Pub Date : 2024-12-30DOI: 10.1016/j.acha.2024.101746
Julia Kostin , Felix Krahmer , Dominik Stöger
In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behaviour of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.
{"title":"How robust is randomized blind deconvolution via nuclear norm minimization against adversarial noise?","authors":"Julia Kostin , Felix Krahmer , Dominik Stöger","doi":"10.1016/j.acha.2024.101746","DOIUrl":"10.1016/j.acha.2024.101746","url":null,"abstract":"<div><div>In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behaviour of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101746"},"PeriodicalIF":2.6,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968152","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 : 2024-12-17DOI: 10.1016/j.acha.2024.101745
Naveen Gupta , S. Sivananthan , Bharath K. Sriperumbudur
Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under Hölder type source condition, which generalizes and sharpens all the known results in the literature.
{"title":"Optimal rates for functional linear regression with general regularization","authors":"Naveen Gupta , S. Sivananthan , Bharath K. Sriperumbudur","doi":"10.1016/j.acha.2024.101745","DOIUrl":"10.1016/j.acha.2024.101745","url":null,"abstract":"<div><div>Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under Hölder type source condition, which generalizes and sharpens all the known results in the literature.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101745"},"PeriodicalIF":2.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874073","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 : 2024-12-09DOI: 10.1016/j.acha.2024.101734
A. Iosevich , A. Mayeli
Let G be a finite abelian group. Let be a signal (i.e. function). The classical uncertainty principle asserts that the product of the size of the support of f and its Fourier transform , and respectively, must satisfy the condition:
In the first part of this paper, we improve the uncertainty principle for signals with Fourier transform supported on generic sets. This improvement is achieved by employing the restriction theory, including Bourgain celebrate result on -sets, and the Salem set mechanism from harmonic analysis. Then we investigate some applications of uncertainty principles that were developed in the first part of this paper, to the problem of unique recovery of finite sparse signals in the absence of some frequencies.
Donoho and Stark ([14]), and, independently, Matolcsi and Szucs ([33]) showed that a signal of length N can be recovered exactly, even if some of the frequencies are unobserved, provided that the product of the size of the number of non-zero entries of the signal and the number of missing frequencies is not too large, leveraging the classical uncertainty principle for vectors. Our results broaden the scope for a natural class of signals in higher-dimensional spaces. In the case when the signal is binary, we provide a very simple exact recovery mechanism through the DRA algorithm.
{"title":"Uncertainty principles, restriction, Bourgain's Λq theorem, and signal recovery","authors":"A. Iosevich , A. Mayeli","doi":"10.1016/j.acha.2024.101734","DOIUrl":"10.1016/j.acha.2024.101734","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group. Let <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>C</mi></math></span> be a signal (i.e. function). The classical uncertainty principle asserts that the product of the size of the support of <em>f</em> and its Fourier transform <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>, <span><math><mtext>supp</mtext><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and <span><math><mtext>supp</mtext><mo>(</mo><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> respectively, must satisfy the condition:<span><span><span><math><mo>|</mo><mtext>supp</mtext><mo>(</mo><mi>f</mi><mo>)</mo><mo>|</mo><mo>⋅</mo><mo>|</mo><mtext>supp</mtext><mo>(</mo><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo><mo>|</mo><mo>≥</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>.</mo></math></span></span></span></div><div>In the first part of this paper, we improve the uncertainty principle for signals with Fourier transform supported on generic sets. This improvement is achieved by employing <em>the restriction theory</em>, including Bourgain celebrate result on <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-sets, and <em>the Salem set</em> mechanism from harmonic analysis. Then we investigate some applications of uncertainty principles that were developed in the first part of this paper, to the problem of unique recovery of finite sparse signals in the absence of some frequencies.</div><div>Donoho and Stark (<span><span>[14]</span></span>), and, independently, Matolcsi and Szucs (<span><span>[33]</span></span>) showed that a signal of length <em>N</em> can be recovered exactly, even if some of the frequencies are unobserved, provided that the product of the size of the number of non-zero entries of the signal and the number of missing frequencies is not too large, leveraging the classical uncertainty principle for vectors. Our results broaden the scope for a natural class of signals in higher-dimensional spaces. In the case when the signal is binary, we provide a very simple exact recovery mechanism through the DRA algorithm.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101734"},"PeriodicalIF":2.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874074","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 : 2024-12-09DOI: 10.1016/j.acha.2024.101735
Yunlong Feng , Qiang Wu
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.
