A convex minimization is formulated to robustly recover a subspace from a contaminated data set, partially sampled around it, and propose a fast iterative algorithm to achieve the corresponding minimum. This disclosure establishes exact recovery by this minimizer, quantifies the effect of noise and regularization, and explains how to take advantage of a known intrinsic dimension and establish linear convergence of the iterative algorithm. The minimizer is an M-estimator. The disclosure demonstrates its significance by adapting it to formulate a convex minimization equivalent to the non-convex total least squares (which is solved by PCA). The technique is compared with many other algorithms for robust PCA on synthetic and real data sets and state-of-the-art speed and accuracy is demonstrated.