Uniform large sample theory of generalized Fréchet means.

Abstract

Motivated by the complexity of modern data and the need to fully take advantage of their geometric structure, in this thesis, we study a flexible notion of centrality for data not necessarily lying on a linear space. In particular, we consider a broad generalization of the Fr\'echet mean. The classical Fr\'echet mean is obtained minimizing a quadratic notion of dispersion. Our generalization instead considers a more general definition of dispersion indexed by an increasing and convex function $\phi$. We start by showing the main properties of these means on linear spaces, and then we study their properties when defined on metric spaces. In this setting, we find necessary and sufficient conditions for $\phi$-means to be well-defined and provide a tight bound for the diameter of the $\phi$-mean set. We also provide sufficient conditions for all the $\phi$-means to coincide in a single point. In one of our main contributions, we prove the consistency of the sample $\phi$-mean of iid observations to its population analogue. Moreover, motivated by the need for robust estimation, we consider a setting in which multiple notions of dispersions are considered jointly. For this purpose, we find conditions under which classes of sample $\phi$-means converge uniformly.Secondly, we quantify more precisely the speed of convergence of the sample $\phi$ means, for this we need to restrict to Riemannian manifolds where the locally linear structure of these space allows us to prove a CLT. Our result is able to capture the slowing impact of the possible presence of probability mass on the cut locus of the population $\phi$-mean. This phenomenon is known in the literature as smeariness and we effectively isolate its impact on the asymptotic variance in our CLT. As in the case of Consistency, we deal with the problem of the joint behaviors of classes of $\phi$-means and provide a functional CLT in this case.We illustrate applications of our results in specific settings that can be of use in practical applications, like when the ambient space is linear, a Sphere, a projective space, or the Lie group of 3d rotations, $\SO_3$. We also briefly describe the algorithms that can be used when dealing with generalized Fr\'echet means.