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Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices

Authors Amit Levi , Yuichi Yoshida

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Author Details

Amit Levi
  • University of Waterloo, Canada
Yuichi Yoshida
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Levi, Amit: Research supported by NSERC Discovery grant and the David R. Cheriton Graduate Scholarship. Part of this work was done while the author was visiting NII Tokyo.
  • Yoshida, Yuichi: Research supported by JSPS KAKENHI Grant Number JP17H04676 and JST ERATO Grant Number JPMJER1201.

Cite AsGet BibTex

Amit Levi and Yuichi Yoshida. Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 17:1-17:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.17

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Probabilistic computation
Keywords
  • Qudratic function minimization
  • Approximation Algorithms
  • Matrix spectral decomposition
  • Graph limits

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