Pub Date : 2025-08-01Epub Date: 2025-05-21DOI: 10.1016/j.acha.2025.101777
Xianchen Zhou , Kun Hu , Hongxia Wang
The generalization capability of Graph Convolutional Networks (GCNs) has been researched recently. The generalization error bound based on algorithmic stability is obtained for various structures of GCN. However, the generalization error bound computed by this method increases rapidly during the iteration since the algorithmic stability exponential depends on the number of iterations, which is not consistent with the performance of GCNs in practice. Based on the fact that the property of loss landscape, such as convex, exp-concave, or Polyak-Lojasiewicz* (PL*) leads to tighter stability and better generalization error bound, this paper focuses on the semi-supervised loss landscape of wide GCN. It shows that a wide GCN has a Hessian matrix with a small norm, which can lead to a positive definite training tangent kernel. Then GCN's loss can satisfy the PL* condition and lead to a tighter uniform stability independent of the iteration compared with previous work. Therefore, the generalization error bound in this paper depends on the graph filter's norm and layers, which is consistent with the experiments' results.
{"title":"A tighter generalization error bound for wide GCN based on loss landscape","authors":"Xianchen Zhou , Kun Hu , Hongxia Wang","doi":"10.1016/j.acha.2025.101777","DOIUrl":"10.1016/j.acha.2025.101777","url":null,"abstract":"<div><div>The generalization capability of Graph Convolutional Networks (GCNs) has been researched recently. The generalization error bound based on algorithmic stability is obtained for various structures of GCN. However, the generalization error bound computed by this method increases rapidly during the iteration since the algorithmic stability exponential depends on the number of iterations, which is not consistent with the performance of GCNs in practice. Based on the fact that the property of loss landscape, such as convex, exp-concave, or Polyak-Lojasiewicz* (PL*) leads to tighter stability and better generalization error bound, this paper focuses on the semi-supervised loss landscape of wide GCN. It shows that a wide GCN has a Hessian matrix with a small norm, which can lead to a positive definite training tangent kernel. Then GCN's loss can satisfy the PL* condition and lead to a tighter uniform stability independent of the iteration compared with previous work. Therefore, the generalization error bound in this paper depends on the graph filter's norm and layers, which is consistent with the experiments' results.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"78 ","pages":"Article 101777"},"PeriodicalIF":2.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144108039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-04-28DOI: 10.1016/j.acha.2025.101766
M. Fanuel, R. Bardenet
In this paper, we consider a -connection graph, that is, a graph where each oriented edge is endowed with a unit modulus complex number that is conjugated under orientation flip. A natural replacement for the combinatorial Laplacian is then the magnetic Laplacian, an Hermitian matrix that includes information about the graph's connection. Magnetic Laplacians appear, e.g., in the problem of angular synchronization. In the context of large and dense graphs, we study here sparsifiers of the magnetic Laplacian Δ, i.e., spectral approximations based on subgraphs with few edges. Our approach relies on sampling multi-type spanning forests (MTSFs) using a custom determinantal point process, a probability distribution over edges that favors diversity. In a word, an MTSF is a spanning subgraph whose connected components are either trees or cycle-rooted trees. The latter partially capture the angular inconsistencies of the connection graph, and thus provide a way to compress the information contained in the connection. Interestingly, when the connection graph has weakly inconsistent cycles, samples from the determinantal point process under consideration can be obtained à la Wilson, using a random walk with cycle popping. We provide statistical guarantees for a choice of natural estimators of the connection Laplacian, and investigate two practical applications of our sparsifiers: ranking with angular synchronization and graph-based semi-supervised learning. From a statistical perspective, a side result of this paper of independent interest is a matrix Chernoff bound with intrinsic dimension, which allows considering the influence of a regularization – of the form with – on sparsification guarantees.
