Minimum $\ell_1-$norm interpolation via basis pursuit is robust to errors

This article studies basis pursuit, i.e. minimum $\ell_1$-norm interpolation, in sparse linear regression with additive errors. No conditions on the errors are imposed. It is assumed that the number of i.i.d. Gaussian features grows superlinear in the number of samples. The main result is that under these conditions the Euclidean error of recovering the true regressor is of the order of the average noise level. Hence, the regressor recovered by basis pursuit is close to the truth if the average noise level is small. Lower bounds that show near optimality of the results complement the analysis. In addition, these results are extended to low rank trace regression. The proofs rely on new lower tail bounds for maxima of Gaussians vectors and the spectral norm of Gaussian matrices, respectively, and might be of independent interest as they are significantly stronger than the corresponding upper tail bounds.