Stable sparse approximations via nonconvex optimization

We present theoretical results pertaining to the ability of lscrp minimization to recover sparse and compressible signals from incomplete and noisy measurements. In particular, we extend the results of Candes, Romberg and Tao (2005) to the p < 1 case. Our results indicate that depending on the restricted isometry constants (see, e.g., Candes and Tao (2006; 2005)) and the noise level, lscrp minimization with certain values of p < 1 provides better theoretical guarantees in terms of stability and robustness than lscr1 minimization does. This is especially true when the restricted isometry constants are relatively large.