Please notice that this question was asked in MO, but it seems that it doesn't interest MO community. So, I have got a comment to post in this community in the hope that I may get some attention to this question.
Background:
For Kernel ridge regression, I have normally come across the data-set given in vector and scalar form, i.e., $\overline{S}:= \{x_i, \overline{y}_i \}$, where $x_i \in M_{n,1}(\mathbb{R})$ and $\overline{y}_i \in \mathbb{R}$. Then, for example, I could do the ridge-regression as given here link.
My question:
Can we do the same Kernel ridge regression but with a data set in matrix-vector form, i.e., $S := \{ X_i, y_i \}_{i=1}^{N}$, where $X_i \in M_{n,p}(\mathbb{R})$ and $y_i \in M_{n,1}(\mathbb{R})$? If yes, can you prove it or give me a reference/paper?
Thank you so much in advance for your help.