  The paper deals with the problem of penalized empirical risk minimization
over a convex set of linear functionals on the space of Hermitian matrices with
convex loss and nuclear norm penalty. Such penalization is often used in low
rank matrix recovery in the cases when the target function can be well
approximated by a linear functional generated by a Hermitian matrix of
relatively small rank (comparing with the size of the matrix). Our goal is to
prove sharp low rank oracle inequalities that involve the excess risk (the
approximation error) with constant equal to one and the random error term with
correct dependence on the rank of the oracle.
