Output-Sparse Matrix Multiplication Using Compressed Sensing

We give two algorithms for output-sparse matrix multiplication (OSMM), the problem of multiplying two $n \times n$ matrices $A, B$ when their product $AB$ is promised to have at most $O(n^{\delta})$ many non-zero entries for a given value $\delta \in [0, 2]$. We then show how to speed up these algorithms in the fully sparse setting, where the input matrices $A, B$ are themselves sparse. All of our algorithms work over arbitrary rings. Our first, deterministic algorithm for OSMM works via a two-pass reduction to compressed sensing. It runs in roughly $n^{\omega(\delta/2, 1, 1)}$ time, where $\omega(\cdot, \cdot, \cdot)$ is the rectangular matrix multiplication exponent. This substantially improves on prior deterministic algorithms for output-sparse matrix multiplication. Our second, randomized algorithm for OSMM works via a reduction to compressed sensing and a variant of matrix multiplication verification, and runs in roughly $n^{\omega(\delta - 1, 1, 1)}$ time. This algorithm and its extension to the fully sparse setting have running times that match those of the (randomized) algorithms for OSMM and FSMM, respectively, in recent work of Abboud, Bringmann, Fischer, and K\"{u}nnemann (SODA, 2024). Our algorithm uses different techniques and is arguably simpler. Finally, we observe that the running time of our randomized algorithm and the algorithm of Abboud et al. are optimal via a simple reduction from rectangular matrix multiplication.