Connectivity Versus Entropy

Yaser S. Abu-Mostafa California Institute of Technology Pasadena, CA 91125 ABSTRACT How does the connectivity of a neural network (number of synapses per neuron) relate to the complexity of the problems it can handle (measured by the entropy)? Switching theory would suggest no relation at all, since all Boolean functions can be implemented using a circuit with very low connectivity (e.g., using two-input NAND gates). However, for a network that learns a problem from examples using a local learning rule, we prove that the entropy of the problem becomes a lower bound for the connectivity of the network. INTRODUCTION The most distinguishing feature of neural networks is their ability to spontaneously learnthe desired function from'training' samples, i.e., their ability to program themselves. Clearly, a given neural network cannot just learn any function, there must be some restrictions on which networks can learn which functions.