  We consider the problem of recovering a low-rank matrix from its clipped
observations. Clipping is conceivable in many scientific areas that obstructs
statistical analyses. On the other hand, matrix completion (MC) methods can
recover a low-rank matrix from various information deficits by using the
principle of low-rank completion. However, the current theoretical guarantees
for low-rank MC do not apply to clipped matrices, as the deficit depends on the
underlying values. Therefore, the feasibility of clipped matrix completion
(CMC) is not trivial. In this paper, we first provide a theoretical guarantee
for the exact recovery of CMC by using a trace-norm minimization algorithm.
Furthermore, we propose practical CMC algorithms by extending ordinary MC
methods. Our extension is to use the squared hinge loss in place of the squared
loss for reducing the penalty of over-estimation on clipped entries. We also
propose a novel regularization term tailored for CMC. It is a combination of
two trace-norm terms, and we theoretically bound the recovery error under the
regularization. We demonstrate the effectiveness of the proposed methods
through experiments using both synthetic and benchmark data for recommendation
systems.
