Low-rank matrix factorization (LRMF) has received much popularity owing to its successful applications in both computer vision and data mining. By assuming the noise term to come from a Gaussian, Laplace or a mixture of Gaussian distributions, significant efforts have been made on optimizing the (weighted) @math or @math -norm loss between an observed matrix and its bilinear factorization. However, the type of noise distribution is generally unknown in real applications and inappropriate assumptions will inevitably deteriorate the behavior of LRMF. On the other hand, real data are often corrupted by skew rather than symmetric noise. To tackle this problem, this paper presents a novel LRMF model called AQ-LRMF by modeling noise with a mixture of asymmetric Laplace distributions. An efficient algorithm based on the expectation-maximization (EM) algorithm is also offered to estimate the parameters involved in AQ-LRMF. The AQ-LRMF model possesses the advantage that it can approximate noise well no matter whether the real noise is symmetric or skew. The core idea of AQ-LRMF lies in solving a weighted @math problem with weights being learned from data. The experiments conducted with synthetic and real datasets show that AQ-LRMF outperforms several state-of-the-art techniques. Furthermore, AQ-LRMF also has the superiority over the other algorithms that it can capture local structural information contained in real images.
We analyse the noise arising in collaborative filtering when formalised as a probabilistic matrix factorisation problem. We show empirically that modelling row- and column-specific variances is important, the noise being in general non-Gaussian and heteroscedastic. We also advocate for the use of a Student-t prior for the latent features as the standard Gaussian is included as a special case. We derive several variational inference algorithms and estimate the hyperparameters by type-II maximum likelihood. Experiments on real data show that the predictive performance is significantly improved. Matrix factorization underlies a large variety of computer vision applications. It is a particularly challenging problem for large-scale applications and when there exist outliers and missing data. In this paper, we propose a novel probabilistic model called Probabilistic Robust Matrix Factorization (PRMF) to solve this problem. In particular, PRMF is formulated with a Laplace error and a Gaussian prior which correspond to an l1 loss and an l2 regularizer, respectively. For model learning, we devise a parallelizable expectation-maximization (EM) algorithm which can potentially be applied to large-scale applications. We also propose an online extension of the algorithm for sequential data to offer further scalability. Experiments conducted on both synthetic data and some practical computer vision applications show that PRMF is comparable to other state-of-the-art robust matrix factorization methods in terms of accuracy and outperforms them particularly for large data matrices. Many computer vision problems can be posed as learning a low-dimensional subspace from high dimensional data. The low rank matrix factorization (LRMF) represents a commonly utilized subspace learning strategy. Most of the current LRMF techniques are constructed on the optimization problem using L_1 norm and L_2 norm, which mainly deal with Laplacian and Gaussian noise, respectively. To make LRMF capable of adapting more complex noise, this paper proposes a new LRMF model by assuming noise as Mixture of Exponential Power (MoEP) distributions and proposes a penalized MoEP model by combining the penalized likelihood method with MoEP distributions. Such setting facilitates the learned LRMF model capable of automatically fitting the real noise through MoEP distributions. Each component in this mixture is adapted from a series of preliminary super-or sub-Gaussian candidates. An Expectation Maximization (EM) algorithm is also designed to infer the parameters involved in the proposed PMoEP model. The advantage of our method is demonstrated by extensive experiments on synthetic data, face modeling and hyperspectral image restoration. 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.