Embedding Inequalities for Barron-type Spaces

• URL: http://arxiv.org/abs/2305.19082v3
• Date: Wed, 27 Dec 2023 08:22:46 GMT
• Title: Embedding Inequalities for Barron-type Spaces
• Authors: Lei Wu
• Abstract summary: We show that the constants do not depend on the input dimension $d$, suggesting that the embedding is effective in high dimensions. We also show that the lower and upper bound are both tight.
• Score: 4.184052796218818
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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.

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