Spectral stability of empirical metric-measure Laplacians

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 $\mu$ on a Polish metric space, we compare the eigenvalues of the empirical weighted Laplacian operator ${\mathrm{\Delta }}_{{\mu }_n}^h$ to those of the population counterpart ${\mathrm{\Delta }}_{\mu }^h$ under a spectral gap condition, bounding the relative error by $1\mathrm{/}\sqrt{n{v}_{\mu }(h)}$ for eigenvalues of order smaller than $h^{-2}$, where $v_{\mu }(h)$ is the smallest mass of a ball of radius $h$. This bound requires very weak regularity conditions on $\mu$: it is satisfied if $\mu$ 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.