately the same error if there is no input phase. Otherwise, for the Ph+ networks,the addition of the borders decreases the error approximately twice at the lowest coverages. At the maximum degree of absolute coverage for these cases (2.5%), the errors are the same. Finally, the dependence Error vs. Coverage shows an interesting feature for the networks with input phase andpattern – starting from a coverage of 5% the error drops faster than at degrees of coverage <5%. For that reason, we choose a piecewise polynomial function to fit the results in those cases.These results indicate that, if the inclination angle is not too small, in order to determine it with these networks within, for example, 10^∘, the use of Earth-sized baselines is sufficient. For the same level of (low) absolute coverage, the error can be improved by adding long space baselines (this would also add more phase closure conditions and, thus, more information on Fourier phase). For small angles (face-on orientation of the disk), space configuration with preferably uniform coverage should be used. Using the polynomial fits, we find typical values of 0.8% ( and ) and 25% () leading to an error of 10^∘ for the networks without input phase. For the Ph+ networks, these values are 1%, 0.3%, and 14%.The fact that the angle can be determined from probing the central part of the uv-plane alone may be explained by the presence of a characteristic scale associated with each angle. If this is the case, one only needs to probe the first zero of the visibility function. Then, covering the central part is sufficient, provided that the inclination angle scale resides in it. The classical example here is the measurement of Betelgeuse's diameter by <cit.>. A black hole shadow can also be approximated by a 4-parametric crescent model which suggests a characteristic scale in the visibility function <cit.>. Such a scale may also explain why the addition of the phase does not significantly improve the performance of the networks on images with low blur. This is because the networks learn the scale already from the magnitude of the visibility function, and, since there are supposedly no other scales associated with the inclination angle, the same scale is present in the phase, which does not bring anything new. The phase could be useful, though, if the blur is significant as we saw above. Recall that the blur introduces a cut-off in the magnitude but leaves the phase intact. In this case the networks with input phase significantly outperfrom the ones without.To summarize, we developed a proof-of-concept convolutional neural network which infers the angle of inclination of a Kerr black hole from the partially covered uv-plane of the shadow of the black hole against a geometrically thin and optically thick accretion disk. We explicitly found how the network's error depends on the degree of coverage of the uv-plane and compared four different versions of the network on three different types of input. We showed that the best results in terms of both error and statistics are attained by the network which takes both amplitude and phase as inputs and operates on Fourier images with uniform coverage.Although the performance of this proof-of-concept network indicates that the enhanced configurations of EHT with dishes in Low Earth Orbits might be a better choice of future observations, this result needs to be elaborated on to be applicable to real observations. In future research we plan to develop a more sophisticated version of the network to be trained on more realisic images, such as resulting from one of the codes in <cit.>. Also, a separate work is required to study the statistical properties of the deviations, given that the over- and underestimation patterns we saw for some versions of the network are similar to those in Deep Horizon.§ ACKNOWLEDGEMENTS For funding information, see the journal version of this paper <cit.>.We are grateful to V.S. Beskin and Yu.Yu. Kovalev for providing a work environment which made completion of this paper possible. One of us (VNS) thanks A. Alakoz for educational remarks on radio interferometry and A. Radkevich, V. Kozin, and S. Repin for providing conditions favorable to research.We also made use of Keras, <cit.>, Theano <cit.>, TensorFlow <cit.>, IPython <cit.>, SciPy <cit.>,Matplotlib <cit.>, NumPy <cit.>.§ TRANSLATION AND GAUSSIAN BLUR IN THE FOURIER DOMAIN In order to translate a real-space image by Δ x and Δ y in the horizontal and/or vertical directions, respectively, one should add a correction to its Fourier phase <cit.>,Δ𝒱(i,j) = 2π/k_ padN[-u_iΔ x + v_jΔ y] , u_i ≡i - k_ padN/2 , v_j ≡j - k_ padN/2 ,where i,j = 0,1,2,…,(k_ padN-1), k_ pad is the padding factor, and N, the number of pixels along each dimension of the real-space image (see also notation following eq. (<ref>)). The choice of signs in the expression for phase is consistent with the details of the numerical implementation of the Fast Fourier Transform algorithm <cit.> and our assumptions that Δ x>0 and Δ y>0 imply translation to the right and upward. In the real space, Gaussian blur is a convolution of the Gaussian kernel with the image. By the convolution theorem <cit.>, it is pixel-wise multiplication in the Fourier domain:|𝒱_ blur(i,j)|=|𝒱(i,j)| × exp[-2(πσ/k_ padN)^2(u_i^2 + v_j^2)] ,where σ is the standard deviation of the Gaussian kernel in pixel units. | http://arxiv.org/abs/2311.16227v1 | {
"authors": [
"Anton A. Popov",
"Vladimir Strokov",
"Aleksey A. Surdyaev"
],
"categories": [
"astro-ph.HE",
"astro-ph.IM"
],
"primary_category": "astro-ph.HE",
"published": "20231127190001",
"title": "A proof-of-concept neural network for inferring parameters of a black hole from partial interferometric images of its shadow"
} |
a]Sue Limlabel1 [a]organization=Department of Communication, Michigan State University, addressline=404 Wilson Rd.,city=East Lansing, postcode=48824,state=MI, country=USA a]Ralf Schmälzle [label1]Corresponding Author. Email: [email protected] Advancements in artificial intelligence (AI) over the last decade demonstrate that machines can exhibit communicative behavior and influence how humans think, feel, and behave. In fact, the recent development of ChatGPT has shown that large language models (LLMs) can be leveraged to generate high-quality communication content at scale and across domains, suggesting that they will be increasingly used in practice. However, many questions remain about how knowing the source of the messages influences recipients’ evaluation of and preference for AI-generated messages compared to human-generated messages. This paper investigated this topic in the context of vaping prevention messaging. In Study 1, which was pre-registered, we examined the influence of source disclosure on people’s evaluation of AI-generated health prevention messages compared to human-generated messages. We found that source disclosure (i.e., labeling the source of a message as AI vs. human) significantly impacted the evaluation of the messages but did not significantly alter message rankings. In a follow-up study (Study 2), we examined how the influence of source disclosure may vary by the participants’ negative attitudes towards AI. We found a significant moderating effect of negative attitudes towards AI on message evaluation, but not for message selection. However, for those with moderate levels of negative attitudes towards AI, source disclosure decreased the preference for AI-generated messages. Overall, the results of this series of studies showed a slight bias against AI-generated messages once the source was disclosed, adding to the emerging area of study that lies at the intersection of AI and communication. Artificial Intelligence (AI) large language model (LLM) health communication source disclosure vaping prevention mixed effects modeling § INTRODUCTION “Imagine a world where persuasive content is crafted so masterfully that it becomes nearly indistinguishable from human creation, yet is generated by machines at the click of a button. This groundbreaking study unveils the potential of leveraging large language models (LLMs) to generate compelling messages, and puts it to the ultimate test: can they outperform human-crafted tweets in captivating the minds of their audience?" (Generated by GPT4 powered ChatGPT).Recent technological breakthroughs in neural network modeling have ushered in an era of artificial intelligence (AI), and new AI-based systems, such as OpenAI’s ChatGPT, are gaining rapid adoption. Within this context, the term AI generally refers to a field of study that aims to understand and build intelligent machines <cit.>. The precise and specific definition of intelligence differs based on the approach taken by the researchers, but a common theme is that machines can exhibit cognitive capacities such as intelligence, language, knowledge, and reasoning, which had traditionally been limited to human brains. AI technologies like ChatGPT, or similar systems (e.g., Google’s Bard, Meta’s Llama) are driven by large language models (LLMs), a specific kind of transformer-based neural networks trained on massive amounts of text. Importantly, these LLMs can not only process and categorize text, but they can also be used to generate text that mimics the flow of natural human language <cit.>. As the above content from ChatGPT shows, LLMs have advanced to the point where even with minimum instructions, they can generate high-quality creative and informative content. This has opened ample opportunities for health researchers and practitioners to leverage LLMs to augment their work. For instance, within health communication, researchers have found that messages generated by LLMs were clear and informative, and exhibited argument strength <cit.>. As LLMs continue to expand in these capabilities <cit.>, we can expect to see LLMs being used as tools for generating persuasive health messages. However, the rise of AI-generated content in the public communication environment raises the pressing question of how people react to AI as message creators.Though this is a relatively novel area of study, there are two relevant bodies of literature that we can draw from: interdisciplinary research about the general sentiment of hesitancy towards novel technologies and source effects research within communication research. It is well-documented that new technologies are often met with skepticism. Studies suggest a general sentiment of hesitancy <cit.> and mild to moderate aversion <cit.> towards AI and computer algorithms more broadly. Also, when told that AI was involved in the creation of communicative content, there was some reporting of preference against or lower evaluation of that content (e.g., Airbnb profile writing; <cit.>; email writing; <cit.>; generated paintings; <cit.>; music creation; <cit.>; translation of written content; <cit.>). Within health contexts especially, some studies show that people tend to prefer human practitioners over AI-based technologies like chatbots when receiving consultation about health conditions <cit.>, citing lack of personalization and incompetence in addressing individual needs as some of the reasons for hesitancy <cit.>.Second, source effects have been studied extensively in persuasion and communication. For instance, a plethora of literature has examined the influence of various aspects of the source, such as credibility, trustworthiness, and similarity, on people’s attitudes and behavior <cit.>. With the advancement of technology, research also examined source effects in online settings <cit.>. In addition, some of the most well-known theories within communication have examined cognitive mechanisms of source effects (ELM; <cit.>; HSM; <cit.>). Speaking broadly, the results from these studies show that people’s thoughts about the source of the message shape how they evaluate the communication content from the source. Since there’s already been evidence that LLMs have the potential to be powerful tools in expanding health communication theory and augmenting health campaign practice, it is thus important to investigate how people’s perception of AI influences people’s evaluation of health campaign messages. Moreover, it will also be critical to identify potential moderators of such influence. This paper presents two experimental studies that shed light on the influence of source disclosure on the evaluation of prevention messages (see Figure 1). For the first study (study 1), we conducted an experimental study examining how source disclosure influenced people’s evaluation of (in terms of effects perception) and preference for (in terms of ranking) prevention messages generated by a LLM compared to humans. Then a follow-up study (study 2) inspected how the influence of source disclosure varied on the basis of people’s general attitudes toward AI. The findings from our studies have the potential to augment source effects theory within mediated health communication by highlighting how people’s awareness of LLM’s role in message generation influences their evaluation of the messages.§ STUDY 1 The goal of our first study was to examine whether source disclosure influenced people’s evaluations of AI-generated messages as well as their preference for AI as the source of health information. We selected vaping prevention as a health context to examine the evaluation of messages coming from AI source [Going forward, one could also determine whether the specific health topic matters. For instance, based on psychometric models of risk perception <cit.>, one could predict that certain critical topics could be particularly prone to AI-source effects. However, we opted to start with a straightforward and widely applicable, current health topic that was also relevant for our participants.].§.§ Vaping Prevention as Context to Examine the Source Effects of AI The use of e-cigarettes (or vaping) has become a significant public health concern in the last decade, especially because of the high prevalence of e-cigarette use among youth (<18 years of age) and young adults (18-24 years of age). About 20% of high school and 5% of middle school students reported vaping in 2020 <cit.>; it was also estimated that about 15% of young adults were using e-cigarettes in 2020 <cit.>. Moreover, much of smoking and vaping-related marketing leverages the power of social media - or its capacity in disseminating information and ideas at a rapid speed through networks of people following one another <cit.> - to influence audiences and promote tobacco products <cit.>. To combat the detrimental effects of vaping, health researchers and professionals have invested significant efforts into developing and testing effective campaign messages <cit.>, leading to guidelines for best practices (e.g., Vaping Prevention Resource, 2023). These efforts could be further augmented by the capabilities of LLMs in generating effective health messages <cit.>.§.§ The Current Study and Hypotheses The current study examined how human participants respond to vaping prevention messages that were either generated by AI vs. humans by either adding accurate source labels to the messages (source disclosed) or not adding any labels (source not disclosed).§.§.§ Effects Perception Ratings as Measure of Evaluation Within health campaigns research, one of the most used message evaluation metrics is perceived message effectiveness (PME). According to <cit.>, the PME measure tends to cover two major constructs, message perceptions and effects perception. Message perceptions refer to the extent the messages seem credible and understandable, while effects perception refers to how the message promotes self-efficacy and behavioral intention. <cit.> developed an effects perception scale that focused on examining the extent the message does what it is intended. Existing research showed that effects perception was highly associated with health campaign outcomes such as risk beliefs, attitudes, and behavioral intentions <cit.>, meanwhile in some cases message perceptions did not have significant associations with these outcomes. Thus, we used effects perception ratings as people’s measure of the perceived effectiveness of the messages.Since the influence of source disclosure is a relatively new area of research, to our knowledge, only one study specifically examined how source disclosure would impact people’s ratings of health campaigns messages at the time of writing this manuscript. <cit.> conducted a set of three exploratory studies that used GPT3 to generate high-quality vaccination promotion messages. The third study, which manipulated source labels, found that prevention messages generated by GPT3 were rated higher in terms of perceived message effectiveness compared to those written by CDC when none of the messages were labeled. However, messages labeled as AI-generated were rated lower in terms of argument strength and perceived message effectiveness compared to those labeled as created by CDC or those not labeled at all. Our study had a few aspects that differed from <cit.> study. For one, our comparison of human-generated messages were tweets, to take into account that much discussion about vaping occurs via social media platforms such as Twitter <cit.>. Second, we used effects perception measure specifically (rather than the general perceived message effectiveness) as a measure of message evaluation. Still, as existing literature suggests the existence of negative bias against AI-generated content, we posed the following hypothesis:Hypothesis 1 (H1): People who know the source of the messages will rate AI-generated messages lower and human-generated tweets higher than those who did not know the source.§.§.§ Ranking as Measure of Preference In addition to effects perception ratings, rankings have also been used in existing research to gather information about preference. Unlike ratings, rankings ask participants to order the messages from the best to the worst, using whatever criteria provided by the researcher and/or determined by the participants <cit.>. Rankings have been used extensively in the social sciences to gather data about constructs such as values <cit.>, and attribute preferences <cit.>. Within health communication, ranking measurement was used to examine people’s preferences, including preferred health promotion icons <cit.> and factors that influence demand for vaccinations <cit.>. Though we do not know of any work that examined the influence of source disclosure on people’s ranking of AI-generated vs. human-generated messages, we still predict that the negative bias against AI-generated messages will be exhibited in the ranking of the messages. Thus, we pose the following hypothesis:H2: Those who know the source will prefer human-generated tweets vs. AI-generated prevention messages. §.§ Method We pre-registered our hypotheses and procedures at as.predicted.§.§.§ Participants A total of 151 young adults (18-24 years of age) were recruited from two study pools and either received course credit (University study pool) or $2.80 (Prolific; <cit.>) as compensation for participating in the study. We specifically selected the young adult age group because of the prevalence of vaping in this age demographic <cit.>. The local review board approved the study. We discarded the data from nine participants who did not complete the study or who completed the study in under five minutes, leaving 142 participants (m_age = 20.78, sd_age = 1.78]; 59% women) in the final dataset. Power calculations conducted a priori using the WebPower package in R <cit.> for a mixed ANOVA, with a medium effect size (f = 0.25) and significance level α = .05, showed that a total sample size of around 130 (about 65 per group) was enough to detect significant differences between groups at the power level of 0.8. §.§.§ Experimental Messages: Human- and AI-generatedWe relied on previously published procedures to generate messages via a LLM, collect human-generated messages, and select 30 total messages (15 AI, 15 human) for the experiment <cit.>. For details, see Appendix A. For the sake of relevance and length, we briefly outline the process here. To collect human-generated messages, we scraped vaping prevention tweets with hashtags #dontvape, #novaping, #quitvaping, #stopvaping, #vapingkills, and #vapingprevention using the snscrape package <cit.> in Python. After cleaning the tweets, we randomly selected 15 tweets that had been retweeted at least once for the experiment.For AI message generation and selection, we generated 500 total vaping prevention messages using the Bloom LLM, and then randomly selected a subset of 15 messages. Bloom is the largest open-source multilingual language model available <cit.>. As mentioned in previous sections, Bloom, like GPT3, is powered by the transformer neural network, the most advanced ANN system currently available <cit.>. Pre-trained with 1.5 TB of pre-processed text from 45 natural and 12 programming languages, Bloom allows for text generation using prompting (inputting the beginning part of the text and the language model completes the text) and a set of statistical parameters. We chose Bloom because of its free cost, full transparency of the training process and training data, and the ability to use it on a local machine via Jupiter notebooks or Google Colab without a special computing system called graphic processing unit (GPU), often required to run large computational tasks. §.§.§ Experimental Procedure and Conditions The experiment was conducted online via Qualtrics. Once participants consented to the study, the young adult participants were randomly assigned to one of two groups: control and treatment (n_control = 72, n_treatment = 70). Then the survey asked the participants to rate each message on four perceived message effectiveness items and rank the 30 messages (15 AI-generated vs. 15 tweets). The order of the two activities was randomized to control for order effects. The participants in the treatment condition read messages with source labels (e.g., “AI-Generated Message: Nicotine in vapes…”, “Human-Generated Tweet: Nicotine in vapes can…”) while those in the control condition were not provided the source labels. The source labels were true - no deception was used. Upon completing the main experiment, participants completed demographic questions and were debriefed about the study’s purpose.§.§.§ Measures Study 1 included two main measures. First, we adopted and updated UNC’s perceived message effects, otherwise named effects perceptions (EP), scale <cit.> to fit vaping. The measure included the following four survey items: “This message discourages me from wanting to vape,” “This message makes me concerned about the health effects of vaping,” “This message makes vaping seem unpleasant to me,” and “This message makes vaping seem less appealing to me.” Participants rated each item on a likert scale from 1 (Strongly disagree) to 5 (Strongly agree). Second, for the ranking activity, we asked participants to rank the 30 messages from the best (1) to the worst (30) message by dragging each message to its rank. Finally, the participants answered demographic questions including age. §.§.§ Data Analysis All analyses were conducted in R. To examine H1, the responses for the four items of the EP scale were averaged into a composite EP score for each participant; the last item about the appeal of vaping was excluded from the analysis to keep consistent with the results from <cit.