  Sparse regression models are increasingly prevalent due to their ease of
interpretability and superior out-of-sample performance. However, the exact
model of sparse regression with an $\ell_0$ constraint restricting the support
of the estimators is a challenging (\NP-hard) non-convex optimization problem.
In this paper, we derive new strong convex relaxations for sparse regression.
These relaxations are based on the ideal (convex-hull) formulations for
rank-one quadratic terms with indicator variables. The new relaxations can be
formulated as semidefinite optimization problems in an extended space and are
stronger and more general than the state-of-the-art formulations, including the
perspective reformulation and formulations with the reverse Huber penalty and
the minimax concave penalty functions. Furthermore, the proposed rank-one
strengthening can be interpreted as a \textit{non-separable, non-convex,
unbiased} sparsity-inducing regularizer, which dynamically adjusts its penalty
according to the shape of the error function without inducing bias for the
sparse solutions. In our computational experiments with benchmark datasets, the
proposed conic formulations are solved within seconds and result in
near-optimal solutions (with 0.4\% optimality gap) for non-convex
$\ell_0$-problems. Moreover, the resulting estimators also outperform
alternative convex approaches from a statistical perspective, achieving high
prediction accuracy and good interpretability.
