Global Universality of the Two-Layer Neural Network with the<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>k</a:mi></a:math>-Rectified Linear Unit

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  • Naoya Hatano
    Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
  • Masahiro Ikeda
    Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
  • Isao Ishikawa
    Center for Advanced Intelligence Project, RIKEN, Japan
  • Yoshihiro Sawano
    Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
  • Andrea Scapellato
    editor

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<jats:p>This paper concerns the universality of the two-layer neural network with the<jats:inline-formula><a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:mi>k</a:mi></a:math></jats:inline-formula>-rectified linear unit activation function with<jats:inline-formula><c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"><c:mi>k</c:mi><c:mo>=</c:mo><c:mn>1</c:mn><c:mo>,</c:mo><c:mn>2</c:mn><c:mo>,</c:mo><c:mo>…</c:mo></c:math></jats:inline-formula>with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for<jats:inline-formula><e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"><e:mi>k</e:mi><e:mo>=</e:mo><e:mn>1</e:mn></e:math></jats:inline-formula>by the present authors. This paper covers<jats:inline-formula><g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"><g:mi>k</g:mi></g:math></jats:inline-formula>-sigmoidal functions as an application of the fundamental result on<jats:inline-formula><i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M6"><i:mi>k</i:mi></i:math></jats:inline-formula>-rectified linear unit functions.</jats:p>

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