  Recently, deep learning has achieved huge successes in many important
applications. In our previous studies, we proposed quadratic/second-order
neurons and deep quadratic neural networks. In a quadratic neuron, the inner
product of a vector of data and the corresponding weights in a conventional
neuron is replaced with a quadratic function. The resultant quadratic neuron
enjoys an enhanced expressive capability over the conventional neuron. However,
how quadratic neurons improve the expressing capability of a deep quadratic
network has not been studied up to now, preferably in relation to that of a
conventional neural network. Regarding this, we ask four basic questions in
this paper: (1) for the one-hidden-layer network structure, is there any
function that a quadratic network can approximate much more efficiently than a
conventional network? (2) for the same multi-layer network structure, is there
any function that can be expressed by a quadratic network but cannot be
expressed with conventional neurons in the same structure? (3) Does a quadratic
network give a new insight into universal approximation? (4) To approximate the
same class of functions with the same error bound, is a quantized quadratic
network able to enjoy a lower number of weights than a quantized conventional
network? Our main contributions are the four interconnected theorems shedding
light upon these four questions and demonstrating the merits of a quadratic
network in terms of expressive efficiency, unique capability, compact
architecture and computational capacity respectively.
