  We consider the problem of reconstructing a rank-one matrix from a revealed
subset of its entries when some of the revealed entries are corrupted with
perturbations that are unknown and can be arbitrarily large. It is not known
which revealed entries are corrupted. We propose a new algorithm combining
alternating minimization with extreme-value filtering and provide sufficient
and necessary conditions to recover the original rank-one matrix. In
particular, we show that our proposed algorithm is optimal when the set of
revealed entries is given by an Erd\H{o}s-R\'enyi random graph. These results
are then applied to the problem of classification from crowdsourced data under
the assumption that while the majority of the workers are governed by the
standard single-coin David-Skene model (i.e., they output the correct answer
with a certain probability), some of the workers can deviate arbitrarily from
this model. In particular, the "adversarial" workers could even make decisions
designed to make the algorithm output an incorrect answer. Extensive
experimental results show our algorithm for this problem, based on rank-one
matrix completion with perturbations, outperforms all other state-of-the-art
methods in such an adversarial scenario.
