  We consider an ensemble of $K$ single-layer perceptrons exposed to random
inputs and investigate the conditions under which the couplings of these
perceptrons can be chosen such that prescribed correlations between the outputs
occur. A general formalism is introduced using a multi-perceptron costfunction
that allows to determine the maximal number of random inputs as a function of
the desired values of the correlations. Replica-symmetric results for $K=2$ and
$K=3$ are compared with properties of two-layer networks of tree-structure and
fixed Boolean function between hidden units and output. The results show which
correlations in the hidden layer of multi-layer neural networks are crucial for
the value of the storage capacity.

Disordered Systems and Neural Networks