skip to content
 

Correlation testing for affine invariant properties on $F_p^n$

Presented by: 
S Lovett [IAS Princeton]
Date: 
Wednesday 4th May 2011 - 13:45 to 14:45
Venue: 
INI Seminar Room 1
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 $f: F_p^n \rightarrow F_p$ with polynomials of degree at most $d \le p$ is non-negligible, while making only a constant number of queries to the function. This is an instance of {\em 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 analog of {\em 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 non-linear 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 an extension of a graph theoretical theorem of Erd\"os, Lov\'asz and Spencer to the context of additive number theory. Joint work with Hamed Hatami.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons