  Sparse recovery is one of the most fundamental and well-studied inverse
problems. Standard statistical formulations of the problem are provably solved
by general convex programming techniques and more practical, fast
(nearly-linear time) iterative methods. However, these latter "fast algorithms"
have previously been observed to be brittle in various real-world settings.
  We investigate the brittleness of fast sparse recovery algorithms to
generative model changes through the lens of studying their robustness to a
"helpful" semi-random adversary, a framework which tests whether an algorithm
overfits to input assumptions. We consider the following basic model: let
$\mathbf{A} \in \mathbb{R}^{n \times d}$ be a measurement matrix which contains
an unknown subset of rows $\mathbf{G} \in \mathbb{R}^{m \times d}$ which are
bounded and satisfy the restricted isometry property (RIP), but is otherwise
arbitrary. Letting $x^\star \in \mathbb{R}^d$ be $s$-sparse, and given either
exact measurements $b = \mathbf{A} x^\star$ or noisy measurements $b =
\mathbf{A} x^\star + \xi$, we design algorithms recovering $x^\star$
information-theoretically optimally in nearly-linear time. We extend our
algorithm to hold for weaker generative models relaxing our planted RIP
assumption to a natural weighted variant, and show that our method's guarantees
naturally interpolate the quality of the measurement matrix to, in some
parameter regimes, run in sublinear time.
  Our approach differs from prior fast iterative methods with provable
guarantees under semi-random generative models: natural conditions on a
submatrix which make sparse recovery tractable are NP-hard to verify. We design
a new iterative method tailored to the geometry of sparse recovery which is
provably robust to our semi-random model. We hope our approach opens the door
to new robust, efficient algorithms for natural statistical inverse problems.
