  We propose a spectral-based approach to analyze how two-layer neural networks
separate from linear methods in terms of approximating high-dimensional
functions. We show that quantifying this separation can be reduced to
estimating the Kolmogorov width of two-layer neural networks, and the latter
can be further characterized by using the spectrum of an associated kernel.
Different from previous work, our approach allows obtaining upper bounds, lower
bounds, and identifying explicit hard functions in a united manner. We provide
a systematic study of how the choice of activation functions affects the
separation, in particular the dependence on the input dimension. Specifically,
for nonsmooth activation functions, we extend known results to more activation
functions with sharper bounds. As concrete examples, we prove that any single
neuron can instantiate the separation between neural networks and random
feature models. For smooth activation functions, one surprising finding is that
the separation is negligible unless the norms of inner-layer weights are
polynomially large with respect to the input dimension. By contrast, the
separation for nonsmooth activation functions is independent of the norms of
inner-layer weights.
