Generalized Fr\'echet means with random minimizing domains and its strong consistency
Abstract
This paper introduces a novel extension of Fr\'echet means, called generalized Fr\'echet means as a comprehensive framework for characterizing features in probability distributions in general topological spaces. The generalized Fr\'echet means are defined as minimizers of a suitably defined cost function. The framework encompasses various extensions of Fr\'echet 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 Fr\'echet mean framework to diverse statistical scenarios, including dimension reduction for manifold-valued data.
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