  LASSO regularization is a popular regression tool to enhance the prediction
accuracy of statistical models by performing variable selection through the
$\ell_1$ penalty, initially formulated for the linear model and its variants.
In this paper, the territory of LASSO is extended to two-layer ReLU neural
networks, a fashionable and powerful nonlinear regression model. Specifically,
given a neural network whose output $y$ depends only on a small subset of input
$\boldsymbol{x}$, denoted by $\mathcal{S}^{\star}$, we prove that the LASSO
estimator can stably reconstruct the neural network and identify
$\mathcal{S}^{\star}$ when the number of samples scales logarithmically with
the input dimension. This challenging regime has been well understood for
linear models while barely studied for neural networks. Our theory lies in an
extended Restricted Isometry Property (RIP)-based analysis framework for
two-layer ReLU neural networks, which may be of independent interest to other
LASSO or neural network settings. Based on the result, we advocate a neural
network-based variable selection method. Experiments on simulated and
real-world datasets show promising performance of the variable selection
approach compared with existing techniques.
