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

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.

Figures (12)

• Figure 1: Illustrative 3D objects with different topologies. Each object is labeled by its Betti numbers $(\beta_0,\beta_1,\beta_2)$: $\beta_0$ (0D holes = connected components), $\beta_1$ (1D holes = loops), and $\beta_2$ (2D holes = voids) stenseke2021persistent.
• Figure 2: (a) Example simplices: 0-simplex (point), 1-simplex (edge), 2-simplex (triangle), 3-simplex (tetrahedron). (b) A valid simplicial complex (right) vs. invalid configurations (left) where faces are missing KovacevNikolic2012. (c) Two individual 1-cycles (one blue and one red) exist in the complex; each cycle independently bounds a 2D chain of simplices dey2022computational
• Figure 3: Illustration of persistent homology computation. (a) A point cloud with growing ball radius $r$, showing emerging simplices (blue: existing, orange: new). (b) Resulting persistence barcode: each bar represents one topological feature (0D or 1D). (c) PD corresponding to (b), where each point $(b,d)$ represents a homological feature born at scale $b$ and dying at scale $d$Wong2021PHGCN.
• Figure 4: Representative constructions along the Complex Construction axis of the 3DPHDL design space. Let $P=\{p_0,\dots,p_n\}\subset\mathbb{R}^d$ and denote by $\mathcal{K}_\theta(P)$ a simplicial (or cubical) complex built on $P$ under construction parameter $\theta$ (e.g., scale, landmark subset, or grid resolution). We use $d(\cdot,\cdot)$ for the Euclidean distance, $B_r(x)=\{y\in\mathbb{R}^d \mid d(x,y)\le r\}$ for the closed ball of radius $r$ centered at $x$, $\mathrm{conv}(\sigma)$ for the convex hull of $\sigma\subseteq P$, $\mathrm{rad}(\sigma)$ for the circumradius of simplex $\sigma$, $L\subseteq P$ for a landmark subset, and $W=P\setminus L$ for the witness set. (a) Delaunay complexdelaunay1934sphere$\mathcal{K}_{\mathrm{Del}}(P)$: simplices whose circumspheres are empty of other points in $P$. (b) $\alpha$-complexedelsbrunner2003shape$\mathcal{K}_{\alpha}(P)$: subcomplex of the Delaunay triangulation containing simplices with $\mathrm{rad}(\sigma)\le \alpha$. (c) Vietoris-Rips complexgromov1987hyperbolicvietoris1927hoheren$\mathcal{K}^{\mathrm{VR}}_{\varepsilon}(P)$: simplices included when $d(p_i,p_j)\le \varepsilon$ for all vertex pairs in $\sigma$. (d) Witness complexde2004topological$\mathcal{K}^{\mathrm{W}}_{\varepsilon}(P,L)$: simplices $\sigma\subseteq L$ supported by witnesses in $W$ under proximity constraints. (e) Čech complexvcech1932theorie$\mathcal{K}^{\mathrm{Cech}}_{r}(P)$: simplices whose balls $\{B_r(p)\}_{p\in\sigma}$ have non-empty common intersection. (f) Spiked Skeleton Complex (SSC)kudeshia2024learning$\mathcal{K}^{\mathrm{SSC}}_{r}(P)$: skeleton nodes connected to nearby points within radius $r$, where skelton nodes are constructed through PCA based clustering. (g) Flood complexgraf2025flood$\mathcal{K}^{\mathrm{Flood}}_{r}(P,L)$: Delaunay simplices over landmarks retained if $\mathrm{conv}(\sigma)\subseteq \bigcup_{p\in P} B_r(p)$. (h) Cubical complexkaczynski2004computing$\mathcal{K}^{\mathrm{Cub}}_{h}(P)$: grid-aligned cubes of side length $h$ intersecting $P$, together with their faces. All constructions are illustrated in $\mathbb{R}^2$ for visualization clarity but admit natural generalizations to $\mathbb{R}^d$.
• Figure 5: A design-space visualization for integrating persistent homology into 3D point-cloud learning pipelines. The horizontal band illustrates the standard PHML workflow (raw data $\rightarrow$ complex $\rightarrow$ filtration $\rightarrow$ PH $\rightarrow$ vectorization $\rightarrow$ neural architectures $\rightarrow$ predictions). Six injection points (IP1--IP6) appear as lateral modules, indicating where PH can reshape sampling (IP1), neighborhood/message passing (IP2), self-supervision/augmentations (IP4), training dynamics (IP3), implicit architectural regularization (IP6), and output calibration (IP5).