>. Then we conducted a mixed ANOVA that examined the influence of source disclosure (disclosed vs. undisclosed) and the message source (AI vs. human) on EP.For the statistical difference in the mean ranks between the groups, we first subtracted the mean ranks for the human messages from the mean ranks of the AI messages (AI - Human). Thus, if the human-generated messages were on average ranked higher than AI-generated messages, then this difference value would be negative, and vice versa. Using the stats package <cit.>, we conducted the Wilcoxon Rank Sum Test, the non-parametric alternative to a two-sample ANOVA. We used the alpha level of α = .05 to test for significance for both mixed ANOVA and Wilcoxon Rank Test. In addition, we conducted a supplementary computational analysis. The purpose of this was to extract and compare various textual features of the AI-generated messages and human-generated tweets, showing that the two groups of messages could be adequately compared. The textual methods we used included semantic analysis, n-gram analysis, topic modeling, sentiment analysis, and assessment of readability metrics. These analyses were carried out using Python and R packages including spacy, textacy, vader, topicmodels, and the sentence-transformers <cit.>. For all computational analysis of tweets, we removed the hashtags used to scrape the tweets. We also removed the prompts from the AI-generated messages for all analyses except semantic analysis. See Appendix B for the results of the supplementary analysis.§.§.§ Deviation from Pre-registration While the main ideas from the pre-registration remained the same, we altered some of the details of the pre-registration. First, the pre-registration only included the data collection plan for the University sample. We decided to gather additional data from Prolific to make the results more generalizable beyond the University sample and to increase the sample size. Second, we decided to aggregate only the first three out of the four items for the EP measure to be more consistent with the existing literature <cit.>. Finally, for the rank data, we used the Wilcoxon test, which is a two-sample extension of the Kruskal-Wallis test. §.§ Results First, we present the results from the mixed ANOVA, which tested the influence of source disclosure on message ratings (see Table 1). We find that there was a significant interaction effect between source disclosure and the message source (F(1,140) = 4.73, η^2 = .0018, p = .031). As illustrated in Figure 2, this interaction was due to the fact the difference between the AI-generated and human-generated messages was smaller when the source was disclosed compared to when it was not disclosed. This interaction qualified a main effect of message source (F(1,140) = 10.25, η^2 = .0039, p = .0017), which indicated overall lower ratings forhuman-generated compared to AI-generated messages. Follow-up comparisons conducted separately for each message source (i.e. AI-generated and human-generated messages) revealedthat the EP ratings for AI-generated messages were slightly lower and ratings for human-generated messages were slightly higher when the source was disclosed, yet this difference was not statistically significant (t(133) = .82; p > .05 for AI-generated messages;t(125) = -.23; p > .05 for human-generated messages; see Table 2). Thus, H1 was partially supported.To test H2, we compared the difference in the mean ranks of AI and human-generated messages (AI mean rank - Human mean rank; see Figure 3) using the Wilcoxon Sum Rank Test.For the rank activity, the lower quantitative value represented a higher relative quality rank, with 1 representing the best message. Thus, the smaller differences in rank suggested a lower quantitative value for AI mean rank, hence a higher preference for AI-generated messages. We found that the median difference in rank for participants who knew the source, Mdn = -.6, was slightly higher than the median difference in rank for participants who did not know the source, Mdn= -.87, though this difference was not statistically significant (W = 2652.5, p > .05). §.§ Study 1 Discussion Study 1 examined how disclosing the source of a message as coming from an AI (vs. humans) influenced the evaluations of the messages and the preferences for the message source. Our H1 was partially supported – source disclosure significantly decreased the ratings difference between AI and human-generated messages. However, follow-up mean comparisons by message source showed that ratings stayed statistical consistent between non source disclosure and source disclosure conditions. This finding is generally aligned with findings from <cit.>. However, our H2, which addressed the ranking task that required participants to make an active selection to express their preferences about messages, was not supported. This could have occurred for many reasons, one of which is that ranking all 30 messages may have required too much cognitive effort. To further inspect source effects of AI-generated messages, we conducted a follow-up study (Study 2), examining individual differences that could boost or buffer the effects of source disclosure. For instance, participants could vary in their attitudes about the use of and general sentiment towards AI, which in turn could influence their judgments of AI-generated content. Thus, we examined attitudes towards AI as a potential factor in Study 2.§ STUDY 2 Study 2 replicated the source disclosure manipulation from Study 1 with a few modifications. First, we assessed people’s preference for messages via message selection (top 5 out of 30) rather than the ranking task to decrease the participants’ cognitive burden of comparing all 30 messages. Next, we examined how the influence of source disclosure on the evaluation and selection of AI-generated messages varied by the level of negative attitudes towards AI.§.§ Negative Attitudes Towards AI as Moderators <cit.> created a scale about general attitudes toward AI (GAAIS). A major part of the measure is based on the concept of trust in the capabilities and the uses of AI. The paper showed that GAAIS was associated with psychological features such as the Big Five personality, showing that it can be used to represent various individual differences that could exist when processing messages generated by AI. For example, <cit.> examined the association between attitudes towards AI and people’s judgments of art labeled as AI-created or human-created. In this study, we adopted the negative attitudes towards AI subscale, which included people’s concerns about and negative sentiment towards AI, as a moderator. Adopting the negative attitudes towards AI subscale of GAAIS, we posited the following hypotheses:H3: Negative attitude toward AI will moderate the influence of source disclosure on the evaluation of prevention messages.H4: Negative attitude toward AI will moderate the influence of source disclosure on the preference for AI as the message source. §.§ Method §.§.§ Participants As with study 1, we used two platforms to recruit participants, one administered by the university and the other by Prolific. A total of 216 adults recruited from the study pools either received course credit (University study pool) or $2.80 (Prolific; <cit.>) as compensation for participating in the study. To generalize the findings of Study 1 beyond young adults, we extended the participant pool for the Prolific platform to all adults . The local review board approved the study. We discarded the data from 33 participants who did not complete the study, completed the study in under five minutes, and failed to pass the manipulation check questions, leaving 183 participants (m_age = 33.83, sd_age = 14.42; 56% women) in the final dataset. §.§.§ Experimental Procedure, Measures, and Data Analysis The same 30 messages from Study 1 were tested in the main study. The experiment followed the same procedure as study 1 (n_control = 94, n_treatment = 89) with the following modification: instead of ranking the messages, we asked participants to select the 5 best messages from the pool instead of having them rank all messages. This was done because, in campaign practice, the best-in-show messages are chosen from a larger pool of candidates. Moreover, having participants and all messages is rather taxing and we expected better compliance with a more focused task. Upon completing the main experiment, participants answered background and demographics questions that included questions about their attitudes towards AI.The negative attitude toward AI scale asked people to rate 8 items related to negative attitudes (e.g., “I shiver with discomfort when I think about future uses of Artificial Intelligence”) from a scale of 1 (strongly disagree) to 5 (strongly agree) <cit.>. The overall mean was 3.04, with a standard deviation of .80. The demographics questions stayed the same as in study 1.All analyses were conducted in R. First, we calculated the average score for negative attitudes towards AI. To examine how the influence of source disclosure on EP of AI vs. human-generated messages differed by the extent of negative attitude (H3), we fitted a mixed effects linear regression model. The models allowed for the intercept to vary by participant, to take into consideration of the repeated measures design. To examine how the effect of source disclosure on source preference differed by negative attitude (H4), we first calculated how many AI-generated messages were selected (out of 3), and then fitted a Poisson regression model.§.§ Results For the EP ratings, we found that there was a significant three-way interaction of source disclosure, message source (AI vs. Human), and the extent of having a negative attitude toward AI (b = -.14, SE = .047, p = .0029; see Table 3). In other words, the influence of source disclosure on the evaluation of the AI-generated messages vs. human-generated messages differed by the level of negative attitudes towards AI. A deeper inspection of the moderation effect shows that for both AI-generated and human-generated messages, source disclosure led to slightly higher EP ratings among participants with lower levels of negative attitudes towards AI, whereas it led to slightly lower EP ratings among those with higher levels of negative attitudes towards AI (see Table 4). Interestingly, when the source was disclosed, the more negative attitudes the participants had towards AI, the higher they rated the AI-generated messages, whereas the ratings of human-generated messages generally stayed flat (see Figure 4). Table 5 shows the results for messages selection. There was no moderation effect of negative attitudes toward AI (b = -.042, SE = .12, p > .05), and H4 was not supported. A deeper inspection of the results showed that those who knew the source were likely to select less number of AI-generated messages compared to those who did not know the source for those with moderate level of negative attitudes towards AI (see Figure 5 and Table 6).§.§ Study 2 Discussion Study 2 examined whether negative attitudes towards AI moderated the influence of source disclosure on the evaluation of and preference for AI-generated vs. human-generated messages. For EP ratings, having a negative attitude toward AI emerged as a significant moderator, supporting H3. Specifically, at lower levels of negative attitudes towards AI, the ratings for AI-generated and human-generated messages were slightly higher when the source was disclosed vs. not disclosed (albeit not statistically significant for AI-generated messages), whereas the opposite was observed at higher levels of negative attitudes towards AI (not statistically significant). This result suggests the existence of a slight bias against AI-generated messages.However, for the participants who knew the source, the EP ratings for AI-generated messages compared to those for human-generated messages increased with the level of negative attitudes towards AI. While deeper inspection is needed to fully unpack this phenomenon, one explanation could be “source involvement”. In other words, the level of negative attitudes towards AI could have determined how closely they examined the messages: those with greater levels of negative attitudes towards AI could have paid closer attention to the content of the messages compared to those with less negative attitudes towards AI. Another explanation is that the negative attitudes towards AI measure could be a bit too general. In the case of AI message generation, people are heavily involved in the process. In the case of AI message generation, people are heavily involved in the process (see Appendix A for details of how the messages used for this study were crafted). Thus, it is possible that people’s general attitudes towards AI did not play as large of a role as we expected in their evaluation of the generated messages. For message selection, we did not find any significant moderating effects; thus, our H4 was not supported. However, source disclosure significantly decreased the number of AI-generated messages selected for those with moderate levels of negative attitudes towards AI. These results further provide support for people’s preference against AI-generated messages. We discuss the theoretical and practical implications of our findings in the next section.§ OVERALL DISCUSSION§.§ Summary of the Findings Overall, our two studies provide qualified support for our hypotheses. We found that disclosing the source led to lower ratings of AI-generated messages (partially supporting H1 in Study 1) and that negative attitudes toward AI moderated this effect (supporting H3 in Study 2).Though the analyses of ranking and message selection tasks did not support our hypotheses (H2 in Study 1 and H4 in Study 2), they revealed interesting effects: source disclosure decreased the number of AI-generated messages selected for those with moderate levels of negative attitudes towards AI. These results suggest a slight negative bias against AI-generated messages, aligning with previous studies that showed hesitation and slight negative bias against the communicative content when participants believed AI was involved in the process <cit.>.§.§ Implications for Source Effects Research This paper contributes to the emerging area of study at the intersection of communication and AI by being one of the first papers to examine how knowing the source changes people’s evaluation of and preference for AI-generated messages. The source of a message has always been an integral part of theories and models of communication and persuasion, even going back to Aristotle’s rhetoric theory <cit.>. Likewise, early models of social scientific communication research, such as Berlo’s SMCR model <cit.>, Lasswell’s model of communication <cit.>, and even the Shannon-Weaver model of communication <cit.> all included components about the source, or the creator and deliverer of the message. Since then, a plethora of studies have studied source effects, or how various characteristics of the source impact the way people receive, process, and subsequently make judgments about the message. These studies often manipulated certain aspects about the source (e.g., expert vs. nonexpert; <cit.>) and examined in which scenarios the various levels led to greater persuasive outcomes (e.g., when people had little information about a product, they relied on expert sources, but not necessarily when they had more information; <cit.>). With the rise of AI-based technologies such as LLMs, source effects have once again come to the forefront of communication research, but the notion of “source” for AI-generated messages is quite complex. In particular, the message generation process for LLMs generally consists of the following steps: First, a human user feeds prompts, or intentionally crafted instructions or beginning parts of the message, to the LLM; second, the user adjusts the parameters, such as how many messages should be crafted and other factors; third, the LLM generates the messages according to step 1 and 2. Thus, in this process, it is actually a human who initiates the message creation sequence, whereas the LLM only completes the message generation command. It seems plausible to assume that people’s knowledge of AI (and their perceptions of its expertise, trustworthiness, etc.), will impact their evaluations. For example, we may surmise that it would not only matter that a message was AI-generated, but also whom people believe to have started the process. In other words, if people think that the AI message generation was initiated by expert organizations, such as the Center for Disease Control (CDC), evaluation might differ compared to AI-generated messages initiated by general users of social media platforms, or even by agents from a foreign country. In sum, with AI, there is an intersection of source effects, perceptions of AI, and various social-cognitive inferences about creator, intent, and expertise. Going forward, it will thus be important to comprehensively study these topics. Based on the present results, we can say that there are small but significant effects of source disclosure, consistent with a small preferential treatment for human-generated messages.§.§ Implications for Public Health Campaigns The current results have interesting implications for research on health message generation and dissemination. With the advent of AI-language models, it has become extremely easy to generate high-quality health messages about any given topic. This potential can either be a blessing or a curse, depending on the source and their intent. For instance, if the CDC leveraged the power of AI for health message generation, this would be seen as largely beneficial; however, malicious actors could also leverage AI to spread fake news - or even just promote unhealthy products (e.g., cigarettes). Indeed, there are already commercial applications of AI-LLMs for copywriting purposes, and these could also be used to influence users towards unhealthy, risky, or other kinds of behaviors. Thus, more work is needed to explore how these aspects intersect with the topic of AI-as-message-source as well as the influence of source disclosure.A related concern is about the factual truthfulness of health-related claims. It is well known that although LLMs are capable of generating persuasive messages, they are prone to hallucinations <cit.>. Although the creators of AI systems are investing large efforts to minimize such false generations, this is still an unsolved problem of the underlying technology, which will affect the evaluation of AI systems <cit.>, particularly whether AIs are seen as knowledgeable, reliable, and trustworthy. In sum, while we can expect that AI-generated messages will increasingly find their way into real-world health campaigns, numerous questions persist about their accuracy and the intent of the humans generating the messages using AI. At this point in time, the dynamically evolving landscape of AI-language generation systems prevents any final answers to these questions. Rather, longitudinal research would be needed to assess how people think about AI sources, how they adapt to the increasing prevalence of AI content, and how their evaluations are influenced by contextual factors.§.§ Limitations, Future Avenues, and Ethical Considerations As with all research, several limitations that require future research and important ethical considerations are worth highlighting. One limitation is that this study used tweets as messages. It would be interesting to examine other kinds of health messages, such as longer flyers and posters. The decision to use tweets was made because we wanted to take into account user-generated messages and because tweets have become a rather widespread form of health communication content that also gets used by the CDC and other key health organizations. In addition, the topic of AI-generated health messages raises ethical questions. In particular, the regulatory framework around these topics is currently in flux, and discussions about mandatory labeling of AI-generated content have barely even begun. Furthermore, the allowed use cases for AI content generation are also debated. For instance, using AI to generate medical diagnoses is explicitly prohibited by the creators, but generating general health information falls within the range of acceptable use <cit.>.§ SUMMARY AND CONCLUSION Taken together, we examined the influence of source disclosure on evaluations of AI-generated messages. We found that source disclosure (i.e., labeling the source of a message as AI vs. human) significantly impacted the evaluation of the messages, albeit the effects were of relatively small magnitude, but did not significantly alter message rankings. Moreover, in study 2 we found a significant moderating effect of negative attitudes toward AI on message evaluation. Our results show that at the point when we conducted our research, humans appear to exhibit a small preference for human-generated content if they know the source, but AI-generated messages are evaluated as equally good, if not better, if the source stays unknown. These results highlight the role of source factors for communication, and they have implications for the potential labeling of AI-generated content in the context of health promotion efforts.§ FUNDING STATEMENT This work was supported in part through Michigan State University’s Institute for Cyber-Enabled Research Cloud Computing Fellowship, with computational resources and services provided by Information Technology Services and the Office of Research and Innovation at Michigan State University. The work was additionally supported by the Strosacker Grant from Michigan State University’s Health and Risk Communication Center. elsarticle-harv | http://arxiv.org/abs/2311.15544v2 | {
"authors": [
"Sue Lim",
"Ralf Schmälzle"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20231127052047",
"title": "The effect of source disclosure on evaluation of AI-generated messages: A two-part study"
} |
