# Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution

arXiv: https://arxiv.org/abs/2204.01188 · HTML: https://mlsanity.com/paper/2204.01188/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 75 references that are in the map.
- First posted on arXiv 4 Apr 2022.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (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)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [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 bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [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 Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [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)
- [Vocabulary Learning via Optimal Transport for Neural Machine Translation](https://mlsanity.com/paper/2012.15671/) (2020)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (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)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)

## Built on by (most cited first, 25 of 25)

- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Population-Aligned Persona Generation for LLM-based Social Simulation](https://mlsanity.com/paper/2509.10127/) (2025)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer](https://mlsanity.com/paper/2301.12811/) (2023)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Lightspeed Geometric Dataset Distance via Sliced Optimal Transport](https://mlsanity.com/paper/2501.18901/) (2025)
- [Hilbert Curve Projection Distance for Distribution Comparison](https://mlsanity.com/paper/2205.15059/) (2022)
- [Validating Climate Models with Spherical Convolutional Wasserstein Distance](https://mlsanity.com/paper/2401.14657/) (2024)
- [Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction](https://mlsanity.com/paper/2301.04791/) (2023)
- [Sliced-Wasserstein normalizing flows: beyond maximum likelihood training](https://mlsanity.com/paper/2207.05468/) (2022)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Improving Generative Flow Networks with Path Regularization](https://mlsanity.com/paper/2209.15092/) (2022)
- [Relative Translation Invariant Wasserstein Distance](https://mlsanity.com/paper/2409.02416/) (2024)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [Partial Transport for Point-Cloud Registration](https://mlsanity.com/paper/2309.15787/) (2023)
- [Fast Approximation of the Generalized Sliced-Wasserstein Distance](https://mlsanity.com/paper/2210.10268/) (2022)
- [Unbiased Sliced Wasserstein Kernels for High-Quality Audio Captioning](https://mlsanity.com/paper/2502.05435/) (2025)
- [ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projection](https://mlsanity.com/paper/2605.12879/) (2026)

## Abstract (arXiv)

The conventional sliced Wasserstein is defined between two probability measures that have realizations as vectors. When comparing two probability measures over images, practitioners first need to vectorize images and then project them to one-dimensional space by using matrix multiplication between the sample matrix and the projection matrix. After that, the sliced Wasserstein is evaluated by averaging the two corresponding one-dimensional projected probability measures. However, this approach has two limitations. The first limitation is that the spatial structure of images is not captured efficiently by the vectorization step; therefore, the later slicing process becomes harder to gather the discrepancy information. The second limitation is memory inefficiency since each slicing direction is a vector that has the same dimension as the images. To address these limitations, we propose novel slicing methods for sliced Wasserstein between probability measures over images that are based on the convolution operators. We derive convolution sliced Wasserstein (CSW) and its variants via incorporating stride, dilation, and non-linear activation function into the convolution operators. We investigate the metricity of CSW as well as its sample complexity, its computational complexity, and its connection to conventional sliced Wasserstein distances. Finally, we demonstrate the favorable performance of CSW over the conventional sliced Wasserstein in comparing probability measures over images and in training deep generative modeling on images.