{"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":"10.1016/j.acha.2024.101735","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101735"},"PeriodicalIF":2.6,"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}
Pub Date : 2024-12-03DOI: 10.1016/j.acha.2024.101736
Antoine Maillard , Afonso S. Bandeira , David Belius , Ivan Dokmanić , Shuta Nakajima
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, , with a random Gaussian matrix W, in a high-dimensional setting where . Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for 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.
{"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":"10.1016/j.acha.2024.101736","url":null,"abstract":"<div><div>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, <span><math><mi>x</mi><mo>↦</mo><mrow><mi>ReLU</mi></mrow><mo>(</mo><mi>W</mi><mi>x</mi><mo>)</mo></math></span>, with a random Gaussian <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>W</em>, in a high-dimensional setting where <span><math><mi>n</mi><mo>,</mo><mi>m</mi><mo>→</mo><mo>∞</mo></math></span>. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for <span><math><mi>α</mi><mo>=</mo><mi>m</mi><mo>/</mo><mi>n</mi></math></span> 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.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101736"},"PeriodicalIF":2.6,"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":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-28DOI: 10.1016/j.acha.2024.101726
Roy Y. He , Haixia Liu , Hao Liu
In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [26] (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526) and extend its application to general tasks of block sparse signal recovery. Given an observation y and sampling matrix A, we focus on minimizing the approximation error with respect to the signal c with block sparsity constraints. We prove that when the sampling matrix A 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.
本文对He et al. [26] （Group Projected Subspace Pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526）提出的Group Projected Subspace Pursuit （GPSP）算法进行了收敛性分析，并将其应用于块稀疏信号恢复的一般任务。给定观测值y和采样矩阵A，我们专注于最小化相对于具有块稀疏性约束的信号c的近似误差‖Ac−y‖22。证明了当采样矩阵A满足块受限等距特性（BRIP）且块受限等距常数（BRIC）足够小时，GPSP能准确地恢复出真实的块稀疏信号。当观测值有噪声时，如果真信号的幅度足够大，GPSP的这种收敛性仍然有效。GPSP通过子空间投影准则（SPC）选择候选包含特征，通过响应大小准则（RMC）选择候选排除特征。我们将这些标准与其他最先进的贪婪算法进行比较。我们的理论分析和数值消融研究表明，SPC是GPSP优越性能的关键，RMC可以增强观测值包含噪声时特征识别的鲁棒性。我们在不同的环境下测试并比较了GPSP和其他方法，包括异构随机块矩阵、不精确观察、人脸识别和PDE识别。我们发现，在大多数情况下，对于不同级别的块稀疏性和块大小，GPSP优于其他算法，证明其在一般应用中的有效性。
{"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}
Pub Date : 2024-11-28DOI: 10.1016/j.acha.2024.101727
Vladimir Yu. Protasov , Tatyana Zaitseva
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 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.
{"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}
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 N. 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 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.
具有静态增量的多尺度非高斯时间序列出现在广泛的应用中，尤其是在金融和物理领域。我们引入的随机模型能捕捉间歇现象，如危机或活动爆发、时间反转不对称，并能从大小为 N 的单一实现中进行估算。非高斯特性通过跨尺度小波系数的依赖性显现出来。我们根据小波系数及其模量在时间和尺度上的联合相关性定义最大熵模型。对角矩阵近似值是用这种联合相关性的小波表示来估算的。由此得到的对角矩阵定义了 O(log3N) 矩，称为散射谱。根据散射谱对缩放的不变性，我们定义了广义自相似性的概念，并可在单个实现上对其进行数值测试。我们研究了分数布朗运动、霍克斯过程、多分形随机游走以及金融和湍流时间序列的最大熵散射谱模型的准确性。
{"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}
Pub Date : 2024-11-19DOI: 10.1016/j.acha.2024.101725
Lexing Ying
This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems, providing a rather unified treatment for general kernels and unstructured sampling grids in both real and complex settings. Numerical results are provided to demonstrate the performance of the proposed method.
{"title":"Multidimensional unstructured sparse recovery via eigenmatrix","authors":"Lexing Ying","doi":"10.1016/j.acha.2024.101725","DOIUrl":"10.1016/j.acha.2024.101725","url":null,"abstract":"<div><div>This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems, providing a rather unified treatment for general kernels and unstructured sampling grids in both real and complex settings. Numerical results are provided to demonstrate the performance of the proposed method.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101725"},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696457","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 : 2024-11-15DOI: 10.1016/j.acha.2024.101723
Tamir Bendory, Dan Edidin, Oscar Mickelin
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.
{"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}
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