Deflation as a Method of Variance Reduction for Estimating the Trace of a Matrix Inverse

TL;DR

This paper investigates how deflation affects variance reduction in trace estimation of matrix inverses, revealing that deflation can increase variance for Hermitian matrices but not for non-Hermitian ones, and demonstrates significant variance reduction in Lattice QCD applications.

Contribution

The paper provides a theoretical analysis of deflation's impact on variance in trace estimation and combines deflation with Hierarchical Probing for large-scale Lattice QCD computations.

Findings

Deflation may increase variance for Hermitian matrices but not for non-Hermitian matrices.

Combining deflation with Hierarchical Probing reduces variance by over 60 times in Lattice QCD.

Precomputing smallest singular values enables efficient large-scale eigenvalue computations.

Abstract

Many fields require computing the trace of the inverse of a large, sparse matrix. The typical method used for such computations is the Hutchinson method which is a Monte Carlo (MC) averaging over matrix quadratures. To improve its convergence, several variance reductions techniques have been proposed. In this paper, we study the effects of deflating the near null singular value space. We make two main contributions. First, we analyze the variance of the Hutchinson method as a function of the deflated singular values and vectors. Although this provides good intuition in general, by assuming additionally that the singular vectors are random unitary matrices, we arrive at concise formulas for the deflated variance that include only the variance and mean of the singular values. We make the remarkable observation that deflation may increase variance for Hermitian matrices but not for non-Hermitian ones. This is a rare, if not unique, property where non-Hermitian matrices outperform Hermitian ones. The theory can be used as a model for predicting the benefits of deflation. Second, we use deflation in the context of a large scale application of "disconnected diagrams" in Lattice QCD. On lattices, Hierarchical Probing (HP) has previously provided an order of magnitude of variance reduction over MC by removing "error" from neighboring nodes of increasing distance in the lattice. Although deflation used directly on MC yields a limited improvement of 30% in our problem, when combined with HP they reduce variance by a factor of over 60 compared to MC. For this, we pre-computated 1000 smallest singular values of an ill-conditioned matrix of size 25 million. Using PRIMME and a domain-specific Algebraic Multigrid preconditioner, we perform one of the largest eigenvalue computations in Lattice QCD at a fraction of the cost of our trace computation.