Applied and Computational Harmonic Analysis最新文献

{"title":"Estimates on learning rates for multi-penalty distribution regression","authors":"Zhan Yu ,&nbsp;Daniel W.C. Ho","doi":"10.1016/j.acha.2023.101609","DOIUrl":"10.1016/j.acha.2023.101609","url":null,"abstract":"<div><p><span><span>This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to real-valued outputs from probability measures. The theoretical analysis of distribution regression is far from maturity and quite challenging since only second-stage samples are observable in practical settings. In our algorithm, to transform information of distribution samples, we embed the distributions to a reproducing kernel </span>Hilbert space </span><span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> associated with Mercer kernel <em>K</em> via mean embedding technique. One of the primary contributions of this work is the introduction of a novel multi-penalty regularization algorithm, which is able to capture more potential features of distribution regression. Optimal learning rates of the algorithm are obtained under mild conditions. The work also derives learning rates for distribution regression in the hard learning scenario <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>∉</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span>, which has not been explored in the existing literature. Moreover, we propose a new distribution-regression-based distributed learning algorithm to face large-scale data or information challenges arising from distribution data. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, the work improves the existing literature in various aspects.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"69 ","pages":"Article 101609"},"PeriodicalIF":2.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138297364","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":"Dilational symmetries of decomposition and coorbit spaces","authors":"Hartmut Führ ,&nbsp;Reihaneh Raisi-Tousi","doi":"10.1016/j.acha.2023.101610","DOIUrl":"10.1016/j.acha.2023.101610","url":null,"abstract":"<div><p><span>We investigate the invariance properties of general wavelet coorbit spaces and Besov-type </span>decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings.</p><p>We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices that are coorbit compatible with a given shearlet dilation group. We explicitly determine the groups of compatible dilations, for a variety of concrete examples.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"69 ","pages":"Article 101610"},"PeriodicalIF":2.5,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293110","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":"Image denoising based on a variable spatially exponent PDE","authors":"Amine Laghrib,&nbsp;Lekbir Afraites","doi":"10.1016/j.acha.2023.101608","DOIUrl":"10.1016/j.acha.2023.101608","url":null,"abstract":"<div><p>Image denoising is always considered an important area of image processing. In this work, we address a new PDE-based model for image denoising that have been contaminated by multiplicative noise<span>, specially the Speckle one. We propose a new class of PDEs whose nonlinear structure depends on a spatially tensor depending quantity attached to the desired solution, which takes into account the gray level information by introducing a gray level indicator function in the diffusion coefficient<span>. We give some theoretical results, discretization and also stability condition for the suggested model. Finally, we carry out some numerical results to approve the effectiveness of our model by comparing the results obtained with some competitive models.</span></span></p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101608"},"PeriodicalIF":2.5,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92158652","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 intermediate value property of spectra for a class of Moran spectral measures","authors":"Jinjun Li,&nbsp;Zhiyi Wu","doi":"10.1016/j.acha.2023.101606","DOIUrl":"10.1016/j.acha.2023.101606","url":null,"abstract":"<div><p>We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are in 0 and their upper entropy dimensions. Moreover, for such a Moran spectral measure <em>μ</em>, we show that the Beurling dimension for the spectra of <em>μ</em> has the intermediate value property: let <em>t</em> be any value in 0 and the upper entropy dimension of <em>μ</em>, then there exists a spectrum whose Beurling dimension is <em>t</em><span>. In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution in Fu et al. (2018) </span><span>[20]</span>. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value in 0 and <span><math><msub><mrow><mover><mrow><mi>dim</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>e</mi></mrow></msub><mspace></mspace><mi>μ</mi></math></span> has the cardinality of the continuum.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101606"},"PeriodicalIF":2.5,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516669","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 metaplectic action on modulation spaces","authors":"Hartmut Führ ,&nbsp;Irina Shafkulovska","doi":"10.1016/j.acha.2023.101604","DOIUrl":"10.1016/j.acha.2023.101604","url":null,"abstract":"<div><p>We study the mapping properties of metaplectic operators <span><math><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>∈</mo><mrow><mi>Mp</mi></mrow><mo>(</mo><mn>2</mn><mi>d</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> on modulation spaces of the type <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Our main result is a full characterization of the pairs <span><math><mo>(</mo><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>,</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span> for which the operator <span><math><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> is <em>(i)</em> well-defined, <em>(ii)</em> bounded. It turns out that these two properties are equivalent, and they entail that <span><math><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of <span><math><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>→</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> transfers to <span><math><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>→</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101604"},"PeriodicalIF":2.5,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S106352032300091X/pdfft?md5=0769848d44f7ddda38eab0321ccdd78e&pid=1-s2.0-S106352032300091X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364681","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}

