Applied and Computational Harmonic Analysis最新文献

英文中文

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions 拉普拉斯-贝尔特拉米特征函数中更高的特征切格比

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-21 DOI: 10.1016/j.acha.2024.101710

Gary Froyland, Christopher P. Rock

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.

Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

本文研究了拉普拉斯-贝尔特拉米算子的特征值和特征函数与光滑黎曼流形（可能是加权流形和/或有边界流形）的高Cheeger常数之间的联系。高阶切格常数给出了流形主要几何特征的松散描述。我们根据任何特征函数的特征值与相应的节点域数，给出了高阶切格常数的构造上界。具体地说，我们证明了对于每一个这样的特征函数，其超水平集合的正量度集合的切格比在相应的特征值上都有上界。有些流形的主要特征缠绕在多个特征函数上，没有一个特征函数包含所有主要特征。在这种情况下，可能存在精心选择的特征函数线性组合，每个特征函数在单个特征上的值较大，而在其他特征上的值较小。然后，我们可以对这些线性组合应用软阈值算子，得到新的函数，每个函数都支持一个特征。我们证明，这些函数的水平集的切格比也给出了拉普拉斯-贝尔特拉米特征值的上限。我们将这些水平集结果扩展到非自主动态系统，并证明动态拉普拉斯特征函数揭示了具有较小动态切格比的水平集。

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引用次数: 0

A perturbative analysis for noisy spectral estimation 噪声频谱估计的扰动分析

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-18 DOI: 10.1016/j.acha.2024.101716

Lexing Ying

Spectral estimation is a fundamental task in signal processing. Recent algorithms in quantum phase estimation are concerned with the large noise, large frequency regime of the spectral estimation problem. The recent work in Ding-Epperly-Lin-Zhang proves that the ESPRIT algorithm exhibits superconvergence behavior for the spike locations in terms of the maximum frequency. This note provides a perturbative analysis to understand this behavior and extends to the case where the noise grows with the sampling frequency. However, this does not imply or explain the rigorous error bound obtained by Ding-Epperly-Lin-Zhang.

频谱估计是信号处理中的一项基本任务。量子相位估计的最新算法关注的是频谱估计问题的大噪声、大频率机制。Ding-Epperly-Lin-Zhang 的最新研究证明，ESPRIT 算法在尖峰位置的最大频率方面表现出超收敛行为。本论文提供了一种扰动分析来理解这种行为，并扩展到噪声随采样频率增长的情况。然而，这并不意味着或解释丁-埃珀利-林-张所获得的严格误差约束。

引用次数: 0

Solving PDEs on spheres with physics-informed convolutional neural networks 用物理信息卷积神经网络求解球面上的 PDEs

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-15 DOI: 10.1016/j.acha.2024.101714

Guanhang Lei , Zhen Lei , Lei Shi , Chenyu Zeng , Ding-Xuan Zhou

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.

物理信息神经网络（PINN）已被证明能从多种实验角度高效地求解偏微分方程（PDE）。最近的一些研究还提出了针对包括球体在内的曲面上的偏微分方程的 PINN 算法。然而，对于 PINN 的数值性能，尤其是曲面或流形上的 PINN，仍然缺乏理论上的了解。在本文中，我们建立了用于求解球面上 PDE 的物理信息卷积神经网络（PICNN）的严格分析。通过使用并改进深度卷积神经网络和球面谐波分析的最新近似结果，我们证明了关于索博列夫规范的近似误差上限。随后，我们将其与创新的定位复杂性分析相结合，建立了 PICNN 的快速收敛率。我们的理论结果也得到了实验的证实和补充。鉴于这些发现，我们探讨了规避高维 PDEs 求解时出现的维度诅咒的潜在策略。

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引用次数: 0

Linearized Wasserstein dimensionality reduction with approximation guarantees 具有近似保证的线性化瓦瑟斯坦降维法

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-15 DOI: 10.1016/j.acha.2024.101718

