Pub Date : 2025-11-04DOI: 10.1016/j.acha.2025.101823
M.H.A. Biswas , P. Massopust , R. Ramakrishnan
In the first part of this paper, we define a deep convolutional neural network connected to the fractional Fourier transform (FrFT) using the -translation operator, the translation operator associated with the FrFT. Subsequently, we study -translation invariant properties of this network. It is well known that the network introduced by Mallat is translation invariant. In general, our network need not be -translation invariant. However, the network can be made asymptotically -translation invariant by choosing suitable pooling factors.
In the second part, we study data approximation problems using the FrFT. More precisely, given a data set , we obtain such thatwhere the minimum is taken over all -shift invariant spaces generated by at most elements. Moreover, we prove the existence of a space of band-limited functions in the FrFT domain which is “closest” to in the above sense.
{"title":"The theory of deep convolutional neural networks and a data approximation problem based on the fractional Fourier transform","authors":"M.H.A. Biswas , P. Massopust , R. Ramakrishnan","doi":"10.1016/j.acha.2025.101823","DOIUrl":"10.1016/j.acha.2025.101823","url":null,"abstract":"<div><div>In the first part of this paper, we define a deep convolutional neural network connected to the fractional Fourier transform (FrFT) using the <span><math><mstyle><mi>Θ</mi></mstyle></math></span>-translation operator, the translation operator associated with the FrFT. Subsequently, we study <span><math><mstyle><mi>Θ</mi></mstyle></math></span>-translation invariant properties of this network. It is well known that the network introduced by Mallat is translation invariant. In general, our network need not be <span><math><mstyle><mi>Θ</mi></mstyle></math></span>-translation invariant. However, the network can be made asymptotically <span><math><mstyle><mi>Θ</mi></mstyle></math></span>-translation invariant by choosing suitable pooling factors.</div><div>In the second part, we study data approximation problems using the FrFT. More precisely, given a data set <span><math><mrow><mi>F</mi><mo>=</mo><mrow><mo>{</mo><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>f</mi><mi>m</mi></msub><mo>}</mo></mrow><mo>⊂</mo><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span>, we obtain <span><math><mrow><mstyle><mi>Φ</mi></mstyle><mo>=</mo><mo>{</mo><msub><mi>ϕ</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>ϕ</mi><mi>ℓ</mi></msub><mo>}</mo></mrow></math></span> such that<span><span><span><math><mrow><msub><mi>V</mi><mstyle><mi>Θ</mi></mstyle></msub><mrow><mo>(</mo><mstyle><mi>Φ</mi></mstyle><mo>)</mo></mrow><mo>=</mo><mi>arg</mi><mi>min</mi><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msup><mrow><mo>∥</mo><msub><mi>f</mi><mi>j</mi></msub><mo>−</mo><msub><mi>P</mi><mi>V</mi></msub><msub><mi>f</mi><mi>j</mi></msub><mo>∥</mo></mrow><mn>2</mn></msup><mo>,</mo></mrow></math></span></span></span>where the minimum is taken over all <span><math><mstyle><mi>Θ</mi></mstyle></math></span>-shift invariant spaces generated by at most <span><math><mi>ℓ</mi></math></span> elements. Moreover, we prove the existence of a space of band-limited functions in the FrFT domain which is “closest” to <span><math><mi>F</mi></math></span> in the above sense.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101823"},"PeriodicalIF":3.2,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145441524","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}
Pub Date : 2025-11-03DOI: 10.1016/j.acha.2025.101822
Ayoub Belhadji , Rémi Bardenet , Pierre Chainais
We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based on two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in L2 norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling improves on i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.
{"title":"Signal reconstruction using determinantal sampling","authors":"Ayoub Belhadji , Rémi Bardenet , Pierre Chainais","doi":"10.1016/j.acha.2025.101822","DOIUrl":"10.1016/j.acha.2025.101822","url":null,"abstract":"<div><div>We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based on two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in <em>L</em><sup>2</sup> norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling improves on i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"82 ","pages":"Article 101822"},"PeriodicalIF":3.2,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145434666","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}
Pub Date : 2025-10-31DOI: 10.1016/j.acha.2025.101820
Richard Chow, James Bremer
It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. We resolve this difficulty by introducing a generalized phase function method for second order linear ordinary differential equations with turning points. More explicitly, we give an efficient numerical algorithm for computing an Airy phase function which efficiently represents the solutions of such an equation. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation’s solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and, unlike step methods that only generate the values of a rapidly-varying solution at the nodes of a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the evaluation of the solution at any point in the equation’s domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.
