Trajectory Inference via Mean-field Langevin in Path SpaceDownload PDF

Published: 31 Oct 2022, Last Modified: 22 Oct 2023NeurIPS 2022 AcceptReaders: Everyone
Keywords: trajectory inference, path space, mean-field dynamics, interacting particle methods, Langevin algorithm, Wiener measure, entropic regularization, optimal transport, Schrödinger bridge
TL;DR: The estimator for trajectory inference that minimizes the entropy relative to Wiener measure can be computed with a Langevin dynamics in path space (convergence guaranteed).
Abstract: 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 in [Lavenant et al., 2021], 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.
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