An analysis on stochastic Lanczos quadrature with asymmetric quadrature nodes

TL;DR

This paper improves the error analysis and parameter optimization of the Stochastic Lanczos Quadrature method for matrix trace approximation, emphasizing asymmetric quadrature rules and efficient resource allocation.

Contribution

It introduces an optimized error reallocation technique and clarifies theoretical discrepancies in asymmetric Lanczos quadrature analysis.

Findings

Reallocation of error budget reduces total matrix-vector multiplications.

Larger Lanczos subspace size (m) is more beneficial than increasing Monte Carlo samples (N).

Numerical experiments confirm tighter bounds and practical parameter guidelines.

Abstract

This paper revisits the error analysis of the Stochastic Lanczos Quadrature (SLQ) method for approximating the trace of matrix functions, with a specific focus on asymmetric Lanczos quadrature rules. We reexplain an existing theoretical discrepancy regarding the necessity of a scaling factor when applying an affine transformation from the reference interval to the physical spectral interval. Furthermore, we introduce an optimized error reallocation technique for log-determinant estimation. Rather than evenly splitting the error tolerance between the Hutchinson trace estimator and the Lanczos quadrature, we formulate an optimization problem to strategically distribute the error budget. This approach minimizes the total number of matrix-vector multiplications (MVMs) required to reach a target accuracy for both Rademacher and Gaussian queries. Numerical experiments validate that this reallocation yields tighter theoretical bounds and provides a concrete rule-of-thumb for parameter configuration: to achieve a target accuracy efficiently, more computational resources should be allocated to the Lanczos process (larger m) rather than Monte Carlo sampling (smaller N).