A short note on non-convex compressed sensing

In this note, we summarize the results we recently proved in [14] on the theoretical performance guarantees of the decoders ∆p. These decoders rely on ` minimization with p ∈ (0, 1) to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. Our guarantees generalize the results of [2] and [16] about decoding by `p minimization with p = 1, to the setting where p ∈ (0, 1) and are obtained under weaker sufficient conditions. We also present novel extensions of our results in [14] that follow from the recent work of DeVore et al. in [8]. Finally, we show some insightful numerical experiments displaying the trade-off in the choice of p ∈ (0, 1] depending on certain properties of the input signal.