Fuzzy Relational Matrix Factorization and Its Granular Characterization in Data Description

This article is concerned with a problem of relational factorization which engages fuzzy relational calculus. It forms an interesting alternative to the method of nonnegative matrix factorization that has been commonly discussed and found in numerous applications. The relational factorization takes original <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional data located in the unit hypercube and factorizes it into data of lower dimensionality and some fuzzy relations. Owing to the logic nature of processing delivered by relational calculus, the dimensionality reduction exhibits transparency as the reduction mechanism itself is described in terms of logic expressions. Two types of factorizations mechanisms are investigated by using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s–t</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t–s</i> composition operators where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> are triangular norms and conorms, respectively. A two-level process of factorization is designed. A gradient-based learning scheme is developed. The quantification of the performance of the factorization process is realized by bringing a concept of information granularity: The obtained fuzzy relations are formed based on granular constructs and the quality of the produced factorization is assessed in terms of the coverage and specificity of the obtained granular results. A collection of experiments is included to present the performance of factorization and its parametric analysis. In addition, the proposed algorithm comes with sound interpretability in terms of both the structure of the model and an intuitive meaning of the fuzzy relations being the result of factorization.