Noisy Demixing: Convex Relaxation Meets Nonconvex Optimization

This paper focuses on the noisy demixing problem for robust recovery of a sequence of source signals and impulse responses from a sum of their convolution with additional noise. There are two main prevalent paradigms for this nonconvex estimation problem. One is leveraging the convex relaxation approach to provide good theoretical sample complexity guarantees. Another one is based on the nonconvex optimization method to enjoy good computational complexity. However, both methods are explored separately at this stage. Instead, we shall develop a method bridging convex relaxation with nonconvex optimization in a rigorous theoretical way. In fact, we find that the solution of the convex relaxation approach and the critical point of nonconvex optimization method can be almost the same. Based on our work, the good theoretical sample complexity guarantee of convex relaxation approach can be applied to nonconvex optimization method. And on the other hand, the certain stability guarantees from nonconvex optimization method can be propagated to convex relaxation approach.