# Understanding MCMC Dynamics as Flows on the Wasserstein Space

arXiv: https://arxiv.org/abs/1902.00282 · HTML: https://mlsanity.com/paper/1902.00282/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 14 of 14)

- [Improving Task-free Continual Learning by Distributionally Robust Memory Evolution](https://mlsanity.com/paper/2207.07256/) (2022)
- [Discrete Langevin Sampler via Wasserstein Gradient Flow](https://mlsanity.com/paper/2206.14897/) (2022)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Coin Sampling: Gradient-Based Bayesian Inference without Learning Rates](https://mlsanity.com/paper/2301.11294/) (2023)
- [DPVI: A Dynamic-Weight Particle-Based Variational Inference Framework](https://mlsanity.com/paper/2112.00945/) (2021)
- [Relative Entropy Gradient Sampler for Unnormalized Distributions](https://mlsanity.com/paper/2110.02787/) (2021)
- [On the Generative Utility of Cyclic Conditionals](https://mlsanity.com/paper/2106.15962/) (2021)
- [A stochastic Stein Variational Newton method](https://mlsanity.com/paper/2204.09039/) (2022)
- [De-randomizing MCMC dynamics with the diffusion Stein operator](https://mlsanity.com/paper/2110.03768/) (2021)
- [Generative Modeling with Flow-Guided Density Ratio Learning](https://mlsanity.com/paper/2303.03714/) (2023)
- [Particle Dynamics for Learning EBMs](https://mlsanity.com/paper/2111.13772/) (2021)
- [Boosting Statistic Learning with Synthetic Data from Pretrained Large Models](https://mlsanity.com/paper/2505.04992/) (2025)

## Abstract (arXiv)

It is known that the Langevin dynamics used in MCMC is the gradient flow of the KL divergence on the Wasserstein space, which helps convergence analysis and inspires recent particle-based variational inference methods (ParVIs). But no more MCMC dynamics is understood in this way. In this work, by developing novel concepts, we propose a theoretical framework that recognizes a general MCMC dynamics as the fiber-gradient Hamiltonian flow on the Wasserstein space of a fiber-Riemannian Poisson manifold. The "conservation + convergence" structure of the flow gives a clear picture on the behavior of general MCMC dynamics. The framework also enables ParVI simulation of MCMC dynamics, which enriches the ParVI family with more efficient dynamics, and also adapts ParVI advantages to MCMCs. We develop two ParVI methods for a particular MCMC dynamics and demonstrate the benefits in experiments.