Sobolev Approximation of Deep ReLU Network in Log-weighted Barron Space

• URL: http://arxiv.org/abs/2601.01295v1
• Date: Sat, 03 Jan 2026 22:03:19 GMT
• Title: Sobolev Approximation of Deep ReLU Network in Log-weighted Barron Space
• Authors: Changhoon Song, Seungchan Ko, Youngjoon Hong,
• Abstract summary: We introduce a log-weighted Barron space $mathscrBlog$.<n>We prove that functions in $mathscrBlog$ can be approximated by deep ReLU networks with explicit depth dependence.
• Score: 8.36932842016193
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $L^2$. Yet classical Barron spaces $\mathscr{B}^{s+1}$ still require stronger regularity than Sobolev spaces $H^s$, and existing depth-sensitive results often assume constraints such as $sL \le 1/2$. In this paper, we introduce a log-weighted Barron space $\mathscr{B}^{\log}$, which requires a strictly weaker assumption than $\mathscr{B}^s$ 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 $\mathscr{B}^{\log}$ can be approximated by deep ReLU networks with explicit depth dependence. We then define a family $\mathscr{B}^{s,\log}$, establish approximation bounds in the $H^1$ 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.

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