Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices

TL;DR

This paper introduces a sublinear-time algorithm for quadratic minimization problems that improves error bounds by leveraging a novel spectral matrix decomposition, with applications to singular value approximation.

Contribution

It presents a new spectral decomposition method that separates matrices into structured and pseudorandom parts, enabling faster approximation algorithms.

Findings

Achieves better error bounds than previous algorithms.

Provides a sublinear-time algorithm for top singular value approximation.

Extends to quadratic minimization over spheres.

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.