# Accelerated Flow for Probability Distributions

arXiv: https://arxiv.org/abs/1901.03317 · HTML: https://mlsanity.com/paper/1901.03317/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2019.

## Built on by (most cited first, 15 of 15)

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- [Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flows](https://mlsanity.com/paper/2107.02951/) (2021)
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- [Non-geodesically-convex optimization in the Wasserstein space](https://mlsanity.com/paper/2406.00502/) (2024)
- [On the Generative Utility of Cyclic Conditionals](https://mlsanity.com/paper/2106.15962/) (2021)
- [Accelerated Markov Chain Monte Carlo Algorithms on Discrete States](https://mlsanity.com/paper/2505.12599/) (2025)
- [Towards understanding Accelerated Stein Variational Gradient Flow -- Analysis of Generalized Bilinear Kernels for Gaussian target distributions](https://mlsanity.com/paper/2509.04008/) (2025)
- [Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points](https://mlsanity.com/paper/2509.16974/) (2025)
- [Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization](https://mlsanity.com/paper/2601.19220/) (2026)
- [SympFormer: Accelerated attention blocks via Inertial Dynamics on Density Manifolds](https://mlsanity.com/paper/2603.16535/) (2026)
- [Accelerated Regularized Wasserstein Proximal Sampling Algorithms](https://mlsanity.com/paper/2601.09848/) (2026)

## Abstract (arXiv)

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (wibisono, et. al. 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton's equations for the optimal gradient flow. The Hamilton's equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton's equations as a system of N interacting particles. The continuous limit of the Nesterov's algorithm is shown to be a special case with N=1. The algorithm is illustrated with numerical examples.