Relational-product architectures for information processing
Information Sciences: an International Journal - Special issue on expert systems
Fuzzy sets and fuzzy logic: theory and applications
Fuzzy sets and fuzzy logic: theory and applications
Formal Concept Analysis: Mathematical Foundations
Formal Concept Analysis: Mathematical Foundations
Fuzzy Relation Equations and Their Applications to Knowledge Engineering
Fuzzy Relation Equations and Their Applications to Knowledge Engineering
Fuzzy Relational Systems: Foundations and Principles
Fuzzy Relational Systems: Foundations and Principles
Logics from Galois connections
International Journal of Approximate Reasoning
Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory
International Journal of Approximate Reasoning
Discovery of optimal factors in binary data via a novel method of matrix decomposition
Journal of Computer and System Sciences
PKDD'06 Proceedings of the 10th European conference on Principle and Practice of Knowledge Discovery in Databases
A new algebraic structure for formal concept analysis
Information Sciences: an International Journal
Factorizing three-way binary data with triadic formal concepts
KES'10 Proceedings of the 14th international conference on Knowledge-based and intelligent information and engineering systems: Part I
Isotone fuzzy Galois connections with hedges
Information Sciences: an International Journal
Possibility theory and formal concept analysis: Characterizing independent sub-contexts
Fuzzy Sets and Systems
Closure spaces of isotone galois connections and their morphisms
AI'11 Proceedings of the 24th international conference on Advances in Artificial Intelligence
Row and Column Spaces of Matrices over Residuated Lattices
Fundamenta Informaticae - Concept Lattices and Their Applications
International Journal of Approximate Reasoning
International Journal of Approximate Reasoning
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We describe optimal decompositions of an nxm matrix I into a triangular product I=A@?B of an nxk matrix A and a kxm matrix B. We assume that the matrix entries are elements of a residuated lattice, which leaves binary matrices or matrices which contain numbers from the unit interval [0,1] as special cases. The entries of I, A, and B represent grades to which objects have attributes, factors apply to objects, and attributes are particular manifestations of factors, respectively. This way, the decomposition provides a model for factor analysis of graded data. We prove that fixpoints of particular operators associated with I, which are studied in formal concept analysis, are optimal factors for decomposition of I in that they provide us with decompositions I=A@?B with the smallest number k of factors possible. Moreover, we describe transformations between the m-dimensional space of original attributes and the k-dimensional space of factors. We provide illustrative examples and remarks on the problem of computing the optimal decompositions. Even though we present the results for matrices, i.e. for relations between finite sets in terms of relations, the arguments behind are valid for relations between infinite sets as well.