Journal of Complexity最新文献

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Worst case tractability of linear problems in the presence of noise: Linear information 存在噪声的线性问题的最坏情况可跟踪性:线性信息
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-07-26 DOI: 10.1016/j.jco.2023.101782
Leszek Plaskota, Paweł Siedlecki
{"title":"Worst case tractability of linear problems in the presence of noise: Linear information","authors":"Leszek Plaskota,&nbsp;Paweł Siedlecki","doi":"10.1016/j.jco.2023.101782","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101782","url":null,"abstract":"<div><p><span>We study the worst case tractability of multivariate linear problems defined on separable Hilbert spaces. Information about a problem instance consists of noisy evaluations of arbitrary bounded </span>linear functionals, where the noise is either deterministic or random. The cost of a single evaluation depends on its precision and is controlled by a cost function. We establish mutual interactions between tractability of a problem with noisy information, the cost function, and tractability of the same problem, but with exact information.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101782"},"PeriodicalIF":1.7,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49876976","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}
引用次数: 0
On the complexity of a unified convergence analysis for iterative methods 关于复杂性的一种统一收敛分析迭代方法
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-07-11 DOI: 10.1016/j.jco.2023.101781
Ioannis K. Argyros , Stepan Shakhno , Samundra Regmi , Halyna Yarmola
{"title":"On the complexity of a unified convergence analysis for iterative methods","authors":"Ioannis K. Argyros ,&nbsp;Stepan Shakhno ,&nbsp;Samundra Regmi ,&nbsp;Halyna Yarmola","doi":"10.1016/j.jco.2023.101781","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101781","url":null,"abstract":"<div><p><span><span>A local and a semi-local convergence of general iterative methods for solving nonlinear operator equations in </span>Banach spaces is developed under </span><em>ω</em>-continuity conditions. Our approach unifies existing results and provides a new way of studying iterative methods. The main idea is to find a more accurate domain containing the iterates. No extra effort is used to obtain this. Also, the results of the numerical experiments are given that confirm obtained theoretical estimates.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101781"},"PeriodicalIF":1.7,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49876979","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}
引用次数: 1
Approximating smooth and sparse functions by deep neural networks: Optimal approximation rates and saturation 用深度神经网络逼近平滑和稀疏函数:最优逼近率和饱和度
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-07-01 DOI: 10.1016/j.jco.2023.101783
Xia Liu
{"title":"Approximating smooth and sparse functions by deep neural networks: Optimal approximation rates and saturation","authors":"Xia Liu","doi":"10.1016/j.jco.2023.101783","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101783","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 1","pages":"101783"},"PeriodicalIF":1.7,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54746300","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}
引用次数: 3
The rate of convergence for sparse and low-rank quantile trace regression 稀疏和低秩分位数迹回归的收敛速度
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-19 DOI: 10.1016/j.jco.2023.101778
Xiangyong Tan , Ling Peng , Peiwen Xiao , Qing Liu , Xiaohui Liu
{"title":"The rate of convergence for sparse and low-rank quantile trace regression","authors":"Xiangyong Tan ,&nbsp;Ling Peng ,&nbsp;Peiwen Xiao ,&nbsp;Qing Liu ,&nbsp;Xiaohui Liu","doi":"10.1016/j.jco.2023.101778","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101778","url":null,"abstract":"<div><p>Trace regression models are widely used in applications involving panel data, images, genomic microarrays, etc., where high-dimensional covariates<span> are often involved. However, the existing research involving high-dimensional covariates focuses mainly on the condition mean model. In this paper, we extend the trace regression model to the quantile trace regression model when the parameter is a matrix of simultaneously low rank and row (column) sparsity. The convergence rate of the penalized estimator is derived under mild conditions. Simulations, as well as a real data application, are also carried out for illustration.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101778"},"PeriodicalIF":1.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49876978","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}
引用次数: 0
Optimal recovery and volume estimates 最佳恢复和容量估计
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-17 DOI: 10.1016/j.jco.2023.101780
Alexander Kushpel
{"title":"Optimal recovery and volume estimates","authors":"Alexander Kushpel","doi":"10.1016/j.jco.2023.101780","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101780","url":null,"abstract":"<div><p>We study volumes of sections of convex origin-symmetric bodies in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span> induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes </span><span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>γ</mi></mrow></msubsup></math></span>, <span><math><mi>γ</mi><mo>&gt;</mo><mn>0</mn></math></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> on two-point homogeneous spaces in the difficult case, i.e. if <span><math><mn>1</mn><mo>&lt;</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101780"},"PeriodicalIF":1.7,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877027","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}
引用次数: 0
Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs Bochner空间中的深度ReLU神经网络逼近及其在参数偏微分方程中的应用
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-15 DOI: 10.1016/j.jco.2023.101779
Dinh Dũng , Van Kien Nguyen , Duong Thanh Pham
