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-Ms and the Jacobi vector field intensity J(Ms). 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. (C) 2010 Elsevier B.V. All rights reserved.