On $\ell_1$-regularized estimation for nonlinear models that have sparse underlying linear structures

TL;DR

This paper demonstrates that in certain nonlinear models with sparse linear structures, -regularized estimation can attain error bounds comparable to those of -regularized methods, simplifying the estimation process.

Contribution

The paper shows that -regularized estimation can match the error bounds of -regularized estimation in specific nonlinear models with sparse structures.

Findings

-regularization achieves similar error bounds as -regularization.

Applicable to important cases of nonlinear models with sparsity.

Simplifies estimation while maintaining accuracy.

Abstract

In a recent work (arXiv:0910.2517), for nonlinear models with sparse underlying linear structures, we studied the error bounds of $\ell_0$-regularized estimation. In this note, we show that $\ell_1$-regularized estimation in some important cases can achieve the same order of error bounds as those in the aforementioned work.