Information Geometry of Smooth Densities on the Gaussian Space: Poincaré Inequalities

Information geometry of dynamics on graphs and hypergraphs

“…This induced structure can be regarded as an extension to discrete manifolds of the Otto structure [65,66]: the formal Riemannian structure induced by the L 2 -Wasserstein distance. This is also related to Pistone's infinitedimensional information geometry [72,121].…”

“…Under this doubly dual flat structure, we can consider the dynamics of densities as a generalized flow, and various previous results can be unified in this framework. We exclusively consider dynamics of densities on finite-dimensional discrete manifolds, i.e., finite graphs or hypergraphs, because the structure introduced here can be explicitly manifested in this setup and also because we do not need the mathematically elaborated setup for infinite-dimensional information geometry on a smooth manifold [72]. For the case of FPE in a continuous state space, the dually flat structure built on the flux space can be reduced to the formal Riemannian geometric structure of L 2 Wasserstein geometry where the convex functions that induce the dually flat structure become quadratic.…”

“…1.2. From the viewpoint of homological algebra, the structure we work on is a modification of the chain and cochain complexes of graphs or hypergraphs, which replace the usual inner product duality [73] on each pair of chains and cochains with Legendre duality. Moreover, the dually flat space built on the flux space is linked to a finite-dimensional version of Orlicz spaces [74], which have been employed for constructing infinite-dimensional information geometry [72]. From the nice properties of the doubly dual flat structures, we can obtain information-geometric extensions of the Helmholtz-Hodge-Kodaira (HHK) decomposition (Theorem 1), the Otto calculus (Theorem 2), and its induction to cycle spaces(Theorem 3).…”

“…64 Second, the scalar function ψ * ( f ) is the N-function. The N-function of the (cosh( f ) − 1)-type and the associated Orlicz space have been employed for establishing the infinite-dimensional information geometry by Pistone [72,135,136]. In functional analysis, the Orlicz space is a generalization of the L p spaces, which arise naturally when we work on the L log + L space for the divergences and large deviation functions.…”

“…On the contrary, by definition, j st ∈ P v (0) is the flux that makes the state x a steady state, i.e., ẋ = 0, and is also induced by the force in the same quotient set of force 72 These decompositions are used in the subsequent sections (Sect. 8 and Sect.…”