Embedding Inequalities for Barron-type Spaces

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)}\lesssims |f|{\mathcal{B}s(\Omega)}\lesssims |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.