My question is can the product of the VC dimensions of two function spaces be less than the VC dimension of the new space formed by the composition of functions from these two spaces?
More specifically, my question is as follows: Suppose we have hypothesis spaces $H_1$ and $H_2$, with VC dimensions $d_1$ and $d_2$ respectively. Then, for $H_3 = \{ y | y = f \circ g, f \in H_1, g \in H_2 \} $, what can be the relationship between the VC dimension $d_3$ of $H_3$ and $d_1, d_2$? There are many examples where $d_3 = d_1 \times d_2$, but is it possible to provide an example where $d_3 > d_1 \times d_2$?
I would be immensely grateful for any help you can provide!