ADMiRA: Atomic Decomposition for Minimum Rank Approximation

In this paper, we address compressed sensing of a low-rank matrix posing 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 providing an analogy between parsimonious representations of a sparse vector and a low-rank matrix and extending efficient greedy algorithms from the vector to the matrix case. In particular, we propose an efficient and guaranteed algorithm named atomic decomposition for minimum rank approximation (ADMiRA) that extends Needell and Tropp's compressive sampling matching pursuit (CoSaMP) algorithm from the sparse vector to the low-rank matrix case. The performance guarantee is given in terms of the rank-restricted isometry property (R-RIP) and bounds both the number of iterations and the error in the approximate solution for the general case of noisy measurements and approximately low-rank solution. With a sparse measurement operator as in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. Numerical experiments for the matrix completion problem show that, although the R-RIP is not satisfied in this case, ADMiRA is a competitive algorithm for matrix completion.