I am coding simplified version of Platt's SMO and it defines a function like this:
$$ \newcommand{\<}{\langle} \newcommand{\>}{\rangle} f(x) = \sum_{i=1}^{m} \alpha_iy^{(i)} \<x^{(i)},x\> + \hspace{0.20cm} b $$
where $x$ is a matrix of dimension $m \times n$
$\alpha$ and $y$ are column vectors of size $m$
I am trying to understand how to calculate $\<x^{(i)},x\>$ since it's apparent that the result of $f(x)$ has to be a scalar.
All the paper says about $\<A,B\>$ is that it's an "inner product". There are many choices of inner products over matrices (with a corresponding choice of kernel), but the most commonly used inner product in this context is the Hilbert-Schmidt inner product, given by $$ \<A,B\> = \operatorname{trace}(A^TB) = \sum_{i = 1}^m\sum_{j=1}^n A_{ij}B_{ij} $$ That is to say, we can simply take the "dot product" of matrices, as one would with vectors.