1.8 KiB
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)