Abstract
It is known that any given probability distribution of the states of the observable units of a Boltzmann machine can be realized if no limit is imposed on the number of hidden units. But very little is known about the number of hidden units necessary for such realization. We consider Boltzmann machines as associative memories and show that there exist vector sets whose memorization on a Boltzmann machine requires a number of hidden units which is exponential in the size of the vectors (i.e., the number of components in each vector). Additional results give tight bounds on the number of hidden units needed in terms of the vector set size (i.e., the number of vectors in the set). Furthermore, we show how to construct Boltzmann machines which realize negation, intersection, and composition of the vector sets memorized by given Boltzmann machines.
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