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We calculate the electromagnetic polarizabilities of the spin-$\frac{1}{2}$ singly heavy baryons in the heavy baryon chiral perturbation theory up to $\mathcal{O}(p^3)$. We estimate the low-energy constants using the magnetic moments of singly charmed baryons from lattice QCD simulations and the experimental decay widths of $\Sigma_c$ and $\Sigma_c^*$. Our results indicate that the long-range chiral corrections make significant contributions to the polarizabilities. Additionally, the magnetic dipole transitions $\mathcal{B}_6^* \to \mathcal{B}_6 +\gamma $ also provide large contribution to the magnetic polarizabilities., Comment: 17 pages, 1 figure, 6 tables

EEG signals have emerged as a powerful tool in affective brain-computer interfaces, playing a crucial role in emotion recognition. However, current deep transfer learning-based methods for EEG recognition face challenges due to the reliance of both source and target data in model learning, which significantly affect model performance and generalization. To overcome this limitation, we propose a novel framework (PL-DCP) and introduce the concepts of feature disentanglement and prototype inference. The dual prototyping mechanism incorporates both domain and class prototypes: domain prototypes capture individual variations across subjects, while class prototypes represent the ideal class distributions within their respective domains. Importantly, the proposed PL-DCP framework operates exclusively with source data during training, meaning that target data remains completely unseen throughout the entire process. To address label noise, we employ a pairwise learning strategy that encodes proximity relationships between sample pairs, effectively reducing the influence of mislabeled data. Experimental validation on the SEED and SEED-IV datasets demonstrates that PL-DCP, despite not utilizing target data during training, achieves performance comparable to deep transfer learning methods that require both source and target data. This highlights the potential of PL-DCP as an effective and robust approach for EEG-based emotion recognition.

Non-suicidal self-injury (NSSI) is a serious threat to the physical and mental health of adolescents, significantly increasing the risk of suicide and attracting widespread public concern. Electroencephalography (EEG), as an objective tool for identifying brain disorders, holds great promise. However, extracting meaningful and reliable features from high-dimensional EEG data, especially by integrating spatiotemporal brain dynamics into informative representations, remains a major challenge. In this study, we introduce an advanced semi-supervised adversarial network, NSSI-Net, to effectively model EEG features related to NSSI. NSSI-Net consists of two key modules: a spatial-temporal feature extraction module and a multi-concept discriminator. In the spatial-temporal feature extraction module, an integrated 2D convolutional neural network (2D-CNN) and a bi-directional Gated Recurrent Unit (BiGRU) are used to capture both spatial and temporal dynamics in EEG data. In the multi-concept discriminator, signal, gender, domain, and disease levels are fully explored to extract meaningful EEG features, considering individual, demographic, disease variations across a diverse population. Based on self-collected NSSI data (n=114), the model's effectiveness and reliability are demonstrated, with a 7.44% improvement in performance compared to existing machine learning and deep learning methods. This study advances the understanding and early diagnosis of NSSI in adolescents with depression, enabling timely intervention. The source code is available at https://github.com/Vesan-yws/NSSINet.

Pressure-sensitive smart textiles are widely applied in the fields of healthcare, sports monitoring, and intelligent homes. The integration of devices embedded with pressure sensing arrays is expected to enable comprehensive scene coverage and multi-device integration. However, the implementation of identity recognition, a fundamental function in this context, relies on extensive device-specific datasets due to variations in pressure distribution across different devices. To address this challenge, we propose a novel user identification method based on contrastive learning. We design two parallel branches to facilitate user identification on both new and existing devices respectively, employing supervised contrastive learning in the feature space to promote domain unification. When encountering new devices, extensive data collection efforts are not required; instead, user identification can be achieved using limited data consisting of only a few simple postures. Through experimentation with two 8-subject pressure datasets (BedPressure and ChrPressure), our proposed method demonstrates the capability to achieve user identification across 12 sitting scenarios using only a dataset containing 2 postures. Our average recognition accuracy reaches 79.05%, representing an improvement of 2.62% over the best baseline model.

