Quasi-optimal hierarchically semi-separable matrix approximation

Abstract: We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\times N$ matrix $A$ using only matrix-vector products with $A$ and $A<sup>T$. We prove that, using $O(k \log(N/k))$ matrix-vector products and ${O}(N k² \log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm error $\mathbb{E}[|A - B|_F<sup>2]$ is at most $O(\log(N/k))$ times worse than the best possible approximation error by an HSS rank-$k$ matrix. In fact, the algorithm we analyze in a simple modification of an empirically effective method proposed by [Levitt & Martinsson, SISC 2024]. As a stepping stone towards our main result, we prove two results that are of independent interest: a similar guarantee for a variant of the algorithm which accesses $A$'s entries directly, and explicit error bounds for near-optimal subspace approximation using projection-cost-preserving sketches. To the best of our knowledge, our analysis constitutes the first polynomial-time quasi-optimality result for HSS matrix approximation, both in the explicit access model and the matrix-vector product query model.