Explicit Information Geometric Calculations of the Canonical Divergence of a Curve

Abstract

Information geometry concerns the study of a dual structure (g, ∇, ∇∗) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g, ∇, ∇∗). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.