Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication

TL;DR

This paper introduces algorithms for approximating matrix multiplication in spectral norm using low-rank sketches, with bounds depending on the matrices' intrinsic dimensionality, and develops a new matrix-valued Chernoff bound.

Contribution

The paper presents novel algorithms for spectral norm approximation of matrix products using low-rank sampling and introduces a new low rank matrix-valued Chernoff bound.

Findings

Bounds depend only on the matrices' rank and stable rank.

Two sampling procedures effectively approximate the product.

New matrix-valued Chernoff inequality enables stable rank-based bounds.

Abstract

In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two down-sampled sketches, \tilde{A}\in\RR^{t\times m} and \tilde{B}\in\RR^{t\times p}, where t\ll n such that \norm{\tilde{A}^\top \tilde{B} - A^\top B} \leq \eps \norm{A}\norm{B} with high probability. We use two different sampling procedures for constructing \tilde{A} and \tilde{B}; one of them is done by i.i.d. non-uniform sampling rows from A and B and the other is done by taking random linear combinations of their rows. We prove bounds that depend only on the intrinsic dimensionality of A and B, that is their rank and their stable rank; namely the squared ratio between their Frobenius and operator norm. For achieving bounds that depend on rank we employ standard tools from high-dimensional geometry such as concentration of measure arguments combined with elaborate \eps-net constructions. For bounds that depend on the smaller parameter of stable rank this technology itself seems weak. However, we show that in combination with a simple truncation argument is amenable to provide such bounds. To handle similar bounds for row sampling, we develop a novel matrix-valued Chernoff bound inequality which we call low rank matrix-valued Chernoff bound. Thanks to this inequality, we are able to give bounds that depend only on the stable rank of the input matrices...