Continuous electroencephalography (EEG) signals are widely used in affective brain-computer interface (aBCI) applications. However, not all continuously collected EEG signals are relevant or meaningful to the task at hand (e.g., wondering thoughts). On the other hand, manually labeling the relevant parts is nearly impossible due to varying engagement patterns across different tasks and individuals. Therefore, effectively and efficiently identifying the important parts from continuous EEG recordings is crucial for downstream BCI tasks, as it directly impacts the accuracy and reliability of the results. In this paper, we propose a novel unsupervised deep reinforcement learning framework, called Emotion-Agent, to automatically identify relevant and informative emotional moments from continuous EEG signals. Specifically, Emotion-Agent involves unsupervised deep reinforcement learning combined with a heuristic algorithm. We first use the heuristic algorithm to perform an initial global search and form prototype representations of the EEG signals, which facilitates the efficient exploration of the signal space and identify potential regions of interest. Then, we design distribution-prototype reward functions to estimate the interactions between samples and prototypes, ensuring that the identified parts are both relevant and representative of the underlying emotional states. Emotion-Agent is trained using Proximal Policy Optimization (PPO) to achieve stable and efficient convergence. Our experiments compare the performance with and without Emotion-Agent. The results demonstrate that selecting relevant and informative emotional parts before inputting them into downstream tasks enhances the accuracy and reliability of aBCI applications., Comment: 11 pages, 4 figures, 4 tables, submitted to AAAI 2025

Emotion recognition using electroencephalography (EEG) signals has garnered widespread attention in recent years. However, existing studies have struggled to develop a sufficiently generalized model suitable for different datasets without re-training (cross-corpus). This difficulty arises because distribution differences across datasets far exceed the intra-dataset variability. To solve this problem, we propose a novel Soft Contrastive Masked Modeling (SCMM) framework. Inspired by emotional continuity, SCMM integrates soft contrastive learning with a new hybrid masking strategy to effectively mine the "short-term continuity" characteristics inherent in human emotions. During the self-supervised learning process, soft weights are assigned to sample pairs, enabling adaptive learning of similarity relationships across samples. Furthermore, we introduce an aggregator that weightedly aggregates complementary information from multiple close samples based on pairwise similarities among samples to enhance fine-grained feature representation, which is then used for original sample reconstruction. Extensive experiments on the SEED, SEED-IV and DEAP datasets show that SCMM achieves state-of-the-art (SOTA) performance, outperforming the second-best method by an average accuracy of 4.26% under two types of cross-corpus conditions (same-class and different-class) for EEG-based emotion recognition., Comment: 16 pages, 8 figures, 15 tables, submitted to AAAI 2025

Affective brain-computer interfaces (aBCIs) are increasingly recognized for their potential in monitoring and interpreting emotional states through electroencephalography (EEG) signals. Current EEG-based emotion recognition methods perform well with short segments of EEG data. However, these methods encounter significant challenges in real-life scenarios where emotional states evolve over extended periods. To address this issue, we propose a Dual Attentive (DuA) transformer framework for long-term continuous EEG emotion analysis. Unlike segment-based approaches, the DuA transformer processes an entire EEG trial as a whole, identifying emotions at the trial level, referred to as trial-based emotion analysis. This framework is designed to adapt to varying signal lengths, providing a substantial advantage over traditional methods. The DuA transformer incorporates three key modules: the spatial-spectral network module, the temporal network module, and the transfer learning module. The spatial-spectral network module simultaneously captures spatial and spectral information from EEG signals, while the temporal network module detects temporal dependencies within long-term EEG data. The transfer learning module enhances the model's adaptability across different subjects and conditions. We extensively evaluate the DuA transformer using a self-constructed long-term EEG emotion database, along with two benchmark EEG emotion databases. On the basis of the trial-based leave-one-subject-out cross-subject cross-validation protocol, our experimental results demonstrate that the proposed DuA transformer significantly outperforms existing methods in long-term continuous EEG emotion analysis, with an average enhancement of 5.28%., Comment: 11 pages, 3 figures

