On the Complete Monotonicity of Rényi Entropy

TL;DR

This work analyzes the complete monotonicity of Rényi entropy along the heat flow, establishing derivative signs up to the fourth order in precise $\alpha$-regimes and linking these properties to entropy power and diffusion-contractivity. The authors extend and contrast results for Tsallis entropy, recovering Hung’s fourth-order Tsallis bound and proving complete monotonicity at $\alpha=2$, while formulating a conjecture for $\alpha\in(1,2)$. Methodologically, they combine integration-by-parts, curve-fitting, and sum-of-squares/semidefinite optimization to construct positive semi-definite representations that certify the signs of higher-order derivatives, and they connect these monotonicity results to $\alpha$-stability and noise stability. The findings delineate regimes of stability and identify counterexamples outside these ranges, contributing to a deeper understanding of entropy dynamics under diffusion and offering a path toward a broader completely monotone theory for Rényi and Tsallis entropies. The results have implications for information-dtheoretic diffusion inequalities and for understanding how different entropy measures evolve under Gaussian perturbations.

Abstract

In this paper, we investigate the complete monotonicity of Rényi entropy along the heat flow. We confirm this property 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 $α\in(1,2)$. Our proofs in this paper are based on the techniques of integration-by-parts, sum-of-squares, and curve-fitting.

Figures (3)

• Figure 1: Curve-fitting for the best $(a(\alpha),b(\alpha),c(\alpha),d(\alpha),e(\alpha))$.
• Figure 2: The sequential principal minors of matrix $\hat{A}$.
• Figure 3: The sequential principal minors of matrix $\tilde{A}$.

Theorems & Definitions (33)

• Theorem 1
• Example 1
• Proposition 1
• Theorem 2
• Corollary 1
• Proposition 2
• Remark 1
• Theorem 3
• Conjecture 1
• Example 2