  Feature subset selection arises in many high-dimensional applications of
statistics, such as compressed sensing and genomics. The $\ell_0$ penalty is
ideal for this task, the caveat being it requires the NP-hard combinatorial
evaluation of all models. A recent area of considerable interest is to develop
efficient algorithms to fit models with a non-convex $\ell_\gamma$ penalty for
$\gamma\in (0,1)$, which results in sparser models than the convex $\ell_1$ or
lasso penalty, but is harder to fit. We propose an alternative, termed the
horseshoe regularization penalty for feature subset selection, and demonstrate
its theoretical and computational advantages. The distinguishing feature from
existing non-convex optimization approaches is a full probabilistic
representation of the penalty as the negative of the logarithm of a suitable
prior, which in turn enables efficient expectation-maximization and local
linear approximation algorithms for optimization and MCMC for uncertainty
quantification. In synthetic and real data, the resulting algorithms provide
better statistical performance, and the computation requires a fraction of time
of state-of-the-art non-convex solvers.
