Relational Factor Analysis with o-Matrix Decomposition

The paper presents results on factorization of matrices describing objects and their fuzzy attributes. Entries of the matrices are truth degrees, e.g., numbers from the real unit interval [0, 1]. In general, matrix entries can be elements from a complete residuated lattice. We propose a novel method to factorize such matrices which is based on using so-called formal concepts as factors. To factorize an n times m object-attribute matrix I means to decompose I into a product A omicron B of an n times k object-factor matrix A and an k times m factor-attribute matrix B. In addition, we want the number k of factors as small as possible. The product o we consider in this paper is the well-known product corresponding to max-t-norm composition of fuzzy relations. We focus on theoretical analysis of the method we propose. We prove several results, e.g., a result which says that our method provides the best factorization in that it leads to the smallest number of factors. In addition, we present an illustrative example.