Optimal decompositions of matrices with grades

We present theoretical results regarding decomposition of matrices with grades, i.e. matrices I with entries from a bounded ordered set L such as the real unit interval [0, 1]. If I is such an n times m matrix, we look for a decomposition of I into a product A omicron B of an ntimesk matrix A and a ktimesm matrix B with entries from L with k as small as possible. This problem generalizes the decomposition problem of Boolean factor analysis which is a particular case when L has just two elements 0 and 1. The product we consider is a supt-norm product of which the well known max-min product of Boolean matrices as well as max-min and max-min product of matrices with entries from [0, 1] are particular examples. I, A, and B can be interpreted as object times attribute, object times factor, and factor times attribute matrices describing degrees of expression of attributes on objects, factors on objects, and attributes on factors. In this interpretation, a decomposition I into A omicron B corresponds to discovery of k factors explaining the original data I. We propose to use formal concepts of I in the sense of formal concept analysis as factors. The formal concepts are fixed points of a particular closure operator and can be seen as particular submatrices of I. We prove several results regarding such a decomposition including a theorem which says that decompositions using formal concepts as factors are optimal in that they provide us with the least number of factors possible. Based on the geometrical insight provided by the theorem, we propose a greedy approximation algorithm for finding optimal decompositions. We provide examples illustrating the concepts and implications of the results.