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

arXiv: https://arxiv.org/abs/2604.04299 · HTML: https://mlsanity.com/paper/2604.04299/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 98 references that are in the map.
- First posted on arXiv 5 Apr 2026.

## Builds on (most cited first, 30 of 50)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [4D Spatio-Temporal ConvNets: Minkowski Convolutional Neural Networks](https://mlsanity.com/paper/1904.08755/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [Point-BERT: Pre-training 3D Point Cloud Transformers with Masked Point Modeling](https://mlsanity.com/paper/2111.14819/) (2021)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [PartNet: A Large-scale Benchmark for Fine-grained and Hierarchical Part-level 3D Object Understanding](https://mlsanity.com/paper/1812.02713/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PointMamba: A Simple State Space Model for Point Cloud Analysis](https://mlsanity.com/paper/2402.10739/) (2024)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Nested Learning: The Illusion of Deep Learning Architectures](https://mlsanity.com/paper/2512.24695/) (2025)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Mamba3D: Enhancing Local Features for 3D Point Cloud Analysis via State Space Model](https://mlsanity.com/paper/2404.14966/) (2024)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Point-JEPA: A Joint Embedding Predictive Architecture for Self-Supervised Learning on Point Cloud](https://mlsanity.com/paper/2404.16432/) (2024)

## Built on by (most cited first, 1 of 1)

- [Auditing Training-Free 3D Shape Retrieval with Diffused Geodesic Moments](https://mlsanity.com/paper/2605.29004/) (2026)

## Abstract (arXiv)

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.