Sparse Recovery by Non-convex Optimization -- Instance Optimality

In this note, we address the theoretical properties of p, a class of compressed sensing decoders that rely on p minimization with p (0, 1) to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Cand`es, Romberg and Tao [3] and Wojtaszczyk [30] regarding the decoder 1, based on 1 minimization, to p with p (0, 1). Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for 1 the decoders p are robust to noise and stable in the sense that they are (2, p) instance optimal. Second, we extend the results of Wojtaszczyk to show that, like 1, the decoders p are (2, 2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution. While the extension of the results of [3] to the setting where p (0, 1) is straightforward, the extension of the instance optimality in probability result...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Rayan Saab, Özgür Yilmaz
• Keywords: Certain Sufficient Conditions, CORR 2008, Decoders, Education, Sufficient Conditions