Junta Distance Approximation with Sub-Exponential Queries

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:±1n → ±1, we give a poly(k, frac1varepsilon) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+varepsilon)-far from k'-juntas, where k' = O(frackvarepsilon2). In the non-relaxed setting, we extend our ideas to give a 2^tildeO(√k/varepsilon) (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+varepsilon)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [AoP, 1994].