Visualization of State Transitions in the Hopfield Network and in Search for a Mechanism of the Virtual Magnetic Diminuendo Method

Abstract

A Hopfield network is a good tool for solving combinatorial optimization problems. But one of its major drawbacks is existence of energy local minima, because iterative state transitions are carried out just reducing the energy defined in advance. In order to avoid such poor solutions, a virtual magnetic diminuendo (VMD) method is proposed recently. Although its effectiveness is confirmed through some computer simulations, its working mechanism has not been clear yet. Then, in order to solve this tough problem, behavior of the Hopfield network is investigated carefully with the help of visualized representation in this paper. As a result, it is confirmed as follows: i) When the virtual magnetic parameter exceeds one of its critical values, easiness of neuron firing chance is changed and it makes improve the score. ii) A configuration of the energy function is changed, e.g. an energy basin becomes a hillside and vice versa in some cases, by controlling the virtual magnetic parameter, and it makes be possible to escape from a trapped energy local minimum. These two facts must be the very essence of the VMD method, and they strongly support the reason of performance improvement by introducing the VMD method.