本文考虑一个U(1)-连接图，即每个有向边都有一个单位模复数，该复数在有向翻转下共轭。组合拉普拉斯的自然替代品是磁拉普拉斯，一个包含图连接信息的厄米矩阵。例如，在角同步问题中出现了磁拉普拉斯算子。在大而密集图的背景下，我们研究了磁拉普拉斯Δ的稀疏化算子，即基于少边子图的谱近似。我们的方法依赖于使用自定义确定性点过程对多类型跨越森林（mtsf）进行采样，这是一种有利于多样性的边缘概率分布。简而言之，MTSF是一个生成子图，其连接的组件要么是树，要么是环根树。后者部分捕获连接图的角度不一致，从而提供一种压缩连接中包含的信息的方法。有趣的是，当连接图具有弱不一致的循环时，可以使用带有循环弹出的随机漫步，从所考虑的确定性点过程中获得样本。我们为连接拉普拉斯的自然估计量的选择提供了统计保证，并研究了我们的稀疏化器的两个实际应用：角同步排序和基于图的半监督学习。从统计的角度来看，本文的一个独立的结果是一个具有固有维数的矩阵Chernoff界，它允许考虑形式为Δ+qI with q>；0的正则化-对稀疏化保证的影响。
{"title":"Sparsification of the regularized magnetic Laplacian with multi-type spanning forests","authors":"M. Fanuel, R. Bardenet","doi":"10.1016/j.acha.2025.101766","DOIUrl":"10.1016/j.acha.2025.101766","url":null,"abstract":"<div><div>In this paper, we consider a <span><math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-connection graph, that is, a graph where each oriented edge is endowed with a unit modulus complex number that is conjugated under orientation flip. A natural replacement for the combinatorial Laplacian is then the <em>magnetic</em> Laplacian, an Hermitian matrix that includes information about the graph's connection. Magnetic Laplacians appear, e.g., in the problem of angular synchronization. In the context of large and dense graphs, we study here sparsifiers of the magnetic Laplacian Δ, i.e., spectral approximations based on subgraphs with few edges. Our approach relies on sampling multi-type spanning forests (MTSFs) using a custom determinantal point process, a probability distribution over edges that favors diversity. In a word, an MTSF is a spanning subgraph whose connected components are either trees or cycle-rooted trees. The latter partially capture the angular inconsistencies of the connection graph, and thus provide a way to compress the information contained in the connection. Interestingly, when the connection graph has weakly inconsistent cycles, samples from the determinantal point process under consideration can be obtained <em>à la Wilson</em>, using a random walk with cycle popping. We provide statistical guarantees for a choice of natural estimators of the connection Laplacian, and investigate two practical applications of our sparsifiers: ranking with angular synchronization and graph-based semi-supervised learning. From a statistical perspective, a side result of this paper of independent interest is a matrix Chernoff bound with intrinsic dimension, which allows considering the influence of a regularization – of the form <span><math><mi>Δ</mi><mo>+</mo><mi>q</mi><mi>I</mi></math></span> with <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span> – on sparsification guarantees.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"78 ","pages":"Article 101766"},"PeriodicalIF":2.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143891370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-05-12DOI: 10.1016/j.acha.2025.101773
Xinliang Liu , Bingxin Zhou , Chutian Zhang , Yu Guang Wang
Graph neural networks have achieved champions in wide applications. Neural message passing is a typical key module for feature propagation by aggregating neighboring features. In this work, we propose a new message passing based on multiscale framelet transforms, called Framelet Message Passing. Different from traditional spatial methods, it integrates framelet representation of neighbor nodes from multiple hops away in node message update. We also propose a continuous message passing using neural ODE solvers. Both discrete and continuous cases can provably mitigate oversmoothing and achieve superior performance. Numerical experiments on real graph datasets show that the continuous version of the framelet message passing significantly outperforms existing methods when learning heterogeneous graphs and achieves state-of-the-art performance on classic node classification tasks with low computational costs.
