Relaxed sparse eigenvalue conditions for sparse estimation via non-convex regularized regression

•We give new weaker conditions for sparse estimation with non-convex regularizers.•Our regularizers are general, including many existing non-convex regularizers.•Our estimation conditions are applicable even when the solutions is suboptimal.•The desired suboptimal solutions can be obtained by coordinate descent.

Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regularized regression, our sparse eigenvalue based conditions are weaker than that of L1-regularization for parameter estimation and sparseness estimation. For the approximate global and approximate stationary (AGAS) solutions, almost the same conditions are also enough. We show that the desired AGAS solutions can be obtained by coordinate descent (CD) based methods. Finally, we perform some experiments to show the performance of CD methods on giving AGAS solutions and the degree of weakness of the estimation conditions required by the sharp concave regularizers.