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.

We present an extension of the discrete basis problem, recently a profoundly studied problem, from the Boolean setting to the setting of fuzzy attributes, i.e., a setting of ordinal data. Our problem consists in finding for a given object-attribute matrix I containing truth degrees and a prescribed number k of factors the best approximate decomposition of I into an object-factor matrix A and a factor-attribute matrix B. Since such matrices represent fuzzy relations, the problem is related to but very different from that of decomposition of fuzzy relations as studied in fuzzy relational equations because neither A nor B are supposed to be known in our problem. We observe that our problem is NP-hard as an optimization problem. Consequently, we provide an approximation algorithm for solving this problem and provide its time complexity in the worst case. The algorithm is inspired by the Asso algorithm, which is known for Boolean attributes and is based on new considerations regarding associations among fuzzy attributes. We provide an experimental evaluation on various datasets and demonstrate that our algorithm is capable of extracting informative factors in data. We conclude with a discussion regarding future research issues.