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Open AccessArticle

Conformal Flattening for Deformed Information Geometries on the Probability Simplex

Department of Electrical and Electronics, University of Fukui, Bunkyo, Fukui 910-8507, Japan
This paper is an extended version of our paper published in SigmaPhi 2014, 2017 and Geometric Science of Information (GSI 2017).
Entropy 2018, 20(3), 186; https://doi.org/10.3390/e20030186
Received: 20 February 2018 / Revised: 8 March 2018 / Accepted: 8 March 2018 / Published: 10 March 2018
(This article belongs to the Special Issue New Trends in Statistical Physics of Complex Systems)
Recent progress of theories and applications regarding statistical models with generalized exponential functions in statistical science is giving an impact on the movement to deform the standard structure of information geometry. For this purpose, various representing functions are playing central roles. In this paper, we consider two important notions in information geometry, i.e., invariance and dual flatness, from a viewpoint of representing functions. We first characterize a pair of representing functions that realizes the invariant geometry by solving a system of ordinary differential equations. Next, by proposing a new transformation technique, i.e., conformal flattening, we construct dually flat geometries from a certain class of non-flat geometries. Finally, we apply the results to demonstrate several properties of gradient flows on the probability simplex. View Full-Text
Keywords: representing functions; affine immersion; nonextensive statistical physics; invariance; dually flat structure; Legendre conjugate; gradient flow representing functions; affine immersion; nonextensive statistical physics; invariance; dually flat structure; Legendre conjugate; gradient flow
MDPI and ACS Style

Ohara, A. Conformal Flattening for Deformed Information Geometries on the Probability Simplex . Entropy 2018, 20, 186. https://doi.org/10.3390/e20030186

AMA Style

Ohara A. Conformal Flattening for Deformed Information Geometries on the Probability Simplex . Entropy. 2018; 20(3):186. https://doi.org/10.3390/e20030186

Chicago/Turabian Style

Ohara, Atsumi. 2018. "Conformal Flattening for Deformed Information Geometries on the Probability Simplex " Entropy 20, no. 3: 186. https://doi.org/10.3390/e20030186

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