Applied and Computational Harmonic Analysis最新文献

Optimal rates for functional linear regression with general regularization

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-12-17 DOI: 10.1016/j.acha.2024.101745

Naveen Gupta, S. Sivananthan, Bharath K. Sriperumbudur

引用次数: 0

Uncertainty principles, restriction, Bourgain's Λq theorem, and signal recovery

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-12-09 DOI: 10.1016/j.acha.2024.101734

A. Iosevich, A. Mayeli

引用次数: 0

Tikhonov regularization for Gaussian empirical gain maximization in RKHS is consistent RKHS 中高斯经验增益最大化的 Tikhonov 正则化是一致的

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-12-09 DOI: 10.1016/j.acha.2024.101735

Yunlong Feng, Qiang Wu

{"title":"Tikhonov regularization for Gaussian empirical gain maximization in RKHS is consistent","authors":"Yunlong Feng, Qiang Wu","doi":"10.1016/j.acha.2024.101735","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101735","url":null,"abstract":"Without imposing light-tailed noise assumptions, we prove that Tikhonov regularization for Gaussian Empirical Gain Maximization (EGM) in a reproducing kernel Hilbert space is consistent and further establish its fast exponential type convergence rates. In the literature, Gaussian EGM was proposed in various contexts to tackle robust estimation problems and has been applied extensively in a great variety of real-world applications. A reproducing kernel Hilbert space is frequently chosen as the hypothesis space, and Tikhonov regularization plays a crucial role in model selection. Although Gaussian EGM has been studied theoretically in a series of papers recently and has been well-understood, theoretical understanding of its Tikhonov regularized variants in RKHS is still limited. Several fundamental challenges remain, especially when light-tailed noise assumptions are absent. To fill the gap and address these challenges, we conduct the present study and make the following contributions. First, under weak moment conditions, we establish a new comparison theorem that enables the investigation of the asymptotic mean calibration properties of regularized Gaussian EGM. Second, under the same weak moment conditions, we show that regularized Gaussian EGM estimators are consistent and further establish their fast exponential-type convergence rates. Our study justifies its feasibility in tackling robust regression problems and explains its robustness from a theoretical viewpoint. Moreover, new technical tools including probabilistic initial upper bounds, confined effective hypothesis spaces, and novel comparison theorems are introduced and developed, which can faciliate the analysis of general regularized empirical gain maximization schemes that fall into the same vein as regularized Gaussian EGM.","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"34 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823231","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

Injectivity of ReLU networks: Perspectives from statistical physics

IF 2.5 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-12-03 DOI: 10.1016/j.acha.2024.101736

Antoine Maillard, Afonso S. Bandeira, David Belius, Ivan Dokmanić, Shuta Nakajima

{"title":"Injectivity of ReLU networks: Perspectives from statistical physics","authors":"Antoine Maillard, Afonso S. Bandeira, David Belius, Ivan Dokmanić, Shuta Nakajima","doi":"10.1016/j.acha.2024.101736","DOIUrl":"https://doi.org/10.1016/j.acha.2024.101736","url":null,"abstract":"When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, <mml:math altimg=\"si1.svg\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">ReLU</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, with a random Gaussian <mml:math altimg=\"si2.svg\"><mml:mi>m</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:math> matrix <ce:italic>W</ce:italic>, in a high-dimensional setting where <mml:math altimg=\"si3.svg\"><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>∞</mml:mo></mml:math>. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for <mml:math altimg=\"si4.svg\"><mml:mi>α</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>n</mml:mi></mml:math> by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min–max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790150","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

Group projected subspace pursuit for block sparse signal reconstruction: Convergence analysis and applications 1

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-28 DOI: 10.1016/j.acha.2024.101726

Roy Y. He , Haixia Liu , Hao Liu

{"title":"Group projected subspace pursuit for block sparse signal reconstruction: Convergence analysis and applications 1","authors":"Roy Y. He , Haixia Liu , Hao Liu","doi":"10.1016/j.acha.2024.101726","DOIUrl":"10.1016/j.acha.2024.101726","url":null,"abstract":"<div><div>In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. <span><span>[26]</span></span> (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), <em>Journal of Computational Physics</em>, 494, 112526) and extend its application to general tasks of block sparse signal recovery. Given an observation <strong>y</strong> and sampling matrix <strong>A</strong>, we focus on minimizing the approximation error <span><math><msubsup><mrow><mo>‖</mo><mi>A</mi><mi>c</mi><mo>−</mo><mi>y</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> with respect to the signal <strong>c</strong> with block sparsity constraints. We prove that when the sampling matrix <strong>A</strong> satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of the true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101726"},"PeriodicalIF":2.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759319","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

Anisotropic refinable functions and the tile B-splines

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-28 DOI: 10.1016/j.acha.2024.101727

Vladimir Yu. Protasov , Tatyana Zaitseva

{"title":"Anisotropic refinable functions and the tile B-splines","authors":"Vladimir Yu. Protasov , Tatyana Zaitseva","doi":"10.1016/j.acha.2024.101727","DOIUrl":"10.1016/j.acha.2024.101727","url":null,"abstract":"<div><div>The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101727"},"PeriodicalIF":2.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759438","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

Scale dependencies and self-similar models with wavelet scattering spectra 小波散射谱的尺度依赖性和自相似模型

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101724

Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat

{"title":"Scale dependencies and self-similar models with wavelet scattering spectra","authors":"Rudy Morel , Gaspar Rochette , Roberto Leonarduzzi , Jean-Philippe Bouchaud , Stéphane Mallat","doi":"10.1016/j.acha.2024.101724","DOIUrl":"10.1016/j.acha.2024.101724","url":null,"abstract":"<div><div>Multi-scale non-Gaussian time-series having stationary increments appear in a wide range of applications, particularly in finance and physics. We introduce stochastic models that capture intermittency phenomena such as crises or bursts of activity, time reversal asymmetries, and that can be estimated from a single realization of size <em>N</em>. Variations at multiple scales are separated with a wavelet transform. Non-Gaussian properties appear through dependencies of wavelet coefficients across scales. We define maximum entropy models from the joint correlation across time and scales of wavelet coefficients and their modulus. Diagonal matrix approximations are estimated with a wavelet representation of this joint correlation. The resulting diagonals define <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mi>N</mi><mo>)</mo></math></span> moments that are called scattering spectra. A notion of wide-sense self-similarity is defined from the invariance of scattering spectra to scaling, which can be tested numerically on a single realization. We study the accuracy of maximum entropy scattering spectra models for fractional Brownian motions, Hawkes processes, multifractal random walks, as well as financial and turbulent time-series.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"75 ","pages":"Article 101724"},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696456","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

Multidimensional unstructured sparse recovery via eigenmatrix 通过特征矩阵进行多维非结构化稀疏恢复

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-19 DOI: 10.1016/j.acha.2024.101725

Lexing Ying

引用次数: 0

The beltway problem over orthogonal groups 正交群上的带路问题

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-15 DOI: 10.1016/j.acha.2024.101723

Tamir Bendory, Dan Edidin, Oscar Mickelin

{"title":"The beltway problem over orthogonal groups","authors":"Tamir Bendory, Dan Edidin, Oscar Mickelin","doi":"10.1016/j.acha.2024.101723","DOIUrl":"10.1016/j.acha.2024.101723","url":null,"abstract":"<div><div>The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101723"},"PeriodicalIF":2.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696470","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

On quadrature for singular integral operators with complex symmetric quadratic forms 关于具有复对称二次形式的奇异积分算子的正交性

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101721

Jeremy Hoskins , Manas Rachh , Bowei Wu

引用次数: 0