Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging

Capturing data from dynamic processes through cross-sectional measurements is seen in many fields, such as computational biology. Trajectory inference deals with the challenge of reconstructing continuous processes from such observations. In this work, we propose methods for B-spline approximation and interpolation of point clouds through consecutive averaging that is intrinsic to the Wasserstein space. Combining subdivision schemes with optimal transport-based geodesic, our methods carry out trajectory inference at a chosen level of precision and smoothness, and can automatically handle scenarios where particles undergo division over time. We prove linear convergence rates and rigorously evaluate our method on cell data characterized by bifurcations, merges, and trajectory splitting scenarios like \emph{supercells}, comparing its performance against state-of-the-art trajectory inference and interpolation methods. The results not only underscore the effectiveness of our method in inferring trajectories but also highlight the benefit of performing interpolation and approximation that respect the inherent geometric properties of the data.