Does the representer theorem hold for the following loss function that needs to be minimized ober $f$ and $A$, $X$ are fixed real-matrices and $X_{i\mathbb{.}}$ denotes the row $i$ of matrix $X$.
The Loss function with a Hilbert norm regularizer is :
$\sum_{i,j}A_{ij}[f(X_{i\mathbb{.}})-f(X_{j\mathbb{.}})]^T[f(X_{i\mathbb{.}})-f(X_{j\mathbb{.}})] +\lambda ||f||_H$
I am attempting to substitute $f(x_{i\mathbb{.}})$ with $\sum_{j=1}^{n}\alpha_j K(X_{i\mathbb{.}},X_{j\mathbb{.}}) $ using a kernel $K(.)$ and solve for the vector $\alpha$ using the representer theorem.
I am concerned on the fact that $f(X_{i\mathbb{.}})$ is a vector instead of a scalar and so what is the corresponding representer theorem instead of $\sum_{j=1}^{n}\alpha_j K(X_{i\mathbb{.}},X_{j\mathbb{.}}) $? Does the representer theorem even exist in this setting!?
Thanks!
References: http://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space http://en.wikipedia.org/wiki/Representer_theorem http://www.amazon.com/Harmonic-Analysis-Semigroups-Functions-Mathematics/dp/0387909257 http://www.ams.org/journals/bull/2002-39-01/S0273-0979-01-00923-5/ , Steve Smale
