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We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened.
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T. Poggio, H. N. Mhaskar, L. Rosasco, B. Miranda, Q. Liao, Why and when can deep—but not shallow—networks avoid the curse of dimensionality: A review, International Journal of Automation and Computing 14 (2017) 503–519
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D. Yarotsky, Error bounds for approximations with deep ReLU networks, Neural Netw. 94 (2017) 103–114
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N. Harvey, C. Liaw, A. Mehrabian, Nearly-tight VC-dimension bounds for piecewise linear neural networks, in: S. Kale, O. Shamir (Eds.), 30th Annual Conference on Learning Theory, Proc. Mach. Learn. Res. 65, 2017, pp. 1–5
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P. Petersen, F. Voigtlaender, Optimal approximation of piecewise smooth functions using deep ReLU neural networks, Neural Netw. 108 (2018) 296–330
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U. Shaham, A. Cloninger, R. R. Coifman, Provable approximation properties for deep neural networks, Appl. Comput. Harm. Anal. 44 (2018) 537–557
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D. Yarotsky, Optimal approximation of continuous functions by very deep ReLU networks, in: S. Bubeck, V. Perchet, P. Rigollet (Eds.), 31st Annual Conference on Learning Theory, Proc. Mach. Learn. Res. 75, 2018, pp. 1–11
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H. Montanelli, Q. Du, New error bounds for deep ReLU networks using sparse grids, SIAM J. Math. Data Sci. 1 (2019) 78–92
2019
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