# A Gradient Flow Perspective on Minimum MMD Estimation

arXiv: https://arxiv.org/abs/2607.03871 · HTML: https://mlsanity.com/paper/2607.03871/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 88 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Builds on (most cited first, 30 of 57)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Revisiting Natural Gradient for Deep Networks](https://mlsanity.com/paper/1301.3584/) (2013)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [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)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [On Graduated Optimization for Stochastic Non-Convex Problems](https://mlsanity.com/paper/1503.03712/) (2015)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence](https://mlsanity.com/paper/1912.05737/) (2019)
- [Particle-based Variational Inference with Preconditioned Functional Gradient Flow](https://mlsanity.com/paper/2211.13954/) (2022)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [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)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Wasserstein Policy Optimization](https://mlsanity.com/paper/2505.00663/) (2025)
- [Discrepancy-based Inference for Intractable Generative Models using Quasi-Monte Carlo](https://mlsanity.com/paper/2106.11561/) (2021)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)

## Built on by (most cited first, 1 of 1)

- [On the Optimisation Landscape of Overparameterised Weighted MMD](https://mlsanity.com/paper/2610.19861/) (2026)

## Abstract (arXiv)

Minimum maximum mean discrepancy (MMD) estimation has emerged as a robust and likelihood-free alternative to maximum likelihood estimation for parameter estimation. Yet, despite its practical success, the associated optimization problem remains poorly understood, with theoretical guarantees for existing algorithms hinging on convexity assumptions that rarely hold in practice. We address this gap by proposing a preconditioned gradient descent (PGD) scheme, establishing its asymptotic global convergence under explicit gradient-dominance and projection-residual conditions. Our approach is inspired by recent progress on MMD gradient flows, a nonparametric descent scheme on the space of probability measures. We provide extensive empirical evidence that our PGD scheme outperforms standard gradient descent across a range of challenging parameter estimation and composite hypothesis testing problems.