An important problem in machine learning theory is to understand the approximation and generalization properties of two-layer neural networks in high dimensions. To this end, researchers have introduced the Barron space $\mathcal{B}_s(\Omega)$ and the spectral Barron space $\mathcal{F}_s(\Omega)$, where the index $s\in [0,\infty)$ indicates the smoothness of functions within these spaces and $\Omega\subset\mathbb{R}^d$ denotes the input domain. However, the precise relationship between the two types of Barron spaces remains unclear. In this paper, we establish a continuous embedding between them 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 \[ \delta \|f\|_{\mathcal{F}_{s-\delta}(\Omega)}\lesssim_s \|f\|_{\mathcal{B}_s(\Omega)}\lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}. \] Importantly, the constants do not depend on the input dimension $d$, suggesting that the embedding is effective in high dimensions. Moreover, we also show that the lower and upper bound are both tight.

翻译：机器学习理论中的一个重要问题是理解高维空间中双层神经网络的逼近与泛化性质。为此，研究者引入了Barron空间$\mathcal{B}_s(\Omega)$和谱Barron空间$\mathcal{F}_s(\Omega)$，其中指标$s\in [0,\infty)$表示这些空间中函数的光滑性，$\Omega\subset\mathbb{R}^d$表示输入域。然而，这两类Barron空间之间的精确关系尚不明确。本文通过以下不等式建立了两者之间的连续嵌入关系：对任意$\delta\in (0,1)$、$s\in \mathbb{N}^{+}$和$f: \Omega \mapsto\mathbb{R}$，有\[ \delta \|f\|_{\mathcal{F}_{s-\delta}(\Omega)}\lesssim_s \|f\|_{\mathcal{B}_s(\Omega)}\lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}. \]重要的是，这些常数不依赖于输入维数$d$，这表明该嵌入在高维情形下仍然有效。此外，我们还证明了上下界均为紧确的。