Embedding Inequalities for Barron-type Spaces
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Abstract (arXiv)
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 Bs(Omega) and the spectral Barron space Fs(Omega), where the index s∈ [0,∞) indicates the smoothness of functions within these spaces and Omegasubsetℝd 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 δ∈ (0,1), s∈ mathbbN+ and f: Omega mapstoℝ, it holds that [ δ |f|_Fs-δ(Omega)lesssims |f|Bs(Omega)lesssims |f|_Fs+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.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 15 of its 26 references that are in the map. First posted on arXiv 30 May 2023.
Builds on
15 of 26 references.
References · 15
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3 more
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Citing papers per year: 2024: 1; 2025: 0; 2026: 1.
2 papers.