A multi-valued Boltzmann machine

The idea of a Hopfield network is based on the Ising spin glass model in which each spin has only two possible states: up and down. By introducing stochastic factors into this network and performing a simulated annealing process on it, it becomes a Boltzmann machine which can escape from local minimum states to achieve the global minimum. This paper generalizes the above ideas to multi-value case based on the XY spin glass model in which each spin can be in any direction in a plane. Simply using the gradient descent method and the analog Hopfield network, two different analog connectionist structures and their corresponding evolving rules are first designed to transform the XY spin glass model to distributed computational models. These two analog computational models are single-layered connectionist structures and multi-layered Hopfield analog networks. The latter network eases the node (neuron) computational requirement of the former at the expense of more neurons and connections. With the proposed evolving rules, the proposed models evolve according to a predefined Hamiltonian (energy function) which will decrease until it reaches a (perhaps local) minimum. Since these two structures can easily get stuck in local minima, a multi-valued Boltzmann machine is proposed which adopts the discrete planar spin glass model for the local minimum problem. Each neuron in the multi-valued Boltzmann machine can only take n discrete directions (states). The stochastic simulated annealing method is introduced to the evolving rules of the multi-valued Boltzmann machine to solve the local minimum problem. The multi-valued Boltzmann machine can be applied to the mobile robot navigation problem by defining proper artificial magnetic field on the traverse terrain. This new artificial magnetic field approach for the mobile robot navigation problem has shown to have several advantages over existing graph search and potential field techniques