mlsanityData to 7 Oct 2026

Towards a mathematical theory of trajectory inference

arXiv 2102.09204 ↗First posted 18 Feb 2021stat.ML cs.LG math.OCcited by 31

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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.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Builds on 19 of its 88 references that are in the map. First posted on arXiv 18 Feb 2021.

Builds on

19 of 88 references.

References · 19

  1. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  2. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  3. 2015Learning with a Wasserstein Losscited by 194
  4. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 155
  5. 2015Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measurescited by 138
  6. 2020Kernel Operations on the GPU, with Autodiff, without Memory Overflowscited by 101
  7. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  8. 2015A Smoothed Dual Approach for Variational Wasserstein Problemscited by 29
  9. 2020Fast and Smooth Interpolation on Wasserstein Spacecited by 21
  10. 2018Second order models for optimal transport and cubic splines on the Wasserstein spacecited by 19
  11. 2018Measure-valued spline curves: an optimal transport viewpointcited by 17
  12. 2018An entropy minimization approach to second-order variational mean-field gamescited by 9
7 more
  1. 2015Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusionscited by 5
  2. 2020Minimizing relative entropy of path measures under marginal constraintscited by 4
  3. 2020A generalized conditional gradient method for dynamic inverse problems with optimal transport regularizationcited by 4
  4. 2018Small noise limit and convexity for generalized incompressible flows, Schr"odinger problems, and optimal transportcited by 1
  5. 2019An optimal transport approach for solving dynamic inverse problems in spaces of measurescited by 1
  6. 2019Dynamic Cell Imaging in PET with Optimal Transport Regularizationcited by 1
  7. 2019On the extremal points of the ball of the Benamou-Brenier energycited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 5; 2023: 5; 2024: 9; 2025: 4; 2026: 8.

31 papers.

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  1. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 94
  2. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  3. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  4. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  5. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  6. 2022Trajectory Inference via Mean-field Langevin in Path Spacecited by 29
  7. 2024Multi-marginal Schr"odinger Bridges with Iterative Reference Refinementcited by 26
  8. 2025Curly Flow Matching for Learning Non-gradient Field Dynamicscited by 17
  9. 2024Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshotscited by 10
  10. 2024Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fittingcited by 9
  11. 2024Inferring biological processes with intrinsic noise from cross-sectional datacited by 9
  12. 2024Plug-in estimation of Schr"odinger bridgescited by 8
19 more
  1. 2025Hierarchical Refinement: Optimal Transport to Infinity and Beyondcited by 8
  2. 2022Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamicscited by 7
  3. 2023Stochastic force inference via density estimationcited by 5
  4. 2023Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergencecited by 4
  5. 2024Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processescited by 4
  6. 2025Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-freecited by 3
  7. 2022Wassersplines for Neural Vector Field--Controlled Animationcited by 2
  8. 2023Manifold Learning with Sparse Regularised Optimal Transportcited by 2
  9. 2024Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernelscited by 2
  10. 2025Learning Explicit Single-Cell Dynamics Using ODE Representationscited by 1
  11. 2026Distribution-Conditioned Transportcited by 1
  12. 2026A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshotscited by 1
  13. 2024Learning to Approximate Particle Smoothing Trajectories via Diffusion Generative Modelscited by 0
  14. 2026Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transportcited by 0
  15. 2026Continuum-marginal optimal transport: a mesh-free kernel methodcited by 0
  16. 2026Learning biophysical models of gene regulation with probability flow matchingcited by 0
  17. 2026PACE: Geometry-Aware Bridge Transport for Single-Cell Trajectory Inferencecited by 0
  18. 2026Trajectory Inference of Human Aging from Cross-Sectional DNA Methylation Datacited by 0
  19. 2026Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshotscited by 0