Computing the log-determinant of symmetric, diagonally dominant matrices in near-linear time

Abstract

We present new algorithms for computing the log-determinant of symmetric, diagonally dominant matrices. Existing algorithms run with cubic complexity with respect to the size of the matrix in the worst case. Our algorithm computes an approximation of the log-determinant in time near-linear with respect to the number of non-zero entries and with high probability. This algorithm builds upon the utra-sparsifiers introduced by Spielman and Teng for Laplacian matrices and ultimately uses their refined versions introduced by Koutis, Miller and Peng in the context of solving linear systems. We also present simpler algorithms that compute upper and lower bounds and that may be of more immediate practical interest.