# Stationary MMD Points

arXiv: https://arxiv.org/abs/2505.20754 · HTML: https://mlsanity.com/paper/2505.20754/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 95 references that are in the map.
- First posted on arXiv 27 May 2025.

## Builds on (most cited first, 30 of 44)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Practical Coreset Constructions for Machine Learning](https://mlsanity.com/paper/1703.06476/) (2017)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [M3D: Dataset Condensation by Minimizing Maximum Mean Discrepancy](https://mlsanity.com/paper/2312.15927/) (2023)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [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)
- [On Graduated Optimization for Stochastic Non-Convex Problems](https://mlsanity.com/paper/1503.03712/) (2015)
- [Generalized Kernel Thinning](https://mlsanity.com/paper/2110.01593/) (2021)
- [On gradient regularizers for MMD GANs](https://mlsanity.com/paper/1805.11565/) (2018)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
- [Monte Carlo with Determinantal Point Processes](https://mlsanity.com/paper/1605.00361/) (2016)
- [A Finite-Particle Convergence Rate for Stein Variational Gradient Descent](https://mlsanity.com/paper/2211.09721/) (2022)
- [Distribution Compression in Near-linear Time](https://mlsanity.com/paper/2111.07941/) (2021)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Optimal quantisation of probability measures using maximum mean discrepancy](https://mlsanity.com/paper/2010.07064/) (2020)
- [Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent](https://mlsanity.com/paper/2409.08469/) (2024)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)

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

- [Thinned Mean Field Langevin Dynamics](https://mlsanity.com/paper/2605.28589/) (2026)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)

## Abstract (arXiv)

Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective typically precludes global minimisation. Instead, we consider the concept of stationary points of the MMD which, in contrast to points globally minimising the MMD, can be accurately computed. Our main contributions are two-fold and theoretical in nature. We first prove the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the numerical integration error of stationary MMD points vanishes faster than the MMD. Motivated by this super-convergence property, we consider MMD gradient flows as a practical strategy for computing stationary points of the MMD. We then prove that MMD gradient flow can indeed compute stationary MMD points, based on a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound.