## Information Geometry and Its Applications

author: Shun-ichi Amari, RIKEN Brain Science Institute
published: Dec. 5, 2008,   recorded: November 2008,   views: 3576
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# Slides

0:00 Slides Information Geometryand its Applications Information Geometry Information Geometry? Invariance Riemannian Structure AffineConnection Duality: two affine connections Dual Affine Connections Alpha affine connection-duality Dually flat manifold Information Geometry --Dually Flat Manifold Dually Flat Manifold Projection Theorem Applications to Statistics Other Applications Linear Programming(cone programming) Multilayer Perceptron Multilayer Perceptron singularities Milnor attracter Information Geometry of Belief Propagation Information Geometry on Positive Arrays dally flat space <-->convex functions space of positive measures : vectors, matrices, arrays Csiszar f-divergence divergence of fS a divergence divergence Invariance ---characterization of f-divergence (1) Invariance ---characterization of f-divergence (2) Invariance Bregman divergence Legendre duality Example a divergence Geometry Pythagorean Theorem Projection Theorem U-divergence β-divergence Geometry α-representation a divergence β-divergence Geometry α-representation Divergence over α-representation (1) Divergence over α-representation (2) Manifold of positive-definite matrices Integration of Stochastic Evidences Various Means Generalized mean: f-mean

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# Description

Information geometry emerged from studies on invariant properties of a manifold of probability distributions. It includes convex analysis and its duality as a special but important part. Here, we begin with a convex function, and construct a dually flat manifold. The manifold possesses a Riemannian metric, two types of geodesics, and a divergence function. The generalized Pythagorean theorem and dual projections theorem are derived therefrom.We construct alpha-geometry, extending this convex analysis. In this review, geometry of a manifold of probability distributions is then given, and a plenty of applications are touched upon. Appendix presents an easily understable introduction to differential geometry and its duality.