Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions

TL;DR

This paper introduces a fast, linear-time randomized algorithm for approximating log-determinants of large positive definite matrices, significantly reducing computation time in large-scale machine learning applications.

Contribution

The authors develop a novel stochastic Chebyshev expansion method combined with trace approximation, enabling efficient and accurate log-determinant computation for massive matrices.

Findings

Achieves high accuracy with orders of magnitude speedup over traditional methods

Can handle matrices with tens of millions of variables

Provides rigorous error bounds based on matrix condition number

Abstract

Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume ellipsoids, metric learning and kernel learning. Log-determinant computation involves the Cholesky decomposition at the cost cubic in the number of variables, i.e., the matrix dimension, which makes it prohibitive for large-scale applications. We propose a linear-time randomized algorithm to approximate log-determinants for very large-scale positive definite and general non-singular matrices using a stochastic trace approximation, called the Hutchinson method, coupled with Chebyshev polynomial expansions that both rely on efficient matrix-vector multiplications. We establish rigorous additive and multiplicative approximation error bounds depending on the condition number of the input matrix. In our experiments, the proposed algorithm can provide very high accuracy solutions at orders of magnitude faster time than the Cholesky decomposition and Schur completion, and enables us to compute log-determinants of matrices involving tens of millions of variables.