Fast Approximate Determinants Using Rational Functions

TL;DR

This paper introduces a fast, rational function-based method for approximating determinants of large matrices, demonstrating superior accuracy and speed over existing stochastic methods when combined with effective preconditioning.

Contribution

It develops a novel approach using rational function approximations for determinants, improving accuracy and efficiency over prior stochastic algorithms.

Findings

Third order rational approximation outperforms stochastic Lanczos quadrature in accuracy.

Method achieves a good speed-accuracy trade-off with proper preconditioning.

Effective for matrices from Gaussian process kernels like Matérn and RBF.

Abstract

We show how rational function approximations to the logarithm, such as $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that when combined with a good preconditioner, the third order rational function approximation offers a very good trade-off between speed and accuracy when measured on matrices coming from Mat\'ern-$5/2$ and radial basis function Gaussian process kernels. In particular, it is significantly more accurate on those matrices than the state-of-the-art stochastic Lanczos quadrature method for approximating determinants while running at about the same speed.