A Novel Approach to Canonical Divergences within Information Geometry

Abstract

Abstract:A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections ∇ and ∇ * on M. When M is dually flat, that is flat with respect to ∇ and ∇ * , a canonical divergence is known, which is uniquely determined from (M, g, ∇, ∇ * ). We propose a natural definition of a canonical divergence for a general, not necessarily flat, M by using the geodesic integration of the inverse exponential map. The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. Finally, we show that the integrability of the inverse exponential map implies the geodesic projection property.

Information geometry of sandwiched Rényiα-divergence

“…This problem raises another interesting question about reconstructing D α (ρ σ) from a purely differential geometrical viewpoint. It is well known that the canonical divergence on a dually flat statistical manifold (M, g, ∇, ∇ * ) is reconstructed by integrating the metric g along either ∇ or ∇ * -geodesic [1,3]. Unfortunately, this method is not applicable to our problem because the quantum statistical manifold (S, g (Dα) , ∇ (Dα) , ∇ (Dα) * ) is not dually flat unless α = 1.…”

Canonical Divergence for Measuring Classical and Quantum Complexity

“…This structure turns out to be induced on the manifold of positive density operators by the Bogoliubov inner product [18]. In this setting, we prove that the divergence introduced in [9] reduces to the quantum relative entropy. In addition, we also show that D(σ, ρ) = Q(σ, ρ) = D * (ρ, σ).…”

Logarithmic Divergences: Geometry and Interpretation of Curvature

“…To conclude this paper, let us remark that for a statistical manifold with non-constant sectional curvature, this cross sectional curvature is not intrinsic as there are different divergences (and hence Kim-McCann metrics) which induce the same dualistic structure. A natural starting point is to analyze the canonical divergence of Ay and Amari constructed in [4]. We leave this as a problem for future research.…”