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- W3137945565 abstract "Abstract In this paper, it is shown that $$C_beta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> -smooth functions can be approximated by deep neural networks with ReLU activation function and with parameters $${0,pm frac{1}{2}, pm 1, 2}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . The $$l_0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $$l_1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> parameter norms of considered networks are thus equivalent. The depth, the width and the number of active parameters of the constructed networks have, up to a logarithmic factor, the same dependence on the approximation error as the networks with parameters in $$[-1,1]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . In particular, this implies that the nonparametric regression estimation with constructed networks achieves, up to logarithmic factors, the same minimax convergence rates as with sparse networks with parameters in $$[-1,1]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> ." @default.
- W3137945565 created "2021-03-29" @default.
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- W3137945565 date "2022-01-19" @default.
- W3137945565 modified "2023-10-14" @default.
- W3137945565 title "Function Approximation by Deep Neural Networks with Parameters $${0,pm frac{1}{2}, pm 1, 2}$$" @default.
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- W3137945565 doi "https://doi.org/10.1007/s42519-021-00229-5" @default.
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