Inability to express the confidence level and detect unseen classes has limited the clinical implementation of artificial intelligence in the real-world. We developed a foundation model with uncertainty estimation (FMUE) to detect 11 retinal conditions on optical coherence tomography (OCT). In the internal test set, FMUE achieved a higher F1 score of 96.76% than two state-of-the-art algorithms, RETFound and UIOS, and got further improvement with thresholding strategy to 98.44%. In the external test sets obtained from other OCT devices, FMUE achieved an accuracy of 88.75% and 92.73% before and after thresholding. Our model is superior to two ophthalmologists with a higher F1 score (95.17% vs. 61.93% &71.72%). Besides, our model correctly predicts high uncertainty scores for samples with ambiguous features, of non-target-category diseases, or with low-quality to prompt manual checks and prevent misdiagnosis. FMUE provides a trustworthy method for automatic retinal anomalies detection in the real-world clinical open set environment., Comment: All codes are available at https://github.com/yuanyuanpeng0129/FMUE

Inner-approximate reachability analysis involves calculating subsets of reachable sets, known as inner-approximations. This analysis is crucial in the fields of dynamic systems analysis and control theory as it provides a reliable estimation of the set of states that a system can reach from given initial states at a specific time instant. In this paper, we study the inner-approximate reachability analysis problem based on the set-boundary reachability method for systems modelled by ordinary differential equations, in which the computed inner-approximations are represented with zonotopes. The set-boundary reachability method computes an inner-approximation by excluding states reached from the initial set's boundary. The effectiveness of this method is highly dependent on the efficient extraction of the exact boundary of the initial set. To address this, we propose methods leveraging boundary and tiling matrices that can efficiently extract and refine the exact boundary of the initial set represented by zonotopes. Additionally, we enhance the exclusion strategy by contracting the outer-approximations in a flexible way, which allows for the computation of less conservative inner-approximations. To evaluate the proposed method, we compare it with state-of-the-art methods against a series of benchmarks. The numerical results demonstrate that our method is not only efficient but also accurate in computing inner-approximations., Comment: the extended version of the paper accepted by CAV 2024

Cross-center data heterogeneity and annotation unreliability significantly challenge the intelligent diagnosis of diseases using brain signals. A notable example is the EEG-based diagnosis of neurodegenerative diseases, which features subtler abnormal neural dynamics typically observed in small-group settings. To advance this area, in this work, we introduce a transferable framework employing Manifold Attention and Confidence Stratification (MACS) to diagnose neurodegenerative disorders based on EEG signals sourced from four centers with unreliable annotations. The MACS framework's effectiveness stems from these features: 1) The Augmentor generates various EEG-represented brain variants to enrich the data space; 2) The Switcher enhances the feature space for trusted samples and reduces overfitting on incorrectly labeled samples; 3) The Encoder uses the Riemannian manifold and Euclidean metrics to capture spatiotemporal variations and dynamic synchronization in EEG; 4) The Projector, equipped with dual heads, monitors consistency across multiple brain variants and ensures diagnostic accuracy; 5) The Stratifier adaptively stratifies learned samples by confidence levels throughout the training process; 6) Forward and backpropagation in MACS are constrained by confidence stratification to stabilize the learning system amid unreliable annotations. Our subject-independent experiments, conducted on both neurocognitive and movement disorders using cross-center corpora, have demonstrated superior performance compared to existing related algorithms. This work not only improves EEG-based diagnostics for cross-center and small-setting brain diseases but also offers insights into extending MACS techniques to other data analyses, tackling data heterogeneity and annotation unreliability in multimedia and multimodal content understanding.

