Alternating projections, remotest projections, and greedy approximation

TL;DR

This paper explores the convergence behaviors of alternating and remotest projections in Hilbert spaces, establishing conditions for exponential or slow convergence, and linking these to greedy approximation methods.

Contribution

It introduces a unified analysis of projection methods and greedy approximation, revealing a dichotomy in convergence rates based on subspace configurations.

Findings

Exponential convergence when the sum of orthogonal complements equals the space.

Arbitrarily slow convergence can occur for certain initial points.

A polynomial decay rate is established for specific projection sequences.

Abstract

Let $L_1,L_2,\dots,L_K$ be a family of closed subspaces of a Hilbert space $H$, $L_1\cap \dots \cap L_K =\{0\}$; let $P_k$ be the orthogonal projection onto $L_k$. We consider two types of consecutive projections of an element $x_0\in H$: alternating projections $T^nx_0$, where $T=P_K\circ\dots\circ P_1$, and remotest projections $x_n$ defined recursively, $x_{n+1}$ being the remotest point for $x_n$ among $P_1x_n,\dots,P_Kx_n$. These $x_n$ can be interpreted as residuals in greedy approximation with respect to a special dictionary associated with $L_1,L_2,\dots,L_K$. We establish parallels between convergence properties separately known for alternating projections, remotest projections, and greedy approximation in $H$. Here are some results. If $L_1^\perp+\dots+L_K^\perp=H$, then $x_n\to 0$ exponentially fast. In case $L_1^\perp+\dots+L_K^\perp\not=H$, the convergence $x_n\to 0$ can be arbitrarily slow for certain $x_0$. Such a dichotomy, exponential rate of convergence everywhere on $H$, or arbitrarily slow convergence for certain starting elements, is valid for greedy approximation with respect to general dictionaries. The dichotomy was known for alternating projections. Using the methods developed for greedy approximation we prove that $|T^nx_0|\le C(x_0,K)n^{-\alpha(K)}$ for certain positive $\alpha(K)$ and all starting points $x_0\in L_1^\perp+\dots+L_K^\perp$.