  This work presents a fast and non-convex algorithm for robust subspace
recovery. The data sets considered include inliers drawn around a
low-dimensional subspace of a higher dimensional ambient space, and a possibly
large portion of outliers that do not lie nearby this subspace. The proposed
algorithm, which we refer to as Fast Median Subspace (FMS), is designed to
robustly determine the underlying subspace of such data sets, while having
lower computational complexity than existing methods. We prove convergence of
the FMS iterates to a stationary point. Further, under a special model of data,
FMS converges to a point which is near to the global minimum with overwhelming
probability. Under this model, we show that the iteration complexity is
globally bounded and locally $r$-linear. The latter theorem holds for any fixed
fraction of outliers (less than 1) and any fixed positive distance between the
limit point and the global minimum. Numerical experiments on synthetic and real
data demonstrate its competitive speed and accuracy.
