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Second order models for optimal transport and cubic splines on the Wasserstein space

arXiv 1801.04144 ↗First posted 12 Jan 2018math.OCcited by 19

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Abstract (arXiv)

On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multi-marginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). First posted on arXiv 12 Jan 2018.

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Citing papers per year: 2021: 2; 2022: 3; 2023: 3; 2024: 2; 2025: 5; 2026: 4.

19 papers.

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  1. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  2. 2023Deep Momentum Multi-Marginal Schr"odinger Bridgecited by 40
  3. 2021Towards a mathematical theory of trajectory inferencecited by 31
  4. 2022Riemannian Metric Learning via Optimal Transportcited by 21
  5. 2025Trajectory Inference with Smooth Schr"odinger Bridgescited by 10
  6. 2024Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averagingcited by 9
  7. 2024Parametric model reduction of mean-field and stochastic systems via higher-order action matchingcited by 8
  8. 2025OAT-FM: Optimal Acceleration Transport for Improved Flow Matchingcited by 7
  9. 2025Learning of Population Dynamics: Inverse Optimization Meets JKO Schemecited by 6
  10. 2025DICE: Discrete inverse continuity equation for learning population dynamicscited by 6
  11. 2022Wassersplines for Neural Vector Field--Controlled Animationcited by 2
  12. 2023Generative modeling of time-dependent densities via optimal transport and projection pursuitcited by 2
7 more
  1. 2022The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splinescited by 1
  2. 2026Two-Parameter Flows for Learning Population Dynamics of Physical Systemscited by 1
  3. 2021Areas on the space of smooth probability density functions on S2cited by 0
  4. 2025From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Propertiescited by 0
  5. 2026A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Spacecited by 0
  6. 2026TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamicscited by 0
  7. 2026VTV-FM: Flow Matching through Variational Terminal-Velocity Closurecited by 0