Sparse Recovery by Non-convex Optimization -- Instance Optimality

Abstract In this note, we address the theoretical properties of Δ 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) [3] and Wojtaszczyk (2009) [30] regarding the decoder Δ 1 , based on l 1 minimization, to Δ 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 Δ 1 the decoders Δ 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 Δ 1 , the decoders Δ p are ( 2 , 2 ) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution.

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