# How to prove there is no finite-dimensional feature space for Gaussian RBF kernel?

How to prove that for the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ there is no finite-dimensional feature space $H$ such that for some $\Phi: \text{R}^n \to H$ we have $k(x, y) = \langle \Phi(x), \Phi(y)\rangle$?

• Is this question more appropriate for Mathematics? – Leo Sep 20 '12 at 0:37
• One possible plan of attack would be to exhibit a subspace of $H$ that is not closed. – Nick Alger Sep 28 '12 at 7:36
• @Nick Alger: maybe this helps: stats.stackexchange.com/questions/80398/… – user83346 Aug 22 '15 at 8:33

Assume that Gaussian RBF kernel $k(x, y)$ is defined on domain $X \times X$ where $X$ contains an infinite number of vectors. One can prove (Gaussian Kernels, Why are they full rank?) that for any set of distinct vectors $x_1, ..., x_m \in X$ matrix $(k(x_i, x_j))_{m \times m}$ is not singular, which means that vectors $\mathrm\Phi(x_1), ..., \mathrm\Phi(x_m)$ are linearly independent. Thus, a feature space $H$ for the kernel $k$ cannot have a finite number of dimensions.
• Here you find a more 'intuitive' explanation that the $\Phi$ can map onto a spave of dimension equal to the size of the training sample, even for an infinite training sample: stats.stackexchange.com/questions/80398/… – user83346 Aug 22 '15 at 8:36