Geometric Factorization of Sufficient Harmonic Representations

2026-06-05

AI Summary

This work addresses the problem of constructing minimal sufficient invariant representations for likelihood families invariant under Lie group actions. By introducing spherical harmonic analysis on compact homogeneous spaces, the authors establish a connection between the spherical Fourier coefficients of finite-bandwidth harmonic exponential families and minimal sufficient statistics. Leveraging Clebsch–Gordan decomposition, they isolate the trivial representation component to derive an algebraic expression for the partition function. The study theoretically proves that empirical harmonic coefficients constitute minimal sufficient statistics, thereby providing—for the first time—an explicit algebraic form of the partition function for this class of models. This result establishes a computable harmonic representation framework for invariant statistical inference.

Abstract

For tasks of likelihood families invariant under the action of a lie group, the quotient is the minimal sufficient invariant representation. On compact homogeneous spaces, this quotient representation admits a harmonic realization through spherical Fourier coefficients; for finite-band harmonic exponential families, the empirical harmonic coefficients are minimal sufficient statistics. The partition function can be expressed algebraically by extracting the trivial representation component through Clebsch-Gordan decomposition.

Problem

• sufficient representation
• Lie group invariance
• harmonic analysis
• homogeneous space
• partition function

Innovation

• geometric factorization
• harmonic representation
• minimal sufficient statistics
• Clebsch-Gordan decomposition
• spherical Fourier coefficients