[Paper Review] Using Sums-of-Squares to Prove Gaussian Product Inequalities

Oliver Russell, Wei Sun|arXiv (Cornell University)|May 4, 2022

Analytic Number Theory Research4 citations

TL;DR

This paper introduces a computational sums-of-squares (SOS) method to rigorously prove new cases of the long-standing Gaussian product inequality (GPI) conjecture for centered Gaussian vectors. By leveraging SOS representations and semi-definite programming, the authors prove two novel GPI inequalities: $\mathbb{E}[X_1^{2m_1}X_2^6X_3^4] \geq \mathbb{E}[X_1^{2m_1}]\mathbb{E}[X_2^6]\mathbb{E}[X_3^4]$ and $\mathbb{E}[X_1^{2m_1}X_2^2X_3^2X_4^2] \geq \mathbb{E}[X_1^{2m_1}]\mathbb{E}[X_2^2]\mathbb{E}[X_3^2]\mathbb{E}[X_4^2]$, with the method extending to cases where one exponent is unbounded.

ABSTRACT

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this paper, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two new GPIs: $E[X_1^{2m_1}X_2^{6}X_3^{4}]\ge E[X_1^{2m_1}]E[X_2^{6}]E[X_3^{4}]$ and $E[X_1^{2m_1}X_2^{2}X_3^{2}X_4^{2}]\ge E[X_1^{2m_1}]E[X_2^{2}]E[X_3^{2}]E[X_4^{2}]$.

Motivation & Objective

• To develop a computational algorithm based on sums-of-squares (SOS) representations to rigorously verify specific cases of the Gaussian product inequality (GPI) conjecture.
• To extend the scope of provable GPI cases beyond known theoretical results, particularly for configurations with mixed or negative correlations.
• To demonstrate the method's power by proving two new GPI inequalities with fixed exponents and extending them to cases with one unbounded exponent.
• To provide computational and algebraic evidence supporting the GPI conjecture, especially in scenarios where traditional analytical methods fail.
• To establish a rigorous link between probability inequalities and algebraic geometry via SOS decompositions, opening new research avenues.

Proposed method

• The method uses sums-of-squares (SOS) representations of multivariate polynomials to prove non-negativity of the difference between the product of marginal moments and the joint moment in the GPI.
• It formulates the GPI inequality as a polynomial inequality in the entries of the covariance matrix, treating them as free real variables.
• The algorithm leverages semi-definite programming (SDP) to search for exact SOS decompositions of the polynomial difference, ensuring rigorous proof.
• The approach relies on the fact that the joint moment $\mathbb{E}[\prod_{j=1}^n X_j^{2m_j}]$ can be expressed as a coefficient in a generating polynomial $G(t_1,\dots,t_n)$ derived from the covariance matrix.
• The method is implemented using symbolic computation software (e.g., 'SumsOfSquares' package), enabling exact decomposition and verification.
• The authors verify that the polynomial difference is SOS by checking its decomposition across multiple correlation regimes, including cases with negative correlations.

Experimental results

Research questions

• RQ1 Can sums-of-squares representations be used to rigorously prove new cases of the Gaussian product inequality (GPI) conjecture?
• RQ2 Can the SOS method handle GPI cases with negative correlations, which are particularly challenging for analytical methods?
• RQ3 Can the method be extended to cases where one exponent is unbounded, thus proving stronger inequalities than previously known?
• RQ4 Is there a systematic computational approach to verify GPI inequalities without relying on restrictive assumptions like non-negative correlations?
• RQ5 Can the SOS method provide computational evidence supporting the full GPI conjecture, even when a general proof remains elusive?

Key findings

• The paper proves the inequality $\mathbb{E}[X_1^{2m_1}X_2^6X_3^4] \geq \mathbb{E}[X_1^{2m_1}]\mathbb{E}[X_2^6]\mathbb{E}[X_3^4]$ for any centered Gaussian vector $(X_1,X_2,X_3)$, resolving a previously unsolved case.
• The paper proves $\mathbb{E}[X_1^{2m_1}X_2^2X_3^2X_4^2] \geq \mathbb{E}[X_1^{2m_1}]\mathbb{E}[X_2^2]\mathbb{E}[X_3^2]\mathbb{E}[X_4^2]$ for any centered Gaussian vector $(X_1,X_2,X_3,X_4)$, establishing a new GPI case.
• The SOS method successfully proves GPI inequalities even when one exponent is unbounded, as demonstrated in Theorems 4.1 and 4.2, extending beyond fixed-exponent results.
• The method provides exact SOS decompositions for the polynomial difference, confirming non-negativity and thus proving the inequality with computational rigor.
• The authors introduce and support a stronger conjecture (Conjecture 5.1) that the GPI inequality corresponds to an SOS polynomial in the correlation parameters, linking probability to algebraic geometry.
• The approach is general and applicable to any GPI of the form (1.2) with fixed dimension and all but one exponent fixed, making it the first universal computational method for the GPI.