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

• URL: http://arxiv.org/abs/2605.24390v1
• Date: Sat, 23 May 2026 04:21:20 GMT
• Title: Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds
• Abstract summary: We introduce the Neuralspace Operator (NEO), a feed-forward framework to predict the spectrum directly from point clouds.<n>NEO circumvents the ill-posed nature of standard eigenvector regression, by learning the stable, invariant low-frequency subspace instead.<n>Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy.
• Score: 13.977358640919462
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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