Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive scalar $\alpha$. Also let $\phi:\mathbb{R}\mapsto \mathbb{R}$ be a function. I am interested in figuring out the most general conditions on the nonlinearity $\phi$ such that the functions $\phi(w^Tx_1), \phi(w^Tx_2), \ldots, \phi(w^Tx_n)$ are linearly independent (viewed as functions of $w\in\mathbb{R}^d$). While I'm after general conditions under which this is true some simple examples which I would like to show this for are $\phi(z)=log(1+e^z), \phi(z)=\max(0,z)$ and $\phi(z)=\frac{1}{1+e^{-z}}$

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for any choice of points $x_1,\dots,x_n$(always assumed not in "co-radial" position)? $\endgroup$ – Pietro Majer Jul 20 at 6:48