Alexander Cloninger , Keaton Hamm , Varun Khurana , Caroline Moosmüller

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in

R^{n}

, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

我们介绍了 LOT Wassmap，这是一种在计算上可行的算法，用于揭示 Wasserstein 空间中的低维结构。该算法的动机是观察到许多数据集被自然地解释为概率度量，而不是 Rn 中的点，要找到这些数据集的低维描述，需要在 Wasserstein 空间中使用流形学习算法。大多数现有算法都基于计算成对的 Wasserstein 距离矩阵，这对于高维度的大型数据集来说，计算难度很大。我们的算法利用 Sinkhorn 距离和线性化最优传输等近似方案来加快计算速度，尤其是避免了计算成对距离矩阵。我们为这种近似方法下的嵌入质量提供了保证，包括在没有明确的概率度量描述而必须处理有限样本的情况下。实验证明，LOT Wassmap 可以获得正确的嵌入，而且质量会随着样本量的增加而提高。我们还展示了与依赖成对距离计算的算法相比，LOT Wassmap 如何显著降低计算成本。

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引用次数: 0

Weighted variation spaces and approximation by shallow ReLU networks 加权变异空间和浅层 ReLU 网络逼近

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-10 DOI: 10.1016/j.acha.2024.101713

Ronald DeVore , Robert D. Nowak , Rahul Parhi , Jonathan W. Siegel

We investigate the approximation of functions f on a bounded domain

Ω \subset R^{d}

by the outputs of single-hidden-layer ReLU neural networks of width n. This form of nonlinear n-term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on Ω whose approximation rates do not grow unbounded with the input dimension. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains Ω. The current definition of these model classes does not depend on the domain Ω. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.

我们研究了宽度为 n 的单隐层 ReLU 神经网络输出对有界域 Ω⊂Rd 上函数 f 的逼近。这种形式的 NNA 有几个著名的逼近结果，它们引入了 Ω 上函数的新模型类，其逼近率不会随着输入维度的增加而无限制地增长。这些新类包括巴伦类，以及基于稀疏性或变化的类，如拉顿域 BV 类。目前这些模型类的定义并不依赖于域 Ω。通过引入加权变异空间的概念，我们给出了关于域上模型类的更恰当的新定义。这些新的模型类是领域本身所固有的。这些新模型类的重要性在于，它们严格来说比经典（与域无关）类大。然而，研究表明它们保持了相同的 NNA 率。

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引用次数: 0

Two subspace methods for frequency sparse graph signals 频率稀疏图信号的两种子空间方法

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-10-02 DOI: 10.1016/j.acha.2024.101711

Tarek Emmrich, Martina Juhnke, Stefan Kunis

We study signals that are sparse in graph spectral domain and develop explicit algorithms to reconstruct the support set as well as partial components from samples on few vertices of the graph. The number of required samples is independent of the total size of the graph and takes only local properties of the graph into account. Our results rely on an operator based framework for subspace methods and become effective when the spectral eigenfunctions are zero-free or linear independent on small sets of the vertices. The latter has recently been addressed using algebraic methods by the first author.

我们研究了图谱域中稀疏的信号，并开发了明确的算法，从图中少数顶点的样本中重建支持集和部分成分。所需的样本数量与图的总大小无关，并且只考虑图的局部属性。我们的结果依赖于基于算子的子空间方法框架，当谱特征函数在小的顶点集上无零或线性独立时，我们的结果就会变得有效。第一作者最近使用代数方法解决了后者的问题。

引用次数: 0

Approximation theory of wavelet frame based image restoration 基于小波帧的图像修复近似理论

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-09-27 DOI: 10.1016/j.acha.2024.101712

Jian-Feng Cai , Jae Kyu Choi , Jianbin Yang

In this paper, we analyze the error estimate of a wavelet frame based image restoration method from degraded and incomplete measurements. We present the error between the underlying original discrete image and the approximate solution which has the minimal

ℓ_{1}

-norm of the canonical wavelet frame coefficients among all possible solutions. Then we further connect the error estimate for the discrete model to the approximation to the underlying function from which the underlying image comes.

