  Because of the widespread existence of noise and data corruption, recovering
the true regression parameters with a certain proportion of corrupted response
variables is an essential task. Methods to overcome this problem often involve
robust least-squares regression, but few methods perform well when confronted
with severe adaptive adversarial attacks. In many applications, prior knowledge
is often available from historical data or engineering experience, and by
incorporating prior information into a robust regression method, we develop an
effective robust regression method that can resist adaptive adversarial
attacks. First, we propose the novel TRIP (hard Thresholding approach to Robust
regression with sImple Prior) algorithm, which improves the breakdown point
when facing adaptive adversarial attacks. Then, to improve the robustness and
reduce the estimation error caused by the inclusion of priors, we use the idea
of Bayesian reweighting to construct the more robust BRHT (robust Bayesian
Reweighting regression via Hard Thresholding) algorithm. We prove the
theoretical convergence of the proposed algorithms under mild conditions, and
extensive experiments show that under different types of dataset attacks, our
algorithms outperform other benchmark ones. Finally, we apply our methods to a
data-recovery problem in a real-world application involving a space solar
array, demonstrating their good applicability.
