In this article we identify a general class of high-dimensional continuous
functions that can be approximated by deep neural networks (DNNs) with the
rectified linear unit (ReLU) activation without the curse of dimensionality. In
other words, the number of DNN parameters grows at most polynomially in the
input dimension and the approximation error. The functions in our class can be
expressed as a potentially unbounded number of compositions of special
functions which include products, maxima, and certain parallelized Lipschitz
continuous functions.

本文发现使用修正线性单元（ReLU）激活的深度神经网络（DNNs）可以近似表示一类高维连续函数，其参数数量与输入维度和近似误差的多项式规模相同，该类函数由多个特殊函数的组合表示，包括乘积，最大值和某些并行的 Lipschitz 连续函数。

深度神经网络逼近复合函数的无维度诅咒

Deep neural network approximation of composite functions without the  curse of dimensionality

We study the implicit regularization effects of deep learning in tensor
factorization. While implicit regularization in deep matrix and 'shallow'
tensor factorization via linear and certain type of non-linear neural networks
promotes low-rank solutions with at most quadratic growth, we show that its
effect in deep tensor factorization grows polynomially with the depth of the
network. This provides a remarkably faithful description of the observed
experimental behaviour. Using numerical experiments, we demonstrate the
benefits of this implicit regularization in yielding a more accurate estimation
and better convergence properties.

研究深度学习在张量因式分解中的隐式正则化效应，通过数值实验证明了这种隐式正则化可促进获得更准确的估计和更好的收敛特性。