Polynomial-time sparse measure recovery

How to recover a probability measure with sparse support from particular moments? This problem has been the focus of research in theoretical computer science and neural computing. However, there is no polynomial-time algorithm for the recovery. The best algorithm for the recovery requires $O(2^{\text{poly}(1/\epsilon)})$ for $\epsilon$-accurate recovery. We propose the first poly-time recovery method from carefully designed moments that only requires $O(\log(1/\epsilon)/\epsilon^2)$ computations for an $\epsilon$-accurate recovery. This method relies on the recovery of a planted two-layer neural network with two-dimensional inputs, a finite width, and zero-one activation. For such networks, we establish the first global convergence of gradient descent and demonstrate its application in sparse measure recovery.