Volume 134, Number 1, April 2021
|Number of page(s)||7|
|Section||Interdisciplinary Physics and Related Areas of Science and Technology|
|Published online||14 May 2021|
Entropy, cross-entropy, relative entropy: Deformation theory(a)
1 University of Michigan - Ann Arbor, USA
2 Nagoya Institute of Technology - Nagoya, Japan
Received: 13 November 2020
Accepted: 22 March 2021
Attempts at generalizing Shannon entropy and Kullback-Leibler divergence (relative entropy) led to a plenthora of deformation models in theoretical physics, including q-model, κ-model, etc. Naudts and Zhang (Inf. Geom., 1 (2018) 79) established that these models can be unified under two notions: deformed ϕ-exponential family (Naudts, J., J. Inequal. Pure Appl. Math., 5 (2004) 102) and conjugate -embedding (Zhang J., Neural Comput., 16 (2004) 159) of probability functions. Conjugate -embedding has a gauge freedom which, upon its fixing, subsumes the U-model of Eguchi (Sugaku Expositions, 19 (2006) 197) proposed in a statistical machine learning context. The generalization by -entropy, -cross-entropy, -divergence, when applied to the ϕ-exponential family, yields either a Hessian structure or a conformal Hessian structure under different gauge selections —this “splitting” is the hallmark when deforming the exponential family with its dually flat (Hessian) geometry. This letter provides a unified information geometric perspective of deformation of the exponential model, with calculations for Tsallis q-model.
© 2021 EPLA
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.