# Sliced Wasserstein Variational Inference

arXiv: https://arxiv.org/abs/2207.13177 · HTML: https://mlsanity.com/paper/2207.13177/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 46 references that are in the map.
- First posted on arXiv 26 Jul 2022.

## Builds on (most cited first, 23 of 23)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Markovian Score Climbing: Variational Inference with KL(p||q)](https://mlsanity.com/paper/2003.10374/) (2020)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [Learning to Draw Samples with Amortized Stein Variational Gradient Descent](https://mlsanity.com/paper/1707.06626/) (2017)
- [f-Divergence Variational Inference](https://mlsanity.com/paper/2009.13093/) (2020)
- [Approximate Inference with Amortised MCMC](https://mlsanity.com/paper/1702.08343/) (2017)
- [A Contrastive Divergence for Combining Variational Inference and MCMC](https://mlsanity.com/paper/1905.04062/) (2019)
- [Wasserstein Variational Inference](https://mlsanity.com/paper/1805.11284/) (2018)
- [Operator Variational Inference](https://mlsanity.com/paper/1610.09033/) (2016)
- [Understanding MCMC Dynamics as Flows on the Wasserstein Space](https://mlsanity.com/paper/1902.00282/) (2019)
- [Estimating Convergence of Markov chains with L-Lag Couplings](https://mlsanity.com/paper/1905.09971/) (2019)
- [MCMC-Interactive Variational Inference](https://mlsanity.com/paper/2010.02029/) (2020)

## Built on by (most cited first, 18 of 18)

- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Sliced Wasserstein Estimation with Control Variates](https://mlsanity.com/paper/2305.00402/) (2023)
- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction](https://mlsanity.com/paper/2301.04791/) (2023)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Spherical Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11249/) (2025)
- [Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach](https://mlsanity.com/paper/2504.20319/) (2025)
- [Fast Approximation of the Generalized Sliced-Wasserstein Distance](https://mlsanity.com/paper/2210.10268/) (2022)
- [Streaming Sliced Optimal Transport](https://mlsanity.com/paper/2505.06835/) (2025)
- [Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance](https://mlsanity.com/paper/2601.16425/) (2026)
- [Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions](https://mlsanity.com/paper/2606.30310/) (2026)
- [Amortised Inference through One-Step Implicit Sampling](https://mlsanity.com/paper/2610.12679/) (2026)

## Abstract (arXiv)

Variational Inference approximates an unnormalized distribution via the minimization of Kullback-Leibler (KL) divergence. Although this divergence is efficient for computation and has been widely used in applications, it suffers from some unreasonable properties. For example, it is not a proper metric, i.e., it is non-symmetric and does not preserve the triangle inequality. On the other hand, optimal transport distances recently have shown some advantages over KL divergence. With the help of these advantages, we propose a new variational inference method by minimizing sliced Wasserstein distance, a valid metric arising from optimal transport. This sliced Wasserstein distance can be approximated simply by running MCMC but without solving any optimization problem. Our approximation also does not require a tractable density function of variational distributions so that approximating families can be amortized by generators like neural networks. Furthermore, we provide an analysis of the theoretical properties of our method. Experiments on synthetic and real data are illustrated to show the performance of the proposed method.