Variable selection in high-dimensional additive models based on norms of projections

TL;DR

This paper introduces a geometric projection-based method for variable selection in high-dimensional sparse additive models with nonparametric components, providing theoretical guarantees for consistency and convergence rates.

Contribution

It develops a novel projection norm approach for variable selection in nonparametric additive models and establishes minimal geometric conditions for consistency.

Findings

Proves concentration inequalities under minimal geometric assumptions.

Derives conditions for consistent variable selection.

Establishes convergence rates for estimating single components.

Abstract

We consider the problem of variable selection in high-dimensional sparse additive models. We focus on the case that the components belong to nonparametric classes of functions. The proposed method is motivated by geometric considerations in Hilbert spaces and consists of comparing the norms of the projections of the data onto various additive subspaces. Under minimal geometric assumptions, we prove concentration inequalities which lead to new conditions under which consistent variable selection is possible. As an application, we establish conditions under which a single component can be estimated with the rate of convergence corresponding to the situation in which the other components are known.