# Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization

arXiv: https://arxiv.org/abs/2606.27767 · HTML: https://mlsanity.com/paper/2606.27767/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 57 of its 91 references that are in the map.
- First posted on arXiv 26 Jun 2026.

## Builds on (most cited first, 30 of 57)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Refining Deep Generative Models via Discriminator Gradient Flow](https://mlsanity.com/paper/2012.00780/) (2020)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Policy Optimization as Wasserstein Gradient Flows](https://mlsanity.com/paper/1808.03030/) (2018)
- [Deep Generative Learning via Variational Gradient Flow](https://mlsanity.com/paper/1901.08469/) (2019)
- [Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds](https://mlsanity.com/paper/2403.06560/) (2024)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Generative Drifting is Secretly Score Matching: a Spectral and Variational Perspective](https://mlsanity.com/paper/2603.09936/) (2026)
- [First-order Conditions for Optimization in the Wasserstein Space](https://mlsanity.com/paper/2209.12197/) (2022)
- [Gradient Flow Drifting: Generative Modeling via Wasserstein Gradient Flows of KDE-Approximated Divergences](https://mlsanity.com/paper/2603.10592/) (2026)
- [Stochastic Optimization for DC Functions and Non-smooth Non-convex Regularizers with Non-asymptotic Convergence](https://mlsanity.com/paper/1811.11829/) (2018)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Wasserstein Policy Optimization](https://mlsanity.com/paper/2505.00663/) (2025)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies](https://mlsanity.com/paper/2603.01977/) (2026)
- [On the rate of convergence of the Difference-of-Convex Algorithm (DCA)](https://mlsanity.com/paper/2109.13566/) (2021)

## Abstract (arXiv)

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.