Learning Significant Persistent Homology Features for 3D Shape Understanding

TL;DR

This work adds a topology-aware layer to 3D point-cloud understanding by computing persistent homology features (in $H_1$ and $H_2$) for ModelNet40 and ShapeNet, and by introducing TopoGAT, a three-branch GNN that learns to select significant PD points via a differentiable, classification-guided framework. A novel TopoLoss composed of Wasserstein-distance, persistent entropy, and reduction components guides the topological feature selection, with learnable weights satisfying $oldsymbol{ extalpha}+oldsymbol{ extbeta}+oldsymbol{ extgamma}=1$. Empirically, TopoGAT outperforms statistical feature-filtering baselines on topology-preserving metrics and yields improvements in both classification and part-segmentation when topological features are integrated, demonstrating the practical potential of topology-aware DL in 3D shape understanding. The topological datasets and learning framework enable systematic evaluation and broader adoption of persistent homology in real-world 3D point-cloud tasks.

Abstract

Geometry and topology constitute complementary descriptors of three-dimensional shape, yet existing benchmark datasets primarily capture geometric information while neglecting topological structure. This work addresses this limitation by introducing topologically-enriched versions of ModelNet40 and ShapeNet, where each point cloud is augmented with its corresponding persistent homology features. These benchmarks with the topological signatures establish a foundation for unified geometry-topology learning and enable systematic evaluation of topology-aware deep learning architectures for 3D shape analysis. Building on this foundation, we propose a deep learning-based significant persistent point selection method, \textit{TopoGAT}, that learns to identify the most informative topological features directly from input data and the corresponding topological signatures, circumventing the limitations of hand-crafted statistical selection criteria. A comparative study verifies the superiority of the proposed method over traditional statistical approaches in terms of stability and discriminative power. Integrating the selected significant persistent points into standard point cloud classification and part-segmentation pipelines yields improvements in both classification accuracy and segmentation metrics. The presented topologically-enriched datasets, coupled with our learnable significant feature selection approach, enable the broader integration of persistent homology into the practical deep learning workflows for 3D point cloud analysis.

Figures (10)

• Figure 1: General framework of Persistent Homology-based point cloud learning based on the proposed work. Main contribution of this work lies in the curation of the Topological-Signatures dataset and Significant Topological Feature selection Network TopoGAT.
• Figure 2: The persistence diagram records the birth and death of topological features (clusters, loops, cavities) as points. A $\delta$-band is drawn around the diagonal: features with short lifetimes, represented by points inside this band (highlighted), are classified as topological noise. Features whose points lie outside the band are highlighted as significant persistent homology features, capturing robust and meaningful structure in the underlying shape (lower-right corner).
• Figure 3: Topological Signatures Dataset: Example $H_{1}$ and $H_{2}$ persistence diagrams of eight objects each from different classes of ModelNet40 wu20153d (in green) and ShapeNet shapenet2015 (in purple) respectively. Generating these PDs took over 860 hours on an Intel Silver 4216 Cascade Lake CPU (187 GB RAM) with NVIDIA V100 Volta GPU (32G HBM2 memory) across ModelNet40 and ShapeNet, reflecting the high computational cost of persistent homology at scale, and underscoring the utility of our precomputed topological signatures. In each diagram, blue points denote $H_{1}$ PD points and green points denote $H_{2}$ PD points.
• Figure 4: TopoGAT Architecture: Three identical GNN feature extraction modules individually construct dynamic KNN graphs of a point cloud along with its topological representations while utilizing GAT layers to extract geometric as well as topological features. Classification MLP head predicts logits for each point cloud, which is further utilized to predict a threshold value for each point cloud using Regression MLP heads for each homological dimension. Classification and regression MLP heads differ in the input size and output layer.
• Figure 5: Example of original PD (blue indicate $H_{1}$ features and green indicate $H_{2}$ features) with significant features selected by TopoGAT (red circle).

Theorems & Definitions (3)

• Theorem 3.1
• Lemma 3.2
• proof