{"title":"Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs","authors":"Dinh Dũng ,&nbsp;Van Kien Nguyen ,&nbsp;Duong Thanh Pham","doi":"10.1016/j.jco.2023.101779","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101779","url":null,"abstract":"<div><p><span><span>We investigate non-adaptive methods of deep ReLU neural network approximation in </span>Bochner spaces </span><span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msup><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>,</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span> of functions on <span><math><msup><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span><span> taking values in a separable Hilbert space </span><em>X</em>, where <span><math><msup><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> is either <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span><span> equipped with the standard Gaussian probability measure, or </span><span><math><msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mo>∞</mo></mrow></msup></math></span> equipped with the Jacobi probability measure. Functions to be approximated are assumed to satisfy a certain weighted <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-summability of the generalized chaos polynomial expansion coefficients with respect to the measure <em>μ</em><span>. We prove the convergence rate of this approximation in terms of the size of approximating deep ReLU neural networks. These results then are applied to approximation of the solution to parametric<span> elliptic PDEs with random inputs for the lognormal and affine cases.</span></span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101779"},"PeriodicalIF":1.7,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877026","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}
引用次数: 0
The curse of dimensionality for the Lp-discrepancy with finite p 有限p的lp差异的维数诅咒
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-08 DOI: 10.1016/j.jco.2023.101769
Erich Novak , Friedrich Pillichshammer
{"title":"The curse of dimensionality for the Lp-discrepancy with finite p","authors":"Erich Novak ,&nbsp;Friedrich Pillichshammer","doi":"10.1016/j.jco.2023.101769","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101769","url":null,"abstract":"<div><p>The <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy is a quantitative measure for the irregularity of distribution of an <em>N</em>-element point set in the <em>d</em>-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension <em>d</em> and error threshold <span><math><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> is the minimal number of points in <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that the minimal normalized <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy is less or equal <em>ε</em>. It is well known, that the inverse of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-discrepancy grows exponentially fast with the dimension <em>d</em>, i.e., we have the curse of dimensionality, whereas the inverse of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-discrepancy depends exactly linearly on <em>d</em>. The behavior of inverse of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy for general <span><math><mi>p</mi><mo>∉</mo><mo>{</mo><mn>2</mn><mo>,</mo><mo>∞</mo><mo>}</mo></math></span> has been an open problem for many years. In this paper we show that the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy suffers from the curse of dimensionality for all <em>p</em> in <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> which are of the form <span><math><mi>p</mi><mo>=</mo><mn>2</mn><mi>ℓ</mi><mo>/</mo><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> with <span><math><mi>ℓ</mi><mo>∈</mo><mi>N</mi></math></span>.</p><p>This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-norm, where <em>q</em> is an even positive integer satisfying <span><math><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo>=</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101769"},"PeriodicalIF":1.7,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877025","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}
引用次数: 1
On the cardinality of lower sets and universal discretization 下集的基数性与泛离散化
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-01 DOI: 10.1016/j.jco.2022.101726
F. Dai , A. Prymak , A. Shadrin , V.N. Temlyakov , S. Tikhonov
{"title":"On the cardinality of lower sets and universal discretization","authors":"F. Dai ,&nbsp;A. Prymak ,&nbsp;A. Shadrin ,&nbsp;V.N. Temlyakov ,&nbsp;S. Tikhonov","doi":"10.1016/j.jco.2022.101726","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101726","url":null,"abstract":"<div><p>A set <em>Q</em> in <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> is a lower set if <span><math><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>Q</mi></math></span> implies <span><math><mo>(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>Q</mi></math></span> whenever <span><math><mn>0</mn><mo>≤</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for all <em>i</em>. We derive new and refine known results regarding the cardinality of the lower sets of size <em>n</em> in <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>d</mi></mrow></msubsup></math></span><span>. Next we apply these results for universal discretization of the </span><span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-norm of elements from <em>n</em><span>-dimensional subspaces of trigonometric polynomials generated by lower sets.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101726"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882271","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}
引用次数: 0
The BMO-discrepancy suffers from the curse of dimensionality bmo差异受到维度的诅咒
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-01 DOI: 10.1016/j.jco.2023.101739
Friedrich Pillichshammer
{"title":"The BMO-discrepancy suffers from the curse of dimensionality","authors":"Friedrich Pillichshammer","doi":"10.1016/j.jco.2023.101739","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101739","url":null,"abstract":"<div><p>We show that the minimal discrepancy of a point set in the <em>d</em><span><span>-dimensional unit cube with respect to the BMO seminorm suffers from the </span>curse of dimensionality.</span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101739"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882273","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}
引用次数: 1
The area of empty axis-parallel boxes amidst 2-dimensional lattice points 在二维晶格点之间的空轴平行盒的面积
IF 1.7 2区 数学
Journal of Complexity Pub Date : 2023-06-01 DOI: 10.1016/j.jco.2022.101724
Thomas Lachmann, Jaspar Wiart
{"title":"The area of empty axis-parallel boxes amidst 2-dimensional lattice points","authors":"Thomas Lachmann,&nbsp;Jaspar Wiart","doi":"10.1016/j.jco.2022.101724","DOIUrl":"https://doi.org/10.1016/j.jco.2022.101724","url":null,"abstract":"<div><p>The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices<span> in the plane, that is, the supremum<span> of the area of the empty axis-parallel boxes amidst the lattice points<span>. We introduce a framework with which to study this based on the continued fractions expansions<span> of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.</span></span></span></span></p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"76 ","pages":"Article 101724"},"PeriodicalIF":1.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882299","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}
引用次数: 1
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