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1. 2: The Elman network. With this network, the hidden unit activation values are fed back to the input layer, to a set of extra neurons called the context units. 2. pattern xt is clamped, the forward calculations are performed once 3. the back-propagation learning rule is applied 4. t t + 1 go to 2. The context units at step t thus always have the activation value of the hidden units at step t ; 1. Example As we mentioned above, the Jordan and Elman networks can be used to train a network on reproducing time sequences.

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. However, by this generalisation of the basic architecture we have also incurred a serious loss: we no longer have a learning rule to determine the optimal weights! 6 Multi-layer perceptrons can do everything In the previous section we showed that by adding an extra hidden unit, the XOR problem can be solved.

A more elegant proof is given in (Minsky & Papert, 1969), but the point is that for complex transformations the number of required units in the hidden layer is exponential in N . 7 Conclusions In this chapter we presented single layer feedforward networks for classi cation tasks and for function approximation tasks. The representational power of single layer feedforward networks was discussed and two learning algorithms for nding the optimal weights were presented. The simple networks presented here have their advantages and disadvantages.

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