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

TL;DR

This paper introduces near-linear time algorithms for approximating the log-determinant of symmetric, diagonally dominant matrices, leveraging advanced sparsification techniques to improve computational efficiency.

Contribution

It presents novel algorithms that approximate the log-determinant in near-linear time, building upon ultra-sparsifiers and refined methods for Laplacian matrices.

Findings

Algorithms achieve near-linear time complexity

Provides bounds that are computationally simpler

Builds on advanced sparsification techniques

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.