A Persistent Homology Design Space for 3D Point Cloud Deep Learning

2026-04-05

AI Summary

This work addresses the insufficient modeling of topological structures in existing deep learning approaches for 3D point clouds, where persistent homology has largely been relegated to peripheral roles. The authors introduce 3DPHDL—the first systematic design space that deeply integrates persistent homology as a structural inductive bias throughout the entire point cloud learning pipeline. This integration encompasses six well-defined injection points spanning simplicial complex construction, filtration strategies, persistence representations, and their coordination with backbone architectures. Through controlled experiments on PointNet, DGCNN, and Point Transformer—augmented with persistence diagrams, images, and landscapes—on ModelNet40 and ShapeNetPart, the approach significantly improves accuracy in classification and segmentation, enhances part consistency, and boosts robustness to noise and sampling variations, while also revealing inherent trade-offs between representational capacity and computational complexity.

Abstract

Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric representations of 3D data. Yet, despite strong theoretical guarantees and increasing empirical adoption, its integration into deep learning for point clouds remains largely ad hoc and architecturally peripheral. In this work, we introduce a unified design space for Persistent-Homology driven learning in 3D point clouds (3DPHDL), formalizing the interplay between complex construction, filtration strategy, persistence representation, neural backbone, and prediction task. Beyond the canonical pipeline of diagram computation and vectorization, we identify six principled injection points through which topology can act as a structural inductive bias reshaping sampling, neighborhood graphs, optimization dynamics, self-supervision, output calibration, and even internal network regularization. We instantiate this framework through a controlled empirical study on ModelNet40 classification and ShapeNetPart segmentation, systematically augmenting representative backbones (PointNet, DGCNN, and Point Transformer) with persistence diagrams, images, and landscapes, and analyzing their impact on accuracy, robustness to noise and sampling variation, and computational scalability. Our results demonstrate consistent improvements in topology-sensitive discrimination and part consistency, while revealing meaningful trade-offs between representational expressiveness and combinatorial complexity. By viewing persistent homology not merely as an auxiliary feature but as a structured component within the learning pipeline, this work provides a systematic framework for incorporating topological reasoning into 3D point cloud learning.

Problem

• Persistent Homology
• 3D Point Cloud
• Deep Learning
• Topological Integration
• Design Space

Innovation

• Persistent Homology
• 3D Point Cloud
• Topological Inductive Bias
• Design Space
• Deep Learning