Matrix recovery from matrix-vector products
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Can one recover a matrix efficiently from only matrix-vector products? If so, how many are needed? This paper describes algorithms to recover structured matrices, 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 × N unknown hierarchical low-rank matrix from only 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.
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