Applied and Computational Harmonic Analysis最新文献

英文中文

On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-01-10 DOI: 10.1016/j.acha.2025.101747

Andreas Horst, Jakob Lemvig, Allan Erlang Videbæk

The frame set conjecture for Hermite functions formulated in [13] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α and modulation rates β that avoid all known obstructions lead to Gabor frames for L2(R). By results in [24,25] and [22], it is known that the conjecture is true for the Gaussian, the 0th order Hermite functions, and false for Hermite functions of order 2,3,6,7,10,11,…, respectively. In this paper we disprove the remaining cases except for the 1st order Hermite function.

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引用次数: 0

Kadec-type theorems for sampled group orbits

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-01-10 DOI: 10.1016/j.acha.2025.101748

Ilya Krishtal, Brendan Miller

We extend the classical Kadec 14 theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.

我们将区间上指数函数系统的经典 Kadec 14 定理扩展到等距群表示下的向量轨道采样所形成的框架和原子分解。

引用次数: 0

How robust is randomized blind deconvolution via nuclear norm minimization against adversarial noise?

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-30 DOI: 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.

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引用次数: 0

Optimal rates for functional linear regression with general regularization

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-17 DOI: 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.

引用次数: 0

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

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-09 DOI: 10.1016/j.acha.2024.101734

A. Iosevich, A. Mayeli

Let G be a finite abelian group. Let f:G→C 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 fˆ, supp(f) and supp(fˆ) respectively, must satisfy the condition:|supp(f)|⋅|supp(fˆ)|≥|G|.

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 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-12-09 DOI: 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.

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引用次数: 0

Injectivity of ReLU networks: Perspectives from statistical physics

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

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

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, x↦ReLU(Wx), with a random Gaussian m×n matrix W, in a high-dimensional setting where n,m→∞. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for α=m/n 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.

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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 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-28 DOI: 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

{‖ A c - y ‖}_{2}^{2}

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.

{"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 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-11-28 DOI: 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

W_{2}^{k} (R^{n})

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.

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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 Q1 MATHEMATICS, APPLIED

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

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

O (\log^{3} N)

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。

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引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

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Applied and Computational Harmonic Analysis

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