Generalized Alternating Projections on Manifolds and Convex Sets

TL;DR

This paper extends convergence analysis of the generalized alternating projections method from subspaces to smooth manifolds and convex sets, showing similar local behavior and convergence rates.

Contribution

It generalizes previous results to smooth manifolds and convex sets, demonstrating local convergence behavior and finite identification properties.

Findings

Convergence rates are similar for manifolds and subspaces.

Finite identification property leads to asymptotic convergence.

Examples illustrate when the identification property holds or fails.

Abstract

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.