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 $k(x, y) = \langle \Phi(x), \Phi(y)\rangle$?

Answer

The Moore-Aronszajn theorem guarantees that a symmetric positive definite kernel is associated to a unique reproducing kernel Hilbert space. (Note that while the RKHS is unique, the mapping itself is not.)

Therefore, your question can be answered by exhibiting an infinite-dimensional RKHS corresponding to the Gaussian kernel (or RBF). You can find an in-depth study of this in “An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels“, Steinwart et al.

Attribution
Source : Link , Question Author : Leo , Answer Author : gung – Reinstate Monica

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