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.