Learning a compressive sensing matrix with structural constraints via maximum mean discrepancy optimization

We introduce a learning-based algorithm to obtain a measurement matrix for compressive sensing related recovery problems. The focus lies on matrices with a constant modulus constraint which, e.g., represent a network of analog phase shifters in hybrid precoding/combining architectures. We interpret a matrix with restricted isometry property as a mapping from a high- to a low-dimensional hypersphere. We argue that points on the low-dimensional hypersphere should be uniformly distributed to combat measurement noise. This notion is formalized as an optimization problem which uses a maximum mean discrepancy metric as objective function. Recent success of such metrics in neural network related topics motivate a solution of the optimization problem based on machine learning. Numerical experiments show a better performance than random matrices that are typical for compressive sensing. Further, we adapt a method from the literature to the constant modulus constraint. This method can also compete with random matrices and harmonizes well with the proposed algorithm if it is used as an initialization. Lastly, we describe how other structural matrix constraints, e.g., a Toeplitz constraint, can be taken into account as well.