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Entropy 2014, 16(4), 2131-2145; doi:10.3390/e16042131
Article

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

Received: 14 February 2014 / Revised: 9 April 2014 / Accepted: 10 April 2014 / Published: 14 April 2014
(This article belongs to the Special Issue Information Geometry)
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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.
Keywords: information geometry; dually flat structure; decomposable divergence; (ρ,τ) -structure information geometry; dually flat structure; decomposable divergence; (ρ,τ) -structure
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Amari, S.-I. Information Geometry of Positive Measures and Positive-Definite Matrices: Decomposable Dually Flat Structure. Entropy 2014, 16, 2131-2145.

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