# Non-geodesically-convex optimization in the Wasserstein space

arXiv: https://arxiv.org/abs/2406.00502 · HTML: https://mlsanity.com/paper/2406.00502/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 78 references that are in the map.
- First posted on arXiv 1 Jun 2024.

## Builds on (most cited first, 30 of 31)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [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)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient](https://mlsanity.com/paper/1803.06573/) (2018)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [First-order Conditions for Optimization in the Wasserstein Space](https://mlsanity.com/paper/2209.12197/) (2022)
- [Accelerated Flow for Probability Distributions](https://mlsanity.com/paper/1901.03317/) (2019)
- [Computing B-Stationary Points of Nonsmooth DC Programs](https://mlsanity.com/paper/1511.01796/) (2015)
- [Composite Difference-Max Programs for Modern Statistical Estimation Problems](https://mlsanity.com/paper/1803.00205/) (2018)
- [DC approximation approaches for sparse optimization](https://mlsanity.com/paper/1407.0286/) (2014)
- [A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems](https://mlsanity.com/paper/1810.13117/) (2018)
- [On the Pervasiveness of Difference-Convexity in Optimization and Statistics](https://mlsanity.com/paper/1704.03535/) (2017)
- [Low-rank matrix recovery with Ky Fan 2-k-norm](https://mlsanity.com/paper/1904.05590/) (2019)
- [A Boosted-DCA with Power-Sum-DC Decomposition for Linearly Constrained Polynomial Programs](https://mlsanity.com/paper/2205.02571/) (2022)

## Built on by (most cited first, 5 of 5)

- [Linear convergence of proximal descent schemes on the Wasserstein space](https://mlsanity.com/paper/2411.15067/) (2024)
- [Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold](https://mlsanity.com/paper/2410.02490/) (2024)
- [Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity](https://mlsanity.com/paper/2501.14993/) (2025)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is nonconvex along generalized geodesics. Specifically, the objective exhibits some difference-of-convex structure along these geodesics. The setting also encompasses sampling problems where the logarithm of the target distribution is difference-of-convex. We derive multiple convergence insights for a novel semi Forward-Backward Euler scheme under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.