Information Geometry of Positive Measures and Positive-Definite Matrices: Decomposable Dually Flat Structure

Abstract

Information geometry studies the dually flat structure of a manifold, highlighted by the generalized Pythagorean theorem. The present paper studies a class of Bregman divergences called the (ρ,τ)-divergence. A (ρ,τ) -divergence generates a dually flat structure in the manifold of positive measures, as well as in the manifold of positive-definite matrices. The class is composed of decomposable divergences, which are written as a sum of componentwise divergences. Conversely, a decomposable dually flat divergence is shown to be a (ρ,τ) -divergence. A (ρ,τ) -divergence is determined from two monotone scalar functions, ρ and τ. The class includes the KL-divergence, α-, β- and (α, β)-divergences as special cases. The transformation between an affine parameter and its dual is easily calculated in the case of a decomposable divergence. Therefore, such a divergence is useful for obtaining the center for a cluster of points, which will be applied to classification and information retrieval in vision. For the manifold of positive-definite matrices, in addition to the dually flatness and decomposability, we require the invariance under linear transformations, in particular under orthogonal transformations. This opens a way to define a new class of divergences, called the (ρ,τ) -structure in the manifold of positive-definite matrices.