jacob_drori comments on DSLT 1. The RLCT Measures the Effective Dimension of Neural Networks

[jacob_drori]

The theorem guarantees the existence of a $d$-dimensional analytic manifold $M$ and a real analytic map

$$g:M\ni u\mapsto w\in W$$

such that for each coordinate $M_{\alpha }$ of $M$ one can write

$$\begin{array}{cc}K\left(g\right(u\left)\right)& =u_1^{2k_1}\dots u_d^{2k_d}\\ ...& \end{array}$$

I’m a bit confused here. First, I take it that $\alpha$ labels coordinate patches? Second, consider the very simple case with $d=2$ and $K\left(w\right)=w_1^2+w_2^2$. What $g$ would put $K$ into the stated form?