Applied and Computational Harmonic Analysis最新文献

{"title":"Large data limit of the MBO scheme for data clustering: Γ-convergence of the thresholding energies","authors":"Tim Laux ,&nbsp;Jona Lelmi","doi":"10.1016/j.acha.2025.101800","DOIUrl":"10.1016/j.acha.2025.101800","url":null,"abstract":"<div><div>In this work we present the first rigorous analysis of the MBO scheme for data clustering in the large data limit. Each iteration of the scheme corresponds to one step of implicit gradient descent for the thresholding energy on the similarity graph of some dataset. For a subset of the nodes of the graph, the thresholding energy at time <em>h</em> measures the amount of heat transferred from the subset to its complement at time <em>h</em>, rescaled by a factor <span><math><msqrt><mrow><mi>h</mi></mrow></msqrt></math></span>. It is then natural to think that outcomes of the MBO scheme are (local) minimizers of this energy. We prove that the algorithm is consistent, in the sense that these (local) minimizers converge to (local) minimizers of a suitably weighted optimal partition problem.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101800"},"PeriodicalIF":3.2,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904067","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}

{"title":"The Wigner distribution of Gaussian tempered generalized stochastic processes","authors":"Patrik Wahlberg","doi":"10.1016/j.acha.2025.101799","DOIUrl":"10.1016/j.acha.2025.101799","url":null,"abstract":"<div><div>We define the Wigner distribution of a tempered generalized stochastic process that is complex-valued symmetric Gaussian. This gives a time-frequency generalized stochastic process defined on the phase space. We study its covariance and our main result is a formula for the Weyl symbol of the covariance operator, expressed in terms of the Weyl symbol of the covariance operator of the original generalized stochastic process.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101799"},"PeriodicalIF":3.2,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144861221","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}

{"title":"Permutation-invariant representations with applications to graph deep learning","authors":"Radu Balan ,&nbsp;Naveed Haghani ,&nbsp;Maneesh Singh","doi":"10.1016/j.acha.2025.101798","DOIUrl":"10.1016/j.acha.2025.101798","url":null,"abstract":"<div><div>This paper presents primarily two Euclidean embeddings of the quotient space generated by matrices that are identified modulo arbitrary row permutations. The original application is in deep learning on graphs where the learning task is invariant to node relabeling. Two embedding schemes are introduced, one based on sorting and the other based on algebras of multivariate polynomials. While both embeddings exhibit a computational complexity exponential in problem size, the sorting based embedding is globally bi-Lipschitz and admits a low dimensional target space. Additionally, an almost everywhere injective scheme can be implemented with minimal redundancy and low computational cost. In turn, this proves that almost any classifier can be implemented with an arbitrary small loss of performance. Numerical experiments are carried out on two datasets, a chemical compound dataset (<span>QM9</span>) and a proteins dataset (<span>PROTEINS_FULL</span>).</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101798"},"PeriodicalIF":3.2,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809570","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}

{"title":"Minibatch and local SGD: Algorithmic stability and linear speedup in generalization","authors":"Yunwen Lei ,&nbsp;Tao Sun ,&nbsp;Mingrui Liu","doi":"10.1016/j.acha.2025.101795","DOIUrl":"10.1016/j.acha.2025.101795","url":null,"abstract":"<div><div>The increasing scale of data propels the popularity of leveraging parallelism to speed up the optimization. Minibatch stochastic gradient descent (minibatch SGD) and local SGD are two popular methods for parallel optimization. The existing theoretical studies show a linear speedup of these methods with respect to the number of machines, which, however, is measured by optimization errors in a multi-pass setting. As a comparison, the stability and generalization of these methods are much less studied. In this paper, we study the stability and generalization analysis of minibatch and local SGD to understand their learnability by introducing an expectation-variance decomposition. We incorporate training errors into the stability analysis, which shows how small training errors help generalization for overparameterized models. We show minibatch and local SGD achieve a linear speedup to attain the optimal risk bounds.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101795"},"PeriodicalIF":2.6,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653251","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}