{"title":"Airy phase functions","authors":"Richard Chow, James Bremer","doi":"10.1016/j.acha.2025.101820","DOIUrl":"10.1016/j.acha.2025.101820","url":null,"abstract":"<div><div>It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. We resolve this difficulty by introducing a generalized phase function method for second order linear ordinary differential equations with turning points. More explicitly, we give an efficient numerical algorithm for computing an Airy phase function which efficiently represents the solutions of such an equation. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation’s solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and, unlike step methods that only generate the values of a rapidly-varying solution at the nodes of a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the evaluation of the solution at any point in the equation’s domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101820"},"PeriodicalIF":3.2,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145412058","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}
Pub Date : 2025-10-30DOI: 10.1016/j.acha.2025.101821
Yousef Qaddura
Given an inner product space and a group of linear isometries, max filtering offers a rich class of convex -invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on , the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space . Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in and may be of independent interest.
As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp. quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp. quaternionic) representations of the group of unit complex numbers (resp. unit quaternions ).
We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.
{"title":"A max filtering local stability theorem with application to weighted phase retrieval and cryo-EM","authors":"Yousef Qaddura","doi":"10.1016/j.acha.2025.101821","DOIUrl":"10.1016/j.acha.2025.101821","url":null,"abstract":"<div><div>Given an inner product space <span><math><mi>V</mi></math></span> and a group <span><math><mi>G</mi></math></span> of linear isometries, max filtering offers a rich class of convex <span><math><mi>G</mi></math></span>-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on <span><math><mrow><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span>, the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space <span><math><mrow><mi>V</mi><mo>/</mo><mi>G</mi></mrow></math></span>. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in <span><math><mrow><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mi>G</mi></mrow></math></span> and may be of independent interest.</div><div>As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp. quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp. quaternionic) representations of the group of unit complex numbers <span><math><mrow><msup><mi>S</mi><mn>1</mn></msup><mo>≅</mo><mi>SO</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> (resp. unit quaternions <span><math><mrow><msup><mi>S</mi><mn>3</mn></msup><mo>≅</mo><mi>SU</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>).</div><div>We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101821"},"PeriodicalIF":3.2,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145383235","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}
Pub Date : 2025-10-26DOI: 10.1016/j.acha.2025.101818
Yu Xia , Zhiqiang Xu
Compressed sensing has demonstrated that a general signal () can be estimated from few linear measurements with an error proportional to the best -term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the -minimization decoder, where , for both real and complex cases. More specifically, we prove that (2,1) and (1,1)-instance optimality of order can be achieved with phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately -sparse signals from phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of (2,2)-instance optimality result in probability applicable to any fixed vector . These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.
{"title":"Instance optimality in phase retrieval","authors":"Yu Xia , Zhiqiang Xu","doi":"10.1016/j.acha.2025.101818","DOIUrl":"10.1016/j.acha.2025.101818","url":null,"abstract":"<div><div>Compressed sensing has demonstrated that a general signal <span><math><mrow><mi>x</mi><mo>∈</mo><msup><mi>F</mi><mi>n</mi></msup></mrow></math></span> (<span><math><mrow><mi>F</mi><mo>∈</mo><mo>{</mo><mi>R</mi><mo>,</mo><mi>C</mi><mo>}</mo></mrow></math></span>) can be estimated from few linear measurements with an error proportional to the best <span><math><mi>k</mi></math></span>-term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the <span><math><msub><mi>ℓ</mi><mi>p</mi></msub></math></span>-minimization decoder, where <span><math><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, for both real and complex cases. More specifically, we prove that (2,1) and (1,1)-instance optimality of order <span><math><mi>k</mi></math></span> can be achieved with <span><math><mrow><mi>m</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>)</mo></mrow></math></span> phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately <span><math><mi>k</mi></math></span>-sparse signals from <span><math><mrow><mi>m</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>)</mo></mrow></math></span> phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of (2,2)-instance optimality result in probability applicable to any fixed vector <span><math><mrow><mi>x</mi><mo>∈</mo><msup><mi>F</mi><mi>n</mi></msup></mrow></math></span>. These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"80 ","pages":"Article 101818"},"PeriodicalIF":3.2,"publicationDate":"2025-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145383223","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}
Pub Date : 2025-10-26DOI: 10.1016/j.acha.2025.101817
Jielun Chen , Michael Lindsey
The quantum Fourier transform (QFT), which can be viewed as a reindexing of the discrete Fourier transform (DFT), has been shown to be compressible as a low-rank matrix product operator (MPO) or quantized tensor train (QTT) operator [1]. However, the original proof of this fact does not furnish a construction of the MPO with a guaranteed error bound. Meanwhile, the existing practical construction of this MPO, based on the compression of a quantum circuit, is not as efficient as possible. We present a simple closed-form construction of the QFT MPO using the interpolative decomposition, with guaranteed near-optimal compression error for a given rank. This construction can speed up the application of the QFT and the DFT, respectively, in quantum circuit simulations and QTT applications. We also connect our interpolative construction to the approximate quantum Fourier transform (AQFT) by demonstrating that the AQFT can be viewed as an MPO constructed using a different interpolation scheme.
