The Discrete Basis Problem and Asso Algorithm for Fuzzy Attributes

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.