Noisy population recovery in polynomial time

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.