The noisy population recovery problem is a basic statistical inference problem. Given an unknown distribution in \{0,1\}^n with support of size k,
and given access only to noisy samples from it, where each bit is flipped independently with probability 1/2-\eps,
estimate the original probability up to an additive error of \eps. We give an algorithm which solves this problem in time polynomial in (k^{\log \log k}, n, 1/\eps).
This improves on the previous algorithm of Wigderson and Yehudayoff [FOCS 2012] which solves the problem in time polynomial in (k^{\log k}, n, 1/\eps).
Our main technical contribution, which facilitates the algorithm, is a new reverse Bonami-Beckner inequality for the L_1 norm of sparse functions.