Relational Factor Analysis with o-Matrix Decomposition

Abstract

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.