Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime
SIAM Journal on Computing(2014)
摘要
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.
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关键词
property testing,higher-order Fourier analysis,68Q87,11B30
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