Spectral Bayesian Regression on the Sphere

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 Matérn 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.

💡 Research Summary

This paper develops a fully intrinsic Bayesian framework for non‑parametric regression on the unit sphere (S^{d}). The authors start by exploiting the rotational invariance of the sphere: isotropic Gaussian random fields on (S^{d}) are completely characterized by an angular power spectrum (C_{\ell}) and admit an exact expansion in spherical harmonics (Y_{\ell,m}), the eigenfunctions of the Laplace–Beltrami operator with eigenvalues (\lambda_{\ell}=\ell(\ell+d-1)). Because the multiplicity of each eigenvalue grows like (M_{d,\ell}\asymp \ell^{d-1}), the high‑frequency part of the model possesses many more degrees of freedom than in Euclidean settings, a fact that fundamentally influences the bias–variance trade‑off of any estimator.

The regression model assumes a uniform random design: observation locations ({x_i}{i=1}^n) are i.i.d. from the normalized surface measure on (S^{d}). Under this design the design matrix built from the covariance kernel diagonalizes exactly in the spherical harmonic basis. Consequently the whole Bayesian model reduces to a Gaussian sequence model: for each pair ((\ell,m)) the coefficient (\theta{\ell,m}) follows a prior (N(0,C_{\ell})) and the noisy observations provide an empirical coefficient (\bar y_{\ell,m}). The posterior for each mode is then \