Adaptive estimation of Sobolev-type energy functionals on the sphere

TL;DR

This paper develops an adaptive, minimax-optimal method for estimating Sobolev-type energy functionals of unknown densities on the sphere using spherical needlet frames and Lepski's method.

Contribution

It introduces a new adaptive estimator for Sobolev functionals on the sphere that achieves optimal rates without nonlinear or sparsity assumptions.

Findings

Achieves minimax-optimal rates over Sobolev classes.

Provides explicit bias-variance risk bounds.

Uses needlet frames for localized multiscale analysis.

Abstract

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods.