Universal Approximation with Quadratic Deep Networks

Recently, deep learning has been playing a central role in machine learning research and applications. Since AlexNet, increasingly more advanced networks have achieved state-of-the-art performance in computer vision, speech recognition, language processing, game playing, medical imaging, and so on. 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 second-order 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. In this paper, we ask three basic questions regarding the expressive capability of a quadratic network: (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? Our main contributions are the three theorems shedding light upon these three questions and demonstrating the merits of a quadratic network in terms of expressive efficiency, unique capability, and compact architecture respectively.