Applied and Computational Harmonic Analysis最新文献

{"title":"The generic crystallographic phase retrieval problem","authors":"Dan Edidin,&nbsp;Arun Suresh","doi":"10.1016/j.acha.2026.101888","DOIUrl":"10.1016/j.acha.2026.101888","url":null,"abstract":"<div><div>In this paper we consider the problem of recovering a signal <span><math><mrow><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup></mrow></math></span> from its power spectrum assuming that the signal is sparse with respect to a generic basis for <span><math><msup><mi>R</mi><mi>N</mi></msup></math></span>. Our main result is that if the sparsity level is at most  ∼ <em>N</em>/2 in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is  ∼ <em>N</em>/4 then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in <span><math><msup><mi>C</mi><mi>N</mi></msup></math></span> which extend earlier results of Wang and Xu [1].</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"84 ","pages":"Article 101888"},"PeriodicalIF":3.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147849902","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":"Approximately dual and pseudo-dual probabilistic frames","authors":"Dongwei Chen,&nbsp;Emily J. King,&nbsp;Clayton Shonkwiler","doi":"10.1016/j.acha.2026.101885","DOIUrl":"10.1016/j.acha.2026.101885","url":null,"abstract":"<div><div>This paper studies properties of dual probabilistic frames—in particular in relation to redundancy—and introduces both approximately dual probabilistic frames and pseudo-dual probabilistic frames. We show that the canonical dual probabilistic frame is the only dual frame of pushforward type of a probabilistic frame with zero redundancy. Furthermore, we show that probabilistic frames with finite redundancy are atomic and finite. Approximately dual probabilistic frames generalize duality, with pseudo-duality being a further generalization. We introduce these concepts and prove certain structural results. In particular, every probabilistic frame has a discrete finite frame as an approximate dual.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"84 ","pages":"Article 101885"},"PeriodicalIF":3.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147849905","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":"Stable low-rank matrix recovery from 3-designs","authors":"Timm Gilles","doi":"10.1016/j.acha.2026.101887","DOIUrl":"10.1016/j.acha.2026.101887","url":null,"abstract":"<div><div>We study the recovery of low-rank Hermitian matrices from rank-one measurements obtained by uniform sampling from complex projective 3-designs, using nuclear-norm minimization. This framework includes phase retrieval as a special case via the PhaseLift method. In general, complex projective <em>t</em>-designs provide a systematic way to partially derandomize Gaussian measurement models. While near-optimal recovery guarantees are known for 4-designs, and it is known that 2-designs do not permit recovery with a subquadratic number of measurements, the case of 3-designs has remained open. In this work, we close this gap by establishing recovery guarantees for (exact and approximate) 3-designs that parallel the best-known results for 4-designs. In particular, we derive bounds on the number of measurements sufficient for stable and robust low-rank recovery via nuclear-norm minimization. Our results are especially relevant in practice, as explicit constructions of 4-designs are significantly more challenging than those of 3-designs.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"84 ","pages":"Article 101887"},"PeriodicalIF":3.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147849903","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 impact of smoothness of kernels and target functions on unsupervised covariate shift adaptation in RKHS","authors":"Elke R. Gizewski ,&nbsp;Shuai Lu ,&nbsp;Stephanie Mangesius ,&nbsp;Hoan D. Nguyen ,&nbsp;Sergiy Pereverzyev Jr.","doi":"10.1016/j.acha.2026.101866","DOIUrl":"10.1016/j.acha.2026.101866","url":null,"abstract":"<div><div>We analyze domain adaptation within the framework of reproducing kernel Hilbert spaces under the covariate shift assumption. In this setting, previously known results concerning the least squares excess risk bounds have mainly been derived for importance-weighted kernel ridge regression and either in terms of the smoothness of the target function or in terms of the capacity of the underlying space. The primary novelty of the current research lies in the analysis of general importance-weighted spectral algorithms in both above terms, which enables a substantial improvement of the excess risk bounds. Furthermore, we delve into the covariate shift adaptation utilizing estimated density ratios and explore its application in the context of imbalanced learning from real clinical data.