# Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy

arXiv: https://arxiv.org/abs/2410.21107 · HTML: https://mlsanity.com/paper/2410.21107/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 120 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Builds on (most cited first, 30 of 46)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Visual Genome: Connecting Language and Vision Using Crowdsourced Dense Image Annotations](https://mlsanity.com/paper/1602.07332/) (2016)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [A cost function for similarity-based hierarchical clustering](https://mlsanity.com/paper/1510.05043/) (2015)
- [Neural Embeddings of Graphs in Hyperbolic Space](https://mlsanity.com/paper/1705.10359/) (2017)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Scalable Nearest Neighbor Search for Optimal Transport](https://mlsanity.com/paper/1910.04126/) (2019)
- [Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation](https://mlsanity.com/paper/1807.10479/) (2018)
- [Diffusion Nets](https://mlsanity.com/paper/1506.07840/) (2015)
- [Diffusion Earth Mover's Distance and Distribution Embeddings](https://mlsanity.com/paper/2102.12833/) (2021)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Dynamical Optimal Transport on Discrete Surfaces](https://mlsanity.com/paper/1809.07083/) (2018)
- [Fast Unbalanced Optimal Transport on a Tree](https://mlsanity.com/paper/2006.02703/) (2020)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces](https://mlsanity.com/paper/1911.02536/) (2019)

## Built on by (most cited first, 5 of 5)

- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [H4G: Unlocking Faithful Inference for Zero-Shot Graph Learning in Hyperbolic Space](https://mlsanity.com/paper/2510.12094/) (2025)
- [Harnessing Data Asymmetry: Manifold Learning in the Finsler World](https://mlsanity.com/paper/2603.11396/) (2026)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)
- [Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions](https://mlsanity.com/paper/2606.30310/) (2026)

## Abstract (arXiv)

Finding meaningful distances between high-dimensional data samples is an important scientific task. To this end, we propose a new tree-Wasserstein distance (TWD) for high-dimensional data with two key aspects. First, our TWD is specifically designed for data with a latent feature hierarchy, i.e., the features lie in a hierarchical space, in contrast to the usual focus on embedding samples in hyperbolic space. Second, while the conventional use of TWD is to speed up the computation of the Wasserstein distance, we use its inherent tree as a means to learn the latent feature hierarchy. The key idea of our method is to embed the features into a multi-scale hyperbolic space using diffusion geometry and then present a new tree decoding method by establishing analogies between the hyperbolic embedding and trees. We show that our TWD computed based on data observations provably recovers the TWD defined with the latent feature hierarchy and that its computation is efficient and scalable. We showcase the usefulness of the proposed TWD in applications to word-document and single-cell RNA-sequencing datasets, demonstrating its advantages over existing TWDs and methods based on pre-trained models.