Penalized Estimation in Additive Regression with High-Dimensional Data

Understand

Additive regression provides an extension of linear regression by modeling the signal of a response as a sum of functions of covariates of relatively low complexity.

• We study penalized estimation in high-dimensional nonparametric additive regression where functional semi-norms are used to induce smoothness of component functions and the empirical $L_2$ norm is used to induce sparsity.
• The functional semi-norms can be of Sobolev or bounded variation types and are allowed to be different amongst individual component functions.
• We establish new oracle inequalities for the predictive performance of such methods under three simple technical conditions: a sub-gaussian condition on the noise, a compatibility condition on the design and the functional classes under consideration, and an entropy condition on the functional classes.