Conformal Flattening for Deformed Information Geometries on the Probability Simplex †
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 r...
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doaj-927cf246cdbc4805b1509e05c93b5daf2020-11-24T23:15:17ZengMDPI AGEntropy1099-43002018-03-0120318610.3390/e20030186e20030186Conformal Flattening for Deformed Information Geometries on the Probability Simplex †Atsumi Ohara0Department of Electrical and Electronics, University of Fukui, Bunkyo, Fukui 910-8507, JapanRecent 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.http://www.mdpi.com/1099-4300/20/3/186representing functionsaffine immersionnonextensive statistical physicsinvariancedually flat structureLegendre conjugategradient flow |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Atsumi Ohara |
spellingShingle |
Atsumi Ohara Conformal Flattening for Deformed Information Geometries on the Probability Simplex † Entropy representing functions affine immersion nonextensive statistical physics invariance dually flat structure Legendre conjugate gradient flow |
author_facet |
Atsumi Ohara |
author_sort |
Atsumi Ohara |
title |
Conformal Flattening for Deformed Information Geometries on the Probability Simplex † |
title_short |
Conformal Flattening for Deformed Information Geometries on the Probability Simplex † |
title_full |
Conformal Flattening for Deformed Information Geometries on the Probability Simplex † |
title_fullStr |
Conformal Flattening for Deformed Information Geometries on the Probability Simplex † |
title_full_unstemmed |
Conformal Flattening for Deformed Information Geometries on the Probability Simplex † |
title_sort |
conformal flattening for deformed information geometries on the probability simplex † |
publisher |
MDPI AG |
series |
Entropy |
issn |
1099-4300 |
publishDate |
2018-03-01 |
description |
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. |
topic |
representing functions affine immersion nonextensive statistical physics invariance dually flat structure Legendre conjugate gradient flow |
url |
http://www.mdpi.com/1099-4300/20/3/186 |
work_keys_str_mv |
AT atsumiohara conformalflatteningfordeformedinformationgeometriesontheprobabilitysimplex |
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1725591230872551424 |