  In the noisy population recovery problem of Dvir et al., the goal is to learn
an unknown distribution $f$ on binary strings of length $n$ from noisy samples.
For some parameter $\mu \in [0,1]$, a noisy sample is generated by flipping
each coordinate of a sample from $f$ independently with probability
$(1-\mu)/2$. We assume an upper bound $k$ on the size of the support of the
distribution, and the goal is to estimate the probability of any string to
within some given error $\varepsilon$. It is known that the algorithmic
complexity and sample complexity of this problem are polynomially related to
each other.
  We show that for $\mu > 0$, the sample complexity (and hence the algorithmic
complexity) is bounded by a polynomial in $k$, $n$ and $1/\varepsilon$
improving upon the previous best result of $\mathsf{poly}(k^{\log\log
k},n,1/\varepsilon)$ due to Lovett and Zhang.
  Our proof combines ideas from Lovett and Zhang with a \emph{noise attenuated}
version of M\"{o}bius inversion. In turn, the latter crucially uses the
construction of \emph{robust local inverse} due to Moitra and Saks.
