# Hierarchical Sliced Wasserstein Distance

arXiv: https://arxiv.org/abs/2209.13570 · HTML: https://mlsanity.com/paper/2209.13570/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 77 references that are in the map.
- First posted on arXiv 27 Sep 2022.

## Builds on (most cited first, 30 of 45)

- [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)
- [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)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (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)
- [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)
- [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)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [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)
- [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)

## Built on by (most cited first, 23 of 23)

- [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)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [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)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction](https://mlsanity.com/paper/2301.04791/) (2023)
- [Importance Sparsification for Sinkhorn Algorithm](https://mlsanity.com/paper/2306.06581/) (2023)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics](https://mlsanity.com/paper/2307.01770/) (2023)
- [Improving Generative Flow Networks with Path Regularization](https://mlsanity.com/paper/2209.15092/) (2022)
- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [Constrained Sliced Wasserstein Embedding](https://mlsanity.com/paper/2506.02203/) (2025)
- [Fast Approximation of the Generalized Sliced-Wasserstein Distance](https://mlsanity.com/paper/2210.10268/) (2022)
- [DeepDRK: Deep Dependency Regularized Knockoff for Feature Selection](https://mlsanity.com/paper/2402.17176/) (2024)
- [Wasserstein Diversity-Enriched Regularizer for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/2308.00989/) (2023)
- [Gaussian-Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2404.03273/) (2024)
- [Relation-Aware Slicing in Cross-Domain Alignment](https://mlsanity.com/paper/2507.13194/) (2025)
- [Synergizing Deconfounding and Temporal Generalization For Time-series Counterfactual Outcome Estimation](https://mlsanity.com/paper/2511.16006/) (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 has been widely used in different application scenarios since it can be scaled to a large number of supports without suffering from the curse of dimensionality. The value of sliced Wasserstein distance is the average of transportation cost between one-dimensional representations (projections) of original measures that are obtained by Radon Transform (RT). Despite its efficiency in the number of supports, estimating the sliced Wasserstein requires a relatively large number of projections in high-dimensional settings. Therefore, for applications where the number of supports is relatively small compared with the dimension, e.g., several deep learning applications where the mini-batch approaches are utilized, the complexities from matrix multiplication of Radon Transform become the main computational bottleneck. To address this issue, we propose to derive projections by linearly and randomly combining a smaller number of projections which are named bottleneck projections. We explain the usage of these projections by introducing Hierarchical Radon Transform (HRT) which is constructed by applying Radon Transform variants recursively. We then formulate the approach into a new metric between measures, named Hierarchical Sliced Wasserstein (HSW) distance. By proving the injectivity of HRT, we derive the metricity of HSW. Moreover, we investigate the theoretical properties of HSW including its connection to SW variants and its computational and sample complexities. Finally, we compare the computational cost and generative quality of HSW with the conventional SW on the task of deep generative modeling using various benchmark datasets including CIFAR10, CelebA, and Tiny ImageNet.