# Sliced Wasserstein Distance for Learning Gaussian Mixture Models

arXiv: https://arxiv.org/abs/1711.05376 · HTML: https://mlsanity.com/paper/1711.05376/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2017.

## Built on by (most cited first, 30 of 59)

- [Voice2Series: Reprogramming Acoustic Models for Time Series Classification](https://mlsanity.com/paper/2106.09296/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Generative Multiplane Images: Making a 2D GAN 3D-Aware](https://mlsanity.com/paper/2207.10642/) (2022)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution](https://mlsanity.com/paper/2204.01188/) (2022)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Wasserstein Unsupervised Reinforcement Learning](https://mlsanity.com/paper/2110.07940/) (2021)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [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)
- [An Optimal Transport Approach to Personalized Federated Learning](https://mlsanity.com/paper/2206.02468/) (2022)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Supervised Tree-Wasserstein Distance](https://mlsanity.com/paper/2101.11520/) (2021)
- [Fixed Support Tree-Sliced Wasserstein Barycenter](https://mlsanity.com/paper/2109.03431/) (2021)
- [Weakly Supervised High-Fidelity Clothing Model Generation](https://mlsanity.com/paper/2112.07200/) (2021)
- [Privacy-preserving data release leveraging optimal transport and particle gradient descent](https://mlsanity.com/paper/2401.17823/) (2024)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [Slicing the Gaussian Mixture Wasserstein Distance](https://mlsanity.com/paper/2504.08544/) (2025)
- [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [PianoMotion10M: Dataset and Benchmark for Hand Motion Generation in Piano Performance](https://mlsanity.com/paper/2406.09326/) (2024)
- [Set Representation Learning with Generalized Sliced-Wasserstein Embeddings](https://mlsanity.com/paper/2103.03892/) (2021)
- [A sampling-based approach for efficient clustering in large datasets](https://mlsanity.com/paper/2112.14793/) (2021)
- [Hybrid-CSR: Coupling Explicit and Implicit Shape Representation for Cortical Surface Reconstruction](https://mlsanity.com/paper/2307.12299/) (2023)
- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)

## Abstract (arXiv)

Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm.