Geometric Information Decomposition for Weighted Empirical Measures on the Sphere

2026-07-03

AI Summary

Existing approaches, such as those based on the von Mises–Fisher (vMF) distribution, model only the mean direction and thus fail to capture complex geometric structures—such as multimodality, axial symmetry, or zonal patterns—in spherical weighted empirical measures. This work proposes a Geometric Information Decomposition (GID) framework that leverages spherical harmonics to construct a nested sequence of maximum-entropy projections, hierarchically quantifying the incremental information-theoretic gaps at each level. For the first time, this enables a layered decomposition of higher-order geometric structures inherent in spherical measures, transcending the limitations of single-parameter models by fully characterizing features ranging from the mean direction to high-order anisotropy and fine angular patterns. Theoretical guarantees include invariance, consistency, and asymptotic normality, along with a quadratic-form zero-calibration test. Experiments on circular and spherical data successfully reveal latent structures invisible to vMF-based methods, demonstrating the approach’s efficacy and practical utility.

Abstract

We study directional uncertainty when the data already represent a weighted probability measure on the unit sphere, as in importance samples, quadrature rules, or attention-weighted embeddings. A standard approach fits a von Mises-Fisher distribution and reports its concentration or entropy. This is principled but incomplete because vMF uses only mean-direction information and can miss antipodal, axial, girdle-like, or multimodal structure. We introduce the geometric information decomposition (GID), which fits a nested sequence of maximum-entropy projections using spherical features and reports the entropy gap added at each level. The first gap recovers vMF information, the second captures Fisher-Bingham/Bingham-type anisotropy, and later gaps capture finer angular structure. We prove invariance, consistency, asymptotic normality away from zero gaps, and a quadratic-form null calibration for deciding whether a new level carries information. Experiments on circular and spherical examples, calibration studies, and a query-weighted digit projection show when vMF uncertainty suffices and when higher-order gaps reveal hidden structure.

Problem

• directional uncertainty
• weighted empirical measures
• spherical data
• geometric structure
• information decomposition

Innovation

• Geometric Information Decomposition
• spherical data
• maximum-entropy projection
• directional uncertainty
• von Mises-Fisher distribution