3.1 Introduction

In this chapter, we present an objective function formulation of the BCM theory that enables us to demonstrate the connection between unsupervised BCM learning and various statistical methods, in particular, that of Projection Pursuit. It provides a general method for stability analysis of the fixed points of the theory and enables us to analyze the behavior and the evolution of the network under various visual rearing conditions. It also allows comparison with many existing unsupervised methods.

In this framework, we extend the single-neuron learning rule to nonlinear neurons and to a network of laterally connected neurons. The formulation via objective function gives a natural form for the learning rule as gradient descent optimization. The objective function formulation of the BCM theory has led us to modify slightly our learning rule resulting in improved stability and statistical properties (Appendix 3A presents some mathematical results concerning the convergence and characterization of the fixed points.) This variant of the BCM rule has some advantages over the original exploratory projection pursuit model [Friedman, 1987]. Due to its computational efficiency, it can extract several features in parallel, taking into account the interaction between the different extracted features via a lateral inhibition network. Feature extraction based on this model has been applied to various real-world problems [Intrator et al., 1996; Huynh et al., 1996; Tankus et al., 1997; Dotan and Intrator, 1998].

3.2 Formulation of the BCM Theory Using an Objective Function

The objective function formulation of the synaptic modification theory of Bienenstock, Cooper...