Solving Overdetermined Systems in lp Quasi-Norms

The theory of compressed sensing has shown that sparse signals can be reconstructed exactly from remarkably few measurements by solving a nonconvex underdetermined lp-regularized quasi-norm problem via an iterative weighted least-squares problem. In this work, we consider the problem of recovering an input signal by solving a nonconvex overdetermined lp-regularized quasi-norm problem. In order to do this, we carry over a fixed-point algorithm, presented in [17], [10] and [1] from a nonconvex underdetermined to a nonconvex overdetermined lp quasi-norm problem. Then, we reformulate this procedure by a sequential quadratic program, and use two alternative algorithms for solving its associated linear systems so called augmented system: a direct method and a projected conjugate gradient. The sequential quadratic program takes into account the signals and its associated error. While the direct method scheme works with a sequence of approximations of the signals and its errors simultaneously, the projected conjugate gradient algorithm finds first an approximation error, and later, using this error, an approximate signal is obtained using just a least-squares problem. The numerical advantage of using a direct method for solving the augmented system is that it allows a sparser and cheaper factorization than the Cholesky factorization for solving the weighted normal equation for dense matrices. Besides, the projected conjugate gradient needs only one matrix factorization in all the optimization procedures which is appealing to solve large-scale problems. We implemented these strategies and compare their capabilities to recover signals. Specifically, our interest is to identify at what rate of corruption each formulation fails to recover the signal exactly for different values 0 ≤ p < 1, and compare with the convex problem when p = 1.