  Numerous learning methods for fuzzy cognitive maps (FCMs), such as the
Hebbian-based and the population-based learning methods, have been developed
for modeling and simulating dynamic systems. However, these methods are faced
with several obvious limitations. Most of these models are extremely time
consuming when learning the large-scale FCMs with hundreds of nodes.
Furthermore, the FCMs learned by those algorithms lack robustness when the
experimental data contain noise. In addition, reasonable distribution of the
weights is rarely considered in these algorithms, which could result in the
reduction of the performance of the resulting FCM. In this article, a
straightforward, rapid, and robust learning method is proposed to learn FCMs
from noisy data, especially, to learn large-scale FCMs. The crux of the
proposed algorithm is to equivalently transform the learning problem of FCMs to
a classic-constrained convex optimization problem in which the least-squares
term ensures the robustness of the well-learned FCM and the maximum entropy
term regularizes the distribution of the weights of the well-learned FCM. A
series of experiments covering two frequently used activation functions (the
sigmoid and hyperbolic tangent functions) are performed on both synthetic
datasets with noise and real-world datasets. The experimental results show that
the proposed method is rapid and robust against data containing noise and that
the well-learned weights have better distribution. In addition, the FCMs
learned by the proposed method also exhibit superior performance in comparison
with the existing methods. Index Terms-Fuzzy cognitive maps (FCMs), maximum
entropy, noisy data, rapid and robust learning.
