Sparse recovery by non-convex optimization - instance optimality

In this note, we address the theoretical properties of Delta(p), a class of compressed sensing decoders that rely on l(P) minimization with 0 < p < 1 to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Candes, Romberg and Tao (2006) 13] and Wojtaszczyk (2009) [30] regarding the decoder Delta(1), based on Pi minimization, to Delta(p) with 0 < p < 1. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for Delta(1) the decoders Delta(p) are robust to noise and stable in the sense that they are (2. p) instance optimal for a large class of encoders. Second, we extend the results of Wojtaszczyk to show that, like Delta(1) the decoders Delta p are (2.2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution. (C) 2009 Elsevier Inc. All rights reserved.