Factorization of Matrices With Grades With Overcovering

We present a novel algorithm for factorization of matrices with grades or, equivalently, for decomposition of fuzzy relations. The algorithm is inspired by a recent factorization method for Boolean matrices and develops two ideas in the setting of ordinal attributes. First, it uses formal concepts associated with the factorized matrix, or fuzzy relation, as essential components around which factors are built. Second, it steps back when computing new factors to check whether some computed factors may be eliminated or improved given the subsequently generated factors. The new algorithm thus uses convenient properties of formal concepts utilized by previous factorization algorithms. Still, unlike the previous algorithms, our algorithm allows for more general and therefore more precise factorizations due to a possible overcovering of the input data, which our new algorithm admits. We provide an experimental evaluation of the new algorithm and compare it to some existing algorithms for factorization of data with grades. The evaluation reveals that our new algorithm outperforms the current algorithms in terms of quality of factorization. We also present observations and improvements for factorization of Boolean matrices. We conclude with a discussion regarding open research topics.