Stable sparse approximations via nonconvex optimization

We present theoretical results pertaining to the ability of lp minimization to recover sparse and compressible signals from incomplete and noisy measurements. In particular, we extend the results of Cande`s, Romberg and Tao [1] to the p < 1 case. Our results indicate that depending on the restricted isometry constants (see, e.g.,[2] and [3]) and the noise level, lp minimization with certain values of p < 1 provides better theoretical guarantees in terms of stability and robustness than l1 minimization does. This is especially true when the restricted isometry constants are relatively large.