55 research outputs found

    K-Formal Concept Analysis as linear algebra over idempotent semifields

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    We report on progress in characterizing K-valued FCA in algebraic terms, where K is an idempotent semifield. In this data mining-inspired approach, incidences are matrices and sets of objects and attributes are vectors. The algebraization allows us to write matrix-calculus formulae describing the polars and the fixpoint equations for extents and intents. Adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra. This allows us to shed some light into Formal Concept Analysis from the point of view of the theory of linear operators over idempotent semimodules. In the opposite direction, we state the importance of FCA-related concepts for dual order homomorphisms of linear spaces over idempotent semifields, specially congruences, the lattices of extents, intents and formal concepts

    Galois Connections between Semimodules and Applications in Data Mining

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    In [1] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, K-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. A fundamental result was missing there, namely the second half of the equivalent of the main theorem of Formal Concept Analysis. In this continuation we introduce the structural lattice of such generalised contexts, providing a limited equivalent to the main theorem of K-Formal Concept Analysis which allows to interpret the standard version as a privileged case in yet another direction. We motivate our results by providing instances of their use to analyse the confusion matrices of multiple-input multiple-output classifiers

    The Singular Value Decomposition over Completed Idempotent Semifields

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    In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.This research was funded by the Spanish Government-MinECo project TEC2017-84395-P and the Dept. of Research and Innovation of Madrid Regional Authority project EMPATIA-CM (Y2018/TCS-5046)

    Towards Galois Connections over Positive Semifields

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    In this paper we try to extend the Galois connection construction of K-Formal Concept Analysis to handle semifields which are not idempotent. Important examples of such algebras are the extended non-negative reals and the extended non-negative rationals, but we provide a construction that suggests that such semifields are much more abundant than suspected. This would broaden enormously the scope and applications of K-Formal Concept Analysis.CPM & FVA have been partially supported by the Spanish Government-MinECo projects TEC2014-53390-P and TEC2014-61729-EX

    Towards the algebraization of Formal Concept Analysis over complete dioids

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    Actas de: XVII Congreso Español sobre Tecnologías y Lógica Fuzzy (ESTYLF 2014). Zaragoza, 5-7 de febrero de 2014.Complete dioids are already complete residuated lattices. Formal contexts with entries in them generate Concept Lattices with the help of the polar maps. Previous work has already established the spectral nature of some formal concepts for contexts over certain kinds of dioids. This paper tries to raise the awareness that linear algebra over exotic semirings should be one place to look to understand the properties of FCA over L-lattices.FJVA was partially supported by EU FP7 project LiMoSINe (contract 288024) for this research. CPM was partially supported by the Spanish Government-Comisión Interministerial de Ciencia y Tecnología project 2011-268007/TEC.Publicad

    Four-fold Formal Concept Analysis based on Complete Idempotent Semifields

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    Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯¯¯¯-four-fold Formal Concept Analysis (K¯¯¯¯-4FCA) where K¯¯¯¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯¯¯¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results

    Activating Generalized Fuzzy Implications from Galois Connections

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    This paper deals with the relation between fuzzy implications and Galois connections, trying to raise the awareness that the fuzzy implications are indispensable to generalise Formal Concept Analysis. The concrete goal of the paper is to make evident that Galois connections, which are at the heart of some of the generalizations of Formal Concept Analysis, can be interpreted as fuzzy incidents. Thus knowledge processing, discovery, exploration and visualization as well as data mining are new research areas for fuzzy implications as they are areas where Formal Concept Analysis has a niche.F.J. Valverde-Albacete—was partially supported by EU FP7 project LiMoSINe, (contract 288024). C. Peláez-Moreno—was partially supported by the Spanish Government-CICYT project 2011-268007/TEC.Publicad

    Spectral Lattices of reducible matrices over completed idempotent semifields

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    Proceedings of: 10th International Conference on Concept Lattices and Their Applications. (CLA 2013). La Rochelle, France, October 15-18, 2013.Previous work has shown a relation between L-valued extensions of FCA and the spectra of some matrices related to L-valued contexts. We investigate the spectra of reducible matrices over completed idempotent semifields in the framework of dioids, naturally-ordered semirings, that encompass several of those extensions. Considering special sets of eigenvectors also brings out complete lattices in the picture and we argue that such structure may be more important than standard eigenspace structure for matrices over completed idempotent semifields.FJVA is supported by EU FP7 project LiMoSINe (contract 288024). CPM has been partially supported by the Spanish Government-ComisiĂłn Interministerial de Ciencia y TecnologĂ­a project TEC2011-26807 for this paper.Publicad
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