Minimizers of sparsity regularized robust loss functions

We study the structure of the local and global minimizers of the Huber loss and the sum of absolute deviations functions regularized with a sparsity penalty L0 norm term. We char-acterize local minimizers for both loss functions, and establish conditions that are necessary and sufficient for local minimizers to be strict. A necessary condition is established for global minimizers, as well as non-emptiness of the set of global minimizers. The sparsity of minimizers is also studied by giving bounds on a regularization parameter controlling sparsity. Results are illustrated in numerical examples.