{"ID":2823155,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2601.01295","arxiv_id":"2601.01295","title":"Sobolev Approximation of Deep ReLU Networks in Log-Barron Space","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 $s\u003e0$. 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.","short_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...","url_abs":"https://arxiv.org/abs/2601.01295","url_pdf":"https://arxiv.org/pdf/2601.01295v2","authors":"[\"Changhoon Song\",\"Seungchan Ko\",\"Youngjoon Hong\"]","published":"2026-01-03T22:03:19Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.NA\"]","methods":"[]","has_code":false}