Fully dynamic quasi-biclique edge covers via Boolean matrix factorizations

Miettinen, Pauli

doi:10.1145/2489247.2489250

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Another interesting avenue for research could be a study on connection between various approximations of formal concepts (fault-tolerant concepts (Besson et al, 2006) and object-attribute biclusters (Ignatov et al, 2012(Ignatov et al, , 2017), Boolean matrix factorization (Miettinen, 2013;Belohlávek et al, 2019) and quasi-bicliques.…”

Section: Results and Conclusionmentioning

confidence: 99%

Mixed Integer Programming for Searching Maximum Quasi-Bicliques

Ignatov¹,

Ivanova²,

Zamaletdinova³

2020

Springer Proceedings in Mathematics &Amp; Statistics

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This paper is related to the problem of finding the maximal quasi-bicliques in a bipartite graph (bigraph). A quasi-biclique in the bigraph is its "almost" complete subgraph. The relaxation of completeness can be understood variously; here, we assume that the subgraph is a γ-quasi-biclique if it lacks a certain number of edges to form a biclique such that its density is at least γ ∈ (0, 1]. For a bigraph and fixed γ, the problem of searching for the maximal quasi-biclique consists of finding a subset of vertices of the bigraph such that the induced subgraph is a quasi-biclique and its size is maximal for a given graph. Several models based on Mixed Integer Programming (MIP) to search for a quasi-biclique are proposed and tested for working efficiency. An alternative model inspired by biclustering is formulated and tested; this model simultaneously maximizes both the size of the quasi-biclique and its density, using the least-square criterion similar to the one exploited by triclustering T B .

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Section: Results and Conclusionmentioning

confidence: 99%

Mixed Integer Programming for Searching Maximum Quasi-Bicliques

Ignatov¹,

Ivanova²,

Zamaletdinova³

2020

Springer Proceedings in Mathematics &Amp; Statistics

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“…Indeed, after the publication of the preliminary version of this article, the proposed encoding length functions have already been adapted to find model orders in joint subspace BMF [Miettinen 2012] and dynamic BMF [Miettinen 2013]. Furthermore, extending our encoding methods to other models for representing binary matrices as a product of binary factor matrices is straightforward.…”

Section: Discussionmentioning

confidence: 99%

Mdl4bmf

Miettinen

Vreeken

2014

ACM Trans. Knowl. Discov. Data

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Matrix factorizations-where a given data matrix is approximated by a product of two or more factor matrices-are powerful data mining tools. Among other tasks, matrix factorizations are often used to separate global structure from noise. This, however, requires solving the "model order selection problem" of determining the proper rank of the factorization, that is, to answer where fine-grained structure stops, and where noise starts.Boolean Matrix Factorization (BMF)-where data, factors, and matrix product are Boolean-has in recent years received increased attention from the data mining community. The technique has desirable properties, such as high interpretability and natural sparsity. Yet, so far no method for selecting the correct model order for BMF has been available. In this article, we propose the use of the Minimum Description Length (MDL) principle for this task. Besides solving the problem, this well-founded approach has numerous benefits; for example, it is automatic, does not require a likelihood function, is fast, and, as experiments show, is highly accurate.We formulate the description length function for BMF in general-making it applicable for any BMF algorithm. We discuss how to construct an appropriate encoding: starting from a simple and intuitive approach, we arrive at a highly efficient data-to-model-based encoding for BMF. We extend an existing algorithm for BMF to use MDL to identify the best Boolean matrix factorization, analyze the complexity of the problem, and perform an extensive experimental evaluation to study its behavior. ACM Reference Format:Pauli Miettinen and Jilles Vreeken. 2014. MDL4BMF: Minimum description length for Boolean matrix factorization.

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“…In this section, numerical methods for approximate Boolean CP decomposition of binary tensors are discussed. A recently developed method called WalkNMerge is an example of how Boolean decomposition provides a framework for new methods with potential advantages over real-valued decompositions (Miettinen 2013).…”

Section: Methodsmentioning

confidence: 99%

“…Erdos et al (1966) proves that any graph on n vertices with no isolated points can be covered by at most ⌈ 2 4 ⁄ ⌉ complete sub-graphs; an analogy for 3-partite, 3-uniform hypergraphs is proven in the next section. The connection between the Boolean rank of matrices/tensors and the covering of a graph by complete sub-graphs has been previously recognized; however, the connection can be exploited further to answer questions about binary tensors (Miettinen 2013).…”

Section: Lemmamentioning

confidence: 99%