Computing symmetric nonnegative rank factorizations

Kalofolias, Vassilis; Gallopoulos, Efstratios

doi:10.1016/j.laa.2011.03.016

Cited by 15 publications

(16 citation statements)

References 34 publications

(59 reference statements)

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php proof. Kalofolias and Gallopoulos [17] extend this result and construct a factorization of completely positive rank-two matrices.…”

mentioning

confidence: 74%

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: 74%

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Dickinson¹,

Duer²

2012

SIAM J. Matrix Anal. & Appl.

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“…The case of rank(A) = 2 has been studied in [4] and [1] where an algorithmic process for an exact NRF of A is proposed but our way is very simple and a part of our general method of matrix factorization. In [5], an exact, symmetric nonnegative rank factorization of A, i.e. A = W W T , is determined in the case where A is a symmetric n × n nonnegative real matrix which contains a diagonal principal submatrix of the same rank with A.…”

Section: The Nmf Problem and Positive Basesmentioning

confidence: 99%

The NMF problem and lattice-subspaces

Polyrakis

2020

Linear Algebra and its Applications

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Suppose that A is a nonnegative n × m real matrix. The NMF problem is the determination of two nonnegative real matrices F , V so that A = F V with intermediate dimension p smaller than min{n, m}. In this article we present a general mathematical method for the determination of two nonnegative real factors F, V of A. During the first steps of this process the intermediate dimension p of F, V is determined, therefore we have an easy criterion for p. This study is based on the theory of lattice-subspaces and positive bases. Also we give the matlab program for the computation of F, V but the mathematical part is the main part of this article.

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“…Algorithm 4 can be easily adapted to handle (19), by replacing the b ij 's with b ij + Λ j . In fact, the derivative of the penalty term only influences the constant part in the gradient; see (12).…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

“…In fact, the derivative of the penalty term only influences the constant part in the gradient; see (12). However, it seems the solutions of (19) are very sensitive to the parameter Λ and hence are difficult to tune. Note that another way to identify sparser factors is simply to increase the factorization rank r, or to sparsify the input matrix A (only keeping the important edges in the graph induced by A; see [1] and the references therein) -in fact, a sparser matrix A induces sparser factors since…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

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Efficient and Non-Convex Coordinate Descent for Symmetric Nonnegative Matrix Factorization

Vandaele

Gillis

Lei

et al. 2016

IEEE Trans. Signal Process.

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Given a symmetric nonnegative matrix A, symmetric nonnegative matrix factorization (sym-NMF) is the problem of finding a nonnegative matrix H, usually with much fewer columns than A, such that A ≈ HH T . SymNMF can be used for data analysis and in particular for various clustering tasks. In this paper, we propose simple and very efficient coordinate descent schemes to solve this problem, and that can handle large and sparse input matrices. The effectiveness of our methods is illustrated on synthetic and real-world data sets, and we show that they perform favorably compared to recent state-of-the-art methods.

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Computing symmetric nonnegative rank factorizations

Cited by 15 publications

References 34 publications

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices

The NMF problem and lattice-subspaces

Efficient and Non-Convex Coordinate Descent for Symmetric Nonnegative Matrix Factorization

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