The Singular Value Decomposition over Completed Idempotent Semifields

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Show simple item record Valverde Albacete, Francisco José Peláez Moreno, Carmen 2021-05-27T11:09:20Z 2021-05-27T11:09:20Z 2020-09
dc.identifier.bibliographicCitation Valverde-Albacete, F. & Peláez-Moreno, C. (2020). The Singular Value Decomposition over Completed Idempotent Semifields. Mathematics, 8(9), 1577.
dc.identifier.issn 2227-7390
dc.description.abstract 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.
dc.description.sponsorship 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).
dc.format.extent 39
dc.language.iso eng
dc.publisher MDPI
dc.rights © 2020 by the authors.
dc.rights Atribución 3.0 España
dc.subject.other Complete idempotent semifields
dc.subject.other Formal concept analysis
dc.subject.other Idempotent singular value decomposition
dc.subject.other Max-plus algebra
dc.subject.other Min-plus algebra
dc.subject.other Schedule algebra
dc.subject.other Tropical algebra
dc.title The Singular Value Decomposition over Completed Idempotent Semifields
dc.type article
dc.subject.eciencia Telecomunicaciones
dc.rights.accessRights openAccess
dc.relation.projectID Gobierno de España. TEC2017-84395-P
dc.relation.projectID Comunidad de Madrid. Y2018/TCS-5046
dc.type.version publishedVersion
dc.identifier.publicationfirstpage 1577
dc.identifier.publicationissue 9
dc.identifier.publicationtitle Mathematics
dc.identifier.publicationvolume 8
dc.identifier.uxxi AR/0000027493
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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