  Subspace recovery from corrupted and missing data is crucial for various
applications in signal processing and information theory. To complete missing
values and detect column corruptions, existing robust Matrix Completion (MC)
methods mostly concentrate on recovering a low-rank matrix from few corrupted
coefficients w.r.t. standard basis, which, however, does not apply to more
general basis, e.g., Fourier basis. In this paper, we prove that the range
space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few
coefficients w.r.t. general basis, though $r$ and the number of corrupted
samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers
previous ones as special cases, and robust MC can recover the intrinsic matrix
with a higher rank. Moreover, we suggest a universal choice of the
regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our
$\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can
further reduce the computational cost of our model. As an application, we also
find that the solutions to extended robust Low-Rank Representation and to our
extended robust MC are mutually expressible, so both our theory and algorithm
can be applied to the subspace clustering problem with missing values under
certain conditions. Experiments verify our theories.
