I'm studying vector valued RKHS and came across this paper where they essential prove the vector valued version of the representer theorem. I would like to apply it to a case, where I'm not sure if it fits it.
Lets first look at the scalar case. Assume we have a vector $x=(x_1,\dots,x_n)$ given and the following equation holds
$$Af(x) = b $$
with $A\in\mathbb{R}^{m\times n}$, $f(x) = (f(x_1), \dots,f(x_n))$ for a scalar valued function $f:X\to\mathbb{R}$ which we would like to learn and $b\in\mathbb{R}^m$. Now we can easily try to apply a kernel ridge regression over a suitable Reproducing Kernel Hilbert space (RKHS) $\mathcal{H}$ for the function $f$
$$\operatorname{argmin}_{f\in\mathcal{H}} \|b-Af(x)\|^2+\lambda \|f\|$$
Now assume we have a vector valued functions, $F=(f_1,f_2)$ with each $f$ same as above and belonging to the same scalar RKHS. We also assume that $x$ is the same across the two $f$'s. In the vector valued case a kernel is matrix valued. I would like to optimize the following problem
$$\operatorname{argmin}_{F=(f_1,f_2)} \|b_1-A_1f_1(x)\|^2+\|b_2-A_2f_2(x)\|^2+\lambda \|F\|_K$$
where we try to fit each component separately but the relationship between the two functions $f_1$ and $f_2$ entering via $\|F\|_K$ for a suitable matrix valued kernel $K$.
My issue is the following. In the version of the vector valued representer theorem one usually sum over the input data and not over the output, i.e. it states problems of the form
$$\operatorname{argmin}_{F=(f_1,f_2)} \sum_{i=1}^n\|y_i-F(x_i)\|+\lambda \|F\|_K$$
for input/output pairs $(x_i,y_i)$ where $y_i\in\mathbb{R}^2$. So can't we apply it in this kind of scenarios?