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 the observed measurement matrix and a bilinear factorization. Most popular loss functions include the L1 and L2 losses. While L1 is optimal for Laplacian distributed noise, L2 is optimal for Gaussian noise. However, real data is often corrupted by an unknown noise distribution, which is unlikely to be purely Gaussian or Laplacian. To address this problem, this paper proposes a low-rank matrix factorization problem with a Mixture of Gaussians (MoG) noise. The MoG model is a universal approximator for any continuous distribution, and hence is able to model a wider range of real 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 synthetic, structure from motion, face modeling and background subtraction experiments.