  Many problems in data science can be treated as estimating a low-rank matrix
from highly incomplete, sometimes even corrupted, observations. One popular
approach is to resort to matrix factorization, where the low-rank matrix
factors are optimized via first-order methods over a smooth loss function, such
as the residual sum of squares. While tremendous progresses have been made in
recent years, the natural smooth formulation suffers from two sources of
ill-conditioning, where the iteration complexity of gradient descent scales
poorly both with the dimension as well as the condition number of the low-rank
matrix. Moreover, the smooth formulation is not robust to corruptions. In this
paper, we propose scaled subgradient methods to minimize a family of nonsmooth
and nonconvex formulations -- in particular, the residual sum of absolute
errors -- which is guaranteed to converge at a fast rate that is almost
dimension-free and independent of the condition number, even in the presence of
corruptions. We illustrate the effectiveness of our approach when the
observation operator satisfies certain mixed-norm restricted isometry
properties, and derive state-of-the-art performance guarantees for a variety of
problems such as robust low-rank matrix sensing and quadratic sampling.
