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Appears in collection : CEMRACS: Transport in Physics, Biology and Urban Traffic / CEMRACS: Transport en physique, biologie et traffic urbain

We consider statistical and computation methods to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data. The goal of this mini-course is to present and understand the estimator proposed by [Lavenant et al. 2020] which searches for the diffusion process that fits the observations with minimal entropy relative to a Wiener process. This estimator comes with consistency guarantees—for a suitable class of ground truth processes—and lends itself to computational methods with global optimality guarantees. Its analysis is the occasion to review important tools from optimal transport and diffusion process theory.

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  • LÉONARD, Christian. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems-Series A, 2014, vol. 34, no 4, p. 1533-1574 - http://dx.doi.org/10.3934/dcds.2014.34.1533
  • LAVENANT, Hugo, ZHANG, Stephen, KIM, Young-Heon, et al. Towards a mathematical theory of trajectory inference. arXiv preprint arXiv:2102.09204, 2021 - https://doi.org/10.48550/arXiv.2102.09204
  • ZHANG, Stephen, CHIZAT, Lénaïc, HEITZ, Matthieu, et al. Trajectory Inference via Mean-field Langevin in Path Space. arXiv preprint arXiv:2205.07146, 2022. - https://doi.org/10.48550/arXiv.2205.07146

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