浅层 ReLU^k 网络逼近速率的几何依赖性与稳定性分析
Geometry-Dependent Approximation and Stability Properties of Shallow ReLU$^k$ Networks on Bounded Euclidean Domains
这篇论文把浅层 ReLU^k 网络的逼近误差和参数几何量 h 与分离距离 h̲ 直接挂上钩，还给出 Gram 矩阵谱和系数稳定性的估计。做理论或想搞懂为什么收敛速率能反映目标函数正则性的人可以看看，比单纯跑实验的逼近论文扎实。
一篇 arXiv 论文研究了有界欧氏域上隐藏参数几何如何影响浅层 ReLU^k 网络。论文以填充距离 h 和对踵分离距离 h̲ 刻画参数几何，给出代表性逼近估计：误差上界为 h^r 乘以目标函数的 H^r 范数，下界约为 h̲^((d+2k+1)/2) 乘以 L^2 范数。
上界可利用目标函数直到临界阶 (d+2k+1)/2 的正则性，下界则针对比该阶更光滑的固定目标成立；上界在内子域 Ω' 上成立，下界涉及更大域 Ω。
技术上采用平滑局部化和二进傅里叶估计，并覆盖 Gram 与刚度矩阵谱、Bernstein 与 Marcinkiewicz-Zygmund 不等式、离散最小二乘恢复及逆逼近。此外论文给出几何依赖的谱估计以量化条件数和系数稳定性，以及从逼近速率反推目标正则性的结论。
对理解浅层网络训练中系数表示的稳定性与收敛速率含义有直接参考价值。
Geometry-Dependent Approximation and Stability Properties of Shallow ReLU$^k$ Networks on Bounded Euclidean Domains
On bounded Euclidean domains, we study how hidden-parameter geometry affects shallow ReLU$^k$ networks through results on approximation rates, saturation-type lower bounds, Gram and stiffness matrix spectra, Bernstein and Marcinkiewicz--Zygmund inequalities, discrete least-squares recovery, and inverse approximation. Our analysis uses smooth localization and dyadic Fourier estimates. We quantify parameter geometry by the fill distance $h$ and antipodal separation distance $\underline h$. For the prescribed network space $L_n^k$, representative approximation estimates take the form \begin{equation*} \begin{aligned} \inf_{g_n\in L_n^k}\|f-g_n\|_{\mathcal L^2(Ω')}&\lesssim h^r\|f\|_{\mathcal H^r(Ω)},\\ \inf_{g_n\in L_n^k}\|f-g_n\|_{\mathcal L^2(Ω)}&\gtrsim \underline h^{\frac{d+2k+1}{2}}\|f\|_{\mathcal L^2(Ω')}. \end{aligned} \end{equation*} The upper bound exploits target regularity up to the critical order $\frac{d+2k+1}{2}$, while the lower bound applies asymptotically to fixed targets smoother than this order. The upper estimate holds on an interior subdomain $Ω'$, whereas the lower estimate concerns the larger domain $Ω$. We also obtain geometry-dependent spectral estimates that quantify conditioning and coefficient stability. Further results establish norm discretization and regularized recovery from function values, and infer target regularity from approximation rates under coefficient or separation conditions. Altogether, these results describe how parameter geometry governs approximation accuracy, the stability of coefficient representations, discrete recovery, and the regularity information contained in convergence rates.
This report is categorized under AITOP's "Papers" desk. Key entities and technical topics under continuous tracking include ReLU网络, 逼近论, 理论分析, arXiv.
Source attribution: curated from arXiv cs.LG (@Yuqing Liu, Tong Mao, Jinchao Xu).