Shallow neural network representation of polynomials. (arXiv:2208.08138v2 [stat.ML] UPDATED)

We show that $d$-variate polynomials of degree $R$ can be represented on
$[0,1]^d$ as shallow neural networks of width
$d+1+\sum_{r=2}^R\binom{r+d-1}{d-1}[\binom{r+d-1}{d-1}+1]$. Also, by SNN
representation of localized Taylor polynomials of univariate $C^\beta$-smooth
functions, we derive for shallow networks the minimax optimal rate of
convergence, up to a logarithmic factor, to unknown univariate regression
function.

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