Neural Graph Laplacian Estimator

• The paper introduces a neural approach that learns optimal discrete Laplacian operators from point clouds, replicating mesh-based Laplace–Beltrami behavior.
• A U-Net-style GNN architecture uses spectral and spatial probes to accurately map local graph structure, achieving order-of-magnitude MSE improvements.
• Parameterized Laplacians allow adaptive control over diffusion scopes in GNNs, effectively handling heterophilic graph scenarios with tunable spectral properties.

A neural graph Laplacian estimator is a data-driven approach to constructing discrete Laplacian operators on graphs or point clouds, leveraging neural networks to learn operator weights so as to replicate the behavior of desirable analytical or geometric Laplacians. Neural estimators address the longstanding problem of defining optimal discrete Laplacians directly from raw, unordered data (such as 3D point clouds), as well as enabling adaptive, task-driven modifications of the diffusion scope in graph neural networks (GNNs). Two representative lines are graph-based Laplacian estimators trained to mimic mesh Laplace–Beltrami operators in geometric settings (Pang et al., 2024), and parameterized Laplacians that accommodate heterophilic graphs by learning flexible diffusion operators (Lu et al., 2024).

1. Discrete Laplacians and the Estimation Problem

Discrete Laplacian operators are fundamental tools for analyzing geometry and function diffusion on graphs, meshes, or point sets. On a triangle mesh discretizing a Riemannian manifold, the Laplacian is commonly defined via the cotangent formula:

$w_{ij}^{gt} = \cot(\alpha_{ij}) + \cot(\beta_{ij})$

with the Laplacian acting as

$\Delta_{gt} f := M_{gt}^{-1} L_{gt} f$

where $L_{gt}$ is the stiffness matrix (encoding cotangent weights), $M_{gt}$ is the diagonal mass matrix (e.g., Voronoi cell areas), and $f$ is a scalar function on vertices. The Laplace–Beltrami operator displays critical properties such as symmetry, positive semidefiniteness, and convergence to the smooth manifold Laplace operator under mesh refinement. However, these mesh-based definitions cannot be directly translated to point clouds lacking explicit triangulations or to irregular graphs, necessitating learned estimators of the Laplacian.

2. Neural Laplacian Estimation for 3D Point Clouds

To address the absence of mesh structure in point clouds, Pang et al. (Pang et al., 2024) proposed NeLo, a neural estimator for graph Laplacians that operates on K-nearest neighbor (KNN) graphs built over point clouds. The KNN graph is constructed as $G=(V,E)$, where each vertex $v_i$ connects to its $K$ nearest Euclidean neighbors, creating a symmetric adjacency structure. The Laplacian is parametrized by edge weights $w_{ij} \geq 0$ and diagonal masses $M_{ii}>0$:

$$L_{ij} = \begin{cases} \sum_{k \neq i} w_{ik}, & i=j \ - w_{ij}, & i \neq j, (i,j) \in E \ 0, & \text{otherwise} \end{cases}$$

with the operator $\hat{\Delta} f = M^{-1}L f$.

A U-Net-style GNN architecture learns $\{w_{ij}\}$ and $\{M_{ii}\}$ from local graph structure, omitting raw coordinates at input for translation-invariance:

• Vertex features: $x_i^{(0)} = [1,1,1, \deg(v_i)]$
• Message passing: Residual blocks aggregate neighbor features and relative positions, followed by GroupNorm and ReLU.
• Weight decoding: Two MLPs map point features (with symmetry enforced by dependence on $(p_i-p_j)^2$) to $w_{ij}$ and $M_{ii}$, with Softplus activations.

3. Training Methodology: Behavioral Loss and Probe Functions

Supervising Laplacian weights directly is infeasible due to graph connectivity mismatches between KNN and mesh-based constructions. Instead, behavioral imitation is employed: the network is trained so that its Laplacian acts on a diverse set of probe functions $F$ similarly to the ground-truth mesh Laplacian. The loss is

$$\mathcal{L}_{lap} = \sum_{f \in F} w_f \| M^{-1}Lf - M_{gt}^{-1}L_{gt}f \|_2^2$$

with a normalization $w_f = 1/(\text{mean}(\Delta_{gt}f)+\varepsilon)$. A secondary mass-matching term penalizes deviations in diagonal masses:

$$\mathcal{L}_{mass} = \| \text{diag}(M)-\text{diag}(M_{gt}) \|_2^2$$

and total loss is $$\mathcal{L} = \mathcal{L}_{lap} + \lambda \mathcal{L}_{mass}$$, $\lambda=0.1$.

Two categories of probe functions are used:

• Spectral probes: First 64 non-constant eigenvectors of $L_{gt}$, filtered to attenuate dominance of low-frequency modes.
• Spatial probes: 48 sinusoids parameterized by direction, frequency, and phase.

The dataset (~12k watertight CAD models remeshed to $\sim$5.2k vertices each) from ShapeNet provides training and evaluation coverage across classes; all meshes are normalized to $[-1,1]^3$. Hyperparameters include $K=8$ neighbors, GNN feature dimension 256, and a training budget of 500 epochs with AdamW optimizer.

