Junta 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 1/2 >c_u>c_ℓ> 0, there is a tester which given oracle access to f:{-1,1}^n →{-1,1}, with query complexity 2^k ·poly(k,1/|c_u-c_ℓ|) and distinguishes between the following cases: 1. The distance of f from any k-junta is at least c_u; 2. There is a k-junta g which has distance at most c_ℓ 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_ℓ> 0. The best previously known results by Blais et al., required c_u > 16 c_ℓ. 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 poly(k, 1/|c_u-c_ℓ|) for the (weaker) task of distinguishing between the following cases: 1. The distance of f from any k'-junta is at least c_u. 2. There is a k-junta g which is at a distance at most c_ℓ from f. Here k'=O(k^2/|c_u-c_ℓ|). Our main tools are Fourier analysis based algorithms that simulate oracle access to influential coordinates of functions.
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