Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators

Sparse Direction-of-Arrival estimators depend on the regularization parameter <tex>$\lambda$</tex> which is often empirically tuned. In this work, conducted under the vectorized covariance matrix model, we are looking for theoretical equivalence between the Maximum Likelihood (ML) and sparse estimators. We show that under mild conditions, <tex>$\lambda$</tex> can be chosen thanks to the distribution of the minimum of the ML criterion in the case of two impinging sources. We derive this distribution under complex non-circular Gaussian noise. The corresponding <tex>$\lambda$</tex> choice is <tex>$\theta{-}$</tex> invariant, only requiring an upper bound on the number of sources. Furthermore, it guarantees the global minimum of the sparse <tex>$\ell_{0}$</tex> -regularized criterion to be the ML solution. Numerical experiments confirm that, for the proposed <tex>$\lambda$</tex>, sparse and ML estimators yield the same statistical performance.