Title: Depth Separation for Neural Networks,
Abstract: Let $f:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}\to\mathbb{S}$ be a function of
the form $f(\mathbf{x},\mathbf{x}') = g(\langle\mathbf{x},\mathbf{x}'\rangle)$
for $g:[-1,1]\to \mathbb{R}$. We give a simple proof that shows that poly-size
depth two neural networks with (exponentially) bounded weights cannot
approximate $f$ whenever $g$ cannot be approximated by a low degree polynomial.
Moreover, for many $g$'s, such as $g(x)=\sin(\pi d^3x)$, the number of neurons
must be $2^{\Omega\left(d\log(d)\right)}$. Furthermore, the result holds
w.r.t.\ the uniform distribution on $\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}$.
As many functions of the above form can be well approximated by poly-size depth
three networks with poly-bounded weights, this establishes a separation between
depth two and depth three networks w.r.t.\ the uniform distribution on
$\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}$.