# Markovian Sliced Wasserstein Distances: Beyond Independent Projections arXiv: https://arxiv.org/abs/2301.03749 · HTML: https://mlsanity.com/paper/2301.03749/ ## Key facts - Cited by 11 papers in the map (data through 14 Oct 2026). - Builds on 44 of its 64 references that are in the map. - First posted on arXiv 10 Jan 2023. ## Builds on (most cited first, 30 of 44) - [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014) - [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014) - [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018) - [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014) - [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017) - [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019) - [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017) - [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019) - [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018) - [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018) - [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019) - [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018) - [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019) - [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019) - [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020) - [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018) - [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020) - [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019) - [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019) - [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017) - [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020) - [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019) - [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (2021) - [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022) - [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020) - [On the Efficiency of Entropic Regularized Algorithms for Optimal Transport](https://mlsanity.com/paper/1906.01437/) (2019) - [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019) - [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022) ## Built on by (most cited first, 11 of 11) - [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023) - [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024) - [Fr'echet Wavelet Distance: A Domain-Agnostic Metric for Image Generation](https://mlsanity.com/paper/2312.15289/) (2023) - [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024) - [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025) - [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023) - [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026) - [Gaussian-Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2404.03273/) (2024) - [Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimization](https://mlsanity.com/paper/2509.17405/) (2025) - [Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS](https://mlsanity.com/paper/2606.29665/) (2026) - [Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions](https://mlsanity.com/paper/2606.30310/) (2026) ## Abstract (arXiv) Sliced Wasserstein (SW) distance suffers from redundant projections due to independent uniform random projecting directions. To partially overcome the issue, max K sliced Wasserstein (Max-K-SW) distance (K≥ 1), seeks the best discriminative orthogonal projecting directions. Despite being able to reduce the number of projections, the metricity of Max-K-SW cannot be guaranteed in practice due to the non-optimality of the optimization. Moreover, the orthogonality constraint is also computationally expensive and might not be effective. To address the problem, we introduce a new family of SW distances, named Markovian sliced Wasserstein (MSW) distance, which imposes a first-order Markov structure on projecting directions. We discuss various members of MSW by specifying the Markov structure including the prior distribution, the transition distribution, and the burning and thinning technique. Moreover, we investigate the theoretical properties of MSW including topological properties (metricity, weak convergence, and connection to other distances), statistical properties (sample complexity, and Monte Carlo estimation error), and computational properties (computational complexity and memory complexity). Finally, we compare MSW distances with previous SW variants in various applications such as gradient flows, color transfer, and deep generative modeling to demonstrate the favorable performance of MSW.