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