Near-Oracle Performance of Basis Pursuit under Random Noise

We consider the problem of estimating a deterministic sparse vector x0 from underdetermined noisy measurements, in which the noise is a Gaussian random vector. Two techniques which are commonly used in this setting are the Dantzig selector and basis pursuit denoising (BPDN). It has previously been shown that, with high probability, the Dantzig selector comes close to the performance of the oracle estimator which knows the locations of the nonzero components of x0, when the performance is measured by the L2 distance between x0 and its estimate. In this paper, we demonstrate that BPDN achieves analogous results, but that the constants involved in the BPDN analysis are significantly better.