# Trajectory Inference via Mean-field Langevin in Path Space

arXiv: https://arxiv.org/abs/2205.07146 · HTML: https://mlsanity.com/paper/2205.07146/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 45 references that are in the map.
- First posted on arXiv 14 May 2022.

## Builds on (most cited first, 19 of 19)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Analysis of a Two-Layer Neural Network via Displacement Convexity](https://mlsanity.com/paper/1901.01375/) (2019)
- [Super-efficiency of automatic differentiation for functions defined as a minimum](https://mlsanity.com/paper/2002.03722/) (2020)
- [Overrelaxed Sinkhorn-Knopp Algorithm for Regularized Optimal Transport](https://mlsanity.com/paper/1711.01851/) (2017)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion](https://mlsanity.com/paper/2111.01666/) (2021)
- [An entropy minimization approach to second-order variational mean-field games](https://mlsanity.com/paper/1807.09078/) (2018)
- [Analysis of k-Nearest Neighbor Distances with Application to Entropy Estimation](https://mlsanity.com/paper/1603.08578/) (2016)

## Built on by (most cited first, 30 of 30)

- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior](https://mlsanity.com/paper/2406.07475/) (2024)
- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Inferring biological processes with intrinsic noise from cross-sectional data](https://mlsanity.com/paper/2410.07501/) (2024)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport](https://mlsanity.com/paper/2601.06810/) (2026)
- [Stochastic force inference via density estimation](https://mlsanity.com/paper/2310.02366/) (2023)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Inferring stochastic dynamics with growth from cross-sectional data](https://mlsanity.com/paper/2505.13197/) (2025)
- [Gradient-flow SDEs have unique transient population dynamics](https://mlsanity.com/paper/2505.21770/) (2025)
- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference](https://mlsanity.com/paper/2604.20775/) (2026)
- [Thinned Mean Field Langevin Dynamics](https://mlsanity.com/paper/2605.28589/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [The Central Spanning Tree Problem](https://mlsanity.com/paper/2404.06447/) (2024)
- [Machine-Learned Sampling of Conditioned Path Measures](https://mlsanity.com/paper/2506.01904/) (2025)
- [Private Continuous-Time Synthetic Trajectory Generation via Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2506.12203/) (2025)
- [Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr"odinger Bridge Estimation](https://mlsanity.com/paper/2510.22560/) (2025)
- [Learning biophysical models of gene regulation with probability flow matching](https://mlsanity.com/paper/2604.25062/) (2026)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (2026)
- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

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 Schr"odinger 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.