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

Prachi Kudeshia; Jiju Poovvancheri; Amr Ghoneim; Dong Chen

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) [ 83 ] .

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 [ 53 ] . (c) Two individual 1-cycles (one blue and one red) exist in the complex; each cycle independently bounds a 2D chain of simplices [ 32 ]

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

[ 92 ] .

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)\leq 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 complex [ 30 ]

\mathcal{K}_{\mathrm{Del}}(P)

: simplices whose circumspheres are empty of other points in

P

. (b)

\alpha

-complex [ 34 ]

\mathcal{K}_{\alpha}(P)

: subcomplex of the Delaunay triangulation containing simplices with

\mathrm{rad}(\sigma)\leq\alpha

. (c) Vietoris-Rips complex [ 40 , 90 ]

\mathcal{K}^{\mathrm{VR}}_{\varepsilon}(P)

: simplices included when

d(p_{i},p_{j})\leq\varepsilon

for all vertex pairs in

\sigma

. (d) Witness complex [ 28 ]

\mathcal{K}^{\mathrm{W}}_{\varepsilon}(P,L)

: simplices

\sigma\subseteq L

supported by witnesses in

W

under proximity constraints. (e) Čech complex [ 16 ]

\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) [ 54 ]

\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 complex [ 38 ]

\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 complex [ 52 ]

\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).

Figure 6: Comparison of point cloud sampling strategies: Random, Farthest Point Sampling (FPS) [ 66 ] , and Persistent Homology (PH)-guided sampling (left to right) with 15 samples per shape. PH-based sampling better preserves structurally and topologically salient features (e.g., boundaries and cycles), motivating its use over geometry-only approaches.

Construction	1K pts		10K pts		50K pts		100K pts		500K pts		1M pts
Construction	Time (s)	#Simplices	Time (s)	#Simplices	Time (s)	#Simplices	Time (s)	#Simplices	Time (s)	#Simplices	Time (s)	#Simplices
SSC	0.44118	147	0.92023	142	0.91501	147	0.92777	144	0.92155	148	0.91768	150
Flood	0.08871	671	0.49493	4146	0.85859	8922	1.62552	17069	4.87386	36829	9.82121	55328
Delaunay	0.11202	8840	0.98786	54895	1.07122	68863	0.98627	68879	1.19278	68879	1.77467	68879
Alpha	0.17922	4545	0.46643	25793	0.65020	31873	0.67728	31884	0.85199	31884	1.30659	31884
Rips	0.01149	3155	0.41204	36985	1.44091	183788	2.41841	367500	10.30217	1844188	20.78028	3678235

Table 1: Comparison of complex construction time and number of simplices across varying point counts of a solid sphere and torus point cloud (Fig. 7 ). For each sampling size, we report construction time and number of simplices.

Figure 7: Visualization of simplicial and cubical complexes constructed on a solid sphere and torus point cloud under varying sampling densities. Rows correspond to 100K points, 33% down-sampled, and 66% down-sampled inputs, while columns show SSC [ 54 ] , Alpha [ 34 ] , Rips (VR) [ 40 ] , Witness [ 28 ] , Cubical [ 52 ] , and Delaunay [ 30 ] and Flood [ 38 ] complexes. The figure highlights qualitative differences in structural fidelity, sparsity, and geometric coherence across complex types as point density decreases.

Figure 8: Scaling behavior of complex construction methods across varying point cloud sizes (1k-1M points). Both plots use logarithmic scales on both axes. (a) Construction time vs point count: Cubical maintains the fastest construction (0.001-0.44s). SSC shows constant time (0.9s) across all scales. Rips shows exponential scaling (0.01→20.78s), becoming impractical beyond 500k points. (b) Number of simplices vs point count: SSC maintains constant minimal simplices (145-150). Cubical shows strong growth (36k→7.9M simplices). Delaunay and Alpha plateau beyond 50k points. Rips demonstrates explosive growth (3k→3.68M simplices). Logarithmic scales reveal distinct scaling regimes: constant (horizontal), sub-linear (gentle slopes), and exponential (steep slopes).

Complex	$\sigma=0.010$		$\sigma=0.020$		$\sigma=0.050$		$\sigma=0.100$
Complex	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$
Flood	0.44	0.30	0.67	0.47	0.69	0.63	0.59	0.72
Alpha	3.11	0.88	4.77	1.57	6.57	2.29	7.86	2.63
Cubical	0.24	0.00	0.33	0.00	0.56	0.00	0.71	0.00
Rips (VR)	0.67	0.37	0.45	0.39	0.57	1.22	2.66	2.06
Witness	0.00	0.11	0.00	0.17	0.00	0.20	0.00	0.20

Table 2: Topological stability measured by the 2-Wasserstein distance

W_{2}^{(k)}=W_{2}(\mathcal{D}_{k}(P),\mathcal{D}_{k}(P_{\sigma}))

between PDs of the original point cloud

P

and its Gaussian-noised version

P_{\sigma}

for different noise levels

\sigma

.

