  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.
