Strong law of large numbers for the generalized Fréchet means with random minimizing domains

This paper introduces a novel extension of Fr\'{e}chet means, called \textit{generalized Fr\'{e}chet means} as a comprehensive framework for characterizing features in probability distributions in general topological spaces. The generalized Fr\'{e}chet means are defined as minimizers of a suitably defined cost function. The framework encompasses various extensions of Fr\'{e}chet 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\'{e}chet mean framework to diverse statistical scenarios, including dimension reduction for manifold-valued data.