[Paper Review] Smoothing Brascamp-Lieb Inequalities and Strong Converses for Common Randomness Generation

Jingbo Liu, Thomas A. Courtade+2 more|arXiv (Cornell University)|Feb 6, 2016

Diffusion and Search Dynamics15 references3 citations

TL;DR

This paper establishes a duality between smooth Brascamp-Lieb inequalities and common randomness (CR) generation, proving strong converses for CR generation in both discrete and Gaussian settings by leveraging single-shot converses derived from functional inequalities. The key contribution is the first strong converse for continuous sources with auxiliary random variables, achieved via a novel smoothing technique on Brascamp-Lieb constants that converges to mutual information bounds.

ABSTRACT

We study the infimum of the best constant in a functional inequality, the Brascamp-Lieb-like inequality, over auxiliary measures within a neighborhood of a product distribution. In the finite alphabet and the Gaussian cases, such an infimum converges to the best constant in a mutual information inequality. Implications for strong converse properties of two common randomness (CR) generation problems are discussed. In particular, we prove the strong converse property of the rate region for the omniscient helper CR generation problem in the discrete and the Gaussian cases. The latter case is perhaps the first instance of a strong converse for a continuous source when the rate region involves auxiliary random variables.

Motivation & Objective

• To establish a duality between smooth Brascamp-Lieb inequalities and common randomness (CR) generation problems.
• To prove strong converse theorems for CR generation in both discrete and Gaussian memoryless sources.
• To develop a method for single-shot converses that avoids reliance on the method of types, especially for continuous sources.
• To demonstrate that smoothing the best constant in Brascamp-Lieb inequalities over measures near product distributions yields mutual information-type bounds in the asymptotic limit.
• To extend the applicability of strong converse proofs to problems involving stochastic encoders and decoders, and to derive second-order asymptotic bounds.

Proposed method

• Introduces a smoothed version of the generalized Brascamp-Lieb (GBLL) inequality by minimizing the best constant over auxiliary measures within a total variation neighborhood of a product measure.
• Applies the GBLL inequality to single-shot CR generation, using indicator functions to model decoding sets and deriving a total variation bound between the generated key distribution and the uniform distribution.
• Uses the duality between CR generation achievability and smooth BLL converse to prove that achievability in CR implies a lower bound on the smooth BLL constant.
• Employs a single-shot converse bound (Theorem 20) that links the deviation of the key distribution from uniformity to the smoothed Brascamp-Lieb constant.
• Refines the analysis in the Gaussian case to derive a second-order converse bound involving a constant term that captures the back-off from the asymptotic rate region.
• Demonstrates that the infimum of the GBLL constant over smoothed measures converges to the mutual information-based constant in the limit, under regularity conditions.

Experimental results

Research questions

• RQ1 Can strong converses be proven for common randomness generation in continuous memoryless sources when auxiliary random variables are involved?
• RQ2 Does the infimum of the best constant in a smoothed Brascamp-Lieb inequality converge to a mutual information-based expression in the asymptotic regime?
• RQ3 Can the duality between CR generation and smooth Brascamp-Lieb inequalities be used to derive strong converses without relying on the method of types?
• RQ4 How does the second-order asymptotic behavior of CR generation relate to the smoothed Brascamp-Lieb constant in the Gaussian case?
• RQ5 Can the proposed method handle stochastic encoders and decoders, unlike traditional methods such as the method of types?

Key findings

• The strong converse holds for the omniscient helper common randomness generation problem in both discrete and Gaussian cases, even when auxiliary random variables are present.
• The smooth Brascamp-Lieb constant converges to the mutual information-based constant in the asymptotic limit, under regularity conditions, enabling strong converse proofs.
• The method applies to stochastic encoders and decoders, extending beyond the limitations of the method of types.
• In the Gaussian case, a second-order converse bound is derived, showing that the back-off from the asymptotic rate region is bounded by a constant term involving the smooth BLL constant.
• The duality between CR generation and smooth BLL is established: achievability in CR generation implies a lower bound on the smooth BLL constant, and vice versa.
• The paper provides the first strong converse for a continuous source with auxiliary variables, marking a significant advancement in network information theory.