Structured matrix recovery from matrix-vector products

TL;DR

This paper introduces efficient algorithms for recovering structured matrices, such as hierarchical low-rank matrices, solely from matrix-vector products, significantly reducing the number of products needed through a novel recursive projection approach.

Contribution

It presents a new randomized recursive projection algorithm for hierarchical low-rank matrix recovery from matrix-vector products, improving efficiency over traditional peeling methods.

Findings

Hierarchical low-rank matrices can be recovered with O((k+p) log N) matrix-vector products.

The proposed method exploits matrix structure through carefully designed randomized input vectors.

The recursive projection approach outperforms existing recursive peeling procedures.

Abstract

Can one recover a matrix efficiently from only matrix-vector products? If so, how many are needed? This paper describes algorithms to recover matrices with known structures, such as tridiagonal, Toeplitz, Toeplitz-like, and hierarchical low-rank, from matrix-vector products. In particular, we derive a randomized algorithm for recovering an $N \times N$ unknown hierarchical low-rank matrix from only $\mathcal{O}((k+p)\log(N))$ matrix-vector products with high probability, where $k$ is the rank of the off-diagonal blocks, and $p$ is a small oversampling parameter. We do this by carefully constructing randomized input vectors for our matrix-vector products that exploit the hierarchical structure of the matrix. While existing algorithms for hierarchical matrix recovery use a recursive "peeling" procedure based on elimination, our approach uses a recursive projection procedure.