  Inspired by neuronal diversity in the biological neural system, a plethora of
studies proposed to design novel types of artificial neurons and introduce
neuronal diversity into artificial neural networks. Recently proposed quadratic
neuron, which replaces the inner-product operation in conventional neurons with
a quadratic one, have achieved great success in many essential tasks. Despite
the promising results of quadratic neurons, there is still an unresolved issue:
\textit{Is the superior performance of quadratic networks simply due to the
increased parameters or due to the intrinsic expressive capability?} Without
clarifying this issue, the performance of quadratic networks is always
suspicious. Additionally, resolving this issue is reduced to finding killer
applications of quadratic networks. In this paper, with theoretical and
empirical studies, we show that quadratic networks enjoy parametric efficiency,
thereby confirming that the superior performance of quadratic networks is due
to the intrinsic expressive capability. This intrinsic expressive ability comes
from that quadratic neurons can easily represent nonlinear interaction, while
it is hard for conventional neurons. Theoretically, we derive the approximation
efficiency of the quadratic network over conventional ones in terms of real
space and manifolds. Moreover, from the perspective of the Barron space, we
demonstrate that there exists a functional space whose functions can be
approximated by quadratic networks in a dimension-free error, but the
approximation error of conventional networks is dependent on dimensions.
Empirically, experimental results on synthetic data, classic benchmarks, and
real-world applications show that quadratic models broadly enjoy parametric
efficiency, and the gain of efficiency depends on the task.
