Pub Date : 2025-03-21DOI: 10.1016/j.acha.2025.101762
Thomas Allard, Helmut Bölcskei
We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.
{"title":"Entropy of compact operators with applications to Landau-Pollak-Slepian theory and Sobolev spaces","authors":"Thomas Allard, Helmut Bölcskei","doi":"10.1016/j.acha.2025.101762","DOIUrl":"10.1016/j.acha.2025.101762","url":null,"abstract":"<div><div>We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101762"},"PeriodicalIF":2.6,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746775","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 : 2025-03-20DOI: 10.1016/j.acha.2025.101761
Lutz Kämmerer
For an arbitrary given span of high dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each polynomial of this span.
The approach presented here combines three different types of efficiency. First, the construction of a spatial discretization should be computationally efficient with respect to the dimension of the span of the Chebyshev polynomials. Second, the constructed discretization should be sample efficient, i.e., the number of sampling nodes within the constructed discretization should be reasonably low. Third, there should be an efficient algorithm for the unique reconstruction of a polynomial from given sampling values at the sampling nodes of the discretization.
The first two mentioned types of efficiency are also present in constructions based on random sampling nodes, but the lack of structure here causes the inefficiency of the reconstruction method. Our approach uses a combination of cosine transformed rank-1 lattices whose structure allows for applications of univariate fast Fourier transforms for the reconstruction algorithm and is thus a priori efficiently realizable.
Besides the theoretical estimates of numbers of sampling nodes and failure probabilities due to a random draw of the used lattices, we present several improvements of the basic design approach that significantly increases its practical applicability. Numerical tests, which discretize spans of multivariate Chebyshev polynomials depending on up to more than 50 spatial variables, corroborate the theoretical results and the significance of the improvements.
{"title":"An efficient spatial discretization of spans of multivariate Chebyshev polynomials","authors":"Lutz Kämmerer","doi":"10.1016/j.acha.2025.101761","DOIUrl":"10.1016/j.acha.2025.101761","url":null,"abstract":"<div><div>For an arbitrary given span of high dimensional multivariate Chebyshev polynomials, an approach to construct spatial discretizations is presented, i.e., the construction of a sampling set that allows for the unique reconstruction of each polynomial of this span.</div><div>The approach presented here combines three different types of efficiency. First, the construction of a spatial discretization should be computationally efficient with respect to the dimension of the span of the Chebyshev polynomials. Second, the constructed discretization should be sample efficient, i.e., the number of sampling nodes within the constructed discretization should be reasonably low. Third, there should be an efficient algorithm for the unique reconstruction of a polynomial from given sampling values at the sampling nodes of the discretization.</div><div>The first two mentioned types of efficiency are also present in constructions based on random sampling nodes, but the lack of structure here causes the inefficiency of the reconstruction method. Our approach uses a combination of cosine transformed rank-1 lattices whose structure allows for applications of univariate fast Fourier transforms for the reconstruction algorithm and is thus a priori efficiently realizable.</div><div>Besides the theoretical estimates of numbers of sampling nodes and failure probabilities due to a random draw of the used lattices, we present several improvements of the basic design approach that significantly increases its practical applicability. Numerical tests, which discretize spans of multivariate Chebyshev polynomials depending on up to more than 50 spatial variables, corroborate the theoretical results and the significance of the improvements.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101761"},"PeriodicalIF":2.6,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697625","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}
Pub Date : 2025-03-20DOI: 10.1016/j.acha.2025.101760
Yu Ping Wang , Yan-Hsiou Cheng
In this paper, we study direct and inverse problems for Dirac systems with complex-valued potentials on p-star-shaped graphs. More precisely, we firstly obtain sharp 2-term asymptotics of the corresponding eigenvalues. We then formulate and address a Horváth-type theorem, specifically, if the potentials on edges of the p-star-shaped graph are predetermined, we demonstrate that the remaining potential on can be uniquely determined by part of its eigenvalues and the given remaining potential on , , under certain conditions.
