# On the rate of convergence of iterated Bregman projections and of the alternating algorithm

## TL;DR

This paper analyzes the convergence rates of the alternating algorithm and iterated Bregman projections in Banach spaces, establishing conditions for linear convergence in uniformly convex and smooth spaces.

## Contribution

It introduces new conditions ensuring linear convergence of the alternating algorithm in certain Banach spaces and links Bregman and metric projections for improved analysis.

## Key findings

- Linear convergence of the alternating algorithm under specific conditions.
- Convergence results for iterated Bregman projections in Banach spaces.
- Connection between metric and Bregman projections enhances analysis.

## Abstract

We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated Bregman projection method.