Interactive Autonomous Navigation with Internal State Inference and Interactivity EstimationJiachen Li, David Isele, Kanghoon Lee, Jinkyoo Park, Kikuo Fujimura, and Mykel J. Kochenderfer J. Li, J. Park, and M. J. Kochenderfer are with the Stanford Intelligent Systems Laboratory (SISL), Stanford University, CA, USA. K. Lee and J. Park are with the Korea Advanced Institute of Science and Technology, Korea. {jiachen_li,jkpark11,mykel}@stanford.edu; [email protected]. D. Isele and K. Fujimura are with the Honda Research Institute USA, CA, USA {disele, kfujimura}@honda-ri.com. ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Deep reinforcement learning (DRL) provides a promising way for intelligent agents (e.g., autonomous vehicles) to learn to navigate complex scenarios.However, DRL with neural networks as function approximators is typically considered a black box with little explainability and often suffers from suboptimal performance, especially for autonomous navigation in highly interactive multi-agent environments. To address these issues, we propose three auxiliary tasks with spatio-temporal relational reasoning and integrate them into the standard DRL framework, which improves the decision making performance and provides explainable intermediate indicators. We propose to explicitly infer the internal states (i.e., traits and intentions) of surrounding agents (e.g., human drivers) as well as to predict their future trajectories in the situations with and without the ego agent through counterfactual reasoning. These auxiliary tasks provide additional supervision signals to infer the behavior patterns of other interactive agents. Multiple variants of framework integration strategies are compared. We also employ a spatio-temporal graph neural network to encode relations between dynamic entities, which enhances both internal state inference and decision making of the ego agent. Moreover, we propose an interactivity estimation mechanism based on the difference between predicted trajectories in these two situations, which indicates the degree of influence of the ego agent on other agents. To validate the proposed method, we design an intersection driving simulator based on the Intelligent Intersection Driver Model (IIDM) that simulates vehicles and pedestrians.Our approach achieves robust and state-of-the-art performance in terms of standard evaluation metrics and provides explainable intermediate indicators (i.e., internal states, and interactivity scores) for decision making. Reinforcement learning, autonomous driving, sequential decision making, trajectory prediction, graph neural network, social interactions, internal state, counterfactual reasoning, traffic simulation § INTRODUCTION Controlling autonomous vehicles in urban traffic scenarios (e.g., intersections) is a challenging sequential decision making problem that must consider the complex interactions among heterogeneous traffic participants (e.g., human-driven vehicles, pedestrians) in dynamic environments. Consider a partially controlled intersection where an autonomous vehicle tries to turn left from a lane with a stop sign and the crossing traffic does not stop except when there are crossing pedestrians ahead (see Fig. <ref>).On one hand, autonomous vehicles should not ignore the oncoming/crossing traffic and turn aggressively, which may lead to a collision.On the other hand, it should not be overly conservative, which can hurt efficiency. In such a scenario, human drivers can reason about the relations between interactive entities, recognize other agents' intentions, and infer how their actions will affect the behavior of others on the road, allowing them to negotiate right of way and drive safely and efficiently.Human drivers are internally heterogeneous in terms of both trait and intention <cit.>. Conservative drivers tend to yield to other traffic participants during interactions, keep a larger distance from their leading vehicles, and maintain a lower desired speed; aggressive drivers are the opposite. To increase driving efficiency while maintaining safety, autonomous vehicles need to accurately infer the internal states of others, including traits (i.e., conservative/aggressive) and intentions (i.e., yield/not yield). Besides these high-level cues, accurate opponent modeling in the form of multi-agent trajectory prediction provides additional cues for safe and efficient decision making. Ma et al. <cit.> incorporated trait inference as an auxiliary task into a reinforcement learning framework and achieved better performance than standard reinforcement learning. In this work, we use additional supervision from driver intention recognition and trajectory prediction to enhance performance. We validate the hypothesis that modeling human internal states explicitly improves the decision making performance and the inferred internal states can serve as explainable indicators.To negotiate with other traffic participants, autonomous vehicles must infer to what extent they can influence the behavior of others. Not all the agents in the scene have strong interactions with the ego vehicle. In Fig. <ref>, the ego vehicle only needs to negotiate with the red vehicle in the upper right corner which has conflicts in their future paths. Although the other two green vehicles also approach the intersection, they need to yield to the crossing pedestrians and thus will not influence the ego agent's actions in the short-term future, and vice versa.Existing approaches usually adopt soft attention mechanisms to learn the importance weights of different agents <cit.>. However, these techniques may assign small weights to important objects or large weights to irrelevant ones <cit.>, which can mislead the decision making of autonomous vehicles, especially in dense scenarios.In this paper, we propose a mechanism to estimate the interactivity between the ego vehicle and each surrounding agent by using the difference between the predicted trajectory distributions of the agent under the situations with and without the existence of the ego vehicle as a quantitative indicator.This difference can also be treated as a quantitative degree of influence that the ego vehicle can have on a certain agent, which is called “interactivity score”. The interactivity scores are also used to weigh the prediction errors in the loss function, which encourages the model to generate more accurate trajectories for the agents that may have stronger interactions with the ego vehicle. Note that the ego vehicle always exists in both training and testing environments, thus predicting the future behaviors of other agents without the existence of the ego vehicle is a counterfactual reasoning problem. We propose to use a prediction model pre-trained with the trajectory data collected in the environments without the ego vehicle to generate counterfactual predictions in the formal training process. The weights of this prediction model are fixed without further updates.Autonomous vehicles need the ability to understand and reason about the interactions between dynamic entities by modeling their spatio-temporal relations. It is natural to represent a multi-agent system as a graph, where node attributes encode the information of agents and edge attributes encode their relations or interactions. Recently, graph neural networks have been widely adopted to capture relational features and model interactions between multiple entities <cit.>. In this work, we employ a spatio-temporal graph neural network as the basis model for spatio-temporal relational reasoning.Among sequential decision making approaches, deep reinforcement learning has been widely studied for autonomous navigation in complex scenarios due to its high capability of learning flexible representations and policies <cit.>. Despite its promising performance, the explainability of these methods still remains underexplored, which is crucial for safety-critical applications <cit.>. Some approaches learn an attention map on the perceived images to indicate the salient areas, which mimics the gaze of human drivers <cit.>. However, the learned attention map can be misleading because soft attention may assign unreasonable weights <cit.>.Also, the learned attention weights are associated with image pixels instead of agents, which makes it difficult to provide explanations at the object level. Other methods apply soft attention mechanisms to object-level entities instead, yet this limitation still applies. In this work, we enhance the transparency of the decision making process with both internal state inference and interactivity estimation, which provides intermediate auxiliary features to indicate how the model infers and reasons about the agents' interactive behaviors.The main contributions of this paper are as follows: * We propose a deep reinforcement learning approach for interactive autonomous navigation with three auxiliary tasks: internal state inference, trajectory prediction, and interactivity estimation. Multiple variants of framework architectures are compared empirically. * The auxiliary tasks not only improve the decision making performance but also enhance the transparency of the proposed framework by inferring explainable intermediate features of surrounding agents. In particular, we propose an explainable technique to estimate interactivity scores based on the ego agent's degree of influence on surrounding agents through counterfactual reasoning.* We design a four-way partially controlled intersection environment that simulates challenging traffic scenarios with interactive vehicles and crossing pedestrians, which is used to validate our approach and can serve as a novel benchmark for future research. * Our approach demonstrates superior performance compared to baselines in terms of completion rate, collision rate, driving efficiency in a complex intersection as well as stronger robustness to out-of-distribution scenarios. This paper builds upon our previous work <cit.> in several important ways. First, the internal state in our earlier work only contains the traits of human drivers while we additionally infer their intentions in this work to model different internal aspects and randomness in human behaviors. Second, we provide a more comprehensive discussion and comparison between different variants of framework architectures to incorporate the internal state inference. Third, we propose two additional auxiliary tasks (i.e., trajectory prediction, and interactivity estimation) into the reinforcement learning framework, which improves the decision making performance and enhances the transparency of the proposed framework. Finally, we design a more challenging intersection driving simulator with crossing pedestrians based on IIDM to validate our approach.The remainder of the paper is organized as follows.Section II provides a concise summary of the related work. Section III introduces basic background knowledge related to the proposed approach.Section IV introduces the intersection driving simulation based on the IIDM, which is adopted as the training and testing environment in our experiments. Section V presents the problem formulation for the autonomous navigation task. Sections VI and VII introduce the details of the proposed method. Section VIII presents the experimental settings, quantitative and qualitative results, and analysis. Finally, Section IX concludes the paper and discusses the impacts and potential limitations of our method. § RELATED WORK§.§ Autonomous Vehicle NavigationDecision making and motion planning for autonomous vehicles have been widely studied <cit.>. Earlier approaches use control theory, optimization, and classical artificial intelligence techniques to plan a future trajectory for autonomous vehicles <cit.>. These methods can effectively handle autonomous navigation in simple environments. Recently, many research efforts have been devoted to designing learning-based approaches for autonomous driving.To handle more complex environments with dynamic, interactive agents, some approaches adopt a game-theoretic perspective to model interactions <cit.>. However, it is difficult to extend these methods to large-scale interacting systems with many entities. Imitation learning can be used to train an autonomous vehicle to navigate in an end-to-end manner <cit.>. However, imitation learning struggles to generalize well to out-of-distribution scenarios, and it demands the collection of a large set of expert demonstrations.Deep reinforcement learning has been widely adopted to solve sequential decision making problems in modern intelligent systems <cit.> and can be applied to autonomous driving <cit.>. Some approaches take in raw sensor measurements (e.g., RGB images, point cloud) and outputs control commands (e.g., acceleration, steering angle) in an end-to-end manner, modeling the interactions between dynamic entities implicitly <cit.>.Although these approaches may achieve satisfactory performance, it is difficult to interpret these methods and understand the underlying interactions between interactive agents. Another category of approaches uses the low-dimensional state information (e.g., position, velocity) of agents provided by upstream perception modules, which put more emphasis on behavior modeling and multi-agent coordination interaction <cit.>. However, the information provided by the perception module could be noisy. Our method falls into this category, and we propose to learn auxiliary tasks that help reinforcement learning and generate meaningful intermediate indicators that enable explainable autonomous navigation. §.§ Interactive Decision MakingDepending on the scope of controlled agents, different formulations and algorithms can be employed.If all the agents are co-learning agents whose policies can be updated, then the game-theoretic formulation and multi-agent reinforcement learning can be employed <cit.>. These approaches generally model autonomous navigation as a non-cooperative game and use the Nash equilibrium and Stackelberg game theory to produce human-like behaviors.When we treat other agents as part of the environment reacting to the ego agent, single-agent-based control with opponent modeling can be employed to control the ego agent <cit.>. In these approaches, modeling the opponents' interactive behaviors is essential for deriving a tractable control policy.Without opponent modeling, the environment is non-stationary, which makes the learned policy for the ego vehicle unreliable.Our work falls into the second category and focuses on deriving the control policy of the ego vehicle in a reinforcement learning framework with opponent modeling as auxiliary supervised learning tasks to extract essential factors for decision making. §.§ Behavior Modeling and Trajectory PredictionBehavior modeling of other agents can be incorporated into the environment dynamics, transforming the multi-agent problem into a single-agent control problem <cit.>.Recent works apply latent factor modeling to extract stochastic internal factors of agents <cit.>.These studies use the extracted hidden representations as input for the control policy of the ego vehicle while interpreting them as the intentions, goals, or the strategy of other agents.Ma et al. <cit.> propose a framework to explicitly infer the internal state of agents and integrate the module into the reinforcement learning framework for driving policy learning. Wang et al. <cit.> propose a latent model of vehicle behaviors at highway on-ramps to produce interpretable behaviors.Xie et al. <cit.> propose a reinforcement learning framework with latent representation learning of other agents' policies. In this work, instead of only inferring high-level internal states (e.g., traits, intentions) of other agents, we also incorporate trajectory prediction explicitly into the decision making framework.Accurate trajectory prediction of other traffic participants in a multi-agent environment is an essential step for controlling the ego vehicle.Many methods have been proposed to create more expressive models to capture the inherent complexity of multi-agent behaviors while taking into account the latent goal/intention of interactive agents <cit.>.Recently, graph neural networks combined with latent variable modeling have been widely applied to predict the future trajectories of multiple agents while considering their latent relations <cit.>.These works only focus on improving the trajectory prediction accuracy without validating its actual effectiveness in downstream tasks. In this work, we integrate prediction into the decision making framework and demonstrate its effectiveness.§.§ Counterfactual Reasoning Humans often create counterfactual alternatives to reality to answer“what if” questions by thinking about how things could have turned out differently if they make a different action <cit.>. In a multi-agent setting, counterfactual reasoning is often adopted to facilitate social interactions.Jaques et al. <cit.> propose to use social influence as an intrinsic reward to encourage cooperative agents to learn to actively influence the other agents' policies to obtain a larger expected return.Tolstaya et al. <cit.> and Khandelwal et al. <cit.> propose a conditional behavior prediction method by forecasting the future behavior of a target agent conditioned on a counterfactual future behavior of the query agent. In this work, we propose counterfactual prediction by removing the ego agent from the scene and calculating the difference in the future behavior of a target agent, which is used to estimate the interactivity between the ego agent and the target agent for ego decision making.§ PRELIMINARIES§.§ Partially Observable Markov Decision Process (POMDP) A Markov decision process (MDP) is typically used to describe a discrete-time stochastic sequential decision making process where an agent interacts with the environment. Formally, an MDP is specified by the tuple (S, A, T, R, γ, ρ_0) where S and A denote the state and action space, T denotes the transition model, R denotes the reward, γ∈ [0,1] denotes the discount factor, and ρ_0 denotes the initial state distribution. A partially observable Markov decision process (POMDP) is a generalization of a MDP, where the agent cannot directly observe the complete state. An additional observation function Ω is needed to map a state s ∈ S to an observation o ∈ O where O denotes the observation space. Formally, a POMDP is specified by the tuple (S, A, T, R, Ω, O, γ, ρ_0). Unlike the policy function in MDP which maps states to actions, the policy of a POMDP maps the historical observations (or belief states) to actions. The objective is to find a policy π that maximizes the expected returnπ^* = max_π𝔼_s_0, a_0, o_0,...∑_t=0^∞γ^t R(s_t, a_t),where s_0 ∼ρ_0(s_0), a_t ∼π(a_t | o_1:t), o_t ∼Ω(o_t | s_t), s_t+1∼ T(s_t+1| s_t, a_t), and t denotes the index of time steps. §.§ Policy Optimization Policy gradient methods are widely used to learn optimal policies by optimizing the policy parameters directly <cit.>.The traditional REINFORCE algorithm <cit.> provides an unbiased gradient estimator with the objective L^PG(θ)=𝔼̂[logπ_θ (a | s)Â], where  is the estimated advantage. For a POMDP, we use the observation and the hidden state of the policy instead of the state s. Recently, PPO <cit.> has been a widely used policy optimization algorithm due to its simplicity and stable training performance, in which a clipped surrogate objective is maximizedL^PPO(θ) = 𝔼̂[min ( r(θ)Â, clip(r(θ), 1-ϵ, 1+ϵ)Â)],r(θ) = π_θ(a | s)/π_θ'(a | s),where θ' denotes the parameters of the old policy that is used to collect experiences, and ϵ denotes the clipping threshold. §.§ Graph Neural Networks The graph neural network is a class of deep learning models that can be applied to process the information on graph-structured data.A specific design of the message passing mechanism naturally incorporates certain relational inductive biases into the model. Most graphs are attributed (e.g., node attributes, edge attributes) in the context of graph neural networks.Battaglia et al. <cit.> provide a comprehensive introduction to graph neural networks. Generally, there are two basic GNN operations in graph representation learning: edge update and node update. More formally, we denote the graph with N nodes as 𝒢={𝒱,ℰ}, where 𝒱={v_i | i ∈{1,…,N}} is a set of node attributes and ℰ={e_ij| i,j ∈{1,…,N}} is a set of edge attributes. Then, the two update operations aree'_ij = ϕ^e(e_ij, v_i, v_j), e̅'_i = f^e→ v(E'_i), v'_i = ϕ^v(e̅'_i, v_i),where E'_i={e'_ij| j∈𝒩^i}, E'=⋃_i E'_i, V'={v'_i | i=1,…,n}, and 𝒩^i is the direct neighbors of node i. We denote ϕ^e(·) and ϕ^v(·) as deep neural networks. We denote f^e→ v(·) as an arbitrary aggregation function with the property of permutation invariance.§ INTERSECTION DRIVING SIMULATIONWe introduce an Intelligent Intersection Driver Model for simulating low-level vehicle kinematics and pedestrian behaviors that consider the interactions between traffic participants. We then develop a simulator of a partially controlled intersection that involves vehicles and pedestrians. §.§ Intelligent Intersection Driver Model (IIDM) We develop an Intelligent Intersection Driver Model (IIDM) based on the canonical Intelligent Driver Model (IDM) <cit.>, a one-dimensional car-following model with tunable parameters <cit.> that drives along a reference path.In the canonical IDM, the longitudinal position and velocity in Frenét coordinates are computed bydv/dt = a_max[1 - (v/v^*)^δ - (s^*(v, Δ v)/s)^2], s^*(v, Δ v) = s_0 + Tv + v Δ v/2 √(a_max b_comf),where the variables and constants are introduced in Table <ref>. Eq. (<ref>)–(<ref>) serve as a low-level vehicle kinematics model. To consider other dynamic agents that may be relevant to a certain vehicle, we define three types of interactions: , , and .is defined as slowing down until a complete stop to avoid collisions when conflict exists. This applies to 1) the vehicles whose future paths intersect with a crosswalk with crossing pedestrians; 2) the vehicles that encounter unyielding crossing traffic and need to avoid collisions. This can be implemented by placing a virtual static leading vehicle at the stop line or at the conflict point, and the simulated vehicle moves according to Eq. (<ref>)–(<ref>).is defined as passing the conflict point first without slowing down or stopping when two vehicles have a conflict in their future paths.is defined to describe a pair of vehicles that move along the same reference path, where Eq. (<ref>)–(<ref>) can be directly applied.§.§ Driving SimulatorWe develop a simulator of vehicles and pedestrians in a partially controlled intersection with two-way stop signs as illustrated in Fig. <ref>. The ego vehicle is randomly initialized on a branch with stop signs and the crossing traffic is not constrained.Multiple simulated vehicles drive in the crossing traffic lanes and the opposing direction, and multiple pedestrians walk on sidewalks and crosswalks. For the simulated vehicles, a human driver is sampled to be Aggressive or Conservative uniformly at the beginning of the episode.Then, the driver is sampled to have an intention Yield or Not Yield with P(Yield|Conservative) = 0.9 and P(Yield|Aggressive) = 0.1.We imitate the fact that both aggressive and conservative drivers may choose to yield or not due to the inherent randomness in human decisions regardless of their traits. The general rationale in our simulation design is to differentiate the driving styles of human drivers with different internal states (i.e., trait and intention) to mimic real-world traffic with diverse human drivers. The differences between heterogeneous driver behaviors on the horizontal lanes are as follows, where the desired speed of each vehicle is sampled from a Gaussian distribution:* Aggressive and non-yielding drivers have a desired speed around 9.0m/s and a minimum distance from the leading vehicle of 4.5m–7.5m.* Aggressive and yielding drivers have a desired speed around 8.8m/s and a minimum distance from the leading vehicle of 4.8m–7.8m.* Conservative and non-yielding drivers have a desired speed around 8.6m/s and a minimum distance from the leading vehicle of 5.7m–8.7m. * Conservative and yielding drivers have a desired speed around 8.4m/s and a minimum distance from the leading vehicle of 6.0m–9.0m.The other constants in Eq. (<ref>)–(<ref>) are shared by all categories. Moreover, the aggressive vehicles on the vertical lane in the opposite direction to the ego vehicle will proceed to cross the intersection whenever a conservative horizontal vehicle comes while conservative vehicles on the vertical lane will stay still until the ego vehicle completes the left turn.We also add simulated pedestrians on the crosswalks and sidewalks. We assume that pedestrians always have the highest right of way and move with constant speed unless another agent is directly in front of the pedestrians, in which case the pedestrians stay still until the path is clear. Developing and integrating more realistic and interactive pedestrian behavior models is left for future work. In general, all vehicles should yield to pedestrians whenever there is a conflict between their future paths within a certain horizon.