Structural interpolation and approximation with fuzzy relations: A study in knowledge reuse

In this study, we are concerned with a problem of structural interpolation and approximation realized in the framework of fuzzy relations. Fuzzy relations are fundamental concepts which represent, reveal and process key dependencies between systems' variables, factors and concepts thus capturing an underlying knowledge about systems. Different relations could deliver various insights into the nature of the system. In particular, such relations could be constructed at different levels of granularity granularity and in this sense may provide a variety of perspectives (views) at the same system. Here we focus attention on the issue of structural knowledge reconstruction and demonstrate how for a given collection of fuzzy relations, one can effectively reconstruct fuzzy relations at some required level of granularity. We also cast this problem in the context of knowledge re-use by showing how the existing fuzzy relations could be effectively employed (re-used) in this setting. The ensuing optimization task is formed and a detailed gradient-based learning scheme is presented. Some numeric illustration is also included.