  Deep neural networks can empirically perform efficient hierarchical learning,
in which the layers learn useful representations of the data. However, how they
make use of the intermediate representations are not explained by recent
theories that relate them to "shallow learners" such as kernels. In this work,
we demonstrate that intermediate neural representations add more flexibility to
neural networks and can be advantageous over raw inputs. We consider a fixed,
randomly initialized neural network as a representation function fed into
another trainable network. When the trainable network is the quadratic Taylor
model of a wide two-layer network, we show that neural representation can
achieve improved sample complexities compared with the raw input: For learning
a low-rank degree-$p$ polynomial ($p \geq 4$) in $d$ dimension, neural
representation requires only $\tilde{O}(d^{\lceil p/2 \rceil})$ samples, while
the best-known sample complexity upper bound for the raw input is
$\tilde{O}(d^{p-1})$. We contrast our result with a lower bound showing that
neural representations do not improve over the raw input (in the infinite width
limit), when the trainable network is instead a neural tangent kernel. Our
results characterize when neural representations are beneficial, and may
provide a new perspective on why depth is important in deep learning.
