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<p>In recent years, neural networks have achieved significant success in function approximation. However, several challenges remain, such as the extraction of overly specific features from composite functions and the generation of non-differentiable output functions. To address these issues, we propose a deep <i>k</i>-power of the rectified linear unit (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\text {ReLU}^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>ReLU</mtext> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>) convolutional neural networks algorithm for approximating functions with additive ridge features. Our approach enables automatic feature extraction without requiring prior knowledge of the composite structure, and the network’s output exhibits higher-order differentiability compared to existing methods. In addition, we prove that deep <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\text {ReLU}^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>ReLU</mtext> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> convolutional neural networks can mitigate the curse of dimensionality for target functions with the low-dimensional structure of additive ridge features. These results highlight the superior learning capability of our network architecture in capturing complex features more effectively.</p>

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Approximating Functions with Additive Ridge Features by Deep \(\text {ReLU}^k\) Convolutional Neural Networks

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Junyan Huang,
  • Jun Xian

摘要

In recent years, neural networks have achieved significant success in function approximation. However, several challenges remain, such as the extraction of overly specific features from composite functions and the generation of non-differentiable output functions. To address these issues, we propose a deep k-power of the rectified linear unit ( \(\text {ReLU}^k\) ReLU k ) convolutional neural networks algorithm for approximating functions with additive ridge features. Our approach enables automatic feature extraction without requiring prior knowledge of the composite structure, and the network’s output exhibits higher-order differentiability compared to existing methods. In addition, we prove that deep \(\text {ReLU}^k\) ReLU k convolutional neural networks can mitigate the curse of dimensionality for target functions with the low-dimensional structure of additive ridge features. These results highlight the superior learning capability of our network architecture in capturing complex features more effectively.