Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging

2024-05-30

arXiv.org

AI Summary

This work addresses the challenge of reconstructing continuous trajectories from cross-sectional point-cloud observations across time in dynamic processes—such as cellular evolution—where topological changes (e.g., splitting and merging) occur. We propose a trajectory reconstruction method in Wasserstein space that explicitly supports such topological transitions. Our core innovation is the first integration of Wasserstein intrinsic continuous averaging with B-splines, yielding an optimal-transport-based geodesic subdivision interpolation framework. This framework ensures geometric awareness, adaptive smoothness, and controllable reconstruction accuracy. We establish theoretical convergence guarantees for the method. Empirical evaluation on simulated single-cell data featuring branching and merging events demonstrates that our approach significantly outperforms state-of-the-art methods, while rigorously preserving the intrinsic geometry of the data and consistency with Wasserstein transport distances.

Technology Category

• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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 instrinsic 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 rigorously evaluate our method by providing convergence guarantees and testing it on simulated cell data characterized by bifurcations and merges, 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.

Problem

• Reconstruct continuous processes from cross-sectional measurements.
• Handle particle division scenarios in trajectory inference.
• Ensure interpolation respects geometric properties of data.

Innovation

• B-spline approximation in Wasserstein space
• Consecutive averaging for trajectory inference
• Optimal transport-based geodesic for smooth interpolation