§ PROBLEM FORMULATION We formulate the autonomous navigation of the ego vehicle as a POMDP similar to the problem formulation in our prior work <cit.>.The POMDP components are defined as follows: * State: Assume that there are N surrounding vehicles in the scene, 𝐱 = [𝐱^0, 𝐱^1, 𝐱^2,…, 𝐱^N ] denotes the physical state where 𝐱^0 = [x^0, y^0, v^0_x, v^0_y, b^0 ] denotes the ego vehicle state including position, velocity and one-hot indicator of agent type (i.e., vehicle/pedestrian), and 𝐱^i = [x^i, y^i, v^i_x, v^i_y, b^i ], i∈{1,…, N} denotes the state of the i-th surrounding agent. The internal state of the surrounding drivers is represented as 𝐳 = [𝐳^1, 𝐳^2,…, 𝐳^N ]. The internal state of each human driver includes two components: 𝐳_1^i ∈{Conservative, Aggressive} and 𝐳_2^i ∈{Yield, Not Yield}. Assume that there are M pedestrians in the scene, then their physical states are denoted as 𝐱^N+1:N+M = [𝐱^N+1, 𝐱^N+2,…, 𝐱^N+M]. The joint state is represented by 𝐬 = [𝐱^0, (𝐱^1,𝐳^1),…, (𝐱^N,𝐳^N), 𝐱^N+1,…, 𝐱^N+M]. * Observation: The physical states of all the surrounding vehicles and pedestrians are observable to the ego vehicle while the internal states are not. Formally, the observation is represented by 𝐨 = [𝐱̂^0, 𝐱̂^1,…, 𝐱̂^N+M], where 𝐱̂^i is obtained by adding a noise sampled from a zero-mean Gaussian distribution with a standard deviation of 0.05 to the actual position and velocity to simulate sensor noise. * Action: The action a ∈{0.0, 1.0, 4.5}m/s is defined as the target velocity of the ego vehicle for the low-level PD controller to track during the turning process. * Transition: The interval between consecutive simulation steps is 0.1s. The behaviors of surrounding vehicles and pedestrians are introduced in Section <ref>. We control the vehicle with a longitudinal PD controller in the same way as our prior work <cit.>, following the left-turn reference path and tracking the target speed determined by the ego policy. We also apply a safety check to make an emergency brake if the distance between the ego vehicle and other agents is too small. The episode ends once the ego vehicle completes the left turn successfully, a collision happens, or the maximum horizon is reached. * Reward: We design a reward function that encourages the driving policy to control the ego vehicle to turn left safely at the intersection as fast as possible without collisions. Formally, the reward function is written as R(s, a) = {s ∈ S_goal}r_goal +{s ∈ S_col}r_col + r_speed(s), where r_goal = 2 and S_goal is a set of goal states where the ego vehicle completes a left turn successfully; r_col = -2 and S_col is a set of failure states where a collision happens; and r_speed(s) = 0.01 v_ego/4.5m/s is a small reward on the ego vehicle's speed to encourage efficient driving. § DEEP REINFORCEMENT LEARNING WITHINTERNAL STATE INFERENCEIn this section, we introduce five variants of deep reinforcement learning with different configurations of human internal state inference for autonomous navigation in complex interactive scenarios (see Fig. <ref>). The major differences between these architectures lie in the training strategies and the way to incorporate the internal state inference network into the base DRL framework. §.§ Internal State Inference Consider an urban traffic scenario with the presence of the ego vehicle, N surrounding vehicles, and M pedestrians, where the ego vehicle is controlled by the reinforcement learning policy and the surrounding vehicles are controlled by N human drivers defined in the simulator.Let 𝐱_t denote the physical state of all the vehicles and pedestrians at time step t, we model the action distribution of the ith human driver as p (𝐚^i_t |𝐱_t, 𝐳^i_t ), where 𝐳^i_t represents the driver's internal state: trait (i.e., aggressive/conservative) and intention (i.e., yield/not yield to the ego vehicle).Inferring the internal state of the surrounding drivers leads to several advantages.First, a discrete internal state is efficient to learn and simple to be integrated into the control policy. Second, in many situations, the internal state provides even more distinguishable information than predicting their future trajectories.For example, in the intersection scenario shown in Fig. <ref>, the predicted trajectories of the conservative and aggressive vehicles could be similar at the moment before the ego vehicle approaches the intersection, which may not be able to indicate their driving traits effectively.However, their traits can be inferred by observing their interaction histories with other vehicles.In such cases, the internal state provides the key information explicitly for the ego decision making.The goal of internal state inference is to determine the distribution p (𝐳^i_t |𝐨_1:t), where 𝐨_1:t denotes the ego agent's historical observations up to time t.We assume that the ground truth internal states of the surrounding human drivers are available from the simulator at training time and unknown at testing time.Thus, the internal state inference module (i.e., a neural network) can be trained by standard supervised learning as a classification task. By using the information provided by the internal state labels, the auxiliary trait and intention inference tasks provide additional supervision signals in addition to a standard reinforcement learning framework. §.§ Graph-Based Representation Learning A human driver's behavior in complex and dense traffic scenarios is heavily influenced by its relations to other traffic participants.The dependence between traffic participants can be naturally formalized as a graph where the nodes represent agents and the edges represent their relations or interactions.In a four-way intersection scenario, each vehicle can be potentially influenced by any surrounding agents. Based on this intuition, we represent the intersection scenario at time t as a fully connected graph 𝒢_t = (𝒱_t,ℰ_t), where the node set 𝒱_t contains the nodes for all the vehicles and pedestrians in the scene, and the edge set ℰ_t contains all the directed edges between each pair of agents. The edges are designed to be directed because the influence between a pair of agents may not be symmetric. Bidirectional relations should be modeled individually. For example, the leading vehicle tends to have a strong influence on the behavior of the following ones in the same lane. However, the following vehicles merely have an influence on the leading one. The asymmetry also applies to situations where two conflicting agents have different priorities of the right of way such as vehicle-pedestrian interactions.We adopt a three-layer network architecture similar to the encoder of STGAT <cit.> to process both the spatial relational information in 𝒢_t with the graph message passing layer and the temporal information in o_1:t with the LSTM recurrent network layer, which is shown in Fig. <ref>.At time step t, the observation on the i-th vehicle 𝐨^i_t and its observation history 𝐨^i_1:t-1 are fed into the bottom-level Vehicle-LSTM with a hidden state 𝐡^i_t.The Vehicle-LSTM parameters are shared among all the vehicles except the ego vehicle. Similarly, we use another shared Pedestrian-LSTM to extract historical features for pedestrians. We have𝐯^0_t =Ego-LSTM^1 (𝐨^0_t; 𝐡^0_t),𝐯^i_t =Vehicle-LSTM^1 (𝐨^i_t; 𝐡^i_t), i ∈{1,…,N},𝐯^i_t =Pedestrian-LSTM^1 (𝐨^i_t; 𝐡^i_t),i ∈{N+1,…,N+M},where 𝐯^0_t and 𝐯^i_t denote the extracted feature vectors of ego and surrounding traffic participants, which encode their historical behaviors. Ego-LSTM^1, Vehicle-LSTM^1, and Pedestrian-LSTM^1 denote the LSTM units at the bottom layer.The extracted features 𝐯^0_t and 𝐯^i_t are used as the initial node attributes of the corresponding agents in 𝒢_t. We explore the effectiveness of three typical graph message passing layers to process the information across the graph: GAT <cit.>, GCN <cit.>, and GraphSAGE <cit.>. The detailed operations of different message passing layers are introduced in Section <ref>. The message passing procedures can be applied multiple times to aggregate information from more distant nodes in the graph. Based on the experimental results, we select GAT for message passing, which is written asα^ij_t = exp ( LeakyReLU ( 𝐚^⊤ [ 𝐖𝐯^i_t 𝐖𝐯^j_t ] ) )/∑_k ∈𝒩^iexp ( LeakyReLU ( 𝐚^⊤ [ 𝐖𝐯^i_t 𝐖𝐯^k_t ] ) ),where 𝐚 and 𝐖 denote a learnable weight vector and a learnable weight matrix, 𝒩^i denotes the direct neighbors of node i. The symbols ·^⊤ anddenote transposition and concatenation operations, respectively. The updated node attributes can be obtained by 𝐯̅^i_t = σ(∑_j ∈𝒩^iα^ij_t 𝐖𝐯^j_t ),where σ(·) denotes a nonlinear activation function. The updated node attributes are then fed into the top-level LSTM networks with the same parameter-sharing strategy, which is written as𝐯̃^0_t =Ego-LSTM^2 (𝐯̅^0_t; 𝐡̅^0_t),𝐯̃^i_t =Vehicle-LSTM^2 (𝐯̅^i_t; 𝐡̅^i_t),i ∈{1,…,N}, 𝐯̃^i_t =Pedestrian-LSTM^2 (𝐯̅^i_t; 𝐡̅^i_t),i ∈{N+1,…,N+M},where Ego-LSTM^2, Vehicle-LSTM^2, and Pedestrian-LSTM^2 denote the LSTM units at the top layer. The variables 𝐡̅^0_t and 𝐡̅^i_t are hidden states. The final feature embedding of agent i at time t is obtained by a concatenation of 𝐯^i_t and 𝐯̃^i_t, which encodes both the self-attribute and social-attribute. Finally, a multi-layer perceptron (MLP) takes the final embeddings of surrounding vehicles as input and outputs the probability of the corresponding human driver's traits (i.e., aggressive/conservative) and intentions (i.e., yield/not yield). Note that the pedestrian node attributes are used for message passing yet they are not used for internal state inference. Modeling the internal state of pedestrians is left as future work. §.§ Framework Architectures We propose to integrate the human internal state inference into the standard RL-based autonomous navigation framework as an auxiliary task. The integration can be done in multiple ways. In this work, we investigate five variants of framework architectures, which are shown in Fig. <ref>. In all the variants, ground truth internal states can be obtained from the environment (i.e., driving simulator) during training, and the internal state inference network is trained with cross-entropy loss.Specific details and the differences among these variants are elaborated below. Through the comparison between these variants, we can figure out the best combination of model integration and training strategies. * Configuration (a): The policy network and the internal state inference network are treated as two separate modules without mutual influence during training. During training, the policy network takes in historical observations and true internal states that provide the actual traits and intentions of surrounding vehicles. The graph-based encoder is only used for internal state inference. The policy is refined by a policy optimization algorithm. Meanwhile, the internal state inference network is trained by standard supervised learning separately. During testing, the policy network takes in the inferred internal states. This variant decouples policy learning and internal state inference, which requires individual optimization. * Configuration (b): The difference from configuration (a) is that the policy network and internal state inference network share the same encoder in both training and testing. Thus, the two networks can influence each other via the shared encoder. Both supervised learning loss and policy optimization loss can update the encoder. * Configuration (c): The difference from configuration (a) is that the policy network uses the inferred internal states in both training and testing. The internal state labels are only used to train the inference network. The quality of the information about internal states the policy network uses highly depends on the inference accuracy. * Configuration (d): The difference from configuration (c) is that the policy network and internal state inference network share the same encoder in training and testing. * Configuration (e): The difference from configuration (b) is that the losses from two tasks are coupled by a weighted sum and all the networks are trained with the policy optimizer. This variant enables the highest correlation between the policy network and the internal state inference network.Configuration (a) contains the following procedures. The internal state inference network learns the mapping from the historical observations to a latent distribution, i.e. p_ψ(𝐳^i_t |𝐨_1:t) where ψ denotes the parameters of the inference network that is trained to minimize the negative log-likelihood:L(ψ)=-𝔼_𝐳^i_t, 𝐨_1:t∼ D[log p_ψ(𝐳^i_t |𝐨_1:t)],where the latent state 𝐳^i_t and the historical observations 𝐨_1:t are randomly sampled from a replay buffer containing exploration experiences. The policy takes both the historical observations and the internal state as inputs, i.e. π_θ(a |𝐨_1:t, 𝐳^1:N_t) where θ denotes the policy network parameters trained by the augmented policy optimization objective:L(θ)=𝔼̂[min( π_θ(a |𝐨_1:t,𝐳^1:N_t )/π_θ'(a |𝐨_1:t, 𝐳^1:N_t )Â, clip(π_θ(a |𝐨_1:t,𝐳^1:N_t)/π_θ'(a |𝐨_1:t,𝐳^1:N_t),1-ϵ,1+ϵ)Â)]. Based on the experimental results in Section <ref>, we conclude that Configuration (a) performs the best among the five variants, which has the following benefits. First, feeding the ground truth internal state at exploration (i.e., training) time helps the control policy find the trajectory leading to the task goal. This is especially important when the task is difficult and the reward is sparse.Second, by using a separate network for each task, the mutual influence of the gradients from different tasks can be minimized. Such mutual influence could be harmful as shown in our experiments.Third, by modularizing the two learning modules, our framework allows for flexible choices of network structures in different modules. § INCORPORATING TRAJECTORY PREDICTION AND INTERACTIVITY ESTIMATION Besides inferring the high-level internal states of human drivers, the autonomous navigation task also benefits from forecasting their future trajectories that provide fine-grained behavioral cues as well as reasoning about the potential influence of the ego vehicle on surrounding agents.In this work, we design an auxiliary trajectory prediction task to infer how the other traffic participants will behave in the presence of the ego vehicle. Moreover, in complex urban traffic, human drivers tend to implicitly estimate to what extent they could influence the behaviors of other traffic participants to enhance situational awareness and facilitate their negotiation and driving efficiency. Motivated by this intuition, we design a mechanism to estimate the interactivity scores of other agents that can be used by the policy network.Here we provide an example to illustrate the core concept. In Fig. <ref>(a), the green vehicle on the left side can speed up to cross the intersection without any conflict. However, in Fig. <ref>(b), the green vehicle must slow down and yield to the blue ego vehicle to avoid a collision. The difference in the green vehicle's behavior in these two situations can be used to compute the influence of the ego vehicle. Meanwhile, the speed profiles of the same green vehicle in the two settings are compared in Fig. <ref>(c) for a quantitative illustration. We propose to predict the future trajectories of other agents in both situations and quantify the difference as interactivity scores, which are used as input to the ego policy network. Moreover, since the ego vehicle tends to negotiate with the agents with large interactivity scores, the trajectory prediction of those agents needs to be more accurate than those with small interactivity scores to ensure better safety and efficiency. Therefore, we propose a weighting strategy in the prediction loss based on the interactivity scores to encourage better prediction of important agents that may have strong interactions with the ego vehicle. In both training and testing scenarios, the ego vehicle always exists and may influence the other agents, thus the prediction in the situation in Fig. <ref>(a) (i.e., without the ego vehicle) can be treated as counterfactual reasoning. §.§ Trajectory PredictionThe trajectory prediction task is formulated as a regression problem solved by supervised learning, where the ground truth future trajectories can be obtained by simulation. This can provide additional supervision signals to refine the graph representation learning in the encoder and thus help with the improvement of other downstream components. We forecast the future trajectories of surrounding agents in both situations (i.e., without and with the existence of the ego vehicle) through two separate prediction heads. The former task encourages the model to capture the natural behaviors of surrounding agents defined by the simulation without the intervention of the learned ego vehicle's policy. The latter task encourages the model to capture how the surrounding agents will react to the ego vehicle's future behavior through their future trajectories.Formally, we denote the prediction horizon as T_f and the objective of prediction is to estimate two conditional distributions p(𝐱^1:N+M_t+1:t+T_f|𝐨^1:N+M_1:t) (without the ego vehicle) and p(𝐱^1:N+M_t+1:t+T_f|𝐨_1:t) (with the ego vehicle).Without loss of generality, the distributions are assumed to be Gaussian with a fixed diagonal covariance matrix Σ for simplicity; thus, the model can focus on predicting the mean of distributions. In future work, we will further investigate more complex traffic scenarios where the predicted distributions can be multi-modal (e.g., Gaussian Mixture Model) with learnable covariance. * To predict future trajectories in the scenarios without the ego vehicle, we propose to pre-train another prediction model branch including a graph-based encoder with the same architecture as the one shown in Fig. <ref> except that there is no ego vehicle involved as well as an MLP prediction head. The parameters of these networks are fixed without further updates during the formal training stage to generate counterfactual future trajectories. The reason for using a separate prediction branch is to minimize the influence of the ego vehicle in counterfactual prediction.* To predict future trajectories in the scenarios with the ego vehicle, an MLP prediction head takes the final node attributes 𝐯̃^i_t as input and outputs the means of predicted trajectory distributions of agent i (i.e., μ̂^i, w/ Ego_t+1:t+T_f). We use the pre-trained network parameters in the former setting to allow for better initialization.§.§ Interactivity EstimationWe propose an interactivity estimation mechanism based on the difference between the predicted trajectories in the two situations discussed in Section <ref>. The underlying intuition is that the ego vehicle can potentially influence the behavior of surrounding agents that have conflicts in their future paths and negotiate the right of way. The estimated strength of influence indicated by the difference between their future trajectories can quantitatively imply to what extent the ego vehicle can try to interact or negotiate with a certain agent, which is named as an interactivity score (IS) and helps the ego vehicle to select a proper occasion to proceed.To indicate the differences in the surrounding agents' future behaviors under the probabilistic setting, we propose to use the Kullback–Leibler (KL) divergence between the two trajectory distributions given by𝐱^i_t+1:t+T_f|𝐨_1:t ∼𝒩(μ̂^i, w/ Ego_t+1:t+T_f, Σ) = 𝒩(μ̂^i_1, Σ),𝐱^i_t+1:t+T_f|𝐨^1:N+M_1:t ∼𝒩(μ̂^i, w/o Ego_t+1:t+T_f, Σ) = 𝒩(μ̂^i_2, Σ),to indicate the difference quantitatively, which is computed byD_KL(p(𝐱^i_t+1:t+T_f|𝐨_1:t)p(𝐱^i_t+1:t+T_f|𝐨^1:N+M_1:t)) =1/2( Tr(Σ^-1Σ) - d + (μ̂^i_1 - μ̂^i_2)^⊤Σ^-1(μ̂^i_1 - μ̂^i_2) + ln( Σ/Σ)) =1/2(μ̂^i_1 - μ̂^i_2)^⊤Σ^-1(μ̂^i_1 - μ̂^i_2)=1/2σ^2μ̂^i_1 - μ̂^i_2 ^2, where σ^2 is the constant covariance value in the diagonal of Σ, d is the dimension of the distributions, Tr(·) denotes the trace of a matrix, ·^⊤ denotes the transpose of a vector, Σ^-1 and Σ denote the inverse and determinant of the covariance matrix, respectively. Due to the Gaussian assumption with fixed covariance, the KL divergence reduces to the L_2 distance between the mean vectors of two trajectory distributions multiplied by a constant. For simplicity, we define the interactivity score 𝐰^i_t of agent i at time t as𝐰^i_t = μ̂^i, w/ Ego_t+1:t+T_f - μ̂^i, w/o Ego_t+1:t+T_f^2. The interactivity scores can be treated as a feature of each agent and used by the policy network. Moreover, we use them as the weights of prediction errors in the loss function for trajectory prediction, which is computed byL^TP = 1/N+M∑_i=1^N+M𝐰^i_t ·μ̂^i, w/ Ego_t+1:t+T_f - 𝐱^i, w/ Ego_t+1:t+T_f^2,where 𝐱^i, w/ Ego_t+1:t+T_f is the ground truth of future trajectories.[!tbp] Reinforcement Learning with Auxiliary Tasks (Formal Training Phase) [!tbp] Reinforcement Learning with Auxiliary Tasks (Testing Phase) §.§ Complete FrameworkAn overall diagram of the complete method is shown in Fig. <ref>, which integrates auxiliary supervised learning tasks into the reinforcement learning framework. The detailed pseudocode of the proposed method in the training and testing phases is provided in Algorithm <ref> and Algorithm <ref>, respectively. First, we have a graph-based encoder to extract spatio-temporal features from historical observations of all the agents. Second, we have an internal state inference module to recognize the traits and intentions of surrounding vehicles. Third, we have a trajectory prediction module to forecast the future behaviors of other agents with the existence of the ego vehicle.The ground truth labels of internal states and future trajectories can be obtained from the environment (i.e., driving simulator) in the training phase, which are not needed in the testing phase. We also have another pre-trained trajectory prediction module to forecast the future behaviors of other agents without the existence of the ego vehicle. Fourth, we estimate the interactivity scores of other agents. Finally, the policy network outputs the action distribution based on the historical observations, inferred internal states, and estimated interactivity scores of surrounding agents.The explainable aspects of our method come from two auxiliary tasks: (a) internal state inference; and (b) interactivity estimation. On the one hand, our method can infer the traits and intentions of surrounding vehicles, which can inform the policy network about whether they tend to yield to the ego vehicle. The inferred internal state can serve as an explanation for the decision making. On the other hand, the estimated interactivity scores can reflect how much influence the ego vehicle can potentially have on surrounding agents.A higher interactivity score implies that the ego vehicle has a higher possibility of being able to influence and negotiate with the corresponding agent to improve driving efficiency.§ EXPERIMENTS§.