{"title":"An inverse problem for Dirac systems on p-star-shaped graphs","authors":"Yu Ping Wang , Yan-Hsiou Cheng","doi":"10.1016/j.acha.2025.101760","DOIUrl":"10.1016/j.acha.2025.101760","url":null,"abstract":"<div><div>In this paper, we study direct and inverse problems for Dirac systems with complex-valued potentials on <em>p</em>-star-shaped graphs. More precisely, we firstly obtain sharp 2-term asymptotics of the corresponding eigenvalues. We then formulate and address a Horváth-type theorem, specifically, if the potentials on <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> edges of the <em>p</em>-star-shaped graph are predetermined, we demonstrate that the remaining potential on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></math></span> can be uniquely determined by part of its eigenvalues and the given remaining potential on <span><math><mo>[</mo><mi>a</mi><mo>,</mo><mi>π</mi><mo>]</mo></math></span>, <span><math><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mi>π</mi></math></span>, under certain conditions.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101760"},"PeriodicalIF":2.6,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679153","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 : 2025-03-14DOI: 10.1016/j.acha.2025.101759
Jiuyang Liang , Zhenli Xu , Qi Zhou
This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians M in the u-series and the base of interpolation nodes b in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as for the energy and for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.
{"title":"Error estimate of the u-series method for molecular dynamics simulations","authors":"Jiuyang Liang , Zhenli Xu , Qi Zhou","doi":"10.1016/j.acha.2025.101759","DOIUrl":"10.1016/j.acha.2025.101759","url":null,"abstract":"<div><div>This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians <em>M</em> in the u-series and the base of interpolation nodes <em>b</em> in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mi>M</mi></mrow></msup><mo>)</mo></math></span> for the energy and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mn>3</mn><mi>M</mi></mrow></msup><mo>)</mo></math></span> for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101759"},"PeriodicalIF":2.6,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143675537","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 : 2025-02-21DOI: 10.1016/j.acha.2025.101756
Mahya Ghandehari , Georgi S. Medvedev
The W-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for W-random graphs from [19], we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the corresponding graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of small-world and bipartite random graphs conditioned on atypical edge counts. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces are affected by large deviations. We discuss the implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing.
W 随机图为大型随机网络建模提供了一个灵活的框架。利用文献[19]中针对 W-随机图的大偏差原理（LDP），我们证明了相应类别的随机对称希尔伯特-施密特积分算子的大偏差原理。我们的主要结果描述了积分算子的特征值和特征空间如何受到底层随机图元大偏差的影响。为了证明 LDP，我们证明了与积分算子相关的谱度量对相应图元的连续依赖性，并使用了收缩原理。为了说明我们的结果，我们获得了以非典型边数为条件的小世界和双方形随机图特征值的前阶渐近性。这些例子说明了特征值和特征空间如何受到大偏差的影响。我们将讨论这些观察结果对动态系统分岔分析和图信号处理的影响。
{"title":"The Large Deviation Principle for W-random spectral measures","authors":"Mahya Ghandehari , Georgi S. Medvedev","doi":"10.1016/j.acha.2025.101756","DOIUrl":"10.1016/j.acha.2025.101756","url":null,"abstract":"<div><div>The <em>W</em>-random graphs provide a flexible framework for modeling large random networks. Using the Large Deviation Principle (LDP) for <em>W</em>-random graphs from <span><span>[19]</span></span>, we prove the LDP for the corresponding class of random symmetric Hilbert-Schmidt integral operators. Our main result describes how the eigenvalues and the eigenspaces of the integral operator are affected by large deviations in the underlying random graphon. To prove the LDP, we demonstrate continuous dependence of the spectral measures associated with integral operators on the corresponding graphons and use the Contraction Principle. To illustrate our results, we obtain leading order asymptotics of the eigenvalues of small-world and bipartite random graphs conditioned on atypical edge counts. These examples suggest several representative scenarios of how the eigenvalues and the eigenspaces are affected by large deviations. We discuss the implications of these observations for bifurcation analysis of Dynamical Systems and Graph Signal Processing.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101756"},"PeriodicalIF":2.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479228","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 : 2025-02-17DOI: 10.1016/j.acha.2025.101749
Massimo Fornasier , Timo Klock , Marco Mondelli , Michael Rauchensteiner
The identification of the parameters of a neural network from finite samples of input-output pairs is often referred to as the teacher-student model, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature – after adding suitable distributional assumptions – has established finite sample identification of two-layer networks with a number of neurons , D being the input dimension. For the range the problem becomes harder, and truly little is known for networks parametrized by biases as well. This paper fills the gap by providing efficient algorithms and rigorous theoretical guarantees of finite sample identification for such wider shallow networks with biases. Our approach is based on a two-step pipeline: first, we recover the direction of the weights, by exploiting second order information; next, we identify the signs by suitable algebraic evaluations, and we recover the biases by empirical risk minimization via gradient descent. Numerical results demonstrate the effectiveness of our approach.
