One of the fundamental problems in deep learning theory is understanding the approximation and generalization properties of two-layer neural networks in high dimensions. In order to tackle this issue, researchers have introduced the Barron space $\mathcal{B}_s(\Omega)$ and the spectral Barron space $\mathcal{F}_s(\Omega)$, where the index $s$ characterizes the smoothness of functions within these spaces and $\Omega\subset\mathbb{R}^d$ represents the input domain. However, it is still not clear what is the relationship between the two types of Barron spaces. In this paper, we establish continuous embeddings between these spaces as implied by the following inequality: for any $\delta\in (0,1), s\in \mathbb{N}^{+}$ and $f: \Omega \mapsto\mathbb{R}$, it holds that \[ \delta\gamma^{\delta-s}_{\Omega}\|f\|_{\mathcal{F}_{s-\delta}(\Omega)}\lesssim_s \|f\|_{\mathcal{B}_s(\Omega)}\lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}, \] where $\gamma_{\Omega}=\sup_{\|v\|_2=1,x\in\Omega}|v^Tx|$ and notably, the hidden constants depend solely on the value of $s$. Furthermore, we provide examples to demonstrate that the lower bound is tight.

本文研究深度学习理论中关于高维度下两层神经网络逼近和泛化特性的问题，引入Barron空间和谱Barron空间来探究函数的光滑度，建立了这两种空间之间的连续嵌入关系，提供了实例证明下界是紧的。

Barron型空间的嵌入不等式