Principal Curves In Metric Spaces And The Space Of Probability Measures

Abstract: We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem). We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting.