Average Case Analysis of Sparse Multivariate Regression under Noise

We consider recovering the jointly sparse multichannel signals from incomplete and noisy measurements. We take the approach of penalized least squares with the penalty given by the ell2,1 norm of the unknown regression coefficients. In contrast to recent efforts on the worstcase analysis, the average case analysis demonstrates more optimistic support recovery results under weaker assumptions. A two-stage procedure is also proposed for estimating the signals across all channels, and is shown to be better on average MSE than the oracle least squares projection. It is indeed proven to be within a small factor to the best procedure possible. The numerical performance of the procedures is also considered.