We provide a novel -- and to the best of our knowledge, the first --
algorithm for high dimensional sparse regression with constant fraction of
corruptions in explanatory and/or response variables. Our algorithm recovers
the true sparse parameters with sub-linear sample complexity, in the presence
of a constant fraction of arbitrary corruptions. Our main contribution is a
robust variant of Iterative Hard Thresholding. Using this, we provide accurate
estimators: when the covariance matrix in sparse regression is identity, our
error guarantee is near information-theoretically optimal. We then deal with
robust sparse regression with unknown structured covariance matrix. We propose
a filtering algorithm which consists of a novel randomized outlier removal
technique for robust sparse mean estimation that may be of interest in its own
right: the filtering algorithm is flexible enough to deal with unknown
covariance. Also, it is orderwise more efficient computationally than the
ellipsoid algorithm. Using sub-linear sample complexity, our algorithm achieves
the best known (and first) error guarantee. We demonstrate the effectiveness on
large-scale sparse regression problems with arbitrary corruptions.

提出了一种用于高维度稀疏回归中具有常量分数的自变量和 / 或响应变量的污染的算法，它是迄今为止的首个这样的算法，利用使用这种算法，我们提供了强健的稀疏回归方法和过滤算法。