The computation of a matrix function $f(A)$ is an important task in scientific computing appearing in machine learning, network analysis and the solution of partial differential equations. In this work, we use only matrix-vector products $x\mapsto Ax$ to approximate functions of sparse matrices and matrices with similar structures such as sparse matrices $A$ themselves or matrices that have a similar decay property as matrix functions. We show that when $A$ is a sparse matrix with an unknown sparsity pattern, techniques from compressed sensing can be used under natural assumptions. Moreover, if $A$ is a banded matrix then certain deterministic matrix-vector products can efficiently recover the large entries of $f(A)$. We describe an algorithm for each of the two cases and give error analysis based on the decay bound for the entries of $f(A)$. We finish with numerical experiments showing the accuracy of our algorithms.

翻译：矩阵函数$f(A)$的计算是科学计算中的重要任务，广泛应用于机器学习、网络分析和偏微分方程求解等领域。本文仅利用矩阵-向量乘积$x\mapsto Ax$来逼近稀疏矩阵及其相似结构矩阵的函数，包括稀疏矩阵$A$本身或具有与矩阵函数相似衰减特性的矩阵。我们证明，当$A$为稀疏模式未知的稀疏矩阵时，可在自然假设下应用压缩感知技术；此外，若$A$为带状矩阵，则特定确定性矩阵-向量乘积能高效恢复$f(A)$的大幅值元素。针对上述两种情况分别给出算法，并基于$f(A)$元素的衰减界进行误差分析。最后通过数值实验验证算法的精度。