论文标题
R和C上的定性神经网络近似：分析和多项式激活的基本证明
Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation
论文作者
论文摘要
在本文中，我们证明了通过基本参数具有分析激活函数的深层神经网络类别中的近似定理。我们证明了具有非多项式激活的真实和复杂网络，即神经网络类别的关闭与多项式空间的关闭相吻合。闭合可以进一步以Stone-Weierstrass定理（在实际情况下）和Mergelyan定理（在复杂情况下）的特征。在实际情况下，我们进一步证明了具有高维谐波激活和正交投影线性图的网络的近似结果。我们进一步表明，具有多项式激活功能的较大深度的完全连接和残留的网络可以在某些宽度要求下近似任何多项式。所有证明都是完全基本的。
In this article, we prove approximation theorems in classes of deep and shallow neural networks with analytic activation functions by elementary arguments. We prove for both real and complex networks with non-polynomial activation that the closure of the class of neural networks coincides with the closure of the space of polynomials. The closure can further be characterized by the Stone-Weierstrass theorem (in the real case) and Mergelyan's theorem (in the complex case). In the real case, we further prove approximation results for networks with higher-dimensional harmonic activation and orthogonally projected linear maps. We further show that fully connected and residual networks of large depth with polynomial activation functions can approximate any polynomial under certain width requirements. All proofs are entirely elementary.
