This letter examines the problem of robust subspace discovery from input data samples (instances) in the presence of overwhelming outliers and corruptions. A typical example is the case where we are given a set of images; each image contains, for example, a face at an unknown location of an unknown size; our goal is to identify or detect the face in the image and simultaneously learn its model. We employ a simple generative subspace model and propose a new formulation to simultaneously infer the label information and learn the model using low-rank optimization. Solving this problem enables us to simultaneously identify the ownership of instances to the subspace and learn the corresponding subspace model. We give an efficient and effective algorithm based on the alternating direction method of multipliers and provide extensive simulations and experiments to verify the effectiveness of our method. The proposed scheme can also be used to tackle many related high-dimensional combinatorial selection problems.