  Statistical and machine learning theory has developed several conditions
ensuring that popular estimators such as the Lasso or the Dantzig selector
perform well in high-dimensional sparse regression, including the restricted
eigenvalue, compatibility, and $\ell_q$ sensitivity properties. However, some
of the central aspects of these conditions are not well understood. For
instance, it is unknown if these conditions can be checked efficiently on any
given data set. This is problematic, because they are at the core of the theory
of sparse regression.
  Here we provide a rigorous proof that these conditions are NP-hard to check.
This shows that the conditions are computationally infeasible to verify, and
raises some questions about their practical applications.
  However, by taking an average-case perspective instead of the worst-case view
of NP-hardness, we show that a particular condition, $\ell_q$ sensitivity, has
certain desirable properties. This condition is weaker and more general than
the others. We show that it holds with high probability in models where the
parent population is well behaved, and that it is robust to certain data
processing steps. These results are desirable, as they provide guidance about
when the condition, and more generally the theory of sparse regression, may be
relevant in the analysis of high-dimensional correlated observational data.
