Applied and Computational Harmonic Analysis最新文献

A parameter-free two-bit covariance estimator with improved operator norm error rate 一种改进算子范数错误率的无参数二位协方差估计器

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-05-02 DOI: 10.1016/j.acha.2025.101774

Junren Chen , Michael K. Ng

{"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":"2025-05-02","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}

引用次数: 0

Sparsification of the regularized magnetic Laplacian with multi-type spanning forests 具有多类型跨林的正则磁拉普拉斯算子的稀疏化

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-04-28 DOI: 10.1016/j.acha.2025.101766

M. Fanuel, R. Bardenet

{"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":"2025-04-28","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}

引用次数: 0

Duality for neural networks through Reproducing Kernel Banach Spaces 利用核Banach空间再现神经网络的对偶性

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-27 DOI: 10.1016/j.acha.2025.101765

Len Spek , Tjeerd Jan Heeringa , Felix Schwenninger , Christoph Brune

引用次数: 0

Controlled learning of pointwise nonlinearities in neural-network-like architectures 类神经网络结构中点非线性的受控学习

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-25 DOI: 10.1016/j.acha.2025.101764

Michael Unser, Alexis Goujon, Stanislas Ducotterd

{"title":"Controlled learning of pointwise nonlinearities in neural-network-like architectures","authors":"Michael Unser, Alexis Goujon, Stanislas Ducotterd","doi":"10.1016/j.acha.2025.101764","DOIUrl":"10.1016/j.acha.2025.101764","url":null,"abstract":"<div><div>We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the second-order total variation of each trainable activation. The slope constraints allow us to impose properties such as 1-Lipschitz stability, firm non-expansiveness, and monotonicity/invertibility. These properties are crucial to ensure the proper functioning of certain classes of signal-processing algorithms (e.g., plug-and-play schemes, unrolled proximal gradient, invertible flows). We prove that the global optimum of the stated constrained-optimization problem is achieved with nonlinearities that are adaptive nonuniform linear splines. We then show how to solve the resulting function-optimization problem numerically by representing the nonlinearities in a suitable (nonuniform) B-spline basis. Finally, we illustrate the use of our framework with the data-driven design of (weakly) convex regularizers for the denoising of images and the resolution of inverse problems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101764"},"PeriodicalIF":2.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738438","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}

引用次数: 0

Mathematical algorithm design for deep learning under societal and judicial constraints: The algorithmic transparency requirement 社会和司法约束下深度学习的数学算法设计：算法透明度要求

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-24 DOI: 10.1016/j.acha.2025.101763

Holger Boche , Adalbert Fono , Gitta Kutyniok

{"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":"2025-03-24","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}

引用次数: 0

Entropy of compact operators with applications to Landau-Pollak-Slepian theory and Sobolev spaces 紧算子的熵及其在Landau-Pollak-Slepian理论和Sobolev空间中的应用

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-21 DOI: 10.1016/j.acha.2025.101762

Thomas Allard, Helmut Bölcskei

引用次数: 0

An efficient spatial discretization of spans of multivariate Chebyshev polynomials 多元切比雪夫多项式跨度的有效空间离散化

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-20 DOI: 10.1016/j.acha.2025.101761

Lutz Kämmerer

{"title":"An efficient spatial discretization of spans of multivariate Chebyshev polynomials","authors":"Lutz Kämmerer","doi":"10.1016/j.acha.2025.101761","DOIUrl":"10.1016/j.acha.2025.101761","url":null,"abstract":"<div><div>For an arbitrary given span of high dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each polynomial of this span.</div><div>The approach presented here combines three different types of efficiency. First, the construction of a spatial discretization should be computationally efficient with respect to the dimension of the span of the Chebyshev polynomials. Second, the constructed discretization should be sample efficient, i.e., the number of sampling nodes within the constructed discretization should be reasonably low. Third, there should be an efficient algorithm for the unique reconstruction of a polynomial from given sampling values at the sampling nodes of the discretization.</div><div>The first two mentioned types of efficiency are also present in constructions based on random sampling nodes, but the lack of structure here causes the inefficiency of the reconstruction method. Our approach uses a combination of cosine transformed rank-1 lattices whose structure allows for applications of univariate fast Fourier transforms for the reconstruction algorithm and is thus a priori efficiently realizable.</div><div>Besides the theoretical estimates of numbers of sampling nodes and failure probabilities due to a random draw of the used lattices, we present several improvements of the basic design approach that significantly increases its practical applicability. Numerical tests, which discretize spans of multivariate Chebyshev polynomials depending on up to more than 50 spatial variables, corroborate the theoretical results and the significance of the improvements.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101761"},"PeriodicalIF":2.6,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697625","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}

引用次数: 0

An inverse problem for Dirac systems on p-star-shaped graphs p星形图上Dirac系统的反问题

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-20 DOI: 10.1016/j.acha.2025.101760

Yu Ping Wang , Yan-Hsiou Cheng

引用次数: 0

Error estimate of the u-series method for molecular dynamics simulations 分子动力学模拟u系列方法的误差估计

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2025-03-14 DOI: 10.1016/j.acha.2025.101759

Jiuyang Liang , Zhenli Xu , Qi Zhou

引用次数: 0

The Large Deviation Principle for W-random spectral measures w -随机谱测量的大偏差原理