Kernel-Matrix Determinant Estimates from stopped Cholesky Decomposition

TL;DR

This paper introduces a probabilistic method to estimate the determinant of kernel matrices efficiently by stopping the Cholesky decomposition early, reducing computational cost with guaranteed accuracy.

Contribution

It proposes an augmentation of the Cholesky decomposition that allows early stopping to estimate determinants with probabilistic guarantees, saving computational resources.

Findings

Early stopping can save significant computation time.

Overhead of the method is less than 5% when not stopping early.

The approach applies to sum approximation with sequentially revealed addends.

Abstract

Algorithms involving Gaussian processes or determinantal point processes typically require computing the determinant of a kernel matrix. Frequently, the latter is computed from the Cholesky decomposition, an algorithm of cubic complexity in the size of the matrix. We show that, under mild assumptions, it is possible to estimate the determinant from only a sub-matrix, with probabilistic guarantee on the relative error. We present an augmentation of the Cholesky decomposition that stops under certain conditions before processing the whole matrix. Experiments demonstrate that this can save a considerable amount of time while having an overhead of less than $5\%$ when not stopping early. More generally, we present a probabilistic stopping strategy for the approximation of a sum of known length where addends are revealed sequentially. We do not assume independence between addends, only that they are bounded from below and decrease in conditional expectation.