# Generative Sliced MMD Flows with Riesz Kernels

arXiv: https://arxiv.org/abs/2305.11463 · HTML: https://mlsanity.com/paper/2305.11463/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 90 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 30 of 52)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [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)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Sub-sampled Cubic Regularization for Non-convex Optimization](https://mlsanity.com/paper/1705.05933/) (2017)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)

## Built on by (most cited first, 30 of 37)

- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [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)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Fast Kernel Summation in High Dimensions via Slicing and Fourier Transforms](https://mlsanity.com/paper/2401.08260/) (2024)
- [Kernel Semi-Implicit Variational Inference](https://mlsanity.com/paper/2405.18997/) (2024)
- [Importance Corrected Neural JKO Sampling](https://mlsanity.com/paper/2407.20444/) (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)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Learning Probabilistic Filters with Strictly Proper Scoring Rules](https://mlsanity.com/paper/2606.26497/) (2026)
- [Paired Wasserstein Autoencoders for Conditional Sampling](https://mlsanity.com/paper/2412.07586/) (2024)
- [Ultra-fast feature learning for the training of two-layer neural networks in the two-timescale regime](https://mlsanity.com/paper/2504.18208/) (2025)
- [DDEQs: Distributional Deep Equilibrium Models through Wasserstein Gradient Flows](https://mlsanity.com/paper/2503.01140/) (2025)
- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Smoothed Distance Kernels for MMDs and Applications in Wasserstein Gradient Flows](https://mlsanity.com/paper/2504.07820/) (2025)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [Sliced ReLU attention: Quasi-linear contextual expressivity via sorting](https://mlsanity.com/paper/2512.11411/) (2025)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [A Kernel Approach for Semi-implicit Variational Inference](https://mlsanity.com/paper/2601.12023/) (2026)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operators](https://mlsanity.com/paper/2507.06055/) (2025)

## Abstract (arXiv)

Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels K(x,y) = - |x-y|r, r ∈ (0,2) have exceptional properties which allow their efficient computation. We prove that the MMD of Riesz kernels, which is also known as energy distance, coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for r=1, a simple sorting algorithm can be applied to reduce the complexity from O(MN+N2) to O((M+N)log(M+N)) for two measures with M and N support points. As another interesting follow-up result, the MMD of compactly supported measures can be estimated from above and below by the Wasserstein-1 distance. For the implementations we approximate the gradient of the sliced MMD by using only a finite number P of slices. We show that the resulting error has complexity O(√d/P), where d is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for image applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.