Classifiers with limited connectivity

Abstract

For many neural network models in which neurons are trained to classify inputs like perceptrons, the number of inputs that can be classified is limited by the connectivity of each neuron, even when the total number of neurons is very large. This poses the problem of how the biological brain can take advantage of its huge number of neurons given that the connectivity is sparse. One solution is to combine multiple perceptrons together, as in committee machines. The number of classifiable random patterns would then grow linearly with the number of perceptrons, even when each perceptron has limited connectivity. However, the problem is moved to the downstream readout neurons, which would need a number of connections that is as large as the number of perceptrons. Here we propose a different approach in which the readout is implemented by connecting multiple perceptrons in a recurrent attractor neural network. We prove analytically that the number of classifiable random patterns can grow unboundedly with the number of perceptrons, even when the connectivity of each perceptron remains finite. Most importantly, both the recurrent connectivity and the connectivity of downstream readouts also remain finite. Our study shows that feed-forward neural classifiers with numerous long range afferent connections can be replaced by recurrent networks with sparse long range connectivity without sacrificing the classification performance. Our strategy could be used to design more general scalable network architectures with limited connectivity, which resemble more closely the brain neural circuits which are dominated by recurrent connectivity.