Boolean Functions: Noise Stability, Non-Interactive Correlation, and Mutual Information

Let Tε be the noise operator acting on Boolean functions f:{0,1}
ⁿ
→{0,1}, where ε ∈ [0,1/2] is the noise parameter. Given and the mean \mathbbEf, which Boolean function f maximizes the p-th moment \mathbbE(Tεf)
^p?
Our findings are: in the low noise scenario, i.e., ε is small, the maximum is achieved by the lexicographical function; in the high noise scenario, i.e., ε is close to 1/2, the maximum is achieved by Boolean functions with the maximal degree-1 Fourier weight; and when p is an integer, the maximum is achieved by some monotone function, and in particular, among balanced Boolean functions, the maximum is achieved by any function which is 0 on all strings with fewer than n/2 1
^,
s when p is large enough. Our results recover Mossel and O'Donnell's results about the problem of non-interactive correlation distillation, and confirm a conjecture of Courtade and Kumar on the most informative Boolean function in the low noise and high noise regimes. We also observe that Courtade and Kumar's conjecture is equivalent to that the dictator function maximizes \mathbbE(Tεf)
^p
for p close to 1.