Polynomial-time sparse measure recovery
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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^poly(1/ϵ)) for ϵ-accurate recovery. We propose the first poly-time recovery method from carefully designed moments that only requires O(log(1/ϵ)/ϵ^2) computations for an ϵ-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.
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