{"title":"Direct interpolative construction of the discrete Fourier transform as a matrix product operator","authors":"Jielun Chen , Michael Lindsey","doi":"10.1016/j.acha.2025.101817","DOIUrl":"10.1016/j.acha.2025.101817","url":null,"abstract":"<div><div>The quantum Fourier transform (QFT), which can be viewed as a reindexing of the discrete Fourier transform (DFT), has been shown to be compressible as a low-rank matrix product operator (MPO) or quantized tensor train (QTT) operator [1]. However, the original proof of this fact does not furnish a construction of the MPO with a guaranteed error bound. Meanwhile, the existing practical construction of this MPO, based on the compression of a quantum circuit, is not as efficient as possible. We present a simple closed-form construction of the QFT MPO using the interpolative decomposition, with guaranteed near-optimal compression error for a given rank. This construction can speed up the application of the QFT and the DFT, respectively, in quantum circuit simulations and QTT applications. We also connect our interpolative construction to the approximate quantum Fourier transform (AQFT) by demonstrating that the AQFT can be viewed as an MPO constructed using a different interpolation scheme.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101817"},"PeriodicalIF":3.2,"publicationDate":"2025-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145383224","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}
Pub Date : 2025-10-22DOI: 10.1016/j.acha.2025.101819
Gao Huang , Song Li , Hang Xu
We consider the problem of recovering an unknown signal from phaseless measurements. In this paper, we study the convex phase retrieval problem via PhaseLift from linear Gaussian measurements perturbed by -bounded noise and sparse outliers that can change an adversarially chosen -fraction of the measurement vector. We show that the Robust-PhaseLift model can successfully reconstruct the ground-truth up to global phase for any with measurements, even in the case where the sparse outliers may depend on the measurement and the observation. The recovery guarantees are based on the robust outlier bound condition, along with an analysis of the product of two Gaussian variables and the minimum balance function. Moreover, we construct adaptive counterexamples to show that the Robust-PhaseLift model fails when with high probability. Finally, we also provide some preliminary discussions on the adversarially robust recovery of complex signals.
{"title":"Robust outlier bound condition to phase retrieval with adversarial sparse outliers","authors":"Gao Huang , Song Li , Hang Xu","doi":"10.1016/j.acha.2025.101819","DOIUrl":"10.1016/j.acha.2025.101819","url":null,"abstract":"<div><div>We consider the problem of recovering an unknown signal <span><math><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></math></span> from phaseless measurements. In this paper, we study the convex phase retrieval problem via PhaseLift from linear Gaussian measurements perturbed by <span><math><msub><mi>ℓ</mi><mn>1</mn></msub></math></span>-bounded noise and sparse outliers that can change an adversarially chosen <span><math><mi>s</mi></math></span>-fraction of the measurement vector. We show that the Robust-PhaseLift model can successfully reconstruct the ground-truth up to global phase for any <span><math><mrow><mi>s</mi><mo><</mo><msup><mi>s</mi><mo>*</mo></msup><mo>≈</mo><mn>0.1185</mn></mrow></math></span> with <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math></span> measurements, even in the case where the sparse outliers may depend on the measurement and the observation. The recovery guarantees are based on the robust outlier bound condition, along with an analysis of the product of two Gaussian variables and the minimum balance function. Moreover, we construct adaptive counterexamples to show that the Robust-PhaseLift model fails when <span><math><mrow><mi>s</mi><mo>></mo><msup><mi>s</mi><mo>*</mo></msup></mrow></math></span> with high probability. Finally, we also provide some preliminary discussions on the adversarially robust recovery of complex signals.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"80 ","pages":"Article 101819"},"PeriodicalIF":3.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416941","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}
Pub Date : 2025-10-02DOI: 10.1016/j.acha.2025.101816
François G. Meyer
The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a “summary graph” that captures the mean topology and connectivity structure of a training dataset of graphs. The construction of a barycentre requires the definition of a metric to quantify distances between pairs of graphs. In this work, we use a multiscale spectral distance that is defined using the eigenvalues of the normalized graph Laplacian. The eigenvalues – but not the eigenvectors – of the normalized Laplacian of the barycentre graph can be determined from the optimization problem that defines the barycentre. In this work, we propose a structural constraint on the eigenvectors of the normalized graph Laplacian of the barycentre graph that guarantees that the barycentre inherits the topological structure of the graphs in the sample dataset. The eigenvectors can be computed using an algorithm that explores the large library of Soules bases. When the graphs are random realizations of a balanced stochastic block model, then our algorithm returns a barycentre that converges asymptotically (in the limit of large graph size) almost-surely to the population mean of the graphs. We perform Monte Carlo simulations to validate the theoretical properties of the estimator; we conduct experiments on real-life graphs that suggest that our approach works beyond the controlled environment of stochastic block models.
