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Recent years have seen a flurry of activity in solving statistical estimation and learning problems via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. This talk presents two recent stories about nonconvex statistical estimation, which highlight the important role of statistical models in…