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Learning Structured Representations by Embedding Class Hierarchy with Fast Optimal Transport
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Replaces the centroid-based l2 distance in CPCC hierarchy regularization with optimal-transport distances (EMD and approximations), introducing a linear-time Fast FlowTree variant that improves hierarchy embedding and fine/coarse generalization.
Problem
Embedding a class hierarchy into feature space via CPCC regularization uses distances between class means, which poorly represent multi-modal class-conditional distributions and cannot exactly embed tree metrics in l2.
Approach
Measure pairwise class distances with Earth Mover's Distance over sample-level features inside the CPCC regularizer (OT-CPCC family), and propose Fast FlowTree, which exploits an augmented label tree so that the tree-based flow reduces to a linear-time 1D greedy matching.
Key results
- On CIFAR100, FastFT raises TestCPCC from 83.08 (l2-CPCC) to 93.85, and on INAT from 67.01 to 73.20.
- OT-CPCC variants achieve the best TestCPCC on 5 of 7 datasets, while l2-CPCC is slightly better on E13 and E30.
- FastFT and other OT-CPCC methods outperform the Flat baseline on coarse-level target accuracy (e.g., CIFAR100 tAcc 44.29 FastFT vs 43.09 Flat).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · CIFAR10 | TestCPCC: 99.94 (0.02) -> 99.95 (0.00) | No clear effect |
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · CIFAR100 | TestCPCC: 83.08 (0.11) -> 93.85 (0.05) | Helped |
| Replace ell2-CPCC with FastFT-CPCC | ImageNet-pretrained ResNet18 · INAT | TestCPCC: 67.01 (0.04) -> 73.20 (0.03) | Helped |
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · L17 | TestCPCC: 92.30 (0.35) -> 91.71 (0.15) | Hurt |
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · N26 | TestCPCC: 93.03 (0.32) -> 92.68 (0.21) | Hurt |
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · E13 | TestCPCC: 92.02 (0.08) -> 91.97 (0.20) | No clear effect |
| Replace ell2-CPCC with FastFT-CPCC | ResNet18 · E30 | TestCPCC: 93.37 (0.19) -> 91.81 (0.44) | Hurt |
| Increase batch size for OT-CPCC (ablation) | OT-CPCC on CIFAR100 · CIFAR100 | TestCPCC | Helped |
| Vary regularization factor lambda (ablation) | FastFT-CPCC, EMD-CPCC, and ell2-CPCC on CIFAR100 · CIFAR100 | TestCPCC | No clear effect |
| Change backbone architecture (ablation) | FastFT-CPCC, EMD-CPCC, and ell2-CPCC on CIFAR100 · CIFAR100 | TestCPCC | No clear effect |
Surprising findings
- On CIFAR100 and BREEDS, plain EMD-CPCC underperforms l2-CPCC on TestCPCC (89.60 vs 83.08 is higher, but on INAT EMD 65.80 is below l2 67.01), indicating the exact EMD is not uniformly better than approximations.
- SWD-CPCC gives the best TestCPCC on L17 (95.10) but worst on CIFAR100 (73.17), showing the approximation's quality is dataset-dependent.
Limitations
- Exact EMD is intractable on BREEDS, so EMD results are missing there.
- Gradients treat the transport plan as constant for approximations, which is not a true derivative of the approximate objective.
- Gains are not uniform; l2-CPCC remains best on E13 and E30 for TestCPCC.
Experimental setup
Datasets: CIFAR10 CIFAR100 BREEDS (LIVING17, NONLIVING26, ENTITY13, ENTITY30) iNaturalist-mini (INAT)
Benchmarks: CIFAR10 CIFAR100 BREEDS INAT
Metrics: accuracy mean average precision TestCPCC
Baselines: Flat cross-entropy l2-CPCC EMD-CPCC Sinkhorn-CPCC SWD-CPCC
Builds on: CPCC regularizer (Zeng et al., 2022) FlowTree Sinkhorn Sliced Wasserstein Distance Tree Wasserstein Distance
Closest papers by meaning
- Learning Structured Representations by Embedding Class Hierarchy (ICLR 2023 · similarity 0.93)
- Learning Structured Representations with Hyperbolic Embeddings (NeurIPS 2024 · similarity 0.86)
- Hyperbolic Embeddings of Supervised Models (NeurIPS 2024 · similarity 0.84)
- Hierarchical Correlation Clustering and Tree Preserving Embedding (CVPR 2024 · similarity 0.84)
- Self-Paced Unified Representation Learning for Hierarchical Multi-Label Classification (AAAI 2024 · similarity 0.84)
- Lightspeed Geometric Dataset Distance via Sliced Optimal Transport (ICML 2025 · similarity 0.83)
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