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194
Robust face recognition via sparse representation
 IEEE TRANS. PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2008
"... We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signa ..."
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Cited by 916 (41 self)
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We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by ℓ 1minimization, we propose a general classification algorithm for (imagebased) object recognition. This new framework provides new insights into two crucial issues in face recognition: feature extraction and robustness to occlusion. For feature extraction, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficiently large and whether the sparse representation is correctly computed. Unconventional features such as downsampled images and random projections perform just as well as conventional features such as Eigenfaces and Laplacianfaces, as long as the dimension of the feature space surpasses certain threshold, predicted by the theory of sparse representation. This framework can handle errors due to occlusion and corruption uniformly, by exploiting the fact that these errors are often sparse w.r.t. to the standard (pixel) basis. The theory of sparse representation helps predict how much occlusion the recognition algorithm can handle and how to choose the training images to maximize robustness to occlusion. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm, and corroborate the above claims.
Nonnegative matrix factorization with sparseness constraints
 Jour. of
, 2004
"... www.cs.helsinki.fi/patrik.hoyer ..."
On the equivalence of nonnegative matrix factorization and spectral clustering
 in SIAM International Conference on Data Mining
, 2005
"... Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X ..."
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Cited by 154 (20 self)
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Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X = FGT is equivalent to simultaneous clustering of rows and columns of a bipartite graph. Algorithms are given for computing these symmetric NMFs. 1
Sparse Representation For Computer Vision and Pattern Recognition
, 2009
"... Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of ..."
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Cited by 142 (9 self)
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Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of dictionary plays a key role in bridging this gap: unconventional dictionaries consisting of, or learned from, the training samples themselves provide the key to obtaining stateoftheart results and to attaching semantic meaning to sparse signal representations. Understanding the good performance of such unconventional dictionaries in turn demands new algorithmic and analytical techniques. This review paper highlights a few representative examples of how the interaction between sparse signal representation and computer vision can enrich both fields, and raises a number of open questions for further study.
Orthogonal nonnegative matrix trifactorizations for clustering
 In SIGKDD
, 2006
"... Currently, most research on nonnegative matrix factorization (NMF) focus on 2factor X = FG T factorization. We provide a systematic analysis of 3factor X = FSG T NMF. While unconstrained 3factor NMF is equivalent to unconstrained 2factor NMF, constrained 3factor NMF brings new features to constr ..."
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Cited by 114 (22 self)
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Currently, most research on nonnegative matrix factorization (NMF) focus on 2factor X = FG T factorization. We provide a systematic analysis of 3factor X = FSG T NMF. While unconstrained 3factor NMF is equivalent to unconstrained 2factor NMF, constrained 3factor NMF brings new features to constrained 2factor NMF. We study the orthogonality constraint because it leads to rigorous clustering interpretation. We provide new rules for updating F,S,G and prove the convergence of these algorithms. Experiments on 5 datasets and a real world case study are performed to show the capability of biorthogonal 3factor NMF on simultaneously clustering rows and columns of the input data matrix. We provide a new approach of evaluating the quality of clustering on words using class aggregate distribution and multipeak distribution. We also provide an overview of various NMF extensions and examine their relationships.
Convex and SemiNonnegative Matrix Factorizations
, 2008
"... We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F GT, we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable ra ..."
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Cited by 109 (9 self)
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We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F GT, we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods.
H: Sparse nonnegative matrix factorizations via alternating nonnegativityconstrained least squares for microarray data analysis
 Bioinformatics
"... Many practical pattern recognition problems require nonnegativity constraints. For example, pixels in digital images and chemical concentrations in bioinformatics are nonnegative. Nonnegative matrix factorization (NMF) is a useful technique in approximating these high dimensional data. Sparse N ..."
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Cited by 93 (11 self)
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Many practical pattern recognition problems require nonnegativity constraints. For example, pixels in digital images and chemical concentrations in bioinformatics are nonnegative. Nonnegative matrix factorization (NMF) is a useful technique in approximating these high dimensional data. Sparse NMFs are also useful when we need to control the degree of sparseness in nonnegative basis vectors or nonnegative lowerdimensional representations. In this paper, we introduce novel sparse NMFs via alternating nonnegativityconstrained least squares. We applied one of the proposed sparse NMFs to cancer class discovery and gene expression data analysis. Our experimental results illustrate that our proposed method achieves better clustering performance than NMF based on multiplicative update rules and sparse NMFs based on the gradient descent method. 1
NONNEGATIVE MATRIX FACTORIZATION BASED ON ALTERNATING NONNEGATIVITY CONSTRAINED LEAST SQUARES AND ACTIVE SET METHOD
"... The nonnegative matrix factorization (NMF) determines a lower rank approximation of a ¢¤£¦¥¨§�©���� �� � matrix where an ������������������ � interger is given and nonnegativity is imposed on all components of the factors applied to numerous data analysis problems. In applications where the compone ..."
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Cited by 87 (7 self)
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The nonnegative matrix factorization (NMF) determines a lower rank approximation of a ¢¤£¦¥¨§�©���� �� � matrix where an ������������������ � interger is given and nonnegativity is imposed on all components of the factors applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative such as chemical concentrations in experimental results or pixels in digital images, the NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for the NMF based on alternating nonnegativity constrained least squares (NMF/ANLS) and the active set based fast algorithm for nonnegativity constrained least squares with multiple right hand side vectors, and discuss its convergence properties and a rigorous convergence criterion based on the KarushKuhnTucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by �� �norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior. £�¥�§�©� � and � £�¥���©� �. The NMF has attracted much attention for over a decade and has been successfully
Graph regularized nonnegative matrix factorization for data representation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2011
"... 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 dat ..."
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Cited by 87 (4 self)
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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 lowdimensional manifold embedded in a highdimensional 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 stateoftheart algorithms on realworld problems.
The relationships among various nonnegative matrix factorization methods for clustering
 In ICDM
, 2006
"... The nonnegative matrix factorization (NMF) has been shown recently to be useful for clustering. Various extensions of NMF have also been proposed. In this paper we present an overview and theoretically analyze the relationships among them. In addition, we clarify previously unaddressed issues, such ..."
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Cited by 45 (10 self)
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The nonnegative matrix factorization (NMF) has been shown recently to be useful for clustering. Various extensions of NMF have also been proposed. In this paper we present an overview and theoretically analyze the relationships among them. In addition, we clarify previously unaddressed issues, such as NMF normalization, cluster posterior probabilty, and NMF algoritm convergence rate. Experiments are also conducted to empirically evaluate and compare various factorization methods.