Robust matrix factorization with unknown noise

Many problems in computer vision can be posed as recovering a low-dimensional subspace from high-dimensional visual data. Factorization approaches to low-rank subspace estimation minimize a loss function between an observed measurement matrix and a bilinear factoriza-tion. Most popular loss functions include the L2 and L1 losses. L2 is optimal for Gaussian noise, while L1 is for Laplacian distributed noise. However, real data is often corrupted by an unknown noise distribution, which is un-likely to be purely Gaussian or Laplacian. To address this problem, this paper proposes a low-rank matrix factoriza-tion problem with a Mixture of Gaussians (MoG) noise model. The MoG model is a universal approximator for any continuous distribution, and hence is able to model a wider range of noise distributions. The parameters of the MoG model can be estimated with a maximum likelihood method, while the subspace is computed with standard approaches. We illustrate the benefits of our approach in extensive syn-thetic and real-world experiments including structure from motion, face modeling and background subtraction. 1