发表机构
Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在拟Banach范围内建立了Besov空间与浅层ReLU^k变分空间之间的尖锐嵌入，确定了最优光滑性阈值，并应用于PDE解的正则性到逼近误差和贪婪算法收敛性的转化。
AI 中文摘要
设 $\mathcal D$ 为由 $\operatorname{ReLU}^k$ 在有界Lipschitz域 $\Omega\subset\mathbb R^d$ 上生成的归一化脊字典。我们在拟Banach范围 $0<p\le 1$ 内建立了各向同性Besov空间与相关变分空间 $\mathcal L_1(\mathcal D)$ 之间的尖锐嵌入。具体地，当 $0<q\le 1$ 且 $s\ge k+d/p$ 时，以及当 $1<q\le\infty$ 且 $s>k+d/p$ 时，有 $B^s_{p,q}(\Omega)\hookrightarrow \mathcal L_1(\mathcal D)$。一种重缩放凸包构造表明该光滑性阈值是尖锐的。反之，对于 $0<p<1$，有 $\mathcal L_1(\mathcal D)\hookrightarrow B^{k+1}_{p,2}(\Omega)$，且光滑性指数 $k+1$ 和精细指标 $2$ 都是最优的。正向嵌入将已知的Besov正则性（特别是偏微分方程解的正则性）转化为受控的逼近误差界以及基于浅层 $\text{ReLU}^k$ 神经网络的贪婪算法的收敛性。证明结合了Littlewood–Paley局部化、Fourier–Radon表示、测度值导数以及向量值奇异积分估计。
英文摘要
Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0<p\leq 1$. Specifically, \[ B^s_{p,q}(Ω)\hookrightarrow \mathcal L_1(\mathbb{D}) \] when $s\geq k+d/p$ for $0<q\leq 1$, and when $s>k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(Ω), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.
Comments28 pages
