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-μ)/2, estimate the original probability up to an additive error of ε. We give an algorithm which solves this problem in time polynomial in (klog log k, n, 1/ε). This improves on the previous algorithm of Wigderson and Yehudayoff [FOCS 2012] which solves the problem in time polynomial in (klog k, n, 1/ε). Our main technical contribution, which facilitates the algorithm, is a new reverse Bonami-Beckner inequality for the L1 norm of sparse functions.