Quantifying The Complexity Of Geodesic Paths On Curved Statistical Manifolds Through Information Geometric Entropies and Jacobi Fields

TL;DR

This paper analyzes the complexity of geodesic paths on curved statistical manifolds using information geometric entropies and Jacobi fields, revealing how correlations influence complexity decay and divergence attenuation.

Contribution

It introduces a novel approach to quantify geodesic complexity on curved statistical manifolds considering correlations via information geometric tools.

Findings

Power law decay of information geometric complexity influenced by correlations

Correlational structure causes asymptotic compression of macrostates

Embedding constraints attenuate exponential divergence of Jacobi fields

Abstract

We characterize the complexity of geodesic paths on a curved statistical manifold M_{s} through the asymptotic computation of the information geometric complexity V_{M_{s}} and the Jacobi vector field intensity J_{M_{s}}. The manifold M_{s} is a 2l-dimensional Gaussian model reproduced by an appropriate embedding in a larger 4l-dimensional Gaussian manifold and endowed with a Fisher-Rao information metric g_{{\mu}{\nu}}({\Theta}) with non-trivial off diagonal terms. These terms emerge due to the presence of a correlational structure (embedding constraints) among the statistical variables on the larger manifold and are characterized by macroscopic correlational coefficients r_{k}. First, we observe a power law decay of the information geometric complexity at a rate determined by the coefficients r_{k} and conclude that the non-trivial off diagonal terms lead to the emergence of an asymptotic information geometric compression of the explored macrostates {\Theta} on M_{s}. Finally, we observe that the presence of such embedding constraints leads to an attenuation of the asymptotic exponential divergence of the Jacobi vector field intensity.