Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds

2026-05-23

AI Summary

This work addresses the high computational cost of computing low-frequency eigenmodes of the Laplace–Beltrami operator on large-scale point clouds by introducing the Neural Eigenbasis Operator (NEO). NEO employs a feedforward network to directly predict the low-frequency eigensubspace, circumventing traditional iterative solvers. It constructs the target subspace using redundant basis functions and recovers accurate eigenpairs via a lightweight Rayleigh–Ritz refinement. By bypassing explicit eigenvector regression, NEO avoids sign-flip and rotational ambiguities, instead learning geometrically invariant and stable subspace representations. The method further incorporates a mass-aware neural operator with attention mechanisms that integrate point-area weights, enhancing robustness to non-uniform sampling and enabling zero-shot generalization across resolutions. NEO achieves near-linear runtime, significantly accelerating computation while preserving accuracy, and supports efficient transfer to high-resolution point clouds, providing a powerful spectral foundation for geometric analysis and downstream tasks.

Abstract

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.

Problem

• Laplacian eigenspace
• point clouds
• eigendecomposition
• spectral geometry
• low-frequency eigenmodes

Innovation

• Neural Eigenspace Operator
• Laplacian eigendecomposition
• mass-aware neural operator
• zero-shot generalization
• Rayleigh-Ritz refinement