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

arXiv: https://arxiv.org/abs/1801.04144 · HTML: https://mlsanity.com/paper/1801.04144/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2018.

## Built on by (most cited first, 20 of 20)

- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging](https://mlsanity.com/paper/2405.19679/) (2024)
- [Parametric model reduction of mean-field and stochastic systems via higher-order action matching](https://mlsanity.com/paper/2410.12000/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [DICE: Discrete inverse continuity equation for learning population dynamics](https://mlsanity.com/paper/2507.05107/) (2025)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)
- [Wassersplines for Neural Vector Field--Controlled Animation](https://mlsanity.com/paper/2201.11940/) (2022)
- [Generative modeling of time-dependent densities via optimal transport and projection pursuit](https://mlsanity.com/paper/2304.09663/) (2023)
- [The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines](https://mlsanity.com/paper/2209.09081/) (2022)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [Areas on the space of smooth probability density functions on S2](https://mlsanity.com/paper/2110.07773/) (2021)
- [From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties](https://mlsanity.com/paper/2507.11387/) (2025)
- [A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space](https://mlsanity.com/paper/2606.07325/) (2026)
- [VTV-FM: Flow Matching through Variational Terminal-Velocity Closure](https://mlsanity.com/paper/2610.00785/) (2026)
- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## 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.