Minimax spectral estimation of weighted Laplace operators
Yann Chaubet, Vincent Divol
TL;DR
This work addresses the problem of estimating the spectrum of weighted Laplace operators Δ_f on a known compact manifold, bridging diffusion-based spectral methods with rigorous minimax theory. The authors develop plug-in estimators based on estimating f, and establish exact minimax rates for eigenfunctions and eigenvalues under Hölder-Zygmund regularity, revealing faster rates for eigenfunctions in L^q-norms and a distinct, s-dependent rate for eigenvalues with an efficiency threshold at s>d/4. A central contribution is a general framework for estimating nonlinear functionals on Hölder-Zygmund spaces via higher-order debiasing, which yields minimax-optimal rates and asymptotic efficiency for a broad class of functionals, including eigenvalues. The paper also provides a robust perturbation theory for functions of Δ_f and spectral projectors, yielding practical guidance for score-based spectral approaches and clarifying the limitations and potential of graph-Laplacian-based estimators in known-manifold settings.
Abstract
Given $n$ i.i.d. observations, we study the problem of estimating the spectrum of weighted Laplace operators of the form $Δ_f=Δ+ α\nabla \log f\cdot \nabla$, where $f$ is a positive probability density on a known compact $d$-dimensional manifold without boundary and $α\in \mathbb{R}$ is a hyperparameter. These operators arise as continuum limits of graph Laplacian matrices and provide valuable geometric information on the underlying data distribution. We establish the exact minimax rates of estimation for this problem, by exhibiting two different rates of convergence for eigenfunctions and eigenvalues. When $f$ belongs to a Hölder-Zygmund class $\mathscr{C}^s$ of regularity $s\geqslant 2$, the eigenfunctions can be estimated with respect to the $\mathrm{L}^q$-norm ($q\geqslant 1$) via plug-in methods at the minimax rate $n^{-\frac{s+1}{2s+d}}$ for $d\geqslant 3$ (with different rates for $d\leqslant 2$). Moreover, eigenvalues can be estimated at the minimax rate $n^{-\frac{4s}{4s+d}}+n^{-\frac 12}$. In the regime $s>\frac d4$, we further show that asymptotically efficient estimators exist. We also present a general framework for estimating nonlinear functionals over Hölder-Zygmund spaces, with potential applications to a broad class of statistical problems.