Spectral stability of empirical metric-measure Laplacians

What the math gives to ML

The paper’s transferable asset is a finite-sample stability law for the spectrum of a fixed-bandwidth metric-measure graph Laplacian: below the operator’s natural frequency cutoff, relative eigenvalue error scales as $(n v_\mu(h))^{-1/2}$ rather than depending strongly on density smoothness. This suggests making spectral neural features conditional on an explicit reliability estimate instead of treating Laplacian eigenvectors or diffusion coordinates as noiseless preprocessing. A practical use is an adaptive spectral positional-encoding module that selects bandwidth, number of eigenvectors, and sample size based on local ball mass and eigengaps, then exposes only spectrally stable coordinates to a GNN or transformer.