Spectral Bayesian Regression on the Sphere

2026-01-28

AI Summary

This work addresses nonparametric regression on the unit sphere by constructing an intrinsic Bayesian framework that leverages spherical harmonic bases induced by the Laplace–Beltrami operator to exactly diagonalize isotropic Gaussian random field priors, yielding a Gaussian sequence representation in the spectral domain. By introducing an optimal spectral truncation strategy and establishing posterior contraction rate theory, the authors demonstrate that the posterior mean coincides with the Laplace–Beltrami smoothing spline and achieves sharp minimax-optimal convergence rates over Sobolev classes. The approach accommodates priors with polynomially decaying angular power spectra—such as spherical Matérn processes—and simultaneously ensures theoretical optimality and geometric adaptivity to the spherical manifold.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We develop a fully intrinsic Bayesian framework for nonparametric regression on the unit sphere based on isotropic Gaussian field priors and the harmonic structure induced by the Laplace-Beltrami operator. Under uniform random design, the regression model admits an exact diagonalization in the spherical harmonic basis, yielding a Gaussian sequence representation with frequency-dependent multiplicities. Exploiting this structure, we derive closed-form posterior distributions, optimal spectral truncation schemes, and sharp posterior contraction rates under integrated squared loss. For Gaussian priors with polynomially decaying angular power spectra, including spherical Mat\'ern priors, we establish posterior contraction rates over Sobolev classes, which are minimax-optimal under correct prior calibration. We further show that the posterior mean admits an exact variational characterization as a geometrically intrinsic penalized least-squares estimator, equivalent to a Laplace-Beltrami smoothing spline.

Problem

• nonparametric regression
• Bayesian framework
• spherical harmonics
• Laplace-Beltrami operator
• posterior contraction

Innovation

• Bayesian nonparametric regression
• spherical harmonics
• Laplace-Beltrami operator
• posterior contraction
• geometrically intrinsic smoothing