Improved Sparse Recovery for Approximate Matrix Multiplication

TL;DR

This paper introduces a fast randomized algorithm for approximate matrix multiplication that achieves error bounds proportional to the output norm, improving efficiency over previous methods by utilizing a novel pseudo-random rotation technique.

Contribution

The paper presents a new, faster randomized algorithm for AMM with error bounds related to the output norm, using a novel pseudo-random rotation method.

Findings

Algorithm runs in $O(n^2(r+ ext{log} n))$ time.

Achieves error bounds comparable to state-of-the-art methods.

Provides both biased and unbiased estimators with controlled error.

Abstract

We present a simple randomized algorithm for approximate matrix multiplication (AMM) whose error scales with the *output* norm $\|AB\|_F$. Given any $n\times n$ matrices $A,B$ and a runtime parameter $r\leq n$, the algorithm produces in $O(n^2(r+\log n))$ time, a matrix $C$ with total squared error $\mathbb{E}[\|C-AB\|_F^2]\le (1-\frac{r}{n})\|AB\|_F^2$, per-entry variance $\|AB\|_F^2/n^2$ and bias $\mathbb{E}[C]=\frac{r}{n}AB$. Alternatively, the algorithm can compute an *unbiased* estimation with expected total squared error $\frac{n}{r}\|{AB}\|_{F}^2$, recovering the state-of-art AMM error obtained by Pagh's TensorSketch algorithm (Pagh, 2013). Our algorithm is a log-factor faster. The key insight in the algorithm is a new variation of pseudo-random rotation of the input matrices (a Fast Hadamard Transform with asymmetric diagonal scaling), which redistributes the Frobenius norm of the *output* $AB$ uniformly across its entries.