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Gaussian Optimality for Derivatives of Differential Entropy Using Linear Matrix Inequalities.
Xiaobing Zhang1,2,3, Venkat Anantharam4, Yanlin Geng2,5
1Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
McKean
Area of Science:
- Information Theory
- Probability Theory
- Stochastic Processes
Background:
- McKean conjectured that Gaussian distributions maximize derivatives of differential entropy.
- This conjecture implies alternating signs for these derivatives.
- Previous work confirmed this for the first four orders.
Purpose of the Study:
- To investigate the validity of McKean's conjecture for higher-order derivatives.
- To explore alternative methods beyond linear matrix inequalities.
- To analyze the conjecture under log-concavity conditions.
Main Methods:
- Linear matrix inequality techniques.
- Analysis of differential entropy derivatives.
- Exploration of log-concave probability distributions.
Main Results:
- Linear matrix inequality methods may not generalize to higher derivative orders.
- McKean's conjecture is validated for orders up to five when the distribution is log-concave.
- A simpler proof for Toscani's result on entropy power derivatives is provided.
Conclusions:
- The study offers new insights into McKean's conjecture on differential entropy.
- Log-concavity is identified as a condition supporting the conjecture.
- The findings contribute to understanding the behavior of stochastic processes and information theory.
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