This paper proposes a method for the localization of correlated sources, through the estimation of the covariance matrix of the sources. In order to deal with the ill-posedness of the estimation and take into account the prior informations on the sources (small number of punctual, possibly correlated, sources), the inversion is based on the sparsity and the low-rankedness of the covariance matrix of the sources. Results show that the performances are greatly improved due to the joint use of sparsity and low rank compared to Tikhonov regularization or the use of only one of these priors. Additionaly, the low rank and sparsity constraints improve the resolution of the localization: the ability to separate close sources is improved when each of the source is correlated with another, sufficiently resolved, source. The estimated matrix is obtained through the minimization of a criterion, sum of an error term with respect to the data, the l_1 norm of the coefficients of the matrix to take in to account its sparsity, and the nuclear norm of the covariance matrix, to penalize matrix with high ranks. This numerical optimization problem is solved by SDMM (Simultaneous Direction Method of Multipliers). Moreover, fast SVD of the low-rank matrices can be used to deal with the high-dimensionality of the problem. The performances are illustrated by numerical results obtained with various scenario of correlated sources.
