2025/05/07 by Andrew Warren, Warren, Andrew, Anton Afanassiev +6
Computer Science · Mathematics · #49Q20 #62R20 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Primary: 62G05 #Secondary: 62P10 #Statistics Theory (math.ST) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2505.04168
openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.