Journal of Complexity最新文献_第2页

Space-filling designs on Riemannian manifolds 黎曼流形上的空间填充设计

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-09-24 DOI: 10.1016/j.jco.2024.101899

Mingyao Ai , Yunfan Yang , Xiangshun Kong

引用次数: 0

On the number of solutions to a random instance of the permuted kernel problem 关于包络核问题随机实例的解数

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-09-24 DOI: 10.1016/j.jco.2024.101898

Carlo Sanna

{"title":"On the number of solutions to a random instance of the permuted kernel problem","authors":"Carlo Sanna","doi":"10.1016/j.jco.2024.101898","DOIUrl":"10.1016/j.jco.2024.101898","url":null,"abstract":"<div><div>The Permuted Kernel Problem (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an <math><mi>ℓ</mi><mo>×</mo><mi>m</mi></math> matrix A and an <math><mi>m</mi><mo>×</mo><mn>1</mn></math> vector b over a finite field of q elements <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, the PKP asks to find an <math><mi>m</mi><mo>×</mo><mi>m</mi></math> permutation matrix π such that <math><mi>π</mi><mi>b</mi></math> belongs to the kernel of A. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters <math><mi>q</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>m</mi></math>. Previous works have heuristically estimated the expected number of solutions to be <math><mi>m</mi><mo>!</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math>.</div><div>We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related Inhomogeneous Permuted Kernel Problem (IPKP), considering two natural ways of generating random instances.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101898"},"PeriodicalIF":1.8,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X/pdfft?md5=939873f4b51043507214927d47f2bb37&pid=1-s2.0-S0885064X2400075X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314359","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

Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces 巴拿赫空间中具有均匀凸约束条件的迭代正则化 Landweber 迭代的收敛性分析

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-09-13 DOI: 10.1016/j.jco.2024.101897

Gaurav Mittal , Harshit Bajpai , Ankik Kumar Giri

引用次数: 0

High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory 求解有内存和无内存非线性方程的高效参数迭代方案

IF 1.8 2区数学

Journal of Complexity Pub Date : 2024-09-11 DOI: 10.1016/j.jco.2024.101896

Raziyeh Erfanifar, Masoud Hajarian

引用次数: 0

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":"86 ","pages":"Article 101887"},"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":"86 ","pages":"Article 101886"},"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":"85 ","pages":"Article 101884"},"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