Classic Fuzzy Cognitive Maps Are Not Universal Approximators

Abstract

Fuzzy cognitive maps (FCMs) are knowledge-based recurrent neural networks that involve neural concepts and causal relationships. Despite being successful in several domains, classic FCMs often fall behind black-box models in terms of their approximation capabilities. However, the literature only reports a few studies devoted to understanding their theoretical foundations and the cause of their limited performance. In this article, we prove that FCMs are not universal approximators and base our proof on recent theoretical findings and theorems related to the dynamic behavior of FCM-based models. Our results hold for activation functions that are bounded and monotonically increasing. These analytical findings and the empirical evidence (from the analysis of covering and proximity measures applied to synthetically generated FCMs) show that there are significant state space regions that are never produced for some problems. Consequently, classic FCM models cannot generally approximate these values, thus hindering their predictive capabilities in machine learning tasks. The same theoretical results that exposed the design weaknesses of FCMs can be used to overcome them. As the second contribution of our article, we propose two enhanced FCM-based classifiers equipped with a quasi-nonlinear reasoning rule, together with a decision-making layer that uses derived analytical results. To fine-tune the classifiers' learnable parameters, we introduce a backpropagationlike algorithm that balances convergence and accuracy. Numerical simulations using real-world datasets indicate that our enhanced FCM-based classifiers significantly outperform the classical model.