本文分析了基于小波帧的图像复原方法从退化和不完整测量中得出的误差估计。我们提出了底层原始离散图像与近似解之间的误差，近似解在所有可能的解中具有最小的小波帧系数 ℓ1-norm 。然后，我们进一步将离散模型的误差估计与底层图像所来自的底层函数的近似值联系起来。

引用次数: 0

Donoho-Logan large sieve principles for the wavelet transform 小波变换的 Donoho-Logan 大筛原理

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-09-26 DOI: 10.1016/j.acha.2024.101709

Luís Daniel Abreu , Michael Speckbacher

In this paper we formulate Donoho and Logan's large sieve principle for the wavelet transform on the Hardy space, adapting the concept of maximum Nyquist density to the hyperbolic geometry of the underlying space. The results provide deterministic guarantees for

L_{1}

-minimization methods and hold for a class of mother wavelets that constitutes an orthonormal basis of the Hardy space and can be associated with higher hyperbolic Landau levels. Explicit calculations of the basis functions reveal a connection with the Zernike polynomials. We prove a novel local reproducing formula for the spaces in consideration and use it to derive concentration estimates of the large sieve type for the corresponding wavelet transforms. We conclude with a discussion of optimality of localization and Lieb inequalities in the analytic case by building on recent results of Kulikov, Ramos and Tilli based on the groundbreaking methods of Nicola and Tilli. This leads to a sharp uncertainty principle and a local Lieb inequality for the wavelet transform.

在本文中，我们针对哈代空间的小波变换提出了 Donoho 和 Logan 的大筛原理，将最大奈奎斯特密度的概念调整为基础空间的双曲几何。这些结果为 L1 最小化方法提供了确定性保证，并适用于构成哈代空间正交基的一类母小波，而且可以与更高的双曲朗道水平相关联。基函数的显式计算揭示了与 Zernike 多项式的联系。我们为所考虑的空间证明了一个新颖的局部重现公式，并利用它推导出相应小波变换的大筛型集中估计。最后，我们以 Kulikov、Ramos 和 Tilli 基于 Nicola 和 Tilli 的开创性方法所取得的最新成果为基础，讨论了解析情况下的局部最优性和李卜不等式。这导致了小波变换的尖锐不确定性原理和局部利布不等式。

引用次数: 0

Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application 具有可分离协方差结构的高维噪声下奇异值的数据驱动优化收缩及其应用

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-09-12 DOI: 10.1016/j.acha.2024.101698

Pei-Chun Su , Hau-Tieng Wu

We develop a data-driven optimal shrinkage algorithm, named extended OptShrink (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.

我们针对具有高维噪声和可分离协方差结构的矩阵去噪，开发了一种数据驱动的最优收缩算法，并将其命名为扩展 OptShrink（eOptShrink）。这种噪声是有颜色的，并依赖于不同的样本。该算法利用了噪声数据随机矩阵奇异值和向量的渐近行为。我们的理论包括非离群奇异值的粘性特性、弱信号奇异向量的脱域以及离群奇异值和向量的频谱行为。我们引入了三种估计器：新颖的秩估计器、纯噪声矩阵频谱分布估计器和最优收缩器 eOptShrink。值得注意的是，eOptShrink 无需估计噪声的可分离协方差结构。我们从理论上保证了这些估计器的收敛速度。通过数值模拟以及与最先进的最优收缩算法的比较，我们展示了 eOptShrink 在从单通道经腹母体心电图中提取母体和胎儿心电图中的应用。

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引用次数: 0

Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups 构建由 LCA 组滤波器生成的成对正交 Parseval 框架

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2024-09-07 DOI: 10.1016/j.acha.2024.101708

Navneet Redhu , Anupam Gumber , Niraj K. Shukla

The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in $L^{2} (G)$ satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in $L^{2} (R)$ and $L^{2} (G)$ , using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

广义平移不变（GTI）系统将广义平移不变系统的离散帧理论与其连续版本（如小波、小剪切、Gabor变换等）统一起来。本文提供了在 L2(G) 中构建满足局部可整性条件（LIC）且卡尔德龙和为一的成对正交 Parseval GTI 框架的充分条件，其中 G 是第二可数局部紧凑非良性群。成对正交性在多址通信、隐藏数据、合成超级帧和帧等方面起着至关重要的作用。此外，我们还提供了一个结果，用于构造 N 个成对正交的 GTI Parseval 帧。因此，我们利用 B-样条函数作为生成函数，在 L2(R) 和 L2(G) 中获得了成对正交 Parseval 帧的显式构造。最后，我们特别讨论了小波系统的结果。

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