文章基本信息

标题：Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices
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作者：Amit Levi ; Yuichi Yoshida
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2018
卷号：116
页码：1-19
DOI：10.4230/LIPIcs.APPROX-RANDOM.2018.17
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.
关键词：Qudratic function minimization; Approximation Algorithms; Matrix spectral decomposition; Graph limits