Abstract
Most correlation inequalities for highdimensional functions in the literature, such as the FortuinKasteleynGinibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have nonnegative correlation. We give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic analysis of functions over product spaces. We instantiate this general approach in several different concrete settings to obtain a range of new and nearoptimal quantitative correlation inequalities, including:
 A {quantitative} version of Royen’s celebrated Gaussian Correlation Inequality [Royen, 2014]. In [Royen, 2014] Royen confirmed a conjecture, open for 40 years, stating that any two symmetric convex sets must be nonnegatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree2 Hermite coefficients of the two convex sets, conceptually similar to Talagrand’s quantitative correlation bound for monotone Boolean functions over {0,1}ⁿ [M. Talagrand, 1996]. We show that our quantitative version of Royen’s theorem is within a logarithmic factor of being optimal.
 A quantitative version of the wellknown FKG inequality for monotone functions over any finite product probability space. This is a broad generalization of Talagrand’s quantitative correlation bound for functions from {0,1}ⁿ to {0,1} under the uniform distribution [M. Talagrand, 1996]; the only prior generalization of which we are aware is due to Keller [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009], which extended [M. Talagrand, 1996] to product distributions over {0,1}ⁿ. In the special case of pbiased distributions over {0,1}ⁿ that was considered by Keller, our new bound essentially saves a factor of p log(1/p) over the quantitative bounds given in [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009]. We also give {a quantitative version of} the FKG inequality for monotone functions over the continuous domain [0,1]ⁿ, answering a question of Keller [Nathan Keller, 2009].
BibTeX  Entry
@InProceedings{de_et_al:LIPIcs.ITCS.2021.69,
author = {Anindya De and Shivam Nadimpalli and Rocco A. Servedio},
title = {{Quantitative Correlation Inequalities via Semigroup Interpolation}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {69:169:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13608},
URN = {urn:nbn:de:0030drops136081},
doi = {10.4230/LIPIcs.ITCS.2021.69},
annote = {Keywords: complex analysis, correlation inequality, FKG inequality, Gaussian correlation inequality, harmonic analysis, Markov semigroups}
}
Keywords: 

complex analysis, correlation inequality, FKG inequality, Gaussian correlation inequality, harmonic analysis, Markov semigroups 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 