  We give safe screening rules to eliminate variables from regression with
$\ell_0$ regularization or cardinality constraint. These rules are based on
guarantees that a feature may or may not be selected in an optimal solution.
The screening rules can be computed from a convex relaxation solution in linear
time, without solving the $\ell_0$ optimization problem. Thus, they can be used
in a preprocessing step to safely remove variables from consideration apriori.
Numerical experiments on real and synthetic data indicate that, on average,
76\% of the variables can be fixed to their optimal values, hence, reducing the
computational burden for optimization substantially. Therefore, the proposed
fast and effective screening rules extend the scope of algorithms for
$\ell_0$-regression to larger data sets.
