Polynomial-time Sparse Deconvolution

How can a probability measure be recovered with sparse support from its generalized moments? This problem, called sparse deconvolution, has been the focus of research in mathematics, theoretical computer science, and neural computing. However, there is no polynomial-time algorithm for the recovery. The best algorithm requires $O\left(\text{dimension}^{\text{poly}(1/\epsilon)}\right)$ for $\epsilon$-accurate recovery. We propose the first poly-time recovery method from carefully designed moments that requires $O\left(\text{dimension}^4\log(1/\epsilon)/\epsilon^2\right)$ computations for an $\epsilon$-accurate recovery. This method relies on learning a planted two-layer neural network with two-dimensional inputs, a finite width, and zero-one activation. For learning such networks, we establish the first poly-time complexity, and demonstrate its application in sparse deconvolution.