  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.
