Improved Sparse Recovery for Approximate Matrix Multiplication
Yahel Uffenheimer, Omri Weinstein
TL;DR
This work introduces a randomized approximate matrix multiplication algorithm with error tied to the output norm $\|AB\|_F$ that runs in $O(n^2(r+\log n))$ time. The core idea is a new matrix-space preconditioning via a Fast Hadamard Transform with asymmetric diagonal scaling, enabling uniform energy distribution across entries and allowing accurate recovery from $rn$ sampled positions. It provides both a biased estimator with $\mathbb{E}[C]=(r/n)AB$ and an unbiased estimator achieving $\mathbb{E}[\|C-AB\|_F^2]=(n/r)\|AB\|_F^2$, matching state-of-the-art results while offering a log-factor speedup. The approach avoids input compression and yields favorable per-entry variance properties, with potential amplification to exact results. This introduces a novel direction in AMM that leverages matrix-space rotations instead of column/row sketching.
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.