Sparse Recovery by Non-convex Optimization -- Instance Optimality

8 years 2 months ago
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...
Rayan Saab, Özgür Yilmaz
Added 09 Dec 2010
Updated 09 Dec 2010
Type Journal
Year 2008
Where CORR
Authors Rayan Saab, Özgür Yilmaz
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