The Difference of L1 and L2 for Compressive Sensing and Image Processing
Abstract: A fundamental problem in compressive sensing (CS) is to reconstruct a sparse signal under a few linear measurements far less than the physical dimension of the signal. Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, in which case conventional methods, such as L1 minimization, do not work well. In this talk, I will present a novel non-convex approach, which is to minimize the difference of L1 and L2 norms (L1-L2) in order to promote sparsity. In addition to theoretical aspects of the L1-L2 approach, I will discuss two minimization algorithms. One is the difference of convex (DC) function methodology, and the other is based on a proximal operator, which makes some L1 algorithms (e.g. ADMM) applicable for L1-L2. Experiments demonstrate that L1-L2 improves L1 consistently and it outperforms Lp (0<p<1) for highly coherent matrices. Finally, a recovery problem of point sources in 1D and 2D signal from a set of low-frequency measurements will be present, showing advantages of L1-L2 over L1 when a necessary condition for perfect reconstruction is not satisfied.