Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields

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 2 l -dimensional Gaussian model reproduced by an appropriate embedding in a larger 4 l -dimensional Gaussian manifold and endowed with a Fisher–Rao information metric g μ ν ( Θ ) 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 Θ 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.