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 VM_s and the Jacobi vector field intensity JM_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 FisherRao information metric gμν(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 rk 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.