Retractions by Alternating Projections

TL;DR

This paper develops a unified framework for analyzing alternating projection methods on intersections of smooth manifolds, introducing retraction concepts and connecting them to Newton schemes for constrained optimization.

Contribution

It introduces a general theory of retractions induced by alternating projections on manifold intersections, extending local convergence analysis and linking to second-order optimization methods.

Findings

The associated alternating mapping admits a well-defined local limiting map on the intersection.

Under smoothness conditions, this map is a retraction, and a second-order retraction if higher smoothness is assumed.

The NewtonSLRA scheme can be interpreted as inducing a second-order retraction, enabling new optimization tools.

Abstract

Alternating projections and their variants are classical tools for computing points in intersections of sets. Existing analyses for smooth manifolds mainly focus on local convergence rates under transversality or related regularity conditions. In this work, we develop a unified framework for a broad class of (possibly inexact) alternating-projection-type methods on intersections of smooth manifolds. Specifically, under the assumption that two $C^{2,1}$ embedded submanifolds $\mathcal{M}_1, \mathcal{M}_2 \subset \mathbb{R}^n$ intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map $\psi$ on the intersection manifold $\mathcal{M}=\mathcal{M}_1\cap \mathcal{M}_2$, and that $\psi$ is a retraction on $\mathcal{M}$. If, in addition, $\mathcal{M}_1$ and $\mathcal{M}_2$ are $C^{3,1}$, then $\psi$ is a second-order retraction. Furthermore, the standard NewtonSLRA scheme, which exhibits quadratic local behavior under transversality, can be understood as inducing a second-order retraction on \(\M\). This framework thus provides new retraction-based optimization tools for problems constrained to the intersection manifold.