Work

On the Complete Monotonicity of Rényi Entropy

Year: 2025

Type: article

Abstract: In this paper, we investigate the completely monotone conjecture along the heat flow for the Rényi entropy. We confirm this conjecture for the order of derivative up to 4, when the order of Rényi entropy is in certain regimes. We also investigate concavity of Rényi entropy power and the complete monotonicity of Tsallis entropy. We recover and slightly extend Hung’s result on the fourth-order derivative of the Tsallis entropy, and observe that the complete monotonicity holds for Tsallis entropy of order 2, which is equivalent to that the noise stability with respect to the heat semigroup is completely monotone. Based on this observation, we conjecture that the complete monotonicity holds for Tsallis entropy of all orders α ∈ (1, 2). Our proofs in this paper are based on the techniques of integration-by-parts, sum-of-squares, and curve-fitting.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2025.3561384

Authors: Hao Wu, Lei Yu, Laigang Guo

Institutions: Nankai University, Beijing Normal University

Language: English

Topic: Mathematical Inequalities and Applications

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Nankai University, Fundamental Research Funds for the Central Universities

Awards: G5776708844, G6795857712, G7535408997