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In this note, we address the theoretical properties of $\Delta_p$, a class ofcompressed sensing decoders that rely on $\ell^p$ minimization with 0<p<1 torecover estimates of sparse and compressible signals from incomplete andinaccurate measurements. In particular, we extend the results of Candes,Romberg and Tao, and Wojtaszczyk regarding the decoder $\Delta_1$, based on$\ell^1$ minimization, to $\Delta_p$ with 0<p<1. Our results are two-fold.First, we show that under certain sufficient conditions that are weaker thanthe analogous sufficient conditions for $\Delta_1$ the decoders $\Delta_p$ arerobust to noise and stable in the sense that they are (2,p) instance optimalfor a large class of encoders. Second, we extend the results of Wojtaszczyk toshow that, like $\Delta_1$, the decoders $\Delta_p$ are (2,2) instance optimalin probability provided the measurement matrix is drawn from an appropriatedistribution.

Sparse Recovery by Non-convex Optimization -- Instance Optimality