Randomized matrix-free quadrature: unified and uniform bounds for stochastic Lanczos quadrature and the kernel polynomial method

TL;DR

This paper provides a unified theoretical analysis and bounds for randomized matrix-free quadrature algorithms, specifically the kernel polynomial method and stochastic Lanczos quadrature, for spectrum and spectral sum approximation, supported by numerical examples.

Contribution

It unifies and simplifies existing analyses of these algorithms and derives spectral sum bounds with high-probability guarantees for all bounded analytic functions.

Findings

Unified bounds for spectrum and spectral sum approximation.

High-probability guarantees for spectral sum accuracy.

Numerical examples illustrating algorithm performance and differences.

Abstract

We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our analysis of spectrum approximation unifies and simplifies several one-off analyses for these algorithms which have appeared over the past decade. In addition, we derive bounds for spectral sum approximation which guarantee that, with high probability, the algorithms are simultaneously accurate on all bounded analytic functions. Finally, we provide comprehensive and complimentary numerical examples. These examples illustrate some of the qualitative similarities and differences between the algorithms, as well as relative drawbacks and benefits to their use on different types of problems.