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On the Complexity of Computing and Learning with Multiplicative Neural Networks
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Startseite » Mitarbeiter » M. Schmitt » On the Complexity of Computing and Learning with Multiplicative Neural Networks

pix pix On the Complexity of Computing and Learning with Multiplicative Neural Networks

In a great variety of neuron models neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks which contain units that multiply their inputs instead of summing them and, thus, allow inputs to interact nonlinearly. The class of multiplicative networks comprises such widely known and well studied network types as higher-order networks and product unit networks.

We investigate the complexity of computing and learning for multiplicative neural networks. In particular, we derive upper and lower bounds on the Vapnik-Chervonenkis (VC) dimension and the pseudo dimension for various types of networks with multiplicative units. As the most general case, we consider feedforward networks consisting of product and sigmoidal units, showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks. Moreover, we show that this bound holds even in the case when the unit type, product or sigmoidal, may be learned. Crucial for these results are calculations of solution set components bounds for new network classes. As to lower bounds we construct product unit networks of fixed depth with superlinear VC dimension.

For higher-order sigmoidal networks we establish polynomial bounds that, in contrast to previous results, do not involve any restriction of the network order. We further consider various classes of higher-order units, also known as sigma-pi units, characterized by connectivity constraints. In terms of these we derive some asymptotically tight bounds.

Multiplication plays an important role both in neural modeling of biological behavior and in applications of artificial neural networks. We also briefly survey research in biology and in applications where multiplication is considered an essential computational element. The results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks.

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