Correction to 'Lower Bounds on the VC-Dimension of Smoothly Parametrized Function Classes'

The paper 3] gives lower bounds on the VC-dimension of various smoothly parametrized function classes. The results were proven by showing a relationship between the uniqueness of decision boundaries and the VC-dimension of smoothly parametrized function classes. The proof is incorrect; there is no such relationship under the conditions stated in 3]. For the case of neural networks with tanh activation functions, we give an alternative proof of a lower bound for the VC dimension proportional to the number of parameters which holds even when the magnitude of the parameters is restricted to be arbitrary small. Theorem 10 from 3] which was used to prove lower bounds on various neural network classes is incorrect. Theorem 10 ((3]) Let A be an open subset of R m , X be an open subset of R n and f : AX ! R be a continuously diierentiable function (in all of its arguments). Let F := ff(a;) : a 2 Ag. If there exists a k-dimensional manifold M A which has unique decision boundaries, then VCdim(F) k, where F is the thresholded function class formed from F. As counterexamples, observe that y = (a ? x) 2 has unique decision boundaries but VC dimension zero. Similarly y = (x ?a)(x?b) 2 has unique decision boundaries but VC dimension 1. In 3], Theorem 10 was used to prove a lower bound for the VC-dimension of neural networks with tanh activation functions. We can prove this lower bound more directly using Lemma 9 from 3].