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101866"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147279165","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":"Model agnostic signal encoding by leaky integrate-and-fire, performance and uncertainty","authors":"Diana Carbajal ,&nbsp;José Luis Romero","doi":"10.1016/j.acha.2026.101856","DOIUrl":"10.1016/j.acha.2026.101856","url":null,"abstract":"<div><div>Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal’s charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete <em>bandwidth-based Ansatz</em> that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover’s (Wasserstein) distance to measure spike discrepancy.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101856"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146110523","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":"Hierarchic flows to estimate and sample high-dimensional probabilities","authors":"Etienne Lempereur ,&nbsp;Stéphane Mallat","doi":"10.1016/j.acha.2026.101854","DOIUrl":"10.1016/j.acha.2026.101854","url":null,"abstract":"<div><div>Finding low-dimensional interpretable models of complex physical fields such as turbulence remains an open question, 80 years after the pioneer work of Kolmogorov. Estimating high-dimensional probability distributions from data samples suffers from an optimization and an approximation curse of dimensionality. It may be avoided by following a hierarchic probability flow from coarse to fine scales. This inverse renormalization group is defined by conditional probabilities across scales, renormalized in a wavelet basis. For a φ⁴ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. In a well chosen wavelet basis, conditional probabilities can be captured with low dimensional parametric models, because interactions between wavelet coefficients are local in space and scales. An outstanding issue is also to approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models of wavelet conditional probabilities with the scattering covariance. It is calculated with a second wavelet transform, which defines interactions over two hierarchies of scales. We estimate and sample these wavelet scattering models to generate 2D vorticity fields of turbulence, and images of dark matter densities.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101854"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146033548","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":"Dictionary learning under symmetries via group representations","authors":"Subhroshekhar Ghosh ,&nbsp;Aaron Y.R. Low ,&nbsp;Yong Sheng Soh ,&nbsp;Zhuohang Feng ,&nbsp;Brendan K.Y. Tan","doi":"10.1016/j.acha.2026.101855","DOIUrl":"10.1016/j.acha.2026.101855","url":null,"abstract":"<div><div>The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101855"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146095742","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":"Approximating sparse matrices and their functions using matrix-vector products","authors":"Taejun Park ,&nbsp;Yuji Nakatsukasa","doi":"10.1016/j.acha.2026.101869","DOIUrl":"10.1016/j.acha.2026.101869","url":null,"abstract":"<div><div>The computation of a matrix function <em>f</em>(<em>A</em>) is an important task in scientific computing appearing in machine learning, network analysis and the solution of partial differential equations. In this work, we use only matrix-vector products <em>x</em>↦<em>Ax</em> to approximate functions of sparse matrices and matrices with similar structures such as sparse matrices <em>A</em> themselves or matrices that have a similar decay property as matrix functions. We show that when <em>A</em> is a sparse matrix with an unknown sparsity pattern, techniques from compressed sensing can be used under natural assumptions. Moreover, if <em>A</em> is a banded matrix then certain deterministic matrix-vector products can efficiently recover the large entries of <em>f</em>(<em>A</em>). We describe an algorithm for each of the two cases and give error analysis based on the decay bound for the entries of <em>f</em>(<em>A</em>). We finish with numerical experiments showing the accuracy of our algorithms.