Feedback stabilization using two-hidden-layer nets

The representational capabilities of one-hidden-layer and two-hidden-layer nets consisting of feedforward interconnections of linear threshold units are compared. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into direct and inverse problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating one-sided inverses of continuous functions, and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not, in general, using just one