On Gaussian Weight matrix and rank

Suppose one has a set of $n$ points in $\mathbb{R}^d$, where the points are represented as $p_1$ through to $p_n$. Define the weight matrix $W$ as follows: let $W_{i,j}$ be $e^{-||p_i - p_j ||^2}$, also sometimes known as a Gaussian weight matrix. My question is, is there a relation of the rank of $W$ to the underlying dimensionality $d$? If not, is there a rank $d$ matrix approximation of $W$? The reason I say this is that in a loose "information theoretic" sense, all the weights can be reconstructed from the $d$ real coordinates of each point.