Some Worked-Out Examples

Abstract

The sharp oracle inequalities for the Lasso, the square-root Lasso and other structured sparsity estimators are completed with a probability statement saying the result holds with confidence at least 1 −α (say), with α > 0 a fixed error level. Also a sharp oracle inequality for projection estimators of a density are given. As a “representative” example for the case of non-linear M-estimation with an ℓ ₁-penalty, a sharp oracle inequality is presented for logistic regression. For trace regression with nuclear norm penalty, a sharp oracle is given when least squares loss is used, and a non-sharp one when least absolute deviations loss is used. In the latter case, the design is taken as in the matrix completion problem. As final example sparse principal component analysis is addressed.