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<p>Tsallis and Rényi’s entropies are examples of generalized entropies. The basic idea of generalization is to replace the underlying natural logarithm in the Shannon’s definition of entropy with a new one. We aim for one such generalization of entropy through the proposed (<i>q</i>,&#xa0;<i>r</i>)-logarithm. We first derive some mathematical properties, such as continuity, differentiability, concavity, monotonicity of the (<i>q</i>,&#xa0;<i>r</i>)-logarithm. The (<i>q</i>,&#xa0;<i>r</i>)-logarithm is then used to define a generalization of the Tsallis entropy. We show that it is expansible, sub-additive, and has maximum value for a uniform distribution. Finally, we demonstrate the applicability of the proposed entropy to a pattern recognition problem.</p>

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On a Possible Generalization of Tsallis Entropy and its Application to Pattern Recognition

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Nammi Venkata Apparao,
  • Naveen Kumar,
  • Ambesh Dixit,
  • Vivek Vijay

摘要

Tsallis and Rényi’s entropies are examples of generalized entropies. The basic idea of generalization is to replace the underlying natural logarithm in the Shannon’s definition of entropy with a new one. We aim for one such generalization of entropy through the proposed (q, r)-logarithm. We first derive some mathematical properties, such as continuity, differentiability, concavity, monotonicity of the (q, r)-logarithm. The (q, r)-logarithm is then used to define a generalization of the Tsallis entropy. We show that it is expansible, sub-additive, and has maximum value for a uniform distribution. Finally, we demonstrate the applicability of the proposed entropy to a pattern recognition problem.