Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems

TL;DR

This paper introduces a Riemannian preconditioning framework and a new accelerated eigensolver method that significantly improves convergence rates for symmetric eigenvalue problems, supported by theoretical analysis and numerical experiments.

Contribution

It develops a novel Riemannian preconditioning approach and an accelerated gradient method for symmetric eigenvalue problems, with proven convergence improvements.

Findings

RAP achieves a convergence rate of 1 - Cκ^{-1/2}

The exponent in κ^{-1/2} is sharp

Numerical experiments confirm theoretical results

Abstract

The analysis of the acceleration behavior of gradient-based eigensolvers with preconditioning presents a substantial theoretical challenge. In this work, we present a novel framework for preconditioning on Riemannian manifolds and introduce a metric, the leading angle, to evaluate preconditioners for symmetric eigenvalue problems. We extend the locally optimal Riemannian accelerated gradient method for Riemannian convex optimization to develop the Riemannian Acceleration with Preconditioning (RAP) method for symmetric eigenvalue problems, thereby providing theoretical evidence to support its acceleration. Our analysis of the Schwarz preconditioner for elliptic eigenvalue problems demonstrates that RAP achieves a convergence rate of $1-C\kappa^{-1/2}$, which is an improvement over the preconditioned steepest descent method's rate of $1-C\kappa^{-1}$. The exponent in $\kappa^{-1/2}$ is sharp, and numerical experiments confirm our theoretical findings.