Emotion decoding using Electroencephalography (EEG)-based affective brain-computer interfaces represents a significant area within the field of affective computing. In the present study, we propose a novel non-deep transfer learning method, termed as Manifold-based Domain adaptation with Dynamic Distribution (MDDD). The proposed MDDD includes four main modules: manifold feature transformation, dynamic distribution alignment, classifier learning, and ensemble learning. The data undergoes a transformation onto an optimal Grassmann manifold space, enabling dynamic alignment of the source and target domains. This process prioritizes both marginal and conditional distributions according to their significance, ensuring enhanced adaptation efficiency across various types of data. In the classifier learning, the principle of structural risk minimization is integrated to develop robust classification models. This is complemented by dynamic distribution alignment, which refines the classifier iteratively. Additionally, the ensemble learning module aggregates the classifiers obtained at different stages of the optimization process, which leverages the diversity of the classifiers to enhance the overall prediction accuracy. The experimental results indicate that MDDD outperforms traditional non-deep learning methods, achieving an average improvement of 3.54%, and is comparable to deep learning methods. This suggests that MDDD could be a promising method for enhancing the utility and applicability of aBCIs in real-world scenarios.

The integration of human emotions into multimedia applications shows great potential for enriching user experiences and enhancing engagement across various digital platforms. Unlike traditional methods such as questionnaires, facial expressions, and voice analysis, brain signals offer a more direct and objective understanding of emotional states. However, in the field of electroencephalography (EEG)-based emotion recognition, previous studies have primarily concentrated on training and testing EEG models within a single dataset, overlooking the variability across different datasets. This oversight leads to significant performance degradation when applying EEG models to cross-corpus scenarios. In this study, we propose a novel Joint Contrastive learning framework with Feature Alignment (JCFA) to address cross-corpus EEG-based emotion recognition. The JCFA model operates in two main stages. In the pre-training stage, a joint domain contrastive learning strategy is introduced to characterize generalizable time-frequency representations of EEG signals, without the use of labeled data. It extracts robust time-based and frequency-based embeddings for each EEG sample, and then aligns them within a shared latent time-frequency space. In the fine-tuning stage, JCFA is refined in conjunction with downstream tasks, where the structural connections among brain electrodes are considered. The model capability could be further enhanced for the application in emotion detection and interpretation. Extensive experimental results on two well-recognized emotional datasets show that the proposed JCFA model achieves state-of-the-art (SOTA) performance, outperforming the second-best method by an average accuracy increase of 4.09% in cross-corpus EEG-based emotion recognition tasks.

Recently, formal verification of deep neural networks (DNNs) has garnered considerable attention, and over-approximation based methods have become popular due to their effectiveness and efficiency. However, these strategies face challenges in addressing the "unknown dilemma" concerning whether the exact output region or the introduced approximation error violates the property in question. To address this, this paper introduces the UR4NNV verification framework, which utilizes under-approximation reachability analysis for DNN verification for the first time. UR4NNV focuses on DNNs with Rectified Linear Unit (ReLU) activations and employs a binary tree branch-based under-approximation algorithm. In each epoch, UR4NNV under-approximates a sub-polytope of the reachable set and verifies this polytope against the given property. Through a trial-and-error approach, UR4NNV effectively falsifies DNN properties while providing confidence levels when reaching verification epoch bounds and failing falsifying properties. Experimental comparisons with existing verification methods demonstrate the effectiveness and efficiency of UR4NNV, significantly reducing the impact of the "unknown dilemma"., Comment: 11 pages, 4 figures

Electroencephalography (EEG) is an objective tool for emotion recognition with promising applications. However, the scarcity of labeled data remains a major challenge in this field, limiting the widespread use of EEG-based emotion recognition. In this paper, a semi-supervised Dual-stream Self-Attentive Adversarial Graph Contrastive learning framework (termed as DS-AGC) is proposed to tackle the challenge of limited labeled data in cross-subject EEG-based emotion recognition. The DS-AGC framework includes two parallel streams for extracting non-structural and structural EEG features. The non-structural stream incorporates a semi-supervised multi-domain adaptation method to alleviate distribution discrepancy among labeled source domain, unlabeled source domain, and unknown target domain. The structural stream develops a graph contrastive learning method to extract effective graph-based feature representation from multiple EEG channels in a semi-supervised manner. Further, a self-attentive fusion module is developed for feature fusion, sample selection, and emotion recognition, which highlights EEG features more relevant to emotions and data samples in the labeled source domain that are closer to the target domain. Extensive experiments conducted on two benchmark databases (SEED and SEED-IV) using a semi-supervised cross-subject leave-one-subject-out cross-validation evaluation scheme show that the proposed model outperforms existing methods under different incomplete label conditions (with an average improvement of 5.83% on SEED and 6.99% on SEED-IV), demonstrating its effectiveness in addressing the label scarcity problem in cross-subject EEG-based emotion recognition., Comment: arXiv admin note: text overlap with arXiv:2304.06496