4. Parameterized Laplacians and Diffusion Scopes in GNNs

Whereas NeLo learns the Laplacian from data to mimic mesh behavior, recent work introduces parameterized Laplacians $L^{(\alpha,\gamma)}$ to provide adaptive control over the spectral (and thus diffusion) properties of the graph operator, addressing limitations in conventional Laplacians for heterophilic graphs (Lu et al., 2024). The parameterized normalized Laplacian is defined as

$$L^{(\alpha,\gamma)} = \gamma \cdot [\gamma D + (1-\gamma)I]^{-\alpha} L [\gamma D + (1-\gamma)I]^{\alpha-1}$$

where $L = D-A$ is the combinatorial Laplacian, $D$ is degree, $A$ is adjacency, and $(\alpha, \gamma)$ are tunable scalars.

This construction interpolates known Laplacians:

• $\alpha=1, \gamma=1$: random walk Laplacian $L_{rw}$
• $\alpha=1/2, \gamma=1$: symmetric normalized Laplacian $L_{sym}$
• $\gamma\to 0$: reduces to unnormalized Laplacian $L$ (up to scaling)

A key theoretical result establishes an order-preserving relationship between diffusion distance and the first spectral distance of the Laplacian; tuning $\gamma$ controls the spectral gap and the rate at which information diffuses across the graph. Lower $\gamma$ accelerates long-range mixing, critical for learning in heterophilic networks.

Topology-guided rewiring is employed to further augment diffusion by explicitly connecting nodes that are "far apart" in spectral coordinates, thereby reducing their effective diffusion distance.

5. Empirical Performance and Properties

NeLo: Point Cloud Laplacian Estimation (Pang et al., 2024)

Evaluation metrics on test point clouds include mean squared error (MSE) per probe, fraction of outlier MSEs ($R_{MSE>1}$), and sparsity (average nonzeros per Laplacian row). NeLo achieves:

• Order-of-magnitude lower MSE ($\approx 0.0049$) and $R_{MSE>1}$ (by $\approx 50\times$) than uniform KNN, local heat-kernel, or intrinsic Delaunay methods.
• Robust generalization across point clouds with thin structures, sharp features, or significant sparsity.
• Preservation of key operator properties: symmetry, KNN-locality, positive semidefiniteness, and non-negative weights.
• Empirical convergence to ground truth is observed under moderate densification of input point clouds.

Ablation studies reveal that omitting spectral or spatial probes, or using standard GNN layers (e.g., GraphSAGE, GATv2), degrades accuracy, confirming the importance of geometry-aware architecture and training protocol.

Parameterized Laplacians: Heterophilic Graph Learning (Lu et al., 2024)

Synthetic and real-world benchmarks on node classification tasks exhibit:

• Superior performance of PD-GCN and PD-GAT architectures over baselines, with top-3 ranking on 6 of 7 heterophilic datasets.
• Strong empirical correlation between the optimal value of $\gamma$ and the level of graph homophily: highly heterophilic graphs require low $\gamma$ (wider diffusion), while homophilic graphs prefer high $\gamma$ (local diffusion).
• No gradient-based learning of $(\alpha, \gamma)$ is performed; these are hyperparameters selected via grid search on validation sets.
• Parameterized spectral embeddings and rewiring further enhance the long-range connectivity essential for heterophilic tasks.

6. Applications in Geometry Processing and Learning

Neural Laplacian estimators enable standard Laplacian-based processing directly on point clouds or graphs, without constructing a mesh:

• Heat diffusion and heat-kernel smoothing (explicit Euler integration) align with classical mesh-based results.
• Geodesic distance computation via the heat method accurately recovers ground-truth distances.
• Laplacian smoothing for denoising is equivalent to mesh counterparts.
• Spectral filtering, including computation of eigenmodes for Fourier analysis or filtering, matches mesh-based outputs.
• As-rigid-as-possible (ARAP) deformation supports direct editing of point sets.

Parameterized Laplacians in heterophilic GNNs allow continuous control of the receptive field, enhancing classification performance and adaptability to varying levels of label homophily.

7. Significance, Limitations, and Generalization

Neural graph Laplacian estimators such as NeLo rigorously establish a learned, data-driven discrete $\Delta$ operator suitable for geometric inference on unordered point clouds (Pang et al., 2024). Operator properties (locality, symmetry, positive semidefiniteness) are guaranteed by design, and empirical results demonstrate quantitative and qualitative fidelity to analytical operators. In learning settings, parameterized Laplacians confer continuous, interpretable control over the spectral behavior of GNNs, with validated advantages on diverse benchmarks (Lu et al., 2024).

While NeLo forgoes mesh construction entirely, a plausible implication is that extension to topologically complex or non-Euclidean settings may require additional architecture innovations or probe designs. For parameterized Laplacians, treating $(\alpha, \gamma)$ as network-learned, rather than hyperparameters, could potentially improve adaptivity and automation of diffusion scope; however, this remains open for further investigation.