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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2021

Approximation of Smoothness Classes by Deep Rectifier Networks

Résumé

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^\alpha_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We show that deep rectifier networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space $B^\alpha_{\tau}(L^\tau)$ on the critical embedding line $1/\tau=\alpha/d+1/p$ for arbitrary smoothness order $\alpha>0$. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.

Dates et versions

hal-02909881 , version 1 (31-07-2020)

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Citer

Mazen Ali, Anthony Nouy. Approximation of Smoothness Classes by Deep Rectifier Networks. SIAM Journal on Numerical Analysis, 2021, 59 (6), pp.3032-3051. ⟨10.1137/20M1360657⟩. ⟨hal-02909881⟩
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