Sparsistency of $\ell_1$-Regularized $M$-Estimators

We consider the model selection consistency or sparsistency of a broad set of $\ell_1$-regularized $M$-estimators for linear and non-linear statistical models in a unified fashion. For this purpose, we propose the local structured smoothness condition (LSSC) on the loss function. We provide a general result giving deterministic sufficient conditions for sparsistency in terms of the regularization parameter, ambient dimension, sparsity level, and number of measurements. We show that several important statistical models have $M$-estimators that indeed satisfy the LSSC, and as a result, the sparsistency guarantees for the corresponding $\ell_1$-regularized $M$-estimators can be derived as simple applications of our main theorem.