提出可显式控制参数量的神经网络，实现高精度函数逼近。
On Explicit Super-Expressive Approximation for Neural Networks
- 用中国剩余定理构造编码机制，显式控制参数大小。
- 对光滑函数逼近误差，参数量对数为O(ε^{-2D/(r+γ)} log(1/ε))。
- 适合关注参数效率与理论保证的研究者参考。
本文研究固定架构神经网络在显式参数约束下的逼近能力，针对传统方法缺乏量化且非渐近的参数规模分析问题，提出利用中国剩余定理作为构造性编码机制。对于[0,1]^D上的Lipschitz连续函数，构建了宽度max{D,4}、深度5的网络，实现参数-误差显式权衡；对于C^{r,γ}_A([0,1]^D)类Hölder光滑函数，固定宽度max{2D, D+5N+1}、深度r+9的网络，其参数量ℙ满足log₂ℙ=𝒪(ε^{-2D/(r+γ)} log(1/ε))。该结果是参数受限、架构不限范式下的对偶成果。
原文摘要 · Abstract (English)
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.
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