{"title":"Framelet message passing","authors":"Xinliang Liu , Bingxin Zhou , Chutian Zhang , Yu Guang Wang","doi":"10.1016/j.acha.2025.101773","DOIUrl":"10.1016/j.acha.2025.101773","url":null,"abstract":"<div><div>Graph neural networks have achieved champions in wide applications. Neural message passing is a typical key module for feature propagation by aggregating neighboring features. In this work, we propose a new message passing based on multiscale framelet transforms, called Framelet Message Passing. Different from traditional spatial methods, it integrates framelet representation of neighbor nodes from multiple hops away in node message update. We also propose a continuous message passing using neural ODE solvers. Both discrete and continuous cases can provably mitigate oversmoothing and achieve superior performance. Numerical experiments on real graph datasets show that the continuous version of the framelet message passing significantly outperforms existing methods when learning heterogeneous graphs and achieves state-of-the-art performance on classic node classification tasks with low computational costs.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"78 ","pages":"Article 101773"},"PeriodicalIF":2.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-05-02DOI: 10.1016/j.acha.2025.101774
Junren Chen , Michael K. Ng
A covariance matrix estimator using two bits per entry was recently developed by Dirksen et al. (2022) [11]. The estimator achieves near minimax operator norm rate for general sub-Gaussian distributions, but also suffers from two downsides: theoretically, there is an essential gap on operator norm error between their estimator and sample covariance when the diagonal of the covariance matrix is dominated by only a few entries; practically, its performance heavily relies on the dithering scale, which needs to be tuned according to some unknown parameters. In this work, we propose a new 2-bit covariance matrix estimator that simultaneously addresses both issues. Unlike the sign quantizer associated with uniform dither in Dirksen et al., we adopt a triangular dither prior to a 2-bit quantizer inspired by the multi-bit uniform quantizer. By employing dithering scales varying across entries, our estimator enjoys an improved operator norm error rate that depends on the effective rank of the underlying covariance matrix rather than the ambient dimension, which is optimal up to logarithmic factors. Moreover, our proposed method eliminates the need of any tuning parameter, as the dithering scales are entirely determined by the data. While our estimator requires a pass of all unquantized samples to determine the dithering scales, it can be adapted to the online setting where the samples arise sequentially. Experimental results are provided to demonstrate the advantages of our estimators over the existing ones.
{"title":"A parameter-free two-bit covariance estimator with improved operator norm error rate","authors":"Junren Chen , Michael K. Ng","doi":"10.1016/j.acha.2025.101774","DOIUrl":"10.1016/j.acha.2025.101774","url":null,"abstract":"<div><div>A covariance matrix estimator using two bits per entry was recently developed by Dirksen et al. (2022) <span><span>[11]</span></span>. The estimator achieves near minimax operator norm rate for general sub-Gaussian distributions, but also suffers from two downsides: theoretically, there is an essential gap on operator norm error between their estimator and sample covariance when the diagonal of the covariance matrix is dominated by only a few entries; practically, its performance heavily relies on the dithering scale, which needs to be tuned according to some unknown parameters. In this work, we propose a new 2-bit covariance matrix estimator that simultaneously addresses both issues. Unlike the sign quantizer associated with uniform dither in Dirksen et al., we adopt a triangular dither prior to a 2-bit quantizer inspired by the multi-bit uniform quantizer. By employing dithering scales varying across entries, our estimator enjoys an improved operator norm error rate that depends on the effective rank of the underlying covariance matrix rather than the ambient dimension, which is optimal up to logarithmic factors. Moreover, our proposed method eliminates the need of <em>any</em> tuning parameter, as the dithering scales are entirely determined by the data. While our estimator requires a pass of all unquantized samples to determine the dithering scales, it can be adapted to the online setting where the samples arise sequentially. Experimental results are provided to demonstrate the advantages of our estimators over the existing ones.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"78 ","pages":"Article 101774"},"PeriodicalIF":2.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-03-14DOI: 10.1016/j.acha.2025.101759
Jiuyang Liang , Zhenli Xu , Qi Zhou
This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians M in the u-series and the base of interpolation nodes b in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as for the energy and for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.
