Recursive Karcher Expectation Estimators And Geometric Law of Large Numbers

In this paper we present a form of law of large numbers on Pn, the space of n× n symmetric positive-definite (SPD) matrices equipped with Fisher-Rao metric. Specifically, we propose a recursive algorithm for estimating the Karcher expectation of an arbitrary distribution defined on Pn, and we show that the estimates computed by the recursive algorithm asymptotically converge in probability to the correct Karcher expectation. The steps in the recursive algorithm mainly consist of making appropriate moves on geodesics in Pn, and the algorithm is simple to implement and it offers a tremendous gain in computation time of several orders of magnitude over existing non-recursive algorithms. We elucidate the connection between the more familiar law of large numbers for real-valued random variables and the asymptotic convergence of the proposed recursive algorithm, and our result provides an example of a new form of law of large numbers for SPD matrix-variate random variables. From a practical viewpoint, the computation of the mean of a collection of symmetric positive-definite (SPD) matrices is a fundamental ingredient in many algorithms in machine learning, computer vision and medical imaging applications. We report experiments using the proposed recursive algorithm for K-means clustering, demonstrating the algorithm’s efficiency and accuracy. Appearing in Proceedings of the 16 International Conference on Artificial Intelligence and Statistics (AISTATS) 2013, Scottsdale, AZ, USA. Volume 31 of JMLR: W&CP 31. Copyright 2013 by the authors.