Super-approximation for spectral Barron functions

Determine whether the super-approximation rate observed for very deep ReLU networks also holds for spectral Barron functions.

Background

The paper establishes a lower bound for approximating spectral Barron functions with fully connected ReLU networks in terms of width and depth. When the width is bounded, the number of parameters is proportional to the depth, and the resulting lower bound can decay faster than the paper’s general approximation upper bound. This suggests that very deep networks might achieve a stronger, so-called super-approximation rate.

The authors note that analogous super-approximation phenomena have been studied for classical smoothness classes, but they do not resolve whether the same phenomenon occurs for spectral Barron functions. Resolving this question would clarify whether increasing depth beyond the logarithmic-depth regime can yield approximation rates substantially better than the established parameter-based upper bound.