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.
Highlights
Lattice Computing (LC) [1] intends to provide “an evolving collection of tools and methodologies that process lattice-ordered data”, as a means of establishing an information processing paradigm belonging to the wider field of Computational Intelligence [2], with explicit aim at modelling CyberPhysical Systems [3].In this paper, we present a fundamental tool for the factorization of matrices by developing the Singular Value Decomposition (SVD), a staple technique of data analysis [4,5], for matrices over idempotent semifields [6,7]
The authors have a long tradition of using spectral results in their papers [31,38,39], but we know of no direct application of the SVD after their initial paper, nor the embedding of the construction in the wider setting of Galois connections
The following explains the pertinence of K-Formal Concept Analysis (FCA) to i-SVD and the names of the factors: Lemma 4
Summary
Lattice Computing (LC) [1] intends to provide “an evolving collection of tools and methodologies that process lattice-ordered data”, as a means of establishing an information processing paradigm belonging to the wider field of Computational Intelligence [2], with explicit aim at modelling Cyber. We present a fundamental tool for the factorization of matrices by developing the Singular Value Decomposition (SVD), a staple technique of data analysis [4,5], for matrices over idempotent semifields [6,7]. In it we both use computing in lattices—since the matrices take values in idempotent semifield, an algebraic structure which is similar, but distinct to a fuzzy semiring—and computing with lattices—since our results highlight the role of the lattices of Formal Concept Analysis (FCA) [8] in reconstructing matrices
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