Section: New Results

Exact continuous penalties for ${\ell }_2$-${\ell }_0$ minimization: Application to Channel and Direction Of Arrival (DOA) estimation problems

Participants : Emmanuel Soubies, Laure Blanc-Féraud.

This work is made in collaboration with Adilson Chinatto, Cynthia Junqueira, João M. T. Romano (University of Campinas, Brazil) and Pascal Larzabal, Jean-Pierre Barbot (ENS Cachan, SATIE Lab).

In this work, we have proposed to extend the Continuous Exact ${\ell }_0$ (CEL0) penalty, which we initially introduced for the real single measurement vector (SMV) case, to complex SMV and complex multiple measurement vector (MMV) situations involving structured sparsity. Such an extension is necessary to address sparse signal processing estimation problems like augmented resolution channel estimation and direction of arrival (DOA) estimation for which the mixture matrix do not verify restrict isometry property (RIP) and incoherence conditions. We thereby have derived a row-structured version of the CEL0 penalty and showed that the relations between minimizers of the resulting relaxation and those of the initial ${\ell }_0$-penalised least squares criteria, that we previously showed in the real SMV case, are still valid for complex SMV and MMV situations using the proposed row-structured CEL0 penalty. Finally, we have employed state of the art nonsmooth nonconvex algorithms to minimize the proposed relaxation and we have compared the results obtained by our method with those provided by the well known iterative hard thresholding (IHT) algorithm as well as some classical algorithms for the studied problems. We have shown that minimizing the row-structured CEL0 relaxation provides better estimation results than IHT, which minimizes directly with the initial ${\ell }_0$-penalized least-squares criteria, and than classical algorithms used for such problems where the mixture matrix is highly correlated. Moreover, the proposed method is able to reach the oracle RMSE in some cases. This work has been submitted to the IEEE Transaction on Signal Processing journal.