Applied and Computational Harmonic Analysis最新文献_第3页

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

Gaussian approximation for the moving averaged modulus wavelet transform and its variants 移动平均模小波变换的高斯近似及其变体

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-13 DOI: 10.1016/j.acha.2024.101722

Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu

{"title":"Gaussian approximation for the moving averaged modulus wavelet transform and its variants","authors":"Gi-Ren Liu , Yuan-Chung Sheu , Hau-Tieng Wu","doi":"10.1016/j.acha.2024.101722","DOIUrl":"10.1016/j.acha.2024.101722","url":null,"abstract":"<div><div>The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos, and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases. We provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101722"},"PeriodicalIF":2.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142658789","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

Naimark-spatial families of equichordal tight fusion frames 等弦紧密融合框架的奈马克空间族

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-11-08 DOI: 10.1016/j.acha.2024.101720

Matthew Fickus, Benjamin R. Mayo, Cody E. Watson

引用次数: 0

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning 基于自动编码器的泛化误差保证非线性模型还原用于算子学习

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-30 DOI: 10.1016/j.acha.2024.101717

Hao Liu , Biraj Dahal , Rongjie Lai , Wenjing Liao

{"title":"Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning","authors":"Hao Liu , Biraj Dahal , Rongjie Lai , Wenjing Liao","doi":"10.1016/j.acha.2024.101717","DOIUrl":"10.1016/j.acha.2024.101717","url":null,"abstract":"<div><div>Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101717"},"PeriodicalIF":2.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578634","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

Unlimited sampling beyond modulo 超出模数的无限采样

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-24 DOI: 10.1016/j.acha.2024.101715

Eyar Azar , Satish Mulleti , Yonina C. Eldar

{"title":"Unlimited sampling beyond modulo","authors":"Eyar Azar , Satish Mulleti , Yonina C. Eldar","doi":"10.1016/j.acha.2024.101715","DOIUrl":"10.1016/j.acha.2024.101715","url":null,"abstract":"<div><div>Analog-to-digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required, or strict constraints on the nonlinear operations are imposed, both of which are not desirable in practice. In this paper, we propose a generalized flexible nonlinear operator which is sampling efficient. Moreover, by carefully choosing its parameters, clipping, modulo, and companding can be seen as special cases of it. We show that bandlimited signals are uniquely identified from the nonlinear samples of the proposed operator when sampled above the Nyquist rate. Furthermore, we propose a robust algorithm to recover the true signal from the nonlinear samples. Compared to the existing methods, our approach has a lower mean-squared error for a given sampling rate, noise level, and dynamic range. Our results lead to less constrained hardware design to address the dynamic range issues while operating at the lowest rate possible.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101715"},"PeriodicalIF":2.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554887","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

Inverse problems are solvable on real number signal processing hardware 实数信号处理硬件可解决逆问题

IF 2.6 2区数学

Applied and Computational Harmonic Analysis Pub Date : 2024-10-24 DOI: 10.1016/j.acha.2024.101719

Holger Boche , Adalbert Fono , Gitta Kutyniok

{"title":"Inverse problems are solvable on real number signal processing hardware","authors":"Holger Boche , Adalbert Fono , Gitta Kutyniok","doi":"10.1016/j.acha.2024.101719","DOIUrl":"10.1016/j.acha.2024.101719","url":null,"abstract":"<div><div>Despite the success of Deep Learning (DL) serious reliability issues such as non-robustness persist. An interesting aspect is, whether these problems arise due to insufficient tools or fundamental limitations of DL. We study this question from the computability perspective by characterizing the limits the applied hardware imposes. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. On digital hardware, a conceptual barrier on the capabilities of DL for solving finite-dimensional inverse problems has in fact already been derived. This paper investigates the general computation framework of Blum-Shub-Smale (BSS) machines, describing the processing and storage of arbitrary real values. Although a corresponding real-world computing device does not exist, research and development towards real number computing hardware, usually referred to by “neuromorphic computing”, has increased in recent years. In this work, we show that the framework of BSS machines does enable the algorithmic solvability of finite dimensional inverse problems. Our results emphasize the influence of the considered computing model in questions of accuracy and reliability.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101719"},"PeriodicalIF":2.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561034","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

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

Applied and Computational Harmonic Analysis Pub Date : 2024-10-21 DOI: 10.1016/j.acha.2024.101710

Gary Froyland, Christopher P. Rock

{"title":"Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions","authors":"Gary Froyland, Christopher P. Rock","doi":"10.1016/j.acha.2024.101710","DOIUrl":"10.1016/j.acha.2024.101710","url":null,"abstract":"<div><div>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.</div><div>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.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101710"},"PeriodicalIF":2.6,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535999","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

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

Applied and Computational Harmonic Analysis Pub Date : 2024-10-18 DOI: 10.1016/j.acha.2024.101716

Lexing Ying

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

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

{"title":"Solving PDEs on spheres with physics-informed convolutional neural networks","authors":"Guanhang Lei , Zhen Lei , Lei Shi , Chenyu Zeng , Ding-Xuan Zhou","doi":"10.1016/j.acha.2024.101714","DOIUrl":"10.1016/j.acha.2024.101714","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101714"},"PeriodicalIF":2.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442786","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