{"title":"Exponential bases for partitions of intervals","authors":"Götz Pfander ,&nbsp;Shauna Revay ,&nbsp;David Walnut","doi":"10.1016/j.acha.2023.101607","DOIUrl":"10.1016/j.acha.2023.101607","url":null,"abstract":"<div><p>For a partition of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> into intervals <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> we prove the existence of a partition of <span><math><mi>Z</mi></math></span> into <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that the complex exponential functions with frequencies in <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> form a Riesz basis for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, and furthermore, that for any <span><math><mi>J</mi><mo>⊆</mo><mo>{</mo><mn>1</mn><mo>,</mo><mspace></mspace><mn>2</mn><mo>,</mo><mspace></mspace><mo>…</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>}</mo></math></span>, the exponential functions with frequencies in <span><math><msub><mrow><mo>⋃</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub><msub><mrow><mi>Λ</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> form a Riesz basis for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo></math></span> for any interval <em>I</em> with length <span><math><mo>|</mo><mi>I</mi><mo>|</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub><mo>|</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo></math></span>. The construction extends to infinite partitions of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, but with size limitations on the subsets <span><math><mi>J</mi><mo>⊆</mo><mi>Z</mi></math></span>; it combines the ergodic properties of subsequences of <span><math><mi>Z</mi></math></span> known as Beatty-Fraenkel sequences with a theorem of Avdonin on exponential Riesz bases.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101607"},"PeriodicalIF":2.5,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514551","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":"A multivariate Riesz basis of ReLU neural networks","authors":"Cornelia Schneider ,&nbsp;Jan Vybíral","doi":"10.1016/j.acha.2023.101605","DOIUrl":"10.1016/j.acha.2023.101605","url":null,"abstract":"<div><p><span>We consider the trigonometric-like system of piecewise linear<span> functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of </span></span><span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>)</mo></math></span><span> based on the Gershgorin theorem. We also generalize this system to higher dimensions </span><span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span> by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><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><span> can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of </span><em>d</em><span>, making it an attractive building block regarding future multivariate analysis of neural networks.</span></p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101605"},"PeriodicalIF":2.5,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71518637","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 relation between Fourier and Walsh–Rademacher spectra for random fields","authors":"Anton Kutsenko ,&nbsp;Sergey Danilov ,&nbsp;Stephan Juricke ,&nbsp;Marcel Oliver","doi":"10.1016/j.acha.2023.101603","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101603","url":null,"abstract":"<div><p>We discuss relations between the expansion coefficients of a discrete random field when analyzed with respect to different hierarchical bases. Our main focus is on the comparison of two such systems: the Walsh–Rademacher basis and the trigonometric Fourier basis. In general, spectra computed with respect to one basis will look different in the other. In this paper, we prove that, in a statistical sense, the rate of spectral decay computed in one basis can be translated to the other. We further provide explicit expressions for this translation on quadrilateral meshes. The results are illustrated with numerical examples for deterministic and random fields.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101603"},"PeriodicalIF":2.5,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49791518","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":"Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness","authors":"Hao Liu ,&nbsp;Alex Havrilla ,&nbsp;Rongjie Lai ,&nbsp;Wenjing Liao","doi":"10.1016/j.acha.2023.101602","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101602","url":null,"abstract":"<div><p>Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering <em>n</em> noisy training samples, along with their noise-free counterparts, on a <em>d</em>-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>⁡</mo><mi>n</mi></math></span>, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.</p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101602"},"PeriodicalIF":2.5,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49778386","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":"Time and band limiting for exceptional polynomials","authors":"M.M. Castro ,&nbsp;F.A. Grünbaum ,&nbsp;I. Zurrián","doi":"10.1016/j.acha.2023.101600","DOIUrl":"https://doi.org/10.1016/j.acha.2023.101600","url":null,"abstract":"<div><p><span>The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in </span>Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.</p><p>Here we give a general result that insures the existence of a commuting differential operator<span> for a given family of exceptional orthogonal polynomials satisfying the “bispectral property”. As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.</span></p></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"68 ","pages":"Article 101600"},"PeriodicalIF":2.5,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49778389","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}