I read recently in the papers [1, 2] that the assumption below is made.
Assumption 1 The unknown function $g$ has bounded norm in the RKHS $\mathcal{H}_k$, induced by the continuously differentiable kernel $k$, i.e. $\|g\|_k \leq B_g$.
They are trying to learn a unknown nonlinear dynamic leveraging GP.
However, in [3], it states that "it is easy to show that a GP sample path $\mathrm{f} \sim \mathcal{G} \mathcal{P}(0, k)$ does not belong to the corresponding RKHS $\mathcal{H}_k$ with probability 1 if $\mathcal{H}_k$ is infinite dimensional".
It means the unknown nonlinear function as a sampling path almost surely not lives in the RKHS corresponding to the kernel $k$; hence, the norm $\|g\|_k$ can actually not be taken.
Am I wrong or should this assumption be modified such that the norm is defined on a corresponding larger RKHS space?
Any hint would be really appreciated! Thanks a lot in advance.
Reference:
[1] Berkenkamp, Felix, et al. "Safe learning of regions of attraction for uncertain, nonlinear systems with gaussian processes." 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, 2016.
[2] Berkenkamp, Felix, et al. "Safe model-based reinforcement learning with stability guarantees." Advances in neural information processing systems 30 (2017).
[3] Kanagawa, Motonobu, et al. "Gaussian processes and kernel methods: A review on connections and equivalences." arXiv preprint arXiv:1807.02582 (2018).