Spectral stability of empirical metric-measure Laplacians
Abstract
The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth h>0. Specifically, given n i.i.d. samples from a probability measure μ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator Δμnh to those of the population counterpart Δμh under a spectral gap condition, bounding the relative error by 1/√(nvμ(h)) for eigenvalues of order smaller than h−2, where vμ(h) is the smallest mass of a ball of radius h. This bound requires very weak regularity conditions on μ: it is satisfied if μ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale O(h). Even for measures having densities of regularity s>2 on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.
Related
- Sharp spectral-scale stability for parabolic equations with measure-valued delay
- Metric dimension and maximum eigenvalue multiplicity of graph Laplacians
- Spectral Reality for Certain Fourth-Order Nonsymmetric Finite-Difference Laplacians
- Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds
- The Sunada method on Metric Measure Spaces
BibTeX
@misc{divol2026spectral,
title = {{Spectral stability of empirical metric-measure Laplacians}},
author = {Vincent Divol},
howpublished = {HAL},
year = {2026},
month = oct,
eprint = {2608.23150},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05783560v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Spectral stability of empirical metric-measure Laplacians
AU - Vincent Divol
PY - 2026
DA - 2026/10/07
PB - HAL
UR - https://hal.science/hal-05783560v1
AB - The variance of nonparametric estimators is typically insensitive to the regularity of the object being estimated. We establish such a property for the spectra of graph Laplacian matrices at a fixed bandwidth h>0. Specifically, given n i.i.d. samples from a probability measure μ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator Δ_μ_n^h to those of the population counterpart Δ_μ^h under a spectral gap condition, bounding the relative error by 1/√(nv_μ(h)) for eigenvalues of order smaller than h^−2, where v_μ(h) is the smallest mass of a ball of radius h. This bound requires very weak regularity conditions on μ: it is satisfied if μ belongs to the class of coarse PI measures that we introduce. This class contains measures on metric graphs, spaces with sufficiently regular boundaries, corners, or branch points, together with discretizations or thickenings of these at scale O(h). Even for measures having densities of regularity s>2 on manifolds (the only known case so far), our bound improves on the state-of-the-art by shaving off logarithmic factors.
ER -