Journal of Complexity最新文献_第9页

Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere 绕过球面上超插值的正交精度假设

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-18 DOI: 10.1016/j.jco.2023.101789

Congpei An , Hao-Ning Wu

{"title":"Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere","authors":"Congpei An , Hao-Ning Wu","doi":"10.1016/j.jco.2023.101789","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101789","url":null,"abstract":"<div>This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math>-orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101789"},"PeriodicalIF":1.7,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887909","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

Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds 黎曼流形上凸复合优化问题的主要条件下高斯-牛顿法的收敛性

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-09 DOI: 10.1016/j.jco.2023.101788

Qamrul Hasan Ansari , Moin Uddin , Jen-Chih Yao

引用次数: 1

The minimal radius of Galerkin information for the problem of numerical differentiation 数值微分问题的伽辽金信息的最小半径

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-05 DOI: 10.1016/j.jco.2023.101787

S.G. Solodky, S.A. Stasyuk

引用次数: 0

Sampling numbers of smoothness classes via ℓ1-minimization 通过1-最小化得到平滑类的采样个数

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-05 DOI: 10.1016/j.jco.2023.101786

Thomas Jahn , Tino Ullrich , Felix Voigtlaender

{"title":"Sampling numbers of smoothness classes via ℓ1-minimization","authors":"Thomas Jahn , Tino Ullrich , Felix Voigtlaender","doi":"10.1016/j.jco.2023.101786","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101786","url":null,"abstract":"<div>Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math> can be upper bounded by best n-term trigonometric widths in <math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math>. We describe a recovery procedure from m function values based on <math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msup></math>-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of <math><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math> (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to <math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mi>W</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math> on the d-torus with a logarithmically better rate of convergence than any linear method can achieve when <math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></math> and d is large. This effect is not present for isotropic Sobolev spaces.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101786"},"PeriodicalIF":1.7,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877022","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

Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration 高维积分的随机素数-固定向量随机格算法

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-02 DOI: 10.1016/j.jco.2023.101785

Frances Y. Kuo , Dirk Nuyens , Laurence Wilkes

{"title":"Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration","authors":"Frances Y. Kuo , Dirk Nuyens , Laurence Wilkes","doi":"10.1016/j.jco.2023.101785","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101785","url":null,"abstract":"<div>We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the d-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget n of a maximum allowed number of function evaluations, we uniformly pick a prime p in the range <math><mi>n</mi><mo>/</mo><mn>2</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>n</mi></math>. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>δ</mi></mrow></msup><mo>)</mo></math>, with <math><mi>δ</mi><mo>></mo><mn>0</mn></math>, for a Korobov space with smoothness <math><mi>α</mi><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></math> and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors offline ahead of time at cost <math><mi>O</mi><mo>(</mo><mi>d</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mi>ln</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math> when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101785"},"PeriodicalIF":1.7,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877023","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

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality 人工神经网络近似的下界：浅层神经网络无法克服维数诅咒的证明

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-01 DOI: 10.1016/j.jco.2023.101746

Philipp Grohs , Shokhrukh Ibragimov , Arnulf Jentzen , Sarah Koppensteiner

{"title":"Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality","authors":"Philipp Grohs , Shokhrukh Ibragimov , Arnulf Jentzen , Sarah Koppensteiner","doi":"10.1016/j.jco.2023.101746","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101746","url":null,"abstract":"<div>Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden layers, have been very successfully used in a series of practical relevant computational problems involving high-dimensional input data ranging from classification tasks in supervised learning to optimal decision problems in reinforcement learning. There are also a number of mathematical results in the scientific literature which study the approximation capacities of ANNs in the context of high-dimensional target functions. In particular, there are a series of mathematical results in the scientific literature which show that sufficiently deep ANNs have the capacity to overcome the curse of dimensionality in the approximation of certain target function classes in the sense that the number of parameters of the approximating ANNs grows at most polynomially in the dimension <math><mi>d</mi><mo>∈</mo><mi>N</mi></math> of the target functions under considerations. In the proofs of several of such high-dimensional approximation results it is crucial that the involved ANNs are sufficiently deep and consist a sufficiently large number of hidden layers which grows in the dimension of the considered target functions. It is the topic of this work to look a bit more detailed to the deepness of the involved ANNs in the approximation of high-dimensional target functions. In particular, the main result of this work proves that there exists a concretely specified sequence of functions which can be approximated without the curse of dimensionality by sufficiently deep ANNs but which cannot be approximated without the curse of dimensionality if the involved ANNs are shallow or not deep enough.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"77 ","pages":"Article 101746"},"PeriodicalIF":1.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50200237","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

On Huber's contaminated model 关于Huber污染模型

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-01 DOI: 10.1016/j.jco.2023.101745

Weiyan Mu , Shifeng Xiong

引用次数: 0

A continuous characterization of PSPACE using polynomial ordinary differential equations 用多项式常微分方程连续刻画PSPACE

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-01 DOI: 10.1016/j.jco.2023.101755

Olivier Bournez , Riccardo Gozzi , Daniel S. Graça , Amaury Pouly

引用次数: 2

Dmitriy Bilyk and Feng Dai are the winners of the 2023 Joseph F. Traub Prize for Achievement in Information-Based Complexity 德米特里·比利克和冯戴是2023年约瑟夫·F·特劳布信息复杂性成就奖的得主

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-08-01 DOI: 10.1016/j.jco.2023.101756

Erich Novak

引用次数: 0

Rates of approximation by ReLU shallow neural networks ReLU浅层神经网络的近似速率

IF 1.7 2区数学

Journal of Complexity Pub Date : 2023-07-31 DOI: 10.1016/j.jco.2023.101784

Tong Mao , Ding-Xuan Zhou

{"title":"Rates of approximation by ReLU shallow neural networks","authors":"Tong Mao , Ding-Xuan Zhou","doi":"10.1016/j.jco.2023.101784","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101784","url":null,"abstract":"<div>Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from Hölder spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with m hidden neurons can uniformly approximate functions from the Hölder space <math><msubsup><mrow><mi>W</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math> with rates <math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>m</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>d</mi></mrow></msup><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mfrac><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>d</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></math> when <math><mi>r</mi><mo><</mo><mi>d</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>2</mn></math>. Such rates are very close to the optimal one <math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></msup><mo>)</mo></math> in the sense that <math><mfrac><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>d</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math> is close to 1, when the dimension d is large.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101784"},"PeriodicalIF":1.7,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49876977","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