Expressivity of Shallow and Deep Neural Networks for Polynomial Approximation
This addresses the curse of dimensionality in neural network expressivity for researchers in approximation theory and machine learning, providing foundational insights into depth vs. complexity trade-offs.
The study tackled the problem of approximating multivariate monomials with ReLU neural networks, finding an exponential lower bound on neuron requirements for shallow networks over general domains, but not for normalized Lipschitz monomials over the unit cube.
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.