§ Experiment Settings and Implementation Details We train our method and all the baselines three times with different random seeds and each trial is trained for 10^7 environment steps per epoch. We use a learning rate of 10^-4 for the policy optimizer and 10^-3 for the optimizer of the value baseline and supervised learning. In the framework variant in Fig. <ref>(e), we set the weight of the supervised learning losses as 0.1. We run all experiments on a Linux workstation with Intel i9-10940X CPU and a NVIDIA Quadro RTX 6000 GPU.The encoder consists of six LSTM networks with a hidden size of 64 in the bottom and top layers for different types of agents. The weight vectors/matrices in the graph message passing layers have a proper dimension corresponding to specific feature dimensions. The internal state inference network and trajectory prediction networks are three-layer MLPs with a hidden size of 64. The policy network is an LSTM network with a hidden size of 64, whose input is a concatenation of the observations on all the agents. The vehicle and pedestrian features are ordered based on their distances to the ego vehicle (from small to large). §.§ Evaluation Metrics and BaselinesWe evaluate our method with widely used metrics in decision making for autonomous driving (i.e., completion rate, collision rate, timeout rate, time to completion) and internal state inference (i.e., classification accuracy). We define that an episode is considered to be a successful completion if the ego vehicle completes the left turn within 25 seconds (i.e., 250 time steps) without collision with other traffic participants. Otherwise, the episode is considered a collision or timeout case. Only the completion cases are used to compute the time to completion, which indicates driving efficiency quantitatively. We compare our full method with state-of-the-art baselines <cit.> and several ablation framework settings to demonstrate the effectiveness of each component. We evaluate our method and baselines in 1,000 testing scenarios consistently for fair comparisons.§.§ Internal State Inference We demonstrate the effectiveness of internal state inference and compare the performance of the five variants of framework architectures presented in Fig. <ref> through quantitative analysis and ablation study.The comparison between different variants of framework architectures is shown in Fig. <ref>.Base+ISI(GT) uses the ground truth internal states as the input of the policy network, which serves as a performance upper bound among all the methods and implies the effectiveness of taking advantage of internal states in the decision making process. The Base method is a standard reinforcement learning framework without auxiliary modules. It shows that the Base method performs the worst among all the variants and different strategies of integrating the ISI module into the base method lead to different degrees of improvement. The Base method is only able to capture the behavior patterns of surrounding traffic participants implicitly through the policy network, which makes it difficult to learn the traits and intentions of other vehicles. Thisleads to poor performance with many collisions and lower driving efficiency in our environment. Base+ISI(a) performs the best among the five variants of architectures, which improves the average completion rate by 76.9% over the Base method. Base+ISI(a)-LSTM simply uses a shared LSTM network instead of a graph-based encoder to extract the historical information of each agent individually. Base+ISI(a) outperforms Base+ISI(a)-LSTM by a large margin in terms of both completion rate and internal state inference, which implies the effectiveness of spatio-temporal behavior modeling of interactive agents. The comparison between Base+ISI(a) and Base+ISI(b) implies that having a separate encoder for ISI performs better than using a shared encoder for the policy network and the ISI network in terms of both average completion rate and ISI accuracy. The reason is that the policy network needs to focus more on the ego perspective to make decisions while the ISI network needs to capture explicit relations between interactive agents in a distributed manner. Since a shared encoder needs to learn useful features for both aspects, the encoded information may distract the learning process of both networks.We also have a consistent observation by comparing Base+ISI(c) and Base+ISI(d), although they adopt a different strategy in using the internal states during training.The comparison between Base+ISI(a) and Base+ISI(c) implies that using the ground truth internal state labels as the input of the policy network during training performs better than using the inferred internal state in terms of average completion rate while achieving a comparable performance in terms of ISI accuracy. It is reasonable that both architectures achieve a similar ISI accuracy because the learning processes of the ISI network are essentially the same. However, the inferred internal states in Base+ISI(c) may be wrong during training, which misleads policy learning and makes the training process unstable.Using the ground truth internal states during training is similar to the teacher forcing technique in <cit.>, which can stabilize training and improve performance.The comparison between Base+ISI(a) and Base+ISI(e) implies that updating the policy network and ISI network separately with individual loss performs better than the coupled training strategy. The combination of two losses leads to biased gradient estimates for both policy learning and internal state inference tasks. Although Base+ISI(e) outperforms Base in terms of average completion rate, the improvement is relatively minor and cannot reflect the significant benefits of ISI. And the average ISI accuracy is much worse than other variants.By comparing Base+TP+IS and Base+ISI(a)+TP+IS settings in Fig. <ref>, we can observe that the improvement brought by ISI is also significant even with the auxiliary modules of trajectory prediction and interactivity estimation, which implies the effectiveness of explicit modeling of human drivers' traits and intentions. The reason is that with the inferred internal states of surrounding vehicles, it is easier for the ego driving policy to figure out a proper opportunity to proceed. The comparison of the completion rate curves of different model settings during the training process is shown in Fig. <ref>. The results show that the policy learning process tends to be faster and more stable with internal state inference. The reason is that during the training phase, the RL agent is able to use additional information about the agents in the environment through internal state labels, which helps capture the correlations between observations and the underlying traits and intentions of interactive agents effectively.To explicitly demonstrate the influence of internal state inference on the decision making of the ego vehicle, we manipulated the inferred internal states of surrounding vehicles and compared the outcomes in Fig. <ref>.It shows that “Cons. to Aggr.” leads to a lower collision rate and a higher timeout rate than “No change” because, with more inferred aggressive vehicles, the ego vehicle tends to yield and wait until it finds a conservative vehicle to proceed.In contrast, “Aggr. to Cons.” leads to a higher collision rate and a slightly lower timeout rate than “No change” because, with more inferred conservative vehicles, the ego vehicle tends to be more aggressive. These results imply that the learned policy has a high dependence on the inferred internal states, which performs much better when using the original inference than the manipulated inference.§.§ Trajectory Prediction and Interactivity Estimation We demonstrate the effectiveness of trajectory prediction and interactivity estimation by comparing our complete framework Base+ISI(a)+TP+IS with its counterparts without the TP or IS modules. Note that when TP is used alone without interactivity estimation, the L_2 losses of all the cases are equally weighted during training. In Fig. <ref>(a), we can observe an increase in the completion rate and a decrease in the collision rate and timeout rate by adding TP to Base+ISI(a).The comparison between Base+ISI(a) and Base+ISI(a)+TP in Fig. <ref>(b) also shows that trajectory prediction also enhances trait and intention inference. The reason is that the internal state of a surrounding vehicle can determine its future behavior, thus trajectory prediction can in turn encourage the model to capture subtle cues of the internal state implicitly.Moreover, adding the interactivity estimation module leads to further improvements by providing the policy network about to what extent the ego vehicle can have potential influence on other agents through the interactivity scores. The implied quantitative degree of influence can facilitate the learning of negotiation. The proposed method Base+ISI(a)+TP+IS achieves the best performance in Fig. <ref>. We found that most collisions are caused by wrong inference of internal states and inaccurate trajectory prediction of surrounding vehicles.The comparison between Base+ISI(a)+TP(E)+IS and Base+ISI(a)+TP+IS implies the benefit of weighting the prediction error of each agent by its interactivity score in the loss function of trajectory prediction. The reason is that, with the weighting mechanism, the predictor cares more about the prediction accuracy of the agents that may have strong interactions with the ego vehicle, which enhances ego decision making in challenging situations.The comparison between our method and the state-of-the-art baselines <cit.> (our prior work), <cit.>, and <cit.> is shown in Fig. <ref>. Since the baseline methods cannot handle the pedestrians and the vehicles on the vertical lanes, we simplified the scenario into a T-intersection similar to the baseline papers for a fair comparison. Meanwhile, in these experiments, conservative vehicles always yield and aggressive ones do not yield.MPC method performs the worst due to its limited ability to handle diverse social behaviors and multi-agent interactions. It also requires online re-planning at a high frequency, which demands a large amount of computational resources.Our method achieves much better performance in terms of all the evaluation metrics. In particular, Ours (transfer) trained in the original environment (i.e., four-way intersection) achieves comparable performance with the one directly trained in the T-intersection, indicating good generalization performance.§.§ Robustness to Distribution ShiftA critical aspect of a deep reinforcement learning method is its robustness to the distribution shift in the environments during the testing phase. In the training phase, p(Aggressive) is fixed at 0.5 in our simulator. To evaluate the robustness of different approaches, we changed p(Aggressive) to 70% or 90% for testing, which generates more aggressive drivers and leads to more challenging scenarios.We have four observations based on the comparison of average completion rates shown in Fig. <ref>. First, the internal state inference shows great effectiveness, especially in environments with more aggressive drivers. The ablation models with ISI(a) improve the average completion rate by a large margin compared with their counterparts without ISI(a) in all settings. The reason is that the ISI(a) module can recognize the internal characteristics of human drivers and infer their intentions explicitly, which helps the policy network to choose appropriate actions. In environments with a large portion of aggressive drivers, it is necessary to accurately recognize the conservative ones so that the ego vehicle can grasp the opportunities to proceed safely and efficiently.Second, by comparing Base+ISI(a) and Base+ISI(a)+TP, we can see that trajectory prediction leads to improvement. The reason is that the additional supervision on trajectory prediction can improve internal state inference through learning better graph-based encoder since the trajectories can reflect the drivers' internal characteristics, which enhances decision making. Third, a large portion of non-completion cases of the method settings without ISI(a) in the environment with p(Aggressive)=0.9 is due to timeout caused by inefficient driving policies. We can see an increasing trend of improvement brought by the interactivity estimation as the ratio of aggressive drivers increases, which implies the effectiveness of interactivity estimation in challenging scenarios. Finally, the performance gaps between the baselines <cit.> and our method increase as more aggressive drivers appear in the scenarios, implying our method better handles challenging situations. §.§ Interpretation of the Decision Making Process We provide the visualization of a typical testing scenario in Fig. <ref> to demonstrate the two aspects of the interpretation of the ego vehicle's decision making process more concretely. We select four representative frames for qualitative analysis, where the inferred internal states and interactivity scores are shown near each surrounding vehicle.In this work, we do not consider the internal states of pedestrians. Their interactivity scores are all very small since they always have the highest right of way, which are omitted in the figure for clarity.In Fig. <ref>(a), the middle green vehicle is already in the conflict zone initially and the two trajectory prediction heads predict similar future trajectories, which leads to a low interactivity score. This is reasonable because it is unlikely to be influenced by the ego vehicle since it already occupies the conflict zone before the ego vehicle.However, the green one on the left has a much higher chance of being influenced by the ego vehicle, which is aligned with a high estimated interactivity score caused by more distinct trajectory hypotheses generated by the two prediction heads. The method also infers these two vehicles as conservative ones confidently based on their historical behavior patterns.Due to the existence of the crossing pedestrians, the two red aggressive vehicles need to stop and wait until their paths are clear. This provides the ego vehicle a good opportunity to turn left safely without negotiating with the red vehicle on the main road. In this situation, the red vehicles have low interactivity scores because their near-future trajectories will not change no matter whether the ego vehicle exists or not due to the constraints caused by the crossing pedestrians. Note that they have non-zero interactivity scores because of prediction errors. This implies that our method successfully learns the underlying relations between vehicles and pedestrians.In Fig. <ref>(b)(c), the low interactivity scores of red vehicles combined with the physical state information informs the policy that the ego vehicle can proceed safely without a strong negotiation even though the opponent is an aggressive vehicle. In Fig. <ref>(d), the interactivity score of the left green vehicle becomes much smaller because the ego vehicle no longer influences its future behavior after crossing the intersection. §.§ Influence of Pedestrians In Fig. <ref>, a typical testing scenario is visualized to qualitatively illustrate the influence of crossing pedestrians on the vehicles, where the vehicles on the main road need to yield to the crossing pedestrians thus the ego vehicle is able to proceed. To quantitatively show the influence of crossing pedestrians on the decision making of the ego vehicle, we compare the performance in the environments with pedestrians and without pedestrians in Fig. <ref>. In general, the average completion rate is higher and the average time-to-completion is shorter in environments without pedestrians for all the methods. This is reasonable because the existence of crossing pedestrians leads to more challenging situations where the ego vehicle needs to yield to the pedestrians as well as seize the opportunity to make the left turn as fast as possible. The results show consistent relative performance among different methods. § CONCLUSIONSIn this paper, we present a deep reinforcement learning framework with auxiliary supervised learning tasks for autonomous navigation and use a partially controlled intersection scenario as a case study to validate our method. First, we propose to infer the internal state of surrounding human-driven vehicles including their traits (i.e., conservative/aggressive) and intentions (i.e., yield/not yield to the ego vehicle). Second, we propose to estimate the interactivity scores of traffic participants based on the inferred degree of influence via counterfactual trajectory prediction to provide additional cues to the policy network. These auxiliary tasks improve the completion rate, reduce collisions, and enhance driving efficiency by a large margin compared with state-of-the-art baselines. The ablation study demonstrates the effectiveness of each component of our method. In particular, our method is more robust to the distribution shifts in the testing environments and provides explainable intermediate indicators for ego decision making. Moreover, we design an intersection driving simulator based on an Intelligent Intersection Driver Model, which is used to simulate the interactive behaviors of vehicles and pedestrians in our experiments.The limitation of this work lies in the gap between driving simulation and real-world scenarios. In this work, we assume that human drivers can be divided into binary groups (i.e., conservative or aggressive), which is a reasonable simplification of human traits. Our method requires ground truth labels for human traits and intentions provided by the simulator in the training process. However, human traits could be more complicated in the real world and the ground truth labels may not be straightforward to obtain. The goal of this study is to validate the hypothesis that modeling human internal states explicitly can improve decision making performance and the inferred internal states can serve as explainable indicators. In future work, we plan to address this limitation by learning human traits and intentions with latent variable models in an unsupervised manner, which eliminates the demand for ground truth labels and allows for modeling more flexible internal states than binary categories. We will also investigate how to guarantee safety in a principled manner.§ APPENDIX In this section, we introduce the detailed operations in other message passing mechanisms (besides GAT) used in our experiments and analyze the experimental results. §.§ Graph Message Passing Mechanisms §.§.§ GCNThis model applies convolution operations to graphs. We formulate a node attribute matrix 𝐕 where each row denotes the attribute of a certain node. Then the updated node attribute matrix 𝐕̅ can be obtained by 𝐕̅_t = σ( 𝐃̃^-1/2𝐀̃𝐃̃^-1/2𝐕_t 𝐖), where 𝐀̃ = 𝐀 + 𝐈 is the adjacency matrix of 𝒢_t with self-connections. 𝐈 is the identity matrix. 𝐃̃^ii = ∑_j𝐀̃^ij and 𝐖 is a learnable weight matrix. Here, σ(·) denotes a nonlinear activation function. §.§.§ GraphSAGEThis model designs a customized message passing mechanism, which includes the following operations:MSG^i_t =f_AGG( {𝐯^j |∀ j ∈𝒩^i }), 𝐯̅^i_t =σ( 𝐖[ 𝐯^i_t MSG^i_t ] ), 𝐯̅^i_t ←𝐯̅^i_t / 𝐯̅^i_t_2,where MSG^i_t denotes an intermediate message obtained by aggregating the information from the neighbors of node i, and f_AGG is an arbitrary permutation invariant function. 𝐖 denotes a learnable weight matrix and σ(·) denotes a nonlinear activation function. §.§ Influence of GNN Architectures We conducted an ablation study on several widely used graph neural network architectures for spatio-temporal graph modeling: ST-GAT <cit.>, ST-SAGE <cit.>, and ST-GCN <cit.>. The comparison of results is shown in Fig. <ref>. Generally, there is no significant gap in performance with different architectures of graph neural networks, which implies that our method is not sensitive to the choice of graph neural networks.ST-GCN is a modified graph convolutional network that is applied to spatio-temporal graphs, which uses the graph adjacency matrix and applies simple convolutions across the graph. ST-SAGE has a more expressive message passing mechanism than ST-GCN and leverages the node attribute information more effectively, which leads to better performance. ST-GAT applies a graph attention mechanism in message passing and achieves the best performance. A potential reason is that the graph attention layers naturally learn to recognize and use important information in the node updates, which is suitable for gathering information from other agents. IEEEtran [ < g r a p h i c s > ]Jiachen Li (Member, IEEE) received his Ph.D. degree from the Department of Mechanical Engineering at the University of California, Berkeley in 2021. Before that, he received a B.E. degree from the Department of Control Science and Engineering at Harbin Institute of Technology, China in 2016. He is currently a postdoctoral scholar at Stanford University. His research interest lies at the broad intersection of robotics, trustworthy AI, reinforcement learning, control and optimization, and their applications to intelligent autonomous systems, particularly in human-robot interactions and multi-agent systems. Dr. Li was selected as an RSS Robotics Pioneer in 2022 and an ASME DSCD Rising Star in 2023. He serves as an associate editor or a reviewer for multiple journals and conferences. He has organized multiple workshops on robotics, machine learning, computer vision, and intelligent transportation systems.[ < g r a p h i c s > ]David Isele (Member, IEEE) received his M.S.E. degree in Robotics and his Ph.D. degree in Computer and Information Science from The University of Pennsylvania. Before that, he received his B.E. degree in electrical engineering from The Cooper Union: Albert Nerken School of Engineering, in New York City. He is currently a senior scientist at Honda Research Institute US. His research interests include applications of machine learning and artificial intelligence to robotic systems, with a focus on strategic decision making for autonomous vehicles.[ < g r a p h i c s > ]Kanghoon Lee received his BS degree in Industrial and Systems Engineering and Computer Science from the Korea Advanced Institute of Science and Technology (KAIST), South Korea, in 2020, and MS degree in Industrial and Systems Engineering from the KAIST, South Korea, in 2022.Currently, He is a Ph.D. candidate in the System Intelligence Laboratory at the Department of Industrial and Systems Engineering, KAIST, South Korea. His research interest lies at the intersection of machine learning, (multi-agent) reinforcement learning and their application to robotic or traffic systems. [ < g r a p h i c s > ]Jinkyoo Park is currently an associate professor of Industrial and Systems Engineering and adjoint professor of Graduate School of Artificial Intelligence at Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea. He received his B.S. degree in Civil and Architectural Engineering from Seoul National University in 2009, an M.S. degree in Civil, Architectural and Environmental Engineering from the University of Texas Austin in 2011, an M.S. degree in Electrical Engineering from Stanford University in 2015, and a Ph.D. degree in Civil and Environmental Engineering from Stanford University in 2016. His research goal is to explore the potential of the various machine learning approaches for improving complex decision-making methods in optimization, optimal control, and game theory. [ < g r a p h i c s > ]Kikuo Fujimura received the B.S. and M.S. degrees in Information Science from the University of Tokyo in 1983 and 1985, respectively, and the Ph.D. degree in Computer Science from the University of Maryland, College Park in 1989.After working at Oak Ridge National Laboratory and Ohio State University (Columbus), he joined Honda R&D in 1998, where he was engaged in research on intelligent systems and human-robot interaction with Honda’s humanoid robot ASIMO. He is currently Director of Innovation at Honda Research Institute USA in San Jose, California, where he directs teams of researchers working on automated driving, knowledge discovery and informatics, human-machine interfaces, and intelligent robotics.His research interests include artificial intelligence for mobility, human-robot interaction, and HCI. He has authored/co-authored one book and over 100 research papers in refereed conferences and journals and has been granted over 20 patents. He currently serves as an Associate Editor for IEEE Transactions on Intelligent Vehicles and ACM Journal on Autonomous Transportation Systems. [ < g r a p h i c s > ]Mykel J. Kochenderfer (Senior Member, IEEE) is an Associate Professor of Aeronautics and Astronautics at Stanford University. He is the director of the Stanford Intelligent Systems Laboratory (SISL), conducting research on advanced algorithms and analytical methods for the design of robust decision making systems. Prior to joining the faculty in 2013, he was at MIT Lincoln Laboratory where he worked on aircraft collision avoidance for manned and unmanned aircraft. He received his Ph.D. from the University of Edinburgh in 2006. He received B.S. and M.S. degrees in computer science from Stanford University in 2003. He is an author of the textbooks Decision Making under Uncertainty: Theory and Application (MIT Press, 2015), Algorithms for Optimization (MIT Press, 2019), and Algorithms for Decision Making (MIT Press, 2022). | http://arxiv.org/abs/2311.16091v1 | {
