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.

翻译：深度学习理论中的基础问题之一，是理解高维双层神经网络的逼近与泛化特性。为应对这一问题，研究者引入了Barron空间 $\mathcal{B}_s(\Omega)$ 与谱Barron空间 $\mathcal{F}_s(\Omega)$，其中指标 $s$ 表征这些空间中函数的平滑度，$\Omega\subset\mathbb{R}^d$ 表示输入域。然而，两类Barron空间之间的关系仍不明确。本文通过以下不等式建立了这些空间之间的连续嵌入：对任意 $\delta\in (0,1)$、$s\in \mathbb{N}^{+}$ 及 $f: \Omega \mapsto\mathbb{R}$，有 \[ \delta\gamma^{\delta-s}_{\Omega}\|f\|_{\mathcal{F}_{s-\delta}(\Omega)}\lesssim_s \|f\|_{\mathcal{B}_s(\Omega)}\lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}, \] 其中 $\gamma_{\Omega}=\sup_{\|v\|_2=1,x\in\Omega}|v^Tx|$，且隐式常数仅取决于 $s$ 的取值。此外，我们通过实例证明了该下界是紧的。