[Paper Review] Grothendieck's inequality and completely correlation preserving functions -- a summary of recent results and an indication of related research problems

Frank Oertel|arXiv (Cornell University)|Oct 2, 2020

Mathematical Inequalities and Applications30 references6 citations

TL;DR

This paper investigates Grothendieck's inequality through entrywise functions on correlation matrices using Schur products, Gaussian copulas, and Taylor series inversion. It proposes a novel framework linking multivariate Gaussian analysis, special functions, and operator theory to bound the Grothendieck constant, offering a path toward determining its exact value in real and complex cases.

ABSTRACT

As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach, primarily built on functions which map correlation matrices entrywise to correlation matrices by means of the Schur product, multivariate Gaussian analysis, copulas and inversion of suitable Taylor series. We summarise first results and point towards related open problems and topics for future research.

Motivation & Objective

• To develop a new analytical framework for determining the exact value of the Grothendieck constant $K_G^\mathbb{R}$ and $K_G^\mathbb{C}$, which remain open since 1953.
• To extend the application of Grothendieck's inequality beyond Hilbert spaces and tensor norms by focusing on entrywise maps on correlation matrices.
• To connect multivariate Gaussian dependence structures, copulas, and special functions to the metric theory of tensor products.
• To address the combinatorial complexity underlying Grothendieck's inequality and assess whether explicit computation of $K_G^\mathbb{R}$ is feasible.

Proposed method

• Utilizes the Schur product to map correlation matrices to correlation matrices via entrywise functions.
• Applies Gaussian copula models to express expectations of sign functions in terms of arcsine laws, linking to the bivariate Gaussian copula at $ (1/2, 1/2) $.
• Employs Taylor series inversion and ordinary partial Bell polynomials to analyze functional transforms of correlation matrices.
• Uses Hermite polynomials and Gaussian hypergeometric functions to model higher-order dependence and spectral properties.
• Applies Schoenberg’s Theorem to characterize completely positive definite functions on correlation matrices.
• Integrates operator ideals and trace duality to reframe Grothendieck’s inequality in terms of Banach space tensor norms.

Experimental results

Research questions

• RQ1 Can the Grothendieck constant $K_G^\mathbb{R}$ be computed exactly using entrywise functions and Gaussian copula models?
• RQ2 To what extent can the Gaussian structure in Grothendieck’s inequality be replaced by heavy-tailed or extreme-value distributions without losing the inequality?
• RQ3 Is there a non-commutative generalization of copulas that captures quantum correlations and relates to the Tsirelson bound?
• RQ4 Can the combinatorial complexity of the Grothendieck problem be overcome via functional-analytic and special function techniques?
• RQ5 Does the use of generalized extreme value (GEV) distributions or infinitely divisible laws yield improved bounds on $K_G^\mathbb{R}$?

Key findings

• The paper provides a short proof of the real Grothendieck inequality using Krivine’s upper bound, leveraging the proposed functional framework.
• It establishes a precise link between the bivariate Gaussian copula evaluated at $ (1/2, 1/2) $ and the arcsine law: $ \mathbb{E}[\text{sign}(X)\text{sign}(Y)] = \frac{2}{\pi}\arcsin(\rho) $, where $ \rho = \mathbb{E}[XY] $.
• The framework reveals that the Grothendieck constant $ K_G^\mathbb{R} $ is intimately tied to the non-commutative nature of quantum correlations, as formalized by the Tsirelson bound.
• The use of Hermite polynomials and hypergeometric functions enables a systematic expansion of correlation-preserving functions, suggesting a path toward series-based approximation of $ K_G^\mathbb{R} $.
• The paper indicates that the Grothendieck constant $ K_G^\mathbb{R} $ is bounded below by the real Grothendieck constant and above by $ \sqrt{2}K_G^\mathbb{C} $, with $ K_G^\mathbb{C} < \pi/2 $, but its exact value remains unknown.
• The study suggests that replacing Gaussian dependence with tail-dependent models like GEV distributions may alter the inequality’s validity, raising open questions about robustness and bounds.