  Sparse linear regression with ill-conditioned Gaussian random designs is
widely believed to exhibit a statistical/computational gap, but there is
surprisingly little formal evidence for this belief, even in the form of
examples that are hard for restricted classes of algorithms. Recent work has
shown that, for certain covariance matrices, the broad class of Preconditioned
Lasso programs provably cannot succeed on polylogarithmically sparse signals
with a sublinear number of samples. However, this lower bound only shows that
for every preconditioner, there exists at least one signal that it fails to
recover successfully. This leaves open the possibility that, for example,
trying multiple different preconditioners solves every sparse linear regression
problem.
  In this work, we prove a stronger lower bound that overcomes this issue. For
an appropriate covariance matrix, we construct a single signal distribution on
which any invertibly-preconditioned Lasso program fails with high probability,
unless it receives a linear number of samples.
  Surprisingly, at the heart of our lower bound is a new positive result in
compressed sensing. We show that standard sparse random designs are with high
probability robust to adversarial measurement erasures, in the sense that if
$b$ measurements are erased, then all but $O(b)$ of the coordinates of the
signal are still information-theoretically identifiable. To our knowledge, this
is the first time that partial recoverability of arbitrary sparse signals under
erasures has been studied in compressed sensing.
