  Kernel-based feature selection is an important tool in nonparametric
statistics. Despite many practical applications of kernel-based feature
selection, there is little statistical theory available to support the method.
A core challenge is the objective function of the optimization problems used to
define kernel-based feature selection are nonconvex. The literature has only
studied the statistical properties of the \emph{global optima}, which is a
mismatch, given that the gradient-based algorithms available for nonconvex
optimization are only able to guarantee convergence to local minima. Studying
the full landscape associated with kernel-based methods, we show that feature
selection objectives using the Laplace kernel (and other $\ell_1$ kernels) come
with statistical guarantees that other kernels, including the ubiquitous
Gaussian kernel (or other $\ell_2$ kernels) do not possess. Based on a sharp
characterization of the gradient of the objective function, we show that
$\ell_1$ kernels eliminate unfavorable stationary points that appear when using
an $\ell_2$ kernel. Armed with this insight, we establish statistical
guarantees for $\ell_1$ kernel-based feature selection which do not require
reaching the global minima. In particular, we establish model-selection
consistency of $\ell_1$-kernel-based feature selection in recovering main
effects and hierarchical interactions in the nonparametric setting with $n \sim
\log p$ samples.
