Optimal decompositions of matrices with entries from residuated lattices

We describe optimal decompositions of matrices whose entries are elements of a residuated lattice L, such as L=[0, 1]. Such matrices represent relationships between objects and attributes with the entries representing degrees to which attributes represented by columns apply to objects represented by rows. Given such an n × m object-attribute matrix I, we look for a decomposition of I into a product A ° B of an n × k object-factor matrix A and a k × m factor-attribute matrix B with entries from L with the number k of factors as small as possible. We show that formal concepts of I, which play a central role in the Port-Royal approach to logic and which are the fixpoints of particular Galois connections associated to I, are optimal factors for decomposition of I in that they provide us with decompositions with the smallest number of factors. Moreover, we describe transformations between the space of original attributes and the space of factors induced by a decomposition I = A ° B. The article contains illustrative examples demonstrating the significance of the presented results for factor analysis of relational data. In addition, we present a general framework for a calculus of matrices with entries from residuated lattices in which both the matrix products and decompositions discussed in this article as well as triangular products and decompositions discussed elsewhere can be regarded as two particular cases of a general type of product and decomposition. We present the results for matrices, i.e. for relations between finite sets in terms of relations, but the arguments behind are valid for relations between infinite sets as well.