# Towards a mathematical theory of trajectory inference

arXiv: https://arxiv.org/abs/2102.09204 · HTML: https://mlsanity.com/paper/2102.09204/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 88 references that are in the map.
- First posted on arXiv 18 Feb 2021.

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

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Second order models for optimal transport and cubic splines on the Wasserstein space](https://mlsanity.com/paper/1801.04144/) (2018)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)
- [An entropy minimization approach to second-order variational mean-field games](https://mlsanity.com/paper/1807.09078/) (2018)
- [Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions](https://mlsanity.com/paper/1510.05526/) (2015)
- [Minimizing relative entropy of path measures under marginal constraints](https://mlsanity.com/paper/2001.10920/) (2020)
- [A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization](https://mlsanity.com/paper/2012.11706/) (2020)
- [Small noise limit and convexity for generalized incompressible flows, Schr"odinger problems, and optimal transport](https://mlsanity.com/paper/1810.12036/) (2018)
- [An optimal transport approach for solving dynamic inverse problems in spaces of measures](https://mlsanity.com/paper/1901.10162/) (2019)
- [Dynamic Cell Imaging in PET with Optimal Transport Regularization](https://mlsanity.com/paper/1902.07521/) (2019)
- [On the extremal points of the ball of the Benamou-Brenier energy](https://mlsanity.com/paper/1907.11589/) (2019)

## Built on by (most cited first, 30 of 31)

- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Inferring biological processes with intrinsic noise from cross-sectional data](https://mlsanity.com/paper/2410.07501/) (2024)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamics](https://mlsanity.com/paper/2204.04853/) (2022)
- [Stochastic force inference via density estimation](https://mlsanity.com/paper/2310.02366/) (2023)
- [A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots](https://mlsanity.com/paper/2605.08550/) (2026)
- [Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence](https://mlsanity.com/paper/2305.15557/) (2023)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Wassersplines for Neural Vector Field--Controlled Animation](https://mlsanity.com/paper/2201.11940/) (2022)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Manifold Learning with Sparse Regularised Optimal Transport](https://mlsanity.com/paper/2307.09816/) (2023)
- [Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels](https://mlsanity.com/paper/2402.08425/) (2024)
- [Distribution-Conditioned Transport](https://mlsanity.com/paper/2603.04736/) (2026)
- [Learning Explicit Single-Cell Dynamics Using ODE Representations](https://mlsanity.com/paper/2510.02903/) (2025)
- [PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inference](https://mlsanity.com/paper/2605.18587/) (2026)
- [Learning to Approximate Particle Smoothing Trajectories via Diffusion Generative Models](https://mlsanity.com/paper/2406.00561/) (2024)
- [Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport](https://mlsanity.com/paper/2603.01771/) (2026)
- [Continuum-marginal optimal transport: a mesh-free kernel method](https://mlsanity.com/paper/2604.24226/) (2026)
- [Learning biophysical models of gene regulation with probability flow matching](https://mlsanity.com/paper/2604.25062/) (2026)
- [Trajectory Inference of Human Aging from Cross-Sectional DNA Methylation Data](https://mlsanity.com/paper/2607.06583/) (2026)

## Abstract (arXiv)

We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from samples of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which provide high dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real datasets.