Deep neural networks, (DNNs, a.k.a. NNs), have been widely used in various tasks and have been proven to be successful. However, the accompanied expensive computing and storage costs make the deployments in resource-constrained devices a significant concern. To solve this issue, quantization has emerged as an effective way to reduce the costs of DNNs with little accuracy degradation by quantizing floating-point numbers to low-width fixed-point representations. Quantized neural networks (QNNs) have been developed, with binarized neural networks (BNNs) restricted to binary values as a special case. Another concern about neural networks is their vulnerability and lack of interpretability. Despite the active research on trustworthy of DNNs, few approaches have been proposed to QNNs. To this end, this paper presents an automata-theoretic approach to synthesizing BNNs that meet designated properties. More specifically, we define a temporal logic, called BLTL, as the specification language. We show that each BLTL formula can be transformed into an automaton on finite words. To deal with the state-explosion problem, we provide a tableau-based approach in real implementation. For the synthesis procedure, we utilize SMT solvers to detect the existence of a model (i.e., a BNN) in the construction process. Notably, synthesis provides a way to determine the hyper-parameters of the network before training.Moreover, we experimentally evaluate our approach and demonstrate its effectiveness in improving the individual fairness and local robustness of BNNs while maintaining accuracy to a great extent.

Neural networks (NNs) are increasingly applied in safety-critical systems such as autonomous vehicles. However, they are fragile and are often ill-behaved. Consequently, their behaviors should undergo rigorous guarantees before deployment in practice. In this paper, we propose a set-boundary reachability method to investigate the safety verification problem of NNs from a topological perspective. Given an NN with an input set and a safe set, the safety verification problem is to determine whether all outputs of the NN resulting from the input set fall within the safe set. In our method, the homeomorphism property and the open map property of NNs are mainly exploited, which establish rigorous guarantees between the boundaries of the input set and the boundaries of the output set. The exploitation of these two properties facilitates reachability computations via extracting subsets of the input set rather than the entire input set, thus controlling the wrapping effect in reachability analysis and facilitating the reduction of computation burdens for safety verification. The homeomorphism property exists in some widely used NNs such as invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs), and the open map is a less strict property and easier to satisfy compared with the homeomorphism property. For NNs establishing either of these properties, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. Moreover, for NNs that do not feature these properties with respect to the input set, we explore subsets of the input set for establishing the local homeomorphism property and then abandon these subsets for reachability computations. Finally, some examples demonstrate the performance of the proposed method., Comment: 25 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:2210.04175

Electrocardiography (ECG) is a non-invasive tool for predicting cardiovascular diseases (CVDs). Current ECG-based diagnosis systems show promising performance owing to the rapid development of deep learning techniques. However, the label scarcity problem, the co-occurrence of multiple CVDs and the poor performance on unseen datasets greatly hinder the widespread application of deep learning-based models. Addressing them in a unified framework remains a significant challenge. To this end, we propose a multi-label semi-supervised model (ECGMatch) to recognize multiple CVDs simultaneously with limited supervision. In the ECGMatch, an ECGAugment module is developed for weak and strong ECG data augmentation, which generates diverse samples for model training. Subsequently, a hyperparameter-efficient framework with neighbor agreement modeling and knowledge distillation is designed for pseudo-label generation and refinement, which mitigates the label scarcity problem. Finally, a label correlation alignment module is proposed to capture the co-occurrence information of different CVDs within labeled samples and propagate this information to unlabeled samples. Extensive experiments on four datasets and three protocols demonstrate the effectiveness and stability of the proposed model, especially on unseen datasets. As such, this model can pave the way for diagnostic systems that achieve robust performance on multi-label CVDs prediction with limited supervision.