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101869"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147334823","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 spectrality of Cantor-Moran measure and Fuglede’s conjecture","authors":"Jinsong Liu ,&nbsp;Zheng-Yi Lu ,&nbsp;Ting Zhou","doi":"10.1016/j.acha.2026.101853","DOIUrl":"10.1016/j.acha.2026.101853","url":null,"abstract":"<div><div>Let <span><math><mrow><mo>{</mo><mrow><mo>(</mo><msub><mi>p</mi><mi>n</mi></msub><mo>,</mo><msub><mi>D</mi><mi>n</mi></msub><mo>,</mo><msub><mi>L</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>}</mo></mrow></math></span> be a sequence of Hadamard triples on <span><math><mi>R</mi></math></span>. Suppose that the associated Cantor-Moran measure<span><span><span><math><mrow><msub><mi>μ</mi><mrow><mo>{</mo><msub><mi>p</mi><mi>n</mi></msub><mo>,</mo><msub><mi>D</mi><mi>n</mi></msub><mo>}</mo></mrow></msub><mo>=</mo><msub><mi>δ</mi><mrow><msubsup><mi>p</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>D</mi><mn>1</mn></msub></mrow></msub><mo>*</mo><msub><mi>δ</mi><mrow><msup><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>D</mi><mn>2</mn></msub></mrow></msub><mo>*</mo><mo>⋯</mo><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><msub><mi>sup</mi><mi>n</mi></msub><mrow><mo>{</mo><mo>|</mo></mrow><msubsup><mi>p</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mi>d</mi><mrow><mo>|</mo><mo>:</mo><mi>d</mi><mo>∈</mo><msub><mi>D</mi><mi>n</mi></msub><mo>}</mo></mrow><mo>&lt;</mo><mi>∞</mi></mrow></math></span> and <span><math><mrow><mi>sup</mi><mo>#</mo><msub><mi>D</mi><mi>n</mi></msub><mo>&lt;</mo><mi>∞</mi></mrow></math></span>. It has been observed that the spectrality of <span><math><msub><mi>μ</mi><mrow><mo>{</mo><msub><mi>p</mi><mi>n</mi></msub><mo>,</mo><msub><mi>D</mi><mi>n</mi></msub><mo>}</mo></mrow></msub></math></span> is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of <em>Double Points Condition Set</em> (<em>DPCS</em>) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede’s Conjecture on Cantor-Moran set. We show that the equi-positivity of <span><math><msub><mi>μ</mi><mrow><mo>{</mo><msub><mi>p</mi><mi>n</mi></msub><mo>,</mo><msub><mi>D</mi><mi>n</mi></msub><mo>}</mo></mrow></msub></math></span> implies the tiling of its support, and the reverse direction holds under certain conditions.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"83 ","pages":"Article 101853"},"PeriodicalIF":3.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145957328","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":"Proximal subgradient norm minimization of ISTA and FISTA","authors":"Bowen Li ,&nbsp;Bin Shi ,&nbsp;Ya-Xiang Yuan","doi":"10.1016/j.acha.2025.101848","DOIUrl":"10.1016/j.acha.2025.101848","url":null,"abstract":"<div><div>The study of acceleration in first-order smooth optimization has a long history. Yet, it was only recently that the mechanism behind acceleration was successfully elucidated through the introduction of the gradient correction term and its equivalent implicit-velocity formulation. Building on the high-resolution differential equation framework, augmented by phase-space representation and Lyapunov analysis, a faster convergence rate has been established for the squared gradient norm of Nesterov’s accelerated gradient descent (<span>NAG</span>) method. Despite this progress, such results do not directly extend to composite optimization problems widely encountered in practice, such as linear inverse problems with sparsity constraints. In this work, we refine a key descent inequality in the smooth setting and generalize it to the composite case, demonstrating that it admits a tighter bound. By incorporating this refined inequality into a carefully constructed Lyapunov function, we derive a proximal subgradient norm minimization result without relying on gradient correction or implicit-velocity scheme. Specifically, we establish that the squared proximal subgradient norm for the iterative shrinkage-thresholding algorithm (<span>ISTA</span>) decays at an inverse square rate, while for its accelerated variant (<span>FISTA</span>), it improves to an inverse cubic rate.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"82 ","pages":"Article 101848"},"PeriodicalIF":3.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145731504","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}