# Projected Langevin dynamics and a gradient flow for entropic optimal transport

arXiv: https://arxiv.org/abs/2309.08598 · HTML: https://mlsanity.com/paper/2309.08598/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 61 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [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)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals](https://mlsanity.com/paper/2304.04451/) (2023)
- [A proof of the Caffarelli contraction theorem via entropic regularization](https://mlsanity.com/paper/1904.06053/) (2019)

## Built on by (most cited first, 17 of 17)

- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations](https://mlsanity.com/paper/2309.13332/) (2023)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Non-asymptotic error bounds for probability flow ODEs under weak log-concavity](https://mlsanity.com/paper/2510.17608/) (2025)
- [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Optimal Transportation by Orthogonal Coupling Dynamics](https://mlsanity.com/paper/2410.08060/) (2024)
- [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026)
- [Stability of the Kim--Milman flow map](https://mlsanity.com/paper/2511.01154/) (2025)
- [Collision-based Dynamics for Multi-Marginal Optimal Transport](https://mlsanity.com/paper/2412.16385/) (2024)
- [Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective](https://mlsanity.com/paper/2507.12246/) (2025)
- [Thermodynamic structure of the Sinkhorn flow](https://mlsanity.com/paper/2510.12639/) (2025)
- [Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees](https://mlsanity.com/paper/2606.16610/) (2026)
- [Near-Lipschitz stability of the Kim--Milman flow map](https://mlsanity.com/paper/2606.23383/) (2026)

## Abstract (arXiv)

The classical (overdamped) Langevin dynamics provide a natural algorithm for sampling from its invariant measure, which uniquely minimizes an energy functional over the space of probability measures, and which concentrates around the minimizer(s) of the associated potential when the noise parameter is small. We introduce analogous diffusion dynamics that sample from an entropy-regularized optimal transport, which uniquely minimizes the same energy functional but constrained to the set Pi(μ,nu) of couplings of two given marginal probability measures μ and nu on ℝd, and which concentrates around the optimal transport coupling(s) for small regularization parameter. More specifically, our process satisfies two key properties: First, the law of the solution at each time stays in Pi(μ,nu) if it is initialized there. Second, the long-time limit is the unique solution of an entropic optimal transport problem. In addition, we show by means of a new log-Sobolev-type inequality that the convergence holds exponentially fast, for sufficiently large regularization parameter and for a class of marginals which strictly includes all strongly log-concave measures. By studying the induced Wasserstein geometry of the submanifold Pi(μ,nu), we argue that the SDE can be viewed as a Wasserstein gradient flow on this space of couplings, at least when d=1, and we identify a conjectural gradient flow for d ≥ 2. The main technical difficulties stems from the appearance of conditional expectation terms which serve to constrain the dynamics to Pi(μ,nu).