Reproduce “Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds” on OpenResearch

About this paper

Zherui Yang, Tao Du, Ligang Liu

NEO predicts low-frequency Laplace–Beltrami eigenspaces on point clouds, rather than individual eigenvectors, then uses a small Rayleigh–Ritz solve to recover eigenpairs. Mass-aware attention weights aggregation by estimated local area. On ShapeNet, span loss is 3.7 × 10⁻³ ± 1.7 × 10⁻³. At 512,000 points, 96 eigenpairs take 0.52 seconds; NEO remains approximate, with degradation on higher modes and fine-scale structures.

Key insights

• Learning the stable, invariant low-frequency eigenspace (a subspace) rather than ambiguous individual eigenvectors provides a robust and mathematically sound approach for spectral prediction.
• Explicitly integrating point-wise mass weights into the neural operator's attention mechanism is crucial for robustness against non-uniform sampling and enables strong zero-shot generalization to unseen resolutions and discretizations.
• Neural operators can effectively amortize the computational cost of LBO eigendecomposition, transforming expensive iterative computations into a rapid, near-linear scaling inference step.

Summary by alphaXiv.