We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized algorithm that returns an approximation with the given sparsity pattern with Frobenius-norm error at most $(1+\varepsilon)$ times the best possible error. When each row of the desired sparsity pattern has at most $s$ nonzero entries, this algorithm requires $O(s/\varepsilon)$ non-adaptive matrix-vector products with $\mathbf{A}$. We also prove a matching lower-bound, showing that, for any sparsity pattern with $\Theta(s)$ nonzeros per row and column, any algorithm achieving $(1+\epsilon)$ approximation requires $\Omega(s/\varepsilon)$ matrix-vector products in the worst case. We thus resolve the matrix-vector product query complexity of the problem up to constant factors, even for the well-studied case of diagonal approximation, for which no previous lower bounds were known.

翻译：我们研究在仅能通过矩阵-向量乘积访问矩阵 $\mathbf{A}$ 时，用固定稀疏模式（例如对角阵、带状阵等）的矩阵近似 $\mathbf{A}$ 的问题。本文描述了一种简单随机算法，该算法能以弗罗贝尼乌斯范数误差不超过最佳可能误差的 $(1+\varepsilon)$ 倍，返回具有给定稀疏模式的近似矩阵。当目标稀疏模式每行最多包含 $s$ 个非零元时，该算法仅需 $O(s/\varepsilon)$ 次针对 $\mathbf{A}$ 的非自适应矩阵-向量乘积。同时我们证明了匹配的下界：对于每行每列均含有 $\Theta(s)$ 个非零元的任意稀疏模式，任何实现 $(1+\epsilon)$ 近似度的算法在最坏情况下均需 $\Omega(s/\varepsilon)$ 次矩阵-向量乘积。因此，我们解决了该问题在矩阵-向量乘积查询复杂度上的常数因子最优性——即使对于此前无下界结论的对角近似这一经典情形亦如此。