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Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy
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Proposes a tree-Wasserstein distance for high-dimensional samples whose features follow a latent hierarchy, decoding a feature tree from multi-scale hyperbolic diffusion embeddings and applying TWD on it.
Problem
Computing meaningful distances between high-dimensional samples (e.g., documents over words, cells over genes) that account for an unobserved hierarchical structure among the features, rather than the sample hierarchy studied in hyperbolic representation learning.
Approach
Embed features into a multi-scale product of Poincaré half-spaces using diffusion densities on a dyadic grid, define a hyperbolic LCA (Fréchet mean) and its relation, decode a binary feature tree bottom-up from these relations, and use the decoded tree in a TWD between samples; the TWD provably approximates the TWD under the latent feature tree.
Key results
- On the synthetic dataset the proposed TWD reaches 99.8% kNN accuracy, above the best baseline Quadtree at 98.2%.
- On word-document benchmarks the method achieves 96.1 on BBCSPORT, 73.4 on TWITTER, and 93.1 on AMAZON, outperforming all baselines including WMD on those three datasets.
- On scRNA-seq, it reaches 89.1 on Zeisel and 84.3 on CBMC, versus 84.2 and 81.4 for GMD.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed latent-feature-hierarchy TWD instead of Quadtree | kNN document classification · Synthetic | classification accuracy: 98.2 -> 99.8 | Helped |
| Use the proposed latent-feature-hierarchy TWD instead of Quadtree | kNN document classification · BBCSPORT | classification accuracy: 95.5 -> 96.1 | Helped |
| Use the proposed latent-feature-hierarchy TWD instead of WMD | kNN document classification · TWITTER | classification accuracy: 71.3 -> 73.4 | Helped |
| Use the proposed latent-feature-hierarchy TWD instead of WMD | kNN document classification · CLASSIC | classification accuracy: 97.2 -> 96.9 | Hurt |
| Use the proposed latent-feature-hierarchy TWD instead of WMD | kNN document classification · AMAZON | classification accuracy: 92.6 -> 93.1 | Helped |
| Use the proposed latent-feature-hierarchy TWD instead of GMD | cell classification · Zeisel | classification accuracy: 84.2 -> 89.1 | Helped |
| Use the proposed latent-feature-hierarchy TWD instead of UltraTree | cell classification · CBMC | classification accuracy: 82.3 -> 84.3 | Helped |
Limitations
- Restricted to samples on the probability simplex (non-negative histograms).
- Naive tree decoding requires building a distance matrix and eigendecomposition, which may limit scale without the stated acceleration.
- Recovers the hidden feature tree only up to bilipschitz equivalence for sufficiently large K_c and m.
Experimental setup
Datasets: synthetic produce word-document data (100 documents, 8 words) BBCSPORT TWITTER CLASSIC AMAZON Zeisel CBMC
Benchmarks: BBCSPORT TWITTER CLASSIC AMAZON Zeisel CBMC synthetic produce dataset
Metrics: kNN classification accuracy
Baselines: Quadtree Flowtree TSWD UltraTree WCTWD WQTWD SWCTWD SWQTWD MST-TWD TR-TWD HHC-TWD gHHC-TWD
Builds on: tree-Wasserstein distance diffusion geometry Word Mover's Distance Gene Mover's Distance tree-sliced Wasserstein distance
Closest papers by meaning
- Joint Hierarchical Representation Learning of Samples and Features via Informed Tree-Wasserstein Distance (NeurIPS 2025 · similarity 0.93)
- UltraTWD: Optimizing Ultrametric Trees for Tree-Wasserstein Distance (ICML 2025 · similarity 0.90)
- Fast unsupervised ground metric learning with tree-Wasserstein distance (ICLR 2025 · similarity 0.89)
- Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformer (ICML 2024 · similarity 0.87)
- A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions (ICML 2024 · similarity 0.86)
- Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances (ICML 2025 · similarity 0.86)
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