Network Characteristics

There are a number of different descriptions of the Boltzmann machine in the literature. The following is based on the paper of Ackley et al. [4].

The Boltzmann machine is a fully connected network with symmetric bidirectional connections. In addition, certain units of the network are designated as input units, certain other units are designated as output units, while the remainder are the hidden units. This division of the network is entirely arbitrary, since the units are indistinguishable.

The units in the network can be in one of two states, which may be denoted as on and off, and 1, or 1 and -1. The network has an energy function of the same form as that of the Hopfield network:

$$E(t) = - \frac{1}{2} \sum_i \sum_{j\neq i} w_{ij} \mu_i(t) \mu_j(t) + \sum_i \mu_i(t) \theta_i$$

where $\mu_i(t)$ is the state of the ith unit at time t, w_ij is the weight on the link between the ith and jth units, and $\theta_i$ is the threshold of the ith unit.

The network also has a control parameter T, the temperature parameter, which controls the simulated annealing process when the network is run.

The figure shows a simple Boltzmann machine.

 

Figure 5.22: A simple Boltzmann machine