Complex	500k		100k		50k		10k		1k
Complex	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$	$W_{2}^{(0)}$	$W_{2}^{(1)}$
Flood	0.046	0.119	0.034	0.123	0.050	0.128	0.086	0.152	0.714	0.684
Alpha	0.000	0.000	0.000	0.000	0.000	0.000	0.832	0.408	3.606	1.703
Cubical	0.000	0.000	0.000	0.000	0.000	0.000	0.134	0.000	0.523	0.000
Rips (VR)	0.433	0.422	0.221	0.418	0.313	0.380	0.211	0.411	0.730	0.293
Witness	0.140	0.068	0.089	0.045	0.085	0.066	0.059	0.051	0.085	0.100

Table 3: Topological stability measured by the 2-Wasserstein distance

W_{2}^{(k)}=W_{2}(\mathcal{D}_{k}(P),\mathcal{D}_{k}(P_{s}))

between PDs of the original point cloud

P

(1M points) and its downsampled versions

P_{s}

for different downsampling factors.

Figure 9: Topological stability measured by the 2-Wasserstein distance

W_{2}^{(k)}=W_{2}(\mathcal{D}_{k}(P),\mathcal{D}_{k}(P_{s}))

for both Gaussian noise (left) and downsampling (right).

Figure 10: Schematic diagram of modified backbone architectures used for the 3D object classification or part segmentation experiments.

Backbone	Complex	Vectorization	Accuracy	mAcc	Precision	Recall	F1	Params (M)	$T_{e}$ (s)	$T_{inf}$ (s)
PointNet [ 78 ]	–	–	89.2	86.1	86.3	85.9	86.1	3.4	3.5	1.9
	Rips	PI	90.4	86.9	87.7	86.2	86.9	26.3	1361.5	273.6
	Rips	Landscape	90.7	87.3	87.4	86.6	87.0	3.6	2672.4	428.1
	Rips	PD	89.2	86.8	86.9	86.0	86.4	3.7	10.7	1.9
DGCNN [ 91 ]	–	–	92.9	90.2	90.8	90.1	90.4	1.8	15.3	15.7
	Rips	PI	93.5	90.9	91.1	90.6	90.8	24.6	1368.6	282.4

Table 4: Effect of topological vectorization choices across point-cloud backbones on ModelNet40 [ 93 ] classification. We compare Rips-complex-based persistence vectorization (PI, landscapes, PD) against non-topological baselines for PointNet [ 78 ] , DGCNN [ 91 ] , and PointTransformer [ 99 ] , reporting accuracy, robustness metrics, and model complexity.

T_{e}

denotes average training time per epoch;

T_{inf}

denotes inference time.

Figure 11: Performance comparison of vectorization methods for each backbone model for classification (left) and part-segmentation (right). Incorporating persistence-based vectorizations (PDs, PIs, PLs) yields consistent gains in accuracy and balance metrics (Precision/Recall/F1), demonstrating that topological summaries capture discriminative shape characteristics not fully represented by point geometry alone.

Backbone	Complex	Vectorization	cat. mIoU	ins. mIoU	Accuracy	Precision	Recall	F1	Params (M)	$T_{e}$ (s)	$T_{inf}$ (s)
PointNet [ 78 ]	–	–	79.7	82.4	92.7	88.5	87.9	88.2	8.3	38.9	18.1
	Rips	PI	80.3	82.7	93.1	89.5	87.7	88.6	31.0	1033.7	264.1
	Rips	Landscape	81.1	83.3	93.8	89.8	88.4	89.1	8.4	2170.4	519.4
	Rips	PD	79.6	82.6	93.2	89.3	88.1	88.7	8.5	40.8	57.7
DGCNN [ 91 ]	–	–	81.4	84.3	94.8	90.8	89.5	90.2	1.4	84.3	47.1
	Rips	PI	82.2	85.1	95.5	91.2	88.3	89.7	24.1	1089.5	283.9

Table 5: Part-segmentation performance on ShapeNetPart [ 95 ] using different topological vectorization across backbones. We compare Rips-complex-based persistence vectorization (PI, Landscape, PD) against non-topological baselines for PointNet [ 78 ] , DGCNN [ 91 ] , and PointTransformer [ 99 ] , reporting accuracy, mean IoUs, robustness metrics, and model complexity.

T_{e}

denotes average training time per epoch;

T_{inf}

denotes inference time.

Figure 12: Qualitative visualization of learned point cloud representations for part segmentation on ShapenetPart across different backbone architectures. Each row presents the same six object categories (airplane, mug, lamp, chair, motorbike, bag) as processed by PointNet (top), DGCNN (middle), and PointTransformer (bottom). Each column corresponds to a different backbone architecture with different input modalities. Only PC refers to the backbone architecture having a single branch with point cloud input only. PC & PI , PC & PL , and PC & PD refer to three-branch architectures where one branch processes the point cloud, and the other two branches process the

H_{1}

and

H_{2}

persistent homology features, vectorized using persistence images (PI), persistence landscapes (PL), and persistence diagrams (PD), respectively.

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

Abstract

Figures & tables

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