# Tree-Sliced Wasserstein Distance with Nonlinear Projection

arXiv: https://arxiv.org/abs/2505.00968 · HTML: https://mlsanity.com/paper/2505.00968/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 71 references that are in the map.
- First posted on arXiv 2 May 2025.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [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)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Optimal Transport Kernels for Sequential and Parallel Neural Architecture Search](https://mlsanity.com/paper/2006.07593/) (2020)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Bending Graphs: Hierarchical Shape Matching using Gated Optimal Transport](https://mlsanity.com/paper/2202.01537/) (2022)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Equivariant Neural Functional Networks for Transformers](https://mlsanity.com/paper/2410.04209/) (2024)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Dynamical Optimal Transport on Discrete Surfaces](https://mlsanity.com/paper/1809.07083/) (2018)
- [Monomial Matrix Group Equivariant Neural Functional Networks](https://mlsanity.com/paper/2409.11697/) (2024)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (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)
- [On the complexity of the optimal transport problem with graph-structured cost](https://mlsanity.com/paper/2110.00627/) (2021)
- [Stereographic Spherical Sliced Wasserstein Distances](https://mlsanity.com/paper/2402.02345/) (2024)
- [Bridging Vision and Language Spaces with Assignment Prediction](https://mlsanity.com/paper/2404.09632/) (2024)

## Built on by (most cited first, 3 of 3)

- [Differentiable Generalized Sliced Wasserstein Plans](https://mlsanity.com/paper/2505.22049/) (2025)
- [Quasi-Equivariant Metanetworks](https://mlsanity.com/paper/2604.23720/) (2026)
- [From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transport](https://mlsanity.com/paper/2606.04695/) (2026)

## Abstract (arXiv)

Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.