TopoHyper: Integrated Topological-Hypergraph Neural Networks for Medical Image Classification
A novel hybrid architecture that integrates Topological Neural Networks (TNNs) with Hypergraph Neural Networks (HGNNs) for medical image classification, achieving 82.0% test accuracy on PathMNIST (9-class colon pathology).
Key Innovation
TopoHyper introduces a three-phase message passing mechanism that combines the strengths of both topological and hypergraph representations:
Phase 2 — Hypergraph Convolution: Spectral propagation via D_v^{-1/2} H W D_e^{-1} Hᵀ D_v^{-1/2}, modeling arbitrary higher-order group relationships
Phase 3 — Cross-Structure Fusion: Attention-gated combination + bridge matrix B = A_sc ⊙ A_hg that propagates information through nodes connected in both views
The bridge matrix turned out to be the most critical component — ablation shows removing it drops accuracy by 3.5%.
The bridge-only variant actually scores highest (83.0%), suggesting simpler fusion may be better
Theoretical Background
Topological Neural Networks (TNNs)
TNNs operate on simplicial/cell complexes using algebraic topology. The fundamental object is the boundary operator B_k: C_k → C_{k-1}, and the Hodge Laplacian L_k = B_kᵀ B_k + B_{k+1} B_{k+1}ᵀ decomposes signals into gradient, curl, and harmonic components.
Reference: Papillon et al., "Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks" (arXiv:2304.10031)
Hypergraph Neural Networks (HGNNs)
HGNNs operate on hypergraphs H=(V,E,W) where hyperedges connect arbitrary subsets of vertices. The spectral convolution uses: X^{(l+1)} = σ(D_v^{-1/2} H W D_e^{-1} Hᵀ D_v^{-1/2} X^{(l)} Θ^{(l)}).
Advantages: Arbitrary higher-order relationships, no closure requirement, efficient V→E→V propagation.
Limitations: No boundary/orientation information, less rich spectral theory, symmetric node treatment within hyperedges.
Reference: Feng et al., "Hypergraph Neural Networks" (arXiv:1809.09401)
Compatibility Resolution
Challenge
Solution
TNN uses signed B_k; HGNN uses unsigned H
Use |B_k| (absolute boundary) for message passing
Different spectral paradigms
Three-phase architecture with parallel branches
Different optimization objectives
Single end-to-end loss with attention-gated fusion
Key insight: |B₁| is an incidence matrix for the simplicial complex viewed as a hypergraph. This duality enables principled integration.
Medical Image → Graph Pipeline
Each 64×64 medical image is converted to a graph:
Patch extraction: 8×8 patches with stride 6 → 100 nodes per image
Feature engineering (38-dim per node):
Color histogram: 24 bins (8 per RGB channel)
Texture: 8 values (gradient statistics at 2 scales)
If you use this work, please cite the foundational papers:
bibtex
1@article{papillon2023architectures,
2 title={Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks},
3 author={Papillon, Mathilde and Sanborn, Sophia and Hajij, Mustafa and Miolane, Nina},
4 journal={arXiv preprint arXiv:2304.10031},
5 year={2023}
6}
78@inproceedings{feng2019hypergraph,
9 title={Hypergraph Neural Networks},
10 author={Feng, Yifan and You, Haoxuan and Zhang, Zizhao and Ji, Rongrong and Gao, Yue},
11 booktitle={AAAI},
12 year={2019}
13}
