Abstract

While there exists an emerging large sample theory for Fréchet means in a general "finite-dimensional" setting, there also exist a few notable results in particular "infinite-dimensional" settings of interest related to optimal transport and functional data analysis. Presently, we introduce a geometric condition for the general infinite-dimensional setting under which one can prove that Fréchet means satisfy the desired limit theorems with the provably minimal moment assumptions. We also show that the collection spaces satisfying this condition is closed under many operations (products, quotients, and more) which allows one to develop the first ever rigorous statistical guarantees theory for Fréchet means used in pattern theory.