Generalized Fréchet means with random minimizing domains and its strong consistency
Jaesung Park, Sungkyu Jung
TL;DR
The paper extends Fréchet means to allow random minimizing domains, framing generalized Fréchet means as $E_0=\\argmin_{m\\in M_0} F(m)$ and $\\hat E_n=\\epsilon_n-\\argmin_{m\\in M_n} F_n(m)$, and proves strong consistency (BP and Ziezold) under Kuratowski convergence of $M_n$ to $M_0$ and standard regularity. It unifies numerous Fréchet-mean generalizations, provides practical sufficient conditions for consistency via cost-function classes such as $\\mathfrak c=G\\{\\rho\\}$ with equicontinuous $\\rho$ and subadditive $G$, and demonstrates the approach with Principal Geodesic Analysis on the hypersphere, establishing convergence of the empirical PGA sequence. The results advance the applicability of Fréchet-mean methods to random-domain optimization and manifold-valued data, enabling reliable dimension-reduction and feature characterization in non-Euclidean spaces. The framework thus offers a principled, versatile foundation for M-estimation in general topological spaces with data-dependent minimization domains.
Abstract
This paper introduces a novel extension of Fréchet means, called \textit{generalized Fréchet means} as a comprehensive framework for characterizing features in probability distributions in general topological spaces. The generalized Fréchet means are defined as minimizers of a suitably defined cost function. The framework encompasses various extensions of Fréchet means in the literature. The most distinctive difference of the new framework from the previous works is that we allow the domain of minimization of the empirical means be random and different from that of the population means. This expands the applicability of the Fréchet mean framework to diverse statistical scenarios, including dimension reduction for manifold-valued data.