Local Linear Convergence of Approximate Projections onto Regularized Sets

TL;DR

This paper analyzes the local linear convergence of approximate projection algorithms onto regularized sets, providing a theoretical foundation and stopping criteria, with applications to phase retrieval in imaging.

Contribution

It introduces a regularization strategy to ensure regularity of set intersections and proves linear convergence of approximate projections, applicable to inverse problems.

Findings

Regularization improves intersection regularity and convergence.

Approximate projections achieve linear convergence under regularization.

The method is validated on phase retrieval with experimental data.

Abstract

The numerical properties of algorithms for finding the intersection of sets depend to some extent on the regularity of the sets, but even more importantly on the regularity of the intersection. The alternating projection algorithm of von Neumann has been shown to converge locally at a linear rate dependent on the regularity modulus of the intersection. In many applications, however, the sets in question come from inexact measurements that are matched to idealized models. It is unlikely that any such problems in applications will enjoy metrically regular intersection, let alone set intersection. We explore a regularization strategy that generates an intersection with the desired regularity properties. The regularization, however, can lead to a significant increase in computational complexity. In a further refinement, we investigate and prove linear convergence of an approximate alternating projection algorithm. The analysis provides a regularization strategy that fits naturally with many ill-posed inverse problems, and a mathematically sound stopping criterion for extrapolated, approximate algorithms. The theory is demonstrated on the phase retrieval problem with experimental data. The conventional early termination applied in practice to unregularized, consistent problems in diffraction imaging can be justified fully in the framework of this analysis providing, for the first time, proof of convergence of alternating approximate projections for finite dimensional, consistent phase retrieval problems.