# Fast and Smooth Interpolation on Wasserstein Space

arXiv: https://arxiv.org/abs/2010.12101 · HTML: https://mlsanity.com/paper/2010.12101/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2020.

## Built on by (most cited first, 22 of 22)

- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (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)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformers](https://mlsanity.com/paper/2404.09411/) (2024)
- [Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior](https://mlsanity.com/paper/2406.07475/) (2024)
- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Multi-Marginal Stochastic Flow Matching for High-Dimensional Snapshot Data at Irregular Time Points](https://mlsanity.com/paper/2508.04351/) (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)
- [On the Latent Holes of VAEs for Text Generation](https://mlsanity.com/paper/2110.03318/) (2021)
- [Wassersplines for Neural Vector Field--Controlled Animation](https://mlsanity.com/paper/2201.11940/) (2022)
- [Squared Wasserstein-2 Distance for Efficient Reconstruction of Stochastic Differential Equations](https://mlsanity.com/paper/2401.11354/) (2024)
- [Generative modeling of time-dependent densities via optimal transport and projection pursuit](https://mlsanity.com/paper/2304.09663/) (2023)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [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)
- [Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport](https://mlsanity.com/paper/2606.21256/) (2026)
- [Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference](https://mlsanity.com/paper/2610.19878/) (2026)

## Abstract (arXiv)

We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines.