Suppose we have a data matrix $X \in \mathbb{R}^{m\times n}$, where $m\gg n$. This matrix represents a series of measurements of a physical system, where each column represent a single measurement. Upon introducing the radial basis function $f(z,y,\gamma)=\exp(-\gamma\|z-y\|^2)$, where $\gamma\in\mathbb{R}$, we can form the Gramian kernel matrix $G\in\mathbb{R}^{n\times n}$ with entries $G_{ij}=f(x_i,x_j,\gamma)$, where $x_i$ and $x_j$ are the $i$:th and $j$:th column of $X$, respectively.
I would like to know if there is any fast and efficient way of choosing $\gamma$ such that the rank of $G$ is as high as possible. Typically, $n\approx 500-1000$.
Edit: For most applications, the columns of $X$ are linearly independent. Could this be helpful?