# Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions

arXiv: https://arxiv.org/abs/2408.07498 · HTML: https://mlsanity.com/paper/2408.07498/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 54 references that are in the map.
- First posted on arXiv 14 Aug 2024.

## Builds on (most cited first, 9 of 9)

- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)
- [Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line](https://mlsanity.com/paper/2301.04441/) (2023)
- [Efficient displacement convex optimization with particle gradient descent](https://mlsanity.com/paper/2302.04753/) (2023)
- [Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals](https://mlsanity.com/paper/2104.06121/) (2021)

## Built on by (most cited first, 7 of 7)

- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows](https://mlsanity.com/paper/2504.07820/) (2025)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Wasserstein gradient flows for Coulomb discrepancies](https://mlsanity.com/paper/2607.12579/) (2026)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Well-Posed KL-Regularized Control via Wasserstein and Kalman-Wasserstein KL Divergences](https://mlsanity.com/paper/2602.02250/) (2026)

## Abstract (arXiv)

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals mathcal F_nu := MMDK2(·, nu) towards given target measures nu on the real line, where we focus on the negative distance kernel K(x,y) := -|x-y|. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone mathcal C(0,1) subset L2(0,1) of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on L2(0,1). Based on the construction of an appropriate counterpart of mathcal F_nu on L2(0,1) and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures nu, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of mathcal C(0,1). For certain mathcal F_nu-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme, which is easily computable by a bisection algorithm. For continuous targets nu, also the explicit Euler scheme can be employed, although with limited convergence guarantees.