Indirect Encoding Evolutionary Learning Algorithm for the Multilayer Morphological Perceptron

This article describes an indirectly encoded evolutionary learning algorithm to train morphological neural networks. The indirect encoding method is an algorithm in which the training of the neural network is done by finding the solution without considering the exact connectivity of the network. Looking for the set of weights and architecture in a reduced search space, this simple, but powerful training algorithm is able to evolve to a feasible solution using up to three layers required to perform the pattern classification. This type of representation provides the necessary compactness required by large networks. The algorithm was tested using Iris Fisher data and a prototype was written using Matlab. Introduction Morphological Neural Networks (MNN) are a new type of neural networks described by Ritter, Sussner, and Beavers (Ritter and Sussner 1996), (Ritter and Sussner 1997), (Sussner 1998), and (Ritter and Beavers 1999). These types of neural networks replace the classical operations of multiplication and addition by addition and maximum or minimum operations. The maximum and minimum operations allow performing a nonlinear operation before the application of the activation or transfer function. MNN utilize algebraic lattice operations structure known as semiring ( , , , , ') ±∞ ∨ ∧ + + R , different from traditional neural networks that are based on the algebraic structure known as ring (R,+,×). The operations ∧ and ∨ denote minimum and maximum binary operations, respectively. Genetic Algorithms (Yao 1999) have proven to be effective to search for an optimal solution in very large, complex, and irregular search spaces such as the neural networks architectures. This article describes a method using genetic algorithms that can be used to train the morphological neural networks introduced by Ritter, Sussner and Beavers. The algorithm can be used to train up to three layers morphological perceptron architectures based on evolutionary computation, which are able to classify most traditional pattern classification problems. Copyright © 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Morphological Neural Networks Morphological neural networks are a new type of neural network, based on lattice operations. The morphological neuron follows the mathematical model described by Equation 1, ( ) 1 n j ij i ij i f p r x w = ⎛ ⎞ ⋅ ∨ + ⎜ ⎟ ⎝ ⎠ (1) where ∨ is the maximum operator (or minimum operator ∧ can be used), xi is the i-th input value for the j-th neuron, wij denotes the synaptic weight associated between the i-th input and the j-th neuron, rij represents the inhibitory or excitatory pre-synaptic value between the i-th input and the j-th neuron, and pj represents the postsynaptic response of the j-th neuron. Both rij and pj can assume values of {+1, -1}. In addition, the morphological perceptron uses a special hard-limit transfer function, as shown in Equation 2: : 0,1 1 if x > 0 0 else f x → ⎧ → ⎨ ⎩ (2) Figure 1 shows a graphical representation of a two-layer morphological neural network.