Weak limits of consecutive projections and of greedy steps

TL;DR

This paper studies the behavior of weak limit points in sequences generated by random projections and greedy algorithms in Hilbert spaces, providing new proofs and highlighting open questions about convergence.

Contribution

It offers a new proof of weak convergence for projections onto subspaces and explores properties of weak limits in more general settings.

Findings

Weak convergence holds for projections onto subspaces.

Properties of weak limit points are characterized for convex sets.

Open questions remain for general convex sets.

Abstract

Let $H$ be a Hilbert space. We investigate the properties of weak limit points of iterates of random projections onto $K\geq 2$ closed convex sets in $H$ and the parallel properties of weak limit points of residuals of random greedy approximation with respect to $K$ dictionaries. In the case of convex sets these properties imply weak convergence in all the cases known so far. In particular, we give a short proof of the theorem of Amemiya and Ando on weak convergence when the convex sets are subspaces. The question of the weak convergence in general remains open.