A Fuzzy Restricted Boltzmann Machine: Novel Learning Algorithms Based on the Crisp Possibilistic Mean Value of Fuzzy Numbers

A fuzzy restricted Boltzmann machine (FRBM) is extended from a restricted Boltzmann machine (RBM) by replacing all the real-valued parameters with fuzzy numbers. A new FRBM that employs the crisp possibilistic mean value of a fuzzy number to defuzzify the fuzzy free energy function is presented. This approach is much clearer and easier to obtain the expression of the defuzzified free energy function and its approximation than the centroid method. Several theorems that discuss the error bounds of the approximation to ensure the rationality and validity are also investigated. Learning algorithms are given for the designed FRBM with symmetric triangular fuzzy numbers (STFNs), asymmetric triangular fuzzy numbers, and Gaussian fuzzy numbers. By appropriately choosing the parameters, a theorem is concluded that all FRBMs with symmetric fuzzy numbers will have identical learning algorithm to that of FRBMs with STFNs. This is illustrated by a case of FRBM with Gaussian fuzzy numbers. Two experiments including the MNIST handwriting recognition and the Bars-and-Stripes benchmark are carried out. The results show that the proposed FRBMs significantly outperform RBMs in learning accuracy and generalization ability, especially when encountering unlearned samples and recovering incomplete images.