Parameterized low-rank binary matrix approximation

Fomin, Fedor V.; Golovach, Petr A.; Panolan, Fahad

doi:10.1007/s10618-019-00669-5

Cited by 21 publications

(30 citation statements)

References 72 publications

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“…Let us now look at algorithms with theoretical guarantees. Gramm et al (2008) and Fomin et al (2020) showed that an exact solution for BMF can be computed in time 2 2 O(k) (mn) O(1) , thus matching the above lower bound.…”

Section: Theoretical Algorithmssupporting

confidence: 59%

Recent Developments in Boolean Matrix Factorization

Miettinen

Neumann

2020

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence

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The goal of Boolean Matrix Factorization (BMF) is to approximate a given binary matrix as the product of two low-rank binary factor matrices, where the product of the factor matrices is computed under the Boolean algebra. While the problem is computationally hard, it is also attractive because the binary nature of the factor matrices makes them highly interpretable. In the last decade, BMF has received a considerable amount of attention in the data mining and formal concept analysis communities and, more recently, the machine learning and the theory communities also started studying BMF. In this survey, we give a concise summary of the efforts of all of these communities and raise some open questions which in our opinion require further investigation.

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Section: Theoretical Algorithmssupporting

confidence: 59%

Recent Developments in Boolean Matrix Factorization

Miettinen

Neumann

2020

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence

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“…Related results are also presented in very recent works which investigated the complexity of different matrix editing and clustering problems from the parameterized and approximation perspectives (Fomin, Golovach, and Panolan 2018;Fomin et al 2019;Eiben et al 2019). All of these papers except for the last one are concerned with the complete data setting.…”

Section: Introductionmentioning

confidence: 90%

On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank

Ganian

Kanj

Ordyniak

et al. 2020

AAAI

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We consider a fundamental matrix completion problem where we are given an incomplete matrix and a set of constraints modeled as a CSP instance. The goal is to complete the matrix subject to the input constraints and in such a way that the complete matrix can be clustered into few subspaces with low rank. This problem generalizes several problems in data mining and machine learning, including the problem of completing a matrix into one with minimum rank. In addition to its ubiquitous applications in machine learning, the problem has strong connections to information theory, related to binary linear codes, and variants of it have been extensively studied from that perspective. We formalize the problem mentioned above and study its classical and parameterized complexity. We draw a detailed landscape of the complexity and parameterized complexity of the problem with respect to several natural parameters that are desirably small and with respect to several well-studied CSP fragments.

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“…Other works which investigate the theoretical aspects of the BMF problem have also been done recently [21], [38]- [40]. A method to solve the joint BMF problems, meaning to find Boolean factors shared by two Boolean data matrices, is also studied in [41].…”

Section: Related Workmentioning

confidence: 99%

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

et al. 2021

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A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with an additional constraint over an auxiliary matrix whose Boolean structure is identical to the initial Boolean data. This additional auxiliary matrix constraint forces the support of the NMF solution to adhere to that of a BMF solution. The solution of the nonnegative auxiliary optimization problem is thresholded to provide a solution for the BMF problem. We provide the proofs for the equivalencies of the two solution spaces under the existence of an exact solution. Moreover, the nonincreasing property of the algorithm is also proven. Experiments on synthetic and real datasets are conducted to show the effectiveness and complexity of the algorithm compared to other current methods. INDEX TERMS Boolean matrix factorization, nonnegative matrix factorization

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Parameterized low-rank binary matrix approximation

Cited by 21 publications

References 72 publications

Recent Developments in Boolean Matrix Factorization

Recent Developments in Boolean Matrix Factorization

On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank

Boolean Matrix Factorization via Nonnegative Auxiliary Optimization

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