Next Article in Journal
Possible Further Evidence for the Thixotropic Phenomenon of Water
Next Article in Special Issue
Computational Information Geometry in Statistics: Theory and Practice
Previous Article in Journal
Wiretap Channel with Information Embedding on Actions
Previous Article in Special Issue
Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes
Open AccessArticle

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

RIKEN Brain Science Institute, Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan
Entropy 2014, 16(4), 2131-2145; https://doi.org/10.3390/e16042131
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)
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. View Full-Text
Keywords: information geometry; dually flat structure; decomposable divergence; (ρ,τ) -structure information geometry; dually flat structure; decomposable divergence; (ρ,τ) -structure
MDPI and ACS Style

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

Show more citation formats Show less citations formats

Article Access Map by Country/Region

1
Only visits after 24 November 2015 are recorded.
Back to TopTop