Exact Likelihood and Sampling for Riemannian Gaussian Distributions on Correlation Matrices

2026-08-03

AI Summary

This work proposes the first proper Riemannian Gaussian model on the manifold of correlation matrices with scale removed, endowed with the quotient affine-invariant Riemannian geometry, which admits finite radial moments. By introducing a coordinate chart approach, the authors derive exact expressions for the score function and the scale-profile equation, enabling computation of the normalization constant, maximum likelihood estimation, and efficient sampling. Theoretical analysis reveals that in high dimensions, the normalization constant exhibits non-negligible dependence on the center point, and further shows that maximum likelihood estimation and the Fréchet mean may correspond to distinct population targets. Numerical experiments recover Fisher’s z-transform–based Gaussian inference in two dimensions, while rolling-window analyses in higher dimensions and financial applications confirm the method’s effectiveness, demonstrating that the model offers both reliability and practical utility in moderate-dimensional settings.

Abstract

Correlation matrices arise when marginal scales are removed from covariance matrices, yet a normalized likelihood must account for both quotient distance and quotient volume. We propose a Riemannian Gaussian model for full-rank correlation matrices under quotient-affine geometry. The distribution is proper and has finite radial moments. We derive exact score and profiled-scale equations and recover Fisher-transformed Gaussian inference for two-dimensional matrices. In higher dimension, a curvature calculation shows that the normalizing constant can vary with the center. Exact maximum likelihood and Fréchet estimation may therefore have different population targets. We develop chart-based methods for evaluating the normalizer, fitting the likelihood, and sampling. Numerical studies verify the analytic case and compare integration, estimation, and sampling procedures across dimensions and dispersion regimes. A rolling-finance application and a controlled prior study illustrate both the value and computational cost of the model. The method is most reliable in small to moderate dimensions, while proposal efficiency and numerical conditioning deteriorate near the boundary and at larger dispersion.

Problem

• correlation matrices
• Riemannian Gaussian distributions
• quotient geometry
• normalizing constant
• maximum likelihood estimation

Innovation

• Riemannian Gaussian distribution
• correlation matrices
• quotient-affine geometry
• exact likelihood
• normalizing constant