Nonnegative rank factorization—a heuristic approach via rank reduction

Dong, Bo; Lin, Matthew M.; Chu, Moody T.

doi:10.1007/s11075-013-9704-0

Cited by 16 publications

(33 citation statements)

References 27 publications

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“…Finally, Dong, Lin, and Chu [11] provide a heuristic method for the so-called (nonsymmetric) nonnegative rank factorization, i.e., finding a decomposition X = U V of X with U, V nonnegative but not necessarily U = V T (which would correspond to our setting). Their procedure can be applied to completely positive matrices and would be able to heuristically check whether cp-rank(X) = rank(X) and, if affirmative, compute a factorization of X.…”

mentioning

confidence: 99%

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Dickinson¹,

Duer²

2012

SIAM J. Matrix Anal. & Appl.

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Abstract.A matrix X is called completely positive if it allows a factorization X = b∈B bb T with nonnegative vectors b. These matrices are of interest in optimization, as it has been found that several combinatorial and quadratic problems can be formulated over the cone of completely positive matrices. The difficulty is that checking complete positivity is N P-hard. Finding a factorization of a general completely positive matrix is also hard. In this paper we study complete positivity of matrices whose underlying graph possesses a specific sparsity pattern, for example, being acyclic or circular, where the underlying graph of a symmetric matrix of order n is defined to be a graph with n vertices and an edge between two vertices if the corresponding entry in the matrix is nonzero. The types of matrices that we analyze include tridiagonal matrices as an example. We show that in these cases checking complete positivity can be done in linear-time. A factorization of such a completely positive matrix can be found in linear-time as well. As a by-product, our method provides insight on the number of different minimal rank-one decompositions of the matrix.

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confidence: 99%

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Dickinson¹,

Duer²

2012

SIAM J. Matrix Anal. & Appl.

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“…The number of components in P Δ is identified with the non-negative rank of P Δ . Determining the non-negative rank of a matrix is computationally difficult however, and is still a subject of on-going research (see, for example, Dong et al (2009)). (Vavasis (2009) has shown that determining the non-negative rank of a matrix is non-deterministic polynomial time hard.)…”

Section: Two-variable Casementioning

confidence: 99%

Non-Parametric Identification and Estimation of the Number of Components in Multivariate Mixtures

Kasahara

Shimotsu

2013

Journal of the Royal Statistical Society Series B: Statistical Methodology

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This article analyzes the identifiability of the number of components in k-variate, Mcomponent finite mixture models in which each component distribution has independent marginals, including models in latent class analysis. Without making parametric assumptions on the component distributions, we investigate how one can identify the number of components from the distribution function of the observed data. When k ≥ 2, a lower bound on the number of components (M ) is nonparametrically identifiable from the rank of a matrix constructed from the distribution function of the observed variables. Building on this identification condition, we develop a procedure to consistently estimate a lower bound on the number of components.

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“…This dataset also contains a word-document matrix of size 3;703 Â 3;312 and a citation matrix of size 3;312 Â 3;312. Another kind of dataset Cora, 8 called Cora-II, is used to demonstrate our approach on classification problems. Cora-II consist of abstracts and references of research papers in the five fields of computer science: data structure (DS), hardware and architecture (HA), ML, operation system (OS), and programming language (PL); each of the fields is of multiple subjects.…”

Section: Numerical Implementationsmentioning

confidence: 99%

“…Practically, it is an NPhard problem to compute the nonnegative rank. One can refer to [8] for a heuristic approach. The traditional SVD or the NMF ignore the possible nonlinearity inherent in data.…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Matrix Approximation with Manifold Regularization

Zhang

Zhao

2013

IEEE Trans. Pattern Anal. Mach. Intell.

152

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This paper proposes a new model of low-rank matrix factorization that incorporates manifold regularization to the matrix factorization. Superior to the graph-regularized nonnegative matrix factorization, this new regularization model has globally optimal and closed-form solutions. A direct algorithm (for data with small number of points) and an alternate iterative algorithm with inexact inner iteration (for large scale data) are proposed to solve the new model. A convergence analysis establishes the global convergence of the iterative algorithm. The efficiency and precision of the algorithm are demonstrated numerically through applications to six real-world datasets on clustering and classification. Performance comparison with existing algorithms shows the effectiveness of the proposed method for low-rank factorization in general.

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Nonnegative rank factorization—a heuristic approach via rank reduction

Cited by 16 publications

References 27 publications

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Non-Parametric Identification and Estimation of the Number of Components in Multivariate Mixtures

Low-Rank Matrix Approximation with Manifold Regularization

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