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Trajectory Inference via Mean-field Langevin in Path Space

Lénaïc Chizat · Stephen Zhang · Matthieu Heitz · Geoffrey Schiebinger

Keywords: [ trajectory inference ] [ mean-field dynamics ] [ Wiener measure ] [ path space ] [ entropic regularization ] [ interacting particle methods ] [ Schrödinger Bridge ] [ optimal transport ] [ Langevin algorithm ]


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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