Tight bounds for the VC-dimension of piecewise polynomial networks

O(ws(s log d + log(dqh/s))) and O(ws((h/s) log q)+ log(dqh/s)) are upper bounds for the VC-dimension of a set of neural networks of units with piecewise polynomial activation functions, where s is the depth of the network, h is the number of hidden units, w is the number of adjustable parameters, q is the maximum of the number of polynomial segments of the activation function, and d is the maximum degree of the polynomials; also Omega(ws log(dqh/s)) is a lower bound for the VC-dimension of such a network set, which are tight for the cases s = Theta(h) and a is constant. For the special case q = 1, the VC-dimension is Theta(ws log d).