NDLI: Correlation Testing for Affine Invariant Properties on $\mathbb{F}

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lovett, Shachar Hatami, Hamed
Copyright Year	2014
Abstract	Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n\to\mathbb{F}_p$ with polynomials of degree at most $d\leq p$ is nonnegligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analogue of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for nonlinear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász, and Spencer to the context of additive number theory.
Starting Page	1417
Ending Page	1455
Page Count	39
File Format	PDF
ISSN	00975397
DOI	10.1137/110831349
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	43
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-07-31
Access Restriction	Subscribed
Subject Keyword	Probability in computer science Arithmetic combinatorics; higher degree uniformity property testing higher-order Fourier analysis
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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