Uniqueness of the weights for minimal feedforward nets with a given input-output map

|{ We show that, for feedforward nets with a single hidden layer, a single output node, and a \transfer function" Tanhs, the net is uniquely determined by its input-output map, up to an obvious nite group of symmetries (permutations of the hidden nodes, and changing the sign of all the weights associated to a particular hidden node), provided that the net is irreducible, i.e. that there does not exist an inner node that makes a zero contribution to the output, and there is no pair of hidden nodes that could be collapsed to a single node without altering the input-output map. The author thanks Eduardo Sontag for suggesting the problem and for his helpful comments and ideas, and an anonymous referee for suggesting how to improve the exposition at several points.