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.

翻译：能否仅通过矩阵-向量乘积高效地恢复一个矩阵？若能，需要多少次乘积？本文描述了从矩阵-向量乘积中恢复具有已知结构（如三对角、Toeplitz、类Toeplitz以及分层低秩）矩阵的算法。特别地，我们推导出一种随机化算法，仅需$\mathcal{O}((k+p)\log(N))$次矩阵-向量乘积即可高概率恢复未知$N \times N$分层低秩矩阵，其中$k$为分块矩阵非对角块的秩，$p$为小过采样参数。该方法通过精心构造利用矩阵分层结构的随机化输入向量来实现。与现有基于消去法的递归"剥离"式分层矩阵恢复算法不同，我们的方法采用递归投影过程。