Matrix Nearness Problems with Off-Block-Diagonal Rank Constraints

This paper explores the problem of finding a rank-structured approximation to a given Hermitian positive definite matrix. We first study a fundamental matrix nearness problem which seeks an approximation whose off-diagonal block must have low rank. Our investigation focuses on three objective functions: the spectral norm, Frobenius divergence, and log-determinant divergence. We derive closed-form solutions for the fundamental nearness problem, extending our analysis to include problems with a range restrictions on the off-diagonal block. Building upon these results, we propose a practical, scalable algorithm to compute a positive definite hierarchical off-diagonal low-rank approximation to a Hermitian positive definite matrix. Through numerical experiments, we explore the use of our approximation as a general-purpose preconditioner for the conjugate gradient method. Our results demonstrate that the log-determinant divergence-based approximation exhibits good performance across a diverse set of test problems.