Data analysis on nonstandard spaces

Abstract

The task to write on data analysis on nonstandard spaces is quite substantial, with a huge body of literature to cover, from parametric to nonparametrics, from shape spaces to Wasserstein spaces. In this survey we convey simple (e.g., Fréchet means) and more complicated ideas (e.g., empirical process theory), common to many approaches with focus on their interaction with one‐another. Indeed, this field is fast growing and it is imperative to develop a mathematical view point, drawing power, and diversity from a higher level of abstraction, for example, by introducing generalized Fréchet means. While many problems have found ingenious solutions (e.g., Procrustes analysis for principal component analysis [PCA] extensions on shape spaces and diffusion on the frame bundle to mimic anisotropic Gaussians), more problems emerge, often more difficult (e.g., topology and geometry influencing limiting rates and defining generic intrinsic PCA extensions). Along this survey, we point out some open problems, that will, as it seems, keep mathematicians, statisticians, computer and data scientists busy for a while.