From low-rank retractions to dynamical low-rank approximation and back

TL;DR

This paper explores the use of retractions in numerical integration on manifold-constrained optimization problems, introducing new methods and a novel retraction, with implications for dynamical low-rank approximation techniques.

Contribution

It introduces a new retraction called KLS retraction, and two novel numerical schemes, AFE and PRH, for differential equations on manifolds, connecting retractions with DLRA.

Findings

KLS retraction derived from unconventional DLRA integrator

AFE and PRH methods achieve third-order local truncation error

Numerical experiments demonstrate advantages of new methods

Abstract

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston--Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.