LGOct 21, 2019
Approximation capabilities of neural networks on unbounded domainsMing-Xi Wang, Yang Qu
In this paper, we prove that a shallow neural network with a monotone sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function can arbitrarily well approximate any L^p(p>=2) integrable functions defined on R*[0,1]^n. We also prove that a shallow neural network with a sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function expresses no nonzero integrable function defined on the Euclidean plane. Together with a recent result that the deep ReLU network can arbitrarily well approximate any integrable function on Euclidean spaces, we provide a new perspective on the advantage of multiple hidden layers in the context of ReLU networks. Lastly, we prove that the ReLU network with depth 3 is a universal approximator in L^p(R^n).
CPOct 2, 2019
The option pricing model based on time values: an application of the universal approximation theory on unbounded domainsYang Qu, Ming-Xi Wang
We propose a time value related decision function to treat a classical option pricing problem raised by Hutchinson-Lo-Poggio. In numerical experiments, the new decision function significantly improves the original model of Hutchinson-Lo-Poggio with faster convergence and better generalization performance. By proving a novel universal approximation theorem, we show that our decision function rather than Hutchinson-Lo-Poggio's can be approximated on the entire domain of definition by neural networks. Thus the experimental results are partially explained by the representation properties of networks.