# Energy-Based Sliced Wasserstein Distance

arXiv: https://arxiv.org/abs/2304.13586 · HTML: https://mlsanity.com/paper/2304.13586/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 57 references that are in the map.
- First posted on arXiv 26 Apr 2023.

## Builds on (most cited first, 30 of 31)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [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)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [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)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (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)
- [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)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019)
- [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)
- [Approximate Bayesian Computation with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1910.12815/) (2019)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [Amortized Projection Optimization for Sliced Wasserstein Generative Models](https://mlsanity.com/paper/2203.13417/) (2022)
- [Improving Relational Regularized Autoencoders with Spherical Sliced Fused Gromov Wasserstein](https://mlsanity.com/paper/2010.01787/) (2020)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Lidar Upsampling with Sliced Wasserstein Distance](https://mlsanity.com/paper/2301.13558/) (2023)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)

## Built on by (most cited first, 30 of 39)

- [LeJEPA: Provable and Scalable Self-Supervised Learning Without the Heuristics](https://mlsanity.com/paper/2511.08544/) (2025)
- [Dataset Distillation via the Wasserstein Metric](https://mlsanity.com/paper/2311.18531/) (2023)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [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)
- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [Efficient Prior Calibration From Indirect Data](https://mlsanity.com/paper/2405.17955/) (2024)
- [Sharp bounds for max-sliced Wasserstein distances](https://mlsanity.com/paper/2403.00666/) (2024)
- [ESPFormer: Doubly-Stochastic Attention with Expected Sliced Transport Plans](https://mlsanity.com/paper/2502.07962/) (2025)
- [Color Conditional Generation with Sliced Wasserstein Guidance](https://mlsanity.com/paper/2503.19034/) (2025)
- [Validating Climate Models with Spherical Convolutional Wasserstein Distance](https://mlsanity.com/paper/2401.14657/) (2024)
- [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [CLOT: Closed Loop Optimal Transport for Unsupervised Action Segmentation](https://mlsanity.com/paper/2507.03539/) (2025)
- [Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances](https://mlsanity.com/paper/2509.20508/) (2025)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Downscaling Extreme Precipitation with Wasserstein Regularized Diffusion](https://mlsanity.com/paper/2410.00381/) (2024)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [Relative Translation Invariant Wasserstein Distance](https://mlsanity.com/paper/2409.02416/) (2024)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [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)
- [Unbiased Sliced Wasserstein Kernels for High-Quality Audio Captioning](https://mlsanity.com/paper/2502.05435/) (2025)
- [Improved 3D Scene Stylization via Text-Guided Generative Image Editing with Region-Based Control](https://mlsanity.com/paper/2509.05285/) (2025)
- [Target-Guided Selective Reweighting for Physics-Informed Neural Network Inverse Problems: A Transfer Learning Approach](https://mlsanity.com/paper/2607.05271/) (2026)
- [Reconstruction of dynamical systems from data without time labels](https://mlsanity.com/paper/2312.04038/) (2023)
- [Overlap-Aware Feature Learning for Robust Unsupervised Domain Adaptation for 3D Semantic Segmentation](https://mlsanity.com/paper/2504.01668/) (2025)

## Abstract (arXiv)

The sliced Wasserstein (SW) distance has been widely recognized as a statistically effective and computationally efficient metric between two probability measures. A key component of the SW distance is the slicing distribution. There are two existing approaches for choosing this distribution. The first approach is using a fixed prior distribution. The second approach is optimizing for the best distribution which belongs to a parametric family of distributions and can maximize the expected distance. However, both approaches have their limitations. A fixed prior distribution is non-informative in terms of highlighting projecting directions that can discriminate two general probability measures. Doing optimization for the best distribution is often expensive and unstable. Moreover, designing the parametric family of the candidate distribution could be easily misspecified. To address the issues, we propose to design the slicing distribution as an energy-based distribution that is parameter-free and has the density proportional to an energy function of the projected one-dimensional Wasserstein distance. We then derive a novel sliced Wasserstein metric, energy-based sliced Waserstein (EBSW) distance, and investigate its topological, statistical, and computational properties via importance sampling, sampling importance resampling, and Markov Chain methods. Finally, we conduct experiments on point-cloud gradient flow, color transfer, and point-cloud reconstruction to show the favorable performance of the EBSW.