scholarly journals The Physical Meaning of the Rényi Relative Entropy

2017 ◽  
Vol 21 (1) ◽  
pp. 292
Author(s):  
Gökhan Barış BAĞCI
Keyword(s):  
Relative Entropy ◽  
Physical Meaning ◽  
Rényi Relative Entropy

scholarly journals Fuzzy Clustering Methods with Rényi Relative Entropy and Cluster Size

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1423
Author(s):  
Javier Bonilla ◽  
Daniel Vélez ◽  
Javier Montero ◽  
J. Tinguaro Rodríguez
Keyword(s):  
Relative Entropy ◽  
Fuzzy Clustering ◽  
Cluster Size ◽  
Computational Study ◽  
Gaussian Kernel ◽  
Clustering Methods ◽  
Rényi Divergence ◽  
Divergence Measures ◽  
Fuzzy Clustering Methods ◽  
Rényi Relative Entropy

In the last two decades, information entropy measures have been relevantly applied in fuzzy clustering problems in order to regularize solutions by avoiding the formation of partitions with excessively overlapping clusters. Following this idea, relative entropy or divergence measures have been similarly applied, particularly to enable that kind of entropy-based regularization to also take into account, as well as interact with, cluster size variables. Particularly, since Rényi divergence generalizes several other divergence measures, its application in fuzzy clustering seems promising for devising more general and potentially more effective methods. However, previous works making use of either Rényi entropy or divergence in fuzzy clustering, respectively, have not considered cluster sizes (thus applying regularization in terms of entropy, not divergence) or employed divergence without a regularization purpose. Then, the main contribution of this work is the introduction of a new regularization term based on Rényi relative entropy between membership degrees and observation ratios per cluster to penalize overlapping solutions in fuzzy clustering analysis. Specifically, such Rényi divergence-based term is added to the variance-based Fuzzy C-means objective function when allowing cluster sizes. This then leads to the development of two new fuzzy clustering methods exhibiting Rényi divergence-based regularization, the second one extending the first by considering a Gaussian kernel metric instead of the Euclidean distance. Iterative expressions for these methods are derived through the explicit application of Lagrange multipliers. An interesting feature of these expressions is that the proposed methods seem to take advantage of a greater amount of information in the updating steps for membership degrees and observations ratios per cluster. Finally, an extensive computational study is presented showing the feasibility and comparatively good performance of the proposed methods.

scholarly journals Relative entropy for von Neumann subalgebras

2020 ◽  
Vol 31 (06) ◽  
pp. 2050046
Author(s):  
Li Gao ◽  
Marius Junge ◽  
Nicholas LaRacuente
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Markov Semigroups ◽  
Von Neumann ◽  
Decoherence Time ◽  
Rényi Relative Entropy ◽  
Finite Von Neumann Algebras

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner–Popa index connects to sandwiched [Formula: see text]-Rényi relative entropy for all [Formula: see text], including Umegaki’s relative entropy at [Formula: see text]. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

Sandwiched Rényi relative entropy on density operators

2020 ◽  
Vol 18 (08) ◽  
pp. 2150003
Author(s):  
Ting Zhang ◽  
Xiaofei Qi
Keyword(s):  
Relative Entropy ◽  
Unitary Operator ◽  
Trace Class ◽  
Quantum Information Theory ◽  
Complex Hilbert Space ◽  
Quantum States ◽  
Infinite Dimensional ◽  
Density Operators ◽  
Trace Class Operators ◽  
Rényi Relative Entropy

Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.

scholarly journals The holographic dual of Rényi relative entropy

2019 ◽  
Vol 2019 (8) ◽  
Author(s):  
Ning Bao ◽  
Mudassir Moosa ◽  
Ibrahim Shehzad
Keyword(s):  
Relative Entropy ◽  
Rényi Relative Entropy
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