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تسجيل مستخدم جديدJunta correlation is testable
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The problem of tolerant junta testing is a natural and challenging problem which asks if the property of a function having some specified correlation with a $k$-Junta is testable. In this paper we give an affirmative answer to this question: We show that given distance parameters $frac{1}{2} >c_u>c_{ell} ge 0$, there is a tester which given oracle access to $f:{-1,1}^n rightarrow {-1,1}$, with query complexity $ 2^k cdot mathsf{poly}(k,1/|c_u-c_{ell}|)$ and distinguishes between the following cases: $mathbf{1.}$ The distance of $f$ from any $k$-junta is at least $c_u$; $mathbf{2.}$ There is a $k$-junta $g$ which has distance at most $c_ell$ from $f$. This is the first non-trivial tester (i.e., query complexity is independent of $n$) which works for all $1/2 > c_u > c_ell ge 0$. The best previously known results by Blais emph{et~ al.}, required $c_u ge 16 c_ell$. In fact, with the same query complexity, we accomplish the stronger goal of identifying the most correlated $k$-junta, up to permutations of the coordinates. We can further improve the query complexity to $mathsf{poly}(k, 1/|c_u-c_{ell}|)$ for the (weaker) task of distinguishing between the following cases: $mathbf{1.}$ The distance of $f$ from any $k$-junta is at least $c_u$. $mathbf{2.}$ There is a $k$-junta $g$ which is at a distance at most $c_ell$ from $f$. Here $k=O(k^2/|c_u-c_ell|)$. Our main tools are Fourier analysis based algorithms that simulate oracle access to influential coordinates of functions.
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