  It is well-known that the statistical performance of Lasso can suffer
significantly when the covariates of interest have strong correlations. In
particular, the prediction error of Lasso becomes much worse than
computationally inefficient alternatives like Best Subset Selection. Due to a
large conjectured computational-statistical tradeoff in the problem of sparse
linear regression, it may be impossible to close this gap in general.
  In this work, we propose a natural sparse linear regression setting where
strong correlations between covariates arise from unobserved latent variables.
In this setting, we analyze the problem caused by strong correlations and
design a surprisingly simple fix. While Lasso with standard normalization of
covariates fails, there exists a heterogeneous scaling of the covariates with
which Lasso will suddenly obtain strong provable guarantees for estimation.
Moreover, we design a simple, efficient procedure for computing such a "smart
scaling."
  The sample complexity of the resulting "rescaled Lasso" algorithm incurs (in
the worst case) quadratic dependence on the sparsity of the underlying signal.
While this dependence is not information-theoretically necessary, we give
evidence that it is optimal among the class of polynomial-time algorithms, via
the method of low-degree polynomials. This argument reveals a new connection
between sparse linear regression and a special version of sparse PCA with a
near-critical negative spike. The latter problem can be thought of as a
real-valued analogue of learning a sparse parity. Using it, we also establish
the first computational-statistical gap for the closely related problem of
learning a Gaussian Graphical Model.
