Linear subspace models have recently been successfully employed to model highly incomplete high-dimensional data, but they are sometimes too restrictive to model the data well. Modeling data as a union of subspaces gives more flexibility and leads to the problem of Subspace Clustering, or clustering vectors into groups that lie in or near the same subspace. Low-rank matrix completion allows one to estimate a single subspace from incomplete data, and this work has recently been extended for the union of subspaces problem [3]. However, the algorithm analyzed there is computationally demanding. Here we present a fast algorithm that combines GROUSE, an incremental matrix completion algorithm, and k-subspaces, the alternating minimization heuristic for solving the subspace clustering problem. k-GROUSE is two orders of magnitude faster than the algorithm proposed in [3] and relies on a slightly more general projection theorem which we present here.