{"title":"Error estimate of the u-series method for molecular dynamics simulations","authors":"Jiuyang Liang , Zhenli Xu , Qi Zhou","doi":"10.1016/j.acha.2025.101759","DOIUrl":"10.1016/j.acha.2025.101759","url":null,"abstract":"<div><div>This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians <em>M</em> in the u-series and the base of interpolation nodes <em>b</em> in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mi>M</mi></mrow></msup><mo>)</mo></math></span> for the energy and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mn>3</mn><mi>M</mi></mrow></msup><mo>)</mo></math></span> for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101759"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143675537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-03-21DOI: 10.1016/j.acha.2025.101762
Thomas Allard, Helmut Bölcskei
We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.
{"title":"Entropy of compact operators with applications to Landau-Pollak-Slepian theory and Sobolev spaces","authors":"Thomas Allard, Helmut Bölcskei","doi":"10.1016/j.acha.2025.101762","DOIUrl":"10.1016/j.acha.2025.101762","url":null,"abstract":"<div><div>We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101762"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-03-24DOI: 10.1016/j.acha.2025.101763
Holger Boche , Adalbert Fono , Gitta Kutyniok
Deep learning still has drawbacks regarding trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated with trustworthiness have been proposed via regulatory guidelines, e.g., in the European AI Act. Therefore, a central question is to what extent trustworthy deep learning can be realized. Establishing the described properties constituting trustworthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework that enables us to analyze whether a transparent implementation in a computing model is feasible. The core idea is to formalize and subsequently relate the properties of a transparent algorithmic implementation to the mathematical model of the computing platform, thereby establishing verifiable criteria.
We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale machines, respectively. Based on previous results, we find that Blum-Shub-Smale machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas Turing machines cannot guarantee trustworthiness to the same degree.
{"title":"Mathematical algorithm design for deep learning under societal and judicial constraints: The algorithmic transparency requirement","authors":"Holger Boche , Adalbert Fono , Gitta Kutyniok","doi":"10.1016/j.acha.2025.101763","DOIUrl":"10.1016/j.acha.2025.101763","url":null,"abstract":"<div><div>Deep learning still has drawbacks regarding trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated with trustworthiness have been proposed via regulatory guidelines, e.g., in the European AI Act. Therefore, a central question is to what extent trustworthy deep learning can be realized. Establishing the described properties constituting trustworthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework that enables us to analyze whether a transparent implementation in a computing model is feasible. The core idea is to formalize and subsequently relate the properties of a transparent algorithmic implementation to the mathematical model of the computing platform, thereby establishing verifiable criteria.</div><div>We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale machines, respectively. Based on previous results, we find that Blum-Shub-Smale machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas Turing machines cannot guarantee trustworthiness to the same degree.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101763"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-02-21DOI: 10.1016/j.acha.2025.101756
Mahya Ghandehari , Georgi S. Medvedev
The W-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for W-random graphs from [19], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the corresponding graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of small-world and bipartite random graphs conditioned on atypical edge counts. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces are affected by large deviations. We discuss the implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing.
