Local And Global Optimality Of Lp Minimization For Sparse Recovery

In solving the problem of sparse recovery, non-convex techniques have been paid much more attention than ever before, among which the most widely used one is l(p), minimization with p is an element of(0, 1). It has been shown that the global optimality of l(p) minimization is guaranteed under weaker conditions than convex l(1) minimization, but little interest is shown in the local optimality, which is also significant since practical non-convex approaches can only get local optimums. In this work, we derive a tight condition in guaranteeing the local optimality of l(p) minimization. For practical purposes, we study the performance of an approximated version of l(p) minimization, and show that its global optimality is equivalent to that of l(p) minimization when the penalty approaches the l(p) "norm". Simulations are implemented to show the recovery performance of the approximated optimization in sparse recovery.