Journal of Complexity最新文献_第2页

On the approximation of vector-valued functions by volume sampling 关于用体积采样法逼近矢量值函数

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-07-31 DOI: 10.1016/j.jco.2024.101887

Daniel Kressner , Tingting Ni , André Uschmajew

{"title":"On the approximation of vector-valued functions by volume sampling","authors":"Daniel Kressner , Tingting Ni , André Uschmajew","doi":"10.1016/j.jco.2024.101887","DOIUrl":"10.1016/j.jco.2024.101887","url":null,"abstract":"<div>Given a Hilbert space <math><mi>H</mi></math> and a finite measure space Ω, the approximation of a vector-valued function <math><mi>f</mi><mo>:</mo><mi>Ω</mi><mo>→</mo><mi>H</mi></math> by a k-dimensional subspace <math><mi>U</mi><mo>⊂</mo><mi>H</mi></math> plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue–Bochner space <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>;</mo><mi>H</mi><mo>)</mo></math>, the best possible subspace approximation error <math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup></math> is characterized by the singular values of f. However, for practical reasons, <math><mi>U</mi></math> is often restricted to be spanned by point samples of f. We show that this restriction only has a mild impact on the attainable error; there always exist k samples such that the resulting error is not larger than <math><msqrt><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>⋅</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup></math>. Our work extends existing results by Binev et al. (2011) [3] on approximation in supremum norm and by Deshpande et al. (2006) [8] on column subset selection for matrices.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000645/pdfft?md5=810287a810b23405b1bc8161d82ba70e&pid=1-s2.0-S0885064X24000645-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935983","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}

引用次数: 0

High probability bounds on AdaGrad for constrained weakly convex optimization 受约束弱凸优化的 AdaGrad 高概率边界

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-07-31 DOI: 10.1016/j.jco.2024.101889

Yusu Hong , Junhong Lin

引用次数: 0

No existence of a linear algorithm for the one-dimensional Fourier phase retrieval 不存在一维傅立叶相位检索的线性算法

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-07-04 DOI: 10.1016/j.jco.2024.101886

Meng Huang , Zhiqiang Xu

{"title":"No existence of a linear algorithm for the one-dimensional Fourier phase retrieval","authors":"Meng Huang , Zhiqiang Xu","doi":"10.1016/j.jco.2024.101886","DOIUrl":"10.1016/j.jco.2024.101886","url":null,"abstract":"<div>Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal <math><mi>x</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>N</mi></mrow></msup></math> in <math><mtext>Poly</mtext><mo>(</mo><mi>N</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math> time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value <math><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo></math>, then <math><mi>P</mi><mo>=</mo><mrow><mi>NP</mi></mrow></math>. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value <math><mo>|</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo></math>, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000633/pdfft?md5=306cc05c455de6efb9f908455c6f3128&pid=1-s2.0-S0885064X24000633-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141637397","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}

引用次数: 0

Interpolation by decomposable univariate polynomials 可分解单变量多项式插值法

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-06-26 DOI: 10.1016/j.jco.2024.101885

Joachim von zur Gathen , Guillermo Matera

引用次数: 0

Sharp lower bounds on the manifold widths of Sobolev and Besov spaces 索波列夫和贝索夫空间流形宽度的尖锐下限

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-06-20 DOI: 10.1016/j.jco.2024.101884

Jonathan W. Siegel

{"title":"Sharp lower bounds on the manifold widths of Sobolev and Besov spaces","authors":"Jonathan W. Siegel","doi":"10.1016/j.jco.2024.101884","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101884","url":null,"abstract":"<div>We study the manifold n-widths of Sobolev and Besov spaces with error measured in the <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-norm. The manifold widths measure how efficiently these spaces can be approximated by continuous non-linear parametric methods. Existing upper and lower bounds only match when the smoothness index q satisfies <math><mi>q</mi><mo>≤</mo><mi>p</mi></math> or <math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></math>. We close this gap, obtaining sharp bounds for all <math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mo>∞</mo></math> for which a compact embedding holds. In the process, we determine the exact value of the manifold widths of finite dimensional <math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>M</mi></mrow></msubsup></math>-balls in the <math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-norm when <math><mi>p</mi><mo>≤</mo><mi>q</mi></math>. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485089","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

Minimal dispersion on the cube and the torus 立方体和环面上的最小色散

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-06-13 DOI: 10.1016/j.jco.2024.101883

A. Arman , A.E. Litvak

引用次数: 0

Adaptive Huber trace regression with low-rank matrix parameter via nonconvex regularization 通过非凸正则化实现低阶矩阵参数的自适应胡贝尔痕量回归

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-06-11 DOI: 10.1016/j.jco.2024.101871

Xiangyong Tan , Ling Peng , Heng Lian , Xiaohui Liu

引用次数: 0

Kinetic Langevin MCMC sampling without gradient Lipschitz continuity - the strongly convex case 无梯度 Lipschitz 连续性的动力学 Langevin MCMC 采样--强凸情况

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-06-11 DOI: 10.1016/j.jco.2024.101873

Tim Johnston , Iosif Lytras , Sotirios Sabanis

引用次数: 0

Randomized complexity of mean computation and the adaption problem 均值计算的随机复杂性与适应问题

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-06-11 DOI: 10.1016/j.jco.2024.101872

Stefan Heinrich

引用次数: 0

On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient 论具有非 Lipschitz 漂移系数的随机微分方程强逼近的复杂性

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-06-03 DOI: 10.1016/j.jco.2024.101870

Thomas Müller-Gronbach , Larisa Yaroslavtseva

{"title":"On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient","authors":"Thomas Müller-Gronbach , Larisa Yaroslavtseva","doi":"10.1016/j.jco.2024.101870","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101870","url":null,"abstract":"<div>We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.</div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000475/pdfft?md5=1abf95a86603ccdc1b342109b28265f5&pid=1-s2.0-S0885064X24000475-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141313488","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}

引用次数: 0