{"ID":6138880,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-10T20:06:54.844448407Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.06781","arxiv_id":"2607.06781","title":"On Explicit Super-Expressive Approximation for Neural Networks","abstract":"In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error. We resolve this issue by introducing the Chinese Remainder Theorem as a constructive encoding mechanism. For Lipschitz continuous functions on $[0,1]^D$, we construct a width-$\\max\\{D,4\\}$, depth-$5$ network with explicit parameter-error trade-offs. For Hölder-smooth functions in $C^{r,γ}_A\\left([0,1]^D\\right)$, our fixed network of width $\\max\\{2D,\\ D+5N+1\\}$ and depth $r + 9$ achieves the parameter magnitude $\\mathcal{P}$ bounded by $\\log_2 \\mathcal{P}=\\mathcal{O}\\bigl(\\varepsilon^{-2D/(r+γ)}\\log(1/\\varepsilon)\\bigr)$. This is the dual result compared to those in the parameter-bounded and architecture-unbounded paradigm.","short_abstract":"In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with resp...","url_abs":"https://arxiv.org/abs/2607.06781","url_pdf":"https://arxiv.org/pdf/2607.06781v1","authors":"[\"Feng-Lei Fan\",\"Ze-Yu Li\",\"Chen-Yu Wang\",\"Jian-Jun Wang\"]","published":"2026-07-07T20:21:57Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}