Suboptimal subspace construction for log-determinant approximation

TL;DR

This paper introduces a probabilistic approach to construct suboptimal subspaces for approximating the log-determinant of large matrices, combining variance reduction and stochastic Lanczos quadrature with theoretical guarantees.

Contribution

It proposes a novel method using probabilistic techniques for subspace construction and provides theoretical error bounds for log-determinant approximation.

Findings

Method effectively approximates log-determinants of large matrices.

Derived error bounds are validated through numerical experiments.

The approach offers insights into parameter selection for improved accuracy.

Abstract

Variance reduction is a crucial idea for Monte Carlo simulation and the stochastic Lanczos quadrature method is a dedicated method to approximate the trace of a matrix function. Inspired by their advantages, we combine these two techniques to approximate the log-determinant of large-scale symmetric positive definite matrices. Key questions to be answered for such a method are how to construct or choose an appropriate projection subspace and derive guaranteed theoretical analysis. This paper applies some probabilistic approaches including the projection-cost-preserving sketch and matrix concentration inequalities to construct a suboptimal subspace. Furthermore, we provide some insights on choosing design parameters in the underlying algorithm by deriving corresponding approximation error and probabilistic error estimations. Numerical experiments demonstrate our method's effectiveness and illustrate the quality of the derived error bounds.