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Entropy, Information Theory, Information Geometry and Bayesian Inference in Data, Signal and Image Processing and Inverse Problems
Article

Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family

1
Department of Electrical and Electronic Engineering, Shinshu University, Nagano, Japan
2
Inria Saclay, Île-de-France, Orsay Cedex, France
3
De Castro Statistics, Collegio Carlo Alberto, Moncalieri, Italy
*
Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in the Proceedings of MaxEnt 2014 Conference on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Amboise, France, 21–26 September 2014.
Academic Editors: Frédéric Barbaresco and Ali Mohammad-Djafari
Entropy 2015, 17(6), 4215-4254; https://doi.org/10.3390/e17064215
Received: 31 January 2015 / Revised: 21 May 2015 / Accepted: 2 June 2015 / Published: 18 June 2015
(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)
In this paper, we study Amari’s natural gradient flows of real functions defined on the densities belonging to an exponential family on a finite sample space. Our main example is the minimization of the expected value of a real function defined on the sample space. In such a case, the natural gradient flow converges to densities with reduced support that belong to the border of the exponential family. We have suggested in previous works to use the natural gradient evaluated in the mixture geometry. Here, we show that in some cases, the differential equation can be extended to a bigger domain in such a way that the densities at the border of the exponential family are actually internal points in the extended problem. The extension is based on the algebraic concept of an exponential variety. We study in full detail a toy example and obtain positive partial results in the important case of a binary sample space. View Full-Text
Keywords: information geometry; stochastic relaxation; natural gradient flow; expectation parameters; toric models information geometry; stochastic relaxation; natural gradient flow; expectation parameters; toric models
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MDPI and ACS Style

Malagò, L.; Pistone, G. Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family. Entropy 2015, 17, 4215-4254. https://doi.org/10.3390/e17064215

AMA Style

Malagò L, Pistone G. Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family. Entropy. 2015; 17(6):4215-4254. https://doi.org/10.3390/e17064215

Chicago/Turabian Style

Malagò, Luigi, and Giovanni Pistone. 2015. "Natural Gradient Flow in the Mixture Geometry of a Discrete Exponential Family" Entropy 17, no. 6: 4215-4254. https://doi.org/10.3390/e17064215

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