A Majorization-Minimization Algorithm for Computing the Karcher Mean of Positive Definite Matrices

TL;DR

This paper introduces a parameter-free majorization-minimization algorithm for efficiently computing the Karcher mean of positive definite matrices, with proven asymptotic linear convergence.

Contribution

It presents a novel MM algorithm that is parameter-free and guarantees linear convergence for calculating the Karcher mean.

Findings

Algorithm converges asymptotically linearly

No need for step size tuning

Effective for large sets of positive definite matrices

Abstract

An algorithm for computing the Karcher mean of $n$ positive definite matrices is proposed, based on the majorization-minimization (MM) principle. The proposed MM algorithm is parameter-free, does not need to choose step sizes, and has a theoretical guarantee of asymptotic linear convergence.