{"title":"Multi-dimensional unlimited sampling and robust reconstruction","authors":"Dorian Florescu,&nbsp;Ayush Bhandari","doi":"10.1016/j.acha.2025.101796","DOIUrl":"10.1016/j.acha.2025.101796","url":null,"abstract":"<div><div>In this paper we introduce a new sampling and reconstruction approach for multi-dimensional analog signals. Building on top of the Unlimited Sensing Framework (USF), we present a new folded sampling operator called the multi-dimensional modulo-hysteresis that is also backwards compatible with the existing one-dimensional modulo operator. Unlike previous approaches, the proposed model is specifically tailored to multi-dimensional signals. In particular, the model uses certain redundancy in dimensions 2 and above, which is exploited for input recovery with robustness. We prove that the new operator is well-defined and its outputs have a bounded dynamic range. For the noiseless case, we derive a theoretically guaranteed input reconstruction approach. When the input is corrupted by Gaussian noise, we exploit redundancy in higher dimensions to provide a bound on the error probability and show this drops to 0 for high enough sampling rates leading to new theoretical guarantees for the noisy case. Our numerical examples corroborate the theoretical results and show that the proposed approach can handle a significantly larger amount of noise compared to USF.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101796"},"PeriodicalIF":2.6,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144665025","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}

{"title":"On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks","authors":"Yunfei Yang","doi":"10.1016/j.acha.2025.101797","DOIUrl":"10.1016/j.acha.2025.101797","url":null,"abstract":"<div><div>This paper studies the problem of how efficiently functions in the Sobolev spaces <span><math><msup><mrow><mi>W</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> and Besov spaces <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>r</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> can be approximated by deep ReLU neural networks with width <em>W</em> and depth <em>L</em>, when the error is measured in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> norm. This problem has been studied by several recent works, which obtained the approximation rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>W</mi><mi>L</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>2</mn><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></math></span> up to logarithmic factors when <span><math><mi>p</mi><mo>=</mo><mi>q</mi><mo>=</mo><mo>∞</mo></math></span>, and the rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></math></span> for networks with fixed width when the Sobolev embedding condition <span><math><mn>1</mn><mo>/</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>/</mo><mi>p</mi><mo>&lt;</mo><mi>s</mi><mo>/</mo><mi>d</mi></math></span> holds. We generalize these results by showing that the rate <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>W</mi><mi>L</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>2</mn><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></math></span> indeed holds under the Sobolev embedding condition. It is known that this rate is optimal up to logarithmic factors. The key tool in our proof is a novel encoding of sparse vectors by using deep ReLU neural networks with varied width and depth, which may be of independent interest.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101797"},"PeriodicalIF":2.6,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653252","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}

{"title":"Nonharmonic multivariate Fourier transforms and matrices: Condition numbers and hyperplane geometry","authors":"Weilin Li","doi":"10.1016/j.acha.2025.101791","DOIUrl":"10.1016/j.acha.2025.101791","url":null,"abstract":"<div><div>Consider an operator that takes the Fourier transform of a discrete measure supported in <span><math><mi>X</mi><mo>⊆</mo><msup><mrow><mo>[</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> and restricts it to a compact <span><math><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We provide lower bounds for its smallest singular value when Ω is either a closed ball of radius <em>m</em> or closed cube of side length 2<em>m</em>, and under different types of geometric assumptions on <span><math><mi>X</mi></math></span>. We first show that if distances between points in <span><math><mi>X</mi></math></span> are lower bounded by a <em>δ</em> that is allowed to be arbitrarily small, then the smallest singular value is at least <span><math><mi>C</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>m</mi><mi>δ</mi><mo>)</mo></mrow><mrow><mi>λ</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, where <em>λ</em> is the maximum number of elements in <span><math><mi>X</mi></math></span> contained within any ball or cube of an explicitly given radius. This estimate communicates a localization effect of the Fourier transform. While it is sharp, the smallest singular value behaves better than expected for many <span><math><mi>X</mi></math></span>, including when we dilate a generic set by parameter <em>δ</em>. We next show that if there is a <em>η</em> such that, for each <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, the set <span><math><mi>X</mi><mo>∖</mo><mo>{</mo><mi>x</mi><mo>}</mo></math></span> locally consists of at most <em>r</em> hyperplanes whose distances to <em>x</em> are at least <em>η</em>, then the smallest singular value is at least <span><math><mi>C</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>m</mi><mi>η</mi><mo>)</mo></mrow><mrow><mi>r</mi></mrow></msup></math></span>. For dilations of a generic set by <em>δ</em>, the lower bound becomes <span><math><mi>C</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>m</mi><mi>δ</mi><mo>)</mo></mrow><mrow><mo>⌈</mo><mo>(</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>d</mi><mo>⌉</mo></mrow></msup></math></span>. The appearance of a <span><math><mn>1</mn><mo>/</mo><mi>d</mi></math></span> factor in the exponent indicates that compared to worst case scenarios, the condition number of nonharmonic Fourier transforms is better than expected for typical sets and improve with higher dimensionality.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101791"},"PeriodicalIF":2.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144595712","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}

