Reducing or eliminating logarithmic factors for ReLU^k approximation

Determine whether the logarithmic factors in the upper bounds for shallow neural networks with the ReLU^k activation can be reduced or eliminated.

Background

The paper establishes nearly optimal algebraic approximation rates for shallow neural networks using the ReLU^k activation in mixed Sobolev spaces. Specifically, the algebraic exponent is identified as min{α,k+1}, with matching lower bounds, but the upper bounds contain additional logarithmic factors arising from the Fourier-block construction, neuron allocation, and related estimates. The authors explicitly leave unresolved whether these logarithmic losses are an artifact of the proof or reflect an intrinsic limitation of the approximation method.