Applied and Computational Harmonic Analysis最新文献_第2页

Anisotropic refinable functions and the tile B-splines 各向异性可细化函数和平铺b样条

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

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