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By Kroese B., van der Smagt P.

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3 Boltzmann machines The Boltzmann machine, as first described by Ackley, Hinton, and Sejnowski in 1985 (Ackley, Hinton, & Sejnowski, 1985) is a neural network that can be seen as an extension to Hopfield networks to include hidden units, and with a stochastic instead of deterministic update rule. The weights are still symmetric. The operation of the network is based on the physics principle of annealing. This is a process whereby a material is heated and then cooled very, very slowly to a freezing point.

2 Hopfield network as associative memory A primary application of the Hopfield network is an associative memory. In this case, the weights of the connections between the neurons have to be thus set that the states of the system corresponding with the patterns which are to be stored in the network are stable. These states can be seen as ‘dips’ in energy space. When the network is cued with a noisy or incomplete test pattern, it will render the incorrect or missing data by iterating to a stable state which is in some sense ‘near’ to the cued pattern.

A) The perceptron of fig. 1 with an extra hidden unit. With the indicated values of the weights wij (next to the connecting lines) and the thresholds θi (in the circles) this perceptron solves the XOR problem. 6 onto the four points indicated here; clearly, separation (by a linear manifold) into the required groups is now possible. a linear manifold (plane) into two groups, as desired. This simple example demonstrates that adding hidden units increases the class of problems that are soluble by feed-forward, perceptronlike networks.

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