Mapper-GIN: Topology-Based 3D Classification

• The paper presents Mapper-GIN, which applies Mapper-based decomposition to split 3D point clouds into overlapping regions for robust graph-based classification.
• The method utilizes PCA for lensing and DBSCAN for clustering, followed by a 4-layer GIN encoder that aggregates region features to recover global shape information.
• Mapper-GIN achieves competitive performance on the ModelNet40-C benchmark under various corruptions while using only 0.5M parameters for efficiency.

Mapper-GIN is a topology-inspired graph-based pipeline designed for robust and efficient classification of corrupted 3D point clouds. Instead of increasing model complexity or relying on specialized data augmentation, Mapper-GIN leverages structural abstraction by decomposing a point cloud into overlapping regions via the Mapper algorithm, then constructing a region graph classified using a Graph Isomorphism Network (GIN). This approach demonstrates strong performance and parameter efficiency under a broad suite of 3D corruptions, as evidenced on the ModelNet40-C benchmark (You et al., 5 Feb 2026).

1. Mapper-Based Decomposition for Point Clouds

Mapper-GIN commences with a structural decomposition of a 3D point cloud $X = \{x_1,\ldots,x_N\} \subset \mathbb{R}^3$ using the classical Mapper pipeline. The decomposition proceeds as follows:

1. Lens Function: A 3-dimensional Principal Component Analysis (PCA) embedding $f_{PCA}: X \to \mathbb{R}^3$ acts as a filter or "lens" function:

$f_{PCA}(x) = W^T(x - \mu),$

where $\mu \in \mathbb{R}^3$ is the sample mean and $W \in \mathbb{R}^{3 \times 3}$ contains the top three principal directions.

2. Cubical Cover: The PCA image $f_{PCA}(X)$ is covered with a uniform cubical grid $\mathcal{U}$ by dividing each coordinate axis into $n_{intervals}=6$ overlapping intervals, each of length

$$L = \frac{\max f_{PCA}(X) - \min f_{PCA}(X)}{6 - 5\gamma},$$

with overlap fraction $\gamma=0.3$. The Cartesian product of these intervals defines up to $6^3$ bins $U_i$, though only bins containing at least one point are retained.

3. Density-Based Clustering (DBSCAN): Each cover cell’s pullback $S_i = f_{PCA}^{-1}(U_i) \subset X$ is subdivided into density-connected components using DBSCAN with parameters $\varepsilon=0.05$ (search radius) and $\text{minPts}=10$ (minimum cluster size). Sparse points are discarded as noise. The resulting clusters in $S_i$ are denoted $$\mathcal{C}_{i,1}, \mathcal{C}_{i,2},\dots,\mathcal{C}_{i,k_i}$$.

This procedure produces a set of overlapping, spatially localized clusters that serve as a basis for higher-level abstractions.

2. Region-Graph Construction

From the collection of clusters $\{\mathcal{C}_{i,j}\}$, Mapper-GIN constructs an undirected region graph $G=(V,E)$:

• Nodes: Each cluster $\mathcal{C}_{i,j}$ forms a node $v = n_{i,j} \in V$.
• Edges: Edges are inserted between nodes if their corresponding clusters overlap in at least one point:

$$E = \{ (n_{i,j}, n_{i',j'}) \mid \mathcal{C}_{i,j} \cap \mathcal{C}_{i',j'} \neq \emptyset \}.$$

The adjacency matrix $A$ is thus binary, $A_{uv}=1$ if $C_u \cap C_v \neq \emptyset$ and zero otherwise.

This "region graph" abstracts away from individual points, encoding the topological organization and redundancy of the point cloud.

3. Graph Isomorphism Network (GIN) Encoder Architecture

After constructing $G=(V,E)$, each node $v$ receives a feature $z_v^{(0)}$ derived from its sub-cloud $\mathcal{C}_v$. Mapper-GIN applies a 4-layer GIN encoder, with layerwise updates:

$$h_v^{(k)} = \mathrm{MLP}^{(k)}\left( (1 + \epsilon^{(k)}) \cdot h_v^{(k-1)} + \sum_{u\in N(v)} h_u^{(k-1)} \right),$$

where $h_v^{(k)} \in \mathbb{R}^{d}$ is the node embedding at layer $k$, $\epsilon^{(k)}$ is a learnable scalar factor, and $\mathrm{MLP}^{(k)}$ is a 2-layer perceptron with BatchNorm and ReLU activations. Per-layer regularization employs DropEdge with $p_{edge} = 0.3$ and feature dropout $p_{feat}=0.3$. Post-aggregation steps include GraphNorm and ReLU.

