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TR14-123 | 7th October 2014 02:39
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#### Improved noisy population recovery, and reverse Bonami-Beckner inequality for sparse functions

**Abstract:**
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.