Prototype Analysis in Hopfield Networks with Hebbian Learning

Abstract

We discuss prototype formation in the Hopfield network. Typically, Hebbian learning with highly correlated states leads to degraded memory performance. We show this type of learning can lead to prototype formation, where unlearned states emerge as representatives of large correlated subsets of states, alleviating capacity woes. This process has similarities to prototype learning in human cognition. We provide a substantial literature review of prototype learning in associative memories, covering contributions from psychology, statistical physics, and computer science. We analyze prototype formation from a theoretical perspective and derive a stability condition for these states based on the number of examples of the prototype presented for learning, the noise in those examples, and the number of non-example states presented. The stability condition is used to construct a probability of stability for a prototype state as the factors of stability change. We also note similarities to traditional network analysis, allowing us to find a prototype capacity. We corroborate these expectations of prototype formation with experiments using a simple Hopfield network with standard Hebbian learning. We extend our experiments to a Hopfield network trained on data with multiple prototypes and find the network is capable of stabilizing multiple prototypes concurrently. We measure the basins of attraction of the multiple prototype states, finding attractor strength grows with the number of examples and the agreement of examples. We link the stability and dominance of prototype states to the energy profile of these states, particularly when comparing the profile shape to target states or other spurious states.