A wonderful triangle in compressed sensing

Abstract

In order to explore some concrete metric relationship between the ‖x‖1/‖x‖∞ and ‖x‖0, we introduce a wonderful triangle whose sides are composed of ‖x‖0,‖x‖1 and ‖x‖∞ for any non-zero vector x∈Rn by delving into the iterative soft-thresholding operator in this paper. Based on the angle β of the triangle corresponding to the side whose length is ‖x‖∞-‖x‖1/‖x‖0, we demonstrate that the sparsity of signal is generally meaningful within a certain exact interval [1,50] without regard to its dimension. Finally, we construct a feasible algorithm for solving ℓ1/ℓ∞ minimization, which further illustrates that it is effective for sparse recovery.