[Paper Review] Improved Hölder and reverse Hölder inequalities for correlated Gaussian random vectors

Wei-Kuo Chen, Nikos Dafnis+1 more|arXiv (Cornell University)|Jun 11, 2013

Point processes and geometric inequalities23 references3 citations

TL;DR

This paper establishes improved Hölder and reverse Hölder inequalities for products of functions of correlated Gaussian random vectors by introducing algebraic criteria based on covariance matrix comparisons. The key contribution is a sharp, covariance-dependent inequality that generalizes classical Hölder and reverse Hölder inequalities, recovers known results like Gaussian hypercontractivity and the Prékopa-Leindler inequality, and provides a unified framework via the geometric Brascamp-Lieb inequality for Gaussian measures.

ABSTRACT

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp-Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian Hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in literatures including the Prékopa-Leindler and Barthe inequalities.

Motivation & Objective

• To derive improved two-sided bounds for the expectation of products of functions of correlated Gaussian random vectors, going beyond classical Hölder inequalities.
• To establish algebraic criteria based on the comparison of the full covariance matrix T with a block-diagonal matrix P = diag(p₁T₁₁, ..., pₘTₘₘ) to determine when Hölder-type inequalities hold.
• To show that the upper bound inequality (1.4) is equivalent to the geometric Brascamp-Lieb inequality for Gaussian measures, thereby unifying two major areas of analysis.
• To recover and generalize known inequalities such as Gaussian hypercontractivity, reverse hypercontractivity, and sharp Young and reverse Young inequalities.
• To present a new framework for entropy inequalities and functional inequalities in the Gaussian setting using the derived bounds.

Proposed method

• Propose a matrix inequality condition: if T ≤ P (where P = diag(p₁T₁₁, ..., pₘTₘₘ)), then 𝔼[∏fᵢ(Xᵢ)] ≤ ∏(𝔼fᵢ(Xᵢ)^{pᵢ})^{1/pᵢ} for pᵢ > 1.
• Use test functions fᵢ(xᵢ) = exp(⟨αᵢ, xᵢ⟩) to show that the inequality conditions T ≤ P and T ≥ P are necessary and sufficient for the respective bounds to hold universally.
• Apply the method of Lagrange multipliers and Gaussian moment computations to derive the critical condition (T − P) ≥ 0 or ≤ 0 for the inequalities.
• Generalize the results to entropy inequalities by differentiating Lᵖ norms of product functions at p = 1, leading to a three-term chain of inequalities involving integrals, norms, and entropy.
• Use Fubini’s theorem and change of variables to relate the Lᵖ norms of multivariate Gaussian functions to entropy expressions.
• Derive a new entropy inequality (6.15) that bounds the entropy of a marginal distribution by a weighted sum of individual entropies and a determinant correction term.

Experimental results

Research questions

• RQ1 Under what conditions on the covariance matrix T and exponents pᵢ does the expectation 𝔼[∏fᵢ(Xᵢ)] satisfy a Hölder-type upper bound that improves upon the classical inequality?
• RQ2 How does the structure of the covariance matrix T relative to the matrix P = diag(p₁T₁₁, ..., pₘTₘₘ) determine whether the upper or lower inequality holds?
• RQ3 Can the improved Hölder inequality be shown to be equivalent to the geometric Brascamp-Lieb inequality for Gaussian measures?
• RQ4 To what extent can the new inequality framework recover or generalize known results such as Gaussian hypercontractivity and the Prékopa-Leindler inequality?
• RQ5 What is the role of the determinant and entropy terms in the derived entropy inequality (6.15), and how do they reflect the dependence structure of the Gaussian vectors?

Key findings

• The upper inequality (1.4) holds if and only if the full covariance matrix T satisfies T ≤ P = diag(p₁T₁₁, ..., pₘTₘₘ), providing a sharp, matrix-based criterion for improved Hölder bounds.
• The reverse inequality (1.5) holds if and only if T ≥ P, and strict inequality occurs when fᵢ are non-constant a.s. and T ≫ P.
• The upper inequality (1.4) is equivalent to the geometric Brascamp-Lieb inequality for Gaussian measures, establishing a deep connection between functional and geometric inequalities.
• The framework recovers Gaussian hypercontractivity and its reverse as special cases when m = 2 and p₁, p₂ satisfy Hölder’s condition.
• The method leads to a generalization of sharp Young and reverse Young inequalities, which in turn recover the Prékopa-Leindler and Barthe inequalities as special cases.
• A new entropy inequality (6.15) is derived, showing that the entropy of a marginal distribution is bounded by a weighted sum of individual entropies and a determinant correction term involving the covariance structure.