I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of [Srinivas et.al 2012](https://ieeexplore.ieee.org/document/6138914). I found on page 3 column 1 like: 



> 
> The induced RKHS norm $||{f}||\_k=\sqrt{<f,f>}\_k$ measures smoothness of $f$ w.r.t. $k$. 
> 
> 
> 


I am new to this field so cannot understand how to calculate $||f||\_k$ numerically. Also, what does $<f,f>\_k$ mean? How it is different from the conventional inner product $<f,f>$ ? 



How to calculate or estimate RKHS norm?
For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://arxiv.org/abs/1807.02582) by Kanagawa et al. (2018).


In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:


**Theorem 2.4** (Kanagawa et al.)


Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$ and assume $k \in C(\mathbb{R}^d) \cap L\_1(\mathbb{R}^d)$.


Then the RKHS of k is given by



> 
> $$
> \mathcal{H}\_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L\_1(\mathbb{R}^d), ~||f||\_{\mathcal{H}\_k} < \infty \Big\rbrace
> $$
> 
> 
> 


Where the inner product is given by



> 
> $$
> \langle f, g \rangle\_{\mathcal{H}\_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega
> $$
> 
> 
> 




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If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan [Reproducing Kernel Hilbert Spaces in Probability and Statistics](https://link.springer.com/book/10.1007/978-1-4419-9096-9).


Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.


Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.


