  This paper proposes a new neural network architecture by introducing an
additional dimension called height beyond width and depth. Neural network
architectures with height, width, and depth as hyper-parameters are called
three-dimensional architectures. It is shown that neural networks with
three-dimensional architectures are significantly more expressive than the ones
with two-dimensional architectures (those with only width and depth as
hyper-parameters), e.g., standard fully connected networks. The new network
architecture is constructed recursively via a nested structure, and hence we
call a network with the new architecture nested network (NestNet). A NestNet of
height $s$ is built with each hidden neuron activated by a NestNet of height
$\le s-1$. When $s=1$, a NestNet degenerates to a standard network with a
two-dimensional architecture. It is proved by construction that height-$s$ ReLU
NestNets with $\mathcal{O}(n)$ parameters can approximate $1$-Lipschitz
continuous functions on $[0,1]^d$ with an error $\mathcal{O}(n^{-(s+1)/d})$,
while the optimal approximation error of standard ReLU networks with
$\mathcal{O}(n)$ parameters is $\mathcal{O}(n^{-2/d})$. Furthermore, such a
result is extended to generic continuous functions on $[0,1]^d$ with the
approximation error characterized by the modulus of continuity. Finally, we use
numerical experimentation to show the advantages of the super-approximation
power of ReLU NestNets.
