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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 matrices withknown structures, such as tridiagonal, Toeplitz, Toeplitz-like, andhierarchical low-rank, from matrix-vector products. In particular, we derive arandomized algorithm for recovering an $N \times N$ unknown hierarchicallow-rank matrix from only $\mathcal{O}((k+p)\log(N))$ matrix-vector productswith high probability, where $k$ is the rank of the off-diagonal blocks, and$p$ is a small oversampling parameter. We do this by carefully constructingrandomized input vectors for our matrix-vector products that exploit thehierarchical structure of the matrix. While existing algorithms forhierarchical matrix recovery use a recursive "peeling" procedure based onelimination, our approach uses a recursive projection procedure.

Structured matrix recovery from matrix-vector products