Stable sparse approximations via nonconvex optimization

We present theoretical results pertaining to the ability of lscr_p 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, lscr_p 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.