The Generalized Likelihood Ratio Test and the Sparse Representations Approach

When sparse representation techniques are used to tentatively recover the true sparse underlying model hidden in an observation vector. they can be seen as solving a joint detection and estimation problem. We consider the l(2) - l(1) regularized criterion, that is probably the most used in the sparse representation community, and show that, from a detection point of view, minimizing this criterion is similar to applying the Generalized Likelihood Ratio Test. More specifically tuning the regularization parameter in the criterion amounts to set the threshold in the Generalized Likelihood Ratio Test.