On the nonnegative rank of Euclidean distance matrices

Lin, Matthew M.; Chu, Moody T.

doi:10.1016/j.laa.2010.03.038

Cited by 20 publications

(10 citation statements)

References 14 publications

(17 reference statements)

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“…Formulations and algorithms based on Kullback-Leibler divergence [67,79], Bregman divergence [24,68], Itakura-Saito divergence [29], and Alpha and Beta divergences [21,22] have been developed. For discussion on nonnegative rank as well as the geometric interpretation of NMF, see Lin and Chu [72], Gillis [34], and Donoho and Stodden [27]. NMF has been also studied from the Bayesian statistics point of view: See Schmidt et al [78] and Zhong and Girolami [88].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

2013

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We review algorithms developed for nonnegative matrix factorization (NMF) and nonnegative tensor factorization (NTF) from a unified view based on the block coordinate descent (BCD) framework. NMF and NTF are low-rank approximation methods for matrices and tensors in which the low-rank factors are constrained to have only nonnegative elements. The nonnegativity constraints have been shown to enable natural interpretations and allow better solutions in numerous applications including text analysis, computer vision, and bioinformatics. However, the computation of NMF and NTF remains challenging and expensive due the constraints. Numerous algorithmic approaches have been proposed to efficiently compute NMF and NTF. The BCD framework in constrained non-linear optimization readily explains the theoretical convergence properties of several efficient NMF and NTF algorithms, which are consistent with experimental observations reported in literature. In addition, we discuss algorithms that do not fit in the BCD framework contrasting them from those based on the BCD framework. With insights acquired from the unified perspective, we also propose efficient algorithms for updating NMF when there is a small change in the reduced dimension or in the data. The effectiveness of the proposed updating algorithms are validated experimentally with synthetic and real-world data sets.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

2013

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“…Lin and Chu [11] claimed a positive answer for Problem 1.3, but their argument has been shown to contain a gap [8,9]. A negative answer for Problem 1.3 has been obtained in [8] for a special case of so-called Euclidean distance matrices.…”

Section: Theorem 12 [10 Theorem 2]mentioning

confidence: 99%

An upper bound for nonnegative rank

Shitov

2014

Journal of Combinatorial Theory, Series A

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Abstract. We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that 6n 7 linear inequalities suffice to describe a convex n-gon up to a linear projection. PreliminariesConsider a convex polytope P ⊂ R n . An extension [5,8] of P is a polytope Q ⊂ R d such that P can be obtained from Q as an image under a linear projection from R d to R n . An extended formulation [8,10] of P is a description of Q by linear equations and linear inequalities (together with the projection). The size [8, 10] of the extended formulation is the number of facets of Q. The extension complexity [8, 10] of a polytope P is the smallest size of any extended formulation of P , that is, the minimal possible number of inequalities in the description of Q. The number of facets of Q can sometimes be significantly smaller [5] than that of P , and this phenomenon can be used to reduce the complexity of linear programming problems useful for numerous applications [3,5,10

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“…This is interesting because it is nontrivial to construct matrices with rank(M) < rank + (M) [37]. In fact, it is easy to check that picking two random nonnegative matrices U and V of dimensions m-by-r and r-by-n respectively, and constructing M = UV will generate with probability one a matrix M of rank r, hence with rank(M) = rank + (M) .…”

Section: Remarkmentioning

confidence: 99%

On the geometric interpretation of the nonnegative rank

Gillis

Glineur

2012

Linear Algebra and its Applications

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The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining and graph theory. In particular, it can be used to characterize the minimal size of any extended reformulation of a given polytope. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in computational geometry, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasley and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.

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On the nonnegative rank of Euclidean distance matrices

Abstract: The Euclidean distance matrix for n distinct points in ℝr is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.

Cited by 20 publications

References 14 publications

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

An upper bound for nonnegative rank

On the geometric interpretation of the nonnegative rank

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