Rates of Convergence for Chains of Expansive Markov Operators

TL;DR

This paper establishes conditions for the local convergence rates of iterated random functions in complex spaces, with applications to optimization, imaging, and phylogenetics.

Contribution

It introduces new convergence rate conditions for expansive Markov operators in Hadamard spaces, extending analysis beyond nonexpansive mappings.

Findings

Convergence rates are characterized for a broad class of Markov operators.

Applications demonstrate the practical relevance in optimization and imaging.

Results provide theoretical guarantees for stochastic algorithms in complex spaces.

Abstract

We provide conditions that guarantee local rates of convergence in distribution of iterated random functions that are not nonexpansive mappings in locally compact Hadamard spaces. Our results are applied to stochastic instances of common algorithms in optimization, stochastic tomography for X-FEL imaging, and a stochastic algorithm for the computation of Fr\'echet means in model spaces for phylogenetic trees.