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Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems

Nian Shao, Wenbin Chen

TL;DR

This paper proposes the Riemannian Acceleration with Preconditioning (RAP) for symmetric eigenvalue problems, and proves its acceleration, and analyzes the convergence of LORAG.

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 $1-Cκ^{-1/2}$, which is an improvement over the preconditioned steepest descent method's rate of $1-Cκ^{-1}$. The exponent in $κ^{-1/2}$ is sharp, and numerical experiments confirm our theoretical findings.

Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems

TL;DR

This paper proposes the Riemannian Acceleration with Preconditioning (RAP) for symmetric eigenvalue problems, and proves its acceleration, and analyzes the convergence of LORAG.

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 , which is an improvement over the preconditioned steepest descent method's rate of . The exponent in is sharp, and numerical experiments confirm our theoretical findings.
Paper Structure (29 sections, 33 theorems, 227 equations, 2 figures, 1 table)

This paper contains 29 sections, 33 theorems, 227 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Geodesic convexity of preconditioned eigensystems
  3. Incorporate preconditioning on Riemannian manifolds
  4. Analysis on $M$-sphere: without preconditioning
  5. Challenge in analysis on $B$-sphere
  6. Leading angle
  7. Analysis on $B$-sphere: with preconditioning
  8. Convex Riemannian optimization
  9. Riemannian accelerated gradient descent method
  10. Locally optimal Riemannian accelerated gradient method
  11. Riemannian acceleration with preconditioning
  12. Second order symmetric elliptic eigenvalue problems with Schwarz preconditioners
  13. Theory of Schwarz and domain decomposition methods
  14. Analysis for leading angle
  15. Analysis for convergence rate
  16. ...and 14 more sections

Key Result

Lemma 1

Consider the following preconditioned gradient flow: The gradient flow $x(t)$ from conflow lies on a $B$-sphere for all $t>0$, i.e., $\lVert x(t)\rVert_{B}=\lVert x_{0}\rVert_{B}$.

Figures (2)

  • Figure 1: Leading angle $\vartheta$. The figure is under $A$ inner-product.
  • Figure 2: Construction of an overlapping domain decomposition.

Theorems & Definitions (71)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • Lemma 5
  • proof
  • Definition 6: Leading Angle
  • ...and 61 more