{"title":"Efficient identification of wide shallow neural networks with biases","authors":"Massimo Fornasier , Timo Klock , Marco Mondelli , Michael Rauchensteiner","doi":"10.1016/j.acha.2025.101749","DOIUrl":"10.1016/j.acha.2025.101749","url":null,"abstract":"<div><div>The identification of the parameters of a neural network from finite samples of input-output pairs is often referred to as the <em>teacher-student model</em>, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature – after adding suitable distributional assumptions – has established finite sample identification of two-layer networks with a number of neurons <span><math><mi>m</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, <em>D</em> being the input dimension. For the range <span><math><mi>D</mi><mo><</mo><mi>m</mi><mo><</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> the problem becomes harder, and truly little is known for networks parametrized by biases as well. This paper fills the gap by providing efficient algorithms and rigorous theoretical guarantees of finite sample identification for such wider shallow networks with biases. Our approach is based on a two-step pipeline: first, we recover the direction of the weights, by exploiting second order information; next, we identify the signs by suitable algebraic evaluations, and we recover the biases by empirical risk minimization via gradient descent. Numerical results demonstrate the effectiveness of our approach.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"77 ","pages":"Article 101749"},"PeriodicalIF":2.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429952","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}
Pub Date : 2025-01-10DOI: 10.1016/j.acha.2025.101748
Ilya Krishtal, Brendan Miller
We extend the classical Kadec theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.
{"title":"Kadec-type theorems for sampled group orbits","authors":"Ilya Krishtal, Brendan Miller","doi":"10.1016/j.acha.2025.101748","DOIUrl":"10.1016/j.acha.2025.101748","url":null,"abstract":"<div><div>We extend the classical Kadec <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101748"},"PeriodicalIF":2.6,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990415","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 : 2025-01-10DOI: 10.1016/j.acha.2025.101747
Andreas Horst , Jakob Lemvig , Allan Erlang Videbæk
The frame set conjecture for Hermite functions formulated in [13] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α and modulation rates β that avoid all known obstructions lead to Gabor frames for . By results in [24], [25] and [22], it is known that the conjecture is true for the Gaussian, the 0th order Hermite functions, and false for Hermite functions of order , respectively. In this paper we disprove the remaining cases except for the 1st order Hermite function.
{"title":"On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture","authors":"Andreas Horst , Jakob Lemvig , Allan Erlang Videbæk","doi":"10.1016/j.acha.2025.101747","DOIUrl":"10.1016/j.acha.2025.101747","url":null,"abstract":"<div><div>The frame set conjecture for Hermite functions formulated in <span><span>[13]</span></span> states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates <em>α</em> and modulation rates <em>β</em> that avoid all known obstructions lead to Gabor frames for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. By results in <span><span>[24]</span></span>, <span><span>[25]</span></span> and <span><span>[22]</span></span>, it is known that the conjecture is true for the Gaussian, the 0th order Hermite functions, and false for Hermite functions of order <span><math><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>,</mo><mo>…</mo></math></span>, respectively. In this paper we disprove the remaining cases <em>except</em> for the 1st order Hermite function.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101747"},"PeriodicalIF":2.6,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990416","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}
Pub Date : 2024-12-30DOI: 10.1016/j.acha.2024.101746
Julia Kostin , Felix Krahmer , Dominik Stöger
In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behaviour of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.
{"title":"How robust is randomized blind deconvolution via nuclear norm minimization against adversarial noise?","authors":"Julia Kostin , Felix Krahmer , Dominik Stöger","doi":"10.1016/j.acha.2024.101746","DOIUrl":"10.1016/j.acha.2024.101746","url":null,"abstract":"<div><div>In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behaviour of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101746"},"PeriodicalIF":2.6,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968152","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 : 2024-12-17DOI: 10.1016/j.acha.2024.101745
Naveen Gupta , S. Sivananthan , Bharath K. Sriperumbudur
Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under Hölder type source condition, which generalizes and sharpens all the known results in the literature.
{"title":"Optimal rates for functional linear regression with general regularization","authors":"Naveen Gupta , S. Sivananthan , Bharath K. Sriperumbudur","doi":"10.1016/j.acha.2024.101745","DOIUrl":"10.1016/j.acha.2024.101745","url":null,"abstract":"<div><div>Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under Hölder type source condition, which generalizes and sharpens all the known results in the literature.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"76 ","pages":"Article 101745"},"PeriodicalIF":2.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874073","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}
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