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 $p \in (0,1)$ 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 $\Delta_1$, based on $\ell^1$ minimization, to $Δp$ with $p \in (0,1)$. 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. 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. While the extension of the results of [3] to the setting where $p \in (0,1)$ 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 $LQ_p$ 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.