Training fuzzy number neural networks with alpha-cut refinements

In a fuzzy number neural network, the inputs, weights, and outputs are general fuzzy numbers. The requirement that F¯^α(1) ⊂F¯^α(2) whenever α(1)>α(2) imposes an enormous number of constraints on the weight parameterizations during training. This problem can be solved through a careful choice of weight representation. This new representation is unconstrained, so that standard neural network training techniques may be applied. Unfortunately, fuzzy number neural networks still have many parameters to pick during training, since each weight is represented by a vector. Thus moderate to large fuzzy number neural networks suffer from the usual maladies of very large neural networks. In this paper, we discuss a method for effectively reducing the dimensionality of networks during training. Each fuzzy number weight is represented by the endpoints of its α-cuts for some discretization 0⩽α₁<α₂<...<α_n ⩽1. To reduce dimensionality, training is first done using only a small subset of the α_i. After successful training, linear interpolation is used to estimate additional α-cut endpoints. The network is then retrained to tune these interpolated values. This refinement is repeated as needed until the network is fully trained at the desired discretization in α