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Joint Hierarchical Representation Learning of Samples and Features via Informed Tree-Wasserstein Distance
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Proposes an iterative alternating scheme that builds hyperbolic/diffusion-based trees for samples and features, computing tree-Wasserstein distances across modes, optionally refined by Haar wavelet filtering, and uses the learned sample tree to initialize HGCNs.
Problem
Hierarchical structure often exists in both samples and features of a data matrix, but existing hierarchical representation methods model only one mode at a time.
Approach
Alternately build a tree for one mode, compute tree-Wasserstein distances for the other mode using that tree, and rebuild the tree from those distances; add data-driven Haar wavelet filtering from the learned trees at each iteration, with convergence guarantees.
Key results
- On word-document and scRNA-seq sparse approximation, the proposed methods achieve the lowest Haar L1 norms, e.g., CBMC sample L1 of 489.4 vs. 926.8 for the best baseline Co-TR-TWD.
- Document and single-cell kNN classification: proposed method reaches 97.3% on BBCSPORT and 94.0% on ZEISEL, above baselines such as QUE at 91.6% and 83.6%.
- Integrated as preprocessing in HGCN, it improves link prediction on DISEASE to 98.4 AUC vs. 90.8 for HGCN and node classification on CORA to 83.9 vs. 79.9.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Haar wavelet filtering at each iteration (ablation) | Joint TWD · BBCSPORT, TWITTER, CLASSIC, AMAZON, ZEISEL, and CBMC sparse approximation | Haar coefficient L1 norm: BBCSPORT samples: 12.9 -> 10.1, features: 7.0 -> 4.8; TWITTER samples: 25.4 -> 22.4, features: 3.7 -> 3.4; CLASSIC sampl | Helped |
| Add Haar wavelet filtering at each iteration (ablation) | Joint TWD · BBCSPORT, TWITTER, CLASSIC, AMAZON, ZEISEL, and CBMC classification | accuracy: BBCSPORT: 96.7$\pm$0.3 -> 97.3$\pm$0.5; TWITTER: 74.1$\pm$0.5 -> 76.7$\pm$0.7; CLASSIC: 97.3$\pm$0.2 -> 97.6$\pm$0.1; AM | Helped |
| Use learned sample tree as HGCN preprocessing | HGCN · DISEASE link prediction and node classification | ROC AUC / F1 score: LP: 90.8$\pm$0.3 -> 93.2$\pm$0.6; NC: 74.5$\pm$0.9 -> 87.9$\pm$0.7 | Helped |
| Add Haar wavelet filtering to joint TWD preprocessing (ablation) | HGCN + Joint TWD · DISEASE link prediction and node classification | ROC AUC / F1 score: LP: 93.2$\pm$0.6 -> 98.4$\pm$0.4; NC: 87.9$\pm$0.7 -> 89.4$\pm$0.3 | Helped |
Surprising findings
- Applying wavelet filters with trees from other TWD baselines does not consistently improve tree quality, the authors argue because those trees approximate Wasserstein distance rather than hierarchy.
- Wavelet filters alone without the iterative scheme do not yield comparable gains for HGCNs.
Limitations
- Trees may not represent data on complex geometries such as spherical manifolds, mixed-curvature spaces, asymmetric data, or general graphs.
- Convergence theorem requires strictly positive regularization (gamma > 0), while practice uses gamma = 0.
- Method is not differentiable, so it cannot be directly integrated into neural architectures.
Experimental setup
Datasets: BBCSPORT TWITTER CLASSIC AMAZON ZEISEL CBMC CORA PUBMED DISEASE AIRPORT
Benchmarks: BBCSPORT TWITTER CLASSIC AMAZON ZEISEL CBMC CORA PUBMED DISEASE AIRPORT
Metrics: L1 norm of Haar coefficients kNN classification accuracy ROC AUC F1 score accuracy
Baselines: Co-Quadtree Co-Flowtree Co-UltraTree Co-TSWD Co-SWCTWD Co-SWQTWD QUE Tree-WSV WSV HGCN H2H-GCN GCN
Builds on: Tree-Wasserstein distance Tree-WSV Wasserstein singular vectors HGCN diffusion geometry tree construction
Closest papers by meaning
- Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy (ICLR 2025 · similarity 0.93)
- Fast unsupervised ground metric learning with tree-Wasserstein distance (ICLR 2025 · similarity 0.88)
- UltraTWD: Optimizing Ultrametric Trees for Tree-Wasserstein Distance (ICML 2025 · similarity 0.87)
- Hierarchical Correlation Clustering and Tree Preserving Embedding (CVPR 2024 · similarity 0.86)
- Learning Structured Representations with Hyperbolic Embeddings (NeurIPS 2024 · similarity 0.86)
- Residual Hyperbolic Graph Convolution Networks (AAAI 2024 · similarity 0.84)
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