# Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows

arXiv: https://arxiv.org/abs/2504.07820 · HTML: https://mlsanity.com/paper/2504.07820/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 59 references that are in the map.
- First posted on arXiv 10 Apr 2025.

## Builds on (most cited first, 18 of 18)

- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (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)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)
- [Stein Variational Gradient Descent: many-particle and long-time asymptotics](https://mlsanity.com/paper/2102.12956/) (2021)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Strictly proper kernel scores and characteristic kernels on compact spaces](https://mlsanity.com/paper/1712.05279/) (2017)
- [Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms](https://mlsanity.com/paper/2401.08260/) (2024)
- [Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions](https://mlsanity.com/paper/2408.07498/) (2024)
- [Fast Summation of Radial Kernels via QMC Slicing](https://mlsanity.com/paper/2410.01316/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Maximum Mean Discrepancy Distributionally Robust Nonlinear Chance-Constrained Optimization with Finite-Sample Guarantee](https://mlsanity.com/paper/2204.11564/) (2022)
- [Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization](https://mlsanity.com/paper/1910.06124/) (2019)
- [Minimizers for an aggregation model with attractive-repulsive interaction](https://mlsanity.com/paper/2307.13769/) (2023)
- [Enhanced Digital Halftoning via Weighted Sigma-Delta Modulation](https://mlsanity.com/paper/2202.04986/) (2022)
- [Generative Feature Training of Thin 2-Layer Networks](https://mlsanity.com/paper/2411.06848/) (2024)

## Built on by (most cited first, 2 of 2)

- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

Negative distance kernels K(x,y) := - |x-y| were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. In particular, so-called slicing techniques for handling high-dimensional kernel summations profit from the simple parameter-free structure of the distance kernel. However, due to its non-smoothness in x=y, most of the classical theoretical results, e.g. on Wasserstein gradient flows of the corresponding MMD functional do not longer hold true. In this paper, we propose a new kernel which keeps the favorable properties of the negative distance kernel as being conditionally positive definite of order one with a nearly linear increase towards infinity and a simple slicing structure, but is Lipschitz differentiable now. Our construction is based on a simple 1D smoothing procedure of the absolute value function followed by a Riemann-Liouville fractional integral transform. Numerical results demonstrate that the new kernel performs similarly well as the negative distance kernel in gradient descent methods, but now with theoretical guarantees.