We give a polynomial time algorithm for the lossy population recovery problem. In this problem, the goal is to approximately learn an unknown distribution on binary strings of length $n$ from lossy samples: for some parameter $μ$ each coordinate of the sample is preserved with probability $μ$ and otherwise is replaced by a `?'. The running time and number of samples needed for our algorithm is polynomial in $n$ and $1/\varepsilon$ for each fixed $μ>0$. This improves on algorithm of Wigderson and Yehudayoff that runs in quasi-polynomial time for any $μ> 0$ and the polynomial time algorithm of Dvir et al which was shown to work for $μ\gtrapprox 0.30$ by Batman et al. In fact, our algorithm also works in the more general framework of Batman et al. in which there is no a priori bound on the size of the support of the distribution. The algorithm we analyze is implicit in previous work; our main contribution is to analyze the algorithm by showing (via linear programming duality and connections to complex analysis) that a certain matrix associated with the problem has a robust local inverse even though its condition number is exponentially small. A corollary of our result is the first polynomial time algorithm for learning DNFs in the restriction access model of Dvir et al.