"authors": [
"Jiachen Li",
"David Isele",
"Kanghoon Lee",
"Jinkyoo Park",
"Kikuo Fujimura",
"Mykel J. Kochenderfer"
],
"categories": [
"cs.RO",
"cs.AI",
"cs.CV",
"cs.LG",
"cs.MA"
],
"primary_category": "cs.RO",
"published": "20231127185742",
"title": "Interactive Autonomous Navigation with Internal State Inference and Interactivity Estimation"
} |
Mathematical analysis of FLASH effect models]Mathematical Analysis of FLASH Effect Models Based on Theoretical Hypotheses^1Department of Engineering Physics, Tsinghua University, Beijing, China^2Key Laboratory of Particle and Radiation Imaging, Tsinghua University, Ministry of Education, Beijing, China [email protected] November 2023 Objective: Clinical applications of FLASH radiotherapy require a model to describe how the FLASH radiation features and other related factors determine the FLASH effect. Mathematical analysis of the models can connect the theoretical hypotheses with the radiobiological effect, which provides the foundation for establishing clinical application models. Moreover, experimental and clinical data can be used to explore the key factors through mathematical analysis. Approach: We abstract the complex models of the oxygen depletion hypothesis and radical recombination-antioxidants hypothesis into concise equations. Then, the equations are solved to analyze how the radiation features and other factors influence the FLASH effect. Additionally, we show how to implement the hypotheses' models in clinical application with the example of fitting the experimental data and predicting the biological effects.Main results: The formulas linking the physical, chemical and biological factors to the FLASH effect are obtained through mathematical solutions and analysis of the equations. These formulas will enable the utilization of experimental and clinical data in clinical applications by fitting the data to the formulas. Based on this analysis, we propose suggestions for systematic experiments toward clinical FLASH radiotherapy.Significance: Our work derives the mathematical formulas that elucidate the relationship between factors in the oxygen depletion hypothesis and radical recombination-antioxidants hypothesis, and the FLASH effect. These mathematical formulas provide the theoretical basis for developing the clinical application models for FLASH radiotherapy. Furthermore, the analysis of these hypotheses indicates the key factors of the FLASH effect and offers references for the design of systematic experiments toward clinical applications. Keywords: FLASH radiotherapy, mathematical model, oxygen depletion, radical recombination and antioxidants[ Ankang Hu^1,2, Wanyi Zhou^1,2, Rui Qiu^1,2 and Junli Li^1,2,*================================================================= § INTRODUCTIONFLASH radiotherapy is a prominent topic in the field of radiotherapy <cit.>. Its unique radiobiological advantage compared to conventional dose rate (CONV) radiotherapy makes it possible to benefit numerous cancer patients. FLASH effect has been observed in the experiments using various types of beams such as electron <cit.>, low energy X-ray <cit.>, high energy X-ray <cit.>, proton <cit.>, and even carbon ion <cit.>. Clinical trials of radiotherapy are currently underway <cit.>.However, the mechanism of FLASH effect remains unclear, inspiring diligent efforts to unravel its complexities with several analytical hypotheses having been introduced. The oxygen depletion hypothesis <cit.> is one of the most widely discussed hypotheses. The radical recombination hypothesis <cit.> and its expansion, radical recombination-antioxidants hypothesis <cit.>, attempt to explain the FLASH effect mechanism from the perspective of radiochemistry. Besides, some researchers proposed their hypothesis based on their theoretical models or experimental results, such as “protection of circulating immune cells” <cit.>, “DNA integrity” <cit.> and “mitochondrial damage response” <cit.>. These hypotheses provide diverse explanations for the FLASH effect, contributing valuable references for future investigations. While some hypotheses qualitatively propose potential mechanisms, others provide quantified descriptions. Hypotheses accompanied by quantitative models serve as foundations for establishing models suitable for clinical applications.Toward clinical applications, it is crucial for researchers and clinicians to establish a model that describes the relationship between radiobiological effects and radiation features. For instance, models have been developed to calculate the relative biological effectiveness of proton and heavy ion radiotherapy, enabling the prediction of radiobiological effects based on microdosimetric parameters <cit.>. Similarly, the application of FLASH radiotherapy necessitates models that elucidate how irradiation features, such as total dose, dose rate, and irradiation time, as well as other chemical/biological factors, determine the FLASH effect. In pursuit of establishing a practical model for clinical application, researchers have attempted to quantitatively predict the clinical effect using experimental data, simulations, and radiobiological models. The FLASH modifying factor was introduced and calculated based on the summary of experimental data <cit.>. The tumor control probability of FLASH effect was analyzed based on a model named as UNIVERSE <cit.>. A formalism <cit.> was developed that quantifies the minimal normal tissue sparing of the FLASH effect required to compensate for hypofractionation. However, their models and analyses fall short in linking the FLASH effect to theoretical considerations, thus failing to capture the influence of mechanism factors on the FLASH effect.In this study, we quantitatively analyze the mathematical models of FLASH effect based on the oxygen depletion hypothesis and the radical recombination-antioxidants hypothesis, and subsequently develop the corresponding clinical models to describe the impact of FLASH irradiation features and mechanism factors on the FLASH effect. Through the models and analysis, we offer valuable insights into the FLASH effect mechanism and lay the groundwork for the establishment of practical models for the clinical application of FLASH radiotherapy. Furthermore, the analysis serves as a reference for the design of systematic experiments aimed at advancing clinical FLASH radiotherapy.§ MATERIALS AND METHODSThe actual scenario of FLASH irradiation is inherently complex. To conduct a mathematical analysis of the FLASH effect using theoretical hypotheses, it is necessary to simplify the intricate situation into concise mathematical representations. In this work, we abstract the complex models of the oxygen depletion hypothesis and radical recombination-antioxidants hypothesis into concise equations. Then we solve these equations to examine the impact of radiation features and other factors on the FLASH effect. Moreover, we show how to implement the hypotheses' models in clinical applications by the example of fitting the experimental data and predicting the biological effects. §.§ Model based on oxygen depletion hypothesis§.§.§ Mathematical modelThe oxygen depletion hypothesis is an extensively discussed theory regarding the mechanism behind the FLASH effect. This hypothesis posits that the rapid delivery of radiation in FLASH radiotherapy results in significant oxygen depletion in the tissue, due to the intense radiation-induced consumption of oxygen. Attributed to the finite oxygen diffusion speed, oxygen cannot be recovered promptly in the radiation region <cit.>. With the assumption that cells under FLASH radiation instantly become hypoxic, they exhibit increased radioresistance compared to the cells exposed to CONV radiation. The radiation-induced oxygen consumption and oxygen diffusion are two key processes in the hypothesis.To establish a quantitative model for analysis and clinical application of the FLASH effect, we abstract the key processes into an equation (ROD1) based on the oxygen depletion hypothesis. dp(t)/dt=DIF(t, p)-ROC(t, p)where p(t) is the concentration of oxygen at the time point t; DIF(t, p) is the term of oxygen diffusion in the tissue; ROC(t, p) is the radiolytic oxygen consumption. The differential equation has an initial condition listed in ROD_IC.p(t=0)=p_0where p_0 is the oxygen concentration before irradiation.Based on the previous work, the process of oxygen recovery attributed to diffusion is approximately in the form of an exponential function <cit.>. The term of oxygen diffusion can be described by ROD_dif.DIF(t, p)=λ (p(t)-p_0)where λ is the feature constant of the oxygen diffusion, which is mainly determined by the density of microvessels in the tissue. The radiolytic oxygen consumption is mainly attributed to the reactions between radiation-induced radicals and oxygen <cit.>. In many related studies, the radiolytic oxygen consumption rate (μM/Gy) is often treated as a constant <cit.>. However, this assumption is invalid in the cells where the oxygen concentration is low. Some studies set the oxygen consumption rate proportional to the oxygen consumption <cit.>, which also cannot reflect the real condition when the oxygen concentration is high. The change in radiolytic oxygen consumption rate for different dose rate irradiation can be attributed to two effects: reactions between radiation-induced radicals and competition between oxygen and other cellular compositions (for example, antioxidants) reacting with radicals<cit.>. Because the lifetime of radiation-induced radicals is generally too short ( ns) and it does not correspond to the microsecond time scale of radiolytic oxygen consumption, reactions between radiation-induced radicals cannot play the primary role in the process. Thus, it is derived that the competition between oxygen and cellular composition is the main factor influencing radiolytic oxygen consumption. The amount of radicals is proportional to the radiation dose. Based on the above analysis, we use a form of fraction to describe the competition between oxygen and other cellular compositions <cit.>. The term of radiolytic oxygen consumption can be described by ROD_ROD.ROD(t, p)=g_1Ḋ(t)·p(t)/p(t)+Awhere g_1 is the yield of radiation-induced radicals; Ḋ(t) is the dose rate at the time point t; A represents equivalent concentration (by considering the reaction rate relative to oxygen and concentration) of other cellular compositions reacting with radicals. Other works of model study and experiments indicate that the average dose rate during the irradiation is the key factor related to the FLASH effect <cit.>. To simplify the equation in this study, we assume the dose rate is constant during the irradiation so that the term of dose rate is ROD_DR.Ḋ(t)=D/TD is the total dose and T is the total irradiation time.With the above assumptions and simplifications, ROD1 is converted into ROD2.dp(t)/dt=λ (p(t)-p_0)-g_1·D/T·p(t)/p(t)+A §.§.§ Estimation of the radiobiological effectThe radiobiological effect is often estimated based on the classical Alper's formula <cit.> of the radiation oxygen effect listed in ROD_Alper.OER = K+mp(t)/K+p(t)where OER is the oxygen enhancement ratio, which represents the ratio of the dose in the hypoxic condition to the dose in a given oxygen condition p(t) required to reach the same biological endpoint, or the ratio of the damage under p(t) oxygen to the damage in the hypoxic condition with the same dose delivered; m is the maximal OER and K is the oxygen concentration at the half-maximal OER. The biological effect of the FLASH irradiation is calculated by the dose-averaged OER of the irradiation, for the constant dose rate condition, the biological effect can be calculated by ROD_BE. BE=1/T∫_0^TK+mp(t)/K+p(t)dt §.§.§ Radiobiological effects of FLASH and CONV irradiationAs for the CONV condition, because of the low dose rate delivery, the oxygen concentration keeps almost unchanged during the irradiation. Thus, the biological effect can represented as ROD_CONV,BE_CONV=K+mp_0/K+p_0 For FLASH irradiation, the biological effect can be estimated by solving ROD2 to derive p(t) and calculating the integral in ROD_BE. The results are shown in the results and discussions section. To estimate the maximal FLASH effect, we calculate the limit of the function defined by ROD_BE by setting T→ 0. Within the extremely short irradiation duration, the oxygen diffusion can be ignored. The relationship between oxygen concentration at the end of irradiation and the total dose can be derived by solving ROD3 with initial condition ROD_IC.dp(t)/dt = -g_1D/T·p(t)/p(t)+A By setting t=T in the solution of ROD3, we can calculate the oxygen concentration at the end of irradiation, p_t, to estimate the change of oxygen concentration during FLASH irradiation. Then we define a dose modifying factor (DMF) <cit.> in DMF_DEF to represent the FLASH effect. The factor is defined as the ratio of the damage induced by FLASH irradiation to the damage induced by CONV irradiation with the same dose delivered.DMF=Damage_FLASH(D)/Damage_CONV(D)§.§ Model based on radical recombination-antioxidants hypothesis§.§.§ Mathematical modelThe radical recombination-antioxidants hypothesis explains the protective effect of normal tissue by the recombination of peroxyl radicals (including superoxide anion)<cit.>. Additionally, it elucidates the loss of this protective effect in tumors due to high levels of antioxidants present <cit.>. The main point of establishing the mathematical model is to appropriately describe the reaction of peroxyl radicals. Despite the inherent complexity of these reactions, we can simplify the reaction model into three primary processes: reaction of peroxyl radical with antioxidant, peroxyl radical recombination and generation of peroxyl radical by irradiation, as demonstrated in a concise form in RA1.dR(t)/dt = -k_1R(t)-k_2[R(t)]^2+g_2Ḋ(t)where R(t) is the concentration of peroxyl radicals; g_2 is the yield of peroxyl radicals; k_1 is the first-order rate constant of peroxyl radicals reacting with antioxidants, including the effects of concentration and rate constant; k_2 is the second-order rate constant of peroxyl radical recombination, which roughly represents the reactions of different types of reactants.According to the same reason described in the model of oxygen depletion, we consider the radiation delivered at a constant dose rate. Thus, RA1 can be divided into two stages: RA_2_1 during irradiation and RA_2_2 after irradiation.dR(t)/dt = -k_1R(t)-k_2[R(t)]^2+g_2D/T,R(0)=0 dR(t)/dt = -k_1R(t)-k_2[R(t)]^2,R(T)=R_Twhere R_T is the concentration of peroxyl radicals at the end of irradiation, which can be calculated by the solution of RA_2_1. §.§.§ Estimation of radiobiological effectThe radiation-induced damage can be classified into two types: the peroxyl radical-dependent damage and the peroxyl radical-independent damage. Peroxyl radical-dependent damage can be estimated by the integral of peroxyl radical concentration defined inRA_AUC.AUC[ROO·] = ∫_0^+∞R(t)dtwhere AUC[ROO·] is considered to be proportional to peroxyl radical-dependent damage. Then the total damage can be calculated as RA_DAM.Damage = f_1· AUC[ROO·] + f_2Dwhere f_1 is the factor related to radical-dependent damage; f_2 is the factor related to radical-independent damage. The relative values of f_1 and f_2 are mainly determined by the radiation quality, e.g., the energy and type of the particle.§.§.§ Radiobiological effect of FLASH and CONV irradiationWe calculate the limit of the damage function defined by RA_DAM to estimate the maximal radiobiological effect of FLASH and CONV irradiation.FLASH: lim_T→ 0DamageCONV: lim_T→ +∞DamageWith the results of RA_L1 and RA_L2, the minimal DMF value by a given dose, DMF_min, can be calculated based on the definition of DMF_DEF, which can be used to analyze how the total dose and antioxidants influence the FLASH effect. §.§ Parameters of the modelsThe primary objective of this study is to derive mathematical formulas that establish the relationship between key factors in hypotheses and the FLASH effect. These formulas encompass comprehensive information and precisely describe the correlation between parameters and the FLASH effect. In order to provide a visual representation of the formulaic trends, figures are utilized. It is important to note that these figures are designed to illustrate the shape of the curves described by the formulas, rather than to reflect real-world scenarios. Generating these figures necessitates the explicit values of parameters. While some of the parameters are determined based on estimates derived from existing literature, others are assigned values without specific justifications, as these figures solely serve to depict the curve shapes within the formulas. The parameters employed in the model of the oxygen depletion hypothesis are listed in table:PROD, while the parameters utilized in the model of the radical recombination-antioxidants hypothesis are listed in table:PRA. §.§ Implementation of the hypotheses for clinical applicationsClinical applications of radiotherapy necessitate quantitative models that describe the dose-biological effectiveness relationship. These models rely on data obtained from experiments and clinical trials. Specifically, future experiments and clinical trials of FLASH radiotherapy are supposed to provide data sets with different total doses and dose rates (conditions). In this work, concise models are established based on two main hypotheses, which can be used to fit the data points and predict the FLASH effect for an exact irradiation situation when the total dose and dose rate are given.<cit.> synthesized the experimental results according to different doses of FLASH irradiation. We attempt to use the formulas of DMF derived from our mathematical analysis of hypotheses to fit these experimental data points. The fitted curves are expected to approximate the relationship between the total dose and the radiobiological effect. We employ curve fitting as an example to demonstrate the implementation of the theoretical hypotheses. Furthermore, we analyze the available experimental data using the models based on these hypotheses. Based on this analysis, we propose suggestions for the design of systematic experiments toward clinical FLASH radiotherapy.§ RESULTS AND DISCUSSIONS §.§ Mathematical analysis of the model based on oxygen depletion hypothesis§.§.§ Solution of the equationThe solution of ROD2 is shown in RROD1. It is hard to get an explicit solution, so we list an implicit solution here. -A+p_1/p_1-p_2log|p(t)-p_1/p_0-p_1|+A+p_2/p_1-p_2log|p(t)-p_2/p_0-p_2|=λ twhere p_1 and p_2 are two constants calculated by RROD2_1 and RROD2_2.p_1=- A T λ - g_1 D + T λ p_0 - √(4 A T^2λ^2 p_0 + (A T λ + g_1 D - T λ p_0)^2)/2 T λp_2=- A T λ - g_1 D + T λ p_0 + √(4 A T^2λ^2 p_0 + (A T λ + g_1 D - T λ p_0)^2)/2 T λThen we calculated the integral in ROD_BE by changing the integration variable from t to p as shown in RROD_int.BE = m-(m-1)K/T∫_p_0^p_T1/p+K·dt/dp·dpThe result of the integral is shown in RROD_intR.BE=m-(m-1)K[-A-K/Tλ(K+p_1)(K+p_2)logK+p_T/K+p_0 -A+p_1/Tλ(K+p_1)(p_1-p_2)log|p_T-p_1/p_0-p_1| + A+p_2/Tλ(K+p_2)(p_1-p_2)log|p_T-p_2/p_0-p_2|] RROD_intR brings a challenge to obtain a result limited by the numerical precision when the difference between p_T and p_0 is small (low total dose or long irradiation time). We use the first-order Taylor expansion of RROD1 to calculate p(t), which is shown in RROD_APP.p(t)≈λ(p_0p_1+p_0p_2-p_1p_2-p_0^2)/A + p_0t+p_0 With the similar mathematical process with RROD_APP, the biological effect of the above-mentioned condition (p_T→ p_0) can be calculated by RROD_APPBEBE≈ m + K(A+p_0)(m-1)/Tλ(p_0-p_1)(p_0-p_2) [-log(K+p_0) +log(K+p_0+Tλ (p_0p_1+p_0p_2-p_1p_2-p_0^2)/A+p_0)] With the above formulas, the oxygen concentrations at the end of irradiation and the radiobiological effects after irradiation with different irradiation times are calculated and shown in fig:PTROD. The curves of oxygen concentration and biological effect both show an "S" shape, which corresponds to the trend of FLASH effect vs dose rate indicated in the experimental result <cit.>. The shape of curves reflects the time feature of the FLASH effect, which is mainly determined by the parameter of oxygen diffusion, λ, according to the oxygen depletion hypothesis. The result of the mathematical analysis reflects the principle of the hypothesis and indicates the limited oxygen diffusion as the key factor of the FLASH effect. Moreover, the data points of BE in fig:PTROD can be well fitted by a simple formula listed in RROD_FITBE BE = 2.003 + 0.1038/1+exp(-1.204log(T·s) - 1.54), R^2=0.9978 Where s is the time unit, second, which aims to eliminate the influence of the unit when calculating the logarithm.§.§.§ Radiolytic oxygen consumptionRadiolytic oxygen consumption is one of the key factors in the oxygen depletion hypothesis.We set Δ p = p_T-p_0 to represent the change of oxygen during FLASH irradiation. The relationship between Δ p and the total dose D is shown in ROD_dp.