# Linear independence of functions

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 $$\phi(z)=\frac{1}{1+e^{-z}}$$

• So the condition on $\phi$ refers to the given points $x_1,\dots,x_n$. Or do you mean a stronger condition, stating that the functions $\phi(w^Tx_1), \phi(w^Tx_2), \ldots, \phi(w^Tx_n)$ are linearly independent for any choice of points $x_1,\dots,x_n$ (always assumed not in "co-radial" position)? – Pietro Majer Jul 20 at 6:48
• If the $x_1,...,x_n$ lie in an affine hyperplane, you have no chance. Otherwise I would anticipate that the answer is any $\phi$ except a polynomial works. This may be morally true rather than literally true. – Anthony Quas Jul 20 at 8:39
• Thanks, Pietro. I mean the latter any choice of points assumed not in "co-radial" position – mohi Jul 20 at 21:08
• Thanks Anthony. I disagree that you need the points not to lie on the same plane. In particular, I think for $\phi$ that is not a polynomial and as long as $x_i=\alpha x_j$ (but $\alpha$ not necessarily positive unlike the question) the above result would hold. – mohi Jul 20 at 21:10