# Tree-Sliced Variants of Wasserstein Distances

arXiv: https://arxiv.org/abs/1902.00342 · HTML: https://mlsanity.com/paper/1902.00342/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 30 of 54)

- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Diffusion Earth Mover's Distance and Distribution Embeddings](https://mlsanity.com/paper/2102.12833/) (2021)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [Supervised Tree-Wasserstein Distance](https://mlsanity.com/paper/2101.11520/) (2021)
- [Re-evaluating Word Mover's Distance](https://mlsanity.com/paper/2105.14403/) (2021)
- [Fixed Support Tree-Sliced Wasserstein Barycenter](https://mlsanity.com/paper/2109.03431/) (2021)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Hilbert Curve Projection Distance for Distribution Comparison](https://mlsanity.com/paper/2205.15059/) (2022)
- [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024)
- [Graph Filtration Kernels](https://mlsanity.com/paper/2110.11862/) (2021)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (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)
- [Mirror Sinkhorn: Fast Online Optimization on Transport Polytopes](https://mlsanity.com/paper/2211.10420/) (2022)
- [SEAL: Simultaneous Label Hierarchy Exploration And Learning](https://mlsanity.com/paper/2304.13374/) (2023)
- [Scalable Counterfactual Distribution Estimation in Multivariate Causal Models](https://mlsanity.com/paper/2311.00927/) (2023)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Iterative Alignment Flows](https://mlsanity.com/paper/2104.07232/) (2021)
- [Embedding Signals on Knowledge Graphs with Unbalanced Diffusion Earth Mover's Distance](https://mlsanity.com/paper/2107.12334/) (2021)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Learning Ultrametric Trees for Optimal Transport Regression](https://mlsanity.com/paper/2210.12288/) (2022)
- [Few-Shot Object Detection via Synthetic Features with Optimal Transport](https://mlsanity.com/paper/2308.15005/) (2023)
- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)
- [Learning Structured Representations by Embedding Class Hierarchy with Fast Optimal Transport](https://mlsanity.com/paper/2410.03052/) (2024)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)

## Abstract (arXiv)

Optimal transport (OT) theory defines a powerful set of tools to compare probability distributions. OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the textitsliced formulation, which exploits the closed-form formula between univariate distributions by projecting high-dimensional measures onto random lines. We consider in this work a more general family of ground metrics, namely textittree metrics, which also yield fast closed-form computations and negative definite, and of which the sliced-Wasserstein distance is a particular case (the tree is a chain). We propose the tree-sliced Wasserstein distance, computed by averaging the Wasserstein distance between these measures using random tree metrics, built adaptively in either low or high-dimensional spaces. Exploiting the negative definiteness of that distance, we also propose a positive definite kernel, and test it against other baselines on a few benchmark tasks.