# A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel

arXiv: https://arxiv.org/abs/2111.10722 · HTML: https://mlsanity.com/paper/2111.10722/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 87 references that are in the map.
- First posted on arXiv 21 Nov 2021.

## Builds on (most cited first, 30 of 39)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium](https://mlsanity.com/paper/1706.08500/) (2017)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [A Survey on Generative Adversarial Networks: Variants, Applications, and Training](https://mlsanity.com/paper/2006.05132/) (2020)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [A Comprehensive guide to Bayesian Convolutional Neural Network with Variational Inference](https://mlsanity.com/paper/1901.02731/) (2019)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [A Unified Particle-Optimization Framework for Scalable Bayesian Sampling](https://mlsanity.com/paper/1805.11659/) (2018)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [Riemannian Stein Variational Gradient Descent for Bayesian Inference](https://mlsanity.com/paper/1711.11216/) (2017)

## Built on by (most cited first, 3 of 3)

- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Accelerating Particle-based Energetic Variational Inference](https://mlsanity.com/paper/2504.03158/) (2025)

## Abstract (arXiv)

We propose a novel deterministic sampling method, EVI-MMD, to approximate a target distribution ρ^* by minimizing the kernel discrepancy, also known as the Maximum Mean Discrepancy (MMD). Leveraging the energetic variational inference framework (Wang et al., 2021), we transform the MMD minimization problem into solving a dynamic system of Ordinary Differential Equations (ODEs) for particles. The implicit Euler scheme is employed to solve the ODE system, leading to a proximal minimization problem at each iteration, which is efficiently addressed using optimization algorithms such as L-BFGS. A key innovation of our method is a dynamic bandwidth selection strategy for the Gaussian kernel, which, although heuristic at this stage, represents a meaningful step toward addressing a long-standing challenge in kernel-based methods. Comprehensive numerical experiments demonstrate that this adaptive bandwidth significantly enhances the performance of EVI-MMD. We apply the EVI-MMD algorithm to two types of sampling problems: (1) when the target distribution is fully specified by a density function, and (2) the ``two-sample problem,'' where only training data are available. In the latter case, EVI-MMD serves as a generative model, producing new samples that faithfully replicate the distribution of the training data. With carefully tuned parameters, EVI-MMD outperforms several existing methods in both scenarios.