On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices

Catral, Minerva; Han, Lixing; Neumann, Michael; Plemmons, Robert J.

doi:10.1016/j.laa.2003.11.024

Cited by 53 publications

(37 citation statements)

References 8 publications

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“…The state-of-art in terms of symmetric NMF algorithms can be found in He et al [29], which proposed three algorithms-the multiplicative update, -SNMF and -SNMF, and showed that the latter two outperform other alternatives. Other papers on symmetric NMF include Catral et al [30] who studied the symmetric NMF problem for that is not necessarily positive semidefinite, and conditions under which asymmetric NMF yields a symmetric approximation; and Ding et al [31], who developed interesting links between symmetric NMF and 'soft' k-means clustering.…”

Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning

confidence: 99%

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Huang

Sidiropoulos

Swami

2014

IEEE Trans. Signal Process.

252

248

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Abstract-Non-negative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms as well. Uniqueness aspects of NMF are revisited here from a geometrical point of view. Both symmetric and asymmetric NMF are considered, the former being tantamount to element-wise non-negative square-root factorization of positive semidefinite matrices. New uniqueness results are derived, e.g., it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second-order cone. Checking this condition is shown to be NP-complete; yet this and other results offer insights on the role of latent sparsity in this context. On the computational side, a new algorithm for symmetric NMF is proposed, which is very different from existing ones. It alternates between Procrustes rotation and projection onto the non-negative orthant to find a non-negative matrix close to the span of the dominant subspace. Simulation results show promising performance with respect to the state-of-art. Finally, the new algorithm is applied to a clustering problem for co-authorship data, yielding meaningful and interpretable results.

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Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning

confidence: 99%

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Huang

Sidiropoulos

Swami

2014

IEEE Trans. Signal Process.

252

248

View full text Add to dashboard Cite

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“…As mentioned in Section 1, the nonnegative factorization of a kernel matrix (1.3) appeared in Ding et al [7], where the kernel matrix K is positive semi-definite and nonnegative. Their algorithm for solving (1.3) is a variation of the multiplicative update rule algorithm for NMF [4]. When K has negative entries, this algorithm for symmetric NMF is not applicable because it introduces negative entries into H and violates the constraint H ≥ 0.…”

Section: Related Workmentioning

confidence: 99%

“…In previous work on symmetric NMF [7,4], A is required to be nonnegative and symmetric positive semidefinite. Our formulation differs in that the matrix A in (2.4) can be any symmetric matrix representing similarity values.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Nonnegative Matrix Factorization for Graph Clustering

Kuang

Ding

Park

2012

Proceedings of the 2012 SIAM International Conference on Data Mining

359

323

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Nonnegative matrix factorization (NMF) provides a lower rank approximation of a nonnegative matrix, and has been successfully used as a clustering method. In this paper, we offer some conceptual understanding for the capabilities and shortcomings of NMF as a clustering method. Then, we propose Symmetric NMF (SymNMF) as a general framework for graph clustering, which inherits the advantages of NMF by enforcing nonnegativity on the clustering assignment matrix. Unlike NMF, however, SymNMF is based on a similarity measure between data points, and factorizes a symmetric matrix containing pairwise similarity values (not necessarily nonnegative). We compare SymNMF with the widely-used spectral clustering methods, and give an intuitive explanation of why SymNMF captures the cluster structure embedded in the graph representation more naturally. In addition, we develop a Newton-like algorithm that exploits second-order information efficiently, so as to show the feasibility of SymNMF as a practical framework for graph clustering. Our experiments on artificial graph data, text data, and image data demonstrate the substantially enhanced clustering quality of SymNMF over spectral clustering and NMF. Therefore, SymNMF is able to achieve better clustering results on both linear and nonlinear manifolds, and serves as a potential basis for many extensions and applications.

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“…This regretfully is not the case as one can easily construct counter examples. A recent study by [4] concludes that NMF would generate a symmetric or CP decomposition only in very specialized situations. Therefore the NMF machinery is of no help as well.…”

Section: The Cp Factorization Algorithmmentioning

confidence: 99%