Clipping (a.k.a. censoring) is a salient type of measurement deficit that appears in various scientific areas such as biology and psychology, and it is known to obstruct the statistical analyses of the data. What can we do to mitigate this problem, especially after we have already made the measurements?
We consider recovering a low-rank matrix from its clipped observations (clipped matrix completion; CMC). Matrix completion (MC) methods are known to provably recover a low-rank matrix from various information deficits such as random missingness. However, the existing theoretical guarantees for low-rank MC do not apply to clipped matrices because the deficit depends on the underlying values. Therefore, the feasibility of 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, and we further propose practical CMC algorithms by extending ordinary MC methods. We also propose a novel regularization term tailored for CMC. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.
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