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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2021.69
URN: urn:nbn:de:0030-drops-136081
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13608/
De, Anindya ;
Nadimpalli, Shivam ;
Servedio, Rocco A.
Quantitative Correlation Inequalities via Semigroup Interpolation
Abstract
Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative 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 near-optimal 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 non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 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 well-known 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 p-biased 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 = {69:1--69:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-177-1},
ISSN = {1868-8969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13608},
URN = {urn:nbn:de:0030-drops-136081},
doi = {10.4230/LIPIcs.ITCS.2021.69},
annote = {Keywords: complex analysis, correlation inequality, FKG inequality, Gaussian correlation inequality, harmonic analysis, Markov semigroups}
}
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Keywords: |
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complex analysis, correlation inequality, FKG inequality, Gaussian correlation inequality, harmonic analysis, Markov semigroups |
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Collection: |
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12th Innovations in Theoretical Computer Science Conference (ITCS 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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04.02.2021 |
