Sparse Recovery by Non - Convex Optimization –

In this note, we address the theoretical properties of ∆p, a class of compressed sensing decoders that rely on ℓ 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 Candès, Romberg and Tao [3] and Wojtaszczyk [30] regarding the decoder ∆ 1 , based on ℓ 1 minimization, to ∆p with p ∈ (0, 1). Our results are twofold. 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 of [30] is non-trivial. In particular, we need to prove that the LQ 1 property, introduced in [30], and shown to hold for Gaussian matrices and matrices whose columns are drawn uniformly from the sphere, generalizes to an LQp property for the same classes of matrices. Our proof is based on a result by Gordon and Kalton [18] about the Banach-Mazur distances of p-convex bodies to their convex hulls.