Statistics on stratified spaces

Although manifold statistics is an established tool for computational anatomy, a number of structures are not well modeled on manifolds. Examples include data with variable topological structure, such as trees and graphs, as well as objects invariant with respect to groups that do not act freely on the space of measurements. Such data can often be represented more faithfully as residing on a stratified space, which consists of multiple manifold components, potentially with different dimensions, joined together in a controlled fashion. In this chapter we give a brief introduction to stratified spaces and geometric tools that are useful for performing statistics in them. We review existing least squares models in stratified spaces along with some unexpected behavior that they exhibit, illustrated in simple stratified spaces. Next, we review two particular examples of stratified spaces given by two different tree-spaces. The first is the Billera–Holmes–Vogtmann space of phylogenetic trees, for which a number of statistical algorithms have been proposed. The second is the space of unlabeled trees, which models more general attributed trees found in computational anatomy. This space has a more complicated geometry, and not much is known about its structure and statistics. We present a novel result connecting the two tree-spaces and their geodesics along with a consequential theorem on uniqueness of geodesics. Finally, we discuss other, less studied applications of stratified spaces as statistical domains, including spaces of graphs, point sets and sequences, and quotient spaces.