Trajectory Inference via Mean-field Langevin in Path Space
This provides an interpretable and theoretically sound method for inferring dynamics from temporal snapshots, particularly useful in computational biology for analyzing single-cell data, though it builds incrementally on prior work.
The paper tackles the problem of trajectory inference from snapshot data by introducing a grid-free algorithm based on mean-field Langevin dynamics, which globally converges to a min-entropy estimator with theoretical guarantees. It also extends the method to handle mass variations, making it applicable to single-cell RNA-sequencing data where cells branch and die.
Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced by Lavenant et al. arXiv:2102.09204, and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.