We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with matrix compressed sensing, including lasso as a partial problem, and matrix completion, and then we obtain sharp estimation error bounds. To obtain the error bounds for different models such as matrix compressed sensing and matrix completion, we propose a simple unified approach based on a combination of the Huber loss function and the nuclear norm penalization, which is a different approach from the conventional ones. Some error bounds obtained in the present paper are sharper than the past ones.

翻译：本文研究当输出被对抗者污染时，作为迹回归的鲁棒低秩矩阵估计问题。对抗者被允许向任意输出添加任意值，且这些值可以依赖于任何样本。我们处理矩阵压缩感知（包括作为其子问题的lasso）与矩阵补全问题，并获得了尖锐的估计误差界。为得到矩阵压缩感知和矩阵补全等不同模型的误差界，我们提出了一种基于Huber损失函数与核范数惩罚相结合的简单统一方法，这与传统方法有所不同。本文获得的某些误差界比以往结果更为尖锐。