# On the Optimisation Landscape of Overparameterised Weighted MMD

arXiv: https://arxiv.org/abs/2610.19861 · HTML: https://mlsanity.com/paper/2610.19861/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 49 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 16 of 16)

- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Optimal Thinning of MCMC Output](https://mlsanity.com/paper/2005.03952/) (2020)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)
- [Optimal quantisation of probability measures using maximum mean discrepancy](https://mlsanity.com/paper/2010.07064/) (2020)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024)
- [Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancy](https://mlsanity.com/paper/2101.07564/) (2021)
- [Superresolution without Separation](https://mlsanity.com/paper/1506.03144/) (2015)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)
- [Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels](https://mlsanity.com/paper/2608.01182/) (2026)
- [New Constraint Qualifications for Optimization Problems in Banach Spaces based on Asymptotic KKT Conditions](https://mlsanity.com/paper/1912.06531/) (2019)
- [On the Optimization Landscape of Maximum Mean Discrepancy](https://mlsanity.com/paper/2110.13452/) (2021)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)

## Abstract (arXiv)

We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.