Optimal decompositions of matrices with grades into binary and graded matrices
We study the problem of decomposition of object-attribute matrices whose entries contain degrees to which objects have attributes. The degrees are taken from a bounded partially ordered scale. Examples of such matrices are binary matrices, matrices with entries from a finite chain, or matrices with entries from the unit interval [0, 1]. We study the problem of decomposition of a given object-attribute matrix I with degrees into an object-factor matrix A with degrees and a binary factor-attribute matrix B, with the number of factors as small as possible. We present a theorem which shows that decompositions which use particular formal concepts of I as factors for the decomposition are optimal in that the number of factors involved is the smallest possible. We show that the problem of computing an optimal decomposition is NP-hard and present two heuristic algorithms for its solution along with their experimental evaluation. For the first algorithm, we provide its approximation ratio. Experiments indicate that the second algorithm, which is considerably faster than the first one, delivers decompositions whose quality is comparable to the decompositions delivered by the first algorithm. We also present an illustrative example demonstrating a factor analysis interpretation of the decomposition studied in this paper.
factor analysis matrix decomposition formal concept analysis fuzzy logic