Building a completely positive factorization

Bomze, Immanuel M.

doi:10.1007/s10100-017-0499-2

Cited by 8 publications

(8 citation statements)

References 53 publications

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“…The next question we should answer is how we can prune constraints from our outer approximation (5). Pruning is often used to reduce the number of constraints defining the outer approximation, which means keeps the computational effort per iteration stable.…”

Section: Pruning Constraintsmentioning

confidence: 99%

“…Let A ∈ R m×n have rows a ⊤ 1 , ..., a ⊤ m , and let b ∈ R m , and r > 0. Let Q be as defined in (5), and assume that it has nonempty interior. Define Φ as in (14), and let x * be the minimizer of Φ.…”

Section: : End Functionmentioning

confidence: 99%

“…An obvious way to verify that C is completely positive is to find a factorization C = BB ⊤ where B ≥ 0. Several authors have done this for specific matrix structures, see Dickinson and Dür [15], Bomze [5], and the references therein. For general matrices, factorization methods have been proposed by Nie [33], and Sponsel and Dür [39], but these methods do not perform well on bigger matrices.…”

Section: Introductionmentioning

confidence: 99%

“….., a ⊤ m , and let b ∈ R m , and r > 0. Let Q be as defined in (5), and assume that it has nonempty interior. Then, x * is the analytic center of Q if and only if there exist…”

mentioning

confidence: 99%

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An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

Badenbroek¹,

Klerk²

2020

Preprint

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We propose an analytic center cutting plane method to determine if a matrix is completely positive, and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman, Dür, and Shaked-Monderer [Electronic Journal of Linear Algebra, 2015]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like O(d 2 ) for d×d matrices. The method is implemented in Julia, and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl.

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Section: Pruning Constraintsmentioning

confidence: 99%

Section: : End Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“….., a ⊤ m , and let b ∈ R m , and r > 0. Let Q be as defined in (5), and assume that it has nonempty interior. Then, x * is the analytic center of Q if and only if there exist…”

mentioning

confidence: 99%

See 2 more Smart Citations

An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

Badenbroek¹,

Klerk²

2020

Preprint

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“…Property (c) reflects the combinatorial difficulty of problem (3.9) -see, e.g., [Bom18] and references therein. Therefore, we next discuss various relaxations.…”

Section: )) (C) Holds Withmentioning

confidence: 99%

Self-Assignment Flows for Unsupervised Data Labeling on Graphs

Zisler¹,

Zern²,

Petra³

et al. 2019

Preprint

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This paper extends the recently introduced assignment flow approach for supervised image labeling to unsupervised scenarios where no labels are given. The resulting self-assignment flow takes a pairwise data affinity matrix as input data and maximizes the correlation with a low-rank matrix that is parametrized by the variables of the assignment flow, which entails an assignment of the data to themselves through the formation of latent labels (feature prototypes). A single user parameter, the neighborhood size for the geometric regularization of assignments, drives the entire process. By smooth geodesic interpolation between different normalizations of self-assignment matrices on the positive definite matrix manifold, a one-parameter family of self-assignment flows is defined. Accordingly, our approach can be characterized from different viewpoints, e.g. as performing spatially regularized, rank-constrained discrete optimal transport, or as computing spatially regularized normalized spectral cuts. Regarding combinatorial optimization, our approach successfully determines completely positive factorizations of self-assignments in large-scale scenarios, subject to spatial regularization. Various experiments including the unsupervised learning of patch dictionaries using a locally invariant distance function, illustrate the properties of the approach.

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Computing the completely positive factorization via alternating minimization

Behling,

Lara,

Oviedo

2023

Numerical Linear Algebra App

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In this article, we propose a novel alternating minimization scheme for finding completely positive factorizations. In each iteration, our method splits the original factorization problem into two optimization subproblems, the first one being a orthogonal procrustes problem, which is taken over the orthogoal group, and the second one over the set of entrywise positive matrices. We present both a convergence analysis of the method and favorable numerical results.

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Building a completely positive factorization

Cited by 8 publications

References 53 publications

An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

Self-Assignment Flows for Unsupervised Data Labeling on Graphs

Computing the completely positive factorization via alternating minimization

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