Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units (Communications in Computational Physics, 2019)
Deep neural networks with rectified linear units (ReLU) are getting moreand more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLUnetwork for functions in Sobolev space and Korobov space have recently been madeby [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du,SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smoothfunctions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convertpolynomials into deep RePU networks of optimal size with no approximation error.Comparing to the results on ReLU networks, the sizes of RePU networks required toapproximate functions in Sobolev space and Korobov space with an error toleranceε, by our constructive proofs, are in general $O$($log\frac{1}{ε}$) times smaller than the sizes ofcorresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993],our constructions use less number of activation functions and numerically more stable,they can be served as good initials of deep RePU networks and further trained to breakthe limit of linear approximation theory. The functions represented by RePU networksare smooth functions, so they naturally fit in the places where derivatives are involvedin the loss function.
Keywords: Deep neural network; high dimensional approximation; sparse grids; rectified linear unit; rectified power unit; rectified quadratic unit.