Polynomial-time sparse measure recovery

Many problems in computer science reduce to the recovery of an $n$-sparse measure from its (generalized) moments. Sparse measure recovery has been the research focus in super-resolution, tensor decomposition, and learning neural networks. The existing methods use either convex relaxations or overparameterization for recovery. Here, we propose recovery with non-convex optimization without overparameterization. Our algorithm is a (sub)gradient descent method optimizing a non-convex energy function studied in physics. We establish the global convergence of gradient descent on the energy function. This result enables us to solve super-resolution in $O(n^2)$ time, which significantly improves upon $O(n^3)$ time for solving convex relaxations. For a particular neural network, we prove the global convergence of subgradient descent on the population loss without overparameterization. The studied network has zero-one activations, and inputs drawn from the unit sphere.