  We tackle the problem of recovering an unknown signal observed in an
ill-posed inverse problem framework. More precisely, we study a procedure
commonly used in numerical analysis or image deblurring: minimizing an
empirical loss function balanced by an $l^1$ penalty, acting as a sparsity
constraint. We prove that, by choosing a proper loss function, this estimation
technique enables to build an adaptive estimator, in the sense that it
converges at the optimal rate of convergence without prior knowledge of the
regularity of the true solution
