Near-optimal hierarchical matrix approximation from matrix-vector products

We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an n×n matrix A, accessible only though matrix-vector products with A and A^T. We prove that, for the rank-k HODLR approximation problem, our method achieves a (1 + β)^log(n⁾-optimal approximation in expected Frobenius norm using O(k log(n)/β³) matrix-vector products. In particular, the algorithm obtains a (1 + ε)-optimal approximation with O(k log⁴(n)/ε³) matrix-vector products, and for any constant c, an n^c-optimal approximation with O(k log(n)) matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just O(n poly(log(n), k, β)). We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least Ω(k log(n) + k/ε) queries to obtain a (1 + ε)-optimal approximation. Our algorithm can be viewed as a robust version of widely used “peeling” methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst-case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm.