text: Non-negative matrix factorization (NMF) minimizes the euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large errors, and develop a Truncated CauchyNMF to robustly learn the subspace on noisy datasets contaminated by outliers. We theoretically analyze the robustness of Truncated CauchyNMF comparing with the competing models and theoretically prove that Truncated CauchyNMF has a generalization bound which converges at a rate of order @math , where @math is the sample size. We evaluate Truncated CauchyNMF by image clustering on both simulated and real datasets. The experimental results on the datasets containing gross corruptions validate the effectiveness and robustness of Truncated CauchyNMF for learning robust subspaces.
  @cite_9: Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation-Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence.
  @cite_16: Nonnegative matrix factorization (NMF) approximates a given data matrix as a product of two low-rank nonnegative matrices, usually by minimizing the L2 or the KL distance between the data matrix and the matrix product. This factorization was shown to be useful for several important computer vision applications. We propose here two new NMF algorithms that minimize the Earth mover's distance (EMD) error between the data and the matrix product. The algorithms (EMD NMF and bilateral EMD NMF) are iterative and based on linear programming methods. We prove their convergence, discuss their numerical difficulties, and propose efficient approximations. Naturally, the matrices obtained with EMD NMF are different from those obtained with L2-NMF. We discuss these differences in the context of two challenging computer vision tasks, texture classification and face recognition, perform actual NMF-based image segmentation for the first time, and demonstrate the advantages of the new methods with common benchmarks.
  @cite_18: Matrix factorization techniques have been frequently applied in information retrieval, computer vision, and pattern recognition. Among them, Nonnegative Matrix Factorization (NMF) has received considerable attention due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts based in the human brain. On the other hand, from the geometric perspective, the data is usually sampled from a low-dimensional manifold embedded in a high-dimensional ambient space. One then hopes to find a compact representation,which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In this paper, we propose a novel algorithm, called Graph Regularized Nonnegative Matrix Factorization (GNMF), for this purpose. In GNMF, an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization, which respects the graph structure. Our empirical study shows encouraging results of the proposed algorithm in comparison to the state-of-the-art algorithms on real-world problems.
  @cite_7: Let A be a real m×n matrix with m≧n. It is well known (cf. [4]) that @math (1) where @math The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T , and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that @math Thus if rank(A)=r, σ r+1 = σ r+2=⋯=σ n = 0. The decomposition (1) is called the singular value decomposition (SVD).
summary: Traditional NMF @cite_9 assumes that noise obeys a Gaussian distribution and derives the following squared @math -norm based objective function: @math , where @math signifies the matrix Frobenius norm. It is commonly known that NMF can be solved by using the multiplicative update rule (MUR, @cite_9 ). Because of the nice mathematical property of squared @math -norm and the efficiency of MUR, NMF has been extended for various applications @cite_18 @cite_7 @cite_16 . However, NMF and its extensions are non-robust because the @math -norm is sensitive to outliers.