{"title":"On exact systems {tα⋅e2πint}n∈Z∖A in L2(0,1) which are weighted lower semi frames but not Schauder bases, and their generalizations","authors":"Elias Zikkos","doi":"10.1016/j.acha.2025.101794","DOIUrl":"10.1016/j.acha.2025.101794","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be an exponential Schauder basis for &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be its dual Schauder basis. Let &lt;em&gt;A&lt;/em&gt; be a non-empty subset of the integers containing exactly &lt;em&gt;M&lt;/em&gt; elements. We prove that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; the weighted system &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is exact in the space &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, that is, it is complete and minimal in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We also show that such a system is not a Riesz basis for &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In particular, the weighted trigonometric system &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is exact in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, but this system is not even a Schauder basis for &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101794"},"PeriodicalIF":2.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596066","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}

{"title":"On the limits of neural network explainability via descrambling","authors":"Shashank Sule ,&nbsp;Richard G. Spencer ,&nbsp;Wojciech Czaja","doi":"10.1016/j.acha.2025.101793","DOIUrl":"10.1016/j.acha.2025.101793","url":null,"abstract":"<div><div>We characterize the exact solutions to <em>neural network descrambling</em>–a mathematical model for explaining the fully connected layers of trained neural networks (NNs). By reformulating the problem to the minimization of the Brockett function arising in graph matching and complexity theory we show that the principal components of the hidden layer preactivations can be characterized as the optimal “explainers” or <em>descramblers</em> for the layer weights, leading to <em>descrambled</em> weight matrices. We show that in typical deep learning contexts these descramblers take diverse and interesting forms including (1) matching largest principal components with the lowest frequency modes of the Fourier basis for isotropic hidden data, (2) discovering the semantic development in two-layer linear NNs for signal recovery problems, and (3) explaining CNNs by optimally permuting the neurons. Our numerical experiments indicate that the eigendecompositions of the hidden layer data–now understood as the descramblers–can also reveal the layer's underlying transformation. These results illustrate that the SVD is more directly related to the explainability of NNs than previously thought and offers a promising avenue for discovering interpretable motifs for the hidden action of NNs, especially in contexts of operator learning or physics-informed NNs, where the input/output data has limited human readability.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101793"},"PeriodicalIF":2.6,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572145","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}

{"title":"Gaussian process regression with log-linear scaling for common non-stationary kernels","authors":"P. Michael Kielstra ,&nbsp;Michael Lindsey","doi":"10.1016/j.acha.2025.101792","DOIUrl":"10.1016/j.acha.2025.101792","url":null,"abstract":"<div><div>We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Matérn kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as <span><math><mi>O</mi><mo>(</mo><mi>N</mi><mi>log</mi><mo>⁡</mo><mi>N</mi><mo>)</mo></math></span>, where <em>N</em> is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to state-of-the-art rank-structured approaches in spatial dimension <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"79 ","pages":"Article 101792"},"PeriodicalIF":2.6,"publicationDate":"2025-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596067","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}