{"title":"The spectral barycentre of a set of graphs with community structure","authors":"François G. Meyer","doi":"10.1016/j.acha.2025.101816","DOIUrl":"10.1016/j.acha.2025.101816","url":null,"abstract":"<div><div>The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a “summary graph” that captures the mean topology and connectivity structure of a training dataset of graphs. The construction of a barycentre requires the definition of a metric to quantify distances between pairs of graphs. In this work, we use a multiscale spectral distance that is defined using the eigenvalues of the normalized graph Laplacian. The eigenvalues – but not the eigenvectors – of the normalized Laplacian of the barycentre graph can be determined from the optimization problem that defines the barycentre. In this work, we propose a structural constraint on the eigenvectors of the normalized graph Laplacian of the barycentre graph that guarantees that the barycentre inherits the topological structure of the graphs in the sample dataset. The eigenvectors can be computed using an algorithm that explores the large library of Soules bases. When the graphs are random realizations of a balanced stochastic block model, then our algorithm returns a barycentre that converges asymptotically (in the limit of large graph size) almost-surely to the population mean of the graphs. We perform Monte Carlo simulations to validate the theoretical properties of the estimator; we conduct experiments on real-life graphs that suggest that our approach works beyond the controlled environment of stochastic block models.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"80 ","pages":"Article 101816"},"PeriodicalIF":3.2,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145266273","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}
Pub Date : 2025-09-18DOI: 10.1016/j.acha.2025.101812
Ethan N. Epperly , Gil Goldshlager , Robert J. Webber
The randomized Kaczmarz (RK) method is a well-known approach for solving linear least-squares problems with a large number of rows. RK accesses and processes just one row at a time, leading to exponentially fast convergence for consistent linear systems. However, RK fails to converge to the least-squares solution for inconsistent systems. This work presents a simple fix: average the RK iterates produced in the tail part of the algorithm. The proposed tail-averaged randomized Kaczmarz (TARK) converges for both consistent and inconsistent least-squares problems at a polynomial rate, which is known to be optimal for any row-access method. An extension of TARK also leads to efficient solutions for ridge-regularized least-squares problems.
{"title":"Randomized Kaczmarz with tail averaging","authors":"Ethan N. Epperly , Gil Goldshlager , Robert J. Webber","doi":"10.1016/j.acha.2025.101812","DOIUrl":"10.1016/j.acha.2025.101812","url":null,"abstract":"<div><div>The randomized Kaczmarz (RK) method is a well-known approach for solving linear least-squares problems with a large number of rows. RK accesses and processes just one row at a time, leading to exponentially fast convergence for consistent linear systems. However, RK fails to converge to the least-squares solution for inconsistent systems. This work presents a simple fix: average the RK iterates produced in the tail part of the algorithm. The proposed tail-averaged randomized Kaczmarz (TARK) converges for both consistent and inconsistent least-squares problems at a polynomial rate, which is known to be optimal for any row-access method. An extension of TARK also leads to efficient solutions for ridge-regularized least-squares problems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"80 ","pages":"Article 101812"},"PeriodicalIF":3.2,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158362","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}
Pub Date : 2025-09-15DOI: 10.1016/j.acha.2025.101815
Diego Castelli Lacunza , Carlos A. Sing Long
Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned.
In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead of performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call -multipliers, that can be used to perform extrapolation in frequency. We establish connections between -multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.
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