Interpolative Decomposition via Proxy Points for Kernel Matrices

Abstract

In the construction of rank-structured matrix representations of dense kernel matrices, a heuristic compression method, called the proxy point method, has been used in practice to efficiently compute the low-rank approximation of certain kernel matrix blocks in the form of an interpolative decomposition. We present a long overdue error analysis for the proxy point method, rigorously proving the effectiveness of the method under specific conditions. The analysis also generalizes the method, allowing it to be applied to the construction of different types of rank-structured matrices with general kernel functions in low-dimensional spaces. Based on the analysis, a systematic and adaptive scheme for selecting the proxy points used in the method is developed, which can guarantee that the method is effective for any given kernel function under specific conditions.