eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-13
17:1
17:19
10.4230/LIPIcs.APPROX-RANDOM.2018.17
article
Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices
Levi, Amit
1
https://orcid.org/0000-0002-8530-5182
Yoshida, Yuichi
2
https://orcid.org/0000-0001-8919-8479
University of Waterloo, Canada
National Institute of Informatics, Tokyo, Japan
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol116-approx-random2018/LIPIcs.APPROX-RANDOM.2018.17/LIPIcs.APPROX-RANDOM.2018.17.pdf
Qudratic function minimization
Approximation Algorithms
Matrix spectral decomposition
Graph limits