Principal Curves In Metric Spaces And The Space Of Probability Measures

TL;DR

This paper introduces principal curves in Wasserstein and general metric spaces, enabling new methods for analyzing evolving populations in probability measure spaces, with applications in trajectory inference and high-density data collection.

Contribution

It develops a framework for principal curves in Wasserstein space, proves consistency of the estimator, and establishes new discretization results for principal curves in metric spaces.

Findings

Proposed a Wasserstein principal curve estimator with proven consistency.

Established validity of numerical discretization schemes for principal curves.

Applied framework to trajectory inference in cellular development studies.

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.