Universal Approximation with Quadratic Deep Networks

TL;DR

This paper investigates the expressive power of quadratic neural networks, demonstrating their advantages over conventional networks in approximation efficiency, capacity, and compactness through four theoretical theorems.

Contribution

It introduces four theorems that reveal the superior expressive efficiency, unique capabilities, and compactness of quadratic neural networks compared to traditional neural networks.

Findings

Quadratic networks can approximate certain functions more efficiently than conventional networks.

There are functions expressible by quadratic networks that are not possible with conventional neurons.

Quadratic networks can achieve similar approximation errors with fewer weights, indicating higher efficiency.

Abstract

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.