The increasing use of deep neural networks (DNNs) in safety-critical systems has raised concerns about their potential for exhibiting ill-behaviors. While DNN verification and testing provide post hoc conclusions regarding unexpected behaviors, they do not prevent the erroneous behaviors from occurring. To address this issue, DNN repair/patch aims to eliminate unexpected predictions generated by defective DNNs. Two typical DNN repair paradigms are retraining and fine-tuning. However, existing methods focus on the high-level abstract interpretation or inference of state spaces, ignoring the underlying neurons' outputs. This renders patch processes computationally prohibitive and limited to piecewise linear (PWL) activation functions to great extent. To address these shortcomings, we propose a behavior-imitation based repair framework, BIRDNN, which integrates the two repair paradigms for the first time. BIRDNN corrects incorrect predictions of negative samples by imitating the closest expected behaviors of positive samples during the retraining repair procedure. For the fine-tuning repair process, BIRDNN analyzes the behavior differences of neurons on positive and negative samples to identify the most responsible neurons for the erroneous behaviors. To tackle more challenging domain-wise repair problems (DRPs), we synthesize BIRDNN with a domain behavior characterization technique to repair buggy DNNs in a probably approximated correct style. We also implement a prototype tool based on BIRDNN and evaluate it on ACAS Xu DNNs. Our experimental results show that BIRDNN can successfully repair buggy DNNs with significantly higher efficiency than state-of-the-art repair tools. Additionally, BIRDNN is highly compatible with different activation functions., Comment: 12 pages, 3 figures

Electroencephalography (EEG) is an objective tool for emotion recognition and shows promising performance. However, the label scarcity problem is a main challenge in this field, which limits the wide application of EEG-based emotion recognition. In this paper, we propose a novel semi-supervised learning framework (EEGMatch) to leverage both labeled and unlabeled EEG data. First, an EEG-Mixup based data augmentation method is developed to generate more valid samples for model learning. Second, a semi-supervised two-step pairwise learning method is proposed to bridge prototype-wise and instance-wise pairwise learning, where the prototype-wise pairwise learning measures the global relationship between EEG data and the prototypical representation of each emotion class and the instance-wise pairwise learning captures the local intrinsic relationship among EEG data. Third, a semi-supervised multi-domain adaptation is introduced to align the data representation among multiple domains (labeled source domain, unlabeled source domain, and target domain), where the distribution mismatch is alleviated. Extensive experiments are conducted on two benchmark databases (SEED and SEED-IV) under a cross-subject leave-one-subject-out cross-validation evaluation protocol. The results show the proposed EEGmatch performs better than the state-of-the-art methods under different incomplete label conditions (with 6.89% improvement on SEED and 1.44% improvement on SEED-IV), which demonstrates the effectiveness of the proposed EEGMatch in dealing with the label scarcity problem in emotion recognition using EEG signals. The source code is available at https://github.com/KAZABANA/EEGMatch.

Body weight, as an essential physiological trait, is of considerable significance in many applications like body management, rehabilitation, and drug dosing for patient-specific treatments. Previous works on the body weight estimation task are mainly vision-based, using 2D/3D, depth, or infrared images, facing problems in illumination, occlusions, and especially privacy issues. The pressure mapping mattress is a non-invasive and privacy-preserving tool to obtain the pressure distribution image over the bed surface, which strongly correlates with the body weight of the lying person. To extract the body weight from this image, we propose a deep learning-based model, including a dual-branch network to extract the deep features and pose features respectively. A contrastive learning module is also combined with the deep-feature branch to help mine the mutual factors across different postures of every single subject. The two groups of features are then concatenated for the body weight regression task. To test the model's performance over different hardware and posture settings, we create a pressure image dataset of 10 subjects and 23 postures, using a self-made pressure-sensing bedsheet. This dataset, which is made public together with this paper, together with a public dataset, are used for the validation. The results show that our model outperforms the state-of-the-art algorithms over both 2 datasets. Our research constitutes an important step toward fully automatic weight estimation in both clinical and at-home practice. Our dataset is available for research purposes at: https://github.com/USTCWzy/MassEstimation.

