Generalized Fréchet means with random minimizing domains and its strong consistency

Summary This paper introduces a novel extension of Fréchet means, referred to as generalized Fréchet means, as a comprehensive framework for describing the characteristics of random elements. The generalized Fréchet mean is defined as the minimizer of a cost function, and the framework encompasses various extensions of Fréchet means that have appeared in the literature. The most distinctive feature of the proposed framework is that it allows the domain of minimization for the empirical generalized Fréchet means to be random and different from that of its population counterpart. This flexibility broadens the applicability of the Fréchet mean framework to various statistical scenarios, including sequential dimension reduction for non-Euclidean data. We establish a strong consistency theorem for generalized Fréchet means and demonstrate the utility of the proposed framework by verifying the consistency of principal geodesic analysis on the hypersphere.