Journal of Complexity最新文献

{"title":"Approximate equality for two sums of roots","authors":"Artūras Dubickas","doi":"10.1016/j.jco.2024.101866","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101866","url":null,"abstract":"<div><p>In this paper, we consider the problem of finding how close two sums of <em>m</em>th roots can be to each other. For integers <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></math></span>, let <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> be the largest exponents such that for infinitely many integers <em>N</em> there exist <em>k</em> positive integers <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>N</mi></math></span> for which two sums of their <em>m</em>th roots <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> and <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> are distinct but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other, or they are distinct modulo 1 but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other modulo 1. Some upper bounds on <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>≥</mo><mi>min</mi><mo>⁡</mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn><mi>s</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>m</mi></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>s</mi><mo><</mo><mi>k</mi></math></span> and that <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101866"},"PeriodicalIF":1.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140905544","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}

{"title":"On the convergence of gradient descent for robust functional linear regression","authors":"Cheng Wang ,&nbsp;Jun Fan","doi":"10.1016/j.jco.2024.101858","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101858","url":null,"abstract":"<div><p>Functional data analysis offers a set of statistical methods concerned with extracting insights from intrinsically infinite-dimensional data and has attracted considerable amount of attentions in the past few decades. In this paper, we study robust functional linear regression model with a scalar response and a functional predictor in the framework of reproducing kernel Hilbert spaces. A gradient descent algorithm with early stopping is introduced to solve the corresponding empirical risk minimization problem associated with robust loss functions. By appropriately selecting the early stopping rule and the scaling parameter of the robust losses, the convergence of the proposed algorithm is established when the response variable is bounded or satisfies a moment condition. Explicit learning rates with respect to both estimation and prediction error are provided in terms of regularity of the regression function and eigenvalue decay rate of the integral operator induced by the reproducing kernel and covariance function.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101858"},"PeriodicalIF":1.7,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140823062","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}

{"title":"David Krieg is the winner of the 2024 Joseph F. Traub Prize for Achievement in Information-Based Complexity","authors":"Erich Novak","doi":"10.1016/j.jco.2024.101857","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101857","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101857"},"PeriodicalIF":1.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000347/pdfft?md5=2ff01990fe6f5661ee24005dadc016f8&pid=1-s2.0-S0885064X24000347-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140633006","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}

{"title":"A unified treatment of tractability for approximation problems defined on Hilbert spaces","authors":"Onyekachi Emenike ,&nbsp;Fred J. Hickernell ,&nbsp;Peter Kritzer","doi":"10.1016/j.jco.2024.101856","DOIUrl":"10.1016/j.jco.2024.101856","url":null,"abstract":"<div><p>A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions <span><math><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span> grows at a moderate rate, i.e., the sequence of problems is <em>tractable</em>. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101856"},"PeriodicalIF":1.7,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799561","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}

{"title":"Enhancing the applicability of Chebyshev-like method","authors":"Santhosh George,&nbsp;Indra Bate,&nbsp;Muniyasamy M,&nbsp;Chandhini G,&nbsp;Kedarnath Senapati","doi":"10.1016/j.jco.2024.101854","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101854","url":null,"abstract":"<div><p>Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101854"},"PeriodicalIF":1.7,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619311","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}

{"title":"Improved bounds for the bracketing number of orthants or revisiting an algorithm of Thiémard to compute bounds for the star discrepancy","authors":"Michael Gnewuch","doi":"10.1016/j.jco.2024.101855","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101855","url":null,"abstract":"<div><p>We improve the best known upper bound for the bracketing number of <em>d</em>-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the <em>d</em>-dimensional unit cube <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>). More precisely, we provide a better estimate for the cardinality of an algorithmic bracketing cover construction due to Eric Thiémard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from Thiémard (2001) <span>[22]</span>. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper estimate for the bracketing number of arbitrary axis-parallel boxes 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>. In our upper bounds all constants are fully explicit.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101855"},"PeriodicalIF":1.7,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000323/pdfft?md5=23c928b5ffffc6732ad1f4739311a07b&pid=1-s2.0-S0885064X24000323-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619310","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}

{"title":"On regularized polynomial functional regression","authors":"Markus Holzleitner ,&nbsp;Sergei V. Pereverzyev","doi":"10.1016/j.jco.2024.101853","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101853","url":null,"abstract":"<div><p>This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101853"},"PeriodicalIF":1.7,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140309915","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}

{"title":"Linear Monte Carlo quadrature with optimal confidence intervals","authors":"Robert J. Kunsch","doi":"10.1016/j.jco.2024.101851","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101851","url":null,"abstract":"<div><p>We study the numerical integration of functions from isotropic Sobolev spaces <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> at confidence level <span><math><mn>1</mn><mo>−</mo><mi>δ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a method called <em>stratified control variates</em> (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty <em>δ</em>. We also analyse a version of SCV in the low smoothness regime where <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> may contain functions with singularities. Here, we observe a polynomial dependence of the error on <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> in contrast to the logarithmic dependence in the high smoothness regime.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101851"},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000281/pdfft?md5=6b29dfcc17f60ddbf6ee19d289e21700&pid=1-s2.0-S0885064X24000281-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140180612","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}

{"title":"Heuristic approaches to obtain low-discrepancy point sets via subset selection","authors":"François Clément ,&nbsp;Carola Doerr ,&nbsp;Luís Paquete","doi":"10.1016/j.jco.2024.101852","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101852","url":null,"abstract":"<div><p>Building upon the exact methods presented in our earlier work (2022) <span>[5]</span>, we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes <span><math><mn>30</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>240</mn></math></span>, we obtain point sets in dimension 6 with <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> star discrepancy up to 35% better than that of the first <em>n</em> points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) <span>[31]</span>, showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101852"},"PeriodicalIF":1.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000293/pdfft?md5=026f36f25d20579c91a0fc64a95356e5&pid=1-s2.0-S0885064X24000293-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140190621","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}

{"title":"Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients","authors":"Chenxu Pang ,&nbsp;Xiaojie Wang ,&nbsp;Yue Wu","doi":"10.1016/j.jco.2024.101842","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101842","url":null,"abstract":"<div><p>This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101842"},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000190/pdfft?md5=196a33f1ce0b753c885d6d05ad1d70a4&pid=1-s2.0-S0885064X24000190-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188091","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}