A method recovers an uncorrupted low-rank matrix, noise in corrupted data and a subspace from the data in a form of a high-dimensional matrix. An objective function minimizes the noise to solve for the low-rank matrix and the subspace without estimating the rank of the low-rank matrix. The method uses group sparsity and the subspace is orthogonal. Random subsampling of the data can recover subspace bases and their coefficients from a much smaller matrix to improve performance. Convergence efficiency can also be improved by applying an augmented Lagrange multiplier, and an alternating stepwise coordinate descent. The Lagrange function is solved by an alternating direction method.