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arXiv · 1806.10626

Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices

Amit Levi·Yuichi Yoshida
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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.

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Amit Levi, Yuichi Yoshida. 2018-06-27. Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices. https://arxiv.org/abs/1806.10626

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