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Spectral Barron space for deep neural network approximation

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arxiv 2309.00788 v2 pith:HIBR6IBG submitted 2023-09-02 math.NA cs.NA

classification math.NAcs.NA
keywords spacebarronspectralapproximationembeddingnetworkneuralprove
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abstract

We prove the sharp embedding between the spectral Barron space and the Besov space with embedding constants independent of the input dimension. Given the spectral Barron space as the target function space, we prove a dimension-free convergence result that if the neural network contains $L$ hidden layers with $N$ units per layer, then the upper and lower bounds of the $L^2$-approximation error are $\mathcal{O}(N^{-sL})$ with $0 < sL\le 1/2$, where $s\ge 0$ is the smoothness index of the spectral Barron space.

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  1. Sharp uniform approximation for spectral Barron functions by deep neural networks

    math.NA 2025-07 conditional novelty 6.0 of 10

    ReLU networks approximate spectral Barron functions of smoothness as low as 1/2 at the N^{-1/2} rate, and L-layer networks achieve sharp N^{-sL} rates for 0<sL<=1/2.

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