{"title":"Admissible subspaces and the subspace iteration method","authors":"","doi":"10.1007/s10543-024-01012-1","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix <em>A</em> by matrices of rank <em>h</em>. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of <em>A</em>. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the <em>h</em>th singular value of <em>A</em> lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the <em>h</em>th singular value of <em>A</em> belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the <em>h</em>th and the <span> <span>\\((h+1)\\)</span> </span>st singular values of <em>A</em> coincide, which does not seem to be covered by previous works in the deterministic setting.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10543-024-01012-1","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix A by matrices of rank h. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of A. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the hth singular value of A lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the hth singular value of A belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the hth and the \((h+1)\)st singular values of A coincide, which does not seem to be covered by previous works in the deterministic setting.
摘要 在这项工作中，我们重新审视了子空间迭代法（SIM）的收敛性分析，该方法用于计算秩为 h 的矩阵 A 的近似值。通常情况下，这些低秩近似值的收敛性分析是通过首先估计 SIM 产生的子空间与 A 的主要子空间之间的（角）距离获得的。我们注意到，如果 A 的第 h 个奇异值位于奇异值群中，这种方法会导致高估近似误差的上限。为了克服这一困难，我们引入了一种主导子空间的替代方法，我们称之为可容许子空间。我们对由 SIM 产生的子空间与可容许子空间进行了接近性分析；反过来，这种分析又使我们能够获得由确定性 SIM 实现的低阶矩阵近似误差的新估计值。我们的结果适用于 A 的第 h 个奇异值属于一个奇异值簇的情况。事实上，我们的方法允许我们考虑 A 的第 h 次奇异值和第（(h+1)\）次奇异值重合的情况，而这似乎并不是之前确定性设置中的工作所涵盖的。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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