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Spectral Bayesian Regression on the Sphere

Claudio Durastanti

TL;DR

This work addresses Bayesian nonparametric regression on the unit sphere by exploiting intrinsic geometry through isotropic Gaussian fields and spherical harmonics. By imposing polynomially decaying angular power spectra, notably spherical Matérn priors, it achieves exact harmonic diagonalization under uniform random design, derives closed-form posteriors, and proves minimax-optimal contraction rates with rate ρ_n = n^{-β/(2α+d)} when f_0 ∈ H^β(S^d) and α≥β. A key contribution is the exact variational interpretation: the posterior mean is the unique minimizer of a geometrically intrinsic penalized least-squares functional, linking Gaussian process regression, Sobolev regularization, and Laplace–Beltrami smoothing splines. The results illuminate the role of sphere geometry in Bayesian inference, connect to kernel ridge and Whittle-type methods, and are complemented by numerical experiments illustrating contraction rates and the impact of prior calibration.

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 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.

Spectral Bayesian Regression on the Sphere

TL;DR

This work addresses Bayesian nonparametric regression on the unit sphere by exploiting intrinsic geometry through isotropic Gaussian fields and spherical harmonics. By imposing polynomially decaying angular power spectra, notably spherical Matérn priors, it achieves exact harmonic diagonalization under uniform random design, derives closed-form posteriors, and proves minimax-optimal contraction rates with rate ρ_n = n^{-β/(2α+d)} when f_0 ∈ H^β(S^d) and α≥β. A key contribution is the exact variational interpretation: the posterior mean is the unique minimizer of a geometrically intrinsic penalized least-squares functional, linking Gaussian process regression, Sobolev regularization, and Laplace–Beltrami smoothing splines. The results illuminate the role of sphere geometry in Bayesian inference, connect to kernel ridge and Whittle-type methods, and are complemented by numerical experiments illustrating contraction rates and the impact of prior calibration.

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 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.
Paper Structure (35 sections, 5 theorems, 173 equations, 3 figures, 2 tables)

This paper contains 35 sections, 5 theorems, 173 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Preliminaries
  4. Gaussian Process Regression and Spectral Regularization
  5. Bayesian nonparametric regression and Gaussian process priors
  6. Spectral and operator-theoretic structure of Gaussian process priors
  7. Gaussian Fields on the Sphere
  8. Geometry and Harmonic Structure
  9. Isotropic Gaussian Fields
  10. Spectral formulation of Gaussian process priors on the sphere
  11. Analogy with Euclidean spectral representations.
  12. Matérn-type spectral priors on the sphere.
  13. Covariance operators and spectral decomposition on $\mathbb{S}^d$.
  14. Functional calculus of the Laplace--Beltrami operator.
  15. Matérn-type priors and Sobolev regularity.
  16. ...and 20 more sections

Key Result

Theorem 3.3

Assume the nonparametric regression model of Section sec:posterior and let the prior satisfy Condition cond:polydecay for some $\alpha>d/2$. Assume that the true regression function satisfies Let the truncation level be chosen as Then the posterior distribution induced by the truncated Gaussian prior contracts around $f_0$ at rate in the $L^2(\mathbb{S}^d)$ norm, in the sense that for every suf

Figures (3)

  • Figure 1: Log--log plot of the empirical RMSE of the posterior mean versus sample size $n$. The dashed line corresponds to the least-squares fit, while the dotted line indicates the theoretical slope $-\beta/(2\alpha+2)=-1/3$.
  • Figure 2: Visualization on $\mathbb{S}^2$ of the true regression function $f_0$ (left), the posterior mean $\hat{f}_{n,L_n}$ (center), and the pointwise error $\hat{f}_{n,L_n}-f_0$ (right), for a representative realization. Color intensity represents function values on the sphere.
  • Figure 3: Log--log plot of the empirical RMSE versus sample size under prior mis-calibration. Solid lines correspond to empirical RMSE curves, while dotted lines indicate theoretical slopes $-\beta/(2\alpha+2)$ for $\alpha=1,2,3$. The correctly calibrated case $\alpha=\beta=2$ achieves the optimal rate.

Theorems & Definitions (25)

  • Remark 2.1: Matérn priors on manifolds
  • Remark 2.2: Matérn priors as elliptic Gaussian measures
  • Remark 3.2: Relation to the literature
  • Theorem 3.3: Posterior contraction under spectral truncation
  • Corollary 3.4: Minimax optimality under correct prior calibration over Sobolev classes
  • Remark 3.5: Interpretation, calibration, and saturation
  • Remark 3.6: Comparison with needlet regression and Besov regularity
  • Remark 3.7: Unknown prior smoothness and Whittle-type adaptation
  • Remark 3.8: Spectral coefficient representation and functional viewpoint
  • Definition 3.9: Spherical Matérn prior
  • ...and 15 more