Probability-Generating Function Kernels for Spherical Data

TL;DR

The paper introduces probability-generating function (PGF) kernels for data on the unit hypersphere, defining them as compositions of PGFs applied to the input correlation and demonstrating their generalization of RBF kernels. It develops a semi-parametric learning approach that truncates PGF expansions and optimizes coefficients, enabling GP regression and deep kernel learning on spherical domains. Theoretical results establish positive-definiteness and rotational stationarity, and relations to common kernels, while empirical studies on circular/spherical GP regression and hyperspherical DKL show competitive or superior performance of PGF kernels. The work outlines future directions in kernel-closure approximations, parameter selection, and practical spherical-data applications like PDE emulation and earth science tasks.

Abstract

Probability-generating function (PGF) kernels are introduced, which constitute a class of kernels supported on the unit hypersphere, for the purposes of spherical data analysis. PGF kernels generalize RBF kernels in the context of spherical data. The properties of PGF kernels are studied. A semi-parametric learning algorithm is introduced to enable the use of PGF kernels with spherical data.

Figures (3)

• Figure 1: Visual demonstration of the effect of PGF kernel width and depth on predictive performance for GP regression fitted to data drawn from the circular von Mises density. The blue line represents the circular von Mises density from which the data are simulated. The orange, pink and red points represent training data, test data and predictions, respectively. Increasing the width or depth brings the predictions (red points) closer to the test data (pink points) and therefore improves predictive performance.
• Figure 2: Noiseless (left) and noisy version (right) of the spherical exp-cos function given by equation \ref{['eq:expcosf']}. For the noisy version, Gaussian noise $\mathcal{N}(\mu = 0, \sigma=0.2)$ is added to the exp-cos function. Training sets are drawn from such noisy realizations, whereas test sets are drawn from the noiseless version of the exp-cos function.
• Figure 3: Embeddings in $\mathbb{R}^3$ generated by training a DKL classification model on a dataset simulated from a hyperspherical Thomas process in $\mathbb{R}^{18}$. Each of the four spheres displays the embeddings generated by training the DKL model with a different kernel. On each sphere, the four colors of the embeddings represent the four classes of the classification problem.

Theorems & Definitions (28)

• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Proposition 3.6
• proof