It is well known that one can store up to 2N unconelated p a m s in a simple perceptron with N input neurons. Changing the architecture of the net by adding a hidden layer may enhance the storage capacity. Leaming in multilayered nehvorlts, however. is difficult and slow compared to perceptron learning. In this work a different approach is taken. A large hidden layer with N' neurons is used onto which the patterns are m p p d according to a one-ta-one wde which is fixed beforehand. Only the wnnections from the hidden layer to the output unit are modified by learning. Here we show how to treat the wnelations which are intraduc+ by the coding. W e find that the storage capacity of such a net cm be made exponentially large. Moreover, our results shed new light on the optimal capacity problem for single-layer perceptrons. We 6nd the optimal capacity to be determined by the dimension of the space spanned by the input palm" rather than by the size of the input layer.
The performance of large neural networks can be judged not only by their storage capacity but also by the time required for learning. A polynomial learning algorithm with learning time ∼ N 2 in a network with N units might be practical whereas a learning time ∼ e N would allow rather small networks only. The question of absolute storage capacity α c and capacity for polynomial learning rules α p is discussed for several feed-forward architectures, the perceptron, the binary perceptron, the committee machine and a perceptron with fixed weights in the first layer and adaptive weights in the second layer. The analysis is based partially on dynamic mean field theory which is valid for N → ∞. Especially for the committee machine a value α p considerably lower than the capacity predicted by replica theory or simulations is found. This discrepancy is resolved by new simulations investigating the learning time dependence and revealing subtleties in the definition of the capacity.
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