# Geometric Information Decomposition for Weighted Empirical Measures on the Sphere (Raw Archive)

- **arXiv**: 2607.02943v1
- **Author**: Kisung You (Baruch College)
- **Published**: July 3, 2026
- **Category**: stat.ME
- **Code**: https://github.com/kisungyou/GeoInfoDec

## 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. Standard approach fits a von Mises-Fisher distribution and reports concentration or entropy — 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. First gap recovers vMF information, 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.

## Key Technical Details

- **Feature spaces**: Nested V₀ ⊂ V₁ ⊂ ... ⊂ V_L ⊂ L²₀(M, ν), with V₀ = {0} (uniform)
- **Maximum-entropy projection**: p_L,m = exp{λ_L(m)ᵀ φ_L(x) - ψ_L(λ_L(m))}, where λ solves ∇ψ_L(λ) = m
- **Entropy deficit**: D_L(P) = KL(p^P_L ν ∥ ν) — non-uniform info explained up to level L
- **Information gap**: I_L(P) = D_L(P) - D_{L-1}(P) — additional info at level L
- **Level 1** = vMF mean-direction, **Level 2** = Fisher-Bingham axial/girdle, **Level 3+** = multimodal
- **Null calibration**: 2a²_n Î_L ⇒ χ²_q under null I_L=0 (Theorem 8)
