Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy.

Area of Science:

• Information Theory
• Probability Theory
• Multivariate Analysis

Background:

• Differential entropy and its derivatives are crucial in information theory.
• Costa's entropy power inequality provides bounds on entropy.
• Previous conjectures by Cheng-Geng and McKean focused on real-valued random variables.

Purpose of the Study:

• To investigate higher-dimensional analogues of conjectures concerning derivatives of differential entropy.
• To prove new multivariate cases for these conjectures.
• To propose and verify a weaker version of McKean's conjecture.

Main Methods:

• Analysis of derivatives of differential entropy for random vectors in Rn.
• Symbolic computation and semidefinite programming.
• Proof techniques for multivariate cases under log-concave conditions.

Main Results:

• New multivariate cases for C1(m,n) and C2(m,n) conjectures are proven (C1(3,i) for i=2,3,4).
• A new conjecture, C3(m,n), is proposed and several multivariate cases are proven (C3(3,i) for i=2,3,4 and C3(4,2)) under log-concavity.
• A systematic procedure for proving Cl(m,n) is established.

Conclusions:

• The study successfully extends entropy power inequality research to higher dimensions.
• The proposed conjecture C3(m,n) offers a more tractable approach to understanding entropy properties.
• The developed methodology provides a robust framework for future research in this area.