Shallow-network approximation beyond the regime n < (d+3)/2 and p >= 2

Extend the approximation results for shallow path-norm-constrained ReLU networks to Sobolev smoothness n >= (d+3)/2 and to integrability exponents 1 <= p < 2, while retaining approximation-error guarantees in the W^{1,p}-norm.

Background

The shallow-network theorem requires dimension d >= 2, smoothness n < s=(d+3)/2, and exponent p >= 2. The remark explains that the restriction p >= 2 is tied to the embedding used by the proof, while the exponent s=(d+3)/2 is sharp in the relevant metric-entropy sense.

Consequently, the paper does not establish whether comparable shallow-network approximation rates hold when the Sobolev smoothness reaches or exceeds (d+3)/2, or when the integrability exponent lies below 2. These are explicitly identified as open problems.