  We address the inverse problem that arises in compressed sensing of a
low-rank matrix. Our approach is to pose the inverse problem as an
approximation problem with a specified target rank of the solution. A simple
search over the target rank then provides the minimum rank solution satisfying
a prescribed data approximation bound. We propose an atomic decomposition that
provides an analogy between parsimonious representations of a sparse vector and
a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for
the vector case are extended to the matrix case through this atomic
decomposition. In particular, we propose an efficient and guaranteed algorithm
named ADMiRA that extends CoSaMP, its analogue for the vector case. The
performance guarantee is given in terms of the rank-restricted isometry
property and bounds both the number of iterations and the error in the
approximate solution for the general case where the solution is approximately
low-rank and the measurements are noisy. With a sparse measurement operator
such as the one arising in the matrix completion problem, the computation in
ADMiRA is linear in the number of measurements. The numerical experiments for
the matrix completion problem show that, although the measurement operator in
this case does not satisfy the rank-restricted isometry property, ADMiRA is a
competitive algorithm for matrix completion.