W 随机图为大型随机网络建模提供了一个灵活的框架。利用文献[19]中针对 W-随机图的大偏差原理（LDP），我们证明了相应类别的随机对称希尔伯特-施密特积分算子的大偏差原理。我们的主要结果描述了积分算子的特征值和特征空间如何受到底层随机图元大偏差的影响。为了证明 LDP，我们证明了与积分算子相关的谱度量对相应图元的连续依赖性，并使用了收缩原理。为了说明我们的结果，我们获得了以非典型边数为条件的小世界和双方形随机图特征值的前阶渐近性。这些例子说明了特征值和特征空间如何受到大偏差的影响。我们将讨论这些观察结果对动态系统分岔分析和图信号处理的影响。
{"title":"The Large Deviation Principle for W-random spectral measures","authors":"Mahya Ghandehari , Georgi S. Medvedev","doi":"10.1016/j.acha.2025.101756","DOIUrl":"10.1016/j.acha.2025.101756","url":null,"abstract":"<div><div>The <em>W</em>-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for <em>W</em>-random graphs from <span><span>[19]</span></span>, we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the corresponding graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of small-world and bipartite random graphs conditioned on atypical edge counts. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces are affected by large deviations. We discuss the implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101756"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-02-17DOI: 10.1016/j.acha.2025.101749
Massimo Fornasier , Timo Klock , Marco Mondelli , Michael Rauchensteiner
The identification of the parameters of a neural network from finite samples of input-output pairs is often referred to as the teacher-student model, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature – after adding suitable distributional assumptions – has established finite sample identification of two-layer networks with a number of neurons , D being the input dimension. For the range the problem becomes harder, and truly little is known for networks parametrized by biases as well. This paper fills the gap by providing efficient algorithms and rigorous theoretical guarantees of finite sample identification for such wider shallow networks with biases. Our approach is based on a two-step pipeline: first, we recover the direction of the weights, by exploiting second order information; next, we identify the signs by suitable algebraic evaluations, and we recover the biases by empirical risk minimization via gradient descent. Numerical results demonstrate the effectiveness of our approach.
{"title":"Efficient identification of wide shallow neural networks with biases","authors":"Massimo Fornasier , Timo Klock , Marco Mondelli , Michael Rauchensteiner","doi":"10.1016/j.acha.2025.101749","DOIUrl":"10.1016/j.acha.2025.101749","url":null,"abstract":"<div><div>The identification of the parameters of a neural network from finite samples of input-output pairs is often referred to as the <em>teacher-student model</em>, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature – after adding suitable distributional assumptions – has established finite sample identification of two-layer networks with a number of neurons <span><math><mi>m</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, <em>D</em> being the input dimension. For the range <span><math><mi>D</mi><mo><</mo><mi>m</mi><mo><</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> the problem becomes harder, and truly little is known for networks parametrized by biases as well. This paper fills the gap by providing efficient algorithms and rigorous theoretical guarantees of finite sample identification for such wider shallow networks with biases. Our approach is based on a two-step pipeline: first, we recover the direction of the weights, by exploiting second order information; next, we identify the signs by suitable algebraic evaluations, and we recover the biases by empirical risk minimization via gradient descent. Numerical results demonstrate the effectiveness of our approach.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101749"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-01Epub Date: 2025-03-20DOI: 10.1016/j.acha.2025.101760
Yu Ping Wang , Yan-Hsiou Cheng
In this paper, we study direct and inverse problems for Dirac systems with complex-valued potentials on p-star-shaped graphs. More precisely, we firstly obtain sharp 2-term asymptotics of the corresponding eigenvalues. We then formulate and address a Horváth-type theorem, specifically, if the potentials on edges of the p-star-shaped graph are predetermined, we demonstrate that the remaining potential on can be uniquely determined by part of its eigenvalues and the given remaining potential on , , under certain conditions.
{"title":"An inverse problem for Dirac systems on p-star-shaped graphs","authors":"Yu Ping Wang , Yan-Hsiou Cheng","doi":"10.1016/j.acha.2025.101760","DOIUrl":"10.1016/j.acha.2025.101760","url":null,"abstract":"<div><div>In this paper, we study direct and inverse problems for Dirac systems with complex-valued potentials on <em>p</em>-star-shaped graphs. More precisely, we firstly obtain sharp 2-term asymptotics of the corresponding eigenvalues. We then formulate and address a Horváth-type theorem, specifically, if the potentials on <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> edges of the <em>p</em>-star-shaped graph are predetermined, we demonstrate that the remaining potential on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></math></span> can be uniquely determined by part of its eigenvalues and the given remaining potential on <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>π</mi><mo>]</mo></math></span>, <span><math><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mi>π</mi></math></span>, under certain conditions.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101760"},"PeriodicalIF":2.6,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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