Improved Noisy Population Recovery, And Reverse Bonami-Beckner Inequality For Sparse Functions

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 - mu)/2, estimate the original probability up to an additive error of epsilon. We give an algorithm which solves this problem in time polynomial in (k(log log k), n, 1/epsilon). This improves on the previous algorithm of Wigderson and Yehudayoff [FOGS 2012] which solves the problem in time polynomial in (k(log k) n, 1/epsilon). Our main technical contribution, which facilitates the algorithm, is a new reverse Bonami-Beckner inequality for the L-1 norm of sparse functions.