This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the product function over a general compact domain. We also demonstrate this lower bound doesn't apply to normalized Lipschitz monomials over the unit cube. These findings suggest that shallow ReLU networks experience the curse of dimensionality when expressing functions with a Lipschitz parameter scaling with the dimension of the input, and that the expressive power of neural networks is more dependent on their depth rather than overall complexity.

翻译：本研究探讨了修正线性单元（ReLU）神经网络逼近多元单项式所需神经元数量的问题。我们证明了在一般紧致域上逼近乘积函数的任意浅层网络复杂度存在指数级下界，同时表明该下界不适用于单位立方体上的归一化Lipschitz单项式。这些发现表明，当表达Lipschitz参数随输入维度缩放的目标函数时，浅层ReLU网络会遭遇维度灾难，且神经网络的表达能力更多取决于其深度而非整体复杂度。