Factorization of matrices with grades

We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a product has a given feature, or a person performs well in a test. We assume that the grades are taken from bounded scales equipped with certain aggregation operators that are involved in the decompositions. Particular cases of the decompositions include the well-known Boolean matrix decomposition, and the sup-t-norm and inf-residuum decompositions. We consider the problem of decomposition of a given matrix into a product of two matrices with grades such that the number of factors, i.e. the inner dimension, be as small as possible. We observe that computing such decompositions is NP-hard and present a greedy approximation algorithm. Our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions. These factors correspond to fixpoints of certain Galois connections associated with the input matrix, which are called formal concepts, and allow an easy interpretation of the decomposition. We present illustrative examples and experimental evaluation of the algorithm.