Sobolev Approximation of Deep ReLU Networks in Log-Barron Space

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical success of deep models on high-dimensional data. Barron space theory addresses this: if a target function belongs to a Barron space, a two-layer network with n parameters achieves an O(n-1/2) approximation error in L2. Yet classical Barron spaces mathscrBs+1 still require stronger regularity than Sobolev spaces Hs, and existing depth-sensitive results often assume constraints such as sL ≤ 1/2. In this paper, we introduce a log-weighted Barron space mathscrBlog, which requires a strictly weaker assumption than mathscrBs for any s>0. For this new function space, we first study embedding properties and carry out a statistical analysis via the Rademacher complexity. Then we prove that functions in mathscrBlog can be approximated by deep ReLU networks with explicit depth dependence. We then define a family mathscrBs,log, establish approximation bounds in the H1 norm, and identify maximal depth scales under which these rates are preserved. Our results clarify how depth reduces regularity requirements for efficient representation, offering a more precise explanation for the performance of deep architectures beyond the classical Barron setting, and for their stable use in high-dimensional problems used today.