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 proceed to prove a matching lower-bound. Specifically, we show that for any $s\geq 1$, there are matrices $\mathbf{A}$ such that, for any sparsity pattern with $\Theta(s)$ nonzeros per row and column, any algorithm which obtains a $(1+\varepsilon)$ accurate approximation of the given sparsity from matrix-vector products requires at least $\Omega(s/\varepsilon)$ matrix-vector products. Our bounds therefore 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：我们研究在仅通过矩阵-向量乘积访问矩阵$\mathbf{A}$时，使用具有固定稀疏模式（例如对角矩阵、带状矩阵等）的矩阵对其进行近似的问题。我们描述了一种简单的随机化算法，该算法返回具有给定稀疏模式的近似解，其Frobenius范数误差不超过最佳可能误差的$(1+\varepsilon)$倍。当所需稀疏模式的每行最多包含$s$个非零元素时，该算法需要$O(s/\varepsilon)$次非自适应矩阵-向量乘积与$\mathbf{A}$交互。随后，我们证明了一个匹配的下界。具体而言，我们表明对于任意$s\geq 1$，存在矩阵$\mathbf{A}$，使得对于任何每行每列具有$\Theta(s)$个非零元素的稀疏模式，任何通过矩阵-向量乘积获得给定稀疏模式的$(1+\varepsilon)$精度近似解的算法至少需要$\Omega(s/\varepsilon)$次矩阵-向量乘积。因此，我们的界限解决了该问题在矩阵-向量乘积查询复杂度上的常数因子层次，即使对于先前无下界已知的对角近似这一经典情形也是如此。