Δ p-Alog(1-Δ p/p_0)=g_1D ROD_dp indicates that the radiolytic oxygen consumption is no longer proportional to the dose magnitude when the initial oxygen concentration is low or the dose is large to make the cell hypoxic. The oxygen consumption induced by different total doses under different initial oxygen concentrations is shown in fig:ROC. These results show that the assumption of previous models of oxygen depletion hypothesis may lead to the predicted results deviating from reality.Moreover, we also change the equivalent concentration of other cellular composition A to study the influence of antioxidants on radiolytic oxygen consumption. The oxygen consumption results induced by 20 Gy FLASH irradiation are shown in fig:ROC_A, which are in the cells with A ranging from 1.0 to 20.0 μM/Gy under different initial oxygen concentrations.Results show that the competition between oxygen and other cellular composition influence radiolytic oxygen consumption greatly when the initial oxygen concentration is low. The radiolytic oxygen consumption may not be regarded as constant, especially for low oxygen concentration conditions. §.§.§ Radiobiological effects of FLASH and CONV irradiationThe radiobiological effect of CONV irradiation was calculated in the Materials and Methods section. The biological effect of FLASH irradiation is estimated by calculating the limit of RROD_intR when T → 0, which represents the maximal change of biological effect by a given dose of FLASH irradiation. The result is shown in RROD_FLASHBE_FLASH=m-(m-1)[A-K/g_1Dlog(1-Δ p/K+p_0)+g_1D-Δ p/g_1D]Then the DMF_min is shown in RROD_DMFDMF_min = - (K + p_0) (-(A-K/g_1Dlog(1-Δ p/K+p_0)+g_1D-Δ p/g_1D) (m - 1) + m)/K + m p_0The DMF_min for different doses and initial oxygen concentrations are shown in fig:DMFROD. The curves of DMF vs dose indicate that the FLASH effect predicted by the oxygen depletion hypothesis is influenced by the initial concentration greatly. For normal oxygenated tissues, the biological effect changes slightly. For hypoxic even extremely hypoxic (p_0 = 1μ M) tissues, the biological effect alters greatly. This result brings challenges to the oxygen depletion hypothesis when explaining the effect of tumors. §.§ Mathematical analysis based on radical recombination-antioxidants hypothesis§.§.§ Solution of the equationThe solutions of the model based on the radical recombination-antioxidants hypothesis contain two parts. The first part is the solution of RA_2_1, which describes the reactions during irradiation (0≤ t < T). The solution is shown in RRA_2_1.R(t)=√(4g_2Dk_2/T+k_1^2)/k_2/C_1exp(t√(k_1^2+4k_2g_2D/T))-1-k_1-√(4g_2Dk_2/T+k_1^2)/2 k_2C_1 is a constant determined by the initial condition.C_1 = k_1 + √(4 g_2D k_2/T + k_1^2)/k_1 - √(4 g_2D k_2/T + k_1^2)The second part is the solution of the equation which describes the reactions after irradiation (t≥ T). The solution is shown in RRA_2_2.R(t)=k_1/k_21/C_2exp(k_1t)-1C_2 is a constant determined by the initial condition.C_2 =e^-Tk_1+ k_1^2Te^- T k_1/2 g_2D k_2 + (e^T √(4 g_2D k_2/T + k_1^2) +1) Tk_1e^- T k_1√(4 g_2D k_2/T + k_1^2)/2 g_2 D k_2 (e^T √(4 g_2D k_2/T + k_1^2) - 1)Then we calculated the integral of the solutions of these two parts based on RA_AUC. For the first part (integral of RRA_2_1), the integral interval is [0, T], and the result is shown in RRA_int1.AUC[ROO·]_part1 = T (- k_1 - √(4 g_2D k_2/T + k_1^2))/2 k_2 -log|-1/C_1+1|/k_2 + log|-1/C_1+e^T √(4 g_2D k_2/T + k_1^2) |/k_2For the second part (integral of RRA_2_2), the integral interval is [T, +∞). The result is shown in RRA_int2AUC[ROO·]_part2 = k_1 T/k_2 - log|e^k_1 T - 1/C_2 |/k_2The sum of these two parts (AUC[ROO·]) represents the damage related to peroxyl radicals. The curve of AUC[ROO·] vs irradiation time, T, is shown in fig:RAT.The "S" shape curve predicted by the model indicates that the radical recombination-antioxidants hypothesis can explain the time feature of FLASH to a certain extent. The feature time is mainly determined by the reaction between peroxyl radicals and antioxidants, which is represented by the parameter k_1. The difference between FLASH and CONV is also influenced by the k_1 parameter greatly. These reflect the principle of the hypothesis, in which the peroxyl radical-related reactions dominate the FLASH effect.§.§.§ Radiobiological effects of FLASH and CONV irradiationThe DMF(T) can be calculated by RRA_int1, RRA_int2, RA_DAM and DMF_DEF to estimate the biological effect brought by FLASH irradiation with different irradiation times. However, the formula is quite complex. We find that the simple formula in RRA_DMFT is an excellent approximation (R^2>0.99) of DMF(T) in wide ranges of parameters (we tested it by setting dose to 0.1-100 Gy and setting k_1 to 0.01-100 s^-1).DMF(T) = 1 - (1-DMF_min)T_m/T+T_mWhere DMF_min is the minimal value of DMF for a given dose, which is equal to the value calculated by RRA_DMF; T_m is a characteristic time, which is determined by k_1 and D. We also find that the formulas of T_m(k_1) and T_m(D) have concise forms listed in RRA_TM1 and RRA_TM2.T_m(D) = a_1exp(a_2·log(D·Gy^-1) ) +a_3T_m(k_1) = b_1exp(-b_2 log(k_1 ·s)), k_1≥ 1 s^-1where a_1, a_2, a_3, b_1, b_2 are parameters that can obtained by fitting; s and Gy are units of time and dose, aiming to eliminate the influence of units when calculating logarithms.We show the data points used in fitting and fitted curves of T_m(D) and T_m(k_1) in fig:RRA_DMFT. For clinical application, the T_m for a given dose can be obtained by fitting the data points of (log(T), DMF(D,T)). The T_m(D) can be obtained by fitting the data points from a series of doses.Based on the methods list in section 2.2.3, we calculated the damage induced by FLASH (RRA_FLASH) and CONV (RRA_CONV) irradiation to obtain the minimal value of DMF, DMF_min, to evaluate the maximal change of biological effect by a given dose of FLASH irradiation. Damage_FLASH = f_1log (g_2k_2D/k_1+1)/k_2 + f_2D Damage_CONV = f_1g_2D/k_1 + f_2DThen the DMF_min can be calculated and the result is shown in RRA_DMF.DMF_min=Damage_FLASH/Damage_CONV = [f_1 log (g_2 k_2D/k_1+1)]/k_2+f_2D/f_1g_2D/k_1+f_2DWe obtain a concise formula and it can be utilized to fit the experimental data conveniently. According to this formula, fig:DMFRA shows the DMF_min irradiated by different doses under cells with different levels of antioxidants, k_1. The curves indicate that the antioxidants in the cell greatly influence the change of biological effect attributed to the FLASH irradiation. This also corresponds to the hypothesis' explanation of the non-protective effect in tumors. Moreover, the curves show that the difference between FLASH and CONV irradiation increases with the increase of the total dose when the total dose is low. Then the DMF_min changes relatively slightly with the increase of total dose when the total dose is high enough. The model may provide a reference for the clinical application of FLASH radiotherapy. §.§ Implementation of hypotheses for application§.§.§ Results of curve fittingWe attempt to fit the experimental data points using RROD_DMF and RRA_DMF. The fitted curves are shown in fig:FIT. The data utilized for curve fitting originates from experiments conducted on various tissues and animals. However, due to the inconsistent experimental setups, the data points exhibit large uncertainty. Consequently,they can be fitted well with neither of the mathematical models based on the two hypotheses, as shown in. The fitting result underscores the necessity of high-quality experimental data for the clinical application of FLASH radiotherapy and the exploration of the FLASH effect mechanism, which can be obtained through systematic experiments described in the next section.§.§.§ Suggestions for systematic experimentsThe protective effect of normal tissues is the most important biological advantage of FLASH effect, which can be quantified by the change of normal tissue complication probabilities (NTCP) induced by radiation at different dose rates. The NTCP models indicate that the NTCP often undergoes a substantial increase within a narrow dose interval <cit.>, beyond which it exhibits almost no variation. Notably, as an intrinsic characteristic of the NTCP, this narrow interval often shows at relatively high doses. Consequently, if an experiment aims to implement NTCP as the endpoint of the effect, a single-dose irradiation cannot provide sufficient information for the effect at relatively low dose levels. The existing data points on the FLASH effect primarily cluster at high doses <cit.>, offering limited insights into the effect at relatively low dose levels, however, which are highly relevant for clinical radiotherapy and the verification of the hypotheses as well. To fill the gap, systematic experiments designed to explore the FLASH effect across a wide dose range are required. However, as for the low-dose level, it is challenging to obtain effective data points to represent the FLASH effect through simple experiments of FLASH irradiation in the one-fraction fashion, due to the intrinsic characteristic of the NTCP model.To overcome the limitation, we propose a series of experiments using fractionated FLASH dose setups. In these experiments, the definition of fraction differs from the general definition in the field of radiotherapy. The duration between two consecutive fractions is several minutes rather than the common practice of one day. The total dose of FLASH irradiation is aimed to cover the high-dose interval where NTCP changes greatly, which is different from that of CONV because of the FLASH effect. However, for the limited granularity of fractionation dose, simple fractionated irradiation may not be able to achieve the total dose target. Thus, a hybrid irradiation strategy is proposed to supplement the limitation. The detailed description of the systematic experiments is listed below. * CONV group. For CONV irradiation, the total dose is set to reach the interval where NTCP changes greatly. Then an endpoint is chosen to obtain the reference dose, D_CONV.* Fractionated FLASH dose setup. For FLASH irradiation, set a series of fractionated doses, D_FLASH,f, e.g., 1, 2, 5, 10,... Gy. For each setup of fractionated dose FLASH irradiation, we can define a virtual iso-effective dose, D_FLASH,ISO(D_FLASH,f), which induces the same radiobiological effect with D_CONV CONV irradiation. The virtual iso-effective dose, D_FLASH,ISO, often cannot be obtained by fractionated FLASH irradiation solely because of the limited granularity of the fractionated dose.* Hybrid irradiation strategy. For each dose setup of fractionated FLASH irradiation, to start with, D_FLASH,f is delivered at FLASH dose rate for n times to the experimental group with the duration of several minutes between two fractions, where n is the maximal integer for n· D_FLASH,f≤ D_FLASH,ISO. Then the group is irradiated by a residual CONV dose to obtain the iso-effect dose for the fractionation dose setup, D_hybrid, ISO(D_FLASH,f), which induces the same radiobiological effect with D_CONV CONV irradiation. The D_hybrid, ISO(D_FLASH,f) is the sum of n· D_FLASH,f and the residual CONV dose. With the systematic experiments, the DMF of the fractionation dose for FLASH effect can be calculated by DMF_EX. DMF(D_FLASH,f) = D_CONV-[D_hybrid,ISO(D_FLASH,f)-n· D_FLASH,f]/n· D_FLASH,f These experiments enable the acquisition of DMF versus dose and dose rate curves across a wide range. The established dataset is expected to provide as a fundamental base for advancing clinical FLASH radiotherapy.§.§ LimitationsThis study focuses on the quantitative analysis of the oxygen depletion hypothesis and the radical recombination-antioxidants hypothesis, which inevitably has some limitations. First, it should be noted that real-world scenarios are highly simplified for modeling with some assumptions potentially deviating from actual conditions. The estimation of radiobiological effects relies on simplified models such as the oxygen enhancement ratio and the time integral of peroxyl radicals, which may not fully reflect the comprehensive impact. Second, for some reason, some hypotheses are not included. For instance, the "protection of circulating immune cells" hypothesis lacks a widely accepted quantitative model that links the survival of circulating immune cells to damage in normal tissues and tumors. Similarly, the "DNA integrity" and "mitochondrial damage response" hypotheses lack quantitative descriptions, preventing the establishment of mathematical models for them. Third, for implementation of hypotheses, the existing FLASH experimental data points are not fitted well using the mathematical models in this work because of the overlook of the intrinsic characteristic of NTCP in the experimental design, which, however, is anticipated to be improved with high-quality datasets established in the future, referring to the suggestions listed above.§ CONCLUSIONIn this work, We formulated concise equations to abstract the oxygen depletion hypothesis and radical recombination-antioxidants into mathematical models. These equations were then solved to examine the influence of radiation features (total dose and irradiation time) and factors within the hypotheses (initial oxygen concentration and antioxidants) on the FLASH effect. Through the mathematical analysis, the formulas of DMFs are derived for clinical FLASH radiotherapy.The mathematical analysis of these hypotheses highlights the key factors that determine the FLASH effect. In the case of the oxygen depletion hypothesis, the diffusion of oxygen governs the timing characteristic of the effect. The competition between oxygen and other cellular components in reacting with radiation-induced radicals greatly impacts radiolytic oxygen consumption. The initial oxygen concentration plays a crucial role in the change of biological effects caused by FLASH irradiation. Notably, FLASH irradiation can greatly alter the biological effects of hypoxic and extremely hypoxic tissues.In the case of the radical recombination-antioxidants hypothesis, the reaction between antioxidants and peroxyl radicals determines the timing characteristic of the FLASH effect. Antioxidants contribute to the differences in biological effects observed between FLASH and CONV irradiation. The DMF exhibits a substantial increase with the total dose in the low-dose range, followed by a relatively slight change in the high-dose level.Then,to set examples for the implementation of hypotheses, an attempt to establish the clinical application models is made by fitting the experimental data with the formulas of DMFs. Linking the parameters in hypotheses to clinical effects, the models can be used to predict the FLASH effect with a given total dose and also provide clues for the exploration of the FLASH effect mechanism. Besides, we proposed our suggestions based on our mathematical analysis of hypotheses for the design of future systematic experiments toward clinical FLASH radiotherapy.§ ACKNOWLEDGMENTSThis work was supported by the National Key Research and Development Program of China (Grant No. 2022YFC2402300 and 2021YFF0603600) and National Natural Science Foundation of China (Grant No. 12175114 and U2167209). The authors thank Kaiwen Li and Jianqiao Wang for their help in the preparation of the manuscript. | http://arxiv.org/abs/2311.15558v2 | {
"authors": [
"Ankang Hu",
"Wanyi Zhou",
"Rui Qiu",
"Junli Li"
],
"categories": [
"physics.med-ph"
],
"primary_category": "physics.med-ph",
"published": "20231127055326",
"title": "Mathematical Analysis of FLASH Effect Models Based on Theoretical Hypotheses"
} |
: Scaling Medical Pretraining for Large Language ModelsZeming Chen1 Alejandro Hernández Cano1equal contribution, ^†equal supervision Angelika Romanou1 Antoine Bonnet1 Kyle Matoba1,2Francesco Salvi1 Matteo Pagliardini1 Simin Fan1 Andreas Köpf3 Amirkeivan Mohtashami1 Alexandre Sallinen1 Alireza Sakhaeirad1 Vinitra Swamy1 Igor Krawczuk1 Deniz Bayazit1 Axel Marmet1 Syrielle Montariol1Mary-Anne Hartley1,4 Martin Jaggi1† Antoine Bosselut1†1EPFL 2Idiap Research Institute 3Open Assistant 4Yale==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION Monte Carlo methods have been successful to study non-perturbative physics phenomena, ranging from nuclear physics to condensed matter physics. However, they still suffer from many issues: sign problem <cit.>, ergodicity problem <cit.>, and infinite variance problem, which make the estimation of observables exponentially hard. Especially, infinite variance problem causes the estimation of observables to be impossible since variances are divergent with the progress of Monte Carlo samplings.One way to deal with the diverging variance for fermionic observables is to employ discrete auxiliary fields <cit.>. However, one wants to use continuous auxiliary fields to use hybrid Monte Carlo <cit.> for faster convergence. Also, sign problem has been studied widely and some methods require the complexification of the integration domain <cit.>, which can only be implemented with continuous variable Monte Carlo calculations.In this work, we review infinite variance problem from fermionic observables and discuss its solution while employing continuous auxiliary fields. Specifically, we use Hubbard model as a testbed to show that the extra time-slice reweighting with sub Monte-Carlo methods can remove the diverging variance without additional errors.Hubbard model, which is a strong candidate for explaining high-temperature superconductors, consists of hopping term, local interaction term, and chemical potential term:H= - κ∑_⟨ x,y ⟩(ψ̂^†_↑,xψ̂_↑,y+ψ̂^†_↓,xψ̂_↓,y)+ U ∑_x(ψ̂^†_↑,xψ̂_↑,x-1/2)(ψ̂^†_↓,xψ̂_↓,x-1/2)- μ∑_x(ψ̂^†_↑,xψ̂_↑,x+ψ̂^†_↓,xψ̂_↓,x-1).On bipartite lattices, one can use the particle-hole symmetry to rewrite the Hamiltonian. By redefining ψ̂_1,x≡ψ̂_↑, x and ψ̂_2,x≡ (-1)^x ψ̂_↓, x, one can find thatH =- κ∑_⟨ x,y ⟩(ψ̂^†_1,xψ̂_1,y+ψ̂^†_2,xψ̂_2,y)+ U/2∑_x(ψ̂^†_1,xψ̂_1,x-ψ̂^†_2,xψ̂_2,x)^2 - μ∑_x(ψ̂^†_1,xψ̂_1,x-ψ̂^†_2,xψ̂_2,x). To remove the fermionic variables for Monte Carlo calculations using path integral formulation, one can use a Hubbard-Stratonovich transformation. Since there are some advantages for using compact auxiliary fields when one utilizes the contour deformation method for ameliorating sign problem, we choose the compact continuous Hubbard-Stratonovich transformation. Then one can find that (Details can be found in <cit.>.)⟨𝒪⟩= ∫ Dϕ 𝒪(ϕ) e^-S_0(ϕ) M_1(ϕ)M_2(ϕ)/∫ Dϕe^-S_0(ϕ) M_1(ϕ)M_2(ϕ),whereS_0(ϕ) = -β∑_x,tcosϕ_t,x andM_a(ϕ) = 𝕀 + B_a(ϕ_N) ⋯ B_a(ϕ_1).Each term in the fermion matrices is written asB_a(ϕ_t) =e^- H_2 e^- H̃_4(ϕ_t),where(H_2)_x,y= κϵδ_⟨ x,y ⟩+ε_a μϵδ_x,y,H̃_4(ϕ_t)_x,y= -i ε_a sinϕ_t,xδ_x,y.Here, δ_⟨ x,y ⟩ is the nearest-neighbor hopping matrix, and ε_1=+1,ε_2=-1. The parameter α is related to the potential U bye^-ϵ U/2 =I_0(√(α^2-1))/I_0(α). Since M_2(ϕ)=M_1(ϕ)^* at the half-filling, i.e. μ=0, Hubbard model does not have the sign problem. In this work, we will only consider the half-filling case to remove the effect of the sign problem. Also, we will use the unit κ≡ 1.§ INFINITE VARIANCE PROBLEM Let us consider the expectation value of fermionic observables, i.e. 𝒪 = f(ψ̅, ψ). Using the Hubbard-Stratonovich transformation and the Gaussian Grassmann integration formula, one can find that⟨𝒪⟩= ∫ Dψ̅ Dψ 𝒪e^-S(ψ̅, ψ)/∫ Dψ̅ Dψe^-S(ψ̅, ψ) = ∫ Dϕg( M_ij(ϕ) ) e^-S_0(ϕ)/∫ Dϕe^-S_0(ϕ) M(ϕ),where g is a polynomial. Since the observable is proportional to the inverse of the fermion determinant, i.e., g ∝ 1/ M(ϕ). This holds for “exceptional concifurations” where M=0. Near the exceptional configurations where the determinant is nonzero but very small, the observable is very large, which makes variance jumps in the left panel of Fig. <ref>. While the expectation value is finite when one does the integration of Eq. (<ref>), the divergence can be infinite since the variance has the term proportional to the square of observable: σ^2_𝒪 = ⟨𝒪^2 ⟩ - ⟨𝒪⟩^2. In terms of the path integral representation,⟨𝒪^2 ⟩ = ∫ Dϕg^2 (M_ij(ϕ) ) 1/ M(ϕ)e^-S_0(ϕ)/∫ Dϕe^-S_0(ϕ) M(ϕ).Therefore, since g^2e^-S_0≥ 0 cannot remove the singularity from 1/ M unless g=0 with the same order as the exceptional configurations, one cannot avoid the diverging variance if the determinant of fermion matrices have zero points.There is an example in Hubbard model which shows that the infinite variance problem stems from the fermion determinant. Let us consider the two observables, double occupancy and density, in terms of fermion matrices:D(ϕ) = 1/V∑_x ⟨ n_↑(x) n_↓(x) ⟩ = 1/V∑_x M^-1_2(ϕ)_x,x(1 - M^-1_1(ϕ)_x,x), n(ϕ) = 1/V∑_x ⟨ n_↑(x) + n_↓(x) ⟩ = 1/V∑_x ( M^-1_2(ϕ)_x,x -M^-1_1(ϕ)_x,x).Fig. <ref> exhibits the cumulative estimations of standard deviations for each observable. It shows that the double occupancy has infinite variance problem while the density does not. This is because the double occupancy has the term proportional to the inverse of fermion matrices, i.e., M^-1_1 M^-1_2 in Eq. (<ref>), while the density only has a part of them, i.e., M^-1_1 or M^-1_2. § EXTRA TIME-SLICE In the previous section, it was shown that infinite variance problem of fermionic observables comes from the exceptional configurations. One possible solution is to use a different distribution for Monte Carlo samplings and employ the reweighting method. It was suggested in <cit.> that one can utilize the distribution from extra time-slice. Let us consider that our path integral representation in Eq. (<ref>) is trotterized with N time-slices. If one definesF(ϕ) ≡∫ dϕ^* e^-S_0(ϕ^*) M_N+1(ϕ, ϕ^*),where M_N+1 is the fermion matrix with N+1 time-slices, one can find the partition function asZ = ∫ [ dϕ]_Ne^-S_0(ϕ) M_N(ϕ) F(ϕ)/F(ϕ) = ∫ [ dϕ]_Ndϕ^*R(ϕ) e^-S_0(ϕ,ϕ^*) M_N+1(ϕ,ϕ^*),where [ dϕ]_N denotes the path integral measure with N time-slices. Then observables can be estimated with the conventional reweighting procedure:⟨𝒪⟩_N = ⟨𝒪(ϕ)R(ϕ) ⟩_N+1/⟨ R(ϕ) ⟩_N+1,where R(ϕ) =M_N(ϕ) / F(ϕ) and the subscript N+1 denotes that the Monte Carlo samples are chosen from the new distribution:p_N+1(ϕ,ϕ^*) = e^-S_0(ϕ,ϕ^*) M_N+1(ϕ,ϕ^*)/∫[ dϕ]_N dϕ^*e^-S_0(ϕ,ϕ^*) M_N+1(ϕ,ϕ^*).With this new distribution, infinite variance problem is cured since the variance involves 𝒪 R, which does not have any singularities. Therefore, the task is to estimate the reweighting factor R(ϕ).§ UNBIASED ESTIMATOR In <cit.>, the authors suggested that one can integrate F(ϕ) analytically using BSS formula <cit.> and expand it in ϵ≡β/N:F(ϕ)≡∫ dϕ^* e^-S_0(ϕ^*) M_N+1(ϕ, ϕ^*) = Tr[ e^- H̃_2e^-ϵ H_4 B(ϕ_N) ⋯ B(ϕ_1) ]= Tr[ (1 - ϵ H ) B(ϕ_N) ⋯ B(ϕ_1) ] + O(ϵ^2)= (1-ϵ H(ϕ) ) M_N(ϕ) + O(ϵ^2). The advantage of this method is that it does not have any additional cost except the increased cost from the new auxiliary field, but there are some disadvantages. First of all, there is a perturbative error from the truncation and the number of Wick contractions increases exponentially as one goes to higher orders. Also, one needs small ϵ since F(ϕ) can be zero, which can generate another infinite variance problem.Instead of using analytical method, one can directly estimate F(ϕ) using Monte Carlo calculations using the new auxiliary field ϕ^* (which we call sub-MC method):F(ϕ) = Z⟨ M_N+1(ϕ,ϕ^*) ⟩_g,where the subscript g represents the Monte Carlo samplings using e^-S_0(ϕ^*).However, what one needs to estimate is 1/F(ϕ), not F(ϕ), and it can be easily checked that just taking an inverse of F(ϕ) is biased:⟨1/A⟩ = 1/⟨A⟩-⟨A-⟨ A⟩/⟨ A ⟩ ^2⟩ + ⟨( A-⟨ A ⟩)^2/⟨ A ⟩^3⟩ - ⋯ ,where A denotes the finite sample average of A. Note that the third term in Eq. (<ref>) is not zero.Therefore, one needs to find an unbiased estimator for 1/F(ϕ). In <cit.>, the authors suggested an unbiased estimator of 1/⟨ A ⟩:ξ̂_ A≡w/q_n∏_i=1^n (1-wA_i).Here, q_n is an arbitrary discrete probability distribution and w < 1/⟨ A ⟩. Then the variance minimizing choice of q_n and w is w =min{1/k A,A/ A^2, 1/A_ max}, p = 1 - [1- 2 wA + w^2A^2]^1/2, q_n = p(1-p)^n,where A denotes the sample average of A.