Approximation of functions from Korobov spaces by deep convolutional neural networks

Abstract

The efficiency of deep convolutional neural networks (DCNNs) has been demonstrated empirically in many practical applications. In this paper, we establish a theory for approximating functions from Korobov spaces by DCNNs. It verifies rigorously the efficiency of DCNNs in approximating functions of many variables with some variable structures and their abilities in overcoming the curse of dimensionality.

Rates of approximation by ReLU shallow neural networks

“…In [39], it was proved that the last layer of any fully-connected network is identical to that of a deep CNN with at most 8 times number of free parameters. For approximating or learning ridge function [10], radial functions [23], and functions from Korobov spaces [24], deep CNNs can be achieve the same accuracy with much smaller number of free parameters than fullyconnected networks. In a recent application of CNNs to readability of Chinese texts [11], it is found that one layer or two is already efficient.…”