A Two-Parameter Generalized Tsallis Entropy and Its Information-Theoretic Properties

Concepts de base

This article proposes a two-parameter generalization of the Tsallis entropy, called the generalized Tsallis entropy, and derives its fundamental information-theoretic properties, including pseudo-additivity, sub-additivity, joint convexity, and information monotonicity.

Résumé

The article introduces a modified version of the Sharma-Mittal entropy, which the authors call the generalized Tsallis entropy. This new entropy measure is a two-parameter generalization of the Tsallis entropy and can be reduced to the Tsallis entropy for specific parameter values.

The key highlights and insights from the article are:

1. The authors derive a product rule for the two-parameter deformed logarithm, which enables them to establish a chain rule for the generalized Tsallis entropy.
2. They prove that the generalized Tsallis entropy satisfies the pseudo-additivity property, where the joint entropy of two independent random variables is the sum of their individual entropies minus a term involving their product.
3. The generalized Tsallis entropy is shown to be sub-additive, meaning the entropy of a joint distribution is less than or equal to the sum of the individual entropies.
4. The authors define a generalized Tsallis relative entropy (or divergence) and demonstrate its pseudo-additivity, joint convexity, and information monotonicity properties.
5. The information-geometric aspects of the generalized Tsallis relative entropy are discussed, including its connection to the Sharma-Mittal entropy and the Tsallis relative entropy.

The article provides a comprehensive analysis of the information-theoretic and information-geometric characteristics of the proposed generalized Tsallis entropy and relative entropy measures.

Stats

The generalized Tsallis entropy is defined as S{k,r}(X) = -Σ_x (p(x))^(r+k+1) ln{k,r}(p(x)), where ln{k,r}(x) = (x^k - x^(-k)) / (2kx^r).
The generalized Tsallis relative entropy is defined as D{k,r}(P||Q) = Σ_x p(x) (p(x)/q(x))^(r-k) ln{k,r}(p(x)/q(x)).
The pseudo-additivity property of the generalized Tsallis entropy is: S{k,r}(X,Y) = S{k,r}(X) + S{k,r}(Y) - 2kS{k,r}(X)S{k,r}(Y).
The sub-additivity property of the generalized Tsallis entropy is: S{k,r}(X1, X2, ..., Xn) ≤ Σ_i S{k,r}(Xi).
The pseudo-additivity property of the generalized Tsallis relative entropy is: D{k,r}(P(1)⊗P(2)||Q(1)⊗Q(2)) = D{k,r}(P(1)||Q(1)) + D{k,r}(P(2)||Q(2)) - 2kD{k,r}(P(1)||Q(1))D{k,r}(P(2)||Q(2)).
The joint convexity property of the generalized Tsallis relative entropy is: D{k,r}(P(1)+λP(2)||Q(1)+λQ(2)) ≤ D{k,r}(P(1)||Q(1)) + λD{k,r}(P(2)||Q(2)).
The information monotonicity property of the generalized Tsallis relative entropy is: D{k,r}(WP||WQ) ≤ D{k,r}(P||Q).

Citations

"This article proposes a modification in the Sharma-Mittal entropy and distinguishes it as generalised Tsallis entropy."
"We derive a product rule (xy)^(r+k) ln{k,r}(xy) = x^(r+k) ln{k,r}(x) + y^(r+k) ln{k,r}(y) + 2kx^(r+k)y^(r+k) ln{k,r}(x) ln{k,r}(y), for the two-parameter deformed logarithm ln{k,r}(x) = (x^r - x^(-k)) / (2k)."
"This article is an exposit investigation on the information-theoretic, and information-geometric characteristics of generalized Tsallis entropy."

Idées clés tirées de

A two-parameter entropy and its fundamental properties

by Supriyo Dutt... à arxiv.org 05-02-2024

https://arxiv.org/pdf/1908.01696.pdf

Questions plus approfondies

How can the generalized Tsallis entropy and relative entropy be applied to real-world problems in areas like machine learning, data analysis, or physics

The generalized Tsallis entropy and relative entropy have various applications in real-world problems across different fields. In machine learning, Tsallis entropy can be used as a measure of uncertainty or diversity in probability distributions. It can help in clustering algorithms, anomaly detection, and feature selection by capturing the non-linear relationships between variables. In data analysis, Tsallis entropy can provide insights into the complexity and structure of data, aiding in dimensionality reduction and pattern recognition. In physics, Tsallis entropy has been applied in the study of complex systems, such as in modeling the behavior of particles in non-extensive systems or in understanding the dynamics of turbulent flows. Overall, the generalized Tsallis entropy offers a versatile tool for analyzing complex systems with non-Gaussian distributions and long-range interactions.

What are the potential limitations or drawbacks of the generalized Tsallis entropy compared to other entropy measures, and how can these be addressed

While the generalized Tsallis entropy has many advantages, it also has some limitations compared to other entropy measures. One limitation is the sensitivity of Tsallis entropy to the choice of parameters k and r, which can affect the results and interpretations. Additionally, the non-extensive nature of Tsallis entropy may not always align with the assumptions of traditional statistical mechanics or information theory. To address these limitations, one approach is to conduct sensitivity analysis to understand the impact of parameter variations on the results. Another strategy is to compare Tsallis entropy with other entropy measures, such as Shannon entropy or Rényi entropy, to gain a more comprehensive understanding of the system under study. By combining different entropy measures, researchers can mitigate the drawbacks of individual measures and obtain a more robust analysis of complex systems.

What are the connections between the generalized Tsallis entropy and other information-theoretic concepts, such as the Rényi entropy or the Kullback-Leibler divergence, and how can these connections be further explored

The generalized Tsallis entropy is closely related to other information-theoretic concepts, such as the Rényi entropy and the Kullback-Leibler divergence. The Rényi entropy is a special case of Tsallis entropy when k = r, providing a connection between the two entropy measures. The Kullback-Leibler divergence, on the other hand, quantifies the difference between two probability distributions and can be generalized using Tsallis relative entropy. Exploring these connections further can lead to a deeper understanding of the relationships between different entropy measures and their applications in various fields. By investigating the interplay between Tsallis entropy, Rényi entropy, and Kullback-Leibler divergence, researchers can uncover new insights into information theory, statistical mechanics, and complex systems analysis.