# Proximal optimal transport divergences

arXiv: https://arxiv.org/abs/2505.12097 · HTML: https://mlsanity.com/paper/2505.12097/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 62 references that are in the map.
- First posted on arXiv 17 May 2025.

## Builds on (most cited first, 28 of 28)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Formulation and properties of a divergence used to compare probability measures without absolute continuity](https://mlsanity.com/paper/1911.07422/) (2019)
- [A kernel formula for regularized Wasserstein proximal operators](https://mlsanity.com/paper/2301.10301/) (2023)
- [Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data](https://mlsanity.com/paper/2210.17230/) (2022)
- [Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions](https://mlsanity.com/paper/2405.13962/) (2024)
- [Combining Wasserstein-1 and Wasserstein-2 proximals: robust manifold learning via well-posed generative flows](https://mlsanity.com/paper/2407.11901/) (2024)

## Built on by (most cited first, 3 of 3)

- [A Unifying View of Variational Generative Wasserstein Flows](https://mlsanity.com/paper/2605.31369/) (2026)
- [A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling](https://mlsanity.com/paper/2606.12710/) (2026)
- [Graph-Informed Adversarial Modeling: Infimal Subadditivity of Interpolative Divergences](https://mlsanity.com/paper/2603.20025/) (2026)

## Abstract (arXiv)

We introduce the proximal optimal transport divergence, a novel discrepancy measure that interpolates between information divergences and optimal transport distances via an infimal convolution formulation. This divergence provides a principled foundation for optimal transport proximals and proximal optimization methods frequently used in generative modeling. We explore its mathematical properties, including smoothness, boundedness, and computational tractability, and establish connections to primal-dual formulations and adversarial learning. The proximal operator associated with the proximal optimal transport divergence can be interpreted as a transport map that pushes a reference distribution toward the optimal generative distribution, which approximates the target distribution that is only accessible through data samples. Building on the Benamou-Brenier dynamic formulation of classical optimal transport, we also establish a dynamic formulation for proximal OT divergences. The resulting dynamic formulation is a first order mean-field game whose optimality conditions are governed by a pair of nonlinear partial differential equations: a backward Hamilton-Jacobi equation and a forward continuity equation. Our framework generalizes existing approaches while offering new insights and computational tools for generative modeling, distributionally robust optimization, and gradient-based learning in probability spaces.