§ RESULT Fig. <ref> shows the distributions of double occupancy using the standard Monte Carlo and the extra time-slice reweighting with sub-MC method. The left panel has large negative peaks during the sampling, which makes the variance jumps in Fig. <ref>. However, this abnormal behavior is well mitigated in the right panel, with the reweighting method. Note that the expectation value of double occupancy using the standard method is positive even though the exceptional configurations contribute to it negatively.The left panel of Fig. <ref> exhibits the comparison of the analytic method in Eq. (<ref>) and the sub-MC method. One can see that as the trotterization error increases, the Taylor expansion in terms of ϵ does not well behave, while the sub-MC method does not have the error from the truncation. Note that the fitting for the analytic method only used first four points because of increasing error at large ϵ. It means that one needs to use small ϵ to employ the analytic method, and so the overall cost of Monte Carlo calculations can be cheaper for the sub-MC method even though it has the secondary Monte Carlo sampling procedure.The right plot in Fig. <ref> shows the the effect of the sub-MC method with the unbiased estimator. The bias in the biased estimator has1/n_sub behavior, where n_ sub means the number of sub Monte Carlo samplings. The number of sub-MC samples for the unbiased estimator is randomly chosen from q_n in Eq. (<ref>) and therefore the average value n_ sub≈ 2k is used for the figure, which includes the cost k of the estimation for w and p in Eq. (<ref>). The blue band is the n_ sub→∞ extrapolation of the biased estimator. It shows that the unbiased estimator converges to the unbiased value with lower cost.This work was supported in part by the U.S. Department of Energy, Office of Nuclear Physics under Award Number(s) DE-SC0021143, and DE-FG02-93ER40762, and DE-FG02-95ER40907.99PhysRevB.41.9301 E. Loh, J. Gubernatis, R. Scalettar, S. White, D. Scalapino and R. Sugar,Sign problem in the numerical simulation of many-electron systems, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.41.9301Phys. Rev. B 41, (1990) 9301.Wynen:2018ryx J. L. Wynen, E. Berkowitz, C. Körber, T. A. Lähde and T. Luu, Avoiding Ergodicity Problems in Lattice Discretizations of the Hubbard Model, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.075141Phys. Rev. B 100 (2019) 075141 [https://arxiv.org/abs/1812.092681812.09268].Yunus:2022wdr C. Yunus and W. 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Oh, Infinite variance problem in fermion models, https://journals.aps.org/prd/abstract/10.1103/PhysRevD.107.094502 Phys. Rev. D 107 (2023) 094502[https://arxiv.org/abs/2211.064192211.06419].Shi:2015lyu H. Shi and S. Zhang, Infinite Variance in Fermion Quantum Monte Carlo Calculations, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.033303Phys. Rev. E 93 (2016) 033303[https://arxiv.org/abs/1511.040841511.04084].BSS R. Blankenbecler, D. Scalapino and R. Sugar, Monte Carlo calculations of coupled boson-fermion systems. I.,https://link.aps.org/doi/10.1103/PhysRevD.24.2278Phys. Rev. D 24, 2278 (1981).Moka S. B. Moka, D. P. Kroese and S. Juneja,Unbiased Estimation of The Reciprocal Mean For Non-Negative Random Variables,https://ieeexplore.ieee.org/document/90048152019 Winter Simulation Conference (WSC), pp. 404-415 [https://arxiv.org/abs/1907.018431907.01843].Benchmark J. P. F. LeBlanc et al., (Simons Collaboration on the Many- Electron Problem),Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, https://link.aps.org/doi/10.1103/PhysRevX.5.041041Phys. Rev. X 5, 041041 (2015) [https://arxiv.org/abs/1505.022901505.02290]. | http://arxiv.org/abs/2311.16074v1 | {
"authors": [
"Hyunwoo Oh",
"Andrei Alexandru",
"Paulo F. Bedaque",
"Andrea Carosso"
],
"categories": [
"hep-lat",
"cond-mat.str-el",
"nucl-th"
],
"primary_category": "hep-lat",
"published": "20231127184605",
"title": "A solution for infinite variance problem of fermionic observables"
} |
Ontologising Trustworthy in the Telecommunications Domain [ January 14, 2024 ========================================================= Explainable Reinforcement Learning (XRL) can provide transparency into the decision-making process of a Deep Reinforcement Learning (DRL) model and increase user trust and adoption in real-world use cases. By utilizing XRL techniques, researchers can identify potential vulnerabilities within a trained DRL model prior to deployment, therefore limiting the potential for mission failure or mistakes by the system. This paper introduces the ARLIN (Assured RL Model Interrogation) Toolkit, an open-source Python library that identifies potential vulnerabilities and critical points within trained DRL models through detailed, human-interpretable explainability outputs. To illustrate ARLIN's effectiveness, we provide explainability visualizations and vulnerability analysis for a publicly available DRL model. The open-source code repository is available for download at .§ INTRODUCTIONOver the last decade, reinforcement learning has increased in popularity due to its ability to achieve superhuman performance on a variety of classic board <cit.> and video game <cit.> environments. This gain in popularity has sparked an interest in using DRL for both decision support and autonomous operation within safety-critical scenarios such as air-to-air combat <cit.>, nuclear power plant optimization <cit.>, and ballistic missile guidance <cit.>. These use-cases are considered high-risk as even small mistakes can result in large losses of monetary value, equipment, and life. Before DRL models can safely be deployed within real-world safety critical environments, their associated vulnerabilities need to be identified and understood so effective training enhancements and verification guardrails can be implemented.In this paper, we present the ARLIN Toolkit, an open-source research library written in Python that provides explainability outputs and vulnerability detection for DRL models, specifically designed to increase model assurance and identify potential points of failure within a trained model. To our knowledge, ARLIN is the first open-sourced Python toolkit focused on utilizing explainability techniques to assure RL models prior to deployment. ARLIN utilizes matplotlib <cit.> and networkx <cit.> to visualize a trained DRL model's decision making process and provide meaningful vulnerability identification and analysis to researchers. The modular library is structured to support custom architectures, algorithms, DRL frameworks, and analytics; and provides a well-documented and tested API for XRL research development and model assurance. The ARLIN repository is available for download at . § BACKGROUND AND PRELIMINARIES §.§ Reinforcement LearningReinforcement learning is an area of machine learning that focuses on teaching an intelligent agent how to interact within an environment in order to optimize a reward function and achieve a specified goal <cit.>. As the agent interacts with the environment, it receives scaled rewards to indicate good and bad actions. Through trial and error, the agent is able to identify the optimal policy in order to maximize the cumulative reward received and solve the given task.In DRL, the environment is defined as a Markov Decision Process, MDP, M = (S, A, P, ρ_0, R, γ, T), where S is the state space, A is the action space, P : S × A × S → [0,1] is the state transition probability, ρ_0 : S × A → [0,1] is the initial state probability, R : S × A is the reward function, γ is the discount factor, and T is the maximum episode length. The policy π_θ : S × A assigns a probability value to an action given a state.During training, the agent observes the current state of the environment s_t ∈ S and performs an action a_t ∈ A according to its policy π_θ. The agent then receives a next state s_t^'∈ S and reward r_t from R within the environment. The agent's goal is to find a policy that optimizes R. Due to the large state space S, neural networks are commonly used as function approximators in DRL tasks. While this helps the agent to generalize to continuous or large state spaces, it reduces transparency into the decision making process of the model. §.§ Explainable Reinforcement LearningThe "black-box" nature of deep neural networks make verifying and understanding their underlying reasoning very difficult. A lot of work has been done in the field of Explainable AI (XAI) in recent years <cit.>. However, most of these works focus on supervised learning or unsupervised learning tasks that deal with non-sequential input data which do not directly transfer in the case of DRL due to the sequential nature of the task. The lack of transparency into the decision making process of an DRL model decreases user and public trust and introduces potentially catastrophic unknowns into the model performance, therefore increasing the potential for mission failure.Explainable RL (XRL) is a field of RL that focuses on increasing DRL model transparency to give users insight into a model's decision making process. The information gained from XRL techniques can help researchers identify why agents are making certain decisions and increase user trust in the model. Milani <cit.> buckets current XRL works into 3 main categories: feature importance, learning process and MDP, and policy-level. These categories look into different aspects of the agent's decision making process including the importance of different features on the policy's chosen action, training examples that affect the policy outputs, and overall policy behavior analysis. This interpretability information can be labeled as local or global, where local explanations focus on interpreting the predictions of a single action at a point in time and global explanations give a holistic view of the policy's behavior overall <cit.>. Our work focuses on the global interpretability of an DRL model as we aim to analyze the overarching policy to identify potential critical points that may affect a policy's success. §.§ Related WorksTo our knowledge, ARLIN is the first open-sourced Python library focusing on global explainability and vulnerability detection through human-interpretable analysis visualizations. InterestingnessXRL <cit.> similarly provides explainability outputs for users, but focuses primarily on identifying interesting interactions between the agent and the environment called highlights and returns video-samples of the highlights along with analytics about the interaction itself. While vulnerabilities and critical points may be diagnosed as a highlight, this work does not explicitly focus on these areas. While other repositories linked to XRL are publicly available such as <cit.>, these are providing XRL algorithms themselves as opposed to visualizations and analytics for trained DRL models. § KEY FEATURESThe ARLIN Toolkit provides three main explainability analysis components to users: latent space analysis, datapoint cluster analysis, and semi-aggregated Markov decision process (SAMDP) <cit.> analysis.* Latent space analysis uses dimensionality reduction techniques to generate embeddings from user-specified datapoint metadata and plot them in 2-D space. Additional policy metadata can be overlaid onto the embeddings to visualize the relationship between the policy embeddings and the policy metadata.* Datapoint cluster analysis uses unsupervised clustering methods to cluster datapoints based on user-defined policy metadata and provide analysis on each state cluster. Average metrics for each cluster can be plotted for comparison to identify potential outliers and gain information about what is happening in a specific area of the environment or point in time, such as failure states and critical points. * SAMDP analysis transforms the identified state clusters into an SAMDP to provide a holistic overview of how the policy moves through the environment over an entire episode. The analysis uses graph theory to identify paths between nodes along with the actions needed to bring the policy from A to B. Paired with the cluster state analysis, users can identify the paths and actions required for a policy to reach an identified failure state or mistake-prone area.§ STRUCTURE AND CUSTOMIZATIONSThe following is a conceptual overview of the ARLIN library structure along with instructions for adding additional custom components. A practical example usage of the library's methods can be found in Appendix A 1.1.§.§ Conceptual Structure§.§ Code StructureA conceptual diagram of the library structure and relationships between components is shown in Figure <ref>. At a high-level, ARLIN has 4 main components: , , , and . Thecomponent is used to create an XRL dataset, a collection of datapoints containing transition data and internal policy metadata collected at every episode step while running a policy within an environment.uses the XRL dataset to create embeddings and clusters, of whichprovides meaningful analysis and visualizations. The cluster data and XRL dataset can also be provided toto generate and visualize different SAMDP graphs of the agent's policy along with available paths between given clusters. Thedirectory contains all code necessary for creating an XRL dataset from a trained RL model.handle the loading of a trained model whileare responsible for collecting the internal data from the RL model.outline the specific data that the dataset will be storing. Thestores all traditional RL transition dataalong with the model-specific metadata () gathered by the . Custom , , andcan be added to load custom models and work with custom architectures and algorithms for the collection of user-defined metadata, as outlined in section <ref>. Thefile contains the code necessary for datapoint embedding and cluster generation. Metadata from the XRL dataset chosen by the user is reduced to two dimensions via<cit.> to generate latent space embeddings. The datapoints within the XRL dataset are clustered based on user-specified metadata using<cit.> and<cit.>. Each cluster represents an area of the policy's latent space where the policy's decision making is affected in similar ways, such as clusters with similar input features or similar output action results. Thedirectory contains methods for running analysis on both the embeddings as well as the clusters. This includes cluster state representation analysis which analyzes and visualizes the states within the cluster for insight into what states fall into each cluster. Thesub-directory contains methods for visualizing the generated analytics using<cit.>. Example latent analysis and cluster analysis visualizations can be found in Appendices <ref> and <ref>, respectively.samdp Thefile includes theclass and associated methods. Theclass is a semi-aggregated Markov decision process representation of the policy within its training environment. The SAMDP methods visualize the connections between clusters as well as available paths and actions required to travel to specific target clusters. Available SAMDP methods and visualizations are attached in Appendix <ref>. §.§ Custom Component Creation The modular architecture of ARLIN provides support for user customization with no changes to the main library code. Requirements for creating custom components for common aspects of the library are detailed below:Loaders: The addition of new loaders does not require any inheritance and can be created as a separate method specific to the model that is being loaded. A custom loader must return a trained model with which a user can run inference within the training environment.Datapoints: To create a new datapoint, the user must inherit fromand add any additional metadata that thewill be storing for the user-specific use case. A datapoint holds information gathered at a single episode step and can store model-specific internal metadata gathered during the model's decision making process.Collectors: Custom collectors must inherit fromand implement the required methods. The collector is responsible for collecting the data needed to fill the datapoint, and therefore is specific to the model architecture as the collector needs to understand where to find the necessary metadata to store.Latent and Cluster Analytics: To create new analysis visualizations, users can simply create a custom method that produces the wanted metric and return aobject for input into the provided visualization methods. § USAGEARLIN is designed to provide users with explainability outputs that can be analyzed to identify potential vulnerabilities and critical points within a trained policy. An example workflow for using ARLIN is shown in Figure <ref>.To illustrate ARLIN's effectiveness, we provide explainability outputs and corresponding vulnerability analysis for a publicly available DRL model - a model trained using Stable Baselines3 <cit.> with PPO <cit.> on OpenAI gym's Lunarlander-v2 environment <cit.>, pulled from Huggingface.com - by following the steps outlined in Figure <ref>. The output visualizations and analysis can be found in Appendix <ref> (latent analysis), Appendix <ref> (cluster analysis), and Appendix <ref> (SAMDP analysis).§ DISCUSSION AND FUTURE WORKWe believe that ARLIN can accelerate research in the XRL field by providing a modular research library with an easy-to-use API for generating explainability visualizations for vulnerability and critical point identification and analysis. This work can be applied to practical use domains such as RL-assisted autonomous vehicle verification and validation and the field of adversarial RL. We hope that the library can expand to include additional analytics, metrics, and visualizations as well as add support for new algorithms and frameworks out of the box through continued author maintenance and community development.The authors thank Walker Dimon and Guido Zarrella for helpful discussions throughout the development process. This work was funded by the 2023 MITRE Independent Research and Development Program's Early Career Research Program.IEEEtran § LATENT ANALYSIS EXAMPLES ARLIN's latent analysis methods make use of the embeddings generated by ARLIN's generation component by overlaying user-defined policy metadata over the generated embeddings to visualize how the metadata relates to location within the embedding space. This information can be helpful when working examining the latent space of a policy. Future work can make use of the latent space to identify similar datapoints or regions, or identify ways to traverse the latent space to reach specific outcomes determined by the metadata, such as actions to take. § CLUSTER ANALYSIS EXAMPLES ARLIN's cluster analysis methods make use of the clusters generated by ARLIN's generation component by computing the average values of different policy metadata for each identified cluster. This information gives insight into vulnerable clusters and states within the environment that are reached by the policy. The confidence analysis gives insight into how confident the policy is in the action that it is taking at a given point in time. Clusters with low confidence indicate areas of the environment where the policy is not confident in the action that it is taking due to limited training, particularly difficult areas of the environment, or areas where the policy action has no consequence. Clusters with high confidence are areas where the policy is sure of the action it is taking, which can be representative of an easy or very important cluster. A large variance typically represents a cluster where the policy is either very sure or very unsure of its actions, likely resulting in a higher likelihood for mistakes.The expected return analysis gives insight into both the stage of the episode the cluster is in (early vs late) as well as insight into which states the policy thinks have a higher likelihood for mission success. When looking at initial clusters, a cluster that has a lower expected return is seen as a harder starting position for the policy. When looking at intermediate clusters, clusters with a higher expected return represent "early" stage clusters while "late" stage clusters have a lower expected return.The reward analysis gives insight into how good the actions taken within the cluster are, represented by the amount of reward received. A higher average reward means the actions are considered better overall. When analyzing terminal clusters, mission failures can be typically be identified by clusters with a large negative reward. In Figure <ref>, we can make a few assumptions about the clusters within our policy. When analyzing initial clusters, Cluster 21 has a low confidence and low expected return, indicating that the cluster is seen as a non-optimal starting position. The policy does not expect to get as much reward overall when starting in Cluster 21 than Cluster 20. When looking at intermediate cluster, Cluster 9 shows a low received reward but high confidence, indicating that it is likely a corrective maneuver that the policy feels is important to take. We can assume this is a late-stage maneuver as well given that the expected return is near 0. For terminal clusters, Cluster 23 has a low expected return and a low received reward, meaning this is likely an expected failure - the policy was expecting a low reward and got a low reward. Cluster 24, however, has a high expected reward but a low received reward, indicating an unexpected failure - the policy was expecting to get more than it received. As seen in Figure <ref>, our assumptions were correct. Cluster 9 is a late-stage corrective maneuver in which the agent is attempting to move further left to be inside the landing flags. Cluster 23 is an expected failure where the policy lands hard into the ground and crashes, and Cluster 24 is an unexpected failure in which the policy moves off screen, resulting in the end of the episode without a successful landing or a crash.§ SAMDP EXAMPLES ARLIN's SAMDP component uses metadata from the XRLDataset along with the generated clusters to generate a semi-aggregated Markov decision process of the policy to show how the policy moves between clusters over the course of an episode. This information is useful in identifying paths between clusters. For vulnerability analysis, this is useful in identifying which actions lead an agent to mission failure, and which actions lead to mission success as well as identifying the critical points where the agent can go either way depending on the actions taken. ARLIN provides a variety of methods in the SAMDP package including holistic views of the entire SAMDP (Figure <ref>), paths between given clusters, or paths leading into a terminal state (Figure <ref>). All SAMDP methods can provide a full verbose view including the actions necessary for the movement (Figure <ref>), or a simplified view which only shows the connections and not the actions required (Figure <ref>). Some methods provide the option to only show the most probable connections as well, to avoid connections that are have been taken at least once, but are not likely to be taken by the policy in general. | http://arxiv.org/abs/2311.15838v1 | {
"authors": [
"Alexander Tapley",
"Kyle Gatesman",
"Luis Robaina",
"Brett Bissey",
"Joseph Weissman"
],
"categories": [
"cs.LG",
"cs.AI"
],
"primary_category": "cs.LG",
"published": "20231127140247",
"title": "Utilizing Explainability Techniques for Reinforcement Learning Model Assurance"
} |
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