For global graph classification, the embeddings from the final layer are aggregated via global max-pooling over nodes:

$g = \max_{v \in V} h_v^{(L)} \in \mathbb{R}^d,$

followed by LayerNorm and a linear classifier mapping $g$ to 40 output classes, with prediction via softmax-cross-entropy.

4. Training Protocol and Implementation Details

Input point clouds are downsampled to $N=1024$ points using farthest-point sampling. All graph construction (including Mapper decomposition) is performed offline, resulting in immutable region graphs. The classification head is trained using standard cross-entropy loss across 40 categories. Optimization employs Adam (initial learning rate $1\times 10^{-3}$, weight decay $1\times 10^{-4}$, StepLR decay with $\gamma=0.9$ every 10 epochs), batch size $512$, and $400$ training epochs. No data augmentation (rotations, jittering, or Mix strategies) is applied. For comparison, PointNet++ uses batch size $256$ due to GPU constraints (You et al., 5 Feb 2026).

5. Robustness Evaluation on ModelNet40-C

Mapper-GIN's robustness is evaluated on ModelNet40-C, containing 15 corruption types in three families (Density, Noise, Transformation) at five severity levels. Metrics include mean accuracy across severities and runs, subdivided by corruption type. The following table summarizes the main results:

Model	Params	Hard (O/L/B)	Density*	Noise*	Transf.	Overall
MLP	0.02M	39.1%	84.6%	83.8%	72.5%	71.2%
PointNet	3.5M	35.2%	85.3%	84.8%	75.5%	71.9%
PointNet++	1.7M	53.8%	82.9%	82.9%	80.9%	76.4%
Mapper-GIN-Base	0.5M	33.9%	64.9%	76.7%	74.9%	65.2%
Mapper-GIN	0.5M	48.3%	82.8%	84.8%	78.7%	75.1%

“Hard” indicates the most destructive corruptions: Occlusion, LiDAR, and Background. Density excludes Occlusion/LiDAR; Noise* excludes Background.

Full per-corruption results indicate Mapper-GIN matches or surpasses PointNet/MLP on Noise* and demonstrates competitiveness under transformations, while employing only 0.5M parameters (≈3× smaller than PointNet++).

6. Structural Abstraction and Robustness Properties

Mapper-GIN's corruption robustness is primarily attributed to region-level structural abstraction:

• Topology Preservation: Region overlaps are stable under global deformations (e.g., rotations, warping), allowing GIN message-passing to recover global shape information.
• Outlier Mitigation: DBSCAN-based clustering removes sparse noise (e.g., impulse, upsampling), and global region-level pooling reduces pointwise jitter effects.
• Parameter Efficiency: Operating on a graph of regions (tens of nodes) instead of $1024$ points yields robustness with significantly fewer parameters.

Such design choices position Mapper-GIN as an interpretable, parameter-efficient model for robust 3D visual recognition.

7. Limitations and Future Prospects

Acknowledged limitations include:

• Sensitivity to Point Removal: Certain corruptions (Density ↓, Cutout) distort the region graph topology more severely than in pointwise baselines, leading to larger performance drops.
• Hyperparameter Dependence: Mapper-GIN requires careful tuning of lens choice, cover resolution/overlap, and clustering parameters for each dataset.
• Offline Graph Construction: The Mapper decomposition is precomputed and fixed; current architecture is not end-to-end differentiable. Future work could address this via learnable lenses or differentiable covers.

Overall, Mapper-GIN demonstrates that region-graph abstraction and a lightweight GIN encoder are effective for achieving strong corruption robustness in 3D point cloud classification at reduced parameter cost (You et al., 5 Feb 2026).