Discovery of factors in matrices with grades

Abstract: 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 form a bounded scale equipped with certain aggregation operators and conforms to the structure of a complete residuated lattice. We present a greedy approximation algorithm for the problem of decomposition of such matrix in a product of two matrices with grades under the restriction that the number of factors be small. 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 formal concepts of the input data and allow an easy interpretation of the decomposition. We present illustrative examples and experimental evaluation.