A multilayer FOCUSS approach for sparse representation

Focal Underdetermined System Solver (FOCUSS) is a powerful method for sparse representation, in which the Lp-norm like cost function is very often used. However, this cost function is not only nondifferentiable but also can be very ill-conditioned in some situations. The local minima problem of FOCUSS is discussed in this paper. Moreover, to solve this problem, we first extend the Lp-norm like cost function to its corresponding Lp-approximation. After this, we analyze the nonconvexity of the new cost function, which results in that FOCUSS algorithm gets stuck in the local minima in many situations, especially when the hidden sources are not very sparse. To reduce the number of the local minima, a multilayer FOCUSS is developed in this paper. Comparing with the conventional FOCUSS, the experiments inclusing MRI reconstruction demonstrate that multilayer FOCUSS can significantly improve the performance. Even for some very challenging cases, where the conventional FOCUSS fails, multilayer FOCUSS still works well.