Recognizing human emotions from complex, multivariate, and non-stationary electroencephalography (EEG) time series is essential in affective brain-computer interface. However, because continuous labeling of ever-changing emotional states is not feasible in practice, existing methods can only assign a fixed label to all EEG timepoints in a continuous emotion-evoking trial, which overlooks the highly dynamic emotional states and highly non-stationary EEG signals. To solve the problems of high reliance on fixed labels and ignorance of time-changing information, in this paper we propose a time-aware sampling network (TAS-Net) using deep reinforcement learning (DRL) for unsupervised emotion recognition, which is able to detect key emotion fragments and disregard irrelevant and misleading parts. Extensive experiments are conducted on three public datasets (SEED, DEAP, and MAHNOB-HCI) for emotion recognition using leave-one-subject-out cross-validation, and the results demonstrate the superiority of the proposed method against previous unsupervised emotion recognition methods.

Credit assignment problem of neural networks refers to evaluating the credit of each network component to the final outputs. For an untrained neural network, approaches to tackling it have made great contributions to parameter update and model revolution during the training phase. This problem on trained neural networks receives rare attention, nevertheless, it plays an increasingly important role in neural network patch, specification and verification. Based on Koopman operator theory, this paper presents an alternative perspective of linear dynamics on dealing with the credit assignment problem for trained neural networks. Regarding a neural network as the composition of sub-dynamics series, we utilize step-delay embedding to capture snapshots of each component, characterizing the established mapping as exactly as possible. To circumvent the dimension-difference problem encountered during the embedding, a composition and decomposition of an auxiliary linear layer, termed minimal linear dimension alignment, is carefully designed with rigorous formal guarantee. Afterwards, each component is approximated by a Koopman operator and we derive the Jacobian matrix and its corresponding determinant, similar to backward propagation. Then, we can define a metric with algebraic interpretability for the credit assignment of each network component. Moreover, experiments conducted on typical neural networks demonstrate the effectiveness of the proposed method., Comment: 9 pages, 4 figures

Topological vacua are a family of degenerate ground states of Yang-Mills fields with zero field strength but nontrivial topological structures. They play a fundamental role in particle physics and quantum field theory, but have not yet been experimentally observed. Here we report the first theoretical proposal and experimental realization of synthetic topological vacua with a cloud of atomic Bose-Einstein condensates. Our setup provides a promising platform to demonstrate the fundamental concept that a vacuum, rather than being empty, has rich spatial structures. The Hamiltonian for the vacuum of topological number n = 1 is synthesized and the related Hopf index is measured. The vacuum of topological number n = 2 is also realized, and we find that vacua with different topological numbers have distinctive spin textures and Hopf links. Our work opens up opportunities for exploring topological vacua and related long-sought-after instantons in tabletop experiments.

Neural networks (NNs) are increasingly applied in safety-critical systems such as autonomous vehicles. However, they are fragile and are often ill-behaved. Consequently, their behaviors should undergo rigorous guarantees before deployment in practice. In this paper we propose a set-boundary reachability method to investigate the safety verification problem of NNs from a topological perspective. Given an NN with an input set and a safe set, the safety verification problem is to determine whether all outputs of the NN resulting from the input set fall within the safe set. In our method, the homeomorphism property of NNs is mainly exploited, which establishes a relationship mapping boundaries to boundaries. The exploitation of this property facilitates reachability computations via extracting subsets of the input set rather than the entire input set, thus controlling the wrapping effect in reachability analysis and facilitating the reduction of computation burdens for safety verification. The homeomorphism property exists in some widely used NNs such as invertible NNs. Notable representations are invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs). For these NNs, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. For NNs which do not feature this property with respect to the input set, we explore subsets of the input set for establishing the local homeomorphism property, and then abandon these subsets for reachability computations. Finally, some examples demonstrate the performance of the proposed method., Comment: 19 pages, 7 figures